[    {        "title": "2007.15226v5.Resonant_subwavelength_control_of_the_phase_of_spin_waves_reflected_from_a_ferromagnetic_film_edge.pdf",        "content": "Resonant subwavelength control of the phase of spin waves reﬂected from a ferromagnetic ﬁlm edge\nKrzysztof Sobucki,1,\u0003Wojciech ´Smigaj,2Justyna Rychły,3Maciej Krawczyk,1and Paweł Gruszecki1, 3, †\n1Faculty of Physics, Adam Mickiewicz University, Uniwersytetu Pozna´ nskiego 2, 61-614 Pozna´ n, Poland\n2Met Ofﬁce, FitzRoy Rd, Exeter, EX1 3PB, UK\n3Institute of Molecular Physics, Polish Academy of Sciences, Mariana Smoluchowskiego 17, 60-179 Pozna´ n, Poland\n(Dated: November 3, 2021)\nUsing frequency-domain ﬁnite element calculations cross-checked with micromagnetic simulations, we\ndemonstrate that the phase of spin waves reﬂected from an interface between a permalloy ﬁlm and a bi-\nlayer can be controlled by changing dimensions of the bilayer. Treating the bilayer formed by the permalloy\nﬁlm and a ferromagnetic stripe as a segment of a multi-mode waveguide, we show that spin-wave Fabry-\nPerot resonances of one of its modes are responsible for the high sensitivity of the phase of reﬂected waves\nto stripe width and the stripe-ﬁlm separation. Thus, the system is a unique realization of a fully magnonic\nGires–Tournois interferometer based on a two-modes resonator, which can be treated as a magnonic coun-\nterpart of a metasurface, since it enables manipulation of the phase of spin waves at subwavelength dis-\ntances. Knowledge gained from these calculations might be used to design magnonic devices such as ﬂat\nlenses or magnetic particle detectors.\nI. INTRODUCTION\nThe recent years have been marked by a rapidly growing\ndemand for interconnected mobile devices. This emerging\necosystem of connected devices, preferably communicating\nwirelessly, is referred to as the Internet of Things. There are\nestimations that within the next few years, the number of\nWiFi-enabled devices will be at least four times larger than\nthe total population of the world [1]. One of the essential\ncomponents of the Internet of Things are small and energeti-\ncally efﬁcient devices processing signals converted from and\nthen back to microwaves. In this ﬁeld, the application of spin\nwaves (SWs), which are collective disturbances of magnetiza-\ntion propagating in the same frequence range as microwaves\nand thus able to couple to them, opens up a new opportu-\nnity to increase the efﬁciency of these devices. Compared to\nexisting microwave devices, SW components offer prospects\nfor increased miniaturization (SWs have wavelengths 3–5 or-\nders of magnitude shorter than microwaves of the same fre-\nquency), easy external control of SW signals, reprogramma-\nbility, and small energy demands due to lack of Joule heating\nrelated with SWs propagation [2–4].\nIn order to use any kind of waves as an information car-\nrier, efﬁcient methods of their excitation and control over\ntheir amplitude and phase must be developed. In modern\nphotonics, a breakthrough in the control of these proper-\nties of reﬂected or transmitted waves at subwavelength dis-\ntances has recently been achieved through the use of arrays of\nnanostructured antennas absorbing and reemitting modiﬁed\nelectromagnetic waves [5, 6]. These arrays, so-called meta-\nsurfaces, are used to obtain anomalous refraction of inci-\ndent waves or to design ﬂat, ultra-narrow lenses able to focus\nwaves. Moreover, such nanostructured antennas can serve as\ncolor ﬁlters that can be used to produce silicon- or metallic-\nbased color pixels for printing purposes as a replacement of\nchemical dyes [7, 8].\nThere are several reports on the coupling of small magnetic\nelements with an uniform ﬁlm. Kruglyak et al. have shown\n\u0003krzsob@st.amu.edu.pl\n†gruszecki@amu.edu.plthat a narrow ferromagnetic element placed on top of a mag-\nnetic waveguide can be used to emit SWs, to control the phase\nof SWs passing below the resonator, and under some condi-\ntions even to absorb the energy of propagating SWs [9–11 ]. Yu\nat al. have demonstrated chiral excitation of SWs in a thin ﬁlm\nthrough its dipolar coupling with a single nanowire or a grat-\ning of nanowires placed in a spatially uniform microwave-\nfrequency magnetic ﬁeld [12, 13 ]. Subsequently, the existence\nof Fano resonances and their inﬂuence on the amplitude and\nphase of transmitted waves in a single-mode waveguide has\nbeen further studied by Al et al. [14], and Zhang et al. have\ndemonstrated the application of a single dynamically tun-\nable resonator in zero bias ﬁeld placed on top of a waveg-\nuide to tune the phase of the transmitted SWs [15]. A grating\ncoupler made up of an array of resonators has been used to\nexcite short-wavelength SWs [16, 17 ], and Graczyk et al. [18]\nhave demonstrated that dynamical coupling of a homoge-\nneous ferromagnetic ﬁlm with a periodic array of ferromag-\nnetic stripes placed underneath can lead to the formation of a\nmagnonic band structure in the ﬁlm. However, the effect of a\nresonator on the phase of the reﬂected wave has not yet been\nstudied in magnonics; moreover, the conditions for the exis-\ntence of Fabry-Perot resonances and their effect on both re-\nﬂected and transmitted SWs remain almost unexplored [19],\nwhile both may be key to creating a magnonic metasurface.\nIn this work, we investigate theoretically the interaction\nof a narrow, subwavelength-width stripe placed above the\nedge of a homogeneously magnetized ﬁlm with propagating\nSWs and its inﬂuence on the phase shift of reﬂected SWs.\nUsing frequency-domain ﬁnite-element calculations and mi-\ncromagnetic simulations we ﬁnd that this shift depends on\nthe width of the stripe in a non-trivial way: an overall slow\nand steady increase of the phase shift with stripe width is re-\npeatedly interrupted by sharp phase jumps by 360\u000e. Treating\nthe stripe and the underlying ﬁlm as a non-reciprocal waveg-\nuide supporting two pairs of counter-propagating modes, we\nformulate a semi-analytical model that explains this behav-\nior as a consequence of Fabry-Perot resonances produced by\none of these mode pairs. We also show that by varying the\nﬁlm-stripe separation it is possible to switch between res-\nonances of different order. Our results point to the impor-\ntance of Fabry-Perot resonances appearing in locally bilay-\nered ferromagnetic elements, with potential applications forarXiv:2007.15226v5  [cond-mat.mes-hall]  22 Sep 20202\nd2\nd1PyH0\nz\nxy\nstripew\ns x = 0 x = w\nFigure 1. Geometry of the system used in simulations. A ferromag-\nnetic stripe of width wand thickness d2is separated from a semi-\ninﬁnite permalloy ﬁlm of thickness d1by a distance s. The right\nedges of both layers are aligned at x=w. The whole system is placed\nin vacuum in an external magnetic ﬁeld \u00160H0=0.1 parallel to the\nyaxis. The geometry of the system is independent from y.\nthe control of SW propagation in magnonic devices.\nThe paper is organized as follows. In Sec. II, we present the\ngeometry of the system under consideration and the numer-\nical methods used in our study. In Secs. III A–III C, we analyze\nthe key physical effects occurring in individual components\nof the system of interest: SW dynamics in inﬁnitely extended\nﬁlms and bilayers, SW resonances in ﬁnite-width stripes and\nSW reﬂection from edges of truncated ﬁlms. In Secs. III D–\nIII F, which form the core of this work, we investigate SW re-\nﬂection at the end of a ﬁlm coupled to a ﬁnite-width stripe.\nWe show how interactions between parts of the system dis-\ncussed in the previous subsections make the phase shift of\nthe reﬂected wave highly sensitive to stripe width and the\nstripe-ﬁlm separation. Our results are summarized in Sec. IV.\nII. STRUCTURE AND METHODS\nA. Structure under consideration\nWe consider a system composed of non-magnetic and fer-\nromagnetic materials. Its geometry is independent of the\nycoordinate and piecewise constant along x, as shown\nschematically in Fig. 1. The system consists of a semi-inﬁnite\npermalloy (Py) ﬁlm of thickness 50 nm and a ferromagnetic\nstripe of thickness 40 nm and ﬁnite width w. Both elements\nare separated by a distance sand their right edges are aligned\ntox=w. Throughout the paper, we will vary the width wof\nthe stripe and its separation sfrom the ﬁlm. We are interested\nin manipulating the phase of the reﬂected SWs using sub-\nwavelength elements; therefore the width of the stripe will\nbe smaller than or comparable to the wavelength of SWs in\nthe Py ﬁlm at the frequency of operation. The system is mag-\nnetized by a uniform in-plane bias magnetic ﬁeld of magni-\ntude\u00160H0=0.1 T directed along the yaxis. In all calcula-\ntions we have taken the saturation magnetization of the ﬁlm\nto be MS=760 kA /m and its exchange constant, A=13 pJ /m.\nThe stripe is made of a material (called FM2 from here on)\nwith MS=525 kA /m and A=30 pJ /m; lowering its saturation\nmagnetization and increasing the exchange constant with re-\nspect to Py will allow us to exploit interactions of local res-onances of the stripe with propagating waves in Py at fre-\nquencies characteristic for the dipolar and dipole-exchange\nSWs. Such a choice of the stripe’s material shifts the SW spec-\ntrum down and ﬂattens the ﬁrst band for lower wavevectors\nwith respect to the Py, as will be discussed in the next section.\nThe gyromagnetic ratio of both ferromagnets is \r=\u0000176 rad\nGHz /T.\nB. Governing equations\nThe magnetization dynamics is described by the Landau-\nLifshitz equation:\n@tM=\u0000j\rj\u00160\n1+\u000b2\nM\u0002Heff+\u000b\nMSM\u0002(M\u0002Heff)\n, (1)\nwhere Mis the magnetization vector, \u00160is the permeability\nof vacuum,\u000bis a dimensionless damping parameter, and\nHeff=H0+Hm+Hex (2)\nis the effective magnetic ﬁeld. The latter is the sum of the ex-\nternal magnetic ﬁeld H0, the magnetostatic ﬁeld Hm, and the\nisotropic Heisenberg exchange ﬁeld Hex=r\u0001(l2rM), where\nl=q\n2A=(\u00160M2\nS)is the exchange length. The magnetostatic\nﬁeld fulﬁls the magnetostatic Maxwell’s equations\nr\u0001(M+Hm) =0, (3a)\nr\u0002Hm=0; (3b)\nthe latter makes it possible to write it as Hm=\u0000r', where\n'is the magnetic scalar potential.\nAssuming a harmonic time dependence [exp(\u0000i!t)], zero\ndamping (\u000b=0) and alignment of the external magnetic\nﬁeld H0with the yaxis, splitting the magnetization M\nand magnetostatic ﬁeld Hminto static ( MSˆy,H0ˆy) and dy-\nnamic (radio-frequency) components ( m= [mx,0,mz],hm=\n[\u0000@x',0,\u0000@z']), linearizing the Landau-Lifshitz equation (1)\n(applicable only in the ferromagnetic layers) and coupling it\nwith the Gauss law for magnetism, Eq. (3a) (applicable every-\nwhere), we arrive at the following system of equations for the\nmagnetic potential 'and the dynamic magnetization com-\nponent m:\n@x(mx\u0000@x')+@z(mz\u0000@z') =0, (4a)\n@x'\u0000r\u0001(l2rmx)+H0\nMSmx\u0000i!\nj\rj\u00160MSmz=0, (4b)\n@z'\u0000r\u0001(l2rmz)+H0\nMSmz+i!\nj\rj\u00160MSmx=0. (4c)\nC. Numerical methods\nWe have used three complementary numerical methods\nto study SW dynamics. First, we use micromagnetic simula-\ntions performed in the open-source environment mumax3\n[20], which solves the full Landau-Lifshitz equation [Eq. (1) ]\nwith the ﬁnite-difference time-domain (FDTD) method. We\nuse this method to calculate the dispersion relations of SWs\nand steady states obtainable after long continuous excitation3\nof SWs by a speciﬁed source. More details can be found in\nAppendix A. The main disadvantage of this method is its high\ncomputational cost. Resonant systems can take a long time to\nreach steady state, and the cost of a single time step is pushed\nup by the need to discretize the whole system on a uniform\ngrid whose resolution is dictated by the size of the smallest\ngeometric features.\nIn order to avoid these limitations, we rely primarily on cal-\nculations using the frequency-domain ﬁnite element method\n(FD-FEM). Its major advantage is the possibility of reﬁn-\ning the mesh locally, e.g. only around small geometric fea-\ntures, rather than globally. In addition, it allows direct and\nfast calculation of the eigenfrequencies and eigenmodes\n(mode proﬁles) of the system, which can be identiﬁed with\nits steady states. To perform the calculations, we have used\nthe COMSOL Multiphysics software [21]to solve the lin-\nearized Landau-Lifshitz equation coupled with the Gauss\nlaw, Eqs. (4), as described in [22]. At the edges of the com-\nputational domain (far from the ferromagnetic materials) the\nDirichlet’s boundary conditions, forcing the magnetic poten-\ntial to vanish, are imposed.\nIn Sec. III E we formulate a semi-analytical model depen-\ndent on the numerical values of scattering matrices asso-\nciated with interfaces separating SW waveguides with dif-\nferent cross-sections. To calculate these matrices, we used\nthe ﬁnite element modal method. In this method, Eq. (4) is\ntransformed into an eigenproblem whose solutions are the\nwavenumbers and proﬁles of propagative and evanescent\nmodes of a particular x-invariant part of the system shown\nin Fig. 1, e.g. the Py ﬁlm or the Py /FM2 bilayer. This eigen-\nproblem is then discretized by expanding the magnetic po-\ntential and dynamic magnetization in a ﬁnite-element basis\nand solved numerically. Fields on each side of the interface\nare represented as superpositions of the eigenmodes of the\nrespective parts calculated in the previous step. Finally, the\nscattering matrix is obtained by imposing the standard conti-\nnuity conditions on these ﬁelds and solving the resulting sys-\ntem of equations for the excitation coefﬁcients of outgoing\nmodes. This method, inspired by similar techniques used in\nphotonics [23], will be described in detail in a forthcoming\npaper [24].\nIII. RESULTS\nA. Dispersion relation of bilayers\nBefore we study the coupling of a ﬁlm with a ﬁnite-width\nstripe, let us ﬁrst analyze the interaction between modes in\ninﬁnitely extended ﬁlms, i.e., in bilayers composed of an in-\nﬁnite Py ﬁlm separated by a non-magnetic gap from another\ninﬁnite ﬁlm made of FM2. Two example dispersion relations\nfor gap widths s=200 nm and s=10 nm are presented in\nFig. 2(a) and (b), respectively. It is clear that for a 200 nm-\nwide gap, the SW modes in Py and stripe almost do not in-\nteract with each other: the calculated dispersion curve coin-\ncides with the analytical dispersion curves of isolated Py and\nFM2 ﬁlms calculated using [25, Appendix C.7 ]:\n!2=!0(!0+!M)+!2\nM\n4\u0002\n1+e\u00002k d\u0003\n, (5)\n5101520frequency(GHz)(a) s=200 nm\nPy FM2\n FEM\n5101520frequency(GHz)(b) s=10 nm\nd\n30 20 10 0 10 20 30\nwavevector (rad/m)0100200300separation, s(nm)(c)\n01\nIntensity (a.u.)\n−10 1020406080100y(nm)\n−10 1−10 1−10 1FM2\nmx (a.u.)Py(d) (e) (f) (g)ef gFigure 2. (a)–(b) Dispersion relations calculated for two inﬁnite\nﬁlms made of Py and FM2 separated by non-magnetic gap of width\n(a) 200 nm and (b) 10 nm. Colormaps in the background present\nresults obtained by means of micromagnetic simulations, whereas\ngreen points correspond to the results of FD-FEM calculations. The\ndashed blue and the dash-dotted red lines are the analytical disper-\nsion relations of SWs supported by isolated FM2 and Py ﬁlms, re-\nspectively. The horizontal line marks the frequency f=11 GHz used\nin further calculations. (c) Dependence of the wavenumber of the\nslow modes, d and g, located predominantly in FM2, on the separa-\ntion between ﬁlms at frequency 11 GHz. (d)–(g) Mode proﬁles of mz:\n(d), (g) slow and (e)–(f) fast modes, in the Py /FM2 bilayer separated\nby a 10 nm non-magnetic gap at frequency 11 GHz [marked in (b) ].\nwhere dis the ﬁlm thickness, !=2\u0019fis the angular fre-\nquency of SWs ( fdenotes the frequency), kis the wavenum-\nber,!0=j\rj\u00160(H0+MSl2k2)[26, Chapter 7.1 ], and!M=\nj\rj\u00160MS.\nThe only visible difference occurs at frequencies above\n14 GHz where we can see a hybridization between the fun-\ndamental SW mode and the ﬁrst mode quantized across the\nthickness, a so-called perpendicular standing SW [25]. This4\nhybridization and perpendicular standing SWs, however, are\nnot considered in the analytical model.\nAt frequencies below 14 GHz the modes in bilayered struc-\nture can be classiﬁed according to their origin and group ve-\nlocity. The fast modes are related to SW dynamics in Py [see\nFig. 2(e) and (f) ]and are characterized by steeper dispersion\n(therefore higher group velocity) and longer wavelengths. In\ncontrast, the slow modes originating in FM2 [see Fig. 2(d)\nand (g) ]have lower group velocity and shorter wavelengths.\nIt is worth noting that the wavelengths of SWs in separated\nlayers do not depend on the direction of propagation, while\nthe dynamic dipolar coupling between SW modes in both lay-\ners combined with the nonreciprocal nature of surface SW\nmodes introduces asymmetry [27–29 ]. For s=200 nm the\ncoupling is still very weak and at the frequency of 11 GHz that\nis used in further analysis and marked with the white hori-\nzontal line in Fig. 2(a), the fast and slow modes have wave-\nlengths of 2660 nm and 590 nm, respectively, for both propa-\ngation directions.\nReduction of the non-magnetic gap width to s=10 nm\ncauses the modes of the individual ﬁlms to interact much\nmore strongly. The wavelength of the slow modes decreases\nsigniﬁcantly and their dispersion relation becomes strongly\nnonreciprocal; at 11 GHz, slow modes propagating leftwards\nhave wavelength 390 nm and those propagating rightwards,\n270 nm. Figure 2(c) shows that the asymmetry of the disper-\nsion relation for slow modes grows as the ﬁlms are brought\ncloser together.\nB. Eigenmodes of ﬁnite-width stripes\nThe dependence of the eigenmode spectrum of isolated\nﬁnite-width FM2 stripe on their width is displayed in Fig. 3(a).\nThese calculations were made with FD-FEM for the ﬁrst thirty\nmodes of stripes with widths up to 2800 nm (note that SW\nwavelength in uniform Py ﬁlms at 11 GHz is 2660 nm). As in-\ntuitively expected, mode frequency decreases with increas-\ning stripe width. The horizontal dashed line in Fig. 3 marks\nf=11 GHz. It is visible that stripes of multiple widths sup-\nport modes at this frequency. Proﬁles of three successive\nmodes (for stripes of width 656 nm, 936 nm, and 1200 nm)\nare shown in Fig. 3(b)–(d). Successive resonances appear for\nstripes of widths differing by approximately half of the wave-\nlength of the eigenmode of a homogeneous FM2 ﬁlm, i.e.,\n\u0001w\u0019280 nm\u00190.5\u0015FM2.\nLateral mode conﬁnement in a reciprocal medium leads\nto formation of standing waves and quantization of the\nwavenumber. The standing waves have the form exp (iknx)+\nexp(\u0000iknx), where nis the mode index, kn=rn\u0019=wand rn=\nn+\u000e(0\u0014\u000e\u00141). For the Dirichlet (magnetic wall) bound-\nary conditions, with the dynamic magnetization vanishing at\nthe edges, we get rn=n+1, whereas for “free spins” at the\nedges, rn=n. However, due to dipolar interactions, in mag-\nnetic stripes neither of these cases is correct and magnetiza-\ntion is partially pinned at the stripe edges, 0 <\u000e< 1[30]. This\ncan also be interpreted as the effective length of the waveg-\nuide being slightly larger than the real one, or in terms of a\nnon-zero phase shift 'being experienced at the stripe edges\nby the SWs forming the standing wave. Thus, resonances oc-\n936 nm 656 nm 1216 nm(a)\n(b) (c) (d)...mn04Figure 3. (a) Dependence of the frequency of stripe modes on stripe\nwidth. The dashed line marks the frequency f=11 GHz. (b), (c), and\n(d) Proﬁles of modes supported by stripes of width 656 nm, 936 nm,\nand 1216 nm at frequency 11 GHz.\nECAFRETNI ECAFRETNI(a)\n(b)<mx><mx>\n2λ λ2λ λ\nmx,incmx,inc\nmx,refmx,refφ=0°\nφ=230°\nFigure 4. (a) Incident and reﬂected plane waves at an arbitrary\ntime, and the intensity of the resulting interference pattern, in the\nzero phase shift case, i.e., '=0\u000e. The dotted orange line and the\ndashed blue line correspond to the incident and reﬂected plane\nwaves, respectively. The solid green line corresponds to the SW in-\ntensity (time-averaged squared dynamic magnetization) of the in-\nterference pattern. (b) The same for a phase shift of '=230\u000e, i.e.,\nthe value obtained for SW reﬂection from the edge of a semi-inﬁnite\n50-nm-thick Py ﬁlm.\ncur when the following condition is met:\nk w +'=\u0019n,n=1,2,... (6)\nAccording to this equation, successive resonances at fre-\nquency 11 GHz should appear for stripes of widths differing\nby ca.\u0015=2\u0019280 nm; this is conﬁrmed by the FD-FEM calcu-\nlations.5\nC. Phase shift of the reﬂected SWs\nBefore studying the inﬂuence of the stripe’s presence on\nthe SW reﬂection, let us ﬁrst discuss SW reﬂection from the\nedge of an isolated truncated ﬁlm. According to Stigloher et\nal.[31], dynamic dipolar interactions induce a phase shift be-\ntween the incident and reﬂected SWs. This phase shift is a\nnatural consequence of the previously discussed dipolar pin-\nning occurring at the boundaries of thin ferromagnetic ﬁlm\n[30]. Interestingly, Verba et al. [32]have recently shown that\na phase shift may also be introduced by a polarization mis-\nmatch between incident and reﬂected SW modes. Regardless\nof the physical mechanism responsible for the phase shift, we\ncan extract its magnitude from steady-state solutions formed\nfar away from the edge.\nThe phase shift manifests itself in the resulting interference\npattern as a displacement of nodes with respect to the inter-\nface from which the waves are reﬂected. If the interface is\nlocated at x=x0and the reﬂection coefﬁcient is ei', where\n'is the phase shift, the standing wave pattern sufﬁciently far\nfrom the interface (at x\u001cx0) will be\nm(t;x) =Re\b\nAe\u0000i2\u0019f t[eik(x\u0000x0)+ei'e\u0000ik(x\u0000x0)]\t\n=a(t)cos[k(x\u0000x0)\u0000'=2],(7)\nwhere Aand a(t)are scaling coefﬁcients independent of x.\nThe change in the standing wave pattern due to varying\nphase shift is illustrated in Figs. 4(a) and (b).\nIn practice, we calculate 'by ﬁtting the expression on the\nright-hand side of Eq. (7) to a snapshot of mxon the symme-\ntry axis of the Py ﬁlm obtained from micromagnetic or FD-\nFEM simulations. To avoid distortions caused by evanescent\nwaves excited at the interface, only points lying at least one\nstripe width to the left of the interface are taken into account.\nThe phase shift occurring at the edge of a 50-nm-thick Py\nﬁlm at frequency 11 GHz is found numerically to be 230\u000e. The\nresulting standing wave pattern is shown in Fig. 4(b).\nD. Phase shift dependence on stripe width\nThe introduction of a stripe over the Py ﬁlm edge locally\nmodiﬁes the environment in which SWs propagate due to dy-\nnamic dipolar interactions between the ﬁlm and the stripe.\nIn consequence, it inﬂuences also the phase shift of reﬂected\nwaves. The variation of this phase shift with stripe width at\nfrequency 11 GHz, calculated using FD-FEM, is plotted in\nFig. 5(a). In general, this phase shift, deﬁned according to\nEq. (7) with interface deﬁned at the left edge of the stripe, at\nx0=0, grows steadily with stripe width, however with peri-\nodic jumps by 360\u000e. These jumps occur approximately every\n160 nm and are accompanied by an increase of the ampli-\ntude of SWs in the Py ﬁlm underneath the stripe, as shown\nin Fig. 5(b). In fact, at these stripe widths SWs are ampliﬁed\nin the whole bilayer, indicating that a resonant mode of the\nbilayer is excited. This can be seen by comparing snapshots\nofmxfor stripes of width 1270 nm (slowly changing phase\nshift) and 1350 nm (rapidly changing phase shift) presented\nin Figs. 5(c) and (d), respectively.\nIn addition to the enhancement of SW amplitude in the\nbilayer, Fig. 5(c) shows that the SWs in the stripe and in\n(d)λPy(a)\n(c)\n(d)1=n\n2=n\n3=n\n4=n\n5=n\n6=n\n7=n\n8=n\n9=n\n01=n\n11=n\n31=n\n41=n\n51=n\n61=n\n71=n21=n\n(b)\n(c)Figure 5. (a) Phase shift of SWs at frequency 11 GHz as a function\nof the stripe width, calculated by the FD-FEM. The stripe-ﬁlm sep-\naration is s=10 nm. Resonances (rapid changes of the phase shift,\nmarked with vertical dotted lines) appear periodically with a period\nof ca. 160 nm. (b) SW ampliﬁcation factor in Py below the stripe, ob-\ntained by dividing the maximum of jmxjin the part of the ﬁlm lying\nbelow the stripe by the maximum of jmxjin the far ﬁeld, i.e., further\nthan half of width of the Py layer to the left of the stripe. (c) Snapshot\nof the dynamic magnetization mxin a plateau region (stripe width:\n1270 nm). (d) Snapshot of the system in resonance at 11 GHz (stripe\nwidth: 1350 nm). The red dashed lines in (c) and (d) denote the SW\nwavelength in a uniform Py ﬁlm at the same frequency.\nthe underlying Py layer have approximately opposite phases.\nIn view of the proﬁles of the fast and slow modes shown\nin Fig. 2(d)–(g), this indicates that the slow modes domi-\nnate. The magnetization pattern in both layers is more com-\nplex than that of a typical standing wave composed of two\ncounter-propagating waves with the same wavelength; in-\ndeed, as discussed in Sec. III A, the bilayer modes have an\nasymmetric dispersion relation [33]. For such a scenario, we\ncan generalize the resonance condition (6) to\n(ku+kd)w+'l+'r=2\u0019n,n=1,2,..., (8)\nwhere kuand kdare the wavenumbers of right- and left-\npropagating modes, and 'land'rare the phase shifts oc-\ncurring at the left and right interfaces of the stripe. For ku=\nkd=kand'r='lthis equation reduces to Eq. (6). Substitut-\ning here the wavelengths of the slow modes of a bilayer with\ns=10 nm given in Sec. III A, we conclude that successive res-\nonances should occur every 160 nm, which matches well the\nresults of FD-FEM calculations shown in Fig. 5.\nWe have cross-checked these results against micromag-\nnetic simulations made with a ﬁnite damping coefﬁcient \u000b=\n0.0001, see Fig. 6. Due to computational demands, these have\nbeen performed for a narrower range of stripe widths, 0–\n490 nm, 1000–1650 nm, and 2350–2720 nm, encompassing\nseveral resonances. The obtained results are consistent with\nthose of FD-FEM calculations: the positions of resonances\nare the same as obtained by FD-FEM and the slope of the6\n(c)(a)\n(c)\n(d)\nλPy(b)\n(d)\nFigure 6. (a) Phase shift of SWs at frequency 11 GHz as a function\nof stripe width, obtained with micromagnetic simulations. (b) SW\nampliﬁcation factor in Py below the stripe, deﬁned as in Fig. 5(b).\n(c) Snapshot of the dynamic magnetization mxin a system without\na resonance at 11 GHz (stripe width: 1270 nm). (d) The same in a\nsystem with a resonance at 11 GHz (stripe width: 1350 nm). The red\ndashed lines in (c) and (d) denote the SW wavelength in a uniform\nPy ﬁlm at the same frequency.\ncurve in intermediate regions is virtually identical. The al-\nmost perfect alignment of those results obtained by two dif-\nferent numerical methods conﬁrms their correctness.\nSnapshots of mxin systems with stripes of width 1270 nm\nand 1350 nm are displayed in Fig. 6(c) and (d), respectively.\nThe former does not have a resonance at the chosen fre-\nquency, whereas the latter does. The obtained magnetization\npatterns are qualitatively similar to those calculated by FD-\nFEM [Figs. 5(c) and (d) ].\nE. Two-mode model analysis\nAs discussed above, the plot of the phase shift vs. stripe\nwidth [Fig. 5(a) ]shows areas of slow but steady growth sepa-\nrated by sharp resonances. The resonances are not identical:\nsome, such as those at n=7 and n=9, are broader than oth-\ners. These and other features can be explained by considering\na semi-analytical model introduced below.\nWave scattering on the interface x=0 separating the ﬁlm\nand the bilayer (see Fig. 1) can be described by a scatter-\ning matrix Slinking the complex amplitudes of the incoming\nand outgoing modes on both sides of the interface. If both\nparts are sufﬁciently long for the amplitudes of all incoming\nevanescent modes to be negligible, the amplitudes of the out-\ngoing propagative modes are given by\n2\n4d1\nu2\nu33\n5=S2\n4u1\nd2\nd33\n5\u00112\n4S11S12S13\nS21S22S23\nS31S32S333\n52\n4u1\nd2\nd33\n5. (9)\nHere, u1and d1are the amplitudes of the right- and left-propagating modes of the Py ﬁlm, u2and d2are the ampli-\ntudes of the right- and left-propagating slow modes of the bi-\nlayer, and u3and d3are the amplitudes of the right- and left-\npropagating fast modes of the bilayer (see dispersion relation\nshown in Fig. (2). All these amplitudes are measured at the in-\nterface between the ﬁlm and the bilayer. The elements of the\nscattering matrix Scan be calculated using the ﬁnite-element\nmodal method [24]. At 11 GHz, their numerical values are\nS=2\n40.100\u00000.011i 0.135 +0.016i 0.986 +0.008i\n\u00000.104\u00000.168i\u00000.290 +0.929i 0.035\u00000.113i\n0.975\u00000.023i 0.119 +0.143i\u00000.118\u00000.0273i3\n5(10)\n(these values are obtained for modes normalized to carry unit\npower, with the phase at the interface chosen so that mxis\nreal and positive on the symmetry axis of the Py layer). It can\nbe seen that the ﬁlm mode is coupled primarily with the fast\nmode of the bilayer. The slow bilayer mode is strongly re-\nﬂected. There is only weak, though non-negligible, coupling\nbetween the fast and slow bilayer modes.\nLikewise, the interface x=wbetween the bilayer and the\nvacuum can be described by a scattering matrix S0:\nd0\n2\nd0\n3\n=S0u0\n2\nu0\n3\n\u0011S0\n22S0\n23\nS0\n32S0\n33u0\n2\nu0\n3\n. (11)\nHere, u0\n2and d0\n2are the amplitudes of the right- and left-\npropagating slow modes of the bilayer, and u0\n3and d0\n3are\nthe amplitudes of the right- and left-propagating fast modes\nof the bilayer, all measured at the bilayer-vacuum interface\n(hence the prime, used to distinguish them from the ampli-\ntudes measured at the ﬁlm-bilayer interface). The numerical\nvalues of these scattering coefﬁcients calculated at 11 GHz\nare\nS0=\u00000.043 +0.975i\u00000.103 +0.190i\n0.189 +0.105i\u00000.561\u00000.799i\n. (12)\nBoth modes are strongly reﬂected and there is only weak\ncross-coupling.\nMode amplitudes at the two interfaces are linked by\nu0\ni=exp(iki uw)ui\u0011\bi uui, (13a)\ndi=exp(\u0000iki dw)d0\ni\u0011\bi dd0\ni,i=2,3, (13b)\nwhere ki uand ki dare the wave numbers of the right- and left-\npropagating modes, numerically determined to be k2u=16.2,\nk3u=2.22, k2d=\u000023.1 and k3d=\u00001.90 rad /mm.\nTogether, (9), (11) and (13) form a system of nine equations\nfor as many unknown mode amplitudes (the amplitude u1of\nthe mode incident from the input ﬁlm is treated as known).\nTo obtain an intelligible expression for the reﬂection coefﬁ-\ncient r\u0011d1=u1, it is advantageous to start by eliminating the\namplitudes u3,d3,u0\n3and d0\n3of the fast bilayer mode, which\nis only weakly reﬂected at the interface with the Py ﬁlm and\nhence will not give rise to strong Fabry-Perot-like resonances.\nThis mimics the approach taken by Lecamp et al. [34]in their\nmodel of pillar microcavities. This reduces the second row of\nEq. (9) and the ﬁrst row of Eq. (11) to\nu2=˜S21u1+˜S22d2, (14a)\nd0\n2=˜S0\n22u0\n2+˜S0\n23\b3uS31u1, (14b)7\nwhere\n\u0002˜S21˜S22\u0003\n\u00111\n1\u0000\u0014S23\b3dS0\n32\b2u\u0002\nS21+\u0014S23\b3dS0\n33\b3uS31S22+\u0014S23\b3dS0\n33\b3uS32\u0003\n,\n(15a)\n\u0002˜S0\n22˜S0\n23\u0003\n\u00111\n1\u0000\u0014S0\n23\b3uS32\b2d\u0002\nS0\n22+\u0014S0\n23\b3uS33\b3dS0\n32\u000bS0\n23\u0003\n(15b)\nand\n\u0014\u0011(1\u0000S33\b3dS0\n33\b3u)\u00001. (16)\nThe fast bilayer mode is only weakly reﬂected at the interface\nwith the ﬁlm:jS33j\u00190.12\u001c1. Therefore multiple reﬂections of\nthe fast mode at bilayer interfaces do not give rise to strong\nFabry-Perot resonances and the coefﬁcient \u0014remains close to\n1 for all bilayer lengths. Together with the fact that the cross-\ncoupling coefﬁcients S23,S32,S0\n23and S0\n32are small, this means\nwe can expect the scattering coefﬁcients with a tilde deﬁned\nin Eq. (15) to be well approximated by\n\u0014˜S21˜S22\n˜S0\n22˜S0\n23\u0015\n\u0019S21+S23\b3dS0\n33\b3uS31S22\nS0\n22S0\n23\n. (17)\nSolving the equations remaining after elimination of the\namplitudes of the fast mode for the amplitudes of the slow\nmode and substituting the resulting expressions to the for-\nmula for d1in the ﬁrst row in Eq. (9), we arrive at the following\nformula for the reﬂection coefﬁcient:\nr\u0011d1=u1= (a+\fb), (18)\nwhere\na\u0011S11+S13\b3d\u0014S0\n33\b3uS31, (19a)\nb\u0011S12\b2d(˜S0\n22\b2u˜S21+˜S0\n23\b3uS31)\n+S13\b3d\u0014\u0002\nS0\n33\b3uS32\b2d(˜S0\n22\b2u˜S21+˜S0\n23\b3uS31)\n+S0\n32\b2u(˜S21+˜S22\b2d˜S0\n23\b3uS31)\u0003(19b)\nand\frepresents the effect of multiple reﬂections of the slow\nmode:\n\f\u0011(1\u0000S22\b2dS0\n22\b2u)\u00001. (20)\nTo facilitate the interpretation of Eqs. (18)–(20), the scattering\ncoefﬁcients with magnitude much smaller than 1 have been\nunderlined.\nIt can be seen that the reﬂection coefﬁcient ris made up\nof two terms. The ﬁrst, a, is dominated by the phase shift ac-\nquired by the fast mode of the bilayer during a single round-\ntrip across it. This term produces the slow but steady in-\ncrease of the phase shift visible in Fig. 5(a) (also in Fig. 7).\nThe second term, \fb, is proportional to b, which is a su-\nperposition of six small terms, each containing a product of\ntwo scattering coefﬁcients of small magnitude. Therefore \fb\nhas an appreciable effect on the reﬂection coefﬁcient bonly\nwhen the factor \f, representing the combined effect of mul-\ntiple reﬂections of the slow mode on both ends of the bi-\nlayer, is much greater than 1. This happens at stripe widths wcorresponding to Fabry-Perot resonances of the slow mode,\nwhere [argS22+argS0\n22+(k2u+k2d)w]is a multiple of 2 \u0019, jus-\ntifying the postulated resonance condition Eq. (8). Since bis\na combination of multiple terms of similar magnitude, its de-\npendence on the stripe width is rather complicated. This ex-\nplains the variability of the shapes of individual resonances\nin Fig. 5(a) [also in Fig. 7(a) ].\nTo conﬁrm this interpretation of the role of the various\nterms in Eq. (18), let us visualize and compare the effects of\napplying successively stronger approximations to it. In Fig. 7,\nthe black symbols show the variation of the phase of the re-\nﬂection coefﬁcient obtained directly from numerical calcu-\nlations made with the ﬁnite-element modal method [in close\nagreement with the FD-FEM results from Fig. 5(a) ]. The red\nsolid curve in 7(a) shows the phase of the reﬂection coefﬁ-\ncient calculated from Eq. (18). The only approximation made\nin its derivation was to neglect evanescent coupling between\nthe left and right end of the bilayer; clearly, this approxima-\ntion is very well satisﬁed everywhere except for stripes nar-\nrower than 250 nm. The blue dashed curve in 7(a) shows the\neffect of applying the approximation (17) and setting \u0014to 1\nin the formula (19b) for b(but not in the formula (19a) for a).\nThis corresponds to neglecting terms proportional to prod-\nucts of more than two small scattering coefﬁcients; the result-\ning curve is almost indistinguishable from the previous one.\nNeglecting the second term \fbin Eq. (18) produces the red\nsolid curve in 7(b). The resonances are gone, but the long-\nterm increase in phase shift with stripe width is still repro-\nduced faithfully. Finally, the blue dashed curve in 7(b) shows\nthe result of approximating \u0014by 1 also in the formula (19a)\nfora. Its small deviation from the red curve conﬁrms the mi-\nnor role played by multiple reﬂections of the fast mode.\nF . Phase shift dependence on the layer separation\nIn Sec. III A we observed that the separation inﬂuences\nthe strength of the dynamical dipolar coupling between the\ninﬁnitely wide stripe and Py ﬁlm. It affects the SW disper-\nsion relation, especially the wavelengths of the slow modes\npropagating leftwards and rightwards. Therefore, according\nto Eq. (8), by varying the separation s, and hence kuand kd,\nwhile keeping the stripe width wconstant, it should be possi-\nble to sweep over resonances of different orders n. Indeed, we\nhave found multiple resonances in dependence of the phase\nshift on separation sfor a stripe of width w=1676 nm, as\nshown in Fig. 8.\nResonances do not appear periodically; the spacing be-\ntween subsequent Fabry-Perot resonances increases with the\naltitude of the stripe, and for the chosen stripe width the\nlast resonance occurs at the separation s=106 nm. This is\nbecause with increasing sthe coupling between the Py ﬁlm\nand the stripe weakens and the wavenumbers kuand kdap-\nproach their asymptotic limits. Snapshots of the magnetiza-\ntion at the resonances found at separations 10 nm, 28 nm,\n50 nm and 106 nm are shown in Figs. 8(c)–(e). These ﬁgures\ndemonstrate a clear enhancement of the SW amplitude be-\nlow the stripe and a decrease in the number of nodal point\nwith increasing separation, in line with the shift of the disper-\nsion relation of the slow mode towards smaller wavenumbers\nshown in Fig. 2(c).8\n(a)\n(b)\nFigure 7. Comparison of the reﬂection coefﬁcient phase calculated\nnumerically using the ﬁnite-element modal method FD-MM (black\npoints) with the semi-analytical model from Eq. (18) at varying de-\ngrees of approximation (color lines on both subplots). Details in the\nplot’s legends and in the text.\n(c)(a)\n(b)(c)(d)\n(d)λPy(e)\n(e)\n s = 106 nm s = 50 nm s = 10 nm\n s = 28 nm(b)\nFigure 8. (a) Phase shift dependence on the separation between the\nFM2 stripe and Py ﬁlm, calculated by the FD-FEM. (b)–(e) Snapshots\nof the dynamic magnetization mxat separations 10 nm, 28 nm,\n50 nm and 106 nm, corresponding to the resonances marked in plot\n(a). The dashed red lines mark the SW wavelength in the Py ﬁlm.\n5 6 7 8 9 10 11\nn101102separation,s(nm)\n17 nm\n10 nm\n5.5 nm106 nm\n50 nm\n28 nm\nn(s)\nresonances, FD-FEMFigure 9. Red line: dependence of the resonance index ncalcu-\nlated from Eq. (8) on the separation s, with'l+'rset to the ﬁtted\nvalue 0.12. Resonances are predicted to occur at integer values of n.\nBlack circles: positions of resonances found in FD-FEM calculations\nshown in Fig. 8.\nCombining the Fabry-Perot resonance condition, Eq. (8),\nwith the numerically calculated dispersion relations and set-\nting'l+'rto the ﬁtted value of 0.12, we calculate the depen-\ndence of non the separation between the stripe and the Py\nﬁlm, plotted with the red line in Fig. 9. In the range from 106\ndown to 5 nm, we ﬁnd six integer values of n, corresponding\nto successive resonances. These values agree well with the re-\nsults of FD-FEM calculations, where six resonances, marked\nwith black points in Fig. 9, are detected in that range of sepa-\nration.\nIV . CONCLUSIONS\nWe have studied theoretically the inﬂuence of a narrow fer-\nromagnetic stripe of subwavelength width placed at the edge\nof a ferromagnetic ﬁlm on the phase shift of reﬂected SWs. At\nthe considered frequency (11 GHz) the bilayer formed by the\nﬁlm and the stripe supports two pairs of slow (short wave-\nlength) and fast (long wavelength) guided SW modes prop-\nagating in opposite directions; these modes couple with the\nSW mode of the Py ﬁlm. This allowed us to interpret the nu-\nmerical results by modelling the system as a series of waveg-\nuides linked by junctions at which waveguide modes are scat-\ntered into each other.\nWe have found a strong nonlinear dependence of the phase\nshift on the stripe width. We showed that the reﬂection coef-\nﬁcient, from which the phase shift can be derived, consists\nof two terms, each having a different origin. One produces\na slow but steady increase of the phase shift with increas-\ning bilayer width and is dominated by the phase accumu-\nlated by the fast mode during a single round-trip across the\nbilayer. The other term has an appreciable effect on the re-\nﬂection coefﬁcient only when multiple reﬂections of the slow\nmode on both edges of the bilayer interfere constructively,\nwhich corresponds to Fabry-Perot resonances of this mode.\nInterestingly, the incoming wave from the Py ﬁlm couples\nstrongly only to the fast mode, but at Fabry-Perot resonances,\nthe phase of the reﬂected SW is controlled by the weakly cou-\npled slow mode. Essentially, this system is a realization of a\nGires–Tournois interferometer [35]operating on SWs. How-9\never, in this design, its width is smaller than the wavelengths\nof the incident waves and the interferometer utilitizes two\nnonreciprocal SW modes present in the bilayer.\nWe have also found that the phase shift of the reﬂected\nSW passes through a series of resonances as the separation\nbetween the stripe and the Py ﬁlm is increased. This unex-\npected effect originates from the dependence of the wave-\nlength of the slow SW in the bilayer on the strength of the\ndipolar coupling between the two layers. As a result, the bi-\nlayer width at which the resonance Fabry-Perot condition is\nsatisﬁed changes with the coupling strength as well, giving\nrise to the separation-dependent resonances.\nOverall, this research shows that SW Gires–Tournois inter-\nferometer can be used to modify the phase of reﬂected SWs\nin a wide range by tiny changes of the bilayer part width\nor stripe-ﬁlm distance. This is signiﬁcant for the further de-\nvelopment of magnonic devices where SW phase control is\nof key importance, in particular in integrated systems with\ncomponents smaller than the SW wavelength. This may in-\nclude the use of arrays of resonators in tunable SW optical\nelements, such as lenses, magnonic metasurfaces and phase\nshifters, as well as the sensing applications of magnonics, for\nexample the development of magnonic counterparts of sen-\nsors utilizing surface plasmon resonances.A. Acknowledgments\nThe research leading to these results has received funding\nfrom the Polish National Science Centre projects No. UMO-\n2015 /17/B/ST3/00118, UMO-2019 /33/B/ST5/02013, and\nUMO-2019 /35/D/ST3/03729. The simulations were partially\nperformed at the Poznan Supercomputing and Networking\nCenter (Grant No. 398).\nAppendix A: Micromagnetic simulations\nMicromagnetic simulations were performed in the\nmumax3environment [20]for the same magnetic param-\neters and geometry as described in Sec. II and damping\n\u000b=0.0001. The simulated structure was discretized on a\nmesh consisting of regular 5 \u0002100\u00025 nm3unit cells. In order\nto model a ﬁlm inﬁnitely extended along the yaxis, we have\nimposed periodic boundary conditions along the yaxis with\nassumed 1024 repetitions of the system along the yaxis.\nAfter stabilizing the system with a magnetic ﬁeld of value\n0.1 T applied along the yaxis, SWs were excited in the ﬁlm\nby a local source of microwave-frequency magnetic ﬁeld of\nfrequency 11 GHz and amplitude 0.1 mT placed at 3.6 mm\nfrom the right edge. To prevent wave reﬂections from the left\nedge of the ﬁlm, an absorbing zone with gradually increasing\ndamping was deﬁned on the left side of the system. A contin-\nuous harmonic SW excitation was maintained for 162.6 ns in\norder to reach a fully evolved (steady-state) interference pat-\ntern of the incident and reﬂected waves.\n[1]I. C. Ng and S. Y. Wakenshaw, The internet-of-things: Review\nand research directions, Int. J. Res. Mark. 34, 3 (2017).\n[2]M. Krawczyk and D. Grundler, Review and prospects of\nmagnonic crystals and devices with reprogrammable band\nstructure, J. Phys. Condens. Matter 26, 123202 (2014).\n[3]A. Chumak, V . Vasyuchka, A. Serga, and B. Hillebrands, Magnon\nspintronics, Nat. Phys. 11, 453 (2015).\n[4]A. V . Chumak, Fundamentals of magnon-based computing\n(2019), arXiv:1901.08934 [cond-mat.mes-hall ].\n[5]N. Yu and F . Capasso, Flat optics with designer metasurfaces,\nNat. Mater. 13, https: //doi.org /10.1038 /nmat3839 (2014).\n[6]N. Yu, P . Genevet, M. A. Kats, F . Aieta, J.-P . Tetienne, F . Capasso,\nand Z. Gaburro, Light propagation with phase discontinuities:\ngeneralized laws of reﬂection and refraction, Science 334, 333\n(2011).\n[7]K. Kumar, H. Duan, R. Hegde, S. C. W. Koh, J. N. Wei, and J. K. W.\nYang, Printing colour at the optical diffraction limit, Nat. Nan-\notechnol. 7, 557 (2012).\n[8]V . Vashistha, G. Vaidya, P . Gruszecki, A. E. Serebryannikov,\nand M. Krawczyk, Polarization tunable all-dielectric color ﬁl-\nters based on cross-shaped Si nanoantennas, Sci. Rep. 7, 8092\n(2017).\n[9]V . Kruglyak, C. Davies, Y. Au, F . Mushenok, G. Hrkac, N. White-\nhead, S. Horsley, T. Philbin, V . Poimanov, R. Dost, et al. , Graded\nmagnonic index and spin wave Fano resonances in magnetic\nstructures: Excite, direct, capture, in Spin Wave Conﬁnement\n(Jenny Stanford Publishing, 2017) pp. 11–46.\n[10]Y. Au, M. Dvornik, O. Dmytriiev, and V . Kruglyak, Nanoscale\nspin wave valve and phase shifter, Appl. Phys. Lett. 100, 172408\n(2012).[11]Y. Au, E. Ahmad, O. Dmytriiev, M. Dvornik, T. Davison, and\nV . Kruglyak, Resonant microwave-to-spin-wave transducer,\nAppl. Phys. Lett. 100, 182404 (2012).\n[12]T. Yu, Y. M. Blanter, and G. E. Bauer, Chiral pumping of spin\nwaves, Phys. Rev. Lett. 123, 247202 (2019).\n[13]T. Yu, C. Liu, H. Yu, Y. M. Blanter, and G. E. Bauer, Chiral excita-\ntion of spin waves in ferromagnetic ﬁlms by magnetic nanowire\ngratings, Phys. Rev. B 99, 134424 (2019).\n[14]H. Al-Wahsh, E. El Boudouti, B. Djafari-Rouhani, A. Akjouj,\nT. Mrabti, and L. Dobrzynski, Evidence of Fano-like resonances\nin mono-mode magnetic circuits, Phys. Rev. B 78, 075401\n(2008).\n[15]Z. Zhang, S. Liu, T. Wen, D. Zhang, L. Jin, Y. Liao, X.-L. Tang,\nand Z. Zhong, Bias-free reconﬁgurable magnonic phase shifter\nbased on a spin-current controlled ferromagnetic resonator, J.\nPhys. D 53, 105002 (2020).\n[16]H. Yu, G. Duerr, R. Huber, M. Bahr, T. Schwarze, F . Brandl, and\nD. Grundler, Omnidirectional spin-wave nanograting coupler,\nNat. Commun. 4, 2702 (2013).\n[17]H. Yu, O. d’ Allivy Kelly, V . Cros, R. Bernard, P . Bortolotti,\nA. Anane, F . Brandl, F . Heimbach, and D. Grundler, Approaching\nsoft x-ray wavelengths in nanomagnet-based microwave tech-\nnology, Nat. Commun. 7, 11255 (2016).\n[18]P . Graczyk, M. Krawczyk, S. Dhuey, W.-G. Yang, H. Schmidt, and\nG. Gubbiotti, Magnonic band gap and mode hybridization in\ncontinuous permalloy ﬁlms induced by vertical dynamic cou-\npling with an array of permalloy ellipses, Phys. Rev. B 98, 174420\n(2018).\n[19]S. Mieszczak, O. Busel, P . Gruszecki, A. N. Kuchko, J. W. Kłos,\nand M. Krawczyk, Anomalous refraction of spin waves as a way10\nto guide signals in curved magnonic multimode waveguides,\nPhys. Rev. Appl. 13, 054038 (2020).\n[20]A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F . Garcia-\nSanchez, and B. Van Waeyenberge, The design and veriﬁcation\nof mumax3, AIP Adv. 4, 107133 (2014).\n[21]COMSOL Multiphysics 5.1a, www.comsol.com, COMSOL AB,\nStockholm, Sweden.\n[22]J. Rychły and J. W. Kłos, Spin wave surface states in 1D planar\nmagnonic crystals, J. Phys. D: Appl. Phys. 50, 164004 (2017).\n[23]Q.-H. Liu and W. C. Chew, Numerical mode-matching method\nfor the multiregion vertically stratiﬁed media, IEEE Trans. An-\ntennas Propag. 38, 498 (1990).\n[24]W.´Smigaj, K. Sobucki, P . Gruszecki, and M. Krawczyk, in prepa-\nration.\n[25]D. D. Stancil and A. Prabhakar, Spin waves: Theory and applica-\ntions (Springer, 2009).\n[26]A. G. Gurevich and G. A. Melkov, Magnetization Oscillations and\nWaves (CRC Press, 1996).\n[27]M. Mruczkiewicz and M. Krawczyk, Nonreciprocal dispersion\nof spin waves in ferromagnetic thin ﬁlms covered with a ﬁnite-\nconductivity metal, J. Appl. Phys. 115, 113909 (2014).\n[28]R. Gallardo, T. Schneider, A. Chaurasiya, A. Oelschlägel,\nS. Arekapudi, A. Roldán-Molina, R. Hübner, K. Lenz, A. Barman,\nJ. Fassbender, et al. , Reconﬁgurable spin-wave nonreciprocity\ninduced by dipolar interaction in a coupled ferromagnetic bi-\nlayer, Phys. Rev. Appl. 12, 034012 (2019).[29]K. An, V . Bhat, M. Mruczkiewicz, C. Dubs, and D. Grundler, Op-\ntimization of spin-wave propagation with enhanced group ve-\nlocities by exchange-coupled ferrimagnet-ferromagnet bilay-\ners, Phys. Rev. Appl. 11, 034065 (2019).\n[30]K. Y. Guslienko, S. O. Demokritov, B. Hillebrands, and A. N.\nSlavin, Effective dipolar boundary conditions for dynamic\nmagnetization in thin magnetic stripes, Phys. Rev. B 66, 132402\n(2002).\n[31]J. Stigloher, T. Taniguchi, H. Körner, M. Decker, T. Moriyama,\nT. Ono, and C. Back, Observation of a Goos-Hänchen-like phase\nshift for magnetostatic spin waves, Phys. Rev. Lett. 121, 137201\n(2018).\n[32]R. Verba, V . Tiberkevich, and A. Slavin, Spin-wave transmission\nthrough an internal boundary: Beyond the scalar approxima-\ntion, Phys. Rev. B 101, 144430 (2020).\n[33]B. W. Zingsem, M. Farle, R. L. Stamps, and R. E. Camley, Un-\nusual nature of conﬁned modes in a chiral system: Directional\ntransport in standing waves, Phys. Rev. B 99, 214429 (2019).\n[34]J. P . H. G. Lecamp, P . Lalanne and J. Gérard, Energy transfer\nthrough laterally conﬁned bragg mirrors and its impact on pil-\nlar microcavities, IEEE J. Quantum Electron. 41, 1323 (2005).\n[35]F . Gires and P . Tournois, Interferometre utilisable pour la com-\npression d’impulsions lumineuses modulees en frequence, C.\nR. Acad. Sci. Paris. 258, 6112–6115 (1964)."    },    {        "title": "2307.16472v2.Gating_ferromagnetic_resonance_of_magnetic_insulators_by_superconductors_via_modulating_electric_field_radiation.pdf",        "content": "Gating ferromagnetic resonance of magnetic insulators by superconductors via\nmodulating electric-field radiation\nXi-Han Zhou and Tao Yu∗\nSchool of Physics, Huazhong University of Science and Technology, Wuhan 430074, China\n(Dated: September 20, 2023)\nWe predict that ferromagnetic resonance in insulating magnetic film with inplane magnetization\nradiates electric fields polarized along the magnetization with opposite amplitudes at two sides of the\nmagnetic insulator, which can be modulated strongly by adjacent superconductors. With a single\nsuperconductor adjacent to the magnetic insulator this radiated electric field is totally reflected with\naπ-phase shift, which thereby vanishes at the superconductor side and causes no influence on the\nferromagnetic resonance. When the magnetic insulator is sandwiched by two superconductors, this\nreflection becomes back and forth, so the electric field exists at both superconductors that drives\nthe Meissner supercurrent, which in turn shifts efficiently the ferromagnetic resonance. We predict\nan ultrastrong coupling between magnons in the yttrium iron garnet and Cooper-pair supercurrent\nin NbN with a frequency shift achieving tens of percent of the bare ferromagnetic resonance.\nI. INTRODUCTION\n“Magnonics” exploits magnetic excitations, i.e., spin\nwaves or their quanta, magnons, as potential infor-\nmation carriers for spin transport in insulators with\nlow-energy consumption [1–10]. Interaction between\nmagnons and Cooper-pair supercurrent in heterostruc-\ntures composed of magnets and superconductors may\nmodulate the transport of spin information [11–20],\nstrongly enhance the magnon-photon interaction [21–\n27], and lead to the emergence of triplet Cooper pair-\ning [28–32], which may bring unprecedented functionali-\nties in spintronics [28–30], quantum information [33–39],\nand topological quantum computation [40]. In this het-\nerostructure, the hybridized quantum states and distri-\nbution of macroscopic electromagnetic fields govern its\nproperties. For example, the “ultrastrong coupling” [41]\nwith the coupling strength close to the ferromagnetic\nresonance (FMR) frequency unveils the importance of\nthe dipolar interaction in the superconductor(S) |metallic\nferromagnet(F) |superconductor(S) heterostructure [22–\n24], where the photon mode with a large mode density\nis localized in the nano-scale between two superconduc-\ntors [42].\nThe importance of the dipolar interaction also mani-\nfests in the superconductor gating effect on magnons [14,\n15, 20, 43–46], in which the frequency of magnons with\nfinite wave number [47–50] can be shifted up to tens of\nGHz, as recently predicted [14, 15] and observed [20]\nin the superconductor(S) |ferromagnet insulator(FI) het-\nerostructure. The stray electric field of magnons drives\nthe supercurrent in the adjacent superconductor which\nin turn generates the Oersted magnetic field that affects\nthe low-frequency magnetization dynamics. This gating\neffect favors the spin diode [10, 51] and magnon trap [52–\n54] in proper gating configurations. The FMR frequency\nin this S |FI bilayer is not affected, however.\n∗taoyuphy@hust.edu.cn\nG)\u0010 G) [\u000bD\f\n),](\nG) G)\u0010 [\u000bE\f\n6 ),](\nG) G)\u0010GG 6 )\u0010 \u0010\nGG 6 )\u000e [\u000bF\f\n6 ), 6](\nG) G)\u0010\nGG 6 )\u0010 \u0010\nGG 6 )\u000e [\u000bG\f\n6 6 ),](FIG. 1. Snapshots of magnetization-radiated electric fields\nin different heterostructure configurations. The electric field\nchanges linearly across the thickness of the ferromagnetic in-\nsulating film. (a) The electric-field amplitude is opposite at\ntwo sides of the thin magnetic insulator. (b) When fabricat-\ning a superconductor thin film on a ferromagnetic insulator,\nthe electric field is suppressed to vanish at the superconduc-\ntor side but enhanced at the other side of the magnet. When\nthe magnet is sandwiched by two superconductors, the elec-\ntric field exists but differs at both sides in both symmetric (c)\nand asymmetric (d) configurations.\nOn the other hand, the FMR of the metallic ferromag-\nnet sandwiched by two superconductors was shifted up\nto 50 mT in the resonant field when the thickness of the\ntwo superconductor layers is larger than the London’s\npenetration depth, as observed in several recent exper-\niments [55–57]. Above the superconducting transition\ntemperature, the FMR frequency recovers to the Kittel\nmode [58], which may be exploited to realize the mag-\nnetic logic gate through a phase transition in the super-\nconductor. This phenomenon may be related to the fre-\nquency splitting induced by spin-triplet superconducting\nstate [55], Meissner screening [57], and giant demagneti-\nzation effects [16, 59]. It appears that this modulation\ncould be absent for the FMR in the ferromagnetic insula-\ntors [16, 55, 57, 59], however, which has not been reportedarXiv:2307.16472v2  [cond-mat.supr-con]  19 Sep 20232\nin the experiments yet [60–62]. Silaev predicted recently\nultrastrong coupling between magnons and microwave\nphotons in a magnetic insulator when sandwiched by\ntwo superconductors of infinite thickness, where the ra-\ndiation of the electric field out of the heterostructure is\ncompletely suppressed [25]. The experiment [55] showed\nthat inserting a thin insulator layer in the heterostruc-\ntures composed of a metallic ferromagnet sandwiched by\ntwo superconductors completely suppresses the shift of\nFMR. This raises the issue of whether the FMR can be\ngated or not in magnetic insulators by adjacent super-\nconductors in proper configurations.\nIn this work, we study this issue by going beyond the\nquasi-static approximation for magnetostatic modes [63]\nand demonstrate that although the stray magnetic field\nof Kittel magnon with uniform magnetization preces-\nsion is vanishingly small outside of the in-plane mag-\nnetized ferromagnetic insulating film, the radiated elec-\ntric field is significant with opposite amplitudes at two\nsides of the magnetic film and polarization parallel to\nthe magnetization direction. This distribution of the ra-\ndiated electric field is sensitive to the adjacent super-\nconductors due to the total reflection, as illustrated in\nFig. 1 for snapshots of the distribution of electric fields\nin different heterostructure configurations. The elec-\ntric field is opposite at two sides of a single thin ferro-\nmagnetic insulator [Fig. 1(a)]; contra-intuitively, in the\nS|FI bilayer this electric field is suppressed to vanish at\nthe superconductor side [Fig. 1(b)], when the supercon-\nductor thickness is larger than a nanometer; neverthe-\nless, when sandwiched by two superconductors, the elec-\ntric field is neither shifted to vanish nor screened com-\npletely, as plotted in Figs. 1(c) and (d) for symmetric\nand asymmetric configurations. These features are well\nunderstood by our mechanism of modulated reflection of\nmagnetization-induced electric fields by superconductors,\nwhich predicts the absence of FMR shift in ferromagnetic\ninsulator |superconductor heterostructure and the ultra-\nstrong modulation of FMR, shifted up to tens of percent\nof the bare frequency when the ferromagnetic insulator\nis sandwiched by two thin superconductors.\nThis paper is organized as follows. We address the\nmodel and general formalism in Sec. II. In Sec. III, IV,\nand V, we analyze the distribution of the electric fields\nfrom FMR of a single ferromagnetic insulator, S |FI bi-\nlayer, and S |FI|S heterostructure, respectively, and ad-\ndress the ultrastrong interaction between the FMR and\nsupercurrent. We conclude and discuss in Sec. VI.\nII. MODEL AND GENERAL FORMALISM\nWe consider a heterostructure composed of a ferromag-\nnetic insulating film of thickness 2 dF∼O(100 nm) with\ninplane magnetization sandwiched by two thin supercon-\nductor layers with thickness dS≲λandd′\nS≲λ, re-\nspectively, as illustrated in Fig. 2. Here λ∼O(100 nm)\nis London’s penetration depth of conventional supercon-ductors. In the ferromagnetic insulators, the dynamics of\nmagnetization M=Mxˆx+Myˆy+M0ˆz, where M0is the\nsaturated magnetization, is phenomenologically governed\nby the Landau-Lifshitz-Gilbert (LLG) equation [64]\n∂M/∂t=−µ0γM×H+αG(M/M0)×∂M/∂t, (1)\nwhere µ0is the vacuum permeability, −γis the electron\ngyromagnetic ratio, and αGis the damping coefficient\nof the magnetic insulator. The magnetization precesses\naround the effective magnetic field H=Happ+Hrthat\ncontains the external static field Happ=H0ˆzand the\nradiated dynamic field Hrgenerated by the “magnetic\ndipole radiation” [14, 65]. The energy flow out of the\nmagnetic insulator then causes the radiation damping\nsince the radiated magnetic field out of phase of the mag-\nnetization can exert a damping-like torque on the mag-\nnetization. The exchange interaction plays no role in the\nFMR since the gradient of Mvanishes for the uniform\nprecession.\nSuperconductor\nFerromagnet Insulatory\nSuperconductor\nzx\nSuperconductor (1)\n(2)sJ\n'sJSd\nFd2\n'Sd0M\nFIG. 2. S(1) |FI|S(2) heterostructure. The thickness of super-\nconductors above and beneath the thin ferromagnetic insula-\ntor of thickness 2 dFisdSandd′\nS, respectively. The driven\nsupercurrents JsandJ′\nsby FMR flow oppositely along the\nmagnetization direction.\nThe oscillating magnetic induction B=µ0(M+H)\ngoverns the radiation of electric fields inside and outside\nthe ferromagnetic insulator according to [65]\n∇ ×E=−∂B\n∂t, ∇ ×H=Js+ε0∂E\n∂t,(2)\nwhere ε0is the vacuum permittivity. When coupled with\nsuperconductors, this electric field drives the supercur-\nrentJsvia London’s equation [66]\n∂Js\n∂t=1\nµ0λ2E, ∇ ×Js=−1\nµ0λ2B.(3)\nHere London’s penetration depth at different tempera-\ntures T < T cfollows the relation [66]\nλ(T) =λ0 \n1−\u0012T\nTc\u00134!−1/2\n, (4)3\nwhere λ0is London’s penetration depth at zero temper-\nature.\nThe boundary condition describes the fields at the in-\nterfaces [65]. For the magnetic induction and field, B⊥\nandH∥are continuous at the boundaries. Since there is\nno surface current or charge accumulation, the electric\nfieldEis continuous at interfaces.\nAt low frequencies and with near fields, the quasi-static\napproximation is usually applied [65], in which situation\nthe radiation damping should be negligibly small. This\nis proved according to the calculation of radiation damp-\ning in Sec. III A. It is then sufficient to express the radi-\nated magnetic field Hras the summation of the dipolar\nfieldHdand the Oersted field Hsfrom the superconduc-\ntor [63, 64]. The dipolar field\nHd,β(M) =1\n4π∂βX\nα∂αZ\ndr′Mα(r′)\n|r−r′|\n=1\n4π∂βZ\ndr′−ρm(r′)\n|r−r′|(5)\nis governed by Coulomb’s law in terms of the magnetic\ncharge ρm=−∇ ·M.\nWith the quasi-static approximation, ∇×B=µ0Jsin\nsuperconductors. Taking the curl of Eq. (2) and substi-\ntuting Eq. (3) into it, the electric field inside the super-\nconductor obeys\n∇2E−E/λ2= 0. (6)\nOn the other hand, taking the curl of ∇×B=µ0Jsand\ncombining with Eq. (3), the magnetic induction inside\nthe superconductor obeys ∇2B−B/λ2= 0. The driven\nsupercurrent then affects the magnetization dynamics.\nFrom Eq. (3), the electric field drives supercurrent inside\nthe superconductor, which then generates the vector po-\ntential. With the uniform magnetization precession, the\nsystem is translational invariant in the y-zplane, so the\nsupercurrent only depends on xand as it for the vector\npotential [65]\nA(x) =µ0\n4πZ\ndr′Js(x′)\n|r−r′|. (7)\nAccordingly, the Oersted magnetic field\nHs= (1/µ0)∇ ×A (8)\nonly contains the y-component Hy=−∂xAz(x)/µ0,\nwhich drives the magnetization.\nIII. SINGLE THIN FERROMAGNETIC\nINSULATOR\nWe start with a single insulating ferromagnetic film\nto address the significant radiated electric fields from the\nuniform magnetization precession. For a single ferromag-\nnetic insulator of thickness 2 dFbiased by a static mag-\nnetic field Happ=H0ˆz, the magnetization Mfor theFMR is uniform inside the ferromagnetic layer by the\nconstant demagnetization factor Nxx=−1. Since the\nmagnetic film is sufficiently thin, we stick to the uniform\nprecession throughout this work. The opposite magnetic\ncharges at the two surfaces of the film generate opposite\nmagnetic field outside, which results in vanished stray\nmagnetic field Hd= 0 outside the ferromagnetic layer,\nas also calculated from Eq. (5); inside the ferromagnet,\nHd={−Mx,0,0}andB={0, µ0My, µ0(H0+M0)},\nin which only the y-component of Boscillates with fre-\nquency ωthat can radiate the electric field.\nA. Full solution\nHere we go beyond the quasi-static approximation and\nsolve the radiated electric field. According to Eq. (2), the\noscillating electromagnetic field is the source for radiating\nmicrowaves in space. Taking the curl of the first equation\nin Eq. (2), the electric field of frequency ωobeys\n∇2E+ε0µ0ω2E=−iωµ0∇ ×M. (9)\nSuch a radiation process is governed by the oscillating\n“magnetization current” JM=∇ ×M, which is anal-\nogous to the radiation caused by the normal oscillating\ncharge current [65].\nVia the Green function technique [65], Eq. (9) has the\nsolution\nE(r) =iµ0ω\n4πZ[∇′×M(r′)]eik|r−r′|\n|r−r′|dr′, (10)\nwhere k=ω/cis the wave number of microwaves. Since\nonly the xandycomponents of Moscillate with fre-\nquency ωandMis uniform inside the ferromagnetic\nlayer, ( ∇×M)x,y= 0 in all space, leading to Ex=Ey= 0\nand\nEz(x) =iµ0ω\n4πZ[∂x′My(r′)]eik|r−r′|\n|r−r′|dr′. (11)\nUsing Weyl identity [10]\neik|r−r′|\n|r−r′|=Z\ndk′\nzdk′\nyieik′\nz(z−z′)+ik′\ny(y−y′)ei√\nk2−k′2z−k′2y|x−x′|\n2πq\nk2−k′2z−k′2y,\n(12)\nwe obtain the electric field\nEz=µ0ωMy\n2k\n\ne−ik(x−dF)−eik(x+dF),−dF< x < d F\neik(x−dF)−eik(x+dF), x > d F\ne−ik(x−dF)−e−ik(x+dF), x < −dF.\n(13)\nFrom Eq. (2), we find the magnetic induction Bx= 0,\nBz=µ0(H0+M0) is static, and By=−∂xEz/(iω) fol-4\nlows\nBy=µ0My\n2\n\neik(x+dF)+e−ik(x−dF), −dF< x < d F\neik(x+dF)−eik(x−dF), x > d F\n−e−ik(x+dF)+e−ik(x−dF), x < −dF.\n(14)\nWe can understand the radiated electric field (13) well\nvia the oscillating “magnetization current” JM. For the\nuniform magnetization precession, JMis located at the\nsurfaces of the ferromagnetic insulator, i.e., the dynamic\ncomponent\nJM(x) = [δ(x+dF)−δ(x−dF)]Myˆz∝My (15)\nhas the same magnitude but opposite sign at two sur-\nfaces x=±dF, as illustrated in Fig. 3. Such oscillating\nmagnetization current then radiates the electromagnetic\nwaves of wave vector kˆxand−kˆxwith k=ω/cinto\ntwo opposite directions. Due to the opposite sign of JM\natx=±dF, the amplitudes of the electric fields radi-\nated by the left and right surfaces are of opposite sign\nEL=−ER≡E0∝My. At the right-hand side of the\nsample, i.e., x > d F, the propagation phases of the ra-\ndiated electric field from the left and right surfaces are\nk(x+dF) and k(x−dF), respectively, resulting in a net\nelectric field E=E0(eik(x+dF)−eik(x−dF)). Similarly,\nwhen x <−dF,E=E0(e−ik(x−dF)−e−ik(x+dF)). These\nrecover exactly the solution (13).\nFI\nxˆk\nMJMJM\nxˆk\nxˆ\nFdFd Οxˆk xˆk\nFIG. 3. Electric field radiated from the surface magnetization\ncurrent at the two surfaces of the magnetic insulator.\nWith the full solutions (13) and (14), we are allowed\nto calculate the radiation damping of the FMR due to\nthe energy radiated out of the magnetic insulator. Ac-\ncording to Eq. (14), the radiated magnetic field inside\nthe magnetic insulating film\nHr\nx=−Mx,\nHr\ny=iωdFMy\nc=−dF\ncdMy\ndt(16)\ndrives the magnetization, leading to the linearized LLG\nequation\n−iωM x+µ0γMyH0=i(αG+αR)ωMy,\niωM y+µ0γH0Mx=−µ0γM0Mx+iαGωMx,(17)where the damping coefficient contributed by the radia-\ntion reads\nαR=µ0γM0dF/c. (18)\nIt is negligibly small: for the YIG film of thickness\n2dF= 120 nm and µ0M0= 0.2 T [67, 68], αR≈\n7.3×10−6≪αG∼5×10−4. However, the radiation\ndamping is enhanced with thicker films.\nWe are interested in the field near the ferromagnet\nwith a distance ∼λ. In ferromagnetic insulators, ω∼\n2π×4 GHz [14], and λ∼100 nm for conventional su-\nperconductors, so kλ∼10−5≪1. When kx→0, we\nhave\nEz(x) =\n\n−iµ0ωMyx,−dF< x < d F\n−iµ0ωMydF, x > d F\niµ0ωMydF, x < −dF, (19)\nas plotted in Fig. 1(a) for a snapshot. The magnetic\ninduction\nBy(x) =\n\nµ0My,−dF< x < d F\n0, x > d F\n0, x < −dF(20)\nrecovers to the results from quasi-static approxima-\ntion [63] with vanishing magnetic field Hyoutside of the\nferromagnet.\nB. Quasi-static approximation\nThe above analysis implies that when focusing on the\nnear-field limit, we may apply the quasi-static approxi-\nmation that sets ∇×H= 0 in Eq. (2). When focusing on\nthe FMR case, Eis translation invariant in the y-zplane.\ni.e.,∂zEx= 0. Taking the y-component of Eq. (1), the\noscillation of Byonly generate Ezparallel to the magne-\ntization:\n−∂xEz=iωµ0My. (21)\nIntegrating along xacross the ferromagnet yields\nEz(x) =−iωµ0My(x+dF) +Ez(x=−dF).(22)\nThereby, Ezdepends linearly on xinside the ferromag-\nnet. Outside the ferromagnet,\nEz(x) =−2iωµ0MydF+Ez(x=−dF) (23)\nis uniform, which is consistent with the vanished mag-\nnetic field Hy|outside = 0 in the quasi-static approxima-\ntion. According to the symmetry, Ez(x= 0) = 0, so the\nelectric field is exactly the same as Eq. (19).5\nIV. S |FI HETEROSTRUCTURE\nWe consider the S |FI heterostructure composed of a\nferromagnetic film of thickness 2 dFand a superconductor\nof thickness dS, as shown in Fig. 4. We demonstrate the\nadjacent superconductors modulate strongly the radiated\nelectric field which explains the absence of the FMR shift\nin this configuration [20, 55].\nS\nikxeE1\nikxeE'1xikeE'\n2'xikeE'\n2ikxeE3FI\nikxeE\n4\nFdFdS Fd dxy\nFIG. 4. Radiated electric field of the FI |S heterostructure.\nA. Full solution\nInside the ferromagnet, since ∇ ×M= 0 for uniform\nM, Eq. (9) has the solution Ez(x) =E1eikx+E′\n1e−ikx.\nInside the superconductor, according to Eqs. (1) and (3),\nthe electric field obeys\n∂2\nxEz+ (ε0µ0ω2−1/λ2)Ez= 0, (24)\nwhich has the solution Ez(x) =E2eik′x+E′\n2e−ik′x, where\nk′=p\n(ω/c)2−1/λ2≈i/λis purely imaginary with\nmicrowave frequencies. For example, with frequency ω∼\n2π×4 GHz, k=ω/c∼83.8 m−1is much smaller than\n1/λ∼107m−1with London’s penetration depth λ∼\n100 nm. Therefore, due to the Meissner effect, the low-\nfrequency electromagnetic waves no longer propagate but\ndecay in the superconductor. Out of the heterostructure,\nthe electric fields E3eikxandE4e−ikxare radiated. These\nradiated electric fields are illustrated in Fig. 4.\nThe amplitudes {E1, E′\n1, E2, E′\n2, E3, E4}are governed\nby the boundary conditions, i.e., EzandHyare continu-\nous at interfaces. The continuous Ezat interface requests\nE1eikdF+E′\n1e−ikdF=E2eik′dF+E′\n2e−ik′dF,\nE2eik′(dF+dS)+E′\n2e−ik′(dF+dS)=E3eik(dF+dS),\nE1e−ikdF+E′\n1eikdF=E4eikdF. (25)\nIn the superconductors, Hy=−1/(iωµ0)∂xEz, while in\nthe ferromagnet, Hy=−1/(iωµ0)∂xEz−My, so the con-tinuous Hyat interfaces leads to\nk′(E2eik′dF−E′\n2e−ik′dF) =k(E1eikdF−E′\n1e−ikdF)\n+ωµ0My,\nk′(E2eik′(dF+dS)−E′\n2e−ik′(dF+dS)) =kE3eik(dF+dS),\nk(E1e−ikdF−E′\n1eikdF) +ωµ0My=−kE4eikdF.(26)\nCombining Eqs. (25) and (26), we obtain all the am-\nplitudes. In the ferromagnetic insulator,\nEz(−dF< x < d F) =RE0e−ik(x−dF)+Esingle(x),\n(27)\nwhere the amplitude E0=−[ωµ0My/(2k)]\u0000\ne2ikdF−1\u0001\n,\nEsingle(x) is the radiated electric field from a single mag-\nnetic insulator [Eq. (13)], and\nR=eik′dS(k2−k′2) +e−ik′dS(k′2−k2)\neik′dS(k−k′)2−e−ik′dS(k+k′)2(28)\nis the reflection coefficient of the electric field at the su-\nperconductor surface.\nWe plot the dependence of Ron the superconductor\nthickness dSin Fig. 5 with different London’s penetration\ndepth λunder the frequency ω∼2π×4 GHz. The\nreflection coefficient saturates to R → − 1 when dS>\n0.1 nm, but is reduced to 0 when dS→0, recovering\nthe solution (13) of the single layer case. We conclude\nthat even with a small dS≪λ, since |k|=ω/cis much\nsmaller than |k′| ≈1/λwhen ω∼2π×4 GHz, R → − 1.\nThis implies the total reflection of the electric fields at\nthe FI |S interface even with an ultrathin conventional\nsuperconductor layer. As shown below, this indicates\nthe absence of FMR shift in all the available experiments\nwith thick superconductors [20, 55].\n/uni00000013 /uni00000013/uni00000011/uni00000018 /uni00000014 /uni00000014/uni00000011/uni00000018\n/uni00000047\n/uni00000036/uni0000000b/uni00000051/uni00000050/uni0000000c/uni000000ed/uni00000014/uni00000011/uni00000013/uni000000ed/uni00000013/uni00000011/uni00000018/uni00000013/uni00000011/uni00000013/uni00000035/uni00000048/uni0000000b /uni0000000c/uni00000003/uni00000226 /uni00000020 /uni00000015/uni0000028c /uni000000ee /uni00000017 /uni0000002a/uni0000002b/uni0000005d/uni0000021c /uni00000020 /uni00000014/uni00000013/uni00000013 /uni00000051/uni00000050\n/uni0000021c /uni00000020 /uni00000014 /uni0000021d/uni00000050\nFIG. 5. Reflection coefficient Re( R) as a function of the su-\nperconductor thickness dSwith different London’s penetra-\ntion depth λ= 100 nm and 1 µm. We take the frequency\nω= 2π×4 GHz.6\nInside the superconductor,\nEz(dF< x < d F+dS) =2kE0\neik′dS(k−k′)2−e−ik′dS(k+k′)2\n×\u0010\n(k−k′)e−ik′(x−dF+dS)−(k+k′)eik′(x−dF−dS)\u0011\n,\n(29)\nwhich is indeed very weak since |k| ≪ | k′|. Out of the\nheterostructure,\nEz=\n\n−4kk′E0eik(x−dF−dS)\neik′dS(k−k′)2−e−ik′dS(k+k′)2, x > d F+dS\nRE0e−ik(x−dF)+Esingle(x), x < −dF.\n(30)\nAt low frequencies and near the heterostructure, kx→0,\nkdF→0, and kdS→0, so the electric fields\nEz(x) =\n\n0, x > d F\n−iωµ0My(x−dF),−dF< x < d F\n2iωµ0MydF, x < −dF,(31)\nwhich is illustrated in Fig. 1(b) for a snapshot. The elec-\ntric field vanishes in the superconductor due to the total\nreflection with a π-phase shift R=−1 that generates no\nsupercurrent and thereby leads to no modulation on the\nFMR.\nB. Quasi-static approximation\nThe full solution clearly shows the absence of elec-\ntric fields at the superconductor side of the S |FI het-\nerostructure, which can be well understood within the\nquasi-static approximation ∇ ×H= 0 or Js. Assum-\ningEz(x=dF) =˜E0at the FI |S interface, according to\nEq. (6) the electric field in the adjacent superconductor\nEz(x) =˜E0cosh (( x−dS−dF)/λ)\ncosh ( dS/λ)(32)\ndrives the supercurrent. For a thin superconducting film\nof thickness O(λ), we are allowed to take an average of the\nsupercurrent Js,z= [Js,z(x=dF)+Js,z(x=dF+dS)]/2,\nand from the first equation of Eq. (3)\nJs,z=i\nµ0ωλ2˜E01 + cosh( dS/λ)\n2 cosh( dS/λ). (33)\nThe supercurrents generate the vector potential (7) and\nthe Oersted magnetic field according to Hy=−∂xAz/µ0.\nTaking k= 0 at low frequencies in the Weyl identity (12),\ni.e., [10]\n1\n|r−r′|=Z\ndk′\nxdk′\nyeik′\nx(x−x′)+ik′\ny(y−y′)e−√\nk′2x+k′2y|z−z′|\n2πq\nk′2x+k′2y,\n(34)we obtain the Oersted magnetic field generated by the\nsupercurrents\nHs,y(x) =\u001a\ndSJs,z/2, x > d F+dS\n−dSJs,z/2, x < d F.(35)\nHowever, constant Hs,yindependent of xshould vanish\nout of the heterostructure within the quasi-static approx-\nimation since a constant magnetic field renders the ra-\ndiated electric field divergent, which requests Js,z= 0\nwhen dS̸= 0 and Ez(x > d F) = 0. Since the electric\nfield is continuous at interfaces, Ez(x=dF) = ˜E0= 0\nand according to Eq. (21) Ez(x=−dF) = 2 idFωµ0My.\nThese simple calculations thereby capture precisely the\nkey physics of the full solution (31).\nV. S |FI|S HETEROSTRUCTURE\nFurther, we consider the S |FI|S heterostructure as il-\nlustrated in Fig. 2 composed of the ferromagnetic insu-\nlator of thickness 2 dFand two adjacent superconductor\nfilms of thickness dSandd′\nS, respectively. In comparison\nto that of the S |FI bilayer, the distribution of the elec-\ntric field in S |FI|S heterostructure changes much due to\nits back-and-forth reflection by the superconductors, as\naddressed in this section.\nA. Full solution\nSimilar to the S |FI heterostructure, inside the ferro-\nmagnet, Ez(x) =E1eikx+E′\n1e−ikx; in the superconduc-\ntor “1”, Ez(x) =E2eik′x+E′\n2e−ik′x; and in the supercon-\nductor “2”, Ez(x) =E3eik′x+E′\n3e−ik′x. Out of the het-\nerostructure, the electric fields E4eikxandE5e−ikxare\nradiated. These electric fields are illustrated in Fig. 6.\nS\nikxeE1\nikxeE'1xikeE'\n2'xikeE'\n2ikxeE4FI\nikxeE\n5\nFdFdS Fd dxy\nxikeE'\n3\nxikeE'\n3'\n'S Fd dS (1) (2)\nFIG. 6. Radiated electric field of the S |FI|S heterostructure.\nThe amplitudes {E1, E′\n1, E2, E′\n2, E3, E′\n3, E4, E5}are\ngoverned by the boundary conditions. The continuous7\nEzat interfaces requests\nE1eikdF+E′\n1e−ikdF=E2eik′dF+E′\n2e−ik′dF,\nE1e−ikdF+E′\n1eikdF=E3e−ik′dF+E′\n3eik′dF,\nE2eik′(dF+dS)+E′\n2e−ik′(dF+dS)=E4eik(dF+dS),\nE3e−ik′(dF+d′\nS)+E′\n3eik′(dF+d′\nS)=E5eik(dF+d′\nS),(36)\nand the continuous Hyat interfaces leads to\nk′(E2eik′dF−E′\n2e−ik′dF) =k(E1eikdF−E′\n1e−ikdF)\n+ωµ0My,\nk′(E3e−ik′dF−E′\n3eik′dF) =k(E1e−ikdF−E′\n1eikdF)\n+ωµ0My,\nk′(E2eik′(dF+dS)−E′\n2e−ik′(dF+dS)) =kE4eik(dF+dS),\nk′(E3e−ik′(dF+d′\nS)−E′\n3eik′(dF+d′\nS)) =−kE5eik(dF+d′\nS).\n(37)\nCombining Eqs. (36) and (37), we obtain the electric-\nfield distribution. In particular, when dS=d′\nS, in the\nferromagnetic film,\nEz(|x|< dF) =−ωµ0Mysinh ( ikx)\nkcosh ( ikdF)−k′f(u) sinh ( ikdF),\n(38)\nwhere u=−[(k+k′)/(k−k′)] exp( −2ik′dS) and\nf(u) =u−1\nu+ 1=k′sinh ( ik′dS)−kcosh ( ik′dS)\nksinh ( ik′dS)−k′cosh ( ik′dS).(39)\nIn the superconductor “1”,\nEz(dF< x < d F+dS)\n=−ωµ0My(ueik′(x−dF)+e−ik′(x−dF))\nk(1 +u) coth( ikdF)−k′(u−1),(40)\nand in the superconductor “2”,\nEz(−dF−dS< x < −dF)\n=ωµ0My(ue−ik′(x+dF)+eik′(x+dF))\nk(1 +u) coth( ikdF)−k′(u−1). (41)\nThey both exist, and Ez(x=−dF) and Ez(x=dF)\nare opposite. This feature may be understood from the\nmagnetic dipole radiation: since the magnetization cur-\nrentJM(15) is opposite at the two surfaces x=±dF\nof the magnetic film, the amplitudes of the electric fields\nradiated by the two surfaces x=±dFare of opposite\nsign, which launches to superconductors and drives the\nopposite supercurrents in them.\nOut of the heterostructure,\nEz(x > d F+dS)\n=−ωµ0My(ueik′dS+e−ik′dS)\nk(1 +u) coth( ikdF)−k′(u−1)eikx,\nEz(x <−dF−dS)\n=ωµ0My(ueik′dS+e−ik′dS)\nk(1 +u) coth( ikdF)−k′(u−1)e−ikx, (42)which, when far away from the heterostructure, is re-\nduced to a simpler form\nEz(x)≈iωµ0dFλMy\nλcosh ( dS/λ) +dFsinh ( dS/λ)\n×(\n−eikx, x≫dF+dS\ne−ikx, x≪ −(dF+dS). (43)\nThe radiation out of the heterostructure is then com-\npletely suppressed when dS≫λ.\nWe refer to Appendix A for the solution of asymmetric\nconfiguration. We illustrate in Fig. 7 the distribution of\nthe electric fields Re( Ez/(iωµ0MydF)) at T= 0.5Tc=\n5.5 K in the symmetric d′\nS=dS= 60 nm and asym-\nmetric d′\nS= 2dS= 120 nm S |FI|S heterostructure, re-\nspectively, in the near-field limit. For NbN, Tc= 11 K,\nthe London penetration depth λ(T= 0) = 85 nm [69]\nandλ(T= 0.5Tc) = 87 .8 nm. The fields are opposite at\nthe two superconductors in the symmetric heterostruc-\nture but are skewed when dS̸=d′\nS. These fields carrying\nenergy are radiated out in the far zone [65]. When the su-\nperconductors are sufficiently thick {dS, d′\nS} ≫λ, these\nelectric fields are confined between them, which corre-\nsponds to an excellent waveguide with small size [42].\nB. Ultrastrong interaction between Kittel magnon\nand Cooper-pair supercurrent\nAbove we address that the dynamics of magnetization\nMgenerates Hr\nyvia the backaction of superconductors,\nwhich, in turn, drives Min the ferromagnet, imposing\na self-consistent problem that is solved by combining\nthe Landau-Lifshitz and Maxwell’s equations. In other\nwords, the precession of the magnetization radiates the\nelectric field that drives the supercurrent in the supercon-\nductor via microscopically generating the center-of-mass\nmomentum of the Cooper pairs. Such a collective motion\nof Cooper pairs, i.e., the supercurrent in turn generates\nthe Oersted magnetic field that affects the dynamics of\nthe magnetization, i.e., a shift in its FMR frequency.\nUsing Eq. (38) and By=−∂xEz/(iω), we find the\nradiated magnetic field inside the ferromagnetic insulator\nof the symmetric S |FI|S heterostructure\nHr\ny(|x|< dF) =Mykcosh ( ikx)\nkcosh ( ikdF)−k′f(u) sinh ( ikdF)−My,\n(44)\nwhich drives the precession of the magnetization. In\nterms of the (linearized) LLG equation (1), we arrive at\n−iωM x+µ0γMyH0=µ0γM0Hr\ny+iαGωMy,\nµ0γH0Mx+iωM y=−µ0γM0Mx+iαGωMx.(45)\nWe see that the real part of the radiated magnetic field\n(44) is in the same phase of My, which provides a field-\nlike torque for the magnetization. Retaining the leading8\nFIG. 7. Distribution of electric fields in symmetric dS=\nd′\nS= 60 nm [(a)] and asymmetric d′\nS= 2dS= 120 nm\n[(b)] S |FI|S heterostructure. The thickness of the ferromag-\nnetic film 2 dF= 120 nm and London’s penetration depth\nλ(T= 0.5Tc) = 87 .8 nm.\norder in k, the homogeneous\nRe(Hr\ny) =−dFtanh( dS/λ)\nλ+dFtanh( dS/λ)My (46)\nrenormalizes the FMR frequency to be\nωK=µ0γs\n(H0+M0)\u0012\nH0+dFtanh( dS/λ)\nλ+dFtanh( dS/λ)M0\u0013\n,\n(47)\nwhich differs from the bare Kittel frequency ˜ ωK=\nµ0γp\nH0(H0+M0) [58]. When dS≫λ, the solution\n(47) recovers that in Ref. [25], where an ultrastrong cou-\npling between magnons and microwave photons is pre-\ndicted in a magnetic insulator when sandwiched by two\nsuperconductors of infinite thickness.\nOn the other hand, the imaginary part of the radiated\nmagnetic field is out of phase of My, which thereby con-\ntributes to a damping-like torque. Retaining the leading\norder in k,\nIm(Hy)≈MykdF\ncosh2(dS/λ)\u0012\n1 +dFtanh ( dS/λ)\nλ\u0013−2contributes to a damping coefficient\nαR=µ0γM0dF\nccosh2(dS/λ)\u0012\n1 +dFtanh ( dS/λ)\nλ\u0013−2\n.\nIn comparison to a single layer of magnetic insulator\n(Sec. III A), the radiation of magnetization is suppressed\nwhen shielded by two superconductors, and the radia-\ntion damping is expected to be reduced. With dS=dF=\n60 nm, λ∼85 nm, and ω∼2π×4 GHz, αR≈2.2×10−6\nis indeed smaller than that of a single magnetic insulator\n(7.3×10−6). When dS≫λ,αR→0 since no field is\nradiated out of the S |FI|S heterostructure.\nThe general solution of ωK[Eq. (A8)] and αR\n[Eq. (A10)] in the asymmetric S |FI|S heterostructure is\ncalculated in Appendix. A. In Appendix B, we calculate\nthem with the quasistatic approximation.\nTo show the FMR shift, we assume an oscillating mag-\nnetic field ˜He−iω0tˆyof frequency ω0applied along the ˆ y-\ndirection (the associated microwave electric field is along\nthe normal ˆx-direction). The wavelength of this mi-\ncrowave is much larger than the thickness of the het-\nerostructure, so it can be treated as uniform across the\nheterostructure thickness. It can penetrate the supercon-\nductor easily when {dS, d′\nS} ∼λ. With the wave vector\n(along ˆz) parallel to the film, it only excites Min the\nferromagnet but does not drive the superconductor.\nIncluding the external pump field ˜He−iω0tˆyinto the\nLLG equation (45), we find when αG≪1\nMy=µ2\n0γ2M0(H0+M0)\nω2\nK−ω2\n0−iΓ˜H,\nMx=−iMy\u0014ω0\nµ0γ(H0+M0)+iαGω2\n0\n(µ0γ(H0+M0))2\u0015\n,\n(48)\nwhere\nΓ =αGω3\n0\nµ0γ(H0+M0)+µ0γ(H0+M0)(αG+αR)ω0.\n(49)\nFrom Eq. (40), we find the average electric field Ez=\n[Ez(x=dF) +Ez(x=dF+dS)]/2 in the thin supercon-\nductor “1” as\nE(1)\nz=−˜H\n2ωµ0(u+ 1 + ueik′dS+e−ik′dS)\nk(1 +u) coth( ikdF)−k′(u−1)\n×µ2\n0γ2M0(H0+M0)\nω2\nK−ω2\n0−iΓ. (50)\nFrom Eq. (3), the corresponding average suppercurrent\ninside the superconductor is\nJ(1)\nz=−i˜H\n2λ2u+ 1 + ueik′dS+e−ik′dS\nk(1 +u) coth( ikdF)−k′(u−1)\n×µ2\n0γ2M0(H0+M0)\nω2\nK−ω2\n0−iΓ. (51)9\nWe illustrate the numerical results considering a yt-\ntrium iron garnet (YIG) film of thickness 2 dF= 120 nm\nsandwiched by two NbN superconductors of thickness\ndS=d′\nS= 60 nm. Insulating EuS thin magnetic\nfilm [70, 71] is also a possible candidate to test our predic-\ntion. For YIG, µ0M0= 0.2 T and αG= 5×10−4[67, 68].\nWe use λ(T= 0.5Tc) = 87 .8 nm for NbN [69] . We take\nthe bias field µ0H0= 0.05 T and the excitation field\nµ0˜H= 0.01 mT. Figure 8 shows the radiated electric\nfield in (one of) the superconductors and the excited am-\nplitudes of Mas a function of the excitation frequency\nω0. The frequency shift is 2 π×1.6 GHz, comparable to\nhalf of the bare FMR frequency ˜ ωK= 2π×3.2 GHz,\ncorresponding to the decrease of the resonant magnetic\nfield as large as 55 mT. This demonstrates the potential\nto achieve ultrastrong interaction between magnons and\nCooper-pair supercurrent even with magnetic insulators.\nBefore we address the temperature dependence of the\nfrequency shift, we first show that the normal current\nmainly provides additional damping to the FMR with a\ntiny frequency shift even when T→Tc. We estimate\nthe contribution of the normal current via the two-fluid\nmodel with the conductivity at low frequencies [66]\n˜σ(ω)≈ρne2τ\nme+iρse2\nme1\nω=σn+i1\nωµ0λ2. (52)\nwhere τis the relaxation time of electrons and ρn(ρs) is\nthe normal fluid (superfluid) density. ρnequals to the\nelectron density newhen T > T c. Incorporating the\nconductivity (52) into Maxwell’s equation, the radiated\nmagnetic field contributed by both the normal and su-\npercurrents in the symmetric S |FI|S heterostructure (to\nthe leading order of k) reads\n˜Hy=Myi˜kdF(˜ktanh ( i˜kdS)−k)\ntanh ( i˜kdS)(k−i˜k2dF) +˜k(ikdF−1),(53)\nwhere ˜k2=iωµ0σn−1/λ2, with which we find the FMR\nfrequency and the additional damping coefficient\nωK=µ0γp\nH0+M0q\nH0−M0Re(˜Hy)/My,\n˜α=µ0γM0Im(˜Hy)/(ωKMy). (54)\nWhen T→Tc, with σn∼1.1×106(Ω·m)−1for NbN [72],\ndS=dF= 60 nm, and ω∼2π×4 GHz, we find the\nfrequency shift δω=ωK−˜ωK∼10−5GHz is negligibly\nsmall, while the additional damping is considerably large\n˜α∼2×10−4for YIG.\nSince the normal current can be disregarded in the fre-\nquency shift, we calculate the temperature dependence of\nthe FMR frequency according to Eq. (4), as plotted in\nFig. 8(c) with the same parameters used in Fig. 8(a) and\n(b). When T→0, the resonance frequency reaches its\nmaximum, while when T→Tc, the resonance frequency\nrecovers to the Kittel bare frequency since the supercon-\nductivity is depleted. We compare the full solution (black\nline) and the quasi-static solution (dashed line) and find\nthe quasi-static approximation is excellent in all the tem-\nperature regimes when dS≲λ.\n/uni00000015/uni00000013 /uni00000015/uni00000018 /uni00000016/uni00000013 /uni00000016/uni00000018 /uni00000017/uni00000013\n/uni00000013/uni0000000b/uni0000002a/uni0000002b/uni0000005d/uni0000000c\n/uni00000013/uni00000018/uni00000014/uni00000013/uni00000014/uni00000018/uni00000028/uni0000005d/uni0000000b/uni00000039/uni00000012/uni00000050/uni0000000c/uni00000003/uni0000000b/uni00000044/uni0000000c\n/uni0000002a/uni00000020/uni00000018/uni000000ee/uni00000014/uni00000013/uni00000017\n/uni00000013/uni00000030/uni00000013/uni00000020/uni00000013/uni00000011/uni00000015/uni00000037\n/uni00000013/uni0000002b/uni00000013/uni00000020/uni00000013/uni00000011/uni00000013/uni00000018/uni00000037\n/uni00000020/uni0000001b/uni0000001a/uni00000011/uni0000001b/uni00000051/uni00000050\n/uni00000014/uni00000013 /uni00000015/uni00000013 /uni00000016/uni00000013 /uni00000017/uni00000013\n/uni00000013/uni0000000b/uni0000002a/uni0000002b/uni0000005d/uni0000000c\n/uni00000013/uni00000014/uni00000013/uni00000015/uni00000013/uni00000016/uni00000013/uni00000017/uni00000013 /uni00000013/uni00000030/uni0000000b/uni00000050/uni00000037/uni0000000c/uni00000003\n/uni0000000b/uni00000045/uni0000000c\n/uni00000056/uni0000004b/uni0000004c/uni00000049/uni00000057\n/uni0000002e/uni00000020/uni00000015/uni000000ee/uni00000016/uni00000011/uni00000015/uni0000002a/uni0000002b/uni0000005d\n/uni0000002e/uni00000020/uni00000015/uni000000ee/uni00000017/uni00000011/uni0000001b/uni0000002a/uni0000002b/uni0000005d\n/uni00000003/uni00000030/uni0000005c/uni00000003/uni0000005a/uni0000004c/uni00000057/uni0000004b/uni00000003/uni00000036\n/uni00000030/uni0000005c/uni00000003/uni0000005a/uni0000004c/uni00000057/uni0000004b/uni00000052/uni00000058/uni00000057/uni00000003/uni00000036\n/uni00000013 /uni00000016 /uni00000019 /uni0000001c /uni00000014/uni00000015\n/uni00000037/uni0000000b/uni0000002e/uni0000000c/uni00000015/uni00000013/uni00000015/uni00000017/uni00000015/uni0000001b/uni00000016/uni00000015 /uni0000002e/uni0000000b/uni0000002a/uni0000002b/uni0000005d/uni0000000c/uni00000003\n/uni0000000b/uni00000046/uni0000000c\n/uni00000014/uni00000014/uni0000002e/uni00000020/uni00000015/uni000000ee/uni00000016/uni00000011/uni00000015/uni0000002a/uni0000002b/uni0000005d\n/uni00000037/uni00000046/uni00000029/uni00000058/uni0000004f/uni0000004f\n/uni00000034/uni00000058/uni00000044/uni00000056/uni0000004c/uni00000010/uni00000056/uni00000057/uni00000044/uni00000057/uni0000004c/uni00000046FIG. 8. FMR spectra with the excitation field µ0˜H=\n0.01 mT. In (a) and (b), we use the temperature T= 0.5Tc=\n5.5 K. (a) plots the excited electric field amplitude in (one of)\nthe superconductors in the symmetric S |FI|S heterostructure.\nThe amplitude of the resonance electric field Ez∼14 V/m.\n(b) is the excited amplitudes of the magnetization Mywith\nand without two adjacent superconductors. Mx≈0.6Myand\n0.5Mywith and without the superconductors. The frequency\nshift is as large as 2 π×1.6 GHz ∼˜ωK/2. (c) is the tempera-\nture dependence of FMR frequency ωKby solutions with the\nfull calculation (black line) and quasi-static approximation\n(dashed line). The bare FMR frequency ˜ ωK= 2π×3.2 GHz.10\nVI. CONCLUSION AND DISCUSSION\nMagnetic insulators are ideal candidates for long-range\nspin transport [1, 2, 5, 6, 8–10], strong coupling between\nmagnons and microwaves [38], and quantum information\nprocessing [33, 34, 36, 37, 39], gating which by supercon-\nductors may bring new control dimensions. In compari-\nson to metallic magnets, the mutual proximity effect may\ndiffer between magnetic insulators and superconductors,\nwhich may be helpful to distinguish different competitive\nmechanisms [30] in future studies. Our model system\ndiffers from the metallic ferromagnets since there are no\nelectric currents flowing in the insulators that, if large,\nmay affect the field distribution via radiation.\nThe formulation of the response in the superconductor\nby London’s equation is phenomenological, which, never-\ntheless, captures the key physics of the interplay between\nFMR in the magnetic insulator and supercurrent in the\nsuperconductor. Some interesting effects, such as the role\nof impurity and finite correlation length of Cooper pairs,\nmay be not precisely taken into account in the classi-\ncal London model, however. Our work can be a starting\npoint for an extension to a fully microscopic model in\nterms of, e.g., the Usadel equation [73], in the future.\nIn conclusion, we analyze the interaction between the\nKittel magnons in insulating magnetic film and Cooper-\npair supercurrent in superconductors mediated by the\nradiated electric fields from the magnetization dynam-\nics. Via highlighting the role of the total reflection of\nthe electric fields at the ferromagnet-superconductor in-\nterface that are solved beyond the quasi-static approxi-\nmation, we provide a comprehensive understanding of the\nabsence of the FMR shift in the FI |S heterostructure and\npredict its existence in the S |FI|S heterostructure with\nthe Meissner screening. The coupling between magnons\nand Cooper-pair supercurrent is ultrastrong with the fre-\nquency shift achieving tens of percent of the bare FMR\nfrequency, which may bring superior advantage in infor-\nmation processing in on-chip magnonics and quantum\nmagnonics.\nACKNOWLEDGMENTS\nWe gratefully acknowledge Prof. Guang Yang and\nProf. Lihui Bai for many inspiring discussions. This work\nis financially supported by the National Natural Science\nFoundation of China under Grant No. 12374109, and\nthe startup grant of Huazhong University of Science and\nTechnology (Grants No. 3004012185 and 3004012198).\nAppendix A: General solution of Ezin S|FI|S\nheterostructure\nHere we list the general solution of Ez(x) in the S |FI|S\nheterostructure when dS̸=d′\nSin Fig. 6. Inside the fer-romagnet,\nEz(−dF< x < d F)\n=−ωµ0My(Geikx+e−ikx)\nk(GeikdF−e−ikdF)−k′f(u)(GeikdF+e−ikdF),\n(A1)\nwhere\nG=−−2ksinh(ikdF) +k′(f(u)e−ikdF+f(u′)eikdF)\n−2ksinh(ikdF) +k′(f(u)eikdF+f(u′)e−ikdF),\n(A2)\nandu′=−[(k+k′)/(k−k′)] exp( −2ik′d′\nS). In the su-\nperconductor “1”,\nEz(dF< x < d F+dS) =ueik′(x−dF)+e−ik′(x−dF)\n1 +u\n×−ωµ0My(GeikdF+e−ikdF)\nk(GeikdF−e−ikdF)−k′f(u)(GeikdF+e−ikdF).\n(A3)\nIn the superconductor “2”,\nEz(−dF−d′\nS< x < −dF) =eik′(x+dF)+u′e−ik′(x+dF)\n1 +u′\n×−ωµ0My(Ge−ikdF+eikdF)\nk(GeikdF−e−ikdF)−k′f(u)(GeikdF+e−ikdF).\n(A4)\nOut of the heterostructure,\nEz(x > d F+dS) =ueik′dS+e−ik′dS\n1 +u\n×−ωµ0My(GeikdF+e−ikdF)eik(x−dF−dS)\nk(GeikdF−e−ikdF)−k′f(u)(GeikdF+e−ikdF),\nEz(x <−dF−d′\nS) =e−ik′d′\nS+u′eik′d′\nS\n1 +u′\n×−ωµ0My(Ge−ikdF+eikdF)e−ik(x+dF+dS)\nk(GeikdF−e−ikdF)−k′f(u)(GeikdF+e−ikdF).\n(A5)\nThe magnetic field follows By=−∂xEz/(iω), which\ninside the magnetic insulator reads\nHy=−(2MydF/λ)f(u)f(u′)\n(f(u) +f(u′)) + 2( dF/λ)f(u)f(u′).(A6)\nRetaining the leading order in k, its real part\nRe(Hy)≈ −2dFMytanh( dS/λ) tanh( d′\nS/λ)\n×[λ(tanh( dS/λ) + tanh( d′\nS/λ))\n+2dFtanh( dS/λ) tanh( d′\nS/λ)]−1(A7)\nleads to the FMR frequency\nωK=µ0γp\nH0+M0q\nH0−M0Re(Hy)/My,(A8)11\nand its imaginary part\nIm(Hy) =2kdFMy \ntanh2(d′\nS/λ)\ncosh2(dS/λ)+tanh2(dS/λ)\ncosh2(d′\nS/λ)!\n×[tanh( dS/λ) + tanh( d′\nS/λ)\n+ 2dF/λtanh( dS/λ) tanh( d′\nS/λ)]−2, (A9)\ncontributes to the damping coefficient\nαR=µ0γM0Im(Hy)/(ωKMy). (A10)\nAppendix B: Quasi-static approximation in S |FI|S\nheterostructure\nAs justified, the quasi-static approximation ∇×H= 0\norJsis allowed when solving the electric fields near the\nheterostructure [65]. In the FMR case, the radiated elec-\ntric field is uniform in the y-zplane, so from ∇×E=iωB,\nthex-component Bx=Hd,x+Mx= 0 generates no elec-\ntric field outside the magnet. On the other hand, in\nthe linear response regime for the magnetization dynam-\nics,Mz=M0, soBz=µ0(H0+Mz) is static, so only\nBy=µ0Myin the magnet radiates the time-dependent\nelectric field according to −∂xEz=iωµ0(My+Hs,y). In-\ntegrating along xacross the ferromagnet yields the net\nelectric field at the interfaces obeying\nEz(x=dF)−Ez(x=−dF) =−2dFiωµ0(My+Hs,y).\n(B1)\nOut of the heterostructure, from the z-component of ∇×\nH= 0, Hy|outside is a constant, which can be proved to\nvanish as in Sec. IV B.\nIn the quasi-static approximation, the electric field in\nthe superconductors “1” and “2” obeys Eq. (6). From\nthe boundary conditions with continuous EzandHyat\ninterfaces and Hy|outside = 0, the electric field in the\nsuperconductors reads\nEz(dF< x < d F+dS)\n=Ez(x=dF)cosh(( x−dS−dF)/λ)\ncosh( dS/λ),\nEz(−dF−dS< x < −dF)\n=Ez(x=−dF)cosh(( x+d′\nS+dF)/λ)\ncosh( d′\nS/λ), (B2)\nwhich drive the supercurrents in the superconductors ad-\njacent to the magnet. For thin superconducting films of\nthickness O(λ), we are allowed to take an average of the\nsupercurrents J(1)\ns= [Js(x=dF) +Js(x=dF+dS)]/2\nandJ(2)\ns= [Js(x=−dF) +Js(x=−dF−dS)]/2, i.e.,\nJ(1)\ns,z=i\nωµ0λ2Ez(x=dF)1 + cosh( dS/λ)\n2 cosh( dS/λ),\nJ(2)\ns,z=i\nωµ0λ2Ez(x=−dF)1 + cosh( d′\nS/λ)\n2 cosh( d′\nS/λ).(B3)The supercurrents generate the vector potential (7)\nand the Oersted magnetic field according to Hs,y=\n−∂xAz/µ0. Using the Weyl identity (34) we obtain\nHs,y(x) =\n\n\u0010\ndSJ(1)\ns,z+d′\nSJ(2)\ns,z\u0011\n/2, x > d F+dS\u0010\n−dSJ(1)\ns,z+d′\nSJ(2)\ns,z\u0011\n/2,−dF< x < d F\u0010\n−dSJ(1)\ns,z−d′\nSJ(2)\ns,z\u0011\n/2, x < −dF−d′\nS.\n(B4)\nHs,y|outside = 0 requests\ndSJ(1)\ns,z+d′\nSJ(2)\ns,z= 0, (B5)\nso the Oersted magnetic field inside the ferromagnetic\nslab is reduced to\nHs,y(−dF< x < d F) =d′\nSJ(2)\ns,z=−dSJ(1)\ns,z. (B6)\nThereby, when dS=d′\nS, the supercurrents are opposite\nin the two superconductors. When d′\nS→0,Hs,yvanishes\nin the magnet.\nSubstituting Eqs. (B3) and (B1) into (B5), we obtain\nthe electric field at the surface of the ferromagnetic film:\nEz(x=−dF) =iµ0ωdSdF(My+Hs,y)cosh( dS/λ) + 1\ncosh( dS/λ)\n×\u0012dS(cosh( dS/λ) + 1)\n2 cosh( dS/λ)+d′\nS(cosh( d′\nS/λ) + 1)\n2 cosh( d′\nS/λ)\u0013−1\n.\n(B7)\nSubstituting it into Eq. (B6), the Oersted magnetic field\nin the ferromagnetic film\nHs,y(−dF< x < d F) =−MydFd′\nSdSG(dS, d′\nS, λ)\nλ2+dFd′\nSdSG(dS, d′\nS, λ),\n(B8)\nwhere\nG(dS, d′\nS, λ) =(cosh( dS/λ) + 1)\ncosh( dS/λ)(cosh( d′\nS/λ) + 1)\ncosh( d′\nS/λ)\n×\u0012dS(cosh( dS/λ) + 1)\ncosh( dS/λ)+d′\nS(cosh( d′\nS/λ) + 1)\ncosh( d′\nS/λ)\u0013−1\n.\n(B9)\nThese results capture precisely the key physics of the\nfull solution and are convenient for the calculation of the\ninteraction between Kittel magnon and Cooper-pair su-\npercurrent.\nIn the linear regime of the magnetization dynamics,\nsubstituting Bx=Mx+Hd,x= 0 into the Landau-\nLifshitz equation\n−iωM x+µ0γMyH0=µ0γM0Hs,y,\niωM y+µ0γMxH0=µ0γM0Hd,x, (B10)12\nwe find Myrelates to Hs,yvia\nMy=µ2\n0γ2M0(H0+M0)\nµ2\n0γ2H0(H0+M0)−ω2Hs,y. (B11)\nWhen d′\nS→0,Hs,y= 0 according to Eq. (B8), and\nthe FMR frequency recovers the Kittel formula ˜ ωK=\nµ0γp\nH0(H0+M0) [58]. With finite dSandd′\nS, the\nFMR frequency is self-consistently solved via combining\nEqs. (B8) and (B11), leading to the modified FMR fre-quency\nωK=µ0γ\n×s\nλ2H0(H0+M0) +dSd′\nSdFG(dS, d′\nS, λ)(H0+M0)2\ndSd′\nSdFG(dS, d′\nS, λ) +λ2.\n(B12)\nIn particular, when dS=d′\nS,\nωK=µ0γ\u00122λ2cosh ( dS/λ)H0(H0+M0)\ndSdF(cosh ( dS/λ) + 1) + 2 λ2cosh ( dS/λ)\n+dSdF(cosh ( dS/λ) + 1)( H0+M0)2\ndSdF(cosh ( dS/λ) + 1) + 2 λ2cosh ( dS/λ)\u00131/2\n.\n(B13)\nApproaching Tc,λ→ ∞ , cosh( dS/λ)→1, so the FMR\nfrequency (B13) recovers the Kittel formula ωK→˜ωK;\notherwise T < T c, it is shifted.\n[1] B. Lenk, H. Ulrichs, F. Garbs, and M. M¨ unzenberg,\nThe building blocks of magnonics, Phys. Rep. 507, 107\n(2011).\n[2] A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B.\nHillebrands, Magnon spintronics, Nat. Phys. 11, 453\n(2015).\n[3] L. J. Cornelissen, J. Liu, R. A. Duine, J. Ben Youssef, and\nB. J. van Wees, Long-distance transport of magnon spin\ninformation in a magnetic insulator at room temperature,\nNat. Phys. 11, 1022 (2015).\n[4] J. Zou, S. Zhang, and Y. Tserkovnyak, Topological\nTransport of Deconfined Hedgehogs in Magnets, Phys.\nRev. Lett. 125, 267201 (2020).\n[5] D. Grundler, Nanomagnonics around the corner, Nat.\nNanotechnol. 11, 407 (2016).\n[6] V. E. Demidov, S. Urazhdin, G. de Loubens, O. Klein,\nV. Cros, A. Anane, and S. O. Demokritov, Magnetization\noscillations and waves driven by pure spin currents, Phys.\nRep.673, 1 (2017).\n[7] X. S. Wang, A. Qaiumzadeh, and A. Brataas, Current-\nDriven Dynamics of Magnetic Hopfions, Phys. Rev. Lett.\n123, 147203 (2019).\n[8] A. Brataas, B. van Wees, O. Klein, G. de Loubens, and\nM. Viret, Spin Insulatronics, Phys. Rep. 885, 1 (2020).\n[9] Barman et al. , The 2021 Magnonics Roadmap, J. Phys.\nCondens. Matter 33, 413001 (2021).\n[10] T. Yu, Z. C. Luo, and G. E. W. Bauer, Chirality as gen-\neralized spin–orbit interaction in spintronics, Phys. Rep.\n1009 , 1 (2023).\n[11] O. V. Dobrovolskity, R. Sachser, T. Br¨ acher, T. B¨ ottcher,\nV. V. Kruglyak, R. V. Vovk, V. A. Shklovskij, M. Huth,\nB. Hillebrands, and A. V. Chumak, Magnon-fluxon inter-\naction in a ferromagnet/superconductor heterostructure,\nNat. Phys. 15, 477 (2019).\n[12] Y. Yao, R. Cai, T. Yu, Y. Ma, W. Xing, Y. Ji, X.-\nC. Xie, S.-H. Yang, and W. Han, Giant oscillatory\nGilbert damping in superconductor/ferromagnet/super-\nconductor junctions, Sci. Adv. 7, eabh3686 (2021).[13] L. G. Johnsen, H. T. Simensen, A. Brataas, and J. Lin-\nder, Magnon Spin Current Induced by Triplet Cooper\nPair Supercurrents, Phys. Rev. Lett. 127, 207001 (2021).\n[14] T. Yu and G. E. W. Bauer, Efficient Gating of Magnons\nby Proximity Superconductors, Phys. Rev. Lett. 129,\n117201 (2022).\n[15] M. A. Kuznetsov and A. A. Fraerman, Temperature-\nsensitive spin-wave nonreciprocity induced by interlayer\ndipolar coupling in ferromagnet/paramagnet and ferro-\nmagnet/superconductor hybrid systems, Phys. Rev. B\n105, 214401 (2022).\n[16] M. Silaev, Anderson-Higgs Mass of Magnons in\nSuperconductor-Ferromagnet-Superconductor Systems,\nPhys. Rev. Appl. 18, L061004 (2022).\n[17] I. V. Bobkova, A. M. Bobkov, A. Kamra, and W. Belzig,\nMagnon-cooparons in magnet-superconductor hybrids,\nCommun. Mater. 3, 95 (2022).\n[18] A. M. Bobkov, S. A. Sorokin, and I. V. Bobkova, Renor-\nmalization of antiferromagnetic magnons by supercon-\nducting condensate and quasiparticles, Phys. Rev. B 107,\n174521 (2023).\n[19] A. S. Ianovskaia, A. M. Bobkov, I. V. Bobkova, Magnon\ninfluence on the superconducting DOS in FI/S bilayers,\narXiv:2307. 03954.\n[20] M. Borst, P. H. Vree, A. Lowther, A. Teepe, S. Kurdi, I.\nBertelli, B. G. Simon, Y. M. Blanter, and T. van der Sar,\nObservation and control of hybrid spin-wave–Meissner-\ncurrent transport modes, arXiv:2307.07581.\n[21] A. T. G. Janssønn, H. T. Simensen, A. Kamra, A.\nBrataas, and S. H. Jacobsen, Macroscale nonlocal trans-\nfer of superconducting signatures to a ferromagnet in a\ncavity, Phys. Rev. B 102, 180506(R) (2020).\n[22] A. F. Volkov and K. B. Efetov, Hybridization of Spin and\nPlasma Waves in Josephson Tunnel Junctions Contain-\ning a Ferromagnetic Layer, Phys. Rev. Lett. 103, 037003\n(2009).\n[23] I. A. Golovchanskiy, N. N. Abramov, V. S. Stolyarov, M.\nWeides, V. V. Ryazanov, A. A. Golubov, A. V. Ustinov,13\nand M. Y. Kupriyanov, Ultrastrong photon-to-magnon\ncoupling in multilayered heterostructures involving su-\nperconducting coherence via ferromagnetic layers, Sci.\nAdv. 7, eabe8638 (2021).\n[24] I. A. Golovchanskiy, N. N. Abramov, V. S. Stolyarov, A.\nA. Golubov, M. Y. Kupriyanov, V. V. Ryazanov, and A.\nV. Ustinov, Approaching Deep-Strong On-Chip Photon-\nTo-Magnon Coupling, Phys. Rev. Appl. 16, 034029\n(2021).\n[25] M. Silaev, Ultrastrong magnon-photon coupling,\nsqueezed vacuum, and entanglement in superconduc-\ntor/ferromagnet nanostructures, Phys. Rev. B 107,\nL180503 (2023).\n[26] A. T. G. Janssønn, H. G. Hugdal, A.\nBrataas, and S. H. Jacobsen, Cavity-mediated\nsuperconductor–ferromagnetic-insulator coupling,\nPhys. Rev. B 107, 035147 (2023).\n[27] A. Ghirri, C. Bonizzoni, M. Maksutoglu, A. Mercu-\nrio, O. D. Stefano, S. Savasta, and M. Affronte, Ul-\ntrastrong magnon-photon coupling achieved by magnetic\nfilms in contact with superconducting resonators, Phys.\nRev. Appl. 20, 024039 (2023).\n[28] M. Eschrig, Spin-polarized supercurrents for spintronics:\na review of current progress, Rep. Prog. Phys. 78, 104501\n(2015).\n[29] J. Linder and J. W. A. Robinson, Superconducting spin-\ntronics, Nat. Phys. 11, 307 (2015).\n[30] F. S. Bergeret, M. Silaev, P. Virtanen, and T. T. Heikkil¨ a,\nColloquium: Nonequilibrium effects in superconductors\nwith a spin-splitting field, Rev. Mod. Phys. 90, 041001\n(2018).\n[31] J. Linder and A. V. Balatsky, Odd-frequency supercon-\nductivity, Rev. Mod. Phys. 91, 045005 (2019).\n[32] M. Amundsen, J. Linder, J. W. A. Robinson, I. ˇZuti´ c,\nand N. Banerjee, Colloquium: Spin-orbit effects in su-\nperconducting hybrid structures, arXiv: 2210.03549.\n[33] Y. Tabuchi, S. Ishino, A. Noguchi, T. Ishikawa, R. Ya-\nmazaki, K. Usami, and Y. Nakamura, Coherent coupling\nbetween a ferromagnetic magnon and a superconducting\nqubit, Science 349, 405 (2015).\n[34] D. L. Quirion, S. P. Wolski, Y. Tabuchi, S. Kono, K.\nUsami, and Y. Nakamura, Entanglement-based single-\nshot detection of a single magnon with a superconducting\nqubit, Science 367, 425 (2020).\n[35] T. Yu, M. Claassen, D. M. Kennes, and M. A. Sentef,\nOptical manipulation of domains in chiral topological su-\nperconductors, Phys. Rev. Research 3, 013253 (2021).\n[36] H. Y. Yuan, Y. Cao, A. Kamra, R. A. Duine, and P. Yan,\nQuantum magnonics: When magnon spintronics meets\nquantum information science, Phys. Rep. 965, 1 (2022).\n[37] Z. Li, M. Ma, Z. Chen, K. Xie, and F. Ma, Interaction be-\ntween magnon and skyrmion: Toward quantum magnon-\nics, J. Appl. Phys. 132, 210702 (2022).\n[38] B. Z. Rameshti, S. V. Kusminskiy, J. A. Haigh, K. Us-\nami, D. Lachance-Quirion, Y. Nakamura, C. -M. Hu, H.\nX. Tang, G. E. W. Bauer, and Y. M. Blanter, Cavity\nmagnonics, Phys. Rep. 979, 1 (2022).\n[39] D. Xu, X. -K. Gu, H. -K. Li, Y. -C. Weng, Y. -P. Wang, J.\nLi, H. Wang, S. -Y. Zhu, and J. Q. You, Quantum Control\nof a Single Magnon in a Macroscopic Spin System, Phys.\nRev. Lett. 130, 193603 (2023).\n[40] K. M. D. Hals, M. Schecter, and M. S. Rudner,\nComposite Topological Excitations in Ferromagnet-\nSuperconductor Heterostructures, Phys. Rev. Lett. 117,017001 (2016).\n[41] A. F. Kockum, A. Miranowicz, S. D. Liberato, S. Savasta,\nand F. Nori, Ultrastrong coupling between light and mat-\nter, Nat. Rev. Phys. 1, 19 (2019).\n[42] J. C. Swihart, Field Solution for a Thin-Film Supercon-\nducting Strip Transmission Line, J. Appl. Phys. 32, 461\n(1961).\n[43] I. A. Golovchanskiy, N. N. Abramov, V. S. Stolyarov, V.\nV. Bolginov, V. V. Ryazanov, A. A. Golubov, and A. V.\nUstinov, Ferromagnet/Superconductor Hybridization for\nMagnonic Applications, Adv. Funct. Mater. 28, 1802375\n(2018).\n[44] I. A. Golovchanskiy, N. N. Abramov, V. S. Stolyarov, V.\nV. Ryazanov, A. A. Golubov, and A. V. Ustinov, Modi-\nfied dispersion law for spin waves coupled to a supercon-\nductor, J. Appl. Phys. 124, 233903 (2018).\n[45] I. A. Golovchanskiy, N. N. Abramov, V. S. Stolyarov,\nP. S. Dzhumaev, O. V. Emelyanova, A. A. Golubov, V.\nV. Ryazanov, and A. V. Ustinov, Ferromagnet/Super-\nconductor Hybrid Magnonic Metamaterials, Adv. Sci. 6,\n1900435 (2019).\n[46] I. A. Golovchanskiy, N. N. Abramov, V. S. Stolyarov, A.\nA. Golubov, V. V. Ryazanov, and A. V. Ustinov, Nonlin-\near spin waves in ferromagnetic/superconductor hybrids,\nJ. Appl. Phys. 127, 093903 (2020).\n[47] Seshadri, Surface magnetostatic modes of a ferrite slab,\nProc. IEEE 58, 506 (1970).\n[48] C. Bayer, J. Jorzick, B. Hillebrands, S. O. Demokritov, R.\nKouba, R. Bozinoski, A. N. Slavin, K. Y. Guslienko, D.\nV. Berkov, N. L. Gorn, and M. P. Kostylev, Spin-wave ex-\ncitations in finite rectangular elements of Ni 80Fe20, Phys.\nRev. B 72, 064427 (2005).\n[49] T. Yu, C. P. Liu, H. M. Yu, Y. M. Blanter, and G. E. W.\nBauer, Chiral excitation of spin waves in ferromagnetic\nfilms by magnetic nanowire gratings, Phys. Rev. B 99,\n134424 (2019).\n[50] T. Yu, Y. M. Blanter, and G. E. W. Bauer, Chiral Pump-\ning of Spin Waves, Phys. Rev. Lett. 123, 247202 (2019).\n[51] T. Yu, J. Zou, B. Zeng, J. W. Rao, and K. Xia, Non-\nHermitian Topological Magnonics, arXiv:2306.04348.\n[52] A. V. Chumak, A. A. Serga, and B. Hillebrands, Nat.\nCommun. 5, 4700 (2014).\n[53] T. Yu, H. Wang, M. A. Sentef, H. Yu, and G. E. W.\nBauer, Magnon trap by chiral spin pumping, Phys. Rev.\nB102, 054429 (2020).\n[54] O. A. Santos and B. J. van Wees, Magnon confinement in\nan all-on-chip YIG cavity resonator using hybrid YIG/Py\nmagnon barriers, arXiv:2306.14029.\n[55] 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[56] 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[57] 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 /Ni80Fe20/Nb Trilayers,\nPhys. Rev. Appl. 11, 014061 (2019).14\n[58] C. Kittel, On the Theory of Ferromagnetic Resonance\nAbsorption, Phys. Rev. 73, 155 (1948).\n[59] S. V. Mironov and A. I. Buzdin, Giant demagnetization\neffects induced by superconducting films, Appl. Phys.\nLett. 119, 102601 (2021).\n[60] F. Stageberg, R. Cantor, A. M. Goldman, and G. B.\nArnold, Electronic density of states at the surface of a su-\nperconductor in contact with a magnetic insulator, Phys.\nRev. B 32, 3292 (1985).\n[61] T. Golod, A. Rydh, V. M. Krasnov, I. Marozau,\nM. A. Uribe-Laverde, D. K. Satapathy, T. Wag-\nner, and C. Bernhard, High bias anomaly in\nYBa 2Cu3O7−x/LaMnO 3+δ/YBa 2Cu3O7−x super-\nconductor/ferromagnetic insulator/superconductor\njunctions: Evidence for a long-range superconducting\nproximity effect through the conduction band of a\nferromagnetic insulator, Phys. Rev. B 87, 134520\n(2013).\n[62] Y. Yao, Q. Song, Y. Takamura, J. P. Cascales, W. Yuan,\nY. Ma, Y. Yun, X. C. Xie, J. S. Moodera, and W. Han,\nProbe of spin dynamics in superconducting NbN thin\nfilms via spin pumping, Phys. Rev. B 97, 224414 (2018).\n[63] S. M. Rezende, Fundamentals of Magnonics (Springer,\nCham, 2020).\n[64] L. D. Landau and E. M. Lifshitz, Electrodynamics of\nContinuous Media , 2nd ed. (Butterworth-Heinenann,\nOxford, U.K., 1984).\n[65] J. D. Jackson, Classical Electrodynamics (Wiley, New\nYork, 1998).\n[66] J. R. Schrieffer, Theory of Superconductivity , (W. A. Ben-jimin, New York, 1964).\n[67] X. Y. Wei, O. A. Santos, C. H. S. Lusero, G. E. W. Bauer,\nJ. B. Youssef, and B. J. van Wees, Giant magnon spin\nconductivity in ultrathin yttrium iron garnet films, Nat.\nMater. 21, 1352 (2022).\n[68] S. Knauer, K. Dav´ ıdkov´ a, D. Schmoll, R. O. Serha, A.\nVoronov, Q. Wang, R. Verba, O. V. Dobrovolskiy, M.\nLindner, T. Reimann, C. Dubs, M. Urb´ anek, and A. V.\nChumak, Propagating spin-wave spectroscopy in a liquid-\nphase epitaxial nanometer-thick YIG film at millikelvin\ntemperatures, J. Appl. Phys. 133, 143905 (2023).\n[69] M. P. Tu, K. Mbaye, L. Wartski, and J. Halbritter, rf\ncharacterization of thermally diffused superconducting\nniobium nitride, J. Appl. Phys. 63, 4586 (1988).\n[70] O. W. Dietrich, A. J. Henderson, Jr., and H. Meyer, Spin-\nwave analysis of specific heat and magnetization in Euo\nand Eus, Phys. Rev. B 12, 2844 (1975).\n[71] Y. Hou, F. Nichele, H. Chi, A. Lodesani, Y. Wu, M. F.\nRitter, D. Z. Haxell, M. Davydova, S. Ili´ c, O. Glezakou-\nElbert, A. Varambally, F. S. Bergeret, A. Kamra, L. Fu,\nP. A. Lee, and J. S. Moodera , Ubiquitous Superconduct-\ning Diode Effect in Superconductor Thin Films, Phys.\nRev. Lett. 131, 027001 (2023).\n[72] S. P. Chockalingam, M. Chand, J. Jesudasan, V. Tri-\npathi, and P. Raychaudhuri, Superconducting properties\nand Hall effect of epitaxial NbN thin films, Phys. Rev. B\n77, 214503 (2008).\n[73] K. D. Usadel, Generalized Diffusion Equation for Super-\nconducting Alloys, Phys. Rev. Lett. 25, 507 (1970)."    },    {        "title": "1502.01816v1.Resonant_indirect_exchange_via_remote_2D_channel.pdf",        "content": "arXiv:1502.01816v1  [cond-mat.mes-hall]  6 Feb 2015Resonant indirect exchange via remote 2D channel\nI. V. Rozhansky,1,2,3,∗I. V. Krainov,1N. S. Averkiev,1B. A. Aronzon,4,5\nA. B. Davydov,4K. I. Kugel,6V. Tripathi,7and E. L¨ ahderanta2\n1Ioﬀe Institute, 194021 St.Petersburg, Russia\n2Lappeenranta University of Technology, FI-53851 Lappeenr anta, Finland\n3St. Petersburg State Polytechnic University, 195251 St. Pe tersburg, Russia\n4P.N. Lebedev Physical Institute, 119991 Moscow, Russia\n5National Research Centre ”Kurchatov Institute”, 123182 Mo scow, Russia\n6Institute of Theoretical and Applied Electrodynamics, 125 412 Moscow, Russia\n7Tata Institute of Fundamental Research, Mumbai 400005, Ind ia\n(Dated: October 26, 2018)\nWe apply the resonant indirect exchange interaction theory to explain the ferromagnetic prop-\nerties of the hybrid heterostructure consisting of a InGaAs -based quantum well (QW) sandwiched\nbetween GaAs barriers with a remote Mn delta-layer. The expe rimentally obtained dependence of\nthe Curie temperature on the QW depth exhibits a maximum rela ted to the region of resonant\nindirect exchange. We suggest the theoretical explanantio n and a ﬁt to this dependence as a result\nof the two contributions to ferromagnetism - the intralayer contribution and the resonant exchange\ncontribution provided by the QW.\nPACS numbers: 75.75.-c, 78.55.Cr, 78.67.De\nI. INTRODUCTION\nDilute magnetic semiconductors (DMS) have been at-\ntracting a lot of attention for quite a while [1]. A lot of\neﬀorts have been put forward to combine the numerous\nadvantages of semiconductors with the spin-related phe-\nnomena introduced by the magnetic impurities. In this\nﬁeld, however, still much remains unclear. For instance,\nthe details of the mechanism responsible for the ferro-\nmagnetic properties of GaAs doped with a small amount\nof Mn has not yet been clariﬁed [2]. It is commonly ac-\ncepted that the ferromagnetism in (Ga,Mn)As is due to\nthe indirect exchange interaction mediated by the holes.\nThe highest Curie temperature achieved for bulk dilute\n(Ga,Mn)As samples is near 200 Kwhich is still far be-\nlow the room temperature [3]. While it is the Mn sol-\nubility limit that basically prevents the increase of Tc\nin bulk samples [4], the indirect exchange depends on\nthe concentration of the holes. In this regard the GaAs\nheterostructures with a Mn layer coupled to a remote\n2D holes channel have gained a considerable interest [5–\n8]. It has been demonstrated that GaAs heterostructure\nwith a Mn δ-layer located in a vicinity of In xGa1−xAs\nquantum well (QW) shows ferromagnetic behavior sim-\nilar to that of the bulk Mn-doped GaAs DMS. It was\ndiscovered, however, that the dependence of the Curie\ntemperature on the QW depth shows a non-monotonic\nbehavior [9]. It was suggested that the non-monotonic\nbehaviororiginatesfromfallingoftheholeboundstateat\nMn ion into the energy range of occupied 2D heavy holes\nsubband of the ﬁrst QW size quantization level [10]. A\ntheory of the indirect exchange via a remote conducting\n∗rozhansky@gmail.comchannel was developed in [10],[11]. The theory predicted\nenhancement of the exchange interaction strength due to\nresonant tunnel coupling of a bound state at magnetic\nion with the continuum of delocalized 2D states in the\nchannel. In this paper we present a comparison with the\nexperimentally observed dependence of the Curie tem-\nperature on the QW depth and analyze how the tem-\nperature aﬀects the indirect exchange interaction in the\nresonant case. As soon as we are talking about 2D struc-\ntures, the QW and the Mn δ- layer, one should be aware\nof what is meant by the Curie temperature as there is no\npossibility ofspontaneousbreakingofa continuous(rota-\ntional) symmetry in our 2D Heisenberg ferromagnet. In\nour theoretical considerations we actually consider the\nmean eﬀective exchange constant, i.e. the energy of the\nindirectexchangeinteractionbetweenthe twoneighbour-\ning Mn ions. For a 3D bulk case it is indeed close to the\ncritical Curie temperature for the system to undergo the\nferromagnetic phase transition. In the 2D case, however\ntoleavethe thingsconsistentitis reasonabletodeﬁne the\nCurie temperature as the one marking the onset of ’lo-\ncal ferromagnetic order’, when the magnetic correlation\nlength well exceeds the distance between the Mn ions.\nIn the experiment the so deﬁned Curie temperature is\nobtained from a maximum (bump) on the dependence\nof in-plane electrical resistance vs temperature which is\nknown to be related to the onset of the ferromagnetic or-\nder [12],[13]. For more detailed discussion on the critical\ntemperature in the system under study see Ref.[14],[15].\nII. THEORY OF RESONANT INDIRECT\nEXCHANGE\nWe consider the problem of resonant indirect exchange\nbetween two magnetic ions iandjvia a remote 2D chan-2\nR\nx\nyz,j j Iε ,iiIε\nid jda. b. \nTδ-Mn \n0ε\nFEz0U\nFIG. 1. The illustration of the indirect exchange interacti on\nvia remote channel (a) and band diagram of a InGaAs-based\nheterostructure with a QW and a remote Mn δ-layer (b).\nnel as shown schematically in Fig 1a. Here Ii,jis the\nspin projection of the ith(jth) magnetic ion, εi,jis the\nions’ bound state energy, di,jthe distances between the\nions and the channel. Indirect exchange interaction me-\ndiated by free carriers is usually described on the basis\nof Ruderman-Kittel-Kasuya-Yosida(RKKY) theory [16].\nSeparation of the magnetic ions from the free carriersgas\nbyapotentialbarrierleadstothesuppressionoftheeﬀect\ndue to weak penetration of the 2D carriers wavefunction\ninto the region containing magnetic centers. However, if\na magnetic ion possess a bound state having the energy\nwithin the 2D gas energy spectra a resonant tunneling\nmay occur. The resonant coupling of a bound state with\nthe2Dcontinuumpreventstheproblemtobe straightfor-\nwardlyattackedwithRKKYapproach. TheHamiltonian\nof the two magnetic ions i,jcoupled to the free electron\ngas can be expressed in the tunneling Hamiltonian for-\nmalism:\nH=H0+HT+HJ, (1)\nwhereH0–theHamiltonianofthesystemwithouttunnel\ncoupling and spin-spin interaction, HT– the tunnelling\nterm,HJ– the exchange interaction term. In the second\nquantization representation:\nH0=εia+\niai+εja+\njaj+/integraldisplay\nελc+\nλcλdλ,\nHT=/integraldisplay/parenleftbig\ntiλa+\nicλ+tjλa+\njcλ+h.c./parenrightbig\ndλ,\nHJ=JA/parenleftbig\nIisa+\niai+Ijsa+\njaj/parenrightbig\n, (2)\nwherea+\ni,aiare the creation and annihilation operators\nfor the bound states at the impurity ion i, character-\nized by the energy level εiand localized wavefunction\nψi,c+\nλ,cλare the creation and annihilation operators for\na continuum state characterized by the quantum num-\nber(s)λ, having the energy ελand the wavefunction ϕλ,\nJis the exchange constant, A is the squared wavefunc-\ntion amplitude at the ions site, sis the 2D carrier’s spin\nprojection, ti,λisthe Bardeen’stunneling matrixelement\ngiven by [10]:\nti(k) =/radicalbigg\n/planckover2pi12Ti\n2πmeikRi, (3)where m is the 2D continuum density of states eﬀective\nmass,Tiis the energy parameter for the tunneling:\nTi=αU0e−2qdi, (4)\nwhereU0is the height of the potential barrier separating\nthe magnetic centers from the channel, q=√2m⊥U0//planckover2pi1,\nm⊥being an eﬀective mass in the direction of the tun-\nneling, the dimensionless parameter αdepends on the\nchannel and magnetic centers details [11]. We obtained\nthe exchange interaction energy between the two ions in\nthe form [10], [17]:\nEij=1\nπ/integraldisplayEF\n0dεarctan/bracketleftbigg8π2j2TiTjJ0(kR)Y0(kR)\n((εi−ε)2−j2)((εj−ε)2−j2)/bracketrightbigg\n,\n(5)\nwherej=JA|I||s|is the exchange interaction strength,\nEFdenotes the Fermi level of the carriers in the channel\n(zero temperature is assumed), k=√\n2mε//planckover2pi1,J0,Y0are\nBessel and Neumann functions of zeroth order,\nThe formula (5) accounts both for the case of resonant\nand non-resonant tunnel coupling between the magnetic\nions and the channel. The resonant case corresponds to\nthe bound states energy lying within the energy range of\nthe occupied states in the 2D channel:\nεi,εj∈[0,EF]. (6)\nIn this regime the main contribution to the exchange en-\nergy(5) comesfromthe poles ofthe arctangentargument\nand can be estimated as\nEij≈γ/radicalbig\njT, ifβ >1\nEij≈γβ/radicalbig\njT, ifβ <1\nβ=√jT\n|εi−εj|, T=/radicalbig\nTiTj. (7)\nHereγis given by:\nγ=√\n2π[J0(kiRij)Y0(kiRij)+J0(kjRij)Y0(kjRij)]1/4,\n(8)\nwhereRijis the distance between the ions. γis the\nparameter that incorporates the oscillating behavior of\nthe indirect exchange in the similar way that stan-\ndard RKKY theory does. Unlike RKKY theory, here\nthe Fermi wavevector kFis replaced by the ’resonant’\nwavevectors corresponding to the bound levels: ki=√2mεi//planckover2pi1. For the experimental situation considered be-\nlow the parameter γappears to be γ≈1 being still far\nfrom the ﬁrst maximum of the oscillations. The approx-\nimation (7) is quite good as illustrated in Fig. 2. The\ndotted curve shows the exchange interaction energy cal-\nculated according to (5) for the case εi=εj=ε0, while\nthe solid curve shows the approximation (7) assuming\nγ= 1. The value of the exchange energy in the resonant\ncase is much larger than in the non-resonant one. The\nlatter case agrees well with the RKKY approach, the in-\ntegration (5) for the case when the arctangent argument3\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53 /s51/s46/s48/s48/s50/s52/s54/s56/s49/s48\n/s32/s69/s120/s97/s99/s116/s32/s99/s97/s108/s99/s117/s108/s97/s116/s105/s111/s110\n/s32/s65/s112/s112/s114/s111/s120/s105/s109/s97/s116/s105/s111/s110\n/s32/s32/s69\n/s101/s120/s32/s40/s109/s101/s86/s41\n/s40/s106/s84/s41/s49/s47/s50\n/s32/s40/s109/s101/s86/s41\nFIG. 2. Approximation for indirect resonant exchange energ y\n(dashed line) compared to exact result (solid line).\nhas no poles and therefore is small as far as the tunneling\nis weak yields [10]:\nEnr=8πT2j2EF\nε4\n0χ(R),\nχ(R) =J0(kFR)Y0(kFR)+J1(kFR)Y1(kFR),(9)\nwhereRdenotes the mean distance between the ions.\nNote that the resonant and non-resonant cases have dif-\nferentparametricdependenceonthetunnelingparameter\nTand the exchange parameter j, this leads to substan-\ntial ampliﬁcation ofthe indirect exchangein the resonant\ncase.\nIII. COMPARISON WITH EXPERIMENT\nWe applied the theory of the resonant exchange to\nthehybridInGaAs-basedsemiconductorhetterostructure\ndoped with a δ-layerof Mn, which have been also studied\nexperimentally. Theenergydiagramforthesystemunder\nstudy is shown schematically in Fig.1b. It is a GaAs/Mn\nδ-layer/GaAs/In xGa1−xAs/GaAs heterostructure. The\nMn content (Mn layer eﬀective thickness) is 0.25-0.3\nmonolayers(ML),thespacerthickness d, betweentheMn\nlayer and the In xGa1−xAs quantum well (QW) is d= 3\nnm, the thickness of the QW is 10 nm and its depth is\ncontrolled by In concentration x. For detailed descrip-\ntion of the structure see [8]. The samples were shown to\nexhibit ferromagnetic properties, which were found to be\nnon-monotonously dependent on the QW depth [9]. The\nCurie temperature derived from the resistance anomaly\nappears to depend on the parameters of the QW thus\nfavouring the idea that the indirect exchange interaction\nis partly due to 2D holes sitting in the QW. The inter-\naction between the Mn ions mediated by the 2D holes\nin the QW must be considered with account for the res-\nonant indirect exchange, because the acceptor binding\nenergy of Mn in GaAs is comparable to the QW depth/s56/s48 /s57/s48 /s49/s48/s48 /s49/s49/s48 /s49/s50/s48 /s49/s51/s48 /s49/s52/s48 /s49/s53/s48/s49/s48/s50/s48/s51/s48/s52/s48\n/s53\n/s54\n/s52/s50\n/s49/s51/s84\n/s67/s32/s40/s75/s41\n/s81/s87/s32/s100/s101/s112/s116/s104/s32/s40/s109/s101/s86/s41\nFIG. 3. Dependence of the Curie temperature on the QW\ndepth. Experimental data (circles) and the theoretical ﬁt\n(solid curves).\nfor the holes. This makes it possible to meet the reso-\nnant condition (6). This very case is shown in Fig. 1b,\nwhereε0denotes the average energy of the bound state\nof a hole at Mn in the δ-layer (zero energy corresponds\nto the ﬁrst heavy holes quantization level in the QW).\nIt is also worth noting that in real samples the the Mn\nδ-layer has a certain width. In fact due to the Mn ions\ndiﬀusion its halfwidth is known to be around 1.5 nm [18].\nIt is thus natural to expect that the exchange via QW is\nnot the onlycontribution, i.e. without QWthe ferromag-\nnetic properties of the Mn layer would be resembling the\nbulk ferromagnetism of a dilute magnetic semiconductor\nsample of a small yet ﬁnite thickness around 3 nm. The\nferromagneticpropertiesoftheMn δlayerembeddedinto\nGaAs matrix has been also studied theoretically [19]. We\nmade a ﬁt to the experimental data [9] (circles in Fig.3)\nassuming that there are two contributions to the indi-\nrect exchange interaction between the Mn ions. The ﬁrst\none is assumed to be itinerant ferromagnetism of the Mn\nδ-layer itself due to the weakly localized holes located\nprimarily in this layer in the same manner as the ferro-\nmagnetism in the bulk dilute GaMnAs semiconductor is\nbelievedtoemerge. Thiscontributiondoesnotdependon\nthe QW properties. The second contribution is the reso-\nnant indirect exchange via the 2D holes of the QW. This\none is treated on the basis of our theoretical result (5).\nWe demonstrate that the maximum of the Curie tem-\nperature corresponds to the resonant indirect exchange\nvia the 2D holes of the QW while its decrease for both\ntoo shallow or too deep QW is explained by driving the\nsystem out of the resonance (6). In our calculation we\nassumed the Curie temperature being the sum of the two\nterms:\nTC=TC1+TC2, (10)\nwhereTC1does not depend on the QW properties. In\ncalculation of the second term TC2we assumed that the\nenergy levels at Mn ions are normally distributed having\nan average value ε0and dispersion σε. The distance be-4\ntween the neighbouring ions was assumed to be constant,\nequal to the mean one R. For the Mn δ-doping of 0.3 ML\none can take R= 1.5 nm. We checked that taking into\naccount some distribution over the distances as well as\nvarying the mean value has little eﬀect on the resonant\nexchange term. This is because γdeﬁned by (8) is a very\nweak function of Rin the vicinity of γ= 1. On the con-\ntrary, the Mn bound levels energy distribution does play\nan important role and must be accounted for. In order\nto introduce the energy levels distribution into the ﬁtting\nprocedure it is convenient to replace the approximation\n(7) by a similar function:\n/tildewideE(εi,εj) =γ2jT\n|εi−εj|+γ√jT(11)\nThe resonant contribution to the Curie temperature TC2\n(10) is calculated using the following expression:\nTC2=2\nkB/integraldisplayEF\n0dε/integraldisplay+∞\n−∞dε′P(ε)P(ε′)/tildewideE(ε,ε′),(12)\nwhere:\nP(ε) =1√\n2πσεe−(ε−ε0)2\n2σ2ε, (13)\nkBis the Boltzmann constant. In the limiting case of a\ndelta-like bound states energy distribution σε→0 the\nexpression (12) yields:\nTC2=/braceleftbigg2\nkBγ√jT, ε0∈[0,EF]\n0,otherwise,(14)\ni.e. the resonant contribution vanishes whenever ε0goes\nbelow or above the energy range occupied by the carriers\nin the QW. The approach was used to ﬁt the experimen-\ntal values of the Curie temperature measured for two\nseries of samples. The samples 1-3 had 0.25 ML of Mn\nand the QW depth for the holes U0varied from 80 to 140\nmeV, the samples 4-6 had 0.3 ML of Mn with U0varied\nfrom 90 to 150 meV. The two theoretical ﬁts for the two\nseries of experimental points are presented in Fig.3. The\naverage value of the Mn bound state for the best ﬁt was\nε0=U0−103 meV, i.e. 103 meV above the top of the\nvalence band for GaAs, which roughly matches the Mn\nacceptorbinding energy( ≈110meV). Thisvalue wasthe\nsame for the two ﬁts. The holes concentration and thus\nthe holes Fermi level was derived independently from the\ntransport experiments [8],[20] and the QW depth from\nthe optical experiments. These values are given in the\nTable 1 along with the other parameters of the ﬁt. Along\nwith the mean bound state energy the ﬁtting parameters\nwere the non-resonant component of the Curie tempera-\ntureTC1, the bound state energy dispersion σεand the\nproductjT, assumed to be the same for all the ions in\nthe layer. As it is seen from Table 1, for the ﬁt covering\nsamples 4-6 TC1appeared to be higher than for the sam-\nples 1-3, this is consistent with the latter having weakerNoTc, KU0, meVp, cm−2EF, meVTC1, K√jT, meVσε, meV\n113805.6·10117.8 9 3.3 18\n2361058.9·101112.5 9 3.3 18\n3251401.8·101225.2 9 3.3 18\n419900.7·10111.0 17 4.1 13\n5361103.0·10114.2 17 4.1 13\n6281502.3·101232.2 17 4.1 13\nTABLE I. The parameters of the ﬁt\n/s50/s46/s48 /s50/s46/s53 /s51/s46/s48 /s51/s46/s53 /s52/s46/s48 /s52/s46/s53 /s53/s46/s48/s48/s49/s48/s50/s48/s51/s48/s52/s48/s53/s48/s32/s81/s87/s32/s99/s111/s110/s116/s114/s105/s98/s117/s116/s105/s111/s110\n/s32/s73/s110/s116/s114/s97/s108/s97/s121/s101/s114/s32/s99/s111/s110/s116/s114/s105/s98/s117/s116/s105/s111/s110\n/s32/s81/s87/s32/s43/s32/s73/s110/s116/s114/s97/s108/s97/s121/s101/s114\n/s32/s32/s84\n/s67/s32/s40/s75/s41\n/s100/s32/s40/s110/s109/s41\nFIG. 4. The calculated dependence of the Curie temperature\non the distance between QW and Mn δ–layer.\nMn doping. The larger jTproduct for the more heavily\ndoped samples can be understood if we recall that the so-\ncalledδ– layer of Mn in reality has rather thick spatial\ndistribution, expected to be thicker for larger Mn con-\ncentration due to Mn diﬀusion [8]. Thus, the minimum\ndistance between the Mn layer and the QW is expected\nto decrease with increase of Mn doping level resulting\nin increase of the tunneling parameter. The diﬀerence\nin the energy levels dispersion σεperhaps cannot be un-\nambiguously explained with the similar plain arguments.\nWhat is read from the ﬁt (Table 1) is that the diagonal\ndisorder is somewhat smaller in the Mn layer with higher\nMn concentration. To summarize we conclude that the\nﬁt demonstrates good agreement with the experimental\ndata and the obtained values of the ﬁtting parameters\nseem quite reasonable. Thus, the experimental data can\nbedescribedbythetwocontributionsasexplainedabove.\nItmightbealsousefultoillustratetheinterplaybetween\nthe two contributions to the ferromagnetism. Shown in\nFig. 4 is the dependence of the two contributions as a\nfunctions of the distance dbetween the Mn δ– layer\nand the QW. TC1term is referred as intralayer contribu-\ntion in Fig. 4. It is, of course, independent of d. The\nQW contribution does depend on dthrough the tunnel-\ning parameter. For the tunneling case this dependence\nappears to be weaker than for the non-resonant one as\nTCroughly follows the√\nTdependence (7) rather than\nT2(9). The illustration presented in Fig. 4 corresponds\nto the parameters of the sample 5 (Table 1) and extrap-5\nolated to spacer thickness other than that of sample 5\n(d= 3 nm). We note here that the dependence of TCon\ndappears to be still too strong compared to the experi-\nmental observations [9]. We attribute this disagreement\nto the uncertainty in determination of the distance dbe-\ntween Mn layer and QW due to the ﬁnite thickness of\nthe Mn layer being around 3 nm. The detailed analysis\nhere requires more experimental data.\nIV. SUMMARY\nOn the basis of the the previously developed theory\nof the resonant indirect exchange interaction we ana-\nlyzed the ferromagnetic properties of the hybrid het-\nerostructure consisting of a InGaAs QW and remote Mn\nlayer. The experimentally observed non-monotonous de-\npendence of the Curie temperature on the QW depth\nwas explained and ﬁt as the result of two contributions\nto ferromagnetism. The ﬁrst one is the intralayer con-tribution stems from the same mechanism as that of the\nferromagnetism in bulk dilute GaMnAs samples. The\nsecond contribution is the resonant indirect exchange via\nthe 2D holes populating the QW. It is this mechanism\nthat is responsible for the non-monotonous behavior of\nTC. As only the second contribution depends on the\ndistance between the QW and Mn layer, further exper-\nimental investigations are required to separate the two\nmechanisms.\nV. ACKNOWLEDGEMENTS\nIVR, IVK, NSA acknowledge support of Russian\nScience Foundation (project 14-12-00255), BAA, KIK,\nABD, VT are grateful for support to Indian-Russian col-\nlaborative grant DST-MSE 14.513.21.0019, RFBR grant\n14-02-00586, VT acknowledges support of DST for Indo-\nRussian collaborative grants RFBR-P-141 and RMES-\n02/14.\n[1] T. Jungwirth, J. Wunderlich, V. Nov´ ak, K. Olejn´ ık, B. L .\nGallagher, R. P. Campion, K. W. Edmonds, A. W. Rush-\nforth, A.J. Ferguson, andP.Nˇ emec, Rev.Mod. Phys. 86,\n855 (2014).\n[2] T. Jungwirth, J. Sinova, J. Maˇ sek, J. Kuˇ cera, and A. H.\nMacDonald, Rev. Mod. Phys. 78, 809 (2006).\n[3] T. Dietl and H. Ohno, Rev. Mod. Phys. 86, 187 (2014).\n[4] H. Ohno, Journal of Magnetism and Magnetic Materials\n200, 110 (1999).\n[5] B. Rupprecht, W. Krenner, U. Wurstbauer, C. Heyn,\nT. Windisch, M. A. Wilde, W. Wegscheider, and\nD. Grundler, J. Appl. Phys. 107, 093711 (2010).\n[6] Y. Nishitani, D. Chiba, M. Endo, M. Sawicki, F. Mat-\nsukara, T. Dietl, and H. Ohno, Phys. Rev. B 81, 045208\n(2010).\n[7] M. A. G. Balanta, M. J. S. P. Brasil, F. Iikawa, J. A.\nBrum, U. C. Mendes, Y. A. Danilov, M. V. Dorokhin,\nO. V. Vikhrova, and B. N. Zvonkov, Journal of Applied\nPhysics116, 203501 (2014).\n[8] B. A. Aronzon, M. A. Pankov, V. V. Rylkov, E. Z. Mei-\nlikhov, A. S. Lagutin, E. M. Pashaev, M. A. Chuev, V. V.\nKvardakov, I. A. Likhachev, O. V. Vihrova, et al., J.\nAppl. Phys. 107, 023905 (2010).\n[9] B. Aronzon, A. Davydov, M. Goiran, B. Raquet,\nA. Lashkul, and E. Lahderanta, Journal of Physics: Con-\nference Series 456, 012001 (2013).\n[10] I. V. Rozhansky, N. S. Averkiev, I. V. Krainov, and\nE.Lahderanta, physicastatussolidi (a) 211, 1048 (2014).\n[11] I. V. Rozhansky, I. V. Krainov, N. S. Averkiev, and\nE. L¨ ahderanta, Phys. Rev. B 88, 155326 (2013).[12] V. Nov´ ak, K. Olejn´ ık, J. Wunderlich, M. Cukr,\nK. V´ yborn´ y, A. W. Rushforth, K. W. Edmonds, R. P.\nCampion, B. L. Gallagher, J. Sinova, et al., Phys. Rev.\nLett.101, 077201 (2008).\n[13] M. Wang, R. A. Marshall, K. W. Edmonds, A. W. Rush-\nforth, R. P. Campion, and B. L. Gallagher, Applied\nPhysics Letters 104, 132406 (2014).\n[14] V. Tripathi, K. Dhochak, B. A. Aronzon, V. V. Rylkov,\nA. B. Davydov, B. Raquet, M. Goiran, and K. I. Kugel,\nPhys. Rev. B 84, 075305 (2011).\n[15] V. Tripathi, K. Dhochak, B. A. Aronzon, B. Raquet,\nV. V. Tugushev, and K. I. Kugel, Phys. Rev. B 85,\n214401 (2012).\n[16] M. A. RudermanandC. Kittel, Phys. Rev. 96, 99(1954).\n[17] I. V. Rozhansky, N. S. Averkiev, I. V. Krainov, and\nE. Lahderanta, Journal of Magnetism and Magnetic Ma-\nterialsin press , (2015).\n[18] B. A. Aronzon, M. V. Kovalchuk, E. M. Pashaev,\nM. A. Chuev, V. V. Kvardakov, I. A. Subbotin, V. V.\nRylkov, M. A. Pankov, I. A. Likhachev, B. N. Zvonkov,\net al., Journal of Physics: Condensed Matter 20, 145207\n(2008).\n[19] V. N. Men’shov, V. V. Tugushev, S. Caprara, P. M.\nEchenique, and E. V. Chulkov, Phys. Rev. B 80, 035315\n(2009).\n[20] S. V. Zaitsev, Low Temperature Physics 38, 399 (2012)."    },    {        "title": "1510.05745v1.Evaluation_of_Spin_Waves_and_Ferromagnetic_Resonance_Contribution_to_the_Spin_Pumping_in_Ta_CoFeB_Structure.pdf",        "content": "Evaluation  of Spin Waves and Ferromagnetic Resonance \nContribution to the Spin Pumping in Ta/CoFeB Structure  \n \nMahdi Jamalia, Angeline K. Smitha, Hongshi Lib and Jian-Ping Wang a \naDepartment  of Electrical and Computer Engineering, University of Minnesota, 4 -174 200 Union \nStreet SE, Minneapolis, MN 55455, USA  \nbDepartment of Chemical Engineering and Materials Science, University of Minnesota, \nMinneapolis, MN 55455      \n \nThe spin  waves and ferromagnetic resonance  (FMR)  contribution to the spin pumping signal is \nstudied in the Ta/CoFeB interface  under different excitation bias fields . Ferromagnetic resonance \nis excited utilizing  a coplanar waveguide  and a microwave generator . Using a narrow waveguide  \nof about 3 μm , magnetostatic surface spin wave s with large wavevector  (k) of about 0.8 1 μm-1 are \nexcited . A large k value res ults in dissociation of spin wave s and FMR frequencies  according to \nthe surface spin wave dispersion relation. Spin waves and FMR  contribution to the spin pumping \nare calculated based on the area under the Lorentzian curve fitting  over experimental results . It is \nfound that the FMR over spin waves contribution is about 1 at large bias field s in Ta/CoFeB \nstructure . Based on our spin pumping results, we propose a method to characterize the spin wave \ndecay constant which is found to be about 5.5±1.27  μm in the Ta/CoFeB structure  at a bias field \nof 600 Oe . \nElectronic address:  jpwang@umn.edu  Spin pumping  as a mechanism of spin current generation due to  dynamical magnetization state s \nhas absorbed vast attention among researcher s recently  1–4. Pumping of spin curren t has been \nsuccessfully demonstrated from a magnetic layer into metallic channel s 2,5–7, semiconductor s 8,9, \ninsulator s 10 and more recently into topological insulators  11,12. In the spin pumping experiment, \nthe magnetization dynamics is usually excited utilizing a microwave source  1,5,13. The injected spin \ncurrent into the nonmagnetic channel is detected by means of  the inverse spin Hall effect of the \nnonmagnetic channel  14,15. Typically, f erromagnetic resonance  (FMR)  is intended to be the main \npart of magnetization dynamics in the spin pumping experiment ; however , due to the geometrical \neffect of the coplanar waveguide  1 or the magnetic layer itself  10,16,17, spin waves are often excited \nas well. Geometrical effect of the waveguides refers to the finite size of the waveguide compared \nto the magnetic layer that results to a non -uniform excitation. Geo metrical effect of the magnetic \nlayer refers to  the spin wave s confinement that results in the spin waves quantization and formation \nof spin wave s standing waves. Estimation of spin waves and FMR contribution to the spin pumping \nsignal is cr ucial  knowing that the spin Hall angle of the nonmagnetic material can be extracted \nfrom the amplitude of the spin pumping signal 5,18,19. In the spin pumping experiments with \nferromagnetic metallic layer s such as  CoFeB5 or NiFe4, spin waves and FMR frequencies are close. \nMoreover, due to broadening of the FMR linewidth which is enhanced by the nonmagnetic heavy \nmetal like Ta or Pt, only a single resonant frequency can be observed in the spin pumping \nexperim ent5. Most of the previous works on the investigation of spin wave contri bution to the spin \npumping are based on  using  wide waveguide s that result in mixing of spin w aves and FMR \nsignals20 and make distinction of the them very complex. Multiple eigenfrequencies in spin \npumping experiments are mostly reported in the experiment s based on the YIG magnetic oxide21,22. In this letter, we have estimated the relative intensity of spin wave and FMR contribution s to the \noutput spin pumping signal in Ta/CoFeB bilayer metallic system s. \nThe output spin pumping signal is the electromotive force generated by the inverse spin Hall \neffect of the Ta channel acting on the spin current generated from the  magnetization dynamics of \nthe CoFeB layer.  By employing a narrow waveguide, surface spin wave s with a wavevector of \nabout 0.81 μm-1 are excited in the CoFeB layer result ing in a large difference between the FMR \nand spin waves frequencies. In addition, based on our results, the authors have proposed a method \nto extract the spin wave s characteristic decay length in ferromagnetic metallic system which is \nfound to be about 5.5±1.27 μm in the Ta/CoFeB structure  at a bias field of 600 Oe . In this study, \nFMR refers to the nonpr opagating magnetization dynamical  mode. Due to the large linewidth of \nspin pumping signal, it also includes the quasi -static spin waves mode with very small wavevector. \nThe propagating modes of magnetization dynamics are spin waves that have a sizable wavevector. \nThe spin waves have diffe rent resonant frequenc ies compared to the FMR mode.  The spin waves \nmodes present in our experiments are mostly magnetostatic surface spin waves  mode s where the \nbias magnetic field and the spin waves wavevector are in the plane of the magnetic film and norm al \nto each other5,23,24.  \nExchange -coupled spin waves  with a large wavevector (short wavelength)  in metal/magnetic \ninsulator structure have been studied using parametric excitation or narrow waveguide s 25–27. \nFerromagnetic metallic layers utilized in the spin pumping experiment s are usually very thin (~ 10 \nnm) and their damping constant is much larger than magnetic insulator s. Since parametric spin \npumping is not easily achievable in  a metallic ferromagnet, to obtain  a sizable difference be tween \nthe frequency of FMR and spin waves, a narrow microwave waveguide must be utilized.  Fig. 1(a) shows a schematic of the device structure and measurement setup. Initially, Ta (5 nm) \nis sputter deposited utilizing a 6 -target Shamrock sputtering system with a built-in argon ion miller \non a thermally oxidized Si substrate with a SiO 2 thickness of about 300 nm. The Ta film is \npatterned  using photolithography  into rectangular shape s with a size of 200 μm× 50 μm using \nnegative resist and subsequent argon ion milling. Next, by lift -off process, Co20Fe60B20 (10 nm) \nwith a size of 30  μm×50 μm is placed on top of the Ta channel. The surface of Ta layer is slightly \netched  (~0.4 nm)  before CoFeB deposition to provide a fresh interface between the Ta and CoF eB. \nThe samples are field-annealed in a vacuum system with a base pressure of less than 1×10-6 Torr \nin the presence of a magnetic field of 0.4 T and temperature of 300°C  for 2hrs . Magnetization \ndynamics is excited using a n asymmetric coplanar waveguide in the GS form and a sinusoidal \nmicrowave source. The waveguide is isolated from the magnetic layer and Ta channel by SiO 2 (50 \nnm) that is deposited by electron beam evaporation. An optical micrograph of the fabricated device \nis given in Fig. 1(b). The waveguide has signal and ground lines with width s of 3 and 9 μm , \nrespectively, and the spacing between them is 3 μm.  \nUpon excitation of the magnetization dynamics by the rf-field ge nerated from the coplanar \nwaveguide, spin current ( Js) is pumped into the Ta layer (in z-direction). B oth FMR and spin waves \nare excited by the waveguide and contribute to the pumping of the spin current. Magnetic field is \napplied along the x-direction during the measurements. Strong spin -orbit interaction of the Ta layer \ntranslates Js into a charge current Jc due to the inverse spin Hall effect  (ISHE).  The electric field \ninduced by the ISHE could be written  as22: \n \nISHE sEJ    (1) \nwhere Js is the spin current injected from CoFeB in Ta and σ is the spin polarization vector of the \nspin current defined by the bias magnetic field. Magnetization dynamics in th e CoFeB generates nonequalibiruim polarized electron in the adjacent nonmagnetic layer28. The pumped spin current \nresults in additional damping of the magnetic layer itself. An electromotive force  is generated  \nacross the Ta channel  in the y-direction that can be detected by  a nano -voltmeter . The spin -orbit \ninteraction is responsible for the inverse spin -Hall effect  (ISHE)  and is a process that converts a \nspin current into an electric voltage. The strong spin -orbit inte raction in heavy metals like Pt  and \nTa23 allows observation of the ISHE at room temperature.  \nThe frequency spectra of the output dc -voltage at ±130 Oe is presented in Fig. 2(a). As \nseen, the output voltage polarity is altered by changing th e magnetic field polarity which is \nconsistent with the spin pumping  experiments  reported by other groups4,5. In most of the previous \nspin pumping works based on metallic ferro magnet s like NiFe or CoFeB , only a single resonant \npeak is observed 5,6,29; however , in this experiment , three  frequencies are present in the output \nvoltage spectra . The main frequency  occurs at 6.2 GHz  which is associated with FMR excitation \nwhile the higher frequencies of 6.7 and 8.3 GHz  are correlated with spin wave s excitation in the \nmagnetic layer. Due to  the narrow width of the waveguide, spin waves with large wavevector s can \nbe excited. Since the magnetization of CoFeB is in -plane and the magn etic field is applied along \nx-direction while spin wave s propagation is along the y-direction, magnetostatic surface spin \nwave s (MSSW) are excited in the CoFeB23,24. It is well known that MSSW shows nonreciprocal \nbehavior for opposite field p olarities23,30. The non -reciprocity of MSSW is indeed observed in our \nexperiments for the spin pump ing signal at positive and negative fields due to asymmetric coplanar \nwaveguide s. The difference between the amplitude of the spin pumping si gnal at positive and \nnegative fields is less than 10% and we have safely neglected it in our calculation. The frequ ency \nspectra of spin pumping is shown in Fig. 2(b) -(c) for ±260 and ±390 Oe, respectively. The s pin \npumping resonan t frequency  corresponding  to FMR is shifted to 8.3 and 10 GHz for the bias magnetic fields of 260 and 390 Oe. The second resonant frequency is shifted to 8.7 GHz at 260 Oe \nand it is merged with the FMR peak at 390 Oe.  This resonant peak could present the \nnonhomogeneous magnetization excitation that disappears at large bias field s due to c omplete \nmagnetization saturation along the field  direction . The third  peak  that corresponds t o MSSW  is \nchanged to 10.1 and 11.5 GHz at the field s of 260 and 390 Oe, respectively. There are also \ncontribution s from the anisotropic magnetoresistance (AMR) and/or the anomalous Hall effect \n(AHE) of the magnetic layer (CoFeB) in the output voltage. Both AMR and AHE have the form \nof asymmetric Lorentzian functions and can be isolated from the spin pumping  signal19,31,32.  \nThe effect of the excitation amplitude on the spin pumping frequency spectra is shown in \nFig. 2(d)  for the bias magnetic field of -200 Oe . By increasing the excitation a mplitude from 0 \ndBm to 7  dBm, the amplitude of the spin pumping increases accordingly. The amplitude of FMR \nand MSSW peaks at 0 dBm  (1 mW)  excitation are 1.5 and 1.7 μV while f or the excitation power \nof 7 dBm  (5 mW) , they are change d to 7.8 and 9.4 μ V, respectively . Moreover, the  resonant  peak \npositions from  spin pumping for the first two peaks at 5.9 and 6.6 GHz are the same upon \nincreasing the power from 0 to 7 dBm showing negligible nonlinear effect due to the input \nexcitation  power. Only the third peak  at 8.6 GH z shows slight red -shift down to 8.3 GHz by the \nincreasing of the input power that could be associated  with the nonlinear behavior of spin waves \nat large input power. This is expected since narrow c oplanar waveguide s can generate large rf -\nfields at high input power .  \nFig. 2(e)  is a schematic image of the device showing the profile of the magnetization \nexcitation in our structure. FMR is mostly excited in the CoFeB layer located under the waveguide. \nSurface spin waves are also excited at the same time in the CoFeB which  propagate toward left \nand right with a wavevector of Ksw. Due to decay of the spin waves along the propagation direction in the magnetic layer, the injected spin current by the spin waves into the Ta layer is also non -\nuniform and de cay accordingly. In Fig. 3(a), the resonant  frequency of the first peak (that is merged \nwith the second peak at high bias field) and the third peak (that is the second peak at high field) in \nthe spin pumping spectrum corresponding to FMR and MSSW at different bias magnetic fields are \ndemonstrated. Both FMR and MSSW peaks show behavior that is consistent with their dispersion \nrelation.  FMR dispersion follow s the Kitt el formula: \n0 ( )( )2a a eff f H H H H M    where \nγ is the gyromagnetic ratio , Meff is the effective saturation magnetization  of thin film, and Ha \naccounts for shape/crystal line anisotropy . Upon curve fitting of  the Kittel formula over the FMR \ndata, the corresponding value for γ and Meff are found to be 2.9×105 m.A-1.s-1 and 1 .3×106 A/m, \nrespectively.  Utilizing the relation\nBg\n , a Lande ʹ g-Factor  (g) of about 2.6 is obtained for the \nCoFeB thin film. This value is slightly  larger than what is reported by another group33 for the \nperpendicular CoFeB thin film. One possible reason could be because the spin pumping effect in \nthe in -plane film enhances the effective damping and increases the effective spin -orbit coupling of \nthe CoFeB layer  at the interface with the  Ta layer . Magnetostatic surface spin wave s also known  \nas Damon -Eshbach  spin waves24 are defined by the dispersion relation: \n2 2 0\n0 ( )( ) ( ) (1 )22SWkd s\na a effMf H H H H M e      \nwhere d is the magnetic layer \nthickness  (10 nm) . From the curve fitting of MSSW dispersion relation over the experimental data, \nthe spin wave wavevector is calculated to be about 0.81 μm-1 corresponding  to a wavelength of \n7.8 μm. \nWe has also performed  a micromagnetic simulation to understand the origin of the peaks \nthat are present in the spin pumping spectra. One dimensional micromagnetic simulation is performed with a cell size of 25 nm×200 μm×10 nm using OOMMF package23,34,35. Magnetization \ndynamics are excited  by a Gaussian field pulse of 50 ps  for different bias field s. Fig. 3(b) shows \nthe spin wave wavevector spectrum extracted from the simulation after 2 ns from the pulse field \nexcitation for the magnetic field of 200 Oe. The wavevector is calculated using t he fast Fourier \ntransform of spatial distribution of magnetization dynamics. As seen  in Fig. 3(b) , the main \nwavevec tor of the surface spin waves happens at 0.98  μm-1 corresponding to the wavelength  of \nabout 6.5 μ m which is close to the experimental value of 7.8 μm.  In addition, the second peak \nwitnessed in the experimental results is not observed in the simulation confirming that it is due to \nnonhomogeneous magnetization excitation.  \nThe relative intensity of spin waves and FMR contribution to the spin pum ping is \ncalculated based on the area of the Lorentzian curve corresponding to FMR and spin waves.  The \ncurve fitting of the Lorenzian curve over the experimental results are presented for the bias field s \nof -600 and 80 Oe in Fig. 4(a) and Fig. 4(b), respect ively.  Having the Lorentzian curve of \n/\n/ 22\n0 ()FMR SW\nFMR SWAVff\n, the intensity of FMR and spin wave contribution to the spin pumping are \ngiven by\n2\n/\n/\n/()FMR SW\nFMR SW\nFMR SWf df\nIfV\n  . In this formula \n/ FMR SWf is the FMR/SW frequency \nlinewidth. At a bias field of -600 Oe, the ratio of the FMR to spin wave contribution to the spin \npumping signal is about 1.0. At low bias fields of 80 Oe, this ratio drops  to 0.8 Therefore, in \nferromagnetic metallic  layers with narrow waveguide s, the spin wave contribution to the spin \npumping signal is equally important compared to  the FMR and it must be consider ed. This is very \nsignificant especially  when the spin Hall angle is estimated from the spin pumping signal.   The ratio of spin waves to  FMR contribution could be utilized to estimate the spin wave \ndecay length in the magnetic layer once a heavy metal is in contact with the magnetic metallic \nlayer. This is especially  useful knowing that  the spin wave decay constant is much shorter in the \npresence of the heavy metal and direct characterization of spin wave s is difficult. The injected spin \ncurrent into the nonmagnetic la yer is proportional to sin2(θ)5,6 where θ is the magnetization \nprecession cone angle. Assuming that the FMR precession cone angle is θ0 which happens only in \nthe area under the waveguide, the spin waves propagate toward left and right wi th an exponential \ndecay constant of Λ. The FMR precession angle can be derived from  the below  formula knowing \nthe input  rf-field (hrf). \n                                         \n( )( 4 )14rf\nr r s\nsh\nH H H H MHHM\n\n\n        (1) \nHere Hr is the resonant field and Δ H is the FMR linewidth. Thus, the maximum precession cone \nangle equals\nrfh\nH . Knowing that \ndHHfdf  , ΔH  can be obtained from the spin pumping \nfrequency spectra utilizing the relation \n0 21 ( )4efffH\nM\nf\n  5,36.  \nAssuming that the spin wave precession angle is the same as FMR at the boundary of the \nwaveguide (± L/2), the spin wave precession angle could be obtained from \n/2\n0xL\ne  for \n2Lx\nand \n/2\n0xL\ne  for \n2Lx  as shown in Fig. 5. This estimation is quite accurate once the spin \nwaves and FMR frequencies are close.  The precession cone angle can be extracted utilizing Eq. \n(1). At the bias field of 600 Oe, the spin waves and FMR resonant fr equencies are 13.6 and 12.36 GHz, respectively. Moreover, the spin waves and FMR frequency linewidths are 0.75 and 0.6 GHz, \nrespectively. The precessional cone angle is proportional to the inverse of the frequency linewidth . \nTherefore , there is about 0.23%  difference in the cone angle of spin waves and FMR at the bias \nfield of 600 Oe in our experiment.  \nThe ratio of FMR to spin wave contribution could be estimated by this formula:  \n/22\n0/2\n2\n/2sin ( )\n2 sin ( )L\nL FMR\nSW\nLdx I\nI dx\n\n\n\n                      (2) \nThis ratio can be solved numerically as a function of Λ. At the bias field of 600 Oe, the spin wave \ndecay constant is found to be about 5.5 ±1.27  μm. The curve fitting of Lorentzian curve over the \nexperiment results has less than 5% error . According to our previous work, t he Gilbert  damping \nconstant increases by a factor of about 2.5 in Ta/CoFeB bilayer structure compared to the CoFeB \nlayer5. Assuming the spin waves decay constant is pro portional to the Gilbert damping, the spin \nwave s decay length in  a CoFeB  thin film  is estimated to be about 13.75 μ m at large bias field s.  \nIn summary, the spin wave contribution to the spin pumping signal in Ta/CoFeB bilayer is \nstudie d experimentally. Using a narrow waveguide of 3 μm width, magnetostatic surface spin \nwaves with a wavevector of about 0.81 μm-1 are excited that result s in large dissociation of spin \nwaves and FMR resonant frequencies. Based on the ratio of spin waves to FMR contribution to \nthe spin pumping signal, a method is proposed to estimate the spin wave decay constant in the \nbilayer heavy metal/magnet structure. Our experimental results and proposal pave the way in better \nunderstanding of the spin wave contribution to the spin pumping signal and it c ould be utilized to \ncharacterize spin waves in metallic system s by means of spin pumping.  Additionally, this shows a \nsignificant contribution of spin waves to the spin pumping signal for narrow co -planar waveguides. This is critical for understanding and correct interpretation of results when using spin pumping \nexperiments for determination of material parameters such as  spin Hall angles.  \n \nAcknowledgement  \nThis work was partially supported by the C -SPIN center, one of six STARnet program research \ncenters , and National Science Foundation Nanoelectronics Beyond 2020 (Grant No.NSF NEB \n1124831 ). Parts of this work were  carried  out in the Minnesota Nano Center which  receives partial \nsupport from NSF through the  NNIN program . \n \nReferences  \n1 M. Costache, M. Sladkov, S. Watts, C. van der Wal, and B. van Wees, Phys. Rev. Lett. 97, 216603 \n(2006).  \n2 Y. Kajiwara, K. Ando, K. Sasage, and E. Saitoh, J. Phys. Conf. Ser. 150, 042080 (2009).  \n3 a. Azevedo, L.H. Vilela Leão, R.L. Rodriguez -Suarez, a. B. Oliveira, and S.M. Rezende, J. Appl. Phys. \n97, 10C715 (2005).  \n4 H.Y. Inoue, K. Harii, K. Ando, K. Sasage, and E. Saitoh, J. Appl. Phys. 102, 083915 (2007).  \n5 M. Jamali, A. Klemm, and J. -P. Wang, Appl. Phys. Lett. 103, 252409 (2013).  \n6 O. Mosendz, J.E. Pearson, F.Y. Fradin, G.E.W. Bauer, S.D. Bader, and a. Hoffmann, Ph ys. Rev. Lett. \n104, 046601 (2010).  \n7 K. Ando, S. Takahashi, J. Ieda, Y. Kajiwara, H. Nakayama, T. Yoshino, K. Harii, Y. Fujikawa, M. \nMatsuo, S. Maekawa, and E. Saitoh, J. Appl. Phys. 109, 103913 (2011).  \n8 K. Ando and E. Saitoh, Nat. Commun. 3, 629 (2012).  \n9 K. Ando, S. Takahashi, J. Ieda, H. Kurebayashi, T. Trypiniotis, C.H.W. Barnes, S. Maekawa, and E. \nSaitoh, Nat. Mater. 10, 655 (2011).  \n10 Z. Qiu, Y. Kajiwara, K. Ando, Y. Fujikawa, K. Uchida, T. Tashiro, K. Harii, T. Yoshino, and E. Saitoh, \nAppl. Phys. Le tt. 100, 022402 (2012).  \n11 Y. Shiomi, K. Nomura, Y. Kajiwara, K. Eto, M. Novak, K. Segawa, Y. Ando, and E. Saitoh, Phys. Rev. \nLett. 113, 196601 (2014).  12 M. Jamali, J.S. Lee, J.S. Jeong, F. Mahfouzi, Y. Lv, Z. Zhao, B.K. Nikolić, K.A. Mkhoyan, N. Samarth,  \nand J. -P. Wang, Nano Lett. 15, 151001063053004 (2015).  \n13 Z. Qiu, Y. Kajiwara, K. Ando, Y. Fujikawa, K. Uchida, T. Tashiro, K. Harii, T. Yoshino, and E. Saitoh, \nAppl. Phys. Lett. 100, 022402 (2012).  \n14 E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Appl.  Phys. Lett. 88, 182509 (2006).  \n15 O. Mosendz, J.E. Pearson, F.Y. Fradin, G.E.W. Bauer, S.D. Bader, and A. Hoffmann, Phys. Rev. Lett. \n104, 046601 (2010).  \n16 C.W. Sandweg, Y. Kajiwara, K. Ando, E. Saitoh, and B. Hillebrands, Appl. Phys. Lett. 97, 252504 \n(2010). \n17 K. Ando, J. Ieda, K. Sasage, S. Takahashi, S. Maekawa, and E. Saitoh, Appl. Phys. Lett. 94, 262505 \n(2009).  \n18 K. Ando and E. Saitoh, J. Appl. Phys. 108, 113925 (2010).  \n19 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).  \n20 S.S. Mukherjee, P. Deorani, J.H. Kwon, and H. Yang, Phys. Rev. B 85, 94416 (2012).  \n21 O. Dzyapko, H. Kurebayashi, V.E. Demido v, M. Evelt, A.J. Ferguson, and S.O. Demokritov, Appl. \nPhys. Lett. 102, 252409 (2013).  \n22 V.E. Demidov, M.P. Kostylev, K. Rott, P. Krzysteczko, G. Reiss, and S.O. Demokritov, Appl. Phys. \nLett. 95, 112509 (2009).  \n23 M. Jamali, J.H. Kwon, S. -M. Seo, K. -J. Le e, and H. Yang, Sci. Rep. 3, 3160 (2013).  \n24 R.W. Damon and J.R. Eshbach, J. Phys. Chem. Sol. 19, 308 (1961).  \n25 H. Kurebayashi, O. Dzyapko, V.E. Demidov, D. Fang, a. J. Ferguson, and S.O. Demokritov, Appl. \nPhys. Lett. 99, 162502 (2011).  \n26 K. Ando, T. An,  and E. Saitoh, Appl. Phys. Lett. 99, 092510 (2011).  \n27 C.W. Sandweg, Y. Kajiwara, A. V. Chumak, A.A. Serga, V.I. Vasyuchka, M.B. Jungfleisch, E. Saitoh, \nand B. Hillebrands, Phys. Rev. Lett. 106, 216601 (2011).  \n28 Y. Tserkovnyak, A. Brataas, and G. Bauer, Phys. Rev. B 66, 224403 (2002).  \n29 K. Ando, Y. Kajiwara, S. Takahashi, S. Maekawa, K. Takemoto, M. Takatsu, and E. Saitoh, Phys. Rev. \nB 78, 014413 (2008).  \n30 K. Sekiguchi, K. Yamada, S.M. Seo, K.J. Lee, D. Chiba, K. Kobayashi, and T. Ono, Appl. Phys. Lett.  \n97, 22508 (2010).  31 K. Kondou, H. Sukegawa, S. Mitani, K. Tsukagoshi, and S. Kasai, Appl. Phys. Express 5, 73002 \n(2012).  \n32 L. Bai, P. Hyde, Y.S. Gui, C. -M. Hu, V. Vlaminck, J.E. Pearson, S.D. Bader, and A. Hoffmann, Phys. \nRev. Lett. 111, 217602 (2013).  \n33 T. Devolder, P. -H. Ducrot, J. -P. Adam, I. Barisic, N. Vernier, J. -V. Kim, B. Ockert, and D. Ravelosona, \nAppl. Phys. Lett. 102, 022407 (2013).  \n34 M. Jamali, K. -J. Lee, and H. Yang, New J. Phys. 14, 033010 (2012).  \n35 M. Jamali, H. Yang, and K. -J. Lee, Appl. Phys. Lett. 96, 242501 (2010).  \n36 S.S. Kalarickal, P. Krivosik, M. Wu, C.E. Patton, M.L. Schneider, P. Kabos, T.J. Silva, and J.P. \nNibarger, J. Appl. Phys. 99, 093909 (2006).  \n \n \n \nFigure Captions  \nFIG. 1.   (a) A schematic of spin pumping characterization device in Ta/CoFeB bilayer structure. \nMagnetization dynamics is excited using a n asymmetric coplanar waveguide and the \noutput electromotive force is characterized by a nanovoltmeter. (b) An optical \nmicrograph of t he actual fabricated device where individual layer is labeled.  \nFIG. 2.   The spin pumping frequency spectra at the bias field of (a) ±130 Oe, (b) ±260 Oe, and (c) \n±390 Oe. (d) The spin pumping spectra characterized at a bias field of -200 Oe and for \nthe e xcitation power of 0, 2, 4, and 7 dBm. (e) A schematic showing how spin waves \nand FMR excite and contribute to the pumping of spin current.  The spin current \nindicating with down arrows are injecting into the Ta channel by both FMR and spin \nwaves.  \nFIG. 3.   (a) The FMR and magnetostatic surface spin waves resonant frequencies obtained at \ndifferent bias fields from the spin pumping experiment. (b) The spin waves wavevector for a bias field of 200 Oe and an excitation field pulse of 50 ps obtained from \nmicroma gnetic simulation.  \nFIG. 4.   (a) The curve fitting of Lorentzian function over the experimental results for bias field s \nof (a) -600 Oe and (b) -80 Oe.  \nFIG. 5.   A schematic showing the magnetization precession cone angle distribution under and \naway from the coplanar waveguide.  \n \n \n \n \n \n \n \n \n \n \n \nFigure 1  \nMICROWAVEV\nHS G\nyx\nzJs\nJc\n(a) (b)H\nxyA\nBG\nS\n4 8 12 1604812\n  Out voltage ( V)\nFrequency (GHz)Field = 165 Oe -15 μV      0.0     15 μVTa\nCoFeB(a) (b) \n \n \nFigure 2  \n \n \n \n \n \n3 6 9 12051015Amplitude ( V)\nFrequency (GHz) 6 dBm\n 4 dBm\n 2 dBm\n 0 dBm(a)\n(b)\n(c)(d)\n(e)\nFMR\nSpin wavesksw\n-10010\n-10010\n4 8 12 16-10010-130 Oe\n 130 Oe\n-260 Oe\n 260 OeAmplitude ( V)\nFrequency (GHz)-390 Oe\n 390 Oe0 200 400 6004812\n0 10 20 30 400.00.51.0\nFMRFrequency (GHz)\nField (Oe) Data\n Curve fittingspin waves(b)Amplitude (a.u.)\nWave vector (( m)-1)(a)Figure 3  \n \n \n \n \n \n \n4 8 12 16048\n4 8 12 16048(b)Amplitude ( V)\nFrequency (GHz) Data\n FMR\n SW\n FMR+SWH = 600 Oe (a)H = 80 OeAmplitude ( V)\nFrequency (GHz) Data\n FMR\n SW1\n SW2\n FMR+SW1+SW2\nFigure 4  \n  \nFigure 5  \nPrecession Angle ( θ)\n-L/2    L/2Decay length: \nWGCoFeB"    },    {        "title": "1410.3005v1.Spin_wave_free_spectrum_and_magnetic_field_gradient_of_nanopatterned_planes_of_ferromagnetic_cobalt_nanoparticles__key_properties_for_magnetic_resonance_based_quantum_computing.pdf",        "content": "arXiv:1410.3005v1  [cond-mat.mes-hall]  11 Oct 2014epl draft\nSpin wave free spectrum and magnetic ﬁeld gradient of\nnanopatterned planes of ferromagnetic cobalt nanoparticl es:\nkey properties for magnetic resonance based quantum comput ing.\nK. Benzid(a), D. Muller(b), P. Turek(c), J. Tribollet(d)\n1/ Institut de Chimie (UMR 7177 CNRS-UDS), Universit ´ede Strasbourg,\n4 rue Blaise pascal, CS 90032, 67081 Strasbourg cedex, Franc e\n2/D´epartement de physique, Laboratoire de Physique Quantique et Syst`emesDynamiques,\nUniversit ´ede Ferhat Abbas S´etif1,Alg´erie\n3/ Laboratoire ICube (UMR 7357 CNRS-UDS), Universit ´ede Strasbourg,\n23 rue du Loess, BP 20, 67 037 Strasbourg cedex 2, France.\nPACS76.30.-v –\nPACS03.67.Lx –\nPACS76.30.Fc –\nAbstract –We present a study by ferromagnetic resonance at microwave Q band of two sheets\nof cobalt nanoparticles obtained by annealing SiO2layers implanted with cobalt ions. This ex-\nperimental study is performed as a function of the applied ma gnetic ﬁeld orientation, tempera-\nture, and dose of implanted cobalt ions. We demonstrate that each of those magnetic sheet of\ncobalt nanoparticles can be well modelled by a nearly two dim ensional ferromagnetic sheet hav-\ning a reduced eﬀective saturation magnetization, compared to a regular thin ﬁlm of cobalt. The\nnanoparticles are found superparamagnetic above around 21 0 K and ferromagnetic below this\nblocking temperature. Magnetostatic calculations show th at a strong magnetic ﬁeld gradient of\naround 0.1 G/nm could be produced by a ferromagnetic nanostr ipe patterned in such magnetic\nsheet of cobalt nanoparticles. Such a strong magnetic ﬁeld g radient combined with electron para-\nmagnetic resonance may be relevant for implementing an inte rmediate scale quantum computer\nbased on arrays of coupled electron spins, as previously rep orted (Eur. Phys. J. B (2014) 87,\n183). However, this new approach only works if no additional spin decoherence is introduced by\nthe spin waves exitations of the ferromagnetic nanostructu re. We thus suggest theoretically some\npossible magnetic anisotropy engineering of cobalt nanopa rticles that could allow to suppress the\nspin qubit decoherence induced by the unwanted collective e xcitation of their spins.\nMagnetic nano-objects have many potential appli-\ncations. Magnetic nanoparticles (NPs) can be used\nas contrats agent in the diagnosis and treatment of\ncancer [1], magnetic nanostripes can be used as a medium\nfor eﬃcient classical data transmission and processing [2],\nand magnetic nanodots can be used as storage elements\nfor high density magnetic data recording [3]. The present\n(a)1-2/ benzid@unistra.fr\n(b)3/ dominique.muller@icube.unistra.fr\n(c)1/ turek@unistra.fr\n(d)1/ tribollet@unistra.fr : corresponding authorwork reports on a new potential application of metal\nmagnetic nanoparticles embedded in dielectric matrix.\nThis is related to quantum information processing and\nelectron paramagnetic resonance (EPR) spectroscopy.\nIt was recently theoretically demonstrated [4] that the\nstrong magnetic ﬁeld gradient produced by a ferromag-\nnetic nanostripe combined with the microwave pulses\ndelivered by a pulsed electron paramagnetic resonance\nspectrometer, could constitute two of the three key\nelements constituting the hardware of a potential small\nscale spin based quantum computer, the third key ele-\np-1K.Benzid1-2/benzid@unistra.fr,D.Muller3/dominique.muller@icube.u nistra.fr,P.Turek1/turek@unistra.fr,J.Tribollet1/tribollet@un istra.fr: correspondingauthor\n/s48/s44/s54 /s48/s44/s56 /s49/s44/s48 /s49/s44/s50 /s49/s44/s52/s45/s49/s48/s45/s53/s48\n/s48/s44/s54 /s48/s44/s56 /s49/s44/s48 /s49/s44/s50 /s49/s44/s52/s45/s54/s44/s48/s45/s52/s44/s53/s49/s52/s48 /s50/s49/s48 /s50/s56/s48/s48/s54/s49/s50 /s99/s47\n/s66\n/s48/s32/s40/s84/s41\n/s32/s69/s80/s82/s32/s115/s105/s103/s110/s97/s108/s32/s40/s97/s46/s117/s41\n/s66\n/s48/s32/s40/s84/s41/s97/s47\n/s98/s47/s69/s80/s82/s32/s115/s105/s103/s110/s97/s108/s32/s40/s97/s46/s117/s41\n/s32/s32\n/s32/s32/s68 /s46/s73/s46/s40/s97 /s46/s117 /s41\n/s84/s32/s40/s75/s41\nFig. 1: Ferromagnetic resonance (FMR) spectrum obtained at Q band (33.95 GHz) with the magnetic ﬁeld applied in the plane\n(0o) of the cobalt magnetic nanoparticles, (1a) for sample I (1 .1017cm−2Co+ions) and (1b) for sample II (0 .51017cm−2Co+\nions). The inset (1c) inside ﬁgure 1a shows the temperature d ependence of the double integration (noted D.I.) of the FMR\nsignal which is proprotionnal to the eﬀective magnetizatio n of the cobalt nanoparticles detected by FMR spectroscopy a t Q\nband. The modulation frequency was 100 kHz. The modulation a mplitude was 4 G. The microwave power used was 2 mW.\nThe narrow linewidth signals seen on the spectrum of the two s amples below 0.8 Tesla are due to impurities in the Q band\nmicrowave cavity used.\nment being the coupled electron spin qubits themselves\nplaced nearby the ferromagnetic nanostripe. In this\nprevious work [4], it was also shown that unwanted spin\ndecoherence could occur during information processing\nby microwave pulses if the paramagnetic resonances of\nthe electron spin qubits spectrally overlapp the spin wave\nresonances existing in a bulk like ferromagnetic nanos-\ntripe. The solution to this problem that was proposed\nconsist in a carefull design of the spin wave spectrum\nexpected for a bulk like ferromagnetic nanostripe. This is\na feasible but rather complex approach. Here, we suggest\nan alternate approach to achieve the same objective. This\nconsist in the patterning of one or several ferromagnetic\nnanostripes in a thin ﬁlm containing magnetic metal\nnanoparticles. One then expects a strong magnetic ﬁeld\ngradient produced by such a diluted magnetic nanostripe,\nalthough it should be somehow reduced as compared to\nthe one produced by a bulk like ferromagnetic nanostripe.\nHowever, such magnetic dilution may be an advantage\nfor this new particular kind of application according to\nthe superparamagnetic behavior of such features. The\nsolely strong spin wave mode expected at relatively low\nmicrowave frequencies (X or Q bands) is the uniform\nmode of precession of magnetization, generally called\nthe ferromagnetic resonance mode in the case of a bulk\nferromagnet. Avoiding the spectral overlap problem in\nthis context thus corresponds to shifting this usually\nbroad ferromagnetic-like resonance mode, suﬃciently farfrom the paramagnetic resonances of the qubits. As it\nwill be shown here, combining experiments and theory,\nthis could be done by an appropriate engineering of the\nmagnetic anisotropy of each magnetic metal nanoparticle\npresent inside the diluted ferromagnetic nanostripe.\nIn the ﬁrst part of the article we present FMR ex-\nperiments, similar to previous ones performed at X\nband [5], but here performed at microwave Q band and\nas a function of the direction of the applied magnetic\nﬁeld, on two cobalt nanoparticles sheets obtained by\nimplantation of cobalt ions in a dielectric matrix of SiO2,\nfollowed by thermal annealing. This FMR experimental\nstudy allow us to demonstrate that each of those magnetic\nsheets of cobalt nanoparticles can be fairly modelled by\na nearly two dimensional ferromagnetic plane having a\nreduced eﬀective saturation magnetization, with respect\nto a true bulk like thin ﬁlm of cobalt. Subsequent calcu-\nlations based on magnetostatics then demonstrates that a\nstrong magnetic ﬁeld gradient of around 0.1 G/nm can be\nproduced by such a magnetic nanostripe made of cobalt\nnanoparticles. Since the key issue of the present work is\nquantum information processing of electron spin qubits,\nwe show that a suitable engineering of the magnetic\nanisotropy of the cobalt nanoparticles may allow the in-\ndividual microwave adressing and coherent manipulation\nof spin qubits state through EPR. This is done upon\nseparating the spin resonance of the nanostripes from the\np-2Spinwavefreespectrumandmagneticﬁeldgradientofnanopatter nedplanesofferromagneticcobaltnanoparticles: keyproperties formagneticresonancebasedquantumcomputing.\n0 20 40 60 80 1001.051.11.151.21.251.3x 104\nAngle (°)Bres (Gauss)\n0 20 40 60 80 1001.081.11.121.141.161.181.21.22x 104\nAngle (°)Bres (Gauss)a/ b/\nFig. 2: Full rotational pattern of the ferromagnetic resona nce mode of sample I (a/), and of sample II(b/), measured at ro om\ntemperature and at Q band. Solid lines are numerical simulat ion obtained using the Smit and Beljers formalism (see text) .\nspin resonances of the qubits.\nTwo amorphous SiO2thin ﬁlms, obtained by oxida-\ntion of Si wafers, were implanted at room temperature\n(295 K) with 160 keV Co+ions at two diﬀerent doses,\n0.5 1017cm−2(sample I) and 1 .1017cm−2(sample II),\nand further annealed at 873 K under hydrogen ﬂow,\nin order to produce the metallic cobalt nanoparticles\ninside the SiO2matrix. Similar samples implanted at\n160 keV with a nominal dose of 1 .1017cm−2Co+ions\nhave been previously investigated by TEM and Squid\nmagnetometry [6]. This previous study has shown that\nthe magnetic cobalt metal nanoparticles have mainly the\nhexagonal phase, with a c axis of magneto-cristalline\nanisotropy randomly oriented in the amorphous SiO2\nmatrix, that the average diameter of those magnetic\ncobalt nanoparticles is around 4.5 nm, and that they\nare superparamagnetic at room temperature, as shown\nby Squid magnetometry. The ferromagnetic resonance\nstudy of sample I and II containing the magnetic cobalt\nnanoparticles is performed using an EMX Bruker contin-\nuous wave electron paramagnetic resonance spectrometer\noperating at Q band (microwave frequency around 34\nGHz). Modulation coils are used to modulate the EPR\nsignal at 100 kHz, which allows to perform a sensitive\nlock in detection of the microwave absorption signal. This\nmodulation also produces spectrum which appear as the\nderivative of a standard microwave absorption spectrum\nwith gaussian or lorentzian lines. The cobalt implanted\nthin ﬁlms of SiO2can be rotated inside the microwave\nresonator in order to vary the direction of the applied\nstatic magnetic ﬁeld, from in plane (angle: 0o) to out of\nplane (angle: 90o). The sample temperature can also be\nvaried from 4 K to 300 K using an Oxford Helium ﬂowcryostat with a temperature controller designed for EPR\nexperiments.\nThe FMR spectra obtained for sample I and II are\nshown on ﬁgure 1. The FMR resonance at Q band for a\nmagnetic ﬁeld applied in the plane (0o) occurs at around\n1,06 T for sample I and at around 1,09 T for sample\nII. As expected, due to the random orientation of the c\naxis of the hexagonal cobalt nanoparticles and also due\nto their size distribution resulting from the implantation\nprocess, the FMR signal is broad (pic to pic linewidth\nof around 0.2 T) for the two samples. Also, the double\nintegration of the FMR signal of sample I is larger than\nthe one of sample II, as expected given the larger ion dose\nin this sample. The in plane (0o) magnetization of sample\nI has also been recorded as a function of temperature\n(inset 1c of ﬁgure 1a). As it was previously shown by\nSQUID magnetometry [6] on similar samples, the cobalt\nnanoparticles of this sample are superparamagnetic at\nroom temperature and ferromagnetic at low temperature.\nWe found a transition between the two regimes occuring\naround a blocking temperature of around TB= 210K.\nExperimentally, the sample was cooled below 100 K with-\nout any magnetic ﬁeld applied. As the applied magnetic\nﬁeld is set to zero between two successive steps, one\nexpects that during FMR experiments performed below\nTB, the magnetization of each nanoparticle is locked\nalong the hexagonal c axis, which is random over the\nensemble of nanoparticles. When the FMR spectrum is\nacquiered, the magnetic ﬁeld rapidly increases, but it does\nnot produce any alignement of the magnetization vectors\nof the nanoparticles due to the too slow magnetization\nrelaxation dynamics below the blocking temperature. For\np-3K.Benzid1-2/benzid@unistra.fr,D.Muller3/dominique.muller@icube.u nistra.fr,P.Turek1/turek@unistra.fr,J.Tribollet1/tribollet@un istra.fr: correspondingauthor\n/MT120\n/MT122\n/MT66/MT48/MT44/MT122/MT97/MT47\n/MT98/MT47/MT82/MT101/MT115/MT105/MT115/MT116\n/MT67/MT111/MT32/MT78/MT80\n/MT83/MT105/MT79 /MT50\n/MT32/MT32/MT32/MT32/MT32/MT83/MT105\n/MT115/MT117/MT98/MT115/MT116/MT114/MT97/MT116/MT101/MT99/MT47\n/MT100/MT47\n/MT101/MT47/MT69/MT116/MT99/MT104/MT105/MT110/MT103\n/MT109/MT97/MT116/MT114/MT105/MT120/MT32/MT99/MT32/MT97/MT120/MT105/MT115\n/MT109/MT97/MT116/MT114/MT105/MT120/MT32/MT99/MT32/MT97/MT120/MT105/MT115\n/MT65/MT110/MT116/MT105/MT70/MT101/MT114/MT114/MT111/MT109/MT97/MT103/MT110/MT101/MT116/MT105/MT99\n/MT115/MT104/MT101/MT108/MT108/MT32/MT111/MT114/MT32/MT109/MT97/MT116/MT114/MT105/MT120/MT97/MT109/MT111/MT114/MT112/MT104/MT111/MT117/MT115/MT32/MT109/MT97/MT116/MT114/MT105/MT120\nFig. 3: 3a: Nanofabrication process, based on etchingthrou ghamask, suggested toproducethe dilutedferromagnetic na nostripe\ndiscussed inthe textandshown on ﬁgure3b. Figures 3c, 3dand 3e illustrate thephysical concepts behindthemagnetic ani stropy\nengineering of the nanoparticles discussed in the text.\ntemperatures far below TB= 210K, this slow relaxation\ndynamics leads to an almost zero magnetization signal\nmeasured over the ensemble of cobalt nanoparticles with\nrandomly oriented c axis. Around TB= 210Kand\nabove, the relaxation time of the magnetic nanopar-\nticles of cobalt becomes much faster than the time\nscale requiered for the full FMR ﬁeld sweep spectrum\nacquisition. As a consequence, around TB= 210Kand\nabove, the magnetic nanoparticle start to align along the\ndirection of the applied magnetic ﬁeld producing a net\nmagnetization. The theoretical magnetic relaxation time\nτ1of an isolated superparamagnetic nanoparticle is given\nby a standard Arrhenius law expression [7]. It takes into\naccount the temperature T and the magnetic anisotropy\nenergyK Vof the nanoparticle of volume V. It is given\nbyτ1=τ0exp/parenleftBig\nK V\nkBT/parenrightBig\n. The intrinsic time scale τ0is\ngenerally in the range of 10−9s≤τ0≤10−13s. Here,\nthe measurement time considered is the time requiered\nfor the acquisition of a full ﬁeld sweep FMR spectrum\nover more than 8000 Gauss, which is around τmes= 100s\nhere. This measurement time roughly correspond to the\ntime window let to the nanoparticle to overcome the\nmagnetic anisotropy barrier and thus to reorient itself\nin the direction of the applied magnetic ﬁeld at the\ntemperature T. Taking τ0= 10−9s, as in the previ-\nous analysis of similar samples investigated by SQUID\nmagnetometry [6], and using τmes= 100sfor the FMR\nexperiments, one obtains the following relation between\nthe magnetic anisotropy energy K Vof the nanoparticle\nof volume V, and its blocking temperature TB(blocking\nenergykBTB):K V≈25kBTB. Using the average\ndiameter of R= 4.5nmobtained by TEM analysis ofsimilar samples implanted by 160 keV cobalt ions at a\ndose of 1 .1017cm−2[6], and using the standard value of\nthe constant of anisotropy for cobalt, K= 2106ergcm−3\n[8], one estimates TB= 218K. This shows that our\nexperimental measurements by FMR spectroscopy are in\nvery good agreement with well known cobalt properties\nand with previous datas obtained by TEM and Squid\nmagnetometry on similar samples.\nNow, using the standard Kittel formula [9] for the FMR\nof a thin ferromagnetic ﬁlm with an in plane applied\nmagnetic ﬁeld, it is possible to estimate the eﬀective\nsaturation magnetizations of those thin ﬁlms of cobalt\nnanoparticles. However, it is more reliable to estimate\nthe eﬀective saturation magnetizations of the ﬁlms by\nrecording the full rotational patterns of their ferromag-\nnetic resonance mode, as it is shown on ﬁgure 2a and\n2b respectively for sample I and II. As a matter of fact,\nthe observed rotational pattern symmetry is due here\nto the shape anisotropy of each magnetic ﬁlm, which is\nrelated to its eﬀective saturation magnetization. Numer-\nical simulations of the rotational patterns, performed\nusing the Smit and Beljers formalism [10] and taking\ninto account shape anisotropy, thus conﬁrms that the\ndipolar couplings between the cobalt nanoparticles can\nnot be neglected in those two magnetic ﬁlms. From this\npoint of view, the two ﬁlms of magnetic nanoparticles\ncan thus be seen as mimicking bulk like magnetic thin\nﬁlms with a reduced eﬀective saturation magnetization\ncompared to the bulk material ( Bsat,bulk ≈1.84T).\nThe results of the simulation made for sample I gives\nBsat, I≈0.15Tand the one made for sample II gives\nBsat, II≈0.079G. The ratio of the eﬀective saturation\nmagnetizations of the two ﬁlms is very close to 2. It nicely\np-4Spinwavefreespectrumandmagneticﬁeldgradientofnanopatter nedplanesofferromagneticcobaltnanoparticles: keyproperties formagneticresonancebasedquantumcomputing.\n−1000 −500 0 500 1000−100−50050100\nz (nm)Bz(z)   (G)\n  \n−1000 −500 0 500 1000−4−3−2−101x 104\nx (nm)C(x)  (G.nm)\n−800−600−400−200 0200400600800−150−100−500\nx (nm)Bz(x, z=0)  (G)\n−1000 −500 0 500 1000−0.2−0.100.10.2\nx (nm)dBz(x, z=0)/dx  (G/nm)a/\nc/ d/b/\nFig. 4: Magnetostatic properties of the dipolar magnetic ﬁe ld produced by a diluted ferromagnetic nanostripe of cobalt nanopar-\nticles. See text for details concerning the parameters used for the calculations.\ncorresponds to the ratio of the dose of implanted Co ions\nbetween sample I and sample II. Also, one notes that\nthe ratio between the eﬀective saturation magnetization\nof each ﬁlm and the bulk saturation magnetization,\nnoted f, is given by fI=0.15\n1.84≈0.081 for sample I\nandfII=0.079\n1.84≈0.043 for sample II. This ratio f\nshould correspond to the averaged atomic fraction of\ncobalt in the SiO2matrix, as it can be measured by\nRutherford BackScattering experiments (RBS) [11] .\nThose RBS measurements were previously performed [11]\non similar samples obtained by implantation of Co+ions\nat 160 keV with a dose of 1 .1017cm−2and revealed an\naveraged atomic fraction of cobalt in the SiO2matrix of\nfRBS,1.1017cm−2≈0.13. The present FMR experiments\ngivefFMR,1.1017cm−2≈0.08. This reduced value shows\nthat part of the cobalt NPs are not metallic nanoparticles\nbut instead oxidized nanoparticles. The ratio of the\nmetallic cobalt amount over the total cobalt amount in\ntheSiO2matrix,RFMR/RBS =fFMR\nfRBS≈0.61, is in\ngood agreement with the ratio RSquid≈0.72 that was\npreviously determined by Squid magnetometry on similar\nsamples [6].\nThe suggested application of the present work to quantum\ncomputing requires a strong magnetic ﬁeld gradient [4].\nThe reported FMR experiments demonstrate that the\ntwo ﬁlms of magnetic nanoparticles can be modelled as\neﬀective continuous magnetic thin ﬁlm. Therefore, to get\na strong magnetic ﬁeld gradient one may elaborate nanos-\ntructures, e.g. nanostripes, within such nano-implanted\nthin ﬁlms. This could be done, as it is shown on ﬁgure\n3a and 3b, by an etching process, following either a deep\nUV optical lithography or electron beam lithography pro-\ncessing of a resist, depending on the targeted nanostripedimensions. One such isolated and magnetically diluted\nferromagnetic nanostripe of cobalt nanoparticles could\nthus produce the required strong magnetic ﬁeld gradient\nas shown in ﬁgure 4 by the theoretical simulations based\non magnetostatic calculations [4]. One has assumed in\nﬁgure 4 a diluted ferromagnetic nanostripe of cobalt\nnanoparticles, having a width W= 1000 nm along the z\naxis, a thickness of T= 100 nm, corresponding roughly\nto the halfwidth of the cobalt ion implantation proﬁle\nin theSiO2matrix [11], and an inﬁnite length along\nthe y axis (100 microns). For the diluted ferromagnetic\nnanostripe of cobalt nanoparticles considered here, the\neﬀective saturation magnetization is Msat, eff, with\nµ0Msat, eff =Bsat, eff = 1840 G=Msat, bulkCo\n10.\nThe ﬁgure 4a plots the dipolar magnetic ﬁeld produced\nby this nanostripe, Bz(z, xoptim= 290nm), at the\noptimal position xoptimfor the qubits discussed below\n(see ﬁgure 3 for the deﬁnition of x and z axis) and\nas a function of the in plane position z of the qubit.\nFigure 4a thus demonstrate the good in plane homo-\ngeneity of the dipolar magnetic ﬁeld gradient produced\nby the nanostripe in a plane placed at the distance\nxoptim= 290n mabove (or below) this nanostripe and\nfor z values close to zero. This in plane homogeneity\nis investigated more quantitatively on ﬁgure 4b using\nthe homogeneity coeﬃcient C(x) previously introduced,\nC(x) =/integraltext+100\n−100dz(Bz(x, z)−Bz(x,0)) (see [4]).\nOne deﬁnes the position xoptimbyC(xoptim) = 0. For\nthe diluted ferromagnetic nanostripe considered here,\none ﬁnds xoptim= 290n m. The ﬁgure 4c then plots\nBz(z= 0, x), which allows to determine the shift of the\nresonant magnetic ﬁeld of the paramagnetic qubits for a\ngiven position x. Finally, the ﬁgure 4d shows the gradient\np-5K.Benzid1-2/benzid@unistra.fr,D.Muller3/dominique.muller@icube.u nistra.fr,P.Turek1/turek@unistra.fr,J.Tribollet1/tribollet@un istra.fr: correspondingauthor\n0 20 40 60 800.511.522.53x 104\nAngle (°)Resonant magnetic field:  Bres (G)\n0 5000 10000 1500000.511.522.533.54x 10−3\nB0 (G)Microwave absorption spectrum (a.u.)a/ b/\nFig. 5: 5a/ The three rotational patterns of the uniform FMR- like mode of the diluted superparamagnetic nanostripe expe cted\nin the three situations described qualitatively on ﬁgure 3c (blue dots: nanostripe with shape anisotropy alone), 3d (da rk dots:\nnanostripe with shape anisotropy and with magneto-cristal line anisotropy of hexagonal cobalt with an eﬀective anisot ropy ﬁeld\nBA, MC= 6826G, obtained from the K value in [8]) and 3e (continuous violet: nanostripe with shape anisotropy , with magneto-\ncristalline anisotropy of hexagonal cobalt with the eﬀecti ve anisotropy ﬁeld BA, MC= 6826G, and with exchange bias anisotropy\ndue to the antiferromagnetic shells, with an eﬀective aniso tropy ﬁeld BA, EB= 7400G, see [13]). 5b/ Full electron spin resonance\nspectrum expected for the hybrid paramagnetic qubits-supe rparamagnetic nanoparticles nanodevice. This hybrid nano device is\nassumed to contain electron spin qubits placed at a distance xoptim= 290nmabove or below the diluted superparamagnetic\nnanostripe of cobalt NPs described on ﬁgure 3e. The linewidt h of the uniform FMR-like mode was assumed to be around 1000\nGauss, and the one of the paramagnetic resonance of the qubit s around 10 Gauss. µ0Msat, eff=Bsat, eff= 1840G.\nalong x of the dipolar magnetic ﬁeld produced by the fer-\nromagnetic nanostripe,d Bz(x)\ndx(assuming z=0 and y=0).\nFigure 4d shows that one could obtain a strong magnetic\nﬁeld gradient having a maximum strength of more than\n0.1 G/nm at the optimal position xoptim= 290n m\nabove this nanostripe. This position is also the one where\nthis nearly one dimensionnal magnetic ﬁeld gradient has\nits maximum in plane homogeneity, as it was discussed\nabove.xoptimis thus the position where many electron\nspin qubits should be placed nearby the nanostripe for\nquantum computing [4]. Note that the dipolar magnetic\nshift of the paramagnetic resonance lines of the qubits is\nexpected to be much smaller at xoptimin the case of such\ndiluted ferromagnetic nanostripe of cobalt nanoparticles,\nthan in the case of a true bulk like nanostripe (see\nﬁgure 4c). As a consequence, and also due to the broad\nferromagnetic resonance mode of the cobalt nanoparticles\na spectral overlap is expected between the broad FMR\nmode of cobalt nanoparticles and the weakly shifted\nparamagnetic resonances of the electron spin qubits of\nthe quantum nanodevice (g=2.00 is assumed here for the\nqubits). This may be adressed by a careful engineering of\nthe magnetic anisotropy of the cobalt nanoparticules, as\nillustrated on ﬁgure 3c, 3d and 3e. The optimal strategy\nis illustrated by ﬁgure 3e. Figure 3c shows hexagonalcobalt nanoparticles with randomly oriented c axis in\nthe amorphous SiO2matrix (this work and [5]). Figure\n3d shows a similar assembly but with NPs having their\nhexagonal c axis oriented along the in plane c axis of an\nanisotropic cristalline matrix, like Al2O3[12]. Figure 3e\nshows another assembly similar to the one of ﬁgure 3d but\nhere with all NPs surrounded and exchange coupled to a\nshell of antiferromagnetic material, whose spins are also\ndirected along the c axis common to the cobalt nanopar-\nticles and to the anisotropic cristalline matrix [13,14].\nA controlled annealing of the sample under an oxygen\natmosphere may allow to control the antiferromagnetic\nshell around the metallic cobalt nanoparticles and thus to\nincrease their resulting in plane magnetic anisotropy. Ion\nimplantation of cobalt ions directly inside an anisotropic\nantiferromagnetic cristalline matrix followed by the same\nnanofabrication process could be another way to produce\nthis strongly anisotropic diluted ferromagnetic nanostripe\nwith an in plane easy axis of magnetization. The increase\nof the uniaxial in plane magnetic anisotropy, from case\n3c to case 3e, is well demonstrated on ﬁgure 5, which\nshows the rotationnal pattern of the uniform FMR-like\nmode of such superparamagnetic implanted nanostripes\nin those three cases. It is observed (ﬁgure 5a), that\nincreasing the in plane uniaxial anisotropy shift to lower\nmagnetic ﬁeld values the uniform FMR-like mode of\np-6Spinwavefreespectrumandmagneticﬁeldgradientofnanopatter nedplanesofferromagneticcobaltnanoparticles: keyproperties formagneticresonancebasedquantumcomputing.\nthe diluted superparamagnetic nanostripe for a in plane\n(0o) applied magnetic ﬁeld. The paramagnetic resonance\nof the electron spin qubits placed at xoptimabove the\nnanostripe is quite insensitive to this magnetic anisotropy\nengineering, whereas the FMR-like mode is much shifted\ntowards lower ﬁelds as previously discussed. The ﬁgure\n5b summarizes the expected situation.\nAs previously discussed [4], the inter- qubits distances\ninside spin chains have also to be well chosen. Given\nthat here the magnetic ﬁeld gradient produced by the\ndiluted ferromagnetic nanostripe is reduced compared to\nthe one produced by a bulk like ferromagnetic nanostripe,\ndipolar coupled electron spin qubits would thus have to\nbe separated by distances larger than the optimal 3.5\nnm distance previously estimated, in order to distin-\nguish their paramagnetic resonance and to coherently\nmanipulate them by microwave pulse sequences [4]. This\nwould be possible but would then lead to a further\nreduced scalability of the small scale quantum computer\nnanodevice. To avoid this new problem occuring in the\nnew nanodevice design proposed presently, one could\nuse stronger spin-spin coupling between successive spin\nqubits inside the spin chains, like exchange couplings,\nthus keeping an interesting number of spin qubits in\nsuch small scale quantum processor [4]. ZnO bulk single\ncrystals are particularly interesting anisotropic single\ncrystals that could be tested for such purpose. The\nﬁrst reason is that ZnO has the wurtzite anisotropic\nstructure (c axis of anisotropy) and that this kind of\nalignement of the anisotropy axis of cobalt NPs has been\nalready observed [15]. The second reason is that ZnO is a\nsemiconductor matrix with a very low spin orbit coupling\nand which can be isotopically puriﬁed. This implies that\nit is an excellent host matrix for electron spin qubits,\nlike the shallow indium donors [16], or transition metal\nions, like the Fe3+spin qubits [17] and the Mn2+spin\nqubits [18], that we previously investigated by pulsed\nEPR spectroscopy. The third reason is that ZnO is a di-\nrect gap semiconductor with excellent optical properties,\nincluding a very large exciton binding energy. This could\nbe used for producing an eﬀective optically induced long\nrange exchange couplings between the qubits present in\nthe ZnO matrix [19, 20], as requiered here due to the\nreduced strength of the available magnetic ﬁeld gradient\nproduced by a superparamagnetic nanostripe.\nIn conclusion, we have demonstrated by a ferromag-\nnetic resonance study at microwave Q band of two cobalt\nnanoparticles sheets obtained by implantation of cobalt\nions in a dielectric matrix of SiO2, followed by thermal\nannealing, that each of those magnetic sheets can be\nwell modelled by a nearly two dimensional ferromagnetic\nplane having a reduced eﬀective saturation magnetization\nwith respect to a true bulk like thin ﬁlm of cobalt.\nMagnetostatic calculations have then shown that a\nstrong magnetic ﬁeld gradient of around 0.1 G/nm could\nbe produced by a ferromagnetic nanostripe patternedin such magnetic plane of cobalt nanoparticles. This\nstrong magnetic ﬁeld gradient combined with electron\nparamagnetic resonance may be usefull for implementing\nan intermediate scale quantum computer based on arrays\nof coupled electron spins(J. Tribollet, Eur. Phys. J. B\n(2014) 87, 183). This is possible as far as the magnetic\nanisotropy engineering of the cobalt nanoparticles allows\nto overcome the problem of the spectral overlap between\nthe narrow shifted paramagnetic resonances of electron\nspin qubits and the broad uniform FMR-like mode of the\ndiluted superparamagnetic nanostripe. This magnetic\nanisotropy engineering of the cobalt nanoparticles could\nthus suppress the spin qubit decoherence induced by the\nunwanted spin waves excitations, and thus represents an\nalternative solution to the previously proposed solution\nrequiring a carefull design of the spin wave spectrum of\na bulk like ferromagnetic nanostripe. The price to pay\nfor using this new design is to chose spin-spin couplings\nbetween qubits which are stronger, over large nanometer\nscale distances, than the dipolar couplings previously\nproposed. We ﬁnally suggested that this new strategy for\nquantum computing may be particularly well suited for\nexciton-mediated exchange coupled electron spin qubits\nin wurtzite zinc oxide, individually and coherently ma-\nnipulated by means of microwave pulses and of the strong\nmagnetic ﬁeld gradient produced by a superparamagnetic\nnanostripe made of implanted cobalt nanoparticles.\nREFERENCES\n[1] J. Gallo et al., Chem. Soc. Rev. 42, 7816 (2013).\n[2] A. Khitun et al., J. Phys. D: Appl. Phys. 43, 264005\n(2010).\n[3] D.A. Thompson et al., IBM J. Res. Develop. 44, 311\n(2000).\n[4] J. Tribollet, Eur. Phys. J. B 87, 183 (2014).\n[5] F. Yildiz et al., J. of the Korean Phys. Soc. 53, 3699\n(2008).\n[6] D. Muller et al., Nucl. Instr. and Meth. in Phys. Res. B\n178, 144 (2001).\n[7] D.E. Madsen et al., J. Phys.: Condens. Matter 20, 345209\n(2008).\n[8] M. Jamet et al., Phys. Rev. Lett. 86, 4676 (2001).\n[9] C. Kittel, Physical Review 73, 155 (1948). *\n[10] J. Smit et al., Philips Res. Rep. 10, 113 (1955).\n[11] C. D’Orleans, PhD Thesis - University of Strasbourg -\nPHASE Laboratory - France (2003).\n[12] A. Meldrum et al., Nucl. Instr. and Meth. in Phys. Res.\nB207, 36 (2003).\n[13] V. Skumryev et al., Nature 423, 850 (2003).\n[14] J.K. Lee et al., Appl. Phys. Lett. 89, 182502 (2006).\n[15] D.P. Norton et al., Appl. Phys. Lett. 83, 5488 (2003).\n[16] J. Tribollet, Eur. Phys. J. B 72, 531 (2009).\n[17] J. Tribollet et al., Europhysics Letters 84, 20009 (2008).\n[18] K. Benzid et al., Europhysics Letters 104, 47005 (2013).\n[19] C. Piermarocchi et al., Phys. Rev. Lett. 89, 167402\n(2002).\n[20] G.F. Quinteiro et al., Phys. Rev. Lett. 97, 097401 (2006).\np-7"    },    {        "title": "1004.2680v1.Ferromagnetism_in_repulsive_Fermi_gases__upper_branch_of_Feshbach_resonance_versus_hard_spheres.pdf",        "content": "arXiv:1004.2680v1  [cond-mat.quant-gas]  15 Apr 2010Ferromagnetism in repulsive Fermi gases:\nupper branch of Feshbach resonance versus hard spheres\nSoon-Yong Chang, Mohit Randeria, and Nandini Trivedi\nDepartment of Physics, The Ohio State University, Columbus , OH 43210, USA\nWe use quantum Monte Carlo, including backﬂow corrections, to investigate a two-component\nFermi gas on the upper branch of a Feshbach resonance and cont rast it with the hard sphere gas.\nWe ﬁnd that, in both cases, the Fermi liquid becomes unstable to ferromagnetism at a kFasmaller\nthan the mean ﬁeld result, where kFis the Fermi wavevector and athe scattering length. Even\nthough the total energies E(kFa) are similar in the two cases, their pair correlations and ki netic\nenergies are completely diﬀerent, reﬂecting the underlyin g potentials. We discuss the extent to\nwhich our calculations shed light on recent experiments.\nPACS numbers: 67.85.-d, 37.10.Jk, 71.27.+a\nIntroduction: Ultracold atomic gases are emerging\nas a unique laboratory for testing quantum many-body\nHamiltonians. A problem of fundamental importance is\nthe ground state of two species of fermions interacting\nviarepulsive interactions. The attractive case is now\nwell-understood and shows the BCS-BEC crossover [1]\nin the superﬂuid ground state. The broken symmetry is\nalready apparent within BCS mean ﬁeld theory (MFT)\nwith an arbitrarily small attraction leading to a paired\nsuperﬂuid. In contrast, we know much less about the re-\npulsive case. The Landau Fermi liquid, known to exist at\nweak repulsion [2], can become unstable only beyond a\ncritical value of the interaction[3]. Thus the phase tran-\nsition is nota weak coupling problem, and the validity\nof MFT in the repulsive case is questionable.\nAn excitingnewdevelopmentisarecentexperiment[4]\nwhich has been interpreted as evidence for a ferromag-\nnetic instability [3, 5–7] in a “repulsive” Fermi gas of\n6Liatoms. A crucial point is that the interactions be-\ntween the atoms are quite diﬀerent from the textbook\nproblem of hard-sphere interactions. In the experiment,\nthe atoms are on the upper branch of a Feshbach reso-\nnance with a positive s-wave scattering length a. The\ntwo-body ground state then is a molecule of size a. But\nin the upper branch, where the wave function is made\nup from scattering states, the atoms feel an eﬀective re-\npulsioncharacterized by a >0, despite the fact that the\nunderlying potential is attractive.\nThe main question we examine in this Letter is the ex-\ntent to which the many-body physicsin the upper branch\nis similar to, or diﬀerent from, that of a purely repul-\nsive Fermi gas. We use quantum Monte Carlo (QMC) to\ncompute the energy, chemical potential and pair distri-\nbution function of the two systems – upper branch and\nrepulsive – to understand the instability of the Fermi liq-\nuid to ferromagnetism. We believe that such a study of\nequilibrium properties is necessary, before one addresses\nnon-equilibrium questions in the upper branch.\nBefore describing our results, we emphasize important\nways in which our work diﬀers from previous studies,which focus on MFT of purely repulsive interactions.\nFirst, we carefully discuss what it means for a many-\nbody wavefunction to be on the upper branch, which is\nessential to describe the experiments. Second, it is cru-\ncial to use QMC for this strong coupling problem. For\ninstance, QMC calculations [8] for the electron gas show\nthat ferromagnetism sets in at a critical density nearly\n3 orders of magnitude smaller than that predicted by\nHartree-Fock MFT. Finally, we include backﬂow correc-\ntions, which can have a nontrivial eﬀect on the nodes\nof the many-body wavefunction, and thus on the ground\nstate energy [9]. Not including backﬂowmay lead to spu-\nrious ferromagnetic instabilities in normal3He [10].\nOur main results are that we ﬁnd Ferromagnetic (FM)\ninstabilitiesinboththeupperbranchandthehardsphere\nFermi gas. For small kFa >0, with kFthe Fermi\nwavevector and athe s-wave scattering length, both sys-\ntems are Landau Fermi liquids. The upper branch be-\ncomes unstable to a FM state at kFa= 0.89(2), indepen-\ndent of the details of the interaction (in the zero-range\nlimit). The critical kFais similar for a purely repulsive\ninteraction, but the result is non-universal and depends\non details the potential; we will focus on hard spheres\nof diameter a. In both cases the critical value is consid-\nerably smaller than the Stoner MFT result ( kFa)MFT=\nπ/2 [3]. Despite similar values of the critical interac-\ntion, the behavior of the kinetic energy and the two-\nbody correlationsare qualitatively diﬀerent for the upper\nbranch and hard spheres. We also discuss the harmoni-\ncally trapped gas using the local density approximation\n(LDA). We conclude with a brief comparison of our re-\nsults with experiments [4]. We ﬁnd that some aspects of\nthe experiment cannot be understood within our equilib-\nrium theory.\nModel: We consider a gas of N= (N↑+N↓) fermions\nof massmwith two species, denoted by “spin” ↑and↓,\nwhich interact via a potential V(r). The Hamiltonian is\nH=/summationdisplay\niσp2\niσ\n2m+1\n2/summationdisplay\ni,jV(rij) (1)2\n0 0.5 1 1.5 2\nkFa0123E/(N 3εF /5)QMC\nPert.\n0.89(2)a)\n0 0.5 1 1.5 2\nkFa0123E/(N 3εF /5)QMC(w/o BF)\nQMC(w. BF)\nPert.\n0.90(2)b)\n0 1 2 3\nkFr01η(r)/ηmaxkFa = 0.5, HS\nkFa = 1.0, HS\nkFa = 0.5, UBc)\nFIG. 1: (Color online) QMC energy per particle for the Fermi l iquid state (for N↑=N↓= 33 particles) as a function of kFa\nfor (a) the upper branch and for (b) hard spheres, compared wi th the perturbative result Eq. (3). Panel (b) shows QMC with\nand without backﬂow (BF) corrections. Backﬂow does not have a signiﬁcant eﬀect on the upper branch results in (a). There\nis a transition to a ferromagnetic state when the QMC energy c rosses that of the fully polarized gas (horizontal lines). ( c) The\nfunction η(r) describing BF correlation, discussed in the text, for the u pper branch (UB) and hard sphere (HS) gas.\nwithrij=|ri↑−rj↓|. For the QMC calculations we\nconsider a cubic box with periodic boundary conditions.\n(At the end, we alsodiscusstrap eﬀects within LDA). We\nmeasure lengths in units of k−1\nF, wherekF= (3π2n)1/3\nfor a free Fermi gas of density n. We measure energies in\nunits ofǫFG= 3ǫF/5 whereǫF=k2\nF/2m(with ¯h= 1).\nWe consider two forms of the interaction potential.\nFor the repulsive case, we use V(r) =V0>0 for\nr < Rand zero elsewhere. In the hard sphere limit\nV0→ ∞and the diameter of the sphere R=a\nscattering length. For the attractive case, we use\nV(r) =−(8/mR2)V0/cosh2(2r/R), extensively used in\nBCS-BECcrossoverstudies[11,12]. We choosethe range\nsuch that kFR≪0.1, so that we obtain universal results\nindependent of the detailed form of V(r). We use V0to\ntune the scattering length a >0, such that V(r) has a\nsingle bound state. We will focus on scattering states to\nconstruct the upper branch wavefunction.\nQMC Results: The many-body wave function for a\n(paramagnetic)Fermi liquid is ofthe Jastrow-Slaterform\nΨ =/productdisplay\ni↑,j↓f(rij)ΦFG↑ΦFG↓. (2)\nThe Slater determinants Φ FGσ’s are constructed from\nplane waves, and the symmetric Jastrow function f(r)\naccounts for interactions.\nWe now argue that the upper branch Jastrow factor\nmust be qualitatively diﬀerent from the f(r)≥0 used\nfor the purely repulsive case. To see this, consider using\na conventional nodeless f(r) for the attractive Fermi gas.\nThis state is a normalFermi liquid, and thus orthogo-\nnal (for large N) to the superﬂuid ground state [1, 11]\nof the BCS-BEC crossover for all kFa. However, the en-\nergy per particle in this state is always lowerthan the\nfree gas 3 ǫF/5, which means that the fermions do feel\nan attraction. In other words, this normal wave function\nnecessarily has some pairing (bound state-like) correla-\ntions, and is therefore noton the upper branch.A necessary condition for a many-body state to be on\nthe upper branch is that its energy per particle must\nbe greater than 3 ǫF/5. We must ensure that every pair\nof particles feels an eﬀective repulsion. We achieve this\nby introducing a node in the Jastrow f(r). To deter-\nminef(r), we use the lowest-order constrained varia-\ntional (LOCV) method [13], which is well known in nu-\nclear physics and has also been used for strongly inter-\nacting quantum gases [12, 14]. The LOCV equation has\nan upper-branch solution [15] f(r) with a node, whose\nlocation tracks the scattering length at small a[i.e.,\nf(r)∼(1−a/r)] but then saturates at large a.\nWe use QMC to calculate the energy for the up-\nper branch [Fig. 1(a)] and the hard sphere Fermi gas\n[Fig. 1(b)] with N↑=N↓. For small kFa, both results\nagree with the well-known perturbative result [2]\nE\nNǫF=3\n5+2\n3πkFa+4\n35π2(11−2ln2)(kFa)2+...(3)\nWe note that Eq. (3) should be taken seriously only for\nkFa≪1; the third order term is known to be non-\nuniversal, and depends on the detailed shape of the po-\ntential and on the p-wave scattering channel [2].\nA suﬃcient criterion for ferromagnetism (FM) is that\nthe energy of the paramagnetic Fermi liquid state exceed\nthat of the fully polarized state ǫP\nFG/(3ǫF/5) = 22/3≃\n1.58. It is instructive to begin with simple analytical ap-\nproximations (even though these involve using Eq. (3)\nbeyond its domain of validity!) The simplest approxima-\ntion is to just keep the ﬁrst term in (3), the mean ﬁeld\nHartree shift. We ﬁnd that this energy crosses that of\nthe fully polarized ǫP\nFGatkFa= (22/3−1)9π/10≃1.66.\nThis is slightly larger than the Stoner estimate of π/2,\nbut still below the hard sphere solidiﬁcation limit kFa=\n(9π/4)1/3≃1.92. Including the second order term in (3)\nincreasesthe energy of the paramagneticFermi-liquid so-\nlution, and thus FM sets in closer to kFa≃1.\nThe QMC energy for both the upper branch and hard3\nspheres implies a FM ground state for kFa>∼0.9. We\nnextaddressbackﬂowtoseehowitaﬀectsourconclusion.\nBackﬂow: It is very important to include backﬂow\nwhich, as noted above, makes nontrivial modiﬁcations to\nthe nodal surfaces and can lead to large quantitative ef-\nfects[9]inthegroundstateenergy. Backﬂowmodiﬁesthe\nsingle-particleplane waveorbitals φk(riσ) = exp[ik·riσ]\nused to construct the Slater determinants in Eq. (2) via\nthe replacement riσ→riσ+/summationtext\njη(rij)rij, wherejlabels\nparticles of the opposite spin ¯ σ.\nThe optimal form of the backﬂow function η(r) must\nbe determined for each problem; it is known to be very\ndiﬀerent for the4He roton [16], for normal3He [17], and\nfor the electron gas [18]. Insight into the form of η(r) for\n3He came from analyzing the problem of a3He impurity\nin4He[17]. Followingthe samelogic, weconsiderasingle\nspin-down impurity in a spin-up Fermi sea. Omitting the\ndetails of our analysis(which will be reported elsewhere),\nwe ﬁnd the η(r)’s shown in Fig. 1(c).\nFor the hard sphere gas the optimal η(r) vanishes in-\nside the hard-core diameter and has a single peak just\nbeyond it, qualitatively similar to the case of3He [9, 17].\nAs in3He, we approximate the form of η(r) by a Gaus-\nsian whose parameters we optimize. We use QMC to\ncompute the energy of hard spheres using Eq. (2) with\na nodeless Jastrow times backﬂow-corrected Slater de-\nterminants. We ﬁnd that backﬂow leads to a signiﬁcant\nlowering of energy [see Fig. 1(b)] that becomes more pro-\nnounced with increasing kFa. For example, there is a\n5.5% reduction in energy at kFa= 1.\nFor the upper branch, we ﬁnd that the form of the op-\ntimalη(r) is qualitatively diﬀerent; see Fig. 1(c). It is\nnonzero at the origin and decreases monotonically, with\na power-law decay at large r. Further, η(r) changes\nvery little with kFacompared with the hard sphere case.\nThe form of upper branch η(r) is similar to systems\nwith a soft-core, long-range repulsion, like the electron\ngas [18]. We use QMC to calculate the energy of the\nupper branch state (2), with a Jastrow with a single\nnode, times backﬂow-corrected Slater determinants. In\nthis case the reduction in energy is small [Fig. 1(a)] and\nfalls within our statistical error of <∼1%.\nWe thus ﬁnd that backﬂow is important for hard\nspheres when k−1\nFis comparable to the hard-core diame-\nterR=a. On the other hand, backﬂow eﬀects are small\nfor the upper branch, where k−1\nF≫R, the range.\nObservables: For both the upper branch and for hard\nspheres, weconcludethatferromagnetismisenergetically\nfavorable, based on the crossing of energies of the para-\nmagnetic Fermi liquid and the fully polarized FM; see\nFig. 1(a,b). For the upper branch, we ﬁnd that FM state\nis stable for kFa≥0.89(2). The order of the transi-\ntion requires a careful ﬁnite-size scaling analysis in the\nvicinity of the phase transition, beyond the scope of our\npresent investigation.\nAlthough the total energies in the Fermi liquid phases0 1 2 3 4 5\nkFr00.511.522.53g(r)00.5 11.5\nkFa0103g(0)\na)\n0 1 2 3 4 5\nkFr00.511.522.53g(r)kFa = 0.5\nkFa = 0.8\nkFa = 1.0\nb)\nFIG. 2: (Color online) Pair distribution function g(r)≡\ng↑↓(r) for (a) the upper branch and (b) the hard sphere gas\nfor various values of kFa. Inset in panel (a) shows g(r= 0)\nas a function of kFa.\nin the upper branch and hard spheres are similar, the\npotential /angbracketleftV/angbracketrightand kinetic energy /angbracketleftK/angbracketrightare very diﬀerent\nin the two cases. To understand this, it is illuminat-\ning to look at the pair distribution function g↑↓(r), de-\nnoted by g(r) for simplicity. In the hard sphere case\n[Fig. 2(b)], g(r) vanishes inside the hard-core and goes\nto unity at large separation. The potential energy /angbracketleftV/angbracketright ∼/integraltext\nd3rg(r)V(r) then vanishes identically and the total en-\nergy [Fig. 1(b)] in the hard sphere case is entirely kinetic.\nIn the upper branch, on the other hand, we ﬁnd a large\ncancellation between a positive /angbracketleftK/angbracketrightand a negative /angbracketleftV/angbracketright.\nIn marked contrast to hard spheres, the upper branch\ng(r) is extremely large at r= 0, has a pronounced dip\nat the node in the Jastrow f(r) and then goes to unity\nat larger; see Fig. 2(a). For the short-range attraction,\nthe potential energy /angbracketleftV/angbracketright ∼g(0)/integraltext\nd3rV(r) is thus large\nand negative, dominated by the growth of g(0) with in-\ncreasing kFa[inset of Fig. 2(a)]. This is compensated by\na large positive kinetic energy /angbracketleftK/angbracketright[Fig. 3(a)] so that we\nﬁnd the total energy shown in Fig. 1(a).\nHarmonic Trap: We ﬁrst obtain from our QMC data\nthe chemical potential µ= (∂E/∂N) as a function of\ndensity [Fig. 3(b)]. We then invert this to ﬁnd the equa-\ntion of state n(µ) of thehomogeneous system.\nWe restrict ourselves to the paramagnetic Fermi liquid\nregime here, and use the LDA µ(r) =µ(0)−Vtrap(r) to\nstudy the eﬀects of the harmonic trap Vtrap(r) with asso-\nciated length scale aHO. To compare with experiments,4\n0 1\nkFa0123\nµ/εFQMC\nPert.\n0 1\nkFa0510152025KE/(N 3εF /5)QMC\nLDA\na) b)\nFIG. 3: (Color online) (a) Kinetic energy (KE) per particle o f\nthe upper branch. Squares represent QMC data as a function\nkFa. The dashed line at kFa= 0.89 shows the ferromagnetic\ntransition at which the KE is greatly suppressed and then\nremains constant. The dot-dash curve is the LDA result for\nthe KE versus k0\nFausing the QMC equation of state. Within\nLDA, ferromagnetism appears at the center of the trap when\nk0\nFa≃1.1. (b) Chemical potential µ(kFa), related to the\nsquare of the LDA radius of the trapped cloud, for the upper\nbranch. The perturbative result for µ= (∂E/∂N) obtained\nfrom Eq. (3) is also shown.\nwe use the parameter k0\nF= (24N)1/6/aHOas a measure\nof the total number of particles N. To ﬁnd the chemical\npotential at the center µ(0), we solve the LDA equa-\ntion (kFa)6= 23/248π/integraltext˜µ(0)\n0d˜µ[˜µ(0)−˜µ]1/2˜n(˜µ). Here\nwe have used dimensionless quantities ˜ µ=µ(0)ma2and\n˜n(˜µ) =n(µ)a3, wheren(µ) is the QMC equation of state.\nWe then ﬁnd the density n(r= 0) at the center of\nthe trap, from which we can determine the interaction\nparameter kF(0)a= [3π2˜n(˜µ(0))]1/3. We ﬁnd that for\nk0\nFa≃1.1, the trap center reaches kF(0)a= 0.89, the\ncritical value in the homogeneous case. At this point the\ncenter of the trap should become unstable to ferromag-\nnetism. We have also calculated within LDA the total\nand kinetic energies in the trapped system as functions\nofk0\nFa; the latter is shown in Fig. 3(a).\nComparison with experiments: While we were mo-\ntivated by the experiments of Ref. [4], we focus only on\n“equilibrium” in the upper branch, and do not address\ndynamical questions. Ifthree-body processes leading to\nmolecule formation can be suppressed, there may be a\nwindow of time-scales where equilibrium physics in the\nupper branch, as described here, would be observed. The\nkFa-dependence of g(r= 0) [inset of Fig. 2(a)] is relevant\nto the loss rate [19] due to molecule formation.\nEven with these caveats, there are some aspects of the\nexperiment which we can understand qualitatively and\nothers we cannot. First, we do ﬁnd a ferromagnetic in-\nstability in the upper branch, but predict that it should\nhappen in a homogeneous system at kFa= 0.89, which\ntranslatesintotheonsetofFMatthecenterofthetrapat\nk0\nFa≃1.1, while the experiment sees interesting features\nonly atk0\nFa≃2. The behavior of the chemical potential\n[Fig. 3(b)], which increases with increases kFaand thensaturates beyond the transition is qualitatively consis-\ntent with the experiment. However, the kFa-dependence\nof the kinetic energy is not; our results in Fig. 3(a) are\nqualitatively diﬀerent from the experiments. Finally, we\nhave not addressed here the question of FM domains and\ntheir sizes, which is important to understand given that\nthey have not been seen in the experiment.\nConclusions: We show using QMC that fermions\nwith eﬀectively repulsive interactions become unstable\nto ferromagnetism beyond a critical interaction strength\nkFa≃0.9. This is true both for fermions in the upper\nbranch (scattering state with positive a) of an attractive\npotential and also for hard sphere repulsion, despite im-\nportant diﬀerences in their short range correlations and\nthe kinetic energy that reﬂect the underlying potentials.\nAcknowledgments: We acknowledge support from\nARO W911NF-08-1-0338 and nsf-dmr 0706203 and the\nuse of computational facilities at the Ohio Supercom-\nputer Center. We thank S. Zhang for discussions.\nNote added: As we werewriting this paper, the workof\nPilatiet al.appeared [20]. Itaddressesthesameproblem\nusing a similar, but not identical, approach. Wherever\nthey overlap, our results are in essential agreement.\n[1] S. Giorgini, L. P. Pitaevskii and S. Stringari, Rev. Mod.\nPhys.80, 1215 (2008).\n[2] A. L. Fetter and J. D. Walecka, “Quantum Theory of\nMany-Particle Systems” (Dover, Mileola, 2003); Sec. 11.\n[3] E. Stoner, Philos. Mag. 15, 1018 (1933). K. Huang, “Sta-\ntistical Mechanics” (Wiley, New York, 1987); Sec. 11.7.\n[4] G.-B. Jo et al, Science 325, 1521 (2009).\n[5] R. A. Duine and A. H. MacDonald, Phys. Rev. Lett. 95,\n230403 (2005).\n[6] L. J. LeBlanc et al, Phys. Rev. A 80, 013607 (2009).\n[7] G. J. Conduit, A. G. Green and B. D. Simons, Phys. Rev.\nLett.103, 207201 (2009).\n[8] F. H. Zong, C. Lin and D. M. Ceperley, Phys. Rev. E 66,\n036703 (2002).\n[9] K. E. Schmidt, M. A. Lee and M. H. Kalos, Phys. Rev.\nLett.47, 807 (1981).\n[10] D. M. Ceperley, (private communication).\n[11] J. Carlson et al, Phys. Rev. Lett. 91050401 (2003).\n[12] S.-Y. Chang et al, Phys. Rev. A 70, 043602 (2004).\n[13] V. Pandharipande and H. Bethe, Phys. Rev. C 7, 1312\n(1973).\n[14] S. Cowell et al, Phys. Rev. Lett. 88, 210403 (2002).\n[15] The upper branch solution ( λ >0) forf(r) with a sin-\ngle node is obtained from the LOCV equation [13, 14]/bracketleftbig\n−(1/m)d2/dr2+V(r)/bracketrightbig\nrf(r) =λrf(r) forr≤“healing\nlength”. For a zero-range potential f(r) = sin(√\nmλ(r−\nb))/r, withbgiven by√\nmλb= tan−1(√\nmλa). For√\nmλa→0+, we thus get f(r)∼(1−a/r).\n[16] R. Feynmanand M. Cohen, Phys. Rev. 102, 1189 (1956).\n[17] V. Pandharipande and N. Itoh, Phys. Rev. A 8, 2564\n(1973); V. Pandharipande, Phys. Rev. B 18, 218 (1978).\n[18] Y. Kwon, D. M. Ceperley and R. M. Martin, Phys. Rev.\nB48, 12037 (1993).\n[19] D. S. Petrov, Phys. Rev. A 67010703(R)(2003).\n[20] S. Pilati et al, arXiv:1004.1169."    },    {        "title": "1508.01410v1.Interface_driven_spin_torque_ferromagnetic_resonance_by_Rashba_coupling_at_the_interface_between_non_magnetic_materials.pdf",        "content": "arXiv:1508.01410v1  [cond-mat.mes-hall]  6 Aug 2015Interface-driven spin-torque ferromagnetic resonance by Rashba coupling at the\ninterface between non-magnetic materials\nM. B. Jungﬂeisch,1,∗W. Zhang,1J. Sklenar,1,2W. Jiang,1J. E. Pearson,1J. B. Ketterson,2and A. Hoﬀmann1\n1Materials Science Division, Argonne National Laboratory, Argonne IL 60439, USA\n2Department of Physics and Astronomy, Northwestern Univers ity, Evanston IL 60208, USA\n(Dated: September 28, 2018)\nThe Rashba-Edelstein eﬀect stems from the interaction betw een the electron’s spin and its mo-\nmentum induced by spin-orbit interaction at an interface or a surface. It was shown that the inverse\nRashba-Edelstein eﬀect can be used to convert a spin- into a c harge current. Here, we demonstrate\nthat a Bi/Ag Rashba interface can even drive an adjacent ferr omagnet to resonance. We employ a\nspin-torque ferromagnetic resonance excitation/detecti on scheme which was developed originally for\na bulk spin-orbital eﬀect, the spin Hall eﬀect. In our experi ment, the direct Rashba-Edelstein eﬀect\ngenerates an oscillating spin current from an alternating c harge current driving the magnetization\nprecession in a neighboring permalloy (Py, Ni 80Fe20) layer. Electrical detection of the magnetiza-\ntion dynamics is achieved by a rectiﬁcation mechanism of the time dependent multilayer resistance\narising from the anisotropic magnetoresistance.\nConventional spintronics relies on the exchange inter-\naction between conduction electrons on one side and lo-\ncalized spins in magnetic materials on the other side [1].\nStimulated by the experimental demonstration of spin-\nto charge current conversion using bulk spin Hall eﬀects\n(SHE), these kind of spin-orbital phenomena were ac-\ntively investigated in the last decade and opened up the\ndoor to the research ﬁeld of spin-orbitronics [2–5]. SHEs\ncan be investigated by means of spin-current injection\nfrom a ferromagnet (FM) into materials with large spin-\norbit coupling, usually normal metals (NM) such as Pt\nor Pd [6], and sensing the generated voltagegenerated by\nmeansoftheinversespinHalleﬀect(ISHE)[7–14]. Other\ninteresting applications of SHEs are the eﬀective magne-\ntization switching of nanomagnets or the movement of\ndomain walls [15–17]. Furthermore, the ferromagnetic\nlinewidth modulation as well as the excitation of spin\nwaves and ferromagnetic resonance by SHE was demon-\nstrated in ferromagnetic metals and insulators [18–22].\nThe SHE is a bulk eﬀect occurring within a certain vol-\nume of the NM determined by the spin-diﬀusion length.\nThe conversioneﬃciency can be expressed by a material-\nspeciﬁc parameter, the spin Hall angle γSHE[4].\nVery recently, it has been shown that the inverse\nRashba-Edelstein eﬀect (IREE) can also be used for\ntransformation of a spin- into a charge current [23–26].\nThe IREE is the inverse process to the Rashba-Edelstein\neﬀect (REE) [27]. The REE originates from spin-orbit\ninteraction in a 2D electron gas at interfaces or surfaces,\nwhich eﬀectively produce a steady non-equilibrium spin\npolarization from a charge current driven by an electric\nﬁeld. TheHamiltonianofthis interactionisgivenby[23]:\nHR=αR(k×ˆez)·σ,whereαRis the Rashba coeﬃcient,\nˆezis the unit vector in z-direction [see Fig. 1(b,c)] and\nσis the vector of Pauli matrices. As a result of this in-\nteraction the dispersion curves of the 2D electron gas are\nspin-split if αR/negationslash= 0, as illustrated in Fig. 1(a). Analo-\ngous to the spin Hall angle, the spin- to charge currentinterconversion parameter can be deﬁned as [23]:\nλREE=αRτS/¯h, (1)\nwhereτSis the eﬀective relaxation time describing the\nratiobetween spin injection and spin-momentum scatter-\ning and ¯his the reduced Planck constant. The spin-split\n2D electron gas dispersions and Fermi contours of many\nRashba surfaces and interfaces have been investigated\nspectroscopically[28]. In general, largeRashbacouplings\noccur at interfaces between heavy elements with strong\nspin-orbit interaction (such as Bi, Pb, and Sb) and other\nnon-magnetic materials with small spin-orbit coupling\nsuch as Ag, Au, and Cu [28, 29]. Even though, the in-\nteraction between a charge current and a non-zero spin\ndensity at a Rashba interface has been demonstrated by\ninjection of a spin-pumping driven spin current at fer-\nromagnetic resonance, the reverse process remains to be\nexplored experimentally until now.\nHere, we demonstrate that a Bi/Ag Rashba inter-\nface can drive spin-torque ferromagnetic resonance (ST-\nFMR) in an adjacent ferromagnetic layer. We interpret\nour results in terms of an excitation by the direct REE,\nwhich drives an oscillating spin current from an alter-\nnating charge current that scatters of the Rashba inter-\nface (Ag/Bi). The generated spin current excites the\nmagnetizationprecessionin aneighboringpermalloy(Py,\nNi80Fe20) layer by the spin-transfer torque eﬀect [20, 31].\nThe precessional magnetization leads to resistance oscil-\nlations on account of the anisotropic magnetoresistance\n(AMR) of Py. The mixing between the applied alter-\nnating current and the oscillating resistance allows for\na direct voltage detection of the induced magnetization\ndynamics [20, 22]. Injecting an additional DC current to\nthe sample results in an additional spin current gener-\nation due to the REE which enables to manipulate the\nferromagnetic resonance linewidth by exerting a torque\non the magnetization.\nWe fabricated the devices using magnetron sputtering2\nFIG. 1. (Color online) (a) Dispersion curves of a 2D electron\ngas are spin-split due to the REE. (b) Scheme of the ST-FMR\nexperimental setup. (c) ST-FMR mechanism in Py/Ag/Bi\nmultilayers. The alternating RF current drives an Oersted\nﬁeldhRFexerting a ﬁeld-like torque τ⊥on the magnetization\nM. At the same time a oscillatory transverse spin accumula-\ntion at the Py/Ag interface generated at the Ag/Bi interface\nby the REE exerts a damping-like torque τ||on the magneti-\nzation.\nand photolithography. The multilayers were prepared in\nthe shape of30 ×5µm2stripes using lithographyand lift-\noﬀ on intrinsic Si substrates with 300-nm thick thermally\ngrown SiO 2. Four diﬀerent types of multilayers were de-\nposited using magnetron sputtering: Py, Py/Bi, Py/Ag\nand Py/Ag/Bi. In the case of the Py/Ag/Bi systems,\nthe Ag thickness was tAg= 2, 4, 6, 10, 15 nm, the Py\nthickness tPy= 9 nm and the Bi thickness tBi= 4 nm.\nThe control samples feature a Py thickness of 7 nm, Ag\nthickness 6 nm and Bi thickness 4 nm. In a subsequent\nprocess step, the coplanar waveguide (CPW) was fabri-\ncated on top of the multilayers. Figure 1(b) illustrates\nthe experimental setup. A bias-T is used to apply a mi-\ncrowave signal and to detect the rectiﬁed DC voltage at\nthe sametime. The applied microwavepoweris keptcon-\nstant at +10 dBm, unless otherwise mentioned. An in-\nplane magnetic ﬁeld is applied at an angle of θ= 45◦[see\nillustration in Fig. 1(b,c)]. While sweeping the external\nmagnetic ﬁeld the DC voltage is detected by a lock-in\nampliﬁer with an amplitude modulation at 3 kHz. All\nmeasurements were performed at room temperature.\nFigure2shows typical spectra at an excitation fre-\nquency of f= 4 GHz. Let’s ﬁrst discuss the trilayers\n[Fig.2(a)]. Inourexperiment, magnetizationdynamicsis\nexcited simultaneously by the Oersted ﬁeld as well as by\nthe REE which generatesan oscillating spin current from\nthe alternating charge current driving the magnetization\nprecession in the neighboring permalloy layer when the\ncondition of ferromagnetic resonance is fulﬁlled,\nf=|γ|\n2π/radicalbig\nH(H+4πMeﬀ). (2)Here,Meﬀis the eﬀective magnetization and |γ|is the\ngyromagnetic ratio. Electrical detection of the magne-\ntization dynamics is achieved by a rectiﬁcation mecha-\nnism of the time dependent multilayer resistance arising\nfrom the AMR of Py. A rectiﬁcation by spin pump-\ning and IREE is a secondary eﬀect in our experiment\n[20]. As apparent from Fig. 2(a), the Py/Ag/Bi samples\nexhibit a superimposed symmetric and antisymmetric\nLorentzian lineshape. The smallest Ag interlayer thick-\nness of 2 nm shows the largest symmetric contribution,\nbut the smallest absolute signal. With increasing tAg\nthe signal tends to be more antisymmetric and the ab-\nsolute value increases. The control samples are depicted\nin Fig.2(b). The pure Py sample features a small, an-\ntisymmetric Lorentzian signal due to a rectiﬁcation by\nAMR. The Py/Bi sample exhibits a very small, mostly\nsymmetric signal. Py/Ag features a reasonably large an-\ntisymmetric signal: The Ag layer is beneﬁcial for the\nabsolute voltage because a larger Oersted ﬁeld is gener-\nated in the Py layer resulting in a higher AMR signal\nmanifested in a substantial antisymmetric lineshape.\nFigures3(a) and (c) illustrate how the resonance ﬁeld\nand linewidth alter for diﬀerent Ag interlayer thicknesses\nat various excitation frequencies. The excitation of fer-\nromagnetic resonance is conﬁrmed by a ﬁt to Eq. ( 2), see\nFig.3(b). Furthermore, the data shown in Fig. 3(d) is\ngoverned by a linear dependence between linewidth ∆ H\n-60-40-2002040Voltage V ( µV)\n500400300200100\nMagnetic field H (Oe)f = 4 GHzControl samples:\n Py (7)\n Py(7)/Bi(4)\n Py(7)/Ag(6)\n \nComparison:\n Py(9)/Ag(10)/Bi(4)-60-40-2002040Voltage V ( µV)\nf = 4 GHzAg thickness variation:\n Py(9)/Ag(2)/Bi(4)\n Py(9)/Ag(4)/Bi(4)\n Py(9)/Ag(6)/Bi(4)\n Py(9)/Ag(10)/Bi(4)\n Py(9)/Ag(15)/Bi(4)(a)\n(b)\nFIG. 2. (Color online) Spectra of REE-driven ST-FMR mea-\nsured at a frequency of 4 GHz and an applied microwave\npower of +10 dBm. Thickness in brackets given in nm. (a)\nAg thickness dependence of the resonance signal. (b) Com-\nparison between control samples and Py(9)/Ag(10)/Bi(4).3\n1200\n1000\n800\n600\n400Resonance field H (Oe)\n1284\nAg thickness tAg (nm) 4 GHz\n 5 GHz\n 6 GHz\n 7 GHz\n 8 GHz\n 9 GHz\n100\n80\n60Linewidth ∆H (Oe)\n1284\nAg thickness tAg  (nm) 4 GHz\n 5 GHz\n 6 GHz\n 7 GHz\n 8 GHz\n 9 GHz(a)\n120\n100\n80\n60\n40Linewidth ∆H (Oe)\n108 6 4\nFrequency f (GHz) Py/Ag(2)/Bi\n Py/Ag(4)/Bi\n Py/Ag(6)/Bi\n Py/Ag(10)/Bi\n Py/Ag(15)/Bi(c)10\n8\n6\n4Frequency f (GHz)\n1200800400\nMagnetic field H (Oe) Py/Ag(2)/Bi\n Py/Ag(4)/Bi\n Py/Ag(6)/Bi\n Py/Ag(10)/Bi\n Py/Ag(15)/Bi(b)\n(d)\nFIG. 3. (Color online) (a) Resonance at various excitation\nfrequencies for diﬀerent Ag thicknesses. (b) Dispersion me a-\nsured for diﬀerent Ag interlayer thicknesses, tPy= 9 nm,\ntBi= 4 nm. A ﬁt to Eq. (2) conﬁrms the excitation of ferro-\nmagnetic resonance; shown as solid lines. (c) Evolution of t he\nFMR linewidth with tAgat diﬀerent excitation frequencies.\n(d) Determination of Gilbert damping parameter α. Solid\nlines show a ﬁt to Eq. (3).\nand the excitation frequency f:\n∆H(f) = ∆H0+4πfα\n|γ|, (3)\nwhere ∆ H0is the inhomogeneous linewidth broaden-\ning given by the zero-frequency intercept and αis the\nGilbert damping parameter. This conﬁrms the excita-\ntion of FMR in our samples.\nThe magnetization dynamics in a macrospin model is\ngovernedby a modiﬁed Landau-Lifshitz-Gilbert equation\n[22]:\ndˆm\ndt=−|γ|ˆm×/vectorHeﬀ+αˆm×dˆm\ndt+|γ|τ/bardblˆm×(ˆy×ˆm)\n+|γ|τ⊥ˆy×ˆm,\n(4)\nwhere ˆmis the magnetization direction, Heﬀis the eﬀec-\ntive magnetic ﬁeld, τ||andτ⊥are the two acting torque\ncomponents, and the coordinate system (ˆ x,ˆy,ˆz) is de-\nﬁned as shown in Fig. 1(b,c).\nThe two vector components of the current-induced\ntorqueτ||,τ⊥can be related to the amplitudes of the\nsymmetric and antisymmetric components of the reso-\nnance lineshape [22]: (1) An in-plane component τ||∼\nˆm×(ˆy×ˆm) results in a symmetric contribution and (2)\nan out-of-plane component τ⊥∼ˆy×ˆmresults in an an-\ntisymmetric contribution, see Fig. 1(c) [22]. Figure 4(a)\nillustratestheAgthicknessdependence oftheamplitudes\nof both contributions, respectively, as a function of thedriving RF frequency. We observe the following trend:\nThe amplitudes increase with increasing tAgup totAg≈\n7 nm. At larger thicknesses, the antisymmetric contribu-\ntion (dashed lines) remains constant up to tAg≈10 nm\nbefore it decreases. The symmetric contribution, how-\never, peaks at tAg≈7 nm and reduces afterwards. In\norder to highlight this observation we plot a torque-ratio\nequivalent T = Vantisymm/(Vantisymm+Vsymm) as a func-\ntion oftAgin Fig.4(b). Clearly, the symmetric contribu-\ntion to the lineshape is greatest at a lower Ag thickness\nand becomes negligible for larger tAg. The reason for\nthis trend is the larger Oersted ﬁeld produced in samples\nwith a thicker Ag layer and, thus, a larger out-of-plane\ntorque contribution τ⊥. As is apparent from Fig. 4(b),\nthistrendisindependent ontheexcitationfrequency. We\nalso show the ratio of the control samples Py/Ag and Py\nin the same plot as a red dot and a green square, respec-\ntively.\nWe interpret our observations in the following way: If\ntheobservedincreaseofthesymmetriccomponent( ∼τ||)\nwith respect to the antisymmetric component ( ∼τ⊥)\nwas caused by the SHE in Ag, we should observe the\nsame ratio for the control sample Py/Ag. As is apparent\nfrom Fig. 4(b), this is not the case. Since Ag features a\nlong spin-diﬀusion length of ∼300 nm [30], it would also\nbe possible that the SHE in Bi generates a spin current\nwhich diﬀuses through the Ag layer. However, since the\ngenerated voltage for the control sample Py/Bi is neg-\n100\n80\n60\n40\n20Amplitude |V| ( µV)\n1412108642\nAg thickness tAg  (nm)Symmetric\ncontribution\n 4 GHz\n 5 GHz\n 6 GHz\n 7 GHz\n 8 GHz\n 9 GHz\n \nAntisymmetric\ncontribution\n 4 GHz\n 5 GHz\n 6 GHz\n 7 GHz\n 8 GHz\n 9 GHz\n0.95\n0.90\n0.85\n0.80\n0.75\n0.70\n0.65Torque ratio T\n1412108642\nAg thickness tAg  (nm)Py(7) @ 5 GHz\nPy(7)/Ag(6) @ 5 GHz\n 4 GHz\n 5 GHz\n 6 GHz\n 7 GHz\n 8 GHz\n 9 GHz(a)\n(b)\nFIG. 4. (Color online) (a) Deconvoluted symmetric\nand antisymmetric contribution to DC voltage amplitude\nfor various Ag interlayer thicknesses. (b) Ratio T =\nVantisymm/(Vantisymm +Vsymm) as function of tAgfor various\nfrequencies.4\n1.10\n1.05\n1.00\n0.95\n0.90Linewidth change \n∆(∆H) (a. u.)\n-10 -5 0 5 10\nDC current IDC (mA) Negative field\n Positive field\n Linear fit, neg. field\n Linear fit, pos. fit\nFIG. 5. (Color online) Manipulation of the FMR linewidth\nby a simultaneous injection of an electrical DC current.\nPy(15)/Ag(4)/Bi(4), f= 4 GHz, PRF= +2 dBm.\nligibly small, see Fig. 2(b), this mechanism can also be\nruled out. We conclude that the magnetization dynam-\nics in our Py/Ag/Bi samples is driven by an interfacial\ncharge-spin conversion due to the REE.\nAccording to the spin-torque theory [31], an addi-\ntional spin current injected into the FM layer will in-\ncrease or decrease the eﬀective magnetic damping, i.e.,\nthe linewidth, depending on its relative orientation with\nrespect to the magnetization [20, 21]. Since Ag features\na very small spin Hall angle [32] and our Bi layer is al-\nmost non-conducting [24], the demonstration of the fer-\nromagnetic linewidth manipulation by an additional DC\ncurrent injection would be an independent manifestation\nof charge- to spin current conversion by the REE. Fig-\nure5shows unambiguously that it is indeed possible to\nmanipulate the resonance lineshape if an additional DC\ncurrent is injected into the sample. For this purpose a\nrather small RF power of +2 dBm is chosen. Appar-\nently, for a positive magnetic ﬁeld polarity, a positive\nDC current leads to an enhanced linewidth, i.e., a damp-\ning enhancement. In contrast, a negative current leads\nto a decreased linewidth, i.e., a damping reduction. Re-\nversingthe ﬁeld polarity results in an opposite trend. We\nﬁnd a relative linewidth change of 0 .8% mA−1.\nAlthough it isn’t physical to speak of a thickness in\ncase of an interface eﬀect, it is still possible to adapt a\nlineshape analysis approach which was presented origi-\nnally in Ref. [20] to relate the spin Hall angle to the ratio\nsymmetric/antisymmetric components of the resonance\nlineshape. We can estimate a spin Hall angle equiva-\nlentγ∗if we hypothetically assume that the charge-spin\nconversion process was a bulk- rather than an interface-\ndriven eﬀect [20]:\nγ∗=S\nAeµ0MStPytNM\n¯h/radicalbigg\n1+4πMeﬀ\nH.(5)\nHere,tNMis the non-magnetic layer thickness. We ﬁnd\nthe spin Hall angle equivalent to be γ∗≈18% for our\nPy/Ag/Bi samples, exceeding most paramagnetic met-\nals. In our previous work we determined the REE con-versionparameter λREE≈0.1nm[24]. Usingtherelation\nλREE= 1/2dγ∗, wheredis theinterface layer thickness\n[23], we obtain d≈1 nm, which is a reasonable estimate.\nIn summary, we demonstrated the conversion of a\ncharge- into a spin current by Rashba coupling of in-\nterface states by adapting a spin-torque ferromagnetic\nresonance excitation/detection technique. The Ag thick-\nness dependence clearly demonstrates that the spin dy-\nnamics in the adjacent Py layer is driven by an interface-\ngenerated spin-polarized electron current that exerts a\ntorque on the magnetization rather than a bulk eﬀect\nsuch as the spin Hall eﬀect. Our conclusions are further\nvalidatedbyaFMRlinewidth modulationduetothespin\ncurrent injection by applying an additional DC charge\ncurrent to the sample stack. Our results will stimulate\nexperimental and theoretical endeavors to explore novel\ninterface- and surface-driven spin-orbital phenomena for\nthe eﬃcient excitation of magnetization dynamics.\nWe thank Roland Winkler for illuminating discussions.\nThis work was supported by the U.S. Department of En-\nergy,OﬃceofScience, MaterialsScienceandEngineering\nDivision. Lithography was carried out at the Center for\nNanoscale Materials, an Oﬃce of Science user facility,\nwhich is supported by DOE, Oﬃce of Science, Basic En-\nergy Science under Contract No. DE-AC02-06CH11357.\n∗jungﬂeisch@anl.gov\n[1]I.ˇZuti´ c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys.\n76, 323 (2004).\n[2]M.I. D’yakonov and V.I. Perel’, Sov. Phys. JETP Lett.\n13, 467 (1971).\n[3]J.E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999).\n[4]A. Hoﬀmann, IEEE Trans. Magn. 49, 5172 (2013).\n[5]Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida,\nM. Mizuguchi, H. Umezawa, H. Kawai, K. Ando,\nK. Takanashi, S. Maekawa, and E. Saitoh, Nature 464,\n262 (2010).\n[6]W. Zhang, M.B. Jungﬂeisch, W. Jiang, Y. Liu, J.E. Pear-\nson, S.G.E. te Velthuis, A. Hoﬀmann, F. Freimuth, and\nY. Mokrousov, Phys. Rev. B 91, 115316 (2015).\n[7]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[8]O. Mosendz, V. Vlaminck, J.E. Pearson, F.Y. Fradin,\nG.E.W. Bauer, S.D. Bader, and A. Hoﬀmann, Phys. Rev.\nB82, 214403 (2010).\n[9]W. Zhang, M.B. Jungﬂeisch, W. Jiang, J.E. Pearson,\nA. Hoﬀmann, F. Freimuth, and Y. Mokrousov, Phys.\nRev. Lett. 113, 196602 (2014).\n[10]E. Saitoh, M. Ueda, H. Miyajima and G. Tatara, Appl.\nPhys. Lett. 88, 182509 (2006).\n[11]A. Azevedo, L.H. Vilela Le˜ ao, R.L. Rodriguez-Suarez,\nA.B. Oliveira, and S.M. Rezende, J. Appl. Phys. 97,\n10C715 (2005).\n[12]W. Zhang, M.B. Jungﬂeisch, W. Jiang, J. Sklenar,\nF.Y. Fradin, J.E. Pearson, J.B. Ketterson, and A. Hoﬀ-\nmann, J. Appl. Phys. 117, 172610 (2015).5\n[13]M.B. Jungﬂeisch, A.V. Chumak, V.I. Vasyuchka,\nA.A. Serga, B. Obry, H. Schultheiss, P.A. Beck,\nA.D. Karenowska, E. Saitoh, and B. Hillebrands, Appl.\nPhys. Lett. 99, 182512 (2011).\n[14]M.B. Jungﬂeisch, A.V. Chumak, A. Kehlberger,\nV. Lauer, D.H. Kim, M.C. Onbasli, C.A. Ross, M. Kl¨ aui,\nand B. Hillebrands, Phys. Rev. B 91, 134407 (2015).\n[15]D.C. Ralphand M.D. Stiles, J. Magn. Magn. Mater. 320,\n1190 (2008).\n[16]A. Hoﬀmann and H. Schultheiß, Curr. Opin. Solid State\nMater. Sci. (2014), doi:10.1016/j.cossms.2014.11.004.\n[17]W. Jiang, P. Upadhyaya, W. Zhang, G. Yu,\nM.B. Jungﬂeisch, F.Y. Fradin, J.E. Pearson,\nY. Tserkovnyak, K.L. Wang, O. Heinonen,\nS.G.E. te Velthuis, A. Hoﬀmann, Science 349, 283\n(2015).\n[18]A. Hamadeh, O. d’Allivy Kelly, C. Hahn, H. Meley,\nR. Bernard, A.H. Molpeceres, V.V. Naletov, M. Viret,\nA. Anane, V. Cros, S.O. Demokritov, J.L. Prieto,\nM. Mu˜ noz, G. de Loubens, and O. Klein, Phys. Rev.\nLett.113, 197203 (2014).\n[19]J. Sklenar, W. Zhang, M.B. Jungﬂeisch, W. Jiang,\nH. Chang, J.E. Pearson, M. Wu, J.B. Ketterson, and\nA. Hoﬀmann, ArXiv e-prints (2015),arXiv:1505.07791\n[cond-mat.meshall].\n[20]L. Liu, T. Moriyama, D.C. Ralph, and R.A.Buhrman,\nPhys. Rev. Lett. 106, 036601 (2011).\n[21]Z. Wang, Y. Sun, M. Wu, V. Tiberkevich, and A. Slavin,Phys. Rev. Lett. 107, 146602 (2011).\n[22]A.R. Mellnik, J.S. Lee, A. Richardella, J.L. Grab,\nP.J. Mintun, M.H. Fischer, A. Vaezi, A. Manchon, E.-\nA. Kim, N. Samarth, and D.C. Ralph, Nature 511, 449\n(2014).\n[23]J.C. Rojas-S´ anchez, L. Vila, G. Desfonds, S. Gambarelli,\nJ.P. Attan´ e, J.M. De Teresa, C. Mag´ en, andA. Fert, Nat.\nCommun. 4, 2944 (2013).\n[24]W. Zhang, M.B. Jungﬂeisch, W. Jiang, J.E. Pearson, and\nA. Hoﬀmann, J. Appl. Phys. 117, 17C727 (2015).\n[25]S. Sangiao, J.M. De Teresa, L. Morellon, I. Lucas,\nM.C. Martinez-Velarte, and M. Viret, Appl. Phys. Lett.\n106, 172403 (2015).\n[26]A. Nomura, T. Tashiro, H. Nakayama and K. Ando,\nAppl. Phys. Lett. 106, 212403 (2015).\n[27]V.M. Edelstein, Solid State Commun. 73, 233 (1990).\n[28]C.R. Ast, J. Henk, A. Ernst, L. Moreschini, M.C. Falub,\nD. Pacil´ e, P. Bruno, K. Kern, and M. Grioni, Phys. Rev.\nLett.98, 186807 (2007).\n[29]Y.M. Koroteev, G. Bihlmayer, J.E. Gayone,\nE.V. Chulkov, S. Bl¨ ugel, P.M. Echenique, and Ph. Hof-\nmann, Phys. Rev. Lett. 93, 046403 (2004).\n[30]Y. Fukuma, L. Wang, H. Idzuchi, S. Takahashi,\nS. Maekawa , and Y. Otani, Nat. Mater. 10, 527 (2011).\n[31]J.C. Slonczewski, J. Magn. Magn. Mater. 159, L1(1996).\n[32]H.L. Wang, C.H. Du, Y. Pu, R. Adur, P.C. Hammel, and\nF.Y. Yang, Phys. Rev. Lett. 112, 197201 (2014)."    },    {        "title": "1710.03079v1.Weak_magnetic_anisotropy_in_GdRh__2_Si__2__studied_by_magnetic_resonance.pdf",        "content": "Weak magnetic anisotropy in GdRh 2Si2studied by magnetic resonance.\nJ.Sichelschmidt,1,\u0003K.Kliemt,2M.Hofmann-Kliemt,3and C.Krellner2\n1Max Planck Institute for Chemical Physics of Solids, 01187 Dresden, Germany\n2Physikalisches Institut, Goethe-Universit at Frankfurt/M, 60438 Frankfurt/M, Germany\n3Fachbereich Mathematik, Technische Universit at Darmstadt, 64289 Darmstadt, Germany\n(Dated: November 12, 2018)\nThe antiferromagnetically (AFM) ordered state of GdRh 2Si2which consists of AFM-stacked fer-\nromagnetic layers is investigated by magnetic resonance spectroscopy. The almost isotropic Gd3+\nparamagnetic resonance becomes anisotropic in the AFM ordered region below 107 K. The emerging\ninternal anisotropic exchange-\felds are still small enough to allow an investigation of their magne-\ntization dynamics by using a standard microwave-frequency magnetic resonance technique. We\ncould characterize this anisotropy in detail in the ferromagnetic layers of the excitation at 9 and 34\nGHz. We derived a resonance condition for the AFM ordered phase to describe the weak in-plane\nanisotropic behaviour in combination with a mean-\feld analysis.\nI. INTRODUCTION\nGdRh 2Si2belongs to the silicides with tetragonal\nThCr 2Si2-structure which show exceptional magnetic\nproperties, e.g. the antiferromagnetic Kondo systems\nYbRh 2Si21and CeRh 2Si22, HoRh 2Si2which exhibits so-\ncalled \\component separated\" magnetic transitions3and\na temperature tunable surface magnetism4, or SmRh 2Si2,\nshowing unusual valence states of the Sm ions at the sur-\nface and in the bulk5. GdRh 2Si2possesses antiferromag-\nnetic (AFM) order of well localized magnetic moments\nappearing below TN= 107 K6which is characterized by\nan AFM propagation vector (001) and a stacking of fer-\nromagnetic layers6,7. In spite of the pure spin ground\nstate of Gd3+a weak in-plane anisotropy occurs which is\nindicated by the magnetization behavior of the ordered\nmoments being aligned in the basal plane. A mean \feld\nmodel could describe the magnetization data with the as-\nsumption that the ordered magnetic moments are aligned\nparallel to the [110] direction8.\nRecent angle-resolved photoelectron spectroscopy re-\nvealed two-dimensional electron states at the Si-\nterminated surface of GdRh 2Si2and their interplay with\nthe Gd-magnetism. These surface states exhibit itinerant\nmagnetism and their spin-splitting arises from the strong\nexchange interaction with the ordered Gd 4 fmoments9.\nMagnetic resonance techniques are widely used to study\nthe dynamic properties of magnetic ordering10. With\nGdRh 2Si2we study a prototypical material which not\nonly exhibits a simple magnetic structure but also allows\nfor the investigation of the magnetically ordered regime\nwith conventional magnetic resonance techniques at low\n\felds and frequencies. We could estimate the anisotropy\n\felds by applying a standard condition for the resonance\nmodes in the ferromagnetic sublattices. However, for the\nresonance anisotropies observed in GdRh 2Si2common\nAFM resonance theories10turned out to be not appli-\ncable. Instead, we utilized a particular mean-\feld model\nfor the AFM ordering to describe the angular dependence\nof the resonance \feld.II. EXPERIMENTAL\nHigh-quality single-crystalline GdRh 2Si2were used in\nthis study. The growth and characterization of which is\ndescribed in Ref. 6. We investigated the paramagnetic\nresonance (above TN) and the magnetic resonance of\nthe ordered moments (below TN) by using a continuous-\nwave Electron Spin Resonance (ESR) spectrometer to-\ngether with helium- and nitrogen-\row cryostats allowing\nfor temperatures between 5 and 300 K. Two frequencies\n!=2\u0019= 9:40 GHz (X-band) and !=2\u0019= 34:07 GHz (Q-\nband) were utilized to evaluate the resonance \feld con-\ndition which in the paramagnetic region simply reads:\n!=\r=Hreswhere\r=g\u0016B=~is the gyromagnetic ratio\nandgis the spectroscopic splitting factor.\nIn general, an ESR spectrometer allows to measure the\nabsorbed power Pof a transversal magnetic microwave\n\feld as a function of a static and external magnetic \feld\n\u00160H. A lock-in technique improves the signal-to-noise ra-\ntio by a \feld modulation which then yields the derivative\nof the resonance signal dP=dH as the measured quan-\ntity. The resulting spectra were \ftted with a Lorentzian\nfunction including the in\ruence of the counter-rotating\ncomponent of the linearly polarized microwave \feld11.\nFrom the \ft we obtained the resonance \feld Hresand\nthe linewidth \u0001 H(half-width at half maximum).\nIII. RESULTS AND DISCUSSION\nA. Paramagnetic regime\nFor the paramagnetic regime, i.e. for T >T N= 107 K,\nthe ESR spectra and their temperature dependence was\ndiscussed in a recent paper12. The spectra display a be-\nhavior as typically expected for well-de\fned local mo-\nments in a metallic environment and with a tempera-\nture dependence as expected for anisotropic exchange-\ncoupled paramagnets13{15. For temperatures nearby\nmagnetic ordering the critical linewidth divergence could\nbe described by a slowing down of in-plane ferromag-arXiv:1710.03079v1  [cond-mat.str-el]  9 Oct 20172\nnetic \ructuations within a model for a 3D Heisenberg\nferromagnet16.\nB. Ordered regime: temperature dependence\nGdRh 2Si2is a layered antiferromagnet below TN=\n107 K. The Gd 4 fmoments are ferromagnetically or-\ndered within the basal plane (with alignment parallel to\nthe [110]-direction) while they stack in antiferromagnetic\norder along the [001]-direction8.\nFigure 1 shows selected spectra for the in-plane direction\nHk100. Upon cooling below TN= 107 K the paramag-\nnetic resonance develops into a resonance mode of the\nmagnetization of ferromagnetic (FM) sublattices. For\ntemperatures below \u001965 K the spectra consist of more\nthan two lines. The spectral structure indicated by open\ncircles appears near the \felds of the spin-\rop transi-\ntion (from magnetization data6,8, indicated by stars).\nBy sweeping across the spin-\rop \feld the internal \feld\nrapidly changes and during this change it also matches\nthe resonance condition (Eqn. (2)) which leads then to\nthe observed structure. In a narrow temperature region\nbetween 55 K and 65 K a component of the easy-direction\nresonance ( Hk110) is observed in the Hk100 { spectra as\nindicated by the open squares. A slight misorientation\nmight explain that.\n0.00.20.40.60.81.0GdRh2Si2   c\n ⊥ H   1\n00 // HdP/H (arb. units)µ\n0H (T)102K5\n2K130K1\n00K7\n0K6\n4K6\n2K6\n0K5\n9K5\n8K5\n5K4\n2K2\n9K2\n0K1\n1K\nFIG. 1. (Color online) X-band (9.4 GHz) magnetic resonance\nspectra at various temperatures, mostly in the magnetically\nordered region ( TN= 107 K), for the external \feld along the\nparticular in-plane direction (100). Open squares and circles\nindicate the resonance \felds of additional lines at \felds below\nthe main line, see also corresponding symbols in Fig. 2. Stars\nindicate the spin-\rop \feld as determined from magnetization\ndata6,8.\n0.40.81.20\n102030405060708090100050100 // H1\n10 // H9\n.4 GHz100 // H1\n10 // HResonancefield (T)3 4 GHz9\n.4 GHz110 // H1\n00 // HLinewidth (mT)T\n (K)GdRh2Si2    H ⊥ c FIG. 2. (Color online) Temperature dependence of resonance\n\feldHresand linewidth \u0001 Hfor the external \feld along two\ndi\u000berent in-plane directions and two microwave frequencies\nas indicated. Solid lines guide the eyes. Open squares and\ncircles indicate Hresof additional lines as shown in Fig. 1.\nThe spectral structures could be well described by\nLorentzian lineshapes which results in resonance \felds\nand linewidths as shown in Figure 2. For the external\n\feld along the easy direction [110], the X-band spectra\ndisappear at temperatures below about 60 K whereas\nthe Q-band spectra are well de\fned down to the lowest\ntemperatures. The reason for this behaviour is a temper-\nature dependent anisotropy energy (\feld) which at T= 0\nis between the X- and Q-band energies (\felds) and which\nmatches the X-band energy at around 60 K. Increasing\nthe temperature towards TNreduces the anisotropy of\nthe line parameters, i.e. the anisotropy \feld decreases\nwith increasing temperature.\nThe anisotropy \feld can be estimated from the reso-\nnance \feld as follows. The conditions of a ferromagnetic\nresonance for a sample with cubic crystal structure may\nbe used for an approach to describe the resonance \felds\nin case of the ferromagnetic in-plane order in GdRh 2Si210\n(for our case with the tetragonal in-plane anisotropy the\nsymmetries are the same as those for the cubic case).\nWith this, we get the resonance condition for a ferro-\nmagnetic sublattice:\neasy directionh110i:\n!=\r=Hres+ 2HA1 (1)\nhard directionh100i:\n!=\r=\u0014\n(Hres\u00002HA1)\u0012\nHres+HA1+1\n2HA2\u0013\u00151=2\n:\n(2)\nHere,HA1;A2=K1;2=Mare anisotropy \felds with K1;23\n0.00.20.40.60.81.00\n2 04060801000.00.40.81.2GdRh2Si2    H ⊥ c 1\n00 // H1\n10 // Hµ0HA1 (T)    µ0Hres (T)9\n.4 GHzT\n (K)100 // H1\n10 // Hµ0HA1 (T)    µ0Hres (T)3\n4 GHz\nFIG. 3. (Color online) Temperature dependence of reso-\nnance \feld Hres(closed triangles) and anisotropy \feld (open\ntriangles, Eqns. 3,4) for the data at 9.4 and 34 GHz.\n\frst (second) order anisotropy constants. From Eqns.\n(1,2) we calculated HA1, neglecting HA2:\neasy directionh110i:\nHA1=1\n2(!=\r\u0000Hres) (3)\nhard directionh100i:\nHA1=\u0000Hres=4 +q\n9\n16H2\nres\u00001\n2(!=\r)2: (4)\nFigure 3 shows the results of Eqns. (3,4) by using the\nexperimental temperature dependent Hres.\nThe anisotropy \feld HA1has to be distinguished from\nthe internal exchange \felds which lead to magnetic order.\nThe antiferromagnetic order corresponds to an internal,\nin-plane exchange \feld which is much too large for an\nAFM resonance mode to be observable at GHz frequen-\ncies. According to Eqn. (A2) from App. A the internal\n\feld which is created by the AFM stacked FM sublattices\nA and B is\nBx\ninterior ;A;B=3kB\n\u00162\ne\u000b\u0002Nr\nM2\nsat(1\u0000T\n\u0002N)\u0000(\u001f?Bz)2:\n(5)\nWith an external \feld component Bz= 0 one obtains for\nT!0,Msat= 7\u0016B,\u0016eff= 8:28\u0016B, \u0002N= 107 K:\nBx\ninterior ;A;B= 48:8 T. Hence, in order to observe an\nantiferromagnetic resonance a resonance frequency of\n\u0017=g\u0016B=h\u0001Bx\ninterior ;A;B= 1:37 THz (g=2) is required.\nThis may hard to be veri\fed because THz spectroscopy\nrequires samples with a good transmission for THz radi-\nation - which is not the case for GdRh 2Si2.\nThe in-plane ferromagnetic order is caused by internal\nexchange \felds allowing for the resonance observation atGHz-frequencies. The z-component of the internal \feld\nis solely determined by the external \feld Bzas\nBz\ninterior ;A;B=3kB\n\u00162\ne\u000b\u0002W\u001f?Bz (6)\nagain using Eqn. (A2) from App. A. One gets with\n\u001f?(T= 78 K) = 0 :1\u0016B=T and \u0002 W= 8 K\nBz\ninterior ;A;B=Bz= 0:052: (7)\nThis means that if an external \feld is applied along the c-\naxis only\u00195% (atT= 78 K) is internally available as an\ne\u000bective \feld for the magnetic resonance. For example,\nusingBz= 6 T from an estimated value \u00160Hk\nres= 6 T\nof the out-of plane uniaxial resonance \feld (Fig. 4, left\nframe) one gets Bz\ninterior ;A;B= 0:31 T. This value is close\nto the value for the X-band resonance \feld of Gd3+in\nthe paramagnetic state12and also close to the resonance\n\feld along the FM ordered direction [110].\nC. Ordered regime: anisotropy at 78 K\nWe investigated the anisotropy of the X-band data at\nT= 78 K where the linewidth for the 110 direction shows\na minimum, see Fig. 2. The anisotropy of resonance \feld\nand linewidth shown in Fig. 4 is considerably stronger for\ntilting the external \feld out of the tetragonal plane (angle\n\u0002, left frame) than rotating it within the plane (angle #,\nright frame). Interestingly, the out-of-plane anisotropy\ncan be nicely described by an uniaxial behavior (solid\nlines, left frame) just like a paramagnetic resonance with\nan uniaxial crystalline \feld anisotropy. This indicates\nthat the e\u000bective internal \feld is always aligned along\nthe external \feld. Also, the above internal exchange-\feld\nestimation, Eq. 7, shows that the value of the e\u000bective\nresonance \feld corresponds to a typical g-value of Gd3+\nas observed in the paramagnetic regime12.\nThe in-plane anisotropy as shown in the right frame\nof Fig. 4 presents a 90\u000eperiodicity of both resonance\n\feld and linewidth which re\rects the fourfold symmetry\nin the tetragonal basal plane. The open symbols show\nthe out-of plane data of the left frame. Obviously the\nangular dependencies of both in-plane and out-of-plane\ndata sets are very similar near the easy direction of mag-\nnetization,h110i. Such a behaviour can be understood\nas follows: Magnetization measurements at T= 78 K on\nsingle crystals yield a spin-\rop \feld Bsf\u0019250 mT for a\n\feld parallel to the [100]-direction and a domain-\rip \feld\nBdf\u0019160 mT for \feld parallel to the [110]-direction8.\nThis implies that for \felds of the order of the resonance\n\feld, applied along a main symmetry direction, the mo-\nments of both magnetic sublattices are in good approxi-\nmation aligned perpendicular to that \feld (Figs. 3 and 5\nin Ref. 8). The magnetic moments can therefore be de-\nscribed as one large domain that extends over the whole\nsingle crystal. Upon rotating the \feld in the basal plane4\n0.300.350.400\n3060901201503236400.51.01.50\n306090120150050100150in-planeH\n ⊥ 001GdRh2Si2   T=78K  \n100 // H1\n10 // Hθ\n (deg.) Resonancefield (T) Linewidth (mT)Θ\n (deg.)110 // HH\n ⊥ cH // c   110 ⊥ H\nFIG. 4. Angle dependence of X-band resonance \feld Hres\nand linewidth \u0001 H, [110] is the easy direction of magnetiza-\ntion. External \feld is oriented by angles \u0002 and \u0012respec-\ntive the indicated crystalline directions. Left frame: Out-of\nplane anisotropy. Solid lines indicate uniaxial behavior with\n\u00160Hk\nres= 6T,\u00160H?\nres= 0:29T and\u00160\u0001Hk= 4T,\u00160\u0001H?=\n0:03T. Right frame: In-plane anisotropy with external \feld\nHin the basal plane (001) ( c?H) at varying directions. Red\nsolid line indicates Eqn. (B9) with \u0018= (\rD=\rM)2!0 leading\nto\rD!0. Open squares indicate the data of the left frame.\naway from a main symmetry direction, the magnetiza-\ntions of the two sublattices are not equivalent anymore\nand a sine-like modulation of the resonance \feld occurs.\nTo model the in-plane behaviour and to describe the\nanisotropy in the ferromagnetic sublattices the solutions\ngiven by the standard theory for an AFM resonance10\nare not su\u000ecient. We are not aware of any published ap-\nproach which would be applicable to GdRh 2Si2. There-\nfore, we derived an antiferromagnetic resonance condi-\ntion for this anisotropy as described in Appendix B. The\nmean-\feld model that describes the magnetization of the\nsystem8together with the resonance condition Eqn. (B2)\npredicts the sine-like modulation with excellent quantita-\ntive consistency as is demonstrated by the red solid line\nin the right frame of Fig. 4 which depicts Eqn. (B9) in\nApp. B.\nThe mean-\feld model8yields, that the values of BD\nandBM(see Appendix, Fig. 5) are di\u000berent for di\u000berent\nAFM domains. When approaching the [110]-direction,\nthe energy di\u000berence between both domains decreases\nand according to the domain distribution estimated by\nthe Ising chain model8both domains coexist. On the\nother hand, by approaching the [100]-direction, the pre-\ndicted values of BDandBMbecome almost equal in\nvalue such that the magnetic resonance frequency of both\ndomains becomes similar, too. The structure in the an-\ngle dependence of the linewidth around #= 45\u000e;135\u000e\nmay be therefore due to a superposition and exchange-narrowing of anisotropic resonance signals arising from\ndi\u000berent domains. A similar behaviour was suggested for\nCdCr 2S4where four resonance \felds are combined via\nexchange narrowing into one line17.\nIV. SUMMARY\nGdRh 2Si2presents an exemplary case for easy-plane\nmagnetic order with a weak in-plane magnetic anisotropy.\nThe presented magnetic resonance data in the magnet-\nically ordered regime depict a ferromagnetic resonance\nmode displaying features similar to a paramagnetic res-\nonance mode. Its anisotropy can nicely be described by\na resonance condition for the ferromagnetic sublattices\nwith weak anisotropy together with a mean-\feld model\nwhich assumes that the ordered magnetic moments are\naligned parallel to the [110]-direction8.\nACKNOWLEDGMENTS\nKK and CK gratefully acknowledge support by the\nDFG through grant KR3831/5-1. We acknowledge help-\nful discussions with Christoph Geibel, Hans-Albrecht\nKrug von Nidda and Zhe Wang. We are particularly\ngrateful to Dieter Ehlers for his generous help and inter-\nest.\nAppendix A: Internal \feld in the AFM phase\nIn Ref. 8 a free energy-based model to describe the\nAFM phase of GdRh 2Si2was introduced. The free en-\nergy is\nF=\u0000TS\u00001\n2(MA+MB)\u0001B+\u001e(MA;MB)\n=\u0000TS\u00001\n2(MA+MB)\u0001B+EFM+EAFM +Fan\nwith the contribution within a plane\nEFM=\u00003kB\n\u00162\ne\u000b(\u0002W+ \u0002N)1\n8(M2\nA+M2\nB)\nand between planes\nEAFM =\u00003kB\n\u00162\ne\u000b(\u0002W\u0000\u0002N)1\n4(MA\u0001MB):\nThe anisotropic part Fanwill be neglected for the dis-\ncussion of the c-direction. We consider the \feld that is\nproduced by a ferromagnetic plane B and acts on an ion\nof the sublattice A\nF=\u00001\n2MA\u0001B\n|{z}\nZeeman Term\u00003kB\n\u00162\ne\u000b(\u0002W\u0000\u0002N)1\n4(MA\u0001MB)\n| {z }\nbetween layers+\u0001\u0001\u0001\n=\u00001\n2MA\u0002\nB+3kB\n\u00162\ne\u000b(\u0002W\u0000\u0002N)1\n2MB)\n|{z}\nBinterior ;B\u0003\n+\u0001\u0001\u0001:5\nIn the following, we determine the magnetization MBof\none ferromagnetic layer. We have\nM2+D2=D2\n0; D 0=Msatr\n1\u0000T\n\u0002N;\nwithMsat= 7\u0016B:For the \feld along the c-direction we\nhaveM?Dand in particular\nM(B) =\u001f?Bz:\nThis results in\nMA= (D;0;M) = (q\nD2\n0\u0000(\u001f?Bz)2;0;\u001f?Bz)\nandMB= (\u0000D;0;M):\nTherefore we have\nM=1\n2(MA+MB) = (0;0;M);\nD=1\n2(MA\u0000MB) = (D;0;0):\nFor the choice of the coordinate system see Fig. 8 in\nRef. 8. The \feld that acts on an ion of the sublattice\nA, which is created by the sublattices A and B, reads\nBinterior ;A;B(T;B z) (A1)\n=\u00002@\n@MA[EFM+EAFM]\n=3kB\n\u00162\ne\u000bh\n\u0002W1\n2(MA+MB) + \u0002 N1\n2(MA\u0000MB)i\n=3kB\n\u00162\ne\u000b\u001a\u0002W\n2M+\u0002N\n2D\u001b\n=3kB\n\u00162\ne\u000b0\n@\u0002Np\nM2\nsat(1\u0000T=\u0002N)\u0000(\u001f?Bz)2\n0\n\u0002W\u001f?Bz1\nA:(A2)\nThe values of \u0002 W= 8 K and \u0002 N= 107 K and \u0016e\u000b=\n8:28\u0016Band\u001f?= 0:149\u0016B=T have been determined by\nmagnetic measurements6,8.\nAppendix B: Inplane anisotropy\nIn the following, we consider the behaviour of one do-\nmain. We use the mean\feld model developed in Ref. 8\nto predict the ESR resonance \feld for an external \feld\nBapplied perpendicular to the crystallographic [001]-\ndirection. The free energy\nF(') =F0(D0)\u0000B2\n4(\u001f?+\u001fk)\u0000B2\n4(\u001f?\u0000\u001fk)u\n+B2\nsf\n8(\u001f?\u0000\u001fk) sin22'\nwithu=\u0000cos(2#\u00002'), Eqn. (6)8, and the magnetiza-\ntion\nM= ^\u001fB=\u001f?(1\u0000eD\neD)B+\u001f?eD\neDB;\nFIG. 5. On the choice of the coordinate system.\nEqn. (4)8serve as the starting point. For the choice of the\ncoordinate system see Fig. 5. For our purpose it is su\u000e-\ncient to ignore \u001fk. The ESR interaction is that fast, that\nwe do not expect an isothermic relaxation. Here, only\n\u001f?is relevant since this keeps the entropy unchanged.\nTherefore, the magnetization becomes\nM=\u001f?BMeM\nwithBM=B\u0001eMforBaligned perpendicular to the\n[001]-direction. The free energy\nF(')\n=F0\u0000\u001f?B2\n4[ 1\u0000cos(2#\u00002') ] +\u001f?B2\nsf\n8sin22'\ncan be minimized with respect to '\n@\n@'F(')\n=\u001f?B2\n2sin(2#\u00002') +\u001f?B2\nsf\n4cos 2'sin 2'\n= 0:\nWithBD=Bcos(#\u0000') andBM=\u0000Bsin(#\u0000') the\nminimum condition reads\n\u0000BDBM+1\n4B2\nsfsin 4'= 0: (B1)\nThe decomposition of B=Bext(Fig. 5) into BDandBM\nis done with respect to one AFM domain which consists\nof two FM sublattices. In the paramagnetic regime, the\nESR frequency !can be decomposed in the following\nway: The square of !is the sum of three parts that arise\nfrom the three components of the external \feld. This\nreads as\n!2=!2\nx+!2\ny+!2\nz\nwith\n!x=\rBx; ! y=\rBy; ! z=\rBz:\nWe deduce a similar ansatz, to describe the ESR fre-\nquency in the ordered regime. In particular, we describe\nthe ESR behaviour of one certain domain. First of all, we\nintroduce!\feldand take into account that the external\nmagnetic \feld decomposes in a parallel BDand orthogo-\nnal component BMwith respect to the ordering param-\neterD. Furthermore, we account for the anisotropy in6\nthe system by utilizing !aniso and add a constant term\n!0. These considerations lead to the ansatz\n!2=!2\n0+!2\naniso+!2\n\feld(B):\nAn arbitrary analytic function that respects the symme-\ntry of one domain has the form\n\u001e(B) =\u001e0+cDB2\nD+cMB2\nM\nand we can write\n!2\n\feld(B) =\r2\nDB2\nD+\r2\nMB2\nM:\nFor symmetry reasons, there is a \u0019=2 periodicity upon\nrotations in the basal plane of the tetragonal lattice, such\nthat\n!2\naniso =!2\nancos 4'\nand summation yields\n!2=!2\n0+!2\nancos 4'+\r2\nDB2\nD+\r2\nMB2\nM (B2)\nfor the resonance frequency. To introduce the ampli-\ntudes into the \ft formula, we use the [100]-direction\nwhere the resonance \feld has its maximum Bmaxand\nthe [110]-direction where the resonance \feld has its min-\nimumBmin. We choose the coordinate system such that\n'is the angle between the [110]-direction and the order-\ning vector D. In both cases, the external \feld is parallel\nto a main symmetry axis, such that B=BM. This leads\nto\n!2=!2\n0\u0000!2\nan+\r2\nDB2\nD+\r2\nMB2\nmax\n!2=!2\n0+!2\nan+\r2\nDB2\nD+\r2\nMB2\nmin:\nFrom these two equations we determine\n!2\u0000!2\n0=\r2\nM1\n2(B2\nmax+B2\nmin)\n!2\nan=\r2\nM1\n2(B2\nmax\u0000B2\nmin):\nWith Eqn. (B2) we get\n1\n2(B2\nmax+B2\nmin)\u00001\n2(B2\nmax\u0000B2\nmin) cos 4'\n=\u0018B2\nD+B2\nM (B3)\nwith a parameter \u0018= (\rD=\rM)2to be determined by \ft-\nting. To parametrize the plot in ', which is the angle be-\ntween the x-axis ([110]-direction) and the direction of the\nordering vector D, we rewrite Eqn. (B1) and Eqn. (B3)\nand get\nB2\nM+\u0018B2\nD=A1 (B4)\nBMBD=A2 (B5)with\nA1:=1\n2(B2\nmax+B2\nmin)\n\u00001\n2(B2\nmax\u0000B2\nmin) cos 4' (B6)\nand\nA2:=1\n4B2\nsfsin 4': (B7)\nTo solve these equations we multiply Eqn. (B4) by B2\nM\nand get a quadratic equation\nB4\nM+\u0018A2\n2=A1B2\nM:\nFrom the two solutions we use the larger one\nB2\nM=A1\n2+r\nA2\n1\n4\u0000\u0018A2\n2\nsuch thatjBDj<jBMjis ful\flled. Now we compute the\ncomponent of the external \feld that is parallel to D\nB2\nD=A2\n2\nB2\nM=1\n\u0018(\nA1\n2\u0000r\nA2\n1\n4\u0000\u0018A2\n2)\n:\nThis yields the square of the external \feld for the reso-\nnance condition:\nB2\nres=B2\nM+B2\nD\n=\u00121\n2+1\n2\u0018\u0013\nA1+\u00121\n2\u00001\n2\u0018\u0013q\nA2\n1\u00004\u0018A2\n2:(B8)\nSince we have\nBM=Brescos(#\u0000'); B D=Bressin(#\u0000')\nwe get\nA2=BMBD=1\n2B2\nressin(2#\u00002'):\nWith this we get a relation between the resonance \feld\nBresand the angle #between the external \feld and the\n[110]-direction\n#='+1\n2arcsin2A2\nB2res(B9)\nusing Eqns. (B6),(B7),(B8). For \u0018!0 (and\rD!0)\nthe ESR data are described well by Eqn. (B9) as shown\nin Fig. 4, right frame.\n\u0003joerg.sichelschmidt@cpfs.mpg.de1O. Trovarelli, C. Geibel, S. Mederle, C. Langhammer,7\nF. M. Grosche, P. Gegenwart, M. Lang, G. Sparn, and\nF. Steglich, Phys. Rev. Lett. 85, 626 (2000).\n2S. Quezel, J. Rossat-Mignod, B. Chevalier, P. Lejay, and\nJ. Etourneau, Solid State Comm. 49, 685 (1984).\n3T. Shigeoka, T. Fujiwara, K. Munakata, K. Matsubayashi,\nand Y. Uwatoko, J. Phys. Conf. Ser. 273, 012127 (2011).\n4A. Generalov, M. M. Otrokov, A. Chikina, K. Kliemt,\nK. Kummer, M. H oppner, M. G uttler, S. Seiro, A. Fedorov,\nS. Schulz, S. Danzenb acher, E. V. Chulkov, C. Geibel,\nC. Laubschat, P. Dudin, M. Hoesch, T. Kim, M. Radovic,\nM. Shi, N. C. Plumb, C. Krellner, and D. V. Vyalikh,\nNano Lett. 17, 811 (2017).\n5A. Chikina, A. Generalov, K. Kummer, M. G uttler,\nV. N. Antonov, Y. Kucherenko, K. Kliemt, C. Krellner,\nS. Danzenb acher, T. Kim, P. Dudin, C. Geibel, C. Laub-\nschat, and D. V. Vyalikh, Phys. Rev. B 95, 155127 (2017).\n6K. Kliemt and C. Krellner, J. Crystal Growth 419, 37\n(2015).\n7G. Czjzek, V. Oestreich, H. Schmidt, K.  Latka, and\nK. Tomala, J. Magn. Mag. Mat. 79, 42 (1989).\n8K. Kliemt, M. Hofmann-Kliemt, K. Kummer, F. Yakhou-\nHarris, C. Krellner, and C. Geibel, Phys. Rev. B 95,\n134403 (2017).\n9M. G uttler, A. Generalov, M. M. Otrokov, K. Kummer,\nK. Kliemt, A. Fedorov, A. Chikina, S. Danzenb acher,\nS. Schulz, E. V. Chulkov, Y. M. Koroteev, N. Caroca-Canales, M. Shi, M. Radovic, C. Geibel, C. Laubschat,\nP. Dudin, T. K. Kim, M. Hoesch, C. Krellner, and D. V.\nVyalikh, Sci. Rep. 6, 24254 (2016).\n10A. Gurevich and G. Melkov, Magnetization Oscillations\nand Waves (CRC Press, Boca Raton, New York, London,\n1996).\n11D. Rauch, M. Kraken, F. J. Litterst, S. S ullow,\nH. Luetkens, M. Brando, T. F orster, J. Sichelschmidt,\nA. Neubauer, C. P\reiderer, W. J. Duncan, and F. M.\nGrosche, Phys. Rev. B 91, 174404 (2015).\n12J. Sichelschmidt, K. Kliemt, C. Krellner, and C. Geibel,\nJ. Phys.: Conf. Ser. 807, 012007 (2017).\n13B. Elschner and A. Loidl, \\Electron-spin resonance on lo-\ncalized magnetic moments in metals,\" (Elsevier Science\nB.V., 1997) Chap. 162, p. 221.\n14D. L. Huber, Mod. Phys. Lett. B 26, 1230021 (2012).\n15E. Kwapuli\u0013 nska, K. Kaczmarska, and A. Szytu la, J. Magn.\nMag. Mat. 73, 65 (1988).\n16H. Benner and J. P. Boucher, \\Spin dynamics in the para-\nmagnetic regime: Nmr and epr in two-dimensional mag-\nnets,\" in Magnetic Properties of Layered Transition Metal\nCompounds , Vol. 9, edited by L. J. De Jongh (Kluwer Aca-\ndemic Publishers, Dordrecht, Boston, London, 1990) pp.\n323{378.\n17D. Ehlers, V. Tsurkan, H.-A. Krug von Nidda, and\nA. Loidl, Phys. Rev. B 86, 174423 (2012)."    },    {        "title": "0801.1191v1.Relevance_of_ferromagnetic_correlations_for_the_Electron_Spin_Resonance_in_Kondo_lattice_systems.pdf",        "content": "arXiv:0801.1191v1  [cond-mat.str-el]  8 Jan 2008Relevance of ferromagnetic correlations for the Electron S pin Resonance in Kondo\nlattice systems\nC. Krellner, T. F¨ orster, H. Jeevan, C. Geibel, and J. Sichelschmidt\nMax Planck Institute for Chemical Physics of Solids, D-0118 7 Dresden, Germany\n(Dated: October 30, 2018)\nElectronSpinResonance(ESR)measurementsoftheferromag netic KondolatticesystemCeRuPO\nshow a well deﬁned ESR signal which is related to the magnetic properties of the Ce3+moment. In\ncontrast, no ESR signal could be observed in the antiferroma gnetic homologue CeOsPO. Addition-\nally, we detect an ESR signal in a further ferromagnetic Yb co mpound, YbRh, while it was absent\nin a number of Ce or Yb intermetallic compounds with dominant antiferromagnetic exchange, in-\ndependently of the presence of a strong Kondo interaction or the proximity to a (quantum) critical\npoint. Thus, the observation of an ESR signal in a Kondo latti ce is neither speciﬁc to Yb nor to the\nproximity of a quantum critical point, but seems to be connec ted to the presence of ferromagnetic\nﬂuctuations. These conclusions not only provide a basic con cept to understand the ESR in Kondo\nlattice systems even well below the Kondo temperature as obs erved in the heavy fermion metal\nYbRh 2Si2but point out ESR as a prime method to investigate directly th e spin dynamics of the\nKondo ion.\nPACS numbers: 71.27.+a, 75.20.Hr, 76.30.-v\nHeavy fermion intermetallic compounds with concen-\ntrated 4f-and 5f-ions showno Electron Spin Resonance\n(ESR) signal. This common belief was experimentally\nwell established and was justiﬁed by the strongelectronic\ncorrelations which arise from the hybridization between\n4f- or 5f-electrons and conduction electrons. An ESR\ninvestigation of the Kondo ions’ spin dynamics seemed\nonly to be possible indirectly by additionally doped ESR\nprobe spins, usually Gd3+[1, 2], analogous to the princi-\nple of NMR in these systems. However, the dense Kondo\nlattice systems YbRh 2Si2and YbIr 2Si2pose a remark-\nable exception. They exhibit an ESR signal with clear\npropertiesoflocal Yb3+moments and, most surprisingly,\nthe signal is observablebelow the Kondotemperature TK\nwhere the local Yb3+moments become screened because\nof pronounced Kondo interactions.[3, 4] As both com-\npounds are located very close to a quantum critical point\n[5, 6] the ESR signal was initially thought to be a direct\nveriﬁcation of the localized moment scenario of quantum\ncriticality [3]. However, the consequences of this signal\nfor understanding the physics close to a quantum critical\npointin particularorforthe Kondoionphysicsin general\nremained unclear. A plausible explanation would be the\ninherent diﬀerence between Yb- and Ce- based Kondo\nlattices, the former one presenting a much stronger lo-\ncal character allowing for a narrow observable ESR line.\nAnother mechanism could be based on the presence of\nferromagnetic ﬂuctuations, since YbRh 2Si2and YbIr 2Si2\nseem to be unique cases of Kondo lattice systems close\nto a critical point with strong ferromagnetic ﬂuctuations\nand a concomitant strongly enhanced spin susceptibility.\nSuch a scenario is supported by previous ESR reports\nabout how exchange enhanced spin susceptibility leads\nto a reduced ESR line broadening [7] compared to the\nline broadening in simple ferromagnetic systems like Fe\nor Ni [8] in their paramagnetic regime. However, untilnow, no results or arguments allowing a discrimination\nbetween these scenarios were reported. Thus, the origin\nfor the observability of the ESR signal in the these two\ncompounds remained a mystery.\nRecently, we have shown the CeTPO (T=Ru, Os) com-\npound series to be an interesting class of Ce-based\nKondo lattice systems at the border between intermetal-\nlic and oxide materials, presenting a layered structure\nand strong tendency to ferromagnetism.[9] For CeRuPO\na pronounced decrease of the electrical resistivity at tem-\nperatures below 50 K indicates the onset of coherent\nKondo scattering which is conﬁrmed by an enhanced\nthermopower. The reference compound LaRuPO neither\nshows signatures of the Kondo eﬀect nor of magnetism\nof the Ru atoms.[9] The temperature and magnetic ﬁeld\ndependence of magnetic susceptibility and speciﬁc heat\nevidence ferromagnetic (FM) order at TC= 15K. Thus,\nCeRuPO ( TK≃10K) seems to be a rare example of\naf-electron based FM Kondo lattice [9], along further\nd-electron based Kondo lattice systems [10]. In con-\ntrast, CeOsPO shows antiferromagnetic (AFM) order at\nTN= 4.4K, despite only minor changes in lattice param-\neters and electronic conﬁguration. Therefore, CeRuPO\nand CeOsPO are ideally suited to study the diﬀerence\nbetween FM and AFM Kondo lattices. We present ESR\nresults on these new Ce-based systems and discuss the\nESR of further Ce- or Yb- based compounds with ei-\nther ferro- or antiferromagnetic exchange and diﬀerent\nstrength of the Kondo interaction. The results demon-\nstrate that the above mentioned unexpected ESR obser-\nvationinheavyfermionmetalsisrestrictedneithertoYb-\nbased compounds nor to compounds close to a quantum\ncritical point, nor to compounds with a strong Kondo\ninteraction. Instead, they indicate that the observability\nof the ESR signal in a dense Kondo lattice is connected\nwith the presence of FM correlations between the Kondo2\nFIG. 1: ESR spectra of polycrystalline samples of CeRuPO\n(TC= 15 K) and CeOsPO ( TN= 4.4 K). Dashed line repre-\nsents a ﬁt to the data with a metallic Lorentzian line shape at\ng= 2.6. Spectra were taken with same instrument settings.\nions.\nWe used polycrystalline samples of CeTPO (T=Ru, Os)\nwhose preparation and characterization was described\nin Ref.[9]. X-ray powder diﬀraction and energy dis-\npersive X-ray measurements conﬁrmed the formation of\nsingle phase CeTPO. The tetragonal crystal structure\n(P4/nmm) contains alternating layers of OCe 4and TP 4\ntetrahedra. It is worth to point out that despite the pres-\nence of oxygen the compounds show pronounced metal-\nlic properties with a residual resistivity ρ0= 1.5µΩcm\nfor CeRuPO. ESR probes the absorbed power Pof a\ntransversal magnetic microwave ﬁeld as a function of an\nexternal, static magnetic ﬁeld B. To improve the signal-\nto-noise ratio, we used a lock-in technique by modulating\nthe static ﬁeld, which yields the derivative of the res-\nonance signal dP/dB. The ESR experiments were per-\nformed at X-band frequencies ( ν≈9.4 GHz) and at tem-\nperatures between 5 K ≤T≤125 K. The measured ESR\nspectra could be ﬁtted with a Lorentzian lineshape. Here\nwe discuss the lineshape parameters linewidth (∆ B) and\nresonance ﬁeld ( Bres) which determines an eﬀective ESR\ng-factor (g=hν/µBBres). In case of a local ESR spin\nprobe in a metallic environment ∆ Bmeasures the spin-\nspin relaxation rate of the local spin, whereas gis deter-\nmined both by the magnitude of the local moment and\nthe static magnetic ﬁeld at the spin site.\nFig. 1 shows the main result of our ESR measurements\non CeTPO (T=Ru, Os) at temperatures just above the\nrespective magnetic ordering. The ESR investigations of\nboth compounds were performed with the same instru-\nment parameters and almost the same sample masses.\nFor CeRuPO, which shows FM correlations, a well de-\nﬁned and intense ESR signal could easily be observed\nwhereas the AFM correlated CeOsPO is absolutely ESR\nsilent, even when using the largest instrument ampliﬁ-cation. The diﬀerent nature of magnetic correlations in\nCeRuPO and CeOsPO seems to be the only diﬀerence\nwhich is relevant for the ESR observation. In fact, many\nof their properties are similar: the preparation method is\nidentical, the lattice parameters are very close, and the\ndisorder eﬀect on the electrical resistivity is comparable.\nThe dashed line in Fig.1 ﬁts the CeRuPO spectrum with\na metallic Lorentzian line shape (including the eﬀects of\nthe counter rotating component of the linearly polarized\nmicrowaveﬁeld). The observed asymmetry of the line in-\ndicates the metallic character of the sample powder and\nreﬂects a penetration depth being smaller than the grain\nsize [11].\nFig. 2 shows the temperature dependence of the ESR\nspectra which is distinctly diﬀerent for temperatures\nabove and below TC. This diﬀerence indicates that the\nobserved ESR reﬂects the bulk magnetic properties of\nCeRuPO. Above TC, the resonance ﬁeld is strongly tem-\nperature dependent and varies between 310 mT at 15 K\nand 210mT at 24K. With decreasingtemperature, when\nentering the FM phase, the ESR signal splits into a low\nﬁeld (LF) and high ﬁeld (HF) component. Both show a\npronounced temperature dependence of their resonance\nﬁeldsBLF,HF\nres(T) which shift in opposite directions. The\ndecrease of BLF\nres(T) with decreasing temperature reﬂects\nthe developing internal magnetic ﬁelds because of the on-\nset of FM order. The HF component is much less intense\nand gets visible below ≈13 K (see dashed line ﬁt in Fig.\n2).BHF\nres(T) shifts to higher ﬁelds with decreasing tem-\nperature, continuing the temperature behaviorofthe res-\nonance ﬁeld at T > T C. This is unusual, since one would\nnotexpectthe resonanceﬁeldtoincreasewhenapproach-\ning and entering the FM ordered state. Instead it should\ninversely follow the temperature dependence of the mag-\nnetization (taken at the ESR X-band ﬁeld) [12]. Mag-\nnetization and preliminary ESR results on a CeRuPO\nsingle crystal show that this unusual temperature depen-\ndence is due to a complex anisotropy, with the single ion\nanisotropybeing oppositeto the exchangeanisotropy.[13]\nA detailed ESR study is going on and shall be presented\nin a dedicated forthcoming paper.[14] In the paramag-\nnetic regime, at T= 20 K, the resonance ﬁeld corre-\nsponds to an eﬀective ESR gfactor of 2.6. This value is\nclose to the expected value from the Γ 6wavefunction of\na Ce3+ion (J= 5/2) in tetragonal point symmetry.\nTo the best of our knowledge an ESR signal due to dense\nCe3+magnetic moments in an environment with conduc-\ntion electrons has only been reported for the Kramers\nsalt CeP [15]. There the Ce3+resonance originates from\ntransitions within the excited Γ 8quartet. Note, that al-\nthough CeP shows AFM correlations with an anomaly in\nmagnetic susceptibility at TN≃9 K FM interactions are\nindicated by a positive Weiss temperature. This is also\nreﬂected in the Curie-Weiss behavior of the ESR gfactor\nwith a positive Weiss temperature. Several ESR investi-\ngations of diluted Ce3+in metals have been performed in3\n/s48/s46/s48 /s48/s46/s52 /s48/s46/s56 /s49/s46/s50 /s49/s46/s54/s49/s53 /s50/s48/s48/s46/s48/s48/s46/s53/s52/s46/s53/s32/s75\n/s56/s46/s53/s32/s75\n/s57/s46/s53/s32/s75\n/s49/s49/s46/s53/s32/s75\n/s49/s50/s46/s53/s32/s75\n/s49/s51/s46/s53/s32/s75\n/s49/s52/s46/s53/s32/s75\n/s49/s53/s46/s53/s32/s75\n/s49/s55/s46/s53/s32/s75\n/s50/s48/s32/s75/s67/s101/s82/s117/s80/s79\n/s32/s100 /s80 /s32/s47/s32/s100 /s66/s32 /s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s66/s32 /s40/s84/s41/s32/s32 /s84\n/s67/s32\n/s32/s32 /s84/s32 /s40/s75/s41/s66/s32 /s40/s84/s41\nFIG.2: ESRspectraofpowdersamplesofCeRuPObelowand\naboveTC= 15 K. Dotted lines represent metallic Lorentzian\nline ﬁts. Below TCa low ﬁeld and a high ﬁeld component\nof the ESR line develops as shown with two Lorentzian lines\n(dashed line). Inset: T-dependence of the ESR linewidth ∆ B\n(squares). The open circles refer to the high ﬁeld component\nwhich appears below TC.\nthe cubic monopnictideswith aΓ 7groundstate multiplet\n[16, 18]. It turned out that Ce3+ESR is hard to observe\nbecause of a small Γ 7−Γ8splitting and the concomitant\nlarge linewidth contribution from the excited CEF level.\nThe inset of Fig.2 shows the temperature dependence\nof the linewidth below TC(solid symbols: LF compo-\nnent; open circles: HF component) and above TCwhere\nonly one single, well deﬁned line (see line-ﬁt in Fig.1)\ncould be observed. Within the temperature range where\nthe HF component can be observed its linewidth changes\nonly weakly whereas the linewidth of the LF component\nshows an extreme broadening towards low temperatures.\nWe therefore suspect that the two components refer to\ndiﬀerent in-plane and out-of-plane magnetic properties\nof CeRuPO. This anisotropybecomes apparent below TC\nand thus the ESR signalof polycrystalline samples splits.\nWithin the FM phase no hysteresis eﬀects could be ob-\nserved in the ESR spectra. This absence is probably due\nto the weakness of the hysteresis eﬀect in magnetization\nmeasurements at 2 K of oriented powder [9].\nThe stronglinewidth increase ofthe LF component whenpassingTCis typical for the onset ofmagnetic orderingof\nthe ESR probe spins: the spin ﬂuctuation time increases,\ntheexchangenarrowingmechanismgetslesseﬀectiveand\ninhomogeneous broadening gains inﬂuence, leading to a\nstrong increase of the linewidth. The relatively sudden\nlinewidth increase immediately below TCindicates that\nthe eﬀect of atomic disorder or short range magnetic\norder is negligible for the ESR linewidth. A broaden-\ning which occurs already at temperatures far above TC\nwas attributed to eﬀects of atomic disorder in various\nrare earth containing FM intermetallic compounds [12].\nTherefore, the observed ∆ B(T) behavior indicates that\nthe investigated CeRuPO sample is a ferromagnet with\na high grade of homogeneity.\nAt temperatures above TC, ∆B(T) reﬂects the typical\nESR behavior of a magnetic moment in a metallic host\n[19]. ∆B(T) increases linearly with ≃10 mT/K in a\nsmall region up to ≈18 K whereas at higher tempera-\ntures an exponential increase dominates. A small ampli-\ntude and a large linewidth makes the signal undetectable\nforT>∼24 K. A very similar behavior of ∆ B(T) is found\nfor diluted Ce3+in cubic LaAs, for instance [18].\nOur comparison of the ESR properties of CeTPO\n(T=Ru,Os) provides a ﬁrst evidence for a connection be-\ntween the observability of the ESR line and the presence\nof ferromagnetic correlations. We obtained more exper-\nimental arguments for this conclusion by investigating\nfurther Yb or Ce based intermetallic compounds with ei-\nther FM or AFM correlations and diﬀerent strength of\nthe Kondo interaction (see Tab. I). Besides the posi-\ntiveESRobservationsinYbRh 2Si2andYbIr 2Si2(I-type)\n[3, 4] which both exhibit strong FM correlations (as in-\ndicated, e.g., by an enhanced Sommerfeld-Wilson ratio)\n[6, 23] we found an ESR signal in yet another Yb-based\ncompound with FM correlations, YbRh. However, in\ncontrast to both above mentioned heavy fermion com-\npounds, Kondo-type features are neither found in the\nelectricalresistivity nor in the speciﬁc heat ofYbRh. FM\norder of Yb3+magnetic moments below TC= 1.2 K is\nconﬁrmed by magnetic susceptibility and magnetization\nmeasurementsaswellasbytheﬁelddependenceoftheor-\ndering temperature. [17] We investigated polycrystalline\nYbRh (CsCl structure) and observed a clean and well\ndeﬁned ESR signal within our accessible temperature\nrange 3-30 K. The linewidth (minimum ∆ B= 220 mT\natT= 3 K) shows a temperature dependence typical\nfor local moments in metals. The ESR g-value reaches a\nconstant value of 2.55 for T≥10 K which is comparable\nto theg-values for Yb3+in cubic monopnictides [18].\nWe looked for an ESR line in further Ce- and Yb-based\ncompounds, with diﬀerent strength of the Kondo inter-\naction, but all with dominant antiferromagnetic correla-\ntions. Although the measurements were performed on\nhigh quality single crystals in a wide temperature range\naroundTKand / or the magnetic ordering temperature,\nwe did not observe an ESR signal in any of these systems4\nTABLE I: Intermetallic compounds with predominating AFM or FM spin correlations investigated by X-band ESR. ”Kondo”\ndenotes a Kondo lattice behavior in ρ(T) or other properties.\nCompound AFM FM Kondo ESR signal\nCeRuPO - /check /check Yes\nCeOsPO /check - /check No\nYbRh - /check - Yes\nYbRh 2Si2[20] - /check /check Yes [3]\nYbIr2Si2(I-type) [6] - /check /check Yes [4]\nYbIr2Si2(P-type) [6] /check - /check No\nYb4Rh7Ge6[21] /check - - No\nYbNi2B2C [22] /check - /check No\nCeCu 2Si2(S/A) /check - /check No\nCeNi2Ge2 /check - /check No\nCeCu 6−xAux(x=0, 0.1) /check - /check No\n(see Tab. I). P-type tetragonal YbIr 2Si2[6] and cu-\nbic Yb 4Rh7Ge6[21], which both have been grown from\nIn-ﬂux as YbRh 2Si2, display AFM ordering and exhibit\nmore stable Yb3+moments than YbRh 2Si2(because of\na weaker Kondo interaction). CeCu 2Si2(S/A type),\nCeNi2Ge2and CeCu 6−xAux(x=0, 0.1) are three well\nestablished heavy fermion systems close to a quantum\ncritical point. However, the former two are expected to\npresent aspin density wavecritical point, while the latter\nis suspected to present a local quantum critical point.\nOur results demonstrate that the observability of the\nESR line in a dense Kondo lattice system is neither con-\nnected with speciﬁc properties of Yb or Ce, nor to the\nstrength of the Kondo interaction, nor to the proxim-\nity of a (quantum) critical point (and thus also not to\nthe character of the critical point). In contrast, these\nresults give a strong evidence for the importance of fer-\nromagnetic correlations for the narrowing of the ESR\nline. This conclusion bears a strong analogy to the sit-\nuation in itinerant transition metal compounds. Itiner-\nantd-systems with FM correlations may show a strong\nenhancement of their conduction electron susceptibil-\nity. This enhancement leads to an enhanced shift of\nthe ESR g-value as well as a reduced Korringa broad-\nening of the linewidth [7]. As a consequence an ESR\nsignal may even be observed from the conduction elec-\ntrons in the form of a collective conduction spin ESR\n(CESR) with a linewidth which is narrowed by electron-\nelectron exchange interactions.[25] Various examples of\nCESR in enhanced spin-susceptibility metals have been\nreported: Pd [26] and TiBe 2[27] which both show en-\nhanced Sommerfeld-Wilson ratios, and the weak itiner-\nantferromagnetZrZn 2[28]. However,wearenotawareof\nanyexample ofa stronglyanisotropicCESR and ananal-\nogy to the ESR in Kondo lattice systems is not straight-\nforward. Phenomenologically, the anisotropy may arise\nfrom the strong coupling between the local magnetic mo-\nments and the conduction electrons while a ”bottleneck”relaxation framework may apply as indicated from ESR\nexperiments on Yb 1−xLaxRh2Si2.[29]\nWe conclude that a substantial amount of experimen-\ntal data indicates the importance of strong FM corre-\nlations among the ESR probed magnetic moments for\nthe observability of an ESR signal in Kondo lattice sys-\ntems in general. This conjecture is most strongly sup-\nported by the presence of an ESR signal in CeRuPO\nonly since the dominant magnetic interaction is switched\nfrom AF (CeOsPO) to FM (CeRuPO) while keeping\nstructural properties and disorder eﬀects practically un-\nchanged. These results not only provide a ﬁrm exper-\nimental basis for the understanding of the unexpected\nESR line in YbRh 2Si2[3] and YbIr 2Si2[4], but more im-\nportantly establish ESR as a new experimental tool for\na direct study of the Kondo ion in dense Kondo systems.\nSince NMR and NQR cannot be performed on Ce and\nwere yet not successful on Yb systems with strong mag-\nnetic correlations, our results establish ESR as a unique\nlocalprobe for adirect investigationofthe spin dynamics\nofthe Kondoion. The standardESR technique (X-band)\nis especially qualiﬁed for this purpose because the mag-\nnetic ﬁeld ( ≪1 T) is small enough to leave the Kondo\nstateundisturbed. Preliminaryinvestigationsontheﬁeld\nand temperature dependence of the ESR response in\nYbRh2Si2evidence e.g. a correlation between the evolu-\ntion of the g-factor and the evolution of thermodynamic\nproperties [30]. Such investigations should give a deeper\ninsight in the way the heavy quasiparticles are formed\nupon lowering the temperature or the ﬁeld.\nWe would like to thank J. Ferstl, M. Deppe, and O.\nStockert for providing us single crystals of Yb 4Rh7Ge6,\nCeNi2Ge2, and CeCu 6−xAux(x=0, 0.1), respectively.5\n[1] B. Elschner and A. Loidl, in Handbook on the Physics\nand Chemistry of Rare Earths 24 , (Elsevier Science B.V.,\nAmsterdam, 1997), Chap. 162.\n[2] H.-A. Krug von Nidda, M.Heinrich, and A. Loidl, in Re-\nlaxation phenomena , (Springer, 2003), Chap. 2.2.\n[3] J. Sichelschmidt, V. Ivanshin, J. Ferstl, C. Geibel, and\nF. Steglich, Phys. Rev. Lett. 91, 156401 (2003).\n[4] J. Sichelschmidt, J. Wykhoﬀ, H.-A. K. von Nidda, I. I.\nFazlishanov, Z. Hossain, C. Krellner, C. Geibel, and\nF. Steglich, J. Phys. Condens. Matter 19, 16211 (2007).\n[5] P. Gegenwart, J. Custers, C. Geibel, K. Neumaier,\nT. Tayama, K. Tenya, O. Trovarelli, and F. Steglich,\nPhys. Rev. Lett. 89, 056402 (2002).\n[6] Z. Hossain, C. Geibel, F. Weickert, T. Radu, Y. Tokiwa,\nH. Jeevan, P. Gegenwart, and F. Steglich, Phys. Rev. B\n72, 094411 (2005).\n[7] M. Peter, J. Dupraz, and H. Cottet, Helv. Phys. Acta\n40, 301 (1967).\n[8] B. R. Cooper and F. Keﬀer, Phys. Rev. 125, 896 (1962).\n[9] C. Krellner, N. S. Kini, E. Br¨ uning, K. Koch, H. Rosner,\nM. Nicklas, M. Baenitz, and C. Geibel, Phys. Rev. B 76,\n104418 (2007).\n[10] K.S. Burch et al., Phys. Rev. Lett. 95, 046401 (2005);\nS. Kondo et al., Phys. Rev. Lett. 78, 3729 (1997); K.\nMiyoshi et al., Phys. Rev. B 69, 132412 (2004); J.W.\nAllenet al., Phys. Rev. B 41, 9013 (1990).\n[11] G. Feher and A. F. Kip, Phys. Rev. 98, 337 (1955).\n[12] R. H. Taylor and B. R. Coles, J. Phys. F: Met. Phys. 5,\n121 (1975).\n[13] C. Krellner and C. Geibel, J. Crystal Growth (2007),\ndoi:10.1016/j.jcrysgro.2007.10.045.\n[14] T. F¨ orster et al., to be published.[15] C. Huang, K. Sugawara, and B. Cooper, in Int. Conf.\non Crystal Field Eﬀects in Metals and Alloys , (Plenum\nPress, New York, London, 1976), pp. 51–60.\n[16] K. Sugawara and C. Huang, J. Phys. Soc. Jap. 40, 295\n(1976).\n[17] H. Jeevan et al., to be published.\n[18] K. H. Wienand, B. Elschner, and F. Steglich, J. Magn.\nMagn. Mat. 28, 234 (1982).\n[19] S. Barnes, Adv. Phys. 20, 801 (1981).\n[20] O. Trovarelli, C. Geibel, S. Mederle, C. Langhammer,\nF. M. Grosche, P. Gegenwart, M. Lang, G. Sparn, and\nF. Steglich, Phys. Rev. Lett. 85, 626 (2000).\n[21] J. Ferstl, F. Weickert, and C. Geibel, J. Magn. Mag. Mat.\n272-276 , E71 (2004).\n[22] P. Bonville, J. Hodges, Z. Hossain, R. Nagarajan,\nS. Dhar, L. Gupta, E. Alleno, and C. Godart, Eur. Phys.\nJ. B11, 377 (1999).\n[23] P. Gegenwart, J. Custers, Y. Tokiwa, C. Geibel, and\nF. Steglich, Phys. Rev. Lett. 94, 076402 (2005).\n[24] P. Gegenwart, J. Custers, T. Tayama, K. Tenya,\nC. Geibel, G. Sparn, N. Harrison, P. Kerschl, D. Eckert,\nK.-H. Muller, et al., J. Low Temp. Phys. 133, 3 (2003).\n[25] D. R. Fredkin and R. Freedman, Phys. Rev. Lett. 29,\n1390 (1972).\n[26] P. Monod, J. Phys. Colloque C6 C6, 1472 (1978).\n[27] D. Shaltiel, D. Ioshpe, A. Grayevsky, V. Zevin, and J. L.\nSmith, Phys. Rev. B 36, 4090 (1987).\n[28] W. M. Walsh, G. S. Knapp, J. L. W. Rupp, and P. H.\nSchmidt, J. Appl. Phys. 41, 1081 (1970).\n[29] J. Wykhoﬀ, J. Sichelschmidt, J. Ferstl, C. Krellner,\nC. Geibel, F. Steglich, I. Fazlishanov, and H.-A. Krug\nvon Nidda, Physica C 460-462 , 686 (2007).\n[30] High ﬁeld ESR experiments of YbRh 2Si2are presently\nunder way."    },    {        "title": "1009.5790v1.AC_Josephson_Effect_Induced_by_Spin_Injection.pdf",        "content": "arXiv:1009.5790v1  [cond-mat.supr-con]  29 Sep 2010AC Josephson Eﬀect Induced by Spin Injection\nA. G. Mal’shukov1and Arne Brataas2\n1Institute of Spectroscopy, Russian Academy of Sciences, 14 2190, Troitsk, Moscow oblast, Russia\n2Department of Physics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway\nPure spin currents can be injected and detected in conductor s via ferromagnetic contacts. We\nconsider the case when the conductors become superconducti ng. A DC pure spin current ﬂowing\nin one superconducting wire towards another superconducto r via a ferromagnet contact induces AC\nvoltage oscillations caused by Josephson tunneling of cond ensate electrons. Quasiparticles simul-\ntaneously counterﬂow resulting in zero total electric curr ent through the contact. The Josephson\noscillations can be accompanied by Carlson-Goldman collec tive modes leading to a resonance in the\nvoltage oscillation amplitude.\nPACS numbers: 72.25.Dc, 71.70.Ej, 73.40.Lq\nI. INTRODUCTION\nElectric and spin transport near ferromagnet-\nparamagnet interfaces received a large attention boost\nwith the discovery of the giant magnetoresistance eﬀect1\nand the subsequent developments in magnetoelectron-\nics and spintronics. Aronov2, and later Johnson and\nSilsbee3theoretically predicted that an electric current\nthrough such an interface leads to an accumulation of\nnonequilibrium spin polarization with an accompanying\nspin current in the paramagnetic metal. A reverse ef-\nfect also takes place, a pure spin current from the nor-\nmal metal gives rise to an electric potential diﬀerence\nin the ferromagnet. The physics of these phenomena\nis quite simple. A suﬃciently large diﬀerence of con-\nductivities of spin-up and spin-down electrons in ferro-\nmagnets induces spin polarization of the electric cur-\nrent therein. Spin polarized currents passing through\nferromagnet-paramagnet boundaries result in an accu-\nmulation of nonequilibrium magnetization near the in-\nterface. Both the spin injection and detection of this\nspin polarization has been experimentally demonstrated\nin Refs. 4–6 in systems containing two or more junctions\nof thin normal metal wires with ferromagnets. One of\nthem acts as a spin injector, while the other is a detec-\ntor, where the voltage created by diﬀusing spins can be\nmeasured. Related spin-polarized transport phenomena\nhave been investigated in many spintronic applications,\nsuch as giant magnetoresistance1, spin Hall eﬀects7, cur-\nrent induced magnetization dynamics8, spin-pumping9,\nand spin caloritronics10.\nIn the case of superconducting systems, spin injec-\ntion and detection within a nonlocal setup similar to\nthe one studied in Ref. 4–6 was investigated both\ntheoretically11–13and experimentally14. These studies\nhave been focused on DC transport. They revealed a\nstrong renormalization of spin-related transport param-\neters as compared to normal systems. These changes\nwere mostly caused by the modiﬁed density of states in\na superconductor. Beyond such quasi-particle transport\nproperties, the macroscopic coherent state of the super-\nconducting condensate can give rise to a quite diﬀerenttransportphenomenonassociatedwiththespin-polarized\ntransport.\nBelow we will consider an AC eﬀect produced by a DC\nspin current towards a thin ferromagnetic contact. The\nDC potential induced by this polarization ﬂux gives rise\nto an AC electric current of condensate electrons. Since\nin the considered experimental setup the total current\nof the superconducting and normal components must be\nzero, the AC condensate oscillations result in an AC po-\ntential diﬀerence between the opposite sides of the con-\ntact. A schematic of a possible experimental setup is\nshown in Fig. 1. A current is passed from a ferromagnet\nto a normal metal generating an associated spin accumu-\nlation and spin current therein. In the non-local geom-\netry, this spin accumulation also diﬀuses transversely in\na contacted normal metal towards another normal metal\nreservoirviaaferromagnetcontact. Thenon-localpoten-\ntialVincreases with the injected DC current Iand the\nnonlocal resistance Rnl=V/Idescribes the spin trans-\nport properties in the device. We will demonstrate that\nwhen the normal metals become superconducting, Rnl\nacquires an AC component in addition to the DC com-\nFIG. 1: A possible setup for observation of Josephson volt-\nage oscillations. A spin current (dashed arrows) is injecte d\nby passing a DC current (solid arrow) from the ferromagnetic\ncontact (brown). The DC spin current through the ferromag-\nnetic contact between the right (R) and left (L) supercon-\nducting electrodes induces periodic oscillations of their elec-\ntric potential diﬀerence V.2\nponent in the normal state.\nThe outline of this paper is as follows. A model sys-\ntem used in our calculation is described in Sec. III. Also\nin this section we present a simple calculation of the\nAC voltage oscillations assuming a local thermodynamic\nequilibrium between quasiparticles and the condensate.\nA microscopic analysis based on coupled kinetic equa-\ntions for the superconducting order parameter and the\nquasiparticle distribution function will be given in Sec.\nIII. A discussion of results will be presented in Sec. IV.\nII. AC VOLTAGE OSCILLATIONS IN A LOCAL\nTHERMODYNAMIC EQUILIBRIUM\nOur model system consists of two superconducting\nwires in contact via a spin-active barrier. We consider\nthis contact to be weak in the form of a thin ferromag-\nnetic layerwith, if necessary, additional insulating layers.\nSuch a barrier can be characterized by two resistances\nR↑andR↓corresponding to two spin eigenstates. We\nassume that a non-equilibrium spin polarization is cre-\nated in the left wire (see Fig.1), either by spin injection,\nas shown in Fig. 1, or by other means. Moreover, we as-\nsume that the electron’s energy relaxation is faster than\ntheir spin relaxation, so that up and down spin distri-\nbutions can be characterized by the respective chemical\npotentials µL\n↑andµL\n↓, resulting in the spin accumulation\npotential δµs=/parenleftBig\nµL\n↑−µL\n↓/parenrightBig\n. In the right ( R) wireδµs\nis much smaller, if the spin relaxation is faster than the\ninﬂux of polarization from the left reservoir through the\nferromagnetic contact. This is satisﬁed when the contact\nresistanceismuchlargerthantheresistanceofthewireof\nthe length ls=√Dτs, whereDis the diﬀusion constant\nandτsis the spin relaxation rate. This is true in many\npractical cases, in particular, in the systems studied in\nRef. 5,6. We, therefore, simplify our model assuming\nµR\n↑=µR\n↓=µ−eV/2 andµL\n↑+µL\n↓= 2µ+eV, whereµ\nis the equilibrium chemical potential and Vis the charge\npotential diﬀerence between two wires.\nWith these deﬁnitions the electric current through the\ncontact in the normal state is\nIn=V\nRc+δµs\n2eRs, (1)\nwherethe inversechargeandspinresistancesaregivenby\nR−1\nc=R−1\n↑+R−1\n↓andR−1\ns=R−1\n↑−R−1\n↓, respectively.\nIn an open circuit, electro-neutrality requires I= 0, and\nEq. (1) gives V=−δµsRc/2eRs=−δµsP/2e, where\nP=R−1\ns/R−1\ncis the spin current polarization of the\ncontact. This is just the voltage induced by the spin\ncurrent trough the contact, as it has been experimentally\ndemonstrated in Ref. 4–6.\nLet us now consider this situation for superconduct-\ning wires. We assume that 2 δµs≪kBTc, so that\nthe nonequilibrium spin polarization does not cause\ndepairing12. The diﬀerence betweenCooper pairenergiesonoppositesidesofthecontactis2 eV. Thispotentialdif-\nference gives rise to the AC Josephson current IJof con-\ndensed electrons. In addition, electro-neutrality causes\nan oppositely directed current Inof quasi-particles, so\nthat the total electric current is zero. This results in DC\nand AC voltage diﬀerences between the left and right\nsuperconductors that we will now compute.\nThe simplest approach to this problem is based on the\nassumption that in the vicinity of the critical temper-\natureTc−T≪Tc,the quasiparticle current remains\nexpressed by Eq. (1). We will discuss in the next sec-\ntion in which regime this approach is valid. Denoting\nthe phase diﬀerence of the order parameters between the\nleft and right wires as φ, and taking into account that\ndφ/dt=−2eV//planckover2pi1, electro-neutrality In+IJ= 0 and Eq.\n(1) dictates\nIcsinφ−/planckover2pi1\n2eRcdφ\ndt+δµs\n2eRs= 0, (2)\nwhereIcis the critical Josephson current. It is easy to\nsee that when Ic≪δµs/2eRsthe Josephson current is\ndominated by harmonic oscillations with the frequency\nω=Pδµs//planckover2pi1≡2eV0//planckover2pi1. Hence, the voltage induced by\nthe spin current is\nV(t) =−V0−IcRcsinωt. (3)\nThe DC component −V0of this voltage is exactly the\nsame as in the case of normal metals. Additionally, V(t)\ncontains a term that oscillates with a frequency deter-\nmined by the DC (normal state) contribution of the non-\nlocal signal. The magnitude of the oscillating voltagecan\nbe estimated by noting that at temperatures close to Tc\nIc=ζ\nRcπ∆2\n4ekBTc, (4)\nwhere ∆ is the superconducting gap and ζis a dimen-\nsionless coeﬃcient that takes into account the depairing\neﬀect inside the ferromagnetic contact layer15, leading\nto exponential suppression of the Josephson current and,\nconsequently to small values of ζ. It should be noted\nthat according to Eqs. (3) and (4) the oscillation ampli-\ntudeIcRcdoes not explicitly depend on the transmission\ncoeﬃcient, apart from the weak dependence through the\ndepairing factor ζ.\nIII. NONEQUILIBRIUM EFFECTS AND\nCOLLECTIVE MODES\nThe above analysis was based on the assumption that\nnear the critical temperature, the current carried by\nquasiparticles can be represented by the expression in\nthe normal state Eq. (1), ignoring small corrections as-\nsociated with the gap in the quasiparticle spectrum. The\nsmall gap alone, however, does not justify this assump-\ntion. In particular, when quasiparticles are transmitted3\nbetweentheleftandrightwirestheymaynotbeinthelo-\ncal thermal equilibrium with the respective condensates,\nthat have been assumed when deriving Eq.(2). To take\nintoaccountnonequilibriumeﬀects, oneneedstoconsider\ntime dependent transport and relaxation of the quasi-\nparticles. There are two physical eﬀects that determine\nkinetics of quasiparticles in the superconducting wires.\nThe ﬁrst one is the so called16–18charge (or branch) im-\nbalance of electron and hole excitations. It is produced\nby quasiparticletunneling between superconducting elec-\ntrodes, leadingtoaquasiparticledistributionwith alocal\nchemical potential diﬀerent from that of the condensate.\nThis diﬀerence relaxes during a time much longer than\nthe electron-phononscatteringtime. Another eﬀect is re-\nlated to condensate space-time oscillations. It dominates\nover the charge imbalance relaxation when ωis large\nenough. We will demonstratethat the spin injection then\nenables detection of collective condensate-quasiparticle\nmodes, Carlson-Goldman modes19which are character-\nized by oppositely directed oscillations of condensate and\nnormal ﬂuids. There is an important diﬀerence with re-\nspect to the usual Josephson eﬀect, since our device re-\nquires no net current I= 0. The usual Josephson eﬀect\ndoes not couple to Carlson-Goldman modes and is not\nreduced at low temperatures, T→0. In contrast, to\nprovide a counterﬂow we need excitations that vanish at\nlow temperatures. The coupling to the collective modes\nis enabled by a spin-driven battery eﬀect induced by the\nspin injection.\nLet us now detail the calculations. Assuming a small\ndeviation from equilibrium we employthe linearized time\ndependent kinetic andGinzburg-Landauequationsin the\ndiﬀusive regime18, when the elastic mean free path is\nmuch less than the superconductor’s coherence length,\nas well as other relevant length scales. In this case,\nthe isotropic quasiparticle distribution function fσ(E,t),\nwhereσis the spin projection, depends only on the en-\nergyEand time t. Within the linear theory the sin-\nglet condensate couples to the spin-independent part\nf(E,t)≡(f↑(E,t)+f↓(E,t))/2 of the distribution func-\ntion. Therefore, after ignoring small terms ( δµs/kBT)2,\ntheunperturbedspin-independentdistributionstakesthe\nform of Fermi equilibrium functions fL/R\n0(E,t) of the left\nand right wires with respective electrochemical poten-\ntialseV/2 +µand−eV/2 +µ. In its turn, the corre-\nsponding gap functions of unperturbed condensates are\n∆exp(iφ/2−2iµt) and ∆exp( −iφ/2−2iµt). It is easy\nto see that in this unperturbed state the spin indepen-\ndent contribution to the quasiparticle current through\nthe contact is given by the ﬁrst term in the right-hand\nside of Eq. (1). Taking into account above condensate\nfunctions one can easy obtain Eqs. (2) and (3). In the\nperturbed state we have f(E,r,t) =f0(E,t)+δf(E,r,t)\n(we will skip here and below the labels LandR). Since\nthe perturbation violates the electron-hole symmetry, it\ngives rise to a spatially dependent potential ϕ(r,t) near\nthe contact. Also, a correction to the order parameter\nδ∆(r,t) appears. In order to simplify the further analy-sis, we assume that ω≪∆ and 1/τE≪∆, where 1 /τE\nis the electron-phonon relaxation rate. Besides that, the\ncritical supercurrent Icis taken small enough, so that the\ntime dependence of all functions is dominated by har-\nmonic oscillations. Accordingly, we introduce the time\nFourier components δfω(r,E),δ∆ω(r) andϕω(r). From\nRefs. 18,21 it follows that fωobeys the kinetic equation\n/parenleftBig\n−iωN1−/tildewideD∇2+2∆N2/parenrightBig\nδfω+\niωN1f0eϕω−ωN2f0δ∆ω=Ist,(5)\nwheref0= 1/4kBTcosh2(E/2kBT) andIstis the\nelectron-phonon scattering integral, whose explicit form\ncan be found in Ref. 18,21. Furthermore, /tildewideD=D(N2\n1+\nN2\n2), where N1andN2are the spectral functions:\n2N1/2=G1/2(E+/tildewideω/2)∓G1/2(E−/tildewideω/2),(6)\nwhereG1(E) = E/√\nE2−∆2andG2(E) =\ni∆/√\nE2−∆2, with/tildewideω=ω+i/τE. In its turn, the lin-\nearized Ginzburg-Landau equation takes the form\n−iωδ∆ω+8ikBT∆\nπ|∆|/integraldisplay\ndEN2δfω=D∇2δ∆ω.(7)\nWe will employ the above equations for the analysis of\nour model in two limiting cases of weak and strong en-\nergy relaxation versus the Josephson frequency, τEω≪1\nandτEω≫1, corresponding to very diﬀerent physi-\ncal situations. In the former case slow time variations\nofδfmay be ignored, so that the quasiparticle kinet-\nics is dominated by the charge imbalance of electron and\nhole excitations. The deviation from the thermodynamic\nequilibrium decreases with increasing distance from the\ncontact on the characteristic length scale√DτR, where\nτR= 4kBTcτE/π∆ is the charge imbalance relaxation\ntime, that is much longer than τE. In the opposite high-\nfrequency regime inelastic collision processes are not im-\nportant, because the quasiparticle distribution oscillates\nfast. Therefore, one can neglect 1 /τEandIstin Eqs.\n(5,7). In this case, since Josephson oscillations of the\ncondensate take place at zero total current, they strongly\ncouple to Carlson-Goldman modes. Therefore, one can\nexpect such modes to be excited near the contact and\npropagate along the left and right wires.\nIn both low-frequency and high-frequency regimes, us-\ning Eqs. (5,7) with a reduced form of Istfrom Refs. 21\nand 18 and taking into account the zero electric current\ncondition, one arrives to the equation for the potential\nκ2(ω)ϕω=∇2ϕω, (8)\nwhereκ2(ω) = 1/DτRatτEω≪1 and\nc2\nsκ2(ω) =−ω2−iπω∆2/4kBT (9)\natτEω≫1, where the sound velocity cs=√\n2D∆. Eq.\n(8) is well known. At τEω≪1 it describes the charge\nimbalance relaxation17,18, while in the opposite limit it\ngives the dispersion of Carlson-Goldman modes20.4\nFor our geometry, when κ−1is much larger than the\nwidth and thickness of the wire, ϕωdepends only on the\ncoordinate xalong the wire. Then, at τEω≪1,ϕω\nexponentiallydecreaseswith increasingdistancefromthe\ncontact, while at τEω≫1 it shows decaying oscillations.\nWe assume that the left and right wires are of the same\nlengthL. Since the system is symmetric with respect\ntox→ −x, the oscillating part of the electrochemical\npotential is −Vω/2+ϕω(x) atx >0 andVω/2−ϕω(−x)\natx <0, where Vωdenotes the Fourier component of\nV(t). The solution of Eq. (8) has the form\nϕω(x) =αeκ(ω)x+βe−κ(ω)x(10)\nwith the boundary conditions ∇xϕω(±L) = 0 and\nVω−2ϕω(0)\nRc=Aσ∇xϕω(0), (11)\nwhereAis the wirecross-sectionareaand σisthe normal\nstate conductivity. These boundary conditions provide a\nzeroelectriccurrentofquasiparticlesatthe wireendsand\nthe current equal to the injected one at x= 0. From Eqs.\n(8),(10)-(11) one obtains a periodic part of the injected\ncurrent\nVω−2ϕω(0)\nRc=Vω\nRc+2Rw≡Vω\nReff(ω),(12)\nwhere\nRw=1\nAσκ(ω)1+e−2κL\n1−e−2κL,Re(κ)>0.(13)\nHence, the result is a renormalization of Rcin Eq. (1),\nsuch that Rc→Rc+ 2Rw. To ﬁnd the voltage Vω,\nthe quasiparticle current (12) must be equated with the\nJosephson current. By this way we obtain a new expres-\nsionfor atime dependent part of V, instead ofthe second\nterm in the right-hand side of Eq. (3):\nV+V0=−Ic[cosωtImReff(ω)+sinωtReReff(ω)] (14)\nIV. DISCUSSION\nLet us analyse above results in some limiting cases.\nSinceκ(ω)→ ∞in both casesofhigh frequencies ω→ ∞\nand strong energy relaxation τE→0, it follows from Eq.\n(13) that Rw→0. We thus obtain Eq. (3), that is an\nexpected result, because in these limits a deviation from\nequilibrium is small. On the other hand, the nonequi-\nlibrium eﬀect of quasiparticle’s kinetics becomes strongwhenRc/lessorsimilar2Rw. The assumed linearization condition,\nhowever, restricts this inequality. This condition can be\nexpressed in the form ζ|Rw/Rc| ≪1. Therefore, the lin-\near theory allows Rc/lessorsimilarRwonly at small ζ. It should be\nnoted that, according to Eq. (13), Rwcan be enhanced\ndue to resonances of Josephson oscillations with collec-\ntive modes at Im κL=πn, if they are not overdamped\n(if ReκL≪1).Rwalso increases at small enough L,\nwhen 2|κ|L≪1. In practice Rwmay be varied in quite\nwide range. In Al wires from Ref. 6 V0= 10−6V, re-\nsulting in ω−1≃0.3·10−9s. Since ω−1∼τE∼10−9s22,\na regime intermediate between charge imbalance relax-\nation and generation of Carlson-Goldman modes will be\nrealized, with κ−1about several µm. Therefore, strong\nresonances in Rware not expected. One can evaluate\nRw≃50Ω, that is much less than Rc= 600Ω. Hence,\nin the considered parameter range Eq. (3) remains valid.\nIn samples with higher polarizations Pand at larger spin\ncurrentthroughthe contactthe Josephsonoscillationfre-\nquency is expected to be large enough to produce notice-\nable collective resonances of Reffin Eq. (14).\nThe above calculations of Josephson voltage oscilla-\ntions have been restricted to ∆ ≪Tc. At the larger gap\nthe oscillationamplitude isexpected to decrease, because\nless excitations are available to compensate the super-\ncurrent through the contact. On the other hand, in this\nrangeone shouldtakeinto accountthat besides quasipar-\nticles the spin transport through the contact can be asso-\nciated with triplet components of Cooper pair states that\nappear due to spin dependent tunneling and nonequilib-\nrium spin polarization of superconducting wires. Further\nstudies are needed to understand the eﬀect of such trans-\nport.\nIn conclusion, we considered an AC Josephson eﬀect\ninduced by a DC spin current through the contact whose\ntransmittance depends on the spin orientation of tunnel-\ning electrons. The oscillations of the voltage across the\ncontactatzeroelectriccurrenthave,incertainparameter\nrange,harmonictimedependencewiththefrequencypro-\nportional to the spin current. The amplitude and phase\nof these oscillations depend on coupled kinetics of quasi-\nparticles and condensate in superconducting wires. The\ncorresponding calculations have been performed within\nlinearized kinetic equations at temperature close to Tc.\nWe predict that at the high enough frequency the mea-\nsured AC voltage will show up the resonance structure\nassociated with excitation of Carlson-Goldman modes.\nA.G.M. gratefully acknowledges hospitality of NTNU.\n1A. Fert, Rev. Mod. Phys. 80, 1517 (2008).\n2A. G. Aronov, Pis’ma Zh. Eksp. Teor. Fiz. 24, 37 (1976),\nJETP Lett. 24, 32 (1976).3M. Johnson and R. H. Silsbee, Phys. Rev. B 35, 4959\n(1987);37, 5312 (1988)\n4M. Johnson and R. H. Silsbee, Phys. Rev. Lett. 55, 17905\n(1985).\n5F. J. Jedema, A. Filip, and B. J. van Wees, Nature (Lon-\ndon)410, 345 (2001).\n6F. J. Jedema, H. B. Heersche, A. T. Filip, J. J. A. Basel-\nmans, and B. J. van Wees: Nature 416, 713 (2002)\n7H.-A. Engel, E. I. Rashba, and B. I. Halperin, Theory\nof Spin Hall Eﬀects in Semiconductors, in Handbook of\nMagnetism and Advanced Magnetic Materials, edited by\nH. Kronm¨ uller and S. Parkin, John Wiley & Sons Ltd.,\nChichester, UK 2007 , pp. 2858.\n8D. C. Ralph and M. D. Stiles, J. Magn. Magn. Mater. 320,\n1190 (2008).\n9Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I.\nHalperin, Rev. Mod. Phys. 77, 1375 (2005).\n10Special Issue on Spin Caloritronics, edited by G. E. W.\nBauer, A. H. MacDonald, and S. Maekawa, Solid State\nCommun. 150, 459-552 (2010)\n11H.L.ZhaoandS.Hershﬁeld, Phys.Rev.B 52, 3632(1995).\n12T. Yamashita, S.Takahashi, H.Imamura, andS.Maekawa,\nPhys. Rev. B 65, 172509 (2002); S. Takahashi and S.\nMaekawa, ibid. 67, 052409 (2003), J. Phys. Soc. of Japan\n77, 031009 (2008)13J. P. Morten, A. Brataas, and W. Belzig, Phys. Rev. B 70,\n212508 (2004); Phys. Rev. B 72, 014510 (2005)\n14M. Johnson, Appl. Phys. Lett. 65, 1460 (1994); N. Poli1,\nJ. P. Morten, M. Urech, Arne Brataas, D. B. Haviland,\nand V. Korenivski, Phys. Rev. Lett. 100, 136601 (2008).\n15F. S. Bergeret, A. F. Volkov, K. B. Efetov, Rev. Mod.\nPhys.77, 1321 (2005); A. I. Buzdin, Rev. Mod. Phys, 77,\n935 (2005)\n16W. J. Skocpol, M. R. Beasley, and M. Tinkham, J. Low\nTemp. Phys. 16, 145 (1974)\n17C. J. Pethick and Smith, Ann. of Phys., 42, 119 (1979)\n18A. Schmid and G. Sch¨ on, J. Low Temp. Phys. 20, 207\n(1975).\n19R. V. Carlson and A. M. Goldman, Phys. Rev. Lett. 34,\n11 (1975), J. Low Temp. Phys. 25, 67 (1976)\n20A. Schmid and G. Sch¨ on, Phys. Rev. Lett. 34, 941 (1975);\nS. N. Artimenko and A. F. Volkov, ZETF 69, 1764 (1975)\n[Soviet Phys. JETP 42, 896 (1976)]\n21G. Sch¨ on, V. Ambegaokar, Phys. Rev. B 19, 3515 (1979)\n22S. Wind, M.J. Rooks, V. Chandrasekhar, and D.E. Prober,\nPhys. Rev. Lett. 57, 633 (1986)"    },    {        "title": "1509.04801v1.Fano_resonance_in_a_normal_metal_ferromagnet_quantum_dot_superconductor_device.pdf",        "content": "arXiv:1509.04801v1  [cond-mat.str-el]  16 Sep 2015Fano resonance in a normal metal/ferromagnet-quantum dot- superconductor device\nLin Li,1Zhan Cao,2Hong-Gang Luo,2,3Fu-Chun Zhang,4,5and Wei-Qiang Chen1\n1Department of physics, Southern University of science and t echnology of China, Shenzhen 518005, China\n2Center of Interdisciplinary Studies and Key Laboratory for Magnetism and Magnetic\nMaterials of the Ministry of Education, Lanzhou University , Lanzhou 730000, China\n3Beijing Computational Science Research Center, Beijing 10 0084, China\n4Department of Physics, Zhejiang University, Hangzhou 31002 7, China\n5Department of Physics and Center of Theoretical and Computa tional Physics,\nThe University of Hong Kong, Hong Kong, China\n(Dated: July 2, 2021)\nWe investigate theoretically the Andreev transport throug h a quantum dot strongly coupled with\na normal metal/ferromagnet and a superconductor (N/F-QD-S ), in which the interplay between the\nKondo resonance and the Andreev bound states (ABSs) has not b een clearly clariﬁed yet. Here\nwe show that the interference between the Kondo resonance an d the ABSs modiﬁes seriously the\nlineshape oftheKondoresonance, which manifests as aFanor esonance. The ferromagnetic lead with\nspin-polarization induces an eﬀective ﬁeld, which leads to splitting both of the Kondo resonance and\nthe ABSs. The spin-polarization together with the magnetic ﬁeld applied provides an alternative\nway to tune the lineshape of the Kondo resonances, which is de pendent of the relative positions of\nthe Kondo resonance and of the ABSs. These results indicate t hat the interplay between the Kondo\nresonance and the ABSs can signiﬁcantly aﬀect the Andreev tr ansport, which could be tested by\nexperiments.\nI. INTRODUCTION\nIn the past years, the interplay between the Kondo\neﬀect and superconductivity has been intensively inves-\ntigated in hybrid superconductor-nanostructures.1–24In\na superconductor-quantum dot-superconductor (S-QD-\nS) device, the Josephsoncurrent showsan interesting 0- π\nphase transition at TK/∆∼1, where TKis the normal\nstate Kondo temperature, and ∆ is the superconduct-\ning gap. At the transition point, the energy gained from\nthe formation of Kondo singlet can exceed the energy\ngap, which leads to the crossing of Andreev bound states\n(ABSs).1–13Furthermore, if one electrode of the QD is\nreplaced by a normal or ferromagnetic lead, namely, for a\nnormal metal/ferromagnet-quantum dot-superconductor\n(N/F-QD-S) device, the subgap transport shows much\nricher features.14,25–27For example, the Kondo eﬀect en-\nhancement of Andreev transport has been observed in\nodd occupation regimes.16,25The coexistence of Kondo\nresonance and ABSs has been proposed and/or observed\nin the superconductor-QD devices.26–29However, the in-\nterplay between the Kondo resonance and the ABSs in\nsuch a device has not been explored in detail.\nIt is well-known that in the N/F-QD-S device, the\nAndreev levels induced by the proximity of the super-\nconductor will be broadened, whose width is roughly\nproportional to the coupling strength with the normal\nor ferromagnetic lead. At the same time, the coupling\nwith the normal or ferromagnetic lead will also lead to\na Kondo resonance due to the screening of the spin in\nthe QD if the system is in the Kondo regime. As men-\ntioned above, both these two subgap features have been\nobserved experimentally.26A natural question arises: is\nthere any interplay between these two subgap features?\nIn this work we explore theoretically the interaction be-\nSC-lead                                                               F-lead \nABS \n \n \n \n \nKondo \nεd EZeff  EZeff  \nEB \nFIG. 1. Schematic diagram of the interplay between the\nKondo resonance and the Andereev bound states (ABSs) in\nan F-QD-S device. The spin-polarization induces an eﬀec-\ntive ﬁeld EZeff, which removes the degeneracy in dot level εd\nand splits the ABSs and Kondo resonance. When the split-\nting ABSs and subpeak of Kondo resonance get close to each\nother, the interference between them modiﬁes the lineshape\nof the Kondo resonance, which aﬀects the Andreev transport.\ntween these two distinguished features and ﬁnd that\nthe lineshape of the Kondo resonance can be modiﬁed\nseriously by the interference between the Kondo reso-\nnance and the ABSs, which can be attributed to Fano\nresonance.30,31\nThe experimental observations that the Fano reso-\nnance modiﬁes the lineshape of the Kondo resonance2\nhavebeen reportedin the experiments abouttwodecades\nago,32–35in which the Kondo resonance has been un-\ncovered by scanning tunneling spectroscopy of a mag-\nnetic adatom on metal surface. There the asymmetric\nlineshape of the Kondo resonance observed are resulted\nfrom the interference between a symmetric Kondo res-\nonance (discrete channel) and the metal surface states\n(continuum channel).36Furthermore, it was found that\nin the strongly coupling cases, for example, Ti/Au and\nTi/Ag,35,37the broadening impurity level has a signif-\nicant contribution to the lineshape of the Kondo reso-\nnance observed experimentally.38These works indicated\nthat the Fano resonanceis popularin such astronglycor-\nrelated impurity system (for an overview one can refer to\nRef.[31]).\nIn this work, we explore the Fano resonance in the\nN/F-QD-S device. It is shown that the interference be-\ntween the Kondo resonance and the broadening ABSs\nhas a signiﬁcant inﬂuence on the Andreev transport.\nIn the N-QD-S case, the Kondo resonance peak devel-\noped in Andreev transport gradually evolves into an\nanti-resonancedip structure with increasing the coupling\nstrength of the normal lead and the QD. This can be as-\ncribed to the interference of the Kondo resonance ﬁxed\naroundtheFermilevelandthe broadeningABSsnearthe\nFermi level. Here the broadening ABSs play a key role\nin changing the lineshape of the Kondo resonance from a\npeaktoadipstructure. IntheF-QD-Scase,thesituation\nismoreinteresting. It isknownthatthespin-polarization\ncan induce an eﬀective magnetic ﬁeld EZeff,39–43which\nremoves the degeneracy of dot level and results in split-\nting of Kondo resonance and the ABSs. The interference\nbetween the sub-peaks of Kondo resonance and splitting\nABSs can lead to ﬁne Fano resonance, as shown schemat-\nically in Fig. 1. The spin-polarization as well as the mag-\nnetic ﬁeld applied providea novelwayto tune the subgap\nphysics of the Andreev transport in such a device, which\ncan be further tested by future experiments.\nThe paper is organized as follows. In Sec. IIwe in-\ntroduce the Anderson impurity model and outline the\nformalism on the Andreev transport. In Sec. III, we\ndiscuss the numerical results of Andreev transport in the\nN/F-QD-S device, and analyze the result obtained by us-\ning the Fano resonance picture. Finally, a brief summary\nis given in Sec. IV.\nII. THE MODEL AND FORMALISM\nThe N/F-QD-S device can be described by Anderson\nimpurity model44\nH=HL+HD+HV, (1)\nwhere\nHL=/summationdisplay\nkασεkασc†\nkασckασ−∆/summationdisplay\nkSσ/parenleftBig\nc†\nkS↑c†\n−kS↓+H.c./parenrightBig\n(2)istheHamiltonianofleads α=S,N/F. Thesecondterm\nis only available in the superconducting lead. The oper-\natorc†\nkασ(ckασ) represents the creation (annihilation) of\nan electron with the energy εkασin superconductor or\nnormal/ferromagnetic leads, respectively.\nHD=/summationdisplay\nσεdσd†\nσdσ+U\n2/summationdisplay\nσnσn¯σ (3)\nis the dot Hamiltonian, in which εdσis the dot level, d†\nσ\n(dσ) is the creation (annihilation) operator of electron in\nQD,nσ=d†\nσdσ, andUis the Coulomb repulsion.\nHV=/summationdisplay\nkασ/parenleftBig\nVαc†\nkασdσ+H.c./parenrightBig\n(4)\nis the Hamiltonian describing the coupling between the\ndot and leads with the tunneling amplitude Vα. The\nspin-polarization in ferromagnetic electrode leads to the\nspin-dependent dot-lead coupling Γ F↑(↓)= ΓF(1±P)/2.\nΓF= ΓF↑+ ΓF↓=π|VF|2ρFis the coupling strength,\nρF=ρF↑+ρF↓is the density of states of the nor-\nmal/ferromagnetic lead. Pis the spin polarization and\nP= 0 denotes a normal electrode. It is known that\nthe ferromagnetic proximity eﬀect induces an eﬀective\nexchange ﬁeld on dot level due to the spin-dependent\ncharge ﬂuctuation.41–43This behavior can be treated by\nthe Haldane’s scaling theory,45and the modiﬁcation on\ndot level is\nδεdσ=/integraldisplaydε\nπ/braceleftbiggΓFσ[1−f(ε)]\nεdσ−ε+ΓF¯σf(ε)\nε−U−εd¯σ/bracerightbigg\n.(5)\nThe ﬁrst (second) term denotes charge ﬂuctuation be-\ntween the single occupation state and empty (double oc-\ncupation) state on dot level.41,42\nThe subgap transports through the device can be de-\nrived by Nambu Green’s function\nGdσ(ε) =/bracketleftbigg∝angbracketleft∝angbracketleftdσ;d†\nσ∝angbracketright∝angbracketrightε∝angbracketleft∝angbracketleftdσ;d¯σ∝angbracketright∝angbracketrightε\n∝angbracketleft∝angbracketleftd†\n¯σ;d†\nσ∝angbracketright∝angbracketrightε∝angbracketleft∝angbracketleftd†\n¯σ;d¯σ∝angbracketright∝angbracketrightε/bracketrightbigg\n.(6)\nAll the componentsin Gdσ(ε) can be calculatedby equa-\ntion of motion approach.13,14,28,46–50In the frame of\nHartree-Fock mean ﬁeld, the dot Green’s function reads\nGHF\ndσ(ε) =/parenleftBig\nεˆI−˜εdσˆσz−ˆΣ0(ε)−ˆΣHF(ε)/parenrightBig−1\n,(7)\nwhere the non-interactingself-energyΣ0\n11(ε) = Σ0\n22(ε) =\n−i/parenleftBig\nΓF+θ(|ε|−∆)|ε|√\nε2−∆2ΓS/parenrightBig\n,Σ0\n12(ε) = Σ0\n21(ε) =∆√\n∆2−ε2ΓS\nwith Γ S=π|VS|2ρS. ˜εdσ=εdσ+δεdσis the renormal-\nized dot level. δεdσplays a role of eﬀective ﬁeld EZeff\non the dot level. ΣHF= ˆσzU∝angbracketleftˆn¯σ∝angbracketright+ ˆσx∆dis the inter-\nacting self-energy, where ∆ d=U∝angbracketleftd†\nσd†\n¯σ∝angbracketrightis the proximity\ninduced order parameter.49,51\nIn order to capture the Kondo physics, the higher-\norderGreen’sfunctionsshouldbetakenintoaccount. We3\ntruncate the hierarchy of Green’s function by Lacroix’s\nscheme.46It qualitatively captures the characters of\nKondo eﬀect even at zero temperature52or particle-hole\nsymmetry.53Although the equation of motion approach\ntends to underestimate the Kondo temperature, it can\nproperly capture the competition between Kondo eﬀect\nand superconductivity.13After some straightforward cal-\nculations, the dot Green’s function obtained is\nGdσ(ε) =/parenleftBig\nεˆI−˜εdσˆσz−ˆΣ0−ˆΣK/parenrightBig−1\n,(8)\nwhereˆΣK= ˆσzU∝angbracketleftˆn¯σ∝angbracketright[ε−˜εdσ−Σ0\n11(ε)−Pσ(ε)]+Qσ(ε)//angbracketleftˆn¯σ/angbracketright\nε−˜εdσ−Σ0\n11(ε)−U(1−/angbracketleftˆn¯σ/angbracketright)−Pσ(ε)+\nˆσx∆dΣ0\n12(ε)\nε+˜εd¯σ−Σ0\n11(ε)is the self-energy containing Kondo\ncorrelation, and the notations PσandQσread\nPσ(ε) = [Ω 1σ(ε)+Ω2σ(ε)]+UA2σ(ε), (9)\nQσ(ε) = (ε−˜εdσ−U∝angbracketleftnd¯σ∝angbracketright)[A1σ(ε)−A2σ(ε)]+∝angbracketleftnd¯σ∝angbracketright\n×[Ω1σ(ε)+Ω2σ(ε)]−[B1σ(ε)+B2σ(ε)],(10)\nwhere\nAiσ(ε)≈1\nπ/integraldisplayΓσ(ε′)f(ε′)Re[Gσ(ε′)]\nε−εiσdε′,(11)\nBiσ(ε)≈1\nπ/integraldisplayΓσ(ε′)f(ε′)\nε−εiσdε′, (12)\nΩiσ(ε) =1\nπ/integraldisplayΓσ(ε′)\nε−εiσdε′. (13)\nIn the above expressions, Γ σ(ε) = ΓFσ+ ΓS|ε|θ(|ε|−∆)\n2√\nε2−∆2,\nε1σ= ˜εdσ+ε′−˜εd¯σ, ε2σ=−ε′+ 2˜εd¯σ+U,i= 1,2.\nf(ε) is the Fermi distribution function. We can calculate\nthe Green’s functions self-consistently and the current\nthrough the quantum dot by using the above formulas.\nIn this work, we study the Andreev transport through\nN/F-QD-S device, in which an electron injects from nor-\nmal/ferromagnetic lead to superconductor through the\nquantum dot and a hole with oppose spin reﬂects to the\nnormal/ferromagnetic lead, then a Cooper pair is cre-\nated in the superconductor lead, and vice versa.28,29,48,50\nTherefore, the Andreev current reads\nIA(ε) =2e\nh/summationdisplay\nσ/integraldisplay\ndεΓFσΓF¯σ|Gr\ndσ(ε)12|2×\n[fF(ε−eVsd)−fF(ε+eVsd)],(14)\nwherefF(ε±eVsd) is the distribution function of elec-\ntrons in normal metal/ferromagnetic lead, Vsdis the bias\napplied.\nIII. NUMERICAL RESULT AND DISCUSSION\nIn this section, we discuss the numerical results of An-\ndreev transport in the N/F-QD-S device. In odd occu-\npation regimes, the Kondo resonance appears around the\nFermi level of N/F lead due to the screening of local mo-\nment. Here, we pay attention to the interplay between/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s48/s48/s46/s52/s48/s46/s56\n/s32/s86\n/s115/s100/s47\n/s32/s32/s100/s73/s47/s100/s86\n/s115/s100/s40/s49/s48/s45/s50\n/s101/s50\n/s47/s104/s41/s32\n/s78/s61\n/s32\n/s78/s61\n/s32\n/s78/s61\n/s32\n/s78/s61\n/s32\n/s78/s61/s40/s98/s41\n/s32/s32/s100/s73/s47/s100/s86\n/s115/s100/s40/s49/s48/s45/s50\n/s101/s50\n/s47/s104/s41/s32\n/s83/s61/s53\n/s32\n/s83/s61/s55\n/s32\n/s83/s61/s56\n/s32\n/s83/s61/s49/s48/s40/s97/s41\n/s69\n/s66\nFIG. 2. (a) The ABSs dominated by the competition be-\ntween Kondo eﬀect and superconductivity in the N-QD-S de-\nvice with thecouplingΓ N= 0.2∆ and Γ S= (5,7,8,10)∆. (b)\nThe coexistence of Kondo resonance and ABSs with the cou-\npling Γ S= 5∆ and Γ N= (0.2,1,1.5,2,2.5)∆. Other param-\neters used: the dot level εd=−5∆, the temperature T= 0,\nthe magnetic ﬁeld EZ= 0, and the half bandwidth D= 20∆.\nthe Kondo resonance and the ABSs, especially, the in-\nterference between them. In the calculations, we assume\nthat the density of states ρα= 1/2Dand the tempera-\ntureT= 0,Dis the half band-width of the N/F lead.\nIn addition, we neglect the inﬂuence of applied magnetic\nﬁeld on the leads and set U→ ∞for simpliﬁcation.\nIn Fig.2, we show the Andreev transport in the N-\nQD-S device, namely, the polarization P= 0. When the\nquantum dot weakly coupled with normal lead (Γ N≪\nΓS,εd), the Kondo resonance peak can not be observed\nin the conductance as shown in Fig. 2(a). In this case,\nthe position of ABSs level EBis determined by the cou-\npling Γ S. ForTK≪∆, the level EBsituates away\nfrom the Fermi level. In the opposite limit, namely,\nTK≫∆, the bound states would position above the\nFermi level. The Andreev level EBgoes cross the Fermi\nlevel about Γ S= 8∆ and TK/∆∼1, which corresponds\nto the quantum phase transition between magnetic dou-\nblet and screened Kondo singlet ground state.6,12,13,54\nThe width of the ABSs, which associates with its life-\ntime, only depends on the coupling Γ N. In Fig. 2(b),\nwe ﬁx the coupling Γ S= 5∆ and check the evolution of\nthe Kondo resonance with increasing Γ N. The Kondo\nresonance peak develops at the Fermi level by increasing\nthe coupling Γ N= ∆,1.5∆. Further increasing Γ N, the\nresonance peak evolves into a dip structure, as shown for\nΓN= 2∆,2.5∆. This change is ascribed to the interfer-\nence between the Kondo resonance and the signiﬁcantly\nbroadening ABSs, as discussed later.4\n/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s50/s48/s48/s52/s48/s48/s54/s48/s48/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48/s48/s53/s48/s49/s48/s48/s49/s53/s48\n/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48/s48/s50/s48/s48/s52/s48/s48/s54/s48/s48/s49/s48/s48/s50/s48/s48/s51/s48/s48\n/s32\n/s32/s76/s68/s79/s83/s32/s80/s61/s48\n/s32/s80/s61/s48/s46/s52\n/s32/s80/s61/s48/s46/s56/s40/s98/s41\n/s32/s32/s76/s68/s79/s83/s32/s115/s112/s105 /s110/s32/s117/s112\n/s32/s115/s112/s105 /s110/s32/s100/s111/s119/s110\n/s32\n/s32/s76/s68/s79/s83/s32/s115/s112/s105 /s110/s32/s117/s112\n/s32/s115/s112/s105 /s110/s32/s100/s111/s119/s110/s40/s97/s41/s32\n/s32/s76/s68/s79/s83/s32/s69\n/s90/s61/s48\n/s32/s69\n/s90/s61/s48/s46/s48/s53\n/s32/s69\n/s90/s61/s48/s46/s49\nFIG. 3. The splitting of the ABSs. (a) The case with diﬀerent\nexternal magnetic ﬁeld EZ= 0,0.05∆,0.1∆ andP= 0. The\ninset is the spin-dependent local density of states (LDOS)\nwithEZ= 0.1∆. (b)Thecase with diﬀerentspin-polarization\nwithP= 0,0.4,0.8 andEZ= 0. The inset shows the spin-\ndependent LDOS with P= 0.8. Other parameters used are\nεd=−5∆, Γ S= 5∆ and Γ N= 0.2∆, and D= 20∆.\nWhen an external magnetic ﬁeld is applied and/or the\nspin polarization in the ferromagneticlead is present, the\nAndreev transport displays some interesting splitting be-\nhaviors. In particular, the spin polarization induces a\nlocal eﬀective ﬁeld which removes the spin degeneracy\nin Kondo eﬀect and ABSs.27In Fig.3(a) and (b), we\nshow the splitting of the ABSs in the presence of Zeeman\nenergyEZinduced by external magnetic ﬁeld and spin-\npolarization P, respectively. Diﬀerently from the split-\nting induced by external magnetic ﬁeld (see Fig. 3(a)),\nthe ferromagnetic polarization induces an imbalance in\nthe local density of states (LDOS) of spin-up and spin-\ndown electrons, as shown in the inset of Fig. 3(b). Due\nto weak coupling with the dot Γ N= 0.2∆, the Kondo\nresonance at the Fermi level does not occur. However,\nwhen the quantum dot is strongly coupled with normal\nor ferromagneticlead, the Kondo resonanceoccurs at the\nFermi level, as shown in Fig. 4(marked by black lines).\nThis Kondo resonance peak becomes splitting when an\nexternal magnetic ﬁeld (see Fig. 4(a)) or a ferromagnetic\nlead (see Fig. 4(b)) is applied. Besides the occurrence of\nthe Kondo resonance, another distinguished eﬀect lead\nby strongly coupling Γ N= 1.5∆ is the broadening of the\nABSs,asshownasabroadpeakmarkedbytheblacklines\nin Fig.4(b), which are also splitting under the external\nmagnetic ﬁeld or the ferromagnetic ﬁeld. The broaden-\ning and splitting of ABSs are found to have a signiﬁcant\ninﬂuence on the lineshape of the Kondo resonance. For/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s50/s48/s52/s48/s54/s48/s56/s48 /s48/s50/s48/s52/s48/s54/s48/s56/s48\n/s32\n/s32/s76/s68/s79/s83/s32/s80/s61/s48\n/s32/s80/s61/s48/s46/s50\n/s32/s80/s61/s48/s46/s52\n/s32/s80/s61/s48/s46/s54/s40/s98/s41/s32\n/s32/s76/s68/s79/s83/s32/s69\n/s90/s61/s48\n/s32/s69\n/s90/s61/s48/s46/s48/s53\n/s32/s69\n/s90/s61/s48/s46/s49\n/s32/s69\n/s90/s61/s48/s46/s50/s40/s97/s41\nFIG. 4. The splitting of the Kondo resonance and the\nABSs. (a) The case for diﬀerent external magnetic ﬁeld\nEZ= 0,0.05∆,0.1∆ and P= 0. (b) The case for diﬀerent\nspin-polarization with P= 0,0.2,0.4 andEZ= 0. Other\nparameters used are the same as those in Fig. 3 except\nΓN= 1.5∆.\nexample, in Fig. 4(b), forP= 0.2∆,0.4∆ the lineshape\nof the splitting Kondo resonance peaks is obviously dif-\nferent. For the sub-peak above the Fermi level, the line-\nshape is a resonance peak, which is similar to that of\nP= 0. On the contrary, the lineshape of the sub-peak\nbelow the Fermi level is quite asymmetric(as presented\nas red line in Fig. 5(a)), showing a dip-peak structure,\nwhichis reminiscentofthe Fanoresonance. Inthe follow-\ning we argue that the broadening ABSs provides a signif-\nicant continuum which leads to the asymmetric Kondo\nsub-peak by interference eﬀect.\nIt iswell-knownthat the subgapfeaturesin the present\ncaseinvolveatmosttwoentities: oneisthe ABSsandthe\nother is the Kondo resonance if the conditions are met.\nThus, the dot Green’s function in the subgap regime can\nbe written by introducing T-matrix as\nGd(ε) =G0\nd(ε)+G0\nd(ε)Td(ε)G0\nd(ε),(15)\nwhereG0\nd(ε) is the dot Green’s function in the subgap,\nwhich is nothing but the ABS described approximately\nby a simple Lorentzian resonance\nG0\nd(ε) =ZB\nε−EB+iΓB, (16)\nwhereEB,ΓB, andZBdenote the position, the width,\nand the weight of the ABSs. It is known that Γ B∝\nΓN/F. TheT-matrixTd(ε) ismainlycomposedbyKondo5\nresonance in the Kondo regime\nTd(ε)≈ΓK\nπρ0\nd(ε)1\nε−εK+iΓK, (17)\nwhich is also approximately described by a Lorentzian\nline-shape.38ΓK=TKis the half-width of Kondo reso-\nnance,εKis its position.\nFromEqs.( 15)-(17), the localdensity ofstates(LDOS)\nof the dot in the subgap ρd(ε) =−1\nπImGd(ε) reads\nρd(ε) =−1\nπIm[Gd(ε)]\n=ρ0\nd(ε)−π/bracketleftbig\nρ0\nd(ε)/bracketrightbig2×\n/bracketleftbig/parenleftbig\nq2\nd(ε)−1/parenrightbig\nImTd(ε)−2qd(ε)ReTd(ε)/bracketrightbig\n,(18)\nwhereρ0\nd(ε) =−1\nπIm/bracketleftbig\nG0\nd(ε)/bracketrightbig\nandqd(ε) =−ReG0\nd(ε)/\nImG0\nd(ε) is the asymmetry factor. Substituting Td(ε)\ninto Eq.( 18), one can obtain a Fano-like formula for the\nLDOS in the subgap regime\nρd(ε) =ρ0\nd(ε)(x+qd(ε))2\nx2+1, (19)\nwithx= (ε−εK)/ΓK, as given in Ref.38. The density\nof states ρ0\nd(ε) can be obtained from Eq.( 16),\nρ0\nd(ε) =1\nπZBΓB\n(ε−EB)2+Γ2\nB. (20)\nEq.(19) can simply capture the in-gap LDOS and the\ncharacteristicsin Andreev transport, which can be recog-\nnized as the coherent superposition of ABSs and Kondo\nresonance.\nIn the absence of Kondo resonance (Γ N≪ |εdσ|), the\nABSs always keep the Lorentzian line shape even the\nfactor|qd| → ∞. By increasing the coupling Γ N, the\nKondo resonance peak appears and evolves into a Fano-\ndipstructureforΓ N∼ |εdσ|asshowninFig. 2(b). Inthis\ncase, the interference between the Kondo resonance and\nthe signiﬁcant broadening ABSs distorts the line shape\nof Kondo resonance. In the presence of magnetic ﬁeld or\nspin-polarization, the interference between the subpeak\nofKondoresonanceandsplitting ABSsalsoleadstoFano\nasymmetric structures as shown in Fig. 4. In Fig. 5(a),\nwe plot the spin-dependent LDOS with P= 0.4. The\nZeeman splitting of the square-root singularity situates\nsymmetrically around the energy gap. This phenomenon\nhas been reported in the previous experimental and theo-\nretical investigations.29,55Interestingly, the Kondo reso-\nnance splits into a resonant sub-peak and a Fano-type\nasymmetric structure for spin-up and spin-down elec-\ntrons, respectively. In Fig. 5(b), we ﬁt the asymmet-\nric structure in the LDOS of spin-down electrons by the\nqualitative theory of Fano resonance[see Eq.( 19)] by us-\ning the following parameters: qd= 1.15,TK= 0.031∆,\nEB=−0.2∆,εK=−0.3∆,ZB= 0.04, andΓ B= 0.12∆,\nand the level EB=E0\nB+EZeffwithE0\nB=−0.5∆ and\nEZeff= 0.3∆, where E0\nBis the bare level of ABS in the/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s49/s48/s50/s48/s51/s48/s52/s48\n/s45/s48/s46/s51/s53 /s45/s48/s46/s51/s48 /s45/s48/s46/s50/s53 /s45/s48/s46/s50/s48/s49/s50/s49/s56/s50/s52/s51/s48/s32\n/s32/s100/s40 /s41\n/s47/s32/s115/s112/s105/s110/s32/s117/s112\n/s32/s115/s112/s105/s110/s32/s100/s111/s119/s110/s40/s97/s41\n/s32\n/s32/s100/s40 /s41\n/s47/s32/s78/s117/s109/s101/s114/s105/s99/s97/s108/s32/s114/s101/s115/s117/s108/s116\n/s32/s70/s105/s116/s116/s101/s100/s32/s108/s105/s110/s101/s40/s98/s41\nFIG. 5. (a) The spin-dependent LDOS with P= 0.4 and\nEZ= 0 as given by the dotted line in Fig.4 (b). (b) The\nasymmetric structure in the LDOS of spin-down electrons is\nﬁtted with the theory of Fano resonance.\nabsence of spin polarization, and EZeffis the Zeeman\nenergy induced by the eﬀective ﬁeld.\nIn Fig.6and Fig. 7, we show the evolution of Kondo\nresonance peak (Γ N= 1.5∆) and Fano dip (Γ N= 2∆)\nstructures in Andreev transport spectrum of F-QD-S de-\nvice. Whentheferromagneticleadisweaklycoupledwith\nquantum dot, the spin polarization induced eﬀective ﬁeld\nEZeffistoo smallto be observedin ABSs. Therefore, we\ndiscussthecasethatthedotisstronglycoupledbothwith\nferromagneticandsuperconductorleads. FromFig. 6(a),\nwecanseethat theABSsaresplit andsuppressedbyspin\npolarization as indicated by the blue dashed line 2. In\naddition, the Kondo resonance is also split into two side-\npeaks by the induced eﬀective ﬁeld.39–43By increasing\nP, the lineshape of the side-peaks evolves from a peak to\na peak-dip, and even to a dip. Eventually, the Kondo ef-\nfect is completely suppressed by the largepolarization P,\nas indicated by the green dotted line 1. As shown above,\nthe Fano resonance originates from the interference be-\ntween the side-peaks of Kondo resonance and the split-\nting ABSs. At the bias Vsd=±∆, the Kondo resonance\nat the Fermi level of ferromagnetic lead also signiﬁcantly\nenhances the Andreev transport as shown by the green\ndotted lines 1′. In Fig. 6(b), we show the compensation\neﬀect in Andreev transport, since external applied ﬁeld\ncan counteract the inﬂuence of eﬀective ﬁeld on electron\nspin. All the splitting structure in ABSs can be com-\npensated by applied magnetic ﬁeld. For the polarization6\n/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54\n/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s32\n/s32/s32/s100/s73/s47/s100/s86\n/s115/s100/s40/s49/s48/s45/s50\n/s101/s50\n/s47/s104/s41/s40/s97/s41\n/s80/s49\n/s50\n/s49/s50\n/s69\n/s90/s40/s98/s41/s32\n/s32/s100/s73/s47/s100/s86\n/s115/s100/s40/s49/s48/s45/s50\n/s101/s50\n/s47/s104/s41\n/s86\n/s115/s100/s47\nFIG. 6. (a) The evolution of ABSs and the splitting of the\nKondo resonance due to the spin polarization of lead, the po-\nlarization Pincreases from P= 0 (the top) to P= 0.9 (the\nbottom) with the step 0 .1, andEZ= 0. (b) The compensa-\ntion eﬀect of the ABSs and the Kondo splitting by external\nmagnetic ﬁelds with P= 0.3. The Zeeman energy varying\nfrom the top EZ= 0 to the bottom EZ= 0.225∆ with a step\n0.025∆. The compensation ﬁeld is Ec\nZ= 0.135∆, as shown\nthe bold dark curve. Other parameters used: the dot level\nεd=−5∆, the coupling Γ S= 5∆ and Γ N= 1.5∆, and the\nhalf bandwidth D= 20∆.\nP= 0.3, the corresponding compensated Zeeman energy\nisEc\nZ= 0.135∆ as shown the bold dark curve in Fig. 6\n(b). By increasing the external magnetic ﬁeld, the asym-\nmetric structures ﬁrstly merge into a Kondo resonance\npeak. For EZ> Ec\nZ, the Kondo resonance peak splits\nagain into two sub-peaks, which become more and more\nsymmetric, because the Fano interference is suppressed\nby increasing the distance between the Kondo resonance\nand ABSs through the Zeeman splitting EZ.\nIn Fig.7(a), we show the inﬂuence of eﬀective ﬁeld\non anti-resonance dip structure in Andreev transport. In\nthe absence of polarization, the anti-resonance dip orig-\ninates from the interference between Kondo resonance\nand ABSs. In the presence of polarization, the inter-\nference between the sub-peaks of Kondo resonance and\nsplitting ABSs can be tuned by increasing the distance\nbetween them. As indicated by the green solid line 1, the\nresonance dip splits into two side-dip structures, which\nevolve into resonance peaks and then are completely sup-\npressed by increasing the polarization P. The ABSs are\nalso split by the eﬀective ﬁeld and shows crossing behav-\nior. In Fig. 7(b), we show the splitting of ABSs and\nKondo resonance can be compensated by applied mag-\nnetic ﬁeld. For the polarization P= 0.2, the splitting in\nABSs can be compensated by the applied magnetic ﬁeld\nwithEc\nZ= 0.1∆. The anti-resonance dips display cross-\ning behavior and evolve into asymmetric structures by/s48/s46/s48/s48/s46/s51/s48/s46/s54\n/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s80\n/s40/s98/s41\n/s32\n/s32/s32/s100/s73/s47/s100/s86\n/s115/s100/s40/s49/s48/s45/s50\n/s101/s50\n/s47/s104/s41/s40/s97/s41\n/s50/s49\n/s49\n/s50\n/s69\n/s90\n/s32/s32/s100/s73/s47/s100/s86\n/s115/s100/s40/s49/s48/s45/s50\n/s101/s50\n/s47/s104/s41\n/s86\n/s115/s100/s47\nFIG. 7. The suppression of ABS and the splitting of Kondo\ndip due to the spin polarization and its compensation eﬀect\nby external magnetic ﬁeld. (a) The polarization Pincreases\nfromP= 0 (the top) to P= 0.9 (the bottom) with the step\n0.1 and the Zeeman energy EZ= 0. (b) The compensation\nof ABS and the Kondo splitting by external magnetic ﬁeld\nvarying from the top EZ= 0 to the bottom EZ= 0.225∆\nwith the step 0 .025∆. For the polarization P= 0.2, the\ncompensation ﬁeld is Ec\nZ= 0.1∆, as shown the bold dark\ncurve. Other parameters used : the dot level εd=−5∆, the\ncoupling Γ S= 5∆ and Γ N= 2∆, and the half bandwidth\nD= 20∆.\nincreasing the magnetic ﬁeld. In Fig. 6(b) and Fig. 7\n(b), the Fano asymmetric structures at ﬁnite-bias are\npronounced when the splitting ABSs gets close to the\nsub-peak of Kondo resonance. The line shape is deter-\nmined by the Fano factor qd. Oppositely, the Kondo res-\nonance recovers the peak structure when it is away from\nthe ABSs. Because the dot level can not be signiﬁcantly\nbroadened by applied magnetic ﬁeld, the Fano eﬀect can\nonly attributes to the interference between the ABSs and\nsub-peak of Kondo resonance.\nIV. SUMMARY\nIn conclusion, we have studied the Fano resonance in\nAndreev transport through the N/F-QD-S device. In\nN-QD-S device, the interference between the Kondo ef-\nfect and the broadening ABSs leads to an anti-resonance\nstructure at zero-bias in Andreev transport. In the pres-\nence of spin-polarization, the ferromagnetic lead induces\nan eﬀective magnetic ﬁeld, which splits the Kondo res-\nonance and ABSs. All the splitting structures can be\ncompensated by external applied magnetic ﬁeld. When\nthe sub-peaks of Kondo resonance and splitting ABSs7\nget close to each other, the interference between them\nresults in Fano-type asymmetric structures at ﬁnite-bias.\nThe Fano resonance in Andreev transport is expected to\nbe observed in future experiments.\nL. Li gratefully acknowledge useful discussions withDr.Hua Chen. This work is supported by NSFC (Grants\nNos. 11174115, 11325417, 11204186, 11274269), PC-\nSIRT (Grant No. IRT1251) of China, and Natural\nScience Foundation of Guangdong Province of China\n2014A030310137.\n1A. V. Rozhkov, and D. P. Arovas, Phys. Rev. Lett. 82,\n2788 (1999).\n2A. A. Clerk and V. Ambegaokar, Phys. Rev. B 61, 9109\n(2000).\n3M. -S. Choi, M. Lee, K. Kang, and W. Belzig, Phys. Rev.\nB70, 020502(R) (2004).\n4F. Siano and R.Egger, Phys. Rev. Lett. 93, 047002 (2004).\n5J. S. Lim and M.-S. Choi, J. Phys.: Condens. Matter 20,\n415225 (2008).\n6G. Sellier, T. Kopp, J. Kroha, and Y. S. Barash, Phys.\nRev. B72, 174502 (2005).\n7J.-P. Cleuziou, W. Wernsdorfer, V. Bouchiat, T. Ondar-\ncuhu, and M. Monthioux, Nat. Nanotechnol. 1, 53 (2006).\n8C. Buizert, A. Oiwa, K. Shibata, K. Hirakawa, and S.\nTarucha, Phys. Rev. Lett. 99, 136806 (2007).\n9C. Karrasch and V. Meden, Phys. Rev. B 79, 045110\n(2009).\n10D. J. Luitz, F. F. Assaad, T. Novotny, C. Karrasch, and\nV. Meden, Phys. Rev. Lett. 108, 227001 (2012).\n11J. -D. Pillet, P. Joyez, R. Zitko, and M. F. Goﬀman, Phys.\nRev. B88, 045101 (2013).\n12B.-K. Kim, Y.-H. Ahn, J.-J. Kim, M.-S. Choi, M.-H. Bae,\nK. Kang, J. S. Lim, R. Lopez, and N. Kim, Phys. Rev.\nLett.110, 076803 (2013).\n13L. Li, B. B. Zheng, W. Q. Chen, H. Chen, H. G. Luo, and\nF. C. Zhang, Phys. Rev. B 89, 245135 (2014).\n14Q. -F. Sun, H. Guo, and T. H. Lin, Phys. Rev. Lett. 87,\n176601 (2001).\n15A. V. Rozhkov and D. P. Arovas, Phys. Rev. B 62, 6687\n(2000).\n16M. R. Buitelaar, T. Nussbaumer, and C. Sch ´’onenberger,\nPhys. Rev. Lett. 89, 256801 (2002).\n17M. R. Buitelaar, W. Belzig, T. Nussbaumer, B. Babic,C.\nBruder, and C. Sch¨ onenberger, Phys. Rev. Lett. 91,\n057005 (2003).\n18M. R. Graber, T. Nussbaumer, W. Belzig and C. Schonen-\nberger, Nanotechnology 15S479-S482 (2004).\n19T. Meng, S. Florens, and P. Simon, Phys. Rev. B 79,\n224521 (2009).\n20V. Koerting, B. M. Andersen, K. Flensberg, and J. Paaske,\nPhys. Rev. B 82, 245108 (2010).\n21R. Zitko, M. Lee, R. L´ opez, R. Aguado, and M.-S. Choi,\nPhys. Rev. Lett. 105, 116803 (2010).\n22A. Mart´ ın-Rodero and A. Levy Yeyati, Adv. Phys. 60, 899\n(2011).\n23Eduardo J. H. Lee, Xiaocheng Jiang, Ramon Aguado,\nGeorgios Katsaros, Charles M. Lieber, and Silvano De\nFranceschi, Phys. Rev. Lett. 109, 186802 (2012).\n24A. Kumar, M. Gaim, D. Steininger, A. Levy Yeyati, A.\nMartin-Rodero, A. K. Huettel, C. Strunk, Phys. Rev. B\n89, 075428 (2014).\n25A.Eichler, et al., Phys.Rev.Lett. 99, 126602 (2007); Phys.\nRev. B79, 161407(R) (2009).\n26R. S. Deacon, Y. Tanaka, A. Oiwa, R. Sakano, K. Yoshida,K. Shibata, K. Hirakawa, and S. Tarucha, Phys. Rev. Lett.\n104, 076805 (2010); Phys. Rev. B 81, 121308 (2010).\n27L. Hofstetter, et al., Phys. Rev. Lett. 104, 246804 (2010).\n28M. Krawiec andK.I.Wysoki´ nski, Supercond.Sci. Technol.\n17, 103 (2004).\n29T. Domanski, A. Donabidowicz, and K. I. Wysokinski,\nPhys. Rev. B 76, 104514 (2007).\n30U. Fano, Phys. Rev. 124, 1866 (1961).\n31A. E. Miroshnichenko, S. Flach, and Y. S. Kivshar, Rev.\nMod. Phys. 82, 2257 (2010).\n32Jintao Li, W.-D. Schneider, R. Berndt, and B. Delley,\nPhys. Rev. Lett. 80, 2893 (1998).\n33V. Madhavan, W. Chen, T. Jamneala, M. F. Crommie,\nand N. S. Wingreen, Science 280, 567 (1998).\n34H. C. Manoharan, C. P. Lutz, and D. M. Eigler, Nature\n(London) 403, 512 (2000).\n35T. Jamneala, V.Madhavan, W. Chen, andM. F.Crommie,\nPhys. Rev. B 61, 9990 (2000).\n36O. Ujsaghy, J. Kroha, L. Szunyogh, and A. Zawadowski,\nPhys. Rev. Lett. 85, 2557 (2000).\n37K. Nagaoka, T. Jamneala, M. Grobis, and M. F. Crommie,\nPhys. Rev. Lett. 88, 077205 (2002).\n38H. G. Luo, T. Xiang, X. Q. Wang, Z. B. Su, and L. Yu,\nPhys. Rev. Lett. 92, 256602 (2004).\n39J. Martinek, et al., Phys. Rev. Lett. 91, 127203 (2003);\nPhys. Rev. Lett. 91, 247202 (2003).\n40M. S. Choi, D. Sanchez, and R. Lopez, Phys. Rev. Lett.\n92, 056601 (2004).\n41R. Swirkowicz, M. Wilczynski, and J. Barnas, J. Phys.\nCondens. Matter 18, 2291 (2006).\n42M. Sindel, L. Borda, J. Martinek, R. Bulla, J. Konig, G.\nSchon, S. Maekawa, and J. von Delft, Phys. Rev. B 76,\n045321 (2007).\n43L. Li, Y.-Y. Ni, T.-F. Fang, and H.-G. Luo, Phys. Rev. B\n84, 235405 (2011).\n44P. W. Anderson, Phys. Rev. 124, 41 (1961).\n45F. D. M. Haldane, Phys. Rev. Lett. 40, 416 (1978).\n46C. Lacroix, J. Phys. F 11, 2389 (1981).\n47H.-G. Luo, Z.-J. Ying, and S.-J. Wang, Phys. Rev. B 59,\n9710 (1999).\n48Q. -F. Sun, J. Wang, and T. H. Lin, Phys. Rev. B 62, 648\n(2001).\n49J. C. Cuevas, A. Levy Yeyati, and A. Martn-Rodero, Phys.\nRev. B63, 094515 (2001).\n50J. Baranski and T. Domanski, Phys. Rev. B 84, 195424\n(2011).\n51E. J. H. Lee, X. C. Jia, M. Houzet, R. Aguado, C. M.\nLieber, and S. D. Franceschi, Nature Nanotechnol. 9, 79-\n84 (2014).\n52V. Kashcheyevs, A. Aharony, and O. Entin-Wohlman,\nPhys. Rev. B 73, 125338 (2006).\n53Y. Qi, J. X. Zhu, and C. S. Ting, Phys. Rev. B 79, 205110\n(2009).\n54W.Chang, V.E.Manucharyan, T.S.Jespersen, J.Nygard,8\nand C. M. Marcus, Phys. Rev. Lett. 110, 217005 (2013).55R. Meservey, P. M. Tedrow, and P. Fulde, Phys. Rev. Lett.\n25, 1270 (1970)."    },    {        "title": "1102.5722v1.Ferromagnetic_resonance_in_epitaxial_films__Effects_of_lattice_strains_and_voltage_control_via_ferroelectric_substrate.pdf",        "content": "Ferromagnetic resonance in epitaxial films: Effects of lattice strains  \n and voltage control via ferroelectric substrate  \nN. A. Pertsev ,1,2 H. Kohlstedt ,1 and R . Knöchel3 \n1Nanoelekt ronik, T echnische Fakultät, Christian -Albrechts -Universität zu Kiel,  D-24143 Kiel, Germany  \n2A. F. Ioffe Physico -Technical Institute, Russian Academy of Sciences, 194021 St. Petersburg, Russia  \n3Hochfrequenztechnik, Technische Fakultät, Christian -Albrechts -Universität zu Kiel, D -24143 Kiel, Germany  \n(arXiv:cond -mat, 1  March 20 11) \n \nThe phenomenon  of ferromagnetic resonance (FMR) provides  fundamental information  on the physic s of \nmagnetic materials and lies at the heart of  a variety of  signal processing microwave devices. Here we \ndemonstr ate theoretically that substrate -induced lattice strains may change the FMR frequency of an epitaxial \nferromagnetic film dramatically, leading to ultralow and ultrahigh resonance frequencies at room temperature. \nRemarkably, the FMR frequency varies with th e epitaxial strain nonmonotonically , reaching minimum at a \ncritical strain corresponding to the strain -induced spin reorientation transition.  Furthermore, b y coupling the \nferromagnetic film to a ferroelectric substrate, it becomes  possible to achieve a n efficient  voltage  control  of \nFMR  parameters . In contrast to previous studies , we found  that the tunability  of FMR fr equency varies  with the \napplied electric field and strongly increase s at critical  field intensity . The revealed feature s open up wide  \nopportun ities for the development of  advanced  tunable magnetoelectric devices  based on strained nanomagnets . \n \nI. INTRODUCTION  \nThe ferromagnetic resonance (FMR), predicted by \nLandau and Lifshitz1 in 1935 and observed \nexperimentally by Griffiths2 in 1946,  arises  from the  \nprecession al motion  of the magnetization vector about \nthe equilibrium direction that is driven  by a microwave \nmagnetic field.  The s tudies of FMR in bulk crystals , \nthin films , and multilayers provided fundamental  \ninformation on the gyromagnetic ratio , magnetic  \nanisotropy , interlayer exchange coupling , and spin \ndynamics in magnetic materials .3-5 The FMR also has a \nsignificant practical importance, being employed in \nsignal processing microwave devices such as tunable \nband -stop filters , phase shifters,  and signal -to-noise \nenhancer s.6-9 \nThe tuning of resonance co ndition  is usually \nrealized via the application of a static magnetic field to \nthe ferromagnet. However  this approach has the \ndisadvantage s of slow tuning speed and high energy \nconsumption  associate d with  magnetic field generation . \nOn the other hand , the existence of a magnetoelastic contribution to the free energy indicates  that the FMR \ncan be also affected by lattice strains .10-12 This effect \nwas employed in ferrite -piezoelectric composites to \ntune the reso -nance magnetic field Hr by an applied \nvoltage ,13-15 which allows high tuning speed with low \nenergy consumption.  It was found that Hr varies with \nvoltage almost linearly, which was attributed  to strains \ninduced in the piezoelectric material  and tra nsmitted to \nthe ferrite via the mechanical interfacial coupling.  The \nexperimental observations were described by a linear \ntheory taking into account small piezoelectric \ndeformations  generated by applied field , which  induc e \nlattice strains  in the ferromagne tic material .14-16 \nAlthough the l inear approximation  seems to be  \nsufficient for the theoretical description of FMR in \nweakly strained bulk magnetic materials and \ncompo sites, it is expected to give poor results for \nepitaxial ferromagnetic  films and nano structures . \nIndeed, considerable strains are usually generated in \nepitaxial heterostructures during growth , especially in \nfilms with nanoscale  thicknesses, where strain 2 \n \n \n relaxation via formation  of misfit dislocations does not \noccur . Moreover, large voltage -induced  strains of ~1% \nmay be created  in a thin ferromagnetic film  attached to \na ferroelectric substrate  with ultrahigh piezoelectric \ncoefficients , such as  Pb(Zn 1/3Nb 2/3)O3–PbTiO 3 (PZN -\nPT) or Pb(Mg 1/3Nb 2/3)O3–PbTiO 3 (PMN -PT). As lattice  \nstrains can affect the  magnetization direction and even \ninduce spin reorientation transitions ,17-23 dramatic \nchanges in the FMR  are expected  for epitaxial hetero -\nstructures.  This expectation is supported by the obser -\nvation of a nonlinear electric -field-induced variation of \nthe F MR frequency  in the FeGaB /PZN -PT hybrid .24 \nIn this paper, we describe the strain effect on FMR \nwith the aid of nonlinear  thermodynamic theory . Both \nthe initial  epitaxial strain imposed on  a ferromagnetic \nfilm and additional variable strain s produced  by the  \nsubstrate piezoelectric defor mations under applied \nelectric field are taken into account in the calculations  \nof FMR frequency and resonance magnetic field .  \n \nII. RESONANCE CONDITIONS IN STRAINED \nFILMS  \nConsider a homogeneously magnetized ferro -\nmagnet ic film  subjected to  a static magnetic field H \nand a microwave magnetic field h(t). The Landau -\nLifshitz -Gilbert torque equation of motion for the \ndynamic magnetization M(t)  = Ms + M(t) may be \nwritten as   \n  dtd\nMG\ndtd MM HMM\n2 eff 0\n,        (1) \nwhere  is the gyromagnetic ratio , 0 is the \npermeability of vacuum, Heff is the effective  magnetic \nfield, and G is the Gilbert damping parameter .1,5 Well \nbelow the Curie temperature, the magnitude M of \ndynamic magnetization can be regarded as a fixed \nquantity .11 Then Eq. (1) may be rewritten in terms of \ndirection cosines mi (i = 1,2,3)  of the magnetization \nvector M and linearized with respect to small \ndeviati ons mi << 1 from the static  orientation  {\n0\n30\n20\n1 ,,mmm }, which are  induced by a weak microwave \nfield with amplitude |h| << | H|. The effective magnetic \nfield involved in Eq. (1) is defined by the relation \n0Heff = F/M, where  F is the Helmholtz free  energy \ndensity of a ferromagnet. Heff is the sum of static field \nH, micro wave field h, demagnetizing  field depending \non the demagnetizing factors Ni of a ferromagnetic \nsample, and additional terms  resulting from  the energy \nof magnetocrystalline anisotropy  and magnetoelastic \nenergy.  \nLet us calculate  the effective field Heff for an \nepitaxial film of a cubic ferromagnet with the (001) \ncrystallographic orientation in the paramagnetic state. \nRestricting our analysis to the film thickness es beyond  \na few -monolaye r range , for which  the surface \ncontribution to the  total energy25 may be neglected , we \ncan consider  the Helmholtz free energy density  F of \ninterior  homogeneous magnetic state  only. As epitaxial \nstrains lower the symmetry of crystal lattice, the \nmagnetic a nisotropy of a strained ferromagnetic film \ndiffers from that of a bulk ferromagnet . This feature \ncan be described by  calculating the energy density F \nwith the account of magnetoelastic and strain energ ies \nand mechanical boundary conditions imposed on an \nepitaxial film .22 Adding the Zeeman energy and th e \ndemagnetizing field energy to the expression  derived \nfor F in Ref. 22 and using the reference  frame ( x1, x2, \nx3) shown in Fig. 1, we obtain  \n216 22\n222\n11 12\n32\n22\n122\n32\n22\n1\n442\n2\n112\n1\n12\n22\n11\n) () (2 2\nmmuB mu muB mmmKmm mcB\ncBK mmKF\nm m m   \n\n\n \n\n(2)                   ), ( )(21\n6\n33 22 11 02\n332\n222\n112\n02\n3 2 1\n1112\n111\n1\nmH mH mHM mNmN mNM m u ucc\ncBB\nss m m\n  \n\n\n \nwhere mi are the direction cosines of magnetization M \nwith respect  to the principal cubic axes , K1 and K2 are \nthe anisotropy constants of fourth and sixth order at \nconstant strains, B1 and B2 are the magne toelastic \ncoefficients, c11, c12, and c44 are the elastic stiffnesses at 3 \n \n \n constant magneti zation , and um1, um2, um6 denote  the in -\nplane film strains u11, u22, u12 imposed by a dissimilar \nthick substrate.  Diffe rentiating Eq. (2), one readily \nfinds  the deri vatives F/mi defining  the sought \neffective field Heff. \nSubstituting Heff into Eq. (1) and taking into \naccount that in equilibrium M  Heff = 0, we de rived a \nsystem of equations for the quantities mi(t) describing  \nthe magnetization precession.  Since \n12\n32\n22\n1  m m m , \none of these quantities can be ex pressed through two \nothers. Focusing o n the case  of \n00\n3m , we used the \nrelation \n0\n3 20\n2 10\n1 3 /) ( m mmmm m     and, neglecting  \nthe loss term in the first approximation , reduce d the \nsystem of linearized equations to  \n,30\n2 0 20\n3 0 2 12 1 111hmM hmM mA mAdtmdM\ns ss \n \n30\n1 0 10\n3 0 2 22 1 212hmM hmM mA mAdtmdM\ns ss \n \n. \n(3) \nHere Aij are polynomials  of direction cosines  \n0\nim of the \nstatic magnetization  Ms, in which  coefficients depend \non components Hi of the static magnetic field , misfit \nstrains um1, um2, um6, and the material parameters \ninvolved in Eq. (2). Since A11 + A22 = 0, the expression  \nfor the FMR frequency r resulting from Eq. (3), which \nis valid to first order in the loss term ,26 takes the form   \n21 12 22 112AA AAMsr  \n .                  (4)\n Using the relatio ns \n  sin sin  ,sin cos0\n20\n1   m m , \nand \ncos0\n3m , it is possible to rewrite Eq. (4) in terms \nof the orientation angles  and  of the static \nmagnetization Ms. Indeed, the se angles must  \ncorrespond to the equilibrium orientation of Ms \ndetermined via minimization of the free energy  given \nby Eq. (2). It should be noted that , at sinthe FMR \nfrequency can be also  calculated from the formula  \n)sin 2/(2  s r M F FF \n, where the quantities \nunder the square root are the second derivatives of energy F with respect to the angles  and .27 However, \nthis formula cannot be used for the magnetization \ndirections  orthogonal to the film surfaces  (sin), \nwhich are important for our study . The procedure \nemployed in Ref. 24, which involves the direct \nsubstitution of the strain -dependent  effect ive magnetic \nfield Heff into the usual  relation for the FMR frequency , \ndoes not provide rigorous  theoretical description of the \ninfluence of lattice strains on the resonance conditions . \n \n \nFIG. 1.  Schematic drawing of a ferromagnetic film \ncoupled to a ferroelectric substrate covered by two \nextended electrodes.  A film with static magnetization \nMs is subjected to the constant magnetic field H and \nthe microwave magnetic field h. The polarization P of \nferroelectric material varies with the applied electric \nfield E, which creates macroscop ic substrate \ndeformations.  Panel (a) shows the top -bottom \narrangement of electrodes, whilst panel (b) describes \ntheir lateral configuration.  4 \n \n \n III. STRAIN EFFECT ON FERROMAGNETIC \nRESONANCE  \nThe theoretical analysis  shows that the strain effect  \non FMR frequency has several remarkable features.  \nImportantly, r varies linearly  only in  the particular \ncase of fully symmetric configuration, where the film \nwith the out -of-plane magnetization Ms (= 0, ) and \nequal in -plane demagnetizing factors ( N1 = N2) is \nsubjected to the orthogonal magnetic field (H1 = H2 = 0, \nH3  0) and isotropic biaxial strain  (um1 = um2 = um, um6 \n= 0). For the resonance frequency r, under these \nconditions we o btain             \n(5)                , 2162)31(21\n2\n1\n1112\n112\n1 *\n13 0 3 0\n\n\n\n\n\n\n\n\n   \nm\nss r\nuBcc\ncBKMN M H \nwhere \n)2/( )2/(442\n2 112\n1 1*\n1 c B c BK K   . Equation ( 5) \nshows  that both the FMR frequency r and field Hr \nlinearly depend  on um. It should be noted that  Hr \nincreases with strain when  r decreases and vice versa . \nAs soon as  the four -fold symmetry about the \nsubstrate normal is broken, the str ain dependence of \nFMR frequency becomes nonlinear . In the particular  \ncase of  = /2, the variation of r with lattice strains \num1, um2, and um6 takes the form   \n\n.)2cos sin ( 2cos) (62 4cos22cos) ( ) cos sin( ) ( 2 2cos) cos31(32cos) sin (2) ( sec2\n2/1\n6 2 1 0 6 2 2 1\n1112\n2\n111\n1*\n13 22\n0 3 2 0 2 1\n1112\n1 1\n442\n22\n112\n1 2 2\n2 1 3 12\n0 3 0\n\n \n\n\n\n\n     \n\n\n\n\n \n   \n           \nm s m m m ms s m m ms s\nsr\nuB HM uB u uccucBB KN NM H HM u uccuBcBcBK K N NM HMM\n          (6)\nIt can be seen that the deviation of  from zero imparts  \nnonlinearity even to the dependence of r on the \nisotropic biaxial strain um. Using Eq. (6) together with \nEq. (2), it is also possible to determine the strain \ndependence of the resonance magnetic field Hr but this  task is more involved as the orientation angle  \ngenerally varies with the field intensity.  \nIn turn, for a film with an in-plane static  \nmagnetization  ( = /2), the calculation gives  \n \n                                               , 2sin ) ( sin cos 262sin2)4cos1(2) cos sin (2sin 2 2cos) (2 4cos2 2cos) (2\n2/1\n6 2 2 1\n1112 2\n22\n1 1112\n1 2 2 1 *\n12\n12\n2 32\n0 || 02/1\n6 2 1 2 1 1 1 22\n0 || 0\n\n  \n         \n            \nm m m m ms sm m m s s\nsr\nuB u uccu uBcB K KK N N NM HMuB u uB K N NM HMM\n            (7)\nwhere H|| = H1 cos + H2 sin is the projection of H on \nthe direct ion of static magnetization . Remarkably, at \nfixed orient ation angle and um1 = um2 = um, um6 = 0, \nEq. (7) reduces  to a square -root dependence  of \nresonance frequency on the isotropic biaxial  strain um.  \nIt should be emphasized that Eq. (5) and other \nanaly tic relations for the strain dependence of FMR  frequency  hold only for certain ranges  of misfit strains , \nwithin which the equilibrium direction  of \nmagnetization remains fixed. In general, the orientation \nof Ms depends on misfit strain s as well ,22 which aff ects \nthe FMR parameters additionally . To study the \ninfluence of strain -induced magnetization \nreorientations on resonance conditions, we performed \nnumerical calculations for epitaxial films of  Fe60Co40 5 \n \n \n and CoFe 2O4.28  Figure 2(a) shows the direction cosines  \nmi of magnetization as a function of biaxial  strain um \ncalculated for  a Fe 60Co40 film subjected to an out-of-\nplane magnetic field H3 = 25 kOe . It can be seen that \nthe magnetization Ms is orthogonal to the  film surfaces \nonly at strains below  \n*\nmu  0.39% . Above this  critical  \nstrain, Ms starts to rotate gradually towards the in -plane \n[010] direction , which is accompanied by a qualitative  \nchange  in the strain dependence of FMR frequency r. \nAs demonstrated by Fig. 2(b), the linear decrease  of r \nwith strain is replaced above \n*\nmu by a nonlinear \nincrease . Remarkably , the FMR frequency goes to zero \nat \n*\nm mu u , where t he second -order  SRT takes place .  \nIn cont rast, the strain -induced SRT in a CoFe 2O4 \nfilm subjected to the magnetic field H3 = 5 kOe is of \nthe first order . This feature is evidenced by an abrupt \nreorientation of magnetization at \n*\nmu   0.06%  \ndemonstrated by Fig. 3(a). As a resul t, the FMR \nfrequency remains finite at the critical strain \n*\nmu , but \nexperiences a step -like drop across the SRT [see Fig. \n3(b)]. It can be also seen that the resonance frequency \nis very sensitive to the tensile strain s in an epitaxial \nCoFe 2O4 film. Remarkably, the FMR can be shifted to \nultrahigh frequencies exceeding 200 GHz by a biaxial \nstrain  of less than  1%. \nSuppose now that the ferromagnetic film is \nmechanically coupled to a thick ferroelectric substrate , \nas shown in Fig. 1. In this case, the film strains uij can \nbe changed by an electric field E applied to the \nsubstrate .22 Indeed, owing  to the converse piezoelectric \neffect inherent in ferroelectric materials , the applied \nfield creates  macroscopic deformations \nk kijs\nij Ed u in \nthe substrate  with piezoelectric coefficients dkij. The \ndeformations \nsu  (,  =1, 2) appearing in the (001) \nplanes parallel to the film/substrate interface  are \ntransmitted to the film via the interfacial bonding . In \nthe  case  of  perfec t transmission expected for epitaxial  \n0.0 0.5 1.0 1.5 2.002468 \n(b)Fe60Co40\nMisfit strain (%)Resonance frequency (GHz)\n   \n0.0 0.5 1.0 1.5 2.00.00.20.40.60.81.0 Fe60Co40\n  \n m2m3\nMisfit strain (%)Direction cosines mi\n(a)\n \n \nFIG. 2.  Magnetization orientation and FMR frequency \ncalculated for strained Fe 60Co40 epitaxial films . The \ndirection cosines mi of the equilibrium magnetization \nMs (a) and the FMR frequency r (b) are shown as a \nfunction of the misfit strain in the film/substrate \nsystem. The film is assumed to be under the static \nmagnetic field H3 = 25 kOe orthogonal to t he film \nsurfaces. The dashed line indicates the strain -induced \nspin reorientation transition.  \n \nhetero structures, the in -plane film strains  become \n)( )(0E Esu u u  \n, where \n0\nu  denote  their initial \nvalues defined by the misfit strains um1, um2, and  um6.  \nWhen the electric field is orthogona l to the interface \nand the film/substrate system has the four -fold \nsymmetry about the substrate normal , we obtain 6 \n \n \n \n3 31 3 22 3 11 )( )( Ed u Eu Eum and \n0)(3 12Eu since  um1 \n= um2 = um, um6 = 0, d311 = d322 = d31, and d312 = 0. \nAccordingly,  the influence of electric field E3 on the \nFMR frequency r becomes  similar to th at of the \nisotropic strain um. In particular,  for the film with out -\nof-plane static magnetization , from Eq. (5) we find that \nthe tunability vr/E3 of FMR frequency is independent \nof the misf it strain in the heterostructure and the \nmagnetic field H3 and equals  \n311\n1112\n321 dBcc\nM Esr\n\n\n\n\n.                 (8) \nTaking the relaxor ferroelectric Pb(Zn 1/3Nb 2/3)O3–\n4.5% PbTiO 3 (PZN -4.5% PT) as a representative \nsubstrate materi al, with d31 = 1000 pm/V  (Ref. 29) we \ncalculate  the tunability to be about  0.18 GHz/(kV/cm) \nfor Fe60Co40, 2.05 GHz/(kV/cm) for CoFe 2O4, and \n0.23 GHz/(kV/cm) for Ni.30 When  the film \nmagnetization deviates  from the [001] direction, the \nmagnitude of vr/E3 becomes smaller than in the out -\nof-plane magnetic state. As follows  from Figs. 2(b) and  \n3(b), the tunability  changes sign at the strain -induced \nSRT and start s to vary  with the misfit strain um. \n Instead of the top-bottom electrode arrangement, \none can al so cover the ferroelectric substrate by lateral \nelectrodes  (see Fig. 1). Since in t his configuration the \nelectric field is parallel to the interface, the four -fold \nsymmetry about the substrate normal becomes broken \neven in heterostructure s with um1 = um2 and um6 = 0. As \na result, the influence of in -plane electric field on FMR \nparameters appears to be  very different from that of the \nisotropic biaxial strain um. In particular , when the field \ndirection  is parallel to  the [100] crystallographic axis of  \nthe fil m, the in-plane  lattice strains become \n1*\n33 1 11 )( Ed u Eum\n and \n1*\n31 1 22 )( Ed u Eum , where \n*\nind \nare the  substrate piezoelectric coefficients defined  in \nthe rotated reference frame \n),,(*\n3*\n2*\n1 xxx  with the \n*\n3x  axis  \n \n-2.0 -1.5 -1.0 -0.5 0.0 0.50.00.20.40.60.81.0\n(a)\n  \n m2\nm3\nMisfit strain (%)Direction cosines mi\nCoFe2O4\n-2.0 -1.5 -1.0 -0.5 0.0 0.5050100150\nCoFe2O4\n  \nMisfit strain (%)Resonance frequency (GHz)(b)  \n \nFIG. 3. Magnetization orientation and FMR frequency \ncalculated for strained CoFe 2O4 epitaxial films.  The \ndirection cosines mi of the spontaneous ma gnetization \nMs (a) and the FMR frequency r (b) are shown as a \nfunction of the misfit strain in the film/substrate \nsystem. The film is assumed to be under the static \nmagnetic field H3 = 5 kOe orthogonal to the film \nsurfaces. The dashed line indicates the s train-induced \nspin reorientation transition.  \n \noriented along the field direction and the \n*\n1x  axis \nparallel to the interface.  Substituting these relations \ninto Eq. (6), we obtain  the following analytic formula \nfor the FMR frequency of a film with the out -of-plane \nmagnetization subject ed to vertical  magnetic field : \n 0 10 20 30 400.00.51.01.52.02.53.0\nNickel\n  Resonance frequency (GHz)\nElectric field (kV/cm)(a)\n0 10 20 30 406.56.66.76.86.97.0\n(b)\n  \nNickelResonance magnetic field (kOe)\nElectric field (kV/cm) \nFIG. 4. FMR frequency r (a) and field Hr (b) of an epitaxial Ni film at zero initial misfit strain ( um = 0). The electric \nfield E applied to the PZN -4.5% PT substrate is parallel to the interface , being directed along the [100] crystallographic \naxis of the film. The FMR frequency is given for the static magnetic field H3 = 7 kOe orthogonal to the film surfaces, \nand the critical magnetic field corresponds to the frequency = 1 GHz of the microwave  field.  \n \n(9)     . ) ( 2162) () ( 2162) (2\n2/1\n1*\n31*\n33\n1112 *\n31 1 1\n1112\n112\n1 *\n1\n03 2 32/1\n1*\n31*\n33\n1112 *\n33 1 1\n1112\n112\n1 *\n1\n03 1 30\n\n\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n\n\n\n\n\n\n\n  \nE d dccdB uBcc\ncBKMN NM HE d dccdB uBcc\ncBKMN NM H\nm\nssm\nss r\n \nSince the piezoelectric coefficients  involved in Eq. (9) \nare different in sign and \n2/ ~*\n33*\n31d d , the field \ndependence of FMR frequency has the form \n) 1)( 1(~1 1 qE pEr \n, where parameters p and q are \npositive. Hence the resonance frequency may vary with \nelectric field nonmonotonically  even at a fixed \norientation of the static magnetization. This remarkable \nfeature is  illustrated in Fig. 4(a), where  the dependence vr(E1) is plotted for the init ially unstrained ( um = 0) Ni \nfilm under the magnetic field H3 = 7 kOe.31 The critical \nmagnetic field Hr(E1) providing FMR in such a film \nalso varies nonmonotonically, as shown in Fig. 4(b). \nSince the magnetization remains orthogonal to the film \nsurfaces, t he dependence Hr(E1) at constant frequency \n of microwave field can be described by an analytic \nrelation   \n  \n   (10)                        . 2 2162     221\n21 2\n1*\n31*\n33\n1112\n01\n112\n1 *\n1\n02 1 32\n1*\n31*\n33\n01\n2 12\n0\nEd d ucc\nMB\ncBKMN N N M Ed dMBN NM H\nm\ns ss\nss r\n\n\n\n \n\n\n  \n  \n\n\n\n  \nIt can be seen  that nonlinearity of Hr(E1) results from \nthe strain anisotropy created by in-plane electric field.  \nThe electric field may also affect the equilibrium \nmagnetization orientation in an epitaxial film .22 In this \nsituation, the field dependence of FMR parameters may \nbe very different from the case  of fixed magnetization orientation . Figure 5 shows the direction cosines mi(E1) \nof static magnetization and the resonance frequency \nvr(E1) calculated for the Ni film strained initially by \n0.07%. At this value of the misfit strain um, the \nmagnetization remains orthogonal to the film surfaces \nup to \n*\n1E  4.55 kV/cm  and rotate s gradually towards 7 8 \n \n \n the in -plane [010] orientation at field intensities  \n*\n1 1E E\n. As is seen from Fig. 5(b), the electric field \ndependence of vr changes dramatically across the SRT , \nwhich  occu rs at the critical intensity  \n*\n1E. Remarkably, \nthe FMR frequency goes to zero at th is second -order  \nphase transition.  Moreover, the tunability vr/E1 of \nresonance frequency has an anomaly at the SRT, \ndiverging at  \n*\n1EE .  \n0 10 20 300.00.20.40.60.81.0\n(a)Nickel\n  Direction cosines mi m3\n m2\nElectric field (kV/cm)\n0 10 20 30012345\nNickel\n(b)\n  Resonance frequency (GHz)\nElectric field (kV/cm)\n \nFIG. 5. Direction cosines mi of the magnetization Ms \n(a) and the FMR frequency r (b) of a  Ni film plotted \nas a function of the electric field applied to the PZN -\n4.5% PT substrate along the interface . The initial misfit \nstrain um is taken to be 0.07%. The film is assumed to \nbe under the static magnetic field H3 = 7 kOe \northogonal to the film surfaces. The dashed line \nindicates the strain -induced SRT. IV. CONCLUSIONS  \nThus , we have shown that substrate -induced lattice  \nstrains represent  an efficient  tool for tuning the FMR \nparameters of ferromagnetic films . Our main \ntheoretical predictions can be summarized as follows :  \n(i) A linear variation of the FMR f requency r appears \nonly in the symmetric case, where the static magnetic \nfield H and magnetization Ms are orthogonal to the \nsurfaces of a film subjected to isotropic biaxial in -plane \nstrain um.  \n(ii) A nonmonotonic  variation of the resonance \nfrequency r with the isotropic strain um is revealed \nacross the strain -induced spin reorientation transitions \n(SRTs). Remarkably, r reaches minimum at a critical \nstrain \n*\nmu at which the SRT takes place.   \n(iii) The FMR frequency of a  ferromagnetic film \ncoup led to a ferroelectric substrate may change \ndramatically under the influence of electric field E \napplied to the substrate. The tunability vr/E of \nresonance frequency strongly depends on the field \nintensity E, displaying a change in sign across the \nfield-induced SRT and a drastic increase in magnitude \nnear the critical field.   \n(iv) The resonance magnetic field Hr measured at a \nconstant microwave frequency may also vary \nnonmonotonically with the electric field applied to the \nferroelectric substrate.  \nThese  predictions provide guidelines for the \nfabrication of advanced linear and nonlinear \nmicrowave devices with improved performances. In \nparticular, owing to strongly enhanced  electric \ntunability of FMR frequency , properly strained thin-\nfilm hybrids should be  superior to bulk ferromagnetic -\nferroelectric composites for applications in electrically  \ntunable microwave devices with high tuning speed and \nlow energy consumption. Moreover, the tunability \nitself can be varied in a controlled manner by a bias \nvoltage appl ied to a ferroelectric substrate. In the linear \nregime,  multiferroic hybrids can be used not only for 9 \n \n \n magnetoelectric  bandpass filters and phase shifters ,32,33 \nbut also  for resonant circuits in tunable oscillators with \nhigh tuning speed and for  frequency t unable resonant \nantennas. If the ferromagnetic films are exploited  in the \nnonlinear regime, they can be applied in parametric \ncircuits, mixers, and frequency multipliers . The se \ncomponents can represent essential building blocks of \nhigh speed reconfigurable  microwave systems, such as \nradar and communication systems or phased antenna \narrays.  We hope that our theoretical results will trigger \nexperimental studies of the strain and voltage effects \non the FMR in ferromagnetic thin films and \nnanostructures  and the  development of  novel \nmagnetoelectric  devices.  Finally, it should be noted \nthat our approach  also opens the way for nonlinear \nthermodynamic calculations of FMR parameters of \nmultiferroic columnar nanostructures .34,35 \n \nACKNOWLEDGMENTS  \nThis work was financial ly supported by the Deutsche \nForschungsgemeinschaft  via the DFG grant INST \n257/343 -1/570236 (Mercator Visiting Professorship \nawarded to N.  A. P.) and the SFB 855 01/10. The \nauthors also thank Franz Faupel and Michael Hambe \nfor critical reading of the manus cript.  \n \nReferences  \n1. L. D. Landau  and E. M. Lifshitz , Phys. Z. \nSowjetunion  8, 153 (1935).  \n2. J. H. E. Griffiths , Nature  158, 670 (1946).  \n3. T. G. Philipps  and H. M. Rosenberg , Rep. Prog. \nPhys . 29, 285 (1966).  \n4. B. Heinrich  and J. F. Cochran, Adv. Phys.  42, 523 \n(1993). \n5. M. Farle, Rep. Prog. Phys.  61, 755 (1998).  \n6. J. D. Adam  and S. N. Stitzer, Appl. Phys. Lett . 36, \n485 (1980).  \n7. W. S. Ishak, Proc. IEEE 76, 171 (1988).  \n8. H. How, W. Hu, C. Vittoria, L. C. Kempel, and K. \nD. Trott, J. Appl. Phys.  85, 4853 (1999) . 9. N. Cramer, D.  Lucic, R. E. Camley, and Z. \nCelinski, J. Appl. Phys. 87, 6911 (2000) . \n10. C. Kittel, Phys. Rev. 73, 155 (1948).  \n11. J. R. MacDonald, Proc. Phys. Soc. A 64, 968 \n(1951 ). \n12. B. Z. Rameev,  A. Gupta, F. Yıldız, L.  R. Tagirov,  \nand B. Aktaş, J. Magn. Magn. Mater. 300, e526  \n(2006) . \n13. M. I. Bichurin, V. M. Petrov, and Yu. V. Kiliba , \nFerroelectrics 204, 311 (1997).   \n14. M. I. Bichurin, V. M. Petrov, Yu. V. Kiliba , and \nG. Srinivasan , Phys. Rev. B 66, 134404 (2002). \n15. S. Shastry , G. Srinivasan , M. I. Bichurin, V. M. \nPetrov, and A. S. Tatarenko , Phys. Rev. B 70, \n064416 (2004) . \n16. M. I. Bichurin, I. A. Kornev, V. M. Petrov, A. S. \nTatarenko, Yu. V. Kiliba, and G. Srinivasan , Phys. \nRev. B 64, 094409  (2001).  \n17. B. Schu lz and K. Baberschke, Phys. Rev. B 50, \n13467 (1994).  \n18. M.-T. Lin, J. Shen, W. Kuch, H. Jenniches, M. \nKlaua, C. M.  Schneider, and J. Kirschner, Phys. \nRev. B 55, 5886  (1997) . \n19. A. Lisfi, C. M. Williams, L. T. Nguyen, J. C. \nLodder, A. Coleman,  H. Corcoran, A. Joh nson, P. \nChang, A. Kumar, and W.  Morgan, Phys. Rev. B \n76, 054405  (2007) . \n20. A. Brandlmaier, S. Geprägs, M. Weiler, A. Boger, \nM. Opel, H. Huebl, C. Bihler, M. S. Brandt, B. \nBotters, D. Grundler, R. Gross, and S. T. B. \nGoennenwein, Phys . Rev. B 77, 104445 (2008 ). \n21. C. Bihler, M. Althammer, A. Brandlmaier, S. \nGeprägs, M. Weiler, M. Opel, W. Schoch, W. \nLimmer, R. Gross, M. S. Brandt, and S. T. B. \nGoennenwein, Phys. Rev. B 78, 045203 (2008).  \n22. N. A. Pertsev, Phys. Rev. B 78, 212102 (2008).  \n23. N. A. Pertsev and H. Kohlste dt, Appl. Phys. Lett. \n95, 163503  (2009) ; Nanotechnology 21, 475202 \n(2010).  10 \n \n \n 24. J. Lou, M . Liu, D . Reed, Y . Ren, and N . X. Sun , \nAdv. Mater. 21, 4711 (2009).  \n25. L. Neel, J. Phys. Rad. 15, 225 (1954).  \n26. M. J. Hurben and C. E. Patton, J. Appl. Phys. 83, \n4344 (1998).  \n27. J. Smit and H. G. Beljers, Philips Res. Rep. 10, \n113 (1955).  \n28. The following sets of material parameters were \nused in the numerical calculations: Ms = 1.8×106 \nA/m [ I. S. Jacobs, IEEE Trans. Magn. MAG -21, \n1306 (1985) ], K1 = 1.3×104 J/m3, K2 = 0, B1 = \n29.4×106 J/m3, B2 = 3×106 J/m3 [R. C. Hall, J. \nAppl. Phys. 31, S157 (1960 )], c11 = 2.8×1011 N/m2, \nc12 = 1.4×1011 N/m2, and c44 = 1×1011 N/m2 [J. P. \nHirth and J. Lothe, Theory of Dislocations  \n(McGraw -Hill, New York, 1968) ] for Fe 60Co40 and \nMs = 3.5×105 A/m [ J.  Smi t and H.  P. J. Wijn, \nFerrites  (Wiley, New York, 1959) ], K1 = 2.9×105 \nJ/m3, K2 = 0 [H. Shenker, Phys. Rev. 107, 1246 \n(1957) ], B1 = 5.9×107 J/m3, B2 = 3.6×107 J/m3 [V. \nJ. Folen, in Magnetic and Other Properties of \nOxides and Related Compounds ; edited by K. -H. \nHellwege and A. M. Hellwege, Landolt -Börnstein, \nvol. 3, part 4b (Springer -Verlag, Berlin, 1970) ], c11 \n= 2.7×1011 N/m2, c12 = 1.6×1011 N/m2, c44 = 1×1011 \nN/m2 [M. I. Bichurin, V. M. Petrov, and G. \nSrinivasan, Phys. Rev. B  68, 054402 (2003 )] for \nCoFeO 3. We also set  = 1.760861011 rad s-1 T-1 and assumed N1 = N2 = 0 and N3 = 1, which \ncorrespond to films with in-plane dimensions \nmuch larger than the film thickness.  \n29. J. Yin and W. Cao, J. Appl. Phys. 92, 444 (2002).  \n30. The material parameters of Ni used in the \nnumerical calculations are listed in Ref. 22. \n31. The piezoelectric coefficients of PZN -4.5%PT \nwere taken to be \n*\n33d  = 2000 pm/V and \n*\n31d  = \n1000 pm/V (Ref. 29). Since the lattice parameter \nof PZN -4.5%PT  differs considerably  from that of \nNi, a suitable buffer layer should be used for the \nfabrication of single -crystalline Ni films on this \nferroelectric substrate.  \n32. G. Srinivasan , A. S. Tatarenko, and M. Bichurin , \nElectron. Lett. 41, 596 (2005).  \n33. A. S. Tatarenko, G. Srinivasan, an d M. I. Bichurin, \nAppl. Phys. Lett. 88, 183507 (2006).  \n34. H. Zheng, J. Wang, S. E. Lofland, Z. Ma, L. \nMohaddes -Ardabili, T. Zhao, L. Salamanca -Riba, \nS. R. Shinde, S. B. Ogale, F. Bai, D. Viehland, Y. \nJia, D. G. Schlom, M. Wuttig, A.  Roytburd, and R. \nRamesh, S cience, 303, 661  (2004 ). \n35. N. Benatmane, S. P. Crane, F. Zavaliche, R. \nRamesh, and T. W. Clinton, Appl. Phys. Lett. 96, \n082503 (2010).  \n  \n "    },    {        "title": "1201.6493v1.Exchange_induced_phase_separation_in_Ni_Cu_films.pdf",        "content": "arXiv:1201.6493v1  [cond-mat.mtrl-sci]  31 Jan 2012Exchange-inducedphaseseparationin Ni-Cu ﬁlms\nA.F. Kravetsa,∗, A.N.Timoshevskiia, B.Z.Yanchitskya,O.Yu.Salyuka, S.O.Yablonovskiia,S. Anderssonb, V. Korenivskib\naInstitute of Magnetism, National Academy ofSciences ofUkr aine, Vernadsky36 b,03142 Kyiv, Ukraine\nbNanostructure Physics, RoyalInstitute ofTechnology, 106 91 Stockholm, Sweden\nAbstract\nMagneto-structuralpropertiesofﬁlmsofdilutedferromag neticalloysNi xCu1−xintheconcentrationrange0 .7<x<1.0arestudied\nexperimentally. Films deposited by magnetron sputtering s how partial phase separation, as evidenced by structural an alysis and\nferromagnetic resonance measurements. The phase diagram o f the Ni xCu1−xbulk system is obtained using numerical theoretical\nanalysis ofthe electronicstructure,takinginto accountt he inter-atomicexchangeinteractions. The resultsconﬁrm the experimen-\ntally found partial phase separation, explain it as magneti c in origin, and indicate an additional metastable region co nnected with\nthe ferromagnetictransitionin thesystem.\nKeywords: diluteferromagnets,thinﬁlms,phasediagram,ferromagne ticresonance,Curie temperature\n1. Introduction\nMagnetic multilayered ﬁlms have been receiving much at-\ntention from the research community in the ﬁeld of spintron-\nics. Recently,adevicehasbeenproposed[1]anddemonstrat ed\n[2, 3], in which the magnetic and spin-transport properties are\ncontrolled thermo-electrically. At the core of the new desi gn\nis a tri-layer ( F/f/F) of two strong ferromagnets ( F) exchange-\ncoupledbya weaklyferromagneticspacer( f) havingthe Curie\npoint at/ornearroomtemperature(300 −500 K).The spacer f,\ncontrollingtheexchangecouplingbetweentheouterlayers ,can\nbe made of a diluted ferromagnetic alloy, such as Ni xCu1−x\n[2,3]. It isdesirablethatthespaceriscompositionallyun iform\nand does not contain multiple phases, which can form as a re-\nsult ofatomicsegregationduringﬁlmdepositionorsubsequ ent\nheat treatment.\nAlloysNi xCu1−xintheirbulkformare fccbinarysubstitution\nalloys(α-phase),with Ni andCu mutuallysolvablein anypro-\nportion up to the temperature of 627 K, where at x=0.67, the\nαphaseseparatesintotwo, α1andα2[4]. Accordingto[4],the\nphase diagram contains a metastable region with phase sepa-\nration at the nickel concentration of 0 .86<x<0.91, which\ncan lead to additional magneto-structural inhomogeneitie s in\nthe system. The results, obtained in this work, however, are\nobtained using an empirical model, based on the experimenta l\ndata. It is desirable to independently simulate the phase eq ui-\nlibria in the system Ni xCu1−x, ﬁrst without using experimental\ninformation and then compare the results with the experimen -\ntal ﬁndings. Of particular interest is the inﬂuence of the ma g-\nneticinteractionsinthesystemonthepotentialstructura land/or\ncompositionalphaseseparation. TheCurietemperatureofb ulk\nNixCu1−xalloys depends linearly on the Ni concentration [4].\n∗Corresponding author\nEmailaddress: anatolii@kth.se (A.F.Kravets)By varying the Ni content from 0.5 to 1 the Curie temperature\nischangedfrom0Kto627K.However,basedonneutronscat-\nteringdata,itwasshownthatbothbulksamples[5,6]andspu t-\ntered ﬁlms of Ni-Cu alloys [7] exhibit partial phase separat ion\nandformNi richclustersoftypicalsize 5 −10Å andmagnetic\nmoment 8−12µB. The formation of such Ni clusters and the\nresulting magnetic inhomogeneity in the material can lead t o\nanomalousmagnetic[8]andtransport[9]properties.\nInthisworkweinvestigatethemechanismsbehindphasefor-\nmations in diluted ferromagnetic alloy ﬁlms of Ni xCu1−xand\nexplain using ﬁrst-principles calculations the enhancing eﬀect\nof the exchange interaction on phase separation in the ferro -\nmagnetic composition range of 0 .7<x<1.0. We discuss op-\ntimumwaystoprepareﬁlmswithenhancedmagneto-structura l\nhomogeneity.\n2. Experimentalmethods\nFilms of diluted ferromagnetic Ni xCu1−xalloys, with 0.7<\nx<0.9 in Ni concentration, 100 nm thick, were deposited at\nroomtemperatureonthermallyoxidizedSisubstratesusing DC\nmagnetronco-sputteringfrom Cu and Ni targets. Substrates of\n150×10mminsize,placedabovetheCuandNitargets,along\nthe line connectingthe centersof the targets,were used to p ro-\nduceacontinuousandessentiallylinearcompositionalgra dient\nalong the substrate strip. The base pressure in the depositi on\nchamber was∼5×10−8Torr and the Ar pressure used dur-\ning deposition was 5 mTorr. The depositionrate for Ni and Cu\nin the center of the substrate was ∼0.5 Å/sec. Samples for\nmagneto-structural measurements were cut from the substra te\nstrip into short sections with dimensions of 5 ×10 mm. In\ntotal, 30 samples of various composition Ni xCu1−xwere pro-\nducedin the same fabricationcycle underthe same condition s.\nThe thicknessof the ﬁlms was determinedusing a surface pro-\nﬁlometer. The composition of the ﬁlms was determined using\nPreprint submitted to Journal of Magnetism and Magnetic Mat erials September 20,2018x-raydispersionspectroscopyanalysis.\nFerromagnetic resonance (FMR) was used to determine the\neﬀective magnetization of the Ni xCu1−xﬁlms (Mef f) and their\nCurie temperature ( Tc). FMR measurements were performed\nat 9.45 GHz using a Bruker ELEXSYS-E500 spectrometer\nequippedwithagoniometerforangle-dependentmeasuremen ts\nand a variable temperature cryostat. The e ﬀective magnetiza-\ntion was obtained from the resonance ﬁelds using the Kittel\nformulae[10]:\nω\nγ=H⊥−4πMef f\nω\nγ=/radicalig\nH/bardbl/parenleftig\nH/bardbl+4πMef f/parenrightig\n, (1)\nwhereH⊥andH/bardblare the measured perpendicularand in-plane\nFMR resonanceﬁelds, respectively, ω– frequency,andγ– the\ngyromagneticratio.\nThe Curie temperatureof the ﬁlms was determinedfromthe\nFMRdataastemperatureatwhicheither Mef fortheFMRsig-\nnalvanished. Thedegreeofmagneto-structuralnon-unifor mity\nin the ﬁlms was estimated from the width of the FMR peaks\n(magneticnon-uniformityleadstobroaderFMRpeaks).\nMechanical stress arising from mismatches in the lattice pa -\nrametersofthesubstrateandtheﬁlmscanleadtoperpendicu lar\nto the plane magnetic anisotropy in the ﬁlms through magne-\ntostriction. This magnetostrictive contribution to Mef fcan be\nestimatedusing[11]:\n4πMef f=4πMs−H∗\n⊥. (2)\nHereMsis the saturation magnetization of the ﬁlm, H∗\n⊥=\n−2λσ/3 – magnetostriction-induced perpendicular-anisotropy\nﬁeld,λ– magnetostrictive constant, σ– mechanical stress in\nthe ﬁlm. It follows from (2) that for a strong magnetostricti ve\ncontribution and/or small saturation magnetization, the e ﬀec-\ntive magnetizationcan become negative, which corresponds to\nits perpendicular-to-the-planeorientation in the absenc e of ex-\nternalﬁelds[10].\n3. Experimentalresultsanddiscussion\nFigure1showsthetemperaturedependenceoftheFMRres-\nonance ﬁelds for Ni xCu1−xﬁlms with diﬀerent Ni content. For\nNi0.71Cu0.29and Ni 0.77Cu0.23ﬁlms,H⊥<H/bardblin a broad tem-\nperature range, which indicates an out-of-plane magnetiza tion\norientation for these compositions. For ﬁlms with higher co n-\ncentrationsof Ni ( x=0.82−0.92),H⊥>H/bardbland the magneti-\nzationisin-planeat zeroﬁeld.\nThe temperature dependence of the e ﬀective magnetiza-\ntion obtained using (1) and the measured resonance ﬁelds\n(see Fig. 1) are shown in Fig. 2. For all compositions, the\neﬀective magnetization ﬁrst increases with increasing tempe r-\nature from 150 K to ∼320 K and subsequently decreases\naboveT>320 K. This indicates a signiﬁcant magnetostric-\ntive contribution to the anisotropy. For low Ni concentra-\ntions, Ni 0.71Cu0.29and Ni 0.77Cu0.23,Mef fis negative, while\nfor Ni 0.82Cu0.18, Ni0.85Cu0.15, Ni0.87Cu0.13, Ni0.89Cu0.11andNi0.92Cu0.08Mef f>0. Thus, in ﬁlms with low Ni concentra-\ntionsandthereforelowsaturationmagnetization,magneto stric-\ntionresultsin perpendicularmagneticanisotropy.\n/s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48/s50/s51/s52/s120 /s32/s61/s32/s48/s46/s55/s49\n/s32/s32/s32/s72\n/s73 /s73 /s32/s32/s32/s32 /s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s32/s72\n/s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48 /s53/s48/s48/s50/s46/s53/s51/s46/s48/s51/s46/s53/s52/s46/s48/s52/s46/s53\n/s120 /s32/s61/s32/s48/s46/s56/s50/s32/s72\n/s114/s101/s115/s32/s32/s40/s107/s71/s41\n/s32/s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48 /s53/s48/s48/s50/s51/s52/s53\n/s120 /s32/s61/s32/s48/s46/s56/s53/s32\n/s32/s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48 /s53/s48/s48/s50/s51/s52/s120 /s32/s61/s32/s48/s46/s55/s55/s32\n/s32/s32\n/s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48 /s53/s48/s48/s50/s52/s54\n/s120 /s32/s61/s32/s48/s46/s56/s57\n/s84/s32/s32/s40/s75/s41/s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48/s50/s52/s54\n/s120 /s32/s61/s32/s48/s46/s57/s50/s32\nFigure 1: FMR resonance ﬁelds for Ni xCu1−xﬁlms versus temperature. ( )\n– external ﬁeld perpendicular to the ﬁlm plane, ( )– ﬁeld in the ﬁlm plane.\n/s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48 /s53/s48/s48 /s54/s48/s48/s45/s49/s48/s48/s48/s49/s48/s48/s50/s48/s48/s51/s48/s48\n/s32/s32/s77\n/s101/s102/s102/s32/s32/s40/s101/s109/s117/s47/s99/s109/s51\n/s41\n/s84/s32/s32/s40/s75/s41/s32/s120 /s61/s48/s46/s55/s49\n/s32/s32 /s48/s46/s55/s55\n/s32/s32 /s48/s46/s56/s50\n/s32/s32 /s48/s46/s56/s53\n/s32/s32 /s48/s46/s56/s55\n/s32/s32 /s48/s46/s56/s57\n/s32/s32 /s48/s46/s57/s50/s78/s105\n/s120 /s67/s117\n/s49/s45/s120 \nFigure2: Temperature dependence ofthee ﬀective magnetization ofNi xCu1−x\nﬁlmswithdiﬀerentNicontent x. Dashedlineistheapproximation for x=0.92.\nThe temperature of the Curie transition from the ferromag-\nnetic to the paramagnetic state ( Tc) for Ni xCu1−xﬁlms with\nx=0.82;0.85;0.87;0.89and0.92was determinedas thetem-\nperature at which Mef f→0. For Ni 0.71Cu0.29and Ni 0.77Cu0.23\n2ﬁlmswithstrongmagnetostrictionandout-of-planemagnet iza-\ntion, it was more appropriateto determine the Tcdirectly from\nthetemperaturedependenceoftherespectiveamplitudesof the\nFMRsignal(Fig.3). Tcinthiscasewasdeterminedasthetem-\nperature at which the FMR signal amplitude approached zero\n(Fig.3).\n/s50/s53/s48/s48 /s51/s48/s48/s48 /s51/s53/s48/s48 /s52/s48/s48/s48/s45/s52/s46/s48/s120 /s49/s48/s53/s45/s50/s46/s48/s120 /s49/s48/s53/s48/s46/s48/s50/s46/s48/s120 /s49/s48/s53/s52/s46/s48/s120 /s49/s48/s53\n/s78/s105\n/s48/s46/s55/s55/s67/s117\n/s48/s46/s50/s51/s70/s77/s82/s32/s32/s115/s105/s103/s110/s97/s108/s32/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121/s32/s32/s40/s97/s46/s117/s46/s41\n/s72/s32/s32/s32/s40/s71/s41/s32/s49/s53/s48/s32/s75\n/s32/s50/s48/s48/s32/s75\n/s32/s50/s53/s48/s32/s75\n/s32/s51/s48/s48/s32/s75\n/s32/s51/s52/s48/s32/s75\n/s32/s51/s56/s48/s32/s75\n/s32/s52/s50/s48/s32/s75/s40/s97/s41\n/s51/s48/s48 /s51/s53/s48 /s52/s48/s48 /s52/s53/s48/s48/s46/s48/s50/s46/s48/s120 /s49/s48/s53/s52/s46/s48/s120 /s49/s48/s53/s54/s46/s48/s120 /s49/s48/s53\n/s40/s98/s41/s32/s32 /s78/s105\n/s48/s46/s55/s49/s67/s117\n/s48/s46/s50/s57\n/s32/s32/s78/s105\n/s48/s46/s55/s55/s67/s117\n/s48/s46/s50/s51/s70/s77/s82/s32/s32/s115/s105/s103/s110/s97/s108/s32/s32/s97/s109/s112/s108/s105/s116/s117/s100/s101/s32/s32/s40/s97/s46/s117/s46/s41\n/s84/s32/s32/s40/s75/s41\nFigure 3: A setof FMR lines for di ﬀerent temperatures for Ni 0.77Cu0.23, mea-\nsured with the in-plane ﬁeld (a). FMR signal amplitude for th e in-plane ﬁeld\nconﬁguration for Ni 0.71Cu0.29and Ni0.77Cu0.23ﬁlms (b).\nTcas a function of concentration in Ni xCu1−xﬁlms, deter-\nmined from the FMR data as described above, is shown in\nFig. 4. The ﬁgure also shows the phase diagram and Tcfor\nthe same system in the bulk form [4]. The phase diagram\nalso shows the lines of the binodal, spinodal, as well as the\nmetastableregions,theexistenceofwhichwaspredictedin [4].\nFig. 4 shows that in the vicinity of the metastable equilibri a\n(linesab−ab′),theconcentrationdependenceofthe Tcforbulk\n[4] and ﬁlm materials (this work) are signiﬁcantly di ﬀerent. In\nﬁlms, the behaviour of Tc(x) is signiﬁcantly non-linear, while\nin thebulkit ispracticallylinear.\nThe non-linear Tcvs. x in ﬁlms indicates that magnetic and\nstructuralinhomogeneitiesincreasewithincreasingCuco ntent,\nspeciﬁcally in the concentration range 0 .7<x<0.9. The factthatTcin ﬁlms is always higher than in the bulk (Fig. 4) sug-\ngests that in sputter deposited ﬁlms regions or clusters ric h in\nNicomparedtothenominalcompositionoftheﬁlmareformed.\nTheTcofsuchNi-richregionsishigherthanthe Tcforthenom-\ninalcomposition.\n 400 450 500 550 600 650\n 0.7 0.75 0.8 0.85 0.9 0.95 1T (K)\nx (atomic fraction Ni)TTbinodal\nb'ab\ncbulkcfilmspinodal\nFigure 4: Phase diagram according to [4] and the Curie temper ature versus\nconcentration for bulk Ni xCu1−x(Tbulk\nc) [4] and thin ﬁlms ( Tﬁlm\nc,, this\nwork). Lines ab−ab′correspond to metastable equilibria.\nAdditional information about the non-uniformities in the\nﬁlms canbe obtainedfromthe analysisof the FMRline width.\nFig.5showsthehalfwidthoftheFMRlinesforNi xCu1−xﬁlms,\nforvarioustemperaturesbetween200and380K.Judgingfrom\nthe relativelynarrowline widths(Fig. 5), the ﬁlms can be co n-\nsidered rather uniform magnetically, at least within the fe rro-\nmagnetic exchange length of ∼10 nm. It is also seen from\nthe data that the degree of non-uniformitydependson the con -\ncentrationinNi xCu1−xﬁlms. Thus,at smallCuconcentrations,\n0.89<x<1.0, the alloy has a narrow FMR line indicating\nrelatively good magneto-structural uniformity. For highe r Cu\nconcentrations,0.70<x<0.89,theFMRlinebroadens,which\nindicates a higher magnetic non-uniformity. This, combine d\nwith the observed enhanced Tcfrom Ni clustering, indicates\nthat in this interesting concentration range near the metas table\nregion of the phase diagram the ﬁlms can become structurally\nnon-uniform, possibly consisting from multiple compositi onal\nphases. Belowweexaminetheoreticallythemicroscopicmec h-\nanismofphaseseparationinthesystem,takingintoaccount the\ninteratomicmagneticinteractions.\n4. Calculationofphase equilibria in Ni xCu1−xsystem\nTherearenodirectexperimentalconﬁrmationsthatmagneti c\ninteractions in Ni xCu1−xcould be responsible for additional\nphaseequilibriainthesystem. Temperaturesrequiredforr each-\ningequilibriumconditionsaredi ﬃculttoattainexperimentally,\nif at all possible. Nevertheless, it was predicted [4] that i n\nNixCu1−xalloyswith(0.86<x<0.91)ametastablemiscibility\ngap should exist due to magnetic interactions. For describi ng\nthe magnetic contribution to the free energy, the authors of [4]\n3usedamodelproposedbyChuang[12,13]. Themodelcontains\nanempiricalexpressionfortheheatcapacityofaferromagn etic\nalloy,withadjustableparametersobtainedfromtheexperi ment.\nInordertoobtainadeeperinsightintowhethersuchmiscibi lity\ngapcouldindeedexistandbeduetomagneticinteractions,i tis\nhighlydesirabletoﬁrst constructasolidstatephasediagr amof\nNixCu1−xsystem using experiment-independent ﬁrst-principle\ncalculations of interatomic interactions, and then compar e the\nresultswiththe experiment.\n/s48/s46/s55/s48 /s48/s46/s55/s53 /s48/s46/s56/s48 /s48/s46/s56/s53 /s48/s46/s57/s48 /s48/s46/s57/s53/s50/s48/s48/s51/s48/s48/s52/s48/s48/s53/s48/s48\n/s32/s32\n/s120 /s32/s32/s32/s40/s97/s116/s111/s109/s105/s99/s32/s102/s114/s97/s99/s116/s105/s111/s110 /s32/s78/s105/s41/s70/s77/s82/s32/s32/s108/s105/s110/s101/s32/s32/s104/s97/s108/s102/s45/s119/s105/s100/s116/s104/s32/s32/s40/s71/s41/s32/s50/s48/s48/s32/s75\n/s32/s50/s53/s48/s32/s75\n/s32/s51/s48/s48/s32/s75\n/s32/s51/s54/s48/s32/s75\n/s32/s51/s56/s48/s32/s75/s78/s105\n/s120 /s67/s117\n/s49/s45/s120 \nFigure 5: FMR line width versus concentration for Ni xCu1−xﬁlms, measured\nat diﬀerent temperatures.\nFor describing the Ni xCu1−xalloy we use a model Hamilto-\nnian taking into account the Ni-Ni exchange interactions. T he\nHamiltonian is similar to the one used in [14] for describing\nFexNi1−xalloys. The approach relies on a lattice model, with\neachsiteiofthefcclatticebeingoccupiedeitherbyanatomof\nNi (ci=1) or Cu ( ci=0). The atomic fraction of Ni ( x) will\nbe taken as c. The magnetic interactionsbetween the Ni atoms\nin the system are described by the classical isotropic Heise n-\nberg model. Since the local magnetic moment of Cu atoms is\nnegligible, the magnetic interactions of Cu-Cu and Cu-Ni ar e\nomitted. Thus,themodelHamiltonianofthesystemis:\nE=1\n2/summationdisplay\nijvijcicj+1\n2/summationdisplay\nijJijcicjsisj, (3)\nwherevijare the mixing potentials responsible for ”chemical”\ninteractions between the atoms, Jij– magnetic exchange con-\nstants for Ni-Ni interactions (independent of concentrati onc),\nsi–vectorofunitlengthalongthedirectionoftheNilocalmag -\nnetic moment. Values of unknowns vijfor ﬁrst 4 coordination\nshellsand Jijfor2shellswereobtainedassolutionsofasystem\nof equations (3) for a set of ordered superstructures repres ent-\ning the Ni xCu1−xalloys. The structures have various distribu-\ntions of nickel atoms and di ﬀerent magnetic ordering. Since\nthemostinterestingregionofthephasediagramisat highco n-\ncentrations of Ni, 5 superstructures of composition CuNi 7(32\natoms per unit cell, ferromagnetic ordering) and 3 structur es\nof pure Ni with ferromagnetic and antiferromagnetic orderi ng\nwere selected.Thetotalfreeenergyofthestructureswerecalculatedwith in\nthe density functionaltheory(DFT) using the FLAPW method\n(Wien2k package [15]). The calculations were performed tak -\ning into accountspin polarization,and the GGA exchangecor -\nrelationpotentialwas takento be of the formof[16]. The MT-\nradii for Cu and Ni were equal to 2.2 a.u., the electron density\nwas calculated using 500 k-points in the ﬁrst Brillouin zone .\nStructural relaxation was performed for the lattice parame ters\nand the atomic positions. The atomic positions were iterate d\nuntil the forces on the nuclei became smaller than 1 mRy /a.u.\nThe accuracy of the total energy was approximately 1 meV.\nThe calculated values of the interatomic interactions (in m eV)\nwere:vij={-16.37,32.55,-7.72,-2.52 },Jij={-6.78,-18.27}.\nThe calculation of the phase equilibria was performed withi n\nthe mean-ﬁeld approximation for the Hamiltonian of (3). All\nNi atoms were taken to have the same value of magnetic mo-\nmentm=(0,0,m).mis obtained as [17] m=L(H/kBT),\nwithH=−(/summationdisplay\njJ1j)mcbeing the eﬀective magnetic ﬁeld, kB–\nBoltzmannconstant, T–absolutetemperature, L(x)–Langevin\nfunction. Thefreeenergyisgivenbytheexpression:\nF=1\n2/summationdisplay\njv1jc2+kBT(clnc+(1−c)ln(1−c))\n+1\n2Hmc−kBTclnsinh(H/kBT)\nH/kBT. (4)\nPhase equilibria were calculated using the standard “com-\nmontangent”method. Theobtainedphasediagramofthesolid\nNixCu1−xis shown in Fig. 6. The lines shown are the bin-\nodal,spinodal,metastableequilibria,andtheCurietempe rature\nTcof the alloy versus composition. The Curie temperature be-\nlow the binodal line correspondsto a homogeneoussolid solu -\ntion. Metastable equilibria b′′d′−bdandbc−b′c′are shown\nasdashedlines.\n 450 500 550 600 650 700 750\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1T (K)\nx (atomic fraction Ni)Tc739K\n629K0.50\nspinodalspinodalb\ndb''\nd' cb'\nc'613K\n0.83a\nFigure 6: Calculated phase diagram for Ni xCu1−x(crystalline state).\nThe phase diagram contains a stable miscibility gap (lines\nab−ab′) due to magnetic transformationsin the alloy. A sim-\nilar miscibility gap exists in the Fe-Ni system (phases γ1,γ2\n4[13]). The presence of this miscibility gap in Ni xCu1−xsystem\nresults in additional equilibria of phases with close compo si-\ntions but diﬀerent magnetic propertiesand, importantly, di ﬀer-\nent Curie temperatures. The presenceofdi ﬀerentphasesin the\nalloy should broaden the magnetic transition, which is inde ed\nobservedinourNi xCu1−xﬁlms.\nPossiblyduetoourNiatomicmagneticmomentstakentobe\nindependentof composition, the obtained theoretical Tcversus\ncomposition (Fig. 6) deviates from the measured one (Fig. 4) .\nIt should also be mentioned, that in contrast to the results o f\n[4], our calculationsshow that the additional miscibility gap is\nstable.\nThus, our ab-initio numerical theoretical analysis, not re ly-\ning on experimental data, shows additional phase equilibri a in\nthe NixCu1−xsystem at high concentrationsof Ni. These addi-\ntionalequilibriaareduetotheferromagneticinteraction sinthe\nsystem.\nUsing the expressionfor the free energyof Eq. (4) it is pos-\nsible to obtain the generalconditionfor the existence of an ad-\nditional miscibility gap. For this, it is convenient to intr oduce\nthe followingquantities: V0=/summationtext\njv1j,J0=/summationtext\njJ1j. Thelinesof\nthe spinodal and magnetic transformations intersect at poi nta\n(Fig.6),whichgivestwo conditions: ∂2F/∂2c=0,T=Tc.\nThe expressions for the spontaneous magnetization near Tc,\nT<Tc:m=√5/3√(Tc−T)/Tc, and that for the Curie tem-\nperatureTc=−J0c/3k, yield the following equation for the\nconcentrationat point a:\n∂2F\n∂2c=V0−1\n3(1−ca)J0+5\n6J0=0. (5)\nThus:\nca=6+3J0/V0\n6+5J0/V0. (6)\nCondition 0<ca<1 yields for the interaction parameters:\nJ0/V0>0, This in turn gives the concentration range for the\nmiscibility gap: 3/5<ca<1. To sum up, for binary alloys\nwith decomposition ( V0<0) and with one element being fer-\nromagnetic ( J0<0), there must exist an additional miscibility\ngapdueto theinter-atomicmagneticinteractions.\nThe obtained phase diagram indicates that the magnetic and\nstructural properties of the samples should strongly depen d on\nthetemperatureoffabrication(substratetemperaturefor ﬁlms).\nSpeciﬁcally,quenchingthematerialontosubstratesisexp ected\nto be beneﬁcial for the uniformity of the ﬁlms. Here, a uni-\nform mixingof the two elementsin the magnetronplasma ﬂux\nis frozen in on the substrate before the magnetic atoms have\ntime to interact by exchange. If annealed at a su ﬃciently high\ntemperatureandthencooledslowly,theexchangeinteracti onin\nthe ﬁlm is expectedto result in atomic di ﬀusionasto favor Ni-\nclustering and therefore partial phase separation into a Ni -rich\nandCu-richphases,asdiscussedabove.\n5. Conclusions\nDilutedferromagneticalloyﬁlmsNi xCu1−xobtainedbymag-\nnetron sputtering exhibit partial phase separation into Ni -richand Cu-rich phases. The phase separation is more pronounced\nfor higher Cu dilution and leads to a broadening of the ferro-\nto paramagnetic phase transition in the system. This proces s\nis modeledusing ab-initiocalculations, takinginto accou ntthe\nmagneticinteractionsinthesystem. Ouranalysisshowstha tthe\nphaseseparationintheconcentrationrangeofinterestisd ueto\nthe Ni-Ni exchange interaction, which favors clustering of Ni\nand thereby compositional gradients in the alloy. This can a d-\nditionally lead to a signiﬁcantly non-linear dependence of the\nCurietemperatureonthealloyconcentration. Ouranalysis fur-\nthersuggeststhatthegeneralnatureoftheobservedphases ep-\narationin Ni-Cu, namelytheexchange-inducedmagneticato m\nclustering in an otherwise perfectly solvable two-compone nt\nsystem, should be found in other diluted alloys of magnetic-\nnonmagnetic or strongly magnetic-weakly magnetic element s\nandlikewiseleadto magneto-structuralinhomogeneitiest here.\nAcknowledgements\nAuthorswouldliketothanktheFP7-FET-STELEprojectfor\nprovidingﬁnancialsupportforthisstudy.\nReferences\n[1] A.M.Kadigrobov,S.Andersson,D.Radi´ c,R.I.Shekhter ,M.Jonson,and\nV. Korenivski, J.Appl. Phys.107, 123706, (2010).\n[2] S.Andersson and V.Korenivski, J. Appl. Phys.107,09D71 1, (2010).\n[3] S.Andersson and V.Korenivski, IEEETrans.Magn. 46, 214 0, (2010).\n[4] D. J. Chakrabarti, D. E. Laughlin, S. W. Chen, and Y. A. Cha ng, in\nPhase Diagrams of Binary Copper Alloys, edited by P. Subrama nian,\nD.Chakrabarti, andD.Laughlin,ASMInternational, Materi als Park,OH,\n(1994), pp. 276 – 286.\n[5] T.J. Hicks, B. Rainford, J. S.Kouvel, G. G. Low,and J. B.C omly, Phys.\nRev. Lett. 22,531, (1969).\n[6] C. G. Robbins, H. Claus, and P. A. Beck, Phys. Rev. Lett. 22 , 1307,\n(1969).\n[7] G. Iannone, D. Zola, A. A. Armenio, M. Polichetti, and C. A ttanasio,\nPhys.Rev. B 75, 064409, (2007).\n[8] A.Kidron and E.Hermon, Phys.Lett. A31, 186, (1970).\n[9] R. W. Houghton, M. P. Sarachik, and J. S. Kouvel, Phys. Rev . Lett. 25,\n238, (1970).\n[10] C. Kittel, Phys.Rev. 73,155, (1948).\n[11] R. M.Bozorth, Ferromagnetism, Wiley-VCH, (1993).\n[12] Y.-Y. Chuang, R. Schmid, and Y. A. Chang, Metall. Trans. A 16, 153,\n(1985).\n[13] Y.-Y. Chuang, Y. A. Chang, R. Schmid, and J.-C. Lin, Meta ll. Trans. A\n17, 1361, (1986).\n[14] M. B. Taylor and B. L. Gyor ﬀy, J. Magn. Magn. Mater. 104-107, 877,\n(1992).\n[15] P. Blaha, K. Schwarz, G. Madsen, D. Kvasnicka, and J. Lui tz, WIEN2k,\nAn Augmented Plane Wave +Local Orbitals Program for Calculating\nCrystal Properties, Karlheinz Schwarz, Techn. Universit¨ at Wien, Austria,\n(2001).\n[16] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett . 77, 3865,\n(1996).\n[17] K.H.J.Buschow and F.R.deBoer, Physics ofMagnetism an d Magnetic\nMaterials, Kluwer Academic Publishers, (2003).\n5"    },    {        "title": "2106.13988v3.Spin_current_injection_via_equal_spin_Cooper_pairs_in_ferromagnet_superconductor_heterostructures.pdf",        "content": "Spin current injection via equal-spin Cooper pairs in ferromagnet/superconductor\nheterostructures\nX. Montiel1,\u0003and M. Eschrig2,y\n1Department of Materials Science & Metallurgy, University of Cambridge, CB3 0FS Cambridge, United Kingdom\n2Institute of Physics, University of Greifswald, Felix-Hausdor\u000b-Strasse 6, 17489 Greifswald, Germany\n(Dated: January 19, 2023)\nEqual-spin Cooper pairs are pivotal building blocks for superconducting spintronics devices. In\nrecent experiments unusual behavior was observed in ferromagnet/ferromagnet/superconductor de-\nvices when a precession of the magnetization was induced by ferromagnetic resonance. By using a\nnon-equilibrium Usadel Green function formalism, we study spin transport for such a setup. We\nsolve for spin-resolved distribution functions and demonstrate that the spin injection process in\nsuperconductors is governed by the inverse proximity e\u000bect in the superconducting layer. We \fnd\nthat equal-spin Cooper pairs, which are produced by the two misaligned ferromagnetic layers, trans-\nport spin inside the S layer. This then results in an increase of the injected spin current below the\nsuperconducting critical temperature. Our calculations provide the \frst evidence of the essential\nrole of equal-spin Cooper pairs on spin-transport properties of S/F devices and pave new avenues\nfor the design of superconducting spintronics devices.\nIntroduction Spin transport in superconductors (S)\nis essential for the development of energy-saving spin-\nbased logic and memory devices1{3. Contrary to con-\nventional singlet Cooper pairs, composed by two elec-\ntrons with opposite spins, equal-spin cooper pairs are\ncomposed by two electrons with parallel spins, and\nare able to carry a spin polarized current through\nstrongly spin polarized ferromagnets (F)1,2. Equal-\nspin Cooper pairs can be produced from singlet Cooper\npairs by proximity e\u000bect with inhomogeneous magnetic\nstructures1{16(e.g. two ferromagnets with non colinear\nmagnetizations17{19or magnetic domains wall16,20{25),\nstrongly spin polarized ferromagnet with spin-active\ninterfaces15,26or material exhibiting spin-orbit couplingv\n(SOC)27{31. Non-equilibrium spin injection techniques\ncan be used together with measurements of transport\nproperties to characterize the presence of equal-spin\nCooper pairs in mesoscopic devices32{38. They a\u000bect spin\nand charge decoupling in superconductors36,37, spin re-\nlaxation time39{41or e\u000bective spin-orbit interaction42but\ntheir e\u000bect on spin current has not been ubiquitously es-\ntablished.\nSpin current injection in a superconductor can be\nachieved either by the injection of a spin-polarized\ncharge current in lateral structures36or by injecting pure\nspin current utilizing ferromagnetic resonance (FMR)\ntechnique43,44. In the latter, spin injection e\u000eciency\nis related to the damping of the F layer magnetization\nprecession i.e. the Gilbert damping45. FMR exper-\niments in Nb/Py/Nb Josephson junctions have shown\nthat the injected spin current magnitude (related to\nGilbert damping45) decreases below the S layer criti-\ncal temperature Tc43. Usual Andreev re\rections occur-\nring at the S/F interfaces imply the transmission of elec-\ntrons in the S layer as singlet Cooper pairs resulting in\na decrease of the injected spin-polarized current46. Nev-\nertheless, FMR experiments in Pt/Nb/Py/Nb/Pt pen-\ntalayers have shown an increase of the injected spin\ncurrent below Tc44. This increase has been explainedby the presence of equal-spin Cooper pairs originat-\ning the strong SOC and Fermi liquid corrections in the\nPt layer47. Several extrinsic sources of damping have\nbeen proposed such as spin-polarized vortices48or space-\ndependent spin susceptibility49. Recent experiments in\nPt/Co/Pt/Nb/Py/Nb/Pt/Co/Pt structure have demon-\nstrated the crucial role of SOC in the increase of spin in-\njection e\u000eciency below Tc50. An increase in the Gilbert\ndamping has been proven as possible in weakly coupled\nferromagnetic insulator/superconductor bilayers42or in\npresence of pair breaking impurities46. In the above-\nmentioned works, the increase of spin current only ex-\nists for temperatures close to Tc. While the role of\nequal-spin Cooper pairs on equilibrium spin current has\nbeen demonstrated47,50, their precise role on the non-\nequilibrium pure spin current is still lacking.\nIn this paper, we calculate non-equilibrium spin cur-\nrent in F1/F2/S devices (see Fig. 1) induced by the mag-\nnetization precession in the presence of equal-spin Cooper\npairs. In contrast to modelling the injection process for\na \fxed value of the injected spin current51,52, we will\ntake another avenue by directly modelling the precess-\ning ferromagnetic magnetization within a rotating frame\nscheme. Our calculations provide evidence that i) the\ninverse proximity e\u000bect in the S layer plays a crucial role\nin the spin injection process, and ii) the injected spin\ncurrent can increase below Tcin presence of equal-spin\nCooper pairs. Our model is consistent with the recent\nobservation of an increase of the Gilbert damping at low\ntemperature observed in Nb/Cr/Fe/Cr/Nb pentalayer53.\nModel We focus on spin-valve systems with a geometry\nshown in Fig.1 where magnetization precession is induced\nin both F1 and F2 layers at the same frequency \n. We\nconsider spin precession occurring in the x-y plane while\nmagnetization is tilted from the z-axis by the angle \u0012F1\n(\u0012F2) in F1 (F2) layer. While having the magnetization\nin the F2 layer non-precessing would certainly be closer\nto recent experiments, the model we study here has the\nadvantage of allowing for a numerically e\u000ecient treat-arXiv:2106.13988v3  [cond-mat.supr-con]  17 Jan 20232\nF1 F2 S\nFIG. 1. (Color online) Schematic geometry of the F 1/F2/S\nspin valve of thickness dF1,dF2anddSrespectively and\nL=dF1+dS+dF2.We assume that that both F layers\nmagnetization precess at the same frequency \n.\nment of the spatio-temporal spin dynamics by utilizing a\nrotating frame scheme. We model the time-dependence\nof the ferromagnetic layers magnetization by54\nJFi(t) =jJFij(sin\u0012Fisin\nt;sin\u0012Ficos\nt;cos\u0012Fi) (1)\nwherei= 1;2 andJFiis the Fi layer exchange-\feld\nstrength. To obtain non-equilibrium and non-stationary\nproperties in di\u000busive superconductors, we use time-\ndependent non-equilibrium Usadel equations54{57which\nare derived from quasi-classical equations58,59. The ex-\nchange \feld time dependency in Eq.(1) allows to trans-\nform non-stationary Usadel equations in the laboratory\nframe into stationary Usadel equations in the rotating\nframe54. Usadel formalism is formulated in terms of\nGreen function \u0014G(x;E) depending on the coordinate x\nand the energy E. The Green function \u0014Gis a 8\u00028 matrix\nin the Keldysh\nspin\nparticle-hole space where \nis the\ntensorial product and Keldysh, spin and particle-hole are\n2\u00022 subspaces (see Appendix A). The stationary non-\nequilibrium Usadel equations in the rotating frame are\n(see Appendix B)54\nD\n\u0019@x\u0000\u0014G@x\u0014G\u0001\n+\u0014\nE^\u001c3\u0000\n2\u001bZ\u0000\u0014\u0006;\u0014G\u0015\n= 0 (2)\nwhereDis the di\u000busion coe\u000ecient, \u001bZ(\u001c3) the third Pauli\nmatrix acting on the spin (particle-hole) subspace, [ ;] is\nthe commutator and \u0014\u0006 describes the normal and anoma-\nlous self-energies in the Keldysh space (see Appendix A).\nThe normal self-energy includes spin-orbit scattering pro-\ncesses (see Appendix A). Spin-orbit scattering processes\ndescribe how spin scatters on non-magnetic impurities\ndue to spin-orbit coupling42,60. The strength of these\nprocesses increases with the atomic number of the scat-tering impurity42,60. Spin-orbit scattering processes ex-\nist at all temperatures and do not a\u000bect the SC critical\ntemperature as no time-reversal symmetry breaking is\ninvolved in these processes. The superconducting self-\nenergy describes the superconducting order parameter\n(see Appendix A). Transformation into a rotating frame\ninduces the onset of an e\u000bective exchange \feld along the\nF1 layer magnetization whose amplitude is proportional\nto the magnetic \feld precession frequency \n. We rotate\nthe axis such that the F1 layer magnetization points in\nthe z-axis direction54implying all results depend on the\ndi\u000berence between the F1 and the F2 layer magnetization\nprecession angle, \u0012=\u0012F2\u0000\u0012F1. Identity matrices are not\nexplicitely written for simplicity of notation. The Usadel\nequations (2) must be supplemented by suitable bound-\nary conditions61{64. For inner interfaces, we assume cur-\nrent conserving Nazarov boundary conditions62,63:\n\u001bl\u0014Gl@x\u0014Gl=\u001br\u0014Gr@x\u0014Gr\n\u001bl\u0014Gl@x\u0014Gl=1\nSRb2\u00192\u001c[\u0014Gl;\u0014Gr]\n4\u00192\u0000\u001c(f\u0014Gl;\u0014Grg+2\u00192)(3)\nwhereGl(r)is the Green function on the left(right) side of\nthe interface, \u001bis the normal state electrical conductivity,\nS is the area of the junction, Rbis the interface resistivity\nand 0< \u001c < 1 the interface transparency63,64. For the\nouter F1 boundary at x= 0, we impose \u0014G(x= 0) = \u0014GF1\nwhere \u0014GF1is the Green function for a bulk ferromag-\nnetic material with a spin-resolved non-equilibrium dis-\ntribution function f\"(#)=fFD(E+ (\u0000)\n0\n2) withfFD\nthe Fermi Dirac distribution. At the outer S boundary\n(x=L), we impose a vanishing-current boundary con-\ndition@x\u0014G\f\f\nx=L= 0. The singlet-superconducting order\nparameter is \fxed by the self-consistency equation\n\u0001 (x) =R1\n\u00001dE\n4i\u0019fK\ns(E;x)R1\n\u00001dE\n2Etanh\u0000E\n2T\u0001\n+ ln\u0000T\nTc\u0001 (4)\nwherefK\nsis the singlet part of the Keldysh anomalous\nGreen function (see Appendix A). In the following, we\ncalculate spin current densities from the following for-\nmula:\nIs=I0\nsZ+1\n\u00001dETrh\n^\u001c3\u001b\u0000\u0014G@x\u0014G\u0001Ki\n(5)\nwithI0\ns=~N0D\n16\u00192L,N0the Fermi level density of state,\nethe electrical charge, ~the reduced Planck constant,\nLthe sample total thickness, and \u001b=\u0000\n\u001bX;\u001bY;\u001bZ\u0001\nthe Pauli matrix vector. The spin current vector Isis\ngiven byIs=\u0000\nIX\ns;IY\ns;IZ\ns\u0001\nin the Pauli matrix basis. A\nspin-dependent distribution function f\"(#)at the outer\nF1 boundary implies the onset of a pure spin current to\n\row in the trilayer while the charge current vanishes since\nno electric potential is applied36. The pure spin current\nis polarized along the F1 layer magnetization IZ\ns. In ab-\nsence of magnetization precession (\n = 0), we recover\nthe equilibrium solution and IZ\ns;eq= 0.3\nThe Gilbert damping Gtcan be expressed in the form\nGt=G0+G\u0011G0+\fIZ\ns (6)\nwhereG0describes an intrinsic Gilbert damping inde-\npendent of temperature and superconducting properties,\nandGdescribes the additional Gilbert damping due to\nspin injection; the latter is proportional to the dissipa-\ntive part of the spin current52, which in itself is propor-\ntional toIZ\nsatx= 0 with a coe\u000ecient depending on\nthe tip angle. The quantity \u000eG=GT>T c=\u000eIZ\ns=IZ\ns;T>T c\nwith\u000eIZ\ns=IZ\ns\u0000IZ\ns;T>T cis independent of the tip-angle\ndependent quantity \fand therefore quanti\fes the addi-\ntional Gilbert damping in our system.\nWe assume that spin di\u000busion originates from\nspin-orbit scattering processes which only a\u000bect\nspin-triplet correlations without a\u000becting spin-singlet\nsuperconductivity42,46,47,60,65,66. We further assume\nthat the FMR process in the F1 layer compensates spin\ndi\u000busion processes i.e. the spin di\u000busion length in the\nF1 layer is in\fnite, \u0015F1!1 . Unless otherwise stated,\nthe magnitude of the F1 and F2 layer exchange \felds\nisJF1=JF2= 20\u0001 0, the spin-di\u000busion length of the\nF2 and S layer is \u0015F2=\u0015S=\u00180with\u00180=q\nD\n\u00010the\nsuperconducting coherence length in bulk S at zero\ntemperature, and we consider \u00180= 30nm in Niobium.\nIn the following, we set the resonance frequency at\nthe experimentally measured value fres= 20GHz43,44\nimplying that ~\n\u00190:1\u00010.\nResults The spin current pro\fle in the F1/F2/S tri-\nlayer for collinear magnetizations ( \u0012= 0) is presented\nin Fig. 2. The magnitude of the spin current decays in\nthe F2 and S layers because of spin-orbit scattering pro-\ncesses. The magnitude of the spin current is higher in the\nnormal state than in the superconducting state because\nof the opening of the superconducting gap. The spin cur-\nrent is constant at the inner interfaces as expected from\nboundary conditions (3). The magnitude of the spin cur-\nrent strongly depends on the inverse proximity e\u000bect and\nthe S layer thickness dS, which a\u000bects the Gilbert damp-\ning parameter \u000eG=GT>T cas shown in Fig. 2 b). For\nsmall S layer thicknesses, dS< \u00180, the Gilbert damping\nparameter is the same above and below Tcsince in both\ncases the superconducting gap vanishes at the F2/S in-\nterfaces (except in the regime \u001bS=\u001bF1= 10) as shown\nin Fig. 2 c). Note that for dS= 0, the Gilbert damp-\ning parameter does not vanish since spins are absorbed\nin the F2 layer. For a thick S layer, dS\u001d\u00180=\u0015S,\nthe Gilbert damping parameter above and below Tcbe-\ncomes constant since the spin is massively absorbed in\nthe S layer close to the F2/S interface on the spin di\u000bu-\nsion length scale \u0015S. In the intermediate regime, dS\u0019\u00180,\nthe Gilbert damping parameter below Tcbecomes sightly\ndi\u000berent from above Tcin conjunction with the supercon-\nducting gap opening at the F2/S interfaces as shown in\nFig. 2 c). The inverse proximity e\u000bect can be tuned\ntheoretically by changing the normal state conductivity\nratio\u001bS=\u001bF1(with\u001bF2=\u001bF1)47. For a weak inverse\na)\nb) c)\n0\n1\n2\n3\n4\n5\n0.00\n0.25\n0.50\n0.75\n∆/∆0\ndS/ξ0\n0.0\n0.5\n1.0\n1.5\n2.0\n0.000\n0.025\n0.050\nT=0.01Tc0\nT=1.05Tc0\nx/ξ0\nI\nZS(x)/I\n0s\n0\n1\n2\n3\n4\n5\n-0.75\n-0.50\n-0.25\n0.00\nσS�σF1=0.1\nσS�σF1=1\nσS�σF1=10\nδG/G\ninfT>Tc\ndS/ξ0s F1 F2FIG. 2. (Color online) a) Spin current pro\fle IZ\ns(x) in the\nF1/F2/S trilayer (in blue, orange and green respectively) at\nlow temperature T= 0:01Tc0(solid line) and in the normal\nstateT >T c0(dashed line) for \u0012F2= 0. The layer thicknesses\naredF1=\u00180,dF2= 0:1\u00180, anddS= 2\u00180whileSRb= 1 and\n\u001c= 1. The vertical lines mark the position of the F1/F2 and\nF2/S interfaces. b) Gilbert damping parameter \u000eG=G T>Tc\nat the outer F1 layer boundary as a function of the S layer\nthicknessdSfor various S layer conductivities (see color leg-\nend) below and above Tc(solid and dashed line respectively)\nc) Magnitude of the superconducting gap at the F2/S inter-\nface as a function of the S layer thickness dSfor various S\nlayer conductivities.\nproximity e\u000bect i.e. \u001bS=\u001bF1= 10, the superconducting\ngap \u0001 is fully established at the F2/S interfaces. In the\nregime ~\n<\u0001, spin currents cannot \fnd any states to\npropagate further in the S layer. Only singlet Andreev\nre\rections occur at the F2/S interface implying a decay\nof the Gilbert damping below Tc43,46. For a strong in-\nverse proximity e\u000bect i.e. \u001bS=\u001bF1= 0:1, the supercon-\nducting gap is strongly suppressed at the F2/S interface\nimplying that non-equilibrium spin current can be in-\njected in the S layer. Therefore, the Gilbert damping\nrecovers the same magnitude above and below Tc. The\ndependence of the Gilbert damping on the inverse prox-\nimity e\u000bect explains why it does not necessarily vanishes4\na)\nb) c)\n0.00\n0.25\n0.50\n0.75\n1.00\n-0.4\n-0.2\n0.0\nθF2=0\nθF2=π/4\nθF2=π/2\nδG/GT>Tc\nT/Tc0\nFIG. 3. (Color online) a) Temperature dependence of the\nGilbert damping parameter \u000eG=G T>TcforT= 0:01Tcfor\nvarious F2 layer magnetization tilting angles \u0012= 0 (black),\n\u0012=\u0019=4 (red),and \u0012=\u0019=2 (blue) and dS= 4\u00180. b) Gilbert\ndamping parameter \u000eG=G T>Tcas a function of \u0012F2for di\u000ber-\nent S layer thicknesses dS.\u0012cis the F2 layer magnetization\nangle where \u0001 IZ\nschanges its sign. c) Gilbert damping pa-\nrameter\u000eG=G T>Tcas a function of \u0012F2for various S layer\nconductivities \u001bSanddS= 4\u00180. The other parameters are\nthe same as in Fig. 2\nat zero temperature43,46.\nThe Gilbert damping varies with temperature as shown\nin Fig. 3 a). This dependency strongly depends on the\nmisalignment angle between the F1 and F2 layer mag-\nnetization\u0012. The Gilbert damping magnitude decreases\nbelow the critical temperature for \u0012= 0 and\u0012=\u0019=4\nwhile it increases for \u0012=\u0019=2 as shown in Fig. 3 a). In\nthis case, an additional damping torque appears below\nTcby the onset of equal-spin Cooper pairs. The angle\ndependency of the Gilbert damping is shown in Fig. 3\nb). For\u0012\u0019\u0019=2, the Gilbert damping is higher at zero\ntemperature, \u000eG=GT>T c>0 while it is smaller for \u0012<\u0012c\nwhere\u0012cis the angle where \u000eGchanges its sign (see Fig.\n3 b). The Gilbert damping is weakly a\u000bected by the\nS layer thickness and becomes constant for dS> \u00180as\nb) a)FIG. 4. (Color online) a) Gilbert damping parameter\n\u000eG=G T>Tcas a function of the interfaces transparencies \u001cfor\nvarious exchange \feld magnitudes and \u0012=\u0019=2. b) Gilbert\ndamping parameter \u000eG=G T>Tcas a function of the interface\ntransparencies \u001cfor exchange \feld J= 10\u0001 0and various S\nlayer conductivities, and \u0012=\u0019=2. The other parameters are\nthe same as in Fig. 2\nshown in Fig. 2 b). Nevertheless, the Gilbert damping\nis a\u000bected by the inverse proximity e\u000bect and the value\nof\u0012cdepends on the quality of the interfaces (see Fig. 3\nc). The generation process for equal-spin Cooper pairs is\na\u000bected by the value of the superconducting gap at the\nF2/S interface which depends on the conductivity ratio\n\u001bS=\u001bF2.\nThe Gilbert damping parameter \u000eG=GT>T cat\u0012=\u0019=2\ndepends non-monotonically on the interfaces transparen-\ncies and exchange \feld amplitude as shown in Fig. 4 a).\nFor small transparencies and exchange \feld J\u001410\u0001 0,\nthe Gilbert damping amplitude is increased below the\ncritical temperature \u000eG=GT>T c>0 while it decreases\n\u000eG=GT>T c<0 for transparencies close to 1 (see Fig.\n4 a). For higher exchange \feld amplitude, J= 20\u0001 0,\nthe injected spin current is increased for all transparen-\ncies. This non-monotonic behavior depends on the in-\nverse proximity e\u000bect as shown in Fig. 4 b). For an ex-\nchange \feld J= 10\u0001 0, the Gilbert damping is reduced\nfor transparency close to 1 when the inverse proximity\ne\u000bect is weak \u001bS=\u001bF1= 10 while it increases for strong\ninverse proximity e\u000bect \u001bS=\u001bF1= 0:1\nDiscussion In an S/F/S Josephson junction, the pre-\ncession of the F layer magnetisation should produce\nequal-spin Cooper pairs originating the misalignment be-\ntween the non-equilibrium magnetization and the F layer\nmagnetization54. This e\u000bect strongly depends on the\nmisalignment angle value and must be negligible when\nit tends to zero. In our calculation, this e\u000bect is negligi-\nble since\u0012F1!0.\nThe behavior of the Gilbert damping below Tccan be\nunderstood as a competition between a decrease origi-\nnating from standard Andreev re\rection processes43,46\nand spin-\rip Andreev re\rection processes31,32. In the\nlatter, one electron is transmitted in the S layer as an\nequal-spin Cooper pair while a hole with the same spin\nis retro-re\rected31,32. We expect this process to exist\nat interfaces with SOC31or spin-polarization32,64. For\nsmall angle, \u0012<\u0012c, the standard Andreev re\rection pro-5\ncess dominates leading to a decrease of the injected spin\ncurrent while for \u0012 > \u0012cthe spin-\rip Andreev re\rection\nprocess is dominant.\nExperimentally, we expect this e\u000bect to be observable\nin interfaces with inhomogeneous spin-polarisation as in\nFe/Cr interfaces67{70.\nIn our calculation, the precession of the F2 layer mag-\nnetization implies that spin may be injected back to\nthe F1 layer reducing the Gilbert damping. We expect\nthat in the presence of a static F2 layer magnetization,\nthe Gilbert damping increase below Tc should be even\nhigher.\nWe can estimate the Gilbert damping from the cal-\nculated spin current density. In equation (5), the spin\ncurrent density is I0\ns\u00191010eVcm\u00002 considering a total\nsample thickness of 2 :15\u00180, a Fermi level density of states\nN0\u00191022eV\u00001cm\u00003and a di\u000busive coe\u000ecient D=\u00182\n0\u00010\n~\nwith \u0001 0= 1:4meV the bulk Niobium superconducting\ngap at zero temperature. In Eq.(6), the proportionality\nfactor is\f=\r0\nMsd F1fres52with\r0the gyromagnetic ra-\ntio andMsthe magnetization saturation. Considering\n\r0\u0019107s\u00001T\u00001andMs\u00191014eVT\u00001cm\u00003, we obtain\n\f\u00191:6\u000210\u000011eV\u00001cm2. From the value of the current\nin the F1 layer plotted in the Fig. 2, we can estimate the\nGilbert damping produced by the injected spin current\nin Eq.(6):G=\fIsZand we \fnd GT=0\u00190:007 at zero\ntemperature and GT>T c\u00190:01 above Tc. These values\nare of the same order of magnitude as the ones measured\nin Pt/Nb/Py/Nb/Pt44and Nb/Cr/Fe/Cr/Nb53penta-\nlayers.\nConclusion We provide the \frst theoretical evidence\nthat equal-spin Cooper pairs can enhance the injected\nspin current in superconducting nanostructures below\nthe superconducting critical temperature. We anticipate\nthat such proof will play a crucial role in the interpre-\ntation of forthcoming experiments and will in\ruence fur-\nther developments for applications in the \feld of super-\nconducting spintronics.\nAppendix A: Green function and Self-energy\nstructure in the Keldysh formalism\n1. Green function and self-energy general structure\nin Keldysh space\nGreen's functions in quasiclassical theory of supercon-\nductivity exhibit internal and external degrees of free-\ndom. The external degrees of freedom describe the mo-\ntion in space (either ballistic along quasiclassical trajec-\ntories, or di\u000busive in the case of Usadel theory), while the\ninternal degrees of freedom (2 \u00022 spin and 2\u00022 particle-\nhole degrees of freedom) are discrete in nature and are\nrepresented by a matrix structure of the Green's func-\ntion. In non-equilibrium the powerful Keldysh formal-\nism adds a further 2 \u00022 matrix structure. In the 8 \u00028\nspin\u0002particle-hole\u0002Keldysh space, the Green's functionsexhibit the following overall matrix structures5,57:\n\u0014G=\u0012^GR^GK\n0^GA\u0013\n^GR;A=\u0012gR;AfR;A\nefR;AegR;A\u0013\n^GK=\u0012gKfK\n\u0000efK\u0000egK\u0013 (A1)\nwhere \u0014:::corresponds to functions written in the full 8 \u00028\nKeldysh\nspin\nparticle-hole space ( \nis the tensorial\nproduct), and ^ :::corresponds to the 4 \u00024 spin\u0002particle-\nhole space. The symbol e:::combines a complex conjuga-\ntion with the change E!\u0000Ein the energy argument,\ni.e., ~f(E) =f\u0003(\u0000E). In the quasiclassical approach,\nboth in the ballistic regime (Eilenberger formalism) and\nin the di\u000busive regime (Usadel formalism), the Green's\nfunction ful\flls the normalization condition :\n\u0014G:\u0014G=\u0000\u00192\u00141: (A2)\nIn both Eilenberger and Usadel equations, there appear\nself-energies whose matrix structure in Keldysh space is\ngiven by\n\u0014\u0006 =\u0012^\u0006R^\u0006K\n0^\u0006A\u0013\n^\u0006R;A=\u0012\u0006R;A\u0001R;A\ne\u0001R;Ae\u0006R;A\u0013\n^\u0006K=\u0012\u0006K\u0001K\n\u0000e\u0001K\u0000e\u0006K\u0013\n(A3)\nwhere \u0014\u0006 is the self energy written in the full 8 \u00028\nKeldysh\nspin\nparticle times space, while ^\u0006 are self-\nenergies written in the spin \nparticle-hole space. \u0006 and\n~\u0006 are normal self-energies in the 2 \u00022 spin space while \u0001\nand ~\u0001 are anomalous self-energies in the spin space. In\nthe following, we refer to the 2 \u00022 spin space through\nthe unit matrix \u001b0and the three spin Pauli matrices\u0000\n\u001bX;\u001bY;\u001bZ\u0001\n, while we refer to the 2 \u00022 particle-hole\nspace via the unit matrix \u001c0and the three Pauli matrices\n(\u001c1;\u001c2;\u001c3).\n2. Self energies in the Usadel equation 2\nThe self-energy \u0014\u0006 including the single-particle terms\ndescribing the exchange splitting of spin bands in the\nferromagnet writes in the full Keldysh space\n\u0014\u0006 = \u0014\u0006imp+\u0014\u0006ex+\u0014\u0001\nwhere \u0014\u0006imp=\u0012^\u0006imp;R ^\u0006imp;K\n0 ^\u0006imp;A\u0013\nis the self-energy pro-\nduced by the spin-\rip on magnetic impurities and spin-\norbit scattering, \u0014\u0006ex=\u0012^\u0006ex;R ^\u0006ex;K\n0 ^\u0006ex;A\u0013\ndescribes the\nspin-splitting of the energy bands produced by the ex-\nchange \feld in the F layer, and \u0014\u0001 =\u0012^\u0001R^\u0001K\n0^\u0001A\u0013\nis the\nself energy describing singlet superconductivity. Note\nthat ^\u0006ex;K=^\u0001K= 0 while in general ^\u0006imp;K6= 0.6\nIn the following, we work only in the retarded subspace\nwhere ^\u0006R=^\u0006imp;R+^\u0006ex;R+^\u0001Rwith\n^\u0006imp;R=\u0012\u0006imp\u0001imp\ne\u0001impe\u0006imp\u0013\n^\u0006ex;R=\u0012\u0006ex0\n0e\u0006ex\u0013\n^\u0001R=\u00120 \u0001SC\ne\u0001SC0\u0013\na. Spin-\rip impurities self-energy\na. Spin \rip due to magnetic impurities The spin-\rip\nscattering self-energy due to magnetic impurities writes\nin the Keldysh space as\n\u0014\u0006m=1\n8\u001cm^\u001b:\u00141:\u0014G:^\u001b:\u00141\nwhere ^\u001bis the spin Pauli matrice vector in the Nambu-\nspin space ^\u001b=\u0012\n\u001b0\n0\u001b\u0003\u0013\n.\nb. Spin \rip due to spin-orbit scattering The spin-\norbit scattering self-energy writes in the Keldysh space\nas\n\u0014\u0006SO=1\n8\u001cso^\u001b:\u00141:^\u001c3:\u0014G:^\u001c3:^\u001b:\u00141:\nb. Exchange \feld\nIn the ferromagnet, we consider an exchange \feld ori-\nented along the zaxis which can be included by formally\nintroducing a self-energy term as\n^\u0006ex;R=\u0012\nJ\u001b 0\n0J\u001b\u0003\u0013\nwhereJis the exchange \feld and \u001bis the vector of Pauli\nmatrices.\nThe symmetries between advanced and retarded func-\ntions are ^\u0006ex;A=\u0010\ne\u0006ex;R\u0011y\n=\u0012\nJ\u001b 0\n0J\u001b\u0003\u0013\n=^\u0006ex.\nc. Superconducting self-energy\nIn a spin-singlet superconductor, the order parameter\nis given by\n^\u0001R=\u00120 \u0001SC\ne\u0001SC0\u0013\nwhere \u0001SC=i\u001by\u0001ei\u001ewith\u001ethe superconducting phase.\nThe symmetries between advanced and retarded func-\ntions are \u0001A=\u0000\u0010\ne\u0001R\u0011y\n=\u00120 \u0001SC\ne\u0001SC0\u0013\n=^\u0001R. Thesinglet superconductivity order parameter is \fxed by the\nself-consistency equation :\n\u0001 (x) =R1\n\u00001dE\n4i\u0019fK\ns(E;x)R1\n\u00001dE\n2Etanh\u0000E\n2T\u0001\n+ ln\u0000T\nTc\u0001 (A4)\nwherefK\nsis the singlet part of the Keldysh anomalous\nGreen function.\nAppendix B: Description of The FMR process with\ntime dependent Usadel equations: From the\nlaboratory frame to the rotating frame\n1. Time dependent Usadel equation\nThe time-dependent Usadel equation in the Keldysh\nspace is given by\ni\u0000\n^\u001c3@t1\u0014G+@t2\u0014G^\u001c3\u0001\n+D\n\u0019rR\u0002\u0014G\u000erR\u0002\u0014G\u0003\u0003\n\u0000\u0002\u0014\u0006;\u000e\u0014G\u0003\n= 0\n(B1)\nwhere \u0014G=\u0014G(t1;t2;R) witht1andt2the time coordi-\nnates and Rthe space coordinate. The symbol \u000edenotes\nthe time convolution product de\fned as :\nA\u000eB(t1;t2) =Z1\n\u00001dt0A(t1;t0)B(t0;t2)\nSolution of Usadel's equation with a time dependence\nis complicated by the evaluation of these time convo-\nlution products. Note that in the above equation, the\nself-energy \u0014\u0006 is time-dependent too. For our purpose\nit is su\u000ecient to consider the case where \u0014\u0006 (t1;t2) =\n\u0014\u0006 (t1)\u000e(t1\u0000t0).\n2. Exchange \feld time dependency in the\nlaboratory frame\nConsidering the ferromagnetic resonance process, we\nmust take into account the time dependency of the F\nlayer exchange \feld. Close to the resonance, we consider\nthat the magnetization precesses around an e\u000bective \feld\ndirection. Assuming an e\u000bective \feld directed along the\nz-axis, the time-dependency of the exchange \feld is given\nby\nh(t) =jhj(sin(\u0012)sin(\nt);sin(\u0012)cos(\nt);cos(\u0012))\n(B2)\nwheretis the time, \u0012is the tilting angle from the zaxis\nand \n is the precession frequency. Note that for t= 0,\nthe magnetization is tilted from the zaxis towards the y\naxis by the angle \u0012. In the Usadel's equation (B1), a term\ndue to the exchange \feld is present. One can separate the\ncorresponding self-energy term \u0014\u0006 (t) in a time-dependent\nand time-independent part \u0014\u0006 (t) =\u0014h(t) +\u0014\u00060where \u0014h(t)\nis the self-energy associated with the exchange \feld. The7\nself-energy \u0014hexhibits the symmetry in the Keldysh space\n:\n\u0014h=\u0012^h0\n0^h\u0013\n;^h=\u0012\nh\u001b0\n0h\u001b\u0003\u0013\n(B3)\nwhere\u001bis the Pauli matrix vector in spin space. The\nUsadel's equation (B1) then reads\ni\u0000\n^\u001c3@t1\u0014G+@t2\u0014G^\u001c3\u0001\n+D\n\u0019\u000erR\u0002\u0014GrR\u0002\u0014G\u0003\u0003\n\u0000\u0002\u0014h(t);\u000e\u0014G\u0003\n\u0000\u0002\u0014\u00060;\u0014G\u0003\n= 0(B4)\nNote that the convolution product disappears from the\nlast term of the equation because the self-energy \u0014\u00060is\ntime-independent.\n3. From the laboratory frame to the rotating frame\nAssuming the time-dependency of the exchange \feld\ndescribed in Eq. (B2), we de\fne a unitary transforma-\ntion which transforms the time-dependent Usadel equa-\ntion in the laboratory frame to a time-independent Us-\nadel equation in the rotating frame. This transforma-\ntion is possible only if the exchange \feld exhibits the\ntime dependency shown in the formula (B2). For another\ntime-dependency, one must solve the time dependent Us-\nadel equation. We can de\fne this unitary transformation\nthrough the unitary operator \u0014Uwhich has the following\nstructure in the Keldysh space,\n\u0014U(t) = \n^U(t) 0\n0^Uy(t)!\n;^U=\u0012\nU0\n0U\u0003\u0013\n(B5)\nwith the operator U=e\u0000i\u001bZ\nt\n2where\u001bZis the third\nPauli matrix. In spin space, the transformation operator\nis given by U=cos\u0000\nt\n2\u0001\n\u0000i\u001bZsin\u0000\nt\n2\u0001\n. Unitarity of the\ntransformation imposes that \u0014U(t1)\u0014Uy(t2) =\u000e(t2\u0000t1)\nwhere\u000eis the Dirac distribution. From this transforma-\ntion, one can relate the Green's function in the rotating\nframe\u0016\u0014Gto the Green's function in the laboratory frame\n\u0014Gvia the transformation \u0014Uby\n\u0016\u0014G(t1;t2;R) =\u0014U(t1)\u0014G(t1;t2;R)\u0014Uy(t2) (B6)\nApplying this transformation to the Usadel equation (B4)\n(multiplying on the left by \u0014U(t1) then on the right by\n\u0014Uy(t2) and considering the unitary relation of \u0014U), we\n\fnd\ni\u0010h\n@t2\u0014G^\u001c3\u00141 + ^\u001c3\u00141@t1\u0014Gi\u0011\n+D\n\u0019rRh\u0016\u0014G\u000erRh\u0016\u0014Gii\n\u0000h\n\n2\u001bz\u00141;\u0014Gi\n\u0000h\n\u0014heff;\u0016\u0014Gi\n\u0000h\n\u0014\u00060;\u0016\u0014Gi\n= 0(B7)\nwhere \u0014heff=\u0014U\u0014h\u0014Uyis the exchange \feld in the ro-\ntating frame with the same structure as described in\nEq.(B3) with a time-independent exchange \feld heff=\njhj(0;sin(\u0012);cos(\u0012)). Working in the rotating frame im-\nposes an additional exchange \feld along the zdirectionthe intensity of which is proportional to \n =2. This ad-\nditional term is produced by the transformation of the\ntime derivative term of Eq.(B1). This term reads\n\u0014U(t1)\u0002\ni\u0000\n^\u001c3@t1\u0014G+@t2\u0014G^\u001c3\u0001\u0003\u0014Uy(t2)\nand reduces to\ni\u0010\n^\u001c3@t1\u0016\u0014G+@t2\u0016\u0014G^\u001c3\u0011\n\u0000\u0014\n2\u001bz\u00141;\u0016\u0014G\u0015\n:\nThe source term in the Usadel equation (B7) (the com-\nmutator term) does not depend on time which implies\nthat the Green's function only depends on the time dif-\nference\u000et=t1\u0000t2,\u0016\u0014G(t1;t2;R) =\u0016\u0014G(\u000et;R). We then\nconsider the Fourier transform:\n\u0014G(\u000et;R) =Z\ndEG (E;R)eiE\u000et:\nApplying this Fourier transform to the Usadel equation\n(B7), we \fnd\nD\n\u0019rRh\u0016\u0014GrRh\u0016\u0014Gii\n+\u0014\nE^\u001c3\u0000\n2\u001bz\u00141\u0000\u0014heff\u0000\u0014\u00060;\u0016\u0014G\u0015\n= 0:\n(B8)\nThis time-independent Usadel equation (B8) describes\nthe superconducting physics in the rotating frame. In\norder to return back to the description in the laboratory\nframe, one has to apply the inverse transformation onto\nthe Green's function.\nAppendix C: Usadel's equations in the gamma\nparametrization\nHere we present the Usadel equation for the matrices\n\rand ~\r. We start with the Usadel equation\nDrR[\u0014grR[\u0014g]] +i\u0002\nE^\u001c3\u0000\u0014\u0006;\u0014g\u0003\n= 0 (C1)\nwhich is the same than the equation (B8), where for sim-\nplicity we write\u0016\u0014G=\u0014G,\u0014G=\u0000i\u0019\u0014gand\u0014\u0006 =\n2\u001bz\u00141+\u0014heff+\n\u0014\u00060. The Usadel equation (C1) is divided in two distinct\nterms : the spatial derivative term rR[\u0014grR[\u0014g]] and the\nnon-derivative term\u0002\u0014\u0006;\u0014g\u0003\n. It is convenient parameterize\nthe Green's function such that it already ful\flls the nor-\nmalisation condition (A2). Here, we use the Riccatti ma-\ntrix parametrization57where the Green's functions are\ngiven in 4\u00024 spin\u0002particle-hole space by\n^GK=\u00002i\u0019:^NR:\u0012 \u0000\nx\u0000\rR:~x:e\rA\u0001\n\u0000\u0000\n\rR:ex\u0000x:\rA\u0001\n\u0000\u0000\ne\rR:x\u0000ex:e\rA\u0001 \u0000\n~x\u0000e\rR:x:\rA\u0001\u0013\n:^NA\n(C2)\nand\n^GR;A=\u0007i\u0019:^NR;A:\u0012\n1 +\rR;Ae\rR;A2\rR;A\n\u00002e\rR;A\u0000\u0000\n1 +e\rR;A\rR;A\u0001\u0013\n:\n(C3)8\n1. Usadel's equation for retarded Green's functions\nIn this section, we focus on the Usadel equation for\nretarded component which reads\nDr\u0002\n^gRr\u0002\n^gR\u0003\u0003\n+ih\nE^\u001c3\u0000^\u0006R;^gRi\n= 0: (C4)\nIt leads to the equation\nDh\nr2\rR+ 2r\rReNRe\rRr\rRi\n+i\u0010\n2E\rR\u0000\u0006R\rR+\rRe\u0006R+ \u0001R\u0000\rRe\u0001R\rR\u0011\n= 0\n(C5)\nwhich corresponds to a di\u000berential equations for \rR.\nSolving the equations for \rRand ~\rRcan be achieved\nby numerical methods as for example relaxation meth-\nods. From the solution, one uses Eq. (C3) to build\nthe retarded Green's function. Using the symmetries be-\ntween retarded and advanced Green's functions described\nin Eq.(D1), we can derive the expression for \rAand ~\rA\nand construct the advanced Green's function.\n2. Usadel's equation for Keldysh Green's function:\nquantum kinetic equation\nTo calculate the properties of di\u000busive superconduct-\ning nanostructures, we consider the equation for non-\nequilibrium distribution functions i.e. the quantum ki-\nnetic equation. In this section, we derive the kinetic equa-\ntion for the distribution function xand ~xfrom Eqs.(C2).\na. Distribution function in the Riccatti matrices\nparametrization\nThe choice for the the distribution functions xand ~xis\nnot unique57. The choice of the functions xand ~xleads\nto a simpli\fcation of the kinetic equations. These distri-\nbution functions are related to the distribution function\nhand~hintroduced by Larkin and Ovchinikov as57by\nh=n=1X\nn=0h\u0000\n\rR~\rR\u0001n\u000e\u0000\nx\u0000\rR\u000e~x\u000e~\rA\u0001\n\u000e\u0000\n\rA~\rA\u0001ni\nwhere ~hcan be deduced by applying the ~ :::transforma-\ntion to the function h. The distribution function hcan\nbe related to the distribution functions for electrons andholesfand ~fby the relation\nf=1\n2(1\u0000h)\n~f=1\n2\u0010\n1 +~h\u0011\nFrom the resolution of the kinetic equation, we can cal-\nculate the space dependency of the distribution function\nin the system.\nb. The kinetic equations for the distribution functions x\nand~x\nWe derive the kinetic equations from the Keldysh part\nof the Usadel equation (B8).\na. Full kinetic equation We \fnd the kinetic equation\nin the form5,57:\nDn\nr2xK+ 2r\rReNRe\rRrxK+ 2rxKNA\rAre\rA\n\u00002r\rReNR\u0002\nexK\u0000e\rRxK\rA\u0003eNAre\rAo\n+i\u0010\n\u0000\u0006R\u0000\rRe\u0001R\u0011\nxK+ixK\u0000\n\u0006A\u0000\u0001Ae\rA\u0001\n+i\u0010\n\u0006K\u0000\u0001Ke\rA\u0000\rRe\u0001K+\rRe\u0006Ke\rA\u0011\n= 0\n(C6)\nThe kinetic equation for ~ xKcan be obtained by apply-\ning the projection ^PR\n\u0000(:::)^PA\n+of the Usadel equation or\napplying the ~ :::transformation to Eq. (C6).\nAppendix D: Symmetries in Keldysh-space\nThe following symmetries connect retarded compo-\nnents with advanced components and express symmetries\nof Keldysh components:57:\n\rA=\u0000\ne\rR\u0001y\u0001A=\u0000\u0010\ne\u0001R\u0011y\n\u0006A=\u0010\ne\u0006R\u0011y\nx= (x)y\u0001K=\u0010\ne\u0001K\u0011y\n\u0006K=\u0000\u0000\n\u0006K\u0001y\n(D1)\nACKNOWLEDGMENTS\nThe authors acknowledge M. Blamire, A. K. Chan, L.\nCohen, K.-R. Jeon, H. Kurebayashi, and J. Robinson for\nfruitful discussions. This work has been supported by\nthe EPSRC Programme Grant EP/N017242/1.\n\u0003xm252@cam.ac.uk\nymatthias.eschrig@uni-greifswald.de\n1M. Eschrig, Phys. Today 64, 43 (2011).\n2M. Eschrig, Rep. Prog. Phys. 78, 104501 (2015).\n3J. Linder and J. W. A. Robinson, Nat. Phys. 11, 307\n(2015).4Y. A. Izyumov, Y. N. Proshin, and M. G. Khusainov,\nPhysics-Uspekhi 45, 109 (2002).\n5M. Eschrig, J. Kopu, A. Konstandin, J. C. Cuevas, M. Fo-\ngelstr om, and G. Sch on, in Advances in Solid State Physics\n(Springer, 2004) pp. 533{545.9\n6A. A. Golubov, M. Y. Kupriyanov, and E. Il'ichev, Rev.\nMod. Phys. 76, 411 (2004).\n7F. S. Bergeret, A. F. Volkov, and K. B. Efetov, Rev. Mod.\nPhys. 77, 1321 (2005).\n8A. I. Buzdin, Rev. Mod. Phys. 77, 935 (2005).\n9I. F. Lyuksyutov and V. L. Pokrovsky, Adv. Phys. 54, 67\n(2005).\n10M. G. Blamire and J. W. A. Robinson, J. Phys. Condens.\nMatter 26, 453201 (2014).\n11N. O. Birge, Philos. Trans. Royal Soc. A 376(2018).\n12T. Tokuyasu, J. A. Sauls, and D. Rainer, Phys. Rev. B\n38, 8823 (1988).\n13M. Fogelstr om, Phys. Rev. B 62, 11812 (2000).\n14Y. S. Barash and I. V. Bobkova, Phys. Rev. B 65, 144502\n(2002).\n15M. Eschrig, J. Kopu, J. C. Cuevas, and G. Sch on, Phys.\nRev. Lett. 90, 137003 (2003).\n16F. S. Bergeret, A. F. Volkov, and K. B. Efetov, Phys. Rev.\nLett. 86, 4096 (2001).\n17T. L ofwander, T. Champel, J. Durst, and M. Eschrig,\nPhys. Rev. Lett. 95, 187003 (2005).\n18M. Houzet and A. I. Buzdin, Phys. Rev. B 76, 060504\n(2007).\n19K. Halterman, O. T. Valls, and P. H. Barsic, Phys. Rev.\nB77, 174511 (2008).\n20T. Champel and M. Eschrig, Phys. Rev. B 71, 220506\n(2005).\n21T. Champel and M. Eschrig, Phys. Rev. B 72, 054523\n(2005).\n22Y. V. Fominov, A. F. Volkov, and K. B. Efetov, Phys.\nRev. B 75, 104509 (2007).\n23B. Crouzy, S. Tollis, and D. A. Ivanov, Phys. Rev. B 76,\n134502 (2007).\n24A. I. Buzdin, A. S. Mel'nikov, and N. G. Pugach, Phys.\nRev. B 83, 144515 (2011).\n25J. N. Kupferschmidt and P. W. Brouwer, Phys. Rev. B 80,\n214537 (2009).\n26M. Eschrig and T. L ofwander, Nat. Phys. 4, 138 (2008).\n27G. Annunziata, D. Manske, and J. Linder, Phys. Rev. B\n86, 174514 (2012).\n28F. S. Bergeret and I. V. Tokatly, Phys. Rev. Lett. 110,\n117003 (2013).\n29F. S. Bergeret and I. V. Tokatly, Phys. Rev. B 89, 134517\n(2014).\n30S. H. Jacobsen, J. A. Ouassou, and J. Linder, Phys. Rev.\nB92, 024510 (2015).\n31P. H ogl, A. Matos-Abiague, I. Zutic, and J. Fabian, Phys.\nRev. Lett. 115, 116601 (2015).\n32R. Grein, T. L ofwander, G. Metalidis, and M. Eschrig,\nPhys. Rev. B 81, 094508 (2010).\n33F. H ubler, J. C. Lemyre, D. Beckmann, and\nH. v. L ohneysen, Phys. Rev. B 81, 184524 (2010).\n34F. H ubler, M. J. Wolf, D. Beckmann, and H. v. L ohneysen,\nPhys. Rev. Lett. 109, 207001 (2012).\n35M. J. Wolf, F. H ubler, S. Kolenda, H. v. L ohneysen, and\nD. Beckmann, Phys. Rev. B 87, 024517 (2013).\n36C. Quay, D. Chevallier, C. Bena, and M. Aprili, Nat. Phys.\n9, 84 (2013).\n37T. Wakamura, N. Hasegawa, K. Ohnishi, Y. Niimi, and\nY. Otani, Phys. Rev. Lett. 112, 036602 (2014).\n38D. Beckmann, J. Phys.: Condens. Matter 28, 163001\n(2016).\n39N. Poli, J. P. Morten, M. Urech, A. Brataas, D. B. Hav-\niland, and V. Korenivski, Phys. Rev. Lett. 100, 136601(2008).\n40H. Yang, S.-H. Yang, S. Takahashi, S. Maekawa, and S. S.\nParkin, Nat. Mater. 9, 586 (2010).\n41T. Wakamura, H. Akaike, Y. Omori, Y. Niimi, S. Taka-\nhashi, A. Fujimaki, S. Maekawa, and Y. Otani, Nat.\nMater. 14, 675 (2015).\n42M. Inoue, M. Ichioka, and H. Adachi, Phys. Rev. B 96,\n024414 (2017).\n43C. Bell, S. Milikisyants, M. Huber, and J. Aarts, Phys.\nRev. Lett. 100, 047002 (2008).\n44K.-R. Jeon, C. Ciccarelli, A. J. Ferguson, H. Kurebayashi,\nL. F. Cohen, X. Montiel, M. Eschrig, J. W. A. Robinson,\nand M. G. Blamire, Nat. Mater. 17, 499 (2018).\n45Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I.\nHalperin, Rev. Mod. Phys. 77, 1375 (2005).\n46J. P. Morten, A. Brataas, G. E. W. Bauer, W. Belzig, and\nY. Tserkovnyak, Europhys. Lett. 84, 57008 (2008).\n47X. Montiel and M. Eschrig, Phys. Rev. B 98, 104513\n(2018).\n48S. K. Kim, R. Myers, and Y. Tserkovnyak, Phys. Rev.\nLett. 121, 187203 (2018).\n49T. Taira, M. Ichioka, S. Takei, and H. Adachi, Phys. Rev.\nB98, 214437 (2018).\n50K. R. Jeon, X. Montiel, S. Komori, C. Ciccarelli, J. Haigh,\nH. Kurebayashi, L. F. Cohen, C. M. Lee, M. G. Blamire,\nand J. W. A. Robinson, arXiv:1908.00873 (2019).\n51M. A. Silaev, Phys. Rev. B 102, 180502(R) (2020).\n52H. T. Simensen, L. G. Johnsen, J. Linder, and A. Brataas,\nPhys. Rev. B 103, 024524 (2021).\n53A. K. Chan, M. Cubukcu, X. Montiel, S. Komori, A. Van-\nstone, J. E. Thompson, G. K. Perkins, M. Blamire,\nJ. Robinson, M. Eschrig, H. Kurebayashi, and L. F. Co-\nhen, arXiv:2202.01520 (20221).\n54M. Houzet, Phys. Rev. Lett. 101, 057009 (2008).\n55K. D. Usadel, Phys. Rev. Lett. 25, 507 (1970).\n56W. Belzig, F. K. Wilhelm, C. Bruder, G. Sch on, and A. D.\nZaikin, Superlattices Microstruc. 25, 1251 (1999).\n57M. Eschrig, Phys. Rev. B 80, 134511 (2009).\n58G. Eilenberger, Z. Phys. 214, 195 (1968).\n59A. Larkin and Y. N. Ovchinnikov, Sov Phys JETP 28,\n1200 (1969).\n60L. Gor'kov and A. Rusinov, Sov. Phys.{JETP 19, 922.[Zh.\nEksp. Teor. Fiz. 46, 1363.] (1964).\n61M. Y. Kuprianov and V. F. Lukichev, Sov. Phys.{JETP\n67,1163 [Zh. Eksp. Teor. Fiz. 94, 139.] (1988).\n62Y. V. Nazarov, Phys. Rev. Lett. 73, 1420 (1994).\n63Y. V. Nazarov, Superlattices Microstruc. 25, 1221 (1999).\n64M. Eschrig, A. Cottet, W. Belzig, and J. Linder, New J.\nPhys. 17, 083037 (2015).\n65S. H. Jacobsen, I. Kulagina, and J. Linder, Sci. Rep. 6,\n23926 (2016).\n66J. A. Ouassou, S. H. Jacobsen, and J. Linder, Phys. Rev.\nB96, 094505 (2017).\n67J. W. A. Robinson, N. Banerjee, and M. G. Blamire, Phys.\nRev. B 89, 104505 (2014).\n68Y. Ishikawa, R. Tournier, and J. Filippi, J. Phys. Chem.\nSolids 26, 1727 (1965).\n69J. O. Strom-Olsen, D. F. Wilford, S. K. Burke, and B. D.\nRainford, J. Phys. F: Met. Phys. 9, L95 (1979).\n70B. Babic, F. Kajzar, and G. Parette, J. Phys. Chem. Solids\n41, 1303 (1980)."    },    {        "title": "1409.4630v2.Contact_potential_instability_in_the_path_integral_description_of_itinerant_ferromagnetism.pdf",        "content": "arXiv:1409.4630v2  [cond-mat.quant-gas]  7 Dec 2015Contact potential instability in the path-integral\ndescription of itinerant ferromagnetism\nE. Vermeyen and J. Tempere\nTQC (Theory of Quantum systems and Complex systems)\nUniversiteit Antwerpen, 2610 Wilrijk, Belgium\n15th December 2014\nAbstract\nIt has long been predicted that a two-component non-localiz ed Fermi\ngas will exhibit spontaneous polarization for suﬃciently s trong repulsive\ninteractions, a phenomenon which is called itinerant ferro magnetism. Re-\ncent experiments with ultracold atomic gases have reached t he interaction\nstrength for which theoretical models have predicted the oc currence of the\nnormal-to-itinerant-ferromagnetic phase transition, bu t so far this tran-\nsition has not been observed. The instability of the repulsi ve branch of\nthe Feshbach resonance prevents the formation of the itiner ant ferromag-\nnetic state, but it is not clear whether this is the only insta bility impeding\nits experimental realization. In this article, we use the pa th-integral for-\nmalism with density ﬁelds in the Hubbard-Stratonovich tran sformation to\nstudy the stability of a homogeneous two-component Fermi ga s with con-\ntact interactions. Within the saddle-point approximation we show that\nnone of the extrema of the action are minima, meaning all extr ema are\nunstable to small density ﬂuctuations. This implies a more g eneral me-\nchanical instability of the polarized (itinerant ferromag netic) and normal\nstates of the system in the path-integral formalism. We ﬁnd t hat it is\nimportant to consider the stability of the system when study ing itinerant\nferromagnetism. Since (mechanical) stability may be inﬂue nced by the\ndetails of the interaction potential, we suggest the use of a more realistic\npotential than the contact potential in future theoretical descriptions.\n1 Introduction\nItinerant ferromagnetism is deﬁned as spontaneous polarization o f non-localized\nparticles. In two-component Fermi gases, we can expect it to occ ur for strongly\nrepulsive interactions: due to the exchange interaction, polarizat ion reduces the\ninteraction energy of the gas, at the cost of an increased kinetic e nergy. It\nwas originally predicted by Bloch in 1929 [1] for electrons and the theo retical\ngroundworkwasreﬁnedby Stoner[2] shortlythereafter. Despit e itslonghistory,\nitinerant ferromagnetism is not yet well understood. The strong c orrelations\ninvolved make it diﬃcult to create an accurate theoretical descript ion and it is\nhardtoﬁndagoodexperimentalsysteminwhichitcanbestudied. In solid-state\nsystems it occurs in d-band transition metals (e.g. Fe, Ni, Co), but t he inﬂuence\nof the exact shape of the band and the mixture of both localized and itinerant\n1ferromagnetism make it hard to verify models for itinerant ferroma gnetism in\nthese systems [3]. Ultracold atomic gases have already proven to be a fertile\ntesting ground for several theoretical models (e.g. the BCS-BEC crossover[4, 5,\n6, 7, 8, 9, 10]). Spin-polarization has been investigated in the super ﬂuid state,\nboth experimentally [11] and theoretically [12]. Several studies have proposed\nthat, also in the normal state, quantum gases could serve as a mod el system for\nitinerant ferromagnetism [13, 14, 15, 16, 17].\nIn2009,sometantalizingexperimentalhintsofthenormal-to-itine rant-ferro-\nmagnetic phase transition were observed [18], but the ﬁnal conﬁrm ation of the\nnature of the observed transition was missing: no magnetic domains could be\nobserved. Interestingly, the observed transition was found at t he dimensionless\nscattering length kFas= 1.9±0.2, a value which is signiﬁcantly larger than the\nmean-ﬁeld prediction kFas=π/2 or the second order prediction kFas= 1.054\n[17]. These results renewed the theoretical interest in itinerant fe rromagnetism\n[19, 20, 21, 22, 23, 24]. In a subsequent experiment [25] it was fou nd that the\nformation of the itinerant ferromagnetic state is prevented by a r apid decay into\nbound pairs due to near-resonant three-body interactions. Exp erimentalists are\ncurrently looking for a way to suppress this decay; the most promis ing proposal\nis the use of a mixture of40K and6Li [26]. There are also several proposals\nthat try to avoid the decay by reducing the critical interaction str ength of the\nnormal-to-itinerant-ferromagnetic phase transition [27, 28]. How ever, it is not\nclearifthisinstabilityofthepositivebranchoftheFeshbachbranch resonanceto\nmolecularpairingisthe onlyinstability preventingthe formationofthe itinerant\nferromagnetic state.\nIn this paper we revisit the itinerant ferromagnetic state using the path-\nintegral formalism, with the goal of studying its stability with respec t to small\n(density) ﬂuctuations within the saddle-point approximation. We st art from a\nhomogeneoustwo-componentFermigaswithcontactinteraction s. Thepartition\nfunction corresponding to this system, written as a path integral, cannot be\ncalculated exactly for a general case. A Hubbard-Stratonovich t ransformation\nis used to rewrite the fermionic path integral into a form that can be calculated\nexactly, at the cost of introducing an auxiliary bosonic ﬁeld. In the s addle-\npoint approximation we assume the auxiliary bosonic ﬁeld to be consta nt and\nwe require its value to be a minimum, maximum or saddle point (hencefor th\ncalled extremum) of the action. In order to study itinerant ferrom agnetism, the\nauxiliary bosonic ﬁeld is chosen to be a density ﬁeld (also called the Hart ree\nchannel). An extremum of the action is stable to small ﬂuctuations if it is also a\nminimum. In order to study their stability, we study the nature of all extrema.\nWe ﬁnd that all extrema of the action for the density ﬁelds are unst able against\nsmall (density) ﬂuctuations and we discuss the possible implications o f this\ninstability.\nThis paper is organized as follows: In Sec. 2, we construct the path -integral\ntreatment of the problem, including a detailed description of the Hub bard-\nStratonovich transformation. In Sec. 3, we introduce the saddle -point approx-\nimation and we look into the details of the resulting expression for the saddle-\npoint thermodynamic grand potential per unit volume. In Sec. 4, th e stability\nofthe extrema ofthe action arestudied and it is shown that none of the extrema\nare minima. In Sec. 5, we discuss the results and their possible implicat ions.\n22 The path-integral treatment\nThe thermodynamic grand potential Ω per unit volume can be determ ined from\nthe grand-canonical partition sum Z,\nΩ =−1\nβVlnZ (1)\nwithβ= 1/kBTthe inversetemperature, Tthe temperature, kBthe Boltzmann\nconstant and Vthe volume. In the path-integralformalism, the grand-canonical\npartition sum of a two-component Fermi gas is given by\nZ=/productdisplay\nσ=↑,↓/parenleftbigg/integraldisplay\nD¯ψσ/integraldisplay\nDψσ/parenrightbigg\nexp/parenleftbig\n−S/bracketleftbig¯ψσ,ψσ/bracketrightbig/parenrightbig\n, (2)\nwhere we take the sum over all possible conﬁgurations of the Grass mann ﬁelds\n¯ψ↑,ψ↑,¯ψ↓andψ↓, weightedbyafactorthatdependsontheaction Sofeachspe-\nciﬁc conﬁguration. The two components are called spin-up ↑and spin-down ↓,\nfor example referringto two diﬀerent hyperﬁne states of an atom (in the context\nof ultracold Fermi gases). For a uniform Fermi gas with contact int eractions,\nthe action (in units /planckover2pi1= 1, 2m= 1 andkB= 1) is given by\nS/bracketleftbig¯ψσ,ψσ/bracketrightbig\n=/summationdisplay\nσ=↑,↓/integraldisplayβ\n0dτ/integraldisplay\ndx¯ψxτσ/bracketleftbigg∂\n∂τ−∇2\nx−µσ/bracketrightbigg\nψxτσ\n+g/integraldisplayβ\n0dτ/integraldisplay\ndx¯ψxτ↑¯ψxτ↓ψxτ↓ψxτ↑, (3)\nwith the Grassmann ﬁelds now expressed as a function of imaginary t imeτ\nand position vector x. In (3),µσis the chemical potential of particles in spin\nstateσandgis the strength of the interaction potential, in 3D related to the\ns-wave scattering length asbyg= 4π/planckover2pi12as/m(org= 8πasin the chosen units).\nDue to the presence of the interaction term in (3), which is of fourt h order\nin the fermionic ﬁelds, the path integral in (2) cannot be solved exac tly for\na general case (although an exact solution is available for the 1D cas e with\ncontact interactions [29]). The Hubbard-Stratonovich transfor mation is used to\ntransform the interaction term into a form which is quadratic in the f ermionic\nﬁelds (and thus can be calculated exactly), at the cost of introduc ing an extra\npath integral over an auxiliary bosonic ﬁeld. This way, the problem is s hifted\nto the bosonic path integral, which usually has to be approximated.\nThere are many diﬀerent ways to decouple the quartic interaction t erm (us-\ning a Hubbard-Stratonovich transformation). They fall into thre e categories,\nreferring to the three possible ways to divide the four fermionic ﬁeld s of the\ninteraction term into two pairs [30]:\n•Bogoliubov: ¯ψxτ↑¯ψxτ↓andψxτ↓ψxτ↑\n•Fock:¯ψxτ↑ψxτ↓and¯ψxτ↓ψxτ↑\n•Hartree: ¯ψxτ↑ψxτ↑and¯ψxτ↓ψxτ↓\nThe Bogoliubov channel is suitable to describe superﬂuidity and focu sses on\nfermionic pair formation [31, 32, 33, 34]. The Fock channel represe nts spin-\nﬂip interactions. In this paper we use the Hartree channel, as the d ensity ﬁelds\n3Ψ/barb2up\nρ/barb2downρ/barb2upΨ/barb2down\nΨ/barb2downρ/barb2down\nΨ/barb2downΨ/barb2up\nΨ/barb2downΨ/barb2up\n= +\nρ/barb2upΨ/barb2up+\nρ/barb2upρ/barb2downΨ/barb2upΨ/barb2up\nρ/barb2upΨ/barb2downΨ/barb2up\nΨ/barb2downΨ/barb2up\n= +ρ/barb2down\nΨ/barb2down\nΨ/barb2down+(A) \n(B) \nFigure 1: The Feynman diagrams corresponding to the two Hubbard -\nStratonovich transformations: (A) corresponds to (4) and (B) corresponds to\n(5). The black arrows represent spin-up ﬁelds, the grey arrows r epresent spin-\ndown ﬁelds. The bosonic ﬁelds ρ↑andρ↓(dashed arrows) are introduced as a\nmediator of the interaction between the fermionic ﬁelds (full arrow s). The right\nhand sides of (A) and (B) show the same process in opposite directio ns, so both\ntransformations are complementary.\ncapturethe essenceofthe interactionsin thenormalandpolarize dstates[35,36,\n37]. The Hubbard-Stratonovich transformation itself is exact for any channel,\nbut the choice of the channel determines the physics that is include d in the\nsaddle-point approximation for the corresponding auxiliary bosonic ﬁeld. That\nis why the choice of the decoupling of the interaction term is a very imp ortant\nstep in our description.\nIn the Hartree channel still two kinds of density ﬁelds appear, cor responding\nto the density in the two diﬀerent spin states. Together with their c onjugated\ncounterparts, we end up with four density ﬁelds: ¯ ρ↑,ρ↑, ¯ρ↓andρ↓. For symme-\ntry reasons two separate transformations are used, each corr esponding to one\nhalf of the interaction energy:\nexp/parenleftigg\n−g\n2/integraldisplayβ\n0dτ/integraldisplay\ndx¯ψxτ↑¯ψxτ↓ψxτ↓ψxτ↑/parenrightigg\n= (4)\n/integraldisplay\nD¯ρ↓/integraldisplay\nDρ↑exp/bracketleftigg\n1\n2/integraldisplayβ\n0dτ/integraldisplay\ndx/parenleftbigg\n−¯ρxτ↓¯ψxτ↓ψxτ↓+¯ρxτ↓ρxτ↑\ng−¯ψxτ↑ψxτ↑ρxτ↑/parenrightbigg/bracketrightigg\n,\nexp/parenleftigg\n−g\n2/integraldisplayβ\n0dτ/integraldisplay\ndx¯ψxτ↑¯ψxτ↓ψxτ↓ψxτ↑/parenrightigg\n= (5)\n/integraldisplay\nD¯ρ↑/integraldisplay\nDρ↓exp/bracketleftigg\n1\n2/integraldisplayβ\n0dτ/integraldisplay\ndx/parenleftbigg\n−¯ρxτ↑¯ψxτ↑ψxτ↑+¯ρxτ↑ρxτ↓\ng−¯ψxτ↓ψxτ↓ρxτ↓/parenrightbigg/bracketrightigg\n.\nThese transformations can be viewed as two directions of the same inter-\naction process (Fig. 1). After the transformation, the partition sum is given\n4by\nZ=/productdisplay\nσ=↑,↓/parenleftbigg/integraldisplay\nD¯ψσ/integraldisplay\nDψσ/integraldisplay\nD¯ρσ/integraldisplay\nDρσ/parenrightbigg\nexp/parenleftbig\n−S/bracketleftbig¯ψσ,ψσ,¯ρσ,ρσ/bracketrightbig/parenrightbig\n(6)\nwith\nS/bracketleftbig¯ψσ,ψσ,¯ρσ,ρσ/bracketrightbig\n=\n/summationdisplay\nσ=↑,↓/integraldisplayβ\n0dτ/integraldisplay\ndx/bracketleftbigg\n¯ψxτσ/parenleftbigg∂\n∂τ−∇2\nx−µσ+¯ρxτ(−σ)+ρxτ(−σ)\n2/parenrightbigg\nψxτσ/bracketrightbigg\n−1\n2/integraldisplayβ\n0dτ/integraldisplay\ndx/parenleftbigg¯ρxτ↓ρxτ↑\ng+¯ρxτ↑ρxτ↓\ng/parenrightbigg\n. (7)\nThe derivatives in (7) are removed by a Fourier transformation. Su bse-\nquently we change to the Nambu spinor notation and ﬁnally the fermio nic path\nintegral is performed. Together with a change in notation for the d ensity ﬁelds\nand the chemical potentials,\n\n\nρ=ρ↑+ρ↓\n2\nφ=ρ↑−ρ↓\n2and\n\nµ=µ↑+µ↓\n2\nζ=µ↑−µ↓\n2, (8)\ncorresponding to the average density (resp. chemical potential) and half the\ndensity (resp. chemical potential) diﬀerence between the spin sta tes, this results\nin\nZ=/productdisplay\nσ=↑,↓/parenleftbigg/integraldisplay\nD¯ρσ/integraldisplay\nDρσ/parenrightbigg\nexp/parenleftigg/summationdisplay\nk/parenleftbig\n¯ρkρk−¯φkφk/parenrightbig\ng+Tr/braceleftig\nln/bracketleftig\n−det\nσ/parenleftig\n−G−1\nk;k′/parenrightig/bracketrightig/bracerightig/parenrightigg\n.\n(9)\nIn this expression, we used the short notation k= (k,ωn) withkthe wavevector\nandωn= (2n+1)π/βthe fermionic Matsubara frequencies ( n∈Z).−G−1is\na matrix in k-space, but each matrix element is in itself a 2 ×2 matrix over the\nspin states:\n−G−1\nk,k′=/parenleftbigg\n−iωn+k2−µ−ζ 0\n0 −iωn−k2+µ−ζ/parenrightbigg\nδ(∆k) (10)\n+1\n2√βV/parenleftbigg\n(¯ρ−∆k+ρ∆k)−/parenleftbig¯φ−∆k+φ∆k/parenrightbig\n0\n0 (¯ ρ∆k+ρ−∆k)+/parenleftbig¯φ∆k+φ−∆k/parenrightbig/parenrightbigg\nwith ∆k=k−k′andδ(∆k) the Dirac delta function. The determinant det σ\nin (9) works on the 2 ×2 matrices that are the matrix elements of −G−1, while\nthe trace in (9) is taken over k-space.\n3 The saddle-point approximation\nThe matrix −G−1is non-diagonal in k-space. Because of this, the bosonic\npath integral cannot be solved exactly, necessitating approximat ions. Without\ninteractions, only the ﬁrst term of (10) would remain, which is diagon al ink.\n5The saddle-point approximation makes the second term in (10) diago nal ink,\nby assuming only the ∆ k= 0 terms are important,\n/braceleftbiggρ∆k=√βVδ(∆k)ρ\n¯ρ∆k=√βVδ(∆k)ρ∗, (11)\n/braceleftbiggφ∆k=√βVδ(∆k)φ\n¯φ∆k=√βVδ(∆k)φ∗. (12)\nThe average density and half the density diﬀerence are assumed to be constant,\nhence we neglect all density ﬂuctuations. The saddle-point thermo dynamic\ngrand potential per unit volume then becomes\nΩsp(β,µ,ζ;ρ,φ) =1\ng/parenleftbig\nφ2\nI−ρ2\nI/parenrightbig\n+1\ng/parenleftbig\nφ2\nR−ρ2\nR/parenrightbig\n−1\nβV/summationdisplay\nk,nln[−(−iωn+Ek−ζ′)(−iωn−Ek−ζ′)]. (13)\nHere, we used the following notations to shorten the expressions. The indices\nRandIrefer to the real and imaginary part of ρandφ.Ek=k2−µ′is the\nsingle-particle dispersion relation. µ′=µ−ρRandζ′=ζ+φRare the eﬀective\nchemicalpotentials. We now seethat the imaginaryparts ρIandφIonlyappear\nin the ﬁrst term of (13) as a shift in the thermodynamic grand poten tial per\nunit volume, so they can be ignored. The values of the real parts ρRandφR\nstill have to be determined by solving the saddle-point equations,\n\n\n∂Ωsp(β,µ,ζ;ρR,φR)\n∂ρR/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nβ,µ,ζ;φR= 0\n∂Ωsp(β,µ,ζ;ρR,φR)\n∂φR/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nβ,µ,ζ;ρR= 0. (14)\nThis way we obtain the extrema of Ω sp(β,µ,ζ;ρR,φR) as a function of ρRand\nφR.\nBefore continuing, it is interesting to look at the physics behind expr ession\n(13) for the thermodynamic grand potential per unit volume. The d ensity ﬁelds\nhave two eﬀects on Ω sp(β,µ,ζ;ρR,φR). First, they result in an extra interac-\ntion energy, given by the second term of (13). Second, the third t erm of (13)\ncorresponds to the thermodynamic grand potential per unit volum e of the non-\ninteractingtwo-componentFermi gas, except for the shifted ch emical potentials.\nAs such, the density ﬁelds can change the polarizationof the gas. N ote also that\nwhen half the density diﬀerence φequals the average density ρ, something we\nwould expect in the fully polarized case, only the chemical potential s hift re-\nmains. This is the expected result for the fully polarized itinerant fer romagnetic\nstate.\nTo ease our notation, from now on we will drop the index RofρRandφR.\nBy performing the Matsubara summation in (13), we ﬁnally ﬁnd\nΩsp(β,µ,ζ;ρ,φ) =1\ng/parenleftbig\nφ2−ρ2/parenrightbig\n−/integraldisplaydDk\n(2π)D/braceleftbigg1\nβln/bracketleftbigg1\n2cosh(βζ′)+1\n2cosh(βEk)/bracketrightbigg\n−Ek/bracerightbigg\n(15)\n6in D dimensions. The integral over kcannot be solved exactly for general cases\n(although it can be solved exactly in some cases, for example for D= 2 and in\nthe zero temperature limit).\n4 The stability of the solution\nThe saddle-point thermodynamic grand potential per unit volume Ω sp(β,µ,ζ)\ncan be calculatedby solvingthe saddle-pointequations(14) foreac hvalue ofthe\nthermodynamic variables β,µandζ. Here, we ask ourselves the following ques-\ntion: assuming that a certain set of parameter values ( β,µ,ζ;ρ,φ) is a solution\nto the saddle-point equations, would that solution be stable to quan tum ﬂuctua-\ntions? For the solution to be stable, it has to be a minimum of Ω sp(β,µ,ζ;ρ,φ)\nas a function of ρandφfor a given set of thermodynamic variables ( β,µ,ζ).\nTostudythenatureofanextremum, weinvestigatetheHessianma trixofthe\nsaddle-point thermodynamic grand potential per unit volume Ω sp(β,µ,ζ;ρ,φ),\nH=\n∂2Ωsp(β,µ,ζ;ρ,φ)\n∂ρ2/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nβ,µ,ζ;φ∂2Ωsp(β,µ,ζ;ρ,φ)\n∂ρ∂φ/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nβ,µ,ζ\n∂2Ωsp(β,µ,ζ;ρ,φ)\n∂φ∂ρ/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nβ,µ,ζ∂2Ωsp(β,µ,ζ;ρ,φ)\n∂φ2/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nβ,µ,ζ;ρ\n. (16)\nCalculating the second derivatives of (15) in (16), we ﬁnd\nH=/parenleftbigg−2\ng−I1I2\nI22\ng−I1/parenrightbigg\n(17)\nwithI1andI2referring to the integrals\nI1=β/integraldisplaydDk\n(2π)D/parenleftigg\n1+cosh(βζ′)cosh(βEk)\n[cosh(βζ′)+cosh(βEk)]2/parenrightigg\n, (18)\nI2=β/integraldisplaydDk\n(2π)D/parenleftigg\nsinh(βEk)sinh(βζ′)\n[cosh(βζ′)+cosh(βEk)]2/parenrightigg\n. (19)\nFor an extremum to be a minimum of the thermodynamic grand potent ial\nper unit volume, the Hessian matrix has to have two positive eigenvalu es. If at\nleast one of its eigenvalues is negative, the solution is unstable. The t race of a\nmatrix is equal to the sum of its eigenvalues. Tr H=−2I1≤0 (because the\nhyperbolic cosine is a positive function), so at least one of the eigenv alues ofH\nis negative for all possible values of the parameters g,β,µ,ζ,ρandφ. If this\nis true for all values, it is also true for the values that solve the sadd le-point\nequations, implying all solutions are unstable to small (density) ﬂuct uations.\n5 Conclusion and discussion\nThe main conclusion of this paper is that none of the extrema of the a ction for\na homogeneous two-component Fermi gas with contact interactio ns are a mini-\nmum, if the Hubbard-Stratonovich transformation is performed w ith only den-\nsity ﬁelds. This means that those extrema do not correspond to a ( meta)stable\n7state of the system, since they are unstable to small (quantum) ﬂ uctuations. Of\ncourse (stable) minima of the action itself do exist, but for contact interactions\nthey cannot be found or approximated in the Hartree channel (us ing density\nﬁelds). For low temperatures, a well-known minimum of the action is fo und in\nthe Bogoliubov channel, which is used to describe superﬂuidity.\nWhen considering density ﬁelds, small ﬂuctuations are in fact densit y ﬂuctu-\nations. Consequently, we have studied the mechanical stability of t he extrema:\nstability against collapse, expansion and spin separation (or a mix of t hese\nthree). The instability we found indicates that the normal and polar ized (itin-\nerant ferromagnetic) states of a homogeneous two-component Fermi gas with\ncontact interactions are mechanically unstable.\nIn experiments with ultracold atomic gases, we already know that th e pres-\nence of the closed scattering channel causes an instability which pr events the\nformation of the itinerant ferromagnetic state (or more generally the formation\nof any equilibrium state in the short time scales of the experiment). I n light\nof the proposals to circumvent this problem, it would be interesting t o know if\nthere areanyotherinstabilities which wouldinhibit the experimentalr ealization\nof the itinerant ferromagnetic state.\nOur work can answer this question for the contact potential: in the path-\nintegral formalism, the itinerant ferromagnetic state is unstable t o density ﬂuc-\ntuations. Since the details of the interaction potential may have a b ig inﬂuence\non the stability of the system, further study is needed to determin e if simi-\nlar instabilities occur for other interaction potentials. We can only go beyond\nmean-ﬁeld if we ﬁnd a regime in which the extrema are stable (minima), s o\nthis stability analysis will be crucial for an improved theoretical desc ription of\nitinerant ferromagnetism.\nIf we use the Hartree channel, we assume that the dominant intera ction\neﬀects in the system are related to density ﬂuctuations. That ass umption fails\nwhen all saddle-points are unstable. In that case there are other interaction\neﬀects present which are more important than the density ﬂuctua tions (e.g.\npair formation, leading to superﬂuidity).\nWhat kind of potential would stabilize the itinerant ferromagnetic st ate?\nWould a short-ranged potential be suﬃcient, or is a long-ranged po tential re-\nquired? UsingTan’srelations,itcanbeshownthatthezero-ranget wo-component\nFermi gas does not have a stable fully polarized ferromagnetic stat e [38]. As\nTan’s relations should be valid to a good approximation for ﬁnite-rang e interac-\ntions of suﬃciently short range, this suggests that a long ranged in teraction is\nneeded to stabilize the itinerant ferromagnetic state. However, o ther work has\npredicted the occurrence of itinerant ferromagnetism for eﬀect ive short-ranged\ninteractions [39, 40].\nZero-rangepotentialsinFermigasesarenecessarilyspin-depend ent: between\ntwo equal spins the strength of a contact potential is always zero . This is no\nlonger strictly true when the potential acquires a ﬁnite range, and (together\nwith the fact that the kinetic energy is diﬀerent) this may explain why Kollaret\nal.[39] ﬁnd that nearest neighbour interactions in a lattice model allow f or itin-\nerant ferromagnetism. On the other hand, long-rangeinteractio ns can overcome\nspatial unmixing eﬀects and increasedomain sizes in the itinerant fer romagnetic\nstate. So, our suggestion would be to look at long-rangepotentials for the Fermi\ngases, combined with a short-range repulsion to enhance the exch ange eﬀects.\nA generalization of our approach to the case of a conﬁning externa l potential\n8can be made by using the Local Density Approximation (LDA). LDA re lies on\nthe assumption that the gas can be described by local thermodyna mic variables\n(chemical potentials). Locally, the gas is approximated by the homo geneous\ngas at constant chemical potentials with the same values of the the rmodynamic\nvariables, which is the case we considered in the paper. Because we f ound an\ninstability for allvalues of the thermodynamic variables in a homogeneous gas\nwith contact interactions, we can rule out itinerant ferromagnetis m for a two-\ncomponent Fermi gas with contact interactions in an external pot ential as long\nas LDA is valid.\nTwo requirements need to be fulﬁlled for LDA to be valid. First, the de nsity\n(and therefore also the external potential) needs to be suﬃcient ly slowly vary-\ning. This assumption is true for many, but not all conﬁning external potentials\nused in experiments with ultracold atomic gases, and it generally brea ks down\nat the edge of the atomic cloud. The second requirement is that the interactions\ncan also be considered to be local, hence that the range of the inter actions is\nsuﬃciently small. The second requirement is always fulﬁlled for contac t inter-\nactions.\nSummarizing, we ﬁnd that stability is an important factor to consider when\nstudying itinerant ferromagnetism. Furthermore, in the context of ultracold\natomic gases we suggest to use a more realistic potential than the c ontact po-\ntential, preferably a long-ranged potential with a short-ranged r epulsion.\nAcknowledgements The authors would like to thank C.A.R. S´ a de Melo and\nA. Pelster for useful discussions, and J.P.A. Devreese for a careful reading of\nthe manuscript. E. Vermeyen gratefully acknowledges suppo rt in the form of a\nPh. D. fellowship of the Research Foundation - Flanders (FWO ). This work was\nsupported by the following Research Programmes of the Resea rch Foundation-\nFlanders (FWO): G.0119.12.N, G.0115.12.N, G.0180.09.N, a nd G.0429.15.N.\nThis work was also supported by the Research Council of Antwe rp University\nvia a GOA grant.\nReferences\n[1] F. Bloch, Z. Phys. 57, 545 (1929)\n[2] E. Stoner, Phil. Mag. 15, 1018 (1933)\n[3] Y. Kakehashi, Adv. Phys. 53, 497 (2004)\n[4] M.W. Zwierlein, C.A. Stan, C.H. Schunck, S.M.F. Raupach, S. Gupta,\nZ. Hadzibabic, W. Ketterle, Phys. Rev. Lett. 91, 250401 (2003)\n[5] S. Jochim, M. Bartenstein, A. Altmeyer, G. Hendl, S. Riedl, C. Chin , J.H.\nDenschlag, R. Grimm, Science 302, 2101 (2003)\n[6] M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, C. Chin, J.H. Den schlag,\nR. Grimm, Phys. Rev. Lett. 92, 120401 (2004)\n[7] C.A. Regal, M. Greiner, D.S. Jin, Phys. Rev. Lett. 92, 040403 (2004)\n9[8] T. Bourdel, L. Khaykovich, J. Cubizolles, J. Zhang, F. Chevy, M. Teich-\nmann, L. Tarruell, S.J.J.M.F. Kokkelmans, C. Salomon, Phys. Rev. Let t.\n93, 050401 (2004)\n[9] C.A.R.S. de Melo, M. Randeria, J. Engelbrecht, Phys. Rev. Lett. 71, 3202\n(1993)\n[10] Y. Ohashi, A. Griﬃn, Phys. Rev. Lett. 89, 130402 (2002)\n[11] C. Sanner, E.J. Su, A. Keshet, W. Huang, J. Gillen, R. Gommers, W. Ket-\nterle, Phys. Rev. Lett. 106, 010402 (2011)\n[12] T. Kashimura, R. Watanabe, Y. Ohashi, Phys. Rev. A 86, 043622 (2012)\n[13] M. Houbiers, R. Ferwerda, H.T.C. Stoof, W.I. McAlexander, C.A. Sackett,\nR.G. Hulet, Phys. Rev. A 56, 4864 (1997)\n[14] M. Amoruso, I. Meccoli, A. Minguzzi, M.P. Tosi, Eur. Phys. J. D. 8, 361\n(1999)\n[15] L. Salasnich, B. Pozzi, A. Parola, L. Reatto, J. Phys. B. 33, 3943 (2000)\n[16] T. Sogo, H. Yabu, Phys. Rev. A 66, 043611 (2002)\n[17] R.A. Duine, A.H. MacDonald, Phys. Rev. Lett. 95, 230403 (2005)\n[18] G.B.Jo, Y.R. Lee, J.H. Choi, C.A. Christensen, T.H. Kim, J.H. Thywis sen,\nD.E. Pritchard, W. Ketterle, Science 325, 1521 (2009)\n[19] G.J. Conduit, B.D. Simons, Phys. Rev. A 79, 0503606 (2009)\n[20] G.J. Conduit, B.D. Simons, Phys. Rev. Lett. 103, 200403 (2009)\n[21] H. Zhai, Phys. Rev. A 80, 051605(R) (2009)\n[22] R.A. Duine, M. Polini, H.T.C. Stoof, G. Vignale, Phys. Rev. Lett. 104,\n220403 (2010)\n[23] D. Pekker, M. Babadi, R. Sensarma, N. Zinner, L. Pollet, M.W. Zw ierlein,\nE. Demler, Phys. Rev. Lett. 106, 050402 (2011)\n[24] I.Sodemann, D.A.Pesin,A.H.MacDonald,Phys.Rev.A 85,033628(2012)\n[25] C. Sanner, E.J. Su, W. Huang, J.G. A. Keshet, W. Ketterle, Phy s. Rev.\nLett.108, 240404 (2012)\n[26] C. Kohlstall, M. Zaccanti, M. Jag, A. Trenkwalder, P. Massignan , G.M.\nBruun, F. Schreck, R. Grimm, Nature 485, 615 (2012)\n[27] S. Pilati, I. Zintchenko, M. Troyer, Phys. Rev. Lett. 122, 015301 (2014)\n[28] S.S. Zhang, J. Ye, W.M. Liu, (2014). ArXiv:1403.7031v3 [cond-ma t.quant-\ngas]\n[29] X.W. Guan, M.T. Batchelor, C. Lee, Rev. Mod. Phys 85, 1633 (2013)\n[30] H. Kleinert, Electron. J. Theor. Phys. 25, 57 (2011)\n10[31] H.T.C. Stoof, K.B. Gubbels, D.B.M. Dickerscheid, Ultracold Quantum\nFields(Springer, 2009)\n[32] J. Tempere, S.N. Klimin, J.T. Devreese, Phys. Rev. A 78, 023626 (2008)\n[33] J. Tempere, M. Wouters, J.T. Devreese, Phys. Rev. B 75, 184526 (2007)\n[34] E. Taylor, A. Griﬃn, N. Fukushima, Y. Ohashi, Phys. Rev. A 74, 063626\n(2006)\n[35] D.R. Hamann, Phys. Rev. Lett. 23, 95 (1969)\n[36] S.Q. Wang, W.E. Evenson, J.R. Schrieﬀer, Phys. Rev. Lett. 23, 92 (1969)\n[37] N. Nagaosa, Quantum Field Theory in Condensed Matter Physics\n(Springer, 1999)\n[38] M. Barth, W. Zwerger, Ann. Phys 326, 2544 (2011)\n[39] M. Kollar, R. Strack, D. Vollhardt, Phys. Rev. B 53, 9225 (1996)\n[40] P. Massignan, M. Zaccanti, G.M. Bruun, Rep. Prog. Phys. 77, 034401\n(2014)\n11"    },    {        "title": "2201.03739v1.Resonant_Precession_of_Magnetization_and_Precession____Induced_DC_voltages_in_FeGaB_Thin_Films.pdf",        "content": "1 \n Resonant Precession of Magnetization and \nPrecession - Induced DC voltages  in FeGaB \nThin Films.  \nPrabesh Bajracharya, Vinay Sharma, Anthony Johnson, and Ramesh C. Budhani * \nDepartment of Physics, Morgan State University, Baltimore, MD, 21251 USA  \n*Ramesh.budhani@morgan.edu  \nAbstract  \nMeasurements of frequency dependent ferromagnetic resonance (FMR) and spin pumping driven dc voltage (V dc) are \nreported for amorphous films of Fe 78Ga13B9 (FeGaB) alloy  to address the phenomenon of self -induced inverse spin \nHall effect (ISHE) in plain films of metallic ferromagnets. The V dc signal, which is antisymmetric on field reversal, \ncomprises of symmetric and asymmetric Lorentzians centered ar ound the resonance field. Dominant role of thin film \nsize effects is seen in setting the magnitude of static magnetization, V dc and dynamics of magnetization precession in \nthinner films (≤  8 nm). The film thickness dependence of magnetization parameters indicates the presence of a \nmagnetically disordered region at the film – substrate interface, which may promote preferential flow of spins \ngenerated by the precessing magnetization towards th e substrate. However, the Vdc signal also draws contributions  \nfrom rectification effects of a  0.4 % anisotropic magnetoresistance and a large ( 54 n.m) anomalous Hall \nresistivity  (AHR)  of these films which ride over the effect  of spin  – orbit coupling driven spin -to-charge conversion \nnear the film – substrate interface.  We have addressed these data in the framework of the existing theories of \nelectrodynamics of a ferromagnetic film subjected to radio -frequency field in a coplanar waveguide geometry. Our \nestimation of the self -induced ISHE for the sample with 54 n m AHR shows that it may contribute significantly (  \n90%) to the measured symmetric voltage.  This study is expected to be very useful for fully understanding the spin \npumping induced dc voltages in metallic ferromagnets with disordered interfaces and large  anomalous Hall effect . \n \nKeywords: Ferromagnetic resonance, Gilbert damping, Spin orbit coupling, Inverse spin Hall effect, Magnetic dead \nlayer , spin rectificatio n. \n \n \n \n \n 2 \n I. Introduction  \nThe observation of an inverse spin Hall effect  (ISHE) – \nlike dc voltage in plain films of metallic ferromagnets \n(FM) at the ferromagnetic resonance (FMR) frequency \nhas generated considerable interest in recent years [1 -6]. \nThe ISHE is a process by which spin currents injected \nfrom a ferromagnet into a  proximate non -magnetic metal \n(NM) of strong spin -orbit -coupling (SOC) are converted \ninto charge currents [7, 8]. This process is driven by spin \nHall effect (SHE) and interfacial Rashba Edelstein effect \n(REE) [9 -11]. While the origin of a dc voltage induce d \nby thermally pumped spin currents is well -understood \n[12-14], such is not the case for spin pumping by radio \nfrequency (RF) assisted resonant precession of \nmagnetization. Some earlier measurements of ISHE on \nFM/NM bilayers at resonance have addressed the  role of \neffects such as spin rectification  due to a large \nanisotropic magnetoresistance (AMR) and anomalous \nHall effect (AHE) , and thermal gradient induced dc \nvoltages in contaminating the true ISHE signal [15 -18]. \nThese studies have also reported measure ments of a dc \nvoltage at resonance in control samples of plain FM \nfilms, and this weak antisymmetric Lorentzian signal has \nbeen attributed to AMR induced spin rectification effect \n(SRE) in the metallic ferromagnet. However, extensive \nstudies of SRE  at reso nance in thin film of metallic \nferromagnets performed in coplanar waveguide as well \nas cavity spectrometers have shown that the rectified \nvoltage has both field symmetric and field asymmetric \ncontributions, depending  on the geometry of the \nexperiment [19, 20 ]. Notwithstanding these results , \nseveral recent measurements of the ISHE – like signal at \nFMR in thin films of metallic ferromagnets present a \nrather different picture. The data of Tsukahara et. al. [1] \non 10 nm t hick film of permalloy (Py) driven to \nresonance in a cavity spectrometer show both symmetric \nand antisymmetric Lorentzian line shapes. They attribute the symmetric part of the signal to ISHE generated by \nthe non -zero SOC in Py films. Measurements on plain \nfilms of Fe and Co also reveal an ISHE -like symmetric \nsignal at resonance [4]. The work of Azevedo et. al. [2] \non Py films of different thicknesses shows a robust and \npredominantly symmetric signal at resonance in the \ndirection of applied magnetic field. T hese authors have \nattributed this signal to the process of magnonic charge \npumping (MCP) wherein the spin waves emitted by \nprecessing magnetization generate a charge current in \nthe FM layer of a non -zero SOC [ 21]. Under open circuit \nconditions, this curren t produces an ac voltage across the \nsample together with a small dc component due to AMR. \nThe MCP results from the Onsager reciprocity theorem \nas the counterpart of spin torque FMR (ST -FMR), where \nthe spin orbit torque produced by ac currents drive the \nmagnetization into precession. Azevedo et. al. [2] argue \nthat surface oxidation of the Py film produces a Rashba \nfield that breaks the inversion symmetry of the system to \nresult in a symmetric dc signal. Tsukahara et. al. [1] have \nattributed the symmetric vol tage signal in Py films at \nresonance to spin pumping induced ISHE in the \ndisordered regions of the film near its interface with the \nsubstrate. However, the theory of SRE clearly shows that \nunder specific configurations of dc and rf fields with \nrespect to t he dc voltage leads both AMR and AHE will \ncontribute to a symmetric rectified signal [ 20]. \nTherefore, the limited studies of the so -called self-\ninduced ISHE in plain films of metallic ferromagnets \nneed to be augmented with the twin objectives of:  (i) \ncorrectly interpreting the ISHE data on metallic FMs \nbilayered with SOC -NMs and various topological \nmaterials. (ii) Understanding the true origin of dc \ncurrents in plain films of metallic ferromagnets driven to \nresonance by ac fields.    \n       Here, we address self-induced ISHE  at FMR in non -\ncrystalline films of a metallic ferromagnet. This material 3 \n is derived from the well -known high magnetostriction \nalloy Fe 80Ga20 [22, 23 ]. The addition of boron in \nFe80Ga20 followed by rapid solidification from liquid or \nvapor phase results in a non -crystalline material [ 24, 25 ]. \nThin films of the alloy Fe 78Ga13B9 display large \nmagneto -elastic coupling together with magnetic \nsoftness characterized by a low Gilbert damping \ncoefficient [ 24, 26, 27 ]. The amorphous nature o f these \nfilms allows us to rule out any possible contributions of \ncrystallographic texture to the rectified dc voltage . We \nhave carried out detailed studies of magnetization \ndynamics with frequency -dependent ferromagnetic \nresonance, vibrating sample magnet ometry (VSM) and \nspin pumping induced ISHE measurements on \nFe78Ga13B9 films as a function of film thickness. Our \nFMR and VSM measurements establish the formation of \na  1.2 nm thick magnetic dead layer at the film - \nsubstrate interface. While the symmetry  and sign of the \ndc signal suggest FMR -driven spin pumping into this \nlayer, we examine this result in the light of possible \nsymmetric voltages emanating from AMR and AHE . \nWhile the ambient temperature AMR in FeGaB  films is \nsmaller by a factor of  5 compared to the AMR in \npermalloy, a large  anomalous Hall resistivity (  54 n \nm) is seen in these films. On the other hand, the AHE in \nNi80Fe20 is very small (  0.96 n m) due to competing \ncontributions from the Ni and Fe derived electronic \nstates  [28]. Our measurements of the dc signal at \nresonance in FeGaB films indicate that notwithstanding  \nthe dead layer which  may promote  unidirectional flow of \nspin current, the AHE related rectification makes a \nsizable contribution to the dc signal.   \n \nII. Experimental details  \nFeGaB  films of thickness ranging from 2.0 nm \nto 12.0 nm were grown by DC sputtering of a Fe 78Ga13B9 \nalloy target on epitaxial quality (0001) plane sapphire substrates at room temperature in 5 x 10-3 torr argon \npressure at the rate of 0.03 nm/s. Static magnetiza tion \nmeasurements were performed at ambient temperature \non  5 x 5 mm2 samples in a vector VSM (MicroSense \nModel 10 Mark II). For the measurements of FMR and \nISHE, we have used a meander line grounded coplanar \nwaveguide (GCPW) -based FMR spectrometer that \noperates in the frequency and field ranges of 5 to 31 GHz \nand 0 to ± 1.5 tesla respectively (Fig. 1(a and b)). The \nISHE was measured by connecting a twisted pair of thin \ncopper wires to the ends of 2 x 5 mm2 film with silver \npaint.  \n \nFigure 1. (a)  Schematic illustration of experimental setup used for \nFMR and ISHE measurements. FMR measurements are performed \nover a frequency range of 5 to 30 GHz by low amplitude modulation \nof the dc field at 650 Hz. FeGaB thin film is placed in a flip -chip \nconfigura tion on the signal line of the grounded coplanar waveguide. \nA twisted pair of copper wires is used to measure the DC voltage \ngenerated across the sample. The DC voltage is measured along y -axis \nwith Y+ contact of the sample connected to the positive termin al of the \nnanovoltmeter . (b) Magnified view of the measurement setup to show \nthe directions of Js, Jc and spin polarization vector ( ), where  𝑱𝒄=\n4 \n 𝑆𝐻( 𝑱𝑠   ) . Direction of  and Jc are controlled by changing the \ndirection of dc magnetic field from  = 00 to 1800. Jrf is the rf current \nflowing in the cpw signal line  \nThe wired sample was then flip chipped on the section of \nthe waveguide where the rf magnetic field is \nperpendicular to the external dc magnetic field. A 100 \nm thick mica sheet separated the sample and signal line \nof GCPW for electrical isolation. The dc v oltage \ngenerated across the sample leads on sweeping the  \nexternal magnetic field across FMR was measured with \na Keithley nanovoltmeter. The FMR signal was recorded \non the same sample in a different experiment with phase \nsensitive detection of absorbed powe r on low frequency \n(650 Hz) modulation of the dc field with a set of \nHelmholtz coils.  The radio frequency (RF) source power \nwas changed from -10 dBm to +15 dBm to ensure linear \nresponse of the system. The measurements of AMR and \nplanar Hall effect ( PHE ) were performed by mounting \nthe films, patterned in a Hall bar geometry, on the \nvertical rotator stage of a physical property measurement \nsystem  (PPMS) . The AHE measurements were \nperformed by mounting the films, patterned in a Hall bar \ngeometry, on the dc resistivity puck of PPMS where dc \nmagnetic field was applied perpendicular to the sample \nsurface.   \n \nIII. Results and Discussion    \nA. Static magnetization characteristics  \n  Fig. 2(a) shows the in -plane magnetic \nhysteresis loops of FeGaB films of different thicknesses \n(tf) measured at room temperature. These samples have \nbeen named as FGBt f, where t f is the film thickness in \nnm. The M(H) loops o f all samples are characterized by \na squareness of low coercive (  0.1 mT) indicating a soft \nmagnetic state of in -plane orientation.  The saturation \nmagnetization ( 0Ms) of these films varies from 0.68 T to 1.43 T on increasing the thickness from 2 nm to 12  \nnm. \n \nFigure 2.  (a) dc magnetization loops (M(H)) of FeGaB thin films of \ndifferent thickness (t f = 2, 4, 6, 8, 10 and 12 nm) measured at room \ntemperature. Magnetic field is applied parallel to the plane of film. \nInset shows saturation magnetization ( 0MS) as a function of film \nthickness. Thin film size effect is seen prominently in the data for t f  \n8nm.  (b) Change in the product of M S and t f as a function of film \nthickness. Red line is a linear fit to show the point of magnetic dead \nlayer formation for t f  1.2 nm. Error bars reflect the uncertainty in the \nmeasurements of sample volume.  \n(See supplementary Fig. S1 for anisotropy calculations)  \nA comparison of the M(H) loops shown in Fig. 2(a) with \nsimilar data for Fe 80Ga20 alloy [ 29, 30 ] reveals that \naddition of boron suppresses the magneto -crystalline \nanisotropy and coercivity of this binary alloy. The \nsaturation magnetization displayed in the inset of Fig. \n2(a) increases linearly with the film thickness till t f   8 \nnm and  reaches a constant value beyond this  critical \nthickness. The saturation magnetization of the thicker \nfilms (t f ≥ 8 nm) agrees with the reported M s of the films \n5 \n of same composition [ 25, 27 ]. In Fig. 2(b) we show the \nplot of magnetization per unit area (M s.tf) as a function \nof film thickness. A linear fit to these data suggests the \nexistence of a magnetically dead layer of thickness 1.2 \nnm at the interface.  \nB. Dynamic magnetization study using \nFMR  \n The results of frequency dependent FMR \nmeasurements with external dc field in the plane of the \nfilm are shown in Fig. 3(a) and 3(b). The field derivative \nof the absorbed RF power (dP/dH) at 10 GHz (the FMR \nline shape) of the 4, 6, 8, 10 and 12 nm thick films is \ndisplayed in the inset of Fig. 3(a). The signal from the 2 \nnm thick film was too weak to detect.  \n \nFigure 3. (a)  Change in FMR resonance field ( 0Hr) with microwave \nfrequency for films of varying thickness. Open symbols in the figure \nare the experimental data calculated from Lorentzian line shape fitting \nand the solid red line the fit to Kitt el equation. Inset shows the \ndifferential FMR signal at 10 GHz with Lorentzian line -shape fitting. \n(b) shows the variation of effective saturation magnetization ( 0Ms) \nand FMR resonance field ( 0Hr) at 10 GHz as a function of sample thickness. As thickness increases, 0Ms increases while 0Hr decreases. \nInset shows the induced surface anisotropy field (H S) and g -factor as a \nfunction of layer thickness.  Red line in the figure is a  linear fit to \nequat ion 𝐻𝑠=4𝐾𝑠/(𝑀𝑠𝑡𝑓). \nThe FMR line shape at all frequencies from 5 to 30 GHz \nhas been analyzed in the framework of a two component \nLorentzian function [ 31]  \n𝑑𝑃\n𝑑𝐻=𝐾14𝐻(𝐻−𝐻𝑟)\n[4(𝐻−𝐻𝑟)2+(𝐻)2]2−𝐾2(𝐻)2−4(𝐻−𝐻𝑟)2\n[4(𝐻−𝐻𝑟)2+(𝐻)2]2     (1) \nwhere K 1 and K 2 are the symmetric and anti -symmetric \ncoefficients, H is the applied dc field, H r the resonance \nfield and H is the full width at half maximum of the \nresonance. The main panel of Fig. 3(a) displays the \nvariation of H r as a function of frequency for all five \nsamples.  \nThese data follow the simple Kittel equation [ 32, 33 ]  \n𝑓= \n20√𝐻(𝐻+𝑀𝑒𝑓𝑓)   (2) \nfor a ferromagnetic film with in -plane magnetization and \nno preferred anisotropy axis.  Here 0Meff is the effective \nsaturation magnetization that considers surface \nanisotropies, f is the excitation frequency,  =𝑔𝐵\nħ is the \ngyromagnetic ratio, g is the Lande’s g -factor, 𝐵 is the \nBohr magneton, and ħ is the Planck’s constant. We used \n0Meff and g values as free parameters while fitting the \nexperimental data to Eq. 2.  The changes in 0Meff and \n0Hr (at f = 10 GHz) as a function of t f are shown in Fig \n3(b). The effective magnetization depends on saturation \nmagnetization and surface anisotropy  field (H s) as,  \n0𝑀𝑒𝑓𝑓=0𝑀𝑠−0𝐻𝑠      (3) \nThe dependence of H s and g -values on FeGaB thickness \nis shown in the inset of Fig. 3(b). Since the surface \nanisotropy field is defined as 𝐻𝑠=4𝐾𝑠/(𝑀𝑠𝑡𝑓), where \nKS is the surface anisotropy constant,  we have calculated \nthe ratio 𝐾𝑠/𝑀𝑠 = 0.168 T.nm from the slope of the H s vs \n1/tf plot (Inset Fig. 3(b)).  The surface anisotropy is \n6 \n maximum at the lower thickness presumably due to \nenhanced pinning of magnetization by surface defects \nand roughness of the surface. The anisotropy field drops \nto zero when M s becomes independent of the film \nthickness. We note that the g -factor which reflec ts the \nratio of orbital and spin magnetic momenta has a reduced \nvalue compared to the g -factor of Fe or FeGa. However, \nwe also see an enhancement in g -factor at lower film \nthicknesses, which is presumably due to degradation of \nmagnetic properties near the interface [ 34]. It is well \nknown that the g -factor is influenced by surface and \ninterface effects as it depends on the local symmetry. \nSuch effects may lead to strong enhancements of the \nratio of orbital and spin momenta [ 35].              \nThe enhanced rol e of surfaces on magnetization \ndynamics of these films is also seen in the behavior of \nthe linewidth H.  Fig 4(a) shows the dependence of H \non excitation frequency. These data have been analyzed \nin the framework of the Landau - Lifshitz - Gilbert \n(LLG) e quation [ 36], \nH=H0+4𝑓\n  (4) \nwhere H0 is the inhomogeneous line broadening caused \nby sample imperfections and surface defects, and  is the \nGilbert damping coefficient. The linear frequency \ndependence of H seen in Fig. 4(a) suggests that the \nmagnetization damping is primarily of the LLG type \nwith negligible contribution from two -magnon processes \n[37].  The  and H0 are dominated by t hin film size \neffects at lower thicknesses as seen in Fig. 4(b).  The \nthickness dependence of  allows us to extract its surface \n(s) and bulk ( B) components as [ 32, 38, 39 ] \n =𝐵+𝑠\n𝑡𝑓  (5) \nA linear fit to the data as shown in the inset of Fig. 4(b) \nyields   B = 2.18 x 10-3 and s = 23.73 x 10-3 nm-1. The  value of B is in excellent agreement with the reported \ndata on FeGaB film [ 27]. \n \nFigure 4. (a)  Change in FMR linewidth ( H) as a function of \nmicrowave frequency for FeGaB films of different thickness. Open \nsymbols are calculated H while red line is a LLG fitting. (b) Variation \nof the Gilbert damping constant (α) and inhomogeneous linewidth \nbroaden ing (H0) as a function of film thickness. Inset shows a plot of \nα vs 1/t f -td (where t d is the magnetic dead layer thickness) where  open \nsymbols are calculated values while red line is a linear fit to Eq. 5. \nIntercept at y -axis shows the α for bulk FeGaB system [2 7].    \nThe large value of s together with the indication of a \nmagnetic dead layer at the interface, suggest that the \nangular momentum of precessing magnetization is lost at \nthe film – substrate interface. This inference is supported \nby our ISHE measurements, which are discussed next . \n \n \nC. Self-Induced ISHE  and rectified  voltage  \nmeasurement  \n7 \n              The flip – chip geometry used for the ISHE \nmeasurements is sketched in Fig. 1(b). The induced dc \nvoltage is measured across the sample with its Y+ end \nconnected to the positive terminal of the nanovoltmeter.  \n \nFigure 5. The measured dc voltage signal (V mix) at 10 GHz for θ = 00 \nand θ = 1800 angles. RF input power used for these measurements is \n+15dBm.    \nThe in -plane external dc magnetic field is directed along \nthe X -axis. Fig. 5 shows the variation of the output \nvoltage of five films at 10 GHz excitation as the dc field \nis scanned across H r. The noteworthy features of these \ndata are: (i) the dc signal appears at the resonance, (ii) \nthe signal is antisymmetric on field reversal, (iii) the line -\nshape is predominantly symmetric around H r, (iv) the \nsignal drops significantly as the film thickness is reduced \nand (v) the linewidth mimics the behavior of H seen in \nFMR measurements.   \nThe dc voltage generated across a thin film of metallic \nferromagnet undergoing FMR derives contributions \nfrom sev eral coherent and incoherent spin and charge \nscattering processes. For example, the RF component of the precessing magnetization results in a dc voltage due \nto AMR,  and AHE in the magnetic layer through the j x \nm term in the force equation. Here j and m are RF current \nand magnetization respectively, which may not be in \nphase. The RF current in our measurement geometry \ncomes  from an inductive coupling between the \nwaveguide and metallic film. The directions of rf current, \nrf magnetic field (h), dc field and voltage leads are \nsketched in Fig . 1(b ). While  the AMR related \nrectification effects in the geometry shown in the Fig. \n1(b) may not contribute to any symmetric dc signal as \nthe rf current is directed orthogonal to the dc voltage \nleads , it does give rise to  an asymmetric voltage  which \nvaries as cos2 HcosH where H is the angle between \ndc magnetic field and rf current  [20]. However , in the \ngeometry of our experiment the AHE does contribute to \na symmetric signal through  the relation  [40], \n 𝑉𝐴𝐻𝐸 =𝜌𝐴𝐻𝐸 .𝑑.𝐽𝑟𝑓\n2.𝑚𝑡𝑧.𝑐𝑜𝑠   (6)  \nwhere ρ AHE is the anomalous Hall resistivity, d is the \ndistance between the Hall contacts, Jrf is the rf current \ndensity , mtz is the perpendicular comp onent of the \ndynamic magnetization and  is the phase lag between \nrf current and dynamic magnetization. For both AMR \nand AHE related rectified voltages  is important in \naddition to the magnitudes  of AMR and AHE in FeGaB \nfilms. We have measured the PHE, AMR and AHE in a \n12 nm thick film of FeGaB patterned in a Hall bar \ngeometry.  Since  these transport phenome na are \ncharacteristic features of spin – orbit interaction \ndominated electronic transport in magnetic alloys  [41], \ntheir magnitude establishes the strength of spin  – orbit \ncoupling  in FeGaB  and their contribution to dc voltage \ngenerated under the FMR condition. The field \nsymmetrized in -plane resistivity ( xx) and PHE \nresistivity ( xy) of a 12 nm thick film measured at 300 K \n8 \n are displayed in Fig.6 as a function of the angle  \nbetween the magn etic field and transport current I.  \n \nFigure 6. Angular dependence of resistivity ρxy and ρxx at 0H = 1 Tesla \napplied in the film plane. Red solid line is a fit to (ρ𝐼𝐼−\nρ⊥).cos .𝑠𝑖𝑛   for PHE and ρ⊥+(ρ𝐼𝐼−ρ⊥).𝑐𝑜𝑠2  for AMR. Inset \nshows the measurement geometry for AMR and PHE. The external dc \nfield is in the plane of the film making angle  with the direction of \ncurrent (I). The field stays in the plane of the film when angle  = \nchanges from 0 to 360 degrees.  \nWe n ote in Fig. 6 that the xx and  xy follow  the predicted  \nangular dependence [ 42, 43 ] of the type cos2 and \ncossin respectively with,  = (ρ𝐼𝐼−ρ⊥) = 8.5 x 10-9 \nOhm.m. This yields a  of  0.4 %, which is five \ntimes smaller than the AMR seen in films of permalloy  \n[41]. This reduced value of AMR suggests that the \ncontribution of PHE to asymmetric signal will be much \nsmaller compared to that in permalloy films.   \nThe Hall resistivity ( xy) of the film measured at ambient \ntemperature ( 300 K) as a function of magnetic field is \nshown in Fig. 7. It follows the magnetic field dependence of the out of plane magnetization as shown in the inset \nof Fig. 7, and reaches a saturation value of (54 n m) at \n 1.4 T. The saturation field for xy is the same as the  \nsaturation field for out-of-plane magnetization. The xy \nfor this FeGaB field is  large compared to the AHE of \npermalloy  ( 0.96 n  m). Such  large Hall resistivity is \nalso seen in several other metallic glasses [ 40, 44].  \n \nFigure 7 . Hall resistivity ( ρxy) measurement of FGB12 sample at room \ntemperature. Inset shows the dc magnetization measurement of FGB12 \nsample in in -plane and out of plane geometry.  \nWe have estimated the possible  value of symmetric dc \nvoltage generated by AHE in FGB12 sample using Eq. 6 \nassuming Cos = 1 (Maximu m), precession angle C  \n0.20 and J rf is same (  1.15 x 106 A/m2) as in the CPW \nsignal line ( See supplementary sec - C). This yields a \npeak value of 0.27 V, which is  7 % of the total \nsymmetric signal  shown in Fig. 8(e).  \nHere we must also mention that in a co -planar waveguide \nconfiguration the rf field would eventually become \nperpendicular to the film plane following the Ampere’s \nLaw. As shown in Ref. 20, the perpendicular component \nwill yield H indep endent  field asymmetric AHE \nvoltage. This component of the rf field ( hrf⊥) also \nproduces a symmetric voltage due to PHE which does \nnot change polarity on field reversal  [45]. However, here \nwe ignore the possible contributions of the out -of-plane \n9 \n rf field due to two reasons: one – the inductive rf current \nis expected to be localized in the region of the sample \nwhich is just above the signal line of the waveguide and \ntherefore this region does  not see much of the h rf⊥. \nSecondly, The PHE contribution of h rf⊥ should act \nagainst the dc signal produced by AHE on field reversal. \nHowever, unlike the behavior seen in Ref. 40, in our case \nthe line shape and its magnitude remain the same when \nthe polari ty of the dc field is reversed.   \nThe second  source of dc voltage is the spin pumping \neffect of  precessing magnetization. In our case, the \nmagnetic dead layer at the film substrate interface may \npromote a unidirectional flow of spin current towards the \nsubstrate  (See supplementary Fig. S 2). This symmetry \nbreaking process would result in a dc voltage through the \ninverse spin Hall effect in the FeGaB near the interface. \nThe charge current produced by the ISHE is given as  \n𝑱𝒄=𝑆𝐻( 𝑱𝑠   )  (7) \nwhere SH is the spin Hall angle which represent s the \nISHE efficiency in a material,  is the spin polarization \nvector determined by the applied dc magnetic field and \nJs is the spin current density. The polarity of the dc \nvoltage measured in our experiment is consistent with \nEq. 7 if it is assumed that the spin Hall angle of FeGaB \nis positive. Since the spin pumping process involves \nincoherent spins their leads to a  volta ge signal that is  \nphase independent and symmetric around H r.   \n          We may also point out here that the rf  current \nflowing through the dead layer may produce an \nalternating spin current because of SHE, which on \nentering the FM will affect the precession of \nmagnetization and produce an antisymmetric rectified \nsignal due to AMR. Finally, heating of the magnetic \nstripe by the absorbed RF power at resonance may produce temperature gradients across the film thickness, \nand hence a symmetric Nernst voltage.  \nWe have decomposed the measured dc voltage into \nsymmetric and anti -symmetric Lorentzian contributions \n[46, 47], \n 𝑉𝑚𝑖𝑥 =𝐾𝑠𝐻2\n(𝐻−𝐻𝑟)2+𝐻2+𝐾𝑎𝑠−2𝐻(𝐻−𝐻𝑟)\n(𝐻−𝐻𝑟)2+𝐻2     (8) \nwhere K s and K as are symmetric and antisymmetric \ncoefficients  and H is the half width at half maxima .  \n \nFigure 8. (a-e) Shows the fitting of experimental data for the samples \nFGB4, FGB6, FGB8, FGB10 and FGB12 using Lorentzian symmetric \nand antiymmetric equation at  = 1800 where V mix = V symm + V asymm .  (f) \nPower dependent symmetric part (V symm) for the sample FGB10 at 10 \nGHz, inset shows the increase of maximum voltage as a function of \ninput RF power at 10 GHz.  \nThe result of this analysis for five films of different \nthicknesses is shown in Fig. 8 (a through e). In Fig. 8(f) \nwe show the symmetric component of the voltage of  10 \nnm thick FeGaB film extracted from the measurements \nVdc performed at different RF power.  \n10 \n A linear power dependence confirms that the data \nreported here have been collected in the linear regime of \nresponse.  In order to ensure that the measured dc signal \nis not an artifact of sample heating,  we have measured \nthe V mix signal of the 10 nm thick film at different scan \nrates of the dc field.  The slower scan rate amounts to a \nlarger dwell time near H r. However, even an order of \nmagnitude change in the sc an rate does not affect the \nposition or amplitude the Vmix voltage. An unperturbed \nvalue of resonance field when measurements were \nperformed at different rf power further confirm absence \nof sample heating in these experiments.  (See \nsupplementary Fig. S 4). The contributions of various \nspin rectification effects to the symmetric and \nantisymmetric part of the measured dc voltage are \ndiscussed in supplementary sec - E. We have estimated \nthese contributions for FGB12 sample.  A similar \nanalysis can be carried out  for thinner films as well with \nthe knowledge of their AHR.  \n  We now quantify the weight of symmetric (W s) and \nantisymmetric (W as) parts of the dc voltage generated in \nthe FeGaB thin films in terms of coefficient K s and K as \nas W s = (K s/ (K s + K as)) and W as = (K as/ (K s + K as)) [46]. \nThe variation of W s and W as as functions of film \nthickness is shown in Fig. 9.  It is interesting to note that \nthe symmetric weight decreases while antisymmetric \nweight increases with increase in thickness of FeGaB.  \nThis behavio r can be understood as follows: (i) The \nsymmetric contribution comes mainly due to the SOC in  \nFeGaB, which is larger in thinner films as suggested by \nthe variation of the g -factor shown in the inset of Fig. 3.  \nAlso, the rectification contribution of AHE may be larger \nas the xy scales with xx which is higher for thinner films \ndue to size effects.  (ii) The antisymmetric contribution  is \ndue to spin rectification effects  derived from PHE , is \nexpected to be larger in thicker films as they allow stronger inductive currents due to their high electrical \nconductivity (see Fig. 9 inset).  \n \n \nFigure 9.  Symmetric (W s) and antisymmetric weight (W as) of the DC \nsignal at f = 10 GHz plotted as a function of FeGaB thickness. W s = \nKs/(K s + K as) and W as = K as/(K s + K as). Inset shows the variation of  \nresistivity of FeGaB as a function of film thickness.  \nIV. Summary  \n We have measured the induced dc  voltage in \namorphous thin films of Fe78Ga13B9 alloy when their \nmagnetization is driven to resonant precession on \nexcitation with a microwave field of variable frequency. \nThis voltage , measured orthogonal to the direction of the \ndc magnetic field, is antis ymmetric on field reversal and \nits line shape has both symmetric and asymmetric \ncomponents centered around the FMR resonance field. \nThe measurements of static magnetization and frequency \ndependent FMR on films of different thicknesses \nindicate a dominant r ole of thin film size effects in \nsetting the values of M s, FMR linewidth, Gilbert \n11 \n damping parameter and g -factor. The thin film size \neffects become increasingly stronger as the film \nthickness drops below  8 nm. The thickness dependence \nof these static and  dynamic magnetization parameters \nalso indicates the formation of a  1.2 nm thick magnetic \ndead layer at the film – substrate interface.  We have \nconsidered several processes that may lead to an ISHE - \nlike voltage at FMR  in these films .  The RF power and  \ndwell -time dependence of V dc and dP/dH rule out any \ncontribution of thermally driven spin currents in \nproducing the V dc. The symmetric component of V dc and \nits polarity, on the other hand, suggest a preferential flow \nof spin currents produced by precessin g magnetization \ntowards the film – substrate interface. We have also \nestimated the contribution of anomalous Hall effect to \nthis dc voltage. The V dc appears to be a compound effect \nof ISHE in the FeGaB  film and its symmetry breaking \ndead -layer  together with spin  rectification effects of \nAHE . Our experiments suggest that the FMR induced \nVdc in plain films of high AHE amorphous alloys needs \nto be considered for correct interpretation of the ISHE \ndata on s pin orbit coupled normal metal – ferromagnet \nheterostructures.  \nAcknowledgements  \nThis research is supported by the Air Force Office of \nScientific Research, Grant # FA9550 -19-1-0082.  \n \nReference  \n[1] Tsukahara A, Ando Y, Kitamura Y, Emoto H, Shikoh \nE, Delmo M P, Shinjo T and Shiraishi M 2014 Self -\ninduced inverse spin Hall effect in permalloy at room \ntemperature Phys. Rev. B  89 235317  \n[2] Azevedo A, Cunha R O, Estrada F, Alves Santos O, \nMendes J B S, Vi lela-Leão L H, Rodriguez -Suárez R \nL and Rezende S M 2015 Electrical detection of \nferromagnetic resonance in single layers of \npermalloy: Evidence of magnonic charge pumping \nPhys. Rev. B  92 24402  [3] Weng Y -C, Luo G Y, Liang C -T and Lin J G 2017 \nThickness -Dependent Self -Induced Spin -Pumping in \nCobalt Thin Films IEEE Trans. Magn.  53 1–4 \n[4] Kanagawa K, Teki Y and Shikoh E 2018 Self -induced \ninverse spin -Hall effect in an iron and a cobalt single -\nlayer films themselves under the ferromagnetic \nresonance AIP Adv.  8 55910  \n[5] Gladii O, Frangou L, Hallal A, Seeger R L, Noël P, \nForestier G, Auffret S, Rubio -Roy M, Warin P, Vila \nL, Wimmer S, Ebert H, Gambarelli S, Chshiev M and \nBaltz V 2019 Self -induced inverse spin Hall effect in \nferromagnets: Demonstration through nonmonotonic \ntemperature dependence in permalloy Phys. Rev. B  \n100 174409  \n[6] Huang Y -H, Weng Y -C, Liang C -T and Lin J G 2020 \nSpin Hall angle and spin diffusion length of permalloy \nAIP Adv.  10 15041  \n[7] Sinova J, Valenzuela S O, Wunderlich J, Back C H \nand Jungwirth T 201 5 Spin Hall effects Rev. Mod. \nPhys.  87 1213 –60 \n[8] Niimi Y and Otani Y 2015 Reciprocal spin Hall \neffects in conductors with strong spinorbit coupling: a \nreview Reports Prog. Phys.  78 124501  \n[9] Hellman F, Hoffmann A, Tserkovnyak Y, Beach G S \nD, Fullerton E E, Leighton C, MacDonald A H, Ralph \nD C, Arena D A , Dürr H A, Fischer P, Grollier J, \nHeremans J P, Jungwirth T, Kimel A V, Koopmans B, \nKrivorotov I N, May S J, Petford -Long A K, \nRondinelli J M, Samarth N, Schuller I K, Slavin A N, \nStiles M D, Tchernyshyov O, Thiaville A and Zink B \nL 2017 Interface -induced  phenomena in magnetism \nRev. Mod. Phys.  89 25006  \n[10] Han W, Otani Y and Maekawa S 2018 Quantum \nmaterials for spin and charge conversion npj Quantum \nMater.  3 27 \n[11] Vedmedenko E Y, Kawakami R K, Sheka D D, \nGambardella P, Kirilyuk A, Hirohata A, Binek C, \nChubykalo -Fesenko O, Sanvito S, Kirby B J, Grollier \nJ, Everschor -Sitte K, Kampfrath T, You C -Y and \nBerger A 2020 The 2020 magnetism roadmap J. Phys. \nD. Appl. Phys.  53 453001  \n[12] Uchida K, Takahashi S, Harii K, Ieda J, Koshibae W, \nAndo K, Maekawa S and Sait oh E 2008 Observation \nof the spin Seebeck effect Nature  455 778–81 \n[13] Jaworski C M, Yang J, Mack S, Awschalom D D, \nHeremans J P and Myers R C 2010 Observation of the \nspin-Seebeck effect in a ferromagnetic semiconductor \nNat. Mater.  9 898–903 \n[14]  Uchida K, Ota T, Harii K, Ando K, Nakayama H and \nSaitoh E 2010 Electric detecti on of the spin -Seebeck \neffect in ferromagnetic metals (invited) J. Appl. Phys.  \n107 09A951  \n[15]  Saitoh E, Ueda M, Miyajima H and Tatara G 2006 12 \n Conversion of spin current into charge current at room \ntemperature: Inverse spin -Hall effect Appl. Phys. Lett.  \n88 182509  \n[16]  Mosendz O, Vlaminck V, Pearson J E, Fradin F Y, \nBauer G E W, Bader S D and Hoffmann A 2010 \nDetection and quantification of inverse spin Hall \neffect from spin pumping in permalloy/normal metal \nbilayers Phys. Rev. B  82 214403  \n[17]  Ando K, Taka hashi S, Ieda J, Kajiwara Y, Nakayama \nH, Yoshino T, Harii K, Fujikawa Y, Matsuo M, \nMaekawa S and Saitoh E 2011 Inverse spin -Hall \neffect induced by spin pumping in metallic system J. \nAppl. Phys.  109 103913  \n[18]  Liu L, Moriyama T, Ralph D C and Buhrman R A \n2011 Spin -Torque Ferromagnetic Resonance Induced \nby the Spin Hall Effect Phys. Rev. Lett.  106 36601  \n[19]  Harder M, Cao Z X, Gui Y S, Fan X L and Hu C -M \n2011 Analysis of the line shape of electrically detected \nferromagnetic resonance Phys. Rev. B  84 54423  \n[20]  Harder M, Gui Y and Hu C -M 2016 Electrical \ndetection of magnetization dynamics via spin \nrectificati on effects Phys. Rep.  661 1 –59 \n[21]  Ciccarelli C, Hals K M D, Irvine A, Novak V, \nTserkovnyak Y, Kurebayashi H, Brataas A and \nFerguson A 2015 Magnonic charge pumping via spin –\norbit coupling Nat. Nanotechnol.  10 50–4 \n[22]  Srisukhumbowornchai N and Guruswamy S 2001 \nLarge magnetostriction in directionally solidified \nFeGa and FeGaAl alloys J. Appl. Phys.  90 5680 –8 \n[23]  Wu R 2002 Origin of large magnetostriction in FeGa \nalloys J. Appl. Phys.  91 7358 –60 \n[24]  Kubota T and Inoue A 2004 Magnetostriction of Fe -\nGa-B Alloys in Amorphous and Crystallized States \nMater. Trans.  45 199–203 \n[25]  Lou J, Insignares R E, Cai Z, Ziemer K S, Liu M and \nSun N X 2007 Soft magnetism, magnetostriction, and \nmicrowave properties of  FeGaB thin films Appl. Phys. \nLett. 91 182504  \n[26]  Gao J, Yang A, Chen Y, Kirkland J P, Lou J, Sun N \nX, Vittoria C and Harris V G 2009 The effect of boron \naddition on the atomic structure and microwave \nmagnetic properties of FeGaB thin films J. Appl. \nPhys . 105 07A323  \n[27]  Emori S, Nan T, Oxholm T M, Boone C T, Jones J G, \nHowe B M, Brown G J, Budil D E and Sun N X 2015 \nQuantification of the spin -Hall anti -damping torque \nwith a resonance spectrometer Appl. Phys. Lett.  106 \n22406  \n[28]  Zhang Y Q, Sun N Y, Shan R, Zhang J W, Zhou S M, \nShi Z and Guo G Y 2013 Anomalous Hall effect in \nepitaxial permalloy thin films J. Appl. Phys.  114 \n163714  [29]  Gopman D B, Sampath V, Ahmad H, \nBandyopadhyay S and Atulasimha J 2017 Static and \nDynamic Magnetic Properties of Sputtered Fe –Ga \nThin Films IEEE Trans. Magn.  53 1–4 \n[30]  Begué A, Proietti M G, Arnaudas J I and Ciria M \n2020 Magnetic ripple domain structure  in FeGa/MgO \nthin films J. Magn. Magn. Mater.  498 166135  \n[31]  Mecking N, Gui Y S and Hu C -M 2007 Microwave \nphotovoltage and photoresistance effects in \nferromagnetic microstrips Phys. Rev. B  76 224430  \n[32]  Farle M 1998 Ferromagnetic resonance of ultrathin \nmetallic layers Reports Prog. Phys.  61 755–826 \n[33]  Budhani R C, Basun S A, McConney M E, Evans D \nR, Page M R and Howe B M 2019 Temperature \ndependent resonant microwave absorption in \nperpendicular magneti c anisotropy epitaxial films of a \nspinel ferrite J. Appl. Phys.  125 243903  \n[34]  Yuan S J, Wang L, Shan R and Zhou S M 2004 \nFerromagnetic resonance linewidth in CoNi/FeMn \nbilayers Appl. Phys. A  79 701 –5 \n[35]  Graët C Le, Spenato D, Beaulieu N, Dekadjevi D T, \nJay J -P, Pogossian S P, Warot -Fonrose B and Youssef \nJ Ben 2016 Driving mechanism for damping and g -\nfactor in non -amorphous ferromagnetic {CoFeZr} \nultrathin films {EPL} (Europhysics Lett.  115 17002  \n[36]  Schoen M A W, Thonig D, Schneider M L, Silva T J, \nNembach H T, Eriksson O, Karis O and Shaw J M \n2016 Ultra -low magnetic damping of a \nmetallic  ferromagnet Nat. Phys.  12 839–42 \n[37]  Lenz K, Wende H, Kuch W, Baberschke K, Nagy K \nand Jánossy A 2006 Two -magnon scattering and \nviscous Gilbert damping in ultrath in ferromagnets \nPhys. Rev. B  73 144424  \n[38]  Barati E, Cinal M, Edwards D M and Umerski A \n2014 Gilbert damping in magnetic layered systems \nPhys. Rev. B  90 14420  \n[39]  Tserkovnyak Y, Brataas A and Bauer G E W 2002 \nEnhanced Gilbert Damping in Thin Ferromagnetic \nFilms Phys. Rev. Lett.  88 117601  \n[40]  Chen H, Fan X, Zhou H, Wang W, Gui Y S, Hu C -M \nand Xue D 2013 Spin rectificati on enabled by \nanomalous Hall effect J. Appl. Phys.  113 17C732  \n[41]  McGuire T and Potter R 1975 Anisotropic \nmagnetoresistance in ferromagnetic 3d alloys IEEE \nTrans. Magn.  11 1018 –38 \n[42]  Tang H X, Kawakami R K, Awschalom D D and \nRoukes M L 2003 Giant Planar Hall Effect in \nEpitaxial (Ga,Mn)As Devices Phys. Rev. Lett.  90 \n107201  \n[43]  Taskin A A, Legg H F, Yang F, Sasaki S, Kanai Y, \nMatsumoto K, Rosch A and Ando Y 2017 Planar Hall 13 \n effect fro m the surface of topological insulators Nat. \nCommun.  8 1340  \n[44]  Wu W, Li J, Liao Z, Jiang H, Shen L, Gu L, Qiu X, \nYao Y and Bai H 2021 Linear magnetization \ndependence and large intrinsic anomalous Hall effect \nin Fe78Si9B13 metallic glasses Phys. Rev. B  104 \n64424  \n[45]  Keller S, Greser J, Schweizer M R, Conca A, Lauer \nV, Dubs C, Hillebrands B and Papaioannou E T 2017 \nRelative weight of the inverse spin -Hall and spin -\nrectification effects for metallic polycrystalline Py/Pt, \nepitaxial Fe/Pt, and insulating YIG/Pt bilayers : \nAngular dependent spin pumping measurements \nPhys. Rev. B  96 24437  \n[46]  Vlaminck V, Pearson J E, Bader S D and Hoffmann \nA 2013 Dependence of spin -pumping spin Hall effect \nmeasurements on layer thicknesses and stacking order \nPhys. Rev. B  88 64414  \n[47]  Kim S -I, Kim D -J, Seo M -S, Park B -G and Park S -Y \n2015 Stacking order depend ence of inverse spin Hall \neffect and anomalous Hall effect in spin pumping \nexperiments J. Appl. Phys.  117 17D901  \n \n \n "    },    {        "title": "0804.1050v1.Ferromagnetic_Josephson_junction_with_precessing_magnetization.pdf",        "content": "arXiv:0804.1050v1  [cond-mat.supr-con]  7 Apr 2008Ferromagnetic Josephson junction with precessing magneti zation\nManuel Houzet1\n1INAC/SPSMS, CEA Grenoble, 17 rue des Martyrs, 38054 Grenobl e Cedex, France\n(Dated: November 3, 2018)\nThe Josephson current in a diﬀusive superconductor/ferrom agnet/superconductor junction with\nprecessing magnetization is calculated within the quasicl assical theory of superconductivity. When\nthe junction is phase-biased, a stationary current (withou t a.c. component) can ﬂow through it\ndespite the non-equilibrium condition. A large critical cu rrent is predicted due to a dynamically\ninduced long range triplet proximity eﬀect. Such eﬀect coul d be observed in a conventional hybrid\ndevice close to the ferromagnetic resonance.\nPACS numbers: 74.45.+c, 76.50.+g.\nThe proximity eﬀect in a ferromagnetic (F) metal in\ncontact with a conventional superconductor (S) is usu-\nallyshortranged[1]. Indeed, the conversionfromCooper\npairsin the superconductortoAndreev pairsin the ferro-\nmagnet involves two electrons with opposite spins. They\nget quickly dephased due the large value of the exchange\nﬁeld acting on their spins. Yet, it was predicted that\na long range proximity eﬀect can take place when the\nmagnetization rotates spatially [2] or when spin-ﬂip pro-\ncesses take place [3] in vicinity of the F/S interface.\nIndeed, Andreev pairs involving electrons with parallel\nspins are then created and can propagate on a long dis-\ntance. The long range proximity eﬀect was invoked to\nexplain the large Josephson current measured through\nhalf-metallic (HM) chromium oxide [4] as well as super-\nconducting phase-periodic conductance oscillations in an\nAndreev interferometermade ofa long metallic wire with\nhelimagneticorder[5]. However,the crossoverfromshort\nrange to long range proximity eﬀect in the same device\nremains to be observed. Recently, it was suggested to\nmeasure the strong variations of the Josephson current\nthrough a trilayer ferromagnetic junction by tuning the\nrelative orientations of the magnetizations [6].\nIn this article, we propose that the long range triplet\nproximity eﬀect can be stimulated by varying in time\n(rather than in space) the orientation of the magnetiza-\ntion in the ferromagnet. Speciﬁcally, we consider a diﬀu-\nsive metallic S/F/S junction with dynamical precession\nof the magnetization, as shown in ﬁg. 1. In the ferro-\nmagnet, the conduction electrons feel the time-varying\nexchange ﬁeld:\nhF(t) =hF(sinθcosΩt,sinθsinΩt,cosθ) (1)\nwhich is proportional to the magnetization and assumed\nto be spatially uniform. Here, Ω /2πis the precession\nfrequency around ˆ z-axis,θis a tilt angle, and the ampli-\ntude of exchange ﬁeld, hF, is constant at temperatures\nwell below the Curie temperature. In the rotating frame,\nsuch precession can be viewed as a diﬀerence between\nspin-resolved chemical potentials in the leads, µs=/planckover2pi1Ωˆz.\nAt ﬁnite tilt angle, hFhas a component transverse to\nµs. Thus, it creates a non-equilibrium situation for theFIG. 1: Geometry of the ferromagnetic Josephson junction\nwith precessing magnetization and conﬁguration of d.c. and\nr.f. magnetic ﬁelds inducing the ferromagnetic resonance.\nconduction electrons. In particular, the device could act\nasaspin pump when the leadsarein the normalstate[7].\nMoreover, in the superconducting state, the non colinear\norientation of hFandµsgenerates a long range proxim-\nity eﬀect similar to the case studied in Ref. [6]. Despite\nthe non-equilibrium condition, we ﬁnd that a stationary\n(d.c. only) Josephson current ﬂows through the junction\nwhen a phase diﬀerence is applied to the leads. In a long\njunction, its amplitude is comparable to that of a normal\n(N) metallic Josephson junction with the same length,\nprovided that /planckover2pi1Ω is comparable with kBTc(Tcis the su-\nperconducting critical temperature of the leads) and θis\nlarge enough, while it would be exponentially suppressed\nin the absence of precession.\nRecently, a Josephson eﬀect was also predicted in\nS/HM/S junctions when magnetic impurities located at\nthe interfaces dynamically precess and mediate spin-ﬂip\ntunneling processes for the conduction electrons [8]. It\nwas also suggested that this eﬀect could be observed at\nthe ferromagneticresonance(FMR). In the present work,\nweaddresstheoppositecasewhentheFermienergymuch\nexceeds the exchange ﬁeld and the spin polarization in\nthe ferromagnet is small. When the junction’s length,\nL, much exceeds the ferromagnetic coherence length,\nξF=/radicalbig\n/planckover2pi1D/hF, whereDis the diﬀusion constant, we\nwill show that the Josephson current has two dominant\ncontributions. Theﬁrstonecomesfromlongrangetriplet2\nproximity eﬀect. The second one arises from the inter-\nference between short range and long range proximity\neﬀects out of equilibrium and it displays a non-analytical\ntemperature dependence close to Tc. Both have an oscil-\nlatory behaviour, depending on the ratios between /planckover2pi1Ω,\nkBTc, and the Thouless energy ET=/planckover2pi1D/L2. This pro-\nvides a mechanism for π-coupling [1] in long ferromag-\nnetic Josephson junctions. In the following, we derive\nboth components, discuss their properties and the con-\nditions to observe them in conventional F/S structures.\nWithin the quasiclassical theory of superconductivity,\nthe current ﬂowing through the junction is [9, 10]:\nI(t) =−πGL\n8eTr[τz(ˇg◦∇ˇg)K(t,t)],(2)\nwhere the quasiclassical Green’s function ˇ g(x,t,t′) is a\nmatrix in spin, Nambu, and Keldysh spaces and it obeys\nthe Usadel equation along the ferromagnetic layer:\n−iD∇(ˇg◦∇ˇg)+[(i∂t+hF(t).σ)τzδ(t−t1)◦,ˇg] = 0.(3)\nHere,Gis the conductance of the ferromagnetic layer,\nthe spatial derivative is taken along ˆ x-axis,eis the el-\nementary charge, σiandτj(i,j=x,y,z) are the Pauli\nmatrices in spin and Nambu spaces, respectively, and ◦\ndenotes the time convolution. (Units with /planckover2pi1=kB= 1\nare adopted from now.) Note also that the orbital ef-\nfect generated by the magnetization has been neglected\nin Eq. (3) as it is usually done [1]. Moreover, ˇ gobeys\nthe properties: Trˇ g= 0 and ˇ g◦ˇg= 1, and it has a\ntriangular structure in Keldysh space with the retarded,\nadvanced and Keldysh components: ˆ gR, ˆgA, and ˆgK, re-\nspectively. The normalisation condition is fulﬁlled pro-\nvided ˆgR/A◦ˆgR/A= 1 and ˆ gK= ˆgR◦ˆf−ˆf◦ˆgA,whereˆf\nis a distribution function matrix which obeys [ ˆf,τz] = 0\n[10].\nIn the present study, we assume that there is a good\nelectric contact at the F/S interfaces and we neglect the\ninverse proximity eﬀect in the leads. This yields the\nboundary conditions:\nˇg(x=±L\n2,t,t′) = ˇgs,∓χ/2(t−t′). (4)\nHere,χis the phase diﬀerence between the leads. The\nquasiclassical Green’s function in a superconductor with\nphaseφis deﬁned in energy space by ˆ gR/A\ns,φ(ε) = [−iετz+\n∆τφ//radicalbig\n∆2−(ε±iΓ)2] and ˆgK\ns,φ= (ˇgR\ns,φ−ˇgA\ns,φ)fT, where\nτφ= cosφτx−sinφτy,fT(ε) = tanh( ε/2T), and ∆ is\nthe conventional superconducting gap at temperature T.\nWe also introduced a small, phenomenological depairing\nparameter Γ which may account for the current-induced\ndepairing, as well as scattering on magnetic impurities in\nthe leads.\nInordertodeterminethe currentthroughthe junction,\none has to solve Eqs. (3)-(4). Despite a time-varying ex-\nchange ﬁeld in Usadel equation (3), the problem is equiv-alent to a stationary (though non-equilibrium) one. In-\ndeed, one may perform the unitary transformation:\nˇg(t,t′)→VU(t)ˇg(t,t′)U(t′)†V†, (5)\nwhereU(t) = exp( iΩtσz/2) transforms from the lab-\noratory frame into a rotating frame and absorbs the\ntime-dependent terms in Usadel equation (3), while V=\nexp(iασy/2), with tan α=hFsinθ/(hFcosθ+Ω/2), ro-\ntatesthe spinquantizationaxis. Speciﬁcally, Valignsthe\neﬀective exchange ﬁeld with amplitude J= [h2\nFsin2θ+\n(hFcosθ+ Ω/2]1/2along ˆz-axis and rotates the preces-\nsion axis away from it. Eq. (3) now takes a simple form\nin energy space:\n−iD∇(ˇg(x,ε)∇ˇg(x,ε))+[(ε+Jσz)τz,ˇg(x,ε)] = 0.(6)\nThe boundary conditions (4) yield: ˇ g(x=±L\n2,ε) =\nVˇgs,∓χ/2(ε+Ωσz/2)V†andthecurrent(2)takestheform\nI=−GL\n16e/integraldisplay\ndεTr[τz(ˇg∇ˇg)K], (7)\nLet us emphasize that the special time-dependence in\nEq. (1) is crucial for the non-equilibrium problem (3) to\nbe formulated as a stationary one, see Eq. (6). For a dif-\nferent time-dependence and/or for applied bias voltage,\none could use the procedure formulated in Ref. [11] to\nstudy numerically the conductance in S/N/S junctions.\nIn spite of the above simpliﬁcation, we are still dealing\nwith a complicated non-linear equation (6). Now, we\nassume that the temperature is close to Tc, so that ∆ ∝\n[Tc(Tc−T)]1/2is vanishingly small. Thus, we can look\nfor a solution ˇ g= ˇg0+ˇg1+ˇg2+...in the seriesexpansion\naround the normal state solution, when ∆ /Tc≪1. In\nzeroth order, one ﬁnds ˆ gR/A\n0=±τzand\nˆf0=f+−f−/bracketleftBigg\nsinα/parenleftBigg\nσxℜchqx\nchqL\n2+σyℑchqx\nchqL\n2/parenrightBigg\n−cosασz/bracketrightBigg\n,\n(8)\nwheref±(ε) = [fT(ε+ Ω/2)±fT(ε−Ω/2)]/2 andq=/radicalbig\n2iJ/D. We note that ˆf0is diagonal in Nambu space\nand the non diagonal components in spin space only ap-\npear at ﬁnite Ω. In the ﬁrst order in ∆, Eq. (6) yields:\n−iD∇2ˆgR/A\n1±{ε+Jσz,ˆgR/A\n1}= 0 (9)\nwhich is solved with ˆ gR\n1= ¯gR\n10+¯gR\n1.σ, where:\n¯gR\n1x=−FR\n−sinα/bracketleftbiggsinhk(L/2−x)\nsinhkLτχ/2\n+sinhk(L/2+x)\nsinhkLτ−χ/2/bracketrightbigg\n, (10a)\n¯gR\n10±¯gR\n1z= (FR\n+±FR\n−cosα)/bracketleftbiggsinhp±(L/2−x)\nsinhp±Lτχ/2\n+sinhp±(L/2+x)\nsinhp±Lτ−χ/2/bracketrightbigg\n, (10b)3\nand ¯gR\n1y= 0 while ˆ gA\n1=−τz(ˆgR\n1)†τz. Here, k=/radicalbig\n−2iε/D,p±=/radicalbig\n−2i(ε±J)/D, andFR\n±(ε) =\n[FR(ε+ Ω/2)±FR(ε−Ω/2)]/2, where FR(ε) =\ni∆//radicalbig\n(ε+iΓ2)−∆2[12]. We note that the component\n¯g1xis long ranged, while ¯ g10and ¯g1zare short ranged. It\nis straightforward to check that the Keldysh component\nis solved with ˆf1= 0 and that the current (7) vanishes\nup to the ﬁrst order in ∆.\nIn the second order in ∆, the current is:\nI=−GL\n16e/integraldisplay\ndεTr/bracketleftBig\nτz/parenleftBig\n2∇ˆf2−ˆgR\n1∇ˆf0ˆgA\n1\n+ℜ/braceleftBig\n[ˆgR\n1∇ˆgR\n1−∇ˆgR\n1ˆgR\n1]ˆf0−[ˆgR\n1]2∇ˆf0/bracerightBig/parenrightBig/bracketrightBig\n.(11)\n(The terms with ˆ gR/A\n2have been eliminated with help\nof the identities: {τz,ˆgR/A\n2}±(ˆgR/A\n1)2= 0 coming from\nthe normalization condition ˇ g2= 1.) The function ˆf2\nvanishes at the boundaries with the leads and solves the\ndiﬀerential equation:\n−2iD∇2ˆf2+2[Jσz,ˆf2] =−iD/parenleftBig\n∇ˆgR\n1ˆgR\n1∇ˆf0\n+∇ˆf0ˆgA\n1∇ˆgA\n1+∇(ˆgR\n1∇ˆf0ˆgA\n1)/parenrightBig\n.(12)\nClose to Tc, the Josephson relation remains sinusoidal,\nI=Icsinχ, and the critical current Iccan be obtained\nby evaluating Eq. (11).\nBefore proceeding, let us make simplifying assump-\ntions. In usual ferromagnets, hFwould exceed Ω and\nTcby several orders of magnitude. Therefore, α≃θ,\nJ≃hFandp−≃q. We also assume that the junction is\nlong (L≫ξF). Thus, we discard the exponentially small\nterms contributing to Icand we ﬁnd that it has two main\nterms:Ic=It\nc+Ist\nc. The ﬁrst one comes from the long\nrange triplet proximity eﬀect (component ¯ g1x), only:\nIt\nc=−Gsin2θ\n2e/integraldisplay\ndεf+ℑ/bracketleftbigg\n(FR\n−)2kL\nsinhkL/bracketrightbigg\n=−GπT∆2sin2θ\neℜ/summationdisplay\nω>0/parenleftbigg1\nω+iΩ−1\nω/parenrightbigg2κωL\nsinhκωL,\n(13)\nwhereω= (2n+ 1)πTare Matsubara frequencies and\nκω=/radicalbig\n2(ω+iΩ/2)/D. In particular, we get\nIt\nc/Ic0≃/braceleftbigg1\n24(Ω/Tc)2sin2θif Ω≪Tc≪ET,\n−1\n2sin2θ ifTc≪Ω≪ET.(14)\nHere,Ic0=πG∆2/4eTcisthecriticalcurrentforaS/N/S\njunction with Tc≪ETat temperatures close to Tc. We\nnotice that It\ncvanishes at Ω = πTc. At larger frequency,\nit changes its sign and a π-coupling between the super-\nconducting leads is realized. When Ω ∼ET, we also ﬁnd\nan oscillatory behaviour of the frequency dependence of\nIt\ncon the scale of ET(see ﬁg. 2) with an alternation of\n0- andπ-couplings./s48 /s50 /s52 /s54 /s56/s45/s48/s44/s53/s45/s48/s44/s52/s45/s48/s44/s51/s45/s48/s44/s50/s45/s48/s44/s49/s48/s44/s48\n/s32/s32/s73\n/s99/s116\n/s32/s47/s32/s40 /s71/s50\n/s32/s115/s105/s110/s50\n/s47/s32/s52/s101/s84\n/s99/s41\n/s47/s50 /s84\n/s99/s32/s69\n/s84/s47/s52 /s84\n/s99/s61/s48/s46/s48/s53\n/s32/s69\n/s84/s47/s52 /s84\n/s99/s61/s48/s46/s49\n/s32/s69\n/s84/s47/s52 /s84\n/s99/s61/s48/s46/s50\n/s32/s69\n/s84/s47/s52 /s84\n/s99/s61/s48/s46/s53\n/s32/s69\n/s84/s47/s52 /s84\n/s99/s61\n/s48 /s50 /s52 /s54 /s56/s45/s48/s44/s50/s48/s44/s48/s48/s44/s50/s48/s44/s52/s48/s44/s54/s48/s44/s56/s49/s44/s48/s73\n/s99/s115/s116\n/s32/s47/s32/s91/s40/s71\n/s70/s47/s51/s50/s101/s76/s41/s32/s115/s105/s110/s50\n/s99/s111/s115 /s108/s110/s40 /s47 /s41/s93/s32/s32\n/s47/s50 /s84\n/s99\nFIG. 2: Precession frequency dependence of long range (left )\nand “anomalous” (right) contributions to the critical curr ent\nfor ferromagnetic Josephson junctions with diﬀerent lengt hs.\nIn contrast to ferromagnetic Josephson junctions with\nspatial variation of the magnetization and S/HM/S junc-\ntions, there arises a mixing of short range and long\nrangeproximityeﬀectsthroughthenon-equilibriumspin-\ndependent terms of the distribution function matrix in\nthe system studied here. Most of the terms in Eq. (11)\nreﬂecting such mixing are suppressed by the small factor\nξF/L, when compared to It\nc. However, the contribution\nIst\nc=GξFsin2θcosθ\n8eL/integraldisplay\ndεf−|FR\n−|2ℜkL\nsinhkL(15)\ncoming from the “anomalous term” ∝ˆgR\n1∇ˆf0ˆgA\n1in\nEqs. (11)-(12) needs more care. Indeed, for this term, a\nﬁnite depairingparameterΓis necessarytoregularizethe\notherwise diverging integral over the energies. Assuming\nΓ≪∆(T) and Ω ≪ET, we get in the logarithmic ap-\nproximation:\nIst\nc≃G∆\n32e/parenleftbiggξF\nL/parenrightbigg\nsin2θcosθln∆\nΓtanhΩ\nTc.(16)\nThis term vanishes like ( Tc−T)1/2ln(Tc−T) close to\nTc. Such non analytic temperature dependence was al-\nready met in the ﬁeld of transport in N/S junctions [13].\nWhen Ω ∼ET, we also get from Eq. (15) an oscillatory\nfrequency dependence of Ist\nc(see ﬁg. 2).\nLet us now compare both main contributions (13) and\n(15) toIc. They have quite diﬀerent dependences on\nthe device parameters. At low precession frequency com-\nparedto Tc,Ist\ncscaleslinearlywith Ω anddominatesover\nIt\ncwhichscalesquadratically. Onthe otherhand, atlarge\nfrequency, It\nctakes over if ∆ /Tc>∼(ξF/L)ln(∆/Γ). At\nlow temperatures, we speculate that the above expres-\nsions still hold qualitatively. As the ratio ∆ /Tcbecomes\nof the order of unity, the critical current at L≫ξFis\ndominated by the long range triplet component only.\nSimplifyingassumptionshavebeenmadetogetanalyt-\nical expressions for the current. In principle, more realis-\ntic conditionscouldalsobe studied. TheUsadel equation4\n(6) may be solved numerically to get the current at all\ntemperatures. Spin-ﬂip diﬀusion in the ferromagnet and\ntunnel barriers at the F/S interfaces could easily be in-\ncorporated in the theory. On the other hand, a ﬁnite\nspin polarization in the ferromagnet, as well as a mean\nfreepath comparablewith thecoherencelengthwouldre-\nquire to go beyond the quasiclassical theory for diﬀusive\nmetals. Nevertheless, we expect that Eqs. (13)-(16) hold\nqualitatively beyond their strict range of validity. Thus,\nour ﬁndings are relevant to the study of F/S junctions\nwith conventional metals. The quasiclassical theory has\nproved its usefulness for the prediction and quantitative\nanalysis of several experiments in these systems [1].\nThe dynamically induced long range proximity eﬀect\nstudied in this work is expected to have special proper-\nties. In particular, it would be of interest to character-\nize the current-voltage characteristics in S/F and S/F/S\njunctions. Wemayexpect ana.c. currentresponseatfre-\nquenciesmixing Ω and the Josephsonfrequency. We note\nthis has been studied in the context of tunneling trans-\nport through a single magnetic impurity placed between\ntwo superconductors [14] as well as S/HM/S junctions\nwith extended interfaces [15].\nThe long range Josephson current may be observed by\nperforming FMR experiments in the microwave regime\n[8] in a planar S/F/S junction. This geometry was used\nin N/F/N junctions [17, 18] to detect electrically the spin\npumping [7] due to a precessing magnetization. A large\nresonance frequency, such that /planckover2pi1Ω0∼kBTc(1K corre-\nsponds to 30GHz), can be reached even by applying a\nmoderate d.c. magnetic ﬁeld H0along the plane of the\nferromagnet [16]. A small transverse r.f. ﬁeld, Hrf, is\nused to induce the magnetization precession. The tilt\nangleθstrongly depends on the precession frequency. At\nresonance, θcan be estimated from the Landau-Lifshitz\nequations: θ∼µHrf/a/planckover2pi1Ω0, whereµis the Bohr magne-\ntonandaistheGilbertdampingparameter. Insoftferro-\nmagnets, a quite large θ≃15ocould be obtained. Away\nfrom resonance, θis much reduced. This would induce\na strong frequency dependence, viaangleθ, in the ex-\npressions for the critical current derived above. We note\nthat FMR was recently performed in an Nb/Permalloy\nbilayer [19]. Below the superconducting critical tempera-\nture, the reduction of the resonance width was observed.\nIt was attributed to an eﬃcient proximity eﬀect leading\nto the reduced eﬃciency of spin-ﬂip processes below Tc\n[20]. In particular, this shows that the proximity eﬀect\nstill persists at FMR. If the same conditions can be met\nin an S/F/S junction, we expect that the large critical\ncurrent predicted in this work will be well measurable.\nIn conclusion, we have proposed a model of ferromag-\nnetic Josephson junction with precessing magnetization\nwhere a large current ﬂows thanks to a dynamically in-\nduced long range proximity eﬀect. The dependence of\nthe current with the precession frequency shows an os-cillatory behavior. We have discussed the conditions for\nthis eﬀect to be observed under the condition of ferro-\nmagnetic resonance in superconductor/ferromagnet de-\nvices with conventional metals.\nThe work presented above has beneﬁted from discus-\nsions with S. Teber, R. M´ elin and D. Feinberg on a re-\nlated problem, as well as with A. Buzdin, M. Aprili and\nT. Champel. Support from ANR-07-NANO011 ELEC-\nEPR is acknowledged.\n[1] A. I. Buzdin, Rev. Mod. Phys. 77, 935 (2005).\n[2] A.Kadigrobov, R.I.Shekhter,andM. Jonson, Europhys.\nLett.54, 394 (2001); F. S. Bergeret, A. F. Volkov, and K.\nB. Efetov, Phys. Rev. Lett. 86, 4096 (2001); Rev. Mod.\nPhys.77, 1321 (2005).\n[3] M. Eschrig, J. Kopu, J. C. Cuevas, and G. Sch¨ on, Phys.\nRev.Lett. 90, 137003 (2003); Y.Asano, Y. Tanaka, A.A.\nGolubov, Phys. Rev. Lett. 98, 107002 (2007); M. Eschrig\nand T. L¨ ofwander, Nature Physics 4, 138 (2008).\n[4] R. S. Keizer, S. T. B. Goennenwein, T. M. Klapwijk, G.\nMiao, G. Xiao, A. Gupta, Nature 439, 825 (2006).\n[5] I. Sosnin, H. Cho, V. T. Petrashov, and A. F. Volkov,\nPhys. Rev. Lett. 96, 157002 (2006).\n[6] V. Braude and Yu. V. Nazarov, Phys. Rev. Lett. 98,\n077003 (2007) ; M. Houzet and A. I. Buzdin, Phys. Rev.\nB76, 060504 (2007).\n[7] Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I.\nHalperin, Rev. Mod. Phys. 77, 1375 (2005).\n[8] S. Takahashi, S. Hikino, M. Mori, J. Martinek, and S.\nMaekawa, Phys. Rev. Lett. 99, 057003 (2007).\n[9] K. D. Usadel, Phys. Rev. Lett. 25, 507 (1970).\n[10] A. I. Larkin and Yu. N. Ovchinnikov, in Nonequilib-\nrium superconductivity (Eds. D. N. Langenberg and A.I.\nLarkin), Elsevier Science Publishers (1986).\n[11] J. C. Cuevas, J. Hammer, J. Kopu, J. K. Viljas, and M.\nEschrig, Phys. Rev. B 73, 184505 (2006).\n[12] The retarded anomalous function reduces to FR(ε) =\ni∆/(ε+i0+) when ∆ ,Γ≪ |ε|. Below, we will see a cir-\ncumstance where the full denominator should be kept,\nstill assuming ∆ ,Γ≪Tc.\n[13] R. Seviour, C. J. Lambert, and A. F. Volkov, Phys. Rev.\nB59, 6031 (1999).\n[14] Jian-Xin Zhu and A. V. Balatsky, Phys. Rev. B 67,\n174505 (2003).\n[15] S. Hikino, M. Mori, S. Takahashi, S. Maekawa,\narXiv:0802.1755.\n[16] C. Kittel, Introduction to solid state physics (John Wiley,\n2005).\n[17] M. V. Costache, M. Sladkov, S. M. Watts, C. H. van der\nWal, and B. J. van Wees, Phys. Rev. Lett. 97, 216603\n(2006).\n[18] T. Moriyama, R. Cao, X. Fan, G. Xuan, B. K. Nikolic,\nY. Tserkovnyak, J. Kolodzey, and John Q. Xiao, Phys.\nRev. Lett. 100, 067602 (2008).\n[19] C. Bell, S. Milikisyants, M. Huber, and J. Aarts, Phys.\nRev. Lett. 100, 047002 (2008).\n[20] J. P. Morten, A. Brataas, G. E. W. Bauer, W. Belzig, Y.\nTserkovnyak, arXiv:0712.2814."    },    {        "title": "1708.02402v2.Spin_orbit_torque_driven_magnetoimpedance_in_Pt_layer_magnetic_ribbon_heterostructures.pdf",        "content": "1 \n Spin-orbit -torque driven magnetoimpedance in  \nPt-layer/magnetic -ribbon heterostructures  \nM. R. Hajiali1, †, S. Morteza Mohseni2, †, &, L. Jamilpanah2, M. Hamdi2,  \nS. E. Roozmeh1, S. Majid. Mohseni 2, * \n1Department of Physics, University of Kashan, 87317 Kashan, Iran  \n2Faculty of Physics, Shahid Beheshti University, Evin, 19839 Tehran, Iran  \n \nWhen a flow of electron passes through a paramagnetic layer with strong spin -orbit -coupling  such as  \nplatinum  (Pt), a net spin current is produced via spin Hall effect (SHE) . This spin current can exert a \ntorque on the magnetization of an adjacent ferromagnetic layer  which can be probed via magnetization \ndynamic response , e.g. spin-torque ferromagnetic resonance  (ST-FMR) . Nevertheless,  that effect in \nlower frequency magnetization dynamic regime  (MHz)  where skin effect occurs in high permeability \nferromagnetic conductors namely the magneto -impedance (MI) effect can be fundamentally important  \nwhich has  not been  studied  so far . Here, by u tilizing the MI effect in magnetic -ribbon/Pt heterostructure  \nwith high transvers magnetic permeability that allow s the ac current effectively confined  at the skin \ndepth of ~100 nm thickness , the effect of spin-orbit -torque (SOT)  induced by  the SHE  probed via  MI \nmeasurement  is investigated . We observe d a systematic MI  frequency shift that increase s by increasing \nthe applied current amplitude and thickness of the Pt layer (varying from 0 nm to 20 nm) . In addi tion, \nthe role of Pt layer in ribbon/Pt heterostructure  is evaluated with ferromagnetic resonance (FMR) effect \nrepresenting standard Gilbert damping increase as the result of presence of the SHE. Our results unveil \nthe role of SOT  in dynamic control of  the transverse magnetic permeability probed with impedance \nspectroscopy  as useful and valuable technique for detection of future SHE devices .  \n \n \n \n \n \n \n \n       \n \n*Corresponding author’s email address: m-mohseni@sbu.ac.ir , majidmohseni@gmail.com  \n† These authors contributed equally.  \n& Current affiliation: Fachbereich Physik and Landesforschungszentrum OPTIMAS, Technische Universität  \nKaiserslautern, 67663 Kaiserslautern, Germany.  2 \n Spin-orbit torques (SOTs) generated by current injection in ferromagnet (FM)/ heavy -metal (HM) \nheterostructures  have attracted considerable attention as a method to effectively manipulate the \nmagnetization of thin FM films1–7. The spin Hall effect (SHE)8 is reported to be the dominant source of \nthe damping -like (DL) SOT  in such heterostructures  that is responsible for magnetization switching9,10, \ndomain  wall (DW)  motion11–13, skyrmion manipulation14,15 and high-frequency magnetization \ndynamics16–18. Rashba -Edelstein effect (REE)  as another source of SOT is also present in FM/HM \nbilayers and depends  on the interface structure of the se bilayer and the ir corresponding thicknesses19. \nThis mechanism can result in  the exertion of field -like (FL) torque on FM.  \nQuantif ication of  the SOTs in FM/HM heterostructures  are based on  spin-torque ferromagnetic \nresonance (ST -FMR)20, planar Hall effect21, low-frequency (~max imum  to few 100 Hz) harmonic Hall \nvoltage6,22, spin Hall magnetoresistance23–25, DW creep velocity26 and magneto -optical effect27. They \nrequires high frequency (few GHz) instruments  or need a complicated assessment process , hence, \ndemonstration of  SOT  materials and techniques  in low frequency regime (MHz) via easy experimental  \nprocess is desirable . \nThe studied heterostructure  in this letter is made of a FM Co68.15Fe4.35Si12.5B15 ribbon and a thin layer of \nplatinum  (Pt). Such ribbon among soft magnetic material is one of the most promising candidates  for \nthe MI effect28-30, primarily because of its application in low -cost and high sensitive magnetic \nsensors31,32 and MI magnetic random access memory (MRAM)33. This effect causes a change in \nelectrical impedance of a conducting FM  with high transverse magnetic permeability  (𝜇𝑡) in the \npresence of a static magnetic field34. By applying an external magnetic field, the skin depth ( 𝛿) changes \ndue to change in 𝜇𝑡, thus varying the impedance of the FM. In the case of the ribbon, with width 𝑙 and \nlength 𝐿, the impedance is approximately35  \n𝑍=(1−𝑖)𝜌𝐿\n2𝑙𝛿=(1−𝑖)𝐿\n2𝑙(𝜋𝜌𝑓𝜇𝑡)1\n2 \n  \n(1) \nwhere 𝜌 is electric resistivity, 𝑓 is frequency of the current and 𝑖 = imaginary unit. Therefor e, the \nimpedance of the ribbon is a function of frequency, driving current and the external dc magnetic field \n(Hdc) through 𝜇𝑡 and 𝛿. \nHere, we study SOT effect on the MI response of the Co-based amorphous ribbon  (~30 µm) /Pt (0-20 \nnm) heterostructure  by measuring its external magnetic field and frequency dependence impedance  \nresponse . It is however noted that , the MI will be studied in a system including a thick FM layer, but \nbased on the aforementioned skin effect , the current distributes at approximately 100 nm thickness close \nto the thin Pt layer  deposited at the interface. This  enables to uncover the dynamic of domain and DW  \nin the present of SOT within a thin region of skin depth . We observe that the impedance response is \nstrongly dependent on the thicknesses of the Pt layer and the applied current amplitude . Our  results \nreveal the possibility of SOT  detection in a FM using the impedance spectroscopy of FM in low \nfrequencies ~MHz  in high transvers magnetic permeability structures. Moreover, we have used 3 \n ferromagnetic resonance  (FMR ) (see section S3 of the supplementary material s) for a better \nunderstanding of the mechanism happening in this system to represent the role of SOT effect based on \nfundamental and standard measurement to confirm the validity of our technique.  \nAmorphous Co-based ribbons (2 mm width , 30 mm length and ~30 µm thickness) were prepared by a \nconventional melt -spinning technique. Before deposition of Pt layer, about 40 nm of ribbons surface \nwere sputter etched via Ar to have clean and oxygen free components. Pt thin layers with thickness o f \n10 nm and 20 nm were deposited on the soft surface (wheel side) of those ribbons in the Ar with gas \npressure of 5 mTorr, base pressure better than 5×10-8 Torr and growth rate of 3 nm/minute. (See \nsupplementary materials including MI measurement , X-ray diffraction (XRD)  analysis  and FMR \nmeasurements)  \nSchematic illustration of a FM/ HM heterostructure system and the definition of the Cartesian coordinate \nsystem in this work are presented in Fig. 1.  The high 𝜇𝑡 of these ribbons allows the skin effect to occur \nin the MHz frequency range with thickness  <100 nm . As shown schematically in Fig. 1(a), an ac charge \ncurrent in the HM layer generates a pure spin -current, oscillating at the same frequency, perpendicular \nto the charge curre nt direction thanks to the SHE. This oscillating spin current flows into the adjacent \nFM layer, exerts two different types of oscillating SOT s5,6,3 6. \n \n \nFIG 1:  (a) Schematic illustration of a FM /HM heterostructure  system. An in -plane charge current Iac generates a perpendicular \nspin current, which in turn generates SOTs acting on ferromagnetic moments.  Oscillations of the magnetization due to ( b) \ndamping -like SOT (T AD) and ( c) the field -like SOT and Oersted field (T FL + T Oe) induced by an ac current.  We should note \nhere that this scenario happen s at the  skin depth 𝛿 of the ribbon.  \nThey are field-like (FL) torque TFL ∼ m × y and damping -like (DL) torque TDL ∼ m × (y × m), where \nm is the magnetization unit vector and y is the in -plane axis perpendicular to current flow direction x \n(Fig. 1 (b, c)). TDL originates from the SHE in the adjacent HM layer. The magnitude of this TDL depends \n4 \n on the transmission of spin current across the FM/HM interface36,37. TFL can be originated  from the REE  \nat the FM/ HM interface due to the structural inversion asymmetry or from the spin current through  HM \nvia the SHE19. When the magnetization lies in -the-plane  of the bilayer sample , the action of TFL is \nequivalent to an in -plane field hFL ∼ y, and that of TDL establishes an out -of-plane field hDL ∼ m × y.  \nAlthough we have  a thick FM layer, a t the studied frequency range ( 1-25 MHz ) in our sample s, the \ncurrent passes through the skin depth 𝛿 of ribbon  (~few nm to 100 nm) , therefore the above mentioned \nscenario is valid in the MI measurement that we introduce here . The magnitude  of TFL varies  \nsignificantly  with thickness of FM  layer5, the type of FM and HM38,39 and the direction of magnetization \nin the FM37. Two origins of TFL are known generally, one due to REE  and the other due to SHE. It is \nadmitted that TFL due to REE, reveals in FM/ HM heterostructures  with 1 -nm-thick FM41,42 and TFL due \nto SHE remains very weak in metallic systems43. Also it is shown that  very large  TFL occur s in magnetic \ntunnel junctions44 and HM/ nonmagnetic /FM/oxide heterostructures45. Therefor e, in our studied \nheterostructure  the contributions of TFL can be neglected  because of the metallic  nature of  layers  and \nlarge thicknesses of  FM layer . From now on, we consider both TDL that comes primarily from the SHE \nand Oersted  torque (TOe) due to Oersted field  generated from the charge current  that depends upon the \nconductivity of each layer and skin depth  𝛿. As the thickness of FM layer is much larger than its skin \ndepth, the Oersted field from FM layer can be important20. In order to detect the effect of these  torque s \nwe carry out  impedance measurement by applying an ac charge current with frequency 𝑓 to the samples , \nand investigate  how the impedance 𝑍 of the bilayer changes as a function of frequency  and field .  \nComparison of frequency sweep MI measurement for 0, 10 and 20 nm Pt is shown in Fig 2(a) where an \nexternal field of 120 Oe was applied to saturate the sample in the plane and an ac current with peak to \npeak amplitude of 66 mA was used to excite the sample.  According to equation 1 and based on \nliteratures arguments the impedance depends on current frequency 𝑓 and transverse magnetic \npermeability  𝜇𝑡(𝑓) with decreas ing trend  at high frequencies46. Therefore, with increasing 𝑓, the \nimpedance of sample increases up to some frequency and further increase of 𝑓 results in the reduction \nof impedance where strong reduction of 𝜇𝑡(𝑓) occurs.   \nWhen an ac current flows through a FM layer the magnetization oscillates about its equilibriu m position , \ny direction , due to Oersted field. Because of the presence of Pt layer, generated spin currents due to the \nSHE  from Pt layer consequences into T DL that derives the  magnetization oscillation in the z direction. \nIt can be seen that the frequency of the maximum impedance of the sample shifts towards high -\nfrequency values, increased from 17 MHz for 0 nm Pt to 18.5 and 19.5 MHz for 10 and 20 nm Pt \ndeposited ribbons, respectively. There is another confirmation for this fact (that will be discussed lat er) \nthat MI versus field shows reduced transverse anisotropy as the magnetization oscillation changed its \norientation toward z direction. We speculate that the angle of precession decreases from that transversal \norientation (without T DL) and the peak posit ion that represents the maximum 𝜇𝑡 goes to higher values 5 \n as shown in Fig. 2(b). The magnitude of h DL can be affected by changing the thickness of the HM and \nFM5 while we have varying thickness of the HM layer. The spin current in FM/HM heterostructures  \nobeys 𝐽𝑠(𝑡) ≈1−sech(𝑡𝐻𝑀𝜆𝑆𝑑⁄)20, where 𝑡𝐻𝑀 is the thickness of HM and 𝜆𝑆𝑑 is the spin -diffusion \nlength. In this relation, as the spin diffusion length of Pt layers is in the Co 75Fe25/Pt bilayer film was \nestimated to be 2.1 ± 0.2 nm47, therefore 𝐽𝑠(𝑡) does not have to change for 10 and 20 nm thickness of \nPt contrary to the frequency shift observed from Pt (10  nm) to that for Pt (20  nm), represented in Fig. \n2(b). However, this effect can be explained based on the resistivity of FM and HM layers. The resistivity  \nof Pt layer is 𝜌 =20 𝜇Ωcm and that for the ribbon is 𝜌 =130 𝜇Ωcm which might pinpoint as a fact that \nat the skin depth region the current in the thicker Pt deposited layer is more than when Pt thickness is \n10 nm. This implies a fact that the current effect and therefore the SHE effect is more significant for \nsample d eposited with 20 nm Pt with enhanced T DL that results in frequency shift.  \n16000000 18000000 200000000.960.981.00\n  \n 0 nm\n 10 nm\n 20 nmZnorm\nFrequency (MHz)(a)\n20 18 16\n \n0 5 10 15 201617181920\n  Frequency Max (MHz)\nPt Thickness (nm)(b)  \nFIG. 2. (a) Frequency sweep of impedance  measurement  of 0, 10 and 20 nm Pt  deposited  ribbon  normalized to maximum \nshowing a shift towards higher f. (b) The maximum frequency vs Pt thickness obtained from (a)  showing higher shift for higher \nthickness of Pt . \nWe measured the frequency sweep  of the MI response with different amplitude  of current app lied to \nthe samples to better elucidate the origin of the frequency shift. Considering the relation between the \nspin current 𝐽𝑠 and the charge current ( 𝐽𝑐), increasing the applied current amplitude results in higher \nspin current generation and higher h DL magnitude (h DL ∝𝐽𝑠)43,48. Frequency sweep impedance \nmeasurement against  ac current with peak to peak amplitude of 33, 66 and 99 mA are shown in Fig. 3 \n(a-c) for 0, 10 and 20 nm Pt deposited ribbons while the ribbons were saturated at 120 Oe. It is cle ar \nfrom Fig. 3(a) that increasing the magnitude of the applied current for 0 nm Pt does not affect the  peak \nposition of the impedance . Whereas for 10 and 20 nm Pt deposited ribbons, increasing the amplitude of \nthe applied current results in a shift in the  maximum impedance frequency. A comparison  between the \nmaximum impedance frequency shift and the applied current for all samples is shown in Fig. 3(d). As \ncan be seen, the role of current for 20 nm Pt is more pronounced with larger frequency -current slop.  6 \n \n15.0M 20.0M0.960.981.00\n18.0M 21.0M0.960.981.0015.0M 20.0M0.960.981.00\n40 60 80 10016182022\n  Znorm \nFrequency (MHz)(b) 10 nm   Znorm \nFrequency (MHz)(c) 20 nm\n  \n 33 mA\n 66 mA\n 99 mAZnorm\nFrequency (MHz)(a) 0 nm\n  Frequency Max  (MHz)\nI (mA) 0 nm \n 10 nm \n 20 nm  (d) \nFIG. 3. Frequency sweep  impedance measurement of (a) 0, (b) 10 and (c) 20 nm Pt at the presence of  an external field of 120  \nOe in different ac current peak to peak amplitude of 33, 66  and 99  mA with a higher frequency shift for higher driving currents . \n(d) Maximum frequency obtained from (a), (b) and (c) versus  ac current amplitude  indicating higher slope of increment for 20 \nnm Pt deposited  ribbon than that for 10 nm Pt deposited  one. \nMI response of a ribbon can give us detailed  information about magnetic anisotropy and transverse \nmagnetic permeability . Therefore, we measured field sweep impedance measurement in an arbitrary \nfrequency of 6 MHz for 0, 10 and 20 nm Pt samples with 66 mA c urrent applied to the samples, as \npresented in Fig. 4(a, b) . It is considered that based on equation S1 in supplement ary materials , the MI \ndecreases from 191% for bare sample to 169% for 10 nm Pt and 152% for 20 nm Pt deposited samples. \nThis behavior is consistent with the TDL tends to reduce the 𝜇𝑡 by exerting a torque perpendicular to \nequilibrium angle of magnetization. As can be seen in Fig. 4(b), the bare ribbon s hows a double peak \nbehavior and Pt deposited ribbons show a single peak behavior.  The observed single - or double -peak \nbehaviors are associated with the longitudinal or transverse magnetic anisotropy with respect to the \nexternal field direction49,50. The disappearance of the transverse anisotropy in ribbon/Pt heterostructures \ncould stem from the T DL which is perpendicular to the plane of the ribbon thus forces the magnetization \nfrom transverse alignment and reduces the transverse  magnetic  permeability.  Furthermore, as another \ntestifier,  we repeated the experiment for ribbon sample coated with 20 nm Copper (Cu) coated layer \nand observed double peak behavior and no frequency shift similar to the bare ribbon ( see FIG. S3 and \nS4 in  supplementary materials). Cu is a representative light metal with weak spin –orbit coupling51 and \nwe expect to see double peak behavior . 7 \n \n-120 -80 -40 040 801200100200\n  MI (%)  \nField (Oe) 0 nm \n 10 nm \n 20 nm  (a) \n-10 -5 0 5 10100125150175200\n  MI (%)\nField (Oe) 0 nm\n 10 nm\n 20 nm(b)  \n0 2 4 6 8 1080120160200\n  MI (%)\nFrequency (MHz) 0 nm \n 10 nm \n 20 nm  (c)  \nFIG. 4.  (a) Comparison of the MI response of the 0, 10 and 20 nm Pt  coated samples in an arb itrary frequency of 6  MHz  and \napplied current of 66  mA. (b) A zoom -in window of MI in the low fields  shows the disappearance of the double peak behavior \nfor Pt deposited  ribbons . (c) MI ratio of 0, 10 and 20 nm Pt deposited  ribbons versus frequency of the applied current with a \nreductive behavior for Pt deposited  ribbons.  \nThe maximum MI ratio of all samples versus frequency are plotted in Fig. 4(c) to better illustrate the \neffect of Pt layer. MI measurements were done at dif ferent frequencies ranging from 1 MHz to 10 MHz. \nIt is noted in Fig. 4(c) that for all investigated samples, with increasing frequency, the maximum MI \nratio first increases, reaches to a maximum at a particular frequency (6 MHz), and then decreases for \nhigher frequencies. This trend can be interpreted by considering the relative contributions of DW \nmotion and moment  rotation to the transverse  magnetic  permeability and hence to the MI52,53. Noted \nthat as frequency increases well above 100 kHz, the contribution of DW motion is damped due to \npresence of the eddy current and moment rotation becomes dominant46,53,54. Thus, t he MI ratio decreases \nat high frequencies. Here, the  𝜇𝑡 decreases, thus resulting in the observed drops of the MI ratio at all \nfrequencies46,52. It is known that DW motion speed increases in the present of SHE12,55,56. MI ratio \nfrequency peak is correlated to the DW relaxation and suggests how DW follows the ac current \nfrequency or correlated with DW speed. Such increase in frequency for 20 nm Pt has same fashion as \nDW does in the present of SHE, dictating another qualitative confirmation.  \nIn summery we have proposed that impedance spectroscopy ca n be used for detection of SOT resulting \nfrom  the SHE  in magnetic -ribbon  /Pt heterostructures . Tunable  impedance response correlated to SOT \ninduced moment re alignments within  FM can be detected . We showed that in a magnetic -ribbon  /Pt \nheterostructure , the acting TDL on FM changes  not only the response of the  MI of the system , but also \ntends to play with the transverse anisotropy of the magnetization  that was probed as frequency shift in \nMI effect . Our results can open a practical  route to study and und erstand the SHE probed via the MI \neffect in FM/HM heterostructures .  \nSee supplementary material for a complete understanding of the detailed procedure of related \nexperiments for the observation of SOT , including FMR measurements  for magnetic -ribbon/Pt  and MI \nfor Cu sputtered ribbon . \nS.M.M. acknowledge s support from the Iran Science Elites Federation (ISEF).  8 \n 1 I. Mihai Miron, G. Gaudin, S. Auffret, B. Rodmacq, A. Schuhl, S. Pizzini, J. Vogel, and P. \nGambardella, Nat. Mater. 9, 230 (2010).  \n2 I.M. Miron, K. Garello, G. Gaudin, P. -J. Zermatten, M. V Costache, S. Auffret, S. Bandiera, B. \nRodmacq, A. Schuhl, and P. Gambardella, Nature 476, 189 (2011).  \n3 L. Liu, C. -F. Pai, Y. Li, H.W. Tseng, D.C. Ralph, and R.A. Buhrman, Science. 336, 555 (2012).  \n4 I.M. Miron, T. Moore, H. Szambolics, L.D. Buda -Prejbeanu, S. Auffret, B. Rodmacq, S. Pizzini, J. \nVogel, M. Bonfim, A. Schuhl, and G. Gaudin, Nat. Mater. 10, 419 (2011).  \n5 J. Kim, J. Sinha, M. Hayashi, M. Yamanouchi, S. Fukami, T. Suzuki, S. Mitani, and H . Ohno, Nat. \nMater. 12, 240 (2012).  \n6 K. Garello, I.M. Miron, C.O. Avci, F. Freimuth, Y. Mokrousov, S. Blügel, S. Auffret, O. Boulle, G. \nGaudin, and P. Gambardella, Nat. Nanotechnol. 8, 587 (2013).  \n7 C.O. Avci, A. Quindeau, C. -F. Pai, M. Mann, L. Caretta, A.S. Tang, M.C. Onbasli, C.A. Ross, and \nG.S.D. Beach, Nat. Mater. 16, 309 (2016).  \n8 J. Sinova, S.O. Valenzuela, J. Wunderlich, C.H. Back, and T. Jungwirth, Rev. Mod. Phys. 87, 1213 \n(2015).  \n9 L. Liu, O.J. Lee, T.J. Gudmundsen, D.C. Ralph, and R.A. Buhrman, Phys. Rev. Lett. 109, 96602 \n(2012).  \n10 C.-F. Pai, L. Liu, Y. Li, H.W. Tseng, D.C. Ralph, and R.A. Buhrman, Appl. Phys. Lett. 101, 122404 \n(2012).  \n11 K.-S. Ryu, L. Thomas, S. -H. Yang, and S. Parkin, Nat. Nanotechnol. 8, 527 (2013).  \n12 P.P.J. Haazen, E. Murè,  J.H. Franken, R. Lavrijsen, H.J.M. Swagten, and B. Koopmans, Nat. Mater. \n12, 299 (2013).  \n13 S. Emori, U. Bauer, S. -M. Ahn, E. Martinez, and G.S.D. Beach, Nat. Mater. 12, 611 (2013).  \n14 G. Yu, P. Upadhyaya, X. Li, W. Li, S.K. Kim, Y. Fan, K.L. Wong, Y. Tse rkovnyak, P.K. Amiri, and \nK.L. Wang, Nano Lett. 16, 1981 (2016).  \n15 S. Woo, K.M. Song, H. -S. Han, M. -S. Jung, M. -Y. Im, K. -S. Lee, K.S. Song, P. Fischer, J. -I. Hong, \nJ.W. Choi, B. -C. Min, H.C. Koo, and J. Chang, Nat. Commun. 8, 15573 (2017).  \n16 V.E. Demidov, S. Urazhdin, H. Ulrichs, V. Tiberkevich, A. Slavin, D. Baither, G. Schmitz, and S.O. \nDemokritov, Nat. Mater. 11, 1028 (2012).  \n17 L. Liu, C. -F. Pai, D.C. Ralph, and R.A. Buhrman, Phys. Rev. Lett. 109, 186602 (2012).  \n18 R.H. Liu, W.L. Lim, and  S. Urazhdin, Phys. Rev. Lett. 110, 147601 (2013).  \n19 A. Manchon, H.C. Koo, J. Nitta, S.M. Frolov, and R.A. Duine, Nat. Mater. 14, 871 (2015).  \n20 L. Liu, T. Moriyama, D.C. Ralph, and R.A. Buhrman, Phys. Rev. Lett. 106, 36601 (2011).  \n21 M. Jamali, Z. Zhao, M. DC, D. Zhang, H. Li, A.K. Smith, and J. -P. Wang, J. Appl. Phys. 119, 133902 \n(2016).  \n22 C.O. Avci, K. Garello, C. Nistor, S. Godey, B. Ballesteros, A. Mugarza, A. Barla, M. Valvidares, E. \nPellegrin, A. Ghosh, I.M. Miron, O. Boulle, S. Auffret, G. Gaudin,  and P. Gambardella, Phys. Rev. B \n89, 214419 (2014).  \n23 H. Nakayama, M. Althammer, Y. -T. Chen, K. Uchida, Y. Kajiwara, D. Kikuchi, T. Ohtani, S. \nGeprägs, M. Opel, S. Takahashi, R. Gross, G.E.W. Bauer, S.T.B. Goennenwein, and E. Saitoh, Phys. \nRev. Lett. 110, 206601 (2013).  \n24 J. Kim, P. Sheng, S. Takahashi, S. Mitani, and M. Hayashi, Phys. Rev. Lett. 116, 97201 (2016).  \n25 Y.-C. Lau, D. Betto, K. Rode, J.M.D. Coey, and P. Stamenov, Nat. Nanotechnol. 11, 758 (2016).  \n26 T. Koyama and D. Chiba, Phys. Rev. B 92, 220402 (2015).  \n27 M. Montazeri, P. Upadhyaya, M.C. Onbasli, G. Yu, K.L. Wong, M. Lang, Y. Fan, X. Li, P. Khalili \nAmiri, R.N. Schwartz, C.A. Ross, and K.L. Wang, Nat. Commun. 6, 8958 (2015).  \n28 M. Knobel, M. Vázquez, and L. Kraus, in Handb. Magn. Mater.  (2003), pp. 497 –563. \n29 V.M. Prida, P. Gorr a, G. V Kurlyandskaya, M.L. S nchez, B. Hernando, and M. Tejedor, \nNanotechnology 14, 231 (2003).  \n30 B. Hernando, M.L. Sánchez, V.M. Prida, M. Tejedor, and M. Vázquez, J. Appl. Phys. 90, 4783 \n(2001).  \n31 M. Hajial i, S.M. Mohseni, S.E. Roozmeh, and M. Moradi, Superlattices Microstruct. 96, 191 (2016).  \n32 L. Jamilpanah, M.R. Hajiali, S.M. Mohseni, S. Erfanifam, S.M. Mohseni, M. Houshiar, and S.E. \nRoozmeh, J. Phys. D. Appl. Phys. 50, 155001 (2017).  \n33 M. Ipatov, V. Zh ukova, A. Zhukov, and J. Gonzalez, Sci. Rep. 6, 36180 (2016).  9 \n 34 V.M. De La Prida, H. García -Miquel, and G. V. Kurlyandskaya, Sensors Actuators A Phys. 142, 496 \n(2008).  \n35 F.L.A. Machado and S.M. Rezende, J. Appl. Phys. 79, 6558 (1996).  \n36 P.M. Haney, H. -W. Lee, K. -J. Lee, A. Manchon, and M.D. Stiles, Phys. Rev. B 87, 174411 (2013).  \n37 C.-F. Pai, Y. Ou, L.H. Vilela -Leão, D.C. Ralph, and R.A. Buhrman, Phys. Rev. B 92, 64426 (2015).  \n38 T.D. Skinner, M. Wang,  a. T. Hindmarch,  a. W. Rushforth ,  a. C. Irvine, D. Heiss, H. Kurebayashi, \nand  a. J. Ferguson, Appl. Phys. Lett. 104, 62401 (2014).  \n39 X. Fan, J. Wu, Y. Chen, M.J. Jerry, H. Zhang, and J.Q. Xiao, Nat. Commun. 4, 1799 (2013).  \n40 X. Qiu, P. Deorani, K. Narayanapillai, K. -S. Lee, K. -J. Lee , H.-W. Lee, and H. Yang, Sci. Rep. 4, \n4491 (2015).  \n41 S. Woo, M. Mann, A.J. Tan, L. Caretta, and G.S.D. Beach, Appl. Phys. Lett. 105, 212404 (2014).  \n42 M.-H. Nguyen, D.C. Ralph, and R.A. Buhrman, Phys. Rev. Lett. 116, 126601 (2016).  \n43 A. V. Khvalkovskiy, V. Cros, D. Apalkov, V. Nikitin, M. Krounbi, K.A. Zvezdin, A. Anane, J. \nGrollier, and A. Fert, Phys. Rev. B 87, 20402 (2013).  \n44 M. D. Stiles and J. Miltat, in Spin Dynamics in Confined MagneticStructures III, edited by B. \nHillebrands and A. Thiaville, Topics in  Applied Physics Vol. 101 (Springer, Berlin, 2006), pp. 225 –\n308. \n45 Y. Ou, C. -F. Pai, S. Shi, D.C. Ralph, and R.A. Buhrman, Phys. Rev. B 94, 140414 (2016).  \n46 F. Qin and H. -X. Peng, Prog. Mater. Sci. 58, 183 (2013).  \n47  a. Gangul y, K. Kondou, H. Sukegawa, S. Mitani, S. Kasai, Y. Niimi, Y. Otani, and A. Barman, Appl. \nPhys. Lett. 104, 72405 (2014).  \n48 S. Emori, U. Bauer, S. -M. Ahn, E. Martinez, and G.S.D. Beach, Nat. Mater. 12, 611 (2013).  \n49 M. Knobel and K.R. Pirota, J. Magn. Magn. Mater. 242–245, 33 (2002).  \n50 J. P. Sinnecker, M. Knobel, K. R. Pirota, J. M. Garcıa, A. Asenjo, M. Vázquez,  J. Appl. Phys.  87, \n4825  (2000).  \n51 H.L. Wang, C.H. Du, Y. Pu, R. Adur, P.C. Hammel, and F.Y. Yang, Phys. Rev. Lett. 112, 197201 \n(2014).  \n52 L.V. Panina, K. Mohri, T. Uchiyama, M. Noda, and K. Bushida, IEEE Trans. Magn. 31, 1249 (1995).  \n53 L. V. Panina and K. Mohri, Appl. Phys. Lett. 65, 1189 (1994).  \n54 M.-H. Phan and H. -X. Peng, Prog. Mater. Sci. 53, 323 (2008).  \n55 S.-M. Seo, K. -W. Kim, J. Ryu, H. -W. Lee, and K. -J. Lee, Appl. Phys. Lett. 101, 22405 (2012).  \n56 M.R. Hajial i, M. Hamdi, S.E. Roozmeh, and S.M. Mohseni, Phys. Rev. B 96, 144406 (2017).  \n "    },    {        "title": "1511.08126v1.Ferromagnetic_resonance_phase_imaging_in_spin_Hall_multilayers.pdf",        "content": "Ferromagnetic resonance phase imaging in spin Hall multilayers\nFeng Guo,1Jason M. Bartell,1and Gregory D. Fuchs1,\u0003\n1School of Applied and Engineering Physics, Cornell University, Ithaca, NY 14853, USA\nWe experimentally image the magnetic precession phase of patterned spin Hall multilayer sam-\nples to study the rf driving \feld vector using time-resolved anomalous Nernst e\u000bect (TRANE)\nmicroscopy. Our ferromagnetic resonance (FMR) measurements quantify the phase and amplitude\nfor both the magnetic precession and the electric current, which allows us to establish the total driv-\ning \feld orientation and the strength of spin Hall e\u000bect. In a channel of uniform width, we observe\nspatial variation of the FMR phase laterally across the channel. We interpret our \fndings in the\ncontext of electrical measurement using the spin-transfer torque ferromagnetic resonance technique\nand show that observed phase variation introduces a systematic correction into the spin Hall angle\nif spatial phase and amplitude variations are not taken into account.\nWhen a spin current traverses the interface between\na normal metal and a ferromagnetic metal, it gener-\nates a spin-transfer torque[1{3] that e\u000eciently manipu-\nlates magnetization. Accurate quanti\fcation of current-\ninduced torques is pivotal to \frst understanding and then\nengineering spintronic devices for future magnetic mem-\nory and information technology. Experimentally, most\nstudies of spin Hall e\u000bect (SHE)-driven torques[4{6] have\nrelied on electrical measurements of devices, which are\ne\u000bective because they provide a large signal-to-noise ra-\ntio. Typical electrical techniques are spin torque fer-\nromagnetic resonance (ST-FMR)[7{9] for in-plane mag-\nnetic moments, and harmonic Hall voltage analysis[10{\n13] for perpendicularly magnetized devices. An essential\nassumption of these methods is that both the driving\ncurrent and the magnetic response are uniform. To gain\ndeeper understanding of SHE-driven torques and go be-\nyond approximate treatments, we quantify the relation-\nship between the driving current and the dynamic mag-\nnetic response using phase sensitive magnetic microscopy.\nOur measurements show that while the assumption of\nuniform driving current is valid, the assumption of uni-\nform magnetic response is not.\nDynamic magnetic imaging provides a method of ver-\nifying the uniformity of a magnetic response and mea-\nsuring spin torques. Several techniques have been\ndeveloped to sense local magnetization dynamics in\nmicro- and nano-structures, including microfocused Bril-\nlouin light scattering (BLS)[14{17], ferromagnetic reso-\nnance force microscopy (FMRFM) dynamics imaging[18{\n21], time-resolved magneto-optical Kerr e\u000bect (MOKE)\nmicroscopy[22, 23], and x-ray magnetic circular dichro-\nism (XMCD)[24{26]. Additionally, an optical technique\nbased on polar MOKE for measuring a dc-driven spin-\ntorque vector was recently introduced[27, 28]. However,\nvery few phase-sensitive imaging techniques provide a full\nset of information | a quantitative image of both drive\nand magnetic response up to gigahertz frequencies.\nIn this work, we use time-resolved anomalous Nernst\ne\u000bect (TRANE) microscopy[29, 30] to simultaneously im-\nage ferromagnetic resonance (FMR) and rf driving cur-\nrent in spin Hall multilayers. By imaging the amplitude\nS\nV\nIrf\nmObj.xyz\nHappl\n𝜑FMR=−57°±4°(a) (b)\n5 μm(c) FMR (d) rfcurrent (e) Reflectivity𝛁𝑻\nFMR signal ( μV)\nApplied field ( G)σFIG. 1: Time-resolved anomalous Nernst e\u000bect\n(TRANE) microscopy measurement concept and ex-\namples of spectroscopical and imaging measurements.\n(a) Schematics of the measurement principle for TRANE mi-\ncroscopy. (b) Example of a ferromagnetic resonance (FMR)\nspectrum using \feld modulation. Images measured at a \fxed\n\feld of 208 G: (c) \feld modulated FMR signal, (d) rf current\nwith chopping signal and (e) laser re\rectivity.\nand phase of the precessing magnetization in relation to\nthe driving current, we \fnd that the driving \feld direc-\ntion in a sample with strong spin torque is di\u000berent than\nin a sample where the spin torque is blocked with a 2\nnm thick Hf spacer. More importantly, we demonstrate\nthat even in a uniform width structure, the FMR phase\nis nonuniform, despite the common assumption of quasi-\nuniformity. We analyze the consequences of spatial vari-\nations in precession phase in terms of device-level mea-\nsurements such as ST-FMR. We show that ST-FMR mea-\nsurements of the spin Hall e\u000eciency can have a sizable\nsystematic error, depending on the details of the sample.\nTo simultaneously probe the local magnetic orientation\nand the rf driving current, we use 3 ps long pulsed laser\nheating, as illustrated in Fig. 1(a). The magnetization\nprojection in ydirection is detected through the anoma-\nlous Nernst e\u000bect. An increase in the local resistivity\ndue to the transient heating also produces a voltage cor-\nresponding to the driving current. Fig. 1 (b) shows an\nexample of an FMR spectrum, from which we obtain thearXiv:1511.08126v1  [cond-mat.mtrl-sci]  25 Nov 20152\nphase of FMR precession. With a \fxed magnetic \feld, we\ncan simultaneously image the FMR signal, the rf current\nsignal, and the laser re\rectivity (Figs. 1 (c) - (e), respec-\ntively). For this study, we fabricate samples with a stack\nstructure of Fe 60Co20B20(4 nm)=Hf(t Hf)=Pt(4 nm). We\nuse two hafnium thicknesses, t Hf=0.3 nm and t Hf=2 nm.\nThe 0.3 nm Hf samples, which we will simply refer to as\nthe \\spin Hall samples\", present a reasonably large spin\nHall e\u000bect while maintain a low damping parameter. In\ncontrast, the samples with 2 nm thick Hf, or the \\non\nspin Hall samples\", have a minimal spin Hall e\u000bect. The\nspin Hall e\u000eciencies of these two sets of samples are also\ncon\frmed with ST-FMR measurements, summarized in\nTable I. As discussed later, we use the non spin Hall sam-\nple to establish the local driving \feld angle in the spin\nHall sample via precesion phase measurements.\nFirst we analyze the e\u000bective driving \feld angle with\nrespect to the sample plane, \u0012e\u000b, from measurements of\nthe FMR precession phase. Speci\fcally, \u0012e\u000b, is directly\nrelated to the di\u000berence between the FMR phase and\nthe driving current phase according to the relation \u0012e\u000b=\n'FMR\u0000'rf+ 90\u000e. In general, the FMR phase can be\nwritten as:\n'\u0006\nFMR=\u0006('rf+\u0012\u0006)\u000090\u000e; (1a)\nIntersection:(\n'int\nrf=\u0000(\u0012++\u0012\u0000)=2;\n'int\nFMR= (\u0012+\u0000\u0012\u0000)=2\u000090\u000e;(1b)\nin which the FMR phase 'FMRsimply follows the current\nphase'rfand the driving \feld angle \u0012. The superscripts\n\\+\" and \\\u0000\" denote the positive and negative \feld di-\nrections respectively.\nTo discuss the physical meanings of the intersection\n('int\nrf,'int\nFMR), we \frst explain the symmetries of various\ntorques. Let us consider a case where the total driving\ntorque has both Oersted and spin torque (anti-damping\nlike) contributions[38]. As illustrated in Figs. 2 (a) and\n(b), the Oersted \feld ^hOedoes not change sign when the\n\feld reverses, while the spin torque driving \feld ^hST=\n^m\u0002^\u001bdoes.\nDue to the di\u000berence in the symmetry of ^hOeand\n^hST, the two coordinates at the intersection in Eq. 1b\nhave di\u000berent physical interpretations. 'int\nrfis the av-\neraged e\u000bective \feld angle. 'int\nFMRis determined by the\ndi\u000berence between \u0012+and\u0012\u0000, which is sensitive to\nthe Oersted \feld orientation. In the Supplementary\nInformation[31] we further show that 'int\nrf\u0019\u0000hST=hk\nOe\nand'int\nFMR\u0019h?\nOe=hk\nOe\u000090\u000e, under the assumption of\nhST=hk\nOe,h?\nOe=hk\nOe\u001c1 (h?\nOeandhk\nOeare the out-of-\nplane and in-plane components of the Oersted \feld, re-\nspectively).\nFigs. 2 (c) and (d) show the phase dependent rf cur-\nrent and FMR signals, measured at the center of the\nchannel. The rf current signal shows a sinusoidal wave-\nform. The FMR phase varies linearly with increasing rf\nTop edge\nCenter\nBo�om edge\nTop edgeCenter\nBo�om edge(c)\n(d) (e)\n�FMRint\n�rfint\ny\nz\nPositive field:\n𝑚\n=\n𝑥\nh\nOe\nh\nST\n𝜃\n+\nh\neff\ny\nz\nNegative field:\n𝑚\n=\n−\n𝑥\nh\nOe\nh\nST\n𝜃\n−\nh\neff(b) (a)\ny\nx\n-270-180-90090180\n-180 -90 0 90 180�FMR (deg)\n�rf(deg)negative field\npositive field-101Norm.Irf(mV)\n-120-90-60\n-60 -30 0�int\nFMR (deg)\n�rfint(deg)spin hall\nnon spin hallFIG. 2: Vector diagrams and FMR phase measure-\nments. Diagrams of spin torque \feld hSTand Oersted \feld\nhOefor (a) positive and (b) negative applied \felds. The\ncharge current is set to the positive (+^ x) for both cases. Nor-\nmalized current signal (c) and FMR phase (d) as functions\nof rf current phase. Both (c) and (d) are measured at the\ncenter the of bar. (e) The points of intersection measured at\nthe top edge, center and bottom edge of the channel. The\nintersection measured at the center of a non spin Hall sample\nis also included (hollow square) in (e).\ncurrent phase, with a slope of +1 ( \u00001) for the positive\n(negative) applied \feld, consistent with Eq. 1a. To inves-\ntigate the spatial dependence of the phase, we repeat the\nmeasurement in Fig. 2 (d) for the top and bottom edges\nof the channel. The points of intersection ( 'int\nrf;'int\nFMR) for\ndi\u000berent positions in the spin Hall sample are plotted in\nFig. 2 (e). We also include the intersection measured at\nthe center of the non spin Hall sample (2 nm Hf spacer)\nas reference. In the absence of the spin Hall e\u000bect, only\nOersted \feld is responsible for the e\u000bective driving \feld,\nand we expect it to be nearly in-plane at the center of\nthe channel. When the spin torque is turned on, given\nthe stack sequence and the positive spin Hall angle for\nplatinum, we expect ^hSTk+^z. As a result, the spin\ntorque tilts \u0012e\u000bout of the sample plane, towards the +^ z\ndirection. As the driving \feld angle increases, 'int\nrfwill\ndecrease, in agreement with Fig. 2 (e).\nBecause the FMR phase and current phase do not\nshare an absolute reference[31], we use the non spin Hall\nsample to de\fne the zero current phase, by assuming\nthat the non spin Hall sample has an in-plane driving\n\feld at the channel center [i.e. 'int\nrf= 0, see the green\npoint in Fig. 2 (e)]. We also assume that the temporal\npro\fles of both the temperature and thermal gradient re-\nmain the same between the two samples, since they are\nnearly identical structures[39]. Finally using the FMR\nphase of the non spin Hall sample, we obtain a driving\n\feld angle of \u00120\ne\u000b= 10:1\u000e\u00064:2\u000efor the spin Hall sample,3\n(a) (b)\n-200-150-100-50050\n-2-1012φFMR(deg)\ny(\u0001m)positivefield\nnegativefield\n-10010(θ+-θ-)/2(deg)\n0204060\n-2-1012(θ++θ-)/2(deg)\ny(\u0001m)(c)\nFIG. 3: A signi\fcant spatial variation in the FMR\nphase along ydirection. (a) FMR phases measured at\npositive (red) and negative (blue) applied \felds as functions\nofyposition. Using Eq. 1, we can also plot (b) ( \u0012+\u0000\u0012\u0000)=2\nand (c) (\u0012++\u0012\u0000)=2 as functions of y.\ncorresponding to a ( Js=Jc)0= 0:048\u00060:020. The com-\nparisons between ST-FMR electrical measurements and\nFMR phase measurements are shown in Table I.\nIn the following, we focus on the position dependent\nFMR phase for the spin Hall sample. As shown in Fig. 2\n(e),'int\nrfnear either the top or bottom edge is less than\nthat measured at the center, indicating a larger \u0012e\u000bat\nthe edges. However 'int\nFMRnear either the top or bottom\nedge shifts towards opposite directions with respect to\nthe center, suggesting a gradual change in h?\nOe=hk\nOe. Fur-\nthermore,h?\nOeis expected to be positive at the top edge\nand negative at the bottom edge, which is consistent with\nthe vertical sequence of the three points in Fig. 2 (e).\nTo further investigate phase variation across the width\nof the channel, we measure the FMR spectrum as a func-\ntion ofyposition for both positive and negative applied\n\felds, with a \fxed rf current phase. The ydependent\nFMR phase is shown in Fig. 3. There is a sizable vari-\nation of\u001850\u000ealong theydirection. We use Eq. 1 to\nplot the di\u000berence and sum of \u0012+and\u0012\u0000, as functions of\ny. Consistent with the previous measurements in Fig. 2\n(e), (\u0012++\u0012\u0000)=2 is larger near the sample edges. While\n(\u0012+\u0000\u0012\u0000)=2 is also in agreement with the vertical posi-\ntions of the points in Fig. 2 (e).\nIn addition, we demonstrate an approach to image the\nFMR phase. Instead of recording the FMR spectrum\nat each location, we combine multiple FMR images to\ncalculate the phase variation. In this example, we \fx\nthe rf current phase corresponding to a FMR phase of\n\u000024\u000eat the channel sample. We then combine 6 FMR\nimages at various applied \felds (from 185 G to 215 G), to\nreconstruct both phase and amplitude images, shown in\nFig. 4 (a) and (b) respectively. The main feature of the\nphase image is that the phase is quasi-uniform near the\ncenter, and it increases near the edges, in quantitative\nagreement with data in Fig. 3. The phase variation is\nmore prominent along the y-direction than that in the x-\nAmplitude ( �V) ��FMR(deg)\n5�m(b)\n (a)FIG. 4: Images of the FMR phase and amplitude. By\n\fxing'FMR=\u000024\u000eand using 6 FMR images at various ap-\nplied \felds, we can decompose (a) the relative FMR phase\nvariation and (b) the FMR amplitude.\ndirection. In contrast, the FMR amplitude is large near\nthe center and decreases towards either edge, as expected.\nNext we speculate on the origin of the phase varia-\ntion, then we evaluate its in\ruence on global electrical\nmeasurements. In a con\fned magnetic structure under\na uniform applied \feld, the internal magnetic \feld is\nhighly nonuniform near the edges due to the demagne-\ntizing \feld. The e\u000bects of nonuniform internal \feld and\nhence nonuniform precession modes are well established\nfor magnetic micro- and nanostructures[32{36]. Simi-\nlarly, for the bar structure samples used in this study, the\ntransverse driving \feld experiences an inhomogeneous\ndemagnetizing \feld. Consequently the in-plane driving\n\feldhk\nOeinside the ferromagnet[40] has a substantial spa-\ntial variation, which plays an important role in the ob-\nserved phase variation. In contrast, the out-of-plane hST\nis uniform across the sample given a uniform current den-\nsity distribution. As a result, hk\nOeis weaker at the channel\nedges and the e\u000bective driving \feld close to the edges has\na larger angle than that at the center, as illustrated in\nthe inset of Fig. 5. We argue that the spatial variation of\nthe rf driving \feld is determined by a number of factors,\nincluding sample dimensions, rf current uniformity, edge\nproperties, and magnetic anisotropy \felds. Therefore the\ndetails of the phase variations are expected to be sample\nspeci\fc.\nNow we examine how a nonunifrom precession phase\ncan a\u000bect the result characterized by ST-FMR. The\nanalysis of ST-FMR relies on two assumptions: one is\nthe uniformity of the precession mode and the other is\nthe uniformity of the rf current. Although the measured\ndriving current is uniform, the assumption of uniform\nprecession breaks down. We start with the signal\nmeasured by ST-FMR under the macrospin spin approx-\nimation. By mixing the rf current with an oscillating\nmagnetoresistence, we get a recti\fed voltage: Vmix/\n\u0012pf\u001f0(H) cos('rf\u0000'FMR) +\u001f00(H) sin('rf\u0000'FMR)g,\nwhere\u0012pis the precession amplitude and \u001f0and\u001f00\nare the real and imaginary susceptibility functions\nrespectively. By \ftting the spectrum Vmix(H) to a\nlinear combination of the symmetric and anti-symmetric4\n(a) (b)\n\u0001eff0\u0001\n0204060\n-2-1012θeff(deg)\ny(\u0002m)hypothetical\ndata\n00.51\n-2-1012Norm.θp\ny(\u0002m)hypothetical\ndata\nFIG. 5: Numerical estimation of the phase correction\ngiven the spatial pro\fles of precession amplitude and\nphase. Spatial distributions of (a) the normalized preces-\nsion amplitude and (b) e\u000bective driving \feld angle. The blue\ncurves in (a) and (b) are the polynomial \fts of the data (gray\npoints). The red line in (b) is the resultant driving \feld angle\nthat would be obtained from ST-FMR, using Eq. 3.\nLorentzian functions, one obtains 'rf\u0000'FMRand thus the\nspin Hall e\u000eciency for the normal metal/ferromagnet\ncombination.\nTo include the e\u000bect of spatial variation in 'FMR, we\nrewrite the averaged Vmixas the integral of the mixing\nvoltage weighted by the precession amplitude ( \u0012p) (see\nSupplementary Information[31] for derivation):\nVmix/\u001f0Z\ndr \u0012p(r) cos\u0002\n'rf\u0000'FMR(r)\u0003\n+\u001f00Z\ndr \u0012p(r) sin\u0002\n'rf\u0000'FMR(r)\u0003\n: (2)\nTherefore, we \fnd the equivalent phase di\u000berence be-\ntween the FMR and rf current that would be obtained\nfrom the global measurement is:\nh'FMR\u0000'rfi=\ntan\u00001\"R\ndr \u0012p(r) sin\u0002\n'FMR(r)\u0000'rf\u0003\nR\ndr \u0012p(r) cos\u0002\n'FMR(r)\u0000'rf\u0003#\n(3)\nAs an interesting point, the phase correction \u0001 is only\ndetermined by the spatial varying component of the FMR\nphase\u000e'FMR(r) and the precession amplitude \u0012p(r), and is\nindependent of the overall o\u000bset '0\nFMR. We can substitute\n'0\nFMR+\u000e'FMR(r) for'FMRin Eq. 3. Using trigonometry,\nwe can rewrite Eq. 3:\nh'FMR\u0000'rfi='0\nFMR\u0000'rf+ \u0001; (4)\n\u0001 = tan\u00001\"R\ndr \u0012p(r) sin\u0002\n\u000e'FMR(r)\u0003\nR\ndr \u0012p(r) cos\u0002\n\u000e'FMR(r)\u0003#\n:\nTherefore the correction for the phase (also driving \feld\nangle) \u0001 is independent of the driving \feld angle near\nthe channel center \u00120\ne\u000b.TABLE I: Comparison between electrical ST-FMR and\nspatially resolved measurements. The spin Hall e\u000e-\nciency,Js=Jc, measured with ST-FMR and calculated from\nthe spatial variation of the FMR phase. \u00120\ne\u000bis the angle of\ne\u000bective driving \feld at the center, and \u0001 is the phase correc-\ntion due to the phase variation. ( Js=Jc)0is the ratio only at\nthe center, corresponding to \u00120\ne\u000b; whilehJs=Jciis integrated\nratio with the phase variation included. The uncertainty of\n\u0001 is calculated using the standard errors of the polynomial\n\fts in Fig. 5.\nSamplespin Hall non spin Hall\n(0.3 nm Hf) (2 nm Hf)\nST-FMR Js=Jc= 0.076\u00060.002 0.010\u00060.003Phase var.\nincluded\u00120\ne\u000b= 10:1\u000e\u00064:2\u000eassume: 0\n\u0001 = 7:5\u000e\u00061:8\u000e3:4\u000e\u00061:9\u000e\n(Js=Jc)0= 0:048\u00060:020 assume: 0\nhJs=Jci= 0:086\u00060:031 0:015\u00060:008\nTo numerically evaluate the correction resulting from\nthe phase nonuniformity, we use the polynomial \fts of the\nprecession amplitude \u0012p(y) and driving \feld angle \u0012e\u000b(y)\nto mimic the experimental results, plotted in Fig. 5. For\nsimplicity we assume both the phase and amplitude of\nthe precession are uniform in the xdirection and we\nonly consider the spatial variation along the ydirection.\nWe get an \\averaged\" value of the e\u000bective driving \feld,\nh\u0012e\u000bi=\u00120\ne\u000b+ \u0001, shown as the red line in Fig. 5 (b). In\nthis example, the phase correction \u0001 = 7 :5\u000e\u00061:8\u000e.\nAs shown in Table I, for the spin Hall sample there is\na substantial discrepancy between ( Js=Jc)0at the chan-\nnel center and the ST-FMR results. At the same time,\nhJs=Jciobtained by including the the spatially varying\nphase is consistent with Js=Jcmeasured from ST-FMR,\nwithin experimental errors. Thus, we argue that the ST-\nFMR technique does not necessarily re\rect the phase\nvalue in the middle of the sample, nor does it sense a\nuniform phase, rather it provides a spatially averaged\nphase. Although the electrical techniques have a supe-\nrior sensitivity, we show an example where it is essential\nto include a correction for spatial variations of both pre-\ncession phase and amplitude to correctly quantify the\nspin Hall e\u000eciency from electrical measurements.\nIn summary, we have studied the FMR phase in uni-\nform width spin hall multilayers. Using TRANE mi-\ncroscopy, we have measured the amplitude and phase of\nboth FMR precession and rf driving current, which en-\nables us to determine the angle of driving \feld vector. In\na sample with substantial spin torque, we found that the\ndriving \feld points around 10\u000eout of the sample plane\nat the center. More importantly, we observe a substan-\ntial precession phase variation across the width of the\nchannel. We expect the phase variation to be important\nin the micro- and nano-structures, depending on device\ndetails. We have also evaluated the correction term in5\ndriving \feld angle \u0001 due to the phase variation. In the\ncase of nonuniform precession phase, we have established\na mechanism by which ST-FMR can yield an inaccurate\nspin Hall e\u000eciency. For the 5 \u0016m wide samples studied,\nwhen including the spatial variations of the precession\namplitude and phase, the integrated spin Hall e\u000eciency\nnearly doubles compared to that in the middle of the\nchannel. Therefore, although electrical measurements\nare very e\u000bective techniques to quantify the spin Hall\ne\u000bect, we conclude the spatial variations of both preces-\nsion amplitude and phase can play an important role and\nshould not be overlooked. Finally, we have shown that\nphase-sensitive imagining techniques such as TRANE mi-\ncroscopy are valuable for quantitative studies of the spin\nHall e\u000bect. The spatial uniformity is also an essential\ningredient for understanding damping and switching dy-\nnamics in magnetic con\fned structures.\nMethods\nBasic principles of time-resolved anomalous Nernst e\u000bect\n(TRANE) microscopy\nWe apply an rf current through a circulator to the\nmultilayer sample to excite magnetic dynamics, shown\nin Fig. 1 (a). We use pulsed (3 ps) laser heating, syn-\nchronized with the phase of the current source to en-\nable stroboscopic measurements of both the magnetiza-\ntion projection myand the rf driving current. The tran-\nsient vertical thermal gradient produces a voltage pulse\ncorresponding to the magnetization projection in ydi-\nrection, through the anomalous Nernst e\u000bect. The tran-\nsient heating also increases the local resistivity, which\nproduces a voltage when a gigahertz driving current is\napplied. An external \feld is applied along ^ xdirection.\nWhen measuring the FMR spectra, the in-plane mag-\nnetization component, my, is recorded as a function of\napplied \feld. We also apply a modulation \feld to distin-\nguish the magnetic signal from the rf current signal. We\nestablish the phase of FMR precession by \ftting the sig-\nnal toA(d\u001f0(H)\ndHcos'FMR+d\u001f00\ndHsin'FMR), where\u001f0and\u001f00\nare the real and imaginary dynamic susceptibility func-\ntions,Ais the local FMR amplitude, and 'FMRis the\nFMR phase at resonance. In the example shown in Fig. 1\n(b), we \fnd 'FMR=\u000057\u000e\u00064\u000e. To image dynamics in this\nsample, we set the applied \feld to 208 G and record the\nFMR, the rf current, and laser re\rectivity, as shown in\nFig. 1 (c), (d) and (e) respectively. A more detailed de-\nscription of the TRANE technique can be found in prior\nwork[30].Sample fabrication and characterization\nThe samples were dc sputtered on the thermally con-\nductive sapphire substrates and subsequently patterned\ninto a 5\u0016m\u000212\u0016m bar geometry using photolithog-\nraphy. The multilayer samples have a stack structure\nof Fe 60Co20B20(4 nm)=Hf(t Hf)=Pt(4 nm). Two hafnium\nthicknesses are used in this study, t Hf=0.3 nm and\ntHf=2 nm. The 0.3 nm Hf samples (\\spin Hall sam-\nples\") have a substantial spin Hall e\u000bect and a low damp-\ning parameter. From a previous study, a thin Hf spacer\nlayer (near 0.5 nm) is helpful to enhance the spin Hall\ne\u000bect e\u000ecacy[37]. The 2 nm thick Hf samples (\\non spin\nHall samples\"), however, have a minimal spin Hall e\u000bect.\nSince the thickness of the Hf spacer already exceeds the\nspin di\u000busion length of 1.5 nm in Hf[12], the Hf layer\nblocks the spin current \rowing from the Pt layer. We\nalso characterize the spin Hall e\u000eciencies of these two\nsets of samples using electrical ST-FMR measurements,\nand the results of which are shown in Table I.\n\u0003Electronic address: gdf9@cornell.edu\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] D. Ralph and M. Stiles, J. Magn. Magn. Mater. 320,\n1190 (2008).\n[4] M. I. Dyakonov and V. I. Perel, Physics Letters A 35,\n459 (1971).\n[5] J. E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999).\n[6] Y. K. Kato, R. C. Myers, A. C. Gossard, and D. D.\nAwschalom, Science 306, 1910 (2004).\n[7] 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[8] J. C. Sankey, P. M. Braganca, A. G. F. Garcia, I. N.\nKrivorotov, R. A. Buhrman, and D. C. Ralph, Phys. Rev.\nLett. 96, 227601 (2006).\n[9] L. Liu, T. Moriyama, D. C. Ralph, and R. A. Buhrman,\nPhys. Rev. Lett. 106, 036601 (2011).\n[10] U. H. Pi, K. Won Kim, J. Y. Bae, S. C. Lee, Y. J.\nCho, K. S. Kim, and S. Seo, Applied Physics Letters\n97, 162507 (2010).\n[11] J. Kim, J. Sinha, M. Hayashi, M. Yamanouchi,\nS. Fukami, T. Suzuki, S. Mitani, and H. Ohno, Nat Mater\n12, 240 (2013).\n[12] C.-F. Pai, M.-H. Nguyen, C. Belvin, L. H. Vilela-Leo,\nD. C. Ralph, and R. A. Buhrman, Applied Physics Let-\nters104, 082407 (2014).\n[13] M. Hayashi, J. Kim, M. Yamanouchi, and H. Ohno, Phys.\nRev. B 89, 144425 (2014).\n[14] K. Perzlmaier, M. Buess, C. H. Back, V. E. Demidov,\nB. Hillebrands, and S. O. Demokritov, Phys. Rev. Lett.\n94, 057202 (2005).\n[15] S. O. Demokritov and V. E. Demidov, Magnetics, IEEE\nTransactions on 44, 6 (2008), ISSN 0018-9464.\n[16] V. E. Demidov, M. Buchmeier, K. Rott, P. Krzysteczko,6\nJ. M unchenberger, G. Reiss, and S. O. Demokritov, Phys.\nRev. Lett. 104, 217203 (2010).\n[17] H. T. Nembach, J. M. Shaw, T. J. Silva, W. L. Johnson,\nS. A. Kim, R. D. McMichael, and P. Kabos, Phys. Rev.\nB83, 094427 (2011).\n[18] O. Klein, G. de Loubens, V. V. Naletov, F. Boust,\nT. Guillet, H. Hurdequint, A. Leksikov, A. N. Slavin,\nV. S. Tiberkevich, and N. Vukadinovic, Phys. Rev. B 78,\n144410 (2008).\n[19] I. Lee, Y. Obukhov, G. Xiang, A. Hauser, F. Yang,\nP. Banerjee, D. V. Pelekhov, and P. C. Hammel, Nature\n466, 845 (2010).\n[20] H.-J. Chia, F. Guo, L. M. Belova, and R. D. McMichael,\nPhys. Rev. Lett. 108, 087206 (2012).\n[21] F. Guo, L. M. Belova, and R. D. McMichael, Phys. Rev.\nLett. 110, 017601 (2013).\n[22] W. K. Hiebert, A. Stankiewicz, and M. R. Freeman,\nPhys. Rev. Lett. 79, 1134 (1997).\n[23] J. P. Park, P. Eames, D. M. Engebretson, J. Berezovsky,\nand P. A. Crowell, Phys. Rev. Lett. 89, 277201 (2002).\n[24] S.-B. Choe, Y. Acremann, A. Scholl, A. Bauer, A. Doran,\nJ. Sthr, and H. A. Padmore, 304, 420 (2004).\n[25] M. K. Marcham, P. S. Keatley, A. Neudert, R. J. Hicken,\nS. A. Cavill, L. R. Shelford, G. van der Laan, N. D.\nTelling, J. R. Childress, J. A. Katine, et al., Journal of\nApplied Physics 109, 07 (2011).\n[26] P. Wessels, J. Ewald, M. Wieland, T. Nisius, A. Vo-\ngel, J. Viefhaus, G. Meier, T. Wilhein, and M. Drescher,\nPhys. Rev. B 90, 184417 (2014).\n[27] X. Fan, H. Celik, J. Wu, C. Ni, K.-J. Lee, V. O. Lorenz,\nand J. Q. Xiao, Nat Commun 5, 3042 (2014).\n[28] X. Fan, A. R. Mellnik, W. Wang, N. Reynolds, T. Wang,\nH. Celik, V. O. Lorenz, D. C. Ralph, and J. Q. Xiao,\nArXiv e-prints (2015), 1509.01266.\n[29] J. M. Bartell, D. H. Ngai, Z. Leng, and G. D. Fuchs, Nat.\nCommun. 6, 8460 (2015).\n[30] F. Guo, J. M. Bartell, D. H. Ngai, and G. D. Fuchs, Phys.\nRev. Applied 4, 044004 (2015).\n[31] Supplemental Information (2015).\n[32] J. Jorzick, S. O. Demokritov, B. Hillebrands, M. Bailleul,\nC. Fermon, K. Y. Guslienko, A. N. Slavin, D. V. Berkov,\nand N. L. Gorn, Phys. Rev. Lett. 88, 047204 (2002).\n[33] C. Bayer, S. O. Demokritov, B. Hillebrands, and A. N.\nSlavin, Applied Physics Letters 82(2003).\n[34] C. Bayer, J. P. Park, H. Wang, M. Yan, C. E. Campbell,\nand P. A. Crowell, Phys. Rev. B 69, 134401 (2004).\n[35] R. D. McMichael and B. B. Maranville, Phys. Rev. B 74,\n024424 (2006).\n[36] H. T. Nembach, J. M. Shaw, C. T. Boone, and T. J.\nSilva, Phys. Rev. Lett. 110, 117201 (2013).\n[37] M.-H. Nguyen, C.-F. Pai, K. X. Nguyen, D. A. Muller,\nD. C. Ralph, and R. A. Buhrman, Applied Physics Let-\nters106, 222402 (2015).\n[38] We do not include a \feld-like spin torque in the discussion\nsince it is indistinguishable from the in-plane Oersted\n\feld using phase measurements only.\n[39] We do not expect the extra 1.3 nm Hf in the non spin\nHall sample to signi\fcantly change the temperature and\nthermal gradient relaxation timescales.\n[40] All the transverse in-plane driving \felds in the system\nwill experience such inhomogeneous demagnetizing \feld,\nfor both in-plane Oersted \feld ( hk\nOe) and possible \feld\nlike contribution ( hFL).Acknowledgments\nThe authors thank Daniel C. Ralph and Minh-Hai\nNguyen for helpful discussions. This work was sup-\nported by AFOSR, under contract No. FA9550-14-1-\n0243. This work made use of the Cornell Center for Ma-\nterials Research Shared Facilities which are supported\nthrough the NSF MRSEC program (DMR-1120296) as\nwell as the Cornell NanoScale Facility, a member of\nthe National Nanotechnology Coordinated Infrastructure\n(NNCI), which is supported by the National Science\nFoundation (Grant ECCS-15420819).\nAuthor contributions\nG.D.F. and J.M.B. developed the measurement tech-\nnique, F.G. and G.D.F. designed the experiments, F.G.\nand J.M.B. performed the experiments, F.G. the devel-\noped analysis method, F.G. and G.D.F. wrote the paper.\nCompeting \fnancial interests\nThe authors declare that they have no competing \f-\nnancial interests1\nSupplemental Information for:\nFerromagnetic resonance phase imaging in spin Hall multilayers\nFeng Guo,1Jason M. Bartell,1and Gregory D. Fuchs1\n1School of Applied and Engineering Physics, Cornell University, Ithaca, NY 14853, USA\nEXPANSIONS OF 'int\nrfAND'int\nFMR\nTo gain more intuition, we expand \u0012+and\u0012\u0000with\nrespect tohST=hk\nOeandh?\nOe=hk\nOe, wherehk\nOeandh?\nOeare\nthe in-plane and out-of-plane components of the Oersted\n\feld respectively. Thus we can rewrite Eq. (1b) to:\n'int\nrf=\u0000hST\nhk\nOe\n+O2\n4 \nhST\nhk\nOe! \nh?\nOe\nhk\nOe!2\n+1\n3 \nhST\nhk\nOe!33\n5;(S1a)\n'int\nFMR=h?\nOe\nhk\nOe\u000090\u000e\n+O2\n4 \nhST\nhk\nOe!2 \nh?\nOe\nhk\nOe!\n+1\n3 \nh?\nOe\nhk\nOe!33\n5:(S1b)\nAs shown above, in the case of hST=hk\nOe,h?\nOe=hk\nOe\u001c1\nwhere the higher order terms become negligible, 'int\nrfis\nprimarily sensitive to hSTwhile'int\nFMRis primarily sensi-\ntive to the Oersted \feld angle, as demonstrated in the\nmain text.\nPHASE REFERENCES OF CURRENT SIGNAL\nAND FMR SIGNAL\nWe raise a subtle point in measuring and de\fning the rf\ncurrent phase. Although it is tempting to simply obtain\nthe current phase from the current signal such as that in\nFig. 2 (c), there is a \fnite o\u000bset between the phase mea-\nsured from the current signal [Fig. 2 (c)] and that mea-\nsured from the FMR signal [Fig. 2 (d)]. To explain that,\nwe need to account for the di\u000berent time scales between\nFMR and current signals[30]. The thermal gradient (cor-\nresponding to the FMR signal) has a faster response to\nthe laser pulse and a quicker decay than these of the\noverall local temperature (corresponding to the current\nsignal)[29]. Therefore the di\u000berence in temporal pro\fles\nof the thermal gradient and the temperature creates an\no\u000bset in the measured current phase.\nxymHappl\n𝜃0𝜃(𝑡)𝑱(𝑡)\nFIG. S1: Diagram of ST-FMR measurement con\fguration.\n\u00120is the in-plane applied \feld angle with respect to ^ xdirec-\ntion.\u0012(t) is the time varying magnetization orientation with\nrespect to ^x.\nDERIVATION OF ST-FMR SIGNAL WITH\nSPATIAL VARYING FMR PHASE\nHere we include the spatial variation for both the FMR\nphase'FMR(r) and amplitude \u0012p(r), while deriving the\nrecti\fed voltage measured with ST-FMR. We use time\ndependent magnetoresistance and rf driving current to\ncompute the recti\fed dc electric \feld as well as the mix-\ning voltage, shown as Eq. 2 in the main text.\nWe start with the Ohm's law\nE=\u001aJ; (S2)\nand we assume the rf current density has a spatially\nuniform distribution and is \rowing along ^ x:J(t) =\nJ0cos(!t+'rf) ^x. To include anisotropic magnetore-\nsistance (AMR) we write the sample's resistivity\n\u001a=\u001a0+ \u0001\u001acos2\u0012(t); (S3)\nwhere\u0012(t) =\u00120+\u0012pcos(!t+'FMR) is the angle between\nthe current (^ x) and magnetization vector, \u0012pthe in-plane\napplied \feld angle and \u0012pis the precession amplitude,\ndepicted in Fig. S2. We combine equations S2 and S3\nand apply trigonometric identities\nE=\b\n\u001a0+ \u0001\u001acos2[\u00120+\u0012pcos(!t+'FMR)]\t\n\u0002J0cos(!t+'rf)\n=\u001a\n\u001a0+1\n2\u0001\u001ah\n1 + cos 2\u00120cos\u0002\n2\u0012pcos(!t+'FMR)\u0003\n\u0000sin 2\u00120sin\u0002\n2\u0012pcos(!t+'FMR)\u0003i\u001b\n\u0002J0cos(!t+'rf): (S4)2\nFor a small precession amplitude \u0012p\u001c1, we can further\nsimplify the Eq. S4 using small angle approximations\nE=\u001a\n\u001a0+1\n2\u0001\u001ah\n1 + cos 2\u00120\u00002\u0012psin 2\u00120cos(!t+'FMR)i\u001b\n\u0002J0cos(!t+'rf): (S5)\nSince ST-FMR measures the mixed dc signal, we can\ndrop all the oscillating terms such as cos( !t+\u0001\u0001\u0001) and\ncos(2!t+\u0001\u0001\u0001). The resulting dc electric \feld is\nEdc=\u00001\n2J0\u0012p(r)\u0001\u001asin 2\u00120cos\u0002\n'rf\u0000'FMR(r)\u0003\n^x:(S6)\nNext, we use the dc current to calculate the mixing\nvoltageVmeasured from ST-FMR\nIdc=Z\ndydzEdc\n\u001a0; (S7)\nVmix=IdcR0\n=\u00001\n2J0\u0001Rsin 2\u00120Z\ndydz\u0012p(r) cos\u0002\n'rf\u0000'FMR(r)\u0003\n;\n(S8)\nwhere \u0001R=R0\u0001\u001a=\u001a0is the resistance change due to\nthe AMR e\u000bect. We only consider the spatial variation\nalong y-direction, so we can rewrite the mixing voltage\nVmix=\u0000I0\u0001Rsin 2\u00120\n2wZ\ndy\u0012p(y) cos\u0002\n'rf\u0000'FMR(y)\u0003\n;\n(S9)\nwhere the rf current amplitude I0=J0=(wt),wis the\nchannel width, and tis the sample thickness.\nIn order to make connection to Eq. 2 in the main text,\nwe include the \feld dependence of the phases. Note that\nwe de\fne'FMRas the precession phase at the resonance\n\feldHapp=Hres. We substitute the FMR phase by\n'FMR!'FMR+\u001e(H), where\u001e(H) is the \feld dependent\nprecession phase. Thus we can rewrite Eq. S9 as\nVmix(H) =\u0000I0\u0001Rsin 2\u00120\n2w\n\u0002Z\ndy\u0012p(y) cos\u0002\n'rf\u0000'FMR(y)\u0000\u001e(H)\u0003\n=\u0000I0\u0001Rsin 2\u00120\n2w\n\u0002n\n\u001f0(H)Z\ndy\u0012p(y) cos['rf\u0000'FMR(y)]\n+\u001f00(H)Z\ndy\u0012p(y) sin['rf\u0000'FMR(y)]o\n;\n(S10)\nin which we use the relations \u001f0(H) = cos\u001e(H) and\n\u001f00(H) = sin\u001e(H). Eq. S10 is essentially Eq. 2 in the\ntext.\n(a) (b)\n0204060\n-2-1012θeff(deg)\ny(\u0001m)hypothetical\ndata\n00.51\n-2-1012Norm.θp\ny(\u0001m)hypothetical\ndataFIG. S2: Spatial distributions for the non spin Hall sample\nwith 2 nm Hf spacer: (a) the normalized precession amplitude\nand (b) e\u000bective driving \feld angle. The gray points in (a)\nand (b) are the data and the solid blue curves are the poly-\nnomial \fts used for computing \u0001. The red line in (b) is the\ndriving \feld angle, calculated from using Eq. 3, that would\nbe obtained from ST-FMR. The resulting phase correction\n\u0001 = (3:4\u00061:9)\u000e.\nFor the uniform precession mode, which is typically\nassumed for ST-FMR analysis, the mixing voltage is re-\nduced to\nVunif\nmix(H) =\u0000I0\u0001R\u0012psin 2\u00120\n2n\n\u001f0(H) cos('rf\u0000'FMR)\n+\u001f00(H) sin('rf\u0000'FMR)o\n: (S11)\nIn the case of an in-plane driving \feld (i.e. in-plane Oer-\nsted \feld without spin torque), the FMR phase and the\nrf current phase have a simple relation 'rf\u0000'FMR= 90\u000e.\nThus in the absence of spin torque, the \u001f0term vanishes\nandVmix(H)/\u001f00(H), corresponding to an antisymmet-\nric spectral line shape. While the spin torque is present,\nthe\u001f0becomes nonzero, and the spectrum is a linear\ncombination of symmetric ( \u001f0) and antisymmetric ( \u001f00)\ncomponents.\nSPATIAL VARIATION OF NON SPIN HALL\nSAMPLE\nFig. S2 presents the yposition dependent FMR am-\nplitude and phase for the sample with 2 nm Hf spacer.\nSimilar to the measurements in Fig. 5, the observed FMR\nphase variation in the non spin Hall sample is nonuni-\nform. The driving \feld titles about 25\u000eout of plane near\nthe top and bottom edges. Despite the phase nonunifor-\nmity, the overall phase variation is smaller than that of\nthe spin Hall sample. Consequently, the corresponding\nphase correction \u0001 = (3 :4\u00061:9)\u000eis smaller than the spin\nHall sample, as summarized in Table I."    },    {        "title": "1610.06661v1.Spin_transport_and_dynamics_in_all_oxide_perovskite_La___2_3__Sr___1_3__MnO__3__SrRuO__3__bilayers_probed_by_ferromagnetic_resonance.pdf",        "content": "Spin transport and dynamics in all-oxide perovskite La 2=3Sr1=3MnO 3/SrRuO 3bilayers\nprobed by ferromagnetic resonance\nSatoru Emori,1,\u0003Urusa S. Alaan,1, 2Matthew T. Gray,1, 2Volker\nSluka,3Yizhang Chen,3Andrew D. Kent,3and Yuri Suzuki1, 4\n1Geballe Laboratory for Advanced Materials, Stanford University, Stanford, CA 94305 USA\n2Department of Materials Science and Engineering, Stanford University, Stanford, CA 94305 USA\n3Department of Physics, New York University, New York, NY 10003, USA\n4Department of Applied Physics, Stanford University, Stanford, CA 94305 USA\n(Dated: November 11, 2021)\nThin \flms of perovskite oxides o\u000ber the possibility of combining emerging concepts of strongly\ncorrelated electron phenomena and spin current in magnetic devices. However, spin transport and\nmagnetization dynamics in these complex oxide materials are not well understood. Here, we ex-\nperimentally quantify spin transport parameters and magnetization damping in epitaxial perovskite\nferromagnet/paramagnet bilayers of La 2=3Sr1=3MnO 3/SrRuO 3(LSMO/SRO) by broadband ferro-\nmagnetic resonance spectroscopy. From the SRO thickness dependence of Gilbert damping, we\nestimate a short spin di\u000busion length of <\u00181 nm in SRO and an interfacial spin-mixing conductance\ncomparable to other ferromagnet/paramagnetic-metal bilayers. Moreover, we \fnd that anisotropic\nnon-Gilbert damping due to two-magnon scattering also increases with the addition of SRO. Our\nresults demonstrate LSMO/SRO as a spin-source/spin-sink system that may be a foundation for\nexamining spin-current transport in various perovskite heterostructures.\nI. INTRODUCTION\nManipulation and transmission of information by spin\ncurrent is a promising route toward energy-e\u000ecient mem-\nory and computation devices1. Such spintronic devices\nmay consist of ferromagnets interfaced with nonmagnetic\nconductors that exhibit spin-Hall and related spin-orbit\ne\u000bects2{4. The direct spin-Hall e\u000bect in the conductor\ncan convert a charge current to a spin current, which ex-\nerts torques on the adjacent magnetization and modi\fes\nthe state of the device5,6. Conversely, the inverse spin-\nHall e\u000bect in the conductor can convert a propagating\nspin current in the magnetic medium to an electric signal\nto read spin-based information packets7. For these device\nschemes, it is essential to understand the transmission of\nspin current between the ferromagnet and the conductor,\nwhich is parameterized by the spin-mixing conductance\nand spin di\u000busion length. These spin transport parame-\nters can be estimated by spin pumping at ferromagnetic\nresonance (FMR), in which a spin current is resonantly\ngenerated in the ferromagnet and absorbed in the adja-\ncent conductor8,9. Spin pumping has been demonstrated\nin various combinations of materials, where the magnetic\nlayer may be an alloy (e.g., permalloy) or insulator (e.g.,\nyttrium iron garnet) and the nonmagnetic conductor may\nbe a transition metal, semiconductor, conductive poly-\nmer, or topological insulator10{16.\nTransition metal oxides, particularly those with the\nperovskite structure, o\u000ber the intriguing prospect of\nintegrating a wide variety of strongly correlated elec-\ntron phenomena17,18with spintronic functionalities19,20.\nAmong these complex oxides, La 2=3Sr1=3MnO 3(LSMO)\nand SrRuO 3(SRO) are attractive materials for epitaxial,\nlattice-matched spin-source/spin-sink heterostructures.\nLSMO, a metallic ferromagnet known for its colossalmagnetoresistance and Curie temperature of >300 K,\ncan be an excellent resonantly-excited spin source be-\ncause of its low magnetization damping21{26. SRO, a\nroom-temperature metallic paramagnet with relatively\nhigh conductivity27, exhibits strong spin-orbit coupling28\nthat may be useful for emerging spintronic applications\nthat leverage spin-orbit e\u000bects2{4.\nA few recent studies have reported dc voltages at FMR\nin LSMO/SRO bilayers that are attributed to the in-\nverse spin-Hall e\u000bect in SRO generated by spin pump-\ning24{26. However, it is generally a challenge to separate\nthe inverse spin-Hall signal from the spin recti\fcation sig-\nnal, which is caused by an oscillating magnetoresistance\nmixing with a microwave current in the conductive mag-\nnetic layer29{31. Moreover, while the spin-mixing con-\nductance is typically estimated from the enhancement in\nthe Gilbert damping parameter \u000b, the quanti\fcation of \u000b\nis not necessarily straightforward in epitaxial thin \flms\nthat exhibit pronounced anisotropic non-Gilbert damp-\ning23,32{37. It has also been unclear how the Gilbert and\nnon-Gilbert components of damping in LSMO are each\nmodi\fed by an adjacent SRO layer. These points above\nhighlight the need for an alternative experimental ap-\nproach for characterizing spin transport and magnetiza-\ntion dynamics in LSMO/SRO.\nIn this work, we quantify spin transport parameters\nand magnetization damping in epitaxial LSMO/SRO bi-\nlayers by broadband FMR spectroscopy with out-of-plane\nandin-plane external magnetic \felds. Out-of-plane FMR\nenables straightforward extraction of Gilbert damping as\na function of SRO overlayer thickness, which is repro-\nduced by a simple \\spin circuit\" model based on di\u000busive\nspin transport38,39. We \fnd that the spin-mixing conduc-\ntance at the LSMO/SRO interface is comparable to other\nferromagnet/conductor interfaces and that spin current\nis absorbed within a short length scale of <\u00181 nm in thearXiv:1610.06661v1  [cond-mat.mtrl-sci]  21 Oct 20162\n42 44 46 48 50LSMO(10)\n  /SRO(18)LSMO(10)\n  log(intensity) (a.u.)\n2 (deg.)LSAT(002)  \nLSMO(002)  \nSRO(002)  \nFigure 1. 2 \u0012-!x-ray di\u000braction scans of a single-layer\nLSMO(10 nm) \flm and LSMO(10 nm)/SRO(18 nm) bilayer.\nconductive SRO layer. From in-plane FMR, we observe\npronounced non-Gilbert damping that is anisotropic and\nscales nonlinearly with excitation frequency, which is ac-\ncounted for by an existing model of two-magnon scat-\ntering40. This two-magnon scattering is also enhanced\nwith the addition of the SRO overlayer possibly due to\nspin pumping. Our \fndings reveal key features of spin\ndynamics and transport in the prototypical perovskite\nferromagnet/conductor bilayer of LSMO/SRO and pro-\nvide a foundation for future all-oxide spintronic devices.\nII. SAMPLE AND EXPERIMENTAL DETAILS\nEpitaxial \flms of LSMO(/SRO) were grown on\nas-received (001)-oriented single-crystal (LaAlO 3)0:3\n(Sr2AlTaO 6)0:7(LSAT) substrates using pulsed laser de-\nposition. LSAT exhibits a lower dielectric constant than\nthe commonly used SrTiO 3substrate and is therefore\nbetter suited for high-frequency FMR measurements.\nThe lattice parameter of LSAT (3.87 \u0017A) is also closely\nmatched to the pseudocubic lattice parameter of LSMO\n(\u00193.88 \u0017A). By using deposition parameters similar to\nthose in previous studies from our group41,42, all \flms\nwere deposited at a substrate temperature of 750\u000eC with\na target-to-substrate separation of 75 mm, laser \ruence\nof\u00191 J/cm2, and repetition rate of 1 Hz. LSMO was de-\nposited in 320 mTorr O 2, followed by SRO in 100 mTorr\nO2. After deposition, the samples were held at 600\u000eC\nfor 15 minutes in \u0019150 Torr O 2and then the substrate\nheater was switched o\u000b to cool to room temperature. The\ndeposition rates were calibrated by x-ray re\rectivity mea-\nsurements. The thickness of LSMO, tLSMO , in this study\nis \fxed at 10 nm, which is close to the minimum thickness\nat which the near-bulk saturation magnetization can be\nattained.\nX-ray di\u000braction results indicate that both the LSMO\n\flms and LSMO/SRO bilayers are highly crystalline\nand epitaxial with the LSAT(001) substrate, with high-\nresolution 2 \u0012-!scans showing distinct Laue fringes\naround the (002) Bragg re\rection (Fig. 1). In this study,\nthe maximum thickness of the LSMO and SRO layers\n770 780 790 800 810 main mode\n sec. mode\n  dIFMR/dH (a.u.)\n0H (mT) data\n fit(a) (b) \n770 780 790 800 810 data\n fit\n  dIFMR/dH (a.u.)\n0H (mT)Figure 2. Exemplary FMR spectra and \ftting curves: (a)\none mode of Lorentzian derivative; (b) superposition of a main\nmode and a small secondary mode due to slight sample inho-\nmogeneity.\ncombined is less than 30 nm and below the threshold\nthickness for the onset of structural relaxation by mis\ft\ndislocation formation41,42. The typical surface roughness\nof LSMO and SRO measured by atomic force microscopy\nis<\u00184\u0017A, comparable to the roughness of the LSAT sub-\nstrate surface.\nSQUID magnetometry con\frms that the Curie tem-\nperature of the LSMO layer is \u0019350 K and the room-\ntemperature saturation magnetization is Ms\u0019300 kA/m\nfor 10-nm thick LSMO \flms. The small LSMO thickness\nis desirable for maximizing the spin-pumping-induced en-\nhancement in damping, since spin pumping scales in-\nversely with the ferromagnetic layer thickness8,9. More-\nover, the thickness of 10 nm is within a factor of \u00192\nof the characteristic exchange lengthp\n2Aex=\u00160M2s\u00195\nnm, assuming an exchange constant of Aex\u00192 pJ/m in\nLSMO (Ref. 43), so standing spin-wave modes are not\nexpected.\nBroadband FMR measurements were performed at\nroom temperature. The \flm sample was placed face-\ndown on a coplanar waveguide with a center conductor\nwidth of 250 \u0016m. Each FMR spectrum was acquired at a\nconstant excitation frequency while sweeping the exter-\nnal magnetic \feld H. The \feld derivative of the FMR\nabsorption intensity (e.g., Fig. 2) was acquired using an\nrf diode combined with an ac (700 Hz) modulation \feld.\nEach FMR spectrum was \ftted with the derivative of the\nsum of the symmetric and antisymmetric Lorentzians, as\nshown in Fig. 2, from which the resonance \feld HFMR\nand half-width-at-half-maximum linewidth \u0001 Hwere ex-\ntracted. In some spectra (e.g., Fig. 2(b)), a small sec-\nondary mode in addition to the main FMR mode was\nobserved. We \ft such a spectrum to a superposition of\ntwo modes, each represented by a generalized Lorentzian\nderivative, and analyze only the HFMR and \u0001Hof the\nlarger-amplitude main FMR mode. The secondary mode\nis not a standing spin-wave mode because it appears\nabove or below the resonance \feld of the main mode\nHFMR with no systematic trend in \feld spacing. We\nattribute the secondary mode to regions in the \flm with3\n0.360.400.440.48\n  0Meff (T)\n0 5 10 15 201.952.002.05\ntSRO (nm)\n  gop\n0 5 10 15 200.00.20.40.60.81.01.2\nLSMO/SRO\nLSMO\n  0HFMR (T)\nf (GHz)(a) (b) \n(c) \nFigure 3. (a) Out-of-plane resonance \feld\nHFMR versus excitation frequency ffor\na single-layer LSMO(10 nm) \flm and a\nLSMO(10 nm)/SRO(3 nm) bilayer. The\nsolid lines indicate \fts to the data using\nEq. 1. (b,c) SRO-thickness dependence of\nthe out-of-plane Land\u0013 e g-factor (b) and ef-\nfective saturation magnetization Me\u000b(c).\nThe dashed lines indicate the values aver-\naged over all the data shown.\nslightly di\u000berent Msor magnetic anisotropy. More pro-\nnounced inhomogeneity-induced secondary FMR modes\nhave been observed in epitaxial magnetic \flms in prior\nreports22,44.\nIII. OUT-OF-PLANE FMR AND ESTIMATION\nOF SPIN TRANSPORT PARAMETERS\nOut-of-plane FMR allows for conceptually simpler ex-\ntraction of the static and dynamic magnetic properties\nof a thin-\flm sample. For \ftting the frequency depen-\ndence ofHFMR, the Land\u0013 e g-factor gopand e\u000bective sat-\nuration magnetization Me\u000bare the only adjustable pa-\nrameters in the out-of-plane Kittel equation. The fre-\nquency dependence of \u0001 Hfor out-of-plane FMR arises\nsolely from Gilbert damping, so that the conventional\nmodel of spin pumping8,9,38,39can be used to analyze the\ndata without complications from non-Gilbert damping.\nThis consideration is particularly important because the\nlinewidths of our LSMO(/SRO) \flms in in-plane FMR\nmeasurements are dominated by highly anisotropic non-\nGilbert damping (as shown in Sec. IV). Furthermore, a\nsimple one-dimensional, time-independent model of spin\npumping outlined by Boone et al.38is applicable in the\nout-of-plane con\fguration, since the precessional orbit of\nthe magnetization is circular to a good approximation.\nThis is in contrast with the in-plane con\fguration with\na highly elliptical orbit from a large shape anisotropy\n\feld. By taking advantage of the simplicity in out-of-\nplane FMR, we \fnd that the Gilbert damping parame-\nter in LSMO is approximately doubled with the addition\nof a su\u000eciently thick SRO overlayer due to spin pump-\ning. Our results indicate that spin-current transmission\nat the LSMO/SRO interface is comparable to previously\nreported ferromagnet/conductor bilayers and that spin\ndi\u000busion length in SRO is <\u00181 nm.\nWe \frst quantify the static magnetic properties of\nLSMO(/SRO) from the frequency dependence of HFMR.\nThe Kittel equation for FMR in the out-of-plane con\fg-\nuration takes a simple linear form,\nf=gop\u0016B\nh\u00160(HFMR\u0000Me\u000b); (1)\nwhere\u00160is the permeability of free space, \u0016Bis the Bohrmagneton, and his the Planck constant. As shown in\nFig. 3(a), we only \ft data points where \u00160HFMR is at\nleast 0.2 T above \u00160Me\u000bto ensure that the \flm is sat-\nurated out-of-plane. Figures 3(b) and (c) plot the ex-\ntractedMe\u000bandgop, respectively, each exhibiting no\nsigni\fcant dependence on SRO thickness tSROto within\nexperimental uncertainty. The SRO overlayer therefore\nevidently does not modify the bulk magnetic proper-\nties of LSMO, and signi\fcant interdi\u000busion across the\nSRO/LSMO interface can be ruled out. The averaged\nMe\u000bof 330\u000610 kA/m (\u00160Me\u000b= 0:42\u00060:01 T) is close\ntoMsobtained from static magnetometery and implies\nnegligible out-of-plane magnetic anisotropy; we thus as-\nsumeMs=Me\u000bin all subsequent analyses. The SRO-\nthickness independence of gop, averaging to 2 :01\u00060:01,\nimplies that the SRO overlayer does not generate a signif-\nicant orbital contribution to magnetism in LSMO. More-\nover, the absence of detectable change in gopwith in-\ncreasingtSRO may indicate that the imaginary compo-\nnent of the spin-mixing conductance8,9is negligible at\nthe LSMO/SRO interface.\nThe Gilbert damping parameter \u000bis extracted from\nthe frequency dependence of \u0001 H(e.g., Figure 4(a)) by\n\ftting the data with the standard linear relation,\n\u0001H= \u0001H0+h\ngop\u0016B\u000bf: (2)\nThe zero-frequency linewidth \u0001 H0is typically attributed\nto sample inhomogeneity. We observe sample-to-sample\nvariation of \u00160\u0001H0in the range\u00191\u00004 mT with no\nsystematic correlation with tSRO or the slope in Eq. 2.\nMoreover, similar to the analysis of HFMR, we only \ft\ndata obtained at \u00150.2 T above \u00160Me\u000bto minimize spu-\nrious broadening of \u0001 Hat low \felds. The linear slope of\n\u0001Hplotted against frequency up to 20 GHz is therefore\na reliable measure of \u000bdecoupled from \u0001 H0in Eq. 2.\nFigure 4(a) shows an LSMO single-layer \flm and an\nLSMO/SRO bilayer with similar \u0001 H0. The slope, which\nis proportional to \u000b, is approximately a factor of 2 greater\nfor LSMO/SRO. Figure 4(b) summarizes the dependence\nof\u000bon SRO-thickness, tSRO. For LSMO single-layer\n\flms we \fnd \u000b= (0:9\u00060:2)\u000210\u00003, which is on the same\norder as previous reports of LSMO thin \flms21{23,26.\nThis low damping is also comparable to the values re-4\n0 5 10 15 200123\n   (10-3)\ntSRO (nm)\n1/Gext 1/G↑↓m LSMO SRO\n0 5 10 15 201.01.52.02.53.0\nLSMO\n  0H (mT)\nf (GHz)LSMO/SRO(a) (b) (c) \nFigure 4. (a) Out-of-plane FMR linewidth \u0001 Hversus excitation frequency for LSMO(10 nm) and LSMO(10 nm)/SRO(3 nm).\nThe solid lines indicate \fts to the data using Eq. 2. (b) Gilbert damping parameter \u000bversus SRO thickness tSRO. The solid\ncurve shows a \ft to the di\u000busive spin pumping model (Eq. 5). (c) Schematic of out-of-plane spin pumping and the equivalent\n\\spin circuit.\"\nported in Heusler alloy thin \flms45,46and may arise from\nthe half-metal-like band structure of LSMO (Ref. 47).\nLSMO can thus be an e\u000ecient source of spin current\ngenerated resonantly by microwave excitation.\nWith a few-nanometer thick overlayer of SRO, \u000bin-\ncreases to\u00192\u000210\u00003(Fig. 4(b)). This enhanced damping\nwith the addition of SRO overlayer may arise from (1)\nspin scattering48,49at the LSMO/SRO interface or (2)\nspin pumping8,9where nonequilibrium spins from LSMO\nare absorbed in the bulk of the SRO layer. Here, we\nassume that interfacial spin scattering is negligible, since\n<\u00181 nm of SRO overlayer does not enhance \u000bsigni\fcantly\n(Fig. 4(b)). This is in contrast with the pronounced in-\nterfacial e\u000bect in ferromagnet/Pt bilayers48,49, in which\neven<1 nm of Pt can increase \u000bby as much as a fac-\ntor of\u00192 (Refs. 50{52). In the following analysis and\ndiscussion, we show that spin pumping alone is su\u000ecient\nfor explaining the enhanced damping in LSMO with an\nSRO overlayer.\nWe now analyze the data in Fig. 4(b) using a one-\ndimensional model of spin pumping based on di\u000busive\nspin transport38,39. The resonantly-excited magnetiza-\ntion precession in LSMO generates non-equilibrium spins\npolarized along ^ m\u0002d ^m=dt, which is transverse to the\nmagnetization unit vector ^ m. This non-equilibrium spin\naccumulation di\u000buses out to the adjacent SRO layer\nand depolarizes exponentially on the characteristic length\nscale\u0015s. The spin current density ~jsat the LSMO/SRO\ninterface can be written as38,53\n~jsjinterface =~2\n2e2^m\u0002d^m\ndt\u0010\n1\nG\"#+1\nGext\u0011; (3)\nwhere ~is the reduced Planck constant, G\"#is the inter-\nfacial spin-mixing conductance per unit area, and Gextis\nthe spin conductance per unit area in the bulk of SRO.\nIn Eq. 3, 1/ G\"#and 1/Gextconstitute spin resistors in\nseries such that the spin transport from LSMO to SRO\ncan be regarded analogously as a \\spin circuit,\" as il-\nlustrated in Fig. 4(c). In literature, these interfacialand bulk spin conductances are sometimes lumped to-\ngether as an \\e\u000bective spin-mixing conductance\" Ge\u000b\n\"#=\n(1=G\"#+ 1=Gext)\u00001(Refs. 10{13, 16, 20, 23, 26, 44). We\nalso note that the alternative form of the (e\u000bective) spin-\nmixing conductance g(e\u000b)\ne\u000b, with units of m\u00002, is related to\nG(e\u000b)\n\"#, with units of \n\u00001m\u00002, byg(e\u000b)\ne\u000b= (h=e2)G(e\u000b)\n\"#\u0019\n26 k\n\u0002G(e\u000b)\n\"#.\nThe functional form of Gextis obtained by solving the\nspin di\u000busion equation with appropriate boundary condi-\ntions38,39,53. In the case of a ferromagnet/nonmagnetic-\nmetal bilayer, we obtain\nGext=1\n2\u001aSRO\u0015stanh\u0012tSRO\n\u0015s\u0013\n; (4)\nwhere\u001aSROis the resistivity of SRO, tSROis the thick-\nness of the SRO layer, and \u0015sis the di\u000busion length of\npumped spins in SRO. Finally, the out\row of spin cur-\nrent (Eq. 3) is equivalent to an enhancement of Gilbert\ndamping9with respect to \u000b0of LSMO with tSRO = 0\nsuch that\n\u000b=\u000b0+gop\u0016B~\n2e2MstLSMO\u00141\nG\"#+ 2\u001aSRO\u0015scoth\u0012tSRO\n\u0015s\u0013\u0015\u00001\n:\n(5)\nThus, two essential parameters governing spin transport\nG\"#and\u0015scan be estimated by \ftting the SRO-thickness\ndependence of \u000b(Fig. 4(b)) with Eq. 5.\nIn carrying out the \ft, we \fx \u000b0= 0:9\u000210\u00003. We note\nthat\u001aSROincreases by an order of magnitude compared\nto the bulk value of \u00192\u000210\u00006\nm astSROis reduced to\na few nm; also, at thicknesses of 3 monolayers ( \u00191.2 nm)\nor below, SRO is known to be insulating54. We there-\nfore use the tSRO-dependent \u001aSRO shown in Appendix\nA while assuming \u0015sis constant. An alternative \ftting\nmodel that assumes a constant \u001aSRO, which is a common\napproach in literature, is discussed in Appendix A.\nThe curve in Fig. 4(b) is generated by Eq. 5 with G\"#=\n1:6\u00021014\n\u00001m\u00002and\u0015s= 0:5 nm. Given the scatter of5\n170175180185\n170\n175\n180\n185[010]\n[110]\n[100] \n  \n0HFMR (mT)\nLSMO\nLSMO/SRO\n0 5 10 15 200100200300400500\n  0HFMR (mT)\nf (GHz)H||[100]\nH||[110](a) (b) (c) \n0 5 10 15 201.952.002.05\ntSRO (nm)\n  gip\n-6-4-20\n  0H||,4 (mT)\n(d) \n14 15330360 \n \n  \nFigure 5. (a) Angular dependence of HFMR at 9 GHz for LSMO(10 nm) and LSMO(10 nm)/SRO(7 nm). The solid curves\nindicate \fts to the data using Eq. 6. (b) Frequency dependence of HFMR for LSMO(10 nm)/SRO(7 nm) with \feld applied in the\n\flm plane along the [100] and [110] directions. Inset: close-up of HFMR versus frequency around 14-15 GHz. In (a) and (b), the\nsolid curves show \fts to the Kittel equation (Eq. 6). (c,d) SRO-thickness dependence of the in-plane cubic magnetocrystalline\nanisotropy \feld (c) and in-plane Land\u0013 e g-factor (d). The dashed lines indicate the values averaged over all the data shown.\nthe experimental data, acceptable \fts are obtained with\nG\"#\u0019(1:2\u00002:5)\u00021014\n\u00001m\u00002and\u0015s\u00190:3\u00000:9\nnm. The estimated ranges of G\"#and\u0015salso depend\nstrongly on the assumptions behind the \ftting model.\nFor example, as shown in Appendix A, the constant- \u001aSRO\nmodel yields G\"#>\u00183\u00021014\n\u00001m\u00002and\u0015s\u00192:5 nm.\nNevertheless, we \fnd that the estimated G\"#\nis on the same order of magnitude as those\nof various ferromagnet/transition-metal heterostruc-\ntures39,55,56, signifying that the LSMO/SRO interface\nis reasonably transparent to spin current. More impor-\ntantly, the short \u0015simplies the presence of strong spin-\norbit coupling that causes rapid spin scattering within\nSRO. This \fnding is consistent with a previous study on\nSRO at low temperature in the ferromagnetic state show-\ning extremely fast spin relaxation with Gilbert damping\n\u000b\u00181 (Ref. 28). The short \u0015sindicates that SRO may be\nsuitable as a spin sink or detector in all-oxide spintronic\ndevices.\nIV. IN-PLANE FMR AND ANISOTROPIC\nTWO-MAGNON SCATTERING\nIn epitaxial thin \flms, the analysis of in-plane FMR\nis generally more complicated than that of out-of-plane\nFMR. High crystallinity of the \flm gives rise to a non-\nnegligible in-plane magnetocrystalline anisotropy \feld,\nwhich manifests in an in-plane angular dependence of\nHFMR and introduces another adjustable parameter in\nthe nonlinear Kittel equation for in-plane FMR. More-\nover, \u0001Hin in-plane FMR of epitaxial thin \flms often\ndepends strongly on the magnetization orientation and\nexhibits nonlinear scaling with respect to frequency due\nto two-magnon scattering, a non-Gilbert mechanism for\ndamping23,32{37. We indeed \fnd that damping of LSMO\nin the in-plane con\fguration is anisotropic and domi-\nnated by two-magnon scattering. We also observe ev-idence of enhanced two-magnon scattering with added\nSRO layers, which may be due to spin pumping from\nnonuniform magnetization precession.\nFigure 5(a) plots HFMR of a single-layer LSMO \flm\nand an LSMO/SRO bilayer as a function of applied \feld\nangle within the \flm plane. For both samples, we observe\nclear four-fold symmetry, which is as expected based on\nthe epitaxial growth of LSMO on the cubic LSAT(001)\nsubstrate. Similar to previous FMR studies of LSMO on\nSrTiO 3(001)57,58, the magnetic hard axes (corresponding\nto the axes of higher HFMR) are alongh100i. The in-\nplane Kittel equation for thin \flms with in-plane cubic\nmagnetic anisotropy is59,\nf=gip\u0016B\nh\u00160\u0002\nHFMR +Hjj;4cos(4\u001e)\u00031\n2\u0002\n\u0014\nHFMR +Me\u000b+1\n4Hjj;4(3 + cos(4\u001e))\u00151\n2\n;\n(6)\nwheregipis the Land\u0013 e g-factor that is obtained from in-\nplane FMR data, Hjj;4is the e\u000bective cubic anisotropy\n\feld, and\u001eis the in-plane \feld angle with respect to\nthe [100] direction. Given that LSMO is magnetically\nvery soft (coercivity on the order of 0.1 mT) at room\ntemperature, we assume that the magnetization is par-\nallel to the \feld direction, particularly with \u00160H\u001d10\nmT. In \ftting the angular dependence (e.g., Fig. 5(a))\nand frequency dependence (e.g., Fig. 5(b)) of HFMR to\nEq. 6, we \fx Me\u000bat the values obtained from out-of-\nplane FMR (Fig. 3(b)) so that Hjj;4andgipare the\nonly \ftting parameters. For the two samples shown in\nFig. 5(a), the \fts to the angular dependence and fre-\nquency dependence data yield consistent values of Hjj;4\nandgip. For the rest of the LSMO(/SRO) samples, we\nuse the frequency dependence data with Hjj[100] and\nHjj[110] to extract these parameters. Figures 5(c) and\n(d) show that Hjj;4andgip, respectively, exhibit no sys-\ntematic dependence on tSRO, similar to the \fndings from6\nout-of-plane FMR (Figs. 3(b),(c)). The in-plane cubic\nmagnetocrystalline anisotropy in LSMO(/SRO) is rela-\ntively small, with \u00160Hjj;4averaging to\u00192.5 mT.gipav-\nerages out to 1 :99\u00060:02, which is consistent with gop\nfound from out-of-plane FMR.\nWhile the magnetocrytalline anisotropy in\nLSMO(/SRO) is found to be modest and indepen-\ndent oftSRO, we observe much more pronounced\nin-plane anisotropy and tSRO dependence in linewidth\n\u0001H, as shown in Figs. 6(a) and (b). Figure 6(a)\nindicates that the in-plane dependence of \u0001 His four-\nfold symmetric for both LSMO(10 nm) and LSMO(10\nnm)/SRO(7 nm). \u0001 His approximately a factor of 2\nlarger when the sample is magnetized along h100icom-\npared to when it is magnetized along h110i. One might\nattribute this pronounced anisotropy to anisotropic\nGilbert damping60, such that the sample magnetized\nalong the hard axes h100imay lead to stronger damp-\ning. However, we \fnd no general correlation between\nmagnetocrystalline anisotropy and anisotropic \u0001 H: As\nwe show in Appendix B, LSMO grown on NdGaO 3(110)\nwith pronounced uniaxial magnetocrystalline anisotropy\nexhibits identical \u0001 Hwhen magnetized along the easy\nand hard axes. Moreover, whereas Gilbert damping\nshould lead to a linear frequency dependence of \u0001 H,\nfor LSMO(/SRO) the observed frequency dependence\nof \u0001His clearly nonlinear as evidenced in Fig. 6(b).\nThe pronounced anisotropy and nonlinear frequency\ndependence of \u0001 Htogether suggest the presence of a\ndi\u000berent damping mechanism.\nA well-known non-Gilbert damping mechanism in\nhighly crystalline ultrathin magnetic \flms is two-magnon\nscattering23,32{37,40,61,62, in which uniformly precessing\nmagnetic moments (a spin wave, or magnon mode, with\nwavevector k= 0) dephase to a k6= 0 magnon mode with\nadjacent moments precessing with a \fnite phase di\u000ber-\nence. By considering both exchange coupling (which re-\nsults in magnon energy proportional to k2) and dipolar\ncoupling (magnon energy proportional to \u0000jkj) among\nprecessing magnetic moments, the k= 0 andk6= 0\nmodes become degenerate in the magnon dispersion re-\nlation61as illustrated in Fig. 6(c).\nThe transition from k= 0 tok6= 0 is activated by\ndefects that break the translational symmetry of the\nmagnetic system by localized dipolar \felds40,61,62. In\nLSMO(/SRO), the activating defects may be faceted such\nthat two-magnon scattering is more pronounced when\nthe magnetization is oriented along h100i. One possibil-\nity is that LSMO thin \flms naturally form pits or islands\nfaceted alongh100iduring growth. However, we are un-\nable to consistently observe signs of such faceted defects\nin LSMO(/SRO) samples with an atomic force micro-\nscope (AFM). It is possible that these crystalline defects\nare smaller than the lateral resolution of our AFM setup\n(<\u001810 nm) or that these defects are not manifested in sur-\nface topography. Such defects may be point defects or\nnanoscale clusters of distinct phases that are known to\nexist intrinsically even in high-quality crystals of LSMO(Ref. 63).\nAlthough the de\fnitive identi\fcation of defects that\ndrive two-magnon scattering would require further in-\nvestigation, we can rule out (1) atomic step terraces\nand (2) mis\ft dislocations as sources of anisotropic two-\nmagnon scattering. (1) AFM shows that the orienta-\ntion and density of atomic step terraces di\u000ber randomly\nfrom sample to sample, whereas the anisotropy in \u0001 H\nis consistently cubic with larger \u0001 HforHjjh100ithan\nHjjh110i. This is in agreement with the recent study\nby Lee et al. , which shows anisotropic two-magnon scat-\ntering in LSMO to be independent of regularly-spaced\nparallel step terraces on a bu\u000bered-oxide etched SrTiO 3\nsubstrate23. (2) Although Woltersdorf and Heinrich have\nfound that mis\ft dislocations in Fe/Pd grown on GaAs\nare responsible for two-magnon scattering33, such dis-\nlocations are expected to be virtually nonexistent in\nfully strained LSMO(/SRO) \flms on the closely-latticed\nmatched LSAT substrates41,42.\nWe assume that the in-plane four-fold anisotropy and\nnonlinear frequency dependence of \u0001 Hare entirely due\nto two-magnon scattering. For a sample magnetized\nalong a given in-plane crystallographic axis hhk0i=h100i\norh110i, the two-magnon scattering contribution to \u0001 H\nis given by40\n\u0001Hhhk0i\n2m = \u0000hhk0i\n2m sin\u00001sp\nf2+ (fM=2)2\u0000fM=2p\nf2+ (fM=2)2+fM=2;(7)\nwherefM= (gip\u0016B=h)\u00160Msand \u0000hhk0i\n2m is the two-\nmagnon scattering parameter. The angular dependence\nof \u0001His \ftted with33\n\u0001H= \u0001H0+h\ngip\u0016B\u000bf\n+ \u0001Hh100i\n2m cos2(2\u001e) + \u0001Hh110i\n2m cos2(2[\u001e\u0000\u0019\n4]):(8)\nSimilarly, the frequency dependence of \u0001 Hwith the sam-\nple magnetized along [100] or [110], i.e., \u001e= 0 or\u0019=4,\nis well described by Eqs. 7 and 8. In principle, it should\nbe possible to \ft the linewidth data with \u0001 H0,\u000b, and\n\u00002mas adjustable parameters. In practice, the \ft car-\nried out this way is overspeci\fed such that wide ranges\nof these parameters appear to \ft the data. We there-\nfore impose a constraint on \u000bby assuming that Gilbert\ndamping for LSMO(/SRO) is isotropic: For each SRO\nthicknesstSRO,\u000bis \fxed to the value estimated from\nthe \ft curve in Fig. 4(c) showing out-of-plane FMR data.\n(This assumption is likely justi\fed, since the damping\nfor LSMO(10 nm) on NdGdO 3(110) with strong uniaxial\nmagnetic anisotropy is identical for the easy and hard\ndirections, as shown in Appendix B.) To account for the\nuncertainty in the Gilbert damping in Fig. 4(c), we vary \u000b\nby\u000625% for \ftting the frequency dependence of in-plane\n\u0001H. Examples of \fts using Eqs. 7 and 8 are shown in\nFig. 6(a),(b).\nFigure 6(d) shows that the SRO overlayer enhances\nthe two-magnon scattering parameter \u0000 2mby up to a7\n0 5 10 15 2004812\nLSMOLSMO/SRO\n  0H (mT)\nf (GHz)(a) (b) \n(c) (d) \n  \nFMR \nfreq. \nk f \nk=0 k≠0 \n0 5 10 15 200102030\n  02m (mT)\ntSRO (nm)H||[100]\nH||[110]\n0510\n0\n5\n10\nLSMO\nLSMO/SRO[110][010]\n[100]\n   0H (mT)\nFigure 6. (a) In-plane angular dependence of\nlinewidth \u0001H at 9 GHz for LSMO(10 nm) and\nLSMO(10 nm)/SRO(7 nm). The solid curves\nindicate \fts to Eq. 8. (b) Frequency depen-\ndence of \u0001H for LSMO(10 nm) and LSMO(10\nnm)/SRO(7 nm) with Happlied along the [100]\ndirection. The solid curves indicate \fts to\nEq. 7. The dashed and dotted curves indicate\nestimated two-magnon and Gilbert damping\ncontributions, respectively. (c) Schematic of a\nspin wave dispersion curve (when the magne-\ntization is in-plane and has a \fnite component\nparallel to the spin wave wavevector k) and two-\nmagnon scattering. (d) Two-magnon scattering\ncoe\u000ecient \u0000 2m, estimated for the cases with H\napplied along the [100] and [110] axes, plotted\nagainst SRO thickness tSRO. The dashed curve\nis the same as that in Fig. 4(c) scaled to serve\nas a guide for the eye for \u0000 2mwith H along\n[100].\nfactor of\u00192 forHjj[100]. By contrast, for Hjj[110], al-\nthough LSMO/SRO exhibits enhanced \u0001 Hcompared to\nLSMO, the enhancement in \u0000 2mis obscured by the un-\ncertainty in Gilbert damping. In Table I, we summa-\nrize the Gilbert and two-magnon contributions to \u0001 H\nfor LSMO single layers and LSMO/SRO (averaged val-\nues for samples with tSRO>4 nm) with Hjj[100] and\nHjj[110]. Comparing the e\u000bective spin relaxation rates,\n(gip\u0016B=h)\u00160Ms\u000band (gip\u0016B=h)\u00160\u00002m, reveals that two-\nmagnon scattering dominates over Gilbert damping.\nWe now speculate on the mechanisms behind the\nenhancement in \u0000 2min LSMO/SRO, particularly for\nHjj[100]. One possibility is that SRO interfaced with\nLSMO directly increases the rate of two-magnon scat-\ntering, perhaps due to formation of additional defects at\nthe surface of LSMO. If this were the case we might ex-\npect a signi\fcant increase and saturation of \u0000 2mat small\ntSRO. However, in reality, \u0000 2mincreases for tSRO>1 nm\n(Fig. 6(d)), which suggests spin scattering in the bulk\nof SRO. We thus speculate another mechanism, where\nk6= 0 magnons in LSMO are scattered by spin pump-\nTable I. Spin relaxation rates extracted from in-plane FMR\n(106s\u00001)\nLSMO LSMO/SRO*\nGilbert:gip\u0016B\nh\u00160Ms\u000b 11\u00062 23\u00064\ntwo-magnon:gip\u0016B\nh\u00160\u00002m(Hjj[100]) 290\u000650 550\u0006100\ntwo-magnon:gip\u0016B\nh\u00160\u00002m(Hjj[110]) 140\u000660 250\u000660\n* Averaged over samples with tSRO>4 nm.ing into SRO. As shown by the guide-for-the-eye curve\nin Fig. 6(d), the tSROdependence of \u0000 2m(forHjj[100])\nmay be qualitatively similar to the tSROdependence of\n\u000bmeasured from out-of-plane FMR (Fig. 4(c)); this cor-\nrespondence would imply that the same spin pumping\nmechanism, which is conventionally modeled to act on\nthek= 0 mode, is also operative in the degenerate k6= 0\nmagnon mode in epitaxial LSMO. Indeed, previous stud-\nies have electrically detected the presence of spin pump-\ning fromk6= 0 magnons by the inverse spin-Hall e\u000bect in\nY3Fe5O12/Pt bilayers64{66. However, we cannot conclu-\nsively attribute the observed FMR linewidth broadening\nin LSMO/SRO to such k6= 0 spin pumping, since it\nis unclear whether faster relaxation of k6= 0 magnons\nshould necessarily cause faster relaxation of the k= 0\nFMR mode. Regardless of its origin, the pronounced\nanisotropic two-magnon scattering introduces additional\ncomplexity to the analysis of damping in LSMO/SRO\nand possibly in other similar ultrathin epitaxial magnetic\nheterostructures.\nV. SUMMARY\nWe have demonstrated all-oxide perovskite bilayers\nof LSMO/SRO that form spin-source/spin-sink systems.\nFrom out-of-plane FMR, we deduce a low Gilbert damp-\ning parameter of \u00191\u000210\u00003for LSMO. The two-fold en-\nhancement in Gilbert damping with an SRO overlayer\nis adequately described by the standard model of spin\npumping based on di\u000busive spin transport. We ar-\nrive at an estimated spin-mixing conductance G\"#\u0019\n(1\u00002)\u00021014\n\u00001m\u00002and spin di\u000busion length \u0015s<\u00181\nnm, which indicate reasonable spin-current transparency\nat the LSMO/SRO interface and strong spin scattering8\nwithin SRO. From in-plane FMR, we reveal pronounced\nnon-Gilbert damping, attributed to two-magnon scatter-\ning, which results in a nonlinear frequency dependence\nand anisotropy in linewidth. The magnitude of two-\nmagnon scattering increases with the addition of an SRO\noverlayer, pointing to the presence of spin pumping from\nnonuniform spin wave modes. Our \fndings lay the foun-\ndation for understanding spin transport and magneti-\nzation dynamics in epitaxial complex oxide heterostruc-\ntures.\nACKNOWLEDGEMENTS\nWe thank Di Yi, Sam Crossley, Adrian Schwartz, Han-\nkyu Lee, and Igor Barsukov for helpful discussions, and\nTianxiang Nan and Nian Sun for the design of the copla-\nnar waveguide. This work was funded by the National\nSecurity Science and Engineering Faculty Fellowship of\nthe Department of Defense under Contract No. N00014-\n15-1-0045.\nAPPENDIX A: SPIN PUMPING AND SRO\nRESISTIVITY\nWhen \ftting the dependence of the Gilbert damping\nparameter\u000bon spin-sink thickness, a constant bulk re-\nsistivity for the spin sink layer is often assumed in lit-\nerature. By setting the resistivity of SRO to the bulk\nvalue\u001aSRO= 2\u000210\u00006\nm and \ftting the \u000b-versus-tSRO\ndata (Fig. 4(c) and reproduced in Fig. 7(a)) to Eq. 5,\nwe arrive at G\"#>\u00183\u00021014\n\u00001m\u00002and\u0015s\u00192:5 nm.\nThe \ft curve is insensitive to larger values of G\"#because\nthe bulk spin resistance 1/ Gext, with the relatively large\nresistivity of SRO, dominates over the interfacial spin re-\nsistance 1/G\"#(see Eqs. 4 and 5). As shown by the dot-\nted curve in Fig. 7, this simple constant- \u001aSROmodel ap-\npears to mostly capture the tSRO-dependence of \u000b. This\nmodel of course indicates \fnite spin pumping at even\nvery small SRO thickness <\u00181 nm, which is likely non-\nphysical since SRO should be insulating in this thickness\nregime54. Indeed,\u0015sestimated with this model should\nprobably be considered a phenomenological parameter:\nAs pointed out by recent studies, strictly speaking, a\nphysically meaningful estimation of \u0015sshould take into\naccount the thickness dependence of the resistivity of the\nspin sink layer39,56,67, especially for SRO whose thickness\ndependence of resistivity is quite pronounced.\nFigure 7(b) plots the SRO-thickness dependence of the\nresistivity of SRO \flms deposited on LSAT(001) mea-\nsured in the four-point van der Pauw geometry. The\ntrend can be described empirically by\n\u001aSRO=\u001ab+\u001as\ntSRO\u0000tth; (9)\nwhere\u001ab= 2\u000210\u00006\nm is the resistivity of SRO in the\nbulk limit,\u001as= 1:4\u000210\u000014\nm2is the surface resistivity\n0 5 10 15 200123\n   (10-3)\ntSRO (nm)\n0 10 20 3010-61x10-51x10-4\n  SRO (m)\ntSRO (nm)(a) (b) Figure 7. (a) Gilbert damping parameter \u000bversus SRO\nthicknesstSRO. The solid curve is a \ft taking into account\nthetSROdependence of SRO resistivity, whereas the dotted\ncurve is a \ft assuming a constant bulk-like SRO resistivity.\n(b) Resistivity of SrRuO 3\flms on LSAT(001) as a function\nof thickness.\n0 5 10 15 20012345\nhard\neasy\n  0H (mT)\nf (GHz)\n0 5 10 15 200123450H (mT)\n  \nf (GHz)hard\neasy(b) (a) \nFigure 8. Frequency dependence of in-plane FMR\nlinewidth \u0001 Hof LSMO(10 nm) on (a) LSAT(001) and (b)\nNdGaO 3(110), with the magnetization along the magnetic\neasy and hard axes. The solid curves are \fts to Eq. 7 with\nthe Gilbert damping parameter \u000b\fxed to 0:9\u000210\u00003.\ncoe\u000ecient, and tth= 1 nm is the threshold thickness\nbelow which the SRO layer is essentially insulating. The\nvalue oftthagrees with literature reporting that SRO\nis insulating at thickness of 3 monolayers ( \u00191.2 nm) or\nbelow54. Given the large deviation of \u001aSROfrom the bulk\nvalue, especially at small tSRO, the trend in Fig. 7(b)\nsuggests that taking into account the tSRO dependence\nof\u001aSROis a sensible approach.\nAPPENDIX B: IN-PLANE DAMPING OF LSMO\nON DIFFERENT SUBSTRATES\nIn Fig. 8, we compare the frequency dependence of \u0001 H\nfor 10-nm thick LSMO \flms deposited on di\u000berent sub-\nstrates: LSAT(001) and NdGaO 3(110). (NdGaO 3is an\northorhombic crystal and has ap\n2-pseudocubic param-\neter of\u00193.86 \u0017A, such that (001)-oriented LSMO grows\non the (110)-oriented surface of NdGaO 3.) As shown\nin Sec. IV, LSMO on LSAT(001) exhibits cubic mag-\nnetic anisotropy within the \flm plane with the h110iand\nh100ias the easy and hard axes, respectively. LSMO\non NdGaO 3(110) exhibits uniaxial magnetic anisotropy\nwithin the \flm plane with [1 \u001610] and [001] of NdGaO 3as9\nthe easy and hard axes, respectively.68Whereas LSMO\non LSAT(001) shows distinct magnitudes of damping\nwhen the \flm is magnetized along the easy and hard\naxes (Figs. 8(a) and 6(a)), in LSMO on NdGaO 3(110)damping is identical for both the easy and hard axes\n(Figs. 8(b)). These results demonstrate that higher\ndamping (wider linewidth) is in general not linked to the\nmagnetic hard axis of LSMO.\n\u0003satorue@stanford.edu\n1R. L. Stamps, S. Breitkreutz, J. \u0017Akerman, A. V. Chu-\nmak, Y. Otani, G. E. W. Bauer, J.-U. Thiele, M. Bowen,\nS. A. Majetich, M. Kl aui, I. L. Prejbeanu, B. Dieny, N. M.\nDempsey, and B. Hillebrands, J. Phys. D. Appl. Phys. 47,\n333001 (2014).\n2A. Ho\u000bmann, IEEE Trans. Magn. 49, 5172 (2013).\n3J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back,\nand T. Jungwirth, Rev. Mod. Phys. 87, 1213 (2015).\n4A. Ho\u000bmann and S. D. Bader, Phys. Rev. Appl. 4, 047001\n(2015).\n5A. Brataas, A. D. Kent, and H. Ohno, Nat. Mater. 11,\n372 (2012).\n6N. Locatelli, V. Cros, and J. Grollier, Nat. Mater. 13, 11\n(2014).\n7A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B. Hille-\nbrands, Nat. Phys. 11, 453 (2015).\n8Y. Tserkovnyak, A. Brataas, and G. Bauer, Phys. Rev.\nLett.88, 117601 (2002).\n9Y. Tserkovnyak, A. Brataas, and G. Bauer, Phys. Rev. B\n66, 224403 (2002).\n10F. D. Czeschka, L. Dreher, M. S. Brandt, M. Weiler, M. Al-\nthammer, I.-M. Imort, G. Reiss, A. Thomas, W. Schoch,\nW. Limmer, H. Huebl, R. Gross, and S. T. B. Goennen-\nwein, Phys. Rev. Lett. 107, 046601 (2011).\n11Y. Sun, H. Chang, M. Kabatek, Y.-Y. Song, Z. Wang,\nM. Jantz, W. Schneider, M. Wu, E. Montoya, B. Kar-\ndasz, B. Heinrich, S. G. E. te Velthuis, H. Schultheiss, and\nA. Ho\u000bmann, Phys. Rev. Lett. 111, 106601 (2013).\n12W. Zhang, M. B. Jung\reisch, W. Jiang, J. Sklenar, F. Y.\nFradin, J. E. Pearson, J. B. Ketterson, and A. Ho\u000bmann,\nJ. Appl. Phys. 117, 172610 (2015).\n13C. Du, H. Wang, P. C. Hammel, and F. Yang, J. Appl.\nPhys. 117, 172603 (2015).\n14K. Ando, S. Takahashi, J. Ieda, H. Kurebayashi, T. Tryp-\niniotis, C. H. W. Barnes, S. Maekawa, and E. Saitoh, Nat.\nMater. 10, 655 (2011).\n15K. Ando, S. Watanabe, S. Mooser, E. Saitoh, and H. Sir-\nringhaus, Nat. Mater. 12, 622 (2013).\n16M. Jamali, J. S. Lee, J. S. Jeong, F. Mahfouzi, Y. Lv,\nZ. Zhao, B. K. Nikoli\u0013 c, K. A. Mkhoyan, N. Samarth, and\nJ.-P. Wang, Nano Lett. 15, 7126 (2015).\n17P. Zubko, S. Gariglio, M. Gabay, P. Ghosez, and J.-M.\nTriscone, Annu. Rev. Condens. Matter Phys. 2, 141 (2011).\n18H. Y. Hwang, Y. Iwasa, M. Kawasaki, B. Keimer, N. Na-\ngaosa, and Y. Tokura, Nat. Mater. 11, 103 (2012).\n19S. Majumdar and S. van Dijken, J. Phys. D. Appl. Phys.\n47, 034010 (2014).\n20E. Lesne, Y. Fu, S. Oyarzun, J. C. Rojas-S\u0013 anchez, D. C.\nVaz, H. Naganuma, G. Sicoli, J.-P. Attan\u0013 e, M. Jamet,\nE. Jacquet, J.-M. George, A. Barth\u0013 el\u0013 emy, H. Ja\u000br\u0012 es,\nA. Fert, M. Bibes, and L. Vila, Nat. Mater. (2016).\n21G. Y. Luo, C. R. Chang, and J. G. Lin, IEEE Trans. Magn.\n49, 4371 (2013).22G. Y. Luo, M. Belmeguenai, Y. Roussign\u0013 e, C. R. Chang,\nJ. G. Lin, and S. M. Ch\u0013 erif, AIP Adv. 5, 097148 (2015).\n23H. K. Lee, I. Barsukov, A. G. Swartz, B. Kim, L. Yang,\nH. Y. Hwang, and I. N. Krivorotov, AIP Adv. 6, 055212\n(2016).\n24S. M. Haidar, Y. Shiomi, J. Lustikova, and E. Saitoh, Appl.\nPhys. Lett. 107, 152408 (2015).\n25V. A. Atsarkin, B. V. Sorokin, I. V. Borisenko, V. V. Demi-\ndov, and G. A. Ovsyannikov, J. Phys. D. Appl. Phys. 49,\n125003 (2016).\n26M. Wahler, N. Homonnay, T. Richter, A. M uller, C. Eisen-\nschmidt, B. Fuhrmann, and G. Schmidt, Sci. Rep. 6, 28727\n(2016).\n27G. Koster, L. Klein, W. Siemons, G. Rijnders, J. S. Dodge,\nC.-B. Eom, D. H. A. Blank, and M. R. Beasley, Rev. Mod.\nPhys. 84, 253 (2012).\n28M. C. Langner, C. L. S. Kantner, Y. H. Chu, L. M. Martin,\nP. Yu, J. Seidel, R. Ramesh, and J. Orenstein, Phys. Rev.\nLett.102, 177601 (2009).\n29A. Azevedo, L. H. Vilela-Le~ ao, R. L. Rodr\u0013 \u0010guez-Su\u0013 arez,\nA. F. Lacerda Santos, and S. M. Rezende, Phys. Rev. B\n83, 144402 (2011).\n30L. Bai, P. Hyde, Y. S. Gui, C.-M. Hu, V. Vlaminck, J. E.\nPearson, S. D. Bader, and A. Ho\u000bmann, Phys. Rev. Lett.\n111, 217602 (2013).\n31M. Obstbaum, M. H artinger, H. G. Bauer, T. Meier,\nF. Swientek, C. H. Back, and G. Woltersdorf, Phys. Rev.\nB89, 060407 (2014).\n32J. Lindner, K. Lenz, E. Kosubek, K. Baberschke, D. Spod-\ndig, R. Meckenstock, J. Pelzl, Z. Frait, and D. L. Mills,\nPhys. Rev. B 68, 060102 (2003).\n33G. Woltersdorf and B. Heinrich, Phys. Rev. B 69, 184417\n(2004).\n34K. Lenz, H. Wende, W. Kuch, K. Baberschke, K. Nagy,\nand A. J\u0013 anossy, Phys. Rev. B 73, 144424 (2006).\n35K. Zakeri, J. Lindner, I. Barsukov, R. Meckenstock,\nM. Farle, U. von H orsten, H. Wende, W. Keune, J. Rocker,\nS. S. Kalarickal, K. Lenz, W. Kuch, K. Baberschke, and\nZ. Frait, Phys. Rev. B 76, 104416 (2007).\n36I. Barsukov, F. M. R omer, R. Meckenstock, K. Lenz,\nJ. Lindner, S. Hemken to Krax, A. Banholzer, M. K orner,\nJ. Grebing, J. Fassbender, and M. Farle, Phys. Rev. B 84,\n140410 (2011).\n37H. Kurebayashi, T. D. Skinner, K. Khazen, K. Olejnik,\nD. Fang, C. Ciccarelli, R. P. Campion, B. L. Gallagher,\nL. Fleet, A. Hirohata, and A. J. Ferguson, Appl. Phys.\nLett.102, 062415 (2013).\n38C. T. Boone, H. T. Nembach, J. M. Shaw, and T. J. Silva,\nJ. Appl. Phys. 113, 153906 (2013).\n39C. T. Boone, J. M. Shaw, H. T. Nembach, and T. J. Silva,\nJ. Appl. Phys. 117, 223910 (2015).\n40R. Arias and D. L. Mills, Phys. Rev. B 60, 7395 (1999).\n41Y. Takamura, R. V. Chopdekar, E. Arenholz, and\nY. Suzuki, Appl. Phys. Lett. 92, 162504 (2008).10\n42A. Grutter, F. Wong, E. Arenholz, M. Liberati, and\nY. Suzuki, J. Appl. Phys. 107, 09E138 (2010).\n43M. Golosovsky, P. Monod, P. K. Muduli, and R. C. Bud-\nhani, Phys. Rev. B 76, 184413 (2007).\n44M. Haertinger, C. H. Back, J. Lotze, M. Weiler, S. Gepr ags,\nH. Huebl, S. T. B. Goennenwein, and G. Woltersdorf,\nPhys. Rev. B 92, 054437 (2015).\n45S. Mizukami, D. Watanabe, M. Oogane, Y. Ando,\nY. Miura, M. Shirai, and T. Miyazaki, J. Appl. Phys.\n105, 07D306 (2009).\n46P. D urrenfeld, F. Gerhard, J. Chico, R. K. Dumas, M. Ran-\njbar, A. Bergman, L. Bergqvist, A. Delin, C. Gould, L. W.\nMolenkamp, and J. \u0017Akerman, Phys. Rev. B 92, 214424\n(2015).\n47C. Liu, C. K. A. Mewes, M. Chshiev, T. Mewes, and W. H.\nButler, Appl. Phys. Lett. 95, 022509 (2009).\n48H. Nguyen, W. Pratt, and J. Bass, J. Magn. Magn. Mater.\n361, 30 (2014).\n49J.-C. Rojas-S\u0013 anchez, N. Reyren, P. Laczkowski, W. Savero,\nJ.-P. Attan\u0013 e, C. Deranlot, M. Jamet, J.-M. George, L. Vila,\nand H. Ja\u000br\u0012 es, Phys. Rev. Lett. 112, 106602 (2014).\n50S. Azzawi, A. Ganguly, M. Toka\u0018 c, R. M. Rowan-Robinson,\nJ. Sinha, A. T. Hindmarch, A. Barman, and D. Atkinson,\nPhys. Rev. B 93, 054402 (2016).\n51M. Caminale, Phys. Rev. B 94, 014414 (2016).\n52S. Emori, T. Nan, A. M. Belkessam, X. Wang, A. D.\nMatyushov, C. J. Babroski, Y. Gao, H. Lin, and N. X.\nSun, Phys. Rev. B 93, 180402 (2016).\n53M. Polianski and P. Brouwer, Phys. Rev. Lett. 92, 026602\n(2004).\n54J. Xia, W. Siemons, G. Koster, M. R. Beasley, and A. Ka-\npitulnik, Phys. Rev. B 79, 140407 (2009).\n55C.-F. Pai, Y. Ou, L. H. Vilela-Le~ ao, D. C. Ralph, and R. A.\nBuhrman, Phys. Rev. B 92, 064426 (2015).\n56E. Montoya, P. Omelchenko, C. Coutts, N. R. Lee-Hone,\nR. H ubner, D. Broun, B. Heinrich, and E. Girt, Phys. Rev.B94, 054416 (2016).\n57M. Belmeguenai, S. Mercone, C. Adamo, L. M\u0013 echin,\nC. Fur, P. Monod, P. Moch, and D. G. Schlom, Phys.\nRev. B 81, 054410 (2010).\n58\u0017A. Monsen, J. E. Boschker, F. Maci\u0012 a, J. W. Wells,\nP. Nordblad, A. D. Kent, R. Mathieu, T. Tybell, and\nE. Wahlstr om, J. Magn. Magn. Mater. 369, 197 (2014).\n59M. Farle, Reports Prog. Phys. 61, 755 (1998).\n60A. A. Baker, A. I. Figueroa, C. J. Love, S. A. Cavill,\nT. Hesjedal, and G. van der Laan, Phys. Rev. Lett. 116,\n047201 (2016).\n61D. Mills and S. M. Rezende, Spin Damping in Ultrathin\nMagnetic Films, in Spin Dyn. Con\fn. Magn. Struct. II ,\nedited by B. Hillebrands and K. Ounadjela, chap. 2, pp.\n27{58, Springer, 2002.\n62R. D. McMichael, M. D. Stiles, P. J. Chen, and W. F.\nEgelho\u000b, J. Appl. Phys. 83, 7037 (1998).\n63E. Dagotto, J. Burgy, and A. Moreo, Solid State Commun.\n126, 9 (2003).\n64C. W. Sandweg, Y. Kajiwara, A. V. Chumak, A. A. Serga,\nV. I. Vasyuchka, M. B. Jung\reisch, E. Saitoh, and B. Hille-\nbrands, Phys. Rev. Lett. 106, 216601 (2011).\n65G. L. da Silva, L. H. Vilela-Leao, S. M. Rezende, and\nA. Azevedo, Appl. Phys. Lett. 102, 012401 (2013).\n66S. A. Manuilov, C. H. Du, R. Adur, H. L. Wang, V. P.\nBhallamudi, F. Y. Yang, and P. C. Hammel, Appl. Phys.\nLett.107, 042405 (2015).\n67M.-H. Nguyen, D. C. Ralph, and R. A. Buhrman, Phys.\nRev. Lett. 116, 126601 (2016).\n68H. Boschker, M. Mathews, E. P. Houwman, H. Nishikawa,\nA. Vailionis, G. Koster, G. Rijnders, and D. H. A. Blank,\nPhys. Rev. B 79, 214425 (2009)."    },    {        "title": "1009.3618v1.Effect_of_spin_transfer_torque_on_the_magnetic_domain_wall_ferromagnetic_resonance_frequency_in_the_nanowires.pdf",        "content": "Effect of spin transfer torque on the magnetic domain wall ferromagnetic resonance \nfrequency in the nanowires \nJungbum Yoon1, Chun-Yeol You1, Younghun Jo2, Seung-Young Park2, and Myung-Hwa Jung3\n1Department of Physics, Inha Univ ersity, Incheon 402-751, Korea \n2Division of Materials Science, Korea Basic Science Institute, Daejeon  305-333, Korea \n3Department of Physics, Sogang University, Seoul 121-742, Korea \nWe investigate the influence of the domain wall ferromagnetic resonance frequency on \nthe spin transfer torque in a ferromagnetic nanowire. By employing micromagnetic simulations with the spin transfer torque, we find that the domain wall resonance frequency decreases with increasing spin polarized current density, when there is no change in the resonance frequency of the domain itself. Surprisingly, the variation of the resonance frequency is remarkable (> 1.6 GHz) with the spin transfer torque even though the domain wall is pinned. Since the presented domain wall ferromagnetic resonance study has been performed for the pinned domain wall, the contributions of extrinsic defects are excluded. It is strong advantages of the present study, since the effects of extrinsic pinning sites are inevitable in the imaging or transport measurements. \nPACS: 72.25.-b, 75.60.Ch, 75.78.Fg, 76.50.+g The spin transfer torque (STT) is the phenomenon that the angular momenta are \ntransferred from the spin-polarized conduction electrons to the localized spins in ferromagnets.\n1,2 Recently, the novel spintronic devices with the STT are actively \ninvestigated such as the magnetic race-track memory,3,4,5 spin-transfer torque \nmagnetoresistive random access memory,6,7,8 and spin torque nano oscillator.9,10 The \nmagnetic race-track memory based on the current-induced domain wall motion (CIDWM) in the ferromagnetic nanowires is one of the candidates for the next generation non-volatile mobile memories. The STT in the nanowire is important not only for the race-track memory applications, where the domain wall (DW) is moved by the STT, but also it has been revealed that many interesting physics are involved in the DW motion.\n4 Since the DW width of the 3 dtransition metal ferromagnetic nanowire is \nmuch larger than other relevant physical length scales such as Fermi wavelength and exchange length, the non-adiabatic contribution has been expected to be small.\n11\nHowever, many experimental observations have been reported that the non-adiabatic contribution is not small and it is important\n12,13,14 even for the Permalloy. Furthermore, \nthere are many controversial theories about magnitude of non-adiabatic contributions.\n15 , 16 , 17 Many experimental efforts have been determined the detail \ninteractions between conducting spins and localized spins inside of the \nDW.18,19,20,21,22,23,24,25 Most experiments can be categorized by two kinds: one is \ntransport measurements20-23 and the other is magnetic domain imaging.24-27 The \ntransport measurements are able to determine whether the DW placed between two electrodes or not by magneto-resistance. Since it is relatively simple and easy to repeat, \nthis method is widely used. However, the transport data cannot determine real DW position in the single ferromagnetic nanowire, and it requires many electrodes.\n18,19 The monitoring DW motion by MOKE (magneto-optical Kerr effect) at the fixed point is \nrelatively simple but it gives limited information whether the DW passed at specific \nposition or not.25 Direct imaging by magnetic force microscope is only possible to \ndetermine the static magnetization configuration.22 And x-ray imaging can measure the \ndynamics of the DW with an assumption of repeated motion of the DW.24 It m ust be \nemphasized that none of them are extracted any physical information without DW movement. However, the DW movement is strongly affected by extrinsic pinning sites. Therefore, any physical quantities extracted from the experimental measurements always include the extrinsic effect. \nIn this study, we would like to introduce the new possibility in the study of STT effect \nin the nanowires, DW-FMR (domain wall ferromagnetic resonance). Magnetic thermal noise, ferromagnetic resonance (FMR), and spin diode effects are widely accepted excellent experimental tools for the study of STT in magnetic tunneling junction geometry.\n8,26,27,28 However, such methods have been applied to only limited study for \nthe DW. According to our previous study, it has been revealed that the DW-FMR frequency is distinguishable from one of domain body itself due to the different effective field in the DW inside.\n29 It can be easily understood that when spin polarized \ncurrents are applied to the nanowire, the spin dynamics of DW will be altered by the \nSTT. The STT acts as an additional torque for each spin, even though the DW is pinned. \nThis is the key idea of this study. We investigate the DW-FMR spectra of the non-moving DW by employing micromagnetic simulations including STT effect.\n30 Even \nthough when the DW distortion is too small to notice with STT, surprisingly, the variation of the resonance frequency is remarkable. Consequently, we can exclude the contributions of extrinsic pinning sites which govern the DW movement. In order to investigate the effect of the spin polarized current on the spin dynamics of \nthe DW, we employed the Landau-Lifshitz-Gilbert (LLG) equation with the STT contribution.\n30\n/g11/g12 /g11/g12 // eff dm dt H m m dm dt u m m u m /g74/g68 /g69 /g170/g186 /g32 /g16 /g117 /g14 /g117 /g16 /g152/g146 /g14 /g117 /g152/g146/g172/g188/g41/g38 /g41/g41 /g38/g41 /g38/g41 /g38 /g41 /g38 /g38/g41 /g38/g41 /g38/g38/g41 /g38\n.        ( 1 )  \nHere /g74,/g68,sM, effH , and m/g41/g38\n are the gyromagnetic ratio, Gilbert damping \nconstant, saturation magnetization, effective magnetic field, and unit vector of the \nmagnetization, respectively. In Eq. (1), the last term with the dimensionless parameter /g69,\ncompared to /g68, presents the non-adiabatic spin torque. The x-direction is the \nlongitudinal direction of the ferromagnetic nanowire as shown in Fig. 1. The velocity u\nis defined parallel to the direction of the conductive electron, + x-direction, with the \namplitude of s BeM JPgu 2//g80 /g32 , where J is the current density and P is the spin \npolarization rate. The majority spin direction is right for u > 0. s BeM g 2//g80  is given by \n11107/g16/g117 m3/C for Permalloy (Py).16 And the material parameters of Py used in our \nsimulation are as follows: the saturation magnetization 5106.8/g117 /g32sM  A/m, the \nexchange stiffness 121013/g16/g117/g32exA  J/m, the gyromagnetic ratio 51021.2/g117 /g32/g74  m/(A½s),\nand we ignore the magnetocrystalline anisotropy. In this simulation, the Gilbert \ndamping parameter 02.0/g32/g68  is fixed. We perform our simulation at a zero \ntemperature and take a cell size of 555/g117/g117  nm3.\nIn order to pin the DW, we introduce a small notch (5 × 10 × 10 nm3) at the center of \nthe 10-nm thick, 80-nm wide, and 2000-nm long Py nanowire as shown in Fig. 1. Initially, two magnetic domains are set up at each side of the nanowire without any external magnetic field and current. A stable tail-to-tail transverse type DW is formed with the energy minimization, and the stable DW is used as an initial magnetization \nconfiguration.31 It must be noted that our study is limited for the transverse type DW. \nIn order to mimic FMR in our micromagnetic simulations, a “sine cardinal (sinc)” \nfunction ) ( 2/)] ( 2sin[ )(0 0 0 ttf ttf HtHH H y /g16 /g16 /g32 /g83 /g83  is applied with 100/g32H  mT and \nthe field frequency 45/g32Hf  GHz to the whole nanowire area.32 Even though H0 is \nsmaller, the results of FMR spectra are the same with smaller amplitude. Since the sinc \nfunction is a Fourier transform of 0 () ( )yHHf H f f /g32/g31 , the Fourier transform of the \nresponse of ()yHt  is the FMR spectra of corresponding frequency. The time varying \ntransverse magnetization component My(x,y,t) configurations are stored in the whole \nPy nanowire at each temporal moment (10-2 ns step) with the simulation time per 10-5 ns \nstep. The results of the simulation are stored with each point for sufficiently long time (100 ns). The FMR spectra due to the RF-magnetic field are obtained by the fast Fourier transform (FFT) of M\ny(x,y,t).\nIn Fig. 2, we plot the typical FMR spectra of the Py nanowire, the FFT of \n/g11 /g12\n,N W,,y\nxyMx y t\n/g143/g166 , without external dc-field and current. Here NW means a whole \nnanowire area. Since there is no external dc-field, the resonance frequencies of two \nopposite domain are the same, but there is clear additional resonance peak at 7.1 GHz (red arrow). According to the simple Kittel’s equation, \n) ) ( () ) ( ()2/(s x y eff s x z eff M N N H M N N H f /g117 /g16 /g14 /g117 /g117 /g16 /g14 /g117 /g32 /g83/g74 , (The Nx,Ny, and Nz\nare the demagnetization factors), we can identify the 10.0 GHz peak (blue arrow) is \ncorresponding to the resonance peak of the two domains. We also confirm that the peak at the 7.1 GHz is originated from the DW.\n29 To clarify the DW contribution, the local magnetization, /g11/g12\n,D W,,y\nxyMx y t\n/g143/g166, where DW means the region around the DW, are only \nconsidered as shown in Fig. 1 with a black rectangle. More details of local spectra \nconcept already reported in our previous report.29 As a result of the local spectra \nanalysis, we find out that the DW-FMR frequency is lower than that of the domain. In the inset of Fig. 2, the local FMR spectra are plotted for the only DW region. Because the direction of each spin inside of the DW is gradually tilted from the one direction to the opposite one, the effective fields of the DW are non-uniform and the demagnetization field of the DW is smaller than the one of the domain. As a result, the resonance frequency decreases with smaller effective field. Even though, we cannot find analytic expression for the effective field of the inside of the DW, but it is clear that the inside effective field of the DW is quite different (smaller) from one of the domain. \nIn order to reveal the effect of STT in DW-FMR spectra, the velocity u representing \nthe spin polarized current density is applied from 0 to 400 m/s with 100 m/s steps for \nthe Py nanowire with \n/g69 = 0.01. First, we apply the spin polarized current and wait for \n100 ns till the DW is stabilized. Due to the STT, the DW is slightly deformed, but it is \nhard to notice with DW images as shown in Fig. 3 (a)~(e). It must be emphasized that all the DWs in our study are almost identical shapes. Therefore, it is impossible to distinguish them by the magnetic imaging or transport measurement experiments. While the spin polarized current is keeping, the FMR simulations are performed. The local spectra for the DW are depicted in Fig. 4 (a). In these spectra, two groups of peaks are found. There are other peaks around 2 GHz (blue circle). We identify that the STT induced whole DW vibration is the source of 2 GHz peaks. These peaks are also found with the STT only without RF-magnetic field. Therefore, we will pay our attention to the peaks around 7 GHz which are indicated by arrows in Fig. 4 (a). Figure 4 (b) shows \nthe dependence of the DW-FMR frequency on the u. It is clearly shown that the DW-\nFMR frequency decreases noticeably with increasing u (from 7.2 to 5.6 GHz). When we \nconsider the distortion of the DW structure is very small, the change of the resonance frequency is dramatic. This is the central results of  our study. It must be noted the DW is \ndepinning when u > 410 m/s. \nNow, let us consider the effect of a non-adiabatic contribution with various \n/g69 in DW-\nFMR spectra. The local spectra for various /g69 (= 0, 0.01, 0.02, and 0.04) are depicted in \nFig. 5 with constant u (= 100 m/s). The dependence of spectra on /g69 is very small, and it \ncan be recognized only enlarged plot in the inset of Fig. 5. It implies that the \ncontribution of non-adiabatic STT term to the DW-FMR spectra is small, and practically \nundetectable by experimentally.  \nLet us discuss about our findings. When the STT is applied, we find the DW-FMR \nfrequency decreases. Since there is no analytic expressi on of susceptibility of the DW \nwith the STT, further detail analysis is unavailable in current stage. However, it is clear \nthat the adiabatic STT, /g11/g12um/g16/g152 /g146/g38/g41 /g38\n, acts an additional torque and it belongs to the \nnanowire ( xy) plane, and its role is somewhat similar to the external magnetic field.16 It \ngives translational motion of the DW as the external field does when the damping is ignored or the damping is canceled with a non-adiabatic term.\n33 With the STT, the DW \nis tilted from a nanowire plane with an angle /g73, which is proportional to the u.16 When \nthe DW is tilted, the effective field is reduced. Therefore, the DW-FMR frequency \ndecreases. It must be note that the frequency shifts are independent on the sign of the u\n(not shown here). The tilting angle /g73 is proportional to the u, but the reduction of the effective field is related with | /g3/g73/g3| so that the change of the DW-FMR frequency depends \non |/g3u/g3|. One more observation is that the dramatic reduction of an amplitude and area of \nthe peaks, which are related with the magnon excitations. Smaller amplitude or area of \nthe peak indicates smaller excitation of magnon. In this stage, the physical origin of the suppression of the magnon excitation is not clear, but it is sure that it is quite different from the magnetic tunneling junction cases.\n6-8\nWhen the STT is applied to the DW, the Gilbert damping term tilts the DW to the out-\nof-plane, and the demagnetization energy prevent the tilting, and the DW will stop. The role of the non-adiabatic STT is suppression of  Gilbert damping, so that DW will move \ncontinuously. Therefore, the contribution of \n/g69 in the DW spectra is clear. It reduces the \nGilbert damping, which is related the line-width of the peak. However, the line-width of \nthe DW-FMR spectra depends on not only the Gilbert damping but also the DW width. The non-uniform magnetization direction in the DW causes extra broadening of the line-width in the DW-FMR spectra. According to our micromagnetic simulations, the \neffect of \n/g69 is very small as shown in Fig. 5. Therefore, unfortunately, our proposed \nmethod will extract the only adiabatic information of the STT. \nFinally, we would like to discuss about the experimental setup. Since the portion of \nthe DW area is much smaller than the domain area, the FMR signal is much smaller than one of the whole sample. Therefore, an extra caution is required in the design of sample and electrodes. Or, BLS (Brillouin Light Scattering) microscopes might be proper tools.\n34,35 Furthermore, the current introduces inevitable Joule heating,36,37 and \nthe higher temperature causes red shift of the resonance frequency, which is the same direction of adiabatic STT contribution. Therefore, a separate resistance measurement to monitoring the temperature of the sample must be considered.\nGIn conclusion, we proposed the DW-FMR spectra measurement in the study of the \nSTT in the nanowire. We found that the DW-FMR frequency is very sensitive on the adiabatic STT, even the DW shape is barely changed. It must be emphasized that the presented DW-FMR is free from the contribution of extrinsic pinning sites, which is inevitable in other measurement methods. \nAcknowledgement \nThis work was supported by Nano R&D program (Grant No. 2008-02553), by KBSI grant (T30405) for Y . Jo, by the IT R&D program of MKE/KEIT [2009-F-004-01], and by the Sogang University Research Grant of 201011014.01. Figure Captions \nFig. 1 Magnetization configuration of the Py nanowire with a transverse Néel type DW. The DW pinned at the notch in a center of the nanowire.  Fig. 2 FMR spectra of the Py nanowire with u = 0 m/s. The red and blue arrows indicate \nthe resonance frequencies of the domain wall and domain, respectively. Inset represents the local DW-FMR spectra. Fig. 3 (a)-(e) Magnetization configuration of the DW with various u = (0 ~ 400 m/s) \nwith 100 m/s steps. The figures are enlarged on the DW with the arrows of macro-magnet directions. Fig. 4 (a) Local DW-FMR spectra for various u. The blue circles indicate the vibration \nof the DW. (b) Dependence of the DW-FMR frequency on u. The velocity u is \nproportional to the current density. \nFig. 5 DW-FMR spectra for various non-adiabatic STT \n/g69/g3 with u= 100 m/s. Inset \nrepresents the enlargement of the resonance peaks.References \nGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG G\n1 J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996).\n2 L. Berger, Phys. Rev. B 54, 9353 (1996). \n3 S. S. P. Parkin, U.S. Patent No. 6834 005, (2004). \n4 S. S. P. Parkin, M. Hayashi, and L. Thomas, Science 320, 190 (2008) \n5 M. Hayashi, L. Thomas, R. Moriya, C. Rettner, and S. S. P. Parkin, Science 320, 209 (2008). \n6 S.-C. Oh, S.-Y . Park, A. Manchon, M. Chshiev, J.-H. Ha n, H.-W. Lee, J.-E. Lee, K.-T. Nam, Y . Jo, Y .-C. \nKong, B. Dieny, and K.-J. Lee, Nature Phys. 5, 898 (2009). \n7 C.-Y . You, Curr. Appl. Phys. 10, 952 (2010). G\n8 M. H. Jung, S. Park, C.-Y . You, and S. Yuasa, Phys. Rev. B 81, 134419 (2010). \n9 A. Bisig, L. Heyne, O. Boulle, and M. Kläui, Appl. Phys. Lett. 95, 162504 (2009). \n10 C.-Y . You, J. Magnetics 14(4) 168 (2009). \n11 J. Xiao, A. Zangwill, and M. D. Stiles, Phys. Rev. B 73, 054428 (2006). \n12 G. S. D. Beach, C. Knutson, C. Nistor, M. Tsoi, and J. L. Erskine, Phys. Rev. Lett. 97, 057203 (2006). \n13 S. Yang and J. L. Erskine, Phys. Rev. B 75, 220403(R) (2007). G\n14 S. Lepadatu, A. Vanhaverbeke, D. Atkinson, R. Allenspach, and C. H. Marrows, Phys. Rev. Lett. 102,\n127103 (2009). \n15 S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204-1 (2004).  \nX]GA. Thiaville, Y . Nakatani, J. Miltat and Y . Suzuki, Europhys. Lett. 69, 990 (2005). \n17 J. He, Z. Li, and S.  Zhang, J. Appl. Phys. 98, 016108 (2005). \n18 M. Kläui, C. A. F. Vaz, J. A. C. Bland, W. Wernsdor fer, G. Faini and E. Cambril, Appl. Phys. Lett. 81,\n108 (2002). \n19 M. Kläui, C. A. F. Vaz, J. A. C. Bland, W. Wernsdorfer, G. Faini, E. Cambril, L. J. Heyderman, F. \nNolting, and U. Rüdig er, Phys. Rev. Lett. 94, 106601 (2005). \n20 M. Hayashi, L. Thomas, Y . B. Bazaliy, C. Rettner, R. Moriya, X.  Jiang, and S. S. P. Parkin, Phys. Rev. \nLett. 96, 197207 (2006). G\nYXGL. Thomas, M. Hayashi, X. Jiang, R. Moriya, C. Rettner , and S. S. P. Parkin, nature 443, 197 (2006). \n22 A. Yamaguchi, T. Ono, S. Nasu, K. Miyake , K. Mibu, and T. Shinjo, Phys. Rev. Lett. 92, 077205-1 \n(2004). \n23 M. Kläui, P.-O. Jubert, R. Allenspach, A. Bischof, J. A. C. Bland, G. Faini, U. Rüdiger, C. A. F. Vaz, L. \nVila, and C. V ouille, Phys. Rev. Lett. 95, 026601 (2005). \n24 G. Meier, M. Bolte, R. Eiselt, B. Krüger, D.-H. Kim, and P. Fischer, Phys. Rev. Lett. 98, 187202 (2007). \n25 K.-J. Kim, J.-C. Lee, S.-M. Ahn, K.-S. Lee, C.-W. Lee, Y . J. Cho, S. Seo, K.-H. Shin, S.-B. Choe and \nH.-W. Lee, nature 458, 740 (2009). \n26 Y . Guan, D. W. Abraham, M. C. Gaidis, G. Hu, E. J. O’Sullivan, J. J. Nowak, P. L. Trouilloud, D. C. \nWorledge, and J. Z. Sun, J. Appl. Phys. 105, 07D127 (2009). G\n27 S. Petit, N. de Mestier, C. Baraduc, C. Thirion, Y . Liu, M. Li, P. Wang, and B. Dieny, Phys. Rev. B 78,\n184420 (2008). \n28 A. A. Tulapurkar, Y . Suzuki, A. Fukushima, H. Kubota, H. Maehara, K. Tsunekawa, D. D. \nDjayaprawira, N. Watanabe, and S. Yuasa, nature 438, 339 (2005). \n29 J. Yoon, C.-Y . You, Y . Jo, S.-Y . Park,  and M.-H. Jung, cond-mat/1006.5763. \n30 See http://www.zurich.ibm.com/s t/magnetism/spintevolve.html \n31 Y . Nakatani, A. Thiaville, and J. Miltat, J. Magn. Magn. Mater. 290–291 , 750 (2005). \n32 K.-S. Lee, D.-S. Han, and S.-K. Kim, Phys. Rev. Lett. 102, 127202 (2009). \n33 M. D. Stiles, W. M. Saslow, M. J. Donahue, and A. Zangwill, Phys. Rev. B 75, 214423 (2007). GG\n34 C. Bayer, J. Jorzick, B. Hillebrands, S. O. Demokritov,  R. Kouba, R. Bozinoski, A. N. Slavin, K. Y . \nGuslienko, D. V . Berkov, N. L. Gorn, and M. P. Kostylev, Phys. Rev. B 72, 064427 (2005). \n35 H. Schultheiss, S. Schäfer, P. Candeloro, B. Leven , B. Hillebrands, and A. N. Slavin, Phys. Rev. Lett. \n100, 047204 (2008). \n36 C.-Y . You, I. M. Sung, and B.-K. Joe, Appl. Phys. Lett. 89, 222513 (2006). \n37 C.-Y . You and S.-S. Ha, Appl. Phys. Lett. 91, 022507 (2007). G"    },    {        "title": "0708.3528v3.Determination_of_Penetration_Depth_of_Transverse_Spin_Current_in_Ferromagnetic_Metals_by_Spin_Pumping.pdf",        "content": "arXiv:0708.3528v3  [cond-mat.mes-hall]  17 Mar 2008Determination of Penetration Depth of Transverse Spin Curr ent in Ferromagnetic\nMetals by Spin Pumping\nTomohiro Taniguchi1,2, Satoshi Yakata3, Hiroshi Imamura2, Yasuo Ando3\n1Institute for Materials Research, Tohoku University, Send ai 980-8577,\n2Nanotechnology Research Institute, National Institute of Advanced Industrial Science and Technology,\n1-1-1 Umezono, Tsukuba, Ibaraki 305-8568, Japan,\n3Department of Applied Physics, Graduate School of Engineer ing, Tohoku University, Sendai\n(Dated: November 10, 2018)\nSpin pumping in nonmagnetic/ferromagnetic metal multilay ers is studied both theoretically and\nexperimentally. We show that the line widths of the ferromag netic resonance (FMR) spectrum\ndepend on the thickness of the ferromagnetic metal layers, w hich must not be in resonance with the\noscillating magnetic ﬁeld. We also show that the penetratio n depths of the transverse spin current\nin ferromagnetic metals can be determined by analyzing the l ine widths of their FMR spectra.\nThe obtained penetration depths in NiFe, CoFe and CoFeB were 3.7 [nm], 2.5 [nm] and 12.0 [nm],\nrespectively.\nPACS numbers: 72.25.Mk, 75.70.Cn, 76.50.+g, 76.60.Es\nThe ﬁeld of current-driven magnetization dynamics\n(CDMD) has drawn enormous attention because of its\npotential applications to non-volatile magnetic random\naccess memory and microwave devices. CDMD is also\nimportant from a scientiﬁc point of view since it pro-\nvides much information about non-equilibrium dynamics\nof the magnetization and physics of spin transport and\nspin relaxation. The concept of CDMD was ﬁrst pro-\nposed by Slonczewski [1] and independently by Berger\n[2] in 1996. In the last decade much eﬀort has been de-\nvoted to studying the physics and applications of CDMD\nboth theoretically and experimentally [3, 4].\nOne of the most important quantities in CDMD is the\npenetration depth of the transverse spin current λt, over\nwhich spin transfer torque is exerted for the magnetiza-\ntion of the free layer. However, there is a controversial\nissue regarding the penetration depth of the transverse\nspin current. One argument is based on the ballistic the-\nory of electron transport, and its λt=π/|k↑\nF−k↓\nF|, which\nis on the order of the lattice constant in conventional fer-\nromagnets such as Fe, Co, Ni, and their alloys [5, 6].\nThe other argument is based on the Boltzmann theory\nof electron transport, and its λtis on the order of a few\nnm [7]. Urazhdin et al.analyzed the CPP-GMR of non-\ncollinear magnetic multilayers using the extended two-\nseries-resistance model and concluded that λt=0.8 [nm]\nfor permalloy [8]. On the other hand, Chen et al.ana-\nlyzed the critical current of the CDMD in the Co/Cu/Co\ntrilayer system and concluded that λt=3.0 [nm] for Co\n[9].\nThe inverse process of CDMD is spin pumping, where\nspin current is generated by precession of magnetiza-\ntion in the ferromagnetic layer [10]. Enhancement of\nthe Gilbert damping constant due to spin pumping has\nbeen extensively studied, and spin diﬀusion lengths, i.e.,\npenetration depths of spin current in nonmagnetic met-\nals, have been obtained by analyzing the dependence of\nthe enhancement of the Gilbert damping constant on\nthe thickness of the nonmagnetic metal layer. In spinFIG. 1: Schematic illustration of a nonmagnetic/ferromag-\nnetic metal ﬁve-layer system. The magnetization of the F 1\nlayer (m1) precesses around the z-axis with angle θ. The\nmagnetization of the F 2layer (m2) is ﬁxed with the z-axis.\nThe precession of the magnetization in the F 1layer pumps\nthe spin current Ipump\ns. The pumped spin current creates\nspin accumulation in the other layers, and the spin accumula -\ntion induces a backﬂow of spin current IN→F\nsacross each N/F\ninterface.\npumping, the direction of the magnetization vector of\nthe pumped spin current is perpendicular to the direc-\ntion of the precessing magnetization vector [10]. Let us\nconsider the nonmagnetic/ferromagnetic metal ﬁve-layer\nsystem shown in Fig. 1. Since the magnetization vec-\ntor of the pumped spin current Ipump\nsis perpendicular\nto the magnetization vector m1of the F 1layer and the\nprecession angle θis very small (about 1 [deg]) in con-\nventional FMR experiments, the dominant component of\nthe pumped spin current is perpendicular to the magne-\ntization vector m2ofthe F 2layer. Therefore, it would be\npossible to determine the penetration depth of the trans-\nverse spin current in the F 2layer if we could analyze the\ndependence of the enhancement of the Gilbert damping\nconstant on the thickness of the F 2layer.\nIn this letter, we study spin pumping in N 1/F1/N2/\nF2/N3ﬁve-layer systems shown in Fig. 1 both theo-\nretically and experimentally. We extend Tserkovnyak’s\ntheory of spin pumping by taking into account the ﬁ-\nnite penetration depth of the transverse spin current and\nshow that the enhancement of the Gilbert damping con-2\nstant due to spin pumping depends on the ratio of the\npenetration depth λtand the thickness d2ofthe F 2layer.\nThe motion of the magnetization vector in the F 2layer\nis not in resonance with an oscillating magnetic ﬁeld;\nhence, the F 2layer plays the role of spin absorber. We\nalso perform FMR experiments in Cu/CoFe/Cu/Py/Cu,\nCu/Py/Cu/CoFe/Cu and Cu/CoFe/Cu/CoFeB/Cu ﬁve-\nlayer systems and measure line widths ∆ Bof energy ab-\nsorption spectra, which are closely related to the Gilbert\ndamping constants. The abbreviations CoFe, CoFeB,\nand Py hereafter refer to Co 75Fe25, (Co50Fe50)80B20and\nNi80Fe20, respectively. Analyzing the dependence of the\nline width on the thickness of the Py, CoFe and CoFeB\nlayers that are not in resonance, we showed that the pen-\netration depths of the transverse spin current in the Py,\nCoFe and CoFeB layers are 3.7 [nm], 2.5 [nm] and 12.0\n[nm], respectively.\nLet us begin with an introduction to the theory of spin\npumping in the nonmagnetic/ferromagnetic metal ﬁve-\nlayer system shown in Fig. 1 with a ﬁnite penetration\ndepth of the transverse spin current. The pumped spin\ncurrent generated by precession of the magnetization m1\nof the F 1layer is given by\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 (imag-\ninary) part of the mixing conductance [10]. The pumped\nspincurrentcreatesspinaccumulationintheotherlayers,\nandthe spin accumulationinducesabackﬂowofspincur-\nrent across each N/F interface. Although the backﬂow\nis obtained from circuit theory [6, 10], the penetration\ndepth of the transverse spin current λtis assumed to be\nzero in this theory. Since we are interested in the eﬀect\nof the penetration depth of the transverse spin current\non spin pumping, we explicitly considerthe diﬀusion pro-\ncess of transversespin accumulation in the ferromagnetic\nlayer. The backﬂow of spin current ﬂowing from the N i\nlayer to the F klayer is expressed as\nINi→Fks=1\n4π/bracketleftbigg2g↑↑g↓↓\ng↑↑+g↓↓{mk·(µNi−µL\nFk)}mk\n+g↑↓\nr(Fk)mk×(µNi×mk)+g↑↓\ni(Fk)µNi×mk\n−t↑↓\nr(Fk)mk×(µT\nFk×mk)−t↑↓\ni(Fk)µT\nFk×mk/bracketrightBig\n,(2)\nwhereg↑↑(↓↓)is the spin up (down) conductance, t↑↓\nr(i)\nis the real (imaginary) part of the transmission mixing\nconductance at the F k/Niinterface and µNiis the spin\naccumulation in the N ilayer [6]. The longitudinal spin\naccumulation in the F klayer is denoted by µL\nFk. The\nlast two terms express contributions from transversespin\naccumulation µT\nFkin the F klayer.\nThe spin accumulation in a ferromagnetic layer is de-\nﬁned by the non-equilibrium distribution matrix at a\ngiven energy ε,ˆf(ε) =f0ˆ1 +f·ˆσ[6], where f=fxt1+fyt2+fzm. Here we introduce the orthogonalunit\nvectors in spin space ( t1,t2,m). The non-equilibrium\ncharge distribution is represented by f0= (f↑+f↓)/2.\nOn the other hand, fz= (f↑−f↓)/2 is the diﬀerence in\nnon-equilibrium distribution between spin-up and spin-\ndown electrons, and fxandfyare the non-equilibrium\ndistributionsofthetransversespincomponents. Thespin\naccumulation is deﬁned as µ=/integraltext\nεFdεTr[ˆσˆf] [10]. The\nspin accumulation in the nonmagnetic layer is deﬁned in\na similar way.\nThe spin accumulation in a nonmagnetic layer, µN,\nobeys the diﬀusion equation [11], and is expressed as a\nlinearcombinationofexp( ±x/λsd(N)), whereλsd(N)isthe\nspin diﬀusion length. The spin current in a nonmagnetic\nlayer is given by\nIN\ns=−∂\n∂x/planckover2pi1SσN\n2e2µN, (3)\nwhereSis the cross section area of the system, σNis\nthe conductivity and eis the absolute value of the elec-\ntron charge. The spin current in the N 3layer is equal to\n−IN3→F2satx=L2+d2because of the continuity of the\nspin current, and vanishes at x=L2+d2+L3(see Fig.\n1). Using above boundary conditions, we obtain the spin\naccumulation in the N 3layer.\nThe longitudinal spin current in a ferromagnetic layer\nis given by\n(m·IF\ns)m=−∂\n∂x/planckover2pi1S\n2e2(σ↑µ↑\nF−σ↓µ↓\nF)m,(4)\nwhereµ↑(↓)\nF=/integraltext\nεFdεf↑(↓)is the electro-chemical poten-\ntial for the spin-up (spin-down) electrons and σ↑(↓)is\nthe conductivity of spin-up (spin-down) electrons. The\npolarization of spin-dependent conductivity is deﬁned as\nβ= (σ↑−σ↓)/(σ↑+σ↓). The longitudinal spin current\nin the F 2layer is equal to m2·IN2→F2satx=L2and\n−m2·IN3→F2satx=L2+d2because of the continuity\nof the spin current. Solving the diﬀusion equation [11]\nwith the above boundary conditions, we obtain longitu-\ndinal spin accumulation in the F 2layer. The longitudinal\nspin accumulation is expressed as a linear combination of\nexp(±x/λsd(FL)), where λsd(FL)is the longitudinal spin\ndiﬀusion length.\nWe assume that the transverse spin accumulation\nobeys the following equation [7]:\n∂2\n∂x2µT\nF=1\nλ2\nJµT\nF×m+1\nλ2\nsd(FT)µT\nF,(5)\nwhereλJ=/radicalbig\n(D↑+D↓)/planckover2pi1/(2J) andλsd(FT)is the\ntransverse spin diﬀusion length. Here Jrepresents\nthe strength of the exchange ﬁeld. The transverse\nspin accumulation is expressed as a linear combina-\ntion of exp( ±x/l+) and exp( ±x/l−), where 1 /l±=/radicalBig\n(1/λ2\nsd(FT))∓(i/λ2\nJ). Therefore, we deﬁne the pene-\ntration depth of the transverse spin current λtby\n1\nλt= Re/bracketleftbigg1\nl+/bracketrightbigg\n. (6)3\nThe transverse spin current in a ferromagnetic layer is\nexpressed as\nm×(IF\ns×m) =−∂\n∂x/planckover2pi1Sσ↑↓\n2e2µT\nF, (7)\nwhereσ↑↓= (1/2)(σ↑/(1 +β′) +σ↓/(1−β′)). Here\nβ′= (D↑−D↓)/(D↑+D↓) is the polarization of the\nspin-dependent diﬀusion constants, D↑andD↓[7]. For\nsimiplicty, we assume that β=β′. The transverse spin\naccumulationinaferromagneticlayerisobtainedbysolv-\ning Eq. (5) with boundary conditions satisfying the con-\ntinuity of the spin current at the N/F interface.\nSolving the diﬀusion equations of the spin accumula-\ntions of the N 3and F2layers, the backﬂow at the N 2/F2\ninterface is re-written as\nIN2→F2s=1\n4π/bracketleftBig\ng∗\n(F2)(m2·µN2)m2\n+ ˜g↑↓\nr(F2)m2×(µN2×m2)+ ˜g↑↓\ni(F2)µN2×m2/bracketrightBig\n,\n(8)\nwhere the conductance g∗\n(F2)is given in Ref. [10], and de-\npends on the ratio d2/λsd(FL). Similarly, the renormal-\nized mixing conductances, ˜ g↑↓\nr,i(F2), depend on the ratio\nd2/l+(F2). If the thickness of the N 3layer is thin enough\ncompared to its spin diﬀusion length, ˜ g↑↓\nr,i(F2)are given\nby\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,(9)\nwhere ∆ = K2\n1+K2\n2andK1andK2are given\nK1= 1+t↑↓\nr(F2)Re/bracketleftbigg1\ngttanh(d2/l+)/bracketrightbigg\n+t↑↓\ni(F2)Im/bracketleftbigg1\ngttanh(d2/l+)/bracketrightbigg\n,(10)\nK2=t↑↓\ni(F2)Re/bracketleftbigg1\ngttanh(d2/l+)/bracketrightbigg\n−t↑↓\nr(F2)Im/bracketleftbigg1\ngttanh(d2/l+)/bracketrightbigg\n,\n(11)\nwheregt/S=h/2e2ρF2l+andρF2is the resistivity of\nthe F2layer. The mixing conductance of the F 1layer\nin Eqs. (1) and (2) is also replaced by the renormalized\nconductance.\nWe assume that spin-ﬂip scattering in the N 2layer\nis so weak that we can neglect the spatial variation\nof the spin current in the N 2layer. Then we have\nIpump\ns−IN1→F1s=IN2→F2s, and the spin accumulation\nin the N 2layer can be determined [10]. The torque act-\ning on the magnetization of the F 1layer is given by\nm1× {(Ipump\ns−IN2→F1s)×m1}, which yields the fol-\nlowing modiﬁed Landau-Lifshitz-Gilbert (LLG) equation\n[10, 12] :\ndm1\ndt=−γeﬀm1×Beﬀ+γeﬀ\nγ(α0+α′)m1×dm1\ndt,(12)FIG. 2: The dependences of the line width of the FMR power\nabsorption spectra, ∆ B, on the thickness of the F 2layer,d2.\nMaterials of the F 2layer are (a) Ni 80Fe20, (b) Co 75Fe25and\n(c) (Co 50F50)80B20, respectively. The ﬁlled circles represent\nexperimental data and the solid lines are ﬁt to the experimen -\ntal data according to the theory with the ﬁnite penetration\ndepth of the transverse spin current λt. The dotted lines rep-\nresent the case of λt= 0.\nwhereBeﬀis the eﬀective magnetic ﬁeld, γis the gy-\nromagnetic ratio, α0is the Gilbert damping constant\nintrinsic to the ferromagnetic metal, and α′is the en-\nhancement of the Gilbert damping constant due to spin\npumping. The Gilbert damping constantis relatedto the\nline width of the FMR absorption spectrum via [13]\n∆B= ∆B0+2ω√\n3γα′, (13)\nwhereω= 2πfis the angular velocity of the oscillating\nmagnetic ﬁeld. We notice that the eﬀects of the N 1and\nN3layers are quite small because, as mentioned below,\nthe thickness of these layers are thin enough compared\nto its spin diﬀusion length in our experiments. Assuming\nthatg↑↓\nr≫g↑↓\ni[10], in the limit of θ→0, we ﬁnd\n∆B−∆B0≃/planckover2pi1ω\n2√\n3πMd1S˜g↑↓\nr(F1)˜g↑↓\nr(F2)\n˜g↑↓\nr(F1)+˜g↑↓\nr(F2),(14)\nwhere ˜g↑↓\nr(Fi)(i= 1,2) is the real part of the renormal-\nized mixing conductance of the i-th ferromagnetic layer.\nWe should note that if we neglect the transverse spin\naccumulation in the ferromagnetic layer the mixing con-\nductances are not renormalized, and that the line width4\n∆Bdoes not depend on the thickness ofthe F 2layer[10].\nThis is due to the fact that the dominant component of\nthe pumped spin current is perpendicular to the magne-\ntization vector m2of the F 2layer in our experiment.\nWe performed FMR experiments on the three diﬀer-\nent N1/F1/ N2/F2/N3ﬁve-layer systems shown in Fig. 1\n[14]. Nonmagnetic layers are made of Cu. The combina-\ntions of the ferromagnetic layers (F 1,F2) of each system\nare (a) (CoFe,Py), (b) (Py,CoFe) and (c) (CoFe,CoFeB).\nThe samples were deposited on Corning 1737 glass sub-\nstrates using an rf magnetron sputtering system in an ul-\ntrahigh vacuum below 4 ×10−6[Pa] and cut to 5 [mm2].\nThe Ar pressure during deposition was 0.077 [Pa]. The\nthickness of all Cu layers are 5 [nm]. The thickness of F 1\nlayers is 5 [nm] for sample (a) and (b), and 10 [nm] for\nsample (c). The FMR measurements were carried out\nusing an X-band microwave source ( f= 9.4[GHz]) at\nroom temperature. The microwave power, modulation\nfrequency, and modulation ﬁeld are 1 [mW], 10 [kHz],\nand 0.1 [mT], respectively. The precession angles of all\nsamples are estimated to be 1 [deg]. The resistivity ρF\nof Py, CoFe and CoFeB are 241 [Ωnm], 94 [Ωnm] and\n1252 [Ωnm], respectively. The magnetizations (4 πM) of\nPy and CoFe are 0.76 [T] and 2.1 [T], respectively. The\ngyromagneticratiois 1 .8467×1011[Hz/T] for all systems\nIn Figs. 2 (a), (b) and (c), the measured line widths of\nthe FMR absorption spectra ∆ Bare plotted with ﬁlled\ncircles against the thickness of the F 2layer,d2. The solid\nlines are ﬁt to the experimental data according to the\ntheory with the ﬁnite penetration depth ofthe transverse\nspin current λt. The dotted lines representthe calculated\n∆Bin the case of λt= 0 [10].\nParameters other than λtare determined as follows.\nThe mixing conductances per unit area of the combina-\ntions (g↑↓\nr(F1)/S,g↑↓\nr(F2)/S) are assumed to be (a) (48.0,\n38.0), (b) (15.2, 17.0) and (c) (48.0, 128.0) [nm−2]. Al-\nthough these values are determined by ﬁtting, they havegood agreement with the ab initio caluclations [6]. For\nsimplicity,weassumethat t↑↓\nr=t↑↓\niwherevaluesof t↑↓\nr,i/S\nof Py, CoFe and CoFeB are taken to be 4.0 [nm−2], 6.0\n[nm−2] and 0.8 [nm−2], respectively. The longitudinal\nspin diﬀusion lengths are 5.5 [nm] for Py and 12 [nm] for\nCoFe and CoFeB, respectively [15, 16]. The polarizations\nof conductance βare 0.73 for Py, 0.65 for CoFe and 0.56\nfor CoFeB, respectively [15, 16, 17]. The transverse spin\ndiﬀusion lengths are given by λsd(FT)=λsd(FL)//radicalbig\n1−β2\n[7]. We take g↑↓\ni/S= 1.0 [nm−2], 2g↑↑g↓↓/(g↑↑+g↓↓)S=\n20.0 [nm−2] [6, 10] for all systems; these are not impor-\ntant parameters for ﬁtting the experimental results. The\nspin diﬀusion length and resistivity of Cu are taken to\nbe 500 [nm] and 21 [Ωnm] [18]. The obtained values of\nλtare 3.7 [nm] for Py, 2.5 [nm] for CoFe and 12.0 [nm]\nforCoFeB, respectively. Ourresults agreequite well with\nthe prediction based on the Boltzmann theory of electron\ntransport [7].\nIn conclusion, we analyzed spin pumping in\nCu/CoFe/Cu/Py/Cu, Cu/Py/Cu/CoFe/Cu and\nCu/CoFe/Cu/CoFeB/Cu ﬁve-layer systems both\ntheoretically and experimentally. We showed that the\nenhancement of the Gilbert dumping constant due to\nspin pumping depends on the ratio of the penetration\ndepthλtand the thickness of the ferromagnetic layers,\nwhich is not in resonance with the oscillating magnetic\nﬁeld. We measured the line widths of FMR absorption\nspectra, which are closely related to the Gilbert dump-\ning constant. Analyzing the experimental results, we\nshowed that the penetration depths of the transverse\nspin current in Py, CoFe and CoFeB are 3.7 [nm], 2.5\n[nm] and 12.0 [nm], respectively. Our results support\nthe Boltzmann theory of transverse spin current [7].\nThe authors would like to acknowledge the valuable\ndiscussions they had with P. M. Levy. This work was\nsupported by CREST and NEDO.\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] J. Z. Sun, Phys. Rev. B 62, 570 (2000).\n[4] S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Em-\nley, R. J. Schoelkopf, R. A. Buhrman, and D. C. Ralph,\nNature425, 380 (2003).\n[5] M. D. Stiles and A. Zangwill, Phys. Rev. B 66, 014407\n(2002).\n[6] A. Brataas, G. E. W. Bauer, and P. J. Kelly, Phys. Rep.\n427, 157 (2006).\n[7] S. Zhang, P. M. Levy, and A. Fert, Phys. Rev. Lett. 88,\n236601 (2002).\n[8] S. Urazhdin, R. Loloee, and W. P. Pratt, Jr, Phys. Rev.\nB71, 100401(R) (2005).\n[9] W. Chen, M. J. Rooks, N. Ruiz, J. Z. Sun, and\nA. D. Kent, Phys. Rev. B 74, 144408 (2006).\n[10] Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I.\nHalperin, Rev. Mod. Phys. 77, 1375 (2005).[11] T. Valet and A. Fert, Phys. Rev. B 48, 7099 (1993).\n[12] T. Taniguchi and H. Imamura, Phys. Rev. B 76, 092402\n(2007).\n[13] S. V. Vonsovskii, FERROMAGNETIC RESONANCE\n(IsraelProgram forScientiﬁcTranslations Ltd., Jersalem ,\n1964).\n[14] S. Mizukami, Y. Ando, and T. Miyazaki, J. Magn. Magn.\nMater.226, 1640 (2001); Jpn. J. Appl. Phys. 1 40, 580\n(2001); Phys. Rev. B 66, 104413 (2002); J. Magn. Magn.\nMater.239, 42 (2002).\n[15] A. C. Reilly, W. Park, R. Slater, B. Ouaglal, R. Loloee,\nW. P. P. Jr., and J. Bass, J. Magn. Magn. Mater. 195,\nL269 (1999).\n[16] A. Fert and L. Piraux, J. Magn. Magn. Mater. 200, 338\n(1999).\n[17] H. Oshima, K. Nagasaka, Y. Seyama, Y. Shimizu,\nS.Eguchi, andA.Tanaka, J. Appl.Phys. 91, 8105(2002).\n[18] J.Bass and W. P. P. Jr, J. Phys.: Condens. Matter 19,\n183201 (2007)."    },    {        "title": "1611.04575v1.Low_temperature_evolution_of_the_spectral_weight_of_a_spin_up_carrier_moving_in_a_ferromagnetic_background.pdf",        "content": "arXiv:1611.04575v1  [cond-mat.str-el]  14 Nov 2016Low-temperature evolution of the spectral weight of a spin- up carrier moving in a\nferromagnetic background\nMirko M. M¨ oller1and Mona Berciu1,2\n1DepartmentofPhysics and Astronomy,University ofBritish Columbia,Vancouver, BritishColumbia,Canada,V6T1Z1\n2Quantum Matter Institute, University of British Columbia, Vancouver, BritishColumbia,Canada,V6T1Z4\n(Dated: September 8, 2018)\nWe derive the lowest-temperature correction to the self-en ergy of a spin-up particle injected in a\nferromagnetic background. The background is modeled with b oth Heisenberg and Ising Hamiltoni-\nans so that diﬀerences due to gapless vs. gapped magnons can b e understood. Beside the expected\nthermal broadening of the quasiparticle peak as it becomes a resonance inside a continuum, we also\nﬁnd that spectral weight is transferred to regions lying out side this continuum. We explain the\norigin of this spectral weight transfer and its low-tempera ture evolution.\nPACS numbers: 71.10.Fd, 75.30.Mb, 71.27.+a, 75.50.Dd\nI. INTRODUCTION\nThe problemofunderstandingthe behaviorofacarrier\ndopedintoamagneticallyorderedinsulatorisrelevantfor\nthe study ofmany materials. Multiple variationsarepos-\nsible: the carriermay enter into the same band that gives\nrise to the magnetic order, or, as if often the case, may\nbe hosted in a diﬀerent band. The background might\nhave antiferromagnetic (AFM) order (parent cuprates1\nbeing the most famous example), or ferromagnetic (FM)\norder (in ferromagnetic chalcogenides like EuO) or more\ncomplicated forms of magnetic order, such as FM lay-\ners that are layered antiferomagnetically in manganite\nperovskites,2,3zig-zag order in iridates,4etc. Finally, the\nmagneticordermaybepresentintheundopedcompound\n(all examples listed above) or may arise as a result of\ndoping, like in diluted magnetic semiconductors such as\nGa1−xMnxAs,5or heavy fermion materials like CeSi x.3,6\nUnderstandingthe propertiesofsuchmaterialshasdirect\ntechnological implications since many of them are candi-\ndates for new spintronic and magnetoelectric devices.7\nThe degree of diﬃculty in solving such problems varies\nwidely. The most diﬃcult problems are those with AFM\nbackgrounds, because of their inherent complexity due to\nthe presence of quantum spin ﬂuctuations – this explains\nwhy what happens when one hole is doped into an AFM\ncuprate layer is still being debated.8\nIn contrast, FM backgrounds are exactly solvable, es-\npecially atT= 0. On the other hand, unlike in the AFM\ncase, here the spectrum of the carrier has a striking de-\npendence on its spin direction. If the carrier is injected\nwith its spin oriented parallel to the local moments, no\nspin-ﬂip excitations are possible and the carrier moves\nfreely. Its spectrum is identical to that of a free carrier\nup to an energy shift due to the zzcomponent of the\nmagnetic exchange. If the carrier is injected with its spin\nanti-parallel to the local moments, the formation of a\ndressed quasi-particle, a so-called spin-polaron, is possi-\nble. This is a state where the carrier continuously emits\nand re-absorbs a magnon while ﬂipping its spin from up\nto down, in a coherent fashion. There are also stateswhere the carrier has spin-up and the magnon is present\n(as required by conservation of the z-component of the\ntotal spin) but not bound to the carrier, giving rise to\na continuum of incoherent states distinct from the spin-\npolaron discrete state.\nThe fact that only one magnon can be emitted by a\nspin-down carrier injected in a T= 0 FM background\n(assuming the carrier has spin-1\n2, which we do here) ex-\nplains why there exists an exactly solvable solution for\nsuch problems. The solution was ﬁrst given by Shastry\nand Mattis9and has recently been generalized to more\ncomplex lattices.10Furthermore, exact analytical deriva-\ntions of the eigenstates and eigenenergies were recently\npresented by Henning et al.11and by Nakano et al.12for\nHamiltonians describing such problems.\nAs far as we know, the only other exact solution for\na generalization of this simplest case is for two carriers\ninjected in the FM background,13because the number of\npossible additional magnons is still very small, resulting\nin a solvable few-body problem. Dealing with ﬁnite car-\nrierconcentrationswhichcaninduceﬁniteconcentrations\nof magnons requires the use of approximations,14except\nin the very trivial case when all carriers have spin-up.\nHere we consider another and in real life more inter-\nesting generalization, namely that of studying the spec-\ntrum of a spin-up carrier injected in a FM background\nat ﬁniteT. An exact solution is no longer possible since\none needs to consider states with arbitrary numbers of\nmagnons when performing temperature averages. A nat-\nural approach for low- Tis to consider states with a small\nnumber of magnons; this is what we do here. As a result,\nthe solution we propose becomes asymptotically exact in\nthe limit of very low temperatures, where “low” means\nwell-below the Curie critical temperature TCof the FM\nbackground.\nAs mentioned, aspin-up carrierhasa verysimple spec-\ntrum atT= 0, mirroring that of the free carrier, with a\nsingle eigenstate for a given momentum. At T/ne}ationslash= 0 ther-\nmally activated magnons are present in the system and\nthe carriercan nowﬂip its spin by absorbingone ofthem.\nInteractionwithevenonesuchmagnontakestheproblem2\nin the Hilbert subspace appropriate for the T= 0 spin-\ndown carrier, which has a very diﬀerent spectrum. As a\nresult, we expect that spectral weight is transferred from\nthe spin-up quasiparticle peak to energies in the spec-\ntrum of the spin-polaron, as Tincreases. How exactly\ndoes this occur at very low T, and what happens to the\ninﬁnitely-lived discrete state that was the only feature in\nthe spectrum at T= 0, is the topic of this work.\nFurthermore, we consider two types of exchange be-\ntween the local moments, namely Heisenberg exchange\nand Ising exchange (in both cases, the characteristic en-\nergy scale is J). For the latter the magnon spectrum is\ngapped, whereas for the former the magnon spectrum is\ngapless. This allows us to contrast the two cases to un-\nderstand the relevance of the magnon’s spectrum on the\nevolution of the up carrier’s spectral function with T.\nFinite temperature studies have been previously car-\nried out by Nolting et al.3for the Kondo lattice model\n(KLM), which is also often referred to as the s-f model.\nThis model accounts for the kinetic energy of the carrier\nas described by a tight-binding model with an energy\nscalet, and for the exchange between the local moments\nand the carrier, described by a Heisenberg exchange with\na couplingJ0. Unlike the models we consider, KLM does\nnot include the exchange Jbetween local moments; this\nis one key diﬀerence between our work and theirs. The\nsecond is the approach employed. While, as mentioned,\nwe use a low- Texpansion to calculate the propagator,\nNolting et al. proposed an ansatz for the self-energy cho-\nsen so as to reproduce asymptotic limits where an ex-\nact solution is available, speciﬁcally the T= 0 solution\nmentioned above and the case of ﬁnite Tbut zero band-\nwidth,t= 0.15(This approach was later generalized to\nﬁnite carrier concentrations as well.14) Their ansatz for\nthe self-energycontainsseveralfree parameterswhichare\nﬁxed by ﬁtting them to a ﬁnite number of exactly calcu-\nlated spectral moments. The temperature dependence is\ncontained implicitly in the magnetization which enters\nthe self-energy as an external parameter. In the limit of\nvery low-Twe consider here, the averagelocal moment is\nessentially unchanged from its T= 0 value, so the eﬀects\nwe uncover are basically absent in the ansatz of Nolting\net al. In other words, besides studying diﬀerent Hamil-\ntonians by very diﬀerent means, our studies also focus\non very diﬀerent regimes: very low T, in our work, vs.\nmedium and high Tin Ref. 15. Needless to say, in the\nabsenceofanexactsolutionit is likelythat acollectionof\napproximations valid in diﬀerent regimes will be needed\nin order to fully understand this problem.\nThe article is organized as follows: in Section II we\nintroduce our models and in Section III we derive the\nlowest-T self-energy correction. Section IV presents our\nresults and Section V contains our conclusions.II. MODELS\nWe consider a single spin-1\n2charge carrier which prop-\nagates on a hypercubic lattice with periodic boundary\nconditions after Nisites in the direction i= 1,d; the\ntotal number of sites is N=/producttextd\ni=1Ni. Our results are\nford= 2 andd= 3. Of course, long-range FM order\nat ﬁnite-T only exists in d= 3. However, we also con-\nsider anisotropic layered compounds, like the mangan-\nites, which have 2D FM layers whose ﬁnite-T long-range\norderisstabilizedbyweakinter-layercoupling,butwhere\none can assume that at very low- Tthe intra-layer carrier\ndynamics determine its properties. In principle, similar\narguments can be employed to study d= 1 chains with\nFM order at ﬁnite- Tmaintained by their immersion in\n3D lattices, but complications due to formation of mag-\nnetic domains would still need to be dealt with.\nThe carrier is an electron in an otherwise empty band\nor a hole in an otherwise full band, described by a tight\nbinding model with nearest neighbor (nn) hopping:\nˆT=/summationdisplay\nk,σǫ(k)c†\nk,σck,σ, (1)\nwithǫ(k) =−2t/summationtextd\ni=1coskifor lattice constant a= 1.\nc†\nk,σcreates a carrier with momentum kand spinσ.\nThe local magnetic moments are described by either a\nHeisenberg or an Ising interaction:\nˆHS=−J/summationdisplay\n/angbracketlefti,j/angbracketright/parenleftbig\nSi·Sj−S2/parenrightbig\n(2)\nfor Heisenberg exchange, while for Ising exchange:\nˆHI=−J/summationdisplay\n/angbracketlefti,j/angbracketright/parenleftbig\nSz\niSz\nj−S2/parenrightbig\n, (3)\nwhereSiisthespin-Smomentlocatedatsite Riandonly\nnn exchange is included in both models. We represent\nlocal moments with a double arrow, eg.⇑, while the\ncarrier spin is represented by a single arrow, eg.↑.\nFor both these models the undoped ground state is\n|FM/an}bracketri}ht=|⇑,⇑,.../an}bracketri}htand has zero energy. The only excited\nstates of interest will be the single magnon states:\n|Φ(q)/an}bracketri}ht=S−\nq√\n2S|FM/an}bracketri}ht=/summationdisplay\njeiqRj\n√\n2SNS−\nj|FM/an}bracketri}ht.(4)\nHereS±\ni=Sx\ni±iSy\niare the raising (+) and lowering\n(−) operators. The key diﬀerence between the Heisen-\nberg and Ising interactions is the dispersion of the sin-\ngle magnon states. For the Heisenberg model this is\nΩq= 4JS/summationtextd\ni=1sin2(qi/2), whereas for the Ising model\nthe magnons are dispersionless, Ω q= Ω = 2dJS.\nThe interaction between the carrier and the local mo-\nments is also a Heisenberg exchange:\nˆHexc=J0/summationdisplay\njsj·Sj, (5)3\nwheresi=/summationtext\nα,βc†\ni,ασα,β\n2ci,βis the carrier spin operator\nandσare the Pauli matrices. The coupling J0can be\neither FM or AFM; we will consider both cases.\nIt is convenient to split ˆHexc=ˆHz\nexc+ˆHx,y\nexc, where\nˆHz\nexc=J0/2/summationtext\nj/parenleftBig\nc†\nj,↑cj,↑−c†\nj,↓cj,↓/parenrightBig\nSz\njandˆHx,y\nexc=\nJ0/2/summationtext\nj/parenleftBig\nc†\nj,↑cj,↓S−\nj+c†\nj,↓cj,↑S+\nj/parenrightBig\n.The ﬁrst term causes\nan energy shift ±J0S/2. The second term is responsible\nfor spin-ﬂip processes, where the carrier ﬂips its spin by\nabsorbing or emitting a magnon.\nThe total Hamiltonian is:\nˆH=ˆT+ˆHS/I+ˆHexc. (6)\nDue to translational invariance, the total momentum is\nconserved. Furthermore, the z−component Sz\ntotof the\ntotal spin (the sum of the carrier spin and lattice spins),\nis also conserved. Therefore, eigenstates ˆH|ψ(m)\nα(k)/an}bracketri}ht=\nE(m)\nα(k)|ψ(m)\nα(k)/an}bracketri}htare indexed by the total momentum of\nthe system, k, by the number mof magnons when the\ncarrier has spin-up so that Sz\ntot=NS+1\n2−m, and by\nαwhich comprises all the other quantum numbers.\nIII. FORMALISM\nWewanttocalculatethelow-TexpressionofZubarev’s\ndouble-time retarded propagator16for a spin-up carrier:\nG↑(k,τ) =−i\nZΘ(τ)Tr[e−βˆHck,↑(τ)c†\nk,↑(0)],(7)\nbut in a canonical (not grand-canonical) ensemble, as-\nsuming that the carrier is injected in the otherwise un-\ndoped FM. As aresult, the traceis overthe eigenstatesof\nˆHS/I(in the absence of carriers, ˆH≡ˆHS/I). Θ(τ) is the\nHeavisidefunction, Z= Tr[e−βˆHS/I] is the partition func-\ntion for the undoped FM, and ck,↑(τ) = eiτˆHck,↑e−iτˆHis\nthe carrier annihilation operator in the Heisenberg pic-\nture. In the frequency domain we have:\nG↑(k,ω) =/integraldisplay∞\n−∞dτeiωτG↑(k,τ).\nAtT= 0, the trace reduces to a trivial expectation\nvalue over |FM/an}bracketri}ht, and we ﬁnd:9\nG(0)\n↑(k,ω) =/an}bracketle{tFM|ck,↑ˆG(ω)c†\nk,↑|FM/an}bracketri}ht=1\nω−E↑(k)+iη.HereˆG(ω) = [ω−ˆH+iη]−1is the resolvent of ˆHandη\nis a small, positive number (we set ¯ h= 1). Physically,\n1/ηsets the carrier lifetime. The eigenenergy is E↑(k) =\nǫ(k) +J0S\n2for both the Heisenberg and Ising models.\nAs discussed, this shows that at T= 0 a spin-up carrier\npropagatesfreely and acquires an energy shift from ˆHz\nexc.\nAt ﬁnite temperature, we expect to ﬁnd:\nG(k,ω) =1\nω−E↑(k)−Σ(k,ω)+iη\n=G(0)\n↑(k,ω)+G(0)\n↑(k,ω)Σ(k,ω)G(0)\n↑(k,ω)+...(8)\nStrictly speaking, the energy shift J0S\n2is part of the self-\nenergy, however it is convenient to separate it as we do\nhere so that Σ( k,ω) contains only the ﬁnite- Tterms.\nSince we are interested in the lowest- Tcontribution to\nΣ(k,ω), we consider only the ﬁrst two terms of Eq. (7):\nG↑(k,ω) =G(0)\n↑(k,ω)+/summationtext\nqe−βΩqG(1)\n↑(k,q,q,ω)+...\n1+/summationtext\nqe−βΩq+...,\n(9)\nwhere we deﬁne the new propagators G(1)\n↑(k,q,q′,ω) =\n−i/integraltext∞\n0dτeiωτ/an}bracketle{tΦ(q′)|ck,↑(τ)c†\nk+q′−q,↑|Φ(q)/an}bracketri}ht.Only diago-\nnalq′=qterms contribute to the trace. After carry-\ning out the Fourier transform we ﬁnd G(1)\n↑(k,q,q′,ω) =\n/an}bracketle{tΦ(q′)|ck,↑ˆG(ω+Ωq′)c†\nk+q′−q,↑|Φ(q)/an}bracketri}ht.Note that the ar-\ngument of the resolvent is shifted by the magnon energy,\nmeaning that the carrier’s energy is measured with re-\nspect to that of the state in which the carrier is injected.\nFollowing calculations detailed in the Appendix, we ﬁnd:\n/summationdisplay\nqe−βΩqG(1)\n↑(k,q,q,ω) =/summationdisplay\nqe−βΩq/braceleftBig\nG(0)\n↑(k,ω)\n−J0\n2N[G(0)\n↑(k,ω)]2\n1+J0SG(0)\n↑(k+q,ω+Ωq)+J0\n2g(k,q,ω)/bracerightBigg\n,\nwhere\ng(k,q,ω) =1\nN/summationdisplay\nQG(0)\n↑(k+q−Q,ω+Ωq−ΩQ)\nis a known function. When this expression is used in Eq.\n(9), we obtain\nG↑(k,ω) =G(0)\n↑(k,ω)(1+/summationtext\nqe−βΩq+...)+[G(0)\n↑(k,ω)]2Σ(k,ω)(1+...)+...\n1+/summationtext\nqe−βΩq+...\n=G(0)\n↑(k,ω)+[G(0)\n↑(k,ω)]2Σ(k,ω)+...,\nsince the terms in the brackets are the expansion of Z(to the order considered here; higher order contributions will\ncome from including many-magnon processes) and cancel with the d enominator. This has the expected form of Eq.4\n(8), so we can identify the lowest- Tcorrection to the self-energy:\nΣ(k,ω) =−J0\n2N/summationdisplay\nqe−βΩq\n1+J0SG(0)\n↑(k+q,ω+Ωq)+J0\n2g(k,q,ω)+.... (10)\nIt is important to mention that although we only con-\nsidered states with zero or one magnon in our derivation,\nwe will see some higher-order eﬀects in our results when\nusingG↑(k,ω) = [ω−E↑(k)−Σ(k,ω)+iη]−1,i.e.when\nthe self-energy is placed in the denominator. These are\nfrom states where multiple magnons are present in the\nsystem but the carrier interacts only with one of them\nwhile the rest are “inert” spectators.\nEq. (10) is the main result of this work. The onlydiﬀerence between Heisenberg and Ising backgrounds is\nthe expression for the magnon energy Ω q. For the Ising\ncase, this energy is independent of momentum, resulting\nin a self-energy Σ( ω) independent of k.\nBefore presenting results, let us consider what the\nspectral weight A↑(k,ω) =−1\nπImG↑(k,ω) should be ex-\npected to reveal. The Lehmann representation of the\npropagator in its expanded form is:\nG↑(k,ω) =1\nZ/bracketleftBigg\n1\nω−E↑(k)+iη+/summationdisplay\nα,qe−βΩq|/an}bracketle{tΦ(q)|ck,↑|Ψ(1)\nα(k+q)/an}bracketri}ht|2\nω+Ωq−E(1)\nα(k+q)+iη+.../bracketrightBigg\n. (11)\nAtT= 0 only the ﬁrst term contributes, giving a single\nquasiparticle peak at ω=E↑(k). The second term has\npoles atω=E(1)\nα(k+q)−Ωq. Them= 1 subspace also\ncorresponds to a spin-down carrier injected in the FM at\nT= 0, thus we can ﬁnd the energies E(1)\nα(k) from the\nspectral weight A(0)\n↓(k,ω) =−1\nπImG(0)\n↓(k,ω) where:\nG(0)\n↓(k,ω) =/an}bracketle{tFM|ck,↓ˆG(ω)c†\nk,↓|FM/an}bracketri}ht\n=/summationdisplay\nn|/an}bracketle{tFM|ck,↓|Ψ(1)\nα(k)/an}bracketri}ht|2\nω−E(1)\nα(k)+iη\n=/braceleftBigg\n[G(0)\n↑(k,ω)]−1+J0S\n1+J0\n2g(k,0,ω)/bracerightBigg−1\n(12)\nThe last result is from Ref. 9. As already mentioned\nand further detailed below, the spectrum E(1)\nα(k) cer-\ntainly contains an up-carrier+magnon continuum span-\nningtheenergies {E↑(k−q′)+Ωq′}q′; intherightcircum-\nstances, a coherent spin-polaron state with the magnon\nbound to the carrier may also appear, see below. Thus,\nforT/ne}ationslash= 0,A↑(k,ω) should have weight at all energies\n{E↑(k+q−q′) + Ωq′−Ωq}q,q′. In the Ising case the\nmagnon energies cancel out so weight should be expected\nat all energies {E↑(q)}qin the spin-up carrier spectrum,\nnot just at E↑(k). This automatically implies that the\nT= 0 inﬁnitely lived quasiparticle of energy E↑(k) ac-\nquiresaﬁnite lifetime at T/ne}ationslash= 0. Thisremainstrueforthe\nHeisenberg case, with the added complication that now,\n{E↑(k+q−q′) + Ωq′−Ωq}q,q′will generally span a\nwider range of energies than {E↑(q)}q. If a spin-polaronappears in the m= 1 sector, additional weight is ex-\npected at energies in its band minus the magnon energy.\nHigher order terms will contribute similarly (remember\nthat our solution for the propagator does include par-\ntial contributions from many-magnon states). To con-\nclude, at ﬁnite Tone can no longer assume that energies\nfor which the spectral weight A↑(k,ω) is non-zero are\nnecessarily in the spectrum of the momentum- kHilbert\nsubspace. This makes the interpretation of the spectral\nweight less straightforward than it is at T= 0.\nIV. RESULTS\nA. Review of T= 0results\nGiven the analysis presented above, it is useful to ﬁrst\nquickly review the dispersion E↑(k) and, more impor-\ntantly, the spectrum E(1)\nα(k) for them= 0 andm= 1\nsectors, respectively. The latter is easiest to see by plot-\nting the spectral weight A(0)\n↓(k,ω). The main focus will\nbe to understand when a spin-polaron state forms in the\nm= 1 sector, but we will also verify the presence of\nthe continuum at the expected location. Since experi-\nmentally this is the most relevant regime, we will assume\nthat|J0|is the largest energy scale and Jis the smallest\none. While realistically one expects J≪t, we will set\nJ/t= 0.5 so that its role can be discerned easily.\nFigure 1 shows E↑(kx,ky= 0) (thick full green line)\nandthespectralweight A(0)\n↓(kx,ky= 0,ω)(contourmap)\nfor the 2D Heisenberg and Ising models, for antiferro-\nmagnetic coupling J0= 3. The spectrum of the m= 15\n-6-4-2 0 2 4 6 8\n 0 0.2 0.4 0.6 0.8  1ω/t\nkx/π\n 0.000 0.005 0.010 0.015 0.020 0.025 0.030\n-6-4-2 0 2 4 6 8\n 0 0.2 0.4 0.6 0.8  1ω/t\nkx/π\n 0.000 0.005 0.010 0.015 0.020 0.025 0.030\nFIG. 1. (color online) Energy E↑(k)(thick full green line) and\nspectral weight A(0)\n↓(k,ω) (contour map) vs. kxatky= 0, for\nthe Heisenberg model (top) and the Ising model (bottom) in\n2D, for AFM coupling J0/t= 3. The dashed red lines mark\nthe expected continuum boundaries in the m= 1 subspace.\nOther parameters are J/t= 0.5,S= 0.5,η/t= 0.01.\nsectorconsistsofadiscretestateatlowenergies,thespin-\npolaron, and the up-carrier + magnon (c+m) continuum\nat higher energies. Because we will encounter a diﬀerent\nspin-polaronlater on, we will refer to this spin polaron as\n“sp1”. To ﬁrst order in perturbation theory its eﬀective\nmass is a factor of (2 S+1) larger than the bare carrier\nmass and its energy is −J0(S+1)/2+O(t,J).10Most of\nthis energy comes from ˆHz\nexcand explains why for AFM\nJ0>0 sp1 is the ground state – states in the continuum\nhave the carrier with spin up and therefore cost ∼J0S/2\nin exchange energy. This also suggests that for FM cou-\nplingJ0<0, the sp1polaronshould be located abovethe\nc+m continuum. This expectation is conﬁrmed below.\nComparing the two panels, we see that the sp1 disper-\nsion is very similar for the two models. This is expected\nbecause this is a coherent state where the magnon is\nlocked into a singlet with the carrier, and this process\nis controlled by J0≫J. A diﬀerence appears in the\nshape of the c+m continuum, however. As mentioned,\nthis must span energies {E↑(k−q)+Ωq}qsince it con-\nsists of up-carrier and magnon scattering states. The-4.00 -3.75 -3.50 -3.25 -3.00\nω/t0.00.10.20.30.40.5A↓(0)(0,0,ω)\n-5.0 -4.5 -4.0 -3.5 -3.0\nω/t0.00.51.01.52.02.5A↓(0)(0,0,ω)Ising\nHeisenberg\n -2.0 -2.5 -3.0 -3.5 -4.0\n  0.1  0.2  0.3ω/t\nkx/π\n 0.0 0.1 0.2 0.3 0.4 0.5\nFIG. 2. (color online) Top: A(0)\n↓(k= 0,ω) for FM J0/t=−2\nin 2D. The lower c+m continuum edge is marked with dashed\nred lines. The Ising model has a discrete state (sp2) below\nthe continuum. Bottom: Spectral weight A(0)\n↓(k,ω) for the\nIsing model in 2D for ky= 0,kx<0.3π. The dashed red\nline marks the lower c+m continuum edge. The sp2 state ap-\npears for small kand then merges into the continuum. Other\nparameters are J/t= 0.5,S= 0.5,η/t= 0.01.\ndashed red lines show the boundaries of this range, in\nagreement with the data (this is more diﬃcult to see for\nthe upper edge, on this scale, due to the reduced spectral\nweight at high energies). Since Ising magnons are disper-\nsionless the continuum boundaries do not change with k.\nIn contrast, the continuum boundaries for the Heisen-\nberg model vary with k, the continuum being wider at\nthe centre of the Brillouin zone than near its edges.\nThis diﬀerence has consequences for a FM coupling\nJ0<0. As mentioned, in this case the c+m continuum\nis expected to be the low-energy feature in the m= 1\nspectrum, with the sp1 state appearing above it. This is\nindeed the case for the Heisenberg model, however in the\nIsing model, for a suﬃciently large J, a second discrete\nstate emerges below the c+m continuum. We will refer\nto this state as “sp2” to distinguish it from sp1. The\ntop panel of Fig. 2 shows its presence (absence) for the\nIsing (Heisenberg) model at k= 0. The bottom panel\nshows that even for the Ising model, the sp2 only exists\nfor small k, at least for these parameters.\nThe origin of the sp2 state is suggested by the ﬁndings6\n-4 -3 -2 -1 0\nJ0/t-4.0-3.8-3.6-3.4Esp2/tsp2 polaron\nlower c+m edge\n-J0S/2-4t\n0.3 0.4 0.50.6 0.7 0.8 0.9 1.0\nJ/t-4.0-3.8-3.6-3.4-3.2Esp2/tsp2 polaron\nlower c+m edge\nFIG. 3. (color online) Ground-state energy of the Ising sp2\npolaron as a function of J0/tforJ/t= 0.5 (top) and as a\nfunction of J/tforJ0/t=−2 (bottom), for S= 0.5.\nofHenning et al. whoshowedthat for J= 0, polaron-like\nstates exist inside the c+m continuum.11We believe that\nthe addition of ˆHIpushes one of them below the contin-\nuum. This is possible because for an Ising coupling, the\nlower continuum edge moves up by Ω = 2 dJS, whereas\nthe polaron-like states experience a smaller energy shift\nsince they include a component with the carrier having\nspin-down. Forthe Heisenbergmodel, on the otherhand,\ninclusion of ˆHSdoes not change the location of the lower\ncontinuum edge at k= 0 since Ω q=0= 0, so the polaron-\nlike state remains a resonance inside the continuum.\nThe ground-state energy of the sp2 polaron is explored\nin Fig. 3. The top panel shows its dependence on J0/t.\nThe sp2 state has weight on both the down-carrier and\non the up-carrier+magnon components. For J0= 0 the\nweightofthe lattercomponentmustvanishsince nospin-\nﬂips are possible and the sp2 state is the same as a free\ndown-carrier, whose energy −J0S/2−4tis also indicated\n(dashedblueline). Theseresultssuggestthatas |J0|/tin-\ncreases, the sp2 state shifts weight from the down-carrier\ncomponent to the up-carrier+magnoncomponent until it\nessentially becomes a continuum-like state.\nThe bottom panelin Fig. 3showsthe sp2ground-state\nenergy vs. J/tfor ﬁxedJ0/t=−2. This value of J0/t\nwas chosen because here the polaronic character of sp2 is\nespecially strong since if we neglect Hx,y\nexc, the energy of\nthe down-carrier component is equal to that of the up-\ncarrier+magnon component. The distance between sp2\nand the continuum increases with J/t, as expected from\nour previous discussion.\nWhile we have only seen the sp2 polaron for the Ising\nmodel, we cannot rule out the possibility that for a very\nnarrow range of momenta and carefully chosen param-\neters, an sp2 state might also appear in the Heisenberg\nmodel. Another important point is that the sp1 state is\nnot guaranteed to exist for all k, either. In Fig. 4 we\nshowA(0)\n↓(k,ω) for the 2D Heisenberg model. No sp2-4-2 0 2 4 6\n 0 0.2 0.4 0.6 0.8  1ω/t\nkx/π\n 0.00 0.02 0.04 0.06 0.08 0.10\n-4-2 0 2 4 6\n 0 0.2 0.4 0.6 0.8  1ω/t\nkx/π\n 0.00 0.02 0.04 0.06 0.08 0.10\nFIG. 4. (color online) Spectral weight A(0)\n↓(k,ω)vskxfor the\n2D Heisenberg model at ky= 0 (top) and ky=π(bottom)\nand FM J0/t=−2. Sp1 appears above the continuum only\nnear the Brillouin zone edge. No sp2 peak is seen below the\ncontinuum. The dashed red lines mark the c+m continuum\nboundariesandthegreenlinemarks E↑(k). Otherparameters\nareJ/t= 0.5,S= 0.5,η/t= 0.01.\nstate appears below the continuum, and sp1 separates\nabove the continuum only near the Brillouin zone edge.\nThis is not a surprise given the rather small value of |J0|,\nsince it controls the separation between sp1 and the con-\ntinuum. For suﬃciently large |J0|, the sp1 polaron splits\noﬀ the continuum in the entire Brillouin zone.9\nTo summarize, the spectrum in the m= 1 (one-\nmagnon) subspace contains the expected c+m contin-\nuum. For AFM J0the low-energy feature is the sp1 po-\nlaron for both the Heisenberg and the Ising models. For\nFMJ0, sp1 becomes the high energy feature and may\nonly appear in a small region of the Brillouin zone if |J0|\nis small. For the Ising model and FM J0, an sp2 polaron\nis also found to appear below the c+m continuum, in a\ncentral region of the Brillouin zone that increases with\nincreasingJ. For the Heisenberg model and FM J0we\ncannot entirely rule out the existence of sp2, although we\nprovided arguments which suggest that this is unlikely.\nWe focused here more on the sp2 polaron because,\nto our knowledge, this solution had not been discussed\nbefore, while the sp1 state has been analyzed in great7\n0.000.010.020.03A↑(0,0,ω) sp1\nc+mE↑(k)\n0.000.020.040.060.080.10A↑(π,π,ω)sp1\nc+mE↑(k)-12 -10 -8 -6-4 -2 0 2 4 68 10\nω/t-30.00.0Σ(0,0,ω)\nIm\nRe\n-12 -10 -8 -6-4 -2 0 2 4 68\nω/t-15.00.015.0Σ(π,π,ω)Im\nRe\nFIG. 5. (color online) Spectral weight A↑(k,ω) and the real\n(solid line) and imaginary (dashed line) part of the self-en ergy\nΣ(k,ω) for the 2D Heisenberg model with AFM J0/t= 10\nandβt= 1, atk= (0,0) (top) and k= (π,π) (bottom). The\nexpected sp1 continuum boundaries are marked with dash-\ndotted blue lines and the expected c+m continuum bound-\naries with dashed red lines. The E↑(k) energy of the T= 0\nδ-peak is marked with a thick green line. Other parameters\nareJ/t= 0.5,S= 0.5,η= 0.02 (top) and η= 0.05 (bottom).\ndetail.9–12We also note that while we presented only\n(computationally less costly to generate) 2D results, we\nﬁnd qualitatively similar results in 3D. This will become\nclear from our ﬁnite-T results shown below.\nB. Low-T results\nWe now presentand analyzelow-Tresults forthe spec-\ntral weight of the spin-up carrier. Since the calculation\nofG↑(k,ω) becomes numerically very expensive in 3D,\nmost of our analysis is in 2D. However, we will also show\na selection of 3D spectra which prove that the 3D results\nare qualitatively similar to the 2D results.\nThe spectral weight A↑(k,ω) =−1\nπImG↑(k,ω) and\nthe self-energy Σ( k,ω) are shown for the Heisenberg and\nIsing models with AFM coupling J0/t= 10 in Figs. 5\nand 6, respectively. In both cases the top panel is for\nk= (0,0) and the bottom one is for k= (π,π). However,\nfortheIsingmodeltheself-energyisindependentof kand\ntherefore in Fig. 6 it is only shown beneath the k= (0,0)0.000.010.02A↑(0,0,ω)sp1\nc+m\nE↑(k)\n-14 -12 -10 -8 -6-4 -2 0 2 4 68 10\nω/t0.000.010.020.030.04A↑(π,π,ω)sp1\nc+m\nE↑(k)-14 -12 -10 -8 -6-4 -2 0 2 4 68 10\nω/t-60.0-30.00.030.0Σ(ω)Im\nRe\nFIG. 6. (color online) Same as Fig. 5 but for the Ising model.\nAll parameters are the same except βt= 0.5 andη= 0.01 in\nboth panels. Note that for the Ising model Σ( ω) is indepen-\ndent ofk. The inset shows a zoom on Σ( ω) at high energies.\nspectral weight. The value of J0/twas chosen so large in\nordertoensurethatthediﬀerentfeaturesinthespectrum\nare well separated, to simplify the analysis. Results for\nsmaller values of J0will be shown below.\nA↑(k,ω), whichatT= 0isthepeak δ(ω−E↑(k))(indi-\ncated by the thick green line), broadensinto a continuum\nat ﬁnite-T. As discussed at the end of Section III, this\ncontinuum has its origin in the c+m continuum of the\nm= 1 sector, thus we continue to call it the “c+m” con-\ntinuum, andshouldspan {E↑(k+q−q′)+Ωq′−Ωq}q,q′.\nThe red dashed lines show the boundaries of this energy\nrange, in excellent agreement with the broadening ob-\nserved inA↑(k,ω). We note that most of the spectral\nweight is still located near E↑(k).\nThis broadening conﬁrms that at ﬁnite-T the quasi-\nparticle acquires a ﬁnite lifetime (the peak at E↑(k) is\nnow a resonance inside a broad continuum, not a dis-\ncrete state). Clearly, this is due to processes where the\nspin-up carrier absorbs a thermal magnon and then re-\nemits it with a diﬀerent momentum, thus scattering out\nof its original state.\nThe ﬁnite lifetime of the carrier in the c+m continuum\nis also evident in the self-energy. The inset in Fig. 6\nshows that for energies within the c+m continuum the\nimaginary part of the self-energy is ﬁnite. The same is8\n-4 -2 0 2 40.000.010.020.03A↑(0,0,ω)\n-4.9 -4.8 -4.7 -4.6-4.5-4.4 -4.3 -4.2 -4.1-4.00.00.51.0A↑(0,0,ω)\n-6 -4 -2 0 2 4 6\nω/t0.000.01A↑(0,0,ω)sp2\nc+msp1\nsp2\nc+msp1(a)\n(b)\n(c)E↑(k)\nE↑(k)\nFIG. 7. (color online) Spectral weight A↑(0,0,ω) for the 2D\nIsing model (panels (a) and (b)) and 2D Heisenberg model\n(panel (c)) for ferromagnetic J0/t=−2 atβt= 0.5,η/t=\n0.01 (Ising) and βt= 1,η/t= 0.02 (Heisenberg). The ex-\npected location of various features are also indicated (see text\nfor more details). Other parameters are J/t= 0.5,S= 0.5.\ntrue for the Heisenberg model (not shown).\nWhile the broadening of the T=0 δ-peak may be\nthought of as quite trivial, Figs. 5 and 6 show that it\nis not the only eﬀect of the ﬁnite-T: spectral weight is\nalso transfered to a new continuum located below the\nc+m continuum. We attribute this continuum to the\nsp1 state. Indeed, if we denote by Esp1(k) the energy\nof the sp1 polaron, we ﬁnd that this continuum spans\n{Esp1(k+q)−Ωq}q(the boundaries of this range are\nmarked by the dashed-dotted blue lines). Its presence\nagrees with the Lehmann representation and reveals this\nspectral weight transfer to be due to processes where the\nspin-up carrier binds a thermal magnon and turns into\nan sp1 polaron.\nThe sp1 continuum is also where both the real and\nimaginary part of Σ( k,ω) take their largest values. Con-\nsequentlythelifetimeofthesestatesisroughlytwoorders\nof magnitude smaller than that of the states within the\nc+m continuum. This is not surprising as the c+m con-\ntinuum stems from a δ-peak with an inﬁnite lifetime at\nT=0, whereas the sp1 continuum vanishes at T=0.\nThere is furthermore a qualitative diﬀerence between\nthe real-part of Σ( k,ω) in the sp1 continuum and in the\nc+m continuum. For the latter the real part falls oﬀ rel-\natively smoothly ( cf.inset in Fig. 6), whereas for the sp1\ncontinuum it is highly singularand almostdiscontinuous.\nNote that there are no major diﬀerences between the\nHeisenberg and Ising models, except for the fact that the\nboundaries of these continua are momentum dependent\nforthe formerand momentum independent forthe latter,\ndue to their diﬀerent magnon dispersions.\nFigures5and 6 alsoshowa verypuzzling discrete stateat low energies. Before we turn our attention to the anal-\nysis of this peak, we quickly discuss the case with FM\ncouplingJ0<0. Ising and Heisenberg results are de-\npicted in Fig. 7 for J0/t=−2 andJ/t= 0.5. From the\ndiscussion of the T= 0 spectrum in the m= 1 Hilbert\nspace, we knowthat for these parametersthe Ising model\nhas an sp2 state below its c+m continuum and there-\nfore expect to ﬁnd its signature in the ﬁnite-T spectrum,\nas well. This is indeed the case, as seen more clearly\nin panel (b) which expands the low-energy part of the\nIsing spectrum shown in (a), revealing weight at ener-\ngies spanning {Esp2(k+q)−Ωq}q(its lower boundary\nis marked by dashed-dotted blue lines). Note that since\nthe sp2 state merges with the c+m continuum (bound-\naries marked by red dashed lines), their corresponding\ncontinua also merge, but panel (b) reveals a clear discon-\ntinuity where they overlap. The high-energy sp1 contin-\nuum is also clearly observed in panel (a), again merged\nwith the c+m continuum since the sp1 state is not fully\nseparated at such a small |J0|, either.\nThe Heisenberg model (panel (c)) only shows the c+m\nand sp1 continua, since there is no sp2 polaron here.\nAgain, agreement with the expected boundaries is ex-\ncellent (the weight seen below the c+m lower edge is due\nto the ﬁnite ηand the fact that we zoomed in close to\nthe axis to make it easier to see the sp1 continuum).\nIt is worth noting that since for small |J0|the vari-\nous features merge, it would be easy to misinterpret the\nthermal broadening as being all ofc+m origin, i.e.to en-\ntirely miss the role played by the spin-polaron solutions\nin them= 1 subspace. This is also illustrated in Fig. 8,\nwhere we return to an AFM J0coupling and show how\nthek= 0 spectra change as J0is decreased. All features\ndiscussed previously can be easily identiﬁed for large J0\nbut merge into one another as J0decreases, so that by\nthe timeJ0/t= 3 there is only one very broad feature,\nalbeit with a non-trivial structure, left in the spectrum\n(apart from the low-energy discrete peak, which we will\ndiscuss later). If one assumed that this is all of c+m\norigin,i.e.scattering of the carrier on individual ther-\nmal magnons, one would infer very wrong values of the\nparameters from the boundaries’ locations.\nThe results shown so far are for large temperatures\nkBT∼t= 2J(for our parameters), where higher or-\nder corrections should certainly become quantitatively\nimportant. On the other hand, from the Lehmann de-\ncomposition we expect that the location of the various\nfeaturesdoesnot depend ontemperature; onlyhowmuch\nspectralweightthey carrycanchangewith T. Foramore\nthorough analysis we return to the case of AFM J0, us-\ning a rather large value so that the various features are\nwell separated, and plot in Fig. 9 the spectral weight\nin the sp1 continuum for several diﬀerent temperatures,\nfor both the Ising and Heisenberg models. This conﬁrms\nthat, indeed, the weight in this continuum decreases fast\nasT→0 while its location is not aﬀected (the location\nof the low-energy peak shifts with T, but as we argue\nbelow, we do not believe that this is a physical feature).9\n0.000.010.020.03A↑(0,0,ω)Ising\nHeisenberg\n0.00.10.20.3A↑(0,0,ω)\n-15 -10 -5 0 5 10\nω/t0.00.10.20.30.40.5A↑(0,0,ω)J0/t=10\nJ0/t=5\nJ0/t=3\nFIG. 8. (color online) Spectral weight A↑(0,0,ω) for the 2D\nIsing (dashed lines) and Heisenberg (full lines) models for\nJ0/t= 10,5,3 in the top, middle and bottom panels, re-\nspectively. Other parameters are J/t= 0.5,S= 0.5 and\nβt= 0.5,η/t= 0.01 (Ising), and βt= 1,η/t= 0.02 (Heisen-\nberg). The oscillations visible especially in the sp1 conti nuum\nare due to ﬁnite-size eﬀects (we used N= 1002andN= 5002\nfor Heisenberg and Ising models, respectively).\nTo quantify the spectral weight transferred, we cal-\nculate/integraltext\nc+mdωA↑(k,ω),i.e.how much is in the c+m\ncontinuum. Since at T= 0 all the weight is in the δ-\npeak atE↑(k) located inside the c+m continuum, this\nvalue starts at 1 and decreases with increasing T, as\nweight is transferred into the sp1 continuum; one can\neasily check that the spectral weight obeys the sum rule/integraltext∞\n−∞dωA↑(k,ω) = 1.\nThe results are shown in Fig. 10 for both models, both\natthecenterandatthecorneroftheBrillouinzone. Note\nthat because of the ﬁnite value of η, some spectral weight\n“leaks” outside the continuum’s boundaries. This prob-\nlem is more severe at lower TbecauseE↑(k) is located\nvery close to an edge of the continuum; this explains why\nthe value saturates below 1 as β→ ∞. This explanation\nis also consistent with the observation that the amount\nof “missing weight” as T→0 is of order η.\nTwo features are immediately apparent. First, there is\na substantial diﬀerence in the amount of spectral weight\ntransferred out of the c+m continuum at k= (0,0) vs.\nk= (π,π). This is expected for the Heisenberg model\nwhere the location of all features changes with k, but\nmay come as a surprise for the Ising model where their\nlocation is independent of k. However, for both models\nE↑(k), where most of the weight is found, moves from\nthe lower edge of the c+m continuum when k= 0, to the\nupper edge for k= (π,π). As a result, it is reasonable\nthat weight is transferred into the low-energy sp1 con-\ntinuum more eﬃciently at k= (0,0) than at k= (π,π),\nsince in the former case the “eﬀective” energy diﬀerence0.000.050.100.15A↑(0,0,ω) βt=2\n0.000.050.10A↑(0,0,ω)βt=8\n0.000.050.10A↑(0,0,ω)βt=14\n-8-7-6-5\nω/t0.000.050.10A↑(0,0,ω)βt=20\nω/t0.000.010.02βt=80.000.010.02βt=60.000.010.02βt=40.000.010.020.03\nβt=2Heisenberg Ising\nFIG. 9. (color online) Spectral weight A↑(k= 0,ω) for the 2D\nHeisenberg(left)andIsing(right)modelswithAFM J0/t= 7,\nat diﬀerent temperatures. Only the sp1 continuum is shown.\nIts edges are indicated with dot-dashed blue lines Other pa-\nrameters are J/t= 0.5,S= 0.5 andη/t= 0.01 and 0 .02 for\nIsing and Heisenberg, respectively.\nbetween the two features is smaller.\nThe secondobservationis that spectralweightistrans-\nferred into the sp1 continuum more eﬃciently in the\nHeisenberg model than in the Ising model. This diﬀer-\nence is also clearly visible in Fig. 9, where the weight\nin the sp1 continuum of the Heisenberg model is still\nrespectable at βt= 20, while for the Ising model this\nweight is already negligible at βt= 8.\nAn explanation for this diﬀerence comes from assum-\ning that the weight in the sp1 continuum is proportional\nto the average number of thermal magnons, since no sp1\npolaron can appear in their absence. Because the Ising\nmagnonspectrum isgapped, atlow- Tthis numberispro-\nportional to the Boltzmann factor e−βΩ. This suggests\nanintegratedweightin the c+mspectrum of a−be−β4JS,\nwherea= 1−O(η) is the limiting value as T→0. We\nﬁtted the datapoints for βt>5 with this form and found\na very good ﬁt (solid lines), which moreover works well10\n0 2 4 6 8 10\nβt0.60.70.80.91.0∫c+mdω A↑(k,ω)\nIsing k=(0,0)\nIsing k=(π,π)\nHeisenberg k=(0,0)\nHeisenberg k=(π,π)\nFIG. 10. (color online) Integrated spectral weight in the c+ m\ncontinuum as a function of β. Lines are ﬁts described in the\ntext. Parameters are: J0/t= 10,J/t= 0.5,S= 1/2,η/t=\n0.01 (Ising), η/t= 0.05 (Heisenberg).\nfor a larger range of βvalues than used in the ﬁt.\nMagnons of the Heisenberg model are gapless so their\nnumber increases much faster with T. A simple estimate\nfor a 2D unbounded parabolic dispersion suggests /an}bracketle{tn/an}bracketri}ht ∼\nkBT.17The lines shown for the Heisenberg model in Fig.\n10 are ﬁts to a−b/βfor the data points with βt >5.\nThe ﬁt is again reasonable over a wider range, and much\nsuperior to other simple functional forms we tried, such\nasa−b/βn,n >1 ora−be−βcJ(the former assuming\nthat we misindentiﬁed the power law, the second to see if\nIsing-like ﬁts might be more appropriate). Of course, one\ncan ﬁnd excellent ﬁts for all data using more complicated\nfunctions with additional parameters, but they are much\nharder to justify physically than our simple hypothesis\nresulting in an eﬀectively one parameter ﬁt.\nLet us now discuss the discrete peak appearing below\nthe sp1 continuum for both models, for AFM J0. Af-\nter carefully investigating many of its properties, such as\nhow its energy and the region in the Brillouin zone where\nit exists depend on various parameters including T,18we\nbelieve that this is an unphysical artefact of our approx-\nimation. Arguments for this are: (i) the temperature\ndependence of its location, clearly visible in Fig. 9 (note\nthat for the Ising model, the peak only separates below\nthe sp1 continuum at higher T. Atβt= 2 one just starts\ntoseeweightpilingupneartheloweredge, inpreparation\nfor this). According to the Lehmann decomposition, the\nranges where ﬁnite spectral weight is seen cannot vary\nwithT; (ii) the fact that the problem is worse at higher-\nT, where we know that higher order corrections ought to\nbe included in the self-energy; these could easily remove\nan unphysical pole; (iii) the fact that this is a discrete\npeak, not a resonance inside a continuum (this can be\neasily veriﬁed by checking that its lifetime is set by η).\nAccordingtothe Lehmanndecomposition, discretepeaks\ncannot appear in the T/ne}ationslash= 0 spectral weight. Even if the\ncarrier binds all thermal magnons in a coherent quasi--10 -5 0 5 100.000.010.020.03A↑(0,0,0,ω)\nc+msp1\n-15 -10 -5 0 5 10\nω/t0.000.020.040.06A↑(0,0,0,ω)\nc+msp1E↑(k)\nE↑(k)\nFIG. 11. (color online) Spectral weight A↑(k= 0,ω) for the\n3D Heisenberg model at βt= 1 for FM J0/t=−3 (top)\nand AFM J0/t= 10 (bottom) couplings. The edges of the\nc+m continuum (dashed red lines) and sp1/sp2 continuum\n(dot-dashed blue lines) are indicated, as is E↑(0) (thick green\nline). Other parameters are J/t= 0.5,S= 0.5,η= 0.1.\nparticle, the ﬁnite- Tspectral weight would reveal only a\ncontinuum associated with it, as is the case for the sp1\nand sp2 polarons. To summarize, we believe that this\ndiscrete peak is an artefact and that in reality, its weight\nis part of the sp1 continuum from which it came.\nIdeally, these arguments would be strengthened by\na calculation of the next correction to the self-energy,\nto check its eﬀects. We found the exact calculation of\nthe two-magnon term to be daunting even for the Ising\nmodel. The diﬃculty is not so much in evaluating diﬀer-\nent terms, but in tracing over all possible contributions\n– so far we did not ﬁnd a suﬃciently eﬃcient way to do\nthis. One can use approximations to speed things up,\nbut that defeats the purpose since it would not be clear\nif the end results are intrinsic or artefacts, as well. Given\nthis, we cannot entirely rule out that the discrete peak\nis a (precursor pointing to a) real feature, but we believe\nthat to be very unlikely.\nSo far we have done the whole analysis in 2D, sim-\nply because the calculation of Σ( k,ω), especially for the\nHeisenberg model, is numerically much faster. However,\nwe did investigate the 3D models and found essentially\nthe same physics. As examples, in Figs. 11, 12 we show\nspectra for both FM J0/t=−3 and AFM J0/t= 10,\nfor both models. These spectra display exactly the same\nfeaturesasthecorresponding2Dspectra. FortheHeisen-\nberg model we chose a larger η= 0.1 and decreased the\nlinear system size drastically to keep the computational\ntime reasonable. Consequently, the continuum edges are\nmore diﬃcult to discern, while ﬁnite size eﬀects are more\npronounced. In any event, the knowledge accumulated\nfrom analysing the 2D data is fully consistent with all\nfeatures we observed in all 3D data we generated.11\n-10 -5 0 50.000.010.02A↑(0,0,0,ω)\n-15 -10 -5 0 5 10\nω/t0.00.10.2A↑(0,0,0,ω)c+msp1\nsp2\nc+msp1E↑(k)E↑(k)\nFIG. 12. (color online) Spectral weight A↑(k= 0,ω) for the\n3D Ising model at βt= 0.5 for FM J0/t=−3 (top) and\nAFMJ0/t= 10 (bottom) couplings. The edges of the c+m\ncontinuum (dashed red lines) and sp1/sp2 continuum (dot-\ndashed blue lines) are indicated, as is E↑(0) (thick green line).\nOther parameters are J/t= 0.5,S= 0.5,η= 0.01.\nV. CONCLUSIONS\nTo summarize, we calculated analytically the lowest- T\ncorrection to the self-energy of a spin-up carrier injected\nin a FM background. We used both Heisenberg and Ising\ncouplings to describe the background, to understand the\nrelevance of gapped vs. gapless magnons. These results\nshowhowthespectralweightevolvesfromadiscretepeak\natT= 0 to a collection of continua for T/ne}ationslash= 0 (these can\nmerge, in the appropriate circumstances), and explain\ntheir origin and how their locations can be inferred.\nWe were aided in this task by the fact that this model\nconserves the z-component of the total spin, allowing us\nto consider the contribution to the spectral weight com-\ning from Hilbert subspaces with diﬀerent numbers mof\nmagnons when the carrier has spin up. Although we\nfocused on the m= 1, lowest- Tcontribution, based on\nthe knowledge we acquired we can extrapolate with some\nconﬁdence to higher- T, as we discuss now.\nOne deﬁnite conclusion of this work is that knowledge\nof theT= 0 carrier spectrum (in the m= 0 sector)\nE↑(k), and of the magnon dispersion, Ω q, is generally\nnot suﬃcient to predict a priori all features of the ﬁnite-\nTspectral weight, although a fairamount can be inferred\nfrom them. To see why, let us assume that magnons do\nnot interact with one another. (This is not true for either\nmodel, for example due to their hard-core repulsion; we\nwill return to possible consequences of their interactions\nbelow.) If magnons were non-interacting, then Lehmann\ndecomposition of the higher-order contributions in Eq.\n(7) would predict ﬁnite- Tspectral weight for all intervals\n{E(m)\nα(k+/summationtextm\ni=1qi)−/summationtextm\ni=1Ωqi}q1+···+qm,m= 0,1,....\nSince we move from the mto them+ 1 subspace by\naddingamagnon, andgiventhattotalmomentum iscon-served, we know that the spectrum in subspace m+ 1\nnecessarily includes the convolution between the spec-\ntrum of the subspace mand the magnon dispersion, i.e.\n{E(m)\nα(k−q)+Ωq}qis part of the spectrum E(m+1)\nα(k)\n(thesearethescatteringstatesbetweentheextramagnon\nand any eigenstate in the mspectrum).\nThis observationallowsus to infer the location of some\nof the ﬁnite- Tspectral weight, by recurrence. E(1)\nα(k)\nmust include all scattering states {E↑(k−q) + Ωq}q,\nso them= 1 contribution to the spectrum must span\n{E(1)\nα(k+q′)−Ωq′}q′={E↑(k−q+q′)−Ωq′+Ωq}q,q′.\nWe called this the c+m continuum and veriﬁed that it is\nindeed seen in the ﬁnite- Tspectral weight. Knowledge of\nthispartofthe m= 1spectrumallowsus toinferscatter-\ning states that are part of the m= 2 spectrum and there-\nfore their Lehmann contribution, etc. The conclusion is\nthatallintervals {E↑(k+/summationtextm\ni=1q′\ni−/summationtextm\ni=1qi)−/summationtextm\ni=1Ωq′\ni+/summationtextm\ni=1Ωqi}q1,...,q′mwill contain some spectral weight at\nﬁnite-T. For the dispersionless Ising magnons this in-\nterval is the same for all m. For dispersive Heisenberg\nmagnons this interval broadens with m. For very small\nJ, the additional broadening as mincreases is very small\nand moreover one would expect little spectral weight in\nthe high-msectors if the Tis not too large. Thus, we\nexpect weight to be visible in the c+m continuum up\nto high(er) temperatures; its boundaries may also slowly\nexpand with T, for a Heisenberg background, as higher\nmsubspaces become thermally activated.\nApart from these scattering states, E(m+1)\nα(k) might\nalso contain bound states where the extra magnon is\ncoherently bound to all the other particles. The exis-\ntence and location of such coherent states cannot be pre-\ndictedapriori, astheydepend onthe detailsofthemodel\n(however, they certainly cannot appear unless coherent\nstates exist in the mspace). An example is the E(1)\nα(k)\nspectrum which indeed contains the scattering states dis-\ncussed above, but also contains the sp1 and/or sp2 dis-\ncrete polarons states. These give rise to their own con-\ntinua of scattering states in higher msubspaces, whose\nlocations can be inferred by recurrence.\nThe question, then, is if it is likely to ﬁnd such new,\nbound coherent states for all values of m, i.e. if the num-\nber of additional continua becomes arbitrarily large with\nincreasingT. Generally, the answer must be “no”, since\nthis requires bound states between arbitrarily large num-\nbers of objects. For the problem at hand, we believe that\nit is quite unlikely that they appear even in the m= 2\nsubspace, since that would involve one carrier binding\ntwo magnons. This is a diﬃcult task given the weak\nnearest-neighbourattractionoforder Jbetweenmagnons\n(due to the breaking of fewer FM bonds), and the fact\nthat the carrier can interact with only one magnon at a\ntime. The exception is likely to be in 1D systems where\nmagnons can coalesce into magnetic domains.\nLet us now consider the role of magnon interactions.\nBecauseofthem, many-magnonstatesarenoteigenstates\nof the Heisenberg Hamiltonian so higher-order terms are12\nnot obtained by tracing over states with many indepen-\ndent magnons (in the Ising model this complication can\nbe avoided by working in real space). If the attraction\nbetween magnons is too weak to bind them, this is not\nan issue since their spectra will still consist of scattering\nstates spanning the same energieslike for non-interacting\nmagnons. As a result, the location of various features is\nnot aﬀected, but the distribution of the spectral weight\ninside them will be since the eigenfunctions are diﬀerent.\nMagnon pairing is unlikely for d>1 unless the exchange\nis strongly anisotropic. However, if it happens and if the\nspectrum of the magnon pairs is known, one could infer\nits eﬀects on the carrier spectral weight just like above.\nBased on these arguments, we expect the higher- T\nspectral weight to show the same features we uncovered\nat low-T(the distribution of the weight between them\nmight be quite diﬀerent, though). These expectations\ncould be veriﬁed with numerical simulations (conversely,\nour low-Tresults can be used to test codes). Such sim-\nulations would also solve the issue of the discrete peak\nthat we observed for AFM J0, and which we argued to\nbe an artefact of our low- Tapproximation.\nTo conclude, although quantitatively our results are\nonly valid at extremely low- T, we believe that this study\nclariﬁesqualitativelyhowthe spectralweightofaspin-up\ncarrier evolves with T. Our arguments can be straight-\nforwardly extended to predict what features appear in\nthe spectral weight of a spin-down carrier, as well.\nA general feature demonstrated by our work is that\nﬁnite-Tdoes not result in just a simple thermal broaden-\ning of the quasiparticle peak, as it becomes a resonance\ninside a continuum. Spectral weight can also be trans-\nferred to quite diﬀerent energies if the quasiparticle can\nbind additional magnons into coherent polarons. When\nthis happens, interpretation of experimentally measured\nand/or of computationally generated spectra could be-\ncome diﬃcult, unless one is aware of this possibility.\nACKNOWLEDGMENTS\nThis work was supported by NSERC and QMI.\nAppendix A: Derivation of the lowest T/negationslash= 0\nself-energy term\nWe present this calculation for the Heisenberg FM; the\nIsing case is treated similarly. To ﬁnd G(1)\n↑(k,q,q′,ω),\nwe divide ˆH=ˆH0+ˆVwhereˆH0=ˆT+ˆHz\nSandˆV=\nˆHx,y\nS+ˆHexc, and use Dyson’s identity ˆG(ω) =ˆG0(ω) +\nˆG(ω)ˆVˆG0(ω) whereˆG0= [ω−ˆH0+iη]−1is the resolvent\nforˆH0. This procedure is similar to that used in Ref. 10for theT= 0 spin-polaron. Applying Dyson’s identity\nonce we obtain:\nG(1)\n↑(k,q,q′,ω) =G(0)\n↑(k+q′−q,ω+Ωq′−Ωq)[δq,q′\n−J0\n2N/summationdisplay\nQG(1)\n↑(k,Q,q′,ω)+J0/radicalbigg\nS\n2NF(k,q,ω)\n(A1)\nThe ﬁrst term on the right-hand side is just the diago-\nnal term. The second term accounts for the energy shift\nthat occurs when the up-carrier is on the same site as\nthe magnon, and the third term contains a new prop-\nagatorF(k,q′,ω) =/an}bracketle{tΦ(q′)|ck,↑ˆG(ω+ Ωq′)c†\nk+q,↓|FM/an}bracketri}ht.\nThis term accounts for spin-ﬂip processes where the up-\ncarrier absorbs the magnon, turning into a down-carrier\nwith momentum k+q. Using Dyson’s identity again, we\nget an equation of motion for F(k,q′,ω):\nF(k,q′,ω) =J0/radicalbigg\nS\n2NG(0)\n↑(k+q,ω+Ωq′+J0S)\n×/summationdisplay\nQG(1)\n↑(k,Q,q′,ω). (A2)\nThe diagonal element vanishes since the bra and ket are\northogonal. The energy shift −J0S/2 of the spin-down\ncarrier is absorbed into the argument of G(0)\n↑, leaving\nonly the spin-ﬂip process which links Fback toG(1)\n↑.\nThese two coupled equations can now be solved as fol-\nlows. We insert Eq. (A2) into Eq. (A1) to obtain:\nG(1)\n↑(k,q,q′,ω) =G(0)\n↑(k+q′−q,ω+Ωq′−Ωq){δq,q′\n−J0\n2f(k,q′,ω)/bracketleftBig\n1−J0SG(0)\n↑(k+q,ω+Ωq′+J0S)/bracketrightBig/bracerightbigg\n,\n(A3)\nwheref(k,q′,ω) =1\nN/summationtext\nQG(1)\n↑(k,Q,q′,ω). Using Eq.\n(A3) in the deﬁnition of f(k,q′,ω) yields:\nf(k,q′,ω) =1\nNG(0)\n↑(k,ω)/bracketleftbigg\n1+J0\n2g(k,q′,ω)\n×/parenleftBig\n1−J0SG(0)\n↑(k+q′,ω+J0S)/parenrightBig/bracketrightBig−1\n,\nwithg(k,q′,ω) =1\nN/summationtext\nQG(0)\n↑(k+q′−Q,ω+Ωq′−ΩQ).\nNote thatg(k,q′,ω) can be calculated numerically since\nG(0)\n↑(k,ω) is a known function.\nAll that is left to do is to insert the above expression\ninto Eq. (A3) and calculate/summationtext\nqe−βΩqG(1)\n↑(k,q,q,ω), to\nﬁnd the expression listed in Section III.\n1J. Z. Bednorz and K. A. M¨ uller, Z. Phys. B 64, 189 (1986);\nP. A. Lee, N. Nagaosa, and X-G Wen, Rev. Mod. Phys. 78,17 (2006).13\n2A. P. Ramirez, J. Phys: Cond. Matt. 9, 8171 (1997).\n3W. Nolting, G. G. Reddy, A. Ramakanth, and D. Meyer,\nPhys. Rev. B 64, 155109 (2001).\n4X. Liu, T. Berlijn, W.-G. Yin, W. Ku, A. Tsvelik, Y.-J.\nKim, H. Gretarsson, Y. Singh, P. Gegenwart, and J.P. Hill,\nPhys. Rev. B 83, 220403 (2011); Y. Singh, S. Manni, J.\nReuther, T. Berlijn, R. Thomale, W. Ku, S. Trebst, and\nP. Gegenwart, Phys. Rev. Lett. 108, 127203 (2012).\n5H. Ohno, A. Shen, F. Matsukura, A. Oiwa, A. Endo, S.\nKatsumoto and Y. Iye, Appl. Phys. Lett. 69, 363 (1996);\nT. Dietl, Nature Materials 2, 646 (2003).\n6W. H. Lee, R. N. Shelton, S. K. Dhar, and K. A. Gschnei-\ndner, Phys. Rev. B 35, 8523 (1987).\n7S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M.\nDaughton, S. v. Molnr, M. L. Roukes, A. Y. Chtchelka-\nnova, and D. M. Treger, Science 294, 1488 (2001), PMID:\n11711666.\n8B. Lau, M. Berciu and G. A. Sawatzky, Phys. Rev. Lett.\n106, 036401 (2011).\n9B. S. Shastry and D. C. Mattis,Phys. Rev. B 24, 5340 (1981).\n10M. Berciu and G. A. Sawatzky,\nPhys. Rev. B 79, 195116 (2009).\n11S. Henning, P. Herrmann, and W. Nolting,\nPhys. Rev. B 86, 085101 (2012).\n12K. Nakano, R. Eder, and Y. Ohta,\nInt. J. Mod. Phys. B 26, 1250154 (2012).\n13M. M¨ oller, G.A. SawatzkyandM. Berciu, Phys.Rev.Lett.\n108, 216403 (2012); ibid, Phys. Rev. B 86, 075128 (2012).\n14W. Nolting, S. Rex, and S. M. Jaya,\nJ. Phys: Cond. Matt. 9, 1301 (1997).\n15W. Nolting and M. Matlak,\nphysica status solidi (b) 123, 155168 (1984).\n16D. N. Zubarev, Sov. Phys. Usp. 3, 320 (1960).\n17The prefactor contains the Riemann number ζ(1) =∞.\nHowever, this approximation ignores non-parabolic correc -\ntions to the dispersion and lacks the equivalent of a “Debye\nenergy”, instead summing over all energies. Including such\ncorrections should remove the singularity.\n18M. M¨ oller (unpublished)."    },    {        "title": "1105.3236v1.On_the_line_shape_of_the_electrically_detected_ferromagnetic_resonance.pdf",        "content": "arXiv:1105.3236v1  [cond-mat.mtrl-sci]  16 May 2011APS/123-QED\nOn the line shape of the electrically detected ferromagneti c resonance\nM. Harder1, Z. X. Cao1,2, Y. S. Gui1, X. L. Fan1,3, and C.-M. Hu1∗\n1Department of Physics and Astronomy, University of Manitob a, Winnipeg, Canada R3T 2N2\n2National Lab for Infrared Physics, Shanghai Institute of Te chnical Physics,\nChinese Academy of Science, Shanghai 200083, People’s Repu blic of China and\n3The Key Lab for Magnetism and Magnetic Materials of Ministry of Education,\nLanzhou University, Lanzhou 730000, People’s Republic of C hina\n(Dated: September 30, 2018)\nThis work reviews and examines two particular issues relate d with the new technique of electrical\ndetection of ferromagnetic resonance (FMR). This powerful technique has been broadly applied for\nstudying magnetization and spin dynamics over the past few y ears. The ﬁrst issue is the relation\nand distinction between diﬀerent mechanisms that give rise to a photovoltage via FMR in composite\nmagnetic structures, and the second is the proper analysis o f the FMR line shape, which remains the\n”Achilles heel” in interpreting experimental results, esp ecially for either studying the spin pumping\neﬀect or quantifying the spin Hall angles via the electrical ly detected FMR.\nPACS numbers: 85.75.-d, 75.40.Gb, 76.50.+g, 42.65.-k\nI. INTRODUCTION\nElectrical detection of ferromagnetic resonance (FMR)\nin ferromagnets (FM) is a powerful new experimen-\ntal tool which has transformed the research on spin\nand magnetization dynamics.1–33Over the past few\nyears, this technique has generated a great deal of in-\nterest in the communities of magnetism, spintronics,\nand microwave technologies. It has been broadly ap-\nplied for studying diverse material structures, rang-\ning from ferromagnetic thin ﬁlms such as Py (permal-\nloy, Ni 80Fe20),4,6,12,13, CrO 2,14Fe3O4,14single crystal\nFe,17GaMnAs,18and La 1−xSrxMnO3,19bilayer devices\nsuch as Py/Pt,7,8,20,21,25,26Py/Au,20,21Py/GaAs,22and\nY3Fe5O12/Pt,23,24to a variety of magnetic tunneling\njunctions (MTJ) based on magnetic multilayers.3,9–11\nFromatechnicalstandpoint, itshighsensitivityhasmade\nit possible to quantitatively determine spin boundary\nconditions27and to directly measure non-linear magne-\ntization damping28–30, the quasiparticle mass for the do-\nmain wall31, the phase diagram of the the spin-transfer\ndriven dynamics2and various kinds of parametric spin\nwave excitation2,32,33. Its capability to probe the in-\nterplay of spins, charges, and photons has been utilized\nfor studying spin rectiﬁcation12,15, spin pumping7, spin\ntorque16, and spin Hall eﬀects20,25,26, which have led to\nthe proposingandrealizationofnoveldynamicspintronic\ndevices such as the spin battery,7,34,35spin diode,3,10,11\nspin dynamo,12,15and spin demodulator36. Very re-\ncently, its ability to detect coherent processes37–39has\nenabled electrical probing of the spin-resonance phase\nand the relative phase of electromagnetic waves37, which\npave new ways for microwave sensing40, non-destructive\nimaging,37and dielectric spectroscopy38. Such a coher-\n∗Electronic address: hu@physics.umanitoba.ca; URL:\nhttp://www.physics.umanitoba.ca/ ∼huent capability is especially exciting as it resembles the\nlatest achievement in semiconductor spintronics, where a\nnew platform for coherent optical control of spin/charge\ncurrents has been developed by using nonresonant quan-\ntum interferences.41–43\nFrom the physical standpoint, many diﬀerent eﬀects\nmay generate a time-independent dc voltage in magnetic\nmaterials via the FMR. Reported mechanisms involve\nspin rectiﬁcation12,15, spin pumping7, spin torque16,\nspin diode3,10,11, spin Hall25and inverse spin Hall\neﬀects8,20,21,26. Two major issues stand out here: (1)\nA uniﬁed picture clarifying the relations and distinctions\nbetween such diverse mechanisms has not been estab-\nlished, which leads to increasing controversy and con-\nfusion in interpreting and understanding experimental\nresults. A stunning example of this issue is found in\nthe very recent studies of the spin Hall eﬀect via elec-\ntrically detected FMR, where two similar experiments\nperformed on similar devices were interpreted completely\ndiﬀerently.20,25(2) When more than one mechanism si-\nmultaneously plays a role in the FMR generated dc volt-\nage, proper interpretation requires a quantitative anal-\nysis of the FMR line shape. In our opinion, this has\nremained the ”Achilles heel” in recent studies of spin\npumping andthe spinHall eﬀect whichutilize electrically\ndetected FMR. The purpose of this article is to address\nthese two critical issues with a brief review of the key\nphysics of this subject, followed by systematically mea-\nsured experimental data with detailed theoretical analy-\nsis.\nThis paper is split into three main sections. First we\nprovidea brief review ofdiﬀerent mechanisms which may\ngenerate the photovoltage via the FMR. Then we use\nthe dynamic susceptibility obtained from a solution of\nthe Landau-Lifshitz-Gilbert equation to derive analyti-\ncal formulae for analyzing the line shape and the sym-\nmetry properties of the photovoltage generated through\nspin rectiﬁcation. Finally we present experimental re-\nsults measured from diﬀerent samples, at diﬀerent fre-2\nFIG. 1: (color online). Dynamic response of magnetic struc-\ntures under microwave irradiation: (a) Single thin ﬁlm laye r\nwhere the spin rectiﬁcation is due to the magnetic ﬁeld torqu e\nas shown in (e). (b) Magnetic bilayer device which has two\nrf currents jandjswith diﬀerent spin polarizations. There-\nfore spin rectiﬁcation is due to both magnetic ﬁeld and spin\ntorques. (c) Magnetic tunneling junction with both jandjs.\n(d) Coordinate system for single ferromagnetic microstrip s\nmeasured in this work under an in-plane applied static mag-\nnetic ﬁeld H. Thez′-axis is ﬁxed along the strip and the\ndirection of current ﬂow, while the z-axis is rotated to follow\nthe direction of H. (e) Components of magnetic ﬁeld torque.\n(f) Spin torque in magnetic tunneling junction.\nquencies, and in diﬀerent experimental conﬁgurations,\nshowing that the FMR line shape is determined by the\nrelative phase of microwaves which is sample and fre-\nquency dependent.\nII. A BRIEF REVIEW OF ELECTRICAL\nDETECTION OF FMR\nUnder microwave excitation at angular frequency ω,\nthe rf electric ( e) and magnetic ( h) ﬁelds inside a fer-\nromagnetic material can be described as e=e0e−iωt\nandh=h0e−i(ωt−Φ), respectively. Note that in general,\ndue to the inevitable losses of microwaves propagating\ninside the ferromagnetic material, there is a phase dif-\nference Φ between the dynamic eandhﬁelds. Such a\nrelative phase is determined by the frequency-dependent\nwave impedance of the materials44. As shown in Fig.\n1, the rf eﬁeld drives a rf current j=σe, while the rf\nhﬁeld exerts a ﬁeld torque on the magnetization anddrives it to precess around its equilibrium direction [Fig.\n1(e)]. Such a magnetization precession is described by\nthe non-equilibrium magnetization m= ˆχh. Hereσand\nˆχare the high-frequency conductivity and Polder ten-\nsor, respectively. Note that due to the resonance na-\nture of the precession, mlagshby a spin resonance\nphase Θ. However, despite the phase of Φ and Θ, the\ndynamic jandmkeep the coherence of their respec-\ntive driving ﬁelds, so that the product of any combi-\nnation of their components may generate a time inde-\npendent signal proportional to ∝angbracketleftRe(˜j)·Re(˜m)∝angbracketright, where∝angbracketleft∝angbracketright\ndenotes the time average. The amplitude of such a sig-\nnal depends on the phase diﬀerence of jandm, which\ncan be easily understood from the trigonometric rela-\ntion: cos( ωt)·cos(ωt−Φ) = [cos(Φ) +cos(2 ωt−Φ)]/2.\nThis is the spin rectiﬁcation12as we highlight in Table\nI. For transport measurements on magnetic structures\nunder microwave irradiation, various magnetoresistance\neﬀects such as anisotropic magnetoresistance (AMR), gi-\nant magnetoresistance(GMR) and tunneling magnetore-\nsistance (TMR) make corrections to Ohm’s law via their\ncorresponding magnetoresistance terms15,45. Such non-\nlinear terms typically lead to the product of jandm.\nSpin rectiﬁcations induced by such magnetoresistance ef-\nfects are listed in Table I by the terms labeled VMR. The\ngeneral feature of VMRis that its amplitude depends on\nboth the relative phase Φ and the spin resonance phase\nΘ, which leads to a characteristic phase signature of the\nFMR line shape37,38.\nSimilar to the eﬀect of the rf hﬁeld torque, a spin\ntorqueinduced by aspin polarizedcurrentmayalsodrive\nmagnetization precession. For example, in a bilayer [Fig.\n1(b)] made of a ferromagnetic layer and a nonmagnetic\nlayerwith spin-orbit coupling25, in addition to the rf cur-\nrentjﬂowingin theferromagneticlayer,the rf eﬁeldalso\ninduces a rf charge current ﬂowing in the nonmagnetic\nlayer. Via the spin Hall eﬀect in such a nonmagnetic\nlayer with spin-orbit coupling, the rf charge current can\nbe converted into a spin current js, which may ﬂow into\nthe ferromagnetic layer and then drive the magnetiza-\ntion precession via the spin torque. Such a spin torque\ninduced non-equilibrium magnetization can be described\nbym= ˆχjjs, where the spin-torque susceptibility tensor\nˆχjintroduces a spin resonance phase ϑthat is diﬀer-\nent from Θ in ˆ χ. Following a similar consideration for\nthe magnetoresistance induced spin rectiﬁcation, a pho-\ntovoltage depending on the spin Hall eﬀect may be gen-\nerated in the ferromagnetic layer. This is the physical\norigin of the spin Hall induced spin rectiﬁcation eﬀect,25\nwhich is listed in Table I by the term labeled VSH. In\nMTJ [Fig. 1(c)], the spin polarized current jscan be\ndirectly generated in the ferromagnetic layer where the\nmagnetization is pinned along a diﬀerent direction than\nthat of the free layer. It tunnels into the free layer and\ndrives the magnetization precession via the spin torque\n[Fig. 1(f)]. The induced spin rectiﬁcationsignalhas been\nmeasured in spin diodes3,10,11, which is listed in Table I\nby the term labeled VSD.3\nOver the past few years, systematic studies on spin\nrectiﬁcations induced by the ﬁeld ( VMR) and spin torque\n(VSH,VSD) have been performed, respectively, at the\nUniversity of Manitoba12,15,17,18,27–29,37,38,40and Cornell\nUniversity2,9,11,16,25,48. It has been found that due to\nthe coherent nature of spin rectiﬁcation, VMR,VSHand\nVSDall depend on the phase diﬀerence between jand\nm. However, only the ﬁeld torque spin rectiﬁcation\n(VMR) can be controlled by the relative phase Φ of the\nmicrowaves.37\nIn addition to such coherent spin rectiﬁcation eﬀects,\nit is known that at the interface between a ferromag-\nnetic and a nonmagneticlayer,microwaveexcitationmay\ngenerate a spin polarized current ﬂowing across the in-\nterface via the spin pumping eﬀect34. This eﬀect has\nbeen observed in a few striking experiments by measur-\ning either transmission electron spin resonance46or en-\nhanced magnetization damping47. It involves FMR, ex-\nchange coupling and non-equilibrium spin diﬀusion. In\nour opinion the physical picture of spin pumping wasbest explained in the classicalpaper ofSilsbee et.al.[Ref.\n46], which highlighted the key mechanism of dynamic\nexchange coupling between the precessing magnetization\nand the spin polarized current. Such a dynamic cou-\npling signiﬁcantly ”ampliﬁes” the eﬀect of the rf hﬁeld\nin generating non-equilibrium spins. It was later pro-\nposed that the spin current generated via spin pumping\nmay also induce a photovoltage, either across the inter-\nface in a spin battery7,34,35, or within the nonmagnetic\nlayer via the inverse spin Hall eﬀect8,20,21,26. Recent ex-\nperiments performed on magnetic bilayers25have found\nthat spin-pumping induced dc voltage (the term VSPin\nTableI) shouldbe about twoordersofmagnitude smaller\nthan spin Hall induced spin rectiﬁcation (the term la-\nbeledVSH). In contrast to phase sensitive coherent spin\nrectiﬁcation eﬀects, the proposed spin-pumping photo-\nvoltage is based on incoherent spin diﬀusion and FMR\nabsorption. Hence, the anticipated FMR line shape is\nsymmetric and phase-independent.\nTABLE I: Relation and distinctions between diﬀerent mech-\nanisms for microwave photovoltages induced by FMR. (For\nsimplicity we consider only one matrix element of ˆ χand ˆχj\nwhich is responsible for the spin rectiﬁcation. ˜jand ˜mde-\nnote a corresponding component of the time-dependent cur-\nrent and magnetization, respectively.)\nac driving ˜ e=e0e−iωt˜j=j0e−iωt˜h=h0e−i(ωt−Φ)˜js=jSe−iωt\nEﬀect Ohm’s law spin Hall ﬁeld torque spin torque spin rectiﬁ cation spin pumping\ndc voltage V∼ /angbracketleftRe(˜j)·Re(˜m)/angbracketright V∼ |˜m|2\nThin ﬁlm ˜j=σ˜e ˜m=χeiΘ˜h V =VMR·(e0h0)\nBilayer ˜j=σ˜e ˜jS ˜m=χeiΘ˜h+χjeiϑ˜jSV=VMR·(e0h0)+VSH·(j0jS) +VSP·|m|2\nMTJ ˜j,˜jS ˜m=χeiΘ˜h+χjeiϑ˜jSV=VMR·(e0h0)+VSD·(j0jS)\nVMR: Spin Rectiﬁcation caused by MagnetoResistances ;12,13\nVSH: Spin Rectiﬁcation caused by Spin Hall eﬀect;25\nVSD: Spin Rectiﬁcation caused by Spin Diode eﬀect;3,10,11\nVSP: Photovoltage caused by Spin Pumping .7,8,20,21,26\nFrom the above discussion, it is clear that the line\nshape analysis plays the essential role in distinguishing\nthe microwavephotovoltagegenerated by diﬀerent mech-\nanisms. This issue has been partially addressed by a\nnumber of theoretical48,49and experimental works3,10,11studying nanostructured MTJs where the photovoltage\nis dominated by the spin torque induced spin rectiﬁca-\ntion. Enlightened by these works and also based on our\nown previous studies15,37, we discuss in the following the\ncritical issue of FMR line shape analysis in microstruc-4\ntured devices, where the ﬁeld and spin torque induced\nspin rectiﬁcation may have comparable strength. Our\ntheoretical consideration and experimental data demon-\nstrate the pivotal role of the relative phase Φ, which\nwas often under-estimated in previous studies. Via sys-\ntematic studies with diﬀerent device structures, mea-\nsurement conﬁgurations and frequency ranges, we ﬁnd\nthat Φ has to be calibrated at diﬀerent microwave fre-\nquencies for each device independently. Hence, our re-\nsults are in strong contradiction with the recent exper-\niment performed on microstructured magnetic bilayers\nfor quantifying the spin Hall angles, where Φ was de-\nclared to be zero for all devices at diﬀerent microwave\nfrequencies20,21.\nIII. FMR LINE SHAPE\nA. The Characteristic Signature\nFrom Table I, the role of the phase in the FMR line\nshape symmetry can be understood by considering the\nspin rectiﬁed voltage V∝ ∝angbracketleftRe(˜j)·Re(˜m)∝angbracketright. For spin\nrectiﬁcation induced by the ﬁeld torque, depending on\nthe experimental conﬁguration, at least one matrix com-\nponentχof the Polder tensor /hatwideχwill drive the FMR;\nwhether an on or oﬀ-diagonal component is responsi-\nble for the magnetization precession depends on the\nmeasurement conﬁguration. Since m=/hatwideχh,Re(˜m)∝\nRe(χ)cos(ωt−Φ) +Im(χ)sin(ωt−Φ). Therefore after\ntime averaging a time independent dc voltage is found\nV(Φ)∝[Re(χ)cos(Φ)−Im(χ)sin(Φ)]. It is well known\nthat for diagonal matrix elements, Re(χ) has a disper-\nsive line shape while Im(χ) has a symmetric line shape.\nHoweversince the on and oﬀ-diagonalsusceptibilities dif-\nfer by a phase of π/2, if the FMR is driven by an oﬀ-\ndiagonal susceptibility, the roles are reversed and Re(χ)\nhas a symmetric line shape while Im(χ) has a dispersive\nline shape.\nBased on the simple argument leading to the above\nV(Φ) expression, one can see that the line shape symme-\ntry has a characteristic dependence on the relative phase\nΦ between electric and magnetic ﬁelds. Thus when mea-\nsuring FMR based on the ﬁeld torque induced spin rec-\ntiﬁcation eﬀect, it is important to consider the relative\nphase, whereas for a spin pumping measurement which\nmeasures |m|2, or for a spin torque induced spin recti-\nﬁcation which involves |j|2, the relative phase does not\ninﬂuence the experiment. In the next two sections, a\ndetailed analysis is given by solving the Landau-Lifshitz-\nGilbert equation, which leads to analytical formulae de-\nscribing the symmetric and dispersive line shapes for dif-\nferent measurement conﬁgurations.B. The Dynamic Susceptibility\nThe Landau-Lifshitz-Gilbert equation provides a phe-\nnomenological description of ferromagnetic dynamics\nbased on a torque provided by the internal magnetic ﬁeld\nHiwhich acts on the magnetization M, causing it to\nprecess50\ndM\ndt=−γ(M×Hi)+α\nM/parenleftbigg\nM×dM\ndt/parenrightbigg\n.(1)\nHereγis the eﬀective electron gyromagnetic ratio and\nαis the Gilbert damping parameter which can be used\nto determine the FMR line width ∆ H, according to\n∆H∼αω/γ. For the case of microwave induced fer-\nromagnetic resonance Eq. (1) can be solved by splitting\nthe internal ﬁeld into dc and rf components and tak-\ning the applied dc ﬁeld H, along the z-axis. We can\nrelate the internal ﬁeld Hi=H0i+hie−iωt, to the\napplied ﬁeld through the demagnetization factors Nk,\nH0iz=H−NzM0,hik=hkeiΦk−Nkmk, where Φ kis\nthe relativephaseshift between the electric and magnetic\nﬁelds in the kthdirection and M0is the dc magnetization\nalso along the z-axis. With the magnetization separated\ninto dc and rf contributions M=M0+me−iωt, the so-\nlution of Eq. (1) yields the dynamic susceptibility tensor\n/hatwideχwhich relates the magnetization mto the externally\napplied rf ﬁeld h\nm=/hatwideχh=\nχxxiχxy0\n−iχxyχyy0\n0 0 0\nh\n=\n|χxx| |χxy|eiπ\n20\n|χxy|e−iπ\n2|χyy|0\n0 0 0\nheiΘ,(2)\nwhere Θ = arctan[∆ H/(H−Hr)] is the spin resonance\nphase37which describes the phase shift between the re-\nsponse and the driving force in terms of the line width\n∆Hand the resonance ﬁeld Hrwhich are constant for a\nﬁxed frequency. Θ will change from 180◦(driving force\nout of phase) to 0◦(driving force in phase) around the\nresonance position, in a range on the order of ∆ H, pass-\ning through 90◦at resonance. This represents the uni-\nversal feature of a resonance; the phase of the dynamic\nresponse always lags behind the driving force.51\nTo emphasize the resonant feature of the susceptibil-\nity tensor elements we deﬁne the symmetric Lorentz line\nshapeL, and the dispersive line shape Das\nL=∆H2\n(H−Hr)2+∆H2,\nD=∆H(H−Hr)\n(H−Hr)2+∆H2. (3)\nClearly the spin resonance phase can also be written\nin terms of LandDas Θ = arctan[∆ H/(H−Hr)] =5\narctan(L/D)sothat L∝sin(Θ)and D∝cos(Θ). There-\nforeLandDcarry the resonant information of the sus-\nceptibility tensor.\nUsingLandDallows the elements of /hatwideχto be written\nas (χxx,χxy,χyy) = (D+iL)(Axx,Axy,Ayy).Axx,Axy\nandAyyare real amplitudes which are related to the\nsample properties\nAxx=γM0(M0Ny+(H−NzM0))\nαω(2(H−NzM0)+M0(Nz+Ny)),\nAxy=−M0\nα(2(H−NzM0)+M0(Nz+Ny)),\nAyy=γM0(M0Nx+(H−NzM0))\nαω(2(H−NzM0)+M0(Nz+Ny)).(4)\nSince these amplitudes are real all components of /hatwideχ\ninclude both a dispersive and a Lorentz line shape deter-\nmined solely from the D+iLterm. However, in a trans-\nmission experiment performed using a resonance cavity\n|m|2∝L2+D2=Lis measured. This product removes\nthe phase dependence carried by LandDand leaves\nonly the Lorentz line shape. For the same reason, the\nmicrowave photovoltage induced by spin pumping (the\nVSPterm in Table I) has a symmetric line shape.\nThe susceptibility for the two cases of in-plane and\nperpendicularly applied dc magnetic ﬁelds can easily be\nfound from Eq. (4) by using the appropriate demagne-\ntization factors. When the lateral dimensions are much\nlarger than the thickness, N x= Nz= 0 and N y= 1\nfor an in-plane ﬁeld and N x= Ny= 0 and N z= 1 for\na ﬁeld applied at a small angle from the perpendicular.\nIn this paper, we focus on the in-plane case. The line\nshape analysis for the perpendicular case can be found\nin Ref. 37. In both cases the form of the susceptibility,\nχ∝D+iL, describes the ferromagnetic resonance line\nshape where each element of /hatwideχis the sum of an antisym-\nmetric and symmetric Lorentzline shape. As we describe\nin the next section, via the VMRterm of the spin recti-\nﬁcation eﬀect, the symmetry properties of the dynamic\nsusceptibility inﬂuence the symmetry of the electrically\ndetected FMR which can be controlled by tuning the rel-\native electromagnetic phase Φ.\nC. Spin Rectiﬁcation Induced by the Field Torque\nThe ﬁeld-torque spin rectiﬁcation eﬀect results in the\nproduction of a dc voltage from the non-linear coupling\nof rf electric and magnetic ﬁelds. For example, it may\nfollow from the generalized Ohm’s law45,52\nJ=σE0−σ∆ρ\nM2(J·M)M+σRHJ×M,(5)\nwhereσis the conductivity, ∆ ρis the resistivity change\nduetoAMR and RHisthe extraordinaryHall coeﬃcient.\nAs shown in Fig. 2, we use two coordinate systems\nto describe a long narrow strip under the rotating in-\nplane magnetic ﬁeld H. The sample coordinate system/s45/s48/s46/s56/s45/s48/s46/s52/s48/s46/s48/s48/s46/s52/s48/s46/s56/s80/s86/s32/s40/s97/s46/s32/s117/s46/s41\n/s51/s54/s48 /s50/s55/s48 /s49/s56/s48/s57/s48/s48\n/s113/s72/s32/s40/s100/s101/s103/s114/s101/s101/s41/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s80/s86/s32/s40/s97/s46/s32/s117/s46/s41\n/s51/s54/s48 /s50/s55/s48 /s49/s56/s48/s57/s48/s48\n/s113/s72/s32/s40/s100/s101/s103/s114/s101/s101/s41/s48/s46/s56\n/s48/s46/s52\n/s48/s46/s48/s80/s86/s32/s40/s97/s46/s32/s117/s46/s41\n/s45/s49/s48/s48 /s48 /s49/s48/s48\n/s72/s32/s40/s97/s46/s32/s117/s46/s41/s45/s48/s46/s52/s48/s46/s48/s48/s46/s52/s80/s86/s32/s40/s97/s46/s117/s46/s41\n/s45/s49/s48/s48 /s48 /s49/s48/s48\n/s72/s32/s40/s97/s46/s32/s117/s46/s41/s40/s98/s41 /s40/s101/s41\n/s40/s99/s41 /s40/s102/s41\n/s40/s97/s41 /s40/s100/s41\nFIG. 2: (color online). Left panel (a) Coordinate system for\nan in-plane dc Hﬁeld applied along the z-axis at an angle θH\nwith respect to the z′-axis, with a rf h-ﬁeld along the x′-axis.\n(b) The calculated photovoltage (PV) spectrum at θH= 45◦\nand (c) the calculated amplitude of the PV spectrum at FMR\nas a function of θHaccording to Eq. (9). Right Panel (d)-\n(f) are the same as (a)-(c), respectively, but with a rf h-ﬁeld\nalong the y-axis, and calculations are according to Eq. (10).\nIn both cases, Φ is assumed to be zero.\n(/hatwidex′,/hatwidey,/hatwidez′) is ﬁxed with the sample length along the z′\ndirection and the sample width in the x′direction. The\nmeasurementcoordinatesystem ( /hatwidex,/hatwidey,/hatwidez) rotateswith the\nHdirection which is along the /hatwidezaxis. We deﬁne θHas\nthe angle between the direction of the strip and the in-\nplane applied static magnetic ﬁeld ( i.e., between the z′\nandzdirections). In both coordinate systems, the /hatwideyaxis\nis along the normal of the sample plane. In the case of a\nsample length much largerthan the width, the rf current,\n˜j=jz′e−iωtﬂows along the strip direction z′. In this\ngeometry the ﬁeld due to the Hall eﬀect will only be in\nthe transverse direction and will not generate a voltage\nalong the strip. Taking the time average of the electric\nﬁeld integrated along the z′direction, the photovoltage\nis found as12,15\nV=∆R\nM0∝angbracketleftRe(˜j)·Re(˜mx)∝angbracketrightsin(2θH),(6)\nwhere ∆Ris the resistancechange due to the AMR eﬀect\nandthe sin(2 θH) term isa resultofthe AMR eﬀect which\ncouplesJandM.\nThe susceptibility tensor given by Eqs. (2) and (4)\ncan be used to write ˜ mxin terms of the rf hﬁeld. Since\nM0andHare both along the z-axis, only the compo-\nnents ofhperpendicular to zwill contribute to m. How-\never, since the rf current ﬂows in the z′direction, to\ncalculate the rectiﬁed voltage, ˜ mxmust be transformed6\ninto the ( x′,y,z′) coordinate system by using the rota-\ntion (/hatwidex,/hatwidey,/hatwidez) = (cos( θH)/hatwidex′−sin(θH)/hatwidez′,/hatwidey,sin(θH)/hatwidex′+\ncos(θH)/hatwidez′), which introduces an additional θHdepen-\ndence into the photovoltage. We ﬁnd that the photo-\nvoltage can be written in terms of the symmetric and\nantisymmetric Lorentz line shapes, LandD, as\nV=∆R\n2M0jz′(ALL+ADD), (7)\nwhere\nAL= sin(2θH)[−Axxhx′cos(θH)sin(Φ x′)\n−Axyhycos(Φy)+Axxhz′sin(θH)sin(Φ z′)],\nAD= sin(2θH)[Axxhx′cos(θH)cos(Φ x′)\n−Axyhysin(Φy)−Axxhz′sin(θH)cos(Φ z′)],\n(8)\nand Φx′,Φyand Φz′are the relative phases between elec-\ntric and magnetic ﬁelds in the x′,yandz′directions,\nrespectively.\nThe amplitudes of the Lorentz and dispersive line\nshape contributions show a complex dependence on the\nrelative phases for the x′,yandz′directions and in gen-\neralbothlineshapeswillbepresent. However,depending\non the experimental conditions, this dependence may be\nsimpliﬁed. For instance when hx′is the dominate driving\nﬁeld as shown in Fig. 2(a), we may take hy=hz′≈0\nand Φ x′= Φ, which results in\nV=−∆R\n2M0jz′Axxhx′cos(θH)sin(2θH)\n[Lsin(Φ)−Dcos(Φ)].(9)\nFrom Eq. (9) we see that the photovoltage line shape\nchanges from purely symmetric to purely antisymmetric\nin 90◦intervals of Φ, being purely antisymmetric when\nΦ =n×180◦and purely symmetric when Φ = (2 n+1)×\n90◦, n= 0,±1,±2....\nAs shown in Fig. 2(b) and (c), the photovoltage in\nEq. (9) also shows symmetries depending on the static\nﬁeld direction θH. SinceH→-Hcorresponds to θH→\nθH+ 180◦,V(H) =−V(−H). Furthermore at θH=\nn×90◦, n= 0,±1,±2...the voltage will be zero.\nSimilarly when hydominates as shown in Fig. 2(c), we\ntakehx′=hz′≈0 and Φ y= Φ which results in a voltage\nV=−∆R\n2M0jz′Axyhysin(2θH)\n[Lcos(Φ)+ Dsin(Φ)].(10)\nThe symmetry properties are now such that the line\nshape is purely symmetric when Φ = n×180◦and\npurely antisymmetric when Φ = (2 n+ 1)×90◦, n=\n0,±1,±2.... Also the photovoltage determined by\nEq. (10) is now symmetric with respect to Hunder\nθH→θH+180◦sothatV(H) =V(−H) asshownin Fig.\n2(e). Therefore, experimentally the diﬀerent symmetryof the FMR at Hand−Hcan be used as an indication\nof which component of the hﬁeld is dominant.\nBoth Eq. (9) and Eq. (10) demonstrate that a change\nin the relativeelectromagneticphase is expected toresult\nin a change in the line shape of the electrically detected\nFMR. It is worth noting that when the relative phase\nΦ = 0, the line shape is purely antisymmetric for FMR\ndriven by hx′and purely symmetric for FMR driven by\nhyas illustrated in Fig. 2(b) and 2(e), respectively. In\nthe general case when ˜ mxis driven by multiple hcompo-\nnents, Eq. (7) must be used in combination with angular\n(θH) dependent measurementsin ordertodistinguish dif-\nferent contributions.\nD. The Physics of Φ\nIt is clear therefore that for ﬁeld torque induced spin\nrectiﬁcation, the relative phase Φ between the microwave\nelectric and magnetic ﬁelds plays the pivotal role in the\nFMR line shape. Note that Φ is a materialand frequency\ndependent property which is related to the losses in the\nsystem.44,54,55When a plane electromagnetic wave prop-\nagates through free space the electric and magnetic ﬁelds\nare in phase and orthogonal to each other.53However\nwhen the same electromagnetic wave travels through a\ndispersive medium where the wave vector is complex, the\nimaginary contribution can create a phase shift between\nelectric and magnetic ﬁelds. The most well known ex-\nample is that of a plane electromagnetic wave moving in\na conductor44where Faraday’s law gives a simple rela-\ntion between electric and magnetic ﬁelds, ωµh=k×e.\nTherefore the complex part of the wave vector kwill in-\nduce a phase shift between electric and magnetic ﬁelds.\nAlthough the ﬁeld will exponentially decay inside a con-\nductor, it will still penetrate a distance on the order of\nthe skin depth, and in a perfect conductor the conduc-\ntivity, which produces an imaginary dielectric constant,\nwill result in a phase shift of 45◦between the electric and\nmagnetic ﬁelds.44\nIn a complex system such as an experimental set up in-\nvolving waveguides, coaxial cables, bonding wires and a\nsample holder, which are required for electrical FMR de-\ntection, the relative phase cannot be simply calculated.\nNevertheless losses in the system which can be charac-\nterized in a variety of ways, such as through the wave\nimpedance,54,55will lead to a phase shift between elec-\ntric and magnetic ﬁelds which will inﬂuence the FMR\nline shape.\nAlthough the physics of Φ is in principle contained in\nMaxwell’s equations, due to the lack of technical tools\nfor simultaneously and coherently probing both eandh\nﬁelds, the eﬀect of the relative phase had often been ig-\nnored until the recent development of spintronic Michel-\nson interferometry37. In the following we provide sys-\ntematically measured data showing the inﬂuence of the\nrelativephaseΦontheline shapeofFMR whichisdriven\nby diﬀerent hﬁeld components.7\nIV. EXPERIMENTAL LINE SHAPE\nMEASUREMENTS\nA. h yDominant FMR\nIn order to use the hyﬁeld to drive FMR a ﬁrst gen-\neration spin dynamo was used where a Cu/Cr coplanar\nwaveguide (CPW) was fabricated beside a Py microstrip\nwith dimension 300 µm×20µm×50 nm on a SiO 2/Si\nsubstrate as shown in Fig. 3(a). A microwave current\nis directly injected into the CPW and ﬂows in the z′di-\nrection inducing a current in the Py strip also along the\nz′-axis. In this geometry the dominant rf hﬁeld in the\nPy will be the Oersted ﬁeld in the ydirection produced\naccording to Amp` ere’s Law. This ﬁeld will induce FMR\nprecession with the same cone angle independent of the\nstaticHorientation.\nThe AMR resistance depends on the orientation of the\nmagnetizationrelativetothecurrentandfollowstherela-\ntionR(H) =R(0)−∆Rsin2(θM), where θM(not shown)\nis the angle between the magnetization and the current\ndirection. ForPytheAMReﬀect, whichisresponsiblefor\nthe spin rectiﬁcation, is observed to produce a resistance\nchange of ∆ R/R(0)∼0.4 %. When His applied along\nthex′-axis,i.e., thein-planehardaxis,themagnetization\nMtends to align toward the static ﬁeld Hand the angle\nθMis determined by sin( θM) =H/HAforH < H A,\nwhereHA=Nx′M0is the in-plane shape anisotropy\nﬁeld. The measured data (symbols) shown in Fig. 3(c) is\nﬁt(solidcurve)accordingto R(H) =R(0)−∆Rsin2(θM)\nwithR(0) = 112 .66 Ω, ∆R= 0.47 Ω,µ0HA= 4.0 mT,\nandNx′=0.004.\nFig. 3(d) shows that the line shape at θH= 120◦\nandω/2π= 5 GHz is almost purely dispersive, in-\ndicating that at this frequency Φ ∼90◦according\nto Eq. (10). The θHdependence of Hris shown in\nFig. 3(e) and can be well ﬁt by the function ω=\nγ/radicalBig\n(|Hr|+HAcos(2θH))[|Hr|+M0−HA(1+sin2(θH)]\nby taking the shape anisotropyﬁeld HAalong the x′-axis\ninto account.56As expected the amplitude of these os-\ncillations is µ0HA= 4.0 mT. The frequency dependence\nofHratθH= 45◦is shown in Fig. 3(f) and is ﬁt using\nω=γ/radicalbig\n|Hr|(|Hr|+M0) withγ/2π= 29.0µ0GHz/T\nandµ0M=1.0 T.\nBy systematically measuring the line shape as a func-\ntion of the microwave frequency, we observe the interest-\ning results of Fig. 4. The FMR line shape is observed\nto change from almost purely dispersive at ω/2π= 5\nGHz to almost purely symmetric at ω/2π= 5.56 GHz.\nAs discussed before, the line shape may be aﬀected by\nthehorientation, i.e., diﬀerent hvector components will\naﬀect the line shape diﬀerently. Hence, if changing the\nmicrowave frequency changes the dominant driving ﬁeld,\nthe line shape may change. To rule out such a possibil-\nity an angular dependent experiment was performed to\nmeasure the line shape at diﬀerent θHfor each frequency\nω. The results are plotted on the right panel of Fig. 4/s49/s49/s50/s46/s54\n/s49/s49/s50/s46/s53\n/s49/s49/s50/s46/s52\n/s49/s49/s50/s46/s51\n/s49/s49/s50/s46/s50/s82/s32/s40/s87/s41\n/s45/s49/s48 /s48 /s49/s48\n/s109/s48/s72/s32/s40/s109/s84/s41/s72\n/s48/s46/s56\n/s48/s46/s52\n/s48/s46/s48/s80/s86/s32/s40/s109/s86/s41\n/s45/s54/s48/s45/s52/s48/s45/s50/s48 /s48/s50/s48/s52/s48/s54/s48\n/s109/s48/s72/s32/s40/s109/s84/s41/s72 /s49/s50/s48/s111\n/s51/s52\n/s51/s50\n/s51/s48\n/s50/s56/s109/s48/s72/s114/s32/s40/s109/s84/s41\n/s51/s54/s48 /s50/s55/s48 /s49/s56/s48 /s57/s48 /s48\n/s113/s72/s32/s40/s100/s101/s103/s114/s101/s101/s41\n/s49/s50\n/s56\n/s52\n/s48/s119/s47/s50/s112 /s32/s40/s71/s72/s122/s41\n/s45/s50/s48/s48 /s45/s49/s48/s48 /s48 /s49/s48/s48 /s50/s48/s48\n/s109/s48/s72/s114/s32/s40/s109/s84/s41/s72 /s52/s53/s111/s40/s99/s41\n/s40/s100/s41\n/s40/s101/s41\n/s40/s102/s41/s40/s97/s41\n/s40/s98/s41\nFIG. 3: (color online). (a) Schematic diagram of the ﬁrst\ngeneration spin dynamo where the Py strip is located be-\nside the CPW. The dominate magnetic ﬁeld in the Py is\nthe Oersted ﬁeld in the ydirection due to the current in\nthe CPW. (b) Micrograph of the device. (c) Magnetoresis-\ntance at θH= 90◦. AMR is seen to be ∼0.4%. Arrows\ndenote the anisotropic ﬁeld, µ0HA= 4.0 mT. Open circles\nare experimental data and solid curve is the ﬁtting result\nusingR(0) = 112 .66 Ω,∆R= 0.47 Ω,HA= 4.0 mT. (d)\nElectrically detected FMR at θH= 120◦andω/2π= 5 GHz\nshowing an almost purely dispersive line shape (Φ ≃90◦).\nFit is according to Eq. (10) with µ0∆H= 3.6 mT, µ0Hr\n= 32.2 mT. (e) Oscillating Hrdependence on the static ﬁeld\ndirection θHwith amplitude 2 HA. (f) Dependence of FMR\nfrequency on the resonant ﬁeld HratθH= 45◦. Open circles\nare experimental data and the solid line is the ﬁt according\ntoω=γ/radicalbig\n|Hr|(|Hr|+M0).\nwhich shows the sinusoidal curves for the Lorentz, AL,\nand dispersive, AD, amplitudes (dashed and solid curves\nrespectively) as a function of the static ﬁeld angle θH.\nBoth the Lorentz and dispersive amplitudes are found to\nfollow a sin(2 θH) dependence on the ﬁeld angle in agree-\nment with Eq. (10) indicating that the magnetization\nprecession is indeed dominantly driven by the hyﬁeld.\nTherefore the line shape change indicates that the rela-\ntive phase Φ is frequency dependent. As shown in Fig.\n5(a), atω/2π= 5 GHz the amplitude of ADis approx-\nimately one order of magnitude larger than AL, while\natω/2π= 5.56 GHz ADis one order of magnitude less\nthanAL. Such a large change in AL/ADshows that in\na microwave frequency range as narrow as 0.6 GHz, the8\n/s45/s49/s46/s48/s48/s46/s48/s49/s46/s48/s80/s86/s32/s40/s109/s86/s41\n/s51/s54/s48 /s50/s55/s48 /s49/s56/s48 /s57/s48 /s48\n/s113/s48/s32/s40/s100/s101/s103/s114/s101/s101/s41/s70/s61/s56/s55/s111/s48/s46/s56\n/s48/s46/s52\n/s48/s46/s48/s80/s86/s32/s40/s109/s86/s41\n/s52/s48 /s48 /s45/s52/s48\n/s109/s48/s72/s32/s40/s109/s84/s41/s53/s46/s48/s71/s72/s122\n/s49/s46/s48\n/s48/s46/s53\n/s48/s46/s48\n/s45/s48/s46/s53\n/s45/s49/s46/s48/s80/s86/s32/s40/s109/s86/s41\n/s51/s54/s48 /s50/s55/s48 /s49/s56/s48 /s57/s48 /s48\n/s113/s72/s32/s40/s100/s101/s103/s114/s101/s101/s41/s70/s61/s55/s56/s111\n/s48/s46/s52\n/s48/s46/s50\n/s48/s46/s48\n/s45/s48/s46/s50\n/s45/s48/s46/s52/s80/s86/s32/s40/s109/s86/s41\n/s45/s52/s48 /s48 /s52/s48\n/s109/s48/s72/s32/s40/s109/s84/s41/s53/s46/s48/s53/s71/s72/s122\n/s45/s50/s45/s49/s48/s49/s50/s80/s86/s32/s40/s109/s86/s41\n/s51/s54/s48 /s50/s55/s48 /s49/s56/s48 /s57/s48 /s48\n/s113/s72/s32/s40/s100/s101/s103/s114/s101/s101/s41/s70/s61/s54/s51/s111\n/s45/s48/s46/s56/s45/s48/s46/s52/s48/s46/s48/s48/s46/s52/s80/s86/s32/s40/s109/s86/s41\n/s52/s48 /s48 /s45/s52/s48\n/s109/s48/s72/s32/s40/s109/s84/s41/s53/s46/s49/s71/s72/s122\n/s45/s50/s45/s49/s48/s49/s50/s80/s86/s32/s40/s109/s86/s41\n/s51/s54/s48 /s50/s55/s48 /s49/s56/s48 /s57/s48 /s48\n/s113/s72/s32/s40/s100/s101/s103/s114/s101/s101/s41/s70/s61/s53/s49/s111\n/s45/s49/s46/s50/s45/s48/s46/s56/s45/s48/s46/s52/s48/s46/s48/s48/s46/s52/s80/s86/s32/s40/s109/s86/s41\n/s52/s48 /s48 /s45/s52/s48\n/s109/s48/s72/s40/s109/s84/s41/s53/s46/s50/s71/s72/s122\n/s45/s51/s45/s50/s45/s49/s48/s49/s50/s80/s86/s32/s40/s109/s86/s41\n/s51/s54/s48 /s50/s55/s48 /s49/s56/s48 /s57/s48 /s48\n/s113/s72/s32/s40/s100/s101/s103/s114/s101/s101/s41/s70/s61/s50/s54/s111\n/s45/s49/s46/s53/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s80/s86/s32/s40/s109/s86/s41\n/s45/s52/s48 /s48 /s52/s48\n/s109/s48/s72/s32/s40/s109/s84/s41/s53/s46/s51/s71/s72/s122\n/s45/s52/s45/s50/s48/s50/s52/s80/s86/s32/s40/s109/s86/s41\n/s51/s54/s48 /s50/s55/s48 /s49/s56/s48 /s57/s48 /s48\n/s113/s72/s32/s40/s100/s101/s103/s114/s101/s101/s41/s70/s61/s49/s52/s111\n/s45/s51/s45/s50/s45/s49/s48/s80/s86/s32/s40/s109/s86/s41\n/s45/s52/s48 /s48 /s52/s48\n/s109/s48/s72/s32/s40/s109/s84/s41/s53/s46/s53/s71/s72/s122\n/s45/s52/s45/s50/s48/s50/s52/s80/s86/s32/s40/s109/s86/s41\n/s51/s54/s48 /s50/s55/s48 /s49/s56/s48 /s57/s48 /s48\n/s113/s72/s32/s40/s100/s101/s103/s114/s101/s101/s41/s70/s61/s45/s57/s111\n/s45/s51/s45/s50/s45/s49/s48/s80/s86/s32/s40/s109/s86/s41\n/s45/s52/s48 /s48 /s52/s48\n/s109/s48/s72/s32/s40/s109/s84/s41/s53/s46/s53/s54/s71/s72/s122/s40/s97/s41 /s40/s98/s41\n/s40/s99/s41 /s40/s100/s41\n/s40/s101/s41 /s40/s102/s41\n/s40/s103/s41 /s40/s104/s41\n/s40/s105/s41 /s40/s106/s41\n/s40/s108/s41 /s40/s107/s41\n/s40/s109/s41 /s40/s110/s41\nFIG. 4: (color online). Data shown for a ﬁrst generation spin\ndynamo. FMR spectra at θH= 120◦for several frequencies\nfrom 5.0 to 5.56 GHz with corresponding Lorentz and dis-\npersive amplitudes as a function of θH. Circles and squares\nindicate the Lorentz and dispersive amplitudes of Eq. (10) r e-\nspectively andshow asin(2 θH) dependenceas expected. Solid\nand dashed curves are sin(2 θH) functions.\nrelative phase Φ can change by 90◦. Fig. 5(b) shows Φ\ndetermined by using Eq. (10), which smoothly changes\nwith microwave frequency except for a feature near 5.18\nGHz, which is possibly caused by a resonant waveguide\nmode at this frequency.\nSuch a large change of Φ within a very narrow range\nof microwave frequency indicates the complexity of wave\nphysics. Note that microwaves at ∼5 GHz have wave-\nlengths on the order of a few centimeters which are much\nlarger than the sub-millimeter sample dimensions. Con-\nsequently the microwave propagation depends strongly\nonthe boundaryconditionsofMaxwell’sequationswhich\nphysically include the bonding wire, chip carrier, as well\nas the sample holder. This is similar to the microwave/s48/s46/s48/s49/s48/s46/s49/s49/s49/s48/s124/s65/s68/s47/s65/s76/s124\n/s53/s46/s54 /s53/s46/s53 /s53/s46/s52 /s53/s46/s51 /s53/s46/s50 /s53/s46/s49 /s53/s46/s48\n/s119/s47/s50/s112 /s32/s40/s71/s72/s122/s41/s49/s48/s48\n/s56/s48\n/s54/s48\n/s52/s48\n/s50/s48\n/s48/s70/s32/s40/s100/s101/s103/s114/s101/s101/s41\n/s53/s46/s54 /s53/s46/s53 /s53/s46/s52 /s53/s46/s51 /s53/s46/s50 /s53/s46/s49 /s53/s46/s48\n/s119/s47/s50/s112 /s32/s40/s71/s72/s122/s41/s40/s97/s41\n/s40/s98/s41/s53/s32/s71/s72/s122/s53/s46/s54/s32/s71/s72/s122\nFIG. 5: (color online). (a) The AD/ALratio as a function of\nω/2πshowing the line shape change from dispersive at 5 GHz\n(left inset) to Lorentz at 5.6 GHz (right inset) with a step si ze\nof 0.01 GHz. (b) Φ dependence on ω/2πover same frequency\ninterval showing the same dependence as AD/AL.\npropagation in a waveguide where the ﬁeld distribution\ni.e.the waveguide modes, are known to depend strongly\non boundary conditions and frequency.57Despite the\ncomplexwaveproperties,thekeymessageofourresultsis\nclear and consistent with the consideration of the physics\nof the relative phase: it shows that in order to properly\nanalyze the FMR line shape, Φ has to be determined for\neach frequency independently.\nB. h x′Dominant FMR\nIn order to drive the FMR using the rf ﬁeld in the x′\ndirection, hx′, a second generation spin dynamo was fab-\nricated with the Py strip underneath the CPW as shown\nin Fig. 6. In this case the 300 µm×7µm×100 nm Py\nstrip is underneath the Cu/Cr coplanar waveguide which\nis fabricated on a SiO 2/Si substrate. Again a microwave\ncurrent is directly injected into the CPW and induces a\ncurrent in the z′direction in the Py strip. The dominant\nrf ﬁeld in the Py is still the Oersted ﬁeld, but due to the\nnew geometry it is in the x′direction.\nDue to the smaller width and larger thickness, the de-\nmagnetization factor, Nx′= 0.008 is twice that in the\nﬁrst generation sample. This corresponds to µ0HA=\n8.0 mT as indicated by the broader AMR curve in Fig.\n6(c). This value is further conﬁrmed by the HrvsθH\nplot shown in Fig. 6(e). Fig. 6(f) shows the frequency\ndependence of Hrfor FMR (circles) and for the ﬁrst per-\npendicular standing spin wave resonance (SWR) (trian-\ngles) measured at θH= 45◦. The frequency dependence\nofHrfollowsω=γ/radicalbig\n(|Hr|+Hex)(|Hr|+M0+Hex)\nwhereHexisthe exchangeﬁeld. In Fig. 6(f) the standing9\n/s49/s50/s49/s46/s52\n/s49/s50/s49/s46/s50\n/s49/s50/s49/s46/s48\n/s49/s50/s48/s46/s56/s82/s32/s40/s87/s41\n/s49/s48 /s48 /s45/s49/s48\n/s109/s48/s72/s32/s40/s109/s84/s41/s72\n/s45/s49/s46/s48/s48/s46/s48/s49/s46/s48/s80/s86/s32/s40/s109/s86/s41\n/s49/s48/s48 /s48 /s45/s49/s48/s48\n/s109/s48/s72/s32/s40/s109/s84/s41/s49/s50/s48/s111\n/s72\n/s55/s54\n/s55/s50\n/s54/s56\n/s54/s52/s109/s48/s72/s114/s32/s40/s109/s84/s41\n/s51/s54/s48 /s50/s55/s48 /s49/s56/s48 /s57/s48 /s48\n/s113/s72/s32/s40/s100/s101/s103/s114/s101/s101/s41\n/s49/s50\n/s56\n/s52\n/s48/s119/s47/s50/s112 /s32/s40/s71/s72/s122/s41\n/s45/s49/s48/s48 /s45/s53/s48 /s48 /s53/s48 /s49/s48/s48\n/s109/s48/s72/s114/s32/s40/s109/s84/s41/s72/s52/s53/s111/s40/s99/s41\n/s40/s100/s41\n/s40/s101/s41/s40/s97/s41\n/s40/s98/s41\n/s40/s102/s41\nFIG. 6: (color online). (a) Schematic diagram of the second\ngeneration spin dynamo where the Py strip is located under-\nneath the CPW. In this case the dominant magnetic ﬁeld in\nthePyistheOerstedﬁeldinthe x′direction duetotheﬁeldin\nthe CPW. (b) Micrograph of the Py CPW device. (c) Magne-\ntoresistance at θH= 90◦. AMR is seen to be ∼0.5%. Arrows\ndenote the anisotropic ﬁeld, µ0HA= 8.0 mT. Open circles are\nexperimental data and solid curve is the ﬁtting result using\nR(0) = 121 .53 Ω and ∆ R= 0.66 Ω. (d) Electrically detected\nFMR at θH= 120◦andω/2π= 8 GHz showing a nearly\nsymmetric Lorentz line shape. Fit is according to Eq. (10)\nwithµ0∆H= 6.0 mT ,µ0Hr= 76.5 mT and Φ = −102◦.\n(e) Oscillating Hrdependence on the static ﬁeld direction θH\nwith amplitude 2 HA. (f) Dependence of FMR frequency on\nthe resonant ﬁeld HratθH= 45◦. Solid circles show the\nFMR frequency dependence while the solid triangles are the\nstanding SWR frequency dependence. The solid line is a ﬁt\ntoω=γ/radicalbig\n|Hr|(|Hr|+M0).\nSWR is ﬁt using γ/2π= 29.0µ0GHz/T , µ0Hex= 30\nmT and µ0M0= 1.0 T.\nSimilar to the results presentedin the previoussection,\nthe line shape of FMR measured on the second genera-\ntion sample is also found to be frequency dependent (not\nshown). Hence, Φ is found to be non-zero in the general\ncase. For example, at ω/2π= 8 GHz, the line shape is\nfound to be nearly symmetric, as shown in Fig. 6(d) for\nthe FMR measured at θH= 120◦, which indicates Φ is\nclose to−90◦at this frequency. Note that our result is\nin direct contrast with the recent study of Ref. 20 and\n21, where experiments were measured in the same con-\nPV (µV)\n-150-100-500\nµ0H (mT)θH=90o\nθH=180oHH\n-1.00.01.0PV (µV)\nθH (degree)\n10\n8\n6\n4µ0∆H (mT)\n43210\nCone angle (degree)10\n8\n6\n4µ0∆H (mT)\n360270180900\nθH (degree)µ0h=0.5 mT(a)\n(b)\n(c)\n(d)\nFIG. 7: (color online). Data shown for a second generation\nspin dynamo. (a) FMR line shape at ﬁxed frequency, ω/2π=\n8 GHz for several θHfrom 90◦to 180◦in steps of 10◦. Open\ncircles are experimental data and solid lines are ﬁts using E q.\n(9) with Φ = −102◦ﬁxed. (b) ADandALshown in squares\nand circles respectively as a function of θH. Fitting curves are\nsin(2θH)cos(θH) functions. (c) ∆ Hfor several values of θH\nshowing an oscillation with θH. (d) Non-linear dependence of\nline width ∆ Hon the cone angle. Dashed line is the expected\nlinear Gilbert dampingwhereas thedatafollows thequadrat ic\ndependence shown by the solid line.\nﬁguration and where it was suggested that Φ = 0◦for all\nsamples at all frequencies.\nWhile the line shape and hence the relative phase is\nfound to be frequency dependent, Φ is expected to be\nindependent of the static ﬁeld direction θH. This is con-\nﬁrmed in Fig. 7(a) which shows the line shape measured\nat several values of θHin 10◦increments. The data can\nbe ﬁt well using Eq. (9) with a constant Φ = −102◦for\nallθH. It conﬁrms that the FMR is driven by a single h\ncomponent, in this case the hx′ﬁeld, and that Φ does not\ndepend on θH. In Fig. 7(b) the θHdependence of AL\nandAD(solid/circles and dashed/squares respectively)\nis shown. The circles and squares are experimental data\nwhile the solid and dashed lines are ﬁtting results using\na sin(2θH)cos(θH) function according to Eq. (9). It pro-\nvides further proof that the hx′ﬁeld is responsible for\ndriving the FMR in this sample.\nWhile the results from both the 1st and 2nd genera-\ntionspindynamosshowconsistentlythatΦissampleand\nfrequency dependent, the 2nd generation spin dynamos\nexhibit special features in comparison with the 1st gen-\neration spin dynamos: the reduced separation between\nthe Py strip and CPW enhances the hx′ﬁeld so that the10\nline width ∆ His enhanced by non-linear magnetization\ndamping28,29,58, whichdepends onthe coneangle θofthe\nprecession via the relation θ∼hx′cos(θH)/∆H(θ). As\nshown in Fig. 7(c), ∆ His found to oscillate between 4.0\nand 9.0mTas θHchanges. At θH= 0◦,θ∼hx′/∆Hand\ntheconeangleisatitslargest(about4◦). AsθHincreases\nfrom 0◦and moves toward 90◦,θdecreases so that the\nnon-linear damping contribution to ∆ Hdecreases. Us-\ning the cone angle calculated from Fig. 7(c), we plot\nin Fig. 7(d) ∆ H(θ) as a function of the cone angle. It\nshows that ∆ Hhas a quadratic dependence on the pre-\ncession cone angle, which is in agreement with our previ-\nous study in the perpendicular H-ﬁeld conﬁguration28,29.\nWe note that for cone angles above only a few degrees,\nthe non-linear damping already dominates the contribu-\ntion to ∆ H. Again, this is in direct contrast with the\nresult of Mosendz et al.,20,21, where ∆ Hwas found to\nbe constant by varying θH, indicating no inﬂuence from\nnon-linear damping, but the cone angle θwas estimated\nto be as high as 15◦based on the line shape analysis\nassuming relative phase Φ = 0.\nC. Arbitrary h Vector\nNext we consider the most general case which is de-\nscribed by Eq. (7) where all components of hmay con-\ntribute to the FMR line shape. The sample used here is a\nsingle Py strip where a waveguide with a horn antennae\nprovided both the electric and magnetic driving ﬁelds.\nThe sample chip is mounted near the centre, at the end\nof a rectangular waveguide and the Py strip is directed\nalong the short axis of the waveguide.\nIn a waveguide, the electromagnetic ﬁelds are well\nknown and in general three components, hx′,hyandhz′\nexist.57Figure 8(a) shows both the FMR and perpen-\ndicular standing SWR at θH= 45◦. Indeed both the\namplitude and the line shape are diﬀerent for the two\nFMR peaks located at Hand−H, which indicates the\nexistence of multiple hﬁeld components and Eq. (7) and\nEq. (8) are needed to separate them.\nThis separation is done using the Lorentz and disper-\nsive amplitudes determined from a ﬁt to the FMR which\nare plotted as a function of θHin Fig. 8(b) and (c) for\nω/2π= 12 and 11.2 GHz, respectively. A ﬁt using Eq.\n(8) allows a separation of the contributions from each\nof thehx′,hyandhz′ﬁelds based on the their diﬀerent\ncontributions to the θHdependence of the line shape.\nThe results of the ﬁt have been tabulated in Table II\nwhereγ/2π= 28.0µ0GHz/T, µ0M0= 0.97 T and µ0Hr\n= 152 mT were used. The amplitudes of the diﬀerent h\nﬁeld components have been normalized with respect to\nthehx′component. At both 11.2 and 12 GHz the hx′\nﬁeld is much larger than hyorhz′, which is expected\nbased on the wave propagation in a horn antennae.\nIn changing from 11.2 to 12 GHz the relative phase for\neach component is seen to change. Therefore even in the\ncase of a complex line shape produced by multiple hﬁeld/s49/s46/s48\n/s48/s46/s53\n/s48/s46/s48\n/s45/s48/s46/s53/s80/s86/s32/s40/s109/s86/s41\n/s50/s48/s48 /s49/s48/s48 /s48 /s45/s49/s48/s48 /s45/s50/s48/s48\n/s109/s48/s72/s32/s40/s109/s84/s41/s119/s47/s50/s112 /s61/s49/s50/s32/s71/s72/s122/s70/s77/s82/s70/s77/s82\n/s83/s87/s82/s83/s87/s82/s104/s120/s39/s104/s121/s104/s122/s39\n/s49/s46/s48\n/s48/s46/s53\n/s48/s46/s48\n/s45/s48/s46/s53\n/s45/s49/s46/s48/s80/s86/s32/s40/s109/s86/s41\n/s51/s54/s48 /s50/s55/s48 /s49/s56/s48 /s57/s48 /s48\n/s113/s72/s32/s40/s100/s101/s103/s114/s101/s101/s41/s119/s47/s50/s112 /s61/s49/s50/s32/s71/s72/s122\n/s45/s49/s46/s48/s48/s46/s48/s49/s46/s48/s80/s86/s32/s40/s109/s86/s41\n/s51/s54/s48 /s50/s55/s48 /s49/s56/s48 /s57/s48 /s48\n/s113/s72/s32/s40/s100/s101/s103/s114/s101/s101/s41/s119/s47/s50/s112 /s61/s49/s49/s46/s50/s32/s71/s72/s122/s40/s97/s41\n/s40/s98/s41\n/s40/s99/s41\nFIG. 8: (color online). Data shown for a single Py strip with\nprecession driven by horn antennae ﬁeld. The strip dimen-\nsions are 3 mm ×50µm×45 nm. (a) Spectra showing dis-\ntinct resonances due to FMR and SWR at ω/2π= 12 GHz.\n(b) Separated Lorentz and dispersive line shapes (circles a nd\nsquares respectively) as a function of θHfrom a ﬁt to Eq. (7)\natω/2π= 12 GHz and (c) ω/2π= 11.2 GHz.\nTABLE II: Angular separation of hﬁeld components for 12\nand 11.2 GHz.\n12 GHz 11.2 GHz\n|hx′|1 1\n|hy|0.02 0.14\n|hz′|0.19 0.37\nΦx′-23◦50◦\nΦy40◦-30◦\nΦz′-33◦82◦\ncomponents, by separating the individual contributions\nof the rf magnetic ﬁeld via angular dependence measure-\nments, the relative phase Φ of each ﬁeld component is\nfound to be frequency dependent.\nD. Additional Inﬂuences on Φ\nIn addition to the frequency and sample dependencies,\nthe relative phase Φ may also depend on the lead conﬁg-\nuration and wiring conditions of a particular device, as\nwe have mentioned in Section A. Here, we address such\nadditional inﬂuences by using the ﬁrst generation spin11\n/s49/s50\n/s56\n/s52\n/s48/s80/s86/s32/s40/s109/s86/s41\n/s45/s53/s48 /s45/s50/s53 /s48 /s50/s53 /s53/s48\n/s109/s48/s72/s32/s40/s109/s84/s41 /s80/s86/s32\n/s52/s53 /s52/s48 /s51/s53 /s51/s48 /s50/s53 /s50/s48\n/s109/s48/s72/s32/s40/s109/s84/s41/s100/s61/s54/s48/s32/s110/s109\n/s32\n/s70/s61/s50/s55/s111/s70/s61/s45/s50/s57/s111\n/s83/s50/s83/s49/s80/s86\n/s52/s53 /s52/s48 /s51/s53 /s51/s48 /s50/s53 /s50/s48\n/s109/s48/s72/s32/s40/s109/s84/s41/s70/s61/s45/s49/s49/s111\n/s70/s61/s50/s50/s111/s100/s61/s49/s48/s48/s32/s110/s109\n/s32\n/s83/s49\n/s83/s50/s40/s97/s41\n/s40/s98/s41 /s40/s99/s41\nFIG. 9: (color online). (a) FMR observed in a ﬁrst generation\nspin dynamo. Inset shows the device structure with two Py\nstrips labeled S1 and S2. (b) FMR for Py thickness d= 100\nnm for both S1 and S2. In S1 Φ = −11◦, while in S2 the line\nshape is slightly more asymmetric and Φ = 22◦. (c) For d=\n60 nm the relative phase is Φ = −29◦for S1 and Φ = 27◦for\nS2.\ndynamos12shown in the inset of Fig. 9(a). Two spin\ndynamos with the same lateral dimensions but diﬀerent\nPy thickness dare studied. Each spin dynamo involves\ntwo identical Py strips denoted by S1 and S2, one in each\ncenter of the G-S strips of the CPW, which are placed\nsymmetrically with respect to the S strip. The current\nand rfhﬁeld are induced in the Py via a microwave\ncurrent directly injected into the CPW. Similar to the\nsample discussed in Section A, hyis the dominant ﬁeld\nwhich drives the FMR.\nAs shown in Fig. 9(a), FMR measured at ω/2π= 5\nGHz on the sample S1 with d= 100 nm shows a nearly\nsymmetric Lorentz line shape and a ﬁeld symmetry of\nV(H) =V(−H). From the FMR line shape ﬁtting, Φ =\n-11◦is found. Interestingly, as shown in Fig. 9(b), the\nFMR of the sample S2 of the same spin dynamo mea-\nsured under the same experimental conditions shows a\ndiﬀerent line shape, from which a diﬀerent Φ = 22◦is\nfound. We can further compare Φ measured on the other\nspin dynamo with a diﬀerent Py thickness of d= 60 nm,\nalso atω/2π= 5 GHz. Here for S1, Φ = -29◦while for\nS2, Φ = 27◦. Again, the relative phase is found diﬀer-\nent for S1 and S2. These results demonstrate that due\nto additional inﬂuences such as a diﬀerent lead conﬁgu-\nration and wiring conditions, even for samples with the\nsame lateral dimensions, Φ in each device is not neces-\nsarily the same. It demonstrates clearly that the relative\nphase Φ can not be simply determined by analyzing theFMR line shape measured on a reference device.\nE. Closing Remarks\nThe experimental data presented above show that re-\ngardless of the FMR driving ﬁeld conﬁguration, the rel-\native phase between the rf electric and magnetic ﬁeld is\nsampleandfrequencydependentandnon-zero. Thisnon-\nzero phase results in both symmetric and antisymmetric\nLorentz line shapes in the FMR detected via ﬁeld-torque\ninduced spin rectiﬁcation. The Φ dependence of the line\nshape symmetry changes based on which component of\nthe rfhﬁeld is responsible for driving the FMR pre-\ncession. For instance a purely antisymmetric line shape\ncould correspond to Φ = 0◦if the FMR is driven by hx′,\nor to Φ = 90◦if the FMR is driven by hy, therefore the\nline shape itself cannot be used to determine Φ directly.\nTo separate the hﬁeld components an angular ( θH) de-\npendence measurement is necessary, which allows both\nhas well as the phase to be determined. Using such a\nmeasurement Φ has been observed to change from 0◦to\n90◦in a narrow frequency range (0.6 GHz) resulting in\na change from an antisymmetric to symmetric line shape\ndemonstrating the large eﬀect the relative phase has on\nthe FMR line shape. Furthermore Φ is not identical even\nin samples with the same geometrical size. Therefore in\nour opinion Φ cannot be simply determined from a ref-\nerence sample but should be calibrated for each sample,\nat each frequency and for each measurement cycle.\nV. CONCLUSION\nSpin rectiﬁcations caused by the coupling between cur-\nrent and magnetization in a ferromagnetic microstrip\nprovide a powerful tool for the study of spin dynamics.\nIn order to distinguish diﬀerent mechanisms which en-\nable the electrical detection of FMR via microwave pho-\ntovoltages, it is essential to properly analyze the FMR\nline shape. For spin rectiﬁcation caused by a microwave\nﬁeld torque, due to the coherent nature of this coupling,\nthe resulting dc voltage depends strongly on the relative\nphase between the rf electric and magnetic ﬁelds used to\ndrivethecurrentandmagnetization, respectively. There-\nfore not only does electrical FMR detection provide a\nroute to study the relative phase, but it also necessi-\ntates calibrating the relative phase prior to performing\nelectrically detected FMR experiments. Based on a sys-\ntematic study of the electrically detected FMR, the line\nshape is observed to depend strongly on the microwave\nfrequency, driving ﬁeld conﬁguration, sample structure\nand even wiring conditions. It is in general a combi-\nnation of Lorentz and dispersive contributions. These\neﬀects have been quantitatively explained by accounting\nfor the relative phase shift between electric and magnetic\nﬁelds. Analytical formula have been established to an-\nalyze the FMR line shape. Our results imply that for12\nelectrically detected FMR which involves both spin Hall\nand spin rectiﬁcation eﬀects, the pivotal relative phase\nmust be calibrated independently in order to properly\nanalyze the FMR line shape and quantify the spin Hall\nangle. This cannot be done by using a reference sample\nbut could be achieved through such techniques as spin-\ntronic Michelson interferometry37.\nACKNOWLEDGEMENTS\nWe would like to thank B. W. Southern, A. Hoﬀmann,\nand S. D. Bader for discussions. This work has been\nfunded by NSERC, CFI, CMC and URGP grants (C.-M.\nH.). ZXC wassupported bythe NationalNaturalScience\nFoundation of China Grant No. 10990100.13\n1M. Tsoi, A. G. M. Jansen, J. Bass, W.-C. Chiang, V. Tsoi,\nand P. Wyder, Nature (London) 406, 46 (2000).\n2S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley,\nR. J. Schoelkopf, R. A. Buhrman, and D. C. Ralph, Nature\n(London) 425, 380 (2003).\n3A.A.Tulapurkar, Y.Suzuki, A.Fukushima, H. Kubota, H.\nMaehara, K. Tsunekawa, D. D. Djayaprawira, N. Watan-\nabe and S. Yuasa, Nature (London) 438, 339 (2005).\n4Y. S. Gui, S. Holland, N. Mecking, and C.-M. Hu, Phys.\nRev. Lett. 95, 056807 (2005).\n5A. Azevedo, L. H. Vilela Leo, R. L. Rodriguez-Suarez, A.\nB. Oliveira, and S. M. Rezende, J. Appl. Phys. 97, 10C715\n(2005).\n6M. V. Costache, S. M. Watts, M. Sladkov, C. H. van der\nWal, and B. J. van Wees, Appl. Phys. Lett. 89, 232115\n(2006).\n7M. V. Costache, M. Sladkov, S. M. Watts, C. H. van der\nWal, and B. J. van Wees, Phys. Rev. Lett. 97, 216603\n(2006).\n8E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Appl.\nPhys. Lett. 88, 182509 (2006).\n9J. C. Sankey, P. M. Braganca, A. G. F. Garcia, I. N. Kriv-\norotov, R. A. Buhrman, and D. C. Ralph, Phys. Rev. Lett.\n96, 227601 (2006).\n10H. Kubota, A. Fukushima, K. Yakushiji, T. Nagahama,\nS. Yuasa, K. Ando, H. Maehara, Y. Nagamine, K.\nTsunekawa, D. D. Djayaprawira, N. Watanabe, and Y.\nSuzuki, Nature Physics 4, 37 (2007).\n11J. C. Sankey, Y. T. Cui, J. Z. Sun, J. C. Slonczewski,\nRobert A. Buhrman, and D. C. Ralph, Nature Physics 4,\n67 (2007).\n12Y. S. Gui, N. Mecking, X. Zhou, G. Williams and C. -M.\nHu, Phys. Rev. Lett. 98, 107602 (2007).\n13A. Yamaguchi, H. Miyajima, T. Ono, Y. Suzuki, S. Yuasa,\nA. Tulapurkar, and Y. Nakatani, Appl. Phys. Lett. 90,\n182507 (2007).\n14S. T. Goennenwein, S. W. Schink, A. Brandlmaier, A.\nBoger, M. Opel, R. Gross, R. S. Keizer, T. M. Klapwijk,\nA. Gupta, H. Huebl, C. Bihler, and M. S. Brandt, Appl.\nPhys. Lett. 90, 162507 (2007).\n15N. Mecking, Y. S. Gui, and C.-M. Hu, Phys. Rev. B 76,\n224430 (2007).\n16J. C. Sankey, Y. T. Cui, J. Z Sun, J. C. Slonczewski, R. A.\nBuhrman and D. C. Ralph, Nature Physics, 4, 67 (2008)\n17X. Hui, A. Wirthmann, Y. S. Gui, Y. Tian, X. F. Jin, Z.\nH. Chen, S. C. Shen, and C. -M. Hu, Appl. Phy. Lett. 93,\n232502 (2008).\n18A. Wirthmann, X. Hui, N. Mecking, Y. S. Gui, T.\nChakraborty, C. -M. Hu, M. Reinwald, C. Sch¨ uller, and\nW. Wegscheider, Appl. Phys. Lett. 92, 232106 (2008).\n19V. A. Atsarkin, V. V. Demidov, L. V. Levkin, and A. M.\nPetrzhik, Phys. Rev. B 82, 144414 (2010).\n20O. Mosendz, J. E. Pearson, F. Y. Fradin, G. E. W. Bauer,\nS. D. Bader, and A. Hoﬀmann, Phys. Rev. Lett. 104,\n046601 (2010).\n21O. Mosendz, V. Vlaminck, J. E. Pearson, F. Y. Fradin, G.\nE. W. Bauer, S. D. Bader, and A. Hoﬀmann, Phys. Rev.\nB82, 214403 (2010).\n22P. Saraiva, A. Nogaret, J. C. Portal, H. E. Beere, and D.\nA. Ritchie, Phys. Rev. B 82, 224417 (2010).\n23Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida,M. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K.\nTakanashi, S. Maekawa, and E. Saitoh, Nature(London)\n464, 262 (2010).\n24C. W. Sandweg, Y. Kajiwara, K. Ando, E. Saitoh, and B.\nHillebrands, Appl. Phys. Lett. 97, 252504 (2010).\n25L. Liu, T. Moriyama, D. C. Ralph, and R. A. Buhrman\nPhys. Rev. Lett. 106, 036601 (2011).\n26A. Azevedo, L. H. Vilela-Le˜ ao, R. L. Rodr´ ıguez-Su´ arez, A .\nF. Lacerda Santos, and S. M. Rezende, Phys. Rev. B 83,\n144402 (2011).\n27Y. S. Gui, N. Mecking, and C.-M. Hu, Phys. Rev. Lett.\n98, 217603 (2007).\n28Y. S. Gui, A. Wirthmann, N. Mecking, and C.-M. Hu,\nPhys. Rev. B 80, 060402(R) (2009).\n29Y. S. Gui, A. Wirthmann, and C.-M. Hu, Phys. Rev. B\n80, 184422 (2009).\n30C. T. Boone, J. A. Katine, J. R. Childress, V. Tiberkevich,\nA. Slavin, J. Zhu, X. Cheng, and I. N. Krivorotov, Phys.\nRev. Lett. 103, 167601 (2009).\n31D. Bedau, M. Kl¨ aui, S. Krzyk, U. R¨ udiger, G. Faini and\nL. Vila, Phys. Rev. Lett. 99, 146601 (2007).\n32S. Bonetti, V. Tiberkevich, G. Consolo, G. Finocchio, P.\nMuduli, F.Mancoﬀ, A.Slavin, andJ. ˚Akerman, Phys.Rev.\nLett.105, 217204 (2010).\n33S. Urazhdin, V. Tiberkevich, and A. Slavin, Phys. Rev.\nLett.105, 237204 (2010).\n34Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I.\nHalperin, Rev. Mod. Phys. 77, 1375 (2005).\n35X. Wang, G. E. W. Bauer, B. J. van Wees, A. Brataas,\nand Y. Tserkovnyak, Phys. Rev. Lett. 97, 216602 (2006).\n36A. Yamaguchi, H. Miyajima, S. Kasai, and T. Ono, Appl.\nPhys. Lett. 90, 212505 (2007).\n37A. Wirthmann, X. Fan, Y. S. Gui, K. Martens, G.\nWilliams, J. Dietrich, G. E. Bridges, and C.-M. Hu, Phys.\nRev. Lett. 105, 017202 (2010).\n38X. F. Zhu, M. Harder, A. Wirthmann, B. Zhang, W. Lu,\nY. S. Gui, and C.-M. Hu, Phys. Rev. B 83, 104407 (2011).\n39X. Fan, S. Kim, X. Kou, J. Kolodzey, H. Zhang, and J. Q.\nXiao, Appl. Phys. Lett. 97, 212501 (2010).\n40L. H. Bai, Y. S. Gui, A. Wirthmann, E. Recksiedler, N.\nMecking, C.-M. Hu, Z. H. Chen, and S. C. Shen, Appl.\nPhys. Lett. 92, 032504 (2008).\n41H. Zhao, E. J. Loren, H. M. van Driel, and A. L. Smirl,\nPhys. Rev. Lett. 96, 246601 (2006).\n42J. Wang, B. F. Zhu, and R. B. Liu, Phys. Rev. Lett. 104,\n256601 (2010).\n43L. K. Werake, and H. Zhao, Nature Physics, 6, 875 (2010).\n44J. D. Jackson, Classical Electrodynamics (John Wiley &\nSons, New York, 1975), 2nd ed.\n45H. J. Juretschke, J. Appl. Phys. 31, 1401 (1960).\n46R. H. Silsbee, A. Janossy, and P. Monod, Phys. Rev. B 19,\n4382 (1979).\n47B. Heinrich, Y. Tserkovnyak, G. Woltersdorf, A. Brataas,\nR. Urban, and G. E. W. Bauer, Phys. Rev. Lett. 90,\n187601 (2003).\n48J. N. Kupferschmidt, S. Adam, and P. W. Brouwer, Phys.\nRev. B74, 134416 (2006).\n49A. A. Kovalev, G. E. W. Bauer, and A. Brataas, Phys.\nRev. B75, 014430 (2007).\n50T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004).\n51L.D.LandauandE.M.Lifshitz, Mechanics , secondedition14\n(Pergamon Press, Oxford, 1969).\n52J. P. Jan, in Solid State Physics , edited by F. Seitz and D.\nTurnbull (Academic, New York, 1957), Vol. 5.\n53M. Born andE. Wolf, Principles of optics: Electromagnetic\ntheory of propagation, interference and diﬀraction , 7th edi-\ntion (Cambridge university press, Cambridge, 1999).\n54W. Heinrich, IEEE Trans. Microwave Theory Tech. 38,\n1468 (1990).\n55W. Heinrich, IEEE Trans. Microwave Theory Tech. 41, 45\n(1993).56S. V. Vonsovskii, Ferromagnetic Resonance: The Phe-\nnomenon of Resonant Absorption of a High-Frequency\nMagnetic Field in Ferromagnetic Substances , (Oxford:\nPergamon, 1966).\n57B. S. Guru, and H. R. Hiziroˇ glu, Electromagnetic Field\nTheory Fundamentals , 2nd ed. (Cambridge University\nPress, Cambridge, England, 2004).\n58V. Tiberkevich and A. Slavin, Phys. Rev. B 75, 014440\n(2007)."    },    {        "title": "2303.15985v2.Exploring_terahertz_scale_exchange_resonances_in_synthetic_ferrimagnets_with_ultrashort_optically_induced_spin_currents.pdf",        "content": "Exploring terahertz-scale exchange resonances in synthetic ferrimagnets with\nultrashort optically induced spin currents\nJulian Hintermayr,∗Youri L. W. van Hees, and Bert Koopmans\nDepartment of Applied Physics, Eindhoven University of Technology,\nP.O. Box 13, 5600 MB Eindhoven, the Netherlands\n(Dated: June 21, 2023)\nUsing spin currents generated by fs laser pulses, we demonstrate excitation of GHz ferromag-\nnetic resonance and THz ferrimagnetic exchange resonances in Co/Gd/Co/Gd multilayers by time-\nresolved magneto-optic Kerr effect measurements. Varying the Gd layer thickness allows for a tuning\nof the resonance spectrum by manipulating the total angular momentum and strength of effective\nexchange fields between the antiferromagnetically coupled layers. Close to the compensation point\nof angular momentum, a minimum in the frequency of the exchange-dominated mode and a max-\nimum in the frequency of the ferromagnetic resonance mode is observed. Finally, to gain better\nunderstanding of the excitation mechanism, we analyze the anomalous variation in the measured\nexchange mode amplitude as a function of its frequency. A peak in this amplitude in the vicin-\nity of the compensation point of angular momentum is explained using a macrospin model, taking\nnonlinear effects at finite precession amplitudes into account.\nI. INTRODUCTION\nFerrimagnets combine a number of advantages of ferro-\nand antiferromagnets, making them a promising platform\nfor fast and easily integratable spintronic devices [1–3].\nStrong, alternating exchange fields are inherent to anti-\nferromagnets, giving rise to high-frequency ( ∼THz) ex-\nchange resonance modes (EXMs) [4, 5]. Their lack of\nnet spin polarization however makes experimental ac-\ncess challenging [6]. Metallic ferrimagnets , in contrast,\ncan exhibit finite conduction spin polarization — even\nat the compensation point of angular momentum — and\ncan thus easily be probed by magneto-optic or magneto-\nresistive effects, given that they consist of sublattices\nwith different atomic species [1].\nWhile the high frequency makes this class of materi-\nals a promising candidate for the development of THz\nspintronic devices, techniques of exciting resonances at\nsuch high frequencies coherently are sparse. In the past,\nEXMs have been excited for instance by thermal laser\nexcitation in combination with an applied field [7, 8]\nand circularly polarized laser pulses [9–11]. The former\napproach only made it possible to excite EXMs in the\n<100 GHz range, whereas the latter approach requires\nmaterials with significant inverse Faraday effect. In ad-\ndition, pulses of intense THz radiation have been shown\nto facilitate excitation of fast spin dynamics [12–14].\nIn this work, we will make use of optically induced\nspin currents that are generated by a neighboring fer-\nromagnet upon ultrafast demagnetization [15–18]. The\noptically induced spin-transfer torques (OSTTs) gener-\nated in this manner have been used to excite exchange-\ndominated standing spin waves with frequencies exceed-\ning 1 THz [19–21]. Moreover, its has been shown that\nsuch spin currents can be used to assist in single-shot\n∗Electronic mail: j.hintermayr@tue.nlall-optical switching of ferrimagnets [22, 23] or even in-\nduce full switching in ferromagnets [23–26]. As we will\nshow, a synthetic ferrimagnetic quadlayer consisting of\nthe transition metal (TM) Co and the rare earth (RE)\nGd can host not only GHz ferromagnetic resonance but\nalso THz exchange resonance modes, in spite of the ar-\nguably weaker exchange coupling compared to CoGd al-\nloys. Co/Gd multilayers are highly relevant for a variety\nof spintronic applications as they exhibit energy-efficient\nsingle-shot all-optical magnetization switching [27]. Ad-\nditionally, ultrafast domain wall motion close to the com-\npensation point of angular momentum has been observed\nvery recently [28]. The coexistence of these properties\nmakes this material a promising candidate for hybrid\nspintronic-photonic memory applications [28, 29].\nIn the following, we will show that OSTTs are an ex-\ncellent tool to excite and study GHz ferromagnetic res-\nonance (FMR) and THz EXMs in Co/Gd multilayers.\nFurthermore, we explain our findings with an analytical\nmacrospin model.\nII. SAMPLE STRUCTURE AND\nCHARACTERIZATION\nThe idea behind the sample design is to layer a ferri-\nmagnet with an in-plane (IP) magnetic easy axis onto\na ferromagnet with perpendicular magnetic anisotropy\n(PMA) that efficiently generates spin currents upon\nultrafast demagnetization [17, 30]. We chose a synthetic\nferrimagnetic quadlayer of Co/Gd/Co/Gd as it offers\nthe possibility of reaching magnetic compensation by\nfine-tuning individual layer thicknesses while maintain-\ning IP anisotropy. The spin current generation layer\nis a Pt/[Co/Ni] 4multilayer which provides PMA at\nrelatively large magnetic volumes [19]. Between the two\nlayers, a Cu spacer is placed to magnetically decouple\nthe two systems while allowing for spin currents to pass.\nThe final material stack used in this study is as follows:arXiv:2303.15985v2  [cond-mat.mes-hall]  19 Jun 20232\nFIG. 1. Cartoon of a, sample architecture showing local arrangements of magnetization and the resonance excitation mechanism\nb, stable spin precession configurations and c, local directions of exchange fields in canted magnetization states between Co\nand Gd, allowing for fast resonance frequencies. d, Demagnetization trace of the [Co/Ni] injection layer, based on which the\noptically induced spin current pulse in eis calculated. f, In-plane hysteresis loops of the Co/Gd/Co/Gd absorption layer for\ndifferent Gd thicknesses measured with L-MOKE. g, Coercive field and MOKE step as a function of Gd thickness. The vertical\ngray line indicates the magnetization compensation point Mcomp.\nSi:B/Ta(4)/Pt(4)/[Co(0.2)/Ni(0.6)] 4/Co(0.2)/Cu(5)/\n[Co(0.8)/Gd( tGd)]2/TaN x(5) (thicknesses in nm) with\nGd thicknesses ranging from 0–3 nm (schematically\nshown in Fig. 1 a). Layers are deposited by dc mag-\nnetron sputter deposition at room temperature. The\ndemagnetization is triggered and measured by ∼100 fs\nlaser pulses with a central wavelength of 780 nm and\n80 MHz repetition rate, using a standard pump-probe\nsetup to measure the time-resolved magneto-optic\nKerr effect (TR-MOKE). As the MOKE of Gd at this\nwavelength is small [31], we mostly probe the signal\narising from Co and Co/Ni layers. Pump and probe\npulses are focused on the sample surface with near\nnormal incidence.\nFirstly, we investigate the demagnetization trace and\nderive the optically induced spin current profile to under-\nstand the excitation mechanism. The resulting trace is\nshown in Fig 1 d, revealing rapid demagnetization within\nthe first 300 fs and a slower remagnetization over several\nps. We assume that the spin current originating from this\nprocess follows the spin-pumping model, also referred to\nas “d M/dt” model [17, 18, 32, 33], where the demagneti-\nzation is explained by excitation of magnons that transfer\ntheir angular momentum to mobile conduction electrons.\nThe shape of such a spin current pulse is thus derived\nfrom the time derivative of the demagnetization curve\n(Fig. 1 a) and shown in Fig. 1 e. De- and remagneti-\nzation give rise to a positive and negative peak in spin\ncurrent respectively, resulting in a bipolar pulse on the\nps timescale. It is thus suited to excite THz-scale dy-namics. To characterize the static magnetic properties of\nthe Co/Gd absorption stack, MOKE hysteresis loops in\nlongitudinal geometry are recorded for different Gd thick-\nnesses. Measurements are shown in Fig. 1 f. We find that\nthe easy axis of the absorption stack lies in-plane (IP),\nwith a magnetic compensation point ( Mcomp) at around\n1.9 nm Gd. At this thickness, the magnetic moments of\nthe antiferromagnetically coupled Co and Gd layers ex-\nactly cancel each other out. The switch in sign of the\nmeasured Kerr ellipticity εmaxis accompanied by a max-\nimum in coercive field, as shown in Fig. 1 g, where Mcomp\nis indicated by a vertical line. The angular momentum\ncompensation point Lcomp is expected to lie at slightly\nlower Gd thicknesses, as gCo> gGd(see Appendix). Due\nto the fact that the exact magnetization profile in the\nstack is not known, it is not possible to precisely de-\ntermine Lcomp. We note that, even though the samples\nare deposited as multilayers, significant intermixing be-\ntween Co and Gd is expected at the interface when sput-\nter depositing at room temperature [34–37]. As the Co\nthickness is only 0.8 nm, the intermixing depth in thin\nRE/TM multilayers is expected to be in a similar order\nof magnitude as the individual layer thickness, leading to\nsignificant alloy-like regions [38].\nIII. SPIN RESONANCE MODES\nHaving discussed the static magnetic properties of the\nsynthetic ferrimagnetic multilayer, we will introduce the3\ntypes of spin resonances that can occur in this system.\nGenerally, two types of precessions are possible within\na two-sublattice spin system with antiferromagnetic cou-\npling. For simplicity, we limit ourselves to the alloy-like\ncase for now, assuming one macrospin per material, and\nneglect magnetic anisotropy. Further discussion on the\nvalidity of this approximation is provided later in this\nwork. In the presence of an external magnetic field,\nFMR-like dynamics are allowed, where the magnetiza-\ntion of Co and Gd sublattices, MCoandMGd, are aligned\nfully antiparallel and precess around the direction of the\nexternal field H(schematically shown in Fig. 1 b). The\nfrequency of this oscillation can be approximated as [39]\nfFMR =µ0\n2πMCo−MGd\nMCo\nγCo−MGd\nγGdH=µ0\n2πγeffH. (1)\nµ0denotes the vacuum permeability, γCo,Gdthe gyro-\nmagnetic ratios of Co and Gd, and γeffthe effective gyro-\nmagnetic ratio. Given that γCo̸=γGd, the magnetization\nand angular momentum of the two sublattices compen-\nsate at different atomic fractions. As we approach the\npoint where the total angular momentum is fully com-\npensated ( Lcomp), the torque which is proportional to\nMeff×Hremains finite, resulting in a divergence in γeff,\nas predicted by eq. (1). Please note that this approxima-\ntion neglects demagnetizing fields and becomes invalid\nvery close to Lcomp. A more involved treatment removes\nthe divergence and leads to only a pronounced maximum\nin FMR frequency at Lcomp [40].\nThe exchange-dominated resonance mode with typi-\ncally much higher frequencies requires MCoto be at an\nangle with the exchange field it experiences. This field\nis proportional to the Weiss constant λ, indicating sign\nand strength of the exchange interaction, and can be ex-\npressed as HGd\nex=λMGd. Figure 1 bandcschematically\nshow the canted alignments of spins and exchange fields\nthat are necessary for this mode to exist. Its frequency\nis given by [39]\nfEXM =µ0λγCoγGd\n2π\u0012MCo\nγCo−MGd\nγGd\u0013\n(2)\nin the limit of small oscillation angles. In contrast to the\nFMR mode, the EXM frequency vanishes at the angu-\nlar momentum compensation point. The change in sign\nin frequency predicted by the dispersion relation indi-\ncates an intriguing change in handedness of the oscilla-\ntion, which has been experimentally verified by Brillouin\nlight scattering [41, 42].\nThe effective damping parameter αeffin this system\nhas been calculated to show a maximum at Lcomp for\nboth FMR and EXM modes [40]. Spin resonances are\ntherefore expected to only be very short-lived close to\ncompensation. If the individual sublattice damping co-\nefficients are identical, that is, αCo=αGd,αeffof FMR\nand EXM will likewise be equal. If they differ from one\nanother, the ratio of damping parameters of FMR and\nEXM will depend on αCo/αGd.We note that the symmetry of the quadlayer system\nallows in principle for higher-order exchange resonances\nwhere, for instance, the two Co layers are precessing out\nof phase. However, the frequencies of such modes are ex-\npected to be far above those of the fundamental EXM.\nWe therefore cannot excite them with our technique, as-\nsuming the wedged Gd does not become thick enough\nto only provide a weak link between the Co layers. Yet\nanother type of exchange-driven modes that can exist in\nthin films are quantum-confined standing spin waves that\nhave been excited and measured with techniques similar\nto those used in this work [19–21], but for a 0.8 nm Co\nlayer expected frequencies ( >10 THz) are too high to be\nexcited by our method. Also, more complex modes de-\nlocalized throughout the whole ferrimagnetic stack have\nnot been identified in our experiments, and we find a de-\nscription in terms of an alloy with only two sublattices\nsufficient to explain all our results.\nIV. RESULTS\nA. Time-resolved magnetization dynamics\nTo investigate the dynamics of the sample, we again\nemploy pump-probe spectroscopy. As mentioned previ-\nously, resonances are excited by injection of spin current\npulses generated in the Co/Ni layer into the Co/Gd ab-\nsorption layer. The current from the injection layer shows\nOOP spin polarization and exerts an OSTT on the IP\nabsorption layer, canting the magnetization OOP away\nfrom its ground state, as illustrated in Fig. 1 a. If the\neffective magnetization and the applied field are at an\nangle in the excited state, the FMR mode is triggered. If\nMCoandMGdare at an angle, the following precession is\ndescribed by the EXM. We note that the ultrafast laser\nexcitation of the Co/Gd quadlayer could lead to an injec-\ntion of an IP-polarized spin current into the Co/Ni layer\nas well. Since the dynamics in the Co/Ni layer induced\nby the following OSTT are expected to be fully IP as\nwell, the dynamics are not probed during the experiment\ndue to the polar measurement geometry.\nBefore the measurement, we saturate the injection\nstack in the positive z-direction and apply an IP field\nfor FMR measurements and no IP field for EXM mea-\nsurements. We then repeat the measurement with the\ninjection stack saturated in the opposite direction and\ncalculate the magnetic signal as the difference of mea-\nsurements with positively and negatively saturated injec-\ntion layer. The sum of the measurements includes non-\nmagnetic artefacts such as the coherence peak that occurs\nduring pump-probe overlap [43, 44] and is disregarded.\nHomogeneous, FMR-like precession modes with an ap-\nplied IP field of 92 mT are shown in Fig. 2 afor a range\nof Gd thicknesses. During the first few ps, the demagne-\ntization of the injection layer dominates the signal. Sub-\nsequently, the oscillations of the absorption layer are ob-\nserved. For a more detailed analysis of the measurements,4\nFIG. 2. aTime-resolved MOKE measurements of homoge-\nneous FMR precessions in the in-plane Co/Gd/Co/Gd layer\nat a magnetic field of 92 mT. Offsets are proportional to the\nGd thickness. The inset shows the precession frequency as\na function of applied field at a fixed Gd thickness of 1.5 nm\nincluding a fit according to eq. (4). bOscillation measure-\nments of exchange modes using complex MOKE. Black lines\nrepresent damped sine fits. Note the different timescales in a\nandb.\nwe fit functions of the form\nAsin(2πft−φ)e−t/τ+Be−t/C+D (3)\nto our data (black lines), allowing an extraction of the\nfrequency fand the effective damping parameter αeff=\n1/2πfτ.Adenotes the oscillation amplitude, φthe\nphase, and τthe decay time of the oscillation. The fol-\nlowing terms capture the remagnetization behavior of the\ninjection layer with an amplitude of B, a decay time of\nCand a constant offset D. A steady increase of the\nFMR frequency as a function of tGdis observed, which\nis explained by an increase in Gd angular momentum,\nwhereby the total angular momentum decreases. In ac-\ncordance to eq. (1) γeffincreases, and with it the fre-\nquency and effective damping, as observed in Fig. 3 a\nandb.\nFor a Gd thickness of 1.5 nm, FMR measurements at\ndifferent fields were recorded. The extracted frequenciesas a function of applied field are plotted in the inset of\nFig. 2 a. As a linear fit according to eq. (1) yielded unsat-\nisfactory results due to neglecting demagnetizing fields,\nwe used the following Kittel-like equation, accurately im-\nplementing the effect of finite demagnetizing effects in\nthin films, where the effective magnetization Meffand\nγeffare fitted:\nfKitt=µ0γeff\n2πp\nH(H+Meff). (4)\nThe best fit is found for γeff/γe= 3.0, with the electron\ngyromagnetic ratio γe. This strong boost of γeffis ex-\npected as Lcomp is being approached. Furthermore, we\nextract Meff= 25 kA /m, which corresponds to ∼2%\nof the saturation magnetization of elemental Co. Ad-\nditional possible anisotropy-related effects arising from\nthe multilayered nature of the sample are also captured\ninMeff. They may stem for instance from interfacial\nanisotropy which could give both positive or negative\ncontributions to Meffas well as shape anisotropy. The\nvery low value of Meffimplies, however, that the sample\nis close to Mcomp and that the anisotropy contributions\nare small.\nUpon increasing the Gd thickness to 2.1 nm, a dras-\ntic discontinuous change in the oscillation frequency is\nobserved (see Fig 2 a). We associate this observation\nwith the onset of the EXM and the simultaneous sup-\npression or over-damping of the FMR mode. To in-\ncrease our sensitivity to the EXM at short time delays,\na quarter-wave plate in the probe beam path is used\nto mostly cancel out the contribution of the demagne-\ntization of the injection layer to the signal. This tech-\nnique is known as complex MOKE, for further informa-\ntion the reader is referred to Refs. [45, 46]. Figure 2 b\nshows oscillation measurements of EXM resonances ac-\nquired in this manner. During the first ps, some un-\navoidable leaking signal from the injection layer and re-\nmanent artefacts from the coherence peak close to zero\ndelay are visible in the signal. Thereafter, damped os-\ncillations are observed. Please note the different scale\nof the x-axis compared to Subfigure a. Increasing Gd\nthicknesses results in a strong decrease of precession fre-\nquency and a variation of the precession amplitude. The\ndecrease of frequency is well explained by an approach\nof the angular momentum compensation point as pre-\ndicted by eq. (2). Plotting the resonance frequencies as\na function of tGd(see Fig. 3 b) shows a highly non-linear\nbehaviour. Two factors give rise to such nonlinearity:\nFirstly, we recall that the Curie temperature of Gd is be-\nlow room temperature. Thus, it only shows a finite mag-\nnetization within the partially intermixed Co/Gd and\nGd/Co interfaces as well as proximity-induced magneti-\nzation in the regions close to the interfaces. As one moves\naway, this induced magnetization—and thus the angular\nmomentum—decreases exponentially [38]. Secondly, in-\ncreasing the Gd thickness decreases the average exchange\nparameter λin eq. (2) which scales inversely proportional\nto the thickness in magnetic multilayers. Please note that5\nthis coupling parameter only influences the EXM and not\nthe FMR frequency. Hence, this effect does not influence\nthe thickness dependence of the FMR mode.\nFIG. 3. aFrequency of the ferromagnetic resonance mode\nin a field of 92 mT and that of the exchange mode in the\nabsence of a field as a function of Gd thickness. Please note\nthe different y-scales. bEffective damping parameter of said\nmodes. Lines are guides for the eye.\nAnother peculiarity of the EXM frequency is that it\ndecreases monotonously, implying either that the angu-\nlar momentum compensation point is being approached,\nyet never crossed, or that oscillation amplitudes are too\nlarge for the small angle approximation to hold that pre-\ndicts a vanishing frequency. The magnetostatic charac-\nterization, on the other hand, clearly revealed a mag-\nnetic compensation point around 1.9 nm Gd. Since\nγCo> γ Gd,Lcomp should lie at even smaller tGdthan\nMcomp. However, the magnetization of Gd is strongly\ntemperature-dependent, especially in TM/Gd multilay-\ners. Shortly after the arrival of the laser pulse it is\ntherefore very well possible that the system is Co dom-\ninant even though the room temperature characteriza-\ntion revealed Gd-dominance. Thus, the absence of an\nangular momentum compensation point in time-resolved\nmeasurements at strongly increased temperatures during\nthe first ps after laser irradiation could result in a Co-\ndominated sample across all investigated Gd thicknesses.\nA dependence of the frequency on the pump energy in\nCoGd alloys was found by Mekonnen et al. [7], which was\nexplained by this effect. The steadily increasing behavior\nofαeffshown in Fig. 3 bis well in line with the expectedmonotonous increase towards Lcomp. Furthermore, non-\nlocal damping effects due to spin-pumping either across\nCo/Gd interfaces or through the Co/Cu interface could\ncontribute to the total effective damping.\nIn terms of the phase of the EXM, a slight variation\nas a function of Gd thickness was observed (not shown),\nwhich could be expected, considering the fact that EXMs\nare excited by a bipolar spin current pulse acting on\ntimescales similar to those of the precessions. A simi-\nlar analysis as in Ref. [32] could be carried out, where\nthe phase of THz standing spinwave excited by the same\nps bipolar spin current pulse is investigated. However,\nthis would go beyond the scope of this work.\nB. Exchange-dominated precession amplitudes\nFinally, we seek to explain the observed variation\nin EXM amplitude for different oscillation frequencies,\nwhich is shown in Figure 4 a, to gain both a better un-\nderstanding of the system and insights into the exact ex-\ncitation mechanism. A naive first guess could be the as-\nsumption that the bipolar spin current resonantly drives\nthe precession mode. The EXM amplitude as a function\nof the frequency should then follow the Fourier Trans-\nform (FT) of the injection pulse. A comparison of said\nFT (continuous black line) to the measured data points\n(orange symbols) does show qualitative agreement in the\nsense that there is a maximum at a certain frequency\nand a decrease in amplitude towards zero and high fre-\nquencies. However, the maxima are located at different\nfrequencies and the nature of the decay does not match at\nall. We note that the shape of the Fourier spectrum of the\nspin current pulse crucially depends on our assumption\nof the d M/dtmodel, which is still a subject of ongoing\ndiscussion. Other models, such as the superdiffusive hot\nelectron model [47, 48], assume generation of even faster\noptically induced spin currents, leading to stronger devi-\nations from experimental data due to theoretical peaks\nat higher frequencies. Our assumption can therefore be\nseen as a conservative estimate for the high-frequency\nbehaviour.\nIn the following, we employ a simple macrospin model\nthat captures the ferrimagnetic nature of the absorption\nlayer and model the OSTT as a canting of the macrospins\nwith respect to the horizontal antiparallel state. We\nagain consider a two sublattice, alloy-like case where the\nGd concentration cGdis varied and both sublattices are\ninitially canted by the same angle δwith respect to the\nfilm plane (shown schematically in Fig. 4 d). While, in\npractice, we are dealing with a quadlayer system, we ar-\ngue that the fact that only two types of spin resonances\nare observed and the strong intermixing between thin\nlayers make the alloy model a reasonable approximation.\nNote that the concentration cGdin the model relates to\nthe magnetically active atomic fraction of Gd only, mak-\ning direct comparisons to Gd thicknesses challenging. By\nusing the same canting angle δfor both lattices, equal6\naverage spin transfer efficiencies for the Co and Gd sub-\nlattice are assumed. While a large fraction of spins are\nabsorbed in the bottom Co layer, the strong intermixing\nwith Gd contributes to a significant absorption of spin an-\ngular momentum by Gd. Furthermore, it has been shown\nthat the spin coherence length in layered ferrimagnets is\nlargely enhanced compared to ferromagnets [49] leading\nto considerable OSTTs even deeper into the Gd layers.\nUsing the model defined above, we aim at understand-\ning the anomalous dependence of the EXM amplitude as\na function of frequency. Following initial canting, con-\nservation of angular momentum dictates the subsequent\nresonance be around the total angular momentum vector\nLtot=LCo+LGd, enclosing an angle of Ω with the film\nplane (Fig. 4 d, green arrow). The vertical precession\namplitude of MCo, which is the one that is probed dur-\ning the experiment, then depends on Ω and the angle θCo\nthat is enclosed by LtotandMCo(schematically shown\nin Fig. 4 d) according to\nAz= 2|sinθCocos Ω|. (5)\nWe consider two cases; in the linear approximation, the\nprecession frequency is determined only by the degree of\nstatic angular momentum compensation of the sample\nwith fEXM given by eq. (2). In other words, we assume\nthe small angle approximation. Azas a function of fre-\nquencies is fitted to the data by first converting from\nfEXM tocGdand optimizing the free parameters λandδ.\nValues for saturation magnetization and g-factors of Co\nand Gd are given in the Appendix. The dashed red line\nin Fig. 4 a, obtained for the linear case with δ= 16.0◦,\nshows good agreement. This result implies that, after\nthe spin current pulse is fully absorbed, the magnetiza-\ntion vectors of Co and Gd are canted out of the film\nplane by this angle. In Figure 4 bandc, the under-\nlying frequency and amplitude dependence on the alloy\nconcentration are shown. To understand the evolution of\nthe amplitude, we start by considering the Co-dominated\ncase, corresponding to low values of cGd(ortGdin the\ncase of the experiment). Due to the strong degree of un-\ncompensation, the oscillation frequency is high and Ltot\nis very close to LCo. As a result, θCois very small, leading\nto small Az. Upon increasing cGd,fEXM decreases and\nLtotis tilted away from MCo, inducing larger precession\namplitudes. When approaching Lcomp, an injection of\nOOP spins tilts Ltotfully OOP. Therefore, the projec-\ntion of the precession onto the z-axis vanishes as cos Ω\napproaches zero (see Figure 4 c). This explains the de-\ncrease of precession amplitude in our experiments at low\nEXM frequencies close to Lcomp.\nSince the model treated thus far is based on excited\nstates far away from the antiparallel equilibrium state,\none may argue that the commonly used small angle ap-\nproximation does not hold here. To account for finite pre-\ncession amplitudes, we derive the dispersion relation for\narbitrary precession angles in the Appendix. The main\ndifference to the linear approximation is that the EXM\nfrequency remains finite for nonzero canting angles, even\nFIG. 4. aAmplitude of measured EXMs, fitted analytical\nmodels, and Fourier intensity of the optically induced spin\ncurrent pulse as a function of frequency. The fits are based on\nthe frequency ( b) and amplitude ( c) dependencies of EXMs\non the alloy concentration. Designated angles are explained\nby the cartoon in d, showing deflected angular momenta of\nCo and Gd for different alloy concentrations.\nat the compensation point of angular momentum. The\nlarger the precession angles are, the higher is the pre-\ndicted frequency. We repeat the fitting procedure and\nobtain the continuous blue curve in Fig. 4 a. Again, the\nfit agrees reasonably with experimental data. For the\nnonlinear case, only a canting of 3 .5◦is required to ex-\nplain the observed trend in amplitude while the frequency\nis not required to cross zero (see Fig. 4 b). To obtain a\nrealistic estimate for the upper bound of the canting an-7\ngle, we consider the amount of angular momentum that\nis dissipated during ultrafast demagnetization and, based\non this, calculate by how much the Co sublattice can cant\nupon full absorption of said angular momentum. From\nMOKE measurements of the injection stack, we extract\na maximum demagnetization of ∼12.5%. Further, we\ntake a thicknesses of 3.4 nm and 1.6 nm for injection and\nthe sum of both Co layers, respectively, and use the satu-\nration magnetizations given in the Appendix. We find a\ntheoretical maximum of the canting angle of 7 .0◦. In light\nof this, the results from the nonlinear model seem more\nrealistic and can explain all experimental observations.\nThe hitherto assumed equal spin transfer efficiency on\nCo and Gd requires further attention. The EXM fre-\nquency predicted by our model is actually independent of\nthe precise ratio of δCo/δGdand thus, differences in spin\ntransfer efficiencies. Instead, the sum δCo+δGdalone is\ndecisive, meaning that a variety of combinations of δCo\nandδGdcan yield the same frequency. This is due to\nthe assumed isotropy of the system. Possible anisotropy\nfields are much lower than exchange fields between Gd\nand Co and would only induce small perturbations. The\nz-component of the precession amplitude, on the other\nhand, does depend on δCo/δCo. We did confirm that the\napplicability of our model is not strictly limited to the\ncase that δCo=δGdby fitting the data with various ra-\ntios of δCo/δGdand obtaining good results. However, it\nwas not possible to reliably extract the best ratio from the\nfits. Furthermore, different efficiencies and variations in\nabsorption length of Co and Gd might have a minor influ-\nence on the excitation efficiency. Since dramatic changes\nin EXM dynamics were observed among adding less than\none monolayer of Gd, we conclude that the mechanisms\ndiscussed thus far dominate.\nAnother interesting aspect of the dynamics at Lcomp\nis that half an oscillation around Ltotcorresponds to\nswitching the magnetization by 180◦in the film plane.\nTheoretical studies on antiferromagnets [50, 51] came to\nthe similar conclusion that a perpendicular spin current\npulse can manipulate the order parameter of a magnetic\nsystem. Since our TR-MOKE setup is insensitive to\nchanges in IP magnetization, we refrain from claims as\nto whether such switching is realistically possible in our\ndevice. We further note that distinct IP easy axes would\nbe required to establish defined states between which the\nmagnetization can be switched.\nFinally, we will put our results into perspective with\nprevious studies on EXMs. While higher resonance fre-\nquencies than the ones we find have been observed in\ninsulating ferri- and antiferromagnets, [52–54] those ma-\nterials are extremely challenging to be implemented into\ndevices due to their poor conductivity and the lack of\nconduction spin polarization. Exchange resonance modes\nin a similar frequency range ( ∼0.4 THz) have not been\nreported in metallic systems at room temperature to the\nbest of our knowledge. Comparable Ru-based synthetic\nantiferromagnetic oscillators only reach frequencies in the\n∼20 GHz range at zero field [55] which is owed to theRKKY coupling being much weaker than the RE-TM ex-\nchange. The sample design used in our study offers not\nonly a high conductivity in general but also ferromagnetic\ninterfacial layers with high conduction spin polarization.\nConsequently, our layer structure could easily be inte-\ngrated into electronic applications, as magnetoresistive\neffects can be used to probe the precession state on-chip.\nFurthermore, our results deepen understanding on intri-\ncacies in the fascinating platform of Co/Gd multilayers\nthat are highly relevant for other types of spintronic ap-\nplications such as magnetic racetrack memory devices.\nIt is worth noting that the coherent excitation of THz\ndynamics by optically induced spin currents is not limited\nto the sample design used in this work. Instead, it can\neasily be adapted to study high-frequency modes in other\ntypes of ferrimagnets or even antiferromagnets.\nV. CONCLUSION\nWe have demonstrated excitation of ferromagnetic res-\nonance and exchange resonance modes in IP synthetic\nferrimagnetic Co/Gd/Co/Gd multilayers using OSTTs\ngenerated in a neighboring perpendicular magnetic layer.\nOptically induced spin currents were proven to be an\nexcellent tool to excite and study those modes. FMR\noscillations in the 10 GHz range and EXM modes with\nfrequencies up to 0.4 THz were observed at room temper-\nature. Varying the thickness of the Gd layers was inves-\ntigated, enabling an easy path for manipulating the spin\nresonance spectrum over orders of magnitude by tuning\nthe total angular momentum in the system. The de-\npendence of the frequency of the exchange mode gives\nunique insight into the excitation mechanism and pos-\nsible nonlinear dynamics close to compensation. Our\nfindings open up new pathways for the development of\nferrimagnetic THz spintronic devices and exploring their\nhigh-frequency response.\nACKNOWLEDGMENTS\nThis project has received funding from the Euro-\npean Union’s Horizon 2020 research and innovation pro-\ngramme under the Marie Sk lodowska-Curie grant agree-\nment No 861300.\nAppendix: Nonlinear exchange resonance dynamics\nThe equations of motion predicting the time evolu-\ntion of two antiferromagnetically coupled magnetic mo-\nments MCo,Gdin the absence of external fields, magnetic\nanisotropy, and Gilbert damping are given by the follow-8\ning coupled Landau-Lifshitz (LL) equations:\ndMCo\ndt=−µ0γCoMCo×λMGd,\ndMGd\ndt=−µ0γGdMGd×λMCo.(A.1)\nFIG. 5. Exchange resonance frequency aas a function of θCo\naccording to eq. (A.6) and bas a function of δas given by\neq. (A.7) for different alloy concentrations. A change in sign\nof the frequency implies a reversal of the mode’s handedness.\nUpon canting the magnetic moments by an angle δ\nwith respect to the film plane, as explained in the main\ntext, conservation of angular momentum requires the\nconsequent precessions to revolve around Ltot, enclosing\nan angle of\nΩ = arctansin(δ)h\nMCo\nγCo+MGd\nγGdi\ncos(δ)h\nMCo\nγCo−MGd\nγGdi (A.2)with the film plane. The angles of MCo,Gdwith respect\ntoLtotare given by θCo= Ω−δandθGd= Ω+ δ(graph-\nically shown in Fig. 4 d). As the angular momentum\nperpendicular to the rotation axis has to vanish, the fol-\nlowing relation between the two angles holds:\nMCo\nγCosinθCo=MGd\nγGdsinθGd. (A.3)\nEquation (A.1) may now be solved for the precession fre-\nquency assuming an oscillatory solution and identifying\nangles between Hex\nGd,CoandMCo,Gd. The LL equations\nthen simplify to\n2πfM CosinθCo=µ0γCoλMCoMGdsin(θGd−θCo),\n(A.4)\n2πfM GdsinθGd=−µ0γGdλMGdMCosin(θCo−θGd).\n(A.5)\nThe constraint given by eq. (A.3) makes the two solutions\nequivalent. A simple rearrangement of the solution for\nthe Co sublattice yields\nf=µ0γCoλMGdsin(θGd−θCo)\n2πsinθCo(A.6)\n=µ0γCoλMGdsin(2δ)\n2πsinθCo. (A.7)\nThe former equation as a function of θCoand the lat-\nter one as a function of the canting angle δare plotted\nin Fig. 5 aandb, respectively for a variety of alloy com-\npositions and an arbitrarily chosen exchange constant of\nλ= 2. Both solutions reproduce the change in sign of\nthe frequency across Lcomp, implying a reversal of hand-\nedness. Increasing θCoandδboth leads to an increase of\n|f|with respect to the value obtained at θCo=δ= 0. In\nthe Co-dominated region, dispersion curves in Fig. 5 a\nshow discontinuities whenever eq. (A.3) has no real value\nsolution for θGd. Figure 5 bis only zero when Lis\ncompensated and δ= 0. Therefore, any canting away\nfrom the antiparallel ground state in a ferrimagnet will\nresult in an exchange-driven spin resonance. One can\neasily show that eq. (A.6) simplifies to eq. (2) from the\nmain text in the small angle approximation by lineariz-\ning trigonometric functions and substituting eq. (A.3) for\nθGd.\nTABLE I. Saturation magnetizations and g-factors of Co and\nGd used in the macrospin model.\nMCo 1.44 MA/m [56]\nMGd 2.06 MA/m [57]\nMCo/Ni 0.66 MA/m [19]\ngCo 2.17 [56]\ngGd 1.95 [56]9\n[1] J. Finley and L. Liu, Spintronics with compensated fer-\nrimagnets, Appl. Phys. Lett. 116, 110501 (2020).\n[2] S. K. Kim, G. S. D. Beach, K.-J. Lee, T. Ono, T. Rasing,\nand H. Yang, Ferrimagnetic spintronics, Nat. Mater. 21,\n24 (2022).\n[3] Y. Zhang, X. Feng, Z. Zheng, Z. Zhang, K. Lin, X. Sun,\nG. Wang, J. Wang, J. Wei, P. Vallobra, Y. He, Z. Wang,\nL. Chen, K. Zhang, Y. Xu, and W. Zhao, Ferrimagnets\nfor spintronic devices: From materials to applications,\nAppl. Phys. Rev. 10, 011301 (2023).\n[4] E. V. Gomonay and V. M. Loktev, Spintronics of anti-\nferromagnetic systems, Low Temp. Phys. 40, 17 (2014).\n[5] F. Keffer and C. Kittel, Theory of antiferromagnetic res-\nonance, Phys. Rev. 85, 329 (1952).\n[6] T. Jungwirth, X. Marti, P. Wadley, and J. Wunderlich,\nAntiferromagnetic spintronics, Nat. Nanotechnol. 11, 231\n(2016).\n[7] A. Mekonnen, M. Cormier, A. V. Kimel, A. Kirilyuk,\nA. Hrabec, L. Ranno, and T. Rasing, Femtosecond laser\nexcitation of spin resonances in amorphous ferrimagnetic\nGd1-xCoxalloys, Phys. Rev. Lett. 107, 117202 (2011).\n[8] C. D. Stanciu, A. V. Kimel, F. Hansteen, A. Tsukamoto,\nA. Itoh, A. Kiriliyuk, and T. Rasing, Ultrafast spin\ndynamics across compensation points in ferrimagnetic\nGdFeCo: The role of angular momentum compensation,\nPhys. Rev. B 73, 220402(R) (2006).\n[9] A. V. Kimel, A. Kiriliyuk, P. A. Usachev, R. V. Pisarev,\nA. M. Balbashov, and T. Rasing, Ultrafast non-thermal\ncontrol of magnetization by instantaneous photomagnetic\npulses, Nature 435, 655 (2005).\n[10] M. Deb, P. Molho, B. Barbara, and J.-Y. Bigot, Tempera-\nture and magnetic field dependence of rare −earth↔iron\nexchange resonance mode in a magnetic oxide studied\nwith femtosecond magneto-optical Kerr effect, Phys. Rev.\nB94, 054422 (2016).\n[11] S. Parchenko, T. Satoh, I. Yoshimine, F. Stobiecki,\nA. Maziewski, and A. Stupakiewicz, Non-thermal opti-\ncal excitation of terahertz-spin precession in a magneto-\noptical insulator, Appl. Phys. Lett. 108, 032404 (2016).\n[12] T. Kampfrath, A. Sell, G. Klatt, A. Pashkin, S. M¨ ahrlein,\nT. Dekorsy, M. Wolf, M. Fiebig, A. Leitenstorfer, and\nR. Huber, Coherent terahertz control of antiferromag-\nnetic spin waves, Nat. Photonics 5, 31 (2011).\n[13] S. Baierl, M. Hohenleutner, T. Kampfrath, A. K.\nZvezdin, A. V. Kimel, R. Huber, and R. V.\nMikhaylovskiy, Nonlinear spin control by terahertz-\ndriven anisotropy fields, Nat. Photonics 10, 715 (2016).\n[14] T. G. H. Blank, K. A. Grishunin, E. A. Mashkovich,\nM. V. Logunov, A. K. Zvezdin, and A. V. Kimel, THz-\nscale field-induced spin dynamics in ferrimagnetic iron\ngarnets, Phys. Rev. Lett. 127, 037203 (2021).\n[15] G. Malinowski, F. Dalla Longa, J. H. H. Rietjens, P. V.\nPaluskar, R. Huijink, H. J. M. Swagten, and B. Koop-\nmans, Control of speed and efficiency of ultrafast demag-\nnetization by direct transfer of spin angular momentum,\nNat. Phys. 4, 855 (2008).\n[16] T. Kampfrath, M. Battiato, P. Maldonado, G. Eil-\ners, J. N¨ otzold, S. M¨ ahrlein, V. Zbarsky, F. Freimuth,\nY. Mokrousov, S. Bl¨ ugel, M. Wolf, I. Radu, P. M. Oppe-\nneer, and M. M¨ unzenberg, Terahertz spin current pulses\ncontrolled by magnetic heterostructures, Nat. Nanotech-nol.8, 256 (2013).\n[17] G.-M. Choi, B.-C. Min, K.-J. Lee, and D. G. Cahill, Spin\ncurrent generated by thermally driven ultrafast demag-\nnetization, Nat. Comm. 5, 4334 (2014).\n[18] G.-M. Choi, C.-H. Moon, B.-C. Min, K.-J. Lee, and D. G.\nCahill, Thermal spin-transfer torque driven by the spin-\ndependent seebeck effect in metallic spin-valves, Nat.\nPhys. 11, 576 (2015).\n[19] M. L. M. Lalieu, P. L. J. Helgers, and B. Koopmans,\nAbsorption and generation of femtosecond laser-pulse\nexcited spin currents in noncollinear magnetic bilayers,\nPhys. Rev. B 96, 014417 (2017).\n[20] M. L. M. Lalieu, R. Lavrijsen, R. A. Duine, and B. Koop-\nmans, Investigating optically excited terahertz standing\nspin waves using noncollinear magnetic bilayers, Phys.\nRev. B 99, 184439 (2019).\n[21] L. Brandt, U. Ritzmann, N. Liebing, M. Ribow, I. Raz-\ndolski, P. Brouwer, A. Melnikov, and G. Woltersdorf, Ef-\nfective exchange interaction for terahertz spin waves in\niron layers, Phys. Rev. B 104, 094415 (2021).\n[22] Y. L. W. van Hees, P. van de Meugheuvel, B. Koopmans,\nand R. Lavrijsen, Deterministic all-optical magnetization\nwriting facilitated by non-local transfer of spin angular\nmomentum, Nat. Comm. 11, 3835 (2020).\n[23] Q. Remy, J. Igarashi, S. Iihama, G. Malinowski, M. Hehn,\nJ. Gorchon, J. Hohlfeld, S. Fukami, H. Ohno, and S. Man-\ngin, Energy efficient control of ultrafast spin current to\ninduce single femtosecond pulse switching of a ferromag-\nnet, Adv. Sci. 7, 2001996 (2020).\n[24] J. Igarashi, Q. Remy, S. Iihama, G. Malinowski, M. Hehn,\nJ. Gorchon, J. Hohlfeld, S. Fukami, H. Ohno, and S. Man-\ngin, Engineering single-shot all-optical switching of ferro-\nmagnetic materials, Nano Letters 20, 8654 (2020), pMID:\n33226825.\n[25] Q. Remy, J. Hohlfeld, M. Verg` es, Y. Le Guen, J. Gor-\nchon, G. Malinowski, S. Mangin, and M. Hehn, Acceler-\nating ultrafast magnetization reversal by non-local spin\ntransfer, Nat. Comm. 14, 445 (2023).\n[26] J. Igarashi, W. Zhang, Q. Remy, E. D´ ıaz, J.-X. Lin,\nJ. Hohlfeld, M. Hehn, S. Mangin, J. Gorchon, and\nG. Malinowski, Optically induced ultrafast magnetiza-\ntion switching in ferromagnetic spin valves, Nat. Mater.\n10.1038/s41563-023-01499-z (2023).\n[27] M. L. M. Lalieu, M. J. G. Peeters, S. R. R. Haenen,\nR. Lavrijsen, and B. Koopmans, Deterministic all-optical\nswitching of synthetic ferrimagnets using single femtosec-\nond laser pulses, Phys. Rev. B 96, 220411(R) (2017).\n[28] P. Li, T. J. Kools, B. Koopmans, and R. Lavrijsen, Ultra-\nfast racetrack based on compensated Co/Gd-based syn-\nthetic ferrimagnet with all-optical switching, Adv. Elec-\ntron. Mater. , 2200613 (2022).\n[29] M. L. M. Lalieu, R. Lavrijsen, and B. Koopmans, In-\ntegrating all-optical switching with spintronics, Nat.\nComm. 10, 110 (2019).\n[30] A. J. Schellekens, K. C. Kuiper, R. R. J. C. de Wit,\nand B. Koopmans, Ultrafast spin-transfer torque driven\nby femtosecond pulsed-laser excitation, Nat. Comm. 5,\n4333 (2014).\n[31] J. L. Erskine and E. A. Stern, Magneto-optic kerr effects\nin gadolinium, Phys. Rev. B 8, 1239 (1973).\n[32] T. Lichtenberg, M. Beens, M. H. Jansen, B. Koopmans,10\nand R. A. Duine, Probing optically induced spin currents\nusing terahertz spin waves in noncollinear magnetic bi-\nlayers, Phys. Rev. B 105, 144416 (2022).\n[33] M. Beens, R. A. Duine, and B. Koopmans, Modeling ul-\ntrafast demagnetization and spin transport: The inter-\nplay of spin-polarized electrons and thermal magnons,\nPhys. Rev. B 105, 144420 (2022).\n[34] J. Colino, J. P. Andr´ es, J. M. Riveiro, J. L. Mart´ ınez,\nC. Prieto, and J. L. Saced´ on, Spin-flop magnetoresistance\nin gd/co multilayers, Phys. Rev. B 60, 6678 (1999).\n[35] J. Pe lka, W. Paszkowicz, A. Wawro, L. Baczewski, and\nO. Seeck, Structural study of Co/Gd multilayers by x-\nray diffraction and GIXR, Journal of Alloys and Com-\npounds 328, 253 (2001), proceedings of the 5th Interna-\ntional School and Symposium on Synchrotron Radiation\nin Natural Science.\n[36] J. P. Andr´ es, L. Chico, J. Colino, and J. M. Riveiro,\nMagnetic behavior of sputtered Gd/Co multilayers, Phys.\nRev. B 66, 094424 (2002).\n[37] S. Singh, M. A. Basha, C. L. Prajapat, H. Bhatt, Y. Ku-\nmar, M. Gupta, C. J. Kinane, J. Cooper, M. R. Gonal,\nS. Langridge, and S. Basu, Antisymmetric magnetoresis-\ntance and helical magnetic structure in a compensated\nGd/Co multilayer, Phys. Rev. B 100, 140405(R) (2019).\n[38] T. J. Kools, M. C. van Gurp, B. Koopmans, and\nR. Lavrijsen, Magnetostatics of room temperature com-\npensated Co/Gd/Co/Gd-based synthetic ferrimagnets,\nAppl. Phys. Lett. 121, 242405 (2022).\n[39] A. Gurevich and G. Melkov, Magnetization Oscillations\nand Waves , 1st ed. (Taylor & Francis, 1996).\n[40] F. Schlickeiser, U. Atxitia, S. Wienholdt, D. Hinzke,\nO. Chubykalo-Fesenko, and U. Nowak, Temperature de-\npendence of the frequencies and effective damping pa-\nrameters of ferrimagnetic resonance, Phys. Rev. B 86,\n214416 (2012).\n[41] C. Kim, S. Lee, H.-G. Kim, J.-H. Park, K.-W. Moon,\nJ. Y. Park, J. M. Yuk, K.-J. Lee, B.-G. Park, S. K. Kim,\nK.-J. Kim, and C. Hwang, Distinct handedness of spin\nwave across the compensation temperatures of ferrimag-\nnets, Nat. Mater. 19, 980 (2020).\n[42] E. Haltz, J. a. Sampaio, S. Krishnia, L. Berges, R. Weil,\nA. Mougin, and A. Thiaville, Quantitative analysis of\nspin wave dynamics in ferrimagnets across compensation\npoints, Phys. Rev. B 105, 104414 (2022).\n[43] H. Eichler, D. Langhans, and F. Massmann, Coherence\npeaks in picosecond sampling experiments, Opt. Com-\nmun. 50, 117 (1984).[44] C. W. Luo, Y. T. Wang, F. W. Chen, H. C. Shih, and\nT. Kobayashi, Eliminate coherence spike in reflection-\ntype pump-probe measurements, Opt. Express 17, 11321\n(2009).\n[45] J. Hamrle, J. Ferr´ e, M. N´ yvlt, and i. c. v. Viˇ sˇ novsk´ y, In-\ndepth resolution of the magneto-optical kerr effect in fer-\nromagnetic multilayers, Phys. Rev. B 66, 224423 (2002).\n[46] A. J. Schellekens, N. de Vries, J. Lucassen, and B. Koop-\nmans, Exploring laser-induced interlayer spin transfer by\nan all-optical method, Phys. Rev. B 90, 104429 (2014).\n[47] M. Battiato, K. Carva, and P. M. Oppeneer, Superdiffu-\nsive spin transport as a mechanism of ultrafast demag-\nnetization, Phys. Rev. Lett. 105, 027203 (2010).\n[48] M. Battiato, K. Carva, and P. M. Oppeneer, Theory of\nlaser-induced ultrafast superdiffusive spin transport in\nlayered heterostructures, Phys. Rev. B 86, 024404 (2012).\n[49] J. Yu, D. Bang, R. Mishra, R. Ramaswamy, J. H. Oh,\nH.-J. Park, Y. Jeong, P. Van Thach, D.-K. Lee, G. Go,\nS.-W. Lee, Y. Wang, S. Shi, X. Qiu, H. Awano, K.-J.\nLee, and H. Yang, Long spin coherence length and bulk-\nlike spin–orbit torque in ferrimagnetic multilayers, Nat.\nMater. 18, 29 (2019).\n[50] R. Cheng, M. W. Daniels, J.-G. Zhu, and D. Xiao, Ul-\ntrafast switching of antiferromagnets via spin-transfer\ntorque, Phys. Rev. B 91, 064423 (2015).\n[51] M. Weißenhofer, F. Foggetti, U. Nowak, and P. M. Op-\npeneer, N´ eel vector switching and terahertz spin-wave\nexcitation in mn 2Au due to femtosecond spin-transfer\ntorques, Phys. Rev. B 107, 174424 (2023).\n[52] P. Zhang, F. Su, S. Zhang, H. Mei, C. Zhang, X. Luo,\nJ. Dai, and L. Pi, Terahertz magnetic circular dichroism\ninduced by exchange resonance in CoCr 2O4single crys-\ntal, Opt. Express 23, 17805 (2015).\n[53] C. Tzschaschel, K. Otani, R. Iida, T. Shimura, H. Ueda,\nS. G¨ unther, M. Fiebig, and T. Satoh, Ultrafast optical\nexcitation of coherent magnons in antiferromagnetic NiO,\nPhys. Rev. B 95, 174407 (2017).\n[54] W.-H. Hsu, K. Shen, Y. Fujii, A. Koreeda, and T. Satoh,\nObservation of terahertz magnon of kaplan-kittel ex-\nchange resonance in yttrium-iron garnet by raman spec-\ntroscopy, Phys. Rev. B 102, 174432 (2020).\n[55] H. J. Waring, N. A. B. Johansson, I. J. Vera-Marun, and\nT. Thomson, Zero-field optic mode beyond 20 GHz in a\nsynthetic antiferromagnet, Phys. Rev. Appl. 13, 034035\n(2020).\n[56] J. M. D. Coey, Magnetism and Magnetic Materials (Cam-\nbridge University Press, 2010).\n[57] R. O’Handley, Modern Magnetic Materials (Wiley, 1999)."    },    {        "title": "0807.1978v1.Micromagnetic_simulations_of_small_arrays_of_submicron_ferromagnetic_particles.pdf",        "content": "arXiv:0807.1978v1  [cond-mat.mtrl-sci]  12 Jul 2008Micromagnetic simulations of small arrays\nof submicron ferromagnetic particles\nChristine C. Dantas∗and Luiz A. de Andrade†\nDivis˜ ao de Materiais (AMR), Instituto de Aeron´ autica e Es pa¸ co (IAE),\nComando-Geral de Tecnologia Aeroespacial (CTA), Brazil\n(Dated: June 5, 2018)\nAbstract\nWe report the results of a set of simulations of small arrays o f submicron ferromagnetic particles.\nThe actions of dipolar and exchange interactions were quali tatively investigated by analysing the\nferromagnetic resonance spectra at 9 .37 GHz resulting from the magnetization response of con-\nnected and unconnected particles in the array as a function o f the applied dcmagnetic ﬁeld. We\nﬁnd that the magnetization precession movement (at resonan ce) observed in individual particles in\nthe array presents a distinctive behaviour (an “amplitude m ismatch”) in comparison to isolated,\none-particle reference simulations, a result that we attri bute to the action of interparticle dipolar\ncouplings. Exchange interactions appear to have an importa nt role in modifying the spectra of\nconnected particles, even through a small contact surface.\nPACS numbers: 75.30.Ds, 76.50.+g, 73.21.-b\n1I. INTRODUCTION\nThe main theoretical framework that deﬁnes micromagnetism has b een established and\nreﬁned several decades ago by Landau, Lifshitz and Gilbert, amon g others1,2,3. Micromag-\nnetism describes magnetic phenomena at scales of ∼10−6−10−9m and oﬀers a formal\nbasis for the comprehension of emerging and collective phenomena in magnetic materials,\nas for instance magnetic domains in ferromagnets3, collective excitations of spins (“spin\nwaves”)4,5,6, etc. It is based on a continuum, semi-classical dynamical model fo r the gyro-\nmagnetic precession, including a phenomenological damping term.\nDuring several years7, micromagnetism has been somewhat overlooked due to diﬃcul-\nties in devising high-resolution (submicrometric) experiments to tes t it. In addition, the\nfundamental equation of micromagnetism, known as the Landau-L ifshitz-Gilbert equation\n(hereon, LLG)3, can only be solved analytically for special cases or supposing consid erable\nsimpliﬁcations for the description of the problem8. This situation changed rapidly in the\npast few years, with the development of improved experimental te chniques and by the use\nof numerical simulations with exponentially increasing performance r ates, allowing a deeper\nunderstanding of the physical phenomena encompassing the realm of magnetism at submi-\ncroscopic scales9,10,11. These studies mainly aim at the application of microelectronics and\ndata storage12. In particular, the study of the role of the dynamics of conﬁned sp in waves\nin patterned arrays of magnetic elements in thin ﬁlms13,14,15,16,17is of great interest in those\napplications, given that the excitations of spin waves eﬀectively limit t he time scale of the\nmagnetization reversal process.\nThe main purpose of this work is to study the ferromagnetic resona nce (hereon, FMR)\nspectra emerging from small arrays of ferromagnetic particles, w ith diﬀerent spatial conﬁg-\nurations and geometries, using as references single, isolated one- particle simulations. The\nidea is to identify qualitatively important contributions fromdipolar an d exchange couplings\nbetween spins as a function of the array conﬁguration. In order t o obtain an overall un-\nderstanding of these contributions, we have studied conﬁguratio ns varying from particles\nseparated by a small distance, to particles slightly touching each ot her, or being arbitrarily\nconnected with ferromagnetic material. The present simulations we re shown to be com-\nputationally demanding, so at this time we have not surveyed the pro blem through an\nexhaustive number of array conﬁgurations, but instead focused on general trends from some\n2representative cases.\nThis paper is organized as follows. A theoretical background summa ry and the time\ndomain micromagnetic simulation setups are given in Sec. I. In Sec. II , we describe the\nFMR spectra of the array conﬁgurations and the equilibrium magnet ization ﬁelds. In Sec.\nIV, we discuss the results.\nII. MICROMAGNETIC SIMULATIONS\nA. Theoretical Background\nLet the magnetization vector /vectorMbe deﬁned as the sum of Nindividual magnetic mo-\nments/vector µj(j= 1,...,N) in a small volume dVat a given position /vector rof a ferromagnetic\nparticle, namely: /vectorM(/vector r)≡PN\nj=1/vector µj\ndV. Micromagnetism assumes that the direction of /vectorMvaries\ncontinuously with position3. The dynamics of the magnetization ﬁeld /vectorM(/vector r,t) under the\naction of a external magnetic ﬁeld /vectorHextis that of a precession movement of /vectorMaround an\neﬀective magnetic ﬁeld /vectorHeff, deﬁned as /vectorHeff≡ −µ−1\n0∂Eeff\n∂/vectorM. In this expression, Eeffrep-\nresents the energy associated with the eﬀective magnetic ﬁeld, an d is given by the sum of\nfour ﬁelds representing diﬀerent interactions among the magnetic moments (spins) of the\nmagnetic material, namely: Eeff=Eexch+Eanis+Emag+EZee. These four terms are,\nrespectively: the exchange energy, the anisotropy energy, the magnetostatic or dipolar en-\nergy and the Zeeman energy (namely, the energy associated with t he external magnetic ﬁeld\n/vectorHext). The equilibrium state of such a ferromagnetic system is that in whic h the total energy\nis minimized.\nThe Landau-Lifshitz-Gilbert equation mathematically describes the dynamics outlined\nabove, with the following additional physical parameters speciﬁed: the saturation magneti-\nzation,Ms(determined by the temperature, here ﬁxed throughout), the g yromagnetic ratio,\nγ, and a phenomenological damping constant, α. The LLG equation is then given by:\nd/vectorM(/vector r,t)\ndt=−γ/vectorM(/vector r,t)×/vectorHeff(/vector r,t)−γα\nMs/vectorM(/vector r,t)×/bracketleftBig\n/vectorM(/vector r,t)×/vectorHeff(/vector r,t)/bracketrightBig\n.(1)\nNote that the magnetization vector precesses around the /vectorHeffﬁeld and looses energy to the\nenvironment in accordance with the damping constant, tending the refore to align with /vectorHeff,\ngiven suﬃcient time span.\n3The magnetization precession movement can proceed uniformly or n ot. The latter case\nleads to spin waves. Their occurence was conﬁrmed experimentally a nd their behavior has\nbeen the object of several numerical investigations and compara tive studies with the initial\ntheoretical predictions6,18,19. We brieﬂy summarize these possible magnetization precession\nmovements in a small ferromagnet. The uniform precession moveme nt of/vectorM(/vector r,t) about the\ndirection of an eﬀective local ﬁeld will occur at a given frequency ω0. The application of\nan oscillating magnetic ﬁeld /vectorHacatω0, perpendicularly to the former ﬁeld, will result in a\ncoupling of /vectorMand/vectorHacwith energy absorption from the acﬁeld by the system (leading to\nan “uniform resonance” or major peak in the FMR spectrum). But t heacﬁeld may also\ncouple to nonuniform (spin wave) modes of precession of the /vectorMﬁeld. Exchange and dipolar\ninteractions contribute to the energy of these modes. The forme r (exchange) interactions are\nexpected to bemoredominant at physically smaller magnetic elements , leading toadditional\nresonancesatfrequencies ωp=ω0+Dk2\np, accordingtotheKittel’smodel5, whereDisafactor\nthat depends on the exchange interaction between adjacent spin s, andkpis the (quantized)\nwave vector corresponding to a given spin wave excitation in the fer romagnet. The resonant\npeaks associated with the exchange interactions lie at the left of th e uniform peak13,14,\nthat is, at smaller values of the dccomponent of the applied ﬁeld. The later (dipolar)\ninteractions, on the other hand, are relatively independent of the size of the ferromagnet,\nand may be important in a lattice of ferromagnetic elements13, leading to an interparticle\ndipolar coupling ﬁeld.\nIn order to probe the spin standing modes of coupled dots, two equ ivalent methods are\navailable. The ﬁrst one ﬁxes the frequency of ansmall amplitude app liedacﬁeld fordiﬀerent\nvalues of a static magnetic ﬁeld13. This is the method adopted in the present work (see next\nsection). The second one ﬁxes the static ﬁeld and probe the frequ encies of all the (quasi-\nuniform and non-uniform) modes. The former method is suitable to c ompare with FMR\nresults, the latter with Brillouin scattering (BLS) results?. The behavior and character of\nthe modes arethe same. The latter method hasbeen developed sinc e about 2000 (pioneering\nwork of Jorzick et al.?) and since then the dipolar-exchange nature and the symmetry of\nthe spin modes of submicrometric dots has been fully understood. I n particular, modes with\nnodal planes either parallel or perpendicular to the static applied ﬁe ld and edge modes have\nbeen identiﬁed, in addition to the quasi-uniform mode (see Gubbiotti et al.17and references\ntherein). It is accepted that the modes with nodal planes parallel t o the magnetization\n4are high frequency modes (or equivalently on the “left” of the FMR p eak) and modes with\nnodal planes perpendicular to the magnetization can exhibit freque ncies lower and higher\nthanthequasi-uniformmode(or, inotherwords, onthe“right”an d“left”oftheFMRpeak),\ndepending onthenumber ofnodesandthebalancebetween thedipo larandexchange eﬀects.\nThedescriptionofspinwaves inamagneticelement canbegiveninanalo gytoavibrating\nmembrane15, butinthemagneticcasetwo additional“restoringforces”takep lace, leading to\na more complicated description of the normal modes than in the memb rane case, where the\ndescription ismadeintermsofsinusoidal standing waves with aunique restoring forceacting\non the membrane. A thorough review of conﬁned spin waves can be f ound in Demokritov\net al.20.\nB. Simulations Setup\nWe have performed micromagnetic simulations based on the numerica l integration of the\nLLG equation (c.f. Eq. 1) using the freely available integrator OOMMF (Object Oriented\nMicromagnetic Framework)21. In the present work, we have focused our investigations on\na set of diﬀerent small ferromagnetic (permalloy Ni80Fe20) circular particle arrays with a\nsmall, ﬁnite thickness. Full (3D) simulations of the magnetization vec tor dynamics were\nperformed, and interparticle magnetostatic or dipolar interaction s were explicitly consid-\nered in the computations. Simulations of isolated, single particles, we re also performed for\ncomparison purposes.\nThe main global parameters of the OOMMF simulator are listed in Tab. I , and were\nﬁxed for all simulations here considered. Only parameters depende nt on the speciﬁcation\nof the particle geometries diﬀer (these can be seen in Fig. 1), as well as the number of\nexternal applied dcmagnetic ﬁelds used to generate the FMR spectrum in each simulation .\nTable II lists speciﬁc data of the simulations that may diﬀer from each other, e.g.: the\ndiameter of the particles ( d), the diameter of the array ( D), the interparticle spacing ( a;\ni.e., the distance between the centers of two adjacent particles, o r lattice spacing) and other\nsimulation parameters, to be discussed in a moment.\nThe arrays hold a small number of particles each, arranged in a regu lar, 2×2 (four\nparticles, B-“family”) or 3 ×3 (nine particles, A-“family”), grid (2D square lattice); see\nFig. 1. In the case of four particle arrays, we have studied three t ypes of conﬁgurations:\n5equally spaced particles (B1 conﬁguration), equally spaced particle s with arbitrary connec-\ntions among them (B2; in this case we deﬁne the interparticle spacing aas that of B1,\nalthough the particles are connected among themselves)22, and particles “touching” each\nother (B3). As a reference, we have considered isolated particles (labelled Z0, A0 and B0),\nwhich diﬀer slightly in diameter, according to the particle diameters of the corresponding\nconﬁgurations, as indicated in Fig. 1. Notice that we have generally p rioritised the deﬁn-\ntion of the array diameter with the choice of more “rounded” values for the discretization\n(deﬁned by the “cell size” parameter; see Tab. II) over individual particle diameters, but\nsuch a choice is immaterial; the eﬀect of diﬀerent relative particle diam eters are considered\nin the analysis. The Z0 particle simulation was included in order to compa re with previous\nwork by Jung et al.13,14.\nLarger arrays of particles were not considered at the present tim e, given the high com-\nputational demand of these simulations. Indeed, in order to obtain accurate results, the\nvalue of the cell size should not exceed the exchange length8, deﬁned as lex=/radicalbig\n2A/(µ0M2s),\nwhich in the present work results in lex∼5.7 nm. Most of the simulations were executed on\na 3 GHz Intel Pentium PC running Kurumin Linux and on a 2GHz Intel Co re Duo running\nMac OS X, and each simulation took between ∼6 to 28 hours of CPU time, depending on\nthe conﬁguration.\nIn order to obtain the ferromagnetic resonance of each of the co nﬁgurations previously\ndescribed, the following prescription was adopted. An external ma gnetic ﬁeld in the plane\nof the particles was applied, formed by two components: a static ( dc) magnetic ﬁeld ( /vectorBdc≡\nµ0/vectorHdc)intheydirection, andavarying( ac)magneticﬁeld( /vectorBac≡µ0/vectorHac)ofsmall amplitude\nin thexdirection, conforming with Jung et al.14:\n/vectorBac= (1−e−λt)/vectorBac,0cos(ωt), (2)\nwith the acﬁeld frequency given by f=ω/(2π) = 9.37 GHz, λ∼f, and/vectorBac,0= 1 mT.\nFig. 2 shows the time dependence and discretization of the applied /vectorBacﬁeld. The values\nof the/vectorBacﬁeld at intervals of 0 .005 ns were used as inputs in the “ﬁeld range” record of\nOOMMF, which in turn were stepped linearly by the simulator. The simula tions were run\nup to 5 ns, resulting in 1000 outputs for each simulation (i.e., one simula tion for each value\nof the/vectorBdcﬁeld), as indicated in Fig. 2. We have performed a set of simulations fo r each\narray conﬁguration by varying the range of the /vectorBdcﬁeld from 0 .00 T to 0 .39 T, at intervals\n6of 0.01 T, resulting typically in 40 simulations for each conﬁguration, as ind icated in the last\ncolumnofTab. II. Thesaturatedregimeisat /vectorBdc>0.5Tandthereforeoutsideouranalysis.\nSome conﬁgurations present a few more than 40 simulations (B0, B1 ); in these cases, the\nadditional simulations were performed more ﬁne-grainly around the resonance peak in order\ntoevaluatewhether signiﬁcantdivergence wasfoundintheresults (seenext section). TheA1\nconﬁguration had a fewer number of simulations due to high computa tional demands; hence\na more coarse-grained “sampling” of the underlying FMR spectrum w as obtained in this\ncase, as compared to the other conﬁgurations. Finally, we point ou t that the magnetization\nﬁeld of the particles was initially aligned to the same direction of the ext ernal/vectorBdcﬁeld.\nIII. RESULTS\nIn Fig. 3, we present the simulated time dependence of the spatially a veraged magneti-\nzation vector /vectorM, in thexdirection, normalized by the saturation magnetization ( Ms). The\nvarious curves represent the resulting time dependence for each array/particle simulation\nevaluated at their respective FMR peak outputs, to be discussed b elow. The curves were\noﬀset for clarity. The A0 simulation is not shown due to the fact of be ing very similar to\nthe B0 one, so it is omitted for clarity. The time dependence of the /vectorBacﬁeld (arbitrarily\nnormalized in the main ﬁgure and in the insets) is also included for compa rison purposes.\nThe insets in Fig. 3 correspond to zoom-in regions of the initial time ev olution (left inset)\nand the steady state regime (right inset). Notice that all magnetiz ation ﬁeld responses are\nout of phase with the applied ﬁeld after transient eﬀects vanish, an d the phase responses\nare practically identical in all cases, only diﬀering in amplitude. Intere stingly, the B3 con-\nﬁguration starts oﬀ at a diﬀerent phase, but catchs up soon afte r the ﬁrst cycle. From that\nﬁgure it is already clear that the magnetization ﬁeld of all array simula tions have, on aver-\nage, a smaller response to the external ﬁeld than the correspond ing reference, one-particle\nsimulations (represented by the B0 simulation in the ﬁgure).\nIn order to obtain the FMR spectra of the conﬁgurations, we proc eeded as follows. The\nﬁrst 3 ns of all data have been excluded. For each simulation in a given conﬁguration\n(i.e., for each applied dcmagnetic ﬁeld), the Fourier transform of the spatially averaged\nmagnetization vector /vectorM, in thexdirection, was obtained (as already mentioned, Fig. 3\nrefers to the results at the FMR peak only). The amplitude of the ma ximum Fourier peak\n7at each/vectorBdcﬁeld was then obtained, resulting in the FMR spectra of Fig. 4. In Fig. 5,\nwe show the derivatives of the FMR spectra. Regarding the main bod y of Fig. 4, we\nhave applied a spline ﬁt to the reference single particle simulations’ da ta (Z0, A0, B0), but\nmaintained the individual data points of the array simulations. In the insets of Fig. 4,\nhowever, all data has been spline ﬁt to facilitate the comparison of t he overall behavior of\nthe curves. The derivative FMR spectra (Fig. 5) were obtained by d erivation (in intervals\nof 0.022 T) of the latter splined curves.\nWe report the following three overall observations. First, the res onance uniform mode\npeaks of the reference single particle simulations show the expecte d trend14, namely, a shift\nin the peak position as a function of the diameter of the particle (the peak position shifts\ntowards smaller values of the external ﬁeld /vectorBdcas the particle diameter increases). The\ntrend is very small, given that the particles diﬀer only slightly in diamete r. Second, (i)the\nresonant uniformmodepeakposition oftheB1 arrayconﬁguration is shiftedtowards smaller\nvalues of the external ﬁeld /vectorBdcas compared to the reference (B0) one-particle simulation,\neven though the particles that compose the B1 array have the sam e diameter of the B0\nparticle. Also, as already observed in Fig. 3, (ii)the amplitude of the uniform mode peak\nof the B1 conﬁguration is smaller than that of B0. Finally, (iii)the secondary peak at the\nright of the uniform mode peak in B0 (also clear in Z0 and A0, see arrow s in Fig. 5) is\nno longer apparent in B1. These statements, namely, (i)-(iii), are also applicable to the\nA1 conﬁguration, although the eﬀects seem to be slightly more pron ouced in this case, as\none can notice by inspecting the derivative FMR spectra of Fig. 5 (se e the dashed lines\nconnecting the corresponding resonance peaks). Third, as the p articles of the B1 array\nare arbitrarily connected (B2) or approach to the point of “touch ing” each other (B3), the\nresulting spectra signiﬁcantly evolve away from the B0-B1 spectra :(a)the amplitude of the\nuniform mode peak of both B2 and B3 decrease and the overall spec trum gets more spread,\nwith the presence of secondary contributions at the left of the un iform peak position (B2\nand B3), as well as to the right of it (in the case of B2). (b)The B2 uniform mode peak\ndoes not appear to shift in position relatively to the B1 peak, wherea s the B3 does, towards\nhighervalues of the /vectorBdcﬁeld.\nIn order to better understand the observed characteristics of the FMR spectra, we anal-\nysed in more detail the aspect of the magnetization ﬁelds at the unif orm resonance peak.\nIn Fig. 6 the “snapshots” of the magnetization vector ﬁeld (at FMR ) at four points of the\n8cycle (ωt= 0,π/2,π,3π/2) are shown, being selected from a cycle around ∼4 ns (when\ntransient eﬀects are over). Diﬀerent pixel tonalities correspond to diﬀerent values of the x\ncomponent of the magnetization vector ﬁeld, which in turn was subs ampled to display an\narrow for the average of 9 vectors per cell element. Notice that w e have included a sinusoidal\nfunction at the top marking the four points corresponding to the s elected “snapshots”, so\nthat immediately below each point of the reference curve the magne tization state at the\ncycle point can be directly observed and compared with that of the r eference single particle\nsimulation.\nThe most obvious feature of Fig. 6 is that the individual particles com posing the array\nconﬁgurations do not oscillate in synchrony with the corresponding single particle simula-\ntions. When averaging out the magnetization ﬁeld over the array pa rticles, the emerging\nresponse is lower than that of single particles, resulting in the obser ved lower amplitude\nof the uniform peak resonance of the spectra (Figs. 4 and 5). Ind eed, one could natu-\nrally expect that the individual particles in the array conﬁgurations B2 and B3 would have\ntheir magnetization ﬁeld evolving somewhat diﬀerently than that of t he single particle sim-\nulations, since these conﬁgurations physically join the particles of t he array, removing the\ncircular geometry characterization of the individual particles (for ming eﬀectively a “larger”\nsingle particle with a diﬀerent geometry). But in the case of the A1 an d B1 conﬁgurations,\nthe most suspicious agent causing the desynchronization would be a n interparticle dipolar\ncoupling ﬁeld.\nIn order to illustrate more clearly the behavior of the oscillating magn etization ﬁelds,\nwe have synthetized the main information of the oscillating pattern in to a “correspondence\nscheme”. We have used the single particle simulations to associate a c ircular symbol of a\ndiameter proportional to a given cycle point aspect (the pixel tona lities distribution) of the\nmagnetization ﬁeld. Fig. 7 illustrates the correspondence scheme a dopted. We have used\nthis correspondence scheme to recast the previous results (Fig. 6) into the diagrammatical\nforms of Fig. 8 (excluding the B2 conﬁguration, which cannot be rep resented in a simple\nmanner in the scheme). Under this new representation, the under lying oscillatory patterns\nare more clearly displayed or synthetized. One can see that the sus pected desynchronization\nis an “amplitude mismatch”: the amplitude of the x-component of the magnetization as a\nfunction of position in the magnet, in each of the particles i n a given array, is not similarly\ndistributed among the particles .\n9Another interesting point is the following. If one recast the above r epresentation into\na matrix representation (B conﬁgurations as 2 ×2 matrices and the A1 conﬁguration as a\n3×3 matrix), one can observe that the B1,jelements behave in opposition to the B2,jones\n(j= 1,2). In the case of the A1 matrix, one can notice a similar trend: the e lements in the\nA11,jandA13,jrows (j= 1,...,3) evolve in opposition to each other. The central row A12,j\n(j= 1,...,3) appears to evolve independently.\nIV. DISCUSSION\nThe simulations of small arrays of ferromagnetic particles here per formed indicate that\ntheir spatial conﬁgurations inﬂuence the resulting FMR spectra in d istinctive ways. Here we\ndiscuss the possible eﬀects of these spectra from the point of view two physical interactions:\ndipolar and exchange mode interactions.\n•Dipolar interactions between the particles:\nA1 and B1 conﬁgurations: since in these cases the particles are not physically in\ndirect contact, we may presume that the dipolar interaction is caus ing the moderate\ndecrease in amplitude of the uniform mode peak in comparison to the o ne-particle\nreference simulations. Indeed, one can suppose that the ﬁnitene ss and symmetry of\nthe arrays over which the particles are distributed favor an interp article dipolar inter-\naction for which the combined eﬀect is that of a mismatching of the loc al amplitudes\nattained by the magnetization ﬁeld precession movement (observe d at the uniform\nmode outputs). Another observation is the shift of the uniform mo de peak towards a\nsmaller value of the applied dc magnetic ﬁeld, in comparison to the one- particle ref-\nerence simulations. It mimmics the expected spectrum of an eﬀectiv e single particle\nof larger diameter. One may notice that in the case of the A1 conﬁgu ration the whole\nspectrum seems to shift accordingly (see dashed lines of Fig. 5); th is does not seem to\nbe the case of the whole B1 spectrum. The eﬀect, clearly seen in the A1 conﬁguration,\nhas already been experimentally measured with BLS and calculated by Gubbiotti et\nal.?with the second alternative method mentioned in Sec. II.A for a similar 3×3\narray. The shift towards lower values of the static ﬁeld of the mode s corresponds to\nthe increase of the mode frequencies in Fig. 3 of Gubbiotti et al. A dir ect compar-\nison between the modes found in the simulation of Gubbiotti et al. and the ones in\n10the present work is diﬃcult because in the former case the dynamic m agnetization is\nshown, while in this work (c.f. Fig. 6) the total magnetization (static plus dynamic)\nis shown. We intend to proceed these investigations in a future work . Regarding the\nnonuniform modes of precession, one can observe the following fac ts. First, the lowest\nenergy mode (at the right of the uniform mode), seen in the refere nce-one particle sim-\nulations (see arrows in Fig. 5 over the derivative spectra of Z0, A0 a nd B0), diminish\nfor theA1 and B1 conﬁgurations(one cannot aﬃrm that they comp letely vanish, given\nthe smothing over a limited resolution in /vectorBdcused to derive the spectra). Second, a\nrelatively larger gain in response of the nonuniform modes at the left of the uniform\nmode peak is found (c. f. main body of Fig 5). This will be discussed in mo re detail\nbelow, since these peaks are thought to arise from exchange inter actions, although will\npossible contributions from dipole interactions.\nB2 and B3 conﬁgurations: In these cases, the particles are directly in contact. In\nthe case of B3 conﬁguration, the “amplitude mismatch” eﬀect seen in the previous\ncases is again clearly observed (see Figs. 6 and 8). Yet, from these ﬁgures one can\nqualitatively observe that, looking at individual particles in the array , the magneti-\nzation ﬁeld precess in smaller amplitude as compared to the B0, B1 con ﬁgurations,\nalthough in similar pattern as the latter, resulting in a smaller relative a mplitudes of\nthe average magnetization ﬁeld, which one can conﬁrm from an inspe ction of Fig. 3.\nAlthough we have not traced an ocillatory pattern scheme for the B 2 conﬁguration, it\nis remarkable that it somewhat seems to follow the pattern observe d in the B1 conﬁg-\nuration (as one can observe looking only at the central parts of th e particles of the B2\narray, excluding the dynamics occuring at their connections, c.f. F ig. 6), from which\nthe latter diﬀers from the former only by the application of arbitrar y connections.\nThe amplitude of the average magnetization ﬁeld of the B2 conﬁgura tion is greater\nthan that of the B3 conﬁguration, possibly because the contribut ion coming from the\ncentral region of the particles of the B2 conﬁguration can follow so mewhat that of the\nB1 ones, yet being signiﬁcatively smaller than the latter ones becaus e of the averaging\nover the ﬁeld behavior of the connected parts. Evidently, the res ulting spectrum will\nbe more complex in this case, with, for instance, the presence of a n ew nonuniform\npeak at the far right of the uniform one (see the arrow over the B2 curve of Fig. 5).\nAnother distinct aspect of the B3 conﬁguration is the shift of the u niform mode peak\n11towards a largervalue of the applied dcmagnetic ﬁeld, in comparison to the one par-\nticle reference simulation, contrary to the trends of the A1 and B1 , B2 conﬁgurations,\nin which the peaks move to the opposite direction. The origin of the B3 shift is also\nunclear at this point and deserves further investigation.\n•Exchange mode interactions:\nA1 and B1 conﬁgurations : In these cases, the ﬁrst nonuniform peak at the left of\nthe uniform one does not appear to suﬀer signiﬁcant changes in amp litude as com-\npared to the reference one-particle simulations. However, as alre ady mentioned, the\nspectrum of the A1 conﬁguration is clearly shifted as a whole to the le ft. The other\nhigher harmonics to the left of the uniform mode from both A1 and B1 seem to have\nslightlyhigher amplitudesincomparisontotheone-particlereferenc e simulation(spec-\ntra between /vectorBdc= 0,...,0.08 T). Since the particles in these conﬁgurations are not in\nphysical contact, the energy of these nonuniform peaks, under stood to be mainly the\nresult of exchange mode interactions, must have been suplemente d by the injection\ncoming from other means. In this case, dipolar interaction is a reaso nable candidate,\nsince the particles do not touch each other. This assertion would ag ree with the hy-\npothesis raised by Jung et al.13on the possible coupling of the dipolar interactions\nwith exchange spin wave modes.\nB2 and B3 conﬁgurations: In these cases, as expected, the spectra evolve in a\ncomplex wayincomparisontothereference simulations. Inparticula r, onecanobserve\na relatively large gain in response of the nonuniform modes (Fig 5). As observed\npreviously, a fraction of this gain should be a result of the coupling of the dipolar and\nexchange interactions. But since the particles of these arrays ar e in physical contact,\nthe exchange interactions should have a more important role. One r emarkable aspect\nis the form of the B3 conﬁguration spectrum which is quite diﬀerent f rom that of\nthe B1 conﬁguration, and in some aspects approach more closely to that of the B2\nconﬁguration (c.f. Fig. 5). This fact implies that propagative eﬀect s arising from\nexchange interactions, even when are able to act through a small c ontact surface, can\nresult in signiﬁcant modiﬁcations in the FMR spectrum.\nFinally, an important remark should be mentioned. According to the w ork of Gubbiotti\net al.?, the eﬀect of the interparticle dipolar coupling is to split the modes, s preading them\n12into bands, in the limit of large arrays. In the case of a 3 ×3 array, each mode should give\nrise to 9 modes, including degeneracy. Gubbiotti et al. have shown t hat this broadening is\nappreciable, even large, for the quasi-uniform mode. However, th is eﬀect is not visible in\nthe present study (c.f. cases A0-A1 and B0-B1; Figs. 4 and 5). Th e reasons for the lack of\nsplitting are currently under investigation.\nIn summary, in the present paper we have presented a set of 3D sim ulations of small\narrays of ferromagnetic particles supposed here to represent s mall isolated sections of a\npatterned thin ﬁlm. We have analysed the resulting FMR spectra and the magnetization\nﬁeld behavior at the resonance modes. We show that the spatial co nﬁgurations and ge-\nometries of the particles in the arrays inﬂuence the resulting FMR sp ectra in distinctive\nand perhaps unanticipated ways. We have attempted to isolate the action of dipolar and\nexchange interactions by studying arrays with particles both conn ected and not connected\namong themselves. These interactions appear to have an interest ing role on the dynamics of\nthe magnetization precession among the particles in the array, as d escribed in detail in this\npaper (synthetized in Fig. 8). Other simulations are intended to be p erformed in a future\nwork. In particular, it would be interesting to perform simulations wit h the applied ﬁeld at\ndiﬀerent nominal angles (specially, at 45◦), using the same arrays here analysed. According\nto measurements of Jung et al.13, the typical low energy peak of one-particle simulations\nappear more pronounced in these cases. Since this low energy peak is suspected to arise\nmainly from dipolar interactions, it would serve as an interesting trac er of the role of these\ninteractions in the array.\nAcknowledgments\nWe would like to thank the attention and technical support of Dr. Mic hael J. Donahue\nin the initial phases of this project. We also wish to acknowledge the s upport and encour-\nagement of Dr. Mirabel C. Rezende throughout this work. Finally, w e thank the referees\nfor useful comments and corrections, which helped to improve the paper considerably.\n13TABLE I: Main global parameters adopted for the OOMMF simula tor, ﬁxed for all simulations\nin the present work, except for the “particle width/height” and “cell size” parameters, which are\nlisted separately in Tab. II, as indicated.\nSimulation Parameter/Option Parameter Value/Option\nSaturation magnetization [A/m] 8 .0×105\nExchange stiﬀness [J/m] 1 .3×10−11\nAnisotropy constant [J/m3] 0 .0\nDamping constant 0 .05a\nGyromagnetic ratio [m/(A.s)] 2 .21×105\nParticle thickness [nm] 85 .0\nParticle width/height see DinTable II\nCell size see Table II\nDemagnetization algorithm type magnetization constant in each cell\naThe value of the damping constant here adopted is far larger than t he real one for Permalloy ( ≤0.01).\nA small value of damping constant would allow a better resolution of th e absorption lines within the FMR\nspectra but would lead to prohibitive computation times.\n14TABLE II: Particular parameter values of the simulations: t he diameter of the particles ( d), the\ndiameter of the array ( D; for individual particles, D=d.), the interparticle spacing ( a, when\napplicable), and cell size. Other data are: the number of fer romagnetic particles (column 2) and\nthe number of external applied dcmagnetic ﬁelds (or number of simulations) used to generate t he\nFMR spectrum (column 7).\nLabel # FM particles d(µm)D(µm)a(µm) Cell size (nm) /vectorBdc(number of simulations)\nZ0 1 0 .500 0 .500 – 5 .00 40\nA0 1 0 .559 0 .559 – 5 .59 40\nB0 1 0 .591 0 .591 – 5 .91 46\nA1 9 0 .559 1 .900 0 .671 5 .00 26\nB1 4 0 .591 1 .300 0 .709 5 .00 43\nB2 4 0 .591 1 .300 0 .709 5 .00 40\nB3 4 0 .591 1 .200 0 .591 6 .00 40\n15FIG. 1: Particle conﬁguration geometries, arranged in “fam ilies” of common characteristics.\n160 1 2 3 4 5\nTime [ns]-0.001-0.000500.00050.001 B_ac [T]\nSampling= 0.005 ns\nFIG. 2: The time dependence and discretization of the applie d/vectorBacﬁeld.\n170 1 2 3 4 5\nTime [ns]-0.400.40.8Amplitude [arb. units.]A1\nB3\nB2\nB1\nB0\nB_ac 0 0.1 0.2 0.3-0.04-0.0200.020.04\n4.5 4.6 4.7 4.8 4.9\nTime [ns]-0.04-0.0200.020.04\nFIG. 3: Time dependence of the amplitude (at FMR) of the spati ally averaged magnetization\nvector/vectorMx/Msof the array conﬁgurations. In the main body of the ﬁgure, the curves were oﬀset\nfor clarity, and are shown at the same presentation order as t he legend (top curve is that of the A1\nconﬁguration). The time dependence of the external /vectorBacﬁeld (arbitrarily normalized; thin line) is\nincluded.\n180 0.1 0.2 0.3 0.4\nB_dc [T]0123456Absorption [arb. units]Z0\nA0\nB0\nA1\nB1\nB2\nB3\n00.20.404812\nZ0A0B000.20.404812B\n0\n1\n2\n3\n00.20.4048A\n0\n1\nFIG. 4: The ferromagnetic spectra of all conﬁgurations.\n190 0.1 0.2 0.3 0.4\nB_dc [T]Absorption [arb. units] B0A0\nB1A1\nB2\nB3Z0\nFIG. 5: The derivative ferromagnetic spectra of all conﬁgur ations.\n20FIG. 6: “Snapshots” of the magnetization vector ﬁeld (at FMR ) at four points of the cycle ( ωt=\n0,π/2,π,3π/2; see sinusoidal curve at the top of the snapshots) after ∼4 ns (when transient eﬀects\nareover). Diﬀerent pixeltonalities correspondtothevalue ofthexcomponentofthemagnetization\nvector ﬁeld at the pixel element. 21FIG. 7: The oscillatory pattern correspondence scheme (see the main text form further details).\nFIG. 8: Same as Fig. 6, recasted according to the scheme of Fig . 7, synthetizing the combined\neﬀect of the local amplitude of the x-component of the magneti zation ﬁeld at each cycle point of\ntheacdriving ﬁeld, for conﬁgurations B1, B3 and A1.\n22∗Electronic address: ccdantas@iae.cta.br\n†Electronic address: andrade@iae.cta.br\n1E. Landau, L. & Lifshitz, Physik. Z. Sowjetunion (1935).\n2T. L. Gilbert, Phys. Rev. (1955).\n3W. F. Brown, Micromagnetics (Interscience Publishers, 1963).\n4F. J. Dyson, Physical Review 102, 1217 (1956).\n5C. Kittel, Physical Review 110, 1295 (1958).\n6K. Baberschke, Physica Status Solidi (b) 245, 174 (2007).\n7A. Aharoni, Physica B (2001).\n8M.d’Aquino, Doctorate ThesisinElectrical Engineering, U niversit` a Degli StudidiNapoli, It´ alia\n(2004).\n9J. Fidler and T. Schreﬂ, Journal of Physics D Applied Physics 33, 135 (2000).\n10S. Mørup, D. E. Madsen, C. Frandsen, C. R. H. Bahl, and M. F. Han sen, Journal of Physics\nCondensed Matter 19, 3202 (2007).\n11S. Gliga, M. Yan, R. Hertel, and C. M. Schneider, Phys. Rev. B 77, 060404(R) (2008).\n12D. M. Newns et al., IBM J. Res. & Dev. 48, 173 (2004).\n13S. Jung, B. Watkins, L. Delong, J. B. Ketterson, and V. Chandr asekhar, Phys. Rev. B 66,\n132401 (2002), arXiv:cond-mat/0109307.\n14S. Jung, J. B. Ketterson, and V. Chandrasekhar, Phys. Rev. B 66, 132405 (2002).\n15M. Bailleul, R. H¨ ollinger, and C. Fermon, Phys. Rev. B 73, 104424 (2006).\n16C. Yu, M. J. Pechan, W. A. Burgei, and G. J. Mankey, Journal of A pplied Physics 95, 6648\n(2004).\n17G. Gubbiotti, G. Carlotti, T. Okuno, M. Grimsditch, L. Giova nnini, F. Montoncello, andF. Niz-\nzoli, Phys. Rev. B 72, 184419 (2005).\n18M. Plihal et al., Phys. Rev. Lett. (1999).\n19I. Neudecker et al., Phys. Rev. B 73, 134426 (2006).\n20S. O. Demokritov, B. Hillebrands, and A. N. Slavin, Phys. Rep .348, 441 (2001).\n21M. Donahue and D. Porter, Interagency Report NISTIR 6376, Na tional Institute of Standards\nand Technology, Gaithersburg, MD (1999), URL http://math.nist.gov/oommf/ .\n2322Ourmotivation for consideringthe B2conﬁguration was the s tudyof therelative contribution of\ndipolar and exchange interactions for complex geometries. In addition, as one can observe in the\npaperofJunget al.13(their Fig. 1), thepatterningprocedureappearsnottobepe rfectandoften\nproduces artifacts between the dots. We have attempted to in clude arbitrary “bridges” between\nelements, with no particular criteria for their geometrica l shape. Finally, the understanding\nof the micromagnetics of such complex geometries is being cu rrently conducted for speciﬁc\napplications of interest at the AMR/IAE/CTA Division.\n24"    },    {        "title": "1703.10630v1.Study_of_spin_pumping_in_Co_thin_film_vis_a_vis_seed_and_capping_layer_using_ferromagnetic_resonance_spectroscopy.pdf",        "content": "1 \n Study of spin pumping in Co thin film vis -à-vis seed and capping layer \nusing ferromagnetic resonance spectroscopy  \n \nBraj Bhusan Singh , Sukant a Kumar Jena, Subhankar Bedanta * \nLaboratory for Nanomagnetism and Magnetic Materials (LNMM), School of Physical \nSciences, National Institute of Science Education and Research (NISER), HBNI, P.O. - \nBhimpur Padanpur, Via – Jatni, Odisha, Pin - 752050 , India   \n \nAbstract : We investigated the dependence of the seed (Ta/Pt, Ta/Au) and capping (Pt/Ta, \nAu/Ta ) layer s on spin pump ing effect in the ferromagnetic 3 nm thick Co thin film using \nferromagnetic resonance spectroscopy . The data is  fitted with Kittel’s equation  to evaluate \ndamping constant and g-factor.  A strong  dependen ce of seed and capping layer s on spin \npumping has been  discussed . The value of damping constant  (α) is found to be relatively \nlarge  i.e. 0.0326±0.0008  for the Ta(3)/Pt(3)/Co(3)/Pt(3)/Ta(3) (nm) multilayer struc ture, \nwhile it is 0.0104±0.0003 for  Ta(3)/Co(3)/Ta(3) (nm) . Increase in α is observed  due to Pt  \nlayer that  works as a good sink for spins due t o high spin orbit coupling. In addition, we \nmeasured the effective spin conductance  \ng = 2.00 ± 0.08 × 1018 m-2 for the tri -layer \nstructure Pt(3)/Co(3)/Pt(3 ) (nm)  as a result of the enhancement in α relative to its bulk value. \nWe observ ed that t he evaluated g-factor decreases as effective demagnetizing m agnetic field \nincreases  in all the studied samples . The azimuthal dependence of magnetic resonance  field \nand line width  showed relatively high  anisotropy in the  tri-layer Ta(3)/Co(3)/Ta(3 ) (nm) \nstructure.   \nPACS numbers:  75.76.+j , 75.78. -n, 75.70.Cn, 76.50.+g   \n*Corresponding author:  \nElectronic mail: sbedanta@niser.ac.in  2 \n INTRODUCTION  \n The generation of pu re spin current and its effect on the switching of the magnetization \nby spin transfer torque via spin orbital torque has been subject of vivid research in last one \ndecade. Apart from basic  research this subject has also implications in potential devices based \non spintronics/spin -orbitro nics  [1–3], spin torque - oscillator   [4,5] , magnetic random \nmemory devices   [6,7] , magnonics  [6] etc. A fundamental understandi ng of the exchange \ninteraction and spin -orbital interaction at the interfaces will increase speed, energy efficiency, \nminiaturization of size and its  multi -function al utilization   [8]. Pure spin cu rrent  is produced \nby asymmetric scattering of the two electrons with opposite spin angular momentum  in the \npresence of spin orbit coupling at the interface  which is known as spin Hall effect   [1,3,9,10] .  \nThis is usually investigated  via ferromagn etic resonance (FMR) in which a microwave \nexcites the spin in ferromagnetic (FM) layers. The spin excitation in the FM layer generates \nand propagates pure spin current into paramagnetic heavy metal (NM) . This phenomenon  is \ncalled as “spin pumping”  and the spin current in to NM is given by  \ndtmdmg sJS \n4ˆ\n ……………………………………………………………………….(1)  \nwhere \nm  is the reduced magnetization (\nSMM ), \n is the reduced Planck’s constant, and \ng  is \nthe effective spin mixing conductance which is governed by the transmission of the spin \ncurrent through FM/NM interfaces  [11]. The understanding of m agnetic dynamics dissipation \nin the material is very important to get sustainable spin current. Normally , magnetic \ndynamics dissipates the pure spin current through Gilbert damping effect, which  is related to \nthe electronic structure  of the ordered material . However, other damping mechanism s e.g. in -\nhomogeneities in the sample, two magnon scattering and interfaces might  also play important \nrole to enhance the damping effect  [12–16]. The interfaces are very importa nt for creating 3 \n pure spin current and its dissipa tion. In addition , it affects  the damping properties critically  by \nvarious seed and capping layers . Recently, research has been focussed  to unde rstand the \neffect of heavy metal as seed  and capping  layers   [17–22]. In a recent report Tokac  et al. [21] \nhave  shown that the  effective spin conductance mixing strongly depends  on the interface s of \nseed (Ta/Cu ) and capping ( Cu or Ir ) layer s. They observed large effective spin mixing \nconductance for Ir as a capping layer  due to its large spin orbit coupling . It is c lear that  \neffective spin mixing conductance critical ly depends on the symmetry of the interfaces with \nrespect to the ferromagnetic layer. In order to understand the effect of seed and capping \nlayers  on spin pumping dynamics  we fabricated  the multilayer str ucture Seed \nlayer /Co/Capping layer . The seed layer and capping layers are the combination of heavy \nmetals  like Ta, Pt, and Au . We studied thin films having symmetric (seed and capping layers \nare same)  and asymmetric (seed and capping layers are different) interfaces around the FM  \nlayer.  \n \nEXPERIMENTAL DETAILS  \nThe samples a re deposited at room temperature in a high vacuum system (Mantis \nDeposition Ltd., UK) having base pressure better than 5.0 × 10-8 mbar. The schematic of the \nsample structure is shown in Fi g. 1(a). The Ta, Cu, and Co  layers are deposited by dc \nsputtering . While Pt and Au thin films are prepared  by rf sputtering and electron beam \nevaporation, respectively. We deposited two sets of samples on Si(100) substrates having \nnative oxide layer. In th e first set of the sample s, we deposited the multilayers \nTa(3)/HMT/Co(3)/ HMB/Ta(3) ( in parenthesis thickness is in nm) in which HMT and HMB \nstands for heavy metal for top and bottom  layers , respectively. Pt and Au are taken as heavy \nmetal s for HMT and HMB due to their high spin orbit coupling.  To compare the results, we  4 \n  \ndeposited second set of samples having tri -layer  structure as shown in Fig. 1(b). The samp le \ndetails are listed in Table I . The deposition rates for Ta, Cu, and Pt are kept to be same and \nare 0.018 nm/sec. Co is deposited at a rate of 0.016 nm/sec. The working pressure during  \n \n \nFIG 1(a) and (b) shows the schematics of generation of spin current in the multilayer and tri -\nlayer sample structures (c) FMR measurement setup in which rf generator creates GHz \nfrequenc y which works as a perturbation field ( hrf) perpendicular to external applied field \n(Hext) and a diode detector through AC magnetic field modulation detects transmitted signal \nvia lock -in based technique.  \nTable I . The list and details of the fabricated samples .  \nSample Name  Sample details  (In parenthesis thicknes s is in nm)  \nS1 Si/Ta(3)/Pt(3)/Co(3 )/Pt(3 )/Ta(3)  \nS2 Si/Ta(3)/Au(3)/Co(3)/Au(3 )/Ta(3)  \nS3 Si/Ta(3)/Au(3)/Co (3)/Pt(3 )/Ta(3)  \nS4 Si/Ta (3)/Pt(3)/Co(3)/Au(3)/Ta(3)  \nS5 Si/Ta(3)/Co(3)/Ta(3)  \nS6 Si/Cu(3)/Co(3)/Cu(3)  \nS7 Si/Pt(3)/Co(3)/Pt(3)  5 \n sputtering deposition was 2.0 × 10-3 mbar. A growth rate of 0.006 nm/sec is used to deposit \nAu thin film at  3.0 × 10-7 mbar working pressure. The deposition rates are measured using  \nquartz crystal monitor. FMR  measurements are performed using NanoOsc Instrument Phase \nFMR in the frequency range of 5 to 17 GHz. The sample i s kept in flip -chip manner on a 200 \nμm wide  coplanar waveguide as shown in the Fig.  1(c). A lock -in amplifier based techniques \nis used to detect the signal in which rf field ( hrf) is perpendicular to the external magnetic \nfield ( Hext). \n \nRESULTS AND DISCUSSIONS  \nFigure 2(a) shows the frequency  dependent re sonance field for the sample S1  to S4.  \nThe open symbols  show  the experimental data  and the solid lines are the fits  with Kittel \nresonance condition  [23]: \n) 4 )( (2eff res K res K M H H H H f   \n……………………………… ………………………  (2)  \nwhere \nBg  is gyromagnetic ratio ; HK, Hres, g, and μB are the in-plane anisotropy field,  the \nresonance magnetic field, Lande g-factor and Bohr magneton, respectively , at the resonance \nfrequency f. The effective demagnetizing field is given by  \nFMSS\nS efftMKM M24 4 \n…………………………………………………………………………. (3) \nwhere Ms is the saturation magnetization, KS is the perpendicular surface anisotropy constant, \nand tFM is the thickness of the ferromagnetic layer.  \nThe fitting with equation (2) to the frequency dependent resonance field graph gives the value \nof g factor, effecti ve demagnetization field  (\neffM4 ) and HK. The data is well fitted with  6 \n \n500 1000 1500 2000 25004681012141618Frequency (GHz)\nH res(Oe)S1 S2 S3 S4\n  Fit  Fit   Fit\nFit  (a) \n4 6 810 12 14 16 18 20100200300400H(Oe)\nFrequency (GHz) S1  S2  S3  S4\n Fit  Fit  Fit\n Fit(b) \nFIG 2(a) Resonance frequencies versus  resonance magnetic field ( Hres) for sample S1  to S4. \nOpen symbols repres ent the experimental values while  solid lines are  fitted with equation (2 ). \n(b) Line width ( H) versus resonance frequencies for sample S1  to S4. Open symbols are \nexperimental data and solid lines are fitted with equation (4).  \n \nreduced chi -square  (χ2) value ~ 10 × 10-4. For fitting we used the procedure described by  \nShaw et al . [24]. The value of damping constant  (α) has been evaluated using the \nequation  [25]  \nfH H4\n0\n……………………………………………………………… ...................... ..…(4) \nwhere H, H0, and  γ are the line width  of the FMR spectra at frequency f, the in -\nhomogeneous line width broadening  and gyromagnetic ratio, respectively.  The g-factor has \nbeen calculated from the gyromagnetic ratio value. It is known that t he value of H0 depends \non the qual ity of the thin film  [25,26] .  \nThe fitted parameters HK, 4πM eff, H0, g-factor , and damping constant  (α) of sample s \nS1 to S4 are shown in the Table II . The sample S1 and S4 has the same seed layer (Ta/Pt ) but 7 \n different capping layer s (Pt/Ta or Au/Ta ) which make  the multilayers structure with \nsymmetric and antisy mmetric  capping layer s. The extracted value  of 4πM eff is higher for the \nsample S4 in compar ison to sample S1.  \n \nA non -linear behaviour  is also observed for sample S1 (Fig. 2(b)) above 12 GHz frequency  \nwhich  may be due to the other extrinsic mechanism like two magnon scattering. It is also \ncorroborated by the  enhancement of in -homogeneous  broadening of sample S1 ( H0 =52.96 \n±5.33 Oe) in comparison to sample S4 ( H0= 22.03 ±3.48 Oe), which indicates the interface \neffect. The value of α in sample S1 increases by approximately 34 % relative to S4 sample. It \nis a cumulative effect of spin pumping , d-d electrons hybridization  and two magnon \nscattering  [12,14,20] . Since we kept  the thickness constant for the Co layer and ot her NM \nlayers, we may assume that d-d electrons hybridization and two -magnon scattering will be \nsimilar contribution  in all layers . Therefore, the spin-pumping  enhancement at the interface of \nPt/Ta is the main cause fo r the higher α. In addition to this, it is noted that the capping layer \nalso affects the value of  HK. By changing the capping layer Pt/Ta (S1) to Au/Ta (S4) , the \nvalue of HK increases by ~ 6 Oe . We cannot also rule out the effect of the phase s of Co (hcp \nand fcc) as show n by Tokac  et al. [21]. Table II : The parameters evaluated from the fitting of experimental data of \nsample S1, S2, S3 and S4 using equation (2) and (4).  \n \nSample  HK (Oe) \n 4πM eff  (Oe)  g -factor  \n         α H0 (Oe)  \nS1 -46.81±1.8  7240±157  2.17±0.01  0.0326±0.0008  52.96±5.33  \nS2 -65.86±3.1  10947±403  2.05±0.01  0.0169±0.0002  22.16±1.89  \nS3 -36.03±2.3  8967±182  2.14±0.01  0.0245±0.0004  57.18±3.39  \nS4 -52.33±2.1  8987±164  2.10±0.0 1 0.0243±0.0004  22.03±3.48  8 \n Now , if we compare the sample S1 to S3 (same capping layer but different seed layer), we \nobserved lower α value and higher value  of 4πM eff in sample S3.  However  α and  4πM eff values \nare similar for S3 and S4. Therefore, when Au is seed and/or capping layer, the Co/Au \ninterface is significantly  affecting the damping mechanism  and hence spin pumping . It should \nbe noted that  both Pt and Au are heavy metal s with comparable  strength of spin orbit \ncoupling  [27]. However, when Pt is seed and/or capping layer the spin pumping efficiency \nincreases e.g. in sample S1.  We also noticed that 4πM eff value increases whe n Ta/Au is taken \nas any of the seed or capping layer which indicates that Sample S2 and S3 have improved \nstructural quality of the film. This is further corroborated by the reduction of α for sample S2.  \nTherefore, it is concluded  that the seed and capping  layers are pla ying important role in the \nspin pumping for dissipating  spin current into the NM layer . In order to understand the role of \nthe combination of the two layers Ta/Pt or Ta/Au , we also fabricated samples in tri -layer \n0 500 1000 1500 20004681012141618 Frequency (GHz)\nHres(Oe)S5 S6 S7\nFit Fit  Fit  (a) \n4 6 8 10 12 14 16 18100200300400H(Oe)\nFrequency (GHz) S5  S6  S7\n Fit  Fit  Fit(b)  \nFIG. 3(a) Frequency versus resonance magnetic field ( Hres) graph for sample S5, S6, S7. (b) It \nshows line width ( H) versus frequ ency for sample S5, S6, S7. Open symbols represents the \nexperimental data and solid lines are fit with equation (2) and (4).  \n 9 \n structures M/Co(3)/M, w here M  stands for  Ta, Cu and Pt for samples S5, S6 and S7, \nrespectively.   \nFigure s 3(a) and 3(b) show the frequency dependent data of Hres and H, respectively . \nThe open symbols show  the experiment al data and solid line s indicate  the fittin g with \nequation (2) and  (4) in Fig s. 3(a) and 3(b) , respectively. The fitted parameters are listed in the \nTable III. It can be seen from the Ta ble III that the  value of  α for S5 and S6  is nearly equal to \nthe bulk value of Co  i.e. 0.011   [26]. It is noted that  Cu has low spin orbit coupling having \nlarge spin diffusion length . Therefore , the spin flipping is very weak which causes backflow \nof accumulated spin towards spin pu mping interface . This spin back flow cancel s the spin \ncurrent  which makes Cu as a bad spin -sink. However , Ta is a high spin orbit coupling  \nmaterial and hence good spin sink , but it shows low α (sample S5). This low value of α may \nbe due to its growth as a mixed crystalline phase  and the low effective spin mixing \nconductance at the interface . In addition to this, the value of 4πM eff is higher in sample S6  \n \nTable III.  The values of the fitted parameters extracted  from the fitting of \nexperimental data of Hres and H sample S5, S6, and S7 using equation (2) and (4).  \n \nSample  \n HK (Oe)  \n 4πM eff (Oe)  g factor  \n      α H0 (Oe)  \nS5 -71.46±3.0  12477±7  2.22±0.05  0.0104±0.0003  141.8±2.37  \nS6 -46.25±0.5  15268±20  2.11±0.01  0.0109±0.0006  60.7±0.47  \nS7 -53.99±2.6  12548±7  2.15±0.04  0.0243±0.0005  89.91±4.33  \n \nwhile it is lower in S5. The higher value of 4πM eff in sample S6  may be due to relatively \nbetter crystalline  quality  as Cu and Co have  good lattice matching . It is known that ultra -thin \nfilms of Ta may grow in mixed phas es (bcc α- phase and tetragonal β -phase)  [28–30]. \nTherefore , lower value of 4πM eff in S5 may be due to mixed phase  growth  of Ta , which may \nlead to  poor structural growth of Co. In addition , it has relatively poor spin scattering 10 \n efficiency due to relatively long spin diffusion length of 10 nm  [31] as compared to Pt (3.5 to \n10 nm)  [19,20] . This is corroborated  by the increase in the value of the inh omogeneous \nbroadening H0. We observe that when both seed and capping layer is Pt  (S7), the value of α \nis high in comparison to the sample s having Ta (S5) and Cu  (S6). This shows the presence of \nlarge effective mixed conductance at the interface due to high spin orbit coupling of Pt. \nHowever,  if we compare sample s S7 with S1 which having symmetric multila yer Ta/Pt layers \nat top and bottom interfaces, the value of α is still higher in S1. This may be explained \nconsidering the growth properties of the Pt is affected by the Ta, which help s to induce (111) \norientation of Pt   [32].  \n Spin orbit coupling is an important parameter for generation of large spin current. The \nlater is also affected by coupling between the spin, orbital angular moment , and crystal field  \n \nin the system. In this context  Lande -g factor w as evaluated from the gyromagnetic ratio that \nobtained by the fitting frequency dependent Hres using equation (2 ). The summary of the g-\nfactor and the 4πM eff are listed  in Table II  and III, respectively,  for sample S1 to S7. The \nhighest value of g-factor ~ 2.22±0.05  is obtained for sample S5  which is higher than  bulk \nvalue of Co (2.18)  [33]. The g-factor is given by the combination of orbital (μL) and s pin (μS) \nangular magnetic moments by the  relation\n12\nSL g\n . Normally,  in the bulk,  the μL is Table IV.  Effec tive spin conductance values for sample S1 to S7 . \nSample  Sample details  \n(In parenthesis thickness is in nm)  Effective spin mixing  \nconductance ( g↑↓) (× 1018 m-2) \nS1 Si/Ta(3)/Pt(3)/Co(3)/Pt(3)/Ta(3)  1.86±0.08  \nS2 Si/Ta(3)/Au(3)/Co(3)/Au(3)/Ta(3)  0.81±0.04  \nS3 Si/Ta(3)/Au(3)/Co(3)/Pt(3)/Ta(3)  1.46±0.05  \nS4 Si/Ta(3)/Pt(3)/Co(3)/Au(3)/Ta(3)  1.47±0.05  \nS5 Si/Ta(3)/Co(3)/Ta(3)  0.15±0.01  \nS6 Si/Cu(3 )/Co(3)/Cu(3)  0.02±0.001  \nS7 Si/Pt(3)/Co(3)/Pt(3)  2.00±0.08  11 \n absorbed by the crystal field of the material , which is protected by c rystal symmetry, but, it \nmay not be fully absorbed in case of the broken crystal symmetry at the interface . It is \ninteresting to note that g-factor and 4πM eff values show opposite behavior . It means  for a \nparticular sample when the g-factor value is higher , the value of 4πM eff is lower and vice \nversa.  \nTo quantify the spin pumping effect, w e estimated the total effective spin mixing \nconductance (g↑↓) for all samples . The value of g↑↓ can be calculated by the below e xpression  \n gtMg\nFMeffB\n40\n  ……………… ……………………………………… ……. (5) \nwhere g is the Lande g factor, μB is Bohr magneton, tFM is the thickness of the  FM layer, and \nwe consider the α0 is the bulk value of Co  for evaluation of g↑↓. The values  are listed in Table \nIV for sample S1 to S7 . It is known that g↑↓ describe s the total spin current and it gets \ndissipate d from the FM film through its interface with NM layer by considering the backflow \nof the spin current. The estimated values of g↑↓ are comparable to the other recent reported \nvalues  [21]. It can be seen from  Table IV  that the value  of g↑↓ is highest  (2.00 ± 0.08 × 1018 \nm-2) for sample S7 in which Pt is used as both seed and capping layers. Sample S1 exhibits a \nlittle lower g↑↓ ~ 1.86  ± 0.08 × 1018 m-2, in which Ta/Pt work as seed and capping layers. This \nmay be due to mixed phase growth of the Ta at such thin layer, which may not work as good \nspin-sink layer as compared to only Pt.  \nTo investigate the in -plane anisotropy we measured th e FMR spectra with the variation of \nazimuthal angle at the step of 10 . Figure s 4(a) and 4 (b) show  the azimuthal  dependence of \nHres and  H, respectively,  for the sample s S1 to S4. It can be observed  from Figs.  4(a) and \n4(b) that all samples show the uniaxi al anisotropy. However, the relative strength of the  12 \n   \nanisotropy  (difference between two extreme values of Hres) is higher for sample S2 having \nTa/Au as seed and capping layers . In contrary s ample S4 having Ta/Pt as a seed layer and \nAu/Ta as a capping layer shows lowest anisotropy . In addition, sample S1 with Ta/Pt as seed \nand capping layers shows low er anisotropy than sample S2. It means T a/Pt seed layer reduces \nanisotropy in Co thin film. In order to understand the effect of bilayer and single seed layer, \nwe measured the angular dependence of Hres and  H for tri-layer samples as shown in Figs.  \n4(c) a nd 4(d), respectively.  It is visible that tri -layer sample S5 having Ta as seed and capping \nlayers has highe st anisotropy  among all the samples . Therefore, it is concluded that the  \nFIG. 4 Angular dependence of the Hres of (a) sample S1 to S4 and (c) sample S5  to S7. \nAngular  dependence of H of (b) sample S1  to S4 and (d) sample S5  to S7. \n0 50 100 150 200 250 300 350500525550575600625650\n S1\n S2\n S3\n S4Hres (Oe)\n Angle ()(a)\n0 50 100 150 200 250 300 350100120140160180200H(Oe)\n Angle () S1  S2  S3  S4 (b)\n0 50 100 150 200 250 300 350360380400420440Hres (Oe)\n Angle () S5  S6  S7 (c)\n0 50 100 150 200 250 300 35080100120140160180200220H(Oe)\n Angle ()S5  S6  S7 (d)13 \n anisotropy of Co thin film is decreased when Ta/Pt is used as seed or cappi ng layer . \nHowever, we obtained higher spin pumping effect with Pt and Ta/Pt as seed and capping \nlayers. Therefore, not only spin pumping damping proper ties, but magnetic anisotropy  is also \nsignificantly affected by  the seed and capping layer s. \nCONCLUSIONS   \nWe presented a detailed study  of magnetic dynamics  of 3 nm  Co thin film using FMR . \nThe damping properties are investigated by analysing  the linewidth vis-à-vis seed and \ncapping layer s in the multilayer structure . We observed higher  value of damping consta nt i.e. \nα = 0.0326±0.0008 for the symmetric multilayers having  Ta/Pt as both seed and capping \nlayer s in comparison to its bulk value  (0.011) . This enhancement is governed by s pin \npumping. By considering bulk value as reference layer,  we evaluated effective  spin mixing \nconductance ~ 2.00 ± 0.08 × 1018 m-2 for Pt/Co/Pt layer which is larger than Ta/Pt/Co/Pt/Ta. \nThese values are in agreement with other report s [21]. It is also observed that the g-factor \ndecreases  with increasing effective demagnetization field with respect to seed and capping \nlayers.  \nACKNOWLEDGEMENT S \nWe acknowledge  National Institute of Science Education and Research, Bhubaneswar, India,  \nDAE  and DST - Nanomission  (SR/NM/NS1088/2011(G)) of the Govt. of India  for financial \nsupport . \n \nREFERENCES  \n[1] J. E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999).  \n[2] T. Kuschel and G. Reiss, Nat. Nanotechnol. 10, 22 (2015).  \n[3] J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back, and T. Jungwirth, Rev. Mod. \nPhys. 87, 1213 (2015).  14 \n [4] D. Ciudad, Nat. Mat er. 15, 127 (2016).  \n[5] D. C. Ralph and M. D. Stiles, J. Magn. Magn. Mater. 320, 1190 (2008).  \n[6] A. D. Kent and D. C. Worledge, Nat. Nanotechnol. 10, 187 (2015).  \n[7] J. Appl. Phys. 115, 172607 (2014).  \n[8] F. Hellman, A. Hoffmann, Y. Tserkovnyak, G. Beach,  E. Fullerton, C. Leighton, A. \nMacDonald, D. Ralph, D. Arena, H. Durr, P. Fischer, J. Grollier, J. Heremans, T. \nJungwirth, A. Kimmel, B. Koopmans, I. Krivorotov, S. May, A. Petford -Long, J. \nRondinelli, N. Samarth, I. Schuller, A. Slavin, M. Stiles, O. Tche rnyshyov, A. Thiaville, \nand B. Zink, ArXiv160700439 Cond -Mat (2016).  \n[9] T. Kimura, Y. Otani, T. Sato, S. Takahashi, and S. Maekawa, Phys. Rev. Lett. 98, \n156601 (2007).  \n[10] S. O. Valenzuela and M. Tinkham, Nature 442, 176 (2006).  \n[11] Y. Tserkovnyak, A. B rataas, G. E. W. Bauer, and B. I. Halperin, Rev. Mod. Phys. 77, \n1375 (2005).  \n[12] E. Barati, M. Cinal, D. M. Edwards, and A. Umerski, EPJ Web Conf. 40, 18003 (2013).  \n[13] M. C. Hickey and J. S. Moodera, Phys. Rev. Lett. 102, 137601 (2009).  \n[14] K. Lenz, H.  Wende, W. Kuch, K. Baberschke, K. Nagy, and A. Jánossy, Phys. Rev. B \n73, 144424 (2006).  \n[15] J. Lindner, I. Barsukov, C. Raeder, C. Hassel, O. Posth, R. Meckenstock, P. Landeros, \nand D. L. Mills, Phys. Rev. B 80, 224421 (2009).  \n[16] Y. Tserkovnyak, A. Bra taas, and G. E. W. Bauer, Phys. Rev. Lett. 88, 117601 (2002).  \n[17] A. Azevedo, L. H. Vilela -Leão, R. L. Rodríguez -Suárez, A. F. Lacerda Santos, and S. \nM. Rezende, Phys. Rev. B 83, 144402 (2011).  \n[18] A. Conca, S. Keller, L. Mihalceanu, T. Kehagias, G. P. D imitrakopulos, B. Hillebrands, \nand E. T. Papaioannou, Phys. Rev. B 93, 134405 (2016).  15 \n [19] O. Mosendz, V. Vlaminck, J. E. Pearson, F. Y. Fradin, G. E. W. Bauer, S. D. Bader, and \nA. Hoffmann, Phys. Rev. B 82, 214403 (2010).  \n[20] J.-C. Rojas -Sánchez, N. Reyr en, P. Laczkowski, W. Savero, J. -P. Attané, C. Deranlot, \nM. Jamet, J. -M. George, L. Vila, and H. Jaffrès, Phys. Rev. Lett. 112, 106602 (2014).  \n[21] M. Tokaç, S. A. Bunyaev, G. N. Kakazei, D. S. Schmool, D. Atkinson, and A. T. \nHindmarch, Phys. Rev. Lett. 115, 056601 (2015).  \n[22] W. Zhang, W. Han, X. Jiang, S. -H. Yang, and S. S. P. Parkin, Nat. Phys. 11, 496 (2015).  \n[23] C. Kittel, Phys. Rev. 73, 155 (1948).  \n[24] J. Appl. Phys. 114, 243906 (2013).  \n[25] J. Appl. Phys. 99, 093909 (2006).  \n[26] S. Azzawi, A. Gang uly, M. Tokaç, R. M. Rowan -Robinson, J. Sinha, A. T. Hindmarch, \nA. Barman, and D. Atkinson, Phys. Rev. B 93, 054402 (2016).  \n[27] T. Tanaka, H. Kontani, M. Naito, T. Naito, D. S. Hirashima, K. Yamada, and J. Inoue, \nPhys. Rev. B 77, 165117 (2008).  \n[28] J. Lin, J. J. Moore, W. D. Sproul, S. L. Lee, and J. Wang, IEEE Trans. Plasma Sci. 38, \n3071 (2010).  \n[29] R. Hoogeveen, M. Moske, H. Geisler, and K. Samwer, Thin Solid Films 275, 203 \n(1996).  \n[30] J. Vac. Sci. Technol. Vac. Surf. Films 5, 3408 (1987).  \n[31] J. App l. Phys. 110, 033910 (2011).  \n[32] J. Appl. Phys. 117, 17A731 (2015).  \n[33] T. G. A. Verhagen, H. N. Tinkey, H. C. Overweg, M. van Son, M. Huber, J. M. van \nRuitenbeek, and J. Aarts, J. Phys. Condens. Matter 28, 056004 (2016).  \n 16 \n FIGURE CAPTIONS  \nFigure 1  \nFIG 1(a) and (b) shows the schematics of generation of spin current in the multilayer and tri -\nlayer sample structures (c) FMR measurement setup in which rf generator creates GHz \nfrequency which works as a perturbation field ( hrf) perpendicular to external app lied field \n(Hext) and a diode detector through AC magnetic field modulation detects transmitted signal \nvia lock -in based technique.  \nFigure 2  \nFIG 2(a) Resonance frequencies versus  resonance magnetic field ( Hres) for sample S1 to S4 . \nOpen symbols repres ent the experimental values while  solid lines are  fitted with equation (2 ). \n(b) Line width ( H) versus resonance frequencies for sample S1 to S4. Open symbols are \nexperimental data and solid lines are fitted with equation (4).  \nFigure 3  \nFIG. 3(a) Frequency versus resonance magnetic field ( Hres) graph for sample S5, S6, S7. (b) \nIt shows line width ( H) versus frequency for sample S5, S6, S7. Open symbols represents \nthe experimental data and solid lines are fit with equation (2) and (4).  \nFigure 4  \nFIG. 4  Angular dependence of the Hres of (a) sample S1 to S4 and (c) s ample S5  to S7. \nAngular  dependence of H of (b) sample S1  to S4 and (d) sample S5 to S7  "    },    {        "title": "2312.06929v1.Interacting_Floquet_topological_magnons_in_laser_irradiated_Heisenberg_honeycomb_ferromagnets.pdf",        "content": "Interacting Floquet topological  magnons in laser-irradiated \nHeisenberg honeyco mb ferromagnets \nHongchao Shia, Heng Zhua, Bing Tang*, Chao Yang \n \nDepartment of Physics, Jishou University, Jishou 416000, China \n \nABSTRACT \nWhen a  Heisenberg honeycomb ferromagnet is irradiated by high-frequenc y \ncircularly polarized light, th e underlying uncharged magnons ac quire a \ntime-dependent Aharonov–Casher phase, which makes it a Floquet topological \nmagnon insulator. In this context, we investigate the many-body  interaction effects of \nFloquet magnons in laser-irradiat ed Heisenberg honeycomb ferrom agnets with \nocontaining Dzyaloshinskii-Mori ya interaction under the applica tion of circularly \npolarized off-resonant light. We  demonstrate that the quantum f erromagnet systems \nperiodically laser-driven exhibi ts temperature-d riven topologic al phase transitions due \nto Floquet magnon-magnon interactions. The thermal Hall effect of Floquet magnons \nserves as a prominent signature for detecting these many-body e ffects near the critical \npoint, enabling experimental investigation into this phenomenon . Our study \ncomplements the lack of previous  theoretical works that the top ological phase \ntransition of the Floquet magnon un der the linear spin wave app roximation is only \ntunable by the light field. Our study presents a novel approach  for constructing \nFloquet topological phases in per iodically driven quantum magne t systems that goes \nbeyond the limitations of the linear spin wave theory. We provi de numerical results \nbased on the well-known van der Waals quantum magnet CrX 3 (X=F, Cl, Br, and I), \ncalling for experimental implementation.  \n  \nⅠ. INTRODUCTION \n                                                 \naThese authors contributed equally to this work \n* bingtangphy@jsu.edu.cn  Over the past few decades, studies of the topological insulator s and topological \nphase have made great progress in the field of condensed matter  physics [1-8].  In \nanalogy to electronic systems, t he topological phases have also  been extended to \nbosonic systems, such as photonic[9-11], phononic [12,13], and magnonic systems \n[14-17]. There is a surge of in terest in utilizing magnons, a l ow-energy collective \nexcitation in magnets [18,19] that are easily manipulated by ma gnetic fields, have \nlow-dissipation and permit a pure spin transport without Joule heating, for spintronics \n[20,21]. \nRecently, the thermal magnon Hall effect has been realized expe rimentally in the \ninsulating quantum kagome ferromagnets Cu(1-3, bdc) [22,23] and  the pyrochlore \nferromagnets Lu 2V2O7, Ho 2V2O7, In 2Mn 2O7 [24,25] following a theoretical proposal \n[26,27]. It is generally believed that thermal magnon Hall effe ct results from the \nnontrivial topology of magnon disp ersions [26-31] encoded in th e Berry curvature \ninduced by the Dzyaloshinskii-Moriya interaction (DMI) [32, 33] , which plays the \nrole of spin orbit coupling. In insulating quantum magnets the DMI is an intrinsic \nanisotropy and it is present due to the lack of inversion symme try of the lattice. For \nhoneycomb magnets, the midpoint between two magnetic ions on th e next-nearest \nneighbour bonds is not an inversion center. Therefore, a DMI is  allowed on the \nhoneycomb lattice and a magnon analogue of the Haldane model [1 ] can be realized \nin honeycomb ferromagnets[34-37]. Thus, the thermal magnon Hall  e f f e c t  a l s o  t o  \nexists in honeycomb magnets. It i s worth noting that the early study of topological \nmagnons mainly based on the linear spin wave theory, where the interactions between \nmagnons can be safely ignored [26-31, 34-36]. \nBut in actuality, as the temperature increases, the influence of ma gnon-magnon \ninteractions becomes more pronounc ed, which are typically treat ed as small terms that \noften expanded using techniques such as Holstein-Primakoff (HP)  [38] or \nDyson-Maleev transformations [39,40].  Recently, the importance of the many-body \ninteractions effect has been rec ognized in magnonic systems, wh ere the interactions \nbetween magnons lead to magnon decays and spectral renormalizat ions [41-50]. \nThese scattering processes can in troduce intriguing momentum or  temperature-dependent behaviors of magnons in magnet systems, w hich have not \nbeen previously explored in the study conducted using the linea r spin-wave theory \n[26-31,34-36]. In particular, there  have been some investigatio ns on the effects of \ninteractions on the Dirac magnons in a honeycomb lattice [45-48 ] show that the phase \ntransition of magnons  at a critical temperature is driven by their interactions, wher e \nthe DMI is essential in determi ning the topological properties of magnon-magnon \ninteraction in honeycom b ferromagnets. \nAlso, laser-irradiation of solid-state materials has attracted considerable attention \nand interest as an alternative wa y for engineering topological nontrivial states from \ntopologically trivial quantum materials recently [51-56]. In th is formalism, \ntopologically trivial systems can be periodically driven to non trivial topological \nsystems termed Floquet topologi cal insulators [53,54]. They hav e an advantage over \ntheir static (equilibrium) topological counterpart, in that the ir intrinsic properties can \nbe manipulated and different topological phases can be achieved . In irradiated \ninsulating quantum magnets with charge-neutral magnons [57-60],  the Floquet \nphysics can emerge from the coupling of the electron spin magne tic dipole moment to \nthe laser electric field through  the time-dependent version of the static \nAharonov-Casher phase [61], whic h acts as a vector potential or  gauge field to the \nspin current. A quantum magnet system driven periodically can b e studied by the \nFloquet-Bloch theory [62]. Previously it has been shown that a tunable DMI by laser \nfield in a two-dimensional (2D) laser-irradiated Heisenberg fer romagnets can induce \nphotoinduced topological phase transition[57,60]. However, to t he best of our \nknowledge, the topological property of the Floquet magnon with many-body \ninteractions effects at finite temperatures involved has not be en studied yet. How the \nmagnon interactions affect the t opology of laser-irradiated ins ulating quantum \nmagnets is still an issue of f undamental interest in topologica l quantum materials   \nIn this work, we investigate the many-body interaction effects of Floquet \nmagnons in a laser-irradiated Heisenberg honeycomb ferromagnet with off-resonant \ncircularly polarized  light field  tunable DMI. Using linear spin wave and magnonic \nFloquet-Bloch theory, we show that when the magnet systems are periodically driven by off-resonant circularly polari zed lights, they effectively m ap onto the \ncorresponding static spin model plus a tunable photoinduced mag netic field along the \nˆz direction, which is perpendicular to the honeycomb plane. In p articular, when the \nmany-body interaction effects of  Floquet magnons are considered , we combine \nmagnonic Floquet-Bloch theory and Green's function method, we f ind that the \ntopological phase transitions in laser-irradiated Heisenberg ho neycomb ferromagnets \nare driven by temperature due  to the Floquet magnon-magnon inte ractions. These \ntransitions are marked by the Floquet magnon band gap closing-r eopening with \nincreasing temperature and the sign change of the Chern number and thermal Hall \nconductivity with increasing temp erature at a finite off-resona nt circularly polarized \nlight field and magnetic field.  \nOur paper is organized as follows. In Sec. II, we sketch the mo del of this work. \nIn Sec. III, we presented the magnonic Floquet-Bloch theory and  Green’s function \nmethod. In Sec. IV, we have shown the magnon band structure of Heisenberg \nhoneycomb ferromagnet under linear spin wave theory and conside r the \nmagnon-magnon interaction,  respectively.  In Sec. V, we discuss the topological \nproperties of interacting Floquet magnons. In Sec. VI, We inves tigate the thermal \ntransport property. The summa ry and outlook are given in Sec. V II. \n  \nⅡ. MODEL  \nIn this article, we take into account a two-dimensional (2D) la ser-irradiated \nHeisenberg honeycomb ferromagnet, whose lattice structure is sh own in Fig. 1. An \nexternal magnetic field is also considered. This ferromagnetic system is described by \nthe following spin Hamiltonian \n\n, ,ij ij i j B i\nij i ijJg        SS SS D B S ,               ( 1 )  \nwhere ,ij and ,ij  represent the summation over the nearest and next-nearest \nneighbor sites, respectively.  J and D represent the pairwise interactions between \nthe nearest neighboring ions and the DMI between the next-neare st neighboring ions, respectively. The DMI vector is pa rallel to the z-direction wit h ˆij ijvDDz , where \n1ijv  for counter-clockwise and clockw ise hopping spin magnetic momen ts on \neach honeycomb layer sublattice.  The applied magnetic field along the z axis with \nˆBBz , and we define B hgB  being the magnetic field strength, g is the spin \ng-factor and  B is the Bohr magneton. \n \nFig. 1.  (Color online) Schematic of the honeycomb lattice structure, w hich is made up \nof two triangular sublattices. \n \n  \n  \nFig. 2.  (Color online) Schematic representation of a honeycomb ferroma gnet being \nilluminated by a circularly polarized laser (perpendicular to t he ferromagnet plane). \n \nW e  c o n s i d e r  a  c i r c u l a r l y  p o l a r i z e d  l a s e r  i r r a d i a t e d  o n t o  t h e  2 D  Heisenberg \nhoneycomb ferromagnet, as shown in Fig. 2. Physically, charge-n eutral magnons can \ninteract with an electromagneti c field through their spin magne tic dipole moment. The \ncorresponding time-dependent ver sion of the Aharonov-Casher pha se emerges explicitly from quantum field theory with the Dirac-Pauli Lagra ngian. We take the \nspin magnetic dipole moment carried by magnons to be along  the \nz-directionBzge , where g is called the g-factor and 2Beem  i s   \nknown as the Bohr magneton. In the presence of a laser (electri c) field ()tE, the \nhopping spin magnetic dipole moments accumulate a time-dependen t version of the \nAharonov-Casher phase [61]  \n 2j\niB\nijgtt dc r\nrΞ l.                       ( 2 )  \nwhere  is the reduced Planck’s constant, and c is the speed of light. \nˆ ()ttΞEz  with () ()t tA tE , ( ) tA  is the time-dependent vector potential \ngiven by  \n0sin , s , )0(c o tt tA    A                  ( 3 )  \nwhere 00AE  is the strength of the time-dependent vector potential. 0E stands \nfor the amplitude of the electric field,  represents the circular frequency of the \nlight wave. The corresponding tim e-dependent oscillating electr ic field is given by \n0sin ,c () o s , 0 Ettt    Ξ .                ( 4 )  \nThe resulting time-dependent Hamiltonian is given by \n \n, ,1.. ..22ij ijit it zz z\nij ij i j ij i\nij i ijDtJ S S e S S H c i v e S S H c h S            . (5) \nNoticing that the direction of  the vector pointing from i t oj defines a relative angle \nijand ij, we get  0sinij ijt t      a n d 0sinij ijt t     , 2Bg\nc\n i s  \nabsorbed in 0 and 00 3 . We focus on the linear spin-wave approximation, \nwhich is reasonable in the large spin value limit and the low-t emperature regime. This \ncan be implemented via recasting the spin operators in the time -dependent \nHamiltonian in terms of the follo wing linearized HP transformat ion [38] \n+2iiSS a , †2iiSS a , † z\nii iSS a a .             ( 6 )  Here, †()iiaa is the magnon creation (annihilation) operator. The resulting l inear \nbosonic Hamiltonian has time periodicity, i.e.,  tT t , where 2T\n  \ncorresponds to the perio d of the laser field. \n  \nⅢ. METHODS   \nA. Magnonic Floquet-Bloch theory \nThe Floquet-Bloch theory[62] is a formalism for studying period ically driven \nquantum systems and it applies to different cases of physical i nterests. The magnonic \nversion describes the interac tion of light with magnonic Bloch states in insulating \nquantum magnets. In the present case, the time-dependent Hamilt onian \n,tk  can \nbe obtained by making the time-dependent Peierls substitution ()tkk A . Note \nthat the ,tk  is periodic due to the time-periodicity of the vector potentia l. \nHence, it can be expanded in Fourier space as \n    ,,in t\nntt T e\n   kk k  ,               ( 7 )  \nwhere    \n0† 1,n\nnn nTitet d tT\nkk k   is the Fourier component. Therefore, \nits eigenvectors in the Floquet- Bloch theory can be written as \n ,, = ,ittt e t\n kkk k ,   ,= , + =it\nntt T e\n    kk k  i s  \nthe time-periodic Floquet-Bl och wave function of magnons and k a r e  t h e  \nmagnon quasi-energies. We define the Floquet operator as  ,= ,tFtt i  kk , \nwhich leads to the Floqu et eigenvalue equation \n ,mn\nnm\nmnmm      kk k k .               ( 8 )  \nB. Green’s function \nIn this section, we will introduce the perturbation methods of the many-body \nGreen’s function[62] in order to study the many-body effect of Floquet magnons and \nits interplay with thermal fluct uation in later study. We defin e a matrix Green’s function as  †0 , kk k , where  is a time-ordering operator for \nthe imaginary time it, and 0  with  1\nBkT . The τ-dependent \noperator is defined as  (0) Oe O e , which is formally obtained by the \nanalytic continuation to imaginary time  of the Heisenberg operator Ot , \nand0i n t is the zeroth-order static tim e-independent effective Floquet \nHamiltonian. The bracket  denotes the thermodynamic average. The first-order \nHartree Feynman diagrams resulting from Floquet magnon-magnon i nteractions are \ndepicted in Fig. 2.  \n \nFig3. The Feynman diagram of the Hartree contributes to 1/S many-body \ncorrections in linear spin-wave theory. The solid line represen ts a Floquet magnon \nand the arrow denotes the propagation direction. With number-co nserving \nfour-Floquet magnon vertex is indicated by a black circle. Vari able q denotes the \nmomentum of thermally excited Floquet magnons.  \n \nBased on the first-order Hartree diagram shown in Fig. 3, to ge t the solution, we \nsolve the Heisenberg equation of motion for the Green’s functio n elements and apply \nthe random phase approximation to  extract the nonlinear self-en ergy corrections from \nFloquet magnon-magnon interactions. A fter a Fourier transformat ion \n 1, ,ni\nn\nne  k k ,                   ( 9 )  \nwhere n is the bosonic Matsubara frequency. The Green's functions ,nk  satisfy a matrix Dyson's equation  1\n1 ,nniH k k  , where 1 i s  t h e  \ninverse of the Green’s function, and 1Hk is the renormalized effective \nHamiltonian. \n  \nⅣ. MAGNONIC FLOQUET BAND \n(i). Linear Hamiltonian for Laser-Irradi ated Heisenberg honeycomb ferromagnets \nIn order to study the periodically  laser driven Heisenberg hone ycomb ferromagnet \nin Eq. (4), we will apply the Floquet theory to transform the p resent time-dependent \nspin model to one static time-independent effective spin model,  which is describe by \nthe Floquet Hamiltonian. Physica lly, this static effective Hami ltonian eff can be \nexpressed in terms of 1, namely, 0mm\neff m\n ,where m is an integer. \nWhen the circular frequency  of the laser is much larger than the magnon \nfrequency bandwidth , i.e., , this way is applicable. In the current work, we \npay attention to the off-resonant regime, thus it suffices to t ake into account the zeroth \norder of the static Floquet Hamiltonian [52,56,59,60]. By makin g use of the discrete \nFourier component of the time-dependent Hamiltonian, we can obt ain a zeroth order \neffective Hamiltonian 0\n0 , where \n01 Tmi m tdte tT . In the momentum \nspace, the effective magnon Hamiltonian can be written as †\n00 ()\nkH  kk k\n , \nwhere † ††(,)abkk k  is a two-component spinor operator and the Bogoliubov \nHamiltonian is given by  \n00 0() + () () ()x xy yz z Hh h h h    kk k k .           ( 1 0 )  \nwhere03hJ S h ,() R e ( )xFhJ S  k k ,( ) I m ( )yFhJ S k k , ( ) 2zFhD S k k , \n3\n1nik\nk\nne\n\n , 3\n1sinn\nn\n  k kζ, and the vectors nδ a n d  nζ are shown in Fig.1.  \n00 FJJ J , and 00 FDD J  , where mJx  is the Bessel function of order mZ. ( , , )iix y z  are Pauli matrices and 0 is identity matrix.  In fact, the light \nintensity of the laser can be characterized via a dimensionless  quantity \n2\n00 /BgEa c . The Floquet magnon bands are given by \n 0\n0h kk ,                       ( 1 1 )  \nwhere    222\nxyzhhh kk k k  and 1( 1)  corresponds to the \nup(down) band, namely, ()ud .  \nThe tunable Floquet magnon energy bands and Density of states a re depicted in \nFig. 4(a) and Fig. 4(b),  respectively. The results show that the magnon band and the \nDensity of states can be tuned by the circularly polarized ligh t field.  \n \nFig. 4.  (Color online)  (a)T he tunable Floquet magnon energy bands alo ng the \nhigh-symmetry lines   . (b) The tunable Floquet Density of states per \nunit cell of the ferromagnet on th e honeycomb lattice. The othe r parameters are set to \n0.1DJ , 0.1hJ , and 12S . \n \nIn fact, the laser field tunable DMI can cause an  tunable effective Haldane mass \nterm, which will open up a tunable non-trivial band gap K63FDS\n  between the \nupper and lower branches at the Dirac points . In the absence of the laser field, \nthe size of the band gap at the Dirac point is 63K DS\n .  A s  t h e  l a s e r  f i e l d  i s  \napplied, the zero energy mode is lifted for 2 , which implies a photoinduced magnetic order with-out a high applied magnetic field [64]. In the presence of laser \nfields, the magnonic excitation energy at  is given by the tunable photoinduced \nmagnetic field and the magnetic field h.  \n \n(ii). The renormalized magnon band for Interacting topological Dirac magnons \nWe relate the spin and boson operators using the HP transformat ion truncated to \nthe first order in 1/S as follows, \n†\n2\n4ii i\ni iaa aSS aS, ††\n†2\n4iii\ni iaaaSS aS , † z\nii iSS a a .     ( 1 2 )  \nThe higher-order terms from the HP transformation in Eq. (8) gi ve the  zeroth-order  \nstatic time-independent effective interaction  Hamiltonian associated with Eq.(4) \n\n\n\n\n4 1 2 3 42 1 2 3 44 1 2 3 42 1 2 3 4\n4 2 12 34 2 4 1 2 3 4 1234* ? †† †† † * ? †\n†† † † ††int4\n442i\niiF\nFJbbba babb aaab abaaN\nD Jabab aaaa bbbbNN\n   \n  \nkk kkk kkkkk k kk kk k k k kk\nk\nkk kk kk k k k k k k kk kk\nkk\n(13) \nwhere ik stands for summation over all ik. In Eq. (8), the conservation of \nmomentum is due to 1234\n12 34 ,1i ir\nieN \n kkkk\nkk kk\n. Similar to a recent new \nstudy[47] ,these intera ction terms are able to be rewritten as a more compact \nexpression  such as \n1, 2\n3 41234† †\nint ,\niV kk\nkk k k k k\nk . We can get the effective \nHamiltonian by using the Green’s function method which is intro duced in Sec. Ⅲ. B, \nwritten as  \n0\n**\n0(1)\n10()\n()() ()F\nFhpm J S M v\nHHJS M v h p m\n   \n  qk k k\nkk q kkk k ,    ( 1 4 )  \nwhere  2\n0112F\nFDpJ JNN      qq q q q q q q\nqq, 2 ,F mD S  kk k  \n   00 00, , () (), () (),2FF\ndu duJDMf f f fNN           k\nk\nqqq q q\nqqq qq\n a n d  212i JveN \n q\nq kq kq\nq. Here, 1/BkT , 0 1() 0() 1 fe\nqq  i s  the Bose-Einstein distribution function, and zhq qq . \n \nFig. 5.  (Color online) The renormalized Floquet magnons band structure s driven by \nperiodically circularly polarized light 2 . (a) The renormalized Floquet magnon \ndispersion curves along the high-symmetry lines   , and the light \nintensity00.15 . (b) The gap  at the Dirac point K as a function of \ntemperature. The parameters are chosen as 0.1DJ , 0.1hJ , and 12S . \n \nIn Fig. 5(a), we show the renormalized Floquet magnon bands of periodically \ncircularly polarized light-d riven honeycomb ferromagnets  at three different \ntemperatures. It is clearly seen that, with the increase of the  temperature,  the \nrenormalized Floquet magnon band gap at the Dirac points K decreases and will \nclose at approximately 0.78cTJ . If the temperature is further elevated, the \nrenormalized Floquet magnon band gap reopens at the Dirac point  K and its width \nincreases with T. To understand the /g3influence of the light intensity on width of the \nband gap, we plot band gaps at the Dirac point K as a function of temperature \ncorresponding to different the light intensity 0, as shown in Fig. 5(b).  It can be \nclearly seen that the critical temperature cT for the gap-closing increases as \nincreasing the light intensity 0. The temperature can derive a gap-closing \nphenomenon, which means a topological phase transition may occu r with the increase of the temperature, and we will discuss it later. \n  \nⅤ. TOPOLOGICAL PROPERTIES OF I NTERACTING FLOQUET MAGNON \nThe Berry curvature is one of the main important quantities in topological \nsystems, which provides an effective gauge field for the bosoni c magnons in the \nk-space and dominates the trans port properties. In Floquet topo logical systems, it is \ncustomary to assume that the qua sienergy levels of the Floquet Hamiltonian are close \nto the equilibrium system, which is realized in the off-resonan t limit [52,59,65].  \nTherefore, the properties of equilibrium topological systems ca n be applied to Floquet \ntopological systems. To investigate the magnon transport in Flo quet topological \nsystems, we define the Berry curvature of the  Floquet magnon bands as \n  FF Fi      kk kk k             ( 1 5 )  \nwith ,ud . Here, F\n k are the Floquet eigenvectors of 1Hk. The \nassociated Chern number is defined as the integration of the Be rry curvature over the \nentire frist Brillouin zone (BZ), \n2 1\n2FF\nBZCdk k                   ( 1 6 )  \n To investigate topological prope rties of interacting Floquet m agnon at finite \ntemperatures, we can put the Floquet magnon band physics on one  Bloch sphere, and \ntopological natures are include d in one total mass term 2F mD S   kk k . For the \nconvenience of numerical calcula tion, one can adopt the Berry c urvature of \npseudospin freedom [66], which is  \n31\n2 x y kk d   ddd k .                   ( 1 7 )  \nwhere, 222\nx yz d ddd  , and  ,,x yzdddd  is the effective field vector of \nEq.(14). By manipulating the te mperature-dependent Floquet magn on population, we \ncan achieve a topological phase t ransition in the upper band of  the renormalized Floquet magnon from 1F\nuC t o  1F\nuC. \n \nFig.6.  (Color online) (a)The Chern number of the upper band of the re normalized \nFloquet magnon, with 00.15 . (b) The phase diagram in the 0T plane for the \nupper band of the renormalized Floquet magnon, and the Chern nu mber of the down \nband of the renormalized Floquet magnon is the opposite. The ot her parameters are \nchosen as 0.1DJ , 0.1hJ , and 12S . \n \nIn Fig. 6, we display the dependence of the Chern number for th e upper band of \nthe renormalized Floquet magnon on the temperature and the phas e diagram in the \nplane for the upper band of the renormalized Floquet magnon. Th ere is a topological \nphase transition driven by temperature, the Chern number for th e upper band changes \nbelow cT and above cT as shown in Fig. 6(a), and the renormalized magnon gaps of \nthe Dirac points K close and reopen near the critical temperature cT. The closing \nof band gap at the transition point is essential to ensure the topological phase \ntransition [7]. To further inves tigate temperature-induced topo logical transitions at the \ntunable light intensity 0, we plot the phase diagram in the 0T plane for the \nupper band of the renormalized Floquet magnon as shown in Fig. 6(b). The results \nshow that the critical temperature cT of temperature-induced topological phase \ntransition increases with the increase of light intensity. It i s easy to understand that \nwith the application of laser f ield, the appearance of photo-in duced magnetic order contributes to the stability of ferromagnetic phase, which mean s that phase transition \nrequires a higher temperature. More generally, in Heisenberg fe rromagnets, the \nexchange interaction between magnetic atoms is one of the main causes of \nferromagnetism. The light intensity can affect the exchange int eraction and thus the \nmagnetism of the ferromagnet. Whe n the light intensity increase s, the exchange \ninteraction may be enhanced, resulting in enhanced magnetism of  the ferromagnet. In \naddition, light intensity can also affect the magnetism of ferr omagnets by affecting the \nthermal motion and lattice vibration of the material. With the increase of light \nintensity, the thermal motion and lattice vibration of the mate rial may be enhanced, \nresulting in changes in magnetic properties. In summary, the li ght intensity has a \ncomplex effect on the magnetism of ferromagnets. At a certain t emperature, with the \nincrease of light intensity, the enhancement of exchange intera ction, thermal motion \nand lattice vibration may cause the magnetic properties of ferr omagnets to change, \nthus affecting the critical temperature of the topological phas e transition induced by \ntemperature. \n  \nⅥ. THERMAL HALL EFFECT OF INTERACTING FLOQUET MAGNON \nThe most interesting aspects of ferromagnetic topological magno ns is that they \nexhibit the thermal Hall effect. Topological magnon systems, su ch as the magnonic \nanalog of spin Hall insulators and Weyl semimetals, have been e xtensively explored \nin recent years. These systems host linear magnon spin Nernst o r thermal Hall \ncurrents, which are induced by non -collinear spin texture or DM I. However, in the \nabsence of DMI the linear thermal Hall signal vanishes and tran sport signatures in the \nHall response appear only in the nonlinear response regime. Thi s has motivated the \nrecent exploration of exciting nonlinear transport phenomena in  bosonic systems, \nespecially the nonlinear thermal Hall effect. In ferromagnetic insulators, the thermal \nHall effect has only been studied in the topological magnon ins ulator phase, when the \nlowest (acoustic) magnon band is well separated and carry a wel l-defined Chern \nnumber. For periodically driven magnon systems, we focus on the  regime where the Bose distribution function is close to thermal equilibrium. In t h i s  r e g i m e ,  t h e  s a m e  \ntheoretical  concept of the thermal Hall e ffect in undriven topological magn on systems \ncan be applied to the driven magnon systems. The transverse com ponent of the \nthermal Hall conductivity [26 , 27] is given explicitly by \n2\n2\n,() ( )B\nxy\nudkTcnV\n\n \nkk                ( 1 8 )  \nwhere V is the volume of the system, Bk and Tare the Boltzmann constant and \nthe temperature respectively. And 1\n, 1 nf e\n    k\nk k  corresponds to \nwell-known Bose-Einstein distribu tion function close to thermal  equilibrium, \n1/BkT , 2\n2\n221() ( 1 ) 2 L i( ) ln lnxcx x x xx  , and 2Li ( )x stands for the \ndilogarithm function. \n \nFig.7. (Color online) (a) The thermal Hall conductivity as a function of temperature at \ndifferent light intensities. (b) Schematic figure of the therma l Hall effect. The \nparameters are chosen as 0.1DJ , 0.1hJ , and 12S . \n \nEvidently, the thermal Hall conductivity is simply the Berry cu rvature weighed \nby the 2c function. Therefore, its dominan t contribution comes from the peaks of the \nBerry curvature. In addition, the thermal Hall transport is det ermined mainly by the \nacoustic (lower) magnon branch due to the bosonic nature of mag nons. In Fig. 7(a), \nwe have shown the trend of the thermal Hall conductivity xy for different light \nintensities as the temperature increase. The main result of thi s report is that intrinsic topological magnon insulators in the honeycomb ferromagnets can  be driven to \ndifferent topological phases with  different Berry curvatures us ing photo-irradiation. \nTherefore, each topological phase is associated with a differen t sign of the thermal \nHall conductivity, which results in a sign reversal of the magn on heat photocurrent. \nThe finite thermal Hall conductivity in these topologically tri vial phases can be \nattributed to the photoinduced B erry curvatures.  We all know th at the thermal Hall \nconductivity vanishes for the ma gnonic Floquet trivial insulato r induced by \nperiodically circularly polarized light 2  without regard to the \nmagnon-magnon interaction, because  the integration of the Berry  curvature vanishes. \nBut for Interacting topological Dirac magnons system, the therm al Hall conductivity \nalways exists at a small light intensity as the Berry curvature  exists. In other words, \nthe thermal Hall effect in 2D ferromagnetic insulators is not n ecessarily a \nconsequence of topological magnon insulator, but it depends sol ely on the Berry \ncurvature of the magnon bands. This is expected as periodically  circularly polarized \nlight will break timereversal symmetry. \nAs shown in Fig.7, the magnon-magnon interaction has an importa nt influence \non the thermal Hall effect in Heisenberg honeycomb ferromagnets . This interaction \ncan occur in two ways: exchange interaction and direct interact ion. Exchange \ninteraction is the main mode of interaction between magnons in ferromagnets. In this \ninteraction, the spin directions of magnons will influence each  other, resulting in a \nchange in the orientation of the spin magnetic moments. This ch ange in orientation \naffects the thermal Hall effect because the heat flow must over come the exchange \ninteraction between the spin magnetic moments as it passes thro ugh the ferromagnet, \nresulting in a reduction in the heat flow. Therefore, the prese nce of exchange \ninteractions results in a positive sign for the thermal Hall co nductivity. Direct \ninteraction is another way of interaction between magnons, whic h can occur through \nthe mechanism of many-body intera ction. In this interaction, th e spin directions of \nmagnons will influence each other, resulting in a change in the  distribution of spin \nmagnetic moments. This change in distribution affects the therm al Hall effect because the heat flow must pass through the distribution region of the spin magnetic moment \nwhen passing through the ferromagnet. When the distribution of spin magnetic \nmoments changes, the transmission mode of heat flow will also c hange, resulting in a \nchange in the sign of the thermal Hall conductivity. In additio n, as the temperature \nincreases, thermal excitation will gradually destroy the ordere d arrangement between \nmagnons, resulting in a change in the interaction between magno ns. This change may \ncause the sign of the thermal H all conductivity to reverse. Thu s, in a Heisenberg \nhoneycomb ferromagnet, the influence of magnon-magnon interacti on on the thermal \nHall effect may change with temperature as our study shown.   \nⅦ. SUMMARY AND OUTLOOK \nIn summary, we have investigated the many-body interaction effe cts of Floquet \nmagnons in laser-irradiated Heisenberg honeycomb ferromagnet wi th off-resonant \ncircularly polarized  light field  tunable DMI. Our results show that  the quantum \nferromagnet systems periodically l aser-driven exhibits temperat ure-driven topological \nphase transitions due to the in teraction between Floquet magnon s. These transitions \nare marked by the Floquet magnon band gap closing-reopening wit h increasing \ntemperature and the sign change o f the Chern number and thermal  Hall conductivity \nwith increasing temperature at a  finite off-resonant circularly  polarized light field and \nmagnetic field. Our work complements the lack of previous theor etical works that the \ntopological phase transition of the Floquet magnon under the li near spin wave \napproximation is only tunable by the light field. The thermal H all effect of Floquet \nmagnons provides a prominent signature of the topological phase  transitions near the \npoint K- allowing for the experimental  investigation of this many-body  effect. Our \nproposal and conclusion are quite  universal for quantum ferroma gnet systems driven \nperiodically and can be extended  to quantum antiferromagnet sys tems driven \nperiodically. In general, we believe that the results in this p aper are pertinent to \nexperiments and will remarkably impact future research in topol ogical magnon insulators, topological insulatin g antiferromagnets and their p otential practical \napplications to photo-magnonics and magnon spintronics. In expe riments, the \nCr-based trihalide CrX 3 (X=F, Cl, Br, and I) may be viewed as a potential candidate to \nconfirm our results. \n  \nACKNOWLEDGMENTS \nThis work was supported by the National Natural Science Foundat ion of China under \nGrant No. 12064011 and the Scientific Research Fund of Hunan Pr ovincial Education \nDepartment under Grant No. 23A0404.   \nREFERENCES: \n[1] F.-D.-M. Haldane, Phys. Rev. Lett.  61, 2015 (1988). \n[2] C.-L. Kane, E.-J. Mele, Phys. Rev. Lett. 95, 226801 (2005). \n[3] B. A. Bernevig, S.-C. Zhang, Phys. Rev. Lett.  96, 106802 (2006). \n[4] M.-Z. Hasan, C.-L. Kane, Rev. Mod. Phys. 82, 3045 (2010). \n[5] X.-L. Qi, S.-C. Zhang, Rev. Mod. Phys.  83, 1057-1110 (2011). \n[6] D. Zhang, W.-K. Lou, M. S. Miao, S. C. Zhang, K. Chang, Phys. Rev. Lett.  111, \n156402 (2013). \n[7] I. Garate, Phys.Rev.Lett.  110, 046402 (2013). \n[8] P.-A Puente, P. Chudzinski, Phys. Rev.B  106, L081103 (2022). \n[9] L. Lu, J.-D. Joannopoulos, M. Soljaˇci´c, Nat. Photon.  8, 821 (2014). \n[10] F. Flamini, N. Spagnolo, F. Sciarrino, Rep. Prog. Phys.  82, 016001 (2018). \n[11] X.-D. Chen, W.-M. Deng, F.- L. Shi, F.-L. Zhao, M. Chen, J. -W. Dong, Phys. Rev. \nLett. 122, 233902 (2019). \n[12]Y. Pennec, J.-O. Vasseur, B.  Djafari-Rouhani, L. Dobrzyński , P. A. Deymier, \nSurf. Sci. Rep.  65, 229291 (2010). \n[13] M. Serra-Garcia, V. Peri, R. Süsstrunk, O.-R. Bilal, T. La rsen, L. G. Villanueva, \nS. D. Huber, Nat. 555, 342-345 (2018). [14] V. V. Kruglya k, R. J. Hicken, J. Magn. Magn. Mater.  306, 191 (2006). \n[15] A. A. Serga, A. V. Chumak, B. Hillebrands, J. Phys. D  43, 264002 (2010). \n[16] K. Nakata, S. K. Kim , J. Klinovaja, D. Loss, Phys. Rev. B  96, 224414 (2017). \n[17] Z.-X. Li, Y. Cao, P. Yan , Phy. Rep.  915, 1-64 (2021). \n[18] F. Bloch, Z. Phys.  61, 206 (1930). \n[19]C. Kittel, Phys. Rev.  73, 155 (1948). \n[20] P. Laurell, G. A. Fiete, Phys.Rev.B  98, 094419 (2018). \n[21] K.-S. Kim, K.-H. Lee , S.-B. Chung, J.-G. Park, Phys. Rev.B  100, 064412 (2019). \n[22] M. Hirschberger, R. Chisnell, Y. -S. Lee, N.-P. Ong, Phys. Rev. Lett.  115, \n106603 (2015). \n[23] R. Chisnell, J.-S. Helton, D.-E. Freedman, D.-K. Singh, R. -I. Bewley, D.-G. \nNocera, Y.-S. Lee,  Phys. Rev. Lett. 115, 147201 (2015). \n[24]Y. Onose, T. Ideue, H. Katsura, Y. Shiomi, N. Nagaosa, Y. T okura, Science  329, \n297 (2010). \n[25]T. Ideue, Y. Onose, H. Katsura, Y. Shiomi, S. Ishiwata, N. Nagaosa, Y. Tokura, \nPhys. Rev. B.  85, 134411 (2012). \n[26]H. Katsura, N. N agaosa, P. A. Lee, Phys. Rev. Lett.  104, 066403 (2010). \n[27]R. Matsumoto, S. Murakami, Phys. Rev. Lett. 106, 197202 (20 11); Phys. Rev. B.  \n84, 184406 (2011). \n[28]L. Zhang, J. Ren, J. S. Wang, B. Li, Phys. Rev. B  87, 144101 (2013). \n[29] H. Lee, J.-H . Han, P.-A. Lee, Phys. Rev. B  91, 125413 (2015). \n[30] S. Murakami, A. Okamoto, J. Phys. Soc. Jpn.  86,011010 (2017). \n[31] F. Zhuo, J. Kang, A. Manchon, Z. Cheng, Advanced Physics Research,  2300054 \n(2023).  \n[32] I. Dzyaloshinsky, J. Phys. Chem. Solids  4, 241 (1958). \n[33] T. Moriya, Phys. Rev.  120, 91 (1960). \n[34] S. A. Owerre,  J. Phys.: Condens. Matter  28, 386001 (2016). \n[35] S. K. Kim, H. Ochoa, R. Zarzuela, Y. Tserkovnyak, Phys. Rev. Lett. 117, 227201 \n(2016). \n[36]S. A. Owerre, J. Appl. Phys.  120, 043903 (2016). [37] W. Feng, B. Tang, L. Wu, P. Kong, C. Yang, L. Wang, K. Den g, J. Magn. Magn. \nMater. 536, 168089 (2021). \n[38]T. Holstein, H. Primakoff,  Phys. Rev. 58, 109 (1940). \n[39] F.-J. Dyson,  Phys. Rev.  102, 1217 (1956); 102, 1230 (1956). \n[40] S.-V. Maleev, Sov. Phys. Jetp.  64, 654 (1958). \n[41]M.-E. Zhitomirsky, A.-L. Chernyshev, Rev. Mod. Phys. 85, 219 (2013). \n[42] P.-A. McClarty, J.-G. Rau, Phys. Rev. B  100, 100405 (2019). \n[43] V. V. Mkhitaryan, L. Ke, Phys. Rev. B  104, 064435 (2021). \n[44] S. S. Pershoguba, S. Banerjee, J. C. Lashley, J. Park, H. Agren, G. Aeppli, A. V. \nBalatsky, Phys. Rev. X  8, 011010 (2018). \n[45] A. Mook, K. Plekhanov , J. Klinovaja, D. Loss, Phys. Rev. X  11, 021061 (2021). \n[46] Y.-S. Lu, J.-L. Li, C.-T. Wu, Phys. Rev. Lett.  127, 217202 (2021). \n[47] H. Sun, D. Bhowmick, B. Yang, P. Sengupta,  Phys. Rev. B  107, 134426 (2023). \n[48] H. Zhu, H.-C. Shi, Eur. Phys. J. Plus  138, 1045 (2023). \n[49] Y.-M. Li , X.- W. Luo, K. Chang, Phys. Rev. B  107, 214417 (2023). \n[50] Q.-H. Chen, F.- J. Huang, Y.-P. Fu Phys. Rev. B  108, 024409(2023) \n[51]T. Oka, H. Aoki, Phys. Rev. B  79, 081406 (2009). \n[52] J. Inoue, A. Tanaka, Phys. Rev. Lett.  105, 017401 (2010). \n[53] N. Lindner., G. Refael. V Gaslitski., Nat. Phys.  7, 490(2011). \n[54] J.Cayssol, B.D´ora, F.Simon, R.Moessner,  Phys. Status Solidi (RRL)-  Rap. Res. \nLett. 7, 101 (2013). \n[55]G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T. Uehlinger ,  D .  G r e i f  ,  T .  \nEsslinger, Nature  515, 237 (2014). \n[ 5 6 ]  L .  D u ,  P . - D .  S c h n a s e ,  A . - D .  B a r r ,  A . - R .  B a r r ,  G . -  A .  F i e t e , Phys. Rev. B  98, \n054203 (2018).  \n[57] S. A. Owerre,  J. Phys. Commun.  1, 021002 (2017). \n[58]S. Kar, B. Basu ,  Phys. Rev. B 98, 245119(2018). \n[59] S.A. Owerre, P. Mellado, G. Baskaran, Eur. Lett. 126, 27002(2019). \n[60] Z. Zhang, W. Fe ng, Y. Yao, B. Tang, Phys. Lett.  A 414, 127630 (2021). \n[61] Y. Aharonov, A. Casher, Phys. Rev. Lett ., 53,319 (1984). [62] J. H.Shirley,  Phys. Rev.  38, B979 (1965). \n[63] G. D. Mahan, Springer Science &Business Media , (2013). \n[64] P. A.McClarty, X.-Y.Dong, M.Gohlke, J. G.Rau, F.Pollmann, R.Moessner, Penc \nK., Phys. Rev. B  98, 060404 (2018). \n[65] L. Du,  Phys. Rev. B  98, 054203 (2018). \n[66] D. Xiao, M.- C. Chang, Q. Niu, Rev. Mod. Phys.  82, 1959 (2010). \n "    },    {        "title": "1907.01303v1.Bias_free_reconfigurable_magnonic_phase_shifter_based_on_a_spin_current_controlled_ferromagnetic_resonator.pdf",        "content": "arXiv:1907.01303v1  [physics.app-ph]  2 Jul 2019Bias-free reconﬁgurable magnonic phase shifter based on a s pin-current controlled\nferromagnetic resonator\nZikangZhang, Shuang Liu, Tianlong Wen, Dainan Zhang, LichuanJin, Y ulong Liao, Xiaoli Tang, and ZhiyongZhong∗\nState Key Laboratory of Electronic Thin Films and Integrate d Devices,\nUniversity of Electronic Science and Technology of China, C hengdu 610054, China\n(Dated: July 3, 2019)\nControllable phase modulation plays a pivotal role in the re searches of magnonic logic gates. Here\nwe propose a reconﬁgurable spin-current controlled magnon ic phase shifter based on a ferromagnetic\nresonator. The proposed phase shifter requires no magnetic bias ﬁeld during operation. The device\nis directly conﬁgured over the waveguide while keeping the o riginal structure of the waveguide\nunaﬀected. Numerical micromagnetic simulations show that the phase shifter could yield either a\nπ-phase or no shift depending on the magnetization status of t he resonator, which can be controlled\nby a current pulse. Moreover, the phase-shifting operation could be aﬀected by spin current. At\ndiﬀerent input current density, the device could be either u sed as a dynamic controlled phase shifter\nor a spin-wave valve. Finally, a XNOR magnonic logic gate is d emonstrated using the proposed\nphase shifter. Our work can be a beneﬁcial step to enhance the functionality and compatibility of\nthe magnonic logic circuits.\nI. INTRODUCTION\nFurtherminiaturizationandintegrationofcomplemen-\ntary metal oxide semiconductor (CMOS) circuitry are\nlimited by the fundamental physical limitation and the\nJoule heating dissipation [1, 2]. This circumstance stim-\nulates a global interest in search for novel alternative\ntechnologies of CMOS [3, 4]. Among these alternative\ntechnologies, using spin waves (or magnons) as informa-\ntion carriers, which could be used to fulﬁll wave-based\ninformation processing [5], is one of the most promising\ntechnologies [6–12]. The typical operational wavelength\nof spin waves is several orders of magnitude shorter than\nthat of the electromagnetic waves at the same frequency\n[9, 11, 13, 14], which allows for a better dimensional scal-\ning of magnonic devices in nano-meter scale. Besides,\ndue to no particle transfer during the transmission of\nspin waves, magnonic devices have extremely low energy\nconsumptions [15], which was estimated to be at the aJ\nlevel by Intels benchmarking of beyond-CMOS devices\n[16,17]. Furthermore,withencodingtheinformationinto\nthe amplitude and phase of spin waves, magnonic devices\ncould even open the way to non-Boolean computing [18–\n22], reversible logic [23–25] and artiﬁcial neural networks\n[5, 26, 27].\nRecent studies have highlighted the promising applica-\ntion ofmagnonicdevicesin logiccircuits [15, 28–33]. His-\ntorically, the most representative prototype of magnonic\nlogic gates was based on the principle of the Mach-\nZenhder-type interferometer (MZI) [18]. In this proto-\ntype logic gate, the spin waves (SWs) propagate along\nthe two branches of the MZI and interfere at the out-\nput terminal. By modulating the phases of the SWs\nin the two branches, the output signal was interfered\neither constructively or destructively depending on the\n∗zzy@uestc.edu.cnrelative phase diﬀerence of the two incident SWs. Here\nthe magnonic phase shifter is responsible for the phase\nmodulation and the interference output, which is crit-\nical for magnonic logic gates. As a result, an eﬃcient\nreconﬁgurable magnonic phase shifter is essential for\nthe eﬃcient implementation of the interferometer-based\nmagnoniclogicgates. Variousphase-shiftingmechanisms\nhavebeenproposedinrecentyears. Themainapproaches\nare based on the micromagnetic structure such as the do-\nmain walls [34–36], magnetic defects [37, 38], magnonic\ncrystals [39] and others [40, 41]. These micromagnetic\nstructure based magnonic phase shifters generally serve\naspassiveelementsto provideconstantphase shifts with-\nout extra energy input, namely, they are ”static”. These\nstatic magnonic phase shifters are not adaptable for the\ndynamic occasions. However, in the construction of\nmagnonic logic gates, we do need the dynamic phase\nshifter, which could be controlled in real time by an ex-\nternal signal. The reportedapproachesto dynamic phase\nshifter includes that applying magnetic ﬁelds [28, 42],\nelectric currents [43], spin-polarized currents [44–46] or\nelectric ﬁelds [33, 47–49] to achieve the externally initi-\nated phase modulation. The main requirements for these\ndynamic phase shifters include simple control mode, low\npower consumption and scalable structure [15]. From\nthis point ofview, most dynamic magnonic phaseshifters\nnow have drawbacks such as the performance degrada-\ntion without bias ﬁeld, tremendous input energy and the\nnarrow application scope.\nIn this paper, we propose a bias-free magnonic phase\nshifter based on a ferromagnetic resonator. The res-\nonator is dynamically modulated by a relatively small\nmagnitude of spin current. The proposed device could\nbe used as a static or dynamic magnonic phase shifter\ndepending on the operation mode.\nThe remainder of this paper is organized as follows. In\nSec. II, the device structure and the simulation methods\narepresented. In Sec. III, we ﬁrst givean overviewofthe\nphase-shifting mechanism, and we then perform numeri-2\nFIG. 1. Device structure. (a) Schematic illustration of the micromagnetic model and perspective image (zoom in) of the\nmagnonic phase shifter. The magnonic phase shifter include s a heavy metal layer with dimensions of 100 nm ×300 nm×10\nnm, a resonator with the dimensions of 150 nm ( l)×50 nm (w)×2 nm (t) and electrodes. An excitation ﬁeld with 200 A/m\namplitude generates spin wave along the x-axis. The length, width and thickness of stripe waveguide are L = 2000 nm, W =\n50 nm and T = 4 nm, respectively. (b) Side view of the waveguide and phase shifter. The resonator is formed beneath the\nheavy metal layer, and the spacing between resonator and wav eguide is d = 2 nm. (c) Detailed parameters of the resonator.\nThe two diagonal corners of the resonator are transformed in to an arc with radius R = 25 nm.\ncalsimulationstodemonstrateacontrollablestaticphase\nshifter mode and a dynamic phase shifter mode. Finally,\nwe highlight the implementation of universal Boolean\nXNOR logic gate and the applicable conditions of the\nproposed magnonic phase shifter. Sec. IV summarizes\nthe key results and advantages of this work.\nII. METHODS\nFig.1 shows the scheme of the proposed magnonic\nphase shifter. The phase shifter is consisted of a narrow\nstrip waveguide with dimensions of 2 µm (L)×50 nm\n(W)×4 nm (T), two blocks of electrode, a heavy metal\nlayer (e.g. Pt, Td, W, etc.) with dimensions of 100 nm\n×300 nm×10 nm and a resonator. Both the waveguide\nand the resonator are made of Permalloy (Py), which is\nsuitable for nano-patterning [15]. The length, width and\nthickness of the resonator are l= 150 nm, w= 50 nm\nandt= 2 nm, respectively. In order to destroy the ax-\nial symmetry, two diagonal corners of the resonator are\ndesigned to be an arc with radius R = 25 nm. The res-\nonator is placed above the center of the waveguide with\nspacing between them of d = 2 nm, which is achieved\nby the supporting of electrodes. We performed the simu-\nlation using Object Oriented Micromagnetic Framework(OOMMF) that numerically solves the Landau-Lifshitz-\nGilbert (LLG) equation [50]. The cell size is set as 5\n×5×2 nm3. The following material parameters of Py\nare used: saturation magnetization Ms= 8×105A/m,\nexchange stiﬀness A= 1.3×10−11J/m and anisotropy\nconstant K= 0 J/m3. The Gilbert damping factor αis\nchosen to be 0.5 for obtaining equilibrium state, 0.005for\ndynamic simulations and 0.05 at the ends of the waveg-\nuide to prevent the reﬂection of spin waves. The ground\nstate is obtained by relaxing the system under no bias\nﬁeld for the waveguide and uniformly magnetized for the\nresonator in the positive x and y axis directions respec-\ntively. We choose a sinusoidal magnetic ﬁeld along the\ny axis with 200 A/m amplitude, 10.50 GHz frequency as\nthe excitation signal. The excitation area is 50 ×50 nm2\nat 500 nm away from the left end of the waveguide.\nIII. RESULTS AND DISCUSSIONS\nA. The phase-shifting mechanism\nThe main part of the proposed magnonic phase shifter\nis the resonator. According to the magnetization status\nof the resonator, the beneath propagating spin waves in3\nFIG. 2. (a) The normal state with the magnetization of the res onator along +y direction. (b) The phase-shifting state wit h\nthe magnetization of the resonator along -y direction. (c) T he spatial distributions of z-component of the magnetizati on (Mz)\nin the waveguide along the x-axis under both states at 10 ns in the dynamic simulations. (d) The time dependent Mz at the\npoint x = 1300 nm of the waveguide from 0.5 to 3.5 ns.\nthe waveguide have diﬀerent temporal and spatial distri-\nbution. When the magnetization of the resonator points\nup as shown in Fig.2(a), the phase and amplitude of the\nSWs are rarely aﬀected and shown by the blue line in\nFig. 2(c). We name this state as normal state. How-\never, when the magnetization of the resonator points\ndown as shown in Fig.2(b), the SWs undergo a π-phase\nshift and an amplitude attenuation, as shown by the\nred line in Fig.2(c). We name this state as phase shift-\ning state. To understand the phase-shifting mechanism\nclearly, Fig.2(d) showsthe out-of-planeMz component as\na function of time at the point x = 1300nm ofthe waveg-\nuide. We propose the phase-shifting process roughly has\nthree stages. Firstly, resonator absorbs energy from the\napproachingSWs to excite oscillation (absorption stage).\nSecondly, when the absorption of SWs is strong enough,\nthe resonance in the resonator occurs, and the phase of\nSWsstarttovary(phaseshiftingstage). Finally,asteady\nstate of resonance is achieved with a determined phase\nshift (stabilization stage).\nThephase-shiftingmechanismcouldbeillustratedbya\nstrikingimbalanceofdispersionrelationsofthe SWs that\npropagate beneath the excited resonator [40]. When the\nresonator is resonantly excited by the SWs, the dynamic\ndipolar stray ﬁeld of the resonator, as shown in Fig.3(a),\nrotates around the surface. And through the dynamic\ndipolar stray ﬁeld, the resonator exerts mutual inﬂuence\non the incident signals. This interplay is aﬀected by the\nchirality of the dipolar stray ﬁeld. In the normal state\n(Fig.3b), the stray ﬁeld on the waveguide is along the\npropagation direction of the SWs. While in the phaseshifting state (Fig.3c), the stray ﬁeld is opposite to the\npropagation direction of the SWs. Namely, the chirality\nof the stray ﬁeld causes an imbalance of dispersion re-\nlations of the SWs. In other words, the propagation of\nSWs beneath the resonator is nonreciprocal, where the\nSWs could only be aﬀect by the excited resonator in one\nmagnetization direction. Remarkably, without varying\nthe resonator,the originalnormalstateasin Fig.2(a)will\nswitch to phase-shifting state when the SWs reversetheir\npropagation directions, and vice versa (see Appendix A).\nHence, when the transmission direction of SWs is deter-\nmined, we could then determine the states according to\nthe magnetization of the resonator as discussed above.\nB. The application to a static phase shifter\nWe ﬁrst consider utilizing this magnonic phase shifter\nas a static controllable phase shifter. The operational\nprinciple is shown in Fig.4(a). To switch the magnetiza-\ntion of the resonator, we exert a current pulse into the\nheavy metal layer along the y-axis directions. When the\ncurrentpulse ﬂowsintothe heavymetal layerwith strong\nspin-orbitcoupling, a spincurrentis generateddue to the\nspin Hall eﬀect [51–54]. The generated spin current is in-\njected into the adjacent Py ﬁlm in the x-axis directions.\nIn the presence of the current-induced spin-orbit torque,\nthe magnetization dynamics of the resonator is described\nby the LLG equation with an additional Slonczewski-like4\nFIG. 3. (a) Scheme of the dipolar stray ﬁeld of the resonator. Simpliﬁed diagram representing the directions of the dynam ic\ndipolar stray ﬁeld and the SWs under (b) normal state and (c) p hase shifting state.\nFIG. 4. (a) Schematic diagram of the operational principle a s a static phase shifter. (b) The switching procedure of the\nmagnetization of resonator. (c) The shape anisotropy.5\ntorque [54]:\n∂m\n∂t=−γ0m×Heff+αm×∂m\n∂t+τSLT(1)\nand\nτSLT=γ0τdm×(m×σ) (2)\nwheremis the unit magnetization vector, Heffis the\neﬀective magnetic ﬁeld, γ0is the gyromagneticratio, αis\nthe Gilbert damping parameter, τSLTis the Slonczewski-\nlike torque term, τdis the magnitude of τSLT, andσis\nthe directionofspinpolarization. This torquetermtends\nto align the magnetization towards the direction of spin\npolarization.\nIt should be noted that the current density Jused in\nthis paper is under the general simulation circumstance\nwith the polarization degree P = 0.4 and the resistance\nmismatch Λ = 2. These parameters give an ordinary po-\nlarization eﬃciency of the ferromagnetic ﬁlms. However,\nthe polarization eﬃciency could be much higher accord-\ning to the spin Hall angle. The spin current density Js\nis related to the applied current density Jin spin-orbit\ncoupling model as follow [53]:\nJs=θSH(σ×J) (3)\nwhereθSHis the spin Hall angle ofthe heavymetal layer,\nand theσis polarization of the electron spin. If the spin\nHall angle of the chosen heavy metal material is large\nenough, the actually required current density Jcan be\nvery small.\nHere we give an example of the switching proce-\ndure based on micromagnetic simulations as shown in\nFig.4(b). The left diagram represents the initial magne-\ntizations of the resonator and the waveguide stabilized\nfrom ground state. The middle diagram is a snapshot\nof the magnetizations of the resonator and the waveg-\nuide after applying a current pulse with a density of 5\n×1012A/m2for 250 ps. Here the polarization direction\nis along the +x direction. After a suﬃciently long relax-\nation ( 20ns), the magnetization of the resonator is even-\ntually toggled to the -y direction as shown in the right\ndiagram. The relaxation process is aﬀected by the shape\nanisotropy as shown in Fig.4(c), which ensure the cer-\ntainty of the ﬁnal magnetization status of the resonator.\nBy reversing the current pulse direction, the magneti-\nzation switching can also be readily accomplished. The\nswitching procedures are determinative stable. This op-\nerational mode is very suitable for the application of pro-\ngrammablecircuits, sincethedeviceisreconﬁgurableand\nthe energy consumption is extremely low.\nC. The application as a dynamic controlled phase\nshifter\nThe proposedmagnonicphaseshiftercould alsobe uti-\nlized as a dynamic phase shifter in magnonic logic gate.The operational principle of the dynamic mode is de-\npicted in Fig.5(a). In this mode, we maintain the mag-\nnetization of the resonator in phase-shifting state and\nexert constant currents (versus current pulse for static\nphase shifter) throughout the whole phase-shifting pro-\ncess. The polarization of generated spin current is along\nthe +x direction. As mentioned above, such a spin cur-\nrent introduces a spin-transfer term to the LLG equa-\ntion and thus alters the original excitation state. We\nobserved a signiﬁcant diﬀerence of the phase shifting ef-\nfects at diﬀerent current densities. Fig.5(b) represents\nthe dependence of the waveforms of the SWs on diﬀerent\napplied current densities J. The amplitudes of the SW\nsignals diﬀer remarkably, and there is a sudden change\nof the relative phase between J= 8×1010A/m2and\nJ = 9×1010A/m2. The detailed relationship between\nthe amplitude and the current density Jis summarized\nin Fig.5(c). The curve shows a single-valley image with a\nsteep descending branch and a slowly ascending branch.\nWe attribute this result to the interaction of the spin-\norbit torque and the resonant excitation. In the ini-\ntial stage, the spin-orbit torque tends to align the mag-\nnetic moments to the polarization direction. However,\nits strength is not enough to overcomethe resonancepro-\ncess. The magnetic moments are easier to be resonantly\nexcited even in presence of spin-orbit torques. As a re-\nsult, the resonant excitation absorbs more energy from\nthe SWs, and the amplitude of the SWs decreases obvi-\nously at the beginning, which is corresponding to the de-\nscending branch in Fig.5(c). As applied current density J\nincreases, the procession of magnetic moments is gradu-\nally dominated by the spin-orbit torque which gradually\ndiminish the resonant excitation in turn. Finally, the\nmagnetic moments are pinned to the polarized direction\nby the current, and resonatorhas minute eﬀect on the in-\ncident SWs as in the normal state. This process requires\nlarge applied current density, which corresponds to the\nslowly ascending branch. To conﬁrm this mechanism, we\ndid simulations by altering the spin current direction or\nmagnetization of the resonator (see Appendix B).\nWe could build a controllable spin-wave phase shifter\nor spin-wave valve based on this phenomenon. Fig.6(a)\nshows the typical results of time dependent waveforms.\nThe corresponding snapshots of the spatial distribution\nof dynamic magnetizations of the waveguide are repre-\nsented in Fig.6(b). At J= 0 A/m2, the incident SWs are\nshifted by π-phase when passing through the resonator.\nAtJ=3×1011A/m2, the phase of the incident SWs is\nunaﬀected with respect to the originalsignal. At J=9×\n1010A/m2, the SWs are fully absorbed by the resonator\nthat referes to an oﬀ-state. Hence, by only changing the\napplied current density, we could utilize this device as a\ndynamic phase shifter or spin-wave valve as illustrated in\nFig.6(c).\nAccordingly, we demonstrate a XNOR gate using\nthe magnonic phase shifter. Fig.7(a) represents the\nschematic diagramof the proposed XNOR gate. Fig.7(b)\nand (c) shows the truth table and the simulated spatial6\nFIG. 5. (a) Schematic diagram of the operational principle a s a dynamic controlled phase shifter. (b) The graph shows the\ndependence of the waveforms of the SWs under diﬀerent applie d current densities. (c) The detailed relationship between the\namplitude and the current density.\nmaps of the logic gate. The spacing between the two\nbranches is wide enough to avoid dynamic magnetostatic\ncoupling. SWs at frequency f= 10.5 GHz are simultane-\nously excited in the left edge to ensure these two signals\nhave the same initial phase. The resonators are the same\nas above mentioned to ﬁt the application frequency. The\ninput logic 0 and 1 represent the applied current density\nJ= 0 A/m2andJ= 3×1011A/m2, respectively. The\noutput logic 0 and 1 are deﬁned as the destructive and\nconstructive interference results, respectively. Both sim-\nulated snapshots and the truth table excellently demon-\nstrate the operation of the XNOR logic.\nD. The application conditions\nTo strengthen the applicability of the proposed\nmagnonic phase shifter, it is necessary to investigate the\napplication conditions of the device. To simplify the\nmodel and ensure the consistency of the arc edges of the\nresonator, we only consider the width of the waveguide\nW and the length of the resonator las the size variables.\nWe ﬁrstly investigate the impact of waveguide width W\non the phase shift under the original conditions. The\nresult is shown in Fig.8(a). In a limited range (approx-\nimately from 35 nm to 55nm), the waveguide width Whas minute eﬀect on the phase shift. However, when W is\neither too narrow or too wide compared to the resonator\nlengthl, the phase shift amount decreases rapidly. Thus\nthe SWs could hardly excite the resonator when W is\ntoo narrow. While the interplay from the resonator is\nnot suﬃciently strong to cause a π-phase shift when W\nis too wide. The results suggest that the ratio of W/ l\nshould be chosen appropriately (about 1:3).\nFor a given waveguide, it is critical to determine a suit-\nable length of the resonator for the desired frequency.\nFig.8(b) represents the frequency dependency of phase\nshift at diﬀerent lengths l(with W = 50 nm). For each\ncurve, there exists a frequency marked by a star in the\nﬁgure that have maximum phase shift. We deﬁne the fre-\nquency and the corresponding phase shift as maximum\nphase-shifting frequency fmaxand maximum phase shift\nPmaxrespectively. The results show that only speciﬁc W\nandlcan achieve a π-phase shift for a frequency. Only\nwhen the Pmaxis equal to π, the corresponding fmax\n(fπ) is the exact application frequency. Fig.9 depicts the\ndependence of Pmaxon the resonator length l. The inset\nrepresents the dependence of fπonl. Due to the limita-\ntion of dispersion relation of the waveguide (the cut-oﬀ\nfrequency is 8 GHz), l >200 nm is not considered. The\nmaximum phase shift Pmaxdecreases sharply near the\npoint at x = 130 nm, due to the incompatibility of the7\nFIG. 6. (a) The typical results of time dependent waveforms f or applications of phase shifter or spin-wave valve. (b) The\ncorresponding snapshots of the spatial distribution of dyn amic magnetizations of the waveguide. (c) The table of the fu nction\nof the device.\nFIG. 7. (a) Schematic diagram of the proposed XNOR gate. (b) T he corresponding truth table of the device. (c) The simulate d\nspatial maps of the logic gate under all logic inputs.\nresonator and the waveguide as also shown in Fig.8(b).\nInthismodel, theworkingfrequency fπislinearlyrelated\ntolwhich could be summarized as follow:\nfπ=k∗l+C (4)\nwherek= 0.04 GHz/nm and C= 4.5 GHz according\nto the simulation. From the above formula, we could\nchoosea suitable length of the resonatorto ﬁt the desired\napplication frequency. The simulation results are in good\nagreement with the prediction equation (see AppendixC).\nIV. CONCLUSIONS\nIn conclusion, we propose a bias-free magnonic phase\nshifter based on a ferromagnetic resonator. This device\nhas outstanding advantages in compatibility and minia-\nturization for its surface mounting structure. We use\nnumerical micromagnetic simulations to show that the8\nFIG. 8. (a) The graph shows the change in SW phase shift with di ﬀerent waveguide width W. (b) The SW phase shifts are\nplotted as a function of the frequency under diﬀerent resona tor length l.\nFIG. 9. The dependence of Pmaxon the resonator length l.\nmagnonic phase shifter could be utilized as a static ele-\nment providingconstant phase shift. Meanwhile, it could\nalso be utilized as dynamic phase shifter controlled by\nthe spin current. This phase shifter requires relatively\nlow current input and provides stable phase shifts. We\ndemonstrate a universal XNOR gate based on this phase\nshifter. Furthermore, the relationship between the ap-\nplication frequency and the size of the phase shifter is\nsummarized. Results indicate that the magnonic phase\nshifter could be applied into a speciﬁc waveguide under\ndesired frequency by selecting an appropriate size pa-\nrameter. We believe this research could provide positive\nexperiences for the development of the magnonic logic\ncircuits.\nAppendix A: The nonreciprocity of the resonator\nSee Fig.10\nFIG. 10. Demonstration of the nonreciprocity of the res-\nonator. The snapshots of dynamic magnetizations of waveg-\nuide (a) without magnonic phase shifter at 5 ns, the directio n\nof the wave vector kxpoints right; (b) with the phase shifter\nunder normal state at 5 ns, the direction of the wave vector\nkxpoints right; (c) with the phase shifter under normal state\nat 5 ns, the direction of the wave vector kxpoints left; (d)\nwith the phase shifter under phase-shifting state at 5 ns, th e\ndirection of the wave vector kxpoints right.\nAppendix B: Comparison of the phase-shifting eﬀect\nunder applying the spin current in diﬀerent\ndirections.\nIn this section, we show that applying the spin current\nwith spin polarization along y direction could get the\nsimilar results under the phase shifting state, as shown\nin Fig.11. In this case, the density of the applied current\nneeded to cut the SWs is less than the +x case. The\nobtained results conform to our expectation. The spin\ncurrent in y case also interacts with the resonant exci-\ntation and causes the similar aﬀection. While the phase\nshifting results are unchanged if we perform this simu-\nlation under the normal state (not shown here). Under\nthe normal state, the resonant excitation has little reac-\ntion to the incident SWs. Even if the input spin current\naﬀects the resonance process, there is no eﬀect on the9\nincident SWs.\nFIG. 11. (a) The snapshots of magnetizations are shown un-\nder diﬀerent current density with the spin polarization alo ng\n+x direction. (b) The snapshots of magnetizations are shown\nwith the spin polarization along -y direction.\nAppendix C: The detailed data of the phase shifting\nsnapshots\nIn this section, we give some simulated results men-\ntioned in Fig.8 and Fig.9, see Fig.12 and Fig.13.\nFIG. 12. The snapshots of dynamic magnetizations represent\nthe points in Fig.8.\nFIG. 13. The snapshots of dynamic magnetizations represent\nthe points in Fig.9.\nACKNOWLEDGMENTS\nThis work is supported by the National Natural\nScience Foundation of China (grant Nos. 61734002,\n61571079,51702042and51827802),theNationalKeyRe-\nsearch and Development Plan (No. 2016YFA0300801),\nand the SichuanScience and TechnologySupport Project\n(No. 2017JY0002).\n[1] W. Haensch, E. Nowak, R. Dennard,\nP. Solomon, A. Bryant, O. Dokumaci, A. Ku-\nmar, X. Wang, J. Johnson, and M. Fis-\nchetti, Silicon cmos devices beyond scaling,\nIBM Journal of Research and Development 50, 339 (2006).\n[2] T. N. Theis and H. . P. Wong, The end of moore’s\nlaw: A new beginning for information technology,\nComputing in Science Engineering 19, 41 (2017).\n[3] J. A. Hutchby, G. I. Bourianoﬀ, V. V. Zhirnov,\nand J. E. Brewer, Extending the road beyond cmos,IEEE Circuits and Devices Magazine 18, 28 (2002).\n[4] K. Bernstein, R. K. Cavin, W. Porod,\nA. Seabaugh, and J. Welser, Device and ar-\nchitecture outlook for beyond cmos switches,\nProceedings of the IEEE 98, 2169 (2010).\n[5] A. V. Chumak, Fundamentals of magnon-based com-\nputing, arXiv e-prints , arXiv:1901.08934 (2019),\narXiv:1901.08934 [cond-mat.mes-hall].\n[6] A. Khitun and K. Wang, Nano scale com-\nputational architectures with spin wave bus,10\nSuperlattices and Microstructures 38, 184 (2005).\n[7] K. Bernstein, R. K. Cavin, W. Porod,\nA. Seabaugh, and J. Welser, Device and ar-\nchitecture outlook for beyond cmos switches,\nProceedings of the IEEE 98, 2169 (2010).\n[8] S. Bader and S. Parkin, Spintronics,\nAnnual Review of Condensed Matter Physics 1, 71 (2010).\n[9] V. Kruglyak, S. O Demokri-\ntov, and D. Grundler, Magnonics,\nJournal of Physics D: Applied Physics 43, 264001 (2010).\n[10] B. Lenk, H. Ulrichs, F. Garbs, and M. Mnzen-\nberg, The building blocks of magnonics,\nPhysics Reports 507, 107 (2011).\n[11] A. Chumak, V. Vasyuchka, A. Serga, and B. Hillebrands,\nMagnon spintronics, Nature Physics 11, 453461 (2015).\n[12] D. Sander, S. O Valenzuela, D. Makarov, C. H Mar-\nrows, E. Fullerton, P. Fischer, J. Mccord,\nP. Vavassori, S. Mangin, P. Pirro, B. Hillebrands,\nA. Kent, T. Jungwirth, O. Gutﬂeisch, C. G Kim,\nand A. Berger, The 2017 magnetism roadmap,\nJournal of Physics D: Applied Physics 50, 363001 (2017).\n[13] A. Serga, A. Chumak, and\nB. Hillebrands, Yig magnonics,\nJournal of Physics D: Applied Physics 43, 264002 (2010).\n[14] Z. Duan, A. Smith, L. Yang, B. Youngblood, and I. Kriv-\norotov, Nanowire spin torque oscillator driven by spin\norbit torques, Nature communications 5(2014).\n[15] A. Khitun, M. Bao, and\nK. Wang, Magnonic logic circuits,\nJournal of Physics D: Applied Physics 43, 264005 (2010).\n[16] D. Nikonov and I. A. Young, Overview of beyond-cmos\ndevices and a uniform methodology for their benchmark-\ning, Proceedings of the IEEE 101, 2498 (2013).\n[17] D. Nikonov and I. A. Young, Benchmarking of beyond-\ncmos exploratory devices for logic integrated circuits,\nIEEE Journal on Exploratory Solid-State Computational Dev ices and Circuits 1, 1 (2015).\n[18] K.-S. Lee and S.-K. Kim, Conceptual design of\nspin wave logic gates based on a machzehnder-type\nspin wave interferometer for universal logic functions,\nJournal of Applied Physics 104, 053909 (2008).\n[19] P. Shabadi, A. Khitun, P. Narayanan, M. Bao,\nI. Koren, K. L. Wang, and C. A. Moritz,\nTowards logic functions as the device, in\n2010 IEEE/ACM International Symposium on Nanoscale Archit ectures\n(2010) pp. 11–16.\n[20] S. Klingler, P. Pirro, T. Brcher, B. Leven,\nB. Hillebrands, and A. Chumak, Design of a spin-\nwave majority gate employing mode selection,\nApplied Physics Letters 105, 152410 (2014).\n[21] O. Zografos, S. Dutta, M. Manfrini, A. Vaysset, B. Sore,\nA. Naeemi, P. Raghavan, R. Lauwereins, and I. P. Radu,\nNon-volatile spin-wave majority gate at the nanoscale,\nAIP Advances 7(2016).\n[22] T. Fischer, M. Kewenig, D. Bozhko, A. Serga, S. Ihor I.,\nF. Ciubotaru, C. Adelmann, B. Hillebrands, and A. Chu-\nmak, Experimental prototype of a spin-wave majority\ngate, Applied Physics Letters 110(2016).\n[23] A. Roohi, H. Khademolhosseini, S. Sayedsalehi,\nand K. Navi, Implementation of reversible logic de-\nsign in nanoelectronics on basis of majority gates, in\nThe 16th CSI International Symposium on Computer Architect ure and Digital Systems (CADS 2012)\n(2012) pp. 1–6.\n[24] J. Gope, S. Mondal, M. Kundu, S. Chowdhury, and\nS. Bhadra, Reversible logic gates based on single spinlogic (2017) pp. 613–619.\n[25] M. Balynskiy, H. Chiang, D. Gutierrez, A. Kozhevnikov,\nY. Filimonov, and A. Khitun, Reversible mag-\nnetic logic gates based on spin wave interference,\nJournal of Applied Physics 123, 144501 (2018).\n[26] N. Locatelli, V. Cros, and J. Grollier, Spin-torque bui ld-\ning blocks, Nature materials 13, 11 (2013).\n[27] C. Pan and A. Naeemi, Non-boolean com-\nputing benchmarking for beyond-cmos de-\nvices based on cellular neural network,\nIEEE Journal on Exploratory Solid-State Computational Dev ices and Circuits PP, 1 (2016).\n[28] M. P. Kostylev, A. Serga, T. Schneider, B. Leven,\nand B. Hillebrands, Spin-wave logical gates,\nApplied Physics Letters 87(2005).\n[29] T. Schneider, A. Serga, B. Leven, B. Hillebrands,\nR. Stamps, and M. P. Kostylev, Realization of spin-wave\nlogic gates, Applied Physics Letters 92(2008).\n[30] A. Khitun and K. Wang, Non-volatile\nmagnonic logic circuits engineering,\nJournal of Applied Physics 110, 034306 (2011).\n[31] A. Khitun, Multi-frequency magnonic\nlogic circuits for parallel data processing,\nJournal of Applied Physics 111(2011).\n[32] A. Chumak, A. Serga, and B. Hillebrands,\nMagnon transistor for all-magnon data processing,\nNature communications 5, 4700 (2014).\n[33] B. Rana and Y. Otani, Voltage-controlled recon-\nﬁgurable spin-wave nanochannels and logic devices,\nPhysical Review Applied 9(2018).\n[34] R. Hertel, W. Wulfhekel, and J. Kirschner,\nDomain-wall induced phase shifts in spin waves,\nPhysical review letters 93, 257202 (2004).\n[35] C. Bayer, H. Schultheiss, B. Hillebrands,\nand R. Stamps, Phase shift of spin waves\ntraveling through a 180◦bloch-domain wall,\nIEEE Transactions on Magnetics - IEEE TRANS MAGN 41, 3094 (2005).\n[36] G. Duerr, R. Huber, and D. Grundler, En-\nhanced functionality in magnonics by domain\nwalls and inhomogeneous spin conﬁgurations,\nJournal of physics. Condensed matter: an Institute of Physi cs journal 24, 024218 (2012).\n[37] S. Louis, I. Lisenkov, S. Nikitov, V. Tyberkevych, and\nA. Slavin, Bias-free spin-wave phase shifter for magnonic\nlogic, AIP Advances 6, 065103 (2016).\n[38] K. Baumgaertl, S. Watanabe, and D. Grundler,\nPhase control of spin waves based on a mag-\nnetic defect in a one-dimensional magnonic crystal,\nApplied Physics Letters 112, 142405 (2018).\n[39] Y. Zhu, K. Chi, and C. Tsai, Magnonic\ncrystals-based tunable microwave phase shifters,\nApplied Physics Letters 105, 022411 (2014).\n[40] Y. Au, M. Dvornik, O. Dmytriiev, and V. Kruglyak,\nNanoscale spin wave valve and phase shifter,\nApplied Physics Letters 100(2012).\n[41] Y. Au, E. Ahmad, O. Dmytriiev, M. Dvornik, T. Davi-\nson, and V. Kruglyak, Resonant microwave-to-spin-wave\ntransducer, Applied Physics Letters 100(2012).\n[42] A. Ustinov, B. A. Kalinikos, and E. Lhderanta, Nonlin-\near phase shifters based on forward volume spin waves,\nJournal of Applied Physics 113(2013).\n[43] U.-H. Hansen, V. E. Demidov, and S. Demokritov, Dual-\nfunction phase shifter for spin-wave logic applications,\nApplied Physics Letters 94(2009).\n[44] Q. Wang, H. Zhang, X. Tang, H. Fangohr,\nF. Bai, and Z. Zhong, Dynamic control of11\nspin wave spectra using spin-polarized currents,\nApplied Physics Letters 105, 112405 (2014).\n[45] X. Chen, Q. Wang, Y. Liao, X. Tang, H. Zhang,\nand Z. Zhong, Control phase shift of spin-wave by\nspin-polarized current and its application in logic gates,\nJournal of Magnetism and Magnetic Materials 394(2015).\n[46] X. Chen, Q. Wang, F. Bai, X. Tang, H. Zhang,\nand Z. Zhong, A methodology to design spin-\nwave-based logic gates in a single ferromag-\nnetic nanostripe using spin-transfer torque eﬀects,\nIEEE Transactions on Magnetics 52, 1 (2016).\n[47] T. Liu and G. Vignale, Electric con-\ntrol of spin currents and spin-wave logic,\nPhysical review letters 106, 247203 (2011).\n[48] X. Zhang, T. Liu, M. Flatt, and H. Tang, Electric-\nﬁeld coupling to spin waves in a centrosymmetric ferrite,\nPhysical review letters 113(2014).\n[49] X.-g. Wang, L. Chotorlishvili, G.-h. Guo, and J. Berak-\ndar, Electric ﬁeld controlled spin waveguide phase shifter\nin YIG, Journal of Applied Physics 124, 073903 (2018).\n[50] M. J. Donahue, D. G. Porter, N. I. of Standards, and\nT. (U.S.), OOMMF user’s guide , version 1.0. ed. (U.S.\nDept. of Commerce, Technology Administration, Na-tional Institute of Standards and Technology Gaithers-\nburg, MD, 1999).\n[51] K. Garello, C. O. Avci, I. M. Miron, M. Baumgartner,\nA. Ghosh, S. Auﬀret, O. Boulle, G. Gaudin, and P. Gam-\nbardella, Ultrafast magnetization switching by spin-orbi t\ntorques, Applied Physics Letters 105, 212402 (2014),\nhttps://doi.org/10.1063/1.4902443.\n[52] C. Zhang, S. Fukami, H. Sato, F. Matsukura,\nand H. Ohno, Spin-orbit torque induced mag-\nnetization switching in nano-scale ta/cofeb/mgo,\nApplied Physics Letters 107(2015).\n[53] X. Fong, Y. Kim, K. Yogendra, D. Fan, A. Sen-\ngupta, A. Raghunathan, and K. Roy, Spin-\ntransfer torque devices for logic and mem-\nory applications : Prospects and perspectives,\nIEEE Transactions on Computer-Aided Design of Integrated C ircuits and Systems 35, 1 (2015).\n[54] K. Wu, D. Su, R. Saha, and J.-P. Wang, Spin-\norbit Torque and Spin Hall Eﬀect-based Cellular\nLevel Therapeutic Neuromodulators: Modulat-\ning Neuron Activities through Spintronic Nan-\nodevices, arXiv e-prints , arXiv:1903.02726 (2019),\narXiv:1903.02726 [physics.app-ph]."    },    {        "title": "1506.05292v1.Continuous_wave_approach_for_simulating_Ferromagnetic_Resonance_in_nanosized_elements.pdf",        "content": "Continuous wave approach for simulating Ferromagnetic Resonance in nanosized\nelements\nK. Wagner\u0003and S. Stienen\nHelmholtz-Zentrum Dresden-Rossendorf,\nInstitute of Ion Beam Physics and Materials Research,\nBautzner Landstraße 400, 01328 Dresden, Germany\nM. Farle\nFaculty of Physics and Center for Nanointegration (CeNIDE),\nUniversity of Duisburg-Essen, Lotharstr. 1, 47057 Duisburg, Germany\n(Dated: November 1, 2016)\nWe present a numerical approach to simulate the Ferromagnetic Resonance (FMR) of micron and\nnanosized magnetic elements by a micromagnetic ﬁnite diﬀerence method. In addition to a static\nmagnetic ﬁeld a linearly polarized oscillating magnetic ﬁeld is utilized to excite and analyze the\nspin wave excitations observed by Ferromagnetic Resonance in the space- and time-domain. Our\ncontinuous wave approach (CW) provides an alternative to the common simulation method, which\nuses a pulsed excitation of the magnetic system. It directly models conventional FMR-experiments\nand permits the determination of the real and imaginary part of the complex dynamic susceptibility\nwithout the need of post-processing. Furthermore not only the resonance ﬁelds, but also linewidths,\nellipticity, phase relations and relative intensities of the excited spin wave modes in a spectrum can\nbe determined and compared to experimental data. The magnetic responses can be plotted as a\nfunction of spatial dimensions yielding a detailed visualization of the spin wave modes and their\nlocalization as a function of external magnetic ﬁeld and frequency. This is illustrated for the case\nof a magnetic micron sized stripe.\nI. INTRODUCTION\nThe detailed understanding of spin wave spectra of\nmagnetic micro- and nanostructures and their magneti-\nzation dynamics has found increasing interest from both\nfundamental and applied points of view for example in\nspin caloritronics and spin torque phenomena1–4. A pow-\nerful tool to investigate these spin wave spectra exper-\nimentally is the Ferromagnetic Resonance (FMR) de-\ntectedinthefrequencydomain5,6. Howeverinmostcases\nthe obtained FMR-spectra are complex in nature fea-\nturing several -often overlapping- resonances and require\ntheoretical descriptions of the nanostructured magnetic\nsystems to extract quantitative information. Micromag-\nnetic simulations of the FMR can be used to model those\nsystems and provide additional information on the char-\nacter of the observed magnetic excitations as well as their\ndependenceonmagneticparameters,geometries,conﬁne-\nment eﬀects or charge currentss7,8. This is especially of\ninterest when the complex geometries and interactions of\nthe nanoscale ferromagnet aggravate quantitative analyt-\nical approaches.\nHere we present a ﬁnite diﬀerence method utilizing a\nhomogeneous oscillating magnetic ﬁeld to simulate FMR\nspectra corresponding to experiments. In addition we\nshow how to further analyze the spectra by visualizing\nthe spatial distribution of the magnetic excitations. We\nstart by describing the problem deﬁnition, initialization\nand recorded data of the simulations in section II. Sub-\nsequently in section III the derivation of FMR-spectra is\ndescribed in detail as well as determination of the FMR\nﬁelds, linewidth and ellipticity of the resonances. In sec-tion IV we investigate the resonances contained in the\nspectra in terms of spin waves and spatial variations.\nII. METHOD\nThe micromagnetic simulations presented here, are\nbased on the public domain 3D-OOMMF (Object Ori-\nentedMicroMagneticFramework)9solver. Thisﬁnitedif-\nference software solves numerically the Landau-Liftshitz\nequation15(LLE)\nd~M\ndt=\u0000\r\u0010\n~M\u0002~Heﬀ\u0011\n\u0000\r\u0001\u000b\nMs\u0010\n~M\u0002\u0010\n~M\u0002~Heﬀ\u0011\u0011\n(1)\nwhere\ris the gyromagnetic ratio, \u000bthe Gilbert damp-\ning constant and ~Heﬀthe eﬀective ﬁeld. As time evolver\na Runge-Kutta method is used. Further details on the\nimplemention can be found in Ref. 5. Our approach to\nsimulate FMR-experiments can be split into three diﬀer-\nent steps: 1. relaxation 2. transient phase 3. dynamic\nequilibrium.\nFor initialization the spatial dimensions of a nanos-\ntructured ferromagnetic system are deﬁned by a grid of\nequal rectangular cells. All cells are assigned with an\nidentical magnetization vector ~M, located at the cen-\nter of the cell. In order to simulate the external ﬁeld of\nFMR-experiments, astaticmagneticﬁeld ~Hisappliedto\nthe system. To obtain the static magnetic ground state\nd~M=dt = 0arelaxationsimulationisperformed, without\napplying any excitation. So the motion of ~Mdamps out\nand~Mwill reorient to an equilibrium direction, givenarXiv:1506.05292v1  [physics.comp-ph]  17 Jun 20152\nby the local eﬀective ﬁeld. For faster convergence the\nprecession term in the LLE may be switched oﬀ and the\ndamping constant \u000bset large. As stopping criterion for\nthe simulation typically values of d^u=dt< 0:001\u000e=ns (^u\nis the unit vector of ~M) are chosen, to achieve the quasi\nstatic state, which is used as the inital state for the sub-\nsequent FMR-simulations.\nIn addition to the static ﬁeld ~Ha linearly polar-\nized oscillating magnetic ﬁeld ~hRF(t) =~h0\u0001sin(!\u0001t)\nis added for continous excitation of the magnetization\n(CW).~hRFis uniform over all cells, oriented perpendic-\nular to the static ﬁeld and corresponds to the magnetic\nmicrowave ﬁeld used in conventional FMR-experiments:\nh0= 398A/m ^ = 0:5mT (satifying the relation h0<<\n~H).\nDue to the interaction with ~hRFa driving torque is ex-\nerted on the magnetization. After a transient phase the\nmagnetization reaches a dynamic equilibrium, precess-\ning around the eﬀective ﬁeld with the angular frequency\n!. In this state ~hRFtransfers power to the magnetic\nsystem to compensate dissipation, induced by damping.\nTo study the dynamic equilibrium and discard transient\neﬀects, a ﬁxed time period is simulated, without gen-\nerating data for analysis. This time period is given by\nts= 2\u0019\u0001N=!, using an integer oscillation number Nof\n~hRF(details described in section III). Nis typically set\nbetween 40 and 60 depending on the magnetic parame-\nters (e.g. damping constant \u000b). This provides a constant\nprecession amplitude between consecutive oscillation cy-\ncles of the magnetization with a deviation of less than\n0:02 %. When the simulation time reaches tsthe actual\nparameters (like magnetization ~M, oscillating ﬁeld ~hRF,\nstatic ﬁeld ~H,...) are stored for each following iteration\nstep of the time evolver. This process continues for one\nfurther cycle of the oscillating ﬁeld ( N+ 1) and consists\nof at least 1000 iteration steps, which provides a time\nresolution in the ps regime. By analyzing these data, it\nis possible to follow the precession trajectory of the cell\nspeciﬁc magnetization in the space- and time-domain.\nIII. ANALYSIS OF THE SIMULATION\nWe now describe the analysis of the simulated data for\nthe dynamic equilibrium. In the linear response regime\ndiscussed here the dynamic magnetization ~ m(t)of each\ncell exposed to the oscillating ﬁeld is described by the\ndynamic susceptibility-tensor \u001f=\u001f0+i\u001f00:\n~ m(t) =\u001f~hRF(t) =\u001f~h0exp\u0000i(!t\u0000\u0019=2)(2)\nWhere we have used a complex representation of the\nsusceptibility and the applied sinusoidal varying mag-\nnetic ﬁeld~hRF.\u001fis a3\u00023-tensor with elements \u001fij.\nTo illustrate the motion of ~ min respect to the ~hRF,\nthe simulated time dependences for the case of an inﬁnite\nFIG. 1. Micromagnetic simulation of the time dependent re-\nsponse of the dynamic magnetization driven by an external\ndynamic ﬁeld ~hRFfor an inﬁnite thin ﬁlm in the xy-plane.\nThe normalized dynamic magnetization in the yz-plane (solid\nlines) are shown together with ~hRF(dashed line) oriented\nalong the ﬁlm-normal in the z-direction. The static ﬁeld ~His\noriented in the ﬁlm plane (x-direction) and is chosen to match\nthe resonance condition.\nﬁlm spanning the xy-plane are shown in ﬁg. 1. ~hRFis\noriented in the z-direction (out of plane) and ~Hin the\nx-direction (in plane), respectively. The displayed time\ndependent components of the oscillating dynamic mag-\nnetization (solid lines) lie in the yz-plane driven by ~hRF\n(dashed line). The oscillation of ~ mand its components\nmy;zis described by its amplitude Ay;z, frequency !and\nphase relation \u001ey;zin respect to ~hRF. Note that the\ndriven component mzexhibits a phase-shift of 90\u000eto the\noscillating ﬁeld as expected for a resonantly driven sys-\ntem. The precessional motion of the magnetization in\nequilibrium as well as the ellipticity of its trajectory can\nbe readily observed by the 90\u000ephase shift between the\ndynamic components my;zand the ratio of their diﬀering\nmaximal amplitudes.\nThe simulated FMR-spectra, which can be quantita-\ntively compared to experimental ones are derived as de-\nscribed in the following. The power Pabsorbed by the\nmagnetic system from ~hRFand consequently the FMR-\nsignalSis proportional to the diagonal element \u001f00\nzzof\nthe imaginary part of the susceptibility16:\nS/P/\u001f00\nzz (3)\nTo determine \u001f00\nzzfrom the simulation it is suﬃcient\nto analyze the component of ~ mparallel to~hRF(in this\ncasemz) after a complete cycle of oscillations as given for\nthe timets(see also section 2). This can be seen when\nconsidering the observable real part of ~ min equation 2:\nmz=<(\u001fzzh0exp (\u0000i(!t\u0000\u0019=2))) (4)\nInsertingts(!ts= 2\u0019N) yields:3\nmz(ts)=\u0000\u001f00\nzzh0 (5)\nHence, a proportional FMR-Signal ( \u001f00\nzz) can be sim-\nulated by directly monitoring \u0000mz(ts)=h0without the\nneed for extensive post-processing. (By a similar logic)\nthe real part of the susceptibility \u001f0\nzz- and therefore the\ncomplete\u001fzz- can be retrieved from the simulation by\nextractingmzat the maximum of ~hRF.\nFig. 2 shows the simulated external ﬁeld dependent\namplitude, phase and imaginary part of the susceptibil-\nity for an inﬁnite thin ﬁlm with the magnetic parameters:\nexchange constant A= 13\u000110\u000012J=m3, saturation mag-\nnetizationMs= 8:3\u0001106A=m, g-factorg= 2:12, Gilbert\ndamping parameter \u000b= 5\u000110\u00003.\nThe magnetic response shows the typical hallmarks of\na driven oscillator in respect to phase and amplitude and\na lorentzian absorption curve17. This enables one to de-\ntermine the resonance positions as well as their linewidth\nand relative signal strength. In the case of multiple res-\nonances a decomposition of the resultant spectra (exper-\niment or simulation) into lorentzian absorption lines is\nneeded to obtain those quantities. Subsequently this can\nbe compared to complex experimental results as for ex-\nample for the case of magnetic micron sized stripes11–13.\nFIG. 2. Micromagnetic simulation of amplitude, phase and\nimaginary part of the susceptibility for an inﬁnite thin ﬁlm\nfor diﬀerent static ﬁelds and a driving frequency of 15 GHz.\nThe magnetic resonance exhibits the classical hallmarks of a\ndriven oscillator, where the imaginary part is proportional to\nthe experimentally detected FMR-Signal.\nIV. CHARACTERIZING THE MAGNETIC\nEXCITATIONS\nTo further investigate the magnetic resonances in the\nsimulated FMR spectra the amplitude, phase and imag-\ninary part of the susceptibility may be analyzed for each\ncell of the magnetic system individually. The retrieved\nspatial distribution of the magnetic response often helpstoidentifyspinwavelikeresonances, theirwavevectorora\nlocalized character of the excitations. Here we exemplar-\nily simulate a FMR Spectrum of a 5\u0016m\u00021\u0016m\u000220 nm\nstripe at 14 GHz. As magnetic parameters we choose:\nexchange constant A= 13\u000110\u000012J=m3, saturation mag-\nnetizationMs= 8:3\u0001106A=m, g-factorg= 2:13, Gilbert\ndamping parameter \u000b= 7\u000110\u00003. The simulated FMR-\nSignal together with the stripe geometry and ﬁeld orien-\ntation is shown in ﬁg. 3.\nFIG. 3. Micromagnetic simulation of the susceptibility \u001f00\nzz\nfor a 5\u0016m\u00021\u0016m\u000220 nmPermalloy stripe and a frequency of\n14 GHz. In the conﬁned system multiple magnetic resonances\nare observed (labeled from 1 to 4)\nIn such a conﬁned system multiple magnetic reso-\nnances (labeled from 1 to 4) with diﬀering positions,\nlinewidths and intensities occur18. The spatial distribu-\ntion of the normalized imaginary part of the susceptibil-\nity for the most intense resonance 1 is shown color coded\nin ﬁg. 4. As can be seen this magnetic resonance ex-\nhibits the strongest excitation in the center of the stripe\nand will here be referred to as a localized quasi-uniform\nmode. Adiﬀerentmodecharactercanbeobservedforthe\nless intense magnetic resonance 2 in ﬁg. 4. \u001f00\nzzshows\na change in sign across the stripe, two nodal lines and a\nwavelike varying dependence along the dashed line.\nThis is very well approximated by a sinusoidal function\nas shown in ﬁg. 5 and resembles the expected modepro-\nﬁle of a standing spinwave with wavelength 734 nmand\ndipolar pinning19at the edges of the stripe, which devi-\nates from simply closed or open pinning conditions. In\na similar consideration the resonances 3, 4 can be as-\nsigned to higher standing spinwaves across the width of\nthe stripe, where the wavelength decreases for smaller\nresonance ﬁelds.\nBy such a spatial analysis of the magnetic excitations\none can for example explore the dependence of the reso-\nnance position, linewidth, mode proﬁle, ellipticity, inten-\nsity and pinning conditions on the magnetic parameters\nas well as on the exact geometry of the magnetic systems.\nThis information can be crucial for planning experiments\nand lead to a deeper understanding of the multiple reso-4\nFIG. 4. Normalized imaginary part of the susceptibility for\nthe magnetic resonances labeled as 1 and 2 respectively. Res-\nonance 1 shows a localized character (quasi uniform mode)\nwith the strongest excitation in the center region of the stripe.\nResonance 2 exhibits a harmonic dependence across the width\nof the stripe as expected for a standing spinwave with a wave-\nlength of 734 nm. The proﬁle along the dashed line is shown\nin ﬁg. 5.\nnances in experimentally observed spectra.\nFIG. 5. Lineproﬁle of the normalized imaginary part of the\nsusceptibility(blackdots)forthemagneticresonanceslabeled\nas 2 along the dashed line of ﬁg. 4. The spatial variation\nis very well approximated by a harmonic dependence of a\nstanding spinwave with a wavelength of 734 nm(red line).V. CONCLUSION\nIn summary, we presented a numerical approach to\nsimulate the FMR of nanosized elements and nanostruc-\ntured systems using a ﬁnite diﬀerence method. The sim-\nulation utilizes a homogeneous linearly polarized oscil-\nlating magnetic ﬁeld to drive the magnetic system. We\nwould like to point out that this method is very easy to\nimplementandhandle. ThederivationofsimulatedFMR\nspectra, which directly correspond to experimentally de-\ntected spectra, is explained in detail. This allows one\nto compare not only the resonance position, but also the\nlinewidth and intensities of experiment and simulation\ndirectly.\nFurthermore the spatial dependence of the magnetic\nexcitations spin waves and localized resonances can be\nidentiﬁed and visualized. Additionally the time depen-\ndent trajectory of the magnetization as well as the accu-\nrate phase relation to the driving ﬁeld can be extracted.\nVI. ACKNOWLEDGMENTS\nWe acknowledge ﬁnancial support by the Deutsche\nForschungsgemeinschaft (SFB 491) as well as C. Hassel,\nR.MeckenstockandJ.Lindnerforfruitfuldiscussionsand\nhelp in the initial stages of work.\n\u0003k.wagner@hzdr.de; Corresponding author\n1V. V. Kruglyak, S. O. Demokritov and D. Grundler, J.\nPhys. D: Appl. Phys. 43, 264001 (2010)\n2B.Lenk, H.Ulrichs, F.GarbsandM.Münzenberg, Physics\nReports 507, 107-136 (2011)\n3M. Madami, S. Bonetti, G. Consolo, S. Tacchi, G. Carlotti,\nG. Gubbiotti, F. B. Mancoﬀ, M. A. Yar and J. Åkerman,\nNature Nanotechnology 6, 635-638 (2011)\n4G. E. W. Bauer, E. Saitoh, B. J. van Wees, Nature Mate-\nrials11, 391-399 (2012)5M.Farle, T. Silva and G. Woltersdorf, eds H. Zabel and M.\nFarle, Springer Tracts in Modern Physics 246, 437 (2013)\n6J. Lindner and M. Farle, Springer Tracts in Modern\nPhysics 227, 45-96 (2008)\n7R. D. McMichael and B. B. Maranville, Phys. Rev. B 74,\n024424 (2006)\n8G. Venkat, D. Kumar, M. Franchin, O. Dmytriiev, M.\nMruczkiewicz, H. Fangohr, A. Barman, M. Krawczyk and\nA. Prabhakar, Magnetics, IEEE Transactions on 49, 524-\n529 (2013)5\n9Code and documentation available at:\nhttp://math.nist.gov/oommf/\n10M. J. Donahue and D. G. Porter, OOMMF user’s guide,\nversion 1.2a3, National Institute of Standards and Tech-\nnology, Gaithersburg, Md, USA, 2010\n11A. Banholzer, R. Narkowicz, C. Hassel, R. Meckenstock,\nS. Stienen, O. Posth, D. Suter, M. Farle and J. Lindner,\nNanotechnology 22, 295713 (2011)\n12C. Schöppner, K. Wagner, S. Stienen, R. Meckenstock, M.\nFarle, R. Narkowicz, D.Suter, J. Lindner J. Appl. Phys.\n116, 033913 (2014)13Z. Duan, C. T. Boone, X. Cheng, I. N. Krivorotov, N.\nReckers, S. Stienen, M. Farle, and J. Lindner, Phys. Rev.\nB90, 024427 (2014)\n14C. Kittel, Phys. Rev. 73, 155-161 (1948)\n15L. Landau and E. Lifshits, Phys. Zeitsch. der Sow. 8, 153-\n169 (1935)\n16S. V. Vonsovskij, Ferromagnetic Resonance, Pergamon\nPress, (1966)\n17C. P. Poole, Electron Spin Resonance, John Wiley & Sons,\n(1983)\n18B. Hillebrands, Spin Dynamics in Conﬁned Magnetic\nStructures, Springer, (2002)\n19K. Yu. Guslienko, S. O. Demokritov, B. Hillebrands and\nA. N. Slavin, Phys. Rev. B 66, 132402 (2002)"    },    {        "title": "0905.4779v2.Ferromagnetic_resonance_linewidth_in_ultrathin_films_with_perpendicular_magnetic_anisotropy.pdf",        "content": "arXiv:0905.4779v2  [cond-mat.mtrl-sci]  15 Jun 2009Beaujour et al.\nFerromagnetic resonance linewidth in ultrathin ﬁlms with p erpendicular magnetic\nanisotropy\nJ-M. Beaujour,1D. Ravelosona,2I. Tudosa,3E. Fullerton,3and A. D. Kent1\n1Department of Physics, New York University, 4 Washington Pl ace, New York, New York 10003, USA\n2Institut d’Electronique Fondamentale, UMR CNRS 8622,\nUniversit ´eParis Sud, 91405 Orsay Cedex, France\n3Center for Magnetic Recording Research, University of Cali fornia,\nSan Diego, La Jolla, California 92093-0401, USA\n(Dated: June 15, 2009)\nTransition metal ferromagnetic ﬁlms with perpendicular ma gnetic anisotropy (PMA) have ferro-\nmagnetic resonance (FMR) linewidths that are one order of ma gnitude larger than soft magnetic\nmaterials, such as pure iron (Fe) and permalloy (NiFe) thin ﬁ lms. A broadband FMR setup has\nbeen used to investigate the origin of the enhanced linewidt h in Ni |Co multilayer ﬁlms with PMA.\nThe FMR linewidth depends linearly on frequency for perpend icular applied ﬁelds and increases sig-\nniﬁcantly when the magnetization is rotated into the ﬁlm pla ne. Irradiation of the ﬁlm with Helium\nions decreases the PMA and the distribution of PMA parameter s. This leads to a great reduction\nof the FMR linewidth for in-plane magnetization. These resu lts suggest that ﬂuctuations in PMA\nlead to a large two magnon scattering contribution to the lin ewidth for in-plane magnetization and\nestablish that the Gilbert damping is enhanced in such mater ials (α≈0.04, compared to α≈0.002\nfor pure Fe).\nPACS numbers: 75.47.-m,85.75.-d,75.70.-i,76.50.+g\nMagnetic materials with perpendicular magnetic\nanisotropy (PMA) are of great interest in information\nstorage technology, oﬀering the possibility of smaller\nmagnetic bits [1] and more eﬃcient magnetic random ac-\ncess memories based on the spin-transfer eﬀect [2]. They\ntypically are multilayers of transition metals (e.g., Co |Pt,\nCo|Pd, Ni |Co) with strong interface contributions to the\nmagnetic anisotropy [3], that render them magnetically\nhard. In contrast to soft magnetic materials which have\nbeen widely studied and modeled [4, 5, 6, 7], such ﬁlms\nare poorly understood. Experiments indicate that there\nare large distributions in their magnetic characteristics ,\nsuch as their switching ﬁelds [1]. An understanding of\nmagnetization relaxation in such materials is of particu-\nlar importance, since magnetization damping determines\nthe performance of magnetic devices, such as the time-\nscale for magnetization reversal and the current required\nfor spin-transfer induced switching [2, 8].\nFerromagnetic resonance (FMR) spectroscopy pro-\nvides information on the magnetic damping through\nstudy of the linewidth of the microwave absorption peak,\n∆H, when the applied ﬁeld is swept at a ﬁxed microwave\nfrequency. FMR studies of thin ﬁlms with PMA show\nvery broad linewidths, several 10’s of mT at low frequen-\ncies (/lessorsimilar10 GHz) for polycrystalline alloy [9], multilayer\n[10] and even epitaxial thin ﬁlms [11]. This is at least one\norder of magnitude larger than the FMR linewidth found\nfor soft magnetic materials, such as pure iron (Fe) and\npermalloy (FeNi) thin ﬁlms [5]. Further, it has recently\nbeen suggested that the FMR linewidth of perpendicu-\nlarly magnetized CoCrPt alloys cannot be explained in\nterms of Landau-Lifshitz equation with Gilbert damping\n[12], the basis for understanding magnetization dynamicsin ferromagnets:\n∂M\n∂t=−γµ0M×Heﬀ+α\nMsM×∂M\n∂t. (1)\nHereMis the magnetization and γ=|gµB//planckover2pi1|is the gy-\nromagnetic ratio. The second term on the right is the\ndamping term, where αis the Gilbert damping constant.\nThis equation describes precessional motion of the mag-\nnetization about an eﬀective ﬁeld Heﬀ, that includes the\napplied and internal (anisotropy) magnetic ﬁelds, which\nis damped out at a rate determined by α. The absorp-\ntion linewidth (FWHM) in a ﬁxed frequency ﬁeld-swept\nFMR experiment is given by µ0∆H= 4παf/γ , i.e., the\nlinewidth is proportional to the frequency with a slope\ndetermined by α. This is the homogeneous or intrinsic\ncontribution to the FMR linewidth. However, experi-\nments show an additional frequency independent contri-\nbution to the linewidth:\n∆H= ∆H0+4πα\nµ0γf, (2)\nwhere ∆H0is referred as the inhomogeneous contribution\nto the linewidth.\nThe inhomogeneous contribution is associated with\ndisorder. First, ﬂuctuations in the materials magnetic\nproperties, such as its anisotropy or magnetization, lead\nto a linewidth that is frequency independent; in a simple\npicture, independent parts of the sample come into res-\nonance at diﬀerent applied magnetic ﬁelds. Second, dis-\norder can couple the uniform precessional mode ( k= 0),\nexcited in an FMR experiment, to degenerate ﬁnite- k\n(k/negationslash= 0) spin-wave modes. This mechanism of relaxation\nof the uniform mode is known as two magnon scattering2\n/s48 /s51/s48 /s54/s48 /s57/s48/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s48 /s49/s40/s97/s41/s40/s98/s41\n/s72\n/s72/s32/s61/s57/s48/s111/s86/s105/s114/s103/s105/s110\n/s73/s114/s114/s97/s100/s105/s97/s116/s101/s100\n/s32/s32\n/s70/s105/s101/s108/s100/s32/s97/s110/s103/s108/s101/s32\n/s72/s32/s32/s40/s100/s101/s103/s46/s41/s50/s48/s32/s71/s72/s122/s32/s72\n/s114/s101/s115/s32/s32/s40/s32/s84/s32/s41/s120/s121/s122\n/s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s50/s48/s52/s48/s54/s48\n/s77\n/s115\n/s72/s32/s32/s102/s32/s32/s40/s71/s72/s122/s41\n/s48/s72\n/s114/s101/s115/s32/s32/s32/s32/s40/s32/s84/s32/s41/s32/s48/s72\n/s32/s32/s40/s84/s41/s32/s70/s77 /s82/s32/s115/s105/s103/s110/s97/s108\nFIG. 1: a) The frequency dependence of the resonance ﬁeld\nwith the applied ﬁeld perpendicular to the ﬁlm plane. The\nsolid lines are ﬁts using Eq. 4. Inset: FMR signal of the\nvirgin and irradated ﬁlms at 21 GHz. b) The resonance ﬁeld\nas a function of applied ﬁeld angle at 20 GHz. The solid lines\nare ﬁts to the experimental data points. The inset shows the\nﬁeld geometry.\n(TMS) [13]. TMS requires a spin-wave dispersion with\nﬁnite-kmodes that are degenerate with the k= 0 mode\nthat only occurs for certain magnetization orientations.\nIn this letter we present FMR results on ultra-thin\nNi|Co multilayer ﬁlms and investigate the origin of\nthe broad FMR lines in ﬁlms with PMA. Ni |Co mul-\ntilayers were deposited between Pd |Co|Pd layers that\nenhance the PMA and enable large variations in the\nPMA with Helium ion irradiation [14]. The ﬁlms are\n|3nm Ta |1nm Pd |0.3nm Co |1nm Pd |[0.8nm Ni |0.14nm\nCo]×3|1nm Pd |0.3nm Co |1nm Pd |0.2nm Co |3nm Ta |de-\nposited on a Si-SiN wafers using dc magnetron sputter-\ning and were irradiated using 20 keV He+ions at a ﬂu-\nence of 1015ions/cm2. The He+ions induce interatomic\ndisplacements that intermix the Ni |Co interfaces lead-\ning to a reduction of interface anisotropy and strain in\nthe ﬁlm. The magnetization was measured at room tem-\nperature with a SQUID magnetometer and found to be\nMs≃4.75×105A/m.\nFMR studies were conducted from 4 to 40 GHz at room\ntemperature with a coplanar waveguide as a function of\nthe ﬁeld angle to the ﬁlm plane. The inset of Fig. 1b\nshows the ﬁeld geometry. The parameters indexed with\n‘⊥’ (perpendicular) and ‘ /bardbl’ (parallel) refer to the applied\nﬁeld direction with respect to the ﬁlm plane. The absorp-\ntion signal was recorded by sweeping the magnetic ﬁeld\nat constant frequency [15]. FMR measurements were per-\nformed on a virgin ﬁlm (not irradiated) and on an irra-\ndiated ﬁlm.\nFig. 1a shows the frequency dependence of the reso-\nnance ﬁeld when the applied ﬁeld is perpendicular to the\nﬁlm plane. The x-intercept enables determination of the\nPMA and the slope is proportional to the gyromagneticratio. We take a magnetic energy density:\nE=−µ0M·H+1\n2µ0M2\nssin2φ\n−(K1+ 2K2)sin2φ+K2sin4φ.(3)\nThe ﬁrst term is the Zeeman energy, the second the mag-\nnetostatic energy and the last two terms include the ﬁrst\nand second order uniaxial PMA constants, K1andK2.\nTakingµ0Heﬀ=−δE/δMin Eq. 1 the resonance con-\ndition is:\nf=γ\n2π/parenleftbigg\nµ0H⊥\nres−µ0Ms+2K1\nMs/parenrightbigg\n. (4)\nFrom thex-intercepts in Fig. 1a, K1= (1.93±0.07)×\n105J/m3for the virgin ﬁlm and (1 .05±0.02)×105J/m3\nfor the irradiated ﬁlm; Helium irradiation reduces the\nmagnetic anisotropy by a factor of two. Note that in\nthe irradiated ﬁlm the x-intercept is positive ( µ0Ms>\n2K1/Ms). This implies that the easy magnetization di-\nrection is in the ﬁlm plane. The angular dependence of\nHres(Fig. 1b) also illustrates this: the maximum res-\nonance ﬁeld shifts from in-plane to out-of-plane on ir-\nradiation. The gyromagnetic ratio is not signiﬁcantly\nchangedγ= 1.996±0.009×10111/(Ts) for the vir-\ngin ﬁlm and γ= 1.973±0.004×10111/(Ts) for the\nirradiated ﬁlm (i.e., g= 2.24±0.01). The second order\nanisotropy constant K2was obtained from the angular\ndependence of the resonance ﬁeld, ﬁtting HresversusφH\nfor magnetization angle φbetween 45oand 90o. For the\nvirgin ﬁlm, K2= 0.11×105J/m3. Note that when K2\nis set to zero, χ2of the ﬁt increases by a factor 30. For\nthe irradiated ﬁlm, K2= 0.03×105J/m3. HenceK2de-\ncreases upon irradiation and remains much smaller than\nK1. The solid line in Fig. 1b is the resulting ﬁt. When\nthe ﬁeld approaches the in-plane direction, the measured\nresonance ﬁeld is higher than the ﬁt. The shift is of the\norder of 0.1 T for the virgin ﬁlm and 0 .025 T for the ir-\nradiated ﬁlm. It is frequency dependent: increasing with\nfrequency. This shift will be discussed further below.\nFig. 2a shows the frequency dependence of the\nlinewidth (FWHM) for two directions of the applied ﬁeld.\n/s48/s49/s48/s48/s50/s48/s48\n/s48 /s49/s48 /s50/s48 /s51/s48/s48/s53/s48/s49/s48/s48/s72/s32\n/s124/s124\n/s32/s32/s32 /s72/s32\n/s32\n/s32/s40/s97/s41 /s32/s86/s105/s114/s103/s105/s110/s72/s32 /s32/s40/s109/s84/s41\n/s32\n/s102/s32/s32/s40/s71/s72/s122/s41/s40/s98/s41 /s32/s73/s114/s114/s97/s100/s105/s97/s116/s101/s100\n/s72\n/s84/s77/s83\nFIG. 2: The frequency dependence of the FMR linewidth with\napplied ﬁeld in-plane and perpendicular to the plane. The\nsolid black lines are linear ﬁts that enable determination o fα\nand ∆ H0from Eq. 2. The dotted lines show the linewidth\nfrom TMS and the red lines is the total linewidth.3\n/s48/s49/s48/s48\n/s48 /s51/s48 /s54/s48 /s57/s48/s48/s49/s48/s48/s72 /s32/s32/s32/s32/s32 /s72\n/s105/s110/s104/s32\n/s72 /s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s72\n/s84/s77/s83/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s40/s98/s41\n/s32/s48/s72 /s32/s40/s32/s109/s84/s32/s41/s40/s97/s41\n/s32/s70/s105/s101/s108/s100/s32/s97/s110/s103/s108/s101/s32\n/s72/s32/s32/s40/s32/s100/s101/s103/s46/s32/s41\n/s32\nFIG. 3: Angular dependence of the linewidth at 20 GHz for\n(a) the virgin and (b) the irradiated ﬁlm. The solid line\n(∆H) is a best ﬁt of the data that includes the Gilbert damp-\ning (∆ Hα) and the inhomogeneous (∆ Hinh) contributions.\nLinewidth broadening from TMS (∆ HTMS) is also shown.\nThe total linewidth is represented by the red line.\n∆H⊥of the virgin ﬁlm increases linearly with frequency\nconsistent with Gilbert damping. Fitting to Eq. 2, we\nﬁndα= 0.044±0.003 andµ0∆H⊥\n0= 15.6±3.6 mT.\nWhen the ﬁeld is applied in the ﬁlm plane, the linewidth\nis signiﬁcantly larger. ∆ H/bardbldecreases with increasing\nfrequency for f≤10 GHz and then is practically inde-\npendent of frequency, at ≈140±20 mT. However, for\nthe irradiated ﬁlm, the linewidth varies linearly with fre-\nquency both for in-plane and out-of-plane applied ﬁelds,\nwith nearly the same slope. The Gilbert damping is\nα= 0.039±0.004. Note that µ0∆H/bardbl\n0is larger than\nµ0∆H⊥\n0by about 15 mT.\nThe angular dependence of the linewidth at 20 GHz\nis shown in Fig. 3. The linewidth of the virgin ﬁlm de-\ncreases signiﬁcantly with increasing ﬁeld angle up 30o,\nand then is nearly constant, independent of ﬁeld angle.\nThe linewidth of the irradiated ﬁlm is nearly indepen-\ndent of the ﬁeld angle, with a relatively small enhance-\nment of ∼15 mT close to the in-plane direction. We\nﬁt this data assuming that the inhomogeneous broad-\nening of the line is associated mainly with spatial vari-\nations of the PMA, speciﬁcally local variation in K1,\n∆Hinh.(φH) =|∂Hres/∂K1|∆K1. ∆K1= 4×103J/m3\nfor the virgin ﬁlm and 3 ×102J/m3for the irradiated ﬁlm,\nwhich corresponds to a variation of K1of 2% and 0.3%\nrespectively. Including variations in K2and anisotropy\nﬁeld direction do not signiﬁcantly improve the quality\nof the ﬁt. Such variations in K1produce a zero fre-\nquency linewidth in the perpendicular ﬁeld direction,\nµ0∆H⊥\n0= 16.8 mT, in excellent agreement with linear\nﬁts to the data in Fig. 2. However, the combination\nof inhomogeneous broadening and Gilbert damping can-\nnotexplain the enhanced FMR linewidth observed for\nin-plane applied ﬁelds.\nThe enhanced linewidth observed with in-plane applied\nﬁelds is consistent with a signiﬁcant TMS contribution to\nthe relaxation of the uniform mode–the linewidth is en-\nhanced only when ﬁnite- kmodes equi-energy with theuniform mode are present. We derive the spin-wave dis-\npersion for these ﬁlms following the approach of [16]:\nω2\nk=ω2\n0−1\n2γ2µ0Mskt(Bx0(cos2φ\n+ sin2φsin2ψk)−By0sin2ψk) +γ2Dk2(Bx0+By0),\n(5)\nwhere:\nBx0=µ0Hcos(φH−φ)−µ0Meﬀsin2φ\nBy0=µ0Hcos(φH−φ) +µ0Meﬀcos2φ\n+2K2\nMssin22φ.(6)\nThe eﬀective demagnetization ﬁeld is µ0Meﬀ= (µ0Ms−\n2K1\nMs−4K2\nMscos2φ).ω0=γ/radicalbig\nBx0By0is the resonance\nfrequency of the uniform mode. Dis the exchange stiﬀ-\nness andtis the ﬁlm thickness. ψkis the direction of\npropagation of the spin-wave in the ﬁlm plane relative\nto the in-plane projection of the magnetization. The in-\nset of Fig. 4 shows the dispersion relation for the virgin\nand the irradiated ﬁlm for an in-plane applied ﬁeld at\n20 GHz. For the virgin ﬁlm, with the easy axis normal\nto the ﬁlm plane ( M/bardbl\neﬀ<0) there are degenerate modes\navailable in all directions in k-space. For the irradiated\nﬁlm (M/bardbl\neﬀ>0) degenerate modes are only available when\nψk/lessorsimilar74o.\nThe spin waves density of states, determined from Eq.\n5, is shown as a function of ﬁeld angle in Fig. 4 at 20\nGHz. The DOS of the virgin ﬁlm is two times larger\nthan that of the irradiated ﬁlm at φH= 0. Note that for\nboth ﬁlms, the DOS vanishes at a critical ﬁeld angle that\ncorresponds to a magnetization angle φ= 45o. For the\nvirgin ﬁlm, the enhancement of ∆ Hoccurs atφH≃30o\n(Fig. 3a), at the critical angle seen in Fig. 4.\nThe TMS linewidth depends on the density of states\nand the disorder, which couples the modes:\n∆HTMS=/parenleftbigg∂Hres\n∂ω/parenrightbigg|A0|2\n2π/integraldisplay\nCk(ξ)δ(ωk−ω0)dk,(7)\nwhereA0is a scattering amplitude. Ck(ξ) = 2πξ2/(1 +\n(kξ)2)3/2is a correlation function, where ξis correlation\nlength, the typical length scale of disorder. Eq. 7 is valid\nin the limit of weak disorder.\nWe assume that the disorder of our ﬁlms is associated\nwith spatial variations of the PMA, K1. Then the mag-\nnetic energy density varies as ∆ E(/vector r) =−k1(/vector r)M2\ny/M2\ns,\nand the scattering probability is [17]:\n|A0|2=γ4\n4ω2\n0(B2\nx0sin4φ+B2\ny0cos22φ\n−2(ω0/γ)2sin2φcos2φ)/parenleftbigg2∆k1\nMs/parenrightbigg2\n.(8)\n∆k1is the rms amplitude of the distribution of PMA,\nk1(r). Therefore the TMS linewidth broadening scales4\n/s48 /s51/s48 /s54/s48 /s57/s48/s48/s50/s48/s52/s48/s54/s48/s50/s48\n/s48 /s53/s50/s48\n/s32/s68/s101/s110/s115/s105/s116/s121/s32/s111/s102/s32/s115/s116/s97/s116/s101/s115/s32/s32/s40/s97/s46/s117/s41\n/s70/s105/s101/s108/s100/s32/s97/s110/s103/s108/s101\n/s72/s32/s32/s40/s100/s101/s103/s46/s41/s107/s61/s57/s48/s111\n/s107/s61/s48/s111/s32/s32 /s32/s32/s102/s32/s32 /s40/s71/s72/s122/s41\n/s32\n/s73/s114/s114/s97/s100/s105/s97/s116/s101/s100\n/s32\n/s107 /s32/s40/s49/s48/s53\n/s32/s114/s97/s100/s47/s99/s109/s41/s86/s105/s114/s103/s105/s110\nFIG. 4: The density of spin-waves states degenerate with the\nuniform mode as a function of ﬁeld angle at 20 GHz for the\nvirgin ﬁlm (solid line) and the irradiated ﬁlm (dashed-dott ed\nline). Inset: Spin wave dispersion when the dc ﬁeld is in the\nﬁlm plane.\nas the square of ∆ k1. Since the variations in PMA of the\nvirgin ﬁlm are larger than that of the irradiated ﬁlm the\nlinewidth broadening from the TMS mechanism is ex-\npected to be much larger in the virgin ﬁlm, qualitatively\nconsistent with the data.\nA best ﬁt of the linewidth data to the TMS model is\nshown in Fig. 3a. For the virgin ﬁlm, we ﬁnd ξ≈44 nm,\napproximately four times the ﬁlm grain size, and ∆ k1=\n9×103J/m3. The exchange stiﬀness, D= 2A/µ0Mswith\nthe exchange constant A= 0.83×10−11J/m, is used in\nthe ﬁttings. The cut-oﬀ ﬁeld angle for the enhancement\nof the ﬁeld linewidth agrees well with the data (Fig. 3a).\nFor the irradiated ﬁlm, a similar analysis gives: ξ= 80±\n40 nm and ∆ k1= (4±2)×103J/m3.\nTMS is also expected to shift the resonance position\n[17]. For applied ﬁelds in-plane and f= 20 GHz we\nestimate the resonance ﬁeld shift to be ≈33 mT. This is\nsmaller than what is observed experimentally ( ≈93 mT).\nThe deviations of the ﬁts in Fig. 1b may be associated\nwith an anisotropy in the gyromagnetic ratio, i.e. a g\nthat is smaller for Min the ﬁlm plane. Note that if we\nassume that the g-factor is slightly anisotropic ( ∼1%),\nwe can ﬁt the full angular dependence of the resonance\nﬁeld of the irradiated ﬁlm.\nWe note that the TMS model cannot explain the en-\nhanced linewidth for small in-plane applied ﬁelds for the\nvirgin ﬁlm (Fig. 2a). The FMR linewidth increases\ndramatically when the frequency and resonance ﬁeld de-\ncreases. When the applied in-plane ﬁeld is less than the\neﬀective demagnetization ﬁeld ( −µ0M||\neﬀ= 0.31 T) the\nmagnetization reorients out of the ﬁlm plane. For fre-\nquencies less than about 8 GHz this leads to two resonant\nabsorption peaks, one with the magnetization having an\nout-of-plane component for Hres<−M||\neﬀand one with\nthe magnetization in-plane for Hres>−M||\neﬀ. It may\nbe that these resonances overlap leading to the enhanced\nFMR linewidth.In sum, these results show that the FMR linewidth in\nNi|Co multilayer ﬁlms is large due to disorder and TMS\nas well as enhanced Gilbert damping. The latter is an\nintrinsic relaxation mechanism, associated with magnon-\nelectron scattering and spin-relaxation due to spin-orbit\nscattering. As these materials contain heavy elements\nsuch as Pd and short electron lifetimes at the Fermi level,\nlarge intrinsic damping rates are not unexpected. The re-\nsults indicate that the FMR linewidth of Ni |Co multilay-\ners can be reduced through light ion-irradiation and fur-\nther demonstrate that the Gilbert damping rate is largely\nunaﬀected by irradiation. These results, including the re-\nduction of the PMA distribution at high irradiation dose,\nhave important implications for the applications of PMA\nmaterials in data storage and spin-electronic application s\nwhich require tight control of the anisotropy, anisotropy\ndistributions and resonant behavior.\nACKNOWLEDGMENTS\nWe thank Gabriel Chaves for help in ﬁtting the data\nto the TMS model. This work was supported by NSF\nGrant No. DMR-0706322.\n[1] T. Thomson, G. Hu, and B. D. Terris, Phys. Rev. Lett.\n96, 257204 (2006).\n[2] S. Mangin et al., Nature Mater. 5, 210 (2006).\n[3] G. H. O. Daalderop, P. J. Kelly, and F. J. A. den Broeder,\nPhys. Rev. Lett. 68, 682 (1992).\n[4] B. Heinrich, Ultrathin Magnetic Structures III (Springer,\nNew York, 2005), p. 143.\n[5] C. Scheck et al., Phys. Rev. Lett. 98, 117601 (2007).\n[6] K. Gilmore, Y. U. Idzerda, and M. D. Stiles, Phys. Rev.\nLett.99, 027204 (2007).\n[7] A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer, Phys.\nRev. Lett. 101, 037207 (2008).\n[8] J. Z. Sun, Phys. Rev. B 62, 570 (2000).\n[9] T. W. Clinton et al., J. Appl. Phys. 103, 07F546 (2008).\n[10] S. J. Yuan et al., Phys. Rev. B 68, 134443 (2003).\n[11] J. BenYoussef et al., J. Magn. Magn. Mater. 202, 277\n(2003).\n[12] N. Mo et al., Appl. Phys. Lett. 92, 022506 (2008).\n[13] M. Sparks, Ferromagnetic-Relaxation Theory (McGraw-\nHill, 1964).\n[14] D. Stanescu et al., J. Magn. Magn. Mater. 103, 07B529\n(2008).\n[15] J.-M. L. Beaujour et al., Eur. Phys. J. B 59, 475 (2007).\n[16] P. Landeros, R. E. Arias, and D. L. Mills, Phys. Rev. B\n77, 214405 (2008).\n[17] R. D. McMichael and P. Krivosik, IEEE Trans. Magn.\n40, 2 (2004)."    },    {        "title": "0907.2415v1.Localized_ferromagnetic_resonance_force_microscopy_in_permalloy_cobalt_films.pdf",        "content": "arXiv:0907.2415v1  [cond-mat.mtrl-sci]  14 Jul 2009Localized ferromagnetic resonance force microscopy in\npermalloy-cobalt ﬁlms\nE. Nazaretski, I. Martin, K. C. Cha, E. A. Akhadov, and R. Movsho vich\nLos Alamos National Laboratory, Los Alamos, NM 87545\nYu. Obukhov, D. V. Pelekhov, and P. C. Hammel\nDepartment of Physics, Ohio State University, Columbus OH 4 3210,\nAbstract\nWe report Ferromagnetic Resonance Force Microscopy (FMRFM ) experiments on a justaposed\ncontinuous ﬁlms of permalloy and cobalt. Our studies demons trate the capability of FMRFM\nto perform local spectroscopy of diﬀerent ferromagnetic mat erials. Theoretical analysis of the\nuniform resonance mode near the edge of the ﬁlm agrees quanti tatively with experimental data.\nOur experiments demonstrate the micron scale lateral resol ution in determining local magnetic\nproperties in continuous ferromagnetic samples.\nPACS numbers: 07.79.Pk, 07.55.-w, 76.50.+g, 75.70.-i\n1Magnetic resonance force microscopy (MRFM) is attracting increa sing attention as a re-\nsult of its high spin sensitivity and excellent spatial resolution in param agnetic and nuclear\nspin systems.[1, 2, 3, 4, 5, 6] MRFM studies on microfabricated and c ontinuous ferromag-\nnetic samples have been also performed. [7, 8, 9, 10, 11] Here we re port FMRFM experi-\nments performed on a non-overlapping permalloy (Py) and cobalt (C o) continuous ﬁlms and\ndemonstrate the capability of FMRFM to spectroscopically identify t he distinct magnetic\nproperties of two adjacent ferromagnetic ﬁlms. We quantitatively model the resulting force\nsignal strength and compare it with the experimental data.\nThe permalloy-cobalt sample is schematically shown in Fig. 1. A 20 nm thic k Ti ﬁlm\nwas uniformly applied onto the surface of a 100 µm thick Si (100) wafer. 20 nm of Co\nwas deposited into a rectangular area (2.5 ×5 mm) deﬁned in photoresist followed by\nthe lift-oﬀ. A complimentary rectangular area of 20 nm thick Py was s imilarly deﬁned\nand deposited. The entire structure was then coated with a 20 nm t hick layer of Ti. The\ninterface between the Co and Py regions was examined in a scanning e lectron microscope\n(SEM) and revealed a gap whose width varies between 3 and 6 µm along the entire length of\nthe sample (see SEM image in Fig. 1). An approximately 1.7 ×1.7 mm2piece was cut and\nglued to the stripline resonator of the FMRFM apparatus and the ﬁlm plane was oriented\nperpendicular to the direction of the external magnetic ﬁeld Hext. For FMRFM studies\nwe used the cantilever with the spherical magnetic tip (see SEM image in Fig. 1) and its\nspatial ﬁeld proﬁle has been carefully characterized [12]. More det ails on the experimental\napparatus can be found in Ref. [13]\nIn Fig. 2 we show the evolution of the FMRFM signal as a function of th e lateral position\nand applied magnetic ﬁeld. The cantilever was scanned across the int erface between Co and\nPy, in the region indicated by arrows in Fig. 1. The FMRFM signal was re corded in two\ndiﬀerent regions of Hextwhich correspond to Py and Co resonance ﬁelds for the microwave\nfrequency of fRF=9.35 GHz. Insets in Fig. 2 show the evolution of the FMRFM spectra a s\na function of lateral position. The signal, reminiscent of those repo rted earlier in [14], is\ncomprised of two distinctive contributions. The ﬁrst, a negative sig nal which occurs at lower\nvalues of Hextis a localized resonance originating from the region of the sample right under\nthe cantilever tip where the probe ﬁeld is strong and positive. The se cond contribution is\npositive and is observed at higher values of Hext. This signal arises from a larger region of\n2sample remote from the tip which, therefore, experience a weak ne gative tip ﬁeld; we will\nlabel this the “uniform resonance”. As seen in Fig. 2, at the beginnin g of the lateral scan\nthe negative (lower ﬁeld) resonance structure is present only in th e Co spectrum (see inset\na)). Near a lateral position of 9 µm we see no localized signals (with negatively shifted Hext)\nfor either the Py or the Co signals. However upon scanning further over the Py ﬁlm, the\nPy resonance begins to show a localized signal, while the Co signal cont inues to show only\na uniform (positively shifted Hext) signal (inset b)).\nWe analyze the uniform contribution to the FMRFM signals considering the case when\nthe entire dynamic magnetization mis constant and the resonance ﬁeld is only weakly\naﬀected by the probe. This approximation is valid for the large probe -sample distances\n(insets b) and c) in Fig. 2). The frequency of the uniform resonanc e in a thin ﬁlm can be\nwritten as ωRF/γ=Hext−4πMs, where 4 πMsis the saturation magnetization and γis the\ngyromagnetic ratio. FMRFM spectra shown in b) and c) insets in Fig. 2 yield the values of\n4πMs= 8052 G for Py and 4 πMs= 15013 G for Co respectively.\nForquantitative analysisoftheFMRFMdataitisimportanttohavean accurateestimate\nof the probe-sample separation. Magnetic Force Microscopy (MFM ) measurements were\nused to calibrate the probe-sample separation. The cantilever was scanned across the Py\n- Co interface and changes in its resonance frequency were recor ded. The gradient of the\nMFM force for a semi-inﬁnite ﬁlm can be written as follows:\n∂F\n∂z= 4mpMsLx(x2−3z2)\n(x2+z2)3, (1)\nwheremp= 7×10−9emu is the probe magnetic moment [12] and Lis the ﬁlm thickness. z\nis the probe-ﬁlm distance and xis the lateral position with respect to the ﬁlm edge ( x≥0).\nMFM data were acquired at H ext= 18255 G, thus, both ﬁlms were saturated. The MFM\ndata and the ﬁt to Eq. 1 are shown in Fig. 3a, yielding the tip-sample se paration z≈4.4\nµm and the ﬁlms boundaries ( x≤8µm for Co and x≥11µm for Py).\nThe tip ﬁeld suppresses the uniform FMR mode in the region under the tip, and ac-\ncording to Obukhov et al.[15] the magnitude of the suppression depends on the tip-sample\nseparation. It is described as partial suppression at distances z≫/radicalBig2mp\nπMsLα0(α0is the ﬁrst\nzero of the Bessel function J0(α0) = 0) and full suppression at z≪/radicalBig2mp\nπMsLα0. The region\nof suppressed magnetization is conﬁned to a region of radius r=√\n2z. FMRFM data dis-\ncussed here were taken at the boundary of these two regions, th us we consider the regime of\n3full suppression, however we introduce the magnitude of the supp ression as a ﬁt parameter.\nFerromagnetic resonance excitation generates a precessing tra nsverse magnetization m, thus\nreducing Mz; the change of MzisδMz=/radicalBig\nM2s−m2−Ms≈ −m2/2Ms. Here we modulate\nthe amplitude of mwith a 100% modulation depth at the cantilever resonance frequenc y.\nThe FMRFM force exerted on a cantilever is F=−/integraltextLm2/2Ms·∂Hp/∂zdr′, where integra-\ntion is performed over the entire ﬁlm area. The total FMRFM force c lose the edge of the\nﬁlm is well approximated by\nF=−m2\n2MsL/parenleftBigg4xzmp\n(x2+z2)2−β/integraldisplay\nSθ(x′)∂Hp\n∂z(x−x′)dr′/parenrightBigg\n, (2)\nwhere the ﬁrst term describes the force between the probe and t he semi-inﬁnite ﬁlm and\nthe second term represents the force between the probe and th e area of radius r=√\n2z\nunder the tip. The Heaviside function θ(x′) represents the fact that the ﬁlm is positioned\natx′≥0 and the dimensionless parameter βquantiﬁes the degree of suppression of the\nuniform FMR mode. In Fig. 3b we plot the experimental data extract ed from Fig. 2 and\ncorresponding ﬁts using Eq. 2. Fig. 3b demonstrates good qualitat ive and quantitative\nagreement between theory and experiment and demonstrates th e validity of the model. It\nis important to mention that in our model we assume the dynamic magn etization mto be\nconstant throughout the ﬁlm. However, mmay vary due to the change of the demagnetizing\nﬁelde.g. −4πMsfarfromtheﬁlmboundaryand −2πMsattheﬁlmboundary. Ourestimates\nshow that mchanges from a constant value in the ﬁlm down to zero at the ﬁlm edge . The\nlength scale of this change is πMsL/∆H≈1µm (∆His the linewidth of the uniform\nresonance), small compared to the probe-sample distance thus o nly weakly aﬀecting the ﬁts\nshown in Fig. 3b.\nThe spatial resolution of the uniform FMR mode shown in Fig. 3b is comp arable to\nthe MFM lateral resolution depicted in Fig. 3a and is determined by the probe-sample\nseparation of z ≈4.4µm. However, it can be further improved by tracking the intensity\nof the FMRFM signal at values of H extlower than that of the uniform FMR mode (insets\na) and d) in Fig. 2). In Fig. 3c we show the FMRFM force acquired at H ext= 17960 G\nfor Co and H ext= 11150 G for Py respectively (values of H extare schematically indicated\nwith dotted lines in Fig. 2). The contribution to the FMRFM signal at low er values of H ext\noriginates from the localized region of the sample under the probe. A s seen in Fig. 3c the\nlateral resolution is on the order of 3 µm (10% - 90% change in localized signal intensity)\n4and is determined by the FMR resonance linewidth and the spatial pro ﬁle of the FMR\nmode under the tip. Further theoretical and numerical analysis is r equired to understand\nthe evolution of the FMR modes excited under the probe in the prese nce of a strongly\ninhomogeneous tip ﬁeld and boundaries of the sample.\nIn conclusion, we have obtained local FMR spectra in justaposed fe rromagnetic samples\nand our quantitative model for the spatial variation of the uniform mode agrees well with\nexperimental data. We have demonstrated spectroscopic imaging of Py and Co semi-inﬁnite\nﬁlms with the spatial resolution for the tip induced resonance of ≈3µm.\nThe work performed at Los Alamos was supported by the US Depart ment of Energy, and\nCenter for Integrated Nanotechnologies at Los Alamos and Sandia National Laboratories.\nThe work at OSU was supported by the US Department of Energy th rough grant DE-FG02-\n03ER46054.\n5[1] D. Rugar, C. S. Yannoni, and J. A. Sidles, Nature 360, 563 (1993)\n[2] D. Rugar and O. Z¨ uger and S. Hoen and C. S. Yannoni and H. M. Vieth and R. D. Kendrick,\nScience264, 1560 (1994)\n[3] Z. Zhang, M. L. Roukes, and P. C. Hammel, J. Appl. Phys. 80, 6931 (1996)\n[4] D. Rugar, R. Budakian, H. J. Mamin, and W. Chui, Nature 430, 329 (2004)\n[5] C. L. Degen, Q. Lin, A. Hunkeler, U. Meier, M. Tomaselli, a nd B. H. Meier, Phys. Rev. Lett.\n94, 207601 (2005)\n[6] C. L. Degen, M. Poggio, H. J. Mamin, C. T. Rettner, and D. Ru gar, Proc. Natl. Acad. Sci.\n106, 1313 (2009)\n[7] Z. Zhang, P. C. Hammel, and P. E. Wigen, Appl. Phys. Lett. 68, 2005 (1996)\n[8] G. de Loubens, V.V. Naletov, O. Klein, J. Ben Youssef, F. B oust, and N. Vukadinovic, Phys.\nRev. Lett. 98, 127601 (2007)\n[9] Yu. Obukhov, D. V. Pelekhov, J. Kim, P. Banerjee, I. Marti n, E. Nazaretski, R. Movshovich,\nS. An, T. J. Gramila, S. Batra, P. C. Hammel, Phys. Rev. Lett. 100, 197601, (2008)\n[10] E. Nazaretski, D. V. Pelekhov, I. Martin, M. Zalalutdin ov, D. Ponarin, A. Smirnov, P. C.\nHammel, and R. Movshovich, Phys. Rev. B 79, 132401 (2009)\n[11] E. Nazaretski, J. D. Thompson, M. Zalalutdinov, J. W. Ba ldwin, B. Houston, T. Mewes, D.\nV. Pelekhov, P. Wigen, P. C. Hammel, and R. Movshovich, J. App l. Phys.101, 074905 (2007)\n[12] E. Nazaretski, E. A. Akhadov, I. Martin, D. V. Pelekhov, P. C. Hammel, and R. Movshovich,\nAppl. Phys. Lett. 92, 214104 (2008).\n[13] E. Nazaretski, T. Mewes, D. Pelekhov, P. C. Hammel, and R . Movshovich, AIP Conf. Proc.\n850, 1641 (2006)\n[14] E. Nazaretski, D. V. Pelekhov, I. Martin, M. Zalalutdin ov, J. W. Baldwin, T. Mewes, B.\nHouston, P. C. Hammel, and R. Movshovich, Appl. Phys. Lett. 90234105 (2007)\n[15] Yu. Obukhov, D. V. Pelekhov, E. Nazaretski, R. Movshovi ch, P. C. Hammel, Appl. Phys.\nLett.94, 172508 (2008)\n6Figure Caption\nFigure 1: Schematic of the Co - Py sample. The arrows mark the scan range for spectra\nshown in Fig. 2. The SEM image on the right shows the gap between Py a nd Co. The SEM\nimage on the left depicts the cantilever tip.\nFigure 2: FMRFM force image as a function of Hextand the lateral position. We show\nthe Co and Py forces in the upper and lower panels respectively. Ins ets a) - d) demonstrate\nthe evolution of the FMRFM signal as a function of lateral position ind icated on the left-\nhand side of each inset. Vertical dashed lines show the boundaries o f the Co and Py ﬁlms.\nThe horizontal dashed-doted lines are drawn through the values o f Hext= 18255 G for\nCo and H ext= 11296 G for Py respectively and correspond to the uniform reson ance (see\nFig. 3b). The horizontal dotted lines at H ext= 17960 G and H ext= 11150 G for Co and\nPy respectively, mark the localized FMRFM signals. Experimental par ameters: T = 11 K,\nfRF=9.35 GHz, probe-sample distance ≈5.6µm.\nFigure 3: a) MFM data acquired at H ext=18255 G, solid line is the ﬁt to Eq. 1. b)\nFMRFM force data for the uniform ferromagnetic resonance (FMR ) modes. H ext= 18255\nG for Co (squares) and H ext= 11296 G for Py (circles). Solid and dashed lines are ﬁts of\nEq. 2 to the data. Fit parameters: m/Ms= 0.0014, β= 0.65 for Co and m/Ms= 0.0028, β\n= 0.5 for Py. c) FMRFM force for the localized (close to the probe) FM R mode acquired at\nHext= 17960 G for Co (squares) and H ext= 11150 G for Py (circles). The lateral resolution\nis better than 3 µm.\n7100 m□Si /c109\nTi\n20□nm□Co\n20□nm□Py‘0 m’/c109 ‘19 m’/c109\n2 m/c109 2 m/c109z\nx\nFIG. 1:\na) b)\nd)\nc)\nFIG. 2:\n8FIG. 3:\n9"    },    {        "title": "1601.02048v1.Current_Control_of_Magnetic_Anisotropy_via_Stress_in_a_Ferromagnetic_Metal_Waveguide.pdf",        "content": " \nCurrent C ontrol of  Magnetic Anisotropy  via Str ess in a Ferromagnetic Metal  Waveguide  \nKyongmo An1, Xin Ma1, Chi -Feng Pai3, Jusang Yang1, Kevin S. Olsson1, James L. Erskine1, Daniel C. Ralph3,4, \nRobert A. Buhrman3 and Xiaoqin Li1,2,* \n \n1Department of Physics,  The University of Texas  at Austin , Austin, Texas 78712, USA  \n2Center for Complex Quantum Systems, The Universit y of Texas at Austin, Austin, Texas  78712 , USA  \n3Cornell  University, Ithaca, New york 14853, USA  \n4Kavli Institute at Cornell, Cornell University, Ithaca, New York 14853, USA  \n \n*Email address: elaineli@physics.utexas.edu  \nWe demonstrate that in -plane charge curre nt can effectively control the spin precession resonance in an \nAl2O3/CoFeB/ Ta heterostructure . Brilloui n Light S cattering (BLS) was used to detect the ferromagnetic \nresonance field under microwave excitation of spin waves at fixed frequencies.  The current control  of \nspin precession resonance  originates  from modification of the  in-plane uniaxia l magnetic anisotropy  \nfield 𝐻k, which  changes symmetric ally with respect to the current direction . Numerical simulation \nsuggest s that the anisotropic stress  introduced by Joule heating plays a n important  role in controlling \n𝐻k. These results  provide new  insights into  current manipulation of magnetic properties  and have broad \nimplications for spintronic devices.  \n \nPACS:   75.76.+j, 75.30.Ds, 75.30.Gw , 76.50.+g  \n \nI. INTRODUCTION  \n \nMagnetic anisotropy plays an important role in the \nperformance of high-density spintronic devices including \nspin valves1,2, magnetic tunnel junctions3-6, and emerging \nmulti -ferroic technologies7. Such anisotropy defines the low -\nenergy orientation o f the magnetization as well as the \nstability  of the magnetization  with respect to external fields, \nelectric currents8, and temperature -induced fluctuations9,10. \nThe control of magnetic anisotropy is typically realized by \ncontrolling the growt h condition of the magnetic layer11, \nswitching substrates12, applying external stress13, heating11, \nor an  external electric field14. Recently, perpendicular \nmagnetic anisotropy has been achieved in \noxide /ferromagnetic metal (FM) heterostructures such as  \nMgO/ CoFeB , leading  to low critical current s for spin \ntransfer torque switching  of tunnel junction s6. Therefore, \napproaches to effectively control magnetic anisotropy  as \nwell as elucidating their  physical origin s become  important \nfor fu rther development of mul ti-functional spintronic \ndevices .  \nCharge current has recently been utilized to manipulate \nmagnetization including control of magnetic domain wall \nmotions and magnetization switching3, 15-19. Efficient control \ncan be achieved using  spin-orbit torques (SOTs) originating  \nfrom either the spin Hall effect in the bulk of a heavy metal20 \nor  the Rashba effect at a magnetic interface21. CoFeB -based \nalloys  have attracted great attention due to their high \nmagneto -resistance22 and they are commonly used as the \nelectrode  material for magnetic tunnel junction s. Although \ncharge -current -induced magnetization manipulation  of CoFeB  has been extensively studied, current -induced \nmagneto -elastic effects  have been r arely discussed, even \nthough CoFeB is known to exhibit a large magneto elastic \nconstant23.  \nIn this letter, we investigat e current -induced  magnetic \nresonance shift s in a CoFeB/ Ta waveguide  deposited on  an \nAl2O3 substrate  with the Brillouin light scattering  (BLS) \ntechnique.  The magnetic resonance shift exhibits both \nsymmetric and asymmetric dependence s when  the direction \nof the direct current (DC)  is re versed. A number of \nmechanisms which can contribute to the asymmetric shift  \nhave been investigated previous ly21,24, including  the Oersted \nfield, the spin Hall effect, and the Rashba effect . In this paper, \nwe focus on the symmetric frequency shift , which can be \nunderstood as  arising from a current -induced change in the  \nin-plane uniaxial magnetic anisotropy field  𝐻k. A \nmodification  of 𝐻k up to ~20% is realized using a moderate  \ncurrent  of 4×106 A/cm2. Numerical simulation s suggest \nthat the current -controlled magnetic  anisotropy origi nates  at \nleast in part  from anisotropic stress in the waveguide , \ngenerated by  Joule heating from the in -plane current flow . \nOur study shows that the effective H field induced by \nanisotropic stress  can play an important role in \nmagnetization control  in addition to the  frequently discussed  \nfield-like SOT from the spin Hall effect  or interfacial Rashba \ntorque in CoFeB/Ta bilayer structure25.  \n \nII. SAMPLE  STRUCTURE AND \nCHARACTERIZATION WITH MOKE  \n \nThe sample s investigated are a  series of  \nCo40Fe40B20(10)/Ta(1 0) films  deposited onto an Al2O3 \nsubstrate by sputtering20, where the numbers in parentheses \nrepresent the layer thicknesses in nanometers.  Following \ndeposition, the bilayer structure was patterned int o a 10 -μm-\nwide and 200 -μm-long waveguide. After the deposition of \n240-nm-thick SiO 2 insulating layer, a 5 -μm wide \nCu(150)/Au(10) antenna was created on top of the bilayer \nwaveguide, as depicted in Fig. 1(a).  From the measured \nresistance of the bilayer stru cture, 1930   Ω, the resistivity of \nbilayer structure of 193  𝜇Ω cm was calculated. These \nbilayer structures have been previously used to investigate \nmagnetic switching20 and spin wave amplification via \nSOT s26. While phenomena driven by SOT  were  observed in \nthis sample , it does not appear  to be the most critical \nmechanism behind the experimental observation of \nresonance  field shifts  discussed in this manuscript .  \n We first characterize the  CoFeB  samples with magneto \noptical Kerr effect ( MOKE ) measurements at room \ntemperature, as presented in Fig s. 1(b , c). Due to the strong \ndemagnetization field, the magnetization lies in the x-y plane , \ni.e., the  plane of  the film.  Angle resolved MOKE \nmeasurements show that there is in -plane anisotropy.  The i n-\nplane easy axis lies along the waveguide  𝜙=0° (parallel to \nthe waveguide axis)  while the in-plane hard axis is \nperpendicular to the waveguide  at 𝜙=90° as shown in Fig. \n1(b). The normalized  remanent magnetization (𝑀r/𝑀s) \nplotted as a function of 𝜙 in Fig. 1(c)  confirms that the in -\nplane magnetic anisotropy is  indeed  uniaxial . To calculate \nthe uniaxial anisotropy field , 𝐻k, we integrated the curve at \n𝜙=90° in Fig. 1(b) , when the magnetic field  is applied \nalong  the in -plane hard axis :27 \n \n \n𝐻k=2∫𝑑𝑚 𝐻(𝑚)1\n0,  (1)  \n \nfrom which  we found  𝐻k= 44±3 Oe, where  𝑚=𝑀/𝑀s \nis the normalized projection of magnetization 𝑀 along the \nexternal field 𝐻, and 𝐻(𝑚) denotes the required external \nmagnetic field to induce the fractional magnetization 𝑚. \n \nIII. BLS EXPERIMENT S \n \nBLS measurements were  then performed to investigate \nspin waves in  the geometry depicted in Fig . 1(a). Because \nthe external magnetic field 𝐻 is much larger than the \nsaturation magnetic field ~44 Oe obtained from MOKE, the \nmagnetization is kept aligned with the external magnetic \nfield 𝐻 in our experiments.  Damon –Eshbach  spin wave  \nmodes28 propagating perpendicular to the magnetization \ndirection  were excited by a microwave current through the \nantenna . A line arly-polarized laser beam  was normally \nincident on the sample surface, and the orthogonal -polarized \ncomponent of the backscattered light was collected  and sent \nto a Sandercock -type multipass tandem Fabry -Perot \ninterferometer. Fig. 1(d) inset shows  a typical  BLS raw spectrum from the spin waves propagating along the CoFeB \nwaveguide with a microwave excitation at f = 8 GHz . The \npeak position s of the measured  Stokes and anti -Stokes peaks \nare determined by the microwave source while the linewidth \nis limited by the frequency resolution of the interferometer. \nThus, very limited information can be obtained from the raw \nBLS spectrum. In the following, we vary the magnitude of \nthe applied magnetic field and the DC to investigate  how the \nDC can modify the magnetic properties of the waveguide .  \nTo begin, we study how  the spin wave intensity , \nproportional to  the integrated BLS intensity , changes with \nthe applied magnetic field at zero DC. The spin wave excited \nby a fixed microwave frequency exhibits a resonance \nbehavior as shown in Fig. 1 (d). The resonance can be wel l-\nfitted with  a Lorentz ian function , from which the peak \nposition 𝐻=𝐻R, or the field corresponding to the maximal \nBLS intensity can be extracted.  The resonance field and the \nfrequency of uniform precession can be related by the Smit -\nSuhl equation29, 30.  \n \n 𝑓=𝛾\n2𝜋√(𝐻R−𝐻k)(𝐻R+4𝜋𝑀eff), (2)  \n \nFig. 1. (a) Schematic illustration of sample geometry us ed \nin the BLS experiment. (b) Measured MOKE data with three \ndifferent magnetic field directions. (c) Polar plot of the \nnormalized remanent magnetization. The solid line shows a \ncosine function fit of the data. (d) Integrated BLS intensity \nas a function of external field H, where the line is a \nLorentzian fit. The inset is the raw BLS spectrum in \nfrequency domain under microwave excitation at a fixed \nfrequency.   \nwhere 𝛾 is the gyromagnetic ratio  and 4𝜋𝑀 eff is the \neffective demagnetization field  which  also includes  the out -\nof-plane anisotropy field .  Strictly speaking, our BLS \nexperiments measure spin waves with small but  finite wave  \nvector s instead of the spatially uniform precession.  This \nwould lead to a constant offset of 𝐻R by ~ 3% from the peak \nin BLS -resonan ce curve, as demonstrated by our  previous \nwork on CoFeB/Ta on Si substrates26. Because this offset is \nsmall, we will approximately equate  𝐻R with the field \ncorresponding to  the peak in the BLS spectra  as shown in Fig. \n1(d).  \nWe then investigate how the resonant magnetic field 𝐻R \nchanges as a DC passes throug h the waveguide.  Our key  \nfinding  is that 𝐻R decreases with increasing  DC as shown in \nFig. 2(a)  at f = 8 GHz . The change in 𝐻R exhibits both \nsymmetric and anti -symmetric behaviors with respect to  the \nDC. The anti-symmetric component can be  attributed  to a \ncombination of Oersted field, spin Hall effect,  and Rashba \neffect21,24. The induced magnetic field  from these effects  lies \nalong the direction of  the external magnetic field, and the \ndirection of the effective field is reversed  by reversing the \nDC direction , leading to  anti-symmetric change in 𝐻R with \nDC. \nWe focus  here on the  symmetric reduction of 𝐻R with \nrespect to  the DC. Joule heatin g is known to cause a \nreduction  of 4𝜋𝑀eff, and hence  a symmetric  shift in 𝐻R. We \nexamine the effect of simple heating by raising the sample \ntemperature uniformly on a heater stage.  This control \nexperiment was performed at  an excitation microwave \nfrequency of  5 GHz.  As shown in Fig. 2 (b), 𝐻R is observed \nto shift upward at a higher temperature, which is opposite to \nthe change in 𝐻R observed in our experiments by  passing \nDCs through the waveguide. Hence, t here must exist other \nmechanisms that overcome  the increase of 𝐻R due to the \ndecrease in  4𝜋𝑀eff by simple heating and reduce 𝐻R at \nhigher DCs.   \n To further investigate the origin of the sy mmetric \nreduction of 𝐻R, H field dependent measurements were \nperformed under different excitation microwave frequenc ies. \nThe maximal  symmetric shift  defined by ∆𝐻symmm≡\n[𝐻R(𝐼=𝐼max)+𝐻R(𝐼=−𝐼max)]/2−𝐻R(𝐼=0)  is \nplotted as a function of 𝐻R(𝐼=0) at each microwave \nfrequency in Fig. 2(c)  with a linear fitting line.  In other words,  \n∆𝐻symmm represents the symmetric shift in the resonant field \n𝐻R at the highest current (𝐼max =8 mA)applied  in our \nexperiments . To understand the correlation between ∆𝐻symmm \nand 𝐻R(𝐼=0), we modify the uniform frequency formula , \nEq. (2), to take into account the DC effect  \nphenomenologically as the following  \n \n \n𝑓=𝛾\n2𝜋√(𝐻R−𝐻k,0+𝐶1𝐼2)\n×(𝐻R+4𝜋𝑀eff,0+𝐶2𝐼2)  (3)  \n \nHere we only keep the lowest -order  even  contribution from the DC, i.e.,  the term proportional to  𝐼2. 𝐻k,0 and \n4𝜋𝑀eff,0 are the uniaxial anisotropy field and the effective \nmagnetization without DC. The symmetric dependence  of \n𝑀eff and 𝐻k with respect to DC  are explicitly written  by \nintroducing 𝐶1𝐼2and 𝐶2𝐼2. With changing DCs, 𝐻R is shifted \nbut 𝑓 remains the same because  of the fixed frequency of the \nmicrowave excitation.  By taking the derivative with respect \nto 𝐼2, we can obtain the desired relationship30 between \n∆𝐻symmm and 𝐻R(𝐼=0). \n \n ∆𝐻symmm=𝐴1𝐻R(𝐼=0)+𝐴2, \n \nwhere,  \n \n𝐴1≡−(𝐶2−𝐶1)𝐼max2\n4𝜋𝑀eff,0, \n𝐴2≡−𝐶1𝐼max2−𝐴1(𝐻k,0−𝐶1𝐼max2 ). (4)  \n \nFig. 2. (a) Measured 𝐻R as a fucntion of DC at f = 8 GHz \n(b) Temperature dependence of 𝐻R at f = 5 GHz for \nuniform heating using a heater stage. (c) The relationship \nbetween ∆𝐻symmm and measured 𝐻R(𝐼=0) at different \nmicrowave frequencies, where the solid line is a fit to Eq. \n(4). The arrow shows the data point at f = 8 GHz. Data were \ntaken by varying microwave frequency  f  in the range of 6 -\n9 GHz with a step size of 0.5 GHz. (d) C urrent dependence \nof the uniaxial anisotropy field 𝐻𝑘 calculated based on the \nfitting parameters from Fig. 2(c).   \nThus,  𝐴1 and 𝐴2 correspond  to the slope and y-intercept  \nof the fitting line  and are determined to be 0.014 ±0.001 \nand − 9.9 ±0.5 Oe, respectively . Using these values,  we \ndetermine 𝐶1𝐼max2=9.4±0.5 Oe and 𝐶2𝐼max2=\n(−0.014 ±0.001 ) 4𝜋𝑀eff|𝐼=0+9.4 Oe. We interpret the \n𝐶2 term as the reduction of  4𝜋𝑀eff caused by Joule heating. \nBased on  the Bloch’s law31, ~1.4% reduction of 4𝜋𝑀eff \ncorresponds to a temperature rise of 22 K. The 𝐶1 term can \nbe interpreted as the change in 𝐻k, which  decreases by 20%  \nat 𝐼=𝐼max. Based on  the 𝐶1 and 𝐶2 values, we plot 𝐻k as a \nfunction of DC using 𝐻k=𝐻k|𝐼=0−𝐶1𝐼2, as shown in Fig. \n2(d).   \n \nIV. SIMULATION RESULTS  \n \nNext, we explore the possibility that the anisotropic \nstress , induced by Joule heating from current flow through \nthe bilayer waveguide , plays an important role in the \nmodification of 𝐻k . We used the thermal stress module of \nCOMSOL software30. We  took the power dissipation \nthrough the waveguide as a heat source  and calculated spatial \nprofiles of stresses. Fig. 3  show s that the calculated stress  \nvalues  for the waveguide along  x (𝜎x) and y (𝜎y) directions  \nat 𝐼=8 mA. The stress values are negativ e, indicating  that \nthe larger thermal expansion of CoFeB/Ta compared to  the \nAl2O3 substrate  leads to compressive stress es on CoFeB . The \nanisotropic stress es arise mainly due to the stripe -like shape \nof the  waveguide , as the  stress  difference  between two axes  \nbecomes zero if the  waveguide ha s a square rather than \nrectangular geometry . Based on the volume averaged  stress \nvalues, we calculated  the magneto -elastic energy 𝐸σ given \nby27  \n \n 𝐸σ=3\n2𝜆 (𝜎x sin2𝜙+𝜎y cos2𝜙), (5)  \n \nwhere λ is the magne to-elastic constant of CoFeB , 20×\n10−6 23. 𝜙 is the angle between x axis and  the magnetization  \nas shown in Fig. 1(a) . The effective magnetic field associated \nwith 𝐸σ can change  the uniform frequency formula. By  \nadding the stress  induced energy 𝐸σ to the total magnetic \nfree energy E and using the Smit -Suhl formula29, 30, we obtain  the modified uniform frequency formula given by   \n \n \n𝑓=𝛾\n2𝜋√(𝐻R−(𝐻k− 3𝜆\n𝑀s(𝜎x−𝜎y))\n×(𝐻R+4𝜋𝑀eff) (6)  \n \nWith  the calculated  stress  difference 𝜎x−𝜎y=1.6 ×\n108 dyn/cm2  and 𝑀s=1273  ±80 emu /cm3 23, we \nobtain  a stress  induced field  of 7.5 ± 0.5 Oe, which is \nreasonably close to  the measured 𝐻k decrease  of 9.5 Oe at \n𝐼=± 8 mA.  \nTo further confirm that anisotropic stress plays a key role \nin the observed  magnetic resonance shift with DCs, we \ncompare the observed  symmetric change in the resonance \nfield defined  by \n \n ∆𝐻symm ≡𝐻R(𝐼)+𝐻R(−𝐼)\n2−𝐻R(0). (7)  \n \nThe data for  Al2O3/CoFeB(10)/Ta(10) and \nSi/SiO 2(500) /CoFeB(10)/Ta(10)  are shown in Fig. 4.  The \nCoFeB waveguide on the Si substrate was 8 μm-wide and \n270 μm-long. ∆𝐻symm  for CoFeB on the Si  substrate \nincreases with DCs, which is consistent with a simple Joule \nheating effect while that of CoFeB on Al 2O3 substrate \ndecreases with DCs.   \nA similar  COMSOL calculation was performed  for the \nSi/SiO 2/CoFeB(10)/Ta(10) structure . The calculated stress \ndifference 𝜎x−𝜎y was only 2.0 ×107 dyn/cm2. Since 𝐸σ \ndepends on the difference in stresses, this leads to a much \nsmaller ∆𝐻symm  compared to the one on the Al2O3 substrate . \nThis small difference between 𝜎x and 𝜎y originates from  the \nfact that SiO 2 has a small thermal expansion coefficient \n(0.6×10−6) compared to that of Al2O3(7.5×10−6). Thus, \nthe stress from the  anisotropic  thermal expansion of CoFeB  \non the Si substrate  is limited and the  isotropic  thermal  stress \nFig. 3. Calculated stress disctribution along (a) x-direction \nand (b) y-direction. The center strip is the CoFeB/Ta \nwaveguide . \nFig. 4. Measured  ∆𝐻symm  as a function of current at 8 \nGHz microwave frequency for a CoFeB waveguide on \nAl2O3 (red) and Si/SiO 2 (blue) substrates.  \n  \ndominates30.  \n \nV. CONCLUSION  \n \nIn conclusion, we have investigated the uniaxial \nmagnetic anisotropy field of  a CoFeB/Ta waveguide on an \nAl2O3 substrate and its dependence on in -plane charge \ncurrent with the BLS technique. The in -plane uniaxial \nmagnetic anisotropy field is modified by 20% at a modest \ncharge cu rrent of 4×106 A/cm2. The modification of 𝐻k is \nsymmetric with respect to the current direction, which \ncannot be explained by either spin Hall or the Rashba effect s. \nOur simulations suggest that anisotropic stress induced by \nJoule heating from DCs passing  the waveguide can cause a \nchange in 𝐻k, which agrees reasonably well with the \nexperimental observation. This Joule heating induced \nanisotropic stress control of magnetic anisotropy may offer \nadditional design flexibility in the development of new \nspintronic devices, such as spin valves and magnetic \ntunneling junctions.  \n \nACKNOWLEDGEMENTS  \n \nThe work at UT -Austin (K. An., X. Ma, K. S. Olsson, X. \nLi) is supported by SHINES, an Energy Frontier Research \nCenter funded by the U.S. Department of Energy (DoE), \nOffice of Science, Basic Energy Science (BES) under award \n# DE -SC0012670.  The work at Corn ell was supported by \nthe NSF/MRSEC program (DMR -1120296) through the \nCornell Center for Materials Research.  \n  \nReference  \n1 B. Dieny, J. Magn. Magn. Mater. 136, 335 (1994).  \n2 C. Chappert, A. Fert, and F. N. Van Dau, Nature \nMater. 6, 813 (2007).  \n3 S. Mangin, D. Ravelosona, J. A. Katine, M. J. Carey, \nB. D. Terris, and E. E. Fullerton, Nature Mater. 5, 210 \n(2006).  \n4 S. S. P. Parkin, C. Kaiser, A. Panchula, P. M. Rice, B. \nHughes, M. Samant, and S. -H. Yang, Nature Mater. 3, \n862 (2004).  \n5 S. Yu asa, T. Nagahama, A. Fukushima, Y. Suzuki, and \nK. Ando, Nature Mater. 3, 868 (2004).  \n6 S. Ikeda, et al., Nature Mater. 9, 721 (2010).  \n7 F. Matsukura, Y. Tokura, and H. Ohno, Nature \nNanotech. 10, 209 (2015).  8 H. Morise and S. Nakamura, Phys. Rev. B 71, 014 439 \n(2005).  \n9 E. B. Myers, F. J. Albert, J. C. Sankey, E. Bonet, R. A. \nBuhrman, and D. C. Ralph, Phys. Rev. Lett. 89, \n196801 (2002).  \n10 R. H. Koch, G. Grinstein, G. A. Keefe, Y. Lu, P. L. \nTrouilloud, W. J. Gallagher, and S. S. P. Parkin, Phys. \nRev. Lett. 84, 5419 (2000).  \n11 F. Michael, Reports on Progress in Physics 61, 755 \n(1998).  \n12 A. T. Hindmarch, et al., Phys. Rev. Lett. 100, 117201 \n(2008).  \n13 G. Yu, et al., Appl. Phys. Lett. 106, 072402 (2015).  \n14 S. Kanai, M. Endo, S. Ikeda, F. Matsukura, and H. \nOhno , J. Phys.: Conf. Ser. 266, 012092 (2011).  \n15 I. M. Miron, et al., Nature 476, 189 (2011).  \n16 Y. Fan, et al., Nature Mater. 13, 699 (2014).  \n17 G. Yu, et al., Nature Nanotech. 9, 548 (2014).  \n18 A. R. Mellnik, et al., Nature 511, 449 (2014).  \n19 J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 \n(1996).  \n20 L. Liu, C. -F. Pai, Y. Li, H. W. Tseng, D. C. Ralph, and \nR. A. Buhrman, Science 336, 555 (2012).  \n21 I. Mihai Miron, G. Gaudin, S. Auffret, B. Rodmacq, A. \nSchuhl, S. Pizzini, J. Vogel, and P. Gambar della, \nNature Mater. 9, 230 (2010).  \n22 S. Ikeda, J. Hayakawa, Y. Ashizawa, Y. M. Lee, K. \nMiura, H. Hasegawa, M. Tsunoda, F. Matsukura, and \nH. Ohno, Appl. Phys. Lett. 93, 082508 (2008).  \n23 R. C. O'Handley, Modern magnetic materials: \nprinciples and applicati ons (Wiley, New York, 2000).  \n24 L. Liu, T. Moriyama, D. C. Ralph, and R. A. Buhrman, \nPhys. Rev. Lett. 106, 036601 (2011).  \n25 P. M. Haney, H. -W. Lee, K. -J. Lee, A. Manchon, and \nM. D. Stiles, Phys. Rev. B 87, 174411 (2013).   \n26 K. An, D. R. Birt, C. -F. Pai, K . Olsson, D. C. Ralph, R. \nA. Buhrman, and X. Li, Phys. Rev. B 89, 140405 \n(2014).  \n27 S. s. Chikazumi, Physics of magnetism  (Wiley, New \nYork, 1964).  \n28 R. W. Damon and J. R. Eshbach, J. Phys. Chem. \nSolids. 19, 308 (1961).  29 H. Suhl, Phys. Rev. 97, 555 (1955).  \n30 See supplemental material at ?? for detailed \ncalculations.  \n31 N. W. Ashcroft and N. D. Mermin, Solid state physics , \nSaunders, Philadelphia, 1976, 1976).  \n \n \n   \nSupplemental Material: Current Control of Magnetic Anisotropy via Stress in a Ferromagnetic Metal \nWaveguide  \n \nKyongmo An1, Xin Ma1, Chi -Feng Pai3, Jusang Yang1, Kevin  S. Olsson1, James L. Erskine1, Daniel C. Ralph3,4, \nRobert A. Buhrman3 and Xiaoqin Li1,2,* \n \n1Department of Physics, University of Texas, Austin, Texas 78712, USA  \n2Center for Complex Quantum Systems, The Universit y of Texas at Austin, Austin, Texas 78712, USA  \n3Cornell University, Ithaca, New york 14853, USA  \n4Kavli Institute at Cornell, Cornell University, Ithaca, New York 14853, USA  \n \n*Email address: elaineli@physics.utexas.edu  \n \nS1. Derivation of uniform frequency formula  \n \nThe uniform precession frequency formula can be derived from the Smit -Suhl formula given by  \n \n 𝑓=𝛾\n2𝜋1\n𝑀ssin𝜃[𝜕2𝐸\n𝜕𝜃2𝜕2𝐸\n𝜕𝜙2−(𝜕2𝐸\n𝜕𝜃𝜕𝜙)2\n], (S1) \n \nwhere 𝛾 is the gyromagnetic ratio and 𝑀s is the saturation magnetization.  𝜃 and 𝜙 are the angles that represent the \nmagnetization direction defined in Fig. 1(a) of the main text. 𝐸 is the  energy associated with magnetization of our system \ngiven by  \n \n 𝐸=−𝑯∙𝑴+1\n2 𝑀𝑠 (4𝜋𝑀eff  cos2𝜃−𝐻k sin2𝜃 cos2𝜙), (S2) \n \nwhere 𝑀eff is the effective magnetization and  𝐻k is the in -plane uniaxial anisotropy field. For an in -plane external \nmagnetic field perpendicular to the waveguide, we found numerically the equilibrium direction of magnetization by \nminimizing the total energy. With the ener gy term and the calculated equilibrium orientation of magnetization, we can \ncalculate the uniform precession frequency using the Smit –Suhl’s formula. Then we obtain  \n \n 𝑓=𝛾\n2𝜋√(𝐻R−𝐻k)(𝐻R+4𝜋𝑀eff). (S3) \n \nS2. Derivation of the relationship between ∆𝑯𝐬𝐲𝐦𝐦𝐦 and 𝑯𝐑  \n \nWe take the derivative of Eq. (3) in the main text with respect to 𝐼2. \n \n 4𝜋2𝑑(𝑓2)\n𝛾𝑑(𝐼2)=0= (𝑑𝐻R\n𝑑𝐼2+𝐶2)(𝐻R−𝐻k,0+𝐶1𝐼2)+(𝐻R+4𝜋𝑀eff,0+𝐶2𝐼2)(𝑑𝐻R\n𝑑𝐼2+𝐶1). (S4) \n \nFurther simplifying, we obtain a formula for the change in 𝐻R with respect to 𝐼2 given by  \n     \n 𝑑𝐻R\n𝑑𝐼2=−𝐶1−1\n4𝜋𝑀eff,0(𝐶2−𝐶1)(𝐻R−𝐻k,0+𝐶1𝐼2)+𝑂[(1\n4𝜋𝑀eff,0)2\n]\n≈−1\n4𝜋𝑀eff,0(𝐶2−𝐶1)𝐻R−𝐶1−1\n4𝜋𝑀eff,0(𝐶2−𝐶1)(−𝐻k,0+𝐶1𝐼2). (S5)  \n \nIn the calculation above, we only keep the first order term of 1/(4𝜋𝑀eff,0). 𝑑𝐻R/𝑑𝐼2 can be approximated to \n∆𝐻symmm/𝐼max2 because the change in 𝐻R is observed to be only 1% of 𝐻R when 𝐼 increases to 𝐼max =8 mA. Then, we \nobtain  \n \n ∆𝐻symmm=𝐴1𝐻R+𝐴2, \n \nwhere,  \n \n𝐴1≡−(𝐶2−𝐶1)𝐼max2\n4𝜋𝑀eff,0 \n𝐴2≡−𝐶1𝐼max2−(𝐶2−𝐶1)(−𝐻𝑘,0+𝐶1𝐼2 )\n4𝜋𝑀eff,0𝐼max2. (S6) \n \nS3. Comsol calculation  \n \nThe thermal stress module of COMSOL software was used to calculate the spatial profiles of stresses and strains for \nboth Al 2O3/CoFeB(10)/Ta(10) and Si/SiO 2(500)/CoFeB(10)/Ta(10) due to the Joule heating at 𝐼=8 mA. The size of the \nwaveguide used in the simulation was the same as the actual sample size. The Al 2O3 substrate was assumed to be 8 mm \n× 8 mm laterally with a thickness of 0.8 mm.  The temperature at the bottom of the substrate was fixed at a tempera ture \nof 293 K. An adiabatic boundary condition was assumed for other surfaces. We chose a mechanical boundary condition \nthat allows free expansions along all directions.  The mesh size was about 3 μm laterally. Five meshes were distributed \nevenly for each l ayer along the thickness direction. The calculated volume averaged stresses and strains over th e \nwaveguide are shown in Table I . The material properties used in the simu lation are summarized in Table II . \n \nTable I.  Calculated volume averaged values over the CoFeB layer for the temperature rise, stress, \nand strain tensors  at I = 8 mA  \n \nSample  ∆𝑇(K) 𝜎𝑥𝑥 \n(MPa)  𝜎𝑦𝑦 \n(MPa)  𝜀𝑥𝑥 𝜀𝑦𝑦 \nAl2O3/CFB/Ta  21 - 42 - 26 4.2×10−5 1.7×10−4 \nSi/SiO 2/CFB/Ta  31 - 83 - 81 3.3×10−6 2×10−5 \n \nTable II. Parameters used in the comsol simulations  \nMaterial  Thermal \nconductivity \n(W/(m ∙K)) Thermal \nexpansion \ncoefficient  \n(10-6) Young’s \nmodulus \n(GPa)  Poisson \nratio Heat \ncapacity \n(J/(k g∙K)) Density \n(kg/m3) \nTa 57 6.3 186 0.34 140 16690  \nCoFeB  90 12 162 0.3 500 8900  \nAl2O3 30 7.5 345 0.27 760 3970  \nSi 130 2.6 150 0.22 700 2330  \nSiO 2 1.4 0.6 70 0.17 700 2200  \n \nS4. Stress -strain relation  \n  \nThe stress tensor 𝜎𝑖𝑗 is related with the strain tensor 𝜀𝑖𝑗 by the following equation.  \n \n 𝜎𝑖𝑗=𝐸\n(1+𝜈)(1−2𝜈)[(1−2𝜈)𝜀𝑖𝑗+∑𝜈𝛿𝑖𝑗𝜀𝑘𝑘\n𝑘]−𝛿𝑖𝑗𝐸𝛼∆𝑇\n(1−2𝜈), (S7) \n \nwhere 𝐸 is the Young’s modulus, 𝜈 is the Poisson ratio, 𝛼 is the thermal expansion coefficient, 𝛿𝑖𝑗 is the Kronecker \ndelta, and ∆𝑇 is the temperature change. The equation consists of a strain dependent part (square bracket) and strain \nindependent part. Thus only the first term (strain dependent term) can give rise to the anisotropic stress. For CoFeB on \nthe Al 2O3 substrate, the strain dependent term is comparable with the other term . However, as shown in Table I , CoFeB \non the Si substrate has strain values about one order of magnitude smaller compared to CoFeB on Al 2O3 due to the small \nthermal expansion coefficient of SiO 2. Thus, it has a negligible contribution from the strain depen dent term and the strain \nindependent term contributes dominantly, leading to the nearly isotropic stress.  \n "    },    {        "title": "1008.2142v1.Rotational_Doppler_Effect_in_Magnetic_Resonance.pdf",        "content": "arXiv:1008.2142v1  [cond-mat.other]  12 Aug 2010Rotational Doppler Eﬀect in Magnetic Resonance\nS. Lend´ ınez1,2, E. M. Chudnovsky1,3, and J. Tejada1,2\n1Departament de F´ ısica Fonamental, Facultat de F´ ısica,\nUniversitat de Barcelona, Avinguda Diagonal 645, 08028 Bar celona, Spain\n2Institut de Nanoci` encia i Nanotecnologia IN2UB,\nUniversitat de Barcelona, c. Mart´ ı i Franqu` es 1, 08028 Bar celona, Spain\n3Physics Department, Lehman College, The City University of New York,\n250 Bedford Park Boulevard West, Bronx, NY 10468-1589, U.S. A.\n(Dated: November 6, 2018)\nWe compute the shift in the frequency of the spin resonance in a solid that rotates in the ﬁeld of\na circularly polarized electromagnetic wave. Electron spi n resonance, nuclear magnetic resonance,\nand ferromagnetic resonance are considered. We show that co ntrary to the case of the rotating LC\ncircuit, the shift in the frequency of the spin resonance has strong dependence on the symmetry\nof the receiver. The shift due to rotation occurs only when ro tational symmetry is broken by the\nanisotropyofthegyromagnetic tensor, bytheshapeofthebo dy,orbymagnetocrystalline anisotropy.\nGeneral expressions for the resonance frequency and power a bsorption are derived and implications\nfor experiment are discussed.\nPACS numbers: 76.30.-v, 76.50.+g, 76.60.-k, 32.70.Jz\nI. INTRODUCTION\nThe term Rotational Doppler Eﬀect (RDE) is used to\ndescribe a frequency shift encountered by a receiver of\nelectromagnetic radiation when either the receiver or the\nsource of the radiation are rotating. The eﬀect is illus-\ntrated in Fig. 1. The frequency of the wave, ω= 2πf,\nmeasured at a given point in space, corresponds to the\nangular velocity of the rotation of the electric (magnetic)\nﬁeld due to the wave. If the receiver is rotating mechan-\nically at an angular velocity Ω about the axis parallel\nto the wave vector k, than the frequency of the wave\nperceived by the receiver equals\nω′=ω±Ω. (1)\nThe sign, plus or minus, depends on the helicity of the\nwave and the direction of the rotation of the receiver.\nThe RDE is less commonly known than the conven-\ntional Doppler eﬀect. One reason is that it is more\ndiﬃcult to observe. M¨ ossbauer technique provides the\nmostsensitivemethod forthe studyofthefrequencyshift\ndue to the conventional Doppler eﬀect, δω= (v/c)ωfor\nv≪c. The limiting velocity has been a fraction of a\nmillimeter per second and is due to the ﬁnite very small\nlinewidth of gamma radiation, δω/ω∼10−13−10−12.\nSuch a small linewidth has even permitted observation of\nthetransverseDopplereﬀect1,2byperformingM¨ ossbauer\nexperiment on a rotating platform. This eﬀect, not to be\nconfused with the RDE, consists of the frequency shift\nδω/ω=−v2/(2c2) due to the relativistic time dilation\nfor a receiver moving tangentially with respect to the\nsource of the radiation. It is easy to see, however, that\nthe frequency shift as little as Ω /ω∼10−13−10−12due\nthe RDE would require angular velocity of the emitter\nor the receiver in the M¨ ossbauer experiment on the order\nof a few MHz or even a few tens of MHz. The latter is\nstill one-two orders of magnitude greater than the angu-FIG. 1: Color online: Rotational Doppler eﬀect. The fre-\nquencyωof the circularly polarized electromagnetic wave\n(ω,k) is the angular velocity of the rotation of the electric\n(magnetic) ﬁeld due to the wave at a given point in space.\nThe rotation of the receiver at an angular velocity Ω, de-\npending on the direction of the rotation and the helicity of\nthe wave, adds or subtracts Ω to the frequency of the wave ω,\nrendering ω′=ω±Ω in the coordinate frame of the receiver.\nlar velocities of high-speed rotors used for magic angle\nspinning in NMR applications.\nThe RDE frequency shift caused by a rotating plate\ninserted into a beam of circularly polarized light was re-\nported in Refs. 3–7. The RDE was predicted for rotat-\ning light beams8and subsequently observed using mil-\nlimeter waves9as well as in the optical range10(see Ref.\n11 for review). In solid state experiments the RDE has\nproved surprisingly elusive. Frequencies of the ferromag-\nnetic resonance (FMR) are typically in the GHz range or\nhigher, which is far above achievable angular velocities of\nmechanical rotation of macroscopic magnets. However,\nsmall magnetic particles in beams12or in nanopores132\nmay rotate very fast. Eq. (1) was recently applied to the\nanalysis of the observed anomalies in the FMR data on\nrotatingnanoparticles13. TheRDEmaybeespeciallyim-\nportant for the NMR technology that uses rapidly spin-\nning samples. Frequency shifts of the quadrupole line in\nthe nuclear magnetic resonance (NMR) experiment with\na rotating sample were reported in Ref. 14 and analyzed\nin terms of Berry phase15. It was never fully explained,\nhowever, why such shifts do not persist in the NMR ex-\nperiments in which the angular velocity of the magic-\nangle-spinning rotor with the sample often exceeds the\nlinewidth by an order of magnitude. Some hint to an-\nswering this question can be found in Ref. 16 that stud-\nied the eﬀect of the rotation on radiation at the atomic\nlevel. The authors of this work correctly argued that\nthe RDE can only be seen in the radiation of atoms and\nmolecules placed in the environment that destroys rota-\ntional symmetry.\nSituation depicted in Fig. 1 rather obviously leads to\nthe frequency shift by Ω when the emitter and the re-\nceiver are based upon LC circuits. This has been tested\nby the GPS for the case of a receiving antenna mak-\ning as little as 8 revolutions per second as compared to\nthe carrier frequency of the electromagnetic waves in the\nGHz range17. Eq. (1) has been also applied to the ex-\nplanation of the frequency shift encountered by NASA\nin the communications with Pioneer spacecrafts18. One\nessential diﬀerence between conventional and rotational\nDoppler eﬀects is that the ﬁrst refers to the inertial sys-\ntems while the second occurs in the non-inertial systems.\nThis prompted works that considered RDE in the con-\ntext of nonlocal quantum mechanics in the accelerated\nframe of reference19. Relativity (or Galilean invariance\nforv≪c) makes the conventional Doppler eﬀect quite\nuniversal. As we shall see below, such a universality\nshould not be expected for the RDE. Indeed, the argu-\nment behind the RDE is based upon perception of a cir-\ncularly polarized wave by a rotating observer. Through\nthe Larmortheorem20the mechanicalrotationofthe sys-\ntem of charges is equivalent to the magnetic ﬁeld. Con-\nsequently, when making the argument, one has to check\nwhether the resonant frequency of the receiver is aﬀected\nby the magnetic ﬁeld. Resonant frequencies of LC cir-\ncuits are known to be insensitive to the magnetic ﬁelds,\nthus making the argument rather solid. On the contrary,\nthe frequency of the receiver based upon magnetic reso-\nnance would be sensitive to the ﬁctitious magnetic ﬁeld\ndue to rotation, thus making the argument incomplete.\nIn this paper we develop a rigorous theory of the RDE\nfor magnetic resonance. We show that the frequency\nshift due to rotation is always diﬀerent from Ω. Bro-\nken rotational symmetry is required for the shift to have\na non-zero value, in which case the magnetic resonance\nsplits into two lines separated by 2Ω. For the electron\nspin resonance (ESR) violation of the rotational sym-\nmetry would naturally arise from the anisotropy of the\ngyromagnetic tensor. In a solid state NMR experiment\nwith a rotating sample, violation of symmetry would bemore common in the presence of the magnetic order that\nprovides anisotropy of the hyperﬁne interaction. For a\nferromagnetic resonance (FMR) the asymmetry comes\nfrom the shape of the sample and from magnetocrys-\ntalline anisotropy. The paper is organized as follows.\nThe physics of spin-rotation coupling is reviewed in Sec-\ntion II. Frequency shift of the ESR in a rotating crys-\ntal with anisotropic gyromagnetic tensor is computed in\nSection III. The eﬀect of rotation on the NMR spectra\nis discussed in Section IV. FMR in a rotating sample is\nstudied in Section V. Power absorption by the rotating\nmagnet is considered in Section VI. Section VII contains\nsome suggestions for experiment and discussion of possi-\nble application of the RDE in solid state physics.\nII. SPIN-ROTATION COUPLING\nIn classical mechanics the Hamiltonian of the system\nin a rotating coordinate frame is given by21\nH′=H−L·Ω. (2)\nHereHis the Hamiltonian at Ω = 0 and Lis the me-\nchanical angular momentum of the system. For a system\nof charges one can write\nL=M\nγ, (3)\nwhereMis the magnetic moment and γis the gyro-\nmagnetic ratio. Eq. (2) then becomes equivalent to the\nHamiltonian,\nH′=H−M·B, (4)\nin the ﬁctitious magnetic ﬁeld,\nB=Ω\nγ, (5)\nwhich is the statement of the Larmor theorem20.\nNeither classical mechanics nor classical ﬁeld theory\ndealswith the conceptofaspin. The questionthen arises\nwhether Eq. (2) should contain spin Salongside with the\norbital angular momentum L. Eq. (4) hints that since\nthe magnetic moment can be of spin origin this should\nbe the case. Also it is known from relativistic physics\nthat the generator of rotations is\nJ=L+S. (6)\nIt should be, therefore, naturally expected that in the\npresence of a spin Eq. (2) should be generalized as\nH′=H−(L+S)·Ω. (7)\nIn quantum theorythis relationcanbe rigorouslyderived\nin the following way. Rotation by an angle φtransforms\nthe Hamiltonian of an isolated system into22\nˆH′= exp/bracketleftbiggi\n¯h(L+S)·φ/bracketrightbigg\nˆHexp/bracketleftbigg\n−i\n¯h(L+S)·φ/bracketrightbigg\n.(8)3\nTo the ﬁrst order on a small rotation φone obtains\nˆH′=ˆH−i\n¯h(L+S)·[ˆH,φ], (9)\nwhere we have taken into account that for an isolated\nsystemJis conserved, that is L+Scommutes with ˆH.\nThis equation becomes Eq. (7) if one takes into account\nthe quantum-mechanical relation\nΩ=dφ\ndt=i\n¯h[ˆH,φ] (10)\nandreplacesoperator Ωbyitsclassicalexpectationvalue.\nFor an electron Eq. (7) can be also formally derived as a\nnon-relativistic limit of the Dirac equation written in the\nmetric of the rotating coordinate frame23. The answer\nfor the corresponding Schr¨ odinger equation reads\ni¯h∂Ψ\n∂t=ˆH′Ψ,ˆH′=ˆp2\n2m−/parenleftbigg\nr×ˆp+1\n2¯hˆσ/parenrightbigg\n·Ω,(11)\nwhererandp=−i¯h∇are the radius-vector and the\nlinear momentum of the electron, respectively, and σx,y,z\nare Pauli matrices.\nThere has been some confusion in literature regarding\nthe term −S·Ωin the Hamiltonian of the body stud-\nied in the coordinate frame that rotates together with\nthe body24–26. To elucidate the physical meaning of this\nterm, let us consider the resulting equation of motion for\na classical spin-vector27\ndS\ndt=−S×δH′\nδS. (12)\nIfHdoes not depend on spin, then the spin cannot be\naﬀected in any way by the rotation of the body. In this\ncaseδH′/δS=−Ωand Eq. (12) simply describes the\nprecession of SaboutΩ:\ndS\ndt=S×Ω. (13)\nIt shows how a constant vector S(or any other vector\nto this matter) is viewed by an observer rotating at an\nangular velocity Ω. This has nothing to do with the\nspin-orbit or any other interaction. Such interactions\nshould be accounted for in the ˆHpart of the Hamilto-\nnianˆH′. The eﬀect of rotations on various magnetic\nresonances is considered in the next sections.\nIII. FREQUENCY SHIFT OF THE ELECTRON\nSPIN RESONANCE DUE TO ROTATION\nIn this Section we consider an electron in a rotating\ncrystal or in a rotating quantum dot characterizedby the\nanisotropic gyromagnetic tensor, gij. The eﬀect of local\nrotations due to transverse phonons on the width of the\nESR has been studied in Ref. 28. Here we are interestedin the eﬀect of the global rotation on the ESR frequency.\nTo deal with the stationary states we shall assume that\nthe axis of rotation Ωis parallel to the applied magnetic\nﬁeldBand will compute the energy levels of the electron\nas measured by the observer rotating together with the\nsystem. In the rotating frame the spin Hamiltonian of\nthe electron is\nˆH′=1\n2µBgijσiBj−1\n2¯hσ·Ω. (14)\nPositive sign of the ﬁrst (Zeeman) term is due to the\nnegativegyromagneticratio γforthe electron( µB= ¯h|γ|\nbeing the Bohr magneton).\nThe geometryofthe problem is illustrated in Fig. 2. In\nthe rotating frame the solid matrix containing the elec-\ntron is stationary. It is convenient to choose the coordi-\nnate axes of that matrix along the principal axes of the\ntensorgij. Thengijis diagonal,\ngij=giδij, (15)\nrepresented by three numbers, gx,gy, andgzthat can be\ndirectly measured when the system is at rest. Eq. (14)\nthen becomes\nˆH′=1\n2[(µBgxBx−¯hΩx)σx+(µBgyBy−¯hΩy)σy\n+ (µBgzBz−¯hΩz)σz]. (16)\nDiagonalization of this Hamiltonian with the account of\nthe fact that Ωwas chosen parallel to Bgives the follow-\ning energy levels of ˆH′:\nE±=±1\n2µBB\n/summationdisplay\ni=x,y,z/parenleftbigg\ngi−¯hΩ\nµBB/parenrightbigg2\nn2\ni\n1/2\n(17)\nHerenis the unit vector in the direction of the axis of\nrotation,\nn=Ω\nΩ=B\nB. (18)\nIn practice, the angular velocity of the mechanical ro-\ntationwill alwaysbe suﬃciently smalltoprovidethe con-\ndition ¯hΩ≪µBB. Contribution of the rotation to the\nESR frequency in the rotating frame,\n¯hω′\nESR=E+−E−, (19)\nwill, therefore, be small compared to the ESR frequency\n¯hωESR=µBB(g2\nxn2\nx+g2\nyn2\ny+g2\nzn2\nz)1/2(20)\nunperturbed by rotation. Expanding Eq. (17) to the ﬁrst\norder in Ω one obtains\nω′\nESR=ωESR−κΩ, (21)\nκ=gxn2\nx+gyn2\ny+gzn2\nz/radicalBig\ng2xn2x+g2yn2y+g2zn2z. (22)4\nFIG. 2: Color online: Spin in the magnetic ﬁeld parallel to\nthe rotation axis of the crystal. The rotating coordinate ax es\nx,y,zare chosen along the principal axes of the gyromagnetic\ntensor.\nHere Ω can be positive or negative depending on the di-\nrection of rotation.\nFew observations are in order. Firstly, according to\nEq. (22), the frequency shift for the observer rotating to-\ngether with the sample containing the electron is never\nzero. Secondly, when the rotation is about one of the\nprincipal axes of the gyromagnetic tensor, Eq. (22) gives\nκ= 1, so that the frequency shift for the rotating ob-\nserver is exactly Ω. The ESR occurs when the frequency\nω′of the circularly polarized electromagnetic wave per-\nceived by the rotating observer and given by Eq. (1) co-\nincides with ω′\nESR. If the rotation is about one of the\nprincipal axes of gij, thenκ= 1 and the angular velocity\nΩ cancels exactly from the equation ω′=ω′\nESRfor the\npolarization of the wave that corresponds to ω′=ω−Ω,\nthus, resulting in no RDE frequency shift for an experi-\nmentalist workingin the laboratoryframe. For the oppo-\nsitepolarizationofthewave,correspondingto ω′=ω+Ω,\nthe shift in the rotationallyinvariantcaseformallyequals\n2Ω. However, such photons would have their spin pro-\njection in the direction opposite to the one necessary to\nproduce the spin transition. They can be absorbed only\nwhen the rotational symmetry is broken so that the elec-\ntron spin in the direction of the wave vector is no longer\na good quantum number (see Section VI).\nIV. FREQUENCY SHIFT OF THE NUCLEAR\nMAGNETIC RESONANCE DUE TO ROTATION\nLet us consider a nuclear spin Iin the magnetic ﬁeld\nparallel to the axis of rotation of the sample. It is clear\nfrom the previous section that the mechanical rotation\ncombined with the rotationally invariant Zeeman inter-\naction of the nuclear magnetic moment with the ﬁeld,\nˆH′=−γngnI·B−I·Ω, (23)(withγn>0 andgnbeing nuclear gyromagnetic ratio\nand gyromagnetic factor, respectively) are not suﬃcient\nto produce the RDE. Isotropic hyperﬁne interaction with\nanatomicspin Softheform −AI·Swouldnotchangethis\neither. However, an anisotropic hyperﬁne interaction,\nˆHhf=−AijIiSj, (24)\nin principle, can do the job. If there is a ferromagnetic\norder in the solid, then Sdevelops a non-zero average,\n/angbracketleftS/angbracketright. Replacing Sjin Eq. (24) with /angbracketleftSj/angbracketrightand adding the\nhyperﬁne interaction to Eq. (23), one obtains\nˆH′=−γngnI·B−AijIi/angbracketleftSj/angbracketright−I·Ω.(25)\nTo work with the stationary energy states in the ro-\ntating frame, we shall assume that all three vectors B,\n/angbracketleftS/angbracketright, andΩare parallel to each other. Let us study the\ncase ofI= 1/2. Choosing the coordinate axes along the\nprincipal axes of tensor Aij=Aiδij, it is easy to see that\nEq. (25) is equivalent to the Zeeman Hamiltonian,\nˆH′=−1\n2µn/bracketleftbig\ngeff\nxσxBx+geff\nyσyBy+geff\nzσzBz/bracketrightbig\n(26)\nwith an eﬀective gyromagnetic tensor whose principal\nvalues are given by ( i=x,y,z)\ngeff\ni=gn+Bhf\ni\nB+¯hΩ\nµnB, (27)\nwhere we have introduced the nuclear magneton, µn=\n¯hγn, and the hyperﬁne ﬁeld, Bhf, with components\nBhf\ni=¯hAi|/angbracketleftS/angbracketright|\nµn. (28)\nThe energy levels of the Hamiltonian (26) are\nE±=±1\n2µnB\n/summationdisplay\ni=x,y,z/parenleftBig\ngeff\ni/parenrightBig2\nn2\ni\n1/2\n,(29)\nwheren=B/B.\nLet us consider the case of small Ω. Making the series\nexpansion of Eq. (29) one obtains to the ﬁrst order on Ω\nω′\nNMR=E+−E−\n¯h=ωNMR+κΩ (30)\nwithκgiven by\nκ=/summationtext\ni=x,y,z/parenleftBig\ngn+Bhf\ni/B/parenrightBig\nn2\ni/radicalbigg\n/summationtext\ni=x,y,z/parenleftBig\ngn+Bhf\ni/B/parenrightBig2\nn2\ni.(31)\nIn the case of the isotropic hyperﬁne interaction, Bhf\nx=\nBhf\ny=Bhf\nz(that is, Ax=Ay=Az), Eq. (31) gives\nκ= 1. Same situation occurs when the direction of the\nﬁeld and the axis of rotation coincide with one of the5\nprincipal axes of the tensor of hyperﬁne interactions. For\narbitrary rotations Eq. (31) gives κ→1 whenB≫Bhf,\nmaking the frequency shift deﬁned by ω′=ω′\nNMRneg-\nligible for the polarization ( ω′=ω+Ω) that is predom-\ninantly absorbed due to the selection rule. Is is likely,\ntherefore, that a signiﬁcant RDE in the NMR can be\nobserved only in magnetically ordered materials, in the\nﬁeld comparable or less than the hyperﬁne ﬁeld, for rota-\ntions about axes that do not coincide with the symmetry\naxes of the crystal. If these conditions are satisﬁed, and\nthe width of the resonance is not very large compared to\nΩ, the NMR produced by linearly polarized waves would\nsplit into two lines of uneven intensity separated by 2Ω.\nIn fact, the existing experimental techniques permit ob-\nservation of this eﬀect (see Section VII).\nV. FREQUENCY SHIFT OF THE\nFERROMAGNETIC RESONANCE DUE TO\nROTATION\nWe now turn to the rotating ferromagnets. We begin\nwith a simplest model of ferromagnetic resonance stud-\nied by Kittel29. In this model one neglects the eﬀects of\nmagnetocrystalline anisotropy and considers a uniformly\nmagnetized ferromagnetic ellipsoid in the external mag-\nnetic ﬁeld B=µ0H(withµ0being the magnetic perme-\nability of vacuum). The energy density of such a ferro-\nmagnet is determined by its Zeeman interaction with the\nexternal ﬁeld and by magnetic dipole-dipole interactions\ninside the ferromagnet:\nH=µ0/bracketleftbigg\n−M·H+1\n2NijMiMj/bracketrightbigg\n.(32)\nHereMis the magnetization and Nijis tensor of demag-\nnetizing coeﬃcients. The principal axes of Nijcoincide\nwith the axes of the ellipsoid. Choosing the coordinate\naxes along the principal axes and taking into account\nthat for a ferromagnet\nM2=M2\nx+M2\ny+M2\nz=M2\n0 (33)\nis a constant, one can rewrite Eq. (32) as\nH=−µ0/bracketleftbigg\nM·H+1\n2(Nx−Nz)M2\nx+1\n2(Ny−Nz)M2\ny)/bracketrightbigg\n,\n(34)\nwhere we have omitted unessential constant. For, e.g.,\nan inﬁnite circular cylinder Nx=Ny= 1/2,Nz= 0. In\ngeneral, for an ellipsoid elongated along the Z-axis one\nhasNx−Nz>0,Ny−Nz>0, so that in the absence\nof the ﬁeld the minimum of Eq. (34) corresponds to M\nin theZ-direction. This will still be true in the external\nﬁeld if the latter is applied in the Z-direction, which is\nthe case we consider here. Note that a ﬁnite ﬁeld is al-\nways needed to prevent the magnet from breaking into\nmagnetic domains.The FMR frequency, ωFMR, can be obtained from ei-\nther classical or quantum mechanical treatment27. Clas-\nsically, itisthe frequencyoftheprecessionof Maboutits\nequilibrium direction. To ﬁnd ωFMRone should linearize\nthe equation,\ndM\ndt=γM×B(eff),B(eff)=−δH\nδM,(35)\naroundM=M0ez(γ <0 being the gyromagneticratio).\nThe answer reads29\nωFMR=√ωxωy, (36)\nwhere\nωx=|γ|[B+(Nx−Nz)µ0M0]\nωy=|γ|[B+(Ny−Nz)µ0M0].(37)\nTo study the RDE we should now solve the same prob-\nlem in the coordinate frame rotating about the Z-axis at\nan angular velocity Ω. In the presence of rotation the\nHamiltonian becomes\nH′=H−M\nγ·Ω. (38)\nIt is easy to see that for Ω= Ωezthis eﬀectively adds\nΩ/γto the external ﬁeld. Consequently, the FMR fre-\nquency in the rotating frame becomes\nω′\nFMR=/radicalBig\nω′xω′y (39)\nwith\nω′\nx=|γ|/bracketleftbigg\nB+Ω\nγ+(Nx−Nz)µ0M0/bracketrightbigg\nω′\ny=|γ|/bracketleftbigg\nB+Ω\nγ+(Ny−Nz)µ0M0/bracketrightbigg\n.(40)\nOur immediate observation is that for a symmetric el-\nlipsoid (Nx=Ny)\nω′\nFMR=ωFMR−Ω, (41)\nso that the RDE frequency shift determined by the equa-\ntionω′=ω−Ω =ω′\nFMRis exactly zero. For an asym-\nmetric ellipsoid ( Nx/negationslash=Ny), expanding Eq. (39) into a\nseries on Ω one obtains to the ﬁrst order\nω′\nFMR=ωFMR−κΩ, (42)\nwith\nκ=1\n2/parenleftbigg/radicalbiggωx\nωy+/radicalbiggωy\nωx/parenrightbigg\n. (43)\nIt is easy to see that κ≥1. At large ﬁelds, B≫µ0M0,\nequations (37) and (43) give κ→1, that is, no frequency\nshift due to the RDE. Sizable frequency shift of the FMR\nobserved in the laboratory frame due to the rotation of6\nFIG.3: Color online: Geometry oftheFMRstudiedinthepa-\nper. Ferromagnet uniformly magnetized by a static magnetic\nﬁeld,B, is rotating at an angular velocity Ωin the radia-\ntion ﬁeld of circularly polarized photons of wave vector kand\nspins. (Due to the negative gyromagnetic ratio, the equilib-\nrium spin of the magnet, S0, is antiparallel to its equilibrium\nmagnetic moment M0.)\nthe sample should occur only at Bnot signiﬁcantly ex-\nceedingµ0M0and only in a sample lacking the rotational\nsymmetry.\nOne can easily generalize the above approach to\ntake into account any type of the magnetocrystalline\nanisotropy. The formulas look especially simple in the\ncase of the second-order anisotropy. Such anisotropy\nadds the term\n−1\n2µ0βijMiMj (44)\nto the Hamiltonian of the magnet, with βijbeing some\ndimensionless symmetric tensor. Consider, e.g., an or-\nthorhombic crystal whose axes ( a,b,c) coincide with the\naxes of the ellipsoid and whose easy magnetization axis,\nc, is parallel to the Z-direction. In this case all the above\nformulas remain valid if one replaces the demagnetizing\nfactors with\nN′\ni=Ni−βi, i=x,y,z, (45)\nwhereβx,βy, andβzare the principal values of βij. Due\nto the orthorhombic anisotropy ( a/negationslash=b→βx/negationslash=βy) the\nRDE may now occur even in a sample of the rotationally\ninvariant shape ( Nx=Ny).\nVI. POWER ABSORPTION BY A ROTATING\nMAGNET\nFor non-relativistic rotations the radiation power ab-\nsorbed by the magnet should be the same in the labora-\ntory frame and in the rotating frame. Calculation in the\nrotatingframe is easier. We shall assume that the dimen-\nsions ofthe sample aresmall comparedto the wavelength\nof the radiation, so that the ﬁeld of the wave at the posi-\ntion of the ferromagnet is nearly uniform. The geometry\nstudied below is illustrated in Fig. 3. Within the model\nof Eq. (38), the rotating magnet placed in the ﬁeld of\na circularly polarized wave feels the oscillating magnetic\nﬁeld that can be represented by a complex function\nh(t) =h0e±iω′t, ω′=ω∓Ω (46)giving the components of the ﬁeld as\nhx= Re(h), hy= Im(h). (47)\nHereh0is the complex amplitude of the wave, ±sign\nin Eq. (46) determines the helicity of the wave, while\nthe sign of Ω determines the direction of rotation of the\nmagnet. Due to the wave the magnetization acquires\na small ac-component m(t) (whose real and imaginary\nparts represent mxandmy, respectively),\nm(t) = ˆχ(ω)h(t), (48)\nwhere ˆχis the susceptibility tensor. The absorbed power\nis given by27\nP=±iµ0ω′h∗\n0(ˆχ−ˆχ†)h0. (49)\nThe problem has, therefore, reduced to the computation\nof the susceptibility in the rotating frame. The latter can\nbe done by solving the Landau-Lifshitz equation,\ndM\ndt=γM×B(eff)−η\nM0|γ|M×/bracketleftBig\nM×B(eff)/bracketrightBig\n,(50)\nin the rotating frame, that is, with B(eff)=−δH′/δM\nand\nH′=H−M\nγ·Ω−M·h. (51)\nThe parameter ηin Eq. (50) is a dimensionless damping\ncoeﬃcient that is responsible for the width of the FMR\nin the absence of inhomogeneous broadening.\nSubstituting M=M0ez+minto Eq. (50) and solving\nfor ˆχone obtains for the power\nP±=1\n2η|γ|M0µ2\n0|h0|2f±(ω′), (52)\nwhere\nf±=ω′2[2(ω′2−ω′2\nFMR)±2ω′(ω′\nx+ω′\ny)+(ω′\nx+ω′\ny)2]\n(ω′2−ω′2\nFMR)2+η2ω′2(ω′x+ω′y)2.\n(53)\nNotice that when there is a full rotational symmetry,\nω′\nx=ω′\ny=ω′\nFMR, the absorbed power at the resonance\nis non-zero only for one polarizationof the wavethat cor-\nresponds to the upper sign in Eqs. (46) and (53). This\nis a consequence of the selection rule due to conserva-\ntion of the Z-component of the total angular momentum\n(absorbed photon + excited magnet).\nLet us now consider a rotating ferromagnet in the ra-\ndiation ﬁeld of a linearly polarized electromagnetic wave.\nIn the rotating frame the complex magnetic ﬁeld of such\na wave is\nh(t) =h0\n2/bracketleftBig\nei(ω−Ω)t+e−i(ω+Ω)t/bracketrightBig\n=h0e−iΩtcos(ωt)\n(54)7\n/SolidCircle\n– –Κ/Equal1—Κ/Equal1.11\n/CapOmega/OverTilde/Equal0.01\nB/OverTilde/Equal0\nΩ/OverTilde\nFMR/Equal0.3162\nΗ/Equal0.003\n0.2900.2950.3000.3050.3100.3150.3200100200300400500600\nΩ/OverTildePower/LParen1a.u/RParen1\nFIG. 4: Color online: Absorption of power of linearly po-\nlarized electromagnetic radiation by a rotating magnet. Fr e-\nquencies are given in the units of γµ0M0. As the rotational\nsymmetry is violated the FMR becomes shifted and the sec-\nond FMR line emerges separated by 2Ω from the ﬁrst line.\nRepeating the above calculation, one obtains for the\npower averaged over the period of rotation\nP=1\n8η|γ|M0µ2\n0|h0|2[f+(ω−Ω)+f−(ω+Ω)].(55)\nWhen the rotational symmetry of the magnet is broken,\nωx/negationslash=ωy,κ >1, the absorption has two maxima of un-\neven height at\nω=ωFMR−(κ∓1)Ω. (56)\nAs the rotational symmetry is gradually restored, ωx→\nωy,κ→1, the rotational shift in the position of the\nmain maximum disappears. In that limit the shift in the\nposition of a smaller maximum approaches 2Ω while the\nheight of that maximum goes to zero, see Fig. 4.\nVII. DISCUSSION\nWe have computed the frequency shift of the magnetic\nresonance due to rotation of the sample. The eﬀect of\nrotation on the ESR, NMR, and FMR has been studied.\nWe found that it is, generally, quite diﬀerent from the ro-\ntational Doppler eﬀect reported in other systems11. The\ndiﬀerences stem from the observation that the spin of an\nelectron or an atom would be insensitive to the rotation\nof the body as whole if not for the relativistic spin-orbit\ncoupling. Even with account of spin-orbit interactions\nthe spin would not simply follow the rotation of the body\nbut would exhibit more complex behavior described by\nthe dynamics of the angular momentum. Everyone who\nwatched the behavior of a gyroscope in a rotating frame\nwould easily appreciate this fact.\nWe found the following common features of the mag-\nnetic resonance in a rotating sample.\n•If the spin Hamiltonian is invariant with respect to\nthe rotation, then the rotation of the body has noeﬀect on the frequency of the resonant absorption\nof a circularly polarized electromagnetic wave.\n•As the rotationalinvariance is violated, the absorp-\ntion line shifts. The shift is diﬀerent from the an-\ngular velocity of rotation, Ω. It depends on the de-\ngree of violation of the rotational symmetry. The\nfrequency shift goes to zero when the symmetry is\nrestored.\n•In the case of a linearly polarized radiation a sec-\nond resonance line emerges, separated by 2Ω from\nthe ﬁrst line. The intensity of that line depends\non the degree of violation of rotational symmetry.\nIt disappears when the rotational symmetry is re-\nstored.\nESR and FMR measurements are usually performed\nin the GHz range, with the width of the resonance\nbeing sometimes as low as a few MHz. Currently\navailable small mechanical rotors can rotate as fast as\n100kHz, which, nevertheless, is still low compared to the\nlinewidths of ESR and FMR. Note, however, that the po-\nsition of the ESR or FMR maximum can be determined\nwith an accuracy of a few hundred kHz. It is then not\nout of question that under appropriate conditions the\nRDE frequency shift and the splitting of the resonance\ncan be observed in high precision ESR and FMR exper-\niments even when the rotation frequency is signiﬁcantly\nlower than the linewidth. Since anisotropy of the sample\nis needed to provide rotational asymmetry, the measure-\nments should be performed on single crystals. Crystals\nwith signiﬁcant anisotropy of the gyromagnetic tensor\nshould be selected for ESR experiments. When the mag-\nnetocrystalline anisotropy is weak, the RDE in FMR can\nbe induced by the asymmetric shape of the sample alone\ndue to the anisotropy of dipole-dipole interactions. Even\nin this case, however, a single crystal would be preferred\nto provide a narrow linewidth. Same applies to experi-\nments on RDE in solid state NMR. The NMR frequency\nrange is much lower than that used in ESR and FMR\nexperiments. The width of the NMR line can be as low\nas a few kHz, that is, well below the available rotational\nangular velocities. The key to the observation of RDE in\na solid state NMR must be the use of a crystal having\nmagnetic order and strong anisotropy of the hyperﬁne\ninteraction.\nA separate interesting question is magnetic resonance\nin small magnetic particles that are free to rotate. Parti-\ncles of size in the nanometer range can easily be excited\ninto rotational states with Ω of hundreds of MHz. Con-\ntrary to the rotational quantum states of molecules that\nhave been studied for decades, analytical solution of the\nproblem of a quantum-mechanical rotator does not exist\nevenwithoutaspin. Presenceofthespininteractingwith\na mechanical rotation complicates this problem even fur-\nther. Rigorous solution has been recently found for the\nlow energy states of a rotator that can be treated as a\ntwo-state spin system30. General solution is very diﬃ-\ncult to obtain. In the case when a particle consists of8\na large number of atoms, one can develop a semiclassi-\ncal approximation in which Ωis replaced with L/I(with\nIbeing the moment of inertia). This suggests that the\nmagnetic resonance in nanoparticles that are free to ro-\ntate would split into many lines related to the quanti-\nzation of L. Some evidence of this eﬀect has been re-\ncently found in the FMR studies of magnetic particles in\nnanopores13. Rapid progress in measurements of single\nmagnetic nanoparticles31may shed further light on their\nquantized rotational states and related spin resonances.VIII. ACKNOWLEDGEMENTS\nS.L. acknowledges ﬁnancial support from Grupo de\nInvestigaci´ on de Magnetismo de la Universitat de\nBarcelona. The work of E.M.C. has been supported by\nthe grant No. DMR-0703639 from the U.S. National Sci-\nence Foundation and by Catalan ICREA Academia. J.T.\nacknowledges ﬁnancial support from ICREA Academia.\n1H. J. Hay, J. P. Schiﬀer, T. E. Cranshaw, and P. A. Egel-\nstaﬀ , Phys. Rev. Lett. 4, 165 (1960); D. C. Champeney\nand P. B. Moon, Proc. Phys. Soc. 77, 350 (1961); W.\nK¨ undig, Phys. Rev. 129, 2371 (1963).\n2A.L.Kholmetskii, T.Yarman, andO.V.Missevitch, Phys.\nScr.77, 035302 (2008); A. L. Kholmetskii, T. Yarman, O.\nV. Missevitch, and B. I. Rogozev, Phys. Scr. 79, 065007\n(2009).\n3B. A. Garetz and S. Arnold, Opt. Commun. 31, 1 (1979).\n4R. Simon, H. J. Kimble, and E. C. G. Sundarshan, Phys.\nRev. Lett. 61, 19 (1988).\n5F. Bretenaker and A. Le Floch, Phys. Rev. Lett. 65, 2316\n(1990).\n6I.V. Basistiy, A.Y. Bekshaev, M.V. Vasnetsov, V.V.\nSlyusar, and M. S. Soskin, JETP Lett. 76, 486 (2002).\n7L. Chen and W. She, Optics Express 16, 14629 (2008).\n8G. Nienhuis, Opt. Commun. 132, 8 (1996).\n9J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and\nM. J. Padgett, Phys. Rev. Lett. 80, 3217 (1998); J. Cour-\ntial, D. A. Robertson, K. Dholakia, L. Allen, and M. J.\nPadgett, Phys. Rev. Lett. 81, 4828 (1998).\n10S. Barreiro, J.W. R. Tabosa, H. Failache, and A. Lezama,\nPhys. Rev. Lett. 97, 113601 (2006).\n11M. Padgett, Nature 443, 924 (2006).\n12See, e.g., X.Xu, S. Yin, R. Moro, and W.A. deHeer, Phys.\nRev. Lett. 95, 237209 (2005), and references therein.\n13J. Tejada, R. D. Zysler, E. Molins, and E. M. Chudnovsky,\nPhys. Rev. Lett. 104, 027202 (2010).\n14R. Tycko, Phys. Rev. Lett. 58, 2281 (1987).\n15M. V. Berry, Proc. Roy. Soc. London, Ser. A 392, 45\n(1984).\n16I. Bialynicki-Birula and Z. Bialynicka-Birula, Phys. Rev.\nLett.78, 2539 (1997).17N. Ashby, Living Rev. Relativity 6, 1 (2003), and refer-\nences therein.\n18J. D. Anderson and B. Mashhoon, Phys. Lett. A 315, 199\n(2003).\n19B. Mashhoon, Phys. Rev. A 474498 (1993). C. Chicone\nand B. Mashhoon, Ann. Phys. (Leipzig) 11, 309 (2002); C.\nChicone and B. Mashhoon, Phys. Lett. A 298, 229 (2002).\n20L. D. Landau and E. M. Lifshitz, The Classical Theory\nof Fields, Vol. 2 of Course of Theoretical Physics, 4th ed.\n(Pergamon, New York, 1975).\n21L. D. Landau and E. M. Lifshitz, Mechanics, Vol. 1 of\nCourse of Theoretical Physics, 3rd ed. (Pergamon, New\nYork, 1982).\n22See, e.g., A. Messiah, Quantum Mechanics , Vol. II (Wiley,\nNew York, 1976).\n23F. W. Hehl and W.-T. Ni, Phys. Rev. D 42, 2045 (1990).\n24E. M. Chudnovsky, Phys. Rev. Lett. 72, 3433 (1994); E.\nM. Chudnovsky and X. Mart´ ınez Hidalgo, Phys. Rev. B\n66, 054412 (2002).\n25F. Hartman-Boutron, P. Politi, and J. Villain, Int. J. Mod.\nPhys. B 10, 2577 (1996).\n26E. M. Chudnovsky, D. A. Garanin, and R. Schilling, Phys.\nRev. B72, 094426 (2005).\n27E. M. Chudnovsky and J. Tejada, Lectures on Magnetism\n(Rinton Press, Princeton, NJ, 2008).\n28C. Calero, E. M. Chudnovsky, an D. A. Garanin, Phys.\nRev. Lett. 95, 166603 (2005).\n29C. Kittel. Phys. Rev. 73, 155 (1948).\n30E. M. Chudnovsky and D. A. Garanin, Phys. Rev. B 81,\n214423 (2010).\n31L. Bogani and W. Wernsdorfer, Nature Materials 7, 179\n(2008)."    },    {        "title": "1912.01331v1.Nanoscale_Tantalum_Layer_Controlling_the_Magnetic_Coupling_between_Two_Ferromagnetic_Electrodes_via_Insulator_of_a_Magnetic_Tunnel_Junction.pdf",        "content": "1 \n Nanoscale Tantalum Layer Controlling the  Magnetic Coupling between \nTwo F erromagnetic Electrodes via Insulator  of a Magnetic Tunnel \nJunction  \nPawan Tyagi1,2* and Tobias Goulet1 \n1Mechanical Engineering, University of the District of Columbia , Washington DC-20008. USA  \n2Chemical and Materials Engineering, University of Kentucky, Lexington, KY -40566, USA  \n*Corresponding Author: ptyagi@udc.edu  \n \nABSTRACT:  Ability to tailor the nature of the magnetic coupling betw een two ferromagnetic electrodes  \ncan enable the realization of new spintronics device systems . This paper discusses our finding that \ndeposition of an ultrathin tantalum (Ta) on the  NiFe top electrode revers ed the nature of inter -\nferromagnetic electrode coupling. W e observed that the deposition of ~ 5 nm Ta on the top of a magnetic \ntunnel junction with Ta( 2 nm)/Co(5 nm )/NiFe (5 nm)/AlOx( 2  nm)/NiFe (10 -15 nm) configuration \nchanged the magnetic coupling between two ferromagnetic electrodes from antiferromagnetic to \nferromagnetic. We investigated Ta effect using multiple magnetic characterizations like ferromagnetic \nresonance, magnetometry, and polarized neutron reflectometry. Ferromagnetic resonance characterization \nwas very sensitive for detecting the changes in mag netic coupling via the insulating spacer. This simple  \napproach of adding Ta film to alter the magnetic coupling can impact the other burgeoning areas like \nmolecular spintronics. We found that preexisting magnetic coupling between two ferromagnetic \nelectrodes impacted the resultant magnetic properties of magnetic tunnel junctions based molecular \nspintronics devices.  \nKey words:  Magnetic tunnel junctions; molecular spintronics; tantalum; exchange  coupling;  \nI INTRODUCTION:   2 \n Tailoring the  nature of the magnetic coupling between two ferromagnetic electrodes has been the \ntopic of intense interest  [1, 2].  Ability to change the inter -ferromagnetic electrode coupling can lead the \ndevelopment of new device forms  and materials [2]. For instance, nanoscale spintronics  devices focus on \nmaneuvering the nature and strength of the inter-ferromagnetic electrode coupling  (IFMEC) [2]. To date, \nthree key approache s have been employed t o tailor the IFMEC. The first approach  involves inserting the \nnanoparticles between two ferromagnetic electrodes  [1]. The second method involves  changing the \nthickness  of nonmagnetic spacers between two ferromagnetic electrodes  [3]. The third method  requires  \nchanging of the nonmagnetic spacer material between two ferromagnetic electrodes . However, these \napproaches are very challenging to implement.  For instance, controlling the distribution and sizes of \nnanoclusters between two ferromagnetic electrodes  is very challenging to exercise and difficult to \nreproduce  [1]. Similarly, tailoring the spacer thickness to sub -nm scale and even changing the spacer \nmaterial altogether requires intensive device optimization [2, 4].  An approach that do es not physically \naffect the spacer between two ferromagnetic electrodes and easy to implement can lead to new \nopportunities. Recently , magnetic tunnel junction based molecular spintronics devices  (MTJMSDs)  were \ndeveloped [5-7]. One can alter the IFMEC by adding Ta top layer before transforming a magnetic tunnel \njunction into a MTJMSD. In this paper , we first discuss the role of Ta on IFMEC. We also discussed the \nimpact  of preexisting IFMEC on the magnetic properties of the MTJMSD .  \nII EXPERIMENTAL DETAILS:  \nTo investigate Ta effect on IFMEC we employed various physical property measurement  \ntechniques  such as Ferromagnetic resonance (FMR) , magnetometry, and polarized neutron reflectivity \n(PNR) .  For the  PNR measurements, unpatterned MTJ and MTJ -Ta samples were employed. The reason \nfor utilizing unpatterned samples was based on the strong effect from t he uncovered substrate that \nstrongly impact ed the measurement and modeling accuracy. The FMR and magnetometry study utilized \npatterned tunnel junctions. The FMR and magnetometry methods  are sensitive towards the magnetic \nproperty of the materials.  Unlike P NR the FMR and magnetometry do not get influenced by the 3 \n nonmagnetic su bstrate . Also, utilizing patterned tunnel junction was necessary for making molecular \ndevices for the FMR and magnetometry study. It is notworthy that e dge effects become prominent for \nsubmicron or nm scale magnetic features  [8]. To avoid the undesirable impact of edges we produced the \nMTJ and MT -Ta with several tens of micron area. We fabricated an array of ~7000  patterned magnetic \ntunnel junctions per sample. Every magnetic tunnel \njunction w as ~5 µm in diameter and ~10 µm distance \nfrom the neighboring magnetic tunnel junctions . We \nutilized an oxidized silicon substrate and performed \nphotolithography to produce a photoresist layer with \nan array of mi cro-cavities. For this study, Shipley®  \n(S1813 ) photoresist was spin coated on an oxidized \nsilicon wafer piece at 3000 rpm speed. This  spin \ncoated  photoresist film was baked at 90 ⁰C for one \nminute. Subsequently, the photoresist was exposed  to \nUV light thro ugh a photomask containing an array pattern. We developed the photoresist film in MF 319 \nMicroposit developer to produce an array of microc avities in the photoresist film . These microcavities \nwere filled with multiple thin  films to create a n array of magnetic tunnel  junctions. Tantalum (Ta), cobalt \n(Co), alumina(AlOx), and the alloy of NiFe  alloy with 80% nickel w ere sputter  deposited with AJA \nInternational sputtering machine. We first sputter depos ited Ta(2 nm)/Co(5 nm)/NiFe (5 nm)/AlOx( 2  \nnm)/NiFe (10 -15 nm) thin film configuration . This magnetic tunnel junction configuration is named as \nMTJ. To prepare the sample for the study of Ta impact on IFMEC MTJ -Ta samples were produced. To \ncreate MTJ -Ta samples , we additionally sputter deposited 5 nm Ta on the top of MTJ configuration . A \nresultant MTJ -Ta sample had Ta(2 nm)/Co(5 nm)/NiFe (5 nm)/AlOx( 2nm)/NiFe (10 -15 nm)/Ta (5 nm) \nconfiguration. The ~ 5 nm Ta top layer thickness ensured a conformal film o n the top NiFe electrode. We \nnoted that deposition of NiFe on the alumina (AlOx)  generally produced as high as 3-5 nm RMS \nroughness.  The high roughness is inevitable due to the amorphous nature of AlOx [8]. With such high \nFig. 1. FMR spectra recorded on MTJ and \nMTJ -Ta samples.  \n4 \n roughness Ta  < 3 nm was unable to form conformal films.  Hence, Ta with thickness around 5 nm was a  \ngood choice. On the other hand increasing Ta thickness also reduced device yield, presumably due to \nhigher mechanical stresses. Mechanical st resses have been found to create tunnel barrier failures [9]. \nThese MTJ and MTJ -Ta samples were characterized by the X -band Bruker EMX300 FMR and  NanoOsc \nPhase FMR over 2 -17 G Hz frequency range at room temperature. Magnetization studies were performed \nwith Quantum Design PPMS SQUID m agnetometer  at 150 K . Low temperature was chosen to avoid \nnoise in the magnetization study. PNR study w as performe d at 150 K and at 150 mT magnetic field at \nNational Institute of Standards and Technology, Gaithersb urg USA. The l ight reflectivity studies were \nperformed with Semiconsoft ® Mprobe thin film measurement sy stem at room temperature .      \nIII RESULTS AND  DISCUSSIONS :  \nWe first studied the IFMEC on MTJ and MTJ -Ta. It is noteworthy that t wo magnetic structu res \nseparated by nm gap exhibit  ferromagnetic or antiferromagnetic couplings  [10]. FMR is a powerful tool to \nstudy the charact eristics of magnetic coupling between two magnetic structures, especially two \nferromagnetic films  [11]. Our FMR studies revealed a striking difference between MTJ and MTJ -Ta (Fig. \n1). Under the identical experimental conditions , both samples showed t wo distinct FMR modes . For both \nsamples, an in -plane DC magnetic field up to 4000 Oe and 9.75G Hz microwave was applied to study the \nresonance modes. Before every measurement , the cavity’s spectra w as checked for the background signal \nat fivefold higher gain than that use d for the MTJ and MTJ -Ta samples. We also investigated  if the \nmanual err or in the sample alignment with respect to  the direction of the magnetic field could impact the \nintensity of FMR modes. The FMR spectra of a sample did not change noticeably within the  ±10º \nvariation on the magnetic field direction. We also utilized the FMR response from the graphite tape  as a \ncontrol sample  to ensure that experimental conditions were identical for the MTJ and MTJ -Ta (Fig. 1). \nDuring the FMR study of MTJ and MTJ -Ta, the  graphite tape produced a delta function type resonance \npeak at 3367±4 Oe (Fig. 1). In our study , we utilized the same graphite tape to mount the MTJ and MTJ -\nTa samples for the FMR study. This sharp resonance peak from the graphite tape was statistically 5 \n identical for the MTJ and MTJ -Ta samples. Invariance of the graphite tape’s signal suggests that \nexperimental conditions for measuring MTJ and MTJ -Ta were identical. The reproducibility of the \ngraphite tape’s resonance signal ensured the robustness and rep roducibility of microwave power, DC \nmagnetic field , and losses due to the measurement system, etc.  \nThe MTJ sample showed acoustic mode (higher intensity resonance peak) before the optical \nmode (lower intensity peak). According to FMR theory [12],[13] two ferromagnetic electrodes  of the MTJ \nsample are antiferromagnetic ally coupl ed. On the other hand, t he two resonance peaks from MTJ -Ta \nsample were significantly different as compared to the FMR peaks from MTJ. It appears that presence of \nTa on the top of NiFe ferromagnetic electrode reduced the intensity of the first resonance peak. As a \nresult , MTJ -Ta exhibited smaller intensity resonance mode (optical mode ) appear ing before the higher \nintensity resonance ( acoustic mode ). This particular form of the FMR spectra from MTJ -Ta is indicative \nof ferromagnetic coupling between the two ferromagnetic electrodes [12], [13].  \nWe estimated the strength of exchange coupling between two ferromagnetic electrodes of the \nMTJ and MTJ -Ta to be of the similar magnitude . \nWe estimated the order of magnitude of the \nmagnetic coupling by  two ways. (a) First , we \nevaluated the slop e of the lines joining the two \nresonance modes of the MTJ and MTJ -Ta. It is \nnoteworthy that zero slop means no interaction \nbetween the two ferromagnetic electrodes. The FMR \nmodes recorded on the isolated top and bottom \nferromagnetic electrodes are uncoupled (Fig. 2). \nHowever, for MTJ and MTJ -Ta the slop e of the line  \nbetween two modes is roughly the same. (b)  The \nestimation of magnetic coupling strength is also possible from the difference in tunnel junction’s mode \nFig. 2. FMR spectra of MTJ and MTJ -Ta samples  with \nrespect to FMR peaks from the isolated Ta/Co/NiFe and \nNiFe electrodes . \n6 \n positions with respect to the mode positions from the isolated ferromagnetic electrodes. It is noteworth y \nthat tunnel junction ’s mode positions shift as a function of the magnetic coupling strength between two \nelectrodes [10, 14]. We noticed that resonance position for the MTJ and MTJ -Ta was only ~45  Oe less as \ncompared to the resonance magnetic field of the Ta/Co/NiFe bottom electrode  grown in isolation. We \nsurmise that addition of Ta only affected the nature of IFMEC, not its magnitude. It is also noteworthy \nthat adding Ta app ears to reduce the intensity of acoustic mode, which appeared close to the top NiFe \nelectrode’s resonance position (Fig. 2).   \nWe also investigated the difference \nin magnetization data obtained from the \nMTJ and MTJ -Ta. We found that both \nsamples produced almost identical \nmagnetization loop s (Fig. 3 ). However, \nMTJ -Ta was relatively less sloped in the \nunsaturation state ( Inset of Fig. 3). This Ta \ninduced subtle difference in the \nmagnetization loop affirms two important \npoints : (i) For MTJ the antiferromagnetic \nIFMEC is not strong otherwise there could be a significant changes in  the magnetization loop  [15]; (ii) \nmagnetization data for MTJ -Ta show ed moderate  increase in the magnetic moment between saturation \nstates as compared t o that of MTJ  (Fig. 3 a). We surmise that t his moderate  increase in magnetization is \ndue to the  emergence of ferromagnetic coupling. If the addition of Ta did not affect the coupling, then \nmagnetization loop should have been the same for the MTJ and MTJ -Ta sample s. On the other hand , if \nthe addition of Ta enhanced the antiferromagnetic coupling, the n the magnetic moment in the \nunsaturation region should have been reduced. Hence, the current form of the magnetization data assert s \nFig.3: .Magnetic field  vs. normalized moment for ±2000 Oe \nrange.  Inset image show zoomed in image of magnetization \ncurve for ± 200 Oe . \n7 \n with the FMR data (Fig. 1) , which indicates that the addition of Ta produced ferromagnetic coupling \nbetween two ferromagnetic electrodes .    \nTo investigate the mechanism \nbehind the Ta effect, we conduc ted FMR \nstudy on tunnel junctions where the \nferromagnetic  electrodes on both sides of \nthe AlOx tunneling barrier were the same. \nWe studied the FMR response from a \ntunnel junction comprising of Ta(2 \nnm)/Co(5 nm)/NiFe (5 nm)/AlOx( 2 \nnm)/NiFe (10 nm)  (Fig. 4 ). This \nconfiguration essentially showed that \nbottom ferromagnetic electrode of the \nMTJ and MTJ -Ta is present on the both \nsides of AlOx  insulator. The FMR spectra \nfor this tunnel junction configuration revealed antiferromagnetic coupling between the two ferromagnetic \nelectrodes via AlOx insulator. The existence of this antiferromagnetic coupling is evident from the \nlocations of acoustic a nd optical modes  [13]. In this case, a coustic mode appeared at 1025 Oe, and optical \nmode existed at 3125 Oe  (Fig. 4) . The resonance peaks for this  configuration are ~2000 Oe apart, whereas \ntwo resonance modes for the MTJ and MTJ -Ta were positioned at a gap of ~340 Oe only.  \nWe also studied FMR on the tunnel junction with NiFe (10 nm)/AlOx (2 nm)/NiFe (1 0 nm) \nconfiguration  (Fig. 4) . It is noteworthy that the top and bottom ferromagnetic electrodes of this tunnel \njunction are similar to the ferromagnetic film utilized in the MTJ and MTJ -Ta. This tunnel junction also \nexhibited antiferromagneti c coupling between two NiFe ferromagnetic electrodes. For this configuration , \nacoustic mode appeared before the optical mode. Acoustic mode appeared at ~1130 Oe, and optical mode \nFig. 4: FMR of magnetic tunnel junctions with  \nTa(2 nm)/Co(5 nm)/NiFe (5 nm)/AlOx( 2 nm)/NiFe (10 nm) and  \nNiFe(10 nm)/AlOx (2 nm)/NiFe (10 nm) configurations.    \n8 \n \nFig. 5: Polarized beam neutron reflectivity study of (a) MTJ and (b) MTJ -Ta samples. Solid line \nshow fitted curve on the experimental d ata. R -- correspond to reflectivity from nuclear profile minus \nreflectivity from magnetic profile. R++ corresponds to reflectivity from nuclear profile plus \nreflectivity from magnetic profile. (c) Nuclear and magnetic scattering length density (ρ) vs. MTJ \nand MTJ -Ta thickness. (d) Comparison of modeled and expected nuclear and magnetic scattering \nlength density data for the MTJ.  \nappeared at 1230 Oe. Hence, the difference between two modes for this tunnel junction was only ~100 Oe \nthat is significantly smaller than that for MTJ and MTJ -Ta, i.e. 340 Oe . This tunnel junction  with similar \nferromagnetic electrodes also possessed antiferromagnetic coupling  [13]. Adding Ta on the top of this \ntunnel junction did not produce the change in the IFMEC.  It appears that impact of Ta is pronounced for \nthe magnetic tunnel junction with the dissimilar magnetic electrodes.  \nTo understand the impact of Ta along the depth of the MTJ , we conducted PNR studies. We \nhypothesized that  magnetic attributes of the MTJ’s top electrode should  be affected by the Ta layer. We \nemployed Polarized Neutron Reflectometry (PNR) to investigate the difference in magnetic attributes of \nthe MTJ and MTJ -Ta. Under this experiment , a polarized beam of neutron s interacted with th e MTJ and \nMTJ -Ta. Subsequently, t he spin and angle of the neutron beam reflected by the samples were analyzed. 9 \n We utilized nuclear reflectivity and n on- spin flip reflectivity  to calculate R -- and R++. Here, R -- \ncorresponds to the difference in reflectivity  from nuclear and magnetic profiles (Fig. 5). Whereas, the \nR++ correspond to the sum of reflectivity  due to nucle ar and magnetic profiles (Fig. 5 ).  The R -- and R++ \nvs. wave  vector graphs for the M TJ and MTJ -Ta are shown in Fig 5a and Fig 5 b, respectively. The \nreflectivity profile for the MTJ (Fig. 5a) is diffe rent than that of MTJ -Ta (Fig. 5 b). We fitted the \nexperimental data  with the scattering length density model to record the depth -wise changes in MTJ  and \nMTJ -Ta magnetization  (Fig.  5c). For MTJ -Ta m agnetic signal in the 220 to 230 nm thickness range \nsuggests that Ta gained magnetic moment. However, the magnetization in the adj acent NiFe’s region \ndecreased ; the NiFe/Ta region for the magnetic sig nal in Fig. 5 c indicates this possibility . Howe ver, for \nthe MTJ sample magnetic signal for the top NiFe electrode has expected profile in the 22 0-230 nm \nthickness range (Fig. 5 c). This PNR observation is in agreement with the prior studie s which reported that \na Ta film deposited on the NiFe gained magn etic moment  [16, 17]. The reduction in magnetic  moment  of \ntop NiFe electrode is a lso in agreement with the decrease  in the intensity of the acoustic mode of MTJ as \ncompared to MTJ -Ta (Fig. 2). Although  Fig 5 c represents the best fit for the nuclear and magnetic data , \nwe do not believe the model ed data is perfectly accurate. To estimate  the degree of deviation we \ncompared the modeled data for the MTJ sampl e with the expected data (Fig. 5 d). Expected data was \ncalculated for the ideal MTJ with perfect interfaces and atomically smoot h films. We attribute the \ndifference between modeled and e xpected data to \nthe significantly high roughness and diffusive \ninterfaces  between AlOx/NiFe (top) and \nNiFe(top)/Ta . Prior study  [8] and our AFM study \nshowed that growth of AlOx induced high \nroughness.  \nIn the quest of getting additional insights, \nwe attempted to fit PNR data by fixing the thin fil m thickness in the  scattering length density  model (Fig. \nFig. 6: Reflectance vs. wavelength graph for \nMTJ and MTJ -Ta. \n10 \n 1S, Supplementary material). The regions of nuclear and magnetic scattering length density for the \nbottom electrode (Ta/Co/NiFe) were in agreement with the expected thickness regions for the individual \nfilms. However, the nuclear and magnetic scattering length density did not show the  good fit  in sections \ncorresponding to top NiFe  and Ta films. This PNR study agrees with our hypothesis that issues  mainly \nstart after the deposition of AlOx. The PNR data provided in  the Fig. 1S of the s upplementary material  \nindicate d that magnetic moment was also prese nt in the Ta region of MTJ -Ta. Future study may \nemphasize on  producing smoothe r top AlOx and improved PNR modeling .    \nEven though roughness in the magnetic tun nel junctions  impacted PNR mode ling, but we do not \nbelieve this roughness level affected  the integrity of magnetic tunnel junctions.  The most delicate part of \nthe magnetic tunnel junction is the AlOx  tunneling barrier. We  found that 3 -5 nm level roughness is not \ndetrim ental to the integrity of tunnel barrier that separates the two ferromagnetic layers. For the \nvalidation, we produced tunnel junctions for the transport study by following the method described \nelsewhere  [7]. The transport study conducted on \ntunnel junctions exhibited excellent tun neling \nbehavior (Supplementary m aterial -Fig.2S). The \npresence of tunneling response confirms that \ntunneling barrier is in good condition and \nunaffected by the level of roughness observed \nin our sample.  \nWe conducted additional experiment to \njudge the quality of MTJ and MTJ -Ta. To make \nsure that top NiFe and Ta films were  continuous \nwe conducted light reflectivity study  on MTJ and MTJ -Ta. We hypothesized that a continuous Ta film on  \nthe top of MTJ -Ta must produce clearly noticeable effects.  The reflectivity data for MTJ and MTJ -Ta \nfollowed the similar trend (Fig. 7). The reflectivity data was consistent with the reflectivity profile from a \nFig. 7 : Magnetic moment of MTJ and MTJ -Ta \nbefore and after treating with OMCs.   \n11 \n continuous NiFe film [18].  However, the reflectivity data below ~450 nm was higher for the MTJ -Ta as \ncompared to the MTJ sample  (Fig. 6). It appears that Ta on top has a  higher reflectivity for light radiation \nbelow ~450 nm only. We also noted that peaks of reflectivity  data for MTJ -Ta were  ~10 nm ahead of the \nMTJ samples  for 200 to 450 nm range. S uch a shift in the position of the reflectivity peaks for short \nwavelength is observed when the top NiFe film is cover ed with a conformal film of different composition \n[18]. This study suggests that although NiFe may be rough , but Ta has formed a conformal film on the \ntop.         \nTo demonstrate the application of maneuvering IFMEC with Ta, w e studied the impact of Ta top \nlayer on the magnetic properties of the magnetic tunnel junction based molecular devices  (MTJMSD) . \nThe MTJMSD  were produced by bridging the paramagnetic molecules between the ferroma gnetic \nelectrodes of the MTJ and MTJ -Ta [19]. These paramagnetic molecules are essentially organometallic \nmolecular clusters (OMCs). For this study, the MTJ and MTJ -Ta samples discussed in Fig. 3, were \nutilized. The method of molecule attachment was described elsewhere [6, 7] . The attributes of  OMC \nparamagnetic molecules have also been published  elsewhere  [20]. An OMC molecule contained an octa -\nnuclear cubic cage with a net spin state. Every corner of the OMC’s cub ic cage possessed an alkane \ntether. At  the end of each alkane tether, a thiol functional group was provided. The thiol functional groups \nhelped to bridge the molecules across the ~ 2 nm AlOx tunnel barrier along the exposed edges of the \ntunnel junctions. Each thiol group had a strong affinity towards the NiFe ferromagnetic electrode of the \nMTJ and MTJ -Ta. The magnetic study showed that OMCs created remarkably stronger exchange \ncoupling as compared to AlOx  tunnel barrier.  It is noteworthy that MTJ and MTJ -Ta samples contained \nseveral thousand tunnel junctions to yield the high signal to noise ratio during magnetic studies. To study \nthe paramagnetic  molecule effect the magnetic moment were  measured for  ±2000 Oe field range at 150 K \ntemperature. The magnetization study showed that OMC channels across the AlOx insulator on the \nexposed sides produced the opposite effect s on MTJ and MTJ -Ta (Fig. 7). The magnetic mom ent of MTJ \ndropped nearly by ~84 % (Fig. 7 ); it must be noted that MTJ had preexisting antiferromagnetic coupling 12 \n between the ferromagnetic electrodes. On the other hand, OMCs increased the magnetic moment of the \nMTJ -Ta by ~116%; it is noteworthy that MTJ -Ta possessed pre -existing ferromagnetic coupling  between \nthe two ferromagnetic electrodes (Fig. 7). This study indicates that preexisting IFMEC  is important in \ndetermining the magnetic properties of the  molecular spintronics devices.  We have conducted Monte \nCarlo study [19] to get qualitative understanding; however, simulation studies with continuous spin \nmodels are recommended to investigate the impact on preexisting IFMEC.     \nHere w e propose  the potential mechanism behind the Ta induced changes in the IFMEC of the \nMTJ. The previous observations of the reversal of IFMEC are re ported due to variation in the property of \nthe tunneling barriers [1]. However, it is noteworthy that in our study the tunneling barrier was grown with \nthe same procedure for MTJ and MT J-Ta. Hence, tunneling barrier is not expected to play a role in \nimpacting IFMEC after the addition of Ta on MTJ. In our case , the magnetic interaction via the tunneling \nbarrier is expected to be governed by the exchange coupling. Prior research showed tha t in the case of \nmagnetic tunn el junctions exchange coupling was the most dominant [21] . The exchange coupling \nstrength decreases exponentially [22] with the tunneling barrier thickness . IFMEC  is also sensitive \ntowards the crystallinity of the tunneling barrier [21, 23]. Our AlOx tunneling barrier growth method is \nbased on prior work that leads to amorphous tunneling barrier [8]. In fact , AlOx tunneling barrier is by \ndefault  amorphous [4, 23] and it is extremely challenging to get crystalline AlOx  tunneling barrier  [8].   \nBased on the experimental studies and prior literature we hypothesized the following mechanism \nbehind the Ta effect on IFMEC. The reduced intensity of NiFe after the deposition of Ta may be due t o \nthe increased damping factor. We surmise that increase in damping may be associated with the creation of \na dead layer at NiFe/Ta interface [16, 17, 24]. In this paper, we also proposed a mechanism based on the \npresences of a dead layer at NiFe/Ta interface  [16, 17, 24]. Ta is found to  acqui re ~0.34 -0.56 µ B magnetic \nmoment when deposited on NiFe surface  14,15. These Ta atoms established antiferromagnetic  coupling \nwith the Ni atoms near the NiFe surface region. This antiferromagnetic coupling between the acquired \nmagnetic moment in Ta  and Ni atoms near the NiFe surface yielded a dead layer  of ~ 2 nm thickness  that 13 \n does not possess any net magnetic moment  [17]. Based on the  prior study14,15 the following mechanism is \nhypothesized  about the Ta effect on IFMEC .   \nOn a MTJ -Ta sample , the addition of Ta layer \nappears to gain magnetic moment by diminishing the spin \ndensity of NiFe  (Fig. 8). Since, Ta layer acquire s a net \nmagnetic moment14,15 hence it is imperative that Ta layer \ncan only pick majority or minority spin density  from NiFe , \nnot both. If it picks both type s of spins , then there may not \nbe any net magnetic moment as reported by previous \nstudies 14,15 and also seen i n our PNR study (Fig. 5c).   Furthermore, we conjecture that FMR  resonance \npeaks for the ferromagnets mainly depend  on the majority spins  only. In fact , FMR theoretical studies  \nhave mainly account ed for majority  types of spin [25, 26] and ignor ed minority spin population . It is also \nwell established that when two ferromagnets are coupled antiferromagnetically , then it means that the \nmajority spins of the two ferromagnets are antiparallel  to each other  (Fig. 8a). For a MTJ sample , with \nantiferromagnetic IFMEC, the majority spin density  (minority spin density) of the top NiFe electrode was \nantiparallel (parallel) to the majority spins of the bottom ferromagnetic electrode  (Fig. 8a). We \nhypothesize that after addition of Ta, a fraction of the NiFe’s majority spins move d into the Ta layer  (Fig. \n8b). As a result, Ni Fe’s major spin density depleted  and became lower than t he minor spin density (Fig. \n8b). Subsequently, the mi nor spin density of NiFe became  new major spin density due to the presence of \nTa (Fig.  8b). It is noteworthy that  NiFe’s  new majority spin after Ta addition is parallel to the majority \nspins of the bottom electrode of the resultant MTJ -Ta. This new configuration is tantamount to \nferromagnetic IFMEC on MTJ -Ta. Hence, the ferromagnetic coupling observed on MTJ -Ta is due to Ta \ninduced rearrange ment of the majority spin density of states  on the top NiFe electrode  (Fig.  8b). One can \nsee that Ta effect is not possible if minority spins from the top NiFe enter in the Ta layer. In that case , \nNiFe’s original majority spin remains unc hanged before and after the ad dition of Ta layer. As a result,  the \nFig.8: Ta effect on spin density of top \nferromagnetic (FM) electrode an d on the \nchange in IFMEC.  \n14 \n nature of IFMEC will also not change. Conceptually , Ta induced IFMEC reversal is only possible when \nmajority spins from the NiFe enter in the Ta layer.  This hypothesis is  in agreement with  the difference in \nthe intensities of the first peak for MTJ and MTJ -Ta. For MTJ -Ta the maximum intensity of the first peak \nis ~15191 around 598 Oe  (Fig. 2) . However, for MTJ  the intensity of the first peak is 62720 at 620 Oe  \n(Fig. 2). The addition of Ta appears to influence the population of spins responsible for producing the \nacoustic mode of the MTJ sample. According to FMR theory peak intensity is directly associated with the \nmagnetic moment of the ferromagnetic  electrode s[10, 14].  \nIV CONCLUSIONS:    \nThis paper discussed the effectiveness of Ta layer in changing the inter -ferromagnetic electrode \nexchange coupling. FMR was found to be us eful in recording the subtle changes due to the addition of Ta \nlayer. The magnetization study was only able to register very small change due to Ta. We also conducted \nneutron scattering studies on MTJ and MTJ -Ta sample s. These neutron studies observed  the moderate \nchange in the magnetic attributes in the top Ta layer and neighboring NiFe region.  We found that \nferromagnetic resonance is extremely sensitive for studying the effect of the change in electrode \ncomposition on the inter -electrode exchange coupling . We found that the ability to change the nature o f \ninter-electrode coupling of a  magnetic tunnel junction  can impact the resulting properties of the molecular \nspintronics devices. We observed that paramagnetic molecules decreased the magnetic moment of th e \nMTJ with pre -existing antiferromagnetic coupling. However, the same paramagnetic molecules increase d \nthe magnetic moment of the MTJ -Ta with preexisting ferromagnetic exchange coupling.  At present w e \nare unsure about the mechanism by which the  nature of Ta and NiFe interaction influences the IFMEC. \nFirst principle calculations are expected to shine light about the underlying mechanism.   \nSUPPLEMENTARY MATERIAL  \nFigure 1S showing the additional PNR results and Figure 2S showing the tunneling type transport via the \nAlOx tunnel barrier is provided in the supplementary material file.  15 \n ACKNOWLEDGEMENTS:   \nPawan Tyagi thank Dr. Bruce Hinds and Department of Chemical and  Materials engineering at the \nUniversity of Kentucky for facilitating experimental work on molecular spintronics during his Ph.D. \nOMC was produced Dr. Stephen Holmes’s group. The preparation of this paper and complementary \nexperiments were in part supporte d by National Science Foundation -Research Initiation Award (Contract \n# HRD -1238802), Department of Energy/ National Nuclear Security Agency (Subaward No. 0007701 -\n1000043016), and  Air Force Office of Sponsored Research (Award #FA9550 -13-1-0152). We also t hank \nCentre of Nanoscience and T echnology, NIST Gaithersburg for allowing the use of microscopy resou rces. \nWe also acknowledge Dr. Brian Kirby of NIST Center of Neutron Reflectivity for the polarized beam \nreflectivity study.  We also thank STEM C enter at UD C for providing the partial funding. Any opinions, \nfindings, and conclusions expressed in this material are those of the author(s) and do not necessarily \nreflect the views of any funding agency and corresponding author’s affiliations and collaborators.    \nREFERENCES:  \n[1] J.J.I. Wong, L. Ramirez, A.G. Swartz, A. Hoff, W. Han, Y. Li, R.K. Kawakami, Tailoring interlayer \nexchange coupling of ferromagnetic films across MgO with Fe nanoclusters, Phys. Rev. B, 81 (2010) \n094406.  \n[2] K. Baberschke, Why are spin  wave excitations all important in nanoscale magnetism?, Physica Status \nSolidi A -App. Res., 245 (2008) 174 -181.  \n[3] J. Linder, K. Baberschke, Ferromagnetic resonance in coupled ultrathin films, J. Phys. -Condens. \nMatter 15 (2003) S465 -S478.  \n[4] J.S. Mooder a, J. Nassar, G. Mathon, Spin -tunneling in ferromagnetic junctions, Ann. Rev. Mater. Sci., \n29 (1999) 381 -432.  \n[5] P. Tyagi, Multilayer edge molecular electronics devices: a review, J. Mater. Chem., 21 (2011) 4733 -\n4742.  \n[6] P. Tyagi, C. Baker, C. D'Angelo, Paramagnetic Molecule Induced Strong Antiferromagnetic Exchange \nCoupling on a Magnetic Tunnel Junction Based Molecular Spintronics Device, Nanotechnology, 26 (2015) \n305602.  \n[7] P. Tyagi, D.F. Li, S.M. Holmes, B.J. Hinds, Molecular electrodes at the exposed  edge of \nmetal/insulator/metal trilayer structures, J. Am. Chem. Soc., 129 (2007) 4929 -4938.  \n[8] J. -G.J. Zhu, C. Park, Magnetic tunnel junctions, Mat. Today, 9 (2006) 36 -45. \n[9] P. Tyagi, B.J. Hinds, Mechanism of Ultrathin Tunnel Barrier Failure Due to Mec hanical Stress Induced \nNano -Sized Hillocks and Voids, J. Vac. Sci. Technol. B, 28 (2010) 517 -521.  \n[10] A. Layadi, J.O. Artman, Study of antiferromagnetic coupling by ferromagnetic resonance (FMR), J. \nMag. Mag. Mat., 176 (1997) 175 -182.  16 \n [11] K. Baberschke, Magnetic anisotropy energy and interlayer exchange coupling in ultrathin \nferromagnets: Experiment versus theory, Philos. Mag., 88 (2008) 2643 -2654.  \n[12] J. Geshev, L.G. Pereira, J.E. Schmidt, Dependence of the ferromagnetic resonance modes on the \ncoupling strength in exchange -coupled trilayer structures, Physica B -Condensed Matter, 320 (2002) 169 -\n171.  \n[13] J. Lindner, K. Baberschke, In situ ferromagnetic resonance: an ultimate tool to investigate the \ncoupling in ultrathin magnetic films, J. Phy. -Cond. Mat.,  15 (2003) R193 -R232.  \n[14] A. Layadi, J.O. Artman, FERROMAGNETIC -RESONANCE IN A COUPLED 2 -LAYER SYSTEM, J. Mag. Mag. \nMat., 92 (1990) 143 -154.  \n[15] S.O. Demokritov, Biquadratic interlayer coupling in layered magnetic systems, J. Phys. D -Appl. Phys., \n31 (199 8) 925 -941.  \n[16] S. Kezilebieke, M. Ali, B. Shadeke, R. Gunnella, Magnetic properties of ultrathin Ni81Fe19 films with \nTa and Ru capping layers, J. Phy. -Cond. Mat., 25 (2013).  \n[17] M. Kowalewski, W.H. Butler, N. Moghadam, G.M. Stocks, T.C. Schulthess, K.J.  Song, J.R. Thompson, \nA.S. Arrott, T. Zhu, J. Drewes, R.R. Katti, M.T. McClure, O. Escorcia, The effect of Ta on the magnetic \nthickness of permalloy (Ni81Fe19) films, J. Appl. Phys., 87 (2000) 5732 -5734.  \n[18] P. Tyagi, E. Friebe, C. Baker, Addressing the c hallenges of using ferromagnetic electrodes in the \nmagnetic tunnel junction -based molecular spintronics devices, J.Nanoparticle Res., 17 (2015) 452.  \n[19] P. Tyagi, C. D'Angelo, C. Baker, Monte carlo and experimental magnetic studies of molecular \nspintronic s devices, NANO, 10 (2015) 1550056.  \n[20] D.F. Li, S. Parkin, G.B. Wang, G.T. Yee, R. Clerac, W. Wernsdorfer, S.M. Holmes, An S=6 cyanide -\nbridged octanuclear (Fe4Ni4II) -Ni-III complex that exhibits slow relaxation of the magnetization, J. Am. \nChem. Soc., 12 8 (2006) 4214 -4215.  \n[21] M.Y. Zhuravlev, E.Y. Tsymbal, A.V. Vedyayev, Impurity -assisted interlayer exchange coupling across \na tunnel barrier, Phys. Rev. Lett., 94 (2005) 026806.  \n[22] L. Xu, Z.S. Ma, D. Wei, Spin orbital torque driven magnetization switchin g in magnetic tunnel \njunction with inter -layer exchange coupling, J. App. Phys., 117 (2015).  \n[23] J.P. Velev, M.Y. Zhuravlev, K.D. Belashchenko, S.S. Jaswal, E.Y. Tsymbal, T. Katayama, S. Yuasa, \nDefect -mediated properties of magnetic tunnel junctions, IEEE  Trans. Magn., 43 (2007) 2770 -2775.  \n[24] M. Kim, W.T. Geng, A.J. Freeman, L.P. Zhong, J. Fernandez -de-Castro, First -principles calculations for \nthe structural and magnetic properties of ordered NiFe(001) thin films with and without a Ta overlayer, \nJ. App. Phys., 87 (2000) 5735 -5737.  \n[25] A. Layadi, Ferromagnetic resonance modes in coupled layers with cubic magnetocrystalline \nanisotropy, J. App. Phys., 83 (1998) 3738 -3743.  \n[26] J. Lindner, K. Baberschke, Ferromagnetic resonance in coupled ultrathin films, Jo urnal of Physics -\nCondensed Matter, 15 (2003) S465 -S478.  \n \n "    },    {        "title": "1902.02665v2.Unidirectional_anisotropy_in_cubic_FeGe_with_antisymmetric_spin_spin_coupling.pdf",        "content": "Dynamic unidirectional anisotropy in cubic FeGe\nwith antisymmetric spin-spin-coupling\nNicolas Josten1, Thomas Feggeler1, Ralf Meckenstock1, Detlef Spoddig1, Marina\nSpasova1, Ke Chai2, Iliya Radulov3, Zi-An Li2, Oliver Gutﬂeisch3, Michael Farle1, and\nBenjamin Zingsem1,4,*\n1Faculty of Physics and Center for Nanointegration (CENIDE), University Duisburg Essen, Duisburg, 47057,\nGermany\n2Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China\n3Department of Material- and Geosciences, Functional Materials, Technische Universit ¨at Darmstadt\n4Ernst Ruska Centre for Microscopy and Spectroscopy with Electrons and Peter Gr ¨unberg Institute,\nForschungszentrum J ¨ulich GmbH, 52425 J ¨ulich, Germany\n*Benjamin.Zingsem@uni-due.de\nABSTRACT\nStrong uni directional anisotropy in bulk polycrystalline B20 FeGe has been measured by ferromagnetic resonance spectroscopy.\nSuch anisotropy is not present in static magnetometry measurements. B20 FeGe exhibits inherent Dzyaloshinskii-Moriya\ninteraction, resulting in a nonreciprocal spin-wave dispersion. Bulk and micron sized samples were produced and characterized.\nBy X-band ferromagnetic resonance at 276K\u00061K, near the Curie temperature, a distribution of resonance modes was observed\nin accordance with the cubic anisotropy of FeGe. This distribution exhibits a uni directional anisotropy, i.e. shift of the resonance\nﬁeld under ﬁeld inversion, of KUD=960J=m3\u000610J=m3, previously unknown in bulk ferromagnets. Additionally, more than 25\nsmall amplitude standing spin wave modes were observed inside a micron sized FeGe wedge, measured at 293 K \u00062 K.\nThese modes also exhibit uni directional anisotropy. This effect, only dynamically measurable and not detectable in static\nmagnetometry measurements, may open new possibilities for directed spin transport in chiral magnetic systems.\nIntroduction\nNon-centrosymmetric crystal structures, such as the B20 phase of FeGe1, 2, can host chiral spin textures like magnetic\nskyrmions3, 4, which have been proposed as new structures for memory storage applications5at room temperature6. Chiral spin\nstructures in general are of signiﬁcant interest in current magnetic research7, 8. Dzyaloshinsky-Morya-interaction (DMI)9, 10\ncauses a chiral symmetry break of the magnetic interaction and inﬂuences the dynamic properties of the magnetic system. For\nexample, the spin wave dispersion becomes non-reciprocal11, 12, as experimentally conﬁrmed by Brillouin spectroscopy13and\nan additional phase shift between neighboring spins of a spin wave affects its resonance intensity14. The space group P 213of\nthe FeGe B20 phase has an inherent broken inversion symmetry, but does not impose chirality. The chirality, in this case, results\nfrom the speciﬁc atomic sites occupied by Fe and Ge inside the unit cell15. The magnetic properties of FeGe were studied\nusing the Mössbauer effect16, vibrating sample magnetometry17and magnetic susceptibility measurements18making FeGe a\nmagnetically well characterized material.\nIn the Heisenberg model of direct nearest neighbour interactions, spin waves (magnons) have a dispersion relation proportional\nto the wave vector ksquared. An antisymmetric contribution to spin-spin interaction results in an additional term in the\ndispersion relation proportional to k11, 12and therefore a shift with regard to the gamma point. Then spin waves propagating in\nopposite directions at the same frequency have different wavelengths leading to complex standing waves with a moving phase\nfront. This allows to detect modes, which would cancel and not be detectable in FMR19.\nWe measured the magnetodynamic properties of a millimeter-sized disk shaped and a micron-sized wedge shaped sample of\nB20 FeGe using ferromagnetic resonance (FMR)20, 21. Previous FMR measurements on this material22–25were performed\nwith millimeter sized single crystalline samples. Solving the Landau-Lifshitz-Gilbert equation (LLG)26, 27for an FMR like\nexcitation28, we determined magnetic material parameters in the usual way29.\nSample preparation\nStoichiometric FeGe was melted, using induction heating and, to guarantee homogeneity, re-melted twice and annealed for\n130 h at1000 K . Cylinders were formed and a high pressure high temperature synthesis inside a Kawai-type30MultianvilarXiv:1902.02665v2  [cond-mat.mtrl-sci]  1 Apr 2020B 90°\n0°B\n90°0° microresonator loop\nsample\n10 µma\nb\ndcf eb)\n a)\n2 µm190 K, 150 mTFigure 1. (a) Lorentz microscopy image at 190 K and 150 mT of an FeGe slice cut from the original sample using standard\nlift-off FIB. The magnetic ﬁelds points perpendicular to the sample. The black and white dots represent an ordered skyrmion\nlattice. (b) Scanning electron micrograph of the specimen inside an R-Type microresonator. The inset shows a schematic\nrepresentation of the geometry and the directions of the magnetic ﬁeld B during the experiment. (Dimension of the sample:\na = 11.3 \u00060:1 µm, b = 10.9 \u00060:1 µm, c = 5.9 \u00060:1 µm, d = 5.0 \u00060:1 µm, e = 0.9 \u00060:1 µm, f = 1.6 \u00060:1 µm).\nApparatus with Walker-type31module was applied. This resulted in 95 % polycrystalline B20 FeGe, conﬁrmed by X-ray\ndiffraction. A maximum of 5 %of the sample material could consist of secondary phase Iron Germanium. Energy-dispersive\nX-ray spectroscopy (EDX) measurements also reveal local composition variations with accumulation of iron (Fe:Ge 55:45).\nFurther investigations with Lorentz microscopy show the formation of helices and skyrmions (ﬁg. 1 (a)) in accordance to32.\nMicron sized samples (ﬁgure 1 (b)) with wedge shaped geometries were cut using a Focused Ion Beam (FIB -– FEI Helios\nnanolab 600) and placed inside an R-Type microresonator33–35using standard lift-off FIB (Omniprobe manipulator with Pt gas\ninsertion system) technique. During the lift-off process a carbon coating with up to 100 nm thickness and up to 15 % platinum\ncontamination36could not be avoided. Furthermore, the lift-off process used Gallium as cutting ions and resulted in a localized\ndeposition of a maximum of 2 :6 % of Ga (as veriﬁed by EDX).\nExperimental\nFMR spectra of a bulk polycrystalline, nearly disc shaped piece of FeGe with a diameter of 3:78 mm and a thickness of\n0:78 mm (2(b) inset) was acquired in a range of 800 mT to0 mT at a frequency of 9:517 GHz \u00060:006 GHz . The ﬁeld was\napplied at angles of 0\u000eto180\u000ein steps of 0:5\u000efrom out-of-plane to in-plane and to the opposite out-of-plane orientation. The\nmeasurement can be seen in ﬁg.2 (a) shown as an amplitude contour plot. The temperature is 276 K\u00061 K, which is below the\nCurie Temperature of T C=280K18, where the sample is ferromagnetic32. The angular precision of our experimental setup is\nbetter than 0 :05\u000eand the precision of the magnetic ﬁeld is better than 0 :5 mT with a relative precision of 0 :005 mT.\nResonance lines in the FMR spectra are identiﬁed by a successive local maximum and minimum amplitude. We observe a\ndistribution of resonances, which is in agreement to previous FMR investigations22of single crystalline FeGe. Each crystallite\nin the sample is contributing to this resonance distribution. They are all inﬂuenced by the applied external ﬁeld and the\ndemagnetization ﬁeld in the sample, due to its general shape. However, their resonance ﬁelds vary with respect to the applied\nmagnetic ﬁeld due to the different symmetry axis of the cubic anisotropy in each crystallite. We simulated the resonance\ndistribution using the known magnetocrystalline anisotropy of FeGe22and a random orientation of crystallites and compared it\nto the measurement. This can be found in the supplementary sec. S1. Figure 2 (a) shows the differentiated angular dependent\nFMR spectra as a grey scale contour plot. The out-of-plane orientations are depicted in detail in ﬁg. 2 (b). The resonance line\nexhibits a unidirectional anisotropy, indicated by a difference in the positon of maximum microwave absorption comparing 0\u000e\nand180\u000e. A similar anisotropy is observed in systems with exchange-bias38. Hence we performed additional magnetometry\nmeasurements, to exclude the presence of exchange bias in our system (ﬁg. 3). No such anisotropic behaviour is observed\n2/9200 400 600 80004590135\nMagnetic Field [mT]Angle [°]0 0.25 0.5 0.75 1Amplitude [arb. units] \n0 200 400 600 8000.00.20.40.6\nMagnetic Field [mT]Amplitude [arb. units]a)\nb)\n180°\n0°~30 mT  diﬀerence180°\n0°\nB0°\nB180°B90°resonance position\nmain mode edge modeFigure 2. (a) Angular dependent out-of-plane (differentiated) FMR spectra shown as an amplitude contour plot at 276 K \u0006\n1 K and f Microwave = 9:517 GHz \u00060:006 GHz. The yellow line marks the angular dependent resonance ﬁeld position. The\ndotted white lines mark the position of the hard direction at 0\u000eand180\u000e. They have been extended to the middle of the ﬁgure\nfor better comparison of the 30 mT ﬁeld difference. (b) shows the spectra of the same FMR measurement at q=0\u000eand\nq=180\u000e. The resonance spectra at 180\u000eappears to consist of two resonance lines. This is due to edge resonances inside the\nsample24, 37. Additionally a schematic representation of the sample can be seen with the most important ﬁeld positions marked.\n3/9-0.6 -0.4 -0.2 0.2 0.4 0.6\nB [T]-2-112M [10 ⁵ A/m] \n-0.02 -0.01 0.01 0.02\nB [T]-0.4-0.20.20.4M [10 ⁵ A/m]T = 276 K  270 280 290 300\nT [K]0.511.5M [10 ⁵ A/m]a) \nb) \n00\n00Figure 3. (a) Temperature dependent measurement of the magnetization at 310 mT external ﬁeld. A part of the original\nsample was used for the measurement. (b) Hysteresis loop measured by vibrating sample magnetometry at 276 K . The sample\nis the same as in (a). The Magnetisation M is plotted against the magnetic ﬁeld B. The hysteresis shows no asymmetry or\nexchange bias.\n4/9200 300 400 500 600-90-4504590\nMagnetic Field [mT ]Angle [°]00.25 0.5 0.75 1Ampli tude [arb . unit]Figure 4. A grey scale contour plot of the (differentiated) FMR signal amplitude of the micron sized wedge shaped sample as\na function of applied magnetic ﬁeld for different orientations of the magnetic ﬁeld between \u000093\u000eand99\u000e(compare ﬁg. 1 (b))\nat9:134 GHz \u00060:006 GHz . The scale bar is depicted on the left. The dotted white lines indicate angular dependent resonance\nﬁelds of spin wave modes.\nin static magnetometry using vibrating sample magnetometry (VSM). We therefore conclude, that this anisotropy must be\ndynamically induced under resonant excitation. Note that it cannot be equated with the linear contribution to the spin wave\ndispersion, as this changes directionality in accordance with the magnetic ﬁeld direction. Due to the skin depth of approximately\n10\u00003mm22one must, to fully reproduce the FMR lineshape, solve the non-uniform LLG24taking the shape of the sample\ninto account. However, we show exemplary in the supplementary sec. S1 that a Dysonian lineshape39, 40and the known\nmagnetocrystalline anisotropy of FeGe17are able to reproduce the measured FMR lineshape satisfyingly, which is sufﬁcient\nfor our needs. The position of resonance was obtained by subtracting the background and locating the zero crossing of the\nresonance line. We analyzed the angular dependent spectra using eq. 1 as a model for the free energy density F. To account\nfor the observed unidirectional symmetry in the angular dependent resonance ﬁeld position, an additional unidiretional ﬁeld\ncontribution needs to be introduced. In this model an additional anisotropy ﬁeld BU=KUD=Mis used. This unidirectional\ncontribution is merely a descriptive model to account for the observed phenomenon. It cannot be seen as an additional\nmagnetocristalline anisotropy but rather as an emergent symmetry contribution which arises under dynamic excitation. In the\nsupplementary sec. S2 the shape of such a unidirectional free energy density is shown. Additionally, a demagnetization and\nZeeman term are considered.\nF=\u0000KUD\u0001cos(q) +m0\n2\u0001~M\u0001N\u0001~M\u0000~M\u0001~B (1)\nThe demagnetisation tensor N(Nzz=0:676;Nxx;yy=0:162) was deduced, using the demagnetisation tensor of a cylinder as\ndescribed in41. The known g-factor of FeGe (g = 2.07)23was used. qis the out-of-plane angle of the magnetisation Mand\nBthe external magnetic ﬁeld. Additionally, the magnetisation Mis considered as a ﬁt parameter. The obtained parameters\nareKUD=960J=m3\u000610J=m3,M=82580Am\u00001\u0006200Am\u00001. The magnetization matches the magnetization measured by\nVSM at 281 K ,5 Kabove the temperature measured by a sensor below the sample. This offset is likely due to microwave\nheating. Following23, we assume this to be the uniform FMR mode of cubic FeGe.\nFigure 4 shows the angular dependent FMR spectra ( 293 K\u00062 K,fMicrowave =9:134 GHz \u00060:006 GHz ) of the wedge shaped\nFeGe sample (ﬁg. 1 (b)) measured inside a microresonator as a grey scale contour plot. Multiple resonances are visible in\nthe spectra, which exhibit anisotropic behavior. The anisotropy is directed such that the resonance ﬁeld increases when the\n5/9-0.2-0.10.00.10.20.3Amplitude [arb. unit]\n-0.2-0.10.00.10.20.3a) b)\n300 320 340 360 380 400 420 440-0.2-0.10.00.10.20.3\nMagnetic Field [mT]Amplitude [arb. unit]\n300 320 340 360 380 400 420 440-0.2-0.10.00.10.20.3\nMagnetic Field [mT]\n200 300 400 500 600 700 800 900-0.100.1\nMagnetic Field [mT]ΔA [arb. unit]\n200 300 400 500 600 700 800 900-0.100.1\nMagnetic Field [mT]B B\n960 9801000 1020 1040 1060 1080-0.100.1\nMagnetic Field [mT]Intensit y [arb . unit]\nDiﬀerence between Amplitude Diﬀerence between Amplitude\nAmplitude [arb. unit] ΔA [arb. unit]\nAmplitude [arb. unit]i) i)\nii) ii)iii) iii)\niv) iv)v)Figure 5. Analysis of the bidirectional (differentiated) FMR measurements ( \u0018293K ) at81\u000eand3:587 GHz \u00060:006 GHz , of\nthe specimen shown in ﬁg. 1 (b). (a,ii) and (b,ii) comparison between two different (differentiated) FMR measurements, the\nformer with the same magnetic ﬁeld direction, the latter at opposite ﬁeld directions. (a,i) and (b,i) show a schematic\nrepresentation of the sample and the respective magnetic ﬁeld directions of the compared measurements. (a,iii) and (b,iii)\ndepict the highlighted areas in (a,ii) and (b,ii) in detail. (a,iv) and (b,iv) are the plotted differences between the compared\nmeasurements. (a,v) noise ﬂoor of the measurement in a magnetic ﬁeld region without resonances.\n6/9static ﬁeld is applied parallel to the long (dipolar-easy) axis of the sample. This suggests that these modes are spinwaves with\nenergies below that of the gamma point (FMR mode), which may be induced by strong dipolar coupling42or DMI. Around\n\u000690\u000e, the number of superimposing resonances and the complex mode intesity distribution19make it difﬁcult to separate\nindividual lines. We assume that these resonances arise due to geometrical conﬁnement of the modes in our specimen (ﬁg.\n1). Consequently, the inclined surface of our wedge results in different geometrical boundary conditions at the same time.\nBidirectional measurements along the \u000681\u000edirection as shown in ﬁg. 5, however, reveal a clear unidirectional shift of the\nresonances under ﬁeld reversal. Figure 5 (a) shows the reproducibility of resonances for ﬁeld sweep up and ﬁeld sweep down,\nwhereas ﬁg. 5 (b) illustrates that under ﬁeld reversal, the resonance position of the spinwaves has shifted. Hence, we ﬁnd a\nunidirectional anisotropy. Figure 5 (c) shows the noise ﬂoor of our spectrometer in a ﬁeld region where no resonances are\nobserved.\nConclusion\nFrom angular dependent ferromagnetic resonance we ﬁnd an unexpected dynamic unidirectional anisotropy (ﬁg. 2) in the\nmagnetic excitation of FeGe just below the Curie temperature. This anisotropy is of a dynamic character, since it is not\ndetectable in static hysteresis measurements. The magnitude of the unidirectional anisotropy of the bulk resonance line is\nKUD=960J=m3\u000610J=m3. Spin waves, detected at 293 K\u00062 Kfor sample sizes with micrometer dimensions, also exhibit\nunidirectional anisotropy (ﬁg. 5).\nMethods\nA conventional Bruker X-band FMR spectrometer was used for FMR measurements on the millimetre sized FeGe sample (see\nﬁg. 2) inside a cylindrical TE 011cavity. FMR measurements on the micron sized FeGe sample (ﬁgure 1 (b), ﬁg. 4 and ﬁg. 5)\nwere performed inside an R-Type microresonator33–35. The resonator was connected to a Varian E102 microwave bridge. The\nmodulated microwave reﬂection was recovered using a SRS SR830DSP lock-in ampliﬁer.\nReferences\n1.Richardson, M. The Partial Equilibrium Diagram of the Fe-Ge System in the Range 40-72 at. % Ge, and the Crystallisation\nof some Iron Germanides by Chemical Transport Reactions. Acta Chem. Scand. 21, 2305–2317, DOI: https://doi.org/10.\n3891/acta.chem.scand.21-2305 (1967).\n2.Richardson, M. Crystal Structure Reﬁnements of the B 20 and Monoclinic (CoGe-type) Polymorphs of FeGe. Acta Chem.\nScand. 21, 753–760, DOI: https://doi.org/10.3891/acta.chem.scand.21-0753 (1967).\n3.Li, Z.-A. et al. Magnetic Skyrmions in an FeGe Nanostripe Revealed by in situ Electron Holography , 974–975 (American\nCancer Society, 2016). https://onlinelibrary.wiley.com/doi/pdf/10.1002/9783527808465.EMC2016.6263.\n4.Mühlbauer, S. et al. Skyrmion lattice in a chiral magnet. Science 323, 915–919, DOI: https://doi.org/10.1126/science.\n1166767 (2009). http://science.sciencemag.org/content/323/5916/915.full.pdf.\n5.Romming, N. et al. Writing and deleting single magnetic skyrmions. Science 341, 636–639, DOI: https://doi.org/10.1126/\nscience.1240573 (2013). http://science.sciencemag.org/content/341/6146/636.full.pdf.\n6.Woo, S. et al. Observation of room-temperature magnetic skyrmions and their current-driven dynamics in ultrathin metallic\nferromagnets. Nat. Mater. 15, 501 EP – (2016).\n7.von Bergmann, K., Kubetzka, A., Pietzsch, O. & Wiesendanger, R. Interface-induced chiral domain walls, spin spirals\nand skyrmions revealed by spin-polarized scanning tunneling microscopy. J. Physics: Condens. Matter 26, 394002, DOI:\nhttps://doi.org/10.1088/0953-8984/26/39/394002 (2014).\n8.Zhang, S. L., van der Laan, G. & Hesjedal, T. Direct experimental determination of spiral spin structures via the dichroism\nextinction effect in resonant elastic soft x-ray scattering. Phys. Rev. B 96, 094401, DOI: https://doi.org/10.1103/PhysRevB.\n96.094401 (2017).\n9.Dzyaloshinsky, I. A thermodynamic theory of “weak” ferromagnetism of antiferromagnetics. J. Phys. Chem. Solids 4,\n241–255, DOI: https://doi.org/10.1016/0022-3697(58)90076-3 (1958).\n10.Moriya, T. Anisotropic superexchange interaction and weak ferromagnetism. Phys. Rev. 120, 91–98, DOI: https:\n//doi.org/10.1103/PhysRev.120.91 (1960).\n11.Melcher, R. L. Linear contribution to spatial dispersion in the spin-wave spectrum of ferromagnets. Phys. Rev. Lett. 30,\n125–128, DOI: https://doi.org/10.1103/PhysRevLett.30.125 (1973).\n7/912.Moon, J.-H. et al. Spin-wave propagation in the presence of interfacial dzyaloshinskii-moriya interaction. Phys. Rev. B 88,\n184404, DOI: https://doi.org/10.1103/PhysRevB.88.184404 (2013).\n13.Di, K. et al. Direct observation of the dzyaloshinskii-moriya interaction in a Pt/Co/Ni ﬁlm. Phys. Rev. Lett. 114, 047201,\nDOI: https://doi.org/10.1103/PhysRevLett.114.047201 (2015).\n14.Puszkarski, H. & Wigen, P. E. Effect of Dzialoshinsky-Moriya Interactions on Propagation of Spin Waves in Ferromagnets:\nDynamical Canting. Phys. Rev. Lett. 35, 1017–1018, DOI: https://doi.org/10.1103/PhysRevLett.35.1017 (1975).\n15.Ullah, A. et al. Crystal structure and dzyaloshinski–moriya micromagnetics. IEEE Transactions on Magn. 55, 1–5, DOI:\nhttps://doi.org/10.1109/TMAG.2018.2890028 (2019).\n16.Wäppling, R., Häggström, L. & Karlsson, E. Magnetic properties of FeGe studied by mössbauer effect. Phys. Scripta 2,\n233 (1970).\n17.Ludgren, L., Beckman, O., Attia, V ., Bhattacheriee, S. P. & Richardson, M. Helical spin arrangement in cubic FeGe. Phys.\nScripta 1, 69–72, DOI: https://doi.org/10.1088/0031-8949/1/1/012 (1970).\n18.Lundgren, L., Blom, K. & Beckman, O. Magnetic susceptibility measurements on cubic FeGe. Phys. Lett. A 28, 175–176,\nDOI: https://doi.org/10.1016/0375-9601(68)90184-9 (1968).\n19.Zingsem, B., Farle, M., Stramps, R. L. & Camley, R. E. The unusual nature of conﬁned modes in a chiral system.\narXiv:1609.03417 [cond-mat.str-el] (2016). https://arxiv.org/abs/1609.03417.\n20.Farle, M. Ferromagnetic resonance of ultrathin metallic layers. Reports on Prog. Phys. 61, 755 (1998).\n21.Lindner, J., Meckenstock, R. & Farle, M. Applications of Ferromagnetic Resonance , 1–20 (American Cancer Society,\n2012). https://onlinelibrary.wiley.com/doi/pdf/10.1002/0471266965.com130.\n22.Haraldson, S., Björn, L., Beckman, O. & Smith, U. Magnetic resonance in cubic FeGe. J. Magn. Reson. (1969) 8, 271–273,\nDOI: https://doi.org/10.1016/0022-2364(72)90009-1 (1972).\n23.Haraldson, S., Pettersson, L. & Bhagat, S. Frequency and temperature dependence of spin resonances in cubic FeGe. J.\nMagn. Reson. (1969) 32, 115–120, DOI: https://doi.org/10.1016/0022-2364(78)90081-1 (1978).\n24.Haraldson, S. & Smith, U. The microwave resonance line shape in ferromagnetic cubic FeGe. J. Phys. Chem. Solids 35,\n1237–1245, DOI: https://doi.org/10.1016/S0022-3697(74)80147-2 (1974).\n25.Smith, U. & Haraldson, S. Spin-wave resonances in cubic FeGe. J. Magn. Reson. (1969) 16, 390–401, DOI: https:\n//doi.org/10.1016/0022-2364(74)90220-0 (1974).\n26.Landau, L. & Lifshitz, E. Anomalous high-frequency resistance of ferromagnetic metals. Phys. Z. Sowjetunion 8, 153–169\n(1935).\n27.Gilbert, T. L. A phenomenological theory of damping in ferromagnetic materials. IEEE Transactions on Magn. 40,\n3443–3449, DOI: https://doi.org/10.1109/TMAG.2004.836740 (2004).\n28.Zingsem, B. W., Winklhofer, M., Meckenstock, R. & Farle, M. Uniﬁed description of collective magnetic excitations.\nPhys. Rev. B 96, 224407, DOI: https://doi.org/10.1103/PhysRevB.96.224407 (2017).\n29.Smit, J. & Beljers, H. G. Ferromagnetic resonance absorption in BaFe12O19, a highly anisotropic crystal. Philips Res.\nReports 10, 113–130 (1955).\n30.Kawai, N. & Endo, S. The generation of ultrahigh hydrostatic pressures by a split sphere apparatus. Rev. Sci. Instruments\n41, 1178–1181, DOI: https://doi.org/10.1063/1.1684753 (1970). https://doi.org/10.1063/1.1684753.\n31.Walker, D., Carpenter, M. A. & Hitch, C. M. Some simpliﬁcations to multianvil devices for high pressure experiments. Am.\nMineral. 75, 1020 (1990).\n32.Wilhelm, H. et al. Precursor phenomena at the magnetic ordering of the cubic helimagnet FeGe. Phys. Rev. Lett. 107, DOI:\nhttps://doi.org/10.1103/PhysRevLett.107.127203 (2011).\n33.Banholzer, A. et al. Visualization of spin dynamics in single nanosized magnetic elements. Nanotechnology 22, 295713\n(2011).\n34.Narkowicz, R., Suter, D. & Niemeyer, I. Scaling of sensitivity and efﬁciency in planar microresonators for electron spin\nresonance. Rev. Sci. Instruments 79, 084702, DOI: https://doi.org/10.1063/1.2964926 (2008).\n35.Narkowicz, R., Suter, D. & Stonies, R. Planar microresonators for epr experiments. J. Magn. Reson. 175, 275 – 284, DOI:\nhttps://doi.org/10.1016/j.jmr.2005.04.014 (2005).\n8/936.Botman, A., Mulders, J. J. L., Weemaes, R. & Mentink, S. Puriﬁcation of platinum and gold structures after electron-beam-\ninduced deposition. Nanotechnology 17, 3779 (2006).\n37.David, P. & Heath, M. Further comments on the shape dependent demagnetizing ﬁeld in ferromagnetic resonance. J. Phys.\nC: Solid State Phys. 4, L282–L286, DOI: 10.1088/0022-3719/4/13/011 (1971).\n38.Stoecklein, W., Parkin, S. S. P. & Scott, J. C. Ferromagnetic resonance studies of exchange-biased permalloy thin ﬁlms.\nPhys. Rev. B 38, 6847–6854, DOI: https://doi.org/10.1103/PhysRevB.38.6847 (1988).\n39.Dyson, F. J. electron spin resonance absorption in metals. II. theory of electron diffusion and the skin effect. Phys. Rev. 98,\n349–359, DOI: https://doi.org/10.1103/PhysRev.98.349 (1955).\n40.Joshi, J. P. & Bhat, S. On the analysis of broad dysonian electron paramagnetic resonance spectra. J. Magn. Reson. 168,\n284 – 287, DOI: https://doi.org/10.1016/j.jmr.2004.03.018 (2004).\n41.Beleggia, M., Graef, M. D. & Millev, Y . T. The equivalent ellipsoid of a magnetized body. J. Phys. D: Appl. Phys. 39, 891\n(2006).\n42.Van Kranendonk, J. & Van Vleck, J. H. Spin waves. Rev. Mod. Phys. 30, 1–23, DOI: https://doi.org/10.1103/RevModPhys.\n30.1 (1958).\nAcknowledgements\nWe like to thank Norimasa Nishiyama and Shrikant Bhat for the support at the sample synthesis and Igor Barsukov and\nKonstantin Skokov for fruitful discussions.\nB Z acknowledges that the research leading to these results has received funding from the European Research Council\nunder the European Union’s Seventh Framework Programme (FP7/2007-2013)/ ERC grant agreement number 320832.\nT F acknowledges ﬁnancial support by German Research Foundation (DFG project: OL513/1-1) and the Austrian Science\nFund (FWF project: I 3050-N36).\nI R and O G acknowledge ﬁnancial support by the German federal state of Hessen through its excellence program LOEWE\n\"RESPONSE\".\nAuthor contributions statement\nN J and T F performed magnetic resonance experiments with support from B Z and R M. Z L and K C performed electron\nmicroscopy. I R and D S prepared the samples and perfomed analytic characterizations with help from O G. M S performed\nmagnetometry. B Z and T F concieved and planned the experiments. B Z and N J analysed the data and wrote the manuscript\nwith support from T F and M F. B Z encouraged N J to investigate the presented samples and supervised the ﬁndings of this\nwork. All authors discussed the results and contributed to the ﬁnal manuscript. B Z supervised the project.\nAdditional information\nThere are no competing interests\n9/9"    },    {        "title": "1804.06796v1.Ferromagnetic_resonance_linewidth_in_coupled_layers_with_easy_plane_and_perpendicular_magnetic_anisotropies.pdf",        "content": "arXiv:1804.06796v1  [cond-mat.mes-hall]  18 Apr 2018Ferromagnetic resonance linewidth in coupled layers with e asy-plane and\nperpendicular magnetic anisotropies\nJun-Wen Xu,1Volker Sluka,1,∗Bartek Kardasz,2Mustafa Pinarbasi,2and Andrew D. Kent1,†\n1Center for Quantum Phenomena, Department of Physics,\nNew York University, New York, NY 10003, USA\n2Spin Transfer Technologies, Inc., Fremont, California 945 38, USA\n(Dated: April 19, 2018)\nMagnetic bilayers with diﬀerent magnetic anisotropy direc tions are interestingfor spintronic appli-\ncations as they oﬀer the possibility to engineer tilted remn ant magnetization states. We investigate\nthe ferromagnetic resonance (FMR) linewidth of modes assoc iated with two interlayer exchange-\ncoupled ferromagnetic layers, the ﬁrst a CoNi multilayer wi th a perpendicular magnetic anisotropy,\nand the second a CoFeB layer with an easy-plane anisotropy. F or antiferromagnetic interlayer ex-\nchange coupling, elevated FMR linewidths are observed belo w a characteristic ﬁeld. This is in\ncontrast to what is found in uncoupled, ferromagnetically c oupled and single ferromagnetic layers\nin which the FMR linewidth increases monotonically with ﬁel d. We show that the characteristic\nﬁeld at which there is a dramatic increase in FMR linewidth ca n be understood using a macrospin\nmodel with Heisenberg-type exchange coupling between the l ayers.\nI. INTRODUCTION\nMagnetic multilayersarewidely used in spintronics de-\nvices for information processing and storage [1]. An es-\nsential means for engineering the coupling between two\nferromagnetic (FM) layers is by interlayer exchange cou-\npling (IEC) mediated by an intervening nonmagnetic\n(NM) layer. The sign and strength of the IEC changes\nwiththethicknessoftheNM;typicallythecouplingoscil-\nlates between FM and AFM while the coupling strength\ndecreases with increasing interlayer thickness [2].\nOne device concept of interest at present is spin\ntransfer torque (STT) magnetic random access memory\n(MRAM) [3]. It oﬀers a high density, non-volatile, short\nswitching time and low power consumption information\nstorage method. The basic structure of the memory cell\nconsists of a magnetic tunnel junction, two FM layers\nseparated by a thin insulating barrier layer. In the con-\nventional layer stacks used in MRAM design, the stable\nmagnetic conﬁgurations exhibit collinear magnetization\nalignment. While these alignments provide maximum\ncontrast in terms of magnetoresistance, and thus optimal\nread-out conditions, such alignment has disadvantages\nfrom the perspective of STT, since the torque exerted is\nproportional to the cross product of the magnetization\nvectors [4]. In other words, small initial deviations from\nperfect alignment are needed to initiate STT-induced\nswitching. Such deviations are present due to thermal\nﬂuctuations, however, the resulting initial torques are\nsmall. For this reason, there is interest in devices with\nnon-collinear magnetic conﬁgurations. Furthermore, it is\nimportant to have a better understanding of the mag-\nnetic relaxation of the ferromagnetic resonance (FMR)\nmodes in such structures, since the time scale of relax-\n∗vs1568@nyu.edu\n†andy.kent@nyu.eduation determines the switching speed of the system. Spin\nrelaxation is described by the phenomenological Gilbert\ndamping and FMR is a convenient way to measure the\ndamping directly from the spectrum [5, 6].\nII. EXPERIMENTAL SETUP\nIn order to investigate magnetization dynamics\nin a system with non-collinear magnetic conﬁgura-\ntions, we have conducted experiments on samples\ncontaining two FM layers exhibiting diﬀerent forms\nof the magnetic anisotropy: one with easy-plane\nanisotropy, the other one with perpendicular magnetic\nanisotropy (PMA). As the PMA material we use a\nCoNi multilayer, while a Co 60Fe20B20(CoFeB) ﬁlm\nserves as the easy-plane magnetic layer. Both layers\nare coupled via IEC by a Ru interlayer: Using dc\nmagnetron sputtering, [Ta(5)/Cu(3)/Ni(0.65)/Co(0.3)/\n[Ni(0.6)/Co(0.2)] 5/Co(0.18)/Ru( tRu)/CoFeB(3)/Ta(3)]\nsamples were deposited on 150mm oxidized silicon\nwafer. In order to investigate diﬀerent strengths and\nsigns of coupling between the CoNi and CoFeB lay-\ners, the thickness of Ru ( tRu), was varied along one\ncoordinate on the wafer in a wedge-like manner, tRu\nranges from 0.71 to 1 .17nm across the wafer. For the\nFMR experiment, we cut 2 ×2mm2-sized pieces from\nthe wafer, where the lateral distance between pieces\nis 8mm. Therefore, to a good approximation, we can\ntreat the samples as having uniform Ru thickness within\neach piece. Besides these coupled-layer samples, we\nalso studied two samples with a single CoNi FM layer\n[Ta(5)/Cu(3)/Ni(0.65)/Co(0.3)/[Ni(0.6)/Co(0.2)] 5/\nCo(0.18)/Ru(3)] and a single CoFeB FM layer [Ta(5)/\nCu(3)/Ru(1)/CoFeB(3)/Ta(3)], as reference.\nIn order to measure the FMR spectra, we apply a dc\nmagnetic ﬁeld parallel to the sample plane. The sample\nis mounted on a waveguide; an ac current through the2\nT a 3nm\nCoFeB 3nm\nRu (t)\nPMA\nCo/Ni\nT a 5nmCu 3nmCo 0.18nm\nCo 0.2nm\nCo 0.3nm\nNi 0.65nmNi 0.6nmx5(a)\nhPort 1\nPort 2H(b)\n(c)\nFigure 1. (a) Multilayer sample, with varying Ru thickness.\n(b) Schematic of FMR measurement using a two-port VNA.\nHis the dc applied magnetic ﬁeld and his ac Oersted ﬁeld.\n(c) Example normalized real and imaginary S-parameter data\nversusappliedﬁeldfor asamplewith tRu= 0.99nmat17GHz\nac frequency.\nwaveguide provides a small oscillating Oersted ﬁeld per-\npendicular to the dc applied ﬁeld and drives the magne-\ntization into a small angle precession, which has its max-\nimum amplitude when the resonance condition is satis-\nﬁed. Weuseavectornetworkanalyzer(VNA)tomeasure\nthe magnetization dynamics by determining the eﬀective\nﬁeld- and frequency-dependent load the sample adds to\nthe waveguide. Fixing the frequency of the ac ﬁeld, the\ndc magnetic ﬁeld µ0His swept from 0 to 1T. We collect\nthe transmission ( S12) and reﬂection ( S11andS22) spec-\ntra with a VNA [7, 8]. For each spectrum, we extract\ndata from the S-parameter providing the highest signal\ntonoiseratio. Byﬁtting both realand imaginarypartsof\nthe S-parameter with a linear combination of symmetric\nand antisymmetric Lorentzian functions, we obtain the\npeak position and full width at half maximum (FWHM),\nwhich correspond to the resonance ﬁeld and linewidth,\nrespectively. This analysis is done for the real and imag-\ninary parts of the chosen S parameter, and the resulting\nvalues for peak position and FWHM are averaged and\nplotted as a function of the applied ﬁeld (Fig. 2).\nIII. EXPERIMENTAL RESULTS\nThe multilayer samples used in the measurement are\ntaken from the same wafer used in previous research [9].\nBased on the previous results, the samples with a Ru\nthickness of 1 .02nm and smaller exhibit AFM coupling\nbetween the layers while for the samples with Ru thick-\nness 1.05nm and larger, the IEC is ferromagnetic. IntRu(nm) 0.95 0.99 1.02 1.05 1.09\nJ(mJ/m2)-0.330 -0.270 -0.116 0.249 0.365\nµ0Hk1(T)0.186 0.179 0.142 0.204 0.181\nµ0Hk2(T)-0.904 -0.911 -0.948 -0.886 -0.909\nTable I. Sample parameters determined in Ref. [9]. tRuis the\nRulayer thickness and Jis the IEC. µ0Hk1andµ0Hk2are the\nCoNi and CoFeB eﬀective perpendicular magnetic anisotropy\nﬁelds, respectively.\norder to investigate how the coupling aﬀects the FMR\nlinewidth, sampleswithRuthicknessesof0.95,0.99,1.02,\n1.05 and 1 .09nm were studied. Their coupling strength\nJis shown in Table I.\nDue to the presence of two FM layers, there exist two\nFMR modes. In the limit of vanishing IEC, these modes\nwill become independent FMR modes of the CoNi and\nCoFeB layers. At high in-plane applied ﬁelds deeply in\nthe saturated regime, the CoFeB mode will have a sig-\nniﬁcantly higher frequency than the CoNi mode. This\nhierarchy is maintained for non-zero IEC, when the two\nmodes hybridize. Therefore, for simplicity, throughout\nthis paper, we will refer to the two modes as the CoNi\nand the CoFeB modes, even in the case of nonzero cou-\npling.\nThe IEC leads to contributions to the eﬀective ﬁeld\nacting on each layer. For high enough applied ﬁelds,\nwhen the layers are saturated, the IEC therefore leads to\na shift in the resonanceﬁelds. Figure 2(a) shows the rela-\n(a)\n(b)\ntRu(nm)\nFigure 2. In-plane FMR measurement of the samples listed\nin Table I. (a) Resonance position of the CoNi mode for dif-\nferent interlayer thicknesses. The dashed lines indicate t he\nintercept of a linear ﬁt to the data with the x-axis, which\nrepresents the in-plane saturation ﬁeld. With increasing I EC,\nthe saturation ﬁeld decreases. (b) Linewidth of the CoFeB\nmode. When the layers are FM coupled, the linewidth de-\ncreases monotonically as the ﬁeld decreases. However, when\nthe layers are AFM coupled, the linewidth ﬁrst decreases wit h\ndecreasing ﬁeld but then increases when the applied ﬁeld is\nless than the saturation ﬁeld.3\ntion between applied ﬁeld and resonancefrequency of the\nCoNi mode. Since CoNi has perpendicular anisotropy,\nwhen the applied ﬁeld is small its magnetization is not\nsaturated in the applied ﬁeld direction. The saturation\nﬁelds correspond to the intercepts of the data sets shown\nin Fig.2(a) with the horizontal axis. The shift of the\nsaturation ﬁeld for diﬀerent samples can be understood\nin the following way: due to the IEC the two magnetic\nlayers experience a coupling ﬁeld from each other. Con-\nsidering the saturated case, for FM coupling, layer iex-\nperiencesanexchangecouplingﬁeldinthesamedirection\nas the magnetization of layer j, while in the case of AFM\ncoupling, the coupling ﬁeld points in the opposite direc-\ntion with respect to the magnetization of layer j. Con-\nsistent with this picture, Fig. 2(a) shows an increasing\nsaturation ﬁeld when the coupling becomes more AFM.\nFigure2(b) displays the linewidth of the CoFeB mode\nas a function of the applied ﬁeld. In the case of FM cou-\npling, the linewidth increases with increasing frequency\nand increasing resonance ﬁeld, which is similar to what\nwould be observed in a single easy-plane layer in in-plane\nﬁeld [10]. However, the ﬁgure also shows that in AFM\ncoupled cases, when going from high to low ﬁelds, be-\nlow a certain sample-dependent characteristic ﬁeld, the\nlinewidth increases signiﬁcantly. This behavior is very\ndiﬀerent from what is typically observed for single lay-\ners.\nIV. MODELING AND SIMULATION\nThe net magnetic ﬁeld acting on a uniformly magne-\ntized layer, known as the eﬀective ﬁeld, is the sum of\napplied ﬁeld, anisotropy ﬁeld and coupling ﬁeld. When\nthe applied ﬁeld is smaller than the saturation ﬁeld, the\nmagnetization of the CoNi layer is not aligned with the\nﬁeld (i.e. an in-plane) direction. Decreasing the applied\nﬁeld further will increase the out-of-plane angle between\nmagnetization of CoNi layer and the applied ﬁeld. Due\nto the AFM coupling, this increasing angle increases the\ncoupling ﬁeld in the applied ﬁeld direction acting on the\nCoFeB layer. Thus, it slows down the decrease of the\neﬀective ﬁeld experienced by the CoFeB layer; the CoNi\nlayercan partially screenthe applied ﬁeld experienced by\nthe CoFeB layer. In the experiment, we ﬁx the frequency\nand sweep the applied ﬁeld to determine the linewidth.\nWhen the applied ﬁeld is smaller than the saturation\nﬁeld, due to this screening eﬀect, a change in the ap-\nplied ﬁeld results in a corresponding smaller change in\ntheCoFeBeﬀectiveﬁeldandthusanincreaseintheFMR\nlinewidth.\nIn order to model the experimental results, we assume\na Heisenberg-type exchange coupling between the layers,with ﬁlm areal energy density [9, 11–13]\nσE(/vector m1, /vector m2) =−µ0Ms1d1/vectorH·/vector m1−1\n2µ0Ms1d1Hk1m′2\nz1\n−µ0Ms2d2/vectorH·/vector m2−1\n2µ0Ms2d2Hk2m′2\nz2\n−J/vector m1·/vector m2,\n(1)\nwhere/vector mi=m′\nxiˆx′+m′\nyiˆy′+m′\nziˆz′is a unit vector in the\nmagnetization direction of each layer in the lab frame,\nwithi= 1 representing the CoNi layer and i= 2 repre-\nsenting the CoFeB layer. Here, we denote our lab frame\nas ˆx′, ˆy′, ˆz′, where ˆx′is aligned with the applied ﬁeld\nand ˆz′is perpendicular to the layer plane. The 1st and\n3rd term, which are proportional to the applied ﬁeld /vectorH,\nrepresent the Zeeman energy; the 2nd and 4th term,\nwhich are proportional to m′2\nzi, represent the magnetic\nanisotropy energy; the last term is the coupling energy.\nThe eﬀective anisotropy is Hki≡H(0)\nki−Msi, whereH(0)\nki\nis the intrinsic anisotropy ﬁeld and Msiis the saturation\nmagnetization. For the CoNi layer, with perpendicular\neasy axis, H(0)\nk1>0 and for CoFeB layer, with in-plane\neasy axis, H(0)\nk2<0.Jis the coupling strength; J >0\nis FM coupling and J <0 AFM coupling. diis the\nthickness of each magnetic layer and µ0is the vacuum\npermeability.\nWhen the applied ﬁeld is small, the magnetization of\neach layer may not be saturated and the precession axis\nneed not be aligned with ˆ x′. Thus, we rotate our lab\nframe by βialong ˆy′, to ˆxi, ˆyi, ˆzi, to the so-called ground\nstate frame, where ˆ xiis the precession axis of each layer.\nWhen minimizing σE, we determine βinumerically. Fig-\nure3shows a numerical solution for βi. Since for the\nCoFeB layer, the coupling strength is smaller compared\nwith the anisotropy ﬁeld, |β2|<|β1|in general. The\nsmaller the coupling strength, the larger the saturation\nﬁeld. When the coupling becomes AFM, the CoNi layer\nprovides a coupling ﬁeld on the CoFeB layer opposite the\ndirection of the CoNi magnetization, which thus requires\na larger applied ﬁeld to compensate.\nThe eﬀective ﬁeld in the lab frame is:\n/vectorHi(m′\nxi,m′\nyi,m′\nzi) =−1\nµ0Msidi∇/vector m′\niσE.(2)\nAfter numerically determining βiat a given applied ﬁeld,\nwe rewrite /vector m′\niin ground state frame, /vector mi=mxiˆxi+\nmyiˆyi+mziˆziby using the rotation matrix shown in the\nappendix. The magnetization precesses around the ˆ xi\naxis. Considering a small angle precession, mxi≈1,myi\nandmzican be expanded around 0. On the other hand,\nwe can rewrite the eﬀective ﬁeld in ground state frame,\n/vectorHi=/vectorHi(mxi,myi,mzi).\nFinally, weuseLLGequationinthegroundstateframe\nto ﬁnd the equation of motion of the magnetization of4\n2J (mJ/m ) \nFigure 3. The CoNi magnetization angle β1(main ﬁgure)\nand CoFeB magnetization angle β2(right inset) obtained by\nminimizing the energy as a function of applied ﬁeld. The\ngreater the AFM coupling strength, the larger ﬁeld required\nto saturate the CoNi layer. The CoFeB magnetization has a\nsmall out of plane component associated with the small ratio\nof the IEC ﬁeld to the in-plane anisotropy ﬁeld. Middle inset :\nschematic showing the βangles.\nboth layers [14]:\nd/vector mi\ndt=−γi(/vector mi×/vectorHi)+αi/parenleftbigg\n/vector mi×d/vector mi\ndt/parenrightbigg\n+hcos(ωt)ˆyi,\n(3)\nwhere, on the right hand side, the ﬁrst term is the preces-\nsionterm, the secondterm is dampingterm andthe third\nterm is oscillating ﬁeld. αiandγirepresent damping co-\neﬃcient and gyromagneticratio, respectively. Bymoving\n2nd term to the left hand side and matrix inversion, we\ncan rewrite Eqn. 3as\ndm\ndt= ΓΛ(Ξm+C), (4)\nwhere\nm(t)≡/parenleftbigmy1(t)mz1(t)my2(t)mz2(t)/parenrightbigT,(5)\nΞ = Ξ a+Ξc+Ξan (6)\nC≡/parenleftbighcos(ω0t) 0hcos(ω0t) 0/parenrightbigT.(7)\nThe modiﬁed gyromagnetic ratio matrix Γ, the Gilbert\ndampingcoeﬃcientmatrixΛandtheelementsofeﬀective\nﬁeld matrices Ξ a, Ξc, Ξanare shown in the appendix.\nInordertosolvethecoupledODEs,wetaketheFourier\ntransformation of Eqn. 4,˜f(ω)≡1√\n2π/integraltext∞\n−∞f(t)e−iωtdt\nto get:\n˜m=−ΓΛ˜C\n(ΓΛΞ−iω1), (8)J (mJ/m ) 2\nFigure 4. Computed CoFeB linewidth versus applied ﬁeld.\nWhen the IEC is FM, the linewidth decreases with decreasing\nﬁeld. When the IEC is AFM, there is an abrupt increase in\nthe linewidth when the ﬁeld decreases below the saturation\nﬁeld.\nwhere the amplitude of the magnetization is dominated\nby 1/|det(ΓΛΞ −iω1)|.\nIn the numerical simulation, we take α1= 0.0108 and\nα2= 0.0053, which come from the single layer sample\nmeasurements. We take γ1andγ2to be the electron’s\ngyromagnetic ratio. We further take µ0H(0)\nk1= 0.779T,\nµ0H(0)\nk2=−0.128T,µ0Ms1= 0.600T and µ0Ms2=\n1.69T, to characterize the magnetic materials [9]. Since\nour goal is to illustrate the mechanism leading to the ob-\nserved increase in linewidth, rather than a quantitiative\nﬁt, we keep these values ﬁxed and only vary the coupling\nstrength J.\nFigure4shows the numerical solution obtained for dif-\nferent coupling strengths J, which matches our experi-\nmental results to a certain degree. When the coupling\nstrength is ferromagnetic, the linewidths increase mono-\ntonically with the applied ﬁeld, which matches our ex-\nperimental results. In the case of AFM coupling, when\nsweeping the ﬁeld from high to low values, the linewidths\ndecrease with decreasing applied ﬁeld, at least initially.\nBut when the applied ﬁeld reaches the saturation ﬁeld,\nthere is a strong enhancement in the linewidth. This\nqualitatively reproduces the behavior seen in the ex-\nperiments. The saturation ﬁelds shown in Fig. 4are\nconsistent with the previous simulation results shown in\nFig.3(a). The saturation ﬁeld increases when the sam-\nples become more AFM coupled. It shows that the basic\ncauseoftheincreasinglinewidthisthescreeningoftheef-\nfective ﬁeld associatedwith the AFM coupling. However,\nwhen further decreasing the applied ﬁeld, the linewidths\nalso decrease, which is not observed experimentally. The\nmodel also does not explain the ﬁnite width of the ﬁeld\nregion, over which linewidth increase takes place. A pos-\nsible reason is that our model assumes two macrospins,5\nwhich means we consider the magnetization to be homo-\ngeneous in the sample plane. When the ﬁeld is small,\nthe magnetization may form domains, which violates the\nmacrospin assumption. In addition, two-magnonscatter-\ning may contribute to the observed linewidth, as well as\ninhomogeneous local ﬁelds arising from defects.\nV. SUMMARY\nIn this work we investigated the linewidth in an in-\nterlayer exchange-coupled bilayer system with mixed\nanisotropies. In samples with AFM coupling, below the\nsaturation ﬁeld, we observed a signiﬁcant increase of the\nlinewidth in the FMR spectra. Analysis in the frame-\nwork of a previously established macrospin model pointstowards a mechanism that can be regarded as an applied\nﬁeld screening eﬀect that results from the combined ac-\ntion of the IEC and the CoNi magnetization tilting due\nto its PMA. The reported screening eﬀect could be ex-\nploited in future spin torqueoscillatorapplications where\nit could help stabilize the oscillator frequency with re-\nspect to changes in the ambient magnetic ﬁeld.\nVI. ACKNOWLEDGEMENTS\nThis research was supported by Spin Transfer Tech-\nnologies, Inc. and in part by National Science Founda-\ntion under Grant No. DMR-1610416.\nVII. APPENDIX\nIn this appendix we provide the matrices used in the main text. The ro tation matrix to transform between the lab\n/vector m′and ground state frame /vector mis:\n\nmxi\nmyi\nmzi\n=\ncos(βi) 0−sin(βi)\n0 1 0\nsin(βi) 0 cos( βi)\n\nm′\nxi\nm′\nyi\nm′\nzi\n. (9)\nThe following are the matrices used to solve the LLG equation Eqn. 4. The modiﬁed gyromagnetic ratio matrix:\nΓ≡\nγ1\n1+α2\n1γ1\n1+α2\n1γ2\n1+α2\n2γ2\n1+α2\n2\n. (10)\nThe Gilbert damping coeﬃcient matrix is:\nΛ≡\n1−α1\nα11\n1−α2\nα21\n. (11)\nFinally, the eﬀective ﬁeld matrices are:\nΞa≡\n−Hcos(β1)\nHcos(β1)\n−Hcos(β2)\nHcos(β2)\n, (12)\nΞc≡\n−J\nµ0Ms1d1cos(β1−β2)J\nµ0Ms1d1cos(β1−β2)\nJ\nµ0Ms1d1cos(β1−β2) −J\nµ0Ms1d1J\nµ0Ms2d2cos(β1−β2) −J\nµ0Ms2d2cos(β1−β2)\n−J\nµ0Ms2d2J\nµ0Ms2d2cos(β1−β2)\n,(13)\nΞan≡\nHk1cos(2β1)\nHk1sin2(β1)\nHk2cos(2β2)\nHk2sin2(β2)\n. (14)\n[1]A. Fert, P. Gr¨ unberg, A. Barth´ el´ emy, F. Petroﬀ, and W.\nZinn. J. Appl. Phys. 140, 1 (1995).[2]P. J. H. Bloemen, H.W. VanKesteren, H. J. M. Swagten,\nand W. J. M. De Jonge. Phys. Rev. B, 50, 13505, (1994).6\n[3]A. D. Kent and D. C. Worledge. Nat. Nanotechnol., 10,\n187, (2015).\n[4]A. Brataas, A. D. Kent, and H. Ohno. Nat. Mater. 11,\n372, (2012).\n[5]S. S. Kalarickal, P. Krivosik, M. Wu, C. E. Patton, M. L.\nSchneider, P. Kabos, and J. P. Nibarger. J. Appl. Phys.,\n99, 093909, (2006).\n[6]J. M. Beaujour, D. Ravelosona, I. Tudosa, E. E. Fuller-\nton, and A. D. Kent. Phys. Rev. B, 80, 180415, (2009).\n[7]Y. Ding, T. J. Klemmer, and T. M. Crawford. J. Appl.\nPhys.,96, 2969, (2004).[8]H. T. Nembach, T. J. Silva, J. M. Shaw, M. L. Schneider,\nM. J. Carey, S. Maat, and J. R. Childress. Phys. Rev. B,\n84, 054424, (2011).\n[9]L. Fallarino, V. Sluka, B. Kardasz, M. Pinarbasi,\nA. Berger, and A. D. Kent. Appl. Phys. Lett., 109,\n082401, (2016).\n[10]C. Kittel. J. Phys. Radium, 12, 291, (1951).\n[11]A. Aharoni. J. Appl. Phys., 83, 3432, (1998).\n[12]J. A. Osborn. Phys. Rev., 67, 351, (1945).\n[13]R. Moskowitz and E. Della Torre. IEEE Trans. Magn, 2,\n739, (1966).\n[14]T. L. Gilbert. IEEE Trans. Magn, 40, 3443, (2004)."    },    {        "title": "1006.0331v1.Magnetization_pinning_in_conducting_films_demonstrated_using_broadband_ferromagnetic_resonance.pdf",        "content": " 1 Magnetization pinning in conducting films demonstra ted using broadband \nferromagnetic resonance \nM. Kostylev 1, A.A. Stashkevich 3, A.O. Adeyeye 2, C. Shakespeare 1, N. Kostylev 1, N. Ross 1,  \nK. Kennewell 1, R. Magaraggia 1, Y. Roussigné 3, and R. L. Stamps 1 \n1School of Physics, University of Western Australia,  Australia  \n2Information Storage Materials Laboratory Department  of Electrical and Computer \nEngineering, National University of Singapore, Sing apore \n3LPMTM CNRS (UPR 9001), Université Paris 13, 93430 V illetaneuse, France  \n \nAbstract \n \n \nThe broadband microstrip ferromagnetic resonance te chnique has been applied for detection \nand characterization of a magnetic inhomogeneity in  a film sample. In the case of a 100nm \nthick Permalloy film an additional magnetically dep leted top subPlayer, practically \nunidentifiable by the conventional ferromagnetic re sonance setup, has been detected and \ncharacterized. These results have been confirmed by  Brillouin light scattering spectroscopy \nrevealing the fact that the optical properties of t he additional subPlayer do not differ much from \nthose of the bulk of the film. Subsequent character ization of a large number of other \npresumably singlePlayer films with thicknesses in t he range 30P100nm using the same \nferromagnetic resonance technique also revealed the  same effect.    \n \n \n \n  2  \nIntroduction \n \nApplications of metallic ferromagnetic materials to  the microwave frequency devices \nmotivate interest in highPfrequency dynamics of con tinuous and patterned ferromagnetic films. \nIn the case of relatively low angle precession, mag netic dynamics manifests itself through bulk \nmagnetic excitations known as spin waves (SW). Brea kthroughs in the technology of single \ncrystal yttrium iron garnet (YIG) ferrite films wit h extremely low losses at microwave \nfrequencies suitable for effective signal processin g, typically in delay lines in the frequency \nrange from 2 GHz to 20 GHz, 1,2  has caused further development of  this concept in  the 1960s – \n1980s. Although soft ferrite materials have found i mportant applications in microwave \nengineering, 3 their frequency performance is severely handicappe d by small values of the \nsaturation magnetization. There is consequently a g rowing research effort in the field of \nmicrowave properties of soft ferromagnetic metals, such as Permalloy or iron, for applications \nto widePband microwave signal processing.  \nIn particular, it has been shown that despite relat ively high insertion losses Permalloy films \ncan be used in delay lines, 4 as well as in parametric ampliﬁers. 5 They are also regarded as \npotential candidates for magnetologic devices, 6,7  emerging as a new interesting field of \napplication of propagating SW, not directly related  to microwave signal processing. The MachP\nZehnder geometry proposed in Ref.8 and whose effici ency in the case of a YIG structure has \nbeen proved in Ref.9, can be regarded as a typical example of the basic physical principle \nunderlying logic operations based on destructive an d constructive SW interferences. However, \nsince metal films are at least one order of magnitu de thinner than their ferrite analogues, they \nallow creation of much smaller microwave and logic components. 8 Moreover sputtering \ntechnology employed in the fabrication of metal fil ms, makes it much easier to incorporate  3 magnetic layers into hybrid integrated structures, than in the case of conventional insulating \nferrites prepared by liquid phase epitaxy. 10P12  \nFinally, one should not underestimate another degre e of freedom characteristic of metal \nfilms and entirely nonPexistent in conventional ins ulating ferrites: the presence of free charge \ncarriers. The recent discovery of the Doppler frequ ency shift of a SW interacting with a current \nin a Permalloy film 13,14  has drawn attention to the importance of the above  mentioned physical \nproperty. \nThe performance of microwave and logic components, employing Permalloy films, \ndrastically depends mostly on two parameters: high frequency damping and pinning \nconditions. Although the high frequency absorption in Permalloy films has been investigated \nsince the 1970s15  this characteristic is still extensively discussed  in recent publications. 16,17   \nThe same applies to the pinning conditions (see e.g . Ref.18) which, as is well known, \nchange significantly the SW spectra. 19  In this paper we report detection of an intermedia te thin \nmagnetic subPlayer causing, via the so called dynam ic pinning, noticeable changes in the \ndynamic behavior of the whole structure. 20  To ensure reliability of the experimental results,  we \nhave adopted a combined technique, employing two co mpletely independent approaches: \nferromagnetic resonance excited by an external wide  microPstrip antenna and Brillouin Light \nScattering (BLS) by thermal magnons.  \nIn the first case we made use, as proposed in Ref.2 1, of the characteristic asymmetry of the \ntwo interacting microwave fields: the field of the exciting structure and that of the excited \nmode. In the complementary BLS approach we appealed  to the dispersion characteristics of \nthermal magnons in a wide range of wavePnumbers whi ch is covered by this powerful \ntechnique. More specifically, we made use of the no nPreciprocity of the DamonPEshbach 22  SW \nexcitations, propagating in an asymmetric ferromagn etic structure, as first reported in Ref.23.  4  \nExperiment \nMicrowave absorption is a common tool in the study of high frequency properties of \nmetallic ferromagnetic films and multilayers. Becau se of the sensitivity of some magnetic \nexcitations to film surfaces and interfaces, this i s a particularly useful technique for \nnondestructive studies of magnetic interfaces.  \nIn our recent experimental 24  and theoretical 21  papers we have shown that magnetic metallic \nbiPlayer films, tailorPmade to optimize the efficie ncy of dynamic pinning, manifest a strong \nasymmetry of microstrip broadband FMR response with  respect to layer ordering. More \nspecifically, in Ref.24 we measured FMR responses f rom a series of CobaltPPermalloy biPlayer \nfilms grown on Silicon substrates, prepared ad hoc to test the efficiency and reliability of FMR \ncharacterization of such structures: for all films the Cobalt (Co) layer thickness was kept at 10 \nnm, while the Permalloy (Py) thickness varied from sample to sample in the range from 40nm \nto 100 nm.  \nThe FMR “signature” defined as the number and as sh ape of the absorption lines depends \ndrastically on the overlap between the exciting hig h frequency field, produced by a microstrip \nand the magnetization profile m(y) in the excited spin wave mode. Both have a pronou nced \nasymmetry across the film. However, while the asymm etry of  m(y)  is fixed with respect to \nthe layer ordering, the external field can be appli ed from both sides by placing the sample on \nthe microstrip differently: with the sample side do wn or with the substrate down. In the latter \ngeometry, to ensure the exposure of the biPlayer to  the exciting field, the width of the \nmicrostrip was made greater then the substrate thic kness. Theoretical simulation (see Ref.21) \nbased on a well defined sample geometry and on typi cal magnetic parameters of both metals \ngave excellent results.   5 In this paper, applying the same approach, we addre ss an inverse problem: detection and \ncharacterization of a supplementary layer in a seem ingly homogeneous Permalloy singlePlayer. \nThe test sample, 100 nm thick, has been grown on a 500 µm thick Si substrates by standard DC \nmagnetron sputtering.  \nIn our broadband FMR experiments we use an Agilent N5230A PNAPL microwave network \nanalyzer to apply a microwave signal to the samples  and to measure magnetic absorption. As a \nmeasure of the absorption we use the microwave scat tering parameter S21  [25]. We keep the \nmicrowave frequency constant and sweep the static m agnetic field H applied in the film plane \nand along the transducer to produce resonance curve s in the form of S21 (H) dependences. This \nis repeated for a number of frequencies. We also me asure S21  for the transducer with no \nsample ( S21 0(H)) to eliminate any fieldPdependent background sign al from the results. The \nresults presented below are Re( S21 (H)/  S21 0(H)).  \nThe driving current is applied to a section of micr ostrip. The sample sits on top of this \nmicrostrip transducer, whose width is 1.5 mm. The e lectromagnetic field in a TEM mode \nsupported by the microstrip line is described by th e Laplace equation. It falls down across the \nsubstrate sufficiently slowly to provide for effici ent SW excitation in the unfavorable geometry \nwhen it is the substrate that faces the microstrip.   \nPreliminary experimental results of the FMR measure ments signaling the presence of a \nsecond subPlayer are given in Fig.1. One can clearl y see that the position of the sample with \nrespect to the transducer (film facing the microstr ip / substrate facing the microstrip), as it has \nbeen outlined in Refs.21, 24, changes drastically t he response: the profile transforms from a \nsingle peak to twin peak curve. The single peak is traditionally identified, according to its field \nposition, as the fundamental “Kittel” mode. Since t he time of numerous classical cavity FMR \nstudies, 26  observing only a fundamental mode is traditionally  considered as an evidence of high  6 homogeneity of film properties across the film thic kness, in particular, the absence of surface \nspin pinning.27   \nImportant here is the fact that when we place the s ample with the substrate facing the wide \nmicrostrip transducer we detect a second intense re sponse (Figure 1, dashed line). The latter \nattests to some sort of pinning on one of the inter faces. The problem is, however, to define on \nwhich surface of the main Py film the additional “p inning” layer is formed. In any case, it is \nplausible to suggest that this layer is magneticall y depleted, either due to oxidation on the free \nsurface or via interPdiffusion at the filmPsubstrat e interface. \nTo identify the nature of this subPlayer and to cha racterize its parameters extensive \nnumerical simulations have been performed. Represen tative theoretical fits of experimental \ndata are given in Fig.2. The following situations a re illustrated: f = 8 GHZ, f = 12 GHZ with \nthe substrate facing the transducer and f = 15 GHZ with the film facing the transducer. By \nfitting responses for both film placements with the  theory 21  one obtains the thickness Ls and \nsaturation magnetization 4 πMs for the subPlayer. The best fit is obtained for Ls=10nm and \n4πMs=4000G (See Fig. 2.) Moreover, numerical simulation s suggest that the depleted subP\nlayer is localized on the free surface.  Similar evidence of the presence of a magnetically  \ndepleted subPlayer was also found for the second pr esumably singlePlayer film from the same \nseries. \nFurther evidence of the presence of this layer was obtained by measuring thermal magnon \ndispersion with Brillouin light scattering (BLS) te chnique at the University ParisPNord. The \nresults obtained are illustrated in Fig.3, where ex perimental points corresponding to BLS peaks \nare superimposed on theoretical dispersion curves. This measurement revealed a considerable \ndifference in frequencies for the Stokes and antiPS tokes BLS peaks, characteristic of a multiP\nlayer with an asymmetric magnetic structure across the film. 23  This difference is well fitted by  7 a nonPreciprocal spin wave dispersion law for a biP layer with the material parameters extracted \nfrom the FMR data above. It should be stressed that  here, once again, we have supposed that \nthe depleted layer is situated on top of the main o ne. Another significant feature pointing to the \npresence of dynamic pinning is the appearance of a pronounced dipolePexchange gap between \nthe DE mode and SWRs, well reproduced both in theor y and experiment. However, this \npeculiarity, while describing the extent of effecti ve pinning, provides no clue as to at which \nsurface it takes place. \n \nDiscussion \nOne of the major points addressed in this paper is how to identify the surface at which the \neffective pinning takes place. Interestingly, in th e case investigated in Ref.24 (a thick Py film \nwith a thin “pinning” Co subPlayer) the larger ampl itudes for the higherPorder modes were \ndetected for the pinning layer facing the transduce r. In other words, in the case when the \nmicrostrip field penetrated the sample through the thin supplementary sub4layer .  At the same \ntime, in our case just the opposite was observed: h igher modes were generated when the \nexciting magnetic field penetrated through the main layer  (see Fig.1). The theory in Ref.21 \ngives indications that this can be related to the r eversal of the magnetization profile in the two \nmagnetic structures studied. In Ref.24 the thin pin ning subPlayer has a higher saturation \nmagnetization 4πM s then the bulk of the sample, while in our case the  depleted pinning subP\nlayer is characterized by a lower value of 4πM s. \nAdditional evidence in favor of this explanation, b ased on our numerical simulations, is \npresented in Fig.4. According to Ref.21 the absorpt ion amplitudes for microstrip broadband \nferromagnetic resonance depend on two major factors : the shape of the thickness profile of the \ntotal microwave magnetic field inside the sample an d the feedback from the excited  8 magnetization to the microstrip. The former determi nes how efficiently magnetization \nprecession is excited; the latter determines how ef ficiently the signal of precessing \nmagnetization is picked up by the transducer.     \nThe total microwave field consists of the microwave  field of the microstrip transducer, \nthe field of eddy currents induced in the sample, a nd the field of precessing magnetization. \nThe latter is usually negligibly small compared to the former two. The feedback from a \nconducting film to the transducer is provided by th e microwave electric field of precessing \nmagnetization,21  so that the absorbed microwave power is given by t he product of the electric \nfield at the film surface and the microwave current  in the transducer [ibid].  \nPanels (a) and (c) in Fig. 4 show mode profiles for  the first higherPorder standingP\nwave mode for a biPlayer film which has a 90nmPthic k layer of Permalloy (4 πMs=10700 Oe)  \nand a 10nmPthick pinning layer exchange coupled to it. In the case of panel (a) the pinning \nlayer has saturation magnetization which is smaller  than that for Permalloy (4 πMs(p) =4000 \nOe). One sees that at the boundary between the two layers y= 90 nm the dynamic \nmagnetization drops, and the magnetization amplitud e in the pinning layer is smaller than for \nthe bulk of the film. Furthermore, one sees that th e mode profile inside the bulk is asymmetric \nwith the maximum of amplitude located at the film s urface y=0 facing away from the pinning \nlayer.  \nObviously, in order to excite this mode most effici ently one has to place the film with \nrespect to the microstrip in such a way that the ma ximum of the mode profile faces the \ntransducer. In this particular case the surface y=0 should face the transducer to meet this \nrequirement. This will form a profile for the total  microwave field shown in Fig. 4b. This \nprofile is characterized by the maximum of the fiel d at y=0 and thus by the maximum overlap \nintegral with the mode profile in Fig.4a. The latte r ensures the maximum excitation \nefficiency.  9 Furthermore, the asymmetric mode profile results in  an asymmetric profile of the \nelectric field with the maximum again at y=0. Thus, it is favourable to place a pick up \ntransducer at the surface y=0 in order to ensure the maximum efficiency of det ection for this \nmode. Thus, one can conclude that the maximum of ef ficiency of excitation of the first \nexchange mode for  Ms(p) < M sb is obtained when the microstrip transducer is plac ed at the film \nsurface facing away from the pinning layer.  \nTurn now to Panels c and d, describing the opposite  situation Ms(p) > M sb. They show \nthe same profiles for the pining layer with saturat ion magnetization larger than for the bulk of \nthe film (4 πMs(p) =16000 Oe). One clearly sees that in this case the mode profile (Fig 4c) is \ndistorted in the opposite way. The maximum of the a mplitude of dynamic magnetization is \nnow in the thin layer (which actually does not pin dynamic magnetization in the bulk of the \nfilm for this particular mode. However, it does pin  magnetization for the fundamental mode, \nas shown in Fig. 2(b) in Ref.21. ) Thus the best wa y to excite this mode is by placing the \nmicrostrip transducer at the surface y=100nm, as it will result in the total microwave ma gnetic \nfield with the maximum at the same surface (see Fig .4d). Furthermore, detection of this mode \nis also maximized by placing the transducer at x=100nm, since the microwave electric field of \nprecessing magnetization is larger at this surface (Fig.4d).). This result is in excellent \nagreement with experiment on Py[40P80nm]/Co[10nm] b ilayer films. 24 .  \nThus, one can conclude that if the pinning layer ha s Ms smaller than the bulk of the \nfilm, in order to detect its presence the microstri p transducer should be placed at the film \nsurface which faces away from the pinning layer.  I f the pinning layer has Ms larger than the \nbulk of the film, the pinning layer should face the  transducer. \nWe also checked our findings on a considerable numb er of other singlePlayer \nPermalloy films of different thicknesses (30P100nm)  grown in different conditions and on \ndifferent setups and compared our broadband FMR res ults with the traditional cavity FMR  10  measurements. All these measurements confirmed our idea: when the cavity FMR data \nshowed a presence of a weak higherPorder resonance,  which is traditionally considered as \nevidence of magnetization pinning at film surfaces,  the broadband FMR measurements for \ntwo different sample placements were able to indica te at which of the sample surfaces the \nmagnetization pinning takes place (Fig. 5), provide d we assume that the saturation \nmagnetization for the spontaneously formed layer is  smaller than for the bulk of the film. \nFitting the broadband data with a theory allows one  to estimate the thickness of the pinning \nsubPlayer. If it is negligibly thin, one can descri be its effect as a surface pinning, while a subP\nlayer of finite thickness can also pin magnetizatio n at the interface of the bulk of the film with \nthe subPlayer. Note that explanations alternative t o a depleted layer may also exist.   Examples \nare different anisotropy fields, large scale roughn ess, or weakened exchange field at a film \nsurface. Considering them was outside the scope of this paper. The key point is that our \nexperiment allows one to detect at which film surfa ce this inhomogeneity is located and to \nextract parameters of this inhomogeneity if the inh omogeneity type is known.  \nTo make our characterization more complete and reli able, we have also carried out \nnumerical simulations of the structure of BLS spect ra. This theoretical analysis was based on \nan ad hoc simplified theory, given in more detail i n the appendix. Representative results for \nthe saturating magnetic field H = 1000 Oe are presented in Fig.6a (angle of incide nce θ = 15°) \nand Fig.6b (angle of incidence θ = 30°).  \nIt should be noted that the amplitudes of the peaks  produced by the DE mode in \nrelatively thick metal films typically exhibit a pr onounced Stokes /antiPStokes asymmetry, \nwhich is our case. Furthermore, the DE mode contrib uting to the Stokes peak (negative \nfrequency shift) is confined to the top of the film . This makes the peak corresponding to \nnegative frequency shifts more pronounced, due to a  better overlapping of the three \ninteracting waves (see the appendix). This means, i n particular, that in our sample the top  11  depleted subPlayer produces a nonPnegligible contri bution to the BLS response. At the same \ntime, strictly speaking, we do not know its optical  parameters. Although we have neglected \npurely optical effects, supposing that the sample m ay be considered optically uniform , the \nagreement between the experiment and theory is stil l good.  \n To check the accuracy of our approximate approach,  we have compared the results \npresented in Fig.6 with those returned by a rigorou s program based on the DissipationP\nFluctuation Theorem and exact expressions for optic al Green’s functions. 28  The differences \nbetween the two sets of curves turned out to be ins ignificant. Moreover, the approximate \napproach provides for a better fit of the peaks pro duced by a hybridized SWR mode. The \nexplanation is straightforward: this mode is very s low and therefore produces an optical \nresponse smeared in the reciprocal space. Our other wise approximate program takes into \naccount a finite numerical aperture while the rigor ous one does not. \nConclusion  \nIn this paper we have demonstrated the efficiency o f the broadband microstrip FMR technique \nfor detection and characterization of a hidden magn etic inhomogeneity in a film sample. \nBased on the nonPreciprocity of the microstrip FMR response with respect to the direction of \npenetration of the exciting microwave field in the sample, it is especially effective in the case \nwhen the hidden layer breaks the symmetry of the fe rromagnetic structure. In the particular \ncase of a 100 nm thick Permalloy film, an additiona l magnetically depleted top subPlayer, \npractically unidentifiable by the conventional cavi ty FMR setup, has been detected and \ncharacterized. These results have been confirmed by  BLS spectroscopy revealing, \nsurprisingly, the fact that the optical properties of the additional subPlayer do not differ much \nfrom those of the bulk of the film.  12  In a broader context, the proposed technique can be  regarded as a nonPdestructive \nexpress method to detect the presence of magnetic i nhomogeneity in conducting \nferromagnetic films. \n \nAcknowledgment \nSupport by Australian Research Council and from the  National Research Foundation, \nSingapore under Grant No. NRFPGPCRP 2007P05 is ackn owledged. \n \n \nAppendix.  Simple model for BLS intensities \nHere we give details of the simple model we used to  calculate intensities of the BLS \npeaks. Since the exact theory of this effect is cum bersome and requires sophisticated \nnumerical models, 29,30  in this work we use a simpler ad hoc approach, pro viding for more \nphysical insight. It can be regarded as an extensio n of the formulas in Refs 31, 32. We are not \ninterested in comparing absolute values  of BLS intensities measured for different  incidence \nangles. Our goal is to derive a tool to compare int ensities for different modes observable for \nthe same incidence angle. Therefore all calculation s are made to a constant which is the same \nfor all spin wave modes seen at the same angle of i ncidence. \nThe BLS intensities are determined by two major con siderations, namely the \nefficiency of the magnetoPoptical (MO) interaction proper and that of the excitation of thermal \nmagnons. The efficiency of the MO interaction with an individual magnon can be represented \nas a product of two separate factors: \nv s I I I = ⋅              (1) \nHere the first one \n 2( ) v i s I= ⋅ × e e m ,     (2)  13  called typically the vector factor, is due to the v ector nature of the threePwave MO interaction \nand it reduces in optically isotropic media to a mi xed product of the polarizations of the \ninteracting waves: ie and se are  those of the incident and scattered light res pectively, and m \ndescribes dynamic magnetization. Its physical sense  is perfectly clear: the vector product, \ncorresponding to the orientation of the polarisatio n induced in the film by the presence of the \nmagnon, reflects the symmetry of the MO Faraday eff ect represented by the unitary antiP\nsymmetric LevyPCivita tensor. The scalar product de scribes the projection of the induced \npolarization on the direction of polarization of th e scattered wave. Strictly speaking, this \nscalar product implies a complex conjugate, but sin ce in our case the scattered wave is “s” \npolarised *s s =e e . \nThe second coefficient sI, which can be called spatial factor, reflects the spatial \noverlapping of the aforePmentioned three interactin g waves. It can be estimated, up to a \nconstant factor, as a convolution, in y=0 , of the induced optical polarization \nexp( ) ( ) yik y m y − ⋅  and the Green’s function exp( ' ) yik y y − − , the latter being an optical \nresponse, in the backscattering geometry, to a delt a layer of polarization ( ') y y δ− : \n2\nexp( 2 ') ( ') ' s y \nLI ik y m y dy = − ∫,     (3) \nwhere 2( ) zk q ip π\nλ= +  is the complex outPof plane wave number ( p<0) and λ = 530 nm is \nthe laser wavelength.   \nThe simplified expression (3) is justified because in our case the thickness L of the \nsamples studied in this work is considerably larger  than the skin depth at the laser wavelength. \nThat is why one can neglect the interference create d by the reflection from the interface y= L . \nAt the same time, it should be noted that the appro ximate expression (1), representing the  14  efficiency as a product of two independent factors,  holds only in the case where the spin mode \npolarization does not depend on y. The latter is correct only for the case of nonPhy bridized \nmodes. However, if a pronounced hybridization takes  place the vector factor vI should kept \nunder the integral. (Since our experimental data su ggest mode hybridization, in our numerical \ncalculation we actually used the latter approach.) \nTo obtain the final formula we also take into accou nt the fact that the lens collects \nscattered light in the range from / 2 θ θ −∆  to / 2 θ θ +∆ . This results in the aperture factor \n(see Eq.(36) in Ref.32 \n \n2atan( / \")/( \") a g g I V k V δ ω ω = ,   (4) \n \nwhere Vg is spinPwave group velocity and ω’’ is the imaginary part of spin wave group \nvelocity responsible for spin wave damping. (We cal culate this factor for the maximum of the \nBLS peak.)  The uncertainty in the transferred wave  number kδ relates to the collection \nangle via 4 cos( ) / kδ π θ θ λ = ∆ . \nWe also add a factor Ifd which arises from the fluctuation dissipation theor em and \nwhich relates amplitudes of thermally excited waves  to the RayleighPJeans distribution: \n2( '') / ' fd Iω ω = ,   (5) \nwhere  ω’ is the real part of the complex spin wave eigenfr equency. Finally, the scattered \nintensity is proportional to the product of the fac tors (2)P(5): \nv s fd I I I I ⋅ ⋅∼ .    (6) \n This expression was used in this work to calculate  BLS intensities shown in Fig.6. For \neach of the layers its own value for Gilbert dampin g parameter was used (0.008 for Permalloy \nand 0.16 for the depleted layer.) By solving the ei genvaluePeigenvector problem for the  15  dipolePexchange operator 34 for the biPlayer complex eigenfrequencies ' '' iω ω + , group \nvelocities, and modal profiles for all spin wave mo des were obtained. These data were \ninserted into (6) to yield the intensities.  16   \nReferences: \n1 D.D. Stancil, Theory of Magnetostatic Waves (Spring er Verlag, 1993). \n2 A.G.Gurevich and G.A.Melkov,  Magnetization Oscillations and Waves  (CRC Press, New \nYork,1996). \n3 M. PardaviPHorvath , J. Mag. Mag. Mat . 215-216 , 171 (2000). \n4 M.Bailleul, D.Olligs, C.Fermon, and S.O.Demokritov,  Europhys. Lett ., 56  741 (2001). \n5 Mingqiang Bao, A. Khitun, YinaWu, JooPYoung Lee, Ka ng L.Wang, and Ajey P.Jacob,  \nAppl. Phys. Lett . 93 ,072509 (2008). \n6 A.Khitun and K.L. Wang, Superlatt. Microstr . 38 , 184 (2005). \n7 S.Bance T.Schrefl, G. Hrkac, A.Goncharov, D.A.Allwo od, and J.Dean, J. Appl. Phys . 103 , \n07E735 (2008). \n8 SPK. Kim and K.PS. Lee, J.Appl.Phys . 104, 053909 (2008). \n9 T.Schneider, A.Serga, B.Leven, B.Hillebrands, R.L.S tamps, and M.Kostylev, \nAppl.Phys.Lett.  92 ,  022505  (2008).  \n10  B. Kuanr, L.Malkinski, R.E.Camley, Z.Celinski, and P.Kabos, J. Appl. Phys . 93 , 8591  \n1(2003). \n11  B. Kuanr, I.R.Harward, R.T. Deiotte, R.E.Camley, an d  Z.Celinski, J. Appl. Phys . 97 , \n10Q103 (2005). \n12  B. Kuanr, D.L.Marvin, T.M. Christensen, R.E.Camley,  and  Z.Celinski, Appl. Phys. Lett . \n87 , 222506 (2005). \n13  R..D. Mc Michael and M.D.Stiles Science , 322, 386 (2008). \n14  V.Vlaminck and M. Bailleul CurrentPInduced SpinPWav e Doppler Shift, Science  322 , 410 \n(2008). \n15  C.E.Patton, Z.Frait, and C.H.Witts, J. Appl. Phys . 46 , 5002 (1975).  17  16  D.J.Twisselmann and R.D.McMichael, J. Appl. Phys.  93 , 6903 (2003). \n17  R.C.Oliveira, R.L.RodrıguezPSuarez, F.M.deAguiar, S .M.Rezende, J.R.Fermin, and \nA.Azevedo, J. Mag. Mag. Mat  272–276 , 795 (2004). \n18  A. A. Stashkevich,  Y. K. Fetisov,  P. Djemia, N. B izière, and C. Fermon , J. Appl. Phys . \n102 , 103905 (2007). \n19  B.A.Kalinikos, “DipolePexchange spinPwave spectrum of magnetic films\" in: M.G.Cottam, \n(Ed.), Linear and nonlinear spin waves in magnetic films a nd Superlattices , World Scientific \nPublishing Company Ltd., Singapore, 1994, p.90P156.  \n20  P.E.Wigen,S.F.Kooi,M.R.Shanabarger,and Th.D.Rossing , Phys.Rev.Lett ., 9, 206 (1962). \n21  M. Kostylev, J. Appl. Phys . 106 , 043903, 2009. \n22  R.W.Damon and J.R.Eshbach J.Phys.Chem. Solids  19 , 308 (1961).  \n23  M.Vohl, P.Barnas, and J.Grünberg, Phys. Rev. B  39 , 12003 (1989). \n24  K. J. Kennewell, M. Kostylev, R. Magaraggia, R. L. Stamps, M. Ali, A. A. Stashkevich, \nD. Greig and B. J. Hickey, arXiv:1001.1837v1 (2010) . \n25  G. Counil, J.PV. Kim, T. Devolder, C. Chappert, K. Shigeto, and Y. Otani,  J. Appl. Phys . \n95 , 5646 (2004).  \n26  B. Lax and K. J. Button, \"Microwave ferrites and fe rrimagnetics\", New York : McGrawP\nHill, (1962). \n27  C. Kittel, Phys. Rev . 73 , 155 (1948). \n28  Y.Roussigné, F. Ganot, C. Dugautier, P. Moch, and D . Renard Phys. Rev. B 52, 350 \n(1995). \n29  R.E.Camley and D.L.Mills , Phys.Rev.B  18 , 4821 (1978). \n30  M.G.Cottam, J.Phys. C: Solid State Phys ., 12  1709 (1979). \n31  R. Zivieri, P. Vavassori, L. Giovannini, F. Nizzoli , E. E. Fullerton, M. Grimsditch and V. \nMetlushko, Phys. Rev. B  65  165406 (2002).  18  32  A. A. Stashkevich, Y. Roussigné, P. Djemia, S. M. C hérif, P. R. Evans, A. P. Murphy, W. \nR. Hendren, R. Atkinson, R. J. Pollard, A. V. Zayat s, G. Chaboussant and F. Ott, Phys. Rev. B  \n80 , 14406 (2009). \n33  M. P. Kostylev and A. A. Stashkevich, Phys. Rev. B  81, 054418 (2010). \n34  B. A. Kalinikos, N. V. Kozhus, M. P. Kostylev, A. N . Slavin, J. Phys. Condens. Matter  2, \n9861 (1990). \n \n \n  19  Figure captions \nFig. 1. Broadband FMR responses for a 100Pnm Permal loy film showing evidence of the \npresence of a magnetically depleted layer. Driving frequency is 10 GHz. Solid line: film faces \nthe microstrip transducer, dashed line: substrate f aces the transducer. \n \nFig. 2. Exemplary theoretical fits of experimental data with theory from Ref.21 for different \nfrequencies. (a): 8 GHz, substrate faces the transd ucer, (b) 12 GHz, substrate faces the \ntransducer, (c) 15 GHz, film faces the transducer. Red solid line lines: experiment, blue \ndashed lines: theory.  \nParameters of calculation. \nTotal film thickness: 100 nm.  \nBulk of the film (90nmPthick): Saturation magnetiza tion: 4 πMs=10700G, exchange constant: \n1.2 ⋅10 P6 erg/cm, conductivity: 4.5 ⋅10 6 Siemens/m. Spontaneously formed layer: Thickness: 10 \nnm, saturation magnetization: 4 πMs=4000G, exchange constant: 0.51 ⋅10 P6 erg/cm, \nconductivity is the same as for the bulk. InterPlay er exchange constant: 60 erg/cm 2.  The thin \nlayer is formed at the free surface of the film.  \n  \nFig.3. Brillouin light scattering data. Different f requencies for the Stokes (black triangles \ndown) and the antiPStokes (red triangles up) are cl early seen. Red solid line: calculated spin \nwave dispersion for the “Stokes” direction of spin wave propagation. Dashed black line: the \nsame for the opposite (“antiPStokes” propagation di rection).  Blue dotted line: dispersion for a \nsinglePlayer film with the same total thickness. \n  20  Fig. 4. Calculated mode profiles for the first high erPorder mode for two different values for \nsaturation magnetization of the pinning layer 4 πMs(p) . (a) and (b): 4 πMs(p)  =4000 KG; (c) and \n(d): 4 πMs(p)  =16000 KG. The other calculation parameters are th e same as for Fig. 2. \n(a) and (c): red solid lines: amplitude of the inPp lane component of dynamic magnetization. \nBlue dashed lines: its phase. (b) and (d): red soli d lines: microwave electric field; blue dotted \nlines: total microwave magnetic field. The correspo nding external microwave field is applied \nfrom the surface which ensures maximum efficiency o f excitation for this mode.  All \nunspecified measurement units are arbitrary. \n \nFig.5. Comparison of the broadband FMR data with th e cavity FMR data for a singlePlayer \n100Pnm Permalloy film grown on a sapphire substrate  and having 10nmPthick gold capping \nand seed layers. Blue dashed line: film facing the broadband transducer. Red dotted line: \nsubstrate facing the transducer. Black solid line: cavity FMR data.  In order to make a valid \ncomparison, in all three cases a microwave diode, s ine wave modulation of the applied field \nand lockPin technique was used to detect absorption . The graph shows antiPderivative of the \nraw data. The data indicate a presence of magnetiza tion pinning at the film interface with the \ngold capping layer. Frequency: 9.527 GHz.   \n \nFig. 6. Measured BLS intensities and their fits wit h the theory in the Appendix. Parameters of \ncalculation are the same as for Fig. 2. The laser l ight is incident from the side of the \nmagnetically depleted layer. a) light incidence ang le θ=15° b) θ=30° , applied field H= 1000 \nOe.  21   \n \n \nApplied field (Oe) 0 500 1000 1500 \nRe(S21/S21 0)\n0.7630 0.7635 0.7640 0.7645 0.7650 0.7655 0.7660 Re(S21/S21 0)\n0.826 0.827 0.828 0.829 0.830 0.831 \n \n \n \nFig. 1  22  (a)\nX Data 0 500 1000 1500 2000 Re(S21) \n0.934 0.936 0.938 0.940 \nRe(S21) \n0.44 0.48 0.52 0.56 \n(b)\nApplied field (Oe)0 500 1000 1500 2000 Re(S21) \nP0.568 P0.564 P0.560 P0.556 P0.552 \nRe(S21) \n0.44 0.48 0.52 0.56 \n(c)\nApplied field (Oe) 0 500 1000 1500 2000 Re(S21) \n0.44 0.48 0.52 \nRe(S21) \n0.692 0.696 0.700 0.704 \n \nFig. 2   23  Wavenumber (10 5 rad/cm) 0.0 0.5 1.0 1.5 Frequency (GHz) \n810 12 14 16 18 20 22 \n \nFig. 3  24   \n(a)\nX Data 0 2e-6 4e-6 6e-6 8e-6 1e-5 Dynamic magnetization \n amplitude | m|\n040 80 120 160 \nPhase (degree) \nP90 P45 045 90 \n(b)\nX Data 0 2e-6 4e-6 6e-6 8e-6 1e-5 Microwave \nelectric field \n0.26 0.28 0.30 0.32 \nTotal microwave \nmagnetic field \n04812 16 \n(c)\nX Data 0 2e-6 4e-6 6e-6 8e-6 1e-5 Dynamic magnetization \n amplitude | m|\n040 80 120 160 \nPhase (degree) \nP90 P45 045 90 \n(d)\nCoordinate across film thickness (nm) 0 20 40 60 80 100 Microwave \nelectric field \n0.26 0.28 0.30 0.32 \nTotal microwave \nmagnetic field \n04812 16 \n \nFig. 4  \n  25  Applied field (Oe) 500 1000 1500 Absorption (arb. unit) \n0.16 0.12 0.08 0.04 0.00 \n \n \nFig. 5  26   \n \n \n(a) \nFrequency (GHz) P20 P15 P10 P5 0 5 10 15 20 BLS intensity \n0510 15 20 \nBLS intensity \n020 40 60 80 100 \n(b) \nFrequency (GHz) P20 P15 P10 P5 0 5 10 15 20 BLS intensity \n0510 15 20 \nBLS intensity \n040 80 120 160 \n \n \nFig. 6 "    },    {        "title": "1312.1404v2.Strong_impact_of_the_eddy_current_shielding_on_ferromagnetic_resonance_response_of_sub_skin_depth_thick_conducting_magnetic_multilayers.pdf",        "content": "Strong impact of the eddy–current shielding on ferromagnetic\nresonance response of sub–skin–depth–thick conducting\nmagnetic multilayers\nIvan S. Maksymov1,*, Zhaoyang Zhang1,2, Crosby Chang1, and Mikhail Kostylev1\n1School of Physics, University of Western Australia, 35 Stirling Highway, Crawley WA 6009,\nAustralia\n2University of Science and Technology of China\n*corresponding author, ivan.maksymov@uwa.edu.au\nAbstract: Exchange–coupled non–magnetic metal (NM) and ferromagnetic metal (FM) multilayers\nare crucial for microwave magnonic and spintronic devices. These layered materials usually have\ntotal  thicknesses  smaller  than  the  microwave  skin  depth.  By  using  a  stripline  broadband\nferromagnetic resonance spectroscopy technique, we experimentally demonstrate that the amplitude\nof the magnetisation precession in the FM layer is strongly diminished by the shielding effect of\nmicrowave (6–12 GHz) eddy currents circulating in the NM capping layers.\n11. Introduction\nMagnonics and spintronics are emerging nanotechnologies offering functionalities beyond\nthe current semiconductor technology.1 Consequently, there is a huge interest in the excitation,\ndetection and control of spin waves at the nano–scale,2 as well as in a number of spintronic effects\nsuch as the spin transfer torque, direct and inverse Spin–Hall effects, and spin pumping.3,4 A deep\nunderstanding of physics of these effects is crucial for the development of novel devices that\ninclude but not limited to: magnetic random access memory (MRAM), spin–torque MRAM, spin–\ntorque nano–oscillators,1,3 frequency–agile left–handed meta-materials5–10, as well as gas sensors.11\nThe  aforementioned  devices  operate  at  microwave  frequencies  and  they  are  made  of\nmultilayers consisting of non–magnetic metal (NM) and ferromagnetic metal (FM) thin films. A\nlarge amount  of literature  is devoted to the investigation  of magnetisation  dynamics  and spin\ntransport parameters of such multilayers (e.g., [1–4, 11–16] to cite just a few recent articles).\nPermalloy (Py = Ni80Fe20) is the material of choice for the FM layer because it exhibits an\noptimal combination of magnetic properties (e.g., the vanishing magnetic anisotropy and one of  the\nsmallest magnetic (Gilbert) losses  G among ferromagnetic metals).2,17–19 NM layers are usually\nmade of Ta, Cu, Au, Pt, or Pd.3,4,11–13 Ta and Cu thin films often act as a seed layer20,21 or as a\ncapping layer protecting the Py film from oxidation.22 Pt and Pd layers are used in spintronic\ndevices exploiting the spin pumping and Spin–Hall effects3,4 as well as in gas sensors.11 The spin\npumping effect in Cu layers is negligible.23 Hence, multilayers with NM  = Cu are often used as\nreference samples.15,24\nFIG. 1 (a) Schematic of a NM/FM/NM multilayer facing a microstrip line with the NM capping\nlayer of thickness  d. (b) Theoretical skin–depth   of Cu and Py as a function of the microwave\n2\nfrequency  f.  (c)  Numerically  simulated  profiles  of  the  in-plane  component  of  the  microwave\nmagnetic  field  across  the  multilayers  with  different  Cu  capping  layer  thicknesses.  The  total\nthickness of the multilayer is counted from the side facing the ML. The microwave frequency\nf = 12 GHz and the thickness of the Cu seed layer is 10  nm in all cases.\nAll the aforementioned applications rely on exposure of the metallic multi–layered materials\nto a microwave electromagnetic field. Most often this field is incident on just one of the two sample\nsurfaces. The most popular way to expose a planar sample to a microwave radiation is by using\nbroadband stripline ferromagnetic resonance (FMR) method (see, e.g., Refs.  [25–29]). In particular,\nthis method is largely used to study the magnetisation dynamics and spin current injection through\ninterfaces (spin pumping and inverse spin Hall effects).\nThe main part of a broadband–FMR (BFMR) setup is a section of a microstrip line (ML).\nThe multilayer under test sits on top of the ML [Fig.  1(a)]. A microwave current flowing through\nthe ML at a fixed frequency f imposes a microwave (Oersted) magnetic field on the multilayer. The\nresonance frequency in the multilayer is determined by a slowly scanned frequency f or external\nstatic magnetic field H. In the latter case, as the value of H is adjusted, the frequency of the natural\nmagnetisation precession resonance eventually equals the frequency of the microwave magnetic\nfield, and significant microwave power absorption occurs.\nAlthough the impact of NM layers on the linewidth ( H) of the FMR response is well–\nknown,14,15,23,30 the  effect  of  their  electrical  conductivity   has  usually  been  neglected  while\ndesigning BFMR experiments. The main reason for this is the belief that metal layers thinner than\nthe microwave skin–depth   [Fig. 1(b)] do not affect magnetisation dynamics.31 However, many\ntheoretical analyses have called this assumption into question.22,32–38 In our Refs. [35, 36], we show\ntheoretically that the in–plane component of the microwave magnetic field vanishes at the far end of\nconducting  multilayers  with  sub-skin-depth  thicknesses  [Fig.  1(c)].39 Most  significantly,  the\namplitude of the microwave magnetic field in the FM layer drops very quickly when the thickness\nof the capping layer d is increased [compare the central sections of the curves in Fig.  1(c)].\nIn  this  work,  we  present  experimental  evidence  of  a  strong and adverse  effect  of  the\nconducting  NM  (Cu) capping  layers  on the strength and profile of the FMR  response of the\nunderlying Py layer. The eddy currents circulating in the capping layer shield the Py layer from\nmicrowave magnetic fields induced by the ML transducer of the BFMR setup. We show that the\n3shielding is very strong even when the thickness of the NM capping layer  is well below the\nmicrowave skin–depth  for the NM material.\nAlthough  in  the  following  we  focus  on  the  stripline  measurements,  our  findings  are\napplicable to a broader class of situations, in which the microwave magnetic field is also incident on\none  film  surface  only.  This  possibility  of  generalisation  was  demonstrated  in  our  recent\nexperiment40 and confirmed theoretically.41 Indeed, in the conditions of the stripline FMR samples\nare exposed to a  near microwave magnetic field of a stripline. However, in Refs. [40, 41] we\nshowed that exposure of metallic ferromagnetic films samples to a  far field results in the same\nbehaviour. For instance, the response in reflection of the samples exposed to a plane  travelling\nelectromagnetic wave incident on a sample surface from free space is very similar to the stripline\nBFMR response. The similarity of the far –  and near–field responses is very important, e.g., for\nexperimental characterisation of magnetic meta–materials.5–10 Note that the far–field and near–field\nresponses are similar,  provided the stripline width is large enough as in our experiment.  The\nrequirement of a large stripline width ensures the absence of the adverse effect of travelling spin\nwave contribution to the FMR response27 and should be fulfilled in any BFMR experiment. \n2. Experimental results and discussion\nWe use a macroscopic 0.33  mm–wide ML transducer and place the sample grown on a Si\nsubstrate on top of it. As we demonstrated previously,22 for macroscopic transducers the eddy–\ncurrent effects should be very strong. We carry out BFMR measurements on magnetron sputter–\ndeposited Si/Cu[10nm]/Py[70nm]/Cu[ d] multilayers with  d = 10, 20, 35 and 70  nm. The capping\nlayer of the samples faces ML [Fig.  1(a)]. We keep the microwave frequency f constant and sweep\nthe magnetic field H applied in the sample plane and along the transducer to record the raw FMR\nabsorption traces. This procedure is repeated for several values of f is in the range 6–12  GHz with\nthe 2 GHz step. To record the traces we use a freshly calibrated microwave network analyser. The\nmeasurements are taken at room temperature.\nAlthough  in  many  experimental  situations  the  thickness  of  the  NM  capping  layers  is\n<10 nm,3,4,11–13 10 nm thick capping layers are also often used, e.g., in Pt–YIG (yttrium–iron–garnet)\nmagnetic  structures.42,43 In  this  work,  we  would  like  to  address  this  very  important  case.\nFurthermore, the impact of the eddy currents strongly depends on the frequency (see Fig.  5 in\nRef. [22]). Therefore, our findings of the effect of the 10  nm–range capping layers in the 6–12  GHz\n4frequency range may be important for experiments at ~40 GHz employing samples with d = 5 nm\nor so.\nIn order to demonstrate  the impact  of the eddy–current  shielding on results  of BFMR\nmeasurements, we study the samples with thicknesses equal to multiples of the technologically\nmeaningful value  d = 10 nm. Since our investigation is comparative in nature, it is important to\ncharacterise samples with well resolved differences in behaviour. Measuring samples with thicker\n(but still sub–skin–depth) capping layers and comparing them to the reference  d = 10 nm sample\nallows us to easily establish the functional dependence of the shielding effect on the layer thickness.\nFor the same reason of maximising measurement accuracy we also use a thick Py layer, although\nthe theory in Ref. [35] predicts a significant impact of eddy currents for much thinner samples, too\n(see Fig. 6 in Ref. [35]).\nThe  absorption  spectrum  of  the  ML  is  obtained  as  a  ratio  of  the  complex  scattering\nparameter S21 for the loaded ML (with a sample on top) to S 21 of the unloaded ML. Figure 2(a)\nshows typical absorption spectra obtained using this post–processing procedure for the multilayer\nwith d = 20 nm. One sees dips in the absorption spectra taken at different f. The amplitude of these\ndips corresponds to the amplitude of the respective FMR response of the multilayer. Note that at\nf = 12 GHz one also observes a smaller dip at a lower applied magnetic field (~1.1  kOe), which can\nbe identified as the first higher–order standing spin wave mode (1st SSWM).35\nFIG. 2 (a) Experimental FMR absorption spectra of the multilayer with d = 20 nm for f = 6, 8, 10\nand 12 GHz (from top to bottom respectively). For clarity, each spectrum is vertically offset by –\n0.015. (b) Experimental  full width at half maximum  line width  H of the FMR response of the\nmultilayer with d = 20 nm as a function of f. The straight line is the best fit of the experimental data.\nNote an extra absorption peak of small amplitude in the lowest trace of Panel (a) (at ~1.1  kOe). This\nextra peak is the response of the 1st SSWM.\n5\nFigure 3 shows the experimental and theoretical amplitudes of the FMR response of the Cu–\nPy samples. All curves are normalised to the amplitude of the sample with d = 10 nm at 6 GHz. We\nobserve that the amplitude drops very quickly as d is increased. This result confirms our previous\ntheoretical predictions.36 The decrease in the amplitude is due to eddy currents circulating in the\ncapping  layers,  which  shield  the  ferromagnetic  Py  film  from  the  microwave  magnetic  field\n[Fig. 1(c)].\nFIG. 3 Experimental (symbols) and theoretical (curves) relative amplitudes of the FMR response of\nthe Cu–Py multilayer  samples  as a function of  f. All curves are normalised to the absorption\namplitude of the sample with d = 10 nm at 6 GHz. Parameters of simulations are from Table 1 and\nthe main text. Some disagreement between the theory and the experiment is attributed to the effects\nof imprecise sample placement on the microstrip line and the sample dimensions29 as well as\nasymmetry of the FMR line shape,44 which could not be taken into account in calculations.\nWe  also  obtain  the  values  of  the  gyromagnetic  ratio  /(2)  and  of  the  saturation\nmagnetisation for the FM layer (4 Ms) by best–fitting experimental data with the Kittel formula45 \ns( 4 )2f H H Mgpp= + . (1)\n6\nThese  values  are  used  to  extract  the  values  of  the  Gilbert  damping  parameter  G.  From  the\nmeasured absorption spectra we extract the full width at half maximum  line width H as a function\nof f and fit the obtained dependence with a straight line46 \n()02\n/ 2fH Ha\ng pD = + D . (2)\nThe parameters extracted from experimental absorption traces are presented in Table 1.\nTable I. Parameters extracted from experimental FMR absorption traces for different values of d.\nd, nm 10203570\n), MHz/Oe 2.9982.9963.0253.030\n4Ms, G8003903386768790\nG0.0075 0.0080.00860.0106\nH(f=10 GHz), Oe 90.780.62100.8120.9\nH0, Oe40.8426.4433.7148.95\nFrom Table I one sees that we have obtained typical values of /(2), 4Ms, and G for Py.\n2,17–19,22 It is not unusual that the saturation magnetisation for Py samples differs considerably from\nthe standard value 4 Ms = 10800 G. Values in the range from 8000  G to the standard value were\nfound by different authors (see, e.g., Refs. [47–49]). Also, we note that for  d = 10 nm we find\nG = 0.0075, which is slightly lower than but very close to the typical value for Py: 0.008.19\nThe damping parameters for the d = 10nm–thick film are somewhat different from the other\nthree samples. Indeed, for the remaining samples one observes a clear trend: an increase in  d\ncorrelates with an increase in G and H0. Furthermore, this trend possibly suggests that the eddy\ncurrents  may  also  contribute  to  the  frequency independent  part  of the  FMR  losses  H0. The\ncorrelation between  G and the thickness of the FM film is well–known.31,38 However, in our\nexperiment the thickness of the FM is constant but the thickness of the NM capping layer is varied.\nThe clear correlation of the loss parameters with d also suggests that the magnetic quality of\nthe Py layer of the d = 10nm–thick film is worse than for the other films. This is seen not only from\nthe large H(f=10 GHz) value for it, but also from the noticeably smaller 4 Ms than for the other\nfilms. This is potentially the reason why the magnetic losses for this film do not follow this trend.\n7We employ the semi–analytical theory from our Ref. [35] to calculate the FMR amplitude as\na function of  d. We use the data from Table 1 as input parameters. Firstly, we use the bulk Cu\nconductivity for both NM layers: bulk = 5.96×107 S/m. This calculation delivers a d–dependence of\nthe FMR amplitude which is significantly steeper than the experimental one.\nHowever,  when  we  assume  that  the  conductivity  of  the  capping  layer  is  0.4 bulk for\nd = 70 nm,  35 nm  and  20 nm,  and  0.42bulk for  d = 10 nm,  where  bulk = 5.96×107 S/m  is  the\nconductivity of bulk Cu, our result is in reasonably good agreement with the experiment (Fig.  3).\nImportantly, the simulated raw traces show shapes very similar to ones in Fig. 2(a) – characterised\nby a very small amplitude of the 1st SSWM with respect to the fundamental mode.\nNote that in the second calculation we need to keep the conductivity of the seed Cu layer\nequal to bulk. Otherwise the calculation delivers much larger relative amplitudes of the 1st SSWM.\nThe  increased  amplitude  of  the  1st SSWM  is  due  to  strong  non–uniformity  of  the  in–plane\nmicrowave magnetic field inside the Py layer. The non–uniformity increases with an increase in the\neddy–current density inside this layer. The smaller the conductivity of the seed layer, the larger is\nthe eddy current inside the Py layer. Hence, the practically vanishing amplitude of the 1st SSWM in\nthe experimental traces may be considered as an evidence of a noticeably larger conductivity of the\nseed layer with respect to the capping layer: a large current flowing inside the seed layer makes the\nmicrowave  eddy–current  field  in  the  Py  layer  more  uniform.  Indeed,  it  has  been  shown\nexperimentally that the electric resistivity of Cu layers strongly depends on the layer on which they\ngrow.50 More precisely, a Cu layer grown on top of a Py layer may have significantly smaller\nconductivity than the one grown on a bare substrate.20\n3. Conclusions\nWe have experimentally investigated the broadband FMR response of metallic magnetic\nmultilayer structures and demonstrated a crucial effect of non–magnetic metallic capping layers of\nsub–skin–depth  thicknesses  on  the  strength  of  this  response.  Eddy  currents  circulating  in  the\ncapping layers shield the ferromagnetic Py film from the microwave magnetic field. The shielding\nleads  to  a  strong  decrease  in  the  amplitude  of  the  FMR  response.  This  finding  has  direct\nimplications for microwave characterisation of magnetic materials, including materials for spin–\ntransport applications. They are also important for applications of ferromagnetic films in frequency\nagile meta–materials. Very often in those experiments, microwave power is incident on just one of\nthe two sample surfaces and is spread over an area at least 100  m×100 mm in size. These are\n8experimental conditions for which our results are relevant. For instance, these conditions are usually\nmet in the case of the broadband stripline FMR.\nAcknowledgements\nThis work was supported by the UPRF scheme of the University of Western Australia and the\nAustralian Research Council.\nReferences\n[1] R. L. Stamps, S. Breitkreutz, J. Åkerman, A. V. Chumak, Y.-C. Otani, G. E W Bauer, J.-U.\nThiele, M. Bowen, S. A. Majetich, M. Kläui, I. L. Prejbeanu, B. Dieny, N. M. Dempsey, and B.\nHillebrands, J. Phys. D: Appl. Phys. 47, 333001 (2014).\n[2] V. V. Kruglyak, S. O. Demokritov, and D. Grundler, J. Phys. D.: Appl. Phys.  43, 264001\n(2010).\n[3] A. Brataas, A. D. Kent, and H. Ohno, Nat. Mater. 11, 372 (2012).\n[4] T. Jungwirth, J. Wunderlich, and K. Olejník, Nature Mater. 11, 382 (2012).\n[5] R. X. Wu, J. Appl. Phys. 97, 076105 (2005).\n[6] J. Chen, D. Tang, B. Zhang, Y. Yang, M. Lu, and H. Lu, J. Phys. Condens. Matt.  18, 6421\n(2006).\n[7] J. Chen, D. Tang, B. Zhang, Y. Yang, M. Lu, and H. Lu, J. Magn. Magn. Mat.  321, 2139\n(2009).\n[8] G. B. G. Stenning, G. J. Bowden, L. C. Maple, S. A. Gregory, A. Sposito, R. E. Eason, N. I.\nZheludev, and P. A. J. de Groot, Opt. Express 21, 1456 (2013).\n[9] S. A. Gregory, G. B. G. Stenning, G. J. Bowden, N. I. Zheludev, and P. A. J. de Groot, New. J.\nPhys. 16, 063002 (2014).\n[10] B. Bhoi, T. Cliff, I. S. Maksymov, M. Kostylev, R. Aiyar, N. Venkataramani, S. Prasad, and R.\nL. Stamps, arXiv:1409.5499 [cond-mat.mtrl-sci].\n[11] C. S. Chang, M. Kostylev, and E. Ivanov, Appl. Phys. Lett. 102, 142405 (2013).\n[12] H. Ulrichs, V. E. Demidov, S. O. Demokritov, W. L. Lim, J. Melander, N. Ebrahim-Zadeh, and\nS. Urazhdin, Appl. Phys. Lett. 102, 132402 (2013)\n[13] B. F. Miao, S. Y. Huang, D. Qu, and C. L. Chien, Phys. Rev. Lett. 111, 066602 (2013)\n[14] M. Harder, Z. X. Cao, Y. S. Gui, X. L. Fan, and C.-M. Hu, Phys. Rev. B 84, 054423 (2011).\n9[15] C. T. Boone, H. T. Nembach, J. M. Shaw, and T. J. Silva, J. Appl. Phys. 113, 153906 (2013).\n[16] W. E. Bailey, C. Cheng, R. Knut, O. Karis, S. Auffret, S. Zohar, D. Keavney, P. Warnicke, J.-\nS. Lee, and D. A. Arena, Nature Commun. 4, 2025 (2013).\n[17] J. P. Park, P. Eames, D. M. Engebretson, J. Berezovsky, and P. A. Crowell, Phys. Rev. Lett.\n89, 277201 (2002).\n[18] T. Verduci, C. Rufo, A. Berger, V. Metlushko, B. Ilic, and P. Vavassori, Appl. Phys. Lett. 99,\n092501 (2011).\n[19] S. Ingvarsson, L. Ritchie, X. Y. Liu, G. Xiao, J. C. Slonczewski, P. L. Trouilloud, and R. H.\nKoch, Phys. Rev. B 66, 214416 (2002).\n[20] M. Kowalewski, W. H. Butler, N. Moghadam, G. M. Stocks, T. C. Schulthess, K. J. Song, J. R.\nThompson, A. S. Arrott, T. Zhu, J. Drewes, R. R. Katti, M. T. McClure, and O. Escorcia, J. Appl.\nPhys. 87, 5732 (2000).\n[21] L. Lu, J. Young, M. Wu, C. Mathieu, M. Hadley, P. Krivosik, and N. Mo, Appl. Phys. Lett.\n100, 022403 (2012).\n[22]  K.  J.  Kennewell,  M.  Kostylev,  N.  Ross,  R.  Magaraggia,  R.  L.  Stamps,  M.  Ali,  A.  A.\nStashkevich, D. Greig, and B. J. Hickey, J. Appl. Phys. 108, 073917 (2010).\n[23] S. Mizukami, Y. Ando, T. Miyazaki, J. Magn. Magn. Mater. 226-230, 1640 (2001).\n[24] D.-H. Kim, H.-H. Kim, and C.-Y. Youa, Appl. Phys. Lett. 99, 072502 (2011).\n[25] A. Fessant, J. Geraltowski, J. Loaec, and H. Legall, J. Magn. Magn. Mater. 83, 557 (1990).\n[26] T. J. Silva, C. S. Lee, T. M. Crawford, and C. T. Rogers, J. Appl. Phys. 85, 7849 (1999).\n[27] G. Counil, J.-V. Kim, T. Devolder, C. Chappert, K. Shigeto, and Y. Otani, J. Appl. Phys. 95,\n5646 (2004).\n[28] S. S. Kalarickal, P. Krivosik, M. Wu, C. E. Patton, M. L. Schneider, P. Kabos, T. J. Silva, and\nJ. P. Nibarger, J. Appl. Phys. 99, 093909 (2006).\n[29] C. Bilzer, T. Devolder, P. Crozat, C. Chappert, S. Cardoso, and P. P. Freitas, J. Appl. Phys.\n101, 074505 (2007).\n10[30] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. Lett. 88, 117601 (2002).\n[31] M. Charilaou, K. Lenz, and W. Kuch, J. Magn. Magn. Mater. 322, 2065 (2010).\n[32] N. S. Almeida and D. L. Mills, Phys. Rev. B 53, 12232 (1996).\n[33] Ngoc Chan, V. Kambersky, and D. Fraitová, J. Magn. Magn. Mater. 214, 93 (2000).\n[34] M. Vroubel and B. Rejaei, J. Appl. Phys. 103, 114906 (2008).\n[35] M. Kostylev, J. Appl. Phys. 106, 043903 (2009).\n[36] I. S. Maksymov and M. Kostylev, J. Appl. Phys. 113, 043927 (2013).\n[37] M. Bailleul, Appl. Phys. Lett. 103, 192405 (2013).\n[38] I. S. Maksymov and M. Kostylev, J. Phys. D: Appl. Phys. 46, 495001 (2013).\n[39] The in–plane wave vector is k = 0 only when the ML transducer is infinitely wide. In practice,\nk ≈ 0 is achieved with macroscopic transducers, such as a 0.33  mm–wide ML used in this work.16, 29\n[40] M. Kostylev, J. Appl. Phys. 113, 053908 (2013).\n[41] M. Kostylev, J. Appl. Phys. 112, 093901 (2012).\n[42] C. W. Sandweg, Y. Kajiwara, A. V. Chumak, A. A. Serga, V. I. Vasyuchka, M. B. Jungfleisch,\nE. Saitoh, and B. Hillebrands, Phys. Rev. Lett. 106, 216601 (2011).\n[43] C. Hahn, G. de Loubens, O. Klein, M. Viret, V. V. Naletov, and J. Ben Youssef, Phys. Rev. B\n87, 174417 (2013).\n[44] A. E. Miroshnichenko, S. Flach, and Yu. S. Kivshar, Rev. Mod. Phys. 82, 2257 (2010).\n[45] D. D. Stancil and A. Prabhakar, Spin Waves: Theory and Application , Springer, Berlin, 2009.\n[46] R. D. McMichael and P. Krivosik, IEEE Trans. Magn. 40, 2 (2004).\n[47] Y. Roussigné, S. M. Cherif, and P. Moch, J. Phys.: Condens. Matter 16, 4591 (2004).\n[48] A. A. Stashkevich, P. Djemia, Y. K. Fetisov, N. Bizière, and C. Fermon, J. Appl. Phys. 102,\n103905 (2007).\n[49] S. Tacchi, M. Madami, G. Gubbiotti, G. Carlotti, S. Goolaup, A. O. Adeyeye, H. T. Nguyen,\nand M. G. Cottam, J. Appl. Phys. 105, 07C102 (2009).\n11[50] V. S. Speriosu, J. P. Nozieres, B. A. Gurney, B. Dieny, T. C. Huang, and H. Lefakis, Phys.\nRev. B 47, 11579 (1993).\n12"    },    {        "title": "1203.0114v1.Current_induced_resonance_in_a_ferromagnet___antiferromagnet_junction.pdf",        "content": "arXiv:1203.0114v1  [cond-mat.mtrl-sci]  1 Mar 2012Current-induced resonance\nin a ferromagnet–antiferromagnet junction\nS. G. Chigarev1, E. M. Epshtein1∗, Yu. V. Gulyaev1,\nV. D. Kotov1, G. M. Mikhailov2, P. E. Zilberman1\n1V. A. Kotelnikov Institute of Radio Engineering and Electronics\nof the Russian Academy of Sciences, Fryazino, 141190, Russia\n2Institute of Microelectronics Technology and High Purity Materials\nof the Russian Academy of Sciences, Chernogolovka, 142432, Rus sia\nAbstract\nWe calculate the response of a ferromagnet–antiferromagne t junction\nto a high-frequency magnetic ﬁeld as a function of the spin-p olarized cur-\nrent through the junction. Conditions are choused under whi ch the re-\nsponse is zero in absence of such a current. It is shown that in creasing\nin the current density leads to proportional increase in the resonance fre-\nquency and resonant absorption. A principal possibility is indicated of\nusing ferromagnet–antiferromagnet junction as a terahert z radiation de-\ntector.\n1 Introduction\nOn a level with “conventional” spintronics studying eﬀects in ferromag-\nnet/ferromagnet junctions, a new direction has emerged whi ch is related\nwith spin-polarized current eﬀect on the antiferromagneti c layer in ferro-\nmagnet/antiferromagnet junction [1]–[14], so that a term “ antiferromag-\nnetic spintronics” has appeared [1, 8].\nThe interest in studying magnetic junctions with antiferro magnetic\nlayers is related with the following features of antiferrom agnets. First,\nthis is low, compared to ferromagnets, magnetization in mag netic ﬁelds\nmuch lower than the exchange ﬁeld. This allows to neglect dem agneti-\nzation eﬀect and, that is more signiﬁcant, leads to substant ially lower\nvalues of magnetic ﬁelds and currents at which switching eﬀe cts occur. In\ncontrast with ferromagnets, where spontaneous magnetizat ion exists even\nin absence of magnetic ﬁelds and currents, so that the role of the latter\nconsists in changing the magnetization direction, in the an tiferromagnets\n∗E-mail: epshtein36@mail.ru\n1/s70/s77 /s65/s70/s77/s78/s77\n/s77\n/s70/s110\n/s120/s121/s122\n/s48/s76\n/s65 /s70/s77/s106/s47/s101/s72 /s40/s116/s41\nFigure 1: Scheme of the ferromagnet (FM)–antiferromagnet (AF M) junction;\nNM being a nonmagnetic layer. The main vector directions are shown.\nwith mutually compensated magnetic sublattices existence of the result-\ning magnetic moment is due to magnetic ﬁelds and/or currents . Second,\nthe eigenfrequency of the magnetic oscillation in antiferr omagnets exceeds\nthe similar frequency in ferromagnets by several orders of m agnitude, so\nthat the range of possible using of antiferromagnetic struc tures extends\nup to terahertz (THz) frequencies.\nOne of the directions in spintronics is studying the spin-po larized cur-\nrent eﬀect on ferromagnetic resonance in magnetic junction s [15]–[24].\nThe current eﬀect on the spectrum and damping of the magnetiz ation\noscillation in antiferromagnets was studied in Refs. [13, 1 4]. It was shown\nthat the spin-polarized current eﬀect leads to decrease in d amping and\nto instability of the antiparallel conﬁguration with switc hing to parallel\none. A possibility was noted of creating canted antiferroma gnet conﬁgura-\ntion with appearance of net magnetization under spin-polar ized electron\ninjection without external magnetic ﬁeld.\nIn present article, we consider forced oscillation of the an tiferromagnet\nmagnetization under high-frequency magnetic ﬁeld in prese nce of spin-\npolarized current. The conditions are choused under which t he response\nto the high-frequency is zero when such a current is absent. U nder such\nconditions, the current exerts a strong eﬀect on the antifer romagnet res-\nonant characteristics. Such a current-driven resonator ma y be used, in\nprinciple, as a THz detector.\n2 The model and basic equations\nLet us consider a magnetic junction consisting of a pinned fe rromagnetic\n(FM) layer, a free antiferromagnetic (AFM) layer, and nonma gnetic (NM)\n2layerclosingtheelectric circuit(Fig.1). Athinspaceris supposedbetween\nFMandAFMlayerstopreventexchangeinteractionthroughth einterface.\nThecurrentﬂowsperpendiculartothelayers(along xaxis)inthedirection\ncorresponding to electron ﬂux from ferromagnet to antiferr omagnet. The\neasy axis of the antiferromagnet lies in the layer plane (alo ngyaxis), the\nferromagnet magnetization vector MFis parallel to zaxis.\nThe AFM layer thickness LAFMis assumed to be small compared to\nthe spin diﬀusion length, so that the macrospin approximati on is valid.\nIn this approximation, the layer magnetization is supposed to be uniform\nin thickness, while the spin current through the interface i s taking into\naccount by means of additional terms in the equations (see Re fs. [25,\n13] for details). A simplest AFM model is considered with two collinear\nequivalent sublattices with equal magnetizations, |M1|=|M2|=M0.\nThe Landau–Lifshitz–Gilbert equations for the AFM layer in the pres-\nence of spin-polarized current and high frequency magnetic ﬁeld take the\nfollowing form (see detailed derivation in Ref. [13]):\ndM\ndt−1\n2κ\nM0/braceleftbigg/bracketleftbigg\nM×dM\ndt/bracketrightbigg\n+/bracketleftbigg\nL×dL\ndt/bracketrightbigg/bracerightbigg\n+γ[M×H(t)]\n+1\n2γ(β+β′)(M·n)[M×n]+1\n2γ(β−β′)(L·n)[L×n]\n+K/bracketleftBig\nM×/bracketleftBig\nM×ˆMF/bracketrightBig/bracketrightBig\n+P/bracketleftBig\nM×ˆMF/bracketrightBig\n= 0, (1)\ndL\ndt−1\n2κ\nM0/braceleftbigg/bracketleftbigg\nL×dM\ndt/bracketrightbigg\n+/bracketleftbigg\nM×dL\ndt/bracketrightbigg/bracerightbigg\n+γ[L×H(t)]−γΛ[L×M]\n+1\n2γ(β+β′)(M·n)[L×n]+1\n2γ(β−β′)(L·n)[M×n]\n+K/bracketleftBig\nL×/bracketleftBig\nM×ˆMF/bracketrightBig/bracketrightBig\n+P/bracketleftBig\nL×ˆMF/bracketrightBig\n= 0. (2)\nHere,M=M1+M2is the AFM magnetization vector, L=M1−M2\nis the antiferromagnetism vector, ˆMFis the unit vector along the FM\nlayer magnetization, nis the unit vector along the anisotropy axis, H(t)\nis the external magnetic ﬁeld, κis the damping coeﬃcient, Λ is the uni-\nform exchange constant, β, β′are the intrasublattice and intersublattice\nanisotropy constants, respectively ( β > β′is assumed), γis the gyromag-\nnetic ratio,\nK=µBQ\neLAFMM2j, (3)\nP=γαsdµBτQ\neLAFMj, (4)\njis the current density, µBis the Bohr magneton, αsdis the (dimension-\nless)sdexchange interaction constant, τis the spin relaxation time in the\nAFM layer, Qis the FM conductivity spin polarization, eis the electron\ncharge.\nThe last two terms in the right-hand sides of Eqs. (1) and (2) d escribe\n(in the macrospin approximation) the spin-polarized curre nt eﬀect on the\nantiferromagnet magnetic conﬁguration. There are two mech anisms of\n3this eﬀect. One of them [26, 27] is due to relaxation of the non collinear\n(with respect to the AFM magnetization) component of the ele ctron spins\nwith transfer corresponding torquetothelattice. This occ urs within a dis-\ntancecomparable withtheFermiwavelengthfromtheFM–AFM i nterface.\nThe injected spins collinear to AFM magnetization with lost transverse\ncomponent remain in nonequilibrium state within much longe r distance\nof the order of the spin diﬀusion length. Such a state is energ etically un-\nfavorable. This can lead to change of the lattice magnetic co nﬁguration\nwith transition to more favorable state. This is the second m echanism\nof the spin-polarized electron interaction with magnetic l attice [28, 29].\nThese mechanisms are described by the terms with KandPcoeﬃcients,\nrespectively. It is seen from Eqs. (1) and (2) that the second mechanism\nis equivalent to the presence of an additional magnetic ﬁeld PˆMF/γpar-\nallel to the FM magnetization vector. The latter circumstan ce leads to\nappearance of a current-induced AFM canted state in absence of external\nmagnetic ﬁeld.\nIn the conﬁguration described, ˆMF={0,0,1},n={0,1,0}. It is\nsuggested that external dc magnetic ﬁeld is absent, while a h igh-frequency\nmagnetic ﬁeld is parallel to the anisotropy axis, H(t) ={0, H0cosωt,0}\nwithH0≪HE, where HE≡ΛM0is the exchange ﬁeld. Under such\nconditions, the AFM magnetization is zero in absence of the c urrent\n(j= 0). Correspondingly, the magnetic susceptibility compon entχyyre-\nsponsible for absorption of the high-frequency ﬁeld with th e polarization\nindicated [30] is zero, too, so that AFM resonance does not oc cur.\n3 Spin-polarized current-driven resonance\nin antiferromagnet\nAs it was mentioned, antiferromagnet magnetization appear s along the\nferromagnet magnetization vector ˆMFunder spin-polarized current. Pre-\ncession of the antiferromagnet magnetization vector makes possible the\nresonance absorption.\nLet us calculate the antiferromagnet magnetization with us ing Eqs. (1)\nand (2). The high-frequency ﬁeld is assumed to be low, is take n into ac-\ncount in scope of the linear approximation, and, hence, does not inﬂuence\nthe static magnetization, which is [13]\nM={0,0,Mz},Mz=P\nγ/parenleftbig\nΛ+1\n2(β−β′)/parenrightbig≈P\nγΛ.(5)\nThe corresponding antiferromagnetism vector is\nL={0,Ly,0},Ly=/radicalBig\n4M2\n0−M2\nz. (6)\nTo calculate the response to the (low) high-frequency ﬁeld, we linearize\nEqs. (1), (2)in small deviations from M,L. The following set of equations\nis obtained:\n∂Mx\n∂t−1\n2κ\nM0/braceleftbigg\n−Mz∂My\n∂t+Ly∂Lz\n∂t/bracerightbigg\n+PMy\n4−1\n2γ(β+β′)MzMy−1\n2γ(β−β′)LyLz+KMzMx\n=γMzH(t), (7)\n∂My\n∂t−1\n2κ\nM0Mz∂Mx\n∂t−PMx+KMzMy= 0, (8)\n∂/tildewiderMz\n∂t+1\n2κ\nM0Ly∂Lx\n∂t+1\n2γ(β−β′)LyLx= 0, (9)\n∂Lx\n∂t−1\n2κ\nM0/braceleftBigg\nLy∂/tildewiderMz\n∂t−Mz∂/tildewideLy\n∂t/bracerightBigg\n−γ/braceleftbigg\nΛ+1\n2(β−β′)/bracerightbigg\nLy/tildewiderMz= 0, (10)\n∂/tildewideLy\n∂t−1\n2κ\nM0Mz∂Lx\n∂t−1\n2γ(β−β′)MzLx= 0, (11)\n∂Lz\n∂t+1\n2κ\nM0Ly∂Mx\n∂t+γ/braceleftbigg\nΛ+1\n2(β−β′)/bracerightbigg\nLyMx+KMzLz= 0,(12)\nwhere/tildewiderMz=Mz−Mz,/tildewideLy=Ly−Ly.\nWe seek a solution in the form of forced oscillation with freq uencyω\nof the external magnetic ﬁeld. We ﬁnd\nMx(ω) =−(−iω+P/η)PH0\nΛ(ω2−Ω2+2iνω)≡χxy(ω)H0, (13)\nMy(ω) =−P2H0\nΛ(ω2−Ω2+2iνω)≡χyy(ω)H0, (14)\nΩ2= 2γ2HAHE+P2+/parenleftbiggγHE\nη/parenrightbigg2\n, (15)\nν=γHE/parenleftbigg\nκ+1\nη/parenrightbigg\n, (16)\nwhereHA= (β−β′)M0is the anisotropy ﬁeld, η=αγM0τ.\nAbsorption of the high-frequency ﬁeld is determined by the i maginary\npart of the diagonal susceptibility\nχ′′\n/bardbl≡Imχyy(ω) =2νωP2\nΛ/bracketleftbig\n(ω2−Ω2)2+4ν2ω2/bracketrightbig. (17)\nThe maximal absorption corresponds to the resonance freque ncy\nωres=/radicalbig\nΩ2−2ν2. (18)\nThe Q-factor of the system is\nQ=Ω\n2ν. (19)\n5/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s48/s44/s48/s48/s49/s48/s44/s48/s48/s50/s48/s44/s48/s48/s51/s48/s44/s48/s48/s52/s48/s44/s48/s48/s53\n/s32 /s50 \n/s48 \n/s48 /s65/s70/s77 \n/s66 /s101/s76 /s77 \n/s106\n/s81 /s32/s106/s47/s106\n/s48/s32/s61/s32/s32/s48/s46/s50/s53 /s32/s40/s49/s41\n/s32/s106/s47/s106\n/s48/s32/s61/s32/s32/s48/s46/s53/s32 /s32/s32/s40/s50/s41\n/s32/s106/s47/s106\n/s48/s32/s61/s32/s32/s49 /s32/s32/s32/s32/s32/s40/s51/s41\n/s32/s106/s47/s106\n/s48/s32/s61/s32/s32/s50 /s32/s32/s32/s32/s32/s40/s52/s41\n/s49/s50/s51/s52\n/s47 /s77\n/s48/s124 /s124 /s39/s39\nFigure 2: Imaginary part of the diagonal susceptibility χ′′\n/bardblas a function of the\ndimensionless frequency ω/γM 0under various (dimensionless) current densities\nj/j0.\nIt follows from Eqs. (15) and (16) that the resonance frequen cy and Q-\nfactor rise under increase in current.\nThe power absorbed in a unit volume is [30]\nW=1\n2ωχ′′\n/bardblH2\n0, (20)\nwhile thelinear absorption coeﬃcient for an electromagnet ic wave incident\non the layer is\nΓ =8πW\ncH2\n0= 4πqχ′′\n/bardbl, (21)\nwherecis the light velocity, q=ω/cis the wavenumber of the incident\nelectromagnetic wave.\n4 Discussion\nLet us make numerical estimates using the following paramet er values:\nM0∼103G, Λ∼103,α∼104,β∼β′∼10−1,κ∼10−2,τ∼10−12\ns,LAFM∼10−6cm. We ﬁnd HE∼106G,HA∼102G,ν∼1011s−1,\nΩ∼1011s−1,η∼102. As a scale of the current density, we choose the\nquantity\nj0=eLAFMγM2\n0\nµBQF, (22)\n6/s48 /s49 /s50 /s51 /s52/s48/s44/s48/s48/s49/s48/s44/s48/s48/s50/s48/s44/s48/s48/s51/s48/s44/s48/s48/s52/s48/s44/s48/s48/s53\n/s32 /s50 \n/s48 \n/s48 /s65/s70/s77 \n/s66 /s101/s76 /s77 \n/s106\n/s81 /s32 /s47 /s77\n/s48/s32/s61/s32/s32/s50/s48 /s32/s40/s49/s41\n/s32 /s47 /s77\n/s48/s32/s61/s32/s32/s53/s48/s32 /s40/s50/s41\n/s32 /s47 /s77\n/s48/s32/s61/s32/s49/s48/s48 /s32/s40/s51/s41\n/s32 /s47 /s77\n/s48/s32/s61/s32/s50/s48/s48 /s32/s40/s52/s41\n/s49/s50/s51/s52/s124 /s124 /s39/s39\n/s106/s47/s106\n/s48\nFigure 3: Imaginary part of the diagonal susceptibility χ′′\n/bardblas a function of\nthe dimensionless current density j/j0at various (dimensionless) frequencies\nω/γM 0.\nso that\nP=ηγM0j\nj0. (23)\nWith indicated parameter values, j0∼107A/cm2. Atj∼j0we have\nΩ≈P, i.e., the eigenfrequency is proportional to the current de nsity. The\nsame applies to the resonant absorption.\nThe absorption spectrum ( χ′′\n/bardblas a function of the dimensionless fre-\nquencyω/γM 0) with various current densities is shown in Fig. 2. It is\nseen, that the resonance frequency and resonant absorption rise propor-\ntionally tothecurrent density. At j∼j0, we havetheresonance frequency\nabout 1012c−1, that corresponds to THz range.\nThe absorption as a function of the current density at variou s fre-\nquencies has the similar form (see Fig. 3 where the same dimen sionless\nvariables are used).\nAt Ω = 1012s−1, ν= 1011s−1we have Q= 5. (For comparison: the\nQ-factor of free oscillation without current\nQ0=1\nκ/radicalbigg\nHA\n2HE(24)\nis less than 1.) The Q-factor rises under increase in the freq uency and/or\ncurrent density.\n7/s48 /s49 /s50 /s51 /s52/s45/s48/s44/s48/s50/s48/s44/s48/s48/s48/s44/s48/s50/s48/s44/s48/s52/s48/s44/s48/s54/s48/s44/s48/s56/s48/s44/s49/s48\n/s106/s47/s106\n/s48/s40/s82\n/s48/s32/s45/s32 /s82 /s41/s47/s40/s49/s45/s32 /s82\n/s48/s41\n/s49/s50/s51/s52\nFigure 4: The current-induced relative change of the reﬂection co eﬃcient as a\nfunction of the (dimensionless) current density. The notations ar e the same as\nin Fig. 3.\nFor THz radiation ( q∼102cm−1) atj∼j0the absorption coeﬃcient\nis Γ∼10 cm−1, so that the absorption within the thickness of the AFM\nlayer is quite small, ∼10−5–10−4. Toovercome this diﬃculty, amultilayer\nstructure of alternating ferromagnet–antiferromagnet la yers may be used\nwith electromagnetic wave incident from the butt side (alon gzaxis). The\nreﬂection coeﬃcient of the normally incident wave is [31]\nR=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/tildewiden−/tildewideµ\n/tildewiden+/tildewideµ/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n, (25)\nwhere/tildewiden=n+ikis the complex index of refraction, /tildewideµ=µ′+iµ′′=\n1+4π(χ′+iχ′′) is the complex magnetic permeability.\nIn long wavelength range [31]\nn≈k≈/radicalbigg\n2πσ0\nω≫1,|µ|, (26)\nwhereσ0is the static conductivity. Therefore, Eq. (25) can written as\nR=R0−4π(1−R0)(χ′+χ′′), (27)\nwhereR0is the reﬂection coeﬃcient in absence of the spin-polarized cur-\nrent when χ′=χ′′= 0 under geometry in consideration (see Eq. (14)).\n8The current-induced relative change of the reﬂection coeﬃc ient (R0−\nR)/(1−R0) = 4π(χ′+χ′′) as a function of the (dimensionless) current\ndensity is shown in Fig. 4. It is seen that ∆ R=R−R0is of the order\nof 10−3–10−2atj∼j0(i.e.∼107A/cm2at chosen parameters). The\ndesired resonance signal can be extracted by the current mod ulation.\n5 Conclusion\nThe results indicate a possibility of a new eﬀect, namely, cu rrent-induced\nresonance in ferromagnet–antiferromagnet junctions. The resonance fre-\nquency and resonant absorption are proportional to the curr ent density\nthroughthejunction. This opensaprincipal possibility of usingsuchjunc-\ntions as current-controlled resonant detectors for THz rad iation. Making\nof corresponding experiments seems to be interesting.\nThe work was supported by the Russian Foundation for Basic Re -\nsearch, Grant No. 10-02-00030-a.\nReferences\n[1] A.S. N´ u˜ nez, R.A. Duine, P. Haney, A.H. MacDonald, Phys . Rev. B\n73, 214426 (2006).\n[2] Z. Wei, A. Sharma, A.S. Nunez, P.M. Haney, R.A. Duine, J.B ass,\nA.H. MacDonald, M. Tsoi, Phys. Rev. Lett. 98, 116603 (2007).\n[3] S. Urazhdin, N. Anthony, Phys. Rev. Lett. 99, 046602 (2007).\n[4] P.M. Haney, R.A. Duine, A.S. N´ u˜ nez, A.H. MacDonald, J. Magn.\nMagn. Mater. 320, 1300 (2008).\n[5] H. Gomonay, V. Loktev, J. Magn. Soc. Jpn. 32, 535 (2008).\n[6] Z. Wei, A. Sharma, J. Bass, M. Tsoi, J. Appl. Phys. 105, 07D113\n(2009).\n[7] H.V. Gomonay, V.M. Loktev, Phys. Rev. B 81, 144127 (2010).\n[8] A.B. Shick, B. Khmelevskyi, O.N. Mryasov, J. Wunderlich , T. Jung-\nwirth, Phys. Rev. B 81, 212409 (2010).\n[9] K.M.D. Hals, Y. Tserkovnyak, A. Brataas, Phys. Rev. Lett .106,\n107206 (2011).\n[10] H.V. Gomonay, R.V. Kunitsyn, V.M. Loktev, arXiv:1106. 4231.v2\n[cond-mat.mtrl-sci].\n[11] Yu.V.Gulyaev, P.E.Zilberman, E.M. Epshtein,J.Commu n.Technol.\nElectron. 56, 863 (2011).\n[12] J. Linder, Phys. Rev. B 84, 094404 (2011).\n[13] Yu.V. Gulyaev, P.E. Zilberman, E.M. Epshtein, J. Exp. T heor. Phys.\n114, 296 (2012).\n[14] Yu.V.Gulyaev, P.E.Zilberman, E.M. Epshtein,J.Commu n.Technol.\nElectron. 57, (2012) (to be published).\n9[15] G.D. Fuchs, J.C. Sankey, V.S. Pribiag, L. Qian, P.M. Bra ganca,\nA.G.F. Garcia, E.M. Ryan, Zhi-Pan Li, O. Ozatay, D.C. Ralph, R.A.\nBuhrman, Appl. Phys. Lett. 91, 062507 (2007).\n[16] S. Petit, N. de Mestier, C. Baraduc, C. Thirion, Y. Liu, M . Li, P.\nWang, B. Dieny, Phys. Rev. B 78, 184420 (2008).\n[17] O. Posth, N. Reckers, R. Meckenstock, G. Dumpich, J. Lin dner, J.\nPhys. D: Appl. Phys. 42, 035003 (2009).\n[18] C.T. Boone, J.A. Katine, J.R. Childress, V. Tiberkevic h, A. Slavin,\nJ. Zhu, X. Cheng, I. N. Krivorotov, Phys. Rev. Lett. 103, 167601\n(2009).\n[19] T. Seki, H. Tomita, A.A. Tulapurkar, M. Shiraishi, T. Sh injo, Y.\nSuzuki, Appl. Phys. Lett. 94, 212505 (2009).\n[20] Y. Guan, J.Z. Sun, X. Jiang, R. Moriya, L. Gao, S.S.P. Par kin, Appl.\nPhys. Lett. 95, 082506 (2009).\n[21] W. Chen, G. de Loubens, J.-M.L. Beaujour, J.Z. Sun, A.D. Kent,\nAppl. Phys. Lett. 95, 172513 (2009).\n[22] R.-X. Wang, P.-B. He, Q.-H. Liu, Z.-D. Li, A.-L. Pan, B.- S. Zou,\nY.-G. Wang, J. Magn. Magn. Mater. 322, 2264 (2010).\n[23] T. Staudacher, M. Tsoi, Thin Solid Films 519, 8260 (2011).\n[24] Y. Okutomi, K. Miyake, M. Doi, H.N. Fuke, H. Iwasaki, M. S ahashi,\nJ. Appl. Phys. 109, 07C727 (2011).\n[25] Yu.V. Gulyaev, P.E. Zilberman, A.I. Panas, E.M. Epshte in, J. Exp.\nTheor. Phys. 107, 1027 (2008).\n[26] J.C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996).\n[27] L. Berger, Phys. Rev. B 54, 9353 (1996).\n[28] C. Heide, P.E. Zilberman, R.J. Elliott, Phys. Rev. B 63, 064424\n(2001).\n[29] Yu.V. Gulyaev, P.E. Zilberman, E.M. Epshtein, R.J. Ell iott, JETP\nLett.76, 155 (2002).\n[30] A.I. Akhiezer, V.G. Baryakhtar, S.V. Peletminskii, Sp in Waves,\nNorth-Holland Publ. Co., Amsterdam, 1968.\n[31] A.V. Sokolov, Optical Properties of Metals, American E lsevier Publ.\nCo., 1967.\n10"    },    {        "title": "0812.0832v1.Observation_of_ferromagnetic_resonance_in_strontium_ruthenate__SrRuO3_.pdf",        "content": "arXiv:0812.0832v1  [cond-mat.mtrl-sci]  3 Dec 2008Observationofferromagnetic resonance instrontium ruthe nate (SrRuO 3)\nM.C. Langner,1,2C.L.S. Kantner,1,2Y.H. Chu,3L.M.\nMartin,2P. Yu,1R. Ramesh,1,4and J. Orenstein1,2\n1Department of Physics, University of California, Berkeley , CA 94720\n2Materials Science Division, Lawrence Berkeley National La boratory, Berkeley, CA 94720\n3Department of Materials Science and Engineering,\nNational Chiao Tung University, HsinChu, Taiwan, 30010\n4Department of Materials Science and Engineering,\nUniversity of California, Berkeley, CA 94720\n(Dated: October 26, 2018)\nAbstract\nWe report the observation of ferromagnetic resonance (FMR) in SrRuO 3using the time-resolved\nmagneto-optical Kerr effect. The FMR oscillations in the ti me-domain appear in response to a sudden,\noptically induced change in the direction of easy-axis anis tropy. The high FMR frequency, 250 GHz, and\nlarge Gilbert damping parameter, α≈1, are consistent with strong spin-orbit coupling. We ﬁnd th at the\nparameters associated with the magnetization dynamics, in cludingα, have a non-monotonic temperature\ndependence, suggestive of alink to theanomalous Hall effec t.\nPACS numbers: 76.50.+g,78.47.-p,75.30.-m\n1Understanding and eventually manipulating the electron’s spin degree of freedom is a major\ngoal of contemporary condensed matter physics. As a means to this end, considerable attention\nis focused on the spin-orbit (SO) interaction, which provid esa mechanism for control of spin po-\nlarization by applied currents or electric ﬁelds [1]. Despi te this attention, many aspects of SO\ncoupling are not fully understood, particularly in itinera nt ferromagnets where the same elec-\ntrons are linked to both rapid current ﬂuctuations and slow s pin dynamics. In these materials,\nSO coupling is responsible for spin-wave damping [2, 3], spi n-current torque [4, 5], the anoma-\nlous Hall effect (AHE) [6], and magnetocrystalline anisotr opy (MCA) [7]. Ongoing research is\naimed toward a quantitative understanding of how bandstruc ture, disorder, and electron-electron\ninteractionsinteracttodeterminethesizeandtemperatur edependenceoftheseSO-driveneffects.\nSrRuO 3(SRO) is a material well known for its dual role as a highly cor related metal and\nan itinerant ferromagnet with properties that reﬂect stron g SO interaction [8, 9, 10]. Despite\nits importance as a model SO-coupled system, there are no pre vious reports of ferromagnetic\nresonance (FMR) in SRO. FMR is a powerful probe of SO coupling in ferromagnets, providing\na means to measure both MCA and the damping of spin waves in the small wavevector regime\n[11]. HerewedescribedetectionofFMRbytime-resolvedmag netoopticmeasurementsperformed\non high-quality SRO thin ﬁlms. We observe a well-deﬁned reso nance at a frequency ΩFMR=\n250 GHz. This resonant frequency is an order of magnitude hig her than in the transition metal\nferromagnets,which accountsforthenonobservationbycon ventionalmicrowavetechniques.\n10-200nmthickSROthinﬁlmsweregrownviapulsedlaserdepo sitionbetween680-700◦Cin\n100 mTorr oxygen. High-pressure reﬂection high-energy ele ctron diffraction (RHEED) was used\nto monitor the growth of the SRO ﬁlm in-situ. By monitoring RH EED oscillations, SRO growth\nwas determined to proceed initially in a layer-by-layer mod e before transitioning to a step-ﬂow\nmode. RHEED patterns and atomic force microscopy imaging co nﬁrmed the presence of pristine\nsurfaces consisting of atomically ﬂat terraces separated b y a single unit cell step ( 3.93 ˚A). X-ray\ndiffractionindicatedfullyepitaxialﬁlmsandx-rayreﬂec tometrywasusedtoverifyﬁlmthickness.\nBulk magnetization measurements using a SQUID magnetomete r indicated a Curie temperature,\nTC, of∼150K.\nSensitive detection of FMR by the time-resolved magnetoopt ic Kerr effect (TRMOKE) has\nbeen demonstrated previously [12, 13, 14]. TRMOKE is an all o ptical pump-probe technique in\nwhichtheabsorptionofan ultrashortlaserpulseperturbst hemagnetization, M, ofaferromagnet.\nThe subsequent time-evolutionof Mis determined from the polarization state of a normally inci -\n2dent, time-delayed probe beam that is reﬂected from the phot oexcited region. The rotation angle\nof the probe polarization caused by absorption of the pump, ∆ΘK(t), is proportional to ∆Mz(t),\nwherezisthedirectionperpendiculartotheplaneoftheﬁlm[15].\nFigs. 1a and 1b show ∆ΘK(t)obtained on an SRO ﬁlm of thickness 200 nm. Very similar\nresults are obtained in ﬁlms with thickness down to 10 nm. Two distinct types of dynamics are\nobserved,dependingonthetemperatureregime. Thecurvesi nFig. 1aweremeasuredattempera-\nturesnearT C. Therelativelyslowdynamicsagreewithpreviousreportsf orthisTregime[16]and\nare consistent with critical slowing down in the neighborho od of the transition [17]. The ampli-\ntudeofthephotoinducedchangeinmagnetizationhasalocal maximumnearT=115K.Distinctly\ndifferentmagnetizationdynamicsareobservedasTisreduc edbelowabout80K,asshowninFig.\n1b. The TRMOKE signal increases again, and damped oscillati ons with a period of about 4 ps\nbecomeclearly resolved.\nFIG. 1: Change in Kerr rotation as a function of time delay fol lowing pulsed photoexcitation, for several\ntemperatures below the Curie temperature of 150 K. Top Panel : Temperature range 100 K <T<150 K.\nBottom panel: Temperature range 5K <T<80 K.Signal amplitude and oscillations grow with decreasin g\nT.\nIn order to test if these oscillations are in fact the signatu re of FMR, as opposed to another\n3photoinduced periodic phenomenon such as strain waves, we m easured the effect of magnetic\nﬁeld on theTRMOKE signals. Fig. 2a shows ∆ΘK(t)for several ﬁelds up to 6 T applied normal\nto the ﬁlm plane. The frequency clearly increases with incre asing magnetic ﬁeld, conﬁrming that\ntheoscillationsare associatedwithFMR.\nThemechanismfortheappearanceofFMRinTRMOKEexperiment siswellunderstood[14].\nBeforephotoexcitation, Misorientedparallelto hA. Perturbationofthesystembythepumppulse\n(by local heating for example) generates a sudden change in t he direction of the easy axis. In the\nresulting nonequilibrium state, MandhAare no longer parallel, generating a torque that induces\nMto precess at the FMR frequency. In the presence of Gilbert da mping,Mspirals towards the\nnewhA, resultingin thedamped oscillationsof Mzthat appearin theTRMOKE signal.\nTo analyze the FMR further we Fourier transform (FT) the time -domain data and attempt to\nextract thereal and imaginary parts of the transversesusce ptibility,χij(ω). Themagnetization in\nthetime-domainisgivenby therelation,\n∆Mi(t) =/integraldisplay∞\n0χij(τ)∆hj\nA(t−τ)dτ, (1)\nwhereχij(τ)is the impulse response function and ∆hA(t)is the change in anisotropy ﬁeld.\nIf∆hA(t)is proportional to the δ-function, ∆Mi(t)is proportional χij(τ)and the FT of the\nTRMOKE signal yields χij(ω)directly. However, for laser-induced precession one expec ts that\n∆hA(t)will be more like the step function than the impulse function , as photoinduced local\nheating can be quite rapid compared with cooling via thermal conduction from the laser-excited\nregion. When ∆hA(t)is proportional to the step function, χij(ω)is proportionalto the FT of the\ntimederivativeof ∆Mi(t), ratherthan ∆Mi(t)itself. Inthiscase, theobservable ωIm{∆ΘK(ω)}\nshouldbecloselyrelated tothereal, ordissipativepart of χij(ω).\nInFig. 2bweplot ωIm{∆ΘK(ω)}foreachofthecurvesshowninFig. 2a. Thespectrashown\nin Fig. 2b do indeed exhibit features that are expected for Re χij(ω)near the FMR frequency. A\nwell-deﬁnedresonancepeakisevident,whosefrequencyinc reaseswithmagneticﬁeldasexpected\nforFMR.TheinsettoFig. 2bshows ΩFMRasafunctionofappliedmagneticﬁeld. Thesolidline\nthroughthedatapointsisaﬁtobtainedwithparameters |hA|=7.2Tandeasyaxisdirectionequal\nto 22 degrees from the ﬁlm normal. These parameter values agr ee well with previous estimates\nbased onequilibriummagnetizationmeasurements[8, 10].\nAlthough the spectra in Fig. 2b are clearly associated with F MR, the sign change at low fre-\nquency is not consistent with Re χij(ω), which is positive deﬁnite. We have veriﬁed that the\n4FIG. 2: Top panel: Change in Kerr rotation as a function of tim e delay following pulsed photoexcitation at\nT=5K,forseveral valuesofappliedmagneticﬁeldranging up to6Tesla. Bottompanel: Fourier transforms\nof signals shown intop panel. Inset: FMRfrequency vs. appli ed ﬁeld.\nnegativecomponentis always present in the spectra and is no t associated with errors in assigning\nthet=0pointinthetime-domaindata. Theoriginofnegativecomp onentoftheFTismadeclearer\nby referring back to the time domain. In Fig. 3a we show typica l time-series data measured in\nzero ﬁeld at 5 K. Forcomparison we show theresponseto a step f unction changein theeasy axis\ndirection predicted by the Landau-Lifschitz-Gilbert (LLG ) equations [18]. It is clear that, if the\nmeasured and simulated responses are constrained to be equa l at large delay times, the observed\n∆ΘK(t)ismuchlarger thantheLLGpredictionat smalldelay.\nWe have found that ∆ΘK(t)can be readily ﬁt by LLG dynamics if we relax the assumption\nthat∆hA(t)is a step-function, in particular by allowing the change in e asy axis direction to\n”overshoot” at short times. The overshoot suggests that the easy-axis direction changes rapidly\nas the photoexcited electrons approach quasiequilibriumw ith the phonon and magnon degrees of\nfreedom. The red line in Fig. 3a shows the best ﬁt obtained by m odeling∆hA(t)byH(t)(φ0+\nφ1e−t/τ), whereH(t)is the step function, φ0+φ1is the change in easy-axis direction at t= 0,\nandτis the time constant determining the rate of approach to the a symptotic value φ0. The ﬁt is\n5FIG. 3: Components of TRMOKE response in time (top panel) and frequency (bottom panel) domain.\nBlacklinesaretheobserved signals. Greenlineinthetoppa nel isthesimulated response toastepfunction\nchange in easy-axis direction. Best ﬁts to the ”overshoot” m odel described in the text are shown in red.\nBluelines are thedifference between the measured and best- ﬁt response.\nclearly much better when the possibility of overshoot dynam ics in∆hA(t)is included. The blue\nline shows the difference between measured and simulated re sponse. With the exception of this\nveryshortpulsecenterednear t=0,theobservedresponseisnowwelldescribedbyLLGdynami cs.\nInprinciple,analternateexplanationforthediscrepancy withthestep-functionassumptionwould\nbe to consider possible changes in the magnitude as well as di rection of M. However, we have\nfound that ﬁtting the data then requires |M(t)|to be larger at t >20ps than|M(t <0)|, a\nphotoinducedincrease thatisunphysicalfora systemin ast ableFM phase.\nIn Fig. 3b we compare data and simulated response in the frequ ency domain. With the al-\nlowance for an overshoot in ∆hA(t)the spectrum clearly resolves into two components. The\npeak at 250 GHz and the sign change at low frequency are the bot h part of the LLG response to\n∆hA(t). The broad peak or shoulder centered near 600 GHz is the FT of t he short pulse compo-\nnentshowninFig. 3a. Wehavefoundthiscomponentisessenti allylinearinpumppulseintensity,\n6and independent of magnetic ﬁeld and temperature - observat ions that clearly distinguish it from\nthe FMR response. Its properties are consistent with a photo induced change in reﬂectivity due to\nband-ﬁlling,whichiswell-knowntocross-coupleintotheT RMOKEsignalofferromagnets [19].\nByincludingovershootdynamicsin ∆hA(t),weareabletodistinguishstimulusfromresponse\nin the observed TRMOKE signals. Assuming LLG dynamics, we ca n extract the two parameters\nthatdescribetheresponse: ΩFMRandα;andthetwoparametersthatdescribethestimulus: φ1/φ0\nandτ. In Fig. 4 we plot all four parameters as a function of tempera ture from 5 to 80 K. The\nT-dependence of the FMR frequency is very weak, with ΩFMRdeviating from 250 GHz by only\nabout 5%overthe range ofthe measurement. TheGilbert damping param eterαis of order unity\nat all temperatures, avaluethatis approximatelyafactor 102largerthan found intransitionmetal\nferromagnets. Over the same T range the decay of the easy axis overshoot varies from about 2\nto 4 ps. We note that the dynamical processes that characteri ze the response all occur in strongly\noverlapping time scales, that is the period and damping time of the FMR, and the decay time of\nthehAovershoot,areeach inthe2-5ps range.\nWhileΩFMRisessentiallyindependentofT,theparameters α,φ1/φ0andτexhibitstructurein\ntheirT-dependencenear40K.Thisstructureisreminiscent oftheT-dependenceoftheanomalous\nHallcoefﬁcient σxythathasbeenobservedinthinﬁlmsofSRO[20,21,22]. Forcom parison,Fig.\n4dreproduces σxy(T)reportedinRef. [20]Thesimilaritybetween theT-dependen ceofAHEand\nparameters related to FMR suggests a correlation between th e two types of response functions.\nRecently Nagaosa and Ono [23] have discussed the possibilit y of a close connection between\ncollective spin dynamics at zero wavevector (FMR) and the of f-diagonal conductivity (AHE). At\na basic level,both effects are nonzero only in the presence o f both SO couplingand time-reversal\nbreaking. However, the possibilityof a more quantitativec onnection is suggested by comparison\nof the Kubo formulas for the two corresponding functions. Th e off-diagonal conductivity can be\nwrittenin theform [24],\nσxy(ω) =i/summationdisplay\nm,n,kJx\nmn(k)Jy\nnm(k)fmn(k)\nǫmn(k)[ǫmn(k)−ω−iγ], (2)\nwhereJi\nmn(k)is current matrix element between quasiparticle states wit h band indices n,mand\nwavevector k. The functions ǫmn(k)andfmn(k)are the energy and occupation difference, re-\nspectively,between such states, and γis a phenomenologicalquasiparticledamping rate. FMR is\nrelated to theimaginary part of theuniformtranverse susce ptibility,with thecorresponding Kubo\n7FIG. 4: Temperature dependence of (a) FMR frequency (triang les) and damping parameter (circles), (b)\novershoot decay time, (c) ratio of overshoot amplitude to st ep-response amplitude ( φ1/φ0), and (d) σxy\n(adapted from [20]).\nform,\nImχxy(ω) =γ/summationdisplay\nm,n,kSx\nmn(k)Sy\nnm(k)fmn(k)\n[ǫmn(k)−ω]2+γ2, (3)\nwhereSi\nmnisthematrixelementofthespinoperator. Ingeneral, σxy(ω)andχxy(ω)areunrelated,\nas they involvecurrent and spin matrix elements respective ly. However, it has been proposed that\nin several ferromagnets, including SRO, the k-space sums in Eqs. 2 and 3 are dominated by a\nsmall number of band-crossings near the Fermi surface [22, 2 5]. If the matrix elements Si\nmnand\nJi\nmnvary sufﬁciently smoothly with k, thenσxy(ω)andχxy(ω)may be closely related, with both\nproperties determined by thepositionofthechemical poten tialrelativeto theenergy at which the\n8bandscross. Furthermore,asGilbertdampingisrelatedtot hezero-frequencylimitof χxy(ω),i.e.,\nα=ΩFMR\nχxy(0)∂\n∂ωlim\nω→∞Imχxy(ω), (4)\nand AHE is the zero-frequency limit of σxy(ω), the band-crossing picture implies a strong corre-\nlationbetween α(T)andσxy(T).\nIn conclusion,we havereported the observationof FMR in the metallictransition-metaloxide\nSrRuO 3. Both the frequency and damping coefﬁcient are signiﬁcantl y larger than observed in\ntransition metal ferromagnets. Correlations between FMR d ynamics and the AHE coefﬁcient\nsuggest that both may be linked to near Fermi surface band-cr ossings. Further study of these\ncorrelations, as Sr is replaced by Ca, or with systematic var iation in residual resistance, could be\na fruitful approach to understanding the dynamics of magnet ization in the presence of strong SO\ninteraction.\nAcknowledgments\nThis research is supported by the US Department of Energy, Of ﬁce of Science, under contract\nNo. DE-AC02-05CH1123. Y.H.C. would also like to acknowledg e the support of the National\nScience Council,R.O.C., underContract No. NSC97-3114-M- 009-001.\n[1] I.ˆZuti´ c, J. Fabian, and S.DasSarma, Rev.Mod. Phys. 76, 323 (2004).\n[2] V. Korenman and R.E.Prange, Phys. Rev.B 6, 2769 (1972).\n[3] V. Kambersk´ y, Can. J. Phys. 48, 2906 (1970).\n[4] J. C.Slonczewski, J. Magn. Magn. Mater. 159, L1(1996).\n[5] L.Berger, Phys. Rev.B 54, 9353 (1996).\n[6] J. M.Luttinger and R. Karplus, Phys. Rev. 94, 782 (1954).\n[7] H.Brooks, Phys. Rev. 58, 909 (1940).\n[8] L. Klein, J. S. Dodge, C. H. Ahn, J. W. Reiner, L. Mieville, T. H. Geballe, M. R.Beasley, and A. Ka-\npitulnik, J.Phys. Cond.-Matt. 8, 10111 (1996).\n[9] P.Kostic,Y.Okada,N.C.Collins,Z.Schlesinger, J.W.R einer,L.Klein,A.Kapitulnik, T.H.Geballe,\nand M. R.Beasley, Phys.Rev. Lett. 81, 2498 (1998).\n9[10] A.F.Marshall, L.Klein, J.S.Dodge, C.H.Ahn,J.W.Rein er, L.Mieville, L.Antagonazza, A.Kapit-\nulnik, T.H.Geballe, and M.R. Beasley, J.Appl. Phys. 85, 4131 (1999).\n[11] B.Heinrich and J. F.Cochran, Adv. Phys. 42, 523 (1993).\n[12] W.K.Hiebert, A.Stankiewicz, and M.R.Freeman, Phys. R ev.Lett.79, 1134 (1997).\n[13] Y. Acremann, C. H.Back, M. Buess, O.Portmann, A. Vaterl aus, D.Pescia, and H.Melchior, Science\n290, 492 (2000).\n[14] M.vanKampen,C.Jozsa, J.T.Kohlhepp, P.LeClair, L.La gae,W.J.M.deJonge, andB.Koopmans,\nPhys. Rev. Lett. 88, 227201 (2002).\n[15] K. Shinagawa, in Magneto-optics , edited by S. Sugano and N. Kojima (Springer-Verlag, Berlin , Ger-\nmany, 2000).\n[16] T. Ogasawara, K. Ohgushi, Y. Tomioka, K. S. Takahashi, H . Okamoto, M. Kawasaki, and Y. Tokura,\nPhys. Rev. Lett. 94, 087202 (2005).\n[17] T. Kise, T. Ogasawara, M. Ashida, Y. Tomioka, Y. Tokura, and M. Kuwata-Gonokami, Phys. Rev.\nLett.85, 1986 (2000).\n[18] W.F.Brown, Micromagnetics (Krieger, 1963).\n[19] B.Koopmans,M.vanKampen,J.T.Kohlhepp,andW.J.M.de Jonge,Phys.Rev.Lett. 85,844(2000).\n[20] R. Mathieu, A. Asamitsu, H. Yamada, K. S. Takahashi, M. K awasaki, Z. Fang, N. Nagaosa, and\nY. Tokura, Phys. Rev. Lett. 93, 016602 (2004).\n[21] L. Klein, J. R. Reiner, T. H. Geballe, M. R. Beasley, and A . Kapitulnik, Phys. Rev. B 61, R7842\n(2000).\n[22] Z. Fang, N. Nagaosa, K. Takahashi, A. Asamitsu, R. Mathi eu, T. Ogasawara, H. Yamada,\nM. Kawasaki, Y. Tokura, and K.Terakura, Science 302, 92(2003).\n[23] M. Onoda, A.S.Mishchenko, and N. Nagaosa, J.Phys. Soc. Jap.77, 013702 (2008).\n[24] M. Onoda and N.Nagaosa, J. Phys.Soc. Jap. 71, 19 (2002).\n[25] X.Wang, J.R. Yates, I. Souza, and D.Vanderbilt, Phys.R ev. B.74, 195118 (2006).\n10"    },    {        "title": "2207.06751v2.Magnetization_dynamics_in_proximity_coupled_superconductor_ferromagnet_superconductor_multilayers__Part_II.pdf",        "content": "arXiv:2207.06751v2  [cond-mat.supr-con]  5 Aug 2022Magnetization dynamics in proximity-coupled\nsuperconductor-ferromagnet-superconductor multilayer s. Part II.\nI. A. Golovchanskiy1,2,3,∗, N. N. Abramov2, V. V. Ryazanov1,2,4, A. A. Golubov5, V. S. Stolyarov1,2,3\n1Moscow Institute of Physics and Technology, State Universi ty,\n9 Institutskiy per., Dolgoprudny, Moscow Region, 141700,\nRussia;2National University of Science and Technology MISIS, 4 Leni nsky prosp., Moscow,\n119049, Russia;3Dukhov Research Institute of Automatics (VNIIA), 127055 Mo scow,\nRussia;4Institute of Solid State Physics (ISSP RAS), Chernogolovka , 142432, Moscow region,\nRussia;5Faculty of Science and Technology and MESA+ Institute for Na notechnology,\nUniversity of Twente, 7500 AE Enschede, The Netherlands.\nIn this work, we the study magnetization dynamics in superco nductor-ferromagnet-\nsuperconductor thin-ﬁlm structures . Results of the broad- band ferromagnetic resonance spec-\ntroscopy are reported for a large set of samples with varied t hickness of both superconducting and\nferromagnetic layers in a wide frequency, ﬁeld, and tempera ture ranges. Experimentally the one-\ndimensional anisotropic action of superconducting torque on magnetization dynamics is established;\nits dependence on thickness of layers is revealed. It is demo nstrated that experimental ﬁndings sup-\nport the recently-proposed mechanism of the superconducti ng torque formation via the interplay\nbetween the superconducting kinetic inductance and magnet ization precession at superconductor-\nferromagnet interfaces.\nI. INTRODUCTION\nAdvantages from hybridization of antagonistic su-\nperconducting and ferromagnetic orders in electron-\nics and spintronics have been repeatedly demonstrated\nin past decades [1]. The interplay between ferro-\nmagnetic and superconducting spin orders enables ma-\nnipulation with spin states and leads to a develop-\nment of various electronic and spintronic elements, in-\ncluding superconductor-ferromagnet-superconductor (S-\nF-S) Josephson junctions [1–3], superconducting phase\nshifters [4, 5], memory elements [6–8], F-S-F–based spin\nvalves [9, 10], Josephson diodes [11] and more complex\nlong-range spin-triplet superconducting systems [12–16].\nRecently application capabilities of S-F hybridization\nhave been expanded by demonstrations of its prospects\nin magnonics. Magnonics is a growing ﬁeld of research\nwhich oﬀers approachesfor the transfer and processingof\ninformation via spin waves. A good overview of various\npotential applications and recent advances in magnon-\nics can be found in Refs. [17–24] and references therein.\nIn development of magnonic systems one of principle re-\nquirements is engineering of appropriate spin-wave dis-\npersion.\nVarious wide-range manipulations with the spin-wave\ndispersion have been demonstrated at cryogenic temper-\natures when coupling magnonic systems with supercon-\nductors. For instance, interaction of a magnonic media\nwith the superconducting vortex matter allows to form\nand tune forbidden bands at sub-micrometer wavelength\nwhich matches the parameter of the vortex lattice [25],\nor to excite exchange spin waves via driving the vortex\nlattice with the electric current [26]. Also, magnetostatic\ninteraction of spin waves with superconducting Meissner\ncurrents in thin-ﬁlm hybrid structures modiﬁes substan-\ntially the spin-wave dispersion [27, 28] and can be used\nto form magnonic crystals [29] or to gate the magnoncurrent [30]. Remarkably, low speed of electromagnetic\npropagation in superconductor-insulator-superconductor\nthin-ﬁlm structures facilitates achievement of the ultra-\nstrong photon-to-magnon coupling in on-chip hybrid de-\nvices [31, 32].\nA striking phenomenon in S-F hybrid structures was\nreported recently in Ref. [33] and studied in more\ndetails in Ref. [34]. In superconductor-ferromagnet-\nsuperconductor trilayer thin-ﬁlm structures a radical in-\ncrease in the ferromagnetic resonance (FMR) frequency\noccurs in the presence of electronic interaction between\nsuperconducting and ferromagnetic layers. The phe-\nnomenon is strong: in some S-F-S structures the high-\nest natural FMR frequencies are reached among in-plane\nmagnetized ferromagnetic systems [34]. Intriguingly, so\nfar, the mechanism behind the phenomenon remains un-\nestablished.\nIn this work, we report a comprehensive experimen-\ntal study of the phenomenon. We report results of\nFMR spectroscopy for a large set of samples with var-\nied thickness of both superconducting and ferromag-\nnetic layers in a wide frequency, ﬁeld, and temperature\nranges. We establish an anisotropic one-dimensional ac-\ntion of hybridization-induced torque acting on magneti-\nzation dynamics and the dependence of this supercon-\nducting torque on the magnetic ﬁeld. Experimental re-\nsults support the recently proposed model by M. Silaev\n[35], which explains the phenomenon in S-F-S structures\nas the outcome of the coupling between magnetization\ndynamics and superconducting kinetic inductance at S-F\ninterfaces.\nII. EXPERIMENTAL DETAILS\nMagnetization dynamics in S-F-S structures is stud-\nied by measuring the ferromagnetic resonance absorp-2\nSample ID ds(Nb), nm dF(Py), nm ds(Nb), nm µ0Ha, mTµ0Meff, Tµ0Hs0, mTTc, Kp\nS1 110 10 110 1.5 1.051 31.8 8.743.79\nS2 120 11 120 -3.3 1.029 39.1 8.563.70\nS3 110 19 110 -0.15 1.100 78.8 8.903.48\nS4 100 35 100 -0.2 1.132 163.6 8.923.80\nS5 140 45 174 -0.4 1.18 193.7 9.084.90\nS6 110 120 110 -0.1 1.123 352.6 8.824.56\nS7 110 350 110 1.6 1.076 617.2 8.748.89\nS8 41 24 41 -0.2 1.131 37.8 7.513.07\nS9 200 30 50 -1 1.146 96.8 8.744.38\nTABLE I. Parameters of studied samples. The dsanddFdenote thickness of superconducting and ferromagnetic lay ers,\nrespectively. The HaandMeffcorrespond to parameters obtained at T > T cwith Eq. 1. The Hs0,Tc, andpare obtained by\nﬁttingHs(T) in Fig. 3 with Eq. 3.\nFIG. 1. Schematic illustration of the investigated chip-\nsample (adopted from Ref. [34]). A series of S-F-S ﬁlm rect-\nangles is placed directly on top of the central transmission line\nof niobium co-planar waveguide. Magnetic ﬁeld His applied\nin-plane along the x-axis.\ntion spectrum using the VNA-FMR approach [36–39]. A\nschematic illustration of investigated samples is shown\nin Fig.1. The chip consists of a thin-ﬁlm supercon-\nducting niobium (Nb) co-planar waveguide with 50 Ohm\nimpedance and 82-150-82 µm center-gap-center dimen-\nsions and a series of niobium-permalloy(Py=Fe 20Ni80)-\nniobium (Nb-Py-Nb) ﬁlm structures with lateral dimen-\nsionsX×Y= 50×140µm and spacing of 25 µm along\nthex−axis that are placed directly on top of the cen-\ntral transmission line of the waveguide. Deposition of\nNb-Py-Nb trilayers is performed in a single vacuum cy-\ncle ensuring the electron transparency at Nb-Py inter-\nfaces. Thin Si orAlO xspacinglayerisdeposited between\nNb co-planar waveguide and the trilayers in order to en-\nsure electrical insulation of the studied samples from the\nwaveguide. As a result, a series of samples has been fab-\nricated and measured with diﬀerent thickness of super-\nconducting (S) and ferromagnetic (F) layers (see Tab. I).\nMicrowave spectroscopy of samples was performed\nby measuring the transmission characteristics S21(f,H)\nin the closed-cycle cryostat Oxford Instruments Triton\n(base temperature 1.2 K) equipped with the home-madesuperconducting solenoid. Spectroscopy was performed\nin the ﬁeld range from -0.22 T to 0.22 T, in the frequency\nrangefrom0 up to 20GHz, and in the temperature range\nfrom 2 to 11 K. Magnetic ﬁeld was applied in-plane along\nthe direction of the waveguide in Fig. 1. FMR spectra\nat diﬀerent temperatures were analysed by ﬁtting S21(f)\ncharacteristics at speciﬁed HandTwith the Lorentz\ncurve and, thus, obtaining the dependencies of the reso-\nnance frequency on magnetic ﬁeld fr(H).\nIII. EXPERIMENTAL RESULTS\n/s48/s46/s48/s48 /s48/s46/s48/s53 /s48/s46/s49/s48 /s48/s46/s49/s53 /s48/s46/s50/s48/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52/s49/s54/s49/s56/s50/s48\n/s32/s50/s46/s48/s32/s75\n/s32/s53/s46/s54/s32/s75\n/s32/s54/s46/s55/s32/s75\n/s32/s55/s46/s53/s32/s75\n/s32/s56/s46/s49/s32/s75\n/s32/s56/s46/s54/s32/s75\n/s32/s57/s46/s53/s32/s75/s82/s101/s115/s111/s110/s97/s110/s99/s101/s32/s102/s114/s101/s113/s117/s101/s110/s99/s121/s32 /s102\n/s114/s44/s32/s71/s72/s122\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32\n/s48/s72 /s44/s32/s84\nFIG. 2. Symbols show experimental dependencies of the\nFMR frequency on magnetic ﬁeld fr(H) for the S5 sample at\ndiﬀerent temperatures. Solid lines show corresponding ﬁt o f\nfr(H) dependencies with Eq. 1 at T > T cand with Eq. 2 at\nT < T c.\nFigure2demonstrates the essence of the studied phe-\nnomenon: it shows dependencies of the FMR frequency\non magnetic ﬁeld fr(H) for the S4 sample (see Tab. I)\nwhich is used as the representative example. The S5\nsample consists of 100 nm thick Nb layers and 35 nm\nthick Py layer. Upon decreasing the temperature be-3\nlow the critical temperature of Nb Tc≈9 K the reso-\nnance curve fr(H) shifts gradually to higher frequencies.\nFor instance, upon decreasing the temperature the fre-\nquency of the natural FMR fr(H= 0) increases from\nabout 0.5 GHz at T≥9 K to about 13 GHz at T= 2 K.\nAtT > T cFMR curves fr(H) in Fig. 2follow the\ntypical Kittel dependence for thin in-plane-magnetized\nferromagnetic ﬁlms at in-plane magnetic ﬁeld:\n(2πfr/µ0γ)2= (H+Ha)(H+Ha+Meff) (1)\nwhereµ0is the vacuum permeability, γ= 1.856×\n1011Hz/T is the gyromagnetic ratio for permalloy, Ha\nis the uniaxial anisotropy ﬁeld that is aligned with the\nexternal ﬁeld, and Meffis the eﬀective magnetization,\nwhich includes the saturation magnetization Msand the\nout-of-plane anisotropy ﬁeld. For all studied samples the\nﬁt of FMR curves at T > T cwith Eq. 1yields neg-\nligible anisotropy ﬁeld Ha, the eﬀective magnetization\nMeff≈1.0−1.2 T, which is close to typical values of the\nsaturation magnetization of permalloy µ0Ms≈1 T, and\nno noticeable dependence of HaandMeffon tempera-\nture. Magnetic parameters HaandMefffor all studied\nsamplesareprovidedinTab. I.Theﬁtof fr(H) curvesfor\nthe S5 sample with Eq. 1at temperature T= 9.5 K> Tc\nis show in Fig. 2with yellow solid line.\nAtT < T cFMR curves obey a diﬀerent expression.\nIn Refs. [31, 34] it was shown that technically by ﬁtting\nfr(H) atT≪Tcwith Eq. 1the action of supercon-\nductivity result in equal but diﬀerent in sign uniaxial\nanisotropy ﬁeld Haand changes of the eﬀective mag-\nnetization ∆ Meff:Ha=−∆Meff. Following the ba-\nsics of derivation of the Kittel formula [40], this equality\nindicates that superconductivity acts on magnetization\nas the one-dimensional restoring torque along the yaxis\nin Fig.1, and the ﬁtting function should take the form\nf2\nr∼(H+Hs)(H+Meff), where Hsis the ﬁeld of\nthe superconducting torque. A satisfactory ﬁt with such\nexpression can be obtained for all samples in Tab Iat\ntemperatures T≪Tc, while for the sample S8, which is\nbased on thin superconducting layers with ds< λL, this\nexpression is valid in the entire temperature range. Here\nλL≈80 nm [31, 41] is the London penetration depth in\nbulk niobium at zero temperature\nHowever, as was also shown in Refs. [31, 34] at tem-\nperatures T/lessorsimilarTcthe superconductivity-induced uniax-\nial anisotropy ﬁeld Hain Eq.1no longer correspond to\n−∆Meff. As it appears, this discrepancy is a rather\nartiﬁcial eﬀect and can be explained by suppression of\nthesuperconductingtorqueuponincreasingthemagnetic\nﬁeld. At higher ﬁelds suppression of the ﬁeld Hsresult\nin reduction of the derivative from the fr(H) curve. The\nﬁt of such resonance line with the conventional Kittel\nformula Eq. 1result in larger reduction of the eﬀective\nmagnetization in comparison to the induced anisotropy\nHa<−∆Meffat temperatures T/lessorsimilarTc. We found that\natT < T cFMR curves fr(H) for all studied samples inTab.Iobey the modiﬁed Kittel dependence:\n(2πfr/µ0γ)2=/parenleftbig\nH+Ha+Hs(1−αsH2)/parenrightbig\n×\n×(H+Ha+Meff)(2)\nwhereHaandMeffare constants and are derived at\nT > T c(see Tab I),Hsis the ﬁeld of one-dimensional\nsuperconducting torque at zero applied ﬁeld, and the pa-\nrameterαsreﬂects the dependence of the superconduct-\ning torque on applied magnetic ﬁeld. We conﬁrm that\nfor all studied samples FMR curves at all temperatures\nT < T ccan be ﬁtted with Eq. 2. Examples of such ﬁt are\nshown for the S4 sample in Fig. 2with solid lines.\nAnalysis of resonance lines fr(H) with Eqs. 1and2\nyields dependencies of the superconducting torque ﬁeld\nand of the ﬁeld-dependence coeﬃcient on temperature,\nHs(T) andαs(T). This data is provided for all studied\nsamples in Fig. 3a-c. Notice that for all samples except\nS8 and S7 the parameter αsgrowth exponentially from\nαs∼1T−2atT≈6Kup to αs∼102T−2atT≈8.5K.\nTemperature dependence of the torque ﬁeld Hs(T) can\nbe characterized by ﬁtting it with the following expres-\nsion [34]:\nHs=Hs0(1−(T/Tc)p) (3)\nwhereHs0is the superconducting spin-torque ﬁeld at\nzero ﬁeld and zero temperature, Tcis the superconduct-\ning critical temperature of S-F-S trilayers, and pis a free\nexponent parameter. The ﬁt of Hs(T) with Eq. 3for all\nsamples is shown in Fig. 3a-b with solid lines and yields\nparameters Hs0andpprovided in Tab I. Notice that for\nallsamplesexceptoftheS7, whichcontainsthethe thick-\nest F-layer, the exponent pis in the range from about 3\nup to 5 with the average value about 4. Also we no-\ntice that due to technical limitations resonance curves of\nS7 sample with the thickest F-layer could be measured\nonly above to 8 K and the value of Hsbelow 8 K are\nobtained via the extrapolation with Eq. 3. This extrapo-\nlationyields the natural FMR frequencyin the S7 sample\nfr(H= 0) = 24 .1 GHz at zero temperature in case if the\nvaluep= 8.89 is correct, or fr(H= 0) = 30 .0 GHz for\np= 4.9 (in the latter case µ0Hs0= 0.96 T), which is the\nmaximum value observed among the rest of samples in\nTab.I.\nFigure3d shows the dependence of the superconduct-\ning torque ﬁeld on the thickness of ferromagnetic layer\nHs0(dF) at zero temperature and at T/Tc= 0.85. This\nis the core result of this work. Figure 3d clearly demon-\nstrates the overall logarithmic-like dependence of Hs0on\nthickness dF, with the linear growth of Hs0at lowdF\n(see the inset in Fig. 3d) and retardation of Hs0growth\nat higher dF. Also, Fig. 3d reveals the dependence of\nthe superconducting torque ﬁeld on the thickness of su-\nperconducting layers: in case is the thickness of one of\nS-layers is reduced Hs0is also reduced (open symbols in\nFig.3d). Notice that while in case of the S8 sample this\nreductioncanbepartiallyexplainedbysuppressedsuper-\nconductivity and smaller Tc≈7.5 K, this is not the case4\n/s48 /s49 /s50 /s51 /s52 /s53 /s54 /s55 /s56 /s57 /s49/s48 /s49/s49/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51\n/s40/s97/s41/s32/s83/s49\n/s32/s83/s50\n/s32/s83/s51\n/s32/s83/s52\n/s32/s83/s53\n/s32/s83/s54\n/s32/s83/s55\n/s32/s83/s56\n/s32/s83/s57/s69/s102/s102/s101/s99/s116/s105/s118/s101/s32/s102/s105/s101/s108/s100/s32\n/s48/s72\n/s115/s44/s32/s84\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32 /s84 /s44/s32/s75/s48 /s49 /s50 /s51 /s52 /s53 /s54 /s55 /s56 /s57 /s49/s48 /s49/s49/s49/s48/s45/s49/s49/s48/s48/s49/s48/s49/s49/s48/s50/s49/s48/s51\n/s40/s99/s41/s67/s111/s101/s102/s102/s105/s99/s105/s101/s110/s116/s32\n/s115/s44/s32/s84/s45/s50\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32 /s84 /s44/s32/s75/s32/s83/s49\n/s32/s83/s50\n/s32/s83/s51\n/s32/s83/s52\n/s32/s83/s53\n/s32/s83/s54\n/s32/s83/s55\n/s32/s83/s56\n/s32/s83/s57\n/s48 /s49 /s50 /s51 /s52 /s53 /s54 /s55 /s56 /s57 /s49/s48 /s49/s49/s48/s50/s48/s52/s48/s54/s48/s56/s48\n/s40/s98/s41/s32/s83/s49\n/s32/s83/s50\n/s32/s83/s51\n/s32/s83/s52\n/s32/s83/s53\n/s32/s83/s54\n/s32/s83/s55\n/s32/s83/s56\n/s32/s83/s57/s69/s102/s102/s101/s99/s116/s105/s118/s101/s32/s102/s105/s101/s108/s100/s32\n/s48/s72\n/s115/s44/s32/s109/s84\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32 /s84 /s44/s32/s75/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48 /s51/s53/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54/s32/s84 /s32/s61/s32/s48\n/s32/s84 /s47/s84\n/s99/s32/s61/s32/s48/s46/s56/s53\n/s32/s84 /s32/s61/s32/s48/s44/s32/s83/s56/s32/s97/s110/s100/s32/s83/s57/s32/s40 /s100\n/s83/s32/s60/s32\n/s76 /s41\n/s32/s84 /s47/s84\n/s99/s32/s61/s32/s48/s46/s56/s53/s44/s32/s83/s56/s32/s97/s110/s100/s32/s83/s57/s32/s40 /s100\n/s83/s32/s60\n/s76 /s41/s69/s102/s102/s101/s99/s116/s105/s118/s101/s32/s102/s105/s101/s108/s100/s32\n/s48/s72\n/s115/s44/s32/s84\n/s84/s104/s105/s99/s107/s110/s101/s115/s115/s32 /s100\n/s70/s32/s111/s102/s32/s116/s104/s101/s32/s70/s32/s108/s97/s121/s101/s114/s44/s32/s110/s109/s79/s98/s116/s97/s105/s110/s101/s100/s32/s118/s105/s97/s32/s101/s120/s116/s114/s97/s112/s111/s108/s97/s116/s105/s111/s110\n/s40/s100/s41/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50\nFIG. 3. a) Dependencies of the superconducting torque ﬁeld o n temperature Hs(T) at zero external ﬁeld. b) Magniﬁcation\nof (a) at µ0Hs<0.1 T. Solid lines in (a) and (b) show the ﬁt of experimental Hs(T) curves with Eq. 3. c) Dependencies\nof the ﬁeld-dependence coeﬃcient of the superconducting to rque ﬁeld in Eq. 2 on temperature αs(T). d) The dependence of\nthe superconducting torque ﬁeld on the thickness of ferroma gnetic layer Hs(dF) at zero ﬁeld at T= 0 (black symbols) and at\nT/Tc= 0.85 red symbols. Solid symbols represent data for samples wit h both superconducting layers ds> λL. Open symbols\nrepresent data for two samples (S8 and S9) with ds< λLat least for one superconducting layer. See Tab I for details . The inset\nin (d) magniﬁes Hs(dF) fordF<45 nm. Solid lines in the inset in (d) show linear ﬁt of Hs(dF) at corresponding temperature.\nfor the S9 sample where Tcis comparable with values for\nthe rest of samples.\nIV. DISCUSSION\nThe original explanation of the phenomenon in S-F-S\ntrilayers suggested the role of spin-polarized spin-triplet\nCooper pairs in formation of the superconductivity-\ninduced anisotropy via the spin-transfer-torque mecha-\nnism [33]. Yet, this mechanism requires the frequency\nof magnetization dynamics of the order of the supercon-\nducting gap [42], can hardly be expected in 350 nm thick\njunctions, and does not yield the one-dimensional torque\non magnetic moment.\nThe ﬁrst hint on inductive origin of the phenomenon\nin S-F-S trilayers was reported in Ref. [43] where the\nmagnetostatic interaction has been considered betweenthe screening currents induced by ferromagnetic stray\nﬁelds in S-layers and the magnetic moment in the F-\nlayer. While only the static case has been considered, it\nwas shown that in case if superconducting currents are\nallowedtolooparoundthe ferromagneticlayeradditional\nDC demagnetisingﬁeldis expected with the followingde-\npendence on magnetic and structural characteristics of\ntrilayers\nHs∝MsdF\nλLdF\nLln2L\ndF, (4)\nwhereλLis the London penetration depth and Lis the\nlength of the structure. The theory in Ref. [43] provides\nan explanation for the one-dimensionality of the demag-\nnetizing ﬁeld along the y-axis in Fig. 1and predicts the\ngrowth of the demagnetizing ﬁeld with increasing thick-\nness of the F-layer. Yet, according to Eq. 4this the-\nory predicts a parabolic dependence of Hs(dF), which5\nis inconsistent with Fig. 3d, and some dependence of\nHson the length of the structure, which was not ob-\nserved experimentally. In fact, this theory leads to a\nrather counter-intuitive outcome that the eﬀect in S-F-S\nstructures where superconducting currentsareallowed to\nloop around the ferromagnetic layer should be measur-\nable with conventional magnetization measurements.\nThe mechanism behind the superconducting torque in\nS-F-S trilayers was revealed in Ref. [35]. The mechanism\nimplies the coupling of the superconducting kinetic in-\nductance with the precessing magnetization at S-F inter-\nfaces and formation of macroscopic circulation currents\ncurling around y-axis in the opposite phase with pre-\ncessing magnetization. The model in Ref. [35] predicts\nexactly the same dependence of the resonance frequency\non the magnetic ﬁeld as in Eq. 2in the limit dF≪λ:\nHs(1−αsH2) =MsdF\nλLtanhdS\nλL. (5)\nIn fact, Eq. 5demonstrates the linear dependence of\nthe superconducting torque on thickness dFat small dF,\nwhich is consistent with Fig. 3d. The reduction of the su-\nperconducting torque upon increasing the magnetic ﬁeld\nin Eq.5is explained by the increase of the penetration\ndepth [41]: (1 −αsH2)∝1/λL(H)tanhdS/λL(H). The de-\npendence of the superconducting torque on the thickness\ndSof S-layers is also captured in Eq. 5as∝tanhdS/λ.\nBoth eﬀects are observed in Fig. 3d. AtdF≫λthe\nmodel in Ref. [35] predicts the saturation of the super-\nconducting torque to a constant value Hs≈Ms, which is\nalsoobservedinFig. 3d. Thus,thekineticinductanceori-\ngin behind the dramatic increase of the FMR frequency\nin S-F-S trilayers with electronic interaction at S-F inter-\nfaces is conﬁrmed.\nAs a ﬁnal remark, it can be expected that supercon-\nductivity in S-F-S structures modiﬁes the dispersion of\nperpendicular standing spin waves (PSSW) [44–48]. In\nthe case of closed PSSW boundary conditions [44, 45, 47]\nmagnetization precession at both S-F interfaces does not\ntake place and, thus, the superconducting torque is not\nformed. In the case of free PSSW boundary conditions[46, 48] magnetization at S-F interfaces precesses at op-\nposite phases for even modes and precesses in-phase for\nodd modes, which corresponds to cancellation of the su-\nperconducting torque for even modes and its presence\nfor odd modes, respectively. In the general case when\nthe F-layers is magnetically non-uniform across its thick-\nness [48] and the spin boundary conditions are aﬀected\nby surface anisotropies [49, 50] the eﬀect of the super-\nconducting torue on the dispersion of PSSWs becomes\nnon-trivial.\nV. CONCLUSION\nSummarising, we report a comprehensive experimen-\ntal study of magnetization dynamics in S-F-S trilayers.\nWe report results of FMR spectroscopy for a large set\nof samples with varied thickness of both superconduct-\ning and ferromagnetic layers in a wide frequency, ﬁeld,\nand temperature ranges. Experimentally we establish\nan anisotropic one-dimensional action of hybridization-\ninduced torque acting on magnetization dynamics and\nthe dependence of this superconducting torque on the\nmagnetic ﬁeld. Experimentalresults conﬁrmthe recently\nproposed model by M. Silaev [35], which explains the\nphenomenon in S-F-S structures as the outcome of the\ncoupling between magnetization dynamics and supercon-\nducting kinetic inductance at S-F interfaces. Our results\nopen wide prospects for application of the superconduct-\ning kinetic inductance in magnonics. In addition, as\nwas suggested in Ref. [35], S-F-S systems provide the\nplayground for Anderson-Higgs mass generation of bo-\nson quasiparticles in high-energy Standard Model and in\ncondensed matter systems.\nVI. ACKNOWLEDGMENTS\nThe authors acknowledge Dr. M. Silaev for fruitful\ndiscussions, critical reading of the manuscript and use-\nful suggestions. The research study was ﬁnancially sup-\nported by the Russian Science Foundation (grant N 22-\n22-00314).\n[1]J. Linder and J. W. A. Robinson, Superconducting spin-\ntronics, Nat. Phys. 11, 307 (2015).\n[2]V. V. Ryazanov, V. A. Oboznov, A. Y. Rusanov, A. V.\nVeretennikov, A. A. Golubov, and J. Aarts, Coupling of\ntwo superconductors through a ferromagnet: Evidence\nfor a pi-junction, Phys. Rev. Lett. 86, 2427 (2001).\n[3]M. Weides, M. Kemmler, E. Goldobin, D. Koelle,\nR. Kleiner, H. Kohlstedt, and A. Buzdin, Appl. Phys.\nLett.89, 122511 (2007).\n[4]T. Yamashita, S. Kim, H. Kato, W. Qiu, K. Semba,\nA. Fujimaki, and H. Terai, Scientiﬁc Reports 10, 13687\n(2020).[5]A. K. Feofanov, V. A. Oboznov, V. V. Bolginov, J. Lisen-\nfeld, S. Poletto, V. V. Ryazanov, A. N. Rossolenko,\nM. Khabipov, D. Balashov, A. B. Zorin, P. N. Dmitriev,\nV. P. Koshelets, and A. V. Ustinov, Implementation of\nsuperconductor/ferromagnet/superconductor pi-shifter s\nin superconducting digital and quantum circuits, Nat.\nPhys.6, 593 (2010).\n[6]I. A. Golovchanskiy, V. V. Bolginov, V. S. Stolyarov,\nN. N. Abramov, A. B. Hamida, O. V. Emelyanova, B. S.\nStolyarov, M. Y. Kupriyanov, A. A. Golubov, and V. V.\nRyazanov, Micromagnetic modeling ofcritical currentos-\ncillations in magnetic josephson junctions, Phys. Rev. B6\n94, 214514 (2016).\n[7]I. V. Vernik, V. V. Bol’ginov, S. V. Bakurskiy, A. A.\nGolubov, M. Y. Kupriyanov, V. V. Ryazanov, and\nO. Mukhanov, Magnetic josephson junctions with super-\nconducting interlayer for cryogenic memory, IEEE Trans.\nAppl. Supercond. 23, 1701208 (2013).\n[8]L. N. Karelina, R. A. Hovhannisyan, I. A. Golovchanskiy,\nV. I. Chichkov, A. B. Hamida, V. S. Stolyarov, L. S.\nUspenskaya, S. A. Erkenov, V. V. Bolginov, and V. V.\nRyazanov, J. Appl. Phys. 130, 173901 (2021).\n[9]E. C. Gingrich, B. M. Niedzielski, J. A. Glick, Y. Wang,\nD. L. Miller, R. Loloee, W. P. P. Jr, and N. O. Birge,\nNat. Phys. 12, 564 (2016).\n[10]D. Lenk, R. Morari, V. I. Zdravkov, A. Ullrich, Y. Khay-\ndukov, G. Obermeier, C. Muller, A. S. Sidorenko, H.-\nA. K. von Nidda, S. Horn, L. R. Tagirov, and R. Tidecks,\nFull-switching fsf-type superconducting spin-triplet ma g-\nnetic random access memory element, Phys. Rev. B 96,\n184521 (2017).\n[11]K.-R. Jeon, J.-K. Kim, J. Yoon, J.-C. Jeon,\nH. Han, A. Cottet, T. Kontos, and S. S. P. Parkin,\nNat. Mater. (2022) .\n[12]J. W. A. Robinson, J. D. S. Witt, and M. G. Blamire,\nControlled injection of spin-triplet supercurrents into a\nstrong ferromagnet, Sci. 329, 59 (2010).\n[13]N. Banerjee, J. Robinson, and M. Blamire, Reversible\ncontrol of spin-polarized supercurrents in ferromagnetic\njosephson junctions, Nat. Comm. 5, 4771 (2014).\n[14]J. Wang, M. Singh, M. Tian, N. Kumar, B. Liu, C. Shi,\nJ. K. Jain, N. Samarth, T. E. Mallouk, and M. H. W.\nChan, Interplay between superconductivity and ferro-\nmagnetism in crystalline nanowires, Nat. Phys. 6, 389\n(2010).\n[15]J. A. Glick, V. Aguilar, A. Gougam, B. M. Niedzielski,\nE. C. Gingrich, R. Loloee, W. P. P. Jr., and N. O. Birge,\nSci. Adv. 4, eaat9457 (2020).\n[16]K.-R. Jeon, B. K. Hazra, K. Cho, A. Chakraborty, J.-C.\nJeon, H. Han, H. L. Meyerheim, T. Kontos, and S. S. P.\nParkin, Nat. Mater. 20, 1358 (2021).\n[17]A. Chumak et al., IEEE Trans. Magn. 58, 0800172\n(2022).\n[18]P. Pirro, V. I. Vasyuchka, A. A. Serga, and B. Hille-\nbrands, Nature Reviews Materials 6, 1114 (2021).\n[19]A. Barman et al., J. Phys.: Condens. Matter 33, 413001\n(2021).\n[20]A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and\nB. Hillebrands, Nat. Phys. 11, 453 (2015).\n[21]B. Lenk, H. Ulrichs, and F. G. M. Munzenberg, The\nbuilding blocks of magnonics, Phys. Rep. 507, 107\n(2011).\n[22]G. Csaba, A. Papp, and W. Porod, Phys. Lett. A 381,\n1471 (2017).\n[23]A. A. Serga, A. V. Chumak, and B. Hillebrands, Yig\nmagnonics, J. Phys. D: Appl. Phys. 43, 264002 (2010).\n[24]V. V. Kruglyak, S. O. Demokritov, and D. Grundler,\nMagnonics, J. Phys. D: Appl. Phys. 43, 264001 (2010).\n[25]O. V. Dobrovolskiy, R. Sachser, T. Bracher, T. Fischer,\nV. V. Kruglyak, R. V. Vovk, V. A. Shklovskij, M. Huth,\nB. Hillebrands, and A. V. Chumak, Magnon-ﬂuxon inter-\naction in a ferromagnet/superconductor heterostructure,\nNat. Phys. 15, 477 (2019).\n[26]O. V. Dobrovolskiy, Q. Wang, D. Y. Vodolazov,\nB. Budinska, R. Sachser, A. V. Chumak, M. Huth, and\nA. I. Buzdin, arXiv , 2103.10156 (2021).[27]I. A. Golovchanskiy, N. N. Abramov, V. S. Stolyarov,\nV. V. Bolginov, V. V. Ryazanov, A. A. Golubov, and\nA. V. Ustinov, Adv. Funct. Mater. 28, 1802375 (2018).\n[28]I. A. Golovchanskiy, N. N. Abramov, V. S. Stolyarov,\nV.V.Ryazanov, A.A.Golubov, andA.V.Ustinov,Mod-\niﬁed dispersion law for spin waves coupled to a supercon-\nductor, J. Appl. Phys. 124, 233903 (2018).\n[29]I. A. Golovchanskiy, N. N. Abramov, V. S. Stol-\nyarov, P. S. Dzhumaev, O. V. Emelyanova, A. A.\nGolubov, V. V. Ryazanov, and A. V. Ustinov, Ferro-\nmagnet/superconductor hybridmagnonic metamaterials,\nAdv. Sci. 6, 1900435 (2019).\n[30]T. Yu and G. E. W. Bauer, arXiv , 2201.09532 (2022).\n[31]I. A. Golovchanskiy, N. N. Abramov, V. S. Stolyarov,\nM. Weides, V. V. Ryazanov, A. A. Golubov, A. V.\nUstinov, and M. Y. Kupriyanov, Science Advances 7,\neabe8638 (2021).\n[32]I. A. Golovchanskiy, N. N. Abramov, V. S. Stolyarov,\nA. A. Golubov, M. Y. Kupriyanov, V. V. Ryazanov,\nand A. V. Ustinov, Physical Review Applied 16, 034029\n(2021).\n[33]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[34]I. A. Golovchanskiy, N. N. Abramov, V. S. Stolyarov,\nV. I. Chichkov, M. Silaev, I. V. Shchetinin, A. A.\nGolubov, V. V. Ryazanov, A. V. Ustinov, and M. Y.\nKupriyanov, Phys. Rev. Appl. 14, 024086 (2020).\n[35]M. A. Silaev, Anderson-higgs mass of magnons\ninsuperconductor/ferromagnet/superconductor systems,\narXiv:2207.13201 (2022).\n[36]I. Neudecker, G. Woltersdorf, B. Heinrich, T. Okuno,\nG. Gubbiotti, and C. Back, Comparison of frequency,\nﬁeld, and time domain ferromagnetic resonance methods,\nJ. Magn. Magn. Mat. 307, 148 (2006).\n[37]S. S. Kalarickal, P. Krivosik, M. Wu, C. E. Patton, M. L.\nSchneider, P. Kabos, T. J. Silva, and J. P. Nibarger,\nFerromagnetic resonance linewidth in metallic thin ﬁlms:\nComparison of measurement methods, J. Appl. Phys. 99,\n093909 (2006).\n[38]Y.-C. Chen, D.-S. Hung, Y.-D. Yao, S.-F. Lee, H.-P. Ji,\nand C. Yu, Ferromagnetic resonance study of thickness-\ndependent magnetization precession in ni 80fe20ﬁlms, J.\nAppl. Phys. 101, 09C104 (2007).\n[39]I. A. Golovchanskiy, V. V. Bolginov, N. N. Abramov,\nV. S. Stolyarov, A. B. Hamida, V. I. Chichkov,\nD. Roditchev, and V. V. Ryazanov, J. Appl. Phys. 120,\n163902 (2016).\n[40]D. Stancil, Theory of Magnetostatic Waves (Springer-\nVerlag New York, Inc., 1993).\n[41]A. I. Gubin, K. S. Ilin, S. A. Vitusevich, M. Siegel, and\nN. Klein, Physical Review B 71, 064503 (2005).\n[42]M. Houzet, Ferromagnetic josephson junction with pre-\ncessing magnetization, Phys. Rev. Lett. 101, 057009\n(2008).\n[43]S. V. Mironov and A. I. Buzdin, Appl. Phys. Lett. 119,\n102601 (2021).\n[44]C. Kittel, Physical Review 100, 1295 (1958).\n[45]M. H. Seavey and P. E. Tannenwald, Journal of Applied\nPhysics30, S227 (1959).\n[46]M. Sparks, Phys. Rev. B 1, 3831 (1970).\n[47]S. Klingler, A. V. Chumak, T. Mewes, B. Khodadadi,\nC. Mewes, C. D. O. S. B. Hillebrands, and A. Conca, J.\nPhys. D: Appl. Phys. 48, 015001 (2015).7\n[48]I.A.Golovchanskiy et al.,Phys.Rev.Materials 6, 064406\n(2022).\n[49]C. H. Bajorek and C. H. Wilts, Journal of Applied\nPhysics42, 4324 (1971).[50]H. Puszkarski, Prog. Surf. Sci. 9, 191 (1979)."    },    {        "title": "1012.1266v1.Local_electrical_characterization_of_resonant_magnetization_motion_in_a_single_ferromagnetic_sub_micrometer_particle_in_lateral_geometry.pdf",        "content": "arXiv:1012.1266v1  [cond-mat.mtrl-sci]  6 Dec 2010Local electrical characterization of resonant magnetizat ion motion in a single\nferromagnetic sub-micrometer particle in lateral geometr y.\nA. Slobodskyy1,2and B. J. van Wees2\n1Karlsruhe Institute of Technology (KIT) Light Technology I nstitute (LTI) Kaiserstrae 12,\n76131 Karlsruhe Germany and Zentrum f¨ ur Sonnenenergie- und\nWasserstoﬀ-Forschung Baden-W¨ urttemberg, Industriestr. 6, 70565 Stuttgart, Germany\n2Physics of Nanodevices Zernike Institute of Advanced Materi als University of Groningen The Netherlands\n(Dated: January 14, 2021)\nInthis article adetailed characterization of amagnetizat ion motion in asingle sub-micrometer and\nmulti-terminal ferromagnetic structure in lateral geomet ry is performed in aGHz regime usingdirect\nDC characterization technique. We have shown applicabilit y of the Stoner-Wohlfarth model to the\nmagnetic nano-structure with large length to with ratio. Ap plying the model to experimental data\nwe are able to extract relevant magnetization motion parame ters and show a correlation between\nhigh frequency inductive currents and local magnetization . Additionally, DC voltage generated over\nthe structure at the resonance, with external magnetic ﬁeld under an angle to the shape anisotropy\naxis, is explained by the model.\nPACS numbers: 85.75.-d, 75.76.+j, 76.50.+g, 75.78.Jp\nMulti-terminal ferromagnetic nanostructures are\nwidely used in spintronics [1–3]. Recent developments\nin the ﬁeld [4–6] brought enhanced interest to the topic,\na specially in combination with ultrafast magnetization\ndynamics [7]. Proper inside into magnetization dynam-\nics of an individual nanostructure is a challenging task\n[8]. Most of the work in this direction is being done\non relatively large structures [9], arrays of structures\n[10], by interpreting magnetization switching events [6],\nor by dynamic response to the magnetization motion\n[11]. This methods are mostly limited to two terminal\nmeasurements. In this article we use multi-terminal\nferromagnetic sub-micrometer structures in lateral\ngeometry, in combination with resonant magnetization\nmotion, for detailed quantitative electrical characteriza-\ntion of the sample magnetization. We also analyze the\nDC voltages generated by the magnetization precession.\nThe sample is prepared by means of electron beam\nlithography and lift-of. Scanning electron microscopy\n(SEM) image of the sample is shown in a Fig. 1(a).\nThe sample consist of a ferromagnetic 3 µm long and\n20 nm thick permalloy (Ni 20Fe80) strip, contacted by\neight 50nm thick copper contacts. Width of the strip\nand the contacts is 100 nm. The strip is placed on 1 µm\ndistancefroma150nmthickand1 µmwidegoldencopla-\nnarwaveguide. Inordertoensuregoodelectricalcontacts\nand uniform magnetization, the permalloy strip and the\ncopper contacts are fabricated by two steps of Electron\nBeam Evaporation under 700angle to the sample plane\nand alongcorrespondinglines, without breakingvacuum.\nThe measurement technique is depicted in a Fig. 1(b).\nCoplanar waveguide is used to apply normal to the sub-\nstrate radiofrequency (RF) magnetic ﬁeld to the permal-\nloy strip. At the resonance, when the RF frequency\nmatches Larmor precession frequency, the ﬁeld induces\nLarmor precession of the strip magnetization around the\u0001\u0002\u0002\u0003\u0004 \u0001\u0002\n\u0003\u0002\n\u0005\u0006\u0007 \n\u0004\b\t\n\u0004\u000b\f\n\r\u000e\u0005 \u0005\u000f\u000e\u0005 \nFIG. 1: a) SEM image of the device, b) and c) evaporation\nunder an angle fabrication technique, b) perpendicular to t he\nstrip and c) along the strip, d) schematic representation of\nthe experiment.\nstatic eﬀective magnetic ﬁeld ( Beff).Beffdetermining\nLarmorprecessionfrequencyisavectorsumofanapplied\nexternal ﬁeld ( Bex) and a constant shape anisotropyﬁeld\n(Ban) pointing along the strips longest (easy) axis. At\nthe resonance the angle of the magnetization precession\nis drastically increased and damped (no precession) oth-\nerwise.2\nDependence of s single-domain ferromagnet resistance\nonrelativeangle Lbetweenmagnetization Mandcurrent\nIiscalledanisotropicmagnetoresistance(AMR)[12], and\ncommonly found to follow the next equation:\nR=R0+(∆R)cos2(L), (1)\nwhereR0is a resistance when magnetization and cur-\nrent are perpendicular, and ∆ Ris a resistance diﬀerence\nbetween parallel and perpendicular conﬁgurations.\nIn presented experiments outer contacts to the strip\nare used as DC current probes and the inner contacts\nas voltage probes. Zero magnetic ﬁeld resistance in this\nconﬁgurationfound tobe R0= 121.5Ω, this value is con-\nsistent with bulk material properties. By ﬁtting AMR in\nthecaseof Bexperpendicularto BanusingEq.(1), values\nofBan= 0.137mTand ∆R= 2.1 Ω are extracted. We\nuse these parameters to extract the average precession\nangle L at the ferromagnetic resonance using Eq. (1).\nIn Fig. 2(a) RF induced change of the resistance (red\nline) and change in DC voltage (black line) as a func-\ntion ofBex, applied along the strip are shown. The\nmeasurements are done at ﬁxed RF frequency while ex-\nternal magnetic ﬁeld is scanned from negative to posi-\ntive direction with amplitude of 300 mT. Magnetization\ntraining procedure is used to ensure monodomain state\nof the magnetization in the strip. Magnetization switch-\ning event is shown by a dashed (vertical) line. RF signal\nis (on, oﬀ) modulated at 17 Hz and DC current is alter-\nnated between -0.1 mA, 0 mA and +0.1 mA every 10 s.\nBackground voltage extracted from the measurements at\nopposite current directions is similar to the voltage at\nzero current. Oﬀset in the resistance and voltage is sub-\ntracted, and the curves are shifted for clarity, by 20 mΩ\nand 6.7 µVrespectively. Clear indication of the reso-\nnance in the magnetization motion is seen as a deep on\nthe resistance curves, but can not be observed in the\nvoltage measurements.\nThe measurements at external magnetic ﬁeld applied\nunder 310angle to the strip easy axis are shown in\nFig. 2(b). Most pronounced diﬀerence in this conﬁgu-\nration from the previous plot, is that the resonance can\nbe observed as a deep in the voltage measurements. The\nDC voltage generated at the resonance can be explained\nbyinterplaybetweenasymmetryinAMR causedbymag-\nnetization precession motion in phase with inductive AC\ncurrents trough the ferromagnetic strip. Although, simi-\nlarresultshavebeenobservedpreviously[4], asystematic\nanalysis overwide range of RF frequencies is done for the\nﬁrst time. Additionally we present a Stoner-Wohlfarth\nparticle model to explain the observed dependence.\nIn the Fig. 2(c) the 310angle data is zoomed in the\nfrequency range from 8 GHz till 11.5 GHz and the ﬁeld\nrange from -100 mT till 100 mT. Resonance in magne-\ntization motion is preserved over zero external magnetic-20 020 40 60 80 \n-0.3 -0.2 -0.1 0 0.1 0.2 0.3 -5 0510 15 20 25 \n13GHz 14GHz 15GHz 16GHz \n12GHz \n-20 020 40 60 80 \n-0.3 -0.2 -0.1 0 0.1 0.2 0.3 -5 0510 15 20 25 \nResistance,□m /c87 DC□Voltage, V \n/c109 13GHz 14GHz 15GHz 16GHz \n12GHz \n020 40 60 80 100 120 140 160 180 \n-100 -50 0 50 100 0510 15 20 25 30 \nField,□mT 8GHz 8.5GHz 9GHz 9.5GHz 10GHz 10.5GHz 11GHz 11.5GHz a)□0° \nb)□31° \nc)□31° \nFIG. 2: RF frequency dependence of resistance (red curve)\nand voltage (black curve) versus external magnetic ﬁeld. a)\nField is parallel to the ferromagnetic strip. b) and c) Field is\nunder 310to the strip easy axis.\nﬁeld and deﬁned by the shape anisotropy ﬁeld. The volt-\nageat the resonance,ischangingfrom adeepfornegative\nexternal magnetic ﬁelds, to a peek for positive external\nmagnetic ﬁelds, just before the magnetization switching.\nThis behavior will be explained further.\nTo describe the observed results, when external mag-\nnetic ﬁeld is applied under an angle βto the easy axis\nof the ferromagnetic strip, we calculate expected angle\nbetween the axis and the magnetization Q, as depicted\nin the Fig. 3(a). We ﬁnd Q, by energy minimization, as\na solution of the following equation: ( Han/2)sin(2Q) +\nHexsin(Q−β) = 0. Solution for 310tilt is shown in\nFig. 3(b). From the ﬁgure it can be seen that the preces-3\nsion axis decline as much as ±200at±200 mT from the\neasy axis. Rapid decrease in the angle at about 50 mT\nis connected with the magnetization switching event.\nUsing the extracted angles and the shape anisotropy\nﬁeldBanwe calculate eﬀective magnetic ﬁeld Heffas a\nfunction of applied external ﬁeld Bex. Than we extract\ncenter of each resonance and full width at half maximum\n(FWHM) by ﬁtting observed resonances in the magne-\ntoresistance curves with Lorentzian [13]. RF frequency\nFof the resonants is shown in Fig. 3(c).\nThe data below 12 GHz is extracted from Fig. 2(c).\nThe resonance positions in the case of external mag-\nnetic ﬁelds parallel to the longest strip axis are shown\nby squares, and the magnetic ﬁelds under an angle of\n310to the axis are shown by triangles. Frequency de-\npendence of the resonance position is well ﬁtted with\nKitel’s equation for small precession angles: F=\nµ0γ/radicalbig\nHeff(Heff+Ms), where µ0is a permeability of\na vacuum, γis the gyromagnetic ratio and Msis a satu-\nration magnetization.\nDespite slight ellipticity expected in the magnetization\nmotion, excellent agreement between ﬁtting and the ex-\ntracted data is achieved with γ= 193.39 GHz and Ms\n= 969000, the ﬁtting is shown in Fig. 3(c) by lines. The\npreviously extracted shape anisotropyﬁeld Ban=137mT\nis used for AMR curve ﬁtting. During magnetization\nswitching 5 mT of the shape anisotropy ﬁeld are pinned\ndue to sample imperfection.\nTaking into account the modulation technique used in\nour experiment, we ﬁnd that the measured value of the\nRF ﬁeld induced change in the resistance is not sensitive\nto the angle of the precession axis Q, and deﬁned only\nby magnetization precession angle L.\nAverage precession angle at the resonance Lextracted\nfrom the data using Eq. (1) is shown in Fig. 3(d). The\ndata at 310is shown by triangles and at 00by squares.\nSolving transverse RF ﬁeld Hy0driven Landau-Lifschitz-\nGilbert equation [14] we ﬁnd that the precession angle\nLat a resonance is inversely proportional to the RF\nfrequency ω:L=A/λω[15], where A=µ2\n0γ2MsHy0\n= 6.95 * 108and the Gilbert damping parameter λ=\n0.0104 similar to the value observed previously [16]. Cal-\nculated value of Lat 310is shown by continuous line and\nthe value at 00by dashed line. Although, the measured\nvalues are scattered around the calculated lines, smear-\ning separation between angular dependence, it is obvious\nthat the general magnetic ﬁeld dependence is well pre-\nserved.\nAtpositiveﬁelds, justbeforethemagnetizationswitch-\ningeventtheaverageprecessionangle Lissaturated,con-\ntrary to expected strong increase. The saturation could\nbe explained by deviation of the magnetization motion\nfrom the Stoner-Wohlfarth particle model.\nTo determine Gilbert damping parameter we use\nknown [17] frequency dependence of FWHMd) \nB,□[T] L,□[Deg] \n5\n467\n-0.2 -0.1 0 0.1 0.2 V,□[ V] \n/c109\n-8 -6 -4 2\n-0.1 0 0.1 0.2 B,□[T] \n-0.2 \ne) 10 11 12 13 14 15 16 \n-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 F,□[GHz] \nB,□[T] c) a) \nB,□[T] Q,□[Deg] \n-20 -15 -10 510 15 20 \n-0.2 -0.1 0 0.1 0.2 b) Han \nmQ\nL\nB/c98\nFIG. 3: External magnetic ﬁeld dependence of experimen-\ntal (squares - 00and triangles - 310) and calculated (lines)\nparameters. a) Sketch of the magnetization motion. b) An-\ngle of displacement of the precession axis Qfrom the shape\nanisotropy ﬁeld. c) Average angle of the precession Lat the\nresonance. d) DC voltage generated over the strip at the res-\nonance. e) Resonance RF frequency.\n∆HFWHM(ω) = ∆Hinhom+2√\n3λ\nγ2Msω,\nwhere ∆ HFWHM(ω) is a frequency dependent value of\nFWHM extracted from Lorentzian ﬁt and ∆ Hinhomis a\nfrequencyindependentinhomogeneousbroadeningdueto\nlocal magnetization variations. Frequency independent\n(within experimental error) value of ∆ HFWHM(ω)≈\n16mT±2mTat negative magnetic ﬁelds (before the\nswitching) and ∆ HFWHM(ω)≈21mT±2mTat posi-\ntive ﬁelds (after the switching) is an indication of domi-\nnating contribution of ∆ Hinhom. Moreover, higher value\nof the inhomogeneous broadening after the magnetiza-\ntion switching, is consistent with an increase in magnetic\ninhomogeneity by the pinned magnetization.\nIn Fig. 2 DC voltages generated over the strip shown\nby black line. Absence of signiﬁcant voltages at 00is an\nindication that the voltages are connected to the magne-\ntization precession symmetry around strip longest axis.\nIn Fig. 3(d) voltages generated at the resonance with the\nexternalmagneticﬁeldtilt310oﬀthestripaxisareshown\nby triangles.4\nTo understand the generated voltages we use already\ndescribed model for the magnetization motion, taking\ninto account that an inductive current generated trough\nthe sample at the resonance is in phase with the magne-\ntization motion. Accordingto the model the diﬀerence in\nAMR at opposite phases in combination with the induc-\ntive current, see Fig 3(a), creates an average DC voltage\nover the ferromagnet. Measured voltage should, there-\nfore,changethesignwhenthemagneticﬁeldcrosseszero,\nsee Fig. 3(b). The voltage changes the sign back to orig-\ninal after 2 πphase shift in the precession motion caused\nby magnetization switching. Calculated magnetic ﬁeld\ndependence of DC voltage, using current of 140 µA, is\nshown by lines in Fig. 3(d). The calculations reproduce\nall the essential features of the measured data.\nIn summary, we have presented a method of a reso-\nnant magnetization motion detection in multi-terminal\nferromagnetic sub-micrometer structures. Change in the\nAMR response at the ferromagnetic resonance is ex-\nploited to investigate the magnetic nano-structure. DC\nresistance response of the structure at the resonance,\nwhen the external magnetic ﬁeld is applied under an an-\ngle between the strip easy axis, is explained. DC voltage\ngeneratedat the resonanceunder anangle is alsowellun-\nderstood, taking into account inductive AC RF currents\ngenerated in the ferromagnetic strip.\nUsing the technique it is now possible to gain new in-\nside on magnetic properties of the ferromagnetic nano-\nmeter scale particles and their diﬀerences from a bulk\nor a layer ferromagnets. Local nature of the measure-\nment technique makes it possible to monitor magnetiza-\ntion motion dynamics in diﬀerent parts of a single fer-\nromagnetic particle. Investigations in this direction are\ncurrently underway.\nThe authors would like to thank to Prof. Caspar H.\nvan der Wal for useful discussions, as well as NanoNed\nfor ﬁnancial support.[1] F.J. Jedema, A.T. Filip, B.J.v. Wees, Nature, 410, 345-\n348, (2001)\n[2] F.J. Jedema, H.B. Heersche, A.T. Filip, J.J.A. Basel-\nmans, B.J.v. Wees, Nature, 416, 713 - 716, (2002)\n[3] X. Lou, C. Adelmann, S. A. Crooker, E. S. Garlid, J.\nZhang, S. M. Reddy, S. D. Flexner, C. J. Palmstrom,\nand P. A. Crowell, Nature Physics, 3, 197 - 202 (2007)\n[4] M. V. Costache, S. M. Watts, M. Sladkov, C. H. van der\nWal, and B. J. van Wees, Appl. Phys. Lett., 89, 232115,\n(2006)\n[5] M. V. Costache, M. Sladkov, S. M. Watts, C. H. van der\nWal, and B. J. van Wees, Phys. Rev. Lett., 97, 216603,\n(2006)\n[6] Masamitsu Hayashi, Luc Thomas, Rai Moriya, Charles\nRettner, Stuart S. P. Parkin, Science, 209 - 211, (2008)\n[7] A. Mourachkine, O. V. Yazyev, C. Ducati and J.-Ph.\nAnsermet, Nano Lett., 8 (11), 3683 - 3687 (2008)\n[8] Christophe Thirion, Wolfgang Wernsdorfer, Dominique\nMailly, Nature Materials, 2, 524 - 527 (2003)\n[9] R. F. Soohoo, J. Appl. Phys., 33, 1276, (1962)\n[10] Ronaldo S. de Biasi and Tessaleno C. Devezas, J. Appl.\nPhys. 49, 2466, (1978)\n[11] S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Em-\nley, R. J. Schoelkopf, R. A. Buhrman and D. C. Ralph,\nNature, 425, 380-383, (2003); H. W. Schumacher, S.\nSerrano-Guisan, K. Rott, and G. Reiss, Appl. Phys.\nLett., 90, 042504, (2007)\n[12] J-E. Wegrowe, D. Kelly, A. Franck, S. E. Gilbert, and\nJ.-Ph. Ansermet, Phys. Rev. Lett. 82, 3681, (1999)\n[13] C. Kittel and Elihu Abrahams, Phys. Rev., 90, 238 - 239,\n(1953)\n[14] T.L. Gilbert, Physical Review, 100, 1243, (1955)\n[15] Y. Guan and W. E. Bailey, E. Vescovo, C.-C. Kao, and\nD. A. Arena, Cond-Mat, 0604610, (2006)\n[16] T. Gerrits, T. J. Silva, J. P. Nibargera, and T. Rasing,\nJ. Appl. Phys. 96, 6023 (2004).\n[17] W. Platow, A. N. Anisimov, G. L. Dunifer, M. Farle, and\nK. Baberschke, Phys. Rev. B 58, 5611 - 5621, (1998)"    },    {        "title": "1809.04372v2.Anomalous_increasing_of_the_intensity_of_field_dependence_optical_mode_ferromagnetic_resonance_in_the_exchange_coupled_bilayer_system.pdf",        "content": "APS/123-QED\nAnomalous increasing of permeability peak value of optical mode\nferromagnetic resonance in the exchange coupled bilayer system\nWenfeng Wang, Lulu Pan, Wenjie Song, Guozhi Chai,\u0003and Desheng Xuey\nKey Laboratory for Magnetism and Magnetic Materials of the Ministry of Education,\nLanzhou University, Lanzhou,730000, Peoples Republic of China.\n(Dated: December 20, 2018)\nAbstract\nAcoustic and optical ferromagnetic resonance (FMR) in the interlayer exchange coupled\nFe20Ni80/Co bilayer have been investigated. In the optical mode, unexpected increasing tendencies\nof peak value at the resonance frequency has been observed under an increasing magnetic \feld. We\npresented analytical calculations with which the exchange coupling between Co and Fe 20Ni80lay-\ners, the magnetization and the in-plane uniaxial anisotropy are taken into account, to interpret the\nincreasing of the maximum values of the optical permeability. Both experimental measurements\nand theoretical calculation show that such tendencies are dependent on the layer thickness t, and\nthat there is a critical \feld above which the optical peak value begins to decrease. These results\nmight help us to understand the mechanism of interlayer exchange coupling induced optical FMR\nand might enlighten us to \fnd new possibility of high frequency applications of magnetic materials.\n1arXiv:1809.04372v2  [cond-mat.mes-hall]  19 Dec 2018I. INTRODUCTION\nFor decades, the magnetic materials possessing excellent soft magnetic properties are most\nwanted in many electronic devices, yet it has been a challenge to alter the magnetic properties\nof a material once it was fabricated. From the points of view of practical use of magnetic\nproperties, the dynamic properties of a magnetic material are governed by the complex\ndynamic permeability1,2:\u0016=\u00160\u0000i\u001600, where\u00160denotes the real part of permeability,\nwhich determines the reorientation angle of the magnetization processional mode, whereas\nthe imaginary permeability \u001600determines the energy dissipation3. Materials with large\n\u00160and small\u001600implicate great performance and small energy consumption, respectively.\nTherefore, a magnetic material with large and controllable \u0016at relative high work frequency\nrange is desperately demanded. However, both real and imaginary permeability of the\nmagnetic materials with traditional magnetization dynamics will reduce to the extent of\nuseless owing to the restriction of Acher's limit4,5. From this point of view, one can have\nonly a descending of \u0016with the increase of the resonance frequency. Consequently, the\nful\fllment of a magnetic material, of which \u0016increase with increasing frequency, is therefore\nvery desirable for modern electronic devices.\nTo meet this demand, new kinds of magnetic material structure consisting of two or more\nmagnetic layers are then carried out by several authors. In these multilayered systems,\ntwo modes of FMR, known as acoustic and optical FMR, respectively, can be observed in\nexperiments6,7, owing to the exist of so called interlayer exchange coupling. The energy of\nthe interaction are described by two exchange coupling parameters, the bilinear coupling J18\nand the biquadratic coupling J29,10. Generally, J1is dominant in the \flms, where moments\nof the two layers are parallel11{15or antiparallel16{18to each other to meet the demand of\nenergy minimization. While some authors have shown that the biquadratic coupling can also\nbecome dominating19,20, leading to a 90\u000e-type coupling, in which the moments of the two\nferromagnetic layers are vertical to each other. It was found that the biquadratic exchange\ncoupling arise from spatial \ructuation of the interlayer thickness in a sandwiched structure9.\nFor theJ1exchange coupling (ferromagnetic or antiferromagnetic corresponding to J1>0\norJ1<0) dominated cases, the exchange energy per unit area at the interface can be written\nas,\nEex=\u0000JM1\u0001M2\nM1M2; (1)\n2whereJis the bilinear exchange coupling coe\u000ecient with the unit of erg/cm2,M1and\nM2are the saturation magnetizations of the individual layers. Note that we neglected the\nsubscript of Jhere for simplicity. It has been found that the exchange coupling strength\ndepend dramatically on the thicknesses of the magnetic \flms6,12, as well as the thickness of\nthe interlayers16. Heinrich and coworkers found out that the magnetic coupling in epitaxial\nbcc Fe(001)/Cu(001) /Fe(001) trilayers changes from ferromagnetic to antiferromagnetic\nas the Cu(001) interlayer thickness changes. The exchange energy can be derived from\nfrequency di\u000berence of the acoustic and optical modes21,22. For the ferromagnetic coupling,\nthe resonance frequency of optical mode is higher than that of acoustic mode, while for the\nantiferromagnetic coupling, the optical resonance frequency is lower than that of acoustic\nmode. The dispersive relations and dependence of resonance intensities on exchange coupling\n\feld, saturation magnetization and anisotropic \feld can be obtained from solving Landau-\nLifshitz-Gilbert equeations12,14,23{25. Note that the resonance intensity mentioned here is\nde\fned as relating to the area under FMR absorption line.\nIn experimental, FMR is one of the most ubiquitous technique to investigate the micro\nwave magnetic properties of interlayer exchange coupled magnetic multi-layers26?{29. By\nmeasuring the permeability spectra as a function of exchange coupling strength, magnetiza-\ntion, anisotropy, damping factor, magnetic layer thickness and even applied magnetic \feld,\none can obtain a lot of basic magnetic properties of the multi-layer systems, and can test\ntheories that describe the mechanisms of these coupling structures.\nIn this paper, we implemented a theoretical calculation and experimental measurements of\npermeability for a exchange coupled bilayer system consisting of two di\u000berent ferromagnetic\nlayers in intimate contact. Both numerical calculation and FMR results for the permeability\nof the bilayer are presented. Di\u000berences of acoustic and optical resonance behaviors under\na increasing external magnetic \feld Hare studied with several parameters are taken into\naccount including layer thickness t, magnetization M, in-plane uniaxial anisotropy (IPUMA)\nHk, damping factor \u000b, and exchange strenth J.\nII. EXPERIMENTAL DETAILS\nA set of Fe 20Ni80/Co bilayer \flms with di\u000berent layer thicknesses are fabricated and\nstudied in this paper. The samples were grown by radio frequency (rf) magnetron sputter\n3\u000bD\f\n \u000bE\f\nFIG. 1. (a) The imaginary permeability of Fe 20Ni80/Co bilayer measured with a increasing mag-\nnetic \feld applied along the easy direction, (b) the peak values of acoustic (black) and optical (red)\nimaginary permeability of the Fe 20Ni80/Co bilayer as functions of applied \feld .\ndeposition on 0.43 mm thick Si (111) substrates, which were attached to oblique sample\nholders with oblique angle of 30\u000eto induce the in-plane uniaxial anisotropy, in a ultrahigh\nvacuum chamber. The base pressure of the chamber prior to sputtering was pumped to\napproximately 6\u000210\u00005Pa. The deposition pressure during the fabrication was maintained\nat 0.3 Pa at Ar ambient with gas rate \row of 10 SCCM (cubic centimetre per minute at\nSTP). The sputter targets of Co metal and Fe 20Ni80alloy are 3 inches in diameter. The rf\npower of 50 W was used to deposit the \flms. The thicknesses of the \flms were controlled\nby controlling the deposition time for each layer. The schematic of the \flm structure is\nshown in Fig. 2, where FM1 represents Co layer and FM2 is Fe 20Ni80layer in this case. For\nsimplicity, the Co layer thickness d1was \fxed to be 28.6 nm, while the Fe 20Ni80layer have\nvarying thicknesses, d2= 33:4;48:8 and 66.8 nm.\nThe FMR measurements of the \flms were performed via vector network analyser (VNA,\nAgilent E8363B, USA) with a home made shorted-circuited microstrip line (MSL) jig con-\nnected to it through a subminiature assembly coaxial connector30,31. The resistance of the\nMSL is 50 \n to meet impedance matching of VNA's test port. During the measurements,\nthe micro magnetic \feld hwas perpendicular to the easy axis (EA) of the samples. For each\nsample, an increasing planer magnetic \feld Halong EA was applied.\nThe experimental results of dc magnetic \feld dependent imaginary permeability of the\nFe20Ni80/Co bilayer are presented in Fig. 1 (a), the applied \feld ( H)lies in the \flm plane\nand along the easy axis of the \flm. Compared to that of the acoustic mode, the resoannce\nfrequency of the optical mode is higher, however the peak value of the imaginary permeability\n4is signi\fcantly small. This is because that in the case of strong ferromagnetic coupling, the\nmoments of the Co and Fe 20Ni80layers precess out-of-phase in optical FMR mode causing\nan o\u000bset of radio-frequency (rf) components of Mi. Hence the peak value of the optical\nimaginary permeability of the bilayer is vastly crippled. A evident downward trend of the\nacoustic mode was observed whereas the absorption peak of the optical mode increased\nunusually with the increase of the applied \feld, which is completely opposite to that of the\nacoustic mode. The di\u000berence of peak values of the two modes decreased from 1098 to 370\nwhenHincreases from 0 to 225 Oe. This abnormal increase of optical peak value might have\nbearing on the out-of-phase precession of magnetic moments. The variation of peak values\nof the acoustic and optical imaginary permeability versus the applied \feld Hare presented\nin Fig. 1 (b), one can see that the two modes have completely opposite variation trend.\nIn the following section, we proposed a theoretical model aimed at numerically calculating\nthe permeability of the Fe 20Ni80/Co bilayers to give a comprehensive understanding on the\neccentric behaviour of optical resonance.\nIII. THEORETICAL MODEL\nThe schematic of the bilayer structure is shown in Fig.2. Considering a bilayer system\nconsisting of FM1 and FM2 layers lies in the x\u0000yplane with the axis znormal to the\n\flm planes. The calculation is based on the LLG equation that the moments of the bilayer\ndeviation from the equilibrium positions, with a microwave magnetic \feld perpendicular to\nthe magnetization in the \flm plane. Only the situation of the external magnetic \feld H\nlying in the \flm plane, at an angle \fwith respect to xaxis, is taken into consideration.\nThe magnetization Miof FMi, is characterized by the angles \u0012iand'i, wherei(i= 1, 2)\ndenotes the FM1 and FM2 respectively. To simplify the calculations, we assume that the\nmagnetization and uniaxial anisotropy of the two ferromagnetic layers are all lie in the \flm\nplanes and that \f= 0, and that the biquadratic coupling is negligible compared to bilinear\ncoupling, i.e., J1\u001dJ2. Note that no magnetocrystalline anisotropies are considered for both\nlayers.\nWith all these assumption above, the total free energy per unit of the system can be\n5FIG. 2. Schematic show of the magnetic bilayer structure and coordinate systems. The axis x\nis chosen to coincide with the uniaxial anisotropy of the sample, and the axis yto be along the\ndirection of micro magnetic \feld h.\nwritten as\nE=t1[\u0000M1Hx1+ 2\u0019M2\n1z2\n1+K1(y2\n1+z2\n1)]\n+t2[\u0000M2Hx2+ 2\u0019M2\n2z2\n2+K2(y2\n2+z2\n2)]\n\u0000J(x1x2+y1y2+z1z2); (2)\nwheretiandKiare the thickness and uniaxial anisotropy constant of FM i(i= 1;2), respec-\ntively.His the applied external magnetic \feld and Jis the exchange coupling constant.\nNote that the biquadratic coupling is not considered here, thus the bilinear coupling J1\nis written to J.xi,yiandziare the direction cosines of the Mito thex,yandzaxes,\nrespectively. They are give by\nx1=sin\u00121cos'1; y1= sin\u00121sin'1\nx2=sin\u00122cos'2; y2= sin\u00122sin'2\nz1=cos\u00121; z2= cos\u00122: (3)\nThe total energy Econsists of the Zeeman energy, the in-plane uniaxial anisotropy24,32\nand the dipolar energy33of FM1 and FM2 layers, as well as the exchange coupling energy\nbetween FM1 and FM2.\nAt equilibrium, the \frst derivatives of Ewith respect to xi,yiandzimust be equal to\nzero. It is apparent that when xi= 1 andyi=zi= 0, the system reach to equilibrium\ncondition, at which the magnetizations, M1andM2, are all lie in the xdirection in the\n\flms plane. Note that in the antiferromagnetic situation, M1andM2are antiparallel, i.e.,\nx1\u0001x2=\u00001.\n6Now we consider the dynamic behaviors of M1andM2in a weak microwave magnetic\n\feldh. It is described by Landau-Lifshitz equation with the Gilbert damping term\ndMi\ndt=\rMi\u0002[rMi(E\nti) +\u000b\n\rMidMi\ndt\u0000hiej!t]: (4)\nHere\ris the gyromagnetic ratio, \u000bis the damping factor and !is the angular frequency of\nhi. Considering that the magnetization vectors excited by hoscillate about the equilibrium\nposition, the Eq. (4) can be linearized by expanding the free energy Ein Taylor series up\nto second order. The motion of the moments then can be written in matrix form as\n0\nBBBBBBB@\rEyy\u0000j\u000b!M 1\rEyz\u0000j!M 1\rEya \rEyb\n\rEzy+j!M 1\rEzz+j\u000b!M 1\rEza \rEzb\n\rEay \rEaz\rEaa\u0000j\u000b!M 1\rEab\u0000j!M 1\n\u0000\rEby\u0000\rEbz\rEba+j!M 1\rEbb+j\u000b!M 11\nCCCCCCCA\u00020\nBBBBBBB@\u0001y\n\u0001z\n\u0001a\n\u0001b1\nCCCCCCCA=0\nBBBBBBB@\rM1hy\n0\n\rM2hy\n01\nCCCCCCCA(5)\nwhereEij=@2E=@i@j are the second partial derivative of energy with respect to iandj\n(i;j=y;z;a;b ) at the equilibrium position. Note that, in Eq. (5), we let y1=y; z 1=\nz; y 2=aandz2=bto make the calculation simple, and \u0001 y;\u0001z;\u0001aand \u0001bdenote the\nsmall variations of y; z; a andb, respectively. By substituting Eq. (2) into Eq. (5) and\nsimplifying the matrix equation, we can obtain\n0\nBBBBB@\nk1\u0000j\u000b!\u0000j!\u0000!J2 0\nj! \nm1+j\u000b! 0\u0000!J2\n\u0000!J1 0 \n k2\u0000j\u000b! j!\n0\u0000!J1j! \nm2+j\u000b!1\nCCCCCA\u00020\nBBBBB@\u0001m1y\n\u0001m1z\n\u0001m2y\n\u0001m2z1\nCCCCCA=0\nBBBBB@!m1hy\n0\n!m2hy\n01\nCCCCCA(6)\nwith\n\nk1=!0+!k1+!J1;\nm1=!0+!k1+!m1+!J1\n\nk2=!0+!k2+!J2;\nm2=!0+!k2+!m2+!J2\nwhere!0=\rH; ! ki=\r2Ki\nMi; !Ji=\rJ\ndiMiand!mi= 4\u0019\rM i(i= 1;2). On solving this\nequation with the help of computer, one can obtain the real and imaginary permeability of\nboth acoustic and optical modes for the bilayer systems numerically.\n712345678-500050010001\n2345678-1000010002000(b)( a)  µ' \nµ''µF\nrequency (GHz)µ\nF\nrequency (GHz) µ' \nµ''FIG. 3. (a) Permeability of Fe 20Ni80/Co bilayer measured at zero external dc magnetic \feld. (b)\nNumerical results of zero-\feld permeability of the bilayer.\n05 01 001 502 002 500246810A\ncoustic modeFrequency (GHz)H\n (Oe)Optical mode4.32 GHz\nFIG. 4. The resonance frequencies vs applied magnetic \feld Hof the Fe 20Ni80/Co bilayer, where\nthe solid black square and circle represent the experimental data and the solid lines denote the\ncalculated dispersion relation.\nIV. RESULTS AND DISCUSSIONS\nFig. 3 (a) shows the permeability spectra of the Fe 20Ni80/Co bilayer at zero external\ndc magnetic \feld, with the thicknesses of Co and Fe 20Ni80layer aret1= 28:6 nm and\nt2= 48:8 nm, respectively. Two FMR modes are observed in the permeability spectra,\none is the acoustic mode ( fac= 1:99 GHz), the other one is the optical mode ( fop= 6:31\nGHz). Numerical results of zero-\feld permeability of the Fe 20Ni80/Co bilayer according to\nEq. (6) are presented in Fig. 3 (b). The parameters used in the calculation are 4 \u0019M 1= 15:2\nkG,Hk1= 44 Oe,\u000b1= 0:028 for Co layer; 4 \u0019M 2= 11:9 kG,Hk2= 36 Oe,\u000b2= 0:008 for\nFe20Ni80layer, and the interlayer exchange coupling strength is J= 0:54 erg/cm2, they are in\naccordance with experimental measurements. A large resonance frequency of fop= 6:31 GHz\nis obtained in the optical mode owing to the presence of the interlayer exchange coupling.\nFig. 4 show the resonance frequencies of the Fe 20Ni80/Co bilayer as the functions of\n80204060806\n7 8 9 -500501000\n5 0100150200-80-60-40406080100120µ'' 0 Oe      40 Oe    80 Oe   \n120 Oe  160 Oe  200 Oe \n240 Oeµ''F\nrequency (GHz)Fe20Ni80 layerCo layerFe20Ni80/Co bilayerµ\n''max(a)F\ne20Ni80/Co bilayerCo layerH\n (Oe)Fe20Ni80 layer(b)FIG. 5. (a) The calculated \feld-dependent optical imaginary permeability of Fe 20Ni80/Co bilayer,\nas well as separated Co and Fe 20Ni80layers respectively. (b) The absorption peak values dependence\non applied magnetic \feld. The parameters are the same with the Fig. 3 (b)\napplied magnetic \feld. The resonance frequency of the optical mode at zero \feld is 4.32 GHz\nhigher than that of the acoustic mode. For the acoustic mode, the magnetization vectors\nof Co and Fe 20Ni80layers precess in phase and the dispersive relation of acoustic mode\ndegenerate with that of single layer because that the interlayer exchange coupling produces\nno dynamic contributions to the resonance. For the optical mode, however, the moments\nof the two layers precess out of phase and therefore the coupling produces e\u000bect exchange\n\felds ofHe1=J=d 1M1= 139 Oe and He2=J=d 2M2= 149 Oe in Co and Fe 20Ni80layers,\nrespectively34,35. The resonance frequencies of both acoustic and optical modes increase as\nHincreased, the black and red line in Fig. 4 denote the calculated dispersive relations of the\ntwo FMR modes, respectively, which are in accordance with the experimental measurements.\nAs previously presented in Fig. 1, the peak value of optical FMR varies abnormally with\nrespect toH, which is di\u000berent from that of acoustic mode. In Fig. 5 (a), we show the\nnumerically calculated optical imaginary permeability spectra of the Fe 20Ni80/Co bilayer\nand of the separated Co and Fe 20Ni80layers, respectively, with applied \feld Hincrease\nfrom 0 to 240 Oe, to understand the abnormal \feld-dependent behavior of the optical res-\nonance. The optical permeability of Co and Fe 20Ni80layers have di\u000berent signs [See Fig. 5\n(a)]corresponding to the in-phase and out-of phase precession of the moments, respectively.\nOne can see that the absolute peak values of separated Co and Fe 20Ni80layers decay at\n951 01 505001000150020006\n8 1012141618020406080100(a)µ''F\nrequency (GHz)(b)µ''F\nrequency (GHz) 0 Oe \n100 Oe \n200 Oe \n300 Oe \n400 Oe \n500 Oe \n600 Oe \n700 Oe \n800 Oe \n900 Oe \n1000 Oe \n1100 Oe \n1200 Oe \n1300 Oe \n1400 Oe \n1500 Oe \n1600 Oe \n1700 OeFIG. 6. The calculated acoustic (a) and optical (b) imaginary permeability of Fe 20Ni80/Co bilayer\nwith the applied \feld increasing from 50 to 1650 Oe. The simulation parameters are the same with\nthe Fig. 3 (b).\ndi\u000berent speeds with the increase of H, leading to an increase of integral peak value of the\nbilayer. The peak values of the bilayer, as shown in Fig. 5 (b), equals numerically to the sum\nof those of separated Co and Fe 20Ni80layers, suggesting that the net rf component of the\nmagnetization vectors in the bilayer comes from those of vector superpositions of Fe 20Ni80\nand Co layers.\nDue to the presence of uniaxial anisotropy Hkand applied \feld H, when applying a\ntransverse microwave magnetic \feld h, the transverse (along the hdirections) rf components\nof the magnetization of Co and Fe 20Ni80layers are always antiparallel to each other resulting\nin a small resonance absorption in optical mode. When His applied along the easy axis,\nthe peak value of acoustic mode decreases with the increase of Howing to reduction of rf\ncomponents of the two layers. While that of the optical mode increases as Hincreases, this\nis related to the di\u000berent reduction speed of rf components of the magnetization vectors in\ntwo layers.\nIn Fig. 6, we plot the simulated imaginary permeability of the Fe 20Ni80/Co bilayer, the\nthickness of Fe 20Ni80is 48.8 nm, the applied \feld Hincreases from 0 to 1700 Oe. A sig-\nni\fcantly downward tendency of the peak value is observed in the acoustic mode when H\nincreases as shown in Fig. 6 (a). For the optical mode, the variation trajectory of peak value\nof the imaginary permeability is divided into two parts. Let us de\fne a critical \feld Hcrt,\nbelow which the peak value increases gradually as Hincreases while above which the peak\nvalue begins to decrease. In this case, the Hcrtis about 750 Oe, as seen in Fig. 6 (b), the\n1004 008 001 2001 60020406080100120µ''maxH\n (Oe) 29.3 nm \n39.1 nm \n48.8 nm \n58.6 nm \n68.4 nmFIG. 7. Calculated peak value of the optical imaginary permeability with respect to Hfor the\nFe20Ni80/Co bilayers with Fe 20Ni80layer thickness increasing from 33.4 nm to 66.8 nm.\npeak value reaches its maximum at this point. This is due to the vector superposition e\u000bect\nof the rf components of the magnetization vectors in two ferromagnetic layers that result in\nan increase of optical mode at a small \feld. However, when H > H crt, the magnetization\nvectors of both layers are completely saturated along the \feld direction leading to small\nprecession angles, which means the value of the optical resonance peak begins to decrease\nwith respect to H.\nIn previous sections we have already discussed that the optical FMR originates from\nthe phase di\u000berence of the magnetic moments precession between Fe 20Ni80and Co layers.\nThe interlayer exchange coupling plays a key role in this process, which can be adjusted\nby changing the layer thickness34,35. Thus one can consequently put in mind that there is\na connection between layer thickness and optical FMR. Fig. 7 shows the optical imaginary\npermeability peak values versus Hfor a series of Fe 20Ni80/Co bilayers with Fe 20Ni80layer\nthicknesst2increasing from 33.4 to 66.8 nm. The Co layer thickness remains unchanged.\nThe measurements of the permeability spectra at higher frequency are limited by our home\nmade shorted-circuited MSL jig, the permeability spectra with applied \feld Hlarger than\n250 Oe are then exceed the test range, therefore, only numerically simulated results are\nshown here. From Fig. 7, one can see how the Fe 20Ni80layer thicknesses a\u000bect the variation\ntendency of the optical mode peak value. For the samples with Fe 20Ni80layer thinner than\n48.8 nm, the peak values change slowly but with high Hcrt, when Fe 20Ni80layer grown\nthicker,Hcrtis reduced and more dramatic changes of the peak values are observed.\n11304 05 06 07 040080012001600Hcrt (Oe)T\nhickness (nm)FIG. 8. The critical \feld Hcrtvs Fe 20Ni80layer thickness.\nFig. 8 plots the critical \feld Hcrtchanges with respect to Fe 20Ni80layer thickness. Hcrtcan\nbe vastly controlled through adjusting the Fe 20Ni80layer thickness of Fe 20Ni80/Co bilayers, a\npossible explanation is that with Fe 20Ni80layer thickness increases, the equivalent exchange\ncoupling \felds He1=J=d 1M1andHe2=J=d 2M2are reduced, which might have in\ruence\nonHcrt.\nV. CONCLUSION\nIn summery, a theoretical model describing dynamic behaviors of the magnetic moments\nof an exchange coupled bilayer system under a microwave magnetic \feld was developed. A\ncomparison of calculated permeability and the experimental data was investigated, and found\nthat for a ferromagnetic coupled bilayer, the FMR frequency of optical mode is much higher\nthan that of acoustic mode, corresponding to an exchange strength of J= 0:54 erg/cm2.\nThe numerical simulations are in well accordance with the experimental results. The e\u000bect\nexchange coupling \feld for the Co and Fe 20Ni80layers are 139 and 149 Oe, respectively.\nThe dispersive relations and the imaginary permeability dependence on external dc \feld\nHwere studied. An upward trend of permeability spectra peak value of the optical mode\nFMR is closely related to the out-of-phase precession of the magnetic moments. When the\napplied \feld Hincreases gradually from 0 to 1700 Oe, there is a critical \feld Hcrtabove\nwhich the peak value begin to decrease. Such a variation of peak value can be regulated via\nchanging the layer thickness. It is thought that the changing of the layer thickness a\u000bects\nthe equivalent exchange \feld. When Fe 20Ni80layer thickness rises from 33.4 to 66.8 nm,\n12Hcrtdrop o\u000b from 1600 Oe to 250 Oe, which can be qualitatively considered as a result of\nreduction of the e\u000bect interlayer exchange coupling \feld.\nACKNOWLEDGMENTS\nThis work is supported by the National Basic Research Program of China (No. 2012CB933101),\nNational Natural Science Foundation of China (NSFC) (Nos. 51471080, 11674143, 51371093,\n51871117) and Program for Changjiang Scholars and Innovative Research Team in University(No.IRT-\n16R35).\n\u0003chaigzh@lzu.edu.cn\nyxueds@lzu.edu.cn\n1T. L. Gilbert, IEEE Transactions on Magnetics 40, 3443 (2004).\n2S. Zhang and S. S.-L. Zhang, Phys. Rev. Lett. 102, 086601 (2009).\n3C. Jiang, C. Jia, F. Wang, C. Zhou, and D. Xue, Phys. Rev. B 97, 060408 (2018).\n4O. Acher and A. L. Adenot, Phys. Rev. B 62, 11324 (2000).\n5G. Chai, D. Xue, X. Fan, X. Li, and D. Guo, Applied Physics Letters 93, 152516 (2008).\n6M. Pomerantz, J. C. Slonczewski, and E. Spiller, J. Appl. Phys. 61, 3747 (1987).\n7J. J. Krebs, P. Lubitz, A. Chaiken, and G. A. Prinz, J. Appl. Phys. 67, 5920 (1990).\n8F. Ho\u000bmann, A. Stanko\u000b, and H. Pascard, J. Appl. Phys. 41, 1022 (1970).\n9J. C. Slonczewski, Phys. Rev. Lett. 67, 3172 (1991).\n10J. Slonczewski, J. Magn. Magn. Mater. 150, 13 (1995).\n11L. Baselgia, M. Warden, F. Waldner, S. L. Hutton, J. E. Drumheller, Y. Q. He, P. E. Wigen,\nand M. Maryko, Phys. Rev. B 38, 2237 (1988).\n12A. Layadi and J. O. Artman, J. Magn. Magn. Mater. 92, 143 (1990).\n13K. Y. Guslienko, N. A. Lesnik, A. I. Mitsek, and B. P. Vozniuk, J. Appl. Phys. 69, 5316 (1991).\n14P. E. Wigen and Z. Zhang, Brazilian J. Phys. 22, 267 (1992).\n15J. Lindner and K. Baberschke, J. Phys. Condens. Matter 15, R193 (2003).\n16P. Gr unberg, R. Schreiber, Y. Pang, U. Walz, M. B. Brodsky, and H. Sowers, Phys. Rev. Lett.\n57, 2442 (1986).\n1317J.J. Krebs, P. Lubitz and A. Chaiken and G. A. Prinz, Phys. Rev. Lett. 63, 1645 (1989).\n18B. Heinrich, Z. Celinski, J. Cochran, W. Muir, J. Rudd, Q. Zhong, A. Arrott, K. Myrtle, and\nJ. Kirschner, Phys. Rev. Lett. 64, 673 (1990).\n19P. Gr unberg, Acta Mater. 48, 239 (2000).\n20A. Layadi, Phys. Rev. B 65, 104422 (2002).\n21Y. Chen, X. Fan, Y. Zhou, Y. Xie, J. Wu, T. Wang, S. T. Chui, and J. Q. Xiao, Adv. Mater.\n27, 1351 (2015).\n22S. Li, Q. Li, J. Xu, S. Yan, G.-X. Miao, S. Kang, Y. Dai, J. Jiao, and Y. L u, Adv. Funct.\nMater. 26, 3738 (2016).\n23B. Heinrich, S. Purcell, J. Dutcher, K. Urquhart, J. Cochran, and A. Arrott, Phys. Rev. B 38,\n12879 (1988).\n24A. Layadi, Phys. Rev. B 63, 174410 (2001).\n25A. Layadi, Phys. Rev. B 72, 024444 (2005).\n26C. Kittel, Phys. Rev. 73, 155 (1948).\n27J. Smit and H. G. Beljers, Philips Res. Rep. 10, 113 (1955).\n28B. Heinrich, K. B. Urquhart, A. S. Arrott, J. F. Cochran, K. Myrtle, and S. T. Purcell, Phys.\nRev. Lett. 59, 1756 (1987).\n29A. Layadi, Spin 6, 1640011 (2016).\n30Y. Liu, L. Chen, C. Y. Tan, H. J. Liu, and C. K. Ong, Rev. Sci. Instrum. 76, 063911 (2005).\n31J. Wei, J. Wang, Q. Liu, X. Li, D. Cao, and X. Sun, Rev. Sci. Instrum. 85, 054705 (2014).\n32A. Layadi, F. W. Ciarallo, and J. O. Artman, IEEE Trans. Magn. 23, 3642 (1987).\n33S. M. Rezende, C. Chesman, M. A. Lucena, A. Azevedo, F. M. de Aguiar, and S. S. P. Parkin,\nJ. Appl. Phys. 84, 958 (1998).\n34Z. Zhang, L. Zhou, P. Wigen, and K. Ounadjela, Phys. Rev. B 50, 6094 (1994).\n35Z. Zhang, L. Zhou, P. Wigen, and K. Ounadjela, Phys. Rev. Lett. 73, 336 (1994).\n36E. Goto, N. Hayashi, T. Miyashita, and K. Nakagawa, Journal of Applied Physics 36, 2951\n(1965).\n37N. Vukadinovic, J. B. Youssef, and H. Le Gall, J. Magn. Magn. Mater. 150, 213 (1995).\n38M. Farle, Rep. Prog. Phys. 61, 755 (1999).\n39S. Li, C. Wang, X.-M. Chu, G.-X. Miao, Q. Xue, W. Zou, M. Liu, J. Xu, Q. Li, Y. Dai, et al. ,\nSci. Rep. 6, 33349 (2016).\n1440W. Wang, G. Chai, and D. Xue, Journal of Physics D: Applied Physics 50, 365003 (2017).\n15"    },    {        "title": "1207.3330v1.Magnetic_relaxation_in_bilayers_of_yttrium_iron_garnet_platinum_due_to_the_dynamic_coupling_at_the_interface.pdf",        "content": " 1 \nMagnetic relaxation in bilayers of yttrium iron gar net/platinum due to the dynamic coupling at \nthe interface  \nS.M. Rezende 1, R. Rodríguez-Suárez 2, M. M. Soares 1, L. H. Vilela-Leão 1, and A. Azevedo 1 \n1Departamento de Física, Universidade Federal de Per nambuco, 50670-901, Recife, PE, Brasil. \n2Facultad de Física, Pontificia Universidad Católica  de Chile, Casilla 306, Santiago, Chile. \n \nWe show that in ferromagnetic (FM)/normal metal (NM ) bilayers the dynamic coupling at the \ninterface transfers an additional magnetic relaxati on from the heavily damped motion of the \nconduction electron spins in the NM layer to the FM  spins. While the FM relaxation rates due \nto two-magnon scattering and spin pumping decrease rapidly with increasing FM film \nthickness, the damping due to the dynamic coupling does not depend on the FM film thickness. \nThe proposed mechanism explains the very large broa dening of ferromagnetic resonance lines \nin thick films of yttrium iron garnet after deposit ion of a Pt layer.  \n \nPACS: 76.20.+q, 76.50.+g, 75.70.Cn, 72.25.Mk \n*Corresponding author: E-mail: rezende@df.ufpe.br  \n \nOne of the fundamental properties of a magnetic sys tem is \nthe manner by which its magnetization relaxes towar ds \nequilibrium. This is governed by the spin interacti ons and \nthe structure of the magnetic system and its detail ed \nunderstanding is important from the point of view o f basic \nphysics and for technological applications. For sev eral \ndecades the magnetic relaxation has been investigat ed \nexperimentally in bulk and thin film materials main ly by \nmeasuring the linewidth of the ferromagnetic resona nce \n(FMR) at microwave frequencies. In bulk magnetic \ninsulators the relaxation occurs through intrinsic \nmechanisms involving magnon-magnon and magnon-\nphonon processes as well as extrinsic mechanisms su ch as \nscattering by impurities. 1,2  In bulk metallic materials the \nrelaxation is dominated by processes involving the \nconduction electrons. 3 In very thin films and multilayers \nnew physical relaxation processes have been discove red in \nthe last fifteen years, the most important ones bei ng two-\nmagnon scattering from the irregularities at the su rfaces or \ninterfaces 4,5  and the spin pumping mechanism.6,7  These \nprocesses contribute with additional relaxation rat es that \nincrease as the magnetic film thickness decreases a nd thus \nbecome very important in ultra-thin films. 5,8 \nIn recent years structures made of bilayers of \nferromagnetic metal (FM) / normal metal (NM) films have \nbeen attracting considerable interest due to the di scoveries \nof the spin Hall effect 9,10  and the inverse spin Hall effect \n(ISHE). 11,12  In a FM/NM bilayer undergoing ferromagnetic \nresonance (FMR) it has been found 11-14  that the precessing \nspins in the FM inject spins into the adjacent NM l ayer \ngenerating a spin-pumping dc voltage by means of th e \nISHE opening immense possibilities in the field of \nspintronics. 15  A very important recent development in this \nfield was the demonstration that the ferrimagnetic insulator \nyttrium iron garnet (YIG) can be used in FM/NM stru ctures to convert charge current into spin current and vic e-versa.16  \nDue to its small magnetic damping, YIG films can be  used \nto transport spin information over much larger dist ances \nthan in FM metals so that YIG/Pt structures have at tracted \nincreasing scientific attention.17-26 However it has been \nobserved that the deposition of a Pt layer on thick  YIG \nfilms produces an unusually large broadening of the  \nmicrowave absorption lines,17,18  which is quite surprising \nbecause one expects the spin pumping mechanism to b e \neffective only at ultra-thin films.  \nIn this paper we show that when a NM layer is \ndeposited on a FM film, in addition to the spin pum ping \nprocess there is another mechanism for magnetic rel axation \nwhich is effective in thick FM films. The mechanism  relies \non the transferred relaxation due to the dynamic co upling \nof the precessing magnetization in the FM with the heavily \ndamped precession of the conduction electron spins in the \nNM layer. This process is effective in FM metallic or \ninsulating films and is independent of the spin-pum ping \nmechanism, although both originate in the spin coup ling at \nthe interface. While the spin-pumping mechanism is due to \nthe flow of angular momentum out of the FM layer in to the \nNM layer and relaxes the longitudinal component of the \nmagnetization, the new mechanism relaxes the transv erse \ncomponents of the magnetization. We show that the \nrelaxation due to dynamic coupling at the interface  explains \nthe observed broadening of the FMR lines in thick Y IG \nfilms with deposition of a Pt layer.  \nWe consider a bilayer of a ferromagnetic material w ith \na nonmagnetic metal in which the precessing magneti c \nmoments of the FM layer interact with the heavily d amped \nspins of the conduction-electron spins in the NM la yer \nthrough the dynamic exchange coupling at the interf ace. In \norder to treat the coupled mode problem we follow t he \nmacroscopic approach of Ref. [16] and consider that  at the  2interface sites i the spins isr of the conduction electrons in \nthe NM layer interact with the spins iSr\n in the FM side \nthrough the s-d exchange interaction, i ii sd sd sSJHrr\n⋅ −=∑, \nwhere Jsd  is the exchange coupling constant. Writing the \nrelation between the magnetization and the spins as  \n)( )(i ii B rrSgrMrrr rr\n− =∑δ µ , where g is the Landé factor and \nBµ is the Bohr magneton, the summation on the interfa ce \nsites i can be written as a surface integral along the \ninterface and the coupling between the magnetizatio n \n),(trMrr\n in the FM side and the magnetization ),(trmNrr of \nthe conduction electrons in the NM side can be writ ten as, \n      ∫∫⋅ −= ),()(),( )/( trmyatrMdy dz dx AJHN eff ex sd rr rr\nδ ,        (1) \nwhere y is the direction perpendicular to the interface pl ane \nx-z with area A at y = 0, )/( MSJJe sd ex γh=  is the \ndimensionless exchange coupling constant, S is an effective \nblock spin per unit cell and M is the magnetization of the \nFM, eγ is the gyromagnetic ratio of the conduction \nelectrons in the NM, 2/se eff ava= is the effective interaction \nrange, ev is the volume per conduction electron, and sa is \nthe lattice constant of the localized spins at the interface on \nthe FM side. In order to make the interface couplin g \ntractable we follow Ref. [16] and consider that the  \nmagnetizations do not vary along the interface plan e and \n),(),( tymtrmN Nrrr=, so that from Eq. (1) we obtain,  \n         ) , 0()( tymatMJHN eff ex sd =⋅ −=rr\nζ ,                             (2) \n where ζ is a factor that accounts for the surface integral : \n1=ζ corresponds to an atomically flat surface and unif orm \nmagnetizations at the interface plane; 1<ζ accounts for \nirregularities at the interface, such as roughness,  or to a \nspatially varying ),(trMvr\n as in a spin wave. Equation (2) \nrepresents the interface energy per unit area and s ince the \nmagnetization is distributed over the whole FM volu me, the \nenergy per unit volume on the FM side is \nFM N eff ex FM \nv dtmatMJ E / ), 0 ( )(rr\n⋅ −=ζ , where FM d is the FM \nlayer thickness. One can write the effective field acting on \nthe FM magnetization due to the interface exchange as, \n      FM Neff ex FM \nv FM \nE dtmaJ\nMEH / ), 0 (rrr\nζ=\n∂∂−= .                           (3) \n Thus one can write the Landau \n     MdamMJHMdt Md\nFM \nFM eff \nN ex r vr rrr\nη ζγ γ − × −×−= )] 0 ( [ ,          (4) \nwhere γ  is the gyromagnetic ratio ( ×=πµ2/hBg 2.8 \nGHz/kOe for YIG) and FM η is the relaxation rate of the FM \nmagnetization in the absence of coupling at the int erface. \nThe equation of motion for the magnetization ),(tymNr of \nthe conduction electrons must take into account the  spin \ndiffusion into the NM side and the exchange couplin g at the \ninterface. The magnetization in the NM can be writt en 16  as )()( )(0 ymyamymN eff Nr rrδδ+ =  where MJ mex Nr rζχ=0  is the \nequilibrium magnetization, Nχ is the paramagnetic \nsusceptibility of the conduction electrons and ),(tymNrδ  \nrepresents the spin accumulation. The equation of m otion \nthen becomes,16 \n     N N Nsf eff N ex e NeN\nmDmyaMmJHmtm\nv vrv rvv\nδ ηδ ζγ γ\n2)()(\n∇+−× −×−=∂∂\n ,          (5) \nwhere sf η is the relaxation rate of the spin accumulation, \nrelated to the electron spin-flip time sf τ by sf sf τη/ 1=, and \nND is the spin diffusion constant. We consider that t he \nstatic field is in a direction parallel to the inte rface plane, \ndesignated the z-direction of a coordinate system that has \nthe y-direction perpendicular to the interface and write  the \ntwo magnetizations as z y x MzmymxM ˆˆˆ ++=r and \nz\nNy\nNx\nN N mzmymxm δδδδ ˆˆˆ ++=r. Equations (4) and (5) \ndescribing the coupled motion of the magnetizations  can be \nsolved for all magnetization components. This has b een \ndone in Ref. [16] for the longitudinal spin accumul ation \n),(tymz\nNδ  from which one can calculate the spin current \ndensity in the NM using z\nN yNBz\ns mDeJ δ µ∇ −= )/( , which \nleads to, in units of angular momentum/(area.time),  \n         )()1 (2) 0 (2 2 22\nzyx\nNBNN z\nSMmmDJΓ+=λµχωh,                               (6) \nwhere 2 / 1)(sf N NDτλ=  is the spin diffusion length, Γ is the \nparameter defined in Ref. [16], with 1=ζ, \n) ( /eff sf sd N aJSτ λh=Γ  and we have assumed circular \nprecession. Comparison of Eq. (6) with the well kno wn \nexpression for the current density in terms of the transverse \ncomponents of the magnetization 6,7  leads to a convenient \nrelation between the exchange coupling parameter an d the \nspin-mixing conductance ↑↓ g, \n            )1 (2\n2 2Γ+=↑↓ \nNBNNDgλµχπh.                                               (7)  \nNotice that this relation differs by a factor of 2 from the one \nin Ref. [16] because we have considered \nyx y x mmmmmm 22 2≈+=−+. The spin current in Eq. (6) \nrepresents a flow of spin angular momentum out of t he FM \nlayer resulting in the relaxation of the FM magneti zation \nthrough the spin pumping mechanism giving rise to a  FMR \nlinewidth (half width at half maximum) given by  6,7     \n                  \nFM SP dMgHπω\n4↑↓ =Δh.                                                  (8) \nwhere Mπ4is the saturation magnetization. Note that the \nspin pumping process relaxes the zMcomponent of the \nmagnetization so that it produces a relaxation time  of the \ntype T1 in the Bloch-Bloembergen formulation 1,2  of the \nLandau-Lifshitz equation. In the derivation of Eq. (6) one  3neglects 16 the reaction of the spin accumulation Nmrδ on \nthe FM magnetization Mr. The full solution of Eqs. (4) and \n(5) for the coupled transverse components of the \nmagnetizations reveals another independent contribu tion to \nthe FM relaxation of the type T2 arising in the dynamic \ncoupling between the spins at the interface. Using for the \ntransverse variables y\nNx\nN N mimm δδδ +=+ and y xmimm+=+ \nwe can solve the equation for the spatial dependenc e of the \ntransverse spin accumulation ),(tymN+δ  obtained from Eq. \n(5). Considering for simplicity a NM layer thicknes s much \nlarger than the spin diffusion length we find \n)/exp( ), 0 ( ),(N N N ytmtym λ δ δ − =+ + so that from Eq. (5) we \nobtain an equation relating )(tm+ and ), 0 (tmN+δ. This \nequation together with the one obtained from Eq. (4 ) form a \nset of two equations for the transverse variables. \nConsidering the time dependence )exp( tiω one obtains the \ncoupled equations,  \n   ) 0 ()/( ])( [+ +−= −−N FM eff ex z FM FM mdaJMmi δ ζγ ηωω  ,       (9) \n   ++\n−−= − −−\nm Mmi\nsf Nz H ex N sf ex H\nηχγωωλδ ηλωω\n]/ ) [( ) 0 (] )1 ( 2 / ) ( [,   (10) \nwhere HeHγω= is the conduction electron spin resonance \nfrequency, )4(MHHFM π γω + = is the FMR frequency and \nΓ=/ζλex  is a dimensionless coupling parameter. From Eqs. \n(9) and (10) one obtains, \n   \n0]) [( 2)] (2 ) [( ]) [( \n=Δ+−−+−Δ+− −−\nCi i\nFM FM ex sf FM FM FM FM \nωωωληωωωηωω\n         (11) \nwhere H FM FM ωωω −=Δ  and )/( 0 FM ex sf eff dM amC λη= . Eq. \n(11) leads to a quadratic equation with complex \ncoefficients, the roots of which are the two comple x \neigenmode frequencies,  \n                    sf Hiηωω 21+≈ ,                                           (12) \n     ] )( )(4 [0 4 4\n2\nFM eff \nsf ex sf ex FM FM da\nMmi ηληληωω +++≈  .             (13) \nThe real and imaginary parts of Eqs. (12) and (13) \ncorrespond respectively to the eigenmode oscillatio n \nfrequencies and relaxation rates. Clearly 1ω is associated \nwith the motion dominated by the conduction electro n spins \nin the NM layer, whereas 2ω is associated with the spin \nprecession in the FM layer. Since 10 10 ~Hωs-1 and \n12 10 ~sf ηs-1 the motion of the spins in Pt is heavily \noverdamped. The important result revealed in Eq. (1 3) is \nthat the relaxation rate of the FM layer has, in ad dition to \nthe intrinsic term FM η, two contributions proportional to the \nfourth power of the exchange coupling parameter and  to the \nconduction electron spin relaxation sf η. Note also that since \n1/0<Mm  and 1/<< FM eff da , for YIG films with µm \nthickness, the third term in the imaginary part of Eq. (13) is \nnegligible compared to the second. As will be shown  in the \nfollowing, the damping transferred from the conduct ion electron spins in the NM layer to the FM magnetizat ion \nrepresented by the second term in the imaginary par t of Eq. \n(13) explains the large broadening of the FMR lines  \nobserved in YIG/Pt. \nThe experiments were carried out at room temperatur e \nwith several samples consisting of a single-crystal  YIG film \nwith thickness in the range 8 to 28 µm, grown by li quid-\nphase epitaxy (LPE) on 0.5 mm thick (111) gadoliniu m \ngallium garnet substrates and cut in an approximate  square \nshape with lateral dimensions of 2 to 4 mm. Measure ments \nwere made with the bare YIG films and with a Pt lay er with \nthickness 6 nm or 20 nm deposited by magnetron \nsputtering. The samples are mounted at the tip of a  plastic \nrod to allow measurements as a function of the angl e and \nplaced in the middle of a shorted rectangular waveg uide \nbetween the poles of an electromagnet. The static m agnetic \nfield with intensity H and the microwave magnetic field are \nkept perpendicular to each other. Field sweep \nmeasurements were made to obtain spectra of the \nmicrowave absorption derivative at several values o f the \nangle Hθ between the field Hr\n the film plane to investigate \nthe out-of-plane angle dependence of the spectra. \nAll samples investigated exhibited similar behavior . \nThe FMR linewidths in the bare YIG film are less th an 1 \nOe with the applied field parallel or perpendicular  to the \nfilm plane. After deposition of a 6 nm or 20 nm Pt layer the \nlinewidths increase to several Oe. We present here only \nsome representative results obtained with microwave  \nfrequency 9.4 GHz and input power less than 10 mW. Fig. \n1 shows the field derivative d P/d H of the microwave \nabsorption spectra for a 28  µm thick YIG film for two \ndirections of the field, parallel and perpendicular  to the film \nplane. The spectra in Figs. 1(a) and 1(b) obtained before \ndeposition of the Pt layer allow a clear identifica tion of the \nabsorption lines. They correspond to standing spin- wave \nmodes that have quantized in-plane wave numbers due  to \nthe boundary conditions at the edges of the film. 17,18,27   The \nstrongest line corresponds to the FMR mode that has  \nfrequency given by 2 / 1 2 / 1\n0 )4(MHH π γω + ≈ . In Fig. 1(a) \nwith the field in the plane, the lines to the left of the FMR \nmode correspond to hybridized surface modes whereas  \nthose to the right are volume magnetostatic modes.  17,18  As \nshown in Figs. 1(c) and (d) the deposition of a 6 n m thick \nPt layer on the YIG film produces a pronounced \nbroadening of the lines for all modes. They have ne arly the \nsame peak-to-peak linewidth pp HΔ of about 6 Oe, which is \nnearly ten times larger than in bare YIG. Similar r esults \nwere obtained with YIG film thickness 8 and 15 µm. In \nboth samples the linewidths in YIG/Pt have nearly t he same \nvalue for θΗ = 0 and 90 o, which are respectively 2.5 and 3.0 \nOe larger than in the corresponding bare YIG film.   \nThe first mechanism one would consider to explain t he \nincreased damping in YIG caused by deposition of Pt  is the \nspin pumping process.  6,7,23 Using for the effective spin-  4 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 1 . Field scan microwave field derivative absorption \nspectra at a frequency 9.4 GHz of a 28 µm thick YIG  film with \nlateral dimensions 2 x 3 mm with the magnetic field  applied \nparallel or perpendicular to the film plane as indi cated. In (a) and \n(b) the YIG film is bare while in (c) and (d) it is  covered with a 6 \nnm Pt layer. \n \nmixing conductance the value 13 10 5 . 1×≈↑↓ g cm -2 determined \nin Ref. [18] from measurements of VSP  using the spin-Hall \nangle 08 . 0=Hγ\n and spin-diffusion length 7 . 3=Nλnm, 14,18  \nand Mπ4= 1.76 kG for YIG, we obtain with Eq. (8) for a \n28 µm thick YIG/Pt bilayer a linewidth of 410 4−×=ΔSP H Oe. \nThis is definitely too small compared to the observ ed line \nbroadening ruling out the spin pumping as the sourc e of the \nadditional relaxation. Another possible origin of t he line \nbroadening with Pt deposition is the enhancement in  the \ninterface two-magnon scattering relaxation similar to the \none observed when a FM layer is in contact with an \nantiferromagnetic material. 5 However, the two-magnon \nlinewidth 4 varies with 2/ 1FM d and like the spin pumping it is \nimportant only in films with thickness of a few nan ometers. \nMoreover, with the field normal to the plane the FM R \nfrequency is at the bottom of the spin-wave manifol d and \nthere are relatively few degenerate states into whi ch the \nFMR mode can decay into, so that the relaxation due  to \ntwo-magnon scattering should be small as demonstrat ed \ntheoretically and observed experimentally. 29,30  As it is clear \nin Fig. 1, the linewidths in YIG/Pt at =Hθ0 and 90 o are \nvery similar, ruling out the two-magnon scattering as the \nmechanism responsible for the line broadening. \nWe will show now that the transferred relaxation \nmechanism proposed here accounts comfortably for th e \nobserved line broadening in YIG/Pt. For FM films wi th \nthickness in the µm range the additional damping in  Eq. \n(13) is dominated by the second term in the imagina ry part \nso one can write the contribution of the transferre d \nrelaxation to the FMR linewidth as,  \n           γηλ/4\nsf ex trans H=Δ ,                                          (14) where the 1/<< Γ=ζλex  exchange parameter is related to \nthe spin mixing conductance by,  \n              \nNNNB\nex Dgχπλµζλh222\n2 ↑↓ = .                                  (15) \nAtomic force microscopy analysis has revealed that \nthe surfaces of the LPE grown YIG films are flat in  the \natomic level, so we consider 1=ζ. Using the spin mixing \nconductance 18  for YIG/Pt, -2 13 cm  10 1.5 ×=↑↓ g , and the \nfollowing parameters for Pt, nm  7 . 3=Nλ (Ref.14), \n12 10 =sf ηs-1 (Ref.16), 1 . 02==sf N NDηλ cm 2s-1, 510 1 . 2−×=Nχ \n(paramagnetic susceptibility of bulk Pt, Ref.31), w e obtain  \n024 . 02=ex λ . This gives for the theoretical additional FMR \nlinewidth due to the transferred relaxation a value  \n32 ≈Δtrans H Oe. This value is six to ten times larger than the \nmeasured ones, revealing that the dynamic interacti on at \nthe interface indeed transfers a considerable dampi ng from \nthe overdamped motion of the conduction electron sp ins in \nPt to the magnetization in YIG. The discrepancy bet ween \ntheory and experiment can be attributed to several factors. \nFirst we note that there is a large discrepancy in the \nparameters for Pt reported by different authors. 32  Second, \nand perhaps more important, we have used in the \ncalculation of ex λ the value of the paramagnetic \nsusceptibility for bulk Pt, which is the only one a vailable in \nthe literature. 31  However, there are experimental evidences \nthat when Pt is deposited on a FM metal, such as Ni  and \nPy, the atomic layers close to interface become spi n \npolarized due to the proximity effect. 33,34  This effect most \ncertainly occurs also in YIG/Pt, producing an enhan cement \nof the Pt susceptibility near the interface, which decreases \nthe value of ex λ and consequently lowers the theoretical \nprediction for trans HΔ. Note that an enhancement by a factor \nof 3 in the Pt susceptibility, which is well within  the \nincrease in the magnetic moment near the interface \nmeasured is Refs. [33,34], is sufficient to produce  a nice \nagreement between theory and experiments. \nIn conclusion, we have demonstrated that in FM/NM \nbilayers, the coupling of the overdamped motion of the \nconduction electron spins in the NM layer to the sp ins in \nthe FM layer gives rise to a transferred damping \nmechanism. This additional damping has an origin th at is \nindependent of the well known spin-pumping \nmechanism.6,7  Both relaxation processes originate in the \ninterface coupling but they are due to quite differ ent \nmechanisms. The conventional spin-pumping is due to  the \nflow of angular momentum out of the FM layer into t he \nNM layer and relaxes the longitudinal component of the \nmagnetization. It does not depend on the conduction  \nelectron spin-flip relaxation rate sf η and varies inversely \nwith the thickness of the FM layer. On the other ha nd, the \nmechanism proposed here relaxes the transverse \ncomponents of the magnetization, resulting in a dam ping 2.50 2.55 2.60 2.65 Signal dP/dH  (arb. units) θH = 0° \nHH(a) \n5.07 5.12 5.17 5.22 (b) θH = 90° \nH\nH\n2.50 2.55 2.60 2.65 (c) Signal dP/dH  (arb. units) \nMagnetic Field H (kOe) θH = 0° HH\n5.07 5.12 5.17 5.22 (d) \nMagnetic Field H (kOe) θH = 90° \n  5rate proportional to sf η and independent of the thickness of \nthe FM layer. The existence of this novel relaxatio n \nmechanism is clearly observed in the broadening of the \nferromagnetic resonance lines in relatively thick Y IG films \nafter deposition of a Pt layer.  \nThis work was supported in Brazil by the agencies \nCNPq, CAPES, FINEP and FACEPE and in Chile by the \nMillennium Science Nucleus \"Basic and Applied \nMagnetism\" No. P10-061-F. \n \nReferences \n1 M. Sparks, Ferromagnetic Relaxation  (Mc Graw-Hill, New \nYork, 1964). \n2 R.M. White, Quantum Theory of Magnetism , 3 rd  Edition \n(Springer-Verlag, Berlin 2007). \n3 V. Koreman and R.E. Prange, Phys. Rev. B 6, 2769 (1972). \n4 R. Arias and D.L. Mills, Phys. Rev. B 60 , 7395 (1999). \n5 S. M. Rezende, A. Azevedo, M.A. Lucena, and F. M. Aguiar, \nPhys.  Rev.  B 63,  214418 (2001). \n6 Y. Tserkovnyak, A. Brataas, G.E.W. Bauer, Phys. Re v. Lett. 88 , \n117601 (2002). \n7 Y. Tserkovnyak, A. Brataas, G. E. Bauer and B.I. H alperin, \nRev. Mod. Phys. 77 , 1375 (2005). \n8 B. Heinrich, Chapter 3  of Ultrathin Magnetic Structures II , \nedited by  B. Heinrich and J.A.C. Bland (Springer V erlag, \nHeidelberg, 1994). \n9 J.E. Hirsch, Phys. Rev. Lett. 83 , 1834 (1999).  \n10  S.O. Valenzuela and M. Tinkham, Nature (London) 442 , 176 \n(2006). \n11  A. Azevedo, L. H. Vilela Leão, R. L. Rodriguez-Sua rez, A. B. \nOliveira, and S. M. Rezende, J. Appl. Phys. 97 , 10C715 (2005). \n12  E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Ap pl. Phys. \nLett. 88 , 182509 (2006). \n13 O. Mosendz, J.E. Pearson, F.Y. Fradin, G.E.W. Baue r, S.D. \nBader, and A. Hoffmann, Phys. Rev. Lett. 104 , 046602 (2010); \nO. Mosendz, V. Vlaminck, J.E. Pearson, F.Y. Fradin,  G.E.W. \nBauer, S.D. Bader, and A. Hoffmann, Phys. Rev. B 82 , 214403 \n(2010). \n14 A. Azevedo, L. H. Vilela-Leão, R. L. Rodríguez-Suár ez, A.F. \nLacerda Santos, and S. M. Rezende, Phys. Rev. B 83 , 144402 \n(2011).  \n15  I. Zutic and H. Dery, Nature Mat. 10 , 647 (2011). \n16  Y. Kajiwara, S. Takahashi, J. Ohe, K. Uchida, M. M izuguchi,  \nH. Umezawa, K. Kawai, K. Ando, K. Takanashi, S. Mae kawa, \nand E. Saitoh, Nature (London) 464 , 262 (2010). 17  C.W. Sandweg, Y. Kajiwara, K. Ando, E. Saitoh, and  B. \nHillebrands, Appl. Phys. Lett. 97 , 252504 (2010). \n18  L.H. Vilela-Leão, C. Salvador, A. Azevedo, and S.M . Rezende,  \nAppl. Phys. Lett. 99 , 102505 (2011). \n19  C.W. Sandweg, Y. Kajiwara, A.V. Chumak, A.A. Serga , V.I. \nVasyuchka, M.B. Jungfleisch, E. Saitoh, and B. Hill ebrands, \nPhys. Rev. Lett. 106 , 216601 (2011). \n20  H. Kurebayashi,  O. Dzyapko, V. E. Demidov, D. Fang, A. J. \nFerguson,and S. O. Demokritov,  Appl. Phys. Lett. 99 , 162502 \n(2011). \n21  B. Heinrich, C. Burrowes, E. Montoya, B. Kardasz, E. Girt, Y-\nY. Song, Y. Sun, and M. Wu, Phys. Rev. Lett. 107 , 066604 \n(2011).   \n22  X. Jia, K. Liu, K. Xia, and G.E.W. Bauer, Eur. Phy s. Lett. 96 , \n17005 (2011).   \n23 Z. Wang, Y. Sun, Y-Y. Song, M. Wu, H. Schultheiss, J.E. \nPearson, and A. Hoffmann, Appl. Phys. Lett. 99 , 162511 (2011). \n24 E. Padrón-Hernández, A. Azevedo, and S.M. Rezende, Phys. \nRev. Lett. 107 , 197203 (2011).  \n25  Z. Qiu, Y. Kajiwara, K. Ando, Y. Fujikawa, K. Uchi da, K. \nHarii, T. Yoshino, and E. Saitoh, Appl. Phys. Lett.  100 , 022402 \n(2012). \n26 J. Xiao and G.E.W. Bauer, Phys. Rev. Lett. 108 , 217204 \n(2012). \n27  R.W. Damon and J.R. Eshbach, J. Phys. Chem. Solids  19 , 308 \n(1961). \n28 D.D. Stancil and A. Prabhakar, Spin Waves- Theory a nd \nApplications, Springer (New York 2009). \n29  P. Landeros, R.E. Arias and D.L. Mills, Phys. Rev.  B 77 , \n214405 (2008). \n30  K. Zakeri, J. Lindner, I. Barsukov, R. Meckenstock , M. Farle, \nU. vonHoersten, H. Wende, W. Keune, J. Rocker, S.S.  \nKalarickal, K. Lenz, W. Kuch, K. Baberschke,and Z. Frait, Phys. \nRev. B 76 , 104416 (2007). \n31 F. E. Hoare and J. C. Matthews, Proc. R. Soc. Lond.  A 212 , \n137 (1952). \n32 L. Liu, T. Moriyama, D.C. Ralph, and R.A. Buhrman, Phys. \nRev. Lett. 106 , 036601(2011); arXiv1111_3702. \n33 F. Wilhelm, P. Poulopoulos, G. Ceballos, H. Wende, K. \nBaberschke, P. Srivastava, D. Benea, H. Ebert, M. A ngelakeris, \nN.K. Flevaris, D. Niarchos, A. Rogalev, and N.B. Br ookes, Phys. \nRev. Lett. 85 , 413 (2000). \n34 F. Meyer, S. Lounis, J. Wiebe, L. Zhou, S. Heers, P . \nMavropoulos, P.H. Dederichs, S. Blugel, and R. Wies endanger, \nPhys. Rev. B 83 , 075407 (2011). \n \n \n \n \n \n "    },    {        "title": "1503.03719v1.Chirp_spectroscopy_applied_to_the_characterization_of_Ferromagnetic_Resonance_in_Magnetic_Tunnel_Junctions.pdf",        "content": "GW-20 \n \nChirp spectroscopy applied to the characterization of Ferromagnetic \nResonance in Magnetic Tunnel Junctions  \nM. Ricci1, P. Burrascano1, M. Carpentieri2, R. Tomasello3, G. Finocchio4 \n{marco.ricci,pietro.burrascano}@unipg.it, mario.carpentieri@poliba.it,  tomasello@deis.unical.it,gfinocchio@unime.it  \n \n1Department of Engineering , Polo Scientifico Didattico di Terni, University of Perugia, Terni, TR, I -50100 Italy  \n2Department of Electrical and Information Engineering, Politecnico of Bari, via E. Orabona 4, I -70125 Bari, Italy  \n3Department of Computer Science, Modelling, Electronics and System Science, University of Calabria, Via P. Bucci, I -87036,        \nRende (CS), Italy  \n4Department of Elect ronic Engineering, Industrial Chemistry and Engineering, University of Messina, C.da di Dio, I -98166, \nMessina, Italy  \nMagnetic Tunnel Junction devices find use in several applications based on the exploitation of the Spin -Transfer Torque \nphenomenon. The Fer romagnetic Resonance curve is a key characteristic of any Magnetic Tunnel Junctions . It is usually characterized \nboth experimentally and numerically by performing a lot of measurements of the magnetic response  to a sinusoidal field or current . \nHere we prop ose the use of a chirp signal as excitation signal to reconstruct the Ferromagnetic resonance curve  with a single \nmeasurement/simulation.  A micromagnetic comparison of the proposed method with the traditional one is shown.  \n \nIndex Terms —Ferromagnetic Resonance, Chirp signal, Magnetic Tunnel Junction, Spin Hall Effect .  \n \nI. INTRODUCTION  \nagnetic Tunnel Junction (MTJ) represents one of the  \nmost interesting structures in the field of spintronics and \nquantum electronics . It finds  many application s such as non -\nvolatile  memory and hard drive read heads for high-density  \ninformation storage. On the other hand, the discovery of the \nSpin-Transfer –Torque phenomen a [1], [2] opened  a plenty of \nnew scenarios for the use of MTJs . In particular MgO -based \nMTJs, beside their application as STT -magnetic memories, are \nat the basis of extremely promising devices for microwave \ngeneration and detection: (a)  the Spin -Transfer -Nano -\nOscillator  [3], [4] and (b) the Spin-Torque -Diode (STD) [5], \n[6],[7],[8],[9]. \nWith respect to the latter application, recent experiments have \ndemonstrated that nanoscale MTJs can rectify microwave \nsignals and can work  as microwave signal detectors. The \nrectiﬁed voltage is produced by the non -linear coupling \nbetween  the magnetoresistance oscillating at microwave \nfrequency due to the precession of the MTJ magnetization and \nthe applied microwave signal. The maximum  rectif ied voltage \nis observed when the frequency of the microwave signal is \nequal to the ferromagnetic resonance (FMR) frequency of the \nMTJ magnetization. This is one reason  to characterize the \nFMR  of MTJ devices, in order to individuate the frequency \nrange of m aximum sensibility.  \nThe FMR is computed as magnetic response to  a radio \nfrequency (RF) magnetic field or spin transfer torque . In the \nlast scenario, an  oscillating current is applied  to resonantly \nexcite the magnetic precession . In MTJs, the oscillation o f the \nmagnetization gives rise to a tunneling magnetoresistive signal \nwhich  oscillates at the same frequency . By sweeping the \nfrequency of the microwave current, the maximum output \nsignal of the rectified voltage produced  is expected when the \nfrequency is swept through the resonance frequency, while a \nvery small signal is expected at  frequencies far from the \nresonance frequency [5], [10]. A similar procedure is \nreproduced in numerical experiments by means of micromagnetic modeling: a sinusoidal excitation is applied to \nthe structure for a certain long time and the  \nsteady-state amplitude of the magnetization oscillation is \nconsidered. By repeating the simulations several times at \ndifferent frequencies the curve of the FMR can be \nreconstructed.  Here , we study the possibility to use a broad \nband excitation, to characteriz e the FMR  in a single \nmeasurement. Precisely,  we test the method by executing \nmicromagnetic analysis of two main structures : a standard  \nMTJ structure , described in [ 11] and [ 12], and a  3-terminal  \nMTJ structure deposited on a Tantalum strip that exhibits giant \nSpin Hall Effect (SHE) , see [ 13],[14],[15]. \nAmong the possible wideband excitation waveform s, the use \nof chirp signal s has been preferred for two main reasons : (a) it \nallows all the frequency range of interest to be spanned \ncontinuously with a single s hot excitation; (b) it allows the \nbehavior  of the MTJ as STD  to be studied in presence of a \ncomplex excitation widely used in microwave radar and \ncommunication for its peculiar characteristics.   \nAs described in the following Section, t he chirp signal is \ncharacterized by an instantaneous frequency that can be \ndefined to follow any continuous path in the time -frequency \nplane. By limiting ourselves to linearly span the range of \ninterest, it is possible to associate at each excitation time a \nsingle frequ ency value and therefore  a frequency analysis, i.e.  \nthe FMR analysis,  can be executed  by a time domain \nprocessing. This method is called chirp spectroscopy and it  \ncould bring to  a drastic reduction of the measurement time \nneeded to characterize the FMR, both fr om an experimental \nand a numerical point of view.  \nThe MTJ -based STDs  have been recently used for on-chip \nmicrowave interferometer s to measure not only the magnitude \nof a microwave field but also the relative phase between the \nelectric field E and the magn etic field  B [16]. The \ninterferometry device developed opens the realization of \nseveral microwaves imaging procedures by using both \nsinusoidal or multi -frequency signals. Chirp excitation could \nbe extremely useful in this perspective for its well -known \nspectral and auto -correlation properties that made chirp one of \nthe most used waveform for pulse -compression applications  M GW-20 \n \n[17]. Moreover the Frequency Modulated Continuous Wave \nprotocol [18] developed for chirp signal well fit with the \nspintronic interferomet ry proposed in [16] and [19]. The \ncharacterization of the magnetization dynamics of an MTJ \ndevice excited by a chirp signal, either current or field, it is \ntherefore of interest.  \nAt the same time , the capability to identify the FMR in several \nstructures re present s a fundamental tool to study the physics of \nmany physical processes, for instance the determination of the \nSHE angle [ 17], [18], the dynamics of spin waves, etc.   \nII. CHIRP SIGNAL  \nA generic chirp signal, defined in the time interval   [   ] , is \nrepresented by the expression:  \n ( )     [ ( )]     *  (      \n ∫ ( )   \n )+        (1) \nwhere: A is the signal amplitude, f0 is the central frequency,    \nis the signal bandwidth, x() is a smooth monotonic signal \ntaking values \nx(0) 1\n and \nx(T) 1\n ,  (t) is the resultant \nphase function.  If  (t) is a linear function , we retrieve the \nstandard definition of a sinusoidal signal, while when  (t) is \nnon-linear , the resulting signal is harmonic , but with a not \nunique oscill ation frequency. In particular, unless a constant \nphase term, any chirp signal is completely characterized by  its \ninstantaneous  frequency  trajectory fist(t), which is related to the \nphase function  (t) by the equation :  \n      ( )  \n    ( )\n        \n  ( )                  (2) \nThe concept of the instantaneous frequency is of utmost \nimportance in chirp design since from the trajectory of fist(t)  \nalso the spectral content of the signal can be derived [ 22].  \n \nFIG. 1 HERE  \n \nWith respect to the measure of  the FMR, we exploit ed linear \nchirp  signal s that exhibit an instantaneous frequency varying \nwith the rule:     ( )     (     ) \n    where f1 is the starting \nfrequency , f2 the stop frequency ,           \n   and  f = f2 – f1.  \nIf f2 > f1 an ―UP‖ chirp is obtained, conversely for f2 < f1 the \nchirp is named ―DOWN‖ . By the definition of  the \ninstantaneous frequency [ 19], the following expression  can be \nderived for a linear chirp :  \n ( )     [  (      \n    )]                          (3) \nfrom which it is straightforward to see that in (1) the linear \nchirp case is retrieved for \n2t x(t) 1.T\n    \nOf course, although the instantaneous frequency is clearly \ndefined, due to the finite length of the signal a perfect \nconfinement of the power spectrum  in the  range  [f1,f2] is \nunrealizable . Nevertheless,  if the product T× f of the duration \nof the signal T and the bandwidth  f is large enough, i.e. \nT× f>>1, the power spectral density is almost flat and well \nconfined in the region [ f1,f2]. An example of a linear chirp of \nduration T=10ns  defined in the rang e 2-10 GHz is reported in \nFig. 1 together with the plot of the instantaneous frequency and \nof the spectral amplitude. Moreover, to illustrate how the \nhighe r the T× f product  is, the higher  is the confinement, the \nspectral amplitudes of two chirps having the same  f, but longer T are shown. Due to the aforementioned properties, \nlinear chirp signals have been used for several years to perform \nspectroscopy analysis: if  a linear chirp is used a s input of a \nlinear system, the output signal envelope represents the \nmagnitude of the transfer function in the spanned range. In the \npresent case, the FMR is an inhere ntly non -linear phenomenon \nbut it is worth to study if the chirp excitation can represen t a \nvalid alternative  to the standard procedure.  \nTo accomplish this aim we executed various full \nmicromagnetic simulations by exciting the structures below \ndescribed with a RF current following a chirp waveform. In \nparticular, we performed simulations by varying the duration, \nthe bandwidth and the amplitude of chirp current waveforms. \nFurthermore, we compare the results obtained by using UP and \nDOWN chirps. This comparison allows highlighting the non -\nlinear feature of the process, indeed, if the system was linear, \nthe same spectral responses should be measured for both UP \nand DOWN chirps . \nIII. NUMERICAL MODEL  \nWe compute the FMR by using the presented method on two \ndifferent strctures:  \n(A): an  MTJ  composed by a synthetic antiferromagnet (SAF) \npinned layer (PL) [IrMn(6.1)/CoFe(1.8)/Ru/CoFeB(2.0)], \ntunnel barrier  [MgO(1.25)], magnetic free layer (FL) \n[CoFe(0.5)/CoFeB(3.4)] (the dimensions are in nm) with \nelliptical cross section of 65x130 nm2 (see Fig. 2a).  \n \nFIG. 2 HERE  \n \n(B): a SHE -MTJ  composed by an MTJ device \nCoFeB(1)/MgO(1.2)/CoFeB(4)/Ta(5)/Ru(5) (thicknesses in \nnm) deposited on a Tantalum strip (6000 x 1200 x 6 nm3). The \nthinner CoFeB (1) is the free layer and it is coupled to the Ta \nstrip, while the thicker CoFeB (4) acts as a pinned layer, with \na magnetization oriented along the –y-direction  (see Fig. 2b) . \nBecause of the ultra -thin free layer , we take into account a \nvery high perpendicular anisotropy ( Ku=0.9x106 J/m3) and the  \ninterfacial Dzyaloshinskii -Moriya Interaction  (DMI) , due the \ncoupling b etween the Tantalum strip (heavy metal with a large \nspin-orbit coupling) and the ferromagnetic free layer  [23], \n[24].   The initial state of the free layer magnetization is out -\nof-plane in the positive z-direction. The structure allows us to \ninject an in -plane current JTa (which gives rise to the spin -Hall \neffect) through the Tantalum strip and a perpendicular current \nJMTJ  via the MTJ  stack . We perform micromagnetic \nsimulations based on the numerical sol ution of the Landau -\nLifshitz -Gilbert -Slonczewski (LLGS) equation  [2], where the \nstandard effective field, the magnetostatic field due to the  \npolarizer, and the Oer sted field due to the current are taken \ninto account. In the case of the structure (A) the equation is \ndescribed in [25] typical parameters for the free-layer were : \nsaturation magnetization MS=1000x103 A/m, damping \nconstant α=0.01, exchange constant A=2.0×10−11 J/m, uniaxial \nanisotropy 4x 103 J/m3, spin  polarization 0.6. The free layer  has \nbeen discreti zed in computational cells of 5x5x 4 nm3. \nRegarding the structure  (B), the spin -orbit torque term driven \nby SHE is also included in the LLGS [15]. Magnetic \nparameters are: MS=1000x103A/m, A=2.0x10-11J/m, =0.015, GW-20 \n \nspin angle\nHα =-0.15,  and spin-polarization \nT =0.66.  The \ndescription of the model lies beyond the scope of the present \npaper and the reader can found exhaustive details on the \nlitera ture, see for example [25] , [26]  . In both cases, the \nintegration time step is 32 fs. The simulations have been \nexecuted by using the time step setting the temperature at \nT=1K and the chirps used spanned the range 4 -8 GHz and 13 -\n20 GHz for the structures (A)  and (B) respectively.  \nIV. RESULTS AND DISCUSSIO N \nScenario (A).  Several chirp current waveform were simulated . \nFig. 3  shows the comparison between the FMR curve obtained \nby using sine excitation (FMR Sine) and the FMR curves \nattained by using UP and DOWN chirps  (FMR UP and \nFMR DOWN ). At a first sight, it can be noted that FMR Sine and \nFMR UP/DOWN  ones are similar  even if , as expected, the \nmagnetization dynamics show s a dependence on the excitation \nfrequency trajectory due to the n on-linear nature of the \nprocess , i.e. the direction of the chirp is important. Indeed the \ncurves for U P and D OWN  chirp reach  the peak for different f \nvalues, (f* UP and f* DOWN ). In particular we found that f*UP and \nf*DOWN  are almost symmetrical displaced with respect to the \nvalue f* SINE calculated by the standard method  but in general  \nChirp UP, with instantaneous frequency increasing with time, \nachieves results closer to the traditional FMR curve  than those \nprovided by Chirp DOWN . We note also that the FMR  \nlinewidth , i.e. the width of the resonance at half of the \nmaximum, is larger for chirp UP and smaller for chirp \nDOWN, i.e. UP >DOWN . These facts suggest that the non -\nlinear nature of the process hampers the rise of the \nmagnetization oscillation s so th at it is necessary a certain \nexcitati on time to reach their maximum. The result is a  delay \nin the Chirp response and therefore a blu -shift of the peak \nfrequency in the UP case and a red -shift in the DOWN case. \nMoreover this phenomenon is stronger (the dela y is longer) for \nthe DOWN chirp, so that the relative linewidth is narrower.     \n \nFIG. 3 HERE  \n \nThe results obtained by varying the chirp duration and the \nexcitation current amplitude are summarized in Fig. 4 -a where \nthe trends of f*SINE, f*UP and f* DOWN  versus the current \namplitude and the duration T are reported.  It can be also noted \nthat the longer the chirp duration  is, the close r are the FMR UP \nand FMR DOWN values  to FMR Sine and a  similar trend is attained \nby increasing t he excitation curren t amplitude.  At the same \ntime, as the current increases, the resonance frequency \ndecreases. To better evaluate how close the FMR UP/DOWN  \ncurves are with respect the FMR Sine, we introduce an  \n―accuracy‖  parameter defined  as the ratio between the \ndisplacement between the FMR UP(DOWN)  and FMR Sine:  \n \n/UP DOWN Sine\nSineFMR FMRAccuracy               (4) \nFIG. 4 HERE  \n \nThe accuracy calculated for Chirp -UP and -DOWN at \ndifferent time duration and current are reported in Fig. 4 -b,c, \nfrom which  we can con clude that, for  the estimation of the FMR for the MTJ structure, Chirp -UP achieves  in general \nbetter results than Chirp -DOWN  and it can represent  a valid \nalternative to t he traditional sine excitation . It allows the \nreduction of computational time as well it provide s a higher \nfrequency resolution in the peak estimation.  Furthermore, if a  \nbetter  accuracy is required, one can estimate the FMR peak \nfrequency by taking the average of the results obtained with \nUP and DOWN chirps : FMR Chirp=( FMR UP + FMR DOWN )/2 . \nFig. 5 reports the values of FMR Chirp derived by the data \nshown in Fig. 4 and the relative accuracy. It is found that \nFMR Chirp  exhibits an accuracy lower than 1 in all cases even \nfor T=10ns. It is therefore possible to estimate the FMR peak \nfrequency with good accuracy w ith two simulations of 10 ns \nemploying a chirp UP and a chirp DOWN respectively.  \nFIG. 5 HERE  \n \nTo further confirm  this result, we performed similar analysis \non the SHE -MTJ structure aforementioned.  The curves of the \nFMR reconstructed by using sine and chirp exc itation are \nreported in Fig. 6. Also in this case  we found that the FMR \ncurves obtained with chirp excitations are symmetrically \nshifted with respect to the resonance frequency individuated \nby means of sine excitation. As for the MTJ case, the UP chirp \n–based FMR curve exhibits a shift toward higher frequencies \nwhile DOWN chirp -based FMR curve is shifted toward lower \nfrequency and, also in this case, the UP chirp FMR curve is \nvery close to the sine -based FMR curve. Moreover, for the UP \ncase, sharp peaks are  found near the FMR, that are not \ndetected by the sine -based method. The presence of these \npeaks will be better investigated in order to link to peculiar \nphysical phenomena.  Fig 6 shows also the comparison of the \nFMR curve obtained by considering or not th e DMI \ninteraction: quite generally the FMR curve depends on this \nterm but, for structures with circular shape as in this case, the \nFMR is not influenced by this term.  \n \nFIG. 6 HERE  \nV. CONCLUSION  \nA numerical procedure for the characterization of the FMR \nrespons e of MTJ s exploiting chirp coded excitation has been \npresented. The results show that the chirp -based approach can \nbe a valid and effective alternative of the traditional procedure \nallowing a significant reduction of the computational time. \nMoreover, the u se of chirp signal has highlighted the peculiar \nnon-linear behavior of the FMR phenomenon and it is of \ninterest to gather further information about this process. In this \nperspective , it has been proposed in literature to use a non -\nlinear exponential chirp as a input signal to c haracterize non -\nlinear systems [ 28] and this method could be applied also for \nthe characterization of the FMR . The use of chirp excitation is \nalso interesting from an experimental point of view since it \ncan find application to charact erize and exploit spin -diode and \nspin-oscillator MTJ -based structures .  \nACKNOWLEDGMENT  \nThis work was supported by  the project PRIN2010ECA8P3  \nfrom  Italian MIUR.  University of Perugia and University of \nCalabria acknowledge financial support from Fondazione GW-20 \n \nCARIT  and CARICAL for the project ― Progettazione e \nsviluppo di sensori spintronici per imaging a microonde con \napplicazione ai test non distruttivi e alla caratterizzazione dei \nmateriali ‖. \n                                                           \n[1] L. Berger, ―Emission of spin waves by a magnetic multil ayer \ntraversed by current‖, Phys. Rev. B , vol. 54, pp. 9353 -9358, 1996.  \n[2] J. C. Slonczewski, ―Current -driven excitation of magnetic \nmultilayers,‖ J. Magn. Magn. Mater. , vol. 159, pp. L1 -L7, 1996.  \n[3] Z.Zeng, , et al, ― Ultralow -current -density and bias -field-free spin -\ntransfer nano -oscillator ,‖ Scientific reports , vol. 3  , 2013.  \n[4] M. Carpentieri, and F. Lattaru lo, ―Spin -Torque Oscillators Using \nPerpendicular Anisotropy in CoFeB —MgO Magnetic Tunnel \nJunctions,‖ IEEE Trans. Magn. , vol.  49 (7), pp. 3151 -3154,  2013.  \n[5] A. A. Tulapurkar, Y. Suzuki, A. Fukushima, H. Kubota, H. \nMaehara, K. Tsunekawa, D. D. Djayaprawira , N. Watanabe, and S. \nYuasa, ―Spin -torque diode effect in magnetic tunnel junctions,‖ \nNature , vol. 438, pp. 339 -342, 2005.  \n[6] J. Y. Tserkovnyak, T. Moriyama, and Q. X. John,  Physical \nReview B , vol. 78(2), p. 020401, 2008.  \n[7] Zhu, Jian, et al. \"Voltage -induced ferromagnetic resonance in \nmagnetic tunnel junctions.\" Physical review letters  vol. 108, (19), \n197203 , 2012 . \n[8] Miwa, S., Ishibashi, S., Tomita, H., Nozaki , T., Tamura, E., \nAndo, K., & Suzuki, Y. ―Highly sensitive nanoscale spin -torque \ndiode ‖, Nature materials , vol.13, pp. 50 -56, 2014 . \n[9] Gonçalves, A. M., Barsukov, I., Chen, Y. J., Yang, L., Katine, J . \nA., & Krivorotov, I. N. ―Spin torque ferromagne tic resonance with \nmagnetic field modulation ‖,  Applied Physics Letters , vol. 103 (17), \n172406, 2013 . \n[10] J. C. Sankey, P. M. Braganca, A. G. F. Garcia, I. N. Krivorotov, \nR. A. Buhrman, and D. C. Ralph, ―Spin -transfer -driven ferromagnetic \nresonance of individual nanomagnets,‖ Phys. Rev. Lett. , vol. 96, p. \n227601, 2006.  \n[11] M. Carpentieri, M. Ricci,  P. Burrascano, L. Torres, and G. \nFinocchio,   ―Noise -like sequences to resonant excite the writing of a \nuniversal memory based on spin -transfer -torque MRAM,‖ IEEE \nTrans. Magn. , vol. 48, 2407, 2012.  \n[12] M. Carpentieri, M. Ricci, P. Burrascano , L. Torres, G. \nFinocchio, ―Wideband microwave signal to trigger fast switching \nproces ses in magnetic tunnel junctions ‖, J. Appl. Phys. , vol. 111, p. \n07C909, 2012.  \n[13] L. Liu, C -F. Pai, Y. Li, H. W. Tseng, D. C. Ralph, R. A. \nBuhrman, ―Spin -Torque Switchin g with the Giant Spin Hall Effect of \nTantalum,‖ Science , vol. 336, pp. 555 -558, 2012.  \n[14] L. Liu, C -F. Pai, D. C. Ralph, and R. A. Buhrman, ―Magnetic \nOscillations Driven by the Spin Hall Effect in 3 -Terminal Magnetic \nTunnel Junction Devices,‖ Phys. Rev. L ett., vol. 109, p. 186602, \n2012 . \n[15] R. Tomasello, M. Carpentieri, and G. Finocchio, ―Dynamical \nproperties of three terminal magnetic tunnel junctions: Spintronics \nmeets Spinorbitronics,‖ Appl. Phys. Lett. , vol. 103, p. 252408, \nDecember 2013 . \n[16] L. Fu, Z. X. Cao, S. Hemour, K. Wu, D. Houssameddine, W. Lu, \nS. Pistorius, Y. S. Gui, and C -M. Hu, ―Microwave reflection imaging \nusing a magnetic tunnel junction based spintronic microwave \nsensor,‖ Appl. Phys. Lett. , vol.  101(23), p. 232406, 2012.  \n[17] J. R. Klauder, A. C. Price, S. Darlington, and W. J. Albersheim, \n―The theory and design of chirp radars,‖ Bell System Technical \nJournal , vol. 39(4), pp. 745 -808, 1960.  \n[18] A. G. Stove, ―Linear FMCW radar techniques,‖ In IEE \nProceedings F (Radar and Signal  Processing) , vol. 139(5), pp. 343 -\n350, 1992.  \n[19] A. Wirthmann, X. Fan, Y. S. Gui, K. Martens, G. Williams, J. \nDietrich, G. E. Bridges, and C -M. Hu, ―Direct phase probing and                                                                                                      \nmapping via spintronic Michelson interferometry,‖ Phys. Rev. Lett. , \nvol. 105(1),  p. 017202, 2010.  \n[20] L. Bai, Lihui, P. Hyde, Y. S. Gui, C -M. Hu, V. Vlaminck, J. E. \nPearson, S. D. Bader, and A. Hoffmann, ―Universal Method for \nSeparating Spin Pumping from Spin Rectification Voltage of \nFerromagnetic Resonance,‖ Phys. Rev. Lett. , vol. 1 11(21), p. 217602, \n2013.  \n[21] G. Finocchio, M. Carpentieri, E. Martinez, and B. Azzerboni, \n―Switching of a single ferromagnetic layer driven by spin Hall \neffect‖, Appl. Phys. Lett. , vol. 102, p. 212410, May 2013.  \n[22] M. Pollakowski, H. Ermert. ―Chirp Sig nal Matching and Signal \nPower Optimization in Pulse -Echo Mode Ultrasonic Non Destructive \nTesting,‖ IEEE Trans. Ultras., Ferr. Freq. Control , vol.  41(5), pp. \n655-659, 1994.  \n[23] T. Moriya, ―New mechanism of anisotropic superexchange \ninteraction,‖ Phys. Rev. Lett., vol. 4, p. 228, 1960.  \n[24] R. Tomasello, M. Carpentieri, G. Finocchio. ―Influence of the \nDzyaloshinskii -Moriya interaction on the spin -torque diode effect,‖ J. \nAppl. Phys ., vol. 115, p.  17C730, 2014.  \n[25] M. Carpentieri, G. Finocchio, B. Azzerboni , and L. Torres, \n―Spin transfer -torque resonant switching and injection locking in the \npresence of a weak external microwave field for spin valves with \nperpendicular materials,‖ Phys. Rev. B , vol. 82, p. 094434, 2010.  \n[26] A. Giordano, G. Finocchio, L. Tor res, M. Carpentieri, and B. \nAzzerboni, ―Semi -implicit integration scheme for Landau –Lifshitz –\nGilbert -Slonczewski equation,‖ J. Appl. Phys ., vol. 111, p.  07D112, \n2012.  \n[27] M. Carpentieri, E. Martinez, and G. Finocchio , ―High frequency \nspin-torque -oscillators with reduced perpendicular torque effect \nbased on asymmetric vortex polarizer,‖ J. Appl. Phys. , vol. 110, p. \n093911, 2011.  \n[28] A. Farina, Audio Engineering Society Convention, vol. 108. \nAudio Engineering Society, 2000.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n GW-20 \n \n                                                                                                     \n \n \n \n \n \n \nFig. 1.  ( a) Example  of linear chirp signal; (b ) its  instantaneous frequency, ( c) \nspectral amplitude of  a linear chirp for T=10, 30 and 50 ns , black,  red and \ngreen curves respectively . \n \n \n \nFig. 2.  Sketch of the structures under investigation (a) MTJ device  and (b) \nSHE -MTJ device.  \n \n \nFig. 3.  Comparison between FMR calcul ated with sine excitation at various \ncurrent amplitudes and with UP and DOWN chirp signals.  \n \n \n \n                                                                                                      \n \nFig. 4.  FMR UP and FMR DOWN peak frequency and relative accuracy  at various \ncurrent amplitudes and chirp duration .  \n \nFig. 5.  FMR Chirp peak frequency and relative accuracy  at various current \namplitudes and chirp duration .  \n \n \n \nFig. 6.  Comparison between FMR curve calcul ated with sine excitation and \nchirp excitation  without ( left) and with ( right )  Dzyaloshinskii -Moriya \nInteraction   \n \n  \n \n \n \n \n \n \n \n \n \n \n"    },    {        "title": "1502.04570v2.Electrical_manipulation_of_a_ferromagnet_by_an_antiferromagnet.pdf",        "content": "Electrical manipulation of a ferromagnet by an antiferromagnet\nV. Tshitoyan,1C. Ciccarelli,1A. P. Mihai,2,\u0003M. Ali,2\nA. C. Irvine,1T. A. Moore,2T. Jungwirth,3, 4and A. J. Ferguson1,y\n1Microelectronics Group, Cavendish Laboratory, University of Cambridge, CB3 0HE, UK\n2School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, UK\n3Institute of Physics ASCR, v.v.i., Cukrovarnick\u0013 a 10, 162 53 Praha 6, Czech Republic\n4School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, UK\n(Dated: November 6, 2018)\nWe demonstrate that an antiferromagnet can be employed for a highly e\u000ecient electrical manip-\nulation of a ferromagnet. In our study we use an electrical detection technique of the ferromagnetic\nresonance driven by an in-plane ac-current in a NiFe/IrMn bilayer. At room temperature, we ob-\nserve antidamping-like spin torque acting on the NiFe ferromagnet, generated by the in-plane current\ndriven through the IrMn antiferromagnet. A large enhancement of the torque, characterized by an\ne\u000bective spin-Hall angle exceeding most heavy transition metals, correlates with the presence of the\nexchange-bias \feld at the NiFe/IrMn interface. It highlights that, in addition to strong spin-orbit\ncoupling, the antiferromagnetic order in IrMn governs the observed phenomenon.\nI. INTRODUCTION\nRecently, a new direction in spintronics has been pro-\nposed based on non-relativistic1{5and relativistic6,7spin-\ntransport phenomena in which antiferromagnets (AFMs)\ncomplement or replace ferromagnets (FMs) in active\nparts of the device. AFMs have for decades played a\npassive role in conventional spin-valve structures where\nthey provide pinning of the reference FM layer8. This im-\nplies that on one hand, incorporation of some AFM ma-\nterials, including IrMn, in common spintronic structures\nis well established. On the other hand, limiting their\nutility to a passive pinning role leaves a broad range of\nspintronic phenomena and functionalities based on AFMs\nvirtually unexplored. In addition to the insensitivity to\nmagnetic \felds and the lack of stray \felds, AFMs are\ncommon among metals, semiconductors, and insulators\nand can have orders of magnitude shorter spin-dynamics\ntimescales, to name a few immediate merits of the fore-\nseen concept of AFM spintronics.\nAFM magneto-resistor and memory functionalities\nhave been demonstrated by manipulation of the AFM\nmoments via a FM sensitive to external magnetic\n\felds9{12. Wadley et al.13showed that in AFMs with\nspeci\fc crystal and magnetic structures AFM moments\ncan be manipulated electrically. Several studies have also\nfocused on transmission and detection of spin-currents\nin AFMs. In FM/AFM/normal-metal (NM) trilayers, a\nspin-current was pumped from the FM, detected by the\ninverse spin-Hall e\u000bect (ISHE) in the NM, and the ob-\nserved robust spin-transport through the interfacial AFM\n(insulating NiO) was ascribed to AFM moment \ructua-\ntions14,15. E\u000ecient spin transmission through an AFM\n(IrMn) was also inferred from an inverse experiment in\nthe FM/AFM/NM structure16in which spin-current was\ngenerated by the spin-Hall e\u000bect (SHE) in the NM and\nabsorbed via the spin transfer torque (STT)17in the FM.\nMeasurements in FM/AFM bilayers have demonstrated\nthat a metallic AFM itself (e.g. IrMn) can act as an e\u000e-cient ISHE detector of the spin-current injected from the\nFM, with comparable spin-Hall angles to heavy NMs18,19.\nOur work makes the next step beyond previous stud-\nies of transmission and detection of spin-currents in\nAFMs by focusing on spin manipulation by AFMs. In a\nNiFe/Cu/IrMn structure we demonstrate that the IrMn\nAFM produces a large SHE spin-current which is trans-\nmitted through Cu and exerts an antidamping-like STT\non the NiFe FM comparable in strength to the SHE-\nSTT generated by Pt. Upon removing the interfacial\nCu layer, we observe that the size of the antidamping-\nlike torque is strongly enhanced and that it correlates\nwith the exchange-bias \feld associated with the \fxed\nAFM moments at the coupled NiFe/IrMn interface. Our\nobservations point to new physics and functionalities\nthat AFMs can bring to the currently highly active re-\nsearch area of relativistic spin-orbit torques induced by\nin-plane currents in inversion asymmetric magnetic struc-\ntures11,20{22,24{28.\nII. MEASUREMENTS\nMultilayers SiO x=Ru(3)=IrMn(d A)=NiFe(4)=Al(2) and\nSiOx=Ru(3)=IrMn(4)=Cu(d N)=NiFe(4)=Al(2) used in our\nmeasurements were grown using dc magnetron sputter-\ning. The numbers represent layer thicknesses in nm,\nIrMn thickness dAin the \frst type of multilayers varies\nfrom 0 - 12 nm, and Cu thickness dNin the second type\nof multilayers is 1 or 2 nm. We apply microwave (MW)\nfrequency electrical current to a bar patterned from the\nmagnetic multilayer. Bars used in our measurements\nvary from 500 nm to 4 \u0016m in width and 5 \u0016m to 240\u0016m\nin length. Torques induced by the oscillating current\nin the bar drive magnetization precession of the NiFe\naround the equilibrium axis de\fned by an applied sat-\nurating magnetic \feld. A diagram of the measurement\nsetup and the device is shown in Fig. 1(a). The bar is\naligned along the x-axis, while the z-axis represents the\nout-of-plane direction. Resonant precession is detectedarXiv:1502.04570v2  [cond-mat.mes-hall]  22 Sep 20152\nhz\nhxhyHeffM\nRuIrMnNiFe\n0.10.20.30.40.5−1−0.500.511.5\nµ0Hext(T)Vdc(µV)\n0.1 0.3 0.5\nμ0H (T)-101VDC (μV)(a) (b)\nFIG. 1. Spin-orbit FMR experiment. (a) Schematic\nrepresentation of the measurement technique. MW current-\ninduced e\u000bective \feld h(hx;hy;hz) drives magnetization pre-\ncession around the total \feld He\u000b. Precessing magnetization\nresults in oscillating resistance due to AMR. This mixes with\noscillating current of the same frequency resulting in a mea-\nsurable DC voltage. (b) Resonance curve decomposed into\nsymmetric and antisymmetric components measured in a bar\nwith 2 nm IrMn at frequency of 17.9 GHz.\nas a recti\fed dc voltage due to anisotropic magnetore-\nsistance (AMR)29. In our studies we keep the frequency\nof the current constant and sweep the in-plane magnetic\n\feld (Fig. 1(b)).\nFrom the decomposition of the resonance into sym-\nmetric and antisymmetric Lorentzians11we deduce the\nout-of-plane and in-plane components of the driving \feld\nas\nVsym=I\u0001R\n2Asymhzsin2\u0012 (1)\nVasy=I\u0001R\n2Aasy(hycos\u0012\u0000hxsin\u0012) sin 2\u0012: (2)\nHereIis the current in the bar, \u0001 Ris the AMR ampli-\ntude,AsymandAasyare coe\u000ecients determined by the\nmagnetic anisotropies, and \u0012is the angle between the\nmagnetization and current directions. Current-induced\n\feldshx;hyandhzcan be obtained from the measured\nangle-dependences of VsymandVasy. We calibrate the\nmicrowave current Iin the bar from the resistance change\ninduced by microwave heating (Supplementary Section\nS1). \u0001Ris obtained from the in-plane AMR measure-\nment using a 1 T magnetic \feld, while the anisotropy\ncoe\u000ecients AsymandAasyare extracted from the angle\ndependence of the resonance \feld (Supplementary Sec-\ntion S4).\nIn Fig. 2(a) we compare resonance curves for samples\nwithout the Cu layer and with 0 and 2 nm thick IrMn.\nThe resonance is predominantly antisymmetric without\nIrMn, indicating a driving \feld in the in-plane direc-\ntion. The resonance then acquires a substantial symmet-\nric component in the presence of the AFM, indicating\nan additional driving \feld in the out-of-plane direction.\nBoth symmetric and antisymmetric components follow\na sin 2\u0012cos\u0012angle dependence (Fig. 2(b)). This meansthat the in-plane e\u000bective \feld is along the y direction\nand is independent on the magnetization direction, re-\nsulting in an out-of-plane \feld-like torque, \u001cz/m\u0002^ y.\nIn contrast, hzdepends on magnetization direction as\ncos\u0012/[j\u0002^ z]\u0002m, thus resulting in an antidamping-like\nin-plane torque \u001cad/m\u0002([j\u0002^ z]\u0002m).\nWe \fnd that for all our samples the magnitude of hy\nis compatible with the magnitude of the Oersted \feld in-\nduced by the current in IrMn and Ru layers. The Oersted\n\feld is calculated using the individual layer resistivities\nextracted from resistance measurements of bars with dif-\nferent IrMn and Ru thicknesses, as described in Supple-\nmentary Sections S2 and S3 ( \u001aIrMn = 20:5\u00063:3\u000210\u00007\n\nm,\u001aRu= 4:0\u00060:3\u000210\u00007\nm, and\u001aNiFe = 5:4\u0006\n0:4\u000210\u00007\nm). From the \fts of the symmetric and\nantisymmetric components to Eqs. (1) and (2) shown\nin Fig. 2(b) we deduce \u00160hz= 1:13\u00060:05 mT and\n\u00160hy= 1:04\u00060:03 mT, while for the Oersted \feld we\n\fnd\u00160hOe= 1:09\u00060:07 mT. All values reported for the\ncurrent-induced \felds are normalised to a current density\nof 107A=cm2in IrMn.\nThe symmetry of hzis compatible both with the\nantidamping-like term of the interface-induced Rashba\n0 90 180 270 360−1.2−0.40.41.2\nθ(°)dc(µ )\nSymmetric\nAntisymmetric\nSymmetric\nAntisymmetric\n0 90 180 270 360-1.2-0.40.41.2Vdc(μV) \n0.1 0.2 0.3 0.4 0.5−0.400.40.8\nµ0Hext(T)dc(µ )\n0 nm IrMn\n2 nm IrMn\nθ (○)0 nm IrMn\n2 nm IrMn\n0.1 0.2 0.3 0.4 0.5Vdc(μV) \n-0.40.00.40.8\nμ0Hext (T)(a)\n(b)\nFIG. 2. AFM-induced torque and its symmetries. (a)\nComparison of resonance curves measured in samples with\nand without the IrMn layer. Both measurements are per-\nformed at 17.9 GHz, \u0012= 45\u000e. Antisymmetric components\nare normalized to 1 \u0016V. (b) Symmetries of VsymandVasyfor\nthe sample with 2 nm IrMn. Solid lines are \fts to equations\n1 and 2.3\nIrMn thickness (nm)Effective field (mT)\n0 2 4 6 810 120.51.52.53.5µ0hz\nµ0hy\nµ0hOe\n0901802703600.2200.2600.300\nθ(°)µ0Hres(T)\n4 nm IrMn\nIrMn thickness (nm)µ0H (mT)\n0 2 4 6 810 120306090Exchange Bias\nRotational Anisotropy\nIrMn thickness (nm)µ0H (mT)\n0 2 4 6 810 120306090Exchange bias (AMR)\nExchange bias (MOKE)\nCoercivity (AMR)\nCoercivity (MOKE)\n,\n0 12 246 8 10\nIrMn Thickness (nm)exchange bias (AMR)\nexchange bias (MOKE)\ncoercivity (AMR)\ncoercivity (MOKE)exchange bias (FMR)\nrotational anisotropy (FMR)\nIrMn Thickness (nm)0 12 2 4 6 8 10Cu spacerCu spacer\n0.51.52.53.5Current-induced Field (mT)0306090\n0306090\n0901802703600.2240.2320.240\nθ(°)µ0Hres(T)\n2 nm IrMn\n0.2240.2320.240 2 nm IrMn\n0.2200.2600.300 4 nm IrMn\n0901802703600.2970.3100.323\nθ(°)µ0Hres(T)\nCu spacer\nCu spacer0.2970.3100.323\n0 90 180 270 360\nθ (○)(a) (b) (c)\nμ0H0 (T)\nμ0H (mT) μ0H (mT)μ0hz\nμ0hy\nμ0hOe\n(d)\nFIG. 3. AFM thickness dependence of current-induced \felds and anisotropies. (a)hz,hyand calculated Oersted\n\feldhOefor 1.8\u0016m wide bars with di\u000berent IrMn thicknesses, as well as the sample with the 2 nm Cu spacer layer. The results\nare normalized to a current density of 107A=cm2in IrMn. The shaded area around hOeis the error due to uncertainties in\nlayer resistivities, whereas the error bars of hzandhyare due to the standard errors from the \ftting of the symmetries, AMR\nand MW current. The systematic uncertainties in layer resistivities have not been included in the error bars of hzandhy,\nhowever this uncertainty, which is approximately 20 %, is included in the values of e\u000bective spin-Hall angles in the main text.\nThe dotted line is the estimated spin-Hall e\u000bect contribution to hzfor\u0015sd= 1 nm. (b) Angle dependences of resonance \feld\nfor the samples with 2 and 4 nm IrMn thicknesses, as well as the sample with the 2 nm Cu spacer layer. Solid lines are \fts\ntaking into account unidirectional, uniaxial and rotational anisotropies. (c) IrMn thickness dependence of the exchange bias\nand the rotational anisotropy extracted from the \fts in (a). (d) IrMn thickness dependence of exchange bias and coercivity\nextracted from hysteresis loops measured using MOKE and AMR switching.\nspin-orbit torque24, as well as with the SHE-STT25,26.\nIn the latter case the spin-current generated in the IrMn\nby the SHE drives magnetization precession in the NiFe\nlayer by STT. Both of these e\u000bects occur in FM/NM\nstructures, however, we show that additional e\u000bects arise\ndue to the AFM nature of IrMn and the exchange cou-\npling at the FM/AFM interface.\nTo separate the contribution of the exchange-coupled\nNiFe/IrMn from the SHE-STT, we performed measure-\nments in samples with 4 nm thick IrMn, and 1 and 2 nm\nthick Cu spacers between IrMn and NiFe. Cu has a\nspin-di\u000busion length of 350 nm30and thus 2 nm of Cu\nwould transfer >99% of the spin-Hall current from IrMn,\nbut eliminate the FM/AFM coupling and the FM/AFM\ninterface-induced e\u000bects.\nResults obtained in samples with the Cu spacer and\nwithout Cu and di\u000berent IrMn thicknesses are summa-\nrized in Fig. 3(a). Firstly, one can see that the hz\feld\ndoes not vanish with the introduction of Cu, indicating\nthe SHE in IrMn. From the value of hzwe can obtain\nthe spin-Hall angle \u0012SHof IrMn from the expression\n\u0012SH=2e\u00160MsdF\n\u0016hJIrMnhz: (3)\nHeredF= 4 nm is the thickness of the NiFe layer,\n\u00160Ms= 1 T is the saturation magnetization of NiFe,\nJIrMn = 107A=cm2is the charge current density in IrMn\nand\u00160hz= 0:58\u00060:02 mT is obtained from the measure-ment. We get \u0012IrMn = 0:056\u00060:009, in good agreement\nwith the expected value for Ir 25Mn7518. Here the uncer-\ntainty also includes the uncertainty of the current density\nin IrMn from the layer resistivity calibration. It is im-\nportant to mention that the same value of \u0012IrMn was\nobtained for both 1 nm and 2 nm Cu spacers, as well\nas bars with 1.8 \u0016m and 500 nm widths. Remarkably,\nin addition to the SHE, we see a large contribution from\nthe FM/AFM interface in samples without Cu, initially\nincreasing with the IrMn thickness and with a peak at\n8 nm of IrMn, with a magnitude corresponding to an\ne\u000bective spin-Hall angle of 0 :22\u00060:04. The values of ef-\nfective spin-Hall angles for two samples, as well as the\ndamping-like nature of hzwere con\frmed by measuring\nthe dc bias dependence of the FMR linewidth12. De-\npending on the direction of DC current with respect to\nFM magnetizaion, an additional damping or antidamp-\ning is induced, thereby increasing or decreasing the FMR\nlinewidth. For the sample with the Cu spacer we obtain\n\u0012SH= 0:043\u00060:001 (Fig. 4(a)) and for the sample with\n2 nm IrMn we get \u0012SH= 0:135\u00060:022 (Fig. 4(b)). We\nuse\n\u0012SH=@(\u00160\u0001H)\n@(jIrMn )\u0002\r\n!2e\n\u0016h(Hres+Meff=2)\u00160MstNiFe\nsin\u0012\n(4)\nwhere the \frst term is the slope of the linear \ft with\nrespect to the current density in IrMn. For comparison,4\nBias Current (mA)μ0ΔH (mT)\n−0.4 −0.2 0 0.2 0.4−0.4−0.200.20.4\nBias Current (mA)µ0∆H (mT)\n45°\n225°\n0.4\n0.2\n0\n-0.2\n-0.4\n0.4 0.2 0 -0.2 -0.445°\n225°\n−2 −1 0 1 2−0.2−0.100.10.2\nBias Current (mA)µ0∆H (mT)\n45°\n225°\n0.2\n0.1\n0\n-0.1\n-0.2\n-2 -1 0 1 2μ0ΔH (mT)\nBias Current (mA)45°\n225°(a)\n(b)\nFIG. 4. DC bias dependence of the FMR linewidth.\n(a) Change of FMR linewidth with DC current for the\nIrMn(4)/Cu(2)/NiFe(4) structure measured at !=2\u0019= 8 GHz\nand (b) IrMn(2)/NiFe(4) structure measured at !=2\u0019= 14:1\nGHz, for two di\u000berent directions of magnetization with re-\nspect to the current. The data points are extracted using\nthe linewidth di\u000berence between positive and negative bias\ncurrents.\nthe values obtained using the magnitude of hzextracted\nfrom our FMR measurements (Fig. 3(a)) are 0 :056\u00060:001\nfor the sample with the Cu spacer and 0 :109\u00060:005 for\nthe 2 nm IrMn sample. The values are in a good agree-\nment if we also include the resistivity calibration error\nof approximately 20 % in addition to the uncertainties\nfrom the \ftting. We note here that in a recent study,\nMoriyama et al.16used similar FM/AFM/NM structures\nbut instead of Ru they had Pt NM. Unlike our results, the\nintroduction of the interfacial IrMn AFM in Moriyama et\nal.structures always reduced the spin torque, compared\nto the reference FM/NM sample without the AFM. The\nauthors concluded that in their case, the SHE in the\nAFM did not play a signi\fcant role and that the ob-\nserved torque was due to the spin-Hall current from Pt\ntransferred to the FM via spin-waves in the AFM. In\nour case, Ru has a small spin-Hall angle32, which we\n\fnd from the control sample without IrMn to be \u00190:009\n(Supplementary Section S5). This, given the current dis-\ntribution in the multilayer, would have a contribution of\nhz\u00190:48 mT in all the samples. Even if we assumed that\nthe spin-angular momentum carried by the spin-Hall cur-\nrent from the Ru layer is fully transferred through IrMn,\nit would still be too small to explain the e\u000bect in sam-ples with IrMn thicknesses larger than 3 nm, as seen in\nFig. 3(a).\nAdditionally, we performed measurements in samples\nwith Ta seed layers instead of Ru, and found a large\npositivehzsimilar to the Ru samples ( hz=hy\u00190:9).\nTa has a large negative spin-Hall angle and one would\nexpect a negative or a largely suppressed hzif the\nseed layer had a signi\fcant contribution (see Supplemen-\ntary Section S6 for the details). The increase of the\nantidamping-like torque in our NiFe/IrMn samples with\nincreasing IrMn thickness cannot be explained by the in-\ncrease in the spin-Hall current, as shown by the dotted\nline in Fig. 3(a), because IrMn has a spin di\u000busion length\nsmaller than 1 nm19,33. It is clearly associated with the\nexchange-coupled NiFe/IrMn interface. The two leading\nanisotropies commonly used to characterise FM/AFM in-\nterfaces are the exchange bias \feld and the rotational\nanisotropy, the latter being the origin of the increased\ncoercivity10,35. Rotational anisotropy can be modelled\nas an additional e\u000bective \feld along the magnetization\ndirection, and thus results in an overall decrease of the\nresonance \feld in FMR measurements. This decrease is\nseen in Fig. 2(a). The anisotropies are quanti\fed from\nthe angle dependence of the resonance \feld, plotted in\nFig. 3(b) for the 2 and 4 nm IrMn samples and the sam-\n0 50 110 170 230 2900.60.81.01.21.41.6\nTemperatureE C Kohz/ E hy\n0 50 110 170 230 290020406080\nTemperatureE C Koµ0HE C mTo\nExchangeE Bias\nCoercivity\n0100 200 3001.11.2\nTe T e m mperatureE C Ko0100 200 300234\nAMRE CΩo\n0100 200 3001.11. 1 1 2\n0100 200 30022324\nAMRE CΩo0 100 200 3001.11.2\n3\n24AMR (Ω)\nTemperature (K)R (kΩ)\n0 50 110 170 230 290\nTemperature (K)0.60.81.01.21.41.6\n0204080\n60μ0H (mT)exchange bias\ncoercivityhz / hy(a)\n(b)\nFIG. 5. Temperature dependence of current-induced\n\felds and anisotropies. (a) Temperature dependence of\nthehz=hyratio for the sample with 2 nm IrMn. The in-\nset shows the temperature dependence of the AMR and total\nresistance of the bar. (b) Temperature dependence of the ex-\nchange bias and the coercivity for the same sample extracted\nfrom AMR switching measurements.5\n0 1000 20000.020.040.06\nHe2(mT2)α\n0.01 0.02 0.03 0.04 0.05 0.060123\nEffective Gilbert Dampinghz(mT)\n10 15 20 25204060\nω(GHz)µ0∆H(mT)\nμ0hz (mT)\n12\n0(b)\nEffective Gilbert Damping (α)0.01 0.02 0.03 0.04 0.05 0.06\n2040\nµ0∆H(mT)\n6 nm IrMn\n10 15 20 25\nω/2π (GHz)μ0ΔH (mT)\n204060(a)\n2 nm3 nm\n4 nm12 nm\n6 nm8 nm\nCu spacer\nμ02He2 (mT2) x 10-3α\n0.020.040.06\n0 1 23 (c)\nFIG. 6. AFM-induced torque and Gilbert damping. (a) Frequency dependence of the FMR linewidth for the sample\nwith 6 nm IrMn. The slope of the linear \ft allows us to extract the e\u000bective Gilbert damping \u000b=\r\u00160(@\u0001H=@! ), where\n\r=2\u0019= 28 GHz/T. (b) Current-induced out-of-plane \feld hzplotted against the e\u000bective Gilbert damping for samples with\ndi\u000berent IrMn thicknesses. (c) Gilbert damping is proportional to the square of the exchange bias, suggesting that one of the\nmain damping mechanisms in our samples is the two-magnon scattering due to the inhomogeneity of the \feld at the FM/AFM\ninterface induced by the exchange anisotropy.\nple with the 2 nm Cu spacer, all measured at 17.9 GHz.\nComparing the top graph (2 nm IrMn with no spacer)\nand the bottom graph (2 nm Cu spacer), we see a smaller\nresonance \feld in the sample with 2 nm IrMn due to the\nrotational anisotropy induced at the FM/AFM interface,\nas discussed earlier. For the thicker IrMn sample (middle\ngraph) a unidirectional contribution due to the exchange\nbias starts to develop.\nThickness dependences of the exchange bias \feld Hex\nand the rotational anisotropy \feld Hrotextracted from\nthe \fts are plotted in Fig. 3(c) and compared to Hexand\nHcextracted from MOKE and AMR switching measure-\nments plotted in Fig. 3(d), showing a good agreement.\nOne can see the onset of exchange bias at 3 nm and a\npeak at 8 nm of IrMn. The rotational anisotropy and\ncoercivity are the largest for the sample with 3 nm IrMn.\nSimilar thickness dependence has been observed experi-\nmentally using di\u000berent techniques36,37. One can see a\ncorrelation between the size of the exchange bias and hz\nby comparing Fig. 3(a) and (c,d). It is worth mentioning\nhere that although the exchange bias has di\u000berent direc-\ntions for 4 - 12 nm IrMn samples the symmetry of hzis\nnot a\u000bected by it (Supplementary Section S7).\nTo con\frm the correlation between the antidamping-\nlike torque and exchange bias in one sample, we per-\nformed temperature dependence measurements of the\nhz=hyratio for the sample with 2 nm IrMn. Although\nthis ratio is not a direct measure of the e\u000bective spin-Hall\nangle due to the possible current redistribution with tem-\nperature, it can help with the qualitative understanding.\nThe results are shown in Fig. 5(a). The monotonous\ndecrease in the hz=hyratio down to 50 K can be ex-\nplained with the current redistribution in the bar. IrMn\nis an alloy, and thus its resistivity decreases less with tem-\nperature compared to Ru, resulting in a smaller propor-\ntion of current \rowing through IrMn, and thus smaller\nhzat lower temperatures. The ratio can also change\nmonotonously with temperature if there are additionaltemperature dependent contributions to hy38. Neverthe-\nless, as one can see the monotonic trend is broken below\n50 K, coinciding with the abrupt increase in the exchange\nbias and decrease in the coercivity (Fig. 5(b). In the inset\nof Fig. 5(a) we plot the change of resistance and AMR\nwith temperature, showing their monotonous behaviour\nfor the whole temperature range. This result is signi\fcant\nbecause it shows dependence of current-induced torques\non AFM-induced anisotropies in a single device. We also\nfound that cooling down the sample from room tempera-\nture to 25 K with applied 1 T magnetic \feld along di\u000ber-\nent directions changes the direction of the exchange bias,\nhowever, this does not signi\fcantly change magnitudes\nand symmetries of the current-induced \felds.\nIII. DISCUSSION\nThe origin of relativistic spin torques induced by an in-\nplane current at FM/NM interfaces is a subject of current\nintense theory discussions. Our results clearly indicate\nthat replacing the NM with an AFM adds to the rich-\nness of these phenomena which inevitably brings more\ncomplexity to their theoretical description. To stimulate\nfuture detailed microscopic analyses we outline here pos-\nsible mechanisms that might be considered as the origin\nof the enhancement of the antidamping-like torque and\nits correlation with the exchange bias. Firstly, the ex-\nchange coupling could increase the transparency at the\nFM/AFM interface resulting in a more e\u000ecient spin-\ntransfer. One can estimate the e\u000eciency of spin-transfer\nthrough FM/NM interface from the frequency depen-\ndence of the FMR linewidth39. This is characterised by\nthe e\u000bective Gilbert damping \u000b, extracted from the slope\nin Fig. 6(a). In Fig. 6(b) we plot hzas a function of \u000bfor\nthe samples with di\u000berent IrMn thicknesses. One can see\na clear linear trend, suggesting that hzis correlated with\nthe spin-angular momentum transfer properties through6\nthe interface. Additionally, in Fig. 6(c) we show that\nthe enhancement of the spin-angular momentum trans-\nfer through the interface is indeed due to the interfacial\nexchange coupling, as \u000bis proportional to the square of\nthe exchange bias. This dependence also suggest that\none of the main damping mechanisms in our samples is\nthe two-magnon scattering at the FM/AFM interface, in\nagreement with the previous studies19,40,41. The exact\nmechanism of the enhancement of hzis of complex ori-\ngin due to the strong spin-orbit coupling in the system\nand the interface magnetic coupling. If we assume that\nthe damping enhancement is merely due to more e\u000ecient\nspin-pumping and try to estimate the value of the trans-\nparency at the interface in the weak spin-orbit coupling\npicture of spin-mixing conductance using39\nGmix=Geff\n1\u00002Geff\u0015SD=\u001bIrMn(5)\nwhere\nGeff=e2\nh4\u0019MstNiFe\ng\u0016B(\u000b\u0000\u000b0) (6)\nusing values \u0015SD= 0:7 nm19and conductivity \u001bIrMn =\n1=\u001aIrMn , we obtain negative values for Gmix, which is\nnon-physical. Here \u000b0= 0:006 is the Gilbert damping\nof bulk NiFe. One would have to assume \u0015SD<0:1\nnm to obtain positive Gmix. This additionally suggests\nthat the mechanism of the damping enhancement, and\nsubsequently the torque enhancement is more complex\nthan just an increase of spin-current transparency at the\ninterface combined with the spin-Hall e\u000bect.\nAnother possibility is that additional torques are in-\nduced directly at the FM/AFM interface, or induced in\nthe AFM and coupled to the FM via the exchange inter-\naction. In this case the level of the magnetic order in the\nAFM layer could be important for the size of the torque.\nWei et al.42and Urazhdin et al.43observed changes in\nexchange bias in current perpendicular-to-plane geome-\ntries, attributed to torques changing the AFM magnetic\nstructure at the FM/AFM interface. We note that our\nmeasurement is not sensitive to the bulk AFM magnetic\norder, except through its correlation with the exchange\nbias at the interface. We also point out that we use 2 -\n3 orders of magnitude lower in-plane currents compared\nto references 42 and 43, avoiding heating e\u000bects and em-\nploying a di\u000berent current path geometry which excludes\nthe possibility of a direct comparison between the exper-\niments.\nIV. CONCLUSIONS\nIn conclusion, we have shown that electrical current\nin the IrMn AFM induces a large torque acting on the\nadjacent NiFe FM. The torque is in-plane and has anantidamping-like symmetry. We have also shown that\nthere are at least two distinct contributions, one coming\nfrom the SHE in IrMn, and the other due to the AFM\norder of IrMn. The spin-Hall angle of IrMn measured in\nthe sample with the Cu spacer between NiFe and IrMn is\nfound to be 0 :056\u00060:009, comparable to that of Pt. An\ne\u000bective spin-Hall angle of 0 :22\u00060:04, almost three times\nlarger than that of Pt, is measured for the sample with\n8 nm IrMn in direct contact with NiFe, exhibiting the\nlargest exchange bias. Our results suggest that electrical\ncurrent in AFMs can induce torques more e\u000eciently than\nin most of the heavy NMs. The AFM-induced torques\nand their correlation with the exchange coupling at the\nFM/AFM interface could lead to novel designs of spin-\ntronic devices.\nV. METHODS AND MATERIALS\nMaterials: The structures were grown using DC mag-\nnetron sputtering on a thermally oxidized Si (100) sub-\nstrate. In-plane magnetic \feld of 200 Oe was applied\nduring growth.\nDevices: The microbars are patterned using electron-\nbeam lithography. In Figs. 2 and 5 we show measure-\nments done in bars with 500 nm width and 5 \u0016m length,\nwhereas the measurements shown in Fig. 3 are done\nin bars with 1.8\u000238\u0016m dimensions. Measurements in\nFigs. 2 and 3 are repeated in at least two bars with dif-\nferent dimensions. The results are consistent across dif-\nferent bars and all the bar dimensions. The resistivity\ncalibration measurements are done in 4 \u0016m wide bars\nwith 40, 80, 120 and 240 \u0016m lengths. Typical resistances\nare on the order of 1 k\n for bars with length to width\nratios of 10.\nExperimental procedure:\nFor more details on the methods related to our SO-\nFMR experiments see Refs. 11 and 28 and the Supple-\nmentary Information therein.\nVI. ADDITIONAL INFORMATION\nCorrespondence and requests for materials should be\naddressed to AJF (ajf1006@cam.ac.uk).\nACKNOWLEDGMENTS\nAuthors would like to acknowledge Dr. Hidekazu\nKurebayashi and Dr. Tim Skinner for useful discus-\nsions. We acknowledge support from the EU European\nResearch Council (ERC) Advanced Grant No. 268066,\nthe Ministry of Education of the Czech Republic Grant\nNo.LM2011026, and the Grant Agency of the Czech Re-\npublic Grant No. 14-37427G. VT would like to acknowl-\nedge Winton Programme for the Physics of Sustainability\nand Cambridge Overseas Trusts for \fnancial support.7\n\u0003Currently at London Centre for Nanotechnology, Depart-\nment of Materials, Imperial College London, SW7 2BP,\nUK\nyajf1006@cam.ac.uk\n1A. N\u0013 u~ nez, R. Duine, P. Haney, and A. MacDonald, Phys-\nical Review B 73, 1 (2006).\n2P. Haney and A. MacDonald, Physical Review Letters 100,\n1 (2008).\n3Y. Xu, S. Wang, and K. Xia, Physical Review Letters 100,\n1 (2008).\n4H. V. Gomonay, Physical Review B 81, 1 (2010).\n5K. Hals, Y. Tserkovnyak, and A. Brataas, Physical Review\nLetters 106, 1 (2011).\n6A. B. Shick, S. Khmelevskyi, O. N. Mryasov, J. Wunder-\nlich, and T. Jungwirth, Physical Review B 81, 212409\n(2010).\n7J.\u0014Zelezn\u0013 y, H. Gao, K. V\u0013 yborn\u0013 y, J. Zemen, J. Ma\u0014 sek,\nA. Manchon, J. Wunderlich, J. Sinova, and T. Jungwirth,\nPhysical Review Letters 113, 157201 (2014).\n8J. Nogu\u0013 es and I. K. Schuller, Journal of Magnetism and\nMagnetic Materials 192, 203 (1999).\n9B. G. Park, J. Wunderlich, X. Mart\u0013 \u0010, V. Hol\u0013 y, Y. Kurosaki,\nM. Yamada, H. Yamamoto, A. Nishide, J. Hayakawa,\nH. Takahashi, A. B. Shick, and T. Jungwirth, Nature ma-\nterials 10, 347 (2011).\n10Y. Y. Wang, C. Song, B. Cui, G. Y. Wang, F. Zeng, and\nF. Pan, Physical Review Letters 109, 137201 (2012).\n11X. Marti, I. Fina, C. Frontera, J. Liu, P. Wadley, Q. He,\nR. J. Paull, J. D. Clarkson, J. Kudrnovsk\u0013 y, I. Turek,\nJ. Kune\u0014 s, D. Yi, J.-H. Chu, C. T. Nelson, L. You, E. Aren-\nholz, S. Salahuddin, J. Fontcuberta, T. Jungwirth, and\nR. Ramesh, Nature materials 13, 367 (2014).\n12I. Fina, X. Marti, D. Yi, J. Liu, J. H. Chu, C. Rayan-\nSerrao, S. Suresha, a. B. Shick, J. Zelezn\u0013 y, T. Jungwirth,\nJ. Fontcuberta, and R. Ramesh, Nature communications\n5, 4671 (2014).\n13P. Wadley, B. Howells, J. Zelezny, C. Andrews, V. Hills,\nR. P. Campion, V. Novak, F. Freimuth, Y. Mokrousov,\nA. W. Rushforth, K. W. Edmonds, B. L. Gallagher, and\nT. Jungwirth, arXiv:1503.03765.\n14H. Wang, C. Du, P. C. Hammel, and F. Yang, Physical\nReview Letters 113, 097202 (2014).\n15C. Hahn, G. de Loubens, V. V. Naletov, J. Ben Youssef,\nO. Klein, and M. Viret, EPL (Europhysics Letters) 108,\n57005 (2014).\n16T. Moriyama, M. Nagata, K. Tanaka, K.-j. Kim, H. Al-\nmasi, and W. G. Wang, arXiv:1411.4100.\n17D. Ralph and M. Stiles, Journal of Magnetism and Mag-\nnetic Materials 320, 1190 (2008).\n18J. B. S. Mendes, R. O. Cunha, O. Alves Santos, P. R. T.\nRibeiro, F. L. A. Machado, R. L. Rodr\u0013 \u0010guez-Su\u0013 arez,\nA. Azevedo, and S. M. Rezende, Physical Review B 89,\n140406 (2014).\n19W. Zhang, M. B. Jung\reisch, W. Jiang, J. E. Pearson,\nA. Ho\u000bmann, F. Freimuth, and Y. Mokrousov, Physical\nReview Letters 113, 196602 (2014).\n20B. Bernevig and O. Vafek, Physical Review B 72, 033203\n(2005).\n21A. Manchon and S. Zhang, Physical Review B 78, 212405\n(2008).22A. Chernyshov, M. Overby, X. Liu, J. K. Furdyna,\nY. Lyanda-Geller, and L. P. Rokhinson, Nature Physics\n5, 656 (2009).\n11D. Fang, H. Kurebayashi, J. Wunderlich, K. V\u0013 yborn\u0013 y,\nL. P. Z^ arbo, R. P. Campion, A. Casiraghi, B. L. Gallagher,\nT. Jungwirth, and A. J. Ferguson, Nature Nanotechnology\n6, 413 (2011).\n24I. M. Miron, G. Gaudin, S. Au\u000bret, B. Rodmacq,\nA. Schuhl, S. Pizzini, J. Vogel, and P. Gambardella, Na-\nture Materials 9, 230 (2010).\n25I. M. Miron, K. Garello, G. Gaudin, P.-J. Zermatten, M. V.\nCostache, S. Au\u000bret, S. Bandiera, B. Rodmacq, A. Schuhl,\nand P. Gambardella, Nature 476, 189 (2011).\n26L. Liu, C.-F. Pai, Y. Li, H. W. Tseng, D. C. Ralph, and\nR. A. Buhrman, Science (New York, N.Y.) 336, 555 (2012).\n27K. Garello, I. M. Miron, C. O. Avci, F. Freimuth,\nY. Mokrousov, S. Bl ugel, S. Au\u000bret, O. Boulle, G. Gaudin,\nand P. Gambardella, Nature Nanotechnology 8, 587\n(2013).\n28H. Kurebayashi, J. Sinova, D. Fang, A. C. Irvine, T. D.\nSkinner, J. Wunderlich, V. Nov\u0013 ak, R. P. Campion, B. L.\nGallagher, E. K. Vehstedt, L. P. Z^ arbo, K. V\u0013 yborn\u0013 y, A. J.\nFerguson, and T. Jungwirth, Nature Nanotechnology 9,\n211 (2014).\n29M. V. Costache, S. M. Watts, M. Sladkov, C. H. van der\nWal, and B. J. van Wees, Applied Physics Letters 89,\n232115 (2006).\n30S. Yakata, Y. Ando, T. Miyazaki, and S. Mizukami,\nJapanese Journal of Applied Physics 45, 3892 (2006).\n12L. Liu, T. Moriyama, D. Ralph, and R. Buhrman, Physical\nReview Letters 106, 1 (2011).\n32T. Tanaka, H. Kontani, M. Naito, T. Naito, D. Hirashima,\nK. Yamada, and J. Inoue, Physical Review B 77, 165117\n(2008).\n33R. Acharyya, H. Y. T. Nguyen, W. P. Pratt, and J. Bass,\nJournal of Applied Physics 109, 07C503 (2011).\n10M. Stiles and R. McMichael, Physical Review B 59, 3722\n(1999).\n35M. Stiles and R. McMichael, Physical Review B 63, 064405\n(2001).\n36M. Ali, C. Marrows, M. Al-Jawad, B. Hickey, a. Misra,\nU. Nowak, and K. Usadel, Physical Review B 68, 214420\n(2003).\n37J. McCord, R. Mattheis, and D. Elefant, Physical Review\nB70, 094420 (2004).\n38J. Kim, J. Sinha, S. Mitani, M. Hayashi, S. Takahashi,\nS. Maekawa, M. Yamanouchi, and H. Ohno, Physical Re-\nview B - Condensed Matter and Materials Physics 89, 1\n(2014), arXiv:1402.6388.\n39Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Physical\nReview B 66, 224403 (2002).\n40S. Rezende, A. Azevedo, M. Lucena, and F. de Aguiar,\nPhysical Review B 63, 214418 (2001).\n41S. Yuan, B. Kang, L. Yu, S. Cao, and X. Zhao, Journal of\nApplied Physics 105, 063902 (2009).\n42Z. Wei, A. Sharma, A. Nunez, P. Haney, R. Duine, J. Bass,\nA. MacDonald, and M. Tsoi, Physical Review Letters 98,\n1 (2007).\n43S. Urazhdin and N. Anthony, Physical Review Letters 99,\n046602 (2007).1\nSupplemental Information: Electrical manipulation of a ferromagnet by an\nantiferromagnet\nV. Tshitoyan,1C. Ciccarelli,1A. P. Mihai,2M. Ali,2A. C. Irvine,1T. A. Moore,2T. Jungwirth,3;4and A. J.\nFerguson1\n1Microelectronics Group, Cavendish Laboratory, University of Cambridge, CB3 0HE, UK\n2School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, UK\n3Institute of Physics ASCR, v.v.i., Cukrovarnick\u0013 a 10, 162 53 Praha 6, Czech Republic\n4School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, UK\nS1. MICROWAVE CURRENT CALIBRATION\nResistances of measured bars vary between a few 100 \n and a few k\n, thus most of the microwave (MW) power\nis re\rected due to the impedance mismatch between the bar and the MW source ( Zout= 50 \n). To calibrate MW\ncurrent we make use of the Joule heating. The amount of heating is measured using the change of resistance. First\nDC current is swept from large negative to large positive values and the di\u000berential resistance is measured, giving\nus the resistance change due to DC heating. Then we measure resistance change with increasing microwave power.\nThese measurements for a 500 nm wide and 5 \u0016m long bar of Ru(3) =IrMn(2)=Py(4)=Al(2) are plotted in Fig. S1(a).\nFor DC the value of current is known because it is all dissipated in the bar, there are no re\rections. We are able to\n\fnd the current for each applied MW power by comparing the MW and DC heatings. In Fig. S1(b) we plot values of\nDC current causing the same amount of heating as MW powers on the x axis. The corresponding MW current isp\n2\ntimes the DC current, because the heating for AC current is given by I2R=2 compared to I2Rfor DC (this is already\ntaken into account in the plot). As expected, MW current is linear with the square root of power (in W). From the\nlinear \ft we can extract the value of MW current per square root of power.\n(a) (b)\nFIG. S1. (a) Comparison of resistance change due to heating caused by DC (left) and MW (right) currents. The DC measure-\nment is symmetric with respect to 0 current. (b) MW current vs square root of applied MW power obtained from the heating\ncalibration.\nS2. LAYER RESISTIVITIES\nTo know the current distribution in our multilayer stack, which is important for calculations of spin hall angles\nas well as estimations of the Oersted \felds, we deduce resistivities of individual metallic layers. One can not take\nbulk resistivities because these values change dramatically for thin layers. In addition, there is always an additional\ncontact resistance which has to be taken into account. These values can be determined by a careful analysis of bars2\nwith di\u000berent dimensions and layer thicknesses. In Fig. S2(a) we plot resistances of 4 \u0016m wide bars of 40, 80 and 120\n\u0016m lengths. The intersection of the linear \ft with y axis is the average contact resistance, Rcont= 235\u000675 \n.\n(a) (b) (c) (d)\nFIG. S2. (a) Resistances of bars with di\u000berent length/width ratios. The \ft to a line yields a contact resistance of 235 \u000675 \n. (b)\nResistances of bars with di\u000berent IrMn thicknesses. Resistances of 3 - 8 nm samples are \ftted to Eqn. S.1. (c) Magnetization\nwas rotated in-plane and resistance of the bar was measured. The \ft to a cos2\u0012allows us to determine the AMR magnitude\nin the multilayer (d) AMR(%) = RNiFe\u0001Rtot\nR2\ntotfor bars with di\u000berent IrMn thickness, giving us the magnitude of AMR in the\nNiFe layer.\nUsing the value of contact resistance we can calculate resistivities of individual layers. In Fig. S2(b) we plot\nresistances of bars with the same dimensional ratio vs the IrMn thickness t in Ru(3)/IrMn(t)/NiFe(4) structures. The\naverage contact resistance has already been subtracted. Thicknesses are given in nm. We neglect the 2 nm Al capping\nlayer as it is the same for all the samples and is believed to be mainly oxidized. It is surprising that the resistance of\nthe sample with 0 nm IrMn is smaller than that of the sample with 2 nm IrMn. We believe this is due to the higher\nresistivity of NiFe grown on IrMn compared to that of NiFe grown on Ru. It is know that NiFe can have di\u000berent\nresistivities depending on the seed layer1{3. The samples with 3 - 8 nm IrMn \ft well to a simple model of parallel\nresistors, given by\nR=Rcont+d\u0001\u001aIrMn\nt+d\u0001\u001aIrMn=r: (S.1)\nHeredis the length/width ratio of the bars (60 for this set of samples), ris the resistance of the multilayer without\nIrMn,tis the thickness of IrMn and \u001aIrMn is its resistivity. The \ft in Fig. S2(b) results in \u001aIrMn = 20:5\u00063:5\u000210\u00007\n\n\u0001m.\nAs already mentioned, resistivity of NiFe is larger if grown on IrMn. This is more prominent for the thinnest\n(2 nm) IrMn sample. We believe this is due to the worse quality of the 2 nm IrMn interface, as this layer is the\nthinnest. To calculate di\u000berent resistivities of NiFe we must know the resistivity of Ru. To estimate the later we\nuse resistances of Ru(3)/IrMn(4)/Ru(2)/NiFe(4) and Ru(3)/NiFe(4), 2624 \n and 3760 \n respectively. Using the\nresistivity of IrMn obtained earlier we get \u001aRu= 4:0\u00060:3\u000210\u00007\n\u0001mand\u001aRu\nNiFe = 4:7\u00060:3\u000210\u00007\n\u0001mfor\nNiFe grown on Ru. Using the resistivity of Ru we can now calculate the resistivity of NiFe grown on IrMn. We\nuse the resistance of Ru(3)/IrMn(2)/NiFe(4) sample and the values obtained from the \ft in Fig. S2(b). We \fnd\n\u001a2nmIrMn\nNiFe = 6:9\u00060:6\u000210\u00007\n\u0001mand\u001aIrMn\nNiFe = 5:4\u00060:4\u000210\u00007\n\u0001m.\nTo verify the parallel resistors approach, we compare values of AMR for layers with di\u000berent IrMn thicknesses.\nChange of the resistance due to AMR is extracted by rotating the direction of the magnetic \feld with respect to the\nsample. A typical measurement result is plotted in Fig. S2(c). The value of measured AMR depends on the proportion\nof the current in the NiFe layer, and the size of its AMR. For example for thicker IrMn samples the proportion of\nthe current in NiFe is smaller and thus smaller total AMR is measured. We deduce the exact relationship using the\nparallel resistors model.\nRtot= (1=RNiFe + 1=Rrest)\u00001=RNiFeRrest\nRNiFe +Rrest(S.2)3\n\u0001Rtot=(RNiFe + \u0001RNiFe)Rrest\nRNiFe + \u0001RNiFe +Rrest\u0000RNiFeRrest\nRNiFe +Rrest=\n=\u0001RNiFeR2\nrest\n(RNiFe + \u0001RNiFe +Rrest)(RNiFe +Rrest)(S.3)\n\u0001Rtot\nRtot=\u0001RNiFeR2\nrest\n(RNiFe + \u0001RNiFe +Rrest)(RNiFe +Rrest)\u0001RNiFe +Rrest\nRNiFeRrest\u0019\u0001RNiFeRtot\nR2\nNiFe(S.4)\n\u0001Rtot\nR2\ntot\u0019\u0001RNiFe\nR2\nNiFe(S.5)\nThus\u0001RNiFe\nRNiFe=RNiFe\u0001Rtot\nR2\ntotgives us the value of AMR in NiFe. In Fig. S2(d) we plot the right hand side of this\nequation for all measured IrMn thicknesses. As one can see it is almost the same for the 3 - 8 nm thickness range\nof IrMn, and is approximately 0.7 %. For the sample with 2 nm IrMn AMR of NiFe is slightly smaller, whereas it\nis slightly larger for NiFe grown on Ru, as expected due to worse and better layer qualities respectively. Decrease of\nintrinsic AMR of NiFe for thin layers, as well as its dependence on the seed layer has been reported previously2,4{6.\nWe believe that the agreement of AMR magnitudes, in combination with the relatively good \ft of IrMn thickness\ndependence of sample resistances, implies that our parallel resistors approach is valid and estimates of layer resis-\ntivities are correct. As yet another additional supporting argument for our calculation, the bulk resistivity ratio is\napproximately 18(IrMn):1(Ru):2(NiFe).(1260 \u000210\u00007\n\u0001m, 71\u000210\u00007\n\u0001m, 140\u000210\u00007\n\u0001m)7{9. The ratios of resis-\ntivities deduced above are 5.1 : 1 : 1.2-1.8. The order is the same, but di\u000berences in resistivities are more moderate\nfor thin \flms because scattering o\u000b interfaces is substantial, and thus resistivity must be less material-dependent.\nThe resistivity of Cu is deduced from the Ru(3)/IrMn(4)/Cu(t Cu)/Py(4) structures , where t Cuis 1, 2 or 4 nm. In\nFig. S3 we plot resistances of bars with di\u000berent Cu thicknesses \ftted to equation S.1, except instead of \u001aIrMn one\nhas\u001aCu. From the \ft we \fnd \u001aCu= 1:55\u000110\u00007\nm.\nFIG. S3. Resistances of bars with di\u000berent Cu spacer thicknesses \ftted to the parallel resistors formula.\nS3. OERSTED FIELD\nCurrent in the IrMn, Ru and Cu layers creates an e\u000bective Oersted \feld in y direction at the centre of the NiFe\nlayer. Current in the NiFe itself generates only a symmetric Oersted \feld with respect to the centre of the layer which\ndoes not contribute to the e\u000bective hyorhz(Fig. S4). From Ampere's law we have\nI\nHOedl=I (S.6)\nWhereIis the current encircled by the integration loop. For our geometry sketched in Fig. S4(a) we can write4\ncenter of current in Ru/IrMn/CuRu/IrMn/CuNiFe\njMW\nHOet\nFIG. S4. Schematic representation of Oersted \felds induced by the current in the multilayer.\n\u00160HOe=\u00160IOe\n2(w+t)\u0019\u00160IOe\n2w: (S.7)\nHereIOeis the current in the Ru, IrMn and Cu layers. We used the fact that the thickness tof the bar (\u001810 nm)\nis very small compared to its width w(500 nm - 4 \u0016m) for all measured devices. This means that the Oersted \feld\ndepends only on the size of the current in Ru, IrMn and Cu layers, and not on layer thicknesses, similar to the case\nof an in\fnite plane.\nS4. MAGNETIC ANISOTROPIES: AsymANDAasy\nTotal magnetic anisotropy is modelled as a combination of unidirectional, uniaxial and rotational anisotropies.\nUnidirectional anisotropy models the exchange bias. Uniaxial anisotropy is a combination of shape anisotropy, crys-\ntalline anisotropy of NiFe and some uniaxial anisotropy due to the exchange bias10. The contribution of each of\nthese towards the cumulative uniaxial anisotropy varies depending on the dimensions of the bar and the thickness\nof the IrMn layer. Rotational anisotropy is due to the partially stable grains of the polycristalline IrMn coupling\nto the NiFe at the interface. These are the same AFM grains responsible for the increased coercivity of magnetic\nhysteresis measurements10. This anisotropy is modelled as an additional in-plane e\u000bective \feld Hrotalong the NiFe\nmagnetizatoin direction. Magnetic free energy per unit area becomes\nF[\u0012;\u001e] =FZeeman [\u0012;\u001e] +Fsurf[\u0012;\u001e] +Fshape [\u0012;\u001e] +FU[\u0012;\u001e] +Fexch[\u0012;\u001e] =\n\u0000\u00160(H+Hrot)MtFM(sin\u001esin\u001eHcos(\u0012\u0000\u0012H) + cos\u001ecos\u001eH)\n+ (\u00160M2tFM=2\u0000KS) cos2\u001e\u0000KUtFMsin2\u001ecos2(\u0012\u0000\u0012uni)\n\u0000\u00160MtFMHexcos(\u0012\u0000\u0012exch) sin\u001e;(S.8)\nwhere (\u0012H,\u001eH) and (\u0012,\u001e) are in and out-of-plane angles of applied \feld Hand magnetization Min spherical\ncoordinates, with \u001e= 90\u000ebeing in the plane of the sample. KSandKUare surface and in-plane uniaxial anisotropy\nconstants,tFMis the thickness of the ferromagnetic layer, Hexis the exchange bias \feld, \u0012uniand\u0012exchare directions\nof the uniaxial anisotropy and the exchange bias respectively. The resonance condition reads\n\u0012!\n\r\u00132\n=1\nM2t2\nFMsin2\u0012\u0001\"\u0012@2F\n@\u00122\u0013\u0012@2F\n@\u001e2\u0013\n\u0000\u0012@2F\n@\u001e@\u0012\u00132#\n; (S.9)\nwhere!is the resonance frequency and \ris the gyromagnetic ratio. Plugging in the expression for F[\u0012;\u001e] into the\nequation above and di\u000berentiating it with respect to \u0012and\u001eone obtains\n\u0012!\n\r\u00132\n=\u00162\n0(H+H1)(H+H2) (S.10)5\nwith\nH1=Hrot+Meff+Hexchcos(\u0012\u0000\u0012exch) +HUcos2(\u0012\u0000\u0012U)\nH2=Hrot+Hexchcos(\u0012\u0000\u0012exch ) +HUcos[2(\u0012\u0000\u0012U)]:(S.11)\nHere we have relabelled variables in the following way\nMeff=M\u00002KS=\u00160MtFM\nHU= 2KU=\u00160M:(S.12)\nWe used these equations to \ft the \u0012dependence of the resonance \feld and extract anisotropies of each sample. In\nthis modelMeffandHrotare correlated, thus we need to know one of these using a di\u000berent method. This correlation\nis easier to see if we rewrite equation S.10 making an approximation Hres+H1\u0019Meff. This is valid because the\nrest of the terms in H1are much smaller than Meff. We write S.10 as\n\u00160Hres=\u0012!\n\r\u001321\n\u00160Meff\u0000\u00160Hrot\u0000\u00160Hexchcos(\u0012\u0000\u0012exch)\u0000\u00160HUcos[2(\u0012\u0000\u0012U)]: (S.13)\nFor the given frequency larger Meffleads to a smaller Hrotand vice versa, so we extract Mefffrom the frequency\ndependence of the resonance \feld and use it to \ft out Hrot(the \ftting is done using the full model and not the\napproximation).\nAsymandAasyentering the expressions for the recti\fed dc voltage are given by\nAsym=\r(Hres+H1)(Hres+H2)\n!\u0001H(2Hres+H1+H2)\nAasy=(Hres+H1)\n\u00160\u0001H(2Hres+H1+H2)(S.14)\nas deduced in reference11, withH1andH2given by equations S.11, and \u0001 Hbeing the resonance linewidth.\nS5. SPIN-HALL ANGLE OF RU\nThe spin hall angle of Ru is calculated using the hz=hyratio measured experimentally in a bar patterned from the\nRu(3)Py(4) bilayer12, assuming that hyis predominantly due to the Oersted \feld. We use\nFIG. S5. (a) A resonance curve measured in the Ru(3)Py(4) bar at 17.9 GHz, decomposed into symmetric and antisymmetric\nLorentzians.6\n\u0012SH=hz\nhy\u0001e\u00160MstRutNiFe\n\u0016h: (S.15)\nHere e is the electron charge, \u00160Ms= 1 T for NiFe, tRu= 3 nm,tNiFe = 4 nm, \u0016his the reduced Planck constant. A\ntypical resonance measurement in this sample is shown in Fig. S5. We \fnd\n\u0012SH= (0:50\u00060:02)\u00011:602\u000110\u000019\u00011\u00013\u00014\u000110\u000018\n1:055\u000110\u000034= 0:0092\u00060:0004: (S.16)\nS6. SAMPLES WITH TA SEED LAYERS\nTo con\frm the fact that the seed layer does not have a major contribution to the observed anti-damping torque\nwe measure SiO x=Ta(4:5)=IrMn(2;3)=NiFe(4)=Al(2) structures with a 4.5 nm Ta seed layer instead of Ru, both at\nroom temperature and at 5 K. Neither 2 nor 3 nm IrMn samples exhibit exchange bias at room temperature. The 2\nnm IrMn sample does not develop any substantial exchange bias even at low temperatures, whereas the 3 nm IrMn\nsample develops an exchange bias of 8 \u00061 mT at 5 K. In Fig. S6(a) we plot resonances measured for the 2 nm IrMn\nsample at room temperature and for the 3 nm IrMn sample at 5 K. Firstly, in both cases the symmetric component\nis positive. Ta has a large negative spin-Hall angle and if the e\u000bect was dominated by the spin-current from Ta one\nwould expect hzand thus the symmetric component to be negative for a positive antisymmetric component. The fact\nthathzis positive means that any e\u000bects due to the spin-Hall e\u000bect in Ta are small compared to the IrMn-induced\ne\u000bects. Additionally, one can see that at low temperature the symmetric component becomes even larger. This can\nalso be clearly seen in the angle dependences of VsymandVasyplotted in Fig. S6(b). This result further supports\nthe argument that the increase of the anti-damping torque with the exchange bias observed in our experiments is not\nrelated to the e\u000eciency of the transfer of the spin-angular momentum generated in the seed layer, as this would lead\nto a decrease of hzfor a seed layer with a negative spin-Hall angle like Ta. We believe that in our experiments the\nspin angular momentum generated in the seed layer is fully absorbed by the \frst few atomic layers of IrMn due to its\nsmall spin di\u000busion length, and the observed anti-damping torque is induced predominantly by the antiferromagnet.\n(a) (b)\nFIG. S6. (a) Resonances measured for the 3 nm IrMn sample at 5 K and for the 2 nm IrMn sample at room temperature (295K),\nat 16.5 and 18.6 GHz microwave frequencies respectively. The antisymmetric components are normalized to 1 \u0016V (not shown),\nand the symmetric components are show with dotted lines. (b) Angle dependences of the symmetric and the antisymmetric\ncomponents for the measurements shown in (a). Solid lines are \fts to hzsin 2\u0012cos\u0012(symmetric) and sin 2 \u0012(hycos\u0012\u0000hxsin\u0012)\n(antisymmetric).7\nS7. INDEPENDENCE OF THE SYMMETRY OF hzON THE EXCHANGE BIAS DIRECTION\n(a) (b)\n(c) (d)\nFIG. S7. (a, b) Angle dependence of the resonance \feld for two di\u000berent bars with 4 nm IrMn and (c, d) corresponding\nsymmetric components of the measured dc voltage. Although the exchange bias is substantial and has di\u000berent directions for\nthe two bars, the angle dependence of the symmetric component of the Lorentzian, which corresponds to hz, is not a\u000bected.\nS8. POWER, FREQUENCY AND DIMENSION DEPENDENCE OF CURRENT-INDUCED TORQUES\nWe present several control measurements to support our interpretation of symmetric and antisymmetric components\nof the resonance. In Fig. S8(a) we show the power dependence of the FMR magnitude for the 2 nm IrMn sample\nat room temperature, measured at 17.9 GHz. As one can see, both symmetric and antisymmetric components scale\nlinearly with power, as expected for the recti\fcation signal (h, I /p\nP, see equations 1 and 2 in the main text). Their\nratio is power independent (Fig. S8(b)).\nIn Fig. S8(c) we show that the hz/hyratio is frequency independent in our devices. The data shown is for the\n3 nm IrMn sample. Note that here the ratio is extracted from single resonances rather than a full angle-dependent\nmeasurement, thus the relatively large \ructuations, although still within about 10 % of each other.\nThe measurements were performed in bars with di\u000berent dimensions to exclude any geometry related e\u000bects. Parts\nof measurements were also performed in two di\u000berent measurement systems, with the same results. Fig. S8(d)\nsummarizes the above stated for the 2 nm IrMn sample.\n\u0003Currently at London Centre for Nanotechnology, Department of Materials, Imperial College London, SW7 2BP, UK\nyajf1006@cam.ac.uk8\n(a) (b)\n(c) (d)\nFIG. S8. (a) Dependence of the magnitudes of symmetric and antisymmetric Lorentzians on applied microwave power for the\n2 nm IrMn sample, \ftted to lines. (b) Microwave power dependence of the ratio of symmetric and antisymmetric Lorentzians\nplotted in (a). (c) Frequency dependence of the hz=hyratio for the 3 nm IrMn sample. (d) hz=hyratio for 2 nm IrMn samples\nwith di\u000berent dimensions and measured in two di\u000berent setups. The bar dimensions are 1.8 \u0016m x 38\u0016m, 4\u0016m x 240\u0016m, 1\n\u0016m x 10\u0016m, 500 nm x 5 \u0016m\n1B Warot, J Imrie, a.K Petford-Long, J.H Nickel, and T.C Anthony. In\ruence of seed layers on the microstructure of NiFe\nlayers. Journal of Magnetism and Magnetic Materials , 272-276:E1495{E1496, May 2004.\n2H. Gong, D. Litvinov, T.J. Klemmer, D.N. Lambeth, and J.K. Howard. Seed layer e\u000bects on the magnetoresistive properties\nof NiFe \flms. IEEE Transactions on Magnetics , 36(5):2963{2965, 2000.\n3Lichuan Jin, Huaiwu Zhang, Xiaoli Tang, Feiming Bai, and Zhiyong Zhong. E\u000bects of ruthenium seed layer on the mi-\ncrostructure and spin dynamics of thin permalloy \flms. Journal of Applied Physics , 113(5):053902, 2013.\n4T Yeh, JM Sivertsen, and JH Judy. Thickness dependence of the magnetoresistance e\u000bect in RF sputtered thin permalloy\n\flms. Magnetics, IEEE Transactions . . . , M(5):2215{2217, 1987.\n5TGSM Rijks and R Coehoorn. Semiclassical calculations of the anisotropic magnetoresistance of NiFe-based thin \flms,\nwires, and multilayers. Physical Review B , 51(1), 1995.\n6G. Choe and M. Steinback. Surface roughness e\u000bects on magnetoresistive and magnetic properties of NiFe thin \flms. Journal\nof Applied Physics , 85(8):5777, 1999.\n7R Acharyya. Spin dependent transport studies in magnetic, non-magnetic, antiferromagnetic, and half metals . PhD thesis,\nMichigan State University, 2012.\n8Jongwan Choi, Youngmin Choi, Jongin Hong, Huyong Tian, Jae-Sung Roh, Younsoo Kim, Taek-Mo Chung, Young Woo\nOh, Yunsoo Kim, Chang Gyun Kim, and Kwangsoo No. Composition and Electrical Properties of Metallic Ru Thin Films\nDeposited Using Ru(C 6 H 6 )(C 6 H 8 ) Precursor. Japanese Journal of Applied Physics , 41(Part 1, No. 11B):6852{6856,\nNovember 2002.9\n9A. F. Mayadas. Resistivity of Permalloy thin \flms. Journal of Applied Physics , 45(6):2780, 1974.\n10M. Stiles and R. McMichael. Model for exchange bias in polycrystalline ferromagnet-antiferromagnet bilayers. Physical\nReview B , 59(5):3722{3733, February 1999.\n11D Fang, H Kurebayashi, J Wunderlich, K V\u0013 yborn\u0013 y, L P Z^ arbo, R P Campion, A Casiraghi, B L Gallagher, T Jungwirth,\nand A J Ferguson. Spin-orbit-driven ferromagnetic resonance. Nature nanotechnology , 6(7):413{7, July 2011.\n12Luqiao Liu, Takahiro Moriyama, D. Ralph, and R. Buhrman. Spin-Torque Ferromagnetic Resonance Induced by the Spin\nHall E\u000bect. Physical Review Letters , 106(3):1{4, January 2011."    },    {        "title": "1101.2137v2.Magnon_Pumping_by_a_Time_Dependent_Transverse_Magnetic_Field_in_Ferromagnetic_Insulators.pdf",        "content": "arXiv:1101.2137v2  [cond-mat.mes-hall]  1 Mar 2011Typeset with jpsj3.cls <ver.1.1> Full Paper\nMagnon Pumping by a Time-Dependent Transverse Magnetic Fie ld in Ferromagnetic\nInsulators\nKouki Nakata∗and Gen Tatara\n∗Yukawa Institute for Theoretical Physics, Kyoto Universit y, Kitashirakawa Oiwake-Cho, Kyoto 606-8502, Japan\nDepartment of Physics, Tokyo Metropolitan University, Hac hioji, Tokyo 192-0397, Japan\nThe magnon pumping eﬀect in ferromagnetic insulators under an external time-dependent\ntransverse magnetic ﬁeld is theoretically studied. Genera tion of a magnon current is discussed\nby calculating the magnon source term in the spin continuity equation. This term represents the\nnon-conservation of magnons arising from an applied transv erse magnetic ﬁeld. The magnon\nsource term has a resonance structure as a function of the ang ular frequency of the transverse\nﬁeld, and this fact is useful to enhance the pumping eﬀect.\nKEYWORDS: magnon pumping, magnon current, magnon source te rm, spintronics, resonance\n1. Introduction\nRecently a new branch of physics and nanotechnology\ncalled spintronics1–3)has emerged and has been attach-\ning special attention from viewpoints of the fundamental\nscience and application. The aim of spintronics is the\ncontrol of the spin as well as charge degrees of freedom\nof electrons, and therefore establishing methods for gen-\neration and observation of a spin current is an urgent\nissue.\nA standard way to generate a spin current is the spin\npumping4–8)eﬀect in ferromagnetic-normal metal junc-\ntions. There the precession of the magnetization caused\nby an external ﬁeld induces a spin current pumped into\na normal metal. This method was theoretically proposed\nby R.H.Silsbee et.al6)and Y.Tserkovnyak et.al,9)and\nwas conﬁrmed experimentally by S.Mizukami et.al.5)In\na spin Hall system, i.e. in a nonmagnetic semiconductor,\nKato et.al10)reported an observation of a spin current\nby measuring optically the spin accumulation which ap-\npears as a result of spin currents at the edge of samples\n(GaAs and InGaAs). A critical issue in the observation\nof a spin current, however, is that a spin current is not\ngenerally conserved and therefore measuring spin accu-\nmulation does not necessarily indicate the detection of a\nspin current, in sharp contrast to the case of charge. Non-\nconservation of spins is represented by a spin relaxation\ntorque,Ts, which appears in the spin continuity equa-\ntion. For a clear interpretation of experimental results\non a spin current, to understand the relaxation torque is\nessential.\nThe spin current means a ﬂow of the spin angular mo-\nmentum in general, and in metals conduction electrons\ncarry a spin current. In insulators, there is no conduc-\ntion electrons, but there exists an other kind of carrier,\nnamely, spin-waves, which are collective motions of mag-\nnetic moments. Experimentally, a spin-wave spin current,\na spin current carried by spin-waves has already been es-\ntablished as a physical quantity. Kajiwara11)et.al have\nshown that a spin-wave spin current in an insulator can\nbe generated and detected using direct and inverse spin-\n∗E-mail address: nakata@yukawa.kyoto-u.ac.jpHall eﬀects. They have revealed the conversion of an elec-\ntric signal into spin-waves, and its subsequent transmis-\nsion through an insulator over macroscopic distance. The\nspin-wave spin current has a novel feature;11)this current\npersists for much greater distance than the conduction\nelectron spin current in metals, which disappears within\nvery short distance (typically micrometers). For exam-\nple in the magnetic insulator (Y 3Fe5O12), the spin-wave\ndecay length can be several centimetres.\nIn contrast to the experimental development, theoreti-\ncal studies so far of magnon transports are not enough to\nexplain the experimental result on bulk systems. Meier\nand Loss12)have investigated the magnon transport in\nboth ferromagnetic and antiferromagnetic materials and\nfound that the spin conductance is quantized in the units\nof order(gµB)2/hin the antiferromagnetic isotropic spin-\n1/2 chains ( gis the gyromagnetic ratio, µBis the Bohr\nmagneton and his Planck constant). Wang et.al13)have\ninvestigated a spin current carried by magnons and de-\nrived a Landauer-B ¨uttiker-type formula for spin current\ntransports. They have also studied the magnon trans-\nport properties of a two-level magnon quantum dot in\nthe presence of the magnon-magnon scattering and ob-\ntained the nonlinear spin current as a function of the\nmagnetochemical potential. These theoretical studies of\nmagnon transports are limited to mesoscopic systems.\nFrom the viewpoint of spintronics, the magnon trans-\nport in a macroscopic scale is an urgent and important\nsubject.\nIn this paper, we focus on three dimensional ferromag-\nnetic insulators. The magnon source term, Tm, arising\nfrom a time-dependent transverse magnetic ﬁeld is de-\nrived microscopically through Heisenberg’s equation of\nmotion. We evaluate it by using Green’s function with-\nout relying on the phenomenological equation, Landau-\nLifshitz-Gilbert equation. This is the main aim of this\npaper. The emergence of this term is in sharp contrast\nto a charge current and represents the non-conservation\nof the magnon number.\nThis paper is structured as follows. In §2.1, we repre-\nsent spin variables of a ferromagnetic insulator by boson\n12 J. Phys. Soc. Jpn. Full Paper Author Name\ncreation/annihilation operators via Holstein-Primakoﬀ\ntransformation. We then apply a time-dependent trans-\nverse magnetic ﬁeld. This magnetic ﬁeld generates the\nmagnon source term, which breaks the magnon conser-\nvation law in the spin continuity equation. In §2.2 and\n§2.3, by evaluating the magnon source term at the low-\ntemperature limit, the dependence of the magnon source\nterm on the angular frequency of a transverse magnetic\nﬁeld is calculated. The magnon source term has a res-\nonating behavior when the angular frequency of an ex-\nternal transverse magnetic ﬁeld is tuned. In §2.4, the\ntemperature dependence of the magnon source term is\nargued. Through the analogy with the usual conduc-\ntion electrons’ spin pumping eﬀect, the possibility for\nthe magnon pumping is discussed in §3.\n2. Magnon Source Term\n2.1 Deﬁnition\nWe consider a ferromagnetic Heisenberg model in\nthree dimensions. It reduces to a free boson system via\nHolstein-Primakoﬀ transformation if we approximate the\nspin as (S: the length of a spin) ; Sz=S−a†a,(S+)†=\nS−= (2S)1/2a†[1−a†a/(2S)]1/2≃(2S)1/2a†. Here op-\neratorsa†/aare magnon creation/annihilation operators\nsatisfying the bosonic commutation relation. Therefore\nin the continuous limit, a three-dimensional ferromag-\nnetic insulator with an external magnetic ﬁeld along the\nquantization ( z) axis is described at low magnon density\nlimit as\nH0=HHeisen+HB\n=/integraldisplay\nd3xa†(xt)/parenleftBig\n−/planckover2pi12∇2\n2mmag+gµB˜B/parenrightBig\na(xt).(1)\nHeremmagis the eﬀective mass of a magnon and it is\nrepresented by a ferromagnetic exchange coupling con-\nstant in the discrete model, J, and the (square) lattice\nconstant, a0, as/planckover2pi12/(2mmag) = 2JSa02. In eq.(1), ˜Bis a\nconstant external magnetic ﬁeld along the quantization\naxis (z-axis),gisg-factor and µBis Bohr magneton.\nFrom now on including g-factor and Bohr magneton, we\nwrite an external magnetic ﬁeld as gµB˜B≡B.We then\napply a time-dependent transverse magnetic ﬁeld with\nan angular frequency, Ω, and a constant ﬁeld strength,\nΓ0, tox-axis asΓ(t) = Γ0cosΩt.\nVΓ(t) = Γ(t)/integraldisplay\nd3xSx(x)\n≃Γ(t)/integraldisplay\nd3x(S\n2)1/2/bracketleftbigg\na(xt)+a†(xt)/bracketrightbigg\n.(2)\nThe total Hamiltonian is H=H0+VΓ(t).\nThe magnon density, ρm(x), of the system is deﬁned as\nthe expectation value of the number operator of magnons\nρm(x,t)≡ ∝angb∇acketlefta†(x,t)a(x,t)∝angb∇acket∇ight. (3)\nThrough Heisenberg’s equation of motion, the magnon\ncurrent density, jm, and the magnon source term, Tm,are deﬁned as\n∂ρm\n∂t=1\ni/planckover2pi1[ρm,H]\n=−∇·jm+Tm.(4)\nHere the magnon current density arises from the free\npart;[ρm,H0]/(i/planckover2pi1)=−∇·jm. It reads\njµ\nm(x,t) =/planckover2pi1\nmmagRe/bracketleftBig\ni <(∂µa†(xt))a(xt)>/bracketrightBig\n,(5)\nwhereµis a direction for a magnon current to ﬂow\n(µ=x,y,z ). The magnon source term, which represents\nthe breaking of magnon conservation, arises from a trans-\nverse magnetic ﬁeld as [ρm,VΓ]/(i/planckover2pi1)≡ Tm, i.e.,\nTm(t) =−(2S)1/2\n/planckover2pi1Im/angbracketleftBig\nΓ(t)a(xt)/angbracketrightBig\n.(6)\nFrom now on, we treat VΓ(t)as a perturbation (i.e. a\nweak transverse magnetic ﬁeld) and study the eﬀects of a\ntime-dependent transverse magnetic ﬁeld to the magnon\nsource term.\n2.2 Evaluation\nThrough the standard procedure of the Keldysh (or\ncontour-ordered) Green’s function,14–16)the Langreth\nmethod,17,18)the magnon source term is evaluated (see\nalso APPENDIX ) as\n<Γ(t)a(t)>=/integraldisplay\nd3x′/integraldisplay\ndt′Γ(t)Γ(t′)(S\n2)1/2Gr(t,t′)\n+O(Γ3).(7)\nHereGris the retarded Green’s function and we have\nneglected terms which are third-order in Γ, which is jus-\ntiﬁed at the low magnon density regime.\nThe retarded Green’s function is Gr(rr′,tt′) =\n(/planckover2pi1/V)/summationtext\nk/integraltext(dω/2π)eik·(r−r′)−iω(t−t′)Gr\nk,ω, and\nGr(k,ω) = [/planckover2pi1ω−ωk+i/planckover2pi1/(2τ)]−1.HereVis a volume\nof the system. The lifetime τrepresents the damping of\nspins (τis inversely proportional to the Gilbert damping\nparameter,17)α). The energy ωkcorresponds to the free\npart,H0, and therefore ωk=Dk2+B,D≡/planckover2pi12/(2mmag).\nThen the magnon source term is calculated as\nTm=Γ2\n0\n4/planckover2pi1VS/bracketleftbigg/planckover2pi1\n2τ+(/planckover2pi1Ω+B)sin2Ωt+/planckover2pi1\n2τcos2Ωt\n(/planckover2pi1Ω+B)2+(/planckover2pi1\n2τ)2\n+/planckover2pi1\n2τ+(/planckover2pi1Ω−B)sin2Ωt+/planckover2pi1\n2τcos2Ωt\n(/planckover2pi1Ω−B)2+(/planckover2pi1\n2τ)2/bracketrightbigg\n.\n(8)\nThe time average of Tmbecomes\n¯Tm=Γ2\n0\n4/planckover2pi1VS/bracketleftbigg/planckover2pi1\n2τ\n(/planckover2pi1Ω+B)2+(/planckover2pi1\n2τ)2+/planckover2pi1\n2τ\n(/planckover2pi1Ω−B)2+(/planckover2pi1\n2τ)2/bracketrightbigg\n.\n(9)\nIt is clear that ¯Tmis positive (for ﬁnite temperature, see\neq.(16) in §2.4).J. Phys. Soc. Jpn. Full Paper Author Name 3\n2.3 Resonance\nWe deﬁne a dimensionless quantity ¯T(Ω)\nmas\n¯T(Ω)\nm≡1\n(2τΩ+2τB\n/planckover2pi1)2+1+1\n(2τΩ−2τB\n/planckover2pi1)2+1,i.e.,\n¯Tm=Γ2\n0\n4/planckover2pi1VS·2τ\n/planckover2pi1¯T(Ω)\nm.\n(10)\nThis shows that the magnon source term has a resonance\nstructure with a time-dependent transverse magnetic\nﬁeld when the angular frequency is tuned as Ω =B//planckover2pi1\n(see Fig.1). This resonance is useful for the enhancement\nof the magnon pumping.\ntT(Ω)\nm\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1\n 0  10  20  30  40  50  60  701/((t+35)^2+1)+1/((t-35)^2+1)\nt1.1 \n1\n0.5 \n0.1 \n0\nFig. 1. A graph of ¯T(Ω)\nm, which represents the angular frequency\ndependence of the magnon source term. Parameters we have use d\nare,2τΩ≡t,τ= 2×10−6[s],˜B= 1[G],g= 1. Therefore 2τB//planckover2pi1\nis35. This¯T(Ω)\nmhas a sharp peak around t= 35. This fact means\nthat the magnon source term has a resonance structure with th e\napplied transverse magnetic ﬁeld.\n2.4 Temperature dependence\nLet us look into the temperature dependence of the\nmagnon source term. To do this, we include the interac-\ntion of third-order in magnon operators. Then VΓ(t)is\nrewritten as\nVΓ(t) = Γ(t)/integraldisplay\nd3xSx(x)\n≃Γ(t)/integraldisplay\nd3x(S\n2)1/2/braceleftbigg\na(xt)+a†(xt)\n−1\n4S/bracketleftBig\na†(xt)a†(xt)a(xt)+a†(xt)a(xt)a(xt)/bracketrightBig/bracerightbigg\n,\n(11)\nand the magnon source term reads\nTm(t) =−(2S)1/2\n/planckover2pi1Im/angbracketleftbigg\nΓ(t)/bracketleftBig\na(xt)+a†(xt)a†(xt)a(xt)\n4S/bracketrightBig/angbracketrightbigg\n.\n(12)Eq.(12) is calculated as\n<Γ(t)a(t)>=/integraldisplay\nd3x′/integraldisplay\ndt′Γ(t)Γ(t′)/bracketleftBig\n(S\n2)1/2Gr(t,t′)\n−i\n2(2S)1/2Gr(t,t′)G<(t′,t′)/bracketrightBig\n+O(Γ3),\n(13)\n<Γ(t)\n4Sa†(t)a†(t)a(t)>=i\n2(2S)1/2/integraldisplay\nd3x′/integraldisplay\ndt′\n·Γ(t)Γ(t′)Ga(t′,t)G<(t,t)+O(Γ3).\n(14)\nHereGaandG<are the advanced and lesser Green’s\nfunctions, respectively. Because we focus on the behav-\nior of the magnon source term at low temperature that\nwe have neglected higher terms than the fourth-order in\nrespect to magnon creation/annihilation operators. The\nFourier transform of the lesser Green’s function satis-\nﬁes,G<(k,ω) =−fB(ωk)[Ga(k,ω)−Gr(k,ω)], where\nGa(k,ω) = [/planckover2pi1ω−ωk−i/planckover2pi1/(2τ)]−1. ThenTmis calculated\nas\nTm=Γ2\n0\n4/planckover2pi1V/bracketleftbigg\nS−/parenleftBig\n1+1\n23/2e−βB/parenrightBig\ne−βB/parenleftBigkBT\n4πD/parenrightBig3/2/bracketrightbigg\n·/bracketleftbigg/planckover2pi1\n2τ+(/planckover2pi1Ω+B)sin2Ωt+/planckover2pi1\n2τcos2Ωt\n(/planckover2pi1Ω+B)2+(/planckover2pi1\n2τ)2\n+/planckover2pi1\n2τ+(/planckover2pi1Ω−B)sin2Ωt+/planckover2pi1\n2τcos2Ωt\n(/planckover2pi1Ω−B)2+(/planckover2pi1\n2τ)2/bracketrightbigg\n,(15)\nwhere β−1iskBT(kB:Boltzmann con-\nstant). Here we have approximated the sum-\nmation over the Bose distribution function as,\n[(2π)3/V]/summationtext\nkfB(ωk) = 2π/integraltext∞\n−∞dkk2[eβ(Dk2+B)−1]−1≃\n2π/integraltext∞\n−∞dk k2e−β(Dk2+B)[1+e−β(Dk2+B)].\nThe time average of Tmbecomes\n¯Tm=Γ2\n0\n4/planckover2pi1V/bracketleftbigg\nS−/parenleftBig\n1+1\n23/2e−βB/parenrightBig\ne−βB/parenleftBigkBT\n4πD/parenrightBig3/2/bracketrightbigg\n·/bracketleftbigg/planckover2pi1\n2τ\n(/planckover2pi1Ω+B)2+(/planckover2pi1\n2τ)2+/planckover2pi1\n2τ\n(/planckover2pi1Ω−B)2+(/planckover2pi1\n2τ)2/bracketrightbigg\n=Γ2\n0\n4/planckover2pi1V2τ\n/planckover2pi1¯T(Ω)\nm¯T(T)\nm,\n(16)\nwhere\n¯T(T)\nm≡S−/parenleftBig\n1+1\n23/2e−βB/parenrightBig\ne−βB/parenleftBigkBT\n4πD/parenrightBig3/2\n,(17)\nis the temperature-dependent part and ¯T(Ω)\nmis deﬁned in\neq.(10). When the temperature gets higher, the magnon\nsource term decreases. This means that quantum and\nthermal ﬂuctuations act in the opposite way, namely to\nincrease and decrease the magnon source term, respec-\ntively. It is clear that eq.(16) reduces at the zero temper-\nature to eq.(9).4 J. Phys. Soc. Jpn. Full Paper Author Name\n3. Magnon Pumping\nThe spin pumping is an eﬀect widely used to cre-\nate a spin current by use of the magnetization preces-\nsion.4–8)Experiments have been carried out in junctions\nof metallic ferromagnets and nonmagnetic metals. Ac-\ncording to the theory by Y.Tserkovnyak and A.Brataas,9)\nthe spin current pumped through the junction reads\nIpump\ns= [/planckover2pi1/(4π)][Arm×(dm/dt)−Ai(dm/dt)], where\nmis the magnetization direction of a localized spin and\nAr,Aiare the interface parameters. This result is un-\nderstood by considering the spin continuity equation,\n∇ ·js=−(∂m/∂t) +Ts, where jsis the spin current\ndensity and Tsis the spin relaxation torque. In fact, the\nspin continuity equation indicates that a spin current\nis generated when the magnetization is dynamic and/or\nwhenTsis ﬁnite. It has been well-known17)that the\nmain term of spin relaxation torque, Ts, has the form\nTs∝m×(dm/dt)in metals. As Brataas et al.19)have\npointed out, this torque Tsis equivalent to the maximum\nspin current that can be drawn from the spin battery.\nThe pumping formula for Ipump\nsin metals is thus under-\nstood from the spin continuity equation. The relaxation\ntorque plays an essential role in spin pumping in met-\nals, and it is expected to be dominant also in the spin\npumping in insulating ferromagnets. Our calculation of\nthe relaxation torque term (i.e. the magnon source term)\nthus describes the spin pumping eﬀect in insulators.\nWe have revealed that the magnon source term has a\nsharp peak around Ω =B//planckover2pi1, as a result of the resonance\nwith a time-dependent transverse magnetic ﬁeld when\nthe angular frequency is appropriately adjusted as Ω =\nB//planckover2pi1. This fact is useful to enhance the magnon pumping\neﬀect because the external magnetic ﬁeld, B, and the\nangular frequency of a transverse magnetic ﬁeld, Ω, is\nunder our control.\nExperimentally, the magnon pumping eﬀect we have\ndiscussed can be identiﬁed by observing the tempera-\nture dependence of the pumped magnon current when B\nis zero;(the pumped magnon current )∝¯T(T)\nm∝(A1−\nA2T3/2), where A 1and A 2are constants.\n4. Summary\nWe have studied theoretically the magnon source term\nwhich represents the breaking of the magnon conser-\nvation law. We have revealed that the magnon source\nterm has a resonance structure with an external time-\ndependent transverse magnetic ﬁeld when the angular\nfrequency of the applied magnetic ﬁeld is tuned. This\nfact will be useful to enhance the magnon pumping ef-\nfect in insulators. This magnon pumping eﬀect is a new\nmethod for a generation of a magnon current (spin-wave\nspin current) without the gradient of an external mag-\nnetic ﬁeld.\nAcknowledgments\nThe author (K.N) would like to thank K.Totsuka,\nM.Oshikawa, Y.Korai and K.Taguchi for useful com-\nments and discussion. One of the author(G.T) is sup-\nported by a Grant-in-Aid for Scientiﬁc Research in Pri-\nority Areas, ”Creation and control of spin current ”underNonmagnetic \nx \n z \nmT\ninsulator \nFerromagnetic \ninsulator Pumped \nmagnon current ( )\nFig. 2. (Color online) A schematic picture of the magnon pump -\ning by a time-dependent transverse magnetic ﬁeld. Thick and\nshort solid arrows represent localized spins, and thin and l ong\nones external magnetic ﬁelds. Circles represent magnons. B y way\nof resonance with a time-dependent transverse magnetic ﬁel d, the\nmagnon source term is enhanced, and an enhanced magnon cur-\nrent is pumped from the ferromagnetic insulator to the adjac ent\nnonmagnetic insulator.\nGrant No. 1948027, and a Grant-in-Aid for Scientiﬁc Re-\nsearch (B) (Grant No. 22340104).\nAppendix: Calculation of Equation.(7)\nIn this section, we brieﬂy show the Langreth method,\nwhich is useful to evaluate the perturbation expansion of\nthe Keldysh ( or contour-ordered) Green’s function. For\nsimplicity here, we evaluate the magnon source term at\nzero temperature, eq.(7), as an example;\nVΓ(t)≃Γ(t)/integraldisplay\nd3x(S\n2)1/2/bracketleftbigg\na(xt)+a†(xt)/bracketrightbigg\n,\nTm(t) =−(2S)1/2\n/planckover2pi1Im/angbracketleftbigg\nΓ(t)a(xt)/angbracketrightbigg\n.(A·1)\nWe have only to estimate < a(xt)>. It is evaluated as\n< a(τ)>=/angbracketleftbigg\nTca(τ)exp/bracketleftBig\n−i/integraldisplay\ncdτ′VΓ(τ′)/bracketrightBig/angbracketrightbigg\n≃ −i(S\n2)1/2/integraldisplay\nd3x′\n·/integraldisplay\ncdτ′Γ(τ′)/angbracketleftBig\nTca(xτ)a†(x′τ′)/angbracketrightBig\n≡ −i(S\n2)1/2/integraldisplay\nd3x′I.(A·2)\nHere T cis the path-ordering operator deﬁned on the\nKeldysh contour, c (see Fig.A·1). We express the Keldysh\ncontour as a sum of the forward path, c→, and the back-\nward path, c←;c=c→+c←. The integral on the Keldysh\ncontour of eq.(A·2), I, is executed by taking an identity\ninto account/integraldisplay\ncdτc=/integraldisplay\nc→dτ→+/integraldisplay\nc←dτ←,\n(A·3)J. Phys. Soc. Jpn. Full Paper Author Name 5\nas\nI=i/integraldisplay∞\n−∞dτ′Γ(τ′)/bracketleftBig\nGt(τ,τ′)−G<(τ,τ′)/bracketrightBig\n.(A·4)\nHereGtis the time-ordered Green’s function. By using\nthe relation, Gr(t,t′) =Gt(t,t′)−G<(t,t′), we obtain\neq.(7).\nFig. A·1. Keldysh contour, c. We have taken τon forward path,\nc→. Even when τis located on backward path, c ←, the result of\nthis calculation is invariant because each Green’s functio n,Gr,\nGa,G<,G>(the greater Green’s function ), is not independent;\nthey obey, Gr−Ga=G>−G<. Both the forward and backward\npaths are actually on the real axis but shifted slightly upwa rds\nand downwards, respectively, to distinguish them clearly.\n1) S.Maekawa: Concepts in Spin Electronics (Oxford science\npublications, 2006).2) I.Zutic, J.Fabian, and S.D.Sarma: Rev.Mod.Phys. 76(2004)\n323.\n3) Y.Tserkovnyak, A.Brataas, G.E.W.Bauer, and B.I.Halper in:\nRev.Mod.Phys. 77(2005) 1375.\n4) A.Takeuchi, K.Hosono, and G.Tatara: Phys.Rev.B. 81(2010)\n144405.\n5) S.Mizukami, Y.Ando, and T.Miyazaki: Phys.Rev.B. 66(2002)\n104413.\n6) R.H.Silsbee, A.Janossy, and P.Monod: Phys.Rev.B. 19(1979)\n4382.\n7) P. W. Brouwer: Phys.Rev.B. 58(1998) R10135.\n8) P.Sharma and C.Chamon: Phys.Rev.Lett. 87(2001) 096401.\n9) Y.Tserkovnyak and A.Brataas: Phys.Rev.Lett. 88(2002)\n117601.\n10) Y.K.Kato, R.C.Myers, A.C.Gossard, and D.D.Awschalom:\nScience. 306(2004) 1910.\n11) Y.Kajiwara, K.Harii, S.Takahashi, J.Ohe, K.Uchida,\nM.Mizuguchi, H.Umezawa, H.Kawai, K.Ando, K.Takanashi,\nS.Maekawa, and E.Saitoh: Nature. 464(2010) 262.\n12) F.Meier and D.Loss: Phys.Rev.Lett. 90(2003) 167204.\n13) B.Wang, J.Wang, J.Wang, and D.Y.Xing: Phys.Rev.B. 69\n(2004) 174403.\n14) T.Kita: Prog.Theor.Phys. 123(2010) 581.\n15) J.Rammer and H.Smith: Rev.Mod.Phys. 58(1986) 323.\n16) A.Kamenev: arXiv:0412296.\n17) G.Tatara, H.Kohno, and J.Shibata: Physics Report. 468\n(2008).\n18) H. Haug and A.P.Jauho: Quantum Kinetics in Transport and\nOptics of Semiconductors . (Springer New York, 2007) p.35.\n19) A.Brataas, Y.Tserkovnyak, G.E.W.Bauer, and B.I.Halpe rin:\nRhys.Rev.B. 66(2002) 060404(R)."    },    {        "title": "1807.11806v1.Spin_absorption_at_ferromagnetic_metal_platinum_oxide_interface.pdf",        "content": "arXiv:1807.11806v1  [cond-mat.mes-hall]  31 Jul 2018Spin absorption at ferromagnetic-metal/platinum-oxide i nterface\nAkio Asami,1Hongyu An,1Akira Musha,1Makoto Kuroda,1and Kazuya Ando1,∗\n1Department of Applied Physics and Physico-Informatics, Ke io University, Yokohama 223-8522, Japan\n(Dated: August 1, 2018)\nWe investigate the absorption of a spin current at a ferromag netic-metal/Pt-oxide interface by\nmeasuring current-induced ferromagnetic resonance. The s pin absorption was characterized by the\nmagnetic damping of the heterostructure. We show that the ma gnetic damping of a Ni 81Fe19ﬁlm\nis clearly enhanced by attaching Pt-oxide on the Ni 81Fe19ﬁlm. The damping enhancement is\ndisappeared by inserting an ultrathin Cu layer between the N i81Fe19and Pt-oxide layers. These\nresults demonstrate an essential role of the direct contact between the Ni 81Fe19and Pt-oxide to\ninduce sizable interface spin-orbit coupling. Furthermor e, the spin-absorption parameter of the\nNi81Fe19/Pt-oxide interface is comparable to that of intensively st udied heterostructures with strong\nspin-orbit coupling, such as an oxide interface, topologic al insulators, metallic junctions with Rashba\nspin-orbit coupling. This result illustrates strong spin- orbit coupling at the ferromagnetic-metal/Pt-\noxide interface, providing an important piece of informati on for quantitative understanding the spin\nabsorption and spin-charge conversion at the ferromagneti c-metal/metallic-oxide interface.\nI. INTRODUCTION\nAn emerging direction in spintronics aims at discover-\ning novel phenomena and functionalities originatingfrom\nspin-orbit coupling (SOC)1. An important aspect of the\nSOC is the ability to convert between charge and spin\ncurrents. The charge-spin conversion results in the gen-\neration of spin-orbit torques in heterostructures with a\nferromagnetic layer, enabling manipulation of magneti-\nzation2–4. Recent studies have revealed that the oxida-\ntion of the heterostructure strongly inﬂuences the gen-\neration of the spin-orbit torques. The oxidation of the\nferromagnetic layer alters the spin-orbit torques, which\ncannot be attributed to the bulk spin Hall mechanism5–7.\nThe oxidation of a nonmagnetic layer in the heterostruc-\nture also oﬀers a route to engineer the spin-orbit devices.\nDemasius et al. reported a signiﬁcant enhancement of\nthe spin-torque generation by incorporating oxygen into\ntungsten, which is attributed to the interfacial eﬀect8.\nThe spin-torque generation eﬃciency was found to be\nsigniﬁcantly enhanced by manipulating the oxidation of\nCu, enablingto turn the light metal into aneﬃcient spin-\ntorque generator, comparable to Pt9. We also reported\nthat the oxidation of Pt turns the heavy metal into an\nelectrically insulating generatorof the spin-orbit torques,\nwhich enables the electrical switching of perpendicular\nmagnetization in a ferrimagnet sandwiched by insulating\noxides10. These studies have provided valuable insights\ninto the oxide spin-orbitronics and shown a promising\nway to develop energy-eﬃcient spintronics devices based\non metal oxides.\nThe SOC in solids is responsible for the relaxation of\nspins, as well as the conversion between charge and spin\ncurrents. The spin relaxation due to the bulk SOC of\nmetals and semiconductors has been studied both ex-\nperimentally and theoretically11–14. The inﬂuence of the\nSOC at interfaces on spin-dependent transport has also\nbeen recognized in the study of giant magnetoresistance\n(GMR). The GMR in Cu/Pt multilayers in the current-perpendicular-to-plane geometry indicated that there\nmust be a signiﬁcant spin-memory loss due to the SOC\nat the Cu/Pt interfaces15. The interface SOC also plays\na crucial role in recent experiments on spin pumping.\nThe spin pumping refers to the phenomenon in which\nprecessing magnetization emits a spin current to the sur-\nrounding nonmagnetic layers12. When the pumped spin\ncurrent is absorbed in the nonmagnetic layer due to the\nbulk SOC or the ferromagnetic/nonmagnetic interface\ndue to the interface SOC, the magnetization damping\nof the ferromagnetic layer is enhanced because the spin-\ncurrent absorption deprives the magnetization of the an-\ngularmomentum16. Althoughthedampingenhancement\ninduced by the spin pumping has been mainly associated\nwith the spin absorption in the bulk of the nonmagnetic\nlayer, recent experimental and theoretical studies have\ndemonstrated that the spin-current absorption at inter-\nfaces also provides a dominant contribution to the damp-\ning enhancement17. Since the absorption of a spin cur-\nrent at interfaces originates from the SOC, quantifying\nthe damping enhancement provides an important infor-\nmation for fundamental understanding of the spin-orbit\nphysics.\nIn this work, we investigate the absorption of a spin\ncurrent at a ferromagnetic-metal/Pt-oxide interface. We\nshow that the magnetic damping of a Ni 81Fe19(Py) ﬁlm\nis clearly enhanced by attaching Pt-oxide, Pt(O), despite\nthe absence of the absorption of the spin current in the\nbulk of the Pt(O) layer. The damping enhancement dis-\nappears by inserting an ultrathin Cu layer between the\nPyand Pt(O)layers. This resultindicates that the direct\ncontact between the ferromagnetic metal and Pt oxide is\nessential to induce the sizable spin-current absorption, or\nthe interface SOC. We further show that the strength of\nthe damping enhancement observedfor the Py/Pt(O) bi-\nlayer is comparable with that reported for other systems\nwith strong SOC, such as two-dimensional electron gas\n(2DEG) at an oxide interface and topological insulators.2\nII. EXPERIMENTAL METHODS\nThree sets of samples, Au/SiO 2/Py,\nAu/SiO 2/Py/Pt(O) and Au/SiO 2/Py/Cu/Pt(O),\nwere deposited on thermally oxidized Si substrates\n(SiO2) by RF magnetron sputtering at room tempera-\nture. To avoid the oxidation of the Py or Cu layer, we\nﬁrst deposited the Pt(O) layer on the SiO 2substrate in\na mixed argon and oxygen atmosphere. After the Pt(O)\ndeposition, the chamber was evacuated to 1 ×10−6Pa,\nand then the Py or Cu layer was deposited on the top\nof the Pt(O) layer in a pure argon atmosphere. For\nthe Pt(O) deposition, the amount of oxygen gas in the\nmixture was ﬁxed as 30%, in which the ﬂow rates of\nargon and oxygen were set as 7.0 and 3.0 standard cubic\ncentimeters per minute (sccm), respectively. The SiO 2\nlayer was deposited from a SiO 2target in the pure argon\natmosphere. The ﬁlm thickness was controlled by the\ndeposition time with a precalibrated deposition rate.\nWe measured the magnetic damping using current-\ninduced ferromagnetic resonance (FMR). For the fabri-\ncation of the devices used in the FMR experiment, the\nphotolithography and lift-oﬀ technique were used to pat-\nterntheﬁlmsintoa10 µm×40µmrectangularshape. A\nblanket Pt(O) ﬁlm on a 1 cm ×1 cm SiO 2substrate was\nfabricatedforthecompositionconﬁrmationbyx-raypho-\ntoelectron spectroscopy (XPS). We also fabricated Pt(O)\nsingle layer and SiO 2/Py/Pt(O) multilayer ﬁlms with a\nHall bar shape to determine the resistivity of the Pt(O)\nand Py using the four-probe method. The resistivity of\nPt(O) (6.3 ×106µΩ cm) is much larger than that of Py\n(106µΩ cm). Because of the semi-insulating nature of\nthe Pt(O) layer, we neglect the injection of a spin cur-\nrent into the Pt(O) layer from the Py layer; only the\nPy/Pt(O) interface can absorb a spin current emitted\nfrom the Py layer. Transmission electron microscopy\n(TEM) was used to directly observe the interface and\nmultilayer structure of the SiO 2/Py/Pt(O) ﬁlm. All the\nmeasurements were conducted at room temperature.\nIII. RESULTS AND DISCUSSION\nFigure 1(a) exhibits the XPS spectrum of the Pt(O)\nﬁlm. Previous XPS studies on Pt(O) show that bind-\ning energies of the Pt 4 f7/2peak for Pt, PtO and PtO 2\nare around 71.3, 72.3 and 74.0 eV, respectively18. Thus,\nthe Pt 4 f7/2peak at 72.3 eV in our Pt(O) ﬁlm indi-\ncates the formation of PtO. By further ﬁtting the XPS\nspectra, we conﬁrm that the Pt(O) ﬁlm is composed of\na dominant structure of PtO with a minor portion of\nPtO2. Figure 1(b) shows the cross-sectional TEM image\nof the SiO 2(4 nm)/Py(8 nm)/Pt(O)(10 nm) ﬁlm. As can\nbe seen, continuous layer morphology with smooth and\ndistinct interfaces is formed in the multilayer ﬁlm. Al-\nthough we deposited the Py layer on the Pt(O) layer to\navoid the oxidation of the Py, it might still be possible\nthat the Py layer is oxidized by the Pt(O) layer. There-\nFIG. 1. (a) The XPS spectrum of the Pt(O) ﬁlm. The gray\ncurve is the experimental data, and the red ﬁttingcurve is th e\nmerged PtO and PtO 2peaks. (b) The cross-sectional TEM\nimage of the SiO 2(4 nm)/Ni 81Fe19(8 nm)/Pt(O)(10 nm) ﬁlm.\nfore, we measured the resistance of the Au/SiO 2/Py and\nAu/SiO 2/Py/Pt(O) samples used in the FMR experi-\nment. The resistance of both samples show the same\nvalue (60 Ω). Furthermore, as described in the follow-\ning section, the saturation magnetization for each device\nwas obtained by using Kittel formula (0.746 T and 0.753\nT for the Au/SiO 2/Py and Au/SiO 2/Py/Pt(O), respec-\ntively). The only 1% diﬀerence indicates that the minor\noxidationofthePylayerdue tothe presenceofthe Pt(O)\nlayer can be neglected.\nNext, we conduct the FMR experiment to investigate\nthe absorption and relaxation of spin currents induced\nby the spin pumping. Figure 2(a) shows a schematic\nof the experimental setup for the current-induced FMR.\nWe applied an RF current to the device, and an in-plane\nexternal magnetic ﬁeld µ0Hwas swept with an angle\nof 45ofrom the longitudinal direction. The RF charge\ncurrentﬂowingintheAulayergeneratesanOerstedﬁeld,\nwhich drives magnetization precession in the Py layer\nat the FMR condition. The magnetization precession\ninduces an oscillation of the resistance of the device due\nto the anisotropic magnetoresistance (AMR) of the Py\nlayer. We measured DC voltage generated by the mixing\nof the RF current and the oscillating resistance using a\nbias tee.\nFigures 2(b), 2(c) and 2(d) show the FMR spec-\ntra for the Au/SiO 2/Py, Au/SiO 2/Py/Pt(O) and\nAu/SiO 2/Py/Cu/Pt(O) ﬁlms, respectively. For the\nFMRmeasurement, asmallRFcurrentpower P= 5mW\nwas applied. Around P= 5 mW, the FMR linewidth is\nindependent of the RF power as shown in the inset to\nFig. 3(a). This conﬁrms that the measured linewidth\nis unaﬀected by additional linewidth broadening due to\nnonlinear damping mechanisms and Joule heating. As\nshown in Fig. 2, clear FMR signals with low noise are\nobtained, allowing us to precisely ﬁt the spectra and ex-\ntract the magnetization damping for the three samples.\nHere, the mixing voltage due to the FMR, Vmix, is ex-3\nFIG. 2. Schematic illustration of the experimental setup\nfor the current-induced FMR. Mis the magnetization in\nthe Py layer. The FMR spectra of the (b) Py(9 nm),\n(c) Py(9 nm)/Pt(O)(7.3 nm), and (d) Py(9 nm)/Cu(3.6\nnm)/Pt(O)(7.3 nm) ﬁlms by changing the RF current fre-\nquency from 4 to 10 GHz. All the ﬁlms are capped with 3\nnm-thick SiO 2and 10 nm-thick Au layers. The RF current\npower was set as 5 mW. The schematic illustrations of the\ncorresponding ﬁlms are also shown.\npressed as\nVmix=Vsym(µ0∆H)2\n(µ0H−µ0HR)2+(µ0∆H)2\n+Vasyµ0∆H(µ0H−µ0HR)\n(µ0H−µ0HR)2+(µ0∆H)2,(1)\nwhereµ0∆Handµ0HRare the spectral width and res-\nonance ﬁeld, respectively19.VsymandVasymare the\nmagnitudes of the symmetric and antisymmetric com-\nponents. The symmetric and antisymmetric components\narise from the spin-orbit torques and Oersted ﬁeld. In\nthe devices used in the present study, the Oersted ﬁeld\ncreated by the top Au layer dominates the RF eﬀective\nﬁelds acting on the magnetization in the Py layer [see\nalso Fig. 2(a)]. The large Oersted ﬁeld enables the elec-\ntric measurement of the FMR even in the absence of the\nspin-orbit torques in the Au/SiO 2/Py ﬁlm.\nThe damping constant αof the Py layer\nin the Au/SiO 2/Py, Au/SiO 2/Py/Pt(O) and\nAu/SiO 2/Py/Cu/Pt(O) ﬁlms can be quantiﬁed byﬁtting the RF current frequency fdependence of the\nFMR spectral width µ0∆Husing\nµ0∆H=µ0∆Hext+2πα\nγf, (2)\nwhere ∆ Hextandγare the inhomogeneous linewidth\nbroadening of the extrinsic contribution and gyromag-\nnetic ratio, respectively19,20. Figure 3(a) shows the f\ndependence of the FMR linewidth µ0∆H, determined by\nﬁtting the spectra shown in Fig. 2 using Eq. (1). As\nshown in Fig. 3(a), the frequency dependence of the\nlinewidth is well ﬁtted by Eq. (2). Importantly, the slope\nof thefdependence of µ0∆Hfor the Py/Pt(O) ﬁlm is\nclearly larger than that for the Py and Py/Cu/Pt(O)\nﬁlms. This indicates larger magnetic damping in the\nPy/Pt(O) ﬁlm. By using Eq. (2), we determined the\ndamping constant αas 0.0126, 0.0169 and 0.0124 for the\nPy, Py/Pt(O) and Py/Cu/Pt(O)ﬁlms, respectively. The\ndiﬀerence in αbetween the Py and Py/Cu/Pt(O)ﬁlms is\nvanishingly small, which is within an experimental error.\nIn contrast, the damping of the Py/Pt(O) ﬁlm is clearly\nlarger than that of the other ﬁlms, indicating an essen-\ntial role of the Py/Pt(O) interface on the magnetization\ndamping.\nThe larger magnetic damping in the Py/Pt(O) ﬁlm\ndemonstrates an important role of the direct contact be-\ntween the Py and Pt(O) layers in the spin-current ab-\nsorption. If the bottom layers inﬂuence the magnetic\nproperties of the Py layer, the diﬀerence in the mag-\nnetic properties can also result in the diﬀerent magnetic\ndamping in the Au/SiO 2/Py, Au/SiO 2/Py/Pt(O) and\nAu/SiO 2/Py/Cu/Pt(O) ﬁlms. However, we have con-\nﬁrmed that the diﬀerence in the magnetic damping is not\ncaused by diﬀerent magnetic properties of the Py layer.\nIn Fig. 3(b), we plot the RF current frequency fdepen-\ndence of the resonance ﬁeld µ0HR. As can be seen, the\nfdependence of µ0HRis almost identical for the diﬀer-\nent samples, indicating the minor change of the magnetic\npropertiesofthe Pylayerdueto the diﬀerent bottomlay-\ners. In fact, by ﬁtting the experimental data using Kittel\nformula21, 2πf/γ=/radicalbig\nµ0HR(µ0HR+µ0Ms), the satura-\ntion magnetizationis obtainedto be µ0Ms= 0.746, 0.753\nand 0.777 T for the Py, Py/Pt(O) and Py/Cu/Pt(O)\nﬁlms, respectively. The minor diﬀerence ( <5%) in the\nsaturation magnetization indicates that the larger damp-\ning of the Py/Pt(O) ﬁlm cannot be attributed to possi-\nble diﬀerent magnetic properties of the Py layer. Thus,\nthe larger magnetic damping of the Py/Pt(O) ﬁlm can\nonly be attributed to the eﬃcient absorption of the spin\ncurrent at the interface. Notable is that the additional\ndamping due to the spin-current absorption disappears\nby inserting the 3.6 nm-thick Cu layer between the Py\nand Pt(O) layers. Here, the thickness of the Cu layer is\nmuchthinnerthanitsspin-diﬀusionlength( ∼500nm)22,\nallowing us to neglect the relaxation of the spin current\nin the Cu layer. This indicates that the directcontactbe-\ntween the Py and Pt(O) layersis essential for the absorp-\ntion of the spin current at the interface, or the interface\nSOC.4 0∆H (mT) \nµ\n3 6 9 12 0246\nf (GHz) (a) (b)\n8\n610 \n0\n 0HR (mT) µf (GHz) \n4\n100150 50 Py/Cu/Pt(O) Py/Pt(O)Py \nPy/Cu/Pt(O) Py/Pt(O) Py \n036\n10 010 110 2\nP (mW)  0∆H (mT) \nµ\nFIG. 3. (a) The RF current frequency fdependence of\nthe half-width at half-maximum µ0∆Hfor the Py, Py/Pt(O)\nand Py/Cu/Pt(O) samples. The solid lines are the linear ﬁt\nto the experimental data. The inset shows RF current power\nPdependence of µ0∆Hfor the Py ﬁlm at f= 7 GHz. (b)\nThe RF currentfrequency fdependenceof the resonance ﬁeld\nµ0HRfor the three samples. The solid curves are the ﬁtting\nresult using the Kittel formula.\nTABLE I. The summarized spin-absorption parameter Γ 0η\nin diﬀerent material systems. In order to directly compare\nthis work with previous works, we used International Sys-\ntem of Units. We used the magnetic permeability in vac-\nuumµ0= 4π×10−7H/m. ∆ αand Γ 0ηfor the Sn 0.02-\nBi1.08Sb0.9Te2S/Ni81Fe19is the values at T <100 K.\nHeterostructure ∆ αΓ0η[1/m2] Ref.\nBi/Ag/Ni 80Fe20 0.015 8.7 ×1018[25]\nBi2O3/Cu/Ni 80Fe20 0.0045 1.5 ×1018[26]\nSrTiO 3/LaAlO 3/Ni81Fe19 0.0013 2.3 ×1018[27]\nPt(O)/Ni 81Fe19 0.0044 2.3 ×1018This work\nα-Sn/Ag/Fe 0.022 1.2 ×1019[28]\nSn0.02-Bi1.08Sb0.9Te2S/Ni81Fe190.013 1.4 ×1019[29]\nBi2Se3/Ni81Fe19 0.0013 2.5 ×1018[30]\nTo quantitatively discuss the spin absorption at the\nPy/Pt(O)interfaceand comparewith othermaterialsys-\ntems, we calculate the spin absorption parameters. In a\nmodel of the spin pumping where the interface SOC is\ntaken into account, the additional damping constant is\nexpressed as23\n∆α=gµBΓ0\nµ0Msd/parenleftbigg1+6ηξ\n1+ξ+η\n2(1+ξ)2/parenrightbigg\n.(3)\nHere,g= 2.11 is the gfactor24,µB= 9.27×10−24Am2\nis the Bohr magneton, dis the thickness of the Py layer,\nandΓ0isthemixingconductanceattheinterface. ξisthe\nback ﬂow factor; no backﬂow refers to ξ= 0 and ξ=∞indicates that the entire spin current pumped into the\nbulk ﬂows back across the interface. ηis the parameter\nthat characterizes the interface SOC. For the Py/Pt(O)\nﬁlm,ξcan be approximated to be ∞because of the spin\npumping into the bulk of the semi-insulating Pt(O) layer\ncan be neglected. Thus, Eq. (3) can be simpliﬁed as\n∆α=6gµBΓ0η\nµ0Msd. (4)\nHere, 6Γ 0ηcorresponds to the eﬀective spin mixing\nconductance g↑↓\neﬀ. From the enhancement of magnetic\ndamping ∆ α, we obtain Γ 0η= 2.3×1018m−2for the\nPy/Pt(O) ﬁlm. We further compared this value with\nΓ0ηfor other systems where eﬃcient interface charge-\nspin conversion has been reported. As shown in Table\nI, the spin-absorption parameter Γ 0ηof the Py/Pt(O)\nﬁlm is comparable with that of the heterostructures with\nthe strong SOC, such as the 2DEG at an oxide interface,\ntopological insulators, as well as metal/oxide and metal-\nlic junctions with the Rashba SOC. This result therefore\ndemonstrates the strong SOC at the Py/Pt(O) interface.\nIV. CONCLUSIONS\nIn summary, we have investigated the spin-current ab-\nsorption and relaxation at the ferromagnetic-metal/Pt-\noxide interface. By measuring the magnetic damping for\nthe Py, Py/Pt(O)and Py/Cu/Pt(O)structures, we show\nthat the direct contact between Py and Pt(O) is essential\nfor the absorption of the spin current, or the sizable in-\nterface SOC. Furthermore, we found that the strength of\nthe spin-absorption parameter at the Py/Pt(O) interface\nis comparable to the value for intensively studied het-\nerostructureswithstrongSOC,suchas2DEGatanoxide\ninterface, topological insulators, metallic junction with\nRashba SOC. The comparable value with these material\nsystems illustrates the strong SOC at the ferromagnetic-\nmetal/Pt-oxide interface. This indicates that the oxida-\ntion of heavy metals provides a novel approach for the\ndevelopment of the energy-eﬃcient spintronics devices\nbased the SOC.\nACKNOWLEDGMENTS\nThis work was supported by JSPS KAKENHI Grant\nNumbers 26220604, 26103004, the Asahi Glass Founda-\ntion, JGC-SScholarshipFoundation, andSpintronicsRe-\nsearch Network of Japan (Spin-RNJ). H.A. is JSPS In-\nternational Research Fellow (No. P17066) and acknowl-\nedges the support from the JSPS Fellowship (Grant No.\n17F17066).\n∗ando@appi.keio.ac.jp1A. Soumyanarayanan, N. Reyren, A. Fert, and\nC. Panagopoulos, Nature (London) 539, 509 (2016).5\n2P. Gambardella and I. M. Miron, Philos. Trans. R. Soc.\nLondon A 369, 3175 (2011).\n3X.Qiu, Z.Shi, W.Fan, S.Zhou, andH.Yang,Adv.Mater.\n30, 1705699 (2018).\n4A. Brataas, A. D. Kent, and H. Ohno, Nat. Mater. 11,\n372 (2012).\n5X. Qiu, K. Narayanapillai, Y. Wu, P. Deorani, D.-H. Yang,\nW.-S.Noh, J.-H. Park, K.-J.Lee, H.-W.Lee, andH.Yang,\nNat. Nanotechnol. 10, 333 (2015).\n6Y. Hibino, T. Hirai, K. Hasegawa, T. Koyama, and\nD. Chiba, Appl. Phys. Lett. 111, 132404 (2017).\n7T. Gao, A. Qaiumzadeh, H. An, A. Musha,\nY. Kageyama, J. Shi, and K. Ando,\nPhys. Rev. Lett. 121, 017202 (2018).\n8K.-U. Demasius, T. Phung, W. Zhang, B. P. Hughes, S.-H.\nYang, A. Kellock, W. Han, A. Pushp, and S. S. Parkin,\nNat. Commun. 7, 10644 (2016).\n9H. An, Y. Kageyama, Y. Kanno, N. Enishi, and K. Ando,\nNat. Commun. 7, 13069 (2016).\n10H. An, T. Ohno, Y. Kanno, Y. Kageyama, Y. Monnai,\nH. Maki, J. Shi, and K. Ando, Sci. Adv. 4, eaar2250\n(2018).\n11S. Mizukami, Y. Ando, and T. Miyazaki, Phys. Rev. B\n66, 104413 (2002).\n12Y. Tserkovnyak, A. Brataas, and G. E. Bauer, Phys. Rev.\nB66, 224403 (2002).\n13L. Chen, F. Matsukura, and H. Ohno, Nat. Commun. 4,\n2055 (2013).\n14H. Nakayama, K. Ando, K. Harii, T. Yoshino, R. Taka-\nhashi, Y. Kajiwara, K.-i. Uchida, Y. Fujikawa, and\nE. Saitoh, Phys. Rev. B 85, 144408 (2012).\n15H. Kurt, R. Loloee, K. Eid, W. Pratt Jr, and J. Bass,\nAppl. Phys. Lett. 81, 4787 (2002).16Y. Tserkovnyak, A. Brataas, and G. E. Bauer, Phys. Rev.\nLett.88, 117601 (2002).\n17J. R. S´ anchez, L. Vila, G. Desfonds, S. Gambarelli, J. At-\ntan´ e, J. De Teresa, C. Mag´ en, and A. Fert, Nat. Commun.\n4, 2944 (2013).\n18Q. Wang, X. Lu, Q. Xin, and G. Sun, Chin. J. of Catal.\n35, 1394 (2014).\n19D.Fang, H.Kurebayashi, J. Wunderlich,K.V` yborn` y,L.P.\nZˆ arbo, R. Campion, A. Casiraghi, B. Gallagher, T. Jung-\nwirth, and A. Ferguson, Nat. Nanotechnol. 6, 413 (2011).\n20B. Heinrich, J. Cochran, and R. Hasegawa, J. Appl. Phys.\n57, 3690 (1985).\n21C. Kittel, Physical Review 73, 155 (1948).\n22H. Wang, C. Du, Y. Pu, R. Adur, P. C. Hammel, and\nF. Yang, Phys. Rev. Lett. 112, 197201 (2014).\n23K. Chen and S. Zhang, Phys. Rev. Lett. 114, 126602\n(2015).\n24J. M. Shaw, H. T. Nembach, T. J. Silva, and C. T. Boone,\nJ. Appl. Phys. 114, 243906 (2013).\n25M. Jungﬂeisch, W. Zhang, J. Sklenar, W. Jiang, J. Pear-\nson, J. Ketterson, and A. Hoﬀmann, Phys. Rev. B 93,\n224419 (2016).\n26S. Karube, K. Kondou, and Y. Otani, Appl. Phys. Express\n9, 033001 (2016).\n27E. Lesne, Y. Fu, S. Oyarzun, J. Rojas-S´ anchez, D. Vaz,\nH. Naganuma, G. Sicoli, J.-P. Attan´ e, M. Jamet,\nE. Jacquet, et al., Nat. Mater. 15, 1261 (2016).\n28J.-C. Rojas-S´ anchez, S. Oyarz´ un, Y. Fu, A. Marty,\nC. Vergnaud, S. Gambarelli, L. Vila, M. Jamet, Y. Oht-\nsubo, A. Taleb-Ibrahimi, et al., Phys. Rev. Lett. 116,\n096602 (2016).\n29A. Nomura, N. Nasaka, T. Tashiro, T. Sasagawa, and\nK. Ando, Phys. Rev. B 96, 214440 (2017).\n30P. Deorani, J. Son, K. Banerjee, N. Koirala, M. Brahlek,\nS. Oh, and H. Yang, Phys. Rev. B 90, 094403 (2014)."    },    {        "title": "2009.03196v3.Spin_pumping_in_d_wave_superconductor_ferromagnet_hybrids.pdf",        "content": " Spin pumping in d -wave superconductor/ferromagnet hybrids  \nS. J. Carreira1,*, D. Sanchez -Manzano1,2, M.-W. Yoo1, K. Seurre1, V. Rouco1, A. Sander1, J. \nSantamaría2, A. Anane1, and J. E. Villegas1 \n \n1Unité Mixte de Physique, CNRS, Thales, Université Paris -Saclay, 91767, Palaiseau, France.  \n2Grupo de Física de Materiales Complejos (GFMC). Dpto. de Física de Materiales. Facultad de \nCiencias Físicas, UCM, Plaza Ciencias, 1 28040, Madrid, Spain.  \nSpin-pumping across  ferromagnet/superconductor (F /S) interfaces has attracted  much \nattention  lately.  Yet the focus has been mainly on  s-wave superconductors -based  systems  \nwhereas (high -temperature) d -wave superconductors such as YBa 2Cu3O7-d (YBCO) have \nreceived scarce attention  despite their fundamental and technological interest .  Here we use \nwideband ferromagnetic resonance to study spin -pumping effects in bilayers that combine a \nsoft metallic Ni80Fe20 (Py) ferromagnet  and YBCO. We evaluate  the spin conductance in YBCO  \nby analyzing the magnetization dynamics in Py. We find that the Gilbert damping exhibits a \ndrastic drop  as the heterostructures are cooled across  the normal -superconducting transition and \nthen, depending on the S/F interface morphology , either stays co nstant or shows a strong upturn . \nThis unique  behavior is explained considering quasiparticle density of states  at the  YBCO  \nsurface , and is a direct consequence of zero-gap nodes for particular directions in the \nmomentum space. Besides showing  the fingerprint of d -wave superconductivity in spin -\npumping, our result s demonstrate the potential of high -temperature superconductors for fine \ntuning of the magnetization dynamics in ferromagnets  using k -space degrees of freedom of d-\nwave/F interface s. \n \n*Corresponding author: santiago.carreira@cnrs -thales.fr  \n  Introduction  \nSpin injection into superconductors constitutes a very active research topic within the \nnascent field of superconducting spintronics , aiming at expanding spintronic functionalities by \nexploiting the dissipationless electron transport and quantum coherence characteristic of \nsuperconductivity   [1–5]. \nTheory and experiments have shown that spin currents can flow into s -wave \nsuperconductors carried by equal -spin triplet Cooper pairs   [1,2,6 –9] or by superconducting \nquasiparticles   [10,11] , whose lifetime can exceed those of spin -polarized electrons in the \nnormal state   [12–16]. Spin-polarized quasiparticles can be  efficiently injected into the \nsuperconductor ( S) using an adjacent ferromagnet (F) by applying across the S/F interface a \nbias voltage that exceeds the superconducting gap   [10,17] . This mechanism has been \nextensively explored in transport experiments with spin valves   [13,18 –21]. Another \nmechanism for inducing a non -equilibrium spin accumulation in superconductors is spin -\npumping   [22] using  the resonant excitation  of the ferromagnet’s magnetization   [23,24]  as \nsource of pure spin current . In these ferromagnetic resonance (FMR) experiments, the \nsupercon ductor‘s efficiency as a spin -sink is evaluated via spin hall effect   [25] or microwave \nabsorption measurements   [8,25 –29], by monitoring the evolution of the resonant peak ’s \nlinewidth across the superconducting transition. The assumption is that the changes of the \nmagnetic damping (which lead to a narrow ing/broadening of the resonance linewidth   [23,24] ) \nreflect variations in the spin relaxation rate  when the superconducting gap opens, because this \nalters both the spin trans mission  across the superconductor/ferromagnet interfac e and the \nrelaxation mechanisms  within the superconductor . Pioneering experiments performed on \nNi80Fe20/Nb (Py/Nb) bilayers have found that the opening of the superconducting gap induces \nan abrupt  FMR linewidth narrowing when temperature is swept across the superconducting \ntransition   [26]. This was explained by considering that the opening of the superconducting gap leads to a vanishing density of states at the Fermi level, thereby hindering the  transmission  of \nspin polarized electrons  across the interface . More recent work on  GdN (F) / NbN ( S) \nmultilayers has found a different behavior, in which the Gilbert damping initially peaks across \nthe superconducting transition, and diminishes below the normal -state value upon further \ntemperature decrease   [30]. That behavior was associated to the presence of spin -orbit scattering \nat the interface   [31]. In contrast to the two examples mentioned above, studies carried out on \nPy/Nb multilayers with an adjacent stron g spin -orbit coupling metal (Pt) found a steady \nbroadening of the linewidth below T C, which was interpreted in terms of enhanced spin \ntransport across the superconductor due to the generation of equal -spin triplet \nsuperconductivity   [7,8] . Adding a new piece to the puzzle, a recent theory  shows that , if the  \nsuperconducting gap is suppressed near  the S/F interface, the presence of quasiparticle surface \nstates can also produce a n enhancement of spin transport in to the superconductor below \nTC  [32]. The strikingly wide variety of observed behaviors illustrate s the complexity of the \nunderlying physics, the importance of  the interfacial properties , and the fact that the conditions \nfor predominance  and interplay of  the different proposed scenarios  (quasiparticles and triplet \nsuperconductivity) is far from being fu lly understood . Beyond raising the se fundamental \nquestions, it is interesting  that the experimental investigations have evidenced that \nsuperconductivity may be exploited for tuning magnetization dynamics.  \nThe experiments discussed so far are based on conventional (low -Tc) s-wave \nsuperconductors, which present an isotropic superconducting gap.  In contrast, in \nunconventional  (high -Tc) d-wave ones the gap is suppressed along particular directions in the \nmomentum space, and there exists a  superconducting -phase shift between d -wave lobes   [33–\n35]. While spin diffusion effects in d -wave superconductors have been discussed in the c ontext \nof electrical measurements   [36–41], to our knowledge spin -pumping and the effects of the \nonset of superconducting pairing on the spin -sink behavior of d-wave cuprates remain unexplored. Notice that, at variance to s -wave superconductors, the presence of zero -gap nod es \nmay provide channels for injection of spin -polarized electrons , even in the superconducting \nstate. Consequently, the effects of superconductivity on spin -pumping and magnetization \ndynamics are expectedly  different  in the case  of s-wave superconductors. Here we \nexperimentally investigate  this issue using c-axis YBCO/Py heterostructures with different \ninterface structure.  In all cases, we observe an abrupt linewidth narrowing across the \nsuperconducting transition, similar to that observed in Py/Nb s -wave system  [26], which \nsuggests that , right below the critical temperature,  the opening of the d -wave gap  significantly \nsuppress spin -pumping. However, upon further temperature decrease , the behavior of the \nlinewidth depends  on the YBCO surface morphology. For the smoother YBCO  films , we \nobserve no further evolution of the linewidth. However,  in the presence of a faceted YBCO \nsurface s, the linewidth  monotonically widens as the temperature is decreased  below Tc. This \nbehavior  can be explained considering the  interfacial density o f quasiparticle state s, which \ndepend s on the YBCO surface morphology due to the anisotropic character of d -wave \nsuperconductivity . These results thereby provide a fingerprint of d-wave superconductivity in \nin the physics of spin-pumping . At the same time, they underline the need of a theor etical  \nframework  that specifically addresses the role of the mechanisms at pl ay (quasiparticle  density \nof states   [32,42] , changes in the spin -imbalance relaxation  [43] and dynamic generation of \ntriplet pairs   [44,45] ) in the context of d -wave superconductivity.  Finally,  this work  \ndemonstrate s the potential of high -temperature superconductors for manipulating the \nmagnetization dy namics of metallic ferromagnets, in a way that could be engineered by \nchoosing the orientation of the d -wave/F interface . \nExperimental  \nWe have studied different multilayers, namely c -axis YBa 2Cu3O7 (30 nm)/Ni 80Fe20 (15 nm)/Al \n(3 nm) grown on (001) SrTiO 3 (one sample) and on (001) NdGaO 3 (two samples)  − respectively  referred to as STO// S/F, NGO/ /S/F #1 and NGO//S/F #2  − and YBa 2Cu3O7 (30 nm)/Au (5 \nnm)/Ni 80Fe20 (15 nm)/Al (3 nm) on STO  −referred to as STO// S/Au/F. The YBCO films were \ngrown by pulsed laser deposition (PLD ) using an excimer laser ( = 305 nm) at a temperature \nof 700 °C and oxygen pressure of 0.36 mbar. Optimum oxygenation was ensured by raising the \nO2 pressure to 760 mbar during cooldown.  Where applicable, the Au interlayer (aimed at \npreventing and assessing the impact of eventual redox reactions between YBCO and Py) was \nsubsequently grown in -situ by PLD, at room temperature and in pure Ar atmosphere. Under \nthese growth condition s, the onset of the superconducting transition  determined  by resistivity \nmeasurements is typically around T c ~ 85 K, regardless of the substrate and presence of an Au \ninterlayer . \n \n   \nFIG. 1. AFM images measured on a 5x5 m2 area of a YBCO thin film grown on (a) STO (001) and (b) NGO \n(001). The height profile shown in (c) was measured along the oblique line 3 m long indicated in (a) and (b) \nrespectively . \n \nThe structural properties of the as-grown YBCO films w ere studied by high-angle  X-\nray diffraction , whic h confirmed c-axis (001) epitaxial growth on both substrates STO and \nNGO , as well as the absence of parasitic phases  (see Fig. S1 in Supplemental  Material).   \nHowever , we found that the YBCO ’s surface morphology is different  depending of the \nsubstrate.  Atomic Force Microscopy (AFM) images displayed in Fig. 1 show that YBCO on \nSTO Fig. 1 (a) present s a relatively  smooth surface  (rms roughness  ~ 2 nm) ,  while YBCO on \n(a) (b) \n(c) NGO  [Fig. 1 (b)] presents a high density of ~ 50 nm tall crystallites [see profile in Fig. 1 (c)] \non top of an otherwise similar background topography . The Py layer  and Al capping (aimed at \npreventing Py surface oxidatio n) were subsequently deposited on the YBCO ex-situ, using rf -\nsputtering in pure Ar atmosphere  at room temperature, without breaking vacuum between each \nlayer deposition.  Control s amples consisting of single Py films grown on both SrTiO 3 and \nNdGaO 3 (labeled  as STO//F, STO/ /Au/F , NGO//F  #1 and NGO//F #2 ) were studied. The \nsamples’ size is in all cases 5 5 mm2.  \n \nFIG. 2. Sketch of the multilayer structure and experimental  \ngeometry for the FMR experiments.  \n \nThe experimental geometry considered for the FMR experiments is sketched in Fig. 2. A DC \nmagnetic field H is applied parallel to the sample plane in order to saturate the magnetization \nof the Py , whose  precession  is excited  by applying and  a radiofrequency  (RF)  magnetic field  \nhRF perpendicular to the DC field, using a coplanar waveguide. A magnetic field modulation at \nlow frequency ( < 2 kHz) is used to measure the derivative of the absorbed power dP/dH with \nrespect to the DC magnetic field  H, as this is swept  around the resonance field H res where  the \ndynamical susceptibility peak s. A typical measurement is shown in Fig. 3 (a). This type of \nmeasurements were done for a number of fixed frequencies in the range 4 GHz  f  40 GHz.  \nFor each fr equency, the peak -to-peak linewidth ΔH pp and the resonance field H res were  \n \n \n \n \nFIG. 3. Typical (a) FMR absorption spectrum and fit, (b) f vs 0Hres and (c) 0Hpp vs f obtained for the sample \nSTO//S/Au/F at 30 K. The fits in (b) and (c) follows the FMR equations (1) and (2).  \n \ndetermined by fitting the dP/dH vs. the applied field H to the derivative of a Lorentzian function, \nas is shown in the example of Fig. 3 (a) . This allows extracting the values of the resonance field \nHres and linewidth ΔH pp versus the frequency, which are shown in Fig s. 3 (b) and (c)  for the \nexample in (a) .  The relationship  between the resonant microwave frequency f and field Hres is \ngiven by the Kittel formula  [46] which, neglecting the small magnetic anisotropy of Py,  is \n𝒇=𝜸𝝁𝟎√𝑯𝒓𝒆𝒔(𝑯𝒓𝒆𝒔+𝑴𝒆𝒇𝒇)    (Eq. 1)  \nwhere  is the gyromagnetic factor and 𝑀𝑒𝑓𝑓 is the effective  magnetization.  The linewidth is \nwell described by the linear expression  [24], \n𝝁𝟎∆𝑯𝒑𝒑=𝟐𝜶𝒇\n√𝟑𝜸+𝝁𝟎∆𝑯𝟎     (Eq. 2)  \nwhere 0H0 is the frequency -independent contribution o r inhomogeneous broadening and  \nis the Gilbert damping factor. Similarly as in the example shown in Fig. 3, the data for all the \nstudied samples is well described by Eq. 1 and Eq. 2. This allowed us to obtain  the te mperature \ndependent   and 0H0 for the series of samples, with error bars calculated from the linear \nregression  of the fits . Notice that, b ased on the linear behavior observed in 0Hpp vs. f for a \n(a) (b) (c) broadband frequency range  in all of the studied samples , we consider that the 2 -magnon \nscattering can be ruled out as a dominant relaxation mechanism [47] in all of them .  \n \n \nResults  \nFig. 4 (a) shows , as an example, a typical  series of the temperature -dependent FMR \nlinewidth  0ΔH pp measured f or different  frequencies , which corresponds to a NGO//F/S sample . \nThe background trend −a steady linewidth broadening  with decreasing temperature , with a drop \nbelow  ~ 20 K  for the measuremen ts at highest frequencies − is similar to that of the NGO//F \nreference samples (see Fig. S2(a) in Supp lemental  Materials ) and to the behavior  observed in  \nearlier  FMR experiments on  single  Py thin films  [47–50]. On top of that background , we \nobserve another feature, a “kink” around T ~ 85 K, which is not present in the reference samples \nand, as discussed below, is related to superconductivity. However, the fact that 0ΔH pp  results  \nfrom the addition of the (frequency independent) inhomogeneous  broadening  and the \n(frequency dependent)  magnetic damping , makes such feature evident  only for f  > 18 GHz . \nFIG. 4. ( a) Temperature dependence of the FMR linewidth, 0Hpp, measured at all frequencies from 4 \nGHz to 40  GHz in steps of 2 GHz for the sample NGO//S/F  #2. (b) 0Hpp - 0H0 vs f for the sample NGO//S/F  \n#2 obtained at temperatures just above (88 K) and below (83 K) the superconducting critical temperature of the \nYBCO. The straight lines correspond to linear fits of the data points.  (a) (b) This feature  indeed corresponds to a drop of the damping factor   across the superconducting \ntransition, as evidenced in Fig. 4 (b) where  the linewidth (after subtraction of the frequency -\nindependent broadening 0H0) is plotted as a function of the frequency. One can see that the \ndamping (slope of the straight  lines) is different above ( 88 K) and below ( 83 K) the \nsuperconducting transition of YBCO.  \nThe above example makes it evident that broadband measurement s are crucial to finely \nquantify the linewidth changes across the superconducting transition, and to univocally  ascribe \nthem to a variation of the damping factor. Thus, in what follows, we will compare samples \nbased o n the temperature dependence of the damping coefficien t (T), which can be obtained together with the temperature dependent inhomogeneous broadening 0H0(T) by applying the \nanalysis described above to a series of 0Hpp vs f measured at different temperatures.   \n \nFIG. 5. Temperature dependence of the [(a) and (b) ] magnetic damping  and [(c) and (d) ] inhomogeneous \nbroadening 0H0 for the samples STO// S/Au/F and STO//Au/F [(a) and (c)]  and NGO// S/F and NGO//F [(b) and \n(d)]. In (b) and (d) we plot the results obtained for two samples with the same nominal composition , #1 (filled \nsymbols) and #2 (open symbols). Data in circles corresponds to the samples with YBCO as a bottom layer and the \ncontrol samples without YBCO are denoted with triangles. The inset in (a) shows  vs T for the sample STO// S/F. \nThe d ash lines are g uides to the eye.  \n \nIn Fig. 5 (a) we show (T) for superconducting multilayers  STO//S /Au/F (red circles, \nmain panel) and STO//S /F (inset) , together with the data (black triangles ) for a single Py film \n(sample STO/ /Au/F) used as reference. One can see that, when Py is combined with the \nsuperconductor, and regardless of the presence of an Au interlayer , (T) drops by ~ 10-15% \nbetween 90 K and 70 K. Upon further temperature decrease (T) stays nearly constant. That \nis, 𝛼 drops across the superconducting  transition , and remains constant thereafter. This \n(a) (b) \n(c) (d) contrast s with the behavior of the STO/ /Au/F  sample used as reference (black dots), which \nshows no clear change of  around that temperature range . Notice also  that the damping  level \n ~ 4.5 10-3 in the temperature range in which  the YBCO is in the normal state (T > 90 K)  is \ncomparable  for the superconducting (STO//S/Au/F) and reference (STO//Au/F ) samples . Fig. 5 \n(c) shows that 0H0(T) behaves very similar ly in the superconducting and reference sample s. \nThis implies  that the presence of the YBCO does not create additional  magnetic  \ninhomogeneities  in Py , and unambiguously demonstrates a decrease of the Gilbert damping \nacross the  superconducting  transition . This effect  can also be observed in the NGO //S/F #1 and \n#2 bilayer s [see Fig. 5 (b)] for which (T) shows a ~ 10% drop across the superconducting \ntransition (red circles ) not observed in the reference NGO//F sample (black triangles ). As was \npointed out for  the STO substrate, the inhomogeneous broadening is not significantly affected \nby the presence of the YBCO layer, see Fig. 5 (d). However , there are two main differences \nwhen comparing samples grown on STO and on NGO.  First,  for NGO//S/F the  damping  level \n ~ 6.5 10-3 in the normal -state (T > 90 K) is significantly higher than for the reference sample \nNGO//F  [Fig. 5 (d)]. Second,  for NGO//S/F the magnetic dumping (T) does  not remain \nconstant  below the superconducting transition,  but show s instead  an upturn with decreasing \ntemperature .  \nDiscussion  \nThe central observation  is that the magnetic damping (T) of  Py in  YBCO/Py \nheterostructures drops across the YBCO superconducting transition  and that, upon further \ntemperature decrease , (T) either  stays constant or shows an upturn depending  on the substrate  \n(STO or NGO)  on which the heterostructures  are grown.  The initial drop across the transition  \nis reminiscent  of that observed  in earlier experiments with s -wave superconductors  [26], which \nwas explained based on the idea that , as the superconducting gap in the electronic density of \nstates opens  [48], the decrease of  electrons states at the Fermi level  impedes  spin injection.  Such blocking effect strengthens as temperature is lowered  further from T C, because this makes  the \nsuperconducting gap widen  and the quasiparticle population diminish  [48]. While such effect  \nis consistent with the behavior of (T) for  heterostructures grown on STO, it can not fully \naccount for the behavior of the samples grown on NGO:  these  show an upturn of the damping \nfactor , which at low temperature reaches values higher than those observed above TC [Fig. 5 \n(b)]. A similar  enhancement of spin -pumping in the superconducting phase was observed in \nS/F interfaces  [7,8]  in the presence of a heavy metal (Pt) , and was explained by the generation \nof equal -spin triplet  pairs. However,  in the present experiments we have no arguments no r \nevidence to  support such scenario. Instead, we have considered a different  situation recently \nstudied theoretically  [32], in which an enhancement of spin -pumping in the superconducting \nphase is explained  the presence of a quasiparticle states (Andreev bound states ) at the interface \nwith the F . In Ref. [32] s-wave superconductors were considered, for which the emergence of \nAndreev bound states stems from  the interfacial suppression of the superconducting gap due to \ninverse proximity effect . However, i n the case of  d-wave superconductors  quasiparticle  \n(Andreev ) surface bound states appear intrinsically , due to  the existence of zero-gap nodes \nalong particular k-space direction s [49]. As we detail below, the quasiparticle density depends \non the interface orientation.  This provides a possible  scenario to  explain  the distinct behaviors  \nof samples grown in STO and NGO  based on their different surface topography.  \nFollwing  [32], the s pin-pumping into the S depends  on the surface  density of \nquasiparticle sta tes: the larger the density of state s, the larger the spin injection  efficiency . \nExtending the full calculations  existing for  s-wave superconductros  [32] to the case of d -wave \nis out of the  present work’s scope . However , a qualitative explanation for experime ntal results  \nis at reach b y considering  the density of quasiparticle states  at d-wave/normal metal interface s \nwith finite tra nsparency . Following  [50], the normalized density of quasiparticle states is:  \n𝜌𝑆0𝜌𝑁⁄ (𝐸)= 1−(𝜎𝑁−1)2|Γ+Γ−|2\n|1+(𝜎𝑁−1)Γ+Γ−exp  (𝑖𝜙−−𝑖𝜙+)|2     (Eq. 3) where N is the normal -state electron density of states,  𝜎𝑁= 1\n1+𝑍2  with Z  the barrier strength \nat the interface,  Γ±=𝐸− √𝐸2−|Δ±|2\n|Δ±|  with E t he quasiparticle energy with respect to the Fermi \nlevel, and 𝜙+(respectively 𝜙−) is the effective phase of the anisotropic pair potential Δ+(Δ−). \nTemperature effects in the quasiparticle population can be taken into account by considering \nthe gap amplitude  Δ(𝑇)= Δ0tanh  (𝑏√𝑇𝐶\n𝑇−1) and by  convoluting  𝜌𝑆0𝜌𝑁⁄ (𝐸) with the \nderivative of the Fermi -Dirac distribution 𝑓𝐹𝐷(𝐸,𝑇)  [51], \n𝜌𝑆/𝜌𝑁 (𝐸,𝑇)= ∫𝜌𝑆0𝜌𝑁⁄ (𝐸′)𝜕𝑓𝐹𝐷\n𝜕𝐸(𝐸−𝐸′,𝑇) 𝑑𝐸′                  (Eq. 4) \nCalculations  of the normalized density of states  𝜌𝑆/𝜌𝑁 (𝐸,𝑇) for interfaces facing a d-\nwave gap  lobe (g = 0) and facing a gap node (g = /4) are shown in Fig. 6 (a) and ( c) \nrespectively , considering a moderate interface transparency Z = 2.5  (Fig. S 3 of the \nSupplemental  Materials demonstrates that, except for very transparent interfaces Z<1, the \neffects discussed thereafter are qualitatively similar for any Z) . The different  behavior s in Fig. \n6(a) and 6( c) result from  the anisotropic nature of the density of states at the YBCO  surface . \nFor a g = 0 surface, we observe  at low energies (𝐸<Δ ) that the opening of the \nsuperconducting gap leads to a fast reduction  of the density of states  upon decreasing \ntemperature, similarly as in s -wave superconductors . On the contrary , for the g = /4 case [ Fig \n6 (c)] we observe the emergence  of Andreev bound states around 𝐸=0, whose population \ngradually increases  upon  decreasing temperature , leading to a peak in the density of state s. In \nour experiments , the microwave energy  ℏ𝑓≪∆, and thus the relevant quantity is  the density \nof states near the Fermi level ( 𝐸~0) [32]. This is shown in Fig. 6 (b) for the two cases g =  \nand g = /4 .   \nFIG. 6. Calculated density of states for an interface (a) facing a d -wave gap lobe g = 0 and (b) facing at d -wave \ngap node g = /4 for different temperatures. The sketches in (a) illustrate the possible directions for the spin \ninjection according to the surface morphology . In (b) we show the temperature depende nce of the density of states \nfor quasiparticles injected along the g = 0 and g = /4 directions and in (d) we plot  the resulting density of states \nwhen 10% / 90% contributions  of the g = 0 and g = /4 are considered for the spin injection.  \n \nBased on th e above , and considering the  different topography of the STO and NGO \nsamples, a  possible interpretation for the different α(T) emerges. As sketched  in the inset of \nFig. 6 (a), in the case of STO the effects along  the out of plane direction  dominate , because of \nthe smoother S/F interface . In this situation, the density of quasiparticle surface states is  as in \nFig. 6 (a) [52] and, as was observed for s-wave supercoductors  [26], we expect that (T) decays  \nacross the superconducting transition , in agreement with our experimental findings  [Fig.  5 (a)]. \nHowever, for samples grown on NGO the presence of crystallites at the surface  allow s spin \npumping into the YBCO  basal (ab) plane [ sketch in the inset of Fig. 6 (a)], which provide s \naccess to a large r density of zero-energy quasiparticle states. If we consider that this results in  \n(a) (b) \n(c) (d) an effective density of state s in which the contribution of directions presenting a large density \nof Andreev bound states weigth s 10%, the calculated ρS/ρN (E,T)  [Fig 6 (d)] qualitatively \nreproduce  the behavior of (T) in the experiments [red in Fig. 5 (b) ]:  an abrup t drop across the \ntransition, followed by an upturn upon further tem perature decrease.  A 10% weight of \ndirections with large zero -energy quasiparticle density is reaso nable for the samples  grown  on \nNGO  considering  the lateral area of the crystallites , which  can be estimated from the AFM \nimages.  As discussed in the Sup plemental Material , the ratio between the lateral  surface area \n(normal to the  ab plane ) and the horizontal one ( normal to the c -axis) is between 1 % and 1.7 \n% depending on the criterion used for the estimate . Their contribution nee ds to be corrected d ue \nthe large electronic anisotropy of YBCO, because the conduc tivity  in the basal ( ab) plane  is up \nto 10 times larger than along the c -axis [53–55]. Thus, the  90%/10% contribution that allows \nreproduc ing the experimetna l results  seem  reasonable.  We stress nevertheles  that the discussed \nmodel aims a provinding a qualitative explanation of the observed behavior, and that th e \nnumerical estimates are made just to verify that the size of the effects are of the right order of \nmagnitude.    \nConsistenly  with the  scenario  discussed above , we observe that for the NGO //S/F  \nsamples  the normal -state damping  is signi ficantly higher than for the reference sample  (see F ig. \n5 (b) for T   90 K ), as Py contacts  the YBCO  not only on the c-axis surface but also on the \nmore conducting basal (ab) plane. This result s in a higher interfacial conductance  than for the \nfilm grown on STO, which enhances the spin absorption and therefore the overall damping.  \nA final word concerning the impact of the Au interlayer. When the Au layer is deposited \non YBCO, w e observe  no major effect on (T), which indicates that its pres ence does not \nsignificantly change the interface transparency and is consistent with the fact the spin the \ndiffusion length of Au (≈ 50 nm at 10 K)   [56] is larger than the Au layer  thickness.  In the \ncontrol (non -superconducting) samples, the presence of an Au interlayer between Py and the insulati ng substrate enhances the magnetic d amping, which reflects that Au is a more efficient \nspin-sink than the substrate.  \nIn summary, we have found that in d-wave superconductor /ferromagnet  YBCO/Py \nmultilayers , the opening of the superconducting gap  reduces  the spin-sinking  efficiency  and \nresults in a significant  drop of the magnetic damping across  the superconducting trans ition. \nHowever, upon further temperature decrease different behavior s are observed (either a plateau \nor an upturn ), which can be associated  with the YBCO’s surface morphology . In particular, the  \nlow-temperature upturn can be explained by  the large density  of quasiparticle bound s tates \ncharacteristic of d -wave superconductivity . Our hypothesis  is that those  states  are accessible \nvia YBCO crystallites at the surface , that directly expose the YBCO ab plane to the interface  \nwith t he ferromagnet .  This suggest s that spin-pumping into quasipartic le bound states could be \nfurther enhanced b y engineering the YBCO surface, for example by growing YBCO in different \ncrystallographic directions, or by creating vicinal surfaces . This, together with further \ntheoretical developments -for instance an extension of Ref .  [32] to the case of d -wave \nsuperconductor s, possibly including  other ingredients  such as changes in the spin relaxation \ntime in the superconducting state  along different crystallographic directions   [10,17] - would \nallow a more accurate quantitative analysis that would  underpin  the proposed scenario.  \n \nAcknowledgements  \nWork supported by ERC grant N° 647100 “SUSPINTRONICS”, French ANR grant ANR -15-\nCE24 -0008 -01 “SUPERTRONICS” , COST action “Nanocohybri”  and EU H2020 Marie Curie \nActions Nº 890964 “SUPERMAGNONICS” . \n \nReferences  \n[1] F. S. Bergeret, A. F. Volkov, and K. B. Efetov, Phys. Rev. Lett. 86, 4096 (2001).  [2] M. Eschrig, Phys. Today 64, 43 (2011).  \n[3] M. G. Blamire and J. W. A. Robinson, J. Phys. Condens. Matter 26, 453201 (2014).  \n[4] J. Linder and J. W. A. Robinson, Nat. Phys . 11, 307 (2015).  \n[5] M. Eschrig, Rep. Prog. Phys. 78, 104501 (2015).  \n[6] A. Kadigrobov, R. I. Shekhter, and M. Jonson, Eur. Lett 54, 394 (2001).  \n[7] K.-R. Jeon, C. Ciccarelli, A. J. Ferguson, H. Kurebayashi, L. F. Cohen, X. Montiel, M. \nEschrig, J. W. A. R obinson, and M. G. Blamire, Nat. Mater. 17, 499 (2018).  \n[8] K.-R. Jeon, C. Ciccarelli, H. Kurebayashi, L. F. Cohen, X. Montiel, M. Eschrig, S. \nKomori, J. W. A. Robinson, and M. G. Blamire, Phys. Rev. B 99, 024507 (2019).  \n[9] F. S. Bergeret, A. F. Volkov, a nd K. B. Efetov, Rev. Mod. Phys. 77, 1321 (2005).  \n[10] S. Takahashi, H. Imamura, and S. Maekawa, Phys. Rev. Lett. 82, 3911 (1999).  \n[11] S. Takahashi and S. Maekawa, Phys. Rev. B 67, 052409 (2003).  \n[12] N. Poli, J. Morten, M. Urech, A. Brataas, D. Haviland, and V. Korenivski, Phys. Rev. \nLett. 100, 136601 (2008).  \n[13] H. Yang, S. -H. Yang, S. Takahashi, S. Maekawa, and S. S. P. Parkin, Nat. Mater. 9, \n586 (2010).  \n[14] H. L. Zhao and S. Hershfield, Phys. Rev.  B 52, 3632 (1995).  \n[15] S. A. Kivelson and D. S. Rokhsar, Phys. Rev. B 41, 11693(R) (1990).  \n[16] C. H. L. Quay, D. Chevallier, C. Bena, and M. Aprili, Nat. Phys. 9, 84 (2013).  \n[17] M. Johnson, Appl. Phys. Lett. 65, 1460 (1994).  \n[18] A. K. Suszka, F. S. Be rgeret, and A. Berger, Phys. Rev. B 85, 24529 (2012).  \n[19] T. Wakamura, N. Hasegawa, K. Ohnishi, Y. Niimi, and Y. Otani, Phys. Rev. Lett. 112, \n1 (2014).  \n[20] K. Ohnishi, Y. Ono, T. Nomura, and T. Kimura, Sci. Rep. 4, 6260 (2014).  \n[21] J. Y. Gu, J. A. Caballero, R. D. Slater, R. Loloee, and W. P. Pratt, Phys. Rev. B 66, 140507 (2002).  \n[22] S. Takahashi, in edited by Y. Xu, D. D. Awschalom, and J. Nitta (Springer \nNetherlands, Dordrecht, 2016), pp. 1445 –1480.  \n[23] E. Landau, Lev Davidovich  Lifshitz, Phys. Z. Sowjet. 8, 153 (1935).  \n[24] T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004).  \n[25] K.-R. Jeon, C. Ciccarelli, H. Kurebayashi, J. Wunderlich, L. F. Cohen, S. Komori, J. \nW. A. Robinson, and M. G. Blamire, Phys. Rev. Appl. 10, 014029 (2018 ). \n[26] C. Bell, S. Milikisyants, M. Huber, and J. Aarts, Phys. Rev. Lett. 100, 47002 (2008).  \n[27] K.-R. Jeon, C. Ciccarelli, H. Kurebayashi, L. F. Cohen, S. Komori, J. W. A. Robinson, \nand M. G. Blamire, Phys. Rev. B 99, 144503 (2019).  \n[28] K.-R. Jeon, C. Ciccarelli, H. Kurebayashi, L. F. Cohen, X. Montiel, M. Eschrig, T. \nWagner, S. Komori, A. Srivastava, J. W. A. Robinson, and M. G. Blamire, Phys. Rev. \nAppl. 11, 014061 (2019).  \n[29] M. Müller, L. Liensberger, L. Flacke, H. Huebl, A. Kamra, W. Belzig, R. Gross, M. \nWeiler, and M. Althammer, Phys. Rev. Lett. 126, 087201 (2021).  \n[30] Y. Yao, Q. Song, Y. Takamura, J. P. Cascales, W. Yuan, Y. Ma, Y. Yun, X. C. Xie, J. \nS. Moodera, and W. Han , Phys. Rev. B 97, 224414 (2018).  \n[31] M. Inoue, M. Ichioka, and H. Adachi, Phys. Rev. B 96, 24414 (2017).  \n[32] M. A. Silaev, Phys. Rev. B 102, 180502 (2020).  \n[33] Z.-X. Shen, D. S. Dessau, B. O. Wells, D. M. King, W. E. Spicer, A. J. Arko, D. \nMarshall, L.  W. Lombardo, A. Kapitulnik, P. Dickinson, S. Doniach, J. DiCarlo, T. \nLoeser, and C. H. Park, Phys. Rev. Lett. 70, 1553 (1993).  \n[34] H. Aubin, K. Behnia, M. Ribault, R. Gagnon, and L. Taillefer, Phys. Rev. Lett. 78, \n2624 (1997).  \n[35] J. R. Kirtley, C. C. T suei, A. Ariando, C. J. M. Verwijs, S. Harkema, and H. Hilgenkamp, Nat. Phys. 2, 190 (2005).  \n[36] V. Pe??a, Z. Sefrioui, D. Arias, C. Leon, J. Santamaria, J. L. Martinez, S. G. E. Te \nVelthuis, and A. Hoffmann, Phys. Rev. Lett. 94, 3 (2005).  \n[37] C. Visani,  Z. Sefrioui, J. Tornos, C. Leon, J. Briatico, M. Bibes,  a. Barthélémy, J. \nSantamaría, and J. E. Villegas, Nat. Phys. 8, 539 (2012).  \n[38] Z. Sefrioui, D. Arias, V. Peña, J. E. Villegas, M. Varela, P. Prieto, C. León, J. L. \nMartinez, and J. Santamaria, Phy s. Rev. B 67, 214511 (2003).  \n[39] Z. P. Niu and D. Y. Xing, Phys. Rev. Lett. 98, 57005 (2007).  \n[40] S. Soltan, J. Albrecht, and H. -U. Habermeier, Phys. Rev. B 70, 144517 (2004).  \n[41] N.-C. Yeh, R. Vasquez, C. Fu,  a. Samoilov, Y. Li, and K. Vakili, Phys. R ev. B 60, \n10522 (1999).  \n[42] T. T. Heikkilä, M. Silaev, P. Virtanen, and F. S. Bergeret, Prog. Surf. Sci. 94, 100540 \n(2019).  \n[43] F. S. Bergeret, M. Silaev, P. Virtanen, and T. T. Heikkilä, Rev. Mod. Phys. 90, 41001 \n(2018).  \n[44] M. Houzet, Phys. Rev. Lett.  101, 1 (2008).  \n[45] T. Yokoyama and Y. Tserkovnyak, Phys. Rev. B 80, 1 (2009).  \n[46] C. Kittel, Phys. Rev. 73, 155 (1948).  \n[47] W. K. Peria, T. A. Peterson, A. P. McFadden, T. Qu, C. Liu, C. J. Palmstrøm, and P. A. \nCrowell, Phys. Rev. B 101, 134430 (2020).  \n[48] M. Tinkham, Introduction to Superconductivity / Michael Tinkham  (McGraw -Hill, \nNew York, 1975).  \n[49] S. Kashiwaya and Y. Tanaka, Reports Prog. Phys. 63, 1641 (2000).  \n[50] Y. Tanaka and S. Kashiwaya, Phys. Rev. B. Condens. Matter 53, 9371 (1996).  \n[51] W. K. Park, L. H. Greene, J. L. Sarrao, and J. D. Thompson, Phys. Rev. B 72, 52509 (2005).  \n[52] J. Y. T. Wei, N. C. Yeh, D. F. Garrigus, and M. Strasik, Phys. Rev. Lett. 81, 2542 \n(1998).  \n[53] M. Tinkham, Introduction to Superconductivity  (2004).  \n[54] P. H.  Kes, J. Aarts, V. M. Vinokur, and C. J. van der Beek, Phys. Rev. Lett. 64, 1063 \n(1990).  \n[55] G. Blatter, V. B. Geshkenbein, and A. I. Larkin, Phys. Rev. Lett. 68, 875 (1992).  \n[56] M. Isasa, E. Villamor, L. E. Hueso, M. Gradhand, and F. Casanova, Phys. Rev. B 91, \n24402 (2015).  \n "    },    {        "title": "1401.5463v1.Coexistence_of_Superconductivity_and_Ferromagnetism_in_P_doped_EuFe2As2.pdf",        "content": "arXiv:1401.5463v1  [cond-mat.supr-con]  22 Jan 2014CoexistenceofSuperconductivity andFerromagnetism inP- doped EuFe 2As2\nS. Nandi,1,2,∗W. T. Jin,1,2Y. Xiao,1Y. Su,2S. Price,1D. K. Shukla,3,4\nJ. Strempfer,3H. S. Jeevan,5P. Gegenwart,5and Th. Brückel1,2\n1Jülich Centre for Neutron Science JCNS and Peter Grünberg In stitut PGI,\nJARA-FIT, Forschungszentrum Jülich GmbH, D-52425 Jülich, Germany\n2Jülich Centre for Neutron Science JCNS, Forschungszentrum Jülich GmbH,\nOutstation at MLZ, Lichtenbergstraße 1, D-85747 Garching, Germany\n3Deutsches Elektronen-Synchrotron DESY, D-22607 Hamburg, Germany\n4UGC DAE Consortium for Scientiﬁc Research, Khandwa Road, In dore 01, India\n5I. Physikalisches Institut, Georg-August-Universität Gö ttingen, D-37077 Göttingen, Germany\nThemagneticstructureoftheEu2+momentsinthesuperconducting EuFe 2(As1−xPx)2samplewith x=0.15\nhas been determined using element speciﬁc x-ray resonant ma gnetic scattering. Combining magnetic, ther-\nmodynamic and scattering measurements, we conclude that th e long range ferromagnetic order of the Eu2+\nmoments alignedprimarilyalongthe caxiscoexists withthe bulksuperconductivity atzeroﬁeld. Atanapplied\nmagnetic ﬁeld ≥0.6T, superconductivity stillcoexists withthe ferromagneti c Eu2+moments whichare polar-\nized along the ﬁeld direction. We propose a spontaneous vort ex state for the coexistence of superconductivity\nandferromagnetism inEuFe 2(As0.85P0.15)2.\nPACS numbers: 74.70.Xa, 75.25.-j, 75.40.Cx, 74.25.Dw\nI. INTRODUCTION\nThe discovery of the iron-based superconductors[ 1] a few\nyearsagohasstimulatedtremendousresearchinterestswor ld-\nwide in unconventionalhigh- TCsuperconductivity[ 2]. Most\nof the research on the Fe-based superconductorshas focused\non mainly four systems, (1) the quaternary “1111” systems,\nRFeAsO 1−xFx(R=La, Nd, Sm, or Pr, etc.) with TCas high\nas 56 K [ 1,3–5], (2) the ternary “122” systems, AFe2As2\n(A=Ba, Ca, Sr, or Eu etc) with TCupto 38 K, [ 6–8], (3) the\nbinary “11” system (e.g. FeSe) [ 9] withTC~18 K and (4)\nthe ternary “245” systems, A2Fe4Se5(A= K, Rb, Cs) with\nTC~30 K [10]. Superconductivitycan be achieved in all the\nabove compounds in different ways, for example, either by\nelectron or hole doping in the Fe-As layers [ 11,12] or by\nisovalent substitution [ 13–15]. Internal chemical pressure by\nisovalent substitution of arsenic with phosphorus [ 14,15] or\nexternal hydrostatic pressure can also give rise to superco n-\nductivity[ 16,17].\nEuFe2As2is an interesting member of the “122” family\nsincethe Asite isoccupiedbyEu2+, whichisan S-staterare-\nearth ion possessing a 4 f7electronic conﬁguration with the\nelectron spin S= 7/2 [18]. EuFe 2As2exhibits a spin density\nwave (SDW) transition in the Fe sublattice concomitant with\na structural phase transition at 190 K. In addition, Eu2+mo-\nments order in an A-type antiferromagnetic (AFM) structure\nat 19 K (ferromagnetic layers ordered antiferromagnetical ly\nalongthe caxis) [19–21]. Superconductivitycan be achieved\ninthissystembysubstitutingEuwithK orNa(Refs.[ 7,22]),\nAswithP(Ref. 23),anduponapplicationofexternalpressure\n(Refs.[16,17,24]).\nSuperconductivity and magnetism are two antagonistic\nphenomena since the superconducting state expels external\nmagnetic ﬂux. Nevertheless, superconductivity in the pnic -\ntidesandcupratesisalwaysfoundincloseproximitytoanan -tiferromagnetic order and the superconducting pairing is b e-\nlievedtobemediatedbytheantiferromagnticspinﬂuctuati ons\n[2]. Mostsurprisingisthecoexistenceofferromagnetismand\nsuperconductivity as recently proposed by many groups for\ntheP-dopedEuFe 2As2samples[ 25–29]. BasedonMössbauer\nstudiesonsuperconductingpolycrystallinesamples,Nowi ket\nal. [27] concluded that the Eu2+moments are aligned fer-\nromagnetically along the caxis with a possible tilting an-\ngle of 20◦from the caxis. Zapf et al. also [28] concluded\nbased on macroscopic measurements that the Eu2+moments\nin EuFe 2(As1−xPx)2order in a canted A-type antiferromag-\nnetic structure with the spin component along the cdirection\nbeing ferromagnetically aligned. The small in plane compo-\nnent of the Eu2+moments in the A-type AFM structure un-\ndergoes a spin glass transition where the moments between\nthelayersaredecoupled[ 29].\nFor a magnetic superconductor with rare-earth moments,\nseveraltheoreticalstudiesclaimthatthesuperconductiv itycan\ncoexist with several forms of the magnetic states, namely,\n(a) “cryptoferromagnetism” (which is a ferromagnetic stat e\nwith small domains, smaller than the superconducting co-\nherence length) [ 30] or (b) transverse amplitude modulated\ncollinear antiferromagnetic structure or (c) spiral antif erro-\nmagnetic structure or (d) with a spontaneous vortex state of\nthe magnetic moments. A spontaneous vortex state or a self-\ninducedvortexstate is a new state of matter in which the two\ncompetingorders,superconductivityandferromagnetism, co-\nexistduetothelowerfreeenergyofthecombinedstatescom-\npared to the individual ones [ 31]. The Pure ferromagnetic\nstate is least preferred. These results clearly show the im-\nportance of the alignment for the rare-earth moments in the\nsuperconductingsamples.\nTo the best of our knowledge, for the superconducting\nEuFe2(As1−xPx)2single crystal samples, direct microscopic\nevidencefortheproposedferromagneticand/orantiferrom ag-2\nnetic structure is still lacking. Due to the strong neutron a b-\nsorption of Eu together with the small sample mass of the\nP-dopedsingle crystals, the magnetic structure determina tion\nin EuFe 2(As1−xPx)2via neutron diffraction is considerably\nmorechallengingthanthatofothermembersofthenewsuper-\nconductors. The only attempt was made on a powder sample\nof the non-superconductingEuFe 2P2where it was concluded\nthattheEu2+momentsorderferromagneticallywithacanting\nangle of 17◦from thecaxis [32]. Here we report on the ﬁrst\nelement-speciﬁc x-ray resonant magnetic scattering (XRMS )\nstudies of the superconducting EuFe 2(As1−xPx)2to explore\nthe details of the magnetic structure of the Eu2+moments.\nOur resonant scattering experimentsshow that the Eu2+mo-\nments order ferromagnetically along the caxis at zero ﬁeld\nand undergo a transition into a ﬁeld induced ferromagnetic\nstate along the applied magnetic ﬁeld direction for applied\nmagnetic ﬁelds ≥0.6T. Both the zero and applied magnetic\nﬁeld ferromagnetic order of the Eu2+moments coexist with\nthebulksuperconductivity.\nII. EXPERIMENTALDETAILS\nSingle crystals of EuFe 2(As1−xPx)2withx= 0.05and\nx= 0.15were grown using FeAs ﬂux [ 33]. For the scatter-\ning measurements and for the superconducting composition\nx= 0.15, an as-grown right isosceles triangular shaped sin-\ngle crystal with a base of approximately 2 mm and a thick-\nness of 0.1 mm was selected. The samecrystal was used for\nall the macroscopic characterizations presented in this co m-\nmunication. For the non-superconducting x= 0.05sample,\na crystal of approximate dimensions of 2 ×2×0.1mm3was\nchosen. The surface of both single crystals were perpendicu -\nlar to the caxis. The XRMS experiments were performed at\ntheEuL 3-edgeatbeamlineP09atthePETRAIIIsynchrotron\nat DESY [ 34]. The incident radiation was linearly polarized\nparallel(π-polarization)andperpendicular( σ-polarization)to\nthe horizontal and vertical scattering planes for the 15% an d\n5% doped samples, respectively. The spatial cross section o f\nthe beam was 0.2(horizontal) ×0.05(vertical) mm2. Copper\nCu(220) was used at the Eu L 3absorption edge as a polar-\nization and energy analyzer to suppress the charge and ﬂuo-\nrescence background relative to the magnetic scattering si g-\nnal. The sample was mounted at the end of the cold ﬁnger\nofa cryomagnetwith [21 0] T-[001] Tplanecoincidentwith\nthe scatteringplaneforthe 15%dopedsample. Themagnetic\nﬁeld was applied along the [1 ¯20] direction which is perpen-\ndicular to the scattering plane. The 5% doped sample was\nmeasuredinsideaclosedcycleDisplexcryogenicrefrigera tor\nwith [1 1 0] T-[0 0 1] Tas the scattering plane. Measurements\nat P09 were performedat temperaturesbetween 5 and 180K.\nFor convenience, we will use tetragonal ( T) notation unless\notherwisespeciﬁed./s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48/s45/s49/s48/s49/s50\n/s45/s56 /s45/s52 /s48 /s52 /s56/s45/s49/s50/s45/s56/s45/s52/s48/s52/s56/s49/s50/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48/s45/s56/s45/s52/s48\n/s45/s56 /s45/s52 /s48 /s52 /s56\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48/s48/s50/s48/s52/s48/s54/s48/s56/s48/s49/s48/s48/s49/s50/s48/s49/s52/s48\n/s50/s50 /s50/s52 /s50/s54 /s50/s56/s49/s52/s49/s54/s49/s56/s48 /s49/s48 /s50/s48 /s51/s48/s48/s50/s48/s48/s46/s48 /s48/s46/s53/s45/s52/s48\n/s45/s53 /s48 /s53/s45/s49/s48/s45/s53/s48/s53/s49/s48\n/s45/s56 /s45/s52 /s48 /s52 /s56/s45/s50/s48/s50\n/s40/s101/s41/s40/s99/s41/s32\n/s84\n/s67/s90/s70/s67\n/s70/s67\n/s72 /s32/s124/s124/s32/s91/s49/s32/s49/s32/s48/s93\n/s84/s118\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32 /s84 /s32/s40/s75/s41/s69/s117/s70/s101\n/s50/s40/s65/s115\n/s49/s45 /s120/s80\n/s120/s41\n/s50/s44/s32 /s120/s32 /s61/s32/s48/s46/s49/s53\n/s84\n/s83/s67/s40/s97/s41\n/s40/s100/s41\n/s72/s32/s124/s124/s32/s91/s49/s32/s49/s32/s48/s93\n/s84\n/s32/s84 /s32/s61/s32/s53/s46/s53/s32/s75\n/s32/s84 /s32/s61/s32/s51/s48/s46/s48/s32/s75/s77/s97/s103/s110/s101/s116/s105/s99/s32/s109/s111/s109/s101/s110/s116/s32/s40\n/s66/s47/s102/s46/s117/s46/s41\n/s70/s105/s101/s108/s100/s32/s40/s84/s101/s115/s108/s97/s41/s40/s98/s41\n/s84\n/s83/s67/s32\n/s118/s32\n/s84\n/s67/s90/s70/s67\n/s70/s67\n/s72 /s32/s124/s124/s32/s91/s48/s32/s48/s32/s49/s93\n/s84\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32 /s84 /s32/s40/s75/s41/s84\n/s83/s67\n/s32/s84 /s32/s61/s32/s51/s48/s46/s48/s32/s75/s72/s32/s124/s124/s32/s91/s48/s32/s48/s32/s49/s93\n/s84/s32/s32/s84 /s32/s61/s32/s53/s46/s53/s32/s75\n/s70/s105/s101/s108/s100/s32/s40/s84/s101/s115/s108/s97/s41/s32/s67\n/s112/s32/s40/s74/s47/s109/s111/s108/s32/s75/s41\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32 /s84 /s32/s40/s75/s41/s32/s67/s112/s32/s40/s74/s47/s109/s111/s108/s32/s75/s41\n/s84 /s32/s40/s75/s41/s84\n/s67/s32/s67\n/s112/s32/s40/s74/s47/s109/s111/s108/s32/s75/s41\n/s32/s84 /s32/s40/s75/s41/s32/s32/s32/s32\n/s32/s32\n/s32/s32\nFigure1: (a)and(b) Temperaturedependencies of themagnet ic sus-\nceptibility measured on heating of the zero-ﬁeld cooled (ZF C) and\nﬁeld cooled (FC) sample at an applied magnetic ﬁeld of 1mT alo ng\nthe crystallographic [1 1 0] Tand [0 0 1] Tdirections, respectively.\n(c) and (d) M-Hcurves for magnetic ﬁelds parallel and perpendic-\nular to the caxis atT= 5 K (below magnetic and superconducting\ntransitions) and 30 K (above superconducting and magnetic t ransi-\ntions). Horizontal dashed lines inboth ﬁgures denote fully saturated\nmoment of Eu2+. Lower insets for both ﬁgures show the hystere-\nsis curves after subtraction of the ferromagnetic contribu tion as de-\nscribed inthe text. The upper inset of the Fig. 1 (d) shows det ails of\ntheM-Hdependence inthelow ﬁledregion. (e) Temperature depen-\ndence of the speciﬁc heat. Upper and lower insets show detail s near\nthe magnetic ordering of the Eu2+and the superconducting transi-\ntion,respectively. The solidcurve represents the ﬁtusing Debye and\nEinstein contributions for the lattice part of the speciﬁc h eat. The\nlattice part was subtracted from the total heat capacity to c alculate\nthe entropy release at TC.3\n/s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s48/s46/s48/s48/s46/s53/s84\n/s83/s67\n/s40/s99/s41/s40/s98/s41/s32/s84\n/s115/s32\n/s32/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32 /s84 /s32/s40/s75/s41/s32/s32 /s32/s61/s32/s40 /s97 /s45/s98 /s41/s47/s40 /s97 /s43 /s98 /s41/s32/s40/s49/s48/s45/s51\n/s41\n/s84\n/s67/s32/s120 /s32/s61/s32/s48/s46/s49/s53/s69/s117/s70/s101\n/s50/s40/s65/s115\n/s49/s45 /s120/s80\n/s120/s41\n/s50/s40/s97/s41\n/s45/s52/s46/s48/s49 /s45/s52/s46/s48/s48 /s45/s51/s46/s57/s57/s48/s53/s49/s48/s49/s53/s50/s48\n/s39\n/s40/s52/s32/s48/s32/s56/s41\n/s79\n/s40/s48/s32/s52/s32/s56/s41\n/s79/s32/s84 /s32/s61/s32/s56/s46/s53/s32/s75\n/s32/s84 /s32/s61/s32/s53/s48/s32/s75\n/s72 /s32/s105/s110/s32/s40 /s72 /s32/s48/s32/s56/s41/s32/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41/s40/s50/s32/s50/s32/s56/s41\n/s84\n/s52/s48/s52/s53/s53/s48/s53/s53\n/s39\n/s39/s39\n/s39/s39\n/s39\n/s32/s32/s73/s110/s116/s46/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41/s72 /s32/s61/s32/s32/s48/s32/s84\n/s39/s84\n/s67/s32/s61/s32/s50/s48/s32/s75\n/s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48 /s51/s53/s49/s50/s49/s52/s49/s54/s49/s56/s50/s48/s69/s117/s32/s76\n/s51/s45/s101/s100/s103/s101\n/s40/s50/s32/s49/s32/s55/s41/s32/s114/s101/s102/s108/s101/s99/s116/s105/s111/s110\n/s39\n/s32/s32/s73/s110/s116/s46/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32 /s84 /s32/s40/s75/s41/s72 /s32/s61/s32/s48/s46/s53/s32/s84/s84\n/s67/s32/s61/s32/s50/s57/s32/s75\nFigure2: (a)Temperaturedependenceoftheorthorhombicdi stortion\nforthex=0.15sample. Theinsetshows( ξξ0)Tscansthroughthe(2\n2 8)Tposition above and below the structural phase transition. T he\nlines represent ﬁts to the data using either one (red) or two ( blue)\nLorentzian squared peaks. (b) Temperature dependence of th e (2 1\n7) reﬂection in both the π→σ′andπ→π′scattering geometries\nat zero ﬁled. The schematic shows the used scattering geomet ry. (c)\nSameas(b)butinanappliedmagneticﬁeldof0.5T.Thetemper ature\ndependenciesweremeasuredatthepeakenergy( ∼6.973keV)ofthe\nresonance enhancement observed inthe energy scans.\nIII. EXPERIMENTALRESULTS\nA. Macroscopic Characterizations\nFigure1(a-b)and(c-d)showmagneticsusceptibility( M-T)\nandisothermalmagnetization( M-H) ofthex= 0.15sample,\nrespectively, measured for magnetic ﬁelds parallel and per -\npendicular to the caxis using a Quantum Design (SQUID)\nmagnetometer. Zeroﬁeldcooledmagnetizationbecomesneg-ative for both ﬁeld directions at TSC= 25 K, signifying a su-\nperconductingtransitionatthistemperature. Uponcoolin gto-\nwardstheonsetofEu2+orderingat TC=19K,thesupercon-\nductingsignal is ﬁrst weakened,before it becomesmore pro-\nnounced at temperatures below TC. Superconductivity wins\nover the Eu2+magnetism if temperature is lowered further.\nThe diamagnetic volume susceptibility for the magnetic ﬁel d\nparallel to the [1 1 0] direction (in this direction demagnet i-\nzation correction is small [ 35]) is greater than -0.5 indicating\nbulk superconductivity [ 51]. Effective diamagnetic suscepti-\nbility close to -1 for the ZFC curve provides an upper limit\nof superconducting volume fraction of 100%. Figures 1(c)\n& (d) show hysteresis loops at T= 5 and 30 K for the two\nﬁelddirections. Theobservedhysteresiscurveslookdiffe rent\nthan a typeII nonmagneticsuperconductor. However,a jump\nin magnetization, which is typical for a type-II supercondu c-\ntor,isclearlyobservedat 7Tmagneticﬁeldbetweentheﬁeld\nincreasing and decreasing cycles. To understand the atypic al\nhysteresis curve, we assume a ferromagnetic contribution o f\ntheEu2+momentsat anappliedmagneticﬁeld H(Tesla)by,\nmEu= (7.0/0.5)×HµB,for|H| ≤0.5(1)\n= 7.0×H/|H|µB,for|H| ≥0.5\nsince very little hysteresis was observed for the ferromag-\nnetic end member EuFe 2P2[36]. Lower insets to Fig. 1(c)\nand(d)showmagnetizationaftersubtractionoftheferroma g-\nneticcontributionfromtheEu2+momentsaccordingtoEq. 1.\nThe hysteresis curves after subtraction look very similar t o\nthe other Fe based superconductors[ 12,37]. The jumpat 7T\nmagnetic ﬁeld is consistent with Bean’s critical state mode l\ntogether with Lenz’s law [ 38–40]. Reversal of the direction\nof change of applied ﬁeld as at 7 T does not remove the\nspecimen from the critical state but merely reverses locall y\nthe direction of the critical current according to Lenz’s la w.\nTherefore,magnetizationmeasurementsstronglyhinttowa rds\na ferromagneticsuperconductorin an applied magnetic ﬁeld .\nTheheatcapacityofthesamesinglecrystalwasmeasuredus-\ningaQuantumDesignphysicalpropertymeasurementsystem\n(PPMS) and is shown in Fig. 1(e). Speciﬁc heat data show a\nclearphasetransitionat TC= 19K indicatingthe onsetof the\nEu2+magnetic order. A speciﬁc heat jump at TSCis clearly\nvisibleandamountsto ∆C≈350mJ/mol.Kwhichis slightly\nless but of the same order of magnitude as that observed for\nthe K-doped BaFe 2As2system [41]. Due to the difﬁculties\nin determination of ∆C as well as \" γ\" as a result of large\nmagnetic contribution at low temperatures, it will be hugel y\nerroneoustoestimatethevalueof ∆C/(γTSC)andmakecom-\nparison with other non-magneticiron based superconductor s.\nHeat capacity measurement down to mK temperature range\nis needed to correctly estimate the value of γ. The entropy\nrelease associated with the magnetic order of the Eu2+mo-\nments amounts to 17.1 J/mol.K which is equal to 99% of the\nexpected theoretical value Rln(2S+ 1)for Eu2+moments\nwith spin S=7/2. Therefore, the speciﬁc heat measurement\nindicates that substantial volume of the sample, if not 100% ,4\n/s48/s49/s50\n/s48/s50/s52\n/s48/s50/s52\n/s54/s57/s52/s48 /s54/s57/s53/s48 /s54/s57/s54/s48 /s54/s57/s55/s48 /s54/s57/s56/s48 /s54/s57/s57/s48 /s55/s48/s48/s48/s48/s50/s48/s52/s48/s54/s48 /s48/s49/s48/s48/s50/s48/s48/s40/s100/s41\n/s32/s32\n/s32/s70/s108/s117/s111/s114/s101/s115/s99/s101/s110/s99/s101/s69/s117/s32/s76\n/s51\n/s32/s40/s50/s32/s49/s32/s55/s41/s44/s32\n/s73/s40/s84 /s32/s61/s32/s54/s32/s75/s41/s45 /s73/s40/s84 /s32/s61/s32/s50/s50/s32/s75/s41\n/s39\n/s40/s99/s41/s40/s98/s41\n/s32/s40/s32/s50/s32/s49/s32/s55/s41/s44/s32\n/s73/s40/s66 /s32/s61/s32/s51/s84/s41/s45 /s73/s40/s66 /s32/s61/s32/s48/s32/s84/s41\n/s39/s73/s110/s116/s46/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41/s40/s97/s41\n/s120 /s32/s61/s32/s48/s46/s49/s53/s120 /s32/s61/s32/s48/s46/s49/s53\n/s84 /s32/s61/s32/s54/s32/s75/s120 /s32/s61/s32/s48/s46/s49/s53/s69/s117/s70/s101\n/s50/s40/s65/s115\n/s49/s45 /s120/s80\n/s120/s41\n/s50\n/s40/s48/s32/s48/s32/s51/s41\n/s114/s101/s102/s108/s101/s99/s116/s105/s111/s110\n/s39/s73/s110/s116/s46/s32/s40/s99/s111/s117/s110/s116/s115/s47/s49/s48/s115/s41\n/s73/s110/s116/s46/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41/s73/s110/s116/s46/s32/s40/s49/s48/s51\n/s32/s99/s111/s117/s110/s116/s115/s47/s115/s41\n/s32/s73/s32/s40/s56/s32/s75/s41\n/s73/s32/s40/s50/s50/s32/s75/s41/s40/s48/s32/s48/s32/s57/s41\n/s114/s101/s102/s108/s101/s99/s116/s105/s111/s110\n/s39\n/s32/s32/s69/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s120 /s32/s61/s32/s48/s46/s48/s53/s40/s101/s41/s73/s32/s40/s54/s32/s75/s41\n/s73/s32/s40/s50/s50/s32/s75/s41\n/s32\nFigure 3: (a) Energy scan of the ﬂuorescence yield. The dashe d line\ndepicts the Eu L 3absorption edge as determined from the inﬂection\npoint of the ﬂuorescence yield. (b) and (c) Energy scans for t he (2\n1 7) reﬂection after subtraction of the non-magnetic backgr ound at\nhigh temperature for (b) and at zero magnetic ﬁeld for (c). (d ) and\n(e) Energy scans through the antiferromagnetic ( 0 0 3) and (0 0 9)\npositions for the 15% and 5% samples, respectively. Lines se rve as\nguides tothe eye.\ncontributes to both the superconductivityand magnetic ord er\nof the Eu2+moments. Moreover,the full momentof Eu2+is\ncompletely orderedat the single phase transition temperat ure\nTCof19K.\nB. X-rayresonant magnetic scattering\nTo determinewhether there is a structural phase transition ,\nasobservedintheparentcompoundEuFe 2As2,(ξξ0)Tscans\nwere performedthroughthe tetragonal(2 2 8) TBraggreﬂec-\ntion as a function of temperature. The inset to Figure 2(a)\nshows a subset of ( ξ ξ0)Tscans through the (2 2 8) Tre-\nﬂection for the 15% doped sample as the sample was cooled\nthroughTS= 49±1 K. The splitting of the (228) TBragg re-\nﬂectionintoorthorhombic( O)(408)Oand(048) OBraggre-\nﬂections below TSis consistent with the structural transition,from space group I4/mmmtoFmmm, with a distortion\nalong the [1 1 0] direction. As the sample is cooled further,\nthe orthorhombicsplitting ( δ)increasesdownto T= 30±1K\nascan beseen fromFig. 2(a). Near TSC,δshowsa localmin-\nimum due to the competition between superconductivity and\nferromagnetism. Lowering the temperature below TCresults\nin a smoothdecrease in δ, reminiscentof that observedin the\nsuperconductingBa(Fe 1−xCox)2As2samples [ 42]. The non-\nsuperconducting5% dopedsampleundergoesa similar struc-\ntural phase transition at TS= 165±1 K but without any de-\ncreaseoftheorthorhombicdistortionforlowertemperatur es.\nBelowTC=20K,amagneticsignalwasobservedwhenthe\nx-ray energy was tuned through the Eu L 3edge at recipro-\ncal lattice points identical to those of the charge reﬂectio ns,\nindicating the onset of the Eu2+magnetic order at the mag-\nnetic propagation vector τ=(000). Figure 2(b) depicts the\ntemperatureevolutionofthe(217)reﬂectionmeasuredatth e\nEu L3edge at resonance ( E=6.973keV). A variation of the\nmagneticintensitywith temperaturewasonlyobservedin th e\nπ→σ′scattering channel whereas the π→π′scattering\nchannel shows no discernible temperature dependence. The\ntransitiontemperatureissimilartothatobservedin thepa rent\nEuFe2As2compoundandconsistentwiththeresultspresented\nin Fig.1. Figure 2(c) shows temperature dependence of the\nsame (2 1 7) reﬂection in an applied magnetic ﬁeld of 0.5 T\nalongthe[1 20]directioninbothscatteringchannels. Itisin-\nterestingtoseethatthetemperaturedependenceappearsin the\noppositescatteringchannelcomparedtothezeroﬁeldandin -\ndicates a possible ﬂop of the magnetic moment in an applied\nmagnetic ﬁeld which will be discussed later. The transition\ntemperatureisincreasedfrom19Katzeroﬁeldto29Kat0.5\nT.\nTo conﬁrm the resonant magnetic behavior of the peaks,\nwe performed energy scans at the Eu L 3absorption edge as\nshown in Fig. 3. We note that for the (217) reﬂection charge\nand magnetic peak coincide. An investigation of the mag-\nnetic signal which is ﬁve to six orders of magnitude weaker\nthan the Thomson charge scattering requires signiﬁcant re-\nduction of the charge background. The charge background\ncan be reduced signiﬁcantly for a reﬂection with scattering\nangle close to 90◦[43,44]. Since the (217) reﬂection has a\nscattering angle of ∼94.5◦at the Eu L 3edge, the investiga-\ntion of the magnetic signal seems feasible for this reﬂectio n.\nFigure3(b)showsanenergyscanthroughthe(217)reﬂection\naftersubtractingthe nonmagneticbackgroundat T= 22 K. A\nclear resonance enhancement can be seen close to the Eu L 3\nedge. A similar resonance enhancement can be observed in\ntheπ→π′scatteringchannelin an appliedmagneticﬁeld of\n3 T. In both energy scans, the resonance peaks appear at and\nabovethe EuL 3absorptionedge,indicatingthedipolenature\nof the transition. Figure 3(d) shows energy scans through\ntheantiferromagnetic(003)position,expectedforanA-ty pe\nAFM structure, for the 15% doped sample in the π→σ′\nscattering channel. For comparison,we also show the energy\nscanthroughthe(009)positioninFig. 3(e)forthe5%doped\nsamplemeasuredundersimilarconditions. Astrongantifer ro-5\nmagneticsignalwasobservedforthe5%dopedsampleat the\nA-type AFM position which is in contrast to the 15% doped\nsample where no magnetic signal was observed. Therefore,\nthe proposed A-type AFM structure [ 28] could not be con-\nﬁrmed for the superconducting 15% P-doped sample. This\nmight be due to the small moment in the A-type AFM struc-\nture together with the glassy freezing of the in-plane compo -\nnentassuggestedbyRef. 29.\nC. Magneticstructureinzeroandappliedmagnetic ﬁelds\nTable I: Basis vectors for the space group Fmmm withτ=(000).\nThe decomposition of the magnetic representation for the Eu site at\n(000)isΓMag= 0Γ1\n1+0Γ1\n2+1Γ1\n3+0Γ1\n4+1Γ1\n5+0Γ1\n6+1Γ1\n7+0Γ1\n8.\nIR Atom BV components Magnetic Intensity\nm/bardblam/bardblbm/bardblc (217)\nπ→σ′π→π′\nΓ31 1 0 0 Yes Yes\nΓ51 0 1 0 Yes Yes\nΓ71 0 0 1 Yes No\nWe now turn to the determinationof the magnetic moment\nconﬁguration for the Eu2+moments in the zero and applied\nmagnetic ﬁelds. For the crystallographic space group Fmmm\nandτ=(000), three independent magnetic representations\n(MRs) are possible [ 45]. Here we note that only ferromag-\nnetic structures with magnetic moments along the three crys -\ntallographic directions a,b,care allowed by symmetry. No\nantiferromagneticstructure with τ=(000) is possible in this\ncaseforsymmetryreasons. AlltheMRsalongwiththecalcu-\nlatedintensitiesfordifferentpolarizationgeometriesa relisted\ninTableI.\nThe resonant scattering of interest, at the Eu L 3absorption\nedge, is due to electric dipole transitions betweenthe core 2p\nstates and the 5 dconduction bands. The 5 dbands are spin\npolarizedthroughthe exchangeinteractionwith the magnet ic\n4felectrons. The resonant magnetic scattering cross-sectio n\nforthedipoleresonancecanbewrittenas[ 46]:\nfXRMS\nnE1= [(ˆǫ′·ˆǫ)F(0)−i(ˆǫ′×ˆǫ)·ˆznF(1)+(ˆǫ′·ˆzn)(ˆǫ·ˆzn)F(2)]\n(2)\nwhereˆznis a unit vector in the direction of the magnetic\nmoment of the nthion. Here ˆǫandˆǫ′are the incident and\nscattered polarization vectors, and F(i)’s are the terms con-\ntaining dipole matrix elements. The ﬁrst term of Eq. 2con-\ntributes to the charge Bragg peak as it does not contain any\ndependence on the magnetic moment. The other two terms\nare sensitive to the magnetic moment. For a ferromagnetic\nstructure, in general all terms contribute to the scatterin g at\neveryBragg reﬂection. However,forthe Eu2+ionswith spin\nonly magnetic moment, the spherical symmetry of the spin-\npolarized5 dbandensuresthat the F(2)term is zero [ 47]. For\ntheπ→σ′scatteringgeometrythescatteringamplitudefrom\nEq.2can be written in a simpliﬁed form as f∝ki·µ, [48]wherekiandµare the wave vector of the incoming photons\nandthemagneticmoment,respectively. Clearly,themagnet ic\nsignalissensitiveto thecomponentoftheorderedmomentin\nthescatteringplanei.e. a/bandccomponents. Forthe π→π′\nscatteringgeometrythescatteringamplitudecanbewritte nas\nf∝(ki×kf)·µ, [48] wherekfis the wave vector of the\noutgoing photons. Therefore, in the π→π′scattering ge-\nometry, the magnetic signal is sensitive to the component of\nthe ordered moment perpendicularto the scattering plane i.e.\nonlya/bcomponents. Since,nomagneticsignalwasobserved\nin theπ→π′scattering channel at zero ﬁeld (see Fig. 2(b)),\nweconcludethatthemagneticmomentsarealignedprimarily\nalongthe caxis. Fortheappliedmagneticﬁeldthesituationis\nreversed. Themagneticsignalisobservedonlyinthe π→π′\nscatteringchannel(see Fig. 2(c))indicatingthe magneticmo-\nments are in the a-bplane. It is most likely that the magnetic\nmomentsarealongtheappliedﬁleddirection i.e.alongthe[1\n¯20] direction. The determined magnetic structures based on\nthepolarizationanalysisofthescatteredsignalispresen tedin\nFig.4(a).\nHaving determinedthe magnetic structures in zero and ap-\nplied magnetic ﬁelds, we have measured the ﬁeld dependen-\ncies of the integrated intensity of the magnetic (2 1 7) reﬂec -\ntion for several temperatures which are presented in Fig. 4\n(b). A clear hysteresis can be seen from the increasing and\ndecreasing ﬁeld cycles at T= 6 K which is typical for a fer-\nromagnet. The critical ﬁeld, H cr, at which the ﬁeld induced\nphase transition occurs, has been determined from the inter -\ncept of the high and low ﬁeld linear interpolation as shown\nfor theT= 11 K measurement in Fig. 4(b). The ﬁeld de-\npendenceof theferromagneticorderingtemperaturehasbee n\ndetermined from the temperature dependence of the (2 1 7)\nreﬂectioninthe π→π′scattering geometryas shownin Fig.\n2(c). Additionally,isothermal magnetization( M-H) at differ-\nent temperatures (not shown) and temperature dependencies\nof magnetization ( M-T) at different magnetic ﬁelds (see Fig.\n4(c)) have been performed to verify the transition tempera-\nturesand critical ﬁelds obtainedfromthe scattering measu re-\nments. A combined phase diagram has been constructed and\nisshowninFig. 4(d). Itcanbeseenthatsuperconductivityco-\nexists with strong ferromagnetic order of the Eu2+moments\nfor a large region of the phase diagram. For B≤0.5T, the\nsuperconducting transition precedes the ferromagnetic tr an-\nsition, whereas the situation is reversed for magnetic ﬁeld s\nhigherthan0.5T.\nIV. DISCUSSIONANDCONCLUSION\nThe most important result of the present study is the ob-\nservationofstrongferromagneticorderoftheEu2+moments\ncoexisting with bulk superconductivity. Magnetization, s pe-\nciﬁcheatandtemperaturedependenceofthestructuraldist or-\ntion indicates bulk nature of the superconducting transiti on.\nIn contrast to the previous studies, we got no indication of\nthe proposed A-type AFM structure or a spiral magnetic or-6\n/s48/s46 /s48 /s48/s46 /s50 /s48/s46 /s52 /s48/s46 /s54 /s48/s46 /s56 /s49/s46 /s48/s57/s48/s49/s48/s48/s49/s49/s48/s49/s50/s48/s49/s51/s48/s49/s52/s48/s48/s46 /s48 /s48/s46 /s53 /s49/s46 /s48 /s49/s46 /s53 /s50/s46 /s48 /s50/s46 /s53\n/s49/s50/s51/s49/s50/s55/s49/s51/s49/s49/s51/s53/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48/s48/s49/s50/s51/s52/s53/s54/s55/s56\n/s72 \n/s99/s114\n/s40/s50/s32 /s49/s32 /s55/s41/s32 /s114/s101/s102 /s108/s101/s99 /s116 /s105/s111/s110/s32 \n/s69/s117/s32 /s76\n/s51 /s32 /s101/s100/s103/s101\n/s39 /s49/s53/s32 /s75/s73 /s110 /s116 /s101 /s110 /s115 /s105/s116 /s121 /s32 /s40 /s97 /s114 /s98 /s46 /s32 /s117 /s110 /s105/s116 /s115 /s41 \n/s77 /s97/s103/s110/s101/s116 /s105/s99 /s32 /s102 /s105/s101/s108/s100/s32 /s66 /s32 /s40/s84 /s101/s115 /s108/s97/s41/s32 /s84 /s32 /s61/s32 /s54/s32 /s75\n/s49/s49/s32 /s75\n/s49/s57/s32 /s75\n/s50/s50/s32 /s75\n/s72 \n/s99/s114\n/s48/s46 /s48 /s48/s46 /s53 /s49/s46 /s48/s48/s49/s48/s50/s48/s51/s48/s52/s48\n/s83/s67 /s70 /s77 \n/s32/s32\n/s80/s77 /s32 \n/s83/s67 /s43/s70 /s77 /s32 /s40/s100/s41\n/s84\n/s67/s84 /s101 /s109 /s112 /s101 /s114 /s97 /s116 /s117 /s114 /s101 /s32 /s84 /s32 /s40 /s75 /s41 \n/s77 /s97/s103/s110/s101/s116 /s105/s99 /s32 /s102 /s105/s101/s108/s100/s32 /s66 /s32 /s40/s84 /s101/s115 /s108/s97/s41/s69/s117/s70 /s101\n/s50 /s40/s65/s115 \n/s49 /s45/s120/s80\n/s120/s41\n/s50 \n/s120 /s32 /s61/s32 /s48/s46 /s49/s53/s40/s97/s41\n/s40/s98/s41\n/s83 /s67/s43/s70/s77 \n/s109 /s32/s124 /s124 /s32/s91/s49 /s32/s50 /s32/s48 /s93/s84 \n/s83/s67 /s40/s99 /s41\n/s32 /s109 /s32 /s124 /s124 /s32 /s91 /s48/s32 /s48/s32 /s49/s93 /s67 /s97/s110/s116 /s101/s100/s32 /s73 /s110 /s116 /s101 /s110 /s115 /s105/s116 /s121 /s32 /s40 /s97 /s114 /s98 /s46 /s32 /s117 /s110 /s105/s116 /s115 /s41 /s32/s72 /s32 /s61/s32 /s48/s32 /s84 \n/s72 /s32 /s124 /s124 /s32 /s91 /s49/s32 /s50/s32 /s48/s93 \n/s54/s46 /s48/s32 /s84 \n/s52/s46 /s48/s32 /s84 \n/s50/s46 /s48/s32 /s84 \n/s49/s46 /s48/s32 /s84 \n/s48 /s46/s53 /s32/s84/s48/s46 /s49/s32 /s84 /s77 /s97 /s103 /s110 /s101 /s116 /s105/s99 /s32 /s109 /s111 /s109 /s101 /s110 /s116 /s32 /s40 \n/s66/s47 /s102 /s46 /s117 /s46 /s41 \n/s84 /s101/s109 /s112/s101/s114/s97/s116 /s117/s114/s101/s32 /s84 /s32 /s32 /s40/s75/s41/s48/s46 /s48/s49/s32 /s84 /s70 /s67 /s90 /s70 /s67 /s72 /s32 /s124 /s124 /s32 /s91 /s49/s32 /s50/s32 /s48/s93 /s72 /s32 /s32 /s48/s46 /s54/s32 /s84 \n/s69/s117\n/s70/s101 \n/s65 /s115 /s69/s117\n/s70/s101 \n/s65 /s115 /s99 \n/s97\n/s98/s99 \n/s97\n/s98\nFigure 4: (a)Magnetic structures of the Eu2+moments inzeroandapplied magnetic ﬁelds. Onlythe Eu2+magnetic moments are shown. (b)\nFielddependence of the intensities of the (217) reﬂectionm easured inthe π→π′scattering geometry after zeroﬁeldcooling of the sample\nfrom 80 K. (c) Temperature dependence of the bulk magnetizat ion at different applied magnetic ﬁelds along the [1 20] direction measured\nusinganMPMS.(d)Magnetic phase diagram forthe15% doped sa mpleconstructedusingmagnetizationandscatteringmeasu rements. Filled\nsymbols are derived from the scattering measurement and the open symbols from M-T(square) and M-H(circles) measurements at different\nﬁeldsandtemperatures,respectively. Thetransitiontemp eratures,TSCandTC,atzeroﬁeldareconsistentwiththepublishedresultsofRe f.33.\nder with propagation vector of the form (0 0 τ) [52]. In the\nFe-Asbasedsuperconductors,itisbelievedthatthesuperc on-\nducting carriers are in the Fe-As layers. Therefore, to unde r-\nstand the phenomena of coexistence, we have calculated the\neffective ﬁeld due the Eu2+moments at the Fe-As layers us-\ningdipoleapproximation. Toaﬁrstapproximation,thedipo le\nﬁeld does not exceed 1T which is much less than the super-\nconductinguppercriticalﬁeld HC2(~40T)[2]buthigherthan\nthelowercritical ﬁeld HC1(~0.02-0.03T)[ 12]. Since thein-\nternal ﬁeld is between HC1andHC2, it is most likely that the\nEuFe2(As1−xPx)2is in a spontaneous vortex state similar to\nwhich have been proposedin Eu(Fe 0.75Ru0.25)2As2[49] andUCoGesuperconductors[ 50]. Atanappliedmagneticﬁeld,it\nismostlikelythatthevorticesinthezero-ﬁeldstate(alon gthe\ncaxis) will graduallychangealongthe appliedﬁeld directio n\ni.e.inthea-bplane.\nInconclusion,themagneticstructureoftheEumomentsin\nsuperconducting EuFe 2(As1−xPx)2withx= 0.15has been\ndetermined using element speciﬁc x-ray resonant magnetic\nscattering. Combining magnetic, thermodynamic and scat-\ntering measurements we conclude that the long range ferro-\nmagneticorderofthe Eu2+momentsalignedprimarilyalong\nthecaxis coexists with the bulk superconductivity. The pro-\nposedcantedantiferromagneticorderorspiralordercould not7\nbeconﬁrmedinthesuperconductingsample. Additionalmea-\nsurementssuch assmall angleneutronscatteringis neededt o\nconﬁrmtheexistenceofaspontaneousvortexstate.\nS. N.likestoacknowledgeB. SchmitzandS. Dasfortech-\nnical assistance, S. Zapf and M. Dressel for fruitful discus -\nsion. Work at Göttingen was supported by the German Sci-\nenceFoundationthroughSPP 1458.\n∗Electronicaddress: s.nandi@fz-juelich.de\n[1] Y. Kamihara, T. Watanabe, M. Hirano, and H. Hosono, J. Am.\nChem. Soc. 130, 3296 (2008).\n[2] D.C.Johnston, Adv. Phys. 59, 803 (2010).\n[3] H. Takahashi, K. Igawa, K. Arii, Y. Kamihara, M. Hirano, a nd\nH.Hosono, Nature (London) 453, 376 (2008).\n[4] X.H.Chen, T.Wu,G.Wu,R.H.Liu,H.Chen,andD.F.Fang,\nNature453, 761 (2008).\n[5] R. Zhi-An, L. Wei, Y. Jie, Y. Wei, S. Xiao-Li, Zheng-Cai,\nC. Guang-Can, D. Xiao-Li, S. Li-Ling, Z. Fang, et al., Chin.\nPhys.Lett. 25, 2215 (2008).\n[6] M. Rotter, M. Tegel, and D. Johrendt, Phys. Rev. Lett. 101,\n107006 (2008).\n[7] H. S. Jeevan, Z.Hossain, D. Kasinathan, H. Rosner, C. Gei bel,\nandP.Gegenwart, Phys.Rev. B 78, 092406 (2008).\n[8] K. Sasmal,B. Lv, B. Lorenz, A.M. Guloy, F.Chen, Y.-Y. Xue ,\nandC.-W.Chu, Phys.Rev. Lett. 101, 107007 (2008).\n[9] F.-C. Hsu, J.-Y. Luo, K.-W. Yeh, T.-K. Chen, T.-W. Huang,\nP.M. Wu, Y.-C.Lee, Y.-L.Huang, Y.-Y. Chu, D.-C.Yan,et al.,\nProc.Natl.Acad. Sci. 105, 14262 (2008).\n[10] J.Guo, S.Jin, G.Wang, S.Wang, K.Zhu, T. Zhou, M. He,and\nX.Chen, Phys.Rev. B 82, 180520 (2010).\n[11] A. Leithe-Jasper, W. Schnelle, C. Geibel, and H. Rosner , Phys.\nRev. Lett. 101, 207004 (2008).\n[12] A. S. Sefat, R. Jin, M. A. McGuire, B. C. Sales, D. J. Singh ,\nandD. Mandrus, Phys.Rev. Lett. 101, 117004 (2008).\n[13] W. Schnelle, A. Leithe-Jasper, R. Gumeniuk, U. Burkhar dt,\nD.Kasinathan,andH.Rosner,Phys.Rev.B 79,214516(2009).\n[14] C.Wang,S.Jiang,Q.Tao,Z.Ren, Y.Li,L.Li,C.Feng, J.D ai,\nG.Cao, andZ.an Xu,Europhys. Lett. 86, 47002 (2009).\n[15] S. Jiang, H. Xing, G. Xuan, C. Wang, Z. Ren, C. Feng, J. Dai ,\nZ.Xu,andG.Cao,J.Phys.: Condens. Mat. 21,382203 (2009).\n[16] C. F.Miclea, M. Nicklas, H. S. Jeevan, D. Kasinathan, Z. Hos-\nsain,H.Rosner,P.Gegenwart, C.Geibel,andF.Steglich,Ph ys.\nRev. B79, 212509 (2009).\n[17] Y. Tokiwa, S.-H.Hübner, O.Beck, H.S.Jeevan, and P.Geg en-\nwart,Phys.Rev. B 86, 220505 (2012).\n[18] R. Marchand and W. Jeitschko, J. Solid State Chem. 24, 351\n(1978).\n[19] J. Herrero-Martín, V. Scagnoli, C. Mazzoli, Y. Su, R. Mi ttal,\nY. Xiao, T. Brueckel, N. Kumar, S. K. Dhar, A. Thamizhavel,\netal.,Phys. Rev. B 80, 134411 (2009).\n[20] Y. Xiao,Y. Su, M. Meven, R.Mittal, C.M. N. Kumar, T.Chat -\nterji, S. Price, J. Persson, N. Kumar, S. K. Dhar, et al., Phys .\nRev. B80, 174424 (2009).\n[21] Y. Xiao, Y. Su, W. Schmidt, K. Schmalzl, C. M. N. Kumar,\nS. Price, T. Chatterji, R. Mittal, L. J. Chang, S. Nandi, et al .,\nPhys.Rev. B 81, 220406 (2010).\n[22] Y.Qi,Z.Gao,L.Wang,D.Wang,X.Zhang,andY.Ma,NewJ.\nPhys.10, 123003 (2008).\n[23] Z.Ren,Q.Tao,S.Jiang, C.Feng,C.Wang,J.Dai,G.Cao,a ndZ.Xu, Phys.Rev. Lett. 102, 137002 (2009).\n[24] T. Terashima, M. Kimata, H. Satsukawa, A. Harada,\nK. Hazama, S. Uji, H. S. Suzuki, T. Matsumoto, and K. Mu-\nrata, J.Phys. Soc.Jpn. 78, 083701 (2009).\n[25] G. Cao, S. Xu, Z. Ren, S. Jiang, C. Feng, and Z. Xu, J. Phys. :\nCondens. Mat. 23, 464204 (2011).\n[26] A.Ahmed,M.Itou,S.Xu,Z.Xu,G.Cao,Y.Sakurai,J.Penn er-\nHahn, andA. Deb,Phys. Rev. Lett. 105, 207003 (2010).\n[27] I. Nowik, I. Felner, Z. Ren, G. H. Cao, and Z. A. Xu, J. Phys .:\nCondens. Mat. 23, 065701 (2011).\n[28] S. Zapf, D. Wu, L. Bogani, H. S. Jeevan, P. Gegenwart, and\nM. Dressel, Phys.Rev. B 84, 140503 (2011).\n[29] S. Zapf, H. S. Jeevan, T. Ivek, F. Pﬁster, F. Klingert, S. Jiang,\nD.Wu,P.Gegenwart,R.K.Kremer,andM.Dressel,Phys.Rev.\nLett.110, 237002 (2013).\n[30] P.W.Anderson and H.Suhl, Phys.Rev. 116, 898 (1959).\n[31] H.S.Greenside,E.I.Blount,andC.M.Varma,Phys.Rev. Lett.\n46, 49(1981).\n[32] D. H. Ryan, J. M. Cadogan, S. Xu, Z. Xu, and G. Cao, Phys.\nRev. B83, 132403 (2011).\n[33] H. S. Jeevan, D. Kasinathan, H. Rosner, and P. Gegenwart ,\nPhys. Rev. B 83, 054511 (2011).\n[34] J. Strempfer, S. Francoual, D. Reuther, D. K. Shukla, A. Skau-\ngen, H. Schulte-Schrepping, T. Kracht, and H. Franz, J. Syn-\nchrotron Rad. 20, 541 (2013).\n[35] J. A.Osborn, Phys.Rev. 67, 351 (1945).\n[36] C. Feng, Z. Ren, S. Xu, S. Jiang, Z. Xu, G. Cao, I. Nowik,\nI. Felner, K. Matsubayashi, and Y. Uwatoko, Phys. Rev. B 82,\n094426 (2010).\n[37] R. Prozorov, N. Ni, M. A. Tanatar, V. G. Kogan, R. T. Gordo n,\nC. Martin, E. C. Blomberg, P. Prommapan, J. Q. Yan, S. L.\nBud’ko, et al.,Phys.Rev. B 78, 224506 (2008).\n[38] C. P.Bean, Phys. Rev. Lett. 8, 250 (1962).\n[39] Y. B. Kim, C. F. Hempstead, and A. R. Strnad, Phys. Rev. 129,\n528 (1963).\n[40] W. A. Fietz, M. R. Beasley, J. Silcox, and W. W. Webb, Phys .\nRev.136, A335 (1964).\n[41] N. Ni, S. L.Bud’ko, A. Kreyssig, S. Nandi, G. E. Rustan, A . I.\nGoldman, S. Gupta, J. D. Corbett, A. Kracher, and P. C. Can-\nﬁeld, Phys.Rev. B 78, 014507 (2008).\n[42] S. Nandi, M. G. Kim, A. Kreyssig, R. M. Fernandes, D. K.\nPratt, A. Thaler, N. Ni, S. L. Bud’ko, P. C. Canﬁeld,\nJ. Schmalian, etal.,Phys. Rev. Lett. 104, 057006 (2010).\n[43] S. Nandi, Y. Su, Y. Xiao, S. Price, X. F. Wang, X. H. Chen,\nJ.Herrero-Martin,C.Mazzoli,H.C.Walker,L.Paolasini,e tal.,\nPhys. Rev. B 84, 054419 (2011).\n[44] J. W. Kim, A. Kreyssig, P. Ryan, E. Mun, P. C. Canﬁeld, and\nA. I.Goldman, Appl. Phys. Lett. 90, 202501 (2007).\n[45] A. S.Wills,Physica B 276-278, 680(2000).\n[46] J. P. Hill and D. F. McMorrow, Acta Crystallogr. A52, 236\n(1996).\n[47] M. D. Hamrick,M. A.Thesis, Rice University(1990).\n[48] S. Nandi, Magnetic x-ray scattering (John Wiley & Sons, New\nYork, 2012).\n[49] W.-H.Jiao,Q.Tao,J.-K.Bao,Y.-L.Sun,C.-M.Feng,Z.- A.Xu,\nI. Nowik, I. Felner, and G.-H. Cao, Europhys. Lett. 95, 67007\n(2011).\n[50] K. Deguchi, E.Osaki,S.Ban,N.Tamura, Y.Simura,T. Sak ak-\nibara, I. Satoh, and N. K. Sato, J. Phys. Soc. Jpn. 79, 083708\n(2010).\n[51]χvis larger than the ideal value of -1. However, considering\nthe ferromagnetic contribution of the Eu2+(as found by the\nscattering measurements), the effective volume susceptib ility\ndue to the superconductivity (SC), ( χSC\nv) =χobserved−χFerro\nEu,8\nmight be very close to -1. The observed ferromagnetic contri -\nbution for the non-superconducting ferromagnet EuFe 2P2[36]\n(in ZFC data) is the same order of magnitude as the χobservedin\nthe present case.[52] Carefulscansalong[00L]directiondonotrevealanyma gnetic\npeak."    },    {        "title": "1403.1485v1.Influence_of_the_Dzyaloshinskii_Moriya_interaction_on_the_spin_torque_diode_effect.pdf",        "content": "1 \n Influence of the Dzyaloshinskii -Moriya interaction  on the  spin-torque diode \neffect  \nR. Tomasello ,1 M. Carpentieri,2 G. Finocchio3  \n1Department  of Computer Science, Model ling, Electronics and System Science, University of \nCalabria, Rende (CS), Italy.  \n \n2Depar tment of Electrical and Information Engineering , Politecnico of Bari, via E. Orabona 4, I -\n70125 Bari, Italy.  \n3Department of Electronic Engineering, Industrial Chemistry and Engineering, University of \nMessina, C.da di Dio, I -98166, Messina, Italy.  \n \nAbstrac t: This paper predicts  the effect of the Dzyaloshinskii -Moriya interaction  (DMI)  and spin \nHall effect in the spin -torque diode response of a Magnetic Tunnel J unction  built over a Ta ntalum \nstrip. Our results indicate that , for a microwave current large enou gh, the DMI can change \nqualitatively the resonant  response by splitting  the ferromagnetic resonance peak . We also find out \nthat the two modes have a non-uniform  spatial distribution .  \n \n \n \n \n \n \n \n \n \n 2 \n MgO -based magnetic tunnel junctions (MTJs) are attracting  much  interest for their several \ntechnological applications, such as spin transfer torque magnetic random access memories (STT -\nMRAM) a nd nano -oscillators1, 2, 3, 4, 5. However,  when a microwave  current JMTJrf flows through a n \nMTJ  with a frequency close to  the ferromagnetic resonance (FMR) frequency of the free -layer (FL), \nthe so called spin -torque diode effect can be observed .6, 7 In oth er words, th e tunnelling \nmagnetoresistance (TMR)  oscillate s at the same frequency of the microwave current and, as a \nresult, a dc voltage can be measured across the MTJ  stack . The spin diode effect  can be also used to \nestimate  the STT-field-like torque term and its voltage dependence .8, 9 In the MTJ configuration \nwhere the FL is coupled to a heavy metal, we can use the additiona l degree of freedom regarding  \nthe spin-Hall effect (SHE)10 (the electrical current  is converted  into a transversal spin current  JSHE) \nin  control ling  the spin -torque diode response . When an ultra-thin ferromagnetic layer is coupled  to \na heavy  metal with s trong spin-orbit coupling, because of  the interfacial Dzyaloshinskii -Moriya \ninteraction  (DMI) , a chiral magnetic field should be  also taken into account . 11, 12, 13, 14, 15 The DMI  \nis an antisymmetric interfacial exchange contribution due to the spin -orbit coupl ing that excites \nspatial  rotational  magnetization configurations , such as spirals, skyrmions and chiral structures .16 \nThe DMI  is relevant  in bulk non-centrosymmetric crystal lattice, and in centrosymmetric ones \nhaving large strains, containing impurities wi th a large spin -orbit coupling and  in ultra -thin \nferromagnet where the inversion -symmetry is broken.17  \nHere , we perform a numerical study of the FMR response  by means of micromagnetic simulations \nby using a full micromagnetic framework  in which both the SHE and the DMI  are implemented. \nWe studied an experimental system similar to the one reported by Liu et al .18 The MTJ stack is \nmade by (CoFeB(1 )/MgO(1.2 )/CoFeB(4 )/Ta(5 )/Ru(5) (thicknesses in nm) ), milled over a Tantalum \n(Ta) strip (6000 nm  x 1200 nm x 6 nm ). We introduce a Cartesian coordinate system w here the x-\naxis is positioned  along the larger dimension  of the Ta strip, the y-and z-axes are consequently  \noriented along the other in -plane direction  and along the thickness  of the Ta strip. The MTJ has an 3 \n elliptical cross section 180 x 50 nm2 with the larger dimension oriented in the  y-direction . The ultra -\nthin CoFeB(1 ) acts as  FL (saturation magnetization Ms=1x106 A/m)  and because of the very low \nthickness , the interfac ial perpendicular anisotropy is large en ough (Ku=7x105 J/m3 related only to \nthe interfacial anisotropy energy contribution )19 to impose an out -of-plane easy axis . The CoFeB(4) \nis the reference layer  (RL)  and its magnetization is in -plane fixed along  the negative y-direction.  We \napply an external magnetic field Hext=8 mT in the negative y-direction to balance the dipolar field \nfrom the  RL. In order to analyse the magnetization dynamics, we numerically solve the following \nnon-linear differential equation, which includes the STT and the spin-orbit to rque from the SHE :20 \n                                      \n22\n0\n22\n00\n2\n01\n(1 ) (1 )\n(1 ) (1 )\n( , ) ( ) ( )( )\ne S\nJJ\nSS\nB MTJ\nT\nsd\nM dt\ndd\nMM\ngJg q V\nMt\n \n\n   \n\n    \n\n    \n\n     EFF EFF\np p pmm h m m h\nmmσ m σ\nm m m m m m m\n \n                                (1) \nbeing  m and mp the magnetization s of the FL and RL  respectively.  \nEFFh  is the FL effective  field \nand it contains, a s well as the standard magnetic fields, the magnetostatic coupling be tween the FL \nand RL and the DMI  contribution . The DMI energy density is  expressed by :17  \n                           \n 2DMI z z D m m   mm\n                                                    (2) \nwhere, because of the ultra -thin free layer , the magnetization spatial variation along the z-direction \nis neglected  \n0zm\n . D is the parameter which takes into account the inten sity of the DMI. From \nthe last equation, we can derive the additional term to the effective field related to the DMI:\n \n01DMI\nSM\nDMIhm. Furth ermore , g is the Landè factor , B is the Bohr magneton , 0 is the vacuum \nmagnetic permeability, e is the electron charge,  0 is the gyromagnetic ratio,   is the Gilbert \ndamping , Ms is the saturation magnetization , t is the FL thickness , JMTJ is the current density  flowing 4 \n through the MT J stack , \n22( , )=1T\nT\nTg\np\npmmmm  characterizes the angular dependence of the spin-\npolarization function for the MTJ as computed by Slonczewski ,21, 22 where  T is the polarization \nefficiency . q(V) is a function which takes into account the  squared voltage  dependen ce of the field -\nlike torque  up to a maximal value equal to the 25% of the in -plane torque .23, 24, 25 The coefficient dj \nis given by \nBH\nJ Ta\nSdJeM t , in which   is the spin Hall angle obtained  by the ratio between the \namplitude of the JSHE and the tantalum current JTa.26, 27  is the direction of the JSHE in the Ta strip. \nThe magnetic parameters for the micromagnetic study are: exchange constant A=2.0 x 10-11 J/m, \nmagnetic damping =0.021  and spin-hall angle  =-0.15.  \nFirstly, we consider th e STT effect  (FMR response with no  JTa) for a JMTJrf \nsen(2π)MAX rfJ f t  with \nan amplitude JMAX=0.5x106 A/cm2 and sw eeping its frequency frf from 3.0  GHz to 7.6 GHz . The \nFMR signal is computed as  the difference between the maximum and the minimum value of the  \noscillating  y-component of the average magnetization . Two scenarios are investigated : the first  one, \nwhere  the DMI  effect  is neglected ( D=0 mJ/m2)  and the other  one when the DMI is relevant  (D=-\n1.2 mJ/m2).17 Witho ut the DMI contribution , the FMR shows only one peak  at 5.8 GHz, (see Fig.1a  \nupper curve) . The insets near to  the peak of Fig. 1a  illustrate the spatial mode distribution s (SMD s) \nfor the y- and z-component of the magnetization at the FMR frequency , as comp uted with the \nmicromagnetic spectral mapping technique .28 As can be noted , a central mode is ex cited for the two \nmagnetization components. By c onsidering the DMI  (see lower curve ), the FMR response displays  \ntwo peaks: the  first one at a frequency of 5.7  GHz  (indicated with 1 in Fig. 1a) and the second one \nat 6.1 GHz  (indicated with 2 in Fig. 1a) . This FMR behavior is clearly due to the effect of the DMI , \nwhich splits the FMR  mode  in two , as also observed  in the SMD s. In fact, while the SMD of the y-\ncomponent  show s a similar central mode,  the SMD  of the z-component  displays the generation of \ntwo edge modes for the first peak and four edge modes for the second pea k. Moreover, observing  5 \n the time domain plot (not represented  here), we note that for the frequency of 5.7  GHz, the TMR  is \nin advance with respect to the injected microwave current; on the contrary, at the frequency of 6.1 \nGHz, the TMR is lagging behind the JMTJrf. The DMI influence s only the z-component  because of \nthe edge non -uniformities  induced by th e dipolar field.  \nFig. 1b  represents the FMR responses for a microwave  current  of JMAX=0.1x106 A/cm2. The FMR \nfrequency increases , either with out DMI (top curve) , reaching 6.1 GHz,  or with DMI (bottom \ncurve) , attain ing 6.2 GHz  and, additionally , both the FM R curves have a single peak . The evidence  \nof only one frequency peak, even  considering the DMI , is ascribed to the use of  a weak microwave  \ncurrent , which keeps  the FMR response in a linear regime . Also in this case, the SMD  of the \nmagnetization  z-component  shows an edge mode . In addition, the FMR frequency changes  (value \nand shape)  with JMAX  because  a higher amplitude of the microwave current generate s non -linear \ndynamics.2, 8 As demonstrated  in Ref. [ 23] (see Figure 3a) , this non -uniform regime can  be \nobserved  by the presence of an a symmetric FMR spectrum.  6 \n \n \nFig. 1: FMR responses for JTa=0 A/cm2. a) JMAX=0.5x 106 A/cm2 without DMI (top curve) and with DMI (bottom curve); \nb) JMAX=0.1x 106 A/cm2 with no DMI (upper curve) and with DMI (lower curve). The insets represent the SMDs for the \ny- and z- components of the magnetization.  \n \nFig. 2 shows the FMR response when a bias JTa=-1.50x107 A/cm2 flows in the Ta str ip with (upper \ncurve) and without ( lower curve) the DMI contribution. A microwave current of JMAX=0.5x106 \nA/cm2 is injected in  the MTJ stack . The top curve shows a similar behavio r with respect to the \ncorresponding  curve without the in -plane current  (Fig. 1a); in fact , a main central mode is visible in \nthe SMDs for both y- and z-component of the magnetization.  A similar behavio r is also obtained in \npresence of the DMI: two FMR peaks are visible  and the SMDs  have a configuration similar  to the \none previous ly investigated  (see for comparison the SMDs  in Fig. 1a lower curve) . Hence,  the FMR \nbehavior is not affected by a  sub-critical JTa, and the DMI effect concerns again  the splitting of the \nmain  mode.  7 \n \n \nFig. 2: FMR responses for  a sub -critical JTa=-1.50x107 A/cm2 and a JMAX=0.5x 106 A/cm2 without DMI (top curve) and \nwith DMI (bottom curve). The insets represent the SMDs for the y- and z- components of the magnetization.  \n \nThe FMR behavior is different  when the JTa is increased.  Fig. 3a shows the computed FMR with \n(upper  curve) and without ( lower  curve) DMI, when  both the JTa=-1.40x108 A/cm2 (this value is \nvery close to the switching one, which leads the FL from out -of-plane to in -plane ) and the \nmicrowave current with amplitude JMAX=0.5x106 A/cm2 are applied. Without the DMI, the \nincreasing  of JTa does not change the FMR qualitatively , but it induces a  reduction of the FMR \nfrequency , from  5.8 GHz  (Fig. 1a)  to 4.8 GHz . Whereas, taking  into account the DMI , a decreasing \nof the FMR frequency and  a low power  peak  at higher frequency are observed . Furthermore , the \nSMDs of the lower frequency peak show that the main excited mode is shifted from the central \nposition. Thus , with a  JTa large enough (that means a relevant SHE contribution), the DMI  move s \nthe SMD of the  main central mode towards the left side of the sample. Changin g the sign of D, the \ncentral mode moves to the right side (not shown). Fig. 3b displays the FMR for JTa=-1.40x108 \nA/cm2 and JMAX=0.1x106 A/cm2. In this case , while the FMR s with and without the DMI are very \nsimilar (top and bottom curves respectively ), a small displacement of the central mode is observed \nin the resonance frequency SMDs including  the DMI.   8 \n \n \nFig. 3:  FMR responses for JTa=-1.40x108 A/cm2. a) JMAX=0.5x 106 A/cm2 without  DMI (top curve) and with DMI \n(bottom curve); b)  JMAX=0.1x 106 A/cm2 with no DMI (upper curve) and with DMI (lower curve). The insets represent \nthe SMDs for the y- and z- components of the magnetization.  \n \nIn summary , the effect of the DMI on the FMR has been  analyzed  in both cases with and wi thout \nthe in -plane Ta  current. We have observe d that, regardless of the JTa, the effect of the DMI is to \nbreak the symmetry of the main central excited mode. However, the way of the symmetry breaking \nhas been  dependent on the Ta  current. If JTa is negligible or it assumes a sub-critical value, we have \nobserve d that DMI breaks the symmetry of the main central mode in two (or more, as function of \nthe peak) different edge modes. For large r JTa, the DMI contribution is not so great  to divide the \nmain excited mode and its effect is the displacement of the main central mode towards a lateral \nposition only.  9 \n References  \n                                                           \n1 S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley, R. J. Schoelkopf, R . A. Buhrman, and \nD. C. Ralph, Nature  425, 380  (2003 ). \n2 Z. Zeng , G. Finocchio , B. Zhang , P. K. Amiri , J. A. Katine , I. N. Krivorotov , Y. Huai , J. Langer , \nB. Azzerboni , K. L. Wang , and  H. Jiang , Scientific Reports  3, 1426 (2013 ). \n3 Z. Zeng, G. Finocchio, H . Jiang , Nanoscale 5, 2219 -2231 (2013) . \n4 A. M. Deac, A. Fukushima, H. Kubota, H. Maehara, Y. Suzuki, S. Yuasa, Y. Nagamine, K. \nTsunekawa, D. D. Djayaprawira, N. Watanabe, Nat. Phys. 4, 804 -809 (2008).  \n5 Z. M. Zeng, P . K. Amiri, I . N. Krivorotov, H . Zhao,  G. Finocchio,  J.-P. Wang, J . A. Katine, Y . \nHuai, J . Langer,  O. K. Galatsis, K . L.Wang, and  H. Jiang , ACS Nano 6, 6115 –6121 (2012).  \n6 C. Wang , Y. T. Cui, J. Z. Sun, J. A. Katine, R. A. Buhrman , and D. C. Ralp h, J. Appl. Phys . 106, \n053905 (2009).  \n7 A. A. Tu lapurkar, Y. Suzuki , A. Fukushim a, H. Kubot a, H. Maehara, K. Tsunekawa,  D. D.  \nDjayaprawira, N.  Watanabe , and  S. Yuasa , Nature 438, 04207 (2006).  \n8 J. C. Sankey, P. M. Braganca, A. G. F. Garcia, I. N. Krivorotov, R. A. Buhrman, and D. C. Ralph, \nPhys. Rev. Lett. 96, 227601  (2006).  \n9 J. Zhu, J. A. Katine, G. E. Rowlands, Y. -J. Chen, Z. Duan, J. G. Alzate, P. Upadhyaya, J. Langer, \nP. K.  Amiri , K. L. Wang, and I.  N. Krivorotov, Phys. Rev. Lett. 108, 197203  (2012).   \n10 L. Liu, C. F. Pai, Y. Li, H. W. Tseng, D. C.  Ralph, and R. A. Buhrman, Science  336, 555 –558 \n(2012).  \n11 T. Moriya, Phys. Rev. Lett.  4, 228 (1960).  \n12 S. X. Huang and C. L. Chien, Phys. Rev. Lett. 108, 267201 (2012).  \n13 S. Emori, U . Bauer, S -M Ahn, E . Martinez , and G . S. D. Beach , Nat. Mater. 12, 611 -616 (2013).  10 \n                                                                                                                                                                                                  \n14 G. Chen, J. Zhu, A. Quesada, J. Li, A. T. N’Diaye, Y. Huo, T. P. Ma, Y. Chen, H. Y. Kwon, C. \nWon, Z. Q. Qiu, A. K. Schmid, and Y. Z. Wu, Phys. Rev. Lett. 110, 177204 (2013).  \n15  K.-S Ryu, L. Thomas, S. -H Yang, and S. Parkin, Nat . NanoTechn. , 8, 527 (2013).  \n16 X. Z. Yu, Y. Onose, N. Kanazawa, J. H. Park, J. H. Han, Y. Matsui, N. Nagaosa , and Y. Tokura ,     \nNature, 465, 901 (2010).  \n17 E. Martinez, S . Emori, and G . S. D. Beach, Appl. Phys. Lett.  103, 072406 (2013).   \n18 L. Liu, C-F. Pai, D. C. Ralph, an d R. A. Buhrman, Phys. Rev. Lett. 109, 186602 (2012).  \n19 P. K. Amiri, Z. M. Zeng, J. Langer, H. Zhao, G. Rowlands, Y. -J. Chen, I. N. Krivorotov, J. \nP.Wang, H. W. Jiang, J. A. Katine, Y. Huai, K. Galatsis, and K. L. Wang, Appl. Phys. Lett. 98, \n112507 (2011).  \n20 G. Finocchio, M. Carpentieri, E. Martinez, and B. Azzerboni, Appl. Phys. Lett. 102, 212410 \n(2013).  \n21 J.C. Slonczewski, Phys. Rev. B 71, 024411 (2005).  \n22 J.C. Slonczewski, J.Z. Sun, J. Magn. Magn. Mater. 310, 169 (2007).  \n23 J. C. Sankey , Y.-T. Cui, J. Z. Sun, J. C. Slonczweski, R. A. Buhrman, and D. C. Ralph, Nat. \nPhys. 4, 67 (2008).  \n24 C. Wang , Y.-T. Cui , J. A. Katine , R. A. Bu hrman , D. C. Ralph , Nat. Phys . 7, 496  (2011) . \n25 M. Carpentieri, E. Ma rtinez, and G. Finocchio, J. Appl. Phys. 110, 093911 (2011 ). \n26 J. E. Hirsch , Phys. Rev. Lett. 83, 1834  (1999). \n27 A. Manchon  and S. Zhang , Phys. Rev. B 79, 094422 (2009) . \n28 M. Carpentieri, G. Finocchio,  B. Azzerboni, and Luis Torres , Phys. Rev. B, 82, 094434 (2010).  \n "    },    {        "title": "1409.5390v1.Interference_effects_induced_by_Andreev_bound_states_in_a_hybrid_nanostructure_composed_by_a_quantum_dot_coupled_to_ferromagnetic_and_superconductor_leads.pdf",        "content": "arXiv:1409.5390v1  [cond-mat.mes-hall]  18 Sep 2014InterferenceeffectsinducedbyAndreevboundstatesin a hy bridnanostructure\ncomposedbya quantumdotcoupledtoferromagneticand super conductorleads\nE.C. Siqueira∗1,2,P.A. Orellana3, A.C.Seridonio2, R.C. Cestari2, M. S.Figueira4,G. G. Cabrera51\n1Departamento de Física, Universidade Tecnológica Federal do Paraná (UTFPR), 84016-210, Ponta Grossa, PR,\nBrasil\n2Departamento de Física e Química, Universidade Estadual Pa ulista (UNESP), 15385-000, Ilha Solteira, SP,\nBrasil\n3Departamento de Física, Universidad Técnica Federico Sant a Maria, Av. Vicuña Mackenna 3939, Santiago,\nChile\n4Institutode Física,Universidade Federal Fluminense, 242 10-340, Niterói,RJ,Brasil\n5Instituto de Física ‘Gleb Wataghin’, Universidade Estadua l de Campinas (UNICAMP), Campinas 13083-859, SP,\nBrasil\nInthiswork,it isconsideredananostructurecomposedbyaq uantumdotcoupledtotwoferromagnetsandasupercon-\nductor. The transport properties of this system are studied within a generalized mean-ﬁeld approximation taking into\naccountproximityeffectsandspin-ﬂipcorrelationswithi nthequantumdot. Itisshownthatthezero-biastransmittan ce\nfor the co-tunneling between the ferromagnetic leads prese nts a dip whose height depends on the relative orientation\nofthe magnetizations. When the superconductoriscoupledt o the system,electron-holecorrelationsbetweendifferen t\nspinstatesleadstoaresonanceintheplaceofthedipappear inginthetransmittance. Suchaneffectisaccompaniedby\ntwo anti-resonancesexplainedby a “leakage” of conduction channelsfrom the co-tunnelingto the Andreev transport.\nInthenon-equilibriumregime,correlationswithinthequa ntumdotintroducea dependenceofthe resonancecondition\non the ﬁnite bias applied to the ferromagnetic leads. Howeve r, it is still possible to observe signatures of the same\ninterferenceeffectintheelectricalcurrent.\nI. INTRODUCTION\nConventional superconductivity ( s-wave) and ferromag-\nnetism present different spin symmetries required by their\norder parameters. In s-wave superconductors, the Cooper\npairs are in the singlet state while exchange interaction in -\nduces a triplet alignment in ferromagnets. As a result, su-\nperconductivity is strongly suppressed or even completely\ndestroyed in bulk compounds in which both order parame-\nters would be present1. On the other hand, the production\nof superconductor/ferromagnet(F/S) layered systems has a l-\nlowed the study of the interplay between superconductivity\nandferromagnetism2experimentally. Inthesesystems,super-\nconductorandferromagnetarespatiallyseparatedbyanarr ow\ninterfacewherebyCooperpairscandiffuseintotheferroma g-\nnet. Conversely,spin polarizationcan be inducedinto the s u-\nperconductor by diffusion of ferromagnetic electrons. The se\neffectsare called proximity effects and are the responsible for\nparticular features of F/S systems. In fact, the thermal, ma g-\nneticandelectricalpropertiesofthesesystemsarecomple tely\ndifferent in comparison to the ferromagnetand superconduc -\ntors in their bulk form2,3. Within the vast set of phenomena,\none can highlight the π-phase transition of the superconduc-\ntor order parameter in S/F/S Josephson junctions4, the non-\nmonotonic behavior of the superconducting critical temper a-\nture(Tc)onthelayersthicknessesinmultilayerssystems5and\noscillationsonelectronicdensityofstates2.\nConcerningtransportproperties,F/Ssystemsyieldcontro l-\nling the charge current through the spin degree of freedom\nwhichisofinterestinareaslikespintronics6,7. Infact,forbias\nvoltageswithin the superconductorgap, the currentis carr ied\nvia Andreev reﬂections (ARs)8,9. In this process, an incident\nelectron of a given energy Eand spin up (down) is reﬂected\nonthe superconductorasa holeofenergy −Eandspin down\nFIG. 1. (Color Online) Schematic diagram for the F1-(QD-S)-F2\nsystem. The magnetization of F1is assumed to be ﬁxed and the\nmagnetization of F2can be varied of an angle θwith respect to the\nF1magnetization. V1andV2are the external potentials applied to\nF1andF2, respectively, while the superconductor is grounded. Vg\nbeingthe gate potential applied tothe QD.\n(up). As a result, the current in the ferromagnet is converte d\ninto a Cooper pair current in the superconductor. The cur-\nrent is different of zero only if there are available states f or\nboth spins around the Fermi level. However, the occupation\nof these states is dependent on the ferromagnet polarizatio n.\nWhentheferromagnetpolarization( P)isequaltounitythere\narenoavailablestatesforholeswithspindownandthesyste m\nbehavesasaninsulator;for P= 0theferromagnetisunpolar-\nizedandthe currentreachesa maximumvalue.\nTheelectronsformingtheCooperpairarehighlycorrelated\ninlargedistancesincomparisontointeratomicdistances. This2\nfeature has been explored by Deutscher and Feinberg10to\npropose a non-local Andreev reﬂection (called crossed AR),\nwhere two electrons of different leads can combine into a\nCooperpairifthedistancebetweentheseleadsissmallerth an\nthesuperconductorcoherencelength11,12. Sincethisproposal,\nthere has been a profusion of works exploring crossed AR in\ndifferent conductionregimes13,14(ballistic and diffusive)and\nother materials like superconductor-induced graphene15and\ncupratesuperconductors16.\nWith the recent technology of production of quantum dots\n(QDs), it is also possible to implement hybridnanoelectron ic\ndevicescombiningsuperconductivityandferromagnetism. In\nthese systems one is able to investigate purely quantum phe-\nnomena and the conduction of electrical current with a dis-\ncrete ﬂux of charges through the QDs17. As a result, new\nphenomena may arise concerning the transport properties of\nthese systems18–33. A particular feature of these hybrid F/S\nsystems is the current dependenceon QD spectral properties ,\nbesides the ferromagnet polarization. Additionally, elec trons\naresqueezedintotheQDthroughwhichthecurrentisinjecte d\ninto the superconductor. As a result, the simplest theoret-\nical treatment for these systems must take into account the\nCoulombcorrelationsinsidetheQDs17,34.\nIn the aforementioned papers, such a correlation is treated\nby using a perturbativeapproachsince its many-bodyfeatur e\nforbids any exact approach to the problem. Many approxi-\nmation schemes to treat the electronic correlationexist in the\nliterature in order to better describe the current ﬂow in the se\nhybridsystems. However,asmentionedintheﬁrstparagraph ,\nsystemsinvolvingsuperconductorsexhibittheso-calledp rox-\nimity effect which induces pair correlations in the materia l\nin contact with the superconductor2. The interplay between\npaircorrelationswithCoulombinteractionmaylead to rath er\ncomplex spectral properties for the QD35. More speciﬁcally,\na multi-peak structure has been observed in a F−QD−S\nnanostructure caused by the interplay between Coulomb cor-\nrelations and Zeeman splitting due to an external magnetic\nﬁeld36.\nIn order to explore the role of the QD spectral properties\non the transport of F/S systems, we consider in this work a\nthree-terminal F1−(QD,S )−F2nanostructureasillustrated\nin Fig. 1. In this system, the QD is coupled to two ferro-\nmagnets in such a way that a voltage bias is applied to F1\nwhile F2isgrounded. Thetransmittancewithandwithoutthe\npresence of the superconductorlead is considered in order t o\ndetermine the role of the AR in the co-tunneling current be-\ntween F1andF2. That orientation of the magnetization for\nthe lead F1is ﬁxed while the magnetizationof F2is directed\nto an angle θwith respect to F1. It may be varied from 0\n(parallel conﬁguration) to π(antiparallel conﬁguration). The\ncorrelationswithin the QD are treated by using a generalize d\nmean-ﬁeld approximation taking into account proximity ef-\nfectsduetothesuperconductorandspin-ﬂipprocesseswith in\nthe QD. While the spin-ﬂip process has been addressed in a\nphenomenological way in previous works37,38, here such an\neffect is a natural result of the interplay between Coulomb\ncorrelationsandthemisalignmentofthemagnetizationsfr om\nthe ferromagnetic leads. It worth mentioning that the geom-etry shown in Fig. 1 has already been studied considering\nthe situation in which the QD is noninteractingand consider -\ning the non-local transport due to crossed AR39,40. More re-\ncently, spin-dependent conductance and thermoelectric pr op-\nerties were addressed for this nanostructurewhere the role of\nARare considered41. InRef. 42, the differentialconductance\nandthemagnetoresistancehavebeenstudiedinwhicha zero-\nbias anomaly in the Andreev conductance is reported. Such\nan anomaly is explained by the spin-accumulation generated\nwithin the QD due to the coupling to ferromagnets. In this\nwork, we focus on the co-tunnelingprocess and how it is af-\nfected by the coupling to the superconductor. We observe a\nresonance appearing in the transmittance due to the interpl ay\nbetweendifferentspin-channelsandtheAndreevboundstat es\nwithintheQD.\nThis paper is organized as follows: in Sec. II we present\nthe model for the system displayed in Fig. 1 and the physical\nquantities are determined by using the formalism of non-\nequilibrium Green’s functions. In Sec. III the results are\npresented and discussed. Finally, a summary and the main\nconclusionsarepresentedinSec. IV.\nII. MODEL ANDFORMULATION\nIn this section we provide a general description of the for-\nmalism to be used to carry out the calculations of the phys-\nical quantities. We have used the Keldysh formalism within\nthe Nambunotation43,44whichallowsusto describespin and\nelectron-hole degrees of freedom in the same footing. This\nis widely used to tackle systems involving ferromagnets and\nsuperconductors.\nA. Hamiltonian\nTheHamiltonianisgivenbyasumoftermsdescribingeach\npartofthesystemillustratedinFig. 1. Theferromagnetica nd\nsuperconductor leads are considered to be non-interacting in\nsuch a way that mean ﬁeld theories can be applied to model\ntheseleads. Thequantumdotisconsideredtobecomposedby\na single levelspin degeneratedwith the presenceof Coulomb\ncorrelations. ThecouplingbetweentheQDandleadsistaken\ninto account phenomenologically by means of a tunneling\nHamiltonian. Inthisway,thefullHamiltonianiswrittenas\nˆH=ˆH1+ˆH2+ˆHS+ˆHC+ˆHT. (1)\nThe terms ˆH1andˆH2are the Hamiltoniansdescribing the\nferromagnets F1andF2, respectively. Explicitly, these are\ngivenby:\nˆH1=/summationdisplay\nkˆΦ†\n1kˆE1,k(0)ˆΦ1k (2)\nand\nˆH2=/summationdisplay\nkˆΦ†\n2kˆE2,k(θ)ˆΦ2k (3)3\nwhere we have deﬁned the Nambu spinor ˆΦηk=\n(ˆf†\nηk↑ˆfηk↓ˆf†\nηk↓ˆfηk↑)†where ˆf†\nηkσand ˆfηkσcreates\nan electron and a hole, respectively with spin σand wave-\nvector kin the ferromagnet Fη,η= 1,2. The matrix ˆEη,k(θ)is written in the 4×4Nambu space resulting from the tensor\nproduct between electron-hole and spin spaces. The general\nformof ˆEη,k(θ)isgivenby:\nˆEη,k(θ) =\nǫk−hηcosθ−µη 0 −hηsinθ 0\n0 −(ǫk+hηcosθ−µη) hηsinθ\n−hηsinθ 0 ǫk+hηcosθ−µη 0\n0 hηsinθ 0 −(ǫk−hcosθ−µη)\n. (4)\nThe ferromagnets are modeled by the Stoner model45in\nwhichthe spinbandsof Fηaresplit byaninternalmean-ﬁeld\nhηproducing a ﬁnite polarization of the electron gas. The\nmagnetizationof F1isconsideredtopointtoaﬁxeddirection\nwhile the magnetization of F2can be rotated by an arbitrary\nangle θ. The chemical potentials of each ferromagnetare de-\nterminedbyanexternalvoltagebias µη=eVηwhichcontrols\ntheFermilevelofeachelectrodeindependently.\nThe superconductor is considered to be a conventional\nsuperconductor ( s-wave) being well described by the BCS\nHamiltonian11. IntheNambunotationthisHamiltonianreads:\nˆHS=/summationdisplay\nkˆΦ†\nskˆES,kˆΦsk (5)\nwith ˆΦsk= (ˆs†\nk↑ˆsk↓ˆs†\nk↓ˆsk↑)†and\nˆES,k=\nǫk−µS ∆∗0 0\n∆ −(ǫk−µS) 0 0\n0 0 ( ǫk−µS) −∆∗\n0 0 −∆ −(ǫk−µS)\n.\nThesuperconductingcorrelationsenterbymeansofthepair\namplitude ∆whichingeneralisacomplexnumberdepending\nonk. Sinceweareusingjustonesuperconductorlead,weuse\nthe well known assumption39,43,44in which ∆is just a con-\nstant real number. In addition, the superconductor chemica l\npotentialisﬁxedtozeroastheground( µS= 0).\nThe quantum dot is considered to be interacting with one\nleveldegeneratedin spin,\nˆHC=ˆΨ†\ndˆEdˆΨd+Uˆnd↑ˆnd↓ (6)\nwhere ˆΨd= (ˆd†\n↑ˆd↓ˆd†\n↓ˆd↑)†,and\nˆEd=\nεd0 0 0\n0−εd0 0\n0 0 εd0\n0 0 0 −εd\n.\nWe consider that the QD level can be displaced by means\nof a gate voltage Vg, thus, εd=ε0−eVgwith ε0being the\nbare QD level (spin degenerated). The Coulomb correlations\nare describedby Uˆnd↑ˆnd↓whoseintensity is controlledby Uwhichisconsideredtobesmallerthanthesuperconductorga p\n∆.\nThe tunneling between the QD and the leads is described\nby\nˆHT=/summationdisplay\nkγ[ˆΦ†\nγkˆVγkˆΨd+ˆΨ†\ndˆV†\nγkˆΦγk](7)\ninwhich\nˆVγk=\nVγk 0 0 0\n0−V∗\nγk0 0\n0 0 Vγk 0\n0 0 0 −V∗\nγk\n\nwhere γ= 1,2,sisthe tunnelingamplitude. Since theenergy\nrangeislimitedtothenarrowsuperconductorgap,itisagoo d\napproximationtoconsider ˆVγkindependenton k.\nB. Green's functions\nIn order to calculate the transport properties we have used\nthenon-equilibriumGreen’sfunctionmethod46. Allthephys-\nical quantities can be cast in terms of the Green’s function o f\ntheQD.IntermsofNambuspinors,the“lesser”( G<)andre-\ntarded/advancedGreen’sfunction( Gr/a) of theQD are writ-\ntenas\nG<(t1,t2) =i/an}bracketle{tˆΨd(t1)⊗ˆΨ†\nd(t2)/an}bracketri}ht (8)\nand\nGr/a(t1,t2) =∓iϑ(±t1∓t2)/an}bracketle{tˆΨd(t1)⊗ˆΨ†\nd(t2)\n+ˆΨ†\nd(t2)⊗ˆΨd(t1)/an}bracketri}ht,(9)\nwherethe symbol ⊗denotesa tensorproduct. Similar deﬁni-\ntions are given for the leads Green’s functions which can be\nexpressedintermsofEqs. (8) and(9).\nByusingtheequationofmotionapproachtechnique,along\nwith the mean-ﬁeld approximation (discussed in Appendix\nsection), we obtain the Dyson’s equation for the retarded\nGreen’sfunction:\nGr/a(ε) =gr/a(ε) +gr/a(ε)Σr/a(ε)Gr/a(ε)(10)4\nwhere gr/aistheGreen’sfunctionforthenon-interactingQD\nisolatedfromtheleads. It iswrittenas\ngr/a=\n(x−εd)−10 0 0\n0 ( x+εd)−10 0\n0 0 ( x−εd)−10\n0 0 0 ( x+εd)−1\n\nwherewehavedeﬁned x=ε±iηandεd=ε0−eVg.\nTheself-energy Σr/aisgivenby\nΣr/a(ε) =Σr/a\n0(ε) +Θ (11)\nwhereΘencodes the electronic correlations and Σr\n0models\nthecouplingbetweentheQD andleads, i.e.,\nΣr\n0(ε) =Σr\ns(ε) +Σr\n1(ε) +Σr\n2(ε)\nwith\nΣr\ns(ε) =−i\n2Γs̺(ε)\n1 −∆/ε 0 0\n−∆/ε 1 0 0\n0 0 1 ∆ /ε\n0 0 ∆ /ε 1\n(12)\nmodelling the coupling to the superconductor where Γs=\n2π|Vs|2Ds(εF)withDs(εF)beingthedensityofstatesofthe\nsuperconductor at the normal state solved at the Fermi level\nandVsis tunnelingamplitude. We have also deﬁned the gen-\neralizedsuperconductordensityofstates,\n̺(ε) =|ε|ϑ(ε−∆)√\nε2−∆2−iεϑ(∆− |ε|)√\n∆2−ε2.\nin which the ﬁrst term is the conventional BCS density of\nstates11,˜̺(ε) =Re[̺(ε)]andthesecondtermaccountsforthe\nAndreevboundstatescorrespondingtoevanescentwavesrep -\nresenting the conversion of quasiparticles into Cooper pai rs\nwithinthesuperconductor12.\nNext, we deﬁne the self-energy due to the coupling with\nF1:\nΣr\n1(ε) =−i\n2\nΓ1↑0 0 0\n0 Γ 1↓0 0\n0 0 Γ 1↓0\n0 0 0 Γ 1↑\n (13)\nwithΓ1σ= 2π|V1|2D1σ(εF)where V1the hoppingtermand\nD1σ(εF)isthedensityofstatesperspinatthe F1Fermilevel.\nThecouplingwith F2exhibitsa similar form,however,the\nF2quantizationaxisisrotatedbyanangle θ:\nΣr,a\n2(ε) =∓i\n2\nA↑0B0\n0A↓0B\nB0A↓0\n0B0A↑\n,(14)\nwith Aσ≡c2Γ2σ+s2Γ2¯σ,B=sc(Γ2↑−Γ2↓),\ns≡sinθ/2and c≡cosθ/2. We also have deﬁned\nΓ2σ= 2π|V2|2D2σ(εF)where V2the hopping term and\nD2σ(εF)is the density of states per spin at the F2Fermilevel. Itisworthmentioningthatwehavetakenthewide-ban d\nlimitinwhichthe Γ1,Γ2andΓsareassumedtobeconstants.\nC. Physical Quantities\nInthissectionwederivethephysicalquantitiesandtherel -\nevant parameters used in the analysis of the results present ed\ninSec. III.\n1. Ferromagnet Polarization\nWithintheStonermodel,theelectronsgaspolarizationisa\nresult of the exchangemean ﬁeld due to the electron-electro n\ninteraction. In this, way we deﬁne the polarization for the\nferromagnet Fαasfollows:\nPα=Γα↑−Γα↑\nΓα↑+ Γα↑(15)\nwhere α= 1,2. Here,the couplingconstants Γασare consid-\neredasindependentparameters.\n2. Electrical Current\nByusingtheequationofmotionmethod,oneisabletocal-\nculate the current between the ferromagnet Fαand the QD.\nBy using the time variation of the average occupation of the\nleadFαweobtainthefollowingequation:\nIα=e\nh/integraldisplay\ndε/bracketleftbig\nGr(ε)Σ<\nα(ε) +G<(ε)Σa\nα(ε) +H.c./bracketrightbig\n11+33\n(16)\nwhere α= 1,2. Thecurrent I1isexplicitlywrittenas\nI1=I12+I1s (17)\nwherewe havedeﬁnedtheco-tunnelingcurrentas\nI12=e\nh/integraldisplay\nT12(f1−f2),dε, (18)\nandthecurrentﬂowingbetween F1andthesuperconductoris\ngivenby\nI1s=e\nh/integraldisplay/bracketleftbig\nTAR,11(f1−¯f1) +TAR,12(f1−¯f2)/bracketrightbig\ndε,(19)\nwhere f1andf2arethecorrespondingFermidistributionsfor\nelectrons in the leads F1andF2and¯f1and¯f2are the cor-\nrespondingones for holes. By comparingthe Fermi distribu-\ntions one is able to determine each contribution in Eq. (17).\nInfact, T12istheco-tunnelingcurrentofelectronsfrom F1to\nF2through the QD; TAR,11accounts for the Andreev reﬂec-\ntionin F1andﬁnally TAR,12isthecrossedAndreevreﬂection\nof anelectronfrom F1as a holein F2. Thetransmittanceex-\npressionsforeachprocessaregivenby5\nTAR,11= Γ 1↑/parenleftbig\n|Gr\n14|2Γ1↑+|Gr\n12|2Γ1↓/parenrightbig\n+ Γ1↓/parenleftbig\n|Gr\n34|2Γ1↑+|Gr\n32|2Γ1↓/parenrightbig\n(20a)\nTAR,12= Γ 1↑[(c2Γ2↑+s2Γ2↓)|Gr\n14|2+ (s2Γ2↑+c2Γ2↓)|Gr\n12|2+sc(Γ2↑−Γ2↓)([Gr\n12]∗Gr\n14+ [Gr\n14]∗Gr\n12)](20b)\n+Γ1↓[(c2Γ2↑+s2Γ2↓)|Gr\n34|2+ (s2Γ2↑+c2Γ2↓)|Gr\n32|2+sc(Γ2↑−Γ2↓)([Gr\n32]∗Gr\n34+ [Gr\n34]∗Gr\n32)]\nT12= Γ 1↓[(s2Γ2↑+c2Γ2↓)|Gr\n33|2+ (c2Γ2↑+s2Γ2↓)|Gr\n31|2+sc(Γ2↑−Γ2↓)([Gr\n33]∗Gr\n31+ [Gr\n31]∗Gr\n33)](20c)\n+Γ1↑[(s2Γ2↑+c2Γ2↓)|Gr\n13|2+ (c2Γ2↑+s2Γ2↓)|Gr\n11|2+sc(Γ2↑−Γ2↓)([Gr\n13]∗Gr\n11+ [Gr\n11]∗Gr\n13)].\nThe corresponding equation for I2can be obtained from I1\njust replacing the 1→2in the previous equations. We point\nout that the expressionfor the currentis the same as obtaine d\nby Y. Zhu et. al.39for a noninteracting QD. In the present\ncase, in spite from the fact of the current formula resembles\nthe one obtained in Ref. 39, it is being considered the pres-\nence of interactionsinto the QD which means that the matrix\nelementsoftheGreen’sfunctionmustbedeterminedinaself -\nconsistent calculation, due to Eq. (25). However, within th e\napproximationusedinthiswork,it isstill possibletoobta ina\nLandauer-like equation for the current. This is an advantag e\nin the sense that one can obtain analytic expressions for the\ntransmittances T12,TAR,11andTAR,12.\nIII. RESULTSANDDISCUSSION\nInthefollowingresults,weconsidertheAndreevregimein\nwhich the applied bias range is boundedby the superconduc-\ntor energygap. Since this quantityis the natural energysca le\nof the problem, all the physical parameters are presented in\nunits of the superconductor gap. We start with the zero-bias\nregimeandthenthe ﬁnite-biascase isconsidered.\nA. Zero-bias regime\nIn order to clarify the effects of the electronic correlatio n\nwithin the QD on transport properties, we consider the zero-\nbias regimeﬁrstly and analyzethe role of the interactionsa p-\npearingintheself-energy,Eq. (11). Asshowin thefollowin g\nresults, the mean-ﬁeld approximation just renormalizes th e\nQD energy level developing the same role in the system as\nthegatevoltages.\n1. Zero-Bias Transmittance for F1−QD−F2system\nWeconsiderthetransmittanceforelectronsbetweenthefer -\nromagnetic leads through the QD. In this case, we start with\nsimplest case in which the electronic correlations are abse nt.\nIn Fig. 2 it is shown the effect of the magnetization on the\nelectronictransport. By settingthe relativeanglebetwee nthemagnetizationsof F1andF2toθ=π/4, an intermediatean-\ngle, we have calculated the transmittance for different val ues\nofP1,thepolarizationof F1. InFig. 2a,for P1= 0thetrans-\nmittance curveis just a resonancewhose width is determined\nbythehybridizationbetweenthediscreteleveloftheQDwit h\nthe continuum of states from the ferromagnet bands. When\nP1is increased a sharp dip emerges for ε= 0whose height\nincreases with P1. For P1= 1this dip reachesthe horizontal\naxisandthetransmittanceiszerofor ε= 0.\nIn Fig. 2b the evolution of the dip is studied by varying\nthe angle θwhile the polarization P1is ﬁxed to the unity as\nthe limit case in which the dip exhibits the most pronounced\nsize. For θ= 0the resonance behavior is recovered but as\nlongas θisdifferentofzerothe dipappearsandthetransmit-\ntance is pushed to zero at ε= 0. As θincreases towards π\nthe transmittance is suppressed and the dip opens up reveal-\ning a two peak structurewhich is illustrated by the curvesfo r\nθ= 1.57andθ= 2.62. Notice that the transmittance is zero\nin the whole range when θ=πand the ferromagnetsare full\npolarized. Thezero-biasresultsofFig. 2 are easily explai ned\nbyconsideringtheexpressionforthetransmittance T12given\nby Eq. (20c) where it can be noted that the effect of the fer-\nromagnetismistocreatetwointerferingchannelsforspins up\nand down. This interferencepattern resulting in the transm it-\ntance curves of Fig. 2 is dependent on the polarizations P1,\nP2and the angle θ. Additionally, the matrix elements of the\nretarded Green’s function encode the processes in which the\nelectronisscatteredwithintheQD.Toillustratethispoin t,we\nnotice that the off-diagonal elements Gr\n13andGr\n31represent\naspin-ﬂipprocessintotheQDduetothemisalignmentofthe\nmagnetizationoftheferromagnets. As θissetto 0orπ,while\ntheferromagnetsarefullpolarized,thesecontributionsa rere-\nmovedandtheinterferenceeffectiscompletelysuppressed . In\nthiscase,thetransmittanceiseitheramaximum(when θ= 0)\nor0 (when θ=π) whenthere areno statesavailable forelec-\ntronsinbothmagnets.\nNext, we consider the effect of the gate voltage on the\ntransmittance curves. In Fig. 3, a contour plot for trans-\nmittance in terms of the energy εand the gate voltage, Vg,\nis shown. In order to explore the interference effect relate d\nto the different spin channels of conduction, we have used\nP1= 0.95andθ=π/4which leads to a small dip on the\ntransmittance. The resulting contour plot exhibits a well l o-\ncalizeddiagonallineconnectingthepoints (ε= 1,eVg=−1)6\n/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s46/s48/s48/s46/s51/s48/s46/s54/s48/s46/s57\n/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48/s40/s97/s41/s84\n/s49/s50/s32/s32/s80\n/s49/s61/s48/s46/s56/s51\n/s32/s80\n/s49/s61/s49/s46/s48/s48/s32/s80\n/s49/s61/s48/s46/s48/s48\n/s32/s80\n/s49/s61/s48/s46/s53/s48/s32\n/s32\n/s32 /s61/s32/s49/s46/s53/s55\n/s32 /s61/s32/s50/s46/s54/s50/s32/s40 /s41\n/s32 /s61/s32/s48/s46/s48/s48\n/s32 /s61/s32/s48/s46/s53/s50/s84\n/s49/s50/s32\n/s40/s98/s41\n/s32/s40 /s41/s32\nFIG. 2. (Color Online) Transmittance curves ( T12) for the system\nF1−QD−F2, i.e., in the absence of the superconductor lead. (a)\nT12curvesfordifferentpolarizationvalues P1oftheferromagnet F1\nwhile the magnetization of F2is aligned at an angle θ=π/4with\nthe respect to the magnetization of F1. (b)T12curves with P1ﬁxed\nto unity and changing the magnetization angle of F2from a parallel\nalignment θ= 0towardsaorientationcloseto θ=πwhere T12= 0,\nseeEq. (21). Fixedparameters: V1=V2= 0,Γ1= 0.40,Γ2= 0.40,\nΓs= 0andP2= 1.0. All the parameters are scaled by the energy\ngapof the superconductor lead.\nand(ε=−1,eVg= +1). Thismeansthattheeffectofthegate\nvoltage is just displace the point at which the completely de -\nstructive interference occurs. In fact, this behavior can a gain\nbe understoodby consideringthe expressionfor T12and not-\ning that the gate voltage just renormalizes the QD level. In\nparticular, for the full polarized case, it is possible to de -\nrivearathercompacttransmittanceexpressionbysubstitu ting\nthecorrespondingGreen’sfunctionsmatrixelementsintoE q.\n(20c). After some algebra, one ends up with the following\nexpression:\nT12(˜ε) =4˜ε2/tildewideΓ2(θ)\n[/tildewideΓ(θ−π/2)]4+ ˜ε2[Γ2\n1+ 2/tildewideΓ2(θ) + Γ2\n2+ ˜ε2](21)\nwhere /tildewideΓ(θ) =√Γ1Γ2cos(θ/2),˜ε=ε−εd,εd=ε0−eVgwith ε0being the bare QD level and Γi= (Γ 1↑+ Γ i↓)/2,\ni= 1,2arethespinaveragedcouplingsoftheQDwith F1and\nF2. Bysettingthecondition T12= 0oneobtainsthatthedipis\nlocatedat ε=−eVgwhere we haveconsideredthe barelevel\nof the QD, ε0= 0and the electronic charge constant e >0.\nAccordingly, the equation ε=−eVgdescribes the diagonal\nline that locates the dip in the transmittance in Fig. 3a. For\npolarization values slight smaller than unity, there would be\nconstantaddedtoright-handsideof ε=−eVgwhichleadsto\nadipwithheightsmallerthanastheoneshowninthecontour\nplot. However,theminimumvalueof T12isstill locatedasin\nthefullpolarizedcase. InFigs. 3band3csomerepresentati ve\ncurvesof T12areillustratedwhoselocationinthecontourplot\nis given by the horizontal lines labeled by A1,B1andC1for\nnegativevaluesof eVgandE1,F1andG1forthecorrespond-\ning negative values of eVg. It can be noted that the gate just\ndisplaces the dip along the ε-axis and the curve still carries a\nsymmetric proﬁle with respect to the dip. In this way, the ef-\nfectofthegatevoltageisjust toproducea rigiddisplaceme nt\nof the point at which the destructiveinterferencebetween t he\nspinchannelsoccurs.\nIn Fig. 3d it is shown the transmittance T12contour plot\nunder the presence of electronic correlations at the QD. The\nstrength of these correlations is given by the parameter U=\n0.80in superconductor gap units. In this case, T12is de-\npendent on the occupation of the QD for both spins and on\nthe spin-ﬂip averages of the form /an}bracketle{tˆd†\nσˆd¯σ/an}bracketri}htwith σ=↑,↓and\n¯σ=−σ, c.f. Eq. (25). As a result, the symmetry with re-\nspect to the sign of both, the gate voltage Vgand energy ε\nis broken as evident from the contour plot in Fig. 3d. The\ncorrelations enter into the expressions for Green’s functi ons\nthrough Eq. (25) which leads to a self-consistent calculati on\nbymeansofthe KeldyshequationgivenbyEq. (27)(see Ap-\npendix). In Figs. 3e and 3f some representative curves are\nalso illustrating an additional lack of symmetry on the tran s-\nmittancecurves. Theadjacentpeakslocatedateachsideoft he\ndip are now presentingdifferentheights. As evidentfrom th e\ncurvesA2andB2the right peak is higher and wider in com-\nparison to the left peak. This trend is inverted for Vg∼0.32\nwhere the right peak is suppressedand transmittanceexhibi ts\na higher value for the left peak. This behavior is maintained\nfrom larger values of Vgas one can see from the curves C2\nup toG2. These results show that the interaction on the QD\njustprovidesminorchangesonthetransportpropertieswit hin\nthe mean-ﬁeld approximation used in this work. In fact the\nphysics is ruled by the coupling constants appearing in Eq.\n(20c) which moderate the role of each matrix element of the\nGreen’sfunctionoftheQD.\n2. Zero-Bias Transmittance for F1−(QD,S )−F2system\nNext, we consider the full system with the presence of the\nsuperconductorleadcoupledtotheQDasillustratedinFig. 1.\nItisworthrecallingthatweareinterestedintheAndreevco n-\nduction regime in which all the parameters are restrict to en -\nergieswithinthe superconductorgap. Inthisrangeofenerg y,\nthesuperconductoracts asa barrierwhichrulesoutthe dire ct7\n/s45/s49 /s46/s48 /s45/s48 /s46/s53 /s48 /s46/s48 /s48 /s46/s53 /s49 /s46/s48 /s45/s49 /s46/s48 /s45/s48 /s46/s53 /s48 /s46/s48 /s48 /s46/s53 /s49 /s46/s48 /s84 \n/s49/s50/s32\n/s70 /s49/s69/s49/s68 /s49/s65 /s49\n/s66 /s49\n/s67 /s49\n/s32/s32\n/s48 /s46/s49 /s48 /s46/s53 /s48 /s46/s57 \n/s65 /s50\n/s66 /s50\n/s67 /s50\n/s68 /s50\n/s69/s50\n/s70 /s50\n/s45/s49 /s46/s48 /s45/s48 /s46/s53 /s48 /s46/s48 /s48 /s46/s53 /s49 /s46/s48 /s45/s49 /s46/s48 /s45/s48 /s46/s53 /s48 /s46/s48 /s48 /s46/s53 /s49 /s46/s48 /s32\n/s32\n/s32/s32\n/s48 /s46/s49 /s48 /s46/s54 /s49 /s46/s48 \n/s45 /s49/s46 /s48 /s45 /s48/s46 /s53 /s48/s46 /s48 /s48/s46 /s53 /s49/s46 /s48/s48/s46 /s48/s48/s46 /s51/s48/s46 /s54/s48/s46 /s57/s84\n/s49/s50\n/s40 /s41 \n/s40 /s41 /s32 /s32 \n/s40 /s41/s101/s86\n/s103 /s40 /s41\n/s84\n/s49/s50/s65 /s49 /s66 /s49 /s67 /s49\n/s45 /s49/s46 /s48 /s45 /s48/s46 /s53 /s48/s46 /s48 /s48/s46 /s53 /s49/s46 /s48/s48/s46 /s48/s48/s46 /s51/s48/s46 /s54/s48/s46 /s57/s40/s100/s41\n/s40/s99/s41/s70 /s49 /s69/s49 /s68 /s49\n/s32 /s32 /s32 /s45 /s49/s46 /s48 /s45 /s48/s46 /s53 /s48/s46 /s48 /s48/s46 /s53 /s49/s46 /s48/s48/s46 /s48/s48/s46 /s51/s48/s46 /s54/s48/s46 /s57\n/s67/s50 \n/s40/s102/s41/s40/s101/s41/s84\n/s49/s50\n/s40 /s41 \n/s40 /s41 /s32 /s32 /s101/s86\n/s103 /s40 /s41\n/s84\n/s49/s50/s65 /s50\n/s66 /s50\n/s45 /s49/s46 /s48 /s45 /s48/s46 /s53 /s48/s46 /s48 /s48/s46 /s53 /s49/s46 /s48/s48/s46 /s48/s48/s46 /s51/s48/s46 /s54/s48/s46 /s57/s40/s98/s41\n/s85 /s61/s48/s46/s56/s48\n/s70 /s50/s69/s50 /s68 /s50\n/s32 /s32 /s32 \n/s40 /s41/s84 \n/s49/s50\n/s85 /s61/s48/s46/s48/s48/s40/s97/s41\nFIG.3. (ColorOnline)Transmittancecurvesforthesystem F1−QD−F2,i.e.,intheabsenceofthesuperconductor lead. (a)Contou rplotfor\nzero-bias transmittance T12in terms of the gate potential Vgand energy εforU= 0. (b)T12curves for positive values of Vg. Their location\nat thecontour plot are indicatedby the horizontal lines lab eledby A1,B1andC1for Vgequal to0.8, 0.5and0.2, respectively. (c) T12curves\nfornegativevaluesof VgwhoselocationinthecontourplotisgivenbyD1,E1andF1lin esfor Vgequalto-0.2,-0.5and-0.8,respectively.(d)\nContour plot for zero-bias transmittance T12in terms of the gate potential Vgand energy εforU= 0.8. (e)T12curves with A2, B2 and C2\ncorresponding to Vgequal to 0.8, 0.5 and 0.2, respectively. (f) T12curves for negative gate voltage values withD2, E2 and F2 cor responding\ntoVgequal to -0.2, -0.5 and -0.8, respectively. Fixed parameter s:θ=π/4,V1=V2= 0,Γ1= 0.40,Γ2= 0.40,Γs= 0,P1= 0.95and\nP2= 1.0. Allthe parameters are scaledbythe energy gapof the superc onductor lead.\ntunneling of quasi-particles from the leads to the supercon -\nductor. In spite of this restrictive condition, it is still p ossible\ntoacurrenttotakeplacefromtheleadstothesuperconducto r\nby means of AR. In the electrical current equation, the direc t\nand crossed ARs contributions are included by means of the\ntransmittance expressions given by Eqs. (20a) and (20b) re-\nspectively. Notice that the Green’s function matrix elemen ts\nappearingin these expressionsare related to conversionof an\nelectron of spin σinto a hole of spin ¯σ. Due to the coupling\nwith the second ferromagnet, whose magnetization may be\npointingin an arbitraryorientation, processesinvolving spin-\nﬂip alsocontributetothefulltransmittance.\nTo contrast the AR with the previous results for the F1−\nQD−F2system,inFig. 4isshownthedirectAndreevtrans-\nmittance TAR,11correspondingtheprocesswhereanelectron\nfrom F1is reﬂected as a hole in the same lead F1. The pat-\ntern observed was obtained for P1= 0.95and in Fig. 4a it is\nshown the dependence of TAR,11with the energy εand gate\nvoltage Vg. As the gate voltage changes from zero, a double\npeak structure emerges with the separation of the peaks in-\ncreasing with eVg. These two peaks represent the so-called\nAndreev bound states which are virtual states of supercon-\nducting quasi-particles formed by a pair of an electron with\na hole which is converted into a Cooper pair as it enters in-\nside the superconductor. These are strongly suppressed by\nthe ferromagnetic polarization once the conventional supe r-\nconductivityrequiresanti-parallelalignmentof the elec tronic\nspins. As a result, in the limit of high polarization, the dir ect\nAndreev contribution has a minor contribution to the trans-\nportandiscompletelyeliminatedwhen Pj= 1,j= 1,2. This\ncan be observed in Figs. 4b and 4c where the transmittance\namplitudeisconﬁnedtovaluesaround0.3for P1= 0.95.\nIn Fig. 4d, the Andreev transmittance is shown for the in-\nteracting case in which the interaction strength U= 0.80. In\nthis case, it can be noted that the transmittance has a simila rpattern as the noninteractingcase except that the intersec tion\npoint where the two peaks merge into a single one is shifted\nalongthegatevoltageaxis. Inaddition,theheightofthepe aks\nare also different as one can observe by comparing the Figs.\n4c and 4d with the curves shown in Figs. 4e and 4f. This\nasymmetry is stronger for positive values of the gate voltag e\nasevidentfromthe A2curveofFig. 4e. Asshowninprevious\nworks47,48, this asymmetry is crucial for the transport since\nthe Cooper pairs are formed by injecting two electrons with\noppositespinsandenergysignsatonceintothesuperconduc -\ntor. In this way, the current being injected into the superco n-\nductorisdeterminedbysmallerpeakofthe transmittance.\nNext, we consider the tunneling between the ferromagnets\ncharacterized by the transmittance T12. The corresponding\ncurvesforbothnoninteractingandinteractingcasesaresh own\nin Fig. 5. The main difference when the superconductor is\ncoupledintotheQDistheemergenceofaseconddiagonaldip\nline in the contour plot shown in Fig. 5a. Thus, the contour\nplot is divided in four triangular regions in which the trans -\nmittance can reach a maximum for particular values of gate\nvoltage and energy. The curves shown in Figs. 5b and 5c\nreveala two dip structure in transmittance with a central we ll\ndeﬁnedpeak. Inthisway,theeffectofthesuperconductorin to\nT12isjusttointroduceasecondstateatwhichthechannelsof\nspinsinterferedestructively. ThisisasignatureoftheAn dreev\nbound states into the transport between the ferromagnets. I n\nfact,bycomparingthecontourplotsofFig. 5aand5dwiththe\ncorrespondingonesofFig. 4aand4d,it isclearthatthese ar e\ncomplementarypatternsofresonances: intheregionsatwhi ch\ntheTAR,11exhibits a maximum value, the transmittance T12\npresentsadip. Thus,thecouplingwiththesuperconductorr e-\nsultsinaleakageofstatesfromthedirectchannelbetweent he\nferromagnetsforthe Andreevstates. This leadsto the patte rn\nobserved in Fig. 5. It is worth mentioning that Calle et. al.49\nhavestudieda three-terminalnanostructurecomposedbytw o8\n/s45/s49 /s46/s48 /s45/s48 /s46/s53 /s48 /s46/s48 /s48 /s46/s53 /s49 /s46/s48 /s45/s49 /s46/s48 /s45/s48 /s46/s53 /s48 /s46/s48 /s48 /s46/s53 /s49 /s46/s48 /s84 \n/s65/s82 /s44 /s49/s49/s32\n/s70 /s49/s69/s49/s68 /s49/s65 /s49\n/s66 /s49\n/s67 /s49\n/s32/s32\n/s49 /s46/s48 /s69 /s45 /s48 /s51 /s48 /s46/s49 /s48 /s46/s50 /s48 /s46/s51 \n/s65 /s50\n/s66 /s50\n/s67 /s50\n/s68 /s50\n/s69/s50\n/s70 /s50\n/s45/s49 /s46/s48 /s45/s48 /s46/s53 /s48 /s46/s48 /s48 /s46/s53 /s49 /s46/s48 /s45/s49 /s46/s48 /s45/s48 /s46/s53 /s48 /s46/s48 /s48 /s46/s53 /s49 /s46/s48 /s32\n/s32\n/s32/s32\n/s49 /s46/s48 /s69 /s45 /s48 /s51 /s48 /s46/s51 /s48 /s46/s52 \n/s45 /s49/s46 /s48 /s45 /s48/s46 /s53 /s48/s46 /s48 /s48/s46 /s53 /s49/s46 /s48/s48/s46 /s48/s48/s46 /s49/s48/s46 /s50/s48/s46 /s51/s48/s46 /s52/s84\n/s65 /s82 /s44/s49/s49\n/s40 /s41 \n/s40 /s41 /s32 /s32 \n/s40 /s41/s101/s86\n/s103 /s40 /s41\n/s84\n/s65 /s82 /s44/s49/s49/s65 /s49/s66 /s49/s67 /s49\n/s45 /s49/s46 /s48 /s45 /s48/s46 /s53 /s48/s46 /s48 /s48/s46 /s53 /s49/s46 /s48/s48/s46 /s48/s48/s46 /s49/s48/s46 /s50/s48/s46 /s51/s48/s46 /s52/s40/s100/s41\n/s40/s99/s41\n/s70 /s49/s69/s49/s68 /s49\n/s32 /s32 /s32 /s45 /s49/s46 /s48 /s45 /s48/s46 /s53 /s48/s46 /s48 /s48/s46 /s53 /s49/s46 /s48/s48/s46 /s48/s48/s46 /s49/s48/s46 /s50/s48/s46 /s51/s48/s46 /s52\n/s67/s50 /s40/s101/s41/s84\n/s65 /s82 /s44/s49/s49\n/s40 /s41 \n/s40 /s41 /s32 /s32 /s101/s86\n/s103 /s40 /s41\n/s84\n/s65 /s82 /s44/s49/s49/s65 /s50/s66 /s50\n/s45 /s49/s46 /s48 /s45 /s48/s46 /s53 /s48/s46 /s48 /s48/s46 /s53 /s49/s46 /s48/s48/s46 /s48/s48/s46 /s49/s48/s46 /s50/s48/s46 /s51/s48/s46 /s52\n/s40/s102/s41/s40/s98/s41\n/s85 /s61/s48/s46/s56/s48\n/s70 /s50/s69/s50/s68 /s50\n/s32 /s32 /s32 \n/s40 /s41/s84 \n/s65/s82 /s44 /s49/s49\n/s85 /s61/s48/s46/s48/s48/s40/s97/s41\nFIG.4. (ColorOnline)Transmittancecurvesforthesystem F1−(QD,S )−F2corresponding todirectAndreevreﬂectionatferromagnet F1,\nTAR,11. (a) Contour plot for zero-bias Andreev transmittance TAR,11in terms of the gate potential Vgand energy εforU= 0. (b)TAR,11\ncurves for positive values of Vg. Theirlocation atthe contour plot are indicatedbythe hori zontal lines labeledbyA1,B1andC1for Vgequal\nto 0.8, 0.50 and 0.2, respectively. (c) TAR,11curves for negative values of Vgwhose location in the contour plot is given by D1, E1 and F1\nlines for Vgequal to -0.2, -0.5 and -0.8, respectively. (d) Contour plot for zero-bias transmittance TAR,11in terms of the gate potential Vg\nand energy εforU= 0.8. (e) TAR,11curves with A2, B2 and C2 corresponding to Vgequal to 0.8, 0.5 and 0.2, respectively. (f) TAR,11\ncurves for negative gate voltage values with D2, E2 and F2 cor responding to Vgequal to -0.2, -0.5 and -0.8, respectively. Fixed parameter s:\nθ=π/4,V1=V2= 0,Γ1= 0.40,Γ2= 0.40,Γs= 0.40,P1= 0.95andP2= 1.0. All the parameters are scaled by the energy gap of the\nsuperconductor lead.\n/s45/s49 /s46/s48 /s45/s48 /s46/s53 /s48 /s46/s48 /s48 /s46/s53 /s49 /s46/s48 /s45/s49 /s46/s48 /s45/s48 /s46/s53 /s48 /s46/s48 /s48 /s46/s53 /s49 /s46/s48 /s84 \n/s49/s50/s32\n/s70 /s49/s69/s49/s68 /s49/s65 /s49\n/s66 /s49\n/s67 /s49\n/s32/s32\n/s48 /s46/s48 /s48 /s46/s53 /s48 /s46/s56 \n/s65 /s50\n/s66 /s50\n/s67 /s50\n/s68 /s50\n/s69/s50\n/s70 /s50\n/s45/s49 /s46/s48 /s45/s48 /s46/s53 /s48 /s46/s48 /s48 /s46/s53 /s49 /s46/s48 /s45/s49 /s46/s48 /s45/s48 /s46/s53 /s48 /s46/s48 /s48 /s46/s53 /s49 /s46/s48 /s32\n/s32\n/s32/s32\n/s48 /s46/s48 /s48 /s46/s53 /s48 /s46/s57 \n/s45 /s49/s46 /s48 /s45 /s48/s46 /s53 /s48/s46 /s48 /s48/s46 /s53 /s49/s46 /s48/s48/s46 /s48/s48/s46 /s51/s48/s46 /s54/s48/s46 /s57/s84\n/s49/s50\n/s40 /s41 \n/s40 /s41 /s32 /s32 \n/s40 /s41/s101/s86\n/s103 /s40 /s41\n/s84\n/s49/s50/s65 /s49\n/s66 /s49/s67 /s49\n/s45 /s49/s46 /s48 /s45 /s48/s46 /s53 /s48/s46 /s48 /s48/s46 /s53 /s49/s46 /s48/s48/s46 /s48/s48/s46 /s51/s48/s46 /s54/s48/s46 /s57/s40/s100/s41\n/s40/s99/s41/s70 /s49\n/s69/s49/s68 /s49\n/s32 /s32 /s32 /s45 /s49/s46 /s48 /s45 /s48/s46 /s53 /s48/s46 /s48 /s48/s46 /s53 /s49/s46 /s48/s48/s46 /s48/s48/s46 /s51/s48/s46 /s54/s48/s46 /s57\n/s67/s50 \n/s40/s102/s41/s40/s101/s41/s84\n/s49/s50\n/s40 /s41 \n/s40 /s41 /s32 /s32 /s101/s86\n/s103 /s40 /s41\n/s84\n/s49/s50/s65 /s50\n/s66 /s50\n/s45 /s49/s46 /s48 /s45 /s48/s46 /s53 /s48/s46 /s48 /s48/s46 /s53 /s49/s46 /s48/s48/s46 /s48/s48/s46 /s51/s48/s46 /s54/s48/s46 /s57/s40/s98/s41\n/s85 /s61/s48/s46/s56/s48\n/s70 /s50\n/s69/s50/s68 /s50\n/s32 /s32 /s32 \n/s40 /s41/s84 \n/s49/s50\n/s85 /s61/s48/s46/s48/s48/s40/s97/s41\nFIG.5. (Color Online) Transmittance curves for the system F1−(QD,S )−F2,i.e.,withthe presence of the superconductor lead coupled to\nthe QD. (a) Contour plot for zero-bias transmittance T12interms of the gate potential Vgand energy εforU= 0. (b)T12curves for positive\nvaluesof Vg. Theirlocationatthecontourplotareindicatedbythehori zontallineslabeledbyA1,B1andC1for Vgequalto0.8,0.50and0.2,\nrespectively. (c) T12curves for negative values of Vgwhose location in the contour plot is given by D1, E1 and F1lin es for Vgequal to -0.2,\n-0.5and-0.8,respectively. (d)Contour plotforzero-bias transmittance T12intermsofthegatepotential Vgandenergy εforU= 0.8. (e)T12\ncurveswithA2,B2andC2corresponding to Vgequalto0.8,0.5and0.2,respectively. (f) T12curvesfornegativegatevoltagevalueswithD2,\nE2 and F2 corresponding to Vgequal to -0.2, -0.5 and -0.8, respectively. Fixed parameter s:θ=π/4,V1=V2= 0,Γ1= 0.40,Γ2= 0.40,\nΓs= 0.40,P1= 0.95andP2= 1.0. Allthe parameters are scaledby the energygapof the superc onductor lead.\nnormal metals coupled by a double quantum dot system and\na superconductor. Inthissystem theyhaveobserveda simila r\npattern as shown in Fig. 5 with two dips and a central peak\ninthetransmittance T12. Theauthorsattributedsuchafeature\nto the Fano effect induced by the second quantum dot. Here,\nthe originof sucha patternisrelated tothe interplaybetwe en\nthe Andreev bound states and spin polarization provided by\nthe ferromagnets. As a result of these correlations, the spe c-\ntral properties of the quantum dot are similar to the double\nquantumdotstructureofRef. 49.B. Finite-bias regime\nIn the ﬁnite bias regime the correlations appearing in Eq.\n(25) couple the transmittance and local density of states\n(LDOS) with the bias applied to F1andF2. In this way, for\neachvalueof V1andV2thereisacorrespondingtransmittance\nandLDOScurve. Inthisway,thedependenceofthesequanti-\nties onthe applied biasbecomesmoreintricate than the zero -\nbias case. In spite of these modiﬁcations, it is also possibl e\nto recognize the signatures of Fano interference in the non-\nequilibriumcase. Inordertoillustratesuchaneffect,weh ave\ncalculated the electrical current for a ﬁnite bias ( eV1= 0.30)9\n/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51\n/s32/s40/s99/s41 /s40/s98/s41\n/s85/s61/s48/s46/s56/s48/s32 /s61/s32/s48/s46/s48/s48/s32/s32/s32/s32 /s32/s61/s32/s48/s46/s54/s51/s32/s32/s32/s32/s32/s32/s32 /s32 /s32/s61/s32/s49/s46/s50/s54/s32/s32/s32/s32/s32/s32/s32 /s32 /s61/s32/s49/s46/s56/s56/s32/s32/s32/s32/s32/s32 /s32 /s61/s32/s50/s46/s53/s49/s73\n/s81/s44/s49/s50/s32/s40/s101 /s47/s104/s41\n/s32 /s32/s85/s61/s48/s46/s48/s48\n/s40/s97/s41\n/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51\n/s101/s86\n/s103/s32/s40 /s41 /s101/s86\n/s103/s32/s40 /s41 /s101/s86\n/s103/s32/s40 /s41/s101/s86\n/s103/s32/s40 /s41 /s101/s86\n/s103/s32/s40 /s41 /s101/s86\n/s103/s32/s40 /s41\n/s73\n/s65/s44/s49/s50/s32/s40/s101 /s47/s104/s41/s73\n/s65/s44/s49/s49/s32/s40/s101 /s47/s104/s41/s32 /s32/s73\n/s81/s44/s49/s50/s32/s40/s101\n/s47/s104/s41\n/s73\n/s65/s44/s49/s49/s32/s40/s101 /s47/s104/s41\n/s73\n/s65/s44/s49/s50/s32/s40/s101 /s47/s104/s41/s40/s100/s41 /s40/s101/s41 /s40/s102/s41/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48/s48/s46/s49/s53/s32 /s32/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48/s32 /s32\n/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s46/s48/s48/s48/s46/s48/s51/s48/s46/s48/s54/s48/s46/s48/s57/s32 /s32/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s46/s48/s48/s48/s46/s48/s50/s48/s46/s48/s52/s48/s46/s48/s54/s32 /s32FIG.6. (Color Online) Currents ﬂowingthrough the lead F1for the system F1−(QD−S)−F2:IQ,12is the co-tunneling current, IA,11is\nthe direct Andreev current and IA,12is the crossed Andreev current. It is considered a ﬁnite bias V1= 0.30applied to F1while F2is kept\ngrounded. Theupperﬁguresshow thecurrentproﬁlesunder th eabsence ofCoulombcorrelationswithintheQD.Forinterme diatevaluesof θ,\nthedirect Andreevcurrent, IA,11exhibits adipat Vg= 0whileacorresponding peak emergesin IQ,12. Asimilarpatternisobserved inFigs.\n(d) and (e) at Vg∼0.40for the interacting case U= 0.80. Notice that for θclose to πthe conduction is ruled by the crossed AR process as\nshown in Figs. (c) and (f). Fixed parameters: V1= 0.30,V2= 0,Γ1= Γ 2= Γ s= 0.4,P1= 0.95,P2= 1.0. All the parameters are scaled\nbythe energy gapof the superconductor lead.\nkeepingtheotherparameterswiththesamevaluesusedinFig .\n5. InFig. 6atheco-tunnelingcurrent IQ,12isshownforsome\nvalues of θ, with U= 0. For θ= 0(solid-black curve), the\ncurrent reaches a maximum value for eVg=−0.25and then\ndecreases for positive values of eVg. As θis increased the\npattern changes with a second peak appearing for eVg= 0.\nThis peak is well pronouncedfor θ= 1.26(dot-dashedgreen\ncurve)andstartsbeingsuppressedas θ >1.88. InFigs. 6band\n6c the currents due to the direct and crossed AR are shown,\nrespectively. In the direct AR, there is a corresponding dip\nateVg= 0corresponding to the peak appearing in the co-\ntunneling current. On the other hand, the crossed AR also\npresents a peak at eVg= 0which increases with θ. This be-\nhavior is a result of the high polarization values which sup-\npressed the available states for local tunneling processes like\nthe direct AR and co-tunneling from F1toF2. As a result,\nthe crossed AR is the dominant process for θclose to πonce\nelectronsofoppositespinsfromdifferentleadscombinein toa\nCooperpairin S. Thisisevidentbycomparingtheamplitude\nofIQ,12,IA,11andIA,12forθ= 2.51. The peak appearing\nateVg= 0forIQ,12asθis changed from 0 to πis the non-equilibrium signature of the Fano-like interference appea ring\nin the zero-bias curves. This effect is also present under th e\npresence of Coulomb correlations within the QD. The corre-\nsponding curves for U= 0.80are plotted in Figs. 6c, 6d and\n6e. It can be noted that the presence of the peak in the co-\ntunneling and crossed AR currents are shifted to eVg∼0.46\nforθ= 2.51. A corresponding dip in the direct AR is also\npresent at the same point which illustrates the fact of the in -\nteraction, within the mean-ﬁeldapproximation,just shift s the\nresonanceconditioninthesame formasthezero-biascase.\n1. Spin-degeneracy\nIn the zero-bias curves shown in Fig. 4, the transmittance\ncurves exhibit a double peak structure related to the Andree v\nresonances. However, it is expected a splitting of these res o-\nnancesduetotheraisingofthespindegeneracycausedbythe\nCoulomb correlation within the QD. In Fig. 7a, it is shown\nthe transmittance curves for both TAR,11andT12for a ﬁnite\nbias voltage V1= 0.95. We also have chosen small values10\nforthe couplingto the ferromagnets, Γ1= 0.1andΓ2= 0.05\nwhich are crucial to allow the resolution of the spin degener -\nacy. In this regime, it is possible to observe such a splittin g\nof the peaks in which the TAR,11curves exhibit a four peak\nstructure.\nBy changing the gate voltage, it is possible to change the\npatternasonecanobservebycomparingthecurvesfor eVg=\n−0.50,eVg= 0.01andeVg= 0.48. For negative values of\nthe gate voltage, the central peaks are suppressed while for\npositive values the pattern is better resolved. The asymmet ry\nwith respect to the signal of the gate voltage is also a result\nof the interaction within the QD. This is clearer in the zero-\nbiasregimeinwhichtheresonanceconditionisshiftedbyth e\npresenceoftheinteraction. Asimilarbehaviorisobserved for\nthe co-tunneling transmittance T12, as illustrated in Fig. 7b.\nNotice that the T12curves do not present the corresponding\nFano-likeresonanceasobservedforzero-biasregime. Infa ct,\nthepositionofthepeaksof T12inFig. 7barecoincidentwith\nthose of TAR,11in Fig. 7a. The values of the parameters to\nobtain the Fano-like resonance in the co-tunneling transmi t-\ntances are different from those that allows for the resoluti on\nof the peaks due to the spin degeneracy. In this way, it is not\npossible to observe both effects with the same set of parame-\nters.\nIV. CONCLUSION\nIn this work, we have studied the interference effects on\nF1−(QD,S )−F2due to the coupling to a conventionalsu-\nperconductor. By varying the angle between the two mag-\nnetization it is possible to obtain a very pronounced dip in\nthe transmittance for ε= 0when the superconductor is de-\ncoupled from the QD. In contrast, the interplay between spin\nimbalance and Andreev bound states gives rise to a central\npeak at ε= 0when the superconductoris coupledto the sys-\ntem. Such an effect is a result of the interference between\nthe different channels of conduction through the QD. Addi-\ntionally,theleakageofstatesforAndreevtransportalsoi ntro-\nduces two anti-resonances in the zero-bias transmittance f or\ntheco-tunnelingofelectronsbetweenthe ferromagnets.\nThe effects of correlationswere taken within a generalized\nmean-ﬁeld approximation also taking into account spin-ﬂip\ncorrelations and proximity effect due to the coupling to the\nsuperconductor. Suchcorrelationsarerelevantsincethep hys-\nical quantities must be determined in self-consistent way f or\neachvalueofgateandbiasvoltages(fornon-equilibriumsi tu-\nation)thusintroducinganontrivialdependenceonthesequ an-\ntities. In fact, as shown in Figs. 3 and 5, the combination of\nthese correlations breaks the symmetry in the transmittanc e\nandshiftstheregionatwhichtheFano-likeinterferenceta kes\nplace. In the non-equilibrium situation, it is also possibl e to\nobserve the signatures of such an interference in the electr i-\ncal current as shown in Fig. 6. The approximation scheme\nused in this work allows us to write the electrical current in\na Landauer-like equation. In this way, it is possible to ob-\ntain analytic expressions for the transmittance for both An -\ndreev and co-tunneling contributions. Additionally, we ha ve/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48/s48/s46/s49/s53/s48/s46/s50/s48\n/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s32/s101/s86\n/s103 /s61/s32/s45/s48/s46/s53/s48/s32/s32/s32 /s32/s101/s86\n/s103 /s61/s32/s48/s46/s52/s56\n/s32/s101/s86\n/s103 /s61/s32/s48/s46/s48/s49\n/s40/s97/s41/s84\n/s49/s50/s32/s32\n/s32/s32/s101/s86\n/s103 /s61/s32/s45/s48/s46/s53/s48/s32\n/s32/s32/s101/s86\n/s103 /s61/s32/s32/s48/s46/s48/s49\n/s32/s32/s101/s86\n/s103 /s61/s32/s32/s48/s46/s52/s56/s32/s40 /s41/s84\n/s65/s82/s44/s49/s49/s32\n/s40/s98/s41\n/s32/s40 /s41/s32\nFIG. 7. (Color Online) Transmittance curves for ﬁnite bias. (a)\nTransmittance curves for direct Andreev reﬂection TAR,11(b) Co-\ntunneling transmittance T12. By reducing the constant coupling to\nthe ferromagnetic leads, it is possible to observe the split ting of the\nAndreev resonances due to the raising of spin degeneracy. Fi xed\nparameters: V1= 0.95,V2= 0,Γ1= 0.10,Γ2= 0.05,Γs= 1.0,\nP1= 0.85,P2= 1.0,U= 0.85,θ= 3π/2,kBT= 0.05. All the\nparameters are scaled bythe energy gapof the superconducto r lead.\nrestricted the calculations for large values of ferromagne tic\npolarization reducing in this way the ﬂuctuation in the oc-\ncupation numbers. Under this condition, the approximation\nyields results in a goodagreement with other approximation s\nschemes. Theresultsabovecanbereproducedinexperiments\nby using half-metal ferromagnets. Additionally, high pola r-\nizations( >90%)valueshavebeen obtainedin ferromagnetic\nﬁlms of CrO 2by Soulen Jr. and co-workers50; polarization\nvalues over 85% have been reported in ferromagnetic semi-\nconductors based on GaMnAs51. Hence, the results above\npresentedarerealisticandmaybeimplementedwiththestat e-\nof-artofexperiments.11\nV. APPENDIX\nIn this section we provide some details used in the cal-\nculation of the Green’s functions. In particular, we discus s\nthe approximation used to determine the Green’s functions\nalong with the determination of the Green’s functions equa-\ntions used to write the correspondingphysical quantities p re-\nsentedintheResults andDiscussionsection.\nA. Generalized Mean-Field Approximation\nIn deriving the Dyson’s equation given by Eq. (10) it is\nnecessary to consider some approximation in order to close\nthe system of equations for the QD Green’s function. In\nfact, the Coulomb correlation at the QD gives rise to an in-\nﬁnite set of equations. Within the Keldysh formalism, both\nretarded/advanced and “lesser\" Green’s functions are dete r-\nminedasanalyticcontinuationsoftime-orderedGreen’sfu nc-\ntionGτ(τ,τ′) =−i//planckover2pi1/an}bracketle{tˆTc{ˆΨd(τ)⊗ˆΨ†\nd(τ′)}/an}bracketri}ht, where ˆTcor-\nders the operators according to their position at the time\ncontour34. The operators are written in the Heisenberg pic-\nture whose dynamicsis given by the full Hamiltonian ˆH, Eq.\n(1). BybuildingtheequationofmotionfortheQDoperator,i t\nispossibletodeterminetheequationofmotionfor Gτ(τ,τ′).\nAfter integratingthe contributionfromthe leads, one ends up\nwiththe followingexpression:\nGτ(τ1,τ2) =gτ(τ1,τ2)\n+/integraldisplay\ncdτ3/integraldisplay\ncdτ4gτ(τ1,τ3)Σ0(τ3,τ4)Gτ(τ4,τ2)\n+/integraldisplay\ncdτ3gτ(τ1,τ3)UGτ(2)(τ3,τ2)(22)\nwhereΣ0carries the information about the coupling to the\nleads whose analytic continuation to real time axis gives th e\nretarded/advanced self-energies Σr/a\n0and the “lesser” self-\nenergyΣ<\n0whose expression will be considered in next sec-\ntion. Notice that the last term is a result of the interaction\nwithintheQD wherethe matrix U\nU=\nU0 0 0\n0−U 0 0\n0 0 U0\n0 0 0 −U\n (23)\ngivesthestrengthoftheinteractionand Gτ(2)(τ3,τ2)isasec-\nond order Green’as function containing four operators of th e\nQD. The equation of motion for this Green’s function would\nresult in a new equationinvolvinga third orderGreen’sfunc -\ntionandsoforth. Inthisway,it isnecessarytoconsidersom e\napproximation to truncate the inﬁnite set of equations gene r-\natedbythistechnique. Toperformsuchanapproximation,westartfromtheexpressionfor Gτ(2)(τ3,τ2),\nGt(2)(τ3,τ2) =\n/an}bracketle{t/an}bracketle{tˆd↑ˆnd↓ˆd†\n↑/an}bracketri}ht/an}bracketri}htc/an}bracketle{t/an}bracketle{tˆd↑ˆnd↓ˆd↓/an}bracketri}ht/an}bracketri}htc/an}bracketle{t/an}bracketle{tˆd↑ˆnd↓ˆd†\n↓/an}bracketri}ht/an}bracketri}htc/an}bracketle{t/an}bracketle{tˆd↑ˆnd↓ˆd↑/an}bracketri}ht/an}bracketri}htc\n/an}bracketle{t/an}bracketle{tˆd†\n↓ˆnd↑ˆd†\n↑/an}bracketri}ht/an}bracketri}htc/an}bracketle{t/an}bracketle{tˆd†\n↓ˆnd↑ˆd↓/an}bracketri}ht/an}bracketri}htc/an}bracketle{t/an}bracketle{tˆd†\n↓ˆnd↑ˆd†\n↓/an}bracketri}ht/an}bracketri}htc/an}bracketle{t/an}bracketle{tˆd†\n↓ˆnd↑ˆd↑/an}bracketri}ht/an}bracketri}htc\n/an}bracketle{t/an}bracketle{tˆd↓ˆnd↑ˆd†\n↑/an}bracketri}ht/an}bracketri}htc/an}bracketle{t/an}bracketle{tˆd↓ˆnd↑ˆd↓/an}bracketri}ht/an}bracketri}htc/an}bracketle{t/an}bracketle{tˆd↓ˆnd↑ˆd†\n↓/an}bracketri}ht/an}bracketri}htc/an}bracketle{t/an}bracketle{tˆd↓ˆnd↑ˆd↑/an}bracketri}ht/an}bracketri}htc\n/an}bracketle{t/an}bracketle{tˆd†\n↑ˆnd↓ˆd†\n↑/an}bracketri}ht/an}bracketri}htc/an}bracketle{t/an}bracketle{tˆd†\n↑ˆnd↓ˆd↓/an}bracketri}ht/an}bracketri}htc/an}bracketle{t/an}bracketle{tˆd†\n↑ˆnd↓ˆd†\n↓/an}bracketri}ht/an}bracketri}htc/an}bracketle{t/an}bracketle{tˆd†\n↑ˆnd↓ˆd↑/an}bracketri}ht/an}bracketri}htc\n\nwhere for compactness we have used the Zubarev notation52\nin which /an}bracketle{t/an}bracketle{tˆAˆBˆC/an}bracketri}ht/an}bracketri}htc=−i/h/an}bracketle{tˆTc{ˆA(τ3)ˆB(τ3)ˆC(τ2)}/an}bracketri}ht. No-\nticethat ˆndσ=ˆd†\nσˆdσistheQD numberoperatorforspin σ.\nIn order to close the system of equations, we use the fol-\nlowingdecouplingscheme53:\n/an}bracketle{t/an}bracketle{tˆdσ1ˆn¯σ1ˆd†\nσ2/an}bracketri}ht/an}bracketri}htc∼ /an}bracketle{tˆdσ1ˆd†\n¯σ1/an}bracketri}ht/an}bracketle{t/an}bracketle{tˆd¯σ1ˆd†\nσ2/an}bracketri}ht/an}bracketri}htc\n− /an}bracketle{tˆdσ1ˆd¯σ1/an}bracketri}ht/an}bracketle{t/an}bracketle{tˆd†\n¯σ1ˆd†\nσ2/an}bracketri}ht/an}bracketri}htc+/an}bracketle{tˆd†\n¯σ1ˆd¯σ1/an}bracketri}ht/an}bracketle{t/an}bracketle{tˆdσ1ˆd†\nσ2/an}bracketri}ht/an}bracketri}htc\n/an}bracketle{t/an}bracketle{tˆdσ1ˆn¯σ1ˆdσ2/an}bracketri}ht/an}bracketri}htc∼ /an}bracketle{tˆdσ1ˆd†\n¯σ1/an}bracketri}ht/an}bracketle{t/an}bracketle{tˆd¯σ1ˆdσ2/an}bracketri}ht/an}bracketri}htc\n− /an}bracketle{tˆdσ1ˆd¯σ1/an}bracketri}ht/an}bracketle{t/an}bracketle{tˆd†\n¯σ1ˆdσ2/an}bracketri}ht/an}bracketri}htc+/an}bracketle{tˆd†\n¯σ1ˆd¯σ1/an}bracketri}ht/an}bracketle{t/an}bracketle{tˆdσ1ˆdσ2/an}bracketri}ht/an}bracketri}htc\nwith a similar decoupling for /an}bracketle{t/an}bracketle{tˆd†\nσ1ˆn¯σ1ˆdσ2/an}bracketri}ht/an}bracketri}htcand\n/an}bracketle{t/an}bracketle{tˆd†\nσ1ˆn¯σ1ˆd†\nσ2/an}bracketri}ht/an}bracketri}htc. In this scheme, it appears three types\nof averages each one related with a speciﬁc feature of the\ncorrelation within the QD. In fact, averages of the form\n/an}bracketle{tˆd†\nσ1ˆdσ1/an}bracketri}htrepresent the Coulomb correlations due to the\nelectron-electron interaction; /an}bracketle{tˆdσ1ˆdσ1/an}bracketri}htand its adjoint are\nanomalous averages being different of zero due to proximity\neffect arising from the coupling with to the superconductor .\nIt represents the amplitude of ﬁnding a superconductor exci -\ntation within the QD. Finally, averages involving /an}bracketle{tˆd†\nσ1ˆd¯σ1/an}bracketri}ht\naccountforspin-ﬂipscatteringwithintheQD.Thisaverage is\nalso non-zero for intermediate values of the angle θbetween\nthemagnetizationvectorsof F1andF2. Inthisway,theelec-\ntronwithinthe QD canﬂip its spin asa result ofthe interplay\nbetween the electronic correlation and the misalignment of\nthemagnetizationoftheferromagneticleads.\nBy substituting the decoupling approximation back into\nGt(2)it ispossibletowrite\nGt(2)(τ3,τ2)∼Θ(τ2)Gt(τ3,τ2) (24)\ninwhichthematrix Θiswrittenas\nΘ=U\n/an}bracketle{tˆd†\n↓ˆd↓/an}bracketri}ht −/an}bracketle{t ˆd↑ˆd↓/an}bracketri}ht /an}bracketle{tˆd↑ˆd†\n↓/an}bracketri}ht 0\n−/an}bracketle{tˆd†\n↓ˆd†\n↑/an}bracketri}ht −/an}bracketle{t ˆd†\n↑ˆd↑/an}bracketri}ht 0 /an}bracketle{tˆd†\n↓ˆd↑/an}bracketri}ht\n/an}bracketle{tˆd↓ˆd†\n↑/an}bracketri}ht 0 /an}bracketle{tˆd†\n↑ˆd↑/an}bracketri}ht −/an}bracketle{t ˆd↓ˆd↑/an}bracketri}ht\n0 /an}bracketle{tˆd†\n↑ˆd↓/an}bracketri}ht −/an}bracketle{t ˆd†\n↑ˆd†\n↓/an}bracketri}ht −/an}bracketle{t ˆd†\n↓ˆd↓/an}bracketri}ht\n(25)\nwhose matrix elements are averagesto be determinedin self-\nconsistent way for each value of the external parameters. In\nthis work we are interested in the stationary regime in which\ntheseaveragesare takenastime-independentquantities.12\nWith the approximationaboveit ispossible to closethe set\nof equations in order to obtain the Dyson’s equation in the\ntime-orderedcontour:\nGτ(τ1,τ2) =gτ(τ1,τ2)\n+/integraldisplay\ncdτ3/integraldisplay\ncdτ4gτ(τ1,τ3)Σ0(τ3,τ4)Gτ(τ4,τ2)\n+/integraldisplay\ncdτ3gτ(τ1,τ3)Θ(τ3)Gτ(τ3,τ2).(26)\nBy analytic continuation of Eq. (26) the relevant Green’s\nfunctionsweredetermined. Theassumptionofstationaryst ate\nallows us to work with Fourier transform of these Green’s\nfunctions.\nB. Self-Consistent Equations\nThe approximation we have used leads to the calculation\nof the averages appearing in Eq. (25). In order to determine\nthese average values, we use the Keldysh equation obtained\nbyanalyticcontinuationofEq. (26). We have\nG<(ε) =Gr(ε)Σ<\n0(ε)Ga(ε) (27)\nwhere\nΣ<\n0(ε) =Σ<\n1(ε) +Σ<\n2(ε) +Σ<\ns(ε)\nsuch that each self-energy is determined by using the\nﬂuctuation-dissipation theorem since the leads are consid -\nered to be in equilibrium. Thus, it is valid to write Σ<\ni=\nFi(Σa\ni−Σr\ni)wherewe havedeﬁnedtheFermimatrix,\nFi(ε) =\nfi0 0 0\n0¯fi0 0\n0 0 fi0\n0 0 0 ¯fi\n, i = 1,2,s(28)\nwith fi=f(ε−eVi)being the electron Fermi function and\n¯fi=f(ε+eVi)is the correspondinghole Fermi distribution.\nOnce the superconductor is grounded, then fs=f(ε)which\nimpliesthat Fsisdiagonal. Consideringthat Σa\ni= [Σr\ni]†one\ncanwrite:\nΣ<\ns(ε) =if(ε)Γs/tildewide̺(ε)\n1 −∆/ε 0 0\n−∆/ε 1 0 0\n0 0 1 ∆ /ε\n0 0 ∆ /ε 1\n\nwhere /tildewide̺(ε) =Re[̺(ε)]whichistheconventionalBCSdensity\nofstatesbeingdifferentofzeroonlyfor |ε|>∆.\nThecontributionfromtheferromagnetsaregivenby:\nΣ<\n1(ε) =i\nf1Γ1↑0 0 0\n0 ¯f1Γ1↓0 0\n0 0 f1Γ1↓0\n0 0 0 ¯f1Γ1↑\n(29)forF1andfor F2oneobtains:\nΣ<\n2(ε) =i\nA↑f20 Bf2 0\n0A↓¯f20 B¯f2\nBf2 0A↓f20\n0 B¯f2 0A↑¯f2\n,(30)\nwithAσandBbeingalreadydeﬁnedin Eq. (14).\nThe“lesser”Green’sfunctioninNambuspace isgivenby:\nG<(t,t′) =i\n/planckover2pi1\n/an}bracketle{tˆd†\n↑(t)ˆd↑(t′)/an}bracketri}ht /an}bracketle{tˆd↓(t)ˆd↑(t′)/an}bracketri}ht /an}bracketle{tˆd†\n↓(t)ˆd↑(t′)/an}bracketri}ht 0\n/an}bracketle{tˆd†\n↑(t)ˆd†\n↓(t′)/an}bracketri}ht /an}bracketle{tˆd↓(t)ˆd†\n↓(t′)/an}bracketri}ht 0 /an}bracketle{tˆd↑(t)ˆd†\n↓(t′)/an}bracketri}ht\n/an}bracketle{tˆd†\n↑(t)ˆd↓(t′)/an}bracketri}ht 0 /an}bracketle{tˆd†\n↓(t)ˆd↓(t′)/an}bracketri}ht /an}bracketle{tˆd↑(t)ˆd↓(t′)/an}bracketri}ht\n0 /an}bracketle{tˆd↓(t)ˆd†\n↑(t′)/an}bracketri}ht /an}bracketle{tˆd†\n↓(t)ˆd†\n↑(t′)/an}bracketri}ht /an}bracketle{tˆd↑(t)ˆd†\n↑(t′)/an}bracketri}ht\n\nand the averages we are looking for in Eq. (25) are obtained\nbysetting t=t′inG<(t,t′)andperformingtheFouriertrans-\nform. Inthiscase, forinstance,theaverageof anytwo opera -\ntors/an}bracketle{tˆdαˆdβ/an}bracketri}htmaybe writtenas\n/an}bracketle{tˆdαˆdβ/an}bracketri}ht=1\n2πi/integraldisplay\nG<\nαβ(ε,/an}bracketle{tˆdαˆdβ/an}bracketri}ht,···)dε\nwith G<\nαβbeing a matrix element of the Keldysh equation\n[Eq. (27)]. Other matrix elements of G<correspond to av-\nerages of /an}bracketle{tˆd†\nαˆdβ/an}bracketri}htand/an}bracketle{tˆd†\nαˆd†\nβ/an}bracketri}ht. Notice that the /planckover2pi1factor was\ncanceled by the one appearing in the Fourier transform. All\nmatrix elements appearing in Eq. (25) may be identiﬁed by\ncomparingwiththe“lesser” Green’sfunctionmatrix.\n1Y. A. Izyumov, Y. N. Proshin, and M. G. Khusainov, Physics-Us pekhi45,\n109 (2002).\n2A.I.Buzdin, Rev. Mod.Phys. 77, 935 (2005).\n3S.deFranceschi, L.Kouwenhoven, C.Schönenberger, andW.W erndorfer,\nNature Nanotech. 5(2010).\n4A.Buzdin, L.Bulaevskii, and S.Panyukov, JETPLetters 35, 178 (1982).\n5J. S. Jiang, D. Davidovi ´c, D. H. Reich, and C. L. Chien,\nPhys.Rev. Lett. 74, 314 (1995).\n6M.Eschrig, Physics Today 64, 43 (2011).\n7I.Žuti´c, J. Fabian, and S.Das Sarma, Rev. Mod.Phys. 76,323 (2004).\n8A.F.Andreev, Sov.Phys.JETP 19, 1228 (1964).\n9C.J.W.Beenakker, cond-mat/9909293v2 (1999).\n10G.Deutscher and D.Feinberg, Appl. Phys.Lett. 76,487 (2000).\n11J.Bardeen, L.N.Cooper, and J.R.Schrieffer, Phys.Rev. 106, 162 (1957).\n12G. E. Blonder, M. Tinkham, and T. M. Klapwijk,\nPhys.Rev. B 25, 4515 (1982).\n13A.Brinkman and A.A.Golubov, Phys.Rev. B 74,214512 (2006).\n14C.J.Lambert, Journal of Physics-Condensed Matter 3,6579 (1991).\n15C.W.J.Beenakker, Rev. Mod. Phys. 80, 1337 (2008).\n16G.Deutscher, Rev. Mod. Phys. 77,109 (2005).\n17S. Gustavsson, R. Leturcq, M. Studer, I. Shorubalko, T. Ihn, K. Ensslin,\nD.Driscoll, and A.Gossard, Surface Science Reports 64, 191 (2009).\n18Z. Chen, B. Wang, D. Y. Xing, and J. Wang,\nApplied Physics Letters 85,2553 (2004).\n19X. Cao, Y. Shi, X. Song, S. Zhou, and H. Chen,\nPhys.Rev. B 70, 235341 (2004).\n20J.-F.Feng and S.-J.Xiong, Phys.Rev. B 67, 045316 (2003).\n21Y.Zhu, Q.-f.Sun, and T.-h.Lin,Phys. Rev. B 65,024516 (2001).\n22F.Dolcini and L.Dell’Anna, Phys.Rev. B 78, 024518 (2008).\n23Physics Letters A 372,6773 (2008).\n24J.P.Morten, A.Brataas, and W.Belzig, Phys.Rev. B 74,214510 (2006).13\n25C.Benjamin, Phys.Rev. B 74, 180503 (2006).\n26R.Mélin and S.Peysson, Phys.Rev. B 68,174515 (2003).\n27F. Giazotto, F. Taddei, F. Beltram, and R. Fazio,\nPhys.Rev. Lett. 97, 087001 (2006).\n28G.Deutscher, Journal of Superconductivity 15, 43 (2002).\n29R.Mélin, Phys.Rev. B 72,054503 (2005).\n30C.Benjamin and R.Citro, Phys.Rev. B 72, 085340 (2005).\n31L.Bai,Z.-Z.Zhang,L.Jiang, andF.-R.Tang,PhysicaE:Low- dimensional\nSystems and Nanostructures 43, 446 (2010).\n32L.Bai,R.Zhang, andC.-L.Duan,PhysicaB:CondensedMatter 405,4875\n(2010).\n33TheEuropean Physical Journal B 86, 35 (2013), 10.1140/epjb/e2012-30569-0.\n34H. Haug and A. P. Jauho, Quantum Kinetics in Transport and Optics of\nSemiconductors (Springer, Berlin, 1996).\n35G. Michałek, B. R. Bułka, T. Doma ´nski, and K. I. Wysoki ´nski,\nPhys.Rev. B 88, 155425 (2013).\n36K.BocianandW.Rudzinski,TheEuropean Physical Journal B 86, 439 (2013), 10.1140/epjb/e2013-40486-3.\n37J.-L.Liand Y.-X. Li,Journal of Physics: Condensed Matter 20, 465202.\n38Z. Chen, J. Wang, B. Wang, and D. Xing,\nPhysics Letters A 334, 436 (2005).\n39Y. Zhu,Q.-f.Sun, and T.H.Lin, Phys.Rev. B 65, 024516 (2001).40J.-F. Feng, X.-S. Wu, and S.-S. Jiang, Journal of Applied Phy sics99, 08\n(2006).\n41K.P.Wójcik and I. Weymann, Phys.Rev. B 89,165303 (2014).\n42I.Weymann and P.Trocha, Phys.Rev. B 89,115305 (2014).\n43Z.Y.Zeng, B.Li, and F.Claro, Phys.Rev. B 68,115319 (2003).\n44Z.Zeng,B.Li, andF.Claro,TheEuropeanPhysicalJournalB- Condensed\nMatter and Complex Systems 32, 401 (2003).\n45P. Fazekas, Lecture notes on Electron Correlations and Magnetism (World\nScientiﬁc, Singapore, 1999).\n46J.Rammer and H.Smith, Rev. Mod. Phys. 58,323 (1986).\n47E.C.Siqueira and G.G.Cabrera, Phys.Rev. B 81, 094526 (2010).\n48E. C. Siqueira and G. G. Cabrera,\nJournal of Applied Physics 111,113905 (2012).\n49A.Calle,M.Pacheco, andP.Orellana,PhysicsLettersA 377,1474 (2013).\n50R.J.SoulenJr.,J.M.Byers,M.S.Osofsky,B.Nadgorny,T.Am brose,S.F.\nCheng, P. R. Broussard, C. T. Tanaka, J. Nowak, J. S. Moodera, A. Barry,\nand J.M. D.Coey, Science 282, 85 (1998).\n51J. G. Braden, J. S. Parker, P. Xiong, S. H. Chun, and N. Samarth ,\nPhys.Rev. Lett. 91, 056602 (2003).\n52D.N.Zubarev, Soviet Physics Uspekhi 3, 320.\n53J.Fransson, Non-EquilibriumNano-Physics: AMany-BodyApproach ,Lec-\nture Notes in Physics (Springer, 2010)."    },    {        "title": "1805.08496v1.FMR_related_phenomena_in_spintronic_devices.pdf",        "content": "1 \n \n  \nFMR -related phenomena in spintronic devices  \n \nYi Wang, Rajagopalan Ramaswamy , and Hyunsoo Yang  \n \nDepartment of Electrical and Computer Engineering, National University of Singapore, 117576, \nSingapore  \n \nSpintronic devices, such as non-volatile magnetic random access memories  and logic devices , have \nattracted considerable attention as potential candidates  for future high efficient data storage and \ncomputing  technology . In a heavy metal or other emerging material  with strong spin-orbit coupling \n(SOC) , the charge current s induce  spin currents or spin accumulations  via SOC . The generated \nspin currents can exert spin-orbit torques (SOTs) on an adjacent ferromagnet, which open s up a \nnew way to realize magnetization dynamics and switching  of the ferromagnetic  layer  for spintronic \ndevices. In the SOT scheme, the charge -to-spin interconversion  efficiency (SOT efficiency) is an \nimportant figure of merit for applications. For the  effective characterization of this efficiency, the \nferromagnetic resonance  (FMR) based methods, such as the spin transfer torque ferromagnetic \nresonance  (ST-FMR) and the spin pumping, are common  utilized in addition to  low frequency  \nharmonic  or dc measurements. In this review, we focus on the ST-FMR measurements for the \nevaluation of the SOT efficiency . We provide  a brief summary of the different  ST-FMR setups and \ndata analysis methods.  We then discuss  ST-FMR and SOT studies in various  materials , including \nheavy metals  and alloys , topological insulators, two dimensional  (2D) materials , interfaces with \nstrong Rashba effect, antiferromagnetic materials , two dimensional electron gas ( 2DEG) in oxide \nmaterials and oxidized nonmagnetic materials.  \nKeywords:  magnetization  dynamics , spin transfer torque ferromagnetic resonance , spin-orbit \ntorque efficiency , charge -to-spin current conversion  \n \n 2 \n \n CONTENTS  \n1. Introduction  ................................................................................................................................. 3 \n2. Magnetization  dynamics  ........ ....................................................................... ............................... 4 \n        2.1. LLG equation  .......... .......................................................................................................... 4 \n        2.2. Ferromagnetic resonance (FMR)  ....................................................... ...............................5  \n        2.3. Spin transfer torque ferromagnetic resonance (ST -FMR)  ................................................. 6 \n3. ST-FMR setups  ............................................................................................................................ 8 \n        3.1. Traditional ST -FMR setup with a dc voltmeter  ................................................................. 8 \n        3.2. Precise ST -FMR setup with a lock -in amplifier  ................................................................ 9 \n        3.3. Design of CPW and impedance match  .............................................. ..............................11  \n4. Methods to evaluate the spin -orbit torqu e (SOT) and SOT efficiency  ........................................12  \n        4.1. Ratio of VS/VA ................................................................................................................. 13 \n        4.2. Only -VS ............................................................................................. ..............................14  \n        4.3. Modulation of damping (MOD)  ........................................................ ..............................15  \n5. Spin Hall effect  (SHE ) in heavy  metals  and alloys  .................................................. ...................17  \n        5.1. Spin Hall effect in Pt  ......................................................................... ..............................19  \n        5.2. Spin Hall effect in Ta  ........................................................................ ..............................2 3 \n        5.3. Spin Hall effect in tungsten (W) and other alloys with heavy metal dopants  ................... 26 \n        5.4. Role of interface transparency of HM and FM in the SOT efficiency  ..............................28  \n        5.5. Difference in the SOT effi ciency between ST -FMR and magnetization switching         \ntechnique  ....................... .................................................................. ..............................30  \n6. SOT in topological insulators (TIs)  .............................................................. ..............................31  \n        6.1. SOT efficiency in TIs  ........................................................... ............. ..............................33  \n        6.2. Anisotropic SOT (efficiency) in Bi 2Se3 MBE films  .......................... .............................. 35 \n        6.3. TSS dominated SOT  ......................................................................... .............................. 38 \n7. SOT beyond HMs and TIs  ................................... ......................................... ..............................40  \n        7.1. Interface between two nonmagnetic materials  ................................... .............................41  \n        7.2. Transition -metal dichalcogenides (TMDs)  ....................................... ..............................4 4 \n        7.3. Antiferromagnetic ( AFM) materials  ................................................. ..............................46  \n        7.4. 2DEG in oxide materials  ................................................................................... ..............48  \n        7.5. Nonmagnetic oxidized materials  ....................................................... ..............................50  \n8. Summary and Perspectives  .......................................................................... ..............................51  \nAcknowledgments  ..................................................................................... ....................................56  \nReferences  ....................................................................................................... ..............................57  \n  3 \n \n 1. Introduction  \nIn the recent years , the electrica l manipulation of the magnetization is a central theme  for \nmodern spintronics  [1-3]. Following the successful application s of magnetic tunnel junctions \n(MTJs)  [4, 5]  in the read head s of hard disk drive s, new function al spintronic  devices, such as non-\nvolatile magnetic random access memories (MRAM) and magnetic logic devices  [6, 7] , are being \npursued enthusiastically by researchers  nowadays . For practical spintronic applications, it is \ndesired to achieve efficient magnetization manipulation  via electrical means  with a higher \noperation speed and lower energy consumption. In order to evaluate the efficiency of the electrical  \nmanipulation of the magnetization as well as to understand underlying  physics  in the magnetic \ndevices , various techniques based on magnetization dynamics under  application of current s have \nbeen  developed.  \nThe study of magnetization dynamics can  be dated back to  the observation of ferromagnetic \nresonance (FMR) in the first half of the 20th century [8, 9] . Originally, FMR was  used to study \nmagnetic  films and the magnetization dynamics  in such films  in the context of external magnetic \nfields only . With the discovery of spin transfer torques (STT) in the year 1996 [10, 11]  and \ndevelopment of the modern nano -fabrication techniques which allow the device dimensions to be \nscaled -down, electrical manipulation o f nanomagnets using spin -polarized current injection gained \npopularity. After the year 2000, the STT driven magnetization switching using electric currents \nwas experimentally observed in giant magnetoresistance (GMR) nanopillars and nanosized MTJ \ndevices [12-14]. In order to get insight i nto the magnetization dynamics of an individual \nnanomagnet  or magnetic device under the application of an electric current, the spin transfer torque \nferromagnetic resonance (ST -FMR) technique was developed and explored in GMR nanopillars \nand nanosized MTJs  [15-18]. 4 \n \n One of the drawbacks of the STT scheme is the requirement of an additional magnet to create \nspin polarized currents. In the year 2004, the  imaging of direct charge -to-spin current conversion  \nin a single semiconductor GaAs  due to the spin Hall effect  (SHE)  was observed [19]. Ever since \nthis observations, p ure spin current  generation  in non-magnets has drawn tremendous attention for \nspintronic  device s [2]. Recently , it was demonstrated that t he magnetization can be switched or \ndriven into precession modes  by only pure spin currents  [20-24], which  open ed up a new route to \nrealize spintronic devices.  In order to achieve highly efficient magnetization manipulation, a key \nchallenge is to identify materials that exhibit a large charge -to-spin interconversion efficie ncy (i.e. \nspin Hall angle or spin -orbit torque  efficiency).  The ST -FMR is  an effective technique to evaluate \nthe spin-orbit torque ( SOT ) efficiency. The first demonstration was reported in a Pt/NiFe  bilayer \nplanar spin Hall device  in 2011 [25], which is based on the spin rectification of the anisotropic \nmagnetoresistance  (AMR) effect. S ubsequently , ST-FMR has been  used to evaluate the SOT \nefficiencies and understand  the underlying physics in many emerging  materials  as presented later  \non. \nThis article  provides a topical review of ST -FMR techn ique and its applications in  study ing \nthe SOT related phenomena in diverse emerging materials. I n Section 2 , we begin with  the basic \nprinciple  of magnetization  dynamics . In Section 3  and 4, we  introduce the ST-FMR setup s and \ndata analysis methods . In S ection 5, 6 and  7, we  review the progress in the SOT studies by using \nST-FMR techniq ue in different material systems , such as heavy metals, topological insulators  and \nother emerging material systems , and  we end with a summary and perspective s in Section 8 .  \n2. M agnetization dynamics  \n2.1. LLG equation  \nUnder macrospin approximation, the time domain dynamics of the magnetization vector ( m) 5 \n \n of a ferromagnet in the presence of an effective magnetic field ( Heff) can be described by the \nLandau -Lifshitz -Gilbert (LLG) equation  [26], \n                                           \n 0tt        effmmm H m                                                      (1) \nwhere  γ, µ0, and α are the gyromagnetic ratio , vacuum  permeability,  the Gilbert damping parameter , \nrespectively . The right hand side of the LLG equation has two torque terms – the first term, \n0eff mH\n, is a precession term that rotates m around  Heff, while the second term, \ntmm , \nis the damping (or energy dissipation) term which aligns  m along  Heff. The value of α determines \nthe rate at which m damps towards the  Heff. When the value of α is large (small), m damps towards \nHeff at a faster (slower) rate.  \nWhen a spin polarized current (polarized along \nˆσ  ) interacts with  m, there are additional \ntorques experienced by m and the LLG equation is modified  as [10, 11, 27]  \n                                     \nDL FL 0tt      effmmm H m ττ                       (2) \nwhere \nDLτ  is of the \n ˆmσm  symmetry and is called the damping -like torque , while  \nFLτ  is of \nthe \nˆσm  symmetry and is called the field -like torque.  \n2.2. Ferromagnetic resonance (FMR)  \nIn Section 2.1, it was discussed that in the presence of  Heff, m precesses about  Heff. Due to \nenergy dissipation (damping) in the ferromagnet , the amplitude of these precessions continuously \ndecay s. An external energy supplied to the ferromagnet in the form of an alternating magnetic field \n(HRF) can compensate this energy dissipation leading to forced precession of  m. If the frequency \nof HRF is tuned to the natural  precession frequency of  m, the amplitude of the force d precession \nwill be maximum due to resonance. This resonance is called the ferromagnetic resonance a nd the 6 \n \n precession frequency during the resonance condition is called the resonance frequency ( f =\nres/2 ). \nf and H0 (the resonant field) are related by the Kittel ’s relation,  \n                                             \n12\n00 eff ( /2 )[ ( 4 )]/f H H M                                                          (3) \nwhere \neff 4M  is the effective demagnetization field o f the ferromagnet. According to  equation  (3), \nthe resonance condition can be matched by fixing H0 and tuning the frequency of HRF or vice versa.  \n2.3. Spin transfer torque ferromagnetic resonance (ST -FMR)  \nIn contrast to the magnetic field driven FMR, it is possible to excite FMR using spin torques \ngenerated by an alternating current , called ST -FMR. T he ST-FMR technique  was first used in a \nPt/NiFe  (Py) bilayer  in the case of a spin Hall scheme  [25], which is proved to be an effective \nmethod to evaluate the charge -to-spin conversion efficiency ( i.e. spin Hall angle ). We first provide \na more  general description  of ST -FMR  process  in the Pt/Py  bilayer as an  example.  \nAs shown in figure 1(a), when an in-plane rf charge current  (IRF) is injected into the Pt/Py \nbilayer , non-equilibrium spins are generated  at the Pt/Py interface due to the spin-orbit coupling \n(SOC)  in the Pt layer. S ubsequently , these generated spins diffuse into the adjacent Py  layer and \nexert SOTs  (damping -like torque, DL and/or field -like torque, FL) on the Py local magnetization. \nIn addition, the  current in the Pt layer induced Oersted field can also exert  the Oersted field torque \n(Oe) on the Py  magnetization . Since the injected charge current is oscillatory,  all the above torques \nare oscillatory and hence  trigger the Py  magnetization into precession around the direction of \neffective magnetic field, which lead s to an oscillat ory AMR in the ST -FMR device. Consequently,  \nthe oscillation of the AMR and the IRF in the ST -FMR device produce a mixing ST -FMR voltage  \nVmix across the Pt/Py  bilayer, which can be detected by a dc voltmeter or a lock-in amplifier.  Since  \nthe ST -FMR signal is the spin rectification of the magnetoresistance  in the device  [28], the ST -\nFMR technique has a high enough sensitivity and can be used  to study  the microsized or  even  7 \n \n nanosized magnetic devices.  \n \nFigure 1.  (a) Illustration of spin current generation in Pt and SOT driven Py magnetization \ndynamics in the ST -FMR measurements . The big blue arrow denotes the electron flow direction \n(opposite to the IRF direction ). The blue with green and red balls denote  the generated spin \npolarizations in the Pt layer. (b)  A typical ST-FMR signal  (open symbols ) from a Pt (6 nm)/Py (4 \nnm) device at 6 GHz with fits (solid curves ), where the blue and green curves represent the \nsymmetric Lorentzian  (VSFS) and antisymmetric Lorentzian  (VAFA) components, respectively.  \nSubsequently, w e present a  derivation  of general  expression for the ST-FMR signal  Vmix. The \napplied rf current to the Pt/Py  ST-FMR device  is written as  \nRF cos( ) I I t   , where I is the \namplitude of IRF with a frequency of ω/2π. The device resistance  is\n2\n0 ( ) cos ( )R t R R t    , where \nR0 is the resistance when  IRF is perpendicular to the Py magnetization direction, ∆R is the AMR .  \nθ(t) is the angle between IRF and Py magnetization, which is a function of time t due to \nmagnetization precession , given by the equation\nHc ( ) cos( )tt        , where θH is the \nconstant  angle between Hext and IRF, θc is the cone angle of Py magnetization precession , which is \na measu re of the total torques  on the Py magnetization . Usually, θc is much  small er than θH. δ is \nthe resonance  phase between the driven force ( i.e. damping -like torque DL, field-like torque FL \nor Oersted field torque Oe) and the magnetization response (i.e. the oscillation of AMR)  [29, 30] .  \n-2000 -1000 1000 2000-40-2002040  Data\n Fit\n VSFS\n VAFA\n  Vmix (V)\nH (Oe)\nx\nyz\nIRFDPM\nHextFL\n(a) (b)Vmix8 \n \n By using  the Taylor ’s expansion, we get \nH H c cos ( ) cos sin cos( ) tt           and  \n2\nc H H H 0 ( ) [cos 2cos sin cos( )]R t R R t           \n . Hence, t he product is\n2\n0 H H c\nHc( ) ( ) ( )\n( cos )cos( ) sin(2 ) cos(2 )/2\nsin(2 ) (cos )/2V t I t R t\nIR I R t I R t\nIR     \n  \n      \n  \n \nThe total voltage V(t) comprises  of three terms at frequencies ω, 2ω, and a time-independent  term. \nThe last dc term is the rectified ST-FMR signal , \nH mix c sin(2 ) (cos )/2 V I R     . The ST -FMR \nsignal  is determined  by the combination of  the amplitude of IRF, AMR  in the device , the angle  of \nHext with respect to IRF, the cone angle  of magnetization precession  θc and the resonance phase  δ. \nFor an in -plane alternative magnetic field driven magnetization  dynamics  of an in -plane \nmagnetization  [30, 31] , the value of δ changes from π to 0 around the resonance field H0 with a \nlinewidth of ∆H (full width at half maximum)  and δ = π/2 at the resonance field H0. Therefore, the \nVmix attributed to the in -plane alternative magnetic field has an antisymmetric Lorentzian  line \nshape, which is the case for the Oe and/or FL driven  magnetization dynamics in a Pt/Py  ST-FMR \ndevice  as shown by the green curve s in figure 1(b) . Whereas for the DL driven magneti zation  \ndynamics, there is an additional  90° phase difference compared to FL or Oe driven magnetization \ndynamics  for an AMR based rectification system . Therefore, the driving force  DL is in-phase ( δ = \n0) at the resonance field H0, leading to a symmetric Lorentzian  line shape  arising from the \ndamping -like torque in a Pt/Py ST -FMR device  as illustrated  by the blue curve s in figure 1(b) . In \nSection 4, we will present three quantitative analys es of the ST -FMR signal in detail.  \n3. ST -FMR setup s \n3.1. Traditional ST -FMR setup with a dc voltmeter  \nAs described in Section 2.3 , the ST -FMR signal is a rectified dc voltage, which can be detected 9 \n \n directly by a dc voltmeter . Figure 2  shows a  schematic of the traditional ST -FMR setup with a dc \nvoltmeter . It depicts a typical  ST-FMR device with a micro -strip of heavy metal (HM) /ferromagnet \n(FM)  bilayer structure.  A ground -signal -ground (G -S-G) coplanar waveguide (CPW) is fabricated \nwith one G and one S electrodes contacting the HM/FM bilayer channel for applying an IRF to the \nST-FMR device which  is supplied by a signal generator (SG) connected to  an rf port of a bias t ee. \nA ST-FMR signal can be simultaneously detected by a voltmeter , such as Keithley 2182A and \nKeithley 2002 , connected to the low frequency port of the bias tee  as shown in the figure 2. The \ntraditional ST -FMR setup with a dc voltmeter  usually  works well for the devices having \nconsiderable amplitude of ST -FMR signals with the value of several tens of µV , such as Pt/Py  \nsystems [25, 32] . \n \nFigure 2 . The schematic of the traditional ST-FMR measurement setup, illustrating the ST -FMR \ndevice and  measuremen t circuit including  a bias t ee, a high frequency signal generator and a dc \nvoltmeter . Adapted from [32], with th e permission of AIP Publishing . \n3.2. Precise ST -FMR setup with a lock-in amplifier  \nFor the traditional ST -FMR setup , the noise level is  usually in the order of ~ 100 nV because \nof the limitation of a dc voltmeter. This  noise level  reduc es the signal -to-noise ratio substantially \nwhile  study ing some of the emerging material systems possessing small er ST-FMR signals with \nthe order of ~ µV , such as the topological insulator /FM bilayers  [33, 34] . In order to decrease the \nnoise  (i.e., increase the signal -to-noise ratio ), a more sensitive  ST-FMR setup  was developed  in \nJc\nJs\nGapW\nGHHext\nHM\nFM\nSSi/SiO2sub.\n～\nBias-Tee\nV\nSG Voltmeter10 \n \n which  the dc voltmeter is  replace d with a lock -in amplifier  [16, 35] . Figure 3(a)  shows the ST -\nFMR setup with lock -in technique.    \n \nFigure 3 . (a) The schematic of the sensitive  ST-FMR setup with a lock-in amplifier , illustrating \nthe ST -FMR device and m easurement circuit including  a bias t ee, a high frequency signal \ngenerator and a lock -in amplifier . Adapted from [32], with th e permission of AIP Publishing . (b) \nIllustration of the amplitude modulation of input IRF from a signal generator by a modulating signal \nwith a much lower frequency, and the modulated IRF is finally injected into the ST -FMR device.  \nThe input IRF is amplitude modulated by a low frequency sinusoidal wave signal (modulating \nsignal) as illustrated in figure 3(b). The low frequency modulating signal is also provided as a \nreference signal into the reference port of the lock -in amplifier and  thus serves as a trigger. The \nmodulated IRF is injected into the ST -FMR device to give rise to the ST -FMR signal. The ST -FMR \nsignal, in this case, is a low frequency voltage (instead of a dc) since the IRF is amplitude modulated. \nThe low frequency ST -FMR signal can be detected by the lock -in amplifier with the phase locking \ntechnique. Therefore, the noise level can be less than ~10 nV , thus providing a better signal -to-\nnoise ratio compared to the traditional ST -FMR setup.  \nJc\nJs\nGapW\nGHHext\nHM\nFM\nSSi/SiO2sub.\n～Bias-Tee\nSG\nRef Signal\nLock -in\nIRF\nModulated IRFModulating signal(a)\n(b)11 \n \n 3.3. Design of CPW  and impedance m atch \nIn order to avoid a large amount of rf power loss or reflection during the rf current transmission \nin the CPW and the HM/FM current channel,  the ST-FMR device should be designed to have an \nimpedance ( Zm) of around 50 Ω. First, the CPW should be designed considering impedance \nmatching.  One example of a typical  CPW  is illustrated in figure 4 (a). The width of S is s and the \ngap between S  and G line is w. The characteristic impedance of the CPW can be tuned by varying  \nthe s and/or w values.  One can use a common  CPW calculator software to select the proper \ncombination of s and w values for the CPW design.  For example, the s is about 60 µm  and w is \nabout 30 µm for the CPW design in Ref. [32]. Further, the HM/FM bilayer  current  channel  should \nalso be  impedance match ed, which can be tuned by changing the channel length  L and width W \n(also see t he current channel denoted by the red circles in the figure s 4(b) and (c) ). For example, \nL is in the range of ~10 – 60 µm and W is in the range of ~10  – 30 µm in Refs. [32-34]. Finally, \nZm of ST-FMR devices can be quantitatively calibrated b y the  microwave -network -analysis \nmeasurement using a vector network analyzer  (VNA) .  \nUsually , there are  two kinds of CPW designs, namely the asymmetric and the symmetric CPW  \ndesign s. The asymmetric CPW design is shown in figure 4 (b), which was initially  used for the \nHM/FM bilayer ST-FMR measurements  [25]. However,  there is a limitation of  the asymmetric \nCPW . To elaborate on this  limitation , consider  the example of  Pt/Py  ST-FMR device as shown in \nfigure  4(d). In addition to the current induced in-plane Oersted field ( HRF) to the Py magnetization , \nan imbalanced current induced out-of-plane Oersted field Hz from the contacting electrodes  can be \nproduced  as IRF is transmitti ng along  the CPW  [32, 36] . Consequently, Hz exerts an in-plane torque \n(–m×Hz) on the Py magnetization and generates  symmetric voltage signal s for both directions  of \nHext, which  super impose  onto the symmetric signals arising from damping -like torque, DL, from 12 \n \n spin currents . Therefore, the Hz induced torque might  lead to  an overestimation or underestimation \nof the spin Hall angle in Pt . However, one can eliminate this contamination by carefully locat ing \nthe GSG probe position in the middle of  the CPW [32]. \n \nFigure 4.  (a) The schematic of a typical  CPW, illustrating the ground -signal -ground (G -S-G) \ntransmission line . ST-FMR devices with (b) asymmetric CPW and (c ) symmetric CPW. (d) The \nPt/Py  microstrip with the charge current JC, spin current  JS, current induced in -plane Oersted field \n(HRF). Hz is the current induced imbalanced out -of-plane Oersted field  in the ST-FMR device with  \nasymmetric CPW design.  Adapted from [32], with th e permission of AIP Publishing . \nIn order to diminish or eliminate the imbalanced Hz and the related influences in the ST -FMR \nmeasurements and spin Hall angle evaluation, the symmetric CPW design was de veloped. As \nshown in figure 4(c ), the current induced Oersted field in the electrodes is balanced as IRF flows in \nthe symmetric CPW, resulting in n o Hz component. Therefore, the symmetric CPW design  is better \nthan the asymmetric CPW for ST -FMR measurements.  \n4. Methods to evaluate the spin -orbit torque (SOT) and SOT efficiency  \nIn Section 2.3, a general derivation of ST -FMR signal  was provided . In this section, w e will \nfurther present three quantitative analys es of the ST -FMR signal . The ST -FMR signals  from a \nHM/FM bilayer device  (see figure 1(b)) can be fitted by\nmix S S ext A A ext ( ) ( ) V V F H V F H  , where \nx\nyz\nNiFe\nPt\nJS\nHRF\nHzJC\nW\nL\n(d)(b) (c)\nG S G\nG S G\nwswG S G(a)13 \n \n \n2 2 2\nS ext ext 0( ) /[( ) ]F H H H H H    is a symmetric Lorentzian function of amplitude VS and \n22\nA ext ext 0 ext 0( ) ( )/[( ) ]F H H H H H H H   \n  is an antisymmetric Lorentzian function of \namplitude VA [25, 32] . The symmetric component has  a maximum value  at the resonance field H0 \nand is centered about  H0, whereas, the a ntisymmetric component has a dispersive curve with the \nvalue of zero at H0. Both components  have the same linewidth  ∆H with respect to  the magnetic \nfield.  \n4.1. Ratio of VS/VA \nIt is noted that VS is proportional to  the amplitude of spin currents JS (i.e. DL) and is written \nas \nS S 0 s /(2 ) V J e M t \n  [18, 25, 37] , where  Ms and t are the  saturation magnetization and thickness  \nof the FM layer , respectively, and \n  is the reduced Planck constant.  While VA is correlated with  \nHRF induced Oersted field torque ( Oe), which is written as \n1/2\nA RF eff ext [1 4 / )] V H ( M H    [18, 25, \n37], where Meff is the e ffective magnetization of the FM  layer. Note that the field-like torque FL \nis assume d to be negligible here . HRF is estimated by Ampere ’s law as\nRF C= /2H J d , where JC is \nthe charge current density in the HM  layer . Note that the charge cu rrent flowing in the FM  layer is \nassumed to  be spatially uniform, therefore, there is no net Oersted field torque on the FM  \nmagnetization itself.  \nIn the spin Hall  scheme , the charge -to-spin conversion efficiency (i.e. spin Hall angle) , defined \nas\nsh S C /JJ , can be found from the ratio of \nA S/VV  [25]. Using the above equations, one obtains \nthe spin Hall angle as\n1/2\nsh S A 0 s eff ext( )( )[ (/ / 1 4 / )] V V e M td M H  \n  . This method is self -\ncalibrated in the sense that the torques on the FM magnetization  only arise  from JC in the Pt layer \nand the spin Hall angle can be easily calculated without knowing the exact  value s of IRF and HRF \nin ST -FMR  devices. The method works well to determine  θsh under the assumption  that the VA 14 \n \n only attributes  to HRF. In other words,  this method may give rise to a wrong estimate  of θsh for the \ncase of a non-negligible  field-like torque  FL, which might arise due to the interfacial effects , such \nas the Rashba effect . Since  FL can also produce an antisymmetric Lorentzian line shape signal \nsimilar to HRF, the value of θsh might be overestimated  or underestimated  from the method of    \nA S/VV\n[38, 39] . Therefore, one should be aware of no significant field -like torque from interfacial \neffect before  using the analysis method of the ratio of  \nA S/VV .   \nNote that for simplicity, hereafter , we refer  the spin Hall angle as the SOT efficiency , unless \notherwise specified , in order to represent the charge -to-spin conversion efficiency in a variety of \nmaterials.  After obtaining the SOT efficiency, we can easily evaluate the damping -like torque using  \nDL sh C s (2/)J eM t\n. \n4.2. Only-VS \nSince  some material  systems  give rise to a significant FL to the adjacent FM magnetization, \nthe SOT efficiency can be better determined by analyzing the symmetric component VS only [33, \n34, 40] . The damping like  torque , τDL, and the associated in-plane SOT efficiency , θsh, can be \nevaluated using  \nRF H\nDL S ext\nHS1()4I cos dRFHdV  , \nS S DL s //J E M t E  , and  \nsh S /    \n[33, 34, 40] , where \nH/dR d  is the angular dependent magnetoresistance at H,  is the linewidth \nof ST -FMR signal  in the frequency domain , E is the microwave electric field across the ST -FMR \ndevice, and S and  are the spin Hall and longitudi nal charge conductivities of the  HM layer , \nrespectively.  In addition , the field -like torque , τFL, can be  derived by \n12\n0 eff ext rf H\nFL Oe asym ext\nHA[1 ( / )]( ) ( )4/MH I cos dFH VR\nd \n [33, 40] . In order to obtain  τFL, \nthe τOe should be evaluated separately and excluded  [33]. Similarly, the out -of-plane SOT 15 \n \n efficiency, θ⊥, can be evaluated accordingly.  \nSince the  resistance  of the HM/FM microstrip  is written as \n2\nH 0 () R R Rcos     due to the \nAMR effect (see Section 2.3) , we obtain \nH H H H\nH2 ( ) ( ) ( ) ( )dRR sin cos sin cosd       . \nConsequently, we find \n2\nmix H H ( ) ( ) V cos sin   from the relation , \nmix H\nH()dRV cosd [32]. Thus, \nit is observed that even  though the change of AMR is maximum at \nH = 45, the largest ST -FMR \nsignals are achieved at \nH ~ 35. \nThe comparison of the in -plane SOT efficiencies evaluated from the ratio of VS/VA method \nand from only -VS method is presented in Section 5.1  for Pt and Section 5.2 for Ta. These results \nsuggest  that the only-VS method  is a more  general  way to evaluate  the SOT efficiency in material s \nwith SOC . However, compared to the VS/VA ratio method , the only -VS technique requires \nadditional measurements to determine the \nH/dR d   and , and also requires quantitative \ndetermination of IRF through a ST-FMR device by considering the device impedance and rf power \nlosses  using the microwave -network analysis measurement [32]. \nNote that the SO T efficiency or in -plane SOT efficiency in the entire review for a variety of \nmaterials is denoted as θsh, and the out-of-plane SOT efficiency is denoted as θ⊥, in order to be \nconsistent.  \n4.3. Modulation of damping (MOD)  \nIn the context of conventional STT , researchers  found that the STT and magnetization \nfluctuation can be modulated  in the vertical MTJs  by applying a dc current bias [41, 42] . Recently , \nthis method was used for  HM/FM bilayer system s in the spin Hall scheme and the magnetization \ndynamics  can be tuned by  applying a dc current in the HM layer [43]. Based on the STT theory  16 \n \n [41], the effec tive magnetic damping (α) and thus the  FMR linewidth  (∆H) will increase or \ndecrease depending on the relative direction of spin polarization s with respect to the magnetization \ndirection.  The relationship can be written as  [25] \nH\n0 sh C sh\next eff 0 ssin 22=( ) +( ) =( 2 ) 2ffH H H JH M M t e       \n, where  (∆H)0 is the linewidth \nat zero dc bias, (∆H)sh is the modulated linewidth due to the dc bias induced spin current s, and f is \nthe rf current frequency.  Therefore, one can perform the ST -FMR measurements to extract  the ∆H \nas a function of JC as shown  in figure 5 . By linearly fitting the data, the SOT efficiency  is obtained \naccordingly .  \n \nFigure 5 . The FMR linewidth  and Gilbert damping coefficient α as a function of dc current, Idc, in \nthe Pt (6 nm)/ Py (4 nm) bilayer device  at f = 8 GHz . Reprinted with permission  from [25]. \nCopyrigh t (2011 ) by the American Physical Society . \nThe modulation of damping (MOD) method is an extension of the ST -FMR measurements. \nSo far, it has been used to evaluate the SOT efficiency commonly  in Pt and Ta [25, 44 -47], which \nusually ha ve considerably large ST -FMR signals. Recently, it has been utilized  for the SOT \nefficiency evaluation of metallic antiferromagnetic materials [48]. However, for the material \n17 \n \n systems possessing smaller ST -FMR signals, it is challenging to employ the MOD method to \nevaluate the SOT efficiency [40].  \nTable 1 summarizes  the advantages and disadvantages of the above  three SOT efficiency \nevaluation methods . The method of VS/VA is simple, however it works only for the material systems \nwhere the VA only attributes to HRF (i.e. there is no significant field -like torque). Therefore, this \nmethod  has limitations for the SOT efficiency evaluation. However, the method of only -VS can \nsuccessfully address this issue and  can be a more general method to extract the SOT efficiency in \na wide range of  material s. \nMethods  VS/VA MOD  Only -VS \nParameters  \nrequired to \nestimate  \nsh VS, VA, Ms, t, d, Meff and \nHext (∆H)sh, f, θH, Hext, Ms, t, \nMeff, and JC VS, IRF, θH, \nH/dR d , ∆, \nMs, t, E and  \nAnalysis \nprocedure  Can directly get \nsh\nfrom ST -FMR lineshape \nanalysis  Need s precise ST -FMR \nlinewidth measurements \nand analysis  Need s additional AMR \n(\nH/dR d ), conductivity \n() measurements and  \nrf power analysis  \nMaterial \nsystems  Limited to material \nsystems without \nsignificant FL Material systems with \nconsiderably large and \nclean ST-FMR signals  Almost  all material \nsystems showing ST -\nFMR signals  \nRemarks  A simple method but \nwith serious limitations  A less simple method \nbut with moderate \nlimitations  A more general method , \nbut need extra measured \nparameters  \nReferences  [25], [32]  [25], [44 -48] [32-34], [40]  \nTable 1.  Summary of the advantages and disadvantages of three SOT efficiency evaluation \nmethods . \n5. Spin Hall effect  (SHE)  in heavy metals  and alloys  \nSpin Hall effect [49-52] is an electrica l technique, whi ch exploits the bulk SOC in a \nnonmagnetic material  (such as HM) to convert charge currents into pure spin currents  without any 18 \n \n external magnetic field . The first theoretical predictions of SHE by Dyakonov and Perel dates back \nto 1971 [53] and, in 1999, the SHE method to generate spin currents was reinvigorated by Hirsch  \n[49]. Subsequently, in 2004 , the SHE was first experimentally observed in the GaAs system  [19]. \nThe phenomenon of SHE relies on the spin -dependent asymmetr ic scattering of the unpolarized \nelectro ns due to the bulk SOC in the nonmagnetic materials  to generate a pure transversal spin \ncurrent density. The SOC effects that give rise to SHE in a nonmagnetic material  may be due to \nthe band structure of the material  (intrinsic SHE) or by addition of high SOC impurities into the \nmaterial  (extrins ic SHE). As illustrated in f igure 6, the spin polarization of the accumulated spins \ndue to SHE is orthogonal to both the directions of the injected charge current as w ell as the spin \ncurrent. Thus, the spin current generation using SHE can be described using the equation \nS sh C () ˆ JJ\n , where \nˆ   is the spin  polarization . Here, \nsh  (\nSC/JJ  ) is spin Hall angle of the \nnonmagnetic material  and quantifies the efficiency of the spin current generation by SHE. For \nfurther details into SHE, the readers can refer to detailed reviews on SHE [51, 52] . \nFor practical spintronic applications, it is desirable to have a higher spin Hall angle for \nefficient spin current generation using SHE. It is generally understood that the in -plane SOT \nefficiency in the ST -FMR technique estimates the spin Hall angle of the NM. Recently, many \nefforts have been devoted to obtain a large SOT efficiency  and efficient  SOT driven magnetization \nswitching or prece ssion via pure spin currents in HM/FM bilayers [2, 20 -24]. Apart from ST -FMR \nmeasurements [25, 32] , several different techniques  have been also employed  to determine the spin \nHall angle , such as  the lateral spin valve method [54-56], spin pumping  [57] and spin Hall \nmagnetoresistance  (SMR)  measurements  [58]. In the following sub -sections, we first review the \nresearch progress of the SOT efficiency evaluation in HMs  and alloys by the ST-FMR  technique.   19 \n \n  \nFigure 6 . Schematic of SHE . The black -colored , blue-colored , and dotted arrows i ndicate the \ndirections of charge current JC, spin current JS, and the motions of spin -up and spin down electrons.  \n5.1. Spin Hall effect in Pt  \nLiu et al. first utilized ST -FMR to evaluate the SOT efficiency in the Pt (6 nm)/Py (4 nm) \nbilayer device at room temperature [25]. The schematic of a Pt/Py bilayer and the main results are \nshown in  figure  7. The value of SOT efficiency  in Pt is determined to be ~ 0.056 . Further , they have  \nalso utilized the MOD method as an independent check to confirm their results.  By taking \nadvantage of the large SOT efficiency, they also demonstrated the  SOT  driven magnetization \nswitching in Pt/Co/Al 2O3 trilayer structures [22]. \n \nCharge current JC\ne-\n(a) (b)\n(c) (d)20 \n \n Figure 7. (a) Schematic of a Pt (6 nm)/Py (4 nm) bilayer illustrating the spin-orbit torque τSTT, the \nOersted field torque τH, and the direction of the damping torque τα. (b) Left-side view of the Pt/Py \nsystem, showing the Oersted field generated by the current flowing just in the Py layer, which \nshould  produce no net effect on the Py ma gnetization . (c) ST-FMR signals on a Pt/Py sample. The \nsample dimension s are 20 µm wide and 110 µm long. The i nset shows the ST -FMR signal of 8 \nGHz in a full magnetic field range . (d) SOT efficiency in Pt /Py at different resonance fields . \nReprinted  with permission  from [25]. Copyright (2011) by the American Physical Society . \nSubsequently, the SOT efficiency in Pt has been  quantified  by different measurement methods \nin different research group s. However, there have been significant disagreements in the reported \nvalues of the SOT efficiency  in Pt, ranging from ~0.012 to ~0.12 [32, 51, 52] . To figure out the \npossible reason s for this discrepancy and determine the intrinsic SOT efficiency value in Pt is of \ngreat importance.  For ST -FMR measurement s, the measured SOT efficiency  is evaluated from the \namount of spins that are absorbed by the FM layer.  There are t hree aspects  that affect  the measured \nSOT efficiency . The first one is the spin diffusion in the Py layer since the spins can transmit into \nPy with a characteristic  length (i.e. spin diffusion length), which is often ignored. Wang et al.  [32] \ncarried out Py thickness dependent ST -FMR measurements in Pt (6 nm)/Py ( t = 2 – 10 nm ) bilayers. \nIt was found that θsh increases when the Py thickness increases as shown in figure 8(a). By taking \ninto account the spin diffusion in the Py layer, t he SOT efficiency of Pt is determined  to be ~0.0 68 \nat room temperature [32]. The second aspect is spin diffusion in the Pt l ayer. JS and the measured \nSOT efficiency θsh in a Pt film of thickness d should be reduced from the bulk value  (d = ) by \nS S S( ) ( ) 1 sech /J d J d    \n and \nsh S 1 sech / d    , respectively  [25, 32, 45, 59] . For \nexample, t he spin d iffusion length , λS of Pt estimated from a Pt/Py device  is ~1.5 nm [32] (figure \n8(b)). The third aspect  is the interfac e transparency as  spin currents  transmit th rough the Pt and Py  \ninterface, which is  discussed  in Section 5.4 .  \nFigure 8 (c) shows a summary of the SOT effici encies  as a function of λS in Pt measured by 21 \n \n different techniques from various  research group s [25, 32, 58, 60 -69]. A clear correlation between \nSOT efficiency θsh and λS is found, which is approximately an inverse  relationship  with θshλS ~ \n0.13 nm  (denoted by the blue thick line)  [32]. The λS in a material is generally proportional to its \nelectrical conductivity  (\n) [32, 70] . Figure 8 (d) shows an approximately linear  relationship \nbetween the reported λS and \n in Pt films [32, 45, 55, 58 -62, 64, 67, 71] . Therefore, the SOT \nefficiency  is inversely related  to the conductivity  of the HMs . All the data in figure s 8(c) and (d) \nwere measured at room temperature, except for those from Refs. [61] and [71] which were \nmeasured at 10 K, as denoted by small black stars.  In addition , it was  found that the  θsh in Pt  \nremains almost constant  as temperature decreases from 300 to 13 K [32, 55] .  \n \nFigure 8. (a) Measured SOT efficiency θsh for Pt/Py (t) devices with different Py thicknesses from \nST-FMR measurements at  8 GHz  and 300 K.  (b) θsh (blue square s) of Pt (d)/Py (4 nm) devices  as \na function of Pt thickness at 300 K . (c) Summarized  SOT efficiency θsh as a function of λS in Pt \nfrom various reports. The ST -FMR, SP, LSV , and SMR represent spin torque ferromagnetic \nresonance, spin pumping, nonlocal measurement in lateral spin valves, and spin Hall \n0 2 4 6 8 10 120.020.040.060.080.100.120.14 STFMR Pt/Py [25,32,60]\n SP Pt/Py [62-64,66,69] \n SP Pt/YIG [68]\n LSV Pt/Py [61] \n SMR Pt/YIG [58,65,67]\n*sh\ns (nm) \n \n0 2 4 6 810 12 140.020.040.060.08m  Data\n FittingT = 300 K \ntPt (nm)S= 1.5 nm\n  (d) (c)(b) (a)\n0.02 0.04 0.06 0.08 0.100510152025\n[32][55]\n[45][71]*\n[62]\n[60][61]*\n[64] [58]\n[59] [67]\n s(nm)\nPt(-1cm-1)\n  \n0.030 0.035 0.0400.040.050.060.07\n250oC\n350oC\n  m\nPt(-1cm-1)As grown\n0 2 4 6 8 100.030.060.09\ncbaT = 300 K  Data\n Fittingm\n \ntNiFe (nm)\n  \nFigure 8_rebuttal22 \n \n magnetoresistance method, respectively. The blue thick curve shows a  correlation  θshλS ~ 0.13 nm . \n(d) λS in Pt as a function of its electrical conductivity  (σPt). The inset shows the measured SOT \nefficiency  in various Pt films with different conductivities. The dashed lines serve as guide for the \neye. Adapted  from [32], with th e permission of AIP Publishing . \nIt is possible that in ST -FMR measurements, spin pumping can also occur due to the inverse \nspin Hall effect by \nC sh S () ˆ JJ . Therefore, an additional dc voltage ( VSP) can also be produced \nand might contaminate the ST -FMR signals. The spin pumping contributions for Pt (6 nm)/Py ( t = \n2 – 10 nm ) bilayers have been estimated in Ref.  [32], by using the following equations in the \nmethod part in Ref. [40].  \neff 2 S\nc H SP 0 0 sh eff\nStanh( )Re( ) ( ) sin( ) /( 4 )22eW R dV g H H M    \n, \n22\nc SA\nH RF12( ) ( )/VVdR d I\n, \nwhere W is the channel width, R is the device resistance, d is the Pt thickness, Re(\neffg ) is the real \npart of the effective spin mixing conductance (  2×1019 m-2), θc is the maximum precession cone \nangle in the device plane, and VS and VA are the symmetric and antisymmetric components of the \nST-FMR signal, respectively. It was found that the estimated spin pumping signals Vsp are at least \none order of magnitude smaller than the ST -FMR symmetric component VS. It is known that Vsp is \nproportional to device resistance [57]. Since t he resistance of the ST -FMR device is usually small \n(~50 Ω for the impedance matching , the spin pumping contributions in the ST -FMR measurements \nare usually much smaller than the ST -FMR signals.  \nIn addition, t he SOT efficiency in Pt/Py bilayers has been quantified by the VS/VA ratio and \nonly-VS methods sepa rately  [32], and no clear  difference between the SOT efficiencies  from these \ntwo methods  was found . This suggests that there is negligible τFL arising from the Pt/ Py interface . 23 \n \n However, it has been also reported that τFL might arise from the Rashba effect at the Pt/ FM \ninterface [20, 38, 72] , which indicates that the characteristics  of the interfaces, such as the interface \nquality and the degree of oxidation of the interfaces , could also affect  the SOT efficiency . On the \nother hand, a  Cu with negligible SOC or other insertion  layers can be  inserted between the HM \nand FM layer to eliminate  or modify the interface SOT effect  [46, 73, 74] .  \n5.2. Spin Hall effect in Ta \nIn 2012, a large  SOT efficiency  and giant spin Hall induced magnetization switching were  \nreported  in a Ta (8 nm)/Co 40Fe40B20 (CoFeB, 4 nm) bilayer at room temperature [21]. Figure s 9(a) \nand (b)  illustrate the  ST-FMR measurements  and results  in a Ta/CoFeB bilayer . A very high SOT \nefficiency of ~ 0.15 was determined by the VS/VA ratio method. The SOT efficiency shows a \nnegative sign compared to the positive sign in Pt, which agrees with the opposite character  of spin \naccumulation  in Pt and Ta.  Moreover, an efficient SOT driven  magnetization switching was  first \nrealized in a three terminal MTJ device  (Ta/CoFeB/MgO/CoFeB ), which open s a new avenue to \nthe SOT based spintronic devices.   \n \nFigure  9. (a) Schematic of sample geometry for a Ta (8 nm)/CoFeB  (4 nm) bilayer ST -FMR device . \n(b) ST-FMR signal  at f = 9 GHz . From [21]. Reprinted with permission from AAAS.  \nThe interface effect between the Ta and C oFeB  layer has not been considered by Liu et al.  [21] \nin the Ta SOT efficiency evaluation . Thus , we compare  the SOT efficiency  in Ta on Si/SiO 2 \n(a) (b)24 \n \n substrate /Co 40Fe40B20 (4 nm)/Ta (8 nm) and Si/SiO 2 substrate /Py (4 nm)/Ta (8 nm)  devices  by ST -\nFMR measurements  at different temperatures.  A significant contribution of field -like torque , τFL, \nis observed , which might arise  from the interface between Ta and CoFeB layers. Figure 10 (a) \nshows ST -FMR signals  and fits (solid lines)  in a CoFeB (4 nm) /Ta (8 nm)  device for frequencies \nbetween 6–10 GHz  at room temperature . The Hext is swept at an angle of θH = 35° with respect to \nthe IRF. The SOT efficiency in Ta is  quantified  by the VS/VA ratio and only -VS methods, respectively.  \nAs shown in  figure 10 (b), the in-plane  SOT efficiency ( θsh) in CoFeB /Ta device is ~ 0.109 from  the \nVS/VA ratio method at room temperature and it remains in the range of ~0.1–0.15 a t most of \ntemperatures. The  θsh value is similar to  the previous report [21]. In addition , we perform  the only-\nVS method  to determine  θsh as well as  the out -of-plane SOT efficiency ( θ⊥) which characterizes  the \nτFL. Surprisingly, t he θsh value from the only -VS method  is ~0.02  at room temperature  as shown  by \nthe red circles  in figure 10 (b). This big inconsistency in θsh indicates a significant  τFL (i.e. θ⊥) \ncontribution in the CoFeB /Ta bilayer.  As shown in figure 10 (c), there is  indeed  a large θ⊥ of ~0.044 \nin a CoFeB/Ta bilayer  at room temperature.  This result  is qualitatively similar to an earlier report \nof a larger τFL in CoFeB /Ta compared to τDL [75].  \nSimilarly, we also  chara cterize the Py (4 nm)/Ta (8 nm) bilayer device . As shown in figure \n10(d), the θsh from the only-VS method is ~ 0.01 at room temperature . The small inconsistency in \nθsh from two analysis methods  indicates that the τFL is not significant  in the Py/Ta bilayer . Finally , \nθsh in Ta is ~ 0.01–0.02 obtained from the FM/Ta bilayer  using ST -FMR  at room temperature , which \nis consistent with other reports [61, 76 -78]. The slightly  different θsh from Py/Ta  and CoFeB/ Ta \ncan be attributed  to the different  interface transparency  [76, 79 -81]. The θsh (from only-VS) almost \nremains constant from 15 to 300 K,  while the θ⊥ shows a fast de crease at low temperature range, \nsuggesting that θsh and θ⊥ might come from different origins.  Moreover, we observed  a similar θsh 25 \n \n of ~0.01–0.015 in Py (20 nm)/ Ta (6–25 nm) and CoFeB (15 nm)/Ta (15 nm) bilayers  by spin \npumping measurements  at room temperature . This similar  SOT efficiencies  from both ST -FMR \nand spin pumping  measurements suggest  the Onsager reciprocity, which has been also \ndemonstrated recently [82].  \n \nFigure 10 . (a) ST-FMR measurements in a CoFeB (4 nm) /Ta (8 nm)  device at different frequencies. \nThe symbols are the measured data, and the solid lines are fits by sum of Lorentzian functions. (b) \nThe in-plane SOT efficiency  and (c) t he out -of-plane SOT efficiency in Ta as a function of \ntemperature obtained fr om ST -FMR measurements on  the CoFeB (4 nm)/Ta (8 nm) device . (d) \nThe in-plane SOT efficiency as a function of temperature  in the Py (4 nm) /Ta (8 nm) device .   \nFigure 11  shows a summary of the SOT efficiencies  in Ta films, ranging from 0.0037 to 0.26, \nmeasured by different techniques from different research groups [61, 75 -77, 83 -89]. The fit of θsh \nas a function  of λS indicates an appro ximately inverse relationship with  θshλS ~ 0.036 nm (denoted \nby the blue thick line). However, it is observed that the correlation between θsh and λS is not as \nclear as  the case of Pt as shown in figure 8 (c), which might be due to multi -phases (such as α-, β-\n0 50 100 150 200 250 3000.000.050.100.150.20 From VS/VA\n From only VS\n T (K)\n  shCoFeB/Ta\n0 50 100 150 200 250 3000.000.020.040.060.080.10\nCoFeB/Ta From only VA\n T (K)\n  \n0 50 100 150 200 250 3000.000.050.100.150.20  From VS/VA\n From only VS\n   sh\nT (K)Py/TaCoFeB /Ta(a) (b)\n(c) (d)300 KVmix(µ V)26 \n \n phases or amorphous film ) in different Ta thin films  [21, 87] . \n \nFigure 11. Summarized SOT efficiency θsh as a function of λS in Ta from various reports. The LSV , \nSP, SMR, dc Hall, ST -FMR, SSE represent nonlocal measurement in lateral spin valves, spin \npumping, spin Hall magnetoresistance, Hall measurements by using  dc current s, spin torque \nferromagnetic resonance, and spin seebeck effect method, respectively. T he blue thick curve shows \na correlation  of θshλS ~ 0.036 nm. The θsh and λS were obtained  at room temperature, except  for the \none by the LSV method which was measured at  10 K  as denoted in the figure.  \n5.3. Spin Hall effect in tungsten ( W) and other alloy s with heavy metal  dopants   \nFor the case of pure HMs , the largest SOT efficiency  is ~0.3 observed in a β-phase W/CoFeB \nbilayer by ST -FMR measurements at room temperature  [35]. This value is about two  times larger \nthan that in Ta, leading to  efficient spintronic devices.  Taking advantage of the highly efficient \nspin current generation, SOT driven  magnetization switching has been demonstrated in  a β-phase \nW based 3 -terminal MTJ device  [35]. The critical switching current density , JC, is in the order of \n~106 A/cm2 in W /CoFeB/MgO  [90], which is almost one order of magnitude smaller than  that in \nPt. \nFigure 12 shows a summary of the SOT efficiencies in W, ranging from 0.0043 to 0.95, \nmeasured by different techniques from different research groups [84, 85, 87, 89 -93]. However, the \n01234567890.000.050.100.150.200.250.30\n LSV Ta/Py [61]\n SP & SMR Ta/YIG [77]\n SP [76,84,86]\n dc Hall Ta/CoFeB [83]\n ST-FMR Ta/CoFeB or Py [88]\n SSE Ta/YIG [85]\n 2nd Harmonic Ta/CoFeB [75]\n SMR Ta/CoFeB [87,89]\n ST-FMR & SP Ta/Py or CoFeB\nOur datash\ns (nm)10 K27 \n \n correlation between θsh and λS is not as clear as the case of  Pt as shown in figure 8 (c). Similar to \nthe case of Ta, the reason can be attributed to the multi -phases (such as α-, β-phases) in various W \nthin films [35, 90] . \n \nFigure 12.  Summary of SOT efficiency θsh as a function of λS in W from various reports. The SOT \nswitching represents current induced magnetization switching via SOT . All data were obtained at \nroom temperature.  \nBesides the pure HMs , alloys  consisting of a light metal  with heavy metal dopants have been \nalso investigated  [94-97]. Ramaswamy  et al.  [97] experi mentally studied  the effects on the SOT \nefficiency  due to a systematic addition of Pt into the light metal Cu by  ST-FMR measuremen ts. As \nshown in figure s 13(a) and (b), the SOT efficiency  of Cu 1−xPtx increases as th e Pt concentration \nincreases  from 0 to 40% . It was  found that only 28%  Pt in Cu 1−xPtx can give rise to a SOT efficiency  \nclose to that of Pt. Since Cu is the most common metallization element in the CMOS platform and \nCu1−xPtx can have a large enough SOT efficiency , the Cu 1−xPtx-based alloy is easier to integrate \nthe spintronic devices into an existing Si fabrication technology.  In addition , from further  analysis \nof the ST -FMR data, it is found that the skew scattering contribution is significant for lower Pt \nconcentrations , while the side-jump contribution is significant  for higher Pt concentrations.  This \n0 1 2 3 40.00.20.40.60.81.0  SP W/YIG [84]\n SSE W/YIG [85]\n SMR [87,89,91]\n 2nd harmonic & SOT switching [92]\n SOT switching W/CoFeB [90]\n 2nd harmonic W/Co2FeAl [93]sh\ns (nm)28 \n \n result can be understood as due to the different scaling of the contributions of skew scattering and \nside-jump with respect to Pt concentrations. As Pt concentration increases, the longitudinal \nresistivity increases. In a simplistic picture, the skew -scattering contribution is independent of this \nincreased resistivity due to Pt, whereas the side -jump contribution is proportional to this increased \nresistivity  [98, 99] . Thus, for higher Pt concentrations, the increase in the resistivity is larger, \nleading to the domination of the side -jump contribution. Furthermore, this result can be also \ncorrelated with an earlier theoretical report [100]  and with experiments in t he context of anomalous \nHall effect due to rare earth impurities in Gd  [101] .  \n \nFigure 13 . Fittings  of ST -FMR signals  from the  Cu1−xPtx (6 nm)/Py (5 nm)  bilayer for x = 0, 6.6, \n13.7, and 19.7% in the negative Hext range for an applied microwave power of 16 dBm and a \nmicrowav e frequency of 8 GHz. As the Pt concentration increases , the amplitude of the symmetric \ncomponent increases (red curve). (b) SOT efficiency , θsh, for different Pt concentrations extracted \nfrom ST -FMR fittings  by using the VS/VA ratio method (blue circles) and only -VS method (red \nsquares), suggesting there is a negligible  τFL at the interface between Cu1−xPtx and Py . Reprinted  \nwith permission from [97]. Copyright (2017 ) by the American Physical Society . \n5.4. Role of i nterface transparency  of HM and FM in the SOT efficiency  \nIt is possible for t he spin loss to occur  at the interfaces  when  the spins transmit into the FM \nlayer from the spin generation sources [70, 79], which can have an important role in the  estimated  \n(a) (b) θsh29 \n \n SOT efficiency . Using the ST -FMR technique, Zhang et al.  [80] studied the transparency of Pt (6 \nnm)/Py  (5.5 nm ) and Pt (6 nm)/Co (5.2 nm) , shown in figures 14(a) and (b), respectively.  Using  \nstandard ST -FMR analysis method, the authors fi nd that the SOT  efficiency in Pt/Co is ~0.11, \nmuch larger than ~0.05 in Pt/Py . They ascribed the discrepancy of the SOT efficiency  to the \ninterface transparency , which is correlated to the  effective  spin-mixing conductance , Geff at the Pt \nand FM interface. They quantified  the Geff to be 3.96× 1019 m-2 (for Pt/Co) and 1.52× 1019 m-2 (for \nPt/Py), and the interface transparencies  to the spin currents are ~0.65 (Pt/Co) and ~0.25 (Pt/Py ), \nrespectively . After taking into account the  transparencies  of these interfaces, they found that the \nintrinsic SOT efficiency in Pt  has a much higher value of 0.17 ± 0.02 in Pt/Co and  0.20 ± 0.03 in \nPt/Py  bilayer.  Further, t his result  also suggests that the spins easily transmit  through  the Pt/Co \ninterface and thus give a large measured SOT efficiency, which might be due to better matching \nof the electronic bands between the Pt and Co layer compared to the Pt and Py  layer .  \n \n(a) (b)\n(c) (d)θsh\nθsh30 \n \n Figure 14 . Illustration of the  interface transparency for (a) Pt/Py and (b) Pt/Co : the red and  blue \narrows represent the up and down spin accumulation. The white  arrows show the electrons \ndiffusing across or being reflected at the  interface.  Frequency dependence of measured SOT \nefficiencies  for (c) Pt (6 nm) /Py (5.5 nm) and (d) Pt (6 nm)/Co (5.2 nm). Insets: measured ST -\nFMR signals for Pt/Py  and Pt /Co at 9 GHz, respectively.  Reprinted by permission from Macmillan \nPublishers Ltd: Nature Physics [80], copyright 2015.   \nSubsequently, Pai et al. [81] also conducted  a systematic study of the interface transparency \nin Pt/Co and Pt/CoFe bilayers. They found that the SOT efficiency can be modulated under \ndifferent interface condition s and a much la rger intrinsic SOT efficiency (~ 0.3) in Pt is obtained \nafter considering the interface transparency  between the Pt and FM layer . This value is much larg er \nthan the measured value of ~ 0.06 [25, 32] , suggesting that there is a large spin memory loss at the \ninterface. Therefore, it is understood that the measured SOT efficiency by ST -FMR technique \nrepresents a lower value  because of the interface transparency. Therefore , modification  or \nenhance ment of  the SOT effici ency via  interface transparency  engineering , such as inserting  layers  \nbetween the HM and FM  [73, 74] , manipulation of the film crystal structures [102]  or using \ndifferent combinations of HM and FM layers  [80, 81] , are some  interesting and important on -going \nresearch activities.  \n5.5. Difference  in the SOT efficiency between ST -FMR and magnetization switching \ntechnique  \nFrom ST-FMR measurements , DL and FL can be separately extract ed due to the 90°  phase  \ndifference in the magnetization dynamics (see Section 2.3) [33, 40] . Subsequently, DL can be used \nto evaluate the in-plane SOT  efficiency θsh. However, for the case of current driven  magnetization  \nswitching  using SOT , it has been reported that both  DL and FL contribute to the magnetization \nswitching and the switching current density JC might  be significantly reduced if there is an \nadditional FL [103-106], which can lower the energy barrier of magnetization switching . Therefore , 31 \n \n the values of the SOT efficiency derived from SOT induced magnetization switching \nmeasurements can be greater  than that from ST -FMR measurements. In addition, during the ST -\nFMR measurements, the magnetization is uniformly aligned in the direction of an applied large \nexterna l magnetic field. Hence , it is a coherent magnetization pr ecession  process . However, the \ncurrent induced magnetization switching proceeds through an thermal ly excited incoherent \nmagnetization switching process [107, 108]  (i.e. domain wall motion with a much lower energy \nbarrier), which can also lead to  a smaller JC and a larger θsh from magnetization switching \nmeasurements compared to the ST -FMR measurements.  Therefore, these aspects should be  taken \ninto account  for the SOT efficiency evaluation or comparison  between ST -FMR and switching \nmeasurements . \n6. SOT in topological insulators (TIs) \nTopological insulators (TIs) are quantum materials t hat have a band gap just like a  normal \ninsulator but have topo logically  protected conducting edge states or surface states  resulting from \nthe inverted conduction and valence band s due to strong SOC [109-112]. TI material s started from  \ntwo-dimensional (2D)  system, called 2D TI . The 2D TI was predicted  [113]  and experimentally \nobserved in HgTe/CdTe quantum wells structur es in 2007  [109] . As shown in figure 15(a), t wo \nconductive edge states  are present  in the edge of  2D TI s, each of which contribute s one quantum \nof conductance  \n2/eh , where \nh  is the Plank’s constant . In the edge states, t he spin polarization  \ndepends  on the electron momentum .  \nA year later, in 2008, the first 3D topological insulator Bi 1-xSbx was experimentally discovered \n[114] . Since then, more 3D TIs  such as Bi2Se3, Bi 2Te3, and Sb 2Te3, having  a larger bandgap and \nsingle Dirac cone at the Γ point,  have been predicted and identified mainly by angle resolved \nphotoemission spectroscopy (ARPES) experiments  [110-112]. The 3D TIs possess spin -32 \n \n momentum -locked topological surface states (TSS)  due to time reversal symmetry protection  [115-\n117] and it can be described by the Dirac Hamiltonian \nkFˆˆ =H z k（）  , where k is electron \nmomentum , \nˆz is the unit vector perpendicular to the T I films and \nF  is the Fermi velocity . This \ndispersion  relationship reveals that  on the TSS, the electron momentum and the spin polarization \ndirections are strongly locked  as shown in figure 15(b). As depicted in figure 15(c ), in real space, \nas charge currents flow on TSS, all t he electron spins are expected to be fully polarized in the \northogonal direction  to the electron moving direction  due to the topologically protection  [118] . \nTherefore, a very efficient spin current generation and thus a giant SOT  efficiency are expected in \nTIs due to  TSS. For further details into TIs and related physics phenomena, the readers can refer \nto detailed reviews on TIs [116, 117] . In the f ollowing part , we will review research work s on 3D \nTIs in the context of  SOT efficiency determination.   \n \nFigure 15.  (a) Schematic of the spin momentum locked edge states in a quantum spin Hall \ninsulator (i.e . 2D TI) . From [109] . Adapted  with permission from AAAS.  (b) Schematic of Dirac \ncone of the  TSS, illustrating the Dirac point  (DP) , Fermi level (EF) and spin momentum locking \nin momentum space . Reprinted  with permission  from  [33]. Copyright (2015 ) by the American \nPhysical Society . (b) Illustration of the electron flow and spin polarizations on the surface of TI \nfor opposite applied charge currents . Adapted with permission  from  [118] . Copyright (2014 ) by \nthe American Physical Society . \n \n(b) (c) (a)\nQuantum \nwellConductance channel with \nup-spin electrons\nConductance channel \nwith down -spin electrons\nFermi surface\nkxky Dirac pointE\nTopological surface state\nSpins33 \n \n 6.1. SOT efficiency in  TIs \nResearch works have been conducted  to demonstrate the TSS and quantify  the SOT efficiency \nin TI/FM heterostructures  [33, 34, 40, 118 -126]. In 2014, Mellnik et al.  [40] studied  the Bi2Se3 (8 \nnm)/Py  (16 nm) bilayers by ST -FMR measurements at room temperature.  Figure s 16(a) and (b) \nshow the schematic of the bilayer structure and the  representative ST -FMR signal. Using the only-\nVS analysis method , the in -plane SOT efficiency , θsh, was determined to be ~2.0–3.5 at room \ntemperature, which is about one to two orders o f magnitude larger than that  in HMs reported \npreviously.  In addition , the out -of-plane SOT efficiency , θ⊥, arising from FL has the similar order \nof amplitude as θsh. The spin configuration is consistent with that expected in the TSS of TIs [115-\n117]. However, the exact mechanisms for this large SOT efficiency observed in Bi 2Se3 was not \nclear  at that time.  \n \nFigure 16 . (a) Schematic diagram of Bi2Se3 (8 nm)/Py  (16 nm) bilayer  structure.  The yellow and \nred arrow s denote spin moment directions . (b) Measured ST -FMR signal  at room temperature with \nf = 8 GHz  and φ = 45° . A fixed microwave power of 5 dBm  is absorbed by  the device, \ncorresponding to IRF = 7.7±1.1 mA. The lines are the fits showing the  symmetric and antisymmetric \ncomponents.  Reprinted by permission from Macmillan Publishers Ltd: Nature  [40], copyright \n2014 . \nSubsequently, Wang et al.  [33] verified that the observed SOTs originate from the TSS in \nBi2Se3 by temperature -dependent ST -FMR measurements on a Bi 2Se3 (20 nm)/CoFeB (5 nm) \n(a) (b)34 \n \n heterostructure . By taking the only -VS and -VA analysis method, the authors determined both θsh \nand θ⊥ as a function of temperature from three representative ST -FMR devices. As shown in figure \n17(a), the θsh was found to increase steeply and nonlinearly, especially below 50 K, reaching the \nvalue of ~ 0.42 , which is almost 10 times larger than that at 300 K . Moreover, a difference in the \nθsh values between two ST -FMR analysis methods indicates a noticeable FL (i.e. θ⊥), which is \nplotted against temperature in figure 17 (b). As it is known that the θsh arising from the bulk spin \nHall mechanism normally shows a very weak temperature dependence [32, 55, 96] , and the θ⊥ \narising from the field -like torque or interfacial Rashba  effect usually shows a smaller value at low \ntemperature in HMs or semiconductors [75, 127, 128] . In addition, the hexagonal warping effect \nin the TSS can account for the out -of-plane spin polarizations and thus affect the  θ⊥ [129-132]. \nTherefore, the significant increase of θsh and θ⊥ in the Bi 2Se3/CoFeB  bilayer as temperature \ndecreases is attributed to the TSS in Bi 2Se3. These results suggest that the TSS with spin \nmomentum locking can play an important role in the spin current generation. This work suggests \nthat the bulk states (BS) in Bi 2Se3 and the 2DEG on top of  Bi2Se3 can lead to a contamination to \nthe SOT effects from TSS. In Section 6.3, we present the role of BS, 2DEG and TSS of Bi 2Se3 in \nthe SOT efficiency by ST -FMR measurements.  \n \nFigure 17 . Temperature dependence of (a) in -plane SOT efficienc y, θsh, and (b) out -of-plane SOT \nefficiency, θ⊥, in Bi2Se3 (20 nm)/CoFeB (5 nm) for three devices: D1, D2, and D3. The SOT \n0 50 100 150 200 250 3000.00.10.20.30.4  D 1\n D 2\n D 3By Vs Only\n D 1\n D 2\n D 3 \n \nT (K)shBy Vs/Va\n0 50 100 150 200 250 3000.00.10.20.30.4 \n \nT (K)  D 1\n D 2\n D 3(a) (b)35 \n \n efficiencies are analyzed by two different methods: by only -VS and by VS/VA ratio. Reprinted  with \npermission  from [33]. Copyright (2015 ) by the American Physical Society . \nThe Fermi level dependent SOT efficiency was recently studied in  a (Bi 1-xSbx)2Te3 (BST, 8 \nnm)/Cu (8 nm)/Py (10 nm) [120] . As shown in figure 18 (a), the Fermi level position relative to the \nDirac point was controlled by using different Sb compositions. From ST -FMR measurements on \neach device, it was observed that the interface  charge -to-spin conversion efficiency,  qICS due to \nthe TSS exhibits a large and nearly constant value for 0 < x < 0.7 (see figure 18 (b)), suggesting the \nhighly efficient charge and spin interconversion in the TSS of TI materials. However, the qICS was \nfound remarkably reduced as the Fermi level  traverses through the Dirac point, which is possibly \ndue to the inhomogeneity of kF and/or instability of the helical spin structure near the Dirac point.  \n \nFigure 18 . (a) Schematic of the energy dispersion and Fermi level position in (Bi 1-xSbx)2Te3 with \ndifferent Sb composition. (b) Interface charge -to-spin conversion efficiency, qICS as a function of \nSb composition. The inset shows the band structure and Fermi level position for each Sb \ncomposition. Bulk in sulating BST with 0.50 ≤ x ≤ 0.90 should have only surface transport. Bi 2Te3 \n(x = 0) and Bi 2Sb3 (x = 1) have both bulk and surface conduction paths. Reprinted by permission \nfrom Macmillan Publishers Ltd: Nature Physics [120] , copyright 2016 . \n6.2. Anisotropic SOT (efficiency) in Bi 2Se3 MBE films  \nBi2Se3 has a three -fold symmetry when viewed from the top (along the z-axis) shown in figures \n19(a) and (b). Therefore, the molecular beam  epitaxy  (MBE) grown Bi 2Se3 films naturally have \n(a) (b)36 \n \n triangular islands as captured by atomic force microscope  in figure 19 (c). It is found that the \norientation of the triangular islands prefers to a certain axis with twin structures due to the six -fold \nsymmetric sapphire substrate. Although the anisotropy of electron transport has been observed in \nBi2Se3 films previously [133] , the anisotropy of SOT in Bi 2Se3 has not been reported.  \nWe have performed ST -FMR to study the anisotropic SOT characteristics in high quality MBE \ngrown Bi 2Se3 films shown in figure 19 (c). Bi 2Se3 Hall bars with different angle φ, the angle \nbetween the channel current ( I) direction with respect to the x-axis denoted in figure 19 (c), were \nfabricated in the same sample. Similarly, using the same Bi 2Se3 wafer, ST -FMR devices of Bi 2Se3 \n(10 QL)/CoFeB (7 nm) devices with different angle φ were also fabricated for the ST -FMR \nmeas urements. As shown in figures 19 (d-e), both the resistivity (BiSe) and sheet carrier \nconcentration ( n2D) show angular dependent behaviors with the oscillation period of 60°. \nFurthermore, the in -plane SOT efficiency, θsh, in figure 19 (f) also shows an oscillation behavior as \nthe current I flows along different crystal axes (i.e. different angle φ). The oscillation period is also \n60°, in which θsh has a low value at φ = 0° but a high value at φ = 30°, and returns to a low value \nat φ = 60°. This is consistent with the six -fold symmetry by considering the Bi 2Se3 twin structures. \nThus, this oscillation of the SOT efficiency is attributed to the hexagonal warping in the Fermi \nsurface of the Bi 2Se3 films [129, 132, 134] . An out-of-plane spin polarization in the TSS has been \nexperimentally observed in Bi 2Se3 [130, 131]  due to the hexagonal  warping effect, which shows a \nfinite amplitude along the Γ-K direction but almost zero along the Γ-M direction in momentum \nspace, leading to an oscillation period of 60° in the out -of-plane spin polarization. If we assume \nthat the total spin polarization is unity in TSS, the in -plane spin polarization should also show the \nsame oscillation period of 60°, which is in line with our observation.  \nNote that since the width of the current channel of the Hall bars and ST -FMR devices is 20 37 \n \n µm, which is much larger  than the triangle size of ~100 –200 nm in figure 19 (c), our data only \nshow the averaged transport results. In addition, there exists nonuniform Bi 2Se3 microstructures \nand devices with different φ are located different places on a sample. Consequently, a fluctuation \nof the high (or low) values at each angle φ can be observed in figures 19 (d-f). Note that the similar \nanisotropy behavior is expected in the range of  φ = 180 – 360° due to the crystal s ymmetry.  Rather \nthan rotating the crystal direction in different devices, the angular dependent ST -FMR \nmeasurements in one device can be combined to characterize the anisotropy of SOTs [135] , which \nwill be discussed in Section 7.2. Therefore, the ST -FMR is a suitable dynamic measurement \nmethod to study the anisotropy of SOTs.  \n \nFigure 19 . (a) Crystal structure of Bi 2Se3 with three primitive lattice vectors denoted as t1;2;3. A \nquintuple layer (1 QL ≈ 1 nm) with Se1 –Bi1–Se2–Bi1’–Se1’ is indicated by the red square. \nAdapted by permission from Macmillan Publishers Ltd: Nature Physics  [111] , copyright  2009 . (b) \nTop view along the z-direction. The triangle lattice in 1 QL has three different positions, denoted \nas Se2, Bi and Se1. Adapted by permission from Macmillan Publishers Ltd: Nature Physics  [111] , \ncopyright  2009 . (c) Atomic force microscope  image of a 10 -QL Bi 2Se3 film with a roughness of \n~0.5 nm. Triangular islands are observed with a  clear terrace step of 1 QL,  indicating a high quality \nBi2Se3 film. (d -f) The Bi 2Se3 resistivity, sheet carrier concentration and in -plane SOT efficiency \n5nm\n-5 nm200 nm\nSe2\nBi\nSe1\n2850304032303420\n1.441.681.922.16\n0 30 60 90 120 150 1800.50.60.70.80.9 \n BiSe (cm) T = 300 K\n n2D (1013/cm2)\n sh\n (o)(a) (b)\n(c)(d)\n(e)\n(f)\nφI\nxy38 \n \n measured with charge currents flowing along different crystalline directions. The angle  φ is \ndenoted in (c).   \n6.3. TSS dominated SOT  \nAs mentioned earlier , the BS, 2DEG, and TSS usually coexist in  TIs such as Bi2Se3. Therefore, \nit is important to understand the role of each channel on the SOT efficiency in TIs, so that one can \nmaximize the SOT efficiency to realize highly efficient SOT driven magnetization switching. \nFigure 20  shows a summary of the reported SOT efficiencies in var ious TIs with different \nthicknesses. The SOT efficiency increases significantly as TI films become thinner even though \nSOT efficiencies are evaluated using different techniques and in different TIs. Recently, the SOT \nefficiency of MBE grown Bi 2Se3 films wi th various thicknesses in the range of  5 to 20 QL has \nbeen quantified by ST-FMR  measurements [34]. The optimum thickness range of Bi2Se3 has been  \nidentified to be 5–8 QL to maximize the SOT effect in Bi 2Se3 devices.  Figure 21 (a) shows the \nschematic of the film stack used for the ST -FMR measurements. First, it was found that the \nresistivity, BiSe rapidly increases when the Bi 2Se3 thickness ( tBiSe) is less than 10 QL as shown in \nfigure 21 (b). On the other hand, the sheet carrier concentration ( n2D) shows an opposite trend, \ndecreasing substantially below 10 QL  as shown in figure 21 (c). The carrier concentrations from \nBS ( n2D-Bulk), 2DEG ( n2DEG) and TSS ( nTSS) were further separated and it was found that in the \nregion of 5–8 QL, the value of nTSS is larger compared to n2DEG and n2D-Bulk, indicating a TSS \ndominated electrical transport in thin Bi 2Se3 films.  \nFigure 21 (d) presents θsh as a function of tBiSe at room temperature by ST -FMR measurements \non Bi2Se3 (tBiSe)/CoFeB  (7 nm)  bilayers . As tBiSe is in the range of 5–8 QL, the Bi 2Se3 films exhibit \na giant θsh of ~1–1.75 at room temperature , which further corroborates the TSS dominated \ntransport in thin  Bi2Se3 film region. In addition, the interface  SOT efficiency from TSS ( TSS) \nusing an interface charge current density JC-TSS (A cm–1) in TSS were estimated for each tBiSe as 39 \n \n shown by squares  in figure 21 (e). Furthermore, after subtracting the opposite 2DEG contribution, \nthe intrinsic TSS was calculated as 0.8 nm-1 for 7, 8 and 10 QL Bi 2Se3, as shown by circles, which \nis similar to recently reported interface SOT efficiency values in (Bi 1-xSbx)2Te3 [120] .  \nThese results suggest that the BS and 2DEG dilute the TSS contribution and weaken the SOT \nefficiency in Bi 2Se3, and thus by using moderately thin TI films (5 -8 QL) one can obtain the TSS \ndominated SOTs for device applications. In addition, the TSS dominated transport has also been \nreported recently [123]  by changing the Bi 2Se3 film thickness with a spin pumping technique, \nwhich is a reverse process (spin -charge conversion) of ST -FMR.  \n \nFigure 20 . Summary of the SOT efficiencies measured in different TIs with different thicknesses. \nThe materials with corresponding measurement te mperature, measurement method and reference \nare indicated. SOT switching represents the SOT driven magnetization switching.  \n10010110210310410510-410-310-210-1100101102103\nBi2Se3, spin pumping, RT [121]SOT efficiency\nTI thickness (nm)(Bi,Sb)2Te3, 1.9 K, SOT switching [119]\n(Bi,Sb)2Te3, <200 K, spin-polarized tunneling [125]\nBi2Se3, RT, ST-FMR [40]\nBi2Se3, RT, ST-FMR and SOT switching [34]\nBi2Se3, <50 K, ST-FMR [33]\nBi2Se3, spin pumping, RT [124]\nBi1.5Sb0.5Te1.7Se1.3, spin pumping, \nlow temp [118]40 \n \n  \nFigure 21 . (a) Schematic of the film stack for ST -FMR devices . (b) Bi2Se3 thickness dependent \nresistivity,  BiSe, in Bi 2Se3 at room temperature.  (c) Estimation of s heet carrier concentration of \nTSS, 2DEG and bulk channels. (d) θsh as a function of Bi 2Se3 thickness at room temperature. Each \nθsh represents the averaged value from three devices. Region I, II and III denoted by  different \ncolours represent the charge -to-spin conversion dominated by different mechanisms . The inset \nshows the schematic of the band structure for each region. (e) Interface SOT efficiency, TSS \n(squares), as a function of tBiSe at room temperature. Amended interface SOT efficiencies from \nTSS after excluding the opposite 2DEG contribution for 7, 8 and 10 QL Bi 2Se3 with circles. \nAdapted from [34]. \nBy taking advantage of this giant θsh, SOT driven  magnetization  switching in the Bi 2Se3 (8 \nQL)/Py (6 nm) heterostructures  was successfully  achieved at room temperature without any \nexternal magnetic field [34]. This work makes a substantial  improvement in  the working \ntemperature  of TI based magnetization switching scheme from 1.9 K  [119]  to 300 K, \ndemonstrating that TI can be an excellent spin generator at room temperature. T he critical \nswitching current density JC in Bi 2Se3 required for the  magnetization  switching is extremely low \n(~6×105 A/cm2), which is almost two orders of magnitude smaller than that in  HMs, such as Pt and \n5 10 15 200.00.20.40.60.81.01.21.4 TSS \n AmendedTSS TSS (nm1)\ntBiSe (QL)\n  \n5 10 15 2002468 n2DEG\n n2D-Bulk n2D data\n 2nTSSn2D (1013 cm2)\n  \n tBiSe (QL) \n5 10 15 200.00.51.01.52.0\nTSSEF\n2DEGBS\nIII II\ntBiSe (QL) sh\n   \nI\n5 10 15 201x1032x1033x1034x103\ntBiSe (QL) \n BiSe (cm)\n (a)(b) (c)\n(d) (e)\nCapping layer\nCoFeB (7 nm) \nBi2Se3 (5–20 QL)\nAl2O3(0001) \nSubstrate41 \n \n Ta [20-22]. The much lower JC for magnetization switching using TIs is promising to address the \noutstanding scalability and power consumption issue in modern magnetic devices. Further more , \nthe magnetization switching scheme  demonstrated in Ref. [34] does not require an assistive  \nmagnetic field, which makes the TI/FM material  systems easy to integrate into the well -established \nindustrial technology for magnetic devices . In addition , the room temperature SOT driven \nmagnetization switching has been also observed recently in the Bi2Se3/ferrimagnetic  CoTb  \nheterostructures  [136]  and in the sputtered BixSe(1-x)/Ta/CoFeB/Gd/CoFeB  heterostructures [137] .  \n7. SOT beyond HMs and TIs  \nBeyond HMs and TIs, other novel material systems with strong SOC are emerging, which also \nhave potential in the future spintronic applications. The following sections discuss some \nrepresentative examples, in which the ST -FMR technique was used to evaluate the SOTs (or SOT \nefficiency), to get insight into the underlying physics of SOTs in these emerging systems.  \n7.1. Interface between two nonmagnetic materials  \nThe Rashba SOC , which was first discussed in the 2DEG systems with spin degeneration \nlifting [138] , can emerge at the interfaces in a variety of material systems  nowadays  [139] . The \nRashba SOC at an interface is explained as follows. For an interface with inversion symmetry \nbreaking in the direction perpendicular to the interface, an electric field E is produced \nperpendicular to the interface. An electron moving within this  interface with the electric field  will \nexperience an effective magnetic field  H due to the relativistic corrections , which  causes a \nmomentum dependent spin splitting  in the bands , analog ous to the Zeeman splitting. In the scheme \nof Rashba  SOC, the interaction between the spin polarization \nˆ  and electron momentum k can \nbe expressed by the Hamiltonian  \nRRˆˆ =H k z（） , where \nR  is the Rashba coefficient, \nˆz  is the \nunit vector perpendicula r to the interface. Figure s 22(a) and (b)  schematically show the dispersion 42 \n \n curves with spin splitting due to the Rashba SOC at the interface and the corresponding Fermi \ncontours  with spin polarization configuration . It is reveal ed that the spins polarizations are  \ntransverse to the electron moving direction.  Since the spins have different chiralities in two \ncontours, they  compensate each  other  and give rise to the net spin polarization s, which can be \nmeasured in the  experiments . \nThe Rashba SOC at the  interface between the HM and FM layer, such as Pt/Co, has been \nstudied previously [20, 72] . Recently, researchers have found that when one nonmagnetic material, \nsuch as Bi, Pb or Sb, contacts with other nonmagnetic material, such a s Ag, their interface can also \nexhibit strong interface SOC [140-147]. In 2015, the charge -to-spin conversion induced by the \nRashba -Edelstein effect was directly observed at the interface of Bi/Ag [146] . Using a spin -\npolarized positron beam, they found an opposi te surface spin polarization between Bi/Ag/Al 2O3 \nand Ag/Bi/Al 2O3 samples. Subsequently, Jungfleisch et al.  [147]  reported the Rashba -Edelstein -\ninduced -spin driven ST -FMR in Bi/Ag/Py heterostructures. They evaluated the SOT efficiency of \nthe Bi/Ag interface to be ~0.18 using the VS/VA ratio method.  \nAnother recent  report shows a very large SOT efficiency at the interface between 10 QL Bi 2Se3 \nTI materials and nonmagnetic material Ag  thin layer at room temperature by ST -FMR \nmeasurements  [148] . As shown in figure 2 2(c), f rom first-principle  calculations, it was found that \nthere is a  pair of  large Rashba splitting bands  emerging at the interface between Bi 2Se3 and Ag. \nMoreover, the Rashba  bands are located outside the TSS linear band s of the Bi2Se3 layer, and t he \nRashba  bands  have the same net spin polarization direction as the TSS of  Bi2Se3. As shown in \nfigure 2 2(d), due to the large interface Rashba SOC at the newly formed Bi 2Se3/Ag interface, the \nmeasured SOT efficiency  shows a significant enhancement as the Ag insertion layer thickness \nincreases to ∼2 nm and rea ches a value of 0.5 for 5 nm Ag. The SOT efficiency in Bi2Se3/Ag (5 43 \n \n nm)/CoFeB is ~ 3 times higher than that from sole TSS in  Bi2Se3/CoFeB at room temperature.  The \nextracted \nR  at the Bi 2Se3/Ag interface has a significant large value about 2 .83−3.83 eV Å and \nthis value is similar to that of Bi/Ag  previous reported [141] . Furthermore , Rashba -induced \nmagnetization switching in Bi 2Se3/Ag/Py with a low current density of 5.8 ×  105 A/cm2 has been \ndemonstrated. As indicated  in Section 6.3, the SOT efficiency in Bi 2Se3/Ag can be further \nenhanced by decreasing the Bi 2Se3 thickness to eliminate the bulk contamination and energy \nconsumption will be further decrease for SOT driven magnetization switching. These reports \nindicate that the nonmagnetic bilayer  interface can be designed to provide a large Rashba spin \nsplitting an d thus giant SOT efficiency, which has the potential  to be used  as efficient spin current \nsources for future spintronic devices.   \n \nFigure 2 2. (a) Typical spin-split dispersion curves of a Rashba 2DEG . (b) Fermi contours with \nspin polarization denoted by the blue and red arrows. Reprinted by permission from Macmillan \nPublishers Ltd: Nature Communications  [143] , copyright 2013 . (c) Spin-orbit -torque efficiency \nobtained for both Bi 2Se3 (10 QL) /Ag (tAg)/CoFeB (7 nm) and Ag  (tAg)/CoFeB (7 nm) ST -FMR \n(a) (b)\n(c) (d)θsh\ntAg44 \n \n devices. (d)  The band structure of Bi 2Se3 (6QL) /Ag (0.95 nm) from first principle calculations. \nReprinted  with permission  from [148] . Copyright (2018 ) by the American Physical Society . \n7.2. Transition -metal dichalcogenides (TMDs)  \nIn the recent years, 2D t ransition -metal dichalcogenide (TMDs) materials  are attracting \ninterests due to the novel physics and potential applications , which were  reviewed in Ref.  [149] . \nSince the TMDs display 2D characteristics, the thickness of 2D -TMDs can be redu ced as thin as a \nmonolayer. Therefore, TMDs provide an interesting platform to observe exotic interface related \nphenomena. Since 2016, the 2D -TMDs have been studied as spin current sources for the SOT \ndevices [135, 150 -153]. Spin current induced ferromagnetic resonance has been observed in \nmonolayer MoS 2/Py structures [151] , where the Py layer was deposited by either magnetron \nsputtering or e -beam evaporation.  It was observed that both DL and FL in the monolayer MoS 2/Py \nsystem were present, with the ratio of  |τFL/τDL| = 0.19. The observed  SOTs were attributed to the \ninterface between the MoS 2 and Py layer.  The ratio of |τFL/τDL| in MoS 2/Py is much smaller than \nthat recently reported in the Bi 2Se3/Py (or CoFeB) systems [33, 40] , whose  value is in the range of \n1.4–2. This implies that the damping -like torque τDL is much larger than τFL in MoS 2/Py system . \nHowever, the real SOT efficiency  of MoS 2/Py bilayer  was not evaluated in Ref. [151]  and the real \nSOT efficiency should be estimated by the me thod only -VS (or τDL) described in Section 4.2  for \nproper comparison with other material systems .  \nSemimetal WTe 2 is also a layered  TMD, which has strong SOC [154, 155]  and has been \nreported to possess exotic topological properties [156] . Compared to other TMDs, such as MoS 2, \nan interesting property of single crystal WTe 2 is the crystal symmetry. As shown in figure 23 (a), \nthe mirror symmetry exists only relative to the bc plane in WTe 2, and there is no mirror symmetry \nin the ac plane. Therefore, there is no 180°  rotation about the c-axis (perpendicular to the sample \nplane). Consequently, due to the lack of two -fold rotational symmetry about the c-axis, there is a 45 \n \n possible existence of the out -of-plane damping -like SOT ( τB) in WTe 2. Recently, MacNeill et al.  \n[135]  have verified the presence of τB in WTe 2/Py bilayer systems by ST -FMR measurements at \nroom temperature shown in figure s 23(b) and (c). It was found that when the charge current flows \nalong the b-axis with mirror symmetry in a WTe 2 (5.5 nm)/Py (6 nm) devic e, both the symmetric \n(VS) and antisymmetric ( VA) ST-FMR components at different in -plane magnetic field angle φ \nfollow the behaviour of cos φ× sin(2 φ), which is similar to the case of the traditional HM/FM \nbilayers. However, as the charge current flows alon g the a-axis where mirror symmetry is broken, \nVA at different φ exhibits a novel behaviour of cos φ ×  sin(2 φ) + sin(2 φ), indicating the presence of \nτB in WTe 2/Py as shown in figure 23 (d). Figure 23 (e) shows the SOTs in WTe 2 with different \nthicknesses ranging from monolayer to ~16 nm [153] . The magnitude of τB shows a very  weak \nWTe 2 thickness dependence, suggesting that it is of the interface origin.  \nMost of the studied interface systems so far only have broken inversion symmetry along their \nvertical structure, which leads to an in -plane damping -like torque or out -of-plane field -like torque \ndue to Rashba -Edelstein effect at the interface. On the contra ry, the results in Ref. [135]   \ndemonstrate the in -plane rotational symmetry breaking induced out -of-plane damping -like torque \nin WTe 2, which is a novel interface related SOT phenomenon and provides a platform for field -\nfree SOT driven magnetization switching of FM with perpendicular m agnetic anisotropy (PMA).  \n 46 \n \n  \nFigure 23 . (a) Crystal structure near the surface of WTe 2. (b) Schematic of the WTe 2/Py sample. \n(c) The device and the circuit used for ST -FMR measurements. (d) Symmetric and antisymmetric \nST-FMR components ( VS and VA) for a WTe 2 (5.5 nm)/Py (6 nm) device as a function of in -plane \nmagnetic -field angle φ denoted in (b). The charge current is applied parallel to the a-axis. The \nmicrowave frequency is 9 GHz and the applied microwave power is 5 dBm. Adapted by permission \nfrom Macmillan Publishers Ltd: Nature Physics  [135] , copyright 2016 . (e) Out -of-plane damping -\nlike torque ( τB, blue circles) and out -of-plane field -like torque ( τA, red circles) per unit interface \ncharge current density ( I0/w) as a function of WTe 2 thickness , where I0 and w are charge current \nand channel width, respectively. Reprinted  with permission  from [153] . Copyright (2017 ) by the \nAmerican Physical Society . \n7.3. Antiferromagnetic (AFM) materials  \nAntiferromagnetic materials have zero net magnetization due to the antiparallel alignment of \nmagnetic moments on the adjacent individual atoms. Therefore, the antiferromagnets are difficult \nto con trol by external magnetic field s. However, an injected charge current induced internal field \ncan perturb the spin structures and thus magnetic ordering in  AFM , as recently reported  in different  \nsingle antiferromagnetic (AFM) material s [157-159]. On the other hand, the devices with \nAFM/FM heterostructures have also been utilized to realize the external -magnetic -field-free SOT \n(a) (b) (c)\n(d)\n(e)47 \n \n driven m agnetization switching due to the presence of exchange bias [160-163], which breaks the \nmagnetization up -down equivalence in the systems with PMA. Furthermore, a single AFM \nmaterial itself can also work as a spin current source [48, 160, 164, 165] , just like a HM layer. It \nhas been found that the SOT efficiency of the  single AFMs, such as  PtMn , IrMn , is in the same \norder of Pt [160, 166, 167] .  \nInterestingly, it was also found  that there is an anisotropy of the ST-FMR results from epitaxial \nPtMn [48] and IrMn 3 [165]  AFM films. For PtMn epitaxial films [48], the modulation of ST -FMR \nlinewidth (MOD) is more significant along the a-axis than c-axis PtMn (10 nm)/Cu (1 nm)/Py (5 \nnm) samples as shown in the figure s 24(a) and (b), indicating the large SOT efficiency in a-axis \nPtMn films. A further only -VS ST-FMR a nalysis method yields a SOT efficiency of ~0 .048 for c-\naxis PtMn and ~0 .089 for a-axis PtMn. For the case of epitaxial IrMn 3/Py bilayer devices [165] , a \nsimilar anisotropic behaviour was found from ST -FMR measurements in the SOT efficiency with \nrespect to the IrMn 3 crystal orientations. As shown in figure 24 (c), the SOT efficiency is much \nlarger (~0.2) in (001) -oriented single -crystalline IrMn 3 than that i n either (111) -oriented or \npolycrystalline -oriented films. Moreover, after perpendicular field annealing, an enhanced SOT \nefficiency up to ~0.35 in (001) -oriented IrMn 3 thin films was demonstrated. Both works discussed \nthat the observed anisotropic SOT eff iciency in PtMn or IrMn 3 epitaxial films originates from the \nintrinsic SHE, which was further corroborated with the first -principles calculations.  48 \n \n  \nFigure 24 . Measured resonance linewidth versus dc -bias current by ST -FMR measurements at 5 \nGHz for (a) a- and (b) c-axis PtMn (10 nm)/Cu (1 nm)/Py (5 nm). Reprinted  with permission  from \n[48]. Copyright (2015 ) by the American Physical Society . (c) SOT efficiency as a function of \nIrMn 3 thickness for different crystal orientation films, i.e. (001) -oriented (olive circles), (111) -\noriented (blue circles), and polycrystalline -oriented (cyan circles). Reprinted from [165] . 2016 © \nThe Authors, some rights reserved;  exclusive licensee American Association for  the Advancement \nof Science. Distributed  under a Creative Commons Attribution NonCommercial License 4.0 (CC \nBY-NC).  \n7.4. 2DEG  in oxide materials  \nRecent advances in film growth and synthesis techniques have enabled the growth of high \nquality oxide heterostructures with very smooth interfaces. This allows for engineering novel \nelectronic properties at the oxide interfaces  [168] . One of the central oxide material systems is the \ninterface between two wide -band -gap insulators, SrTiO 3 (STO) and LaAlO 3 (LAO) as a 2DEG is \nformed at the interface and possesses exotic properties including superconductivity  [169, 170]  and \nmagnetism  [171-173]. Further, due to the broken interfacial inversion symmetry, the 2DEG \nconfined in the vicinity of polar (LAO)/non -polar (STO) interface experiences a strong electric \nfield directed perpendicular to the conduction plane  [174] . Consequently, the LAO/STO interface \npossesses a strong Rashba SOC  that lea ds to a strong coupling between the orbital and spin degrees \nof freedom . Due to the dielectric properties of STO, the strength of the Rashba SOC is tunable by \nPtMn /Cu/ Py IrMn3/Py\n(a) (b) (c)θsh49 \n \n applying an external gate voltage [175] .   \nThe presence of strong Rashba SOC in the d-bands of the 2DEG at the STO/LAO interface \nhas been reported in various earlier studies  [175-177]. Further, the Rashba SOC allows for \ngeneration of spin accumulation in the LAO/STO interface when a charge current flows in the \n2DEG [138, 178] . Subsequently, the existence of current induced SOTs at the STO/LAO interface  \nhas been verified experimentally by angular dependent magnetoresistance measurements [179] . \nRecently, Wang  et al. [180]  have experimentally shown a giant  room temperature charge -to-spin \nconversion efficiency (i.e. θsh) in the STO/LAO/CoFeB  structure  by the ST -FMR technique shown \nin figure 25 (a). The θsh value was estimated to be ~6.3 at room temperature denoted by squares in \nfigure 25 (b), which is almost two orders of magnitude larger than that in HMs, such as Pt and Ta \n[21, 25] . However, the θsh decreases rapidly as the temperature decreases to ~150 K. Finally, it \nbecomes negligible at ~ 50 K. F rom the temperature dependent θsh, it was suggested that inelastic \ntunneling via localized states, such as oxygen vacancies  in the LAO band gap, accounts for the \nspin transmission through the LAO layer, as schematically  shown in the inset of figure 25 (b).  \nIn addition to the ST -FMR measurements, a spin-to-charge  conversion efficiency at the \nSTO/LAO interface has been also reported by the spin pumping technique  [181-183]. Additionally, \na long spin diffusion length over 300 nm in the STO/LAO 2DEG channel has been reported  [184, \n185]. Therefore, the oxide materials, such as STO/ LAO heterostructures, can enable potential \napplications  in the oxide based spintronic  devices .  50 \n \n  \nFigure 25 . (a) Illustration of a STO/LAO/CoFeB ST -FMR device  with the SOT induced \nmagnetization dynamics . (b) SOT efficiency, ǀǀ, as a function of temp erature  (squares)  and the fit \n(red line). The inset is the schematic of the spin -polarized electron inelastic tunneling process via \nthe localized states ( LS), such as oxygen vacancies, V O. The fitting lines with the black, blue and \ngreen colours denote the contributions of the respective direct elastic tunneling & 1 - localized state \n(LS), 2-LS and 3 -LS chains.  Reprint ed with  permission  from [180]. Copyright (2017 ) American \nChemical Society . \n7.5. Nonmagnetic oxidized materials  \nIn conventional HMs, the reported largest SOT efficiency is ~0.3 observed in  β-phase W [35]. \nHowever, the  α-phase W exhibits a much smaller SOT efficiency. Therefore, the SOTs can be \ncontrolled via the HM microstructure engineering . Recently, it has been reported that the SOT \nefficiency can be significantly enhanced in a W/Py bilayer by oxygen incorporation in the W layer \n[186] . It was found that as the oxygen concentration increases, the grain size decreases, bu t the \nresistivity increases, in which β-phase W is well stabilized. Consequently, an enhanced SOT \nefficiency up to ~0.5 was achieved with an oxygen concentration of 12.1% compared to ~0.14 for \nzero oxygen incorporation. For higher oxygen incorporations up to 44%, the bulk properties further \nchanged, however the SOT efficiency showed a very weak oxygen concentration dependence, \nsuggesting that the mechanism responsible  for the observed highly SOT efficiency may originate  \nfrom the W (oxygen)/Py interface.  \nx\nyz  M\nǁIRFH\n\nEC\nEVEF2DEGVO\nPhononSTO CFB LAO0 100 200 3000369\nT (K) Data\n Fit\n    Direct and 1-LS2-LS\n   sh\n3-LS(a) (b)51 \n \n Recently, An et al.  [187]  found that by naturally oxidizing a light metal Cu with very weak \nSOC, the SOT efficiency in a Cu/Py bilayer was enhanced to be ~0.08, which is similar as Pt and \nmore than two orders of magnitude larger than that in pure Cu. These results indicate that \nengineering materials with oxidization opens an intriguing path to enhance the SOT and realize \nefficient spintronic devices  [188] . \n8. Summary and Perspectives  \nIn this review , we first describe d the ST -FMR technique, including two ST -FMR setups and \nthree ST -FMR analysis methods.  As ST -FMR is a high frequency measurement in the GHz \nfrequency range, several aspects should be carefully considered during the ST -FMR measurements \nand data analysis, such as the symmetric CPW design, impedance matching, rf power absorption \nand device cur rent correction as well as possible contamination considerations (field -like torque \nfrom the interface effect and spin pumping). Then the latest SOT research progresses using ST -\nFMR are presented in diverse materials including normal HMs and alloys, TIs, 2 D materials, \ninterfaces with strong Rashba SOC, AFM materials, 2DEG in oxide materials as well as \nnonmagnetic oxidized materials. We find that the ST-FMR measurement is a powerful technique \nto determine the SOT efficiency including the recently observed ou t-of-plane damping -like SOT \nefficiency in a WTe 2 Weyl semimetal. We hope this review can further promote the ST -FMR \ntechnique  in fundamental SOT researches as well as exploiting  novel materials for future spin -\nbased device applications.  \nIn figure 26  we have summarized the representative SOT efficiencies in various materials as \nwell as the power consumption for switching an unit magnetization of a FM layer which is \nproportional to 1/( σ×(θsh)2) [136] . Here we assume tha t the switching time is same in  all the \nmaterials  for simplicity . It shows that the TI materials generally show a larger SOT efficiency as 52 \n \n well as low power due to the  efficient charge -spin conversion in the spin -momentum -locked \nsurface states compared to HMs. Some other materials, apart from HMs and TIs, also exhibit a \nhigh performance in terms of both SOT efficiency and power consumption  and thus  are worth for \nfurther  studies.  \n \nFigure 26 . Summary of the SOT efficiencies (left y-axis) and normalized power consumption of \na variety of materials (right y-axis). The x-axis denotes the materials with their corresponding \nmeasurement temperature and reference. RT means room t emperature. The Bi xSe(1-x) represents \nthe film grown by the magnetron sputtering , the STO/LAO represents SrTiO 3/LaAlO 3 bilayer, and \nCu (O) and W(O) represent the metal films with oxygen incorporation. The star represents an \nextremely small normalized power  consumption value ( ≈ 5.5× 10-8) for (Bi,Sb) 2Te3 at a \ntemperature of 1.9 K. The effective conductivity  σ (~0.17 µ Ω–1· cm–1) of Bi 4 (nm)/Ag (8 nm) \nbilayer, used for the power consumption estimation, is calculated using a parallel circuit model \nwith the given  conductivity values of ~2.1× 10–4 µΩ–1· cm–1 (for Bi) and ~0.25 µ Ω–1· cm–1 (for Ag). \nThe σ (~0.0125 µ Ω–1· cm–1) of Cu (O) layer, used for the power consumption estimation, is \n53 \n \n extracted from the figure 1(b) in Ref. [187]  by using the given value of the pure Cu conductivity \n(~0.025 µ Ω–1· cm–1). \nFigure 27 shows the correlation of SOT e fficiency and power consumption for different \nmaterials, obtained by reorg anising and grouping the data in figure 26 based on their corresponding \nmaterial system. O verall , we find that  the TI materials  (wrapped by the red dashed curve)  shows a \nbetter  performance in terms of the SOT efficiency and the power con sumption  even thoug h the \nvalues are scattered  due to the influence of conducting bulk . The HMs  and their alloys  (wrapped  \nby the dashed blue curve ) possess a similar orders of SOT efficiency and comparable power \nconsumptions as the AFM systems but are inferior to the TI systems . Finally, the Weyl semimetal  \nWTe 2 seems to show a higher power consumption based on  the single  report in Ref [135] . Since , \nWeyl semimetals are a very interesting topological materials, more research works are expected \nin the near future and one can obtain a more conclusive view about the performance of Weyl \nsemimetal s in the power consum ption and  SOT driven switching  efficiency .    \n \nMMM\nM\nMM\nMTI\nTI TI\nTI\nTITI\nTI\nTIIF IFWS\n0.01 0.1 1 10 100 10001E-81E-71E-61E-51E-40.0010.010.1110\nAFM\nAFMPower consumption (a.u.)\n|SOT efficiency | Metals and alloys\n Topological insulators\n Rashba interface\n Weyl semimetal\n Antiferromagnet\nAFM54 \n \n Figure 2 7. Summary of the normalized power consumption as a function of SOT efficiencies for \na variety of materials. ‘Metals and alloys ’ represent Pt, Ta, W , and CuPt  alloy  and nonmagnetic \noxidized materials , ‘Topological insulators ’ represent Bi 2Se3 and (Bi,Sb) 2Te3, ‘Rashba interface ’ \nincludes STO/LAO interface and Bi/Ag interface , ‘Weyl semimetal ’ means WTe 2, and \n‘Antiferromagnet ’ includ es PtMn and IrMn 3. \nMoving towards to the real SOT application s, many reports  recently have studied  the SOT \ndevices i n the nanosecond (ns ) or even picosecond (ps)  regime  [104, 108, 189 -192]. A very high \nefficient SOT driven magnetization switching has been realized  recently [191] , where t he \nswitching energy is only about 60 fJ for each bit writing with the current ~120 µ A and time of 1.2 \nns. The energy con sumption for each bit  writing  is much smaller than the values  known in STT  \ndriven magnetization switching, which is between ~150 fJ  to ~4 pJ  [193-195]. Therefore, the  SOT \ndevices are  promising and potential candidates for the future high efficient spintronic devices. \nThus, t he exploration of novel materials with even higher SOT efficiency w ill be of great \nimportance  for the future spin -based device applications  and, for thi s purpose, the ST -FMR can be \nan easy and a powerful technique.   \nSo far, we have seen that the  origins of the SOT can be  from the  bulk, surface and /or interface \nin a variety of materials. Based upon the origins of the SOT, the material systems will show \ndifferent thickness dependent behaviour s. Figure 2 8 shows the schematic of the SOT efficiency as \na function of thickness in representative TI material Bi 2Se3, HM material Pt and Weyl semimetal \nWTe 2. For HMs (blue curve), the dominant mechanism for the SOT is SHE, which is a bulk \nproperty. Therefore, the SOT efficiency in HMs usually  increases and then saturates  as HMs \nthickness increases. The thickness corresponding to the saturation point of SOT efficiency is \ncorrelated to the spin diffusion length in HMs.  For the TIs (red curve), the TSS serve as  the main \nmechanism for the SOT  and should have almost a constant SOT efficiency against the TI thickness \nas reported in Ref. [34] (also see figure 2 1(e)). However, as mentioned in Section 6.3, the 55 \n \n contamination from the conducti ng bulk might be  unavoidable . Consequently, the SOT efficiency \nshows much larger value s as the TI thickness becomes thinner as TSS becomes domi nant. As \nreported in WTe 2 [153] , the out-of-plane damping -like SOT efficiency , due to a broken lateral \nmirror symmetry , remains almost  constant with respect to  the WTe 2 thickness, which is claimed \nto arise from  interface mechanism s. The weak thickness dependence is illustrated by the green \ncurve  in figure 28 . Therefore, the thickness dependent SOT efficiency (or SOT) measurements by \nST-FMR or even other reliable techniques are very useful to get insight into the underlying p hysics \nof an emerging materials in the future.   \n0 5 10 15 20 Pt\n WTe2T = 300 KSOT efficiency  (a.u.)\n \nThickness (nm) Bi2Se3\n \nFigure 2 8. Schematic of the SOT efficiency as a function of thickness in representative materials \nBi2Se3, Pt and WTe 2, replotted by using the data from figure  21(d) and (e) , 8(a) and 23 (e) in this \nreview article , respectively.    \nThe central physical effect used in the ST -FMR measurement  can be further explored besides \nthe  GMR, tunneling  magnetoresistance ( TMR) or AMR rectification [196]  in the present ST-FMR \ndevices . The other magnetoresistance effects,  such as the SMR, or Hall effects such as the \nanomalous Hall effect (AHE) and planar Hall effect ( PHE), with both in -plane and out -of-plane 56 \n \n measurement schemes , might  also be developed in the near future to extend the ST -FMR technique \nto different material systems , especially for the materials with perpendicular magnetic anisotropy . \nMeanwhile, the ST-FMR an alysis theory  should be  also modified accordingly. Subsequently, the \nSOT efficiency from ST -FMR measurements can provide a much better  comparison with that from \nother measurements technique s. Good examples for this are the recent reports where the ST -FMR \nrectification signals were obtained based on the SMR effect in the Pt/yttrium iron garnet (YIG) \nmagnetic insulator bilayer [197-199].  \nIn addition, ST -FMR measurements on nontrivial topological materials, such as Weyl \nsemimetals (WTe 2 [154]  or MoTe 2 [200] ), can be an future interesting topic. It would be significant  \ndiscovery  if one can observe the Fermi -arc surface states and/or Weyl node related topological \nphenomena using the ST -FMR technique. Apart from the above materials, t he AFM materials are \ninteresting to be further explored using ST -FMR measurements, as  the AFMs are predicted to have  \na long spin coherence length due to the staggered spin order on an atomic scale and thus possess \npossible bulk -torque -like characteristics [201-203].  \nBesides the applications in the SOT studies, it has been demonstrated recently that TMR based \nST-FMR devices have a very high efficiency in converting an rf frequency signal to dc signal, even \nbetter than the conventional Schottky diode [15, 204 -206]. This high rectification efficiency of \nTMR based ST -FMR devices can be utilized for wireless energy harvesting that is becoming more \nrelevant to the technologies such as internet of things (IoT).  \n \nAcknowledgments  \nThis work was partially supported by the National Research Foundation (NRF), Prime Minister ’s \nOffice, Singapore, under its Competitive Research Programme (CRP Award No. NRFCRP12 -\n2013 -01), NRF Industry -IHL Partnership (IIP) grant (R-263-000-C26-281), and A*STAR ’s Pharos \nProg ramme on Topological Insulators . \n 57 \n \n References:  \n[1] Brataas A, Kent A D and Ohno H 2012 Current -induced torques in magnetic materials Nat. Mater.  11 \n372 \n[2] Jungwirth T, Wunderlich J and Olejník K 2012 Spin Hall effect devices Nat. Mater.  11 382 \n[3] Sander D, Valenzuela S, Makarov D, Marrows C, Fullerton E, Fischer P , McCord J, Vavassori P , Mangin \nS and Pirro P 2017 The 2017 Magnetism Roadmap J. Phys. D: Appl.  Phys.  50 363001  \n[4] Yuasa S and Djayaprawira D 2007 Giant tunnel magnetoresistance in magnetic tunnel junctions with \na crystalline MgO (0 0 1) barrier J. Phys. D: Appl. Phys.  40 R337  \n[5] Ikeda S, Miura K, Yamamoto H, Mizunuma K, Gan H, Endo M, Kanai S, Ha yakawa J, Matsukura F and \nOhno H 2010 A perpendicular -anisotropy CoFeB –MgO magnetic tunnel junction Nat. Mater.  9 721-\n4 \n[6] Thomas L, Jan G, Zhu J, Liu H, Lee Y -J, Le S, Tong R -Y , Pi K, Wang Y -J and Shen D 2014 Perpendicular \nspin transfer torque magnetic r andom access memories with high spin torque efficiency and thermal \nstability for embedded applications J. Appl. Phys.  115 172615  \n[7] Han X, Wen Z, Wang Y , Liu H, Wei H and Liu D 2011 Nanoelliptic ring -shaped magnetic tunnel \njunction and its application in MRAM design with spin -polarized current switching IEEE Trans. Magn.  \n47 2957 -61 \n[8] Griffiths J H E 1946 Anomalous High -frequency Resistance of Ferromagnetic Metals Nature  158 670 \n[9] Kittel C 1948 On the theory of ferromagnetic resonance absorption Phys. R ev. 73 155 \n[10] Slonczewski J C 1996 Current -driven excitation of magnetic multilayers J. Magn. Magn. Mater.  159 \nL1-L7 \n[11] Berger L 1996 Emission of spin waves by a magnetic multilayer traversed by a current Phys. Rev. B  \n54 9353  \n[12] Katine J, Albert F, Buhrman R, Myers E and Ralph D 2000 Current -driven magnetization reversal and \nspin-wave excitations in Co/Cu/Co pillars Phys. Rev. Lett.  84 3149  \n[13] Huai Y , Albert F, Nguyen P , Pakala M and Valet T 2004 Observation of spin -transfer swi tching in deep \nsubmicron -sized and low -resistance magnetic tunnel junctions Appl. Phys. Lett.  84 3118 -20 \n[14] Mangin S, Ravelosona D, Katine J, Carey M, Terris B and Fullerton E E 2006 Current -induced \nmagnetization reversal in nanopillars with perpendicula r anisotropy Nature Materials  5 210-5 \n[15] Tulapurkar A, Suzuki Y , Fukushima A, Kubota H, Maehara H, Tsunekawa K, Djayaprawira D, Watanabe \nN and Yuasa S 2005 Spin -torque diode effect in magnetic tunnel junctions Nature  438 339-42 \n[16] Sankey J, Braganca P , Garcia A, Krivorotov I, Buhrman R and Ralph D 2006 Spin -transfer -driven \nferromagnetic resonance of individual nanomagnets Phys. Rev. Lett.  96 227601  \n[17] Fuchs G, Sankey J, Pribiag V, Qian L, Braganca P , Garcia A, Ryan E, Li Z -P , Ozat ay O and Ralph D 2007 \nSpin -torque ferromagnetic resonance measurements of damping in nanomagnets Appl. Phys. Lett.  \n91 062507  \n[18] Sankey J C, Cui Y -T, Sun J Z, Slonczewski J C, Buhrman R A and Ralph D C 2008 Measurement of the \nspin-transfer -torque vector i n magnetic tunnel junctions Nat. Phys.  4 67-71 \n[19] Kato Y , Myers R, Gossard A and Awschalom D 2004 Observation of the spin Hall effect in \nsemiconductors science  306 1910 -3 \n[20] Miron I M, Garello K, Gaudin G, Zermatten P -J, Costache M V, Auffret S, Bandie ra S, Rodmacq B, \nSchuhl A and Gambardella P 2011 Perpendicular switching of a single ferromagnetic layer induced \nby in -plane current injection Nature  476 189-93 \n[21] Liu L, Pai C -F, Li Y , Tseng H W , Ralph D C and Buhrman R A 2012 Spin -Torque Switching with  the Giant \nSpin Hall Effect of Tantalum Science  336 555 \n[22] Liu L, Lee O J, Gudmundsen T J, Ralph D C and Buhrman R A 2012 Current -Induced Switching of \nPerpendicularly Magnetized Magnetic Layers Using Spin Torque from the Spin Hall Effect Phys. Rev. 58 \n \n Lett.  109 096602  \n[23] Demidov V E, Urazhdin S, Ulrichs H, Tiberkevich V, Slavin A, Baither D, Schmitz G and Demokritov S \nO 2012 Magnetic nano -oscillator driven by pure spin current Nat. Mater.  11 1028 -31 \n[24] Seo S -M, Kim K -W, Ryu J, Lee H -W and Lee K -J 2012 Cu rrent -induced motion of a transverse magnetic \ndomain wall in the presence of spin Hall effect Appl. Phys. Lett.  101 022405  \n[25] Liu L, Moriyama T, Ralph D C and Buhrman R A 2011 Spin -Torque Ferromagnetic Resonance Induced \nby the Spin Hall Effect Phys. Rev.  Lett.  106 036601  \n[26] Landau L and Lifshitz E 1935 On the Theory of the disperison of magnetic permeability in \nferromagnetic bodies Phys. Z. Sowjetunion  169 14 \n[27] Zhang S, Levy P and Fert A 2002 Mechanisms of spin -polarized current -driven magnetization \nswitching Physical review letters  88 236601  \n[28] Gui Y , Mecking N, Zhou X, Williams G and Hu C -M 2007 Realization of a room -temperature spin \ndynamo: The spin rectification effect Phys. Rev. Lett.  98 107602  \n[29] Wirthmann A, Fan X, Gui Y , Martens K, William s G, Dietrich J, Bridges G and Hu C -M 2010 Direct \nphase probing and mapping via spintronic michelson interferometry Phys. Rev. Lett.  105 017202  \n[30] Harder M, Cao Z, Gui Y , Fan X and Hu C -M 2011 Analysis of the line shape of electrically detected \nferromagn etic resonance Phys. Rev. B  84 054423  \n[31] Landau L D and Lifshitz E M Mechanics, 2nd ed. (Pergamon, Oxford, 1969)  \n[32] Wang Y , Deorani P , Qiu X, Kwon J H and Yang H 2014 Determination of intrinsic spin Hall angle in Pt \nAppl. Phys. Lett.  105 152412  \n[33] Wang Y , Deorani P , Banerjee K, Koirala N, Brahlek M, Oh S and Yang H 2015 Topological Surface States \nOriginated Spin -Orbit Torques in Bi 2Se3 Phys. Rev. Lett.  114 257202  \n[34] Wang Y , Zhu D, Wu Y , Yang Y , Yu J, Ramaswamy R, Mishra R, Shi S, Elyasi M, Teo K -L, Wu Y and Yang \nH 2017 Room temperature magnetization switching in topological insulator -ferromagnet \nheterostructures by spin -orbit torques Nat. Commun.  8 1364  \n[35] Pai C -F, Liu L, Li Y , Tseng H, Ralph D and Buhrman R 2012 Spin transfer torque devices utilizing the \ngiant spin Hall effect of tungsten Appl. Phys. Lett.  101 122404  \n[36] Thiaville A and Nakatani Y 2008 Electrical rectification effect in single domain magne tic microstrips: \nA micromagnetics -based analysis J. Appl. Phys.  104 093701  \n[37] Kubota H, Fukushima A, Yakushiji K, Nagahama T, Yuasa S, Ando K, Maehara H, Nagamine Y , \nTsunekawa K and Djayaprawira D D 2008 Quantitative measurement of voltage dependence of spin-\ntransfer torque in MgO -based magnetic tunnel junctions Nat. Phys.  4 37 \n[38] Skinner T, Wang M, Hindmarch A, Rushforth A, Irvine A, Heiss D, Kurebayashi H and Ferguson A 2014 \nSpin -orbit torque opposing the Oersted torque in ultrathin Co/Pt bilayers Appl. Phys. Lett.  104 \n062401  \n[39] Allen G, Manipatruni S, Nikonov D E, Doczy M and Young I A 2015 Experimental demonstration of \nthe coexistence of spin Hall and Rashba effects in β− tantalum/ferromagnet bilayers Phys. Rev. B  91 \n144412  \n[40] Mellnik A R, Lee J S, Richardella A, Grab J L, Mintun P J, Fischer M H, Vaezi A, Manchon A, Kim E A, \nSamarth N and Ralph D C 2014 Spin -transfer torque generated by a topological insulator Nature  511 \n449 \n[41] Petit S, Baraduc C, Thirion C, Ebels U, Liu Y , L i M, Wang P and Dieny B 2007 Spin -Torque Influence on \nthe High -Frequency Magnetization Fluctuations in Magnetic Tunnel Junctions Phys. Rev. Lett.  98 \n077203  \n[42] Wang C, Cui Y -T, Katine J A, Buhrman R A and Ralph D C 2011 Time -resolved measurement of spin -\ntransfer -driven ferromagnetic resonance and spin torque in magnetic tunnel junctions Nat. Phys.  7 \n496-501 \n[43] Ando K, Takahashi S, Harii K, Sasage K, Ieda J, Maekawa S and Saitoh E 2008 Electric manipulation of 59 \n \n spin relaxation using the spin Hall effect Phys. Rev. Lett.  101 036601  \n[44] Kasai S, Kondou K, Sukegawa H, Mitani S, Tsukagoshi K and Otani Y 2014 Modulation of effective \ndamping constant using spin Hall effect Appl. Phys. Lett.  104 092408  \n[45] Ganguly A, Kondou K, Sukegawa H, Mitani S, Kasai S, Niim i Y , Otani Y and Barman A 2014 Thickness \ndependence of spin torque ferromagnetic resonance in Co 75Fe25/Pt bilayer films Appl. Phys. Lett.  \n104 072405  \n[46] Nan T, Emori S, Boone C T, Wang X, Oxholm T M, Jones J G, Howe B M, Brown G J and Sun N X 2015 \nCompari son of spin -orbit torques and spin pumping across NiFe/Pt and NiFe/Cu/Pt interfaces Phys. \nRev. B  91 214416  \n[47] Tiwari D, Behera N, Kumar A, Dürrenfeld P , Chaudhary S, Pandya D, Åkerman J and Muduli P 2017 \nAntidamping spin -orbit torques in epitaxial -Py (10 0)/β -Ta Appl. Phys. Lett.  111 232407  \n[48] Zhang W, Jungfleisch M B, Freimuth F, Jiang W, Sklenar J, Pearson J E, Ketterson J B, Mokrousov Y \nand Hoffmann A 2015 All -electrical manipulation of magnetization dynamics in a ferromagnet by \nantiferromagnets with anisotropic spin Hall effects Phys. Rev. B  92 144405  \n[49] Hirsch J E 1999 Spin Hall Effect Phys. Rev. Lett.  83 1834 -7 \n[50] Zhang S 2000 Spin Hall Effect in the Presence of Spin Diffusion Phys. Rev. Lett.  85 393-6 \n[51] Hoffmann A 2013 Spin Hall Effects in M etals IEEE Trans. Magn.  49 5172 -93 \n[52] Sinova J, Valenzuela S O, Wunderlich J, Back C H and Jungwirth T 2015 Spin Hall effects Rev. Mod. \nPhys.  87 1213 -60 \n[53] D'yakonov M and Perel V 1971 Possibility of orienting electron spins with current Soviet Journal  of \nExperimental and Theoretical Physics Letters  13 467 \n[54] Valenzuela S O and Tinkham M 2006 Direct electronic measurement of the spin Hall effect Nature  \n442 176 \n[55] Vila L, Kimura T and Otani Y 2007 Evolution of the spin Hall effect in Pt nanowires: si ze and \ntemperature effects Phys. Rev. Lett.  99 226604  \n[56] Seki T, Hasegawa Y , Mitani S, Takahashi S, Imamura H, Maekawa S, Nitta J and Takanashi K 2008 Giant \nspin Hall effect in perpendicularly spin -polarized FePt/Au devices Nat. Mater.  7 125-9 \n[57] Saitoh E, Ueda M, Miyajima H and Tatara G 2006 Conversion of spin current into charge current at \nroom temperature: Inverse spin -Hall effect Appl. Phys. Lett.  88 182509  \n[58] Nakayama H, Althammer M, Chen Y -T, Uchida K, Kajiwara Y , Kikuchi D, Ohtani T, Geprä gs S, Opel M \nand Takahashi S 2013 Spin Hall magnetoresistance induced by a nonequilibrium proximity effect \nPhys. Rev. Lett.  110 206601  \n[59] Kondou K, Sukegawa H, Mitani S, Tsukagoshi K and Kasai S 2012 Evaluation of spin Hall angle and \nspin diffusion lengt h by using spin current -induced ferromagnetic resonance Appl. Phys. Express  5 \n073002  \n[60] Liu L, Buhrman R and Ralph D 2011 Review and analysis of measurements of the spin Hall effect in \nplatinum arXiv preprint arXiv:1111.3702   \n[61] Morota M, Niimi Y , Ohni shi K, Wei D, Tanaka T, Kontani H, Kimura T and Otani Y 2011 Indication of \nintrinsic spin Hall effect in 4 d and 5 d transition metals Phys. Rev. B  83 174405  \n[62] Feng Z, Hu J, Sun L, You B, Wu D, Du J, Zhang W, Hu A, Yang Y and Tang D 2012 Spin Hall angle  \nquantification from spin pumping and microwave photoresistance Phys. Rev. B  85 214423  \n[63] Nakayama H, Ando K, Harii K, Yoshino T, Takahashi R, Kajiwara Y , Uchida K -i, Fujikawa Y and Saitoh E \n2012 Geometry dependence on inverse spin Hall effect induced by  spin pumping in Ni 81Fe19/Pt films \nPhys. Rev. B  85 144408  \n[64] Vlaminck V, Pearson J E, Bader S D and Hoffmann A 2013 Dependence of spin -pumping spin Hall \neffect measurements on layer thicknesses and stacking order Phys. Rev. B  88 064414  \n[65] Vlietstra N, Shan J, Castel V, Van Wees B and Youssef J B 2013 Spin -Hall magnetoresistance in \nplatinum on yttrium iron garnet: Dependence on platinum thickness and in -plane/out -of-plane 60 \n \n magnetization Phys. Rev. B  87 184421  \n[66] Zhang W, Vlaminck V, Pearson J E, Divan R , Bader S D and Hoffmann A 2013 Determination of the Pt \nspin diffusion length by spin -pumping and spin Hall effect Appl. Phys. Lett.  103 242414  \n[67] Althammer M, Meyer S, Nakayama H, Schreier M, Altmannshofer S, Weiler M, Huebl H, Geprägs S, \nOpel M and Gro ss R 2013 Quantitative study of the spin Hall magnetoresistance in ferromagnetic \ninsulator/normal metal hybrids Phys. Rev. B  87 224401  \n[68] Weiler M, Althammer M, Schreier M, Lotze J, Pernpeintner M, Meyer S, Huebl H, Gross R, Kamra A \nand Xiao J 2013 Experimental test of the spin mixing interface conductivity concept Phys. Rev. Lett.  \n111 176601  \n[69] Obstbaum M, Härtinger M, Bauer H, Meier T, S wientek F, Back C and Woltersdorf G 2014 Inverse \nspin Hall effect in Ni 81Fe19/normal -metal bilayers Phys. Rev. B  89 060407  \n[70] Bass J and Pratt Jr W P 2007 Spin -diffusion lengths in metals and alloys, and spin -flipping at \nmetal/metal interfaces: an experi mentalist ’s critical review J. Phys. Condens. Matter  19 183201  \n[71] Niimi Y , Wei D, Idzuchi H, Wakamura T, Kato T and Otani Y 2013 Experimental verification of \ncomparability between spin -orbit and spin -diffusion lengths Phys. Rev. Lett.  110 016805  \n[72] Mihai Miron I, Gaudin G, Auffret S, Rodmacq B, Schuhl A, Pizzini S, Vogel J and Gambardella P 2010 \nCurrent -driven spin torque induced by the Rashba effect in a ferromagnetic metal layer Nat. Mater.  \n9 230 \n[73] Pai C -F, Nguyen M -H, Belvin C, Vilela -Leão L H, Ra lph D and Buhrman R 2014 Enhancement of \nperpendicular magnetic anisotropy and transmission of spin -Hall-effect -induced spin currents by a \nHf spacer layer in W/Hf/CoFeB/MgO layer structures Appl. Phys. Lett.  104 082407  \n[74] Nguyen M -H, Pai C -F, Nguyen K X, Muller D A, Ralph D and Buhrman R 2015 Enhancement of the \nanti-damping spin torque efficacy of platinum by interface modification Appl. Phys. Lett.  106 222402  \n[75] Qiu X, Deorani P , Narayanapillai K, Lee K -S, Lee K -J, Lee H -W and Yang H 2014 Angular and \ntemperature dependence of current induced spin -orbit effective fields in Ta/CoFeB/MgO nanowires \nSci. Rep.  4 4491  \n[76] Deorani P and Yang H 2013 Role of spin mixing conductance in spin pumping: Enhancement of spin \npumping efficiency in Ta/Cu/Py structures Appl. Phys. Lett.  103 232408  \n[77] Hahn C, De Loubens G, Klein O, Viret M, Naletov V V and Youssef J B 2013 Comparative \nmeasurements of inverse spin Hall effects and magnetoresistance in YIG/Pt and YIG/Ta Phys. Rev. B  \n87 174417  \n[78] Pham V T, Vila L, Zahnd G, Marty A, Savero -Torres W, Jamet M and Attané J -P 2016 \nFerromagnetic/nonmagnetic nanostructures for the electrical measurement of the Spin Hall effect \nNano Lett.  16 6755 -60 \n[79] Rojas -Sánchez J -C, Reyren N, Laczkowski P , Savero W,  Attané J -P , Deranlot C, Jamet M, George J -M, \nVila L and Jaffrès H 2014 Spin pumping and inverse spin Hall effect in platinum: the essential role of \nspin-memory loss at metallic interfaces Phys. Rev. Lett.  112 106602  \n[80] Zhang W, Han W, Jiang X, Yang S -H and Parkin S S 2015 Role of transparency of platinum -ferromagnet \ninterfaces in determining the intrinsic magnitude of the spin Hall effect Nat. Phys.  11 496-502 \n[81] Pai C -F, Ou Y , Vilela -Leão L H, Ralph D and Buhrman R 2015 Dependence of the efficiency of  spin Hall \ntorque on the transparency of Pt/ferromagnetic layer interfaces Phys. Rev. B  92 064426  \n[82] Berger A J, Edwards E R, Nembach H T, Karis O, Weiler M and Silva T 2017 Determination of spin Hall \neffect and spin diffusion length of Pt from self -cons istent fitting of damping enhancement and \ninverse spin -orbit torque measurements arXiv preprint arXiv:1711.07654   \n[83] Zhang C, Yamanouchi M, Sato H, Fukami S, Ikeda S, Matsukura F and Ohno H 2013 Magnetotransport \nmeasurements of current induced effective fields in Ta/CoFeB/MgO Appl. Phys. Lett.  103 262407  \n[84] Wang H, Du C, Pu Y , Adur R, Hammel P C and Yang F 2014 Scaling of spin Hall angle in 3d, 4d, and 5d \nmetals from Y 3 Fe 5 O 12/metal spin pumping Phys. Rev. Lett.  112 197201  61 \n \n [85] Qu D, Huang S, Miao B , Huang S and Chien C 2014 Self -consistent determination of spin Hall angles \nin selected 5 d metals by thermal spin injection Phys. Rev. B  89 140407  \n[86] Gómez J, Tedlla B Z, Álvarez N, Alejandro G, Goovaerts E and Butera A 2014 Spin transport \nparameters i n Ni 80Fe20/Ru and Ni 80Fe20/Ta bilayers Phys. Rev. B  90 184401  \n[87] Liu J, Ohkubo T, Mitani S, Hono K and Hayashi M 2015 Correlation between the spin Hall angle and \nthe structural phases of early 5 d transition metals Appl. Phys. Lett.  107 232408  \n[88] Kondo u K, Sukegawa H, Kasai S, Mitani S, Niimi Y and Otani Y 2016 Influence of inverse spin Hall \neffect in spin -torque ferromagnetic resonance measurements Appl. Phys. Express  9 023002  \n[89] Kim J, Sheng P , Takahashi S, Mitani S and Hayashi M 2016 Spin Hall magn etoresistance in metallic \nbilayers Phys. Rev. Lett.  116 097201  \n[90] Hao Q and Xiao G 2015 Giant spin Hall effect and switching induced by spin -transfer torque in a \nW/Co 40Fe40B20/MgO structure with perpendicular magnetic anisotropy Phys. Rev. Applied  3 0340 09 \n[91] Cho S, Baek S -h C, Lee K -D, Jo Y and Park B -G 2015 Large spin Hall magnetoresistance and its \ncorrelation to the spin -orbit torque in W/CoFeB/MgO structures Sci. Rep.  5 14668  \n[92] Yu J, Qiu X, Legrand W and Yang H 2016 Large spin -orbit torques in Pt /Co-Ni/W heterostructures \nAppl. Phys. Lett.  109 042403  \n[93] Gabor M, Petrisor Jr T, Mos R, Mesaros A, Nasui M, Belmeguenai M, Zighem F and Tiusan C 2016 \nSpin –orbit torques and magnetization switching in W/Co 2FeAl/MgO structures J. Phys. D: Appl. Phys.  \n49 365003  \n[94] Niimi Y , Morota M, Wei D H, Deranlot C, Basletic M, Hamzic A, Fert A and Otani Y 2011 Extrinsic Spin \nHall Effect Induced by Iridium Impurities in Copper Phys. Rev. Lett.  106 126601  \n[95] Niimi Y , Kawanishi Y , Wei D H, Deranlot C, Yang H X, Chshiev M, Valet T, Fert A and Otani Y 2012 Giant \nSpin Hall Effect Induced by Skew Scattering from Bismuth Impurities inside Thin Film CuBi Alloys \nPhys. Rev. Lett.  109 156602  \n[96] Niimi Y , Suzuki H, Kawanishi Y , Omori Y , Valet T, Fert A and Otani Y 2014 Extrinsic spin Hall effects \nmeasured with lateral spin valve structures Phys. Rev. B  89 054401  \n[97] Ramaswamy R, Wang Y , Elyasi M, Motapothula M, Venkatesan T, Qiu X and Yang H 2017 Extrinsic \nSpin Hall Effect in Cu 1−x Ptx Phys. Rev. Applied  8 024034  \n[98] Niimi Y and Otani Y 2015 Reciprocal spin Hall effects in conductors with strong spin –orbit coupling: \na review Rep. Prog. Phys.  78 124501  \n[99] Tian Y , Ye L and Jin X 2009 Proper scalin g of the anomalous Hall effect Phys. Rev. Lett.  103 087206  \n[100]  Fert A and Levy P M 2011 Spin Hall effect induced by resonant scattering on impurities in metals \nPhys. Rev. Lett.  106 157208  \n[101]  Asomoza R, Fert A and Reich R 1983 Gadolinium -heavy rare ear th alloys: preparation, \nmetallographic study and extraordinary hall effect J. Less Common Metals  90 177-201 \n[102]  Tokaç M, Bunyaev S, Kakazei G, Schmool D, Atkinson D and Hindmarch A 2015 Interfacial structure \ndependent spin mixing conductance in cobalt th in films Phys. Rev. Lett.  115 056601  \n[103]  Perez N, Martinez E, Torres L, Woo S -H, Emori S and Beach G 2014 Chiral magnetization textures \nstabilized by the Dzyaloshinskii -Moriya interaction during spin -orbit torque switching Appl. Phys. \nLett.  104 092403  \n[104] Zhang C, Fukami S, Sato H, Matsukura F and Ohno H 2015 Spin -orbit torque induced magnetization \nswitching in nano -scale Ta/CoFeB/MgO Appl. Phys. Lett.  107 012401  \n[105]  Fukami S, Anekawa T, Zhang C and Ohno H 2016 A spin –orbit torque switching scheme wit h collinear \nmagnetic easy axis and current configuration Nat. Nanotech.  11 621-5 \n[106]  Legrand W, Ramaswamy R, Mishra R and Yang H 2015 Coherent subnanosecond switching of \nperpendicular magnetization by the fieldlike spin -orbit torque without an external m agnetic field \nPhysical Review Applied  3 064012  \n[107]  Lee O, Liu L, Pai C, Li Y , Tseng H, Gowtham P , Park J, Ralph D and Buhrman R A 2014 Central role of 62 \n \n domain wall depinning for perpendicular magnetization switching driven by spin torque from the \nspin Hal l effect Phys. Rev. B  89 024418  \n[108]  Zhang C, Fukami S, DuttaGupta S, Sato H and Ohno H 2018 Time and spatial evolution of spin –orbit \ntorque -induced magnetization switching in W/CoFeB/MgO structures with various sizes Jpn. J. Appl. \nPhys  57 04FN2  \n[109]  König M, Wiedmann S, Brüne C, Roth A, Buhmann H, Molenkamp L W, Qi X -L and Zhang S -C 2007 \nQuantum spin Hall insulator state in HgTe quantum wells Science  318 766-70 \n[110]  Xia Y , Qian D, Hsieh D, Wray L, Pal A, Lin H, Bansil A, Grauer D, Hor Y S, Cava R J and Hasan M Z 2009 \nObservation of a large -gap topological -insulator class with a single Dirac cone on the surface Nat. \nPhys.  5 398-402 \n[111]  Zhang H, Liu C -X, Qi X -L, Dai X, Fang Z and Zhang S -C 2009 Topological insulators in Bi 2Se3, Bi 2Te3 and \nSb2Te3 with a s ingle Dirac cone on the surface Nat. Phys.  5 438 \n[112]  Chen Y , Analytis J G, Chu J -H, Liu Z, Mo S -K, Qi X -L, Zhang H, Lu D, Dai X and Fang Z 2009 Experimental \nrealization of a three -dimensional topological insulator, Bi2Te3 science  325 178-81 \n[113]  Bernevig B A, Hughes T L and Zhang S -C 2006 Quantum spin Hall effect and topological phase \ntransition in HgTe quantum wells Science  314 1757 -61 \n[114]  Hsieh D, Qian D, Wray L, Xia Y , Hor Y S, Cava R J and Hasan M Z 2008 A topological Dirac insulator in \na qu antum spin Hall phase Nature  452 970 \n[115]  Moore J E 2010 The birth of topological insulators Nature  464 194-8 \n[116]  Hasan M Z and Kane C L 2010 Colloquium: Topological insulators Rev. Mod. Phys.  82 3045 -67 \n[117]  Qi X-L and Zhang S -C 2011 Topological insul ators and superconductors Rev. Mod. Phys.  83 1057 -110 \n[118]  Shiomi Y , Nomura K, Kajiwara Y , Eto K, Novak M, Segawa K, Ando Y and Saitoh E 2014 Spin -electricity \nconversion induced by spin injection into topological insulators Phys. Rev. Lett.  113 196601  \n[119] Fan Y , Upadhyaya P , Kou X, Lang M, Takei S, Wang Z, Tang J, He L, Chang L -T and Montazeri M 2014 \nMagnetization switching through giant spin –orbit torque in a magnetically doped topological \ninsulator heterostructure Nat. Mater.  13 699-704 \n[120]  Kondou K,  Yoshimi R, Tsukazaki A, Fukuma Y , Matsuno J, Takahashi K S, Kawasaki M, Tokura Y and \nOtani Y 2016 Fermi -level -dependent charge -to-spin current conversion by Dirac surface states of \ntopological insulators Nat. Phys.  12 1027 -31 \n[121]  Deorani P , Son J, Baner jee K, Koirala N, Brahlek M, Oh S and Yang H 2014 Observation of inverse spin \nHall effect in bismuth selenide Phys. Rev. B  90 094403  \n[122]  Fan Y and Wang K L 2016 Spintronics Based on Topological Insulators Spin  6 1640001  \n[123]  Wang H, Kally J, Lee J S, Li u T, Chang H, Hickey D R, Mkhoyan K A, Wu M, Richardella A and Samarth \nN 2016 Surface -State -Dominated Spin -Charge Current Conversion in Topological -Insulator -\nFerromagnetic -Insulator Heterostructures Phys. Rev. Lett.  117 076601  \n[124]  Jamali M, Lee J S, Jeong J S, Mahfouzi F, Lv Y , Zhao Z, Nikolić B K, Mkhoyan K A, Samarth N and Wang \nJ-P 2015 Giant Spin Pumping and Inverse Spin Hall Effect in the Presence of Surface and Bulk \nSpin −Orbit Coupling of Topological Insulator Bi 2Se3 Nano Lett.  15 7126 -32 \n[125]  Liu L, Richardella A, Garate I, Zhu Y , Samarth N and Chen C -T 2015 Spin -polarized tunneling study of \nspin-momentum locking in topological insulators Phys. Rev. B  91 235437  \n[126]  Lee J S, Richardella A, Hickey D R, Mkhoyan K A and Samarth N 2015 Mapping the chemical potential \ndependence of current -induced spin polarization in a topological insulator Phys. Rev. B  92 155312  \n[127]  Kim J, Sinha J, Mitani S, Hayashi M, Takahashi S, Maekawa S, Yamanouchi M and Ohno H 2014 \nAnomalous temperature dependence of current -induced torques in CoFeB/MgO heterostructures \nwith Ta -based underlayers Phys. Rev. B  89 174424  \n[128]  Eldridge P , Leyland W, Lagoudakis P , Karimov O, Henini M, Taylor D, Phillips R and Harley R 2008 All -\noptical measurement of Rashba coefficient in quantum wells Phys. Rev. B  77 125344  \n[129]  Fu L 2009 Hexagonal warping effects in the surface states of the topological insulator Bi 2Te3 Phys. 63 \n \n Rev. Lett.  103 266801  \n[130]  Wang Y H, Hsieh D, Pilon D, Fu L, Gardner D R, Lee Y S and Gedik N 2011 Observation of a Warp ed \nHelical Spin Texture in Bi 2Se3 from Circular Dichroism Angle -Resolved Photoemission Spectroscopy \nPhys. Rev. Lett.  107 207602  \n[131]  Nomura M, Souma S, Takayama A, Sato T, Takahashi T, Eto K, Segawa K and Ando Y 2014 Relationship \nbetween Fermi surface war ping and out -of-plane spin polarization in topological insulators: A view \nfrom spin -and angle -resolved photoemission Phys. Rev. B  89 045134  \n[132]  Kuroda K, Arita M, Miyamoto K, Ye M, Jiang J, Kimura A, Krasovskii E E, Chulkov E V, Iwasawa H, \nOkuda T, Shimada K, Ueda Y , Namatame H and Taniguchi M 2010 Hexagonally Deformed Fermi \nSurface of the 3D Topological Insulator Bi 2Se3 Phys. Rev. Lett.  105 076802  \n[133]  Xu Z, Guo X, Yao M, He H, Miao L, Jiao  L, Liu H, Wang J, Qian D and Jia J 2013 Anisotropic Topolo gical \nSurface States on High ‐Index Bi2Se3 Films Adv. Mater.  25 1557 -62 \n[134]  Yazyev O V, Moore J E and Louie S G 2010 Spin polarization and transport of surface states in the \ntopological insulators Bi 2Se3 and Bi 2Te3 from first principles Phys. Rev. Lett.  105 266806  \n[135]  MacNeill D, Stiehl G M, Guimaraes M H D, Buhrman R A, Park J and Ralph D C 2016 Control of spin –\norbit torques through crystal symmetry in WTe 2/ferromagnet bilayers Nat. Phys.  13 300 \n[136]  Han J, Richardella A, Siddiqui S A, Finley J, Samart h N and Liu L 2017 Room -Temperature Spin -Orbit \nTorque Switching Induced by a Topological Insulator Phys. Rev. Lett.  119 077702  \n[137]  Jamali M, Chen J -Y , Hickey D R, Zhang D, Zhao Z, Li H, Quarterman P , Lv Y , Li M and Mkhoyan K A 2017 \nRoom -temperature perpe ndicular magnetization switching through giant spin -orbit torque from \nsputtered Bi XSe1-x topological insulator material arXiv preprint arXiv:1703.03822   \n[138]  Bychkov Y A and Rashba E 1984 Properties of a 2D electron gas with lifted spectral degeneracy JETP \nlett 39 78 \n[139]  Manchon A, Koo H C, Nitta J, Frolov S M and Duine R A 2015 New perspectives for Rashba spin –orbit \ncoupling Nat. Mater.  14 871 \n[140]  Koroteev Y M, Bihlmayer G, Gayone J, Chulkov E, Blügel S, Echenique P M and Hofmann P 2004 \nStrong spin -orbit splitting on Bi surfaces Phys. Rev. Lett.  93 046403  \n[141]  Ast C R, Henk J, Ernst A, Moreschini L, Falub M C, Pacilé D, Bruno P , Kern K and Grioni M 2007 Giant \nspin splitting through surface alloying Phys. Rev. Lett.  98 186807  \n[142]  Meier F, Dil H, Lobo -Checa J, Patthey L and Osterwalder J 2008 Quantitative vectorial spin analysis \nin angle -resolved photoemission: Bi ∕Ag (111) and Pb ∕Ag (111) Phys. Rev. B  77 165431  \n[143]  Sánchez J R, Vila L, Desfonds G, Gambarelli S, Attané J, De Teresa J, Magén C and Fert  A 2013 Spin -\nto-charge conversion using Rashba coupling at the interface between non -magnetic materials Nat. \nCommun.  4 2944  \n[144]  Nomura A, Tashiro T, Nakayama H and Ando K 2015 Temperature dependence of inverse Rashba -\nEdelstein effect at metallic interfac e Appl. Phys. Lett.  106 212403  \n[145]  Zhang W, Jungfleisch M B, Jiang W, Pearson J E and Hoffmann A 2015 Spin pumping and inverse \nRashba -Edelstein effect in NiFe/Ag/Bi and NiFe/Ag/Sb J. Appl. Phys.  117 17C727  \n[146]  Zhang H, Yamamoto S, Gu B, Li H, Maekawa M , Fukaya Y and Kawasuso A 2015 Charge -to-spin \nconversion and spin diffusion in Bi/Ag bilayers observed by spin -polarized positron beam Phys. Rev. \nLett.  114 166602  \n[147]  Jungfleisch M, Zhang W, Sklenar J, Jiang W, Pearson J, Ketterson J and Hoffmann A 2016 Interface -\ndriven spin -torque ferromagnetic resonance by Rashba coupling at the interface between \nnonmagnetic materials Phys. Rev. B  93 224419  \n[148]  Shi S, Wang A, Wang Y , Ramaswamy R, Shen L, Moon J, Zhu D, Yu J, Oh S, Feng Y and Yang H 2018 \nEfficient Charge -Spin Conversion and Magnetization Switching though Rashba Effect at Topological \nInsulator/Ag Interface Phys. Rev. B  97 041115 (R)  64 \n \n [149]  Manzeli S, Ovchinnikov D, Pasq uier D, Yazyev O V and Kis A 2017 2D transition metal dichalcogenides \nNat. Rev. Mater.  2 17033  \n[150]  Shao Q, Yu G, Lan Y -W, Shi Y , Li M -Y , Zheng C, Zhu X, Li L -J, Amiri P K and Wang K L 2016 Strong \nRashba -Edelstein Effect -Induced Spin –Orbit Torques in Mono layer Transition Metal \nDichalcogenide/Ferromagnet Bilayers Nano Lett.  16 7514 -20 \n[151]  Zhang W, Sklenar J, Hsu B, Jiang W, Jungfleisch M B, Xiao J, Fradin F Y , Liu Y , Pearson J E and Ketterson \nJ B 2016 Research Update: Spin transfer torques in permalloy on  monolayer MoS2 APL Mater.  4 \n032302  \n[152]  Sklenar J, Zhang W, Jungfleisch M B, Jiang W, Saglam H, Pearson J E, Ketterson J B and Hoffmann A \n2016 Perspective: Interface generation of spin -orbit torques J. Appl. Phys.  120 180901  \n[153]  MacNeill D, Stiehl G M,  Guimarães M H, Reynolds N D, Buhrman R A and Ralph D C 2017 Thickness \ndependence of spin -orbit torques generated by WTe 2 Phys. Rev. B  96 054450  \n[154]  Ali M N, Xiong J, Flynn S, Tao J, Gibson Q D, Schoop L M, Liang T, Haldolaarachchige N, Hirschberger \nM an d Ong N 2014 Large, non -saturating magnetoresistance in WTe 2 Nature  514 205-8 \n[155]  Jiang J, Tang F, Pan X, Liu H, Niu X, Wang Y , Xu D, Yang H, Xie B and Song F 2015 Signature of strong \nspin-orbital coupling in the large nonsaturating magnetoresistance mat erial WTe 2 Phys. Rev. Lett.  \n115 166601  \n[156]  Soluyanov A A, Gresch D, Wang Z, Wu Q, Troyer M, Dai X and Bernevig B A 2015 Type -II Weyl \nsemimetals Nature  527 495 \n[157]  Wadley P , Howells B, Železný J, Andrews C, Hills V, Campion R P , Novák V, Olejník K, Macc herozzi F \nand Dhesi S 2016 Electrical switching of an antiferromagnet Science  351 587-90 \n[158]  Yang Y , Xu Y , Zhang X, Wang Y , Zhang S, Li R -W, Mirshekarloo M S, Yao K and Wu Y 2016 Fieldlike spin -\norbit torque in ultrathin polycrystalline FeMn films Phys. R ev. B  93 094402  \n[159]  Tsai M -H, Lin P -H, Huang K -F, Lin H -H and Lai C -H 2017 Spin -orbit -torque MRAM: from uniaxial to \nunidirectional switching arXiv preprint arXiv:1706.01639   \n[160]  Fukami S, Zhang C, DuttaGupta S, Kurenkov A and Ohno H 2016 Magnetization switching by spin -\norbit torque in an antiferromagnet -ferromagnet bilayer system Nat. Mater.  15 535-41 \n[161]  van den Brink A, Vermijs G, Solignac A, Koo J, Kohlhepp J T, Swagten H J M and Koopmans B 2016 \nField -free magnetization reversal by spin -Hall effect  and exchange bias Nat. Commun.  7 10854  \n[162]  Lau Y -C, Betto D, Rode K, Coey J M D and Stamenov P 2016 Spin –orbit torque switching without an \nexternal field using interlayer exchange coupling Nat. Nanotech.  11 758 \n[163]  Oh Y -W, Baek S -h C, Kim Y , Lee H Y , Lee K -D, Yang C -G, Park E -S, Lee K -S, Kim K -W and Go G 2016 \nField -free switching of perpendicular magnetization through spin –orbit torque in \nantiferromagnet/ferromagnet/oxide structures Nature nanotechnology  11 878-84 \n[164]  Sklenar J, Zhang W, Jungfleisch M B, Jiang W, Saglam H, Pearson J E, Ketterson J B and Hoffmann A \n2016 Spin Hall effects in metallic antiferromagnets –perspectives for future spin -orbitronics AIP Adv.  \n6 055603  \n[165]  Zhang W, Han W, Yang S -H, Sun Y , Zhang Y , Yan B and Parkin S S 2016 Giant  facet -dependent spin -\norbit torque and spin Hall conductivity in the triangular antiferromagnet IrMn 3 Sci. Adv.  2 e1600759  \n[166]  Mendes J, Cunha R, Santos O A, Ribeiro P , Machado F, Rodríguez -Suárez R, Azevedo A and Rezende \nS 2014 Large inverse spin Hall e ffect in the antiferromagnetic metal Ir 20Mn 80 Phys. Rev. B  89 140406  \n[167]  Zhang W, Jungfleisch M B, Jiang W, Pearson J E, Hoffmann A, Freimuth F and Mokrousov Y 2014 Spin \nHall effects in metallic antiferromagnets Phys. Rev. Lett.  113 196602  \n[168]  Hwang H Y , Iwasa Y , Kawasaki M, Keimer B, Nagaosa N and Tokura Y 2012 Emergent phenomena at \noxide interfaces Nat. Mater.  11 103 \n[169]  Reyren N, Thiel S, Caviglia A, Kourkoutis L F, Hammerl G, Richter C, Schneider C, Kopp T, Rüetschi A -\nS and Jaccard D 2007 Superconducting interfaces between insulating oxides Science  317 1196 -9 \n[170]  Caviglia A, Gariglio S, Reyren N, Jaccard D, Schneider T, Gabay M, Thiel S, Hammerl G, Mannhart J 65 \n \n and Triscone J -M 2008 Electric field control of the LaAlO 3/SrTiO 3 interface gr ound state Nature  \n456 624 \n[171]  Brinkman A, Huijben M, Van Zalk M, Huijben J, Zeitler U, Maan J, van der Wiel W G, Rijnders G, Blank \nD H and Hilgenkamp H 2007 Magnetic effects at the interface between non -magnetic oxides Nat. \nMater.  6 493 \n[172]  Kalisky B, Bert J A, Klopfer B B, Bell C, Sato H K, Hosoda M, Hikita Y , Hwang H Y and Moler K A 2012 \nCritical thickness for ferromagnetism in LaAlO 3/SrTiO 3 heterostructures Nat. Commun.  3 922 \n[173]  Lee J -S, Xie Y , Sato H, Bell C, Hikita Y , Hwang H and Kao C -C 2013 Titanium d xy ferromagnetism at \nthe LaAlO 3/SrTiO 3 interface Nat. Mater.  12 703 \n[174]  Ohtomo A and Hwang H Y 2004 A high -mobility electron gas at the LaAlO 3/SrTiO 3 heterointerface \nNature  427 423-6 \n[175]  Caviglia A, Gabay M, Gariglio S, Reyren N, Cancellie ri C and Triscone J -M 2010 Tunable Rashba spin -\norbit interaction at oxide interfaces Phys. Rev. Lett.  104 126803  \n[176]  Joshua A, Pecker S, Ruhman J, Altman E and Ilani S 2012 A universal critical density underlying the \nphysics of electrons at the LaAlO 3/SrTiO 3 interface Nat. Commun.  3 2116  \n[177]  Zhong Z, Tóth A and Held K 2013 Theory of spin -orbit coupling at LaAlO 3/SrTiO 3 interfaces and SrTiO 3 \nsurfaces Phys. Rev. B  87 161102  \n[178]  Edelstein V M 1990 Spin polarization of conduction electrons induced by elec tric current in two -\ndimensional asymmetric electron systems Solid State Commun.  73 233-5 \n[179]  Narayanapillai K, Gopinadhan K, Qiu X, Annadi A, Ariando, Venkatesan T and Yang H 2014 Current -\ndriven spin orbit field in LaAlO 3/SrTiO 3 heterostructures Appl. Ph ys. Lett.  105 162405  \n[180]  Wang Y , Ramaswamy R, Motapothula M, Narayanapillai K, Zhu D, Yu J, Venkatesan T and Yang H 2017 \nRoom Temperature Giant Charge -to-Spin Conversion at SrTiO 3/LaAlO 3 Oxide Interface Nano Lett.  17 \n7659  \n[181]  Lesne E, Fu Y , Oyarzun S, Rojas -Sanchez J C, Vaz D C, Naganuma H, Sicoli G, Attane J P , Jamet M, \nJacquet E, George J M, Barthelemy A, Jaffres H, Fert A, Bibes M and Vila L 2016 Highly efficient and \ntunable spin -to-charge conversion through Rashba coupling at oxide interfaces Nat Mater  15 1261 -\n6 \n[182]  Chauleau J -Y , Boselli M, Gariglio S, Weil R, De Loubens G, Triscone J -M and Viret M 2016 Efficient \nspin-to-charge conversion in the 2D electron liquid at the LAO/STO interface EPL 116 17006  \n[183]  Song Q, Zhang H, Su T, Yuan W, Chen Y , Xing W, Shi J, Sun J and Han W 2017 Observation of inverse \nEdelstein effect in Rashba -split 2DEG between SrTiO 3 and LaAlO 3 at room temperature Sci. Adv.  3 \ne1602312  \n[184]  Jin M -J, Moon S Y , Park J, Modepalli V, Jo J, Kim S -I, Koo H C, Min B -C, Lee H -W and Baek S -H 2016 \nNonlocal Spin Diffusion Driven by Giant Spin Hall Effect at Oxide Heterointerfaces Nano Lett.  17 36-\n43 \n[185]  Ohshima R, Ando Y , Matsuzaki K, Susaki T, Weiler M, Klingler S, Huebl H, Shikoh E, Shinjo T and \nGoennenwein S  T 2017 Strong evidence for d -electron spin transport at room temperature at a \nLaAlO 3/SrTiO 3 interface Nat. Mater.  16 609–14 \n[186]  Demasius K -U, Phung T, Zhang W, Hughes B P , Yang S -H, Kellock A, Han W , Pushp A and Parkin S S \n2016 Enhanced spin –orbit torqu es by oxygen incorporation in tungsten films Nat. Commun.  7 10644  \n[187]  An H, Kageyama Y , Kanno Y , Enishi N and Ando K 2016 Spin –torque generator engineered by natural \noxidation of Cu Nat. Commun.  7 13069  \n[188]  Qiu X, Narayanapillai K, Wu Y , Deorani P , Yan g D-H, Noh W -S, Park J -H, Lee K -J, Lee H -W and Yang H \n2015 Spin –orbit -torque engineering via oxygen manipulation Nat. Nanotech.  10 333-8 \n[189]  Garello K, Avci C O, Miron I M, Baumgartner M, Ghosh A, Auffret S, Boulle O, Gaudin G and \nGambardella P 2014 Ultr afast magnetization switching by spin -orbit torques Appl. Phys. Lett.  105 \n212402  66 \n \n [190]  Yoon J, Lee S -W, Kwon J H, Lee J M, Son J, Qiu X, Lee K -J and Yang H 2017 Anomalous spin -orbit \ntorque switching due to field -like torque –assisted domain wall reflection Sci. Adv.  3 e1603099  \n[191]  Shi S, Ou Y , Aradhya S, Ralph D and Buhrman R 2018 Fast Low -Current Spin -Orbit -Torque Switching \nof Magnetic Tunnel Junctions through Atomic Modifications of the Free -Layer Interfaces Phys. Rev. \nApplied  9 011002  \n[192]  Lee J M, Kwo n J H, Ramaswamy R, Yoon J, Son J, Qiu X, Mishra R, Srivastava S, Cai K and Yang H 2018 \nOscillatory spin -orbit torque switching induced by field -like torques Communications Physics  1 2 \n[193]  Amiri P K, Zeng Z, Upadhyaya P , Rowlands G, Zhao H, Krivorotov I,  Wang J -P , Jiang H, Katine J, Langer \nJ, Galatsis K and Wang K L 2011 Low write -energy magnetic tunnel junctions for high -speed spin -\ntransfer -torque MRAM IEEE Electron Device Letters  32 57-9 \n[194]  Chen E, Apalkov D, Driskill -Smith A, Khvalkovskiy A, Lottis D, Moon K, Nikitin V, Ong A, Tang X and \nWatts S 2012 Progress and prospects of spin transfer torque random access memory IEEE Trans. \nMagn.  48 3025 -30 \n[195]  Cai H, Kang W , Wang Y , Naviner L A D B, Yang J and Zhao W 2017 High Performance MRAM with \nSpin -Trans fer-Torque and Voltage -Controlled Magnetic Anisotropy Effects Applied Sciences  7 929 \n[196]  Harder M, Gui Y and Hu C -M 2016 Electrical detection of magnetization dynamics via spin \nrectification effects Phys. Rep.  661 1-59 \n[197]  Chiba T, Bauer G E and Takaha shi S 2014 Current -Induced Spin -Torque Resonance of Magnetic \nInsulators Phys. Rev. Applied  2 034003  \n[198]  Schreier M, Chiba T, Niedermayr A, Lotze J, Huebl H, Geprägs S, Takahashi S, Bauer G E W, Gross R \nand Goennenwein S T B 2015 Current -induced spin torq ue resonance of a magnetic insulator Phys. \nRev. B  92 144411  \n[199]  Wang P , Jiang S, Luan Z, Zhou L, Ding H, Zhou Y , Tao X and Wu D 2016 Spin rectification induced by \nspin Hall magnetoresistance at room temperature Appl. Phys. Lett.  109 112406  \n[200]  Wang Z, Gresch D, Soluyanov A A, Xie W, Kushwaha S, Dai X, Troyer M, Cava R J and Bernevig B A \n2016 MoTe 2: a type -II Weyl topological metal Phys. Rev. Lett.  117 056805  \n[201]  Núnez A S, Duine R, Haney P and MacDonald A 2006 Theory of spin torques and giant \nmagneto resistance in antiferromagnetic metals Phys. Rev. B  73 214426  \n[202]  Haney P M and MacDonald A 2008 Current -induced torques due to compensated antiferromagnets \nPhys. Rev. Lett.  100 196801  \n[203]  Mishra R, Yu J, Qiu X, Motapothula M, Venkatesan T and Yang H 2017 Anomalous current -induced \nspin torques in ferrimagnets near compensation Physical Review Letters  118 167201  \n[204]  Cheng X, Katine J, Rowlands G and Krivorotov I 2013 Nonlinear ferromagnetic  resonance induced by \nspin torque in nanoscale magnetic tunnel junctions Appl. Phys. Lett.  103 082402  \n[205]  Miwa S, Ishibashi S, Tomita H, Nozaki T, Tamura E, Ando K, Mizuochi N, Saruya T, Kubota H and \nYakushiji K 2014 Highly sensitive nanoscale spin -torqu e diode Nat. Mater.  13 50-6 \n[206]  Fang B, Carpentieri M, Hao X, Jiang H, Katine J A, Krivorotov I N, Ocker B, Langer J, Wang K L and \nZhang B 2016 Giant spin -torque diode sensitivity in the absence of bias magnetic field Nat. Commun.  \n7 11259  \n \n "    },    {        "title": "1009.3551v3.Composite_excitation_of_Josephson_phase_and_spin_waves_in_Josephson_junctions_with_ferromagnetic_insulator.pdf",        "content": "arXiv:1009.3551v3  [cond-mat.supr-con]  14 Apr 2011Typeset with jpsj2.cls <ver.1.2.2> Full Paper\nComposite excitation of Josephson phase and spin waves in Jo sephson\njunctions with ferromagnetic insulator\nShin-ichi HIKINO1, Michiyasu MORI2,4, Saburo TAKAHASHI3,4, and Sadamichi\nMAEKAWA2,4\n1Computational Condensed Matter Physics Laboratory, RIKEN , Wako, Saitama 351-0198, Japan\n2Advanced Science Research Center, Japan Atomic Energy Agen cy, Tokai-mura, Ibaraki 319-1195,\nJapan\n3Institute for Materials Research, Tohoku University, Send ai 980-8577, Japan\n4CREST, Japan Science and Technology Agency, Tokyo 100-0075 , Japan\nCoupling of Josephson-phase and spin-waves is theoretically studie d in a superconduc-\ntor/ferromagnetic insulator/superconductor (S/FI/S) junct ion. Electromagnetic (EM) ﬁeld\ninside the junction and the Josephson current coupled with spin-wa ves in FI are calculated\nby combining Maxwell and Landau-Lifshitz-Gilbert equations. In the S/FI/S junction, it\nis found that the current-voltage ( I-V) characteristic shows tworesonant peaks. Voltages\nat the resonant peaks are obtained as a function of the normal mo des of EM ﬁeld, which\nindicates a composite excitation of the EM ﬁeld and spin-waves in the S /FI/S junction. We\nalso examine another type of junction, in which a nonmagnetic insulat or (I) is located at one\nof interfaces between S and FI. In such a S/I/FI/S junction, threeresonant peaks appear in\ntheI-Vcurve, since the Josephson-phase couples to the EM ﬁeld in the I lay er.\nKEYWORDS: Fiske resonance, Josephson junction, supercond uctor, ferromagnetic insulator,\nspin-wave\n1. Introduction\nThe dc Josephson eﬀect is characterized by the zero-voltage c urrent through a thin in-\nsulating barrier sandwiched by two superconductors.1)This eﬀect is a macroscopic quan-\ntum phenomenon involving phase coherence between two super conductors. When a ﬁnite\nvoltage(V)-drop appears in the junction, the diﬀerence in the phase of s uperconducting order\nparameter, i.e. Josephson-phase (θ), oscillates with time according to ∂θ/∂t= (2e//planckover2pi1)V, and\nthe alternating current with frequency (2 e//planckover2pi1)Vﬂows in the junction. This ac Josephson eﬀect\nis derived by the gauge invariance including θ. The electromagnetic response dominated by θ\nshows a resonant behavior in the junction. When a dc magnetic ﬁeld and the dc voltage are\napplied to the junction, the electromagnetic (EM) ﬁeld is ge nerated by spatially modulated\nac Josephson current. In this case, the current-voltage ( I-V) curve exhibits resonant peaks\ndue to the resonance between the ac Josephson current and the EM ﬁeld generated by the\nspatially modulated ac Josephson current itself. This is ca lledFiske resonance .2–6)\nIn recent years, a ferromagnetic Josephson junction compos ed of ferromagnetic metal (F)\n1/16J. Phys. Soc. Jpn. Full Paper\nand superconductors (S’s), i.e., S/F/S junction, has recei ved much attention.7–10)One of the\ninteresting eﬀects is the formation of πstate arising from the Zeeman splitting in F. In ad-\ndition, the interaction between Cooper pairs and spin waves in F is also of importance in\nthe transport properties in the S/F/S junction.11–19)In a small junction, where the junction\nwidth is smaller than the Josephson penetration depth, the s pin-wave excitation induced by\nthe ac Josephson eﬀect is observed.18)In the recent experiment in a S/F/S junction includ-\ning a nonmagnetic insulator (I) in one of interfaces between S and F, it has been reported\nthat the Fiske resonance has multiple structures that must b e associated with the spin-wave\nexcitation.20)Volkovet al. have theoretically studied collective excitations in suc h a junc-\ntion and reported an additional structure in the Fiske reson ance induced by spin-waves.19)In\ntheir theory, a nonmagnetic insulator is crucial to obtain t he Fiske resonance coupled with\nspin-waves. On the other hand, another type of ferromagneti c Josephson junction composed\nof ferromagnetic insulator (FI) and two S’s, i.e., S/FI/S ju nction, is also expected to show the\nsimilar multiple structures in the Fiske resonance. It has b een reported that the dissipation\neﬀect in the S/FI/S junction is smaller than that in the S/F/S j unction.21,22)Such a small\ndissipation in theS/FI/S junction is dueto thesmall probab ility of quasi-particle excitation in\nthe FI.21,22)The damping of spin-waves induced by the similar mechanism i s also very small\nin the FI compared to the case in F.23,24)Therefore, the coupling between Josephson-phase\nand spin-waves can be observed more clearly in the S/FI/S jun ction.\nIn this paper, we theoretically study a composite excitatio n of the Josephson-phase\nand spin-waves in the S/FI/S and S/I/FI/S junctions. First, we calculate the dynamics\nof Josephson-phase coupled with spin waves by using Maxwell and Landau-Lifshitz-Gilbert\n(LLG) equations. Second, we derive the dc Josephson current induced by the Fiske resonance.\nIn the S/FI/S junction, two resonant peaks appear in a curren t-voltage curve for each mode\nof the EM ﬁeld. These two resonant peaks may be associated wit h the direct coupling between\nspin-waves and the EM ﬁeld inside the junction. We also discu ss the Fiske resonance in the\nS/I/FI/S junction. The non-magnetic high resistive layer i s sometimes important, since the\nmagnetic dead layer exist in the ferromagnetic insulator ne ar the S/FI interface. Our results\nclearly show the diﬀerence between S/FI/S and S/I/FI/S junct ions in the dispersion relations\nof the Fiske resonance. In such a S/I/FI/S junction, we show t hat three resonant peaks appear\nin theI-Vcurve for each mode of the EM ﬁeld.\nThe rest of this paper is organized as follows. In Sec. II, by c ombining the Maxwell and\nLLG equations in a S/FI/S junction, we formulate the dc Josep hson current induced by the\nFiskeresonance. InSec.III,theFiskeresonanceisdiscuss edinS/FI/SandS/I/FI/Sjunctions.\nSummary is given in Sec. IV.\n2/16J. Phys. Soc. Jpn. Full Paper\n2. Formulation of Fiske resonance in S/FI/S junction\nThe system considered is a Josephson junction with a FI sandw iched by two s-wave\nsuperconductors (S’s) as shown in Fig. 1. The magnetization in the FI is parallel to the\nz-direction.18)A uniform dc magnetic ﬁeld is applied in the x-direction. In the measurement\nof the Fiske resonance, the dc magnetic ﬁeld is smaller than s everal tens of gauss. Therefore,\nwe can neglect the in-plane magnetization induced by the app lied dc magnetic ﬁeld. Here, we\nconsider that theac electric and magnetic ﬁeldsare in the z- andx-direction respectively, both\nof which are uniform in the x-direction. We consider the situation, in which the z-dependence\nof the electric and magnetic ﬁelds in the FI is negligible due to the very thin thickness of\nthe FI (dFI). In the S regions, it is assumed that the magnetic ﬁeld depen ds ony- andz-\ncomponent. The current density has a nonzero y-component in the superconducting regions\n(Meissner current) and a nonzero z-component in the ferromagnetic region (quasi-particle an d\nJosephson currents). Based on the above assumptions, the Ma xwell equation in each region\nis given by\nrot[Ez(y,t)ez]=−∂\n∂t[µ0Hx(y,z,t)+Mx(y,t)]ex, (1)\nrot[Hx(y,z,t)ex]=Jy\nM(y,t)ey, (2)\nrot[Hx(y,t)ex]−∂\n∂t[Dz(y,t)ez]=Jz\nJ(y,t)ez+Jz\nQ(y,t)ez, (3)\nMx(y,t)=/integraldisplay∞\n−∞dy′dt′χx(y−y′,t−t′)Hx(y′,t′), (4)\nJz\nJ(y,t)=Jcsinθ(y,t), (5)\nJz\nQ(y,t)=1\nRFIEz(y,t). (6)\nHere,ei(i=x,y,z) is a unit vector, Ez(y,t) is the electric ﬁeld in the FI, Hx(y,z,t)\nandHx(y,t) are the magnetic ﬁelds in the S and the FI, respectively. The electrical ﬂux\ndensity,Dz(y,t), in FI is given by Dz(y,t) =ǫFIEz(y,t), where ǫFIis the dielectric constant\nin FI.Jz\nJ(y,t) andJcare the Josephson current and the Josephson critical curren t densities,\nrespectively. Jz\nQ(y,t) andRFIare the quasi-particle current density and the resistivity of FI,\nrespectively. The x-component of the magnetization, Mx(y,t), in the FI is given by Eq. (4).\nThe motion of magnetization is described by the Landau-Lifs hitz-Gilbert (LLG) equation,23)\ndM\ndt=−γM×Heﬀ+α\nM/bracketleftbigg\nM×dM\ndt/bracketrightbigg\n, (7)\nwhereMisthemagnetization ofFI, γisthegyromagnetic ratio, and αistheGilbertdamping.\nThe eﬀective ﬁeld, to which Mresponds, is given by Heﬀ.\nBy using Maxwell and LLG equations, we can obtain the voltage coupled with spin-waves\n(the detail of derivation for Eq. (8) is given in Appendix A.) as follows:\n∂2V(y,t)\n∂y2=1\nc2\nFI/bracketleftbigg∂2V(y,t)\n∂t2+dFI\ndFI+2λL1\nµ0/integraldisplay∞\n−∞dy′dt′χ(y−y′,t−t′)∂2V(y′,t′)\n∂t2\n3/16J. Phys. Soc. Jpn. Full Paper\nWdFI \nLz// M\ny\nx// Hex \nMS\nSFI\nFig. 1. (Coloronline)SchematicﬁgureofaS/FI/Sjunctionwithferr omagneticinsulator(FI)between\ntwo superconductors (S’s). dFIis the thickness of FI. LandWare the widths of the junction. M\nandHexare the magnetization in the FI and the applied dc magnetic ﬁeld, resp ectively.\n+ ΓFI∂V(y,t)\n∂t+ΓFIdFI\ndFI+2λL1\nµ0/integraldisplay∞\n−∞dy′dt′χ(y−y′,t−t′)∂V(y′,t′)\n∂t/bracketrightbigg\n+1\nλ2\nJJcJz\nJ(y,t)+1\nλ2\nJJcdFI\ndFI+2λL1\nµ0/integraldisplay∞\n−∞dy′dt′χ(y−y′,t−t′)Jz\nJ(y′,t′),(8)\nwherecFI=/radicalbig\ndFI/[(dFI+2λJ)ǫFIµ0],λJ=/radicalbig\n/planckover2pi1/[2eµ0(dFI+2λJ)Jc], and Γ FI= (ǫFIRFI)−1\nare the eﬀective velocity of light in the FI, the Josephson pen etration depth, and the damping\nfactor caused by quasi-particle resistivity, RFI, in the FI, respectively.\nWe look for the solution of Eq. (8) in the form\nV(y,t) =V0+v(y,t), (9)\nwhereV0andv(y,t) are the dc bias voltage and ac voltage induced by the ac Josep hson\ncurrent, respectively. In this case, the phase diﬀerence, θ(y,t), between two S’s is given by\nθ(y,t) =ωJt−kHy+θ1(y,t), (10)\nwhereωJ= (2e//planckover2pi1)V0is the Josephson frequency, kH= 2πµ0dFIHex/Φ0depends on the exter-\nnal magnetic ﬁeld, Hex, and Φ 0is the magnetic ﬂux quantum. θ1(y,t) is related to v(y,t) by\nthe equation,\nv(y,t) =/planckover2pi1\n2e∂θ1(y,t)\n∂t. (11)\nSubstituting Eq. (9) and Eq. (11) into Eq. (8), we obtain the e quation for θ1(y,t) as follows:\n∂2θ1(y,t)\n∂y2=1\nc2\nFI/bracketleftbigg∂2θ1(y,t)\n∂t2+dFI\ndFI+2λL1\nµ0/integraldisplay∞\n−∞dy′dt′χ(y−y′,t−t′)∂2θ1(y′,t′)\n∂t2\n4/16J. Phys. Soc. Jpn. Full Paper\n+ ΓFI∂θ1(y,t)\n∂t+ΓFIdFI\ndFI+2λL1\nµ0/integraldisplay∞\n−∞dy′dt′χ(y−y′,t−t′)∂θ1(y′,t′)\n∂t/bracketrightbigg\n+1\nλ2\nJJcJz\nJ(y,t)+1\nλ2\nJJcdFI\ndFI+2λL1\nµ0/integraldisplay∞\n−∞dy′dt′χ(y−y′,t−t′)Jz\nJ(y′,t′).(12)\nWe expand θ1(y,t) in terms of the normal modes of the electromagnetic ﬁeld gen erated by\nthe ac Josephson current,\nθ1(y,t) = Im/bracketleftBigg∞/summationdisplay\nn=1gneiωJtcos(kny)/bracketrightBigg\n, (13)\nwheregnis acomplex numberand kn=nπ/L. Thisequation of θ1(y,t) satisﬁes [ ∂θ1/∂y]y=0=\n[∂θ1/∂y]y=L= 0, which corresponds to the open-ended boundary condition forv(y,t). We\nconsider θ1(y,t) to be a small perturbation and solve Eq. (13) by taking Jz\nJ(y,t) to be\nJcsin(ωJt−kHy). Substituting Eq. (13) into Eq. (12), gnbecomes (see Appendix B )\ngn=−c2\nFI\nλJµ(kH,−ωJ)Bn−iCn\nω2n−µ(kn,−ωJ)ω2\nJ+iΓFIµ(kn,−ωJ), (14)\nBn=2\nL/integraldisplayL\n0dycos(kny)cos(kHy), (15)\nCn=2\nL/integraldisplayL\n0dycos(kny)sin(kHy), (16)\nµ(q,−ωJ)=1+χx(q,−ωJ)dFI/[(dFI+2λJ)µ0], (17)\nwhereωn= (cFIπ/L)n, andqmeanskHorkn. In the linearized LLG equation, the magnetic\nsusceptibility in the FI is given by (see Appendix C)\nχx(q,ωJ) =γMzΩS+iαωJ\nΩ2\nS−(1+α2)ω2\nJ+i2αΩSωJ. (18)\nHere, Ω Sis spin wave frequency whose dispersion relation is given by\nΩS=ΩB+η\n/planckover2pi1q2, (19)\nwhere Ω B=γ(HK−Mz/µ0).HKandηare the anisotropic ﬁeld and the stiﬀness of spin waves\nin the FI, respectively.\nNext, we calculate the dc Josephson current coupled with spi n waves as a function of\nthe dc voltage and of the external magnetic ﬁeld. The functio n, sin(ωJt−kHy+θ1(y,t)), is\nexpanded with respect to θ1(y,t) and the dc Josephson current is given by\nJdc≈lim\nT→∞1\nT/integraldisplayT\n0dt1\nL/integraldisplayL\n0dyJccos(ωJt−kHy)θ1(y,t). (20)\nIntroducingEqs. (13) and (14) into Eq. (20), the analytic fo rmula of the dc Josephson current\nis obtained as,\nJdc=Jcc2\nFI\n4λ2\nJ∞/summationdisplay\nn=1ΨnF2\nn(φ), (21)\nΨn=Re[µ(kH,ωJ)X], (22)\n5/16J. Phys. Soc. Jpn. Full Paper\n()0 L / / 2 V e ωℏJdc /Jc\n1 2 3012\n n = 1 \n n = 2 \n n = 3 \n n = 4 \n n = 5 \nFig. 2. (Color online) Dc Josephson current density, Jdc, as a function of dc voltage, V0, in a S/FI/S\njunction. The solid line is the total dc Josephson current. Red, blue , green, purple, and light blue\nlines are the dc Josephson current of each mode number, n, of electromagnetic ﬁeld. The applied\ndc magnetic ﬁeld determines nvia Eq. (25).\nX=1\nω2n−µ′(kn,ωJ)ω2\nJ+µ′′(kn,ωJ)ΓFIωJ+i/bracketleftbig\nµ′(kn,ωJ)ΓFIωJ+µ′′(kn,ωJ)ω2\nJ/bracketrightbig,(23)\nµ(q,ωJ)=µ′(q,ωJ)+iµ′′(q,ωJ), (24)\nF2\nn(φ)=/bracketleftbigg2φ\nφ+n/2sin(πφ−nπ/2)\nπφ−nπ/2/bracketrightbigg\n, (25)\nwhereφis equal to Φ /Φ0and Φ = µ0HexdFIL.µ′(q,ωJ) = Re[µ(q,ωJ)] andµ′′(q,ωJ) =\nIm[µ(q,ωJ)] (See Eqs. (17) and (18)).\n3. Results and discussion\nIn this section, we examine the numerical solution for Eq. (2 1). Figure 2 shows the dc\nJosephson current density induced by the Fiske resonance as a function of the dc voltage\nforλJ/L= 1, Ω B/ωL= 3,η/(/planckover2pi1ωL) = 3×10−16m2,γMz/(µ0ωL) = 1,25)α= 1×10−4,24)\ndFI/(dFI+ 2λL) = 0.1, ΓFI/ωL= 3×10−1,26)andωL=cFIπ/L.φis ﬁxed as n/2 in the\nFn(φ) function. In the solid line of Fig. 2, the normalized dc Jose phson current density Jdc\nin Eq. (21) is shown as a function of normalized dc voltage.27)Red, blue, green, purple, and\nlight blue lines are the dc Josephson current of each mode num ber of electromagnetic ﬁeld,\nn. In this Figure, the resonant behavior of Jdcis due to the Fiske resonance in the S/FI/S\njunction. However, at V0/(ωL/planckover2pi1/2e)≈3, it is found that additional structures of Jdcappear.\nFrom Fig. 2, it is found that two resonant peaks appear for eac hn. The appearance of the\ntwo resonant peaks in the S/FI/S junction are very diﬀerent fr om conventional Josephson\njunctions, in which a single resonant peak appears for the ea ch mode of EM ﬁeld.2–6)Large\npeakaround V0/(ωL/planckover2pi1/2e) = 3is duetothesummation of n >8becausedcJosephsoncurrents\nof contribution from large nappear around V0/(ωL/planckover2pi1/2e) = 3 in a manner to be described.\nTo elucidate the origin of the resonant structures in the S/F I/S junction, we analyze\n6/16J. Phys. Soc. Jpn. Full Paper\nn()0 L / / 2 V e ωℏ0V+\n0V−\n12345670246\nFig. 3. (Color online) Dc voltage, V0, as a function of mode number, n, of electromagnetic ﬁeld in\nS/FI/S junction.\nEqs. (21) and (22). When the denominator of Ψ nin Eq. (22) is minimum with respect to\nωJ, Ψntakes a maximum, so that the dc Josephson current shows the re sonant behavior as\nshown in Fig. 2. The dc voltage at which the resonance occurs i s determined by neglecting\nthe damping term in Eq. (22) as α= ΓFI= 0. Setting the denominator of Ψ nto be zero, the\nvoltage is given by\nV±\n0=/planckover2pi1\n2e/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbt1\n2\nω2n+Ω2\nS+dFI\ndFI+2λLγMzΩS\nµ0±/radicalBigg/parenleftbigg\nω2n+Ω2\nS+dFI\ndFI+2λLγMzΩS\nµ0/parenrightbigg2\n−4ω2nΩ2\nS\n,\n(26)\nwhereωnis the frequency of the EM ﬁeld in the FI, and Ω SandγMzΩS/µ0are the frequency\nof spin-waves and the real part of the magnetic susceptibili ty withα= 0 in the FI. We have\ntwodcvoltages, V+\n0andV−\n0, at whichtheFiske resonanceoccursforeach n.From theanalytic\nformula in Eq. (26), it is found that two dispersions result f rom the coupling between the EM\nﬁeld and spin waves in the FI. Figure 3 shows a V0-ncurves obtained by Eq. (26). The vertical\naxis is the dc voltage normalized by ωL/planckover2pi1/2eand the horizontal axis is the mode number of\nEM ﬁeld. In Fig. 3, V+\n0andV−\n0are shown by open circles and open squares, respectively.\nForn <3,V+\n0is nearly constant as a function of n, whereas V−\n0is linear with n. InV+\n0for\nn <3, the voltage is nearly equal to Ω S/planckover2pi1/2e, which relates to the spin wave energy in Fig. 3.\nForn≥3,V+\n0increases as a function of n, whereas V−\n0becomes ﬂat with increasing n. For\nn≥3,V−\n0is nearly equal to Ω S/planckover2pi1/2ein Fig. 3. Therefore, it is found that the ﬂat behavior\n7/16J. Phys. Soc. Jpn. Full Paper\n()0 L / / 2 V e ωℏJdc /Jc(a) \n()0 L / / 2 V e ωℏJdc /Jc(b) \n2.6 2.8 3 3.2 3.4 012\n ΓFI /ωL = 0.5 \n ΓFI /ωL = 1 \n2.6 2.8 3 3.2 3.4 012\n α = 1 ×10 -4 \n α = 1 ×10 -2 \nFig. 4. (Color online) (a) Dc Josephsoncurrent density, Jdc, as a function ofdc voltage, V0, by chang-\ning ΓFI/ωLin a S/FI/S junction. (b) Jdcas a function of V0, by changing αin a S/FI/S junction.\nWhere parameters used by numerical calculation are λJ/L= 1, Ω B/ωL= 3,γMz/(µ0ωL) = 1,\ndFI/(dFI+2λL) = 0.1.\nof the voltage comes from the spin-wave excitation in the FI. The spin-wave excitation in the\nFI is induced by the EM ﬁeld generated by the ac Josephson curr ent and the eﬀect of the\nspin-wave excitation is reﬂected in the Fiske resonance in t he S/FI/S junction.\nHere, we examine the Γ FI- andα-dependence of Jdc. Figure 4 (a) shows Jdcas a function\nofV0by changing Γ FI/ωL. From this ﬁgure, it is found that the Fiske resonance withou t\nvery sharp peaks around V0/(ωL/planckover2pi1/2e) = 3 exhibits strong damping by increasing Γ FI/ωL.\nOn the other hand, very sharp peaks around V0/(ωL/planckover2pi1/2e) = 3 almost does not depends on\nΓFI/ωLunlikeanotherresonantpeaks.Next,wefocusonsharppeaks aroundV0/(ωL/planckover2pi1/2e) = 3.\nFigure4(b) shows Jdcas afunctionof V0by changing α. Thesepeaks around V0/(ωL/planckover2pi1/2e) = 3\ninFig.4(b)stronglydependon αbecausetheseresonantpeaksmainlycomesfromspin-waves.\nFrom Fig. 4, we can easily obtain the Fiske resonance coupled with spin-waves in the S/FI/S\njunction due to the small Γ FIandα.\nThe eﬀect of spin-waves having a ﬁnite wave number qis neglected in the Fiske resonance\nbecause of the following reason: In Eq. (19), the ﬁrst term Ω Bis caused by the anisotropic\nand demagnetizing ﬁelds and ﬁnite wave number qis given by nπ/L. In a conventional FI,\n/planckover2pi1ΩBis about tens of µeV.23)On the other hand, ηq2is of the order peV due to the small\nstiﬀness of spin-waves28)when the width ( L) of the junction is a few mm.\nNext, we consider the Fiske resonance in the S/I/FI/S juncti on. The details of the calcu-\n8/16J. Phys. Soc. Jpn. Full Paper\n12345670246\nn()0 L / /2 V e ωℏ0V+\n0V−\nI\n0V\nFig. 5. (Color online) Dc voltage, V0, as a function of mode number, n, of electromagnetic ﬁeld\nin S/I/FI/S junction. We take parameters as Ω B/ωL= 3,cI/cFI= 0.5,γMz/(µ0ωL) = 1 and\ndFI/(dFI+λL) = 0.2.\nlation are given by Appendix D. To analyze the origin of the Fi ske resonance accompanied\nby the spin-wave excitation in the S/I/FI/S junction, we exa mine Eqs. (D ·15) and (D ·17).\nThe condition of the resonance is given by the minimum in the d enominator. To ﬁnd out the\nvoltage at which the Fiske resonance occurs, we neglect the d amping term in the denominator\nof Eqs. (D ·15) and (D ·17). As a result, the voltages at the Fiske resonance are give n by\nVI\n0=/planckover2pi1\n2eωI\nn, (27)\nV±\n0=/planckover2pi1\n2e/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbt1\n2\nω2n+Ω2\nS+dFI\ndFI+λLγMzΩS\nµ0±/radicalBigg/bracketleftbigg\nω2n+Ω2\nS+dFI\ndFI+λLγMzΩS\nµ0/bracketrightbigg2\n−4ω2nΩ2\nS\n,\n(28)\nindicating that there are three dc voltages resulting in the Fiske resonance for each nin the\nS/I/FI/S junction. Figure 5 shows the V0-ncharacteristic obtained by Eqs. (27) and (28).\nThe vertical axis is the dc voltage normalized by ωL/planckover2pi1/2eand the horizontal axis is the mode\nnumber of EM ﬁeld. In Fig. 5, VI\n0,V+\n0andV−\n0are shown by open triangles, open circles\nand open squares, respectively. VI\n0linearly increases as a function of nas shown in Fig. 5\nand comes from the resonance between the EM ﬁeld in the I and th e ac Josephson current.\nForn <3,V+\n0is nearly ﬂat as a function of n, whereas V−\n0shows linear behavior with n.\nForn≥3,V+\n0increases as a function of n, whereas V−\n0becomes ﬂat with increasing n. The\nvoltage of ﬂat region as a function of nis nearly equal to Ω S/planckover2pi1/2ein Fig. 5.\nIn this paper , we assumed that the magnetization of FI is a sin gle domain structure.\n9/16J. Phys. Soc. Jpn. Full Paper\nUsually, magnetization structure in the FI possesses a comp licated domain structure. The\nferromagnetwithmultidomainstructurehasmultiplemagne ticresonancemodesdiﬀerentfrom\nthesingledomainsystem.29)Therefore,wecanexpectseveral additional Fiskeresonanc epeaks\nintheI-VcurvearisingfromthedomainstructureintheFI.Thosedoma instructureandtheir\nspin dynamics, which depend on our choice of material, have r ich variety and wide spectrum.\nHowever, it is diﬃcult to include the multidomain structure of FI in the present theory. The\nFiske resonance in the ferromagnetic Josephson junction wi th the magnetic domains is beyond\nthe scope of the present paper, and will be studied in another one.\n4. Summary\nWe have theoretically studied the coupling of Josephson-ph ase and spin-waves in the\nS/FI/S junction. The dc Josephson current induced by the Fis ke resonance is calculated by\ncombining the Maxwell and the Landau-Lifshitz-Gilbert (LL G) equations. The dc Josephson\ncurrent shows the resonant behavior as a function of the appl ied dc voltage. We derived the\nanalytic formula of the resonant condition in the Fiske reso nance in the S/FI/S junction. Two\nresonant peaks appear in the I-Vcurve for each mode number of EM ﬁeld. We found that\nthe two resonant peaks are generated by the coupling between spin-waves and the EM ﬁeld\nin the FI. We have also studied the Fiske resonance in the S/I/ FI/S junction and found that\nadditional resonant structures appear due to the coupling b etween the ac Josephson current\nand the EM ﬁeld in the I layer.\nAcknowledgments\nThe authors thank M. Aprili and T. Koyama for valuable discus sions and comments.\nThis work is supported by Grant in Aid for Scientiﬁc Research and the next Generation\nSupercomputer Project from MEXT.\n10/16J. Phys. Soc. Jpn. Full Paper\nAppendix A: Derivation of Eq. (8)\nWe integrate Eq. (1) with a narrow stripe Awith inﬁnitesimal width ( dy) in theyz-plane,\n/integraldisplay\nAdS∂\n∂yEz(y,t) =−/integraldisplay\nAdS∂\n∂t[µ0Hx(y,z,t)+Mx(y,t)], (A·1)\nwhereEz(y,t) is conﬁned to the ferromagnetic layer due to vanishing Ez(y,t) in the S. Inte-\ngrating Eq. (A ·1) with respect to yand introducing the London penetration depth λLdeﬁned\nby\nλL=1\nHx(y,t)/integraldisplay±∞\n±dFI/2dzHx(y,z,t), d FI≪λL, (A·2)\nEq. (A·1) becomes\n−dFI∂\n∂yEz(y,t) =µ0∂\n∂tHx(y,t)(dFI+2λL)+dFI∂\n∂tMx(y,t). (A·3)\nIn the same way, we integrate Eq. (2) over the cross-section a reaS′of the junction in the\nxz-plane and obtain\nHx(y,t) =1\nWIM(y,t), I M(y,t)≡/integraldisplay\nS′dSJy\nM(y,t), (A·4)\nwhereWis the width along the x-axis of the junction and IM(y,t) is the current at position\nyin the superconducting electrode. Substituting Eq. (A ·4) into Eq. (A ·3) and Eq. (4), we\nobtain the partial diﬀerential equation\n∂\n∂yV(y,t) =−µ0\nW(dFI+2λL)∂\n∂tIM(y,t)+dFI\nW∂\n∂t/integraldisplay∞\n−∞dy′dt′χ(y−y′,t−t′)IM(y′,t′),(A·5)\nwhereV(y,t)≡dFEz(y,t) is the voltage across the FI. The Ampere’s law in the FI is\n∂\n∂yIM(y,t) =−ǫFI\ndFIW∂\n∂tV(y,t)−WJz\nJ(y,t)−WJz\nQ(y,t) (A ·6)\nDiﬀerentiating partially Eq. (A ·5) with respect to y, Eq. (A·5) becomes\n∂2\n∂y2V(y,t) =−L′∂\n∂t∂\n∂yIM(y,t)−L′dFI\ndFI+2λL1\nµ0∂\n∂tF(y,t). (A·7)\nwhere\nF(y,t) =∂\n∂y/integraldisplay∞\n−∞dy′dt′χ(y−y′,t−t′)IM(y′,t′). (A·8)\nTo obtain the voltage equation coupled with spin waves in the FI, we transform the function\nF(y,t). When we execute the Fourier transformation on χ(y−y′,t−t′) andIM(y′,t′) referring\ntoy′,F(y,t) is given by\nF(y,t)=1\n2π/integraldisplay∞\n−∞dqdt′eiqyχ(q,t−t′)iqIM(q,t′). (A·9)\nMaking use of the Fourier transformation of yin Eq. (A ·6),\niqIM(q,t) =−C′∂\n∂tV(q,t)−WJz\nJ(q,t)−WJz\nQ(q,t), (A·10)\n11/16J. Phys. Soc. Jpn. Full Paper\nand substituting Eq. (A ·10) into Eq. (A ·9),F(y,t) is given by\nF(y,t)=−/integraldisplay∞\n−∞dy′dt′χ(y−y′,t−t′)/bracketleftbigg\nC′∂\n∂t′V(y′,t′)+WJz\nJ(y′,t′)+WJz\nQ(y′,t′)/bracketrightbigg\n,(A·11)\nWhereC′=ǫFIW/dFI. Substituting Eq. (A ·11) into Eq. (A ·7) and using the relation V(y,t) =\ndFIEz(y,t), we have the partial diﬀerential equation\n∂2V(y,t)\n∂y2=1\nc2\nFI/bracketleftbigg∂2V(y,t)\n∂t2+dFI\ndFI+2λL1\nµ0∂\n∂t/integraldisplay∞\n−∞dy′dt′χ(y−y′,t−t′)∂V(y′,t′)\n∂t′\n+ ΓFI∂V(y,t)\n∂t+ΓFIdFI\ndFI+2λL1\nµ0/integraldisplay∞\n−∞dy′dt′χ(y−y′,t−t′)V(y′,t′)/bracketrightbigg\n+/planckover2pi1\n2eλ2\nJJcJz\nJ(y,t)+/planckover2pi1\n2eλ2\nJJcdFI\ndFI+2λL1\nµ0/integraldisplay∞\n−∞dy′dt′χ(y−y′,t−t′)Jz\nJ(y′,t′),(A·12)\nAppendix B: Calculation of factor gn\nTo obtain the complex coeﬃcient gn, substituting Eq. (13) and JJ≈Jcsin(ωJt−kHy) into\nEq. (12), we obtain the equation for gn,\n∞/summationdisplay\nn=1gneiωJtcos(kny)/bracketleftbig\n−ω2\nn+µ(kn,−ωJ)ω2\nJ−iΓFIµ(kn,−ωJ)ωJ/bracketrightbig\n=c2\nFI\nλ2\nJµ(kH,−ωJ)eiωJt−ikHy,(B·1)\nwhereωn= (cFIπ/L)nandµ(q,−ωJ) = 1+χ(q,−ωJ)dFI/[(dFI+2λJ)µ0] andqdenoteskHor\nkn. For both sides of Eq. (B ·1), multiplying cos( kny) and performing integration with respect\ntoyfrom 0 to L, we getgnas follows:\ngn=−c2\nFI\nλJµ(kH,−ωJ)Bn−iCn\nω2n−µ(kn,−ωJ)ω2\nJ+iΓFIµ(kn,−ωJ), (B·2)\nBn=2\nL/integraldisplayL\n0dycos(kny)cos(kHy), (B·3)\nCn=2\nL/integraldisplayL\n0dycos(kny)sin(kHy). (B·4)\nAppendix C: Linearized solution of LLG equation\nWe outline the derivation of the magnetic susceptibility in the FI by using the LLG\nequation. We consider the situation, in which the direction of magnetization is perpendicular\nto the junction. In the experimental measurement of the Fisk e resonance, the dc magnetic\nﬁeld is smaller than several tens of gauss. Therefore, we can neglect the gradient of the\nmagnetization due to the applied dc magnetic ﬁeld. Since the precessional angle of spin in\nthe FI is usually very small even at the magnetic resonance, w e linearize the LLG equation\nas follows:\ndMx\ndt=−γMy(HK−Mz/µ0)+γMzΛ∂2\nyMy−αdMy\ndt, (C·1)\ndMy\ndt=γMx(HK−Mz/µ0)−γMzhx−γMzΛ∂2\nyMx+αdMx\ndt, (C·2)\n12/16J. Phys. Soc. Jpn. Full Paper\nwhereHKandMz/µ0are an anisotropic ﬁeld and a demagnetization ﬁeld in the FI, respec-\ntively. We assume the solutions of Eq. (C ·1) and (C ·2) of the form\nMi=χi(q,ωJ)hxeiωJte−iqy, i=x,y, (C·3)\nwhereχi(q,ωJ) is a magnetic susceptibility of FI and hxis the amplitude of x-component of\nac magnetic ﬁeld and qis a wave number of spin wave. Substituting Eq. (C ·3) into Eq. (C ·1)\nand (C·2), the solutions of the linearized LLG equation are\nMx=χx(q,ωJ)hx, (C·4)\nχx(q,ωJ)=γMzΩS+iαωJ\nΩ2\nS−(1+α2)ω2\nJ+i2αΩSωJ, (C·5)\nMy=χy(q,ωJ)hx, (C·6)\nχy(q,ωJ)=−γMziωJ\nΩ2\nS−(1+α2)ω2\nJ+i2αΩSωJ, (C·7)\nΩS=ΩB+η\n/planckover2pi1q2,ΩB=γ(HK−Mz/µ0), (C·8)\nwhereHKis an anisotropic ﬁeld in the FI and ηis a stiﬀness of spin waves.\nAppendix D: Derivation of dc Josephson current in S/I/FI/S junction\nIn this Appendix, we consider the Josephson junction includ ing a nonmagnetic insulator\n(I) between the S and the FI, that is the S/I/FI/S junction.\nTo calculate the Josephson current in the S/I/FI/S junction , we solve the Maxwell equa-\ntion as shown in the section II to add to equations describing the I as follows:\nrot/bracketleftbig\nEI\nz(y,t)ez/bracketrightbig\n=−µ0∂\n∂t/bracketleftbig\nHI\nx(y,z,t)/bracketrightbig\nex, (D·1)\nrot/bracketleftbig\nHI\nx(y,t)ex/bracketrightbig\n−ǫI∂\n∂t/bracketleftbig\nEI\nz(y,t)ez/bracketrightbig\n=Jz\nJ(y,t)ez+σIEI\nz(y,t)ez, (D·2)\nwhere the subscript I in the above equations indicates ﬁelds in the non-magnetic insulator. ǫI\nandσIare a dielectric constant and a conductivity in the I, respec tively.Jz\nJis a Josephson\ncurrent. In the same manner as the calculation in the section II, we integrate the Faraday’s\nlaw with respect to the plane parallel to the yz-plane and the Ampere’s law with respect to\nthe plane parallel to the xz-plane. As a result, Ampere’s and Faraday’s laws are given by\n−∂EFI\nz(y,t)\n∂ydFI−∂EI\nz(y,t)\n∂ydI=µ0∂HFI\nx(y,t)\n∂t(dFI+λL)+∂Mx(t)\n∂tdFI\n+µ0∂HI\nx(y,t)\n∂t(dI+λL), (D·3)\nHi\nx(y,t)=1\nWIi\nL, i= FI or I . (D·4)\nThe deﬁnition of Ii\nLis same as that of Eq. (A ·4). Performing the procedure of calculation to\nobtain Eq. (8) for Eqs. (D ·3) and (D ·4), we obtain equations describing ac voltages induced\n13/16J. Phys. Soc. Jpn. Full Paper\nby the ac Josephson current in the I and the FI layers as,\n∂2VI(y,t)\n∂y2=1\nc2\nI∂2VI(y,t)\n∂2t+1\nc2\nIΓI∂VI(y,t)\n∂t+/planckover2pi1\n2e1\nλ2\nIJc∂JJ(y,t)\n∂t, (D·5)\nc−2\nI=dI+λL\ndIǫIµ0, λ−2\nI=2e\n/planckover2pi1µ0(dI+λL),ΓI=1\nǫIRI, (D·6)\n∂2VFI(y,t)\n∂y2=1\nc2\nFI∂2VFI(y,t)\n∂2t+1\nc2\nFIdFI\ndFI+λL∂\n∂t/integraldisplay∞\n−∞dt′χ(t−t′)∂VFI(y,t′)\n∂t\n+1\nc2\nFIΓFI∂VFI(y,t)\n∂t+∂\n∂t/integraldisplay∞\n−∞dt′χ(t−t′)VFI(y,t′)\n+/planckover2pi1\n2e1\nλ2\nFIJc∂JJ(y,t)\n∂t+/planckover2pi1\n2e1\nλ2\nFIJcdFI\ndFI+λL∂\n∂t/integraldisplay∞\n−∞dt′χ(t−t′)JJ(y,t′),(D·7)\nc−2\nFI=dFI+λL\ndFIǫFIµ0, λ−2\nFI=2e\n/planckover2pi1µ0(dFI+λL),ΓFI=1\nǫFIRFI, (D·8)\nwheredIis the thickness of I. ǫIandRIare a dielectric constant and resistance per unit\nlength in the I. Here, we adopt the iterative calculation to o btain the dc Josephson current\ndue to the resonance between the electromagnetic ﬁeld and Jo sephson current as shown in\nprevious section. First, we calculate the voltage induced b y the ac Josephson current in each\nlayers. Josephson current is characterized by the phase diﬀe rence between two S’s. Therefore,\nwe assume that the Josephson current ﬂowing layers is given b yJJ(y,t) =Jcsin(ωJt−kHy),\nwherekH= 2πµ0Hex(dFI+dI)/Φ0andωJis the Josephson frequency. For the boundary\ncondition, we adopt the open-ended boundary condition for t he reason mentioned in section\nII. In this case, the voltage expression satisfying the open -ended boundary condition is given\nby\nVj(y,t) = Im/bracketleftBigg∞/summationdisplay\nn=1vj\nneiωJtcos(kny)/bracketrightBigg\n, j= I or FI , (D·9)\nwherevj\nnare complex numbers and kn=nπ/L. Substituting Eq. (D ·9) into Eqs. (D ·5) and\n(D·7), we can obtain the voltage in I and F layers as follows:\nVI(y,t)=/planckover2pi1\neL/parenleftbiggcI\nλI/parenrightbigg2∞/summationdisplay\nn=1/bracketleftBigg\n−ΓIω2\nJBn+ωJ/bracketleftbig\nω2\nJ−(ωI\nn)2/bracketrightbig\nCn/bracketleftbig\nω2\nJ−(ωIn)2/bracketrightbig2+(ΓIωJ)2sin(ωJt)\n+ωJ/bracketleftbig\nω2\nJ−(ωI\nn)2/bracketrightbig\nBn+ΓIω2\nJCn\n(ωIn−ωJ)2+(ΓIωJ)2cos(ωJt)/bracketrightBigg\ncos(kny), (D·10)\nVFI(y,t)=−/planckover2pi1\neL/parenleftbiggcFI\nλI/parenrightbigg2∞/summationdisplay\nn=1[ℜ[g(ωJ)sin(ωJ)+ℑ[g(ωJ)]cos(ωJt))]cos(kny),(D·11)\ngn=−/parenleftbiggcFI\nλJ/parenrightbigg2\nµ(−ωJ)Bn−iCn\nω2n−µ(−ωJ)ω2\nJ+iΓFIµ(−ωJ), (D·12)\nwhereωI\nnandωnare given by ( cIπ/L)nand (cFIπ/L)n, respectively. BnandCnare same\nexpressions as Eq. (15) and Eq. (16), respectively.\n14/16J. Phys. Soc. Jpn. Full Paper\nNext we calculate the dc Josephson current coupled with spin waves as a function of the\ndc voltage and of the external magnetic ﬁeld. Since we consid erVI(FI)(y,t) as a perturbation,\nwe can expand the sine function in terms of VI(FI)(y,t). Within the ﬁrst order term with\nrespect to VI(FI)(y,t), the dc Josephson current is approximately given by\nJdc≈lim\nT→∞1\nT/integraldisplayT\n0dt1\nL/integraldisplayL\n0dyJccos(ωJt−kHy)2π\nΦ0/integraldisplay\ndt/bracketleftbig\nVI(y,t)+VFI(y,t)/bracketrightbig\n.(D·13)\nSubstituting Eqs. (D ·10) and (D ·11) into Eq. (D ·13), we can obtain the analytic formula of\nthe dc Josephson current in the S/I/FI/S junction as follows :\nJdc=JI\ndc+JFI\ndc, (D·14)\nJI\ndc=Jcc2\nI\n4λ2\nI∞/summationdisplay\nn=1ΓIωJ/bracketleftbig\n(ωIn)2−ω2\nJ/bracketrightbig2+(ΓIωJ)2F2\nn(φ), (D·15)\nJFI\ndc=Jcc2\nFI\n4λ2\nFI∞/summationdisplay\nn=1ΨF\nnF2\nn(φ), (D·16)\nΨF\nn=µ′(ωJ)/bracketleftbig\nµ′(ωJ)ΓFIωJ+µ′′(ωJ)ω2\nJ/bracketrightbig\n+µ′′(ωJ)/bracketleftbig\nω2\nn−µ′(ωJ)ω2\nJ+µ′′(ωJ)ΓFIωJ/bracketrightbig\n/bracketleftbig\nω2n−µ′(ωJ)ω2\nJ+µ′′(ωJ)ΓFIωJ/bracketrightbig2+/bracketleftbig\nµ′(ωJ)ΓFIωJ+µ′′(ωJ)ω2\nJ/bracketrightbig2,\n(D·17)\nwhereωI\nnandωnare given by ( cIπ/L)nand (cFIπ/L)n, respectively. Fn(φ) is same function as\nEq. (25). Jdc\nJIcomes from the resonance between the electromagnetic ﬁeld g enerated inside the\nI and the ac Josephson current. Jdc\nJFIoriginates in the resonance between the electromagnetic\nﬁeld generated inside the FI and the ac Josephson current.\n15/16J. 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) Z. Nussinov, A. Shnirman, D. P. Arovas, A. V. Balatsky, and J. X. Zhu, Phys. Rev. B 71(2005)\n214520.\n12) S.Takahashi, S.Hikino, M.Mori, J.Martinek and S.Maekawa, Ph ys.Rev.Lett. 99(2007) 057003.\n13) C. Bell, S. Milikisyants, M. Huber, and J. Aarts, Phys. Rev. Lett .100(2008) 047002.\n14) M. Houzet, Phys. Rev. Lett. 101(2008) 057009.\n15) S. Hikino, M. Mori, S. Takahashi, and S. Maekawa, J. Phys. Soc. Jpn.77(2008) 053707.\n16) F. Konschelle and A. Buzdin, Phys. Rev. Lett. 102(2009) 017001.\n17) T. Yokoyama and Y. Tserkovnyak, Phys. Rev. B 80(2009) 104416.\n18) I. Petkovi´ c, M. Aprili, S. E. Barnes, F. Beuneu, and S. Maekawa, Phys. Rev. B 80(2009) 220502.\n19) A. F. Volkov and K. B. Efetov, Phys. Rev. Lett. 103(2009) 037003.\n20) M. Aprili, Spin and Phase dynamics in Ferromagnetic Josephson junctio ns, KN-3-2, ”9th\nInternational Conference on Materials and Mechanisms of Su perconductivity ”, Tokyo Japan.\n21) S.Kawabata, Y.Asano, Y.Tanaka, A.A.Golubov, and S.Kash iwaya,Phys.Rev.Lett. 104(2010)\n117002.\n22) S.Kawabata, S.Kashiwaya, Y.Asano, Y.Tanaka, and A.Golub ov, Phys.Rev.B 74 (2006) 180502\n(R).\n23) B. Hillebrands and K. Ounadjela, Spin dynamics in conﬁned magnetic structures II (Springer-\nVerlag Berlin Heidelberg, New York, 2003)\n24) 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.\n25) D.D.Stancil andA.Prabhakar, Spin Waves Theory and Applications (SpringerScience+Business\nMedia, LLC 2009)\n26) M. P. Lisitskiy and M. V. Fistul, Phys. Rev. B 81 (2010) 184505.\n27) ForL= 1mmandcFI= 107m/s, the magnitude of dc voltage ( V0) is about tens of µV.\n28) M. Pajda, J. Kudrnovsk´ y, I. Turek, V. Drchal, and P. Bruno , Phys. Rev. B 64(2001) 174402.\n29) U. Ebels, P. E. Wigen, and K. Ounadjela, Europhys. Lett. 46(1999) 94.\n16/16"    },    {        "title": "1709.09830v1.Electromagnon_on_the_Surface_of_Magnetic_Topological_Insulator.pdf",        "content": "arXiv:1709.09830v1  [cond-mat.mes-hall]  28 Sep 2017Electromagnon on the Surface of Magnetic Topological Insul ator\nYusuke Hama1and Naoto Nagaosa2, 3\n1National Institute of Informatics, 2-1-2 Hitotubashi, Chi yoda-ku, Tokyo 101-8430, Japan\n2RIKEN Center for Emergent Matter Science (CEMS), Wako, Sait ama 351-0198, Japan\n3Department of Applied Physics, The University of Tokyo, Bun kyo-ku, Tokyo 113-8656, Japan\n(Dated: September 29, 2017)\nWe investigate theoretically the electromagnon on the surf aces of the magnetic topological insulator thin ﬁlms.\nIt is found that when the magnetic asymmetry between the top a nd bottom surfaces is there, the ferromagnetic\nresonance is driven by the electric ﬁeld which is two orders o f magnitude more efﬁcient compared with that by\nthe magnetic ﬁeld. The resonant frequency of the electromag non is also estimated.\nPACS numbers: 75.70.-i, 75.85.+t, 76.50.+g\nI. INTRODUCTION\nNowadays topological insulator (TI) is one of the central\ntopics in condensed matter physics [1–4]. The two major fea-\ntures are topologically nontrivial bulk band structures an d the\nexistence of the gapless surface state whose low-energy eff ec-\ntive Hamiltonian is described by the spin-momentum locked\nWeyl Hamiltonian [2, 5, 6]. The electromagnetic response\nowing to the topological bulk band is called topological mag -\nnetoelectric effect [1–4, 7, 8]. It is described by the axion\nLagrangian density L=θ(α/4π2)E·Bwithα=e2//planckover2pi1c\nthe ﬁne structure constant, θ=π(0)for topologically non-\ntrivial (trivial) insulator. The associated electromagne tism is\ndescribed such that a magnetization is induced by an electri c\nﬁeld as4πM=αEwhile a magnetic ﬁeld generates an elec-\ntric polarization as 4πP=αB. This indicates that the bulk\nTI exhibits the electromagnetism as multiferroics, which a re\nthe materials where the magnetism and ferroelectricity coe x-\nist [9–11]. It clearly represents that the topological magn eto-\nelectric effect is nothing but the linear magnetoelectric e ffect\nwhich is an inherent property of multiferroics. The multife r-\nroics also have interesting magneto-electric optical prop erties\nsuch as the emergence of the electromagnon having a natural\nfrequency in the GHz to THz regime and directional dichro-\nism [11]. Likewise the bulk TI, the surface state of the TI\nalso exibits the physical multi-functionality, e.g., spin -charge\ntransport phenomena, the magnetization dynamics in ferro-\nmagnet coupled TI, and optical phenomena intrinsic to the\nspin-momentum locking [12–24]. We then naturally expect\nthat these rich TI surface properties enable to realize the m ul-\ntiferroic magneto-electric optical phenomena.\nIn this paper, we study the magnon of the ferromagnet on\nthe surface of magnetic TI. Especially, we focus on the thin\nﬁlms of magnetic TI, which are now experimentally realized\nby magnetic doping, for instance, by Cr, V , and Mn ions [25–\n35], or coating the ferromagnetic insulator (FMI) such as Eu S\nto TI (TI/FMI heterosturcture) [34, 36]. The examples of\nthree dimensional TI materials are Bi 2Se3, Bi2Te3, their com-\npounds Bi 2(SexTe1−x)3, and Sb 2Te3.\nWe demonstrate that when an electromagnetic ﬁeld (emf)\nis applied to the magnetic TI and the surface state is in the\nquantized anomalous Hall state, the magnetization couples not\nonly to the magnetic ﬁeld but also to the electric ﬁeld so thatthe magnetization behaves as the electric polarization. In such\ncircumstance, the ferromagnetic resonance (FMR) due to the\nelectric ﬁeld is much stronger than that by the magnetic ﬁeld .\nThis indicates that the magnon as the ﬂuctuation of the elec-\ntric polarization, i.e., the electromagnon, is induced. Su ch\nelectromagnon can be created when the exchange coupling\nbetween the magnetization and surface state is large enough\nand a magnetic anisotropy along the out-of surface plane is\nstrong. We then analyze in what conditions do this electro-\nmagnon emerges in terms of the helicities of the surface stat es,\nmagnetization direction, the exchange coupling signs, and the\nhelicity of the emf. Here we use the terminology “helicity”\nfor the TI surface states to describe how the spin and momen-\ntum are locked. Mathematically, the helicity of the TI surfa ce\nstate is deﬁned by the inner product between momentum vec-\ntor and the vector product of an unit vector perpendicular to\nthe surface and a spin vector whose components are described\nby the Pauli matrices. Two TI surface states, top and bottom\nsurface states, have opposite helicities. For emf ﬁeld we us e\nit to distinguish whether the light is left or right circular ly po-\nlarized.\nWe show that the relevant quantities to create the electro-\nmagnon are the helicities of the surface states, the magne-\ntization direction, and the emf helicity. In other words, th e\nelectromagnon emerges when two surfaces have asymmetric\nmagnetization conﬁguration and the emf applied with the ap-\npropriate combination of the emf helicity and magnetizatio n\ndirection. Such situation can be realized by making the syst em\ninto a semi-magnetic thin ﬁlm or by surface doping using two\ndifferent types of magnetic ions. Furthermore, we analyze t he\nresonant frequency of the electromagnon.\nII. MODEL\nWe ﬁrst focus on one of the surface, say the top surface in\nthe two-dimensional plane x= (x,y)located at z= 0. The\nsurface state is coupled to an applied emf and the magnetiza-\ntion through the exchange coupling. The Hamiltonian of this\nsystem is\nH=vF(σyΠx−σxΠy)−JSna(x,t)σa, (1)\nwherevFis the Fermi velocity of the surface state, σa(a=\nx,y,z)is the Pauli matrices describing the spin, Πj=2\n−i/planckover2pi1∂j+eAj(j=x,y) withpj=−i/planckover2pi1∂jthe momentum\noperator, −e <0is the electron charge, and Ajthe external\nelectromagnetic vector potential, respectively. The summ a-\ntion for the spin index ais taken here. J=J∗n0is the ex-\nchange coupling with J∗being intrinsic to the ferromagnetic\nmaterials and n0=d/(a2az)is the averaged two-dimensional\nsheet density of magnetic ions with athe separation between\nthe localized spin (the lattice constant) in the xyplane,azthe\nlattice constant for the zaxis, and dis the interaction range.\nHere we set az=d.The vector nais a unit vector for the lo-\ncalized spin ﬁeld with Sits spin magnitude. We assume that\nthe ferromagnet has an easy axis parallel to the zdirection\nand treat Szas a constant. We analyze the ﬂuctuations of the\nin-plane magnetization nx,ygenerated by the applied emf.\nFrom Eq. (1), one can deﬁne the generalized vector poten-\ntial as\nAi(x,t) =Ai(x,t)+ai(x,t),(i=x,y) (2)\nwhere we have deﬁned the emergent vector potential\nax(x,t) =−JS\nevFny(x,t), ay(x,t) =JS\nevFnx(x,t).(3)\nWhen the Fermi energy is inside the gap 2∆ M= 2|JSnz|in-\nduced by the z-component of the magnetization, the system\nshows the quantized anomalous Hall effect, and the electro-\nmagnetic response of the system is characterized by the half -\nquantized Hall conductance σH=e2\n2hsgn(Jnz)such that the\nelectric current density takes the form\nJi(x,t) =ǫijσH(Ej+ej), (4)\nwhereEj=−∂tAj(=∂tAj) andej=−∂taj(=∂taj)\nare the external and emergent electric ﬁelds, respectively .\nǫijis the antisymmetric tensor with ǫxy=−ǫyx= 1.\nThe above quantized Hall current originates from the Chern-\nSimons term.\nNow let us focus on the second term of the Hall current\nin Eq. (4) which is described as the response to the emer-\ngent electric ﬁeld. From the relation between the electric p o-\nlarization and the polarization current P=∂jP/∂t, we see\nthat the in-plane magnetization nx,ycan be identiﬁed with the\ntwo-dimensional electric polarization [13]\nPM\ni(x,t) =JSσH\nevFni(x,t). (5)\nThus, Eq. (5) reﬂects that the surface of the magnetic TI ex-\nhibits both the ferromagnetism and the ferroelectricity, i .e., the\nmultiferroicity. Putting the Fermi velocity vF=4×105m/sec\nandJ∗=145meV ·nm2[28], this electric polarization is esti-\nmated as PM\ni∼=7.02×10−30n0SniC/m3when the density\nof the magnetic ions n0is measured in the unit of m−2.\nIII. FERROMAGNETIC RESONANCE AND\nELECTROMAGNON\nWe now study the FMR on the surface of the magnetic TI.\nFor doing so, we apply both the left and right circular polar-\nized light (LCPL and RCPL). We show that the resonance dueto the electric ﬁeld is stronger than that by the magnetic ﬁel d\nimplying the electromagnon creation. We also study the con-\ndition for the emergence of this electromagnon. The resonan t\nfrequency of this magnon is also being estimated.\nThe magnetization in Eq. (5) couples to the applied emf as\nHem=−n0/parenleftbig\nPM\niEi+/planckover2pi1γeSniBi/parenrightbig\n=−n0S/parenleftbiggJ∗σH\nevFEi+/planckover2pi1γeBi/parenrightbigg\nni, (6)\nwhereγe= 1.761×1011rad/T·s is the gyromagnetic ra-\ntio of the electron. Let us estimate the coupling strength be -\ntween the electric ﬁeld and the magnetization for the case of\nmagnetic doping by using the same values for the exchange\ncoupling J∗and the Fermi velocity vFpresented previously.\nThen the coupling strength between the electric ﬁeld and a\nsingle localized spin is estimated as 2.11 ×10−21n0SJ/T·m2\nwhere we used the relation E0x=±cB0yandE0y=∓cB0y\n(the sign - (+) corresponds to the LCPL (RCPL)). In con-\ntrast, the Zeeman interaction strength in Eq. (6) is equal\nto 1.86×10−23n0SJ/T·m2. Thus, we see that the coupling\nstrength due to the electric ﬁeld is much stronger than that\nowing to the magnetic ﬁeld. This indicates that when the emf\nis applied to the surface of the magnetic TI, instead of the\nordinary magnon created by the magnetic ﬁeld the electro-\nmagnon can be dominantly generated due to the nature of the\nspin-momentum locking. Here we note that in Ref. [37], the\nmagnetic surface gap induced by the out-of plane magnetiza-\ntion has been estimated for Mn-doped TI by the ﬁrst princi-\nple calculation which is 16meV . The surface gap reported in\nRef. [28] is around 60meV , and therefore, they are compara-\nble implying that the coupling between the electric ﬁeld and\nthe magnetization niexceeding the Zeeman coupling can also\nbe realized for Mn-doped TI.\nNext we study the dynamics of the magnetization.\nThe Zeeman coupling term is going to be neglected\nsince it is much smaller than the electric-ﬁeld chan-\nnel coupling. When the transverse emf E(x,z,t) =\n(E0x,E0y,0)ei(kz∓ωt),B(x,z,t) = (B0x,B0y,0)ei(kz∓ωt)\nis applied ( kis the wavenumber and ωthe dispersion satis-\nfyingω=ckwithc= 3.0×108m/s the speed of light in\nthe vacuum. The minus (positive) sign in the plane wave de-\nscribes the LCPL (RCPL)), the equation of motion for the ma-\ngentization niis given by\n˙n+−iω0n+=iJeE+\n4π/planckover2pi1vFSC1sgn(Jnz), (7)\nwhere the dot “ ·” represents the time derivative, n+≡nx+\niny,E+≡Ex+iEy, and\nω0=Ka2azn0\nC1, (8)\nC1=/planckover2pi1n0nz\nS+J2\n4π/planckover2pi1v2\nFsgn(Jnz), (9)\nwithKa magnetic anisotropy constant. Equation of motion\n(7) describes a forced oscillation of the in-plane magnetiz a-\ntionniwhere the sum between the Berry phase term and3\nanisotropic energy term play a role of oscillator with a reso -\nnant frequency given by Eq. (8), while the Chern-Simons term\nact as a time dependent external force in terms of the electri c\nﬁeld. By using the physical parameters shown in the previous\ndiscussion, we have /planckover2pi1n0nz/S≃1.29×10−16·(nz/S)J·s/m2\nandJ2/4π/planckover2pi1v2\nF≃3.80×10−18J·s/m2. Here we have used\nproximate values of data for lattice constants of tetradymi tes\nasa=4.2×101/3≃9.05Å, andaz=30×101/3≃6.46×\n10Å[38] with assuming 10% Cr-concentration. Thus, the co-\nefﬁcientC1in Eq. (9) is approximated as C1≈/planckover2pi1n0nz/S,\nand subsequently, for the frequency (8) we obtain ω0≈\nKa2azS//planckover2pi1nz. The LCPL is expressed by E+=E0e−iωt\nand we take the electomagnetic ﬁeld amplitude E0as a time-\nindependent quantity. By assuming the solution for Eq. (7)\nin a form n+= ¯ne−iωt(¯nis time independent) we have the\nsolution\nnt\n+,L=Ct\n2e−iωt\nvF(ω+ωt\n0),withCt\n2=−|J|eE0\n4π/planckover2pi12n0|nz|,(10)\nwhere we introduced the superscript t and the subscript L rep -\nresenting the top surface and the left circular polarizatio n, re-\nspectively. The solution for the RCPL can be obtained by the\nreplacement ω→ −ω.Saying the external frequency ω >0,\nas we see from the approximated form of the resonant fre-\nquency shown above, whether the resonance on the surface\nmagnetic TI occurs or not depends on the emf helicity of the\nlight and the magnetization direction: the FMR by the electr ic\nﬁeld is triggered by the LCPL (RCPL) when the magnetiza-\ntion is pointing toward negative (positive) zdirection.\nNext, let us estimate the resonant frequency of the elec-\ntromagnon f0=ω0/2π. In Ref. [27], the out-of plane\nanisotropic magnetic ﬁeld BKand the saturation magneti-\nzationMSwere measured in the Cr-doped TI thin ﬁlms as\nBK=0.9T and MS=16×103J/T·m3. By using the relation\nK=BKMS/2, the magnetic anisotropic constant becomes\nK=7.2×103J/m3, and the resonant frequency is estimated as\nf0≈5.75×10·(S/nz)GHz.\nUsually the magnetic impurities are doped in the bulk TI\nuniformly, and the magnetization is also realized for the bo t-\ntom surface [25–35]. Thus, we need to discuss whether the\nFMR is triggered or not by the sum between the top and bot-\ntom magnetization, and indeed, this is what we observe in\nthe experiment. The above argument for the top surface can\nbe similarly applied to the analysis for the bottom surface.\nThe magnetization for the bottom surface corresponding to\nEq. (10) is obtained by ﬁrst vF→ −vFsince the helici-\nties of top and bottom surfaces are oppposite. Second, the\nresonant frequency is in the order of 10GHz, and thus, we\nhavekdz=ωdz/c <10−5≪1,and the plane-wave part\nof the emf on the bottom surface can be approximated as\nei(kdz∓ωt)≈e∓iωt.Here we took the thickness of the TI\nthin ﬁlm dz= 100 Å. Then the magnetization on the bottom\ninduced by the emf is obtained by replacing the quantities-\nsuch as magnetic anisotropy energy, the exchange coupling,\nfor the bottom surface ones. The total in-plane magnetizati onni=nt\ni+nf\nifor the LCPL becomes\nn+,L=/bracketleftbiggCt\n2\nω+ωt\n0−Cb\n2\nω+ωb\n0/bracketrightbigge−iωt\nvF, (11)\nwhilen+,Ris obtained by the replacement ω→ −ω.\nLet us study the conditions for the electric-ﬁeld induced\nsurface FMR based on Eq. (11). First, consider the case when\nthe single type of magnetic ions are dopped to the whole TI\nsample uniformly. Then the same magnetization is generated\nfor the top and bottom surfaces where we have Ct\n2=Cb\n2and\nωt\n0=ωb\n0. In this case, the cancellation of the magnetization\noccurs owing to the opposite helicities between the top and\nbottom surface states, and thus, the FMR and the associated\nelectromagnon are not generated. To avoid this circumstanc es\nand generate the FMR and the electromagnon, we next con-\nsider the different ways of magnetic doping. One way is to\ndope the magnetic ions only to the single surface, i.e., the\nsemi-magnetic thin ﬁlm conﬁguration [30]. Then one of the\nsurfaces shows the FMR described by Eq. (10). Another way\nis to perform the magnetic doping using two types of ions, for\ninstance, doping V ions to the top surface while Cr ions to the\nbottom surface [35]. In such circumstance, since Ct\n2/negationslash=Cb\n2\nandωt\n0/negationslash=ωb\n0,there is no magnetization cancellation between\nthe top and bottom surfaces, and we will observe the FMR as\nwell as the electromagnon.\nWe also examine the case for TI/FMI heterosturcture set-\ntings. When EuS, EuO, or YIG are used as FMI, it seems\nto be difﬁcult to realize the FMR and the associated electro-\nmagnon because the magnetic anisotropy is along the in-plan e\ndirection [34, 36] and the exchange coupling might be weak.\nInstead of them, using MnSe might be a good choice. It has\nbeen investigated by the ﬁrst-principle calculation that t he sur-\nface gap about 54meV can be induced with Mn ion having an\neasy axis along the out-of plane [39], while in Ref. [40] the\ngap of the Weyl surface state is 8.5meV . Both values are com-\nparable with the magnetic surface gap obtained in [28] which\nis around 60meV . Although the electronic states and the asso -\nciated ferromagnetism at the interface between the FMI and\nTI are quite complicated in TI/FMI heterosturctured system s,\nwe still believe that the electromagnon on the surface TI can\nalso be created for these systems with such choice.\nConsequently, in order to generate the electric-ﬁeld induc ed\nFMR and electromagnon, we ﬁrst have to prepare the asym-\nmetric magnetization conﬁgurations between two surfaces s o\nas to prevent from the cancellation originating from the op-\nposite TI surface-state helicities. This can be done by eith er\nthe magnetic doping or coating the FMI on one of the surfaces\nof TI. Once such magnetization conﬁgurations are setup, the\nelectromagnon emerges with the proper combinations of the\nmagnetization direction and emf helicity.\nIV . CONCLUSION\nIn this paper we have investigated the FMR on the sur-\nface of TI. We have found that when the exchange coupling\nis large enough around 10meV , the magnetization behaves as4\ntwo-dimensional electric polarization and couples to the e lec-\ntric ﬁeld so that such coupling exceeds the Zeeman coupling.\nThis reﬂects that the surface of the magnetic TI exhibits the\nmultiferroics. As a result, the electric-ﬁeld induced FMR\nand the associated electromagnon as a ﬂuctuation of the two-\ndimensional electric polarization is generated.\nBy deriving the equation of motion (7) and its solution (11),\nwe have carefully analyzed in what conditions does the elec-\ntromagnon emerges by focusing on the magnetization conﬁg-\nuration, the exchange-coupling sign, the surface-state he lici-\nties, and the emf helicity. We demonstrated that to generate\nthe electromagnon we have to create the asymmetry between\nthe top and bottom surfaces. This could be done, for instance ,\nby doping the magnetic ions only to the single surface or usin g\ntwo different kinds of ions. Besides the magnetic doping, th is\nelectric-ﬁeld induced FMR and the associated electromagno n\ncan also be realized by the TI/FMI heterostructure with the\nproper choice of FMI. Finally, we have estimated the resonan t\nfrequency of the electromagnon for a Cr-dopping case with\nthe 10% concentration, which was in the order of 10 GHz. By\nincreasing the magnetic anisotropic energy or the saturati on\nmagnetization about one or two orders than the above case,\nwe may create the electromagnon in the THz regime, imply-\ning that we can perform the ultrafast manipulation of the mag -\nnetization by the electric ﬁeld owing to the spin-momentum\nlocking of the surface TI.\nWe note that in Ref. [14], the Hall-current (or the elec-\ntric ﬁeld) induced magnetization switching, i.e. the inver se\nspin-Galvanic effect, and the associated magnon was pointe d\nout. Further, in Ref. [22] the effect on conductivity due to\nthe electric-ﬁeld induced magnon has been presented. Al-\nthough the electric-ﬁeld induced magnon has been referred\nto in these articles, in our paper we have found some new\ninsights. First we have numerically shown that the couplingbetween the electric ﬁeld and magnetization mediated by the\nTI surface state is stronger than that between the magnetic\nﬁeld and magnetization, i.e. the possibilty of the electric -ﬁeld\ninduced FMR. For doing this we have used the recent experi-\nmental data and compared with the results due to the ﬁrst prin -\nciple caluculation. Second, we then have focused on two sur-\nfaces and examined in what conditions in terms of the magne-\ntization conﬁguration and its direction as well as the surfa ce-\nstate and emf-ﬁeld helicities does the electromagnon emerg e\non TI surfaces. Our result indicates not only the posibility of\nthe electromagnon emergence but may guide to explore the\nhidden electromagnetic and optical properties of surfaces of\nTI as multiferroics, or those of other two-dimensional mult i-\nferroic materials.\nRecently, the FMR has been performed in magnetic TI\n[21, 31]. The condition adopted in these studies, however, a re\ndifferent from ours: The magnetic TI in Ref. [21] consists of\npermalloy (Ni 81Fe19) and TIs where the surface states are in\nthe metallic regime. In Ref. [31], the magnetization induce d\nby the Mn-doping has an easy axis paralell to the surface-\nplane alignment. Once the conditions for the magnetization\nand TI surface states proposed in our paper are applied exper -\nimentally, the electromagnon may be observed by the FMR in\nthe near future.\nACKNOWLEDGMENTS\nY . H thanks Makiko Nio, Ryutaro Yoshimi, Kenji Yasuda,\nand Masataka Mogi for fruitful discussion and comments.\nThis work was supported by RIKEN Special Postdoctoral Re-\nsearcher Program (Y . H) and by JSPS Grant-in-Aid for Sci-\nentiﬁc Research (No. 24224009, and No. 26103006) from\nMEXT, Japan (N. N).\n[1] M. Z. Hansan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010).\n[2] X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011).\n[3] Y . Ando, J. Phys. Soc. Jpn. 82, 102001 (2013).\n[4] M. Franz and L. Molenkamp, Topological Insulators Volume\n6:Contemporary Concepts of Condensed Matter Science (El-\nsevier, Oxford, 2013).\n[5] W. Y . Shan, H. Z. Lu, S. Q. Shen, New. J. Phys. 12, 043048\n(2010).\n[6] H. Z. Lu, W. Y . Shan, W. Yao, Q. Niu, S. Q. Shen, Phys. Rev. B\n81, 115407 (2010).\n[7] X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Phys. Rev. B 78,\n195424 (2008).\n[8] A. M. Essin, J. E. Moore, and D. Vanderbilt. Phys. Rev. Let t.\n102, 146805 (2009).\n[9] S.-W. Cheong and M. Mostovoy, Nat. Mat. 613 (2007).\n[10] R. Ramesh and N. A. Spaldin, Nat. Mat. 621 (2007).\n[11] Y . Tokura, S. Seki, and N. Nagaosa, Rep. Prog. Phys. 77(2014)\n076501.\n[12] T. Yokoyama, J. Zang, and N. Nagaosa, Phys. Rev. B 81,\n241410(R) (2010).\n[13] K. Nomura and N. Nagaosa, Phys. Rev. B 82, 161401(R)\n(2010).[14] I. Garate and M. Franz, Phys. Rev. Lett. 104, 146802 (2010).\n[15] S. Raghu, S. B. Chung, X.-L. Qi, and S.-C. Zhang, Phys. Re v.\nLett. 104, 116401 (2010).\n[16] W.-K. Tse and A. H. MacDonald, Phys. Rev. Lett. 105, 057401\n(2010).\n[17] P. Hosur, Phys. Rev. B 83, 035309 (2011).\n[18] J. W. McIver, D. Hsieh, H. Steinberg, P. Jarillo-Herrer o, and\nN. Gedik, Nat. Nanotechnol. Lett. 7, 96 (2011).\n[19] Y . Tserkovnyak and D. Loss, Phys. Rev. Lett. 108, 187201\n(2012).\n[20] A. R. Mellnik, J. S. Lee, A. Richardella, J. L. Grab, P. J. Mintun,\nM. H. Fischer, A. Vaezi, A. Manchon, E. -A. Kim, N. Samarth,\nand D. C. Ralph, Nature (London) 511, 449 (2014).\n[21] Y . Shiomi, K. Nomura, Y . Kajiwara, K. Eto, M. Novak,\nK. Segawa, Y . Ando, and E. Saitoh, Phys. Rev. Lett. 113,\n196601 (2014).\n[22] A. Sakai and H. Kohno, Phys. Rev. B 89, 165307 (2014).\n[23] K. Taguchi, K. Shintani, and Y . Tanaka, Phys. Rev. B 92,\n035425 (2015).\n[24] N. Ogawa, R. Yoshimi, K. Yasuda, A. Tsukazaki, M. Kawasa ki,\nand Y . Tokura, Nat. Commun. 7, 12246 (2015).5\n[25] J. G. Checkelsky, J. Ye, Y . Onose, Y . Iwasa, and Y . Tokura , Nat.\nPhys. 8729 (2012).\n[26] C.-Z. Chang, J. Zhang, X. Feng, J. Shen, Z. Zhang,1 M. Guo ,\nK. Li, Y . Ou, P. Wei, L.-L. Wang, Z.-Q. Ji, Y . Feng, S. Ji,\nX. Chen, J. Jia, X. Dai, Z. Fang, S.-C. Zhang, K. He, Y . Wang,\nL. Lu, X.-C. Ma, Q.-K. Xue, Science 340, 167 (2013).\n[27] Y . Fan, P. Upadhyaya1, X. Kou, M. Lang, S. Takei, Z. Wang,\nJ. Tang, L. He, L.-T. Chang, M. Montazeri, G. Yu, W. Jiang,\nT. Nie, R. N. Schwartz, Y . Tserkovnyak, and K. L. Wang, Nat.\nMat. 13, 699-704 (2014).\n[28] I. Lee, C-K. Kima, J. Lee, S. J. L. Billinge, R. Zhong,\nJ. A. Schneeloch, T. Liu, T. Valla, J. M. Tranquada, G. Gu, and\nJ. C. S. Davis, PNAS 112, 1316-1321 (2015).\n[29] X. Kou, Y . Fan, M. Lang, P. Upadhyaya, K. L. Wang, Solid\nState Communications 215-216 , (2015) 34-35.\n[30] M. Mogi, R. Yoshimi, A. Tsukazaki, K. Yasuda, Y . Kozuka,\nK. S. Takahashi, M. Kawasaki, and Y . Tokura, Appl. Phys. Lett .\n107, 182401 (2015).\n[31] H. J. von Bardeleben, J. L. Cantin, D. M. Zhang, A. Richar della,\nD. W. Rench, N. Samarth, and J. A. Borchers, Phys. Rev. B 88,\n075149 (2013).[32] C.-Z. Chang, W. Zhao, D. Y . Kim, H. Zhang, B. A. As-\nsaf, D. Heiman, S.-C. Zhang, C. Liu, M. H. W. Chan, and\nJ. S. Moodera, Nat. Mater. 14, 473 (2015).\n[33] S. Grauer, S. Schreyeck, M. Winnerlein, K. Brunner, C. G ould,\nand L. W. Molenkamp, Phys. Rev. B 92, 201304(R) (2015).\n[34] C.-Z. Chang and M. Li, J. Phys.: Condens. Matter 28(2016)\n123002.\n[35] M. Mogi, M. Kawamura, A. Tsukazaki, R. Yoshimi, K. S. Tak a-\nhashi, M. Kawasaki, and Y . Tokura, arXiv:1708.05387.\n[36] P. Wei, F. Katmis, B. A. Assaf, H. Steinberg, P. Jarillo- Herrero,\nD. Heiman, and J. S. Moodera, Phys. Rev. Lett. 110, 186807\n(2013).\n[37] J. Henk, M. Flieger, I. V . Maznichenko, I. Mertig, A. Ern st,\nS. V . Eremeev, and E. V . Chulkov, Phys. Rev. Lett. 109, 076801\n(2012).\n[38] R. J. Cava, H. Ji, M. K. Fuccillo, Q. D. Gibson, and Y . S. Ho r,\nJ. Mater. Chem. B, 2013, 1, 3176-3189.\n[39] W. Luo and X. L. Qi, Phys. Rev. B 87, 085431 (2013).\n[40] S. V . Eremeev, V . N. Men’shov, V . V . Tugushev,\nP. M. Echenique, and E. V . Chulkov, Phys. Rev. B 88,\n144430 (2013)."    },    {        "title": "0705.2944v1.Evidence_for_ferromagnetic_spin_pairing_superconductivity_in_UGe__2___A____73__Ge_NQR_study_under_pressure.pdf",        "content": "arXiv:0705.2944v1  [cond-mat.supr-con]  21 May 2007Evidence for ferromagnetic spin-pairing superconductivi ty in UGe 2:\nA73Ge-NQR study under pressure\nA. Harada,1,∗S. Kawasaki,1H. Mukuda,1Y. Kitaoka,1Y. Haga,2\nE. Yamamoto,2Y.¯Onuki,2,3K. M. Itoh,4E. E. Haller,5and H. Harima6\n1Department of Materials Engineering Science, Osaka Univer sity, Osaka 560-8531, Japan\n2Advanced Science Research Center, Japan Atomic Energy Rese arch Institute, Tokai, Ibaraki 319-1195, Japan\n3Department of Physics, Osaka University, Osaka 560-0043, J apan\n4Department of Applied Physics and Physico-Informatics, Ke io University, Yokohama 223-8522, Japan\n5University of California at Berkeley and Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA\n6Department of Physics, Faculty of Science, Kobe University , Nada, Kobe 657-8501, Japan\n(Dated: November 3, 2018)\nWe report that a novel type of superconducting order paramet er has been realized in the fer-\nromagnetic states in UGe 2via73Ge nuclear-quadrupole-resonance (NQR) experiments perfo rmed\nunderpressure ( P). Measurements of the nuclear spin-lattice relaxation rat e (1/T1) have revealed an\nunconventional nature of superconductivity such that the u p-spin band is gapped with line nodes,\nbut the down-spin band remains gapless at the Fermi level. Th is result is consistent with that\nof a ferromagnetic spin-pairing model in which Cooper pairs are formed among ferromagnetically\npolarized electrons. The present experiment has shed new li ght on a possible origin of ferromag-\nnetic superconductivity, which is mediated by ferromagnet ic spin-density ﬂuctuations relevant to\nthe ﬁrst-order transition inside the ferromagnetic states .\nThe coexistence of magnetism and superconductiv-\nity (SC) has recently become an important topic in\ncondensed-matter physics. The recent discovery of SC\nin ferromagnets UGe 2[1, 2] and URhGe [3] has been a\ngreat surprise because the Cooper pairs are inﬂuenced by\na non-vanishing internal ﬁeld due to the onset of ferro-\nmagnetism (FM), which is believed to prevent the onset\nof SC. In the ferromagnet UGe 2with a Curie temper-\natureTCurie= 52K at ambient pressure ( P= 0), the\nemergence of P-induced SC has observed in the Prange\nof 1.0-1.6GPa [1, 2]. It is noteworthy that the SC in\nUGe2disappears above Pc∼1.6GPa, beyond which FM\nis suppressed. The SC and FM in this compound have\nbeen shown to be cooperative phenomena [4]. The su-\nperconducting transition temperature ( Tsc) is the high-\nest atPx∼1.2GPa, where a ﬁrst-order transition occurs\nfrom FM2 to FM1 as Pincreases. Here, it should be\nnoted that ferromagnetic moments are increased in the\nﬁrst-order transition from FM1 to FM2 as functions of\ntemperature and pressure, as shown in Fig. 1(a) [5, 6, 7].\nTheP-induced SC in UGe 2coexists with FM1 and FM2\nexhibiting the large magnetization of an order 1 µB/U\neven for the case of TCurie∼30K [6]. Therefore, the\nonset of SC is proposed to be suitable for the formation\nof a spin-triplet pairing state rather than a spin-singlet\npairing state [2]. However, there are few reports that\naddress the type of order parameter is realized in FM1\nand FM2. In a previous study, an unconventional nature\nof the SC has been suggested from the measurement of\nthe73Ge-NQR nuclear spin-lattice relaxation rate 1 /T1\n[6]. However, it has not been well understood whether\nthe presence of the residual density of states (RDOS) atFIG. 1: (color online). (a) Pressure versus temperature phase\ndiagram of UGe 2near the superconducting phase [6]. The Tsc\nvalues of FM1 (open squares) and FM2 (open triangles), and Px\nvalue determined in this study are plotted. (b) Crystal stru cture of\nUGe2with the ferromagnetic moment at the U site below TCurie.\nthe Fermi level in the SC state is intrinsic or not, sug-\ngestingthe occurrenceofa possible extrinsic eﬀect due to\nthe presence of any impurity and/or imperfection in the\nsample [6]. In particular, it is unclear why SC emerges\nwith the highest Tscwhen the ﬁrst-order transition oc-\ncurs from FM1 to FM2 at Px∼1.2GPa. In order to gain\ninsight into this issue, further experiments are required\nforunderstandinga P-inducedevolutionintheFMstates\nand a novel order-parameter symmetry emerging in the\nFM states in UGe 2.\nIn this letter, by performing the73Ge-NQR measure-\nments under pressure at zero ﬁeld ( H= 0) on a newly\npreparedsample, wereportthattheSCinthiscompound\niscausedbythe formationofup-spinCooperpairs, where\nthe gap opens only at the up-spin band in FM1 and FM22\nbut not at the down-spin band [8, 9, 10]. The ferro-\nmagnetic spin-pairing SC is considered to be mediated\nbyferromagneticspin-densityﬂuctuations relevanttothe\nﬁrst-order transition inside the ferromagnetic states.\npresent  \nprevious FM1 (b) (a) Paramagnetic\n8.0 8.4 8.8 9.2Echo-Intensity (arb. units)\nFrequency (MHz)8.0 8.4 8.8 9.2state\nFIG. 2: (color online). Comparison of the73Ge-NQR spectra\nof the present and previous samples of UGe 2in (a) paramagnetic\nstate and (b) FM1. The spectra at (a) P= 1.9GPa and (b) P=\n1.41 and 1.3GPa are shown by solid and open circles, respecti vely,\ndemonstrating that the present sample has better quality th an the\nprevious samples [6].\nApolycrystallinesample enrichedby73Ge wascrushed\ninto coarse powder for the NQR measurement and an-\nnealed to maintain its quality. The NQR experiments\nwere performed by the conventional spin-echo method at\nH= 0 in the frequency ( f) range of 5-11MHz at P=\n1.17, 1.2, 1.24, and 1.41GPa. Hydrostatic pressure was\napplied by utilizing a NiCrAl-BeCu piston-cylinder cell\nﬁlled with Daphne oil (7373) as a pressure-transmitting\nmedium. The value of Pat low temperatures was deter-\nminedfromthe TscofSnmeasuredbytheresistivitymea-\nsurement. Thepossibledistributionofthepressureinside\nthe sample was less than 3% in the present experimental\nsetup. A3He-4He dilution refrigerator was used to ob-\ntain the lowest temperature of 50mK. Figures 2(a) and\n2(b) show the NQR spectra in paramagnetic (PM) phase\nand FM1 phase, respectively. The linewidths in these\nNQR spectra are narrower for the present sample than\nfor the previous sample, demonstrating that the quality\nof the present sample is signiﬁcantly higher than that\nof the previous sample. Moreover, the NQR- T1measure-\nments reveal that the present sample exhibits the highest\nvalue ofTsc= 0.75K obtained thus far at P= 1.24GPa,\nensuring higher quality than before.\nFigure 3(a) shows the NQR spectra for the PM phase\nat 4.2K and P= 1.9GPa where FM1 is completely sup-\npressed. They reveal a structure consisting of separated\npeaks associated with three inequivalent Ge sites in one\nunit cell in the crystal structure illustrated in Fig. 1(b)\n[4, 6]. The number of Ge1 sites is twice that of Ge2 and\nGe3 sites in one unit cell. The Ge1 site is closely located\nalong the uranium (U)-zigzag chain, while the other two\nsites Ge2 and Ge3 are located outside this zigzag chain.\nFromtheanalysisoftheNQRspectraforFM1in thepre-\nvious experiment [6], it was demonstrated that the onset\nof FM1 induces an internal ﬁeld Hint= 0.9T at the Ge\nsites that additionally causes about the Zeeman split-1.24  GPa\n1.17  GPa1.2  GPa(c)(b)(a)\n(d)Ge1\nGe2Ge3 1.9  GPaEcho-Intensity (arb. units)\n5 6 7 8 9 10 11\nFrequency (MHz)(e)1.41  GPa\nFIG. 3: (color online). (a)73Ge-NQR spectra at 4.2K in the\nP-induced paramagnetic phase. The NQR spectra in (b), (c), (d ),\nand (e) represent in the ferromagnetic phases at 1.4K and P=\n1.41, 1.24, 1.2 and 1.17GPa, respectively. The dashed lines in the\nﬁgures indicate the simulated results (see text).\nting in each Ge-NQR spectrum. Furthermore, the an-\ngle between the principal axis for the nuclear quadrupole\nHamiltonian and a direction of Hintwas determined as\nθ∼π/3. When the ﬁrst-order transition occurs from\nFM1 to FM2, the spectral shape changes signiﬁcantly\nfrom the spectra at P= 1.24 and 1.41GPa to those\natP= 1.17 and 1.2GPa, as shown in Figs. 3(b)-(e).\nFrom the analysis of the spectra, it is estimated that\nHint= 0.9T for FM1 increases to Hint= 1.8T for FM2\n(seeFig.5(b)). Thesuddenincreasein Hintinthenarrow\nPrange should be relevant to the ﬁrst-ordertransition at\nPx. In such a case, the spectra near Pxare expected to\nreveala mixture of both domains of FM1 and FM2 in the\nnarrow range of Pclose toPxdue to an inevitable dis-\ntribution of P. In fact, the respective spectra at P= 1.2\nand 1.24GPa are composed of spectra arising from FM1\n(P= 1.41GPa) and FM2 ( P= 1.17GPa) with the ra-\ntios of 1:9 and 7:3, respectively, as shown by the dashed\nlines in Figs. 3(b) and 3(c). By considering an inevitable\nPdistribution (∆ P= 0.04GPa) as a Gaussian func-\ntiongivenbyexp[ −([P−P0]/[∆P/(2√\nln2)])2], weobtain\nPx= 1.23GPa. The present experimental results reveal\nthat the ﬁrst-order transition occurs at Px= 1.23GPa.\nFigure 4(a) shows the Tdependences of 1 /T1for FM1\nand FM2 at pressures that are slightly lower and higher\nthanPx= 1.23GPa, respectively. Clearly, 1 /T1for FM1\nand FM2 decreases without any indication of coherence\npeak just below Tsc, which provides evidence for the uni-\nform coexistence of the unconventionalSC and ferromag-\nnetism. In the previous study [6], the line-node gap\nmodel with RDOS Nresat the Fermi level was applied\nto interpret a systematic evolution in the superconduct-3\ning energy gap ∆ and a fraction of RDOS Nres/N0. Here\nN0is the density of state (DOS) at the Fermi level in\nthe FM phases. It should be noted that all the data of\n1/T1are uniquely determined in the present sample, but\nnot in the previous sample [6]. Therefore, we could not\nexclude the fact that the RDOS is present in the previ-\nous sample due to some impurity eﬀect. Similarly for the\npresent sample with higher quality than that of the pre-\nvious sample, the application of the line-node gap model\nwith the RDOS allows us to estimate Nres/N0= 0.50,\n0.48, 0.29, and 0.30and Tsc= 0.45, 0.55, 0.75, and 0.25K\n(±0.05K) at P= 1.17, 1.2, 1.24, and 1.41GPa, respec-\ntively. It should be noted that Tscdecreases from 0.45K\nto 0.25K although Nres/N0does decrease from 0.50 to\n0.30 atP= 1.2GPa in FM2 and at P= 1.41GPa in\nFM1. These results demonstrate that the presence of\nthe RDOS in the superconducting state is not due to the\nimpurity eﬀect but intrinsicin origin. Althoughsomeim-\npurity and/or imperfection based eﬀects, if any, are not\ncompletely ruled out, we state that the observationofthe\nhighestTsc= 0.75K ensures that the present sample is\none of the best quality samples reported thus far.\nFirst, we address whether or not the RDOS is asso-\nciated with a self-induced vortex state in the SC + FM\nuniformly coexisting state. By assuming the Abrikosov\ntriangular vortex lattice, a coherence length ξ∼130˚A\n[7], and an internal magnetic ﬁeld H= 0.125T [11], we\nconsider that only 3% of Nres/N0arises from the normal\nstate inside the self-induced vortex core in the SC+FM\nstate, which does not agree with the experimental re-\n1.17  GPa 1.2  GPa 1.41  GPa 1.24  GPa \nN \nTemperature (K)FM2 FM2 FM1 FM1( 1/T1T )1/2\nN N N \n0.1 1 0.1 1 0.1 10.20.40.60.8\n0.1 1TscTscTsc\nTsc1.17  GPa 1.2 GPa 1.41  GPa 1.24 GPa \nFM2 FM2 FM1 FM1\nTscTscTsc\nTsc\n(b)(a)\n0 0 0 0 (sec-1/2)  K0.1 1 0.1 110-310-210-11001011 / T1 (sec-1)\n0.1 1 0.1 1~T ~T~T ~T-1/2\nFIG. 4: (color online). (a) Temperature dependences of 1 /T1for\nFM2 at P= 1.17 and 1.2GPa measured at f= 7.68MHz and for\nFM1 at P= 1.24 and 1.41GPa measured at 7.09 and 7.07MHz,\nrespectively. (b) Temperature dependences of (1 /T1T)1/2related\nto the DOS in either the SC state or the normal state at each\nP. The solid curves represent the results calculated based on the\nferromagnetic spin-pairing model (see the text).sult. Alternatively, in another promising scenario that\nexplainsthe RDOS, weconsidera nonunitaryspin-triplet\npairing model [8]. In this model, the superconducting\nenergy gap opens only in the up-spin band parallel to\nthe magnetization of FM phases, but not in the down-\nspin band that remains gapless. We begin by assigning\npossible nuclear relaxation processes of Ge-NQR T1in\nthe ferromagnetic states. One of them is caused by the\ntransversal component of ﬂuctuations of internal mag-\nnetic ﬁeldsat the Gesites originatingfrom intrabandand\ninterband transitions across the Fermi level at each up-\nspin band and down-spin band. Another one is causedby\nonly the interband spin-ﬂip transition across the Fermi\nlevel between the up-spin band and down-spin band. By\nconsidering these relaxation processes, 1 /T1Tin the FM\nstate is expressed as\n1\nT1T∝2t2(T)cos2θ+[t1(T)+2t2(T)+t3(T)]sin2θ,\nt1(T) =1\nkBT/integraldisplay∞\n0dEN2\n↓(E)f(E)[1−f(E)],\nt2(T) =1\nkBT/integraldisplay∞\n0dEN↑(E)N↓(E)f(E)[1−f(E)],\nt3(T) =1\nkBT/integraldisplay∞\n0dEN2\n↑(E)f(E)[1−f(E)],\nwheret1(T),t2(T), andt3(T) indicate the former con-\ntributions and t2(T) represents the latter contribution\nwhich is only possible for θ= 0. When the energy de-\npendence of the DOS is neglected near the Fermi level,\nall contributions of t1=N2\n0↓,t2=N0↑N0↓, andt3=N2\n0↑\nare independent of temperature. Here, N0↑andN0↓are\nthe DOS at the up-spin and the down-spin bands at the\nFermi level in the normal FM state, respectively. θis an\nangle between the quantization axis of the73Ge-nuclear-\nquadrupole Hamiltonian and that of Hintat the Ge site\nin the FM state, which is estimated as θ∼π/3 from the\nanalysis of the NQR spectra in the FM state, and f(E)\nis a Fermi distribution function. In the ferromagnetic\nspin-pairing model, the line-node gap of ∆( φ) = ∆0cosφ\nis assumed only for the density of states Ns↑(E) at the\nup-spin band, but not for Ns↓(E) at the down-spin band.\nIt should be noted that if θ= 0,t2(T) should behave as\n1/T1∝T2well below Tsc. In the present case, because\nofθ∼π/3, the gapless term t1gives rise to the RDOS\nat the Fermi level in the superconducting state, as shown\nin Fig. 4(b). In fact the experimental results are actually\nin good agreement with this theoretical model, as indi-\ncated by the solid lines in Figs. 4(a) and 4(b). There-\nfore, the SC energy gap ∆ and N0↑/N0are estimated as\n2∆/kBTsc∼3.7, 3.8, 4.0, and 3.7 with N0↑/N0= 0.57,4\n0.57, 0.82, and 0.80 at P= 1.17, 1.2, 1.24, and 1.41GPa,\nrespectively. Here, N0(P) =N0↑(P)+N0↓(P).\nIn order to gain further insight into the novel SC state,\nFig. 4(b) shows the Tdependence of (1 /T1T)1/2related\nto the DOS at the Fermi level in either the SC or the\nnormal FM state. As shown in Fig. 5(c), the most in-\nteresting ﬁnding is that N0↑(P) dramatically increases\nasPincreases slightly from P= 1.2 to 1.24GPa across\nPx= 1.23GPa, accompanying the sudden reduction of\nHintshown in Fig. 5(b). By contrast, N0↓(P) remains\nalmost constant and the T-linear coeﬃcient of the spe-\nciﬁc heat γgradually increases with PacrossPx[13].\nThese results reveal that the Fermi level in FM2 is lo-\ncated just above a sharp peak in the majority up-spin\nband, and as Pincreases across Px, it shifts down to-\nward the peak when it enters the FM1 phase. In the\nferromagnetic spin-pairing SC state, the large DOS in\nthe up-spin band in FM1 enhances Tsc, leading to the\nhighest value of Tsc= 0.75K, whereas its reduction in\nFM2 decreases Tsc, asshownin Figs. 5(a)and 5(c). How-\never, it should be noted that as Pincreases further up\ntoP= 1.41GPa, even though N0↑(P) for FM1 remains\nrather larger than that for FM2, Tsc= 0.25±0.05K for\nFM1 atP= 1.41GPa becomes lower than Tsc= 0.45K\nfor FM2 at P= 1.17GPa . In this context, the large\nincrease in N0↑(P) is not always a main factor that in-\ncreasesTsc. Rather, the ﬁrst-order transition from FM2\nto FM1 at Pxis responsible for the mediation of the up-\nspin Cooper pairing. The longitudinal FM spin ﬂuctua-\ntions along the a-axis [14] softens in energy at the critical\nend point for the ﬁrst-order transition at Px. Therefore,\nwe suggest that this longitudinal FM ﬂuctuations along\nthea-axiswould be a mediator of the ferromagneticspin-\npairing SC where the majority up-spin band in the FM\nphases is gapped, while the minority down-spin band is\nnot.\nIn another context, it is predicted that Txcould be\nidentiﬁed with the formation of a simultaneous charge-\nand spin-density wave (CSDW) induced by the imper-\nfect nesting of the Fermi surface for the up-spin band\nand hence the superconducting pairing is mediated by\nCSDW ﬂuctuations around Px[15]. Although the NQR\nspectrum does not directly evidence an onset of static\nCSDW states, the remarkable increase in N0↑(P) across\nPxis relevant to the nesting at the Fermi surface for the\nup-spin band below Tx.\nIn conclusion, the73Ge-NQR measurements under\npressure on well characterized UGe 2have revealed that\nthe superconducting energygap opens only with the line-\nnode at the Fermi level in the majorityup-spin band, but\nthe down-band remains gapless. It is therefore concluded\nthat ferromagnetic spin-pairing SC occurs in UGe 2. We\nhave also shown that the ﬁrst-order transition from FM1\nto FM2 at Px= 1.23GPa occurs because the Fermi\nlevel is located just on the peak in the DOS of the up-\nspin band. The ferromagnetic spin-density ﬂuctuationsFIG. 5: (color online). Pressure dependence of (a) Tsc; (b) inter-\nnal magnetic ﬁeld, Hint, at the Ge1 site; and (c) relative Pdepen-\ndence of N0↑(solid circles) and N0↓(solid squares) estimated from\nthe ferromagnetic spin-pairing model on a scale of (1 /T1T)1/2and\ntheT-linear coeﬃcient of the speciﬁc heat γ[13] (open squares)\nacrossPx(see text). It should be noted that N0↑dramatically in-\ncreases when the ﬁrst-order transition from FM2 at P= 1.2GPa\nto FM1 at 1.24GPa occurs across Px= 1.23GPa.\nemerging in the vicinity of the critical end point for this\nﬁrst-order transition are considered to be the mediator\nof the onset of novel SC realized in ferromagnet UGe 2.\nWe thank H. Kotegawa, N. Tateiwa, S. Watanabe,\nand K. Miyake for fruitful discussions and comments.\nThis study was supported by Grant-in-Aid for Creative\nScientiﬁc Researchi15GS0213); the Ministry of Educa-\ntion, Culture, Sports, Science and Technology (MEXT);\nand the 21st Century COE Program (G18) supported by\nthe Japan Society for the Promotion of Science (JSPS).\nA.H. wasﬁnanciallysupportedbyaGrant-in-AidforEx-\nploratory Research of MEXT (No. 17654066).\n∗aharada@nmr.mp.es.osaka-u.ac.jp\n[1] S.S.Saxena, P.Agarwal, K.Ahilan, F.M.Grosche, R.K.\nW. Haselwimmer, M. J. Steiner, E. Pugh, I. R. Walker,\nS. R. Julian, P. Monthoux, G. G. Lonzarich, A. Huxley,\nI. Sheikin, D. Braithwaite, and J. Flouquet, Nature 406,\n587 (2000).\n[2] A. Huxley, I. Sheikin, E. Ressouche, N. Kernavanois,\nD. Braithwaite, R. Calemczuk, and J. Flouquet, Phys.\nRev. B63, 144519 (2001).\n[3] D. Aoki, A. D. Huxley, E. Ressouche, D. Braithwaite,\nJ. Flouquet, J. P. Brison, E. Lhotel, and C. Paulsen,\nNature413, 613 (2001).\n[4] A. Harada, S. Kawasaki, H. Kotegawa, Y. Kitaoka,\nY. Haga, E. Yamamoto, Y. ¯Onuki, K. M. Itoh,\nE. E. Haller, and H. Harima, J. Phys. Soc. Jpn. 74, 2675\n(2005).\n[5] C. Pﬂeiderer and A. D. Huxley, Phys. Rev. Lett. 89,\n147005 (2002).\n[6] H. Kotegawa, A. Harada, S. Kawasaki, Y. Kawasaki,\nY. Kitaoka, Y. Haga, E. Yamamoto, Y. ¯Onuki,5\nK. M. Itoh, E. E. Haller, and H. Harima, J. Phys. Soc.\nJpn.74, 705 (2005).\n[7] N. Tateiwa, T. C. Kobayashi, K. Hanazono, K. Amaya,\nY. Haga, R. Settai, and Y. ¯Onuki, J. Phys.: Condens.\nMatter13, L17 (2001).\n[8] T. Ohmi and K. Machida, Phys. Rev. Lett. 71, 625\n(1993).\n[9] K. Machida and T. Ohmi, Phys. Rev. Lett. 86, 850\n(2001).\n[10] I. A. Fomin, JETP Letters 74, 111 (2001).\n[11] The internal magnetic ﬁeld His derived from H= (1−\nn)M, where nis the demagnetization factor and Mis\nthe magnetization per volume. Here, the ferromagnetic\nmomentsis givenby1 µB/U[12]andthedemagnetizationfactor forthissamplegeometry(approximatelyspherical)\nisn= 1/3.\n[12] N. Tateiwa, K. Hanazono, T. C. Kobayashi, K. Amaya,\nT. Inoue, K. Kindo, Y. Koike, N. Metoki, Y. Haga,\nR. Settai, and Y. ¯Onuki, J. Phys. Soc. Jpn. 70, 2876\n(2001).\n[13] N.Tateiwa, T.C. Kobayashi, K.Amaya, Y.Haga, R.Set-\ntai, and Y. ¯Onuki, Physica B 312-313 , 109-111 (2002).\n[14] A. D. Huxley, S. Raymond, and E. Ressouche, Phys. Rev.\nLett.91, 207201 (2003).\n[15] S. Watanabe and K. Miyake, J. Phys. Soc. Jpn. 71, 2489\n(2002)."    },    {        "title": "1604.07767v1.Fingerprints_of_entangled_spin_and_orbital_physics_in_itinerant_ferromagnets_via_angle_resolved__resonant__photoemission.pdf",        "content": "arXiv:1604.07767v1  [cond-mat.str-el]  17 Nov 2015Fingerprints of entangled spin and orbital physics in itine rant ferromagnets\nvia angle resolved resonant photoemission\nF. Da Pieve1\n1Laboratoire des Solides Irradi´ es, UMR 7642, CNRS-CEA/DSM ,´Ecole Polytechnique,\nF-91128 Palaiseau, France and European Theoretical Spectr oscopy Facility (ETSF)\n(Dated: June 16, 2021)\nA novel method for mapping the local spin and orbital nature o f the ground state of a system\nvia corresponding ﬂip excitations in both sectors is propos ed based on angle resolved resonant\nphotoemission and related diﬀraction patterns, presented here for the ﬁrst time via an ab-initio\nmodiﬁed one-step theory of photoemission. The analysis is done on the paradigmatic w eak itinerant\nferromagnet bccFe, whose magnetism, seen as a correlation phenomenon given by the coexistence\nof localized moments and itinerant electrons, and the non-F ermi liquid behaviour at ambient and\nextreme conditions both remain unclear. The results oﬀer a r eal space imaging of local pure spin ﬂip\nand entangled spin ﬂip-orbital ﬂip excitations (even at ene rgies where spin ﬂip transitions are hidden\nin quasiparticle peaks) and of chiral, vortex-like wavefro nts of excited electrons, depending on the\norbital characterofthebandsandthedirection ofthelocal magneticmoment. Sucheﬀects, mediated\nbythehole polarization, makeresonant photoemission apro mising tool toperform afull tomography\nof the local magnetic properties of a system with a high sensi tivity to localization/correlation, even\nin itinerant or macroscopically non magnetic systems.\nPACS numbers: 78.20.Bh, 78.20.Ls, 78.70.-g, 79.60.-i\nI. INTRODUCTION\nSpin and orbital degrees of freedom, their ﬂuctu-\nations, entanglement and textures, play a relevant\nrole in many fascinating correlated and/or spin orbit-\ndriven systems, like Mott insulators1–3, non conventional\nsuperconductors4–6and topological phases of quantum\nmatter7–9. In the last two decades, it has become\nclear however that peculiar orbital textures and spin-\norbital coupling are found even without relevant spin\norbit and/or without relevant electron-electron correla-\ntion, like in low-dimensional materials exhibiting Peierls\ntransitions and charge density waves10–12, in some lowly\ncorrelated insulators doped with 3 dions developing long\nrange magnetic order13, correlated metals14and even\nweak itinerant ferromagnets15,16, whose behaviour might\nsometimes challenge the standard model of the metallic\nstate, the (ferromagnetic) Fermi Liquid theory. How-\never, probing simultaneously spin and orbital degrees\nof freedom with high sensitivity to spatial localization\nis complicated, as the orbital angular momentum is\noften quenched by the crystal ﬁeld in many relevant\ncompounds and as the distinction between low energy\nspin and orbital excitations of diﬀerent nature (inco-\nherent particle-hole and collective modes) is not always\nobvious17,18. Finding a strategy to improve the capabili-\nties ofwidely used techniques, like angle resolvedphotoe-\nmission (ARPES)19and resonant inelastic X-ray scatter-\ning (RIXS)20, whose sensitivity to spatial localization is\nlimited due to the linear dependence of the dipole opera-\ntor on the spatial coordinate /vector r, would boost the advance\nfor an atomic-scale mapping of the magnetic properties\neven in macroscopically non magnetic systems.\nOrbital-resolved contributions to ARPES spectra are\noften studied either by analyzing the self-energy enteringthe expression of the one-body spectral function describ-\ning photoemission21or analyzing related dichroism sig-\nnals induced by circular or linearly polarized light22–25.\nOther more explorative works have considered Auger\nemission, in particular in time coincidence with photo-\nelectrons, and unravelled the two-electron (and the cor-\nrespondingtwo-hole)orbitalcontributionstobothenergy\nspectra26and angular polar scans27,28. Earlier works\nhavealsostudiedtheorbital-resolvedcontributionstofull\ntwo-dimensional angular patterns (via the anisotropy of\nthe excited ”source wave” at the absorber) in core level\nphotoemission29,30and Auger spectroscopy31–33. The\nanisotropy of the charge density of such source wave(s)\nand the one of the core hole state (core hole polariza-\ntion,Pc) are inﬂuenced by the polarization (and direc-\ntion) of the impinging light and the polarization of the\nvalencestates. Theyarecharacterizedbyevenmultipoles\n(quadrupole, etc), describing the alignment (i.e., the de-\nviation from sphericity, given by a diﬀerent occupation\namong the diﬀerent mlstates, with a symmetry between\n±ml), and odd multipoles (dipole, etc), describing the\norientation (i.e. the rotation of the charge density, given\nby a preferential occupation of mlstates over- mlstates).\nRecently, pioneering diﬀraction patterns have also\nbeen reported34–36for resonant photoemission (RPES),\nthe participator channel of the resonant Auger eﬀect,\nthe non radiative decay channel following X-ray ab-\nsorption degenerate with usual ARPES. However, ear-\nlier theoretical descriptions of the resonant Auger ef-\nfect, formulated on the basis of the interaction between\ndiscrete and continuum states37, Keldysh formalism38,\nor via time-independent resonant scattering theory39,40\nhave not been accompanied by realistic implemented\nschemes. The existing, practical calculation schemes\n(model hamiltonian-based)41–44onlyfocus onthe specta-2\ntor channels of the resonantAuger eﬀect, with two-holes-\nlike ﬁnal states, and not on the participator ones, where\nthe decay occurrs before the excited electron has delocal-\nized, leading to one-hole ﬁnal states linearly dispersing\nwith photon energy (Raman shift) , visible before and at\nthe edge43. Also, retriving information on local magnetic\nproperties remains diﬃcult, and some eﬀects observed in\nRIXS, like spin ﬂip-orbital ﬂip excitations18,43,45–48have\nnever been reported in RPES.\nIn this work, it is shown that the yet largely unex-\nplored spin polarized angle resolved RPES (AR-RPES)\nis a promising tool for performing a full localspin and\norbital tomography of the ground state of a system, by\nproviding access to local spin ﬂip, orbital ﬂip and chiral\nexcitations. The study is based on a recently presented\nab-initio description for extended systems50, based on a\nmodified one-step theory of photoemission, which is re-\nanalyzed to elucidate matrix elements eﬀects and mixed\nwith an auxiliary analysis of convoluted partial densities\nof states (DOS) to elucidate the connection with local\nspin and orbital properties. The paradigmatic case of\nthe weak itinerant ferromagnet bcc Fe, whose origin of\nferromagnetism is nowadays seen as a correlation phe-\nnomenon, given by the coexistence of localized moments\nassociated to electrons in a narrow egband and itiner-\nant electrons in the t2gbands, is considered. Yet unex-\nplained correlations in the paramagnetic phase eventu-\nally determine the localization of the egstates15and the\nformation of localized moments. Instabilities at extreme\nPTconditions and tendency of egstates to a non-FL\nbehaviour even for ambient conditions16have been re-\nported. Analysis of ARPES spectra at diﬀerent levels of\ntheory other than DFT (which does not contain static\nspin ﬂuctuations)49suggests the importance of non lo-\ncal correlations and the necessity to improve the descrip-\ntion of (orbital-dependent) mass renormalizations. The\nab-initio RPES energy spectra and diﬀraction patterns\npresented here for excitation at the L3edges by circu-\nlarly polarized light show the possibility of mapping the\nspinpolarizationandlocalvalenceorbitalsymmetrywith\nhigh sensitivity to spatial localization by analyzing spin-\nconserving and spin-ﬂip exchange excitations. The re-\nsults show the occurrence of pure spin ﬂip excitations far\nfrom the Fermi level ( EF) and coupled spin ﬂip-orbital\nﬂip excitations in correspondance of a narrow peak in\nthe local partial DOS near EFassociated to the elon-\ngatedeglevels. Similarities and diﬀerences with RIXS\nare discussed, as well as the practical and fundamental\nimplications concerning possible full tomographic stud-\nies of local magnetic properties and studies of spin and\norbital physics in more complex systems.\nII. THEORETICAL SECTION\nThe cross section for resonant photoemission is pro-\nportional to the Kramers-Heisenberg formula for secondorder processes\n∂2σ\n∂Ωp∂ω∝\n/summationdisplay\nf|∝angbracketleftf|Dq|0∝angbracketright+/summationdisplay\nj∝angbracketleftf|V|j∝angbracketright∝angbracketleftj|Dq|i∝angbracketright\nE0−Ej+iΓj\n2|2δ(/planckover2pi1ω+E0−EF)\n(Γjis the core level lifetime-induced width).\nThe ﬁrst term is the dipole matrix element\nDvp=∝angbracketleftiǫpLpσp|Dq|iǫLvσv∝angbracketrightwhich describes, in an eﬀetive\nsingle particle approach, direct valence band photoemis-\nsion (v(p) denotes the valence state (photoelectron) and\nLp= (lp,mp)). The second term represents the resonant\nprocess, described by the product of the core-absorption\ndipole matrix elements Dckand the decay (direct\nand exchange) matrix elements VdandVx, i.e.Rd=\nVd·Dck=∝angbracketleftiǫpLpσp,j′c′|V|iǫLvσv,j′ǫkLkσk∝angbracketright ·Dckand\nRx=Vx·Dck=∝angbracketleftjǫpLpσp,ic′|V|jǫkLkσp,iǫLvσv∝angbracketright ·Dck\n(kdenotes the conduction state where the electron gets\nexcited and c′the quantum numbers m′\nc,σ′\ncto which\nthe initial hole c=mc,σcmight scatter). For the more\nlocalized participator decays, in the direct term the\ncore hole is ﬁlled by the excited electron and a valence\nelectron is emitted, and in the exchange one the two are\nexchanged. In principle, the energy detuning from the\nabsorption edge and a narrow bandwidth of the photons\ncan act as a shutter between diﬀerent channels, although\nonly looking at energy spectra exhibiting the Raman\nshift (as often done) might not always allow the dis-\ntinction between localized and delocalized excitations51,\nwhich remains an open issue for both RIXS and RPES.\nAll delocalized states can be described conveniently\nvia real space multiple scattering, which describes the\npropagation of a wave in a solid as repeated scattering\nevents52and which allows to keep explicit dependence\non the local quantum numbers. The cross section can\nthen be cast in a compact form as:\n∂2σ\n∂Ωp∂ω=/summationdisplay\nqq′εqεq′∗σqq′\nwhereεqare the light polarization tensors and the her-\nmitian 3×3-matrixσqq′is given by\nσqq′=/summationdisplay\nN,N′K(N,q)Imτv(N,N′)K∗(N′,q′),\nK(iLvσv,q) =/summationdisplay\njLpB∗\njLp(kp)(δijδσvσp(Dvp+Rd)+Rx)\nwithN,N′labellingi(atomicsite) and L(=l,m). The\nphotoelectron scattering amplitudes BjLp(kp) can be re-\nsumed asB∗\njLp(kp) =YLp(kp)i−lpeiδlp, i.e., (the source\nwave) + all the scattering contributions. The orbital\nand spin contribution to the outgoing electron wavefunc-\ntions are then determined by the parity and Coulomb3\nselection rules of the whole process. They impose that\n|lc−|lv−lk|| ≤lp≤lc+lv+lk,lc+lv+lk+lp=even and\nmc+mp=mv+mk. For the spin, one has σc=σk=σc′\nfor the direct term (the spin of the core hole does not\nﬂip) andσc=σk=σp,σc′=σvfor the exchange term\n(allowingalsofor possible core hole spin ﬂip leading to\nsimultaneous ﬂip of the orbital projection mc).\nThe connection with ground state properties is high-\nlighted via an auxiliary description, obtained by modi-\nfying an often used expression for normal Auger emis-\nsion (i.e., a convolution of the DOS for the two ﬁnal\nholes,53). By considering now the DOS of the emitted\nelectronD(E−ǫ) and the DOS of the electron dropping\ninto the core hole D(ǫ), weighted by the core hole polar-\nization, the intensity becomes:\nI↑(↓)(E) =M↑↑(↓↓)P+(−)/integraldisplay\nD↑(↓)(E−ǫ)D↑(↓)(ǫ)dǫ+\nM↑↓(↓↑)P−(+)/integraldisplay\nD↑(↓)(E−ǫ)D↓(↑)(ǫ)dǫ\nwhereP±= (1±Pc)/2 takes into account the modi-\nﬁcations of the DOS of the electron ﬁlling the hole by\nthe core hole polarization, and M↑↑(↓↓)andM↑↓(↓↑)are\nrespectively the sum of the modulus squares of the spin\nconserving (direct and exchange) decay matrix elements\nandthe modulus square of the spin ﬂip (exchange) decay\nmatrix element:\nM↑↑(↓↓)=|Vd,↑↑(↓↓)|2+|Vx,↑↑(↓↓)|2,\nM↑↓(↓↑)=|Vx,↑↓(↓↑)|2\nPcisinarangefrom-154(asinaferromagnetwithspin\ndown holes, and light impinging parallel to the magneti-\nzation), to some other values <1 when the hole ﬂips or\nthephotonpolarizationandthelocalvalencepolarization\nform a generic angle (in this latter case, both even and\nodd multipoles contribute to Pc55, and dicroism occurrs\nin both absorption and decay).\nThe important theoretical prediction can then be\nmade that the occurrence of spin ﬂip transitions and\ntheir entanglement with orbital ones are determined by\nthe (geometry-dependent) core hole polarization, how\nit aﬀects excited states of diﬀerent degree of localiza-\ntion/delocalization, and how it weights the decay ex-\nchange matrix elements. Also, orbital ﬂips should be\nmore visible when perturbing a highly symmetric (with\nrespect to relevant quantization axis) intermediate-state\norbital population (alignment), rather than an asymmet-\nric one. Given the inﬂuence of matrix elements on diﬀer-\nent allowed source waves and the high energy of the pho-\ntoelectrons (which reduces the importance of ﬁnal-state\neﬀects), it can be expected that a selective real-space\nmapping of (local) spin and spin-orbital excitations is\npossible by looking at two-dimensional angular patterns.III. COMPUTATIONAL DETAILS\nExcitation at the 2 p3\n2edge of the itinerant weak fer-\nromagnet Fe by circularly polarized light is investigated\nto proof the unique capabilities of RPES. A semispheri-\ncal Fe(010) cluster (with 184 atoms and in-plane magne-\ntization along <001>), and DFT spin polarized poten-\ntials obtained by a scalar relativistic LMTO57calcula-\ntion for bulk Fe bcc in DFT-local spin density approxi-\nmation (LSDA) are used as input for a multiple scatter-\ning code developed by the author, which can calculate\nusual ARPES and RPES from cluster type objects. The\nspectra and full hemispherical patterns are obtained tak-\ning into account the interference due to emission from\ndiﬀerent atomic sites when exciting at resonance. The\ncalculated magnetic moment of 2.26 µBfrom the self-\nconsistent calculation is in good agreement with exper-\niment. Core states are calculated atomically by solving\nthe Dirac equation, while delocalized states (bound and\nunbound) are developed, as mentioned before, via multi-\nple scattering. The photoelectron is described as a time-\nreversed LEED state, i.e. a plane wave with linear mo-\nmentum kplus incoming spherical waveson all atoms. A\nrealinnerpotential(10eV)isusedwhichservesasarefer-\nence energy inside the solid with respect to the vacuum\nlevel and inelastic damping is included via a constant\nimaginary potential (4.5 eV). For the optical transitions,\nthedipoleapproximationintheaccelerationformisused,\nsince the length form is not well deﬁned for delocalized\nstate. The weak spin-orbit (SO) coupling of the valence\nand continuum states has been neglected.\nFrom a theoretical viewpoint, non radiative decays are\ncomplicated dynamical processes which include atomic\nrelaxation and electron screening in response to the core\nhole. However, reasonable approximations can be made\nfor Fe. Electron-corehole interaction is generallyweak in\nmetals because of eﬃcient screening of the Coulomb in-\nteraction and its only observable eﬀect is the reduced\nbranching ratio between the L2andL2edges of the\nisotropic x-ray absorption spectra, with respect to what\nobtained within the independent particle approximation.\nHowever, such reduction is generally smaller for spin-\npolarized and dichroic spectra, and more importantly,\nin RPES it only aﬀects the intermediate states, which\nare not directly observed. For Fe, the deviation of the\nbranching ratio from the statistical value is actually very\nsmall58, indicating a reasonable description in terms of a\nsingle particle approach. Also, as a consequence of be-\ning a weak ferromagnet, both minority and majority spin\nstates can be populated to screen the core hole, leading\nto no drastic change in the local moment60. When the\ndecay takes place, with a valence electron ﬁlling the hole\nand the excited electron emitted, either the eﬀective po-\ntential seen by the valence electrons is restored to its\ninitial form or, as the electron is emitted with high ki-\nnetic energy, a sudden response of the valence electrons\noccurrs due to the destruction of the core hole, with no\ntime for electrons to readjust. Thus the spin polarization4\nFIG. 1: a) DOS of the Fe(010) cluster; b) ARPES and AR-RPES spe ctra (from56) for parallel geometry and normal emission.\nRest of the panel: PED, RPED for initial binding energy corre sponding to the main peak and the spin ﬂip peak in the spin up\nAR-RPES spectrum, and “source waves” patterns (the emitter is embedded in the cluster but no scattering events take plac e).\n(the plotted function is χ=I[θ,φ,ǫ]/I0[θ,ǫ]−1,I0being the intensity averaged over all φ-dependent values. Scans are around\nthe surface normal.)\nof the emitted electron results to be approximately the\none of the intermediate state, very similar though, for\nFe, to the one of the initial ground state59. Dipole and\nAuger-like matrix elements are then calculated here us-\ning ground state scalar relativistic wave functions. The\nrobustness of the approach is demonstrated by earlier\nsuccessful comparisons between calculated spin polariza-\ntion, energy spectra and photoemission diﬀraction pat-\nterns and experiments50,61.\nIV. RESULTS\nFig. 1a, 1b show the d-DOS of the wholecluster and\nthe ARPES and AR-RPES spectra for a photon energy\nat the maximum of the resonance for normal emission\nandparallel geometry (light impinging along the mag-\nnetization, along which spin is measured). The ARPES\nspectra show each one main peak, absence of other sharp\nfeatures as for a genuine lowly correlated system, in\nagreement with experiments62, and null dichroism, due\nto non chiral geometry and neglected SO in delocal-\nized states. In contrast, the resonant spectra exhibit\ndichroism (in this geometry only related to the absorp-\ntion step as the orientation of the core hole is unaﬀected\nby reversal of helicity63) and, more importantly, new\npeaks. Going towards higher binding energies, the spin\nup RPES spectra show a ﬁrst (second) peak for emis-\nsion frome↑\ng(t↑\n2g) states, while the spin down spectraexhibit a ﬁrst peak for emission from t↓\n2gstates and then\nan unexpected peak at an energy where there are almost\nno spin down states in the DOS, and which thus cor-\nresponds to spin up valence states. This means that\nthe spin of the photoelectron is opposite to the one of\nthe ﬁnal valence hole, and thus it is a spin ﬂip tran-\nsition. Such (exchange-induced) spin ﬂip can only oc-\ncur for 2p3/2eigenstates with mixedspin character due\nto SO (the mj=±1/2 sublevels, |3/2,1/2(−1/2)>=/radicalbig\n2/3|Y↑\n10(Y↓\n10)>+/radicalbig\n1/3|Y↓\n11(Y↑\n1−1)>).\nWe now move to the more explorative resonant diﬀrac-\ntion patterns. Ab-initio spin polarized resonant and di-\nrect photoemission diﬀraction patterns (RPED, PED)\nare reported in Fig. 1, for initial energies correspond-\ning to the two peaks in the spin up AR-RPES spectra\n(the main peak near EFand the one at higher binding\nenergy, corresponding to the spin ﬂip excitations in the\nspin down channel). It is clear that, while almost all\nRPED patterns resemble the corresponding direct ones,\na net 90otwist occurrs for right circular polarization for\nthe RPED pattern of the spin down channel, the one al-\nlowing for spin ﬂip transitions, a clear signature of an\naccompanying orbitalﬂip of the photoelectron wave. In-\nterestingly, the eﬀect is actually mainly visible at the\nmain peak, revealingspin ﬂip transitions hidden by dom-\ninating spin-conserving ones in the quasiparticle peak.\nThis orbital ﬂip phenomenon can be understood via\nthe two models described in the theoretical section, by\nanalyzing the exchange matrix elements and the local\npartial DOS. The selection rules dictate lp=1,3,5 (with5\nTABLE I: Exchange transitions at core states with mixedspin\ncharacter, for left (right) polarization ∆ m= +1(−1).\n∆m edge m c;σcmk;σkm′\nc;σ′\nc mp;σpmv;σv\n+13\n2;-1\n20;-1\n21;-1\n2-1;1\n23,4,2,1,0;−1\n21,2,0,-1,-2;1\n2\n+13\n2;1\n21;-1\n22;-1\n20;1\n23,4,2,1,0; −1\n21,2,0,-1,-2;1\n2\n-13\n2;-1\n20;-1\n2-1;-1\n2-1;1\n21,2,0,-1,-2;−1\n21,2,0,-1,-2;1\n2\n-13\n2;1\n21;-1\n20;-1\n20;1\n21,2,0,-1,-2;−1\n21,2,0,-1,-2;1\n2\nFIG. 2: Local partial DOS( l,m-resolved) around aFe central\nion in the cluster\n3 numerically found as the most probable wave, in line\nwith previous works on similar transitions33,64). Table\nI reports the exchange transitions occurring at core hole\nstateswith mixedspincharacter(attheirspindowncom-\nponents,asavailableemptystatesarespindowncorehole\nstates will also be mainly spin down). These are mixed\nspin ﬂip-orbital ﬂip transitions, in which both the ml\nandσzcomponents of the same m jsubstate ﬂip. Tran-\nsitions mixing diﬀerent mjs, likemj= 1/2 ﬂipping to\nmj=−1/2, are also possible, being the mjsublevels\nseparated by 0.32 eV, but these imply only spin ﬂip. We\nrecall that the relevant irreducible representations here\nare:t2g:dxy=1√\n2(ψ2−ψ−2),dyz=1√\n2(ψ1−ψ−1),dzx=\n1√\n2(ψ1+ψ−1);eg:dx2−y2=1√\n2(ψ2+ψ−2),d3z2−r2=ψ0.\nTheir contribution to the partial DOS around a central\nabsorber ion in the the cluster is shown in Fig. 2.\nFor left-handed light (∆ m= +1 here), the excitation\nto amk= 1,↓state (t↓\n2g) (ﬁrst row in Table I) is more\nprobable than photoexcitation of the other spin down\ncomponent of the other sublevel63. The numerical evalu-\nation of the decay matrix elements for diﬀerent orbital\ncontributions, similarly to earlier investigations60,65,66,\nallows to select the dominant transitions (in bold in Ta-\nble I), and it partially reﬂects the reasonable result that\nthe decay is more favourable if the two involved valence\nand conduction electrons have the maximum number of\nequal quantum numbers, as in this case they will repeal\nmore. The decayleadingto a t↑\n2gﬁnalhole with mv=±1\n(dxz,dyz) gives the strongest contribution, making a dis-\ntinction between diﬀerent orbitals in the DOS around\nthe absorber ion. Indeed, considering the localized na-ture of the recombination, such DOS unravels the or-\nbital character of the decaying states better than the\nDOS of the whole cluster, revealing narrow and prou-\nnounced peaks from diﬀerent orbitals of the two irre-\nducible representations in the spin up main peak, re-\nminding of Van Hove singularities in the extended elec-\ntronic structure67,68. Angular momentum conservation\nrules then dictate a Y↓\n33emitted wave, with strong in-\ntensity reduction along the quantization axis, similarly\nto the one expected in direct photoemission from a d-\nshell (Clebsch-Gordan coeﬃcients indeed give the high-\nest probablity for a ﬁnal m=±3 state generated by pho-\nton absorption at the mv=±2 states) and in line with\nprevious reports on aligned f±3emitted waves for diﬀer-\nent compounds32. For right-handed light (∆ m=−1),\nthe absorption is equally probable at the two spin down\ncomponents of the two mixed spin character63sublevels.\nHowever, again the numerical evaluation of the product\nof the matrix elements suggests distinct contributions to\nthe decay, notably a decreasing contribution from the\ndxzvalence states and a stronger one from the e↑\ngstates\nwithmv= 0 (d3z2−1). This leads to a ∼Y↓\n30emitted\nwave, indeed twisted by 90owith respect to the ∼Y3±3\nbehaviour expected in usual photoemission by left/right\npolarization. At the spin ﬂip energy, the eﬀect seems\nabsent, due to a stronger e↑\ng-t↑\n2ghybridization and the\ncontribution from more than one orbital of the same ir-\nreducible representation (the dxz,dyzorbitals of the t↑\n2g).\nThis leads to more balanced contributions of mlwaves\nand to a petal-like structure.\nThe results are the ﬁrst demonstration that RPES is\nsensitivetothe veryorbitalnatureofthe groundstate, as\nforelongatedorbitals( d3z2−1) a diﬀerent type ofspin-ﬂip\ntransition (mixed with an orbital ﬂip) is allowed, con-\ntrary to the planar x2−y2and interaxial t2gorbitals,\nsimilarly to what previously observed in RIXS69. The\nphenomenon indeed reminds of the (local) orbital exci-\ntations (local ddexcitations) often studied by RIXS via\nchanges in the polarization of the scattered light. Here\nsuch excitations manifest themselves as deviations from\nthe anisotropy expected in usual photoemission and can\naccompany spin ﬂip satellites in the spectra, even when\nhidden in the quasiparticle peak. Contrary to ARPES,\nthe photoelectron wave then reﬂects exactly the orbital\ncharacter of the valence state, allowing to map the va-\nlence orbital symmetries via monitoring the angular dis-\ntribution of the resonant current of opposite spin.\nFor the aim of accessing correlated orbitals and under-\nstanding the very nature of their resonant excitations,\nan important observation has to be done: the (exchange-\ninduced) spin ﬂip-orbital ﬂip excitations involve an e↑\ng\nhole which, being in a completely ﬁlled majority spin\nband, is more localized than those in the partially ﬁlled\nminority spin. These more localized valence ﬂip excita-\ntions are then transferred to the photoelectron. The visi-\nble orbital ﬂip eﬀect is thus a manifestation of a diﬀerent\ncorrelation in the two bands with diﬀerent spin, estab-\nlished recently on a quantitative basis by experimental6\nand theoretical studies on Auger emission26, and of dif-\nferent orbital character, as earlier suggested70. Orbitals\nappear nearly as quenched far from EF, where only spin\nﬂip excitations are clear, while spin and orbital degrees\nof freedom are entangled and both active at low energy.\nThis has three fundamental implications. First, it is\nrelevant to underline that, at least in the normal Auger\ndecay, spin ﬂip transitions are not expected to remem-\nber of the photon angular momentum in a two step pro-\ncess and should be alwaysbalanced by an orbital ﬂip to\nconserve the total angular momentum ∆ Jz= 0 due to\nthe scalar nature of the Coulomb interaction. The re-\nsults here suggest that, at resonance and in a one-step\napproach, spin ﬂip transitions might not be always ac-\ncompanied by orbital ﬂip (as it occurrs at the energy\nof the spin ﬂip satellite) and that, even when occurring\nwith orbital ﬂip, as in correspondance of the elongated\n(and more localized) d3z2−1, there is a memory on the\nphoton polarization. This suggests that both the Raman\nshift and the possible memoryon the polarizationas seen\nin the angular distributions should be considered when\ntrying to make a distinction between localized and delo-\ncalized excitations. Second, despite the local crystal ﬁeld\ndescription used here, the results suggest that in a gen-\neral more complex superexchange scenario, the counter-\npart collective excitations (magnons and orbital waves)\nmight also be accessed. This however would require a\nmapping of two-dimensional patterns for diﬀerent detun-\ning energies from the resonance, such to distinguish inco-\nherent particle-hole excitations from collective modes via\ntheir dependence/independence on the photon energy17.\nThird, the observed entangled spin-orbital physics in the\negband of Fe due to enhanced correlations suggests that\nprecursor traces of the non-Fermi liquid behaviour ob-\nserved at extreme PT15and ambient16conditions can\nbe traced even in the phase of ideal PTconditions, often\nthough of insigniﬁcant correlations. Notably, the entan-\ngled spin and orbital degrees of freedom get active at\nthe narrow egpeak nearEF, reminiscent of a Van Hove\nsingularity67,68in the electronic structure, indeed earlier\ninvoked to be partially responsable for the above men-\ntioned instabilities.\nAt last, an important practical implication is brought\nby the fact that the ﬂip eﬀect has an atomic nature, as\nshown by the spin down source waves patterns (Fig. 1),\nand it disappears for the spin unpolarized phase (Fig.3).\nThis demonstrates the sensitivity of RPES to spatial lo-\ncalization, due to the dominance of on-site transitions50\ncaused by the 1/ rbehaviour of the Coulomb operator\nand by the localization of the excited core orbital, open-\ningthepathforelementallysensitiveimagingofmagnetic\ndomains. Practical implementations might well involve\ncutting-edge techniques such as spectromicroscopy71,\nwith energy, angle and high lateral resolution, opening\nthe route for magnetic tomographic photoemission.\nThe situation changes drastically when the core hole\npolarization changes, i.e. when the photon helicity and\nthelocalmagneticmomentareorienteddiﬀerently. Fig.4\nFIG. 3: Spin polarized PED and RPED patterns for paral-\nlel geometry, for excitation at the L3edge for paramagnetic\nFe(010), photon energy at the maximum of the resonance and\ninitial state energy corresponding tothe main peak in thesp in\nup channel for the ferromagnetic phase.\nFIG. 4: PED (P) and RPED (R) patterns for two perpendic-\nular geometries for left (-l) and right (-r) polarization.\nreports the patterns for twodifferent perpendicular ge-\nometries (light impinging perpendicularly to the magne-\ntization), for which the dichroism in absorption is nulla\nbut the core hole polarization (now both the deviation\nfrom sphericity of the charge density and its rotation)\ndoes inﬂuence diﬀerently the emission for left and right\nhanded light. As the incident light direction is rotated7\naway from the quantization axis, the selection rules will\nactually now allow a mixture of ∆ m= 0,±1 transitions\nand thus a detailed microscopic analysis oforbital contri-\nbutionsismorecomplicated. However,someclearfetures\ncan be observed. For grazing incidence, (only the main\npeak energy is considered), the spin down RPED pat-\nterns again deviate from the direct ones, and exhibit a\nrotation between the two polarizations, though diﬀerent\nfrom the previous 90oﬂip. Interestingly, when the light\nis impinging perpendicularly to the surface, and thus\nthe scan around the surface normal coincides with a scan\naround the photon incidence direction, vortex-like fea-\ntures, given by crosses of higher intensity with bending\narms following the counterclockwise (clockwise) rotation\nof the electric ﬁeld for left (right) handed light, appear\nfor speciﬁc channels. Such eﬀect, called circular dichro-\nism in angular distributions and previously observed in\ndirect photoemission even from non magnetic and non\nchiral structures31,72,73, is due to forwardscattering peak\n”rotations” related to the mlof the emitted wave, and is\nhere unveiled to be correlated with local valence orbital\nsymmetries. Emissionfromthe t2g(spin down(up) emis-\nsion for the main (spin ﬂip) peak energy), diﬀerentiating\nfrom theegstates by non isotropic combinations of mls,\ncan easily favour non balanced combinations with prefer-\nence towards ±mlin the continuum wave, according to\nphoton’s helicity. Chirality in the patterns thus remains,\nas the emitted wave is now oriented (the asymmetries do\nnot cancel when summing over its mlcomponents). At\nthe spin ﬂip energy, the spin down channel corresponds\nto emission from mixed eg-t2gstates, and again a petal-\nlike pattern appears. For the resonant patterns, orbital\ntwists are weakened or absent, suggesting smaller contri-\nbutions of spin ﬂip terms and a delocalized valence hole.\nV. CONCLUSIONS\nIn summary, this work presents the exciting prospect\nof a new generation of resonant photoemission experi-\nments, capable to probe simultaneously the spin polar-ization, the (energy resolved) local valence orbital sym-\nmetries and the orientation of local magnetic moments,\nexploiting the core hole polarization as a prism to access\nspin and orbital excitations.\nThe results suggest that the combined analysis of\nangle-resolved resonant photoemission energy spectra\nand diﬀraction patterns can give profund insights into\nthe physics of many fascinating materials. In case of\nFe, a coupling between spin and orbital degrees of free-\ndom near the Fermi level is reported, suggesting it as\ncrucial element in the developement of a uniﬁed theory\nof magnetism encompassing both the localized moments\nand the itinerant behaviour picture for this system. The\naccess to the corresponding diﬀerent excitations accord-\ning to the local orbital symmetry and degree of local-\nization would allow for example to probe metal-oxygen\nand metal-metal orbital hybridizations for diﬀerent en-\nergies in oxides, and to probe the competition between\nelectron localization and delocalization in Mott insula-\ntors and correlated metals. The work obviously also sug-\ngests that matrix elements eﬀects have to be considered\nin the description of resonant photoemission, which nec-\nessarily has to go beyond interpretations based on sole\nspectral functions or estimations of matrix elements av-\neraged over the full valence region. Last, the results also\nchallenge the more conventional use RIXS to probe spin\nand orbital physics, opening the doors for a possible ex-\nploration of both incoherent particle-hole and collective\nmagnetic excitations also via the non radiative resonant\nchannel.\nAcknowledgments\nThe author acknowledges fruitful discussions with P.\nKr¨ uger at the very early stage of this work, the ﬁ-\nnancial support from the EU (Marie Curie Fellowship,\nFP7/2007-2013,Proposal No 627569) and the COST ac-\ntion MP1306: Modern Tools for Spectroscopy on Ad-\nvanced Materials.\n1Y. Tokura and N. Nagaosa, Science 288, 462 (2000)\n2K.I. Kugel and D.I. Khomskii, Sov. Phys. JETP 52, 501\n(1981)\n3W.Brzezicki, J.DziarmagaandA.M.OlesPhys.Rev.Lett.\n109, 237201 (2012)\n4Z.P. Yin, K. Haule and G. Kotliar, Nature Physics 10, 845\n(2014)\n5R.B. Laughlin, Phys. Rev. Lett. 112, 017004 (2014)\n6V. Aji and C.M. Varma, Phys. Rev. B 75, 224511 (2007)\n7W. Witczak-Krempa, G. Chen, Y.-B. Kim and L. Balents,\nAnnual Review of Condensed Matter Physics 5, 57 (2014)\n8H. Zhang, C.-X. Liu, and S.-C. Zhang Phys. Rev. Lett.\n111, 066801 (2013)\n9I. Zeljkovic, Y. Okada, C.-Y. Huang et al., Nature Physics10, 572 (2014)\n10J. van Wezel, Europhys. Lett. 96, 67011 (2011)\n11T. Ritschel, J. Trinckauf, K. Koepernik, B. Bchner, M.v.\nZimmermann, H. Berger, Y.I. Joe, P. Abbamonte and J.\nGeck, Nature Physics 11, 328 (2015)\n12P.A. Bhobe et al. Phys. Rev. X 5, 041004 (2015)\n13F. Da Pieve, S. Di Matteo, T. Rangel, M. Giantomassi,\nD. Lamoen, G.-M. Rignanese, and X. Gonze, Phys. Rev.\nLett. 110, 136402 (2013)\n14A. Georges, L. de’ Medici, J. Mravlje, Annual Reviews of\nCondensed Matter Physics 4, 137-178 (2013)\n15A.A. Katanin, A.I. Poteryaev, A.V. Efremov et al., Phys.\nRev. B 81, 045117 (2010)\n16L.V. Pourovskii et al., Phys. Rev. B 87, 115130 (2013)8\n17M. Minola, G. Dellea, H. Gretarsson, Y.Y. Peng, Y. Lu, J.\nPorras, T. Loew, F. Yakhou, N.B. Brookes, Y.B. Huang,\nJ. Pelliciari, T. Schmitt, G. Ghiringhelli, B. Keimer, L.\nBraicovich, and M. Le Tacon, Phys. Rev. Lett. 114, 217003\n(2015)\n18D.Benjamin, I.KlichandEugeneDemler, Phys.Rev.Lett.\n112, 247002 (2014)\n19A.Damascelli, Z.HussainandZ.-X.Shen, Rev.Mod.Phys.\n75, 473 (2003)\n20L.J.P. Ament, M. van Veenendaal, T.P. Devereaux, J.P.\nHill and J. van den Brink, Rev. Mod. Phys. 83, 705 (2011)\n21M. Aichhorn, S. Biermann, T. Miyake, A. Georges, and M.\nImada, Phys. Rev. B 82, 064504 (2010)\n22C.-Z. Xu, Y. Liu, R. Yukawa, L.-X. Zhang, I. Matsuda,\nT. Miller, and T.-C. Chiang Phys. Rev. Lett. 115, 016801\n(2015)\n23Z.-H. Zhu, C. N. Veenstra, G. Levy, A. Ubaldini, P. Syers,\nN. P. Butch, J. Paglione, M. W. Haverkort, I. S. Elﬁmov,\nand A. Damascelli Phys. Rev. Lett. 110, 216401 (2013)\n24J. Sanchez-Barriga, A. Varykhalov, J. Braun, et al., Phys.\nRev. X 011046 (2014)\n25C.-H. Park and S.G. Louie, Phys. Rev. Lett. 109, 097601\n(2012)\n26R. Gotter, G. Fratesi, R.A. Bartynski et al,. Phys. Rev.\nLett.109, 126401 (2012)\n27R. Gotter, F. Oﬃ, F. Da Pieve, A. Ruocco, G. Stefani, S.\nUgenti, M. Trioni, et al., Journ. Electr. Spectr. Rel. Phen.\n161, 128 (2007)\n28F. Da Pieve, D. S´ ebilleau, S. Di Matteo, R. Gunnella, R.\nGotter, A. Ruocco, G. Stefani and C.R. Natoli, Phys. Rev.\nB78, 035122 (2008)\n29J. Wider, F.Baumberger, M. Sambi, et al. Phys.Rev.Lett.\n86, 2337 (2001)\n30T. Greber and J. Osterwalder, Chem. Phys. Lett. 256, 653\n(1996)\n31H. Daimon, T. Nakatani, S. Imada and S. Suga, J. Electr.\nSpectr. Rel. Phen. 76, 55 (1995)\n32D. E. Ramaker, H. Yang and Y.U. Idzerda, J. Electr.\nSpectr. Rel. Phen. 68, 63 (1994)\n33T. Greber, J. Osterwalder, D. Naumovic, A. Stuck,\nS.H¨ ufner and L. Schlapbah, Phys. Rev. Lett. 69, 1947\n(1992)\n34P. Kr¨ uger, J. Jupille, S. Bourgeois, B. Domenichini, A.\nVerdini, L. Floreano and A. Morgante, Phys. Rev. Lett.\n108, 126803 (2012)\n35H. Magnan et al., Phys. Rev. B 81, 085121 (2010)\n36M. Morscher, F. Nolting, T. Brugger and T. Greber, Phys.\nRev. B84, 140406 (2011)\n37L.C. Davis and L.A. Feldkamp, Phys. Rev. B 23, 6239\n(1981)\n38H. Arai and T. Fujikawa, Phys. Rev. B 72, 075102 (2005)\n39T. Aberg, Physica Scripta 21(1980) 495\n40F. Gel’mukhanov, H. ˚Agren, Physics Reports 312, 87\n(1999)\n41A. Tanaka and T. Jo, J. Phys. Soc. Jpn. 63, 2788 (1994)\n42A. Kotani, J. Appl. Phys. 57, 3632 (1985)\n43F.M.F de Groot, J. Electr. Spectr. Rel. Phen. 92, 207(1998)\n44G. van der Laan and B.T. Thole, J.Phys. Condens. Matter\n7, 9947 (1995)\n45L. Ament, G. Ghiringhelli, M. Moretti Sala, L. Braicovich\nand J. van den Brink, Phys. Rev. Lett. 103, 117003 (2009)\n46M. van Veenendaal, Phys. Rev. Lett. 96, 117404 (2006)\n47F.M.F. de Groot, P.Kuiper and G.A. Sawatzky, Phys. Rev.\nB57, 14584 (1998)\n48M.W. Haverkort, Phys. Rev. Lett. 105, 167404 (2010)\n49J. Sachez-Barriga, J. Fink, V. Boni, I. Di Marco, J. Braun,\nJ. Minar, A. Varykhalov, O. Rader, V. Bellini, F. Manghi,\nH. Ebert, M. I. Katsnelson, A. I. Lichtenstein, O. Eriks-\nson, W. Eberhardt, and H. A. D¨ urr, Phys. Rev. Lett. 103,\n267203 (2009)\n50F. Da Pieve and P. Kr¨ uger, Phys. Rev. Lett. 110, 127401\n(2013)\n51P. Glatzel, A. Mirone, S. G.Eeckhout, M. Sikora, and G.\nGiuli, Phys. Rev. B 77, 115133 (2008)\n52D. S´ ebilleau, R. Gunnella, Z.-Y. Wu, S. Di Matteo and C.\nR. Natoli, J. Phys.: Condens. Matter 18, R175 (2006)\n53see G. Fratesi, et al., Phys. Rev. B 78, 205111 (2008) and\nreferences therein\n54B. Sinkovic, E. Shekel, and S.L. Hulbert, Phys. Rev. B 52,\nR15703(R) (1995)\n55F. Da Pieve, S. Fritzsche, G. Stefani, N.M. Kabachnik,\nJourn. Phys. B: At. Mol. Opt. Phys. 40, 329 (2007)\n56F. DaPieveandP. Kr¨ uger, Phys.Rev.B 88, 115121 (2013)\n57O.K. Andersen and O. Jepsen, Phys. Rev. Lett. 53, 2571\n(1984)\n58J. Schwitalla and H. Ebert, Phys. Rev. Lett. 80, 4586\n(1998)\n59T. Wegner, M. Potthoﬀ, and W. Nolting, Acta Physica\nPolonica A 97, 567 (2000)\n60Yu Kucherenko and P. Rennert, J. Phys. Condens. Matter\n9, 5003 (1997)\n61P. Kr¨ uger, F. Da Pieve, and J. Osterwalder, Phys. Rev. B\n83, 115437 (2011)\n62A.K. See and L.E. Klebanoﬀ, Surf. Sci. 340, 309 (1995)\n63G. van der Laan, Appl. Phys. A 65, 135 (1997) The oppo-\nsite convention for polarization is used in this work.\n64L.J. Terminello and J.J. Barton, Science 251, 1218 (1991)\n65R. Gotter, F. Da Pieve, F. Oﬃ et al., Phys. Rev. B 79,\n075108 (2009)\n66R. Gotter, F. Oﬃ, A. Ruocco, F. Da Pieve, R. Bartynski,\nM. Cini, G. Stefani, Europhys. Lett. 94, 37008 (2011)\n67R. Maglic, Phys. Rev. Lett. 31, 546 (1973)\n68V.Y. Irkhin et al., J. Phys. Cond. Matt. 5, 8763 (1993)\n69P. Marra, AIP Conference Proceedings 1485, 297 (2012)\n70J.B. Goodenough, Phys. Rev. 120, 67 (1960)\n71J. Avila and M.C. Asensio, Synchrotron Radiation News\n27, 24 (2014)\n72F. Matsui, T. Matsushita, F.Z.Guo and H. Daimon, Surf.\nRev. Lett. 14, 1 (2007)\n73N. Mannella, S.-H. Yang, B.S. Mun, L. Zhao, A.T. Young,\nE. Arenholz, and C.S. Fadley, Phys. Rev. B. 74, 165106\n(2006)"    },    {        "title": "0910.2442v1.Resonant_coupling_of_coplanar_waveguides_with_ferromagnetic_tubes.pdf",        "content": "A. Kozhanov et al. October 2009 1\nResonant coupling of coplanar wave guides with ferromagnetic tubes. \n \nA. Kozhanov1, D. Ouellette1, M. Rodwell1, D. W. Lee2, S. X. Wang2 and S. J. Allen1 \n1California Nanosystems Institute, University of Ca lifornia at Santa Barbara, Santa Barbara, CA, 93106 \n2Department of Materials Science and Engineer ing, Sanford Univers ity, Stanford, CA, 94305 \n \n(Received      ) \n \nResonant coupling of coplanar waveguides is explored by wrapping proximate shorted ends of the waveguides with micron \nsize ferromagnetic Co 90Ta5Zr5 tubes.  Ferromagnetic resonance and up to 7 outer surface modes are identified. Experimental \nresults for these contorted rectangular tubes are in good agreem ent with micromagnetic simula tions and model calculations \nof magnetostatic modes for an elliptical fe rromagnetic tube.   These results indicate that the modes are largely determined by \ntube topology and dimensions but less so by the detailed shape. (PACS: 76.50.+g) \n \nMonolithic micro and nanoscale filters, delay lines and \nresonators are potentially important for high frequency \nelectronics.  Here we explor e metallic ferromagnetic micron \nscale magnetostatic wave structures in the form of tubes \nwrapped around coplanar wave guides.  Microwave devices \nbased on magneto-static spin waves in insulating ferrimagnetic \nmaterials like yttrium-iron garnet (YIG)1 have long been \nexplored and developed.  Howeve r, future micro and nano scale \nspin wave based devices ma y benefit from exploiting \nferromagnetic metals that are more easily deposited, processed \nand nanofabricated than ferrim agnetic oxides.  Further, \nferromagnetic metals like CoTaZr, CoFe and CoFeB have nearly \nan order of magnitude larger  saturation magnetization than \ntypical ferrimagnets.2  As a result, they will support higher, \nshape defined, zero magnetic fiel d resonances and consequently \nintrinsically faster response. \nRecent theoretical3,4,5 and experimental6,7,8 work has \nfocused on the magnetization dynamics in ferromagnetic \nnanotubes. Ferromagnetic nanot ubes could serve as magnetic \ncores in nano scale transformers as well as active elements of \ntunable high frequency filters. Several theoretical models predict \nexistence of quantized surface modes of magnetostatic \noscillations in the ferromagnetic nano tubes magnetized along \nthe axis of the tube4,5. The experiments of Mendach et al. \nobserved ferromagnetic resonance in a Permalloy tube, the lowest order and essentially spa tially uniform mode.  They did \nnot report on the rich spectrum of  standing magneto-static waves \nthat circulate around the tube.\n8. \nThis letter describes excitati on and detection of quantized \nsurface magnetostatic oscillations in the ferromagnetic tubes \nformed by wrapping metallic fe rromagnetic film  around exciting \nand detecting coupling loops at e nds of co-planar waveguides.  \nResonances are displayed that are defined by the magnetostatic \nmodes indexed by periodic boundary conditions around the tube \ndespite their contorted geometry. Such a structure could form \nthe building block for filtering elements at microwave \nfrequencies.    \nThe ferromagnetic tube coupler was fabricated in the \nfollowing manner.  200nm thick ferromagnetic Co 90Ta5Zr5 films \nwere sputtered onto Si/SiO 2 wafers, lithographically patterned \ninto two xxx x xxx micron rectangles and covered with an \ninsulating SiO 2 layer. (A saturation magnetization of M s=1.2T \nand a coercive field H c~2 Oe were measured on an unpatterned \nCo90Ta5Zr5 film using a vibrating sa mple magnetometer.)   \nCoupling loops formed by the shorte d ends of a pair of coplanar \nwaveguides were positioned over Co 90Ta5Zr5 rectangles.  The structure was then covered with a 100 nm thick SiO 2 insulating \nlayer and holes etched to allow a subsequent top Co 90Ta5Zr5 \nlayer to complete the magnetic circuit. The top Co 90Ta5Zr5 layer \nwas sputtered on the resist covered structure and lifted-off. This \nprocess resulted in two shorte d coplanar waveguides wrapped \ntogether by a Co 90Ta5Zr5 film (Fig.1). The “tubes” thus formed \nare of course not circular but  topologically equivalent to a \ncylinder and provide a closed ma gnetic circuit (Fig. 1b,c) that \ncaptures the magnetic fields produced by the high frequency \ncurrents flowing in the shorted ends of the coplanar waveguides.  \nFocused ion beam etching (Fig.1 c) exposes the cross section \nimaged in the SEM micrograph.  \nThe closed magnetic circuit effectively couples the two \nshorted ends of the coplanar waveguides only when the tube \nmagnetization is oriented along the axis of the tubes, along the \nshorting lines of the coplanar waveguides.  Only then can the \nmicrowave magnetic fields induce changes in the magnetization \nand couple to the magnetostatic oscillations in the tubes. \n \n \n \nFig. 1.  Fabricated structure SEM micrograph (a), profile scheme(b), \nSEM micrograph of structure cross section.(c) A. Kozhanov et al. October 2009 2\nS-parameters were measured at room temperature using \nAgilent 8720ES vector network analyzer operating from 0.05 to \n20 GHz.  Only S 21, the ratio of high frequency voltage at \nterminals 2 to the input high freque ncy voltage at terminals 1, is \nanalyzed in the following discu ssion.  The test devices were \npositioned on the narrow gap of  small electro-magnet that \nprovided magnetic field bias up to 1000 Oe. By comparing the \nS-parameters at disparate bias  magnetic fields, the magnetic \nfield independent instrument response can be effectively \nremoved to expose the S-parameters related to the magneto-\nstatic mode coupling of exciting a nd detecting wires. See Fig. 2.  \n \n \nIn the absence of external magnetic field, () 21Sf  has a \nnumber of irreproducible peaks that are strongly dependent on \nthe history of the bias magnetic field. With increase of the \nexternal magnetic field directed along the axis of the tube ( Hx), \nthe magnitude of these peaks decreases and related magnetic \ncoupling disappears at Hx~100 Oe.  \nAt Hx ≥ 200Oe we detect a strong peak followed by a series \nof smaller peaks. These peaks shift towards higher frequencies \nand grow in magnitude with increase of Hx. Transmission at the \nlowest frequency peak reaches | S21|~0.012  at Hx=988 Oe. \nThese results suggest that at ~ 100-200 Oe there is a change in \nthe magnetization distribution.  Micromagnetic simulations were \ncarried out to investigate the magnetization alignment within our \nstructures at magnetic field values |H| ≤1kOe. We used a \nrectangular Co 90Ta5Zr5 tube as the model with dimensions \nsimilar to the dimensions of the fabricated ferromagnetic tubes.  \nMicromagnetic structure was si mulated by solving the Landau-\nLivshitz-Gilbert equation using LLG Micromagnetics Simulator. \nThe results of the simulations are shown on Figures 3 \nand 4. The ground state at H=0 is described by a double vortex7 \nstate. The magnetization is circ ularly oriented around the tube \nperimeter pointing in a clockwise direction on one end, counter-\nclock wise direction on the other end, with a domain wall in the \ncenter. This is similar to what was found by Lee et al. for \ncircular ferromagnetic nanotubes7.  For the rectangular tube \nsome of the quasi stable states apparent in the in  circular tube at H=0 are not observed  (The “ferromagnetic” state7 with \nmagnetization aligned along the tube is not stable at H=0 in the \nparticular rectangular tube used  in these experiments.)      \n \n \n \n \nIn our experiment the paired vortex state of our \nferromagnetic tube will result in a complex configuration in \nwhich most of the magnetic moments are parallel to the exciting \nhigh frequency magnetic fields produced by the RF currents in \nthe waveguide. Only the very narrow area of the domain wall \nbetween the paired vortex states magnetic moments will have \nsome alignment along the tube axis and couple to the exciting \nmagnetic fields.  We speculate th at they are responsible for the \nirreproducible low amplitude peaks in () 21Sf  at H=0.  \nThe simulation indicates that with sufficient field, Hx, the \ntube is magnetized along the axis ; this is indicated by the \nsaturation in the hysteresis curve (Fig.4). This proceeds either by \ndomain wall widening or formati on of in plane magnetization \nvortex.  The details are not impo rtant for the present discussion \nbut following the jump of the M(H)  curve at H≈200 Oe we \nassume the magnetic moments are largely pointing along the \ntube axis. Then the high frequency magnetic field is \nperpendicular to th e magnetic moments and can excite the \nvarious standing magnetostatic waves.  \nFig. 3.  Schematic of magnetization alignment in the tube at H x=0. \nFig. 4.  Results of micromagnetic simulations: hysteresis curve of the \nrectangular ferromagnetic tube. \nFig. 2.  Frequency and magnetic field dependence of |S21| measured\nwith the fabricated structure. A. Kozhanov et al. October 2009 3\nStarting from magnetic field valu es of ~200 Oe we detect a \nseries of resonant peaks in () 21Sf  whose frequency increases \nwith increasing Hx (Fig.6). The development of the systematic \nstructure above 200 Oe shown in Fig. 6 confirms the results of \nthe simulation: above 200Oe, the magnetization of the \nrectangular tube is well al igned with the tube axis. \nWe identify up to 8 peaks in () 21Sf  and display them \nversus external magnetic field in Figure 6. We use model by \nPopov and Zavislyak5 to analyze the experimental data. This \nmodel calculates standing magne tostatic wave modes in a \nferromagnetic tube with elliptic al profile magnetized along the \ntube axis. Two classes of solution appear: standing spin waves \non the inner and on the outer surfaces of the tube. We assign the \nobserved peaks to the 7 outer  standing magnetost atic modes plus \nthe lowest, and essentially unif orm, ferromagnetic resonance. \nThe frequency for the latter uniform mode of the elliptical \nnanotubes coincides with the fr equency of the ferromagnetic \nresonance8 of an infinite ferromagnetic film and described \nby1/2~( 4 )fHMγπ+ . The other modes describe standing \nmagnetostatic waves with a whole number of wavelengths \nwithin the outer perimeter of the tube. The inner modes shown \nwith the dashed lines in the Fig. 6 do not seem to appear in the \nexperimental data.  We can offer no explanation. \n \n \nThe profile of the fabricated ferromagnetic tubes is not at \nall elliptical but has gross distortions where the CoTaZr film \ngoes over the coupling loops.  Re markably, we conclude that \nresponse is controlled by the to pology and scale: the detailed \nshape is not critical.  \nIn summary, we fabricated and measured microwave \ntransmission through coplanar waveguides coupled by novel \nrectangular ferromagnetic tubes.  We identified ferromagnetic \nresonance and up to 7 outer surf ace magnetostatic oscillation \nmodes guided by models of the magnetostatic oscillations in \nferromagnetic ellipti cal nanotubes. \nThese structures are potentially  important as high frequency \ntunable filters. The frequency of th e lowest and strongest peak is \nthe ferromagnetic resonance as described earlier. The frequency \nof the higher modes are defined by  the tube geometry (diameter, \nwall widths). As interesting as th e surface modes are, the strong \nferromagnetic mode is probably the most useful.    In order to make the filter work efficiently we will increase \nthe input inductance of the filter either by increasing the number \nof exciting wires inside the magnetic tube (winding) or more \nconveniently by increasing the t ube length while keeping other \ndimensions the same. Lengthening the tubes will introduce \nstrong shape anisotropy.  Th e ground state will have the \nmagnetization along the tube without external bias and a strong \nresonance transmission at H=0 .  Resonance could be fine tuned \nby “on circuit board” fields or self fields produced by DC \ncurrents flowing in the wires encased by the tubes. \nThe authors are grateful to Andrew Cleland for the use \nof the vector network analyzer and hosting aspects of this work \nin his laboratory.  This work is supported by NERC via the \nNanoelectronics Research Initiative (NRI), by Intel Corp. and \nUC Discovery at the Western Institute of Nanoelectronics \n(WIN) Center. \n                                                                 \n1 A review: J.D. Adam, L.E. Davis,  G.F. Dionne, E.F. Schloemann and \nS.N. Stitzer, IEEE. Trans.  Microwave Theory Tech. 50, 721 (2002). \n2 B.Kuanr, I.R. Harward, D.L. Marvin, T, Fal, R.E. Camley, D.L. Mills, \nand Z. Celinski, IEEE Trans. Magnetics, 41, 3538 (2005). \n3 H.Leblond, V.Veerakumar, Phys. Rev.B 70, 134413 (2004) \n4 T.M. Nguyen, M.G. Cottam Surface Science 600, 4151 (2006) \n5 M.A. Popov, I.V. Zavislyak ISSN 0503-1265, Ukr. J. Phys., 53, 7, 702 \n(2008) \n6 F.S.Li, D. Zhou, T. Wang, L.J. Song, and C.T. Xu, J. Appl. Phys, 101, \n014309 (2007) \n7 J. Lee, D. Suess, T. Schrefl, K. Oh, J. Fidler JMMM 310 (2007) 2445-\n2447. \n8 S. Mendach, J. Podbielski, J. Topp,  W. Hansen and D. Heitmann, Appl. \nPhys. Lett. 93, 262501 (2008) \n0 200 400 600 800 100002468101214161820f, GHz\nH, Oeinnerouter\nFig. 6.  |S 21| peaks frequency dependence on the applied magnetic field: \nexperiment (circles), theory for t ubes with elliptical profile (lines) 5. "    },    {        "title": "1301.3802v2.Dynamics_of_localized_modes_in_a_composite_multiferroic_chain.pdf",        "content": "arXiv:1301.3802v2  [cond-mat.mes-hall]  9 Sep 2013Dynamics of localized modes in a composite multiferroic cha in\nL. Chotorlishvili1, R. Khomeriki2,3, A. Sukhov1, S. Ruﬀo4and J. Berakdar1\n1Institut f¨ ur Physik, Martin-Luther Universit¨ at Halle-W ittenberg, D-06120 Halle/Saale, Germany\n2Physics Department, Tbilisi State University, 0128 Tbilis i, Georgia\n2Max-Planck Institute for the Physics of Complex Systems, N¨ othnitzer Str. 38, 01187 Dresden, Germany\n4Dipartimento di Fisica e Astronomia and CSDC, Universit` a d i Firenze,\nCNISM and INFN, via G. Sansone, 1, Sesto Fiorentino, Italy\n(Dated: November 5, 2018)\nIn a coupled ferroelectric/ferromagnetic system, i.e. a co mposite multiferroic, the propagation\nof magnetic or ferroelectric excitations across the whole s tructure is a key issue for applications.\nOf a special interest is the dynamics of localized magnetic o r ferroelectric modes (LM) across the\nferroelectric-ferromagnetic interface, particularly wh en the LM’s carrier frequency is in the band\nof the ferroelectric and in the band gap of the ferromagnet. F or a proper choice of the system’s\nparameters, we ﬁnd that there is a threshold amplitude above which the interface becomes trans-\nparent and a band gap ferroelectric LM penetrates the ferrom agnetic array. Below that threshold,\nthe LM is fully reﬂected. Slightly below this transmission t hreshold, the addition of noise may lead\nto energy transmission, provided that the noise level is not too low nor too high, an eﬀect that\nresembles stochastic resonance. These ﬁndings represent a n important step towards the application\nof ferroelectric and/or ferromagnetic LM-based logic.\nPACS numbers: 85.80.Jm, 75.78.-n, 77.80.Fm\nIntroduction .- Multiferroics (MF) possess coupled fer-\nroic (magnetic, electric, or elastic) ordering [1–3]. The\ncurrent high interest in MF is fueled by the impres-\nsive advances in synthesizing composite ferroelectric\n(FE)/ferromagnetic (FM) nano and multi layer struc-\ntures. These show a substantially larger multiferroic\ncoupling strength [1–5] as compared to bulk matter, so-\ncalled single-phase multiferroics [1, 6] such as Cr 2O3[7].\nMFs are important for addressing fundamental questions\nregarding the connection between electronic correla-\ntion, symmetry, magnetism, and polarization. They also\nhold the promise for qualitatively new device concepts\nbasedonexploitingthemagnetoelectric(ME)couplingto\nsteer magnetism (ferroelectricity) via electric (magnetic)\nﬁelds. Potential applications are wide and range from\nsensoricsand magnetoelectric spintronics to environmen-\ntally friendly devices with ultra low heat dissipation [8–\n10]. Thereby, a key issue is how eﬃciently magnetic\nor ferroelectric information, i.e. an initial excitation, is\ntransmitted in a system with a MF coupling. For in-\nstance, in a two-phase or composite MF [1, 6, 11] such as\nBaTiO 3/CoFe 2O4[12], PbZr 1−xTixO3/ferrites [13, 14],\nBaTiO 3/Fe [15], PbTiO 3/Fe [16, 17] or BaTiO 3/Ni\nthe MF coupling is strongest at the FE/FM interface,\nwhereas away from it the FE or FM order is only\nmarginallyaﬀected. Thus, weexpect that a ferroelectric\nsignal triggered by an electric ﬁeld in the FE part may\nor may not be converted into a magnetic signal depend-\ning on the dynamics taking place at the interface. How\nthis transport of information depends on the properties\nof the system is rarely studied and will be addressed in\nthis Letter. The outcome of such a study would not only\nuncover the conditions for optimal signal handling but\nalso holds the potential for new insights into the multi-ferroic coupling retrieved by tracing the signal dynamics.\nWe will focus on weakly nonlinear localized modes (LM)\nwhich are formed by a modulation of linear excitations\nof the ferroelectric and the ferromagnetic systems. Such\nnonlinear modes have in isolated FM or FE phases a se-\nries of applications in magnetic logic, microwave signal\nprocessing, and spin electronic devices. A clear advan-\ntage is that LMs of a large number of elementary excita-\ntions are very robust and have a particle-like nature [18].\nIn that sense, LMs are very similar to their topological\ncounterparts (magnetic solitons) that have been consid-\nered for logic operations [19–21]. Multiferroics oﬀer new\nfascinating mechanisms for LM dynamics [23]. For ex-\n00\n00\nqsπωsDipolesx\n−π −π~B2\nqpπN−1 12 N 3 N+1 N+3\nSpins\nωpM−1M\nzN+2\nFIG. 1. (color online) Schematics of a chain consisting of a\nferroelectric and a ferromagnetic part coupled at the inter -\nface. The lower panel shows a particular choice of frequency\nfor which aconventional localized mode is formed in the ferr o-\nelectric. In the ferromagnetic part a bandgap localized exc i-\ntation develops with a nonlinear frequency shift proportio nal\ntoB2(Bis the amplitude of a magnetic band-gap localized\nmode). The chosen mutual alignment of the ferroelectric po-\nlarization and the magnetization at the interface resemble s\nthe realized ferroelectric (BaTiO 3) tunnel junction with fer-\nromagnetic (Fe) electrodes [22].2\nample, due to discretenessand/ornonlinearityofthe sys-\ntem, it may happen that the large-amplitude excitation\nfrequency falls within the gap of the linear oscillations\nspectrum, as illustrated in Fig.1. Then, the energy of\nthe excitation would not spread over the lattice. As well-\nestablished in studies on intrinsic LMs, e.g. [24–31], we\nknow that, in spite of the localized energy proﬁle, such\nmodes maymovealongthe whole chain. This meansthat\nexcitationscreatedin the ferroelectricpart viaan electric\nﬁeld can be transmitted to the magnetic part and move\nthere. Moreover, as it will be shown below, the creation\nof these band-gap LMs can be enhanced by noise, man-\nifesting thus some analogy with the stochastic resonance\nphenomenon [32, 33].\nModel.- For our purposes, a large ME coupling is nec-\nessary. In this respect, new fabrication methods [14]\nfor the so-called two-phase or composite multiferroics\n[1, 6, 11], as well as the realization of ferroelectric wires,\n[34] are encouraging. Examples of composite multifer-\nroics BaTiO 3/CoFe 2O4[12] or PbZr 1−xTixO3/ferrites\n[14] are still popular. Major research is focused on\nBaTiO 3/Fe [15], PbTiO 3/Fe [16, 17] or BaTiO 3/Ni com-\nposite multiferroics, to name but a few, since their bulk\nparameters are very well known (Ref. [35] for BaTiO 3\nand Ref. [36] for Fe or Ni) as well as the misﬁt of the\nlattices is relatively low [16]. Relatively high ME con-\nstants [16, 17] were predicted for these materials at room\ntemperatures. A possible mechanism for ME coupling at\nthe FE/FM interface is based on screening eﬀects [37].\nWe assume here the presence of a similar mechanism\nbased microscopically on the rearrangement of charges\nand spins at the FM/FE interface, as conﬁrmed by other\nstudies [5, 15]. The spin-polarized charge density formed\nin the FM in the vicinity of the FM/FE interface [37]\nacts with a torque on the magnetic moments in the FM,\nresulting in a non-collinear magnetic ordering (similar as\nin[38]). Hence, electricpolarizationemergesthatcouples\nthe FM to the FE part. This picture yields a linear ME\ncoupling with a pseudoscalar coupling constant . Tech-\nnically, we describe the bulk unstrained BaTiO 3by the\nGinzburg-Landau-Devonshire (GLD) potential [35]. For\nthe discretized FE polarization ( Pn) in a coarse-grained\napproach, the form of the GLD potential for a general\nphase and arbitrary temperatures is quite involved [39].\nHowever, at room temperature the BaTiO 3-crystal has\nan axis along which the polarization switches (tetragonal\nphase). Consequently, the form of the GLD potential re-\nduces to the one dimensional biquadratic potential. For\nthe description of the magnetization ( Sk) dynamics in\nthe FM, we employ the classical Heisenberg model. Skis\ndiscretized and normalized to the saturation value of the\ncoarse-grained magnetization vector. With the aim of\nexploring the feasibility of conversion of the electric exci-\ntation formed in FE part of the sample into a localized\nspin magnetic excitation in FM part, we thus employthe multiferroic model (cf. Fig. 1)\nH=HP+HS+VSP, (1)\nHP=N/summationtext\nn=1/parenleftBig\nα0\n2/parenleftbigdPn\ndt/parenrightbig2+α1\n2P2\nn+α2\n4P4\nn+κ\n2(Pn+1−Pn)2/parenrightBig\n,\nHS=M/summationtext\nk=N+1/parenleftBig\n−J1/vectorSk/vectorSk+1−J2(Sz\nk)2/parenrightBig\n, VSP=−gPNSx\n1,\nwhereHPis the Hamiltonian of the FE part of the\nmultiferroic system, describing N-interacting FE dipoles\n[30, 39] ( PnanddPn/dtare conjugated variables).\nPnand/vectorSkstand respectively for the deviations from\nthe equilibrium positions of the n-th dipole and the k-\nth spin vector. At room temperature, we can choose\nthe polarization vector to be directed along the xaxis\n/vectorPn= (Pn,0,0), n= 1,...,N.α0is a kinetic coeﬃcient,\nα1,2are potential constants and κis the nearest neigh-\nbor coupling constant. HSdescribes the ferromagnetic\nchain[40], where J1isthenearestneighborexchangecou-\npling in the FM part and J2is the uniaxial anisotropy\nconstant. Interface eﬀects between the spin and the FE\ndipole systems are described by the dipole-spin interac-\ntion Hamiltonian VSP.\nIn our numerical simulations we operate with dimen-\nsionless quantities upon introducing pn=Pn/P0,/vector sk=\n/vectorSk/Sand deﬁning a dimensionless time as t→ω0t\n(ω0=/radicalbig\nκ/α0∼1012Hz). The equations governing the\ntime evolution of the dipoles and the spins (except for\nthe sites near the interface) read\nd2pn\ndt2=−αpn−βp3\nn+(pn−1−2pn+pn+1) (2)\n∂s±\nk\n∂t=±iJ/bracketleftbig\ns±\nk/parenleftbig\nsz\nk−1+sz\nk+1/parenrightbig\n−sz\nk/parenleftbig\ns±\nk−1+s±\nk+1/parenrightbig/bracketrightbig\n±\n±2iDs±\nksz\nk (3)\nwheren/negationslash=Nandk/negationslash=N+1. We haveintroducedthe fol-\nlowing dimensionless constants α=α1/κ,β=α2P2\n0/κ,\nJ=J1S/ω0andD=J2S/ω0. For the dipole pNand the\nspin/vector s1at the interface the following equations hold\nd2pN\ndt2=−αpN−βp3\nN+(pN−1−2pN+gssx\n1),(4)\n∂s±\n1\n∂t=±iJ/bracketleftbig\ns±\n1sz\n2−sz\n1s±\n2/bracketrightbig\n±i/bracketleftbig\n2Ds±\n1sz\n1−gppNsz\n1/bracketrightbig\n.\nHeres±\nk≡sx\nk±isy\nk,gs=gS/(κP0) andgp=gP0/(Sω0).\nThe evolutionaccordingtoEqs. (2-4) proceedsunder the\nconstraint ( sx\nk)2+(sy\nk)2+(sz\nk)2= 1. For the derivation of\ntheweaklynonlinearenvelopesolutionsfromeqs. (2)and\n(3) one can rely on the reductive perturbation theory\ndeveloped in Ref. [41, 42]. One obtains the solutions\nfor the dipoles and the spins separately in the following\nform (a detailed derivation is provided as supplementary\nmaterial to this paper):\npn=Acos[ωpt−qpn+δωpt]\ncosh[(n−Vpt)/Λp]s±\nk=Be±i(ωst−qsk+δωst)\ncosh[(k−Vst)/Λs]\n(5)3\nwhereAandBare the amplitudes of the dipolar and the\nmagnetic localized excitations, respectively; ωpandωs\nare the frequencies of the linear excitations which obey\nthe following dispersion relations\nωp=/radicalBig\nα+2(1−cosqp), ωs= 2[D+J(1−cosqs)],\n(6)\nqpandqsare the carrierwavenumbers of the dipolar and\nthe spin excitations; Vp= sinqp/ωpandVs= 2Jsinqs\nare the group velocities of the corresponding LMs. The\nwidth of the dipolar and spin LMs are\nΛp=1\nA/radicalBigg\n2/parenleftbig\nω4p−α2−4α/parenrightbig\n3ω2pβ,Λs=1\nB/radicalbigg\n4Jcosqs\nωs.(7)\nThe nonlinear frequency shifts are deﬁned as\nδωp=A23β\n16ωp, δω s=−B2ωs\n4.(8)\nNote that, for wave packet transmission, the following\nmatching condition between the frequencies has to be\nfulﬁlled [ ?]\nωp+δωp=ωs+δωs. (9)\nFor an eﬃcient transmission of the LM from the FE\ninto the FM part, the widths of the LM should be the\nsame in both parts, i.e. Λ p= Λswith the restriction\nB≤gpA. If one excites the LM with a carrier fre-\nquencyωwhich is located within the band of both the\ndipolar and the spin wave spectrum, then the localiza-\ntion will safely penetrate from the FE to the FM part,\nbut some portion of the energy will be reﬂected by the\ninterface. By changing the amplitude of the LM, one\ncan manipulate the ratio between the transmitted and\nreﬂected parts of the LM. In addition, the transmission\nis very sensitive to the coupling constant gbetween the\nFE and the FM parts. We have investigated this de-\npendence by varying only the coupling constant gsand\nﬁxing the values of the dimensionless parameters as fol-\nlows:α= 0.2, β= 0.1, J= 1, D= 0.6. We assume\nfor simplicity gp=gs(in general these constants diﬀer,\ndepending on the material of the samples, but this is not\nan obstacle for the theory).\nNumerical Results .- Realistic material parameters are\ntabulated in full detail in the supplementary material\nsection. There, we provide explicitly the relation to the\nnormalized units which we use below. The essential pa-\nrameters entering eq. (1) are: the FE potential coeﬃ-\ncientsα1/(a3\nFE) = 2.77·107[Vm/C], α2/(a3\nFE) = 1.7·108\n[Vm5/C3], the FE coupling coeﬃcient κ/(a3\nFE) = 1.3·108\n[Vm/C], the equilibrium polarization P0= 0.265 [C/m2]\nand the coarse-grained FE cell size aFE= 1 [nm]. The\nFM exchange interaction strength is J1= 3.15·10−20[J],\nthe FM anisotropy constant is J2= 6.75·10−21[J], and\nthe ME coupling strength is g≈10−21[Vm2]. Fig. 2\nFIG. 2. (color online) Insets a), b) and c) show the time and\nsite dependence of the local energy. For dipoles this energy is\ngiven by the local values of HP, and for the spins by the local\nvalues of gpis a coupling constant indicating the strength of\nthe ME interface interaction. The graphs point out the LM\nreﬂection and transmission at the FE/FM interface (white\ndashed line). The dipolar localization carrier wave number is\nchosen as qp= 0.4πand the dipolar localization amplitude is\nchosen as A= 0.2. Dipoles and spins (separated by the white\ndashedline)occupythesites n= 1...150andk= 151...300,\nrespectively. d) Dependenceoftherelativeenergytransfe rred\nto the FM part (i.e. ratio of the energy in the FM part to the\ntotal injected energy) on the coupling constant strength gp.\na), b), c) show the localized energy evolution along the\nlattice for diﬀerent values of the coupling constant gp. In\ngraph d) the dependence of the transmitted energy on\nthe coupling constant is displayed, pointing out that the\ntransmission is maximal when gpis in between the spin\nand the dipolar coupling constants (in reduced units it is\nequal to 1). Spins alignments, the topology of the excita-\ntion, and its propagation in the chain at diﬀerent times\nare displayed in the supplementary material. Further in-\nteresting eﬀects arise when a band localized excitation\nforms with a carrier frequency ωin the band of the dipo-\nlar spectrum (see bottom panel of Fig. 1) and slightly\nbelow the zone boundary ωs(qs= 0) of the spin wave\nspectrum. Then, for small amplitudes, the dipolar LM\nis totally reﬂected by the interface because it does not\nresonate with any mode in the spin array. However, with\nincreasing the amplitude, there is a threshold value Acr\n(due to the nonlinear frequency shift) above which the\nLM is transmitted towards the FM part of the multifer-\nroic chain, forming thus a magnetic band-gap localiza-\ntion. Using Eqs. (8) and (9) and assuming B=A, one\ncan infer the relation deﬁning this threshold amplitude4\nFIG. 3. (color online) This ﬁgure illustrates the dependenc e\non the amplitude Aof the LM reﬂection and transmission at\nthe FE/FM interface. To this end we plot for diﬀerent values\nofAthe same quantity and choose the same parameters as\nin graphs a)-c) of Fig. 2 with gp=gs= 1. The scale is as in\nFig. 2.\nto be\nωs(qs= 0)−ω=/parenleftBigg\ng2\npω\n4+3β\n16ω/parenrightBigg\nA2\ncr.(10)\nBased on this observation, we proceed with the simu-\nlations according to Eqs. (2)-(4) with the set of param-\neters given above. We choose gp=gs= 1 and start\natt= 0 with a LM in the form of the ﬁrst expression\nin Eq. (5) with a carrier wave number qp= 0.37. For\nsuch a wave number the corresponding linear frequency\nisω= 1.1856. This frequency is located in the band\ngap of the spin wave spectrum and no localization trans-\nmission occurs in the case of small amplitudes, as it is\nseen from graph a) of Fig. 3. According to relation (10)\nwe can calculate the threshold amplitude for which lo-\ncalization transmission emerges and ﬁnd Acr= 0.22. In\nthe numerical results, transmission occurs for the inci-\ndent LM amplitudes A >0.27. This discrepancy can be\nexplained by the fact that localization amplitudes in dif-\nferent parts of the multiferroic structure do not exactly\ncoincide. In panel b) of Fig. 3 we display the dynamics\nFIG. 4. (color online) Inﬂuence of noise on the LM reﬂection\nand transmission at the FE/FM interface illustrated by re-\nalizing similar simulations as in Fig. 3 but for A= 0.265 and\nincluding diﬀerent noise levels as indicated on the graphs.\nScale as in Fig.2.\nfor a larger LM amplitude, i.e. A= 0.33, and ﬁnd that\nlocalizedexcitationsareformed in the ferromagneticpart\nas well. By further increasing the LM amplitude, the\ntransmitted localization takes over almost all the energy\nof the incident one (graph c) of the same ﬁgure). If the\namplitude of the incident LM is slightly below threshold\n(hereA= 0.265), even a small perturbation may cause\na transmission to the FM part. Thus, we add a term\n/vectorf(t)/vectorS1to the Hamiltonian (2) describing the action of\na random magnetic ﬁeld at the interface spins. f(t) is\nuncorrelated in time and randomly distributed in the in-\nterval [−f,f]. For small random ﬁelds, f= 0.05, the\npicture is almost the same as for zero noise (cf. upper\ngraphs of Figs. 3 and 4). Increasingthe noise strength to\na moderate level, energy transmission in FM part takes\nplace (see graph b) of Fig. 4). This stochastic resonance\nlike behavior is displayed in graph c) of Fig. 4.\nSummary .- As shown by analytical and numerical re-\nsults, in a two-phase multiferroic the magnetoelectric\ncoupling at the interface determines the conversion of\nan initial ferroelectric LM into a ferromagnetic signal,\npaving thus the way for FE and/or FM LM-based logic5\nin multiferroics. As anessentialstep in this direction, we\nhave identiﬁed the conditions under which a FE signal is\nconverted into a FM one.\nAcknowledgements .- Consultations with Marin Alexe\non the experimental realization are gratefully acknowl-\nedged. L.Ch., A.S. and J.B. are supported by DFG\nthrough SFB 762 and SU 690/1-1. R.Kh. and S.R are\nfunded by a joint Grant from French CNRS and Geor-\ngian SRNSF (Grant No 09/08), R. Kh. is supported by\ngrant 30/12 from SRNSF and S. R. acknowledge support\nof the contract LORIS (ANR-10-CEXC-010-01).\n[1] W.Eerenstein, N.D.Mathur, andJ.F. Scott, Nature 442,\n759 (2006).\n[2] Y. Tokura and S. Seki, Adv. Mater. 22, 1554 (2010).\n[3] C.A.F. Vaz, J. Hoﬀman, Ch. H. Ahn, and R. Ramesh,\nAdv. Mater. 22, 2900 (2010).\n[4] F. Zavaliche, T. Zhao, H. Zheng, F. Straub, M.P. Cruz,\nP.-L. Yang, D. Hao, and R. Ramesh, Nano. Lett. 7, 1586\n(2007).\n[5] H.L. Meyerheim, F. Klimenta, A. Ernst, K. Mohseni, S.\nOstanin, M. Fechner, S. Parihar, I.V. Maznichenko, I.\nMertig, and J. Kirschner, Phys. Rev. Lett. 106, 087203\n(2011).\n[6] R. Ramesh, N.A. Spaldin, Nat. Mater. 6, 21 (2007).\n[7] I.E. Dzyaloshinkskii, Sov. Phys. JETP 10, 628 (1959).\n[8] M. Bibes, A. Barth´ el´ emy, Nature Mater. 7(2008) 425 .\n[9] M. Gajek, M. Bibe, S. Fusil, K. Bouzehouane, J. Fontcu-\nberta, A. Barth´ el´ emy, and A. Fert, Nature Mater. 6\n(2007) 296.\n[10] D. Pantel, S. Goetze, D. Hesse and M. Alexe, Nature\nMater.11(2012) 289.\n[11] C.-W. Nan, M.I. Bichurin, S. Dong, D. Viehland, G.\nSrinivasan, J. Appl. Phys. 103, 031101 (2008).\n[12] J. van den Boomgaard, A.M.J.G. van Run, and J. van\nSuchtelen, Ferroelectrics 10, 295 (1976).\n[13] N. Spaldin, M. Fiebig, Science 309, 391 (2005).\n[14] M. Fiebig, J. Phys. D: Appl. Phys. 38, R123 (2005).\n[15] C.-G. Duan, S.S. Jaswal, E.Y. Tsymbal, Phys. Rev. Lett.\n97, 047201 (2006).\n[16] M. Fechner, I.V. Maznichenko, S. Ostanin, A. Ernst, J.\nHenk, I. Mertig, Phys. Stat. Solidi B 247, 1600 (2010).\n[17] J.-W. Lee, N. Sai, T. Cai, Q. Niu, A.A. Demkov, Phys.\nRev. B81, 144425 (2010).\n[18] A.M. Kosevich, B.A. Ivanov, and A.S. Kovalev, Phys.\nRep. 194 117, (1990).[19] R.P. Cowburn and M.E. Welland, Science 287, 1466\n(2000); R.P. Cowburn and M.E. Welland, Appl. Phys.\nLett.72, 2041 (1998).\n[20] P. Wadhwa and M.B.A. Jalil, Appl. Phys. Lett. 85, 2367\n(2004).\n[21] S. M. Mohseni et al., Science 339, 1295 (2013).\n[22] V. Garcia, M. Bibes, L. Bocher, S. Valencia, F. Kronast,\nA. Crassous, X. Moya, S. Enouz-Vedrenne, A. Gloter, D.\nImhoﬀ, C. Deranlot, N.D. Mathur, S. Fusil, K. Bouze-\nhouane, A. Barth´ el´ emy, Science 327, 1106 (2010).\n[23] F.Kh. Abdullaev, A.A. Abdumalikov, B.A. Umarov,\nPhys.Lett. A 171, 125-128 (1992).\n[24] B.I. Swanson, J.A. Brozik, S.P. Love, G.F. Strouse, A.P .\nShreve, A.R. Bishop, W.Z. Wang, M.I. Salkola, Phys.\nRev. Lett. 82, 3288 (1999).\n[25] U.T. Schwarz, L.Q. English, A.J. Sievers, Phys. Rev.\nLett.83, 223 (1999).\n[26] G. Kopidakis, S. Aubry, G.P. Tsironis, Phys. Rev. Lett.\n87, 165501 (2001).\n[27] S. Flach, K. Kladko, Physica D 127, 61 (1999).\n[28] A.K. Bandyopadhyay, P.C. Ray, V. Gopalan, J. Phys.:\nCondens. Matter 18, 4093 (2006).\n[29] P. Giri, K. Choudhary, A. Sengupta, A.K. Bandyopad-\nhyay, P.C. Ray, J. Appl. Phys. 109, 054105 (2011).\n[30] P. Giri, K. Choudhary, A.S. Gupta, A.K. Bandyopad-\nhyay, A.R. McGurn, Phys. Rev. B 84, 155429 (2011).\n[31] A.K. Bandyopadhyay, P.C. Ray, L. Vu-Quoc, A.R.\nMcGurn, Phys. Rev. B 81, 064104 (2010).\n[32] R. Benzi, A. Sutera, A. Vulpiani, J. Phys.: Math. Gen.\n14, L453 (1981).\n[33] L. Gammaitoni, P. H¨ anggi, P. Jung, F. Marchesoni, Rev.\nMod. Phys. 70, 224 (1998).\n[34] M. Alexe, and D. Hesse, Ferroelectrics 433, 53 (2012).\n[35]Physics of Ferroelectrics , K. Rabe, Ch.H. Ahn and J.-M.\nTriscone (Eds.), (Springer, Berlin 2007).\n[36]Magnetism and magnetic materials , J.M.D. Coey, (Cam-\nbridge University Press, Cambridge 2010).\n[37] T. Cai, S. Ju, J. Lee, N. Sai, A. A. Demkov, Q. Niu, Z.\nLi, J. Shi, and E. Wang, Phys. Rev. B 80, 140415(R)\n(2009).\n[38] N. Sedlmayr, V.K. Dugaev and J. Berakdar Phys. Rev.\nB79, 174422 (2009).\n[39] A. Sukhov, C.-L. Jia, P.P. Horley, and J. Berakdar, J.\nPhys.: Cond. Matt. 22, 352201 (2010); Phys. Rev. B 85,\n054401 (2012); EPL 99, 17004 (2012); Ferroelectrics 428,\n109 (2012); J. Appl. Phys. 113, 013908 (2013).\n[40]Physics of Ferromagnetism , S. Chikazumi, (Oxford Uni-\nversity Press Inc., New York 2002).\n[41] M. Oikawa, N. Yajima, Progr. Theor. Phys. 55, 36\n(1974).\n[42] N. Giorgadze, R. Khomeriki, Phys. Stat. Sol. (b) 207,\n249 (1998)."    },    {        "title": "2103.05871v3.Anisotropic_superconducting_spin_transport_at_magnetic_interfaces.pdf",        "content": "Anisotropic superconducting spin transport at magnetic interfaces\nYuya Ominato1, Ai Yamakage2, and Mamoru Matsuo1;3;4;5\n1Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China\n2Department of Physics, Nagoya University, Nagoya 464-8602, Japan\n3CAS Center for Excellence in Topological Quantum Computation,\nUniversity of Chinese Academy of Sciences, Beijing 100190, China\n4Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, 319-1195, Japan and\n5RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan\n(Dated: October 18, 2022)\nWe present a theoretical investigation of anisotropic superconducting spin transport at a magnetic\ninterface between a p-wave superconductor and a ferromagnetic insulator. Our formulation describes\nthe ferromagnetic resonance modulations due to spin current generation depending on spin-triplet\nCooper pair, including the frequency shift and enhanced Gilbert damping, in a uni\fed manner. We\n\fnd that the Cooper pair symmetry is detectable from the qualitative behavior of the ferromagnetic\nresonance modulation. Our theory paves the way toward anisotropic superconducting spintronics.\nIntroduction.| Use of spin-triplet Cooper pairs as car-\nriers for spin currents in the emergent \feld of super-\nconducting spintronics is challenging1,2. Previous stud-\nies have demonstrated spin transport mediated by spin-\ntriplet Cooper pairs that formed at the s-wave supercon-\nductor (SC)/ferromagnet interfaces of Josephson junc-\ntions. The spin-singlet pairs in SCs are converted into\nspin-triplet pairs in half-metallic CrO 23. However, pre-\nvious studies on spin-triplet pairs at magnetic interfaces\nhave been limited to cases induced by the proximity ef-\nfect.\nOne promising candidate material system for investi-\ngation of spin-triplet currents to enable more active use\nof spin-triplet pairs is the p-wave SC/ferromagnetic in-\nsulator (FI) bilayer thin \flm system4,5. Tunneling of the\nspins is driven by the magnetization dynamics excited\nby ferromagnetic resonance (FMR) in the ferromagnetic\nmaterial via interfacial exchange coupling between the\nmagnetization in the FI and the electron spins in the\np-wave SC, and a spin-triplet current is expected to be\ngenerated. Furthermore, as a backaction of spin injec-\ntion, both the FMR frequency and the Gilbert damping\nof the FI should be modulated6{8. Although similar sce-\nnarios have already been studied vigorously in s-wave\nSC/ferromagnet systems, most previous studies have fo-\ncused on the Gilbert damping modulation due to spin\ninjection9{22. To gain an in-depth understanding of the\nspin-triplet transport mechanisms, the FMR modulation\nprocesses, including both the frequency shift and the en-\nhanced Gilbert damping, should be formulated micro-\nscopically in a systematic manner.\nDetermination of the pairing symmetry of the spin-\ntripletp-wave SCs within the same framework is also\ndesirable. Despite many years of research based on sev-\neral experimental techniques that detect the pairing sym-\nmetry, including nuclear magnetic resonance23, polar-\nized neutron scattering24{26, and muon-spin resonance\ntechniques27, there are few established candidate systems\nfor spin-triplet SCs28{32. The FMR modulation has been\nobserved in various nanoscale magnetic multilayers. Ac-\ncordingly, the technique is widely used to investigate a\n(c) FMR modulation due to the coupling between\n     spin-triplet Cooper pair and magnetization\nH\nH0H0+ δH\n(b) Spin-triplet Cooper pair\n(i) Chiral p-wave (ii) Helical p-wave\nα+ δα\nH\nH0αH0H0+ δHFISC\nFIxz\nY, yθ(a) System\nθZ\nXS-HFISC\nFIG. 1. Mechanism of FMR modulation due to anisotropic\nsuperconducting spin transport at magnetic interfaces. (a)\nPrecession axis located on the x-zplane, where the angle\nbetween the precession axis and the zaxis is\u0012(where 0\u0014\u0012\u0014\n\u0019=2). (b) Two types of spin-triplet Cooper pairs considered\nin this work. (c) FMR signal modulation in the SC/FI bilayer\nsystem compared with the signal in the FI monolayer.\nspin transport property in a variety of nanoscale thin\n\flm systems because it is highly sensitive. Thus one can\nexpect that the FMR measurements in p-wave SC/FI bi-\nlayer systems provide useful information about pairing\nsymmetry.\nIn this Letter, we investigate anisotropic superconduct-\ning spin transport at the magnetic interfaces of hybrid\nsystems composed of p-wave SC/FI thin \flms theoret-\nically, as illustrated in Fig. 1(a). The two-dimensional\nbulk SC is placed on the FI, where the FMR occurs. The\nprecession axis is rotated by an angle \u0012from the direc-\ntion perpendicular to the interface. Here, we use two\ncoordinate systems: ( x;y;z ) and (X;Y;Z ). Thezaxis is\nperpendicular to the interface and the xandyaxes arearXiv:2103.05871v3  [cond-mat.supr-con]  15 Oct 20222\nalong the interface. The ( X;Y;Z ) coordinate is obtained\nby rotating the angle \u0012around the yaxis, so that the\nprecession axis and the Zaxis are parallel. Figure 1(b)\nshows a schematic image of the spin-triplet Cooper pairs\nfor the chiral and helical p-wave SCs considered in this\nwork. Figure 1(c) shows a schematic image of the FMR\nsignal in the FI monolayer and the SC/FI bilayer. The\nFMR frequency and linewidth in the SC/FI bilayer are\nboth modulated because of the spin transfer occurring at\nthe interface.\nUsing the nonequilibrium Green's function method,\nwe formulate the FMR modulations due to the back\naction of the spin-triplet transport process systemati-\ncally. The main advantage of using the nonequilibrium\nGreen's function is dealing with both a spectral function\nand a nonequilibrium distribution function. Indeed, the\ninterface spin current is given by the expression using\nthe nonequilibrium distribution function, which shows\nthat the interface spin current by the spin pumping and\nthe enhanced Gilbert damping are proportional to each\nother. Furthermore, as an advantage of \feld theoretical\ntreatment, the frequency shift and the enhanced Gilbert\ndamping are both described in a uni\fed manner. Addi-\ntionally, it is shown that the symmetry of the spin-triplet\npairs can be extracted from the FMR modulations. The\nresults presented here o\u000ber a pathway toward develop-\nment of anisotropic superconducting spintronics.\nModel Hamiltonian.| The FMR modulation due to the\nSC adjacent to the FI is calculated microscopically using\nthe spin tunneling Hamiltonian method9{11,33{38. The\ne\u000bect of the SC on the FI is treated as a perturbation\nand suppression of ferromagnetism with the onset of su-\nperconductivity is assumed to be negligible, which is con-\nsistent with the results of spin pumping experiments in\nmagnetic multilayer thin \flms. The details of the model\nHamiltonians and the formulations are described in the\nSupplemental Material39. In the main text, we focus on\ngiving an overview of the model Hamiltonians and the\nformulations.\nThe total Hamiltonian H(t) comprises three terms\nH(t) =HFI(t) +HSC+Hex: (1)\nThe \frst term HFI(t) describes the bulk FI,\nHFI(t) =X\nk~!kby\nkbk\u0000h+\nac(t)by\nk=0\u0000h\u0000\nac(t)bk=0;(2)\nwhereby\nkandbkdenote the creation and annihilation op-\nerators of magnons with the wave vector k= (kx;ky;kz),\nrespectively. We assume the parabolic dispersion ~!k=\nDk2\u0000~\rH, where\r(<0) is the electron gyromagnetic ra-\ntio. The coupling between the microwave radiation and\nthe magnons is given by h\u0006\nac(t) = ~\rhacp\nSN=2e\u0007i!t,\nwherehacand!are the amplitude and the frequency of\nthe microwave radiation, respectively. Sis the magni-\ntude of the localized spin and Nis the number of sites\nin the FI. Note that the precession axis for the localized\nspin is \fxed along the Zaxis [see Fig. 1(a)].The second term HSCdescribes the two-dimensional\nbulk SCs,\nHSC=1\n2X\nkcy\nkHBdGck; (3)\nwhere we use the four-component notations\ncy\nk= (cy\nk\";cy\nk#;c\u0000k\";c\u0000k#); (4)\nck= (ck\";ck#;cy\n\u0000k\";cy\n\u0000k#)T: (5)\nHere,cy\nksandcksdenote creation and annihilation op-\nerators, respectively, of electrons with the wave vector\nk= (kx;ky) and thezcomponent of the spin s=\";#.\nThe Bogoliubov-de Gennes Hamiltonian HBdGis a 4\u00024\nmatrix given by\nHBdG=\u0012\n\u0018k\u001b0\u0001k\n\u0000\u0001\u0003\n\u0000k\u0000\u0018k\u001b0\u0013\n; (6)\nwhere\u0018krepresents the energy of the electrons as mea-\nsured from their chemical potential, \u001b0is a 2\u00022 unit\nmatrix, and the pairing potential \u0001 kis also a 2\u00022 ma-\ntrix. We consider three pairing potential types, including\nthe spin-singlet s-wave pairing \u0001 k= \u0001i\u001byand two spin-\ntripletp-wave pairings \u0001 k= (dk\u0001\u001b)i\u001by, where their d\nvectors are given by\ndk=(\n\u0001(0;0;ei\u001ek) : Chiral p\u0000wave\n\u0001(\u0000sin\u001ek;cos\u001ek;0) : Helical p\u0000wave(7)\nwhere\u001ek= arctan(ky=kx) is an azimuth angle. The\nphenomenological form of the gap function is assumed\n\u0001 = 1:76kBTctanh\u0010\n1:74p\nTc=T\u00001\u0011\n; (8)\nwithTcthe superconducting transition temperature. By\ndiagonalizing HBdG, the quasiparticle energy is given by\nEk=p\n\u00182\nk+ \u00012for all SCs considered here. There-\nfore, one cannot distinguish them by the energy spectrum\nalone, and they are simple models suitable for studying\nthe di\u000berence of the magnetic responses due to the pair-\ning symmetry40.\nThe third term Hexrepresents the proximity exchange\ncoupling that occurs at the interface, which describes the\nspin transfer between the SC and the FI10,33,\nHex=X\nq;k\u0000\nJq;k\u001b+\nqS\u0000\nk+ h:c:\u0001\n; (9)\nwhereJq;kis the matrix element for the spin transfer pro-\ncesses,\u001b\u0006\nq= (\u001bX\nq\u0006i\u001bY\nq)=2 represent the spin-\rip opera-\ntors for the electron spins in the SCs, and S\u0000\n\u0000k=p\n2Sby\nk\nandS+\nk=p\n2Sbkrepresent the Fourier component of the\nlocalized spin in the FI. Note that the precession axis is\nalong theZaxis, so that the Zcomponent of the spin\nis injected into the SC when the FMR occurs. Using3\nthe creation and annihilation operators of electrons and\nmagnons,Hexis written as\nHex=X\nq;k;k0;s;s0\u0010p\n2SJq;k\u001b+\nss0cy\nk0sck0+qs0by\n\u0000k+ h:c:\u0011\n:\n(10)\nFrom the above expression, one can see that Hexde-\nscribes electron scattering processes with magnon emis-\nsion and absorption.\nModulation of FMR.| The FMR modulation can be\nread from the retarded component of the magnon Green's\nfunction33, which is given by\nGR\nk(!) =2S=~\n!\u0000!k+i\u000b!\u0000(2S=~)\u0006R\nk(!); (11)\nwhere the Gilbert damping constant \u000bis introduced\nphenomenologically41{43. In the second-order perturba-\ntion calculation with respect to the matrix element Jq;k,\nthe self-energy caused by proximity exchange coupling is\ngiven by\n\u0006R\nk(!) =\u0000X\nqjJq;kj2\u001fR\nq(!); (12)\nwhere the dynamic spin susceptibility of the SCs is de-\n\fned as\n\u001fR\nq(!) :=Z\ndtei(!+i0)ti\n~\u0012(t)h[\u001b+\nq(t);\u001b\u0000\n\u0000q(0)]i:(13)\nThe pole of GR\nk(!) indicates the FMR modulation, i.e.,\nthe shift of resonance frequency and the enhancement of\nthe Gilbert damping. By solving the equation\n!\u0000!k=0\u0000(2S=~)Re\u0006R\nk=0(!) = 0; (14)\nat a \fxed microwave frequency !, one obtains the mag-\nnetic \feld at which the FMR occurs. The imaginary part\nof the self-energy gives the enhancement of the Gilbert\ndamping. Consequently, the frequency shift and the en-\nhanced Gilbert damping are given by\n\u000eH=2S\n\r~Re\u0006R\nk=0(!); \u000e\u000b =\u00002S\n~!Im\u0006R\nk=0(!):(15)\nFrom the above equations and Eq. (12), one can see that\nthe FMR modulation provides information about both\nthe interface coupling properties and the dynamic spin\nsusceptibility of the SCs.\nThe form of matrix element Jq;k=0depends on the\ndetails of the interface. In this work, we assume the\ninterface with uncorrelated roughness. jJq;k=0j2is given\nby\njJq;k=0j2=J2\n1\nN\u000eq;0+J2\n2l2\nNA; (16)\nwhere the \frst and second terms describe averaged uni-\nform contribution and uncorrelated roughness contribu-\ntion, respectively39.J1andJ2correspond to the meanvalue and variance, respectively. Ais the area of the in-\nterface, which is equal to the system size of the SC. lis\nan atomic scale length. Using Eq. (16), the self-energy\nfor the uniform magnon mode is given by\n\u0006R\nk=0(!) =\u0000J2\n1\nN\u001fR\nuni(!)\u0000J2\n2l2\nNA\u001fR\nloc(!); (17)\nwhere the uniform and local spin susceptibilities are de-\n\fned as\n\u001fR\nuni(!) := lim\njqj!0\u001fR\nq(!); \u001fR\nloc(!) :=X\nq\u001fR\nq(!):(18)\nThe self-energy \u0006R\nk=0(!) consists of two terms originating\nfrom the uniform and roughness contributions, so that\nboth\u001fR\nuni(!) and\u001fR\nloc(!) contribute to \u000eHand\u000e\u000b.\nHere, we discuss the FI thickness dependence on the\nFMR modulation44. From Eqs. (15), and (17), one can\nsee that the FMR modulation is inversely proportional\nto the FI thickness ( /A=N ) because\u001fR\nuni(!)/Aand\n\u001fR\nloc(!)/A2. This is consistent with the experiments on\nthe spin pumping in Y 3Fe5O12=Pt heterostructures45. In\norder to observe the FMR modulation experimentally, it\nis necessary to prepare a sample that is su\u000eciently thin,\ne.g., typically, the thickness of several tens of nanometers.\nNumerical results.| In the following, we consider a \rat\ninterface where J2= 0, so that the behavior of the FMR\nmodulation is determined by \u001fR\nuni(!). The roughness\ncontribution proportional to \u001fR\nloc(!) is discussed later.\nFigure 2 shows the frequency shift \u000eHand the enhanced\nGilbert damping \u000e\u000bas a function of temperature and fre-\nquency. Here, we set \u0012= 0 and \u0000=kBTc= 0:05, where \u0000\nis a constant level broadening of the quasiparticle intro-\nduced phenomenologically39.\nFirst, we explain the qualitative properties of \u000eHand\n\u000e\u000bfor the chiral p-wave SC. In the low frequency re-\ngion, where ~!=kBTc\u00141,\u000eHis \fnite and remains al-\nmost independent of !near the zero temperature and\n\u000e\u000bdecreases and becomes exponentially small with the\ndecrease of the temperature. In the high frequency re-\ngion, where ~!=kBTc\u00151, a resonance peak occurs at\n~!= 2\u0001 for both \u000eHand\u000e\u000b. The qualitative proper-\nties of\u000eHand\u000e\u000bfor the helical p-wave SC are the same\nas those of the chiral p-wave SC.\nNext, we explain the qualitative properties of \u000eHand\n\u000e\u000bfor thes-wave SC. In the low frequency region, where\n~!=kBTc\u00141, both\u000eHand\u000e\u000bdecrease and become\nexponentially small with the decrease of the temperature.\nIn the high frequency region, where ~!=kBTc\u00151, both\n\u000eHand\u000e\u000bvanish.\nThep-wave SCs show two characteristic properties\nthat thes-wave SC does not show: a \fnite \u000eHatT= 0\nand a resonance peak of \u000eHand\u000e\u000b. These properties\ncan be understood by the analogy between SCs and band\ninsulators as follows. The uniform dynamic spin suscepti-\nbility consists of contributions from intraband transitions\nwithin particle (hole) bands and interband transitions\nbetween particles and holes. In the low temperature or4\n(a) (b)Γ/kBTc=0.05 Chiral p-wave\n(c) (d)Γ/kBTc=0.05 Helical p-wave\n(e) (f)Γ/kBTc=0.05 s-waveT/T\nc\nhω/kBTcδα/δα1\n0.00.51.0 0\n2\n410\n5\n0\nT/T\nc\nhω/kBTcδH/δH1\n0.00.51.0 0\n2\n410\n5\n0\nT/T\nc\nhω/kBTcδα/δα1\n0.00.51.0 0\n2\n410\n5\n0\nT/T\nc\nhω/kBTcδH/δH1\n0.00.51.0 0\n2\n410\n5\n0\nT/T\nc\nhω/kBTcδH/δH1\n0.00.51.0 0\n2\n42\n1\n0\nT/T\nc\nhω/kBTcδα/δα1\n0.00.51.0 0\n2\n410\n5\n0\nFIG. 2. The frequency shift \u000eHand the enhanced Gilbert\ndamping\u000e\u000bas a function of temperature and frequency nor-\nmalized by the characteristic values \u000eH1=\u0000SJ2\n1DF=(N\r~)\nand\u000e\u000b1=SJ2\n1DF=(NkBTc) in the normal state. DF(/A)\nis the density of states at the Fermi level in the normal state.\nWe set\u0012= 0 and \u0000=kBTc= 0:05. The sign of \u000eHcorresponds\nto the sign of Re \u001fR\nuni(!), which can be positive and negative\nat low and high frequencies, respectively. In contrast, \u000e\u000bis\npositive at any frequency.\nhigh frequency region, the intraband contribution is neg-\nligible and the interband contribution is dominant. In\nthe case of the s-wave SC, the interband transitions are\nforbidden because the Hamiltonian and the spin operator\ncommute. As a result, there is no spin response in the\nlow-temperature or high-frequency regions. In contrast,\nthe Hamiltonian for the p-wave SCs and the spin operator\ndo not commute. Therefore, \u000eHhas a \fnite value near-\nzero temperature due to the interband contribution. In\naddition, a resonance peak occurs when ~!= 2\u0001 because\nthe density of states diverges at the band edge E=\u0006\u0001.\nA detailed proof of the above statement is given in the\nSupplemental Material39.\nThe angle dependences of \u000eHand\u000e\u000bare distinct for\nchiral and helical p-wave SCs, as shown in Fig. 3. In\nboth cases, we set ~!=kBTc= 3:0 as the typical values\nat high frequencies, where the main contribution of the\nuniform spin susceptibility is the interband transitions.\nIn the chiral p-wave SC,\u000eHand\u000e\u000btend to decrease and\nare halved at a \fxed temperature when \u0012increases from\n0 to\u0019=2. Conversely, in the helical p-wave SC, the qual-\n0.4\nT/Tc1.0 0.2 0.08\n4\n0\n0.8 0.6δH/δH1\n2\n-210\n60.4\nT/Tc1.0 0.2 0.08\n4\n0\n0.8 0.6δH/δH1\n2\n-210\n6\nHelical p-waveChiral p-wave\nθ=0\nπ/4\nπ/2\nθ=0π/4π/2\n0.4\nT/Tc1.0 0.2 0.02\n0\n0.8 0.6δα/δα1\n13θ=0\nπ/4\nπ/2\nθ=0π/4π/20.4\nT/Tc1.0 0.2 0.02\n0\n0.8 0.6δα/δα1\n13Γ/kBTc=0.05, hω/kBTc=3.0\n(a) (b)\n(c) (d)Γ/kBTc=0.05, hω/kBTc=3.0FIG. 3. Frequency shift and the enhanced Gilbert damping\nas a function of temperature at angles of \u0012= 0;\u0019=4;\u0019=2. The\nupper and lower panels show the characteristics for the chiral\nand helical p-wave SCs, respectively.\nitative behavior shows the opposite trend. \u000eHand\u000e\u000b\nboth tend to increase and become 1 :5 times larger at a\n\fxed temperature when \u0012increases from 0 to \u0019=2. In\nfact, the angle dependences are approximately obtained\nto be/1 +cos2\u0012and 1 +(sin2\u0012)=2 for chiral and helical\np-wave SCs, respectively39. Therefore, the spin con\fg-\nuration of the Cooper pair can be detected from the \u0012\ndependence data for the FMR modulation.\nThe FMR modulation properties of the three SCs are\nsummarized in Table I. All SCs considered here can be\ndistinguished based on three properties: the frequency\nshift in the low temperature limit, the presence of their\nresonance peak, and their \u0012dependence. For the s-wave\nSC,\u000eHbecomes exponentially small in T!0, while for\nthep-wave SCs, \u000eHis \fnite in T!0. For the s-wave\nSC,\u000eHand\u000e\u000bshow no resonance and no \u0012dependence,\nwhile for the chiral and helical p-wave SCs, both \u000eHand\n\u000e\u000bexhibit a resonance at ~!= 2\u0001 and a \u0012dependence.\nIn addition, these two p-wave SCs can be distinguished\nfrom their\u0012dependences of \u000eHand\u000e\u000b, which are char-\nacterized by @\u0012(\u000eH) and@\u0012(\u000e\u000b), respectively. Here, it\nshould be emphasized that the pairing symmetry can be\ncharacterized by the sign of @\u0012(\u000eH) and@\u0012(\u000e\u000b). These\nproperties are summarized in the Table I.\nSpin-triplet current generation.| The relationship be-\ntween the enhanced Gilbert damping discussed above5\nand the spin-triplet current generation must also be dis-\ncussed. The enhancement of the Gilbert damping is\nknown to originate from the spin current generation at\nthe magnetic interface6,33. The interface spin current in-\nduced by FMRhISiSPis given by39\nhISiSP=N(~\rhac)2\n2\u000b\u0002\n\u0000ImGR\nk=0(!)\u0003\n\u000e\u000b: (19)\nOne can see that hISiSPand\u000e\u000bare proportional to each\nother. In our setup, the enhanced Gilbert damping \u000e\u000b\nwill lead to the generation of both the Cooper pair spin-\ntriplet current and the quasiparticle spin current. Since\nthe angular dependence of \u000e\u000bre\rects the direction of\nthe Cooper pair spins, it is expected that the spin-triplet\ncurrent can be controlled by varying the magnetization\ndirection of the FI.\nDiscussion.| We have considered a \rat SC/FI inter-\nface. In the presence of roughness, the correction term\nproportional to \u001fR\nloc(!) contributes to the FMR mod-\nulation, as shown in Eq. (17). In the rough limit,\nJ2\n1\u001cJ2\n2,\u001fR\nloc(!) dominates to make the FMR modu-\nlation isotropic, due to the angle average by summation\noverq. Namely, the anisotropy peculiar to p-wave SC\nis smeared by the roughness. The detailed behavior of\n\u001fR\nloc(!) is shown in the Supplemental Material39. This\nresult implies that it is crucial to control the interface\nroughness. In principle, the roughness of the interface\ncan be observed using transmission electron microscopy\nof interfaces46{48and it is possible to detect whether the\ninterface of the sample is \rat or rough. More detailed\nspectroscopy can be obtained from the FMR modulation\nby using a \rat interface.\nOur results show that the pairing symmetry can be\ndetected by the sign of @\u0012(\u000eH) and@\u0012(\u000e\u000b) around the\nin-plane magnetic \feld ( \u0012\u0018\u0019=2), where the vortices are\nnegligible. When the external magnetic \feld has a large\nout-of-plane component, the vortex formation may cause\nproblems in observing the angular dependence. The qual-\nitative behavior is expected to change when the out-of-\nplane magnetic \feld approaches the upper critical \feld\n(H\u0018Hc2\u00181T). This is because the coherence length\nof the Cooper pair and the distance between the vor-\ntices can become comparable. Indeed, it has been exper-\nimentally reported that the vortex formation suppresses\nthe characteristic properties in the spin pumping into\nSCs20. Therefore, the out-of-plane magnetic \feld should\nbe as small as possible when FMR measurements are per-\nformed for H\u0018Hc2.\nTABLE I. FMR modulation properties for the \rat SC/FI in-\nterface where J16= 0 andJ2= 0.\nPairing symmetry s Chiral Helical\n\u000eHin the limit of T!0 0 \fnite \fnite\nResonance peak of \u000eH,\u000e\u000b { X X\n@\u0012(\u000eH),@\u0012(\u000e\u000b) 0 negative positiveRecent experiments have reported that UTe 2is a can-\ndidate material for spin-triplet p-wave SCs31, which has\nattracted a great deal of attention. Various experi-\nments, including spectroscopic measurements, are now\nin progress to investigate the pairing symmetry of UTe 2,\nand indicated that the superconducting transition tem-\nperature is about 1K \u001830 GHz. Therefore, the resonance\ncondition ~!= 2\u0001 shown above is accessible to recent\nbroadband FMR measurements.\nIn addition, experiments on spin pumping into d-wave\nSCs have recently been reported49and a theoretical in-\nvestigation of the enhancement of the Gilbert damp-\ning in ad-wave SC/FI bilayer system has recently been\npresented50. Thus anisotropic superconducting spintron-\nics can be expected to develop as a new research direc-\ntion.\nWe should emphasize two important aspects of the\nFMR method presented here: the spectroscopic probe\nmethod for the p-wave SC thin \flms and the versa-\ntile spin injection method. First, the FMR measure-\nment procedure can provide a new spin-sensitive mea-\nsurement method that will complement other measure-\nment methods to enable a breakthrough in the discovery\nof spin-triplet SCs. Second, the FMR method represents\na promising way to generate spin-triplet currents in p-\nwave SC thin \flms.\nConclusions.| We have investigated the anisotropic\nsuperconducting spin transport at magnetic interfaces\ncomposed of a p-wave SC and an FI based on a micro-\nscopic model Hamiltonian. The FMR signal in these p-\nwave SC/FI bilayer systems is modulated via spin trans-\nfer at the interface, which generates spin-triplet currents.\nWe have shown that the pairing symmetry of the SCs\ncan be extracted from the FMR modulation character-\nistics. Our approach provides a unique way to explore\nanisotropic superconducting spintronics, which will be\nuseful for application to emerging device technologies.\nNote added.| After the submission of this manuscript,\nwe became aware of a closely related work, where a way\nto convert spin-triplet currents to magnon spin currents\nin SC/FI bilayer systems is discussed51.\nWe thank R. Ohshima, M. Shiraishi, H. Chudo, G.\nOkano, K. Yamanoi, and Y. Nozaki for helpful discus-\nsions. This work was supported by the Priority Pro-\ngram of the Chinese Academy of Sciences under Grant\nNo. XDB28000000, and by JSPS KAKENHI under\nGrants Nos. JP20K03835, JP20H04635, JP20H01863,\nJP21H04565, and JP21H01800.6\nSUPPLEMENTAL MATERIAL\nI. MODEL HAMILTONIAN\nIn this section, we describe the derivation and details of the model Hamiltonian used in the main text.\nA. Ferromagnetic Heisenberg model\nThe ferromagnetic Heisenberg model with the transverse AC magnetic \feld due to the microwave radiation is given\nby\nHFI(t) =\u0000JX\nhi;jiSi\u0001Sj+~\rHX\njSZ\nj\u0000~\rhacX\nj\u0000\nSX\njcos!t\u0000SY\njsin!t\u0001\n; (S.1)\nwhereJ >0 is the exchange coupling constant, hi;jirepresents summation over all nearest-neighbor sites, Sjis the\nlocalized spin at site jin the ferromagnetic insulator (FI), \r(<0) is the gyromagnetic ratio, His a static magnetic\n\feld,hacis an amplitude of an transverse oscillating magnetic \feld due to the microwave radiation with a frequency\n!. The rotated coordinates ( X;Y;Z ) are shown in Fig. 1(a).\nIt is convenient to introduce the boson creation and annihilation operators in order to formulate the problem in\nterms of the quantum \feld theory. In the current problem, we perturbatively treat the excitation of the FI. In this\ncase, the Holstein-Primako\u000b transformation is useful, where the localized spin can be described using boson creation\nand annihilation operators bj;by\njin Hilbert space constrained to 2 S+ 1 dimensions. The spin operators are written as\nS+\nj=SX\nj+iSY\nj=\u0010\n2S\u0000by\njbj\u00111=2\nbj; (S.2)\nS\u0000\nj=SX\nj\u0000iSY\nj=by\nj\u0010\n2S\u0000by\njbj\u00111=2\n; (S.3)\nSZ\nj=S\u0000by\njbj; (S.4)\nwhere we require [ bi;by\nj] =\u000ei;j;in order that the S+\nj,S\u0000\nj, andSZ\njsatisfy the commutation relation of angular\nmomentum. The deviation of SZ\njfrom its ground-state value Sis quanti\fed by the boson particle number.\nWe consider low-energy excitation in the FI, where the deviation of SZ\njfrom the ground state is small hby\njbji=S\u001c1.\nThe ladder operators S\u0006\njare approximated as\nS+\nj\u0019(2S)1=2bj; (S.5)\nS\u0000\nj\u0019(2S)1=2by\nj; (S.6)\nwhich is called spin-wave approximation. Here, we de\fne the magnon operators\nbk=1p\nNX\nje\u0000ik\u0001rjbj; (S.7)\nby\nk=1p\nNX\njeik\u0001rjby\nj; (S.8)\nwhereNis the number of sites and k= (kx;ky;kz). The inverse transformation is then given by\nbj=1p\nNX\nkeik\u0001rjbk; (S.9)\nby\nj=1p\nNX\nke\u0000ik\u0001rjby\nk: (S.10)\nThe magnon operators satisfy [ bk;by\nk0] =\u000ek;k0and describe the quantized collective excitations. Using the spin-wave\napproximation and the magnon operators, the Hamiltonian HFI(t) is written as\nHFI(t)\u0019X\nk~!kby\nkbk\u0000h+\nac(t)by\nk=0\u0000h\u0000\nac(t)bk=0; (S.11)7\nwhere ~!k=Dk2\u0000~\rHwithD= 2JSa2and the lattice constant a,h\u0006\nac(t) =~\rhacp\nSN=2e\u0007i!t, and constant\nterms are omitted.\nB. BCS Hamiltonian\nWe derive a mean-\feld Hamiltonian, which describes a bulk superconductor (SC), and we diagonalize the mean-\feld\nHamiltonian with the Bogoliubov transformation. At the end of this section, the spin density operators of the SC are\nwritten in terms of the Bogoliubov quasiparticle creation and annihilation operators.\nWe start with the e\u000bective Hamiltonian in momentum space\nHSC=X\nk;s\u0018kcy\nkscks+1\n2X\nk;k0;s1;s2;s3;s4Vs1;s2;s3;s4(k;k0)cy\n\u0000ks1cy\nks2ck0s3c\u0000k0s4; (S.12)\nwhere\u0018kis the band energy measured relative to the chemical potential, and cy\nksandcksare the creation and\nannihilation operators of electrons with the wave vector k= (kx;ky) and thezcomponent of the spin s=\";#. The\nmatrix elements satisfy\nVs1;s2;s3;s4(k;k0) =\u0000Vs2;s1;s3;s4(\u0000k;k0); (S.13)\nVs1;s2;s3;s4(k;k0) =\u0000Vs1;s2;s4;s3(k;\u0000k0); (S.14)\nbecause of the anticommutation relation of fermions, and\nVs1;s2;s3;s4(k;k0) =V\u0003\ns4;s3;s2;s1(k0;k); (S.15)\nbecause of the Hermitianity of the Hamiltonian. We consider a mean-\feld, which is called a pair potential\n\u0001k;ss0=\u0000X\nk0;s3;s4Vs0;s;s 3;s4(k;k0)hck0s3c\u0000k0s4i; (S.16)\nand its conjugate\n\u0001\u0003\n\u0000k;ss0=X\nk0;s1;s2Vs1;s2;s0;s(k0;k)hcy\n\u0000k0s1cy\nk0s2i: (S.17)\nHere, we consider a mean-\feld approximation where the interaction term is replaced as follows\ncy\n\u0000ks1cy\nks2ck0s3c\u0000k0s4!cy\n\u0000ks1cy\nks2hck0s3c\u0000k0s4i+hcy\n\u0000ks1cy\nks2ick0s3c\u0000k0s4\u0000hcy\n\u0000ks1cy\nks2ihck0s3c\u0000k0s4i; (S.18)\nso that the interaction term is rewritten as\nX\nk;k0;s1;s2;s3;s4Vs1;s2;s3;s4(k;k0)cy\n\u0000ks1cy\nks2ck0s3c\u0000k0s4!X\nk;s1;s2h\n\u0001k;s1s2cy\nks1cy\n\u0000ks2\u0000\u0001\u0003\n\u0000k;s1s2c\u0000ks1cks2i\n; (S.19)\nwhere an constant term is omitted. Consequently, we derive a mean-\feld Hamiltonian\nHSC=X\nk;s\u0018kcy\nkscks+1\n2X\nk;s1;s2\u0002\n\u0001k;s1s2cy\nks1cy\n\u0000ks2\u0000\u0001\u0003\n\u0000k;s1s2c\u0000ks1cks2\u0003\n: (S.20)\nUsing a four-component notation\ncy\nk= (cy\nk\";cy\nk#;c\u0000k\";c\u0000k#); (S.21)\nck= (ck\";ck#;cy\n\u0000k\";cy\n\u0000k#)T; (S.22)\nthe mean-\feld Hamiltonian is written as\nHSC=1\n2X\nkcy\nkHBdGck: (S.23)8\nHBdGis the 4\u00024 matrix\nHBdG= \n\u0018k\u001b0\u0001k\n\u0000\u0001\u0003\n\u0000k\u0000\u0018k\u001b0!\n; (S.24)\nwhere\u001b0is the 2\u00022 unit matrix and \u0001 kis the 2\u00022 matrix given as\n\u0001k= \n\u0001k;\"\"\u0001k;\"#\n\u0001k;#\"\u0001k;##!\n: (S.25)\nIn principle, the pair potential is obtained by solving the gap equation self-consistently for an explicit form of the\nmatrix elements Vs1;s2;s3;s4(k;k0). In this work, we do not solve the gap equation, but instead assume an explicit\nform of the pair potential and perform calculations using a phenomenological gap function. For the singlet pairing,\nthe pair potential is given by\n\u0001k= ki\u001by; (S.26)\nwith an even function  k= \u0000k. For ans-wave SC, the pair potential is given by\n\u0001k= \u0001 \n0 1\n\u00001 0!\n: (S.27)\nFor the triplet pairing, the pair potential is given by\n\u0001k= [dk\u0001\u001b]i\u001by; (S.28)\nwith an odd vectorial function dk=\u0000d\u0000k. For a chiral p-wave SC and a helical p-wave SC,dkis given by\ndk=(\n\u0001(0;0;ei\u001ek) : chiral p\u0000wave\n\u0001(\u0000sin\u001ek;cos\u001ek;0) : helical p\u0000wave(S.29)\nwith\u001ek= arctan(ky=kx), so that the pair potential is given by\n\u0001k=8\n>>>><\n>>>>:\u0001 \n0ei\u001ek\nei\u001ek0!\n: chiralp\u0000wave\n\u0001 \nie\u0000i\u001ek0\n0iei\u001ek!\n: helicalp\u0000wave(S.30)\nThe phenomenological gap function is given by\n\u0001 = 1:76kBTctanh\u0010\n1:74p\nTc=T\u00001\u0011\n: (S.31)\nThe Bogoliubov transformation to diagonalize HBdGis given by\nUk= \nukvk\nv\u0003\n\u0000ku\u0003\n\u0000k!\n; (S.32)\nUy\nk= \nuk\u0000vk\n\u0000v\u0003\n\u0000ku\u0003\n\u0000k!\n; (S.33)\nwith the 2\u00022 matricesukandvkgiven by\nuk=s\n1\n2\u0012\n1 +\u0018k\nEk\u0013\n\u001b0; (S.34)\nvk=\u0000s\n1\n2\u0012\n1\u0000\u0018k\nEk\u0013\u0001k\n\u0001; (S.35)9\nwhereEkis the eigenenergy\nEk=q\n\u00182\nk+ \u00012: (S.36)\nUsing the Bogoliubov transformation Uk, the 4\u00024 matrixHBdGis diagonalized as\nUy\nkHBdGUk=0\nBBB@Ek0 0 0\n0Ek0 0\n0 0\u0000Ek0\n0 0 0\u0000Ek1\nCCCA: (S.37)\nThe excitation of HSCis described by the creation and annihilation operators of the Bogoliubov quasiparticles \r(y)\nk\n\ry\nk= (\ry\nk\";\ry\nk#;\r\u0000k\";\r\u0000k#); (S.38)\n\rk= (\rk\";\rk#;\ry\n\u0000k\";\ry\n\u0000k#)T; (S.39)\nwhere they are obtained by the Bogoliubov transformation\n\rk=Uy\nkck; (S.40)\n\ry\nk=cy\nkUk: (S.41)\nThe spin density operators \u001ba(r) (a=x;y;z ) is de\fned as\n\u001ba(r) :=1\nAX\nk;k0;s;s0e\u0000i(k\u0000k0)\u0001r\u001ba\nss0cy\nksck0s0; (S.42)\nwhereAis the area of the system. \u001ba(r) (a=x;y;z ) is expanded in Fourier series\n\u001ba(r) =1\nAX\nqeiq\u0001r\u001ba\nq; (S.43)\nand the Fourier coe\u000ecient is given by\n\u001ba\nq=Z\ndre\u0000iq\u0001r\u001ba(r) =X\nk;s;s0\u001ba\nss0cy\nksck+qs0: (S.44)\nUsing the Bogoliubov transformation Uk, the above expression is rewritten as\n\u001ba\nq=X\nk;s;s0\"\u0010\nsa(1)\nk;k+q\u0011\ns;s0\ry\nks\rk+qs0+\u0010\nsa(2)\nk;k+q\u0011\ns;s0\r\u0000ks\ry\n\u0000k\u0000qs0+\u0010\nsa(3)\nk;k+q\u0011\ns;s0\ry\nks\ry\n\u0000k\u0000qs0+\u0010\nsa(4)\nk;k+q\u0011\ns;s0\r\u0000ks\rk+qs0#\n;\n(S.45)\nwith the 2\u00022 matricessa(i)\nk;k+qgiven by\nsa(1)\nk;k+q=uy\nk\u001bauk+q; (S.46)\nsa(2)\nk;k+q=vy\nk\u001bavk+q; (S.47)\nsa(3)\nk;k+q=uy\nk\u001bavk+q; (S.48)\nsa(4)\nk;k+q=vy\nk\u001bauk+q: (S.49)\nThe \frst and second terms describe the intraband transition from particle-to-particle and from hole-to-hole, respec-\ntively. The third and fourth terms describe the interband transition from hole-to-particle and from particle-to-hole,\nrespectively.10\nC. Proximity exchange coupling at interface\nWe start with a model for the proximity exchange coupling given by\nHex=Z\ndrX\njJ(r;rj)\u001b(r)\u0001Sj: (S.50)\nWe rewrite the above expression in the real space into the expression in the wave space. The proximity exchange\ncoupling is rewritten as\nHex=Z\ndrX\njJ(r;rj)1\nAp\nNX\nq;kei(q\u0001r+k\u0001rj)\u0000\n\u001b+\nqS\u0000\nk+\u001b\u0000\nqS+\nk\u0001\n+Z\ndrX\njJ(r;rj)\u001bZ(r)SZ\nj; (S.51)\nwhere the Fourier series are given by\n\u001b(r) =1\nAX\nqeiq\u0001r\u001bq; (S.52)\nSj=1p\nNX\nkeik\u0001rjSk; (S.53)\nwith the area of the SC, A, and the number of sites in the FI, N, and the ladder operators are given by\n\u001b\u0006=1\n2(\u001bX\u0006i\u001bY); (S.54)\nS\u0006=SX\u0006iSY: (S.55)\nThe matrix element is given by\nJq;k=1\nAp\nNZ\ndrX\njJ(r;rj)ei(q\u0001r+k\u0001rj): (S.56)\nConsequently, the exchange coupling which we use in the main text is derived as\nHex=X\nq;k\u0000\nJq;k\u001b+\nqS\u0000\nk+J\u0003\nq;k\u001b\u0000\n\u0000qS+\n\u0000k\u0001\n; (S.57)\nwhere we use a relation J\u0000q;\u0000k=J\u0003\nq;k, and we omit the last term\nZ\ndrX\njJ(r;rj)\u001bZ(r)SZ\nj; (S.58)\nin order to focus on the spin transfer at the interface. For the uniform magnon mode jkj= 0, the matrix element is\ngiven by\nJq;k=0=1\nAp\nNZ\ndrX\njJ(r;rj)eiq\u0001r: (S.59)\nII. TIME DEPENDENT QUANTUM AVERAGE\nIn this section, we show that the ferromagnetic resonance (FMR) frequency and linewidth are read from the\nmagnon Green's function. We consider the Hamiltonian H(t) composed of the unperturbed Hamiltonian H0and the\nperturbation V(t)\nH(t) =H0+V(t): (S.60)\nThe time-dependent quantum average of a physical quantity Ois calculated as\nhO(t)i=hSy(t;\u00001)~O(t)S(t;\u00001)i; (S.61)11\nwhere ~O(t) is the interaction picture and the S matrix S(t;t0) is given by\nS(t;t0) =Texp Zt\nt0dt0~V(t0)\ni~!\n: (S.62)\nThe time-dependent quantum average hO(t)iis written as\nhO(t)i=hOieq+\u000ehO(t)i; (S.63)\nwherehOieq= Tr (\u001aeqO) is the equilibrium value and \u000ehO(t)iis deviation from the equilibrium. When the perturbation\nis written as V(t) =\u0000AF(t), the \frst order perturbation calculation gives\n\u000ehO(t)i=\u0000Zt\n\u00001dt01\ni~h[~O(t);~A(t0)]iF(t0)\n=\u0000Z1\n\u00001dt0GR(t0)F(t\u0000t0); (S.64)\nwhere we de\fne the retarded Green's function\nGR(t) =1\ni~\u0012(t)h[~O(t);~A(0)]i: (S.65)\nWhen the external force is written as F(t) =Fe\u0000i(!+i0)t,\u000ehO(t)iis written as\n\u000ehO(t)i=\u0000Fe\u0000i(!+i0)tZ1\n\u00001dt0ei(!+i0)t0GR(t0)\n=\u0000Fe\u0000i!tGR(!): (S.66)\nUsing the above formula, the dynamics of \u000ehS+\nk=0(t)iis written as\n\u000ehS+\nk=0(t)i=\u0000~\rhacp\nN\n2e\u0000i!tGR\nk=0(!); (S.67)\nwhereGR\nk(!) is the Fourier transform of the retarded component of the magnon Green's function GR\nk(t). They are\nde\fned as\nGR\nk(t) :=1\ni~\u0012(t)h[S+\nk(t);S\u0000\n\u0000k(0)]i; (S.68)\nGR\nk(!) :=Z1\n\u00001dt0ei(!+i0)t0GR\nk(t0): (S.69)\nFrom Eq. (S.67), one can see that the FMR frequency and linewidth are read from GR\nk(!).\nIII. MAGNON GREEN'S FUNCTION\nIn this section, we perform perturbative calculation for the magnon Green's function. We treat the proximity\nexchange coupling as a perturbation. The Hamiltonian is written as\nH=H0+V; (S.70)\nwhereH0is the unperturbed Hamiltonian\nH0=X\nk~!kby\nkbk+X\nk;sEk\ry\nks\rks; (S.71)\nandVis the perturbation\nV=X\nq;k\u0000\nJq;k\u001b+\nqS\u0000\nk+ h:c:\u0001\n: (S.72)12\n(e) Vertex(b) Keldysh contour\n(d) Self-energytime(a) Magnon Green’s function (c) Dyson equation\nk/uni2032 +qs/uni2032 −k\n−k/uni2032 −qs/uni2032 /uni03C3+\nqS−\nk= + + +−k −k −k−k/uni2032 s k/uni2032 s k/uni2032 s\n−k/uni2032 −qs/uni2032 k/uni2032 +qs/uni2032 −k/uni2032 sGk(/uni03C4,/uni03C4/uni2032 )= = + /uni03A3\n/uni03A3 = + + +\nk/uni2032 +qs/uni2032 −k/uni2032 −qs/uni2032 −k/uni2032 s k/uni2032 s k/uni2032 s\n−k/uni2032 −qs/uni2032 k/uni2032 +qs/uni2032 −k/uni2032 s\nBogoliubov quasiparticle: Magnon:\nFIG. 4. (a) The Feynman diagram for the magnon Green's function. (b) Keldysh contour to perform perturbative calculations.\n(c) The Feynman diagram for the Dyson equation. (d) The self-energy within the second-order perturbation is given by the\ndynamic spin susceptibility of the SCs. (e) The Feynman diagrams for the vertex \u001b+\nqS\u0000\nk, which represent scattering of a\nBogoliubov quasiparticle with magnon emission. The solid and wavy lines represent a Bogoliubov quasiparticle and a magnon,\nrespectively.\nWe de\fne the magnon Green's function\nGk(\u001c;\u001c0) :=1\ni~hTCS+\nk(\u001c)S\u0000\n\u0000k(\u001c0)i; (S.73)\nwhereTCis the time-ordering operator on the Keldysh contour (see Figs. 4(a) and (b)). To perform the perturbative\ncalculation, we introduce interaction picture. The perturbation is written as\n~V(t) =X\nq;k\u0010\nJq;k~\u001b+\nq(t)~S\u0000\nk(t) + h:c:\u0011\n: (S.74)\nThe magnon Green's function is given by\nGk(\u001c;\u001c0) =1\ni~hTCSC~S+\nk(\u001c)~S\u0000\n\u0000k(\u001c0)iconn; (S.75)\nwhereh\u0001\u0001\u0001i connmeans the connected diagrams and the S matrix is given by\nSC=TCexp Z\nCd\u001c~V(\u001c)\ni~!\n: (S.76)\nThe above expressions lead to the Dyson equation (see Fig. 4(c))\nGk(\u001c;\u001c0) =G(0)\nk(\u001c;\u001c0) +Z\nCd\u001c1Z\nCd\u001c2G(0)\nk(\u001c;\u001c1)\u0006k(\u001c1;\u001c2)Gk(\u001c2;\u001c0); (S.77)\nwhereG(0)\nk(\u001c;\u001c0) is the unperturbed magnon Green's function\nG(0)\nk(\u001c;\u001c0) =1\ni~hTC~S+\nk(\u001c)~S\u0000\n\u0000k(\u001c0)i; (S.78)\nand \u0006 k(\u001c1;\u001c2) is the self-energy. Within the second-order perturbation, the self-energy is given by (see Fig. 4(d))\n\u0006k(\u001c;\u001c0) =1\ni~X\nqjJq;kj2hTC~\u001b+\nq(\u001c)~\u001b\u0000\n\u0000q(\u001c0)i: (S.79)\nThe Feynman diagram for the vertex is shown in Fig. 4(e). Substituting the ladder operators expressed in terms of13\n\r(y)\nks, the self-energy is written as\n\u0006k(\u001c;\u001c0) =\u0000i~X\nqjJq;kj2X\nk0;s;s0\"\u0012\f\f\f(s+(1)\nk0;k0+q)s;s0\f\f\f2\n\u0000(s+(1)\nk0;k0+q)s;s0(s\u0000(2)\n\u0000k0\u0000q;\u0000k0)\u0003\ns0;s\u0013\ngk0;s(\u001c0;\u001c)gk0+q;s0(\u001c;\u001c0)\n+\u0012\f\f\f(s+(2)\nk0;k0+q)s;s0\f\f\f2\n\u0000(s+(2)\nk0;k0+q)s;s0(s\u0000(1)\n\u0000k0\u0000q;\u0000k0)\u0003\ns0;s\u0013\ng\u0000k0;s(\u001c;\u001c0)g\u0000k0\u0000q;s0(\u001c0;\u001c)\n\u0000\u0012\f\f\f(s+(3)\nk0;k0+q)s;s0\f\f\f2\n\u0000(s+(3)\nk0;k0+q)s;s0(s\u0000(3)\n\u0000k0\u0000q;\u0000k0)\u0003\ns0;s\u0013\ngk0;s(\u001c0;\u001c)g\u0000k0\u0000q;s0(\u001c0;\u001c)\n\u0000\u0012\f\f\f(s+(4)\nk0;k0+q)s;s0\f\f\f2\n\u0000(s+(4)\nk0;k0+q)s;s0(s\u0000(4)\n\u0000k0\u0000q;\u0000k0)\u0003\ns0;s\u0013\ng\u0000k0;s(\u001c;\u001c0)gk0+q;s0(\u001c;\u001c0)#\n;\n(S.80)\nwhere the quasiparticle Green's function is de\fned as\ngk;s(\u001c;\u001c0) :=1\ni~hTC~\rks(\u001c)~\ry\nks(\u001c0)i: (S.81)\nThe \frst and second terms give the intraband contribution, and the third and fourth terms give the interband\ncontribution. Evaluating the Dyson equation, the retarded component of the magnon Green's function is given by\nGR\nk(!) =1\nh\nG(0)R\nk(!)i\u00001\n\u0000\u0006R\nk(!); (S.82)\nwhere the unperturbed Green's function is written as\nG(0)R\nk(!) =2S=~\n!\u0000!k+i\u000b!: (S.83)\nHere, we introduce the phenomenological dimensionless damping parameter \u000b. Using Eq.(S.83), the retarded Green's\nfunction is written as\nGR\nk(!) =2S=~\n!\u0000!k+i\u000b!\u0000(2S=~)\u0006R\nk(!): (S.84)\nFrom the above expression, the frequency shift at a \fxed !is given by\n\u000eH=2S\n\r~Re\u0006R\nk(!); (S.85)\nand the enhanced Gilbert damping is given by\n\u000e\u000b=\u00002S\n~!Im\u0006R\nk(!): (S.86)\nThe Fourier transform of the self-energy is given as\n\u0006R\nk(!) =Z\ndtei(!+i0)t\u0006R\nk(t) =\u0000X\nqjJq;kj2\u001fR\nq(!); (S.87)\nwhere the dynamic spin susceptibility of the SC is de\fned as\n\u001fR\nq(!) :=Z\ndtei(!+i0)ti\n~\u0012(t)h[~\u001b+\nq(t);~\u001b\u0000\n\u0000q(0)]i: (S.88)\nEvaluating the self-energy Eq. (S.87), one can obtain the information of the FMR modulation, \u000eHand\u000e\u000b. Using the\nsystem's symmetry, the dynamic spin susceptibility \u001fR\nq(!) can be written as\n\u001fR\nq(!) = cos2\u0012\u001fxx\nq(!) +\u001fyy\nq(!) + sin2\u0012\u001fzz\nq(!); (S.89)\nwhich means that both \u000eHand\u000e\u000bshow a dependence on \u0012when the dynamic spin susceptibility is anisotropic.14\nIV. SPIN CURRENT AT THE INTERFACE\nIn this section, we derive the general expression of spin current at the interface. We treat the tunneling Hamiltonian\nas a perturbation and the other terms as the unperturbed Hamiltonian\nH(t) =H0(t) +Hex; (S.90)\nH0(t) =HFI(t) +HSC: (S.91)\nThe operator of spin current \rowing from the SC to the FI at the interface is de\fned by\nIS:=\u0000~\n2_\u001bZ\ntot=\u0000~\n21\ni~[\u001bZ\ntot;Hex] =i\n2[\u001bZ\ntot;Hex]; (S.92)\nwhere\u001bZ\ntotis given by\n\u001bZ\ntot=Z\ndr\u001bZ(r): (S.93)\nCalculating the commutation relation, we obtain the following expression\nIS=iX\nq;k\u0000\nJq;k\u001b+\nqS\u0000\nk\u0000h:c:\u0001\n: (S.94)\nThe time-dependent quantum average of ISis written as\nhIS(t)i= Re2\n42iX\nq;kJq;kh\u001b+\nq(t)S\u0000\nk(t)i3\n5; (S.95)\nwhereh\u0001\u0001\u0001i = Tr[\u001a0\u0001\u0001\u0001] denotes the statistical average with an initial density matrix \u001a0. In order to develop the\nperturbation expansion, we introduce the interaction picture\nhIS(\u001c1;\u001c2)i= Re2\n42iX\nq;kJq;khTCSC~\u001b+\nq(\u001c1)~S\u0000\nk(\u001c2)i3\n5: (S.96)\nSCand ~O(t) are given by\nSC=TCexp Z\nCd\u001c~Hex(\u001c)\ni~!\n; (S.97)\nand\n~O(t) =Uy\n0(t;t0)OU0(t;t0); (S.98)\nwhere\nU0(t;t0) =Texp\u0012Zt\nt0dt0H0(t0)\ni~\u0013\n: (S.99)\nExpandingSCas\nSC\u00191 +Z\nCd\u001cTC~Hex(\u001c)\ni~; (S.100)\nthe spin current is given by\nhIS(\u001c1;\u001c2)i=X\nq;kjJq;kj2Re\"\n2\n~Z\nCd\u001chTC~\u001b+\nq(\u001c1)~\u001b\u0000\n\u0000q(\u001c)ihTC~S+\n\u0000k(\u001c)~S\u0000\nk(\u001c2)i#\n: (S.101)15\nUsing the contour ordered Green's functions\n\u001fq(\u001c1;\u001c) =\u00001\ni~hTC~\u001b+\nq(\u001c1)~\u001b\u0000\n\u0000q(\u001c)i; (S.102)\nGk(\u001c;\u001c2) =1\ni~hTC~S+\nk(\u001c)~S\u0000\n\u0000k(\u001c2)i; (S.103)\nthe above equation is rewritten as\nhIS(\u001c1;\u001c2)i=X\nq;kjJq;kj2Re\"\n2~Z\nCd\u001c\u001fq(\u001c1;\u001c)G\u0000k(\u001c;\u001c2)#\n: (S.104)\nWe put\u001c2on the forward contour and \u001c1on the backward contour to describe spin transfer at the interface in\nappropriate time order. Assuming a steady state, the spin current is written as\nhISi= 2~X\nq;kjJq;kj2Re\"Z1\n\u00001d!0\n2\u0019\u0010\n\u001fR\nq(!0)G<\n\u0000k(!0) +\u001f<\nq(!0)GA\n\u0000k(!0)\u0011#\n: (S.105)\nWe introduce the distribution functions as\n\u001f<\nq(!) =fSC\nq(!)\u0002\n2iIm\u001fR\nq(!)\u0003\n; (S.106)\nG<\nk(!) =fFI\nk(!)\u0002\n2iImGR\nk(!)\u0003\n: (S.107)\nThe formula of the spin current at the interface is derived as\nhISi= 4~X\nq;kjJq;kj2Z1\n\u00001d!0\n2\u0019Im\u001fR\nq(!0)\u0002\n\u0000ImGR\n\u0000k(!0)\u0003\u0002\nfFI\n\u0000k(!0)\u0000fSC\nq(!0)\u0003\n: (S.108)\nWhen both the SC and the FI are in equilibrium, the di\u000berence of the distribution functions is zero (i.e. fFI\n\u0000k(!0)\u0000\nfSC\nq(!0) = 0), so that no spin current is generated. Under the microwave irradiation, the distribution function of\nthe FI deviates from equilibrium, which generates the interface spin current. Performing a second-order perturbation\ncalculation, the deviation of the distribution function of the FI, \u000efFI\n\u0000k(!0), is given by\n\u000efFI\n\u0000k(!0) =2\u0019NS (\rhac=2)2\n\u000b!0\u000ek;0\u000e(!0\u0000!): (S.109)\nConsequently, the interface spin current is written as\nhISiSP= 4~X\nq;kjJq;kj2Z1\n\u00001d!0\n2\u0019Im\u001fR\nq(!0)\u0002\n\u0000ImGR\n\u0000k(!0)\u0003\n\u000efFI\n\u0000k(!0): (S.110)\nFinally, one can show that the spin current is proportional to the enhanced Gilbert damping\nhISiSP= 4~NS(\rhac=2)2\n\u000b!\u0002\n\u0000ImGR\nk=0(!)\u0003X\nqjJq;k=0j2Im\u001fR\nq(!);\n=N(~\rhac)2\n2\u000b\u0002\n\u0000ImGR\nk=0(!)\u0003\n\u000e\u000b: (S.111)\nV. MODEL FOR INTERFACE CONFIGURATIONS\nIn order to calculate Eq. (S.87), one needs to set up an explicit expression for jJq;k=0j2. We consider an interface\nwith uncorrelated roughness. To model this interface, we assume that J(r;rj) satis\fes\nDX\njJ(r;rj)E\nave=J1; (S.112)\nDX\nj;j0J(r;rj)J(r0;rj0)E\nave=J2\n1+J2\n2l2\u000e(r\u0000r0); (S.113)16\nwhereh\u0001\u0001\u0001i avemeans interface con\fguration average. The spatially averaged J(r;rj) is given by a constant J1as\nshown in Eq. (S.112). Equation (S.113) means that the interface roughness is uncorrelated and J2\n2l2is a variance.\nJ1andJ2are coupling constants with dimension of energy, and are independent of the system size. lis introduced\nbecause the Hamiltonian of the SCs is treated as a continuum model. Performing the interface con\fguration average,\nand using Eq. (S.112) and (S.113), one can obtain the expression for jJq;k=0j2in the main text.\nVI. DYNAMIC SPIN SUSCEPTIBILITY OF SC\nEvaluating the retarded component of the self-energy Eq. (S.80), the dynamic spin susceptibility of the SC is given\nby\n\u001fR\nq(!) =\u0000Z1\n\u00001dEf(E)X\n\u0015;k(\nM\u0015;\u0015(a)\nk;k+q\u0014\n\u00001\n\u0019ImgR\n\u0015;k(E)gR\n\u0015;k+q(E+~!)\u00001\n\u0019ImgR\n\u0015;k+q(E)gA\n\u0015;k(E\u0000~!)\u0015\n+M\u0015;\u0000\u0015(a)\nk;k+q\u0014\n\u00001\n\u0019ImgR\n\u0015;k(E)gR\n\u0000\u0015;k+q(E+~!)\u00001\n\u0019ImgR\n\u0000\u0015;k+q(E)gA\n\u0015;k(E\u0000~!)\u0015)\n;\n(S.114)\nwhereM\u0015;\u00150(a)\nk;k+qwitha=s;c;andhare given by\nM\u0015;\u00150(s)\nk;k+q=(\u0018k+\u0015Ek)(\u0018k+q+\u00150Ek+q)\n4\u0015Ek\u00150Ek+q+\u00012\n4\u0015Ek\u00150Ek+q; (S.115)\nM\u0015;\u00150(c)\nk;k+q=(\u0018k+\u0015Ek)(\u0018k+q+\u00150Ek+q)\n4\u0015Ek\u00150Ek+q\u0000\u00012e\u0000i(\u001ek\u0000\u001ek+q)\n4\u0015Ek\u00150Ek+qcos2\u0012; (S.116)\nM\u0015;\u00150(h)\nk;k+q=(\u0018k+\u0015Ek)(\u0018k+q+\u00150Ek+q)\n4\u0015Ek\u00150Ek+q\u0000\u00012sin\u001eksin\u001ek+q\n4\u0015Ek\u00150Ek+qsin2\u0012: (S.117)\n\u0015;\u00150=\u0006give a sign, and a=s;c;andhcorrespond to matrix elements for s-wave, chiral p-wave, and helical p-wave\nSCs, respectively. In Eq. (S.114), the terms multiplied by M\u0015;\u0015(a)\nk;k+qdescribe the intraband transition processes, i.e.,\ntransition processes from particle to particle and from hole to hole, and the terms multiplied by M\u0015;\u0000\u0015(a)\nk;k+qdescribe\nthe interband transition processes, i.e., transition processes from particle to hole and vice versa. The retarded and\nadvanced Green's functions of the quasiparticles gR=A\n\u0015;k(E) are given by\ngR\n\u0015;k(E) =1\nE\u0000\u0015Ek+i\u0000; (S.118)\ngA\n\u0015;k(E) =1\nE\u0000\u0015Ek\u0000i\u0000; (S.119)\nwhere \u0000 is a constant level broadening introduced phenomenologically. \u0000 is introduced to incorporate the intraband\ncontribution in the calculation of the uniform spin susceptibility. The details are explained in the next section.\nThe sum over kis replaced by the integral near the Fermi energy\nX\nkF(k)!DFZ1\n0dEDs(E)X\n\u0011=\u0006F\u0011(E); (S.120)\nX\nkF(k) sin2\u001ek!DFZ1\n0dEDs(E)X\n\u0011=\u00061\n2F\u0011(E); (S.121)\nwhereDFis the density of states near the Fermi energy in the normal state and Ds(E) is the density of states of\nquasiparticles\nDs(E) =jEjp\nE2\u0000\u00012\u0012(jEj\u0000\u0001): (S.122)\nF\u0011(E) means to assign \u0011p\nE2\u0000\u00012to\u0018contained in F(k).17\nVII. UNIFORM SPIN SUSCEPTIBILITY\nIn this section, we explain three properties related to the calculation of the uniform spin susceptibility. First, the\nmatrix element's properties are explained, which is essential to understand the qualitative di\u000berence between spin-\nsinglets-wave and spin-triplet p-wave SCs. Second, the reason to introduce the constant level broadening \u0000. Third,\nthe analytical expression for the uniform spin susceptibility of the p-wave SCs is given.\nPerforming the angular integral and replacing the sum over kby theEintegral, the matrix elements are replaced\nby\nM\u0015;\u00150(s)\nk;k!1 +\u0015\u00150\n4\u0015\u00150; (S.123)\nM\u0015;\u00150(c)\nk;k!(1 +\u0015\u00150)E2\u0000(1 + cos2\u0012)\u00012\n4\u0015\u00150E2; (S.124)\nM\u0015;\u00150(h)\nk;k!(1 +\u0015\u00150)E2\u0000(1 +1\n2sin2\u0012)\u00012\n4\u0015\u00150E2: (S.125)\nHere, the \frst-order terms in \u0018kare omitted because they vanish in the Eintegral. From the above expressions, the\nintraband matrix elements become \fnite for all SCs considered here, while the interband matrix elements vanish in\nthes-wave SC and becomes \fnite in the p-wave SCs. The above properties of the intraband and interband matrix\nelements can be understood using the commutation relation between the Hamiltonian and the spin operators. We\nintroduce the BdG form of the spin operators \u001ba\nBdG(a=x;y;z ) as below\n\u001ba\nBdG= \n\u001ba0\n0\u0000(\u001ba)T!\n: (S.126)\nThe commutation relation of HBdGand\u001ba\nBdGis given by\n[HBdG;\u001ba\nBdG] = 0 :s\u0000wave; (S.127)\n[HBdG;\u001ba\nBdG]6= 0 :p\u0000wave: (S.128)\nEquation (S.127) means that both the Hamiltonian and the spin operator are diagonalized simultaneously, so that the\nmatrix elements of the spin operator between a particle and a hole with the same wave-number vanish. This is because\nthes-wave SC is spin singlet. Therefore, the interband matrix elements vanishes in the s-wave SC. In contrast, in\nthep-wave SCs, the commutation relation between the Hamiltonian and the spin operator is \fnite as shown in Eq.\n(S.128), so that the matrix elements of the spin operator between a particle and a hole with the same wave-number\nis \fnite. This is because the p-wave SCs are spin triplet. As a result, the interband matrix elements are \fnite.\nHere, we explain the reason to introduce the constant level broadening \u0000 for gR=A\n\u0015;k(E). The intraband and interband\ntransitions are schematically shown in Fig. 5. The quasiparticles are scattered due to the magnon emission or\nabsorption. The scattering process conserves the wave-number. Consequently, in the case of the intraband transition,\nthe transition process is forbidden when \u0000 = 0. In order to incorporate the intraband processes, one needs to introduce\n\u0000, otherwise the intraband contribution vanishes, which can be directly shown by calculating Eq. (S.114).\nWhen \u0000 = 0, the uniform spin susceptibility for the chiral p-wave SCs is given by\nRe\u001fR\nuni(!) =2DFZ1\n\u0001dEEp\nE2\u0000\u00012(1 + cos2\u0012)\u00012\n4E2(f(E)\u0000f(\u0000E))\u00121\n2E+~!+1\n2E\u0000~!\u0013\n; (S.129)\nand\nIm\u001fR\nuni(!) =2\u0019DFj~!=2jp\n(~!=2)2\u0000\u00012(1 + cos2\u0012)\u00012\n(~!)2(f(\u0000~!=2)\u0000f(~!=2)): (S.130)\nFrom the above expressions, one can show that both the real part and imaginary part of the uniform spin susceptibility\ndiverge at ~!= 2\u0001, leading a resonance peak. The expressions for the helical p-wave SC can be obtained by replacing\ncos2\u0012with1\n2sin2\u0012. Therefore, \u0012dependence of \u001fR\nuni(!) explained in the main text is obtained from the above\nexpressions.18\nΓintra\ninter\nk+Ek\n−EkE E\nSpectral\nfunction\n2/uni0394/uni210F/uni03C9\n/uni210F/uni03C9Γ\nFIG. 5. Schematic image of intraband transition and interband transitions. The intraband transition gives contribution to the\nuniform spin susceptibility when the excitation energy is comparable to or smaller than the level broadening, ~!.\u0000. The\ninterband contribution is dominant when the excitation energy is comparable to the superconucting gap, ~!\u00192\u0001.\nVIII. LOCAL SPIN SUSCEPTIBILITY\nPerforming the angular integral and replacing the sum over k;qby theE;E0integral, the matrix elements are\nreplaced by\nM\u0015\u00150(s)\nk;q!1\n4+\u00012\n4\u0015E\u00150E0; (S.131)\nM\u0015\u00150(c)\nk;q!1\n4; (S.132)\nM\u0015\u00150(h)\nk;q!1\n4: (S.133)\nThe matrix elements for the chiral and helical p-wave SCs are identical. From the above expressions, one can see\nthat the interband contribution in the s-wave SC is suppressed. Unlike the uniform spin susceptibility, the intraband\ncontribution for the local spin susceptibility is \fnite even when \u0000 = 0. This is because the transition processes\nconsidered here leads to momentum transfer and the intraband transition is not forbidden. Therefore, we calculate\nthe local spin susceptibility at \u0000 = 0. The local spin susceptibility for the s-wave SC is given by\n\u001fR\nloc(!) =\u0000D2\nFZ1\n\u00001dEZ1\n\u00001dE0Ds(E)Ds(E0)\u0012\n1 +\u00012\nEE0\u0013f(E)\u0000f(E0)\nE\u0000E0+~!+i0; (S.134)\nand the local spin susceptibility for the p-wave SCs is given by\n\u001fR\nloc(!) =\u0000D2\nFZ1\n\u00001dEZ1\n\u00001dE0Ds(E)Ds(E0)f(E)\u0000f(E0)\nE\u0000E0+~!+i0: (S.135)\nIX. FMR MODULATION: ROUGH INTERFACE\nIn this section, we show the numerical results and summarize the characteristic properties of the FMR modulation\nfor the rough interface limit. In the following calculations, we set J1= 0 and assume that only \u001fR\nloc(!) contributes to\n\u000eHand\u000e\u000b.\nFigures 6 show (a) \u000eHand (b)\u000e\u000bfor the chiral and helical p-wave SCs as a function of frequency and temperature.\n\u000eHis \fnite inT!0 and has a resonance peak at ~!= 2\u0001.\u000e\u000bexhibits a coherence peak just below the transition\ntemperature in the su\u000eciently low frequency region, where ~!=kBTc\u001c1.\u000e\u000bdrops abruptly at ~!= 2\u0001.\u000e\u000bis\nalmost independent of both frequency and temperature when ~!>2\u0001.\nFigures 6 show (c) \u000eHand (d)\u000e\u000bfor thes-wave SC as a function of frequency and temperature. In the low\nfrequency region, where ~!=kBTc\u00141,\u000eHat a \fxed frequency decreases by about thirty percent with the decrease of\nthe temperature, and \u000eHis \fnite inT!0. As the frequency increases, \u000eHis almost independent of the temperature.\n\u000e\u000bshows a coherence peak just below the transition temperature in the su\u000eciently low frequency, where ~!=kBTc\u001c1.19\nThe coherence peak in the s-wave SC is larger than the corresponding coherence peak in the p-wave SCs. \u000e\u000bhas a\nkink structure at ~!= 2\u0001.\nNote that the cuto\u000b energy Ecwas introduced here to cause the integral for Re \u001fR\nloc(!) to converge. Although\nRe\u001fR\nloc(!) is approximately proportional to Ec, the qualitative properties explained above are independent of Ec.\nThe FMR modulation properties of the three SCs are summarized in Table II. In the case of the rough interface\nlimit, the pairing symmetry can be detected from either the absence or the existence of the resonance peak of \u000eH. The\npairing symmetry may also be detected from the properties of \u000e\u000b, the height of the coherence peak, and the structure\nat~!= 2\u0001. When compared with the resonance peak for \u000eH, however, the properties of \u000e\u000bare too ambiguous to\nallow the pairing symmetry to be distinguished clearly.\n(c) (d)s-wave(a) (b)Chiral & Helical p-wave\nT/T\nchω/kBTcδH/δH2\n0.00.51.00\n2\n410\n8\n6\nT/T\nchω/kBTcδα/δα2\n0.00.51.00\n2\n42\n1\n0\nT/T\nchω/kBTcδH/δH2\n0.00.51.00\n2\n410\n8\n6\nT/T\nchω/kBTcδα/δα2\n0.00.51.00\n2\n42\n1\n0\nFIG. 6. (a) The frequency shift and (b) the enhanced Gilbert damping as a function of both frequency and temperature for the\np-wave SCs. (c) The frequency shift and (d) the enhanced Gilbert damping as a function of both frequency and temperature\nfor thes-wave SC. The terms \u000eH2and\u000e\u000b2are given by \u000eH2=\u00002\u0019SJ2\n2l2D2\nFkBTc=(NA\r ~) and\u000e\u000b2= 2\u0019SJ2\n2l2D2\nF=(NA),\nwhere they are characteristic values in the normal state. The cuto\u000b energy is set to be Ec=kBTc= 10.\nTABLE II. FMR modulation properties for the rough SC/FI interface where J1= 0 andJ26= 0.\nPairing symmetry s Chiral Helical\nResonance peak of \u000eH { X X\nCoherence peak of \u000e\u000b large small small\n\u000e\u000bat~!= 2\u0001 kink jump jump\n1J. Linder and J. W. A. Robinson, Nature Physics 11, 307\n(2015).\n2W. Han, S. Maekawa, and X.-C. Xie, Nature Materials\n19, 139 (2020).\n3R. S. Keizer, S. T. B. Goennenwein, T. M. Klapwijk,\nG. Miao, G. Xiao, and A. Gupta, Nature 439, 825 (2006).\n4Y. Tanaka, T. Yokoyama, A. V. Balatsky, and N. Nagaosa,\nPhysical Review B 79, 060505 (2009).\n5P. M. R. Brydon, Physical Review B 80, 224520 (2009).\n6Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Physical\nReview Letters 88, 117601 (2002).7Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I.\nHalperin, Rev. Mod. Phys. 77, 47 (2005).\n8E.\u0014Sim\u0013 anek and B. Heinrich, Physical Review B 67, 144418\n(2003).\n9M. Inoue, M. Ichioka, and H. Adachi, Physical Review B\n96, 024414 (2017).\n10T. Kato, Y. Ohnuma, M. Matsuo, J. Rech, T. Jonckheere,\nand T. Martin, Physical Review B 99, 144411 (2019).\n11M. A. Silaev, Physical Review B 102, 144521 (2020).\n12M. A. Silaev, Physical Review B 102, 180502 (2020).\n13R. Ojaj arvi, J. Manninen, T. T. Heikkil a, and P. Virtanen,20\nPhysical Review B 101, 115406 (2020).\n14H. T. Simensen, L. G. Johnsen, J. Linder, and A. Brataas,\nPhysical Review B 103, 024524 (2021).\n15C. Bell, S. Milikisyants, M. Huber, and J. Aarts, Physical\nReview Letters 100, 047002 (2008).\n16K.-R. Jeon, C. Ciccarelli, A. J. Ferguson, H. Kurebayashi,\nL. F. Cohen, X. Montiel, M. Eschrig, J. W. A. Robinson,\nand M. G. Blamire, Nature Materials 17, 499 (2018).\n17Y. Yao, Q. Song, Y. Takamura, J. P. Cascales, W. Yuan,\nY. Ma, Y. Yun, X. C. Xie, J. S. Moodera, and W. Han,\nPhysical Review B 97, 224414 (2018).\n18L.-L. Li, Y.-L. Zhao, X.-X. Zhang, and Y. Sun, Chinese\nPhysics Letters 35, 077401 (2018).\n19K.-R. Jeon, C. Ciccarelli, H. Kurebayashi, L. F. Cohen,\nX. Montiel, M. Eschrig, T. Wagner, S. Komori, A. Srivas-\ntava, J. W. Robinson, and M. G. Blamire, Physical Review\nApplied 11, 014061 (2019).\n20K.-R. Jeon, C. Ciccarelli, H. Kurebayashi, L. F. Cohen,\nS. Komori, J. W. A. Robinson, and M. G. Blamire, Phys-\nical Review B 99, 144503 (2019).\n21I. Golovchanskiy, N. Abramov, V. Stolyarov, V. Chichkov,\nM. Silaev, I. Shchetinin, A. Golubov, V. Ryazanov,\nA. Ustinov, and M. Kupriyanov, Physical Review Applied\n14, 024086 (2020).\n22Y. Zhao, Y. Yuan, K. Fan, and Y. Zhou, Applied Physics\nExpress 13, 033002 (2020).\n23A. J. Leggett, Reviews of Modern Physics 47, 331 (1975).\n24C. G. Shull and R. P. Ferrier, Physical Review Letters 10,\n295 (1963).\n25C. G. Shull, Physical Review Letters 10, 297 (1963).\n26C. G. Shull and F. A. Wedgwood, Physical Review Letters\n16, 513 (1966).\n27G. M. Luke, A. Keren, L. P. Le, W. D. Wu, Y. J. Uemura,\nD. A. Bonn, L. Taillefer, and J. D. Garrett, Physical Re-\nview Letters 71, 1466 (1993).\n28S. S. Saxena, P. Agarwal, K. Ahilan, F. M. Grosche,\nR. K. W. Haselwimmer, M. J. Steiner, E. Pugh, I. R.\nWalker, S. R. Julian, P. Monthoux, G. G. Lonzarich,\nA. Huxley, I. Sheikin, D. Braithwaite, and J. Flouquet,\nNature 406, 587 (2000).\n29D. Aoki, A. Huxley, E. Ressouche, D. Braithwaite, J. Flou-\nquet, J.-P. Brison, E. Lhotel, and C. Paulsen, Nature 413,\n613 (2001).\n30N. T. Huy, A. Gasparini, D. E. de Nijs, Y. Huang, J. C. P.\nKlaasse, T. Gortenmulder, A. de Visser, A. Hamann,\nT. G orlach, and H. v. L ohneysen, Physical Review Letters\n99, 067006 (2007).\n31S. Ran, C. Eckberg, Q.-P. Ding, Y. Furukawa, T. Metz,S. R. Saha, I.-L. Liu, M. Zic, H. Kim, J. Paglione, and\nN. P. Butch, Science 365, 684 (2019).\n32J. Yang, J. Luo, C. Yi, Y. Shi, Y. Zhou, and G.-q. Zheng,\nScience advances 7, eabl4432 (2021).\n33Y. Ohnuma, H. Adachi, E. Saitoh, and S. Maekawa, Phys-\nical Review B 89, 174417 (2014).\n34Y. Ohnuma, M. Matsuo, and S. Maekawa, Physical Re-\nview B 96, 134412 (2017).\n35G. Tatara and S. Mizukami, Physical Review B 96, 064423\n(2017).\n36M. Matsuo, Y. Ohnuma, T. Kato, and S. Maekawa, Phys-\nical Review Letters 120, 037201 (2018).\n37Y. Ominato and M. Matsuo, Journal of the Physical Soci-\nety of Japan 89, 053704 (2020).\n38Y. Ominato, J. Fujimoto, and M. Matsuo, Physical Review\nLetters 124, 166803 (2020).\n39See Supplemental Material for details of the model deriva-\ntion, formulation, and calculations.\n40M. Sigrist and K. Ueda, Reviews of Modern Physics 63,\n239 (1991).\n41T. Kasuya and R. C. LeCraw, Physical Review Letters 6,\n223 (1961).\n42V. Cherepanov, I. Kolokolov, and V. L'vov, Physics Re-\nports 229, 81 (1993).\n43L. Jin, Y. Wang, G. Lu, J. Li, Y. He, Z. Zhong, and\nH. Zhang, AIP Advances 9, 025301 (2019).\n44W. Chen, M. Sigrist, J. Sinova, and D. Manske, Physical\nReview Letters 115, 217203 (2015).\n45M. B. Jung\reisch, A. V. Chumak, A. Kehlberger, V. Lauer,\nD. H. Kim, M. C. Onbasli, C. A. Ross, M. Kl aui, and\nB. Hillebrands, Physical Review B 91, 134407 (2015).\n46Z. Qiu, K. Ando, K. Uchida, Y. Kajiwara, R. Takahashi,\nH. Nakayama, T. An, Y. Fujikawa, and E. Saitoh, Applied\nPhysics Letters 103, 092404 (2013).\n47L. Mihalceanu, S. Keller, J. Greser, D. Karfaridis, K. Sime-\nonidis, G. Vourlias, T. Kehagias, A. Conca, B. Hillebrands,\nand E. T. Papaioannou, Applied Physics Letters 110,\n252406 (2017).\n48Ikhtiar, H. Sukegawa, X. Xu, M. Belmoubarik, H. Lee,\nS. Kasai, and K. Hono, Applied Physics Letters 112,\n022408 (2018).\n49S. Carreira, D. Sanchez-Manzano, M.-W. Yoo, K. Seurre,\nV. Rouco, A. Sander, J. Santamar\u0013 \u0010a, A. Anane, and J. Vil-\nlegas, Physical Review B 104, 144428 (2021).\n50Y. Ominato, A. Yamakage, T. Kato, and M. Matsuo,\nPhysical Review B 105, 205406 (2022).\n51L. G. Johnsen, H. T. Simensen, A. Brataas, and J. Linder,\nPhysical Review Letters 127, 207001 (2021)."    },    {        "title": "1010.3858v1.Ferromagnetic_resonance_with_a_magnetic_Josephson_junction.pdf",        "content": "arXiv:1010.3858v1  [cond-mat.supr-con]  19 Oct 2010Ferromagnetic resonance with a magnetic Josephson junctio n\nS. E. Barnes1, M. Aprili2, I. Petkovi´ c3, and S. Maekawa4\n1Physics Department, University of Miami, Coral Gables,\n33124 FL, USA.2Laboratoire de Physique des Solides,\nBˆ at. 510, Universit´ e Paris-Sud, 91405 Orsay Cedex,\nFrance.3Service de Physique de l‘Etat Condens´ e/IRAMIS/DSM (CNRS U RA 2464),\nCEA Saclay, F-91191 Gif-sur-Yvette, France.4Advanced Science Research Center,\nJapan Atomic Energy Agency, Tokai, Ibaraki 319-1195, and CR EST,\nJapan Science and Technology Agency, Tokyo 102-0075, Japan .\n(Dated: November 18, 2018)\nWe show experimentally and theoretically that there is a cou pling via the Aharonov–Bohm phase\nbetween the order parameter of a ferromagnet and a singlet, s-wave, Josephson supercurrent. We\nhave investigated the possibility of measuring the dispers ion of such spin waves by varying the\nmagnetic ﬁeld applied in the plane of the junction and demons trated the electromagnetic nature\nof the coupling by the observation of magnetic resonance sid e-bands to microwave induced Shapiro\nsteps.\nRotation symmetry associated with the O(3) orthog-\nonal group forbids a coupling between a scalar s-wave\norder parameter and the vector order parameter /vectorMof\nthe ferromagnet. However a Josephson junction [1] de-\nﬁnes a plane, the O(3) symmetry is broken, and such a\ncoupling is possible. Here we describe a part of the rich\nspectroscopic magnetic resonance possibilities that this\nobservation implies. It is possible to perform a “photon\nfree” FMR experiment[2] on about 107Ni atoms, some-\nthing infeasible with standard FMR techniques.\nInteractions in nature reﬂect certain gauge groups and\nthe associate phases which generate a vector potential\n/vectorA, called the Berry connection[3]. Interactions via elec-\ntromagnetic ﬁelds generated by the conserved electrical\ncurrents reﬂect the U(1) gauge group and the Aharonov–\nBohm (AB) phase[4]. Associated with angular moment\nis the gauge symmetry SU(2) and the familiar Lie al-\ngebra, i.e., the spin commutation rules. The AB is re-\nplaced by the spin Berry phase. It is often imagined\nthat magnetic moments might interact with the Joseph-\nson current by direct spin-ﬂips, which would involve the\nspin Berry phase[5]. In this paper it will be shown\nthat quantitatively the interaction between the Joseph-\nson current and the order parameter in superconduc-\ntor/ferromagnet/superconductor (SFS) junctions can be\nexplained in terms of the AB-phase and regular electro-\ndynamics. It will be shown such an experiment measures\nrather directly the magnet correlation function.\nWe fabricated Nb /Pd0.9Ni0.1/Nb Josephson junctions\nbyin-situangle evaporation through a resist mask\nand subsequent lift-oﬀ. The mask is deﬁned by e-\nbeam lithography on a Polyether Sulphone - PES (500\nnm)/Si 3N4(60nm)/Polymethyl Methacrylate - PMMA\n(350 nm) trilayer [6] , and etched in a Reactive Ion Etch-\ning (RIE) chamber. The Si 3N4is etched 1 minute 30\nseconds in SF 6and the PES in Oxygen plasma for 10\nminutes, giving an undercut of 500 nm. The mask fabri-\ncation process is schematically presented in Fig. 1a. The\nSi3N4suspended bridge allows shadow evaporation. A\nscanning-electron-microscope(SEM) pictureofthemaskincluding the suspended bridge is reported in Fig. 1c.\nThe ﬁrst Nb layer is evaporated at -45 degrees with re-\nspect tozaxis while the PdNi is evaporatedat 45 degrees\nandthe secondNb layerat 47degrees. The overlapin the\nydirection deﬁnes the junction area. The shadow evapo-\nrationisillustratedinthelowerdrawingarrayofFig. 1a.\nEight junctions are evaporated on the sample chip. A\nSEM image of one of the junctions after lift-oﬀ is shown\nin Fig. 1d. The electron-gun evaporation is carried out\nin ultra-high-vacuum (UHV) with a base pressure lower\nthan 10−9mbar. The two Nb superconducting ﬁlms are\n50 nm thick and the ferromagnetic layer of Pd 0.9Ni0.1\nFIG. 1: Description of the sample. (a)The fabrication proce-\ndure. Trilayer mask is etched to obtain the suspended bridge\nand the undercut, then the junction is fabricated by shadow\nevaporation and liftoﬀ. (b)The principle of the experiment.\nAC Josephson current across the junction excites the spin\nwavemodeswhichinturncoupletotheJosephson currentand\nrectiﬁcation takes place. (c)Scanning Electron Microscopy\n(SEM) picture of the Si 3N4/PES mask before angle evapora-\ntion.(d)SEM picture of a junction after lift oﬀ.2\nis 20 nm thick. The Ni concentration is measured by\nRutherford Backscattering (RBS) on a test sample. The\nmagnetization loops obtained by SQUID magnetometry\nwith in-plane and out-of-plane magnetic ﬁeld show a pre-\ndominant perpendicular anisotropy. Finally, a schematic\nview of the junction including the principle of the exper-\niment is to be found in Fig. 1b.\nTypical current-voltage( IV) characteristicsare shown\nin Fig. 2 as function of temperature. Well below the\ncritical temperature, the critical current Ic≈7µA and\nnormal resistance Rn= 1Ω give a Josephson coupling\nof 7µV, consistent with early studies on highly under-\ndamped PdNi-based Josephson junctions [7]. The IV\ncharacteristics are not hysteretic conﬁrming overdamped\nphase dynamics and are well described by the resistively-\nshunted-junction (RSJ) model [1]. The critical current\nversus temperature (see inset of Fig. 2) shows the typ-\nical linear behavior expected when the Thouless energy\nof the ferromagnetic layer is larger than the Nb super-\nconducting energy gap. This linear dependence has been\nobserved previously in highly underdamped junctions[7].\nThe junction critical temperature is about 7.0 K while\nthe critical temperature of the Nb leads is 7 .6 K and\ntheir criticalcurrentover500 µAat lowtemperature. We\nhave measured four ferromagnetic junctions on the same\nwafer, the dispersion of the critical current from junc-\ntion to junction is about ±1µA, ∆Rnis 0.15 Ω while the\nIcRnvaries by less than 3% from junction to junction.\nWe have also fabricated non-magnetic junctions by the\nsame process but replacing the PdNi with a thicker 70\nnm Pd layer. These junctions have a much larger critical\ncurrent of about 44 µA.\nFIG. 2: Current-voltage characteristics with increasing t em-\nperature. The curves have been shifted vertically for clari ty.\nThey do not show any hysteresis as expected when the phase\ndynamics is strongly damped. The critical current as a func-\ntion of the temperature is shown in the insert, the dashed lin e\nis a linear ﬁt.For a square junction of side L, the total super-current\nis given by the integral [1]\nIs=Jc/integraldisplayL/2\n−L/2dx/integraldisplayL/2\n−L/2dysinφ(x,y,t) (1)\nwith\nφ(/vector r,t) =φ0+ωJt−2e\n¯h/integraldisplay\n/vectorA·d/vector r, (2)\nwhereφ0is an arbitrary phase, ωJ= (2e/¯h)V0is the\nJosephson frequency, and the last term is the AB phase\n[8], involving the vector potential /vectorA. We use a gauge\nwhere/vectorA=A(/vector r,t)ˆ z, the direction ˆ zbeing perpendicular\nto the junction surface. Therefore φ(/vector r,t) =φ0+kx+\nωJt−φ1, where φ1reﬂects time dependent ﬁelds and\nk= (4ed/¯h)µ0H+ (4ea/¯h)µ0M0y. HereM0yis they-\ncomponent of the static magnetisation /vectorM0, the applied\nﬁeld/vectorHis in the ydirection, 2 aand 2d=2(a+λ) are the\nactual and magnetic thickness of barrier and λthe Lon-\ndon penetration depth. These equations describe both\nthe statics and the dynamics of our junctions.\nThe approach is similar to that used for junctions with\nmagnetic impurities in a normal metal barrier[9]. It is\nnecessary to determine the appropriate boundary condi-\ntions for the solutions of Maxwell’s equations. For rea-\nsons of transparency, it is not at all useful to solve the\nvery diﬃcult problem in which the solution within the\nbarrier is matched to that in the exterior region to the\njunction. Within the junction we can ignore the dis-\nplacement and transport currents since the wavelength\nof lightλand the skin depth δare both larger than the\ndimensionsofthe junction attheJosephsonfrequency ωJ\nrelevant for the FMR. We observe that the impedance of\nthe junction of ∼1Ω is much smaller than 377 Ω of free\nspace and as a consequence there is essentially no radi-\nation from the junction. The displacement current, and\nevidently, the transport current can therefore also be ig-\nnoredin theexteriorregion. It isthereforeonlynecessary\nto integrate Amp` ere’s circulation law\n/vector∇×/vectorH=Js(/vector r,t)ˆz;Js=Jcsinφ(/vector r,t) (3)\nwhereφ0is absorbed by a time translation, and φ(/vector r,t) =\nkx+ωJt−φ1. Required is the additional AB phase shift\nφ1=2e\n¯h/integraldisplay/vector r+2aˆz\n/vector r/vectorA1·d/vector r=4ae\n¯hA1\nz, (4)\nwith the magnetic system reﬂected in /vectorA1. In linear re-\nsponseφ1is considered as a perturbation and the dc sig-\nnal is\nI1=−4ae\n¯hωJ1\nT/integraldisplayT\n0dt/integraldisplayL/2\n−L/2dx/integraldisplayL/2\n−L/2dy∂Js\n∂tA1\nz(5)3\nwhichincludesatime averageoverasingleperiod T. The\ndetermination of /vectorA1requires ﬁrst the vector integration\nof/vector∇ ×/vectorH=Js(/vector r,t)ˆzand then /vector∇ ×/vectorA=/vectorBwith/vectorB=\nµ0(/vectorH+/vectorM). Even with the simpliﬁcations of the previous\nparagraph, this is an involved calculation. It is useful to\nmake some formal manipulations in order to avoid this\ndouble integration. First I1is written as\nI1=−4ae\n¯hωJ1\nT/integraldisplayT\n0dt/integraldisplayL/2\n−L/2dx/integraldisplayL/2\n−L/2dy/parenleftbigg∂\n∂t/vector∇×/vectorH/parenrightbigg\n·/vectorA\n(6)\nusing Amp` ere’s law /vectorJ=/vector∇×/vectorH. Performing an integra-\ntion by parts on time we have\nI1=4ae\n¯hωJ1\nT/integraldisplayT\n0dt/integraldisplayL/2\n−L/2dx/integraldisplayL/2\n−L/2dy/parenleftBig\n/vector∇×/vectorH/parenrightBig\n·∂/vectorA\n∂t.\n(7)\nUsing the fact that the Poynting vector, and hence /vectorH×\n∂/vectorA/∂t= 0 on the surface, this is integrated again by\nparts using /vector∇·(/vectorH×(∂/vectorA/∂t)) = (∂/vectorA/∂t)·/vector∇×/vectorH−/vectorH·\n/vector∇×(∂/vectorA/∂t), to give\nI1=4ae\n¯hωJ1\nT/integraldisplayT\n0/integraldisplayL/2\n−L/2dx/integraldisplayL/2\n−L/2dy/vectorH·∂\n∂t/vector∇×/vectorA.(8)\nThen, since /vector∇×/vectorA=/vectorBand/vectorB=µ0(/vectorH+/vectorM), the signal\nI1=−1\nV0/integraldisplay\ndv/vectorH·∂/vectorM\n∂t(9)\nwheredv= 2adxdyis the elementary volume, the inte-\ngral is over the volume of the magnetic layer, and the\naverage is indicated by the bar. The resonance of the\nferromagnetic layer is contained in χi(t), the dynamic\nsusceptibility, and\nMi(/vector r,t) =/integraldisplay\nd/vector r′/integraldisplay\ndt′χi(/vector r−/vector r′;t−t′)Hi(/vector r′,t′),(10)\nwherei=x,y,z. The simple expressions Eqs. (9) and\n(10) are a principal theoretical result presented here and\nhave an obvious interpretation in terms of the magnetic\nenergy. They demonstrate that Josephson junction mag-\nnetic spectroscopy measures very directly the magnetic\nsusceptibility correlation function χi(/vector r−/vector r′;t−t′), and\nall the other excitations to which that couples, in much\nthe same manner as does neutron scattering. As will be\nseen below, the advantage is that this technique couples\npreferentially to small qwave vectors.\nSinceHi(/vector r,t′) isperiodicin time, the timeconvolution\nEq. (10), reduces to a product and ifthe susceptibility\nis suﬃciently local and thenMi(/vector r,ωJ) = (χ′\ni(ωJ)+iχ′′\ni(ωJ))Hi(/vector r,ωJ) (11)\nin the usual complex notion. If rather the response is\nnon- local then\nMi(/vectork,ωJ) = (χ′\ni(/vectork,ωJ)+iχ′′\ni(/vectork,ωJ))Hi(/vectork,ωJ) (12)\nand it requires a Fourier expansion of the spatially de-\npendent Hi(/vector r,ωJ). Finally, when ωJ≈IcR, as for the\nlowest voltage experimental signals, the linear response\napproximation is not strictly applicable and high har-\nmonics of ωJmust be accounted for. Similar expressions\napply but now, in particular, “half-harmonic” signals oc-\ncur since the super-current contains higher harmonics\nand can, corresponding to the second harmonic, excite\na resonance at ωswhenωJ/2 =ωs.\nNowallthat is required is a singleintegration of\n/vector∇×/vectorH=Jcsin[kx+ωJt]ˆz, (13)\nbut which is not a simple task. It is trivial to ver-\nify by diﬀerentiation that such an integral is /vectorHp=\n−(Jc/k)cos(kx+ωJt)ˆy. However this does not satisfy\nthe boundary condition that the current density is zero\noutside the square junction region. With the present\ngauge/vectorA=Azˆz, it is necessary to ﬁnd a solution of\nLaplace’s equation ∇2Az= 0 such that /vectorH=/vectorHp+/vectorHi\nwhere/vectorHi=/vector∇×/vectorAis such that /vectorHsatisﬁes Eq. (13) inside\nthe square but has /vector∇×/vectorH= 0 outside. A little reﬂection\nsuggests there are two contributions to /vectorHiwhich must\nbe accounted for. First, in general, reﬂecting the even\npartJccoskxsinωJtof the current density there is a net\noscillating super-current Ic(H) which causes a circulat-\ningmagnetic ﬁeld about ˆz, and second, associated with\nthe spatially odd part of Jcsin[kx+ωJt], there is a uni-\nform component of the ﬁeld in the ˆy-direction. The ﬁrst\ncontribution is determined by considering the problem\nwithk= 0, i.e., with a uniform current density Js. The\nsolution is /vectorHs=1\n2Js(xˆy−yˆx)cosωJt+...where the el-\nlipsis reﬂects the relatively small corrections for a square\nas compared with a circular cross section. In what fol-\nlows this correction is ignored. The corresponding vector\npotential has Az= (1/4)Js(x2+y2). For ﬁnite H, inte-\ngratingthe evenpart ofthe currentdensity givesanaver-\nage super-current density of Js=Jcsin(kL/2)/(kL/2).\nBy inspection it is observed Az= (1/4)Js(−x2+y2)\nsatisﬁes ∇2Az= 0. The corresponding odd /vectorHo\ni=\nxˆy−yˆx)cosωJt+....The sum /vectorHp+/vectorHo\nicorrectly re-\nduces to /vectorHsin the limit k→0. It is the case that /vectorHp\n(and/vectorHp+/vectorHs\ni) implies a uniform oscillating ﬁeld but one\nwhich diverges as k→0, whereas physically, the even\npart of /vectorHpshould be proportional to kreﬂecting the\nJosephson screening of ﬁelds. That the tangential ap-\nplied ﬁeld Hbe continuous requires the current, induced4\nby the even part of /vectorH, to be zero at the surface. The\neven part of /vectorHpis−Jccoskx\nksinωJtˆy(coskx/k)sinωJtˆy\nThe required even part of Hiis now /vectorHe\ni=\nJccoskL\n2\nksinωJtˆy(coskL\n2/k)sinωJtˆy, which is equally di-\nvergent as k→0. The net result of integrating Eq. (13)\nis therefore\n/vectorH=Jc\nk[cos(kx+ωJt)−coskL\n2sinωJt]ˆy\n+1\n2JcsinkL\n2\nkL\n2(−xˆy−yˆx)cosωJt+...(14)\nImagine that the magnetic layeris composed of a num-\nber of independent crystallites so that the response is\nlocal and Eq. (11) would apply. This local assumption\nalso has the merit of being an useful illustration of the\ntheory since it leads to a relatively simple prediction of\ntheHdependence of the signal which can be compared\nwith experiment. This helps determine if the response is\nindeed local, or extended. There are some complicated\nintegrals involved in the evaluation of Eq. (9). The result\nis written, in closed form, as\nIm= 2πIc(0)Φrf\nΦ0/bracketleftbig\nFxχ′′\nx(ωJ)+Fyχ′′\ny(ωJ)/bracketrightbig\n,(15)\nwhere Φ rf= (2aL)Brf= (2aL)µ0Ic(0)/Lis the ﬂux due\nto the radio frequency ﬁeld and where\nFx=1\n48/bracketleftbiggIc(B0)\nIc/bracketrightbigg2\nand\nFy=2\nx2{1−1\nxsinx\n2cosx\n2−[11\n24+2\nx2]sin2x\n2},\nFIG. 3: In blue is Fyand in pink Fx. These curves satisfy\nthe evident requirement that the coeﬃcient of the xandy\nresponses be the same when B= 0. Markers denote the\nexperimental values of the resonance amplitude as function\nof the applied ﬁeld, with the normalization constant as the\nﬁtting parameter.withx=kL, reﬂect the geometrical structure of the\ncoupling. The equilibrium magnetization is along the z\naxis, and the magnetic resonance signal is contained in\nχ′′\nx(ωJ) andχ′′\ny(ωJ), the Fourier transforms of the imag-\ninary part of the susceptibility. The two functions Fx\nandFyare plotted in Fig. 3. As required by symmetry\nFx=Fywhenk= 0. The χ′′\nx(ωJ) response near the ﬁrst\nzero inIc(B0) =Icis about four times the maximum re-\nsponse to χ′′\ny(ωJ). While there is a clear reﬂection of the\nFraunhofer diﬀraction pattern in the χ′′\ny(ωJ) response,\nthe 1/k2, i.e., 1/H2response dominates that to χ′′\nx(ωJ)\nwith only modulations due to diﬀraction eﬀects. With a\nsingle ﬂux quantum threading thorough the junction the\nFxis zero reﬂecting the net absence of a circulating cur-\nrent. In contrast Fyis a maximum since the current is\nodd andthe junction constitutes asmallﬂat solenoidcar-\nrying the critical current density and hence a maximum\nﬁeld internal to the junction.\nGiven a static magnetisation /vectorM=Mzˆzthe magnetic\nsusceptibility might be approximated by\nχ′′\nx(/vectork,ω) =χ′′\ny(/vectork,ω)≈γeMz/summationdisplay\n±(1\nτ)\n(ωs+ak2∓ω)2(16)\nwhereτis the relaxation time, ωsthe frequency of the\nFMR mode and athe spin-wave stiﬀness. Here γe=\nµB/¯hwithµBthe Bohr magneton.\nIn Fig. 4 we report the dynamical conductance at zero\napplied magnetic ﬁeld (solid line) and the theoretical\nexpectation (dotted line) from Eq. (15) and (16) with\na= 0, using ωsand 1/τas a parameters. We ob-\ntainedωs=23µV as expected from the Kittel’s formula\nωs=γe/radicalbig\n(HK−4πMS)2−H2, where the anisotropy\nﬁeldHKand saturation magnetisation MShave been\nmeasured separately by SQUID magnetometry [2]. The\nFMR frequency is consistent with the value obtained in\na reference sample by EPR spectroscopy [2], i.e., the ob-\nserved signals are entirely consistent with the coupling of\nthe FMR resonanceto the superconductivity via the AB-\nphase. The resonance at 11.5 µV is a subharmonic of the\nmain mode. Kittel’s formula predicts that ωsdecreases\nwith increasing Hwhereas the half-harmonic signal of\nFig. 4 actually seems to increase (see inset of Fig. 4 ).\nThis leads us to believe that this signal corresponds to a\nﬁnite value of aandkin Eq. (16). That the spin-wave\ndispersion might be measured in our type of experiment\nisanexcitingpossibilitywarrantingfurtherinvestigation.\nMoreover, in Fig. 3 we also report the amplitude of the\nFMR signal (markers) as a function of the applied mag-\nnetic ﬁeld. The experimental data follow the Fxcoupling\nfunction obtained above.\nTheelectromagneticnatureofthe responsecanbe con-\nﬁrmed by a study of Shapiro steps[1, 10]. In the absence\nofamagneticresonancemode, anappliedradiofrequency\nﬁeld gives riseto such steps. In orderto accountfor these\nwe write for the bias voltage across the junction\nV=V0+vcosΩt5\nwherevand Ω are the amplitude and frequency of the\napplied microwave ﬁeld while the constant V0, as usual,\ncorresponds to a Josephson frequency ωJ= 2eV0/¯h. The\nJosephson current is now Jctimes\nsin[kx+ωt+2ev\n¯hΩsinΩt].\nExpanding this sine with the Jacobi-Anger identity gives\n/bracketleftBigg\nJ0(2ev\n¯hΩ)S0+∞/summationdisplay\nn=1/parenleftbigg\nJ2n(2ev\n¯hΩ)S2n+J2n−1(2ev\n¯hΩ)S2n−1/parenrightbigg/bracketrightBigg\nwithS2n=/summationtext\n±sin(kz+ω0t+ 2nΩt) andS2n−1=/summationtext\n±±sin(kx±ω0t+ (2n−1)Ωt), where S0=S2n=0.\nEach term is of the form\nJn\ncsin[kz+ω0t±nΩt]→Jn\ncsin[kz+ωnt] (17)\nwhere, in well known fashion[1], Jn\nc=JcJn(2ev/¯hΩ) in-\nvolves the appropriate Bessel function Jn(2ev/¯hΩ). For\nthe voltage at which ωn= 0 there is a direct contri-\nbutionJn\ncsin(φ0+kx) to the average current which is\nequivalent to the zero voltage critical current step but\ndisplaced to Vn=n¯hΩ/2eand reduced from Ic(H) to\nIc(H)Jn(2ev/¯hΩ). This corresponds to the principal\nShapiro steps, shown on the top panel of Fig. 5. For\nthe magnetic response, in the linear response regime, the\ntheory developed without an applied microwave ﬁeld can\nbe adopted. All that is needed is to replace JcbyJn\ncand\nωJwithωnin the appropriate places as described above.\nWe have investigated the Shapiro steps for diﬀerent\nmicrowave power. The top part of Fig. 5 shows the ap-\npearance of the Shapiro steps in the current-bias char-\nacteristics for increasing microwave power. The ampli-\ntude of each step follows the appropriate Bessel func-\ntionJn(2ev/¯hΩ), as expected. In the bottom panel of\nFIG. 4: Dynamical resistance of a ferromagnetic Josephson\njunction showing ferromagnetic resonances with ﬁrst and se c-\nond harmonic. Insert: the magnetic ﬁeld dependence of the\nresonance at the second harmonic. The ﬁt takes into account\nthe spatial dispersion of the spin wave mode, see text.\nFIG. 5: Top panel: IVcurves showing the Shapiro steps,\ntaken at T=1.3 K and with the microwave frequency of 1 GHz\nwith diﬀerent incident power. Bottom panel: the dynamical\nresistance showing Shapiro steps and side-band resonances ,\ntaken at T=35 mK and with microwave frequency of 17.35\nGHz. In passing from the bottom to top the power increases\nfrom 0 dBm to 20 dBm, in steps of 5 dBm.\nFig. 5 is presented the dependence of the magnetiza-\ntion induced side-bands on the micro-wave power. To\nmake these bands more evident, shown is the dynami-\ncal resistance as a function of the voltage. It is clear\nfrom the data that the amplitude of the side-band reso-\nnances follows the Bessel function of the Shapiro steps,\nas expected from the theory described above. Indeed\nthe Shapiro steps can be seen as “replica” of the critical\ncurrent at ﬁnite bias and hence the side-band amplitude\nfollows that of the steps. The two sidebands correspond\nto the two poles of the dynamical susceptibility, Eq. (16).\nIn conclusion, we have described experiments and de-\nvelopedtheorytodemonstratethattherelativeABphase\nofthe superconductorswhich comprise a Josephsonjunc-\ntion couples to the magnetic order parameter of a ferro-\nmagnet. WehavetherebyperformedanFMRexperiment\nwith a sensitivity which greatly exceeds that of conven-\ntionalcavityFMR. Since the coupling isviathe magnetic\nﬁeld it is not necessary to have the current pass through6\nthe magnetic material. It might be imagined that the\nmagnetic layer be the top layer of a FSIS structure in\nwhich the adjacent S-layer has a thickness of the order\nof, or less than, the London penetration length. The\npossibility of measuring the dispersion of spin-wave exci-tations has also be investigated. Our method[2, 9, 11] of\ncoupling superconductivity to magnetism measures di-\nrectly the dynamic susceptibility χ′′(/vector q,ω) with an en-\nhancedsensitivityforsmallwave-vectors,complementary\nto neutron scattering.\n[1] See e.g., Barone, A. and Paterno, G. Physics and Ap-\nplications of the Josephson Eﬀect (John Wiley & Sons,\nNew York, 1982).\n[2] I. Petkovi´ c, et al, Phys. Rev. B 80, 220502(R) (2009).\n[3] M. V. Berry Proc. R. Soc. Lond. A 39245?57 (1984).\n[4] Y. Aharonov, D. Bohm, Phys. Rev. 115485?491 (1959).\n[5] Z. Nussinov, A. Shnirman, D. P. Arovas, A. V. Balatsky,\nand J.-X. Zhu, Phys. Rev. B 71, 214520 (2005); J.-X.\nZhu, Z. Nussinov, A. Shnirman, and A. V. Balatsky,\nPhys. Rev. Lett. 92, 107001 (2004).\n[6] P. Dubos, H. Courtois, B. Pannetier, F. K. Wilhelm, A.\nD. Zaikin A. D., and G. Schon, Phys. Rev. B 63, 064502(2001).\n[7] T. Kontos et al. Phys. Rev. Lett. 81, 301 (2001)\n[8] P. W. Anderson, and J. M. Rowell, Phys. Rev. Lett. 10,\n230 (1963).\n[9] K. Baberschke, K. D. Bures, and S. E. Barnes, Phys.\nRev. Lett. 53, 98 1984; S. E. Barnes and F. Mehran,\nPhys. Rev. B 34, 4537 1986.\n[10] S. Shapiro, Phys. Rev. Lett. 11, 80 (1963).\n[11] S.E. Barnes, J.L. Cohn, F. Zuo, Phys. Rev. Lett. 77,\n3252 (1996)"    },    {        "title": "1808.08379v1.Resonance_spin_transfer_torque_in_ferromagnetic_normal_ferromagnetic_spin_valve_structure_of_topological_insulators.pdf",        "content": "Resonance spin transfer torque in ferromagnetic/normal/ferromagnetic spin-valve\nstructure of topological insulators\nMoslem Zare1\n1Department of Physics, Yasouj University, Yasouj, Iran 75914-353, Iran.\nWe theoretically study the spin current and spin-transfer torque generation in a conventional spin-\nvalve hybrid structure of type ferromagnetic/normal metal/ferromagnetic (FM/NM/FM) made of\nthe topological insulator (TI), in which a gate voltage is attached to the normal layer. We demon-\nstrate the penetration of the spin-transfer torque into the right ferromagnetic layer and show that,\nunlike graphene spin-valve junction, the spin-transfer torque in TI is very sensitive to the chemical\npotential of the NM region. As an important result, by changing the chemical potential of the NM\nspacer and magnetization directions, one can control all components of the STT. Interestingly, both\nthe resonance spin current and the resonance spin-transfer torque appear for energies determined\nfrom a resonance equation. By increasing the chemical potential of the NM spacer, the amplitude of\nthe STTs decreases while at large chemical potentials of \u0016Nthere are intervals of chemical potential\nin which both the spin current and the spin-transfer torque become zero. These \fndings could\nopen new perspectives for applications in spin-transfer torque magnetic random access memory\n(STT-MRAM) devices based on TI.\nI. INTRODUCTION\nThe electric current modulation of the magnetic prop-\nerties of magnetic materials instead of externally applied\nmagnetic \felds has paved the way to integrate magnetic\nfunctionalities into electric-current-controlled spintronics\ndevices with reduced dimensions and energy consump-\ntion compared with conventional magnetic \feld actua-\ntion. The conservation of angular momentum between\nitinerant electrons and localized magnetization in mag-\nnetic heterostructures leads to the of particular inter-\nest concept of spin-transfer torque (STT) [1, 2], plays\na major role in spintronic devices [3{6]. In this phe-\nnomenon, the spin angular momentum of electrons in a\nspin-polarized current, generated by passing an electri-\ncal current through a ferromagnet layer, exerts a torque\non the second magnetization, enabling magnetization\nswitching or precession [7, 8], for su\u000eciently large cur-\nrents without the need for an external \feld. It is found to\nbe important because of its potential for applications in\nspin-torque diode e\u000bect [9], microwave-assisted recording\nof hard-disk drives [10, 11], high-performance, and high-\ndensity magnetic storage devices [12{14]. Compared with\ncurrent memory devices which use magnetic \felds to re-\norient magnetization to store information ,Spin-transfer\ntorque magnetic random access memory (STT-MRAM)\ndevices, which store information in the magnetization of\na nanoscale magnet, is a promising candidate for the last\ntwo decades [7, 8, 15{17]. Magnetic-nonmagnetic mul-\ntilayers such as magnetic tunnel junctions, spin valves,\npoint contacts, nanopillars, and nanowires [8] are com-\nmon structures and device geometries that are applica-\nble for STT proposal. Among them, as originally pro-\nposed [18, 19], magnetic tunnel junctions were used as\na high-performance, non-volatile magnetic memory cells\nin MRAMs [20]. As a large current is needed for current-\ninduced magnetization dynamics, for creating the current\ndensities required for the onset of magnetic instabilities(108A=cm2) nanometre-scale devices should be used. De-\nspite the explosive growth of the \feld of STT in three-\ndimensional materials, only a few works have studied the\nspin-transfer torque of two-dimensional heterostructures.\nSTT generation in ferromagnetic-normal-ferromagnetic\nbulk graphene junctions has been studied theoretically\nin Ref. [21], then possibility of current-induced STT in\nferromagnetic-normal-ferromagnetic graphene nanorib-\nbon junction studied by Ding et al. [22]. Very recently in\na detailed study, we theoretically investigated the trans-\nport and STT in phosphorene-based multilayers with\nnoncollinear magnetizations[23, 24]. In the present work,\nmotivated by the recent measurements of the STT in-\nduced by a topological insulator [25], we theoretically\nstudy the generation of the spin currents and STT in\nF/N/F trilayer heterostructures of TI. Within the scat-\ntering formalism, we \fnd that the application of a local\ngate voltage to the N region of the FM/NM/FM struc-\nture leads to both the spin current and the spin-transfer\ntorque resonance. Depending on the chemical potential\nof the NM region ( \u0016N), and the con\fguration of the mag-\nnetization vectors one can has STTs.\n𝐦𝟐 𝛗𝟐 𝛗𝟏 \n \nz \nTop gate  \n𝐦𝟏 \nTopological Insulator  \nx L Ferromagnetic (    )  \n1F\n Ferromagnetic (     )  \n2F\n y \nFIG. 1. (Color online) Schematic illustration of a FM-\nTI/NM-TI/FM-TI heterostructure, where the total charge\ncurrent is \rowing along the x axis through the left ferro-\nmagnetic layer ( F1) to the right ferromagnetic one ( F2). The\ngreen arrows represent the local magnetic moments with over-\nall magnetization directions m1;m2.\nThis paper is organized in the following way: In Sec. II,\nwe introduce the low-energy e\u000bective Hamiltonian of thearXiv:1808.08379v1  [cond-mat.mes-hall]  25 Aug 20182\nferromagnetic topological insulator and establish the the-\noretical framework which is used to calculate the spin cur-\nrent and spin-transfer torque generation in a conventional\nspin-valve hybrid structure of type ferromagnetic/normal\nmetal/ferromagnetic (FM/NM/FM) made of the topo-\nlogical insulator (TI), in which a gate voltage is attached\nto the normal layer. In Sec. III, we discuss our numeri-\ncal results for the proposed FM/NM/FM hetrostructure.\nFinally, our conclusions are summarized in Sec. IV.\nII. MODEL AND BASIC FORMALISM\nAs illustrated in Fig. 1, we consider a conven-\ntional spin-valve hybrid structure of type ferromag-\nnetic (F1)/normal metal (NM)/ferromagnetic ( F2) made\nof the topological insulator, with a normal spacer of\nwidthL. In general m1andm2in which m=\n(mx;my;mz) =jmj(sin\u0012cos\u001e;sin\u0012sin\u001e;cos\u0012) are vec-\ntors along the magnetization of the left and right layers,\nrespectively which are uniform and can point along any\ngeneral direction. \u0012;\u001edenote polar and azimuthal an-\ngles in the spherical coordinate, respectively. Flowing a\ncurrent from layer F1into region F2induces a spin ac-\ncumulation in NM=F 2interface, exserted a spin-transfer\ntorque on the magnetization F2[1, 2]. We suppose a nor-\nmal metal spacer much thinner than the spin relaxation\nlength in TI. It is worth mentioning that in order to real-ize ferromagnetic TI, one may utilize either doping the TI\nwith magnetic impurities [26{28] or using the proximity\ne\u000bect by coating it with a ferromagnetic insulator[25, 29].\nThe low-energy e\u000bective Hamiltonian of the ferromag-\nnetic TI near the Dirac point, can be written as [30]\n^H=~v^\u001b\u0001(k\u0002z) +^\u001b\u0001m\u0000\"F; (1)\nwhere ^\u001bandvare the spin space Pauli matrices and the\nFermi velocity, respectively and zdenotes the unit vector\nin thezdirection. For simplicity, hereafter we set ~v= 1\nThe electron transport is con\fned in the x-yplane and\nk= (kx;ky;0) =k(cos\u001ek;sin\u001ek;0). The second term in\nEq. (1) is the exchange coupling between itinerant and\nlocal spins and \"Fis the Fermi energy.\nFigure 2 shows the band structure of the pristine and\nferromagnetic TI, (a) mx=my=mz= 0, (b)mx=\nmy= 0;mz= 0:016, (c)my=mz= 0;mx= 0:15\nand (d)mx=mz= 0;my= 0:15. In the absence of\nthe magnetization (the case of (a)), the band structure\nconsists of a massless Dirac cone. An energy gap can be\nopened in the spectrum of TI when the magnetization lies\nout of plane ( mz6= 0 (b)). It is worth mentioning that the\nxandycomponents of the magnetization, have no e\u000bect\non the gap modi\fcation of the TI band structure and\nonly shift the Dirac cone along the yandx-momentum\naxis, respectively (c,d).\nHaL mx=my=mz=0\n-0.2\n0.0\n0.2kx -0.20.00.2\nky-0.10-0.050.000.050.10\nEnergy\n.\nHbL mx=my=0,mz=0.016\n-0.2\n0.0\n0.2kx\n-0.20.00.2\nky-0.10-0.050.000.050.10\nEnergy .\nHcL my=mz=0,mx=0.15\n-0.2\n0.0\n0.2kx-0.20.00.2ky\n-0.10-0.050.000.050.10\nEnergy .\nHdL mx=mz=0,my=0.15\n-0.2\n0.0\n0.2kx-0.20.00.2\nky-0.10-0.050.000.050.10\nEnergy .\nFIG. 2. (Color online) The band structure of the topological insulator (a)in the absence of the magnetization ( mx=my=\nmz= 0), (b)mx=my= 0;mz= 0:016, (c)my=mz= 0;mx= 0:15 and (d) mx=mz= 0;my= 0:15, in the presence of the\nmagnetization.\nThe wave functions that diagonalize the unperturbed\nDirac Hamiltonian ^H(Eqn.1) are explicitly given as\njuF\n+i=\u0012\nei\u000bkcos\fk\n2\nsin\fk\n2\u0013\n;juF\n\u0000i=\u0012\n\u0000ei\u000bksin\fk\n2\ncos\fk\n2\u0013\n;(2)\nwith\u000bk= tan\u00001h\n~vkcos\u001ek\u0000msin\u001esin\u0012\n~vksin\u001ek+mcos\u001esin\u0012i\n; \fk=\ncos\u00001h\nm\nj\"kjcos\u0012i\n; and \"\u0006\nk=\n\u0006p\n~2v2k2+m2+ 2~vkm sin\u0012sin(\u001ek\u0000\u001e). For thenormal region the spinors are as\njuN\n\u0006i=\u0012\n\u0006ei\u000bN\nk\n1\u0013\n; (3)\nin which,\u000bN\nk= tan\u00001h\nkx\nkyi\n. When a spin-polarized\ncurrent interacts with a ferromagnetic layer due to the\nspin \fltering, a spin transfer torque is applied to the\nmagnetic layer. Supposing that there is no spin-\ripping\nprocesses, overall transmission and re\rection amplitudes\nfor spin-up electrons ( t\",r\") are di\u000berent from those of\nspin-down electrons ( t#,r#). Total wave functions in the3\ntwo ferromagnetic regions are as\n F1\nin=eikxx\np\n\n\u0010\nei\u000b+\nkcos(\fk=2)j\"i+ sin(\fk=2)j#i\u0011\n:(4)\n F1\nref=e\u0000ikxx\np\n\n\u0010\nr\"ei\u000b\u0000\nkcos(\fk=2)j\"i+r#sin(\fk=2)j#i\u0011\n:(5)\n F2\ntran=eikxx\np\n\n\u0010\nt\"ei\u000b+\nkcos(\fk=2)j\"i+t#sin(\fk=2)j#i\u0011\n:(6)\nThe corresponding eigenvectors in the normal regioncan be written as\n \u0006N=e\u0006ikN\nxx\np\n2\n\u0010\naei\u000b\u0006N\nkj\"i+bj#i\u0011\n(7)\nHere,\u000b\u0006N\nk=\u000bN\nk(\u0006\u001ek) and \n is a normalization area.\nThe two propagation directions along the xaxis are de-\nnoted by\u0006in \t\u0006N. By matching the wave functions and\ntheir \frst derivatives at the interfaces x= 0 andx=L,\nwe obtain the coe\u000ecients in the wave functions as\nr\"=ei(\u000b+\n1k\u0000\u000b\u0000\n1k)(e2ikN\nxL(kF2\nx\u0000kN\nx)(kF1\nx+kN\nx)\u0000(kF1\nx\u0000kN\nx)(kF2\nx+kN\nx)\ne2ikN\nxL(kNx\u0000kF1x)(kNx\u0000kF2x)\u0000(kNx+kF1x)(kNx+kF2x)\nr#=\u0000e2ikN\nxL(kN\nx+kF1\nx)(kN\nx\u0000kF2\nx) + (kF1\nx\u0000kN\nx)(kF2\nx+kN\nx)\ne2ikNxL(kNx\u0000kF1x)(kNx\u0000kF2x)\u0000(kNx+kF1x)(kNx+kF2x)(8)\nt\"=ei(\u000b+\n1k\u0000\u000b+\n2k+(kN\nx\u0000kF2x)L)kF1\nxkN\nxcos(\f1k=2) sec(\f2k=2)\ne2ikNxL(kNx\u0000kF1x)(kNx\u0000kF2x)\u0000(kNx+kF1x)(kNx+kF2x)\nt#=\u0000ei(kN\nx\u0000kF2x)LkF1\nxkN\nxsin(\f1k=2) csc(\f2k=2)\ne2ikNxL(kNx\u0000kF1x)(kNx\u0000kF2x)\u0000(kNx+kF1x)(kNx+kF2x)(9)\nIn the steady state, the spin transfer torque acting on\na volumeVof material (by conservation of angular mo-\nmentum) can be computed simply by determining the\nnet \rux of non-equilibrium spin current JSthrough the\nsurfaces of that volume as\n\u001cstt=\u0000Z\nVdVr\u0001JS; (10)\nNote that since JSis a tensor, its dot product with a\nvector in real space leaves a vector in spin space. For\na single-electron wavefunction  , similar to the more-\nfamiliar probability current density ( ~=m)Im( \u0003r ), the\nspin current density can be rewritten as\nJS\nij=~\nmIm( \u0003Si\n@j ); (11)\nHerei;j=x;y, withiindicating the spin component\nandjthe transport direction. mis the electron mass,\nandSrepresents the Pauli matrices Sx,Sy, andSz. The\nthree spin current density components can be determined\nsubstituting Eqs. (4-7) into Eq.11 as\nJS\nxx(y);trans =~2k\n2m\n2Re(Im)[t\"ei\u000b+\n2kcos(\f2k=2)t#sin(\f2k=2)]\nJS\nxz;trans =~2k\n2m\n[jt\"j2cos2(\f2k=2)\u0000jt#j2sin2(\f2k=2)]:JS\nxx(y);in=~2k\n2m\n2Re(Im)[ei\u000b+\n1kcos(\f1k=2) sin(\f1k=2)]\nJS\nxz;in =~2k\n2m\ncos(\f1k):\nJS\nxx(y);ref=~2k\n2m\n2Re(Im)[r\"ei\u000b\u0000\n1kcos(\f1k=2)r#sin(\f1k=2)]\nJS\nxz;ref =~2k\n2m\n[jr\"j2cos2(\f1k=2)\u0000jr#j2sin2(\f1k=2)]:\nIt is clear that the total spin current is not conserved\nduring the \fltering process because the spin current den-\nsity \rowing on the left of the magnet JS\nin+JS\nre\ris not\nequal to the spin current density on the right JS\ntrans. Us-\ning Eq. (10), the spin transfer torque \u001cstton an area A\nof the ferromagnet is equal to the net spin current trans-\nferred from the electron to the ferromagnet, and is given\nby\u001cstt=A^ x\u0001(JS\nin+JS\nre\r\u0000JS\ntrans). Using the scattering\ntheory as well as the incoherency of spin-up and -down\nstates inside the ferromagnet, the STT can be formulated\nin terms of the spin dependence of the transmission and\nre\rection coe\u000ecients as4\n\u001cx(y)\nst=A\n\n~2k\nmRe(Im)h\nk1cos(\f1k=2) sin(\f1k=2)(ei\u000b+\n1k\n\u0000r\"r\u0003\n#ei\u000b\u0000\n1k)\u0000k2t\"t\u0003\n#ei\u000b+\n2kcos(\f2k=2) sin(\f2k=2)i\n\u001cz\nst=A\n\n~2k\nmh\njt\"j2(k1cos2(\f1k=2)\u0000k2cos2(\f2k=2))\n\u0000jt#j2(k1sin2(\f1k=2)\u0000k2sin2(\f2k=2))i\n(12)\nWe have used the fact that jt\"j2+jr\"j2= 1 andjt#j2+\njr#j2= 1. It is worth mentioning that for a symmetric\nF/N/F junction there is no component of spin torque\nin the ^ zdirection and the other two components are as\nfollow\n\u001cx(y)\nst=A\n\n~2k\n2msin(\fk)Re(Im)h\nei\u000b+\nk(1\u0000t\"t\u0003\n#)\u0000ei\u000b\u0000\nkr\"r\u0003\n#i\n(13)\nBy including all transverse modes, the total STT of the\nproposed structure at zero temperature is given by\n\u001ci\ntot(E) =Zkmax\ny(E)\n0\u001ci(E;ky)dky; (14)kmax\ny(E) is the maximum value of the transverse momen-\ntum.\nIII. NSUMERICAL RESULTS\nIn this section, we present our numerical re-\nsults. As described in the introduction, a conven-\ntional spin-valve structure has the general ferromag-\nnetic/normal/ferromagnetic structure. We study the\ngeneration of the spin currents and the spin-transfer\ntorque in a spin-valve hybrid structure of type topologi-\ncal insulator junctions. As we are interested in both the\nmetallic (where the chemical potential stands away from\nthe charge neutrality point) and the zero energy regime,\na gate voltage is attached to the normal layer. We set\nm1=m2=m= 0:1 eV in all \fgures and results pre-\nsented in this section.\n02 4 6 8 1 0-0.0030-0.0025-0.0020-0.0015-0.0010-0.00050.00000\n2 4 6 8 1 0-5-4-3-2-100\n2 4 6 8 1 0-3-2-100\n2 4 6 8 1 0-8-6-4-20×10-3×10-3×10-3(\na) \nJSx\nx \nJSx\nxµ\nN[eV]µ N[eV]µ N[eV](b) \nJSx\ny \nJSx\ny(c)JSx\nzJSx\nyJSx\nx \nJSx\nz \nJSx\nz\nFIG. 3. (Color online) The transmitted spin current density (in units of ~2=m\n) versus the chemical potential of the NM spacer\n(\u0016N), for when the \frst and second magnetizations are \fxed along the z(\u00121= 0) and \u0000z(\u00122=\u0019) axes, respectively. (a) JS\nxx\n(b)JS\nxy(c)JS\nxz. The red curves are for L= 10 and the blue ones are for L= 100. The other parameters are taken as m= 0:1\neV,\u0016F= 0:2 eV.\nFigure 3 shows the transmitted spin current densities\n(a)JS\nxx(b)JS\nxy(c)JS\nxz(in units of ~2=m\n) versus the\nchemical potential of the NM spacer ( \u0016N), when the \frst\nand second magnetizations are \fxed along the z(\u00121= 0)\nand\u0000z(\u00122=\u0019) axes, respectively. The red curves are\nforL= 10 and the blue ones are for L= 100. The\nother parameters are taken as m= 0:1 eV,\u0016F= 0:2 eV.\nAs can be seen, the spin current density is an oscillatory\nfunction of the chemical potential of the NM spacer andamplitude of the oscillations drops with increasing the\nchemical potential of the NM region. The spin current\ndensity of a junction with a thicker normal region exhibits\nfaster oscillations. Interestingly, resonant spin current\npeaks appear at the chemical potentials of the NM region\nthat satisfy the equation kN\nxL= 2n\u0019withnthe positive\ninteger where kN\nxandLare the wavevector and width of\nthe NM region, respectively.\nThe dependence of the STT components on the polar angle of the second magnetization ( \u00122), for various chem-5\n0.00.20.40.60.81.0-0.0020.0000.0020.0040.0060.0080.0100.0120.014\n0.000 .250 .500 .751 .000.0000.0020.0040.0060.0080.0100.0120.0140.0160.018 \nµN= 0 \nµN= 0.5 eV \nµN= 1    eV \nµN= 5    eV(f)τz\n0.000 .250 .500 .751 .000.00000.00050.00100.00150.00200.00250.00300.0035 \nµN= 0 \nµN= 0.5 eV \nµN= 1    eV \nµN= 5    eV(e)τy\n0.000 .250 .500 .751 .000.0000.0010.0020.0030.0040.0050.006 \nµN= 0 \nµN= 0.5 eV \nµN= 1    eV \nµN= 5    eV(d)τx\n0.000 .250 .500 .751 .00-0.015-0.010-0.0050.0000.0050.0100.0150.0200.0250.030 \nµN= 0 \nµN= 0.5 eV \nµN= 1    eV \nµN= 5    eV(c)τz\n0.000 .250 .500 .751 .00-0.00050.00000.00050.00100.00150.00200.00250.00300.00350.0040 \nµN= 0 \nµN= 0.5 eV \nµN= 1    eV \nµN= 5    eV(b)τy\n0.000 .250 .500 .751 .000.0000.0010.0020.0030.0040.0050.006 \nµN= 0 \nµN= 0.5 eV \nµN= 1    eV \nµN= 5    eVτx(a)θ\n2 / πθ 2 / πθ 2 / π\nFIG. 4. (Color online) The spin-transfer torques (in units of A~2=m\n) versus the polar angle of the second magnetization\nvector m2(\u00122) for various chemical potential of the NM region ( \u0016N). (a,d)\u001cx\nSTT and (b,e)\u001cy\nSTT and (c,f)\u001cz\nSTT. Top and\nbottom panels are for \u001e1(2)= 0(\u0019) and\u001e1(2)=\u0019=2(3\u0019=2), respectively. The other parameters are taken as \u00121= 0,\u0016F= 0:2\neV,m= 0:1 eV andL= 10.\nical potentials of the NM region \u0016Nis presented in Fig.4,\n(a,d)\u001cx\nSTTand (b,e)\u001cy\nSTTand (c,f)\u001cz\nSTT. Top and bot-\ntom panels are for \u001e1(2)= 0(\u0019) and\u001e1(2)=\u0019=2(3\u0019=2),\nrespectively. The other parameters in this \fgure are\ntaken as\u00121= 0,\u0016F= 0:2 eV,m= 0:1 eV andL= 10.\nIn both panels the \frst magnetization \fxed along the z-\naxis. The second magnetization rotates from the z-axis\nto the\u0000x-axis, (inside the x-zplane) in the top panel\nand from the z-axis to the\u0000y-axis, (inside the y\u0000z\nplane) in the bottom panel. As a whole, we see that\nthe STT components decrease with increasing the chem-\nical potential of the NM region. Maximum STTs are\nrelated to the zero gate voltage. For the con\fguration\n(\u00121(2);\u001e1(2)) = (0(\u00122);0(\u0019)), in a certain angle, STTs\nreached to the maximum value. At the con\fguration\n(\u00121(2);\u001e1(2)) = (0(\u00122);\u0019=2(3\u0019=2)),\u001cx\nSTT and\u001cy\nSTT are\nsymmetric for the interval [0 ;\u0019]. Thezcomponent of the\nSTT (\u001cz\nSTT) in the parallel con\fguration ( \u00121=\u00122= 0),\nfor each chemical potential \u0016Nbecomes zero. The x;ycomponents of the STT reach their maximum value at\n\u00122=\u0019=2 (parallel con\fguration), while the \u001cz\nSTTcompo-\nnent obtains its maximum at \u00122=\u0019(antiparallel con\fg-\nuration).\nThe dependence of the STTs on the azimuthal angle, for\ndi\u000berent values of the chemical potential of the NM re-\ngion (\u0016N), is shown in Fig.5. As the sign tunability in\nthe STTs devices is crucial, we see that one can simply\ncontrol STTs both in terms of sign and magnitude. As\nseen, regardless of the magnetization con\fguration, the\ndependence on the azimuthal angle essentially follows the\nusual sinusoidal behavior, in agreement with Ref. [25].\nExcept\u0016N= 0, the oscillations amplitude of the STTs\ndecreases with increasing chemical potential of the nor-\nmal TI. STTs of these two con\fgurations have a phase\ndi\u000berence of 180 degrees. In both con\fgurations, the case\nof\u0016N= 0 has a phase shift of 180 degrees in \u001cx\nSTT, rela-\ntive to other chemical potentials of the NM region.\nIn Figure 6, we show the e\u000bect of chemical potential\nof the NM region ( \u0016N) on the spin-transfer torque com-\nponents, (a,d) \u001cx\nSTTand (b,e)\u001cy\nSTTand (c,f)\u001cz\nSTT. The\nresults are shown for di\u000berent con\fgurations of the mag-\nnetizations m1andm2. The other parameters are taken\nas\u0016F= 0:2 eV,m= 0:1 eV andL= 10. As an impor-tant result, we see that by changing \u0016N, one can control\nthe STT that could be a useful consequence for the appli-\ncations in TI-based nano-electronic devices. Also, note\nthat the amplitude of the STT oscillations decreases as\nthe chemical potential of the NM region increases. Fur-\nthermore, the formation of resonant-STT in the right fer-6\n-1.0- 0.50 .00 .51 .00.000040.000050.000060.000070.000080.000090.000100.000110.000120.00013-1.0- 0.50 .00 .51 .0024681012-\n1.0- 0.50 .00 .51 .0024681012-\n1.0- 0.50 .00 .51 .0-8-6-4-20246-\n1.0- 0.50 .00 .51 .0-8-6-4-2024×10-4(\na)τx(b)τx×10-4 \nµN= 0 \nµN= 0.5 eV \nµN= 1    eV \nµN= 5    eV×10-3×10-3φ\n2/πφ 2/π(c)τz \nµN= 0 \nµN= 0.5 eV \nµN= 1    eV \nµN= 5    eV µN= 0 \nµN= 0.5 eV \nµN= 1    eV \nµN= 5    eV µN= 0 \nµN= 0.5 eV \nµN= 1    eV \nµN= 5    eV(\nd)τz\nFIG. 5. (Color online) The spin-transfer torques (in units of\nA~2=m\n) versus the azimuthal angle of the second magnetiza-\ntion vector m2(\u001e2), for various chemical potential of the NM\nspacer (\u0016N).(a,b)\u001cx\nSTT(c,d)\u001cz\nSTT. Left (a,c) and right (b,d)\npanels are for \u00121(2)=\u0019=2(\u0019=2) and\u00121(2)=\u0019=2(3\u0019=2), respec-\ntively. The other parameters are taken as \u001e1= 0,\u0016F= 0:2\neV,m= 0:1 eV andL= 10. In both con\fgurations, the\ny-component of the STT ( \u001cy\nSTT) becomes zero.\nromagnetic region is achievable by changing the \u0016N. In\nexcellent agreement with Ref. [31], a clear oscillatory be-\nhavior with sharp peaks in STTs is observed . It is also\nfound that more peaks appear in a same Fermi energy\nregion with enhancing the width of the NM spacer. In-\ncreasing the chemical potential of the NM region leads to\nthe STT resonance occurs at values of the energies deter-\nmined from the equation kN\nxL= 2n\u0019. Interestingly, all of\nthe components of the STT are symmetric with respect\nto the sign reversal of the chemical potential. It is fur-\nther seen that at large chemical potentials \u0016Nthere are\nintervals of potential in which the spin-transfer torques\nbecome zero.\nIn Figure 7, we plot the spin-transfer torque versusthe thickness of the central NM layer, when the \frst and\nsecond magnetizations \fxed along the yandz-axis, re-\nspectively. The STTs display rapid oscillations as a func-\ntion of normal TI (spacer) width. It is easily seen that\nfor each con\fguration, the amplitude of the STT oscil-\nlations decays with the spacer width. The magnitude of\nthe STTs oscillations decreases as the chemical potential\nof the NM region increases.\nIV. SUMMARY\nIn summary, we theoretically study the spin current\nand spin-transfer torque generation in a conventional\nspin-valve hybrid structure of type ferromagnetic/normal\nmetal/ferromagnetic (FM/NM/FM) made of the topo-\nlogical insulator (TI), in which a gate voltage is at-\ntached to the normal layer. We demonstrate the pen-\netration of the spin current and the spin-transfer torque\ninto the right ferromagnetic region and show that, unlike\ngraphene spin-valve junction, the spin-transfer torque in\nTI is very sensitive to the chemical potential of the NM\nregion. As an important result, by changing the chemical\npotential of the NM spacer and magnetization directions,\none can control all components of the STT. It is inter-\nesting to note that both the resonance spin current and\nthe resonance spin-transfer torque appear for the ener-\ngies determined from the equation kN\nxL= 2n\u0019, where\nkN\nxandLare the wavevector and width of the NM re-\ngion, respectively. By increasing the chemical potential\nof the NM spacer, the amplitude of the STTs decreases\nwhile at large chemical potentials of \u0016Nthere are inter-\nvals of chemical potential in which both the spin current\nand the spin-transfer torque become zero. Moreover, we\n\fnd that the spin-transfer torques versus the thickness\nof the central NM layer, display rapid oscillations as a\nfunction of the normal TI width. It is easily seen that\nfor each con\fguration, the amplitude of the STT oscil-\nlations decays with the spacer width. The magnitude of\nthe STT oscillations decreases as the chemical potential\nof the NM region increases. These \fndings could open\nnew perspectives for applications in spin-transfer torque\nmagnetic random access memory (STT-MRAM) devices\nbased on TI.\n[1] L. Berger., Phys. Rev. B 54, (1996) 9353.\n[2] J. C. Slonczewski., J. Magn. Magn. Mater. 159, L1\n(1996).\n[3] M. Tsoi, A. G. M. Jansen, J. Bass, W.-C. Chiang, M.\nSeck, V. Tsoi, and P. Wyder., Phys. Rev. Lett. 80, 4281\n(1998).\n[4] J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers,\nand D. C. Ralph., Phys. Rev. Lett. 843149 (2000).\n[5] Y. Huai, F. Albert, P. Nguyen, M. Pakala, and T. Valet.,\nAppl. Phys. Lett. 84, 3118 (2004) .[6] Z. Diao, D. Apalkov, M. Pakala, Y. Ding, A. Panchula,\nand Y. Huaia., Appl. Phys. Lett. 87, 232502 (2005).\n[7] D. C. Ralph, M. D. Stiles., J. Magn. Magn. Mater. 320,\n1190 (2008).\n[8] H. O. A. Brataas, A. D. Kent., Nat. Mater. 11, 372\n(2012).\n[9] A. A. Tulapurkar, Y. Suzuki, A. Fukushima, H. Kub-\nota, H. Maehara, K. Tsunekawa, D. D. Djayaprawira, N.\nWatanabe, and S. Yuasa., Nature 438, 339 (2005).\n[10] Y. W. J.-G. Zhu., IEEE Trans. Magn. 44, 125 (2008).7\n01 2 3 4 5 0.0000.0010.0020.0030.0040.0050.0060\n1 2 3 4 5 02468100\n1 2 3 4 5 024680\n1 2 3 4 5 -20-10010203040×10-3µ\nN[eV] θ1,2= 0,π        φ1,2= 0, π \nθ1,2= π/2,0     φ1,2= 0,0 \nθ1,2= π/2,π/2  φ1,2= 0,π/2 \nθ1,2= π/2,π/2  φ1,2= 0,πτxµ\nN[eV] θ1,2= 0,π        φ1,2= 0, π \nθ1,2= π/2,0     φ1,2= 0,0 \nθ1,2= π/2,π/2  φ1,2= 0,π/2 \nθ1,2= π/2,π/2  φ1,2= 0,πτyµ\nN[eV]×10-3×10-3(\nc)( a) θ1,2= 0,π        φ1,2= 0, π \nθ1,2= π/2,0     φ1,2= 0,0 \nθ1,2= π/2,π/2  φ1,2= 0,π/2 \nθ1,2= π/2,π/2  φ1,2= 0,πτz(b)\nFIG. 6. (Color online) The spin-transfer torques (a) \u001cx\nSTTand (b)\u001cy\nSTTand (c)\u001cz\nSTT(in units of A~2=m\n) versus the chemical\npotential of the NM spacer ( \u0016N), for various con\fgurations of the magnetizations. The other parameters are taken as \u0016F= 0:2\neV,m= 0:1 eV andL= 10.\n02 0004 0006 0008 0001 00000.0080.0090.0100.0110.0120.0130.0140\n1 00020003000400050000246810120\n1 0002000300040005000024680\n1 0002000300040005000-100-80-60-40-200 \nµN= 0 \nµN= 0.5 eV \nµN= 1    eV \nµN= 2    eVL\n×10-3×10-3×10-3(\na)τxL\n µN= 0 \nµN= 0.5 eV \nµN= 1    eV \nµN= 2    eV(b)τy \nµN= 0 \nµN= 0.5 eV \nµN= 1    eV \nµN= 2    eVL\n(c)τz\nFIG. 7. (Color online) The spin-transfer torques (a) \u001cx\nSTTand (b)\u001cy\nSTTand (c)\u001cz\nSTT(in units of A~2=m\n) versus the length\nof the NM region ( L), for various chemical potential of the NM spacer ( \u0016N). The left ( m1) and right ( m2) magnetizations are\n\fxed along the zandy-axis, respectively.\n[11] X. Z. J.-G. Zhu, Y. Tang., IEEE Trans. Magn. 46, 751\n(2010).\n[12] I. \u0014Zuti\u0013 c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys.\n76, 323 (2004)\n[13] G. E. W. Bauer, E. Saitoh, and B. J. van Wees., Nat.\nMater. 11, 391 (2012).\n[14] J. Linder, J. W. A. Robinson., Nat. Phys. 11, 307 (2015).\n[15] A. D. Kent, D. C. Worledge., Nat. Nanotech. 10, 187\n(2015).\n[16] D. Apalkov, B. Dieny, and J. M. Slaughter., Proc. IEEE\n104, 1796 (2016).\n[17] S.-W. Chung, T. Kishi, J.W. Park, M. Yoshikawa, K. S.\nPark, T. Nagase, K. Sunouchi, H. Kanaya, G.C. Kim,\nK. Noma, M. S. Lee, A. Yamamoto, K. M. Rho, K.\nTsuchida, S. J. Chung, J. Y. Yi, H. S. Kim, Y.S. Chun,\nH. Oyamatsu and S. J. Hong, SK Hynix., IEDM Tech.\nDig.27, 27.1.1 (2016).\n[18] D. D. Tang, P. K. Wang, V. S. Speriosu, S. Le, and K.\nK.Kung., IEEE Trans. Magn. 31, 3206 (1995).\n[19] T. Miyazaki, N. Tezuka., J. Magn. Magn. Mater. 139,\nL231 (1995).\n[20] S. S. P. Parkin, K. P. Roche, M. G. Samant, P. M. Rice,\nand R. B. Beyers,R. E. Scheuerlein, E. J. OSullivan, S.\nL. Brown, J. Bucchigano, D. W. Abraham, Yu Lu, M.\nRooks, P. L. Trouilloud, R. A. Wanner, and W. J. Gal-lagher., J. Appl. Phys. 85, 5825 (1999).\n[21] T. Yokoyama, J. Linder., Phys. Rev. B 83, (2011)\n081418(R).\n[22] G. S. K.-H. Ding, Z.-G. Zhu., Phys. Rev. B 89, 195443\n(2014).\n[23] L. Majidi, M. Zare, R. Asgari., Phys. Rev. B 95, 115426\n(2017).\n[24] M. Zare, L. Majidi, R. Asgari., Physica C 549, 77 (2018).\n[25] A. R. Mellnik, J. S. Lee, A. Richardella, J. L. Grab, P.\nJ. Mintun, M. H. Fischer, A. Vaezi, A. Manchon, E.-A.\nKim, N. Samarth, and D. C. Ralph., Nature (London)\n511, 449 (2014).\n[26] Y. Fan, P. Upadhyaya, X. Kou, M. Lang, S. Takei, Z.\nWang, J. Tang, L. He, L.-T. Chang, M. Montazeri,G.\nYu, W. Jiang, T. Nie, R. N. Schwartz, Y. Tserkovnyak\nand K. L. Wang., Nat. Mater. 13, 699 (2014).\n[27] J. Henk, A. Ernst, S. V. Eremeev, E. V. Chulkov, I.\nV. Maznichenko, and I. Mertig., Phys. Rev. Lett. 108,\n206801 (2012).\n[28] H.-H. Duo, X. L. Qi, S. K. Mo, R. G. Moore, D. H. Lu,\nM. Hashimoto, T. Sasagawa, S. C. Zhang, I. R. Fisher,\nZ. Hussain, Z. X. Shen., Science 329, 5992 (2010).\n[29] M. R. Scholz, J. S\u0013 anchez-Barriga, D. Marchenko, A.\nVarykhalov, A. Volykhov, L. V. Yashina, and O. Rader.,\nPhys. Rev. Lett. 108, 256810 (2012).8\n[30] B. P. Ndiaye, C. A. Akosa, M. H. Fischer, A. Vaezi, E.-A.\nKim, and A. Manchon., Phys. Rev. B 96, 014408 (2017).\n[31] Q. Zhang, K. S. Chan, and J. Li., Sci. Rep. 8, 4343\n(2017)."    },    {        "title": "1311.5767v2.Spin_Wave_Resonance_Model_of_Surface_Pinning_in_Ferromagnetic_Semiconductor__Ga_Mn_As_Thin_Films.pdf",        "content": "arXiv:1311.5767v2  [cond-mat.mtrl-sci]  8 Jan 2014Spin-Wave Resonance Model of Surface Pinning\nin Ferromagnetic Semiconductor (Ga,Mn)As Thin Films\nH. Puszkarski∗\nSurface Physics Division, Faculty of Physics,\nAdam Mickiewicz University\nul. Umultowska 85, 61-614 Pozna´ n, Poland\nP. Tomczak†\nQuantum Physics Division, Faculty of Physics,\nAdam Mickiewicz University\nul. Umultowska 85, 61-614 Pozna´ n, Poland\n(Dated: July 14, 2021)\nThe source of spin-wave resonance (SWR) in thin ﬁlms of the fe rromagnetic semiconductor\n(Ga,Mn)As is still under debate: does SWR stem from the surfaceanisotropy (in which case the\nsurface inhomogeneity (SI) model would apply), or does it or iginate in the bulkinhomogeneity of\nthe magnetic structure of the sample (and thus requires the u se of the volume inhomogeneity (VI)\nmodel)? This paper outlines the ground on which the controve rsy arose and shows why in diﬀerent\nconditions a resonance sample may meet the assumptions of ei ther the SI or the VI model.\nPACS numbers: 75.50.Pp 76.50.+g 75.70.-i 75.30.Ds\nKeywords: ferromagnetic semiconductors, (Ga,Mn)As thin ﬁ lms, spin-wave resonance, surface anisotropy,\nsurface spin pinning, surface exchange length\nI. INTRODUCTION\nDilute ferromagnetic semiconductors are a class of\nvery promising materials of the future.1–6Gallium man-\nganese arsenide (Ga,Mn)As, created on the basis of the\nsemiconductor gallium arsenide by the addition of a\nsmall percentage of manganese as a magnetic dopant,\nis one of the most intensively studied compounds in\nthis class.7–15The free motion of positive charge carriers\n(holes) throughout the crystal results in the ferromag-\nnetic order of the manganese ions. The basic magnetic\ncharacteristics of the material depend on the amount\nof the dopant ions and the spatial distribution of the\nconcentration of the charge carriers (holes) transmitting\nmagnetic information between the Mn ions. A particu-\nlarly interesting situation occurs in thin ﬁlms, in which\nmagnetic characteristics (magnetic anisotropy, magneti-\nzation, exchange length and stiﬀness constant, damp-\ning constant, etc.) are in general nonuniform along the\ngrowth direction perpendicular to the ﬁlm surface. The\ncharacter of this nonuniformity reﬂects the distribution\nproﬁle of the charge carrier concentration in the ﬁlm.\nThe spatial magnetic proﬁles in thin ﬁlms can be de-\ntermined by means of ferromagnetic resonance, which re-\nveals its ﬁne structure in a multi-peak resonance spec-\ntrum in thin-ﬁlm systems; this type of ferromagnetic res-\nonance is referred to as spin-wave resonance (SWR), as\neach peak in the resonance spectrum corresponds to the\nexcitation of a speciﬁc spin wave. On the other hand,\nthe spectrum of allowed spin-wave excitations is deter-\nmined by the shape of the magnon potential of the sys-\ntem. Since the position of each SWR peak corresponds\nto a spin-wave energy level resulting from the prevailing\nmagnon potential, an experimental SWR spectrum canbe turned into the corresponding proﬁle of the magnon\npotential by an appropriatecalculation procedure. Thus,\nproviding information on the spatial distribution of the\nbasic magnetic characteristics, including the charge car-\nrier concentration in the ﬁlm, resonance measurements\nare of vital importance for the elucidation of the origins\nof ferromagnetism in the material under investigation.\nSpin-waveresonanceinthinﬁlmshasbeenstudiedpar-\nticularlyintensivelyin galliummanganesearsenidein the\npast decade.16–28Especially rich resonance spectra were\nobtained in studies with a variable conﬁguration of the\nstatic ﬁeld with respect to the ﬁlm surface. The ﬁeld was\nrotated both perpendicularly to the ﬁlm surface (which\ncorresponds to variable polar angle θHbetween the di-\nrection of the external ﬁeld and the surface normal) and\nin the plane of the ﬁlm (variable azimuth angle φHbe-\ntween the external ﬁeld and a reference direction in the\nﬁlm plane). The resultsofthese measurementsclearlyin-\ndicate that the evolution of the SWR spectrum with the\nﬁeld conﬁguration is correlated with that of the spatial\ndistribution of the spontaneous magnetization and the\nanisotropy; thus, conﬁguration and space dependence of\nthe magnon potential should be assumed as well.\nIn the present paper we shall only analyze SWR mea-\nsurementdataconcerningtheout-of-planerotationofthe\nmagnetic ﬁeld, mainly because of the controversy that\narose in the interpretation of these results over an issue\nwhich therefore requires elucidation (in a separate pa-\nper we intend to analyze measurement data obtained in\nSWR studies with in-plane rotation of the magnetic ﬁeld\nas well). If researchers tend to agree on the interpreta-\ntion of SWR spectra in two extreme conﬁgurations – the\nperpendicular and parallel conﬁgurations, corresponding\ntoθH= 0 and θH= 90◦, respectively – the interpreta-2\ntion of results obtained in intermediate conﬁgurations is\nunder debate. Almost as a rule, a particular conﬁgura-\ntion of the external ﬁeld tends to occur in this range at\na critical angle θc\nH, for which the multi-peak SWR spec-\ntrum collapses to a single-peak FMR spectrum. There\nare two schools of thought regarding the interpretation\nof the occurrence of this critical angle. These two preva-\nlent opinions agree on the physical state of the thin ﬁlm\nin the critical conﬁguration, but diﬀer in the interpre-\ntation of the conﬁguration-related processes that accom-\npany the rotation. Both schools agree that in the critical\nconﬁguration the thin ﬁlm (its magnon potential, to be\nprecise) is magnetically homogeneous, and the boundary\nconditions (speciﬁcally, the surface spin pinning) corre-\nspond to the natural conditions, only resulting from the\nreduced neighborhoodof the surface spins (a precise deﬁ-\nnition ofthe natural pinning conditionsis providedin the\nnextSection). Thediﬀerenceofopinionconcernsthecon-\nﬁguration evolution leading to the above-described “nat-\nurally homogeneous” magnetic state. One school22uses\nthe surface inhomogeneity (SI) model and assumes that\nrotation of the magnetic ﬁeld does not modify the proﬁle\nof the bulk magnon potential, which remains homoge-\nneous acrossthe ﬁlm; only the surface pinning conditions\nchange, diverging from the natural conditions as the an-\ngle grows above or decreases below the critical conﬁgu-\nration (with the surface pinning decreasing or increas-\ning). In contrast, the other school,28using the volume\ninhomogeneity (VI) model, claims that it is the bulk pro-\nﬁle of the magnon potential that changes with the ﬁeld\nconﬁguration: remaining linear, but inclined at diﬀer-\nent angles with respect to the surface of the ﬁlm, the\nmagnonpotential increasesor decreasesinside the ﬁlm as\nthe conﬁguration diverges from the critical angle, while\nthe natural conditions prevail invariably on the surface.\nIn this paper we opt for the interpretation based on the\nSI model and propose a theoretical model of the conﬁgu-\nration evolution of the surface spin pinning in agreement\nwith the experimental data. Our interpretation leads to\nsome physical conclusions, which provide new insights\ninto the surface properties of ferromagnetic semiconduc-\ntor (Ga,Mn)As thin ﬁlms.\nII. THE GOAL OF THE STUDY AND THE\nCONCEPT OF SWR SURFACE PINNING\nPARAMETER\nOur discussion of the state of the art of the research in\nthecriticalangleeﬀectinSWRinferromagneticsemicon-\nductor (Ga,Mn)As thin ﬁlms will rely on the representa-\ntive study performed by Liu et al., reported in Ref. 22,\npresenting SWR spectra measured for intermediate an-\nglesθHbetweentheexternalﬁeldandthesurfacenormal.\nCharacteristically, in the out-of-plane conﬁguration , with\nthe ﬁeld vector rotated in a plane perpendicular to the\nsurface, the SWR spectrum, consisting of multiple peaks\nin the perpendicular ( θH= 0) and parallel ( θH= 90◦)conﬁgurations, is found to collapse to a single-peak FMR\nspectrum in an intermediate conﬁguration corresponding\nto a critical angle θc\nH(19◦in the studied sample).\nThe critical angle eﬀect in SWR has been known for\nyears, but that observed in (Ga,Mn)As samples is very\nunusual. The peculiarity is that the critical angle θc\nH\ncoincides with the border between two conﬁguration do-\nmains in which the SWR spectrum fulﬁlls the assump-\ntions of diﬀerent models: the surface inhomogeneity\nmodel29forθH> θc\nH(in which range the spacing be-\ntween the resonance peaks is proportional to n2, wheren\nis the spin-wave mode number), and the volume inho-\nmogeneity model,30which applies for θH< θc\nH(where\nthe spacing between the resonance modes is proportional\nton). A question arises: what mechanism underlies the\noccurrence of the inhomogeneity, if surface inhomogene-\nity prevails for θH> θc\nH, and volume inhomogeneity\nforθH< θc\nH? And what particular surface mechanism\nleads to the occurrence of the critical angle θc\nHat which\nthese two types of inhomogeneity fail to be “seen” in the\nresonance?\nIt shouldbe notedthat the SWR studiesof(Ga,Mn)As\nconducted so far tended to focus on volumecharacteris-\ntics only, such as the uniaxial anisotropy or the exchange\nconstantof the studied material. The aim ofthis paper is\ntouseSWR forgettingabetterinsightintotheferromag-\nnetism of dilute semiconductors in terms of their surface\ncharacteristics, the current knowledge of which is scarce.\nFor this reason, in the analysis presented in this paper,\nwe refer to our earlier quantum theory of SWR,31–40in\nwhichwehaveintroducedtheconceptof surface spin pin-\nning parameter , a quantity that measures the degree of\npinning of the surface spins and reveals explicitly diﬀer-\nent surface magnetic anisotropies present in thin ﬁlms.\nTheconceptofsurfacepinningisrelatedtothedescrip-\ntiof he energy status of surface spins, speciﬁcally to the\ndegree of freedom of their precession. In a very simpli-\nﬁed image introduced in Refs.32,37, besides the eﬀective\nmagnetic ﬁeld present throughout the sample, an addi-\ntional magnetic ﬁeld Ksurf, referred to as the eﬀective\nsurface anisotropy ﬁeld, acts on the surface spins. As we\nhave shown, the boundary conditions to be fulﬁlled by\nthe precession of the surface spins can be expressed by\nthe surface pinning parameter, deﬁned:\nA= 1−a2\nDexKsurf·m, (2.1)\nwhereais the lattice constant, Dexis the exchange stiﬀ-\nnessconstant, and mdenotesaunitvectororientedalong\nthe magnetization Mof the sample. Note that a com-\nplete lack of anisotropy ﬁeld on the surface corresponds\nto the surface parameter value one; the freedom of the\nsurface spins in this situation will be referred to as the\nnatural freedom . In the case of nonzero anisotropy ﬁeld\nthree situations, substantially diﬀerent from the physical\npoint of view, may occur depending on the angle be-\ntween the magnetization Mand the surface anisotropy\nﬁeldKsurf. If the surface spins are aligned perpendicu-3\nlarly toKsurf, their freedom remains natural(A= 1);\notherwise, the surface spins are pinned (and A <1) or\nunpinned (and A >1) for the above-mentioned angle\nacute or obtuse, respectively. All three pinning regimes\nare schematically depicted in Fig. 1.\n \n K surfM\n             FILMA = 1\n(a)Natural freedom\n \n K surf\n             FILMM           A < 1\n(b)Pinned surface spins\n \n K surf\n             FILMM            A > 1\n(c)Unpinned surface spins\nFIG.1. Schematicrepresentationofthreesurfacespinpinn ing\nregimes which prevail in a thin ﬁlm depending on the conﬁg-\nuration of its magnetization Mwith respect to the eﬀective\nsurface anisotropy ﬁeld Ksurf(see (2.1)). When aligned as\nin (a), the surface spins do not feel the anisotropy ﬁeld and\nA= 1, which corresponds to their natural freedom. In the\nconﬁgurations (b)and(c)thesurfacespinsarepinned( A <1)\nand unpinned ( A >1), respectively, due to the anisotropy\nﬁeld.\nIn the rigorous theory of SWR the surface pin-\nning parameter can be represented (Cracknell and\nPuszkarski35,36) as a series expansion in spherical har-monicsYlm(θ,φ):\nA(θ,φ) = 1−a2\nDexKsurf(θ,φ)·m\n=∞/summationdisplay\nl=0l/summationdisplay\nm=−lAlmYlm(θ,φ)\n=∞/summationdisplay\nl=0/bracketleftBigg\nalP0\nl(cosθ)+l/summationdisplay\nm=−lPm\nl(cosθ)\n×(αlmcosmφ+βlmsinmφ)],(2.2)\nwhereθandφare the out-of-plane polar angle and the\nin-plane azimuth angle, respectively, of the magnetiza-\ntionM. The coeﬃcients al,αlmandβlm(which can be\nfound experimentally) determine the respective energy\ncontributions brought to the eﬀective surface pinning by\ndiﬀerent surface interactions. As established in Ref. 36,\nin the caseofsurfacecut (100)–which isthat ofthe thin-\nﬁlm samples considered in Ref. 22 – all the terms with\nodd values of lvanish, and the only values allowed to m\nare 0, 4, 8, .... In our research we have also observed37\nthat in the case of thin ﬁlms the series (2.2) can be cut\nto only include terms up to l= 4, since further contribu-\ntions tend to be minor. Thus, we propose the following\nangular dependence of the surface parameter as appro-\npriateforthe interpretationoftheSWR spectraobtained\nin Ref. 22:\nA(θ,φ) = 1−a0−a2(θ,φ)/parenleftbig\n3cos2θ−1/parenrightbig\n−a4(θ,φ)cos4φ.(2.3)\nThe above formula provides the basis for the elucidation\nof the most important surface mechanisms behind the\nSWR surface dynamics in (Ga,Mn)As thin ﬁlms, which\nis the main goal of the present paper.\nIII. OUT-OF-PLANE ANGLE DEPENDENCE\nOF THE SURFACE PARAMETER IN (Ga,Mn)As\nTHIN FILMS\nInthepresentpaperweshallfocusontheconﬁguration\ndependence of the SWR spectrum of (Ga,Mn)As thin\nﬁlms with the external ﬁeld Honly rotating in a plane\nperpendicular to the surface of the sample from the di-\nrection along the surface normal ( θH= 0) to the in-plane\ndirection ( θH= 90◦). According to the formula (2.3), in\nthis case the surface parameter of a (Ga,Mn)As thin ﬁlm\ncan be represented as the series:\nA(θM) = 1−a0−a2(θM)/parenleftbig\n3cos2θM−1/parenrightbig\n,(3.1)\nwhereθMis the angle between the surface normal and\nthe magnetization Mof the ﬁlm (let us remark in ad-\nvancethat, exceptfortwoextremeconﬁgurations,ingen-\neralθH/negationslash=θM; the relation between θHandθMwill\nbe discussed in detail in the next Section). Note that4\nthe adoption of the formula (3.1) implies taking into ac-\ncount only two mechanisms of surface spin pinning: the\nisotropic pinning component a0, the inﬂuence of which\non the freedom of the spins is independent of their con-\nﬁguration with respect to the surface of the ﬁlm, and the\nuniaxial factora2(θM) representing the contribution of\nthe uniaxial symmetry, with the surface normal as the\nsymmetry axis, to the surface pinning.\nAlready at this stage interesting conclusions regarding\nthe properties of the surface pinning can be drawn from\nthe equation (3.1) despite its rather general formulation.\nLetus deﬁnetwospecialangles: the critical angle θc\nM, for\nwhichnaturalpinning conditions prevail on the surface\nof the ﬁlm, i.e. A(θc\nM)≡1, and the uniaxial pinning\nannihilation angle θu\nM, for which the uniaxial pinning\nvanishes, i.e. 3cos2θM−1≡0. The following equations\napply to these special angles:\nA(θc\nM)≡1, (3.2a)\nA(θu\nM)≡1−a0. (3.2b)\nThe latter equation provides a simple formula for the\ndetermination of the isotropic component a0of the sur-\nface pinning, only necessitating the value of the sur-\nface parameter in the external ﬁeld conﬁguration corre-\nsponding to the uniaxial pinning annihilation angle θu\nM.\nWitha0known, the conﬁguration dependence of the uni-\naxial factor a2(θM) can be determined by the measure-\nment of the surface parameter A(θM) vs.θM(see the\nequation (3.1)). (We shall refer in this regard to the\npaper by Liu et al.22providing measurement data which\nwill allowusto plot the experimental A(θM) dependence;\nsee Section IV below.) On the other hand, theoreti-\ncal considerations within the model used for describing\nthe surface anisotropy in (Ga,Mn)As samples will lead\nus to an equation, formulated in the next Section, in\nwhicha2(θM) is expressed by magnetic characteristics of\nthe (Ga,Mn)As thin ﬁlm. In Section V very interest-\ning conclusions regarding the interrelation between the\nranges of the exchange interaction on the surface and in\nthe bulk of (Ga,Mn)As thin ﬁlms will be drawn from the\nconfrontation of the theory with the experiment.\nIV. MODEL OF THE UNIAXIAL SURFACE\nANISOTROPY IN (Ga,Mn)As THIN FILMS\nWe shall derive a phenomenological formula for the\ncoeﬃcient a2on the basis of our calculations presented\nin Appendix B, in which the model of the uniaxial\nanisotropy is considered in both the microscopic and\nmacroscopic approaches. From the equation (B10) in\nAppendix B (see also Ref. 40) it follows that the coef-\nﬁcienta2(θM) in the equation (3.1) can be expressed as:\na2(θM) =1\n2/bracketleftbigg\n4π/parenleftBig\nMbulk\neff−Msurface\neff/parenrightBiga2\nDex/bracketrightbigg\n,(4.1)where4πMeff≡4πM−H2⊥,Misthesaturationmagne-\ntization, H2⊥the eﬀective uniaxial anisotropyﬁeld, athe\nlattice constant (the average Mn-Mn distance), and Dex\nthe exchange stiﬀness constant. The above equation in-\ndicates that both the intrinsic uniaxial anisotropy and\nthe demagnetizing ﬁeld contribute to the total uniaxial\nanisotropy in our model.\nAs we will see later, extremely informative for the\nphysical interpretation of the experiments performed by\nLiuet al.22is the expression of the latter contribution by\nthe exchange length λ, deﬁned:\nλb≡/radicalbigg\nDex\n4πMbulk, λs≡/radicalbigg\nDex\n4πMsurface; (4.2)\nwe have introduced here a locallydeﬁned exchange\nlength, diﬀerent for the bulk and the surface. From the\nphysical point of view it is reasonable to assume here\nthat the lattice constant ain the equation (4.1) is iden-\ntical with the exchange length λbthat characterizes the\ninteraction in the wholesample except for its surface.\nUnder these assumptions (4.1) becomes:\na2(θM) =a0\n2+a1\n2(θM), (4.3a)\na0\n2≡1\n2λ2\nb\nDex/parenleftBig\nHsurface\n2⊥−Hbulk\n2⊥/parenrightBig\n, (4.3b)\na1\n2(θM) =1\n2/bracketleftbigg\n1−/parenleftBigλb\nλs/parenrightBig2/bracketrightbigg\n. (4.3c)\nIn the equations (4.3a)–(4.3c) we have indicated in ad-\nvance what will follow from the confrontation of these\nformulas with the experimental data: that only the\nterma1\n2(θM) isconﬁguration-dependent !\nV. CONFRONTATION OF OUR SURFACE\nPINNING MODEL WITH THE SWR STUDY BY\nLIUET AL.22\nFinally, the formula for the surface parameter takes\nthe form:\nA(θM) = 1−a0−/bracketleftbig\na0\n2+a1\n2(θM)/bracketrightbig/parenleftbig\n3cos2θM−1/parenrightbig\n,(5.1)\nwhere the coeﬃcients a0\n2anda1\n2(θM) are as deﬁned\nin (4.3b) and (4.3c). Note that in the surface inhomo-\ngeneity model the surface parameter (8.1) measures the\ndegreeofpinning ofthe surfacespins and describes quan-\ntitatively the degree of the dynamic freedom with which\nthey participate in the motion of the whole system of\nspins. The value A= 1 corresponds to a special case\nreferred to as the naturalfreedom of the surface spins.\nAcquiredby the surface spinsas aresult ofbreakingtheir\ninteraction with those of their neighbors which are elim-\ninated by the introduction of the surface, this freedom\nstemssolelyfrom the broken symmetry in the vicinity5\nof the surface spins. Thus, absolute natural freedom of\nthe surface spins only occurs when all the energy contri-\nbutions in the equation (8.1) vanish simultaneously , i.e.:\na0≡0, (5.2a)\nHsurface\n2⊥=Hbulk\n2⊥, (5.2b)\nλb=λs. (5.2c)\nHowever, as conﬁrmed experimentally, the natural free-\ndom of the surface spins is possible also in a particu-\nlar situation in which the surface parameter value is one\neventhoughthe conditions(5.2) arenot allfulﬁlled. This\nparticular situation may occur when there exists such a\ncritical angle θc\nMthatA(θc\nM)≡1 because all the energy\ncontributions in (8.1) annihilate each other. Further in\nthis Section we shall analyze this situation in detail.\nOn the basis of their SWR study of (Ga,Mn)As thin\nﬁlms Liu et al.22plotted the conﬁguration dependence\nof the surface parameter A(θH) with the magnetic ﬁeld\nrotating from the perpendicular ( θH= 0) to parallel\n(θH= 90◦) conﬁguration (see Fig. 9 in Ref. 22). As\nour formula (8.1) concerns the conﬁguration dependence\nof the surface parameter versus θM, i.e. with rotat-\ningmagnetization of the sample, the ﬁrst thing neces-\nsary for proper interpretation of the measurements of\nLiuet al.was to ﬁnd the dependence θM=θM(θH) in\nequilibrium conditions. The determination of the equi-\nlibrium conditions and the derivation of the sought rela-\ntionθM=θM(θH) between the two conﬁguration angles\nare presented in Appendix A. Figure 2 shows the recalcu-\nlated conﬁguration dependence of the surface parameter,\nwithAplotted versus the new variable θM; the plot cor-\nresponds to the measurementdata of Liu et al.presented\nin Ref. 22, Fig. 9. The natural surface pinning is seen to\noccur for the critical angle θM= 35◦(which corresponds\nto the experimental angle θH= 19◦). Also, the new plot\nreveals the occurrence of a local maximum in the A(θM)\ndependence around the angle θu\nM= 54.73◦, for which\nthe term/parenleftbig\n3cos2θu\nM−1/parenrightbig\nequals zero (we shall take ad-\nvantage of this ﬁnding below in further analysis of the\nexperimental data of Liu et al..22)\nNow we will demonstrate that the experimental curve\nshownin Fig.2canbedescribedbythefunction resulting\nfrom our SI model:\nA(θM) = 1−a0−a2(θM)/parenleftbig\n3cos2θM−1/parenrightbig\n.(5.3)\nKnowing the maximal value of the surface parameter,\nA(θu\nM) = 1.1068, we obtain immediately the value of the\nisotropic term in the series (5.3):\na0=−0.1068. (5.4)\nOn the other hand, the condition of occurrence of the\nlocal maximum at θu\nMimplies that the coeﬃcient a2(θM)\nis zero at this point:\na2(θu\nM) = 0. (5.5) 0.85 0.9 0.95 1 1.05 1.1 1.15\n 20 30 40 50 60 70 80 90Surface spin pining parameter, A( θM)\nMagnetization angle, θM [deg]θu\nM=54.7o\nFIG. 2. Magnetization angle dependence of the surface pin-\nning parameter A(θM) according to the experimental data\nobtained by Liu et al.22in their SWR study of a (Ga,Mn)As\nthin ﬁlm; the plot corresponds to that shown in Fig. 9a in the\ncited paper, presenting the dependence on the magnetic ﬁeld\nangleθH. The applied transformation between the angles θH\nandθMis based on our determination of the equilibrium\ndirection of the magnetization, presented in Appendix A.\n-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7\n 0 10 20 30 40 50 60 70 80 90Surface spin pinning coefficient, a2(θM)\nMagnetization angle, θM [deg]\nFIG. 3. Magnetization angle dependence of the surface pin-\nning coeﬃcient a2(θM) calculated from Eq. (5.3).\nBoth conditions allow to determine explicitly the func-\ntiona2(θM)thatreproducestheexperimentalplotshown\nin Fig. 2 via the series (5.3). The determined function\na2(θM) is presented in Fig. 3.\nIn the next step we shall refer to the formula (4.3a)\npostulated in our model and representing the coeﬃ-\ncienta2(θM) as the sum of a constant component a0\n2\nand afunction a1\n2(θM). This implies that a2(θM) and\na1\n2(θM) have the same angular dependence, and their\nplots only diﬀer by a shift a0\n2along the ordinate axis.\nHowever, we do not know the value of a0\n2! This is a very\nsensitive point of our considerations, since in order to es-\ntablish the value of a0\n2we have to refer to the physical\nassumptions that are the very basis of our model of sur-\nface anisotropy. It seems reasonable to assume that of6\n-0.2-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6\n 0 10 20 30 40 50 60 70 80 90Surface spin pinning coefficient, a12(θM)\nMagnetization angle, θM [deg]θc\nM\nFIG. 4. Magnetization angle dependence of the surface pin-\nning coeﬃcient a1\n2(θM) calculated from Eq. (8.1). (See the\ntext for detailed discussion.)\nthe three conditions (5.2) only (5.2c) is fulﬁlled in the\ncritical angle conﬁguration; the other two energy contri-\nbutions do not vanish, but compensate each other. This\nassumption means that by virtue of the equation (4.3c)\nthe coeﬃcient a1\n2vanishes in the critical angle conﬁgura-\ntion:\na1\n2(θc\nM)≡0, (5.6)\nwhich implies the equality:\na0\n2=a2(θc\nM). (5.7)\nHaving established the value of the component a0\n2we can\nalready determine explicitly the function a1\n2(θM). The\nresult is shown in Fig. 4.\nFrom (5.2c) it follows that:\nλs\nλb=1/radicalbig\n1−2a1\n2(θM), (5.8)\nand,onthebasisofFig.4, wecanﬁndthe θMdependence\nof theλs/λbratio. The obtained dependence is shown\nin Fig. 5. Its analysis leads to very interesting physical\nconclusions.\nNote that in the plot in Fig. 5 the surface exchange\nlengthλsis only slightly smaller than the bulk exchange\nlengthλbfor any angle θMbetween the critical angle θc\nM\nand the parallel conﬁguration angle θM= 90◦:\nθc\nM< θM≤90◦. (5.9)\nThus, in this angle rangea surface disturbance will not\ngo beyond the ﬁrst sub-surface plane formed by the spins\ndirectly under the surface. This means that the assump-\ntions of the SI model are fulﬁlled very well in the angle\nrange deﬁned by (5.9)! In contrast, for angles θM< θc\nM\nλsis greater than λband grows steeply as the perpen-\ndicular conﬁguration θM= 0 is approached. This means 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5\n 0 10 20 30 40 50 60 70 80 90Surface to bulk exchange length ratio, λs/λb\nMagnetization angle, θM [deg] VI Model  SI Model\nθc\nM\nFIG. 5. Magnetization angle dependence of the λs/λbratio\nresulting from our model of surface pinning in (Ga,Mn)As\nthin ﬁlms (see Eq. (5.8)); λsandλbdenote the surface and\nbulk exchange length, respectively; θc\nMis the SWR critical\nangle.\nthatinthisanglerangeasurfacedisturbance,ratherthan\nbeing localized at the surface, penetrates into the bulk,\naﬀecting deeper sub-surfaceplanes. Thus, the applicabil-\nity ofthe SI model is verylimited in this anglerange, and\nthe volume inhomogeneity model will be more adequate.\nThis conclusion is fully conﬁrmed by the experimental\nstudy by Liu et al.22\nVI. FURTHER PHYSICAL IMPLICATIONS OF\nTHE MODEL\nNow let us consider the component a0\n2, which we\nhave found to have a constant value, speciﬁed in (5.7),\nthroughout the angle range θM∈(0,π/2). From the\nderived formula (4.3b) for a0\n2it follows that its con-\nstant value implies the invariance of λbin the rotation\nof the magnetization of the sample (we have already\ntaken advantage of this fact, interpreting the angular de-\npendence of the λs/λbratio as only due to λsin the\npreceding Section). The measurements performed by\nLiuet al.indicate that the material parameter values\nin the studied (Ga,Mn)As sample are Dex= 3.79 T·nm2\nand 4πMeff= 4588 Oe, implying λb≈3 nm. On the\nother hand, for the critical angle θM=θc\nMfrom the for-\nmula (8.1) we get the equality:\na0+a0\n2/parenleftbig\n3cos2θc\nM−1/parenrightbig\n= 0, (6.1)\nwhich, after the substitution of a0=−0.1068 and\nθc\nM= 35◦, yields the sought value:\na0\n2∼=0.108. (6.2)\nNow, getting back to (4.3b), with the above-determined\nvalue of a0\n2we can estimate the diﬀerence between the7\n-0.3-0.2-0.1 0 0.1 0.2\n 10 20 30 40 50 60 70 80 90Surface anisotropy energy, Es(θM) [MDex/ λb]\nMagnetization angle, θM [deg]θu\nM\nFIG. 6. Magnetization angle dependence of the surface\nanisotropy energy Es(θM) resulting from our theory (see\nEq. (6.4)) for the (Ga,Mn)As thin ﬁlm investigated by\nLiuet al.22\neﬀective uniaxial anisotropy ﬁeld values on the surface\nand in the bulk:\n∆H2⊥≡Hsurface\n2⊥−Hbulk\n2⊥≈913 Oe.(6.3)\nTo our best knowledge, this is the ﬁrst quantitative\nestimate of the surface uniaxial anisotropy ﬁeld in\n(Ga,Mn)As thin ﬁlms to be reported in the literature.\nAs a measure of surface spin pinning experimentalists\ntend to use the surface anisotropyenergy Es(θM), a phe-\nnomenological quantity thus related to the surface pin-\nning parameter Aused by us for describing the same\nfeature:\nEs(θM) =MDex\nλb[A(θM)−1].(6.4)\nThe above relation indicates that the character of the\nangular dependence of both quantities used for de-\nscribing the surface pinning is identical, though in the\nequation (6.4) the reference level of the measure of\nthe surface pinning is the zero value of the surface\nanisotropy energy, corresponding to our natural pin-\nningA= 1. For Es(θM)>0 the surface spins\nareunpinned , while for Es(θM)<0 their freedom is\nconstrained, which means that the surface spins are\npinned. Plotted in Fig. 6, Es(θM) has a maximum\nforθM=θu\nM; according to our estimate its maximal\nvalue isEs(θu\nM)≈0.07 erg/cm2. Note that this maximal\nsurface anisotropy value is solely related to the free com-\nponenta0, only responsible for the isotropic part of the\nsurface spin pinning; the other surface anisotropy com-\nponents only reduce this (maximal) value as the angle\ndiverges from θu\nMin either direction.\nVII. FINAL REMARKS\nInourmodeltheSWRcriticalangleisdeterminedfrom\nthe condition that the exchange length must be the sameon the surface and in the bulk:\nλs=λb. (7.1)\nOn the other hand, the experimental studies indicate\nthatλbis conﬁguration-independent, and only the sur-\nfaceexchangelength λsissensitivetotheconﬁgurationof\nthe magnetization of the ﬁlm with respect to its surface.\nWesuggestthatthismightberelatedtothefactthatalso\nthe charge carrier (hole) concentration on the surface is\ndiﬀerent than in the bulk in the studied material.41,42\nIf this hypothesis of ours is true, then any experimental\ntreatment modifying the charge carrier concentration on\nthesurfaceof the studied sample should alter the SWR\ncritical angle! This may be performed for instance by\nhydrogenation ofthe sample, since short-time hydrogena-\ntion has been shown27to provide an eﬃcient tool for ma-\nnipulating the eﬀective surface spin pinning by changing\nthe hole concentration proﬁle of the sample. The sug-\ngested experiment would provide a direct proof that the\nrange of the exchange interaction in the ferromagnetic\nsemiconductor (Ga,Mn)As is correlated with the charge\ncarrier concentration.\nVIII. SUMMARY\nIn this paper we show why in diﬀerent conditions a\nresonance (Ga,Mn)As thin ﬁlm sample may meet the as-\nsumptions of either the Surface Inhomogeneity (SI) or\nthe Volume Inhomogeneity (VI) model. In our consider-\nations we refer to the spin-wave resonance (SWR) spec-\ntra measured by X. Liu et al.22in (Ga,Mn)As thin ﬁlms\nin diﬀerent conﬁgurations of the static magnetic ﬁeld H\nwith respect to the surface. We demonstrated that the\nobservedconﬁgurationdependence ofthe SWR spectrum\nof the studied material can be described with the use of\nthe surface pinning parameter expressed by the formula:\nA(θM) = 1−a0−/bracketleftbig\na0\n2+a1\n2(θM)/bracketrightbig/parenleftbig\n3cos2θM−1/parenrightbig\n,\nwhereθMis the angle between the surface normal and\nthe magnetization Mof the sample. The values of the\ncoeﬃcients are estimated on the basis of the experimen-\ntal data; the estimated value of the isotropic compo-\nnent of the surface pinning, a0=−0.1068, allows to de-\ntermine the maximal surface anisotropy energy density,\nEs≈0.07 erg/cm2. The intrinsic uniaxial anisotropy\nterma0\n2is of the order of 0.1, which implies that the\nuniaxial anisotropy ﬁeld H2⊥on the surface exceeds the\nbulk value by ca. 0.1 T. We postulated that the coef-\nﬁcienta1\n2(θM) is related to the diﬀerence between the\nsurface and bulk exchange lengths ( λsandλb, respec-\ntively), which, when confronted with the measurements,\nimplies that (unlike λb) onlyλsdepends on θM, or the\nmagnetization conﬁguration with respect to the surface.\nFor a critical angle θc\nM, at which the SWR spectrum col-\nlapses to a singlepeak,λs=λb. For angles θM> θc\nM\nthesurfaceexchangelength λsisslightlysmallerthanthe8\nbulk exchange length λb:λs< λb, whereas for θM< θc\nM\nλsis greater than λband grows steeply as the perpen-\ndicular conﬁguration ( θM= 0) is approached. This ﬁnd-\ning shows that the critical angle θc\nMseparates two angle\nranges in which the resonance properties are diﬀerent:\nforθM> θc\nMthe SI model applies, since λs≈λb, and for\nθM< θc\nMthe VI model is adequate due to the domina-\ntion of the surface exchange length ( λs≫λb). Seeking\nthephysicalgroundsofthisresult, weproposedaworking\nhypothesisthatthediscoveredpropertyiscorrelatedwith\ninhomogeneous distribution of the concentration of holes\nmediating the long-range magnetic interaction between\nlocalized spins along the surface normal. We suggested\nfurther experiments to verify this hypothesis.\nACKNOWLEDGMENTS\nThis study is a part of a project ﬁnanced by\nNarodowe Centrum Nauki (National Science Centre\nof Poland), Grant no. DEC-2013/08/M/ST3/00967.\nHenryk Puszkarski would like to address special thanks\nto Prof. J. K. Furdyna of Notre Dame University for the\nsupport given to the project and his kind interest in this\nwork. The authors are also much indebted to Profes-\nsor A.R. Ferchmin for highly usefull discussions and for\ncritical reading the manuscript.\nAppendix A: Determination of the equilibrium\ndirection of magnetization in (Ga,Mn)As thin ﬁlms\nThe experimental SWR spectra analyzed in this pa-\nper were measured in the “out-of-plane geometry”, as re-\nferred to by the Authors of Ref. 22. In this out-of-plane\ngeometry, the (Ga,Mn)As layer was cemented to a par-\nallelepiped of GaAs (100) substrate material, the [110]\nedge of the specimen oriented vertically. The external\nmagnetic ﬁeld Hwas conﬁned to the horizontal plane\n(i.e. perpendicular to the ﬁlm surface) allowing SWR\nmeasurements with Hin any intermediate orientation\nbetween the normal to the ﬁlm surface, H/bardbl[001], and\nthe in-plane orientation, H/bardbl[1¯10]. In this particular\ngeometry the magnetization Mof the sample lies in the\nsame horizontal plane as the ﬁeld H. Thus, the spatial\norientation of the vectors HandMis deﬁned by two\npolar angles, θHandθM, between the respective vectors\nand the normalto the surfaceofthe ﬁlm. For(Ga,Mn)As\nsamples in this particular geometry of the external ﬁeld\nthe free energy density F⊥of the system has the form:25\nF⊥=1\n2M×/bracketleftbigg\n−2H(cosθMcosθH+sinθMsinθH)\n+(4πM−H2⊥)cos2θM−1\n2H4⊥cos4θM\n−1\n4H4/bardblsin4θM−H2/bardblsin2θM/bracketrightbigg\n,(A1)whereH2⊥andH4⊥are the uniaxial and cubic\nanisotropyﬁelds, respectively, perpendicular to the plane\nof the sample; H2/bardblandH4/bardblare the in-plane uniaxial and\ncubic anisotropy ﬁelds, respectively. In the investigated\n(Ga,Mn)As sample these four bulk parameters have the\nvalues:22\n4πMeff≡4πM−H2⊥= 4588 Oe ,(A2a)\nH4⊥= 0,H4/bardbl= 197 Oe ,H2/bardbl= 77 Oe.(A2b)\nLet us determine now the equilibrium direction of the\nmagnetization of the sample, i.e. the equilibrium an-\ngleθM. We will use the condition of equilibrium of the\nsystem, which requires the ﬁrst derivative of its free en-\nergyF⊥to vanish:\n∂F⊥\n∂θM= 0; (A3)\nthis condition allows to determine the sought rela-\ntionθM=θM(θH).\n 0 10 20 30 40 50 60 70 80 90\n 0 10 20 30 40 50 60 70 80 90Equilibrium magnetization angle, θM\nMagnetic field angle, θHθc\nH = 19oθcM = 35o\nFIG. 7. Equilibrium magnetization angle θMvs. the external\nﬁeld angle θHas determined from the condition (A3) for the\n(Ga,Mn)As thin ﬁlm studied by Liu et al.;22θc\nMandθc\nHare\nthe respective critical SWR angles.\nSince the condition (A3) must be fulﬁlled when reso-\nnance occurs, the magnetic ﬁeld Hin (A1) is the res-\nonance ﬁeld, H≡Hres; we read its value from Fig. 5\nin Ref. 22, identifying it with the resonance ﬁeld of the\nfundamental mode (n= 1). The θM=θM(θH) relation\ndetermined numerically on the basis of the above con-\nsiderations is shown in Fig. 7; we refer to this relation\nmany times in this paper when analyzing the experimen-\ntal SWR spectra reported by Liu et al.22\nAppendix B: Surface vs. bulk uniaxial anisotropy\nIn this Appendix we shall consider the case in which\nonly the perpendicular uniaxial anisotropy H2⊥enters9\nthe formula (A1) for the free energy. In that case the\nfree energy reads:\nF⊥=1\n2M×/bracketleftbigg\n−2H(cosθMcosθH+sinθMsinθH)\n+4πMeffcos2θM/bracketrightbigg\n,(B1)\nand the use ofthe well-knownSmit-Beljers resonancefor-\nmula:\n/parenleftbiggω\nγ/parenrightbigg2\n=1\nM2sin2θM/bracketleftBigg\n∂2F⊥\n∂φ2\nM∂2F⊥\n∂θ2\nM\n−/parenleftbigg∂2F⊥\n∂φM∂θM/parenrightbigg2/bracketrightBigg\n(B2)\nleads to the following conﬁguration resonance condition,\nonly applying to the uniform mode k⊥≡0 in the case\nconsidered:\n/parenleftbiggω\nγ/parenrightbigg2\n= [Hcos(θM−θH)−4πMeffcos2θM]\n×/bracketleftbig\nHcos(θM−θH)−4πMeffcos2θM/bracketrightbig\n.(B3)\nIt will be very informative to derive the same condition\nin the microscopic approach, in which the energy of the\nsystem is expressed by the Hamiltonian:\nˆH=−J/summationdisplay\nlj/negationslash=l′j′ˆSlj·ˆSl′j′−gµB/summationdisplay\nljH·ˆSlj\n−D/summationdisplay\nlj/parenleftBig\nˆSz\nlj/parenrightBig2\n; (B4)\nits successive terms account for the isotropic exchange\ninteraction, the Zeeman energy of the spins, and the per-\npendicular uniaxial anisotropy energy. The subscript lj\ndeﬁnes the position of the given spin, with llabeling the\nlayer and the two-dimensional vector jdeﬁning the po-\nsition of the spin ˆSljin thel-th layer. The energy of a\nstanding spin wave with a wave number k⊥in this model\nis given by the expression:43\n/parenleftbiggω\nγ/parenrightbigg2\n=/bracketleftbigg\nHcos(θM−θH)+2DS\ngµBcos2θM\n+2Sz⊥Ja2\ngµBk2\n⊥/bracketrightbigg\n×/bracketleftbigg\nHcos(θM−θH)+2DS\ngµBcos2θM\n+2Sz⊥Ja2\ngµBk2\n⊥/bracketrightbigg\n.(B5)This condition is the counterpart of the condition (B3)\nobtainedinthemacroscopicapproach(for k⊥/negationslash= 0). From\nthe comparison of these two formulas it follows that:\n4πMeff≡ −2DS\ngµB(B6a)\nand the coeﬃcient at k2\n⊥can be identiﬁed as:\nDex≡2Sz⊥Ja2\ngµB. (B6b)\nTo obtain the formula for the surface parameter ex-\npressed by the surface perpendicular uniaxial anisotropy,\nwe must yet rewrite the third term in the Hamilto-\nnian (B4) in the generalized form:\nˆHa=−/summationdisplay\nljDl/parenleftBig\nˆSz\nlj/parenrightBig2\n, (B7)\nwhere the uniaxial anisotropy constant Dlis assumed to\nbe:\nDl=/braceleftbigg\nDsfor surface spins ,\nDbfor bulk spins .(B8)\nOn the basis of our earlier papers40,44it can be demon-\nstrated that in the approximation assuming circular spin\nprecessionthe followingexpressionforthesurfaceparam-\neter results from this model:\nA= 1−Db−Ds\n2z⊥J/parenleftbig\n1−3cos2θM/parenrightbig\n,(B9)\nwhereDbandDs, as indicated above, denote the bulk\nand surface values, respectively, of the microscopic uni-\naxial anisotropy constant. Now, using the identity rela-\ntions (B6) we obtain the sought ﬁnal formula in which\nthe surface parameter is expressed by macroscopic quan-\ntities:\nA(θM) = 1−1\n2/bracketleftbigg\n4π/parenleftBig\nMsurface\neff−Mbulk\neff/parenrightBiga2\nDex/bracketrightbigg\n×/parenleftbig\n1−3cos2θM/parenrightbig\n.(B10)10\n∗Corresponding author, email: henpusz@amu.edu.pl\n†email: ptomczak@amu.edu.pl\n1J. K. Furdyna, Diluted magnetic semiconductors , J. Appl.\nPhys.64, R29 (1988).\n2H. Ohno, Making nonmagnetic semiconductors magnetic ,\nScience281, 951–956 (1998).\n3T. Dietl, H. Ohno, F. Matsukura, J. Cibert, D. Fer-\nrand,Zener Model Description of Ferromagnetism in\nZinc-Blende Magnetic Semiconductors , Science 287,\n1019 (2000).\n4T.Dietl, Hole-mediated ferromagnetism in tetrahedrally co-\nordinated semiconductors , Phys. Rev.B 63, 195205 (2001).\n5T. Jungwirth, Jairo Sinova, J. Maˇ sek, J. Kuˇ cera, and\nA. H. MacDonald, Theory of ferromagnetic (III,Mn)V\nsemiconductors , Rev. Mod. Phys. 78, 809 (2006).\n6T. Dietl, A ten-year perspective on dilute magnetic semi-\nconductors and oxides , Nature Materials 9, 965 (2010).\n7H. Ohno, A. Shen, F. Matsukura, A. Oiwa, A. Endo,\nS. Katsumoto, and Y. Iye, (Ga,Mn)As: a new diluted mag-\nnetic semiconductor based on GaAs , Appl. Phys. Lett. 69,\n363–365 (1996).\n8J. K¨ onig, T. Jungwirth, and A. H. MacDonald, Theory\nof magnetic properties and spin-wave dispersion for ferro-\nmagnetic (Ga,Mn)As , Phys. Rev. B 64, 184423 (2001).\n9M. Sawicki, F. Matsukura, T. Dietl, G. M. Schott,\nC. Ruester, G. Schmidt, L. W. Molenkamp and G. Kar-\nczewski, Temperature Peculiarities of Magnetic Anisotropy\nin (Ga,Mn)As: The Role of the Hole Concentration , Jour-\nnal of Superconductivity 16, 7 (2003).\n10M. Sawicki, K.-Y. Wang, K. W. Edmonds, R. P. Campion,\nC. R. Staddon, N. R. S. Farley, C. T. Foxon, E. Papis,\nE. Kami´ nska, A.Piotrowska, T. Dietl, andB. L. Gallagher,\nIn-plane uniaxial anisotropy rotations in (Ga,Mn)As thin\nﬁlms, Phys. Rev. B 71, 121302(R) (2005).\n11L. V. Titova, M. Kutrowski, X. Liu, R. Chakar-\nvorty, W. L. Lim, T. Wojtowicz, J. K. Furdyna, and\nM. Dobrowolska, Competition between cubic and uniaxial\nanisotropy in Ga1−xMnxAsin the low-Mn-concentration\nlimit, Phys. Rev. B 72, 165205 (2005).\n12X. Liu and J. K. Furdyna, Ferromagnetic resonance in\nGa1−xMnxAsdilute magnetic semiconductors , J. Phys.:\nCondens. Matter 18, R245 (2006).\n13G. Bouzerar, Magnetic spin excitations in diluted ferro-\nmagnetic systems: the case of Ga1−xMnxAs, Europhys.\nLett.79, 57007 (2007).\n14A. Werpachowska and T. Dietl, Theory of spin waves\nin ferromagnetic (Ga,Mn)As , Phys. Rev. B 82, 085204\n(2010).\n15P. Nemec, V. Novak, N. Tesarova, E. Rozkotova, H. Re-\nichlova, D. Butkovicova, F. Trojanek, K. Olejnik, P. Maly,\nR.P. Campion, B. L. Gallagher, Jairo Sinova, andT. Jung-\nwirth,The essential role of carefully optimized synthesis\nfor elucidating intrinsic material properties of (Ga,Mn)A s,\nNat. Commun. 4:1422 doi: 10.1038/ncomms2426 (2012).\n16Y. Sasaki, X. Liu, J. K. Furdyna, M. Palczewska,\nJ. Szczytko, and A. Twardowski, Ferromagnetic resonance\nin GaMnAs , J. Appl. Phys. 91, 7484 (2002).\n17Y. Sasaki, X. Liu, T. Wojtowicz, and J. K. Furdyna, Spin\nWave Resonance in GaMnAs , Journal of Superconductiv-\nity: Incorporating Novel Magnetism 16, 143 (2003).\n18X. Liu, Y. Sasaki, and J. K. Furdyna, Ferromagnetic res-onance in Ga1−xMnxAs: Eﬀects of magnetic anisotropy ,\nPhys. Rev. B 67, 205204 (2003).\n19T. G. Rappoport, P. Redli´ nski, X. Liu, G. Zar´ and,\nJ. K. Furdyna, and B. Jank´ o, Anomalous behavior of spin-\nwave resonances in Ga1−xMnxAsthin ﬁlms , Phys. Rev.\nB69, 125213 (2004).\n20X. Liu, W. L. Lim, M. Dobrowolska, and J. K. Furdyna,\nFerromagnetic resonance study of the free-hole contributi on\nto magnetization and magnetic anisotropy in modulation-\ndopedGa1−xMnxAs/Ga 1−yAlyAs:Be, Phys. Rev. B 71,\n035307 (2005).\n21X. Liu and J. K. Furdyna, Ferromagnetic resonance in\nGa1−xMnxAsdilute magnetic semiconductors , J. Phys.:\nCondens. Matter 18, R245 (2006).\n22X. Liu, Y.-Y. Zhou, and J. K. Furdyna, Angular depen-\ndence of spin-wave resonances and surface spin pinning in\nferromagnetic (Ga,Mn)As ﬁlms , Phys. Rev. B 75, 195220\n(2007).\n23Y.-Y. Zhou, Y.-J. Cho, Z. Ge, X. Liu, M. Dobrowolska,\nJ. K. Furdyna, Magnetic anisotropy, spin pinning, and ex-\nchange constants of (Ga,Mn)As ﬁlms , IEEE Trans. Magn.\n43, 30193021 (2007).\n24Y.-Y.Zhou, Y.-J. Cho, Z.Ge, X.Liu, M.Dobrowolska, and\nJ. K. Furdyna, Ferromagnetic Resonance Study of Ultra-\nthinGa1−xMnxAsFilms as a Function of Layer Thick-\nness, AIP Conf. Proc., 893, 1213 (2007).\n25Y.-Y. Zhou, X. Liu, J. K. Furdyna, M. A. Scarpulla, and\nO. D. Dubon, Ferromagnetic resonance investigation of\nmagnetic anisotropy in Ga1−xMnxAssynthesized by ion\nimplantation and pulsed laser melting , Phys. Rev. B 80,\n224403 (2009).\n26S. T. B. Goennenwein, T. Graf, T. Wassner, M. S. Brandt,\nM. Stutzmann, J. B. Philipp, R. Gross, M. Krieger,\nK. Z¨ urn, P. Ziemann, A. Koeder, S. Frank, W. Schoch, and\nA. Waag, Spin wave resonance in Ga1−xMnxAs, Appl.\nPhys. Lett. 82, 730 (2003).\n27C. Bihler, W. Schoch, W. Limmer, S. T. B. Goennen-\nwein, and M. S. Brandt, Spin-wave resonances and surface\nspin pinning in Ga1−xMnxAsthin ﬁlms , Phys. Rev. B 79,\n045205 (2009).\n28L. Dreher, C. Bihler, E. Peiner, A. Waag, W. Schoch,\nW. Limmer, S. T. B. Goennenwein, and M. S. Brandt,\nAngle-dependent spin-wave resonance spectroscopy of\n(Ga,Mn)As ﬁlms , Phys. Rev. B 87, 224422 (2013).\n29C. Kittel, Excitation of Spin Waves in a Ferromagnet by\na Uniform rf Field , Phys. Rev. 110, 1295 (1958).\n30A. M. Portis, Low-Lying Spin Wave Modes in Ferromag-\nnetic Films , Appl. Phys. Lett. 2, 69 (1963).\n31A. R. Ferchmin, A quantum theory of spin-wave resonance\nin cubic ferromagnets , Phys. Lett. 1, 281 (1962).\n32H. Puszkarski, Quantum Theory of Spin Wave Resonance\nin Thin Ferromagnetic Films. Part I: Spin Waves in Thin\nFilms, Part 2: Spin-Wave Resonance Spectrum , Acta\nPhysica Polonica A 38, 217 and 899 (1970).\n33J. T. Yu, R. A. Turk, and P. E. Wigen, Exchange-\ndominated surface spin waves in thin yttrium-iron-garnet\nﬁlms, Phys. Rev. B 11, 420 (1975).\n34P. E. Wigen, H. Puszkarski, Microscopic Model for the\nTensorial Surface Anisotropy Field Observed in Thin\nYttrium-Iron Garnet Film Spin-Wave Resonance , Solid\nState Comm. 18, 363 (1976).11\n35H. Puszkarski, Complex Angle Dependence of the Surface\nPinning Parameter in Magnetic Thin Films , Solid State\nComm.22, 563-564 (1977).\n36A. P. Cracknell, H. Puszkarski, Symmetry Properties of the\nSurface Pinning Parameter in Magnetic Thin Films , Solid\nState Comm. 28, 891 (1978).\n37H. Puszkarski, Theory of Surface States in Spin Wave Res-\nonance, Progr. Surf. Sci. 9, 191 (1979).\n38H. Puszkarski, Spin Wave Resonance - a Tool for the Study\nof Magnetic Surface Interactions , Chapter IX in Mag-\nnetism in Solids (Some Current Topics), Proceedings of\nthe Twenty-Second Scottish Universities Summer School\nin Physics, Dundee, August 1981.\n39A. R. Ferchmin and H. Puszkarski, Existence conditions\nof surface spin waves in ferromagnetic nanowires and\nnanoparticles , J. Appl. Phys. 90, 5335 (2001).\n40H. Puszkarski and M. Kasperski, On the interpretation ofthe angular dependence of the main FMR/SWR line in fer-\nromagnetic thin ﬁlms , Acta Phys. Polon. 121, 1165 (2012).\n41T. Ohms, K. Hiebbner, H. C. Schneider, and M. Aeschli-\nmann,Spin- and Energy Relaxation of Hot Electrons at\nGaAs Surfaces , inSpin dynamics in conﬁned magnetic\nstructures III , Eds. B. Hillebrands, A. Thiaville (Springer-\nVerlag, Berlin Heidelberg, 2006).\n42Y. Nishitani, D. Chiba, M. Endo, M. Sawicki, F. Mat-\nsukura, T. Dietl, and H. Ohno, Curie temperature\nversus hole concentration in ﬁeld-eﬀect structures of\nGa1−xMnxAs, Phys. Rev. B 81, 045208 (2010).\n43H. Puszkarski, Elliptical Spin Precession Correction Factor\nin the Theory of Spin-Wave Resonace , phys. stat. sol. (b)\n96, 61 (1979).\n44Diep-The-Hung, H. Puszkarski, and J.-C. S. Levy, Eﬀect of\nElliptical Spin Wave Polarization on the Surface Pinning\nParameter , Le Vide, les Couches Minces 204(1980)."    },    {        "title": "0906.4107v1.Kondo_resonance_narrowing_in_d__and_f_electron_systems.pdf",        "content": "arXiv:0906.4107v1  [cond-mat.str-el]  22 Jun 2009Kondoresonance narrowing in d-andf-electron systems\nAndriy H. Nevidomskyy∗and P. Coleman\nDepartment of Physics and Astronomy, Rutgers University, P iscataway, N.J. 08854, USA\n(Dated: April 25, 2022)\nBydevelopingasimplescalingtheoryfortheeffectofHund’ sinteractionsontheKondoeffect,weshowhow\nanexponential narrowingoftheKondo resonance develops in magnetic ions withlargeHund’s interaction. Our\ntheorypredictsanexponentialreductionoftheKondotempe raturewithspinSoftheHund’scoupledmoment,a\nlittle-knowneffect ﬁrstobserved ind-electronalloys int he 1960’s, andmore recentlyencountered innumerical\ncalculations on multi-band Hubbard models with Hund’s inte ractions. We discuss the consequences of Kondo\nresonancenarrowingfortheMotttransitionind-bandmater ials,particularlyironpnictides,andthenarrowESR\nlinewidthrecentlyobserved inferromagnetically correla tedf-electron materials.\nPACS numbers: 75.20.Hr, 71.27.+a, 71.20.Be\nThe theory of the Kondoeffect formsa cornerstonein cur-\nrent understanding of correlated electron systems [1]. Mor e\nthan forty years ago, experiments on d-electron materials\nfoundthatthecharacteristicscaleofspinﬂuctuationsofm ag-\nnetic impurities, known as the Kondo temperature, narrows\nexponentially with the size Sof the impurity spin [2] (Fig\n1). An explanation of this effect was proposed [2] based on\nan early theory of the Kondo effect by Schrieffer [3], who\nfound that strong Hund’s coupling leads to a 2S-fold reduc-\ntion of the Kondo coupling constant. Surprisingly, interes t in\nthis phenomenon waned after the 1960’s. Motivated by a re-\ncentresurgenceofinterestin f-andd-electronsystems,espe-\ncially quantum critical heavy electron systems [4], and pni c-\ntide superconductors [5], this paper revisits this little- known\nphenomenon,whichwerefertoas“Kondoresonancenarrow-\ning”,placingit ina moderncontext.\nThe consequences of Kondo resonance narrowing have\nrecently been re-discovered in calculations on multi-orbi tal\nHubbard and Anderson models [6, 7]. Numerical renormal-\nization group studies found that the introduction of Hund’s\ncoupling into the Anderson model causes an exponential re-\nduction in the Kondo temperature [6]. The importance of\nHund’s effect has also arisen in the context of iron pnictide\nsuperconductors[8, 9], where it appears to play a key role in\nthedevelopmentof“badmetal”stateinwhichthe d-moments\nremainunquencheddownto lowtemperatures.\nInthispaper,weshowthatKondoresonancenarrowingcan\nbe simply understood within a scaling theory description of\nthe multi-channelKondomodelwith Hund’sinteraction. The\nmain result is an exponential decrease of the Kondo temper-\nature that develops when localized electrons lock together to\nforma largespin S,givenbytheformula\nlnT∗\nK(S) = lnΛ 0−(2S)ln/parenleftbiggΛ0\nTK/parenrightbigg\n. (1)\nHere,TKisthe“bare”spin 1/2Kondotemperatureand Λ0=\nmin(JHS,U+Ed,|Ed|)isthescaleatwhichthelockedspin\nSdevelopsundertheinﬂuenceofaHund’scoupling,while U\nandEdaretheinteractionstrengthandpositionofthebare d-\nlevel. Althoughthis result is implicitly containedin the e arly\nworksofSchrieffer[3]andHirst[10],adetailedtreatment has\nTK\nFe\nFe\nCr\nFe\nMnMn\nMn\nMnHosts:\nSTi\nNi\nTi Co\nV\nCo\nFeCr\nFIG. 1: Measured values [11] of the Kondo temperature T∗\nKin host\nalloys Au, Cu, Zn, Ag, Mo, and Cd containing transition metal im-\npurities, plotted vs. the nominal size Sof the spin. Solid line is the\nﬁttoEq.(1) with Λ0≡JHS.\ntoourknowledge,notpreviouslybeengiven.\nTo develop our theory, we consider Kspin1/2impurity\nspinsatasinglesite,ferromagneticallyinteractingviaH und’s\ncoupling JH, each coupled to a conduction electron channel\nofband-width Dvia anantiferromagneticinteraction J:\nH=/summationdisplay\nk,σ,µεkc†\nkσµckσµ−JH/parenleftBiggK/summationdisplay\nµ=1sµ/parenrightBigg2\n+JK/summationdisplay\nµ=1sµ·σµ,(2)\nwhereεkis the conduction electron energy, µ= 1,Kis the\nchannelindexand σµ=/summationtext\nkc†\nkαµσαβckβµistheconduction\nelectronspindensityin channel µat theorigin. We implicitly\nassume that Hund’s scale KJHis smaller than D. When de-\nrivedfromanAndersonmodelof Kspin-1/2impurities,then\nD=min(Ed+U,|Ed|)isthecross-overscaleatwhichlocal\nmomentsformwhile J=|VkF|2(1/(Ed+U)+1/|Ed|)isthe\nSchrieffer–Wolff form for the Kondo coupling constant [1],\nwhereVkFis the Anderson hybridization averaged over the\nFermisurfaceand Ed<0isnegative.\nThe behavior of this model is well understood in the two2\nµ2\nIµ2\nIITχ(  )TII III I\neff\n(b)(a)\nNozieres FL Locked large spins PM\nTK*TK* TK HJS\nHJSg/ 1(Λ)\nTΛ\nFIG. 2: (a) Schematic showing the behavior of the running cou pling\nconstantgeff(Λ) =J(Λ)ρKeffon a logarithmic scale, with Keffthe\neffective number of conduction electron channels per impur ity spin\n(Keff= 1in region I and Kin regions II and III). (b) Schematic\nshowing effective moment µ2\neff(T) =Tχ(T)intermsof thesuscep-\ntibilityχ(T),showingtheenhancement (15)inregionIIandtheloss\nof localizedmoments due toKondo screening inregionIII.\nextreme limits [12]: for JH=∞, theKspins lock to-\ngether, forming a K-channel spin S=K/2Kondo model.\nThe opposite limit JH= 0describes Kreplicas of the spin-\n1/2Kondomodel. Paradoxically,the leadingexponentialde-\npendence of the Kondo temperatureon the coupling constant\nTK∼De−1/2Jρinthesetwolimitsisindependentofthesize\nof the spin. However, as we shall see in the cross-over be-\ntween the two limits, the projection of the Hamiltonian into\nthe space of maximum spin leads to a (2S)-fold reduction in\ntheKondocouplingconstant.\nWe now study the propertiesof this model as a functionof\nenergy scale or cut-off Λ. Qualitatively, we expect three dis-\ntinctregionsdepictedin Fig.2(a):\n(I)Λ≫JHS: a spin-1/2disordered paramagnet character-\nizedbyahightemperatureCuriemagneticsusceptibility\nχI(T) =K(3/4)(gµB)2\n3kBT(3)\n(wheregistheg-factoroftheelectron),witheffectivemoment\n(µI\neff)2= 3K/4;\n(II)T∗\nK≪Λ≪JHS: an unscreened big spin S=K/2is\nformedaboveanemergentKondoenergyscale T∗\nK;\n(III)Λ≪T∗\nK: the Nozi` eres Fermi liquid ground state of the\nK-channelS=K/2Kondoproblem.\nWe employ the “Poor Man’s scaling” approach [1, 13], in\nwhich the leading renormalization ﬂows are followed as the\nelectronsintheconductionbandaresystematicallydecima ted\nfrom the Hilbert space. By computing the diagrams depicted\ninFig.3,weobtainthefollowingrenormalizationgroup(RG )\nFIG.3: Thediagramsappearingin(a)one-loopand(b)two-lo opRG\nequationsforKondocouplingJ(opencircles),withsolidli nesdenot-\ning the conduction electron propagators and dashed line – th e impu-\nrity spin. (c) The lowest order diagrams in the RG ﬂow of Hund’ s\ncoupling (a square vertex denotes bare JH).\nequationsinregionI:\n(I):d(Jρ)\ndlnΛ=−2(Jρ)2+2(Jρ)3(4)\nd(JHρ)\ndlnΛ= 4(Jρ)2JHρ (5)\nwhereρis the density of states of the conduction electrons\nat the Fermi level. The ﬁrst equation is the well-known beta\nfunction for the Kondo model, which to this order is inde-\npendent of Hund’s coupling. As we decimate the states of\nthe conduction sea, reducing the band-width Λdown to the\nHund’sscale JHS,to leadinglogarithmicorderweobtain\nρJ(Λ) =1\nln/parenleftBig\nΛ\nTK/parenrightBig/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nΛ=JHS≡ρJI. (6)\nThere is a weak downward-renormalization of the Hund’s\ncouplingJHdescribedbyEq.(5),whichoriginatesinthetwo-\nloop diagrams (Fig. 3c)). In the leading logarithmic approx i-\nmation,wemayapproximate JHbyaconstant.\nOnceΛis reduced below JHS, the individual local mo-\nmentsbecome locked into a spin S=K/2, similar to the ef-\nfectdiscussedinRef.14forthecaseoftwoimpuritiescoupl ed\nbyferromagneticRKKY interaction. Thelow-energyproper-\ntiesofthesysteminregionIIaredescribedbyaKondomodel\nofspinK/2withKconductionelectronchannels:\nHII\neff=J∗(Λ)K/summationdisplay\nµ=1S·σµ, (7)\nToobtainthe valueof J∗, we mustprojecttheoriginalmodel\ninto the subspace of maximumspin S. By the Wigner-Eckart\ntheorem, the matrix element of any vector operator acting in\nthe basis of states |Sz∝angbracketrightof big spin S=K/2is related by a\nconstantprefactorto Sitself,i.e.\n∝angbracketleftSSz|sµ|SSz∝angbracketright=gS∝angbracketleftSz|S|Sz∝angbracketright (8)\nSummingboth sides ofthe equationoverimpurityindex µ=\n1,...K, one obtains ∝angbracketleftSSz|/summationtext\nµsµ|SSz∝angbracketright=gSKSz. How-\never since/summationtext\nµsµ=Ks≡S, one arrives at the conclusion3\nthatgSK= 1,thereforedeterminingthevalueoftheconstant\ncoefﬁcient gS= 1/KinEq.(8). ComparingEqs.(2)and(7),\nwe arriveatthe effectiveKondocoupling:\nJ∗=J/K. (9)\nThisequationcapturesthekeyeffectofcrossoverfromregi on\nItoregionIIinFig.2. Thisresultwasﬁrstderivedintheear ly\nwork on the multi-channel Kondo problem by Schrieffer [3],\nwherethelimitof JH→ ∞wasimplicitlyassumed,andalso\nappears for the particular case of K= 2in the study of the\ntwo-impurityKondoproblembyJayaprakash etal.[14].\nToonelooporder,thescalingequationfor J∗(Λ)inregion\nIIisidenticaltothatofregionI(4),namely d(J∗ρ)/dlnΛ≈\n−2(J∗ρ)2, though its size is Ktimes smaller. To avoid the\ndiscontinuous jump in coupling constant at the crossover, i t\nis more convenient to consider geff≡J(Λ)ρKeff, where the\neffective number of channels Keff= 1andKin regions I\nandIIrespectively. Thiscontinuousvariablesatisﬁes\n(II):dgeff\ndlnΛ=−2\nKg2\neff+2\nKg3\neff.(10)\nso the speed at which it scales to strong coupling becomes\nKtimes smaller in region II (see Fig. 2a). Solving this RG\nequation to leading order, and setting geff(Λ =T∗\nK)∼1, we\nobtainT∗\nK∼(JHS)(D/JHS)Ke−K\n2Jρfor the renormalized\nKondoscale. ComparingthiswiththebareKondoscale TK∼\nDe−1/2Jρ,we deduce\nT∗\nK∼JHS/parenleftbiggTK\nJHS/parenrightbiggK\n≡TK/parenleftbiggTK\nJHS/parenrightbiggK−1\n,(11)\nfromwhichformula(1)follows. Thisexponentialsuppressi on\nofthespintunnelingratecanbeunderstoodasaresultofa 2S-\nfoldincreaseintheclassicalactionassociatedwithaspin -ﬂip.\nTheseresultsareslightlymodiﬁedwhenthetwo-loopterms\nin the scaling are taken into account. The expression for TK\nnow acquires a pre-factor, TK=D√Jρe−1/2JρandJHis\nweaklyrenormalizedso that\nT∗\nK= (˜JHS)/parenleftbiggTK√\nK˜JHS/parenrightbiggK\n, (12)\nwhere˜JHisdeterminedfromthequadraticequation\nx2−x/parenleftbigg\nx0+4\nln(D/TK)/parenrightbigg\n+4 = 0.(13)\nwherex= ln(˜JHS/TK)andx0≡ln(JHS/TK).\nThemagneticimpuritysusceptibilityin regionIIbecomes\nχ∗\nimp=(gµB)2\n3kBTS(S+1)\n1−1\nln/parenleftBig\nT\nT∗\nK/parenrightBig+O\n1\nln2/parenleftBig\nT\nT∗\nK/parenrightBig\n\n,\n(14)\nfromwhichweseethattheenhancementofthemagneticmo-\nmentatthe crossoverisgivenby(seeFig. 2b))\n/parenleftbiggµII\neff\nµI\neff/parenrightbigg2\n=K+2\n3. (15)When the temperature is ultimately reduced below the ex-\nponentially suppressed Kondo scale T∗\nK, the big spins Sbe-\ncomescreenedtoformaNozi` eresFermiliquid [15]. Aphase-\nshift description of the Fermi liquid predicts that [12, 16] the\nWilsonratio W≡χimp\nχ0/γimp\nγ0isgivenby\nWK=2(K+2)\n3≡2/parenleftbiggµII\neff\nµI\neff/parenrightbigg2\n, (16)\nwhich, compared with the classic result W1=2for the one-\nchannelmodel[15],containsafactorofthemomentenhance-\nment. This result holds in the extreme limit JH≫TK. More\ngenerally, Wdependson the ratio η=U∗/J∗\nHof a channel-\nconserving interaction U∗to an inter-channel Hund’s coupl-\ningJ∗\nHin theFermiliquidphase-shiftanalysis,givingriseto\nWK(η) = 2/parenleftbigg\n1+K−1\n2(1+η)+1/parenrightbigg\n. (17)\nOngeneralgroundswe expect η∼TK/JH.\nWe end with a discussion of the broader implications of\nKondo resonance narrowing for d- andf-electron materials.\nThis phenomenon provides a simple explanation of the dras-\ntic reductionsin spin ﬂuctuation scale observed in the clas sic\nexperiments of the sixties [2]. Our treatment brings out the\nimportantrole of Hund’scouplingin this process. One of the\nuntestedpredictionsofthistheoryisalinearriseoftheWi lson\nratioWwith spin S=K/2(16), from a value W[1] =2.7 in\nTi andNi, to W[5/2] =4.7in Mnimpurities. Takingtogether\nwith the early data, Fig. 1, we are able to essentially conﬁrm\nthe early speculation [3] that were Hund’s coupling absent,\nthe Kondo effect would take place at such high temperatures\nthat dilute d-electron magnetic moments would be unobserv-\nable. Thisis,inessence,thesituationforTiimpuritiesin gold,\nwhere the Kondo temperature of S= 1moments becomes\nso high that magnetic behavior is absent below the melting\ntemperatureof gold. On theotherhand,the Kondoresonance\nnarrowingeffectsofHund’sinteractioncanbecomesosever e,\nthatthere-entryfromregionIIintothequenchedFermiliqu id\nbecomestoolowtoobserve. Thisisthecasefor S= 5/2Mn\nin gold, where T∗\nKis so low that it has never been observed;\ntherecentobservationofa“spinfrozenphase”inDMFTstud-\nies[7]maybea numericalcounterpart.\nWhat then, are the possible implications for dense d-\nelectronsystems? In those materials, the ratio of Kondotem -\nperaturetotheHund’scouplingwillbestronglydependento n\nstructure,screeningand chemistry. In caseswhere JH≪TK,\nthe physics of localized magnetic moments will be lost and\nthed-electronswill be intinerant. On theotherhand,the situ-\nationwhere JH≫TKwillalmostcertainlyleadtolongrange\nmagneticorderwithlocalized d-electrons. Thusinmulti-band\nsystems, the criterion JH∼TKdeterminesthe boundarybe-\ntween localized and itinerant behavior, playing the same ro le\nasthecondition U/D∼1inone-bandMottinsulators.\nTheseissuesmaybeofparticularimportancetotheongoing\ndebate about the strength of electroncorrelationsin the Fe As\nfamily of high-temperature superconductors [5, 17, 18, 19] .4\nCurrent wisdom arguesthat in a multi-bandsystem, the criti -\ncalinteraction UcnecessaryfortheMottmetal-insulatortran-\nsition grows linearly with the number of bands N[20, 21],\nfavoring a viewpoint that iron pnictide materials are itine rant\nmetalslyingfarfromtheMott regime.\nInessence, Hund’scouplingconvertsa onechannelKondo\nmodeltoa K-channelmodel(7). Large- Ntreatmentsofthese\nmodels show that the relevant control parameter is the ratio\nK/N[22], rather than 1/N. By repeating the large- Nargu-\nmentofFlorens etal.[20],weconcludethatthecriticalvalue\noftheon-siteinteraction UfortheMotttransitionis\nUc∝(N/K)V2\nkFρ. (18)\nThus Hund’s coupling compensates for multi-band behavior,\nrestoringUctoavaluecomparablewithone-bandmodels. Re-\ncentDMFTcalculationsonthetwo-orbitalHubbardmodel[6]\nsupportthisview,ﬁndingthat Ucisreducedfrom Uc≈3Dto\nU∗\nc≈1.1DwhenJH/U= 1/4.\nLDA+DMFTstudiesofironpnictidematerials[9]conclude\nthatinordertoreproducetheincoherentbadmetalfeatures of\nthenormalstate, avalueof JH∼0.4eV isrequired,resulting\ninT∗\nKof the order of 200 K. Fitting this with Eq. (11) re-\nsults in a nominal TK∼3000K anda ratio TK/JHS∼0.4.\nBy contrast, for dilute Fe impurities in Cu [11] T∗\nK≈20K,\nfrom which we extract a bare ratio TK/JHS∼0.2and\nTK∼3500K. The bare Kondo temperature is essentially the\nsame in both cases, but TK/JHSis signiﬁcantly increased\ndue to screening of JHin the iron pnictides, placing them\nmore or less at the crossover JH∼TK. A further sign of\nstrong correlations in iron pnictides derives from the Wils on\nratio, known to be ∼1.8 in SmFeAsO [23] and about 4–5 in\nFeCrAs[24],whereasEq.(16)wouldpredict W=4.\nFinally,wediscussheavy f-electronmaterials,whichlieat\nthecrossoverbetweenlocalizedanditinerantbehavior[25 ]. In\nthese materials, spin orbit and crystal ﬁeld interactions d om-\ninate over Hund’s interaction. In fact, crystal ﬁelds are al so\nknown to suppress the Kondo temperature in f-electron sys-\ntems [26], but the suppression mechanism differs, involvin g\na reduction in the spin symmetry rather a projective renor-\nmalization of the coupling constant. But the main reason\nthat Hund’s coupling is unimportantat the single-ion level in\nheavyf-electron materials, is because most of them involve\nonef-electron(e.g. Ce) or one f-holein a ﬁlled f-shell(Yb,\nPu),forwhichHund’sinteractionsareabsent.\nPerhaps the most interesting application of Kondo reso-\nnancenarrowingto f-electronsystemsisin thecontextofin-\ntersite interactions. Indeed, (2) may serve as a useful mode l\nfor a subset of ferromagnetically correlated f-electron mate-\nrials,suchasCeRuPO[27],where JHwouldcharacterizethe\nscaleofferromagneticRKKYinteractionsbetweenmoments,\nasinRef.14. Inthesesystems,ourmodelpredictstheforma-\ntionofmicroscopicclustersofspinswhichremainunscreen ed\nin region II down to an exponentially small scale ∼T∗\nK.\nThis exponential narrowing of the Kondo scale may providea clue to the observation [28, 29] of very narrow ESR ab-\nsorptionlines in a numberof Yb and Ce heavy fermion com-\npounds with enhanced Wilson ratios. In particular, our the-\norywouldpredictthattheKnightshiftoftheelectrong-fac tor\nin region II is proportional to the running coupling constan t\nK(T)∝geff(T)∼1/ln(T/T∗\nK),whereT∗\nKistheresonance-\nnarrowed Kondo temperature. A detailed study of the ESR\nlineshapeinthiscontextwill bea subjectoffuturework.\nWe would like to acknowledge discussions with Elihu\nAbrahams,NatanAndrei,KristjanHaule,GabrielKotliaran d\nAndrew Millis in connection with this work. This research\nwassupportedbyNSF grantno. DMR0605935.\n∗Electronic address: nevidomskyy@cantab.net\n[1] A. C. Hewson, The Kondo Problem to Heavy Fermions (Cam-\nbridge Univ. Press,1993).\n[2] M. D. Daybell and W. A. Steyert, Rev. Mod. Phys. 40, 380\n(1968).\n[3] J. R.Schrieffer,J. Appl. Phys. 38, 1143 (1967).\n[4] Ph. Gegenwart, F. Steglich, and Q. Si, Nature Physics 4, 186\n(2008).\n[5] M. R.Norman, Physics 1, 21(2008).\n[6] T. Pruschke andR. Bulla,Eur.Phys.J. B 44, 217 (2005).\n[7] P.Werner,E.Gull,M. Troyer, andA.J.Millis,Phys.Rev. Lett.\n101, 166405 (2008).\n[8] K. Haule, J. H. Shim, and G. Kotliar, Phys. Rev. Lett. 100,\n226402 (2008).\n[9] K. Haule andG. Kotliar,New J. Phys. 11, 025021 (2009).\n[10] L.L.Hirst,Z.Physik 244, 230(1971).\n[11] M. Daybell, in Magnetism , Eds. G. Rado and H. Suhl (Aca-\ndemic Press,New York,1973), vol.5, pp. 122–147.\n[12] P.Nozi` eres and A.Blandin, J. Physique 41, 193 (1980).\n[13] P.W.Anderson, J.Phys. C 3, 2439 (1970).\n[14] C. Jayaprakash, H. R.Krishnamurthy, and J. W.Wilkins, Phys.\nRev. Lett. 47, 737 (1981).\n[15] P.Nozi` eres, J.Low Temp. Phys. 17, 31(1974).\n[16] A. Yoshimori, J.Phys.C 15, 5241 (1976).\n[17] Y. Kamihara, T. Watanabe, H. Hirano, and H. Hosono, J. Am .\nChem. Soc. 130, 3296 (2008).\n[18] X. H.Chen et al.,Nature (London) 453, 761(2008).\n[19] C. de la Cruz et al.,Nature (London) 453, 899(2008).\n[20] S.Florens, A.Georges, G.Kotliar,and O.Parcollet,Ph ys.Rev.\nB66, 205102 (2002).\n[21] Y.¯Ono, M. Potthoff, and R. Bulla, Phys. Rev. B 67, 035119\n(2003).\n[22] E.Lebanon andP.Coleman, Phys.Rev. B 76, 085117 (2007).\n[23] M. R.Cimberle et al.,J.Mag. Mag. Materials (inpress, 2009).\n[24] W.Wu et al.,Eur.Phys. Lett. 85, 17009 (2009). 17009 (2009)\n[25] P. Coleman, in Handbook of Magnetism and Advanced Mag-\nnetic Materials , Eds. H. Kronmuller and S. Parkin, vol. 1, pp.\n95-148 (John WileyandSons, 2007).\n[26] K. Hanzawa, K. Yamada, and K. Yosida, J. Mag. Mag. Materi -\nals47, 357 (1985).\n[27] C. Krellneret al.,Phys.Rev. B 76, 104418 (2007).\n[28] C. Krellner, T. Farster, H. Jeevan, C. Geibel, and\nJ. Sichelschmidt, Phys.Rev. Lett 100, 066401 (2008).\n[29] J. Sichelschmidt, V. A. Ivanshin, J. Ferstl, C. Geibel, and\nF.Steglich, Phys.Rev. Lett. 91, 156401 (2003)."    },    {        "title": "1310.4840v1.Electrical_Detection_of_Direct_and_Alternating_Spin_Current_Injected_from_a_Ferromagnetic_Insulator_into_a_Ferromagnetic_Metal.pdf",        "content": "arXiv:1310.4840v1  [cond-mat.mtrl-sci]  17 Oct 2013Electrical Detection of Direct and Alternating Spin Curren t Injected from a\nFerromagnetic Insulator into a Ferromagnetic Metal\nP. Hyde, Lihui Bai, D.M.J. Kumar, B.W. Southern, and C.-M. Hu∗\nDepartment of Physics and Astronomy, University of Manitob a, Winnipeg, Canada R3T 2N2\nS. Y. Huang, B. F. Miao, and C. L. Chien\nDepartment of Physics and Astronomy, Johns Hopkins Univers ity, Baltimore, MD 21218, USA\n(Dated: June 8, 2021)\nWe report room temperature electrical detection of spin inj ection from a ferromagnetic insulator\n(YIG) into a ferromagnetic metal (Permalloy, Py). Non-equi librium spins with both static and\nprecessional spin polarizations are dynamically generate d by the ferromagnetic resonance of YIG\nmagnetization, and electrically detected by Py as dc and ac s pin currents, respectively. The dc spin\ncurrent is electrically detected via the inverse spin Hall e ﬀect of Py, while the ac spin current is\nconverted to a dc voltage via the spin rectiﬁcation eﬀect of P y which is resonantly enhanced by\ndynamic exchange interaction between the ac spin current an d the Py magnetization. Our results\nreveal a new path for developing insulator spintronics, whi ch is distinct from the prevalent but\ncontroversial approach of using Pt as the spin current detec tor.\nDeveloping new methods for generating and detecting\nspin currents has been the central task of spintronics.\nIn the pioneering work of Johnson and Silsbee [1], the\ngeneration and detection of spin-polarized currents were\nboth achieved through the use of ferromagnetic metals\n(FM). Recent breakthroughs reveal ferromagnetic insu-\nlators (FI) to be promising spin current sources, in which\nspin currents can be generated without the presence of\nany charge current [2, 3]. In the ground-breaking exper-\niment performed 3 years ago by Kajiwara et al.[2], elec-\ntrical detection of the spin current generated by yttrium\niron garnet (Y 3Fe5O12, YIG) was achieved by utilizing\nthe heavynormalmetal platinum (Pt), in whichspin cur-\nrent was detected via the inverse spin Hall eﬀect (ISHE).\nSince then, nearly the entire insulator-spintronics com-\nmunityhasfollowedsuitandusedPtasthe standardspin\ndetector. But so far, consensus has not yet been achieved\non a few critical spin-dependent material issues of Pt [4–\n7]. Given the fact that ferromagnetic metals are broadly\nused as spin detectors in both semiconductor [8, 9] and\nmetallic spintronics devices [1, 10, 11], it is noteworthy\nthat the appealing topic of how a FM material may de-\ntect the spin current generated by a FI has barely been\ninvestigated. Elucidating this issue is of broad interest\nfor making insulator-spintronics device compatible with\nboth semiconductor and metallic spintronics devices.\nIn this letter, we report room temperature detection\nof spin current generated in YIG by feromagnetic res-\nonance (FMR). Distinct from the popular approach of\nusing Pt as the spin detector, we use the ferromagnetic\nmetal Permalloy (Py) instead, and demonstrate that Py\nnot only detects the dc spin current from YIG, but most\nstrikingly, it also detects the recently predicted ac spin\ncurrent [12] by directly converting it into a dc voltage,\nwhich makes Py a superior spin detector compared to\nPt. Two very recent experiments make this work possi-\nble: (i) the discovery of the ISHE in Py [13], and (ii) theestablishment of a universal method for clearly separat-\ning spin rectiﬁcation from spin pumping [14].\nWe begin by highlighting the basic ideas. As shown in\nFig. 1, let us consideraPy/YIGbilayerunder microwave\nirradiation in an external magnetic ﬁeld H. Choosing the\nxaxis as the longitudinal direction for measuring the dc\nvoltages, and the zaxis as perpendicular to the interface,\nthe direction of His described by the polar (with respect\nto thezaxis) and azimuth (with respect to the xaxis)\nangles of θandφ, respectively, as shown in Fig. 1(c).\nAt the FMR frequency ωYIGof YIG, the magnetization\nof YIG precesses about its saturation magnetization M,\nwhich pumps non-equilibrium spins diﬀusing across the\nPy/YIG interface. Hence, a dc spin current jscarries\nstatic non-equilibrium spin angular momentum which is\nantiparallel to M, while an ac spin current js(ωYIG) car-\nries dynamic non-equilibrium spin angular momentum\nwhich is precessing about M[12]. Both spin currents\nﬂow along the zdirection, as shown in Fig. 1(c).\nBased on the recently discovered ISHE in Py [13], the\nidea of using Py to detect jsis straightforward as shown\nin Fig. 1(a). It can be detected by the dc voltage VSP\nin Py produced through spin pumping and the ISHE,\ni.e.,VSPis proportional to js. In contrast, detecting the\nhigh-frequency ac spin current js(ωYIG) is nontrivial and\nis currently of great interest. Two groups have very re-\ncently developed very smart methods to solve this prob-\nlem [15, 16]. Both use a microwave detector for measur-\ning the ac current in Pt induced by js(ωYIG). Diﬀerent\nfrom the two methods [15, 16], our idea is inspired by\nthe pioneering work of the forgotten masters Silsbee et\nal., who performed 35 years ago the ﬁrst spin pumping\nexperiment via the enhanced spin resonance [17]. And\nwe utilize the spin rectiﬁcation eﬀect in Py which we\nhave systematically studied [18–21]. At the Py FMR fre-\nquencyωPy, the precessing magnetization leads to the\nspin rectiﬁcation which induces a dc voltage VSRpropor-2\n(a)\n(b)\n(c)\nFIG. 1: (Colour online) (a) At φ=90◦, the dc spin current\npumped by YIG FMR can be detected in Py via the ISHE\ninduced dc voltage VSP. (b) At φ=0◦, the Py FMR can be\ndetected by VSRvia the spin rectiﬁcation eﬀect. (c) At the\nequal-resonance condition, the ac spin current pumped from\nthe YIG enhances the FMR of Py, which can be detected by\nthe increased VSR. Correspondingly, enhanced YIG FMR can\nbe detected via the increased VSP.\ntional to the precession angle, as shown in Fig. 1(b). At\nthe equal-resonance condition where ωYIG=ωPy[shown\nin Fig. 1(c)] the ac spin current precessing at ωYIGmay\nenhance the FMR of Py via dynamic exchange interac-\ntion, in a process similar to the enhanced electron spin\nresonances discovered by Silsbee et al.[17]. Thus, mea-\nsuring the enhanced VSRin Py may permit direct elec-\ntrical detection of the ac spin current without the use of\nany microwave detectors.\nSuch a method needs two prerequisites: (i) a clear pro-\ncedure for distinguishing VSPfromVSR, and (ii) a prac-\ntical way for setting the equal-resonance condition where\nωYIG=ωPyatthesamemagneticﬁeld H, orequivalently,\nsettingthe FMR resonanceﬁeld HYIG=HPyatthe same\nmicrowave frequency ω. The required procedure has re-\ncently been established [14] so that we may use the fol-\nlowing angular condition and symmetries to clearly sep-\narate and identify the dc voltages induced by pure spinpumping ( VSP) and pure spin rectiﬁcation ( VSR):\nAt φ= 90◦,VSP(θ,H) =−VSP(θ,−H) =−VSP(−θ,H);\nAt φ= 0◦,VSR(θ,H) =VSR(θ,−H) =−VSR(−θ,H).\n(1)\nTheequal-resonancecondition, aswedemonstratebelow,\ncan be set by adjusting the Hﬁeld direction, making use\nof the diﬀerent magnetic anisotropies of Py and YIG.\nSamples were prepared by magnetron sputtering and\npatternedusingaphoto-lithographyandliftoﬀtechnique.\nA 10-nm thick Py thin ﬁlm was deposited on a YIG sub-\nstrate (10 mm ×4 mm in area) and patterned into Hall\nbar structure with lateral dimensions of 5 mm ×0.2 mm.\nA 100-mW microwave was applied to excite FMR in the\nbilayer through a rectangular waveguide. By sweeping\ntheHﬁeldataﬁxedmicrowavefrequency, dcvoltagesin-\nduced by FMR were detected along the xaxis of the Hall\nbar using a lock-in ampliﬁcation. Here, the microwave\npower was modulated at a frequency of 8.33 kHz.\nFigure 2 shows typical voltage signals measured at\nω/2π= 11 GHz. While sweeping the Hﬁeld applied\natφ= 90◦, we observe a background signal of ±0.3µV\nand sharp resonances at µ0HR=±0.484 T with a line\nwidth of 10.0 mT as shown in Fig. 2(a). At the lower\n(inner) ﬁeld side ofthe sharp resonance, there is a weaker\nresonance together with a series of resonances too weak\nto be accuratelydistinguished. Both the backgroundand\nresonance signals have an odd symmetry with respect to\ntheHﬁeld direction, i.e,V(H) =−V(−H). The data\nplotted in Fig. 2(a) was taken at θ= 25◦, but data with\nan odd symmetry was measured at other angles of θ(not\nshown), provided φ= 90◦. In contrast, by setting φ=\n0◦, both the background and the two sharp resonances\nnearlydisappear, asshownin Fig. 2(b). Instead, broader\nresonances at ±1.137 T with a line width of 17.5 mT are\nobserved, which have an asymmetric line shape but even\nﬁeld symmetry of V(H) =V(−H). Again, as long as φ\n= 0◦, the broad resonances with even ﬁeld symmetry are\nobserved at arbitrary angle of θ, but note that they do\nnot appear in the spectrum measured at φ= 90◦.\nSimilarbackgroundvoltage Vbghasbeenfoundinother\nbilayer devices such as Pt/YIG under microwave excita-\ntion [22]. In general, for devices with a thin metallic\nlayer deposited on a thick substrate, microwave heating\nis known to cause a temperature gradient perpendicular\nto the interface [23]. Hence, a simple interpretation is\nthat such a vertical temperature gradient may drive a dc\nspin current via the spin Seebeck eﬀect [7], which may\nbe detected via the ISHE as Vbg. Indeed, we ﬁnd that\nVbg∝sin(φ) as expected from the spin Seebeck eﬀect.\nHowever, we note that such an angular dependence can\nnot irrefutably rule out the possibility that Vbgis caused\nby the anomalous Nernst eﬀect [5] which leads to the\nsame relation of Vbg∝sin(φ). Hence, we leave the in-\ntriguing origin of Vbgto a future study, and focus in this3\n-1.00.01.0VSR (µV)\n-1.4 -0.7 0.0 0.7 1.4\nµ0H (T)-0.50.00.5VSP (µV)\nθ = 25o   φ =  90o \nθ =  2o  φ = 0oω/2π =   11 GHz (a)\n(b)\n12\n8\n4ωr/2π  (GHz)\n1.5 1.2 0.9 0.6 0.3 0.0\nµ0HR (T) PyFMR\n YIGFMRθ = 90o φ = 45o\nθ = 1o φ = 45o(c)\nFIG. 2: (Colour online) (a) The YIG and (b) the Py FMR\nelectrically detected via VSPatφ= 90◦andVSRatφ= 0◦,\nrespectively. (c) ωr−HRdispersions ofthePyandYIGFMRs\nmeasured at in-plane ( θ= 90◦) and out-of-plane ( θ≈0◦) ﬁeld\nconﬁgurations. Curves are calculated theoretically.\npaper on the detection of spin currents via FMR, which\ncan be conclusively veriﬁed.\nWhenφ∝negationslash=n×π/2 where nis an integer, we ﬁnd\nthat both the sharp and broad resonances appear in the\nsame voltage trace. Although their relative strength de-\npends on φ, as we have discussed, neither of their res-\nonance ﬁelds is sensitive to this angle; both depend on\nthe polar angle, θ. Setting φ= 45◦, the dispersions for\nboth resonances were measured at θ= 1◦and 90◦, cor-\nresponding to perpendicular and in-plane Hﬁeld direc-\ntions, respectively. They are plotted in Fig. 2(c) for\ncomparison. To identify these resonances, we have cal-\nculated the FMR conditions for the Py/YIG bilayer by\nlinearizing the Landau-Lifshitz-Gilbert equations about\nthe equilibrium determined by the Hﬁeld strength and\ndirection. Because of the macroscopic lateral size of the\ndevice, wemakethe simplestapproximationtomodel the\nmagnetic anisotropy by using a perpendicular demagne-\ntization ﬁeld µ0Mdas the ﬁtting parameter. From the\nbest ﬁts we ﬁnd µ0Md= 0.147 and 0.910 T for YIG and\nPy, respectively. The gyromagnetic factor is found to be\nγ= 27.0 and 26.2 GHz/T for YIG and Py, respectively.\nNote that the thin Py ﬁlm has a much larger perpendic-\nular anisotropy than YIG, as expected.\nThe calculated dispersions are plotted in Fig. 2(c) assolid curves. The good agreement allows us to identify\nthe sharp and broad resonances in Fig. 2(a) and (b) as\nthe FMR of YIG and Py, respectively. Their diﬀerent\nline widths are consistent with the fact that the damping\nconstant of YIG is much smaller than that of Py. To\nkeep the focus, our simple model includes neither the\nexchange coupling nor the high-order anisotropy of YIG,\nhence it does not explicitly explain the origin of the weak\nresonance in Fig. 2(a), which could be the spin wave\nobserved previously [2]. Following Eq. 1, at φ= 90◦, the\nmeasured ﬁeld symmetry of V(HYIG)≃ −V(−HYIG) as\nshown in Fig. 2(a) allows us to identify the dc voltage of\nthe YIG FMR as VSP[14]. Hence, the dc spin current js\ninjected from the YIG into the Pyis electricallydetected.\nSimilarly, at φ= 0◦, the measured ﬁeld symmetry of\nV(HPy)≃V(−HPy) as shown in Fig. 2(b) conﬁrms\nthat the Py FMR is electrically detected via pure spin\nrectiﬁcation [14, 18], which we now use to detect the ac\nspin current js(ωYIG).\nAs shown in Fig. 2(c), at the same microwave fre-\nquency, the Py FMR measured in the in-plane conﬁgu-\nration with θ= 90◦appears on the low ﬁeld side of the\nYIG FMR. Due to the larger perpendicular anisotropy\nof Py, in the perpendicular conﬁguration with θ= 1◦,\nthe Py FMR moves to the high ﬁeld side. Hence, the\nequal-resonance condition of Py and YIG can be set by\ntuning the polar angle θ. With the obtained parameters\nwe have calculated and found that it occurs at θ= 12◦.\nWe thus proceeded to study the ac spin current en-\nhanced FMR signal near θ= 12◦. Following Eq. 1 by\nsettingφ= 0◦, we can trace the electrically detected\nFMR of Py when θis tuned through 12◦, as shown in\nFig. 3(a). The shaded areas are the approximate calcu-\nlated FMR ﬁelds of Py. When θis tuned from 9◦to 12◦,\nthe peak-to-peak amplitude of the FMR signal is seen\nto increase by more than a factor of 4 (from below 0.5\nµV to above 2 µV). When θis further tuned from 12◦to\n19◦, the FMR signal amplitude drops back below 0.5 µV.\nNote that the detailed line shape of the FMR signal de-\npends sensitively on the external ﬁeld direction [21], but\natφ= 0◦the amplitude of the FMR signal, electrically\ndetected via spin rectiﬁcation, provides a good measure\nof the cone angle of the magnetization precession [18].\nIn order to rule out the possibility that the dramati-\ncally enhanced Py FMR signal is just due to the static\ninterlayer exchange coupling [24], we monitor the θde-\npendence ofthe YIG FMR signaldetected by spin pump-\ning atφ= 90◦. For the static coupling of Py and YIG\nmagnetizations, one would only observe an anti-crossing\nof their FMRs, with the enhancement of one mode ac-\ncompanied by the suppression of the other [24]. In con-\ntrast, as shown in Fig. 3(b), the FMR signal of YIG is\nalso found to be enhanced dramatically when θis tuned\nthrough 12◦.\nSuchasimultaneousenhancementofbothFMRsignals\nis more clearly seen from the systematic data measured4V\n0.6 0.4 0.2\nµ0H (T)0.2 µVθ  = 9o\nθ  = 12o\nθ  = 19oφ = 90oV\n0.6 0.4 0.2\nµ0H (T)2.0 µVθ = 9o\nθ = 12o\nθ = 19oφ = 0o(a) (b)\nFIG. 3: (Colour online) At θ= 12◦, both amplitudes of (a)\nthe Py FMR measured by VSR(φ= 0◦) and (b) the YIG\nFMR measured by VSP(φ= 90◦) are greatly enhanced. In\nboth (a) and (b) ω/2π= 7GHz.\natω/2π= 7 GHz. As shown in Fig. 4(a), going from\nthe perpendicular down to the in-plane conﬁguration by\nincreasing θ, the FMR ﬁeld of Py deceases much faster\nthan that of YIG due to their diﬀerent perpendicular\nanisotropies. It crosses ﬁrst at θ= 12◦with the YIG\nFMR (as calculated), then it crosses at about 14◦with\nthe weak resonance mode YIG WR. Fig. 4(b) shows the\namplitude of Py FMR signal measured at φ= 0◦via\nspin rectiﬁcation, which is normalized by the maximum\namplitude of 2.67 µV atθ= 12◦. For comparison, the\namplitude of the YIG FMR measured at φ= 90◦via\nspin pumping is plotted in Fig. 4(c), which is normal-\nized by the maximum amplitude of 0.35 µV, also at θ=\n12◦. Clearly, at the equal-resonance condition, the am-\nplitudes of both the Py and YIG FMR voltages increase\ndramatically and simultaneously.\nIt is intriguing to compare the simultaneously en-\nhanced FMRs electrically detected in Py/YIG bilayer\nwith the simultaneously narrowing of the FMRs mea-\nsured by absorption spectroscopy on Fe/Au/Fe layers\n[25]. The absorption experiment performed by Heinrich\net al.is enlightening since it reveals the exact cancel-\nlation of the spin currents ﬂowing in opposite directions\nat equal-resonancecondition, which reduces the damping\nof spin pumping. In our experiment, the dc voltage de-\ntected via the spin rectiﬁcation eﬀect measures the cone\nangle of Py FMR. At the equal-resonance condition, the\nac spin current pumped by YIG FMR injects into Py,\nwhich reduces the damping and therefore enhances the\ncone angle of the Py FMR. In the phenomenological the-\norydevelopedbySilsbee et al.[17], suchan enhancement0.6\n0.4\n0.2µ0HR (T)\n20 16 12 8\nθ  (degree) PyFMR\n YIGFMR\n YIGWRω/2π =   7 GHz\n1.0\n0.5\n0.0\n20 16 12 8\nθ  (degree) PyFMR\n1.0\n0.5\n0.0\n20 16 12 8\nθ  (degree) YIGFMR\n YIGWR(a)\n(b)\n(c) Normalized Amplitude\nFIG. 4: (Colour online) (a) The polar angular dependence\nof the resonance ﬁelds measured at ω/2π= 7GHz, showing\nthe Py FMR crosses the YIG resonances at θ= 12◦and 14◦.\nThe normalized amplitudes of (b) the Py FMR and (c) the\nYIG resonances showing the simultaneous enhancement at\nequal-resonance conditions. Solid curves in (a) are calcul ated\ntheoretically, dashed curves in (b) and (c) are guide to eyes .\nof spin resonance is caused by the dynamic exchange in-\nteraction between the ac spin current and the spin an-\ngular momentum. Either of these two pictures allow us\nto conclude that, by using the spin rectiﬁcation of Py,\nthe ac spin current of YIG can be electrically detected\nat the equal-resonance condition, as demonstrated in our\nexperiment.\nIn summary, we have demonstrated new methods for\nthe electrical detection ofdc and ac spin currents in YIG.\nBothareachievedby usingPyas the spin detector. Since\nthe magnetization in Py is very easy to control by either\ntuning anexternalmagneticﬁeld orbytailoringitsshape\nanisotropy, we expect that our straightforward methods\npermit the advancement of insulator spintronics in a dis-\ntinct new path, setting it free from relying on Pt as the\nspin detector, in which the pivotal spin Hall eﬀect is still\ncontroversial and is very diﬃcult to tune.\nWork in Manitoba has been funded by NSERC, CFI,\nURGP, and UMGF grants (C.-M.H. and B.W.S.). Work\nat JHU has been funded by NSF (DMR-1262253).\nD.M.J.K was supported by Mitacs Globalink Program.\nS.Y.H was partially supported by STARnet sponsored\nby MARCO and DARPA.5\n∗Electronic address: hu@physics.umanitoba.ca\n[1] M. Johnson, R. H. Silsbee, Phys. Rev. Lett. 55, 1790\n(1985).\n[2] Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida,\nM. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K.\nTakanashi, S. Maekawa, and E. Saitoh, Nature 464, 262\n(2010).\n[3] Ken-ichi Uchida, Hiroto Adachi, Takeru Ota, Hiroyasu\nNakayama, Sadamichi Maekawa, and Eiji Saitoh, Appl.\nPhys. Lett. 97, 172505 (2010).\n[4] S.Y. Huang, W. G. Wang, S. F. Lee, J. Kwo, and C. L.\nChien, Phys. Rev. Lett. 107, 216604 (2011).\n[5] S.Y. Huang, X. Fan, D. Qu, Y. P. Chen, W. G. Wang, J.\nWu, T.Y. Chen, J. Q. Xiao, and C. L. Chien, Phys. Rev.\nLett.109, 107204 (2012).\n[6] D. Qu, S.Y. Huang, Jun Hu, Ruqian Wu, and C. L.\nChien, Phys. Rev. Lett. 110, 067206 (2013).\n[7] T. Kikkawa, K. Uchida, Y. Shiomi, Z. Qiu, D. Hou, D.\nTian, H. Nakayama, X.-F. Jin, and E. Saitoh, Phys. Rev.\nLett.110, 067207 (2013).\n[8] C.-M. Hu, J. Nitta, A. Jensen, J. B. Hansen, and H.\nTakayanagi, Phys. Rev. B 63, 125333 (2001).\n[9] Xiaohua Lou, et al., Nature Physics 3, 197 (2007).\n[10] F. J. Jedema, A. T. Filip, and B. J. van Wees, Nature.\n410, 345, (2001).\n[11] F. J. Jedema, H. B. Heersche, A. T. Filip, J. J. A. Basel-\nmans and B. J. van Wees, Nature 416, 713, (2002).\n[12] H.J. Jiao and G. E. W. Bauer, Phys. Rev. Lett. 110,\n217602 (2013).\n[13] B. F. Miao, S. Y. Huang, D. Qu, and C. L. Chien, Phys.Rev. Lett. 111, 066602, (2013).\n[14] Lihui Bai, P. Hyde, Y. S. Gui, C.-M. Hu, V. Vlam-\ninck, J. E. Pearson, S. D. Bader, and A. Hoﬀmann,\narXiv:1308.4041.\n[15] Dahai Wei, Martin Obstbaum, Christian Back, and\nGeorg Woltersdorf, arXiv:1307.2961.\n[16] C. Hahn, G. de Loubens, M. Viret, and O. Klein,\narXiv:1308.3433.\n[17] R. H. Silsbee, A. Janossy, and P. Monod, Phys. Rev. B.\n19, 4382, (1979).\n[18] Y. S. Gui, N. Mecking, X. Zhou, Gwyn Williams, and\nC.-M. Hu, Phys. Rev. Lett. 98, 107602 (2007).\n[19] N. Mecking, Y. S. Gui, and C.-M. Hu, Phys. Rev. B 76,\n224430 (2007).\n[20] A. Wirthmann, X. Fan, Y. S. Gui, K. Martens, G.\nWilliams, J. Dietrich, G.E. Bridges, andC.-M. Hu, Phys.\nRev. Lett. 105, 017202 (2010).\n[21] M. Harder, Z. X. Cao, Y. S. Gui, X. L. Fan, and C.-M.\nHu, Phys. Rev. B 84, 054423 (2011).\n[22] B. Hillebrands, S. Goennenwein, and J. Q. Xiao, private\ncommunication at Gordon Research Conference, Hong\nKong, August, 2013.\n[23] Z. H. Zhang, Y. S. Gui, L. Fu, X. L. Fan, J. W. Cao,\nD. S. Xue, P. P. Freitas, D. Houssameddine, S. Hemour,\nK. Wu, and C.-M. Hu, Phys. Rev. Lett., 109, 037206\n(2012).\n[24] B. Heinrich, Z. Celinski, J. F. Cochran, W. B. Muir, J.\nRudd, Q. M. Zhong, A. S. Arrott, K. Myrtle, and J.\nKirschner, Phys. Rev. Lett. 64, 673, (1990).\n[25] B. Heinrich, Y.Tserkovnyak, G. Woltersdorf, A.Brataa s,\nR. Urban, and G. E. W. Bauer, Phys. Rev. Lett. 90,\n187601 (2003)."    },    {        "title": "0807.4445v1.Magnetoelectric_bistabilities_in_ferromagnetic_resonant_tunneling_structures.pdf",        "content": "arXiv:0807.4445v1  [cond-mat.other]  28 Jul 2008Magnetoelectric bistabilities in ferromagnetic resonant tunneling structures\nChristian Ertler∗\nInstitute for Theoretical Physics, University of Regensbu rg,\nUniversit¨ atsstrasse 31, D-93040 Regensburg, Germany\nThe conditions for the occurrence of pronounced magnetoele ctric bistabilities in the resonant\ntunneling through a ferromagnetic quantum well are theoret ically investigated. The bistability\nappears due to the mutual feedback of the carriers Coulomb in teraction and the carriers exchange\ncoupling with magnetic impurities in the well. It is shown th at the well Curie temperature depends\nstrongly on the relative alignment of the quantum well level and the reservoirs chemical potentials,\nwhich can be modiﬁed electrically. Switching between a ”cur rent-on/magnetism-oﬀ“ and a ”current-\noﬀ/magnetism-on“modebecomes possible, ifthewell temper atureliesin-betweenthebistablevalues\nof the well Curie temperature.\nInultimatemagnetoelectricdevicesthemagneticprop-\nerties should be ideally controllable to a vast extent by\nexternal bias or gate ﬁelds. For this purpose, band-\nengineered magnetic resonant tunneling structures are\nvery promising, since they exhibit a rich variety of tun-\nable magneto-transport properties [1]. Especially, the\nimpetuous development of novel dilute magnetic semi-\nconductors (DMSs) [2, 3, 4] in the last decades, which\nare made magnetic by randomly doping with transition\nmetal elements, e.g., by incorporating Mn in a GaAs\ncrystal host, has considerably enriched the possibilities\nofgrowingdiﬀerent magnetic semiconductor heterostruc-\nture systems. In DMSs the ferromagnetism can depend\nstrongly on the actual particle density, which has been\nconﬁrmed in several experiments, in which ferromag-\nnetism has been generated by tailoring the particle den-\nsity by electrical or optical means [5, 6].\nIn magnetic resonant tunneling structures made of\npara- or ferromagnetic DMSs even small energetic spin\nsplittings of the well subbands can become observable\nin the transport characteristics [7, 8, 9]. Based on their\nspin-dependent transmission magnetic resonant tunnel-\ning structures have been proposed for realizing eﬃcient\nspin valves and spin ﬁltering devices [1, 10], or for dig-\nital magnetoresistance [11, 12]. The magnetic proper-\nties of ferromagnetic quantum wells made of DMSs are\nwell described in the framework of a mean ﬁeld model\n[13, 14, 15, 16], which reveals that the Curie tempera-\nture of the well depends on (i) the 2D-spin susceptibility\nof the carriers and (ii) on the overlap of the subband\nwave function with the magnetic impurity density pro-\nﬁle. Both parameters should be in principle tuneable by\nthe applied bias, which would provide a purely electri-\ncalcontrol of the ferromagnetism in magnetic quantum\nwells.\nInconventionalnonmagneticresonanttunnelingdiodes\n(RTDs) it is well known, that an intrinsic hysteresis in\nthe negative-diﬀerential-resistance (NDR) region of the\ncurrent-voltage (IV) characteristics can occur [17]. This\n∗email:christian.ertler@physik.uni-regensburg.deemitter \nde dc\nw\nRewell\nCe Cc UwVappl(a) \n(b) µeµc\nRccollector well\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. (b) Equivalent circuit mode l\nof the resonant tunneling structure introducing the emitte r\nand collector capacitances Ce,Ccand resistances Re,Rc, re-\nspectively.\nbistability of the tunneling current has been explained\nto result from the nonlinear feedback of Coulomb inter-\naction of the stored well charge [18, 19]. In magnetic\nRTDs this naturally suggests the possibility of hysteretic\nmagnetic states, as has been predicted in [20, 21].\nIn this article a detailed study of possible magneto-\nelectric bistabilities in magnetic RTDs is provided. The\ncarriersdynamicsisdescribedbyaself-consistentsequen-\ntial tunneling model, which includes the feedback eﬀects\nof both the carriers Coulomb interaction and the mag-\nnetic exchange coupling with the magnetic ions. The\nmodel yields a simple expression for the steady state 2D-\nspin susceptibility, which allows to calculate the critical\ntemperature Tcof the quantum well depending on the\napplied bias and the relative alignment of the quantum\nwell level with respect to the chemical potentials of the\nemitter and collector reservoirs. If the well is operated\nat a temperature, which lies between the bistable values\nof the well Curie temperature, the magnetic RTD can\nbe switched between a ”current-on/magnetism-oﬀ“ and\na ”current-oﬀ/magnetism-on“ mode.2\nThe band proﬁle of a generic double-barrier resonant\ntunneling structure with a ferromagnetic quantum well\nmadeofaDMS, e.g., ofGaMnAs, issketchedin Fig.1(a).\nThe vertical transport through the structure can be de-\nscribed by a sequential tunneling model, since the high\ndensityofmagneticimpuritiesin the wellwilllikelycause\ndecoherence processes. By using the transfer Hamilto-\nnian formalism a Pauli master equation for the statisti-\ncal distribution of the particles in the well can be derived\n[1, 22]. In the case that only a single resonant level Ew\nresides in the energy window of interest, which is deﬁned\nby the diﬀerence of the emitter’s and collector’s chemi-\ncal potentials, simple rate equations for the spin-resolved\nwell particle densities Nσ(t) with (σ=↑,↓) are obtained\ndNσ\ndt= Γe(Eσ)Ne,σ+Γc(Eσ)Nc,σ−Γ(Eσ)Nσ.(1)\nHere,Nσ,{e,c}are the densities of particles with the res-\nonant longitudinal energy Eσin the emitter (e) and col-\nlector (c) reservoir, respectively. The energy-dependent\ntunneling rates Γ {e,c},Γ = Γ e+ Γccan be calculated\nby Bardeen’s formula [23], which essentially evaluates\nthe overlap of the lead and well wave functions in the\nbarriers. For high barriers these tunneling rates be-\ncome proportional to the longitudinal momentum pzof\nthe incident particles [22], i.e., Γ e,c∝(Ez)1/2withEz\ndenoting the longitudinal energy. By assuming that\nthe particle reservoirs are described by Fermi-Dirac dis-\ntributions the particle densities are given by Ni,σ=\nD0kBTln{1+exp[(µi−Eσ)/kBT]},i= (e,c),with\nD0=m/2π/planckover2pi12is the two-dimensional density of states\nper spin for carriers with the eﬀective mass m,kBde-\nnotes Boltzmanns’ constant, Tis the lead temperature,\nandµiare the emitter and collector chemical potentials\nwithµc=µe−eVapplwhereVapplis the applied bias.\nIn the framework of a mean ﬁeld model an analytic\nexpression for the steady state exchange splitting ∆ of\nthe well level can be derived [1, 13, 15, 16]\n∆ =Jpd/integraldisplay\ndznimp(z)|ψ0(z)|2\n×SBS/bracketleftBigg\nSJpds(N↓−N↑)|ψ0(z)|2\nkBT/bracketrightBigg\n,(2)\nwhereJpdis the coupling strength between the impurity\nspin and the carrier spin density (in case of GaMnAs p-\nlike holes couple to the d-like impurity electrons), zis the\nlongitudinal (growth) direction of the structure, nimp(z)\nis the impurity density proﬁle, ψ0(z) labels the well wave\nfunction, and s= 1/2 is the particles spin. The Bril-\nlouin function of order Sis denoted by BS, whereSis\nthe impurity spin, which for Mn equals 5/2. By consider-\ning a homogenous impurity distribution ∆ is eﬀectively\ndetermined by the voltage dependent spin polarization\nξ=s(N↑−N↓).\nThe nonlinear feedback of the Coulomb interaction of\nthe well charges is approximately taken into account by\ncalculating the electrostatic well potential in terms of\nEw(meV)\nVoltage (mV)  \n020406080100120−50−250255075\n051015\nAB\nCD\nA\nµeµc\nΕ0e Ε0c B\nµe\nµc Ε0e \nΕ0c\nD\nµe\nµcΕ0e \nΕ0cC\nµe\nµcΕ0e \nΕ0c (a)\n(b)\nFIG. 2: (Color online) (a) Contour plot of the well Curie\ntemperature Tc(K) as a function of the well level position\nEwand the applied bias. (b) Schematic illustration of the\ndiﬀerent occupation probabilities of the quantum well leve l\nfrom the reservoirs for the regions A-D, as indicated in the\ncontour plot (a). The emitter and collector band edges are\ndenoted by E0eandE0c, respectively.\nan equivalent circuit model of the resonant tunneling\ndiode, as shown in Fig. 1(b), where the capacitances Ce\nandCcare determined by the geometrical dimensions\nof the barriers and the well [22]. The potential results in\nUw= [e2(N−Nback)−CceVappl]/C, whereN=N↑+N↓,\nC=Ce+Cc,edenotes the elementary charge, and Nback\nis the positive backgroundchargein the well, which origi-\nnatesfromthemagneticdonors. Sincetheactualposition\nof the quantum well levels Eσ=E0+Uw−σ∆ depends\non both the magnetic exchange splitting ∆ and the elec-\ntrostatic potential Uwall equations become nonlinearly\ncoupled, making a selfconsistent numerical solution nec-\nessary.\nIn order to ﬁnd criterions for the occurrence of mag-\nnetic bistabilities and to interpret the numerical results\nin the following it is very useful to study the dependence\nof the well Curie temperature Tcon the applied bias and\nthe well level position. The mean ﬁeld model yields an\nanalytic expression for the collective Curie temperature\nof a magnetic quantum well\nkBTc=S(S+1)\n3J2\npdχ2D/integraldisplay\ndzni(z)|ψ(z)|4,(3)\nwhere the two-dimensional spin susceptibility is deﬁned3Ew(meV)\nVoltage (mV)  \n04080120−50−250255075\n051015\nAB\nCD\n02040608010001234\nVoltage (mV)j (a.u.)\n  \nF\nR\n0204060801000246810\nVoltage (mV)Tc (K)\n  \nF\nR\nTwell\n02040608010005101520\nVoltage(mV)Δ (meV)\n  \nF\nR(a) (b)\n(c)(d)\nFIG. 3: (Color online) The (a) quantum well level position\nEw, (b) current j, (c) Curie temperature Tc, and (d) well\nsplitting ∆ as a function of the applied bias. The solid lines\nindicate the voltage up-sweep values (F), whereas the dashe d\nlines correspond to the voltage down-sweep values (R). In (a )\ntheEw-voltage curvesare embeddedin the contour plot of the\nwell Curie temperature Tc(K) and in (d) the solid red line\ncorresponds to the actual well temperature Twell= 4.2 K.\nby\nχ2D= lim\n∆→0s(N↑−N↓)\nE↓−E↑. (4)\nWithin the introduced sequential tunneling model\nEq. (1) the steady state spin susceptibility simpliﬁes to\nχ2D(E) =−s(∂N0/∂E), whereN0= (ΓeNe+ΓcNc)/Γ is\nthesteadystatesolutionoftherateequations(1). Hence,\nthe dimensionless susceptibility can be written as\n˜χ=χ2D(E)\nsD0=/summationdisplay\ni=e,cΓi\nΓfi\nFD−Ni\nD0∂\n∂E/parenleftbiggΓi\nΓ/parenrightbigg\n(5)\nwithfi\nFD,i= (e,c) denoting the Fermi-Dirac function\nfor the emitter and collector reservoir, respectively. This\nallows to calculate the Curie temperature Tcas a func-\ntion of the applied voltage (note that Tcdepends via fc\nFD\nexplicitly on the voltage, since µc=µe−eVappl) and\nthe well level position, as displayed in Fig. 2(a). For the\nsimulations I used generic parameters corresponding to\na GaMnAs well: m= 0.5m0,εr= 12.9,de=dc= 20\n˚A,w= 10˚A,µe= 70 meV, nimp= 1.5×1020cm−3,\nJpd= 0.06 eV nm3, wherede,dcandware the emit-\nter barrier, collector barrier and quantum well widths,\nm0denotes the free electron mass, and εris the rel-\native permittivity of the well. The background charge\nnback= 0.1nimpis considered to be only of about 10%\nof the nominal Mn doping density [24] and the lattice\ntemperature is set to T= 4.2 K.\nThe Curie temperature contour plot can be divided\ninto four qualitatively diﬀerent regions A-D, which arecharacterized by diﬀerent probabilities for occupying the\nquantum well level from the reservoirs, as schematically\nillustrated in Fig. 2(b). In regionA, for instance, the well\nlevel can be occupied by particles originating from both\nreservoirs. The Curie temperatures T(A,C)\ncin regions A\nand C diﬀer roughly by a factor 2 compared to TB\ncof\nregion B. This sudden change of Tccan be explained\nas follows: by assuming energy-independent tunneling\nrates and nearly symmetric barriers, i.e., Γ e≈Γcthe\ndimensionless spin susceptibility of Eq. (5) simpliﬁes for\nregion A to ˜ χA= 1/2(fe\nFD+f2\nFD)≈1, whereas in the\nother regions one obtains: ˜ χB=fe\nFD/2≈1/2,˜χC=\nfc\nFD≈1, and ˜χD= 0. This simple estimation, hence,\nyields the desired result T(A,C)\nc/TB\nc≈2.\nThese diﬀerences in the Curie-temperatures of the var-\nious regions can now be exploited to realize hysteretic\nmagnetoelectric states. According to the nonlinear feed-\nback of the stored well charge, the resonant level Ewand\nthe IV-characteristic show a hysteretic behavior, as dis-\nplayed in Fig. 3(a) and (b). For the up-sweep (F) of\nthe applied bias the well is charged before the Ewbe-\ncomes oﬀ-resonant, i.e., when it drops below the emit-\nter band edge, whereas for the voltage down-sweep (R)\nthe well is almost uncharged before Ewbecomes reso-\nnant again. This leads to diﬀerent self-consistent elec-\ntrostatic potentials for up- and down-sweeping voltages,\nexplaining the occurrence of the intrinsic bistability. If\nthe hysteresis of Ewnow switches exactly between the\nTc-regions B and C, as it is the case in Fig. 3(a), then\nalso the voltage-dependent Curie temperature will ex-\nhibit a pronounced hysteresis, as shown in Fig. 3(c). The\nelectric hysteresis will then be accompanied by a mag-\nnetic hysteresis if the actual lattice temperature Tof the\nquantum well, which is displayed as straight solid line\nin Fig. 3(c), fulﬁlls the condition TB\nc< T < TC\nc. This\nis illustrated in Fig. 3(d): as long as the resonant level\nstays in region B the well is nonmagnetic (∆ = 0, since\nT > TB\nc) but when Ewenters the region C the well be-\ncomes immediately magnetic (∆ ∝ne}ationslash= 0). Also notice, that\nat low voltages the well is always magnetic. At roughly\nVappl≈30 mV the well becomes nonmagnetic, since\nEwcrosses the boundary between the regions A and B,\nwhichprovidesapurelyelectricalcontrolofthe wellmag-\nnetism. As a whole, the magnetic well switches between\na ”current-on/magnetism-oﬀ“state for the up-sweep and\na ”current-oﬀ/magnetism-on“-state for the down-sweep\nof the applied voltage. Moreover, the Tccontour plot\nin Fig. 2(a) also suggest the possibility for realizing the\nswitching between a ”current-on/magnetism-on“ and a\n”current-oﬀ/magnetism-oﬀ” mode in the case that the\nhysteresis of Ewswitches between region B and D and if\n0<T <TB\nc.\nIn summary, I have shown by using a selfconsistent se-\nquential tunneling model that the Curie temperature of\na magnetic quantum wells strongly depends on the rela-\ntive alignment of the well level and the reservoirs chemi-\ncal potentials, which can be modiﬁed by external bias or\ngate ﬁelds. Magnetoelectric bistabilities become possible4\nif the hysteresis of the well level position Ewswitches\nbetween regions of diﬀerent well Curie temperatures and\nif the actual well temperature lies in-between these two\nvalues.This work has been supported by the DFG SFB 689.\nThe author thanks J. Fabian for very valuable and in-\nspiring discussions.\n[1] J. Fabian, A. Matos-Abiague, C. Ertler, P. Stano, and\nˇZuti´ c, Acta Phys. Slov. 57, 565 (2007).\n[2] H. Ohno, Science281, 951 (1998).\n[3] T. Dietl, in Modern Aspects of Spin Physics , edited by\nW. P¨ otz, J. Fabian, and U. Hohenester (Springer, Berlin,\n2007), pp. 1–46.\n[4] T. Jungwirth, J. Sinova, J. Maˇ sek, J. Kuˇ cera, and A. H.\nMacDonald, Rev. Mod. Phys. 78, 809 (2006).\n[5] H.Ohno,D.Chiba, F.Matsukura, T.O.E.Abe, T.Dietl,\nY. Ohno, and K. Ohtani, Nature408, 944 (2000).\n[6] H.Boukari, P. Kossacki, M. Bertolini, D. Ferrand, J. Cib -\nert, S. Tatarenko, A. Wasiela, J. A. Gaj, and T. Dietl,\nPhys. Rev. Lett. 88, 207204 (2002).\n[7] A. Slobodskyy, C. Gould, T. Slobodskyy, C. R. Becker,\nG. Schmidt, and L. W. Molenkamp, Phys. Rev. Lett. 90,\n246601 (2003).\n[8] S. Ohya, P. N. Hai, Y. Mizuno, and M. Tanaka, Phys.\nRev. B75, 155328 (2007).\n[9] A. Oiwa, R. Moriya, Y. Kashimura, and H. Munekata, J.\nMagn. Magn. Mater. 272-276 , 2016 (2004).\n[10] A. G. Petukhov, A. N. Chantis, and D. O. Demchenko,\nPhys. Rev. Lett. 89, 107205 (2002).\n[11] C. Ertler and J. Fabian, Phys. Rev. B 75, 195323 (2007).\n[12] C. Ertler and J. Fabian, Appl. Phys. Lett. 89, 242101\n(2006).[13] B. Lee, T. Jungwirth, and A. H. MacDonald, Semi. Sci.\nTechn.17, 393 (2002).\n[14] S. Ganguly, L. F. Register, S. Banerjee, and A. H. Mac-\nDonald, Phys. Rev. B 71, 245306 (2005).\n[15] T. Dietl, A. Haury, and Y. M. d’Aubign´ e, Phys. Rev. B\n55, R3347 (1997).\n[16] T. Jungwirth, W. A. Atkinson, B. H. Lee, and A. H.\nMacDonald, Phys. Rev. B 59, 9818 (1999).\n[17] V. J. Goldman, D. C. Tsui, and J. E. Cunningham, Phys.\nRev. Lett. 58, 1256 (1987).\n[18] F. W. Sheard and G. A. Toombs, Appl. Phys. Lett. 52,\n1228 (1988).\n[19] N. C. Kluksdahl, A. M. Kriman, D. K. Ferry, and\nC. Ringhofer, Phys. Rev. B 39, 7720 (1989).\n[20] D. S´ anchez, A. H. MacDonald, and G. Platero, Phys.\nRev. B65, 35301 (2001).\n[21] S. Ganguly, A. H. MacDonald, L. Register, and S. Baner-\njee, Phys. Rev. B 73, 33310 (2006).\n[22] D. V. Averin, A. N. Korotkov, and K. K. Likharev, Phys.\nRev. B44, 6199 (1991).\n[23] J. Bardeen, Phys. Rev. Lett. 6, 57 (1961).\n[24] S. Das Sarma, E. H. Hwang, and A. Kaminski, Phys.\nRev. B67, 155201 (2003)."    },    {        "title": "0902.2389v2.Charge_pumping_and_the_colored_thermal_voltage_noise_in_spin_valves.pdf",        "content": "arXiv:0902.2389v2  [cond-mat.mes-hall]  29 Apr 2009Charge pumping and the colored thermal voltage noise in spin valves\nJiang Xiao,1Gerrit E.W. Bauer,1Sadamichi Maekawa,2,3and Arne Brataas4\n1Kavli Institute of NanoScience, Delft University of Techno logy, 2628 CJ Delft, The Netherlands\n2Institute for Materials Research, Tohoku University, Send ai, Japan\n3CREST, Japan Science and Technology Agency, Tokyo 100-0075 , Japan\n4Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway\nSpin pumping by a moving magnetization gives rise to an elect ric voltage over a spin valve.\nThermal ﬂuctuations of the magnetization manifest themsel ves as increased thermal voltage noise\nwith absorption lines at the ferromagnetic resonance frequ ency and/or zero frequency. The eﬀect\ndepends on the magnetization conﬁguration and can be of the s ame order of magnitude as the\nJohnson-Nyquist thermal noise. Measuring colored voltage noise is an alternative to ferromagnetic\nresonance experiments for nano-scale ferromagnetic circu its.\nI. INTRODUCTION\nA spin valve consists of a thin non-magnetic metal-\nlic layer (NM) sandwiched by two ferromagnetic layers\n(FM) with variable magnetization direction. One of the\nFM layers is usually thick and its magnetization is ﬁxed,\nwhile the other is thin and its magnetization direction is\nfree to move. Spin valves have a wide range of interest-\ning static and dynamic properties,1,2,3,4,5,6,7,8,9,10,11,12,13\nmany of which are related to the current-induced spin-\ntransfer torque,1,2which can excite magnetization dy-\nnamics (and reversal). Inversely, magnetization dynam-\nics generates a current ﬂow or a voltage output. Berger\nﬁrst discussed the induced voltage in an FM/vextendsingle/vextendsingleNM/vextendsingle/vextendsingleFM\nstructure by magnetization dynamics.14He posited that\na voltage of order /planckover2pi1ω/ecan be generated when the mag-\nnetization of one ferromagnet precesses at frequency ω.\nSimilar dynamically induced voltages have been studied\ntheoretically15and observed16in simple FM/vextendsingle/vextendsingleNM junc-\ntions and in magnetic tunnel junctions (MTJs).17In spin\nvalves and MTJs, voltage induced by the magnetization\ndynamics can be understood as a two-step process: i) the\nmoving magnetization of the free layer generates a spin\ncurrent; ii) the static magnetization of the ﬁxed layer ﬁl-\nters the “pumped” spin current and converts it into a\ncharge current or, in an open circuit, a voltage output.\nThe electrical voltage induced by moving domain walls\ncan be explained analogously.18,19,20,21,22In the ﬁrst part\nof the present paper, we derive a simple formula for the\ncharge pumping voltage in a spin valve by circuit theory\nin which magnetization dynamics is taken into account.\nWe ﬁnd that the magnitude of the voltage is governed\nby the spin-transfer torque in the same structure. We\ntherefore propose to measure the angular dependence of\nthe spin-transfer torque (or torkance, i.e.the torque di-\nvided by the voltage bias) by the angular dependence of\nthe charge pumping voltage.\nThe charge pumping voltage consists of a DC and an\nAC component, even when induced by a steady mag-\nnetization dynamics such as ferromagnetic resonance\n(FMR). The concept can be extended to the case of ther-\nmally activated, i.e.ﬂuctuating, magnetization dynam-\nics, which is an extra source of thermal voltage noisethat only appears in magnetic structures. Johnson23\nand Nyquist24showed that in non-magnetic conductors\nthe voltage noise is associated with thermal agitation\nof charge carriers (driven by ﬂuctuating electromagnetic\nmodes). The power spectrum of this noise is white\nand proportional to the temperature Tand resistance\nR:SJN(ω) = 4kBTRup to frequencies of kBT//planckover2pi1∼104\nGHz at room temperature.23,24In magnetic structures\nsuch as spin valves thermal ﬂuctuations of the magne-\ntization direction have to be considered.25Some conse-\nquences of thermal ﬂuctuation in spin valves, such as\nnoise-facilitated magnetization switching26,27,28and re-\nsistance ﬂuctuations,29,30have been studied before. In\nthe so called thermal ferromagnetic resonance, frequen-\ncies are studied by means of resistance ﬂuctuations with-\nout applied magnetic ﬁelds.31Foroset al.30have shown\nthat the time-averaged auto-correlator of the resistance\nﬂuctuations is signiﬁcantly aﬀected by dynamical ex-\nchange coupling between the magnetic layers.\nIn the second part of this paper, we show that a mag-\nnetization ﬂuctuation-related voltage noise can be of the\nsame order of magnitude as the conventional thermal\nnoise in non-magnetic conductors. This noise is not\n“white” but displays spectral features related to the fer-\nromagnetic resonance (FMR). The noise spectrum there-\nfore contains information comparable to that obtained\nby FMR. For nano-scale ferromagnetic circuits the noise\nmeasurements might be easier to perform than conven-\ntional FMR experiments. Compared to the resistance\nnoise measurement, the pumping voltage noise measure-\nment is non-intrusive because it does not require appli-\ncation of current, which may disturb the system.\nThe paper is organizedas follows: Section II presents a\ngeneral theoretical frame work that combines the magne-\ntoelectric circuit theory and the Landau-Lifshitz-Gilbert\n(LLG) equation. In Section III, we derive a formula\nfor the charge pumping voltage in spin valves. In Sec-\ntion IV, by using the magnetic susceptibility function,\nwe calculate the voltage noise spectrum due to mag-\nnetization ﬂuctuations for two diﬀerent magnetic con-\nﬁgurations. Section V contains some general remarks\non the calculation. In Appendix A we calculate the\nangular-dependence of the magnetic susceptibility for a2\nspin valve, and Appendix B presents an alternative cal-\nculation of the magnetization-related thermal noise by\ncomputing the frequency dependent impedance of a spin\nvalve.\nII. CIRCUIT THEORY WITH DYNAMICS\nFig. 1(a) shows a spin valve structure under considera-\ntion. ThemagnetizationintheleftFMwithdirection m0\nisassumedtobestaticand m, theoneofthe rightFM, to\nbe free, which can be realized by making the right layer\nmuch thinner than the left one. For electron transport,\nwe assume for simplicity that the spin valve is symmet-\nric. Such an assumption may be invoked when both FM\nlayers are of the same material and thicker than the spin\nﬂip diﬀusion length. In that regime, the resistances of\nthe bulk ferromagnet over the spin-ﬂip diﬀusion length\nare in series with the interface resistances, whereas the\nremoter parts of the ferromagnets are magnetically inert\nseries resistances. The regime in which the layers be-\ncome thinner than the spin ﬂip diﬀusion length has been\ntreated by Kovalev et. al.32In this and the next Sec-\ntion, we focus on the spin-active region in the spin valve,\nwhich includes the NM spacer and small part (of the or-\nder of the spin-ﬂip diﬀusion length) of the FM layers as\nindicated by the dashed box in Fig. 1(a).\nWhenmdepends on time, a spin current is pumped\ninto the metallic spacer layer through the F/vextendsingle/vextendsingleN interface.\nThemagnitudeandpolarizationofthespinpumpingcur-\nrent reads:33\nIsp\ns=/planckover2pi1\n4π/parenleftbigg\ngrm×dm\ndt+gidm\ndt/parenrightbigg\n, (1)\nwheregrandgiare the real and imaginary part of the\ndimensionlesstransversespin-mixingconductance.33The\nFIG. 1: (Color online) Spinand charge currentsin spinvalve s.\n(a) For the steady state case studied in Section II and III. (b )\nFor the thermal magnetization ﬂuctuations studied in Secti on\nIV.ﬁrst term in Isp\nscorresponds to a loss of angular momen-\ntum of the free layer magnetization to the adjacent NM\nlayers, thus providing an extra damping torque.33When\nthe adjacent normal metal is an ideal spin sink, the spin\npumping current loss can be represented by a Gilbert\ndamping coeﬃcient (see below), and each interface con-\ntributestothedampingconstantby α′= (γ/planckover2pi1/4πMtot)gr,\nintroducing the gyromagnetic ratio γand the total mag-\nnetization of the right FM ﬁlm Mtot. The imaginary part\ngieﬀectivelymodiﬁesthegyromagneticratioforthemag-\nnetization under consideration.\nIn order to use magnetoelectronic circuit theory, the\nstructurehasﬁrsttobe decomposedintonodes(for bulk)\nand contacts (for interfaces). For each node we may de-\nﬁneachargechemicalpotentialanda(vector)spinchem-\nical potential: let µL,R,NandµL,R,Nbe the charge and\nspin chemical potentials in left, right FM leads, and the\nNM spacer. In the ferromagnet, we may assume that the\nspin accumulations are aligned with the magnetization,\ni.e.µL=µs\nLm0andµR=µs\nRm. The charge current\nIcand the spin current ILthrough the left interface con-\nnectingtheleft FMandthe spacerlayeraregivenby:34,35\nIc=eg\n2h[2(µL−µN)+p(µs\nL−µN·m0)],(2a)\nIL=−eg\n2h[2p(µL−µN)+(µs\nL−µN·m0)]m0\n+e\nhgrm0×µN×m0+e\nhgiµN×m0, (2b)\nwhereg=g↑+g↓is the total conductance and p=\n(g↑−g↓)/gis the polarization of the F/vextendsingle/vextendsingleN interface and\nthe (longitudinal) active regions of the FMs.32Similarly\nfor the right FM lead:\nIc=−eg\n2h[2(µR−µN)+p(µs\nR−µN·m)],(3a)\nIR=−eg\n2h[2p(µR−µN)+(µs\nR−µN·m)]m\n+e\nhgrm×µN×m+e\nhgiµN×m+2e\n/planckover2pi1Isp\ns,(3b)\nwhere the spin current IRat the right interface is modi-\nﬁed due to the additional spin pumping current Isp\nsemit-\nted by the moving magnetization m. Additionally, the\nspin current conservation in the presence of the spin ﬂips\nin the NM spacer requires:\nIL+IR=e\nhh\nDτsfµN≡e\nhgsfµN, (4)\nwhereDthe energy density of states at the Fermi energy\nandτsfthe spin ﬂip relaxation time in the NM.\nThe spin current IRentering the free layer exerts a\nspin-transfertorqueon m, whichisequaltoitstransverse\ncomponent absorbed at the interface:11,12\nNst=/planckover2pi1\n2e[IR−(IR·m)m)]. (5)3\nThe LLG equation is therefore modiﬁed as:\ndm\ndt=−γm×Heﬀ+αm×dm\ndt+γ\nMtotNst,(6)\nwhereHeﬀis the total eﬀective magnetic ﬁeld acting on\nm, andα=α0+ 2α′is the total magnetic damping\nincluding both the bulk damping and the spin pumping\nenhanced damping from both interfaces.\nThe set of equations Eqs. (1-6) describes the\ncharge/spin transport and magnetization dynamics in\nthe metallic magnetic heterostructures. In many cases,\nthe transport equations Eqs. (1-5) and the dynamical\nLLG equation Eq. (6) can be solved separately by ig-\nnoring the spin pumping contribution Isp\nsinIR, in which\ncase the transport only depends on the instantaneous m\nbut not on ˙m. However, for thin magnetic layersthe spin\npumping modiﬁcation cannot be neglected. It has pos-\nsibly important consequences, such as a voltage induced\nby magnetization dynamics, anisotropic magnetic damp-\ning and susceptibility tensor, and colored thermal noise,\nas will become clear from the discussion below.\nAt ﬁrst, let us calculate the static ( ˙m= 0) mag-\nneto conductance of a spin valve. When a bias volt-\nageV= (µL−µR)/eis applied, we can calculate the\ncharge current I=IL=IRfrom Eqs. (2-4), hence the\nmagneto-conductance G=I/V. By setting µN= 0 in\nthe spacer and assuming strong spin ﬂips in the ferro-\nmagnets ( µs\nL=µs\nR= 0), we ﬁnd (with m·m0= cosθ)\nG(θ) =G0g\n4/bracketleftbigg\n1−4pη(θ)sin2θ\n2/bracketrightbigg\n, (7)\nwhereG0= 2e2/his the conductance quantum and\nη(θ) =pg/4\ngsin2θ\n2+2˜grcos2θ\n2+gsf(8)\nis the angular dependent spin current polarization with\n˜gr≡gr+2g2\ni/(2gr+gsf). TheG(θ) above agrees with Eq.\n(160) in Ref. 36.\nIII. CHARGE PUMPING IN SPIN VALVES\nWhen a voltage diﬀerence ∆ Vis applied over a spin\nvalve which does not excite magnetization dynamics\n(˙m= 0), Eqs. (2-5) lead to the spin-transfer torque:1\nNst(θ) = ∆V[τip(θ)m×m0+τop(θ)m0]×m,(9)\nwhereτipandτopare the so-called (angular-dependent)\ntorkances37for the in-plane (Slonczewski’s) component\nand out-of-plane (eﬀective ﬁeld) component:\nτip(θ) =eη(θ)\n2π˜grandτop(θ) =eη(θ)\n2πgigsf\n2gr+gsf.(10)\nWhen the bias polarity is chosen such that the in-plane\ntorque in Eq. (9) works against the magnetic damping,the current ﬂow can excite magnetization dynamics, oth-\nerwise they are suppressed.\nInversely, magnetization dynamics can induce a cur-\nrent ﬂow by the spin-pumping: a moving magnetization\n(m) pumps a spin current (with zero chargecurrent) into\nadjacent contacts, and the pumped spin current is con-\nverted into a charge current IP(or pumping voltage VP)\nby a static ferromagnetic ﬁlter ( m0).17In the following,\nwe use the circuit theory described in Section II to derive\na simple expression for the charge pumping voltage (cur-\nrent) induced by FMR in a spin valve. We shall study\ntwo diﬀerent cases, i) when the spin valve is open, so no\ncurrent ﬂow is allowed ( Ic= 0), and a pumping voltage\nVPis built up; ii) when the spin valve is closed, i.e.the\ntwo ends of the spin valve are short-circuited, so no volt-\nage diﬀerence is allowed at the two ends ( µL=µR), and\na pumping current IPﬂows.\ni) open circuit - For an open circuit, the charge current\nvanishes: Ic= 0. By solving Eqs. (1-4), we ﬁnd an elec-\ntric voltage VP= (µL−µR)/edue to the spin pumping\ncurrentIsp\ns:\nVP(θ) =R(θ)[τip(θ)m×˙m+τop(θ)˙m]·m0,(11)\nwith the magneto-resistance R(θ) = 1/G(θ). Both the\nspin-transfer torque in Eq. (9) and the charge pump-\ning voltage in Eq. (11) are governed by the torkances.\nEq. (11) conﬁrms the two-step process for the charge\npumping: 1) spin current pumped by ˙m, 2) charge cur-\nrent generated by projecting on m0. Note that Eq. (11)\nentails all multiple scattering in the spacer.\nIn Eq. (11), the charge pumping voltage is related to\nthe torkances, which alsogovernthe spin-transfertorque.\nCurrently, the latter can be accessed only indirectly by\nits eﬀect on the current induced magnetization dynamics\nfor MTJs.38,39Eq. (11) can be employed to measure the\nspin-transfer torque or torkances in spin valves via FMR\ninducedvoltageswhenthemagneto-resistance R(θ) isob-\ntained alongside as done by Urazhdin et al.40\nii) closed circuit - When the spin valve is short-\ncircuited, µL=µR, Eqs. (1-4) yield a pumping current\nIP(θ) = [τip(θ)m×˙m+τop(θ)˙m]·m0.(12)\nFor comparison, in the presence of an applied current\ncurrentI,\nNst(θ) =R(θ)I[τip(θ)m×m0+τop(θ)m0]×m.(13)\nUsually, there is a passive series resistance in addition\nto the magneto-resistance R(θ). The charge pumping\nvoltagefortheopen circuitis insensitivetosucha passive\nresistance (for an ideal voltage meter).\nIn transition-metal ferromagnets, gi≤0.1gr,36thus\nfrom now on we neglect the imaginary part of the mixing\nconductance, i.e.gi= 0,τop= 0, and τip=eη(θ)gr/2π.4\nIV. MAGNETIZATION RELATED VOLTAGE\nNOISE IN SPIN VALVES\nAccording to the Fluctuation-Dissipation theorem\n(FDT), the electrical voltage ﬂuctuations across a non-\nmagnetic conductor is associated with the electron lin-\near momentum dissipation by the electrical resistance,\nwhich causes Joule heating. In ferromagnets, there are\nalso magnetization ﬂuctuations associated with the an-\ngular momentum dissipation or magnetic damping. In\nmagnetic heterostructures such as spin valves, the two\nﬂuctuations (electric and magnetic) are coupled by the\ndynamical exchange of spin currents. Electronic noise\nincreases the magnetic ﬂuctuations via the spin-transfer\neﬀect. Inversely, the magnetization noise increases elec-\ntronic ﬂuctuations via spin/charge pumping.\nWe discussed in the previous section the pumping volt-\nage (current) induced by an arbitrary motion of magne-\ntization. Here the formalism is applied to the stochas-\ntic magnetization motion at thermal equilibrium: the\nthermal ﬂuctuations of magnetization induce a pumping\nvoltage (current), which on ﬁltering by the static layer\nbecomes a noisy voltage signal. In this section, we dis-\ncuss this magnetizationﬂuctuation-induced voltagenoise\nVM(t) in a spin valve at thermal equilibrium, the power\nspectrum of which is the Fourier transform of its time\ncorrelation function:\nSM(ω) = 2/integraldisplay\n∝an}bracketle{tVM(0)VM(t)∝an}bracketri}hte−iωtdt. (14)\nAs shown by Johnson and Nyquist,23,24the\nFluctuation-Dissipation theorem (FDT) relates the\nnoise power spectrum S(ω) to the real part of the\nimpedance Z(ω) which characterizes the dissipation:\nS(ω) = 4kBTRe[Z(ω)]. (15)\nWe may calculate the noise spectrum from both sides\nof the FDT, 1) by computing the time-correlation\n∝an}bracketle{tVM(0)VM(t)∝an}bracketri}htfrom the response function, then using\nEq. (14); 2) by computing the frequency-dependent\nimpedance Z(ω) of a spin valve, which consists of an\nelectrical and a magnetic contribution: Z(ω) =RE+\nZM(ω), then making use of the Johnson-Nyquist for-\nmula Eq. (15). The electric part REgives rise to a white\nJohnson-Nyquistnoise of SE= 4kBTRE. In this section,\nwe focus on method 1), and calculate the voltage noise\nspectrum for two special cases with m0∝bardblˆx(perpendicu-\nlar case) and m0∝bardblˆz(parallel case). In Appendix B, we\nreproduce the spectrum for m0∝bardblˆxby using method 2).\nIn bulk ferromagnets, magnetic moment dissipation\nis parameterized by the Gilbert damping constant α0,\nwhich is associated with thermal ﬂuctuations of the di-\nrection ofthe magnetizationvector.25The magnetization\nﬂuctuations are caused by a ﬂuctuating torque from the\nlattice, which is represented by a thermal random mag-\nnetic ﬁeld h0:−Mtotm×h0. The auto-correlator of his25\n∝an}bracketle{tγh0\ni(t)γh0\nj(0)∝an}bracketri}ht=2γα0kBT\nMtotδijδ(t) = Σ0δijδ(t)\nwithi,j=x,y(assuming that the easy axis is along\nz). Similarly, the ferromagnet loses energy and angular\nmomentum by spin pumping. The magnetic damping in-\ncrement α′must be accompanied by a ﬂuctuating trans-\nversespin current(torque) Ifl\nsfromthe contacts,29which\ncan be represented by another random magnetic ﬁeld h′:\nIfl\ns=−Mtotm×h′with auto-correlator29\n∝an}bracketle{tγh′\ni(t)γh′\nj(0)∝an}bracketri}ht=2γα′kBT\nMtotδijδ(t) = Σ′δijδ(t).\nh′andh0are statistically independent: ∝an}bracketle{th′\nih0\nj∝an}bracketri}ht= 0.\nIncluding spin pumping from the magnetization ﬂuc-\ntuations and the ﬂuctuating spin current from the con-\ntacts, the total instantaneous spin current through the\nF/vextendsingle/vextendsingleN interface between the spacer and the free layer is\n(see Fig. 1(b)):\nIs(t) =Isp\ns+Ifl\ns=Mtot\nγ(α′m×˙m−γm×h′).(16)\nDue to the ﬁltering by the static layer magnetization\nm0, the spin current Is(t) is converted into a charge cur-\nrentIc(t) with eﬃciency η(θ). If the imaginary part\nof the mixing conductance is disregarded ( gi= 0 and\nτop= 0), an electrical voltage VM(t) is given by the\nsame expression as Eq. (11) with m×˙mreplaced by\nm×˙m−(γ/α′)m×h′:\nVM(t) =R(θ)η(θ)2e\n/planckover2pi1[m0·Is(t)] =W(θ)m0·f(t) (17)\nwithW(θ) = 2eR(θ)η(θ)eMtot/γ/planckover2pi1. Assuming that m\nﬂuctuates around the ˆz-axis (m≃ˆz),freads (to the lead-\ning order in mandh′)\nfx(t) =γh′\ny−α′˙my, (18a)\nfy(t) =−γh′\nx+α′˙mx, (18b)\nfz(t) =γ(myh′\nx−mxh′\ny)+α′(mx˙my−my˙mx).(18c)\nThe spectrum SMdepends on the direction of the po-\nlarizer. We need to compute, e.g.Fx(t)≡∝an}bracketle{tfx(0)fx(t)∝an}bracketri}ht\nwhenm0∝bardblˆxandFz(t)≡∝an}bracketle{tfz(0)fz(t)∝an}bracketri}htwhenm0∝bardblˆz. The\ncorrelatorsof fare composed of those between ˙mand/or\nh′, whichinturn canbe expressedbythetransversemag-\nnetic susceptibility χ(ω) (in frequency domain) as the re-\nsponse to the magnetic ﬁeld h=h0+h′+h′′(h′and\nh′′account for the random ﬁelds from the left and right\ninterface of the free layer):\n/bracketleftbigg\nmx(ω)\nmy(ω)/bracketrightbigg\n=χ(ω)/bracketleftbigg\nγhx(ω)\nγhy(ω)/bracketrightbigg\n. (19)5\nAll correlators can be calculated from Eq. (19):41\n∝an}bracketle{tmi(t)mj(0)∝an}bracketri}ht=Σ\nα/integraldisplay1\nωχ−\nij(ω)e−iωtdω\n2π,(20a)\n∝an}bracketle{t˙mi(t)mj(0)∝an}bracketri}ht=−Σ\nα/integraldisplay\ni χ−\nij(ω)e−iωtdω\n2π,(20b)\n∝an}bracketle{t˙mi(t) ˙mj(0)∝an}bracketri}ht=Σ\nα/integraldisplay\nω χ−\nij(ω)e−iωtdω\n2π,(20c)\nwith Σ = Σ 0+ 2Σ′(the factor 2 comes from the two\npumping interfaces) and χ−\nij(ω) = [χij(ω)−χ∗\nji(ω)]/2i,\nand\n∝an}bracketle{tmi(t)γh′\nj(0)∝an}bracketri}ht= Σ′/integraldisplay\nχij(ω)e−iωtdω\n2π, (21a)\n∝an}bracketle{t˙mi(t)γh′\nj(0)∝an}bracketri}ht=−Σ′/integraldisplay\niω χij(ω)e−iωtdω\n2π,(21b)\nBy taking all correlators between m,˙m, andh′into ac-\ncount, we conﬁrm that the DC spin/charge current van-\nishes at thermal equilibrium: ∝an}bracketle{tIs(t)∝an}bracketri}ht=∝an}bracketle{tIc(t)∝an}bracketri}ht= 0 as\nrequired by the second law of thermodynamics.\nWhenm0∝bardblˆx, the voltage noise power spectrum reads\nSx\nM(ω) = 2W2(π/2)/integraldisplay∞\n−∞dteiωtFx(t),(22)\nUsing Eqs. (20-21), we have\nFx(t) = Σ′/bracketleftbigg\nδ(t)−α′/integraldisplay\nω χ−\nyy(ω)e−iωtdω\n2π/bracketrightbigg\n.(23)\nFrom Eq. (23) and Eq. (22),\nSx\nM(ω) = 2W2(π/2)Σ′{1−α′ωIm[χyy(ω)]},(24)\nin terms of the imaginary part of the dynamic suscepti-\nbilities,i.e.the magnetic dissipation. A measurement of\nthe former therefore determines the latter, serving as an\nalternative to e.g. FMR measurements.\nWhenm0∝bardblˆz,Sz\nM(ω) follows from Eq. (22) by the re-\nplacement Fx(t)→Fz(t) andW(π/2)→W(0). Ac-\ncording to Eq. (18c), this involves 4-point correlators,\nwhich can be reduced to 2-point correlators by Wick’s\ntheorem:30,42∝an}bracketle{tabcd∝an}bracketri}ht=∝an}bracketle{tab∝an}bracketri}ht∝an}bracketle{tcd∝an}bracketri}ht+∝an}bracketle{tac∝an}bracketri}ht∝an}bracketle{tbd∝an}bracketri}ht+∝an}bracketle{tad∝an}bracketri}ht∝an}bracketle{tbc∝an}bracketri}ht. Af-\nter some tedious algebra, we reach\nSz\nM(ω) = 2W2(0)Σ′2/braceleftbigg1\nα′/summationdisplay\niRe[χii(0)]−\n/integraldisplaydω′\n2πω−2ω′\nω′/summationdisplay\ni,j(−1)δijχ−\nij(ω′)χ−\n¯i¯j(ω−ω′)/bracerightbigg\n(25)\nwith ¯x=yand ¯y=x. For the anti-parallel conﬁguration\n(m0∝bardbl−ˆz), the formulais identical to Eq. (25) except that\nW(0) is replaced by W(π). Note that χfor parallel and\nanti-parallel cases are diﬀerent, as discussed in Appendix\nA.\nWith Eq.(24) andEq.(25), thecalculationofthe noise\npower spectrum reduces to that of the magnetic suscep-\ntibilityχfor the free layer magnetization. Similar tothe Gilbert damping for the free layer magnetization in a\nspin valve,43χin general depends on the magnetization\nconﬁguration of the spin valve. We derive the angular\ndependent χin Appendix A.\nFor simplicity we continue with an isotropic form of\nthe magnetic susceptibility for the free layer magneti-\nzationχ, which includes the eﬀect of the spin pumping\nenhanced damping, but not the multiple scatteringof the\nspin pumping current within the spacer:\nχ(ω) =1\n(ω0−iαω)2−ω2/parenleftbigg\nω0−iαω−iω\niω ω 0−iαω/parenrightbigg\n(26)\nwithα=α0+2α′andω0=γHeﬀ. Using this χ, we ﬁnd\nSx\nM(ω) = 2W2(π/2)Σ′\n/braceleftbigg\n1−α′α(1+α2)ω4+ω2\n0ω2\n[(1+α2)ω2−ω2\n0]2+4α2ω2ω2\n0/bracerightbigg\n.(27)\nA more accurate form of Sx\nM(ω) (Eq. (B2)) as calculated\nin Appendix B using circuit theory is recovered by the\nmethod here by using the angle-dependent susceptibility\ntensor Eq. (A11) instead of Eq. (26) in Eq. (24).\nThe square root of Sx\nM(ω) is plotted in Fig. 2 for\nthe parameters in Table I, where ηis replaced by its\nballistic limit p/2 (left ordinate). Assuming that the\nspin valve resistance Ris dominated by the interface\nresistances, R∝1/A, but does not depend on the free\nlayer thickness d. Considering the volume Ω = Adand\nα′∝1/d, Σ′∝α′/Ω∝1/(Ad2), the white noise background\ninSx\nM(ω) scales like R2Ω2Σ′∝1/A, and thus does not\ndepend on d. The dip in Fig. 2 at the FMR frequency\nis remarkable. Its depth is proportional to α′hence in-\nversely proportional to the free layer thickness, whereas\nits width is proportional to αω0. The constant back-\nground of the spectrum is/radicalbig\nSx\nM(ω)≃50 nV/√\nMHz,\nwhereas the dip is about 4 nV/√\nMHz. For comparison,\nthe root-mean-squareof the electrical contribution to the\n 0 10 20 30 40 50 60\n 0  0.5  1  1.5  2 0 1 2 3 4 5 6[SxM(ω)]1/2 (nV/MHz1/2)\n[SzM(ω)]1/2 (nV/MHz1/2)\nω/ω0T  = 300K\nα  = 0.02, α0 = 2α’  = 0.01\nFIG. 2: (Color online)p\nSx\nM(ω) (solid line with left scale)\nandp\nSz\nM(ω) (dashed line with right scale).6\nQuantity Values Ref.\nMs(Co)1.42×106A m−14\nγ(Co)1.9×1011(T s)−144\nα0(Co)0.01 45\n2α′(Co˛˛Cu)0.01 45\nω010 GHz\np0.35 46,47\nR≃RP0.57 Ω Derivedafrom Ref. 4\nRSample1.6 Ω 4\nT300 K\nΩ =A×d(130·130 nm2)×2.5 nm 4\na(RAP−RP)/RAP≈p2andRAP−RP= 0.073Ω4, where P/AP\nstands for parallel/anti-parallel.\nTABLE I: Typical spin valve parameters (see text).\nnoise is/radicalbig\nSE(ω) =/radicalbig\n4kBTR−Sx\nM(0)≃87 nV/√\nMHz.\nWhenm0∝bardblˆz, Eq. (25) and Eq. (26) lead to\nSz\nM(ω) = 2W2(0)Σ′2\nω0/bracketleftbigg1\nα′−α(1+α2)ω2+4ω2\n0\n(1+α2)2ω2+4α2ω2\n0/bracketrightbigg\n.\n(28)\nThe square root of Sz\nM(ω) is plotted as the dashed curve\nin Fig. 2 (right ordinate). In contrast to Sx\nM(ω), the pre-\nfactor in Sz\nM(ω)∝R2Ω2Σ′2/α′∝1/(A2d), therefore the\nnoise decreases with increasing d. This diﬀers from the\ncasem0∝bardblˆxbecause the projection on the ˆzaxis involves\nthe average deviation of mfrom the equilibrium direc-\ntion, which is inversely proportional to the volume. The\ndivergenceat vanishingthickness is causedby the neglect\nof the ﬁnite transparency of very thin magnetic layers for\ntransverse spin currents. Similar to Sx\nM(ω), the depth of\nthe dip at ω= 0 is proportional to α′, hence inversely\nproportional to the layer thickness and has a width pro-\nportional to αω0.\nV. DISCUSSION\nThe spectrum Sx\nM(ω) consists of three contributions,\nwhich can be seen from the decomposition of Fx(t) =\n∝an}bracketle{tfx(t)fx(0)∝an}bracketri}ht=Fsp\nx+Ffl\nx+Fab\nxwith\nFsp\nx(t) =α′2∝an}bracketle{t˙my(t) ˙my(0)∝an}bracketri}ht, (29a)\nFfl\nx(t) =γ2∝an}bracketle{th′\ny(t)h′\ny(0)∝an}bracketri}ht, (29b)\nFab\nx(t) =−α′γ/bracketleftbig\n∝an}bracketle{t˙my(t)h′\ny(0)∝an}bracketri}ht+∝an}bracketle{t˙my(0)h′\ny(t)∝an}bracketri}ht/bracketrightbig\n.(29c)\nThese three contributions can be interpreted as: i) a\nspin pumping current Fsp\nx, which produces a peak at\nω=ω0, ii) a random torque (spin current) from the\ncontactFfl\nx, whose spectrum is white, and iii) the ab-\nsorption of the random torque from the contact by the\nmagnetization Fab\nx=−2Fsp\nx, which givesa dip at ω=ω0\nwith twice the magnitude of i). The contacts thereforeprovide a white-noise random torque over the ferromag-\nnetic ﬁlm, whereas the magnetization absorbs the noise\npower around ω0. The spectrum Sz\nM(ω) also consists\nof three contributions, but the absorption line of the ˆz-\ncomponentis centeredatzerofrequencybecausethe ﬂuc-\ntuations of the ˆz-component magnetization do not have\na characteristic frequency such as the ˆx,ˆy-components.\nIn inhomogeneous FM ﬁlms, the single macrospin mode\nbreaks up into diﬀerent eigenmodes. The noise power\nspectrum can then provide a “ﬁngerprint” of the various\neigenmode frequencies.\nThe three-point correlators arising in ∝an}bracketle{tVM(0)VM(t)∝an}bracketri}ht\nwhenm0is at arbitrary angles in the ˆx-ˆzplane vanish\nfor normal distributions. The power spectrum is then\na linear combination of Sx\nMandSz\nM, depending on the\nangle with dips at both the FMR and zero frequencies.\nThe modeling of magnetic anisotropies by an easy axis\nis appropriate when the free layer magnetization is ori-\nented normalto the plane in axiallysymmetricpillars. In\nstandard pillars the dominant anisotropy is easy-plane,\nwhich leads to anisotropic ﬂuctuations of the magneti-\nzation. The results remain qualitatively similar, but be-\ncome anisotropic in the ˆx-ˆyplane. For example, when\na strong anisotropy constrains the ﬂuctuations of mto\ntheˆy-ˆzplane,Sy\nMvanishes. We disregarded the imagi-\nnary part of the mixing conductance in our calculation\nfor the noise power spectrum. When it is included, the\nsymmetric dip in the power spectrum in Fig. 2 is skewed\nbygisimilar to for the spin diode eﬀect discussed by\nKupferschmidt et al.48and Kovalev et al.32\nFor asymmetric spin valves, a non-monotonic angu-\nlar dependence of the magnetoresistance and a vanish-\ning torkance at a non-collinear magnetization conﬁgura-\ntion has been demonstrated.49,50,51,52,53A sign change in\ntorkance leads to a sign change in the charge pumping\nvoltage. The magnetic contribution to the thermal noise\nvanishes at the zero torkance point.\nVI. SUMMARY\nIn conclusion, we ﬁnd that a pumping voltage arises\nin a spin valve when the free layer magnetization is in\nmotion. The angular dependence of pumping voltage\nunder FMR condition provides detailed information of\nthe spin transport in spin valves. The pumping voltage\ninduced by the thermal ﬂuctuation of the free layer mag-\nnetization gives rise to additional voltage noise, which\nis associated with the magnetization dissipation. Thus\nthe equilibrium electronic noise in a spin valve consists\nof two contributions: the Johnson-Nyquist noise associ-\nated with the ﬂuctuations of the charge, and magneti-\nzation related noise associated with the ﬂuctuations of\nthe spins. The magnitude of these two contributions can\nbe comparable. Unlike the white Johnson-Nyquist noise,\nthe latter is found to contain an absorption line at the\nFMR frequency (and at zero frequency depending on the\nconﬁguration) on top of an enhanced white noise back-7\nground. The noise spectrum can provide a ﬁngerprint of\nthe magnetic eigenmodes in inhomogeneous structures.\nAcknowledgment\nThis work is supported by EC Contract IST-033749\n“DynaMax”. JX and GB thank SM for his hospitality\nat Tohoku University. JX acknowledges support by H. J.\nGao and S. Yi in Beijing and X. F. Jin in Shanghai.\nAPPENDIX A: ANGULAR DEPENDENCE OF\nTHE TRANSVERSE MAGNETIC\nSUSCEPTIBILITY IN SPIN VALVES\nIn this Appendix we calculate the transverse magnetic\nsusceptibility for the free layer magnetization in a spin\nvalve when no external bias is applied ( Ic= 0),i.e.the\nonly driving force is the thermal random ﬁeld h.\nFor an isolated magnet, the dynamics are described by\nthe Landau-Lifshitz-Gilbert equation:\n˙m=−γm×(Heﬀ+h)+α0m×˙m,(A1)\nwith the thermal magnetic ﬁeld hand bulk damping\nparameter α0. When we consider small amplitude pre-\ncession around the ˆzdirection (assuming Heﬀ∝bardblˆz), the\nFourier transform of the linearized LLG equation be-\ncomes\n/bracketleftBigg\nmx(ω)\nmy(ω)/bracketrightBigg\n=χ0(ω)/bracketleftBigg\nγhx(ω)\nγhy(ω)/bracketrightBigg\n, (A2)\nwith\nχ0(ω) =1\n(ω0−iα0ω)2−ω2/parenleftBigg\nω0−iα0ω−iω\niω ω 0−iα0ω/parenrightBigg\n.\n(A3)\nIn spin valves, χfor the free layer magnetization de-\npends on the relative orientation of m0andmbecause of\nthe multiple scattering within the spacer, which depends\non the orientation of m0, acts as spin-transfer torque on\nm. WenowlinearizeEq.(6), againassumingthat mﬂuc-\ntuates around the ˆz-axis with small amplitude ( m≃ˆz):\n˙mx=−ω0my−α˙my+γ\nMtotNx\nst+γhy,(A4a)\n˙my= +ω0mx+α˙mx+γ\nMtotNy\nst−γhx,(A4b)\nwhereα=α0+ 2α′and 2α′is the enhanced damping\nfrom the two interfaces of the free layer.\nThe circuit theoryEqs.(1-4) arecoupled with the LLG\nequation in Eq. (A4) through ˙min Eq. (3) and IRin\nEq. (A4) and they have to be solved self-consistently.\nWe assume that m0is static and tilted by an angle θ\nfromˆz,i.e.ˆz·m0= cosθ. Eqs. (1-4) can be converted\nto scalar equations by taking dot products with m,m0,andm×=m×m0. Introducing the projections of an\narbitrary vector q: (q0,qm,q×)≡q·(m0,m,m×), setting\ngsf= 0 for simplicity, Eqs. (1-4) become\n0 =I=eg\n2h(2µL−pµ0\nN) =−eg\n2h(2µR−pµm\nN),(A5a)\nI0\nL=eg\n2h(2pµL−µ0\nN), (A5b)\nIm\nL=eg\n2h(2pµL−µ0\nN)cosθ−egr\nh(µm\nN−µ0\nNcosθ),\n(A5c)\nI×\nL=−egr\nhµ×\nN, (A5d)\nI0\nR=−eg\n2h(2pµR−gµm\nN)cosθ\n+egr\nh(µ0\nN−µm\nNcosθ)+egr\n2π˙m×, (A5e)\nIm\nR=−eg\n2h(2pµR−µm\nN), (A5f)\nI×\nR=e\nhgrµ×\nN−egr\n2π˙m0, (A5g)\n0 =I0\nL−I0\nR=Im\nL−Im\nR=I×\nL−I×\nR, (A5h)\nwhere in the third and ﬁfth equation above, we disre-\ngard the time-dependence of m0(t) =m(t)·m0because\nm≃ˆzto leading order in the deviations. The solutions to\nEq. (A5) are\nI0\nR=ξ0egr\n4π˙m×, Im\nR=ξmegr\n4π˙m×, I×\nR=−egr\n4π˙m0,\n(A6)\nwith\nξ0=1−ν\n1−ν2cos2θandξm=−(1−ν)νcosθ\n1−ν2cos2θ(A7)\nandν= [2gr−g(1−p2)]/[2gr+g(1−p2)]. We now deﬁne\nthe coordinate system in the plane normal to ˆz:\nˆx≡m0−ˆzcosθ\nsinθandˆy≡ˆz×ˆx≈m×\nsinθ.(A8)\nTherefore ˙ m0≈˙mxsinθ, ˙m×≈˙mysinθ, and\nγ\nMtotNx\nst≈γ/planckover2pi1\n2eMtotI0\nR−Im\nRcosθ\nsinθ=1\n2ξα′˙my,(A9a)\nγ\nMtotNy\nst≈γ/planckover2pi1\n2eMtotI×\nR\nsinθ=−1\n2α′˙mx, (A9b)\nwithξ=ξ0−ξmcosθ, which is related to the angular\ndependent magnetic damping in Ref. 43.\nPlugging Eq. (A9) into Eq. (A4), after linearization in\nterms of small ﬂuctuations about the ˆz-axis and Fourier\ntransformation, we have\n/bracketleftBigg\nγhx(ω)\nγhy(ω)/bracketrightBigg\n=/parenleftBigg\nω0−iαyω iω\n−iω ω 0−iαxω/parenrightBigg/bracketleftBigg\nmx(ω)\nmy(ω)/bracketrightBigg\n,\n(A10)\nwhereαx=α−1\n2ξα′andαy=α−1\n2α′reﬂect the damp-\ning anisotropy. From Eq. (A10),8\nχ(ω) =/parenleftBigg\nω0−iαyω iω\n−iω ω 0−iαxω/parenrightBigg−1\n=1\n(1+αxαy)ω2−ω2\n0+i(αx+αy)ω0ω/parenleftBigg\nω0−iαxω−iω\niω ω 0−iαyω/parenrightBigg\n.(A11)\nBecause αydepends on angle θthrough ξ,χ(ω) also be-\ncomes angle dependent, i.e.the magnetic susceptibility\nfunction for the free layer magnetization in a spin valve\nis in general angular dependent, for the same reason as\nthemagneticdampinginspinvalves.43Eq.(A11)reduces\nto Eq. (26) when we identify αx≃αy≃αby ignoring the\nback-ﬂow correction to the damping.\nAPPENDIX B: SPIN VALVE IMPEDANCE Z(ω)\nFOR m 0/bardblˆx\nHere, we calculate the frequency dependence of the\nimpedance of a spin valve by applying a small AC cur-\nrent at frequency ω:I(ω). We consider the perpendicu-\nlar case here, i.e.m0∝bardblˆxor cosθ= 0, so that the circuit\ntheory equations equal Eq. (A5) except that we allow\nfor anon-vanishing charge current I∝ne}ationslash=0. Since we are\nnow interested in the deterministic response, the ther-\nmal random ﬁelds hmay be ignored. We can then\nsolve Eqs. (A4, A5) self-consistently in the frequencydomain. We ﬁnd that the impedance of the spin valve\nZ(ω) = [µL(ω)−µR(ω)]/eI(ω) consists of two parts, an\nelectric part REand a magnetic part ZM(ω):Z(ω) =\nRE+ZM(ω):\nRE=4\nG01\ng+η2R2G0g(1−p2)+η2R2G0gr,(B1a)\nZx\nM=η2R2G0gr×/bracketleftbigg\n1−α′ω(αyω+iω0)\n(1+αxαy)ω2−ω2\n0+i(αx+αy)ωω0/bracketrightbigg\n,(B1b)\nwhereη=η(π/2) andR=R(π/2) are the polarization\nfactorandthe DC resistanceforthe spinvalveat θ=π/2\norm0∝bardblˆx.REconsists of the resistances associated with\nthe electrical dissipation and half of the magnetic dissi-\npation from the interface with the static magnetization\nwhich does not emit a spin pumping current.\nBy the Johnson-Nyquist formula Eq. (15), the noise\nspectrum Sx\nM(ω) is given by\nSx\nM= 4kBTRe[Zx\nM(ω)] = 2W2(π/2)Σ′/braceleftbigg\n1−α′αy(1+αxαy)ω4+αxω2ω2\n0\n[(1+αxαy)ω2−ω0]2+(αx+αy)2ω2ω2\n0/bracerightbigg\n. (B2)\nThis equation is identical to Eq. (27) when we that the\nlimitαx≃αy≃α. This diﬀerence comes from the approx-\nimate form of χin Eq. (26). If we use Eq. (A11), then\ntheSx\nMcalculated from Eq. (24) will be exactly the same\nas Eq. (B2).The frequency dependent impedance for m0∝bardblˆzis sec-\nond order in mx,y, therefore it is not so straightforward\nto calculate, which can also be seen from the non-trivial\nconvolution in Eq. (25) calculated from magnetic suscep-\ntibility functions.\n1J. C. Slonczewski, J. Magn. 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).\n4J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers,\nand D. C. Ralph, Phys. Rev. Lett. 84, 3149 (2000).\n5M. Stiles and J. Miltat, Spin-Transfer Torque and Dynam-\nics(2006), pp. 225–308.\n6S. Urazhdin, N. O. Birge, W. P. Pratt, and J. Bass, Phys.\nRev. Lett. 91, 146803 (2003).\n7W. H. Rippard, M. R. Pufall, S. Kaka, S. E. Russek, and\nT. J. Silva, Phys. Rev. Lett. 92, 027201 (2004).8A. Fert, V. Cros, J. M. George, J. Grollier, H. Jaﬀres,\nA. Hamzic, A. Vaures, G. Faini, J. B. Youssef, and H. L.\nGall, J. Magn. Magn. Mater. 272-276 , 1706 (2004).\n9J. Z. Sun, Phys. Rev. B 62, 570 (2000).\n10D. H. Hernando, Y. V. Nazarov, A. Brataas, and G. E. W.\nBauer, Phys. Rev. B 62, 5700 (2000).\n11X. Waintal, E. B. Myers, P. W. Brouwer, and D. C. Ralph,\nPhys. Rev. B 62, 12317 (2000).\n12M. D. Stiles and A. Zangwill, Phys. Rev. B 66, 014407\n(2002).\n13Z. Li and S. Zhang, Phys. Rev. B 68, 024404 (2003).\n14L. Berger, Phys. Rev. B 59, 11465 (1999).\n15X. Wang, G. E. W. Bauer, B. J. van Wees, A. Brataas,9\nand Y. Tserkovnyak, Phys. Rev. Lett. 97, 216602 (2006).\n16M. V. Costache, M. Sladkov, S. M. Watts, C. H. van der\nWal, and B. J. van Wees, Phys. Rev. Lett. 97, 216603\n(2006).\n17J. Xiao, G. E. W. Bauer, and A. Brataas, Phys. Rev. B\n77, 180407(R) (2008).\n18S. E. Barnes and S. Maekawa, Phys. Rev. Lett. 98, 246601\n(2007).\n19W. M. Saslow, Phys. Rev. B 76, 184434 (2007).\n20R. A. Duine, Phys. Rev. B) 77, 014409 (2008).\n21Y. Tserkovnyak and M. Mecklenburg, Phys. Rev. B 77,\n134407 (2008).\n22S. A. Yang, G. S. D. Beach, C. Knutson, D. Xiao, Q. Niu,\nM. Tsoi, and J. L. Erskine, Phys. Rev. Lett. 102, 067201\n(2009).\n23J. B. Johnson, Phys. Rev. 32, 97 (1928).\n24H. Nyquist, Phys. Rev. 32, 110 (1928).\n25W. F. Brown, Phys. Rev. 130, 1677 (1963).\n26R. H. Koch, G. Grinstein, G. A. Keefe, Y. Lu, P. L. Trouil-\nloud, W. J. Gallagher, andS. S.P. Parkin, Phys.Rev.Lett.\n84, 5419 (2000).\n27W. Wetzels, G. E. W. Bauer, and O. N. Jouravlev, Phys.\nRev. Lett. 96, 127203 (2006).\n28X. Wang, W. Zhu, and D. Dimitrov, Phys. Rev. B 78,\n024417 (2008).\n29J. Foros, A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer,\nPhys. Rev. Lett. 95, 016601 (2005).\n30J. Foros, A. Brataas, G. E. W. Bauer, and Y. Tserkovnyak,\nPhys. Rev. B 75, 092405 (2007).\n31A. M. Deac, A. Fukushima, H. Kubota, H. Maehara,\nY. Suzuki, S. Yuasa, Y. Nagamine, K. Tsunekawa, D. D.\nDjayaprawira, and N. Watanabe, Nature Physics 4, 803\n(2008).\n32A. A. Kovalev, G. E. W. Bauer, and A. Brataas, Phys.\nRev. B75, 014430 (2007).\n33Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. Lett. 88, 117601 (2002).\n34A. Brataas, Y. V. Nazarov, and G. E. W. Bauer, Phys.\nRev. Lett. 84, 2481 (2000).\n35A. Brataas, Y. Nazarov, and G. Bauer, Euro. Phys. J. B22, 99 (2001).\n36A. Brataas, G. E. Bauer, and P. J. Kelly, Physics Reports\n427, 157 (2006).\n37J. Slonczewski and J. Sun, J. Magn. Magn. Mater. 310,\n169 (2007).\n38H. Kubota, A. Fukushima, K. Yakushiji, T. Naga-\nhama, S. Yuasa, K. Ando, H. Maehara, Y. Nagamine,\nK. Tsunekawa, D. D. Djayaprawira, et al., Nature Physics\n4, 37 (2008).\n39J. C. Sankey, Y. Cui, J. Z. Sun, J. C. Slonczewski, R. A.\nBuhrman, and D. C. Ralph, Nature Physics 4, 67 (2008).\n40S. Urazhdin, N. O. Birge, J. Pratt, and J. Bass, Appl.\nPhys. Lett. 84, 1516 (2004).\n41N. Smith, J. Appl. Phys. 90, 5768 (2001).\n42K. Triantafyllopoulos, The Mathematical Scientist 28, 125\n(2003).\n43Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. B67, 140404(R) (2003).\n44S. M. Bhagat and P. Lubitz, Phys. Rev. B 10, 179 (1974).\n45J. L. Beaujour, J. H. Lee, A. D. Kent, K. Krycka, and\nC. Kao, Phys. Rev. B 74, 214405 (2006).\n46S. K. Upadhyay, A. Palanisami, R. N. Louie, and R. A.\nBuhrman, Phys. Rev. Lett. 81, 3247 (1998).\n47J. Bass and W. P. Pratt, J. Magn. Magn. Mater. 200, 274\n(1999).\n48J. N. Kupferschmidt, S. Adam, and P. W. Brouwer, Phys.\nRev. B74, 134416 (2006).\n49A. A. Kovalev, A. Brataas, and G. E. W. Bauer, Phys.\nRev. B66, 224424 (2002).\n50J. Manschot, A. Brataas, and G. E. W. Bauer, Phys. Rev.\nB69, 092407 (2004).\n51J. Xiao, A. Zangwill, and M. D. Stiles, Phys. Rev. B 70,\n172405 (2004).\n52O. Boulle, V. Cros, J. Grollier, L. G. Pereira, C. Deranlot,\nF. Petroﬀ, G. Faini, J. Barnas, andA. Fert, Nature Physics\n3, 492 (2007).\n53V. S. Rychkov, S. Borlenghi, H. Jaﬀres, A. Fert, and\nX. Waintal, arXiv:0902.4360 (2009)."    },    {        "title": "2110.05136v2.Spin_pumping_at_terahertz_nutation_resonances.pdf",        "content": "Spin pumping at terahertz nutation resonances\nRitwik Mondal1, 2,\u0003and Akashdeep Kamra3,y\n1Department of Spintronics and Nanoelectronics, Institute of Physics of the Czech Academy of Sciences,\nCukrovarnická 10, CZ - 162 00 Praha 6, Czech Republic\n2Department of Physics and Astronomy, Uppsala University, Box 516, Uppsala, SE-75120, Sweden\n3Condensed Matter Physics Center (IFIMAC) and Departamento de Física Teórica de la Materia Condensada,\nUniversidad Autónoma de Madrid, E-28049 Madrid, Spain\nWe investigate spin pumping current injected by the nutation resonances of a ferromagnet or an\nantiferromagnetintoanadjacentmetal. Comparingthedcspinpumpingcurrentbetweenthenormal\nprecession and nutation resonances, we ﬁnd that the ratio of spin pumping current at the nutation\nresonance to the precession resonance is more pronounced in antiferromagnets. We further show\nthat the spin pumping current injected by the nutation resonance is opposite in sign as compared\nto the normal precession mode. This could oﬀer a useful experimental signature for identifying\nsuch nutation resonances. Analyzing the nature of the nutational eigenmodes, we show that the\nsign change in spin current is rooted in a reversal of the precession sense for the nutation mode(s).\nFurthermore, the nutational modes in antiferromagnets are found to be dominated by precession of\none of the two sublattices only.\nI. INTRODUCTION\nThe ultrafast manipulation of spins at the terahertz fre-\nquencies has paramount scientiﬁc and technological inter-\nest. Most spin dynamics experiments have been explained\nso far by the traditional Landau-Lifshitz-Gilbert (LLG)\nphenomenology [1–7], which describes a precessional mo-\ntion of spins around an eﬀective ﬁeld and an energy dis-\nsipation via a viscous damping term [8, 9]. However, at\nultrafast timescales, the LLG equation has been found to\nbe insuﬃcient [10, 11].\nTo address this shortcoming, the LLG phenomenology\nhas been extended to account for magnetization dynamics\nin the inertial regime [12, 13]. Essentially, the inclusion\nof magnetic inertia leads to a spin nutation at the ultra-\nshort timescales and can be incorporated via a double time-\nderivative of the magnetization i.e., M\u0002M[14, 15]. The\ninertial LLG (ILLG) equation of motion for two-sublattice\nsystems has the form\n_Mi=\u0000\ri\u0000\nMi\u0002He\u000b\ni\u0001\n+Mi\nMi0\u0002h\n\u000bi_Mi+\u0011iMii\n;(1)\nwith the gyromagnetic ratio \r, the eﬀective ﬁeld He\u000b,\nground state magnetic moment M0, the Gilbert damping\nparameter \u000band the inertial relaxation time \u0011. The in-\ndex “i” denotes the sublattice. In general, Gilbert damp-\ning in a two-sublattice and the inertial relaxation time are\ntensors [16–20]. Here, we treat them as scalars for sim-\nplicity and in an attempt to capture the leading order ef-\nfects. The emergence of spin nutation has been attributed\nto several proposed mechanisms [21–28]. The characteris-\ntic timescales of the nutation \u0011have been predicted to be\nin a range of 1\u0000100fs [29, 30]. Experimentally inertial\nrelaxation time \u0011is found to be about hundreds fs in two\nsublattice ferromagnets [31].\nThe spin nutation additionally introduces a second res-\nonance in the ferromagnetic resonance (FMR) spectrum,\n\u0003mondal@fzu.cz\nyakashdeep.kamra@uam.eshowever, at a higher frequency in THz range. Such a reso-\nnance is called ferromagnetic nutation resonance (FMNR)\n[32, 33]. Moreover, the precession resonance frequencies are\ndecreased due to the spin nutation [30, 32, 34]. In a more\nrecentexperiment,severalhigher-ordernutationresonances\nhave been observed [35]. With rapid recent progress, new\npathways for the role of nutation in practical devices have\nalready started to emerge [36].\nDespite several signatures in ultrafast spin dynamics ex-\nperiments attributed to nutation modes [29, 31, 35], further\nsmoking-gun validations are needed. Complementary to a\ntime-resolved tracking of the magnetization dynamics, the\nspin pumping current [37, 38] injected by the latter into\nan adjacent metal has emerged as a powerful probe. It\nhas already been employed in investigating a broad range\nof phenomena from spin Seebeck eﬀect [39–44] to antifer-\nromagnetic resonance [45–47], and over a broad range of\ntime scales. With the anticipation that spin pumping cur-\nrent driven by the nutation mode might be very diﬀerent\nfrom the conventional resonance modes, we theoretically\ninvestigate this hypothesis in both ferromagnets (FM) and\nantiferromagnets (AFM) here ﬁnding valuable results and\ninsights.\nIn this article, we theoretically investigate the dc spin\npumping current injected by a magnet driven by an oscil-\nlating magnetic ﬁeld into an adjacent metal. We consider\nboth ferro- and antiferromagnets, and focus on their nuta-\ntion modes. We ﬁnd that the spin current at the nutation\nresonance is negative, while the spin current at the pre-\ncession resonance is positive. Our results also show that\nthe spin current at the nutation resonance increases with\nincreasing inertial relaxation time \u0011in FM and AFM. The\ncomputed ratio of spin currents at the nutation to the pre-\ncession resonance show that the nutation spin current is\nmorepronouncedinAFM.Wedelineatethenutationeigen-\nmodes ﬁnding the ferromagnetic nutation to entail magne-\ntization precession in the opposite sense, compared to the\nnormal precession mode. In AFM, the nutation modes are\ncharacterized by a larger precession cone for one of the\ntwo sublattice magnetizations, thereby departing from the\nnearly collinear dynamics in the conventional antiferromag-\nnetic resonance. These features may enable a clear distinc-arXiv:2110.05136v2  [cond-mat.mes-hall]  27 Oct 20212\ntion of the nutation modes in experiments.\nII. SPIN PUMPING IN FERROMAGNETS\nWe consider a FM with a single sublattice as M=M0^z\nat the ground state, that is under the inﬂuence of an exter-\nnal Zeeman ﬁeld H=H0^z. Such a FM can be described\nby the following free energy F(M) =\u0000H0Mz\u0000KM2\nz=M2\n0,\nwhereKis the uniaxial anisotropy energy and M0is the\nground state magnetic moment. The eﬀective ﬁeld that en-\nters into the ILLG equation can thus be calculated using\nHe\u000b=\u0000@F=@M.\nWhenasmalloscillatingtransversalﬁeld h(t) =hx(t)^x+\nhy(t)^yis applied, the small magnetization oscillations are\ninducedsuchthattime-dependentmagnetizationis M(t) =\nmx(t)^x+my(t)^y+M0^z. Within the linear response theory\nin the circular basis described by h\u0006=hx\u0006ihy=he\u0006i!t\nandm\u0006=mx\u0006imy=me\u0006i!t,thecalculatedsusceptibility\nexpression is [48]\nm\u0006=\rM0\n\n0\u0000\u0011!2\u0000!\u0006i!\u000bh\u0006=\u001f\u0006h\u0006;(2)\nwhere \n0=\r\nM0[H0M0+ 2K]. The poles of the suscep-\ntibility determine the resonance frequencies. Without the\nnutation term \u0011, only a FMR frequency is obtained. How-\never, thespinnutationadditionallyintroducesasecondres-\nonance FMNR frequency. These two frequencies are [48]\n!FMR =p1 + 4\u0011\n0\u00001\n2\u0011; (3)\n!FMNR =\u0000p1 + 4\u0011\n0+ 1\n2\u0011: (4)\nThe negative frequency in the nutation resonance dictates\nthe fact that the nutation resonance has opposite handed-\nness of rotation compared to the FMR [48, 49]. This has\nbeen further corroborated by an analysis of the nutation\neigenmode presented in the Appendix A.\n-2 -1-101234js×1025¯h[m−2-s−1]\n0.025 0.05 0.075 0.1\nω/2π(THz)η= 0\nη= 10−13s\nFigure 1. The calculated spin pumping dc current for inertial\nrelaxation times \u0011= 0s and\u0011= 10\u000013s. The used parameters\nareM0= 2\u0016B,K= 10\u000023J,\r= 1:76\u00021011T\u00001-s\u00001,\u000b= 0:05,\nH0= 1T,jhj= 10\u00003T,g\"#\nr= 1019m\u00002.The dc component of a generated spin current density\ncan be expressed as [37, 50]\njs=!\n2\u0019Z2\u0019=!\n0~\n4\u0019g\"#\nr1\nM2\n0h\nM(t)\u0002_M(t)i\nzdt:(5)\nWe calculate the spin current in the circular basis andh\nM(t)\u0002_M(t)i\nz=i\n2[m+_m\u0000\u0000m\u0000_m+]:Using Eq. (2),\nthe calculated spin current has the expression\njs=~\n4\u0019g\"#\nr\u0014!\r2\n(\n0\u0000\u0011!2\u0000!)2+\u000b2!2jhj2\u0015\n:(6)\nWecomputesuchspincurrentswithinertialrelaxationtime\n\u0011= 10\u000013s in Fig. 1. We denote the calculated spin\ncurrent at FMR frequencies as jFMR\nsand at nutation fre-\nquencies as jFMNR\ns. Note that the spin current at the FMR\nfrequencies has positive sign, however, the calculated spin\ncurrent at the nutation resonance frequencies has the op-\nposite sign. The reason is that while the precession mode\nrotates anticlockwise, the nutation mode rotates clockwise.\n10−1610−1510−1410−1310−12\nη(s)10−410−310−210−1(jFMNR\ns/jFMR\ns)×(−1)\nFigure 2. The ratio of spin current for ferromagnets at the nuta-\ntion resonance to the precession resonance vs inertial relaxation\ntime\u0011. The used parameters are M0= 2\u0016B,\r= 1:76\u00021011\nT\u00001-s\u00001,\u000b= 0:05,K= 10\u000023J andH0= 0T.\nNonetheless, we calculate the ratio of spin currents cal-\nculated at several inertial relaxation times \u0011in Fig. 2. Note\nthat the ratio of spin currents is independent of several pa-\nrameters including spin mixing conductance g\"#\nr, small ﬁeld\nhetc. We emphasize that the spin current at the nutation\nresonance increases linearly with \u0011, while the current at\nthe precession resonance stays almost constant. Such an\nobservation can easily be understood from Eq. (6). The\nFMR frequency lies in the GHz regime and its shift due\nto nutation is small in ferromagnets. Therefore, the calcu-\nlated spin current at the FMR is roughly proportional to\n!\u00001\nFMR\u00191=\n0+\u0011. However, the dominant spin current\ncontribution at the FMNR is !\u00001\nFMNR\u0019\u0000\u0011. Therefore, at\nsmaller\u0011, the ratio of spin currents is linear in \u0011, however,\nthe ratio deviates from linearity at larger \u0011.\nIII. SPIN PUMPING IN ANTIFERROMAGNETS\nWe consider an AFM having two sublattices namely A\nandB. The ground state of such AFM is MA=MA0^z3\nandMB=\u0000MB0^zunder an inﬂuence of external applied\nZeeman ﬁeld H=H0^z. The free energy of the system\n[18, 48]\nF(MA;MB) =\u0000H0(MAz+MBz)\n\u0000KA\nM2\nA0M2\nAz\u0000KB\nM2\nB0M2\nBz+J\nMA0MB0MA\u0001MB;(7)containing the Zeeman ﬁeld, uniaxial anisotropy energies\nfor individual sublattices in terms of KAandKB, and the\ninter-sublattice magnetic exchange energy J. The eﬀec-\ntive ﬁeld in the ILLG equation can be calculated using\nthe following deﬁnition: He\u000b\nA=\u0000@F=@MAandHe\u000b\nB=\n\u0000@F=@MB. Similar to our consideration of FM, we cal-\nculate the AFM susceptibility assuming hA=hBand\nMA(t) =MA0^z+mA(t)andMB(t) =\u0000MB0^z+mB(t).\nIn the circular basis, the inverse susceptibility expression\nfor AFM is obtained as [48]\n\u0012\nhA\u0006\nhB\u0006\u0013\n=0\nB@1\n\rAMA0\u0000\n\nA\u0006i!\u000bA\u0000\u0011A!2\u0000!\u0001 J\nMA0MB0J\nMA0MB01\n\rBMB0\u0000\n\nB\u0006i!\u000bB\u0000\u0011B!2+!\u00011\nCA\u0012\nmA\u0006\nmB\u0006\u0013\n; (8)\nwith the following deﬁnitions \nA=\rA\nMA0(J+ 2KA+\nH0MA0),\nB=\rB\nMB0(J+2KB\u0000H0MB0). For an AFM, we\nconsider,\rA=\rB=\r,MA0=MB0=M0,KA=KB=\nK,\u0011A=\u0011B=\u0011. With the assumption that J\u001dH0M0\nandJ\u001dK, we have \nA\u0019\nB\u0019\rJ=M 0. Therefore,\nthe obtained approximate frequencies of the antiferromag-\nnetic precession resonance (AFMR) and antiferromagnetic\nnutation resonance (AFMNR) are [48, 51]\n!AFMR\u0019\u0006\r\nM0vuuut4JK\n1 +2\u0011\rJ\nM0; (9)\n!AFMNR\u0019\u0006r\n1 +2\u0011\rJ\nM0\n\u0011: (10)\nNote that unlike FM, the AFM has two sublattices mean-\ning there are two precession resonance frequencies and cor-\nresponding two nutation frequencies. However, we also\nmention that in a two sublattice FM there are two of each\nprecession and nutation resonance frequencies e.g., CoFeB\n[31, 34].\nFirst we examine and compare the conventional and\nnutational eigenmodes in an AFM. To this end, we set\nhA\u0006=hB\u0006= 0and\u000b= 0in Eq. (8) and obtain two\nfollowing equation for mA+andmB+:\n\u0000\n\nA\u0000\u0011!2\u0000!\u0001\nmA++\rJ\nM0mB+= 0;(11)\n\u0000\n\nB\u0000\u0011!2+!\u0001\nmB++\rJ\nM0mA+= 0:(12)\nTwo of such similar equation can also be obtained for mA\u0000\nandmB\u0000. At AFMR, we drop the nutation term and the\nratio between mA+andmB+is obtained as\nmA+\nmB+=\u0000\rJ\nM0\n\nA\u0000!AFMR\u0019\u00001\n1\u0000!AFMRM0\n\rJ:(13)\nOne can easily calculate that!AFMRM0\n\rJ\u001c1and can be\nneglected. Therefore, we obtain mA+\u0019\u0000mB+implying\nNutation modesPrecession modes\nθAθBMAMAMBMBθAθBFigure 3. The precession and nutation modes in antiferromag-\nnets. The precession modes behave as normal antiferromagnets,\nbut for the nutation mode, the spins are not antiparallel giving\nrise to faster dynamics in one sublattice, while the dynamics is\nnot experienced by the other sublattice.\nthat the two sublattice magnetizations remain nearly an-\ntiparallel subtending almost equal cone angles \u0012A\u0019\u0012Bin\nAFMR, as depicted in Fig. 3 (left ﬁgure).\nFor nutation mode, we similarly obtain the ratio with the\nleading term in AFMNR frequency as !AFMNR\u00191=\u0011as\nmA+\nmB+=\u0000\rJ\nM0\n\nA\u0000\u0011!2\nAFMNR\u0000!AFMNR=1\n2M0\n\u0011\rJ\u00001:(14)\nEmployingM0\n\u0011\rJ\u001d1, we ﬁndmA+\u001cmB+. This means at\nthe AFMNR, the two sublattices do not align exactly an-\ntiparallel, giving rise to large cone angle in one sublattice,\nwhile the other sublattice precesses with a much smaller\ncone angle. These characteristics of the nutation eigen-\nmodes have been shown in Fig. 3 (right ﬁgure).\nNext, we calculate the spin pumping current follow-\ning Refs. [52, 53]. To this end, we consider the eﬀects\nof intra-sublattice and also cross-sublattice terms in the\nspin pumping current. The spin mixing conductance for4\n-3 -2 -1 0 1 2 3\nω/2π(THz)−0.75−0.50−0.250.000.250.500.75js,Intra×1024¯h[m−2-s−1](a)\nη= 0\nη= 10−13s\n-3 -2 -1 0 1 2 3\nω/2π(THz)−0.4−0.20.00.20.4js,Cross×1024¯h[m−2-s−1](b)\nη= 0\nη= 10−13s\n-3 -2 -1 0 1 2 3\nω/2π(THz)−1.0−0.50.00.51.0js,Total×1024¯h[m−2-s−1](c)\nη= 0\nη= 10−13s\nFigure 4. The spin current contributions for (a) intra-sublattice terms: MA(t)\u0002_MA(t)+MB(t)\u0002_MB(t)(b) cross-sublattice terms:\nMA(t)\u0002_MB(t) +MB(t)\u0002_MA(t)and (c) total with both intra and cross-sublattice terms calculated without and with the inertial\nrelaxation time i.e., \u0011= 0and\u0011= 10\u000013s. The used parameters are: MA0=MB0=M0= 2\u0016B,\rA=\rB=\r= 1:76\u00021011\nT\u00001-s\u00001,~= 1:05\u000210\u000034m2-kg-s\u00001,g\"#\nr= 10\u000019m\u00002,H0= 0T,hA=hB= 10\u00003T,KA=KB=K= 10\u000023J,J= 10\u000021J,\n\u000bA=\u000bB=\u000b= 0:05and\u0011A=\u0011B=\u0011.\nan AFM interfaces with a metal is a 2\u00022tensor and\ndepends sensitively on the interface [52, 54]. A disor-\ndered interface (expected to be common) eﬀectively be-\nhaves as an uncompensated interface [52, 54]. Here, we\nreport the contributions due to the intra-sublattice and\ncross-sublattice spin pumping separately, keeping in mind\nthat the ﬁnal result is a weighted sum of the two, where\nthe weight depends on the interface [52, 54]. The intra-\nsublattice contributions can be calculated as: MA(t)\u0002\n_MA(t)+MB(t)\u0002_MB(t) =!(mA+mA\u0000+mB+mB\u0000)and\nthe cross-sublattice contributions as: MA(t)\u0002_MB(t) +\nMB(t)\u0002_MA(t) =!(mA+mB\u0000+mB+mA\u0000). The com-\nputed spin current due to intra and cross-sublattice contri-\nbutions become\njs;Intra=~\n4\u0019g\"#\nr\u0002!\u0012mA\u0000mA+\nM2\nA0+mB\u0000mB+\nM2\nB0\u0013\n(15)\njs;Cross =~\n4\u0019g\"#\nr\u0002!\u0012mA\u0000mB+\nMA0MB0+mB\u0000mA+\nMB0MA0\u0013\n(16)\nThe total spin current is calculated from both the intra and\ncross-sublattice contributions as\njs;Total =js;Intra+js;Cross (17)\nThe computed spin currents are shown for intra, cross,\nand total sublattice terms at the inertial relaxation time\n\u0011= 100fs in Fig. 4. Without the application of a static\nmagnetic ﬁeld, the two antiferromagnetic precession res-\nonance modes have exactly the same frequency, however,\nopposite in sign. Therefore, the spin currents appear at\nthe resonance frequencies. Due to the nutation resonance,\nadditional spin current contributions can be observed at\nthe higher THz nutation frequencies. Following the results\nof FM case, the sign of nutation spin currents will be op-\nposite to the corresponding spin current at the precession\nresonance mode, for intra-sublattice contributions.\nAs observed, the magnitude of spin currents at the nu-\ntation resonances is small compared to the spin currents\nat the precession resonance at the lower \u0011e.g., 1 fs. How-\never, it increases rapidly for higher \u0011and surpasses the\nspin currents at the precession resonance already at \u0011\u001810\nfs. Such observation in AFMs is in contrast to the FM,where the nutation spin current is smaller even at \u0011= 1\nps. Note that the antiferromagnetic precession resonance\nis suppressed and thus the spin current at the precession\nresonance is smaller compared to that of the ferromagnetic\nresonance.\nFor intra-sublattice contributions, the nutation spin cur-\nrenthasoppositesigntotheprecessionspincurrent, consis-\ntentwithourﬁndingsforferromagnets[seeFig.4(a)]. How-\never, for the cross-sublattice contributions, the precession\nspin current changes sign, while the nutation spin current\ndoes not change sign compared to the intra-sublattice con-\ntributions [see Fig. 4(b)]. This is due to the fact that the\ntwo sublattices in the antiferromagnetic precession mode\nremain almost antiparallel to each other. On the other\nhand, for the nutation modes in AFM, the two sublattices\ndo not align antiparallel to each other and hence the nu-\ntation spin current do not change sign for intra and cross-\nsublattice contribution. Due to such properties, the total\nspincurrentsattheprecessionresonancealmostcancelwith\neach other, however, the total nutation spin currents add\nup[seeFig.4(c)], assumingequalintra-andcross-sublattice\nspin mixing conductances.\n10−1610−1510−1410−1310−12\nη(s)10−410−2100102104(jAFMNR\ns/jAFMR\ns )\nIntra×(−1)\nCross×(+1)\nTotal×(−1)\nFigure 5. The ratio of spin current for AFMs at the nutation\nresonancetotheprecessionresonancevsinertialrelaxationtime.\nThe used parameters are: MA0=MB0=M0= 2\u0016B,\rA=\n\rB=\r= 1:76\u00021011T\u00001-s\u00001,H0= 0T,KA=KB=K=\n10\u000023J,J= 10\u000021J,\u000bA=\u000bB=\u000b= 0:05and\u0011A=\u0011B=\u0011.5\nTo understand the enhancement of spin currents at the\nnutation resonance, we compute the ratio of spin currents\nat the nutation resoance to the precession resonance in Fig.\n5. By doing so, the results are independent of several pa-\nrameters used in obtaining Fig. 4. For example, the ratio\nof spin currents is independent of spin mixing conductance\ng\"#\nr, small ﬁeld hetc. Compared to the FM spin current in\nFig. 1, the such ratio already crosses unity within \u0011= 10\nfs for AFM. As explained earlier, due to the cancellation of\nthe spin current at the precession resonance, the ratio for\ntotal contribution increases rapidly. In fact, the total spin\ncurrent at the nutation resonance is already higher even\nbelow\u0011= 1fs.\nIV. CONCLUSIONS\nToconclude,wepresentatheoreticalinvestigationofspin\npumping with the magnetic inertial dynamics for one and\ntwo-sublattice systems. The magnetic inertial dynamics\nadditionally introduces a spin pumping current at the THz\nnutation resonance frequencies. However, due to the op-\nposite sense of rotation in precession and nutation modes,\nthe spin pumping current has an opposite sign at the nu-\ntation resonance compared to the one at precession reso-\nnance. Such scenario remains the same for intra-sublattice\nspin current in AFM, while the cross-sublattice spin cur-\nrent only changes sign at the precession resonance. Thus\nthe spin current ratio is negative and positive for AFM in-\ntra and cross-sublattice terms, respectively. While the two\nsublattices remain almost antiparallel in AFM at the pre-\ncessional resonance, the antiparallel alignment is broken at\nthe nutational resonance resulting a large cone angle in one\nof the sublattices. Our obtained results should motivate\nto the experimental search of magnetic inertial dynamics\nin magnets via detection of spin pumping currents.\nACKNOWLEDGMENTS\nWe acknowledge the Swedish Research Council (VR\nGrant No. 2019-06313) and the Spanish Ministry for\nScience and Innovation - AEI Grant CEX2018-000805-M\n(through the “Maria de Maeztu” Programme for Units of\nExcellence in R&D).\nAppendix A: The characteristics of nutation resonance\nin ferromagnet\nFirst, we investigate the characteristics of precession\nand nutation resonance modes in ferromagnet. Following\nRef. [48], we write the equation for inverse susceptibility as\nin Eq. (B6) of Ref. [48]\n\u0012\nhx\nhy\u0013\n=1\n\rM00\nB@\n0+\u000b@\n@t+\u0011@2\n@t2\u0000@\n@t\n@\n@t\n0+\u000b@\n@t+\u0011@2\n@t21\nCA\u0012\nmx\nmy\u0013\n(A1)We replace mx=my/ei!tsuch that@\n@t!i!and@2\n@t2!\n\u0000!2. Therefore, Eq. (A1) can be recast as\n\u0012\nhx\nhy\u0013\n=1\n\rM0\u0012\n\n0+i\u000b!\u0000\u0011!2\u0000i!\ni! \n0+i\u000b!\u0000\u0011!2\u0013\u0012\nmx\nmy\u0013\n(A2)\nNow we employ hx=hy= 0and\u000b= 0in order to examine\nthenatureoftheeigenmodes, ﬁndingarelationbetween mx\nandmy. We obtain two following equations:\n\u0000\n\n0\u0000\u0011!2\u0001\nmx\u0000i!my= 0 (A3)\ni!mx+\u0000\n\n0\u0000\u0011!2\u0001\nmy= 0 (A4)\nThe Eqs. (A3) and (A4) can be written in concise forms as\nmx=i!\n\n0\u0000\u0011!2my (A5)\nmy=\u0000i!\n\n0\u0000\u0011!2mx (A6)\nThe precession and nutation resonance frequencies that\nwere obtained previously in Ref. [48] are:\n!FMR =p1 + 4\u0011\n0\u00001\n2\u0011\u0019\n0\u0000\u0011\n2\n0 (A7)\n!FMNR =\u0000p1 + 4\u0011\n0+ 1\n2\u0011\u0019\u00001\n\u0011\u0000\n0(1\u0000\u0011\n0)(A8)\nmxmymxmy\nmx=cosΩ0t;my=sinΩ0tmx=costη;my=−sintηΩ0t=π/2Ω0t=0tη=π/2tη=0PrecessionmodeNutationmode\nFigure 6. The precession and nutation mode in ferromagnet.\nNotice that the precession and nutation modes have opposite\nsense of rotation.\nWe replace the leading order precession resonance term\ni.e.!FMR\u0019\n0in Eqs. (A5) and (A6). Therefore, at the\nprecession resonance we ﬁnd the relation between mxand\nmyas follows\nmx=i\n0\n\n0my=imy (A9)\nmy=\u0000i\n0\n\n0mx=\u0000imx (A10)\nHowever, the obtained mxandmyare complex quantities.\nIn order to obtain the real and time-dependent parts at the\nprecession resonance, we compute:\nmx=Re\u0002\nmxei!t\u0003\n=Re\u0002\nmxei\n0t\u0003\n=mxcos \n 0t(A11)\nmy=Re\u0002\nmyei!t\u0003\n=Re\u0002\n\u0000imxei\n0t\u0003\n=mxsin \n 0t(A12)6\nAt the nutation resonance, however, we replace the leading\norder frequency term i.e., !FMNR\u0019\u00001=\u0011and ﬁnd\nmx=i(\u00001=\u0011)\n\n0\u00001=\u0011my=\u0000i\n\u0011\n0\u00001my=imy(A13)\nmy=\u0000i(\u00001=\u0011)\n\n0\u00001=\u0011mx=i\n\u0011\n0\u00001mx=\u0000imx(A14)\nWe again calculate the real and time-dependent parts atthe nutation resonance as\nmx=Re\u0002\nmxei!t\u0003\n=Re2\n64mxe\u0000it\n\u00113\n75=mxcost\n\u0011\n(A15)\nmy=Re\u0002\nmyei!t\u0003\n=Re2\n64\u0000imxe\u0000it\n\u00113\n75=\u0000mxsint\n\u0011\n(A16)\nSuch characteristics of precession and nutation resonance\nhave been shown in Fig. 6.\n[1] E. Beaurepaire, J.-C. Merle, A. Daunois, and J.-Y. Bigot,\nPhys. Rev. Lett. 76, 4250 (1996).\n[2] T. Kampfrath, M. Battiato, P. Maldonado, G. Eil-\ners, J. Notzold, S. Mahrlein, V. Zbarsky, F. Freimuth,\nY. Mokrousov, S. Blügel, M. Wolf, I. Radu, P. M. Oppe-\nneer, and M. Münzenberg, Nat. Nano. 8, 256 (2013).\n[3] J. Walowski and M. Münzenberg, J. Appl. Phys. 120,\n140901 (2016).\n[4] R. John, M. Berritta, D. Hinzke, C. Müller, T. San-\ntos, H. Ulrichs, P. Nieves, J. Walowski, R. Mondal,\nO. Chubykalo-Fesenko, J. McCord, P. M. Oppeneer,\nU. Nowak, and M. Münzenberg, Sci. Rep. 7, 4114 (2017).\n[5] T. A. Ostler, J. Barker, R. F. L. Evans, R. W. Chantrell,\nU. Atxitia, O. Chubykalo-Fesenko, S. El Moussaoui,\nL. Le Guyader, E. Mengotti, L. J. Heyderman, F. Nolt-\ning, A. Tsukamoto, A. Itoh, D. Afanasiev, B. A. Ivanov,\nA. M. Kalashnikova, K. Vahaplar, J. Mentink, A. Kirilyuk,\nTh.Rasing, andA.V.Kimel,Nat.Commun. 3,666(2012).\n[6] S. Wienholdt, D. Hinzke, K. Carva, P. M. Oppeneer, and\nU. Nowak, Phys. Rev. B 88, 020406 (2013).\n[7] S. Wienholdt, D. Hinzke, and U. Nowak, Phys. Rev. Lett.\n108, 247207 (2012).\n[8] L. D. Landau and E. M. Lifshitz, Phys. Z. Sowjetunion 8,\n153 (1935).\n[9] T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004).\n[10] T. G. H. Blank, K. A. Grishunin, E. A. Mashkovich, M. V.\nLogunov, A. K. Zvezdin, and A. V. Kimel, Phys. Rev. Lett.\n127, 037203 (2021).\n[11] R. Mondal, A. Donges, U. Ritzmann, P. M. Oppeneer, and\nU. Nowak, Phys. Rev. B 100, 060409(R) (2019).\n[12] M.-C. Ciornei, J. M. Rubí, and J.-E. Wegrowe, Phys. Rev.\nB83, 020410 (2011).\n[13] J.-E. Wegrowe and M.-C. Ciornei, Am. J. Phys. 80, 607\n(2012).\n[14] H. Suhl, IEEE Transactions on Magnetics 34, 1834 (1998).\n[15] D. Böttcher and J. Henk, Phys. Rev. B 86, 020404 (2012).\n[16] R. Mondal, M. Berritta, and P. M. Oppeneer, Phys. Rev.\nB94, 144419 (2016).\n[17] R. Mondal, M. Berritta, and P. M. Oppeneer, Phys. Rev.\nB98, 214429 (2018).\n[18] A. Kamra, R. E. Troncoso, W. Belzig, and A. Brataas,\nPhys. Rev. B 98, 184402 (2018).\n[19] P. Nieves, U. Atxitia, R. W. Chantrell, and O. Chubykalo-\nFesenko, Low Temp. Phys. 41, 739 (2015).\n[20] H. Y. Yuan, Q. Liu, K. Xia, Z. Yuan, and X. R. Wang,\nEPL126, 67006 (2019).\n[21] M. Fähnle and C. Illg, J. Phys.: Condens. Matter 23,493201 (2011).\n[22] M.Fähnle, D.Steiauf, andC.Illg,Phys.Rev.B 84,172403\n(2011).\n[23] S. Bhattacharjee, L. Nordström, and J. Fransson, Phys.\nRev. Lett. 108, 057204 (2012).\n[24] U. Bajpai and B. K. Nikolić, Phys. Rev. B 99, 134409\n(2019).\n[25] R.Mondal, M.Berritta, A.K.Nandy, andP.M.Oppeneer,\nPhys. Rev. B 96, 024425 (2017).\n[26] R. Mondal, M. Berritta, and P. M. Oppeneer, J. Phys.:\nCondens. Matter 30, 265801 (2018).\n[27] S. Giordano and P.-M. Déjardin, Phys. Rev. B 102, 214406\n(2020).\n[28] S. V. Titov, W. T. Coﬀey, Y. P. Kalmykov, M. Zarifakis,\nand A. S. Titov, Phys. Rev. B 103, 144433 (2021).\n[29] Y. Li, A.-L. Barra, S. Auﬀret, U. Ebels, and W. E. Bailey,\nPhys. Rev. B 92, 140413 (2015).\n[30] I. Makhfudz, E. Olive, and S. Nicolis, Appl. Phys. Lett.\n117, 132403 (2020).\n[31] K. Neeraj, N. Awari, S. Kovalev, D. Polley,\nN. Zhou Hagström, S. S. P. K. Arekapudi, A. Semisalova,\nK. Lenz, B. Green, J.-C. Deinert, I. Ilyakov, M. Chen,\nM. Bawatna, V. Scalera, M. d’Aquino, C. Serpico, O. Hell-\nwig, J.-E. Wegrowe, M. Gensch, and S. Bonetti, Nat.\nPhys. 17, 245 (2021).\n[32] E. Olive, Y. Lansac, M. Meyer, M. Hayoun, and J.-E.\nWegrowe, J. Appl. Phys. 117, 213904 (2015).\n[33] M. Cherkasskii, M. Farle, and A. Semisalova, Phys. Rev.\nB102, 184432 (2020).\n[34] R. Mondal, J. Phys.: Condens. Matter 33, 275804 (2021).\n[35] V. Unikandanunni, R. Medapalli, M. Asa, E. Albisetti,\nD. Petti, R. Bertacco, E. E. Fullerton, and S. Bonetti,\n“Inertial spin dynamics in epitaxial cobalt ﬁlms,” (2021),\narXiv:2109.03076 [cond-mat.mes-hall].\n[36] R. Rahman and S. Bandyopadhyay, J. Phys.: Condens.\nMatter 33, 355801 (2021).\n[37] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. B 66, 224403 (2002).\n[38] E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Appl.\nPhys. Lett. 88, 182509 (2006).\n[39] K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae,\nK. Ando, S. Maekawa, and E. Saitoh, Nature 455, 778\n(2008).\n[40] G.E.W.Bauer, E.Saitoh, andB.J.vanWees,Nat.Mater.\n11, 391 (2012).\n[41] J.BarkerandG.E.W.Bauer,Phys.Rev.Lett. 117,217201\n(2016).7\n[42] T. S. Seifert, S. Jaiswal, J. Barker, S. T. Weber, I. Raz-\ndolski, J. Cramer, O. Gueckstock, S. F. Maehrlein, L. Nad-\nvornik, S. Watanabe, C. Ciccarelli, A. Melnikov, G. Jakob,\nM. Münzenberg, S. T. B. Goennenwein, G. Woltersdorf,\nB. Rethfeld, P. W. Brouwer, M. Wolf, M. Kläui, and\nT. Kampfrath, Nat. Commun. 9, 2899 (2018).\n[43] D. Hirobe, M. Sato, T. Kawamata, Y. Shiomi, K. I. Uchida,\nR. Iguchi, Y. Koike, S. Maekawa, and E. Saitoh, Nat. Phys.\n13, 30 (2017).\n[44] M. 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, Phys. Rev. Lett. 111, 176601 (2013).\n[45] R. Cheng, J. Xiao, Q. Niu, and A. Brataas, Phys. Rev.\nLett.113, 057601 (2014).\n[46] P. Vaidya, S. A. Morley, J. van Tol, Y. Liu, R. Cheng,\nA. Brataas, D. Lederman, and E. del Barco, Science 368,160 (2020).\n[47] J. Li, C. B. Wilson, R. Cheng, M. Lohmann, M. Kavand,\nW. Yuan, M. Aldosary, N. Agladze, P. Wei, M. S. Sherwin,\nand J. Shi, Nature 578, 70 (2020).\n[48] R. Mondal, S. Großenbach, L. Rózsa, and U. Nowak, Phys.\nRev. B 103, 104404 (2021).\n[49] T. Kikuchi and G. Tatara, Phys. Rev. B 92, 184410 (2015).\n[50] K. Ando, S. Takahashi, J. Ieda, Y. Kajiwara, H. Nakayama,\nT. Yoshino, K. Harii, Y. Fujikawa, M. Matsuo, S. Maekawa,\nand E. Saitoh, J. Appl. Phys. 109, 103913 (2011).\n[51] R. Mondal and P. M. Oppeneer, Phys. Rev. B 104, 104405\n(2021).\n[52] A. Kamra and W. Belzig, Phys. Rev. Lett. 119, 197201\n(2017).\n[53] Q. Liu, H. Y. Yuan, K. Xia, and Z. Yuan, Phys. Rev.\nMaterials 1, 061401 (2017).\n[54] R. E. Troncoso, M. A. Lund, A. Brataas, and A. Kamra,\nPhys. Rev. B 103, 144422 (2021)."    },    {        "title": "2207.10349v1.Influence_of_nonuniform_magnetization_reorientation_on_spin_orbit_torque_measurements.pdf",        "content": "Influence of nonuniform magnetization reorientation on spin -orbit torque measurement s  \nRyan W. Greening1, Xin Fan1 \n1. Department of Physics and Astronomy, University of Denver, Denver, C olorado 80210,  USA  \n \nAbstract  \nMeasurement s of spin -orbit torque s in a ferromagnetic/nonmagnetic multilayer are typically based on \nan assumption that the entire ferro magnetic layer uniformly responds to the spin -orbit torque. This \nassumption breaks down when the thickness of the ferromagnetic layer  is comparable to the dynamic  \nexchange  coupling  length, which can be  as short as a few nanometers  in certain measurement geometr ies. \nThe nonuniform magnetization reorientation coupl ed with nonuniform contribution of each magnetic \nsublayer to the magnetores istance or the Kerr effect may impact the accuracy in the extrapolation of spin -\norbit torque , particularly if a thick ferromagnetic layer is used . In this paper, we use numerical model s to \ninvestigate such an impact  in three different  techniques: the magneto -optic -Kerr-effect method, the second -\nharmonic method  and the spin torque ferromagnetic resonance method. We show that the second -harmonic \nand magneto -optic -Kerr-effect methods are prone to be influenced by the nonuniform magnetizati on \nreorientation, while the spin torque ferromagnetic resonance method is much less impacted.  \n \n1. Introduction  \nIn recent years, research on spin -orbit interaction has unveiled numerous phenomena with spin charge \ninterconversion s [1-5]. Among these interesting phenomena, spin -orbit torque – a spin torque generated in \na magnetic multilay er film by an in -plane electric current via the spin -orbit interaction – has attracted \nconsiderable attention. Due to the high efficiency in switching magnetization  [6, 7] , moving magnetic \ndomains and skyrmions  [8-10], the spin -orbit torque holds potential in the development of future m agnetic \nrandom access memories [11]. Various measurement techniques have been developed to quantify the spin -\norbit torque, many of which are based on the perturbation method . Typically , a small applied in -plane \nelectric current generates a spin -orbit torque that reorients the magnetization of the magnetic layer. The \nmagnetization reorientation is detected through measurements of the magnetoresistance , Hall effect  or the \nKerr effect [12-19]. In these measurements, it is usually assumed that the magnetization reorientation due \nto the spin -orbit torque is uniform throughout the magnetic film.  However, in a typical spin -orbit device \nconsisting of a FM/NM bilayer, where the FM is a ferromagnetic material and the NM is a nonmagnetic \nmaterial, the spin -orbit torque is only generated at the two surfaces of the FM  due to strong spi n dephasing \n[20]. If the FM thickness is comparable to its dynamic  exchange  coupling  length, the magnetization  \nreorientation due to the spin -orbit torque is not necessarily uniform across the film thickness.  On the other \nhand, the calibration procedures in these experiments are often conducted by applying a magnetic field that \nrotates the magnetization uniformly. It is therefore important to take into consideration the nonuniform \nmagnetization reori entation when quantifying the magnitude of the spin -orbit torque.  \n 2. Model  \nIn this paper we  use a FM/NM bilayer  with in -plane magnetization as an example. An in -plane current \ncan generate spin-orbit torques, possib ly from the spin Hall effect of the NM, the spin -orbit coupling at the \nFM/NM interface or the transverse spin Hall effect of the FM itself  [21]. Regardless of the microscopic \norigin, due to the strong spin dephasing  [20, 22] , spin torque s shall arise only at the top and bottom surfaces \nof the FM layer, denoted as 𝜏T and 𝜏B, respectively  [23], as illustrated in Fig. 1(a) . While the torque s should \nonly generate effective field s on the FM layers near the surfaces, the resultant magnetization tilting \npropagates through  the FM bulk via exchange coupli ng. The propagation length, which is referred to as the \ndynamic  exchange coupling length  here, depends on the exchange constant as well as the susceptibility of \nthe FM.  \nFigure 1 Illustration of magnetization response to surface spin torques for (a) non -uniform magnetization \nreorientation under finite exchange coupling and (b) uniform magnetization reorientation under infinite \nexchange coupling. Here red arrows are magnetization vectors. Blue arrows are spin orientations, yellow \narrows represent spin curren t flow directions and green arrows denote spin torques  at the top and bottom \ninterfaces . (c) Illustration of how the ferromagnetic layer is divided into sublayers for numerical simulations \nin this paper.  \n \n2.1 Dynamic exchange coupling length  \nWe model the ferromagnet by dividing it into many thin layers, each with thickness of a. The \nmagnetization is initially uniform across the film. Driven by su rface spin torques, the non -uniform \nmagnetization tilting gives rise to an effective exchange coupling field from nearest neighbor layers,  \n𝐡exc=2𝐴exc\n𝜇0𝑀s𝑎2[(𝐦𝑖−1−𝐦𝑖)+(𝐦𝑖+1−𝐦𝑖)]=2𝐴exc\n𝜇0𝑀s𝑑2δ𝐦𝑖\n𝑑𝑧2   (1) \nwhere Aexc is the exchange stiffness , 𝑀s is the saturation magnetization,  𝐦 is the unit magnetization vector \nwith subscript as the layer index , δ𝐦𝑖 is the deviation of magnetization from its initial equilibrium direction.  \nTherefore, f or FM layers within the bulk, the magnetizati on reorientation is only due to the effective \nexchange coupling field,  \n δ𝐦𝑖=𝜒𝐡exc\n𝑀s=2𝜒𝐴exc\n𝜇0𝑀s2𝑑2δ𝐦𝑖\n𝑑𝑧2    (2) \n, where 𝜒 is the susceptibility.  \nFor the top surface layer, the magnetization reorientation is due to the combination of effective \nexchange coupling field as well as the effective field due to the spin torque.  \nδ𝐦1=𝜒\n𝑀s[2𝐴exc\n𝜇0𝑀s𝑎2(𝐦2−𝐦1)+𝜏T\n𝜇0𝑀s𝑎].   (3) \nWhen  we approximate  𝑎→0, Eq. (3) can be simplified to  \n𝑑δ𝐦1\n𝑑𝑧=𝜏T\n2𝐴exc     (4) \nCombining Eqs. (2) and (4), we can derive the magnetization reorientation due to  surface torques 𝜏T and \n𝜏B as \nδ𝐦(z)=𝜒𝜏Tcosh𝑧\n𝜆+𝜏Bcosh𝑑−𝑧\n𝜆\n𝜇0𝑀s2𝜆sinh𝑑\n𝜆,    (5) \nWhere d is the magnetic film thickness,  and 𝜆=√2𝜒𝐴exc\n𝜇0𝑀s2 is the dynamic  exchange coupling length. We \nassumed 𝜒 as a scalar in the above derivation , however i n the case of ferromagnetic resonance, 𝜒 should \nbe expressed as a 2 x 2 matrix . The equations (2 -5) are still applicable when 𝜒 and 𝜆 are treated as matrices,  \n𝜏T and 𝜏B are treated as vectors.  All relevant matrices in Eq. (5) are commut ative . \nIt is worth noting that the dynamic  exchange coupling length is different from the magnetostatic \nexchange length √2𝐴exc\n𝜇0𝑀s2 [24], which  is mainly for describing domain walls and  only depends on the \nexchange stiffness. Instead, the dynamic  exchange coupling length 𝜆 in our study also depends on the \nmagnetic susceptibility. This can be understood from Eq. (2) that the influence on magnetiz ation from \nneighboring layer s increases with susceptibility, and therefore propagates further.  As a result, the dynamic  \nexchange coupling length varies depending on the measurement geometry and even measurement \ntechnique. Taking  a Py/Pt bilayer with in -plane magnetization as an example, the damping -like torque is \nequivalent to an out -of-plane magnetic field that tilts the magnetization out of plane. In a DC measurement, \nthe corresponding susceptibility is 𝜒=𝑀s\n|𝐻ext|+𝑀eff, where Hext is the in -plane applied magnetic field, Meff \nis the effective demagnetizing field. Using 𝐴exc=10−11J∙m, 𝜇0𝐻ext=0.01 T 𝜇0𝑀s=1 T, we estimate  \n𝜒≈1 and 𝜆≈5 nm. However, for field-like torque that rotates magnetization in the film plane, the \nrelevant su sceptibility is 𝜒=𝑀s\n|𝐻ext|=100. The dynamic  exchange  coupling  length  for this in -plane magnetization rotation  is estimated to be about 50 nm.  In a spin -torque ferromagnetic resonance \nmeasurement, 𝜒 is much enhanced by resonance, thus leading  to a long dynamic  exchange  coupling  length. \nTherefore, as we will see in a numerical simulation later, the magnetization distribution in a spin -torque \nferromagnetic resonance measurement is much more uniform tha n that in a DC /AC measurement.  \n2.2 Simulation of spin -orbit torque measurements responses  \nIn this numerical study, without loss of generality, we will use Py /NM bilayer as an example and \ncompare results from two different Py thicknesses: 3 nm and 8 nm. The 3 nm Py thickness is shorter than \nthe dynamic  exchange coupling length in all measurements, while the 8 nm Py may be thicker than the \ndynamic  exchange coupl ing length in some measurements.  For each sample, we will assume two scenarios \nfor the surfaces spin -orbit torques: (1) |𝜏T|<|𝜏B| with an exemplary case of Py/Pt [23] (2) |𝜏T|>|𝜏B| \nwith an exemplary case of Py/Ta.   \nThe damping -like torque m easurement results of these samples will be simulated via three different \ntechniques: (1) the magneto -optic -Kerr-effect  (MOKE)  based spin -orbit torque measurement , (2) the \nsecond -harmonic measurement, and (3) the spin-torque ferromagnetic resonance measurement. We will \nfirst present  the magnetization reorientation by damping -like torque in these measurement s. The resultant \nMOKE and electrical signals will be simula ted and compared to th ose under the assumption of uniform \nmagnetization reorientation .  \n2.2.1 Simulation of the MOKE -based measurement  \nIn the MOKE -based spin -orbit torque measurement [17], an in -plane electric current generates a \ndamping -like torque, that tilts magnetization out of plane. The out -of-plane magnetization is detected via \nthe polar MOKE response.  The polar MOKE signal due to the damping -like torque is expected to resemble \nmagnetization hysteresis. Therefore, the applied external magnetic field can be small, e.g. 𝜇0𝐻ext  = 0.01T. \nThe dynamic  exchange coupling length is calculated to be about 5 nm. Under the influence of damping -\nlike torques  𝜏T and 𝜏B at the surfaces, we simulate the out -of-plane magnetization tilting  𝑚z. As a \ncomparison, we also simulate an extreme case with  infinite exchange stiffness 𝐴exc→∞, where \nmagnetization uniformly tilts 𝑚𝑧0 responding to the total torque,  𝜏T+𝜏B. In this case, the uniform \nmagnetization reorientation can be calculated as 𝑚𝑧0=𝜏T+𝜏B\n𝜇0𝑀s2𝑑. In Fig. 2, we plot the deviation of 𝑚z from \n𝑚𝑧0, ∆𝑚z=𝑚z−𝑚𝑧0 , normalized by 𝑚𝑧0,  as a function of layer position z. The four plots correspond to \ncombinational conditions of two different Py thicknesses (3 nm and 8 nm) and two different surface t orque \nratios ( 𝜏T/𝜏B = 0.2 and 𝜏T/𝜏B = -1.3). For 3 nm Py, which is thinner than the dynamic  exchange coupling \nlength, the magnetization reorientation is relatively uniform. On the other hand, the nonuniform \nmagnetization reorientation in the 8 nm Py  is much more pronounced . The 𝜏T/𝜏B = -1.3 case promotes \nstronger nonuniform magnetization reorientation than the  𝜏T/𝜏B = 0.2  case, because the former has \nrelatively weaker total spin -orbit torque  compared to individual surface torque .  \nFigure 2  Simu lation of magnetization reorientation due to surface spin -orbit torques for 8 nm Py ((a) and (c)) and 3 \nnm Py ((b) and (d))  in a typical MOKE measurement . Two types of torques are considered: 𝜏T/𝜏B = 0.2, where the \nheavy metal has a spin Hall angle like Pt for cases (a) and (b), and 𝜏T/𝜏B = -1.3, where the heavy metal has a spin \nHall angle like Ta for cases (c) and (d).  \nSince light has a finite penetration depth in typical metals, the MOKE respons e of each magnetic \nsublayer is not a constant. To the first order approximation, the total polar MOKE signal can be expressed \nas 𝜃k=∫𝜗(𝑧)𝑚z(𝑧)𝑑\n0𝑑𝑧, where 𝜗(𝑧) is the MOKE contribution from the layer at height z, 𝑚z(𝑧) is the \nout-of-plane magneti zation rotation due to the spin torques .  We model  the MOKE response using the \npropagation matrix method  [25]. Detailed simulation and parameters used are discussed in Appendix A1.  \nDue to the wave nature of light, t he Kerr angle is generally a complex number. Figure 3(a) and (b) shows \nthe real and imagin ary part of 𝜗(𝑧) as a function of layer height z for 8 nm Py and 3 nm cases respectively. \nThere are considerable variations of 𝜗(𝑧) as a function of z. The variation of 𝜗(𝑧) couples with the non -\nuniform magnetization reorientation 𝑚z(𝑧) will lead to a different overall MOKE signal from what is \nexpected in a uniform magnetization reorientation  (assuming 𝐴exc→∞). In Fig s.3 (c) and ( e), we plot the \nreal part of the MOKE signals as a function of Py thickness  d for both the realistic non -uniform \nmagnetization reorientation and the assumed uniform magnetization reorientation in two surface torques \nscenarios ( 𝜏T/𝜏B = 0.2 and 𝜏T/𝜏B = -1.3). Fig ures 3(d) and (f) display  the corresponding percentage \ndifference  in MOKE response between the uniform and non -uniform magnetization reorientation cases , \n𝜀MOKE =𝜃Kerrnon −uniform−𝜃Kerruniform\n𝜃Kerruniform ×100% . The deviation is very small when Py is comparable to the \ndynamic exchange coupling length (~ 5 nm ), but  incre ases dramatically as Py gets thicker than 15 nm  due \nto the combination of non-uniform magnetization reorientation as well as the finite light penetration depth .  \nBased on th ese simulation result s, we conclude that when using MOKE to measure the net spin-orbit \ntorque, the thickness of the ferromagnetic metal should be chosen to be shorter or comparable to the \ndynamic exchange coupling length.  In this case , the deviation from the assumed uniform magnetization \nreorientation will be  negligibly small.  On the other hand, if only one surface spin torque is of interest, one \ncan choose the ferromagnet to be considerably thicker than the dynamic exchange coupling length and \nlight penetration depth to , for example,  observe the anomalous spin -orbit torque in a si ngle-layer \nferromagnet  [23].  \n \nFigure 3 MOKE response simulation. (a -b) Kerr angle contribution from each magnetic layer as a function of z-\nposition for 8 nm (a) and 3 nm (b) Py. (c,e) Real part of the  MOKE response under uniform and nonuniform \nmagnetization tilting with 𝜏T/𝜏B = 0.2 (c)   and 𝜏T/𝜏B = -1.3 (e) as a function of film thickness . (d,f) Percent \ndifference in MOKE response ( 𝜀MOKE ) between the uniform and non -uniform magnetization reorientatio n. \n2.2.2 Simulation of the second -harmonic measurement  \nFor the second -harmonic measurement of the damping -like torque  on in -plane magnetized film , we \nadopt  the protocol developed by Avci et al.  [26]. An in -plane electric current is applied through the  sample \nand a second -harmonic transverse voltage is measured. The second -harmonic voltage signal consists of \nthe planar Hall voltage due to field-like torque , the anomalous Hall voltage due to damping -like torque  \nand the anomalous Nernst effect due to Joule’s heating. The three signals can be distinguished via external \nmagnetic field dependence. To suppress the planar Hall voltage contribution, a large in -plane magnetic \nfield shall be applied. In this simulation, we ch oose 𝜇0𝐻ext  = 1T. The dynamic  exchange coupling length \nis about 3.5 nm. The magnetization reorientations, as shown in Fig. 4 have very similar behavior as th ose \nin Fig. 2 , but with larger deviation from uniform magnetization due to even shorter dynamic exchange \ncoupling length.  \n \nFigure 4 Simulation of magnetization reorientation due to surface spin -orbit torques for 8 nm Py ((a) and (c)) and 3 \nnm Py  ((b) and (d)) in a typical second -harmonic measurement. Two types of torques are considered: 𝜏T/𝜏B = 0.2, \nwhere the heavy metal has a spin Hall angle like Pt for cases (a) and (b), and 𝜏T/𝜏B = -1.3, where the heavy metal \nhas a spin Hall angle like Ta for cases (c) and (d).  \n \nThe second -harmonic anomalous Hall voltage can be expressed as  𝑉AH=∫𝑅𝑗e(𝑧)𝜃AH(𝑧)𝑚z(𝑧)𝑑\n0𝑑𝑧, \nwhere R is the total resistance in the transverse direction, je(z) is the electric current density, and θAH(z) is \nthe anomalous Hall angle. The electric current density je(z) is z-dependent because of interface scattering . \nGenerally speaking the current density is lower near the surfaces, and higher in the center of the film. The \nscattering rate depends on the details of the interfaces. For simplicity, we will assume that the NM in the \nPy/NM multilayer has the same electricity as Py, and ignores the scattering at the Py/NM interface. In the \nsimulation of  the electric current density distribution, as illustrated in the inset of Fig. 5(a), we treat the \nPy/NM as a single “extended”  Py film with the thickness equals to the total thickness of the original \nheterostructure , 𝑑̃=𝑑+𝑑NM, where dNM is the NM thickness . Only the electric current within 0≤𝑧≤\n𝑑 contributes to the anomalous Hall signal.  We use the Fuchs -Sondheimer method [27] and assume \ninfinite scattering at both surfaces of the “extended” Py film. The thickness -dependent current density \ndistribution is given as  \n𝑗[𝑧]∝∫ sin3𝛽[1−𝑒−𝑑̃/2\n𝜆mfcos 𝛽cosh[𝑧−𝑑̃/2\n𝜆mfcos 𝛽]]𝑑𝛽𝜋/2\n0,    (6) \nwhere 𝜆mf is the mean free path of the “extended” Py film. Using 𝜆mf=5 nm , dNM = 3 nm, w e simulate \nthe distribution of the current density as in Fig. 5 (a-b), where asymmetric current distribution is observed.  \nAssuming there is no distribution of the anomalous Hall angle, i.e. 𝜃AH(𝑧)=𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 , the second -\nharmonic anomalous Hall voltage can be simulated  as a function of Py thickness d. As shown in Fig s 5 \n(c,e), deviation s are observed between the case of nonuniform magnetization tilting (the one taking into \naccount the finite dynamic exchange length) and the case of uniform magnetization tilting (assuming \ninfinite exchange coupling) . Figures 5( d) and (f) display the correspond ing percentage difference in the \nanomalous Hall response between the uniform and non -uniform magnetization reorientation cases, 𝜀AH=\n𝑉AHnon −uniform−𝑉AHuniform\n𝑉AHuniform ×100% . The deviation becomes prominent when the Py thickness is a few times  \nthe dynamic exchange length, suggesting the uniform magnetization tilting assumption may yield a large \nerror in the extrapolation of the spin -orbit torque when the ferromagnetic layer is relatively thick.  \n \n \n \n \n \n \n  \n \nFigure 5 Simulation of the second -order anoma lous Hall voltage response. (a -b) Current distribution \nwithin Py for 8 nm and 3nm Py samples. Inset of Fig. (a) illustrates the approximation taken for this \nsimulation. The Py/NM bilayer is treated as an extended single Py layer. Here dNM is chosen to be 3  nm. \n(c,e) Simulated anomalous Hall voltage  under uniform and nonuniform magnetization tilting  with 𝜏T/𝜏B = 0.2 (c)   \nand 𝜏T/𝜏B = -1.3 (e) as a function of film thickness. (d,f) P ercent difference in anomalous Hall voltage  response \n(𝜀AH) between the uniform and non -uniform magnetization reorientatio n. \n \nIt should be emphasized that the simulation of the current density is oversimplified compar ed to a \nrealistic situation. However, we believe it conveys a  qualitatively  accurate  picture  that a nonuniform \ncurrent distribution coupl ed with  a nonuniform magnetization tilting will lead to an overall anomalous \nHall voltage different from what one might expect  under  uniform magnetization tilting.  In addition, the \nassumption of a constan t anomalous Hall angle may also be challenged. An anomalous Hall -like signal \ncan also arise from the spin Hall magnetoresistance and the imaginary part of the spin mixing conductance \n[28, 29] . This mechanism will give rise to a n additional anomalous Hall angle that only depend s on the \nmagnetization at the Py/NM interface.  \n \n \n2.2.3 Simulation of the spin -torque ferromagnetic resonanc e measurement  \nFor the spin -torque ferromagnetic resonance measurement [12], a rf current is applied through the \nsample . The spin-orbit torque drives in -plane magnetization precession, resulting in an oscillation of the \nanisotropic magnetoresistance.  The resistance oscillation couples with the rf current, giving rise to a \nrectifying voltage. The voltage signal that resembles a symmetric Lorentzian -dependence on external field \nis attributed to the damping -like torque.  In this simulation, we choose the rf frequency to be 6 GHz, with \nan external magnetic field 𝜇0𝐻ext≈0.05 T, slightly off resonance.  When other magnetic field values near \nresonance are chosen, we observe qualitatively the same results.  We first simulate the in -plane \nmagnetization precessio n as a result of damping -like torques  at the surfaces . Shown in Fig. 6, the non -\nuniform ity of  magnetization precession in both 3 nm Py and 8 nm Py sample is negligibly small. This can \nbe explained  by the long dynamic exchange length, enhanced by the large susceptibility at resonance.  \n \nFigure 6 Simulation of magnetization reorientation due to surface spin -orbit torques for 8 nm Py ((a) and (c)) and 3 \nnm Py ((b) and (d)) in a typical spin torque ferromagnetic r esonance measurement. Two types of torques are \nconsidered: 𝜏T/𝜏B = 0.2 for cases (a) and (b), and 𝜏T/𝜏B = -1.3 for cases (c) and (d).  \n \nThe spin -torque ferromagnetic resonance measures the rectifying voltage due to the anisotropic \nmagnetoresistance, which can be expressed as  𝑉ST−FMR =∫𝑅𝑗e(𝑧)𝜃AMR (𝑧)∆𝑚x(𝑧)𝑑\n0𝑑𝑧, where R is the \ntotal resistance in the transverse direction, je(z) is t he electric current density, and θAMR(z) is the anisotropic \nmagnetoresistance ratio , ∆𝑚x(𝑧) is the in -plane magnetization reorientation due to the spin -orbit torques . \nWe assume the same layer -dependent current density je(z) as that in section 2.2.2 . Figure 7 (a ,c) shows the \nrectified 𝑉ST−FMR signals for both non -uniform and uniform magnetization reorientations in the two \ndifferent spin torque configurations. Figures 7( b) and ( d) display the corresponding percentage differen ce \nin the rectifying voltage  between the uniform and non -uniform magnetization reorientation cases, \n𝜀ST−FMR =𝑉ST−FMRnon −uniform−𝑉ST−FMRuniform\n𝑉ST−FMRuniform ×100% . Because there is nearly no layer -dependent magnetization \nreorientation, the deviation of 𝑉ST−FMR due to nonuniform magnetization tilting from that due to uniform \nmagnetization tilting is negligibly small even for relatively thick Py films.  \n \nFigure 7 Simulation of the ST -FMR voltage response . (a,c) Rectified 𝑉ST−FMR signals for both non -uniform \nand uniform magnetization reorientations tilting  with 𝜏T/𝜏B = 0.2 (a)   and 𝜏T/𝜏B = -1.3 (b)  as a function of film \nthickness. (b,d) Percent difference in ST-FMR  voltage response ( 𝜀𝑆𝑇−𝐹𝑀𝑅) between the uniform and non -uniform \nmagnetization  reorientatio n. \n \n3. Discussion  \nSince the spin -orbit torque arises where inversion symmetry is broken, in a typical \nferromagnetic/nonmagnetic bilayer film, the spin -orbit torques are usually located at the two surfaces of \nthe ferromagnet. The influence of the spin -orbit torque propagates  within the ferromagnet via exchange \ncoupling, but not indefinitely. The dynamic exchange coupling length depends not only on the exchange \nconstant of the ferromagnet, but also the magnetic susceptibility of the specific measurement technique. \nWhen the fer romagnet is thinner than the dynamic exchange coupling length, it is a reasonable \napproximation that the magnetization uniformly responds to the total spin -orbit torques. However, when \nthe ferromagnet thickness exceeds the dynamic exchange coupling length,  magnetization reorientation is \nnonuniform. The nonuniform magnetization tilting coupled with nonuniform response from different \nprobing method may give rise to a signal that is different from what is commonly anticipated from a \nuniform magnetization reori entation.   \nOf the three techniques that we studied, the spin -torque ferromagnetic resonance generally yields the \nlongest dynamic exchange coupling length , making it more suitable for measuring total spin -orbit torques \nin thick ferromagnets. The second -harm onic method with the smallest magnetic susceptibility  tends to \nhave the shortest dynamic exchange coupling length. Moreover, since the magnetic susceptibility is a \nfunction of external magnetic field, the dynamic exchange coupling length can vary with the sweeping of \nthe external magnetic field. When these techniques are applied to a ferromagnet thicker than the dynamic \nexchange coupling length, special caution  should be taken in the analysis of the spin -orbit torques. On the \nother hand, the short exchange coupling length may allow  the second -harmonic method  to probe spin -\norbit torque at an individual surface , similarly to the anomalous spin -orbit torque s observed via MOKE \n[23].  \n \nAcknowledgement   \nThe authors would like to express their great appreciation for Prof. Chia -ling Chien’s inspirations on many \nscientists around the world. This work was supported by the National Science Foundation under Grant No. \n2105218 . \nDeclaration of Competing Interest   \nThe authors declare that t hey have no known competing financial interests.  \n \nReference s \n[1] S.O. Valenzuela, M. Tinkham, Direct electronic measurement of the spin Hall effect, Nature, 442 (2006) \n176-179.  \n[2] E. Saitoh, M. Ueda, H. Miyajima, G. Tatara, Conversion of spin current into charge current at room \ntemperature: Inverse spin -Hall effect, Applied physics letters, 88 (2006) 182509.  [3] D. MacNeill, G. Stiehl, M. Guimaraes, R. Buhrman, J. Park, D. Ralph, Control of spin –orbit torques \nthrough crystal symmetry in WTe2/ferromagnet bilayers, Nature Physics, 13 (2017) 300 -305.  \n[4] A.M. Humphries, T. Wang, E.R. Edwards, S.R. Allen, J.M. Shaw, H.T. Nembach, J.Q. Xiao, T.J. Silva, X. \nFan, Observation of spin -orbit effects with spin rotation symmetry, Nature c ommunications, 8 (2017) 911  \n[5] M. Kimata, H. Chen, K. Kondou, S. Sugimoto, P.K. Muduli, M. Ikhlas, Y. Omori, T. Tomita, A. MacDonald, \nS. Nakatsuji, Magnetic and magnetic inverse spin Hall effects in a non -collinear antiferromagnet, Nature, \n565 (2019) 627 -630.  \n[6] I.M. Miron, K. Garello, G. Gaudin, P. -J. Zermatten, M.V. Costache, S. Auffret, S. Bandiera, B. Rodmacq, \nA. Schuhl, P. Gambardella, Perpendicular switching of a single ferromagnetic layer induced by in -plane \ncurrent injection, Nature, 476 (2011) 1 89-193.  \n[7] L. Liu, C. -F. Pai, Y. Li, H. Tseng, D. Ralph, R. Buhrman, Spin -torque switching with the giant spin Hall \neffect of tantalum, Science, 336 (2012) 555 -558.  \n[8] I.M. Miron, T. Moore, H. Szambolics, L.D. Buda -Prejbeanu, S. Auffret, B. Rodmacq, S. P izzini, J. Vogel, \nM. Bonfim, A. Schuhl, Fast current -induced domain -wall motion controlled by the Rashba effect, Nature \nmaterials, 10 (2011) 419 -423.  \n[9] S. Emori, U. Bauer, S. -M. Ahn, E. Martinez, G.S. Beach, Current -driven dynamics of chiral ferromagneti c \ndomain walls, Nature materials, 12 (2013) 611 -616.  \n[10] W. Jiang, P. Upadhyaya, W. Zhang, G. Yu, M.B. Jungfleisch, F.Y. Fradin, J.E. Pearson, Y. Tserkovnyak, \nK.L. Wang, O. Heinonen, Blowing magnetic skyrmion bubbles, Science, 349 (2015) 283 -286.  \n[11] S. -W. Lee, K. -J. Lee, Emerging three -terminal magnetic memory devices, Proceedings of the IEEE, 104 \n(2016) 1831 -1843.  \n[12] L. Liu, T. Moriyama, D. Ralph, R. Buhrman, Spin -torque ferromagnetic resonance induced by the spin \nHall effect, Physical review letters,  106 (2011) 036601.  \n[13] S. Karimeddiny, J.A. Mittelstaedt, R.A. Buhrman, D.C. Ralph, Transverse and longitudinal spin -torque \nferromagnetic resonance for improved measurement of spin -orbit torque, Physical Review Applied, 14 \n(2020) 024024.  \n[14] J. Kim, J. Sinha, M. Hayashi, M. Yamanouchi, S. Fukami, T. Suzuki, S. Mitani, H. Ohno, Layer thickness \ndependence of the current -induced effective field vector in Ta| CoFeB| MgO, Nature materials, 12 (2013) \n240-245.  \n[15] X. Fan, J. Wu, Y. Chen, M.J. Jerry, H. Zhang, J.Q. Xiao, Observation of the nonlocal spin -orbital effective \nfield, Nature communications, 4 (2013) 1799.  \n[16] K. Garello, I.M. Miron, C.O. Avci, F. Freimuth, Y. Mokrousov, S. Blügel, S. Auffret, O. Boulle, G. Gaudin, \nP. Gambardella, Symmetry and magnitud e of spin –orbit torques in ferromagnetic heterostructures, \nNature nanotechnology, 8 (2013) 587 -593.  \n[17] X. Fan, H. Celik, J. Wu, C. Ni, K. -J. Lee, V.O. Lorenz, J.Q. Xiao, Quantifying interface and bulk \ncontributions to spin –orbit torque in magnetic bilaye rs, Nature communications, 5 (2014) 3042.  \n[18] X. Fan, A.R. Mellnik, W. Wang, N. Reynolds, T. Wang, H. Celik, V.O. Lorenz, D.C. Ralph, J.Q. Xiao, All -\noptical vector measurement of spin -orbit -induced torques using both polar and quadratic magneto -optic \nKerr  effects, Applied Physics Letters, 109 (2016) 122406.  \n[19] M. -H. Nguyen, C. -F. Pai, Spin –orbit torque characterization in a nutshell, APL Materials, 9 (2021) \n030902.  \n[20] D.C. Ralph, M.D. Stiles, Spin transfer torques, Journal of Magnetism and Magnetic Mat erials, 320 \n(2008) 1190 -1216.  \n[21] A. Davidson, V.P. Amin, W.S. Aljuaid, P.M. Haney, X. Fan, Perspectives of electrically generated spin \ncurrents in ferromagnetic materials, Physics Letters A, 384 (2020) 126228.  \n[22] A. Ghosh, S. Auffret, U. Ebels, W. Bail ey, Penetration depth of transverse spin current in ultrathin \nferromagnets, Physical review letters, 109 (2012) 127202.  [23] W. Wang, T. Wang, V.P. Amin, Y. Wang, A. Radhakrishnan, A. Davidson, S.R. Allen, T.J. Silva, H. Ohldag, \nD. Balzar, Anomalous spin –orbit torques in magnetic single -layer films, Nature nanotechnology, 14 (2019) \n819-824.  \n[24] G.S. Abo, Y. -K. Hong, J. Park, J. Lee, W. Lee, B. -C. Choi, Definition of magnetic exchange length, IEEE \nTransactions on Magnetics, 49 (2013) 4937 -4939.  \n[25] Z. Qiu,  S.D. Bader, Surface magneto -optic Kerr effect (SMOKE), Journal of magnetism and magnetic \nmaterials, 200 (1999) 664 -678.  \n[26] C.O. Avci, K. Garello, M. Gabureac, A. Ghosh, A. Fuhrer, S.F. Alvarado, P. Gambardella, Interplay of \nspin-orbit torque and thermoe lectric effects in ferromagnet/normal -metal bilayers, Physical Review B, 90 \n(2014) 224427.  \n[27] E.H. Sondheimer, The mean free path of electrons in metals, Advances in physics, 50 (2001) 499 -537.  \n[28] S. -Y. Huang, X. Fan, D. Qu, Y. Chen, W. Wang, J. Wu, T.  Chen, J. Xiao, C. Chien, Transport magnetic \nproximity effects in platinum, Physical review letters, 109 (2012) 107204.  \n[29] H. Nakayama, M. Althammer, Y. -T. Chen, K. -i. Uchida, Y. Kajiwara, D. Kikuchi, T. Ohtani, S. Geprägs, \nM. Opel, S. Takahashi, Spin Hall magnetoresistance induced by a nonequilibrium proximity effect, Physical \nreview letters, 110 (2013) 206601.  \n \nAppendices  \nA. Propagation matrix method to compute the MOKE response  \n \nThe MOKE response through the heterostructure is derived using the propagation method.  The \nlinearly polarized light is decomposed into a superposition of a left -handed and a right -handed \ncircularly polarized light with corresponding index of refractions in the magnetic materials as 𝑛L=\n𝑛+𝑄𝑚z, and 𝑛R=𝑛−𝑄𝑚z respectively. There is no difference in index of refractions for left -\nhanded and right -handed circularly polarized lights in other nonmagnetic layers.  We’ll calculate \nthe left -handed circularly polarized light as an example.   \nFor the i-th layer, we consider the light propagating in the + z direction having an electric field \ndescribed as 𝐸i+𝑒𝑗𝑛i𝜔𝑧/𝑐, where 𝑛i is the index of refraction of the i-th layer, 𝜔 is the angular \nfrequency, and c is the speed of light . Similarly, t he electric field propagating in the -z direction is \ndescribed as 𝐸i−𝑒−𝑗𝑛i𝜔𝑧/𝑐. At an interface between the i-th and ( i+1)-th layers,  assuming z = 0, the \nMaxwell’s boundary conditions dictate that  \n  {𝐸i++𝐸i−=𝐸i+1++𝐸i+1−\n𝐸i+−𝐸i−\n𝑛i=𝐸i+1+−𝐸i+1−\n𝑛i+1,      (A1)  \nwhich can be written as  a matrix form,  (𝐸i+1+\n𝐸i+1−)=[𝑛i+𝑛i+1\n2𝑛i𝑛i−𝑛i+1\n2𝑛i\n𝑛i−𝑛i+1\n2𝑛i𝑛i+𝑛i+1\n2𝑛i](𝐸i+\n𝐸i−)=Г(i,i+1)(𝐸i+\n𝐸i−). Here \nthe matrix Г(i,i+1) is a propagation matrix describing how the electric fields of the light change \nat the interface. Similarly, a propagation matrix describing how the electric fields change in the bulk of one uniform layer can be written as T(i)=[𝑒𝑗𝑛i𝜔𝑑/𝑐0\n0 𝑒−𝑗𝑛i𝜔𝑑/𝑐]. Therefore, the initial \nincident light and the final transmitted light after the n-th layer can be related by  the multiplication \nof a series of these propagation matrices,  \n(𝐸n+1+\n0)=[∏ Г(i,i+1) T(i)𝑛\n𝑖=1]Г(0,1)(𝐸0+\n𝐸0−)=P(𝐸0+\n𝐸0−),   (A2)  \nwhere we use P=[𝑃11(𝑛L)𝑃12(𝑛L)\n𝑃21(𝑛L)𝑃22(𝑛L)] to denote the total propagation matrix. It should be \npointed out that each matrix element here is a function of nL. When right -handed circularly \npolarized light is used, the matrix element should be changed by replacing nL with nR. \nSince the MOKE measurement measures reflection, i.e. 𝐸0−\n𝐸0+, the overall Kerr signal  from the \nheterostructure can be derived as 𝜃kerr +𝑗𝜅kerr =𝑗𝑃21(𝑛L)\n𝑃22(𝑛L) − 𝑃21(𝑛R)\n𝑃22(𝑛R)\n𝑃21(𝑛L)\n𝑃22(𝑛L) + 𝑃21(𝑛R)\n𝑃22(𝑛R), where the real part 𝜃kerr is the \nKerr rotation, and the imaginary part 𝜅kerr is the Kerr ellipticity.  \nIn the simulation,  we use 780 nm for laser wavelength and  the rest of parameters used are  listed in the \ntable below.  \n n Thickness (nm)  Q \nAir 1 ∞ NA \nPy 2.38 + 4.36j  varies  0.0036 - 0.011j  \nPt 2.76 + 4.84j  3 NA \nSiO 2 1.45 1000  NA \nSi wafer  3.71 + 0.01j  ∞ NA \nTable A1 Parameters used in the MOKE simulation.  \n \n \n "    },    {        "title": "2206.05770v3.Interference_phenomena_in_Josephson_junctions_with_ferromagnetic_bilayers__Spin_triplet_correlations_and_resonances.pdf",        "content": "Interference phenomena in Josephson junctions with ferromagnetic bilayers:\nSpin-triplet correlations and resonances\nDanilo Nikoli\u0013 c,1, 2Mihajlo Vanevi\u0013 c,1Alexander I. Buzdin,3, 4and Zoran Radovi\u0013 c1, 5\n1Department of Physics, University of Belgrade, Studentski trg 12, 11158 Belgrade, Serbia\n2Fachbereich Physik, Universit at Konstanz, D-78457 Konstanz, Germany\n3University Bordeaux, LOMA UMR-CNRS 5798, F-33405 Talence Cedex, France\n4World-Class Research Center \\Digital Biodesign and Personalized Healthcare\",\nSechenov First Moscow State Medical University, Moscow 119991, Russia\n5Serbian Academy of Sciences and Arts, Kneza Mihaila 35, 11000, Belgrade, Serbia\n(Dated: August 31, 2022)\nWe study the Josephson e\u000bect in planar SF 1F2S junctions that consist of conventional s-wave\nsuperconductors (S) connected by two metallic monodomain ferromagnets (F 1and F 2) with arbi-\ntrary transparency of interfaces. We solve the scattering problem in the clean limit based on the\nBogoliubov - de Gennes equation for both spin-singlet and odd in frequency spin-triplet pairing\ncorrelations. We calculate numerically the Josephson current-phase relation I(\u001e). While the \frst\nharmonic of I(\u001e) is completely generated by spin-singlet and short-range spin-triplet superconduct-\ning correlations, for noncollinear magnetizations of ferromagnetic layers the second harmonic has\nan additional long-range spin-triplet component. Therefore, for a strong ferromagnetic in\ruence,\nthe long-range spin-triplet contribution to the second harmonic dominates. We \fnd an exception\ndue to the geometric resonance for equal ferromagnetic layers when the \frst harmonic is strongly\nenhanced. Both \frst and second harmonic amplitudes oscillate with ferromagnetic layer thicknesses\ndue to 0\u0000\u0019transitions. We study the in\ruence of interface transparencies and \fnd additional\nresonances for \fnite transparency of the interface between ferromagnetic layers.\nI. INTRODUCTION\nThe interplay between superconductivity and mag-\nnetism in proximity heterostructures [1{4] has been at-\ntracting considerable interest for decades, see for exam-\nple Refs. [5{16]. Remarkably, odd in frequency spin-\ntriplet pairing correlations may occur in SFS Joseph-\nson structures comprised of superconductors with spin-\nsinglet pairing and a metallic ferromagnet [17, 18]. In\nthe case of a homogeneous ferromagnet, the triplet pair\namplitude has zero total spin projection on the magne-\ntization axis. This amplitude, as well as the spin-singlet\namplitude, decay over the short length scale determined\nby the exchange energy hin the ferromagnet. The char-\nacteristic coherence length in ferromagnet is given by\n\u0018F=~vF=hand\u0018F=p\n~D=h (D=vF`=3 is the dif-\nfusion coe\u000ecient with `being the electronic mean free\npath) in the clean and di\u000busive limit, respectively. The\nsituation is quite di\u000berent for an inhomogeneous ferro-\nmagnet where spin-triplet pair amplitudes with \u00061 total\nspin projection on the magnetization axis emerge [17].\nThese amplitudes decay on substantially larger length\nscales determined by temperature, \u0018F=~vF=(kBT)\nin the clean and \u0018F=p\n~D=(kBT) in the di\u000busive\nlimit [12].\nA simple realization of a Josephson junction with an in-\nhomogeneous ferromagnet is the SF 1F2S heterostructure\nwith two monodomain ferromagnets having noncollinear\nin-plane magnetizations [19{30]. However, in such prox-\nimity structures the long-range spin-triplet component\nof the supercurrent consists only of even harmonic am-\nplitudes [22{25]. In the case of strong ferromagnets the\nshort-range components are suppressed and the secondharmonic is dominant in the current-phase relation. Odd\nharmonic amplitudes can be long-ranged only in hetero-\njunctions with three or more ferromagnetic layers [31{43].\nNote that the anharmonic current-phase relation can\nbe expanded as I(\u001e) =I1sin\u001e+I2sin 2\u001e+:::, where\nthenth harmonic amplitude Incorresponds to the phase-\ncoherent transport of nCooper pairs [23]. Junctions\nwith a pure second harmonic exhibit degenerated ground\nstates for\u001e= 0 and\u0019at the so-called 0 \u0000\u0019transi-\ntion [4, 44]. A small contribution of other harmonics lifts\ndegeneracy and leads to the coexistence of stable and\nmetastable 0 and \u0019states [45{48].\nA long-ranged supercurrent has been observed in Nb\nJosephson junctions with Ni- and Co-based ferromag-\nnetic multilayers [49{53]. An enhanced second harmonic\nin the long-ranged supercurrent has been observed in\nmesa-heterostructures of cuprate superconductors and\nferromagnetic bilayers of manganite and ruthenate [54],\nwhile a pure second harmonic has been observed in\nNbN=GdN=NbN junctions [55].\nA dominant second harmonic, I2\u001dI1, can be realized\nin the regime of highly asymmetric SF 1F2S junctions [22{\n25]. The physical picture behind this e\u000bect is the fol-\nlowing: At the SF interface the exchange \feld generates\nspin-triplet correlations with 0 spin projection. Penetrat-\ning into the next ferromagnetic layer with misoriented\nmagnetization they mix forming long-range spin-triplet\ncorrelations with \u00061 spin projection. Therefore, for a\nfully developed spin-triplet proximity e\u000bect one of two\nferromagnetic layers should be su\u000eciently thin to pro-\nvide a large short-range spin-triplet amplitude with zero\nspin projection at the interface between ferromagnetic\nlayers in order to generate large long-range spin-tripletarXiv:2206.05770v3  [cond-mat.supr-con]  30 Aug 20222\namplitudes with \u00061 spin projection. The other ferro-\nmagnetic layer should be su\u000eciently thick to \flter out\nthe short-range correlations [56].\nIn this paper we study the Josephson e\u000bect in clean\nplanar (three-dimensional) SF 1F2S junctions that consist\nof conventional s-wave superconductors and two metal-\nlic monodomain ferromagnets (equal strength and di\u000ber-\nent thicknesses) with arbitrary transparency of the inter-\nfaces. We calculate numerically the Josephson current-\nphase relation I(\u001e) by using the Bogoliubov-de Gennes\nformalism. In particular, we calculate the \frst and sec-\nond harmonic amplitudes, I1andI2. The long-range\nsecond harmonic is well-pronounced for an overall strong\nferromagnetic in\ruence due to the contribution of the\nodd-frequency spin-triplet correlations with \u00061 spin pro-\njection. However, for equal thicknesses of ferromagnetic\nlayers the spin-singlet contribution to the \frst harmonic\nis dominant due to geometric resonances even for a strong\nferromagnetic in\ruence. In a previous paper the results\nfor linear (one-dimensional) SF 1F2S structures were illus-\ntrated only for equal ferromagnetic layers and the in\ru-\nence of the long-range spin-triplet correlations was com-\npletely hidden [20]. In subsequent papers using the same\napproach [29, 30], the interplay between the geometric\nresonances and spin-triplet correlations was not studied\nexplicitly. Here, we focus on this subject.\nBoth the \frst and the second harmonic oscillate with\nferromagnetic layer thicknesses due to the 0 \u0000\u0019tran-\nsitions. For \fnite transparency of interfaces the super-\ncurrent is suppressed, with higher harmonics being more\na\u000bected. A lower transparency of the interface between\nferromagnetic layers, where the long-range spin-triplet\ncorrelations are generated, has a nontrivial impact on\nthe interference phenomena: For certain thicknesses of\nthe ferromagnetic layers we \fnd additional geometric res-\nonances.\nThe paper is organized as follows. In Sec. II we present\nthe model and the solution. In Sec. III we present and\ndiscuss the numerical results for the Josephson current\nand harmonic amplitudes. Finally, the concluding re-\nmarks are given in Sec. IV.\nII. MODEL\nA. The Bogoliubov-de Gennes equations for SF1F2S\nheterojunctions\nWe consider a clean planar (three-dimensional)\nSI1F1I2F2I3S heterojunction that consists of two super-\nconductors (S), two uniform monodomain ferromagnetic\nlayers (F 1and F 2), and three nonmagnetic interfacial po-\ntential barriers between metallic layers ( I1\u0000I3), depicted\nin Fig. 1. Superconductors are described in the frame-\nwork of BCS formalism, while for ferromagnets we use\nthe Stoner model with a spatially-dependent energy shift\n2h(r) between the spin subbands. The model and meth-\nods are the same as in the previous papers [20, 29, 30].\nSS𝑥𝑦𝑧𝛼1𝑥𝑦𝑧𝛼2F1F2\n𝑑!0𝑑!+𝑑\"FIG. 1. Schematic representation of an SF 1F2S junction.\nTwo ferromagnetic layers F 1and F 2of thicknesses d1andd2,\nrespectively, are coupled to two superconducting electrodes\n(S). The magnetization vectors lie in the yzplane at angles\n\u000b1and\u000b2with respect to the zaxis. The insulating inter-\nfaces between the superconducting and ferromagnetic layers\nare denoted asI1\u0000I3.\nElectronlike and holelike quasiparticles with energy\nEand spin projection \u001b=\";#are described by u\u001b(r)\nandv\u001b(r), respectively, where ris the spatial coordi-\nnate. Using the four-component wave function \t( r) =\n[u\"(r);u#(r);v\"(r);v#(r)]T, the Bogoliubov-de Gennes\nequation has the following form:\n\u0014H\t(r) =E\t(r); (1)\nwith \u0014Hbeing a 2\u00022 matrix in particle-hole space\n\u0014H=\u0012^H(r)^\u0001\n^\u0001\u0003\u0000^H(r)\u0013\n; (2)\nwhere each block itself is a 2 \u00022 matrix in spin space, such\nthat, ^H(r) =H0(r)^1\u0000h(r) sin[\u000b(r)]^\u001c2\u0000h(r) cos[\u000b(r)]^\u001c3\nand ^\u0001(r) = \u0001( r)^\u001c1. Here, ^\u001ciare Pauli matrices, ^1 is\nthe unity matrix, and H0(r) =\u0000~2r2=2m+W(r) +\nU(r)\u0000\u0016. The chemical potential is denoted by \u0016,W(r) =P\niWi\u000e(x\u0000xi) is the potential of the barriers at the\ninterfaces, and U(r) is the electrostatic potential. The x\naxis is chosen to be perpendicular to the layers, whereas\nx1= 0,x2=d1, andx3=d1+d2are coordinates of the\ninterfaces. At zero temperature the di\u000berence \u0016\u0000U(r)\nis equal to the Fermi energy EF. The in-plane ( y\u0000z)\nmagnetizations of the two F layers are not collinear in\ngeneral, and the magnetization orientation is de\fned by\nthe angle\u000b(r) with respect to the zaxis. We choose\n\u000b(r) =\u000b1for 0< x < d 1in F 1, and\u000b(r) =\u000b2for\nd1<x<d 1+d2in F 2(see Fig. 1).\nFor simplicity, we assume equal magnitudes of the ex-\nchange \feld in the ferromagnets, h(r) =h\u0002(x)\u0002(d1+\nd2\u0000x) where \u0002( x) stands for the Heaviside step func-\ntion. We also assume that the e\u000bective electron mass\nmis constant throughout the layers and that the mean\nFermi energy of the ferromagnets, EF= (E\"\nF+E#\nF)=2,\nis equal to the Fermi energy of superconductors. How-\never, the in\ruence of the mismatch of e\u000bective electron3\nmasses and Fermi energies is similar to the in\ruence of\n\fnite interfacial transparency: an increase of the normal\nrefection [46, 57, 58].\nWe take the pair potential \u0001( r) in the form\n\u0001(r) = \u0001[ei\u001e1\u0002(\u0000x) +ei\u001e2\u0002(x\u0000d1\u0000d2)];(3)\nwhere \u0001 is the bulk superconducting gap. The macro-\nscopic phase di\u000berence across the junction is \u001e=\u001e1\u0000\u001e2.\nThe temperature dependence of \u0001 is given by \u0001( T) =\n\u0001(0) tanh(1 :74p\nTc=T\u00001) with \u0001(0) being the pair po-\ntential at zero temperature and Tcthe critical tempera-\nture [59]. In general, \u0001( r) should be determined self-\nconsistently [29, 42]. However, for simplicity we use\nthe stepwise pair potential given in Eq. (3), since self-\nconsistency will not alter our results qualitatively [26].\nB. Solution of the scattering problem\nThe parallel component of the wave vector, kk, is con-\nserved due to the translational invariance of the junc-\ntion in the direction perpendicular to the xaxis. Conse-\nquently, the wave function can be written in the form\n\t(r) = (x)eikk\u0001r; (4)\nwhere (x) = [u\"(x);u#(x);v\"(x);v#(x)]Tsatis\fes the\nfollowing boundary conditions:\n (x)jxi+0= (x)jxi\u00000= (xi); (5)\nd (x)\ndx\f\f\f\f\nxi+0\u0000d (x)\ndx\f\f\f\f\nxi\u00000=ZikF (x): (6)\nHere,Zi= 2mWi=~2kF(i= 1;2;3) are parameters [60]\nthat measure the transparency of the interfaces (i.e.,\nbarrier strengths) located at xi= 0;d1;d1+d2, and\nkF=p\n2mEF=~2is the Fermi wave vector.\nThe four independent solutions of the scattering prob-\nlem for Eq. (1) correspond to the four types of quasi-\nparticle injection processes: an electronlike or a holelike\nquasiparticle injected from either the left or the right su-\nperconducting electrode. When an electronlike quasipar-\nticle is injected from the left, the solutions of Eq. (1) for\nthe left superconductor ( x <0) written in the compact\nform are\n\u0012\nu\u001b\nv\u0016\u001b\u0013\n=\u0012\n\u0016uei\u001e1=2\n\u0016ve\u0000i\u001e1=2\u0013\neik+x+b1\u001b\u0012\n\u0016uei\u001e1=2\n\u0016ve\u0000i\u001e1=2\u0013\ne\u0000ik+x\n+a1\u001b\u0012\n\u0016vei\u001e1=2\n\u0016ue\u0000i\u001e1=2\u0013\neik\u0000x; (7)\nand for the right superconductor ( x>d 1+d2)\n\u0012\nu\u001b\nv\u0016\u001b\u0013\n=c1\u001b\u0012\n\u0016uei\u001e2=2\n\u0016ve\u0000i\u001e2=2\u0013\neik+x+d1\u001b\u0012\n\u0016vei\u001e2=2\n\u0016ue\u0000i\u001e2=2\u0013\ne\u0000ik\u0000x;\n(8)where \u0016\u001bis opposite to \u001b=\"#, \u0016u=p\n(1 + \n=E)=2,\n\u0016v=p\n(1\u0000\n=E)=2, and \n =p\nE2\u0000\u00012. Constants\na1\u001b;b1\u001b;c1\u001b, andd1\u001bcorrespond to Andreev and normal\nre\rections, direct transmission, and nonlocal Andreev re-\n\rection, respectively. For the left ferromagnetic layer\nF1(0<x<d 1) solutions of Eq. (1) are\n\u0012\nu\"\nu#\u0013\n=c1\u0012\nicos\u000b1\n2\n\u0000sin\u000b1\n2\u0013\neiq+\n\"x+c2\u0012\nicos\u000b1\n2\n\u0000sin\u000b1\n2\u0013\ne\u0000iq+\n\"x\n+c3\u0012\nisin\u000b1\n2\ncos\u000b1\n2\u0013\neiq+\n#x+c4\u0012\nisin\u000b1\n2\ncos\u000b1\n2\u0013\ne\u0000iq+\n#x;\n(9)\n\u0012\nv\"\nv#\u0013\n=c5\u0012\nicos\u000b1\n2\n\u0000sin\u000b1\n2\u0013\neiq\u0000\n\"x+c6\u0012\nicos\u000b1\n2\n\u0000sin\u000b1\n2\u0013\ne\u0000iq\u0000\n\"x\n+c7\u0012\nisin\u000b1\n2\ncos\u000b1\n2\u0013\neiq\u0000\n#x+c8\u0012\nisin\u000b1\n2\ncos\u000b1\n2\u0013\ne\u0000iq\u0000\n#x:\n(10)\nSolutions for the right ferromagnetic layer F 2(d1<x<\nd1+d2) can be obtained by substituting \u000b1!\u000b2, with\na new set of constants c0\n1;:::;c0\n8.\nWhen a holelike quasiparticle is injected from the left,\nthe solutions of Eq. (1) for the left superconductor ( x<\n0) are given by\n\u0012\nu\u001b\nv\u0016\u001b\u0013\n=\u0012\n\u0016vei\u001e1=2\n\u0016ue\u0000i\u001e1=2\u0013\ne\u0000ik\u0000x+b2\u001b\u0012\n\u0016vei\u001e1=2\n\u0016ue\u0000i\u001e1=2\u0013\neik\u0000x\n+a2\u001b\u0012\n\u0016uei\u001e1=2\n\u0016ve\u0000i\u001e1=2\u0013\ne\u0000ik+x; (11)\nwhile for the right superconductor ( x>d 1+d2)\n\u0012\nu\u001b\nv\u0016\u001b\u0013\n=c2\u001b\u0012\n\u0016vei\u001e2=2\n\u0016ue\u0000i\u001e2=2\u0013\ne\u0000ik\u0000x+d2\u001b\u0012\n\u0016uei\u001e2=2\n\u0016ve\u0000i\u001e2=2\u0013\neik+x:\n(12)\nConstantsa2\u001b;b2\u001b;c2\u001b, andd2\u001bdescribe analogous pro-\ncesses as in the case of an injected electronlike quasipar-\nticle given earlier.\nSolutions for ferromagnetic layers in the case a hole-\nlike quasiparticle injected from the left can be obtained\nby substituting ci!Ciandc0\ni!C0\niin solutions for\nthe case of an injected electronlike quasiparticle. The\nlongitudinal x- components of the wave vectors in the\nsuperconductors are given by\nk\u0006=r\n2m\n~2(EF\u0006\n)\u0000k2\nk; (13)\nwhile their counterparts in the ferromagnetic layers read\nq\u0006\n\u001b=r\n2m\n~2(EF\u0006E+\u001a\u001bh)\u0000k2\nk: (14)\nThe sign\u0006in the superscript corresponds to the sign of\nthe quasiparticle energy, whereas \u001a\u001b= +1 (\u00001) is related\nto the spin projection \u001b=\"(#).4\nSolutions for quasiparticles with opposite spin orien-\ntations are nontrivially coupled: in the superconductors,\nEqs. (7) and (8) and Eqs. (11) and (12), as well as in the\nferromagnets, Eqs. (9) and (10). In that manner, both\nthe usual and spin-\rip Andreev re\rections are taken into\naccount.\nAll the unknown 48 coe\u000ecients in the above solutions,\nin both electronlike and holelike scattering problems, are\ndetermined from the boundary conditions, Eqs. (5) and\n(6), at the three interfaces.\nC. The Josephson current\nThe Josephson current can be calculated from the lin-\near superposition of amplitudes the normal and anoma-\nlous Andreev re\rections [61], a1\u001banda2\u001b,\nI(\u001e) =e\u0001\n2~X\n\u001b;kk;!nkBT\n2\nn(k+\nn+k\u0000\nn)\u0014a1\u001bn(\u001e)\nk+n\u0000a2\u001bn(\u001e)\nk\u0000n\u0015\n:\n(15)\nHere,\u001e=\u001e1\u0000\u001e2is the superconducting phase di\u000berence,\nanda1\u001bn;a2\u001bn,k\u0006\nn, and \nn=p\n!2n+ \u00012are obtained\nfrom the corresponding quantities shown in the previous\nsection by performing the analytic continuation, E!\ni!n. The Matsubara frequencies are !n= (2n+1)\u0019kBT,\nwithn= 0;\u00061;\u00062;:::and the temperature T.\nFor nonmagnetic (SNS and SIS) Josephson junctions\na1\u001banda2\u001bare\u001bindependent and related by particle-\nhole symmetry, a1(\u001e) =a2(\u0000\u001e). However, for SFS\njunctions (with homogeneous/inhomogeneous magneti-\nzation), when the odd-frequency spin-triplet correlations\n(short/long range) are generated, the amplitudes a1\u001band\na2\u001bare\u001bdependent. In that way, the spin-mixing pro-\ncesses are included.\nPerforming a summation over kkby employingP\nkk!\nA(2\u0019)\u00002R\nd2k, we obtain\nI(\u001e) =\u0001\u0019\nRNekBTZ\u0019=2\n0d\u0012sin\u0012cos\u0012\u0002\n\u0002X\n\u001b!nk+\nn+k\u0000\nn\n4\nn\u0012a1\u001bn(\u001e)\nk+n\u0000a2\u001bn(\u001e)\nk\u0000n\u0013\n:(16)\nHere,RN= 2\u00192~=Ae2k2\nFwithAbeing the cross section\nof the junction and we assume kk=kFsin\u0012, since we deal\nwith standard BCS superconductors, where \u0001 =EF\u0018\n10\u00003\u000010\u00004.\nIn general, the current-phase relation is an anharmonic\n2\u0019-periodic function and can be expanded as\nI(\u001e) =I1sin\u001e+I2sin 2\u001e+:::; (17)\nwhere thenth harmonic amplitude Incorresponds to the\nphase-coherent transport of nCooper pairs.\n0.0 0.2 0.4 0.6 0.8 1.0-0.050.000.050.00.10.2\nφ/πeRNI/π∆\n0ππ\nπ/4π/4\nπ/2π/2\n3π/43π/4\n0SNS\nkFd1= 500\nkFd2= 500\nkFd1= 10\nkFd2= 990(a)\n(b)FIG. 2. The Josephson current in SF 1F2S junctions as a\nfunction of the superconducting phase di\u000berence \u001efor the\nferromagnetic layer thicknesses (a) d1=d2= 500k\u00001\nFand (b)\nd1= 10k\u00001\nF,d2= 990k\u00001\nF, forh=EF= 0:1,T=Tc= 0:1, and\ndi\u000berent relative angles between the magnetizations: \u000br= 0,\n\u0019=4,\u0019=2, 3\u0019=4,\u0019. The Josephson current for SNS junction\n(h= 0) with the thickness d1+d2= 1000k\u00001\nFis shown in the\npanel (a) for comparison (dotted line).\nIII. RESULTS AND DISCUSSION\nWe illustrate our results on SF 1F2S planar junc-\ntions with relatively weak ferromagnets h=EF= 0:1\nat low temperature T=Tc= 0:1. The ferromagnetic\ncoherence length is \u0018F=~vF=h= 20k\u00001\nF. Super-\nconductors are characterized by the bulk pair poten-\ntial at zero temperature \u0001(0) =EF= 10\u00003which corre-\nsponds to the superconducting coherence length \u0018S(0) =\n~vF=\u0019\u0001(0) = 636 k\u00001\nF. The Josephson current is nor-\nmalized to \u0019\u0001=eRNas usual [see Eq. (16)]. The total\nthickness of the ferromagnetic bilayer is kept constant,\nd1+d2= 1000k\u00001\nF= 50\u0018F= 1:57\u0018S(0):\nA. Fully transparent interfaces\nThe current-phase relation in a junction with fully\ntransparent interfaces, Z1=Z2=Z3= 0, for vari-\nous values of the relative angle between magnetizations,\n\u000br=\u000b1\u0000\u000b2, and equal thicknesses of the ferromag-\nnetic layers is shown in Fig. 2(a). It can be seen that the5\n20 40 60 80-0.050.000.05\n4004204404604805000.000.050.10\n(a) (b)\n(c) (d)\nkFd1 kFd1eRNIc/π∆ eRNI1,2/π∆\nI1 I1I2 I2\nFIG. 3. The critical current in SF 1F2S junctions with mutu-\nally orthogonal magnetizations \u000br=\u0019=2 and fully transpar-\nent interfaces, Z1=Z2=Z3= 0, shown as a function of the\nF1layer thickness d1: (a) thin and (b) thick F 1layer. The\ntotal thickness is d1+d2= 1000k\u00001\nF. The amplitudes of the\n\frst (solid line) and the second harmonic (dashed line) of the\nJosephson current are shown in (c) and (d).\ncurrent is completely suppressed for the parallel mag-\nnetizations, \u000br= 0, and increases almost monotonously\nwith a misorientation of magnetizations up to I(\u001e) of the\ncorresponding SNS junction ( h= 0) for the antiparallel\nmagnetizations, \u000br=\u0019, which has been observed experi-\nmentally [62]. The current-phase relation is a practically\nuniversal function of the ferromagnetic in\ruence, which\nis measured by the product of thickness and the exchange\n\feld strength, d\u0001h[46]. This explains the cancellation of\nferromagnetic in\ruence in the case of equal thicknesses\nand equal strengths of the ferromagnets. However, in\nthat case no signi\fcant in\ruence of the triplet correla-\ntions was found even for noncollinear magnetizations [20].\nThis we explain now by a dominant \frst harmonic due\nto the geometric resonance e\u000bect [see Fig. 3(d)].\nA dominant second harmonic can be seen in Fig. 2(b)\nfor highly unequal thicknesses of the ferromagnetic layers,\nkFd1= 10 andkFd2= 990, and noncollinear magneti-\nzations. In contrast to the case of equal ferromagnetic\nlayers, the critical current is not a monotonous function\nof the misorientation angle \u000br. It almost vanishes for\n\u000br= 0;\u0019and reaches the maximum for \u000br=\u0019=2. This\nis a manifestation of the long-range spin-triplet proximity\ne\u000bect in ferromagnetic bilayers where the \frst harmonic\nis suppressed and the phase-coherent transport of two\nCooper pairs becomes dominant [22{26].\nTo illustrate the role of ferromagnetic bilayer asymme-\ntry, we calculate the critical current Icand the ampli-\ntudes of the \frst and the second harmonic, I1andI2,\nas functions of the F 1layer thickness, d1, keeping the\n0.0 0.2 0.4 0.6 0.8-0.0050.0000.005\n0.0 0.2 0.4 0.6 0.8 1.0-0.0050.0000.0050.000.050.10\n-0.04-0.020.000.020.04\nφ/π φ/πeRNI/π∆\n0 0π ππ\n0(a) (b)\n(c) (d)SN 1N2S SF1F2S\nSF1F2S SF1F2Sπ/2\nπ/2\nπ/2(0,1,0)(0,1,0)\n(1,0,1)\n(1,0,1)(1,1,1)\n(1,1,1)FIG. 4. The Josephson current in (a) SN 1N2S and (b){\n(d) SF 1F2S junctions with layer thicknesses d1= 10k\u00001\nF,\nd2= 990k\u00001\nF, and di\u000berent barrier strengths ( Z1;Z2;Z3) at\nthe interfaces. Relative angles between the magnetizations\n\u000br= 0,\u0019=2,\u0019in SF 1F2S junctions are indicated in the plots.\ntotal thickness constant, kF(d1+d2) = 1000. The rela-\ntive angle between the magnetizations is \u000br=\u0019=2 (the\nstrongest e\u000bect of spin-triplet correlations) and the inter-\nfaces are fully transparent, Z1=Z2=Z3= 0. Results\nare shown in Fig. 3. When d1approaches d2we can see\nthe rise of the I1amplitude due to the geometric reso-\nnance. Because of that, the \frst harmonic is dominant\nfor equal ferromagnetic layers and the spin singlet and\nspin triplet with zero spin projection correlations prac-\ntically generate the supercurrent [20]. In ferromagnetic\nbilayers only even harmonics (the second is the largest)\ncan be generated by long-range spin-triplet correlations\nwith\u00061 spin projections [23].\nThe characteristic oscillations of I1(d1);I2(d1), and\nIc(d1) are due to 0\u0000\u0019transitions with the period\npractically equal to the ferromagnetic coherence length\n\u0018F= 20k\u00001\nF. Note that in the clean limit the critical\ncurrentIcis minimum but not zero at the 0 \u0000\u0019transi-\ntion [8, 9, 46].\nB. Finite interfacial transparencies\nThe role of \fnite interfacial transparencies is illus-\ntrated in Figs. 4-6. For comparison, the current-phase re-\nlation for a clean SN 1N2S (h= 0) junction with kFd1=\n10,kFd2= 990 is shown for various interfacial barrier\nstrengths, see Fig. 4(a). With decreasing transparency\nthe supercurrent is suppressed in comparison to the fully\ntransparent case [see the dotted curve in Fig. 2(a)]. In\nthis case the \frst harmonic is dominant. The supercur-\nrent of SF 1F2S junctions with \u000br= 0;\u0019=2;\u0019and di\u000ber-6\nFIG. 5. The \frst harmonic amplitude (solid line) and the\nsecond harmonic amplitude (dashed line) of the Josephson\ncurrent-phase relation in SF 1F2S junctions with orthogonal\nmagnetizations \u000br=\u0019=2 as a function of the F 1layer thick-\nnessd1, for total thickness d1+d2= 1000k\u00001\nF, and for dif-\nferent barrier strengths at the interfaces Z= (Z1;Z2;Z3):\n(a)Z= (0;0;0), (b)Z= (0;1;0), (c)Z= (0;3;0), and (d)\nZ= (1;0;1). Additional geometric resonances are pointed to\nby arrows: (b) and (c).\nent interfacial transparencies is shown in Figs. 4(b)-(d).\nNote that for collinear magnetizations, \u000br= 0;\u0019, the\ncurrent is short ranged and for orthogonal magnetiza-\ntions,\u000br=\u0019=2, the dominant second harmonic is due\nto the long-range spin-triplet correlations. It can be seen\nthat a lower transparency of the interface between ferro-\nmagnets is less destructive than lower transparencies of\nthe interfaces between superconductors and neighboring\nferromagnetic layers. The depairing e\u000bect of normal re-\n\rection at the SF interfaces is stronger due to the direct\nsuppression of the Andreev process.\nThe in\ruence of \fnite interfacial transparencies on the\n\frst and the second harmonics is quite di\u000berent. A \frst\nharmonic is generated by the phase-coherent transport\nof one Cooper pair, while the second harmonic is de-\ntermined by the phase-coherent transport of two Cooper\npairs. In Fig. 5 the \frst harmonic amplitude (solid curve)\nand the second harmonic amplitude (dashed curve) are\nshown as functions of d1forkF(d1+d2) = 1000,\n\u000br=\u0019=2, and di\u000berent transparencies of the interfaces,\n0.0 0.2 0.4 0.6 0.8 1.00.000.050.10\nφ/πeRNI/π∆SF1F2S\nZ= (0,1,0)\nkFd1= 500\nkFd2= 500\n0π/2π\nSN1N2SFIG. 6. The Josephson current in SF 1F2S junctions with\nequal layer thicknesses d1=d2= 500k\u00001\nFand interfacial\nbarrier strengths Z2= 1,Z1=Z3= 0, shown for di\u000ber-\nent relative angles between magnetizations \u000br= 0,\u0019=2,\u0019.\nThe Josephson current in SN 1N2S junction with the same\nlayer thicknesses and barrier strengths is shown for compari-\nson (dotted line).\nZ= (Z1;Z2;Z3). It can be seen that both I1andI2am-\nplitudes are suppressed by decreasing the transparency of\nthe interfaces, the \frst harmonic amplitude being much\nless a\u000bected.\nNew geometric resonances and ampli\fcations of I1\nemerge for a \fnite transparency of the interface between\nferromagnetic layers [see Figs. 5(b) and 5(c)]. Besides\nthe resonant ampli\fcation of I1ford1=d2, we \fnd reso-\nnant ampli\fcations at d1=d2=3;d2=5;:::. This e\u000bect is\nrelated to the multiple re\rections that lead to the emer-\ngence of electron and hole quasiclassical trajectories with\na canceled phase accumulation.\nThe current-phase relations for equal ferromagnetic\nlayers and \fnite interfacial transparency between them\nare shown in Fig. 6. The critical currents are approxi-\nmately two times smaller than in the fully transparent\ncase [see Fig. 2(a)]. We can see a peculiar ampli\fca-\ntion of the Josephson current for antiparallel magnetiza-\ntions,\u000br=\u0019, in comparison with nonmagnetic layers.\nThis e\u000bect was previously reported for SFIFS Joseph-\nson junctions with antiparallel orientations of magneti-\nzations [63], and for junctions between superconductors\nwith ferromagnetic exchange \felds [64, 65].\nIV. CONCLUSIONS\nWe have studied the Josephson e\u000bect in clean planar\nSF1F1S junctions with arbitrary transparencies of the\ninterfaces between the layers. By solving the scatter-\ning problem for the Bogoliubov-de Gennes equation, we\nhave calculated numerically the current-phase relation,7\nthe critical current, and \frst and second harmonic ampli-\ntudes. For relatively a weak exchange \feld, h=EF= 0:1,\nmutually orthogonal magnetizations, \u000br=\u0019=2, and very\nunequal thicknesses of the ferromagnetic layers, d1\u001cd2,\na well-pronounced second harmonic is obtained as a sig-\nnature of the long-range spin-triplet correlations. On\nthe other hand, for equally thick ferromagnetic layers,\nd1=d2, the spin-singlet contribution to the \frst har-\nmonic is enhanced due to the geometric resonance, and\ndominates even for thick layer junctions (strong ferro-\nmagnetic in\ruence) with orthogonal magnetizations.\nBoth resonant and spin-triplet e\u000bects qualitatively\npersist in the presence of impurities or moderate dis-\norder (see, for example, the quasiclassical analysis in\nRefs. [22, 25, 26]). In experiments the resonances can\nbe recognized as more sinusoidal I(\u001e), while the long-\nrange spin-triplet correlations in the Josephson junctions\nwith ferromagnetic bilayers lead to the more anharmonic\ncurrent-phase relation due to the dominant second har-\nmonic.\nBoth the \frst and the second harmonic amplitude show\ncharacteristic oscillations with varying thicknesses of the\nferromagnetic layers. The critical current oscillates in the\nsame manner. This is due to the 0 \u0000\u0019transitions and\nthe period of oscillations is the ferromagnetic coherence\nlength\u0018F.For a \fnite transparency of interfaces the supercur-\nrent is suppressed, with higher harmonics being more\na\u000bected. A low transparency of the F 1=F2interface,\nwhere the long-range spin-triplet correlations are gener-\nated, has a nontrivial impact on the interference phe-\nnomena and consequently to the current-phase relation.\nFor certain thicknesses of the ferromagnetic layers in ad-\ndition tod1=d2new geometric resonances occur at\nd1=d2=3;d2=5;:::, making the \frst harmonic dominant\neven in asymmetric junctions.\nACKNOWLEDGMENTS\nThe work was supported by Serbian Ministry of Edu-\ncation, Science and Technological Development, Project\nNo. 171027. Z.R. also acknowledges the support of the\nSerbian Academy of Sciences and Arts, Grant No. F87.\nA.I.B. acknowledges support by the Ministry of Science\nand Higher Education of the Russian Federation within\nthe framework of state funding for the creation and devel-\nopment of World-Class Research Center \\Digital Biode-\nsign and Personalized Healthcare\", Grant No. 075-15-\n2022-304. D.N. acknowledges support from Deutsche\nForschungsgemeinschaft (German Research Foundation)\nvia SFB 1432 (Project No. 425217212).\n[1] L. N. Bulaevskii and V. V. Kuzii and A. A. Sobyanin,\nSuperconducting system with weak coupling with a cur-\nrent in the ground state, Pis'ma Zh. Eksp. Teor. Fiz. 25,\n314 (1977) [JETP Lett. 25, 290 (1977)].\n[2] A. I. Buzdin, L. N. Bulaevskii, and S. V. Panjukov, Exis-\ntence of superconducting domain walls in ferromagnets,\nZh. Eksp. Teor. Fiz. 87, 299 (1984) [Sov. Phys. { JETP\n60, 174 (1984)].\n[3] Z. Radovi\u0013 c, L. Dobrosavljevi\u0013 c-Gruji\u0013 c, A. I. Buzdin, and\nJ. R. Clem, Upper critical \feld of superconductor- ferro-\nmagnet multilayers, Phys. Rev. B 38, 2388 (1988).\n[4] Z. Radovi\u0013 c, M. Ledvij, L. Dobrosavljevi\u0013 c-Gruji\u0013 c, A.\nI. Buzdin, and J. R. Clem, Transition temperature of\nsuperconductor-ferromagnet superlattices, Phys. Rev. B\n44, 759 (1991).\n[5] P. M. Tedrow and R. Meservey, Spin-polarized electron\ntunneling, Phys. Rep. 238, 173 (1994).\n[6] J. S. Jiang, D. Davidovi\u0013 c, D. H. Reich, and C. L. Chien,\nOscillatory Superconducting Transition Temperature in\nNb/Gd Multilayers, Phys. Rev. Lett. 74, 314 (1995).\n[7] Y. Obi, M. Ikebe, T. Kubo, and H. Fujimori, Oscillation\nphenomenon of transition temperatures in Nb/Co and\nV/Co superconductor/ferromagnet multilayers, Physica\nC Supercond. 317-318 , 149 (1999).\n[8] V. V. Ryazanov, V. A. Oboznov, A. Y. Rusanov, A. V.\nVeretennikov, A. A. Golubov, and J. Aarts, Coupling of\nTwo Superconductors Through a Ferromagnet: Evidence\nfor a\u0019Junction, Phys. Rev. Lett. 86, 2427 (2001).\n[9] T. Kontos, M. Aprili, J. Lesueur, and X. Grison, Inho-\nmogeneous Superconductivity Induced in a Ferromagnet\nby Proximity E\u000bect, Phys. Rev. Lett. 86, 304 (2001).[10] Y. Obi, M. Ikebe, and H. Fujishiro, Evidence for\nZero- and\u0019-Phase Order Parameters of Superconducting\nNb/Co Tri- and Pentalayers from the Oscillatory Behav-\nior of the Transition Temperature, Phys. Rev. Lett. 94,\n057008 (2005).\n[11] A. I. Buzdin, Proximity e\u000bects in superconductor-\nferromagnet heterostructures, Rev. Mod. Phys. 77, 935\n(2005).\n[12] F. S. Bergeret, A. F. Volkov, and K. B. Efetov, Odd\ntriplet superconductivity and related phenomena in\nsuperconductor-ferromagnet structures, Rev. Mod. Phys.\n77, 1321 (2005).\n[13] I. F. Lyuksyutov and V. L. Pokrovsky, Ferromag-\nnet{superconductor hybrids, Adv. Phys. 54, 67 (2005).\n[14] A. A. Golubov, M. Y. Kupriyanov, and E. Il'ichev, The\ncurrent-phase relation in Josephson junctions, Rev. Mod.\nPhys. 76, 411 (2004).\n[15] J. Linder and J. W. A. Robinson, Superconducting spin-\ntronics, Nat. Phys. 11, 307 (2015).\n[16] M. Eschrig, Spin-polarized supercurrents for spintronics:\na review of current progress, Rep. Prog. Phys. 78, 104501\n(2015).\n[17] F. S. Bergeret, A. F. Volkov, and K. B. Efetov, Enhance-\nment of the Josephson Current by an Exchange \feld\nin Superconductor-Ferromagnet Structures, Phys. Rev.\nLett. 86, 4096 (2001).\n[18] A. Kadigrobov, R. I. Shekhter, and M. Jonson, Quantum\nspin \ructuations as a source of long-range proximity ef-\nfects in di\u000busive ferromagnet-super conductor structures,\nEurophys. Lett. 54, 394 (2001).8\n[19] Y. M. Blanter and F. W. J. Hekking, Supercurrent in long\nSFFS junctions with antiparallel domain con\fguration,\nPhys. Rev. B 69, 024525 (2004).\n[20] Z. Pajovi\u0013 c, M. Bo\u0014 zovi\u0013 c, Z. Radovi\u0013 c, J. Cayssol, and\nA. Buzdin, Josephson coupling through ferromagnetic\nheterojunctions with noncollinear magnetizations, Phys.\nRev. B 74, 184509 (2006).\n[21] B. Crouzy, S. Tollis, and D. A. Ivanov, Josephson cur-\nrent in a superconductor-ferromagnet junction with two\nnoncollinear magnetic domains, Phys. Rev. B 75, 054503\n(2007).\n[22] L. Trifunovic, Z. Popovi\u0013 c, and Z. Radovi\u0013 c, Josephson ef-\nfect and spin-triplet pairing correlations in SF 1F2S junc-\ntions, Phys. Rev. B 84, 064511 (2011).\n[23] L. Trifunovic, Long-range Superharmonic Josephson\nCurrent, Phys. Rev. Lett. 107, 047001 (2011).\n[24] A. S. Mel'nikov, A. V. Samokhvalov, S. M. Kuznetsova,\nand A. I. Buzdin, Interference Phenomena and Long-\nrange Proximity E\u000bect in Clean Superconductor-\nFerromagnet Systems, Phys. Rev. Lett. 109, 237006\n(2012).\n[25] C. Richard, M. Houzet, and J. S. Meyer, Superharmonic\nLong-range Triplet current in a Di\u000busive Josephson Junc-\ntion, Phys. Rev. Lett. 110, 217004 (2013).\n[26] M. Kne\u0014 zevi\u0013 c, L. Trifunovic, and Z. Radovi\u0013 c, Signature of\nthe long range triplet proximity e\u000bect in the density of\nstates, Phys. Rev. B 85, 094517 (2012).\n[27] S. Kawabata, Y. Asano, Y. Tanaka, and A. A. Golubov,\nRobustness of spin-triplet pairing and singlet{triplet\npairing crossover in superconductor/ferromagnet hy-\nbrids, J. Phys. Soc. Jpn. 82, 124702 (2013).\n[28] S. Hikino and S. Yunoki, Long-range Spin Current Driven\nby Superconducting Phase Di\u000berence in a Josephson\nJunction with Double Layer Ferromagnets, Phys. Rev.\nLett. 110, 237003 (2013).\n[29] H. Meng, J. Wu, X. Wu, M. Ren, and Y. Ren, Long-\nrange superharmonic Josephson current and spin-triplet\npairing correlations in a junction with ferromagnetic bi-\nlayers, Sci. Rep. 6, 21308 (2016).\n[30] H. Meng, Y. Ren, J. E. Villegas, and A. I. Buzdin,\nJosephson current through a ferromagnetic bilayer: Be-\nyond the quasiclassical approximation, Phys. Rev. B 100,\n224514 (2019).\n[31] A. F. Volkov, F. S. Bergeret, and K. B. Efe-\ntov, Odd Triplet Superconductivity in Superconductor-\nFerromagnet Multilayered Structures, Phys. Rev. Lett.\n90, 117006 (2003).\n[32] T. L ofwander, T. Champel, J. Durst, and M. Eschrig, In-\nterplay of Magnetic and Superconducting Proximity Ef-\nfects in Ferromagnet-Superconductor-Ferromagnet Tri-\nlayers, Phys. Rev. Lett. 95, 187003 (2005).\n[33] V. Braude and Yu. V. Nazarov, Fully Developed Triplet\nProximity E\u000bect, Phys. Rev. Lett. 98, 077003 (2007).\n[34] M. Houzet and A. I. Buzdin, Long range triplet Joseph-\nson e\u000bect through a ferromagnetic trilayer, Phys. Rev. B\n76, 060504(R) (2007).\n[35] Ya. V. Fominov, A. F. Volkov, and K. B. Efetov, Joseph-\nson e\u000bect due to the long-range odd-frequency triplet su-\nperconductivity in SFS junctions with N\u0013 eel domain walls,\nPhys. Rev. B 75, 104509 (2007).\n[36] I. B. Sperstad, J. Linder, and A. Sudb\u001c, Joseph-\nson current in di\u000busive multilayer superconduc-\ntor/ferromagnet/superconductor junctions, Phys. Rev.\nB78, 104509 (2008).[37] F. Konschelle, J. Cayssol, and A. I. Buzdin, Nonsinu-\nsoidal current-phase relation in strongly ferromagnetic\nand moderately disordered SFS junctions, Phys. Rev. B\n78, 134505 (2008).\n[38] L. Trifunovic and Z. Radovi\u0013 c, Long-range spin-triplet\nproximity e\u000bect in Josephson junctions with multilayered\nferromagnets, Phys. Rev. B 82, 020505(R) (2010).\n[39] A. F. Volkov and K. B. Efetov, Odd spin-triplet\nsuperconductivity in a multilayered superconductor-\nferromagnet Josephson junction, Phys. Rev. B 81,\n144522 (2010).\n[40] M. Alidoust and J. Linder, Spin-triplet supercurrent\nthrough inhomogeneous ferromagnetic trilayers, Phys.\nRev. B 82, 224504 (2010).\n[41] A. V. Samokhvalov, R. I. Shekhter, and A. I. Buzdin,\nStimulation of a singlet superconductivity in SFS weak\nlinks by spin{exchange scattering of Cooper pairs, Sci.\nRep.4, 5671 (2014).\n[42] K. Halterman, O. T. Valls, and C.-T. Wu, Charge and\nspin currents in ferromagnetic Josephson junctions, Phys.\nRev. B 92, 174516 (2015).\n[43] C. Chen and B. Jin, Triplet proximity e\u000bect in super-\nconductor/ferromagnet/ferromagnet/ ferromagnet/ su-\nperconductor Josephson junctions, Physica C Supercond.\n588, 1353921 (2021).\n[44] M. J. A. Stoutimore, A. N. Rossolenko, V. V. Bolgi-\nnov, V. A. Oboznov, A. Y. Rusanov, D. S. Baranov, N.\nPugach, S. M. Frolov, V. V. Ryazanov, and D. J. Van\nHarlingen, Second-Harmonic Current-Phase relation in\nJosephson Junctions with Ferromagnetic Barriers, Phys.\nRev. Lett. 121, 177702 (2018).\n[45] Z. Radovi\u0013 c, L. Dobrosavljevi\u0013 c-Gruji\u0013 c, and B. Vuji\u0014 ci\u0013 c, Co-\nexistence of stable and metastable 0 and \u0019states in\nJosephson junctions, Phys. Rev. B 63, 214512 (2001).\n[46] Z. Radovi\u0013 c, N. Lazarides, and N. Flytzanis, Josephson ef-\nfect in double-barrier superconductor-ferromagnet junc-\ntions, Phys. Rev. B 68, 014501 (2003).\n[47] A. Buzdin, Peculiar properties of the Josephson junction\nat the transition from 0 to \u0019state, Phys. Rev. B 72,\n100501(R) (2005).\n[48] P. H. Barsic, O. T. Valls, and K. Halterman, First-order\nphase transitions in ferromagnet/superconductor layered\nstructures, Phys. Rev. B 73, 144514 (2006).\n[49] T. S. Khaire, M. A. Khasawneh, W. P. Pratt, and N.\nO. Birge, Observation of Spin-Triplet Superconductivity\nin Co-Based Josephson Junctions, Phys. Rev. Lett. 104,\n137002 (2010).\n[50] K. Lahabi, M. Amundsen, J. A. Ouassou, E. Beukers, M.\nPleijster, J. Linder, P. Alkemade, and J. Aarts, Control-\nling supercurrents and their spatial distribution in ferro-\nmagnets, Nat. Commun. 8, 2056 (2017).\n[51] J. A. Glick, V. Aguilar, A. B. Gougam, B. M. Niedzielski,\nE. C. Gingrich, R. Loloee, J. W. P. Pratt, and N. O.\nBirge, Phase control in a spin-triplet SQUID, Sci. Adv.\n4, eaat9457 (2018).\n[52] O. M. Kapran, A. Iovan, T. Golod, and V. M. Krasnov,\nObservation of the dominant spin-triplet supercurrent in\nJosephson spin valves with strong Ni ferromagnets, Phys.\nRev. Research 2, 013167 (2020).\n[53] V. Aguilar, D. Korucu, J. A. Glick, R. Loloee, W. P.\nPratt, and N. O. Birge, Spin-polarized triplet supercur-\nrent in Josephson junctions with perpendicular ferromag-\nnetic layers, Phys. Rev. B 102, 024518 (2020).9\n[54] G. A. Ovsyannikov, A. E. Sheyerman, A. V. Shadrin,\nYu. V. Kislinskii, K. Y. Constantinian, and A. Kal-\nabukhov, Triplet superconducting correlations in oxide\nheterostructures with a composite ferromagnetic inter-\nlayer, JETP Lett. 97, 145 (2013).\n[55] A. Pal, Z. H. Barber, J. W. A. Robinson, and M. G.\nBlamire, Pure second harmonic current-phase relation in\nspin-\flter Josephson junctions, Nat. Commun. 5, 3340\n(2014).\n[56] A. F. Volkov, Signatures of a long-range spin-triplet com-\nponent in an Andreev interferometer, Phys. Rev. B 102,\n094517 (2020).\n[57] I. \u0014Zuti\u0013 c and O. T. Valls, Tunneling spectroscopy for fer-\nromagnet/superconductor junctions, Phys. Rev. B 61,\n1555 (2000).\n[58] M. Yu. Kuprianov and V. F. Lukichev, In\ruence of\nboundary transparency on the critical current of \\dirty\"\nSS'S structures, Zh. Eksp. Teor. Fiz. 94, 139 (1988) [Sov.\nPhys. { JETP 67, 1163 (1988)].\n[59] B. M uhlschlegel, Die thermodynamischen Funktionen des\nSupraleiters, Z. Phys. 155, 313 (1959).[60] G. E. Blonder, M. Tinkham, and T. M. Klapwijk, Tran-\nsition from metallic to tunneling regimes in supercon-\nducting microconstrictions: Excess current, charge im-\nbalance, and supercurrent conversion, Phys. Rev. B 25,\n4515 (1982).\n[61] A. Furusaki and M. Tsukada, Dc Josephson e\u000bect and\nAndreev re\rection, Solid State Commun. 78, 299 (1991).\n[62] J. W. A. Robinson, G. B. Hal\u0013 asz, A. I. Buzdin, and M.\nG. Blamire, Enhanced Supercurrents in Josephson Junc-\ntions Containing Nonparallel Ferromagnetic Domains,\nPhys. Rev. Lett. 104, 207001 (2010).\n[63] V. N. Krivoruchko and E. A. Koshina, From inversion to\nenhancement of the dc Josephson current in S/F-I-F/S\ntunnel structures, Phys. Rev. B 64, 172511 (2001).\n[64] F. S. Bergeret, A. F. Volkov, and K. B. Efetov, Enhance-\nment of the Josephson Current by an Exchange Field\nin Superconductor-Ferromagnet Structures, Phys. Rev.\nLett. 86, 3140 (2001).\n[65] N. M. Chtchelkatchev, W. Belzig, and C. Bruder, Joseph-\nson e\u000bect in S FXSFjunctions, JETP Lett. 75, 646 (2002)."    },    {        "title": "1608.01136v1.Evolution_of_magnetic_fluctuations_through_the_Fe_induced_paramagnetic_to_ferromagnetic_transition_in_Cr__2_B.pdf",        "content": "arXiv:1608.01136v1  [cond-mat.str-el]  3 Aug 2016Evolution of magnetic ﬂuctuations through the Fe-induced p aramagnetic to\nferromagnetic transition in Cr 2B\nD. Arˇ con,1,2,∗L. M. Schoop,3,4R. J. Cava,3and C. Felser4\n1Joˇ zef Stefan Institute, Jamova c. 39, 1000 Ljubljana, Slov enia\n2Faculty of Mathematics and Physics, University of Ljubljan a, Jadranska c. 19, 1000 Ljubljana, Slovenia\n3Department of Chemistry, Princeton University, Princeton , New Jersey 08544, USA\n4Max-Planck-Institut f¨ ur Chemische Physik fester Stoﬀe, 01 187 Dresden, Germany\n(Dated: October 15, 2018)\nIn itinerant ferromagnets, the quenched disorder is predic ted to dramatically aﬀect the ferro-\nmagnetic to paramagnetic quantum phase transition driven by external control parameters at zero\ntemperature . Here we report a study on Fe-doped Cr 2B, which, starting from the paramagnetic\nparent, orders ferromagnetically for Fe-doping concentra tionsxlarger than xc= 2.5%. In parent\nCr2B,11B nuclear magnetic resonance data reveal the presence of bot h ferromagnetic and antifer-\nromagnetic ﬂuctuations. The latter are suppressed with Fe- doping, before the ferromagnetic ones\nﬁnally prevail for x > x c. Indications for non-Fermi liquid behavior, usually assoc iated with the\nproximity of a quantum critical point, were found for all sam ples, including undoped Cr 2B. The\nsharpness of the ferromagnetic-like transition changes on movingaway from xc, indicating signiﬁcant\nchanges in the nature of the magnetic transitions in the vici nity of the quantum critical point. Our\ndata provide constraints for understanding quantum phase t ransitions in itinerant ferromagnets in\nthe limit of weak quenched disorder.\nPACS numbers: 76.60.-k, 75.50.Cc, 73.43.Nq, 76.50.+g\nI. INTRODUCTION\nItinerant ferromagnets are a class of materials for\nwhich the transition from the paramagnetic to ferromag-\nnetic state is considered as a canonical example of a sec-\nond order phase transition. When an external control\nparameter such as pressure suppresses the ferromagnetic\nstate, then these materials approach a quantum critical\npoint1(QCP) that separates the itinerant ferromagnetic\nstate from its paramagnetic counterpart.2In such cases,\nthe transition between two nearly degenerate magnetic\nground states is driven by nonthermal ﬂuctuations and\nthe system undergoes a quantum phase transition at zero\ntemperature. In an external magnetic ﬁeld, however, the\ntransition, before being completely suppressed, changes\ntoﬁrstorderbelowatricriticaltemperature.3,4Moreover,\nwhile itinerant ferromagnets are usually well described\nwithin the framework of standard Fermi liquid theory, in\nthe proximity of a QCP a non-Fermi liquid state can of-\nten be inferred5,6from resistivity measurements, i.e. the\nresistivity displays a power-law temperature dependence\nρ∝Tnwith an unconventional exponent of n <2.7\nThe existence of a QCP, the quantum critical region at\nﬁnite temperatures and non-Fermi liquid behavior has\nbeen experimentally veriﬁed on a number of clean itiner-\nant ferromagnets, such as 3 d-based MnSi,7,8ZrZn2(Ref.\n9) or NbFe 2(Ref. 10) and 5 f-based UGe 2, URhGe or\nUCoGe.11–14\nIn contrasttothe establishedcaseofQCPsin the clean\nitinerant ferromagnets mentioned above, the behavior in\nsystems with quenched disorder is less clear. Quenched\ndisorder is predicted to suppress the tricritical tempera-\nture until it vanishes at a critical value of disorder and\nthe transition changes back to second order with non-mean-ﬁeld exponents.2However, examples of itinerant\nferromagnets where the ferromagnetic transition is sup-\npressed by quenched disorderat the QCP are remarkably\nscarce. Therefore, there is a need for new model systems\nwhere the nature of the phase transition and possible\ndeviations from the Fermi-liquid state can be systemat-\nically investigated close to the QCP in the presence of\ndisorder.\nRecently, a paramagnetic to ferromagnetic phase tran-\nsition has been reported in lightly Fe-doped Cr 2B.15The\nparent Cr 2B is an intermetallic compound that crystal-\nlizes in an orthorhombic structure16,17(Fig. 1) with\nmany bands crossing the Fermi energy.18Due to the\nglide planes in the nonsymmorphic space group Fddd,\nthe presence of three-dimensional Dirac points in the\nelectronic structure is symmetry dictated and may ac-\ncount for the n−type carriers with high mobility mea-\nsured in Hall eﬀect experiments.18According to ﬁrst\nprinciplecalculations, thegroundstateofCr 2Bshouldbe\nantiferromagnetic.19Experimentally, antiferromagnetic\ncorrelations have indeed been inferred from the magne-\ntization data, although no transition to a long-range or-\ndered phase in Cr 2B has been found.15Upon Fe-doping,\ndetailed Arrott analysis of the temperature and the ﬁeld\ndependences of magnetization data revealed that a fer-\nromagnetic phase emerges at a critical Fe-concentration\nnearx= 0.02.15Moreover,the observedlogarithmic con-\ntribution to the heat capacity was taken as a hallmark of\nquantum criticality, and the power-law exponent n∼1\nin the temperature dependence of resistivity was taken\nto indicate a non-Fermi liquid state in doped samples.\nHowever, whether Fe-doped Cr 2B does indeed represent\na new family of intermetallics where quenched disorder\ndrives a ferromagnetic quantum phase transition calls for2\nadditional independent experimental conﬁrmation.\nMagnetic resonance techniques, such as nuclear mag-\nnetic resonance (NMR) and electron spin resonance\n(ESR), have proven to be extremely powerful probes of\nlocal spin susceptibilities and spin ﬂuctuations close to a\nQCP.20–26Here we report detailed11B NMR and ESR\nmeasurements on polycrystalline Fe-doped Cr 2B at dif-\nferent doping levels acrossthe paramagneticto ferromag-\nnetic transition. Surprisingly, we ﬁnd a strong tempera-\nture dependent shift of the11B NMR spectra that gen-\nerally follows a Curie-Weiss-like dependence, revealing\na crossover from predominantly antiferromagnetic corre-\nlations to predominantly ferromagnetic correlations at a\ncriticalFe-dopinglevelof x∼2.5%. Oncooling,weﬁnda\ncrossoverto a low-temperaturestate where the11B NMR\nshift displays a non-Fermi-liquid T−1/2-dependence, thus\ncomplyingwithwhatisexpectedinthevicinityofaQCP.\nSimultaneously, the appearance of a strong electron spin\nresonance signal and a substantial broadening of the11B\nNMR spectra provide evidence for ferromagnetic-like or-\ndering at low-temperatures that is completely absent at\nlower Fe-doping levels. The data presented here reveal\nhighly unusual behavior for Fe-doped Cr 2B, which can-\nnotbe simplyrationalizedwithin astandardFermi-liquid\nmodel. Moreover, the systematic evolution of the sharp-\nness of the ferromagnetic-like transition with Fe-doping\nlevelimpliessigniﬁcantchangesinthenatureofthephase\ntransition as the quenched disorder varies across the crit-\nical Fe-doping value for the quantum phase transition\n(QPT).\nFIG.1. (color online). (a)Theorthorhombiccrystal struct ure\nof Cr2B. (b) The local coordination of each boron site (green\npolyhedron) has eight nearest Cr atoms (large blue spheres)\nand two more B atoms (small green spheres) at apical posi-\ntions.II. EXPERIMENTAL METHODS\nFor this study, the polycrystalline Fe-doped Cr 2B sam-\nples used in Ref. 15 to characterize the magnetic and\ntransport properties of the Cr 2−xFexB system were em-\nployed. The Fe-doping range is between 0 and 5%. Ac-\ncording to powder x-ray diﬀraction, only very small frac-\ntions of non-magnetic Cr metal were identiﬁed in the\ndiﬀraction proﬁles in addition to the doped Cr 2B phase.\nThe samples were also characterized in detail by high\nresolution transmission electron microscopy and high an-\ngle annular dark-ﬁeld scattering transmission microscopy\nto exclude the possibility of Fe clustering. The mag-\nnetic susceptibility data excluded the presence of mag-\nnetic impurities.15\nFor the temperature-dependent ESR experiments, the\nsamples were sealed under helium in a standard 4 mm-\ndiameter silica tube (Wilmad Lab Glass) whereas for the\n11B NMR experiments, samples with a mass of around\n90 mg were directly inserted into the NMR coil. A con-\nventional continuous wave (cw) electron paramagnetic\nresonance spectrometer operating at a Larmor frequency\nESRνL= 9.6 GHz was employed to detect the electron\nspin resonance. The spectrometer is equipped with a\nstandard Varian E-101 microwave bridge, a Varian rect-\nangular TE102 resonance cavity, and an Oxford Cryo-\ngenicscontinuous-ﬂowhelium cryostat. The temperature\nstability was better than ±0.1 K over the entire temper-\nature range of measurements (4 −300 K).\nThe11B (nuclear spin I= 3/2) NMR spectra and\nthe spin-lattice relaxation rates were measured between\n5 and 300 K in a magnetic ﬁeld of 4 .7 T. The11B NMR\nshifts are determined relative to the Larmor frequency\n11νL= 64.167MHz, deﬁnedbyaBF 3Et2Ostandard. For\n11B NMR line shape measurements, asolid-echopulse se-\nquence,π/2−τ−π/2−τ−echo, was employed, with\na pulse length tw(π/2) = 1.9µs and an interpulse delay\nτ= 40µs. The complete polycrystalline NMR spec-\ntrum was obtained by summing the real part of spec-\ntra measured step-by-step at resonance frequencies sep-\narated by ∆ ν= 50 kHz. Since it was impossible to com-\npletely invert the11B nuclear magnetization, we used the\nsaturation-recoverypulse sequence for the spin-lattice re-\nlaxation rate measurements. Typically, the saturation\ntrain consisted of a sequence of 20 π/2 pulses separated\nby 50µs.\nBecause11B is a quadrupole nucleus, we model the\n11B NMR spectra with the general spin Hamiltonian\nH=HZ+HQ+HScomprisingthe nuclearZeeman ( HZ),\nnuclear quadrupole ( HQ) and11B shift ( HS) terms, re-\nspectively. The later gives rise to the11B NMR lineshift\nof the central −1/2↔1/2 transition, K, and includes\nthe temperature independent chemical shift, σ, and the\nKnightshift, Ks,whichoriginatesfromthehyperﬁnecou-\npling toitinerant electrons. The mostimportant term for\nour study is Ks, as it is directly proportional to the local\nelectronic susceptibility χ, e.g., its isotropic part is given\nbyKiso=as\nNAµ0χ, whereasis a hyperﬁne coupling con-3\n/s40/s98/s41\n/s53/s46/s48/s37/s32/s70/s101/s50/s46/s53/s37/s32/s70/s101/s50/s46/s48/s37/s32/s70/s101/s48/s46/s53/s37/s32/s70/s101\n/s32/s32/s49/s49\n/s66/s32/s78/s77/s82/s32/s115/s105/s103/s110/s97/s108/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s49/s49\n/s66/s32/s78/s77/s82/s32/s115/s104/s105/s102/s116/s32/s40/s37/s41/s67/s114\n/s50/s66/s84 /s32/s61/s32/s51/s48/s48/s32/s75/s40/s97/s41\n/s48 /s49 /s50 /s51 /s52 /s53 /s54/s48/s50/s48/s52/s48/s54/s48/s56/s48/s32/s49/s47/s50/s32/s40/s107/s72/s122/s41\n/s70/s101/s45/s100/s111/s112/s105/s110/s103/s32/s40/s37/s41/s52/s48/s48/s52/s50/s53/s52/s53/s48/s52/s55/s53/s53/s48/s48\n/s81/s32/s40/s107/s72/s122/s41\nFIG. 2. (color online). (a) Comparison of the room-\ntemperature11B NMR spectra (gray shaded area) measured\non polycrystalline Fe-doped Cr 2B samples at various Fe-\ndoping levels. Solid red lines are the powder lineshape ﬁts\nto a model that includes anisotropic shift and quadrupole ef -\nfects up to the second order. A single11B NMR component\nwas suﬃcient in all cases. (b) Fe-doping dependence of the\n11B NMRlinewidth broadeningparameter, δ1/2, (solid circles,\nleft scale) and of the quadrupole frequency νQ, (open circles,\nright scale). The solid and dashedblack lines are guides tot he\neye for the Fe-doping dependence of δ1/2andνQ, respectively.\nstant,NAis Avogadro’s number and µ0is the magnetic\npermeability of vacuum. When analyzing the11B NMR\nspectra of Cr 2B we include the11B anisotropic shift and\nthe quadrupole eﬀects up to the second order. Fitting of\nthe spectra thus yields, in addition to K, the quadrupole\nsplitting frequency νQ=3eVzzQ\nh2I(2I−1)and the asymmetry\nparameter η= (Vxx−Vyy)/Vzz, both deﬁned by the com-\nponents of the electric ﬁeld gradient (EFG) Vijat the\n11B site. Here Qandhare the11B quadrupole moment\nand the Planck constant, respectively. The powder NMR\nspectrum is computed as a histogram of resonance fre-\nquenciesobtainedbysummingoveruniformlydistributed\npolar and azimuthal angles of the magnetic ﬁeld orien-\ntation with respect to the quadrupole tensor principal\naxes.27,28We included the homogeneous broadening of\nthe line through the convolution of the computed spectra\nwith a Lorentzian line with a full-width-half-maximum\nδ1/2. The eﬀect of quadrupole splitting frequency distri-\nbution on11B NMR spectra was alsotested by using nor-\nmally distributed values of quadrupole frequencies with\nthe center at νQand width of ∆ νQ.\nIII. RESULTS AND DISCUSSION\nA. Homogeneity of the Cr 2B samples after Fe\ndoping\nThe room temperature11B NMR spectrum of Cr 2B\nsample [Fig. 2(a)] displays a characteristic quadrupolepowder lineshape with a clearly pronounced satellite\ntransition ( ±3/2↔ ±1/2) singularities ﬂanking the cen-\ntral peak that corresponds to a −1/2↔1/2 transi-\ntion. The spectrum is considerably shifted by K=\n−228(3) ppm relative to the11B NMR reference fre-\nquency. The central peak has a nearly Lorentzian\nlineshape thus implying that the broadening due to\nthe anisotropic shift interactions and the second-order\nquadrupole corrections is negligible. A lineshape ﬁt of\nthe spectrum yields a quadrupole splitting frequency\nofνQ= 471(8) kHz and an asymmetry parameter of\nη= 0.02(1). Surprisingly, we ﬁnd η≈0, although\nthe11B site symmetry – B is at the low-symmetry 16 g\n(0.125, 0.125, 0.4993) site [Fig. 1(b)]16,17– does not re-\nquire such a restriction. The homogeneous broadening is\nδ1/2= 24(1) kHz.\nLight doping of Cr 2B with 0.5% Fe induces almost no\nchange to the11B NMR lineshape [Fig. 2(a)] and the\nunconstrained ﬁt returns, compared to parent undoped\nCr2B, nearly the same νQ= 483(9) kHz, η= 0.02(1) and\nδ1/2= 24(1) kHz. As the level of Fe-doping increases,\nhowever, the11B NMR spectra drastically broaden, and\nfor 5% Fe doping the central peak also becomes slightly\nanisotropic. On extracting the parameters from the11B\nNMR lineshapeﬁts, weﬁrstnoticethat νQmarginallyre-\nduces with increasingFe concentration[Fig. 2(b)], i.e. to\n467(8) kHz, 465(8) kHz and 461(10) kHz in the 2%, 2.5%\nand 5% doped samples, respectively. This insensitivity\nofνQto the Fe-doping implies that the lattice around\nthe B-sites contributing to the measured spectra is not\nmarkedly perturbed for Fe-doping levels up to 5%. (We\nnote that it is possible that the B atoms sitting next to\nFe-dopant atoms experience signiﬁcantly diﬀerent EFG\nand hyperﬁne ﬁelds that shift and broaden the spectra\nbeyond the sensitivity of the present NMR experiments.)\nNevertheless, the local hyperﬁne ﬁelds at “weakly per-\nturbed”11B sites do change, judging from the Fe-doping\nvariationinthelinewidthparameter δ1/2, whichincreases\nto 31(1) kHz, 34(1) kHz and 64(2) kHz for 2%, 2.5% and\n5% Fe doping, respectively [Fig. 2(b)]. Simultaneously,\nsatellite transition singularities become signiﬁcantly less\npronounced, which explains the larger η= 0.08(1) for\nthe largest Fe-doping level. (We note that for the 5%\ndoped sample a11B NMR lineshape ﬁt that includes a\nsmall distribution of νQ, i.e. ∆νQ/νQ= 6%, describes\nthe experimental spectrum equally well.) Therefore we\nﬁnd that Fe-doping indeed introduces some local-site dis-\norder, which is rather sensitively picked up by the NMR\nparameters. However, because all11B NMR spectra can\nstill be simulated with a single well-deﬁned value of νQ,\nwe conclude that the Fe-doping must be rather homoge-\nneous for all samples, consistent with previous chemical\nand structural characterization.154\nB. The paramagnetic state of parent Cr 2B\nFig. 3a shows the temperature dependence of the cen-\ntral transition peak in parent Cr 2B. This peak retains\nits Lorentzian lineshape at all temperatures and shows\nalmost no broadening between room temperature and\nT= 4 K. The complete absence of broadening of the\npowder11B NMR spectra is a ﬁrm evidence that no\nstatic magnetic order is established. However, the spec-\ntra monotonically shift towards higher frequencies with\ndecreasing temperature. Between room temperature and\n50 K the11B NMR shift Kshows [Fig. 3(b)] a Curie-\nWeiss-like dependence\nK(T) =σ+B/(T−Tcw). (1)\nHereweidentifythe constant σ=−264(3)ppm asatem-\nperature independent chemical shift. On the other hand,\nthe second temperature-dependent Knight-shift contri-\nbution originates from the hyperﬁne interactions with\nthe unpaired electronic moments. The extracted nega-\ntive Curie-Weiss temperature Tcw=−12(3) K implies\nthat the involved itinerant electronic states are antiferro-\nmagnetically correlated and thus corroborates the mag-\nnetization data15and the ﬁrst principle calculations.19\nAtT≈35 K we notice a small discontinuity in K,\nbut for lower temperatures Kstill continues to increase\nwith decreasing temperature. Plotting the temperature\ndependence of the Knight shift Ks(T) =K(T)−σon\na log-log plot [inset to Fig. 3(b)], we notice a gradual\nchange of slope at around 35 K. Whereas at higher tem-\nperaturestheslopeof Ksindeedﬁtstoa T−1dependence,\nas established above, we ﬁnd that for lower temperatures\nKsdevelops approximately a T−1/2dependence before\nleveling oﬀ at the lowest temperatures. A very similar\nsequence of power-laws in the temperature dependence\nof the Knight shift has been reported for YbRh 2Si2,20\nand attributed to non-Fermi-liquid behavior in the vicin-\nity of a QCP.\nNext, the spin dynamics in the paramagnetic state of\nparent Cr 2B were probed through the11B spin-lattice re-\nlaxation rate 1 /T1. In striking contrast to the strongly\ntemperature dependent K, the spin-lattice relaxation\nrate divided by temperature, 1 /T1T, shows a much\nweaker temperature dependence [Fig. 3(c)]. Between\nroom temperature and 35 K, 1 /T1Tgradually decreases\nby∼30%, but then on further cooling it starts to in-\ncrease again.\nForcorrelatedmetals whereelectron-electronexchange\nenhancement eﬀects are important, the Korringarelation\nis29\n11T1TK2\ns=¯h\n4πkBγ2\ne\nγ2\n11β. (2)\nHereγeandγ11are the electronic and11B gyromag-\nnetic ratios, respectively. The Korringa factor βis intro-\nduced to account for the electron-electron exchange.29–31\nInserting the room-temperature value of 1 /T1T= 2.9·10−3s−1K−1andKs= 25 ppm into Eq. (2) we calcu-\nlateβ= 3.3 thus complying with the enhancement of\ndynamic spin susceptibility. In general, β >1 implies\nferromagnetic ﬂuctuations,30,31seemingly contradicting\nthe analysis of the Knight-shift data, which disclosed the\npresence of antiferromagnetic correlations. We addition-\nally note that, although the precise value of the Korringa\nfactorβdepends on the choice of σ, we still ﬁnd that the\nincrease of βwith decreasing temperature further cor-\nroborates the presence of ferromagnetic correlations. In\norder to explain this apparent contradiction, we suggest\nthat both ferromagnetic and antiferromagnetic spin ﬂuc-\ntuations are present, but the later are ﬁltered out at the\n11B site.Namely, the local coordination of11B site has\neight nearest neighboring Cr atoms and two more11B\nsites at the apical positions [Fig. 1(b)]. We next point\nout that near the Fermi level, the calculated density of\nstates is dominated by Cr dbands.19Therefore, the ap-\npropriate electron-nuclear Hamiltonian consists of a sum\nof transferred hyperﬁne coupling of11B at site kto the\neight nearestneighborCr electronspins at k. As aresult,\nthe contribution of antiferromagneticspin ﬂuctuations to\nthe spin-lattice relaxation may be suppressed.29,32The\nimportant conclusion from this part of our study is thus\nthatalthoughthemulti-bandCr 2Bparentcompounddis-\nplays both antiferromagnetic and ferromagnetic correla-\ntions, they are not suﬃciently strong to establish a long-\nrange magnetically ordered state. The observed anoma-\nlies at∼35 K in both K(T) and 1/T1Tdata imply the\npresence of subtle electronic changes that may preclude\nmagnetic ordering at low temperatures, but additional\nmore detailed experiments in this temperature range are\nrequired to unveil the nature of the undoped Cr 2B sys-\ntem.\nC. The ferromagnetic state in Fe-doped Cr 2B\nCr2B becomes ferromagnetic when 2% or more of the\nCr atoms are replaced by Fe.15In order to investigate\nthe ferromagnetic state we ﬁrst employ the electron spin\nresonance technique. For 5% Fe-doped Cr 2B in the high\ntemperature metallic paramagneticphase, no conducting\nelectron spin resonance(CESR) signal could be detected.\nHowever, as the temperature is decreased below ∼70 K,\nanasymmetricDyson-likeresonance33appearsat a g≈2\nresonance ﬁeld [Fig. 4(a)]. The Dyson lineshape of the\nresonance is a hallmark of a metallic state, thus ruling\nout previously undetected insulating iron-oxide impuri-\ntiesasapossiblesourceoftheobservedESRsignal. With\ndecreasing temperature, the signal gradually grows in in-\ntensity and shifts to lower resonance ﬁelds, signaling the\nonset and growth of internal magnetic ﬁelds in the ma-\nterial. The observed behavior is consistent with what\nis seen for ferromagnetic resonance in the metallic state,\nand thus agrees with the proposed ferromagnetic ground\nstate for 5% Fe-doped Cr 2B.\nAs the Fe-dopinglevel decreases,the appearanceofthe5\n/s45/s54/s48/s48 /s45/s52/s48/s48 /s45/s50/s48/s48 /s48 /s50/s48/s48 /s52/s48/s48/s45/s50/s46/s48/s45/s49/s46/s53/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s67/s114\n/s50/s66\n/s52/s32/s75/s49/s54/s32/s75/s50/s55/s32/s75/s53/s48/s32/s75/s56/s48/s32/s75/s49/s49/s48/s32/s75/s49/s51/s48/s32/s75/s49/s54/s48/s32/s75/s50/s48/s48/s32/s75\n/s32/s32/s78/s77/s82/s32/s115/s105/s103/s110/s97/s108/s32/s40/s97/s114/s98/s46/s32/s117/s46/s41\n/s49/s49\n/s66/s32/s78/s77/s82/s32/s115/s104/s105/s102/s116/s32/s40/s112/s112/s109/s41/s51/s48/s48/s32/s75/s40/s97/s41 /s67/s114\n/s50/s66\n/s45/s51/s48/s48/s45/s50/s48/s48/s45/s49/s48/s48/s48\n/s40/s99/s41/s40/s98/s41\n/s32/s32/s75 /s32/s40/s112/s112/s109/s41\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s48/s50/s52\n/s67/s114\n/s50/s66\n/s32/s32/s49/s47 /s84\n/s49/s84 /s32/s40/s49/s48/s45/s51\n/s32/s115/s45/s49\n/s75/s45/s49\n/s41/s32\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s49/s48 /s49/s48/s48/s49/s48/s49/s48/s48/s84/s45/s49/s47/s50/s75/s45 /s32/s40/s112/s112/s109/s41\n/s84/s32/s40/s75/s41/s84/s45/s49\nFIG. 3. (color online). Temperature dependences of (a) the\n11B central transition ( −1/2↔1/2) peak, (b) the shift of\nthe central transition peak, K, and the spin-lattice relaxation\nrate 1/T1divided by Tmeasured in parent undoped Cr 2B.\nThe vertical and horizontal dashed lines in (a) and (c) mark\nroom temperature values. The solid line in (b) is a ﬁt with a\nCurie-Weiss dependence to a negative Curie-Weiss tempera-\nture ofTcw=−12(3) K. The inset to (b) shows the temper-\nature dependence of the Knight shift, Ks, on a log-log scale,\nobtained after subtracting the chemical shift σfromK. Solid\nlines indicate the high-temperature slope of Ks∝T−1and\nthe low-temperature slope of Ks∝T−1/2.\nferromagnetic-like resonance is systematically shifted to\nlower temperatures. For instance, whereas at 40 K in 5%\nFe-doped sample the resonance signal is very strong, no\nsuch line was detected at the same temperature for the\n4% and 3.5% Fe-doping levels [Fig. 4(b)]. It is, however,\ndetected at lower temperatures for those compositions.\nIn order to quantitatively follow the ferromagnetic-like\nresonance, we next ﬁt the spectra to a Dyson lineshape33\n[Fig. 4(a)]. The extracted temperature dependences of\nthe intensity ofthe resonancepeak, which is proportional\nto the sample’s magnetization, aresummarized for diﬀer-\nent Fe-doping levels in Fig. 4(c). The magnetic order-\ning onset temperature Tcsystematically decreases from\n∼70 K at 5% doping to ∼30 K and then to ∼18 K\nfor 4% and 3.5% Fe-doped samples, respectively. In ad-\ndition, whereas the transition in the 5% doped sample\nis smeared over a large temperature interval, it is much\nsharper at lower doping-levels, which are closer to the\nQCP. This experimental observation may imply signiﬁ-\ncant changes in the nature of magnetic transition in the\nvicinity of the QCP.\nFurther insight into the development of local mag-\nnetic ﬁelds is provided by the11B NMR spectra mea-\nsured for 5% Fe-doped Cr 2B [Fig. 5(a)]. Whereas the\nspectra retain a characteristic quadrupole I= 3/2 pow-\nder lineshape with small anisotropic Knight shift inter-\naction [e.g., similar as at 300 K (Fig. 2)] on cooling\ndown to ∼140 K, broadening due to the anisotropic\nKnight shift interaction gradually begins to dominate\nthe spectra at lower temperatures. Below∼50 K the/s48 /s50/s48/s48 /s52/s48/s48 /s54/s48/s48/s45/s49/s48/s49/s50\n/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48/s48/s51/s54/s57\n/s48 /s50/s48/s48 /s52/s48/s48 /s54/s48/s48 /s56/s48/s48/s45/s54/s45/s52/s45/s50/s48/s50/s52/s54\n/s53/s50/s32/s75\n/s52/s48/s32/s75\n/s51/s50/s32/s75\n/s50/s52/s32/s75\n/s49/s50/s32/s75/s69/s80/s82/s32/s115/s105/s103/s110/s97/s108/s32/s32/s40/s97/s114/s98/s46/s117/s46/s41\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s32/s40/s109/s84/s41/s53/s32/s75/s40/s97/s41\n/s53/s37/s32/s70/s101/s215/s53\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s32/s40/s109/s84/s41/s83/s105/s103/s110/s97/s108/s32/s32/s40/s97/s114/s98/s46/s117/s46/s41 /s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s32/s40/s97/s114/s98/s46/s117/s46/s41/s40/s98/s41\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s32/s40/s75/s41/s84 /s32/s61/s32/s52/s48/s32/s75\n/s32/s53/s37/s32/s70/s101\n/s32/s52/s37/s32/s70/s101\n/s32/s51/s46/s53/s37/s32/s70/s101/s40/s99/s41\nFIG. 4. (color online). (a) Temperature dependence of fer-\nromagnetic resonance in 5% Fe doped Cr 2B (black circles)\nmeasured at X-band (ESRνL= 9.6 GHz) frequencies. Solid\nred lines are ﬁts of the spectra to the Dyson lineshape. The\ndotted vertical line at 330 mTmarks theresonance ﬁeld of the\ng= 2 electron paramagnetic resonance signal. (b) Compari-\nson of ferromagnetic resonance spectra measured at 40 K for\n5% (black circles), 4% (red triangles) and 3.5% (blue square s)\nFe-doped Cr 2B, respectively. (c) Temperature dependence of\nthe intensities of the ferromagnetic resonance spectra for 5%\n(black circles), 4% (red triangles) and 3.5% (blue squares)\nFe-doped Cr 2B, respectively.\nspectra already display a lineshape that is reminiscent\nof anisotropic Knight shift interactions. Alternatively,\nthe lineshape broadening originating from the distribu-\ntion of Knight shifts is less probable because it is not\naccompanied also by the large quadrupole frequency dis-\ntribution. Therefore, for the ﬁts to the data, we assumed\ntemperature independent νQ= 461 kHz and η= 0.08,\nboth extracted from the room temperature spectra, and\nthat the isotropic Kand anisotropic Kanisoparts of the\nKnight-shift tensor are temperature dependent parame-\nters. The temperature dependences of KandKanisothus\nobtained are summarized in Fig. 5(b). Both parameters\nexhibit a very strong temperature dependence that can\nagain be described as a Curie-Weiss-like dependence be-\ntween room temperature and 70 K. Fitting the isotropic\nshiftK(T) to Eq. (1) yields σ=−270(8) ppm, which\nis nearly identical to the corresponding chemical shift of\nthe parent Cr 2B. On the other hand, we now ﬁnd a pos-\nitive Curie-Weiss temperature, TCW= 21(4) K, which\nis thus fully consistent with the dominant ferromagnetic\ncorrelations in heavily Fe-doped Cr 2B. The temperature\ndependence of Kanisoalso supports this ﬁnding.\nLarge broadening of the11B NMR spectra at low tem-\nperatures, reﬂected in the enhanced Kaniso, clearly ev-\nidences the development of static local magnetic ﬁelds\nand thus of spin-freezing. The ferromagnetic TCWsug-\ngests that the magnetic moments induced by Fe-doping\nfreeze into a spin state where ferromagnetic correlations\nprevail. However, what may be surprising is that the\nshift ofK(T) remains relatively small, i.e. on the order\nof∼300 ppm, which is thus only a fraction of Kaniso. We6\n/s53/s37/s32/s70/s101\n/s49/s56/s48/s32/s75\n/s32/s32\n/s50/s49/s32/s75/s53/s48/s32/s75/s55/s53/s32/s75/s57/s53/s32/s75/s49/s52/s48/s32/s75/s49/s49\n/s66/s32/s78/s77/s82/s32/s115/s105/s103/s110/s97/s108/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s49/s49\n/s66/s32/s78/s77/s82/s32/s115/s104/s105/s102/s116/s32/s40/s37/s41/s40/s97/s41\n/s45/s50/s48/s48/s45/s49/s48/s48/s48/s49/s48/s48/s32/s75 /s32/s40/s112/s112/s109/s41/s40/s98/s41\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s50/s52\n/s32/s32/s49/s47 /s84\n/s49/s84 /s32/s40/s49/s48/s45/s51\n/s32/s115/s45/s49\n/s75/s45/s49\n/s41\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s40/s99/s41/s45/s48/s46/s51/s45/s48/s46/s50/s45/s48/s46/s49/s48/s46/s48/s75\n/s97/s110/s105/s115/s111/s32/s40/s37/s41\nFIG. 5. (color online). (a) The temperature evolution of\nthe11B NMR spectra for 5% Fe-doped Cr 2B. The solid red\nlines are ﬁts to a model with quadrupole and anisotropic\nKnight-shift interactions. In the model, we assumed tem-\nperature independent νQ= 461 kHz and η= 0.08. (b)\nThe temperature dependences of the isotropic (red circles,\nleft scale) and anisotropic (blue squares, right scale) par ts of\nthe Knight shift. A ﬁt of the11B NMR shift, K(T), to Eq.\n1 (solid red line) yields the chemical shift σ=−270(8) ppm,\nB= 9.7(9)·103ppm K and the ferromagnetic Curie-Weiss\ntemperature TCW= 21(4) K. The dashed blue line is a guide\nto the eye. (c) Temperature dependence of the11B NMR\nspin-lattice relaxation rate divided by temperature, 1 /T1T.\nthus conclude that the contact hyperﬁne (or the isotropic\npart of the transferred hyperﬁne) interaction with itiner-\nant electrons is nearly the same in Fe-doped Cr 2B com-\npared to parent undoped Cr 2B. The reason for this is\ncurrently unknown, but one of the possibilities is that\nwhen Fe is introduced into the lattice it creates localized\nstates that interact with11B mostly via long-range, e.g.\ndipolar, interactions. This may also explain the temper-\nature dependence of the spin-lattice relaxation rate [Fig.\n5(c)]. Compared to parent Cr 2B, the slightly shorter T1\nat room temperature yields 1 /T1T= 1.3·10−3s−1K−1,\nand therefore a ferromagnetically enhanced β≈8. On\ncooling below ∼70 K, 1/T1Tindeed starts to increase as\nexpected when close to the magnetic ordering. However,\nthe absence of a divergence in 1 /T1T, normally found at\nthe magnetic ordering temperature, and the broadening\nof the ferromagnetic-like transition observed in the ESR\ndata (Fig. 4) may be signatures of a distribution of fer-\nromagnetic freezing temperatures, and thus of a smeared\ntransitiontoaferromagnetic-likestatewith ahighdegree\nof disorder.\nD. Behavior close to the critical Fe doping\ncomposition\nFinally, we focus on Fe-doping levels lower than 2.5%,\ni.e. the samples with suppressed ferromagnetic order-ing temperature being thus close to the QCP.15Inspect-\ning the low-temperature11B NMR spectra of these sam-\nples we ﬁnd that they retain a characteristic powder\nquadrupole lineshape at all temperatures (insets to Fig.\n6). This proves the absence of the strong internal ﬁelds\nthatwouldbroadenthe11BNMRspectra,aswasthecase\nfor ferromagnetic 5% Fe-doped Cr 2B at higher temper-\natures. The absence of the ferromagnetic-like resonance\nsignal in these samples down to 4 K provides additional\nevidence for the absence of any magnetic order. On the\nother hand, the temperature dependence of K(T) is a\nstrong indication of a non-Fermi liquid state. Namely,\nbetween room temperature and ∼70 K the shift follows\na Curie-Weiss-like dependence (Fig. 6), which seems to\nbe a general characteristic of all the Cr 2B samples stud-\nied, both doped and undoped.\nFittingK(T) in the temperature interval between 300\nand 70 K reveals a signiﬁcant Fe-doping dependence of\nthe NMR-determined Curie-Weiss temperature. While\nwe ﬁnd a small but positive TCW= 8(3) K for 2.5%\ndoped samples and above, it is negative (antiferromag-\nnetic),TCW=−20(3) K, for the 2% doped material and\nremains negative to lower Fe-dopeing levels. Similar to\nthe case for parent Cr 2B, at lower temperatures Ks(T)\nincreaseswith a smallerpower-lawexponent, i.e. roughly\nasT−1/4, before leveling oﬀ at lowest temperatures.\nCompared to parent Cr 2B, the anomaly in the tem-\nperature dependence of K(T) shifts from 35 K to 24 K\nin the 2% Fe-doped sample (Fig. 6). Surprisingly, this\nanomaly is not clearly observed in the spin-lattice re-\nlaxation rates, which are temperature independent, i.e.\n1/T1T= 2.9·10−3s−1K−1, between 300 and 5 K. In\ncontrast to the 5% Fe-doped sample, there is no en-\nhancement in 1 /T1Tthat would suggest the develop-\nment of a ferromagnetic-like state at low temperatures.\nThe11B NMR shift and the spin-lattice relaxation rate\ndata thus unambiguously show that Fe-doped Cr 2B ma-\nterials at concentrations lower than the critical 2.5%\nvalue lack magnetic order, while systematically showing\nelectronic characteristics that deviate from conventional\nFermi-liquid behavior.\nIV. DISCUSSION AND CONCLUSIONS\nThe insensitivity at room-temperature of the\nquadrapole frequencies νQ[Fig. 2(b)] and11B NMR\nshifts [Fig. 2(a)] clearly demonstrates that Fe doping of\nCr2B at levels up to 5% does not signiﬁcantly perturb\nthe local structural and electronic environment at the\nB-sites. Yet, the emergence of the ferromagnetic-like\nresonance (Fig. 4) and the large broadening of the\n11B NMR spectra (Fig. 5) observed in 5% Fe-doped\nCr2B are consistent with the presence of magnetic\nordering at low temperatures when the Fe concentration\nexceeds the critical value of xc≈2.5%. Therefore, the\nquantum paramagnetic to ferromagnetic transition at\nzero temperature is indeed triggered by Fe doping at7\n/s32/s32\n/s49/s49\n/s66/s32/s78/s77/s82/s32/s115/s104/s105/s102/s116/s32/s40/s37/s41/s50/s37/s32/s70/s101\n/s53/s32/s75/s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53/s50/s46/s53/s37/s32/s70/s101\n/s49/s53/s32/s75\n/s32/s32\n/s49/s49\n/s66/s32/s78/s77/s82/s32/s115/s104/s105/s102/s116/s32/s40/s37/s41\n/s52 /s49/s48 /s49/s48/s48 /s51/s53/s48/s49/s48/s49/s48/s48/s49/s48/s48/s48\n/s50/s37/s32/s70/s101\n/s84/s45/s49/s47/s52\n/s84/s45/s49\n/s32/s75/s32 /s45/s32 /s40/s112/s112/s109/s41\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41\nFIG. 6. (color online). The temperature dependence of the\n11B NMR Knight shift, Ks=K−σ, for 2% Fe-doped Cr 2B\n(open circles). The Knight shifts are extracted directly fr om\nthe shifts of the11B NMR spectra, K, obtained by subtrac-\ntion of the chemical shift σ=−253 ppm. The solid red lines\nindicate that Ks∝T−1at high temperatures, and then grad-\nually changes to Ks∝T−1/4at low temperatures. Insets:\nLow-temperature11B NMR spectra (gray shaded area) mea-\nsured for 2% Fe-doped Cr 2B (T= 5 K) and 2.5% Fe-doped\nCr2B (T= 15 K). The solid red lines are lineshape ﬁts with\nK= 54 ppm, νQ= 496 kHz and δ1/2= 155 kHz (Cr 2B sam-\nple with 2% Fe doping) and K= 167 ppm, νQ= 450 kHz and\nδ1/2= 243 kHz (Cr 2B sample with 2.5% Fe doping).\nxc.So, what really changes after Fe-doping that drives\nsuch transition? Our11B NMR and ESR data highlight\ntwo primary factors that constrain the discussion of\nthe paramagnetic to ferromagnetic transition in this\nmaterial: the non-Fermi-liquid behavior and the simul-\ntaneous presence of antiferromagnetic and ferromagnetic\ncorrelations.\nThe non-Fermi-liquid behavior is revealed through the\nstrong temperature dependence of the11B Knight-shift\nfound across the entire phase diagram (Fig. 7). All sam-\nples surprisingly show a Curie-Weiss-like dependence of\nKsat high temperatures. One possible explanation for\nsuch a dependence is the presence of localized states in\nthe samples, originating either from defects (i.e. a slight\nnon-stoichiometry in case of the parent Cr 2B) or from\nan orbitally selective Mott transition34,35in this multi-\nband system. However, a crossover to a low-temperature\nstate where KsfollowsT−nwith a power-law exponent\nn≤1/2 contradicts both these possibilities. Rather we\nconclude that the Fe-doped Cr 2B materials family is in-\ndeed close to a QCP as originally suggested.15A very\nsimilar temperature dependence of the Knight-shift has\nbeen found in other archetypal materials close to a QCP,\nwhere such dependence was attributed to the existence\nof strong spin correlations in the metallic state.20\nThe second important ﬁnding relates to the origin of\n/s48 /s49 /s50 /s120\n/s67/s51 /s52 /s53 /s54/s45/s52/s48/s45/s50/s48/s48/s50/s48/s52/s48/s54/s48/s56/s48\n/s70/s101/s45/s100/s111/s112/s101/s100/s32/s67/s114\n/s50/s66\n/s70/s77\n/s32/s84\n/s67/s87 \n/s32/s84\n/s99\n/s32/s32/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41\n/s70/s101/s45/s100/s111/s112/s105/s110/s103/s32/s40/s37/s41/s78/s70/s76\nFIG. 7. (color online). Dependence of the NMR-determined\nCurie-Weiss temperature TCW(open black circles) and mag-\nnetic ordering onset temperature Tc(solid blue circles) on the\nof Fe-doping concentration of Cr 2B. The dotted black and the\ndashed blue lines are a guides to the eye. A transition is seen\nfrom a paramagnetic non-Fermi liquid metal (NFL, yellow\nshading) with predominant antiferromagnetic correlation s to\na ferromagnetic metal (FM, blue shading) at the critical con -\ncentration xc= 2.5%.\nspinﬂuctuationsandtheirevolutionwithFe-doping. The\nanalysis of the11B spin-lattice relaxation data is consis-\ntent with ferromagnetic ﬂuctuations. These ferromag-\nnetic ﬂuctuations coexist with antiferromagnetic corre-\nlations deduced from the negative Curie-Weiss temper-\nature. Fe-doping does not signiﬁcantly aﬀect the fer-\nromagnetic ﬂuctuations. On the other hand, the Fe-\ndependence of TCW(Fig. 7) suggests that the antifer-\nromagnetic correlations gradually vanish as the doping\nlevel approachesthe critical Fe concentration. For higher\nFe-doping levels, ferromagnetic correlations prevail and\nas a result the magnetic ordering temperature monoton-\nically and rapidly increases with x.What remains to be\nanswered in future work is why Fe-doping aﬀects the fer-\nromagnetic correlations to a much lesser degree than the\nantiferromagnetic correlations.\nIn conclusion, we have systematically investigated the\neﬀect of Fe-doping on the magnetism in the intermetallic\ncompound Cr 2B.11B NMR and ESR data suggest that\nthese materials may indeed be close to a quantum crit-\nical point at the critical Fe-doping level of xc≈2.5%.\nThe data also reveal that antiferromagnetic and ferro-\nmagnetic correlations coexist in these materials, but are\ndiﬀerently aﬀected by Fe-doping. At xc, ferromagnetic\ncorrelations prevail and magnetic ordering is observed\nat higher doping levels. Our characterization thus indi-\ncates that intermetallic Cr 2B as an interesting material\nsystem where the eﬀect of weak quenched disorder on a\nferromagnetic quantum phase transition can be system-\natically studied.8\nACKNOWLEDGMENTS\nD.A. and C.F. acknowledge the ﬁnancial support by\nthe European Union FP7-NMP-2011-EU-Japan project\nLEMSUPER under contract no. 283214. D.A. also ac-\nknowledges discussions with Peter Jegliˇ c about the11BNMR spectra. D.A. and C.F. also acknowledge discus-\nsions with Peter Adler regarding the magnetic ground\nstate. The research at Princeton University was sup-\nported by the US Depaertment of energy, grant DE-\nFG02-98ER45706.\n∗e-mail: denis.arcon@ijs.si\n1M. Lavagna, Philos. Mag. B 81, 1469 (2001), URL\nhttp://www.tandfonline.com/doi/abs/10.1080/13642810 108208565 .\n2Y. Sang, D. Belitz, and T. R. Kirkpatrick,\nPhys. Rev. Lett. 113, 207201 (2014), URL\nhttp://link.aps.org/doi/10.1103/PhysRevLett.113.207 201.\n3D. Belitz, T. R. Kirkpatrick, and T. Vo-\njta, Phys. Rev. Lett. 82, 4707 (1999), URL\nhttp://link.aps.org/doi/10.1103/PhysRevLett.82.4707 .\n4D. Belitz and T. R. Kirkpatrick, Phys.\nRev. Lett. 89, 247202 (2002), URL\nhttp://link.aps.org/doi/10.1103/PhysRevLett.89.2472 02.\n5C. Varma, Z. Nussinov, and W. van Saar-\nloos, Phys. Rep. 361, 267 (2002), URL\nhttp://www.sciencedirect.com/science/article/pii/S0 370157301000606 .\n6H. v. L¨ ohneysen, A. Rosch, M. Vojta, and\nP. W¨ olﬂe, Rev. Mod. Phys. 79, 1015 (2007), URL\nhttp://link.aps.org/doi/10.1103/RevModPhys.79.1015 .\n7C. C. Pﬂeiderer, S. R. Julian, and G. G.\nLonzarich, Nature 414, 427 (2001), URL\nhttp://www.nature.com/nature/journal/v414/n6862/ful l/414427a0.html .\n8C. Pﬂeiderer, G. J. McMullan, S. R. Julian, and\nG. G. Lonzarich, Phys. Rev. B 55, 8330 (1997), URL\nhttp://link.aps.org/doi/10.1103/PhysRevB.55.8330 .\n9M. Uhlarz, C. Pﬂeiderer, and S. M. Hay-\nden, Phys. Rev. Lett. 93, 256404 (2004), URL\nhttp://link.aps.org/doi/10.1103/PhysRevLett.93.2564 04.\n10M. Brando, W. J. Duncan, D. Moroni-Klementowicz,\nC. Albrecht, D. Gr¨ uner, R. Ballou, and F. M.\nGrosche, Phys. Rev. Lett. 101, 026401 (2008), URL\nhttp://link.aps.org/doi/10.1103/PhysRevLett.101.026 401.\n11A. Huxley, I. Sheikin, E. Ressouche, N. Kerna-\nvanois, D. Braithwaite, R. Calemczuk, and J. Flou-\nquet, Phys. Rev. B 63, 144519 (2001), URL\nhttp://link.aps.org/doi/10.1103/PhysRevB.63.144519 .\n12V. Taufour, D. Aoki, G. Knebel, and J. Flou-\nquet, Phys. Rev. Lett. 105, 217201 (2010), URL\nhttp://link.aps.org/doi/10.1103/PhysRevLett.105.217 201.\n13H. Kotegawa, V. Taufour, D. Aoki, G. Knebel, and\nJ. Flouquet, J. Phys. Soc. Jpn. 80, 083703 (2011), URL\nhttp://dx.doi.org/10.1143/JPSJ.80.083703 .\n14Y. Ihara, T. Hattori, K. Ishida, Y. Nakai, E. Os-\naki, K. Deguchi, N. K. Sato, and I. Satoh,\nPhys. Rev. Lett. 105, 206403 (2010), URL\nhttp://link.aps.org/doi/10.1103/PhysRevLett.105.206 403.\n15L. Schoop, M. Hirschberger, J. Tao, C. Felser, N. P. Ong,\nand R. J. Cava, Phys. Rev. B 89, 224417 (2014), URL\nhttp://link.aps.org/doi/10.1103/PhysRevB.89.224417 .\n16D. Kotzott, M. Ade, and H. Hillebrecht,\nSolid State Sci. 10, 291 (2008), URL\nhttp://www.sciencedirect.com/science/article/pii/S1 293255807002683 .\n17F. Kayser and G. Kayser, J. Mater. Sci. 34, 1271 (1999),\nURLhttp://dx.doi.org/10.1023/A%3A1004581511232 .18Q. D. Gibson, L. M. Schoop, L. Muechler, L. S.\nXie, M. Hirschberger, N. P. Ong, R. Car, and\nR. J. Cava, Phys. Rev. B 91, 205128 (2015), URL\nhttp://link.aps.org/doi/10.1103/PhysRevB.91.205128 .\n19C. Zhou, J. Xing, B. Xiao, J. Feng, X. Xie, and\nY. Chen, Comp. Mater. Sci. 44, 1056 (2009), URL\nhttp://www.sciencedirect.com/science/article/pii/S0 927025608003571 .\n20K. Ishida, K. Okamoto, Y. Kawasaki, Y. Ki-\ntaoka, O. Trovarelli, C. Geibel, and F. Steglich,\nPhys. Rev. Lett. 89, 107202 (2002), URL\nhttp://link.aps.org/doi/10.1103/PhysRevLett.89.1072 02.\n21R. Sarkar, P. Khuntia, C. Krellner, C. Geibel, F. Steglich,\nand M. Baenitz, Phys. Rev. B 85, 140409 (2012), URL\nhttp://link.aps.org/doi/10.1103/PhysRevB.85.140409 .\n22P. Khuntia, A. M. Strydom, L. S. Wu,\nM. C. Aronson, F. Steglich, and M. Baenitz,\nPhys. Rev. B 86, 220401 (2012), URL\nhttp://link.aps.org/doi/10.1103/PhysRevB.86.220401 .\n23T. Misawa, Y. Yamaji, and M. Imada, J.\nPhys. Soc. Jpn. 78, 084707 (2009), URL\nhttp://dx.doi.org/10.1143/JPSJ.78.084707 .\n24A. Ishigaki and T. Moriya, J. Phys.\nSoc. Jpn. 65, 3402 (1996), URL\nhttp://dx.doi.org/10.1143/JPSJ.65.3402 .\n25C. Krellner, T. F¨ orster, H. Jeevan,\nC. Geibel, and J. Sichelschmidt, Phys.\nRev. Lett. 100, 066401 (2008), URL\nhttp://link.aps.org/doi/10.1103/PhysRevLett.100.066 401.\n26T. F¨ orster, J. Sichelschmidt, D. Gr¨ uner,\nM. Brando, N. Kimura, and F. Steglich, J.\nPhys.: Conf. Series 200, 012035 (2010), URL\nhttp://stacks.iop.org/1742-6596/200/i=1/a=012035 .\n27A. Potoˇ cnik, P. Jegliˇ c, K. Kobayashi, K. Kawashima,\nS. Kuchida, J. Akimitsu, and D. Arˇ con,\nPhys. Rev. B 90, 104507 (2014), URL\nhttp://link.aps.org/doi/10.1103/PhysRevB.90.104507 .\n28P. Jegliˇ c, J.-W. G. Bos, A. Zorko, M. Brunelli,\nK. Koch, H. Rosner, S. Margadonna, and\nD. Arˇ con, Phys. Rev. B 79, 094515 (2009), URL\nhttp://link.aps.org/doi/10.1103/PhysRevB.79.094515 .\n29R. E. Walstedt, The NMR Probe of High-Tc Ma-\nterials, vol. 228 of Springer Tracts in Modern\nPhysics (Springer Berlin Heidelberg, 2008), URL\nhttp://link.springer.com/10.1007/978-3-540-75565-4 .\n30P. Jegliˇ c, A. Potoˇ cnik, M. Klanjˇ sek, M. Bobnar,\nM. Jagodiˇ c, K. Koch, H. Rosner, S. Margadonna, B. Lv,\nA. M. Guloy, et al., Phys. Rev. B 81, 140511 (2010), URL\nhttp://link.aps.org/doi/10.1103/PhysRevB.81.140511 .\n31C. H. Pennington and V. A. Stenger,\nRev. Mod. Phys. 68, 855 (1996), URL\nhttp://link.aps.org/doi/10.1103/RevModPhys.68.855 .\n32T. Moriya, J. Phys. Soc. Jpn. 18, 516 (1963), URL\nhttp://dx.doi.org/10.1143/JPSJ.18.516 .9\n33F. J. Dyson, Phys. Rev. 98, 349 (1955), URL\nhttp://link.aps.org/doi/10.1103/PhysRev.98.349 .\n34L. de’ Medici, S. R. Hassan, M. Capone, and\nX. Dai, Phys. Rev. Lett. 102, 126401 (2009), URLhttp://link.aps.org/doi/10.1103/PhysRevLett.102.126 401.\n35D. Arˇ con, P. Jegliˇ c, A. Zorko, A. Potoˇ cnik, A. Y.\nGanin, Y. Takabayashi, M. J. Rosseinsky, and\nK. Prassides, Phys. Rev. B 82, 140508 (2010), URL\nhttp://link.aps.org/doi/10.1103/PhysRevB.82.140508 ."    },    {        "title": "0912.1237v1.High_temperature_ferromagnetism_of_Li_doped_vanadium_oxide_nanotubes.pdf",        "content": "arXiv:0912.1237v1  [cond-mat.str-el]  7 Dec 2009High temperature ferromagnetism of Li-doped vanadium oxid e nanotubes∗\nA.I. Popa,1E. Vavilova,1,2Y.C. Arango,1V. Kataev,1,†C. T¨ aschner,1\nH-H. Klauss,3H. Maeter,3H. Luetkens,4B. B¨ uchner,1and R. Klingeler1\n1Institute for Solid State Research, IFW Dresden, 01171 Dres den, Germany\n2Zavoisky Physical-Technical Institute of RAS, 420029 Kazan , Russia\n3IFP, TU Dresden, D-01069 Dresden, Germany\n4Laboratory for Muon-Spin Spectroscopy, Paul Scherrer Inst itut, CH-5232 Villigen PSI, Switzerland\nThe nature of a puzzling high temperature ferromagnetism of doped mixed-valent vanadium ox-\nide nanotubes reported earlier by Krusin-Elbaum et al.,Nature431(2004) 672, has been addressed\nby static magnetization, muon spin relaxation, nuclear mag netic and electron spin resonance spec-\ntroscopy techniques. A precise control of the charge doping was achieved by electrochemical Li\nintercalation. We ﬁnd that it provides excess electrons, th ereby increasing the number of interact-\ning magnetic vanadium sites, and, at a certain doping level, yields a ferromagnetic-like response\npersisting up to room temperature. Thus we conﬁrm the surpri sing previous results on the samples\nprepared by a completely diﬀerent intercalation method. Mo reover our spectroscopic data provide\nﬁrst ample evidence for the bulknature of the eﬀect. In particular, they enable a conclusion that\nthe Li nucleates superparamagnetic nanosize spin clusters around the intercalation site which are\nresponsible for the unusual high temperature ferromagneti sm of vanadium oxide nanotubes.\nPACS numbers: 75.20.-g; 75.75.+a; 73.22.-f; 76.60.-k; 76. 75.+; 76.30.-v\nThe quest for new materials with novel physical\nproperties and functionalities continuously pushes for-\nward materials’s science eﬀorts to structure ’old known’\nbulk solids into nanoscaled low-dimensional objects (see,\ne.g., [1]). The magnetic and electronic properties of such\nnanostructuresaremainly governedby their surfaces and\ninterfaceswhichrendersthempromisingmaterialsfortai-\nlored functionalization. For oxide heterostructures the\ngreat potential of such an approach is, e.g., highlighted\nby recent ﬁndings of novel eﬀects at interfaces between\nnonmagnetic oxide insulators where magnetism, metal-\nlic behavior and even superconductivity have been re-\nported [2]. In order to tailor nanostructured functional\nelements, diﬀerent design strategies are being used, such\nas fabrication of multilayers, heterostructures with quan-\ntum conﬁnement (quantum wells, wires and dots) etc.\nAnother promising approach is based on self-assembly of\nelementary nano- and even atomic-size building blocks.\nIn particular, two-dimensional layers of transition metal\n(TM) oxides rolled up into nanosized multiwalled tubes\nor scrolls attract a rapidly growing attention owing to\nunique physical properties not occurring in the bulk par-\nent materials (see, e.g., [3]). A recent example of such\na nanostructured TM oxide is multiwalled vanadium ox-\nide nanotubes VO x-NTs [4, 5]. Vanadium ions are mixed\nvalent in this material with an average valence count of\n∼+4.4 (i.e.x≈2.2) [6, 7]. This results in a roughly\nequal number of magnetic V4+(3d1,S= 1/2) and non-\nmagnetic V5+(3d0,S= 0) sites in the VO xlayers. The\n∗with some amendments published in\nEurophysics Letters (EPL) 88(2009) 57002\nhttp://epljournal.edpsciences.org/spins associated with the former sites act either as indi-\nvidual spins or strongly gapped antiferromagnetic (AF)\ndimers or trimers [8]. As reported by Krusin-Elbaum et\nal.[9], doping of VO x-NTs with either holes or electrons\nvia iodine or lithium intercalation yields a nonlinear and\nhysteretic magnetization response to applied magnetic\nﬁelds suggesting the occurrence of ferromagnetism that\npersists even above room temperature. Such high tem-\nperature ferromagnetism (HTFM) is very surprising and\nunexpected for an oxide material comprising TM ions\nwith a small spin-1/2 and provides a remarkable exam-\nple of novel functionalities in nanostructured oxides.\nSince a conclusion on the HTFM in the doped VO x-\nNTs is based in Ref. [9] only on the results of the SQUID\nmagnetometry, where artefacts due to uncontrollable fer-\nromagnetic impurities cannot be always excluded, we\nhave conducted a new experimental study using a com-\nbination of diﬀerent techniques. Our objectives were: (i)\nto obtain an independent evidence of HTFM in Li doped\nVOx-NTs, and, if conﬁrmed, (ii) to obtain insights into\nthe origin of this unusual phenomenon. To achieve these\ngoals we have prepared a series of Li-VO x-NTs samples\nbyacompletelydiﬀerentelectrochemicalmethodofLiin-\ntercalation that as compared with the chemical method\nused in Ref. [9] enables an accurate control of the doping\nlevel. We have performed a complex experimental inves-\ntigation of these samples with three diﬀerent local spin\nprobetechniques, namely, electronspinresonance(ESR),\nnuclear magnetic resonance (NMR) and muon spin re-\nlaxation/rotation ( µSR) spectroscopies, along with mea-\nsurements of the bulk static magnetization. We ﬁnd that\nfor a particular concentration of the Li-dopant a large\nmagnetization Mwhich can be easily saturated even at\nroomtemperaturebyamagneticﬁeld µ0Hofabout1Tis2\npresentinthesample. Therebywereproduceandconﬁrm\nthe results of Ref. [9] on the samples prepared by a dif-\nferent method. Moreover, µSR and NMR measurements\ngive evidence for a bulk nature of the eﬀect. NMR data\nsuggestthatthemagnetizationisnotuniformthroughout\nthesampleandthatstronglymagneticregionsareformed\naround the intercalated Li sites. ESR experiments reveal\na sharp signal that bears essential features of a super-\nparamagnetic resonance. We argue that Li intercalation\naﬀects the charge disproportionation and hence the spin\nstates and magnetic interactions in the rolled-up VO x\nlayers thereby promoting, for particular doping levels,\nthe formation of nanosized interacting spin clusters that\nbehave similar to superparamagnetic nanoparticles.\nPristine multiwalled vanadium oxide nanotubes were\nsynthesized by a hydrothermal technique, previously de-\nscribed in [6]. Electrochemical treatment that enables\na precise control of the Li doping was done by means\nof two-electrode Swagelok-type cells, each including 83.3\nwt% of active material (VO x-NT), 16.7 wt% of Carbon\nSP (Timcal), a Li metal anode, and 1:1 ethylene car-\nbonate/dimethyl carbonate solution of 1M LiPF 6(LP30-\nFerro) that was used as electrolyte. The electrochemical\ndopingwasdoneusingaVMPcontroller(AmetekPrince-\nton Applied Research), in galvanostatic mode, with a\ndischarge rate of 1 Li per formula unit in 100h. By ap-\nplying the constantcurrent mode Li ions havebeen inter-\ncalated to the active material until a desired composition\nLiyVOx-NT was achieved: For the present study samples\nwithy= 0, 0.05, 0.10, 0.15 and 0.6 have been prepared.\nFor all samples, magnetization has been measured in the\ntemperature range T= 2−350K and in ﬁelds up to 5 T\nby means of a Quantum Design MPMS.51V NMR data\nwere recorded on a Tecmag pulse solid-state NMR spec-\ntrometer at T= 4.2−300K. The NMR spectra were ac-\nquired by a point-by-point magnetic ﬁeld sweeping. For\nESR experiments we used an X-band EMX Bruker ESR\nspectrometer operational at T= 3.5−300K. Muon spin\nrelaxation( µ+SR)dataonundopedanddopedVO x-NTs\nwere obtained at the GPS spectrometer of PSI.\nOur static magnetic data show an increase of the mag-\nnetization in Li yVOx-NT with 0 < y≤0.6 as compared\nwith the undoped material. This observation is con-\nsistent with a higher number of paramagnetic V sites\nsuggested by the NMR data (see below). Note, that\na spin-gap feature in the T-dependence of the suscep-\ntibility in pristine VO x-NT attributed to the presence of\ndimers [8, 9] vanishes upon Li doping. The main result of\nthemagnetizationstudyistheobservationofanon-linear\nbehavior for the doping level y= 0.1 which strongly dif-\nfers from the magnetic response at other Li contents. At\n300K, the M(H) curve practically saturates in a ﬁeld\nof∼1T at a value of ∼0.1µBper V site with a small\nsubsequent linear increase at higher ﬁelds (see Fig.1).\nThe data at 50K and 2K (Fig.1) illustrate that at low T\nthis saturation is superimposed by a stronger linear con-/s48 /s49 /s50 /s51 /s52 /s53/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52\n/s45/s48/s46/s50 /s45/s48/s46/s49 /s48/s46/s48 /s48/s46/s49 /s48/s46/s50/s45/s52/s45/s50/s48/s50/s52\n/s86/s79\n/s120/s45/s78/s84/s44/s32/s84/s32/s61/s32/s51/s48/s48/s75\n/s32/s32\n/s32/s51/s48/s48/s32/s75/s50/s32/s75\n/s53/s48/s32/s75/s32\n/s32/s40/s84/s41\n/s77/s97/s103/s110/s101/s116/s105/s122/s97/s116/s105/s111/s110/s32/s40/s49/s48\n/s45/s50\n/s66\n/s47/s86\n/s41/s76/s105\n/s48/s46/s49/s86/s79\n/s88 /s45/s78/s84\n/s32/s77/s97/s103/s110/s101/s116/s105/s122/s97/s116/s105/s111/s110/s32/s40/s49/s48/s45/s50\n/s41/s32\n/s66/s47/s86\n/s121/s32/s61/s32/s48/s46/s49\n/s84/s32/s61/s32/s51/s48/s48/s32/s75\n/s32/s32\nFIG. 1: H-dependence of the magnetization of Li yVOx-NT\nwithy= 0.1 (T= 300K, 50K, 2K) and y= 0 (T= 300K).\nInset: Part of the M(H) loop, at 300K, for y= 0.1.\ntribution to M(H). Note, that the data exhibit a small\nhysteresis with the coercivity 2 Hc∼23mT at 300K (see\nFig.1, inset).\nFurther insight into the magnetic properties is pro-\nvided by zero ﬁeld (ZF) µSR data. Selected ZF- µSR\nspectra obtained on Li yVOx-NT with y= 0 and y= 0.1,\ni.e. for the pristine and the magnetic materials, (see\nFig.2). For y= 0, both at 300K and at 20K the data\nshow only a small decrease of the asymmetry signal A(t)\nat short times. This clearly implies the absence of mag-\nnetic order in this temperature range. On a longer time\nscale of µs, there is a decrease due to the slow relax-\nation which we attribute to nuclear and fast ﬂuctuating\nelectronic magnetic moments.\nThe most important result of the ZF- µSR study on\nLi0.1VOx-NT is the observation of a signiﬁcant and rapid\nloss of asymmetry at early times. As displayed (see\nFig.2(b)), most of the relaxation occurs already dur-\ning the dead time of the spectrometer ( ≈5 ns). The\nfull asymmetry scale was deﬁned by a subsequent mea-\nsurement of a nonmagnetic compound. Such a rapid re-\nlaxation clearly indicates that a signiﬁcant fraction of\nthe muons experiences a local quasi-static magnetic ﬁeld.\nThe absence of an oscillating signal proves a broad static\nmagnetic ﬁeld distribution within the compound.\nAssuming for a quantitative analysis that a fraction\nofamuons experiences both the nuclear and the a elec-\ntronic static magnetic ﬁeld, the data are described by\nA(t) = (1−a)·e−0.5(σnuct)2+a·(2/3e−λTt+1/3e−λLt).\nHereσnucandλT,Lare muon relaxation rates due to\ninteraction with nuclear and electron spins, respectively\n[11]. This analysis implies that ∼2/3 of the muons in\nLi0.1VOx-NT experience a static magnetic ﬁeld originat-\ning from a magnetic ordered volume fraction at or in the\nvicinity of the muon stopping site(s) [10]. The µSR data\nhence unambiguously proof the bulk, and notimpurity\nrelated, character of the magnetism found in the static\nmagnetization.3\n/s48/s46/s50 /s48/s46/s52 /s49/s46/s53 /s51/s46/s48 /s52/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s32/s110/s111/s114/s109 /s97/s108/s105/s122/s101/s100/s32/s97/s115/s121/s109 /s109 /s101/s116/s114/s121/s32/s51/s48/s48/s32/s75\n/s50/s48/s32/s75\n/s40/s98/s41/s32/s76/s105\n/s48/s46/s49/s86/s79\n/s120/s45/s78/s84\n/s32/s40/s97/s41/s32/s117/s110/s100/s111/s112/s101/s100/s32/s86/s79\n/s120/s45/s78/s84\n/s48/s46/s50 /s48/s46/s52 /s49/s46/s53 /s51/s46/s48 /s52/s46/s53\n/s32/s116/s105/s109/s101/s32/s40/s181/s115/s41\nFIG. 2: Selected ZF- µSR spectra at 300K and 20K of\nLiyVOx-NT with (a) y= 0 and (b) y= 0.1. The solid lines\ndenote the evaluation of the data with a function as describe d\nin the text. Note the change of the timescale at 0.5 µs.\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48/s53/s46/s57/s51/s48/s53/s46/s57/s51/s50/s53/s46/s57/s51/s52/s53/s46/s57/s51/s54\n/s57/s46/s48 /s57/s46/s49 /s57/s46/s50 /s57/s46/s51/s48/s50/s52/s54/s56/s49/s48\n/s40/s98/s41/s121/s32/s61/s32/s48/s46/s49/s48/s121/s32/s61/s32/s48/s46/s49/s53\n/s32/s32/s114/s101/s115/s111/s110/s97/s110/s99/s101/s32/s102/s105/s101/s108/s100/s32/s32/s40/s49/s48/s52\n/s32/s71/s41\n/s84/s32/s40/s75/s41/s55\n/s76/s105/s32/s78/s77/s82\n/s56/s52/s46/s52/s53/s32/s77/s72/s122/s121/s32/s61/s32/s48/s46/s48/s53/s32\n/s32/s70/s105/s101/s108/s100/s32/s40/s84/s41/s101/s99/s104/s111/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121\n/s32/s32\n/s53/s49\n/s86/s32/s78/s77/s82\n/s49/s48/s51/s46/s50/s32/s77/s72/s122 /s32/s32\n/s32/s32/s84/s32/s61/s32/s49/s53/s75/s40/s97/s41\nFIG. 3: Low- T51V NMR spectra (a) and T-dependencies of\nthe resonance ﬁeld of7Li NMR (b) for Li yVOx-NT:y=0 (◦),\n0.05 (/triangledownsld), 0.1 (•) and 0.15 ( △)). Lines are guides to the eye.\nNMR studies on51V and7Li shed further light on the\neﬀect of Li doping on the local magnetic properties. The\n51VNMRdata(see Fig.3(a))revealagradualincreaseof\nthe relative intensity of the low-ﬁeld fast-relaxing shoul-\nder of the51V signal at ∼9.16T upon doping. Since this\npart of the spectrum is associated with the response of\nthe magnetic ions’ nuclei, shifted due to hyperﬁne inter-\naction from the central slow-relaxing nonmagnetic V5+\npeak at ∼9.22T [8], its growth indicates an increas-\ning fraction of magnetic vanadium ions upon Li inter-\ncalation and conﬁrms that the doping process aﬀects the\nwhole sample. Though magnetic ordering usually creates\na shift or splitting of the NMR spectrum, neither was ob-\nserved for the51V signal for the Li 0.1VOx-NT sample in\na suﬃciently wide ﬁeld range. The absence of the signif-\nicant loss of its intensity suggests that the large part of\nV ions does sense the charge doping but not the internal\nﬁeld.\nThe situation with7Li NMR is diﬀerent: The reso-/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s55/s48/s55/s53/s56/s48/s56/s53/s57/s48/s57/s53/s49/s48/s48\n/s49/s46/s57/s55/s50/s49/s46/s57/s55/s52/s49/s46/s57/s55/s54/s49/s46/s57/s55/s56/s49/s46/s57/s56/s48/s49/s46/s57/s56/s50/s49/s46/s57/s56/s52\n/s50/s56/s48/s48 /s51/s50/s48/s48 /s51/s54/s48/s48 /s52/s48/s48/s48\n/s32/s32\n/s84/s32/s40/s75/s41/s103 /s45/s102/s97/s99/s116/s111/s114/s32/s32/s76/s105/s110/s101/s119/s105/s100/s116/s104/s32 /s72 /s32/s40/s71/s41\n/s72/s32\n/s32/s76/s105/s48/s46/s49/s76/s105/s48/s46/s54\n/s32/s100 /s80 /s40/s72 /s41/s47/s100 /s72 /s32/s40/s97 /s114/s98 /s46/s117 /s41\n/s72/s32/s40/s71/s41/s68/s105/s102/s102/s101/s114/s101/s110/s99/s101\n/s76/s105/s48/s46/s49/s32/s45/s32/s76/s105/s48/s46/s54/s67/s97/s114/s98/s111/s110\n/s34/s101/s120/s116/s114/s97/s34/s32/s108/s105/s110/s101\n/s103 /s45/s102/s97/s99/s116/s111/s114\nFIG. 4: Inset: ESR spectra (ﬁeld derivative of absorption) o f\nLiyVOx-NT fory= 0.6 (top) and 0.1 (middle) at T= 300K.\nThe”extra” narrowline whichis speciﬁc for thestrongly mag -\nnetic sample with y= 0.1 (marked by the arrow in the middle\nspectrum) is singled out by subtracting the top from the mid-\ndle spectrum (bottom curve). The central sharp line is due to\nthe carbon additive. Main panel: T-dependence of the width\n∆Hand theg-factor of the ”extra” line in the ESR spectrum\nofy= 0.1. Dash lines are guides for the eye.\nnance ﬁeld of the signal steadily changes with doping.\nHowever, the line shift for y= 0.1 is much larger com-\npared to y= 0.05 andy= 0.15 (see Fig.3(b)). This\nsuggests the presence of an internal ﬁeld at the Li sites\nonly for the strongly magnetic Li 0.1VOx-NT. The ab-\nsence of a second unshifted7Li line at the expected place\nbetween the line positions of Li 0.05and Li 0.15samples in-\ndicates that practically allLi nuclei experience internal\nmagnetic ﬁelds.\nThe ESR study of undoped VO x-NTs at a frequency\nν= 9.5GHz reveals, similar to other results [12], a spec-\ntrum comprising two overlapping resonance lines with\nslightly diﬀerent g-factors of ∼2.0 and∼1.97 (not\nshown). The ﬁrst line can be assigned to quasi-free spins\nassociated with V4+(S= 1/2) ions in the tetrahedral po-\nsition. The secondline is due to V4+ions in the distorted\noctahedral coordination which are coupled magnetically\ninto dimers and trimers [8].\nIn the samples of Li yVOx-NTs with y= 0.05,0.1,0.15\nand 0.6 a systematic evolution of the lines in the ESR\nspectrum with increasing the Li doping gives evidence\nfor the increasing number of magnetic V4+ions, and\na reduction of the contribution related to spin dimers\nand trimers, in agreement with the magnetization data\n(see above). We note that the carbon black used as a\nconductive additive for the preparation of Li-doped sam-\nples partially screens the penetration of microwaves into\nthe interior of the sample thereby reducing the signal-to-\nnoise ratio and superimposes an additional ESR line (see\nFig.4, inset). The latter can be identiﬁed by measuring\nthe carbon additive separately and accurately subtracted4\nfrom the spectra. The central result of the ESR study is\nthe observation of an ’extra’ narrow resonance line in the\nstronglymagnetic y= 0.1sample. Despitethedisturbing\neﬀect of the carbon line, the ’extra’ line is clearly visible\nin the raw data (see Fig.4, inset, middle curve) and can\nbe accurately singled out, e.g., by subtracting from the\ny= 0.1 spectrum the spectrum of a weakly magnetic\nsample with a diﬀerent Li content (see Fig.4, inset, bot-\ntom curve). The width of this ’extra’ line ∆ H∼70G is\nsigniﬁcantlysmallercomparedtothe signalsfromparam-\nagneticV4+ionswhereasthe g-factorof ∼1.98issimilar.\nBoth ∆Handgchange little in a broad T-region above\n∼40−70K and become temperature dependent at lower\nT(see Fig.4). As will be discussed below, this narrow\nsignal is directly related with the ferromagnetic-like be-\nhavior of the y= 0.1 sample.\nUniform long-range magnetic order in the low-spin\n(S= 1/2) vanadium oxides is not competitive at high\ntemperatureswith thermalﬂuctuationsand/orinsulator-\nto-metal transitions (see, e.g., [13]). A ferromagnetic-like\nmagnetization up to room and even higher temperatures\nis in particularunexpected. However, in the caseofVO x-\nNTs an alternative cluster scenario of ferromagnetism is\nclearly corroborated by our µSR and NMR results. µSR\ndata yield about 2/3 of the magnetic volume fraction\nand a broad distribution of static magnetic ﬁelds within\nthe sample. The absence of an appreciable shift of the\n51V NMR signal in the ferromagnetic sample suggests\nthat it originates from the regions outside ferromagnetic\nclusters. The51V-NMR line shift from the nuclei inside\nthe clusters is expected to be as big as in magnetically\nordered compounds, hence displacing the signal out of\nthe observation range (see e.g., [14]). Furthermore, ow-\ning to a possible distribution of the clusters’ size, as also\nsuggested by the µSR data, this signal could be very\nbroad and thus practically undetectable. On the other\nhand, the7Li-NMR data give clear evidence that ferro-\nmagnetism is conﬁned to regions around the lithium site\nwhich yields a shift of the7Li-NMR line by the internal\nmagnetic ﬁeld inside the cluster.\nThus one can argue that the Li which is intercalated\nbetween the vanadium oxide layers in the walls of VO x-\nNTs [7] may play a role of nucleating centers for spin\nclusters of diﬀerent size. In fact, the magnetization\ndata (Fig.1) bear features characteristic of a magnetic\nresponse of superparamagnetic particles with a broad\nsize distribution, such as, e.g., samples of ultra-ﬁne γ-\nFe2O3particles studied in [15]. Speciﬁcally, the small\nhysteresis in M(H) at room temperature suggests the\npresence of large clusters with a blocking temperature\nTB>300K. At higher external ﬁelds they are already\nsaturated and the observed ﬁnite slope of M(H) in this\nﬁeld regime is mainly determined by smaller unblocked\nsuperparamagnetic clusters ( TB≪300K) [15]. The\nsharp ESR line with the spin-only paramagnetic reso-\nnance ﬁeld Hpar\nres=hν/gµ Bobserved in the stronglymagnetic Li 0.1VOx-NT sample ( refaesr) can be straight-\nforwardlyassigned to a resonance response of those small\nunblocked clusters. Above TBthe anisotropy ﬁeld, that\notherwise produces a shift and broadens the signal, is\naveraged due to thermal ﬂuctuations yielding a narrow\nline atHpar\nres(see, e.g., [16]). At low temperatures one\ncan consider the shift of the eﬀective g-factor from the\nspin-only value and the increase of the width of this sig-\nnal (see Fig.4) as an indication of approaching the TB\nof the resonating superparamagnetic clusters. Accord-\ning to the study of the eﬀect of the particle size on the\nESR response in [17], for small particle size ( ≈5nm) the\nspectrum is deﬁned by an isotropic and unshifted narrow\nline. In contrast, for a large particle size ( ≈10nm) the\nassociated anisotropy ﬁeld is much stronger, therefore,\nthermal ﬂuctuations even on a room temperature scale\ncannot overcome the anisotropic orientation of the mag-\nnetic moments. As a result, the spectrum is broad and\nshifted towards lower ﬁelds due to the inﬂuence of the re-\nmaining orientational anisotropy. Thus, in the case of a\nbroad size distribution, the ESR response of the blocked\nclusters in Li 0.1VOx-NT could be smeared out and be-\ncome unobservable, in particular, also due to the limited\nsensitivity caused by the carbon additive (see above).\nObviously, local charge and structural distortions\naroundanintercalatedLi+ionaswellasnanostructuriza-\ntionofVO xmaybecrucialforthenucleationofferromag-\nnetic clusters in VO x-NTs. In this respect one can ﬁnd a\nstrikingsimilaritywithhightemperatureferromagnetism\n(HTFM) with Tc>300K recently observed in nanos-\ntructured diluted magnetic semiconductors (DMS) (see,\ne.g.,[18, 19]). AddingasmallpercentageofmagneticTM\nions to nanocrystals [18] or nanowires [19] of nonmag-\nnetic ZnO yields a robust HTFM that was not achieved\nby doping the bulk ZnO. The occurrence of structural in-\nhomogeneities on the nanometer scale concomitant with\nthe charge localization are the key prerequisites for this\nremarkable eﬀect. For example, in Ni:ZnO nanocrystals,\njust by tuning the aggregation of nanocrystals, one ob-\ntains HTFM with a T-independent saturation value and\na small coercivity, very similar to our ﬁndings [18]. The\nstabilization of HTFM in DMS has been discussed theo-\nretically in terms of collective polaronic eﬀects [20, 21],\nnamely that bound interacting ferromagnetic polarons\nmay be formed due to exchange interaction of localized\ncharges with magnetic impurities, in particular in the\npresence of defects. One can conjecture a possible rele-\nvance of this scenario to VO x-NTs in view of some ap-\nparent similarities with DMS: (i) - the current-voltage\ncharacteristics of individual tubes reveals a semiconduct-\ning behavior with conductivity decreasing upon Li dop-\ning [9]; (ii) - electron doping due to the Li intercalation\ncreates additional spin centers and (iii) - locally distorts\nthe structure. A delicate balance between these factors\ncontrolled by the Li intercalation may be the reason for a\nstrong sensitivity of the observed eﬀect to the Li content.5\nAt small doping levels the amount of nonmagnetic V5+\nions (which are the ’holes’ in the magnetic subsystem)\nis big enough to prevent the formation of spin clusters.\nOn the other hand at large Li dopings VO x-NTs turn to\na uniform rolled up spin-1/2 plane with predominantly\nantiferromagnetic interactions which could be much less\nsensitive to a perturbing inﬂuence of Li-caused defects.\nIn summary, we have studied by means of NMR, µSR\nand ESR spectroscopies combined with static magnetic\nmeasurements the inﬂuence of Li intercalation on the\nmagnetic properties of Li yVOx-nanotubes. We ﬁnd a\nparticular concentration of the Li dopant which turns\nthis compound into a stronglymagnetic materialexhibit-\ning ferromagnetism on the room temperature scale. The\ndata give evidence that this very unusual for a low-spin\nvanadium oxide behavioris due to the formationofnano-\nsize interacting ferromagnetic spin clusters around inter-\ncalated Li ions. Such clusters behave as an ensemble of\nsuperparamagnetic particles with a broad size distribu-\ntion whose big magnetic moments can be easily aligned\nby a moderate magnetic ﬁeld even at room temperature.\nThe robustness of the ferromagnetic spin structure may\nbe suggestive of its collective polaronic nature.\nSupport from the DFG (KL 1824/2, 436 RUS\n113/936/0-1) and of the RFBR (08-02-91952-NNIO-a,\n07-02-01184-a) is gratefully acknowledged. YCA ac-\nknowledges support of the Programme Alban, the Eu-\nropean Union Programme of High Level Scholarships for\nLatin America, scholarship No. E04D049329CO.\n†v.kataev@ifw-dresden.de[1]Advanced Magnetic Nanostructures , ed. by D. Sellmyer\nand R. Skomski (Springer, New York, 2006).\n[2] S. Thiele et al., Science 313, 1942 (2006),; N. Reyren et\nal., Science 317, 1196 (2007); A. Brinkman et al., Nature\nMaterials 6, 493 (2007).\n[3] X. Wang and Y. Li, Pure Appl. Chem. 78, 45 (2006).\n[4] F. Krumeich et al., J. Am. Chem. Soc. 121, 8324 (1999).\n[5] M. W¨ orle et al., Z. Anorg. Allg. Chem. 628, 2778 (2002).\n[6] X. Liu et al., Phys. Rev. B. 72, 115407 (2005).\n[7] I. Hellmann et al., J. Chem. Phys. 128, 224701 (2008).\n[8] E. Vavilova et al., Phys. Rev. B. 73, 144417 (2006).\n[9] L. Krusin-Elbaum et al., Nature 431, 672 (2004).\n[10] Due to the electrochemical doping process, ∼17% of the\nsample mass is paramagnetic carbon which contributes\nonly to the slowly relaxing signal.\n[11] S.J. Blundell, Contemp. Phys. 40, 175 (1999).\n[12] H. Kweon et al., Phys. Rev. B. 76, 045434 (2007).\n[13] J. B. Goodenough, Annu. Rev. Mater. Sci. 1, 101 (1971).\n[14] T. Kiyama et al., Phys. Rev. B. 73, 184422 (2006).\n[15] J.M.D. Coey and D. Khalafalla, phys. stat. sol. (a) 11,\n229 (1972).\n[16] R. Berger et al., J. Magn. Magn. Mat. 234, 535 (2001).\n[17] F. Gazeau et al., J. Magn. Magn. Mat. 202, 535 (1999).\n[18] P. V. Radovanovic and D. R. Gamelin, Phys. Rev. Lett.\n91, 157202 (2003).\n[19] G. Z. Xing et al., Adv. Mater. 20, 3521 (2008).\n[20] A. Kaminski and S. Das Sarma, Phys. Rev. Lett. 88,\n247202 (2002).\n[21] A. C. Durst et al., Phys. Rev. B 65, 235205 (2002)."    },    {        "title": "1612.06785v1.Ferromagnetic_resonance_and_interlayer_exchange_coupling_in_magnetic_multilayers_with_compositional_gradients.pdf",        "content": "arXiv:1612.06785v1  [cond-mat.mtrl-sci]  20 Dec 2016Ferromagnetic resonance and interlayer exchange coupling in magnetic\nmultilayers with compositional gradients\nD. M. Polishchuk,1,2,a)A. F. Kravets,1,2Yu. O. Tykhonenko-Polishchuk,1,2A. I. Tovstolytkin,2and V.\nKorenivski1\n1)Nanostructure Physics, Royal Institute of Technology, Sto ckholm, Sweden\n2)Institute of Magnetism, NAS of Ukraine, Kyiv, Ukraine\n(Dated: 16 August 2021)\nFerromagnetic resonance (FMR) in magnetic multilayers of type F1/ f/F2, where two strongly ferromagnetic\nlayersF1andF2areseparatedbyaweaklymagneticspacerfwithac ompositionalgradientalongitsthickness,\nis investigated. The method allows to detect the weak signal from th e spacer in additional to the more\npronounced and readily measured signal from the outer strongly- magnetic layers, and thereby study the\nproperties of the spacer as well as the interlayer exchange intera ction it mediates. Variable temperature\nFMR measurements, especially near the relevant Curie points, reve al a rich set of properties of the exchange\ninteractions in the system. The obtained results are useful for de signing and optimizing nanostructures with\nthermally-controlled magnetic properties.\nPACS numbers: 75.20.En, 75.30.Et, 75.70.Cn\nKeywords: Magnetic multilayers, exchange coupling, ferromagnet ic resonance, diluted ferromagnetic alloys,\nCurie-switch\nI. INTRODUCTION\nMagnetic multilayer structures of the spin-valve type\nF1/NM/F2,wheretwoferromagnetic(FM) layersF1and\nF2 are separated by a nonmagnetic (NM) spacer, exhibit\nrich physics including spin-dependent scattering,1,2in-\nterlayer exchange coupling,3,4spin-torque eﬀects,5,6etc.\nDue to this functional versatility, spin-valve type nanos-\ntructures have become the foundation for many spin-\ntronicapplications.7–9Furthertechnologicalprogresswill\nbe based on the ability to further expand the function-\nality, with the use of external electric ﬁeld, temperature,\netc., to control the magnetic state in the nanostructure.\nRecent studies have demonstrated that modiﬁed\nF1/f/F2 systems, where the weakly FM spacer (f) has\na Curie temperature ( Tf\nC) much lower than that of the\nstrongly FM outer layers (F1 and F2), can expand the\nfunctionality of the spin-valve family, yielding nanostruc-\ntures with thermally-controlled magnetic properties.10,11\nDepending on whether spacer f is in FM or PM (param-\nagnetic) state, two strongly FM layers can be exchange\ncoupled(for T <Tf\nC)oruncoupled(for T >Tf\nC). Forsuch\nnanostructure placed in an appropriately chosen mag-\nnetic ﬁeld, temperature variation may lead to switching\nbetweenparallelandantiparallelmutual alignmentofthe\nmagnetic moments of the F1 and F2 layers. Thus, a vari-\nation in temperature and/or ﬁeld can produce switching\nbetween the P and AP conﬁgurations of the nanostruc-\nture (the so-called Curie switch, CS, or Curie valve ac-\ntion), which can be used in spin-switch sensors, oscil-\nlators, and memory elements with intrinsic thermoelec-\ntronic control.10,12\na)Electronic mail: dpol@kth.se.The experiments described in Ref. 13 and 14conﬁrmed\nthe concept of parallel-antiparallel switching in F1/f/F2\nnanostructures, in particular incorporating Ni xCu100−x\nspacers enclosed by exchange-pinned Co 90Fe10and free\nNi80Fe20(Py) layers. A serious disadvantage of such\nstructures is a relatively wide temperature range of the\ntransitional state between the P and AP conﬁgurations.\nThis mainly occurs due to the high nonuniformity of\nthe eﬀective interatomic exchange coupling and the mag-\nnetization distribution throughout the spacer thickness,\nespecially in the proximity to the strongly ferromag-\nnetic interfaces. The use of a gradient-type spacer f*\n= PM/weak FM/PM can greatly improve the eﬀective-\nexchange uniformity and lead to a signiﬁcantly nar-\nrower magnetic transition range, as was demonstrated in\nRef. 11. Unfortunately, traditional measurement tools,\nsuch as magnetometry, are unable to probe and charac-\nterize the state of the f* spacer, which impedes the de-\nvelopment of competitive CSs with reliably predictable\nproperties. Ferromagnetic resonance (FMR), which was\nparticularly useful for characterization of CS structures\nwith uniform spacers,14should be expected to alsodetect\nthe key magnetic characteristicsof speciﬁcally f* spacers,\nas well as the interlayer exchange properties they medi-\nate.\nIn this work we study FMR in CS-multilayers contain-\ning gradient spacers. We directly detect the magnetic\nresponse from the spacer and conduct variable tempera-\nture studies of the interlayer exchange in the system near\nthe spacer’s Curie point.\nII. SAMPLES AND MEASUREMENTS\nThe conﬁguration used in magnetic resonance mea-\nsurements and the layout of the multilayer system2\nF1/f/F2/AFM, where AFM is the pinning antiferro-\nmagnetic layer, are illustrated in Fig. 1. Thin-ﬁlm\nmagnetic multilayers Py(6 nm)/f(14 nm)/Py(2 nm)/\nCo90Fe10(2 nm)/Ir 20Mn80(12 nm) (hereafter –\nF1/f/F2/AFM system) were deposited at room tem-\nperature on thermally oxidized silicon substrates using\nmagnetron sputtering in an AJA Orion 8-target system\n(details of the fabrication are described in Ref. 11\nand 13). The exchange pinning between the Py(2\nnm)/CoFe(2 nm) bilayer and Ir 20Mn80(12 nm) was set\nin during the deposition of the multilayers using an\nin-plane magnetic ﬁeld Hdep≈1 kOe. Three samples\nin the series studied here, labelled S-gr, S-fm and S-pm,\nhave diﬀerent spacer compositions. The S-gr sample\nhas a gradient-type PM/weak FM/PM spacer: f* =\nNi50Cu50(4 nm)/Ni 72Cu28(6 nm)/Ni 50Cu50(4 nm).\nThe S-fm and S-pm samples have uniform spacers f =\nNi72Cu28(14 nm) and f = Ni 56Cu44(14 nm), respectively.\nDiluted alloys Ni 72Cu28and Ni 56Cu44in the bulk form\nare, respectively, ferromagnetic and paramagnetic in the\ntemperature interval of measurements used in this work,\nT= 120–380 K.11\nThe FMR measurements were performed using an\nX-band Bruker ELEXSYS E500 spectrometer equipped\nwith an automatic goniometer. The operating frequency\nwasf= 9.46 GHz. In-plane FMR spectra were ob-\ntainedwith theexternalmagneticﬁeldapplied Hparallel\n(P) and antiparallel (AP) to the AFM pinning direction\n(Fig. 1) at T= 120–380 K.\nIII. RESULTS AND DISCUSSION\nA. Experimental FMR spectra\nFig.2showstypicalFMRspectrameasuredatdiﬀerent\ntemperatures for the P and AP orientations of ﬁeld H.\nThe signal of the highest intensity in all panels originates\nfrom the free layer, Py(6 nm). For the S-gr sample, at\nhigher temperatures, the position of this signal is weakly\ndependent on the orientation of the magnetic ﬁeld, while\nat lower temperatures the dependence becomes notice-\nable (Fig. 2(a)). In addition to the Py resonance line,\nthe spectra for S-gr contain the line (labeled sf), with\nthe resonance ﬁeld close to 2 kOe, which is not observed\nin the spectra for the S-pm or S-fm samples. The inten-\nsity of the sf signal is highly dependent on temperature:\nit decreases with the increase in Tand vanishes as Tex-\nceeds 320 K. The P and AP resonance ﬁelds of the sf\nsignal are diﬀerent at lower temperatures, but the lines\napproach each other and eventually merge as tempera-\nture increases. At the same time, their average position\nis shifted to higher ﬁelds compared to the signal from Py,\nwhichmeansthattheeﬀectivemagnetizationofthiscom-\nponent is lower than that of Py.14Since bulk Ni 72Cu28\nalloy has lower magnetization than Py, the sf signal can\nbe ascribed to the Ni 72Cu28(6 nm) layer in the gradi-\nent spacer PM/weak FM/PM. The Curie temperature ofbulk Ni 72Cu28is relatively low( ≈330K), which explains\nthe strong temperature variation of the sf resonance ﬁeld\nin the studied temperature region.\nThe highly intensive resonance line from the free layer\nF1 in the S-fm sample (Fig. 2(b)) has diﬀerent resonance\nﬁelds for the P and AP measurement conﬁgurations ( HP\nr1\nandHAP\nr1) in the whole temperature range. The nonzero\ndiﬀerence ( HAP\nr1−HP\nr1) indicates that the exchange pin-\nning ﬁeld from AFM, which acts on the F2 layer, trans-\nmits to the F1 layer due to a relatively strong interlayer\ncoupling in the S-fm stack.14At higher temperatures, an\nadditional low-ﬁeld signal is observed, but only for the\nAP orientation of the magnetic ﬁeld. This signal should\nbe assigned to the pinned F2 layer, as detailed in Ref. 14.\nThefactthat thissignalisalmostindistinguishableinthe\nAP spectra for the S-pm and S-gr samples needs addi-\ntional investigation.\nThe same position of the P and AP resonance lines\nfor the S-pm sample (Fig. 2(c)) means that the F1 and\nF2 layers are fully decoupled, which is the case at all\nmeasurement temperatures.\nB. Analysis of experimental data and discussion\nIn our earlier FMR studies,14,15a phenomenological\ntheory was developed16–18for modelling the dynamics\nof CS-systems containing compositionally uniform spac-\ners. Based on this theory, a comprehensive quantitative\nanalysis of the experimental FMR results for CSs with\nspacers of diﬀerent alloy dilution and thickness was per-\nformed, and the key magnetic parameters were deter-\nmined. In this work, we will show that the same the-\noretical approach forms a reliable basis for understand-\ningthe temperature-dependent behaviorofthe resonance\nparameters for multilayers with gradient spacers (such as\nS-gr).\nFig. 3(a) presents the temperature dependences of\nthe resonance-ﬁeld diﬀerence indicative of the exchange\nthroughthe spacer, ∆ Hr1=HAP\nr1−HP\nr1, whereHP\nr1,HAP\nr1\nare the resonance ﬁelds for the F1 layer (in our case Py)\nwith the external ﬁeld applied parallel and antiparallel\nto the AF pinning direction, respectively. According to\neq. (14) in Ref. 14, ∆ Hr1∼κ2·Hb, whereκis the con-\nstantofinterlayercoupling, whichdependsonthemagne-\ntizations and thicknesses ofthe F1 and F2 layersand also\nincludes the eﬀective magnetization ( m) and magnetic\nexchange length (Λ) of the weakly ferromagnetic spacer;\nHbis the eﬀective biasing ﬁeld acting on magnetization\nM2 of the F2 layer and is the measure of AFM pinning.\nThus, foraconstant Hb, ∆Hr1reﬂectsthe strengthofthe\ninterlayer coupling between strongly FM layers through\nthe weakly FM spacer.\nIn our case, ∆ Hr1is relatively high for S-fm (F1 and\nF2 are fully coupled) and zero for S-pm (F1 and F2 fully\ndecoupled) in the whole temperature range, which is in\nagreement with the expected behavior. The decrease of\nboth ∆Hr1and ∆Hsf\nrfor the S-gr sample with increasing3\nFIG. 1. Conﬁguration used in magnetic resonance measuremen ts (left panel) and layout of multilayers studied (right pan el).\n/s49 /s50\n/s72 /s32/s40/s107/s79/s101/s41/s49 /s50/s40/s99/s41\n/s83/s45/s112/s109\n/s49/s50/s48/s32/s75\n/s72 /s32/s40/s107/s79/s101/s41/s83/s45/s102/s109\n/s49/s50/s48/s32/s75/s32/s32/s83/s45/s103/s114/s32/s32\n/s40/s97/s41\n/s32/s80\n/s32/s65/s80\n/s115/s102/s51/s50/s48/s32/s75/s100/s80/s47/s100/s72/s32/s40/s97/s46/s117/s46/s41/s40/s98/s41\n/s83/s45/s102/s109\n/s51/s50/s48/s32/s75\n/s49/s50/s48/s32/s75/s51/s48/s48/s32/s75\n/s50/s54/s48/s32/s75\n/s50/s50/s48/s32/s75\n/s49/s56/s48/s32/s75\nFIG. 2. Typical FMR spectra measured with the external magne tic ﬁeld applied parallel (P) and antiparallel (AP) to the\ndirection of the exchange pinning for (a) S-gs, (b) S-fm, and (c) S-pm samples. Panel (a) shows only the sf resonance signa l\nfor temperatures between 120 and 320 K.\ntemperature implies that the interlayer coupling, being\nsigniﬁcant at low temperatures, weakens as the temper-\nature increases.\nThe nontrivial eﬀect that should be pointed out is the\nincrease of ∆ Hr1observed in the S-fm sample as temper-\nature is increased (Fig. 3(a)). When the interlayer ex-\nchange coupling between F1 and F2 is weak, the temper-\nature behavior of ∆ Hr1is manly governed by κ2∼m4.\nmdecreases when temperature increases, which should\nlead to a decrease in ∆ Hr1. However, when the inter-\nlayer coupling is suﬃciently strong, the assumption of a\nconstant Hbbecomes invalid. In this case, the F1/f/F2\nstack behaves like a single magnetic layer, and the eﬀec-\ntive biasing ﬁeld from AFM acts on the whole F1/f/F2\nstructure, rather than only on F2. When the interlayercoupling between F1 and F2 becomes weaker, the action\nof the biasing ﬁeld becomes localized to F2. As a result,\nthe biasing eﬀect becomes stronger and ∆ Hr1increases.\nIt is this situation that is believed to manifest in the S-fm\nsample as temperature is increased.\nThe resonance signal from the weakly FM spacer was\nundetectable in our earlier studies on CS-structures hav-\ning spacers of uniform composition.14A possible reason\nfor this is a strong inﬂuence of the adjacent FM layers\non the magnetic state of the spacer. The gradient-type\nspacer studied herein suppresses the ferromagnetic prox-\nimity eﬀect at the interface with the strongly ferromag-\nnetic layers, and its core is therefore much more uni-\nform magnetically, with presumably a much better de-\nﬁnedFMR.Thisshouldmakeiteasiertodetectthesignal4\n/s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48/s48/s49/s50\n/s83/s45/s112/s109\n/s83/s45/s103/s114/s83/s45/s102/s109\n/s115/s112/s44 /s32/s83/s45/s102/s109/s115/s102 /s44/s32/s83/s45/s103/s114/s40/s98/s41/s72/s65/s80 /s114/s49/s32/s40/s107/s79/s101/s41\n/s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52\n/s32/s84 /s32/s40/s75/s41/s83/s45/s102/s109\n/s32/s115/s102 /s44/s32 /s83/s45/s103/s114\n/s32/s83/s45/s103/s114\n/s32/s83/s45/s112/s109/s40/s97/s41/s72\n/s114/s49/s44 /s72/s115/s102 /s114/s32/s40/s107/s79/s101/s41\n/s84 /s32/s40/s75/s41\nFIG. 3. (a) Resonance ﬁeld diﬀerence ∆ Hr1=HAP\nr1−HP\nr1versus temperature; ∆ Hsf\nrrelates to the sf signal (see text for\ndetails). (b) Temperature dependences of the AP resonance ﬁ eld for all detected FMR signals. Shown additionally are the F1\nresonance signals for all samples, the sf signal for the S-gr sample, and the F2 signal (labeled sp) for the S-fm sample.\nfrom the inner part of the spacer (sf signal in Fig. 2(a)).\nAt the same time, the nonzero diﬀerence ∆ Hr1(T) for F1\natT <250 K (Fig. 3(a)) indicates that the gradient-type\nspacer is still fully capable to mediate interlayer coupling\nat lower temperatures.\nThe value of ∆ Hsf\nr(T) is greater than ∆ Hr1(T) at low\ntemperatures ( T <250 K). This demonstrates that the\neﬀective transfer of the exchange pinning via the spacer\ncan be considered to take place in two steps. First, the\npinning is induced in the inner FM part of the spacer via\nthe exchange coupling at the F2/f* interface. Then, the\nfreeF1layersensesthepinning viatheexchangecoupling\nat the f*/F1 interface.\nThe temperature dependence of the AP resonance\nﬁelds, shown in Fig. 3(b), contain additional information\nabout the magnetic state of the studied CSs. Signiﬁcant\nchangesin HAP\nr1versustemperaturecanbe causedbytwo\nfactors: (i) changes in the magnetization of FM, and (ii)\nchanges in the strength of the interlayer coupling (due to\nchanges in the magnetization of the spacer).\nThe resonance ﬁeld of the free layer (F1) in the S-\npm sample slightly increases with increasing tempera-\nture (Fig. 3(b)). In the FMR conﬁguration used, an\nincrease of Hr1should correspond to a decrease in the\nF1 magnetization.18TheH0\nr1(T) dependence for S-pm is\nused as a reference temperature dependence.\nHAP\nr1(T) for the S-gr and S-fm samples diﬀer from\nH0\nr1(T) for S-pm. The observed changes here should be\npredominantlyduetochangesinthestrengthoftheinter-\nlayer exchange. The strong interlayer coupling inherent\nto S-fm leads to a clear separation of HP\nr1andHAP\nr1from\nH0\nr1(T). This is why HAP\nr1(T) for S-fm is so high above\nthat for S-pm. The weaker interlayer coupling inherent\nto S-gr aﬀects the resonance ﬁeld of F1 to a lesser extent.\nOn the other hand, the HAP\nr,sf(T) dependence of the sf sig-\nnal fromthe spacerofS-grcannot be deﬁned byonly one\ndominant mechanism: strong changes with temperature\nof both the magnetization and the interlayer exchange\ncoupling are at play.Finally, the sp signal from the pinned F2 layer is dis-\ntinguishable only in the AP spectra for S-fm at higher\ntemperatures(Fig.2(b) andFig.3(b)). Duetothestrong\nAFM pinning, F2 must have negative resonance ﬁeld for\nthePorientation, whichisnotobservable.14Itisinterest-\ning that this signal has a signiﬁcantly reduced intensity\nin the spectra for the S-gr and S-pm samples.\nIV. SUMMARY\nFMR spectroscopy and its theoretical analysis provide\nan excellent tool for describing the magnetism in mul-\ntilayers of F1/f/F2 type, and determining the key mag-\nnetic parameters of the nanostructure as a whole and\ntheCurie-spacerinparticular. F1/f*/F2multilayerswith\ngradient-typespacersexhibitclearpeaksdue tothe FMR\nin the spacer, which are used to characterize this critical\nlayerin terms of its magnetization and the exchange cou-\npling it mediates between the outer ferromagnetic layers.\nACKNOWLEDGMENTS\nSupport from the Swedish Research Council (VR\nGrant No. 2014-4548), the Swedish Stiftelse Olle En-\ngkvist Byggm¨ astare, the Science and Technology Cen-\nter in Ukraine (project P646), and the National\nAcademy of Sciences of Ukraine (project 0115U003536\nand 0115U00974) is gratefully acknowledged.\n1S. S. P. Parkin, Phys. Rev. Lett. 71, 1641 (1993).\n2C. Chappert, A. Fert, and F. N. V. Dau,\nNat. Mater. 6, 813 (2007).\n3S. S. P. Parkin, Phys. Rev. Lett. 67, 3598 (1991).\n4A. Fert, P. Gr¨ unberg, A. Barth´ el´ emy, F. Petroﬀ, and W. Zin n,\nJ. Magn. Magn. Mater. 140-144 , 1 (1995).\n5J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996).\n6A. Brataas, A. D. Kent, and H. Ohno,\nNat. Mater. 11, 372 (2012).5\n7B. Dieny, V. S. Speriosu, S. S. P. Parkin, B. A. Gurney, D. R.\nWilhoit, and D. Mauri, Phys. Rev. B. 43, 1297 (1991).\n8G. A. Prinz, Science 282, 1660–1663 (1998).\n9T. Shinjo, ed., Nanomagnetism and Spintronics (Oxford; Ams-\nterdam: Elsevier, 2009).\n10A.M.Kadigrobov, S.Andersson, D.Radi,R.I.Shekhter, M.Jo n-\nson, and V. Korenivski, J. Appl. Phys. 107, 123706 (2010).\n11A. F. Kravets, A. N. Timoshevskii, B. Z. Yanchitsky, M. A.\nBergmann, J. Buhler, S. Andersson, and V. Korenivski,\nPhys. Rev. B. 86, 214413 (2012).\n12A. M. Kadigrobov, S. Andersson, H. C. Park, D. Radi,\nR. I. Shekhter, M. Jonson, and V. Korenivski,\nJ. Appl. Phys. 111, 044315 (2012).\n13A. F. Kravets, Y. I. Dzhezherya, A. I. Tovstolytkin,\nI. M. Kozak, A. Gryshchuk, Y. O. Savina, V. A.Pashchenko, S. L. Gnatchenko, B. Koop, and V. Korenivski,\nPhys. Rev. B. 90, 104427 (2014).\n14A. F. Kravets, A. I. Tovstolytkin, Y. I. Dzhezherya,\nD. M. Polishchuk, I. M. Kozak, and V. Korenivski,\nMatter., J. Phys.: Condens. 27, 446003 (2015).\n15A. F. Kravets, D. M. Polishchuk, Y. I. Dzhezherya,\nA. I. Tovstolytkin, V. O. Golub, and V. Korenivski,\nPhys. Rev., B. 94, 064429 (2016).\n16L. D. Landau and L. M. Lifshitz, Physik. Zeits. Sowjetunion 8,\n153 (1935).\n17J. Smit and H. G. Beljers, Phillips Res. Rep. 10, 113 (1955).\n18C. Kittel, Phys. Rev. 73, 155 (1948)."    },    {        "title": "1607.06137v2.A_microwave_interferometer_of_the_Michelson_type_to_improve_the_dynamic_range_of_broadband_ferromagnetic_resonance_measurements.pdf",        "content": "A\tmicrowave\tinterferometer\tof\tthe\tMichelson-type\tto\timprove\tthe\tdynamic\trange\tof\tbroadband\tferromagnetic\tresonance\tmeasurements\t\tE.\tR.\tJ.\tEdwards1*,\tA.\tB.\tKos1,\tM.\tWeiler2,3,\tT.\tJ.\tSilva1\t\t1- Quantum\tElectromagnetics\tDivision,\tNational\tInstitute\tof\tStandards\tand\tTechnology,\tBoulder,\tCO\t80305\t2- Walther-Meißner-Institut,\tBayerische\tAkademie\tder\tWissenschaften,\t85748\tGarching,\tGermany\t3- Physik-Department,\tTechnische\tUniversität\tMünchen,\t85748\tGarching,\tGermany\t\t\tAbstract\t\t\tWe\tpresent\ta\tMichelson-type\tmicrowave\tinterferometer\tfor\tuse\tin\tferromagnetic\tresonance\texperiments.\tThe\tinterferometer\tis\tcapable\tof\tbroadband\toperation\twithout\tmanual\tadjustment\tof\tphase\tdelay\tor\tamplitude\tattenuation.\tA\tprototype\tof\tthe\tdesign\tshows\tsignificant\timprovement\tof\tthe\tsignal-to-noise\tratio\twhen\tcompared\tto\tnon-interferometric\tferromagnetic\tresonance\texperiments.\tWe\tdemonstrate\tthat\tthis\tincrease\tin\tsensitivity\tcan\tlead\tto\ta\tdrastic\tincrease\tin\tthe\tdata\tacquisition\trate\tfor\thard-to-measure\tthin\tfilms\tthat\totherwise\twould\trequire\tlong\tintegration\ttimes.\t\tIntroduction\t\t\tVector\tnetwork\tanalyzer-based\tferromagnetic\tresonance\t(VNA-FMR)\tis\ta\twell-established\ttechnique\tfor\tthe\tcharacterization\tof\tmagnetic\tthin\tfilms\t[Kalarickal\t2006,\tMaksymov\t2014,\tSilva\t2016].\tIn\tVNA-FMR,\ta\tmagnetic\tthin\tfilm\tis\tloaded\tonto\ta\tplanar\ttransmission\tline\tthat\tis\tthen\tplaced\tin\tan\texternal\tmagnetic\tfield.\tA\tVNA\tis\tused\tto\tmeasure\tthe\tS-parameters\tof\tthe\tloaded\ttransmission\tline\tas\ta\tfunction\tof\texternal\tfield\tand\tfrequency.\tThe\thigh\tsensitivity\tof\tVNA-FMR\t[Neudecker\t2006]\tderives\tfrom\tthe\tlarge\tdynamic\trange\tof\tcommercially\tavailable\tVNAs,\tusually\tat\tleast\t120\tdB\tin\ta\t10\tHz\tmeasurement\tbandwidth\tat\tmicrowave\tfrequencies.\tAt\tthe\tsame\ttime,\tsuch\ta\tlarge\tdynamic\trange\tis\trequired\tdue\tto\tthe\tsmall\tfilling\tfactors\t[Pozar\t2011]\tfor\tthis\tmeasurement\tgeometry;\tmost\tof\tthe\tenergy\tof\tthe\tdriving\tmicrowave\tmagnetic\tfield\tdoes\tnot\tinteract\twith\tthe\tmagnetic\tthin\tfilm.\t\tWhile\tit\tis\tpossible\tto\tmeasure\tand\textract\timportant\tmaterial\tinformation\tfrom\ta\twide\tvariety\tof\tsamples\tdespite\tthe\tsmall\tfilling\tfactor\t[Maksymov\t2014,\tShaw\t2011,\tShaw\t2013,\tBoone\t2013]\tthe\tpresence\tof\ta\tlarge\tbackground\tsignal\tat\tthe\tVNA\treceiver\tintroduces\ta\tlarge\toffset\tas\twell\tas\tbackground\tnoise.\tThe\toffset\tdecreases\tthe\tdynamic\trange\tavailable\tfor\tsignal\tdetection,\twhile\tthe\tbackground\tnoise\tslows\tdown\tthe\tmeasurement.\tEliminating\tthis\tbackground\tsignal\tduring\tmeasurement\twould\tboth\tincrease\tthe\tavailable\tdynamic\trange\tof\tthe\tsetup\tand\tdecrease\tacquisition\ttimes.\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t*\tCorresponding\tauthor:\tEric\tR.\tJ.\tEdwards\t(eric.edwards@nist.gov)\tContribution\tof\tNIST,\tnot\tsubject\tto\tcopyright.\t\tIn\tfact,\tit\thas\tlong\tbeen\tknown\tthat\tinterferometric\ttechniques\tcan\tbe\tused\tto\trealize\ta\tstrong\tsuppression\tof\tthe\tbackground\tsignal\tin\tmagnetic\tresonance\texperiments\t[Poole\t1983].\tRecently,\tseveral\tworks\thave\treported\tlarge\tincreases\tin\tthe\tsensitivity\tof\tVNA-FMR\tmeasurements\tusing\tthese\ttechniques\t[Zhang\t2011,\tTamaru\t2014,\tIvanov\t2014].\tIn\torder\tto\trealize\tthese\tgains,\thowever,\tthese\tsetups\thave\tbeen\teither\tnarrowband\t[Zhang\t2011,\tIvanov\t2014]\tor\trequired\treadjustment\tat\teach\tmeasurement\tfrequency\t[Tamaru\t2014].\t\t\tWe\thave\tdeveloped\ta\tMichelson-type\tmicrowave\tinterferometer\t(MMI)\tfor\tuse\tin\tVNA-FMR\texperiments.\tThe\tMMI\tis\ta\tpassive,\ttwo-port\tnetwork\tconsisting\tof\ta\tplanar\tcross\tjunction\tin\ta\tcoplanar\twaveguide\ttransmission\tline\twith\ttwo\tshort-terminated\tarms\tof\tidentical\tlength,\tas\tpresented\tschematically\tin\tFig.\t(a).\tThe\tnominal\tcharacteristic\timpedance\tof\tall\ttransmission\tlines\tin\tthe\tdevice\tis\t50\tW,\tand\tthe\tcenter\tconductor\twidth\tis\t360\tµm.\tAt\tdiscrete\tfrequencies\tdetermined\tby\tthe\tarm\tlength\tL,\tthe\treflected\twave\tfrom\tthe\ttwo\tarms\tdestructively\tinterfere\tat\tport\t1\tor\tport\t2\tas\tschematically\tshown\tin\tFig.\t1(b).\tAt\tthese\tfrequencies\tmeasurements\tof\tvery\thigh\tsensitivity\tbecome\tpossible,\tdrastically\treducing\tthe\tmeasurement\ttime.\tThe\tscattering\tproperties\tof\ta\tsymmetric\tplanar\tcross\tjunction\tcan\tbe\teasily\tunderstood\tin\tterms\tof\tsimple\tnodal\tanalysis\tbased\ton\tKirchoff’s\tLaws,\tunder\tthe\tassumption\tthat\tthe\tfeature\tdimensions\tof\tthe\tcross\tjunction\tare\tmuch\tsmaller\tthan\tthe\twavelength\tof\tthe\tmicrowaves\temployed.\tIf\tEi\tis\tthe\tscalar\tvalue\tof\tthe\telectric\tfield\tincident\ton\tone\tarm\tof\tthe\tcross\tjunction,\tEb\tis\tthe\tbackscattered\telectric\tfield\tthat\tpropagates\tback\tinto\tthe\tarm\tof\tthe\tincident\telectric\tfield,\tand\tEf\tis\tthe\tforward\tscattered\telectric\tfield\tin\tthe\tthree\tother\tarms,\tthen\tnode\tanalysis\trequires\tthat\tEf=−Eb=Ei2.\tBased\ton\tthis\tfundamental\tresult,\twe\tcan\tthen\tcalculate\tthe\ttransmitted\tand\treflected\telectric\tfield\tamplitudes\tin\tthe\tcase\twhere\ttwo\tof\tthe\tcross-junction\tlegs\tare\tterminated\twith\tshorted\ttransmission\tlines\tof\tlength\tL.\tIn\tthat\tcase,\tthe\tcomplex\tvalue\tof\tthe\ttransmitted\telectric\tfield\t !Et\ta\tdistance\tx\tfrom\tthe\tcross-junction\talong\tthe\tremaining\tun-terminated\tleg\tis\tgiven\tby\t\t𝐸\"=12𝐸&1−𝑒&)*+𝑒&*,1\t\twhere\t β!2πnefffc\tis\tthe\ttransmission\tline\tphase\tconstant,\tneff\tis\tthe\teffective\trefractive\tindex\tof\tthe\ttransmission\tline,\tf\tis\tthe\tfrequency,\tand\tc\tis\tthe\tspeed\tof\tlight\tin\tvacuum.\tWe\ttherefore\tsee\tthat\tthe\ttransmission\tfrom\tport\tP1\tto\tport\tP2\tis\tnulled\twhen\tthe\tlength\tof\tthe\tterminated\tjunctions\tis\tchosen\tsuch\tthat\tβL=nπ\twhere\tn\tis\tan\tinteger.\tSimilarly,\tthe\tbackscattered\tfield\tis\tgiven\tby\t\t𝐸-=\t−12𝐸&1+𝑒&)*+𝑒&*,2\t\tHence,\tthe\treflection\tis\tnulled\tfor\tthe\tcomplementary\tcondition\tof\t2βL=2n+1()π.\t\tTo\tmeasure\tFMR\twith\tsuch\tan\tinterferometer,\twe\tcan\tplace\ta\tsample\tonto\tone\tof\tthe\tinterferometer\tarms\twith\ta\tshorted\ttermination\tsuch\tthat\tthe\tsample\tis\texcited\tby\tthe\tac\tmagnetic\tfields\tproximate\tto\tthe\twaveguide\tas\tsketched\tin\tFig.\t1(a).\tWe\tthen\tmake\tuse\tof\tthe\tfact\tthat\tthe\tinductive\tvoltage\tgenerated\tin\tthe\twaveguide\tby\tthe\tmagnetization\tprecession\tin\tthe\tsample\tunder\tthe\tcondition\tof\tferromagnetic\tresonance\tis\tantisymmetric\twith\tregard\tto\tdirection\tof\tpropagation\t[Silva\t1999].\tThis\tantisymmetric\tproperty\tresults\tin\tan\topposite\tsign\tfor\tthe\tbackscattered\tand\tforward\tscattered\twaves\tradiating\tfrom\tthe\tsample.\tThe\tforward\tscattered\twave\tis\tshifted\tin\tphase\tby\t180\tdegrees\tupon\treflection\tfrom\tthe\tshorted\ttermination\tat\tthe\tend\tof\tthe\ttransmission\tline.\tThus\tboth\tinductive\tsignals\tconstructively\tinterfere\twhen\tthey\tarrive\tback\tat\tthe\tcross\tjunction.\t\tExperiment\t\t\tWe\thave\tfabricated\tsuch\ta\tdevice,\tand\tcharacterized\tits\tperformance\tby\tuse\tof\ta\ttwo-port\tVNA.\tThe\tmagnitude\tof\tthe\tmeasured\tcomplex\tS11\tand\tS21-parameters\tof\tthe\tMMI\tare\tplotted\tin\tFig.\t1.\tWe\tobserve\ta\tset\tof\tnotches\twith\ta\tfrequency\tspacing\tof\tapproximately\t400MHz.\tAt\tthe\tnotch\tfrequencies\tthe\tsignals\tfrom\tthe\ttwo\tarms\tinterfere\tdestructively,\tstrongly\tsuppressing\tthe\tamount\tof\tpower\tat\tPort\t2\tof\tthe\tVNA.\tFrom\tthe\tcancellation\tconditions\tfor\tthe\treflected\tand\ttransmitted\telectric\tfield\tamplitudes,\tthis\tspacing\tagrees\twith\tthe\tchosen\tarm\tlength\tof\tL=0.25\tm.\tThe\tdegree\tof\tthe\tbackground\tsuppression\tis\ta\tfunction\tof\tboth\tthe\tphase\tand\tamplitude\timbalance\tbetween\tthe\ttwo\tinterferometer\tarms,\twhich\twe\thave\tplotted\tin\tFig.\t3.\t\tTo\tmeasure\tFMR,\twe\tplace\tthe\tferromagnetic\t(FM)\tfilm\ton\tone\tof\tthe\ttwo\tarms\tand\tapply\tan\texternal\tmagnetic\tfield\t𝐻1\tas\tshown\tin\tFig.\t1(a).\tThe\tfilm\tis\tTa(3)/Ni80Fe20(5)/Ta(3)\tgrown\tby\tsputter\tdeposition\ton\tan\toxidized\tsilicon\tsubstrate,\twhere\tnumbers\tin\tparentheses\tare\tthe\tnominal\tlayer\tthickness\tin\tnanometers.\tIn\torder\tto\toptimize\tthe\tphase\tand\tamplitude\tbalance\tof\tthe\tinterferometer,\ta\tbare\tcoupon\tof\tthe\tsubstrate\tthat\thas\tidentical\tdimensions\tas\tthe\tFM-covered\tsubstrate\tis\tplaced\ton\tthe\treference\tarm,\tthus\tcompensating\tthe\tdielectric\tresponse\tof\tthe\tcommon\tsubstrate\tmaterial.\tThe\tpower\tapplied\tat\tport\t1\tis\t0\tdBm\tin\tall\tmeasurements.\tNo\ttrace\taveraging\twas\tutilized,\tand\tthe\tintermediate\tfrequency\t(IF)\tbandwidth\tis\tset\tto\t500\tHz.\tWe\tfix\tthe\tfrequency\tto\tthe\tnotch\tin\tS21\t(i.e.\tthe\ttransmitted\tsignal)\tat\t1.2682\tGHz,\tand\tthen\trecord\tthe\tS-parameters\tas\ta\tfunction\tof\tswept\tmagnetic\tfield\tH0\t.\tWe\tplot\tthe\tS11\tparameter\tvs.\t𝐻1\t(i.e.\tthe\treflected\tsignal)\t\tin\tFig.\t4(a).\tThis\tsignal\tdoes\tnot\tbenefit\tfrom\tthe\tadvantage\tof\tany\tbackground\tsuppression\t(S11(1.2682GHz)\t=\t-\t2.5\tdB).\tFor\tcomparison,\twe\tplot\tS21\tvs.\t𝐻1\t(i.e.\tthe\ttransmitted\tsignal)\tin\tFig.\t4(b).\tIn\tthis\tcase,\tthe\tbackground\tsignal\tis\tstrongly\tsuppressed\t(S21(1.2682GHz)\t=\t-\t36\tdB\tfrom\tFig.\t2).\tThe\treduction\tin\tnoise\tis\tevident\tfrom\tinspection\tof\tFig.\t4(a)\tand\t4(b).\t\tTo\tdetermine\tthe\tsignal-to-noise\tratio\t(SNR),\twe\tfit\tthe\tmeasured\tcurves\tto\tthe\tcomplex\t𝜒33\tcomponent\tof\tthe\tmagnetic\tsusceptibility\t𝜒\tfor\tan\tin-plane\tbias\tfield\tgeometry\t[Weiler\t2014].\tThe\tdimensionless\tsignal\tamplitude\tAij\tfor\tS-parameter\tSij\tis\ttaken\tas\tthe\tamplitude\tof\tthe\tsusceptibility\tfit\tto\tthe\tmeasured\tresonance,\t\t𝐴&5≐\t𝑆&5𝐻1−𝑂𝜒33𝐻1\t3\t\twhere\tO\tis\tthe\tfitted\tfield-independent\toffset\tof\tthe\tscattering\tparameter.\tThe\tdimensionless\tnoise\tamplitude\tNij\tis\tdefined\tas\tthe\troot-mean-square\tof\tthe\tresiduals\tbetween\tthe\tmeasured\tspectra\tand\tthe\tsusceptibility\tfit,\t\t𝑁&5≐𝐴&5𝜒33𝐻1−𝑆&5𝐻1−𝑂)4\t\twhere\twe\thave\tverified\tthat\tthe\tnoise\tamplitude\tis\tindependent\tof\tmagnetic\tfield.\t\t\tIn\tFig.\t4(c),\twe\tplot\tthe\tSNR\tdefined\tas\t<=>?=>\tor\t20log<=>?=>\tin\tdB\tof\tthe\tbackground-suppressed\tS21\tand\tbackground-full\tS11\tFMR\tmeasurements\tas\ta\tfunction\tof\tthe\tnumber\tof\tfield-sweep\taverages\tin\tS11.\tAs\tis\tevident\tfrom\tFig.\t4(d),\tthe\tbackground-suppressed\tmeasurements\texhibit\ta\t20\tdB\tincrease\tin\tthe\tSNR\tfor\ta\tsingle\tmeasurement\tand\tthe\tbackground-full\tS11\tmeasurements\trequire\t129\taverages\tto\tobtain\tthe\tsame\tSNR\tas\tthat\tobtained\tvia\tthe\tS21\tmeasurements.\tIn\tother\twords,\twe\tobserve\ta\tfactor\tof\t129\tspeedup\tof\tdata\tacquisition\trate\tdue\tto\tthe\tbackground\tsuppression\tof\tthe\tinterferometer.\tIn\torder\tto\tvalidate\tthe\taccuracy\tof\tVNA-FMR\tmeasurements\twith\tthe\tMMI,\twe\tmeasured\tFMR\tspectra\twith\ta\tconventional\tthru-line\tcoplanar\twaveguide\ttransmission\tline.\tBy\tfitting\tthese\tspectra\tto\tthe\tin-plane\tsusceptibility\t𝜒33\twe\tcan\tmake\ta\tdirect\tcomparison\twith\tthe\textracted\tparameters\tfrom\tthe\tmeasurements\twith\tthe\tMMI\tand\tnon-interferometric\tmeasurements.\tWe\tplot\tthe\tresult\tin\tFig.\t5.\tWithin\tthe\tlimits\tof\tthe\tprototype\twe\tfind\tgood\tagreement\tbetween\tthe\ttwo\ttechniques.\t\tDiscussion\t\t\tThe\tSNR\timprovement\tin\tour\tmeasurements\tis\ta\tdirect\tconsequence\tof\tthe\tbackground-suppression\tachieved\twith\tthe\tMMI.\tIn\ttypical\tVNA-FMR\tmeasurement\tof\tthin\tmagnetic\tfilms\twith\tlow-loss\tcoplanar\twaveguides,\tthe\tbackground\tis\tmuch\tlarger\tthan\tthe\tsignal.\tIn\tsuch\tcases,\ttrace\taveraging\tcan\tbe\trequired\tto\teliminate\tartifacts\tdue\tto\tphase\tand\tamplitude\tdrift.\tWhile\ta\tquantitative\tcalculation\tof\tthe\tfilling\tfactor\tfor\tthe\tVNA-FMR\tgeometry\t–\tespecially\tfor\tconducting\tfilms\t–\tis\tbeyond\tthe\tscope\tof\tthis\twork,\twe\tare\table\tto\tcalculate\tthe\trelative\tchange\tin\tthe\tS-parameters\tunder\tcertainly\tsimplifying\tassumptions.\tFor\ta\ttransmission\tmeasurement\tin\twhich\ta\tuniform\texcitation\tfield\tfrom\tthe\tcoplanar\twaveguide\texcites\tuniform\tprecession,\tthe\tchange\tin\tthe\tscattering\tparameters\tof\tthe\tloaded\ttransmission\tline\tat\tthe\tferromagnetic\tresonance\tfield\t𝐻DEF\tis\tgiven\tby\t[Silva\t2016]\t\t𝛥𝑆)H𝐻DEF=𝛾𝜇1𝑀F𝑙𝑑N8𝑍1𝛼𝑤5\t\twhere\t𝑀F\tthe\tsaturation\tmagnetization,\t𝑙\tis\tthe\tlength\tof\tthe\tfilm\talong\tthe\twaveguide,\t𝑑N\tthe\tfilm\tthickness,\t𝑍1\tthe\tcharacteristic\timpedance\tof\tthe\ttransmission\tline,\tand\t𝑤\tis\tthe\twidth\tof\tthe\tcenter\tline\tof\tthe\tcoplanar\twaveguide.\tFor\ttypical\tfilm\tparameters\t[Silva\t2016],\t𝛥𝑆)H𝐻DEF\tis\tsmall,\ton\tthe\torder\tof\t-50\tdB.\tIn\tother\twords,\tthe\tsignal\tpower\tcan\tbe\tas\tsmall\tas\t0.001\t%\tof\tthe\tbackground\tpower\tat\tthe\treceiver.\tBy\tsuppression\tof\tthe\tbackground\tby\ta\tcertain\tamount\tby\tuse\tof\tthe\tMMI,\tthe\tdynamic\trange\tis\tincreased\tby\tthe\tsame\tamount\tuntil\tthe\tbackground\tis\tsuppressed\tbelow\tthe\tsignal\tlevel.\tHowever,\tthe\tquantifiable\treduction\tin\tnoise\tdue\tto\tsuppression\tof\tthe\tbackground\twill\tbe\tspecific\tto\tthe\tVNA.\t\tIn\tconclusion,\twe\thave\tdeveloped\ta\tbroadband\tmicrowave\tinterferometer\tsuitable\tfor\tferromagnetic\tresonance\tmeasurements.\tThe\tinterferometer\tdoes\tnot\trequire\tadjustment\tas\tthe\tmeasurement\tfrequency\tis\tchanged,\tand\twe\thave\tdemonstrated\ta\tbackground\tsuppression\tof\tup\tto\t35\tdB,\tand\ta\tcommensurate\tSNR\timprovement\tof\t20\tdB\tfor\ta\tsingle\tfield-sweep\tmeasurement.\tImproved\tsensitivity\tmay\tbe\texploited\tto\tattain\tfaster\tmeasurement\ttimes\tor\tto\tmeasure\tsamples\tof\tlower\tmagnetic\tvolume.\tWe\thave\tdemonstrated\ta\tspeed-up\tof\ta\tfactor\tof\t129\tcompared\tto\tnon-interferometric\tmeasurements.\tSignificantly,\tthe\tinterferometer\tis\tcompatible\twith\texisting\tVNA-FMR\tsetups\tas\ta\tdrop-in\treplacement\tfor\tthe\tcoplanar\twaveguide\ttransmission\tline,\twith\tonly\tminor\tchanges\tto\tthe\tmeasurement\tprotocol.\tFuture\twork\twill\tfocus\ton\tincreasing\tthe\tbandwidth\tof\tthe\tinterferometer\tby\tlithographic\tpatterning\tand\tproper\tchannelization\tof\tthe\tcoplanar\twaveguide\ttransmission\tline\t[Simons\t2004].\t\tAcknowledgements\t\tERJE\tacknowledges\tsupport\tfrom\tthe\tNational\tResearch\tCouncil\tPostdoctoral\tResearch\tAssociates\tprogram.\t\tReferences\t\t[Boone\t2013]\tBoone,\tC\tT,\tNembach\tH\tT,\tShaw\tJ\tM,\tSilva\tT\tJ\t(2013)\t“Spin\ttransport\tparameters\tin\tmetallic\tmultilayers\tdetermined\tby\tferromagnetic\tresonance\tmeasurements\tof\tspin-pumping,”\tJ.\tAppl.\tPhys.,\tvol.\t113,\t153906,\tdoi:\t10.1063/1.4801799\t[Ivanov\t2014]\t\tIvanov\tE\tN,\tKostylev\tM\t(2014)\t“Extremely\thigh-resolution\tmeasurements\tof\tmicrowave\tmagnetisation\tdynamics\tin\tmagnetic\tthin\tfilms\tand\tnanostructures,”\tarXiv:1402.3459\t[Kalarickal\t2006]\tKalarickal\tS\tS,\tKrivosik\tP,\tWu\tM,\tPatton\tC\tE,\tSchneider\tM\tL,\tKabos\tP,\tSilva\tT\tJ,\tNibarger\tJ\tP\t(2006)\t“Ferromagnetic\tresonance\tlinewidth\tin\tmetallic\tthin\tfilms:\tComparison\tof\tmeasurement\tmethods,”\tJ.\tAppl.\tPhys.,\tvol.\t99,\t093909,\tdoi:\t10.1063/1.2197087\t[Maksymov\t2014]\tMaksymov\tI\tS,\tKostylev\tM\t(2014)\t“Broadband\tstripline\tferromagnetic\tresonance\tspectroscopy\tof\tferromagnetic\tfilms,\tmultilayers\tand\tnanostructures,”\tPhysica\tE,\tvol.\t69,\tpp.\t253-293,\tdoi:\t10.1016/j.physe.2014.12.027\t[Nembach\t2011]\tNembach\tH\tT,\tSilva\tT\tJ,\tShaw\tJ\tM,\tSchneider\tM\tL,\tCarey\tM\tJ,\tMaat\tS,\tChildress\tJ\tR\t(2011)\t“Perpendicular\tferromagnetic\tresonance\tmeasurements\tof\tdamping\tand\tLandé\tg−factor\tin\tsputtered\t(Co2Mn)1−xGex\tthin\tfilms,”\tPhys.\tRev.\tB,\tvol.\t84,\t054424,\tdoi:\t10.1103/PhysRevB.84.054424\t[Neudecker\t2006]\tNeudecker\tI,\tWoltersdorf\tG,\tHeinrich\tB,\tOkuno\tT,\tGubbiotti\tG,\tBack\tC\tH,\t(2006)\t“Comparison\tof\tfrequency,\tfield,\tand\ttime\tdomain\tferromagnetic\tresonance\tmethods,”\tJ.\tMagn.\tMag.\tMat.,\tvol.\t307,\tpp.\t148-156,\tdoi:\t10.1016/j.jmmm.2006.03.060\t[Pozar\t2011]\tPozar\tD\tM\t(2011),\tMicrowave\tEngineering.\tNew\tYork,\tNY,\tUSA:\tJohn\tWiley\t&\tSons\t[Poole\t1983]\tPoole\tC\tP\t(1983),\tElectron\tSpin\tResonance:\tA\tComprehensive\tTreatise\ton\tExperimental\tTechniques.\tNew\tYork,\tNY,\tUSA:\tJohn\tWiley\t&\tSons\t[Shaw\t2011]\tShaw\tJ\tM,\tNembach\tH\tT,\tSilva\tT\tJ\t(2011)\t“Damping\tphenomena\tin\tCo90Fe10/Ni\tmultilayers\tand\talloys,”\tAppl.\tPhys.\tLett.,\tvol.\t99,\t012503,\tdoi:\t10.1063/1.3607278\t[Shaw\t2013]\tShaw\tJ\tM,\tNembach\tH\tT,\tSilva\tT\tJ\t(2013)\t“Measurement\tof\torbital\tasymmetry\tand\tstrain\tin\tCo90Fe10/Ni\tmultilayers\tand\talloys:\tOrigins\tof\tperpendicular\tanisotropy,”\tPhys.\tRev.\tB,\tvol.\t87,\t054416,\tdoi:\t10.1103/PhysRevB.87.054416\t[Silva\t1999]\tSilva,\tT\tJ,\tLee\tC\tS,\tCrawford\tT\tM,\tRogers\tC\tT\t(1999)\t“Inductive\tmeasurement\tof\tultrafast\tmagnetization\tdynamics\tin\tthin-film\tPermalloy,”\tJ.\tAppl.\tPhys.,\tvol.\t85,\t7849,\tdoi:\t10.1063/1.370596\t[Silva\t2016]\tSilva\tT\tJ,\tNembach\tH\tT,\tShaw\tJ\tM,\tDoyle\tB,\tOguz\tK,\tO’brien\tK,\tDoczy\tM\t(2016),\t“Characterization\tof\tMagnetic\tNanostructures\tfor\tSpin-Torque\tMemory\tApplications\twith\tMacro-\tand\tMicro-Scale\tFerromagnetic\tResonance”\tin\tCharacterization\tand\tMetrology\tfor\tNanoelectronics.\tPan\tStanford\tPublishing\tPte.\tLtd.\t[Simons\t2004]\tSimons\tR\t(2001)\tCoplanar\tWaveguide\tCircuits,\tComponents,\tand\tSystems.\tNew\tYork,\tNY,\tUSA:\tWiley-IEEE\tPress.\t[Tamaru\t2014]\tTamaru\tS,\tYakushiji\tK,\tFukushima\tA,\tYuasa\tS,\tKubota\tH\t(2014)\t“Ultrahigh\tSensitivity\tFerromagnetic\tResonance\tMeasurement\tBased\ton\tMicrowave\tInterferometer,”\tIEEE\tMagn.\tLett.,\tvol.\t5,\t3700304,\tdoi:\t10.1109/LMAG.2014.2365435\t[Weiler\t2014]\tWeiler\tM,\tShaw\tJ\tM,\tNembach\tH\tT,\tSilva\tT\tJ\t(2014)\t“Phase-Sensitive\tDetection\tof\tSpin\tPumping\tvia\tthe\tac\tInverse\tSpin\tHall\tEffect,”\tPhys.\tRev.\tLett.,\tvol.\t113,\t157204,\tdoi:\t10.1103/PhysRevLett.113.157204\t[Zhang\t2011]\tZhang\tH,\tDivan\tR,\tWang\tP\t(2011)\t“Ferromagnetic\tresonance\tof\ta\tsingle\tmagnetic\tnanowire\tmeasured\twith\tan\ton-chip\tmi-\tcrowave\tinterferometer,”\tRev.\tSci.\tInstrum.,\tvol.\t82,\t054704,\tdoi:\t10.1063/1.3593502\t\t\t\t\t\t\t\t\t\t\t\t\t\tFig.\t1\t(a)\tSchematic\tof\tthe\tMMI\tshowing\tthe\tdc\tapplied\tfield\t𝑯𝟎,\tthe\tmicrowave\tapplied\tfield\t𝒉𝟏,\tthe\tpositions\tof\tthe\tsample\tand\treference,\tand\tthe\tinterferometer\tarm\tlength\t𝑳.\t(b)\tSchematic\tdepiction\tof\tscattering\tprocesses\tin\tthe\tinterferometer\tfor\t𝜷𝑳=𝒏𝝅.\tArrows\trepresent\ttraveling\telectromagnetic\twaves.\tArrow\twidth/color\tencodes\telectric\tfield\tamplitude\tand\tarrow\toutline\tencodes\tsign.\tThe\tfirst\torder\tscattering\tprocess\t(𝑬𝒊\t->\t𝑬𝒊𝟐,\twide\tto\tmedium\twide\tarrows)\tresults\tin\tscattered\twaves\tof\topposite\tsign\tat\tP1\t(backscattering,\tdashed\toutline)\tand\tP2\tand\tterminated\tarms\t(forward\tscattering,\tsolid\toutline).\tAt\tthe\tterminations,\tthese\twaves\tare\treflected\tand\tinvert\tsign.\tAt\tthe\tcross,\tthe\tsecond\torder\tscattering\tprocess\t(𝑬𝒊𝟐,\t->𝑬𝒊𝟒\t,\tmedium\twide\tto\tnarrow\tarrows)\tresults\tin\ttwo\tforward\tscattered\twaves\twith\tamplitude\t−𝑬𝒊𝟒\tper\tport.\tThe\tthree\toutgoing\twaves\tconstructively\t(destructively)\tinterfere\tat\tPort\t1\t(Port\t2).\tP1a)b)LsamplereferenceLh1H0P2P1P2shortshortEiEi/2Ei/4E>0E<0xy\tFig.\t1.\tMagnitude\tof\tthe\tcomplex\tS11\tand\tS21\tS-parameters\tof\tthe\tunloaded\tMMI\tin\tzero\texternal\tfield\tmeasured\twith\ta\ttwo-port\tVNA.\t\n\tFig.\t3.\tThe\tamount\tof\tbackground\tsuppression\trealized\tfrom\tthe\tlinear\tsuperposition\tof\ttwo\twaves\tas\ta\tfunction\tof\tthe\tphase\timbalance\tfrom\t180\tdegrees\tand\tamplitude\timbalance\tfrom\tequal\tamplitude\tgiven\tin\tdB.\tThe\tblack\tcontour\tlines\tare\tlabeled\twith\tconstant\tvalues\tof\tthe\tsuppression\tin\tdB.\t\n\t\tFig.\t4.\t(a)\tSample-loaded\tS11\tas\tmeasured\twith\ta\ttwo-port\tVNA\tas\ta\tfunction\tof\tbias\tmagnetic\tfield.\t(b)\tSample-loaded\tS21\tas\tmeasured\twith\ta\ttwo-port\tVNA\tas\ta\tfunction\tof\tbias\tmagnetic\tfield.\t(c)\tThe\tSNR\tplotted\tin\tdB\tfor\tthe\tnumber\tof\tdata\taverages.\tThe\tS21\tSNR\tis\tplotted\twithout\taveraging\twith\tthe\tdotted\tline\tfor\tcomparison\tto\tS11\tdata.\t(d)\tThe\tdifference\tin\tthe\tSNR\tof\tS21\tand\tS11\tmeasurements\tshows\tthat\tit\ttakes\t129\taverages\tof\tS11\tdata\tto\tachieve\tthe\tunaveraged\tS21\tSNR.\t\t\n\tFig.\t5.\tA\tcomparison\tof\tinterferometric\t(“MMI”)\tand\tconventional\t(“Thru”)\tFMR\tparameters\textracted\tfrom\tthe\tcorresponding\tspectra.\t\t\n"    },    {        "title": "1607.07485v1.Optically_Detected_Ferromagnetic_Resonance_in_Metallic_Ferromagnets_via_Nitrogen_Vacancy_Centers_in_Diamond.pdf",        "content": "Optically Detected Ferromagnetic Resonance in Metallic Ferromagnets via Nitrogen\nVacancy Centers in Diamond\nM. R. Page*,1F. Guo*,2C. M. Purser,1J. G. Schulze,1T. M. Nakatani,3C. S.\nWolfe,1J. R. Childress,3P. C. Hammel,1,∗G. D. Fuchs,4,†and V. P. Bhallamudi1\n1Department of Physics, The Ohio State University, Columbus, Ohio 43210, USA\n2School of Applied and Engineering Physics, Cornell University, Ithaca NY 14853\n3San Jose Research Center, HGST, a Western Digital company, San Jose, California 95135, USA\n4School of Applied and Engineering Physics, Cornell University, Ithaca NY 14850\n(Dated: July 27, 2016)\nWe report quantitative measurements of optically detected ferromagnetic resonance (ODFMR) of\nferromagnetic thin ﬁlms that use nitrogen-vacancy (NV) centers in diamonds to transduce FMR into\na ﬂuorescence intensity variation. To uncover the mechanism responsible for these signals, we study\nODFMR as we 1) vary the separation of the NV centers from the ferromagnet (FM), 2) record the NV\ncenter longitudinal relaxation time T1during FMR, and 3) vary the material properties of the FM.\nBased on the results, we propose the following mechanism for ODFMR. Decay and scattering of the\ndriven, uniform FMR mode results in spinwaves that produce ﬂuctuating dipolar ﬁelds in a spectrum\nof frequencies. When the spinwave spectrum overlaps the NV center ground-state spin resonance\nfrequencies, the dipolar ﬁelds from these resonant spinwaves relax the NV center spins, resulting in\nan ODFMR signal. These results lay the foundation for an approach to NV center spin relaxometry\nto study FM dynamics without the constraint of directly matching the NV center spin-transition\nfrequency to the magnetic system of interest, thus enabling an alternate modality for scanned-probe\nmagnetic microscopy that can sense ferromagnetic resonance with nanoscale resolution.\nUnderstanding magnetic dynamics in future storage\nand information processing technologies will be a key to\ntheir development [1, 2]. In particular, it will be nec-\nessary to measure and understand relaxation [3, 4], an-\ngular momentum transfer [5–8] and spinwave propaga-\ntion [9–11], not only in extended magnetic ﬁlms, but also\nin nanoscale devices [12]. In addition, establishing new\nmechanisms for imaging magnetization dynamics in con-\nﬁned structures will aid in improving current magnetic\ntechnologies [13–16] and enhance them using emerging\nmaterials such as those featuring magnetic textures [17–\n19].\nThe nitrogen-vacancy (NV) center in diamond has\nemerged as a ﬂexible and sensitive platform for nanoscale\nmagnetic sensing [20–22] due to its atomic-scale size and\nits spin-sensitive ﬂuorescence, enabling optical detection\nof magnetic dynamics [23–25]. NV-based magnetometry\naimed at dynamic magnetic ﬁelds have typically required\neither spin-echo protocols [26], which are constrained to\nfrequencies that are quasi-static compared to FMR (e.g.\n∼MHz or below), or it requires direct resonance with an\nNV center spin transition [27, 28].\nIn contrast, we have recently demonstrated an alter-\nnate modality [29, 30] for detecting ferromagnetic reso-\nnance with diamond NV centers placed in nanoscale prox-\nimity to Yttrium Iron Garnet (YIG) that uses a simple,\ncontinuous wave (CW) protocol. A surprising observa-\n* M. R. Page and F. Guo contributed equally to this work.\nFigure 1. A schematic of the experiment. The sample is\na 20 nm Py ferromagnetic ﬁlm deposited on a single crystal\ndiamond with an implanted layer of NV centers 25 nm - 100\nnm from the surface. In order to apply microwave magnetic\nﬁelds to the sample, a microwire (5 nm Ti/ 300 nm Ag) is\npatterned on an insulating SiO 2layer. Green laser light is\nfocused through the back of the diamond and the resulting\nﬂuorescence changes of the NV centers are monitored. The\nstatic ﬁeldH0can be applied either in the ﬁlm plane or along\na/angbracketleft111/angbracketrightaxis of NV symmetry.\ntion was the change of NV center ﬂuorescence due to fer-\nromagnetic resonance at frequencies that were well sepa-\nrated from the NV center spin resonance. Conveniently,\nthese signals were acquired with no direct resonant ma-\nnipulation of the NV center spins, making them ideal\nfor integration with future NV center-based scanned-\nprobe microscopy of magnetic resonance. Establishing\nthe mechanism of these signals is key to their future use.\nWe posit that the oﬀ-resonant, FMR-induced change in\nNV center spin-state – and thus its ﬂuorescence variationarXiv:1607.07485v1  [cond-mat.mes-hall]  25 Jul 20162\n– must result from ﬂuctuating dipolar ﬁelds produced by\nthe ferromagnetic excitation. Since the uniform FMR\nmode of a continuous ﬁlm cannot produce a ﬂuctuating\ndipolar ﬁeld outside its boundaries, we suggest that a\nspectrum of spatially-inhomogeneous dipole ﬁelds from\nspinwaves are generated during the decay of the uniform\nmode that relax the NV center spins [31]. The eﬀect will\nbe largest when the wavelength of a spinwave is compa-\nrable to the separation between the surface of the fer-\nromagnet and the NV centers, and when the spinwave\nfrequency matches the NV center spin resonance. Here\nwe present experimental data which supports this idea.\nThe experimental results and related analysis are orga-\nnized as follows: In Section I, we ﬁrst show an ODFMR\nsignal from a permalloy (Ni 80Fe20) thin ﬁlm measured\nusing a single crystal diamond which has well-deﬁned\nNV orientations. In contrast to previous ODFMR mea-\nsurements using randomly oriented NV centers in nan-\nodiamonds, the single crystal samples containing a thin\nlayer of implanted NV centers allow us to control the NV\ncenter-ferromagnet (NV-FM) separation, d. We study\nthe variation of the ODFMR signal as a function of d.\nThis data suggests that the signal is optimized at a sep-\naration matching the wavelength of spinwaves whose fre-\nquency matches the NV spin resonance. In Section II, we\nquantitatively measure the longitudinal spin relaxation\nlifetime (T1) of the ground-state NV center spin, which\nis reduced as we drive the FM on resonance. This directly\ndemonstrates that ODFMR arises due to interactions be-\ntween the ground-state NV center spin and a driven fer-\nromagnetic system. In Section III, we demonstrate the\ngenerality of the ODFMR eﬀect by measuring it in three\nferromagnets: permalloy (Py), cobalt (Co), and cobalt\nmanganese iron germanium (CMFG) [32], a Heusler al-\nloy of interest for read heads; this helps us probe the\nmechanism by measuring the variation of the signal with\nmaterial properties. Finally, in Section IV, we discuss\nour conclusions and key future experiments.\nI: DEMONSTRATION OF ODFMR IN SINGLE\nCRYSTAL NV-PY SYSTEM AND DEPENDENCE\nOF SIGNAL ON NV-FM SEPARATION\nTo unravel the origin of the ODFMR signals, we study\nits dependence on the controlled separation dbetween\nthe NV center spins and the surface of a continuous FM\nﬁlm under microwave drive. If a dipolar spinwave mech-\nanism is relevant, then we expect that spinwaves with\na wavevector on the order of 2 π/dwill have the largest\nstray ﬁeld at the NV center position. Fig. 1 presents\na schematic of the sample and the geometry used for\nmeasurements presented in Figs. 2 and 3. The key ele-\nments are a single crystal diamond substrate with a layer\nof NV centers implanted a distance dfrom the surface.\nThere are also NV centers at a lower density through-out the volume of the diamond, but can be experimen-\ntally distinguished from the implanted NV centers, as\ndiscussed in the supplementary information (SI). We de-\nposited Py on the diamond surface and then patterned\nan electrically isolated microstripline antenna to produce\na microwave magnetic ﬁeld, H1. To record ODFMR, we\nmonitor the ﬂuorescence from implanted NV centers as\na function of the static magnetic ﬁeld H0and the mi-\ncrowave frequency, fmw, ofH1.H0can be oriented in\nthe plane of the Py ﬁlm or along one of the NV center\naxes. In addition, we also record the microwave power\nreﬂected by the microstripline, S11, providing an alter-\nnative, spatially averaged measure of FMR. Additional\ndetails of each measurement are provided in the SI.\nIn Fig. 2(a)-(c), we show ODFMR measurements of\nPy ﬁlms. The continuous wave (CW), normalized ﬂuo-\nrescence change (∆FL) is measured using a lock-in am-\npliﬁer that is referenced to amplitude modulation of H1.\nWe recorded ∆FL as a function of H0andfmwfor three\nsingle crystal (100) diamond substrates with d= 25 nm,\n50 nm, and 100 nm. Black circles overlaid on the 2D color\nplots show FMR peaks detected by S11. The features in\n∆FL appearing near 2 .9 GHz are the directly-driven\nground-state resonances of the NV centers, marked with\na guide-to-the-eye. With H0aligned in the plane of the\nsample along the /angbracketleft100/angbracketrightaxes, the four orientations of NV\ncenter/angbracketleft111/angbracketrightsymmetry axes are degenerate in this mea-\nsurement set-up. The feature around 1.2 GHz-1.6 GHz\nand extending over the measured ﬁeld range is due to\nthe NV center excited-state spin resonances, which are\nbroader than ground-state, and thus overlap [33]. The\nODFMR peaks due to the Py FMR appear most clearly\nin the low-ﬁeld, low-frequency regime. Fig. 2(d) also\nshows that as dincreases, the intensity of the ODFMR\nsignal also increases. Furthermore, the ODFMR signal\nis consistent with the direct measurement of S11, shown\nalong the black points. However, as a subtle but intrigu-\ning point, the ODFMR peaks are centered at lower ﬁelds\nthan theS11detected FMR peaks. This may merely\nbe related to diﬀerences in the local and global FMR,\nsince we probe NV centers at the narrowest point of the\nstripline, and thus they experience a larger H1than the\nmajority of the FM. Alternatively, this may provide valu-\nable clues as to the spectral eﬃciency of spinwave gener-\nation that produce ODFMR.\nAs discussed above, we hypothesize that ODFMR re-\nsults from ﬂuctuating dipolar ﬁelds due to spinwaves that\ndecay from the driven, uniform FMR mode. Here we\ndiscuss the spinwave dispersion in relation to FMR fre-\nquency. Fig. 2(e) shows representative dispersions for\nthe spin waves in Py for uniform mode FMR frequen-\ncies of 1 GHz, 1.5 GHz, and 2 GHz. The uniform mode\ncan decay into spinwaves with frequencies and wavevec-\ntors that are allowed by this dispersion [34]. We use this\ndispersion to estimate the wavelength (see the SI for de-\ntails),λres, of the spinwave whose frequency is resonant3\nODFMR Peak Intensity\n2.5 2.0 1.5 1.0 0.5 0.0\nFrequency (GHz)d = 100 nm \nd = 50 nm \nd = 25 nm (d)\n1086420\nField (mT)d = 100 nm (c)\n1086420\nField (mT)d = 50 nm (b)\n3\n2\n1Frequency (GHz)\n1086420\nField (mT)d = 25 nm (a)\n /s68FL\n3.0\n2.5\n2.0\n1.5\n1.0\n0.5Frequency (GHz)\n25x106 20 15 10 5 0\nWavevector (m-1)ωFMR = 1 GHzωFMR = 1.5 GHzωFMR = 2 GHz\nλres = 2π/kresNV GS (e)\n0  ODFMR Peak Intensity\n280 260 240 220 200\nλres - d (nm)0.811.21.41.61.822.2Frequency (GHz)\nVarying d  at constant\n        λres = 302 nm\nVarying  λres via fmw \n        (see top axis); d  = 100 nm(f)\nd = 100 nm \nd = 50 nm \nd = 25 nm \nFigure 2. ODFMR in Py-single crystal diamond and its dependence on d.CW NV center ∆FL vs H0andfmw\nsignals from three single crystal diamonds with NV centers implanted at (a) 25 nm, (b) 50 nm, (c) 100 nm, and capped by\na Py ﬁlm. Black dots indicate the FMR resonance measured inductively. The black dashed lines are a guide to the eye for\nthe ground-state resonances of the NV centers. (d) ∆FL at the ODFMR peak as a function of frequency for three values of\nd. The dashed line indicates the position of the peaks in ∆FL observed at 1 GHz in the three diamond samples, which are\nre-plotted in panel (f) as red points. Additional ODFMR spectra are shown in the SI. (e) The spinwave dispersion for a 20\nnm Py ﬁlm at the FMR ﬁelds for fmw= 1 GHz, 1.5 GHz, and 2 GHz. The solid horizontal lines indicate the frequency of the\nNV center resonance at a ﬁeld corresponding to the respective fmw. The intersection of the spin wave dispersion with the NV\ncenter ground state frequency determines the spinwave wavevectors which most eﬃciently couple to NV centers under FMR\nexcitation. The wavelength of these spin waves, λres, changes as a function of the FMR frequency. See main text for more\ndetails. (f) ODFMR peak intensity at the FMR condition of the ﬁlm versus the normalized distance λres−d, which is varied\neither by adjusting the FMR frequency (and thus λres, blue), or by varying din a series of samples (red). The red data show\nthe ODFMR intensity at 1 GHz (see the dashed line in panel (d)), corresponding to λres= 302 nm, plotted as the implantation\ndepth,dis varied. The blue data are the ODFMR intensity at ﬁxed d= 100 nm (the same data as the blue line in panel (d)),\nplotted versus the distance λres−d, which is varied by changing the frequency of FMR. λresis calculated from the dispersion\nas in panel (e), and the corresponding frequencies are plotted on the top axis. Horizontal error bars are given by the FWHM\nof ion straggle, extracted from SRIM calculations. All points in blue have the horizontal error bar of the d = 100 nm straggle.4\nwith the|0/angbracketright→|− 1/angbracketrightNV center spin transition for each\nfrequency of the uniform mode FMR.\nTo summarize the importance of matching λrestod,\nin Fig. 2(f) we present the measured ODFMR peak\nintensity from Fig. 2(d), re-plotted as a function of\nλres−d. This diﬀerence can be varied either through\nλres, by changing the uniform FMR resonance frequency\n(e.g. by changing the magnetic ﬁeld), or through d, by\nchanging the NV center implantation depth. The blue\npoints show the intensity at d= 100 nm, but plotted\nversusλres−dusing the spinwave dispersion to convert\nthe horizontal coordinate. The correlation between λres\nand frequency is shown in the top axis. The red points\nare measured using a 1 GHz microwave drive, so λres=\n302 nm for each of the three samples ( d= 25 nm, 50 nm,\n100 nm). The variation of the ODFMR signal obtained\nby these two methods shows the qualitative trend that\nwe expect from our hypothesis: the signal decreases as\nthe diﬀerence between λresanddgrows, regardless of the\nmethod by which λres−dis changed. Note that we have\nremoved the data in the extremely low-ﬁeld regime in\nwhich the ﬁlm is in a multi-domain state. Additionally,\nwe point out other eﬀects that appear in the ODFMR\nsignal. For example, the peak around 210 nm arises from\nthe coincidence of the NV center excited-state resonance\nand the FMR frequency, while the dip around 230 nm\nis an artifact due to reduced H1arising from a standing\nwave resonance in our microwave circuit.\nII: EFFECT OF FMR ON NV CENTER\nLONGITUDINAL SPIN-RELAXATION TIME\nAND T 1SPECTROSCOPY\nFluctuating dipolar ﬁelds of spinwaves will enhance the\nNV spin relaxation rate [24]. Here we present T1mea-\nsurements that demonstrate a reduction of the NV center\nT1in response to excitation of FMR.\nWe expect that the ﬂuorescence change in ODFMR\ncan be interpreted as a change in the NV center spin\npopulations as we drive FMR. By measuring the longi-\ntudinal spin-relaxation lifetime of the NV center ground\nspin state with and without FMR, we can quantify how\ndriving FMR modiﬁes the NV spin populations. Using\nthed= 50 nm sample described above, we ﬁrst char-\nacterize CW ODFMR in a diﬀerent measurement set-up\nthat is conﬁgured to measure the lifetimes of the NV cen-\nters, shown in Fig. 3. In this setup, H0is applied along\none of the/angbracketleft111/angbracketrightNV center axes and the ﬂuorescence is\nmonitored using a single photon counter. For this ﬁeld\norientation the four NV center ground state branches are\nvisible, in contrast to the two for in-plane ﬁeld (Fig. 2).\nNext we perform a T1spectroscopy measurement using\nthe pulse sequence shown in Fig. 3(c). The NV cen-\nters are ﬁrst initialized in the |0/angbracketrightspin state with a laser\npulse, and following a ﬁxed delay time of 3 µs, we pulse\n(a)\n(b)(c)\n(d)\n(e)Normaliz ed Fluor escence Frequenc y (GHz) Frequency (GHz)\nField (mT) Delay (μs)Vary MW Freq\n3 μsT1spectroscopy\nInitialization Read out MW\nT1measurem ent\nVary DelayInitialization Read out MWNormaliz ed \nFluorescence Normaliz ed \nFluorescenc e CW-NV\nT1 spectroscopyFigure 3. Eﬀect of FMR on longitudinal spin-\nrelaxation time. (a) CW-NV center ﬂuorescence as a func-\ntion of applied ﬁeld. The white dots show FMR measured\nwith reﬂected microwave power ( S11). (b)T1spectroscopy\n(3µs dark time) as a function of applied ﬁeld. The white\nopen circles show the locations of the measurements in (e).\nPanel (c) shows pulse sequences for T1spectroscopy and (d)\nT1measurement. (e) T1measurements with microwave at the\nFMR frequency on (red) and oﬀ (blue).\nthe laser again for ﬂuorescence readout. During the 3 µs\ndark time, a microwave pulse is applied with the same\npower as in Fig. 3(a). We sweep the microwave fre-\nquency at ﬁxed ﬁeld, then repeat the T1spectroscopy\nmeasurement as a function of applied ﬁeld, as shown in\nFig. 3(b).\nT1spectroscopy is an eﬀective approach to quickly ﬁnd\nthe conditions in which T1changes. Two main features\nare displayed in Fig. 3(b). First, when the FMR oc-\ncurs, we observe a signiﬁcant reduction in the NV center\nT1. As in Fig. 3(a), the FMR signal is pronounced in\nthe low ﬁeld/frequency region. At high ﬁeld/frequency\nwhere the FMR frequency exceeds the NV center reso-\nnance frequency, the FMR signal gradually diminishes.\nThe second feature in Fig. 3(b) is that the background\nofT1spectroscopy increases with increasing ﬁeld. The\nﬁeld dependent T1spectroscopy background is consistent\nwith a separate measurement of T1vs. ﬁeld without the\nmicrowave drive. The eﬀect has been previously stud-\nied in an undriven Py/diamond system where it was at-\ntributed to relaxation by thermal ﬂuctuations of the Py\nmagnetization that produces wide spectrum spin noise\n[24]. These measurements directly demonstrate that the5\n2.5\n2.0\n1.5\n1.0\n0.5Frequency (GHz)\n5.0 2.5 0.0\n Field (mT)(c) Permalloy\n5.0 2.5 0.0\nField (mT)(b) CMFG\n5.0 2.5 0.0\nField (mT)(d) Cobalt\n600\n400\n200\n0λres (nm)\n3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0\nFrequency (GHz)Co\nCMFGPy\nYIG(e)\nIncreasing fd\n3020100\nField (mT)2.5\n2.0\n1.5\n1.0\n0.5Frequency (GHz)YIG (a)\n ΔFL\nFigure 4. The eﬀect of material properties on ODFMR.\n∆FL as a function of ﬁeld and frequency of NV centers in a\nnanodiamond (ND) ﬁlm on top of various ferromagnetic ﬁlms.\nInset: the device schematic for the samples measured using\nnanodiamonds; see SI for details. In (a)-(d), four diﬀerent fer-\nromagnetic ﬁlms are shown, YIG, CMFG, Py, and Co respec-\ntively. The dashed white boxes are a guide for the frequency,\nfd, for which the ODFMR signal decays. The data for YIG\nare from our previous publication [29], and no decay frequency\nis seen in the measured frequency range. (e) The optimal dis-\ntance for the NV centers determined from matching of the\nspinwave modes to the NV resonance condition as a function\nof frequency for Co, Py, CMFG, and YIG. This is calculated\nas in Fig. 2 (e). Changes in the saturation magnetization and\nthe exchange stiﬀness aﬀect the spin wave dispersion, which\nin turns aﬀects the spin wave wavelength resonant with the\nNV ground state. The horizontal black dashed line indicates\nthe largest NV-FM separation, d, encountered in our nanodi-\namond ﬁlms. For λreslarger than the maximum d, the signal\nintensity will decay; the corresponding calculated frequency\nmaximum is shown by the vertical dashed lines. These data\nare suggestive of qualitative agreement between the decay fre-\nquency and the frequency at which λressurpassesd.\nODFMR signal arises due to ﬂuctuating dipolar ﬁelds\ngenerated by a driven ferromagnetic system that relax\nthe ground-state NV center spins, without requiring par-\nticipation of the NV center excited state.\nTo better quantify the T1change during FMR, we also\ndirectly measure T1with and without a microwave drive.We choose three points in Fig. 3(b) for T1measurement,\nas shown in Fig. 3 (e). When the FMR condition is\nmet, a large reduction in T1is observed, in agreement\nwith ourT1spectroscopy results. For example, at 40 G\n(curve (i) in Fig. 3(e)) the measured T1= 3.8±0.1µs\nwith no microwave applied. However, when a microwave\nof 2.19 GHz is applied to satisfy the FMR condition,\nT1= 2.1±0.2µs, which has reduced nearly by a fac-\ntor of 2. Despite the fact that the FMR frequency is\nnot resonant with the NV center spin resonances, the T1\nreduction during FMR suggests an incoherent coupling\nmechanism between NV centers and the nearby driven\nFM. This hypothesis pertains to the ∆FL as a result of\ndriving the uniform mode. Changes in the ﬂuorescence of\nNV centers as a result of thermal equilibrium spin waves\nin a nearby Py disk in the undriven case have already\nbeen investigated [24].\nIII: DEMONSTRATION OF ODFMR IN\nDIFFERENT FERROMAGNETS AND\nDEPENDENCE OF THE SIGNAL ON MATERIAL\nPROPERTIES\nThe spinwave dispersion is sensitive to saturation mag-\nnetization, Ms, and exchange stiﬀness, A, making this\nphenomenon sensitive to material-speciﬁc details of the\nFM. Here we describe diﬀerences in ODFMR as a func-\ntion of material properties, and their relationship to our\nproposed mechanism.\nThe sample structure is shown in the inset of Fig. 4 and\nadditional details can be found in the SI. ∆FL collected\nfrom a nanodiamond ﬁlm was measured using a lock-\nin ampliﬁer and is presented in Fig. 4 as a function of\nin-planeH0andfmwfor four samples: a 30 nm YIG\nﬁlm on GGG from [29], a 20 nm Py ﬁlm on Si, a 5 nm\nCMFG ﬁlm on glass, and a 20 nm Co ﬁlm on Si. The\nwhite dashed boxes highlight the frequencies, fd, above\nwhich the ODFMR signal decays for each material. This\nfrequency is lowest for the Co ODFMR signal, with fd\n∼0.7 GHz, and it is larger for Py and CMFG which\nexhibitfd∼1.8 GHz, while the YIG ODFMR signal\ndoes not strongly decay in this frequency range. Note\nthat the ﬁeld scale in panel (b)-(d) is smaller than in (a).\nIn Fig. 4(e) the calculated λresis plotted as a function\nof the FMR frequency for each material. The values of\nMsandAin the various materials aﬀect their spinwave\ndispersions, and hence the wavelengths of the spinwaves\nthat are resonant with the NV center ground state spin\ntransition. The horizontal black dashed lines indicate\nthe value of dthat is expected based on the thickness\nof the nanodiamond ﬁlms. When λresbecomes larger\nthan this limiting value of d, we expect that the signal\nintensity will decay. The estimated decay frequencies are\nshown by the vertical dashed lines. We ﬁnd that fd(Co)\n< f d(Py)≈(CMFG)< f d(YIG), which matches the6\nexperimental trend for the ODFMR cut-oﬀ frequency in\neach material.\nIV: CONCLUSIONS AND OUTLOOK\nWhile our hypothesis qualitatively explains the re-\nsults presented here and in the previous work [29], a\nmore quantitative theory of the NV-FM coupling is still\nneeded. Noise arising from the incoherent bath of spin-\nwaves reduces the NV center spin lifetime as shown in\nFig. 3. Thus measuring the spin lifetime T1of the NV\ncenters on and oﬀ FMR as a function of dis an important\nfuture step. The change of T1is related to the strength\nof the ﬂuctuating dipolar ﬁelds generated by these spin-\nwaves, which can be calculated theoretically and be quan-\ntitatively compared with our measurements. A more\ndetailed theory will also include the spectral density of\nspinwaves generated by the decay of the uniform mode.\nAn important aspect of this work is the possible avenues\nit highlights for experimentally characterizing damping\nprocesses in ferromagnets, such as magnon-magnon inter-\nactions, which are likely responsible for generating higher\nenergy spinwaves from the uniform mode. Understand-\ning this phenomenon could lead to a new method for\ndeducing the rate at which spinwaves are generated by\nmagnetization dynamics of ferromagnets, with a sensi-\ntivity that can be selected by controlling the separation\nbetween the NV centers and FM surface to match the de-\nsired properties of spinwave to be measured. Extending\nthis idea to scanned-probe sensing in dynamical magnetic\nsystems will oﬀer a unique modality for nanoscale mag-\nnetic imaging.\nIn conclusion, we have demonstrated the optical detec-\ntion of ferromagnetic resonance in a variety of ferromag-\nnets using NV centers. We hypothesize that spinwaves\ngenerated by decay of the uniform mode that match the\nNV center resonance are responsible for altering the NV\ncenter spin lifetime, thus generating an ODFMR signal.\nAlthough our data qualitatively support this hypothesis,\nit requires further measurements of the NV spin lifetime\nand a detailed theoretical analysis.\nWe thank Dr. Sergei Manuilov for helpful discussion.\nFunding for this research at The Ohio State Univer-\nsity was provided by the ARO through award number\nW911NF-12-1-0587 and the Center for Emergent Ma-\nterials at the Ohio State University, an NSF MRSEC\nthrough award Number DMR-1420451. We acknowledge\nuse of Ohio State Nanosystems Laboratory shared facil-\nities. Research at Cornell is supported in part by the\nAFOSR (grant # FA9550-14-1-0243). We acknowledge\nuse of the shared facilities of the Cornell Center for Ma-\nterials Research under grant DMR-1120296.∗hammel@physics.osu.edu\n†gdf9@cornell.edu\n[1] S. D. Bader and S. S. P. Parkin, Annual Review of Con-\ndensed Matter Physics 1, 71 (2010).\n[2] I. ˇZuti´ c, J. Fabian, and S. Das Sarma, Reviews of Mod-\nern Physics 76, 323 (2004).\n[3] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys-\nical Review Letters 88, 117601 (2002).\n[4] J. C. Slonczewski, Journal of Magnetism and Magnetic\nMaterials 159, L1 (1996).\n[5] D. C. Ralph and M. D. Stiles, Journal of Magnetism and\nMagnetic Materials 320, 1190 (2008).\n[6] Z. Qiu, K. Ando, K. Uchida, Y. Kajiwara, R. Takahashi,\nH. Nakayama, T. An, Y. Fujikawa, and E. Saitoh, Ap-\nplied Physics Letters 103, 092404 (2013).\n[7] R. Adur, C. Du, H. Wang, S. A. Manuilov, V. P. Bhal-\nlamudi, C. Zhang, D. V. Pelekhov, F. Yang, and P. C.\nHammel, Physical Review Letters 113, 176601 (2014).\n[8] R. Adur, C. Du, J. Cardellino, N. Scozzaro, C. S. Wolfe,\nH. Wang, M. Herman, V. P. Bhallamudi, D. V. Pelekhov,\nF. Yang, and P. C. Hammel, Journal of Applied Physics\n117, 172604 (2015).\n[9] A. V. Vashkovskii, ´E. G. Lokk, and V. I. Shcheglov,\nJournal of Experimental and Theoretical Physics Letters\n63, 572 (1996).\n[10] Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida,\nM. Mizuguchi, H. Umezawa, H. Kawai, K. Ando,\nK. Takanashi, S. Maekawa, and E. Saitoh, Nature 464,\n262 (2010).\n[11] S. O. Demokritov, Spin Wave Conﬁnement (Pan Stan-\nford Publishing, 2009).\n[12] V. N. Krivoruchko, Low Temperature Physics 41, 670\n(2015).\n[13] J. R. Childress, M. J. Carey, S. Maat, N. Smith, R. E.\nFontana, D. Druist, K. Carey, J. A. Katine, N. Robert-\nson, T. D. Boone, M. Alex, J. Moore, and C. H. Tsang,\nIEEE Transactions on Magnetics 44, 90 (2008).\n[14] S. S. P. Parkin, K. P. Roche, M. G. Samant, P. M.\nRice, R. B. Beyers, R. E. Scheuerlein, E. J. OSullivan,\nS. L. Brown, J. Bucchigano, D. W. Abraham, Y. Lu,\nM. Rooks, P. L. Trouilloud, R. A. Wanner, and W. J.\nGallagher, Journal of Applied Physics 85, 5828 (1999).\n[15] S. Ikeda, K. Miura, H. Yamamoto, K. Mizunuma, H. D.\nGan, M. Endo, S. Kanai, J. Hayakawa, F. Matsukura,\nand H. Ohno, Nature Materials 9, 721 (2010).\n[16] S. S. P. Parkin, M. Hayashi, and L. Thomas, Science\n320, 190 (2008).\n[17] N. Nagaosa and Y. Tokura, Nature Nanotechnology 8,\n899 (2013).\n[18] Y. Tokunaga, X. Z. Yu, J. S. White, H. M. Rnnow,\nD. Morikawa, Y. Taguchi, and Y. Tokura, Nature Com-\nmunications 6, 7638 (2015).\n[19] A. Fert, V. Cros, and J. Sampaio, Nature Nanotechnol-\nogy8, 152 (2013).\n[20] R. Schirhagl, K. Chang, M. Loretz, and C. L. Degen,\nAnnual Review of Physical Chemistry 65, 83 (2014).\n[21] G. Balasubramanian, I. Y. Chan, R. Kolesov, M. Al-\nHmoud, J. Tisler, C. Shin, C. Kim, A. Wojcik, P. R. Hem-\nmer, A. Krueger, T. Hanke, A. Leitenstorfer, R. Brats-\nchitsch, F. Jelezko, and J. Wrachtrup, Nature 455, 648\n(2008).7\n[22] J. R. Maze, P. L. Stanwix, J. S. Hodges, S. Hong, J. M.\nTaylor, P. Cappellaro, L. Jiang, M. V. G. Dutt, E. Togan,\nA. S. Zibrov, A. Yacoby, R. L. Walsworth, and M. D.\nLukin, Nature 455, 644 (2008).\n[23] J.-P. Tetienne, T. Hingant, J.-V. Kim, L. H. Diez, J.-\nP. Adam, K. Garcia, J.-F. Roch, S. Rohart, A. Thiav-\nille, D. Ravelosona, and V. Jacques, Science 344, 1366\n(2014).\n[24] T. van der Sar, F. Casola, R. Walsworth, and A. Yacoby,\nNature Communications 6, 7886 (2015).\n[25] A. Dussaux, P. Schoenherr, K. Chang, N. Kanazawa,\nY. Tokura, C. L. Degen, and D. Meier, arXiv:1503.06622\n[cond-mat] (2015).\n[26] J. M. Taylor, P. Cappellaro, L. Childress, L. Jiang,\nD. Budker, P. R. Hemmer, A. Yacoby, R. Walsworth,\nand M. D. Lukin, Nature Physics 4, 810 (2008).\n[27] P. Wang, Z. Yuan, P. Huang, X. Rong, M. Wang, X. Xu,\nC. Duan, C. Ju, F. Shi, and J. Du, Nature Communica-\ntions 6, 6631 (2015).[28] P. Appel, M. Ganzhorn, E. Neu, and P. Maletinsky, New\nJournal of Physics 17, 112001 (2015).\n[29] C. S. Wolfe, V. P. Bhallamudi, H. L. Wang, C. H. Du,\nS. Manuilov, R. M. Teeling-Smith, A. J. Berger, R. Adur,\nF. Y. Yang, and P. C. Hammel, Physical Review B 89,\n180406 (2014).\n[30] C. S. Wolfe, S. A. Manuilov, C. M. Purser, R. Teeling-\nSmith, C. Dubs, P. C. Hammel, and V. P. Bhallamudi,\nApplied Physics Letters 108, 232409 (2016).\n[31] H. Schultheiss, K. Vogt, and B. Hillebrands, Physical\nReview B 86, 054414 (2012).\n[32] M. R. Page, T. M. Nakatani, D. A. Stewart, B. R. York,\nJ. C. Read, Y.-S. Choi, and J. R. Childress, Journal of\nApplied Physics 119, 153903 (2016).\n[33] G. D. Fuchs, V. V. Dobrovitski, R. Hanson, A. Batra,\nC. D. Weis, T. Schenkel, and D. D. Awschalom, Physical\nReview Letters 101, 117601 (2008).\n[34] R. A. Lukaszew, ed., Handbook of Nanomagnetism: Ap-\nplications and Tools (2015).Supplementary Material: Optically Detected Ferromagnetic Resonance in Metallic\nFerromagnets via Nitrogen Vacancy Centers in Diamond\nM. R. Page*,1F. Guo*,2C. M. Purser,1J. G. Schulze,1T. M. Nakatani,3C. S.\nWolfe,1J. R. Childress,3P. C. Hammel,1,∗G. D. Fuchs,2,†and V. P. Bhallamudi1\n1Department of Physics, The Ohio State University, Columbus, Ohio 43210, USA\n2School of Applied and Engineering Physics, Cornell University, Ithaca NY 14850\n3San Jose Research Center, HGST, a Western Digital company, San Jose, California 95135, USA\n(Dated: July 27, 2016)\nI. CREATION OF SINGLE CRYSTAL\nSUBSTRATES WITH THIN IMPLANTED LAYER\nOF NV CENTERS\nThe single crystal diamond samples are 3 mm x 3 mm\nx 0.3 mm optical grade, CVD diamonds from Element\n6 with [N]<1ppm, [B]<0.05, (100) oriented faces, and\n<100>oriented edges. The diamonds were implanted\nwith a thin layer of NV centers by bombardment with\n15N at 18 keV, 40 keV, and 87 keV to achieve mean\ndepths of 25±8 nm, 50±13 nm, and 100 ±20 nm\naccording to SRIM simulations. The 25 nm-deep sam-\nple was implanted with ﬂuence 3.53 x1013atoms/cm2\nto achieve a peak nitrogen density of 100 ppm. The 50\nand 100 nm sample doses were 1.07x1014atoms/cm2and\n1.54x1014atoms/cm2to achieve peak nitrogen densities\nof 200 ppm. In order to convert implanted nitrogen cen-\nters into NV- centers, the samples were annealed at 915\nC for 2 hours in an atmosphere of approximately 350\nmTorr Ar and 50 mTorr H 2gas.\nWe note that there are native NV centers throughout\nthe entire diamond crystal prior to the implantation. The\nion implantation introduced an additional layer of NV\ncenters. The implanted and native NV centers can be\ndistinguished by measuring the depth dependent ﬂuores-\ncence, as shown in Fig. S1. The normalized ﬂuorescence\nintensity of NV centers is recorded as a function of the\nobjective lens position. When the green pump laser is\nfocused well inside the bulk of the diamond crystal, cor-\nresponding to a negative position of the objective lens,\nwe detect a nearly constant ﬂuorescence independent of\nthe objective’s position due to the uniform distribution of\nnative NV centers inside the diamond crystal. While the\nfocus of the green light approaches the diamond surface\nwhere the additional layer of NV centers is implanted,\na ﬂuorescence peak is observed. As the objective lens\ncontinues to move towards positive direction, the ﬂuores-\ncence level quickly decays to zero as the pump laser is\nfocused outside of the diamond. We point out that the\nwidth of the ﬂuorescence peak is limited by the depth\n* M. R. Page and F. Guo contributed equally to this work.\n∗hammel@physics.osu.edu\n†gdf9@cornell.edu\n 0 0.2 0.4 0.6 0.8 1\n-30 -20 -10  0  10  20Normalized Fluorescence\nObjective Lens Position (µm)Figure S1. Fluorescence measured as a function of objective\nlens position.\nof focus, and it is much wider than the true distribution\nof the implanted NV centers. For the data of Fig. 3 in\nthe main text, the position of the objective is ﬁxed at 0.\nTherefore the majority of the ﬂuorescence signal comes\nfrom the implanted layer of the NV centers.\nII. FERROMAGNETIC FILM DEPOSITION\nAND FABRICATION OF MICROWAVE\nANTENNA\nA schematic of the single crystal diamond samples is\nshown Fig. S2, and an optical image of the diamond\nwithd= 50 nm is shown in Fig. S3. For these samples,\na 20 nm thin layer of Py is ﬁrst deposited by electron\nbeam evaporation on the surface of the diamond near the\nimplanted NV center layer. After using RF sputtering for\nthe deposition of 150 nm of an insulating SiO 2layer on\ntop of the Py, a microstrip is lithographically patterned\non the sample to apply a microwave frequency magnetic\nﬁeld. The microstripline consists of 5 nm Ti, 285 nm\nAg, and 10 nm of Au all deposited by electron beam\nevaporation. For the nanodiamond samples, the metallic\nferromagnetic ﬁlms are deposited on a suitable substrate.\nPy and Co are deposited by electron beam evaporation\nto a thickness of 20 nm on Si and CMFG is sputtered\nto a thickness of 5 nm on glass. The 150 nm SiO 2layer\nand the 300 nm antenna are patterned on top of the\nferromagnet. A ﬁlm of nanodiamonds of approximatelyarXiv:1607.07485v1  [cond-mat.mes-hall]  25 Jul 20162\nFigure S2. Experimental schematic: The sample is a 20 nm\nPy ferromagnetic ﬁlm deposited on a single crystal diamond\nwith an implanted layer of NV centers 25-100 nm from the\nsurface. In order to apply microwave magnetic ﬁelds to the\nsample, a microwire (5 nm Ti/ 300 nm Ag) is grown on an\ninsulating SiO 2layer. Green laser light is focused through the\nback of the diamond and the changes in the resulting ﬂuores-\ncence of the NV centers is monitored as a function of applied\nexternal magnetic ﬁeld and microwave excitation frequency.\nThe applied ﬁeld H0can be applied either in the ﬁlm plane\nor along one axis of the NV center.\nFigure S3. Optical image of the d= 50 nm single crystal\ndiamond sample with Py and miccrostripline. The Py ﬁlm\nand SiO 2layer cover the top surface of the diamond. The\npatterned microstripline can be seen in gold.\n500 nm in thickness is drop-cast on the sample above the\nferromagnetic ﬁlm.\nIII. CONTINUOUS WAVE LOCK-IN BASED\nMEASUREMENTS\nFor the CW measurement of ODFMR in the single\ncrystal and nanodiamond samples from Figs. 2 and 4 in\nthe main text, ﬂuorescence is excited in the NV centers\nusing a 520 nm laser from Coherent model OBIS 520 LX,\nfocused down <2µm spot using a Nikon CFI Plan Fluor\n40x objective with 0.75 N.A. with 35 mW of incident\npower, and measured by a Laser Components A-CUBE\nS1500-10 avalanche photodiode. The NV center ﬂuores-\ncence is recorded at a spot under the microstrip for the\nsingle crystal samples, and next to the microstripline for\nthe nanodiamonds, as a function of the strength of the\nstatic magnetic ﬁeld H0, applied in-plane and parallel toTable S1. Optical components and measurement hardware\nLetter Description\nA pump laser\nB beam expander\nC dichroic mirror\nD objective lens\nE magnet\nF sample\nG color ﬁlters\nH 50 /50 mirror\nI lens\nJ camera\nK photodiode\nL DAQ\nM lock-in\nN MW source\nO signal generator\nP MW diode\nQ computer\nthe antenna, and the frequency, fmw, of the applied mi-\ncrowave ﬁeld H1. A shorted microstrip geometry is used\nto produce a microwave ﬁeld with a Wiltron 68147B sig-\nnal generator. For the single crystal diamonds, the mi-\ncrostripline is shorted and connected to a 50 Ω transmis-\nsion line by indium pressed wires, for the nanodiamonds,\nthe wires are wirebonded. We measure a lock-in signal\nusing a Signal Recovery 7256 for both the ﬂuorescence\nand the reﬂected microwave power ( S11) by modulating\nthe amplitude of H1at∼1 kHz.S11allows simultane-\nous inductive measurement of the FMR using a Krytar\n303BK diode, albeit spatially averaged. The raw voltage\nfrom the photodiode lock-in is normalized by dividing the\nvoltage at the NV center ground state resonance peak at\n0 ﬁeld at 2.9 GHz. A block diagram of the various optical\nand measurement hardware components used is shown in\nFig. S4 with description of each component in Table S1.\nIV. NV CENTER LIFETIME MEASUREMENTS\nThe measurements in Fig. 3 from the main text are\nperformed using a home-built confocal microscopy, as il-\nlustrated in Fig. S5. We use a 532 nm green laser for\ninitialization and read out. An ISOMET 1250C acousto-\noptic modulator (AOM) is used as an optical shutter.\nBoth excitation laser and NV center ﬂuorescence are\nfocused through an Olympus LCPLFLN 50x objective.\nThe ﬂuorescence emitted from NV centers is detected\nwith an Excelitas SPCM-AQR-H-13-FC avalanche pho-\ntodiode (APD) and a home made RF switch box. The\nRF magnetic microwave is generated by a SRS SG384\nsignal generator using IQ modulation. We use a perma-\nnent magnet to produce the magnetic ﬁeld. The magnet\nis mounted on a transitional stage and a home-built go-\nniometer to control the strength and orientation of the\nmagnetic ﬁeld.3\nFigure S4. Block diagram for the optical components and instrumentation used in the CW lock-in measurements. Figure\nadapted from [1]. See Table S1 for description of each component.\nFigure S5. Block diagram for the optical components used in the lifetime measurements.\nV. ODFMR SPECTRA\nSample ODFMR spectra are shown in Fig. S6. These\nspectra are line cuts through the 2D plots presented in\nthe main text. In addition, the ODFMR peak, marked\nwith triangles, is recorded as a function of frequency for\neach of the implantation depths for the analysis in the\nmain text. The position of the triangle is determined\nfrom Lorentzian ﬁts to the S 11detected FMR peak.\nVI. CALCULATION OF THE OPTIMAL\nWAVELENGTH\nAs discussed in the main text, we use the spinwave dis-\npersion to theoretically determine the wavelength, λres,\nof the spinwave whose frequency is resonant with the NVcenter for each frequency of the uniform mode FMR. This\nis calculated as follows. The dispersion of spin waves for\nthe uniform mode is given by [2] :\n/parenleftbiggω(k/bardbl)\nγ/parenrightbigg2\n= (ωFMR\nγ)2−HdπMstFk/bardbl+HeDsk2\n/bardbl(S1)\n(ωFMR\nγ)2=H(H+ 4πMeff) (S2)\nHd= 2(H−(H+ 4πMeff) sin2φk) (S3)\nHe= 2H+ 4πMeff (S4)4\nFigure S6. ∆FL as a function of ﬁeld for several frequencies for the diamond with d= 25 nm (a), d= 50 nm (b), and d=\n100 nm (c). The inverted triangles indicate the position of the ODFMR peak as determined from Lorentzian ﬁts to the S 11\ndetected FMR peak.\nwhereω(k/bardbl) is the frequency of the spin wave, γis the gy-\nromagnetic ratio, ωFMR is the frequency of the uniform\nmode,His the external magnetic ﬁeld, Msis the satu-\nration magnetization, tFis the ﬁlm thickness, Meﬀis the\neﬀective magnetization, k/bardblis the spinwave wavevector,\nDs= 2A/M swhereAis the exchange stiﬀness and φkis\nthe angle between the spinwave wavevector and magne-\ntization.\nFor each uniform mode FMR frequency, the Kittel equa-\ntion shown in Eqn. S2 above is used to determine the\nresonant ﬁeld. Using this resonant ﬁeld, the NV cen-\nter ground state resonance condition is determined by\nﬁnding the eigenvalues of the NV center Hamiltonian [3]\nbelow.\nHNV=γH·S+D(Sz·Sz) (S5)whereDis the 0 ﬁeld crystal splitting equal to approx-\nimately 2.87 GHz at room temperature, Sis the spin 1\nmatrix with components Sx,SyandSz, and/vectorHis the\nvector external magnetic ﬁeld deﬁned as\n/vectorH=H0(cos(θ) sin(φ)ˆi,sin(θ) sin(φ)ˆj,cos(φ)ˆk) (S6)\nThe constants θandφare the relative angles between\nthe NV center crystal directions and the applied mag-\nnetic ﬁeld which change with respect to each crystal axis.\nThere are four NV center axes that create four energy\nlevels that are degenerate with the ﬁeld in plane of the\nsingle crystal diamond. The frequency corresponding to\nthe 0 to -1 transition is used, since this is lower frequency\nand energy and thus more likely to occur. Using this\ncalculated NV center resonance frequency, the spinwave\nwavevector is calculated according to Eqn. S1 and the\nλresis taken to be equal to the associated wavelength.\n[1] C. S. Wolfe, Novel Techniques for Detection and Imag-\ning of Spin Related Phenomena: Towards Sub-Diﬀraction\nLimited Resolution , Ph.D. thesis, The Ohio State Univer-\nsity (2015).[2] R. A. Lukaszew, ed., Handbook of Nanomagnetism: Ap-\nplications and Tools (2015).\n[3] R. Schirhagl, K. Chang, M. Loretz, and C. L. Degen,\nAnnual Review of Physical Chemistry 65, 83 (2014)."    },    {        "title": "1905.06772v6.Element_specific_visualization_of_dynamic_magnetic_coupling_in_a_Co_Py_bilayer_microstructure.pdf",        "content": "Element-speci\fc visualization of dynamic magnetic\ncoupling in a Co/Py bilayer microstructure\nT. Feggeler1;\u0003, R. Meckenstock1, D. Spoddig1, C. Sch oppner1,\nB. Zingsem1;2, T. Scha\u000bers3, H. Ohldag4;\u0003\u0003, H. Wende1, M.\nFarle1;5, A. Ney3and K. Ollefs1\n1Faculty of Physics and Center for Nanointegration Duisburg-Essen (CENIDE),\nUniversity of Duisburg-Essen, 47048 Duisburg, Germany\n*Present address: Advanced Light Source, Lawrence Berkeley National Laboratory,\nBerkeley, CA, United States\n2Ernst Ruska Centre for Microscopy and Spectroscopy with Electrons and Peter\nGr unberg Institute, Forschungszentrum J ulich GmbH, 52425 J ulich, Germany\n3Institute of Semiconductor and Solid State Physics, Johannes Kepler University\nLinz, 4040 Linz, Austria\n4SLAC National Accelerator Laboratory, 94025 Menlo Park, CA, United States\nDepartment of Physics, University of California Santa Cruz, Santa Cruz CA 95064,\nUnited States\n**Present address: Advanced Light Source, Lawrence Berkeley National Laboratory,\nBerkeley, CA, United States and Department of Material Sciences and Engineering,\nStanford University, Stanford CA 94305, United States\n5Kirensky Institute of Physics, Federal Research Center KSC SB RAS, Russia\nE-mail: thomas.feggeler@uni-due.de\nAbstract. We present the element-speci\fc and time resolved visualization of\nuniform ferromagnetic resonance excitations of a Permalloy (Py) disk - Cobalt (Co)\nstripe bilayer microstructure. The transverse high frequency component of the\nresonantly excited magnetization is sampled in the ps regime by a combination of\nferromagnetic resonance (FMR) and scanning transmission X-ray microscopy (STXM-\nFMR) recording snapshots of the local magnetization precession of Py and Co with\nnanometer spatial resolution. The approach allows us to individually image the\nresonant dynamic response of each element, and we \fnd that angular momentum\nis transferred from the Py disk to the Co stripe and vice versa at their respective\nresonances. The integral (cavity) FMR spectrum of our sample shows an unexpected\nadditional third resonance. This resonance is observed in the STXM-FMR experiments\nas well and our microscopic \fndings suggest that it is governed by magnetic exchange\nbetween Py and Co, showing for the Co stripe a di\u000berence in relative phase of the\nmagnetization due to stray \feld in\ruence.arXiv:1905.06772v6  [cond-mat.mes-hall]  10 Feb 2022Element-speci\fc visualization of dynamic magnetic coupling in a Co/Py bilayer microstruct. 2\n1. Introduction\nFor future information technology new con-\ncepts are needed involving the charge of the\nelectron as well as its spin as information unit\n[1]. Several approaches for magnetism based\nlogic have been introduced, ranging from soli-\nton based concepts [2], to magnonics in the\nform of e.g. genetically engineered magnonic\ncomputing [3, 4] to overcome the various lim-\nitations, e.g. thermal load and energy needs,\nencountered by modern computer technology.\nThis \feld of spintronics and magnonics re-\nquires to study even smaller magnetic struc-\ntures in the gigahertz and terahertz regime.\nSpin-based devices usually consist of more\nthan one material, requiring the understand-\ning of the element-speci\fc dynamic magnetic\nproperties and the resulting spin wave modes\non the nanometer scale. X-ray detected ferro-\nmagnetic resonance (XFMR) [7, 8, 9, 10, 11,\n12, 13, 14, 15], combining ferromagnetic res-\nonance (FMR) with element speci\fc magne-\ntometry by means of X-ray Magnetic Circular\nDichroism (XMCD) (see [5, 6] and references\ntherein) is a unique tool to address this chal-\nlenge.\nIn this study Scanning Transmission X-\nray Microscopy detected FMR (STXM-FMR)\n[17] has been used, o\u000bering temporal sampling\ndown to 17 ps and nominal sub 50 nm lateral\nresolution in transverse XFMR geometry with\na continuous wave excitation of the sample\n[16, 17, 18, 19, 20]. Uniform and non uniform\nresonant responses on the micro- [21, 22]\nand sub 50 nm nanometer [23] scale have\nbeen monitored and analysed. Here we\ninvestigate resonant excitations of a bilayered\nmicrostructure consisting of a Cobalt (Co)\nstripe deposited on a Permalloy (Py) disk\nwith element speci\fcity. Earlier studies of\nequally dimensioned ultra-thin ferromagneticbilayers (thickness usually about or below\n10 nm) showed two uniform resonance modes,\ntypically explained as an in-phase and out-of-\nphase optical or acoustical modes e.g. [24]. In\nthe conventional FMR measurements of our\nbilayer microstructure with a total thickness\nof 60 nm the individual resonances of the\nPy and Co microstructures are identi\fed.\nIn addition, a third resonance in both\nmaterials is seen, which cannot be explained\nby the aforementioned approach for equally\ndimensioned ultra-thin bilayers, but by Py\nand Co resonating in phase as an entity,\nmediated by exchange coupling. Thus, by our\nspatially, time and element speci\fc STXM-\nFMR the origin of the three resonances is\nrevealed, visualizing as well local phase and\namplitude variations, which are not visible in\nconventional FMR spectra\n2. Experimental details\nWe measure FMR excitations in their linear\nregime using a micro-resonator based, element-\nspeci\fc and spatially resolved STXM-FMR\nsetup realized at Stanford Synchrotron Radi-\nation Lightsource (SSRL)[17, 18]. The sample\nis a polycrystalline Co stripe (2.0 \u0016m length,\n0.5\u0016m width, 30 nm thickness) deposited\non a polycrystalline Permalloy(Py)-disk with\n2.5\u0016m diameter and 30 nm thickness (see Fig.\n1a)). It is fabricated by a three step lithogra-\nphy and electron-beam deposition of the ferro-\nmagnetic material [25] on a 200 nm thick Si 3N4\nmembrane. To measure the FMR spectrum the\nsample is positioned in the omega-shaped loop\nof a micro-resonator o\u000bering a sensitivity of 106\n\u0016B[26, 27, 28]. The sample is excited by a ho-\nmogeneous linearly polarized microwave \feld\nwith an amplitude of \u00141.5 mT. A STXM im-\nage of the sample using a step size of 100 nm\nis shown in Fig. 1b).Element-speci\fc visualization of dynamic magnetic coupling in a Co/Py bilayer microstruct. 3\nFig. 1c) pictures the conventional\nFMR spectrum of the sample obtained at\na microwave frequency of 9.27 GHz in a\nmagnetic \feld B Ext=0 - 200 mT featuring\nthree main resonances, the \frst resonance\nat B Ext,1=58.3 mT, the second resonance at\nBExt,2=84.9 mT, and the third resonance at\nBExt,3=112.7 mT. Resonances 1 and 3 are\nmodes of the Py disk and the Co stripe,\nrespectively. Magneto-crystalline anisotropy\nis neglible in both samples due to their\npolycrystallinity. The Co stripe exhibits\nthe highest resonance \feld due to shape\nanisotropy despite its high M sat= 1420 kA/m\n[31] considering the stripe geometry and the\nperpendicular orientation of its long side to\nBExt(Fig. 1a), while Py with M sat= 860 kA/m\n[31] shows the lowest resonance \feld. This\nis con\frmed by angular dependent FMR\nmeasurements of a Co stripe [28]. The origin\nof the intermediate resonance 2, however, can\nnot readily be understood, because one would\nonly expect the two individual resonances.\nThe presence of a third resonance at an\nintermediate \feld strongly suggests that Co\nand Py resonate as one entity resulting in\na coupled uniform resonance. Although it\nappears reasonable to assume that this is due\nto exchange coupling across the interface, it is\nnot possible to directly deduce the microscopic\nmechanism behind our observation from the\nclassical FMR spectrum. To elucidate this, we\nuse STXM-FMR.\nIn STXM-FMR the sample is mapped\nby a focussed X-ray beam (energy tunable\nbetween 200 eV and 1200 eV at the SSRL),\nwhile the transmitted intensity is detected by\nan avalanche X-ray photodiode. B extis applied\nin the sample plane along the short axis of\nthe Co stripe (Fig. 1a)) with perpendicular\norientation to the incident circularly polarized\nX-rays. The time-dependent transverse\n0.5 µm2 µm20 µm2.5 µmCoPySEMa)\n-1001234\n255015010075125175200b)STXMc)Figure 1. a) Scanning electron microscopy (SEM)\nimage of the Co stripe/Py disk bilayer on a Si 3N4\nmembrane. The orientation of B extis indicated; b)\nSTXM image of the sample in the micro-resonator\nloop; The high-frequency magnetic \feld oscillates in\nthe out-of-plane direction c) FMR spectrum of the\nsample shown in a) with four major resonances, 1: Py\nresonance, 2: Coupled resonance, 3: Co stripe center\nresonance, 4: Resonance of the long sides of the Co\nstripe.\ncomponent of the magnetization at 9.129 GHz\nis probed by means of the XMCD e\u000bect, for\ndetails see [17]. The magnetization oscillation\nis sampled with 6 consecutive images separated\nby a static phase di\u000berence of 60\u000e(18 ps),\neach with and without applied microwave\nexcitation. To extract the microwave induced\nX-ray absorption the respective di\u000berence of\nboth datasets is taken. Fig. 2a) shows the\nresulting 6 STXM-FMR images at the Co\nL3-edge with an applied external magnetic\n\feld of 112.7 mT (Fig. 1b)). Brighter and\ndarker contrast indicates a lower/higher X-ray\nabsorption than the average. The contrastElement-speci\fc visualization of dynamic magnetic coupling in a Co/Py bilayer microstruct. 4\nwithin the area of the Co stripe indicates\na microwave induced response. Thus, the\nbright and dark contrast in Fig. 2a) shows\nthe deviations of the magnetization from its\nequilibrium orientation along the oscillation\naxis of the high frequency magnetic \feld.\nFig. 2b) shows the oscillation of the STXM-\nFMR signal at the position of the Co stripe. Its\nmaximum is visible at a relative phase of about\n90\u000e. The black curve in Fig. 2b) was recorded\nat an o\u000b resonance \feld of 30 mT and thus\nthe Co is only driven by the microwave \feld.\nThe red STXM-FMR signal is shifted by 90\u000e,\nas generally expected for a resonant response\n[30].\n3. Results and discussion\nThe element-speci\fc and spatially resolved\nmeasurements depicted in Fig. 3 show\nthe STXM-FMR images at the Ni L 3-edge\n(852 eV) (Fig. 3a)-c)), and Co L 3-edge\n(779 eV) (Fig. 3d)-f)) taken at a relative\nphase of 300\u000eexhibiting the highest contrast,\nwith a 100 nm step size and a dwell time of\n5000 ms. Grey contrast corresponds to an\naverage contrast value, which is set to the\nsame background color level for all images\nof the \fgure. In Fig. 3a)-3c) the complete\nspherical area of the Py disk shows STXM-\nFMR contrast at all three resonance \felds,\nindicating a resonant response of the Py disk.\nEach of STXM-FMR images in Fig. 3a)-\n3c) reveals a darker colored contrast area at\nthe location of the on-top lying Co stripe.\nFig. 3d)-f) picture the STXM-FMR contrast\noriginating from Co stripe while the Py disk\nis almost invisible with the The STXM-FMR\nimage at resonance 3 (Fig. 3f)) showing the\ndarkest coloured contrast of all the images\n(Fig. 3d)- f)).\nThe STXM-FMR image taken at reso-\n40002000-20000-4000060180120240300360/0b)300°0°/ 360°60°120°180°240°018375574920\nRelative microwave phase [deg]Time [ps]FMR inducedX-ray transmission [arb. u.]Lines represent ﬁts using: A sin(2πft+  )a)\n-453Intensity [arb. u.]-11753Figure 2. a) 6 STXM-FMR di\u000berence images\nobtained from six microwave on and six microwave o\u000b\nimages recorded every 60\u000e(18 ps) at the Co L 3-edge at\nBExt,3. b) FMR induced X-ray transmission signal (red\ndots: at Co resonance 3, black squares: o\u000b resonance)\nas function of time.\nnance 1 shown in Fig. 3a) (Ni L 3-edge) shows\na uniform contrast distribution within the disk\narea with a higher intensity contrast area at\nthe position of the Co stripe. This is corre-\nsponding to a homogeneous uniform resonant\nresponse of the Py disk as expected from the\nconventional FMR spectrum (Fig. 1a)). The\ncontrast visible in Fig. 3d) within the Co stripe\noriginates from Co being driven by the Py in\nresonance, inducing a slight increased preces-\nsional motion in the Py disk (higher intensityElement-speci\fc visualization of dynamic magnetic coupling in a Co/Py bilayer microstruct. 5\na)d)e)f)b)c)Resonance 1Resonance 1Resonance 2Resonance 2Resonance 3Resonance 3300°300°300°300°300°300°\n-3518Intensity [arb. u.]-15622\nFigure 3. STXM-FMR images taken at the respective\nBextfor resonance 1, 2, and 3. a)-c) are recorded at\nthe Ni L 3-edge, d)-e) are recorded at the Co L 3-edge.\nThe STXM-FMR images correspond to the time slot\nat 92 ps (300\u000e).\ncontrast area in Fig. 3a)) mediated by ex-\nchange coupling. The corresponding observa-\ntion is made for the Py disk at B ext,3, where\nthe Py magnetic moments are less agile due\nto their alignment along the direction of B ext,\nresulting in the Py disk getting only slightly\ndriven by the Co in resonance. In consequence\nthe contrast of the driven Co in Fig. 3d) is\nmore intense than the one of the driven Py\nin Fig. 3c) as the Co moments are not com-\npletely aligned along B ext,1and therefore are\nmore agile and easily driven compared to the\nPy moments of the disk at B ext,3. This exci-\ntation between the two constituents across the\ninterface illustrates a transfer of angular mo-\nmentum (spin current) between the two mag-\nnetic materials due to exchange coupling.\nAt resonance 2 the STXM-FMR images\nat both absorption edges show uniformly\ndistributed contrast at the location of both of\nthe sample constituents, indicating a coupled\nresonance originating from exchange coupling\nbetween Py and Co both being in resonance\nand contributing to the STXM-FMR signal.\nSuch modes have been observed before in\nmultilayer \flms and originate in the interfaceexchange between both constituents. The\nexchange length in Co and Py is several\nnanometers, thus in ferromagnetic resonance\nthe sample behaves in this area alloy-like.\nThis can be seen for example in [35],\nwhere spin wave spectra exist at an e\u000bective\nmagnetization of FeNi as an alloy-like entity in\naddition to the individual spin-wave resonance\nof Fe. Our STXM-FMR measurements of\nresonance 2 show at the Ni L 3edge that the\nresonance can be observed in the whole area of\nthe disc with a darker contrast at the position\nof the Co stripe/interface, broader than the\ncontrast seen at the Co L 3edge, since the edge\nspins of Co are still not aligned along B ext,2\nand thus not in resonance. This is due to that\nbeside the exchange length for this excitation\nthe coherence length of the FMR precession\nis important, which ranges depending on the\nmaterial up to several millimeters (e.g. 7 mm\nfor YIG [36]). In consequence the intensities\nof the three resonance modes shown in Fig. 2\nb) consistent to this interpretation. The Py\nresonance 1 shows the highest intensity due\nto the largest sample volume, therefore the\ncoupling resonance 2 shows less intensity as\nit originates as described above only from a\npart of the sample, while the Co resonance 3\nexhibits the lowest intensity, corresponding to\nthe smallest excited volume.\nAn amplitude and phase analysis of\nthe recorded 6 STXM-FMR images [34] at\nresonance 2, shown in Fig. 4a)-b) at the\nNi and Co L 3-edge, reveals further details\non the origin of the FMR excitations, not\ndirectly visible in the grayscale plots. After\nnormalising the STXM-FMR data to the\naverage intensity of each image, a sine \ft is\napplied to the time evolution of each pixel.\nThus, the pixels of the STXM-FMR image can\nbe color coded representing amplitude, phase\nand \ft accuracy obtained from the sinusoidalElement-speci\fc visualization of dynamic magnetic coupling in a Co/Py bilayer microstruct. 6\n\fts. The color coding was chosen as such\nthat bright pixels represent a large amplitude,\nthe phase is represented as the hue value and\npixels with very high saturation indicate a high\n\ft accuracy by encoding the p-value obtained\nfrom the \ft as color saturation. Thus, Fig. 4a)\nindicates a homogeneously distributed relative\nphase of approximately 90 degrees to the\nexciting microwave inside the whole Py disk,\nwhereas the amplitude of the Py excitation is\nlargest at the position of the upper and lower\nedge of the Co stripe, which is not directly\nvisible in Fig. 3b). Fig. 4b) shows bright and\nsaturated colored pixels only at the position of\nthe Co stripe, depicting di\u000berent phase values\nbetween the center (approx. 90 degrees as with\nthe Py) and the upper and lower edges (poles)\nof the Co stripe (approx. 60 degrees), due to,\nas for a typical bar magnet, the stray \feld\nin\ruence. This leads to a di\u000berent phase at\nthe top and bottom edges of the Co stripe.\nThe local phase change is only resolvable with\nour technique. Fig. 4a)-b) prove that the\nexchange coupled resonance 2 is mainly excited\nat the directly overlapping areas of disk and\nstripe and is not due to either an optical or\nacoustical mode excitation, Both Py and Co\nare resonating at the \feld value corresponding\nto the one of an alloy-like entity, which is\nproven by similar phases.\nThe STXM-FMR observation of the three\nresonances is in agreement with the line widths\nobserved in Fig. 1c. The largest peak to peak\nline width of 15 mT is observed at resonance\n1, where the whole Py disk is in resonance\nbut drives the Co moments at the area of the\nCo stripe. This yields an additional damping\nfor the Py and an additional line distribution\nresulting from the area outside and below the\nstripe. The same is valid for the Co resonance\n3 (peak to peak line width of 10 mT), there\nthe Co is driving the Py underneath, which is\n0 1\n0423\n45\n43\n27\n42Amplitude\nPhase arb. u.\nb) Resonance 2 a) Resonance 2\nRelative Phase [degree]Amplitude\n45901351802252703153601 0\n0Figure 4. a), b) Result of the pixel-wise \ft analysis\nof the STXM-FMR images, the color coding of the\namplitude and relative phase is displayed by the color\nbar. All images represent resonance 2.\nalready completely aligned with the external\n\feld and thus provides a stronger damping.\nThe coupling resonance 2 has the smallest peak\nto peak line width (about 5 mT). In addition,\nan asymmetric line shape of the resonance\nis visible indicating a distribution of di\u000berent\nexcitations.\n4. Conclusion\nThe magnetization dynamics of a coupled\nPy disk Co stripe bilayer microstructure has\nbeen analyzed in the linear response regime\nwith element speci\fcity, spatial, time and\nphase resolution. At the Py resonance the\nCo magnetization is driven into precession\nby angular momentum transfer mediated by\nexchange coupling of the precessing Py. We\nshow in our experiment that a coherently\nprecessing spin polarization is transferred via\ninter-material exchange at the interface to\nthe ferromagnetic material, which is not in\nresonance.\nIn earlier investigations of extended fer-\nromagnetic ultra-thin bilayers two main res-\nonances have been observed attributed to an\nin-phase and out-of-phase optical or acoustical\nmode [24]. In contrast here we revealed in the\nbilayered microstructure the occurrence of aElement-speci\fc visualization of dynamic magnetic coupling in a Co/Py bilayer microstruct. 7\nthird main resonance mode, which is explained\nby Py and Co resonating as an exchanged cou-\npled entity. Using an amplitude and phase\nanalysis method an inhomogeneous excitation\nof the Co stripe at the coupled resonance is re-\nvealed, due to the stray \feld e\u000bects at the poles\nof the stripe, whose in\ruence is visualized by\nmicromagnetic simulations. Thus, this mode\nis identi\fed as an exchange coupled and dipo-\nlarly in\ruenced excitation of the Co/Py disk\nstripe microstructure.\n5. Acknowledgement\nThe authors would like to thank the Ger-\nman Research Foundation (DFG project:\n321560838 (OL513/1-1)) and the Austrian Sci-\nence Fund (FWF project: I 3050-N36) for \f-\nnancial support. We gratefully acknowledge\nthe experimental assistance of S. Bonetti dur-\ning beamline setup. The use of the Stanford\nSynchrotron Radiation Lightsource, SLAC Na-\ntional Accelerator Laboratory, is supported by\nthe U.S. Department of Energy, O\u000ece of Sci-\nence, O\u000ece of Basic Energy Sciences under\nContract No. DE-AC02-76SF00515.\n6. References\n[1] Wolf S A, Awschalom D D, Buhrman R A,\nDaughton J M, von Moln\u0013 ar S, Roukes M L,\nChtchelkanova A Y, and Treger D M 2001\nScience 2941488.\n[2] Cowburn R P and Welland M E 2000 Science 287\n1466.\n[3] Zingsem B W, Feggeler T, Terwey A, Ghaisari S,\nSpoddig D, Faivre D, Meckenstock R, Farle M\nand Winklhofer M 2019 Nat. Commun. 854345.\n[4] Barman A, Gubbiotti G, Ladak, S, et al. 2021 J.\nPhys.: Condens. Matter 33413001.\n[5] D urr H A, Eim uller T, Elmers H-J, Eisebitt S,\nFarle M, Kuch W, Matthes F, Mertins M,\nMertins H-C, Oppeneer P M, Plucinski L,\nSchneider C M, Wende H, Wurth W and Zabel\nH. 2009 IEEE Trans. Magn. 4515.[6] van der Laan G and Figueroa A I 2014 Coordin.\nChem. Rev. 95277.\n[7] Bailey W E, Cheng L, Keavney D J, Kao C-C,\nVescovo E and Arena D A 2004 Phys. Rev. B\n70172403.\n[8] Marcham M K, Keatley P S, Neudert A, Hicken\nR J, Cavill S A, Shelford L R, van der Laan\nG, Telling N D, Childress J R, Katine J A,\nShafer P and Arenholz E 2011 J. Appl. Phys.\n10907D353.\n[9] Bailey W E, Cheng C, Knut R, Karis O, Au\u000bret\nS, Zohar S, Keavney D, Warnicke P, Lee J-S,\nand Arena D A 2013 Nat. Commun. 42025.\n[10] Goulon J, Rogalev A, Wilhelm F, Jaouen N,\nGoulon-Ginet C, Goujon G, Ben Youssef J and\nIndenbom M V 2005 JETP Lett. 8211.\n[11] Boero G, Rusponi S, Bencok P, Popovic R S,\nBrune H and Gambardella P 2005 Appl. Phys.\nLett.87152503.\n[12] Goulon J, Rogalev A, Wilhelm F, Goujon G,\nBrouder Ch, Yaresko A, Ben Youssef J,\nIndenbom M V 2010 J. Magn. Magn. Mater.\n3222308.\n[13] Boero G, Rusponi S, Kavich J, Lodi Rizzini A,\nPiamonteze C, Nolting F, Tieg C, Thiele J-U\nand Gambardella P 2009 Rev. Sci. Instrum. 80\n123902.\n[14] Boero G, Rusponi S, Bencok P, Meckenstock R,\nThiele J-U, Nolting F and Gambardella P 2009\nPhys. Rev. B 79224425.\n[15] Ollefs K, Meckenstock R, Spoddig D, R omer F M,\nHassel Ch, Sch oppner Ch, Ney V, Farle M and\nNey A 2015 J. Appl. Phys. 117223906.\n[16] Cheng Ch and Bailey W E 2012 Appl. Phys. Lett.\n101182407.\n[17] Bonetti S, Kukreja R, Chen Z, Spoddig D, Ollefs\nK, Sch oppner C, Meckenstock R, Ney A, Pinto\nJ, Houanche R, Frisch J, St ohr J, D urr H and\nOhldag H 2015 Rev. Sci. Instr. 86093703.\n[18] Scha\u000bers T, Meckenstock R, Spoddig D, Feggeler\nT, Ollefs K, Sch oppner Ch, Bonetti S, Ohldag\nH, Farle M and Ney A 2017 Rev. Sci. Instrum.\n88093703.\n[19] Gro\u0019F, Tr ager N, F orster J, Weigand M, Sch utz G\nand Gr afe J 2019 Appl. Phys. Lett. 114012406.\n[20] Sluka V, Schneider T, Gallardo R A, K\u0013 akay A,\nWeigand M, Warnatz T, Mattheis R, Rold\u0013 an-\nMolina A, Landeros P, Tiberkevich V, Slavin\nA, Sch utz G, Erbe A, Deac A, Lindner J,\nRaabe J, Fassbender J and Wintz S 2019 Nat.\nNanotechnol. 14328.Element-speci\fc visualization of dynamic magnetic coupling in a Co/Py bilayer microstruct. 8\n[21] Pile S, Feggeler T, Scha\u000bers T, Meckenstock R,\nBuchner M, Spoddig D, Zingsem B, Ney V,\nWende H, Ohldag H, Ney A, Ollefs K 2020 Appl.\nPhys. Lett. 116072401.\n[22] Pile S, Buchner M, Ney V, Scha\u000bers T, Lumet-\nzberger J, Lenz K, Narkowicz R, Lindner J,\nOhldag H, Ney A 2020 Phys. Rev. Applied 14\n034005.\n[23] Feggeler T, Meckenstock R, Spoddig D, Zingsem\nB, Ohldag H, Wende H, Farle M, Winklhofer M,\nOllefs K 2021 Phys. Rev. Res. 3033036.\n[24] Heinrich B, Purcell S T, Dutcher J R, Urquhart\nK B, Cochran J F and Arrott A S 1988-II Phys.\nRev. B 38(18) 12879.\n[25] Posth O, Reckers N, Meckenstock R, Dumpich G\nand Lindner J 2009 J. Phys. D.: Appl. Phys. 42\n035003.\n[26] Banholzer A, Narkowicz R, Hassel C, Meckenstock\nR, Stienen S, Posth O, Suter D, Farle M and\nLindner J 2011 Nanotechnology 22295713.\n[27] Meckenstock R 2008 Rev. Sci. Instrum. 79\n041101.\n[28] Sch oppner C, Wagner K, Stienen S, Meckenstock\nR, Farle M, Narkowicz R, Suter D and Lindner\nJ 2014 J. Appl. Phys. 116033913.\n[29] Scha\u000bers T, Feggeler T, Pile S, Meckenstock R,\nBuchner M, Spoddig D, Ney V, Farle M, Wende\nH, Wintz S, Weigand M, Ohldag H, Ollefs K and\nNey A 2019 Nanomaterials-Basel 9940.\n[30] Pain H J 2005 The Physics of vibrations and waves\n(West Sussex: John Wiley & Sons).\n[31] Jiles D 2016 Introduction to Magnetism and\nMagnetic Materials 3rd Edition (Boca Raton\nCRC Press Taylor and Francis Group).\n[32] Vansteenkiste A, Leliaert J, Dvornik M, Helsen\nM, Garcia-Sanchez F and Van Waeyenberge B\n2014 AIP Adv. 4107133.\n[33] Donahue M J and Porter D G Sept. 1999 OOMMF\nUser's Guide, Version 1.0, Interagency Report\nNISTIR 6376 (Gaithersburg, MD: National\nInstitute of Standards and Technology).\n[34] Zingsem B, Feggeler T, Meckenstock R, Scha\u000bers\nT, Pile S, Ohldag H, Farle M, Wende H, Ney A,\nOllefs K 2019 arXiv:1901.10595 [cond-mat.str-\nel].\n[35] Kordecki R, Meckenstock R, Pelzl J, Becker E,\nDumpich G, Suran G 1991 J. Magn. Magn.\nMater. 93281.\n[36] J. Pelzl J, von Geisau O, 1994 Vol. 2 Spatially\nResolved Detection of Microwave Absorption\nin Ferrimagnetic Materials, in: Progress inPhotothermal and Photoacoustic Science and\nTechnology Non-Destructive Evaluation, edited\nby A. Mandelis (Prentice Hall)."    },    {        "title": "1801.09437v1.Antiferromagnetic_rare_region_effect_in__Pr__0_5_Ca__0_5_MnO_3_.pdf",        "content": "Antiferromagnetic rare region effect in Pr 0:5Ca0:5MnO 3\nVinay Kumar Shukla\u0003and Soumik Mukhopadhyayy\nDepartment of Physics, Indian Institute of Technology, Kanpur, 208016, India\nWe present evidence of coexistence of electron paramagnetic resonance signal and anti-ferromagnetic reso-\nnance signals above the antiferromagnetic (AFM) transition ( TN) inPr0:5Ca0:5MnO 3. We identify the latter\nwith AFM rare regions within the ‘Grifﬁths-like’ phase scenario with the associated temperature scale T\u0003ex-\ntending above room temperature.\nAfter going through several decades of intense experimen-\ntal and theoretical scrutiny, there is a general acceptance of\ncertain features which are near universal in mixed-valent man-\nganites [1, 2] with far reaching implications for complex sys-\ntems in general [3]: 1) Prevalence of the nanoscale phase\nseparation; 2) Existence of a higher temperature scale T\u0003,\nwhere short range ‘clustering’ sets in, in addition to the long\nrange ordering temperature. However, the underlying physi-\ncal model that describes such a scenario and consequently the\norigin of colossal magnetoresistance (CMR) in manganites is\nstill intensely debated [1, 4]. At present there are primarily\ntwo hypotheses: 1) The existence of Grifﬁths-like phase with\npercolation type metal-insulator (MI) transition [5–7]; 2) Al-\nternatively, that the Grifﬁths phase (GP) in itself is insufﬁcient\nto cause CMR, and the formation of ferromagnetic polarons\njust above the long range ordering temperature needs to be\ntaken into account [4, 8, 9]. However, for low band width sys-\ntems near half doping, the debate is mainly centered around\nthe following competing pictures of charge ordering [10]: 1)\nCE type charge and magnetic ordering, originally proposed\nby Goodenough [11]; 2) Zener polaron (ZP) ordering with an-\ntiferromagnetic interaction between strong ferromagnetically\ncoupled dimers [12]; 3) Coexistence of both CE type and ZP\ntype ordering [13] or between correlated and uncorrelated po-\nlarons [14]. The last two frameworks have the additional ad-\nvantage of being able to explain away the emergence of ferro-\nelectricity in charge-ordered manganites [13, 15–17].\nThe existence of a phase similar to the Grifﬁths phase in\nmanganites is conﬁrmed by the following experimental signa-\ntures: 1) Presence of a weak, usually ferromagnetic resonance\n(FMR) signal above T Cagainst the background of an electron\nparamagnetic resonance signal [18]; 2) Deviation from typi-\ncal behavior predicted by the standard theory of second order\nphase transitions so far as the critical indices in the tempera-\nture dependence of the magnetic susceptibility in the param-\nagnetic region is concerned [6], 3) The downturn in the inverse\nsusceptibility as compared to the paramagnetic Curie back-\nground is suppressed by increasing the magnetic ﬁeld due to\nthe increased magnetization of the paramagnetic matrix en-\nclosing the rare region.\nQuenched disorder is a prerequisite for the formation of\nthe GP, although the physical picture of short-range-correlated\ndisorder creating large scale spin and charge inhomogeneities\nin manganites is only applicable to a narrow window at low\ndoping and should be absent near half doping [19]. However,\nrecently there have been a few experimental studies whichclaim existence of GP in half doped systems with intermedi-\nate band width [20, 21]. It has been predicted that the coexis-\ntence of two competing phases separated by a ﬁrst order tran-\nsition enhances the formation of a ‘Grifﬁths-like’ clustered\nphase below a characteristic temperature T\u0003[22]. The coex-\nistence of ferromagnetic metallic (FM) and antiferromagnetic\n(AFM) insulating phase is ubiquitous in manganites [23, 24]\nwith AFM phase dominating close to half doping [25]. Given\nsuch a scenario, it is surprising that occurrence of AFM rare\nregions in manganites has not been reported so far. In this let-\nter, we provide experimental evidence of AFM rare regions\nin a narrow band width manganite at half doping, namely,\nPr0:5Ca0:5MnO 3.\nPr1\u0000xCaxMnO 3(0:3< x < 0:5) in bulk form shows\ntransition from paramagnetic (PM) to AFM phase at low\ntemperature (T N) intermediated by the onset of charge or-\ndering at a higher temperature ( TCO> TN). Poly-\ncrystalline Pr 1\u0000xCaxMnO 3(PCMO) samples with x=\n0:5;0:45;0:4;0:33were prepared by the standard method de-\nscribed elsewhere [16]. The structural characterization of\nall the samples were done by x-ray diffraction \u0012- 2\u0012scans\nat room temperature using PANalytical X’pert diffractometer\nwithCu\u0000K\u000bradiation having wavelength of 1.54 ˚A. The\nRietveld reﬁnement analysis done by using full prof suite re-\nveals that the room temperature phase of all the samples has\northorhombic structure having Pbnm space group symmetry.\nThe microstructure, crystallite size and its distribution were\nstudied by ﬁeld emission scanning electron microscope (FE-\nSEM, Jeol, JSM-7100F). The chemical composition of all the\nsamples were conﬁrmed by energy dispersive spectroscopy\n(EDS) and the x-ray photoemission spectroscopy using PHI\n5000 Versa Prob II, FEI Inc. See Ref. [26] for details of struc-\ntural and chemical characterization. The magnetic measure-\nments were carried out in a Quantum Design PPMS. The tem-\nperature dependent EPR spectroscopy was done done using\nBruker EPR EMX spectrometer in the X-band.\nThe differential EPR signals were recorded at different tem-\nperatures from 120 K up to room temperature for the PCMO\nsamples. The samples were exposed to microwave radiation at\nconstant frequency of 9.46 GHz (X-band) and external mag-\nnetic ﬁeld was varied from 0 to 8000 Gauss. The power (P) ab-\nsorbed by the sample from the transverse magnetic microwave\nﬁeld is captured in the form of its ﬁrst derivative (dP/dH)\nby the standard lock-in technique [27, 28]. Fig. 1A shows\nthe EPR signals of Pr0:5Ca0:5MnO 3at some representa-\ntive temperatures between 120-300 K. In general, the line-arXiv:1801.09437v1  [cond-mat.str-el]  29 Jan 20182\nFIG. 1: (A) ESR signals shown for PCMO(x=0.5, bulk) from 120 K\nto 300 K, (B) The area highlighted in A) is zoomed in to show the\nadditional resonance peaks emerging from the main paramagnetic\nsignal. (C) Low ﬁeld (LF) resonance ﬁeld shifts towards lower mag-\nnetic ﬁeld value with increasing the temperature from 120 K to 220\nK (indicated by solid black arrow), and (D) Shift of high ﬁeld (HF)\nresonance position towards higher H value with increasing the tem-\nperature from 120 K to 220 K is shown. Above 220 K, HF signals\ncould not be distinguished anymore.\nshape is symmetric Lorentzian. The origin of EPR signals in\nthese systems is generally attributed to the combined effect of\nMn3+andMn4+states (which are coupled through double\nexchange interaction) and the lattice [29–32]. Strikingly, we\nobserve the appearance of a pair of additional resonance peaks\nas shown in Fig. 1B. It is clear that the low ﬁeld (LF) and high\nﬁeld (HF) resonance positions approach each other with low-\nering of temperature (Fig. 1C, D). In order to understand the\nnature of these resonance peaks and to accurately calculate\ncorresponding intensities, linewidth and resonance ﬁelds, we\nhave ﬁtted the ESR signals by the following equation [33, 34]-\ndP\ndH/d\ndH\u0012\u0001H\n(H\u0000Hr)2+ \u0001H2+\u0001H\n(H+Hr)2+ \u0001H2\u0013\n(1)\nwhereHris the resonance ﬁeld and \u0001His the linewidth. The\nresonance peaks are extracted by subtracting the main PM res-\nonance signal described by the Lorentzian in equation 1 from\nthe raw data. As a result, we obtain two sets of peaks, namely,\nLF and HF peaks on either side of the positive half maximum\n(around H\u00182180 Oe) of the main resonance peak (Fig. 2A,B).\nThe intensity, linewidth and resonance ﬁelds of LF and HF\nsignals have been calculated by integrating the LF and HF\nspectra and ﬁtting the integrated signals with Lorentzian line\nshape function given in the equation 1. The product of inten-\nsities and linewidth squared ( Imax\u0002\u0001H2) for the LF and HF\nspectra and main resonance peak are shown in Fig. 2D. While\ntheImax\u0002\u0001H2for the LF resonance shows a sharp anomaly\nnear220K, the signal for HF resonance is not observable be-\nyond 220K with the corresponding resonance ﬁeld merging\nFIG. 2: (A) and (B) represents the temperature evolution of LF and\nHF signals after extraction from the main resonance peak. LF and\nHF peaks are obtained by subtracting the main Lorentzian signal\nfrom the raw data as shown in inset of (B). (C) ratio of area under\nthe LF (A LF) and HF (A HF) signals normalized with respect to the\ncorresponding main resonance signal (A main ) is shown for PCMO\n(x=0.5, bulk) and PCMO (x=0.33, bulk). The area under the LF peak\nof PCMO (x=0.33, bulk) is signiﬁcantly reduced as compared to that\nof PCMO (x=0.5, bulk) (shown in inset of (C)). (D) Temperature de-\npendence of the product of intensity and square of linewidth for LF\n(I1\u0002(\u0001H1)2) and HF (I2\u0002(\u0001H2)2) peaks along with main reso-\nnance (Imain\u0002(\u0001Hmain)2) is shown. The temperature dependence\nof heat capacity and its derivative (calculated from the standard data\nin Ref. [35]) are also plotted. (E) The line-widths of LF ( \u0001H1) and\nHF (\u0001H2) signals are compared with the main ( \u0001Hmain ). (F) The\ncorresponding resonance ﬁelds of LF ( Hr1) and HF (Hr2) signals\nalong with the main ( Hmain ) resonance are shown. The inverse of dc\nmagnetic susceptibility ( 1=\u001f) along with its ﬁrst derivative (d \u001f/dT)\nfor PCMO (x=0.5, bulk) are also plotted in (F). The vertical dotted\nblue line represents the temperature 220K.\nwith the paramagnetic backbone. Curiously, the intensity for\nmain resonance shows a maximum again at the same temper-\nature. The temperature dependence of line-width, resonance\nﬁelds for the three signals along with the temperature deriva-\ntive of dc susceptibility (d\u001f\ndT) in the same temperature range\nare plotted in Fig. 2E, F. Strikingly, the maximum ind\u001f\ndT, too,\nappears at 220K (Fig. 2F). We plot the temperature deriva-\ntive of total heat capacity ( C), calculated from the standard\nliterature data [35], in Fig. 2D where the global maximum in\ndC\ndTagain appears around 220K, serving as further evidence\nof a thermodynamic phase transition. We observe similar res-\nonance signals in the EPR spectra for other PCMO samples\n(x=0.45,0.4,0.33) as well. However, the signals are weaker\ncompared to that observed in Pr 0:5Ca0:5MnO 3. For PCMO\n(x=0.4, bulk), we can precisely determine the temperature de-\npendence of \u0001H and intensities only for the LF signal al-3\n109101110131\n021031\n50200250300102103(A) Ix(ΔH)2 (a.u.) \nI\n1x(ΔH1)2I\nmx(ΔHm)2(\nB) ΔH(Oe) \n Δ\nHmainΔ\nH1(\nC) Hr(Oe) \n H\nr mainH\nr 1H\nr 2T\n(K)\n1051071\n021031\n50200250300102103(D)1/χdχ/dT  \n  (\nE)  \n (\nF)  \n T\n(K)\nFIG. 3: (A) The product of the ESR intensity and square of linewidth\n(I x ( \u0001H)2) for LF and main resonance peak for PCMO (x=0.40,\nbulk). (B), (C) Corresponding linewidths and resonance ﬁelds, re-\nspectively. (D), (E), and (F) Plots of the same for PCMO (x=0.33,\nbulk).\nthough the H rvalues of the LF as well as the HF peaks can\nbe estimated. The temperature dependence of Imax\u0002\u0001H2\nand Hrfor the LF peak of PCMO (x=0.40, bulk) and PCMO\n(x=0.33, bulk) as well as the anomalies observed in the tem-\nperature dependence of linewidth are similar to that of the half\ndoped PCMO (Fig. 3). The striking correlation between the\ntemperature dependence of H r,\u0001H and Imaxon one hand and\nthe macroscopic dc susceptibility data on the other observed\nfor the half doped system (Fig. 2), is missing away from half\ndoping (Fig. 3E). Moreover, for x= 0:33, the HF signal is\ncompletely suppressed.\nThe coexistence of LF and HF signal in the intermediate\ntemperature regime with the former extending up to the room\ntemperature suggests existence of a complex magnetic phase\nunlike ever reported before in manganites. To the best of\nour knowledge, in existing reports on manganites, the num-\nber of additional resonance signal observed other than the PM\nsignal is restricted to one and that, too, for low doped sys-\ntems [18]. We shall take up the issue of emergence of these\nsignals in a more elaborate manner shortly hereafter. Let us,\nfor the moment, turn our attention to the dc susceptibility data\naboveTCO(Fig. 4A). Generally one expects a downturn in the\ntemperature dependence of inverse susceptibility due to the\ngrowth of ferromagnetic clusters with lowering of temperature\nin the Grifﬁths phase. However we observe that the downturn\naboveTCOis only marginal which is followed by a sharp up-\nturn just below TCO. The marginal downturn above TCOis\ncompletely suppressed at slightly higher magnetic ﬁeld even-\ntually leading to an upward deviation over the paramagnetic\nCurie background. Prima facie, this suggests existence of\nAFM rare regions above TCOas the AFM susceptibility for\nFIG. 4: (A) The log-log plots of ﬁeld cooled (FC) inverse dc mag-\nnetic susceptibility (1/ \u001f) plotted against reduced temperature (T-\nTR\nC)/TR\nCat different applied magnetic ﬁeld ranging from 100 Oe to\n10 kOe. The Curie-Weiss ﬁts are shown by dashed red line and star\nsymbols represents the location of corresponding T\u0003for different\nmagnetic ﬁelds. (B) Temperature dependence of the ﬁrst derivative\nof inverse dc susceptibility ( d(1=\u001f)=dT) at different applied mag-\nnetic ﬁelds. The charge ordering temperature TCOis indicated by\nthe arrow.(C) The temperature dependence of reduced magnetization\n(M(T)/M(300K)) (FC and ZFC) at different temperature sweep rates\nranging from 0.2 K/min to 9.0. K/min. (D) Variation of ﬁtting pa-\nrametersT\u0003,TCO, andTR\nCas a function of applied magnetic ﬁeld.\nthe rare regions should be less than or comparable to the para-\nmagnetic susceptibility such that when they add up, the down-\nturn is not so pronounced as observed in case of FM Grifﬁths\nphase. Moreover, it is difﬁcult to envisage a ﬁnite FM rare\nregion, since half doped PCMO is known to exhibit electronic\nphase separation with a spatial distribution of hole concentra-\ntion only, without introducing any FM phase [25, 36].\nThe identiﬁcation of the LF and HF signals with AFM clus-\nter phase is supported by the following observations: 1) The\ncoexistence of two symmetrically placed resonances typical of\nAFM resonance spectra where application of external ﬁeld in-\ncreases the effective ﬁeld of one component while decreasing\nthe same for the other. In the present case, the two resonance\nﬁelds have opposite temperature dependence (Fig. 1, Fig. 2F);\n2) For the LF signal in EPR spectra, the resonance ﬁeld (H r)\ndecreases marginally with lowering of temperature with con-\ncomitant sharp reduction in LF signal intensity down to 220K.\nThe corresponding line-width for the LF signal shows a mini-\nmum at the same temperature suggesting exchange narrowing\neffect. 3) Although it is rare to observe AFM resonance signal\nin the X band, for small clusters, the anisotropy ﬁeld HAcould\nbe low enough to push the AFM resonance towards the X-\nband. Indeed, multiple resonance signals, excluding the para-\nmagnetic one in the X-band, have been associated with short\nrange AFM correlations [37, 38]. Below 220K, a sharp in-4\nFIG. 5: (A) The inverse ﬁeld cooled (FC) magnetization normalized\nwith respect to its value at 300 K is shown for different crystallite\nsizes of PCMO (x=0.5) at H=1 kOe. The solid red and dashed black\nlines represents the ﬁts to curie law. (B) The ESR signals of PCMO\n(x=0.5, 80 nm) at some representative temperatures along with LF\nand HF peaks marked by black down and red up arrows respectively.\nThe solid red lines represents the corresponding ﬁts to ESR data.\n(C) Temperature dependence of the product of intensity and square\nof line-width for LF ( I1\u0002(\u0001H1)2) and HF (I2\u0002(\u0001H2)2) sig-\nnals along with that of the main resonance ( Imain\u0002(\u0001Hmain)2)\nare plotted. The inverse of dc magnetic susceptibility ( 1=\u001f) along\nwith its ﬁrst derivative (d \u001f/dT) for PCMO (x=0.5, 80 nm) are also\nplotted over the same temperature scale. The resonance ﬁelds of LF\n(Hr1), HF (Hr2) and main signals ( Hmain ) along with correspond-\ning linewidths (ﬁlled symbols) are shown in (D).\ncrease in the LF and HF signal intensity is observed (Fig. 2D),\nwhich suggests some canting instability in the AFM phase.\nThe effect of canting instability is stronger at low tempera-\nture as supported by the gradual reduction in the difference\nbetweenHrvalues in the two spectra with lowering of tem-\nperature accompanied by an increase in the linewidth. If we\nlook at the ratio of area under the LF signal to that of the main\nsignal (Fig. 2C), the ratio ﬁrst slowly decreases with lowering\nof temperature followed by a signiﬁcant upturn below T CO,\nsuggesting the growing contribution of the LF signal at the\nexpense of the main PM signal with lowering of temperature\nin this temperature regime. The HF signal, too, grows at the\nexpense of PM signal, eventually catching up with the LF sig-\nnal at low temperature. Away from half doping, however, the\ncontribution of LF signal vis- \u0012a-vis PM signal is considerably\nreduced (Fig. 2C). A rough estimation by comparing LF/HF\nsignals with PM signal gives the fraction of spins contributing\nto LF signal for half doped PCMO to be \u00141.3%, whereas the\nfraction of spins contributing to HF signals is \u00141.6%. Away\nfrom half doping, the corresponding fraction of LF signals to\nthe main signals are estimated to be \u00140.15% forx= 0:4and\n\u00140.05% forx= 0:33, respectively. This is consistent with\nour observation that rare regions are not dominant away from\nhalf doping.In order to check whether the macroscopic susceptibility is\ninﬂuenced by cluster effects so clearly observed in the EPR\nspectra, one needs to correlate the temperature dependence of\nvarious parameters obtained from the resonance signals with\nthe temperature dependence of macroscopic inverse suscepti-\nbility. The marginal downturn in the temperature dependence\nof inverse susceptibility above T COis suppressed at higher\nmagnetic ﬁeld due to the growth of PM signal. Below TCO,\nthe strengthening of the AFM contribution at the expense of\nthe PM signal leads to the upturn in inverse susceptibility.\nThat there is a strengthening of AFM cluster phase is further\nsupported by the fact that the linewidth decreases as temper-\nature is lowered towards 220K. Interestingly, the ﬁrst order\nderivative of inverse susceptibility with respect to the temper-\nature is completely insensitive to the application of magnetic\nﬁeld only at T COwhereas above T CO,d(1=\u001f)\ndTshows a local\nmaximum at low ﬁeld reafﬁrming the downturn of inverse sus-\nceptibility which is progressively suppressed at higher mag-\nnetic ﬁeld (Fig. 4B). Moreover, as discussed earlier, the global\nmaxima in the temperature dependence ofd(1=\u001f)\ndTanddC\ndTco-\nincide exactly with the sharp anomaly in the LF signal in-\ntensity, the minima in \u0001Hand the maximum and minimum\ninHr1andHr2respectively suggesting strong inﬂuence of\nthe AFM rare region on the macroscopic susceptibility and a\nphase transition at 220K associated with the rare region. Al-\nthough the LF and HF signals lie below the PM signal, the LF\nand HF resonance ﬁelds should ideally be compared with the\nPM signal above T\u0003which is outside the temperature range for\nEPR measurements. Since the HF signal is very close to the\nPM signal in the same temperature range, it is possible that\nresonance ﬁeld for HF signal might actually be higher than\nthat for the PM signal above T\u0003. There could be an alternative\nscenario as the total number of AFMR signals might be more\nthan two as observed for orthorhombic symmetry before [39].\nIn that case the two observable AFMR signals can lie below\nthe resonance ﬁeld for the paramagnetic signal.\nWe further analyze the data by ﬁtting the temperature de-\npendence of inverse dc magnetic susceptibility using the rela-\ntion [40–44]\n\u001f/1\n(T\u0000TR\nC)1\u0000\u0015(2)\nwhere\u0015lies between 0 and 1. The temperature below which\nthere is deviation from the Curie-Weiss ﬁt determines the tem-\nperature scale T\u0003for cluster formation as shown in the log-log\nplots of 1/\u001fagainst the reduced temperature (T- TCR)/TCR\nfor different magnetic ﬁelds in Fig. 4A. We identify the value\nofTCRfor which the slope (1- \u0015) of ﬁtted data above T\u0003is\nunity (i.e.\u0015= 0). Once we obtain the value of TCR, the\nvalue of\u0015can be extracted from the slope of low temperature\nside of the data below T\u0003. However we do not ﬁnd any distinct\npower law behavior in that regime. The extracted parameters\nTCR, TCOandT\u0003are plotted with respect to the magnetic\nﬁeld in Fig. 4D. While TCOis independent of the magnetic\nﬁeld, the value of T\u0003initially decreases at low magnetic ﬁeld\nbefore increasing at higher H.5\nTo distinguish between conventional second order magnetic\ntransition described by the polaron picture and the Grifﬁth’s\nlike scenario, we study the time relaxation of zero ﬁeld cooled\n(ZFC) and ﬁeld cooled (FC) magnetization, since Grifﬁth’s\nlike state is prone to exhibiting out-of-equilibrium features\ndue to anomalously slow relaxation of magnetization [45].\nThe temperature dependence of dc magnetization (FC and\nZFC) is shown for different sweep rates ranging from 0.2\nK/min to 9.0. K/min in Fig. 4C. With increasing the temper-\nature sweep rate from 0.2 K/min to 9.0 K/min, the magnetic\nanomaly associated with charge ordering shifts towards higher\ntemperature for both FC and ZFC magnetization. Such sensi-\ntivity of FC magnetization to the temperature sweep rate is not\nexpected for a conventional second order magnetic transition.\nInterestingly, for FM Grifﬁths phase, we should expect the\nanomaly to shift to lower temperature with increasing sweep\nrate [46, 47], exactly opposite to our observation. Although,\na theoretical treatment is lacking, we emphasize that such a\nresponse could be due to the AFM nature of the rare region.\nIn the end, an important point remains to be addressed. If\nthe LF and HF signals are attributed to the AFM rare region,\nthen one should expect a common temperature range for both,\nwhich is clearly not the case for the bulk half-doped PCMO.\nOne possibility is that as the HF signal shifts towards higher\nresonance ﬁeld with increasing temperature, it eventually ap-\nproaches the main resonance asymptotically, thus making it\nimpossible to distinguish between the two. Fig. 5A shows\nthe inverse susceptibility data for poly-crystalline PCMO with\ndifferent average grain size along with the bulk. Except for\nthe lowest grain size, the magnetic anomaly related to charge\nordering survives in all other samples. And indeed, as the\naverage grain size is lowered to 80 nm, we ﬁnd that both\nLF and HF signals extend at least up to room temperature\n(Fig. 5B,C,D). On further reduction of grain size, the addi-\ntional resonance signals disappear altogether (not shown in\nFigure). The variation of different parameters extracted from\nthe LF and HF signal is similar to the bulk poly-crystalline\nsample although the striking correlation withd\u001f\ndTis missing\n(Fig. 5C,D). The anomalies in \u0001H,Imax\u0002\u0001H2, etc are in-\nstead conﬁned within a broad temperature region around the\nminima ind\u001f\ndT. A comparison of the LF/HF signals and PM\nsignals shows that the fraction of spins contributing to the LF\nand HF resonances are \u00141:7%and\u00141:65%, respectively,\nwhich are slightly higher than the corresponding bulk sample.\nThe mismatch of the anomalies in the macroscopic suscepti-\nbility and the LF/HF signal parameters can be attributed to the\nincreased FM correlation in the main signal due to reduction\nin grain size [48] as well as the distribution of the grain size\nin the nanocrystalline sample.\nTo conclude, we present direct experimental evidence of\nAFM rare region effects above T Nin half doped narrow band\nwidth PCMO with the associated temperature scale T\u0003ex-\ntending above room temperature. In a nutshell, the various\nﬁndings are as follows: 1) Observation of a pair of resonance\nsignals (extending at least up to room temperature) other than\nthe main paramagnetic resonance; 2) Marginal downward de-viation at low magnetic ﬁeld from the Curie background in the\ninverse susceptibility, which is suppressed at higher magnetic\nﬁeld withT\u0003showing strong non-monotonic magnetic ﬁeld\ndependence; 3) Slow time relaxation in the ﬁeld cooled mag-\nnetization even far above the AFM ordering temperature; 4)\nAlthough we fail to ascertain any power law behaviour well\nbelowT\u0003, there is a strong correlation between the tempera-\nture evolution of the independently measured AFM resonance\nsignals and the macroscopic susceptibility. It seems highly\nprobable that the physics of AFM rare regions within the Grif-\nﬁths phase scenario is applicable to other low and intermedi-\nate bandwidth manganites near half doping as well, something\nwhich needs to be veriﬁed experimentally in future.\n\u0003Electronic address: vkshukla@iitk.ac.in\nyElectronic address: soumikm@iitk.ac.in\n[1] E. Dagotto, New J. Phys. 7, 67 (2005).\n[2] Y . Tokura, Rep. Prog. Phys. 69, 797 (2006).\n[3] E. Dagotto, Science 309, 257 (2005).\n[4] J. A. Souza, J. J. Neumeier, and Yi-Kuo Yu, Phys. Rev. B 78,\n014436 (2008).\n[5] M. B. Salamon, P. Lin, and S. H. Chun, Phys. Rev. lett. 88,\n197203 (2002).\n[6] V . N. Krivoruchko, Low Temperature Physics 40, 586 (2014).\n[7] V . N. Krivoruchko and M. A. Marchenko, J. Exp. Theor. Phys.\n115, 125 (2012).\n[8] Jiyu Fan, Li Pi, Yan He, Langsheng Ling, Jixia Dai, and Yuheng\nZhang, J. Appl. Phys. 101, 123910 (2007).\n[9] V .A. Ivanshin, J. Deisenhofer, H.-A. Krug von Nidda, and A.\nLoidl, J. Magn. Magn. Mater. 310, 1966 (2007).\n[10] R. A. Ewings, T. G. Perring, O. Sikora, D. L. Abernathy, Y .\nTomioka, and Y . Tokura, Phys. Rev. B 94, 014405 (2016).\n[11] J. S. Zhou and J. B. Goodenough, Phys. Rev. B 62, 3834 (2000).\n[12] A. Daoud-Aladine, J. Rodriguez-Carvajal, L. Pinsard-Gaudart,\nM. T. Fernandez-Diaz, and A. Revcolevschi, Phys. Rev. Lett.\n89, 097205 (2002).\n[13] D. V . Efremov, J. Van den Brink, and D. I. Khomskii, Nat.\nMater. 3, 853 (2004).\n[14] Ch. Jooss, et al. ,Proc. Natl. Acad. Sci. 104, 13597 (2007).\n[15] K. Yamauchi, and S. Picozzi, J. Phys. Soc. Jpn. 82, 113703\n(2013).\n[16] V . K. Shukla, S. Mukhopadhyay, K. Das, A. Sarma, and I. Das,\nPhys. Rev. B 90, 245126 (2014).\n[17] V . K. Shukla, and S. Mukhopadhyay, RSC Adv. 6, 93130 (2016).\n[18] J. Deisenhofer, et al. ,Phys. Rev. Lett. 95, 257202 (2005).\n[19] G. Bouzerar and O. Cepas Phys. Rev. B 76, 020401(R) (2007).\n[20] A. K. Pramanik and A. Banerjee, Phys. Rev. B 81, 024431\n(2010).\n[21] A. K. Pramanik and A Banerjee, J. Phys.: Condens. Matter 28,\n35LT02 (2016).\n[22] J. Burgy, M. Mayr, V . Martin-Mayor, A. Moreo, and E. Dagotto,\nPhys. Rev. Lett. 87, 277202 (2001).\n[23] S. Mukhopadhyay and I. Das Europhys. Lett. 83, 27003 (2008).\n[24] S. Mukhopadhyay, I. Das and S. Banerjee J. of Phys.: Condens.\nMatter 21, 026017 (2008).\n[25] J. van den Brink, G. Khaliullin, and D. Khomskii, Phys. Rev.\nLett.83, 5118 (1999).\n[26] See supplementary material.\n[27] Charles P. Poole, Electron Spin Resonance: A comprehensive6\ntreatise on experimental techniques , (Dover, New York, 1983),\n2ndEd..\n[28] John A. Weil and James R. Bolton, Electron Paramagnetic Res-\nonance: Elementary theory and Practical applications ,(Wiley,\nNew Jersey, 2007), 2ndEd..\n[29] M. T. Causa, et al. ,Phys. Rev. B 58, 3233 (1998).\n[30] A. Shengelaya, Guo-meng Zhao, H. Keller, and K. A. Muller,\nPhys. Rev. Lett. 77, 5296 (1996).\n[31] A. Shengelaya, Guo-meng Zhao, H. Keller, K. A. Muller, and\nB. I. Kochelaev, Phys. Rev. B 61, 5888 (2000).\n[32] S. E. Loﬂand, et al. ,Phys. Lett. A 233, 476 (1997).\n[33] V . A. Ivanshin, J. Deisenhofer, H.-A. Krug von Nidda, A. Loidl,\nA. A. Mukhin, A. M. Balbashov, and M. V . Eremin, Phys. Rev.\nB61, 6213 (2000).\n[34] Y . Liu, S.L. Wan, and X.G. Li, J.Phys.:Condens. Matter. 19,\n196213 (2007).\n[35] V . Hardy, A. Maignan, S. Hebert, and C. Martin, Phys. Rev. B\n67, 024401 (2003).\n[36] Y . Tomioka, A. Asamitsu, H. Kuwahara, and Y . Moritomo, and\nY . Tokura Phys. Rev. B 53, R1689 (1996).[37] Gareth R. Eaton, Sandra S.Eaton and Kev M. Salikhov, Foun-\ndations of Modern EPR , (World Scientiﬁc, Singapore, 1998).\n[38] J. Yang, et al. ,Phys. Rev. B 86, 104410 (2012).\n[39] J. Yamauchi, J. Chem. Phys. 67, 2850 (1977).\n[40] R.B. Grifﬁths, Phys. Rev. Lett. 23, 17(1969).\n[41] A.J. Bray, Phys. Rev. Lett. 59, 586 (1987).\n[42] A.H. Castro Neto, G. Castilla and B.A. Jones, Phys. Rev. Lett.\n81, 3531 (1998).\n[43] A.J. Bray and M.A. Moore, J. Phys. C 15, L765 (1982).\n[44] W. Jiang, X. Zhou, and G. Williams, Euro Phys. Lett. 84, 47009\n(2008).\n[45] F. Vazquez, J. A. Bonachela, C. Lopez, and M. A. Munoz Phys.\nRev. Lett. 106, 235702 (2011).\n[46] V . N. Krivoruchko, and M. A. Marchenko, J. Magn. Magn.\nMater. 412, 30 (2016).\n[47] V . N. Krivoruchko, M. A.Marchenko, and Y . Melikhov, Phys.\nRev. B 82, 064419 (2010).\n[48] V . K. Shukla, and S. Mukhopadhyay, J. Appl. Phys. 121,\n093901 (2017)."    },    {        "title": "1809.07302v2.Occurrence_of_Mixed_Phase_in_Bi0_5Sr0_5Mn0_9Cr0_1O3_bulk_sample__Electron_Paramagnetic_Resonance_and_Magnetization_Studies.pdf",        "content": "Occurrence of Mixed Phase in Bi0:5Sr0:5Mn 0:9Cr0:1O3bulk sample: Electron\nParamagnetic Resonance and Magnetization Studies\nBhagyashree K. S., Lora Rita Goveas\u0003, and S. V. Bhat\nDepartment of Physics, Indian Institute of Science, Bangalore-560012,\nIndia and\n\u0003St. Josephs College (Autonomous), Bangalore 560027, India\nWe study the effects of 10% Cr substitution in Mn sites of Bi0:5Sr0:5MnO 3on the an-\ntiferromagnetic (AFM) ( TN\u0018110 K) transition using structural, magnetic and electron\nparamagnetic resonance (EPR) techniques. Field cooled (FC) and zero ﬁeld cooled (ZFC)\nmagnetization measurements done from 400 K down to 4 K show that the compound is\nin the paramagnetic (PM) phase till 50 K where it undergoes a transition to a short range\nferromagnetic phase (FM). Electron paramagnetic resonance measurements performed in\nthe temperature range 300 K till 80 K conform with the magnetization measurements as\nsymmetric signals are observed owing to the paramagnetic phase. Below 80 K, signals\nbecome asymmetric. Electron paramagnetic resonance intensity peaks at \u0018110 K, the de-\ncreasing intensity below this temperature conﬁrming the presence of antiferromagnetism.\nWe conclude that below 50 K the magnetization and EPR results are consistent with a clus-\nter glass phase of BSMCO, where ferromagnetic clusters coexist with an antiferromagnetic\nbackground.\nKeywords: EPR, Manganites, CO, AFM, FM\n1arXiv:1809.07302v2  [cond-mat.str-el]  16 Mar 2019I. INTRODUCTION\nManganites of the form T1\u0000xDxMnO 3(where T is a trivalent rare-earth ion or Bi ion and D is\na divalent alkaline earth ion) display competition between ferromagnetic double exchange (Zener\nexchange) interaction (between Mn3+andMn4+ions leading to ferromagnetic metallic (FMM)\nstate) and antiferromagnetic (AFM) super-exchange interaction (between Mn3+orMn4+ions\nleading to charge ordered insulating (COI) AFM state)1–3. This competition leads to the formation\nof domains with contrasting properties, i.e. the system stabilizes by breaking into regions with\nhigh electron density (this region becomes AFM) and low electronic density (this region becomes\nFM). This phenomena is called phase separation4. Coulomb repulsion between electrons suggests\nthat these regions are of a few nanometers in size. The end members of the phase diagram in\nthese compounds ( TMnO 3andDMnO 3) are stabilized in AFM state; when the end members are\ndoped with divalent ions in case of TMnO 3and trivalent ions in DMnO 3, FM state starts to build\nup. This leads to the formation of mixed phase with AFM phase and FM phase existing together.\nEven in the half doped T0:5D0:5MnO 3, COI-AFM state is stabilized due to the presence of equal\nnumber ofMn3+andMn4+ions. Interestingly, it has been found that a few methods like external\npressure, magnetic ﬁeld, reducing the size of the bulk particle to nano and substitution of the Cr3+\nions in sites of Mn3+ions have resulted in suppression of the CO-AFM phase and enhancement of\nFMM phase5–8.Cr3+breaks the long range CO state due to its unoccupied egorbital, suppressing\nthe AFM phase and enhancing the double exchange interaction resulting in the stabilization of\nFMM phase.\nA few investigations on Cr doped rare-earth manganites have been reported previously7,9–11.\nStudies done on Nd 0:5Ca 0:5Mn 1\u0000yCryO3(y=0.03, 0.05, 0.1)7show that with increasing Cr substi-\ntution CO AFM phase diminishes and FMM phase emerges. Studies done on La0:46Sr0:54Mn 1\u0000yCryO3\n(0\u0014y\u00140:08)9show that at y = 0 the sample undergoes PM to FM and FM to AFM transi-\ntions at 272 K and 190 K respectively. With increasing Cr substitution the FM phase becomes\ndominant over AFM phase. At y = 0.08 AFM phase disappears completely. Studies done on\nNd 0:6Ca 0:4Mn 1\u0000yCryO3system10show that, the parent compound which is CO AFM changes\nto a mixed magnetic state with AFM and FM domains coexisting in the region 0:015\u0014y\u00140:04\nof Cr substitution. In the region 0:04<y < 0:4the sample is in the ferrimagnetic state. Neutron\ndiffraction and magnetization studies done on Nd 0:6Ca 0:4Mn 0:5Cr0:5O311show that manganese\nand chromium ions order in the lattice in a similar way to Mn3+andMn4+ordering in the CO\n2state ofNd 0:6Ca 0:4MnO 3. Below 100 K the magnetic moment increases with decreasing temper-\nature and does not saturate, indicating the presence of FM component. Takuro et.al.12studied the\neffects of Cr substitution as an impurity in Pr1\u0000xCaxMn 0:97Cr0:03O3,La1\u0000xCaxMn 0:97Cr0:03O3\nandNd 1\u0000xSrxMn 0:97Cr0:03O3. In here just 3% of Cr substitution has induced a short range CO\nstate, but the FM and layered AFM phase remain as long range ordered.\nBismuth based manganites of the formula Bi1\u0000xDxMnO 3(where D=Ca,Sr) are different from\ndoped rare-earth manganites due to the presence of highly polarizable 6s2lone pair of elec-\ntrons present on the Bi ion. It signiﬁcantly restricts the mobility of egelectrons and favors\ncharge ordering. Very few investigations have been reported on Cr substituted Bi manganites8,13.\nBi0:5Sr0:5MnO 3showsTCO\u0018525 K andTN\u0018110 K14,15. CO in this composition is very robust\nand this is also followed by the AFM phase. Rao et.al16analyzedBi0:5Sr0:5MnO 3nanoparti-\ncles and found that both CO and AFM phases are unaffected unlike the other doped rare-earth\nmanganites5,6. Aim of this work is to study if 10% of Cr substitution is sufﬁcient to modify the\nexisting properties and give rise to new properties in Bi0:5Sr0:5MnO 3system. We discovered that\nCr substitution has led to the creation of short range FM phase embedded in the matrix of AFM\nphase in the back ground.\nII. PREPARATION AND CHARACTERIZATION\nPolycrystalline powder of Bi0:5Sr0:5Mn 0:9Cr0:1O3(BSMCO) bulk sample was prepared via\nthe solid state route. Calculated stoichiometric amounts of 99.9% pure Bi2O3,SrCO 3,MnCO 3\nandCr2O3were taken and mixed in an agate mortar and ground vigorously for an hour. This\npowder was then kept in a furnace for sintering at 9000C for 12 hours (hrs). Sintered powder was\ntaken out and ground again for an hour, pelleted and kept in the furnace twice, ﬁrst at 11000C and\nthen at 13000C (with intermediate grinding and pelleting) each for a duration of 24 hrs. The phase\npurity and the crystal structure of the sample was determined using X-ray diffractometer. Elec-\ntron probe micro analysis (EPMA) was done to conﬁrm the stoichiometric ratios. Magnetization\nmeasurements were done using PPMS facility. EPR measurements were performed using Bruker\nEMX spectrometer at a nominal frequency of 9.4 GHz.\nRoom temperature X-ray diffraction pattern for BSMCO is shown in Figure 1. Rietveld reﬁne-\nment was done on this XRD pattern taking reference data17using the GSAS computer program.\nThe sample crystallizes in tetragonal structure with space group P4mm, with cell parameters a=\nb= 3:9067 ˚A,c= 3:8010 ˚A\u000b=\f=\r= 900, with\u001f2= 8:563;Rp= 0:0219;wRp = 0:0509 .\n320 40 60 80 100020000400006000080000100000\n(213)(222)(311)(310)(202)(112)(211)(210)(002)(200)(111)(101)(110)(001)(100) Expt\n Fitted\n background\n Expt-Fitted\n Bragg peakCounts(A.U)\n2(degrees)FIG. 1. Rietveld reﬁned XRD of BSMCO sample; Here black cross represents experimental data, red\ncircles represents the ﬁt, blue triangle represents background, green line represents the difference between\nexperimental and ﬁtted data, and the vertical bars are the Bragg reﬂection positions.\nStoichiometric ratios calculated using EPMA match satisfactorily with the expected values. The\ncalculated stoichiometric ratios for elements are Bi=0.510, Sr=0.489, Mn=0.894, Cr=0.106.\nIII. MAGNETIZATION STUDIES\nThe ﬁeld cooled (FC) and the zero ﬁeld cooled (ZFC) magnetization curves for BSMCO, mea-\nsured under 100 Oe ﬁeld, from 400 K till 4 K are shown in Figure 2. Both FC and ZFC curves\nmatch with each other from 400 K till 50 K indicating the presence of PM phase; at \u001850 K there\nis a sharp increase in the FC magnetization curve indicating a building up of FM ordering and at\nthe same temperature ZFC curve undergoes a peak. The large gap between the FC and ZFC curve\npoints to short range FM ordering in the system18. We further note that the observed behavior\n41/dc(emu/g/Oe)-1\n0 50 100 150 200 250 300 350 4000.00.10.20.30.40.50.6\n FC\n ZFC\n 1/dc\nT(K)M(emu/g)\nZFC1/dc\nCW fit\n0.05.0x1031.0x1041.5x1042.0x1042.5x104\n0 50 100 150 200 250 300 350 400-0.08-0.06-0.04-0.020.00dM/dT(A.U)\nT(K)FC at 100 OeFCFIG. 2. FC, ZFC and 1=\u001fbehavior of BSMCO sample in the presence of 100 Oe ﬁeld; The dashed line is\nthe Curie-Weiss ﬁt to the experimental data which look like a continuous line because of the large number\nof data points. Inset shows the dM/dT plot.\nof ZFC and FC magnetizations is very similar to that observed in Bi1\u0000xCaxMnO 3(x=0.125) by\nWoo et al.19and inCa 0:9Sm 0:1MnO 3by Maignan et al.20. Both these materials are argued to be\nexhibiting cluster glass phases.The inset to Figure 2 shows a plot of dM/dT vs. T on FC curve.\nHere a sharp dip at 50 K is due to the FM transition.\nFigure 2 also presents the plot of 1=\u001fdcvs T ﬁtted to the Curie-Weiss law \u001f=C=(T+\u0012). We\nget a value of C = 0.0148. Using this value of C we get \u0016eff= 0.345\u0016BfromC=NA(\u0016eff)2=3kB.\nThis is just\u00188% of the ideal value of 4.282 \u0016Bexpected if all the spins are ferromagnetically\naligned. From this we infer that there is no long range FM in the sample and the sample consists\nof short range FM clusters. While the ﬁt is seen to be good, the intercept on the x-axis is at\n+25 K, instead of a negative value expected if long range AF order is present. This result is\n5-6 -4 -2 0 2 4 6-12-8-404812\n-1 0 1-4045 K\nCooling Field=2TeslaM(emu/g)\nH(Tesla)\nM(emu/g)\nH(Tesla)FIG. 3. Magnetization (M) vs. applied ﬁeld (H) for BSMCO sample at 5 K cooled in the presence of 2Tesla\nﬁeld; inset of the ﬁgure shows an expanded view showing the absence of exchange bias.\nunderstood, following Woo et al.19, who present an argument to explain the wrong sign in terms of\nthe coexistence of AF and FM phases. According to their model the sign could be either positive\nor negative depending on the relative fractions of FM and AFM phases.\nWe performed FC magnetization (M vs. H (applied ﬁeld)) measurement at 5 K in the cooling\nﬁeld of 2 T. The plot is shown in Figure 3. Closer inspection of the hysteresis plot (see the inset\nof Figure 3) clearly indicates the presence of FM phase. Also note that the magnetization curve\ndoes not saturate even till 5Tesla ﬁeld indicating the presence of non-ferromagnetic regions in the\nsystem, which may be due to either the presence of antiferromagnetically aligned spins or due to\nparamagnetic regions. The EPR results to be presented in the next section show that these regions\nare indeed antiferromagnetic.\n60.0 2.0k 4.0k 6.0k 8.0k 10.0k 0.0 2.0k 4.0k 6.0k 8.0k 10.0k\n \n  30K\n \n  50K  70K\n Fit   100K\n Fit\n   130K\n Fit\n  \n   160K\n Fit\n   190K\n Fit\n \n   210K\n Fit\n \n   230K\n Fit\n  250K\n Fit\n \n \n dP/dH(A.U)\n 280K\n Fit\n   \nH(Oe) 300K\n FitFIG. 4. EPR spectra of BSMCO at different temperatures\nIV . EPR STUDIES\nEPR spectra were recorded using a commercial X-band EPR spectrometer in the temperature\nrange of 4 K to 300 K. A speck of DPPH was used as a ﬁeld marker. The signals were ﬁtted to the\nbroad Lorentzian lineshape function in a derivative form21,\ndP\ndH=Ad\ndH(\u0001H\n4(H\u0000Hr)2+ (\u0001H)2+\u0001H\n4(H+Hr)2+ (\u0001H)2) (1)\nwhere P is the microwave power absorbed by the sample at resonance, \u0001His the linewidth, A\nis a quantity proportional to the intensity of the signal and Hris the resonance ﬁeld, to extract the\nlineshape parameters intensity, g-value and linewidth.\nFigure 4 shows the EPR spectra at different temperatures for the BSMCO sample. Here the red\nline represents the double Lorentzian ﬁt. Below 75 K we could not ﬁt to the double Lorentzian\n7FIG. 5. Temperature dependence of resonance ﬁeld Hrand g-value plot for the BSMCO sample (error\nbars are plotted at selected temperatures for both Hrand g-value plots)\nlineshape because the signals were asymmetric owing to the presence of FM phase. But unlike\nbulk FM, where the signals become stronger here the signals get weaker and weaker with decreas-\ning temperature. This suggests that the FM is not macroscopic but of short range in nature and\nalso there is a competition between the ordered (FM) and disordered state; which conforms with\nour magnetization data. The lineshape parameters intensity I, linewidth \u0001Hpp(= \u0001H=p\n3)and\nresonance ﬁeld ( Hr) can be extracted from the ﬁtted equation; but only for the symmetric signals.\nIn case of asymmetric signals (given there is no loss of the signal below zero ﬁeld), one can still\nmeasure the resonance ﬁeld Hrand peak to peak linewidth \u0001Hppand intensity (I) directly from\nthe signal as described later.\nThe temperature dependence of the resonance ﬁeld Hrand the g-value obtained from it are\nshown in Figure 5. Starting from room temperature, Hrshows a slow increase with decreasing\ntemperature till\u001875 K where it begins to decrease suddenly heralding the onset of ferromagnetic\nordering and the development of local ﬁelds. The g-value, as expected shows a behavior opposite\nto this.\nThe EPR linewidth behavior is as shown in Figure 6. As can be seen the linewidth increases\n8100 150 200 250 30030003500400045005000H(Oe)\nT(K)FIG. 6. Temperature dependence of linewidth \u0001Hof the BSMCO sample\nwith decreasing temperature going from PM phase to FM phase. This is unlike usual manganite\nsystems where the linewidth decreases with decreasing temperature towards Tc, reaches a mini-\nmum atTmin\u00181.1Tcand begins to increase again as the temperature is decreased further22–25. To\nunderstand this atypical behavior better we tried ﬁtting the linewidth variation to different known\nmodels23,24,26–29. All these ﬁttings resulted in the poor agreement with our data. The minimum\nin the linewidth observed at \u00181.1Tcin many manganites is a consequence of competing effects\nof spin-phonon interaction and FM/AFM ﬂuctuations which lead to opposite dependence of the\nlinewidth on the temperatures. An absence of an increase in the linewidth with increasing temper-\nature indicates negligible contribution of spin-phonon interaction to the linewidth in our sample.\nWe calculated the intensity by double integrating the EPR signal. First we integrated the deriva-\ntive curve to get the absorption signal; then we integrated the region from center of the absorption\ncurve to the highest ﬁeld in order to ﬁnd the area under the curve. This value is multiplied twice\n950 100 150 200 250 30023456Intensity(A.U)\nT(K)TNFIG. 7. Temperature dependence of Intensity; arrow indicates the transition to an AFM phase at \u0018110 K\nto get the intensity of the EPR signal at different temperatures. Figure 7 shows the intensity plot\nagainst temperature. The EPR intensity is usually proportional to the DC susceptibility30. Intensity\nvs. temperature is plotted and it is observed that intensity increases with decreasing temperature,\nclosely resembling the FC magnetization trend as can be seen in the main Figure 2. Intensity plot\ndistinctly shows that as the temperature decreases intensity increases gradually peaking at around\n\u0018110 K. This peak is very close to the AFM ordering temperature in an undoped system which\nis at\u0018110 K14,15. Earlier reports have shown that a peak is observed in an intensity plot if there\nexists an AFM ordering in the system16,31.\nV . DISCUSSION\n10% Cr ion substitution has resulted in considerable changes in the system. Cr3+is the main\ndriving force behind the appearance of FM phase as well as weakening of AFM phase in our\nsystem. We observe that our results are in agreement with earlier reports on Cr doped rare-earth\n10manganites7,9–12as well as Bi manganites8,13. Below we discuss the results of earlier reports on Cr\nsubstituted Bi manganite systems in comparison with our system.\nAccording to Yamada et.al13inBi1\u0000xSrxMn 1\u0000yCryO3where x=0.25, 0.3, 0.4 and 0\u0014y\u0014\n0:1, for all the samples Curie temperature is positive suggesting ferromagnetic interaction between\nMn spin moments. The magnetization in the AFM phase increases with Cr doping upto 10% and\nTNdisappears for y=0.1. Thus the degree of AFM state by Cr doping decreases with increasing\nx. Authors also observe that upon Cr substitution the AFM phase in the system transforms into\nPM state rather than FM state. According to Xiong et.a8inBi0:5Ca 0:5Mn 1\u0000yCryO3where 0\u0014\ny\u00140:15with increasing Cr content; here for y=0, TN\u0018130 K. With increasing Cr content,\nAFM ordering shifts slightly to lower temperature but it is unaffected till y\u00140:24. Above this\nthere is a depression in the AFM ordering. The reciprocal susceptibilities of all the samples show\npositive\u0012cw(curie constant) value, indicating the presence of local FM correlation and weakening\nof AFM ordering. Our results are in agreement with earlier reports as in our sample we observe\nthe weakening of AFM phase, emergence of FM phase, and below certain temperature both FM\nand AFM coexist together. According to magnetization short range FM ordering takes place below\n50 K. EPR signals become asymmetric below 80 K indicating the presence of local ﬁeld arising\ndue to the presence of FM ordering, which is reﬂected in g-value plot. These asymmetric signals\nbecoming weaker with decreasing temperature and absence of the minimum at \u00181:1Tcin the\nlinewidth plot conﬁrms the very nature of FM being short range. Magnetization does not display\nthe presence of AFM phase but peak in the EPR intensity at \u0018110 K exhibits the presence of\nAFM in the system.\nWe infer that at higher temperatures the sample is in the PM phase and as the temperature\ndecreases the sample undergoes a transition to an AFM phase at \u0018110 K due to the interaction\nbetween Mn ion moments. Further down the temperature at \u001850 K, FM clusters are formed\nnucleating around the Cr ion moments leading to the formation of mixed phase. This causes\nincrease in the magnetization. In this work we observe that 10% of Cr substitution has resulted in\nthe emergence of short range FM phase and weakening of AFM phase.\nVI. CONCLUSION\nIn summary, we prepared Bi0:5Sr0:5Mn 0:9Cr0:1O3bulk sample and performed magnetization\nand EPR measurements. The FC and ZFC magnetization measurements show that the sample\nundergoes transition to a short range FM or cluster glass phase at 50 K. The EPR data are consistent\n11with the magnetization data by indicating the presence of short range FM. Further, EPR intensity\ngives the evidence of the presence of AFM below 110 K. We conclude that the sample below 50\nK is in the mixed phase made up of FM clusters embedded in an AFM matrix.\nAcknowledgment\nSVB thanks the National Academy of Sciences, India for the award of a Senior Scientist Plat-\ninum Jubilee Fellowship.\nReferences\n1Y . Tokura, Colossal magnetoresistive oxides (CRC Press, 2000).\n2C. N. R. Rao and B. Raveau, Colossal magnetoresistance, charge ordering and related properties\nof manganese oxides (World Scientiﬁc, Singapore, 1998).\n3T. V . Ramakrishnan, Journal of Physics: Condensed Matter 19, 125211 (2007).\n4E. Dagotto, T. Hotta, and A. Moreo, Physics Reports 344, 1 (2001).\n5S. S. Rao, K. N. Anuradha, S. Sarangi, and S. V . Bhat, Applied Physics Letters 87, 182503\n(2005).\n6S. S. Rao, S. Tripathi, D. Pandey, and S. V . Bhat, Phys. Rev. B 74, 144416 (2006).\n7A. Sharma and S. V . Bhat, Applied Magnetic Resonance 33, 11 (2008).\n8C. M. Xiong, J. R. Sun, R. W. Li, S. Y . Zhang, T. Y . Zhao, and B. G. Shen, Journal of Applied\nPhysics 95, 1336 (2004).\n9J. Dho, W. S. Kim, and N. H. Hur, Phys. Rev. Lett. 89, 027202 (2002).\n10I. O. Troyanchuk, M. V . Bushinsky, V . V . Eremenko, V . A. Sirenko, and H. Szymczak, Low\nTemperature Physics 28, 45 (2002).\n11I. Troyanchuk, N. Pushkarev, M. Bushinski, and E. Hamari-Sile, Physics of the Solid State 48,\n1315 (2006).\n12T. Katsufuji, S.-W. Cheong, S. Mori, and C.-H. Chen, Journal of the Physical Society of Japan\n68, 1090 (1999).\n13S. Yamada, E. Sugano, and T. hisa Arima, Journal of Solid State Chemistry 179, 3121 (2006).\n14M. Hervieu, A. Maignan, C. Martin, N. Nguyen, and B. Raveau, Chemistry of Materials 13,\n1356 (2001).\n15A. Kirste, M. Goiran, M. Respaud, J. Vanaken, J. M. Broto, H. Rakoto, M. von Ortenberg,\nC. Frontera, and J. L. Garc ´ıa-Mu ˜noz, Phys. Rev. B 67, 134413 (2003).\n16S. S. Rao and S. V . Bhat, Journal of Nanoscience and Nanotechnology 7, 2025 (2007).\n1217S. Thakur, O. Pandey, and K. Singh, Ceramics International 39, 6165 (2013).\n18M. Itoh, I. Natori, S. Kubota, and K. Motoya, Journal of the Physical Society of Japan 63, 1486\n(1994).\n19H. Woo, T. A. Tyson, M. Croft, and S.-W. Cheong, Journal of Physics: Condensed Matter 16,\n2689 (2004).\n20A. Maignan, C. Martin, F. Damay, B. Raveau, and J. Hejtmanek, Phys. Rev. B 58, 2758 (1998).\n21J. P. Joshi and S. V . Bhat, Journal of Magnetic Resonance 168, 284 (2004).\n22A. Shengelaya, G.-m. Zhao, H. Keller, and K. A. M ¨uller, Phys. Rev. Lett. 77, 5296 (1996).\n23M. T. Causa, M. Tovar, A. Caneiro, F. Prado, G. Iba ˜nez, C. A. Ramos, A. Butera, B. Alas-\ncio, X. Obradors, S. Pi ˜nol, F. Rivadulla, C. V ´azquez-V ´azquez, M. A. L ´opez-Quintela, J. Rivas,\nY . Tokura, and S. B. Oseroff, Phys. Rev. B 58, 3233 (1998).\n24D. L. Huber, G. Alejandro, A. Caneiro, M. T. Causa, F. Prado, M. Tovar, and S. B. Oseroff,\nPhys. Rev. B 60, 12155 (1999).\n25S. L. Yuan, G. Li, Y . Jiang, J. Q. Li, X. Y . Zeng, Y . P. Yang, Z. Huang, and S. Z. Jin, Journal of\nPhysics: Condensed Matter 12, L109 (2000).\n26M. S. Seehra and T. C. Jr., Solid State Communications 8, 787 (1970).\n27D. L. Huber, Phys. Rev. B 6, 3180 (1972).\n28E. A. Zvereva, M. I. Stratan, Y . A. Ovchenkov, V . B. Nalbandyan, J.-Y . Lin, E. L. Vavilova, M. F.\nIakovleva, M. Abdel-Haﬁez, A. V . Silhanek, X.-J. Chen, A. Stroppa, S. Picozzi, H. O. Jeschke,\nR. Valent ´ı, and A. N. Vasiliev, Phys. Rev. B 92, 144401 (2015).\n29S. Bhagat, M. Spano, and J. Lloyd, Solid State Communications 38, 261 (1981).\n30A. Abragam, Principles of nuclear magnetic resonance (Clarendon Press, Oxford, 2006) pp.\n40–42.\n31J. P. Joshi, R. Gupta, A. K. Sood, S. V . Bhat, A. R. Raju, and C. N. R. Rao, Phys. Rev. B 65,\n024410 (2001).\n13"    },    {        "title": "1502.02870v1.Disentangling_relativistic_spin_torques_in_a_ferromagnet_semiconductor_bilayer.pdf",        "content": "Disentangling relativistic spin torques in a ferromagnet/semiconductor bilayer\nT. D. Skinner,1,a)K. Olejn \u0013 \u0010k,2L. K. Cunningham,1H. Kurebayashi,1, 3,b)R. P.\nCampion,4B. L. Gallagher,4T. Jungwirth,2, 4and A. J. Ferguson1,c)\n1)Cavendish Laboratory, University of Cambridge, CB3 0HE,\nUnited Kingdom\n2)Institute of Physics, ASCR, v.v.i., Cukrovarnicka 10, 16253 Praha 6,\nCzech Republic\n3)PRESTO, Japan Science and Technology Agency, Kawaguchi 332-0012,\nJapan\n4)School of Physics and Astronomy, University of Nottingham,\nNottingham NG7 2RD, United Kingdom\n(Dated: 5 April 2022)\na)email: timothy.skinner@ucl.ac.uk; Present address: London Centre for Nanotechnology, University College\nLondon, WC1H 0AH, United Kingdom\nb)Present address: London Centre for Nanotechnology, University College London, WC1H 0AH, United\nKingdom\nc)email: ajf1006@cam.ac.uk\n1arXiv:1502.02870v1  [cond-mat.mes-hall]  10 Feb 2015Recently discovered relativistic spin torques induced by a lateral current at\na ferromagnet/paramagnet interface are a candidate spintronic technology for\na new generation of electrically-controlled magnetic memory devices1,2. Phe-\nnomenologically, the torques have \feld-like and antidamping-like components\nwith distinct symmetries. Microscopically, they are considered to have two\npossible origins. In one picture, a spin-current generated in the paramag-\nnet via the relativistic spin Hall e\u000bect3(SHE) is absorbed in the ferromag-\nnet and induces the spin transfer torque4(STT). In the other picture, a non-\nequilibrium spin-density is generated via the relativistic inverse spin galvanic\ne\u000bect5(ISGE) and induces the spin-orbit torque6{8(SOT) in the ferromagnet.\nFrom the early observations in paramagnetic semiconductors, SHE and ISGE\nare known as companion phenomena that can both allow for electrically align-\ning spins in the same structure9{11. It is essential for our basic physical under-\nstanding of the spin torques at the ferromagnet/paramagnet interface to ex-\nperimentally disentangle the SHE and ISGE contributions. To achieve this we\nprepared an epitaxial transition-metal-ferromagnet/semiconductor-paramagnet\nsingle-crystal structure and performed a room-temperature vector analysis of\nthe relativistic spin torques by means of the all-electrical ferromagnetic reso-\nnance (FMR) technique. By design, the \feld-like torque is governed by the\nISGE-based mechanism in our structure while the antidamping-like torque is\ndue to the SHE-based mechanism.\nThe splitting of the two microscopic mechanisms between the \feld-like and the antidamping-\nlike torque components has not been previously achieved for several conceptual reasons. The\noriginal theoretical proposals12{14and experimental observations10,11,15,16of the ISGE were\nmade in paramagnets with no ferromagnetic component in the structure. The corresponding\nnon-equilibrium spin-density, generated in the ISGE by inversion-asymmetry terms in the\nrelativistic Hamiltonian, has naturally no dependence on magnetization. Hence, in the con-\ntext of magnetic semiconductors6,8,17,18or ferromagnet/paramagnet structures1,7,19{21, the\nISGE may be expected to yield only the \feld-like component of the torque \u0018^M\u0002^\u0010, where\nthe vector ^\u0010is independent of the magnetization vector ^M(see Fig. 1a). However, when car-\nriers experience both the spin-orbit coupling and magnetic exchange coupling, the inversion\nasymmetry can generate a non-equilibrium spin density component of extrinsic, scattering-\n2related22,23or intrinsic, Berry-curvature24{26origin which is magnetization dependent and\nyields an antidamping-like torque \u0018^M\u0002(^M\u0002^\u0010). Experiments in (Ga,Mn)As con\frmed\nthe presence of the ISGE-based mechanism8,17,18and demonstrated that the \feld-like and\nthe Berry-curvature antidamping-like SOT components can have comparable magnitudes25.\nThe STT is dominated by the antidamping-like component4in weakly spin-orbit coupled\nferromagnets with \u001cex\u001c\u001cs, where\u001cexis the precession time of the carrier spins in the\nexchange \feld of the ferromagnet and \u001csis the spin life-time in the ferromagnet. This, in\nprinciple, applies also to the case when the spin current is injected to the ferromagnet from\na paramagnet via the SHE (see Fig. 1b). However, at \fnite \u001cs, the STT also acquires a\n\feld-like component4. Experiments in W/Hf/CoFeB structures con\frmed the presence of\nthe SHE-based mechanism in the observed torques and showed that the SHE-STT can have\nboth antidamping-like and \feld-like components of comparable magnitudes27.\nIn the commonly studied polycrystalline transition-metal-ferromagnet/heavy-metal-\nparamagnet samples, the dependence of the torques on the angle of the driving in-plane\ncurrent also does not provide the direct means to disentangle the two microscopic origins.\nThe lowest order inversion-asymmetry spin-orbit terms in the Hamiltonian have the Rashba\nform for which the vector ^\u0010is in the plane parallel to the interface and perpendicular to\nthe current, independent of the current direction. The same applies to the spin-polarization\nof the SHE spin-current propagating from the paramagnet to the ferromagnet. The ^Mand\n^\u0010functional form of the \feld-like and antidamping-like SHE-STTs is the same as of the\ncorresponding SOT components. In the observed lowest order torque terms in Pt/Co and\nTa/CoFeB structures28the ISGE-based and the SHE-based mechanism remained, therefore,\nindistinguishable. The simultaneous observation of higher order torque terms in these sam-\nples pointed to SOTs due to structural inversion-asymmetry terms beyond the basic Rashba\nmodel. From the Ta thickness dependence measurements in the Ta/CoFeB structure it\nwas concluded that in these samples both the ISGE-based and the SHE-based mechanisms\ncontributed to both the \feld-like and the antidamping-like torques29.\nThe SHE and ISGE were originally discovered in III-V semiconductors9{11,15,16but for\nmaximizing the relativistic spin torques in common transition-metal ferromagnets it turned\nout more suitable to interface them with the highly conductive heavy-metal paramagnets\nlike Pt, Ta, or W1,2,27{29. Rather than enhancing the magnitude of the torques, the aim of\nour study is to clearly separate the two microscopic origins for which returning to the III-V\n3semiconductor paramagnets is instrumental. In experiments presented below we succeed\nto disentangle the torques into the ISGE driven \feld-like component and the SHE driven\nantidamping-like component using a 2 nm thick single crystal Fe grown epitaxially without\nbreaking vacuum on top of a 20 nm thick p-doped GaAs epilayer (see Methods).\nAmong the inversion-asymmetry terms characteristic of our system, the Rashba term\ndue to the structural asymmetry of the bilayer plays a minor role. Instead, the leading role\nis played by the broken inversion symmetry in the crystal structure of the semiconductor\nparamagnet. Independent of the interface, holes in the strained zinc-blende lattice of the\nIII-V semiconductor experience a linear-in-wavevector Dresselhaus spin-orbit coupling. The\nISGE of the corresponding symmetry generated in the semiconductor induces the SOT in the\nadjacent Fe \flm with ^\u0010perpendicular to the current for current along [110] or [1 \u001610] crystal\naxes of the semiconductor, while ^\u0010is parallel to the current for the [100] or [010] current\ndirections. Moreover, the ISGE with this characteristic Dresselhaus symmetry generated in\nthe semiconductor paramagnet induces only the \feld-like torque in the adjacent Fe while\nthe antidamping-like SOT component with this symmetry is absent by design. On the other\nhand, the SHE spin-current can be readily generated inside the paramagnetic p-doped GaAs\nlayer and absorbed in the weakly spin-orbit coupled Fe in the form of the antidamping-like\nSTT.\nTo obtain measurable torques in a metal-ferromagnet/semiconductor-paramagnet struc-\nture requires at least partially matched conductances of the semiconductor and metal layers\nwhich we achieved by doping the GaAs host with \u00183% of substitutional Mn Gaacceptors. Mn\nallows us to achieve this exceptionally high charge-doping. Simultaneously, at 3% Mn, hole\nbands within the (Ga,Mn)As semiconductor do not experience a sizeable enough magnetic\nexchange splitting at room temperature that would generate an observable Dresselhaus-\nsymmetry antidamping-like SOT in our structure.\nWe have measured the relativistic spin torques using the electrically induced and detected\nFMR technique1,18(see Fig. 2a). In this method, a microwave current \rowing in the device\ninduces FMR when the externally applied magnetic \feld matches the resonant condition.\nThe resonance of the Fe magnetisation can be detected in the dc voltage induced across\nthe bar,Vdc. This is due to the homodyne mixing of the microwave current with the\noscillating component of magnetoresistance caused by the magnetisation precession. In\nthese measurements we increase the microwave current coupled into the sample, with a\n4typical resistance of 8 k\n, by using an impedance matching network1.\nCurrent-induced torques were measured in 10 \u0002200\u0016m patterned bars. For a series of\nexternal magnetic \feld directions in the plane of the sample, FMR sweeps were recorded\nusing a microwave frequency of close to 16 GHz. From the magnetisation angle-dependence\nof the resonance \feld, we obtained the magnetisation amplitude value of \u00160Ms= 1:85\u0006\n0:03 T which is close to the literature value of 1.7 T for Fe31, and we found an in-plane\nuniaxial anisotropy of \u00160HU= 0:101\u00060:001 T which is typical for thin \flms of Fe grown\non GaAs32. By solving the Landau-Lifshitz-Gilbert equation for a small current induced\nexcitation \feld, ( hx;hy;hz)ej!t,Vdcis found to be comprised of symmetric and antisymmetric\nLorentzian functions with coe\u000ecients VsymandVasyrespectively. Here hxis the excitation\n\feld component parallel to the current and hyis the component perpendicular to the current.\nThe coe\u000ecients depend on the angle, \u0012, of the magnetisation vector relative to the current\nand are given by\nVsym=VmixAyzsin 2\u0012hz; (1)\nVasy=VmixAyysin 2\u0012(hycos\u0012\u0000hxsin\u0012):\nHere,Vmixis the sensitivity of the mixing detection and is given by Vmix=\u00001\n2I0\u0001R, where\nI0(ej!t;0;0) is the microwave current in the device and \u0001 Ris the coe\u000ecient of the anisotropic\nmagnetoresistance of the sample. AyyandAyzare the diagonal and o\u000b-diagonal components\nof the ac magnetic susceptibility, which depend on the magnetic anisotropies and Gilbert\ndamping of the sample. In our devices, \u0001 Ris typically 17 \n which, assuming Fe carries the\nmajority of the current in the bilayer, is consistent in sign and magnitude with literature\nvalues of 0.2% anisotropic magnetoresistance in Fe.33We estimate the proportion of total\nbilayer current in the Fe layer to be 79% by resistance measurements of Hall bars before and\nafter removing the Fe and capping Al (see supplementary information).\nFMR measurements were made using devices patterned in four crystal directions. The\nmicrowave power for all devices, incident on the impedance matching network, was 24 dBm.\nFor each angle, the resonances were \ftted by symmetric and antisymmetric Lorentzian\nfunctions. A typical curve is shown in Fig. 2b. The in-plane uniaxial magnetic anisotropy\nof Fe implies that, in general, the magnetisation does not lie along the external \feld. The\nactual magnetisation angles and uniaxial anisotropy are self-consistently calculated from\nthe dependence of the resonant \feld on the external magnetic \feld angle34. This also allows\n5the susceptibilities, AyzandAyy, to be calculated. The magnetisation angle dependence of\nVsym=AyzandVasy=Ayyis plotted in Fig. 2c for a bar patterned in the [010] crystal direction.\nOur analysis of the current-induced torques using equation (1) is not necessarily valid if\nthe torques do not act in phase with the microwave current. In comparison to electrically\ndetected FMR measurements where the microwave current is capacitively or inductively\ncoupled into the sample35, we do not expect a phase-shift between the microwave current\nand induced \felds as the current is conducted ohmically. Nevertheless, we might worry that\nsome part of our microwave resonator circuit leads to a phase shift. To test this, we repeated\nour measurements with a [100] device over a frequency range (11.8 to 14.4 GHz) using a\nmicrostrip resonator with a fundamental frequency close to 13 GHz (Fig. 3a). If there were\nsome frequency dependent phase shift, we would expect the lineshape to oscillate between\nan antisymmetric and symmetric Lorentzian over this frequency range. However, the ratio\nofVsymtoVasyremains constant in this frequency range to within experimental error (Fig.\n3b), con\frming that our analysis is correct.\nAs shown in Fig. 4a, the in-plane current-induced \feld depends strongly on the crys-\ntal direction of the current and can be well \ftted by the Dresselhaus-symmetry ISGE\n\feld, hISGE\u0018\u0000\ncos 2\u001e[100];\u0000sin 2\u001e[100];0\u0001\n, where\u001e[100]is the angle between the current\nand the [100] crystal direction. This is the expected symmetry of the current-induced\nnon-equilibrium spin-polarisation of carriers in the semiconductor due to the inversion-\nasymmetric crystal structure of the strained zinc-blende lattice of (Ga,Mn)As. The interface\nexchange coupling of these polarized carriers with the adjacent Fe moments induces the \feld-\nlike SOT in Fe with the Dresselhaus symmetry. We note that other torque terms with the\nsymmetry common to the Rashba ISGE, the \feld-like component of the SHE-STT, or the\ntorque due to an Oersted \feld have only a minor contribution to the total measured \feld-like\ntorque.\nTo highlight that carriers in the semiconductor layer are responsible for the Dresselhaus-\nsymmetry ISGE \feld, we compare Fig. 4a with previous measurements in which the in-plane\ncurrent induced \felds were measured in a bare (Ga,Mn)As epilayer25without the Fe \flm.\nTo observe the corresponding SOT in this sample, a larger concentration of magnetic Mn-\nmoments was used, and the measurements were performed at low temperatures where the\nMn moments are ferromagnetic in equilibrium. Instead of the interfacial exchange coupling\nto Fe, the current induced non-equilibrium spin-polarisation of carriers in the semiconduc-\n6tor due to the Dresselhaus-symmetry ISGE is exchange-coupled directly to the ferromag-\nnetic moments on which it exerts the \feld-like SOT. In both the Fe/(Ga,Mn)As and the\n(Ga,Mn)As samples the same crystal-symmetry \feld-like SOT is observed which con\frms\ntheir common Dresselhaus ISGE origin.\nIn contrast to the in-plane \feld, the out-of-plane current induced \feld is independent of\nthe crystal direction of the current but depends on the magnetisation angle. It is dominated\nby a term hSHE\u0018^M\u0002^ y(^ yis the direction perpendicular to the current) which generates\nthe antidamping-like torque. As shown in Fig. 4b, the amplitudes of the \feld-like and\nantidamping-like torques are comparable in our Fe/(Ga,Mn)As structure. The underlying\nmicroscopic mechanism of the antidamping-component can only be of the SHE-STT origin.\nIn previous measurements in the bare (Ga,Mn)As epilayer,25the antidamping-like SOT\nwas dominated by the counterpart microscopic mechanism to the Dresselhaus ISGE. This\nDresselhaus-symmetry antidamping-like SOT is clearly missing in our measured data. It is\nsuppressed in our Fe/(Ga,Mn)As structure by design because carriers in the semiconductor\nare not su\u000eciently magnetized at equilibrium due to the low Mn moment density and high\ntemperature of the experiment.\nA Rashba-symmetry antidamping-like SOT due to the carriers in the Fe experiencing\nthe inversion-asymmetry of the interface could in principle also explain our measured data.\nThis antidamping SOT would have the same symmetry as the antidamping SHE-STT. This\npossibility is, however, ruled out by our control experiment in which we perform electri-\ncally detected FMR in a similar MBE-grown Fe (1 nm)/insulating GaAs structure at room\ntemperature. In this case, we do not observe the anti-damping torque in the recti\fca-\ntion e\u000bect, despite the sample possessing a similar magnetoresistance ratio ( \u00180.2%) to our\nFe/(Ga,Mn)As. This is consistent with the carriers being removed from the semiconductor\nwhich eliminates the SHE source of the spin-current. We note that also consistently with\nthe absence of carriers in the semiconductor in the Fe/insulating-GaAs structure, we do not\nobserve the Dresselhaus-symmetry \feld-like SOT in this control sample.\nTo calibrate the microwave current in the sample we used a bolometric technique (see sup-\nplementary information). Using this calibration, we estimate amplitudes of j\u00160hISGE=JGaAsj=\n16\u000610\u0016T/106Acm\u00002andj\u00160hSHE=JGaAsj= 20\u00069\u0016T/106Acm\u00002. The error is found from\nthe statistical variation from all of the devices measured. To verify the bolometric calibra-\ntion, we also perform an additional check with a single device by measuring the change in\n7Q-factor of the microstrip resonator loaded with and without a sample (see supplementary\ninformation). This calibration yields values of j\u00160hISGE=JGaAsj= 37\u0016T/106Acm\u00002and\nj\u00160hSHE=JGaAsj= 47\u0016T/106Acm\u00002, close to the values of the bolometric technique.\nFrom the measured hSHEin our Fe/(Ga,Mn)As structure we can infer the room-temperature\nspin Hall angle, \u0012SH, in the paramagnetic (Ga,Mn)As using the expression based on the\nantidamping-like STT2,\n\u0012SH=2e\n\u0016h\u00160MsdFehSHE\nJGaAs: (2)\nHere it is assumed that the thickness of the semiconductor is much larger than its spin\ndi\u000busion length (5.6 nm in p-GaAs36) anddFeis the thickness of the Fe layer. Eq. (2) yields\nvalues of\u0012SH= 1:7\u00060:9% (bolometric calibration) and 4% (Q-factor calibration), similar\nto spin Hall angles previously reported for carriers with p-orbital character in GaAs36,37. To\ncheck these spin-Hall values we compare our torque e\u000eciencies in terms of \feld per total\ncurrent density with those of transition-metal/ferromagnet bilayers. For instance, Garello\net al. measured an antidamping-like torque of j\u00160h=Jj= 690\u0016T/106Acm\u00002in layers with\n0.6 nm Co and 3 nm Pt and an equivalent spin-Hall angle of 16%.28Although our spin-Hall\nangle is only 4-8 times smaller, the \feld per total current density is 85-170 times smaller\nbecause the total magnetic moment of the Fe is \u00184 times higher than of the Co and \u001880%\nof the total current is shunted through the Fe.\nTo conclude, we have experimentally disentangled the two archetype microscopic mecha-\nnisms that can drive relativistic current-induced torques in ferromagnet/paramagnet struc-\ntures. In our epitaxial Fe/(Ga,Mn)As bilayer we simultaneously observed ISGE-based and\nSHE-based torques of comparable amplitudes. Designed magnetization-angle and current-\nangle symmetries of our single-crystal structure allowed us to split the two microscopic\norigins between the \feld-like and the antidamping-like torque components. Experimentally\nestablishing the microscopic physics of the relativistic spin torques should stimulate both the\nfundamental and applied research of these intriguing and practical spintronic phenomena.\nMethods\nThe semiconductor (Ga,Mn)As layer of thickness 20 nm was deposited on a GaAs(001)\nsubstrate at a temperature of 260\u000eC. The substrate temperature was then reduced to 0\u000eC,\nbefore depositing a 2 nm Fe layer, plus a 2 nm Al capping layer. In-situ re\rection high energy\nelectron di\u000braction and ex-situ x-ray re\rectivity and di\u000braction measurements con\frmed\n8that the layers are single-crystalline with sub-nm interface roughness.\nTo improve the sensitivity of the FMR measurement, the sample is embedded in a mi-\ncrostrip resonator circuit1which acts to impedance match the \u00188 k\n sample to the external\n50 \n transmission line at the fundamental frequency (in this case \u00188 GHz) or harmonic\nfrequencies of the resonator. To allow measurement of Vdc, the resonator contains an on-\nboard bias-T. In this experiment, FMR measurements are made at the the 2ndharmonic\nfrequency of the resonator ( \u001816 GHz).\nAcknowledgements\nThe authors acknowledge support from EU European Research Council (ERC) ad-\nvanced grant no. 268066, from the Ministry of Education of the Czech Republic grant no.\nLM2011026, from the Grant Agency of the Czech Republic grant no. 14-37427G and the\nAcademy of Sciences of the Czech Republic Praemium Academiae. A.J.F. acknowledges\nsupport from a Hitachi research fellowship. H.K. acknowledges \fnancial support from the\nJapan Science and Technology Agency (JST).\nAuthor Contributions\nK.O., R.P.C., B.L.G. and H.K. grew the materials. K.O. and H.K. prepared the samples.\nT.D.S., L.K.C. and H.K. performed the experiments and T.D.S. analysed the data. T.D.S.,\nT.J. and A.J.F. prepared the manuscript. T.D.S. and A.J.F. planned the project.\nREFERENCES\n1Miron, I. M. et al. Perpendicular switching of a single ferromagnetic layer induced by\nin-plane current injection. Nature 476, 189 (2011).\n2Liu, L. et al. Spin-torque switching with the giant spin Hall e\u000bect of tantalum. Science\n336, 555 (2012).\n3Jungwirth, T., Wunderlich, J. & Olejnik, K. Spin Hall e\u000bect devices. Nature Mater. 11,\n382 (2012).\n4Ralph, D. C. & Stiles, M. D. Spin transfer torques. J. Magn. Magn. Mater. 320, 1190\n(2008).\n5Ivchenko, E. L. & Ganichev, S. Spin-photogalvanics. In Dyakonov, M. I. (ed.) Spin Physics\nin Semiconductors , vol. 157 of Springer Series in Solid-State Sciences , 245 (Springer, New\nYork, 2008).\n96Bernevig, B. A. & Vafek, O. Piezo-magnetoelectric e\u000bects in p-doped semiconductors.\nPhys. Rev. B 72 , 033203 (2005).\n7Manchon, A. & Zhang, S. Theory of nonequilibrium intrinsic spin torque in a single nano-\nmagnet. Phys. Rev. B 78 , 212405 (2008).\n8Chernyshov, A. et al. Evidence for reversible control of magnetization in a ferromagnetic\nmaterial by means of spin-orbit magnetic \feld. Nature Phys. 5, 656 (2009).\n9Kato, Y. K., Myers, R. C., Gossard, A. C. & Awschalom, D. D. Observation of the spin\nHall e\u000bect in semiconductors. Science 306, 1910 (2004).\n10Kato, Y., Myers, R. C., Gossard, A. C. & Awschalom, D. D. Current induced electron\nspin polarization in strained semiconductors. Phys. Rev. Lett. 93, 176601 (2004).\n11Wunderlich, J., Kaestner, B., Sinova, J. & Jungwirth, T. Experimental observation of the\nspin Hall e\u000bect in a two dimensional spin-orbit coupled semiconductor system. Phys. Rev.\nLett.94, 047204 (2005).\n12Aronov, A. G. & Lyanda-Geller, Y. B. Nuclear electric resonance and orientation of carrier\nspins by an electric \feld. JETP Lett. 50, 431 (1989).\n13Edelstein, V. M. Spin polarization of conduction electrons induced by electric current in\ntwo-dimensional asymmetric electron systems. Solid State Commun. 73, 233 (1990).\n14Mal'shukov, A. G. & Chao, K. A. Optoelectric spin injection in semiconductor heterostruc-\ntures without a ferromagnet. Phys. Rev. B 65 , 241308 (2002).\n15Silov, A. Y. et al. Current-induced spin polarization at a single heterojunction. Appl.\nPhys. Lett. 85, 5929 (2004).\n16Ganichev, S. D. et al. Can an electric current orient spins in quantum wells (2004).\narXiv:cond-mat/0403641.\n17Endo, M., Matsukura, F. & Ohno, H. Current induced e\u000bective magnetic \feld and mag-\nnetization reversal in uniaxial anisotropy (Ga,Mn)As. Appl. Phys. Lett. 97, 222501 (2010).\n18Fang, D. et al. Spin-orbit driven ferromagnetic resonance: A nanoscale magnetic charac-\nterisation technique. Nature Nanotech. 6, 413 (2011).\n19Miron, I. M. et al. Current-driven spin torque induced by the Rashba e\u000bect in a ferromag-\nnetic metal layer. Nature Mater. 9, 230 (2010).\n20Pi, U. H. et al. Tilting of the spin orientation induced by Rashba e\u000bect in ferromagnetic\nmetal layer. Appl. Phys. Lett. 97, 162507 (2010).\n21Suzuki, T. et al. Current-induced e\u000bective \feld in perpendicularly magnetized\n10Ta/CoFeB/MgO wire. Appl. Phys. Lett. 98, 142505 (2011).\n22Pesin, D. A. & MacDonald, A. H. Quantum kinetic theory of current-induced torques in\nrashba ferromagnets. Phys. Rev. B 86 , 014416 (2012).\n23Wang, X. & Manchon, A. Di\u000busive spin dynamics in ferromagnetic thin \flms with a\nRashba interaction. Phys. Rev. Lett. 108, 117201 (2012).\n24Garate, I. & MacDonald, A. H. In\ruence of a transport current on magnetic anisotropy\nin gyrotropic ferromagnets. Phys. Rev. B 80 , 134403 (2009).\n25Kurebayashi, H. et al. An anti-damping spinorbit torque originating from the berry cur-\nvature. Nature Nanotech. 9, 211 (2014).\n26Freimuth, F., Bl ugel, S. & Mokrousov, Y. Spin-orbit torques in Pt/Co \flms from \frst\nprinciples (2013). arXiv:1305.4873.\n27Pai, C. et al. Enhancement of perpendicular magnetic anisotropy and transmission of\nspin-hall-e\u000bect-induced spin currents by a Hf spacer layer in W/Hf/CoFeB/MgO layer\nstructures. Appl. Phys. Lett. 104, 082407 (2014).\n28Garello, K. et al. Symmetry and magnitude of spin-orbit torques in ferromagnetic het-\nerostructures. Nature Nanotech. 8, 587 (2013).\n29Kim, J. et al. Layer thickness dependence of the current-induced e\u000bective \feld vector in\nTa|CoFeB|MgO. Nature Mater. 12, 240 (2013).\n30Fang, D. et al. Electrical excitation and detection of magnetic dynamics with impedance\nmatching. Appl. Phys. Lett. 101, 182402 (2012).\n31Danan, H., Herr, A. & Meyer, A. J. P. New determinations of the saturation magnetization\nof nickel and iron. J. Appl. Phys. 39, 669 (1968).\n32Brockmann, M., Zol\r, M., Miethaner, S. & Bayreuther, G. In-plane volume and interface\nmagnetic anisotropies in epitaxial Fe \flms on GaAs(001). J. Magn. Magn. Mater. 198-199 ,\n384 (1999).\n33McGuire, T. & Potter, R. Anisotropic magnetoresistance in ferromagnetic 3d alloys. IEEE\nTrans. Magn. 11, 1018 (1975).\n34Ando, K. et al. Angular dependence of inverse spin-Hall e\u000bect induced by spin pumping\ninvestigated in a NiFe/Pt thin \flm. Phys. Rev. Lett. 78, 014413 (2008).\n35Harder, M., Cao, Z. X., Gui, Y. S., Fan, X. L. & Hu, C. M. Analysis of the line shape of\nelectrically detected ferromagnetic resonance. Phys. Rev. B 84 , 054423 (2011).\n36Chen, L., Matsukura, F. & Ohno, H. Direct-current voltages in (Ga,Mn)As structures\n11induced by ferromagnetic resonance. Nat. Commun. 4, 2055 (2013).\n37Okamoto, N. et al. Electric control of the spin hall e\u000bect by intervalley transitions. Nature\nMater. 13(2014, advance online publication).\n(a)\nJc\nJshISGE\n(b)FM\nPM\nFM\nPM\nJcM\nhSHE\nM\nFIG. 1. (a) In the GaAs layer, as charge carriers are accelerated by an applied electric \feld\nin the inversion-asymmetric crystal potential, scattering events lead to a non-equilibrium spin-\npolarisation. The spin-polarisation depends on the direction of the current with respect to crystal\ndirection. Through exchange coupling at the interface, the magnetisation experiences an e\u000bec-\ntive \feld,hISGE, parallel to the spin-polarisation. (b) In the GaAs layer, when a longitudinal\ncharge current is applied, a transverse spin-current is induced by the SHE, which \rows into the Fe\nlayer. The spin-current exerts a torque that depends on the magnetisation angle. For an in-plane\nmagnetisation this is described by an out-of-plane e\u000bective \feld, hSHE.\n12Fe\n(Ga,Mn)Asxz\nyθ\nθ (deg)V/A ij (μVT)\nVdc (μV)\nμ0H (T)(a)\n(b)\n(c)\n0.04\n0.06\n0.08\n0.10\n0.12\n-2\n0\n2\n4\n6\n0\n90\n180\n270\n360\n-0.04\n-0.02\n0.00\n0.02\n0.04\nVasy\nVsymFIG. 2. (a) Schematic of the measurement showing magnetisation precession. A microwave current\npasses through the capacitor of the bias-T and into the Fe/GaAs bilayer sample where the current-\ninduced torques causes the magnetisation of the Fe to precess around the external \feld. The\nprecession is detected in the dc voltage measured across the device by a magnetoresistance mixing\ne\u000bect. (b) A typical SO-FMR curve detected in Vdcas a function of external \feld, induced by a\n16.245 GHz microwave current. Vdcis \ftted by a combination of symmetric (red dotted line) and\nantisymmetric (blue dashed line) Lorentzians. (c) The in-plane magnetisation angle dependence of\nthe \ftted Lorentzian amplitudes for a device with current in the [010] crystal direction. The full\nexpression for the angle-dependence of the \ftted data is given in the supplementary information.\n1312\n13\n14\n-0.2\n-0.1\n0.0\nf9(GHz)Vdc9(normalised) Vsym/Vasy(a)\n(b)\n0.00\n0.04\n0.08\n-0.6\n-0.3\n0.0\n0.3\n0.6\n14.39GHz\n11.9FIG. 3. (a) Normalised, \ftted, resonant peaks for a series of microwave frequencies detected in\nVdc. The line-shape is dominated by an antisymmetric Lorentzian for every frequency. (b) The\nratio ofVsymtoVasy(blue circles) is constant to within the the standard error of a linear \ft (red\ndashed line) for the measured frequency range. The error in the ratio is bigger away from the\nimpedance-matched frequency (13 GHz) of the microstrip resonator used as the detected peak in\nVdcis smaller.\n14current]direction\nVasy]/Ayy]](μVT)\nVsym]/Ayz]](μVT)VmixhSHE/cos(θ)-0.04-0.020.000.02\nVmixhxISGE\nVmixhyISGE\n[100] [110] [010] [110]0.000.020.040.06(a)\n(b)\nFIG. 4. (a) The \ftted in-plane \feld coe\u000ecients for a set of devices in di\u000berent crystal directions.\n(b) The \ftted out of plane \feld coe\u000ecient (representing the antidamping torque) for the same\ndevices.\n15Supplementary Information\nS1. IMPEDANCE MATCHING NETWORK\nWe use a gap-coupled microstrip resonator, similar to one previously reportedS1, to\nimpedance match the \u00188 k\n sample to the 50 \n transmission line. In contrast to the\nnetwork previously reported, the sample is attached to the opposite end of the resonator\nwith a wirebond. At the resonant frequency of the resonator, the impedance of the network\nbecomes real and is given by\nZ\u00191\nR!2C2\nk: (S1)\nCkis the capacitance of the interdigitated gap capacitor and R is the resistance of the\nsample. By choosing a suitable value of Ck, at the resonant frequency Z\u001950\n and most\nof the microwave power is transmitted from the transmission line to the matched network.\nFor the FMR measurements, we use the resonator at twice its fundamental frequency, \u0018\n16 GHz, where the power is also matched to the sample. To measure Vdc, a wirebond is\nattached at1\n4of the length of the resonator. This causes little perturbation of the resonator\nmode, as at this point a node of electric \feld exists.\nS2. Q FACTOR CALIBRATION OF MICROWAVE CURRENT\nClose to the resonance, the microstrip resonator can be approximated as a series resonator.\nThe quality factor, Q, of the resonator can then be described by\n1\nQ=1\nQsample+1\nQboard; (S2)\nwhereQsample representsQdue to power dissipated in the sample and Qboard represents\ntheQdue to power dissipated through losses in the PCB. When connected to an external\nnetwork, the e\u000bect of dissipation externally can be described by a total Qfactor\nQtotal=Q\n(1 +g); (S3)\nwhere g is a coupling constant that describes the impedance matching of the resonator, given\nbyg=Z0=ZandZ0= 50 \n is the the characteristic impedance of the transmission line.\n1directional\ncouplerLO\nRFIQ\nmicrostrip\nresonatormicrowave \ndiodea bFIG. S1. Microwave set-ups for measuring (a) the re\rected power and (b) the re\rected phase from\nthe microstrip resonator network.\nBoth Q and g can be found from measuring the re\rected power from the resonator network\naround the resonant frequency. At resonance, the re\rection coe\u000ecient is given by\nj\u0000j2=\u00121\u0000g\n1 +g\u00132\n: (S4)\nWe can also determine the dissipation of the resonant circuit through the width of the\nabsorption peak, with Qtotalgiven by\nQtotal=!\n\u0001!FWHM(S5)\nwhere \u0001!FWHM is the full width at half maximum of the absorption peak.\nTo perform a calibration of the microwave current in a sample, we use a directional\ncoupler and calibrated microwave diode (Fig. S1a) to measure the re\rected signal with and\nwithout the resonator loaded by the sample. If we replace the diode with a microwave mixer\n(Fig. S1b), we can measure the phase change of the re\rected signal around the resonance\nof the microstrip resonator. We use the result that at resonance, the gradient of phase is\ngiven by\n@\u001e\n@!\f\f\f\f\n!0=4Q\n!0g\n(1\u0000g2); (S6)\nwhich allows us to determine the value of Qwhen the resonator is not loaded by the sample\nand the resonance is hard to observe in re\rected power. We calibrate a microstrip resonator\n20.0\n0.5\n1.0\n8\n12\n16\n0\n3\n6\nfrequency (GHz)a\nb\nunloaded\nloaded\nfit\n|Γ|2\nϕ (radians)FIG. S2. (a) magnitude and (b) phase of the re\rected power showing a resonance at around 13\nGHz (blue dashed line) for the loaded and unloaded resonator. The full 2 \u0019phase change shows\nin both cases the resonator is overcoupled ( g>1). The visible absorption peak in re\rected power\nwhen the resonator is loaded with the sample shows that the microwave power is better matched\nat resonance.\nat its fundamental frequency, as this has previously been studied. To achieve a high enough\nfrequency for our FMR measurements, we reduce the length of the resonator so that the\nfundamental frequency is around 13 GHz.\nThe measured magnitude and phase of the re\rected signal is shown in Fig. S2a and b\nrespectively. From the peak in absorption we estimate that for the loaded resonator, g= 4:3.\nAlthough we cannot make out a clear peak from the noise for the unloaded case, we can put\na lower limit on g>20. From the gradient of the phase and the calculated values for g, we\n\fnd thatQsample<< Q board and so assume all of the power transmitted to the microstrip\nresonator is dissipated in the sample. On resonance, using the re\rection measurements we\nthen estimate, for a 10 dBm source power, that the microwave current in the device is 0.74\nmA.\n30.0\n0.1\n0.2\n0.3\n8420\n8440\n8460\n8480\n8500\ndc\nmwPower (W)Resistance (Ω)FIG. S3. Joule heating calibration for a [010] bar with 20 nm (Ga,Mn)As, showing the dependence\nof device resistance on dc and source microwave power. The ratio of the gradients (dashed lines)\nallows the fraction of transmitted microwave power to be calculated.\nWe perform FMR measurements at 10 dBm in a [100] bar as discussed in the main text\nand, using the value of the calibrated microwave current, \fnd values of j\u00160hISGE=JGaAsj= 37\n\u0016T/106Acm\u00002andj\u00160hSHE=JGaAsj= 47\u0016T/106Acm\u00002.\nS3. JOULE HEATING CALIBRATION OF MICROWAVE CURRENT\nThe microwave current can also be calibrated by a simple Joule heating method as the re-\nsistance of the semiconductor is sensitive to temperature. This calibration was performed for\neach device measured. To determine the microwave current in the device for a given source\npower, the change in resistance due to the heating of the microwave current is compared to\nthe change due to heating of dc current.\nThe dc current is applied from the bias-T of the resonator, through the sample, to\nground. An applied dc voltage is held for 10 seconds at increasing values before the current\nis recorded. The resistance is then calculated as a function of dc current. Microwave power\nis then applied at increasing increments and held for 10 seconds at each value as before, be-\nfore the resistance is measured from a small dc bias applied concurrently. We then compare\nthe gradients of resistance with microwave and dc power (Fig. S3). From the ratio of these\n4gradients we estimate the proportion of microwave power dissipated in the sample.\nS4. FMR FITTING COEFFICIENTS\nOur analysis in the main text only considers the dominant \feld components found from\n\ftting the magnetisation angle dependence of Vdc. The actual expression for the \ftted angle\ndependence for all the samples is given by\nVsym\nAyz=Csin\u0012sin\u0012+Vmix\u0000\nh0\nz+hcos\u0012\nzcos\u0012+hsin\u0012\nzsin\u0012\u0001\nsin 2\u0012; (S7)\nVasy\nAyy=C0+Vmix(hycos\u0012\u0000hxsin\u0012) sin 2\u0012+Csin 2\u0012sin 2\u0012:\nCsin\u0012,C0andCsin 2\u0012are small additional coe\u000ecients which are empirically needed to \ft the\ndata. Furthermore, hcos\u0012\nzandhsin\u0012\nzare magnetisation angle dependent coe\u000ecients of the hz\ne\u000bective \feld. The dominant e\u000bective \felds in each of the samples are the in-plane \felds,\nhxandhy, originating from the ISGE and the magnetisation angle dependent out-of-plane\n\feld,hcos\u0012\nz, which corresponds to the anti-damping SHE-STT. The analysis in the main text\nonly considers these terms. Compared to the notation in the main text, jhSHEj\u0011hcos\u0012\nzcos\u0012\nandhxandhyare the vector components of hISGE.\nFor completeness we present here a table of all the components as well as the AMR\ncoe\u000ecient and calibrated microwave current used to determine Vmix. Table S1 shows the\nvalues for all of the devices.\n5Direction\n[100] [110] [010] [1 \u001610]\nVasy=Ayy(nVT)\u00160hISGE\nxVmix 28.8 2.4 -22.3 0.8\n\u00160hISGE\nyVmix -11.8 -47.0 -7.1 28.5\nC0 -2.6 -4.7 -1.2 -1.1\nCsin 2\u0012 5.0 -15.8 0.2 0.8\nVsym=Ayz(nVT)\u00160hSHEVmix=cos\u001239.9 43.1 32.5 49.6\n\u00160h0\nzVmix 16.4 -4.3 -3.7 2.2\n\u00160hsin\u0012\nzVmix 4.4 12.9 13.6 -2.9\nCsin\u0012 -19.8 -4.5 -16.1 -16.1\n\u0001R(\n) 17.7 15.4 17.5 16.5\nI0(mA) 3.5 2.2 1.8 1.8\nTABLE S1. Measurement parameters and all \ftted detection coe\u000ecients of the\nFe(2 nm)/Ga 0:097Mn0:03As (20 nm) devices.\nS5. DETERMINING THE RESISTIVITY OF EACH LAYER\nTo \fnd the resistivity of the individual layers, the resistivity of a Fe (2 nm)/(Ga,Mn)As\n(10 nm) patterned Hall bar was measured before and after the Fe and Al capping layer were\nchemically removed. To remove the oxidised Al capping layer, the sample was immersed\nfor 15 s in a solution of 1:10 HCl:H 2O. The Fe layer was then subsequently dissolved by\nimmersion for 15 s in MF319 photodeveloper. The change in sheet resistance from 523 \n =sq\nto 4534 \n=sq after removal of the metal layers, where the (Ga,Mn)As layer is 10 nm thick,\nindicates that only 12% of the microwave current \rows through the (Ga,Mn)As layer. The\nequivalent proportion for the 20 nm layers is 21%.\nREFERENCES\n[S1]Fang, D. et al. Electrical excitation and detection of magnetic dynamics with impedance\nmatching. Appl. Phys. Lett. 101, 182402 (2012).\n6"    },    {        "title": "1505.01729v1.Resonant_driving_of_magnetization_precession_in_a_ferromagnetic_layer_by_coherent_monochromatic_phonons.pdf",        "content": "1 \n Resonant driving of magnetization precession in a ferromagnetic layer  \nby coherent monochromatic phonons  \nJ. V. Jäger1, A. V. Scherbakov2, B. A. Glavin3, A. S. Salasyuk2, R. P. Campion4,  \nA. W. Rushforth4, D. R. Yakovlev1,2, A. V. Akimov4, and M. Bayer1,2 \n1Experimentelle Physik 2, Technische Universität Dortmund, D -44227 Dortmund, Germany  \n2Ioffe Physical -Technical Institute, Russian Academy of Science, 194021 St. Petersburg, Russia  \n3Department of Theoretical Physics, Lashkaryov  Institute of Semiconductor Physics,  \n03028 Kyiv, Ukraine  \n4School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, UK  \n \n \n \n \nABSTRACT  \nWe realize resonant driving of the magnetization precession by monochromatic phonons in a thin \nferromagnetic layer embedded into a phononic Fabry -Perot resonator. A femtosecond laser pulse \nexcites resonant phonon modes of the structure in the 10 -40 GHz fre quency range. By applying an \nexternal magnetic field, we tune the precession frequency relative to the frequency of the phonons \nlocalized in the cavity and observe the enormous increase in the amplitude of the magnetization \nprecession when the frequencies of free magnetization precession and phonons localized in the \ncavity are equal.  \n \nPACS: 75.78.Jp , 63.22. -m \n  2 \n The continual miniaturization of magnetic devices down to the nanometer scale has opened new \nhorizons in data storage [1], computing [ 2,3], sensing [4,5], and medical technologies [ 6]. Progress \nin nanomagnetism is stimulated by emerging technologies, where methods to control magnetic \nexcitations on the nanometer spatial and picosecond temporal  scales include optical [ 7,8], \nelectrical [ 8], and micro mechanical [ 9] techniques. To realize ultrafast nanomagnetism on the \ntechnological level, new physical principles to efficiently induce and control magnetic excitations \nare required, and this remains a challenging task. A new basic approach to this problem  would be \nto explore nanoscale magnetic resonance phenomena: resonant driving and monitoring of \nmagnetic excitation s, which is widely used nowadays in traditional magnetism for microscopy, \nmedicine and spectroscopy. The typical frequencies , fM, of magnetic  resonances  [e.g. the \nferromagnetic resonance (FMR) in ferro - and ferrimagnetic materials], ,  are in the GHz and sub -\nTHz frequency range s. The traditional methods to scan magnetic excitations at these frequencies \nuse microwaves, but due to the requirement of massive microwave resonators  providing long \nwavelength radiation , they cannot provide high -speed control of magnetization locally on the \nnanoscale.  \nAmong various  emerging  techniques  in nanoma gnetism , the application of  stress to \nmagnetostrictive  ferromagnetic layers  has been  show n to be an  effective, low power  method  for \ncontrolling magnetization:  applying in plane  stress  in stationary experiments enables  irreversible \nswitch ing of the  magnetization  vector [ 10]; the injection of pic osecond strain pulses induce s free \nprecession of the magnetization [ 11]; excitation of quantized elastic waves in a membrane  enables \ndriving of the magnetization at GHz phonon frequencies  [12]; and surface acoustic waves can be \nused to control the magnetic dynamics in periodic ferromagnetic nan ostructure s [13,14 ]. In the \npresent work , we examine the interaction of a high-frequency  (10-40 GHz)  magnetic resonance in \na magnetostrictive ferromagnetic film with an elastic  harmonic  excitation  in the form of a localized \nphonon mode , and demonstrate how this interaction becomes significantly stronger  at the \nresonance conditions .  \nOur device consists of a ferromagnetic layer  embedded  into a phonon  Fabry -Perot  (FP) \ncavity . Such a cavity  possesses quantized resonances for elastic waves  (i.e. phonons)  at frequencies \nifR\n (I is the order of the phonon resonance). I n the experiments , we excite the se resonant  modes \noptically by the methods of p icosecond acoustics [15 ]. As it was shown earlier in experiments with \npicosecond strain pulses , coherent  phonons induce free precession of the magnetization  at the \nferromagnetic resonance  frequency  fM [11,16]. By the application of an  external magnetic field, B, \nwe tune the  frequency fM into resonan ce with  the phonon mode ,\nif fR M , and monitor the 3 \n precession of the mag netization in the ferromagnetic layer . We observe an enormous increase of \nthe magnetization precession lifetime and the spectral density at \nMf  when  B correspond s to \nmatching the resonant conditions  of the magnetization precession and the phonon cavity  mode . \nThe studied structure  is shown  schematically  in Fig. 1  (a). A 59 nm thick f erromagnetic \nlayer  of Galfenol (alloy of 81%Fe and 19%Ga [17,18]) capped  by a 3 -nm Al layer ( to prevent \noxidation) is deposited by magnetron sputtering on to acoustic Bragg mirrors. These  mirror s are \nformed by two superlattices (SL1 and SL2) each consisting of 10 periods  of GaAs/AlAs bilayers  \nwith thicknesses (in nm) 59/71 and 42/49 in SL1 and SL2 , respectively.  The SLs are grown by \nmolecular beam epitaxy on a semi -insulating  (001) GaAs  substrate. The Galfenol layer plays the \nrole of a FP cavity between two flat phonon mirrors: one mirror is the free surface and another  is \nthe corresponding SL1 or SL2  Bragg mirror . As SL2 is positioned below SL1 , the cavity with the \nSL2 Bragg mirror  therefore includes also all layers of SL1. The studied  multi layer structure \npossess es a number of localized phonon modes  with frequencies\nifR , which fall into the stop -bands \nof the SLs  [19-21]. Figure s 1 (b) and 1 (c) show the dispersion curves  calculated  for longitudinal \nphonons in  the studied SL1 and SL2 , respectively. The lowest  phonon  stop-bands in SL1 and SL2 \nare centered at 20 GHz and 2 8 GHz , respectively. The horizontal bars in the zoomed fragments  of \nFigs. 1 (b) and 1 (c) indicate the  calculated  frequenc ies of the lowest phonon  FP modes  for the \nSL1 cavit y (\n1Rf  20.0 GHz)  and the SL2 cavity (\n2Rf 28.0 GHz and \n3Rf 29.5 GHz) . The \ncalculation of the FP phonon modes  is described elsewhere [22]. \nFigure 1 (d) shows the se phonon modes as well as  magnetization  precession frequency vs \napplied magnetic field for  the studied sample . The s quares show the measured dependence of the \nprecession frequency in the studied Galfenol film on B, interpolated linear ly by the solid line , \nwhich  is in  good agreement with previous experiments [ 18,23]. The h orizontal dashed lines \nindicate the frequencies \nifR  of the phonon modes  in the SL1 and SL2 cavities . The intersections  \nof the solid and dashed lines give the resonances\ni M f BfR , which occur  at particular resonance \nmagnetic field s B=BRi marked in Fig. 1 (d) by the vertical arrows.  \nThe experimental scheme is shown in Fig. 1 ( a). The FP phonon cavity with the Galfenol \nlayer  was excited by  optical pump pulses from an  amplified Ti:Sapphire laser ( duration 200 fs, \nwavelength of 800 nm, repetition rate 100 kHz ) focused to a spot with diameter 100 μm at the \nsample  surface . The maximum e nergy density of the pump pulse was ~ 10 mJ/cm2. The probe \npulses of lower density  (~ 20 μJ/cm2) split from the same laser and passed through an optical delay \nline were used to measure  the transient magneto -optical Kerr rotation  for monitoring  the temporal 4 \n evolution of the  changes  ΔMz(t) of the z-projection of the macroscopic magnetization  M of the \nGalfenol layer, i.e. of the magnetization component normal to the surface  [24]. The sample was \nmounted in a helium cryostat with superconducting magnet. The experiments were performed at \nvarious temperature s up to room temperature  and most of  the results presented in the paper were \nobtained at T=170 K.  The external magnetic field up to 700 mT was applied in the plane of the \nGalfenol layer , B||[100], which is close to the easy magnetization axis  [10,18 ]. \nThe temporal evolution of the detected signals \ntMz  measure d for three  values of B are \nshown in Fig. 2  (a). The signals possess  oscillatory behavior with the period, amplitude and \nlifetime of the oscillations dependent on B. The most important result is the existence of a high-\namplitude long -lived tail in the middle curve of  Fig. 2 (a) taken  at B=190 mT.  This tail cannot be  \nobserved  above  the noise level at lower and higher fields  (see the lower and upper curves , \nrespectively).  \nTo further inspect the data , we use a s pectral domain presentation . Fast Fourier transform  \nspectra  (FFTs)  of \ntMz  for various  B are shown in Fig. 2 (b) in a frequency range 15 - 25 GHz . \nThese spectra show a main spectral line centered at f1=20.5 GHz which , taking into account the \naccuracy of determining the SL parameters, may be unambiguously associated with the localized \nphonon mode  in the  FP cavity  at \n1Rf =20.0 GHz. The amplitude of th is spectral line changes \nconsiderably with  B: Fig. 2 (c) shows the  B-dependence of th is amplitude where  we clearly see  \nthat the amplitude maximum takes place at B=180 mT . Fig ure2 (b) also show s a broader spectral \nline with lower amplitude whose  position  shifts to higher frequencies with the increase of B. This \nline corresponds to the  fast decaying free precession of the magnetization with frequency  close to\nMf\n, which B-dependence  for our Galfenol film is shown i n Fig. 1 ( d). Comparing  the coordinates \nof the first intersection in  Fig. 1 (d) with the data i n Figs. 2 (b) and 2 (c), we conclude  that the \nmaximum amplitude in the spectra is obtained at B≈BR1. This observation is the main result of the \npresent work  and demonstrates a  resonan ce in the  magnetization  precession  that is driven by \ncoherent phonons with frequency\n1Rf . \nWe now  discuss th e observed increase  of the spectral density at \nif fR M  in more detail. \nThe optical pump pulse absorbed in the Galfenol  layer  results in an instantaneous rise  of the  layer \ntemperature,  generating a broad spectrum of coherent phonons in the form of a  picosecond  strain \npulse [1 5]. The major fraction  of generated phonons leave s the Galfenol film with the sound \nvelocity on a timescale of  ~10 ps, but phonons with frequencies \niffR  remain  localized in the \nFP cavities for a long er time. The calculations for the cavity formed by SL1 at \n1Rf  gives a 5 \n remarkabl y high value for the  decay  time of  \n1R ~10 ns , which  is three orders of magnitude longer \nthan the escape time for non -resonant phonons  from the cavity and two orders of magnitude longer \nthan the lifetime  of the free magnetization precession in this experimenta l geometry  [18,23]. The \nlocalized  phonons drive the magnetization precession  at \niffR  and this driving  force  will last \nduring the leakage time\n1R . The amplitude of the precession amplitude  will incr ease when the \nresonance condition  \nMf =\n1Rf is fulfilled . This is clearly observed in our experiments at \nB=B R1=180 mT for the signals measured in the temporal  and spectral domains . The width Δ f of \nthe resonant curve  in Fig. 2 (c) is the same as  the width  of the spectral line for magnetization \nprecession  [23], which is in full agreement with the expectation for driving an  object by a harmonic \nforce . For non-resonant conditions, when B differs  remarkably from BR1, the low intensity spectral \nline at \nMff  corresponds to the  quickly decaying free oscillations  due to excitation  of magnetic \nprecession  by phonons from the initially generated  broad spectrum and instantaneous temperature \nrise [ 16,25].  \nThus , we conclude that the spectral amplitude  of the magnetic precession  at the frequency \nof the driving force rapidly  increase s at the resonance condition  \nMf=\n1Rf  for B≈BR1. To show the \ngeneral  valid ity of this statement we per form further experiments for B-values at which  \nMf falls \ninto the region of  the other two  FP phonon modes  of the SL2 cavity . In this case , the cavity , which \ncompris es the Galfenol layer and SL1 , has a length 1 362 nm. The calculations for infinitely long \nSL2 predict  two localized phonon states as shown in Fig. 1 ( c). Figure 3 (a) show s the temporal \nevolution of \ntMz  at B=400 mT. The signal lasts longer than 6 ns (the time interval available \nfor the experimental measurements) and possesses pronounced beatings due to simultaneous \nexcitation of several spectral components . The corresponding spectral lines  are clearly  seen in the \nfrequency  domain presentation in Fig. 3 (b) . The frequencies  of the peaks are independent o f B, \nbut their intensities vary strongly with B. The spectral lines with \n2f =28.4 GHz and \n3f =30.0 GHz \nshow absolute maximal  intensities  in the magnetic field  B interval between 350 and 450 mT . The \nfrequencies of these  two spectral line s are in good agreement with th ose of the calculated phonon \nmodes  \n2Rf =28.0 GHz and \n3Rf =29.5 GHz and their maximum amplitude s are detected at the B–\nvalues of the  intersections  in Fig. 1 (d) demarking  the resonance condition \nMf\n2Rf\n3Rf . The \norigin of other lower amplitude spectral lines in Fig.2 (b) is the finite  length of SL2. To confirm \nthis we have calculated the spectrum of the lattice displacement , G() near the surface assuming \nthat light  forming the excitation pulse  is absorbed within  a thickness x0=30 nm of the Galfenol \nfilm:   6 \n \n\n\n\n111\n11~\n00\n22\n00\n2\n02\n0dikdik\ne Re R kxikxG    (1) \nwhere  =2f, k0 is the phonon wave vector in Galfenol, d=59 nm is the thickness of the Galfenol \nfilm, and R(ω) is the complex reflectivity of the acoustic wave incident from the Galfe nol on SL1 \nand SL2 grown on the GaAs substrate . R() is calculated according to  Ref. [ 22]. The result of the \ncalculations  based on  Eq. (1)  is shown i n Fig. 3 (c). Excellent agreement with the experiment is \nobserved: there is a single line at f=fR1=20 GHz and  a number of spectral lines in the regio n of 25 -\n32 GHz exactly as observed experimentally. In fact, the appearance of the additional spectral lines  \nis due to the phonon reflection resonances in finite -thickness SL existing at frequencies beyond \nbut close to the edges of the phonon stop -bands in a SL.  Thus , we c an conclude that resonant \ndriving of the magnetization by coherent phonons also takes place for the SL2 cavity.  The \nspectrum also consists visible nonzero background in the whole frequency range, which supplies \nthe excitation of free precession at non -resonant magnetic fields.  \nFinally , we discuss  the possible role of transverse (TA) phonons in the studied \nnanostructure. In the experiments , we do not see any resonant effects at B–values that  correspond \nto the resonance of  \nMfbeing equal to the frequency of localized TA phonons. This is not surprising \nbecause the optical pump  excitation does not excite shear strain and corresponding ly TA phonons \nin the present high symmetry geometry [ 26,27]. However, we could expect generation of LA and \nTA phonons  from the magnetization precession , as has been observed in conventional  microwave -\ndriven  magneto -acoustic experiments [28]. Then the magnetic and elastic resonance would form \na couple d magneto -elastic  excitation , resulting in renormalization  of magnetic and phonon \neigenstates  [29]. The search for such excitations in ferromagnetic nanostructures is an extremely \nattractive  field in nanomagnetism  and nanomechanics . \nIn conclusion , we have demonstrated a  new method to drive resonantly magnetization \nprecession by sub -THz coherent phonons when the ferromagnetic layer is ins erted into a phonon \ncavity. Resonant driving by coherent phonons can be used locally on the nanoscale and does not \nrequire external resonator s in contra st to microwave excitation. Nowadays it is possible to generate \ncoherent monochroma tic phonons optically up to freque ncies of ~ 1 THz [ 30-32], thus, opening \nappealing  perspective s for resonant driving of magnetic excitation s possessing resonances at \nhigher frequencies, e.g. in antiferromagnetic nanostructures. Interaction of high -frequency \nphonons and magnons on the  nanoscale can lead to the development of a new class  of devices. For \ninstance, the enhanced amplitude of the magnetization precession during resonant driving by 7 \n coherent phonons shows the feasibility of prece ssion al switching o f the magnetization between \nmetastable states [ 33]. \nWe acknowledge the support of the work by the Deutsche Forschungsgemeinschaft  and the \nRussian Foundation for Basic Research  in the frame of the ICRC TRR 160 and by the grant BA \n1549/14 -1, the Government of Russia through the program P220 (14.B25.31.0025) ,  the BMBF \nwithin the RELQUSA project (FKZ: 13N12462) , and the Russian Academy of Science.  AWR \nacknowledges support from a Career Acceleration Fellowship (EP/H003487/1), EPSRC, UK ; \nAVA acknowledge s the Alexander von Humboldt Foundation.  \n \nREFERENCES  \n[1] Y. Shiroishi, K. Fukuda, I. Tagawa, H. Iwasaki, S. Takenoiri, H. Tanaka, H. Mutoh, and \nN. Yoshikawa,  IEEE Trans . Magn . 45, 3816 ( 2009 ) \n[2] A. Imre, G. Csaba, L. Ji, A. Orlov, G. H. Bernstein, W. Porod, Science 311, 205 (2006) .  \n[3] D. Bhowmik, L. You, and S. Salahuddin, Nature Nanotech. 9, 59 (2014).  \n[4] J. R. Maze, P. L. Stanwix , J. S. Hodges, S. Hong, J. M. Taylor, P. Cappellaro , L. Jiang, \nM. V. Gurudev -Dutt, E. Toga 1, A. S. Zibrov, A. Yacoby , R. L. Walsworth , and  M. D. \nLukin , Nature 455, 644 (2008).  \n[5] I. K. Kominis, T. W. Kornack, J. C. Allred, and M. V. Romalis, Nature 422, 596 (2003)  \n[6] E. A. Vitol, V. Novosad, and E. A. Rozhkova, Nanomedicine 7, 1611 (2015).  \n[7] A. Kirilyuk, A. V. Kimel, and T. Rasing, Rev. Mod. Phys. 82, 2731 (2010), and \nreferences therein.  \n[8] N. Locatelli, V. Cros and J. Grollier, Nature Mat. 13, 11 (2014).  \n[9] E. R. MacQuarrie, T. A. Gosavi, N. R. Jungwirth, S. A. Bhave, and G. D. Fuchs, Phys. \nRev. Lett. 111, 227602 (2013).  \n[10]  D. E. Parkes, S. A. Cavill, A. T. Hindmarch, P. Wadley, F. McGee, C. R. Staddon, K. \nW. Edmonds, R. P. Campion, B. L. Gallagher, and A. W. Rushfo rth, Appl. Phys. Lett. \n101, 072402 (2012).  \n[11] A.V. Scherbakov, A.S. Salasyuk, A.V. Akimov, X. Liu, M. Bombeck, C. Bruggemann, \nD.R. Yakovlev, V.F.Sapega, J.K. Furdyna, and M. Bayer,  Phys. Rev. Lett. 105, 117204 \n(2010).  \n[12] D. Afanasiev, I. Razdolski, K.M. Skibinsky, D. Bolotin, S.V. Yagupov, M.B. \nStrugatsky, A. Kirilyuk, Th. Rasing, and A. V. Kimel, Phys. Rev. Lett. 112, 147403 \n(2014).  8 \n [13] S. Davis, A. Baruth, and S. Adenwalla, Appl. Phys. Lett. 97, 232507 (2010) . \n[14] Y. Yahagi, B. Harteneck, S. Cabrini, and H. Schmidt, Phys. Rev. B 90, 140405(R) \n(2014).  \n[15] C. Thomsen, H. T. Grahn, H. J. Maris, and J. Tauc, Phys. Rev. B 34, 4129 (1986).  \n[16] J.-W. Kim, M. Vomir, and J. -Y. Bigot, Phys. Rev. Lett. 109, 166601 (2012).  \n[17] A. E. Clark, K. B. Hathaway, M. Wun -Fogle, J. B. Restorff, Thomas A. Lograsso, V. \nM. Keppens, G. Petculescu, and R. A. Taylor, J. Appl. Phys. 93, 8621 (2003).  \n[18] D. E. Parkes,  L. R. Shelford,  P. Wadley,  V. Holý,  M. Wang,  A. T. Hindmarch,  G. van \nder Laan,  R. P. Campion,  K. W. Edmonds,  S. A. Cavill, and A. W . Rushforth, Sci. \nReports 3, 2220 (2013).  \n[19] W. Chen, Y. L u, H. J. Maris, and G. Xiao, Phys. Rev. B 50, 14506 (1994).  \n[20] M. Trigo, T. A. Eckhause, M. Reason, R. S. Goldman, and R. Merlin, Phys. Rev. Lett. \n97, 124301 (2006) . \n[21] M. F. Pascual Winter, G. Rozas, A. Fainstein, B. Jusserand, B. Perrin, A. Huynh, P. O. \nVaccaro, and S. Saravanan, Phys. Rev. Lett. 98, 265501 (2007).  \n[22] S. Mizuno and S .-I. Tamura , Phys. Rev. B 45, 734  (1992).  \n[23] J. V. Jäger, A. V. Scherbakov, T.L. Linnik , D.R. Yakovlev, M. Wang, P. Wadley, V. \nHoly, S.A. Cavill, A.V. Akimov, A. W. Rushforth, and M. Bayer, Appl. Phys. Lett. 103, \n032409 (2013).  \n[24] W. K. Hiebert, A. Stankiewicz, and M. R. Freeman, Phys. Rev. Lett. 79, 1134 (1997).  \n[25] H. B. Zhao, D. Talbayev, Q. G. Yang, G. Lüpkea,A. T. Hanbicki, C. H. Li, O. M. J. van \n’t Erve, G. Kioseoglou, and B. T. Jonker, Appl. Phys. Lett. 86, 152512 (2005).  \n[26] T. Pezeril, P. Ruello, S. Gougeon, N. Chigarev, D. Mounier, J. -M. Breteau, P. Picart, \nand V. Gusev, Phys. Rev. B 75, 174307 (2007).  \n[27] O. Matsuda, O. B. Wright, D. Hurley, V. Gusev, and K. Shimizu, Phys. Rev. B 77, \n224110 (2008).  \n[28] J. W. Tucker, V. W. Rampton, Microwave Ultrasonics in Solid State Physics (North \nHolland Pub., Amsterdam, 1972) .  \n[29] H. Matthews and R. C. LeCraw, Ph ys. Rev. Lett. 8, 397 (1962).  \n[30] A. Bartels, T. Dekorsy, H. Kurz, and K. Köhler, Phys. Rev. Lett. 82, 1044 (1999).  \n[31] C.-K. Sun, J. -C. Liang, and X. -Y. Yu, Phys. Rev. Lett 84, 179 (2000).  \n[32] A. Huynh, B. Perrin, and A. Lemaître, Ultrasonics 56, 66 (2015).  \n[33] L. Thevenard, J. -Y. Duquesne, E. Peronne, H. J. von Bardeleben, H. Jaffres, S. Ruttala, \nJ-M. George, A. Lemaître, and C. Gourdon, Phys. Rev. B 87, 144402 (2013) .  9 \n  \n \n \n  \n  \nFigure 1.   (a) The scheme of excitation and detection of magnetization precession in \nGalfenol layer grown on two phonon Bragg mirrors formed by GaAs/AlAs superlattices \nSL1 and SL2. (b), (c) Calculated phonon dispersion curves for two SL1 and SL2 Bragg \nmirrors, respectively. Horizontal bars in the zoomed insets indicate the frequency fRi of the \nphonon localized modes formed by the free surface and the corresponding SL Bragg mirror. \n(d) The magnetic field dependence of the free precession frequency fM measured (symbols) \nand linear fit (solid line). Horizontal lines show the spectral positi ons of the cavity phonon \nmodes f Ri and vertical arrows indicate the corresponding resonance fields BRi    \n \n100 200 300 400 500 600 700152025303540\nBR3 BR2fR3\nfR2 Frequency (GHz)\n \nMagnetic field (mT)(d)\nfR1 BR1\n0102030\n SL1\n(b)0.95 1.00192021\n  fR1 \n0.0 0.5 1.00102030\n \nWavevector ( /d)SL2\n(c)0.95 1.0027282930\n  fR3\nfR2 \n \nFrequency (GHz)(a)10 \n   \nFigure 2.  (a) Temporal evolution of the magnetization measured for three values of B.  \n(b) The  spectral density of the measured signals. Dashed vertical and oblique lines \ncorrespond to the maximum of the spectral line at f1=20.5 GHz and estimated position of the \nfree magnetic precession, fM, respectively. The vertical arrow shows the calculated pos ition \nof the lowest cavity mode,  fR1. (c) Measured (symbols) dependence of the spectral amplitudes \non the magnetic field for the spectral line centered at f1=20.5 GHz as indicated by dashed \nvertical line in (b); dotted line is corresponding Gaussian fit of  the experimental data. Vertical \narrow shows the value of the resonant magnetic field BR1=180 mT.  \n \n16 18 20 22 24T=170 K\nfMf1\n100 mT\n Frequency (GHz) Amplitude\n B=300 mT(b)\nfR1\n100 200 300f =20.5 GHz Amplitude\n \nMagnetic field (mT)(c)\nBR10.0 0.5 1.0 1.5 2.0T=170 K  Mz(t)\nTime (ns)(a)B=250 mT\n190 mT\n100 mT11 \n  \n \nFigure 3 .  (a) Temporal evolution of the magnetization measured at B=400 mT, when the \nlong living tail with beatings has the maximum amplitude. The inset shows the zoomed \nfragment of the measured signal. (b) Spectral density of the measured signals for three \nvalues of magnetic field. Vertical lines indicate the spectral positions f2 and f3 in the \nspectra of measured signals and vertical arrows show th e calculated frequencies fR2 and \nfR3; (c) Calculated phonon spectrum for the studied structure for impulsive femtosecond \noptical excitation.    \n \n \n24 26 28 30 32 34fR3fR2f3\n300 mT\n Frequency (GHz) Amplitude\n B=500 mT\n400 mT(b)\nf2\n15 20 25 30 35 Amplitude\n \nFrequency (GHz)(c)\n00 1 2 3 4 5 6B=400 mT  Mz(t)\nTime (ns)(a)\n1.61.82.02.22.4\n \nTime (ns)"    },    {        "title": "1604.05835v1.Resonance_based_Detection_of_Magnetic_Nanoparticles_and_Microbeads_Using_Nanopatterned_Ferromagnets.pdf",        "content": "Resonance-based Detection of Magnetic Nanoparticles and\nMicrobeads Using Nanopatterned Ferromagnets\nManu Sushruth1,*, Junjia Ding2, Jeremy Duczynski3, Robert C. Woodward1, Ryan\nA. Begley1, Hans Fangohr4, Rebecca O. Fuller3, Adekunle O. Adeyeye2, Mikhail\nKostylev1and Peter J. Metaxas1,**\n1School of Physics, M013, University of Western Australia, 35 Stirling Hwy, Crawley\nWA 6009, Australia.\n2Information Storage Materials Laboratory, Department of Electrical and Computer\nEngineering, National University of Singapore, Singapore-117576, Singapore\n3School of Chemistry and Biochemistry, M310, University of Western Australia, 35\nStirling Hwy, Crawley WA 6009, Australia.\n4Engineering and the Environment, University of Southampton, Southampton, SO17\n1BJ, United Kingdom\n*manu.sushruth@research.uwa.edu.au\n**peter.metaxas@uwa.edu.au\nJune 18, 2021\nAbstract\nBiosensing with ferromagnet-based magnetoresistive devices has been dominated by elec-\ntrical detection of particle-induced changes to the devices' static magnetic con\fguration.\nThere are however potential advantages to be gained from using \feld dependent, high fre-\nquency magnetization dynamics for magnetic particle detection. Here we demonstrate the\nuse of nano-con\fned ferromagnetic resonances in periodically patterned magnetic \flms for\nthe detection of adsorbed magnetic particles with diameters ranging from 6 nm to 4 \u0016m.\nThe nanopatterned \flms contain arrays of holes which can act as preferential adsorption sites\nfor small particles. Hole-localized particles act in unison to shift the resonant frequencies\nof the various modes of the patterned layer with shift polarities determined by the local-\nization of each mode within the nanopattern's repeating unit cell. The same polarity shifts\n1arXiv:1604.05835v1  [cond-mat.mes-hall]  20 Apr 2016are observed for a large range of coverages, even when hole-localized particles are covered\nby quasi-continuous particle sheets. For large particles however, preferential adsorption no\nlonger occurs, leading to resonance shifts with polarities which are independent of the mode\nlocalization. Analogous shifts are seen in continuous layers where, for small particles, the\nshift of the layer's fundamental mode is typically about 10 times less than in patterned sys-\ntems and induced by relatively weak \felds emanating beyond the particle in the direction of\nthe static applied \feld. This highlights the importance of having con\fned modes consistently\npositioned with respect to nearby particles.\n1 Introduction\nMagnetic biosensing techniques have shown great promise in terms of providing a matrix-\ninsensitive biosensing platform for future use in point-of-care medical diagnostics.1,2These\ntechniques center on the detection of magnetic particles which are used as tags for analytes of\ninterest in biological \ruids (Fig. 1(a)). While numerous methods for detection of nanoparti-\ncles exist,3{12magnetoresistive structures, exploiting giant or tunneling magnetoresistance phe-\nnomena have garnered signi\fcant interest.13{18Conventional sensing with such structures is\nachieved via the detection of magnetic-particle-induced changes to the static magnetic con\fgu-\nration within these (typically multilayered) ferromagnetic devices (Fig. 1(b)), a process which\nexploits their magnetic-\feld dependent resistivity.19Thus, like Hall e\u000bect sensors,8,9they o\u000ber\na sensing method based on detecting changes to a d.c. voltage level.\nRecently, attention has begun to turn towards the possibility to exploit the magnetic \feld\ndependence20,21of resonant magnetization dynamics for particle sensing.22{25The consequence\nof this dependence is that the precession frequency of the magnetization, typically in the giga-\nHertz range, will be altered by particle-generated magnetic \felds (Fig. 1(c)), potentially enabling\nan intrinsically frequency-based rather than amplitude-based sensing technique.23Notably, the\nstray \felds generated by the particles act directly on the resonance, not requiring particle-\ninduced changes to the magnetic ground state of the system for detection. Indeed, there are\nnumber of reasons to motivate the investigation of such methods. For example, dynamics can\nbe induced and measured even at large external \felds26where particle moments, and thus\nparticle-generated \felds (the basis of particle detection) can be maximized. Furthermore, using\n(e.g.) spin torque oscillators,23it is possible to e\u000eciently probe and read out such dynamics\nelectrically in real time27{30(potentially enabling high speed sensing for applications such as\n2Figure 1: (a) Schematic of a sandwich immunoassay showing a magnetically labeled biological entity\nspeci\fcally bound to a chemically functionalized sensor. The nanoparticle stray \feld can induce changes\nto the (b) static magnetization con\fguration of a ferromagnetic sensing element or (c) its precessional\nmagnetization dynamics. (d) Shows a magnetic particle (red) within a hole (white circles) in a hole-based,\nperiodic magnonic crystal with geometry equivalent to that studied in Sec. 2.2 (300 nm wide holes). Both\nthe particle and patterned \flm are magnetized towards the top of the page. The particle stray \feld locally\nmodi\fes the amplitude of precessional magnetization dynamics around it. The dynamics correspond to\na side mode resonance which is localized between the holes. The color encodes the amplitude of the\nmagnetization precession in the MC (red = high; blue = low). Except for the single nanoparticle, there\nis no magnetic material within the holes.\ncytometry31{33). Furthermore, in the latter devices, favorable signal to noise ratios can be\nretained down to sub-100 nm device length scales.23Indeed here we will demonstrate particle\ndetection using a particular resonance con\fned laterally to a region with size on the order of\n100\u0002100 nm2.\nMagnonic crystals (MCs) are attractive systems to explore the fundamentals of such a sens-\ning technique. MCs are magnetic materials which have been arti\fcially patterned (typically at\nthe nanoscale) to control spin-wave (magnon) propagation or enable periodic con\fnement of\nnano-localized ferromagnetic resonances34(e.g. Fig. 1(d)). High levels of periodicity over large\nlength scales mean that these localized resonant modes (analogous to those excited in isolated\nnanostructures) can be probed easily in laboratory settings in macroscopic samples. Beyond\ntheir envisioned applications in data and signal processing,35we have previously demonstrated\nthat MCs can be used to understand the in\ruence of magnetic \felds generated by magnetic\nnanoparticles on highly localized (or `con\fned') ferromagnetic resonances.25Notably, using a\nhole-based structure enables the localization of particles within the holes with a subsequent\npredictable local modi\fcation of the resonant precession of the magnetization in the neighbor-\nhood of the captured particle. An example of this can be seen in Fig. 1(d) as a change in the\ncolor-coded precession amplitude to the left and right of the hole-localized nanoparticle. More\n3recently, similar e\u000bects were con\frmed numerically in another hole-based MC geometry.36\nIn this paper we demonstrate the successful use of nano-localized ferromagnetic resonances for\nthe detection of magnetic particles with a range of diameters spanning 3 orders of magnitude:\nfrom sub-10 nm superparamagnetic iron-oxide nanoparticles to 4 \u0016m wide magnetic beads.\nThis particle size range also approximates a correspondingly large range of biological length\nscales, from single proteins to cells. We will explicitly show the advantage of using hole based\nstructures for the localization of both modes and particles, the latter ensuring a common, and\nthus reinforcing e\u000bect, from individual particles. Indeed we show that the sensitivity is an\norder of magnitude less when using an unpatterned continuous ferromagnetic layer. There, the\ndominant e\u000bect is from the stray \feld extending far outside the particles which is weak compared\nto that beneath or directly neighboring each particle. When particles are within the MC's holes\nthe localization of di\u000berent modes around each hole in the patterned case de\fnes the sign of the\nfrequency shift. This shift polarity is maintained and its amplitude increased at high coverages\nwhere quasi-continuous particle sheets are formed. However by increasing the particle size to a\ndegree in which the particle cannot enter the holes, we lose the mode-dependent shift polarities\nwith all modes behaving similar to the fundamental mode of a continuous layer in that their\nfrequencies all increase. Varying the particle size can thus enable a transition to a \flm-like\nbehavior, albeit with multiple modes exisiting in the MC. Note that all particles studied in this\nwork exhibit a quasi-null magnetic moment in zero \feld (as checked via magnetometry) and\nthus minimal agglomeration.\n2 Results and discussion\n2.1 Continuous layers\nWe \frst consider the case of particle-induced resonance shifts in continuous, unpatterned mag-\nnetic layers where resonant dynamics are typically laterally uniform across the layer. Fig. 2(a)\nshows a schematic of the dipole \feld of a magnetic particle acting on an underlying magnetic\n\flm subject to an external in-plane static \feld, Hext. The in-plane component of the particle's\n\feld,HP, is strong and opposes Hextdirectly below the particle (in region `2', Fig. 2(a), and to\nthe sides of the particle, the latter due to the symmetry of the dipolar HP). Elsewhere (regions\n`1' and `3', Fig. 2(a)) the in-plane component of HPis weaker but reinforces Hext. Subsequently,\nalthough a bare, thin magnetic layer has one primary resonance mode (which corresponds to the\n4Figure 2: (a) Schematic side view of a particle placed on an underlying magnetic \flm. The in-plane stray\nmagnetic \feld generated by a particle, HP, is strong and opposes the applied in-plane static \feld ( Hext)\ndirectly below it, resulting in a strong downward shift in the resonance frequency in that region (marked\nas `2'). In regions `1' and `3', HPis weaker (blue dashed lines) but reinforces Hext, thus slightly increasing\nthe resonance frequency. (b) Fourier transformed time domain simulation data for an un-patterned \flm\nshowing a small upward shift in fundamental mode's frequency in presence of a particle. The upper-left\ninset shows experimental FMR traces, obtained at 12 GHz, with (solid red line) and without (dotted\ngreen line) particles (concentration is 0.675 \u0016g.\u0016L\u00001) on the surface of unpatterned \flm. Lower insets\nshow the simulated mode pro\fles of the two main modes that exist in the presence of a particle. (c)\nExperimentally obtained result for fundamental mode shifts of an unpatterned \flm as a function of\nparticle concentrations. The measured shifts for each concentration are the average shift for frequencies\nranging 11.5 - 16 GHz. (d-e) SEM images showing the distribution of particles on the unpatterned \flm's\nupper surface. White scale bars are 1 \u0016m long.\n5spatially uniform fundamental mode, shown as a solid line in Fig. 2(b)), an extra, low frequency\nmode appears in simulation when isolated particles are on top of the \flm. Note that these\nsimulation results assume one 150 nm particle 10 nm above every 450 \u0002450 nm2of \flm (further\nsimulation details are given in experimental section).\nThe low frequency mode is located beneath the particle and to its sides where HPopposes\nHext(see bottom left inset in Fig. 2(b) while comparing to region `2' in Fig. 2(a)). Since\nthe total \feld is reduced at that position, so is the resonance frequency. The majority of the\ndynamics within the layer are however concentrated in the remainder of the computation region\nas seen in the bottom right inset in Fig. 2(b) (these areas correspond to regions `1' and `3' in\nFig. 2(a)). This mode is slightly upshifted due to the weak HPin those regions reinforcing Hext,\nthus increasing the total \feld. Notably, at a particle-layer separation of 10 nm, the maximum\nin-planeHPdirectly below a single, perfectly spherical particle will be approximately 15 times\nhigher than the maximum in-plane HPat the upper/lower boundaries of the computation cell.\nThis explains the disparity in the shift amplitudes seen for the dynamics concentrated in region\n`2' versus those concentrated in regions `1' and `3'.\nIn a \feld-resolved experimental FMR trace (obtained at a \fxed frequency, upper-left in-\nset of Fig. 2(b)), the particle-induced upshift of the primary resonance mode manifests as a\ndownshifted resonance \feld for the fundamental mode since the reinforcing e\u000bect of Hpreduces\nthe magnitude of the external \feld which must be applied to obtain the resonance condition.\nIn Fig. 2c we show the consistently negative shift of the fundamental resonance for increas-\ning particle coverages. Di\u000berent particle coverages were obtained by applying consecutively\nhigher concentrations of particle-containing solutions to the \flm and measuring the FMR spec-\ntra between each application. Note that the shift is highest for intermediate particle coverages\n(Figs. 2(f-g)) where there are large numbers of isolated particles or clusters of particles on top\nof the layer. In contrast, low particle coverages (Figs. 2(d-e)) lead to a low reinforcing HP\felds\nwhen averaged across the \flm (and thus low shifts). The quasi-continuous particle layers at high\ncoverages also generate relatively small shifts. This is likely because they approximate contin-\nuous layers which, ignoring the e\u000bects of roughness, generate negligible stray \felds (except at\nthe layer boundaries).\nThe simulated resonance frequency shift is 34 MHz ( \u00110:7 mT given an experimentally\nmeasured slope of 49.4 GHz/T for the fundamental mode, see Supporting Fig. 1). This is\ncomparable to the maximum experimentally observed \feld shift in the continuous layer (1 :3\u00060:7\n6mT). Thus, we can conclude that the weak shift observed for the continuous layer is not a result\nof a low \feld sensitivity of the resonance (since it is actually quite high at almost 50 GHz =T),\nbut rather due to the dominant signal coming from portions of the \flm subject to the relatively\nweak reinforcing HP\felds surrounding the particles (i.e. regions `1' and '3' in Fig. 2(a)) rather\nthan the more intense HP\felds located directly beneath the particles .\n2.2 Patterned \flms: hole arrays\nFigure 3: (a) Simulated frequency resolved FMR spectra obtained at Hext= 180 mT applied in y-direction\nfor a 300 nm MC with (dotted blue lines) and without (solid black line) a particle inside the hole. The\nupper inset shows an SEM image (1.3 \u0016m x 1.3\u0016m ) of the MC and the lower inset shows the simulated\nresonance shift of the EdM in the presence of a particle. (b) Spatial concentration of dynamics for a\nnumber of resonance modes within a unit cell of the MC. A schematic of a particle and its stray \feld has\nbeen added to the EM and SM pro\fles.\nWe now turn to the hole-containing nano-patterned MC where it is possible to have local-\nized resonances in regions where HPis large. The main panel of Fig. 3(a) shows a simulated\nferromagnetic resonance spectrum at Hext= 180 mT for a MC with 300 nm wide holes on a\n450 nm square lattice. The excitation spectrum is clearly much richer than the continuous layer\nwith a number of modes, each having di\u000berent localizations within the unit cell (Fig. 3(b)). The\nsimulated traces have been di\u000berentiated with respect to frequency for more natural compari-\nson to experimental FMR traces. We will focus predominantly on the side mode (SM, shown\nalso in Fig. 1(d)) and extended mode (EM) with intermediate (IM) and edge modes (EdM)\ndiscussed only brie\ry. While the SM is largely localized between horizontally neighboring holes,\nthe EM occurs over extended bands running between rows of holes orthogonal to the applied\n7\feld (Fig. 3(b)). IM1-3 have similar localizations to that of the EM. Good agreement is found\nbetween the simulation and experiment for both the overall mode spectrum and the frequencies\nof the EM and SM (Supporting Figure 2).\nDepending on the spatial localization of each mode, the stray magnetic \feld from magne-\ntized particles within the holes has a y\u0000component which can locally reinforce or oppose the\ny\u0000oriented external \feld.25,37This is shown schematically for the SM and EM in Fig. 3(b)\n(one can consider the more localized modes as being nano-scale dynamic probes for the particle\nstray \felds). Since the y\u0000component of HPopposesHextwhere the SM is localized, the mode's\nfrequency reduces due to a reduced net \feld at that location. This is seen in Fig. 3(a) where we\nhave also included the simulated resonance spectrum in the presence of a 150 nm wide particle\nat the lateral center of an anti-dot. In contrast, the EM (and EdM) resonances shift upwards\nin frequency since HPreinforcesHextat the upper/lower parts of the unit cell. Note that the\ncon\fnement of the modes in well-de\fned regions close to the particle where the stray \felds are\nstrong (i.e. directly to the sides of the particle ( \u0006x) and directly in front of/behind it ( \u0006y))\nmean that signi\fcant shifts ( \u00180:1 GHz) are induced for both the EM and SM (even though\nthe sensitivities of the EM, SM and fundamental modes to static, uniform external \felds are of\nthe same order). The EdM and IMs are also subject to clear positive shifts in the presence of\na hole-localized particle, consistent with the fact that they, like the EM, have their dynamics\nconcentrated in the upper/lower portions of the unit cell (subject to HP>0).\nFollowing the same protocol as for the continuous layer in the previous section, we measured\nthe SM and EM shifts versus particle concentration in a 300 nm MC experimentally. The EM\nresonance \feld decreases since HPincreases the local \feld (Fig. 4(a)). The consequence of this is\nthat the resonance can be attained experimentally at a lower Hext. The resonance \felds of IM1-\n3, which have an EM-like localization, also shift downward in experiment (Supporting Figure\n4) In contrast, the SM resonance \feld increases because HPlocally shields the SM from Hext\n(Fig. 4b). This means that a larger Hextmust be applied to attain the resonance condition. As\nseen previously for low coverages,25we observe a continuing increase of the shift magnitudes with\nparticle coverage over a very wide range of coverages (Fig. 4(c)). For lower nanoparticle solution\nconcentrations, the majority of particles are found within the holes (Figs. 4(d,e)) with both the\npercentage of \flled holes and the percentage of particles lying within the holes increasing with the\nconcentration of the particle solution (Supporting Figure 3). Saturation of the shifts commences\nat the penultimate concentration where a quasi-continuous layer of particles form (Figs. 4(g,h)).\n8Figure 4: Experimental FMR traces (obtained at 12 GHz) showing shifts in (a) EM and (b) SM resonance\nlines with increase in particle concentration.(c) Experimentally obtained result for EM and SM resonant\nshifts for a 300 nm MC as a function of particle concentrations. Horizontal (dashed) lines show the\nsimulated EM and SM resonant shifts for (1 particle/hole) a 150 nm thick sheet of particles on the\nsurface. The error bars are a measure of the spread of shifts across the measured frequency range (11.5 -\n16 GHz) and also includes the uncertainties related to slight variations in sample placement ( \u00180:5 mT\nat most). The continuous \flm's fundamental mode shift versus particle concentration (Fig. 2c) is plotted\n(blue open circles) for direct comparison with the patterned \flm. (d-h) SEM images showing clustered\nshaped nanoparticles inside holes and on the upper surface of the MC. White scale bars are 1 \u0016m long.\nThe \frst four data points for the EM and SM in (c) as well as the bare and 2 lowest concentration traces\nin (a) have been presented previously.25\nFigure 5: Resonance \feld shifts versus \feld for (a) the EM and (b) SM in the 300 nm MC. The numbers\nshown on the plots give the r.f. excitation frequency in GHz. Simulated EM and SM mode shifts for the\ncase of one particle/hole and a disk with an overlying 150 nm thick layer of particles on the crystal's\nsurface are shown on the plots as joined points. The simulated frequency shifts obtained were converted\nto \felds using respective mode's slopes (refer Supporting Figure 2(a)). Qualitatively good agreement is\nseen between the trend of simulated and experimental resonance shifts over the measured \feld range.\n9Notably, the maximum shifts in the MC are, as expected, signi\fcantly higher (about 10 \u0002) than\nthe maximum shift observed in the continuous layer. Note that the continuous layer data from\nFig. 2(a) has been plotted with the MC data in Fig. 4(c) to enable direct comparison. Another\npoint to note is that while the continuous layer resonance \feld shifts decreases when a quasi-\ncontinuous particle sheet forms, in the MC, sheet formation increases the observed shifts, an\ne\u000bect which is reproduced below.\nWe \frst discuss the reproduction of the observed shift for the EM and SM (Fig. 4(c)) at\n0.225\u0016g.\u0016L\u00001where there is approximately 1 particle in each hole (e.g. Fig. 4(e)). This could\nbe done by adding a 150 nm wide particle to the simulation at the hole center and observing the\nresultant resonance shifts. A good reproduction of the shifts observed at high coverages could\nalso be obtained. There, we replaced the particle with a disk (diameter 40 nm less than the\nhole diameter) which was covered by a contiguous 150 nm continuous layer. This successfully\napproximated the e\u000bect of the well \flled-holes (e.g. Figs. 4(g,h)) covered by a semi-complete\nparticle sheet, as observed at high particle solution concentrations. The agreement between\nsimulation and experiment for the EM and SM can be seen in Fig. 4(c) which shows the \feld-\naveraged simulated shifts for the EM and SM for one particle per hole (horizontal dashed line)\nand full particle coverage (solid dashed line). There is excellent agreement between these values\nand the shifts experimentally obtained respectively at the fourth and \fnal coverages.\nExperimental and simulated shifts obtained at di\u000berent \felds are shown in Fig. 5. Experi-\nmentally, di\u000berent resonance \felds are obtained by changing the r.f. frequency and it is the data\nin this \fgure which has been averaged to obtain the points in Fig. 4(c). Here one can again see\nexplicitly that the resonance \felds decrease with coverage at each frequency for the EM (leading\nto leftward slanting data in Fig. 5(a)) but increase for the SM (leading to rightward slanting\ndata in Fig. 5(b)). The simulated shifts obtained explicitly for di\u000berent Hextvalues, taking\ninto account the \feld-dependent moment of the particles25could notably also reproduce the\nexperimentally observed \feld-dependencies of the resonance \feld shifts at the fourth and \fnal\ncoverages (Fig. 5). The most intuitive of these dependencies is a reduced shift at very low \felds\nfor the SM mode where the \feld-dependent particle moment (and thus resultant stray \feld) is\nweak (Fig. 5(b)). In general though, increasing the \feld does not strongly change the size of\nthe shifts meaning that the external \feld can be incnreased to maximize the particle moments\nwithout compromising the frequency shift.\nWe \fnally note that particle sensing was also demonstrated with a second MC having smaller\n10Figure 6: Experimentally obtained EM FMR traces (excited at 12 GHz) showing decrease in resonance\n\feld with increase in concentration of (a) 6 nm nanoparticles, (b) 50 nm nanoparticles, (c) 0.88 \u0016m\nmagnetic beads and (d) 4 \u0016m magnetic beads. (e) EM and SM resonant shifts as a function of applied\n50 nm particle concentrations. SEM images showing the distribution of 50 nm particles for (f) lower\nand (g) higher concentrations. (h) EM and SM resonant shifts as a function of applied 0.88 \u0016m bead\nconcentrations. SEM images showing the distribution of 0.88 \u0016m wide magnetic beads for (i) lower and\n(j) higher concentrations. The error bars in (e,h) are a measure of the spread of shifts across the measured\nfrequency range (11.5 - 16 GHz). White scale bars are 1 \u0016m long.\n11holes (240 nm) but the same lattice pitch. Analogous e\u000bects were observed except that the shifts\nwere reduced. The shifts could albeit be reproduced by assuming a \flling proportional to the\nhole size (Supporting Figure 5). This is consistent with the reduced shifts being due to a reduced\ncapability of \flling rather than an intrinsically lower \feld sensitivity of the modes. Indeed the\nintrinsic \feld sensitivities are similar to the MC with 300 nm holes (Supporting Figure 2).\n2.3 The role of particle size and localization\nIn Figs. 6(a-d), we show shifts in the EM resonance \feld induced by particles with (average or\nsupplier-stated) diameters of 6 nm, 50 nm, 0.88 \u0016m and 4\u0016m. Clear shifts are observed in all\ncases, demonstrating the compatibility of this sensing method with a very large range of particle\nsizes. As explained below however, the polarities of the observed shifts depends on the size of\nthe particles relative to the size of the holes. Before concentrating on the shift polarities, we\nbrie\ry note that as seen in Figs. 4(a,b) and 6(a-d), the dominant e\u000bect of both nanoparticles and\nbeads is typically a resonance shift, albeit accompanied by a relatively weak degree of linewidth\nbroadening (discussed below). Exceptions to this are the 0.88 \u0016m particles and the highest\ncoverage scenario for the 6 nm particles where broadening and shifts are more comparable.\nThe same opposing SM and EM shift directions seen in Fig. 4(c) for the 130 nm particles\nwere observed for two other particle types, the common aspect of these particles being that\nthey could enter the MC's holes thanks to their small size: 6 nm wide particles (Fig. 6(a) and\nSupporting Figure 6) and 50 nm wide particles (Figs. 6(b,e-g)). In both cases, the overall trend\nis again a stronger shift with increasing particle coverage. This is seen clearly in Figs. 6(a,b)\nshowing consecutively larger shifts of the EM resonance for both particle sizes. Note that a clear\nSM shift was seen for the 6 nm particles only at the highest concentration (Supporting Figure\n6).\nAs already highlighted for the 130 nm particles (and seen clearly in Fig. 6(g) for the 50\nnm particles) there can also be signi\fcant numbers of isolated (groups of) small particles on\nthe MC surface which can of course also generate resonance shifts. Indeed, these shifts can\nbe comparable in magnitude to those observed for hole-localized particles, especially when the\nparticles are directly above the position where a mode's dynamics are concentrated. In contrast\nto the fundamental mode in the continuous \flm, due to geometrical con\fnement, the modes in\nthe MC are inherently localized and indeed can be localized directly beneath a particle. In such\na situation, the whole region containing the mode will be subject to the strong HPbelow the\n12particle which opposes Hext(position `2' in Fig. 2(a)), thus decreasing the resonance frequency.\nThis has been shown explicitly via simulation for three di\u000berent surface positions in Fig. 7(a)\nwhere simulated shifts of the EM and SM mode due to surface-located particles are compared\nto the shifts induced by a particle at the center of the hole. For example, when a particle is\nabove the SM mode, shown as case I in Fig. 7(a), its frequency is strongly downshifted due to\nHPlocally opposing Hext(the SM is localized in region `2' in Fig. 2(a)). In this case the EM\nfrequency will be slightly upshifted (the EM will be localized in regions `1' and '3' in Fig. 2(a)\nwhereHPis weaker but reinforces Hext). Similarly, when the particle is lying above the EM,\nthe EM frequency is strongly downshifted whereas the SM frequency is slightly upshifted (cases\nIII and IV in Fig. 7(a)). The critical point to remember here though is that the placement of\nsmall particles on the MC's upper surface is quite random within the unit cell. As a result,\nand as observed in experiment, the observed shifts will be dominated by hole-localized particles\ndue to their common, preferential in-hole localization. As already shown though, the e\u000bect of a\nquasi-continuous layer of particles is such that it reinforces the e\u000bect of hole-localized particles.\nIn Fig. 7(b), we show the linewidth broadening observed for the SM and EM due to appli-\ncation of the 130 nm particles onto the 300 nm MC. Notably, broadening is highest (increase of\n\u0018 \u00022) around the 6th coverage (Fig. 4(g)) where there is a large number of isolated particles (or\ngroups of particles) on the MC's upper surface. These mid-to-high coverages lead to relatively\nhigh numbers of isolated particles on the surface which can locally modify the underlying modes,\nas discussed above. Given the position-dependence of the shifts (Fig. 7(a)), linewidth broadening\nis not unexpected since surface-located particles can both increase and decrease the resonant\n\felds, depending on their location (Fig. 7(a)). The formation of a quasi-continuous layer of\nparticles at higher coverages however, would be expected to reduce the strong localized \felds\ngenerated underneath the particles. Indeed, at these higher coverages, there is a slight reduction\nof the linewidth (Fig. 7(b)) consistent with a reduced contribution from isolated surface-located\nparticles. We also note that distributions in the particle sizes (as can be identi\fed in Figs. 4(d-h))\nwill also contribute to linewidth broadening since smaller (larger) particles will generate smaller\n(larger) shifts (Supporting Figure 7). This broadening will be in a common direction however\nsince both smaller and larger hole-localized particles generate shifts of the polarity shift.\nWe now turn to particle diameters which are larger than the hole diameter, 0.88 \u0016m (Fig. 6(c,h-\nj)) and 4\u0016m (Fig. 6(d) and Supporting Figure 6), we see that both the SM and EM modes\nare characterized by negative polarity \feld shifts. This mirrors the behavior observed for the\n13Figure 7: (a) Simulated SM and EM frequency shifts measured with respect to the bare frequency (shown\nas dotted lines) for a 150 nm particle located within the MC hole (I) and at various positions on the\nMC surface (II: above the SM; III, IV: above the EM). Signi\fcant reductions in a mode's frequency\nare observed when a particle is above the region of the MC containing that mode since HPstrongly\nopposesHext(position `2' in Fig. 2(a)). Weaker increases in the other mode are observed since HP\nwill weakly reinforce Hext(positions `1' and `3' in Fig. 2(a)). Although the induced shifts can be large,\nthe observation of such a shift will be highly dependent on having this particular particle placement.\nNote that, vertical distances in the schematic are proportional to the simulated frequency shifts. (b)\nExperimentally observed changes in EM and SM resonance linewidths as a function of 130 nm particle\nconcentrations applied to 300 nm MC. The error bars are a measure of the spread of linewidth broadening\nacross the measured frequency range, 11.5 - 16 GHz. Data for the lowest four concentrations are taken\nfrom.25\n14fundamental mode in the continuous layer, suggesting that, for large particles which are not pre-\ndominantly hole-localized, the dominant shift, as for the continuous layer, comes from in-plane\n\feld components, here generated by large microbeads quasi-randomly distributed on the MC\nsurface. Indeed, despite some centering of the 0.88 \u0016m particles on top of holes at low concen-\ntrations (two bottom right particles in Fig. 6(i)), they are indeed typically randomly distributed\nacross the MC. Unlike the case of the small particles however, there are no hole-localized parti-\ncles which can act in unison to induce di\u000berent polarity shifts on the EM and SM. As a result,\nfor purely surface-localized beads, both the EM and SM shift in the same direction.\n3 Conclusions\nMagnetic particle detection is critical for magnetic biosensing techniques which have envisioned\nuse in point of care medical diagnostics. This detection is often carried out using conventional\nmagnetoresistive sensors. Looking towards the development of alternative frequency-based sens-\ning methods however, we have demonstrated here the use of nano-con\fned ferromagnetic reso-\nnance modes for magnetic particle sensing. This has been carried out for a range of magnetic\nparticles with sizes from 6 nm to 4 \u0016m. The stray \felds of the particles act directly on the pre-\ncessing moments generating signi\fcant shifts in the resonance \felds and frequencies. As such,\nthe observed resonance shifts in our hole-based patterned ferromagnet can be maintained over\nlarge ranges of \felds, enabling resonance-based nanoparticle detection in strong \felds where\nparticle moments can be maximized. Furthermore, the dominant e\u000bect is typically a resonance\nshift rather than a broadening of the resonance linewidth, a result which is encouraging for the\nimplementation of a sensing method based on detecting changes to resonance frequencies.\nIn this work, we also identi\fed di\u000berent characteristics in the resonance shifts for small\nand large particles. For small particles, their preferential capture in our system's nano-holes\nmeans that the majority of them act on the resonance modes in unison, generating clear, mode-\ndependent resonance shifts and mode-dependent shift polarities which persist up to extremely\nhigh particle coverages. Shifts at intermediate and large coverages of small particles can be well\nreproduced via simulation. In contrast to the case of the small particles however, the e\u000bect of\nmicrobeads is analogous to that seen in continuous \flms with the two major resonance modes\nboth shifting weakly in the same direction. This is due to a lack of particle localization with\nrespect to the spatially periodic nano-scale regions where the modes are localized. Although\n15sub-optimal, detection was nevertheless achieved in such geometries, albeit with low resonance\nshifts which were similar to those seen in continuous \flms. Note that the ability to detect\nparticles is determined not only by how sensitive the frequency is to changes in magnetic \feld\nbut by the ability to generate clear modi\fcations to these frequencies due to favorable particle\npositioning. A proper choice of particle size and reliable positioning of particles close to well\nlocalized modes is thus a critical factor in resonance based sensing.\n4 Experimental section\nThe nanopatterned \flms consist of square arrays of circular holes (`anti-dots') with an array\npitch of 450 nm and hole diameters of 240 nm or 300 nm in a 30 nm thick Permalloy (Ni 80Fe20)\ncontinuous \flm with a 8 or 10 nm gold (Au) capping layer. The large area (4 \u00024 mm2) MCs\nwere fabricated on silicon (Si) substrates using deep ultraviolet lithography, e-beam deposition\nand lifto\u000b.38\nFMR was measured using broadband microwave stripline-based FMR spectroscopy, a tech-\nnique where resonant modes are excited in magnetic materials using a radio-frequency (r.f.)\n\feld, here generated by an r.f. signal passing through an underlying stripline. An absorption\nof microwave power by the magnetic sample is measured when the frequency of the r.f. \feld\nmatches that of a FMR mode. Here we used \feld-modulated FMR spectroscopy wherein a\nlock-in ampli\fer measures the external \feld-( Hext-)derivative of the FMR response at a con-\nstant frequency while sweeping Hext. An interferometric receiver was used to maximize signal\namplitudes.39This is particularly important in our measurement since the sample is separated\nfrom the stripline by a microscope coverslip (rather than sitting directly on the stripline). The\ncoverslip ensures that particles do not rub o\u000b onto the micro-stripline but reduces the signal\namplitude. Hextwas typically swept from 0-350 mT. We note that an overall decrease in the\ndi\u000berential FMR signal was seen when adding increasing concentrations of particles. To enable\ncomparison of traces, the traces were vertically scaled. A small vertical o\u000bset (typically on the\norder of a few tens of \u0016V at most) at times also had to be corrected for.\nParticle detection was carried out for a range of particles: (i) 6 \u00061 nm wide iron-oxide\nnanoparticles (ii) nanomag-D(-spio) cluster-shaped particles from Micromod Partikeltechnologie\nGmbH with stated diameters of 50 nm and 130 nm; (iii) 880 nm wide magnetic beads from Bangs\nLaboratories consisting of iron-oxide nanoparticles within a polymer matrix; and (iv) 4 \u0016m beads\n16from SpheroTech consisting of a polystyrene core coated with a mixture of polystyrene and\nmagnetic nanoparticles. Particles were applied in solution to the \flms' upper surfaces in ambient\nlaboratory conditions in the absence of an external magnetic \feld ( Hext= 0). The solutions\nwere then allowed to dry before re-measurement and imaging. Concentrations of commercial\nparticle solutions (all aqueous) were determined from the manufacturer's speci\fcations after\ndilution in puri\fed water. The 6 nm iron-oxide nanoparticles were synthesized under standard\nSchlenk conditions using the methodology developed by Sun et al. .40Brie\ry, Fe(acac)3(0.7\ng, 2 mmol) and 1,2-hexadecanediol (2.5 g, 10 mmol) were dissolved in benzyl ether (20 mL)\ncontaining oleylamine (6 mmol) and oleic acid (6 mmol). The resulting mixture was heated at\n200\u000eC for 2 hours, increased to 260\u000eC, held for 1 hour then cooled to room temperature. The\nparticles were precipitated by the addition of ethanol (40 mL), centrifuged (5000 rpm, 10 mins)\nand redispersed in 1,2-dichlorobenzene to the required concentration. The particles have been\nfully characterized by a range of routine techniques. The results are contained in Supporting\nFigure 8.\nMicromagnetic simulations were run for a single unit cell of the MC (or an equivalently sized\nregion of conintuous \flm), employing periodic boundary conditions and a tiled macro-geometry41\n(33\u000233 unit cells). For nanoparticles within the holes of the MC, the lower surface of the particle\nwas always located at the lower surface of the MC (i.e. resting on the underlying Si wafer). For\nparticles on top of the MC surface or above the layer, the lower surface of the particle was set at\n10 nm above the upper surface of the magnetic media (i.e. sitting on the upper surface of the non-\nmagnetic capping layer). The following parameters were used for the MC: damping \u000b= 0.008,\nnil intrinsic anisotropy, gyromagnetic ratio 2 \u0019\r= 1:85\u00021011rad/T.s, saturation magnetization\nMs= 8\u0002105A/m and exchange sti\u000bness Aex= 13 pJ/m. The parameters for the particle\nwere: damping \u000b= 0.05, gyromagnetic ratio \ras in the MC and Mstaken from previously\npresented nanoparticle magnetization data.25Micromagnetic results were obtained from time\ndomain simulations run using MuMax3.42For these simulations, the system was initialized\nwith a uniform magnetization in the (1,1) direction and allowed to relax in the presence of an\nexternal \feld using MuMax3's internal relaxation routine. Post-relaxation, an excitation sinc\npulse of 0.5mT (cut-o\u000b freq 30 Ghz, 300 ps o\u000bset) was applied along the x-axis43at a given\nHext(applied along the y-axis). The precessional ringdown data was then Fourier analyzed. All\nresultant mode pro\fle visualizations were determined by extracting the spatially resolved mx\nFourier amplitudes at each identi\fed resonant frequency across the simulation region. Obtained\n17results (e.g. Fig. 3(b)) are shown as intensity plots with the brightest regions corresponding to\nthe highest Fourier amplitude for mxat that frequency. matplotlib44was used for visualization of\nsimulation data. As done previously,25the eigensolver (see e.g.45) in the FinMag micromagnetic\nsimulation package (based on Nmag46) was also used for certain test cases with good levels of\nagreement.\n5 Acknowledgments\nThis research was supported by the Australian Research Council's Discovery Early Career Re-\nsearcher Award scheme (DE120100155) and Discovery Projects scheme (DP110103980), the\nUnited States Air Force (Asian O\u000ece of Aerospace Research and Development, AOARD) an\nEPSRC DTC grant (EP/G03690X/1), the University of Western Australia's (UWA) RDA, RCA,\nECRFS, SIRF, UPAIS, Re-Entry Fellowship, Teaching Relief and Vacation Scholarship schemes\nand by resources provided by the Pawsey Supercomputing Centre with funding from the Aus-\ntralian Government and the Government of Western Australia. A.O.A. was supported by the\nNational Research Foundation, Prime Minister's O\u000ece, Singapore under its Competitive Re-\nsearch Programme (CRP Award No. NRF-CRP 10-2012-03). The authors thank C. Lueng,\nM. Albert, W. Wang, A. Suvorova, A. Dodd, C. Yang, D. Schibeci, A. Chew and R.C. Bording\nfor their assistance. The authors acknowledge access to the UWA's Biomagnetics Wet Labo-\nratory and Magnetic Characterisation Facility as well as the facilities, and the scienti\fc and\ntechnical assistance of the Australian Microscopy & Microanalysis Research Facility at the Cen-\ntre for Microscopy, Characterisation & Analysis, The University of Western Australia, a facility\nfunded by the University, State and Commonwealth Governments.\nReferences\n[1] D. R. Baselt, G. U. Lee, M. Natesan, S. W. Metzger, P. E. Sheehan, and R. J. Colton, \\A\nbiosensor based on magnetoresistance technology\", Biosens. Bioelectron. 13, 731 (1998).\n[2] R. S. Gaster, D. A. Hall, C. H. Nielsen, S. J. Osterfeld, H. Yu, K. E. Mach, R. J. Wilson,\nB. Murmann, J. C. Liao, S. S. Gambhir, et al. , \\Matrix-insensitive protein assays push the\nlimits of biosensors in medicine\", Nat. Med. 15, 1327 (2009).\n18[3] Y. R. Chemla, H. L. Grossman, Y. Poon, R. McDermott, R. Stevens, M. D. Alper, and\nJ. Clarke, \\Ultrasensitive magnetic biosensor for homogeneous immunoassay\", Proc. Natl.\nAcad. Sci. U.S.A. 97, 14268 (2000).\n[4] P.-A. Besse, G. Boero, M. Demierre, V. Pott, and R. Popovic, \\Detection of a single\nmagnetic microbead using a miniaturized silicon Hall sensor\", Appl. Phys. Lett. 80, 4199\n(2002).\n[5] M. M. Miller, G. A. Prinz, S.-F. Cheng, and S. Bounnak, \\Detection of a micron-sized\nmagnetic sphere using a ring-shaped anisotropic magnetoresistance-based sensor: A model\nfor a magnetoresistance-based biosensor\", Appl. Phys. Lett. 81, 2211 (2002).\n[6] L. Ejsing, M. F. Hansen, A. K. Menon, H. A. Ferreira, D. L. Graham, and P. P. Freitas,\n\\Magnetic microbead detection using the planar Hall e\u000bect\", Journal of Magnetism and\nMagnetic Materials 293, (2005).\n[7] M. Donolato, M. Gobbi, P. Vavassori, M. Leone, M. Cantoni, V. Metlushko, B. Ilic, M.\nZhang, S. X. Wang, and R. Bertacco, \\Nanosized corners for trapping and detecting mag-\nnetic nanoparticles\", Nanotechnol 20, 385501 (2009).\n[8] L. Di Michele, C. Shelly, P. de Marco, P. See, D. Cox, and O. Kazakova, \\Detection and\nsusceptibility measurements of a single Dynal bead\", J. Appl. Phys. 110, 063916 (2011).\n[9] S. M. Hira, K. Aledealat, K.-S. Chen, M. Field, G. J. Sullivan, P. B. Chase, P. Xiong,\nS. von Moln\u0013 ar, and G. F. Strouse, \\Detection of target ssDNA using a microfabricated\nHall magnetometer with correlated optical readout\", J. Biomed. Biotechnol. 2012 , 492730\n(2012).\n[10] J. Devkota, A. Ruiz, P. Mukherjee, H. Srikanth, and M.-H. Phan, \\Magneto-Impedance\nBiosensor With Enhanced Sensitivity for Highly Sensitive Detection of Nanomag-D Beads\",\nIEEE Trans. Magn. 49, 4060 (2013).\n[11] H. J. Chung, C. M. Castro, H. Im, H. Lee, and R. Weissleder, \\A magneto-DNA nanopar-\nticle system for rapid detection and phenotyping of bacteria\", Nat. Nanotechnol. 8, 369\n(2013).\n[12] P. I. Nikitin, P. M. Vetoshko, and T. I. Ksenevich, \\New type of biosensor based on magnetic\nnanoparticle detection\", J. Magn. Magn. Mater 311, 445 (2007).\n19[13] J. Llandro, T. J. Hayward, D. Morecroft, J. A. C. Bland, F. J. Castano, I. A. Colin, and\nC. A. Ross, \\Quantitative digital detection of magnetic beads using pseudo-spin-valve rings\nfor multiplexed bioassays\", Appl. Phys. Lett. 91, 203904 (2007).\n[14] B. Srinivasan, Y. Li, Y. Jing, Y. Xu, X. Yao, C. Xing, and J.-P. Wang, \\A Detection\nSystem Based on Giant Magnetoresistive Sensors and High-Moment Magnetic Nanoparticles\nDemonstrates Zeptomole Sensitivity: Potential for Personalized Medicine\", Angew. Chem.\nInt. Ed. 48, 2764 (2009).\n[15] S. J. Osterfeld, H. Yu, R. S. Gaster, S. Caramuta, L. Xu, S.-J. Han, D. A. Hall, R. J.\nWilson, S. Sun, R. L. White, and et al., \\Multiplex protein assays based on real-time\nmagnetic nanotag sensing\", Proc. Natl. Acad. Sci. U.S.A. 105, 20637 (2008).\n[16] D. Hall, R. Gaster, T. Lin, S. Osterfeld, S. Han, B. Murmann, and S. Wang, \\GMR\nbiosensor arrays: A system perspective\", Biosens. Bioelectron. 25, 2051 (2010).\n[17] B. Srinivasan, Y. Li, Y. Jing, C. Xing, J. Slaton, and J.-P. Wang, \\A Three-Layer\nCompetition-Based Giant Magnetoresistive Assay for Direct Quanti\fcation of Endoglin\nfrom Human Urine\", Anal. Chem. 83, 2996 (2011).\n[18] P. P. Freitas, F. A. Cardoso, V. C. Martins, S. A. M. Martins, J. Loureiro, J. Amaral, R. C.\nChaves, S. Cardoso, L. P. Fonseca, A. M. Sebasti~ ao, and et al., \\Spintronic platforms for\nbiomedical applications\", Lab Chip 12, 546 (2012).\n[19] J.-R. Lee, N. Sato, D. J. B. Bechstein, S. J. Osterfeld, J. Wang, A. W. Gani, D. A. Hall,\nand S. X. Wang, \\Experimental and theoretical investigation of the precise transduction\nmechanism in giant magnetoresistive biosensors\", Scienti\fc Reports 6, 18692 (2016).\n[20] K. Mizushima, K. Kudo, T. Nagasawa, and R. Sato, \\Signal-to-noise ratios in high-signal-\ntransfer-rate read heads composed of spin-torque oscillators\", J. Appl. Phys. 107, 063904\n(2010).\n[21] M. Inoue, A. Baryshev, H. Takagi, P. B. Lim, K. Hatafuku, J. Noda, and K. Togo, \\Inves-\ntigating the use of magnonic crystals as extremely sensitive magnetic \feld sensors at room\ntemperature\", Appl. Phys. Lett. 98, 132511 (2011).\n[22] R. J. Ryan, H. Xi, and I. Jin, \\Magnetic oscillator based biosensor\", Patent US8053244\nB2, (2011).\n20[23] P. M. Braganca, B. A. Gurney, B. A. Wilson, J. A. Katine, S. Maat, and J. R. Childress,\n\\Nanoscale magnetic \feld detection using a spin torque oscillator\", Nanotechnol. 21, 235202\n(2010).\n[24] J. P. Fried and P. J. Metaxas, \\Localized magnetic \felds enhance the \feld sensitivity of the\ngyrotropic resonance frequency of a magnetic vortex\", Phys. Rev. B 93, 064422 (2016).\n[25] P. J. Metaxas, M. Sushruth, R. Begley, J. Ding, R. C. Woodward, I. Maksymov, M. Albert,\nW. Wang, H. Fangohr, A. Adeyeye, and M. Kostylev, \\Sensing magnetic nanoparticles\nusing nano-con\fned ferromagnetic resonances in a magnonic crystal\", Appl. Phys. Lett.\n106, 232406 (2015).\n[26] J. R. Petrie, S. Urazhdin, K. A. Wieland, G. A. Fischer, and A. S. Edelstein, \\Using a spin\ntorque nano-oscillator to read memory based on the magnetic permeability\", J. Phys. D:\nAppl. Phys. 47, 055002 (2014).\n[27] I. N. Krivorotov, N. C. Emley, J. C. Sankey, S. I. Kiselev, D. C. Ralph, and R. A. Buhrman,\n\\Time-Domain Measurements of Nanomagnet Dynamics Driven by Spin-Transfer Torques\",\nScience 307, 228 (2005).\n[28] H. Suto, T. Nagasawa, K. Kudo, K. Mizushima, and R. Sato, \\Real-Time Measurement of\nTemporal Response of a Spin-Torque Oscillator to Magnetic Pulses\", Appl. Phys. Express\n4, 013003 (2010).\n[29] Z. Zeng, P. K. Amiri, I. N. Krivorotov, H. Zhao, G. Finocchio, J.-P. Wang, J. A. Katine,\nY. Huai, J. Langer, K. Galatsis, et al. , \\High-power coherent microwave emission from\nmagnetic tunnel Junction nano-oscillators with perpendicular anisotropy\", ACS Nano 6,\n6115 (2012).\n[30] E. Grimaldi, A. Dussaux, P. Bortolotti, J. Grollier, G. Pillet, A. Fukushima, H. Kubota,\nK. Yakushiji, S. Yuasa, and V. Cros, \\Response to noise of a vortex based spin transfer\nnano-oscillator\", Phys. Rev. B 89, (2014).\n[31] J. Loureiro, R. Ferreira, S. Cardoso, P. P. Freitas, J. Germano, C. Fermon, G. Arrias, M.\nPannetier-Lecoeur, F. Rivadulla, and J. Rivas, \\Toward a magnetoresistive chip cytometer:\nIntegrated detection of magnetic beads \rowing at cm/s velocities in micro\ruidic channels\",\nAppl. Phys. Lett. 95, 034104 (2009).\n21[32] J. Loureiro, P. Z. Andrade, S. Cardoso, C. L. da Silva, J. M. Cabral, and P. P. Freitas,\n\\Spintronic chip cytometer\", Appl. Phys. Lett. 109, 07B311 (2011).\n[33] M. Helou, M. Reisbeck, S. F. Tedde, L. Richter, L. B ar, J. J. Bosch, R. H. Stauber, E.\nQuandt, and O. Hayden, \\Time-of-\right magnetic \row cytometry in whole blood with\nintegrated sample preparation\", Lab Chip 13, 1035 (2013).\n[34] M. Krawczyk and D. Grundler, \\Review and prospects of magnonic crystals and devices\nwith reprogrammable band structure\", J. Phys.: Condens. Matter 26, 123202 (2014).\n[35] B. Lenk, H. Ulrichs, F. Garbs, and M. M unzenberg, \\The building blocks of magnonics\",\nPhysics Reports 507, 107 (2011).\n[36] A. Manzin, G. Barrera, F. Celegato, M. Co \u0010sson, and P. Tiberto, \\In\ruence of lattice\ndefects on the ferromagnetic resonance behaviour of 2D magnonic crystals\", Sci. Rep. 6,\n22004 (2016).\n[37] J. Ding, N. Singh, M. Kostylev, and A. O. Adeyeye, \\Static and dynamic magnetic prop-\nerties of Ni 80Fe20anti-ring nanostructures\", Phys. Rev. B 88, 014301 (2013).\n[38] N. Singh, S. Goolaup, and A. O. Adeyeye, \\Fabrication of large area nanomagnets\", Nan-\notechnol 15, 1539 (2004).\n[39] E. N. Ivanov and M. Kostylev, \\Extremely high-resolution measurements of microwave mag-\nnetisation dynamics in magnetic thin \flms and nanostructures\", Pre-print: arXiv:1402.3459\n[cond-mat.mes-hall] (vol.: n/a) , (page: n/a) (2014).\n[40] S. Sun, H. Zeng, D. B. Robinson, S. Raoux, P. M. Rice, S. X. Wang, and G. Li, \\Monodis-\nperse MFe 2O4(M = Fe, Co, Mn) Nanoparticles\", J. Am. Chem. Soc. 126, 273 (2004).\n[41] H. Fangohr, G. Bordignon, M. Franchin, A. Knittel, P. A. J. de Groot, and T. Fischbacher,\n\\A new approach to (quasi) periodic boundary conditions in micromagnetics: The macro-\ngeometry\", J. Appl. Phys 105, 07D529 (2009).\n[42] A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F. Garcia-Sanchez, and B. Van\nWaeyenberg, \\The design and veri\fcation of MuMax3\", AIP Adv. 4, 107133 (2014).\n22[43] G. Venkat, D. Kumar, M. Franchin, O. Dmytriiev, M. Mruczkiewicz, H. Fangohr, A. Bar-\nman, M. Krawczyk, and A. Prabhakar, \\Proposal for a Standard Micromagnetic Problem:\nSpin Wave Dispersion in a Magnonic Waveguide\", IEEE Trans. Magn. 49, 524 (2013).\n[44] J. D. Hunter, \\Matplotlib: A 2D graphics environment\", Comput. Sci. Eng. 9, 90 (2007).\n[45] P. J. Metaxas, M. Albert, S. Lequeux, V. Cros, J. Grollier, P. Bortolotti, A. Anane, and\nH. Fangohr, \\Resonant translational, breathing, and twisting modes of transverse magnetic\ndomain walls pinned at notches\", Phys. Rev. B 93, 054414 (2016).\n[46] T. Fischbacher, M. Franchin, G. Bordignon, and H. Fangohr, \\A Systematic Approach\nto Multiphysics Extensions of Finite-Element-Based Micromagnetic Simulations: Nmag\",\nIEEE Trans. Mag. 43, 2896 (2007).\n23Supporting \fgure 1\nFundamental mode resonance frequencies of the bare continuous magnetic \flm plotted versus\nHext. An average slope of 49.4 GHz/T was used to convert between particle-induced \feld shifts\nand frequency shifts.\n24Supporting \fgure 2\n(a,c) show comparisons between the simulated and experimental mode structures (EM and SM)\nfor both studied MC geometries (240 nm and 300 nm holes on a 450 nm array pitch). The slopes\nof the data were used to convert between particle-induced \feld shifts and frequency shifts. (b,d)\nshows experimentally obtained FMR traces of bare MCs at a \fxed r.f. excitation frequency of\n12 GHz. Intermediate modes are clearly resolved in the plots, lying between the SM and EM.\n25Supporting \fgure 3\n(a-d) SEM images showing the four lowest concentrations of 130 nm particles on the 300 MC.\nThese images were used to extract statistics regarding the number of particles inside holes and\non the MC surface. (e) shows the ratio of \flled holes versus particle solution concentrations.\n(f) shows that increasing the particle solution concentration also increases the proportion of\nhole-localized particles.\n26Supporting \fgure 4\nExperimentally obtained resonance shifts for intermediate modes (IM1-3) as a function of particle\nsolution concentration (130 nm particles). The solid line shows the EM shifts for comparison.\n27Supporting \fgure 5\nUpper: SEM images of 130 nm particles on a 240 nm MC. Particle concentrations are given below\nthe images. The white scale bar represents 1 \u0016m. Lower: Experimentally obtained result for\nEM and SM resonant shifts for a 240 nm MC as a function of particle concentration. Horizontal\ndashed and solid lines resepctively show the average simulated EM and SM resonant shifts for 1\nparticle/hole and for a disk plus a 150 nm thick sheet of particles on the surface (as described in\nthe manuscript). The error bars are a measure of the spread of experimentally obtained shifts\nacross this frequency range and also includes the uncertainties related to slight variations in\nsample placement.\n28Supporting \fgure 6\nExperimentally obtained EM and SM resonant shifts for a 240 nm MC versus solution concen-\ntration for (a) 6 nm particles and (b) 4 \u0016m wide magnetic beads.\n29Supporting \fgure 7\nSimulated (a) EM and (b) SM resonance shifts in a 300 nm MC as a function of particle size for\na hole-located particle laterally centered within the hole.\n30Supporting \fgure 8\n(a) Bright \feld TEM image of as-synthesized iron-oxide particles. (b) Particle size distribution\n(obtained from 7096 particles). (c) Selected area electron di\u000braction pattern (SAED). (d) X-ray\npowder di\u000braction (XRD) spectrum. Labels correspond to the following indices: a (111); b\n(220); c (311); d (400); e (422); f (511) and g (440). (e) Magnetic moment of freeze-dried 6 nm\nparticles m versus applied magnetic \feld.\n31"    },    {        "title": "2009.01977v1.Detection_of_Ferromagnetic_Resonance_from_1_nm_thick_Co.pdf",        "content": "1  Detection of Ferromagnetic Resonance from 1 nm-thick Co  Shugo Yoshii, Ryo Ohshima, Yuichiro Ando, Teruya Shinjo and Masashi Shiraishi †  Department of Electronic Science and Engineering, Kyoto University, Nishikyo-ku, Kyoto 615-8510, Japan  †Corresponding author (shiraishi.masashi.4w@kyoto-u.ac.jp)  Abstract To explore the further possibilities of nanometer-thick ferromagnetic films (ultrathin ferromagnetic films), we investigated the ferromagnetic resonance (FMR) of 1 nm-thick Co film. Whilst an FMR signal was not observed for the Co film grown on a SiO2 substrate, the insertion of a 3 nm-thick amorphous Ta buffer layer beneath the Co enabled the detection of a salient FMR signal, which was attributed to the smooth surface of the amorphous Ta. This result implies the excitation of FMR in an ultrathin ferromagnetic film, which can pave the way to controlling magnons in ultrathin ferromagnetic films.   2  Introduction Nanometer-thick films, so-called ultrathin films, have been collecting broad attention because of their abundant spintronics nature [1-3]. Chiba and co-workers carried out a pioneering study, in which 1 nm-thick ferromagnetic Co film exhibited sizable modulation of its magnetization under an applied gate voltage [1]. Indeed, the Curie temperature of the ultrathin Co was lowered to ca. 320 K and was modulated by gating, which gave a great impact to the spintronics field. A subsequent study using 0.4 nm-thick ultrathin Co revealed greater potential of magnetization control by combining ionic gating [2]. More recently, studies on the spintronic nature of ultrathin films have been expanded to include nonmagnetic metals, and the discovery of a gate-tunable inverse spin Hall effect in ultrathin Pt (2 nm thickness) has opened the field of tunable spin-orbit interaction (SOI) [3]. A common physics among the magnetization control of ultrathin ferromagnetic films and the tunable SOI in ultrathin nonmagnetic films is a substantial shift of the Fermi level under a strong gate electric field [4,5].  Investigation of spin physics using ferromagnetic resonance (FMR) is pivotal in modern spintronics. It has been used in a wide range of studies from both fundamental and applied physics, including those related to the generation of spin current [6] and a spin-torque diode effect [7]. FMR takes place under the simultaneous application of a static external magnetic field and microwave, which enables uniform rotation of magnetization. The uniform magnetization rotation generates a spin wave, a quantized state of which is magnon. Whilst magnons have attracted tremendous attention in spintronics because of their spin current propagation ability [8], they have also attracted great attention in the field of hybrid quantum systems because of the potential of creating strong coupling states with another magnon [9], a photon [10] and a superconducting qubit [11].  Despite the strong potential of magnons in spintronics and other condensed-matter fields, generation of FMR, of which excitation is indispensable to generate magnons in ferromagnets, in ultrathin ferromagnetic metals is not easy. In fact, the thickness of ferromagnetic metal films often used in FMR studies in spintronics is typically greater than 5 nm, because the surface roughness of the substrate beneath the ferromagnetic metal hampers the observation of a clear FMR signal. However, given the success of the substantial magnetization control in ultrathin Co by gating [1,2], efficient and tunable magnon creation in ultrathin ferromagnetic metals under FMR can provide a path to electric-field control of magnons, because the number of magnons is proportional to the square root of the total 3  magnetization and the coupling strength between magnons and photons is collectively enhanced by square root of number of magnons [12]. For the achievement, the first milestone is excitation of FMR with a sufficiently small resonance field and the half-width at half-maximum in ultrathin ferromagnetic metals, resulting in magnon excitation, which has not been sufficiently achieved. In this study, we report the realization of FMR in a 1 nm-thick Co film, which has not been previously demonstrated, by depositing a Ta buffer layer beneath the Co layer. The key to achieving FMR is utilization of smooth surface of amorphous Ta layer.   Results  Figures 1(a) and (b) show sample structures and measuring setup, respectively. We prepared two different types of samples: One is SiO2/Co (type-A), and the other is SiO2/Co/Ta (type-B) (see Fig. 1(a)). In type-A samples, a Co thin film of 1,2,3 or 5 nm thickness was deposited onto a SiO2 (300 nm)/Si substrate using radio-frequency magnetron sputtering. In type-B samples, a Ta buffer layer of 3 nm thickness was deposited beneath the 1 nm-thick Co. FMR measurements were performed using a TE011 (transverse electric mode) cavity of an electron spin resonance system. Figure 2(a) shows the FMR spectra of the four type-A samples, where the thickness of the Co film was varied. The resonance field and the linewidth of the spectra changed dramatically with the Co thickness. More importantly, an FMR signal was barely observed when the thickness of the Co was less than 2 nm. To circumvent the problem of the missing FMR in thin Co films, we introduced Ta as a buffer layer (type-B sample). Amorphous Ta is widely recognized to possess a quite smooth surface [1,2,13], although direct observation of the smooth surface of Ta was difficult without exposing the surface to air in our experimental setup and structural analyses such as x-ray diffraction cannot be implemented due to the thin Ta layer (3 nm).  Hence, we expected that the insertion of Ta beneath ultrathin Co enables formation of a flat and continuous Co film with nm-thick, yielding a sharp FMR spectrum from the Co. In fact, Chiba and co-workers observed the anomalous Hall effect from 0.4 nm-thick Co grown on 3 nm-thick Ta, which is compelling evidence for the formation of a continuous film [1,2]. Figure 2(b) shows a comparison of the FMR spectra of the Co (1 nm) and Co (1 nm)/Ta (3 nm) samples. The difference in the FMR spectra is clearly discernable, and the FMR spectrum of the Co (1 nm)/Ta (3 nm) sample is, in fact, quite obvious. Given that amorphous Ta possesses a smooth surface, the clear FMR spectrum of the Co is 4  attributed to the insertion of the Ta buffer layer with a smooth surface beneath the 1 nm-thick Co. Figure 2(c) shows the resonance fields, the half-width at half-maximum (∆µ0H) of the FMR spectra, and the Gilbert damping constant a as functions of the Co layer thickness of the type-A and -B samples. The resonance field and the ∆µ0H were obtained by deconvolution of the integral form of the FMR spectra (for more detail, see Methods and Supplementary Information). The Gilbert damping constant a was calculated using the following equation, 𝛼=∆𝜇!𝐻\t∙𝛾2𝜋𝑓\"#$,\t\t\t\t\t\t\t\t\t\t\t\t\t(1) where 𝛾 is the gyromagnetic ratio of Co and fres is the applied microwave frequency [14]. Whilst the resonance field, the half-width at half-maximum of the FMR spectra, and the Gilbert damping constant monotonically increased with decreasing the Co thickness, the 1 nm-thick Co layer overlying a Ta buffer layer exhibited noticeable suppression of them. Thus, it is corroborated that the insertion of an amorphous Ta layer beneath an ultrathin Co layer is an efficient approach to excite FMR in the 1 nm-thick Co.  Since the thickness of the amorphous Ta layer in the previous studies was fixed at 3 nm [1,2], the thickness of the Ta buffer layer was varied from 2 to 5 nm in increments of 1 nm and the FMR of the 1 nm-thick Co on Ta buffer layers of various thickness was measured. The FMR spectra of the Co layers with Ta buffer layers of various thickness and the Gilbert damping constant of each sample are shown in Figs. 3(a) and 3(b), respectively. Neither the FMR spectra nor the magnitude of the Gilbert damping constant is dependent on the Ta thickness, which suggests that the insertion of a 3 nm-thick Ta buffer layer is sufficient to induce formation of a flat Co layer. Notably, an FMR signal was not observed from the Co (1 nm)/ Ta (1 nm) sample (see Supplementary Information), which directly indicates that an excessively thin Ta buffer layer does not allow exciting the FMR in a 1 nm-thick Co layer.   To better understanding the aforementioned phenomena, we prepared a Co (1 nm)/\t𝕆 /Ta (3 nm) sample, where the surface of the Ta was intentionally oxidized (𝕆 denotes that the sample was exposed to air at this sample fabrication step). Figure 4(a) shows a comparison of the FMR spectra of the Co (1 nm)/Ta (3 nm) and Co (1 nm)/\t𝕆 /Ta (3 nm) samples. A substantial difference in the FMR spectra is observed; an FMR signal was not observed from the Co (1 nm)/\t𝕆 /Ta (3 nm) sample albeit a 3 nm-thick Ta buffer layer was introduced. The lack of an FMR signal 5  from the Co (1 nm)/\t𝕆 /Ta (3 nm) sample is thus attributed to the oxidized surface of the Ta. As aforementioned, the insertion of an amorphous Ta layer beneath the 1 nm Co facilitates the formation of a flat and continuous Co film because the surface of the amorphous Ta is smooth. Meanwhile, the surface roughness of the oxidized 3 nm-thick Ta was measured to be almost the same as that of the SiO2 substrate; a roughness of ca. 1 nm was observed for both samples by atomic force microscopy (AFM), whilst the grain sizes of these two samples slightly differed (see Figs. 4(b)). Here, to note is that surface of amorphous Ta cannot be measured by AFM without exposing the sample surface to air in our measuring setup. Hence, we deduce that the surface of the oxidized Ta loses sufficient smoothness, which can hinder the formation of a flat and continuous 1 nm-thick Co film. These results unequivocally rationalize that inserting a Ta buffer layer and maintaining its smooth surface play crucial roles in the growth of an ultrathin Co layer that can generate salient FMR.   Discussion The upshift of the resonance field as a function of the Co thickness in type-A samples (see Figs. 2 (a) and (c)) is attributed to a decrease of the total magnetization. Chiba and co-workers observed strong suppression of magnetization in 1 nm-thick Co, resulting in a substantial decrease of the Curie temperature [1]. Hence, the upshift of the resonance field is due to weaker magnetization of the ultrathin Co, consistent with the previous study [1]. Meanwhile, the missing FMR spectra is attributed to roughness of the Co film. We used SiO2/Si substrates, of which surface is not sufficiently smooth. Indeed, it was previously found that the FMR spectra of Ni80Fe20 (Py) were strongly dependent on the substrates and that the FMR linewidth of Py grown on a SiO2 substrate was greater than the linewidths of Py grown on yttrium-iron-garnet and non-doped diamond substrates [15]. The broader FMR spectrum of Py on a SiO2 substrate is ascribed to the roughness of the a SiO2 substrate, which hampers isotropic magnetization and uniform magnetization precession under FMR. The results obtained in the present study are consistent with those reported in the literature [15].  In chronicle of FMR studies of thin ferromagnetic films, a couple of studies were implemented about detection of FMR spectra from ultrathin Co [16,17] almost two decades ago. The thickness of the Co films in those 6  previous studies is comparable to that in our present study. Meanwhile, the resonance field and the FMR linewidth in those studies were roughly 500 mT and 60 mT (note that the linewidth of 60 mT is equivalent to the half-width at half-maximum of roughly 70 mT) [16], respectively (the similarly large resonance field was also reported in the literature [17], whereas the linewidth was not discussed in the study). Such the large resonance field and the broad FMR linewidth are attributed to a fact that the ferromagnetism was quite weak and the magnetization precession under FMR was not uniform, i.e., the quality of the Co was poor. Furthermore, the FMR spectrum exhibited the single branch only when the thickness of the Co was greater than 2 nm in that study [16]. Given that the resonance field and the half-width at half-maximum of the FMR spectra in our study are smaller and roughly 80 mT and 15 mT, respectively, and that the single FMR spectrum can be seen from the 1 nm-thick Co, the ultrathin Co used in our study simultaneously possesses sufficiently strong ferromagnetism and uniform FMR unlike in the previous studies, i.e. we experimentally demonstrated that the quality of the ultrathin Co with a Ta buffer layer is sufficiently good and the ultrathin Co is quite available for future magnon spintronics and quantum hybrid systems.  In summary, we achieved FMR excitation from an ultrathin Co film with a thickness of 1 nm by inserting an amorphous Ta buffer layer. The smoothness of the amorphous Ta played a crucial role in a formation of a flat and ultrathin Co layer. These findings provide a new pathway to the electric-field control of magnons using ultrathin ferromagnetic metals.  Methods In type-A samples, a Co thin film of 1,2,3 or 5 nm thickness was deposited using radio-frequency magnetron sputtering. The base pressure of the sputtering system was kept to be lower than 2.5×10-5 Pa, and the flow rate and partial pressure of the Ar gas were set to be 5 sccm and 0.5 Pa, respectively; and the sputtering temperature was room temperature (RT). The deposition rate of the Co was 0.8 nm/min. A SiO2 layer (10 nm) was deposited onto the Co film to prevent from oxidation of the Co layer. In type-B samples, a Ta buffer layer of 3 nm thickness was deposited, where the deposition rate of the Ta was 3.83 nm/min. The substrate was the same as that used for the type-A samples. During the deposition of Co and Ta, the sample holder was rotated at 20 rpm.  7  FMR measurements were performed using a TE011 (transverse electric mode) cavity of an electron spin resonance system (JEOL JES-FA 200); an external magnetic field under microwave irradiation was applied parallel to the sample plain, and the microwave frequency and power were set to be 9.12 GHz and 10 mW, respectively (see Fig. 1(b)). All of the FMR measurements were carried out at RT.  Deconvolution of the FMR spectra was carried out by using the integral form of the FMR spectra. In addition to the Lorentzian (symmetric) component, an asymmetric component was taken into account. The fitting function used is 𝐹(𝜇!𝐻)=𝐴$%&(∆)!*)\"()!*,)!*#$%)\"-(∆)!*)\"−2𝐴.$%&/()!*,)!*#$%)()!*,)!*#$%)\"-(∆)!*)\"+𝑎(𝜇!𝐻)+𝑏, where Asym and Aasym are symmetric and asymmetric components, respectively, and, a and b are constants.    8  References 1. Chiba, D., Fukami, S., Shimamura, K., Ishiwata, N., Kobayashi K., & Ono, T. Electrical control of the ferromagnetic phase transition in cobalt at room temperature. Nature Mater. 10, 853-856 (2011).  2. Shimamura, K., Chiba, D., Ono, S., Fukami, S., Ishiwata, N., Kawaguchi, M., Kobayashi K. & Ono, T. Electrical control of Curie temperature in cobalt using an ionic liquid film. Appl. Phys. Lett. 100, 122402 (2012).  3. Dushenko, S., Hokazono, M., Nakamura, K., Ando, Y., Shinjo T. & Shiraishi, M. Tunable inverse spin Hall effect in nanometer-thick platinum films by ionic gating. Nature Commun. 9, 3118 (2018).  4. Oba, M., Nakamura, K., Akiyama, T., Ito, T., Weinert M. & Freeman, A.J. Electric-field-induced modification of the magnon energy, exchange interaction and Curie temperature of transition-metal thin films. Phys. Rev. Lett. 114, 107202 (2015).  5. Guo, G., Murakami, S., Chen, T.-W.  & Nagaosa, N. Intrinsic spin Hall effect in Platinum: First-principle calculations. Phys. Rev. Lett. 100, 096401 (2008).  6. Saitoh, E., Ueda, M., Miyajima H. & Tatara, G. Conversion of spin current into charge current at room temperature: Inverse spin Hall effect. Appl. Phys. Lett. 88, 182509 (2006).  7. Tulapurkar, A.A., Suzuki, Y., Fukushima, A., Kubota, H., Maehara, H., Tsunekawa, K., Djayaprawira, D.D., Watanabe N. & Yuasa, S. Spin-torque diode effect in magnetic tunnel junctions. Nature 438, 339-342 (2005).  8. Kajiwara, Y., Harii, K., Takahashi, S., Ohe, J., Uchida, K., Mizuguchi, M., Umezawa, H., Kawai, H., Ando, K., Takanashi, K., Maekawa S. & Saitoh, E. Transmission of electrical signals by spin-wave interconversion in a magnetic insulator. Nature 464, 262-266 (2010).  9. Huebl, H., Zollitsch, C.W., Lotze, J., Hocke, F., Greifenstein, M., Marx, A., Gross R. & Gönnenwein, S.T.B. High cooperativity in coupled microwave resonator ferrimagnetic insulator hybrids. Phys. Rev. Lett. 111, 127003 (2013).  9  10. Osada, A., Hisatomi, R., Noguchi, A., Tabuchi, Y., Yamazaki, R., Usami, K., Sadgrove, M., Yalla, R., Nomura M. & Nakamura, Y. Cavity optomagnonics with spin-orbit coupled photons. Phys. Rev. Lett. 116, 223601 (2016).  11. Tabuchi, Y. Ishino, S., Noguchi, A., Ishikawa, T., Yamazaki, R., Usami K. & Nakamura, Y. Coherent coupling between a ferromagnetic magnon and a superconducting qubit. Science 349, 405-408 (2015).  12. Tabuchi, Y., Ishino, S. Ishikawa, T., Yamazaki, R., Usami K. & Nakamura, Y. Hybridizing ferromagnetic magnons and microwave photons in the quantum limit. Phys. Rev. Lett. 113, 083603 (2014).  13. Vahaplar, K., Tari, S., Tokuc H. & Okur, S. Effect of Ta buffer layer and thickness on the structural and magnetic properties of Co thin films. J. Vac. Sci. Technol. B 27, 2112 -2116 (2009).  14. Nembach, H.T., Silva, T.J., Shaw, J.M., Schneider, M.L., Carey, M.J., Maat S. & Childress, J.R. Perpendicular ferromagnetic resonance measurements of damping and Lande g-factor in sputtered (Co2Mn)1-xGex thin films. Phys. Rev. B 84, 054424 (2011). 15. Tsukahara, A., Ando, Y., Kitamura, Y., Emoto, H., Shikoh, E., Delmo, M.P., Shinjo T. & Shiriashi, M. Self-induced inverse spin Hall effect in permalloy at room temperature. Phys. Rev. B 89, 235317 (2014).  16. Purcell, S.T., van Kestere, H.W., Cosman, E.C., Zeper W.B. & Hoving, W. Magnetic properties of ultrathin epitaxial Co films on a Pd (111) single crystal. J. Appl. Phys. 69, 5640-5642 (1991).  17. Gieniusz, R., Stupakiewicz, A., Liedke, O., Maziewski, A., Gogol P. & Beauvillain, P. FMR study of ultrathin Co magnetic films on vicinal Si (111) substrates. J. Magn. Magn. Mater. 272-276, e911-e912 (2004).     10  Acknowledgement This work is supported in part by a Grant-in-Aid for Scientific Research (S), “Semiconductor spincurrentronics” (No. 16H06330) and Izumi Science and Technology Foundation. The authors thank Prof. Daichi Chiba of Osaka University, Japan, for his fruitful discussion and suggestions about the growth of an amorphous Ta layer. Author contributions M. S., Y. A. and R. O. conceived the experiments. S.Y. fabricated samples, collected data and analyzed results. R.O. helped in the experiments. S.Y., R.O. and M.S. wrote the manuscript. All authors discussed the results.  Competing interest The authors declare no competing interests.  Additional information  Supplementary information is available for this paper at XXX. Correspondence and requests for materials should be addressed to M.S.     11  Figure captions Figure 1. (a) Schematics of the sample structures of the type-A and the type-B samples. The thickness of the Co (tCo) in the type-A samples was 1, 2, 3, or 5 nm. The samples were capped with 10 nm-thick SiO2 to prevent oxidization of the Co. (b) Schematic of the setup used for FMR measurements.  Figure 2. (a) FMR spectra of sample-A Co of 1, 2, 3, and 5 nm in thickness. (b) Comparison of the MR spectra of the Co (1 nm) and Co (1 nm)/Ta (3 nm) samples. (c) The Co thickness dependence of the resonance field µ0Hres (upper panel), the half-width at half-maximum of the FMR spectra DH (middle panel), and the Gilbert damping constant a (lower panel).  Figure 3. (a) FMR spectra from the Co (1 nm)/Ta (tTa nm) samples. The thickness of the Ta (tTa) was changed from 2 to 5 nm. (b) The Ta thickness dependence of the Gilbert damping constant a.  Figure 4. (a) Comparison of the FMR spectra of the Co (1 nm)/Ta (3 nm) and Co (1 nm)/\t𝕆 /Ta (3 nm) samples. (b) Atomic force microscopic views of the surfaces of  the SiO2 substrate and the oxidized Ta.    12  Figures    \n Fig. 1(a) Yoshii et al  \n Fig. 1(b) Yoshii et al   \n13    Fig. 2(a)(b)(c) Yoshii et al.    \n14    \n  Fig. 3(a)(b) Yoshii et al.   \n15   Fig 4(a). Yoshii et al.   \n Fig 4(b) Yoshii et al  \n"    },    {        "title": "2111.07773v3.Nonstanding_spin_waves_in_a_single_rectangular_permalloy_microstrip_under_uniform_magnetic_excitation.pdf",        "content": "Nonstanding spin waves in a single rectangular permalloy microstrip under uniform magnetic\nexcitation\nSanta Pile,1Sven Stienen,2Kilian Lenz,2Ryszard Narkowicz,2Sebastian Wintz,3Johannes F ¨orster,3\nSina Mayr,4, 5Martin Buchner,1Markus Weigand,6Verena Ney,1J¨urgen Lindner,2and Andreas Ney1\n1Institute of Semiconductor and Solid State Physics, Johannes Kepler University Linz, 4040 Linz, Austria\n2Helmholtz-Zentrum Dresden-Rossendorf, Institute of Ion Beam Physics and Materials Research, 01328 Dresden, Germany\n3Max Planck Institute for Intelligent Systems, 70569 Stuttgart, Germany\n4Paul Scherrer Institut, 5232 Villigen PSI, Switzerland\n5Laboratory for Mesoscopic Systems, Department of Materials, ETH Zurich, 8093 Zurich, Switzerland\n6Helmholtz-Zentrum Berlin f ¨ur Materialien und Energie, 12489 Berlin, Germany\n(Dated: February 16, 2022)\nFerromagnetic resonance modes in a single rectangular Ni 80Fe20microstrip were directly imaged using time-\nresolved scanning transmission x-ray microscopy combined with a phase-locked ferromagnetic resonance ex-\ncitation scheme and the ﬁndings were corroborated by micromagnetic simulations. Although under uniform\nexcitation in a single conﬁned microstructure typically standing spin waves are expected, all imaged spin\nwaves showed a nonstanding character both, at and off resonance, the latter being additionally detected with\nmicroantenna-based ferromagnetic resonance. The effect of the edge quality on the spin waves was observed in\nmicromagnetic simulations.\nI. INTRODUCTION\nOver the past decades charge-based computing devices\ndecreased drastically in size, coinciding with an increasing\nand even limiting heat dissipation, which triggered an active\nsearch for new ways of information processing. In that re-\ngard, magnons or spin waves are to be one of the options\nto replace the transfer of electronic charges in logic devices\n[1–4]. Therefore, the corresponding ﬁeld of magnon spin-\ntronics nowadays attracts increasing attention as a promis-\ning direction of research [5–8]. A wide range of geometrical\nmagnonic systems of various materials has been investigated\nat this point, including thin ﬁlms [9–11], multilayer nanostruc-\ntures [12], magnonic crystals [13–16], magnonic waveguides\n[17–19] and conﬁned microstructures [20–22].\nIn this work the focus is put on a fundamental understand-\ning of the dynamic magnetic properties of conﬁned structures,\nas this is a prerquisite for the development of nanoscale com-\nputational devices. It was observed experimentally that the\nspins near the edges of a conﬁned microstructure often be-\nhave as if they are “pinned” and, thus, are hindered to precess\n[19]. Reﬂections at the edges cause spin-wave resonances,\nwhenever the distance between the edges equals an integer\nnumber of half wavelengths, in other words, quantization of\nthe spin-wave modes occurs [7]. The spin-wave spectrum of\na saturated ellipsoidal magnetic element can be calculated an-\nalytically [23, 24]. However, in most cases the magnetic ele-\nments considered for applications have a nonellipsoidal shape\nand are not saturated. It has been shown that both the nonellip-\nsoidal shape and quality of the edges of these elements drasti-\ncally affect their dynamic magnetic properties [21, 22]. Often\nspin waves are investigated using a nonuniform excitation of\nthe structure [1, 3, 11, 13]. However, in micro- or nanoscaledevices a uniform or close to uniform excitation ﬁeld can be\neasier realized, e.g., placing an element in close proximity\nto an antenna. In a rectangular microstrip the pinning effect\ncan be exploited to excite spin waves with an odd number of\nhalf wavelengths within the length of the strip using uniform\nradio-frequency/microwave (MW) driving ﬁelds [7, 25]. The\nquantization conditions for the modes of a rectangular con-\nﬁned structure were described in detail in [20]. Due to the\nhigh inhomogeneity of the effective ﬁeld inside the structure\nalong the external static magnetic ﬁeld, the quantization con-\nditions for a k-vector aligned in the direction of the external\nﬁeld are complicated [26]. Therefore, analytical calculations\nof the spin-wave dispersion and consequent analysis are com-\nplex [27]. A different approach that can be used for the spin-\nwave analysis of such structures is an experimental study in\ncombination with micromagnetic simulations.\nExperimental investigations on a micron to sub-micron\nscale require sensitive experimental techniques. The possi-\nbility to excite and image spin waves in micron-sized struc-\ntures was shown to be possible using techniques such as Bril-\nlouin light scattering [21, 28], time-resolved Kerr-microscopy\n[29] or spatially resolved ferromagnetic resonance force mi-\ncroscopy [30]. On the other hand, time-resolved scanning\ntransmission x-ray microscopy (TR-STXM) [31–36] can be\napplied using a phase-locked ferromagnetic resonance (FMR)\nexcitation scheme (STXM-FMR) [22, 37–41]. This STXM-\nFMR technique enables direct, time-dependent imaging of the\nspatial distribution of the precessing magnetization across the\nsample during FMR excitation with elemental selectivity.\nThe development of planar microresonators/microantennas\nallows for measuring FMR of a single ferromagnetic mi-\ncrostrip including resonance lines corresponding to spin-\nwave excitations [39, 42–45]. The presence of spin waves\nin a single Ni 80Fe20(permalloy, Py) microstrip under uni-arXiv:2111.07773v3  [cond-mat.mes-hall]  15 Feb 20222\nform microwave excitation was evidenced earlier using\nmicroresonator-based FMR and supported by micromagnetic\nsimulations [44]. Those ﬁndings were corroborated by de-\ntailed investigations of comparable spin-wave excitations in a\nCo microstrip with similar dimensions [45]. The spin waves\nin a conﬁned microstructure, which are detectable with FMR,\nwere expected to have a standing character [46], meaning that\ntheir amplitude minima should have remained at the same po-\nsition in space over time. However, a nonstanding behavior of\nsuch spin waves was observed using STXM-FMR in a system\nof two rectangular Py microstrips arranged perpendicular to\neach other with a distance of 2 µm between them [22]. The\nreason for the nonstanding character of the spin waves in one\nof the strips, as was reported in [22], may be inhomogeneities\nproduced by the internal magnetization landscape of the strip\nand/or inhomogeneous external magnetic stray ﬁelds, emerg-\ning from the second strip. Both can lead to a superposition\nof spin-wave eigenmodes possibly resulting in the observed\nmotion of the amplitude minima. In this work it is shown that\nnonstanding spin waves can be excited and directly imaged\nusing STXM-FMR in a single Py microstrip. Furthermore,\nthe spin waves, that exhibit a nonstanding character, can be\ndetected with conventional microantenna FMR.\nII. EXPERIMENTAL DETAILS\nSingle rectangular Py strips with a nominal lateral di-\nmension of 5 \u00021 µm2and 30 nm thickness [see Fig. 1 (a)]\nwere fabricated on highly insulating Si substrates for the\nmicroantenna-based FMR measurements and on commer-\ncial 200-nm-thick 0 :25\u00020:25 mm2SiN-membranes for the\nSTXM-FMR measurements using standard e-beam lithogra-\nphy and magnetron sputtering with subsequent lift-off. Sput-\ntering was carried out at a process pressure of 4 \u000210\u00003mbar\nin an ultrahigh vacuum chamber with a base pressure of\n2\u000210\u00009mbar. Py with a nominal thickness of 30 nm was\nsputtered at room temperature using 10 standard cubic cen-\ntimeters per minute of Ar as process gas. To prevent oxida-\ntion the Py layer was covered with a 5 nm thick Al capping\nlayer grown using pulsed laser deposition. An example of the\nfabricated Py rectangular microstrip is shown in Fig. 1 (b) im-\naged by a scanning electron microscope. In a second step, a\nplanar microantenna [40] with a Au thickness of 600 nm was\nfabricated by photolithography, electron beam physical vapor\ndeposition and a lift-off. In order to improve an adhesion of\nthe Au layer, a 5 nm Ti layer was placed between the Au and\na substrate in the process. The two different designs of mi-\ncroantennas used in this study are shown in Fig. 2. Design (a)\nwas used for the microantenna-based FMR and (b) for STXM-\nFMR measurements.\nFor the microantenna-based FMR measurements a home-\nbuilt MW spectrometer with ﬁeld modulation at 78 kHz and\nlock-in technique was used [47, 48]. The spectrometer was\noperating in ﬁeld-sweep mode. The sample was placed in the\nmicroloop of a planar broadband microantenna with an inner\ndiameter of 20 µm as can be seen in the optical microscope\nFIG. 1. (a) Schematic representation of the rectangular magnetic el-\nement with indication of the directions of the coordinate axes, an\nexample of in-plane external magnetic ﬁeld ~Horientation and dimen-\nsions of the strip. (b) SEM image of the lithographically patterned\nPy rectangular microstrip.\nimage in Fig. 2 (a). The microloop allows for the concentra-\ntion of the MW ﬁeld at the sample area [39, 42–44], thus,\nenhancing the sensitivity of the system compared to conven-\ntional MW resonators. The MW ﬁeld component was ori-\nented perpendicular to the sample plane. The spectrometer\noperates in a Mach-Zender type interferometer scheme [49]\ncovering three frequency bands: 2–4 GHz, 4–8 GHz and 8–\n18 GHz. The FMR absorption and dispersion signals were\ndetected and rectiﬁed using homodyne mixing and fed into\nlock-in ampliﬁers working at the ﬁeld-modulation frequency.\nFor the measurements the external static magnetic ﬁeld ~Hwas\napplied in the plane of the microstrip.\nThe STXM-FMR measurements were carried out at the\nMAXYMUS endstation of the UE46 undulator beam-line at\nthe Helmholtz-Zentrum Berlin during the low-alpha opera-\ntion mode of the BESSY II synchrotron. In STXM the x-ray\nbeam is focused using a diffractive zone plate in combina-\ntion with an order sorting aperture to remove perturbing un-\ndiffracted light and diffraction orders >2. The images are cre-\nated by scanning the sample through the focused x-ray beam\nand detecting the respective x-ray transmission. For the mea-\nsurements presented here the sample was scanned in steps of\n50 nm. The STXM-FMR technique exploits the x-ray mag-\nnetic circular dichroism effect as magnetic contrast mecha-\nnism. The latter allows for probing the dynamic out-of-plane\nmagnetization component across the area of the microstrip\nwhen the circularly polarized x-rays are directed perpendic-\nular to the sample surface. The photon energy was tuned to\nthe Fe L 3-edge ( \u0018708 eV).\nWhile scanning the sample through the focused x-rays,\nthe magnetization dynamics were excited by applying a\nstatic magnetic ﬁeld (in the range of 32–260 mT) and a\nsmall MW ﬁeld ( \u00180:5 mT) in the same geometry as for the\nmicroantenna-based FMR measurements but using the mi-\ncroantenna design shown in Fig. 2 (b). The time resolution\nto probe the precessing magnetization at several intermedi-\nate points of its period can be obtained by a pump-and-probe\nmeasurement scheme, meaning that the MW frequency fMW\ncan be chosen depending on the frequency fsof the x-ray\nbunches impinging on the sample (the so-called “ring fre-\nquency”) [38]:\nfMW=fs\nN\u0001M; (1)3\nFIG. 2. Schematics of the planar microantenna designs with close-\nup optical images of the loops: (a) multi-frequency microantenna,\n(b) microantenna for transmission excitation.\nwhere Nis the number of time channels or, in other words,\nthe number of points over the excitation period at which the\nsample will be probed and Mis the number of excitation pe-\nriods over the observation period ( Ntime spacings between\nx-ray pulses). In order to be able to probe the excitation pe-\nriod at Ndifferent phases, MandNshould not have common\nfactors. The reported results were obtained at the excitation\nfrequency of 9.43 GHz ( fs=500 MHz, M=132, N=7). The\nmagnetic contrast was extracted by dividing the counted x-ray\nphoton signal at each time point by the time averaged value\n[22, 37, 38, 41].\nIII. RESULTS\nA. Micromagnetic Simulations\nAlong with the experiment, simulations of the FMR spec-\ntra and the spatial distribution of the dynamic magnetiza-\ntion of the samples were carried out using M UMAX3 [50].\nIn a ﬁrst step the simulations were run for a rectangular\nstrip [see Fig. 1 (a)] with lateral dimension of 5 \u00021 µm2and\na thickness of 30 nm. The cell size was chosen to be \u0018\n10\u000210\u0002(sample thickness )nm3, which was sufﬁcient for\nthe dipolar-dominated spin waves imaged in this work. A\nsaturation magnetization of Ms=700 kA/m [51], a Gilbert\ndamping parameter a=0:006, and an exchange stiffness con-\nstant of Aex=13\u000210\u000012J/m were used for the simulations.\nThe magnetocrystalline anisotropy was set to zero. The fre-\nquency of the uniform MW ﬁeld aligned along the z-axis [seeFig. 1 (a)] was set to fMW=9:43 GHz, which was used previ-\nously for the STXM-FMR measurements of similar samples\n[22, 40, 41], and its amplitude was set to 0.5 mT. The resulting\nsimulated FMR spectra for the in-plane ﬁeld being oriented\nparallel to the long edge of the strip (easy axis orientation,\ne.a. orientation) and parallel to the short edge of the strip (hard\naxis orientation, h.a. orientation) are shown in Figs. 3 (a) and\n(b), respectively. The simulated FMR spectrum was obtained\nby derivation and normalization of the spectra of the out-of-\nplane magnetization component integrated over the strip area.\nThe direction of the external magnetic ﬁeld in both orienta-\ntions is indicated in the insets of the ﬁgure. The magnitude of\nthe external magnetic ﬁeld was varied from 50 mT to 150 mT\nin 0.25 mT steps. At each ﬁeld the simulation was run for 50\nexcitation cycles in order to reach a steady magnetization pre-\ncession state. The magnetization snapshots for each ﬁeld were\nsaved at the end of the 50thexcitation cycle, where the excita-\ntion ﬁeld amplitude is 0 and, therefore, the resonance response\nshifted byp\n2with respect to the excitation ﬁeld is maximum.\nThe dashed vertical lines indicate some of the resonance ﬁeld\nvalues. Blue-white-red images below the spectra show the\ncalculated characteristic snapshots of the spatial distribution\nof the out-of-plane magnetization component mz(t)at reso-\nnance. The value of mz(t)changes across the area of the strip\ndue to the variation of the precession angle and the phase vari-\nation of the magnetization [25]. At the amplitude minima of\nthe wave the precession angle approaches zero, consequently,\nmz(t)\u00180, which is represented by the white color in the blue-\nwhite-red color scale provided in Fig. 3 (c).\nIn order to make an overview of the simulated mz(t)proﬁles\nalong the length and the width of the strip over the range of\nexternal ﬁeld values at the ﬁxed MW frequency, the averaged\nmiddle part of each spatial map was stacked into color plots\nas shown in Figs. 3 (c-f). In the ﬁgures the overviews of the\nspin-wave proﬁles along the strip (c,d) as well as across the\nstrip (e,f) are shown for e.a. orientation (c,e) and h.a. orienta-\ntion (d,f), respectively. The x-axis of the plots corresponds to\nthe external magnetic ﬁeld in mT. The y-axis shows either the\nposition along ( y-position) or across ( x-position) the strip in\nµm. The color and its intensity denote mz(t)of the spin-wave\nproﬁle on the same scale as in the blue-white-red images in\nFigs. 3 (a,b). The red and the blue rectangles placed over the\nPy strip schematic representation in Figs. 3 (c,d) indicate the\nregions of the Py strip that were used for averaging the data at\neach ﬁeld for the color plots along and across the strip, respec-\ntively. The spatial distribution variation of mz(t)in the direc-\ntions parallel and perpendicular to the external ﬁeld includes\nquantized k-vectors of the spin-wave eigenmodes excited in\nthose directions. Therefore, the color plots allow for visual\nevaluation of the spin-wave eigenmodes, parallel and perpen-\ndicular to the strip, forming the resulting spin-wave pattern\nat each ﬁeld. As a result, the overview visualization together\nwith the FMR spectra as plotted in Figs. 3 (a-f) allows for ana-\nlyzing the excitable spin waves in the strip and helps revealing\nthe relation between the FMR spectrum and the spin-wave ex-\ncitations along and across the strip as a function of the external\nﬁeld.4\nFIG. 3. (a,b) Simulated FMR spectra of the rectangular Py strip. (c-f) Simulated FMR spectra of the Py strip at 9.43 GHz. Overview of the\nspin-wave proﬁles (c,d) along and (e,f) across the strip in (a,c,e) easy and (b,d,f) h.a. orientations.\nIn Figs. 3 (a,b) for each orientation of the strip one can ob-\nserve the main FMR line with the largest intensity and sev-\neral smaller signals, some of which are magniﬁed in the in-\nsets of the plots. The spatial distributions of the out-of-plane\ndynamic magnetization component mz(t)shown in the plots\nconﬁrm that the main FMR signal in both orientations is the\nquasi-uniform excitation of the strip and smaller signals cor-\nrespond to spin waves [44, 45]. The quasi-uniform excitation\nis the one, at which almost all magnetic moments across the\nstrip area precess in phase. A nonuniform mz(t)spatial dis-\ntribution at the edges arises from the inhomogeneity of the\neffective ﬁeld within the strip [20, 52]. When comparing the\nspectra in Figs. 3 (a) and (b) it is visible that due to the shape\nanisotropy the main FMR signal in e.a. orientation appears at\nlower ﬁelds (81.7 mT) than in the h.a. orientation (121.7 mT).\nThe overview analysis of the simulated spin-wave proﬁles\nof a perfectly rectangular Py microstrip plotted in Figs. 3 (c-f)\nshows that a rich spectrum of spin waves with an odd num-\nber of amplitude maxima can be excited using a uniform MW\nﬁeld in both considered orientations of the strip, e.a. and h.a.\nThis observation is in agreement with previous research on the\ntopic [44, 45]. Moreover, resonance ﬁeld differences betweenneighboring odd spin-wave orders in the e.a. orientation [see\nFigs. 3 (a,c,e)] are much smaller than in the h.a. orientation\n[see Figs. 3 (b,d,f)]. For example, in e.a. orientation the res-\nonance ﬁeld difference between the waves with 3 and 5 am-\nplitude maxima along the strip [Fig. 3 (c)] is 4.8 mT, whereas\nin the h.a. orientation it is 12.6 mT [Fig. 3 (d)]. The differ-\nence in the spin-wave mode separation along the strip for the\ntwo different ﬁeld geometries can be attributed to the underly-\ning fundamental dispersion relations: Backward-volume (BV)\nmodes in e.a. orientation and Damon-Eshbach (DE) modes in\nh.a. orientation. For the e.a. orientation, the spin-wave modes\nare occurring at higher ﬁelds than the main FMR mode and\ntheir ﬁeld-separation is relatively small. This corresponds\nto the magnetostatic BV dispersion relation lying below the\nFMR frequency and exhibiting a relatively high slope ( k\nchanges strongly with frequency and, thus, ﬁeld) [7]. For the\nh.a. orientation, the spin-wave modes show up at ﬁelds below\nthe main FMR mode and their ﬁeld separation is relatively\nwide. This corresponds to the magnetostatic DE dispersion\nover frequencies above the FMR and showing a relatively high\nslope (k changes weakly with frequency and, thus, ﬁeld) [53].\nThe FMR simulations were run and further adjusted in or-5\nder to match the FMR measurements as will be discussed in\nsection ”FMR Measurements“. In order to match the STXM-\nFMR measurements time-resolved micromagnetic simula-\ntions were performed providing information on the mz(t)evo-\nlution over the time at each of the measured ﬁeld values as\nwill be discussed further in section ”STXM-FMR Imaging“.\nB. FMR Measurements\nThe results of the FMR measurements with the use of\nthe multifrequency microantenna [see Fig. 2 (a)] at fMW =\n9:4 GHz are plotted as black and red dashed lines in Fig. 4 for\nthe fabricated single Py microstrip in e.a. and h.a. orientations,\nrespectively. The nominal dimensions of the strip are the same\nas for the simulations of the rectangular strip. The FMR spec-\ntra in both orientations include the quasi-uniform and spin-\nwave FMR signals. Additionally, the ﬁeld gap between the\nmain FMR signal positions in e.a. and in h.a. orientations\nis 29.5 mT for the measurements, which is 10.5 mT smaller\nthan for the initial simulations shown in Figs. 3 (a,b). A pos-\nsible reason for these discrepancies could be a deviation of\nthe saturation magnetization from the value used in simula-\ntions, which can cause a shift in the resonance ﬁeld. Further-\nmore, the quality (roughness) of the edges of the fabricated\nstrip can cause not only the resonance ﬁeld to shift due to\nshape anisotropy, but also a change in the effective damping\nparameter. In addition, the lateral shape of the produced strip\nis not perfectly rectangular, but has slightly rounded corners\nand edges [see Fig. 1 (b)], which can change the relative res-\nonance ﬁeld positions of all modes. Finally, a deviation from\nthe initial thickness of the strip can be a reason for an altered\nresonance ﬁeld, as well.\nIn order to match the simulated with the measured FMR\nspectra, the simulations were adjusted taking into account\nmeasuring settings and possible effects from the discrepan-\ncies described above. As a starting point, the frequency was\nset to fMW=9:4 GHz that was used for the actual FMR mea-\nsurements. Lowering the frequency shifted all resonances to\nsmaller ﬁelds. Further, several simulation parameters were\nvaried and set to new values. As a result, the saturation\nmagnetization was set to Ms=730 kA/m, compared to Ms=\n700 kA/m in Figs. 3 (a,b), which shifted the resonances of both\norientations to lower ﬁelds and increased the gap between the\ne.a. and h.a. main resonances. The damping parameter was set\ntoa=0:013 for the e.a. and a=0:008 for the h.a. orientation,\ncompared to a=0:06 for both orientations in Fig. 3, which\nbroadened the linewidth of all resonances. The reason for the\nincreased damping parameter in e.a. orientation was an op-\ntimal match to the measured FMR linewidth. However, in\nthe measurements most probably an angle-dependent inhomo-\ngeneous broadening is observed rather than a change of the\nGilbert damping. The lateral shape of the strip was rounded\n(see the inset in Fig. 4) to be as close to the SEM image of the\nsample and its thickness was set to 25 nm, which decreased\nthe gap between the e.a. and h.a. resonances and slightly the\ngaps between the resonances within one orientation, respec-tively. The cell size was kept the same as for the rectangular\nstrip simulations. In Fig. 4 the simulated FMR spectra using\nadjusted parameters for the strip in e.a. and h.a. orientations\nare plotted as black and red solid lines, respectively. The ad-\njusted simulations show a good match to the measured spectra\nfor the main FMR line and spin-wave signals. The spin-wave\nsignals are much less visible in the simulated FMR spectrum\nusing adjusted parameters for the strip in e.a. orientation com-\npared to the initial simulations plotted in Fig. 3 (a). The reason\nfor that can be an increased damping in the sample as the in-\ncreased linewidth causes an overlap of the quasi-uniform and\nthe spin-wave excitations. Another reason can be the change\nin shape anisotropy due to the rounded edges of the strip in\nthe adjusted simulations, which decreases the resonance ﬁeld\ndifference between the spin waves.\nIn the FMR spectrum measured in h.a. orientation several\nspin-wave signals below and above the main line are recog-\nnizable (see red dashed line in Fig. 4). The signals below the\nmain FMR line ﬁt the simulations nicely, whereas the signals\nabove show some deviation. A possible reason for that is the\nquality of the edges, i.e. the presence of defects etc., which\nshifts the resonance ﬁeld of the localized modes to lower val-\nues [54]. However, edge quality is not taken into account in\nthe adjusted simulations apart from the adapted damping pa-\nrameter. The inﬂuence of the edge quality is more notice-\nable when the spin wave is localized closer to it. As can\nbe seen from Fig. 3 (f) spin waves observed above the quasi-\nuniform excitation are formed along the longer edges of the\nstrip. Thus, the discrepancy is stronger for these spin waves\nand as a result their FMR signals. The adjusted simulations\nconﬁrmed by the FMR measurements show that a variety of\nspin waves can be excited at 9.4 GHz in the single Py strip\nwith nonperfect rectangular shape. Spin waves can be ex-\ncited in both investigated orientations, e.a. and h.a. Due\nto the shape anisotropy and increased damping parameter in\ne.a. orientation spin waves are less pronounced compared to\nthe h.a. orientation.\nC. STXM-FMR Imaging\nThe measured STXM-FMR data consists of seven images\nrepresenting seven equidistant time-steps of the the mz(t)dis-\ntribution across the sample depicting dynamics over one exci-\ntation and, thus, magnetization precession cycle [22, 55]. By\nusing a temporal discrete Fourier transform at each point in\nthe scan the amplitude and the phase at the MW frequency was\nextracted from the measured dynamics [55, 56]. Via a simi-\nlar approach as for the simulations (see Fig. 3), the amplitude\nand the phase data extracted from the measurements at sev-\neral external ﬁeld values was combined into proﬁle overview\namplitude and phase sets (not shown here). In order to be\nable to compare those overview sets of the STXM-FMR re-\nsults with the simulated spin-wave proﬁles, another step of\ndata processing was performed: the amplitude and the phase\noverview data were combined into one plot by multiplying the\namplitude data with the sine of the phase data and normalizing6\nHe.a.\norientation\nh.a.\norientation HPy9.4 GHz\nFIG. 4. Measured and simulated FMR spectra of the slightly rounded\nrectangular Py microstrip in e.a. and h.a. orientations. In order to ﬁt\nthe measurements, different damping parameters were used for the\nsimulations in two different ﬁeld orientations indicated in the plot.\nthe result (to the [-1;1] range).\nThe resulting overview plots of the STXM-FMR data and\nthe simulations along (a,c) and across (b,d) the strip in\ne.a. (a,b) and in h.a. (c,d) orientations, respectively, are shown\nin Fig. 5. The vertical gray dashed lines indicate the correla-\ntion between the measurements and the simulations at reso-\nnances. For comparison the adjusted simulations (see Fig. 5)\nare used with the damping parameter a=0:008 for both ori-\nentations. This value of the damping parameter was used for\nthe h.a. orientation of the strip in adjusted simulations, as it\ngives a good match between the simulations and the STXM-\nFMR measurements in both orientations. The latter conﬁrms\nthis way the previous assumption that the line broadening ob-\nserved in the FMR measurements in e.a. orientation was due\nto angle-dependent inhomogeneous broadening rather than\na change of Gilbert damping. In Fig. 5 for better visual\nmatching in the ﬁgure, the simulations are plotted with the\nsame ﬁeld step size as for the measurements, i.e. 1 mT in\ne.a. orientation and 2 mT in h.a. orientation, respectively.\nIn Fig. 5 by comparing the main FMR signal positions (in\nboth orientations) between the simulations and the combined\nand normalized amplitude/phase data extracted from STXM-\nFMR measurements, a ﬁeld offset is observed. This ﬁeld off-\nset can be a result of the ﬁeld calibration error of the mea-\nsurement setup, and/or a possible saturation magnetization\ndifference between the samples fabricated for the FMR and\nSTXM-FMR measurements. In e.a. orientation the offset is\napproximately 5 mT and in the h.a. orientation it is 9 mT. The\nreason for the different offset in the two orientations is that\nthe change of sample orientation involves remounting and re-\nconnecting the sample with the microantenna. Another evi-\ndence of the offset not being a result of the sample shape dif-\nference between the measurements and the simulations is the\nrelative position of the resonances in each orientation. As can\nbe seen from the initial (see Fig. 3) versus adjusted (see Fig. 4)simulations, the change in the sample shape also changes the\nrelative positions of the resonance lines within one orienta-\ntion. In Fig. 5 the ﬁeld gaps between the resonances mea-\nsured with STXM-FMR are the same as in the simulations,\nmeaning that the shape and the thickness were simulated very\nclosely to the real sample. Hereafter, spin-wave proﬁles cor-\nresponding to the FMR lines will be referred to using the ﬁeld\nvalues from the simulations. The overall comparison of the\nspin-wave overviews in the range of ﬁelds in both orientations\nof the strip along and across the strip in Fig. 5 reveals a good\nagreement between the simulations and the STXM-FMR mea-\nsurements. In the h.a. orientation clear spin-wave patterns can\nbe observed along and across the strip at different static exter-\nnal ﬁeld values [see Figs. 5 (c,d)]. Additionally, the spin-wave\nproﬁle overview plot of the h.a. orientation shows a clear sep-\naration between the spin waves [Fig. 5 (c)] in contrast to the\ne.a. orientation [Fig. 5 (a)].\nTime-dependent images (snapshots) from simulations and\nSTXM-FMR scans of the spatial distribution of the out-of-\nplane dynamic magnetization component mz(t)within the\nentire strip in the e.a. and the h.a. orientations are shown in\nFigs. 6 (a) and (b), respectively. The quasi-uniform excita-\ntion and the three ﬁeld-adjacent spin waves formed along the\nlonger edge of the strip, which correspond to measured FMR\nlines above the main resonance in e.a. orientation and below\nthe main resonance in h.a. orientation, are shown in the ﬁg-\nure. The time-series of images are stacked in rows. The\nSTXM-FMR scans in a row depict 7 measured snapshots of\none magnetization precession period, while the simulations\nrepresent 14 points of the same period. The simulated images\nﬁt well the measured spin-wave conﬁguration and dynamics\nover the entire precession period. Both, measurements and\nsimulations, unexpectedly reveal a nonstanding character of\nthe spin waves at resonance in a single conﬁned microstrip.\nThat can be concluded from tracking the position of the am-\nplitude minima of the waves, which in the images are the\nwhite borders between the red and blue regions. For example,\nin the e.a. orientation at 87.8 mT [see Fig. 6 (a)] one can see\nthat the amplitude minima’s positions move from the center\nof the strip to the shorter edges during half of a precession pe-\nriod. The same is true for the h.a. orientation at 103.6 mT [see\nFig. 6 (b)], but in the opposite direction, the amplitude min-\nima change their position from the shorter edges of the strip\nto the center within half a period. Similar spin-wave behavior\nwas observed in the simulations of the perfectly rectangular\nstrip in both orientations at the ﬁelds corresponding to spin\nwaves similar to the shown ones in Fig. 6. The observed non-\nstanding character of the spin waves shown in Fig. 6 is most\nprobably a result of the inhomogeneity of the internal ﬁeld as\nwas shown in [20], i.e. its gradual decrease closer to the edges\nof the strip along the direction of an external static magnetic\nﬁeld. The same nonstanding character is observed at other\nresonance ﬁelds shown in Fig. 6, at 91.8 mT and 94.8 mT in\ne.a. orientation and 92.3 mT and 80.8 mT in h.a. orientation,\nrespectively. In general the STXM-FMR measurements show\na good agreement with the adjusted simulations both, in the\nsnapshot spin-wave analysis using overview plots and in the\ntime-dependent spin-wave behavior.7\nFIG. 5. Overview of the simulated spin-wave proﬁles and the combined and normalized amplitude/phase data extracted from the STXM-FMR\nmeasurements: (a,c) along and (b,d) across the Py strip in (a,b) e.a. and (c,d) h.a. orientations.\nIV . DISCUSSION AND CONCLUSION\nIt was shown before that in a rectangular microstrip exci-\ntation of a variety of resonances, from quasi-uniform exci-\ntation to spin waves with a uniform periodic ﬁeld, is possi-\nble [22, 44, 45]. Moreover, when an external ﬁeld is applied\nin the plane, a strong demagnetizing ﬁeld gradient causes a\nmagnetization inhomogeneity throughout the strip area and\na rotation of the magnetic orientation closer to the edges of\nthe strip, which are perpendicular to the external ﬁeld [20].\nConsequently, the effective ﬁeld in a rectangular microstrip ishighly inhomogeneous [57]. As a result, the spin-wave disper-\nsion also changes from the center of the strip towards its edges\n[58]. Previously reported measurements using time-resolved\nKerr microscopy had a limited spatial resolution, insufﬁcient\nto observe the time evolution of more localized spin waves\n[29]. In our study STXM-FMR measurements in combina-\ntion with micromagnetic simulations allow us to investigate\nthe time-evolution of spin waves over a static magnetic ﬁeld\nrange within a single thin rectangular Py microstrip. The ex-\nperimental results are in good agreement with micromagnetic\nsimulations. Somewhat unexpectedly, the results reveal a non-8\nFIG. 6. Simulated and STXM-FMR-measured time evolution of mz(t)over one period for the entire strip in the (a) e.a. and (b) h.a. orientations\nat four different resonance ﬁeld values for each orientation.\nstanding character of the excited spin waves in the single con-\nﬁned microstrip at and off resonance, the latter being detected\nwith microantenna-based FMR as well. A non-standing char-\nacter was originally not expected, because only a uniform ex-\ncitation ﬁeld is applied to the specimen during the measure-\nments, suggesting standing spin waves [46], and also no ad-\nditional magnetic microstructures are located in close vicinity\nof the strip, as opposed to the sample system in [22].\nA reason for the non-standing spin-wave character could\nbe the effective ﬁeld gradients in regions closer to the edgesof the strip, which act as centers of inhomogeneous excita-\ntion of the waves. When analyzing the spin-wave excitations\nwithin the conﬁned microstructure, it should be taken into ac-\ncount that the eigenmodes’ dispersion and, thus, the resulting\nspin waves depend strongly on the internal ﬁeld distribution\n[9, 20, 26]. The internal ﬁeld within the microstrip is highly\ninhomogeneous due to its conﬁned size, especially in the di-\nrection parallel to the external static magnetic ﬁeld. There-\nfore, the spin-wave dispersion varies along this direction, i.e.,\nspin-wave eigenmodes change their wavelength/phase during\npropagation. Another reason could be related to the propaga-9\ntion length of the spin waves, which can be smaller than the\nconﬁnement size of the strip, meaning that the spin waves do\nnot/only insufﬁciently reach the opposite edge of the strip and,\nthus, do not/only insufﬁciently get reﬂected to form a standing\nwave.\nMoreover, the inﬂuence of the internal ﬁeld distribution on\nthe spin-wave behavior was observed, when the lateral shape\nof the strip was changed from a perfect rectangle to a rectangle\nwith rounded corners and edges, and, additionally, when the\nthickness of the strip was decreased in the adjusted micromag-\nnetic simulations, exploiting that the internal ﬁeld distribution\ndepends on the overall shape of the strip. That demonstrates\nthat by changing the overall shape of the strip it is possible to\nshift the mutual arrangement of the spin waves in ﬁeld at the\nsame MW frequency. The latter allows to modify, for exam-\nple, the spin conﬁguration of some particular spin waves. Our\nﬁndings provide important insights into the spin-wave dynam-ics in rectangular conﬁned microstructures and their evolution\nduring the magnetization precession period.\nACKNOWLEDGMENTS\nWe thank the HZB for the allocation of synchrotron radi-\nation beamtime. We also thank T. Feggeler, H. Stoll, and J.\nGr¨afe for their help during the STXM-FMR measurements\nand A. Halilovic for her valuable contribution to the lithog-\nraphy process. Additionally, we would like to thank M. Bech-\ntel for technical support at the beamline. The authors would\nlike to acknowledge ﬁnancial support by the Austrian Science\nFoundation (FWF) via project No I-3050. S. Mayr would\nlike to acknowledge funding from the Swiss National Science\nFoundation under Grant No. 172517.\n[1] P. Clausen, K. V ogt, H. Schultheiss, S. Sch ¨afer, B. Obry,\nG. Wolf, P. Pirro, B. Leven, and B. Hillebrands, Mode conver-\nsion by symmetry breaking of propagating spin waves, Appl.\nPhys. Lett. 99, 162505 (2011).\n[2] D. Grundler, Reconﬁgurable magnonics heats up, Nat. Phys. 11,\n438–441 (2015).\n[3] M. V ogel, R. Aßmann, P. Pirro, A. V . Chumak, B. Hillebrands,\nand G. Freymann, Control of Spin-Wave propagation using\nmagnetisation gradients, Sci. Rep. 8, 11099 (2018).\n[4] A. S. Savchenko and V . N. Krivoruchko, Electric-ﬁeld con-\ntrol of nonreciprocity of spin wave excitation in ferromagnetic\nnanostripes, Journal of Magnetism and Magnetic Materials 474,\n9 (2019).\n[5] A. V . Chumak, V . I. Vasyuchka, A. A. Serga, and B. Hille-\nbrands, magnon spintronics, Nat. Phys. 11, 453 (2015).\n[6] Q. Wang, P. Pirro, R. Verba, A. Slavin, B. Hillebrands, and\nA. Chumak, Reconﬁgurable nanoscale spin-wave directional\ncoupler, Sci. Adv. 4, 1 (2018).\n[7] T. Br ¨acher, P. Pirro, and B. Hillebrands, Parallel pumping\nfor magnon spintronics: Ampliﬁcation and manipulation of\nmagnon spin currents on the micron-scale, Phys. Rep. 699,\n1–34 (2017).\n[8] V . Vlaminck and M. Bailleul, Current-Induced Spin-Wave\nDoppler Shift, Science 322, 410 (2008).\n[9] B. A. Kalinikos and A. N. Slavin, Theory of dipole-exchange\nspin wave spectrum for ferromagnetic ﬁlms with mixed ex-\nchange boundary conditions, J. Phys. C: Sol. Stat. Phys. 19,\n7013 (1986).\n[10] H. Yu, R. Huber, T. Schwarze, F. Brand, T. Rapp, P. Berberich,\nG. Duerr, and D. Grundler, High propagating velocity of spin\nwaves and temperature dependent damping in a CoFeB thin\nﬁlm, Appl. Phys. Lett. 100, 262412 (2012).\n[11] O. Gladii, M. Haidar, Y . Henry, M. Kostylev, and M. Bailleul,\nFrequency nonreciprocity of surface spin wave in permalloy\nthin ﬁlms, Phys. Rev. B 93, 054430 (2016).\n[12] M. Madami, S. Bonetti, G. Consolo, S. Tacchi, G. Carlotti,\nG. Gubbiotti, F. B. Mancoff, and J. ¨A. M. A. Yar, Direct ob-\nservation of a propagating spin wave induced by spin-transfer\ntorque, Nat. Nano. 6, 635–638 (2011).\n[13] A. V . Chumak, P. Pirro, A. A. Serga, M. P. Kostylev, R. L.Stamps, H. Schultheiss, K. V ogt, S. J. Hermsdoerfer, B. Laegel,\nP. A. Beck, and B. Hillebrands, Spin-wave propagation in a mi-\ncrostructured magnonic crystal, Appl. Phys. Lett. 95, 262508\n(2009).\n[14] S. Neusser, G. Duerr, S. Tacchi, M. Madami, M. L.\nSokolovskyy, G. Gubbiotti, M. Krawczyk, and D. Grundler,\nMagnonic minibands in antidot lattices with large spin-wave\npropagation velocities, Phys. Rev. B 84, 094454 (2011).\n[15] F. Lisiecki, J. Rychły, P. Ku ´swik, H. Głowi ´nski, J. W. Kłos,\nF. Groß, N. Tr ¨ager, I. Bykova, M. Weigand, M. Zelent, E. J.\nGoering, G. Sch ¨utz, M. Krawczyk, F. Stobiecki, J. Dubowik,\nand J. Gr ¨afe, Magnons in a Quasicrystal: Propagation, Extinc-\ntion, and Localization of Spin Waves in Fibonacci Structures,\nPhys. Rev. Appl. 11, 054061 (2019).\n[16] M. Langer, R. A. Gallardo, T. Schneider, S. Stienen, A. Rold ´an-\nMolina, Y . Yuan, K. Lenz, J. Lindner, P. Landeros, and J. Fass-\nbender, Spin-wave modes in transition from a thin ﬁlm to a full\nmagnonic crystal, Phys. Rev. B 99, 024426 (2019).\n[17] V . E. Demidov, M. P. Kostylev, K. Rott, J. Munchenberger,\nG. Reiss, and S. O. Demokritov, Excitation of short-wavelength\nspin waves in magnonic waveguides, Appl. Phys. Lett. 99,\n082507 (2011).\n[18] P. Klein, R. Varga, and M. V ´azquez, Enhancing the velocity of\nthe single domain wall by current annealing in nanocrystalline\nFeCoMoB microwires, J. Phys. D: Appl. Phys. 47, 255001\n(2014).\n[19] Z. Duan, I. N. Krivorotov, R. E. Arias, N. Reckers, S. Stienen,\nand J. Lindner, Spin wave eigenmodes in transversely magne-\ntized thin ﬁlm ferromagnetic wires, Phys. Rev. B 92, 104424\n(2015).\n[20] C. Bayer, J. Jorzick, B. Hillebrands, S. O. Demokritov,\nR. Kouba, R. Bozinoski, A. N. Slavin, K. Y . Guslienko, D. V .\nBerkov, N. L. Gorn, and M. P. Kostylev, Spin-wave excitations\nin ﬁnite rectangular elements of Ni 80Fe20, Phys. Rev. B 72,\n064427 (2005).\n[21] V . E. Demidov, U. H. Hansen, and S. O. Demokritov, Spin-\nWave Eigenmodes of a Saturated Magnetic Square at Different\nPrecession Angles, Phys. Rev. Let. 98, 157203 (2007).\n[22] S. Pile, T. Feggeler, T. Schaffers, R. Meckenstock, M. Buch-\nner, D. Spoddig, B. Zingsem, V . Ney, M. Farle, H. Wende,10\nH. Ohldag, A. Ney, and K. Ollefs, Non-standing spin-waves in\nconﬁned micrometer-sized ferromagnetic structures under uni-\nform excitation, Appl. Phys. Lett. 116, 072401 (2020).\n[23] C. Kittel, On the Theory of Ferromagnetic Resonance Absorp-\ntion, Phys. Rev. 73, 155 (1948).\n[24] L. Walker, Magnetostatic Modes in Ferromagnetic Resonance,\nPhys. Rev. 105, 309 (1957).\n[25] D. D. Stancil and A. Prabhakar, Spin Waves (Springer, Boston,\nMA, 2009).\n[26] K. Y . Guslienko, R. W. Chantrell, , and A. N. Slavin, Dipolar\nlocalization of quantized spin-wave modes in thin rectangular\nmagnetic elements, Phys. Rev. B 68, 024422 (2003).\n[27] K. Y . Guslienko and A. N. Slavin, Magnetostatic Green’s func-\ntions for the description of spin waves in ﬁnite rectangular mag-\nnetic dots and stripes, JMMM 323, 2418 (2011).\n[28] T. Sebastian, K. Schultheiss, B. Obry, B. Hillebrands, and\nH. Schultheiss, Micro-focused Brillouin light scattering: imag-\ning spin waves at the nanoscale, Front. Phys. 3, 35 (2015).\n[29] J. St ¨ohr and H. C. Siegmann, Magnetism. From Fundamen-\ntals to Nanoscale Dynamics (Springer-Verlag Berlin Heidel-\nberg, 2006).\n[30] T. Mewes, J. Kim, D. V . Pelekhov, G. N. Kakazei, P. E. Wigen,\nS. Batra, , and P. C. Hammel, Ferromagnetic resonance force\nmicroscopy studies of arrays of micron size permalloy dots,\nPhys. Rev. B 74, 144424 (2006).\n[31] H. Stoll, M. Noske, M. Weigand, K. Richter, B. Kr ¨uger, R. M.\nReeve, M. H ¨anze, C. F. Adolff, F.-U. Stein, G. Meier, M. Kl ¨aui,\nand G. Sch ¨utz, Imaging spin dynamics on the nanoscale using\nX-Ray microscopy, Front. Phys. 3, 26 (2015).\n[32] J. Gr ¨afe, M. W. M.D., B. V . Waeyenberge, A. Gangwar, F. Groß,\nF. Lisiecki, J. Rychly, H. Stoll, N. Tr ¨ager, J. F ¨orster, F. Sto-\nbiecki, J. Dubowik, J. Klos, M. Krawczyk, C. H. Back, E. J.\nGoering, and G. Sch ¨utz, Visualizing nanoscale spin waves using\nmaxymus, Proc. SPIE 11090 Spintronics XII , 1109025 (2019).\n[33] C. Cheng and W. E. Bailey, Sub-micron mapping of GHz\nmagnetic susceptibility using scanning transmission x-ray mi-\ncroscopy, Appl. Phys. Lett. 101, 182407 (2012).\n[34] B. V . Waeyenberge, A. Puzic, H. Stoll, K. W. Chou,\nT. Tyliszczak, R. Hertel, M. F ¨ahnle, H. Br ¨uckl, K. Rott,\nG. Reiss, I. Neudecker, D. Weiss, C. H. Back, and G. Sch ¨utz,\nMagnetic vortex core reversal by excitation with short bursts of\nan alternating ﬁeld, Nature 444, 461 (2006).\n[35] Y . Acremann, J. P. Strachan, V . Chembrolu, S. D. Andrews,\nT. Tyliszczak, J. A. Katine, M. J. Carey, B. M. Clemens, H. C.\nSiegmann, and J. St ¨ohr, Time-Resolved Imaging of Spin Trans-\nfer Switching: Beyond the Macrospin Concept, Phys. Rev. Lett.\n96, 217202 (2006).\n[36] A. Puzic, T. Korhonen, B. Kalantari, J. Raabe, C. Quitmann,\nP. J¨ullig, L. Bommer, D. Goll, G. Sch ¨utz, S. Wintz, T. Stra-\nche, M. K ¨orner, D. Mark ´o, C. Bunce, and J. Fassbender, Pho-\nton Counting System for Time-resolved Experiments in Multi-\nbunch Mode, Synchrotron Radiation News 23:2, 226 (2010).\n[37] S. Bonetti, R. Kukreja, Z. Chen, D. Spoddig, K. Ollefs,\nC. Sch ¨oppner, R. Meckenstock, A. Ney, J. Pinto, R. Houanche,\nJ. Frisch, J. St ¨ohr, H. D ¨urr, and H. Ohldag, Microwave soft x-\nray microscopy for nanoscale magnetization dynamics in the\n5–10 GHz frequency range, Rev. Sci. Instrum. 86, 093703\n(2015).\n[38] M. Weigand, Realization of a new Magnetic Scanning X-\nray Microscope and Investigation of Landau Structures under\nPulsed Field Excitation (Cuvillier Verlag, G ¨ottingen, 2015).\n[39] T. Schaffers, R. Meckenstock, D. Spoddig, T. Feggeler,\nK. Ollefs, C. Sch ¨oppner, S. Bonetti, H. Ohldag, M. Farle, and\nA. Ney, The combination of micro-resonators with spatially re-solved ferromagnetic resonance, Rev. Sci. Instrum. 88, 093703\n(2017).\n[40] T. Schaffers, T. Feggeler, S. Pile, R. Meckenstock, M. Buchner,\nD. Spoddig, V . Ney, M. Farle, H. Wende, S. Wintz, M. Weigand,\nH. Ohldag, K. Ollefs, and A. Ney, Extracting the Dynamic\nMagnetic Contrast in Time-Resolved X-Ray Transmission Mi-\ncroscopy, Nanomaterials 9(7), 940 (2019).\n[41] S. Pile, M. Buchner, V . Ney, T. Schaffers, J. Lumetzberger,\nK. Lenz, R. Narkowicz, J. Lindner, H. Ohldag, and A. Ney,\nDirect Imaging of the ac Component of Pumped Spin Polariza-\ntion with Element Speciﬁcity, Phys. Rev. Applied 14, 034005\n(2020).\n[42] R. Narkowicz, D. Suter, and R. Stonies, Planar microresonators\nfor EPR experiments, J. Magn. Reson 175, 275–284 (2005).\n[43] R. Narkowicz, D. Suter, and I. Niemeyer, Scaling of sensitivity\nand efﬁciency in planar microresonators for electron spin reso-\nnance, Rev. Sci. Instrum. 79, 084702 (2008).\n[44] A. Banholzer, R. Narkowicz, C. Hassel, R. Meckenstock,\nS. Stienen, O. Posth, D. Suter, M. Farle, and J. Lindner, Visual-\nization of spin dynamics in single nanosized magnetic elements,\nNanotechnology 22, 295713 (2011).\n[45] C. Sch ¨oppner, K. Wagner, S. Stienen, R. Meckenstock,\nM. Farle, R. Narkowicz, D. Suter, and J. Lindner, Angular de-\npendent ferromagnetic resonance analysis in a single micron\nsized cobalt stripe, J. Appl. Phys. 116, 033913 (2014).\n[46] C. Kittel, Excitation of Spin Waves in a Ferromagnet by a Uni-\nform rf Field, Phys. Rev. 110, 6 (1958).\n[47] J. Lumetzberger, M. Buchner, S. Pile, V . Ney, W. Gader-\nbauer, N. Daff ´e, M. V . Moro, D. Primetzhofer, K. Lenz, and\nA. Ney, Inﬂuence of structure and cation distribution on mag-\nnetic anisotropy and damping in Zn/Al doped nickel ferrites,\nPhys. Rev. B 102, 054402 (2020).\n[48] H. Cansever, R. Narkowicz, K. Lenz, C. Fowley, L. Rama-\nsubramanian, O. Yildirim, A. Niesen, T. Huebner, G. Reiss,\nJ. Lindner, J. Fassbender, and A. M. Deac, Investigating spin-\ntransfer torques induced by thermal gradients in magnetic tun-\nnel junctions by using micro-cavity ferromagnetic resonance,\nAppl. Phys. 51, 224009 (2018).\n[49] H. Cansever and J. Lindner, Microresonators and Microan-\ntennas—Tools to Explore Magnetization Dynamics in Single\nNanostructures, Magnetochemistry 7, 28 (2021).\n[50] A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F. Garcia-\nSanchez, and B. V . Waeyenberge, The design and veriﬁcation of\nMuMax3, AIP ADV ANCES 4, 107133 (2014).\n[51] M. Buchner, J. Lumetzberger, V . Ney, T. Schaffers, N. Daff ´e,\nand A. Ney, Spin pumping from permalloy into uncompensated\nantiferromagnetic Co doped zinc oxide, J. Appl. Phys. 127,\n043901 (2020).\n[52] C. S. Davies, V . D. Poimanov, and V . V . Kruglyak, Mapping\nthe magnonic landscape in patterned magnetic structures, Phys.\nRev. B 96, 094430 (2017).\n[53] R. W. Damon and J. R. Eshbach, Magnetostatic Modes of a\nFerromagnetic Slab, J. Appl. Phys. 31, S104 (1960).\n[54] R. D. McMichael and B. B. Maranville, Edge saturation ﬁelds\nand dynamic edge modes in ideal and nonideal magnetic ﬁlm\nedges, Phys. Rev. B 74, 024424 (2006).\n[55] F. Groß, N. Tr ¨ager, J. F ¨orster, M. Weigand, G. Sch ¨utz, and\nJ. Gr ¨afe, Nanoscale detection of spin wave deﬂection angles\nin permalloy, Appl. Phys. Lett. 114, 012406 (2019).\n[56] B. Zingsem, T. Feggeler, R. Meckenstock, T. Schaffers, S. Pile,\nH. Ohldag, M. Farle, H. Wende, A. Ney, and K. Ollefs, Evalu-\nation protocol for revealing magnonic contrast in STXM-FMR\nmeasurements (2019), arXiv:1901.10595 [cond-mat.str-el].\n[57] A. Anders, J. Pelletier, D. M. Goebel, B. P. Wood, I. G. Brown,11\nW. Ensinger, and M. Tuszewski, Plasma Sources, in Handbook\nof Plasma Immersion Ion Implantation and Deposition , edited\nby A. Anders (John Wiley and Sons, London, 2000) Chap. 7,\npp. 381–465.[58] C. Bayer, J. P. Park, H. Wang, M. Yan, C. E. Campbell, and\nP. A. Crowell, Spin waves in an inhomogeneously magnetized\nstripe, Phys. Rev. B 69, 134401 (2004)."    },    {        "title": "1407.0635v1.Spin_Waves_in_Ferromagnetic_Insulators_Coupled_via_a_Normal_Metal.pdf",        "content": "Spin Waves in Ferromagnetic Insulators Coupled via a Normal Metal\nHans Skarsv\u0017 ag,\u0003Andr\u0013 e Kapelrud, and Arne Brataas\nDepartment of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway\n(Dated: May 27, 2022)\nHerein, we study the spin-wave dispersion and dissipation in a ferromagnetic insulator{normal\nmetal{ferromagnetic insulator system. Long-range dynamic coupling because of spin pumping and\nspin transfer lead to collective magnetic excitations in the two thin-\flm ferromagnets. In addition,\nthe dynamic dipolar \feld contributes to the interlayer coupling. By solving the Landau-Lifshitz-\nGilbert-Slonczewski equation for macrospin excitations and the exchange-dipole volume as well as\nsurface spin waves, we compute the e\u000bect of the dynamic coupling on the resonance frequencies and\nlinewidths of the various modes. The long-wavelength modes may couple acoustically or optically.\nIn the absence of spin-memory loss in the normal metal, the spin-pumping-induced Gilbert damp-\ning enhancement of the acoustic mode vanishes, whereas the optical mode acquires a signi\fcant\nGilbert damping enhancement, comparable to that of a system attached to a perfect spin sink. The\ndynamic coupling is reduced for short-wavelength spin waves, and there is no synchronization. For\nintermediate wavelengths, the coupling can be increased by the dipolar \feld such that the modes\nin the two ferromagnetic insulators can couple despite possible small frequency asymmetries. The\nsurface waves induced by an easy-axis surface anisotropy exhibit much greater Gilbert damping\nenhancement. These modes also may acoustically or optically couple, but they are una\u000bected by\nthickness asymmetries.\nPACS numbers: 76.50.+g,75.30.Ds,75.70.-i,75.76.+j\nI. INTRODUCTION\nThe dynamic magnetic properties of thin-\flm fer-\nromagnets have been extensively studied for several\ndecades.1,2Thin-\flm ferromagnets exhibit a rich vari-\nety of spin-wave modes because of the intricate inter-\nplay among the exchange and dipole interactions and the\nmaterial anisotropies. In ferromagnetic insulators (FIs),\nthese modes are especially visible; the absence of disturb-\ning electric currents leads to a clear separation of the\nmagnetic behavior. Furthermore, the dissipation rates\nin insulators are orders of magnitude lower than those\nin their metallic counterparts; these low dissipation rates\nenable superior control of travelling spin waves and facil-\nitate the design of magnonic devices.3\nIn spintronics, there has long been considerable in-\nterest in giant magnetoresistance, spin-transfer torques,\nand spin pumping in hybrid systems of normal met-\nals and metallic ferromagnets (MFs).4{7The experimen-\ntal demonstration that spin transfer and spin pumping\nare also active in normal metals in contact with insu-\nlating ferromagnets has generated a renewed interest in\nand refocused attention on insulating ferromagnets, of\nwhich yttrium iron garnet (YIG) continues to be the\nprime example.8{19In ferromagnetic insulators, current-\ninduced spin-transfer torques from a neighboring normal\nmetal (NM) that exhibits out-of-equilibrium spin accu-\nmulation may manipulate the magnetization of the insu-\nlator and excite spin waves.8,20,21The out-of-equilibrium\nspin accumulation of the normal metal may be induced\nvia the spin Hall e\u000bect or by currents passing through\nother adjacent conducting ferromagnets. Conversely, ex-\ncited spin waves pump spins into adjacent NMs, and this\nspin current may be measured in terms of the inverse spinHall voltages or by other conducting ferromagnets.8{14\nThe magnetic state may also be measured via the spin\nHall magnetoresistance.16{19,23,24Because of these devel-\nopments, magnetic information in ferromagnetic insula-\ntors may be electrically injected, manipulated, and de-\ntected. Importantly, an FI-based spintronic device may\ne\u000eciently transport electric information carried by spin\nwaves over long distances15without any excessive heat-\ning. The spin-wave decay length can be as long as cen-\ntimeters in YIG \flms.22These properties make FI{NM\nsystems ideal devices for the exploration of novel spin-\ntronic phenomena and possibly also important for future\nspintronic applications. Magnonic devices also o\u000ber ad-\nvantages such as rapid spin-wave propagation, frequen-\ncies ranging from GHz to THz, and the feasibility of cre-\nating spin-wave logic devices and magnonic crystals with\ntailored spin-wave dispersions.25\nTo utilize the desirable properties of FI{NM systems,\nsuch as the exceptionally low magnetization-damping\nrate of FIs, it is necessary to understand how the mag-\nnetization dynamics couple to spin transport in adjacent\nnormal metals. The e\u000bective damping of the uniform\nmagnetic mode of a thin-\flm FI is known to signi\f-\ncantly increase when the FI is placed in contact with\nan NM. This damping enhancement is caused by the loss\nof angular momentum through spin pumping.26{30Re-\ncent theoretical work has also predicted the manner in\nwhich the Gilbert damping for other spin-wave modes\nshould become renormalized.31For long-wavelength spin\nwaves, the Gilbert damping enhancement is twice as\nlarge for transverse volume waves as for the macrospin\nmode, and for surface modes, the enhancement can be ten\ntimes stronger or more. Spin pumping has been demon-\nstrated, both experimentally9and theoretically,31to be\nsuppressed for short-wavelength exchange spin waves.arXiv:1407.0635v1  [cond-mat.mes-hall]  2 Jul 20142\nA natural next step is to investigate the magnetization\ndynamics of more complicated FI{NM heterostructures.\nIn ferromagnetic metals, it is known that spin pumping\nand spin-transfer torques generate a long-range dynamic\ninteraction between magnetic \flms separated by normal\nmetal layers.32The e\u000bect of this long-range dynamic in-\nteraction on homogeneous macrospin excitations can be\nmeasured by ferromagnetic resonance. The combined ef-\nfects of spin pumping and spin-transfer torque lead to\nan appreciable increase in the resonant linewidth when\nthe resonance \felds of the two \flms are far apart and\nto a dramatic narrowing of the linewidth when the reso-\nnant \felds approach each other.32This behavior occurs\nbecause the excitations in the two \flms couple acous-\ntically (in phase) or optically (out of phase). We will\ndemonstrate that similar, though richer because of the\ncomplex magnetic modes, phenomena exist in magnetic\ninsulators.\nIn the present paper, we investigate the magnetization\ndynamics in a thin-\flm stack consisting of two FIs that\nare in contact via an NM. The macrospin dynamics in\na similar system with metallic ferromagnets have been\nstudied both theoretically and experimentally.32We ex-\npand on that work by focusing on inhomogeneous mag-\nnetization excitations in FIs.\nFor long-wavelength spin waves travelling in-plane in\na ferromagnetic thin \flm, the frequency as a function\nof the in-plane wave number Qstrongly depends on the\ndirection of the external magnetic \feld with respect to\nthe propagation direction. If the external \feld is in-\nplane and the spin waves are travelling parallel to this\ndirection, the waves have a negative group velocity. Be-\ncause the magnetization precession amplitudes are usu-\nally evenly distributed across the \flm in this geometry,\nthese modes are known as backward volume magneto-\nstatic spin waves (BVMSW). Similarly, spin waves that\ncorrespond to out-of-plane external \felds are known as\nforward volume magnetostatic spin waves (FVMSW),\ni.e., the group velocity is positive, and the precession\namplitudes are evenly distributed across the \flm. When\nthe external \feld is in-plane and perpendicular to the\npropagation direction, the precession amplitudes of the\nspin waves become inhomogeneous across the \flm, ex-\nperiencing localization to one of the interfaces. These\nspin waves are thus known as magnetostatic surface spin\nwaves (MSSW).33,34\nWhen two ferromagnetic \flms are coupled via a normal\nmetal, the spin waves in the two \flms become coupled\nthrough two di\u000berent mechanisms. First, the dynamic,\nnonlocal dipole-dipole interaction causes an interlayer\ncoupling to arise that is independent of the properties\nof the normal metal. This coupling is weaker for larger\nthicknesses of the normal metal. Second, spin pumping\nfrom one ferromagnetic insulator induces a spin accu-\nmulation in the normal metal, which in turn gives rise\nto a spin-transfer torque on the other ferromagnetic in-\nsulator, and vice versa. This dynamic coupling, is in\ncontrast to the static exchange coupling35rather long-ranged and is limited only by the spin-di\u000busion length.\nThis type of coupling is known to strongly couple the\nmacrospin modes. When two ferromagnetic \flms become\ncoupled, the characterization of the spin waves in terms\nof FVMSW, BVMSW, and MSSW still holds, but the\ndispersion relations are modi\fed. It is also clear that the\ndamping renormalization caused by spin pumping into\nthe NM may di\u000ber greatly from that in a simpler FI jN\nbilayer system. To understand this phenomenon, we per-\nform a detailed analytical and numerical analysis of a\ntrilayer system, with the hope that our \fndings may be\nused as a guide for experimentalists.\nThis paper is organized as follows. Section II intro-\nduces the model. The details of the dynamic dipolar\n\feld are discussed, and the boundary conditions associ-\nated with spin pumping and spin transfer at the FI jN\ninterfaces are calculated. Sec. III provides the analyti-\ncal solutions of these equations in the long-wavelength\nregime dominated by the dynamic coupling attributable\nto spin pumping and spin transfer. To create a more\ncomplete picture of the dynamic behavior of this system,\nwe perform a numerical analysis for the entire spin-wave\nspectrum of this system, which is presented in Sec. IV.\nWe conclude our work in Sec. V.\nII. EQUATIONS OF MOTION\nConsider a thin-\flm heterostructure composed of two\nferromagnetic insulators (FI1 and FI2) that are in elec-\ntrical contact via an NM layer. The ferromagnetic in-\nsulators FI1 and FI2 may have di\u000berent thicknesses and\nmaterial properties. We denote the thicknesses by L1,\ndN, andL2for the FI1, NM, and FI2 layers, respectively\n(see Fig. 1(a)). The in-plane coordinates are \u0010;\u0011, and the\ntransverse coordinate is \u0018(see Fig. 1(b)). We will \frst\ndiscuss the magnetization dynamics in isolated FIs and\nwill then incorporate the spin-memory losses and the cou-\npling between the FIs via spin currents passing through\nthe NM.\nA. Magnetization Dynamics in Isolated FIs\nThe magnetization dynamics in the ferromagnetic in-\nsulators can be described by using the Landau-Lifshitz-\nGilbert (LLG) equation,\n_Mi=\u0000\rMi\u0002He\u000b+\u000bMi\u0002_Mi; (1)\nwhere Miis the unit vector in the direction of the mag-\nnetization in layer i= 1;2,\ris the gyromagnetic ratio,\n\u000bis the dimensionless damping parameter, and He\u000bis\nthe space-time-dependent e\u000bective magnetic \feld. The\ne\u000bective magnetic \feld is\nHe\u000b=Hint+hex+hd+hsurface; (2)\nwhere Hintis the internal \feld attributable to an external\nmagnetic \feld and the static demagnetization \feld, hex=3\ndN2+L2\ndN2\n-dN2\n-dN2-L1NFI2\nFI1\nSUBx\n(a)\n (b)\nFIG. 1: (Color online) a) A cross section of the FI1 jNjFI2 het-\nerostructure. The ferromagnetic insulators FI1 and FI2 are\nin contact via the normal metal N. The transverse coordinate\n\u0018is indicated along with the thicknesses L1,dN, andL2of\nFI1, N, and FI2, respectively. b) The coordinate system of\nthe internal \feld (blue) with respect to the coordinate system\nof the FI1jNjFI2 structure (red). \u0012denotes the angle between\nthe \flm normal and the internal \feld, and \u001eis the angle be-\ntween the in-plane component of the magnetic \feld and the\nin-plane wave vector.\n2Ar2M=MSis the exchange \feld ( Ais the exchange\nconstant), hdis the dynamic demagnetization \feld, and\nhsurface =2KS\nM2\nS(Mi\u0001^n)\u000e(\u0018\u0000\u0018i)^n (3)\nis the surface anisotropy \feld located at the FI jN in-\nterfaces. In this work, hsurface is assumed to exist only\nat the FIjN interfaces and not at the interfaces between\nthe FIs and the substrate or vacuum. It is straightfor-\nward to generalize the discussion to include these surface\nanisotropies as well. We consider two scenarios: one with\nan easy-axis surface anisotropy ( KS>0) and one with no\nsurface anisotropy ( KS= 0). Note that a negative value\nofKS\u0018 \u0000 0:03 erg=cm2, which implies an easy-plane\nsurface anisotropy, has also been observed for sputtered\nYIGjAu bilayers.36In general, the e\u000bective \feld He\u000bmay\ndi\u000ber in the two FIs. We assume the two FIs consist of\nthe same material and consider external \felds that are\neither in-plane or out-of-plane. Furthermore, we consider\ndevices in which the internal magnetic \felds in the two\nFI layers are aligned and of equal magnitude.\nIn equilibrium, the magnetization inside the FIs is ori-\nented along the internal magnetic \feld, Mi=M0. In the\nlinear response regime, Mi=M0+mi, where the \frst-\norder correction miis small and perpendicular to M0.The magnetization vanishes outside of the FIs. Because\nthe system is translationally invariant in the \u0011and\u0010di-\nrections, we may, without loss of generality, assume that\nmconsists of plane waves travelling in the \u0010direction,\nmi(\u0010;\u0011;\u0018 ) =miQ(\u0018)ei(!t\u0000Q\u0010): (4)\nLinearizing Maxwell's equations in miimplies that the\ndynamic dipolar \feld must be of the same form,\nhd(\u0010;\u0011;\u0018 ) =hdQ(\u0018)ei(!t\u0000Q\u0010): (5)\nFurthermore, the total dipolar \feld (the sum of the static\nand the dynamic dipolar \felds) must satisfy Maxwell's\nequations, which, in the magnetostatic limit, are\nr\u0001(hd+ 4\u0019MSm) = 0; (6a)\nr\u0002hd= 0; (6b)\nwith the boundary equations\n(hd+ 4\u0019MSm)?;in= (hd)?;out; (7a)\n(hd)k;in= (hd)k;out; (7b)\nwhere the subscript in (out) denotes the value on the FI\n(NM, vacuum or substrate) side of the FI interface and ?\n(k) denotes the component(s) perpendicular (parallel) to\nthe FI{NM interfaces. Solving Maxwell's equations (6)\nwith the boundary conditions of Eq. (7) yields33\nhdQ(\u0018) =Z\nd\u00180^G(\u0018\u0000\u00180)mQ(\u00180); (8)\nwhere ^G(r\u0000r0) is a 3\u00023 matrix acting on min the (\u0011;\u0010;\u0018 )\nbasis,\n^G(\u0018) =0\n@GP(\u0018)\u0000\u000e(\u0018) 0\u0000iGQ(\u0018)\n0 0 0\n\u0000iGQ(\u0018) 0\u0000GP(\u0018)1\nA: (9)\nHere,GP(\u0018) =Qe\u0000Qj\u0018j=2, andGQ(\u0018) =\u0000sign(\u0018)GP.\nNote that the dynamic dipolar \feld of Eq. (8) accounts\nfor both the interlayer and intralayer dipole-dipole cou-\nplings because the magnetization varies across the two\nmagnetic insulator bilayers and vanishes outside these\nmaterials.\nIt is now convenient to perform a transformation from\nthe\u0010-\u0011-\u0018coordinate system de\fned by the sample geome-\ntry to thex-y-zcoordinate system de\fned by the internal\n\feld (see Fig. 1(b)). In the linear response regime, the\ndynamic magnetization milies in thex-yplane, and the\nlinearized equations of motion become33\n\u0014\ni!\u0012\n\u000b\u00001\n1\u000b\u0013\n+11\u0012\n!H+2A\nMS\u0014\nQ2\u0000d2\nd\u00182\u0015\u0013\u0015\nmiQxy(\u0018) =2X\ni=1Z\nd\u00180^Gxy(\u0018\u0000\u00180)miQxy(\u00180): (10)4\nN\nm1,QFI1m2,QFI2\nee\nFIG. 2: (Color online) Two coupled spin waves with ampli-\ntudem1Qin ferromagnet FI1 and amplitude m2Qin ferro-\nmagnet FI2. The spin-waves inject a spin current into the nor-\nmal metal (NM) via spin pumping. In the NM, the spins dif-\nfuse and partially relax, inducing a spin accumulation therein.\nIn turn, the spin accumulation causes spin-transfer torques to\narise on FI1 and FI2. The combined e\u000bect of spin transfer and\nspin pumping leads to a dynamic exchange coupling that, to-\ngether with the dynamic demagnetization \feld, couples the\nspin waves in the two FIs.\nHere, miQxy = (miQx;miQy) is the Fourier transform of\nthe dynamic component of the magnetization in the x-\nyplane and ^Gxy(\u0018) is the 2\u00022 matrix that results from\nrotating ^G(\u0018) into thex-y-zcoordinate system (see Ap-\npendix A), and considering only the xx,xy,yxandyy-\ncomponents.\nB. Boundary Conditions and Spin Accumulation\nThe linearized equations of motion (10) must be sup-\nplemented with boundary conditions for the dynamic\nmagnetization at the FI jN interfaces. A precessing mag-\nnetization at the FI jN boundaries injects a spin-polarized\ncurrent, jSP, into the NM, an e\u000bect known as spin\npumping .8,28{30The emitted spin currents at the lower\nand upper interfaces ( i= 1;2) are\njSP\ni=~\neg?Mi\u0002_Mi\f\f\f\f\n\u0018=\u0018i; (11)\nwhere\u0018i=\u0007dN=2 at the lower and upper interfaces,\nrespectively, and g?is the real part of the transverse spin-\nmixing conductance per unit area.37We disregard the\nimaginary part of the spin-mixing conductance because\nit has been found to be small at FI jN interfaces.38The\nreciprocal e\u000bect of spin pumping is spin transfer into the\nFIs because of a spin accumulation \u0016Sin the NM. In the\nnormal metal at the lower and upper interfaces ( i=1,2),the associated spin-accumulation-induced spin current is\njST\ni=\u00001\neg?Mi\u0002(Mi\u0002\u0016S)\f\f\f\f\n\u0018=\u0018i: (12)\nThe signs of the pumped and spin-accumulation-induced\nspin currents in Eqs. (11) and (12) were chosen such that\nthey are positive when there is a \row of spins from the\nNM toward the FIs.\nThe pumped and spin-accumulation-induced spin cur-\nrents of Eqs. (11) and (12) lead to magnetic torques act-\ning on the FI interfaces. The torques that correspond to\nthe spin pumping and spin transfer localized at the FI jN\ninterfaces are\n\u001cSP\ni=\r~2\n2e2g?\u000e(\u0018\u0000\u0018i)Mi\u0002_Mi; (13a)\n\u001cST\ni=\u0000\r~\n2e2g?Mi\u0002(Mi\u0002\u0016S)\u000e(\u0018\u0000\u0018i);(13b)\nrespectively. In the presence of spin currents to and from\nthe normal metal, the magnetization dynamics in the\nFIs is then governed by the modi\fed Landau-Lifshitz-\nGilbert-Slonczewski (LLGS) equation,\n_M=\u0000\rMi\u0002He\u000b+\u000bMi\u0002_Mi+X\ni=1;2\u001cSP\ni+\u001cST\ni:(14)\nBy integrating Eq. (14) over the FI jN interfaces and the\ninterfaces between the FI and vacuum/substrate, we \fnd5\nthatmimust satisfy the boundary conditions21,31\n\u0012\n\u0006Lidmi\nd\u0018+\u001fi\u0014\n_mi\u00001\n~M0\u0002\u0016\u0015\n+LiKS\nAcos (2\u0012)mi\u0013\nx\f\f\f\f\n\u0018=\u0007dN=2= 0;(15a)\n\u0012\n\u0006Lidmi\nd\u0018+\u001fi\u0014\n_mi\u00001\n~M0\u0002\u0016\u0015\n+LiKs\nAcos2(\u0012)mi\u0013\ny\f\f\f\f\f\n\u0018=\u0007dN=2= 0;(15b)\ndm1\nd\u0018\f\f\f\f\n\u0018=\u0000dN=2\u0000L1= 0;dm2\nd\u0018\f\f\f\f\n\u0018=dN=2+L2= 0:(15c)\nHere, we have introduced the timescale \u001fi=\nLi~2g?=4Ae2. The subscripts xandyin Eqs. (15a) and\n(15b) denote the xandycomponents, respectively. In\nour expressions for the boundary conditions (15), we have\nalso accounted for the possibility of a surface anisotropy\narising from the e\u000bective \feld described by Eq. (3),\nwhereKS>0 indicates an easy-axis surface anisotropy\n(EASA). The boundary conditions of Eq. (15), in combi-\nnation with the transport equations in the NM , which we\nwill discuss next, determine the spin accumulation in the\nNM and the subsequent torques caused by spin transfer.\nIn the normal metal, the spins di\u000buse, creating a spa-\ntially dependent spin-accumulation potential \u0016Q, and\nthey relax on the spin-di\u000busion length scale lsf. The\nspin accumulation for an FI jNjFI system has been cal-\nculated in the macrospin model.39The result of this\ncalculation can be directly generalized to the present\nsituation of spatially inhomogeneous spin waves by re-\nplacing the macrospin magnetization in each layer with\nthe interface magnetization and substituting the spin-\ndi\u000busion length with a wave-vector-dependent e\u000bective\nspin-di\u000busion length lsf!~lsf(Q) such that\n\u0016Q=\u0000~\n2M0\u0002[(_mQ(\u00181) +_mQ(\u00182))\u00001(\u0018)\n\u0000(_mQ(\u00181)\u0000_mQ(\u00182))\u00002(\u0018)]:(16)\nSee Appendix B for the details of the functions \u0000 1and\n\u00002. The e\u000bective spin-di\u000busion length is found by Fouriertransforming the spin-di\u000busion equation (see Appendix\nC), resulting in\n~lsf=lsf=p\n1 + (Qlsf)2: (17)\nWe thus have all the necessary equations to de-\nscribe the linear response dynamics of spin waves in the\nFI1jNjFI2 system. We now provide analytical solutions\nof the spin-wave modes in the long-wavelength limit and\nthen complement these solutions with an extensive nu-\nmerical analysis that is valid for any wavelength.\nIII. ANALYTIC SOLUTIONS FOR THE SPIN\nWAVE SPECTRUM\nThe e\u000bect that the exchange and dipolar \felds have\non the spin-wave spectrum depends on the in-plane wave\nnumberQ. WhenQLi\u001c1, the dipolar \feld dominates\nover the exchange \feld. In the opposite regime, when\nQLi\u001d1, the exchange \feld dominates over the dipo-\nlar \feld. The intermediate regime is the dipole-exchange\nregime. Another length scale is set by the spin-di\u000busion\nlength. When Qlsf\u001d1, the e\u000bective spin-relaxation\nlength ~lsfof Eq. (17) becomes small, and the NM acts\nas a perfect spin sink. In this case, only the relatively\nshort-ranged dipolar \feld couples the FIs. We therefore\nfocus our attention on the dipole-dominated regime, in\nwhich the interchange of spin information between the\ntwo FIs remains active.\nIn the limit QLi\u001c1, the magnetization is homoge-\nneous in the in-plane direction. We may then use the\nansatz that the deviation from equilibrium is a sum of\ntransverse travelling waves. Using the boundary condi-\ntions on the outer boundaries of the stack, Eq. (15c), we\n\fnd\nmiQxy(\u0018) =\u0012\nXi\nYi\u0013\ncos\u001a\nki\u0014\n\u0018\u0006(Li+dN\n2)\u0015\u001b\n;(18)\nwherei= 1 when\u0018is inside FI1 and i= 2 when\u0018is inside\nFI2.k1andk2are the out-of-plane wave vectors of the\nlower and upper \flms, respectively. The eigenfrequencies\nof Eq. (10) depend on ki. To \frst order in the damping\nparameter\u000b, we have\n!(ki) =!M\"\n\u0006s\u0012!H\n!M+A\n2\u0019M2\nSk2\ni\u0013\u0012!H\n!M+A\n2\u0019M2\nSk2\ni+ sin2\u0012\u0013\n+i\u000b\u0012!H\n!M+A\n2\u0019M2\nSk2\ni+1\n2sin2\u0012\u0013#\n: (19)\nWe can, without loss of generality, consider only those frequencies that have a positive real part. The eigen-6\nfrequency!is a characteristic feature of the entire sys-\ntem, so we must require !(k1) =!(k2), which implies\nthatk1=\u0006k2. We will discuss the cases of symmetric\n(L1=L2) and asymmetric ( L16=L2) geometries sepa-\nrately.\nA. Symmetric FI \flms without EASA\nConsider a symmetric system in which the FIs are of\nidentical thickness and material properties. We assume\nthat the e\u000bect of the EASA is negligible, which is the\ncase for thin \flms and/or weak surface anisotropy ener-\ngies such that KSL=A\u001c1, whereL=L1=L2. The\nother two boundary conditions, (15a) and (15b), cou-\nple the amplitude vectors\u0000X1Y1\u0001Tand\u0000X2Y2\u0001Tof\nEq. (18). A non-trivial solution implies that the deter-\nminant that contains the coe\u000ecients of the resulting 4 \u00024\nmatrix equation vanishes. Solving the secular equation,\nwe \fnd the following constraints on k,\ni\u001fA!A=kLtan(kL); (20a)\ni\u001fO!O=kLtan(kL); (20b)\nwhere\n\u001fA=\u001f \n1\u0000\u0014\n1 +2g?lsf\n\u001btanh(dN=2lsf)\u0015\u00001!\n;(21a)\n\u001fO=\u001f \n1\u0000\u0014\n1 +2g?lsf\n\u001bcoth(dN=2lsf)\u0015\u00001!\n;(21b)\nand\u001f=L~2g?=4Ae2. The two solutions correspond\nto a symmetric mode (acoustic) and an antisymmetric\nmode (optical). This result can be understood in terms\nof the eigenvectors that correspond to the eigenvalues of\nEqs. (20), which are m1= +m2andm1=\u0000m2for\nthe acoustic and optical modes, respectively. Typically,\nbecause spin pumping only weakly a\u000bects the magne-\ntization dynamics, the timescale \u001fthat is proportional\nto the mixing conductance g?is much smaller than the\nFMR precession period. In this limit, kLtan(kL)\u001c1.\nThis result allows us to expand the secular equations (20)\naroundkL=n\u0019, wherenis an integral number, which\nyields\ni\u001f\u0017!\u0017;n\u0019(kL+\u0019n)kL; (22)\nwhere\u0017= A;O. This result can be reinserted into the\nbulk dispersion relation of Eq. (19), from which we can\ndetermine the renormalization of the Gilbert damping\ncoe\u000ecient attributable to spin pumping, \u0001 \u000b. We de\fne\n\u0001\u000b=\u000b\u0010\nIm[!(SP)]\u0000Im[!(0)]\u0011\n=Im[!(0)] (23)\nas a measure of the spin-pumping-enhanced Gilbert\ndamping, where !(0)and!(SP)are the frequencies of\nthe same system without and with spin pumping, respec-\ntively.Similar to the case of a single-layer ferromagnetic\ninsulator,31we \fnd that all higher transverse volume\nmodes exhibit an enhanced magnetization dissipation\nthat is twice that of the macrospin mode. The enhance-\nment of the Gilbert damping for the macrospin mode\n(n= 0) is\n\u0001\u000b\u0017;macro =\r~2g?\n2LMSe2\u001f\u0017\n\u001f; (24)\nand for the other modes, we obtain\n\u0001\u000b\u0017;n6=0= 2\u0001\u000b\u0017;macro: (25)\nCompared with single-FI systems, the additional fea-\nture of systems with two FIs is that the spin-pumping-\nenhanced Gilbert damping di\u000bers signi\fcantly between\nthe acoustic and optical modes via the mode-dependent\nratio\u001f\u0017=\u001f. This phenomenon has been explored both\nexperimentally and theoretically in Ref. 32 for the\nmacrospin modes n= 0 when there is no loss of spin\ntransfer between the FIs, lsf!1 . Our results repre-\nsented by Eqs. (24) and (25) are generalizations of these\nresults for the case of other transverse volume modes and\naccount for spin-memory loss. Furthermore, in Sec. IV,\nwe present the numerical results for the various spin-wave\nmodes when the in-plane momentum Qis \fnite. When\nthe NM is a perfect spin sink, there is no transfer of spins\nbetween the two FIs, and we recover the result for a sin-\ngle FIjN system with vanishing back \row, \u001f\u0017!\u001f.31\nNaturally, in this case, the FI jNjFI system acts as two\nindependent FIjN systems with respect to magnetiza-\ntion dissipation. The dynamical interlayer dipole cou-\npling is negligible in the considered limit of this section\n(QL\u001c1).\nIn the opposite regime, when the NM \flm is much thin-\nner than the spin-di\u000busion length and the spin conductiv-\nity of the NM is su\u000eciently large such that g?dN=\u001b\u001c1,\nthen\u001fA!0 and\u001fO!\u001f. This result implies that for\nthe optical mode, the damping is the same as for a sin-\ngle FI in contact with a perfect spin sink, even though\nthe spin-di\u000busion length is very large. The reason for\nthis phenomenon is that when the optical mode is ex-\ncited, the magnetizations of the two \flms oscillate out\nof phase such that one layer acts as a perfect spin sink\nfor the other layer. By contrast, there is no enhance-\nment of the Gilbert damping coe\u000ecient for the acoustic\nmode; when the \flm is very thin and the magnetizations\nof the two layers are in phase, there is no net spin \row or\nloss in the NM \flm and no spin-transfer-induced losses\nin the ferromagnets. Finally, when the NM is a poor con-\nductor despite exhibiting low spin-memory loss such that\ng?dN=\u001b\u001d(lsf=dN)\u001d1, then\u001f\u0017!0 because there is no\nexchange of spin information. For the macrospin modes\nin the absence of spin-memory loss, these results are in\nexact agreement with Ref. 32. Beyond these results, we\n\fnd that regardless of how much spin memory is lost, it\nis also the case that in trilayer systems, all higher trans-\nverse modes experience a doubling of the spin-pumping-\ninduced damping. Furthermore, these modes can still7\nbe classi\fed as optical and acoustic modes with di\u000berent\ndamping coe\u000ecients.\nB. Symmetric Films with EASA\nMagnetic surface anisotropy is important when the\nspin-orbit interaction at the interfaces is strong. In this\ncase, the excited mode with the lowest energy becomes\ninhomogeneous in the transverse direction. For a \fnite\nKS, the equations for the xandycomponents of the\nmagnetization in the boundary condition (15) di\u000ber, re-\nsulting in di\u000berent transverse wave vectors for the two\ncomponents, kxandky, respectively. Taking this situa-\ntion into account, we construct the ansatz\nmiQxy(\u0018) =\u0012\nXicos (kx;i\u0018\u0006kx;i(L+dN=2))\nYicos (ky;i\u0018\u0006ky;i(L+dN=2))\u0013\n;(26)\nwhich, when inserted into the boundary conditions of\nEqs. (15a) and (15b), yields\ni\u001f\u0017!\u0017+LKS\nAcos (2\u0012) =kxdtan (kxd);(27a)\ni\u001f\u0017!\u0017+LKS\nAcos2(\u0012) =kydtan (kyd);(27b)\nwhere\u0017continues to denote an acoustic (A) or optical\n(O) mode, \u0017= A;O. Depending on the sign of KSand\nthe angle\u0012, the resulting solutions kxandkycan be-\ncome complex numbers, which implies that the modes\nare evanescent. Let us consider the case of KS>0 and\nan in-plane magnetization ( \u0012=\u0019=2). Although kyis\nunchanged by the EASA, with LKS=A> 1\u001d\u001f\u0017!\u0017,kx\nis almost purely imaginary, \u0014=ik=KS=A\u0000i!\u0017\u001f\u0017, so\nthat\nmiQx(\u0018) =Xcosh(\u0014\u0018\u0006\u0014(d+dN=2)): (28)\nThe magnetization along the xdirection is exponentially\nlocalized at the FI jN surfaces. Following the same proce-\ndure as in Sec. III A for the KS= 0 case, we insert this\nsolution into the dispersion relation (19) and extract the\nrenormalization of the e\u000bective Gilbert damping:\n\u0001\u000bEASA\n\u0017 =\r~2g?\n2LMSe2\u001f\u0017\n\u001f1 +!H\n!M\u0002\n1 +2LKS\nA\u0003\n\u0000K2\nS\n2\u0019M2\nSA\n1 + 2!H\n!M\u0000K2s\n2\u0019M2\nSA:\n(29)\nIn the presence of EASA, the damping coe\u000ecient is a ten-\nsor; thus, the e\u000bective damping of Eq. (29) is an average,\nas de\fned in Eq. (23). This Gilbert damping enhance-\nment may become orders of magnitude larger than the\n\u0001\u000bmacro of Eq. (24). For thick \flms, \u0001 \u000bmacro\u0018L\u00001,\nwhereas \u0001\u000bEASA\n\u0017 reaches a constant value that is in-\nversely proportional to the localization length at the FI jN\ninterface. Note that for large EASA, the equilibrium\nmagnetization is no longer oriented along the external\n\feld, and Eq. (29) for \u0001 \u000bEASA\n\u0017 becomes invalid.C. Asymmetric FI Films\nLet us now consider an asymmetric system in which\nL16=L2. In this con\fguration, we will \frst consider\nKS= 0, but we will also comment on the case of a \f-\nniteKSat the end of the section. Because the analytical\nexpressions for the eigenfrequencies and damping coe\u000e-\ncients are lengthy, we focus on the most interesting case:\nthat in which the spin-relaxation rate is slow.\nAs in the case of the symmetric \flms, the dispersion\nrelation of Eq. (10) dictates that the wave numbers in the\ntwo layers must be the same. To satisfy the boundary\nequations (15), we construct the ansatz\nmiQxy(\u0018) =\u0012\nXicos (k\u0018\u0006k(L+dN=2))\nYicos (k\u0018\u0006k(L+dN=2))\u0013\n: (30)\nThe di\u000berence between this ansatz and the one for the\nsymmetric case represented by Eq. (26) is that the mag-\nnitudes of the amplitudes, XiandYi, of the two layers,\ni= 1;2, that appear in Eq. (30) is no longer expected to\nbe equal.\nWhen the two ferromagnets FI( L1) and FI(L2) are\ncompletely disconnected, the transverse wave vectors\nmust be equivalent to standing waves, qn;1=\u0019n=L 1and\nqm;2=\u0019m=L 2in the two \flms, respectively, where nand\nmmay be any integral numbers. Because spin pumping\nis weak, the eigenfrequencies of the coupled system are\nclose to the eigenfrequencies of the isolated FIs. This\n\fnding implies that the wave vector kof the coupled sys-\ntem is close to either qn;1orqm;2. The solutions of the\nlinearized equations of motion are then\nk=kn;1=qn;1+\u000ekn;1or (31a)\nk=km;2=qm;2+\u000ekm;2; (31b)\nwhere\u000ekn;1and\u000ekm;2are small corrections attributable\nto spin pumping and spin transfer, respectively. Here,\nthe indices 1 and 2 represent the di\u000berent modes rather\nthan the layers. However, one should still expect that\nmode 1(2) is predominantly localized in \flm 1(2). In\nthis manner, we map the solutions of the wave vectors in\nthe coupled system to the solutions of the wave vectors\nin the isolated FIs. Next, we will present solutions that\ncorrespond to the qn;1of Eq. (31a). The other family of\nsolutions, corresponding to qm;2, is determined by inter-\nchangingL1$L2and making the replacement n!m.\nInserting Eq. (31a) into the boundary conditions of\nEq. (15) and linearizing the resulting expression in the\nweak spin-pumping-induced coupling, we \fnd, for the\nmacrospin modes,\ni!~\u001fA,O\n1;macro = (L1\u000ek0;1)2; (32)\nwhere\n~\u001fA\n1;macro\u00191\n2dN\nlsf\u001b\ng?lsfL1\nL1+L2\u001f1; (33a)\n~\u001fO\n1;macro\u00191\n2L1+L2\nL2\u001f1: (33b)8\nHere,\u001f1=L1~2g?=4Ae2. Inserting this parameter into\nthe dispersion relation of Eq. (19), we obtain the follow-\ning damping renormalizations:\n\u0001\u000bA\nmacro =\r~2g?\n2MSe21\n2dN\nlsf\u001b\ng?lsf1\nL1+L2;(34a)\n\u0001\u000bO\nmacro =\r~2g?\n2MSe21\n2\u00121\nL1+1\nL2\u0013\n: (34b)\nThese two solutions correspond to an acoustic mode\nand an optical mode, respectively. The corresponding\neigenvectors are m1=m2for the acoustic mode and\nL1m1=\u0000L2m2for the optical mode. As in the sym-\nmetric case, the damping enhancement of the acoustic\nmode vanishes in the thin-NM limit. In this limit, the\nbehavior of the acoustic mode resembles that of a single\nFI of thickness L1+L2. It is the total thickness that\ndetermines the leading-order contribution of the damp-\ning renormalization. The optical mode, however, experi-\nences substantial damping enhancement. For this mode,\nthe damping renormalization is the average of two sepa-\nrate FIs that are in contact with a perfect spin sink. The\ncause of this result is as follows. When there is no spin-\nmemory loss in the NM, half of the spins that are pumped\nout from one side return and rectify half of the angular-\nmomentum loss attributable to spin pumping. Because\nthe magnetization precessions of the two \flms are com-\npletely out of phase, the other half of the spin current\ncauses a dissipative torque on the opposite layer. In ef-\nfect, spin pumping leads to a loss of angular momentum,\nand the net sum of the spin pumping across the NM and\nthe back \row is zero. The total dissipation is not a\u000bected\nby spin transfer, and thus, the result resembles a system\nin which the NM is a perfect spin sink.\nFor the higher excited transverse modes, there are two\nscenarios, which we treat separately. I. The allowed wave\nnumber for one layer matches a wave number for the\nother layer. Then, for some integer n > 0,qn;1=qm;2\nfor some integer m. In this case, we expect a coupling\nof the two layers. II. The allowed wave number for one\nlayer does not match any of the wave numbers for the\nother layer, and thus, for some integer n > 0, we have\nqn;16=qm;2for all integers m. We then expect that the\ntwo layers will not couple.\nI. In this case, we \fnd two solutions that correspond\nto acoustic and optical modes. These modes behave very\nmuch like the macrospin modes; however, as in the sym-\nmetric case, the damping renormalization is greater by a\nfactor of 2:\n\u0001\u000bA,O\nn6=0= 2\u0001\u000bA,O\nmacro;Case I: (35)\nThe eigenvectors of these coupled modes have the same\nform as for the macrospin modes, such that m1=m2\nandL1m1=\u0000L2m2for the acoustic and optical modes,\nrespectively.\nII. In this case, the two layers are completely decou-pled. To the leading order in dN=lsf, we \fnd\n\u0001\u000bn6=0=\r~2g?\n2L1MSe2;Case II; (36)\nfor all modes that correspond to excitations in FI1.\nThe damping renormalization is thus half that of the\nFI(L1)jN(lsf= 0) system.31This result can be explained\nby the zero loss of spin memory in the NM. Although half\nof the spins are lost to the static FI2, half of the spins\nreturn and rectify half of the dissipation attributable\nto spin pumping. The amplitudes of these modes are\nstrongly suppressed in FI2 (or FI1, upon the interchange\nof FI1$FI2), such thatjm2j=jm1j\u0018!\u001f2.\nFinally, let us discuss the case in which EASA is\npresent. In the limit KSLi=A\u001d1, the excitation en-\nergies of the surface modes are independent of the FI\nthicknesses. However, the surface modes do not behave\nlike the macrospin modes for the asymmetric stack. The\nexcitation volume of these modes is determined by the\ndecay length A=KSin accordance with Eq. (28). This\n\fnding is in contrast to the result for the macrospin\nmodes, where the excitation volume spans the entire FI.\nThus, the surface modes couple in the same manner as in\nthe symmetric case. With a good experimental control\nof surface anisotropy, the coupling of the surface modes\nis thus robust to thickness variations. The higher ex-\ncited transverse modes, in the presence of EASA, have\nthickness-dependent frequencies, which means that these\nmodes behave similarly to the n>0 modes in the KS= 0\ncase.\nIV. NUMERICAL RESULTS\nWhen the spin-wave wavelength becomes comparable\nto the \flm thickness, the dipolar \feld becomes a compli-\ncated function of the wavelength. We study the proper-\nties of the system in this regime by numerically solving\nthe linearized equations of motion (10) with the bound-\nary conditions (15). We use the method presented in\nRef. 31, which solves the spin-wave excitation spectrum\nfor an FIjN system, and extend this approach to the\npresent trilayer system. The physical parameters used\nin the numerical calculations are listed in Table I. We\ninvestigate two geometries: I. the BWMSW geometry, in\nwhich the spin wave propagates parallel to the external\n\feld, and II. the MSSW geometry, in which the spin wave\npropagates perpendicular to the external \feld.\nTo calculate the renormalization of the Gilbert damp-\ning, we perform one computation without spin pumping\nand one computation with spin pumping, in which the\nintrinsic Gilbert damping is excluded. Numerically, the\nrenormalization can then be determined by calculating\n\u0001\u000b=\u000bIm[!(SP)]\u000b=0=Im[!(0)], where!(0)is the eigenfre-\nquency obtained for the computation without spin pump-\ning and!(SP)is the frequency obtained for the compu-\ntation with spin pumping.319\nTABLE I: Physical parameters used in the numerical calcu-\nlations\nConstant Value Units\ng?a3:4\u00011015cm\u00002e2=h\n\u001bb5:4\u00011017s\u00001\n4\u0019MSc1750 G\nAc3:7\u000110\u00007erg=cm\nHint 0:58\u00014\u0019MS\n\u000bc3\u000110\u00004\nKS 0;d0:05 erg=cm2\na) Ref. [47], b) Ref. [48], c) Ref. [34]\nd) Reported to be in the range of 0 :1\u00000:01 erg=cm2in\nRef. [21]\nA. BVMSW\nFIG. 3: (Color online) FI(100nm) jN(50nm)jFI(101nm): a)\nSpin-pumping-enhanced Gilbert damping \u0001 \u000bas a function\nofQL1of the uniform modes and the n= 1 modes. The inset\npresents the corresponding dispersion relation. b) Relative\nphase and c) amplitude between the out-of-plane magnetiza-\ntions along xat the edges of FI1 jN and FI2jN. The apparent\ndiscontinuity in the green line in c) appears because the phase\nis de\fned on the interval \u0000\u0019to\u0019.\nLet us \frst discuss the BVMSW geometry. The cou-\npling of the uniform modes in the two \flms is robust;it is not sensitive to possible thickness asymmetries. In\ncontrast, at Q= 0, the sensitivity to the ratio between\nthe thickness and the rather weak dynamic coupling at-\ntributable to spin pumping implies that the coupling of\nthe higher transverse modes in the two bilayers is fragile.\nSmall asymmetries in the thicknesses destroy the cou-\npling. This e\u000bect can best be observed through the renor-\nmalization of the damping. However, we will demon-\nstrate that a \fnite wave number Qcan compensate for\nthis e\u000bect such that the higher transverse modes also\nbecome coupled. To explicitly demonstrate this result,\nwe numerically compute the real and imaginary parts\nof the eigenfrequencies of a slightly asymmetric system,\nFI(100nm)jN(50nm)jFI(101nm) with lsf= 350 nm. The\nasymmetry between the thicknesses of the ferromagnetic\ninsulators is only 1%. The surface anisotropy is consid-\nered to be small compared with the ratio Li=A, and we\nsetKS= 0.\nIn Fig. 3, the numerical results for the e\u000bective Gilbert\ndamping, the dispersion of the modes, and the relative\nphase and amplitude between the magnetizations in the\ntwo FIs are presented. As observed in the relative phase\nresults depicted in Fig. 3(c), the two uniform modes in\nwidely separated FIs split into an acoustic mode and\nan optical mode when the bilayers are coupled via spin\npumping and spin transfer. Figure 3(a) also demon-\nstrates that the acoustic mode has a very low renor-\nmalization of the Gilbert damping compared with the\noptical mode. Furthermore, there is no phase di\u000berence\nbetween the two modes with a transverse node ( n= 1) in\nFig. 3(a), which indicates that the modes are decoupled.\nThesen= 1 modes are strongly localized in one of the\ntwo \flms; see Fig. 3(b). For small QL1, Fig. 3(a) demon-\nstrates that these modes have approximately the same\nrenormalization as the optical mode, which is in agree-\nment with the analytical results. Because the magnetiza-\ntion in the layer with the smallest amplitude is only a re-\nsponse to the spin current from the other layer, the phase\ndi\u000berence is \u0019=2 (Fig. 3(b)). When Qincreases, the dipo-\nlar and exchange interactions become more signi\fcant.\nThe interlayer coupling is then no longer attributable\nonly to spin pumping but is also caused by the long-range\ndipole-dipole interaction. This additional contribution to\nthe coupling is su\u000ecient to synchronize the n= 1 modes.\nThe relative amplitude between the two layers then be-\ncomes closer to 1 (see Fig. 3(b)). Again, we obtain an\nacoustic mode and an optical n= 1 mode, which can be\nobserved from the phase di\u000berence between the two lay-\ners in Fig. 3(c). The spin-pumping-induced coupling only\noccurs as long as the e\u000bective spin-di\u000busion length ~lsfis\nlarge or on the order of dN. Once this is no longer the\ncase, the modes rapidly decouple, and the system reduces\nto two separate FI jN systems with a relatively weak in-\nterlayer dipole coupling. In the limit of large QL1, the\nexchange interaction becomes dominant. The energy of\nthe wave is then predominantly attributable to the mo-\nmentum in the longitudinal direction, and the dynamic\npart of the magnetization goes to zero at the FI jN inter-10\nfaces, causing the renormalization attributable to spin\npumping to vanish.31\nWe also note that the dispersion relation depicted in\nthe inset of Fig. 3(a) reveals that the acoustic mode (blue\nline) exhibits a dip in energy at lower QL1than does the\noptical mode (red line). We suggest that this feature\ncan be understood as follows: The shift in the position\nof the energy dip can be interpreted as an increase in\nthe e\u000bective FI thickness for the acoustic mode with re-\nspect to that for the optical mode. When ~lsfis larger\nthan the NM thickness, the uniform mode behaves as\nif the NM were absent and the two \flms were joined.\nThis result indicates that the dispersion relation for the\nacoustic mode exhibits frequency behavior as a function\nofQ~L=2, where the e\u000bective total thickness of the \flm is\n~L=L1+L2. The optical mode, however, \\sees\" the NM\nand thus behaves as if ~L=L1. Consequently, the dip in\nthe dispersion occurs at lower QL1for the acoustic mode\nthan for the optical mode.\nB. MSSW\nFinally, let us study the dynamic coupling of mag-\nnetostatic surface spin waves (MSSWs). We now con-\nsider a perfectly symmetric system, FI(1000 nm) jN(200\nnm)jFI(1000 nm), with lsf= 350 nm. For such thick\n\flms, surface anisotropies may play an important role.\nWe therefore discuss a case in which we include a surface\nanisotropy of KS= 0:05 erg=cm2. According to the an-\nalytical result presented in Eq. (28), the lowest-energy\nmodes with QL1\u001c1 are exponentially localized at the\nFIjN surfaces, with a decay length of A=KS\u0018200 nm.\nWe now compute the eigenfrequencies, !, as a function\nof the wave vector in the range 10\u00004< QL 1<103. In\nFig. 4(a), we present the real part of the frequency for\nthe six lowest-energy modes with a positive real part, and\nin Fig. 4(b), we present the corresponding renormaliza-\ntions of the Gilbert damping for the four lowest-energy\nmodes. The dispersion relations indicate that the mode\npairs that are degenerate at QL1\u001c1 rapidly split in\nenergy when QL1approaches 10\u00002. Strong anticrossings\ncan be observed between the n= 1 andn= 2 modes.\nSuch anticrossings are also present between the surface\nmode and the n= 1 mode; they are almost too strong to\nbe recognized as anticrossings. The enhanced damping\nrenormalizations exhibit very di\u000berent behavior for the\ndi\u000berent modes. We recognize the large-\u0001 \u000bmode of one\npair as the surface optical mode and the low-\u0001 \u000bmode\nas the volume n= 1 acoustic mode. Without EASA,\nthe anticrossings in Fig. 4(a) would become crossings.\nThe lowest-energy modes at QL1\u001c1 would then cut\nstraight through the other modes. In the case considered\nhere, this behavior is now observed only as steep lines at\nQL1\u00180:05 and atQL1\u00180:5.\nWhenQis increased, the e\u000bective spin-di\u000busion\nlength decreases (see Eq. (17)), which reduces the spin-\npumping-induced coupling between the modes at largeQ. WhenQL1\u0018100, the coupling becomes so weak\nthat the two FIs decouple. This phenomenon can be ob-\nserved from the behavior of \u0001 \u000bin Fig. 4(b), where the\ndamping of the acoustic modes become the same as for\nthe optical modes.\nFIG. 4: (Color online) FI(1000nm) jN(200nm)jFI(1000nm)\nlsf= 350 nm, KS= 0:05 erg=cm2: a) The dispersion rela-\ntion as a function of QL1for the six lowest positive-real-part\nmodes. b) The renormalization of the damping attributable\nto spin pumping for the four lowest modes with frequencies\nwith positive real parts as a function of QL1. At largeQL1,\nthe computation becomes increasingly demanding, and the\npoint density of the plot becomes sparse. We have therefore\nindividually marked the plotted points in this region.\nIn the MSSW geometry, an isolated FI has magneto-\nstatic waves that are localized near one of the two sur-\nfaces, depending on the direction of propagation with\nrespect to the internal \feld.34Asymmetries in the exci-\ntation volume are therefore also expected for the trilayer\nin this geometry. In Fig. 5, we present the eigenvectors\nof the surface modes as functions of the transverse co-\nordinate\u0018for increasing values of the wave vector Q.\nAtQL1= 0:5, the modes have already begun to ex-\nhibit some asymmetry. Note that the renormalization\nof the damping observed in Fig. 4(b) is approximately\none order of magnitude larger than the intrinsic Gilbert\ndamping for the optical mode and that the damping of\nany one mode may vary by several orders of magnitude\nas a function of QL1.31Therefore, these e\u000bects should\nbe experimentally observable. The greatest damping oc-\ncurs when the two layers are completely decoupled; see\nFigs. Fig. 4(b) and 5. Because the damping of the opti-\ncal mode is equivalent to that of a system with a perfect\nspin sink, one might expect that the greatest damping11\nFIG. 5: (Color online)FI(1000nm) jN(200nm)jFI(1000nm),\nlsf= 350nm, KS= 0:05 erg=cm2: a) and b) present the\nreal parts of the xcomponents of the out-of-equilibrium mag-\nnetization vectors for the acoustic and optical surface modes,\nrespectively, for several values of QL1. For values of QL1&1,\nthe modes decouple and become localized in one of the two\nlayers. For large values of QL1\u0018100, the two modes are\nstrongly localized at one of the two FI jN interfaces, which\ncorrespond to the peaks in the damping that are apparent in\nFig. 4(b).\nshould occur for this mode. However, the large localiza-\ntion, which is achieved only at large QL1, in combination\nwith the vanishing of the e\u000bective spin-di\u000busion length\nleads to damping that is much greater than that of the\nsynchronized optical mode.\nV. CONCLUSIONS\nWe investigated the dynamic coupling of spin-wave ex-\ncitations, which are present in single FI thin \flms, pri-\nmarily through spin pumping and spin transfer but also\nthrough the dynamic demagnetization \feld created when\ntwo FI thin \flms are in contact via an NM layer. Because\nof this coupling, the modes are split into acoustical and\noptical excitations. When the NM is thin compared with\nlsf, the renormalization of the Gilbert damping vanishes\nfor the acoustic modes, whereas for the optical modes,\nthe renormalization is equally as large as for a single-\nFIjN system in which the NM is a perfect spin sink. A\nspin current pumped by a travelling magnetic wave has a\nwavelength of equal magnitude, which leads to traversal\npaths across the NM that are longer than the thickness\nof the NM. Consequently, the spin-memory loss is greater\nfor short-wavelength spin currents. This phenomenonleads to an e\u000bective spin-di\u000busion length in the NM that\ndecreases for increasing values of Q. As a result, the dy-\nnamic coupling strength is reduced for short-wavelength\nspin waves. At some critical value of Q, the coupling be-\ncomes so weak that the acoustic- and optical-mode con-\n\fgurations are lost in favor of modes that are localized\nin one of the two FIs. At these values of Q, the inter-\nlayer dipole coupling is also dominated by the intralayer\nexchange coupling. For these high-wave-number modes,\nthe system behaves similar to two separate FI jN(lsf= 0)\nsystems.\nWhen the two \flms are of di\u000berent thicknesses, the\nexchange energies of the higher-order transverse n > 1\nmodes di\u000ber between the two layers. Because of the rel-\natively small coupling attributable to spin pumping, the\nsynchronization of these modes at QL1\u001c1 requires that\nthe FI thicknesses be very similar. A small asymmetry\nbreaks the synchronization; however, for larger QL1\u00181,\nthe modes can again become coupled through interlayer\ndipole interaction. This coupling arises in addition to\nthe spin-pumping- induced coupling. For even larger Q,\nthe e\u000bective spin-di\u000busion length becomes small, and the\ncoupling attributable to spin pumping vanishes. The rel-\natively small dipole coupling alone is not su\u000ecient to\ncouple the modes when there is a \fnite di\u000berence in \flm\nthickness , and the synchronization breaks down.\nDepending on the quality of the interface between the\nFIs and the strength of the spin-orbit coupling in the\nNM , additional e\u000bective surface \felds may be present\nbecause of surface anisotropy energies. For the EASA\ncase, the lowest-energy modes are localized at the FI jN\nsurfaces. These modes couple in the same manner as the\nmacrospin modes. For \flms that are much thicker than\nthe decay length A=KS, the energies of the surface modes\ndo not depend on the \flm thickness. Consequently, the\ncoupling of these modes is independent of the thickness\nof the two FIs. Similar to the simpler FI jN system, the\ndamping enhancement may attain values as high as an or-\nder of magnitude larger than the intrinsic Gilbert damp-\ning. However, in the trilayer system, the presence of both\nacoustic and optical modes results in large variations in\nthe e\u000bective damping within the same physical sample.\nBecause of this wide range of e\u000bective damping, which\nspans a di\u000berence in \u0001 \u000bof several orders of magnitude\nas a function of Q, we suggest that trilayer modes should\nbe measurable in an experimental setting.\nWith more complicated FI structures in mind, we be-\nlieve that this work may serve as a guide for experimen-\ntalists. The large variations in e\u000bective damping for dif-\nferent modes make the magnetic properties of the system\ndetectable both with and without EASA. For spin waves,\ndipole-dipole interactions assist spin pumping in inter-\nlayer synchronization, which may facilitate the design of\nfuture spintronic devices.12\nAcknowledgments\nWe acknowledge support from InSpin 612759 and the\nResearch Council of Norway, project number 216700.\n\u0003Electronic address: hans.skarsvag@ntnu.no\n1C. Kittel, Phys. Rev. 73, 155 (1948).\n2R.W. Damon, J.R. Eshbach, J. Phys. Chem. Solids 19, 308\n(1961).\n3V. Cherepanov, I. Kolokolov and V. L'vov, Phys. Rep. 229,\n81 (1993).\n4A. Brataas, A. D. Kent, and H. Ohno, Nature Mat., 373\n(2012).\n5G. Binasch, P. Gr unberg, F. Saurenbach, and W. Zinn,\nPhys. Rev. B 394828 (1989).\n6L. Berger, Phys. Rev. B 54, 9353 (1996).\n7J.C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996).\n8Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida,\nM. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K.\nTakanashi, S. Maekawa, and E. Saito, Nature (London)\n464, 262 (2010).\n9C. W. Sandweg, Y. Kajiwara, K. Ando, E. Saitoh, and B.\nHillebrands, Appl. Phys. Lett. 97, 252504 (2010).\n10C. W. Sandweg, Y. Kajiwara, A. V. Chumak, A. A. Serga,\nV. I. Vasyuchka, M. B. Jung\reisch, E. Saitoh, and B. Hille-\nbrands, Phys. Rev. Lett. 106, 216601 (2011).\n11B. Heinrich, C. Burrowes, E. Montoya, B. Kardasz, E. Girt,\nY.-Y. Song, Y. Sun, and M. Wu, Phys. Rev. Lett. 107,\n066604 (2011).\n12L. H. Vilela-Leao, C. Salvador, A. Azevedo, S. M. Rezende,\nAppl. Phys. Lett. 99, 102505 (2011).\n13C. Burrowes, B. Heinrich, B. Kardasz, E. A. Montoya, E.\nGirt, Yiyan Sun, Young-Yeal Song, and Mingzhong Wu,\nAppl. Phys. Lett. 100, 092403 (2012).\n14S. M. Rezende, R. L. Rodr\u0013 \u0010guez-Su\u0013 arez, M. M. Soares, L.\nH. Vilela-Le~ ao, D. Ley Dom\u0013 \u0010nguez, and A. Azevedo, Appl.\nPhys. Lett. 102, 012402 (2013).\n15P. Pirro, T. Br acher, A. V. Chumak, B. L agel, C. Dubs,\nO. Surzhenko, P. G ornert, B. Leven and B. Hillebrands,\nAppl. Phys. Lett. 104, 012402 (2014).\n16H. Nakayama, M. Althammer, Y.-T. Chen, K. Uchida, Y.\nKajiwara, D. Kikuchi, T. Ohtani, S. Gepr ags, M. Opel, S.\nTakahashi, R. Gross, G. E. W. Bauer, S. T. B. Goennen-\nwein and E. Saitoh, Phys. Rev. Lett. 110, 206601 (2013).\n17C. Hahn, G. de Loubens, O. Klein, M. Viret, V. V. Naletov,\nand J. Ben Youssef, Phys. Rev. B 87, 174417 (2013).\n18N. Vlietstra, J. Shan, V. Castel, B. J. van Wees, and J.\nBen Youssef, Phys. Rev. B 87, 184421 (2013).\n19A. Brataas, Physics 6, 56 (2013).\n20A. Hamadeh, O. d'Allivy Kelly, C. Hahn, H. Meley, R.\nBernard, A.H. Molpeceres, V. V. Naletov, M. Viret, A.\nAnane, V. Cros, S. O. Demokritov, J. L. Prieto, M. Mu~ noz,\nG. de Loubens and O. Klein, arXiv:1405.7415v1.\n21J. Xiao and G. E. W. Bauer, Phys. Rev. Lett. 108, 217204\n(2012).\n22T. Schneider, A. A. Serga, B. Leven, B. Hillebrands, R. L.\nStamps and M. P. Kostylev, Appl. Phys. Lett. 92, 022505\n(2008).\n23Y.-T. Chen, S. Takahashi, H. Nakayama, M. Althammer,\nS. T. B. Goennenwein, E. Saitoh and G. E. W. Bauer,Phys. Rev. B 87, 144411 (2013).\n24Y. M. Lu, J. W. Cai, S. Y. Huang, D. Qu, B. F. Miao, and\nC. L. Chien, Phys. Rev. B 87, 220409 (2013).\n25V. V. Kruglyak, S. O. Demokritov, and D. Grundler, J.\nPhys. D. Appl. Phys. 43, 264001 (2010).\n26R. Urban, G. Woltersdorf, and B. Heinrich, Phys. Rev.\nLett. 87, 217204 (2001).\n27S. Mizukami and Y. Ando and T. Miyazaki, J. Mag. Mag.\nMater. 239, 42 (2002).\n28Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. Lett. 88, 117601 (2002).\n29A. Brataas, Y. Tserkovnyak, G. E. W. Bauer, and B. I.\nHalperin, Phys. Rev. B 66, 060404 (2002).\n30Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I.\nHalperin, Rev. Mod. Phys. 77, 1375 (2005).\n31A. Kapelrud and A. Brataas, Phys. Rev. Lett. 111, 097602\n(2013).\n32B. Heinrich, Y. Tserkovnyak, G. Woltersdorf, A. Brataas,\nR. Urban and G. E. W. Bauer, Phys. Rev. Lett. 90, 187601\n(2003).\n33B. A. Kalinikos and A. N. Slavin, J. Phys. C 19, 7013\n(1986).\n34A. A. Serga, A. V. Chumak and B. Hillebrands, J. Phys.\nD: App. Phys. 43, 264002 (2010).\n35M. Vohl, J. Barna\u0013 s and P. Gr unberg, Phys. Rev. B 39,\n12003 (1989).\n36B. Heinrich, private communication.\n37A. Brataas, Yu. V. Nazarov, and G. E. W. Bauer, Phys.\nRev. Lett. 84, 2481 (2000).\n38M. Althammer, S. Meyer, H. Nakayama, M. Schreier, S.\nAltmannshofer, M. Weiler, H. Huebl, S. Gepr ags, M. Opel,\nR. Gross, D. Meier, C. Klewe, T. Kuschel, J.-M. Schmal-\nhorst, G. Reiss, L. Shen, A. Gupta, Y.-T. Chen, G. E. W.\nBauer, E. Saitoh and S. T. B. Goennenwein, Phys. Rev. B\n87, 224401 (2013).\n39H. Skarsv\u0017 ag, G. E. W. Bauer, and A. Brataas,\narXiv:1405.2267v1.\n40J. Xiao, G. E. W. Bauer, K.-C. Uchida, E. Saitoh, and S.\nMaekawa, Phys. Rev. B 81, 214418 (2010).\n41J. C. Slonczewski, Phys. Rev. B 82, 054403 (2010).\n42J. Xingtao, L. Kai, K. Xia, and G. E. W. Bauer, EPL 96,\n17005 (2011).\n43A. Kapelrud and A. Brataas, (unpublished).\n44Y. Zhou, H.J. Jiao, Y.T. Chen, G. E. W. Bauer, and J.\nXiao, Phys. Rev. B 88, 184403 (2013).\n45Z. Q. Qiu , J. Pearson and S. D. Bader Phys. Rev. B 46,\n8659 (1992).\n46J.-M.L. Beaujour, W. Chen, A. D. Kent and J. Z. Sun, J.\nAppl. Phys. 99, 08N503 (2006).\n47M. B. Jung\reisch, V. Lauer, R. Neb, A. V. Chumak and\nB. Hillebrands, Appl. Phys. Lett. 103, 022411 (2013).\n48D. Giancoli, \"25. Electric Currents and Resistance\". In Jo-\ncelyn Phillips. Physics for Scientists and Engineers with\nModern Physics (4th ed.), (2009) [1984].13\nAppendix A: Dipole Tensor\nThe dipole tensor in the \u0010\u0011\u0018coordinate system, ^G(\u0018)\nfrom Eq. (9) can be rotated by the xyzcoordinate system\nwith the rotation matrix\nR=0\n@s\u0012\u0000c\u0012s\u0012\u0000c\u0012c\u001e\n0c\u001e\u0000s\u001e\nc\u0012s\u0012s\u001es\u0012c\u001e1\nA; (A1)where we have introduced the shorthand notation s\u0012\u0011\nsin\u0012,c\u0012\u0011cos\u0012and so on. We then get that\n(*\n^Gxyz=R^GRT\n=0\n@s2\n\u0012G\u0018\u0018\u0000c\u001es2\u0012G\u0018\u0010+c2\n\u0012c2\n\u001eG\u0010\u0010\u0000s\u001es\u0012G\u0018\u0010+s\u001ec\u001ec\u0012G\u0010\u0010s\u0012c\u0012G\u0018\u0018\u0000s\u0012c\u0012c2\n\u001eG\u0010\u0010+c\u001e(s2\n\u0012\u0000c2\n\u0012)G\u0018\u0010\n\u0000s\u001es\u0012G\u0018\u0010+s\u001ec\u001ec\u0012G\u0010\u0010 s2\n\u001eG\u0010\u0010 \u0000s\u001ec\u0012G\u0018\u0010+s\u001es\u0012c\u001eG\u0010\u0010\ns\u0012c\u0012G\u0018\u0018\u0000s\u0012c\u0012c2\n\u001eG\u0010\u0010+c\u001e(s2\n\u0012\u0000c2\n\u0012)G\u0018\u0010\u0000s\u001ec\u0012G\u0018\u0010+s\u001es\u0012c\u001eG\u0010\u0010c2\n\u0012G\u0018\u0018+s2\u0012c\u001eG\u0018\u0010+c2\n\u001es2\n\u0012G\u0010\u00101\nA:\n(A2)\nBecause we work in the linear respons regime the equilibrium magnetization should be orthogonal to the dynamic\ndeviation, mi\u0001^z= 0, it is therefor su\u000ecient to only keep the xypart of ^Gxyz. We then \fnd\n^Gxy=\u0012s2\n\u0012G\u0018\u0018\u0000c\u001es2\u0012G\u0018\u0010+c2\n\u0012c2\n\u001eG\u0010\u0010\u0000s\u001es\u0012G\u0018\u0010+s\u001ec\u001ec\u0012G\u0010\u0010\n\u0000s\u001es\u0012G\u0018\u0010+s\u001ec\u001ec\u0012G\u0010\u0010 s2\n\u001eG\u0010\u0010\u0013\n: (A3)\nAppendix B: Spin Accumulation\nThe functions \u0000 1(\u0018) and \u0000 2(\u0018) are taken directly from\nRef.39, and modi\fed to cover the more complicated mag-\nnetic texture model. We then have\n\u00001(\u0018)\u0011cosh\u0010\n\u0018=~lsf\u0011\ncosh\u0010\n\u0018=~lsf\u0011\n+\u001bsinh\u0010\n\u0018=~lsf\u0011\n=2g?~lsf;\n\u00002(\u0018)\u0011sinh\u0010\n\u0018=~lsf\u0011\nsinh\u0010\n\u0018=~lsf\u0011\n+\u001bcosh\u0010\n\u0018=~lsf\u0011\n=2g?~lsf:(B1)\nForQlsf\u001d1 the e\u000bective spin di\u000busion length becomes\nshort, \u0000 1!1 and \u0000 2!0 at the FIjN interfaces.\nAppendix C: E\u000bective spin di\u000busion length\nThe di\u000busion in the NM reads\n@t\u0016S=Dr2\u0016S\u00001\n\u001csf\u0016S; (C1)whereDis the di\u000busion constant and \u001csfis the spin \rip\nrelaxation time. We assume that the FMR frequency is\nmuch smaller than the electron traversal time, D=d2\nN, and\nthe spin-\rip relaxation rate, 1 =\u001csf.39This means the LHS\nof Eq. (C1) can be disregarded. In linear response the\nspin accumulation, which is a direct consequence of spin\npumping, must be proportional to the rate of change of\nmagnetization at the FI jN interfaces. We do the same\nFourier transform, as for the magnetization, so that \u0016\u0018\nexpfi(!t\u0000Q\u0010)g. The spin di\u000busion equation then takes\nthe form\n@2\n\u0018\u0016S=\u0012\nQ2+1\nD\u001csf\u0013\n\u0016S: (C2)\nThe spin di\u000busion length is then lsf=pD\u001csf, and\nby introducing the e\u000bective spin di\u000busion length ~lsf=\nlsf=q\n1 + (Qlsf)2one gets\n@2\n\u0018\u0016S=1\n~l2\nsf\u0016S: (C3)"    },    {        "title": "2008.09857v1.Pseudospin_resonances_reveal_synthetic_spin_orbit_interaction.pdf",        "content": "Pseudospin resonances reveal synthetic spin-orbit interaction\nChristoph Rohrmeier∗and Andrea Donarini\nInstitute of Theoretical Physics, University of Regensburg, 93053 Regensburg, Germany\n(Dated: August 25, 2020)\nWe investigate a spin-full double quantum dot (DQD) coupled to the leads in a pseudospin valve\nconﬁguration. The interplay of interaction and interference produces in the stability diagram a rich\nvariety of resonances, modulated by the system parameters. In presence of ferromagnetic leads and\npseudospin anisotropy, those resonances split, turn into dips and acquire a Fano shape thus revealing\na synthetic spin-orbit coupling induced on the DQD. A set of rate equations derived for a minimal\nmodel captures those features. The model accurately matches the numerical results obtained for the\nfull system in the framework of a generalized master equation and calculated within the cotunneling\napproximation.\nQuantum dots (QDs) are characterized by a charging\nenergy and by a discrete energy spectrum, both originat-\ning from the spatial conﬁnement of their electronic wave-\nfunctions. The many-body spectrum of QDs is probed\nin great detail by coupling them weakly to metallic leads\nand measuring their transport characteristics as a func-\ntion of bias and gate voltage. The sequential tunneling\n(ST) of electrons, hopping from source to drain through\nthe dots, typically produces a diﬀerential conductance\nwith Coulomb diamonds decorated by parallel resonant\nlines which are the spectroscopic signatures of the charg-\ning energy and the discrete many-body spectrum.\nDegeneracies [1] enrich the ST dynamics with inter-\nference eﬀects. The latter originate from the coher-\nent superposition of the degenerate states, which arise\nin this coherent-sequential-tunneling (CST) regime and\nare modulated by the external parameters like the bias\nand gate voltage. For a spin-full level coupled to non-\ncollinearly polarized ferromagnetic leads (a QD spin\nvalve), interference between the degenerate spin states in-\nduces spin accumulation, precession and relaxation, with\na resulting non-equilibrium spin polarization of the dot\n[2–9] and spin torque on the leads [10].\nFor a QD spin valve with almost antiparallel lead po-\nlarization, a novel spin resonance has been predicted [11]\nwithin the one-particle Coulomb diamond. A crucial role\nin this phenomenon is played by the exchange magnetic\nﬁeld [2] generated by virtual electronic charge ﬂuctua-\ntions between the dot and the leads, i.e. the Lamb shift\ncorrection to the dot Hamiltonian. Also, orbitally degen-\nerate states support naturally interference if combined\nwith couplings to the leads which mix the tunneling chan-\nnels [12, 13], as it has been demonstrated for semiconduc-\ntor wires [14, 15], QD molecules [16–21], single-molecule\njunctions [19, 22, 23] and suspended carbon nanotubes\n(CNTs) [24]. The control of a QD spin valve is a paradig-\nmatic example of spintronics .Valleytronics concerns in-\nstead the manipulation of a state living in a twofold or-\nbitally degenerate space. Very recently, this concept has\nbeen further extended to the one of ﬂavortronics [25], for\ninteracting systems with n-fold degeneracy.\nIn this Letter, we investigate the interplay between\nFIG. 1. Schematic setup of a DQD in a pseudospin valve\nconﬁguration: The left/right lead (L/R) is more strongly cou-\npled to the bottom/top dot (B/T). The angle θ/lessorsimilarπbetween\nthe pseudospin polarization of the leads ensures the mixing\nof the pseudospin states. A bias voltage ( Vb) applied to the\nleads and a gate voltage ( Vg) control the transport character-\nistics of the DQD. The blue arrows indicate the parallel spin\npolarization of the leads.\nvalleytronics and spintronics, between the pseudospin of\na DQD with orbital degeneracy and the spin polariza-\ntion of the ferromagnetic leads. The spatial decay of the\nCoulomb interaction implies a pseudospin anisotropy on\nthe DQD. In presence of ferromagnetic leads, synthetic\nspin-orbit interaction emerges. The latter intertwines the\nspin and the pseudospin degrees of freedom and is re-\nvealed by a set of resonances in the stability diagram,\nwhich split, turn into dips and acquire a Fano shape by\nchanging the spin polarization of the leads.\nModel - The spin-full DQD coupled to ferromagnetic\nleads schematically shown in Fig. 1 is described by the\nsystem-bath Hamiltonian: H=HB+HS+HT. The\nbath component reads HB=/summationtext\nlσkεlσkc†\nlσkclσkwhere\nl= L/R labels the left/right lead, σthe spin index and\nk the momentum both in the lead energy level εlσkas\nwell as in the operators clσk. The system Hamiltonian\nHS=/summationtext\nr[(eVg+ε∗)nr+Unr(nr−1)/2] +Vntopnbot,\nin whichnrcounts the electron number on the top or\nbottom dot, contains the on-site energy ε∗shifted by a\ngate voltage Vgas well asUandV, respectively the local\nand the inter-dot Coulomb interaction. The pseudospin\nformulation of the system Hamiltonian (cf. Supplemen-\ntal Material) is characterized by a pseudospin anisotropy\nproportional to U−V, essential for the synthetic spin-arXiv:2008.09857v1  [cond-mat.mes-hall]  22 Aug 20202\norbit eﬀects described below. The tunneling Hamiltonian\nHT=/summationtext\nlσkntl,nc†\nlσkdn+ h.c. combines via the tunneling\namplitudes tl,nthe bath operators with system operators\ndnwherenlabels a single-particle basis for the DQD.\nThe CST dynamics of a system with quasi-degenerate\nmany-body spectrum is expressed in terms of tunneling\nrate matrices [26, 27]. The latter are deduced from HTas\n(Γl)nm= 2π//planckover2pi1/summationtext\nlσkt∗\nl,ntl,mδ(ε−εlσk) and they factorize,\nin absence of intrinsic spin-orbit coupling, into a spin (s)\nand an orbital (o) or pseudospin component:\nΓl= Γ0\nl(112+Psns\nl·σ)⊗(112+Pono\nl·σ) (1)\nwhere Γ0\nlis the bare tunneling rate for the l-lead,Ps(o)\nandns(o)\nlare the strength and the direction vector of the\nspin (pseudospin) polarization of the lead and σis the\nvector of the Pauli matrices σx,σyandσz.\nWe choose parallel spin and almost antiparallel pseu-\ndospin directions no\nL/R=/parenleftbig\ncosθ\n2,0,∓sinθ\n2/parenrightbig\nwithθ=\n0.95π(cf. Fig. 1). Moreover, we consider high pseudospin\npolarizations ( Po≈1) to achieve an essentially closed\npseudospin valve [11]. Comparable pseudospin polariza-\ntion strengths have been observed recently in suspended\nCNTs [24].\nMethods - The transport characteristics are calculated\nwith two complementary approaches. On the one side,\na next-to-leading-order expansion in the tunneling cou-\npling is performed using a generalized master equation.\nTo this end, the kinetic equation for the reduced density\nmatrixρred= Tr B{ρ}, i.e. the trace over the bath of\nthe total density matrix, is obtained with the Nakajima-\nZwanzig projector technique [28, 29]. The steady state is\ndeﬁned by ˙ ρ∞\nred= 0 = (LS+K)ρ∞\nredwhere the Liouville\nsuperoperator, in general deﬁned as Lρ=−i\n/planckover2pi1[H,ρ], is\ntaken here with respect of HS. The Kernel superopera-\ntorKreads\nKρ∞\nred= Tr B/braceleftBigg\nLT∞/summationdisplay\nn=0/parenleftBig\n˜G0QLTQ/parenrightBig2n˜G0LTρ∞\nred⊗ρB/bracerightBigg\n(2)\nwhereQ= 1−P withP= Tr B{•}⊗ρBis a Nakajima-\nZwanzig projector, ρBis the equilibrium density operator\nfor the bath and ˜G0is the Liouville space propagator in\nabsence of tunneling coupling [30–33]. The ﬁrst term of\nthe sum in (2) reproduces the ST regime. We consider\nhere a truncation up to the cotunneling regime ( n= 1; cf.\nSupplemental Material). Eventually, from the stationary\ndensity matrix ρ∞\nred, we calculate the stationary current\nat leadlasIl= Tr S{Klρ∞\nred}where the current Kernel\nis obtained from the propagator kernel in (2) by chang-\ning the leftmost tunneling Liouvillean with the current\noperator [30–33]. A novel treatment of the cotunneling\nintegrals founded on the work of [30, 32, 34, 35] allowed\nus for the implementation of a transport code which in-\ncludes all coherences necessary to capture the interfer-\nence eﬀects in our system. Moreover, a systematic testof robustness for such eﬀects beyond the ST approxima-\ntion has been achieved.\nIn a complementary approach, we set up a minimal\nmodel in the regime of CST (cf. [25]). As we focus on\nthe resonance between zero and one particle, we restrict\nhere to the coupled dynamics of the populations p0and\npσ(empty and singly occupied DQD with spin σ) com-\nplemented by one of the pseudospin vectors Tσ:\n˙p0=−4γ+p0+/summationdisplay\nσDσ/bracketleftbig\nγ−pσ+ 2γ−·Tσ/bracketrightbig\n, (3)\n˙pσ=Dσ/bracketleftbig\n2γ+p0−γ−pσ−2γ−·Tσ/bracketrightbig\n, (4)\n˙Tσ=Dσ/bracketleftbig\n−γ−Tσ+γ+p0−1\n2γ−pσ/bracketrightbig\n+Bσ×Tσ(5)\nwhereγ±=/summationtext\nlΓ0\nlf±\nl(ε)no\nl,γ±=/summationtext\nlΓ0\nlf±\nl(ε) and\nD↑(↓)= 1±Ps. The Fermi-functions are dependent on\nthe temperature TwithkBthe Boltzmann constant, the\nchemical potential µlof the leadlandp=±which indi-\ncates in- or out-tunneling: fp\nl(ε) = 1/(ep(ε−µl)/(kBT)+1).\nThe term 2γ−·Tσin (3)-(4) ensures the coupling of the\npopulations and the accumulated pseudospin. Three con-\nceptually diﬀerent mechanisms yield the time evolution\nof the pseudospin: the ﬁrst term in (5) describes relax-\nation, accumulation due to changes in the populations\ncharacterizes the following two terms. The last term con-\ntains the spin dependent pseudo exchange ﬁeld Bσwhich,\nanalogously to magnetic ﬁelds, generates pseudospin pre-\ncession. The exchange ﬁeld is deﬁned as\nBσ=/summationdisplay\nl2PoΓ0\nl[Dσ(pl(E1−E0)−pl(E2g−E1))no\nl\n+D¯σ(pl(E2e−E1)−pl(E2g−E1)) (no\nl·ez)ez] (6)\nwithpl(x) = ReΨ(0)(1\n2+i(eVg+x−µl)\n2πkBT) where Ψ(0)(z) is\nthe digamma-function. The subscript of the energy Ex\nlabels the one-particle state (1) and the two-particle ex-\ncited/ground state (2e /2g). It is crucial to include in the\nexchange ﬁeld the two-particle energies containing Uand\nV, even though we do not account for the populations of\nthose states. Also, energy levels far from the ST reso-\nnance (∆E/kBT/greatermuch1) do inﬂuence the exchange ﬁeld\ndue to the logarithmic tails of the digamma-functions.\nResults - In Fig. 2 stability diagrams of a DQD in\nthe cotunneling regime are displayed for several spin po-\nlarizations of the leads. We focus on the one-particle\nCoulomb diamond, highlighted in panel (a) by the dot-\nted white lines. Here we would normally expect an es-\nsentially ﬁxed particle number and, due to Coulomb re-\npulsion, only an exponentially suppressed current. An\nexception to this rule can be clearly seen in panel (a)\nwhere a distinctive resonance, highlighted by the dashed\nblack line, is cutting through the Coulomb diamond. In-\ncreasing the spin polarization Ps(Fig. 2 (b)-(d)) leads to\na splitting of this resonance, marked by the dashed lines.\nIn the upper right corner of Fig. 2 (d), a resonance can be\nobserved even outside the diamond. This transport eﬀect\nis explained by pseudospin resonances in analogy to the3\nFIG. 2. Diﬀerential conductance shows pseudospin resonances in a DQD and is tuned by spin polarization Ps: The one-\nparticle diamond is highlighted by the dotted white lines in panel (a). The three vertical black lines ( ⋆,/squaresolid,/trianglesolid) indicate the bias\ntraces of Fig. 4. The dashed magenta (black) line is the resonance condition of the ↑(↓)-electrons (cf. (7)). The solid white line\nindicates the minimum of Bσ,⊥which matches perfectly a local minimum within the pseudospin resonance. The parameters\nare the following: U= 2V,kBT= 0.05V,Po= 0.99,θ= 0.95π, ΓR= 2.5×10−3V= 2ΓL,ε∗=−2VandW= 250V.\nspin resonances reported in [11]. The pseudospin is asso-\nciated with the orbital degree of freedom of the DQD. In\nour setup, the orbital polarizations of the leads are almost\nantiparallel thus resulting in an almost closed pseudospin\nvalve. The latter is indicated in Fig. 1 by the diﬀerent\nsizes of the arrows connecting the leads and the dots.\nSolely varying the coupling strength would correspond\nto a sweep of the lead polarization along the z-direction.\nPseudospin resonances require, instead, non-collinear or-\nbital polarizations as well as an asymmetry in the bare\ncoupling strength Γ0\nlbetween the right and left lead. The\nlatter shifts the resonance away from the zero bias line\n[11]. The necessary σxorσyorbital polarization of the\nleads translates into non-diagonal Γ l-matrices, which can\nbe interpreted as tunneling to a coherent superposition\nof two diﬀerent orbitals. Experimental evidence of such\ncoherent superpositions for QDs in the weak tunneling\nregime has been reported [14, 24]. In the framework of\n(3)-(5), vectorial resonance conditions can be formulated\nsimilarly to the ansatz in [11, 25]:\nBσ·(no\nL−no\nR) = 0. (7)\nThe spin dependent exchange ﬁeld generates two dis-\ntinct conditions, each determining the position of the\ncorresponding resonance in the Vg-Vb-plane: the magenta\n(black) dashed line in Fig. 2 for the ↑(↓)-electrons. The\naccuracy of (7) in determining the resonance positions re-\nduces asθis chosen further away from antiparallel align-\nment. In contrast to the resonance conditions formulated\nin [11] and in [25], we choose (7), where the drain and the\nsource equally participate, since it matches the numeri-\ncal resonances on a broader parameter range. Despite the\nsubtle diﬀerences, though, all three conditions mentioned\nabove can only predict the position of the resonances, butnot their character . The same resonance condition corre-\nsponds to a dip in the current ( ⋆in Fig. 2), or to a peak\n(/trianglesolid) and even to a Fano-like asymmetric peak-dip ( /squaresolid). Fi-\nnally, the current peak is strongly modulated along the\nsame resonance line and it can even disappear, as ex-\nemplary highlighted in panel (a) of Fig. 2 with the solid\nwhite line. The discovery and explanation of such qual-\nitative diﬀerences in the pseudospin resonances, which\noriginate from the intertwining of spin and pseudospin,\nrepresent the main result presented in this Letter.\nFor a deeper understanding of the numerical data of\nFig. 2, we further elaborate on the equations of motion\nof (3)-(5). Solving (5) in the stationary limit leads to\nTσ=aσ\na2σ+B2σ/parenleftBig\nbσ+Bσ·bσ\na2σBσ+Bσ×bσ\naσ/parenrightBig\nwithaσ=Dσγ−\nandb=Dσ/parenleftbig\nγ+p0−1\n2γ−pσ/parenrightbig\n. By substituting Tσinto\n(3)-(4), the problem is reduced to a set of eﬀective rate\nequations for the populations p0andpσwith the transi-\ntion rates schematically indicated in Fig. 3. The station-\nary current reads, correspondingly,\nIL= 4γ+\nLp0+/summationdisplay\nσDσ/parenleftbig\n−γ−\nLpσ−2γ−\nL·Tσ/parenrightbig\n.(8)\nThe panels (a), (c) and (e) of Fig. 4 show a direct compar-\nison between the absolute value of the current as obtained\nform the full numerical calculation (orange) and the an-\nalytical approach (blue) of (8). In all three cases, the\nanalytical result well reproduces the qualitative behavior\nof the current and the position of its extrema.\nIn a simple physical picture, we expect a peak in the\ncurrent whenever the pseudospin precession caused by\nthe exchange ﬁeld releases the blockade induced by the\npseudospin valve. A dip arises, instead, whenever this\nmechanism is locally suppressed. Both phenomena hap-\npen in close vicinity to the aforementioned resonance con-4\nFIG. 3. Rate scheme of the three populations p0,p↑andp↓:\nThe four arrows indicate the rates between the populations\nwhile their size speciﬁes the strength of them. The dashed\nrates for the minority spin are furthermore lowered by the\nmajority spin polarization of the leads.\ndition (7). Only the analysis of the eﬀective rates repre-\nsented in Fig. 3 allows, though, to distinguish them. The\nincoherent superposition of a minority and majority spin\nchannel yields the current. Its modulation is determined\nby the depopulation rates /epsilon1andη. Thus, as conﬁrmed\nby the resemblance between panels (a) and (b) in Fig. 4,\nthe shape of a↓-resonance, is given by the bottleneck rate\nη=D↓γ−\n1−|γ−|2\n(γ−)21\n1 +B2\n↓,⊥\na2\n↓+B2\n↓,/bardbl\n (9)\nwithB2\n↓,/bardbl= (B↓·γ−)2/|γ−|2andB2\n↓,⊥=B2\n↓−B2\n↓,/bardbl\nthe exchange ﬁeld components parallel and perpendicu-\nlar toγ−. In itself,ηis strongly inﬂuenced by the ratio\nΩ =B2\n↓,⊥/(a2\n↓+B2\n↓,/bardbl) in which the proposed physical ex-\nplanation based on the precession dynamics is encoded.\nIn absence of the perpendicular pseudo magnetic ﬁeld\ncomponent, no precession occurs and the bare pseudospin\nvalve factor|γ−|2/(γ−)2reduces the rate. The other ex-\ntreme is reached when the ratio Ω peaks, therefore sup-\npressing the pseuodspin valve factor. Such phenomenon\nonly occurs if the absolute value of the parallel compo-\nnent|B↓,/bardbl|is minimized, since the dephasing rate a↓is\nproportional to a Fermi-function, which varies smoothly\nwithin the Coulomb diamond.\nThe dashed lines in Fig. 4 show the accuracy of the\nprecession argument in determining the position of the\ncurrent extrema. The rate /epsilon1, obtained by replacing ↓\nwith↑in (9), is used for the panels (c) and (d) of Fig. 4.\nIn Fig. 4 (e), both the suppression and the enhancing of\nthe current appear in close vicinity and form a Fano-like\nline shape. In order to emphasize the rather weak dip,\nwe depicted in Fig. 4 (f) the logarithm of the ratio Ω.\nThe ratio Ω has two extrema which stem from minima\nof the corresponding exchange ﬁeld components |B↓,⊥|\nand|B↓,/bardbl|. Despite its superﬁcial resemblance to a Fano\nresonance, the origin of this peak-dip current resonances\ncannot be ascribed to the interference processes typical\nof Fano resonances, also seen in QD setups [36–40].\nMoreover, the relevance of Ω decreases if aσ/greatermuch|Bσ|,\ni.e. when the dephasing rate exceeds the precession fre-\nquency and the direction of the exchange ﬁeld becomes\nFIG. 4. Eﬀective rate analysis of the bias traces from\nFig. 2 (d): The absolute value of the current shows (a) a dip\nat eVg= 1.9V, (c) a peak at e Vg= 1.8Vand (e) a Fano-like\nshape at eVg= 1.58V. The analytic solution of the eﬀective\nST model is depicted in blue whereas the orange line shows\nthe full cotunneling calculations. The black (red) dashed lines\nindicate the position of the minimum of |Bσ,⊥|(|Bσ,/bardbl|) and\ncorrespond to a minimum (maximum) of the current. (b)\nThe rateηstrongly correlates to the current. (d) The absolute\nvalue of the spin of our system |S|is following the trend of the\ncurrent. (f) The logarithm of the ratio Ω = B2\n↓,⊥//parenleftbig\na2\n↓+B2\n↓,/bardbl/parenrightbig\nhighlights the two extrema of Ω which result in a peak and a\ndip in the current.\nirrelevant for the transport. Thus, no resonances appear\non the left upper corner in correspondence to the black\nand magenta dashed lines of the panels (a)-(d) of Fig. 2\neven if they would be predicted by the resonance condi-\ntion (7).\nConclusion - A DQD weakly coupled to ferromagnetic\nleads in pseudospin valve conﬁguration is characterized\nby a rich variety of pseudospin resonances. They decorate\nthe Coulomb diamonds with novel features which range\nfrom a peak to a dip to a Fano shape in the current.\nThese transport characteristics reveal the synthetic spin-\norbit interaction induced on the system by the interplay\nof leads polarization and pseudospin anisotropy on the\nDQD.5\nThe cotunneling calculations ensure the robustness of\nsuch eﬀect beyond the CST limit. Moreover, with the\nhelp of a minimal model, we give an accurate physical\npicture of the resonances and relate their position and\ncharacter to a precession dynamics which modulates the\npseudospin valve eﬀect. The generality of the model al-\nlows for its applicability to the wide class of nanoscale\njunctions with orbital degeneracy, including e.g. single-\nmolecules junctions or CNT-QDs. Particularly, coherent\npopulation trapping and signatures of pseudospin preces-\nsion have been recently demonstrated in a CNT with a\ntunneling coupling similar to the one proposed here [24].\nAcknowledgments - The authors acknowledge ﬁnancial\nsupport from the Elite Netzwerk Bayern via the IGK\nTopological Insulators and the Deutsche Forschungsge-\nmeinschaft via the SFB 1277 (subprojects B02 and B04).\nWe thank moreover M. Grifoni for fruitful discussions.\n∗christoph.rohrmeier@ur.de\n[1] “Eﬀectively degenerate are all states with an energy sepa-\nration smaller than the tunnelling induced broadening.”.\n[2] J. K¨ onig and J. Martinek, Phys. Rev. Lett. 90, 166602\n(2003).\n[3] M. Braun, J. K¨ onig, and J. Martinek, Phys. Rev. B 70,\n195345 (2004).\n[4] S. Braig and P. W. Brouwer, Phys. Rev. B 71, 195324\n(2005).\n[5] W. Rudzi´ nski, J. Barna´ s, R. ´Swirkowicz, and\nM. Wilczy´ nski, Phys. Rev. B 71, 205307 (2005).\n[6] I. Weymann, J. K¨ onig, J. Martinek, J. Barna´ s, and\nG. Sch¨ on, Phys. Rev. B 72, 115334 (2005).\n[7] R. Hornberger, S. Koller, G. Begemann, A. Donarini,\nand M. Grifoni, Phys. Rev. B 77, 245313 (2008).\n[8] L. Y. Zhang, C. Y. Wang, Y. G. Wei, X. Y. Liu, and\nD. Davidovi´ c, Phys. Rev. B 72, 155445 (2005).\n[9] K. Hamaya, M. Kitabatake, K. Shibata, M. Jung,\nM. Kawamura, K. Hirakawa, T. Machida, T. Taniyama,\nS. Ishida, and Y. Arakawa, Appl. Phys. Lett. 91, 022107\n(2007).\n[10] N. M. Gergs, S. A. Bender, R. A. Duine, and\nD. Schuricht, Phys. Rev. Lett. 120, 017701 (2018).\n[11] M. Hell, B. Sothmann, M. Leijnse, M. R. Wegewijs, and\nJ. K¨ onig, Phys. Rev. B 91, 195404 (2015).\n[12] D. Darau, G. Begemann, A. Donarini, and M. Grifoni,\nPhys. Rev. B 79, 235404 (2009).\n[13] M. G. Schultz, Phys. Rev. B 82, 155408 (2010).\n[14] H. A. Nilsson, O. Karlstr¨ om, M. Larsson, P. Caroﬀ, J. N.\nPedersen, L. Samuelson, A. Wacker, L.-E. Wernersson,\nand H. Q. Xu, Phys. Rev. Lett. 104, 186804 (2010).[15] O. Karlstr¨ om, J. N. Pedersen, P. Samuelsson, and\nA. Wacker, Phys. Rev. B 83, 205412 (2011).\n[16] J. K¨ onig and Y. Gefen, Phys. Rev. B 65, 045316 (2002).\n[17] B. Michaelis, C. Emary, and C. W. J. Beenakker, Euro-\nphys. Lett. 73, 677 (2006).\n[18] S. Gustavsson, R. Leturcq, M. Studer, T. Ihn, K. Ensslin,\nD. C. Driscoll, and A. C. Gossard, Nano Lett. 8, 2547\n(2008).\n[19] A. Donarini, G. Begemann, and M. Grifoni, Phys. Rev.\nB82, 125451 (2010).\n[20] T. Hatano, T. Kubo, Y. Tokura, S. Amaha, S. Teraoka,\nand S. Tarucha, Phys. Rev. Lett. 106, 076801 (2011).\n[21] M. Niklas, A. Trottmann, A. Donarini, and M. Grifoni,\nPhys. Rev. B 95, 115133 (2017).\n[22] M. H. Hettler, W. Wenzel, M. R. Wegewijs, and\nH. Schoeller, Phys. Rev. Lett. 90, 076805 (2003).\n[23] G. Begemann, D. Darau, A. Donarini, and M. Grifoni,\nPhys. Rev. B 77, 201406 (2008).\n[24] A. Donarini, M. Niklas, M. Schafberger, N. Paradiso,\nC. Strunk, and M. Grifoni, Nat. Commun. 10, 381\n(2019).\n[25] M. T. Maurer, J. Knig, and H. Schoeller, (2020),\narXiv:2004.05661.\n[26] A. Donarini, Many-Body Methods for Real Materials ,\nedited by E. Pavarini, E. Koch, and S. Zhang, Vol. 9\n(Verlag des Forschungszentrum J¨ ulich, J¨ ulich, 2019) pp.\n393–419.\n[27] “In another context, these matrices are called tunnelling\nself-energies.”.\n[28] S. Nakajima, Prog. Theor. Phys. 20, 948 (1958).\n[29] R. Zwanzig, J. Chem. Phys. 33, 1338 (1960).\n[30] S. Koller, M. Grifoni, M. Leijnse, and M. R. Wegewijs,\nPhys. Rev. B 82, 235307 (2010).\n[31] M. Leijnse, Transport spectroscopy and control of molec-\nular quantum dots (2010).\n[32] S. Koller, Spin phenomena and higher order eﬀects in\ntransport across interacting quantum-dots (2010).\n[33] M. Niklas, Current and noise properties of interacting\nnanojunctions (2018).\n[34] D. Mantelli, Analytical and numerical study of quantum\nimpurity systems in the intermediate and strong coupling\nregimes (2016).\n[35] M. Milgram, (2004), arXiv:math/0406338.\n[36] J. Bara´ nski, T. Zienkiewicz, M. Bara´ nska, and K. J.\nKapcia, Sci. Rep. 10, 2881 (2020).\n[37] M. E. Torio, K. Hallberg, S. Flach, A. E. Miroshnichenko,\nand M. Titov, Eur. Phys. J. B 37, 399 (2004).\n[38] A. B¨ arnthaler, S. Rotter, F. Libisch, J. Burgd¨ orfer,\nS. Gehler, U. Kuhl, and H.-J. St¨ ockmann, Phys. Rev.\nLett. 105, 056801 (2010).\n[39] Y. Joe, J. Kim, E. Hedin, R. Cosby, and A. Satanin, J.\nComput. Electron. 4, 129 (2005).\n[40] A. C. Johnson, C. M. Marcus, M. P. Hanson, and A. C.\nGossard, Phys. Rev. Lett. 93, 106803 (2004).Supplemental Material: Pseudospin resonances reveal synthetic spin-orbit interaction\nChristoph Rohrmeier∗and Andrea Donarini\nInstitute of Theoretical Physics, University of Regensburg, 93053 Regensburg, Germany\n(Dated: August 25, 2020)\nPSEUDOSPIN ANISOTROPY\nThe interference eﬀects presented in the main text are inherently tied to the anisotropy of the pseudospin in the\ndouble quantum dot. In this section, we reformulate the system Hamiltonian to highlight such anisotropy. The system\nHamiltonian is deﬁned as HS=/summationtext\nr[(eVg+ε∗)nr+Unr(nr−1)/2] +Vntopnbotwherenr=/summationtext\nσd†\nrσdrσrepresents\nthe number operator for the r-dot withdrσthe annihilation operator for an electron on the top or bottom dot with\nspinσ,Uthe local Coulomb repulsion, Vthe inter-site Coulomb repulsion, ε∗the on-site energy, Vgthe gate voltage\nand e the elementary charge. The collective index nof the system is here explicitly written in terms of its orbital and\nspin components randσ. We want to express this Hamiltonian in terms of the pseudospin of the system. We use\ntherefore the following deﬁnitions for the total number operator N=/summationtext\nrnrand thez-component for the pseudospin\nTz=/summationtext\nσrr/prime1\n2d†\nr/primeσ(σz)rr/primedrσwhereσzis thez-Pauli matrix:\nntop=ntop+nbot\n2+ntop−nbot\n2:=N\n2+Tz, (1)\nnbot=ntop+nbot\n2−ntop−nbot\n2:=N\n2−Tz (2)\nto obtain\nHS=/parenleftbigg\n¯ε−U\n2/parenrightbigg\nN+U+V\n4N2+ (U−V)T2\nz (3)\nwhere ¯ε= eVg+ε∗. In this representation of the Hamiltonian, it is evident that the diﬀerence of the local and\ninter-site Coulomb repulsion translates into an easy-plane anisotropy of the pseudospin, i.e. it is energetically more\nfavorable for the pseudospin vector to point in the σx-σy-plane than to point in the σz-direction where one has to pay\nextra energy to localize the electrons on one dot. Interestingly, a top-bottom tunneling tin the Hamiltonian could be\nseen in the framework of pseudospin as a pseudo-magnetic ﬁeld:\nHS=/parenleftbigg\n¯ε−U\n2/parenrightbigg\nN+U+V\n4N2+ (U−V)T2\nz+Bt·T (4)\nwithBt,x= 2Ret,Bt,y= 2ImtandBt,z= 0 associated to a top-bottom tunnelling amplitude t. This tunnelling\nprocess would lift the orbital degeneracy of our system and thus destroy the pseudospin resonances if the magnitude\nof such Zeeman-like splitting is big enough. We argue that a small hopping |t|</planckover2pi1Γ is not detrimental as the coupling\nto the leads Γ can not resolve the lifted degeneracies. Therefore, one still expects interference eﬀects to appear and\nthe pseudo-magnetic ﬁeld Btwould simply add to the exchange one generated by the tunnelling to the leads.\nLAMB SHIFT HAMILTONIAN\nIn this section, we deduce the Lamb shift Hamiltonian for the one-particle subspace with the help of the reformulated\nsystem Hamiltonian. The α-component of the spin Sand the pseudospin Toperators in the one-particle subspace\nare deﬁned by\n(P1SαP1)rσ,r/primeσ/prime=1\n2δrr/prime(σα)σσ/prime, (5)\n(P1TαP1)rσ,r/primeσ/prime=1\n2δσσ/prime(σα)rr/prime (6)\nwhereσαare the Pauli matrices and Px=/summationtext\n/lscriptσ|/lscriptσ/angbracketright/angbracketleft/lscriptσ|are the projector operators of the x-particle subspace. For\nexampleP1runs over the four states |top↑/angbracketright,|bot↑/angbracketright,|top↓/angbracketrightand|bot↓/angbracketright. The principal values stemming from thearXiv:2008.09857v1  [cond-mat.mes-hall]  22 Aug 20202\nenergy integration are denoted in the following by pl(x) = ReΨ(0)(1\n2+i(eVg+x−µl)\n2πkBT) where Ψ(0)(z) is the digamma-\nfunction. The Lamb shift Hamiltonian reads then\nHLS=/summationdisplay\nlrσr/primeσ/prime(Γl)rσ,r/primeσ/primeP1/bracketleftbig\nd†\nrσpl(E1−HS)dr/primeσ/prime+dr/primeσ/primepl(HS−E1)d†\nrσ/bracketrightbig\nP1. (7)\nWe insert the system Hamiltonian (cf. (3)), and perform a Taylor expansion with respect to the anisotropy component.\nExploiting the relation P2T2\nzP2=/parenleftbig\nP2T2\nzP2/parenrightbignforn≥1, we can simplify the terms containing T2\nz:\nP2pl/parenleftbig\n¯ε+V−µl+ (U−V)T2\nz/parenrightbig\nP2=P2pl(¯ε+V−µl) +∞/summationdisplay\nn=11\nn!p(n)\nl(¯ε+V−µl) (U−V)nP2T2\nzP2\n=P2/bracketleftbig\npl(¯ε+V−µl) +T2\nz(pl(¯ε+U−µl)−pl(¯ε+V−µl))/bracketrightbig\nP2\n=P2/bracketleftbig\npl(E2g−E1) +T2\nz(pl(E2e−E1)−pl(E2g−E1))/bracketrightbig\nP2. (8)\nWe further simplify the term containing T2\nzusing the relation\nP1dr/primeσ/primeT2\nzd†\nrσP1=1\n2P1dr/primeσ/primed†\nrσP1+/summationdisplay\nk(σz)krP1dr/primeσ/primed†\nkσTzP1. (9)\nSome algebra leads eventually to the following formulation of the Lamb shift Hamiltonian, obtained under the addi-\ntional assumption of parallel spin polarisation of the leads:\nHLS=/summationdisplay\nlΓ0\nl[pl(E1−E0) + 2pl(E2g−E1) +pl(E2e−E1)]P1\n+/summationdisplay\nlΓ0\nl[pl(E1−E0)−pl(E2g−E1)]Psns\nl·P1SP1\n+/summationdisplay\nl(1 +Ps)Γ0\nl(pl(E1−E0)−pl(E2g−E1))Po\nlno\nl·P1TP1\n+/summationdisplay\nl(1−Ps)Γ0\nl(pl(E2e−E1)−pl(E2g−E1))Po\nl(no\nl)zP1TzP1. (10)\nThe ﬁrst two terms do not contribute to the time evolution of the reduced density matrix in the one-particle subspace,\nas the parallel spin polarization deﬁnes a common quantization axis. Finally, we transform the Lamb shift induced\ndynamics for the reduced density matrix into equations of motion for the pseudospins:\n/parenleftbig\n˙ρ∞\nred,1/parenrightbig\nLS=i\n2π/bracketleftbig\nHLS,ρ∞\nred,1/bracketrightbig\n⇐⇒/parenleftBig\n˙Tσ/parenrightBig\nLS=Bσ×Tσ. (11)\nAssuming constant interaction ( U=V), the energies E2eandE2gcoincide. This implies that also the last term of\n(10) vanishes. Consequently, only one pseudo spin resonance is present in the stability diagram (cf. Fig. 1).\nTRANSPORT THEORY\nWe outline in this section the transport theory used for the calculation of the cotunneling current presented in\nFig. 2 and Fig. 4 of the main text. The starting point of the derivation is the Liouville-von Neumann equation which,\nwritten in terms of the Liouville superoperator L(t), reads:\nL(t)X:=−i\n/planckover2pi1[H(t),X] =⇒˙ρ(t) =L(t)ρ(t) (12)\nwhereH(t) is the - in general - time-dependant Hamiltonian and ˙ ρ(t) is time-derivative of our density matrix. The\nLiouville superoperator acts on a generic matrix Xin analogy to a matrix which acts on a vector. In the case of the\ntime-independent Hamiltonian Hwhich we consider in this Letter, the Liouvillian can be written as L=LB+LS+LT.\nBased on (12), we can deduce a generalized master equation (GME) which can accurately describe the time evolution\nof our system while rigorously taking into account not only the populations of the underlying density matrix but also\nall its coherences. Only then it is possible to capture interference eﬀects in all their facets. The method of choice to3\nFIG. 1. Diﬀerential conductance for Ps= 0.99 and constant interaction ( U=V): The stability diagram of a double quantum\ndot whereU=V= 1 shows only one pseudospin resonance in comparison with Fig. 2 (d) of the main text where two resonances\nfor the diﬀerent spin species appear. The plot should highlight the fact that the pseudospin anisotropy is the main cause for the\nsynthetic spin-orbit eﬀect. Changing the spin polarisation at constant interaction does not alter the diﬀerential conductance\nin agreement with (7) of the main text. The calculation is performed with the same cotunneling code as in Fig. 2 of the main\ntext. The chosen parameters are the following: U= 1V,kBT= 0.05V,Po= 0.99,θ= 0.95π, ΓR= 2.5×10−3V= 2Γ L,\nε∗=−2VandW= 250V.\nderive a GME is the Nakajima-Zwanzig projection operator technique [1, 2]. The main idea of Nakajima and Zwanzig\nis to split the total density into two parts: one where the quantum dot system and the leads are separated ( Pρtot)\nand one where the entanglement of the quantum dot system and the leads is captured ( Qρtot). This is done by the\ntwo projectors\nPX:= Tr B{X}⊗ρB, (13)\nQX:= (1−P)X (14)\nwithXan arbitrary density matrix and Tr Bis the trace over the bath. The Nakajima-Zwanzig equation reads\nP˙ρ(t) =LSPρ(t) +/integraldisplayt\n0dsK(t−s)Pρ(s) (15)\nwith the Kernel superoperator K(t) =PLT¯GQ(t)LTP. The propagator for the entangled part is deﬁned as ¯GQ(t) =\ne(LS+LB+QLTQ)t. The Nakajima-Zwanzig equation is so far exact to all orders in the tunneling Hamiltonian HTand\nthe Markovian approximation is not performed so that the time evolution, not only the steady-state, is captured\nexactly via the propagator ¯GQ(t). In this Letter, however, only the steady-state is of interest. The steady-state of\nthe reduced density matrix is reached at an inﬁnite time and is per deﬁnition ρred(t→∞ ) = Tr B{ρ(t→∞ )}:=ρ∞\nred.\nThe steady-state of the reduced density matrix allows us to calculate the expectation value of any observable Oof\nthe system in the steady-state like the current or the spin: /angbracketleftO/angbracketright∞= Tr{ρ∞O}= Tr S{ρ∞\nredO}. With the help of a\nLaplace transformation, the convolutive form of the Kernel and the ﬁnal value theorem, we can simplify (15) to\n˙ρ∞\nred= 0 =Lρ∞\nred= (LS+K)ρ∞\nred (16)\nwith\nKρ∞\nred= Tr B/braceleftBigg\nLT∞/summationdisplay\nn=0/parenleftBig\n˜G0QLTQ/parenrightBig2n˜G0LTρ∞\nred⊗ρB/bracerightBigg\n. (17)\n˜G0= limλ→0+˜G0(λ) = limλ→0+[λ−LS−LB]−1is the free propagator of the system and the bath in Laplace space.\nIt should be noted that the limit lim λ→0+should be performed at the very end of the calculation and not in the4\nfree propagator alone. In the following, this fact that the limit still has to be performed is marked with 0+in the\npropagators. The form of the propagator ˜GQ(λ) stems from a Dyson equation. A perturbative approximation is only\nvalid in the so-called weak coupling limit where the tunneling rate is small compared to the temperature ( /planckover2pi1Γ/lessmuchkBT).\nSequential tunneling\nConsidering the full Kernel from (17) to the lowest non-vanishing order, K=K(2)+O(H4\nT), we get the following\nsequential tunneling Kernel\nK(2)ρ∞\nred= Tr B/braceleftbigg\nLT1\n0+−LS−LBLTρ∞\nred⊗ρB/bracerightbigg\n. (18)\nThis Kernel will be also called second order Kernel due to the appearance of two Louvillians LTand therefore denoted\nby the superscript ”(2)”. To simplify the notation of a superoperator X, let us introduce the parameter αwhich is\ndeﬁned by [ X,ρ] =Xρ−ρX:=X+ρ−X−ρ=/summationtext\nααXαρ. Using this notation for LTyields\nLTX=−i\n/planckover2pi1/summationdisplay\np=±\nα=±/summationdisplay\nlσk\nnpcp,α\nlσkt¯p\nl,nd¯p,α\nnX. (19)\nIn order to simplify the notation, we introduce the index p. For example, the superscript p= + indicates the conjugate\ntranspose of the matrix, d+\nn:=d†\nn, or in the case of the tunneling amplitudes, the complex of it, t+\nl,n:=t∗\nl,n. The\nsuperscript p=−indicates an annihilation operator, d−\nn:=dn, or a bare tunneling amplitude t−\nl,n:=tl,nwith the\nproperty of the p-index: ¯p:=−p. Furthermore, the operators transform to Liouville space superoperators like cp\nlσk→\ncp,α\nlσkwhereαindicates an operator which acts from the left ( α= +) or right ( α=−) side. Applying this on the second\norder Kernel of (18) and using the Wick contraction to get the Fermi-function via Tr {c†\nlσkclσkρ}=/angbracketleftn/angbracketright=f+\nl(εlσk) we\nobtain\nK(2)ρ∞\nred=/summationdisplay\nnmp\nlαα/primeαα/prime¯Γp\nl,nmd¯p,α\nndp,α/prime\nmYα\n+(χ)ρ∞\nred. (20)\nIn this short hand notation of the sequential tunneling Kernel several deﬁnitions apply. Firstly, we use now a\ndiﬀerent deﬁnition of the tunneling rate matrix ¯Γp\nl,nm:= 2π//planckover2pi1/summationtext\nσlg0\nσt∗\nl,ntl,mwith thep-index and with the sum over\nσjust running over the constant part of the density of states g0\nσin order to simplify the expressions. Generally,\nwe deﬁne the density of states of the leads as glσ(ε) =ρ0\nσW2/((ε−µl)2+W2) with the chemical potential µlfor\nthel-lead. The Lorentzian cut-oﬀ-function is needed for the energy integral evaluation but we argue that it is also\njustiﬁed to introduce it because in real metals the bands are not inﬁnite in terms of energy. The argument of Yn\nmis\nχ= (∆Empα/prime−pµl)/(kBT) whereTis the temperature, kBis the Boltzmann constant and ∆ Empα/primeis the energy\ndiﬀerence between the steady-state reduced density matrix and a virtual state where the operator dp,α/prime\nmis applied\non the former. The energy integration itself is contained in the Yn\nm-function with its dimensionless variables µand\nx, the dimensionless Fermi-function f(n)(x) and the Lorentzian cut-oﬀ-function L(˜W,x ) = ˜W2/(x2+˜W2) with the\ndimensionless high energy band limit ˜W=W/(kBT) to ensure the convergence of the integration:\nYn\nm(µ):=−i\n2π/integraldisplay\ndxf(n)(x)L(˜W,x )\nm(x−µ) +i0+. (21)\nApplying the residuum theorem one gets for Yn\n+(µ) [3],\nYn\n+(µ) =−1\n2fn(µ)−in\n2π/bracketleftbigg\nReΨ(0)/parenleftbigg1\n2+iµ\n2π/parenrightbigg\n−C/bracketrightbigg\n=−1\n4−in\n2π/bracketleftbigg\nΨ(0)/parenleftbigg1\n2+iµ\n2π/parenrightbigg\n−C/bracketrightbigg\nwith the constant C= Ψ(0)(1\n2+˜W\n2π) deﬁned by the renormalized wide band constant ˜W. We furthermore introduced\nthe digamma-function\nΨ(0)(z):=−∞/summationdisplay\nn=01\nn+z+∞/summationdisplay\nn=1log/parenleftbigg\n1 +1\nn/parenrightbigg\n, z∈C. (22)\nThe constant Calways disappears when summing over the α-indices. Therefore, we can drop Cfrom the sequential\ntunneling Kernel calculation.5\nCotunneling and pair tunneling\nIf we include the next leading order in the expansion of the Kernel, we get a Kernel which is valid up to the fourth\norder:K=K(2)+K(4)+O/parenleftbig\nH6\nT/parenrightbig\n. This regime of tunneling events up to fourth order in LTis better known as the\ncotunneling transport regime. In this regime two new processes are included, namely the cotunneling ones and pair\ntunneling ones. For the fourth order Kernel, we obtain according to (17):\nK(4)=PLT˜G0QLTQ˜G0QLTQ˜G0LTP\n=PLT˜G0LT˜G0LT˜G0LTP−PL T˜G0LTP˜G0PLT˜G0LTP.\nHere we appliedPL2n+1\nTP= 0 forn∈Nand the fact that PrespectiveQcommutes with ˜G0so that the outermost\nQ-operators vanish and the innermost square to Q. If we letK(4)act on a density matrix, we get:\nK(4)ρ∞\nred=/bracketleftBig\nK(4,D)+K(4,X)/bracketrightBig\nρ∞\nred=/summationdisplay\n{αi}{l}/summationdisplay\n{n}{m}{p}α1α4\nkBT¯Γp\nl,nm¯Γp/prime\nl/prime,n/primem/prime\n/bracketleftBigg\nDα1α2\n++(/square,⊙,•)d¯p,α4\nnd¯p/prime,α3\nn/primedp/prime,α2\nm/primedp,α1\nm+Xα1α2\n++(/square,⋆,•)d¯p,α4\nnd¯p/prime,α3\nn/primedp,α2\nmdp/prime,α1\nm/prime/bracketrightBigg\nρ∞\nred (23)\nwith/square= (µj3−pµl)/(kBT),⊙= (µ/prime\nj1−pµl)/(kBT),•= (∆j2−pµl−p/primeµl/prime)/(kBT) and⋆= (µ/prime\nj1−p/primeµl/prime)/(kBT).\nThe subscripts{j1,j2,j3}of the energy diﬀerences µ/prime,∆ andµindicate that these energies depend on the variables\nof the ﬁrst ( α1,p,m ), two ﬁrst respective three ﬁrst d-superoperators. More details on the derivation of this result\ncan be found for example in chapter 2 of [4]. The D-function is deﬁned by\nDnn/prime\npp/prime(µ,µ/prime,∆) =−i/planckover2pi1\n4π2∞/integraldisplay\n−∞dx∞/integraldisplay\n−∞dx/primef(n)(x)\ni0++p(x−µ)1\ni0++px+p/primex/prime−∆f(n/prime)(x/prime)\ni0++p(x−µ/prime)\n=2π2n(iπ+ 2Cn/prime)\ni/planckover2pi1(µ−µ/prime)/bracketleftbigg\nΨ(0)/parenleftbigg1\n2+iµ\n2π/parenrightbigg\n−Ψ(0)/parenleftbigg1\n2+iµ/prime\n2π/parenrightbigg/bracketrightbigg\n−2πnn/prime\n/planckover2pi1∞/summationdisplay\nk=0Ψ(0)/parenleftbig\n1 +k+i∆\n2π/parenrightbig\n/parenleftbig\nk+1\n2+iµ\n2π/parenrightbig/parenleftBig\nk+1\n2+iµ/prime\n2π/parenrightBig (24)\nand theX-function reads\nXnn/prime\npp/prime(µ,µ/prime,∆) =−i/planckover2pi1\n4π2∞/integraldisplay\n−∞dx∞/integraldisplay\n−∞dx/primef(n)(x)\ni0++p(x−µ)1\ni0++px+p/primex/prime−∆f(n/prime)(x/prime)\ni0++p/prime(x/prime−µ/prime)\n=−4π2\ni/planckover2pi1nn/prime\nµ+µ/prime−∆Ψ(0)/parenleftbigg1\n2+iµ\n2π/parenrightbigg/bracketleftbigg\nΨ(0)/parenleftbigg1\n2+iµ/prime\n2π/parenrightbigg\n−Ψ(0)/parenleftbigg1\n2+i(∆−µ)\n2π/parenrightbigg/bracketrightbigg\n+2πnn/prime\n/planckover2pi1∞/summationdisplay\nk=0Ψ(0)/parenleftbig\n1 +k+i∆\n2π/parenrightbig\n/parenleftBig\nk+1\n2+iµ/prime\n2π/parenrightBig/parenleftBig\nk+1\n2+i(∆−µ)\n2π/parenrightBig. (25)\nIt should be noted that there are many special cases (like ∆ = 0) where the D- andX-functions become fully analytical.\nIt can be shown that this superoperator formalism is equivalent to the ansatz in the dissertation of S. Koller [5]. The\nsuperoperator formalism, though, treats the problem in the Liouville space and maps into a more compact single-path\ndiagrammatics [6]. Koller’s approach is performed in the Hilbert space and maps into a double-path diagrammatics,\ncloser to the real time diagrammatics [7, 8]. We formulated here also an expression for the imaginary part of the\nenergy integrals (24) and (25) of the fourth order Kernel which are needed for the time evolution of fourth order\ncoherences.\n∗christoph.rohrmeier@ur.de\n[1] S. Nakajima, Prog. Theor. Phys. 20, 948 (1958).\n[2] R. Zwanzig, J. Chem. Phys. 33, 1338 (1960).\n[3] D. Mantelli, Analytical and numerical study of quantum impurity systems in the intermediate and strong coupling regimes\n(2016).\n[4] M. Niklas, Current and noise properties of interacting nanojunctions (2018).6\n[5] S. Koller, Spin phenomena and higher order eﬀects in transport across interacting quantum-dots (2010).\n[6] S. Koller, M. Grifoni, M. Leijnse, and M. R. Wegewijs, Phys. Rev. B 82, 235307 (2010).\n[7] H. Schoeller and G. Sch¨ on, Phys. Rev. B 50, 18436 (1994).\n[8] J. Knig, H. Schoeller, and G. Schn, Europhys. Lett. 31, 31 (1995)."    },    {        "title": "2105.03479v1.Magnetic_properties_of_the_itinerant_ferromagnet_LaCrGe3_under_pressure_studied_by_139La_NMR.pdf",        "content": "arXiv:2105.03479v1  [cond-mat.str-el]  7 May 2021Magnetic properties of the itinerant ferromagnet LaCrGe 3under pressure studied by\n139La NMR\nK. Rana,1,2H. Kotegawa,3R. R. Ullah,4E. Gati∗,1,2S. L. Bud’ko,1,2\nP. C. Canﬁeld,1,2H. Tou,3V. Taufour,4and Y. Furukawa1,2\n1Ames Laboratory, U.S. DOE, Iowa State University, Ames, Iow a 50011, USA\n2Department of Physics and Astronomy, Iowa State University , Ames, Iowa 50011, USA\n3Department of Physics, Graduate School of Science, Kobe Uni versity, Kobe 657-8501, Japan\n4Department of Physics, University of California, Davis, CA 95616, USA\n(Dated: May 11, 2021)\n139La nuclear magnetic resonance (NMR) measurements under pre ssure (p= 0−2.64 GPa)\nhave been carried out to investigate the static and dynamic m agnetic properties of the itinerant\nferromagnet LaCrGe 3.139La-NMR spectra for all measured pressures in the ferromagne tically\norderedstate showalarge shift duetotheinternal ﬁeldindu ction|Bint| ∼4TattheLasite produced\nby Cr ordered moments. The change in Bintby less than 5% with pup to 2.64 GPa indicates that\nthe Cr 3dmoments are robust under pressure. The temperature depende nce of NMR shift and Bint\nsuggest that the ferromagnetic order develops below ∼50 K under higher pressures in a magnetic\nﬁeld of∼7.2 T. Based on the analysis of NMR data using the self-consis tent-renormalization (SCR)\ntheory, the spin ﬂuctuations in the paramagnetic state well aboveTCare revealed to be three\ndimensional ferromagnetic throughout the measured pregion.\nI. INTRODUCTION\nA quantum critical point (QCP), deﬁned as a second\norder phase transition at absolute zero temperature T\n= 0 K [1] in itinerant ferromagnets has attracted much\nattention owing to the observation of a wide variety of\nintriguing phenomena such as unconventional supercon-\nductivity[2–6], non-Fermiliquidbehavior[7–9]andpecu-\nliar magnetic properties originating from quantum criti-\ncality [10, 11] which can be controlled by tuning param-\neters such as pressure ( p) and external magnetic ﬁeld\n(H). Interestingly, it is now generally known that the\nferromagnetic (FM) QCP in clean itinerant ferromag-\nnets are avoided [11], although exceptions with noncen-\ntrosymmetric metals having strong spin-orbit interaction\nare pointed out by the recent theoretical [12] and exper-\nimental [13, 14] studies.\nIn most itinerant ferromagnetic systems, when the sec-\nond order paramagnetic (PM)-FM phase transition tem-\nperature ( TC) is suppressed by the application of p, the\norder of the phase transition changes to the ﬁrst order at\nthe tricritical point (TCP) before TCreaches 0 K at the\nquantum phase transition (QPT), known as the avoided\nQCP [15–17]. When the PM-FM transition becomes of\nthe ﬁrst order at the TCP in the p-Tplane, the applica-\ntion ofHleads to a tricritical wing (TCW) structure in\ntheT-p-Hthree dimensional phase diagram as found in\nUGe2[15, 16] and ZrZn 2[17]. A PM-FM QCP can also\nbe avoided by the appearance of an antiferromagnetic\n(AFM) ordered state under pnear the putative QCP, as\nactually observed in CeRuPO[18, 19], MnP[20, 21] and\nLa5Co2Ge3[22]. In this case, no wing structure has been\n∗Present address: Max Planck Institute for Chemical Physics of\nSolids, 01187 Dresden, Germanyreported and the AFM state is suppressed by the appli-\ncation of moderate H.\nIn this context, LaCrGe 3with the hexagonal BaNiO 3-\ntype structure [space group P63/mmc(194)] [23] is a pe-\nculiar itinerant ferromagnet because it was suggested to\nshow both tricritical wings [24] and AFM state [25] in\nthep-T-Hphase diagram as shown in Fig. 1(a). At\nambient p, LaCrGe 3is FM below the Curie tempera-\ntureTC∼85 K with an ordered magnetic moment of\n1.25µB/Cr aligned along the caxis at low temperatures\n[26, 27]. As shown in the phase diagram, upon reduc-\ning the temperature below TC, a crossover to a second\nFM state is observed around 70 K [24, 25]. The two\nFM states, labeled FM1 and FM2 respectively, were also\nfound in the itinerant ferromagnet UGe 2[15] which were\nattributed to lower and higher values of saturated mag-\nnetic moment ( µs) respectively. When pis increased, TC\nis suppressed and the second order FM transition be-\ncomes of the ﬁrst order at p∼1.8 GPa and Taround\n40 K. This yields a TCW structure under H. The FM1-\nFM2 crossover temperature is also suppressed under p\nyielding a second TCW structure [24]. It was also re-\nported that, with increasing p, the system exhibits a new\nphase above 1.5 GPa [25], which was considered as a the-\noretically predicted AFM state. A second possible mod-\nulated AFM state wasalsosuggestedtoappearwhen fur-\nther lowering temperatures at pressures higher than the\ncritical pressure pc∼2.1 GPa [25]. These new phases\nare suppressed under H. Quite recently, the detailed\nstudies on LaCrGe 3using thermodynamic, transport, x-\nray, neutron scattering and µSR measurements reported\na modiﬁed zero-ﬁeld phase diagram [ T-pphase diagram,\nsee Fig. 1(b)] [28]. Similar to the previous report [25],\na clear tricritical point has been reported with a slightly\ndiﬀerent position [ p= 1.5(1) GPa, T= 53(3) K]. Accord-\ning to Gati et al.[28], the phase appearing under high\npis likely not an AFM state but a form of disordered2\nFIG. 1: (a) Three dimensional temperature-pressure-\nmagnetic ﬁeld ( T-p-H) phase diagram for LaCrGe 3taken\nfrom Ref. [24] where His applied parallel to the caxis. Two\ntricritical wings due to the FM1 and FM2 states emerge from\nthetricritical point(TCP)undermagnetic ﬁeldandtermina te\nat quantum wing critical points (QWCP). The high-pressure\nmagnetic (HP-MAG) state is represented by the yellow col-\nored area. It is noted that the HP-MAG state was initially\nconsidered as an antiferromagnetic state [25], but the rece nt\nmeasurements suggest a short-range ferromagnetic ordered\nphase as shown in (b) [28]. (b) Recent T-pphase diagram\nin zero magnetic ﬁeld for LaCrGe 3after Ref. [28]. The blue-\nshaded region corresponds to the region of FM order, which is\nschematically depictedintheinsets byspins(arrows) poin ting\nalong the caxis. The thin and thick blue lines represent sec-\nond and ﬁrst order phase transition boundaries, respective ly.\nDarkand light yellow shaded regions relate to short-range F M\nordered cluster phases that occur for p >1.5 GPa. The inset\nin thispregion visualizes a possible short-range FM ordered\nphase.\nshort-range FM clusters. These results suggest that the\navoided FM criticality in LaCrGe 3has two features: (1)\nthe change in the transition character from second order\nto ﬁrst order and (2) the appearance of short-range fer-\nromagnetic order rather than AFM order. Although the\nﬁrst one is a well-known mechanism for clean itinerant\nferromagnets, the second one contradictorily suggests a\nsort of disorder in systems, making the system peculiar.\nTherefore,tounderstandthemechanismoftheavoidanceof FM criticality in LaCrGe 3, it is important to investi-\ngate the new phase which is reported to appear under\nhigh pressures greater than ∼1.5 GPa.\nWith the motivation to investigate the evolution of\nmagnetic properties of LaCrGe 3underptowards the pu-\ntative QCP as well as to obtain more insights into the\nnew phase under high pressures, here we carried out nu-\nclear magnetic resonance (NMR) measurements under\npup to 2.64 GPa. Our previous139La-NMR study at\nambient pevidenced the presence of 3 dimensional (3D)\nisotropic ferromagnetic ﬂuctuations in this system [29].\nFurthermore, LaCrGe 3was found to follow the general-\nized Rhodes-Wohlfarth (GRW) relation for 3D itinerant\nferromagnets,anditslocationin theGRWplotsuggested\na high degree of localization in Cr 3 delectrons compared\nto other itinerant ferromagnets that exhibit the tricriti-\ncal wings structure [26, 29]. Our present NMR data show\nthat the 3D FM ﬂuctuations persist to dominate in the\nparamagnetic state well above TCin LaCrGe 3through-\nout the measured pregion and suggest a possible ferro-\nmagnetic order developing below ∼50 K under higher\npressures in a magnetic ﬁeld of ∼7.2 T.\nII. EXPERIMENTAL DETAILS\nRod-like shaped LaCrGe 3single crystals were grown\nfrom high temperature solutions as detailed in Ref. [26].\nThe crystalline caxis is parallel to the rod direction.\nNMR measurements of139La nuclei ( I= 7/2,γN/2π=\n6.0146 MHz/T, Q= 0.21 barns) were carried out us-\ning a lab-built phase-coherent spin-echo pulse spectrom-\neter. Three large single crystals with a total mass of\n∼150 mg were aligned along the caxis and were inserted\nwith an NMR coil in a NiCrAl/CuBe piston-cylinder for\npmeasurements up to 2.64 GPa. Here the crystals were\nwell separated by Teﬂon tapes and were placed in the\nNMR coil to make any demagnetization eﬀects negligi-\nble. Daphne 7474wasusedasthe ptransmitting medium\nand the pcalibration was carried out by measuring the\nsuperconducting transition temperature of lead through\nresistivity measurements [30]. The139La NMR spectra\nwere obtained by sweeping Hparallel to the caxis at\nﬁxed resonant frequencies ( f). The zero-shift position\ncorresponding to the Larmor ﬁeld for each resonance fre-\nquency was determined by31P NMR in H 3PO4solution\nor63Cu NMR in Cu metal.\nThe139La nuclear spin-lattice relaxation rate (1/ T1)\nwas measured using a saturation recovery method. 1 /T1\nat each temperature was determined by ﬁtting the nu-\nclear magnetization ( M) versus time ( t) using the ex-\nponential function 1 −M(t)/M(∞) = 0.012e−t/T1+\n0.068e−6t/T1+0.206e−15t/T1+0.714e−28t/T1, whereM(t)\nandM(∞) are the nuclear magnetization at time tafter\nthesaturationandtheequilibrium nuclearmagnetization\natt→∞, respectively,forthecaseofmagneticrelaxation\n[31]. The observed recovery data in the paramagnetic\nstate were well ﬁtted by the function, indicating that the3\n/s55/s46/s48 /s55/s46/s53 /s56/s46/s48 /s56/s46/s53 /s57/s46/s48 /s54/s46/s53 /s55/s46/s48 /s55/s46/s53 /s56/s46/s48 /s56/s46/s53/s40/s98/s41\n/s49/s46/s54/s53/s32/s71/s80/s97/s44/s32/s52/s50/s46/s53/s32/s77/s72/s122\n/s49/s55/s53/s32/s75/s50/s48/s48/s32/s75\n/s48/s72 /s32/s40/s84/s41/s50/s51/s48/s32/s75\n/s49/s53/s53/s32/s75\n/s49/s50/s53/s32/s75\n/s49/s48/s48/s32/s75\n/s56/s53/s32/s75\n/s48/s72 /s32/s40/s84/s41/s83/s112/s105/s110/s32/s101/s99/s104/s111/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41/s48/s32/s71/s80/s97/s44/s32/s52/s50/s46/s53/s32/s77/s72/s122/s50/s53/s48/s32/s75\n/s32/s50/s51/s48/s32/s75/s32\n/s49/s57/s53/s32/s75\n/s49/s53/s53/s32/s75/s40/s97/s41\nFIG. 2: Temperature dependence of ﬁeld-swept139La-NMR\nspectrum ( H||c) under (a) ambient pressure and (b) 1.65\nGPa measured at f= 42.5 MHz. The black curves show the\nobserved spectra and the red curves are the calculated spect ra\nwithνQ=0.66 MHz and η= 0(see text). The vertical dashed\nblack lines in (a) and (b) represent the zero-shift (Larmor)\nposition ( µ0H0= 2πf/γN).\nnuclear relaxation is mainly induced by ﬂuctuations of\nthe hyperﬁne ﬁeld at the139La site.\nIII. RESULTS AND DISCUSSION\nA.139La NMR spectrum\n139La-NMR spectra in LaCrGe 3at ambient pressure\nhave been reported to show typical quadrupolar-split\nlines [29] which are well explained by the combination\nof a large Zeeman interaction due to magnetic ﬁeld and\nasmallquadrupoleinteractionwhosenuclearspin Hamil-\ntonianisgivenby H=−γ¯hI·Heﬀ+hνQ\n6[3I2\nz−I2+1\n2η(I2\n++\nI2\n−)]. Here Heﬀis the eﬀective magnetic ﬁeld at the La\nsite,his Planck’s constant, νQis nuclear quadrupole fre-\nquency deﬁned by νQ= 3e2QVZZ/2I(2I−1)hwhere\nQis the electric quadrupole moment of the La nucleus,\nVZZis the electric ﬁeld gradient (EFG) at the La site,\nandηis the asymmetry parameter of EFG at the La\nsite. From the angle dependence of139La NMR spectra\nin the paramagnetic state at ambient pressure, the prin-\ncipal axis of the EFG was determined to be parallel to\nthecaxis and νQis estimated to be 0.66 MHz with η∼0\n[29]. In fact the observed spectra are well reproduced by\nthe calculated spectra [red curves in Fig. 2(a)] from the\nHamiltonian with those parameters [29]. As in the case\nof the ambient pressure,139La-NMR spectra in LaCrGe 3\nunder pressure show the typical I= 7/2 quadrupole split\nlines with a nearly pressure independent value of νQ=/s51 /s52 /s53/s48 /s49 /s50 /s51/s48/s45/s49/s45/s50/s45/s51/s45/s52/s45/s53\n/s50 /s51/s48/s53/s49/s48/s49/s53/s50/s48/s83/s112/s105/s110/s32/s101/s99/s104/s111/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s48/s72 /s32/s45/s32\n/s48/s72\n/s48/s40/s84/s41/s32\n/s70/s77/s49/s63/s70/s77/s50/s70/s77/s50/s70/s77/s50/s70/s77/s50\n/s40/s50/s46/s52/s55/s32/s71/s80/s97/s44/s32/s52/s50/s46/s53/s32/s77/s72/s122/s41\n/s40/s50/s46/s54/s52/s32/s71/s80/s97/s44/s32/s52/s50/s46/s53/s32/s77/s72/s122/s41/s40/s50/s46/s50/s51/s32/s71/s80/s97/s44/s32/s32/s50/s57/s46/s50/s53/s32/s77/s72/s122/s41/s40/s49/s46/s54/s53/s32/s71/s80/s97/s44/s32/s51/s49/s32/s77/s72/s122/s41/s40/s112 /s44/s32 /s102/s41/s32/s61 /s32\n/s40/s48/s32/s71/s80/s97/s44/s32/s52/s50/s46/s53/s32/s77/s72/s122/s41/s84 /s32/s61/s32/s49/s46/s54/s32/s75\n/s32/s40/s98/s41/s40/s99/s41\n/s32/s32/s66\n/s105/s110/s116 /s32/s32/s40/s84/s41\n/s112 /s32/s40/s71/s80/s97/s41/s32\n/s72/s80/s45/s77/s65 /s71/s70/s77/s50/s70/s77/s49\n/s32/s78/s77/s82/s32/s83/s112/s101/s99/s116/s114/s97/s48/s72 /s32/s40/s84/s41\n/s112 /s32/s40/s71/s80/s97/s41/s80/s77\n/s40/s97/s41/s84 /s61/s50/s32/s75\nFIG. 3: (a)Field-swept139La-NMRspectraofLaCrGe 3at 1.6\nK for various pressures up to 2.64 GPa, as a function of the\ndiﬀerence in external magnetic ﬁeld ( H) and resonance ﬁeld\n(f/γ). The red curves are the calculated spectra with appro-\npriate magnetic broadening. The downward arrows represent\nthe peak position of the central peak for each pressure deter -\nmined from the calculated spectrum. Inset: (b) H-pphase\ndiagram at T= 2 K taken from Ref. [24]. The red crosses\nrepresent the positions of the central peak of the NMR spec-\ntra. (c) Internal magnetic induction ( Bint) at the La site as a\nfunction of p. The black line represents a linear ﬁt.\n0.66MHz and η∼0 in the paramagneticstate well above\nthe magnetic ordering temperature under pressure up to\nthe highest measured pof 2.64 GPa.\nAs shown in the typical temperature dependence of\nthe ﬁeld-swept139La NMR spectrum at p= 1.65 GPa\nforHparallel to the caxis (H||c) [Fig. 2(b)], with\ndecreasing temperature, each line becomes broader due\nto inhomogeneous magnetic broadening and the spectra\nshow less clear features of the quadrupolar split lines be-\nlowT∼100 K. A similar broadening of NMR lines was\nreported for the case of ambient pressure [29] as shown\nin Fig. 2(a). It is noted that one can see the well-split\nlines even at 155K at p= 1.65GPaalthoughthe feature-\nless spectrum was observed at the same temperature at\nambient pressure. This indicates that TCdecreases with\nthe application of pressure, consistent with the reduc-\ntion inTCfrom 85 K at ambient pressure to ∼50 K atp4\n= 1.65 GPa determined by the resistivity measurements\n[25]. The broadening of the spectra at a wide range of\ntemperatures close to TCmake the NMR spectrum mea-\nsurements diﬃcult around TC. However, when the tem-\nperature is decreased down to 1.6 K, well below TC, we\nwere able to observethe139La NMR spectrum in the FM\nstate. Figure 3 shows the ﬁeld-swept139La-NMR spec-\ntra of LaCrGe 3atT= 1.6 K ( H||c) at various pressures\nfromp= 0 GPaup to 2.64GPa, where all the spectraare\nlargely shifted from the Larmor ﬁeld (2 πf/γN=µ0H0)\nto a higher magnetic ﬁeld by ∼4 T. The shift is due\nto the internal magnetic induction ( Bint∼-4 T) at the\nLa site produced by the Cr ordered moments in the FM\nstate, as reported from the NMR measurements at ambi-\nent pressure in Ref. [29]. Note that the horizontal axis of\neachspectrum is shifted by µ0H0and nowcorrespondsto\nthe negative value of Bint. As shown in Fig. 3, although\nthe clear feature of quadrupole split lines of the spec-\ntrum can be seen at ambient pressure, the peak struc-\ntures become less prominent with increasing pwhere the\nlinewidth[determinedbythefull widthathalfmaximum\n(FWHM)] increases from ∼0.67 T at p= 0 to∼0.9 T at\nthe high pressure region ( P >2.23 GPa). This suggests\nthat a sort of inhomogeneity is induced in the system\nby the application of pressure. It would be interesting if\nthe inhomogeneity observed in the NMR measurements\ncorresponds to the possible disorder in LaCrGe 3under\npressure inferred from the short-range magnetic ordered\nstate [28]. As can be seen in the ﬁgure, the peak posi-\ntions of the spectra, marked by downward arrows, shift\nslightly to lower magnetic ﬁelds due to the decrease in\n|Bint|. The pressure dependence of Bintis shown in Fig.\n3(c) as a function of p. The|Bint|decreases slightly by\nonly∼5% up to 2.64GPa. These results seem to be con-\nsistent with the results of µSR measurements where Bint\nat the muon site in the FM state is nearly independent of\npressure [25]. According to the H−pphase diagram at\n2 K reported in Ref. [24] shown in Fig. 3(b), our139La\nNMR spectrum at p= 2.64GPa would correspondto the\none in the FM1 state, whereas other spectra with diﬀer-\nent pressures represent the ones in the FM2 state. As\nseen in Fig. 3, we observed no signiﬁcant change in Bint\nbetween 2.47 GPa and 2.64 GPa. Since the internal ﬁeld\nBintis proportional to the spontaneous magnetization,\nthe results suggest that either µsdoes not show promi-\nnent change between FM1 and FM2, or that we did not\ncross the FM1-FM2 boundary in our experiment, as will\nbe further discussed later.\nSinceBintalso depends on a hyperﬁne coupling con-\nstant (A) which may change by the application of pres-\nsure, one could have a chance to accidentally observe no\nchange in Bintdue to a compensation of the changes in\nbothµsandA, although the large change in Ais un-\nlikely since we do not see any signiﬁcant change at high\ntemperatures above ∼200 K at various pin nuclear spin-\nlattice relaxation times ( T1) which are proportional to\nthe square of A, as will be described below. Therefore\nwehavemeasuredthemagneticﬁeld dependence ofNMR/s51 /s52 /s53/s56 /s57 /s49/s48 /s49/s49/s48/s45/s49/s45/s50/s45/s51/s45/s52/s45/s53\n/s50 /s51/s48/s53/s49/s48/s49/s53/s50/s48\n/s70/s77/s50\n/s70/s77/s50/s70/s77/s49/s63\n/s48/s72/s32 /s45\n/s48/s72\n/s48/s32/s40/s84/s41/s32\n/s84 /s32/s61/s32/s49/s46/s54/s32/s75/s44/s32\n/s112 /s32/s61/s32/s50/s46/s52/s55/s32/s71/s80/s97\n/s32/s102/s32 /s61/s32/s52/s50/s46/s53/s32/s77/s72/s122/s32/s32\n/s40\n/s48/s72\n/s48/s32/s61/s32/s55/s46/s48/s54/s54/s49/s32/s84/s41/s32/s102/s32 /s61/s32/s51/s56/s46/s50/s32/s77/s72/s122/s32/s32\n/s40\n/s48/s72\n/s48/s32/s61/s32/s54/s46/s51/s53/s49/s50/s32/s84/s41/s32/s102/s32 /s61/s32/s50/s55/s46/s48/s32/s77/s72/s122/s32\n/s40\n/s48/s72\n/s48/s32/s61/s32/s52/s46/s52/s56/s57/s32/s84/s41/s83/s112/s105/s110/s32/s101/s99/s104/s111/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s66\n/s105/s110/s116 /s32/s32/s40/s84/s41\n/s48/s72\n/s112/s101/s97/s107/s32/s40/s84/s41\n/s40/s97/s41/s40/s99/s41 /s40/s98/s41\n/s72/s80/s45/s77/s65/s71/s80/s77/s70/s77/s49\n/s32/s32/s48/s72 /s32/s32/s40/s84/s41\n/s112 /s32/s40/s71/s80/s97/s41/s32/s72\n/s70/s77/s49\n/s32/s72\n/s70/s77/s50\n/s32/s72\n/s78/s81/s39\n/s32/s78/s77/s82/s32/s83/s112/s101/s99/s116/s114/s97\n/s70/s77/s50/s84/s61 /s50/s32/s75\nFIG. 4: (a)139La NMR spectra at p= 2.47 GPa taken at\nvarious frequencies, as a function of the diﬀerence in exter nal\nﬁeld (µ0H) and the zero shift position ( µ0H0= 2πf/γN) for\nH||cdirection at T= 1.6 K. The downward arrows show the\npeak positions of the central transition lines of the spectr a\nand the red lines which serve as guides for the eye have the\nsame full width at half maximum (FWHM). Inset: (b) H-p\nphase diagram at T= 2 K taken from Ref. [24] with the\npositions (red crosses) where the central peak of the NMR\nspectra was measured. (c) Hpeakdependence of BintforH||c\nfor the three diﬀerent resonance frequencies.\nspectra at a constant pressure of 2.47 GPa by changing\nthe resonance frequencies from 27 MHz to 42.5 MHz. In\nthis case, any change in Awould not be expected. The\nobserved spectra are shown in Fig. 4 where the peak\npositions of the spectra increase from µ0Hpeak= 8.3 T\nat the lowest resonance frequency of f= 27 MHz up till\nµ0Hpeak= 10.9 T at f= 42.5 MHz. Note that the hor-\nizontal axis of each spectrum is shifted by the Larmor\nﬁeld as in the case of Fig. 3. From the reported H−p\nphase diagram at T= 2 K [24] [(Fig. 4(b)] where one\nexpects the FM1-FM2 crossover around 9.2 T, the NMR\nspectrum at 27.0 MHz may be considered to represent\nthe FM1 state whereas those measured at the higher fre-\nquencies represent the FM2 state. It is found that Bint\n∼ −3.82 T is nearly independent of the external ﬁelds\nas shown in Fig. 4(c) even though Hcrosses the border\nbetween the FM1 and FM2 states at 9.2 T in Ref. [24].5\nWe also do not observeany clear change in the line width\nwhich reﬂects the distribution ofboth the magnitude and\ndirection of µs. While the signal intensity decreases with\ndecreasingtheresonancefrequency, theline width at27.0\nMHz is similar to that at 38.2 MHz within our experi-\nmental uncertainty, as shown by the red curves describ-\ning the broad peak with the same line width. Thus, ei-\nther no clear change in the magnetic states between FM1\nand FM2 is detected, or we did not cross the FM1-FM2\nboundary in our experiments, as will be discussed below.\nThe above discussions are based on the phase diagram\nreported in Refs. [24, 25, 32] [see, Fig. 1(a)], however, it\nis important to point out that the FM1-FM2 crossover\nﬁeld could vary depending on diﬀerent experimental set\nup as well as sample quality. The H-pphase diagram\nshownin Figs. 3(b) and 4(b) wasdetermined bythe mea-\nsurements with modiﬁed Bridgman pressure cells which\nmay produce larger pressure gradients compared with\npiston-cylinder pressure cells. The ferromagnetic insta-\nbility in LaCrGe 3has been shown to be primarily driven\nby the Cr-Cr distance along the caxis [33], which would\nbe the direction of higher pressure gradients in the previ-\nous measurements in Bridgman cells [24, 32]. Therefore,\nthe results of the NMR measurements by utilizing the\npiston-cylinder pressure cells should be compared with\nthe phase diagram of Fig. 1(a) with some caution. It\nis possible that our NMR experiments did not reach the\nFM1-FM2 crossover ﬁeld. To discuss the magnetic prop-\nerties of the FM1 and FM2 phases, it is important to\nmakesurethatthesystemisactuallyineachphase. Since\nthe FM1-FM2 crossover in the compound has been de-\ntected by resistivity measurements [25], one should mea-\nsure the resistivity of the same sample in the same exper-\nimental set up (i.e., pressure cells). This would require\nperforming simultaneous measurements of NMR and in-\nsitu resistivity. This is beyond the scope of the present\nwork and is planned for a future work.\nB.139La-NMR shifts\nTheTdependence of139La NMR Knight shift ( K) is\nshown in Fig. 5(a) for the measured pressures with the\nH||cdirection. The Ks show the Curie-Weiss (CW) type\ntemperature dependence and Kdecreases monotonically\nwith decreasingtemperature. With increasing p, the CW\ntype behavior of Kshifts to lower temperatures, again\nconsistent with the suppression of TC.\nIn general, Khas contributions from the temperature\ndependent spin shift Ks(T) andTindependent chem-\nical shift K0:K(T) =Ks(T) +K0whereKs(T) is\nproportional to the spin part of magnetic susceptibil-\nityχs(T) via hyperﬁne coupling constant ( A),Ks(T) =\nAχs(T)/NAwhereNAis Avogadro’s number. At ambi-\nent pressure, from the Kvs.χplot analysis, the Awas\nestimated to be −27 kOe/µBforH||c[29]. In addition,\nK0was reported to be close to zero [29], indicating that\nthe observed Kis mainly attributed to Ks(T).\nFIG. 5: (a) Temperature dependence of139La Knight shift\nKfor various pressures in the H||cdirection. (b) The tem-\nperature dependence of 1/ |K|for various pvalues. The inset\nshows the zero-ﬁeld phase diagram [Fig. 1(b)] together with\nthe estimated TCatH∼7 T shown by the red circles. (c)\nTemperature dependence of ∆ µ0Hfor various pressures. The\nred curve shows the temperature dependence of M/Hmea-\nsured at 7 T at ambient pressure. Other curves (solid lines)\nshow the M/Hdata estimated from the ambient pressure 7\nT data shown (see text for details). The black dashed curve\nshows the expected temperature dependence of M/HwithTC\n= 15 K under a magnetic ﬁeld of 7 T.\nSince the χsat ambient pressure has been reported\nto follow the CW-type behavior even though the system\nis itinerant [23, 26, 29, 32], one may estimate TCfrom\nthe temperature dependence of KsinceKat high tem-\nperatures is proportional to χswhich is proportional to\nC/(T−TC) (C: Curie constant). Fig. 5(b) shows the\ntemperature dependence of 1/ |K|where the intercepts\nof thexaxis provide an estimate of TCfor each pres-\nsure. The intercept for the case of ambient pressure is6\nestimated to be ∼85±3 K which is in excellent agree-\nment with TC= 85 K reported previously. From the\nintercepts, we estimated the pdependence of TCwhich\ndecreases from 85 K at ambient pressure to 63 ±3 K at\n1.65 GPa, to 53 ±5 K at 2.23 GPa, and to 45 ±5 K\nat 2.64 GPa. It is noted that the slopes in the 1/ Kvs.\nTplots slightly increase with increasing p. Assuming\nthe change were only due to the change in Curie con-\nstant (that is, no change in A), the eﬀective moments\nare estimated to slightly decrease by at most 14 % at p\n= 2.64 GPa from the value at ambient pressure. It is also\nnoted that no clear signature of antiferromagnetic inter-\naction can be found from the temperature dependence of\nKwhere all intercepts are positive for the measured p\nregion.\nWe also checked the estimated TCwith the tempera-\nturedependenceof Bintaswellasthetemperaturedepen-\ndence of H-induced hyperﬁne ﬁeld at the La site by using\nmagnetization ( M/H) data. The red circles in Fig. 5(c)\nshow the temperature dependence of |∆µ0H|=µ0Hpeak\n-µ0H0(corresponding to Bintin the FM state and to the\nHinduced hyperﬁne ﬁeld at the La site in the PM state)\nat ambient pressure which is expected to be proportional\nto the magnetization Min both the PM and FM states\nin ferromagnets. In fact, the temperature dependence of\n|∆µ0H|measured at ambient pressure seems to be well\nreproduced by the M/Hcurve (red curve) measured at\nµ0H= 7.0 T, although no data points are available in\na wide temperature range around TC. The temperature\ndependence of |∆µ0H|for other pressures shown in the\nﬁgure was also reasonably reproduced for not only the\nhigh temperature region but also the low temperature\nregion by the corresponding solid M/Hcurves. Since no\nM/Hdataunder7Tareavailableforpressureotherthan\nambient at present, which will be requested to measure\nin the future, we estimated the M/Hbehavior for other\npressuresbasedon the ambient dataunder 7 T. The solid\ncurves in Fig. 5(c) are the expected M/Hcurves where\nthe temperature for each pressure is normalized to the\nestimated TC[i.e.T(TC/TC(p=0))] and the magnetiza-\ntion is also scaled to the lowest temperature ∆ µ0Hdata\npoint. Thus we conclude that the estimation of TCfrom\nNMR measurements seems to be reasonable.\nAccording to the phase diagram shown in Fig. 1(b),\nthelong-rangeFMstateappearsunder H=0atT∼30K\nand 15 K for p= 2.26 and 2.64 GPa, respectively. Thus\nit is important to check whether or not the observedtem-\nperature dependence of M(∝ |∆H|) can be explained by\ntheTCunder magnetic ﬁeld, as the application of mag-\nnetic ﬁeld produces a large tail of magnetization even\naboveTC. As shown in Fig. 5(c), the observed tem-\nperature dependence of |∆µ0H|atp= 2.64 GPa cannot\nbe explained at all by the black dotted curve which is\nthe expected M/Hbehavior under 7 T for the reported\nzero-ﬁeld TC= 15 K at this pressure [28]. Therefore, the\nhigherTCobtained by the NMR measurements under\nhigh pressures cannot be explained by an artiﬁcial eﬀect\nofthe application ofmagnetic ﬁeld, and those values maysuggest the increase of TCby the application of magnetic\nﬁeld. It is also interesting to note that the estimated\nTCs from the NMR measurements are close to the tran-\nsition temperatures for the short-range FM order phase\nor a mixed state of the FM cluster and PM states deter-\nmined at H= 0 [see the inset of Fig. 5(b)]. Therefore,\nanother possibility is that the NMR spectrum measure-\nments detect the short-range FM ordered state. Since\nwe apply magnetic ﬁeld for the NMR measurements, a\nferromanetic phase transition is smeared, therefore, TC\ncannot be well deﬁned and only the crossover tempera-\nture from a paramagnetic state to a ferromagnetic state\nin magnetic ﬁeld can be inferred. This makes it diﬃ-\ncult to distinguish between long-range and short-range\nordered states by measuring NMR spectra. Neverthe-\nless, we may conclude that our NMR data suggest that\nFM orderings start to develop below T∼50 K under p\n>∼1.5 GPa and µ0H∼7.2 T.\nC. Magnetic ﬂuctuations\nIn order to investigate the magnetic ﬂuctuations in\nLaCrGe 3and their evolution with p, we measured the\ntemperature dependence of139La spin-lattice relaxation\nrate (1/T1) at the peak position of the spectra at vari-\nous pressures with the H||cdirection. Figure 6(a) shows\n1/T1Tas a function of Tfor various pvalues. At all pres-\nsures, 1/ T1Tincreases with lowering temperature from\nroom temperature down towards TC.\nIn our previous paper [29], we discussed FM mag-\nnetic ﬂuctuations in LaCrGe 3based on the T1andK\ndata at ambient pressure using the self-consistent renor-\nmalization (SCR) theory. Here we analyze the present\nNMR data under pressurewith the same theory. Accord-\ning to the SCR theory for weak itinerant ferromagnets,\n1/(T1TKs)and1/( T1TK3/2\ns)areexpected tobe indepen-\ndent ofTfor three dimensional (3D) or two-dimensional\n(2D) FM spin ﬂuctuations, respectively [34, 35]. Utiliz-\ning the 1 /T1Tdata in the PM state well above TCfor\nvarious pressures shown in Fig. 6(a), we plotted the\nTdependence of 1/( T1TK) and 1/( T1TK3/2) for various\npressuresin Fig. 6(b). As in the caseofambientpressure,\n1/T1TKfor each pressure seems to be nearly constant\nin the high temperature region, while 1 /(T1TK3/2) de-\ncreases slightly with decreasing Tat allpvalues. There-\nfore, we conclude that the magnetic ﬂuctuations in this\nsystem are dominated by 3D FM ﬂuctuations through\nthe measured pressures up to 2.64 GPa, indicative of the\nrobust nature of ferromagnetism in LaCrGe 3. It is noted\nthatthe1 /T1TKdatabelow100Kfor p=2.64GPaseem\ntodeviatefromtheconstantvalueattemperatureshigher\n125 K. Although the reason for the deviation is not clear\natpresent, thedecreasein thevalueofthe1/ T1TKbelow\n125 K suggests the suppression of 3D FM spin ﬂuctua-\ntions. It is also noted that, in the recent reported phase\ndiagram under zero magnetic ﬁeld [28], p= 2.64 GPa is\nclose to the pressure where the ﬁrst order FM transition7\n/s40/s98/s41\n/s49/s47/s40 /s84\n/s49/s84/s75 /s41/s32/s32/s32/s32/s58/s32 /s32/s97/s109/s98/s105/s101/s110/s116/s44/s32 /s32/s49/s46/s54/s53/s32/s71/s80/s97/s44/s32 /s32/s50/s46/s50/s51/s32/s71/s80/s97/s44/s32 /s32/s50/s46/s54/s52/s32/s71/s80/s97/s32\n/s49/s47/s40 /s84\n/s49/s84/s75/s51/s47/s50\n/s41/s32/s58/s32 /s32/s97/s109/s98/s105/s101/s110/s116/s44/s32 /s32/s49/s46/s54/s53/s32/s71/s80/s97/s44/s32 /s32/s50/s46/s50/s51/s32/s71/s80/s97/s44/s32 /s32/s50/s46/s54/s52/s32/s71/s80/s97/s32\n/s32/s32/s49/s47/s84\n/s49/s84/s75/s32 /s44/s32 /s49/s47/s84\n/s49/s84/s75/s51/s47/s50\n/s32/s32/s40/s115/s75/s41/s45/s49\n/s84 /s32/s32/s40/s75/s41/s32/s97/s109/s98/s105/s101/s110/s116/s32\n/s32/s49/s46/s54/s53/s32/s71/s80/s97\n/s32/s50/s46/s50/s51/s32/s71/s80/s97\n/s32/s50/s46/s52/s55/s32/s71/s80/s97\n/s32/s50/s46/s54/s52/s32/s71/s80/s97\n/s32/s32/s49/s47 /s84\n/s49/s84 /s32/s32/s40/s115/s75/s41/s45/s49\n/s84 /s32/s32/s40/s75/s41/s40/s97/s41\nFIG. 6: (a) Temperature dependence of139La 1/T1Tunder\nvarious pressures in the PM state measured at µ0H∼7.2 T.\n(b) Temperature dependence of 1/( T1TK) and 1/( T1TK3/2)\nin the PM state measured at µ0H∼7.2 T.\nis completely suppressed. Further studies are required\nto see how magnetic ﬂuctuations change with pbeyond\n2.64 GPa.\nIV. SUMMARY\nIn conclusion, we carried out139La nuclear magnetic\nresonance measurements in the itinerant ferromagnet\nLaCrGe 3under pressure to investigate its static and dy-\nnamic magnetic properties. From the pressure depen-\ndence of139La-NMR spectra at T=1.6 K, the internalmagnetic induction Bintat the La site in the ferromag-\nnetic ordered state is found to decrease very slightly by\nless than 5% with pfrom ambient pressure to 2.64 GPa.\nThisindicatesthattheCr3 dorderedmomentsarerobust\nunder pressure. In the ferromagnetic state, we observed\nthe broadening of NMR spectra under high pressures\nabovep >2.23 GPa. This suggests that inhomogeneity\nis induced by the application of pressure, which could be\nconsistent with a possible disorder in this system under\npressureaspointedoutinRef. [28]. Inaddition, fromthe\ntemperaturedependence of Bintand the Knightshift, the\nferromagnetic state is revealed to exist below ∼50 K at\np= 2.23 and 2.64 GPa under a magnetic ﬁeld of ∼7.2 T\nalthough we could not distinguish between long-range or\nshort-range magnetic order states. Based on the analy-\nsisofNMRdatausingthe self-consistent-renormalization\ntheory, the spin ﬂuctuations in the paramagnetic state\nwell above TCare revealed to be three dimensional fer-\nromagnetic throughout the measured pregion. In this\nsense, as pointed out in Ref. [28], LaCrGe 3might stand\nas a peculiar system having a new route to avoid a ferro-\nmagneticquantumcriticalpointbynotonlychangingthe\norderofthephasetransitionbutalsothroughtheappear-\nance of the high-pressure magnetic phase probably dom-\ninated by ferromagnetic interactions. To understand the\nnature of the avoidance of ferromagnetic quantum crit-\nicality in LaCrGe 3, further detailed studies under lower\nmagnetic ﬁelds as well as higher pressures greater than\n∼3 GPa will be required.\nV. ACKNOWLEDGMENTS\nThe authors would like to thank Y. Kuwata and Y.\nNoma for their help in conducting experiments and Q.-\nP. Ding for valuable discussions. The research was\nsupported by the U.S. Department of Energy, Oﬃce\nof Basic Energy Sciences, Division of Materials Sci-\nences and Engineering. Ames Laboratory is oper-\nated for the U.S. Department of Energy by Iowa State\nUniversity under Contract No. DE-AC02-07CH11358.\nPart of the work was supported by the Japan So-\nciety for the Promotion of Science KAKENHI Grant\nNumbers JP15H05882, JP15H05885, JP15K21732, and\nJP18H04321 (J-Physics). K. R. also thanks the KAK-\nENHI: J-Physics for the ﬁnancial support that provided\nan opportunity to be a visiting scholar at Kobe Univer-\nsity.\n[1] S. Sachdev, Quantum Phase Transitions , (Cambridge\nUniversity Press, New York, 2011).\n[2] E. A. Yelland, J.M. Barraclough, W. Wang, K.V.\nKamenev, and A.D. Huxley, High-ﬁeld superconductiv-\nity at an electronic topological transition in URhGe, Nat.\nPhys.7, 890 (2011).[3] S.S.Saxena, P.Agarwal, K.Ahilan, F.M.Grosche, R.K.\nW. Haselwimmer, M. J. Steiner, E. Pugh, I. R. Walker,\nS. R. Julian, P. Monthoux, G. G. Lonzarich, A. Huxley, I.\nSheikin, D. Braithwaite, and J. Flouquet, Superconduc-\ntivity on the border of itinerant-electron ferromagnetism\nin UGe 2, Nature 406, 587 (2000).8\n[4] E. Hassinger, D. Aoki, G. Knebel, and J. Flouquet,\nPressure–temperature Phase Diagram of Polycrystalline\nUCoGe Studied by Resistivity Measurement, J. Phys.\nSoc. Jpn. 77, 073703 (2008).\n[5] D. Aoki, A.D. Huxley, E. Ressouche, D. Braithwaite,\nJ. Flouquet, J. Brison, E. Lhotel, and C. Paulen, Co-\nexistence of superconductivity and ferromagnetism in\nURhGe, Nature 413, 613 (2001).\n[6] N. T. Huy, A. Gasparini, D..E. de Nijs, Y. Huang, J.C.P.\nKlaasse, T. Gortenmulder, A. de Visser, A. Hamann,\nT. G¨ orlach, and H. v. L¨ ohneysen, Superconductivity on\nthe Border of Weak Itinerant Ferromagnetism in UCoGe,\nPhys. Rev. Lett. 99, 067006 (2007).\n[7] C. Pﬂeiderer, S. R. Julian, and G. G. Lonzarich, Non-\nFermi-liquid nature of the normal state of itinerant-\nelectron ferromagnets, Nature 414, 427 (2001).\n[8] S. Takashima, M. Nohara, H. Ueda, N. Takeshita, C.\nTerakura, F. Sakai, and H. Takagi, Robustness of Non-\nFermi-Liquid Behavior near the Ferromagnetic Critical\nPoint in Clean ZrZn 2, J. Phys. Soc. Jpn. 76, 043704\n(2007).\n[9] M. Maeda, A. V. Andreev, K. Shirasaki, T. Yamamura\nand N. Kimura, Quantum phase transition in Itinerant-\nElectron Ferromagnet System U(Co 1−xOsx)Al, J. Phys.\nSoc. Jpn. 87, 094713 (2018).\n[10] P. C. Canﬁeld, and S. L. Bud’ko, Preserved entropy and\nfragile magnetism, Rep. Prog. Phys. 79, 084506 (2016).\n[11] M. Brando, D. Belitz, F.M. Grosche, and T. R. Kirk-\npatrick, Metallic quantum ferromagnets, Rev. Mod.\nPhys.88, 025006 (2016).\n[12] T. R.Kirkpatrickand D.Belitz, Ferromagnetic Quantum\nCritical Point in Noncentrosymmetric Systems, Phys.\nRev. Lett. 124, 147201 (2020).\n[13] H. Kotegawa, E. Matsuoka, T. Uga, M. Takemura, M.\nManago, N. Chikuchi, H. Sugawara, H. Tou, and H.\nHarima, Indication of ferromagnetic quantum critical\npoint in Kondo lattice CeRh 6Ge4, J. Phys. Soc. Jpn. 88,\n093702 (2019).\n[14] B.Shen, Y.Zhang, Y.Komijani, M.Nicklas, R.Borth, A.\nWang, Y. Chen, Z. Nie, R. Li, X. Lu, H. Lee, M. Smid-\nman, F. Steglich, P. Coleman, and H. Yuan, Strange-\nmetal behavior in a pure ferromagnetic Kondo lattice,\nNature (London) 579, 51 (2020).\n[15] V. Taufour, D. Aoki, G. Knebel, and J. Flouquet, Tri-\ncritical Point and Wing Structure in the Itinerant Ferro-\nmagnet UGe 2, Phys. Rev. Lett. 105, 217201 (2010).\n[16] H. Kotegawa, V. Taufour, D. Aoki, G. Knebel, and J.\nFlouquet, Evolution toward Quantum Critical end Point\nin UGe 2, J. Phys. Soc. Jpn. 80, 083703 (2011).\n[17] N. Kabeya, H. Maekawa, K. Deguchi, N. Kimura, H.\nAoki, and N. K. Sato, Non-Fermi Liquid State Bounded\nby a Possible Electronic Topological Transition in ZrZn 2,\nJ. Phys. Soc. Jpn. 81, 073706 (2012).\n[18] H. Kotegawa, T. Toyama, S. Kitagawa, H. Tou, R.\nYamauchi, E. Matsuoka, and H. Sugawara, Pressure-\nTemperature-Magnetic Field Phase Diagram of Ferro-\nmagnetic Kondo Lattice CeRuPO, J. Phys. Soc. Jpn. 82,\n123711 (2013).\n[19] E. Lengyel, M. E. Macovei, A. Jesche, C. Krellner, C.\nGeibel, and M. Nicklas, Avoided ferromagnetic quan-\ntum critical point in CeRuPO, Phys. Rev. B 91, 035130\n(2015).\n[20] J.G. Cheng, K. Matsubayashi, W. Wu, J.P. Sun, F. K.\nLin, J. L. Luo, and Y. Uwatoko, Pressure Induced Super-conductivity on the border of Magnetic Order in MnP,\nPhys. Rev. Lett. 114, 117001 (2015).\n[21] M. Matsuda, F. Ye, S. E. Dissanayake, J.-G. Cheng, S.\nChi, J. Ma, H. D. Zhou, J.-Q. Yan, S. Kasamatsu, O.\nSugino, T. Kato, K. Matsubayashi, T. Okada, and Y.\nUwatoko, Pressure dependence of the magnetic ground\nstates in MnP, Phys. Rev. B 93, 100405(R) (2016).\n[22] L.Xiang, E.Gati, S.L.Bud’ko, S.M.Saunders, andP.C.\nCanﬁeld, Avoided ferromagnetic quantum critical point\nin pressurized La 5Co2Ge3, Phys. Rev. B103, 054419\n(2021).\n[23] H. Bie, O. Y. Zelinska, A. V. Tkachuk, and A.\nMar, Structures and Physical Properties of Rare-Earth\nChromium Germanides RECrGe 3(RE = La-Nd, Sm),\nChem. Mater. 19, 4613 (2007).\n[24] U. S. Kaluarachchi, S. L. Bud’ko, P. C. Canﬁeld, and\nV. Taufour, Tricritical wings and modulated magnetic\nphases in LaCrGe 3under pressure, Nat. Commum. 8,\n546 (2017).\n[25] V. Taufour, U. S. Kaluarachchi, R. Khasanov, M. C.\nNguyen, Z. Guguchia, P. K. Biswas, P. Bonf` a, Roberto\nDe Renzi, X. Lin, S. K. Kim, E. D. Mun, H. Kim, Y.\nFurukawa, C.-Z. Wang, K.M. Ho, S. L. Bud’ko, and\nP. C. Canﬁeld, Ferromagnetic Quantum Critical Point\nAvoided by the Appearance of Another Magnetic Phase\ninLaCrGe 3underPressure, Phys.Rev.Lett. 117, 037207\n(2016).\n[26] X.Lin, V.Taufour, S.L.Bud’ko, andP.C. Canﬁeld, Sup-\npression offerromagnetism intheLaV xCr1−xGe3system,\nPhys. Rev. B 88, 094405 (2013).\n[27] J. M. Cadogan, P. Lemoine, B.R. Slater, A. Mar, and\nM. Avdeev, Neutron Diﬀraction Study of the Hexagonal\nPerovskite-Type Compound LaCrGe 3, Solid State Phe-\nnom.194, 71 (2013).\n[28] E. Gati, J. M. Wilde, R. Khasanov, L. Xiang, S. Dis-\nsanayake, R. Gupta, M. Matsuda, F. Ye, B. Haberl,\nU. Kaluarachchi, R. J. McQueeney, A. Kreyssig, S. L.\nBud’ko, P. C. Canﬁeld, Formation of short-range mag-\nnetic order and avoided ferromagnetic quantum critical-\nity in pressurized LaCrGe 3, Phys. Rev. B 130, 075111\n(2021).\n[29] K. Rana, H. Kotegawa, R. R. Ullah, J. S. Harvey, S.\nL. Bud’ko, P. C. Canﬁeld, H. Tou, V. Taufour, and Y.\nFurukawa, Magnetic ﬂuctuations in the itinerant ferro-\nmagnet LaCrGe 3studied by139La NMR, Phys. Rev. B\n99, 214417 (2019).\n[30] A Eiling and J S Schilling, Pressure and temperature\ndependence of electrical resistivity of Pb and Sn from 1-\n300 K and 0-10 GPa use as continuous resistive pressure\nmonitor accurate over wide temperature range; super-\nconductivity under pressure in Pb, Sn and In, J. Phys.\nF: Met. Phys. 11, 623 (1981)\n[31] A. Narath, Nuclear Spin-Lattice Relaxation in Hexago-\nnal Transition Metals: Titanium, Phys. Rev. 162, 320\n(1967).\n[32] V. Taufour, U. S. Kaluarachchi, S. L. Bud’ko, and P.\nC. Canﬁeld, Ferromagnetic quantum criticality: New as-\npects from the phase diagram of LaCrGe 3, Physica B:\nCondensed Matter 536, 483 (2018)\n[33] M. C. Nguyen, V. Taufour, S. L. Bud’ko, P. C. Canﬁeld,\nV. P. Antropov, C.-Z. Wang, and K.-M. Ho, Using ﬁrst-\nprinciples calculations to screen for fragile magnetism:\nCase study of LaCrGe 3and LaCrSb 3, Phys. Rev. B 97,\n184401 (2018).9\n[34] T. Moriya and K. Ueda, Nuclear magnetic relaxation in\nweakly ferro-and antiferromagnetic metals, Solid State\nCommun. 15, 169 (1974).\n[35] M. Hatatani and T. Moriya, Ferromagnetic Spin Fluc-tuations in Two-Dimensional Metals, J. Phys. Soc. Jpn.\n64, 3434 (1995)."    },    {        "title": "0709.3204v2.Spin_wave_resonances_in_La__0_7_Sr__0_3_MnO__3__films__measurement_of_spin_wave_stiffness_and_anisotropy_field.pdf",        "content": "arXiv:0709.3204v2  [cond-mat.str-el]  23 Sep 2007Spin wave resonances in La 0.7Sr0.3MnO3ﬁlms: measurement of\nspin wave stiﬀness and anisotropy ﬁeld\nM. Golosovsky∗, P. Monod,\nLaboratoire de Physique du Solide, ESPCI,\n10 rue Vauquelin, 75231 Paris Cedex 05, France\nP.K. Muduli, R.C. Budhani\nDepartment of Physics,\nIndian Institute of Technology,\nKanpur, 208016, India\n(Dated: November 1, 2018)\nAbstract\nWe studied magnetic ﬁeld dependent microwave absorption in epitaxial La 0.7Sr0.3MnO3ﬁlms\nusing an X-band Bruker ESR spectrometer. By analyzing angul ar and temperature dependence of\nthe ferromagnetic and spin-wave resonances we determine sp in-wave stiﬀness and anisotropy ﬁeld.\nThe spin-wave stiﬀness as found from the spectrum of the stand ing spin-wave resonances in thin\nﬁlms is in fair agreement with the results of inelastic neutr on scattering studies on a single crystal\nof the same composition [Vasiliu-Doloc et al., J. Appl. Phys .83, 7343 (1998)].\nPACS numbers: 76.50.+g,75.30.Ds, 75.40.Gb, 75.47.Lx\nKeywords: spin-wave resonance, magnon, manganite, thin ﬁlm, mic rowave absorption, spin-wave stiﬀness\n∗Permanentaddress: theRacahInstituteofPhysics,theHebrew UniversityofJerusalem,91904,Jerusalem,\nIsrael\n1INTRODUCTION\nSpin-wavestiﬀness, D,isanimportantparameterofmagneticmaterialsthatcharacteriz es\nthe magnon dispersion law, ω=Dk2. In the mean-ﬁeld approximation it is directly related\nto exchange integral J, namely,\nD=2JSa2\n/planckover2pi1(1)\nwhereγisthegyromagneticratio, aisthelatticeconstant, and Sisthespin.[1]Inelasticneu-\ntron scattering studies of the spin-wave stiﬀness in manganite sing le crystals [2, 3, 4, 5, 6, 7]\nrevealed important information on the nature of their ferromagne tic transition. Since this\nissue is under debate [7, 8] it is important to study spin-wave stiﬀnes s more closely, in partic-\nular, to measure Din thin ﬁlms. It is also important to explore complementary techniques .\nInprinciple, spin-waveexcitationscanbemeasuredusingmagneto- opticalKerreﬀectorBril-\nlouin light scattering [9] although existing studies of manganites by th ese methods [10, 11]\ndo not focus on spin waves.\nSpin-wave stiﬀness has been traditionally measured by the microwav e absorption tech-\nnique: using aﬁxedfrequency ESR spectrometer [9, 12]orbroadb andtechniques.[13]Several\ngroups reported ﬁxed frequency microwave studies of standing s pin-wave resonances (SWR)\nin La0.67Ba0.33MnO3(Ref. [14]), and La 0.7Mn1.3O3(Refs. [15, 16, 17]) thin ﬁlms. Although\nspin-wave excitation at microwave frequencies was observed in La 0.75Sr0.11Ca0.14MnO3(Ref.\n[18]), La 0.7Ca0.3MnO3(Ref. [19]) and La 0.7Sr0.3MnO3(Ref. [20]) thin ﬁlms, it was not\nstudied in detail.\nIn contrast to inelastic neutron scattering that (i) measures tra velling spin waves with\nlargewavevectors, k∼1-0.1˚A−1and(ii) requires largesingle crystals; the microwave absorp-\ntion technique measures spectrum of standing spin-wave resonan ces with small wavevectors,\nk∼10−2−10−3˚A−1and operates mostly with thin ﬁlms. The recent overview of spin-wav e\nexcitations in manganites [21] compares thin-ﬁlm microwave absorpt ion measurements to\ninelastic neutron scattering studies of single crystals. It turns ou t that there is no man-\nganite compound that was studied simultaneously by both technique s. Our present work\nﬁlls this gap. We study spin-wave spectrum in epitaxial La 0.7Sr0.3MnO3ﬁlms of diﬀerent\nthicknesses and on diﬀerent substrates by the microwave absorp tion technique at 9.4 GHz\nand compare our data to the inelastic neutron scattering studies o n single crystals of the\nsame composition.\n2STANDING SPIN-WAVE RESONANCES IN A THIN FERROMAGNETIC FILM\nConsider a thin ferromagnetic ﬁlm with an ”easy-plane” magnetic anis otropy. Magnetic\nﬁeldisorientedatobliqueangleΨwithrespect totheﬁlmnormal. Orient ationofmagnetiza-\ntion, Θ, is determined by the interplay between the external ﬁeld, H, and the perpendicular\nanisotropy ﬁeld, Ha, and is found from the following equation:[12, 22]\nHasinΘcosΘ = Hsin(Θ−Ψ) (2)\nwhere in-plane anisotropy has been neglected. In the presence of the microwave magnetic\nﬁeld with frequency ω, whereas hmw⊥H, the spin-wave resonances (SWR) are excited.\nThe resonance ﬁeld His found from the following condition:[12, 22]\nω2= [γHcos(Θ−Ψ)−γHacos2Θ+Dk2]\n×[γHcos(Θ−Ψ)−γHacos2Θ+ Dk2] (3)\nIf surface spins are completely pinned or completely unpinned, then\nk=πn/d (4)\nwheredis the ﬁlm thickness and nis an integer. The uniform FMR mode corresponds to\nn= 0, while thespin-wave resonances correspondto n∝negationslash=0. Fortheperpendicular orientation\n(Ψ = Θ = 0) Eq. (3) yields\nHn−H0=Ha−π2n2\nγd2D (5)\nwhereH0=ω\nγandωis the microwave frequency. To determine D, one measures microwave\nabsorption in dependence of magnetic ﬁeld and notices a sequence o f resonances. By ana-\nlyzing their spectrum using Eq. (5) one identiﬁes mode numbers n. The slope of the linear\ndependence, Hnvsn2, yieldsD, while the intercept with the H-axis yields Ha. To enable\nsuch procedure the ﬁlm thickness should lie in certain limits,\nδ >> d >> π/bracketleftbiggD\nγ(H0+Ha)/bracketrightbigg1/2\n(6)\nwhereδis the skin-depth. Indeed, for eﬃcient excitation of the spin-wave resonances the\nﬁlm thickness should be smaller than the skin-depth. On the other ha nd, the ﬁlm should be\nthick enough to support several resonances.\n3EXPERIMENTAL\nOur experiments were performed with a bipolar X-band Bruker ESR spectrometer, a\nTE102resonant cavity, and an Oxford cryostat. We studied La 0.7Sr0.3MnO3ﬁlms (d= 50,\n100, 150 and 200 nm) grown on the (001) SrTiO 3substrate (STO), and La 0.67Sr0.33MnO3\nﬁlms (d= 50 and 150 nm) on the NdGaO 3substrate (NGO). The samples were fabricated\nby the pulsed laser deposition [24] and were cut to small mm-size piece s in order to keep the\nreasonable value of the cavity Q-factor. We measured magnetization and resistivity of these\nﬁlms by SQUID magnetometry and four-point technique, correspo ndingly. The skin-depth\nat 9.4 GHz estimated from our resistivity measurements [23] - 22 µm at 295 K and 5 µm at\n50 K- considerably exceeds the ﬁlm thickness.\nThe ﬁlm uniformity could be estimated from the FMR spectrum. A unifo rm ﬁlm is char-\nacterized by a narrow FMR peak, corresponding to a well-deﬁned an isotropy ﬁeld, while a\nnonuniform ﬁlm usually exhibits a broad FMR peakindicating wide spread of theanisotropy\nﬁelds. Most part of our samples demonstrated narrow FMR peaks a t ambient temperature,\nalthough some samples showed several narrow FMR peaks corresp onding to discrete values\nof anisotropy ﬁeld.\nEXPERIMENTAL RESULTS\nLa0.7Sr0.3MnO3ﬁlms on the SrTiO 3substrate\nFigure 1 shows microwave absorption derivative in the perpendicular ﬁeld for a 200 nm\nthick La 0.7Sr0.3MnO3ﬁlm on the SrTiO 3substrate. We observe a series of slightly asymmet-\nric narrow peaks. The asymmetry arises fromthe coupling to the die lectric resonances in the\nSTO substrate and was observed in other studies as well. [20, 23] T he angular dependence\nof the resonant ﬁelds (not shown here) follows Eq. (5), hence we a ttribute these peaks to the\nspin-wave resonances. We cut the original 5 ×5 mm2ﬁlm to ﬁve pieces with the size of 1 ×\n0.5 mm2and all these pieces demonstrated fairly identical spectra. The ve ry fact that we\nobserve sharp and reproducible resonances proves that the ﬁlm is highly uniform. Another\nindication of the high quality of the ﬁlm is the narrow peak-to-peak line width (16 Oe at\nambient temperature) and low coercive ﬁeld (11-16 Oe at 295 K).\nThe mode numbers were established as follows. We assigned consecu tive numbers to the\n4peaks in Fig. 1 and checked whether linear dependence, Hn∝n2, predicted by Eq. (5),\nholds. The best correspondence to Eq. (3) was achieved for the s equence n=1,2,3.. or\nn=0,2,3... There is some ambiguity in whether the strongest peak corre sponds to n= 0,\nton= 1 or to their sum, since the splitting between the n= 0 and the n= 1 modes as\npredicted by Eq. (5), is only 25 Oe and this is comparable to the linewidt h.\nFigure 2 shows SWR intensities and linewidths. The linewidth increases a lmost linearly\nwith mode number, while intensity decreases. Odd modes have gener ally higher intensities\nthan even modes (see also Fig. 1).\nFigure3 shows dependence of the resonance ﬁelds on the mode num ber. The higher-order\nmodes obey quadratic dependence, Hn∝n2, while the modes with low nshow tendency to\nlinear spacing that is quite common for the ﬁlms with surface pinning. [2 5] We exclude from\nour analysis the ﬁrst two modes that should be most strongly aﬀect ed by surface pinning\nand determine spin-wave stiﬀness from the slope of Hnvsn2dependencies using Eq. (5).\nThe results are shown in Fig. 4.\nTo ﬁnd the perpendicular anisotropy ﬁeld we extrapolate the Hnvsn2dependencies to\nn= 0. Equation (7) and Fig. 3 yield Ha=0.4 T at 295 K and Ha=1 T at 4.2 K. To analyze\nthese values we note that the perpendicular anisotropy ﬁeld of a th in ﬁlm,\nHa=Hdemag+Hcryst+Hstress, (7)\nconsists of demagnetizing ﬁeld- 4 πM, crystalline anisotropy- Hcryst, and stress-induced\nanisotropy- Hstress. The crystalline anisotropy of La 0.7Sr0.3MnO3is very small [26, 27] and\ndoes not exceed 0.03 T.[28] Demagnetization ﬁeld as estimated from m agnetization [26, 27]\nisHdemag=0.74 T at 4.2 K. Since the lattice mismatch between La 0.7Sr0.3MnO3and SrTiO 3\nis only 1.4 % and the ﬁlm is suﬃciently thick, the Hstressis not high and achieves consider-\nable magnitude only at low temperatures.[29] Equation (7) yields the s tress anisotropy ﬁeld,\nHstress= 0.26 T at 4.2 K. This means that even at low temperatures the demag netization\nﬁeld is the dominant contribution to the anisotropy ﬁeld. This is consis tent with other\nmeasurements. Indeed, magnetization studies of the Ref. [27] fo r the ﬁlms of comparable\nthickness found that Hdemag= 0.8Haat 10 K, while Ref. [26] found that Hdemag= 0.95Ha\nat 295 K.\n5La0.67Sr0.33MnO3ﬁlms on the NdGaO 3substrate\nFigure 5 shows microwave absorption spectrum for a 150 nm thick La 0.67Sr0.33MnO3ﬁlm\non the NdGaO 3substrate at 108 K. We observe a strong and narrow peak at 1044 5 Oe and\na series of low-ﬁeld satellites. The peak-to-peak linewidth of the dom inant resonance is very\nsmall (12 Oe at ambient temperature and 33 Oe at 108 K) and this pro ves high quality of\nthe ﬁlm. The coercive ﬁeld at ambient temperature is only 4 Oe. We cut the ﬁlm to several\npieces and they showed consistent spectra.\nTo identify the resonances we measured temperature and angular dependencies of the\nresonant ﬁeld and came to conclusion that the peaks designated wit h integer numbers in\nFig. 5 are spin-wave satellites of the peak at 10445 Oe, while the stro ng peak at H= 7470\nOe does not belong to this series. Indeed, the angular dependencie s of resonant ﬁeld of the\nnumbered peaks are very similar (not shown here) and diﬀerent fro m that for the peak at\n7470 Oe. The same is true with respect to the temperature depend encies. We attribute\nthe peak at 7470 Oe to the region with a diﬀerent discrete value of th e anisotropy ﬁeld and\nexclude it from the subsequent analysis.\nFigure 6 shows that the resonant ﬁelds of higher modes follow Hn∝n2dependence. The\nslope of this dependence (the ﬁrst mode excluded) yields the spin-w ave stiﬀness D. The\nresults are plotted in Fig. 4. The same data yield the perpendicular an isotropy ﬁeld. We\nﬁndHa=0.73 T at 4.2 K. This is almost equal to the demagnetizing ﬁeld, 4 πM=0.74 T,\nand means that the stress-induced anisotropy is negligible here, as expected for the lattice-\nmatched substrate. The linewidth (see inset) steadily increases wit hn.\nOther ﬁlms\nWe observed spin-wave resonances in several ﬁlms with diﬀerent th ickness. In a very thin\n(d= 50 nm) La 0.67Sr0.33MnO3ﬁlm on SrTiO 3we observed only one spin-wave resonance\nwhich was displaced down by 0.23 T from the dominant FMR resonance [w hile Eq. (5) pre-\ndicts 0.08 T]. Such pronounced displacement is obviously related to st rong surface pinning,\nhence it is not possible to measure Dthere. In several 150 nm thick La 0.67Sr0.33MnO3ﬁlms\non SrTiO 3we observed two overlapping series of spin-wave resonances, so t hat determina-\ntion ofDhere was ambiguous. Ref. [17] observed such split spin-wave reso nances in a 350\n6nm thick La 0.7Mn1.3O3ﬁlm on LaAlO 3substrate and showed that the splitting disappears\nafter annealing in oxygen. Following Ref. [17] we studied the eﬀect o f oxygen annealing\non the SWR spectra in our ﬁlms. Contrary to Ref. [17] we found that oxygen annealing\nat diﬀerent temperatures from 6000C to 9000C does not ameliorate the SWR spectrum in\nour ﬁlms but introduces additional splitting. In particular, the reso nant peaks in annealed\nﬁlms split into many narrow lines with the spacing of ∼15 Oe. The diﬀerence between our\nresults and those of Ref. [17] is probably related to the fact that we operate with the ﬁlm\nof diﬀerent composition and on diﬀerent substrate.\nDISCUSSION\nZero-temperature spin-wave stiﬀness\nFigure 4 compares the spin-wave stiﬀness found in our microwave ab sorption studies of\nLa0.7Sr0.3Mn0.7O3thin ﬁlms to inelastic neutron scattering measurements on single cry stals\nof the same composition.[3, 4, 5] Consider ﬁrst the limit T= 0. Our measurements for\ntwo ﬁlms of diﬀerent thickness yield Dfilm\n0=210 meV ˚A2while the corresponding neutron\nscattering data show lower values, Dcryst\n0= 188 (Ref. [3]), 170 (Ref. [2]), 176 (Ref. [4]), and\n190 meV ˚A2(Ref. [5]). The discrepancy between Dfilm\n0andDcryst\n0may arise from the fact\nthat few atomic layers adjacent to the ﬁlm-substrate interface a re nonmagnetic. In LSMO\nthis ”dead layer” may be up to 5 nm thick,[30, 31] hence ”magnetic” th ickness that appears\nin Eq. (5) is smaller than the nominal thickness. The correction for t he ”dead layer” can\nbring down Dfilm\n0by 5%.\nThe spin-wave stiﬀness at T= 0 can be also estimated from the temperature dependence\nof magnetization and the T3\n2-Bloch law (see also Ref. [32]). For the La 0.7Sr0.3MnO3single\ncrystals this yields DBloch\n0=154 (Ref. [33]), while for ceramics of the same composition,\nDBloch\n0=197 meV ˚A2(Ref. [34]).\nTemperature-dependence\nConsider now the temperature dependence of the spin-wave stiﬀn ess. Figure 4 shows\nthat the results for two ﬁlms with diﬀerent thicknesses and on diﬀer ent substrates (one\nwith tensile stress-STO and another with a weak compressive stres s-NGO) are very close,\n7as expected for intrinsic property. At ambient temperature, we ﬁ nd for these two ﬁlms\nDfilm=104 and 114 meV ˚A2, correspondingly. This is almost identical to the single crystal\ndata at ambient temperature- Dcryst=114 (Ref. [3]) and 100 meV ˚A2(Ref. [4]). However,\none should take into account the diﬀerence in TCas well.\nComparison between the samples\nIn order to compare data for the samples with diﬀerent TCwe consider D(T) vsM(T)\nplots where the temperature is an implicit variable. A similar plot was use d earlier by\nRef. [35] to compare spin-wave stiﬀness of the Fe-Cr alloys with diﬀe rent composition and\ndiﬀerent TC. The rationale behind such plot is the mean-ﬁeld expression -Eq. (1) - that\nin fact relates the spin-wave stiﬀness to magnetization, M=gµBS/Va. Here,µBis the\nBohr magneton, gis theg-factor, and Va∝a3is the atomic volume. Equation (1) yields\ndirect proportionality, D∝JMa. Strictly speaking, this proportionality is expected only\nfor the Heisenberg model and at T=0. However, since MandDboth go to zero at TC,\nwhileJ, in general, varies continuously across the ferromagnetic transit ion; then we expect\nthat the D∝Mproportionality holds up to TC. Indeed, as the Ref. [36] shows, empirical\ndata on many ferromagnetic compounds suggest that DandMhave the same temperature\ndependence in the whole range from T= 0 toT=TC.\nTo eﬀectuate this approach we plot DvsM(Fig. 7) where DandMare measured\nat the same temperature.[37] We ﬁnd Dfrom the microwave absorption spectra, while\nthe magnetization is estimated indirectly, from the anisotropy ﬁeld, Ha. Indeed, since the\ndominant contribution to Hain our ﬁlms is the demagnetizing ﬁeld 4 πM(especially for the\nﬁlms on NGO), hence the anisotropy ﬁeld is a measure of magnetizatio n.\nFigure 7 plots the spin-wave stiﬀness versus anisotropy ﬁeld where the temperature is an\nimplicit parameter. It also plots the corresponding single crystal ne utron-scattering data of\nRef. [4] where magnetization was estimated from the intensity of th e electronic Bragg peak.\nThe upper horizontal scale in Fig. 7 was chosen in such a way that the low-temperature\nlimit of the electronic Bragg peak intensity (∆ IB= 1.85×104) corresponds to saturation\nmagnetization of La 0.7Sr0.3MnO3, i.e. 4πM= 0.74 T. The horizontal error bars in our thin\nﬁlm measurements take into account the possible diﬀerence betwee n the anisotropy ﬁeld and\nmagnetization arising from the stress anisotropy ﬁeld [Eq. (7)]. We a ssume extreme values\n8ofHstress/4πM=-0.06 and 0.2 for the ﬁlms on NGO and STO, correspondingly.\nWe observe that above T=295 K the data for both our ﬁlms and for the single crystal\ncollapse. The resulting D(M) dependence is quaisilinear that indicates the same critical\nindices of DandMatTC. This is not obvious since the neutron-scattering measurement\nwere performed in zero magnetic ﬁeld while the microwave measureme nts were performed\nin ﬁnite ﬁeld of 0.4 T to 1 T. Using Eq. (1) we ﬁnd the same value of the ex change integral,\nJ=3.6 meV (for S=1.85 and a= 3.88˚A) for the ﬁlm and single crystal. This is also not\nobvious since thin ﬁlms are strained.\nWeprocessedinthesamewaythemicrowaveabsorptiondataofRef . [15,16]fortheﬁlmof\ndiﬀerent composition -La 0.7Mn1.3O3(see inset in Fig. 7). While D(M) proportionality holds\nat low temperatures, there is a an upward deviation as T→TC. This suggests discontinuous\nvariation of Dacross the ferromagnetic transition. Our data for the La 0.7Sr0.3MnO3ﬁlms\ndo not indicate such discontinuity.\nLinewidth\nThe linewidth that steadily increases with mode number (Figs. 2,6) is no t frequently ob-\nserved in microwave absorption studies of spin-wave resonances. Previous studies of permal-\nloy ﬁlms and other highly conducting ferromagnets found weak and n onmonotonous depen-\ndence ∆H(n). (Refs. [38, 39, 40, 41, 42]) Such nonmonotonous dependence a rises from the\nsum of several sources such as eddy-current damping, surface roughness, and ﬂuctuations\n(exchange energy, anisotropy ﬁeld, thickness). [43]\nThek-dependent linewidth observed in our studies could be hardly intrinsic . Indeed,\ntheoretical prediction for the magnon-magnon scattering in mang anites yields inverse spin\nlifetimeΓ ∝k4(Ref. [44]). TheneutronscatteringstudiesofLa 0.85Sr0.15MnO3singlecrystals\n[6] indeed yield Γ = 3050 k4(herekis in˚A−1and Γ is in meV). Extrapolation of this k4\ndependence (obtained for 10 K and k=0.02-0.16 ˚A−1) down to the range of wavevectors\noccurring in our microwave studies ( kmax= 0.014˚A−1) yields Γ = 1 .1×10−4. This is much\nsmallerthanwhatweobserveinFig. 2-Γ = ∆ H/γ= 4.7×10−3(notehowever thatourﬁlms\nhave diﬀerent composition as compared to those studied in Ref. [6]) . This indicates that\nthek- dependent linewidth found in our studies in thin ﬁlms is extrinsic. It mo st probably\narises either from the inhomogeneous broadening associated with t hickness nonuniformity,\n9[41] or from the spin-wave scattering on surface and bulk disorder . [45, 46, 47]\nIf we take an extreme approach and assume only thickness nonunif ormity, then Ref. [41]\nyields\n∆Hd= ∆H0+2Dπ2n2\nd3∆d, (8)\nwhere ∆H0is the linewidth of the uniform precession mode and ∆ dis the average thickness\nvariation. Analysis of our data according to Eq. (8) (dashed lines in F igs. 2,6) yields ∆ d=\n4 nm (2%) and 6 nm (4%) for the ﬁlms on STO and on NGO, correspondin gly.\nIf we go to another extreme and assume spin-wave scattering to b e the only source of line\nbroadening, then according to the transit-time treatment [47] we ﬁnd\n∆Hsc−∆H0=vg\nγlsc=2Dπn\nγdlsw(9)\nwherevg= 2Dkis the group velocity and lswis the mean free path. Spin-wave scattering\non bulk disorder is characterized by the combination of linear and qua dratick-dependence,\n[45, 46] whereas the linear k-dependence implies constant mean free path. Analysis of our\ndata according to Eq. (9) and assuming constant lsw, yieldslsw/d= 2 for the LSMO/STO\nﬁlm at ambient temperature (Fig. 2). For the LSMO/NGO ﬁlm (Fig. 6) w e ﬁndlsw/d= 1.4\nand 1.3 at 108 K and at 187 K, correspondingly. The spin-wave mean- free-path being on the\norder of ﬁlm thickness, suggests that the scattering occurs pre dominantly at ﬁlm interfaces.\nSuch scattering may result from surface roughness [48] or from t he localized Mn4+-ions [31]\nor Mn2+-ions [49] at ﬁlm surface.\nThe observed k-dependence of the linewidth is most probably due to both mechanism s:\nthickness nonuniformity and scattering. Note however, that bot h extreme approaches [Eq.\n(8) and Eq. (9)] yield long mean free path that practically excludes b ulk scattering. This\nis surprising, since the stoichiometry of La 0.7Sr0.3MnO3suggests two possible states of Mn-\nion: Mn3+with spin S=3/2 and Mn4+with spin S=2. If La 0.7Sr0.3MnO3were Heisenberg\nferromagnet this would certainly result in magnetic inhomogeneity. H owever, observation of\nstanding spin-wave resonances with high mode numbers indicates ne gligible bulk scattering\nand a single magnetic state of Mn-ions in the bulk, as was already esta blished by the NMR\nstudies of Mn55in La0.67Sr0.63MnO3(Ref. [31]).\n10Other manganite compounds\nThestandingspin-waveresonanceshavebeenobservedsofarint hemanganitecompounds\nof the general formula La 1−xAxMnO3where A is Sr,Ca,Ba,Mn and x∼0.3 (Refs. [14, 15,\n16, 17, 18, 19, 20] and present work). The x∼0.3 composition corresponds to highest\nconductivity and is the most distant from phase boundaries, [50] in other words, the phase\nseparation eﬀects should be least insigniﬁcant here. This compositio n has also the highest\nspin-wave stiﬀness [5, 21] that indicates increased length of the ex change interaction. The\nvery fact that so far there is no indication of the standing spin-wav e resonances in manganite\ncompounds with x∝negationslash= 0.3 (we also didn’t ﬁnd any spin-wave resonances in La 0.8Sr0.2MnO3\nﬁlms) may serve as an indirect evidence of strong spin-wave scatte ring there.\nCONCLUSIONS\nWe measured temperature dependence of the spin-wave stiﬀness in thin La 0.7Sr0.3MnO3\nﬁlms as determined from the standing spin-wave resonances at micr owave frequencies. At\nambient temperature, the spin-wave stiﬀness of thin ﬁlms and of sin gle crystals is the same,\nwhile at low temperatures the spin-wave stiﬀness of thin ﬁlms is enhan ced with respect to\nthat of a single crystal.\nThe spin-wave linewidth in our ﬁlms is limited by the scattering at ﬁlm inter faces. The\nvery fact that we are able to observe spin-wave resonance in La 0.7Sr0.3MnO3up to eighth\norder implies high degree of coherence and very low bulk spin-wave sc attering. This is\nimportant for spintronics and means that La 0.7Sr0.3MnO3can be used as a spin conserving\ncomponent.\nWe are grateful to Denis Golosov for illuminating discussions, to Xiang zhen Xu for the\nhelp with the handling of the samples, to Oscar Arnache who initiated t hese experiments,\nand to anonymous referee for instructive comments.\n11[1] C. Kittel, Phys. Rev. 110, 1295 (1958).\n[2] J.W. Lynn, R.W. Erwin, J.A. Borchers, Q. Huang, A.A. Sant oro, J.L. Peng, Z.Y. Li, Phys.\nRev. Lett. 76, 4046 (1996).\n[3] M.C. Martin, G. Shirane, Y. Endoh, K. Hirota, Y. Moritomo , Y. Tokura, Phys. Rev. B 53,\n14285 (1996).\n[4] L. Vasiliu-Doloc, J.W. Lynn, Y.M. Mukovskii, A.A. Arsen ov, and D.A. Shulyatev, J. Appl.\nPhys.83, 7342 (1998).\n[5] A.H. Moudden, L. Vasiliu-Doloc, L. Pinsard, A. Revcolev schi, Physica B 241, 276 (1998).\n[6] L. Vasiliu-Doloc, J. W. Lynn, A. H. Moudden, A. M. de Leon- Guevara, and A. Revcolevschi,\nPhys. Rev. B 58, 14913 (1998).\n[7] J. Zhang, F. Ye, Hao Sha, P. Dai, J. A. Fernandez-Baca, and E.W.Plummer, J. Phys.: Con-\ndens. Matter 19, 315204 (2007).\n[8] M.B. Salamon and M. Jaime, Rev. Mod. Phys. 73, 583 (2001).\n[9] C. E. Patton, Phys. Rep. 103, 251 (1984).\n[10] F.L.A. Machado, M.A. Lucena, A.E.P. de Araujo, A. Azeve do,F.M. de Aguiar, S.M. Rezende,\nA.K. Nigam, Physica B 320, 119 (2002).\n[11] D. Talbayev, H. Zhao, G. Lupke, J. Chen, Qi Li, Appl. Phys . Lett.86, 182501 (2005).\n[12] M. Farle, Rep. Prog. Phys. 61, 755 (1998).\n[13] V.P. Denysenkov, A.M. Grishin, Rev. Sci. Instrum. 74, 3400 (2003).\n[14] S.E. Loﬂand, S.M. Bhagat, C.Kwon, M.C.Robson, R.P. Sha rma, R. Ramesh, T. Venkatesan,\nPhys. Lett. A 209, 246 (1995).\n[15] V. Dyakonov, A. Prohorov, V. Shapovalov, V. Krivoruchk o, V. Pashchenko, E. Zubov, V.\nMihailov, P. Aleshkevych, M. Berkowski, S. Piechota, H. Szy mczak, J. Phys.:Cond. Matter\n13, 4049 (2001).\n[16] V. Dyakonov, A. Prohorov, V. Shapovalov, S. Khartsev, V . Krivoruchko, V. Mihailov, V.\nPashchenko, E. Zubov, P. Aleshkevych, K. Dyakonov, S. Piech ota, H. Szymczak, Phys. Lett.\nA268, 202 (2000).\n[17] P.Aleshkevych, M.Baran, V. Dyakonov, R. Szymczak, H. S zymczak, K. Baberschke, J. Lind-\nner, K. Lenz, Phys. St. Sol. (a) 203, 1586 (2006).\n12[18] J. Yin, Y.X. Sui, J.H. Du, Y.X. Zhang, X.Y. Liu, Z.G. Liu, Phys. Stat. Sol. (a) 174, 499\n(1999).\n[19] A.I. Shames, E. Rozenberg, G. Gorodetsky, M. Ziese, in: Nanostructured Magnetic Materials\nand their Apllications , pp.205-212, Kluwer Acad. Publ. (2004).\n[20] D.L. Lyfar, S.M. Ryabchenko, V.N. Krivoruchko, S.I. Kh artsev, A.M. Grishin, Phys. Rev. B\n69, 100409(R) (2004).\n[21] P.Aleshkevych, M.Baran, and H.Szymczak, Acta Physica Polonica 106, 593 (2004).\n[22] A.G. Gurevich and G.A. Melkov, Magnetization Oscillat ions and Waves, CRC Press, Boca\nRaton, 1996, p.19.\n[23] M. Golosovsky, M.Abu-Teir, D. Davidov, O. Arnache, P. M onod, N. Bontemps, R.C.Budhani,\nJ. Appl. Phys. 98, 084902 (2005).\n[24] K. Senapati and R. C. Budhani, Phys. Rev. B 71, 224507 (2005).\n[25] T.G. Phillips, Proc. Royal Soc. A 292, issue 1429, p.224 (1966).\n[26] Y. Suzuki, H.Y. Hwang, S.W. Cheong, R.B. van Dover, Appl . Phys. Lett. 71, 140 (1997).\n[27] M. Ziese, H.C. Semmelhack, P. Busch, J. Magn. Magn. Mat. 246, 327 (2002).\n[28] S.E. Loﬂand, S.M. Bhagat, C.Kwon, S.D. Tyagi, Y.M. Muko vskii, S.G. Karabashev, A.M.\nBalbashov, J. Appl. Phys. 81, 5737 (1997).\n[29] L. Ranno, A. Llobet, R. Tiron, E. Favre-Nicolin, Appl. S urf. Science 188, 170 (2002).\n[30] B. Wiedenhorst, C. Hofener, Yafeng Lu, J. Klein, L. Alﬀ, R . Gross, B.H. Freitag, W. Mader,\nAppl. Phys. Lett. 74, 3636 (1999).\n[31] A.A. Sidorenko, G.Allodi, R. De Renzi, G. Balestrino, M .Angeloni, Phys. Rev. B 73, 054406\n(2006).\n[32] N. Rosov, J.W. Lynn, J. Kastner, E.F. Wassermann, T. Cha ttopadhyay, H. Bach, J. Appl.\nPhys.75, 6072 (1994).\n[33] V.N. Smolyaninova, J.J. Hamilton, R.L. Greene, Y.M. Mu kovskii, S.G. Karabashev, A.M.\nBalbashov, Phys. Rev. B 55, 5640 (1997).\n[34] S. Budak, M. Ozdemir, B. Aktas, Physica B 339, 45 (2003).\n[35] T. Rossing and T.G. Phillips, J. Appl. Phys. 39, 1375 (1968).\n[36] U. Kobler, A. Hoser, W. Schafer, Physica B-cond.matt. 364, 55 (2005).\n[37] Our measurements were performed with the thermocouple mounted at some distance from the\nsample.Sincethermalization isslow, torestrictthedurat ionoftheexperimenttoseveralhours,\n13in some experimental runs we slowly heated the sample from lo w temperature and measured\nthe spectra in a continuous way without waiting for thermali zation. In such experiments we\ndetermine only DandHa.\n[38] P. Lubitz, S.M. Bhagat, G.C. Bailey, C. Vittoria, Phys. Rev. B11, 3585 (1975).\n[39] G. Suran and R.J. Gambino, J. Appl. Phys. 50, 7671 (1979).\n[40] F. Schreiber and Z. Frait, Phys. Rev. B 54, 6473 (1996).\n[41] C.G. Bailey and C. Vittoria, Phys. Rev. Lett. 28, 100 (1972).\n[42] T.G. Phillips and H.M. Rosenberg, Rep. Prog. Phys. 29, 285 (1966).\n[43] J.C.S. Levy, Phys. Rev. B 25, 2893 (1982).\n[44] D.I. Golosov, Phys. Rev. Lett. 84, 3974 (2000).\n[45] Y. Motome and N. Furukawa, Phys. Rev. B 71, 014446 (2005).\n[46] U. Hoeppe and H. Benner, Phys. Rev. B 71, 144403 (2005).\n[47] D.G. Scotter, J. Phys. D: Appl. Phys. 5, L93 (1972).\n[48] A. Azevedo, A. B. Oliveira, F. M. de Aguiar, and S. M. Reze nde, Phys. Rev. B 62, 5331\n(2000).\n[49] M.P. de Jong, I. Bergenti, V.A. Dediu, M. Fahlman, M. Mar si, C. Taliani, Phys. Rev. B 71,\n014434 (2005).\n[50] M. Imada, A.Fujimori, Y.Tokura, Rev. Mod. Phys. 70, 1039 (1998).\nFIG. 1: Absorption derivative spectrum for a 200 nm thick La 0.7Sr0.3MnO3ﬁlm on the SrTiO 3\nsubstrate in the perpendicular magnetic ﬁeld. The mode numb er is shown at each peak. The\nmicrowave frequency is 9.4 GHz, the incident power is Pmw= 0.2 mW (-30 dB), the modulation\nﬁeld is 10 Oe.\nFIG. 2: Theﬁlled symbols indicate the SWR linewidth, ∆ Hn, the red solid line shows linear ﬁt [Eq.\n(9)], and the red dashed line shows quadratic ﬁt [Eq. (8)]. Th e open symbols indicate integrated\nSWR intensity corrected for the linewidth, I=Ipp(∆H)2. Here,Ippis the peak-to-peak magnitude\nof the absorption derivative. The black solid line here is th e guide to the eye, the dashed black\nline shows Kittel’s 1 /n2prediction [1]. The sample is a 200 nm thick La 0.7Sr0.3MnO3ﬁlm on the\nSrTiO3substrate at ambient temperature.\n14FIG. 3: The resonance ﬁeld of diﬀerent modes for the ﬁlm shown i n Fig.1. Filled symbols indicate\nthe data measured at ﬁxed and known temperatures and at low mi crowave power, Pmw= 0.2\nmW. Open symbols indicate the data measured at ambient tempe rature and increased microwave\npower (62.5 mW and 200 mW, correspondingly). Here, the sampl e temperature is higher than the\nambient temperature due to self-heating. The lines show Hn∝n2approximation.\nFIG. 4: Temperature dependence of the spin-wave stiﬀness D(T). The circles stand for our mi-\ncrowave measurements on thin ﬁlms on substrate. The squares show inelastic neutron scattering\ndata for the La 0.7Sr0.3MnO3single crystals. The triangles stand for the DBloch(T= 0) estimated\nfrom the temperature dependence of magnetization and T3\n2-Bloch law. The lines are the guide to\nthe eye.\nFIG. 5: Absorption derivative spectrum for a 150 nm thick La 0.67Sr0.33MnO3ﬁlm on the NdGaO 3\nsubstrate. The mode number is shown at each peak. A strong pea k atH=7470 Oe originates\nfrom the ferromagnetic resonance in a region of the ﬁlm with t he diﬀerent anisotropy ﬁeld.\nFIG. 6: Resonance ﬁeld of the SWR modes at several temperatur es for the sample shown in Fig. 5.\nThe lines show Hn∝n2approximation. The inset shows linewidth vs mode number. Th e dashed\nlines in the inset show quadratic approximation [Eq. (8)].\nFIG. 7: Spin-wave stiﬀness versus magnetization at varying t emperature. The circles stand for our\nmicrowave measurements on thin ﬁlms. As a measure of magneti zation we take the perpendicular\nanisotropy ﬁeld, Ha(lower horizontal scale). Thesquaresstandfor thesinglec rystal datameasured\nby the inelastic neutron scattering technique (Ref. [4]). A s a measure of magnetization in this case\nwe take the integrated intensity of the electronic Bragg pea k (upper horizontal scale). The dashed\nline shows linear approximation. The inset shows similar D(M) dependence for the La 0.7Mn1.3O3\nﬁlm as extracted from the microwave absorption measurement s of Ref. [15, 16].\n155000 6000 7000 8000 \nH  (Oe) 1\n23\n45\n67x50 \nLSMO/STO 295 K dP  \ndH (arb.units) 1x10 6\n0\n-1x10 6\n-2x10 6\nFig.1 050 100 150 \n10 010 110 210 310 4\n0 2 4 6 8 10 \nIntensity  (arb.units) Linewidth  (Oe) \nnLSMO/STO \n∆HI\nFig.2 4000 6000 8000 10000 \n0 20 40 60 80 100 221 K \n270 K \n289 K LSMO/STO H  (Oe) \nn2\nFig.3 050 100 150 200 250 \n0 100 200 300 400 LSMO/STO \nLSMO/NGO \nsingle crystal-Ref.[4] \nsingle crystal- Ref.[3] \nsingle crystal -Ref.[5] \nsingle crystal- Ref.[32] \nceramics- Ref.[33] D  (meV A 2)\nT  (K) \nFig.4 7000 8000 9000 10000 -1 10 6-5 10 505 10 5\nLSMO/NGO 1\n2\n3 4\n6\nx10 3\n4\n5spurious phase 108 K \nH  (Oe) dP  \ndH (arb.units) 5x10 5\n0\n-5x10 5\n-1x10 6\nFig.5 050 100 150 200 \n0 1 2 3 4 5 6 7108 K \n152 K \n187 K \nn\n\u0001H  (Oe) \n6000 8000 10000 \n0 10 20 30 40 50 60 70 LSMO/NGO 108 K \n236K \n292 K H  (Oe) \nn2180 K \nFig.6 050 100 150 200 \n0 2000 4000 6000 8000 LSMO/STO \nLSMO/NGO \nsingle crystal-Ref.[4] D (meV A 2)\nHa  (Oe) ∆IB  (x10 4)\n295 K 28 K \n197 K \n248 K 200 K \n296 K 221 K 0                  0.5                    1                   1.5                   2 \n108 K \n295 K \n346 K \nFig.7 050 100 150 200 \n0 2000 4000 6000 8000 Ref. [15] D (meV A 2)\nHa  (Oe) "    },    {        "title": "2007.03257v2.Current_Fluctuations_Driven_by_Ferromagnetic_and_Antiferromagnetic_Resonance.pdf",        "content": "Current Fluctuations Driven by Ferromagnetic and Antiferromagnetic Resonance\nArne Brataas\nCenter for Quantum Spintronics, Department of Physics,\nNorwegian University of Science and Technology, NO-7491 Trondheim, Norway\u0003\n(Dated: August 17, 2020)\nWe consider electron transport in ferromagnets or antiferromagnets sandwiched between metals.\nWhen spins in the magnetic materials precess, they emit currents into the surrounding conductors.\nGenerally, adiabatic pumping in mesoscopic systems also enhances current \ructuations. We gener-\nalize the description of current \ructuations driven by spin dynamics in three ways using scattering\ntheory. First, our theory describes a general junction with any given electron scattering properties.\nSecond, we consider antiferromagnets as well as ferromagnets. Third, we treat multiterminal de-\nvices. Using shot noise-induced current \ructuations to reveal antiferromagnetic resonance appears\nto be easier than using them to reveal ferromagnetic resonance. The origin of this result is that the\nassociated energies are much higher as compared to the thermal energy. The thermal energy governs\nthe Johnson-Nyquist noise that is independent of the spin dynamics. We give results for various\njunctions, such as ballistic and disordered contacts. Finally, we discuss experimental consequences.\nI. INTRODUCTION\nIn conductors, a bias voltage generates a net current.\nHowever, the current also \ructuates. Noise exists even\nat equilibrium when the bias voltage is zero. At equilib-\nrium, the current \ructuations are related to the conduc-\ntance via the \ructuation-dissipation theorem as Johnson-\nNyquist noise1,2. When there are non-zero bias voltages\ncomparable to or larger than the thermal energy, the\n\ructuation-dissipation theorem does not apply. Instead,\nthe current \ructuates due to shot noise since the elec-\ntron \row is in discrete quanta of the elementary charge\n\u0000e. The shot noise reveals quantum transport features\nin nanostructures3.\nIn electron transport, at low temperatures, the trans-\nmission probabilities of waveguide eigenmodes fTngde-\ntermine the shot noise of phase-coherent conductors bi-\nased by a voltage V:\nS=2e2\nheVX\nnTn(1\u0000Tn): (1)\nThe shot noise expression (1) is general and captures the\nnature of many contacts, such as di\u000busive, ballistic, and\ntunnel junctions. The factor 1 \u0000Tnarises from the Pauli\nexclusion principle; two electrons cannot simultaneously\noccupy the same waveguide mode. The sum is over the\nwaveguide modes labeled by n.\nFerromagnets have intriguing transport properties\ncaused by the coupling between electric currents, elec-\ntron spin currents, and localized spin dynamics. Currents\ncan induce spin dynamics by spin-transfer torques4{10\nand spin-orbit torques11{18. The magnetization direction\ncan be switched, or magnetic oscillations can be induced.\nThese phenomena are of a fundamental importance and\nmight be utilized in magnetic random access memories,\n\u0003Arne.Brataas@ntnu.nospin-torque oscillators or spin-logic devices. These de-\nvelopments have been reviewed in Refs. 19{22. The phe-\nnomenon reciprocal to spin-transfer torque is spin pump-\ning, the emission of spin currents into metals induced by\nspin excitations in adjacent magnets23{29. Spin pump-\ning exposes details of the transport properties and spin\ndynamics.\nRecently, antiferromagnetic spintronics has attracted\nconsiderable interest because of the intrinsic high fre-\nquencies, new features in spin dynamics, and robustness\nwith respect to external magnetic \feld disturbances30{36.\nMany of the phenomena in ferromagnets have similar\nor richer behavior in antiferromagnets. For instance,\ncurrents can switch the spin con\fgurations31,37,38, and\nantiferromagnetic resonance excitations can pump spin\ncurrents35,36,39{41.\nUsually, bias voltages induce electric currents and shot\nnoise as in Eq. (1). However, out-of-equilibrium currents\ncan be sustained by other methods using temporal exter-\nnal or internal drivers that modify the conductor proper-\nties. Oscillating electric and magnetic \felds can induce\nnet currents. Such drivers also enhance the electric cur-\nrent noise. In magnetic systems, dynamical spin excita-\ntions produce spin currents23{29,35,36,39{41.\nSpin pumping also causes additional magnetization\ndissipation27,42{45. Through the \ructuation-dissipation\ntheorem, this implies that \ructuating spin currents as-\nsociated with spin pumping and spin transfer as well\nexist46. In a recent study, Ref. 47 obtained an expression\nfor the electric (charge) current noise caused by ferro-\nmagnetic resonance excitations in ferromagnetic-normal\nmetal-ferromagnetic double tunnel barrier systems. The\nparticularly nice feature is that the mechanism does not\nrequire spin-orbit-induced spin-to-charge conversion such\nas the spin Hall and inverse spin Hall e\u000bects. It is also a\nnew channel for detecting and characterizing ferromag-\nnetic resonance and electron transport in magnetic con-\nductors.\nTheoretically, scattering matrices capture electron\ntransport in nanostructures well48. They can also de-arXiv:2007.03257v2  [cond-mat.mes-hall]  14 Aug 20202\nscribe current-induced torques49{54. Scattering matrices\nalso capture e\u000bects due to temporal external or inter-\nnal drivers. To the lowest order in the driver frequency,\nthe pumped current is related to the stationary scat-\ntering properties55. This feature considerably simpli-\n\fes the description of adiabatic pumping such as spin-\npumping27,56,57. In general, scattering properties can\nalso describe the enhanced electric current noise due to\nperiodic drivers. For the case when only one waveguide\nmode is linked to each reservoir, Ref. 58 obtained an ex-\npression for the current noise in terms of the dynamical\nscattering properties of the device.\nWe consider a magnet that is in contact with normal\nmetal leads. Our purpose is to obtain a general expres-\nsion for how spin excitations in ferromagnetic and anti-\nferromagnetic structures generate the thermal and shot\nnoise of the electric current. To this end, we generalize\nthe results of Ref. 47 in three ways: 1) the formalism is\nvalid for arbitrary junctions, 2) the theory applies to spin\ndynamics in ferromagnets and antiferromagnets, and 3)\nmultiterminal devices are treated. In this way, we obtain\ngeneral results for electric current noise driven by spin\ndynamics in magnetic materials in arbitrary junctions.\nWe will \fnd that, when spin angular momentum is con-\nserved, the noise vanishes when the magnet is insulating.\nOur results are therefore most relevant for conducting\nsystems.\nIn antiferromagnets, the deduced expressions for the\nnoise are entirely new to the best of our knowledge. Our\ngeneral results in the case of ferromagnetic excitations\nare also new. In the limited case of two-terminal double-\nbarrier tunnel ferromagnetic junctions, our general re-\nsults agree with the results of Ref. 47 by taking into ac-\ncount random disorder in our formulation. Since we use\nan entirely di\u000berent approach, this agreement establishes\nthe consistency of both treatments in this limit. We dis-\ncuss the fact that other junctions have di\u000berent behaviors\nin ferromagnets.\nWe have organized the presentation as follows. Our pa-\nper \frst gives the main results and consequences before\nproceeding section by section with more details of the\nderivations. The next section II introduces the model\nand presents the main results. We will \fnd that four fac-\ntors determine the shot noise: i) The electron-transport-\nrelated shot noise coe\u000ecients, ii) the driver frequency,\niii) the thermal energy, and iv) the spin-dynamics fac-\ntor. Section III discusses the speci\fcs of the in\ruence\nof ferromagnetic and antiferromagnetic dynamics driven\nby magnetic \felds that govern the spin-dynamics factor.\nThen, in section IV, we discuss the shot noise coe\u000ecients\nin various junctions, such as ballistic and disordered con-\ntacts, both in antiferromagnets and ferromagnets. We\npresent the general theory of adiabatic pumping-induced\nelectric current noise in section V. Section VI applies the\ngeneral theory in section V to derive the spin dynamics-\ndriven shot noise in section III. We conclude our pre-\nsentation in section VII. Finally, we derive the general\nscattering theory of adiabatic driven enhanced electriccurrent noise in Appendix A.\nII. MODEL AND MAIN RESULTS\nWe consider a magnet embedded between metals (or\nsemiconductors) in an open circuit. At equilibrium, elec-\ntric currents \ructuate in the metals. In the magnet, there\ncan be thermally induced spin \ructuations or coherent\nspin precessions caused by external forces. We consider\nthe latter case that the spin dynamics is coherent and\ndominated by external drivers as in ferromagnetic reso-\nnance or antiferromagnetic resonance. It is straightfor-\nward to generalize our results to explicitly include con-\ntributions from incoherent spin dynamics relevant when\nthe external drive is weak or absent.\nWhen the spins in the magnet precess, the \ructuations\nare enhanced. Fig. 1 schematically depicts the system in\na two-terminal con\fguration. Our results are also valid\nfor many terminals. The unit vector aligned with the\nMetal Metal Magnet\nδjl(t) δjr(t) n(t)\nFIG. 1. Schematic description of a metal-magnet-metal sys-\ntem. An open circuit (not shown) is connected to the system.\nThe electric currents \ructuate. The spin dynamics in the\nmagnet described by precession of the temporal unit vector\nalong the order parameter n(t) enhance the current \ructua-\ntions.\norder parameter, the magnetization in ferromagnets and\nthe staggered \feld in antiferromagnets, n, is homoge-\nneous. When external magnetic \felds or currents drive\nthe system, the order parameter nprecesses around an\nequilibrium direction. While we subsequently develop\na formulation describing general junctions that may in-\nclude the spin-orbit interaction and magnetic impurity\nscattering, our \frst and primary focus is on systems with\nthe conservation of spin angular momentum. Giant mag-\nnetoresistance, tunnel magnetoresistance, spin-transfer\ntorques, and spin pumping are examples of central phe-\nnomena in such systems.\nIn systems with the conservation of spin angular mo-\nmentum, two independent scattering matrices, S\"and\nS#, for spin-up and spin-down electrons govern electron\ntransport. The scattering matrices contain all details of\nthe junctions related to the interfaces between the metals\nand magnets, band structure, and bulk and surface im-\npurity scattering. We evaluate the electric current and3\nthe associated noise in the metallic leads. The electric\ncurrent direction is towards the magnet. While our for-\nmalism is valid irrespective of the magnet`s conducting\nproperties, it is most relevant for metallic systems since\nwe will demonstrate that, in the absence of spin-orbit\ncoupling, the electric noise vanishes when there is no \row\nof electric charge between the leads. Furthermore, we\nconsider systems where itinerant electrons carry the cur-\nrent and spin currents carried by localized spins can be\ndisregarded. When the spin angular momentum is con-\nserved, our main result is that the low-frequency electric\ncurrent noise in the presence of coherent spin excitations\nhas two contributions:\np\u0010\u0011=p(th)\n\u0010\u0011+p(sn)\n\u0010\u0011; (2)\nwhere\u0010and\u0011label the leads. Electric current conserva-\ntion ensures thatP\n\u0010p\u0010\u0011= 0 =P\n\u0011p\u0010\u0011.\nIn Eq. (2), the \frst term describes thermal Johnson-\nNyquist noise, which is independent of the spin dynamics\nand determined by the conductance tensor Gand the\nthermal energy kBT:\np(th)\n\u0010\u0011= (G\u0010\u0011+G\u0011\u0010)kBT: (3)\nMicroscopically, the conductance tensor is a sum over the\nscattering properties of two spin components:\nG\u0010\u0011=e2\nhTroh\n\u000e\u0010\u0011\u0000S\"y\n\u0011\u0010S\"\n\u0010\u0011i\n+\ne2\nhTroh\n\u000e\u0010\u0011\u0000S#y\n\u0011\u0010S#\n\u0010\u0011i\n: (4)\nThe trace, Tr o, is over orbital (\"o\") degrees of freedom\nonly, a sum over the waveguide modes in the leads.\nThe second and more interesting contribution to the\nnoise in Eq. (2) is the shot noise driven by the coherent\nspin dynamics. We obtain our main result for the zero-\nfrequency shot noise:\np(sn)\n\u0010\u0011=A\u0010\u0011+A\u0011\u0010\n8\u0014\n~!coth~!\n2kBT\u00002kBT\u0015\nD(!);(5)\nwhere the spin-dynamics control the spin-dynamics fac-\ntorD(!) that is independent of the electron transport\nproperties. We discuss D(!) further below. D(!) is a\npositive de\fnite quantity. !is the frequency of the spin\nexcitations. The diagonal components of the shot noise\nof Eq. (5) are positive de\fnite quantities as are the di-\nagonal components of the shot noise coe\u000ecients A. The\nshot noise coe\u000ecients A\u0010\u0011depend on the electron trans-\nport via products of the scattering matrices of spin-up\nand spin-down electrons:\nA\u0010\u0011=e2\nhTro2\n4\u000e\u0010\u0011\u0000X\n\u000b\fS\"\n\u0010\u000bS#y\n\u000b\u0011S#\n\u0011\fS\"y\n\f\u00103\n5+\ne2\nhTro2\n4\u000e\u0010\u0011\u0000X\n\u000b\fS#\n\u0010\u000bS\"y\n\u000b\u0011S\"\n\u0011\fS#y\n\f\u00103\n5: (6)In the case of a two-terminal device, as in Fig. 1:\nAll=2e2\nhTroh\n1\u0000(r\"\nllr#y\nll+t\"\nltt#y\nrl)(r#\nllr\"y\nll+t#\nltt\"y\nrl)i\n;(7)\nwherelmeans left and rmeans right, ris a re\rection\ncoe\u000ecient matrix, and tis a transmission coe\u000ecient ma-\ntrix.\nIn general, the shot noise parameter Aof Eq. (6) di\u000bers\nfrom the conductance Gof Eq. (4), as does the voltage-\nbiased shot noise of Eq. (5) compared to the average cur-\nrent governed by the conductance G. As is well known\nfor the latter case3, signatures of the junctions and con-\nductors can be distinguished by the ratio between the\nvoltage-biased shot noise parameter and the conductance\nvia the so-called Fano factor, F=P\nnTn(1\u0000Tn)=P\nnTn.\nFor instance, in tunnel junctions F= 1, and in di\u000busive\nwiresF= 1=33. The spin dynamics-driven shot noise\nreveals more aspects of the electron transport in spin\nmaterials. We will compute the central shot noise pa-\nrameter of Eq. (7) for ballistic and disordered junctions\nin ferromagnets and antiferromagnets in Section IV.\nRef. 58 found that the adiabatic pumping driven en-\nhanced noise was related to the behavior of two parti-\ncles injected into the system. In agreement with this,\nthe shot noise parameter Aof Eq. (6) contains products\nof four scattering matrices describing two-particle pro-\ncesses. The new aspect of the shot noise parameter A\nof Eq. (6) is that two of the scattering matrices relate\nto spin-up electrons, and two relate to spin-down elec-\ntrons. In contrast, spin pumping is a one-particle process.\nThe spin-mixing conductance27,50,51,56is a product of one\nspin-up scattering matrix and one spin-down scattering\nmatrix, two scattering matrices in total. This is because\nthe pumped spin current has spin along the direction\ntransverse to the magnetization direction, a linear com-\nbination of spin-up and spin-down states along the spin\nquantization axis that is parallel to the order parame-\nter. Similarly, we note that the shot noise coe\u000ecients\nof Eq. (6) have combinations of spin-up and spin-down\nproperties related to the same lead. Since spin dynamics\nproduce electric (charge) current noise, a natural inter-\npretation is that the \ructuations arise due to temporal\n\ructuations of the emissions of spin currents. While the\nemitted spin currents are instantaneously transverse to\nthe order parameter, they can be reconverted to electric\n(charge) currents at later times due to the spin-\fltering\ne\u000bect in magnetic materials.\nWe observe that the shot noise parameter Avanishes\nwhen no transmission occurs between the left and right\nreservoirs. This can be seen by letting t\"!0 and\nt#!0 in Eq. (7) and using the unitarity of the scat-\ntering matrices. This behavior implies that no electric\nnoise will occur in spin dynamics-driven metal-magnetic\ninsulator-metal junctions when spin angular momentum\nis conserved. In contrast, spin pumping and spin-transfer\ntorques can be as e\u000ecient in metal-magnetic insulator bi-\nlayers as in metal-magnetic conductor bilayers59. Beyond\nthe formulation in this section that is based on spin con-4\nservation, spin-orbit coupling in heavy metals provides a\nconversion between charge and spin currents so that even\nmagnetic insulators can become noisy60. Such small ef-\nfects are proportional to the square of the small spin Hall\nangle.\nIn the expression for the shot noise (5), the spin dy-\nnamics solely determine the spin-dynamics factor D(!).\nThis quantity is small and related to the power ab-\nsorbed in resonance experiments42, which implies that\nit can be independently measured. At equilibrium and\nat su\u000eciently low temperatures, n(t) =n0. Oscillat-\ning transverse magnetic \felds at frequency !,H(t) =\nH+expi!t+H\u0000exp\u0000i!t, induce small transverse ex-\ncitations of the order parameter, \u000en=n+expi!t+\nn\u0000exp\u0000i!t. In the linear response, the changes in the\norder parameter and the (external or current-induced)\nmagnetic \felds are related by the frequency-dependent\nspin susceptibility \u001f(!), a 2\u00022 matrix in the basis of\nthe transverse coordinates labeled by i, so thatni\u0006=\n\u001fij\u0006Hj\u0006. In terms of the spin susceptibilities and the\noscillating magnetic \felds:\nD(!) =X\nini+ni\u0000=X\nijk\u001fij+\u001fik\u0000Hj+Hk\u0000: (8)\nThe spin susceptibilities \u001f(!) have peaks at the reso-\nnance frequencies, as does D(!). In section III, we give\ngeneric examples for central classes of anisotropies in fer-\nromagnets and antiferromagnets. In the linear response,\nthe transverse excitations are small. Therefore, the factor\nDis small. Nevertheless, distinguishing the shot noise\nfrom the thermal noise should be possible because the\nformer has a strong dependence on the frequency of the\ndriver, while the latter has no such features. Subtract-\ning the frequency-independent background thermal noise\nreveals the shot noise.\nThe shot noise of Eq. (5) takes a di\u000berent form depend-\ning on the ratio between the energy quantum associated\nwith the time dynamics, ~!, and the thermal energy,\nkBT. At low temperatures, when ~!\u001dkBT, the shot\nnoise becomes:\np(sn)\n\u0010\u0011\u0019A\u0010\u0011+A\u0011\u0010\n8j~!jD(!): (9)\nThe shot noise can be distinguished from direct heat-\ning by the di\u000berent frequency dependence. The low-\ntemperature shot noise of Eq. (9) is linear in the absolute\nvalue of the excitation frequency !relative to the spin-\ndynamics factor D(!) that can be independently mea-\nsured.\nWe \fnd below that A\u00182Gin many systems. The\nratio between the shot noise of Eq. (5) and the thermal\nnoise of Eq. (3) at low temperatures is then psn\n\u0010\u0011=pth\n\u0010\u0011\u0018\nj~!jD(!)=kBT. Since the transverse precession angle is\nsmall, typically D(!)\u001810\u00004at resonance, but the pos-\nsibly large prefactor j~!j=kBTwill increase the ratio be-\ntween the shot noise and the thermal noise from this\nvalue. Stronger external drives can also enhance D(!).In contrast, at high temperatures, kBT\u001d~!, the shot\nnoise is smaller. We can expand the shot noise in the\nsmall parameter ~!and obtain:\np(sn)\n\u0010\u0011\u0019A\u0010\u0011+A\u0011\u0010\n8(~!)2\n6kBTD(!): (10)\nAt high temperatures, the shot noise of Eq. (10) is sup-\npressed by a factor j~!j=6kBTwith respect to the low\ntemperature limit of the shot noise of Eq. (9).\nFerromagnets typically have resonance frequencies less\nthan 100 GHz. These frequencies correspond to a low\ntemperature of less than 1 K. Transport measurements\nin this temperature range can reveal the low-temperature\nshot noise (9). Such and considerably lower-temperature\nmeasurements are standard in the study of the fractional\nquantum Hall e\u000bect and require sophisticated cryogenic\ninstrumentation. At the temperature of liquid helium,\napproximately 4 K, the ratio between the resonance en-\nergy and the thermal energy is approximately 0.2.\nThe resonance frequencies in antiferromagnets can be\none to two orders of magnitude higher than those in\nferromagnets. Therefore, detecting the low-temperature\nlimit of the shot noise of Eq. (9) appears to be easier\nfor antiferromagnets. Antiferromagnets can have reso-\nnance frequencies in the THz range. We can then expect\nto observe low-temperature shot noise (9) at tempera-\ntures below 10 K when an antiferromagnet precesses at\nits resonance frequency. At room temperature, the ra-\ntio between the high-temperature shot noise of Eq. (10)\nand the low-temperature shot noise of Eq. (9) is on the\norder 2\u000210\u00004. Such corrections are small, but their mea-\nsurement might be possible since corrections due to, e.g.,\nthe spin Hall magnetoresistance (SMR), are of a similar\nmagnitude and routinely probed61.\nWe conclude that detection of low-temperature shot\nnoise (9) should be possible in antiferromagnets and, with\ncryogenic techniques, in ferromagnets. Measurement of\nthe high-temperature shot noise (10) is possible in both\nsystems.\nIII. SPIN DYNAMICS\nIn this section, we will compute the spin-dynamics fac-\ntorD(!) in ferromagnets and antiferromagnets.\nConsider a uniaxial ferromagnet with the easy axis\nalong thezdirection. The free energy density is\nfF=\u0000M\n2\r!Am2\nz+\u000efF; (11)\nwhere mis a unit vector along the magnetization with\nmagnitude Mand!Ais the anisotropy energy. A trans-\nverse oscillating magnetic \feld drives the spin dynam-\nics via the additional contribution to the free energy,\n\u000efF=!H?(mxcos!t+mysin!t)M=\r , where!H?is\nthe magnitude of transverse magnetic \feld in units of fre-\nquency. We compute the spin susceptibility that governs5\nthe spin dynamics factor (8) from the Landau-Lifshitz-\nGilbert equation\n@m\n@t=\u0000m\u0002!e\u000b+\u000bm\u0002@m\n@t; (12)\nwhere the e\u000bective \feld !e\u000bdepends on the free energy\ndensity (11) as !e\u000b=\u0000\r\u000efF=M\u000emand\u000bis the Gilbert\ndamping constant. In linear response, the spin dynamics\nfactor (8) then becomes\nDF(!) =!2\nH?\n2 [(!\u0000!A)2+\u000b2!2]: (13)\nAs in Eq. (8), the spin dynamics factor of Eq. (13) is\nquadratic in the transverse \felds, represented by their\nmagnitudes !H?in units of frequency. At resonance,\nDF(!A) = (!H?=\u000b!A)2=2.\nSimilarly, we can consider a uniaxial antiferromagnets\nwith the easy axis along the zdirection. The free energy\ndensity is:\nfAF=L\n2\r\u0002\n!Em2\u0000!An2\nz\u0003\n+\u000efAF; (14)\nwhere nis a unit vector along the staggered \feld,\nmis the dimensionless small magnetic moment, Lis\nthe magnitude of the staggered magnetization, \ris\nthe gyromagnetic ratio, !Eis the exchange energy,\nand!Ais the anisotropy energy. A transverse os-\ncillating magnetic \feld drives the spin dynamics via\nthe additional contribution to the free energy, \u000efAF=\n!H?(mxcos!t+mysin!t)L=\r. The coupled Landau-\nLifshitz-Gilbert equations for the staggered \feld nand\nthe magnetic moment mare\n@n\n@t=\u0000n\u0002!m,e\u000b\u0000m\u0002!n,e\u000b\n+\u000bn\u0002@m\n@t+\u000bm\u0002@n\n@t; (15)\n@m\n@t=\u0000n\u0002!n,e\u000b\u0000m\u0002!m,e\u000b\n+\u000bn\u0002@n\n@t+\u000bm\u0002@m\n@t; (16)\nwhere the e\u000bective \felds !n,e\u000band!m,e\u000b depend on the\nfree energy density (14) as !n,e\u000b=\u0000\r\u000efAF=L\u000enand\n!m,e\u000b=\u0000\r\u000efAF=L\u000em.\nIn linear response, and in the exchange approximation,\n!E\u001d!A, the spin dynamics factor (8) becomes:\nDAFM (!) =!2!2\nH?\n2(!2\u0000!2r)2+ 8\u000b2!2!2\nE; (17)\nwhere!r=p2!A!Eis the resonance energy and \u000b\nis the Gilbert damping constant. As in Eqs. (8) and\n(13), the spin dynamics factor of Eq. (17) is quadratic\nin the transverse \felds, represented by their magnitudes\n!H?in units of frequency. At resonance, DAFM (!r) =\n(!H?=\u000b!E)2=8.Generalizations to other anisotropies and the inclu-\nsion of e\u000bects arising from external magnetic \felds are\nstraightforward in both antiferromagnets and ferromag-\nnets.\nIV. JUNCTIONS\nIn this section, we compute the shot noise coe\u000ecients\nAfor simple models of ballistic and disordered junc-\ntions. Beyond the scope of the present paper, exten-\nsions of these calculations are feasible. Generalizations\nto consider the e\u000bects of the band structure with ab ini-\ntio calculations and more complicated models of junc-\ntions and disorder are possible. Similar calculations have\nbeen successfully carried out for interface resistances62,\nspin-transfer torques63, spin pumping64, and Gilbert\ndamping43,45.\nIn ferromagnets, the potential landscapes for spin-up\nand spin-down electrons strongly di\u000ber. Therefore, the\nre\rection and transmission amplitudes as well as prob-\nabilities are spin dependent. In antiferromagnets, the\nre\rection and transmission probabilities are the same for\nspin-up and spin-down electrons under compensation of\nthe localized spins. Nevertheless, the quantum mechan-\nical phases associated with re\rection and transmission\ndi\u000ber for the two spin directions.\nA. Clean metal\nIn clean, ballistic systems, the waveguide modes ex-\nperience either perfect transmission or perfect re\rec-\ntion. In a simple semiclassical model of a normal metal-\nferromagnet-normal metal junction, we can assume N\"\npropagating channels for spin-up electrons and N#prop-\nagating channels for spin-down electrons. Then,\nAll=2e2\nhPN; (18)\nwhereP= (N\"\u0000N#)=(N\"+N#) is the polarization and\nN=N\"+N#is the total number of conducting channels.\nSimilarly, the two-terminal conductance becomes Gll=\ne2N=h so that the ratio between the shot-noise coe\u000ecient\nand conductance is All=Gll= 2P.\nIn a similar model of compensated antiferromagnets,\nN\"=N#, and thus,\nAll= 0 (19)\nwhileGll=e2N=h so thatAll=Gll= 0.\nTherefore, for this simple semiclassical model, the shot\nnoise vanishes in antiferromagnets. However, this is\ngenerically not the case for other kinds of junctions. More\nrealistic models of clean junctions will probably result in\na small but \fnite shot noise coe\u000ecient in antiferromag-\nnets as well.6\nThe semiclassical results for clean junctions illustrate\nthat the shot noise coe\u000ecients can strongly di\u000ber in an-\ntiferromagnets and ferromagnets.\nB. Disordered Metals\nWhen su\u000ecient disorder exists, either because of bulk\nimpurity scattering or scattering at boundaries, we can\nuse random matrix theory to evaluate the average of scat-\ntering matrices. In the semiclassical regime, the phases of\nthe re\rection and transmission coe\u000ecients are random.\nThey are also statistically independent for spin-up and\nspin-down electrons. The averages of the transmission\nand re\rection probabilities are62:\nT\u001b\nij=1\nN1\n1 +\u0019\u001b(20)\nand\nR\u001b\nij=1\nN\u0019\u001b\n1 +\u0019\u001b; (21)\nwhere\u0019\u001b=\u001a\u001bdNe2=Ah,Nis the number of waveguide\nmodes,\u001a\u001bis the resistivity for each spin direction, dis\nthe width of the junction, Ais the cross section of the\njunction, and the spin directions are \u001b=\"and\u001b=#. We\ncan then compute that the spin-dependent conductance\nisG\u001b= (e2=h)P\nijTij=Gd\u001b=(1 +Gd\u001b=Gsh), where the\nconductance of a di\u000busive conductor is Gd\u001b=A=\u001a\u001bd\nand the Sharvin conductance is Gsh=e2N=h. A more\nintuitive expression is that the resistance consists of the\nSharvin resistance in series with the di\u000busive resistance,\n1=G\u001b= 1=Gsh+ 1=Gd\u001b. The total conductance is G=\nG\"+G#.\nIn ferromagnets, the conductances for spin-up and\nspin-down electrons di\u000ber. Based on Eqs. (20) and (21),\nwe can now obtain the average:\nhAlli= 2\u0012\nG\"+G#\u00002G\"G#\nGsh\u0013\n: (22)\nIn the di\u000busive regime, G\";G#\u001cGsh, and we obtain\nAll= 2G. This result agrees with the results computed\nin Ref. 47 for a double barrier tunnel junction system.\nWhile the transport regimes in our approach and Ref.\n47 are not identical, the treatments seem to share the\ncommon feature that strong randomization of the elec-\ntron trajectories occurs. It is, therefore, natural that the\nresults agree in this limited case.\nIn compensated antiferromagnets, spin-up and spin-\ndown electrons have the same conductance, G\"=G#=\nG=2. However, the phases of the re\rection and trans-\nmission coe\u000ecients for the spin-up and spin-down elec-\ntrons remain statistically independent, as in ferromag-\nnets. Then, the shot noise coe\u000ecient is:\nhAlli= 2e2\nh\u0012\nG\u0000G2\n2Gsh\u0013\n; (23)and in the di\u000busive limit, we obtain the same result as\nfor a ferromagnet, All= 2G.\nWe conclude that for disordered ferromagnets and anti-\nferromagnets, the ratio between the shot noise coe\u000ecient\nand the two-terminal conductance is All=Gll= 2.\nV. THEORY OF PUMPING-INDUCED NOISE\nWe will, in this section, present our general results\nfor noise enhancements by adiabatic pumping. We de-\nrive these results from the general scattering theory with\nmulti-terminals and an arbitrary number of waveguide\nmodes in Appendix A. The results in this section are\nvalid for any periodic drive and are not limited to spin-\ndynamics driven noise discussed in section II. We will,\nin the next section VI, use the results in this section to\nobtain the results for the spin-dynamics drive noise that\nwe presented in section II.\nWe consider phase-coherent conductors attached to\nreservoirs via leads. Within the conductors, scattering\nby spin-conserving impurities, the spin-orbit interaction,\nand the exchange \feld arising from localized spins can\noccur. Above, in section II, we have assumed that spin\nangular momentum is conserved and that the magneti-\nzation in ferromagnets or staggered \felds in antiferro-\nmagnets is homogeneous. However, we do not use these\nassumptions here when presenting the general formula\nfor pumping-induced noise. Appendix A gives details of\nthe derivation of the formulas presented in this section.\nWhile we consider a general setup with many reser-\nvoirs, we give an example of a three-terminal device in\nFig. 2. Currents can \row between the reservoirs, arising\nlead leadleadscattering\nregion\nreservoirreservoir\nreservoir\nFIG. 2. Schematic example of a three-terminal device. A scat-\ntering region (red area) is connected via leads (green areas)\nto particle reservoirs (blue areas). Currents can \row between\nthe reservoirs.\nfrom either di\u000berences in bias voltages therein or time-\ndependent changes within the scattering region. Above,7\nwe have considered the latter case when spin excitations\ndrive the scattering region. Our focus is on the current\n\ructuations when all of the reservoirs are at equilibrium.\nWe consider a general phase-coherent conductor. Scat-\ntering matrices then describe transport between the\nreservoirs. All orbital waveguide modes and spin quan-\ntum numbers span these scattering matrices. In general,\nthe matrices have diagonal and o\u000b-diagonal components\nin orbit and spin. In our case, since the scattering region\nchanges in time, the scattering matrices also have a com-\nplex temporal dependence. However, when the temporal\nchanges are slow compared to the typical electron trans-\nport time, knowing the temporal behavior of the frozen\nscattering matrix is su\u000ecient (see Appendix A). We eval-\nuate the frozen scattering matrix at a snapshot in time\nwhen the driver has a constant value. This scattering\nmatrix isS\u000bn\rl(t;\u000f), where\u000bis the outgoing lead, nis\nthe outgoing waveguide mode (including spin), \ris the\nincoming lead, lis the incoming waveguide mode (includ-\ning spin),tis the time, and \u000fis the electron energy.\nThe current \ructuations are\nP\u0010\u0011(t1;t2) =1\n2h\u0001I\u0010(t1)\u0001I\u0011(t2)+\u0001I\u0011(t2)\u0001I\u0010(t1)i;(24)\nwhere \u0001I\u0010(t) =I\u0010(t)\u0000hI\u0010i(t) is the deviation of the\ncurrentI\u0010(t) from its expectation value hI\u0010(t)iin lead\u0010.\nThe period of the driver is T= 2\u0019=!. Following Ref.\n58, apart from a factor of 2, we de\fne the zero frequency\nnoise as:\np\u0010\u0011=ZT\n01\nTZ1\n\u00001d\u001cP\u0010\u0011(t+\u001c=2;t\u0000\u001c=2): (25)\nOur \frst central step is that we compute a general ex-\npression for the noise induced by a slowly and periodic\nvarying change in the scattering region. When the elas-\ntic transport properties are weakly energy dependent, the\ncurrent cross correlations are:\np\u0010\u0011=X\nqX(s)\n\u0010\u0011(~!q)kBT\n+1\n2X\nqX(s)\n\u0010\u0011(~!q)\u0014\n~!qcoth~!q\n2kBT\u00002kBT\u0015\n;(26)\nwhere the \frst term represents the thermal noise contri-\nbution and the second represents the shot noise contribu-\ntion. The frequency quantum ~!qrelates to the period\nTof the driver by ~!q= 2\u0019q=T , whereqis an inte-\ngral number. The coe\u000ecients X(s)\n\u0010\u0011(~!) = [X\u0010\u0011(~!) +\nX\u0010\u0011(\u0000!q)]=2 are determined by the scattering matrices:\nX\u0010\u0011(~!q) =e2\nhX\nn\u0010n\u0011X\n\fm\rl\b\u0010n\u0010\fm\rl(!q)\b\u0011n\u0011\rl\fm(\u0000!q);\n(27)\nwhere\n\b\u0010n\u0010\fm\rl(!q) =1\nTZT\n0dte\u0000i!qt\b\u0010n\u0010\fm\rl(t); (28)\b\u000bn\fm\rl (t) =\u0002\n\u000e\u000bn\fm\u000e\u000bn\rl\u0000S\u0003\n\u000bn\fm (t)S\u000bn\rl(t)\u0003\n;(29)\nand the static (\"frozen\") scattering matrices are to be\nevaluated at the Fermi energy.\nThe result for the thermal and shot noise of Eq. (26)\nare general for any drivers and valid when the elastic\ntransport properties are weakly energy-dependent. In\nthe next section VI, we use this general result to \fnd the\nnoise driven by spin excitations.\nVI. SPIN DYNAMICS-DRIVEN NOISE\nIn this section, we explain how we can use the general\nresult of the pumping-driven noise in the previous sec-\ntion V to obtain the shot noise when the pumping is due\nto spin dynamics. We consider homogeneous spin dy-\nnamics relevant to ferromagnetic resonance and antifer-\nromagnetic resonance. Now, we assume the conservation\nof spin angular momentum as in the phenomena of spin-\ntransfer torques and spin pumping. We do not explicitly\nconsider the spin-orbit coupling instrumental relevant for\ne.g. spin-orbit torques, but further investigations using\nthe same formalism can elucidate its role.\nSince the degrees of freedom of the orbital are indepen-\ndent of the spin degrees of freedom, we use the notation\nthat the states nconsist of orbital quantum numbers no\nand spin quantum numbers s,n!nos. When spin an-\ngular momentum is conserved, we separate the frozen\nS-matrix into spin-independent (labelled by superscript\n\"(c)\") and spin-dependent terms (labelled by superscript\n\"(s)\"):\nS\u0011nos\u0010mos`=S(c)\n\u0011no\u0010mo\u000ess`+\u001bss`\u0001n(t)S(s)\n\u0011no\u0010mo;(30)\nwhere nis a unit vector in the direction of the order\nparameter, n2= 1. The calculations in this section are\nvalid for both ferromagnets where the order parameter\nis the magnetization and for antiferromagnets where the\norder parameter is the staggered \feld.\nAs the spins precess, only the spin-dependent part of\nthe scattering matrix acts as a pump. Inserting the spin-\ndependent scattering matrix into Eq. (27) and using the\nunitarity of the scattering matrices and the normalization\nn2= 1, after considerable algebra, we obtain that the\nfactor that appears in the general expression for the noise\nof Eq. (26) becomes:\nX(s)\n\u0010\u0011=ZT\n0dt1\nTe\u0000i!qtqZT\n0dt2\nTei!qt2\u0002\n\u0014\nG\u0010\u0011+G\u0011\u0010+A\u0010\u0011+A\u0011\u0010\n8[n(t1)\u0000n(t2)]2\u0015\n;(31)\nwhere the conductance tensor Gis de\fned in Eq. (4) and\nthe shot noise coe\u000ecients Aare de\fned in Eq. (6).\nTo proceed to \fnd the expression for the spin-dynamics\ndriven shot noise of Eq. (2) with the thermal contribution\nof Eq. (3) and the shot noise contribution of Eq. (5), we8\nneed to evaluate the following integral appearing in the\nlast term of Eq. (31):\nW=Zdt1\nTe\u0000i!qtqZdt2\nTei!qt2[n(t1)\u0000n(t2)]2(32)\nto the second order in the deviation of the order parame-\nter from equilibrium. To this end, expanding to the linear\norder is su\u000ecient:\nn(t1)\u0000n(t2) =X\n\u0006\u000en\u0006\u0002\ne\u0006i!t1\u0000e\u0006i!t2\u0003\n; (33)\nwhere\u000en+ and\u000en\u0000are transverse to the equilibrium\nspin directions n0.\nInserting the linear expansion of Eq. (33) into Eq. (32),\nwe then obtain that W(~!q= 0) = 2\u000en+\u0001\u000en\u0000and\nW(~!q=\u0006~!) =\u0000\u000en+\u0001\u000en\u0000. As a consequence, we\n\fnd\nX(s)\n\u0010\u0011(~!q= 0) = (G\u0010\u0011+G\u0011\u0010) +A\u0010\u0011+A\u0011\u0010\n82\u000en+\u0001\u000en\u0000\n(34)\nand\nX(s)\n\u0010\u0011(~!q=\u0006~!) =\u0000A\u0010\u0011+A\u0011\u0010\n8\u000en+\u0001\u000en\u0000:(35)\nFor both ferromagnets and antiferromagnetrs, we can\nnow insert the expressions for X(s)of Eqs. (34) and (35)\ninto the general expression for the noise of Eq. (26). The\nthermal contribution to the noise is then Eq. (3) and\nis independent of the spin oscillations. The shot noise\ncontribution is given in Eq. (5).\nVII. CONCLUSIONS\nIn conclusion, we have presented general expressions\nfor the noise driven by ferromagnetic and antiferromag-\nnetic resonance. The noise consists of thermal and shot\nnoise contributions. Conductances determine the ther-\nmal noise. Shot noise coe\u000ecients and the frequency-\ndependent magnitude of the spin excitations determine\nthe shot noise. The shot noise attains its maximum at\nferromagnetic resonance in ferromagnets and antiferro-\nmagnetic resonance in antiferromagnets.\nThe shot noise parameter can be evaluated for arbi-\ntrarily junctions. We have given examples for ballistic\nsystems and disordered systems. The ratio between the\nspin dynamics-driven shot noise parameter and the con-\nductance is smaller for ballistic systems than for disor-\ndered systems. This feature is similar to the behavior of\nthe Fano factor associated with voltage-driven shot noise.\nOur formalism can be generalized to treat the spin-\norbit coupling related to spin-orbit torques and electric\n(charge) pumping54,65. Such extensions will shed further\nlight on spin-charge conversions related to spin dynamics.ACKNOWLEDGMENTS\nThis work was supported by the Research Council\nof Norway through its Centres of Excellence funding\nscheme, project number 262633, \"QuSpin\". We would\nlike to thank Akashdeep Kamra, Sebastian Goennenwein,\nand Thomas Tybell for comments on the manuscript.\nAppendix A: Derivation of General Theory of Noise\ndue to Adiabatic Pumping\nUsing Floquet scattering theory, Ref. 58 considered\npumping-driven noise in a multiterminal con\fguration\nwith one-dimensional leads. In other words, each lead\nonly had one waveguide mode. The purpose of the\npresent section is to generalize this description to \fnd\nequations for arbitrary two- and three-dimensional leads\nthat can also capture the e\u000bects of impurities and bound-\nary scattering. To this end, we include many waveguide\nmodes. In our derivation, we also found that an alter-\nnative path without explicitly using Floquet scattering\nstates could be easily followed. We will demonstrate\nthat our results agree with the results in Ref. 58 for one-\ndimensional leads.\nIn metallic systems, the energy quantum associated\nwith the pump oscillations is typically much smaller than\nthe Fermi energy. In this regime, the relevant previous\nresult is Eq. (27) in Ref. 58 when the scattering matrix\nis weakly energy dependent due to the noise:\npM\u0010\u0011=p(th)\n\u0010\u0011+p(sh)\nM\u0010\u0011: (A1)\nThe expression for the thermal noise p(th)\n\u0010\u0011is the same\nas that in Eq. (3) in the limit of only one mode in all\nleads. The one-dimensional shot noise contribution in\nthe notation of Ref. 58 is:\np(sh)\nM\u0010\u0011=2e2\nh1X\nq=1C(sym)\n\u0010\u0011;q\u0014\n~!coth~!q\n2kBT\u00002kBT\u0015\n;(A2)\nwhereC(sym)\n\u0010\u0011q= [C\u0010\u0011q+C\u0010\u0011\u0000q]=2,\nC\u000b\fq=X\n\r\u000e\u0002\nS\u0003\n\u000b\rS\u000b\u000e\u0003\nq\u0002\nS\u0003\n\f\u000eS\f\r\u0003\n\u0000q; (A3)\nand the (frozen) scattering matrices should be evaluated\nat the Fermi energy EF. The Fourier transform of the\nproduct of the (frozen) scattering matrices (at the Fermi\nenergy) is de\fned as follows:\n\u0002\nS\u0003\n\u000b\rS\u000b\u000e\u0003\nq=ZT\n0dt\nTeiq!t\u0002\nS\u0003\n\u000b\r(t)S\u000b\u000e(t)\u0003\n: (A4)\nWe reproduce the result in Ref. 58 represented by Eqs.\n(A1), (A2), (A3), and (A4) for one-dimensional leads and\nobtain generalizations to leads with an arbitrary number\nof waveguide modes.9\nThe starting point for our derivation is the expression\nfor the current operator in lead \u000b:\n^I\u000b(t) = 2\u0019~eX\nnh\n^ay\n\u000bn(t)^a\u000bn(t)\u0000^by\n\u000bn(t)^b\u000bn(t)i\n;(A5)\nwhere\u000bdenotes the lead and ndenotes the transverse\nwaveguide mode (orbital and spin). The outgoing oper-\nators ^bare related to the incoming operators ^ avia the\ntime-dependent scattering matrix S:\n^b\u000bn(t1) =X\n\fmZ1\n\u00001dt2S\u000bn\fm (t1;t2)^a\fm(t2):(A6)\nWe use the Fourier transform as follows:\n^a\fm(t) =1\n2\u0019~Z\nd\u000fe\u0000i\u000ft^a\fm(\u000f) (A7)\nand the corresponding inverse Fourier transform. At\nthermal equilibrium, the thermal averages are:\nh^ay\n\u000bn(\u000f2)^a\fm(\u000f1)ieq=\u000e\u000b\f\u000enm\u000e(\u000f2\u0000\u000f1)f(\u000f1);(A8)\nwheref(\u000f) is the Fermi-Dirac distribution function that\ndepends on the chemical potential \u0016and the thermal\nenergykBT. The \ructuations are:\nh^ay\n\u000bk(\u000f1)^a\fl(\u000f2)^ay\n\rm(\u000f3)^a\u000en(\u000f4)i\u0000\nh^ay\n\u000bk(\u000f1)^a\fl(\u000f2)ih^ay\n\rm(\u000f3)^a\u000en(\u000f4)i\n=\u000e\u000bk\u000en\u000e\fl\rmf(\u000f1)[1\u0000f(\u000f2)]\u000e(\u000f1\u0000\u000f4)\u000e(\u000f2\u0000\u000f3):\n(A9)\nWe express the scattering matrix in terms of the\nWigner representation66:\nS(t;t0) =1\n2\u0019~Z1\n\u00001d\u000fS\u0012t+t0\n2;\u000f\u0013\ne\u0000i\u000f(t\u0000t0)=~:(A10)\nThe inverse transform is:\nS(t;\u000f) =Z1\n\u00001d\u001cS(t+\u001c=2;t\u0000\u001c=2)ei\u000f\u001c=~: (A11)\nBy Taylor expanding the S-matrix S((t+t0)=2;\u000f) around\nS(t;\u000f) in the Wigner representation of Eq. (A10), we ob-\ntain:\nS(t;t0) =1\n2\u0019~Z1\n\u00001d\u000fe\u0000i\u000f(t\u0000t0)=~ei~@\u000f@t=2S(t;\u000f):(A12)\nThe current operator of Eq. (A6) can then be expressed\nas:\nI\u000b(t) =e\n2\u0019~X\nn\fm\rlZ\nd\u000f1Z\nd\u000f2ei(\u000f1\u0000\u000f2)t=~\u0002\n\u001e\u000bn\fm\rl (t;\u000f1;\u000f2)^ay\n\fm(\u000f1)^a\rl(\u000f2); (A13)\nwhere\n\u001e\u000bn\fm\rl (t;\u000f1;\u000f2) =\u000e\u000bn\fm\u000e\u000bn\rl\n\u0000e\u0000i~@\u000f1@t=2S\u0003\n\u000bn\fm (t;\u000f1)ei~@\u000f2@t=2S\u000bn\rl(t;\u000f2):(A14)The current \ructuations are de\fned in Eq. (24) and\ncan be expressed as:\nP\u0010\u0011(t1;t2) =1\n2[F\u0010\u0011(t1;t2) +F\u0011\u0010(t2;t1)] (A15)\nin terms of\nF\u0010\u0011(t1;t2) =hI\u0010(t1)I\u0011(t2)i\u0000hI\u0010(t1)ihI\u0011(t2)i:(A16)\nUsing the expectation value of the \ructuations of Eq.\n(A9), we \fnd:\nF\u0010\u0011=e2\n(2\u0019~)2X\nn\u0010n\u0011\fm\rlZ\nd\u000f1Z\nd\u000f2ei(\u000f1\u0000\u000f2)(t1\u0000t2)=~\u0002\n\u001e\u0010n\u0010\fm\rl(t1;\u000f1;\u000f2)\u001e\u0011n\u0011\rl\fm(t2;\u000f2;\u000f1)f(\u000f1)[1\u0000f(\u000f2)]:\n(A17)\nWe follow Ref. 58 (Eq. (9)), apart from a factor of 2,\nand de\fne the zero-frequency noise as in Eq. (25). We\ntherefore introduce:\nf\u0010\u0011=ZT\n0dt\nTZ1\n\u00001d\u001cF\u0010\u0011(t+\u001c=2;t\u0000\u001c=2) (A18)\nso that\np\u0010\u0011= (f\u0010\u0011+f\u0011\u0010)=2: (A19)\nWe therefore \frst consider quantities of the form:\n\u0015=ZT\n0dt\nTZ1\n\u00001d\u001cei(\u000f1\u0000\u000f2)\u001c=~A(t+\u001c=2)B(t\u0000\u001c=2);\n(A20)\nwhereA(t+\u001c=2) andB(t\u0000\u001c=2) are periodic functions\nwith period Tthat depend on the energies \u000f1and\u000f2. The\nFourier transforms of the periodic functions are:\nA(t) =X\nnei!ntAn (A21)\nand similarly for B(t), where!n=n2\u0019=T andnis an\nintegral number. The inverse transforms are de\fned in\ncorresponding ways. We then obtain:\n\u0015= 2\u0019~X\nnAnB\u0000n\u000e(~!n+ (\u000f1\u0000\u000f2)): (A22)\nUsing Eqs. (A17) and (A18), the low-frequency noise is\nof the form:\n\u0014=Z\nd\u000f1Z\nd\u000f2f(\u000f1) [1\u0000f(\u000f2)] 2\u0019~\u0002\nX\nnAnB\u0000n\u000e(~!n+ (\u000f1\u0000\u000f2); (A23)\nwhereAnandB\u0000ndepend on the energies \u000f1and\u000f2.\nCarrying out the integral over the energy \u000f2:\n\u0014= 2\u0019~X\nnZ\nd\u000f1f(\u000f1) [1\u0000f(\u000f1+~!n)]AnB\u0000n;\n(A24)10\nwhere the energy \u000f2=\u000f1+~!nin the coe\u000ecients Anand\nB\u0000n.\nIn metallic systems, the Fermi energy and exchange in-\nteraction are typically much larger than the driving fre-\nquency. In this case, we can approximate the scattering\nmatrix as independent of the driving frequency ~!nand\nthe temperature kBT. We can then evaluate the scatter-\ning matrices at the Fermi energy and obtain:\n\u0014n=An(\u000fF)B\u0000n(\u000fF)2\u0019~~!n[1 +fBE(~!n;kBT)];\n(A25)\nwhere the Bose-Einstein distribution function is:\nfBE=1\nexp~!n=kBT\u00001: (A26)\nWe then obtain:\nf\u0010\u0011=X\nqY\u0010\u0011(!q)~!q[1 +fBE(~!q)]; (A27)\nwhere\nY\u0010\u0011(~!q) =e2\nhX\nn\u0010n\u0011X\n\fm\rl\u001e\u0010n\u0010\fm\rl(!q)\u001e\u0011n\u0011\rl\fm(\u0000!q);\n(A28)\nand we have de\fned the Fourier transform as:\n\u001e\u0010n\u0010\fm\rl(!q) =1\nTZT\n0dte\u0000i!qt\u001e\u0010n\u0010\fm\rl(t;\u000fF;\u000fF):\n(A29)We see that Y\u0010\u0011(\u0000~!q) =Y\u0011\u0010(~!q). Consequently, using\nEq. (A19) and rewriting the Bose-Einstein distribution,\nwe obtain:\np\u0010\u0011=X\nqY(s)\n\u0010\u0011(~!q)kBT\n+1\n2X\nqY(s)\n\u0010\u0011(~!q)\u0014\n~!qcoth~!q\n2kBT\u00002kBT\u0015\n;\n(A30)\nwhere the \frst term represents the thermal noise contri-\nbution, the second represents the shot noise contribution,\nand\nY(s)\n\u0010\u0011(~!q) =h\nY(s)\n\u0010\u0011(~!q) +Y(s)\n\u0010\u0011(\u0000~!q)i\n=2:(A31)\nAt this point, consistent with our assumption that the\nscattering matrix is energy independent on the scale of\nthe frequency and temperature, we can use the static\nscattering matrix in the evaluation of Y,Y!X, where\nXis de\fned in Eq. (27). Hence, we obtain Eq. (26) with\nthe quantities introduced in Eqs. (27), (28), and (29).\n1J. B. Johnson, Physical Review 32, 97 (1928).\n2H. Nyquist, Physical Review 32, 110 (1928).\n3Y. M. Blanter and M. B uttiker, Physics Reports 336, 1\n(2000).\n4L. Berger, Physical Review B 54, 9353 (1996).\n5J. C. Slonczewski, Journal of Magnetism and Magnetic Ma-\nterials 159, L1 (1996).\n6M. Tsoi, A. G. M. Jansen, J. Bass, W. C. Chiang, M. Seck,\nV. Tsoi, and P. Wyder, Physical Review Letters 80, 4281\n(1998).\n7E. B. Myers, D. C. Ralph, J. A. Katine, R. N. Louie, and\nR. A. Buhrman, Science 285, 867 (1999).\n8J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers,\nand D. C. Ralph, Physical Review Letters 84, 3149 (2000).\n9S. 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).\n10I. N. Krivorotov, N. C. Emley, J. C. Sankey, S. I. Kiselev,\nD. C. Ralph, and R. A. Buhrman, Science 307, 228 (2005).\n11A. Manchon and S. Zhang, Physical Review B (Condensed\nMatter and Materials Physics) 78, 212405 (2008).\n12I. M. Miron, G. Gaudin, S. Au\u000bret, B. Rodmacq,\nA. Schuhl, S. Pizzini, J. Vogel, and P. Gambardella, Na-\nture Materials 9, 230 (2010).\n13I. M. Miron, K. Garello, G. Gaudin, P.-J. Zermatten, M. V.\nCostache, S. Au\u000bret, S. Bandiera, B. Rodmacq, A. Schuhl,and P. Gambardella, Nature 476, 189 (2011).\n14L. Liu, C.-F. Pai, Y. Li, H. W. Tseng, D. C. Ralph, and\nR. A. Buhrman, Science 336, 555 (2012).\n15K. Garello, I. M. Miron, C. O. Avci, F. Freimuth,\nY. Mokrousov, S. Blugel, S. Au\u000bret, O. Boulle, G. Gaudin,\nand P. Gambardella, Nature Nanotechnology 8, 587\n(2013).\n16A. R. Mellnik, J. S. Lee, A. Richardella, J. L. Grab, P. J.\nMintun, M. H. Fischer, A. Vaezi, A. Manchon, E. A. Kim,\nN. Samarth, and D. C. Ralph, Nature 511, 449 (2014).\n17D. MacNeill, G. M. Stiehl, M. H. D. Guimaraes, R. A.\nBuhrman, J. Park, and D. C. Ralph, Nature Physics 13,\n300 (2017).\n18L. Zhu, D. C. Ralph, and R. A. Buhrman, Physical Review\nLetters 122, 077201 (2019).\n19T. J. Silva and W. H. Rippard, Journal of Magnetism and\nMagnetic Materials 320, 1260 (2008).\n20D. C. Ralph and M. D. Stiles, Journal of Magnetism and\nMagnetic Materials 320, 1190 (2008).\n21A. Brataas, A. D. Kent, and H. Ohno, Nature Materials\n11, 372 (2012).\n22A. Manchon, J. \u0014Zelezn\u0012 y, I. M. Miron, T. Jungwirth,\nJ. Sinova, A. Thiaville, K. Garello, and P. Gambardella,\nReviews of Modern Physics 91, 035004 (2019).\n23P. Monod, H. Hurdequint, A. Janossy, J. Obert, and\nJ. Chaumont, Physical Review Letters 29, 1327 (1972).11\n24R. H. Silsbee, A. Janossy, and P. Monod, Physical Review\nB19, 4382 (1979).\n25R. Urban, G. Woltersdorf, and B. Heinrich, Physical Re-\nview Letters 87, 217204 (2001).\n26S. Mizukami, Y. Ando, and T. Miazaki, Jpn. J. Appl.\nPhys. 40, 580 (2001).\n27Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Physical\nReview Letters 88, 117601 (2002).\n28B. Heinrich, Y. Tserkovnyak, G. Woltersdorf, A. Brataas,\nR. Urban, and G. E. W. Bauer, Physical Review Letters\n90, 187601 (2003).\n29O. Mosendz, V. Vlaminck, J. E. Pearson, F. Y. Fradin,\nG. E. W. Bauer, S. D. Bader, and A. Ho\u000bmann, Physical\nReview B 82, 214403 (2010).\n30T. Jungwirth, X. Marti, P. Wadley, and J. Wunderlich,\nNat Nano 11, 231 (2016).\n31S. Y. Bodnar, L. Smejkal, I. Turek, T. Jungwirth,\nO. Gomonay, J. Sinova, A. A. Sapozhnik, H. J. Elmers,\nM. Kl aui, and M. Jourdan, Nature Communications 9,\n348 (2018).\n32R. Lebrun, A. Ross, S. A. Bender, A. Qaiumzadeh, L. Bal-\ndrati, J. Cramer, A. Brataas, R. A. Duine, and M. Kl aui,\nNature 561, 222 (2018).\n33V. Baltz, A. Manchon, M. Tsoi, T. Moriyama, T. Ono, and\nY. Tserkovnyak, Reviews of Modern Physics 90, 015005\n(2018).\n34O. Gomonay, V. Baltz, A. Brataas, and Y. Tserkovnyak,\nNature Physics 14, 213 (2018).\n35J. Li, C. B. Wilson, R. Cheng, M. Lohmann, M. Kavand,\nW. Yuan, M. Aldosary, N. Agladze, P. Wei, M. S. Sherwin,\nand J. Shi, Nature 578, 70 (2020).\n36P. Vaidya, S. A. Morley, J. van Tol, Y. Liu, R. Cheng,\nA. Brataas, D. Lederman, and E. del Barco, Science 368,\n160 (2020).\n37P. Wadley, B. Howells, J. \u0014Zelezn\u0012 y, C. Andrews, V. Hills,\nR. P. Campion, V. Novk, K. Olejnk, F. Maccherozzi,\nS. S. Dhesi, S. Y. Martin, T. Wagner, J. Wunderlich,\nF. Freimuth, Y. Mokrousov, J. Kune, J. S. Chauhan, M. J.\nGrzybowski, A. W. Rushforth, K. W. Edmonds, B. L. Gal-\nlagher, and T. Jungwirth, Science (2016).\n38Y. Cheng, S. Yu, M. Zhu, J. Hwang, and F. Yang, Physical\nReview Letters 124, 027202 (2020).\n39R. Cheng, J. Xiao, Q. Niu, and A. Brataas, Physical Re-\nview Letters 113, 057601 (2014).\n40A. Kamra and W. Belzig, Physical Review Letters 119,\n197201 (2017).\n41O. Johansen and A. Brataas, Physical Review B 95, 220408\n(2017).\n42A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer, Physical\nReview Letters 101, 037207 (2008).\n43A. A. Starikov, P. J. Kelly, A. Brataas, Y. Tserkovnyak,and G. E. W. Bauer, Physical Review Letters 105, 236601\n(2010).\n44A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer, Physical\nReview B 84, 054416 (2011).\n45Y. Liu, A. A. Starikov, Z. Yuan, and P. J. Kelly, Physical\nReview B 84, 014412 (2011).\n46J. Foros, A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer,\nPhysical Review Letters 95, 016601 (2005).\n47T. Ludwig, I. S. Burmistrov, Y. Gefen, and A. Shnirman,\nPhysical Review Research 2, 023221 (2020).\n48M. B uttiker, A. Prtre, and H. Thomas, Physical Review\nLetters 70, 4114 (1993).\n49X. Waintal, E. B. Myers, P. W. Brouwer, and D. C. Ralph,\nPhysical Review B 62, 12317 (2000).\n50A. Brataas, Y. V. Nazarov, and G. E. W. Bauer, Physical\nReview Letters 84, 2481 (2000).\n51A. Brataas, Y. Nazarov, and G. Bauer, European Physical\nJournal B 22, 99 (2001).\n52M. D. Stiles and A. Zangwill, Physical Review B 66,\n014407 (2002).\n53V. S. Rychkov, S. Borlenghi, H. Ja\u000bres, A. Fert, and\nX. Waintal, Physical Review Letters 103, 066602 (2009).\n54K. M. D. Hals, A. Brataas, and Y. Tserkovnyak, Epl 90\n(2010).\n55P. W. Brouwer, Physical Review B 58, R10135 (1998).\n56A. Brataas, Y. Tserkovnyak, G. E. W. Bauer, and B. I.\nHalperin, Physical Review B 66, 060404 (2002).\n57Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I.\nHalperin, Reviews of Modern Physics 77, 1375 (2005).\n58M. Moskalets and M. B uttiker, Physical Review B (Con-\ndensed Matter and Materials Physics) 70, 245305 (2004).\n59A. Brataas, G. E. W. Bauer, and P. J. Kelly, Physics\nReports 427, 157 (2006).\n60S. A. Bender, A. Kamra, W. Belzig, and R. A. Duine,\nPhysical Review Letters 122, 187701 (2019).\n61H. Nakayama, M. Althammer, Y. T. Chen, K. Uchida,\nY. Kajiwara, D. Kikuchi, T. Ohtani, S. Geprgs, M. Opel,\nS. Takahashi, R. Gross, G. E. W. Bauer, S. T. B. Goennen-\nwein, and E. Saitoh, Physical Review Letters 110, 206601\n(2013).\n62K. M. Schep, J. B. A. N. van Hoof, P. J. Kelly, G. E. W.\nBauer, and J. E. Ingles\feld, Physical Review B 56, 10805\n(1997).\n63K. Xia, P. J. Kelly, G. E. W. Bauer, A. Brataas, and\nI. Turek, Physical Review B 65, 220401 (2002).\n64M. Zwierzycki, Y. Tserkovnyak, P. J. Kelly, A. Brataas,\nand G. E. W. Bauer, Physical Review B 71, 064420 (2005).\n65C. Ciccarelli, K. M. D. Hals, A. Irvine, V. Novak,\nY. Tserkovnyak, H. Kurebayashi, A. Brataas, and A. Fer-\nguson, Nature Nanotechnology 10, 50 (2015).\n66J. Rammer and H. Smith, Reviews of Modern Physics 58,\n323 (1986)."    },    {        "title": "1909.02204v2.Detection_of_magnetic_nanoparticles__MNPs__using_spin_current_nano_oscillator__SCNO__biosensor__A_frequency_based_rapid__ultra_sensitive__magnetic_bioassay.pdf",        "content": "1 \n Detection of magnetic nanoparticles  (MNPs)  using spin \ncurrent nano-oscillator (SC NO) biosensor : A frequency -\nbased rapid, ultra-sensitive , magnetic  bioassay   \n \nRenata Saha1, Kai Wu1, *, Diqing Su2, and Jian -Ping Wang1, * \n1Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, \nMinnesota 55455, USA \n2Department of Chemical Engineering and Material Science, University of Minnesota, \nMinneapolis, Minnesota 55455, USA  \n*Corresponding author E -mails : wuxx0803@umn.edu  (K. W.) and jpwang@umn.edu  (J.-P. W.)  \n \nThis Letter is a micromagnetic simula tion-based study on the GHz -frequency ferromagnetic \nresonances for the detection of  magnetic nanoparticles  (MNPs)  using spin current nano -oscillator \n(SCNO) operating in precession mode as a spintronic biosensor . The magnetic stray  fields from \nthe MNPs  in an antibody -antigen -MNP complex on the SCNO surface modify the ferromagnetic \nresonance peaks and generat e measurable resonance peak shift s. Moreover, our results strongly \nindicate the position -sensitive behavior  of the SCNO biosensor  and ways to eradicate this effect \nto facilitate better bio-sensing perform ance. Addition ally, a  study  has been made on  how \nnanoparticles  with different sizes  can alter the SCNO device performance . This simulation -based \nstudy on the SCNO device  shows a promise  of frequency -based nano -biosensor  with a sensitivity \nof detecting eve n a single MNP , even in presence of thermal noise.  \n \nIt has been two decades since Baselt et. al1 has designed the Bead Array Counter (BARC) \nthat first show ed the experiment al possibility of bio -detection for multilayer giant \nmagnetoresistance (GMR)  with MNPs  as biomarkers . Ever since then, magnetic bi osensing  for \npoint -of-care (POC) detection of diseases using magneto -resistive  (MR)  sensors2–13 have been \nexplored intensively  and has been subjected to extensive review s14–16. Sandwich -based bio assay9,17, \nflow cytometry18,19 and microfluidic channel20 are the most common techniques  for magnetic \nbiosensing . The most attractive part about biosensing with spintronic sensors lies in the fact that \nbiomedical samples exhibit negligible m agnetic background which  suppr esses noise from cellular \nmatrix  to a great extent21. These spin-valve sensors have the sensitivity of detecting miniscule \nchange in magnetic field  even from those of surface functionalized MNPs9,11 or from a single \nmagnetic bead22,23 by translating the presence of MNP (s) to cause a variation in the static magnetic \nconfiguration of the sensor’s active sensing layer. This alters  the device resistance  which is \nmanifested and measured as the change in voltage . \nHowever, these MR sensors tend to suffer from  high background noise levels at room \ntemperature performance which causes to compromise the sensitivity  of the device . Thus , recently, \nfrequency -based approach for detection of magnetic field24–27 and MNPs28–30 have been  2 \n imple mented through micromagnetic simulation s and validated through experimental analysis . \nHowever, majority of them  were using magnonic crystals24,29, ferromagnetic nanodots28 and/or \nnanodis cs30. The ferromagnetic resonance  (FMR)  frequency of the device interacts directly  with \nthe stray field of the MNP31 and thereby causes a sh ift in the  peak frequency of the device . The \nshift in the peak frequency has been experimentally demonstrated to depend on the concentration \nof the MNPs28, the size of the MNPs28,30 and even the position of the MNPs30 on the magnonic \ncrystal surface. The main advantage of a frequency -based, dynamic approach  over the static MR-\nbased  sensing is that the device response is  linear over a large range of the externally applied \nmagnetic field  leading to more prominent frequency shifts. This frequency  is typically of the orders \nof several GHz , way too higher compared to the low frequency1/f noise , and hence  devoid  of DC \nvoltage -level drift . \nIn this regard, spin torque nano -oscillators (STNO)32–36 driven b y spin transfer torque \n(STT)37,38 deserve  special mention for frequency -based magnetic biosensing39,40.The stack \nstructure of these STNO is similar to a spin -valve structure that is composed of a nonmagnetic \nmetallic layer sandwiched between  two ferromagnetic layers . The spin -polarized electric current \npassing through a thin ferromagnetic  layer dynamically excites the magnetic moment of that layer \nthrough a  transfer  of spin angular momentum. However, STNO devices are limited by \nconsumption of large currents as the electrons are limited by a total angular momentum  of ћ/241. \nEven more , larger the magnetic moment, larger is the current required to operate. But the trade -off \nis that it yields larger thermal stability with larger magnetic moment.  In this respect, in-plane \nmagnetized spin Hall nano -oscillators ( SHNO )41–48 consisting of a heavy metal \n(HM)/ferromagnetic metal  (FM) stack structure  do not have that limitation of angular momentum \nas constant scattering takes place at HM/FM interface. Besid es, no electron is required to flow  \nthrough the active FM. Consequently,  unpredicted damage due to electromigration and ohmic \nheating is prevented in spin Hall effect ( SHE ) devices.  Unlike STT devices, SHE devices support \nmagneto -optical measurements in direct contact with the active area of the device . Even more,  \nfrom fabrication point -of-view, spin Hall nano -oscillators  (SHNO) are easier to fabricate .  \nThere are two conditions to trigger self -oscillations in SHNO devices: f irst, the d ynamic \ndamping must be completely balanced  by spin current ; second, the current to balance the spin \ntorque and the damping torque in the self -oscillation state should be larger than the critical current \nto destabilize the initial state. Since the first condition  is not satisfied, SHNO s with perpendicular \nmagnetized anisotropy ( PMA ) are not feasible  as was proved theoretically by Tomohiro \nTaniguchi49. The spin -orbit effects from the HM/ PMA -FM bilayer system that contribute to the \ncurrent -induced phenomena including  the spin Hall effect, the  Rashba effect50, and the \nDzyaloshinskii -Moriya  interaction (DMI)51, all of which originate  from the broken  inversion \nsymmetry  at the HM/ PMA -FM interface . A combination of all these interacti ons contribute to a \nfar more stable oscillator system in comparison to in -plane SHNO systems.  It is this possibility to \ninduce dynamical states of HM/FM bilayer or alter their static configuration by the  current -induced \nspin torque (ST)37,38 has triggered extensive experimental and theoretical research . In this respect  \nof operation , one of the promising devices is the PMA -spin current nano -oscillators (SCNO)52 3 \n devices . Howev er, the detailed investigation of SCNOs in terms of frequency -based nano -\nbiosensors have not been  made . \nAs per our best knowledge, for th e first time, we investigate  the feasibility  of spin current \nnano -oscillator (SCNO) device as a frequency -based spintronic biosensor  through micromagnetic \nsimulations  on Mumax353. The SCNO device has been  simulated numerically by solving t he \nLandau –Lifshitz –Gilbert (LLG) equation (1) in addition to a spin orbit torque (SOT) : \n                        𝑑𝒎\n𝑑𝑡=𝛾0𝒉𝒆𝒇𝒇×𝒎+𝛼𝒎×𝑑𝒎\n𝑑𝑡+𝑢\n𝑡𝒎×(𝒎𝑝×𝒎),                                    (1) \nwhere, 𝒎 = 𝑴/𝑀s  is the normalized magnetization, 𝛾0=1.85×1011 rad T−1s−1 is the \ngyromagnetic ratio, 𝒉eff = 𝑯eff /𝑀s is the reduced effective field, 𝑡 is the thickness of the \nferromagnetic (FM) layer,  𝒎𝑝 is the current polarization vector, 𝑢=𝛾0(ђ𝑗𝑃\n2𝑒𝑀𝑠),  and j is the density \nof the spin current. The values of parameters Ms, P, α along with dimensions of the magnetic thin \nfilm are listed in Table 1 . All parameters that define the FM layer are adopt ed from Ref.54  and that  \nto define the MNP are adopted  from Ref55.  \n \nTable I. Micromagnetic simulation parameters for SCNO biosensor  \nParameters  Description  Values  \nFM nanopillar Dimension  Length × Width × Thickness  160 nm × 80 nm × 5 nm  \nCell Size  Length × Width × Thickness  2.5 nm × 2.5 nm × 5 nm  \n Gilbert damping factor  0.015  \nA Exchange constant  13×10−12 J/m \n𝑷 Spin Hall Angle  0.6 \n𝑴𝒔 Saturation magnetization  1200 ×103A/m \n𝑲𝒖 𝟏 First order uniaxial \nanisotropy constant  0.7×10−6  Jm-3 \nDMI  Dzyaloshinskii -Moriya \ninteraction  0.7×10−4  Jm-2 \n𝝁𝟎 Permeability of free space  4𝜋×10−7 WbA−1m−1 \n \nFor a bilayer of PMA ferromagnet ( FM) and heavy metal (HM), under  an externally applied \ncurrent and uniform DC magnetic field, the device operates in precession mode. It is with a \nprecession frequency that the  bare SCNO  device oscillates  (see Supplementary Movie SM1 ), \nreferred to in this Letter as the ‘peak frequency’ or in other words the frequency which has the \nmaximum intensity . We have d emonstrated how the peak frequency shifts with respect to regularly \nand/or randomly spaced single and/or a cluster of MNP(s). In addition, how different sized MNPs \ncan affect the peak frequency shift of the SCNO device  have been investigated  (see Supplementary \nMovie SM2 & SM3 ). Finally, discussions follow on how we can optimize the SCNO device \nperformance for it to be best fit in magnetic biosensing application.  \nFigure 1(a) gives a schematic view  of the SCNO biosensor array with the ferromagnetic \nnanopillar of dimensions 160 nm × 80 nm × 5 nm located  0.5 μm apart such that the stray fields \nof adjacent PMA -FM nanopillars do not influence the device performance (see Supplementary 4 \n Information S3 ). Figure 1(a)  also shows  the mechanism in which the SCNO biosensor would \nfacilitate magnetic biosensing through formation of target antibody -antigen -MNP complex. The \nmagnetic material parameters to define the SCNO biosensor in micromagnetic simulations are \nlisted  in Table I . The p roperties of the MNP used in th is simulation work are specified in Table II . \nFigure 1(b)  is a zoomed in image of a single  FM nanopillar . On passing  a charge current (Jc, A/m2) \nthrough the HM layer  along -x direction , it causes spin accumulation along ±y and generation of a \nspin current along z direction (see Figure 1(b) ). When a magnetic field, Hdc (in Oe)  is externally \napplied along +y direction, the spin current causes the FM nanopillar to operate in precession.  The \neffects of a reversed direction of Jc and Hdc on SCNO device performance have been explored in \nSupplementary Information S2 . The color and symbol codes for antigen, antibody, MNP, target \nantibody – antigen – MNP complex, FM layer, HM  layer & substrate of the SCNO biosensor used \nthroughout the figures in this Letter are specified in a separate column  in Figure 1 . The \nperformance of the designed SCNO biosensor in this L etter has been reported at T = 0  K. However, \nthe arguments concerning the thermal effects on its performance have been made in Supplementary \nInformation S1 . \n \n \nFigure 1. (a) Schematic of the spin current nano -oscillator (SC NO) biosensor array with the \nmechanism of target antibody - magnetic nanoparticle (MNP)  - antigen complex demonstra ted. (b) \nA single 160 nm × 80 nm × 5 nm , perpendicularly magnetized (PMA)  ferromagnetic (FM) \nnanopillar  of the SC NO biosensor array zoomed in. The charge current density ( Jc) to the heavy \nmetal (HM) layer is along -x direction and the externally applied magnetic field (Hdc) is directed \n5 \n along +y direction. The black -dashed arrow in the ferromagnetic nanopillar demonstrates  \nprecession mode operation of the device . \n \nTable II. Micromagnetic simulation parameters for MNP (s) \nParameters  Description  Values  \n Gilbert damping factor  0.1 \nA Exchange constant  2.64×10−11 J/m \n𝑷 Spin Hall Angle  0.6 \n𝑴𝒔 Saturation magnetization  3.5×105A/m \n𝑲𝒖 𝟏 First order  uniaxial  \nanisotropy constant  1.25 ×104 Jm−3 \n \nFor an externally applied magnetic field, Hdc = 1.1 kOe  in Figure 2(a)  the p eak frequency \nfor the SCNO biosensor changes with variation of externally applied current  through the heavy \nmetal (i, mA) . As reported in previous literatures  for in -plane s pin Hall nano -osillators (SHNO)41,42, \nwith increase in c urrent for a constant magnetic field, the peak frequency (in GHz) decreases while \nthe intensity of the main frequency  component  of the SCNO device (in arbitrary units ( a.u.)) \nincreases as is expressed in Figure 2(b) . Again, for an externally applied curren t of i = 15 mA  \n(current density, 1.5×108A/cm2) for varying DC magnetic field, we observe a clear shift in the \npeak frequency value in Figure 2(c) . The black diamonds represent the calculated ferromagnetic \nresonance frequency (fFMR) value for a particular externally applied magnetic field as calculated \nby the normal magnetization Kittel Equation,  𝑓𝐹𝑀𝑅 =𝛾(𝐻𝑑𝑐−4𝜋𝑀𝑠)+(4𝜋𝛾𝑀𝑠). As reported \nearlier52, the precession frequency for the designed SCNO biosensor  always l ies below  the \ncalculated FMR frequency.  The trajectory of the magnetization vectors ( mx, my and mz) due to \nprecessional  motion of the FM nanopillar has been demonstrated in Figure 2(d) . Discussions \nrelating power consumption and device performance have been made in Supplementary \nInformation S5 . 6 \n  \nFigure 2. (a) Variation of peak frequency shift of a SC NO biosensor with current  (i, mA) at  Hdc = \n1.1 kOe. (b) As a follow -up from part (a), representation of the v ariation of peak intensity  and \nmagnitude of peak frequency with current  (i, mA) . (c) Variation of the peak frequency with applied \nmagnetic field (H dc, Oe)  at i = 15 mA. The black diamonds represent the value of the ferromagnetic \nresonant (FMR) frequency at the corresponding frequency calculated by the Kittel Equation. (d) \nThe magnetization vector (normalized values mx, m y and mz) trajectory due to  their precessional \nmotion due to i = 15 mA and Hdc = 1.1 kOe.  \n7 \n  \nIn Figure 3 , a SCNO device devoid of any MNP  positioned on the sensor surface  has been \nreferred to as the  ‘bare’ SCNO device.  Figure 3(a) shows 5 independent positions  of a single MNP \nof 20 nm diameter  on a SCNO biosensor. The 5 independent positions correspond to the following \nco-ordinates: (i) = (0, 0), center of the SCNO device; (ii) = (40, 20), first quadrant; (iii) = ( -40, 20), \nsecond quadrant; (iv) = ( -40, -20), t hird quadrant; (v) = (40, -20), fourth quadrant. Figure 3(b) \ncorrespond to  4 independent positions of 6 MNPs, each of 20 nm in diameter and the center of \neach MNP spaced regularly at 30 nm apart from each other. The 4 independent positions \ncorrespond to th e following co -ordinates: (i) = first quadrant; (ii) = second quadrant; (iii) = third \nquadrant; (v) = fourth quadrant. Figure 3(c) corresponds to 8 & 10 MNPs  positioned at the center \nof the SCNO biosensor , each of 20 nm in diameter and center of each MNP  regularly spaced at 30 \nnm from each  other.   \nCorresponding to the highlighted picture background color -codes, the peak frequencies for \ncases in Figure 3(a) , (b) & (c) have been displayed in Figure 3(d), (e)  & (f), respectively. With \nrespect to a bare SCNO bi osensor, Figure 3(d)  shows peak frequency shift for the 5 -independent \npositions of a single 20 nm MNP while Figure 3( e) shows peak frequency shift for the 4 -\nindependent positions of a six, 20 nm MNPs with their centers separated by 30 nm distance. Both \nFigure 3(d) & (e) show that the peak frequency varies for different positions of the MNPs. \nFurthermore, for both the cases of a single MNP and for the  case of 6 MNPs, the peak frequency \nfor the positions in the first & fourth quadrant and for the positions in second & third quadrant are \nsame. Therefore, one can conclude that the two halves of the SCNO device work differently due \nto unique magnetization distribution (see Supplementary Information S4 ). Figure 3(e)  \ndemonstrates the variation in peak frequency due to presence of 8 and 10 MNPs on the SCNO \nbiosensor surface with respect to a bare SCNO device. In summary, Figure 3( a)-(f) validates  the \nfact that  SCNO biosensor  performance is  position specific, precisely, the two identical halve s of \nthe device work uniquely . As much as this position sensitivity of the designed SCNO biosensor is \ndetrimental to biosensing performance for magnetic biosensors, one can not deny the fact that the \nother magnetic biosensors, including the most celebrated biosensor in magnetic biosensing, GMRs \nare position sensitive  too. For instance, analytical studies by Klein et. al56 had shown how the \nedges of the GMR sensors are more sensitive towards detection of MNPs than the remaining part \nof the sensor.  \n 8 \n  \nFigure 3. Demonstration of p ositional sensitivity along with MNP detection capability of the \nSCNO biosensor. Top view of the FM nanopillar containing (a) one MNP at 5 different positions \non the SC NO biosensor marked (i), (ii), (iii) , (iv) & (v) . Each of the 5 cases are independent of \neach other ; (b) 6 MNPs at 4 different positions on the SCNO biosens or marked (i), (ii), (iii) & (iv) . \nEach of the 4 cases are independent of each other ; (c) 8 MNPs & 10 MNPs positioned at the center \nof the SC NO biosensor.  In all the cases (a)-(c), each of the MNPs were identical of 20 nm in \ndiameter and the distance between centers being 30 nm.  With respect to a bare SCNO device, (d) \nPeak frequency for the condition s of MNP position demonstrated in (a). (e) Peak frequencies  for \nthe condition s of MNP position  demonstrated in (b). (f) Peak frequenc ies for the condition s of \nMNP position d emonstrated in (c). (g) Variation of the peak frequencies  with 1, 6, 8 & 10 no. of \nMNPs of 20 nm diameter situated at random , uncontrolled  positions  on the SCNO biosensor. (h) \n9 \n Comparison between the peak frequency (in GHz) variation for regular and random position of \nMNP on the biosensor. (i) The peak frequency shift (in MHz) for values in (h) from that of the \nbare SCNO device.  \n \nHowever, in real platforms for  magnetic biosensors, the MNPs are unlikely to be  uniformly \nspaced as was the case in Figure 3(a) -(c). To demonstrate a more realistic SCNO performance, we \nhave further investigat ed the cases of 1, 6, 8 and 10 MNPs, but this time for 5 random  arrangement s \non the SCNO surface  for each of the 4 number of MNPs . Figure 3(g)  demonstrates t he box-whisker \nplots for the  peak frequencies  of 5 cases of randomly positioned 1, 6, 8 & 10 MNPs , each . The \nmean values for 1, 6, 8 & 10 MNPs have been found to be  2.378 GHz, 2.556 GHz, 2.862 GHz and \n3.314 GHz  respectively and are symbolized in varied colored diamonds  in Figure 3(g) . \nFurthermore, as the number of MNPs on the SCNO biosensor surface increases, the deviations  \ndecreases, that is, the length of the box in the box -whisker plot decreases.  Figure 3(h) shows the \ncomparison of peak frequencies  of regularly spaced 1,  6, 8 & 10 MNPs as discussed in Figure 3(a) -\n(f) to that of the mean peak frequency values  for randomly spaced 1, 6, 8 & 10 MNPs as discussed \nin Figure 3(g) . From Figure 3(g) , it is evident that the trend for both regularly and randomly spaced \nMNPs are the same , which is , the peak frequency  (in GHz)  value increases with increase in number \nof the MNPs from 1, 6, 8 to 10 MNPs.  This fact is validated from Figure 3(h)  by the shift in peak \nfrequency (in MHz) from the bare SCNO device.  The demonstration of the  cases for uniformly  \nspaced MNPs validates t he fact that an SCNO biosensor is position sensitive . The similar trend in \nthe peak frequency shifts between randomly spaced MNPs and regularly spaced MNPs draw s a \nmore realistic picture towards SCNO biosensor performance because in real experiments, the  \nMNPs would be quiet randomly  positioned . \nIn Figure 4 , we have defined a single  MNP  of 7 different diameters at random positions on \nthe SCNO device surface.  For 6 ra ndom positions on the SCNO device surface, MNPs of diameters, \n10 nm, 20 nm, 25 nm, 30 nm, 35 nm, 40 nm and 45 nm show a mean peak frequencies 2.418 GHz, \n2.43 GHz, 2.514 GHz, 2.554 GHz, 2.562 GHz, 2.647 GHz and 2.56 GHz, respectively. With \nincrease in the d iameter of the MNPs, the mean peak frequency increases gradually until at 45 nm \nwhere a sudden drop in mean peak frequency is observed. Analogous results, in terms of the GMR \nsignal level, were experimentally observed by Wang et. al57 for the purpose of GMR biosensing. \nIn comparison between large and small size of MNPs, the latter encourage greater degree of \nBrownian motion which in turn facilitates greater diffusion and binding capacity of MNPs with \nthe SCNO sensor surface. Therefore,  with increased diameter of the MNP, the binding tendency \nto the SCNO sensor surface decreases significantly thereby leading to a decrease in peak frequency.  \n 10 \n  \nFigure 4. SCNO performance for varied sizes of a single MNP  at random positions  on the sensor \nsurface.   \n \nIn conclusion,  we have proposed and investigated the feasibility of a spin current nano -\noscillator ( SCNO ) biosensor with perpendicular magnetic anisotropy  (PMA)  as a frequency -based \nbiosensor . Through micromagnetic simulations, we have demonstrated that the SCNO biosensor \nhas the  sensitivity  of detecting even a single MNP  and its performance varies with the position of \nthe MNP.  That the performance of a SCNO biosensor varies with the position of the MNP (s). \nHowever, in real experiments, with random position of the MNP (s), the position specific behavior \nof the SCNO biosensor can be eliminated to a great extent.  Unlike MR sensors, the SCNO \nbiosensor performance is not noisy at room temperature yielding a more realistic device \nperformance.  Finally, in order to  observe a distinct peak shift on addition of MNPs, a standard \nbinding process is required to be initiated . Therefore, optimizing the size of the MNPs  to facilitate \nbinding  to observe a clear shift of frequency  is extremely essential .  \n \nThis study was financially supported by the Institute of Engineering in Medicine of the \nUniversity  of Minnesota through FY18 IEM Seed Grant Funding Program, the National Science \nFoundation MRSEC facility program, the Distinguished McKnight University Professorship, the \nCentennial Chair Professorship, and the Robert F Hartmann Endowed Chair from the University \nof Minnesota.  \n \nREFERENCES  \n1 D.R. Baselt, G.U. Lee, M. Natesan, S.W. Metzger, P.E. Sheehan, and R.J. Colton, Biosens. Bioelectron. 13, 731 \n(1998).  \n2 M. Miller, G. Prinz, S. -F. Cheng, and S. Bounna k, Appl. Phys. Lett. 81, 2211 (2002).  \n11 \n 3 J. Rife, M. Miller, P. Sheehan, C. Tamanaha, M. Tondra, and L. Whitman, Sens. Actuators Phys. 107, 209 (2003).  \n4 W. Dittmer, P. De Kievit, M. Prins, J. Vissers, M. Mersch, and M. Martens, J. Immunol. Methods 338, 40 (2008).  \n5 B. Srinivasan, Y. Li, Y. Jing, Y. Xu, X. Yao, C. Xing, and J. Wang, Angew. Chem. Int. Ed. 48, 2764 (2009).  \n6 R.S. Gaster, L. Xu, S. -J. Han, R.J. Wilson, D.A. Hall, S.J. Osterfeld, H. Yu, and S.X. Wang, Nat. Nanotechnol. 6, \n314 (2011).  \n7 C. Marqui na, J. De Teresa, D. Serrate, J. Marzo, F. Cardoso, D. Saurel, S. Cardoso, P. Freitas, and M. Ibarra, J. \nMagn. Magn. Mater. 324, 3495 (2012).  \n8 E. Aronoff -Spencer, A. Venkatesh, A. Sun, H. Brickner, D. Looney, and D.A. Hall, Biosens. Bioelectron. 86, 690 \n(2016).  \n9 K. Wu, T. Klein, V.D. Krishna, D. Su, A.M. Perez, and J. -P. Wang, ACS Sens. 2, 1594 (2017).  \n10 T.G. Barroso, R.C. Martins, E. Fernandes, S. Cardoso, J. Rivas, and P.P. Freitas, Biosens. Bioelectron. 100, 259 \n(2018).  \n11 D. Su, K. Wu, V. Krishna, T.  Klein, J. Liu, Y. Feng, A.M. Perez, M.C. Cheeran, and J. -P. Wang, Front. \nMicrobiol. 10, 1077 (2019).  \n12 N. Ravi, G. Rizzi, S.E. Chang, P. Cheung, P.J. Utz, and S.X. Wang, Biosens. Bioelectron. 130, 338 (2019).  \n13 E. Ng, C. Yao, T.O. Shultz, S. Ross -Howe, and S.X. Wang, Nanomedicine Nanotechnol. Biol. Med. 16, 10 \n(2019).  \n14 S.X. Wang and G. Li, IEEE Trans. Magn. 44, 1687 (2008).  \n15 J. Llandro, J. Palfreyman, A. Ionescu, and C. Barnes, Med. Biol. Eng. Comput. 48, 977 (2010).  \n16 K. Wu, D. Su, J. Liu, R. Saha, and J. -P. Wang, Nanotechnology (2019).  \n17 T. Klein, W. Wang, L. Yu, K. Wu, K.L. Boylan, R.I. Vogel, A.P. Skubitz, and J. -P. Wang, Biosens. Bioelectron. \n126, 301 (2019).  \n18 A. Chicharo, L. Barnsley, M. Martins, S. Cardoso, L.  Diéguez, B. Espiña, and P. Freitas, IEEE Trans. Magn. 54, 1 \n(2018).  \n19 A. Chícharo, M. Martins, L.C. Barnsley, A. Taouallah, J. Fernandes, B.F. Silva, S. Cardoso, L. Diéguez, B. \nEspiña, and P.P. Freitas, Lab. Chip 18, 2593 (2018).  \n20 G. Kokkinis, S. Cardo so, F. Keplinger, and I. Giouroudi, Sens. Actuators B Chem. 241, 438 (2017).  \n21 M. Colombo, S. Carregal -Romero, M.F. Casula, L. Gutiérrez, M.P. Morales, I.B. Böhm, J.T. Heverhagen, D. \nProsperi, and W.J. Parak, Chem. Soc. Rev. 41, 4306 (2012).  \n22 G. Li and S.X. Wang, IEEE Trans. Magn. 39, 3313 (2003).  \n23 Y. Liu, W. Jin, Y. Yang, and Z. Wang, J. Appl. Phys. 99, 08G102 (2006).  \n24 M. Inoue, A. Baryshev, H. Takagi, P.B. Lim, K. Hatafuku, J. Noda, and K. Togo, Appl. Phys. Lett. 98, 132511 \n(2011).  \n25 S. Atalay, A. Kaya, V. Kolat, H. Gencer, and T. Izgi, J. Supercond. Nov. Magn. 28, 2071 (2015).  \n26 K. Mizushima, K. Kudo, T. Nagasawa, and R. Sato, J. Appl. Phys. 107, 063904 (2010).  \n27 J.P. Fried and P.J. Metaxas, Phys. Rev. B 93, 064422 (2016).  \n28 M. Su shruth, J. Ding, J. Duczynski, R.C. Woodward, R.A. Begley, H. Fangohr, R.O. Fuller, A.O. Adeyeye, M. \nKostylev, and P.J. Metaxas, Phys. Rev. Appl. 6, 044005 (2016).  \n29 P.J. Metaxas, M. Sushruth, R.A. Begley, J. Ding, R.C. Woodward, I.S. Maksymov, M. Albert,  W. Wang, H. \nFangohr, and A.O. Adeyeye, Appl. Phys. Lett. 106, 232406 (2015).  \n30 M. Albert, M. Beg, D. Chernyshenko, M. -A. Bisotti, R.L. Carey, H. Fangohr, and P.J. Metaxas, Nanotechnology \n27, 455502 (2016).  \n31 I. Lee, Y. Obukhov, G. Xiang, A. Hauser, F. Y ang, P. Banerjee, D.V. Pelekhov, and P.C. Hammel, Nature 466, \n845 (2010).  \n32 J.-V. Kim, V. Tiberkevich, and A.N. Slavin, Phys. Rev. Lett. 100, 017207 (2008).  \n33 P. Braganca, B. Gurney, B. Wilson, J. Katine, S. Maat, and J. Childress, Nanotechnology 21, 235 202 (2010).  \n34 O. Prokopenko, E. Bankowski, T. Meitzler, V. Tiberkevich, and A. Slavin, IEEE Magn. Lett. 2, 3000104 (2011).  \n35 G.E. Rowlands and I.N. Krivorotov, Phys. Rev. B 86, 094425 (2012).  \n36 C. Zheng, H. -H. Chen, X. Zhang, Z. Zhang, and Y. Liu, Chin.  Phys. B 28, 037503 (2019).  12 \n 37 L. Berger, Phys. Rev. B 54, 9353 (1996).  \n38 J.C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996).  \n39 H. Xia, Q. Zheng, C. Mu, C. Song, C. Jin, Q. Liu, and J. Wang, J. Magn. Magn. Mater. 432, 387 (2017).  \n40 P.J. Ryan, H. Xi, and I. Jin, (2011).  \n41 R. Liu, W. Lim, and S. Urazhdin, Phys. Rev. Lett. 110, 147601 (2013).  \n42 V.E. Demidov, S. Urazhdin, H. Ulrich s, V. Tiberkevich, A. Slavin, D. Baither, G. Schmitz, and S.O. Demokritov, \nNat. Mater. 11, 1028 (2012).  \n43 H. Ulrichs, V. Demidov, and S. Demokritov, Appl. Phys. Lett. 104, 042407 (2014).  \n44 M. Ranjbar, P. Drrenfeld, M. Haidar, E. Iacocca, M. Balinskiy, T.  Le, M. Fazlali, A. Houshang, A. Awad, and R. \nDumas, IEEE Magn. Lett. 5, 1 (2014).  \n45 V. Demidov, H. Ulrichs, S. Gurevich, S. Demokritov, V. Tiberkevich, A. Slavin, A. Zholud, and S. Urazhdin, Nat. \nCommun. 5, 3179 (2014).  \n46 A. Awad, P. Dürrenfeld, A. Hous hang, M. Dvornik, E. Iacocca, R. Dumas, and J. Åkerman, Nat. Phys. 13, 292 \n(2017).  \n47 V. Demidov, S. Urazhdin, A. Zholud, A. Sadovnikov, and S. Demokritov, Appl. Phys. Lett. 105, 172410 (2014).  \n48 M. Zahedinejad, H. Mazraati, H. Fulara, J. Yue, S. Jiang, A . Awad, and J. Åkerman, Appl. Phys. Lett. 112, \n132404 (2018).  \n49 T. Taniguchi, Phys. Rev. B 91, 104406 (2015).  \n50 Y.A. Bychkov and E.I. Rashba, J. Phys. C Solid State Phys. 17, 6039 (1984).  \n51 T. Moriya, Phys. Rev. 120, 91 (1960).  \n52 R. Liu, W. Lim, and S.  Urazhdin, Phys. Rev. Lett. 114, 137201 (2015).  \n53 A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F. Garcia -Sanchez, and B. Van Waeyenberge, AIP Adv. 4, \n107133 (2014).  \n54 C. Yoshida, H. Noshiro, Y. Yamazaki, T. Sugii, A. Furuya, T. Ataka, T. Tanaka, and Y. Uehara, AIP Adv. 6, \n055816 (2016).  \n55 K. W u, D. Su, R. Saha, J. Liu, and J. -P. Wang, J. Phys. Appl. Phys. (2019).  \n56 T. Klein, Y. Wang, L. Tu, L. Yu, Y. Feng, W. Wang, and J. -P. Wang, Sens. Actuators Phys. 216, 349 (2014).  \n57 W. Wang, Y. Wang, L. Tu, Y. Feng, T. Klein, and J. -P. Wang, Sci. Rep. 4, 5716 (2014).  \n \n S1 \n Supplemental Information  \n \n \n \nDetection of magnetic nanoparticles (MNPs) using spin \ncurrent nano -oscillator (SCNO) biosensor: A frequency -\nbased rapid, ultra -sensitive, magnetic bioassay  \n \nRenata Saha1, Kai Wu1, *, Diqing Su2, and Jian -Ping Wang1, * \n1Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, \nMinnesota 554 55, USA \n2Department of Chemical Engineering and Material Science, University of Minnesota, \nMinneapolis, Minnesota 55455, USA  \n*Corresponding author E -mails: wuxx0803@umn.edu  (K. W.) and jpwang@umn.edu  (J.-P. W.)  \n  S2 \n Supp lemental  Information S1. Thermal Effects on the spin current nano -oscillator (SCNO) \nperformance  as a biosensor  \n \nIn the  Letter , the performance of the SCNO biosensor was demonstrated at T =  0 K. In Figure S1(a) \n& (b) the performance of a bare SCNO device has been compared to that of the SCNO biosensor \nwith a single MNP positioned at the center of the device  in presence of thermal perturbation . It has \nbeen a known fact that for room -temperature performance of magnetoresistive (MR) biosensors, \nthe real -time sensitivity is high ly compromised by background noise . Thus , to validate the fact that \nSCNO devices are better  in this respect , we have carri ed out the theoretical studies at different \ntemperatures ranging from T = 0  K, 60  K, 200  K, 300  K, 400  K and 500  K. The FFT peaks in \nFigure S1(a) demonstrate that with the gradual increase in temperature, the intensity of the peaks \nfor a bare SCNO device decreases  implying the plot  becomes noisy at 500K but still the peaks are \ndetectable. Figure S1(b) gives a detailed analysis of the intensity and peak frequency of the bare \nSCNO biosensor in comparison to that of a SCNO biosensor with a single MNP s ituated at the \ncenter.  In both cases, with increase in temperature, the value of peak frequency increases  but the \nintensity decreases  significantly . It is evident that at T = 500  K, apart from the intensity (in a.u.) \nbeing low, there is no significant shift in frequency for presence of a single MNP (fpeak = 3.067 \nGHz) in comparison to the frequency of a bare SCNO device  (fpeak = 3.061GHz ). \n It is unlikely that the real -time biosensing experiments using SCNO will be conducted at \nT = 500K. A t room temperature (T = 300 K), the difference in peaks for a bare SCNO device and \na device with only a single MNP positioned at the center is discernible, both in terms of frequency \nand intensity (see Figure S1(b) ; color -symbol code: cyan -stars).  \n S3 \n  \nFigure  S1(a)  The effect of thermal perturbation on the performance of bare spin current nano -\noscillator (SCNO) device.  \nS4 \n  \nFigure S1(b)  Comparison of the peak frequency (in GHz) and intensity (in a.u.)  for thermal \nperturbation between a bare SCNO device and a single MNP situated at the center of the SCNO \ndevice.  \n \n \n \n \n \n \n \nS5 \n Supp lemental  Information S 2. SCNO biosensor performance concerning reversal of current \nand applied magnetic field direction  \n \nIn the Letter , the SCNO device performance has been demonstrated using the current direction to \nbe along -x direction and the uniform external DC magnetic field to be directed along +y direction.  \nIn Figure S2(a) -(c), we observe a case where for a ferromagnetic  (FM)  nanopillar  of dimension \n160 nm × 80 nm × 5 nm , the direction  of the current is along +x direction and the uniform external \nmagnetic field is along -y direction.  The bare device shows a peak frequency at 2.35 GHz in \ncontrast to 2.387 GHz for the reversed directions.  The position dependent performance of the \nSCNO biosensor remains the same where the cases (ii) & (v) and (iii) & (iv) shows the same peaks  \n(see Figure  S2(c) ). This too confirms  the ca se for the reversed directions that the two halves of the \nSCNO biosensor works uniquely.  \n \nFigure S2 . Position dependent sensitivity of the SCNO biosensor with reversed directions of \ncurrent and applied magnetic field.  (a) Schematic of the reversed directions of applied magnetic \nfield and current to the SCNO biosensor. (b) Schematic demonstration of a single MNP on the \nsurface of the biosensor  at 5 different positions (i), (ii), (iii), (iv) & (v). Each of the 5 ca ses are \nindependent of each other.  (c) Peak frequency  (in GHz)  for the 5 different positions of the MNP \non the SCNO biosensor with respect to the bare SCNO surface . \n  \nS6 \n Supp lemental  Information S 3. Magnetic i nteraction between adjacent ferromagnetic (FM) \nnanopillars  of the SCNO  biosensor  \n \nFigure S3. (a) Spatial distribution of the magnetization of the FM nanopillar of dimensions 160 \nnm × 80 nm × 5 nm  when the SCNO was operating in precession mode under a current of 15 mA \nand a uniform DC magnetic field of 1.1kOe. (b) The spatial distributio n of the stray field of the \n160 nm × 80 nm × 5 nm  FM nanopillar  situated exactly at the center of the 200 nm × 120 nm \ngrid space . It is seen that the stray field decays significantly up to a distance of 185 nm along the \nX axes and up to 120 nm along the Y axes. Therefor e, to avoid any interaction b etween the \nadjacent FM nanopillars, it is a requirement for them to be situated away from the stray field \ninterference. Hence, we have chosen a safe distance of 0.5μm (for both along X and Y) axes  \nbetween two adjacent nanopillars (see Figure 1(a) ).   \nS7 \n Supp lemental  Information S 4. Comparison of m agnetization distribution of FM nanopillar  \nof the SCNO biosensor  with and without presence of MNPs  \n \nFigure S4 (a) -(c) represents the magnetization distribution under different conditions of the bare \ndevice with or without MNPs  when the SCNO device operates in precession mode at a total current \nof 15 mA along –x direction and a uniform DC magnetic field of 1.1 kOe along +y direction . The \ncorresponding videos for Figure S4  (a), (b) & (c) are attached in Supplementary Movie SM 1 , \nSupplementary Movie SM 2 and Supplementary Movie SM 3 , respectively.  The peak frequency \nfor each of the cases (a), (b) & (c) has been reported in the Letter as 2.3 87 GHz, 2.509 GHz and \n3.278GHz respectively (see Figure 3(d) & (f) ). \n \n \n \nFigure S4 . Magnetization distribution of FM nanopillar of the designed 160 nm × 80 nm × 5 nm  \nSCNO biosensor operating in precession mode, under a current of 15 mA and DC magnetic field \nof 1.1kOe, when (a) bare, (b) only one MNP of size 20 nm present  at the center of the device , (c) \n10 MNPs, each of size 20 nm, regularly spaced with each of their centers separated by a dis tance \nof 30 nm . \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nS8 \n Supp lemental  Information S 5. Power consumption and device performance  \n \nTo co mment on the power efficiency of the SCNO biosensor and detrimental effect on its sensor \nperformance due to Joule Heating, we have a calculation for the total power consumption and the \narea overhead in Table S1 . The HM/FM layers have been considered to hav e resistances in parallel. \nThe terms current density (J, Am-2) refers to current through the HM only. The total current (I, μA) \nrefers to the current through the entire device. These calculations suggest that SCNO biosensor \nhas a very low power consumption . \nTable S1. Power consumption of the  SCNO nanopillar  \nParameters  Description  Values  \nFM nanopillar  Dimension  Length × Width × Thickness  160 nm × 80 nm × 5 nm \nHM Dimension  Length × Width × Thickness  1μm × 80 nm × 9 nm  \nJ Electrical current density  1.5×1012A/m2 \nArea of HM  Width × Thickness  80 nm × 9 nm \nArea of FM  Width × Thickness  160 nm × 80 nm × π/4  \n𝛒𝐇𝐌 Resistivity of HM  10.6 ×10−8 ohm-m \nRHM Resistance of HM  147.22 ohm \n𝛒𝐂𝐨𝐅𝐞𝐁  Resistivity of FM  5.6 ×10−8 ohm-m \nRCoFeB Resistance of FM  4.319  ohm \nReq Equivalent resistance  4.196  ohm \nI Total current  0.0526 A \nP Power consumption  0.0116 W \n \n "    },    {        "title": "2008.12221v3.Nutation_Resonance_in_Ferromagnets.pdf",        "content": "1 \n Nutation  Resonance in Ferromagnets  \nMikhail Cherkasskii1,*, Michael Farle2,3, and Anna Semisalova2 \n1 Department of General Physics 1 , St. Petersburg State University , St. Petersburg , 199034, Russia  \n2 Faculty of Physics and Center of Nanointegration (CENIDE), University of Duisburg -Essen, Duisburg, 47057, Germany  \n3 Kirensky Institute of Physics, Federal Research Center KSC SB RAS, Russia  \n \n* Corresponding author: macherkasskii@hotmail.co m \n \n \nThe inertial  dynamic s of magnetization in a ferromagnet is investigated  theoreticall y. The analytically derived dynamic \nresponse upon microwave excitation shows two pea ks: ferromagnetic and nutation resonances. The exact analytical expressions \nof frequency and linewidth of  the magnetic  nutation resonance are deduced from the frequency dependent susceptibility \ndetermined by the i nertial Landau -Lifshitz -Gilbert equation.  The study  shows that the dependence of nutation linewidth on the \nGilbert precess ion damping  has a minimum , which becomes more expressive with increas e of the applied magnetic field.  \n \nPACS numbers: 76.50.+g, 78.47.jp, 75.50. -y \n \nI. INTRODUCTION  \nRecently, the effects of inertia in the spin dynamics of \nferromag nets were reported to cause nutation resonance  [1-\n12] at frequencies  higher  than the conventional ferromagnetic \nresonance . It was shown  that inertia  is responsible for the \nnutation , and that this type of motion should be considered  \ntogether with magnetization precession in the applied \nmagnetic field . Nutation in ferromagnets was confirmed \nexperimentally only recently  [2], since  nutation and \nprecession operate at substantially different time scales , and \nconventi onal microwave ferromagnetic resonance (FMR) \nspectroscopy techniques do not easily reach the high -\nfrequency (sub-Terahertz) regime re quired to observe the \ninertia effect which in addition yields a much weaker signal . \nSimilar to any other oscillatory system, t he magnetiza tion \nin a ferromagnet has resonant frequencies  usually studied by  \nferromagnetic resonance  [13,14]. The resonant \neigenfrequency is determined by  the magnetic parameters  of \nthe material  and applied magnetic field . Including inertia of \nthe magnetization in th e model description shows that  nutation \nand precession are complementary to each other  and several \nresonances can be generated . In this Letter , we concentrate on \nthe investigation of the resonance characteristics of nutation.  \nThe investigation of nutation is connected to the progress \nmade  in studies of the  spin dynamics at ultrashort  time \nscales  [15,16] . These successes led to the rapi d development \nof a new scientific field , the so -called ultrafast magnetism  [17-\n25]. The experimental as well as theoretical investigation of \nthe inertial  spin dynamics is at the very beginning , although it \nmight be of significance for future high speed spintronics \napplications  including ultrafast magnetic switching .  Besides nutation driven by magnetization inertia, several  \nother origins  of nutation have been reported . Transient \nnutations (Rabi oscillations) have been widely investigate d in \nnuclear magnetic resonance  [26] and electron spin \nresonance  [27-29], they were  recently addressed in \nferromagnets  [30]. A complex dynamics and Josephson \nnutation of a local spin \n1/ 2s  as well as large spin cluster \nembedded in the tunnel junction between ferromagnetic leads \nwas shown to occur  due to  a coupling to Josephson \ncurrent  [31-33]. Low-frequency nutation was observ ed in \nnanomagnets exhibiting a non -linear FMR  with the large -\nangle precession of magnetization where the onset of spin \nwave instabilities can be delay ed due to geometric \nconfinement  [34]. Nutation dynamics due to inertia of \nmagnetization in ferromagnetic thin films was observed for \nthe first time by Neeraj et al.  [2]. \nThe microscopic derivation of the magnetization  inertia \nwas performed in  ref. [3-7]. A relation between the Gilbert \ndamping constant  and the inertia l regime  characteristic time  \nwas elaborated in ref. [3]. The exchange interaction, damping, \nand moment of inertia can be calculated from first principles \nas shown in  [7]. The study of inertia spin dynamics  with a \nquantum approach  in metallic ferromagnets  was performed \nin [8]. In addition, nutation  was theoretically analyzed as a \npart of magnetization dynamics in ferromagnetic \nnanostructure  [9,10] and nanoparticles  [11]. Despite these \nadvances, exact analytical expressions for the high-frequency \nsusceptibility including inertia had not been derived yet.  \nIn [35], the inertial regime  was introduced in the \nframework of the mesoscopic nonequilibrium \nthermodynamics theory , and it was shown to be responsible \nfor the nutation superimposed on the precession of \nmagnetization . Wegrowe and Ciornei  [1] discussed  the  \n2 \n equivalence between the inertia in the dynamics of uniform \nprecession and a spinning top within the framework of the  \nLandau –Lifshitz –Gilbert equation generalized to the inertial \nregime. This equation was  studied  analytical ly and \nnumerical ly [12,36]. Although the se reports provide \nnumerical tools for obtaining resonance characteristics, the \ncomplexity of the numerical solution of differential equations \ndid not allow to estimate the nutation frequency and linewidth \naccurately . Also in a recent  remarkable paper  [37] a novel \ncollective excitation  – the nutation wave  – was reported,  and \nthe dispersion characteristics  were derived wit hout discussion \nof the nutation  resonance lineshapes  and intensities.  \nThus, at present, there is a necessity to study  the resonance \nproperties of nutation in ferromagnet s, and this paper is \ndevoted to this study. We performed the investigation based \non the Landau -Lifshitz -Gilbert equation with  the addition al \ninertia term  and provide an  analytical solution.   \nIt is well known that the Landau -Lifshitz -Gilbert equation \nallow s finding the susceptibility as the ratio between  the time-\nvarying magnetization and the time-varying driving magnetic \nfield (see for exampl e [38,39] and references therein). This \nsusceptibility describes  well the magnetic response of a \nferromagnet in the linear regime, that is  a small cone angle of \nthe precession . In this description , the ferromagnet usually is \nplaced in a  magnetic field  big enough to align all  atomic \nmagnetic moments along the field , i.e., the ferromagnet is in \nthe saturated state and the  magnetization  precess es. The \napplied  driving magnetic field  allows one to obser ve FMR  as \nsoon as the driving field frequency coincides with  \neigenfrequency of precession. Using the expression for \nsusceptibility, one can elaborate  such resonance \ncharacteristics as eigenfrequency and linewidth.  We will \npresent similar expressions for the dynamic susceptibility, \ntaking nutation into account.  \nII. SUSCEPTIBILITY  \nThe ferromagnet  is subjected to a uniform bias magnetic \nfield \n0H  acting along the z -axis and being strong enough to \ninitiate  the magnetic saturation  state. The small time -varying \nmagnetic field \nh  is superimposed on the bias field. The \ncoupling between impact and response, taking into account \nprecession, damping, and nutation, is given by the Inertial  \nLandau -Lifshitz -Gilbert ( ILLG)  equation  \n \n2\n2\n0,effd d d\ndt M dt dt       M M MMH   (1) \nwhere \n  is the gyromagnetic ratio, \nM  the magnetization  \nvector , \n0M  the magnetization at saturation, \neffH  the vector \nsum of all magnetic fields, external and internal, acting upon \nthe magnetization , \n the Gilbert damping , and \n  the inertial \nrelaxatio n time. For simplicity, we assume that the \nferromagnet is infinite, i.e. there is no demagnetization  correction , with negligible magnetocrystalline anisotropy , and \nonly the  externally applied field s contribute  to the total  field. \nThus, the bias magnetic field \n0H  and signal field \nh  are \nincluded in \neffH . We assume that the signal is small \n0,hH\n hence the magnetization is directed along \n0.H  \nOur interest is to study the correlated  dynamics of nutation \nand precession  simultaneously;  therefore we write the \nmagnetization  and magnetic field  in the general ized form  \nusing  the Fourier transformation  \n \n 01ˆ  ,\n2itt M z d e\n\n\n Mm   (2) \n \n 01ˆ  ,\n2it\nefft H z d e\n\n\n Hh   (3) \nwhere \nˆz  is the unit vector along the z -axis. If we  substitute  \nthese expressions  in the ILLG equation  and neglect the small \nterms, it leads to  \n \n\n \n 00\n211 \n22\nˆˆ\nˆˆ .i t i td i e d e\nM z H z\ni z z   \n\n   \n   \n\n     \n    \n        m\nhm\nmm   (4) \nBy performing the Fourier transform and changing the order \nof integration , equation  (4) becomes   \n \n\n\n \n00\n21 2\n1 2\nˆˆ\nˆˆ ,it\nitd dt i e\nd dt e\nM z H z\ni z z\n  \n\n   \n    \n\n \n\n\n   \n \n     \n       \nm\nhm\nmm   (5) \nwhere the integral representation of the Dirac delta  function  \ncan be found. With the delta function,  the equation (5)\nsimplifies to  \n \n  \n 00\n2ˆˆ\nˆˆ .i M z H z\ni z z     \n      \n   m h m\nmm   (6) \nBy projecting to Cartesian coordinates and  introducing the \ncircular variables  for positive and negative circular \npolarization  \n,xy m m im  \n,xy h h ih one obtains  \n \n  \n 2\n20,\n0,HM\nHMm m i m m h\nm m i m m h   \n\n        \n          \n        (7) \nwhere \n0 H H  is the precession frequency and \n0.M M\n The small -signal susceptibility follows  from \nthese equations :  \n3 \n  \n2\n2,\n,\n.M\nH\nM\nHim\nih\n\n   \n\n  \n   \n\n  \n  \n\n   (8) \nIt is seen that the susceptibility (8) is identical with  the \nsusceptibility for LLG equation , if one drops  the inertial term , \nthat is  \n0.  \nLet us separate dispersive and d issipative  parts  of the \nsusceptibility \n,i       \n \n \n 2\n2,\n,\n,\n,MH\nM\nMH\nMD\nD\nD\nD\n\n   \n\n   \n\n\n\n\n\n\n  \n\n\n   (9) \n \n 2 2 4 3\n2 2 22\n22 , 1   \n     \n    H H HD   (10) \n \n 2 2 4 3\n2 2 22\n22 . 1   \n     \n    H H HD   (11) \n \nThe frequency dependence of the dissipative parts of \nsusceptibilit ies \n and \n  is shown in the Fig. 1. The plus \nand minus subscripts correspond to right -hand and left -hand \ndirection of rotation. Since the denominators \nD  and \nD  are \nquartic polynomials, four critical points  for either  \n or \n \ncan be expected . Two of them  that are extrema with  a clear \nphysical meaning are plotted.  In Fig. 1(a) the extremum , \ncorresponding  to FMR  at \n0 H H  is shown . Due to the \ncontribution  of nutation , the frequency and linewidth of this \nresonance are slightly different from the ones of usual FMR . \nThe resonance occurs  for right -hand precession, i.e. positive \npolarization.   \nIn Fig. 1(b) the nutation resonance  possessing negative \npolarization is presented.  Note  that the polarizations of \nferromagnetic and nutation resonances are reverse d. \nIII. APPROXI MAT ION FOR  NUTATION \nFREQUENCY  \nLet us  turn to the description  of an approximation  of the \nnutation resonance frequency. If we equate the denominator \nD\n to zero, solve the resulting equation, we obtain the \napproximation from the real part of  the roots. This is reasonable , since the numerator of \n  is the linear function of \n\n , and we  are interested in \n1.  Indeed , the equation  \n \n 2 2 4 3 2 2\n202 2 1\n2H\nHH      \n     \n    (12) \nhas four roots that are complex conjugate in pairs  \n \n1,221 1 4 2,2H\nFMRiiw   \n       (13) \n \n1,221 1 4 2.2H\nNiiw   \n       (14) \n \nFIG. 1. (Color online) (a) The FMR peak with nutation. (b) \nThe nutation resonance. The calculation was performed for \n1/ 2 28  GHz T ,\n \n00 1 T, M  \n00 100 mT, H  \n0.0065\n and \n1110  s.  \n \nOne should choose  the same sign from the \n  symbol  in each \nformula , simultaneously . The real part of expression  (13) \ngives the approximate frequency for FMR , but in negative \nnumbers, so the sign should be inversed . The approximate \nfrequency of FMR in positive numbers can be derived from \nequation \n0. D  The approximate nutation frequency  is \nobtained  by the real part of  the expression (14). One takes half \nthe sum of two conjugate roots  \n1,2,Nw neglect s the high \n  \nterms , and obtains  the nutation resonance frequency  \n \n1 1 2\n.2NH\nw \n\n   (15) \nNote  that the expression of \nNw  is close  to the approximation  \ngiven in  [36] at \n1/ , H    namely  \n \nweak\nnu1\n.H \n\n   (16) \nThe similarity of both approximations b ecomes clear , if we \nperform a Taylor series expansion  and return to the notation \n,H\n \n\n2\n2 2 3 3 3\n2\nweak\nnu\n2 2 3 3 31 1 2 1\n2 2 4\n1,4\n1 1\n2\n1.18\n6H HH\nN\nH\nH HH\nHw\nO\nO  \n \n    \n   \n       \n\n   \n\n \n \n4 \n IV. PRECISE  EXPRESSIONS FOR  FREQUENCY \nAND LINEWIDTH OF NUT ATION  \nThe analytical approach proposed in this Letter yields  \nprecise values of the frequency  of nutation resonance  and the \nfull width at half maximum (FWHM) of the peak . The \nfrequency is found by extremum, when the derivative of the \ndissipative part of susceptibilities (9) is zero \n \n0.\n   (17) \nIt is enough to determine  zeros of the n umerator of th e \nderivative , that are given by  \n \n 2 2 4 3 2 2 23 4 2 1 0.HH                (18) \nLet us use Ferrari's solution  for this quartic equation  and \nintroduce the notation:  \n \n22\n2\n2\n2\n2\n3\n23\n24\n343\n4,\n2 1,  \n3,8\n,28,\n3.16 25,\n6rH\nrH\nrr\nr r\nr r r\nrr\nr r r r\nr\nr rrr\nr\nr\nrC\nE\nC\nC\nCEcA\nB\nBaA A\nBBbAA\nBB\nA AA\n\n \n\n\n\n\n\n\n  \n\n   (19) \nIn Ferrari's method , one should determine a root of the nested \ndepressed cubic equation . In the investigated case , the root is \nwritten  \n \n5,6r\nr r ray U V     (20) \nwhere  \n \n32\n3\n2\n32,27 4 2\n,3\n12\n1,\n.3 108 8r r r\nr\nr\nr\nr\nr\nrr\nrr\nr r rP Q QU\nPVU\nPc\nQaa\nabc   \n\n \n     (21) \nThus, the precise value of the nutation frequency is given by  \n \n2\n42\n2 13 2 .2 2rr r\nN\nr\nr\nrr\nrry\nA\nbaa\nay\nyB  \n   \n   (22) \nThe performed analysis shows that approximate value of \nnutation resonance frequency is close to precise value.  The linewidth of the nutation resonance is found at  a half \npeak height. If one denotes the maximum by \n,N X \n the equation  which determines \nfrequencies at half magnitude  is \n \n 2 2 4 3 2 2\n212 2 12\n2 0.H\nH H MX      \n      \n\n   (23) \nWe repeat the procedure for finding solu tions with Ferrari's \nmethod  introducing the  new notations  \n \n 22\n2\n2\n2\n2\n3\n23\n24\n3 4 21\n2\n,\n12 1 ,2\n1\n2\n3,8\n,28\n3.16 2,\n56,\n4lw\nlw\nlw\nlw\nlw\nlw lw\nlw\nlw lw\nlw lw lw\nlw\nlw lw\nlw lw lw lw lw lw\nlw\nlw lwH\nHM\nH\nlw\nlw\nlw lwA\nB\nBaA A\nBBbAA\nB B B D\nA AX\nX\nCX\nDX\nEX\nC\nCD\nA\nCEcA A\n\n \n \n\n\n\n\n\n\n  \n\n\n\n  \n  \n\n\n   (24) \nA root of the nested depressed cubic equation \nlwy  must be \nfound in the same way as provided in (20) with the \ncorresponding replacement of variables, i.e. subscript r is \nreplaced by lw. The difference between two adjacent roots \ngives the nutation linewidth  \n \n23 2 .\n2lw\nN lw lw\nlw lwbay\na y   \n   (25) \nThe explicit expression for the linewidth can be written using \nthe equations  (19)-(25). \n \n \nFIG. 2. (Color online) The dependence of the nutation \nlinewidth on the inertial relaxat ion time for \n00 100 mT, H\n \n00 1 T, M  and \n0.0065.  \n \n5 \n  \nThe effect of the inertial relaxation time on the nutation \nlinewidth is shown in Fig.  2. One can see that increasing \ninertial relaxation time leads to narrowing of the linewidth. \nThis behavior is expected and is consistent with the traditional \nview that decreasing of losses results in narrowing of \nlinewidth.  \n \n \nFIG. 3. (Color online) The dependence of nutation \nresonance linewidth on precession Gilbert damping \nparameter at different magnetic fields \n0H  for \n00 1 T M\nand \n1110  s.  \n \nSince the investig ated oscillatory system implemen ts \nsimultaneous two types of  motions , it is of interest to study the \ninfluence of the Gilbert precession  damping parameter \n  on \nthe nutation  resonance linewidth. The result is presented in \nFig. 3 and is valid for ferromagnets with vanishing anisotropy.  \nOne sees that the dependence of \nN  on \n  shows a \nminimum that becomes more expressive with increasing  bias \nmagnetic field.  In other words, t he linewidth is parametrized \nby the magnitude of field. This non-trivial  behavior of \nlinewidth relates with the nature of th is oscillatory system, \nwhich performs  two coupled  motions.  \nTo elucidate the non-trivial behavior , one can consider  the \nsusceptibility  (9) in the same way as it is usually performed \nfor the forced harmonic oscillator with damping  [40]. For this \noscillator , the linewidth can be direct ly calculated from the \ndenominator of the response expression once the driving \nfrequency is equal to eigenfreq uency. In the investigated case \nof magnetization with inertia , the response expression is (9) \nwith denominator s (10) and (11) written as  \n \n 2 2 4 3\n2 2 22\n21 . 2H H HD   \n      \n   \n   (26) \nSince the applied magnetic field is included in this expression  \nas \n0,H H the linewi dth depends on the field.  \nThe obtained result can be generalized to a fin ite sample \nwith magnetocrystalline anisotropy with method of effective \ndemagnetizing factors  [41,42] . In this case the bias magnetic field \n0H  denotes an external field and in the final expressions \nthis field should be replaced by \n 0 0 0ˆˆ ,i a d NN  H H M  \nwhere \nˆ\naN  is the anisotropy demagnetizing tensor and \nˆ\ndN  is \nthe shape demagnetizing tensor.  \nV. CONCLUSION  \nIn summary, we derived a general analytical expression for \nthe linewidth and f requency of nutation resonance in \nferromagnets, depending on magnetization, the Gilbert \ndamping, the inertial relaxation time and applied magnetic \nfield.  We show the nutation linewidth can be tuned by the \napplied magnetic field , and this tunability breaks the direct \nrelation between losses and the linewidth. This for example  \nleads to the appearance of a minimum  in the nutation \nresonance linewidth for the damping parameter  around  \n0.15.\n The obtained results are valid for ferromagnets with \nvanishing anisotropy . \nACKNOWLEDGEMENTS  \nWe thank  Benjamin Zingsem for helpful discussions . In \npart funded by Research Grant No. 075 -15-2019 -1886 from \nthe Government of the Russian Federation, the Deutsche \nForschungsgemeinschaft (DFG, German Research \nFoundation) – projects  405553726 (CRC/ TRR 270 ), and \n392402498 (SE 2853/1 -1).  \n \n[1] J.-E. Wegrowe and M. -C. Ciornei, Am. J. Phys. 80, 607 \n(2012).  \n[2] K. Neeraj, N. Awari, S. Kovalev, D. Polley, N. Zhou \nHagström, S. S. P. K. Arekapudi, A. Semisalova, K. Lenz, \nB. Green, J. C. Deinert, I. Ilyakov, M. Chen, M. Bawatna, \nV. Scalera, M. d’Aquino, C. Serpico, O. Hellwig, J. E. \nWegrowe, M. Gensch, and S. Bonetti, Nat. Phys. (2020).  \n[3] M. Fähnle, D. Steiauf, and C. Illg, P hys. Rev. B 84, \n172403 (2011).  \n[4] D. Thonig, O. Eriksson, and M. Pereiro, Sci. Rep. 7, 931 \n(2017).  \n[5] R. Mondal, M. Berritta, and P. M. Oppeneer, J. Phys. \nCondens. Matter 30, 265801 (2018).  \n[6] R. Mondal, M. Berritta, A. K. Nandy, and P. M. Oppeneer, \nPhys. Rev. B 96, 024425 (2017).  \n[7] S. Bhattacharjee, L. Nordström, and J. Fransson, Phys. \nRev. Lett. 108, 057204 (2012).  \n[8] T. Kikuchi and G. Tatara, Phys. Rev. B 92, 184410 (2015).  \n[9] D. Böttcher, A. Ernst, and J. Henk, J. Phys. Condens. \nMatter 23, 296003  (2011).  \n[10] D. Böttcher and J. Henk, Phys. Rev. B 86, 020404 (2012).  \n[11] R. Bastardis, F. Vernay, and H. Kachkachi, Phys. Rev. B \n98, 165444 (2018).  \n[12] E. Olive, Y. Lansac, and J. -E. Wegrowe, Appl. Phys. Lett. \n100, 192407 (2012).  \n[13] M. Farle, Reports  Prog. Phys. 61, 755 (1998).  \n[14] M. Farle, T. Silva, and G. Wolte rsdorf, in Springer Tracts \nin Modern Physics , edited by H. Zabel and M. Farle \n(Springer -Verlag, Berlin, Heidelberg, 2013), pp. 37 –83.. \n[15] A. V Kimel, A. Kirilyuk, P. A. Usachev, R. V Pisarev, A. \nM. Balbashov, and T. Rasing, Nature 435, 655 (2005).  \n \n6 \n [16] C. D. Stanciu, F. Hansteen, A. V Kimel, A. Kirilyuk, A. \nTsukamoto, A. Itoh, and T. Rasing, Phys. Rev. Lett. 99, \n047601 (2007).  \n[17] B. Koopmans, M. van Kampen, J. T. Kohlhepp, and W. J. \nM. de Jonge, Phys. Rev. Lett. 85, 844 (2000).  \n[18] B. Koopmans, J. J. M. Ruigrok, F. D. Longa, and W. J. M. \nde Jonge, Phys. Rev. Lett. 95, 267207 (2005).  \n[19] A. Kirilyuk, A. V Kimel, and T. Rasing, Rev. Mod. Phys. \n82, 2731 (2010).  \n[20] M. Battiato, K. Carva, and P. M. Oppeneer, Phys. Rev. \nLett. 105, 27203 (2010).  \n[21] I. Radu, K. Vahaplar, C. Stamm, T. Kachel, N.  Pontius, H. \nA. 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[22] C. Stamm, T. Kachel, N. Pontius, R. Mitzner, T. Quast, K. \nHolldack, S. Khan, C. Lupulescu, E. F. Aziz, and M. \nWietstruk, Nat. Mater. 6, 740 (2007).  \n[23] B. Koopmans, G. Malinowski, F. Dalla Longa, D. Steiauf, \nM. Fähnle, T. Roth, M. Cinchetti, and M. Aeschlimann, \nNat. Mater. 9, 259 (2010).  \n[24] C.-H. Lambert, S. Mangin, B. S. D. C. S. Varaprasad, Y. \nK. Takahashi, M. Hehn, M. Cinchetti, G. Malinowski, K. \nHono, Y. Fainman, and M. Aeschlimann, Science (80 -. ). \n345, 1337 (2014).  \n[25] C. D. Stanciu, A. Tsukamoto, A. V. Kimel, F. Hansteen, A. \nKirilyuk, A. Itoh, and T. Rasing, Phys. Rev. Lett. 99, \n217204 (2007).  \n[26] H. C. Torrey, Phys. Rev. 76, 1059 (1949).  \n[27] N. C. Verma and R. W. Fessenden, J. Chem. Phys. 58, \n2501 (1973).  \n[28] P. W. Atkins, A. J. Dobbs, and K. A. McLauchlan, Chem. \nPhys. Lett. 25, 105 (1974).  \n[29] G. G. Fedoruk, J. Appl. Spectrosc. 69, 161 (2002).  \n[30] A. Capua, C. Rettner, S. -H. Yang, T. Phung,  and S. S. P. \nParkin, Nat. Commun. 8, 16004 (2017).  \n[31] J. Fransson, Nanotechnology 19, 285714 (2008).  \n[32] Z. Nussinov, A. Shnirman, D. P. Arovas, A. V. Balatsky, \nand J. X. Zhu, Phys. Rev. B 71, 214520 (2005).  \n[33] J.-X. Zhu, Z. Nussinov, A. Shnirman, an d A. V Balatsky, \nPhys. Rev. Lett. 92, 107001 (2004).  \n[34] Y. Li, V. V. Naletov, O. Klein, J. L. Prieto, M. Muñoz, V. \nCros, P. Bortolotti, A. Anane, C. Serpico, and G. de \nLoubens, Phys. Rev. X 9, 041036 (2019).  \n[35] M.-C. Ciornei, J. M. Rubí, and J. -E. Wegr owe, Phys. Rev. \nB 83, 020410 (2011).  \n[36] E. Olive, Y. Lansac, M. Meyer, M. Hayoun, and J. -E. \nWegrowe, J. Appl. Phys. 117, 213904 (2015).  \n[37] I. Makhfudz, E. Olive, and S. Nicolis, Appl. Phys. Lett. \n117, 132403 (2020).  \n[38] B. W. Zingsem, M. Winklhofer, R. Meckenstock, and M. \nFarle, Phys. Rev. B 96, 224407 (2017).  \n[39] A. G. Gurevich and G. A. Melkov, Magnetization \nOscillations and Waves  (CRC press, 1996).  \n[40] R. P. Feynman, R. B. Leighton, and M. Sands, The \nFeynman Lectures  on Physics, Vol. I: The New Millennium \nEdition: Mainly Mechanics, Radiation, and Heat  (Basic \nbooks, 2011).  \n[41] C. Kittel, Phys. Rev. 73, 155 (1948).  \n[42] J. R. Macdonald, Proc. Phys. Soc. Sect. A 64, 968 (1951).  \n "    },    {        "title": "1603.01142v1.Cantilever_detected_ferromagnetic_resonance_in_thin_Fe___50__Ni___50____Co__2_FeAl___0_5__Si___0_5___and_Sr__2_FeMoO__6__films_using_a_double_modulation_technique.pdf",        "content": "arXiv:1603.01142v1  [cond-mat.str-el]  3 Mar 2016Cantilever detected ferromagnetic resonance in thin Fe 50Ni50, Co2FeAl0.5Si0.5and\nSr2FeMoO 6ﬁlms using a double modulation technique.\nAlexey Alfonsova,∗, Eiji Ohmichib, Pavel Leksinc, Ahmad Omarc, Hailong Wangd, Sabine Wurmehlc,e, Fengyuan\nYangd, Hitoshi Ohtaa,b\naMolecular Photoscience Research Center, Kobe University, Kobe 657-8501, Japan\nbGraduate School of Science, Kobe University, 1-1 Rokkodai- cho, Nada, Kobe 657-8501, Japan\ncLeibniz Institute for Solid State and Materials Research Dr esden, IFW, D-01171 Dresden, Germany\ndDepartment of Physics, The Ohio State University, Columbus , Ohio 43210, USA\neInstitut fur Festk¨ orperphysik, Technische Universit¨ at Dresden, D-01062 Dresden, Germany\nAbstract\nIn this work we introduce a new method of a ferromagnetic resonan ce (FMR) detection from thin, nm-size, ﬁlms. Our\nsetup is based on the commercial piezo-cantilever,used for atomic force microscopy. It has an option to rotate the sample\nin the magnetic ﬁeld and it operates up to the high microwave frequen cies of 160 GHz. Using our cantilever based FMR\nspectrometer we have investigated a set of samples, namely quasi- bulk and 84 nm ﬁlm Co 2FeAl0.5Si0.5samples, 16\nnm Fe 50Ni50ﬁlm and 150 nm Sr 2FeMoO 6ﬁlm. The high frequency ferromagnetic resonance (FMR) respons e from an\nextremely thin Fe 50Ni50ﬁlm we have ﬁtted with the conventional model for the magnetizatio n dynamics. The cantilever\ndetected FMR experiments on Sr 2FeMoO 6ﬁlm reveal an inability of the conventional model to ﬁt frequency an d angular\ndependences with the same set of parameters, which suggests th at one has to take into account much more complicated\nnature of the magnetization precession in the Sr 2FeMoO 6at low temperatures and high frequencies. Moreover, the\ncomplicated dynamics of the magnetization apparent in all investigat ed samples is suggested by a drastic increase of\nthe linewidths with increasing microwave frequency, and by an emerg ence of the second line with an opposite angular\ndependence.\nKeywords: Ferromagnetic resonance, FMR, Cantilever, Halfmetals\n2010 MSC: 00-01, 99-00\n1. Introduction\nA key property of many intensively studied materials\nis a magnetic anisotropy, which is deﬁned by the com-\nplex interplay of diﬀerent degrees of freedom, such as spin\nor/and orbital moments, charge and lattice. In particular,\nin the ferromagnets promising for spintronic application,\nsuch as Sr 2FeMoO 6[1, 2] and Co 2FeAl0.5Si0.5[3, 4], mag-\nnetic anisotropy deﬁnes the thermal stability of the mag-\nnetization. One of the most appropriate methods to study\nmagnetic anisotropies, as well as gyromagnetic ratios and\nmagnetization dynamics in ferromagnets is ferromagnetic\nresonance (FMR).\nPotentialhalfmetallicferromagneticmaterialshasbeen\nalready studied by means of ferromagnetic resonance. In\nmost cases the study was limited to room temperatures\nand low frequencies of standard spectrometers [5, 6, 7],\nwhereas there are also measurements performed using vec-\ntor network analysers and microstrip resonators at fre-\nquencies up to 70 GHz [8, 9, 10]. Additionally there are\n∗Corresponding author\nEmail address: a.alfonsov@ifw-dresden.de (Alexey Alfonsov)few reports of measurements at even higher frequencies\n[11]. Increasing a measurement frequency in the FMR ex-\nperiment is very important, since it yields a higher res-\nolution and, therefore, better determination of g-factors\nand magnetic anisotropies. Unfortunately high-frequency\nmeasurements are associated with several problems in the\ndetection of the FMR signal, especially when performing\nmeasurements on thin, nm-size, ﬁlms. Namely, in order to\nincrease the sensitivity one has to apply restrictionson the\nmicrowave frequency, or strength and orientation of mag-\nnetic ﬁeld. For instance, standard resonators that amplify\nthemicrowavepoweratthesampleanddrasticallyincrease\nthe sensitivity, are often used only in a narrow frequency\nrange.\nIn this work ﬁrst we introduce a new method of de-\ntecting ferromagnetic resonance from thin, nm-size, ﬁlms,\nwhere all the restrictions named above are lifted. Our\nsetup, which is based on the measurement of the deﬂec-\ntion of a µm-size piezo-cantilever, has an option to change\nthe angle between magnetic ﬁeld and the ﬁlm plane, and\nit works up to the high microwave frequencies of 160 GHz.\nUsingthis setupweinvestigatedaset ofselected materials ,\nnamely quasi-bulk Co 2FeAl0.5Si0.5, 84 nm Co 2FeAl0.5Si0.5\nPreprint submitted to Journal of Magnetic Resonance May 30, 2022ﬁlm, 16 nm Fe 50Ni50ﬁlm and 150 nm Sr 2FeMoO 6ﬁlm.\nWehavemeasuredhighfrequencyferromagneticresonance\nresponse from an extremely thin Fe 50Ni50ﬁlm and ﬁtted\nit with the conventional model for the magnetization dy-\nnamics [12, 13]. The cantileverdetected FMR experiments\non Sr2FeMoO 6ﬁlm reveal an inability of the conventional\nmodel to ﬁt frequency and angular dependences with the\nsame set of parameters, which suggests that one has to\ntake into account much more complicated nature of the\nmagnetization precession in the Sr 2FeMoO 6at low tem-\nperaturesandhighfrequencies. Moreover,thecomplicated\ndynamics of the magnetization apparent in all the inves-\ntigated samples is supported by a drastic increase of the\nlinewidths with increasing microwave frequency, and by\nthe emergence of the second line with an opposite angular\ndependence.\n2. Investigated samples\nPolycrystalline Co 2FeAl0.5Si0.5bulk sample was cast\nbyarc-meltingstoichiometricquantitiesofatleast4Npure\nconstituents. Before melting the sample itself, the cham-\nber wasevacuatedto 10−5mbarpressurebefore backﬁlling\nwith argon followed by melting a Ti piece in order to min-\nimize oxygen. In order to achieve good homogeneity, the\nsample was ﬂipped and re-melted 4 times. Resulting cast\nsample was then sealed in an evacuated quartz ampoule\nand subsequently annealed at 1400 K for 3 days followed\nby slow-cooling. The annealed sample was mechanically\ngrinded to a thin plate of ∼17µm.\nEpitaxial 84 nm Co 2FeAl0.5Si0.5ﬁlm was grown on\nMgAl2O4(001) substrate by oﬀ-axis sputtering in a UHV\nsystem with a base pressure as low as 9.5 ·10−11mbar us-\ning ultra-pure Ar (99.9999%) as sputtering gas. Optimal\nquality Co 2FeAl0.5Si0.5epitaxial ﬁlm was obtained at an\nAr pressure of 6 ·10−3mbar, a substrate temperature of\n600◦C, and DC sputtering at a constant current of 12 mA,\nwhich results in a deposition rate of 5.6 ˚A/min [14, 15].\nThe deposition of 16 nm Fe 50Ni50ﬁlm was performed\nusing an e-gun in ultra high vacuum (UHV) with pressure\n10−9mbar on the MgO(100) substrate at a temperature\nTsub= 300 K [16]. The deposition rate was set to be 30\n˚A/min.\nEpitaxial Sr 2FeMoO 6ﬁlms of 150 nm thickness were\ngrown on SrTiO 3(001) substrates in an ultrahigh vacuum\nsputtering system with a base pressure of 6 .5·10−10mbar\nusing a stoichiometric Sr 2FeMoO 6targets [17, 7]. Direct-\ncurrent (DC) magnetron sputtering was used for ﬁlm de-\nposition with a constant current of 5 mA in a pure Ar gas\n(99.9995%) of 9 ·10−3mbar, which resulted in a growth\nrate of 0.84 ˚A/min. The ﬁlms were deposited in 90ooﬀ-\naxis geometry with the substrate temperature maintained\nat 800◦C.DA+reference\nfrequencyRref\nV2 Lock-In\nV1-V2\nV1Rcanti\nGR1 R2\nV2-V1\n-a) b)\nxθz\nyM\nφθH\nφHH\nFigure 1: a) Schematic diagram of a piezocantilever resista nce de-\ntection system based on the Wheatstone bridge. b) Deﬁnition of the\nmagnetization and the magnetic ﬁeld directions with x-y bei ng a ﬁlm\nplane.\n3. Experimental setup\nThe present experimental setup is based on the can-\ntilever detected ESR setup developed in the Molecular\nPhotoscience Research Center in Kobe University, Kobe,\nJapan [18, 19, 20, 21] . The sample is glued directly to a\ncommercial piezo-cantilever [19, 21, 22]. The deﬂection of\nthe cantilever is measured by means of a bias box based on\nthe Wheatstone bridge, alternating voltage generator (G)\nanddiﬀerentialampliﬁer(DA)(Fig.1(a)). Thealternating\nvoltage generator (G) provides a voltage which feeds the\nbridge. R cantiis the cantilever with the sample, whereas\nRrefis a reference unloaded cantilever [18, 19], which is\nneededtocancelunwantedtemperatureandmagneticﬁeld\neﬀects. Two variable resistances R 1and R 2are tuned in\norder to equalize all the bridge resistances so that the volt-\nage diﬀerence V 2-V1is equal to zero. When the ESR or\nFMR absorption occurs, the cantilever is pulled by the\nsample, which leads to a change of R canti, and therefore\nto a change of the diﬀerence V 2-V1. This change is de-\ntected byahighlysensitiveLock-Inampliﬁer, which tracks\nthe signal at the reference frequency equal to a frequency\nof the voltage generator G (Fig. 1(a)). Our experiments\nshowed that this frequency for the voltage modulation has\nto be near the half of the eigenfrequency of the cantilever\nto maximize the signal to noise ratio.\nIn order to check the sensitivity of the cantilever based\nspectrometer we performed the following experiment. The\n84 nm Co 2FeAl0.5Si0.5thin ﬁlm sample glued to a can-\ntilever was inserted into a cavity of the highly sensitive\ncommercial Bruker X-Band (9.56 GHz) EPR spectrome-\nter. In this experiment, the signal at the X-Band spec-\ntrometer detector was recorded together with the signal\nfrom piezocantilever while sweeping the magnetic ﬁeld at\nroom temperature. As can be seen in Fig. 2(a) the exper-\niment showed that both detection systems yield identical\nFMR responses, and that the sensitivity of the piezocan-\ntilever detection system is comparable to that of the X-\nBand spectrometer from Bruker. Note that the Bruker\nX-Band spectrometer is measuring the absorption deriva-\n2/s48/s46/s56 /s48/s46/s57 /s49/s46/s48 /s48/s46/s48/s48 /s48/s46/s48/s53 /s48/s46/s49/s48 /s48/s46/s49/s53\n/s32/s32/s67/s97/s110/s116/s105/s108/s101/s118/s101/s114/s32/s115/s105/s103/s110/s97/s108/s32/s40/s97/s114/s98/s46/s32/s117/s46/s41\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s40/s84/s41/s97/s41 /s98/s41\n/s99/s97/s110/s116/s105/s108/s101/s118/s101/s114/s32/s100/s101/s116/s101/s99/s116/s105/s111/s110\n/s32/s32/s70/s77/s82/s32/s115/s105/s103/s110/s97/s108/s32/s40/s97/s114/s98/s46/s32/s117/s46/s41\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s40/s84/s41/s32/s32/s32/s32/s88/s45/s98/s97/s110/s100/s32/s100/s101/s116/s101/s99/s116/s105/s111/s110\nFigure 2: a)Comparison of the FMR signal from 84 nm ﬁlm of\nCo2FeAl0.5Si0.5measured at room temperature by the commercial\nX-Band spectrometer from Bruker (bottom line) and by our cus tom\nmade setup with a piezocantilever acting as a detector (top l ine). The\nmeasurements were carried out simultaneously. b)Signal from the\ncantilever as the function of the magnetic ﬁeld measured by d ouble\nmodulation technique. Modulated parameters are the freque ncy of\nthe voltage applied to the Wheatstone bridge and the power of the\nmicrowave.\ntive, whereas the cantilever deﬂection represents a pure\nabsorption, which is depicted in Fig. 2(a).\nIt is important to bear in mind that during the can-\ntilever detected FMR experiment we also measure a de-\nﬂection of the cantilever due to the magnetic anisotropy\nof the sample. This is a very strong non-resonance ef-\nfect, which depends on the sample magnetization and the\nangle between ﬁlm plane and the applied magnetic ﬁeld.\nSuch cantilever deﬂection is the same with or without\nthe microwave radiation. Therefore there is a need to\nsubtract the background signal from such non-resonant\nmagnetic response. First option is to measure the can-\ntileverresponsetwice, ﬁrsttimewith the microwavesource\nswitched on and second time with it switched oﬀ. Then\nthe latter measurement can be subtracted from the for-\nmer one to get the pure FMR signal. In this case the\ndrawback is that any drift of the background during both\nmeasurements will be seen in the result of the subtrac-\ntion. To avoid this and to make these two measurements\nquasi-simultaneous, we have implemented a second modu-\nlation, in this case the modulation of the microwavepower\n(on/oﬀ) at a low frequency of ∼0.2 Hz. The raw measure-\nment data are shown in Figure 2(b). As can be seen the\nsignal is oscillating with the second modulation frequency\n(∼0.2 Hz), and the pure FMR response can be obtained\nby subtracting the lower envelope of this signal from the\nupper one.\nHighmicrowavefrequencyradiationwasgeneratedbya\nset of Gunn oscillators from Millitech. The cantilever was\nmounted on the probehead with in-situsample rotation\nmechanism, described elsewhere [20]. The probehead was\ninserted into a magneto-cryostatfrom Oxford Instruments\nwith maximum magnetic ﬁeld of 15 T. The temperatureof all the high frequency experiments was 8 K.\n4. Model for analysis of the FMR results\nTo analyze the frequency and the angular dependences\nof the measured spectra we haveused a standard approach\n[12, 13], where the resonance frequency is given by the\nfollowing equation:\nω2=γ2\nM2ssin2θ/parenleftbigg∂2E\n∂θ2∂2E\n∂φ2−/parenleftBig∂2E\n∂θ∂φ/parenrightBig2/parenrightbigg\n(1)\nHereγisthegyromagneticratioandtheenergydensity\nEis shown below:\nE=−(# –H·# –M)+2πM2\nscos2θ−KUcos2θ\n+Kc(α2\nxα2\ny+α2\nxα2\nz+α2\nyα2\nz) (2)\nwhere\nαx= sinθcosφ;αy= sinθsinφ;αz= cosθ\nIn Eq.2 the ﬁrst term represents the Zeeman energy\nwith# –H(H,θH,φH) being the applied magnetic ﬁeld and# –M(Ms,θ,φ) being the magnetization, M sis the satura-\ntion magnetization. Second term is a shape anisotropyen-\nergy for a thin ﬁlm. Third and fourth terms are uniaxial\nand cubic anisotropies with KUandKcbeing the out-of-\nplaneuniaxial anisotropy and cubic anisotropy constants,\nrespectively. θ(θH) isthe anglebetweenthe perpendicular\nto the ﬁlm plane and the magnetization direction (applied\nmagnetic ﬁeld# –H).φ(φH) is the angle which deﬁnes the\ndirection of the magnetization (applied magnetic ﬁeld# –H)\nin the ﬁlm plane (see Fig. 1(b)). External magnetic ﬁeld# –His deﬁned as it is applied in the experiment, and the\nanglesθandφare found by numerical minimization of the\nenergy density (Eq. 2) for each given condition. As the\nresult of the ﬁtting the Eq. 1 to the experimental data we\nobtainγ,KUandKc.\n5. Experimental results\n5.1. Overview of the high-frequency FMR response from\nall investigated samples\nAs can be seen in Fig. 3(a) all the investigated thin\nﬁlms exhibit a rather strong FMR response at 8 K and\nat the measurement microwave frequency of 120 GHz. All\nthe FMR signals consist of 2 lines, low ( A) and high ( B)\nﬁeld peaks. At this frequency lines are rather broad, with\nthe widths ranging from ∼1 T to∼1.5 T for the ﬁrst peak,\nand from ∼1.2 T to ∼2.1 T for the second peak, respec-\ntively. These values were obtained by ﬁtting FMR signals\nwith two Lorentzian lines, respective ﬁts are shown in the\nsame Fig. 3(a) as solid lines. Interestingly, the linewidths\nmeasured at high frequencies using cantilever detection\n3/s48 /s49 /s50 /s51 /s48 /s49 /s50 /s51 /s52/s83/s114\n/s50/s70/s101/s77/s111/s79\n/s54/s32\n/s56/s52/s32/s110/s109/s32/s102/s105/s108/s109\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s40/s84/s41/s67/s111\n/s50/s70/s101/s65/s108\n/s48/s46/s53/s83/s105\n/s48/s46/s53/s32\n/s98/s117/s108/s107\n/s72/s32/s124/s124/s32/s102/s105/s108/s109/s32/s112/s108/s97/s110/s101\n/s56/s48/s32/s71/s72/s122/s66\n/s49/s50/s48/s32/s71/s72/s122/s99/s41 /s98/s41\n/s72/s32/s124/s124/s32/s102/s105/s108/s109/s32/s112/s108/s97/s110/s101\n/s49/s50/s48/s32/s71/s72/s122\n/s70/s101\n/s53/s48/s78/s105\n/s53/s48\n/s83/s114\n/s50/s70/s101/s77/s111/s79\n/s54\n/s67/s111\n/s50/s70/s101/s65/s108\n/s48/s46/s53/s83/s105\n/s48/s46/s53/s32/s56/s52/s32/s110/s109/s32/s102/s105/s108/s109/s67/s97/s110/s116/s105/s108/s101/s118/s101/s114/s32/s100/s101/s116/s101/s99/s116/s101/s100/s32/s115/s105/s103/s110/s97/s108/s32/s40/s97/s114/s98/s46/s32/s117/s46/s41\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s40/s84/s41/s97/s41\n/s65\n/s48 /s49 /s50 /s51 /s52/s72/s32/s61/s32/s48/s111\n/s72/s32/s61/s32/s51/s48/s111\n/s72/s32/s61/s32/s54/s48/s111\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s40/s84/s41/s72/s32/s61/s32/s57/s48/s111/s65/s65/s65/s65\nFigure 3: a) Comparison of the cantilever detected FMR spect ra\nof Co2FeAl0.5Si0.5ﬁlm (bottom line), Fe 50Ni50ﬁlm (top line) and\nSr2FeMoO 6ﬁlm (middle line), measured at the temperature of 8\nK, at the frequency of 120 GHz and with magnetic ﬁeld lying in\nthe ﬁlms plane. The solid lines over measured spectra repres ent the\ntwo Lorentzian lines ﬁt. b) Angular dependence of the cantil ever\ndetected FMR signal measured at the temperature of 8 K and at t he\nfrequency of 120 GHz. θHis the angle between perpendicular of the\nﬁlm and the applied magnetic ﬁeld. c) Comparison of the canti lever\ndetected FMR spectra of Co 2FeAl0.5Si0.5bulk (17 µm thick plate)\nsample (bottom line) and Co 2FeAl0.5Si0.5ﬁlm (top line) measured\nat the temperature of 8 K and at the frequency of 80 GHz.\nare much larger then those measured at low frequencies\nusing Bruker X-Band spectrometer, this is especially no-\nticeable in the case of Fe 50Ni50ﬁlm (see spectra in Fig.\n4). Such a drastic increase of the linewidths with increas-\ning microwave frequency at low temperatures remains an\nopen question, but most likely points to the complicated\ndynamics of the magnetization apparent in all the inves-\ntigated samples. As has been shown, the linewidth in the\ncase of FMR is deﬁned by the complex interplay of dif-\nferent mechanisms, including Gilbert damping, mosaicity,\ntwo magnon scaterring, etc. (Ref. [8, 23] and references\ntherein), and therefore needs a further investigationswith-\nout restrictions in frequencies and temperatures.\nThe angular dependence of the FMR response shows\na peculiar behavior for the second line B(Fig. 3(b)).\nWhereas the line Ashifts to higher ﬁelds, as expected for\nFMR signal when magnetic ﬁeld is rotated from the plane\nof the ﬁlm ( θH= 90o, Fig. 1(b)) towards perpendicular\ndirection ( θH= 0o), the second one shifts to lower ﬁelds.\nTherefore the more intensive ﬁrst line ( A), or at least its\nresonance position, we can attribute to a uniform preces-\nsionmode, andthesecondline( B)isanothermodeexcited\nin the sample, possibly perpendicular standing spin wave\nmode [24] excited at such high frequencies. Due to the\nunclear nature of the high ﬁeld peak B, we will focus on\nthe analysis of the low ﬁeld line Awith expected angular\ndependence.\nComparing the Co 2FeAl0.5Si0.5thin ﬁlm FMR signal1\nto the one from a quasi-bulk (17 µm - thick) sample (see\nFig. 3(c)), we see that qualitatively the shape of the FMR\nsignal remains the same, although the details, such as res-/s48 /s49 /s50 /s51 /s52 /s53 /s54/s48/s50/s53/s53/s48/s55/s53/s49/s48/s48/s49/s50/s53/s49/s53/s48/s49/s55/s53/s50/s48/s48\n/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48 /s56/s48 /s57/s48/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s71/s72/s122/s41\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s40/s84/s41/s97/s41/s32 /s32 /s32\n/s32\n/s72/s32/s111\n/s32/s50/s57/s53/s32/s75\n/s32/s115/s105/s109/s117/s108/s97/s116/s105/s111/s110/s32\n/s32/s51/s46/s52/s55/s32/s75\n/s32/s115/s105/s109/s117/s108/s97/s116/s105/s111/s110/s82/s101/s115/s111/s110/s97/s110/s99/s101/s32/s102/s105/s101/s108/s100/s32/s40/s84/s41\nFigure 4: Frequency as a function of the resonance ﬁeld for Fe 50Ni50\n16 nm ﬁlm, shown together with measured spectra. Spectra mea -\nsured at 120 GHz and 160 GHz were detected by the cantilever de -\nﬂection. Spectrum at 9.56 GHz was measured using Bruker X-ba nd\nspectrometer, here the integral is shown. The angular depen dences\nof the resonance ﬁelds measured at 9.56 GHz together with the sim-\nulated curves are shown in the inset.\nonance positions and the linewidths of the individual lines\nAandBarediﬀerent. Interestingly, thelinewidth issome-\nwhat smaller in the case of 17 µm - thick plate. This\nsuggests that such double-line shape is not only due to\na small thickness of the measured ﬁlms, but also repre-\nsents an intrinsic eﬀect related to the high frequency of\nthe excitation. As can be seen in Fig. 5(b) in the case\nof Sr2FeMoO 6sample, the frequency when the second line\nbecomes apparent should be above ∼60 GHz.\n5.2. FMR measurements on Fe 50Ni5016 nm ﬁlm\nIn the case of Fe 50Ni50ﬁlm two spectra were mea-\nsured at 120 GHz and 160 GHz at the temperature of\n8 K and with external magnetic ﬁeld lying in the ﬁlm\nplane (Fig. 4). In addition two angular dependences were\nmeasured by Bruker X-Band spectrometer at 9.56 GHz,\none at room temperature, and another at 3.47 K (see in-\nset in Fig. 4). Fitting the high frequency FMR response\nwith two Lorentzian lines we were able to extract the res-\nonance positions of both peaks, which are shown together\nwith respective spectra in Fig. 4. Using the model de-\nscribed in the Section 4, we have ﬁtted the frequency vs\nresonance ﬁeld dependence for the magnetic ﬁeld being in\nthe ﬁlm plane and the angular dependence measured at\nthe X-band frequency of 9.56 GHz and T = 3.47 K si-\nmultaneously. The ﬁt, which is shown in Fig. 4 as solid\nlines, yields γ/2π= 28 GHz/T, cubic anisotropy constant\nKc=−5·105ergcm−3and uniaxial anisotropy constant\nKU=−125·106ergcm−3. The negative sign and sig-\nniﬁcant value of KUpoints to the fact that Fe 50Ni50has\nan easy plane anisotropy, which can be explained by the\nnoticeable lattice mismatch between Fe 50Ni50(a= 3.578\n1The study of the Co 2FeAl0.5Si0.5thin ﬁlm samples with diﬀerent\nthickness is published elsewhere.\n4/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53 /s51/s46/s48/s48/s50/s48/s52/s48/s54/s48/s56/s48/s49/s48/s48/s49/s50/s48/s49/s52/s48/s49/s54/s48/s49/s56/s48\n/s32/s80/s101/s97/s107/s32/s65\n/s32/s80/s101/s97/s107/s32/s66\n/s32/s115/s105/s109/s117/s108/s97/s116/s105/s111/s110\n/s72/s32/s124/s124/s32/s102/s105/s108/s109/s32/s112/s108/s97/s110/s101\n/s84/s32/s61/s32/s56/s32/s75/s70/s101/s113/s117/s101/s110/s99/s121/s32/s40/s71/s72/s122/s41\n/s82/s101/s115/s111/s110/s97/s110/s99/s101/s32/s102/s105/s101/s108/s100/s32/s40/s84/s41/s48 /s49 /s50 /s51 /s52 /s53 /s54/s57/s46/s53/s54/s32/s71/s72/s122/s44 /s32\n/s105/s110/s116/s101/s103/s114/s97/s116/s101/s100/s32/s88/s45/s98/s97/s110/s100/s32/s115/s105/s103/s110/s97/s108/s72/s32/s124/s124/s32/s102/s105/s108/s109/s32/s112/s108/s97/s110/s101\n/s84/s32/s61/s32/s56/s32/s75\n/s49/s54/s48/s32/s71/s72/s122\n/s49/s52/s48/s32/s71/s72/s122\n/s49/s50/s48/s32/s71/s72/s122\n/s56/s48/s32/s71/s72/s122\n/s54/s48/s32/s71/s72/s122/s70/s77/s82/s32/s114/s101/s115/s112/s111/s110/s115/s101/s32/s40/s97/s114/s98/s46/s32/s117/s46/s41\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s40/s84/s41/s52/s48/s32/s71/s72/s122/s97/s41 /s98/s41\nFigure 5: a) Frequency dependence of the Sr 2FeMoO 6FMR re-\nsponse, measured at T= 8Kand with magnetic ﬁeld lying in the\nﬁlm plane. The lowest line represents an integrated spectru m mea-\nsured in the Bruker X-band spectrometer, the other lines are the\nspectra measured by the cantilever deﬂection. b) Frequency as a\nfunction of the resonance ﬁeld for two Lorentzian lines cons tituting\nthe spectrum (Squares for peak Aand circles for peak B). Solid line\nrepresents a ﬁt using model described in the section 4 (see te xt for\ndetails).\n[25]) and MgO substrate ( a= 4.212 [26]), especially con-\nsidering a rather small ﬁlm thickness of 16 nm. Impor-\ntantly, reducing only the uniaxial anisotropy constant to\nthe vale of KU=−10·106ergcm−3in our simulations,\nwe can ﬁt the room temperature angular dependence mea-\nsured at the frequency of 9.56 GHz. Such drastic increase\nof the anisotropy constants with decreasing temperature\nhas been reported before [27] and explained by the collec-\ntive change of the lattice parameters and the total magne-\ntization with changing temperature .\n5.3. FMR measurements on Sr 2FeMoO 6150 nm ﬁlm\nBesides the angular dependence measured at a fre-\nquency of 120 GHz and temperature of 8 K (see Fig. 3(b))\nwe havemeasuredafrequencydependence ofthe FMR sig-\nnal with magnetic ﬁeld lying in the ﬁlm plane (Fig.5(a)).\nThe ﬁt with two Lorentzian lines described before yields a\nfrequencyvsresonanceﬁelddiagramforbothpeaks,shown\nin Fig.5(b). Usingthemodel describedin theSection 4, we\nhave ﬁtted this frequency vs resonance ﬁeld dependence of\nthe low ﬁeld peak. The result of the ﬁt is depicted as solid\nline. The saturation magnetization value for the equations\n1 and 2 was taken from the static magnetometry measure-\nments and is equal Ms= 1.5µB/F.U. This ﬁt yields the\ngyromagnetic ratio γ/2π= 29.4 GHz/T, cubic anisotropy\nconstant Kc=−1.5·105ergcm−3and uniaxial anisotropy\nconstant KU=−22.5·106ergcm−3. Theγvalue is typical\nfor Sr2FeMoO 6and agrees well with previous reports [6].\nInaddition,wehavemeasuredtheangulardependences\nof the FMR response at two temperatures, T= 4 K and\nT= 295 K, using commercial Bruker X-band spectrome-\nter (ν= 9.56 GHz). Representative spectra are shown in/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48 /s56/s48 /s57/s48/s48/s46/s50/s53/s48/s46/s53/s48/s48/s46/s55/s53/s49/s46/s48/s48/s49/s46/s50/s53/s49/s46/s53/s48/s49/s46/s55/s53\n/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53 /s48/s46/s54 /s48/s46/s55 /s48/s46/s56 /s48/s46/s57/s32 /s32/s61/s32/s49/s50/s48/s32/s71/s72/s122/s44/s32/s84/s32/s61/s32/s56/s32/s75\n/s32 /s32/s61/s32/s57/s46/s53/s54/s32/s71/s72/s122/s44/s32/s84/s32/s61/s32/s52/s32/s75\n/s32 /s32/s61/s32/s57/s46/s53/s54/s32/s71/s72/s122/s44/s32/s84/s32/s61/s32/s50/s57/s53/s32/s75/s82/s101/s115/s111/s110/s97/s110/s99/s101/s32/s102/s105/s101/s108/s100/s32/s40/s84/s41\n/s72/s32/s111/s72/s32/s61/s32/s48/s111\n/s72/s32/s61/s32/s51/s48/s111\n/s72/s32/s61/s32/s54/s48/s111\n/s72/s32/s61/s32/s57/s48/s111/s72/s32/s61/s32/s48/s111\n/s72/s32/s61/s32/s51/s48/s111\n/s72/s32/s61/s32/s54/s48/s111/s65/s98/s115/s111/s114/s112/s116/s105/s111/s110/s32/s100/s101/s114/s105/s118/s97/s116/s105/s118/s101/s32/s40/s97/s114/s98/s46/s32/s117/s46/s41\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s40/s84/s41/s72/s32/s61/s32/s57/s48/s111/s97/s41 /s98/s41\nFigure 6: a) Sr 2FeMoO 6spectra measured in Bruker X-band spec-\ntrometer at ν= 9.56 GHz, at diﬀerent angles ( θH) between ﬁlm\nplane and applied magnetic ﬁeld. Four bottom lines represen t the\nmeasurements performed at T= 4 K, four top lines are the ones\natT= 295 K. b) The angular dependences of the resonance ﬁelds\n(symbols) with ﬁts (solid lines) using model described in th e section 4\n(see text for details). Squares represent the measurement p erformed\natν= 120 GHz, at T= 8 K using cantilever based setup. Cir-\ncles and triangles represent the measurement using Bruker X -band\nspectrometer at T= 4 K and at T= 295K, respectively.\nFig. 6(a). The lines are rather broad, especially at T= 4\nK. To obtain the resonance positions we have integrated\nall the spectra and picked all the resonance ﬁelds at the\nmaxima of these integrals. These resonance positions are\nshown in Fig. 6(b) as a function of the angle θHtogether\nwith the resonance positions measured at ν= 120 GHz\nusing cantilever detected FMR setup.\nTaking the γvalue and the anisotropy constants ob-\ntained from the frequency vs resonance ﬁeld dependence\nﬁt, we have simulated the low temperature angular depen-\ndences measured at ν= 9.56 GHz and ν= 120 GHz. As\ncan be seen in Fig. 6(b), the simulated curves reproduce\nrather well the measured dependences at the angles near\nθH= 90◦,but strongly deviate at smaller angles . Interest-\ningly, the angular dependence measured at ν= 9.56 GHz,\nat the room temperature, can be almost perfectly ﬁtted\nwith the gyromagnetic ratio γ/2π= 29.4 GHz/T and uni-\naxial anisotropy constant KU=−4·106ergcm−3, which\nis noticeably smaller then its low temperature value. Such\ninability to ﬁnd common parameters to ﬁt all the data,\nlike in the case of Fe 50Ni50ﬁlm, suggests that the model\nhas to be reconsidered, taking into account much more\ncomplicated nature of the magnetization precession in the\nSr2FeMoO 6at low temperatures and high microwave fre-\nquencies.\n6. Summary\nIn the present work we have introduced a new method\nfor the detection of the ferromagnetic resonance of the\nsamples with the thickness from 17 µm down to 16 nm. It\nis based on the deﬂection of the cantilever, which occursat\n5theferromagneticresonance. Inordertoincreasethesensi-\ntivity we have implemented a double modulation, namely,\nwe modulate the voltageatthe Wheatstonebridge andthe\noutput of the microwave source at the same time. Using\nthissetupwehaveinvestigatedasetofsamples: quasi-bulk\nand 84 nm ﬁlm Co 2FeAl0.5Si0.5sample, 16 nm Fe 50Ni50\nﬁlm and 150 nm Sr 2FeMoO 6ﬁlm. Low frequency test of\nour setup using a standard Bruker X-Band spectrometer\nshowed identical FMR signals detected by the cantilever\ndeﬂection and by the Bruker X-band detector simultane-\nously.We have measured a high frequency ferromagnetic\nresonance (FMR) response from an quite thin Fe 50Ni50\nﬁlm and successfully ﬁtted the resonance positions of the\nlow ﬁeld FMR lines together with the low frequency mea-\nsurementsusingtheconventionalmodel forthe magnetiza-\ntion dynamics. The cantilever detected FMR experiments\non Sr2FeMoO 6ﬁlm revealed an inability of the conven-\ntional model to perfectly ﬁt both frequency and angular\ndependences of the FMR signal with the same values of\nγ,KUandKc, which suggests that one has to take into\naccount much more complicated nature of the magnetiza-\ntion precessionin the Sr 2FeMoO 6at low temperatures and\nhigh frequencies. Moreover, the complicated dynamics of\nthe magnetization apparent in all the investigated samples\nis supportedbyadrasticincreaseofthelinewidths with in-\ncreasingmicrowavefrequency, and bythe emergenceofthe\nsecond line with an opposite angular dependence. These\nlarge linewidths and the emergence of the second lines re-\nmain an open question.\nAcknowledgment\nThis research has been supported by a DFG interna-\ntional research grant, project numbers AL 1771/1-1 and\nAL 1771/2-1, and by the Emmy Noether WU595/3-1 and\nthe materialsworldnetworkWU595/5-1. Wewould liketo\nthank Dr. T. Sakurai and Dr. S. Okubo for support dur-\ning cantileverdetected FMR experiments. We alsoexpress\nour gratitude to Dr. V. Kataev for fruitful discussions.\nReferences\nReferences\n[1] K.-I. Kobayashi, T. Kimura, H. Sawada, K. Terakura,\nY. Tokura, Nature 395 (1998) 677.\n[2] M. Retuerto, J. A. Alonso, M. J. Mart´ ınez-Lope, J. L. Mar t´ ınez,\nM. Garc´ ıa-Hern´ andez, Appl. Phys. Lett. 85 (2004) 266.\n[3] N. Tezuka, N. Ikeda, S. Sugimoto, K. Inomata, Jpn. J. Appl .\nPhys. 46 (2007) L454.\n[4] C. Felser, G. H. Fecher, B. Balke, Angew. Chem. Int. Ed. 46\n(2007) 668.\n[5] R. Yilgin, M. Oogane, Y. Ando, T. Miyazaki, J. Appl. Phys.\n310 (2007) 2322.\n[6] T. Nosach, G. Mullady, N. Leifer, V. Adyam, Q. Li, S. Green -\nbaum, Y. Ren, J. Appl. Phys. 103 (2008) 07E311.\n[7] C. H. Du, R. Adur, H. L. Wang, A. J. Hauser, F. Y. Yang, P. C.\nHammel, Phys. Rev. Lett. 110 (2013) 147204.[8] K. Zakeri, J. Lindner, I. Barsukov, R. Meckenstock, M. Fa rle,\nU. von H¨ orsten, H. Wende, W. Keune, J. Rocker, S. S. Kalar-\nickal, K. Lenz, W. Kuch, K. Baberschke, Z. Frait, Phys. Rev. B\n76 (2007) 104416.\n[9] S. Mizukami, D. Watanabe, M. Oogane, Y. Ando, Y. Miura,\nM. Shirai, T. Miyazaki, J. Appl. Phys. 105 (2009) 07D306.\n[10] M. S. Gabor, M. Belmeguenai, T. Petrisor, Jr., C. Ulhaq-\nBouillet, S. Colis, C. Tiusan, Phys. Rev. B 92 (2015) 054433.\n[11] H. Ohta, S. Imagawa, M. Motokawa, E. Kita, J. Phys. Soc. J pn.\n62 (1993) 4467.\n[12] J. Smit, H. G. Beljers, Philips Res. Rep. 10 (1955) 113.\n[13] M. Farle, Rep. Prog. Phys. 61 (1998) 755.\n[14] B. Peters, A. Alfonsov, C. G. F. Blum, P. M. Woodward,\nS. Wurmehl, B. B¨ uchner, F. Y. Yang, Appl. Phys. Lett. 103\n(2013) 162404.\n[15] A. Alfonsov, B. Peters, F. Y. Yang, B. B¨ uchner, S. Wurme hl,\nPhys. Rev. B 91 (2015) 064421.\n[16] P. V. Leksin, A. A. Kamashev, J. Schumann, V. Kataev,\nJ. Thomas, B. B¨ uchner, I. A. Garifullin, arXiv.org (2015)\n1510.04846[link].\nURLhttp://arxiv.org/abs/1510.04846\n[17] A. J. Hauser, R. E. A. Williams, R. A. Ricciardo, A. Genc,\nM. Dixit, J. M. Lucy, P. M. Woodward, H. L. Fraser, F. Y.\nYang, Phys. Rev. B 83 (2011) 014407.\n[18] E. Ohmichi, N. Mizuno, M. Kimata, H. Ohta, Rev. Sci. Inst rum.\n79 (2008) 103903.\n[19] H. Ohta, E. Ohmichi, Appl. Magn. reson. 37 (2010) 881.\n[20] E. Ohmichi, S. Hirano, H. Ohta, Journal of Magnetic reso nance\n227 (2013) 9.\n[21] H. Takahashi, E. Ohmichi, H. Ohta, Appl. Phys. Lett. 107\n(2015) 182405.\n[22] E. Ohmichi, T. Osada, Rev. Sci. Instrum. 73 (2002) 3022.\n[23] B. Heinrich, Ultrathin Magnetic Structures, Vol. III, Springer,\nBerlin, 2005.\n[24] M. Belmeguenai, F. Zighem, Y. Roussign´ e, S.-M. Ch´ eri f,\nP. Moch, K. Westerholt, G. Woltersdorf, G. Bayreuther, Phys .\nRev. B 79 (2009) 024419.\n[25] M. Kadzio/suppress lka-Gawe/suppress l, W. Zarek, E. Popiel, A. Chrobak, ACTA\nPHYSICA POLONICA A 117 (2010) 412.\n[26] D. K. Smith, H. R. Leider, J. Appl. Cryst. 1 (1968) 246.\n[27] X. Liu, Y. Sasaki, J. K. Furdyna, Phys. Rev. B 67 (2003) 20 5204.\n6"    },    {        "title": "1904.13275v1.Tunable_ferromagnetic_resonance_in_coupled_trilayers_with_crossed_in_plane_and_perpendicular_magnetic_anisotropies.pdf",        "content": "Tunable ferromagnetic resonance in coupled trilayers with crossed\nin-plane and perpendicular magnetic anisotropies\nDaniel Markó,1,a)Fernando Valdés-Bango,2Carlos Quirós,2, 3Aurelio Hierro-Rodríguez,4María Vélez,2, 3José\nIgnacio Martín,2, 3José María Alameda,2, 3David S. Schmool,1and Luis Manuel Álvarez-Prado2, 3, b)\n1)Groupe d’Etude de la Matière Condensée (GEMaC), UMR8635, CNRS and Université de Versailles/Saint-Quentin-en-Yvelines,\nUniversité Paris-Saclay, 45 ave des Etats-Unis, 78035 Versailles, France\n2)Departamento de Física, Facultad de Ciencias, Universidad de Oviedo, C/ Federico García Lorca n◦18, 33007 Oviedo,\nSpain\n3)Centro de Investigación en Nanomateriales y Nanotecnología (CINN), CSIC-Universidad de Oviedo,\nSpain\n4)SUPA, University of Glasgow, School of Physics and Astronomy, G128QQ Glasgow,\nUK\n(Dated: 1 May 2019)\nA novel approach to tune the ferromagnetic resonance frequency of a soft magnetic Ni 80Fe20(Permalloy = Py) ﬁlm with\nin-plane magnetic anisotropy (IMA) based on the controlled coupling to a hard magnetic NdCo xﬁlm with perpendicular\nmagnetic anisotropy (PMA) through a non-magnetic Al spacer is studied. Using transverse magneto-optical Kerr effect\n(TMOKE), alternating gradient magnetometry (AGM) as well as vector network analyzer ferromagnetic resonance\n(VNA-FMR) spectroscopy, the inﬂuence of both Co concentration and Al spacer thickness on the static and dynamic\nmagnetic properties of the coupled IMA/PMA system is investigated. Compared to a single Py ﬁlm, two striking\neffects of the coupling between IMA and PMA layers can be observed in their FMR spectra. First, there is a signiﬁcant\nincrease in the zero-ﬁeld resonance frequency from 1.3 GHz up to 6.6 GHz, and second, an additional frequency\nhysteresis occurs at low magnetic ﬁelds applied along the hard axis. The maximum frequency difference between the\nfrequency branches for increasing and decreasing magnetic ﬁeld is as high as 1 GHz, corresponding to a tunability of\nabout 20% at external ﬁelds of typically less than ±70 mT. The origin of the observed features in the FMR spectra is\ndiscussed by means of magnetization reversal curves.\nThe magnetic properties of thin ﬁlms and multilayers ex-\nhibiting stripe domains have been investigated extensively in\nboth experiment and theory since their discovery more than\nhalf a century ago1. In recent years, research results on\nstripe domains have triggered the prospect of employing their\nunique properties in future microwave, magnonic, and spin-\ntronic devices with novel functionalities. The formation of\nstripe domains is the result of energy minimization as well\nas the competition between PMA ( K⊥) and shape anisotropy\n(1\n2µ0M2\nS), which favor out-of-plane and in-plane magnetiza-\ntion, respectively. The ratio Q=2K⊥/µ0M2\nS, known as re-\nduced anisotropy or quality factor2, is commonly used to de-\nscribe the extent of stripe domains. For moderate ( Q<1)\nto weak ( Q/lessmuch1) PMA, the magnetization tends to lie in the\nplane, but above a critical ﬁlm thickness dcr, a ground state\nwith stripe domains emerges. The latter is characterized by a\nperpendicular magnetization component alternating between\nup and down within a period λ. The critical thickness dcris\ntypically in the range of 20 – 40 nm for moderate Qvalue ma-\nterials such as amorphous NdCo alloys3,4, whereas for mate-\nrials like Py with small values of Q, generally larger values of\ndcr= 170 – 300 nm are found5–8. Intimately linked to the pres-\nence of stripe domains is the occurrence of a pseudo-uniaxial\nor rotatable anisotropy9,10, which is the result of the in-plane\nmagnetization being aligned along the stripe direction. The\nlatter, however, is not ﬁxed as it can be reoriented by apply-\na)Electronic mail: daniel.marko@uvsq.fr\nb)Electronic mail: lmap@uniovi.esing a saturating ﬁeld along an arbitrary in-plane direction.\nThis particular property of stripe domains has been shown\nrecently to enable tunable and reconﬁgurable dynamic mag-\nnetic properties11,12even after sample preparation and hence\nin contrast to other approaches of increasing FMR and spin\nwave frequencies in soft magnetic thin ﬁlms13–15. Another\npossibility to create stripe domains in soft magnetic materi-\nals even far below the critical thickness dcrstems from the\ncoupling to another magnetic thin ﬁlm or multilayer stack ex-\nhibiting PMA. Here, the inﬂuence of stray ﬁeld and exchange\ninteraction on the soft magnetic layer has been shown to lead\nto a multitude of intriguing effects such as, for example, im-\nprinted topological spin textures16–18, deterministic propaga-\ntion of vortex-antivortex pairs19, and spin wave propagation in\ndomain wall-like magnetic channels20. Though magnetization\ndynamics of stripe domains in uncoupled thin ﬁlms has been\nstudied extensively5–8,10,21–28, the dynamic magnetic proper-\nties of coupled IMA/PMA systems have so far only been in-\nvestigated in a few studies29–31.\nIn this Letter, a novel approach to tune the FMR frequency\nof a soft magnetic thin ﬁlm based on the controlled coupling\nof two magnetic ﬁlms with different types of anisotropies, in-\nplane and perpendicular, is investigated experimentally. Mak-\ning use of the stripe domains’ unique dynamic properties, a\nreconﬁgurable FMR response at low magnetic ﬁelds has been\nachieved. A sketch of the samples fabricated for this work is\nshown in Fig. 1(a). The central element is a trilayer consist-\ning of a 64 nm thick amorphous NdCo xﬁlm with PMA and\na 10 nm thick polycrystalline Py ﬁlm with IMA, which are\ncoupled through a non-magnetic Al spacer. The trilayer struc-arXiv:1904.13275v1  [cond-mat.mes-hall]  30 Apr 20192\n(b) (d)(a)\nAl\nNdCo X\nAl\nSi/SiO 2Al\nPy\n-20 -10 0 10 20-1.0-0.500.51.0\n Easy Axis\n Hard AxisM/MS\nH (mT)Py\n-200 -100 0 100 200\nH (mT) Easy Axis\n Hard AxisNdCo 5\n-1.0-0.500.51.0\nM/MS\n-200 -100 0 100 200\nH (mT) X5T0\n X5T2.5\n X5T5\n X5T10Easy Axis(c)\nFIG. 1. (Color online) (a) Sketch of a coupled trilayer with arrows indicating the anisotropy directions in the magnetic ﬁlms. (b) In-plane\nEA and HA hysteresis loops of a single 10 nm thick Py ﬁlm measured by T-MOKE. (c) In-plane EA and HA hysteresis loops of a single 64\nnm thick NdCo 5ﬁlm measured by AGM. (d) In-plane EA hysteresis loops of X5 series samples for all values of the Al spacer thickness T\nmeasured by AGM.\nture itself is sandwiched between Al seed and capping layers,\nall of which have been grown on a Si/SiO 2substrate using\nmagnetron sputtering. The magnetic properties of the coupled\nthin ﬁlms can be controlled by two independent parameters.\nOn the one hand, varying the Co concentration ( X=5,7.5,9)\nin the NdCo xﬁlm allows the modiﬁcation of the strength of\nits PMA. A maximum has been found for X=5, whereas\nhigher or lower Co concentrations lead to a gradually weaker\nPMA, respectively32,33. On the other hand, by adjusting the\nAl spacer thickness ( T= 0 nm, 2.5 nm, 5 nm, 10 nm), the\ntype of coupling between the two magnetic layers can be set\nto either direct exchange coupling ( T≤1.5 nm) or stray ﬁeld\ncoupling ( T≥2.5 nm). In addition to the coupled bi- and tri-\nlayers, a series of reference samples, consisting of a single 10\nnm thick Py ﬁlm as well as single 64 nm thick NdCo xﬁlms\nwith varying Co concentrations, have also been prepared. For\nthe remainder of the paper, the coupled bi- and trilayers will\nbe named according to their Co concentration and Al spacer\nthickness as, e.g., X5T10 for a sample based on a NdCo 5ﬁlm\nand a 10 nm thick Al spacer.\nThe static magnetic properties of the samples have been\ninvestigated using T-MOKE and AGM. In Fig. 1(b) and (c),\nboth in-plane easy axis (EA) and hard axis (HA) hysteresis\nloops of single Py and NdCo 5ﬁlms obtained by T-MOKE\nand AGM, respectively, are shown. The magnetization re-\nversal loops of the Py ﬁlm show the typical features of a\nsoft magnetic material such as very low coercivity, low sat-\nuration ﬁeld, and, for the EA, almost perfect squareness of\nthe loop. In contrast, the hysteresis loops of the NdCo 5ﬁlm\nshow a much larger coercivity and a higher saturation ﬁeld,\nas expected for a high-anisotropy material. The reason for\nthe higher in-plane remanence Mrof the NdCo 5loop in the\nEA conﬁguration is a smaller out-of-plane component of the\nmagnetization compared to the HA conﬁguration. Upon cou-\npling these two magnetic ﬁlms to form either bilayers (without\nAl spacer) or trilayers (with Al spacer of variable thickness)\nwith crossed anisotropies, respectively, the resulting magnetic\nproperties are different from those of the individual ﬁlms, yet\nthey do not simply constitute a superposition or averaging due\nto the magnetic coupling between the layers. As an example,\nin-plane EA hysteresis loops of the X5 sample series for all\nfour values of the Al spacer thickness measured by TMOKEare depicted in Fig. 1(d). For the bilayer system with direct\nexchange coupling due to a very thin Al spacer, the result-\ning hysteresis loop is very similar to that of the single NdCo 5\nﬁlm. This indicates that the Py layer effectively behaves like\nthe NdCo 5ﬁlm and can be considered almost as an exten-\nsion of the hard magnetic layer. However, for increasing Al\nspacer thicknesses, the magnetic coupling reduces, meaning\nthe Py acts more and more as a soft magnetic ﬁlm, which is\nimportant for its dynamic behavior. The dynamic magnetic\nproperties of the samples have been investigated by means of\nroom temperature broadband VNA-FMR using the ﬂip-chip\nmethod, in which the sample is placed upside-down on top\nof a coplanar waveguide with a 50 µm wide center conductor.\nThe VNA was operated in frequency sweep mode while an in-\nplane dc magnetic ﬁeld H, applied either along the EA or HA\nof the samples, was swept in the following sequence: 0 T →\n0.9 T→− 0.9 T→0 T. Prior to each measurement, the sam-\nples were saturated in order to ensure that at the beginning of\nthe actual FMR experiment the dc magnetic ﬁeld His parallel\nand the rf magnetic ﬁeld Hrf, generated by the CPW, is per-\npendicular to the stripe domains, respectively. The magnitude\nof the forward transmission parameter S21was used to extract\nthe resonance frequencies fafter a reference spectrum taken\nat zero-ﬁeld was subtracted from all of the recorded spectra.\nIn Fig. 2, the fvs.Hdependency for the dc magnetic ﬁeld ap-\nplied along the HA of the coupled trilayers is displayed. In the\nupper row (a−c), the thickness of the Al spacer Tincreases\nfrom the left to the right panel, while the Co concentration X\nvaries for every ﬁxed value of Tin each of the panels. Accord-\ningly, in the bottom row (d −f), the Co concentration of the\nNdCo xﬁlms increases from the left to the right panel, while\nthe Al spacer thickness Tvaries for every ﬁxed value of X. As\nsuch, the same data is shown in (a −c) and (d−f). For compar-\nison, the FMR spectrum of a single 10 nm thick Py reference\nﬁlm has been included in all panels. Due to the phenomeno-\nlogical damping and the corresponding large linewidth, it was\nnot possible to extract any data from the FMR spectra of any\nsingle NdCo xﬁlm as well as any of the directly exchange-\ncoupled bilayers. However, the insertion of the Al spacer with\nits variable thickness Tleads to a gradual decoupling of the\nIMA/PMA stack, thereby effectively enabling the observation\nof the FMR of the soft Py ﬁlm, whose magnetic properties are3\n(a)\n(d)Increasing A l Spacer Thick ness T\nIncreasing Cobalt Cont ent X\n-0.2 -0.1 0 0.1 0.202468101214f (GHz )\nH (T) T2.5X5\n T5X5\n T10X5\n Py(b)  X5.0 T5\n X7.5 T5\n X9.0 T5\n Py (c) X5.0 T10\nX7.5 T10\nX9.0 T10\nPy\n(e) X7.5T2.5\nX7.5 T5\nX7.5 T10\nPyX5.0 T2.5\nX7.5 T2.5\nX9.0 T2.5\nPy\n(f)\n02468101214\nf (GHz ) T2.5X9\n T5X9\n T10X9\n Py\n-0.2 -0.1 0 0.1 0.2\nH (T)-0.2 -0.1 0 0.1 0.2\nH (T)-0.2 -0.1 0 0.1 0.2\nH (T)-0.2 -0.1 0 0.1 0.2\nH (T)-0.2 -0.1 0 0.1 0.2\nH (T)02468101214f (GHz )\n02468101214\nf (GHz )\n \nFIG. 2. (Color online) fvs.Hde-\npendency of the coupled trilayers for\nthe in-plane dc magnetic ﬁeld Happlied\nlong the HA. In the top row (a −c), the\nFMR spectra of trilayers having the same\nAl spacer thickness ( T) are compared,\nwhereas in the bottom row (d −f), the re-\nsults for trilayers with identical Co con-\ncentrations ( X) are displayed. In both\nrows, the values of XandTincrease from\nleft to right, respectively. For compari-\nson, the spectrum of a single 10 nm thick\nPy ﬁlm has been added to each plot. Prior\nto each measurement, stripe domains and\nmagnetization have been aligned parallel\nby the application and removal of a pos-\nitive in-plane saturating magnetic ﬁeld,\nwhich was then swept from 0 T →0.9 T\n→− 0.9 T→0 T during the actual FMR\nexperiment.\nmodiﬁed by the proximity to the hard NdCo xlayer, resulting\nin both an induced rotatable anisotropy and a stripe domain\npattern. All FMR spectra in Fig. 2 show exactly one single res-\nonance: either the uniform FMR mode in the case of the single\nPy ﬁlm or an acoustic mode in the coupled trilayers, which\nbecomes the uniform mode when the samples are saturated\nand the stripe domains are erased. The origin of the acous-\ntic mode is the in-phase precession of spins in adjacent stripe\ndomains. At lower ﬁelds, where the stripe domains in the cou-\npled IMA/PMA samples are nucleated, a signiﬁcant deviation\nfrom the single Py frequencies can be seen, which manifests\nitself by two very distinct features. First, there is a strong in-\ncrease in the zero-ﬁeld resonance frequencies from about 1.3\nGHz for Py up to a maximum of 6.6 GHz for the X7.5T2.5\ntrilayer and second, there is also a frequency hysteresis with\ndifferences between the two ﬁeld sweep directions as high as\n1 GHz in the case of the X7.5T10 trilayer. Within the hys-\nteretic part of the FMR spectra, the lower frequency branch\nat negative ﬁelds and the higher frequency branch at positive\nﬁelds can be accessed when increasing the value of the ap-\nplied magnetic ﬁeld. Conversely, the lower frequency branch\nat positive ﬁelds and the higher frequency branch at negative\nﬁelds can be accessed when decreasing the value of the ap-\nplied magnetic ﬁeld. Although the hysteretic behavior of the\nfvs.Hdependency is a rather rare phenomenon, it has been\nobserved in a variety of materials including, e.g., exchange-\nbiased bilayers34, BaFe 12O9ﬁlms35,36, thick Py ﬁlms25,37,\nartiﬁcial spin ice38, and patterned nanostructures based on\nPy39,40. This effect allows the resonance frequency to be tuned\nas a function of the magnetic history, leading to a reconﬁg-\nurable functionality in a Py ﬁlm exhibiting stripe domains at a\nthickness of just 10 nm. From the top panels (a −c) in Fig. 2,\nin which the results for samples with ﬁxed Al spacer thickness\nare shown, it can be seen that an increase of the Co concentra-\ntionXin the NdCo xalloys leads to a decrease of the resonance\nfrequencies due to its reduced PMA, resulting in a gradualconvergence of the frequencies within the hysteretic part of\nthe spectra to the frequencies of the single Py ﬁlm. Similarly,\nas depicted in the lower panels (d −f) in Fig. 2, an increase of\nthe Al spacer thickness for a constant Co concentration leads\nto a decrease of the FMR frequencies and their gradual con-\nvergence towards the single Py ﬁlm frequencies. The reason\nfor this is the weaker inﬂuence of the NdCo xstray ﬁeld on the\nPy ﬁlm with increasing distance between both these two ﬁlms.\nIn Fig. 3, the simulated stripe domain pattern in a X5T2.5\ntrilayer at remanence after saturation with a magnetic ﬁeld ap-\nplied along the y-direction is depicted. In the NdCo 5layer, mz\nis alternatingly pointing up or down, forming stripe domains\nof periodicity λthat are separated by Bloch walls in which\nmyis maximum. In order to minimize the stray ﬁeld energy,\nthex-component of the magnetization forms closure domains,\nindicated by black/white arrows pointing left/right, at both top\nand bottom of the NdCo 5layer. This closure domain pattern\nis also imprinted and hence extended across the thin Al spacer\ninto the Py layer, where regions with opposite values of mxare\nseparated by Néel walls in which myis maximum. The repli-\ncation of the weak stripe pattern in the Py layer also leads to a\ntransfer of the rotatable anisotropy, allowing the Py ﬁlm in the\ncoupled trilayers to have a much larger IMA than a single Py\nﬁlm. Moreover, it is interesting to note that the lines of maxi-\nmum mywithin the Bloch walls in the PMA layer are shifted\nbyλ/4 with respect to the ones within the Néel walls in the\nIMA layer. In addition, it can be seen that the stripe domain\nperiodicity λin the Py layer is given by d1+d2+2d, i.e., the\nsum of the width of two closure domains (d 1,2) with opposite\nmagnetization ( M1,2) as well as the width of two Néel walls\n(2d) separating them. Typical values of λfor single NdCo x\nﬁlms as well as coupled IMA/PMA samples are in the range\nfrom 145−180 nm and 130−145 nm, respectively, as deter-\nmined from magnetic force microscopy images.\nIn the following, a possible explanation for the observed\nfrequency hysteresis will be discussed using the FMR spec-4\nz\nxAl(2.5) NdCo (64) 5d1d2l\nd d\nyPy(10)/ /\nFIG. 3. (Color online) Simulated magnetic domain conﬁguration in\na X5T2.5 trilayer at remanence after application of a saturating mag-\nnetic ﬁeld along the y-direction. The stray ﬁeld of the stripe domains\nin the NdCo 5layer creates a closure domain pattern in the Py layer\neven across the thin non-magnetic Al spacer.\ntra and magnetization reversal curves of the X7.5T10 trilayer\nmeasured with the magnetic ﬁeld Happlied along EA and\nHA, respectively. In both sets of data, depicted in Fig. 4(a)\nand (b), respectively, three characteristic ﬁelds can be iden-\ntiﬁed, whose values are in excellent agreement. Those are\nthe saturation ﬁeld Hsat, the coercive ﬁeld Hc, as well as the\ncritical ﬁeld Hcrit, at which the splitting/merging of the hys-\nteresis loop and FMR frequency branches occurs. While there\nis a sizeable frequency hysteresis when His applied along the\nHA, with the maximum frequency difference between both\nﬁeld sweep directions occurring at Hc, there is typically no or\nonly a much less pronounced frequency hysteresis observed\nwhen His applied along the EA. The two hysteresis loops\nshown in Fig. 4(b) differ in three important points: the EA\nloop has (i) a 20% higher coercivity Hc, (ii) a 20% higher re-\nmanence Mr, but (iii) an almost 50% lower critical ﬁeld Hcrit\ncompared to the HA loop.\nFor FMR measurements, the inﬂuence of different relative\norientations of stripe domains and rf magnetic ﬁeld Hrfhas\nbeen shown to lead to higher (lower) resonance frequencies\nin case of parallel (perpendicular) alignment as a result of\nthe excitation of optical (acoustic) modes due to out-of-phase\n(in-phase) precession of the magnetization in adjacent stripe\ndomains8,10,12,22,23,37. However, the way the FMR measure-\nments in this work have been performed, stripe domains and\nrf magnetic ﬁeld Hrf(dc magnetic ﬁeld H) are always perpen-\ndicular (parallel) during the entire hysteresis cycle and inde-\npendent of the ﬁeld sweep direction as simulated in Ref. [39]\nfor a single 200 nm thick Py ﬁlm. This means that both fre-\nquency branches in the HA FMR spectra of the coupled tri-\nlayers at low ﬁelds always correspond to an acoustic mode.\nInstead, the fact that generally no or only a minor frequency\nhysteresis can be observed in the EA conﬁguration suggests\nthat the IMA of both the Py and NdCo xlayer and their rela-\ntive orientation with respect to the in-plane dc magnetic ﬁeld\nHare at the origin of the observed dynamic properties. Co-\nsputtering generally induces an IMA in the NdCo xﬁlms of\naround 104J/m3, which is about one order of magnitude larger\nthan the IMA of the Py layer even after rescaling the energy\ndensity with the corresponding values of MS. Moreover, the\nIMA in the NdCo xalloys creates a huge asymmetry in the clo-\n(a)\n-150 -100 -50 0 50 100 150\nH (mT)345678910f (GHz )\nEA\nHA ±Hsat\n±Hcea±Hcritha±Hcrit\n-50 -25 0 25 503456f (GHz )\nH (mT)X7.5T10\n(b)\nM/M\n-150 -100 -50 0 50 100 150-1.0-0.500.51.0\nS\nH (mT)EA\nHA ±Hcha±HcriteaFIG. 4. (Color online) In-plane EA and HA FMR spectra (a) and\nmagnetization reversal curves measured by AGM (b) of the X7.5T10\ntrilayer. The arrows designate the ﬁeld sweep directions, whereas the\ndashed, dashdotted, and dotted lines indicate saturation ﬁeld ( Hsat),\ncritical ﬁelds ( Hcrit), and coercive ﬁeld ( Hc), respectively. The small\ninset at the top of (a) is a zoom into the area bound by the dashed\nrectangle to better see the minor frequency hysteresis in the EA con-\nﬁguration.\nsure domain structure when the stripes are oriented along EA\nor HA. Thus, there is a relevant difference between the stray\nﬁelds generated by the NdCo xﬁlm and the Py layer, respec-\ntively, depending on the relative directions of the IMA and\nthe magnetization components of the closure/stripe domains.\nHowever, to gain further insight into this complex interplay,\nadditional measurements of the azimuthal angle dependency\nof the FMR are necessary to quantify the value of the IMA\nand, in particular, its rotatable anisotropy contribution.\nIn summary, a novel approach to boost the FMR frequency\nof a soft magnetic Py ﬁlm with IMA based on the controlled\ncoupling to a hard magnetic NdCo xﬁlm with PMA through\na non-magnetic Al spacer of variable thickness has been in-\nvestigated experimentally. The two most striking effects ob-\nserved, compared to a single Py ﬁlm, are a signiﬁcant increase\nin the zero-ﬁeld FMR frequency from 1.3 GHz up to 6.6 GHz,\nand a frequency hysteresis at HA ﬁelds below ±70 mT with a\ndifference between the frequency branches for increasing and\ndecreasing ﬁeld of up to 1 GHz, both of which can clearly\nbe attributed to the imprinted stripe domain pattern in the Py\nlayer below saturation. The possibility to control anisotropy\nand coupling strength in this IMA/PMA system by adjusting\nthe Co concentration in the PMA ﬁlm and the Al spacer thick-\nness, respectively, during sample fabrication allows the sys-\ntem to be predeﬁned with respect to the value of the zero-ﬁeld\nresonance frequency. In addition, the FMR frequencies can\nfurther be tuned and reconﬁgured by simply erasing and nu-\ncleating a stripe domain pattern in the Py layer upon applica-\ntion of an in-plane magnetic ﬁeld along its HA, opening new\nperspectives for the development of future microwave, spin-\ntronic or magnonic devices.\nD. S. S. and L. M. Á.-P. acknowledge CNRS for ﬁnan-\ncial support. A. H.-R. acknowledges European Union’s Hori-\nzon 2020 research and innovation program under the Marie\nSkłodowska-Curie Action H2020-MSCA-IF-2016-746958.\nThis work is supported by Spanish MINECO under project\nFIS2016-76058 (AEI/FEDER, UE).5\n1H. Fujiwara, Y . Sugita, and N. Saito, Appl. Phys. Lett. 4, 199–200 (1964).\n2A. Hubert and R. Schäfer, Magnetic Domains: The Analysis of Magnetic\nMicrostructures (Springer, Berlin, 1998).\n3A. Hierro-Rodriguez, R. Cid, M. Vélez, G. Rodriguez-Rodriguez, J. I.\nMartín, L. M. Álvarez-Prado, and J. M. Alameda, Phys. Rev. Lett. 109,\n117202 (2012).\n4A. Hierro-Rodriguez, M. Vélez, R. Morales, N. Soriano, G. Rodríguez-\nRodríguez, L. M. Álvarez-Prado, J. I. Martín, and J. M. Alameda, Phys.\nRev. B 88, 174411 (2013).\n5J. Ben Youssef, N. Vukadinovic, D. Billet, and M. Labrune, Phys. Rev. B\n69, 174402 (2004).\n6C. Ramos, E. V . Brigneti, J. Gómez, and A. Butera, Physica B: Condens.\nMatter 404, 2784 – 2786 (2009).\n7W. Tee Soh, N. N. Phuoc, C. Y . Tan, and C. K. Ong, J. Appl. Phys. 114,\n053908 (2013).\n8J. Wei, Z. Zhu, H. Feng, J. Du, Q. Liu, and J. Wang, J. Phys. D: Appl. Phys.\n48, 465001 (2015).\n9L. M. Álvarez Prado, G. T. Pérez, R. Morales, F. H. Salas, and J. M.\nAlameda, Phys. Rev. B 56, 3306–3316 (1997).\n10S. Tacchi, S. Fin, G. Carlotti, G. Gubbiotti, M. Madami, M. Barturen,\nM. Marangolo, M. Eddrief, D. Bisero, A. Rettori, and M. G. Pini, Phys.\nRev. B 89, 024411 (2014).\n11C. Banerjee, P. Gruszecki, J. W. Klos, O. Hellwig, M. Krawczyk, and\nA. Barman, Phys. Rev. B 96, 024421 (2017).\n12L. Pan, H. Xie, X. Cheng, C. Zhao, H. Feng, D. Cao, J. Wang, and Q. Liu,\nJ. Magn. Magn. Mater. 457, 46 – 51 (2018).\n13G. Chai, N. N. Phuoc, and C. K. Ong, Sci. Rep. 2, 832 (2012).\n14C. Li, G. Chai, C. Yang, W. Wang, and D. Xue, Sci. Rep. 5, 17023 (2015).\n15S. Li, C. Wang, X.-M. Chu, G.-X. Miao, Q. Xue, W. Zou, M. Liu, J. Xu,\nQ. Li, Y . Dai, S. Yan, S. Kang, Y . Long, and Y . Lü, Sci. Rep. 6, 33349\n(2016).\n16D. A. Gilbert, B. B. Maranville, A. L. Balk, B. J. Kirby, P. Fischer, D. T.\nPierce, J. Unguris, J. A. Borchers, and K. Liu, Nat. Commun. 6, 8462\n(2015).\n17C. Blanco-Roldán, C. Quirós, A. Sorrentino, A. Hierro-Rodríguez, L. M.\nÁlvarez Prado, R. Valcárcel, M. Duch, N. Torras, J. Esteve, J. I. Martín,\nM. Vélez, J. M. Alameda, E. Pereiro, and S. Ferrer, Nat. Commun. 6, 8196\n(2015).\n18A. Hierro-Rodriguez, C. Quirós, A. Sorrentino, C. Blanco-Roldán, L. M.\nAlvarez-Prado, J. I. Martín, J. M. Alameda, E. Pereiro, M. Vélez, and\nS. Ferrer, Phys. Rev. B 95, 014430 (2017).\n19A. Hierro-Rodriguez, C. Quirós, A. Sorrentino, R. Valcárcel, I. Estébanez,L. M. Alvarez-Prado, J. I. Martín, J. M. Alameda, E. Pereiro, M. Vélez,\nand S. Ferrer, Appl. Phys. Lett. 110, 262402 (2017).\n20L. Wang, L. Gao, L. Jin, Y . Liao, T. Wen, X. Tang, H. Zhang, and Z. Zhong,\nAIP Adv. 8, 055103 (2018).\n21O. Acher, C. Boscher, B. Brulé, G. Perrin, N. Vukadinovic, G. Suran, and\nH. Joisten, J. Appl. Phys. 81, 4057–4059 (1997).\n22U. Ebels, L. Buda, K. Ounadjela, and P. E. Wigen, Phys. Rev. B 63, 174437\n(2001).\n23N. Vukadinovic, M. Labrune, J. B. Youssef, A. Marty, J. C. Toussaint, and\nH. Le Gall, Phys. Rev. B 65, 054403 (2001).\n24N. A. Buznikov, A. L. Rakhmanov, and K. N. Rozanov, IEEE Trans. Magn.\n38, 3123–3125 (2002).\n25O. Acher, S. Queste, M. Ledieu, K.-U. Barholz, and R. Mattheis, Phys.\nRev. B 68, 184414 (2003).\n26S.-S. Ha, J. Yoon, S. Lee, C.-Y . You, M.-H. Jung, and Y . K. Kim, J. Appl.\nPhys. 105, 07D544 (2009).\n27G. Chai, N. N. Phuoc, and C. K. Ong, Appl. Phys. Lett. 103, 042412\n(2013).\n28J. Wei, Z. Zhu, C. Song, H. Feng, P. Jing, X. Wang, Q. Liu, and J. Wang, J.\nPhys. D: Appl. Phys. 49, 265002 (2016).\n29B. M. S. Teixeira, A. A. Timopheev, R. Schmidt, M. R. Soares, M. Seifert,\nV . Neu, and N. A. Sobolev, J. Phys. D: Appl. Phys. 49, 315002 (2016).\n30L. Fallarino, V . Sluka, B. Kardasz, M. Pinarbasi, A. Berger, and A. D. Kent,\nAppl. Phys. Lett. 109, 082401 (2016).\n31J.-W. Xu, V . Sluka, B. Kardasz, M. Pinarbasi, and A. D. Kent, J. Appl.\nPhys. 124, 063902 (2018).\n32D. Mergel, H. Heitmann, and P. Hansen, Phys. Rev. B 47, 882–891 (1993).\n33R. Cid, J. M. Alameda, S. M. Valvidares, J. C. Cezar, P. Bencok, N. B.\nBrookes, and J. Díaz, Phys. Rev. B 95, 224402 (2017).\n34D. Spenato and S. Pogossian, J. Magn. Magn. Mater. 285, 79 – 87 (2005).\n35S. G. Wang, S. D. Yoon, and C. Vittoria, J. Appl. Phys. 92, 6728–6732\n(2002).\n36V . I. Kostenko, A. M. Sorochak, T. G. Chamor, and L. V . Chevnyuk, Tech.\nPhys. 53, 1379–1381 (2008).\n37D. Cao, C. Song, H. Feng, Y . Song, L. Zhong, L. Pan, C. Zhao, Q. Li, J. Xu,\nS. Li, J. Wang, and Q. Liu, e-print arXiv:1903.00656 (2019).\n38M. Jungﬂeisch, W. Zhang, E. Iacocca, J. Sklenar, J. Ding, W. Jiang,\nS. Zhang, J. Pearson, V . Novosad, J. B. Ketterson, O. Heinonen, and\nA. Hoffmann, Phys. Rev. B 93, 100401(R) (2016).\n39F. Giesen, J. Podbielski, T. Korn, M. Steiner, A. van Staa, and D. Grundler,\nAppl. Phys. Lett. 86, 112510 (2005).\n40Y . Nozaki, K. Tateishi, S. Taharazako, S. Yoshimura, and K. Matsuyama,\nAppl. Phys. Lett. 92, 161903 (2008).Supplement of \"Tunable ferromagnetic resonance in coupled trilayers with\ncrossed in-plane and perpendicular magnetic anisotropies\"\nDaniel Markó,1Fernando Valdés-Bango,2Carlos Quirós,2, 3Aurelio Hierro-Rodríguez,4María Vélez,2, 3José\nIgnacio Martín,2, 3José María Alameda,2, 3David S. Schmool,1and Luis Manuel Álvarez-Prado2, 3\n1)Groupe d’Etude de la Matière Condensée (GEMaC), UMR8635, CNRS and Université de Versailles/Saint-Quentin-en-Yvelines,\nUniversité Paris-Saclay, 45 ave des Etats-Unis, 78035 Versailles, France\n2)Departamento de Física, Facultad de Ciencias, Universidad de Oviedo, C/ Federico García Lorca n◦18, 33007 Oviedo,\nSpain\n3)Centro de Investigación en Nanomateriales y Nanotecnología (CINN), CSIC-Universidad de Oviedo,\nSpain\n4)SUPA, University of Glasgow, School of Physics and Astronomy, G128QQ Glasgow,\nUK\n(Dated: 1 May 2019)\nI. MAGNETIC FORCE MICROSCOPY\nThe magnetic stripe domain structure of the samples has\nbeen imaged by means of magnetic force microscopy (MFM)\nas described in Ref. [1] and the images taken were used to de-\ntermine the corresponding stripe domain periodicity λ. For\neach samples series, X5 to X9, MFM images were taken both\nbefore and after the deposition of the Py layer and for the in-\nplane dc magnetic ﬁeld applied along EA and HA, respec-\ntively. As an example, the stripe domains in a X5T10 sam-\nple before and after the deposition of the Py ﬁlm are shown\nin Fig. 1(a) and (b), respectively. As result of the coupling\nbetween the soft and hard magnetic layers, the stripe domain\npattern of the NdCo 5ﬁlm is replicated in the Py ﬁlm with only\na small difference in λ: 145 nm in the PMA ﬁlm and 140 nm\nin the PMA/IMA trilayer for a dc magnetic ﬁeld applied along\nthe HA of the samples.\na) NdCo /Al (10)5 b) NdCo /Al (10)/P y(10) 5\nFIG. 1. (Color online) MFM images of a X5T10 sample (a) before\nand (b) after the deposition of the 10 nm thin Py layer. The stripe\ndomain pattern of the NdCo 5ﬁlm is replicated in the Py layer of the\nPMA/IMA trilayer with only a minor difference in the periodicity:\nλ= 145 nm in (a) and λ= 140 nm in (b) for the dc magnetic ﬁeld\napplied along the HA.\nII. MICROMAGNETIC SIMULATIONS\nMicromagnetic simulations using the MuMax3 code2have\nbeen performed with the purpose of calculating the FMR fre-\nquencies of the Py layer and simulating the magnetic do-\nmain conﬁguration of the coupled trilayers. The follow-\ning set of material parameters was used for the simulations:γ=15.9×1010rad/T s, NdCo 5:K⊥=16.2×104J/m3,MS\n= 106A/m, A= 0.7×10−11J/m, Py: Ku= 423 J/m3,MS=\n846×103A/m, A= 1.2×10−11J/m, where γis the gyromag-\nnetic ratio, K⊥the PMA, MSthe saturation magnetization, A\nthe exchange stiffness constant, and Kuthe uniaxial IMA. The\ntrilayer structures were modelled by cells with lateral dimen-\nsions of 2.5 nm in-plane as well as 1.25 nm, 1 nm and 2 nm\nout-of-plane for an aluminum spacer thickness Tof 2.5 nm,\n5 nm, and 10 nm, respectively. The dynamic response of the\ntrilayers was micromagnetically simulated as follows: Once\nthe steady state of magnetization has been reached, a mag-\nnetic ﬁeld pulse is applied perpendicular to the in-plane dc\nmagnetic ﬁeld H. The pulse amplitude is 0.1 mT and its tem-\nporal dependence is of type exp (−a(t−t0))for t > t 0and 0\nelsewhere. The parameter awas set to 0.5 ns−1. The damping\nparameter is reduced from α= 1 used for the static simula-\ntions to a smaller value of 0.08 for NdCo 5and 0.01 for Py3.\nAfter the pulse was applied, the total magnetization has been\nevolved for 20 ns and the average out-of-plane magnetization\nof Py was recorded. Its value was fast Fourier transformed\nand the frequency of the resulting peak is determined as His\nvaried.\nT = 2.5 nm\nT =    5 nm\nT =  10 nmNdCo5/Al(T)/Py(a) (b)\nSim. Exp.\n0 0.1 0.22468101214f (GHz )\nH (T)X5T2.5\nX5T5\nX5T10 \n \n \n0 0.2 0.4 0.6 0.805101520253035\nH (T) \n f (GHz )\nFIG. 2. (Color online) (a) Simulated FMR frequencies of the Py ﬁlm\nin NdCo 5-based trilayers with varying Al spacer thickness Tfor a\npositive, decreasing in-plane magnetic ﬁeld Happlied along the HA.\n(b) Comparison between simulated (ﬁlled symbols) and measured\n(open symbols) resonance frequencies within the gray-shaded area\nof (a).\nIn Fig. 2(a), the fvs.Hdependence of the Py ﬁlm in a\nNdCo 5-based trilayer for a positive, decreasing in-plane mag-\nnetic ﬁeld applied along the HA is shown. In excellent agree-arXiv:1904.13275v1  [cond-mat.mes-hall]  30 Apr 20192\nment with experimental results, the increased zero-ﬁeld reso-\nnance frequencies as well as the overall applied ﬁeld and Al\nspacer thickness dependency are reproduced. From the over-\nlay of experimental and simulated data in the frequency and\nﬁeld range marked by the gray-shaded area in Fig. 2(a), de-\npicted in Fig. 2(b), it can be seen that there is generally a very\ngood quantitative agreement, except that the simulations yield\na 1 GHz higher zero-ﬁeld frequency for the X5T5 sample as\nwell as slightly increased values of ffor the X5T5 and X5T10trilayers at higher ﬁelds.\nREFERENCES\n1A. Hierro-Rodriguez, R. Cid, M. Vélez, G. Rodriguez-Rodriguez, J. I.\nMartín, L. M. Álvarez-Prado, and J. M. Alameda, Phys. Rev. Lett. 109,\n117202 (2012).\n2A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F. Garcia-Sanchez,\nand B. Van Waeyenberge, AIP Adv. 4, 107133 (2014).\n3Y . Zhao, Q. Song, S.-H. Yang, T. Su, W. Yuan, S. S. P. Parkin, J. Shi, and\nW. Han, Sci. Rep. 6, 22890 (2016)."    },    {        "title": "2201.11189v2.Coexisting_Kondo_hybridization_and_itinerant_f_electron_ferromagnetism_in_UGe2.pdf",        "content": "Coexistin g Kondo hybridization and itinerant f -electron ferromagnetism in UGe 2 \nIoannis Giannakis1, Divyanshi Sar1, Joel Friedman1, Chang -Jong Kang2,3, Marc Janoschek4,a, Pinaki Das4,b, \nEric D. Bauer4, Gabriel Kotliar2,7, and Pegor Aynajian1*  \n \n1Department  of Physics, Applied Physics and Astronomy, Binghamton University, Binghamton, New York 13902, USA  \n2Department of Physics and Astronomy, Rutgers University, New Jersey 08854, USA  \n3Department of Physics, Chungnam National University, Daejeon 34134, South Ko rea \n4Los Alamos National Laboratory , Los Alamos, 87545, New Mexico, USA  \n5Condensed Matter Physics and Materials Science Department, Brookhaven National Laboratory, Upton, New York \n11973, USA  \naPresent address: Laboratory for Neutron and Muon Instrumentation, Paul Scherrer Institute, Villigen PSI, \nSwitzerland  \nbPresent address: Advanced Photon Source, Argonne National Laboratory, 9700 S. Cass Ave, Lemont, IL 60439  \n \n* To whom correspondence should be addressed : aynajian@binghamton.edu  \n \n \n \n \nKondo hybridization in partially filled f-electron systems conveys  significant amount of \nelectronic states sharply near the Fermi energy leading to various instabilities from \nsuperconductivity to exotic electronic orders. UG e2 is a 5 f heavy fermion system,  where the \nKondo hybridization is interrupted by the formation of two ferromagnetic phases  below a \n2nd order transition T c ~ 52 K and a crossover transition T x ~ 32 K. These two ferromagnetic \nphases are concomitantly related to a spin -triplet superconductivity that only emerges and \npersists  inside the magnetically ordered phase at high pressure. The origin of the two \nferromagnetic phases and how they form within a Kondo -lattice remain ambiguous. Using \nscanning tunneling microscopy and s pectroscopy, we probe the spatial electronic states in \nthe UGe 2 as a function of temperature. We find a Kondo resonance  and sharp  5f-electron \nstates  near  the chemical potential that form at high temperatures above  Tc in accordance \nwith  our density function al theory (DFT) + Gutzwiller  calculations. As temperature is \nlowered  below T c, the resonance narrows and eventually splits below T x dumping itinerant \nf-electron spectral weight right at the Fermi energy. Our findings suggest  a Stoner \nmechanism forming the highly polarized ferromagnetic phase below T x that itself sets the \nstage for the emergence of unconventional superconductivity at high pressure.  \n \n \n \n \n Over the past decade , interest in exploiting ferromagnetic superconductiv ity with non -trivial \ntopology  dominated the field of quantum matter due to their  robust  functionalities in quantum \ninformation1–3. Yet, quantum materials that exhibit natural coexistence of ferromagnetism and \nsuperconductivity remain  rare. To date, only  a handful of such synthesized single crystalline \nmaterials exist, with the majority being  uranium -based heavy fermion compound s including  \nUGe 24, URhGe5–7, UCoGe8 and the recently discovered UTe 29. In these compounds, the 5 f-\nelectrons play a critical  role in the emergence of the exotic superconductivity10, making it \nparticularly crucial  to understand their controversial  normal state behavior.  \nf-electrons in heavy fermion compounds exhibit dual characteristics of being itinerant and \nlocalized , driving an electronic competition between magnetism , very often  of antiferromagnetic \ncharacter,  and heavy Fermi liquid  behavior  with quenched magnetic moments11–15. Recent \nexperimental and computational work demonstrated the dual nature of 5 f electrons in USb216,17, \nan antiferromagnetic heavy fermion system, through orbital selectivity , providing a natural \nexplanation of how localized magnetism and itinerant heavy fermions of the same  uranium  5f \nelectrons coexist. The emergence of f -electron ferromagnetism  and its interplay with Kondo \ncoherence  remains much less explored.  \nThe ferromagne tic heavy fermion UGe 2 displays  an interesting  phase diagram5,18,19. At ambient \npressure, a second order paramagnetic -to-ferromagnetic phase transition at a relatively high T c ~ \n52 K 18 is followed by a crossover meta -magnetic transition from a weakly polari zed ferromagnetic \nstate, FM1 , to a strongly polari zed ferromagnetic state FM2 20 at T x ~ 32 K. Transport \nmeasur ements show  the emergence of the Kondo -lattice effect at T K ~ 110 K, well above T c19. \nHow the Kondo effect is impacted by the emergent ferromagnetism at and below T c remains a \nquestion to be answered.  Below T c, specific heat  measurements display  a broad hump centered \naround T x 19,21 which, along with magnetization measureme nts 19,22 and neut ron scattering23 shows \nthe presence of  itinerant and localized subset of the uranium 5 f electrons. In the same temperature \nrange, Hall effec t studies reveal a rapid increase of charge carriers below T x suggestive of some \nsort of Fermi surface reconstruction 24. This reconst ruction is argued to be cause d by the sudden \ndelocalization of the uranium 5 f electrons. The microscopic origins of T c and T x are particularly \nimportant for the mechanism of emergent exotic superconductivity  in UGe 2. With  the application \nof hydrostatic pressure, the pressure -dependent FM2 transition line  Tx(P) decreases and terminates \nat the maximum  of the emergent superconducting dome  at Px ∼ 1.2 GPa , suggesting its fluctuations \nand destruction are directly related to the  mechanism of superconductivity7. Furthermore , the \nsuperconducting  dome only persists  inside the FM1 phase , where both phases simultaneously \ndisappear at the exact same pressure  Pc ∼ 1.5 GPa  indicating  an intimate re lation between the \nferromagnetism and superconductivity25,26. Therefore, the emergence of itinerant 5 f-electrons \nthrough the Kondo effect in UGe 2 and their evolution across T c and T x forms the low temperature \nnormal state near the Fermi energy  (EF) out of which ferromagnetic superconductivity develops .  \nTheoretical understanding  of the nature of the ferromagnetic state below T x is controversial. One \nmechanism  comes from  the phenomenological ideas  following the rigid -band  Stoner approach, where two sufficiently sharp and narrow ly separated density -of-state peaks located near the Fermi \nenergy form the majority and minority spin bands27. Another idea  involve s charge and spin density \nwaves emerging below T x18. In either case, direct experimental signature  of a double peak structure  \nor de nsity waves have not been observed to date.  \nA sharp resonance in the density of states  can naturally arise in heavy fermion compounds, whose \nenergy relative to E F depends on the valence of the f-electrons  in the material system28,29. Scanning \ntunneling microscopy (STM) has the spatial and energy resolution to probe the sharp resonance \nand it’s possible splitting30. Yet, due to the lack of a natural cleaving plane in UGe 2, which is \ncrucial to obtain clean surface s for STM, such an experiment have so far not been carried out.  Here \nwe use STM to probe the local electronic states  and their temperature evolution  near EF in single \ncrystal UGe 2. We find multiple peaks  in the density of states locat ed near the chemical po tential  \nabove T c. The finding is in qualitative agreement with our DFT + Gutzwiller  calculations presented \nhere, attributing their origin to  the different  5f-electronic orbital characters . With lowering o f \ntemperature, the two peaks located nearest to E F strengthen and narrow . Below T c and particularly \nnear T x ~ 32 K, the peaks split, forming additional kink s that further develop and evolve  with \ntemperature  suggesting a Stoner mechanism of the ferromagnetic order . Our finding indicates a \nsignificant degree of itinerant character  of the f-electrons through Kondo hybridization and the \nitinerant nature of the ferromagnetism involving the same f-electrons. At the lowest temperature, \na sharp  f-electronic  density of states is formed at EF setting the stage for the emergence of \nferromagnetic spin-polarized superconductivity  at higher pressure . \nFigure 1 a, b shows STM topographs of the (0 10) surface of single crystal UGe 2, in-situ cleaved in \nour ultra -high vacuum, variable temperature STM. Cleaving exposes  alternating terraces of two \nchemically different surfaces,  termed  A and B in Fig.1c. While surface A displays  spatial \nuniform ity with no atomic corrugation indicating the extended nature of the electronic states , \nsurface B undergoes  a surface reconstruction, whose quasiperiodic structure differs between \ndifferent cleaves , as seen in Fig. 1a, b. The asymmetry in the s tep height  (A→B > B→A, with \nA↔A = B↔B ≡ b -axis unit cell ) between the different terraces allows us to compare the results \nto the crystal structure . Assuming only a single chemical bond breaking during the cleaving \nprocess leads us to identify surfaces A and B as  uranium a nd germanium terminated, respectively  \n(Fig.1c) . Such an assumption is justified by the fact that only two surfaces have been observed \nbased on ten different sample cleaves.  The Ge surface with two Ge atoms per ac -plane unit cell , \nas compared to a single U or Ge atom per ac-plane for all other  layers  (see F ig.1), is also more \nlikely to undergo  surface relaxation and buckling. Nevertheless, the surface assignment here does \nnot change the conclusions reached below.   \nSTM spectra probe d on the two surfaces just above T c reveal two asymmetric low -energy peaks \nin the density of states in close proximity to the chemical potential, whose intensities are different \non the two surfaces (Fig.1 d). An additional broader high -energy peak near ~100  meV is also \nobserved.  Note that the intensity of the peaks on surface -B are spatially non -uniform and is further elaborated below.  At low temperatures  (8 K), the low energy peaks undergo a splitting with the \nemergence  of a kink/shoulder (Fig.1e) . The high-energy peak remains  mostly  unchanged.  \nPeaks in the density of states near EF is the hallmark of itinerant f-electron systems and have been \nseen in previous STM experiments in various 4 f and 5f heavy fermions31. They are the results of \nthe Kondo hybridization of  localized f-orbitals  with conduction electrons . To corroborate  this \nobservation , we compute the electronic structure of UGe 2 by employing the generalized gradient \napproximation to density functional theory (DFT) in combination with the Gutzwiller \napproximation (DFT+Gutzwiller)32. This method captures electronic correlations beyond the \nsingle -particle picture of DFT and has been successfully applied to other f-electron systems  such \nas UO 233. The local Coulomb interaction strength and the Hund’s coupling constant are U = 6.0 \neV and J = 0.57 eV, respectively, for the correlated U 5 f-orbital. The DFT+Gutzwiller calculations \nwere performed at T = 0 K within the paramag netic phase. Within DFT there is a significant mixing \nbetween U 5 f5/2 and 5 f7/2 states, so their spin -orbit splitting is not apparent. X -ray photoemission \nspectroscopy34 and x -ray magnetic circular dichroism35 measurements indicate that there is a clear \nspin-orbit splitting between U 5 f5/2 and 5 f7/2. Furthermore, the 5 f5/2 level lies below the 5 f7/2 level \nand the magnitude of the splitting is around 1.1 eV. This feature is well captured by the \nDFT+Gutzwiller calculations, which give s a spin -orbit splitting of ~1.5 eV. Our calculations \nindeed show multiple uranium 5 f-electron peaks  with different orbi tal character  to reside near E F \n(Fig.1 f). More specifically, three major peaks  located at energies of -18meV, +35meV and \n+66meV  that have characters of U (J = 5/2, m J = ±1/2, ±5/2, ±1/2), respectively,  are qualitatively \nconsistent with the high temperature experimental data observed on surface A and/or B  at ~ -\n20meV, ~ +25meV , and ~ +100 meV  (Fig.1d, e) .  \nWhile the relative widths and weights  of the spectral lineshapes  are spatially uniform on surface \nA (see Fig.2a) , they vary significantly on surface B due to the structural inhomogeneity induced \nby surface reconstruction, as seen in Fig. 2b. For example, looking at the different peaks, one can \nsee that their intensity can be dramatically suppressed depen ding on the spatial location on the \nsurface. This also applies to the spectra in Fig.1d , where at high temperature only the negative \npeak is observed with almost no peak intensity on the positive side. The disappearance of the \npositive bias peaks is due to the particular location of the spectrum on the surface and other \nlocations (not shown h ere) do reveal a finite peak  at positive and negative biases .  This spatial \nvariation renders studying the detailed temperature dependent evolution  unreliable on surface B.  \nWe therefore focus on surface A  (U surface)  to probe the temperature dependence  of the spectra  \nacross the two ferromagnetic transitions . Figure 3 shows our temperature dependent spectroscopy \nmeasurements  carried out on surface A . The dI/dV conductance  were measured  in a constant \ncurrent mode , Iset, and a bias voltage applied to the sample , Vbias, with a bias modulation of V mod. \nTwo sets of data with different experimental settings  (energy -resolution)  of I set = 150 pA, V bias = \n500 meV, V mod = 5 meV (Figure 3a) and I set = 1 nA, V bias = 200 meV, V mod = 1 meV (Figure 3b) \nare shown . The spectra display  an asymmetric resonance analogous to that seen in other  heavy \nfermion systems31,36 –41. The observed resonance is the manifestation of Kondo hybridization , delocalizing the f-electrons and merging them into the Fermi sea  starting already at temperatures \nabove T c. As temperature is lowered below T c, we observe the sharp kink at E 1 (near the Fermi \nenergy) starting to develop particularly below ~3 5 K (see insets  in Fig.3a,  b). At the lowest \nmeasured temperature of 8  K, a clear double peak structure can be resolved with a peak separation \n(E1 – E2) of ~  16 meV.  \nThe strong temperature broadening of the spectral lineshapes makes it difficult to pin -point the \nonset of the E 1 kink in the raw data. In Fig.3c, d, we  show the 2nd derivative of the spectral \nlineshapes , which highlights  the kink -structure near the Fermi energy. While the temperature \nevolution is weak above ~  35 K, it  becomes more pronounced  at lower temperatures. To better \nvisualize this behavior, we contrast in Fig.3e, the high resolution spectra with a model Fano \nlineshape . A Fano lineshape  in STM spectra  resembles an asymmetric resonance peak due to \ninterference between the two tunneling paths from the tip to the heavy  (resonance)  and light  \n(continuum)  electronic states of a Kondo lattice  and has been widely used in STM analysis of \nheavy fermion systems42–46. The equation below  represents the Fano lineshape   \n𝑑𝐼\n𝑑𝑉∝𝐴(𝑉−𝐸\n𝛤+𝑞)2\n1+(𝑉−𝐸\n𝛤)2  \nwhere E characterizes the resonance energy, Γ the resonance linewidth expressed as Half Width at \nHalf Maximum (HWHM) and q is the tip -sample coupling, also k nown as the asymmetry \nparameter . A is related to the amplitud e of the resonance . Figure 3 e shows the data and the \ncorresponding fi t to a single Fano lineshape . We observe  at 55K  (T>T c) that the data can be nicely \nmodeled by a single resonance. A t 35 K however, we can see that the  data deviates from a single \nFano lineshape particularly in the energy range of ±10 meV, where a second resonance develops  \nand grows stronger with further cooling .  We therefore use a two Fano lineshape  model  (one \ncentered at E 1 and another at E 2) to fit the temperature dependent data. Figure 4 a shows the data \ntogether with their corresponding fit to the summation of two Fano resonances. For all \ntemperatures, the model fit shows an excellent agreement with the data.  No additional background \nis used  in the model.  The extracted resonance amplitude, widths, and energies  are displayed in \nFig.4 b,  c, d respectively.  \nLooking at the amplitude of the resonances (Fig.4b), we first note that both resonance s (E 1 and E 2) \nweaken  with increasing  temperature . While the E 2 resonance amplitude remains finite and large \nwith no apparent anomaly at T c, the E 1 resonance fades  and within the experimental resolution \nbecomes negligible above 35  K, as is reflected by the diverging error bars, which renders their \nvalues m eaningless  above 35  K. The extracted linewidths of the resonances  also paint a similar \npicture . At the lowest  temperature, the linewidth s of the E1 and E 2 resonance s saturate  at values of \n~ 7meV and ~13meV, respectively. With increasing temperature, the E 2 resonance increases and \nwithin the experimental resolution follows the conventional temperature dependence expected in \nKondo lattice systems (T) = √(𝜋𝑘𝐵𝑇)2+2(𝑘𝐵𝑇𝐾)2 38,39. Plotting (T) for a T K of 110  K \nextracted from transpo rt measurements (blue line in Fig.4c) reveals a good agreement with the experimental data. On the other hand, the E1 linewidths grow rapidly  and diverge above 35  K, \nwhere the error bars span the entire y -range  and the data are therefore omitted for T  > 35 K from \nFig.4c . The rapid decrease of spectral weight of the E 1 resonance together with its diverging \nlinewidths with increasing temperature makes it difficult to ascertain its high-temperature  \nevolution , particularly above 35  K. The extracted energies below 35  K show no significant \ntemperature dependence .  \nWe now turn to identify the origin of the observed double -peak structure  at low temperatures. One \npossibility is the indirect Kondo hybridization gap47. However, this scenario can be discarded  in \nUGe 2 for two reasons. First, the observed band splitting occurs far below the Kondo  lattice  \ntemperature of 110  K extracted from transport measurements. In fact, looking at the temperature \ndependence of the E 1 linewidth, one can see that it deviates dramatically from thermal broadening \nand (T) (red line in Fig.4c) and diverges near 35  K. Second, c ontrasting  our observation  in UGe 2 \nwith the Kondo resonance  in antiferromagnetic USb 216, non -magnetic UTe 29, and URu 2Si238,40 \nabove its hidden order temperature, we see that all these uranium -based heavy fe rmion systems \ndisplay a single Fano resonance above the chemical potential. Yet, none  show a splitting or an \nindirect Kondo hybridization gap opening  at low temperature , regardless of the ir magnetic  or \nKondo -coherence temperatures . Therefore, the splitting  that we observe  is likely not  due to an \nindirect hybridization gap, which should show up similarly in these other U -based systems as well. \nWe therefore turn to ferromagnetism as a possible  origin of the band splitting  (~ 16 meV)  that we \nobserve.  \nIn itinerant ferromagnets, below their magnetic transition, the spin -majority and minority bands \nsplit due to the  ferromagnetic exchange . Our ob servation is consistent with this  Stoner mechanics \nof itinerant ferromagnetism.  Indeed, the observation of a Kondo resonance above T c and its further \nevolution below T c provides spectroscopic evidence of the itinerant character  of the f-electrons and \ntherefore of the ferromagnetism in UGe 2. This agrees with the fact that the ordered moment in the \nferromagnetic phase is  1.4 B/U, much smaller than the effective  paramagnetic moment of 2.7 B \n/U19. Evidence of itinerant ferromagnetism and band  splitting  is also seen  in the optical \nspectroscopy of UGe 2 below T c, where  low energy excitations with an energy of  E ~ 13.6 meV48 \nhave been observed . The latter is comparable to the separation of the E 1-E2 resonances seen here  \n(Fig.4d) . Similarly, a Stoner gap of the order of 40  K has also been inferr ed from magnetic neutron \ndiffraction49. \nOverall,  our data reveal a Kondo resona nce with a characteristic Kondo -lattice temperature of 110  \nK, consistent with transport measurements19, that survives in the ferromagnetic phase. In heavy \nfermion systems,  the magnetic ground state is generally in competition with the Kondo quenching \nof the magnetic moments that lead to the famous Doniach diagram. This holds true in Ce -based \nheavy fermions as seen in CeRh In5 and Ce CoIn 550,51. In UGe 2 however, the two phenomena seem \nto coexist with no apparent competition, as seen by the largely unaffected Kondo resonance \ncrossing T c. A similar co nclusion was reached in USb 216,17. Gradually below ~35 K, an additional \nkink/shoulder  (E1 resonance)  develops  that coincides with the highly polarized FM2 phase that emerges as a crossover below T x ~ 32 K. The E 1 resonance shifts the  f-spectral weight closer to E F, \nwhich leads to an enhanced electronic effective mass . The latter has been observed in the optical \nspectroscopy measurements48. Hall measurements are also consistent with this picture, where a \nrapid increase of charge carriers indi cating some sort of Fermi surface reconstruction is observed \nbelow T x 24. Altoge ther, these observations indicate the splitting of the resonance that likely onsets \nat T c and becomes more pronounced  below the crossover temperature T x following the rigid band \nStoner model27,49, leading to a spin polarized f-electron band (E 1) at the Fermi energy, which sets \nthe stage for a spin polarized superconductivity to emerges at high pressure.  Future  spin polarized \nSTM study on the relative peak intensities of E 1 and E 2 will confirm the findings here, but is \nbeyond the scope of this study.  \n \n \n1. Mackenzie, A. P. & Maeno, Y. The superconductivity of Sr2RuO4 and the physics of \nspin-triplet pairing. Rev. Mod. Phys.  75, 657 –712 (2003).  \n2. Sato, M. & Ando, Y. Topological superconductors: A review. Reports Prog. Phys.  80, \n076501  (2017).  \n3. Beenakker, C. W. J. Search for majorana fermions in superconductors. Annu. Rev. \nCondens. Matter Phys.  4, 113 –136 (2013).  \n4. Huxley, A. et al.  UGe2: A ferromagnetic spin -triplet superconductor. Phys. Rev. B  63, \n(2001).  \n5. Saxena, S. S. et al.  Superconductivity on the border of itinerant -electron ferromagnetism \nin UGe2. Nature  406, 587 –592 (2000).  \n6. Aoki, D. et al.  Coexistence of superconductivity and ferromagnetism in URhGe. Nature  \n413, 613 –616 (2001).  \n7. Aoki, D. & Flouquet, J. Superconductivity and Ferromagnetic Quantum Criticality in \nUranium Compounds. J. Phys. Soc. Japan  83, 61011 (2014).  \n8. Huy, N. T. et al.  Superconductivity on the border of weak itinerant ferromagnetism in \nUCoGe. Phys. Rev. Lett.  99, 067006  (2007).  \n9. Jiao, L. et al.  Chiral superconductivity in heavy -fermion metal UTe2. Nature  579, 523 –\n527 (2020).  \n10. Pfleiderer, C. Superconducting phases of f -electron compounds. Rev. Mod. Phys.  81, \n1551 –1624 (2009).  \n11. von Lohneysen, H., Rosch, A., Vojta, M. & Wolfle, P. Fermi -liquid instabilities at \nmagnetic quantum phase transitions. Rev. Mod. Phys.  79, 1015 –1075 (2007).  \n12. Si, Q. M. Quantum criticality and global phase diagram of magne tic heavy fermions. \nPhys. Status Solidi B -Basic Solid State Phys.  247, 476 –484 (2010).  13. Schroder, A. et al.  Onset of antiferromagnetism in heavy -fermion metals. Nature  407, \n351–355 (2000).  \n14. Schröder, A., Aeppli, G., Bucher, E., Ramazashvili, R. & Col eman, P. Scaling of \nMagnetic Fluctuations near a Quantum Phase Transition. Phys. Rev. Lett.  80, 5623 –5626 \n(1998).  \n15. Zwicknagl, G. & Reese, M. Dual nature of strongly correlated 5f electrons. J. Magn. \nMagn. Mater.  310, 201 –206 (2007).  \n16. Ioannis Giannaki s, Justin Leshen, Mariam Kavai, Sheng Ran, Chang -Jong Kang, Shanta \nR. Saha, Y. Zhao, Z. Xu, J. W. Lynn, Lin Miao, L. Andrew Wray, Gabriel Kotliar, \nNicholas P. Butch, P. A. Orbital -selective Kondo lattice and enigmatic f electrons \nemerging from inside the a ntiferromagnetic phase of a heavy fermion. Sci. Adv.  5, \neaaw9061 (2019).  \n17. Chen, Q. Y. et al.  Orbital -selective Kondo entanglement and antiferromagnetic order in \nUSb2. Phys. Rev. Lett.  123, 106402  (2019).  \n18. Huxley, A. et al.  UGe 2: A ferromagnetic spin -triplet superconductor. Phys. Rev. B  63, \n144519 (2001).  \n19. Troc, R., Gajek, Z. & Pikul, A. Dualism of the 5f electrons of the ferromagnetic \nsuperconductor UGe2 as seen in magnetic, transport, and specific -heat data. Phys. Rev. B  \n86, 224403 (2012).  \n20. Hardy, F. et al.  Two magnetic Grüneisen parameters in the ferromagnetic superconductor \nUGe 2. Phys. Rev. B  80, 174521 (2009).  \n21. Palacio Morales, A. et al.  Thermoelectric power quantum oscillations in the ferromagnet \nUGe2. Phys. Rev. B  93, 155120  (2016).  \n22. Zhang, W., Liu, Y., Wang, X., Zhang, Y. & Xie, D. Strong coupling between localized 5f \nmoments and itinerant quasiparticles in the ferromagnetic superconductor UGe2. Chinese \nPhys. B  27, 037501  (2018).  \n23. Haslbeck, F. et al.  Ultrahigh -resolution neutron spectroscopy of low -energy spin \ndynamics in UGe2. Phys. Rev. B  99, 14429 (2019).  \n24. Tran, V. H., Paschen, S., Troć, R., Baenitz, M. & Steglich, F. Hall effect in the \nferromagnet UGe2. Phys. Rev. B  69, 195314  (2004).  \n25. Pflei derer, C. & Huxley, A. D. Pressure Dependence of the Magnetization in the \nFerromagnetic Superconductor UGe2 . Phys. Rev. Lett.  89, 147005  (2002).  \n26. Tateiwa, N. et al.  Magnetic properties of a pressure -induced superconductor UGe2. J. \nPhys. Soc. Japan  70, 2876–2879 (2001).  \n27. Sandeman, K. G., Lonzarich, G. G. & Schofield, A. J. Ferromagnetic superconductivity \ndriven by changing Fermi surface topology. Phys. Rev. Lett.  90, 167005 (2003).  \n28. Fisk, Z., Sarrao, J. L., Smith, J. L. & Thompson, J. D. The physics and chemistry of heavy \nfermions. Proc. Natl. Acad. Sci. U.S.A.  92, 6663 –6667 (1995).  29. Coleman, P. Heavy Fermions: electrons at the edge of magnetism. in Handbook of \nMagnetism and Advanc ed Magnetic Materials  1, (J. Wiley and Sons, 2007).  \n30. Yazdani, A., Eduardo, H., Neto, S. & Aynajian, P. Spectroscopic Imaging of Strongly \nCorrelated Electronic States. Annu. Rev. Condens. Matter Phys.  7, 11–33 (2016).  \n31. Aynajian, P. et al.  Visualizing Heavy Fermion Formation and their Unconventional \nSuperconductivity in f -Electron Materials. J. Phys. Soc. Japan  83, 061008  (2014) . \n32. Lanatà, N., Yao, Y., Wang, C., Ho, K. & Kotliar, G. Phase Diagram and Electronic \nStructure of Praseodymium and Plutonium . Phys. Rev. X 5, 011008  (2015).  \n33. Yao, Y., Deng, X. & Kotliar, G. Slave Boson Theory of Orbital Differentiation with \nCrystal Field Effects : Application to UO 2. Phys. Rev. Lett. 118, 126401  (2017) . \n34. Szajek, A., Che, G., Tro, R., Samsel -czeka, M. & Werwi, M. Intermetallics Electronic \nstructure of UGe 2 at ambient pressure: Comparison with X -ray photoemission spectra. \n19, 1411 –1419 (2011).  \n35. T. Okane ., Kamoto, J. O., Ami ya, K. M.  Ujimori, S. F.  et al Soft X -ray Absorption \nMagnetic Circular Dichroism Study of F erromagnetic Superconductor UGe 2. JSPJ  75, \n024704  (2006) . \n36. Madhavan, V., Chen, W., Jamneala, T., Crommie, M. F. & Wingreen, N. S. Tunneling \ninto a Single Magnetic Atom: Spectroscopic Evi dence of the Kondo Resonance. Science \n280, 567 –569 (1998) . \n37. Aynajian, P. et al.  Visualizing heavy fermions emerging in a quantum critical Kondo \nlattice. Nature  486, 201 –206 (2012).  \n38. Aynajian, P. et al.  Visualizing the formation of the Kondo lattice a nd the hidden order in \nURu2Si2. Proc. Natl. Acad. Sci. U.S.A.  107, 10383 (2010).  \n39. Ernst, S. et al.  Emerging local Kondo screening and spatial coherence in the heavy -\nfermion metal YbRh2Si2. Nature  474, 362 –366 (2011).  \n40. Schmidt, A. R. et al.  Imaging the Fano lattice to ‘hidden order’ transition in URu2Si2. \nNature  465, 570 –576 (2010).  \n41. Allan, M. P. et al.  Imaging Cooper pairing of heavy fermions in CeCoIn5. Nat. Phys.  9, \n468–473 (2013).  \n42. Maltseva, M., Dzero, M. & Coleman, P. Electron Cot unneling into a Kondo Lattice. Phys. \nRev. Lett.  103, 206402 (2009).  \n43. Figgins, J. & Morr, D. K. Differential Conductance and Quantum Interference in Kondo \nSystems. Phys. Rev. Lett.  104, 187202 (2010).  \n44. Wölfle, P., Dubi, Y. & Balatsky, A. V. Tunneling into Clean Heavy Fermion Compounds: \nOrigin of the Fano Line Shape. Phys. Rev. Lett.  105, 246401 (2010).  \n45. Akbari, A., Thalmeier, P. & Eremin, I. Quasiparticle interference in the heav y-fermion \nsuperconductor CeCoIn 5. Phys. Rev. B 84, 134505  (2011).  46. Coleman, P., Pépin, C., Si, Q. & Ramazashvili, R. How do Fermi liquids get heavy and \ndie? J. Phys. Cond. Matter  13, R723 –R738 (2001).  \n47. Guritanu, V. et al.  Optical spectra of the heavy fermion uniaxial ferromagnet UGe2. Phys. \nRev. B  78, 172406  (2008).  \n48. Aso, N. et al.  Stoner gap in the superconducting ferromagnet UGe . Phys. Rev. B 73, \n054512  (2006).  \n49. Knebel, G., Aoki, D. & Flouquet, J. Antiferromagnetism and Superconductivity in Cerium \nbased Heavy Fermion Compounds. C R Physique 12, 542 (2011).  \n50. Oppeneer, P. M. et al.  Fermi surface changes due to localized –delocalized f -state \ntransitions in Ce -115 and Pu -115 compounds. J. Magn. Magn. Mater.  310, 1684 –1690 \n(2007).  \n \nAcknowledgments  \nWork at Binghamton University is supported by the U.S. National S cience Foundation (NSF) CAREER under \naward No. DMR -1654482. Work at Los Alamos National Laboratory was performed under the auspices of \nthe US Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and \nEngineering.  \n \nAuthor Contributions:  \nI.G. performed the STM measurements . I.G. and P.A. performed STM  data analysis  with help from D.S . \nand J.F. . C-J.K. and G.K. performed DFT + Gutzwiller calculations. M.J., P.D. and E.B. synthesized and \ncharacterized the materials. P. A. wrote the manuscript.  \n \n \n \n \n \n \n \n \n \n \n Figure captions:  \nFigure 1  \n \nFigure 1: (a, b)Topographic  image of cleaved UGe 2 revealing subatomic terraces  with  alternating  chemically different surface \ntermination s, one of which undergoes  a surface reconstruction. (c) C omparison of the terrace heights extracted  from the blue line -\ncut in (b)  with the crystal structure of UGe 2 suggesting the breaking of  a single bond (U -Ge2) that exposes the reconstructed Ge2 \nsurface (B) and U surface (A).  (d, e) STM dI/dV spectra on the two surfaces at high (d) and low (e) temperatures.  Note that the \ndata in (d, e) are not taken on the same spatial location . (f) DFT +Gutzwiller  calculation of the electronic density of states in UGe 2 \nin the non -magnetic phase . Note that U 5f 7/2 due to spin -orbit coupling are at much higher energies (1 – 2 eV). \n \n \n \n \n \n \n \n \n \n \n \nFigure 2  \n \nFigure 2: STM spect ra taken at  different locations on surfaces A (a) and B (b) at 8 K. The spectra in (a ) reveal spatial uniformity , \nwhereas those in (b) are spatially inhomogeneous due to the surface inhomogeneity. The lines correspond to the energies of th e \nobserved peaks that are similar  on both surfaces.  \n \n \nFigure 3  \n \nFigure 3: (a, b) STM spectroscopy on surfac e-A as a function of temperature  with two different experimental settings described in \nthe text . The spectra reveal a Fano lineshape (E 2 resonance) located above the chemical potential. As temperature is lowered \nbelow T c, a second resonance emerges near th e Fermi energy (E 1 resonance) below ~ 35 K. (c, d) Second derivative of the spectra \nin (a , b) s howing the evolution of the E 1 and E 2 resonances as a function of temperature . (e) Fitting of the data in (b) to a  single \nFano lineshape . The spectrum above T c can be explained by a single resonance, while at lower  temperatures , the data deviates \nfrom a single Fano lineshape.  \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 4  \n \nFigure 4: (a) STM spectroscopy and corresponding two Fano -lineshape model fit  at different temperatures . (b,c,d) extracted \nFano amplitude, width and energy as a function of temperature. Red data points correspond to the E 1 resonance near E F \nwhereas the blue data points correspond to the E 2 resonance above the chemical potential. The lines in (c) represent (T) for a \nTK of 110K (blue) and TK of 55K (red).  \n"    },    {        "title": "2103.08704v1.Ferromagnetic_Resonance_in_Permalloy_Metasurfaces.pdf",        "content": " Ferromagnetic Resonance in Permalloy Metasurfaces N. Noginova1*, V. Gubanov2, M. Shahabuddin1, Yu. Gubanova2, S. Nesbit1, V. V. Demidov3, V.A. Atsarkin3,  E. N. Beginin2, A. V. Sadovnikov2 1Norfolk State University, Norfolk VA 23504 USA 2Laboratory “Metamaterials\", Saratov State University, Saratov 410012, Russia 3Kotel’nikov Institute of Radio Engineering and Electronics, RAS, Moscow 125009, Russia  *corresponding author. email: noginova@nsu.edu  Abstract.  Permalloy films with one-dimensional (1D) profile modulation of submicron periodicity are fabricated based on commercially available DVD-R discs and studied using ferromagnetic resonance (FMR) method and micromagnetic numerical simulations. The main resonance position shows in-plane angular dependence which is strongly reminiscent of that in ferromagnetic films with uniaxial magnetic anisotropy. The main signal and additional low field lines are attributed to multiple standing spin wave resonances defined by the grating period.  The results may present interest in magnetic metamaterials and magnonics applications.  Introduction.  Advances in nanofabrication and development of metamaterial concepts bring to life a new class of composite materials whose properties are artificially engineered via responses of nano-features to the electric component of electromagnetic field [1,2]. Nanoinclusions made from plasmonic metal (silver, gold, aluminum) are commonly used in optical metamaterials due to their strong response to illumination in the range of plasmon resonances [2]. Using magnetic inclusions which respond to the magnetic component of the electromagnetic radiation, one can design magnetic properties and propagation of electromagnetic waves in a material as well [3-8]. In similarity with optical metamaterials [2], such systems can be described in terms of the effective media approximation [9] with the material constants: dielectric permittivity, e, and magnetic permeability, µ.  However, magnetic metamaterials are restricted to the radio frequency range since the dynamic of the magnetic response is relatively slow, with typical frequencies of magnetic resonance in GHz range. On the other hand, the use of magnetic components enables easy tunability of such systems with the external magnetic field. The effective permeability of composites with single-domain small magnetic nanoparticles in polymer matrices can be tuned from negative to positive values [5, 6], presenting interesting opportunities for temporal and spatial control of the wave propagation. Via mutual arrangement of magnetic nanofeatures and their shape, one can design materials with certain anisotropy of magnetic and microwave properties [10-15].  With increased sizes of magnetic features, the effective medium approximation using a single µ might be no longer sufficient as the response to the illumination becomes more complicated due to the excitation and propagation of spin waves. Structures with periodic arrangement of magnetic elements are commonly considered in terms of magnonic crystals [16-23] in analogy with photonic crystals [24, 25] exhibiting effective photonic forbidden and allowed bands.  In this work we study grating-like permalloy structures with the submicron periodicity. Permalloy is a soft ferromagnetic with high magnetic permeability; it is a common material for various magnetic and magnonic [13,22] studies. In addition, it exhibits plasmonic behavior [26] and an interesting coupling between plasmonic, magnetic and electric properties [27], presenting interest for plasmon-induced magnetization switching and magnetically controlled plasmonics. In our work, we employ the FMR method and micromagnetic numerical simulations in order to better understand magnetic and magnetic resonance behavior of such systems, applicability of metamaterial description, and a role of spin wave-related effects.  Experimental Our experimental structures are permalloy (Ni-Fe alloy with 80% of Ni and 20% of Fe) thin films with one-dimensional (1D) profile modulation based on the substrates derived from commercially available DVD-R discs. The fabrication starts with obtaining polycarbonate grating substrates from disassembling commercial DVD-R by carefully taking out the polymer, plastic, silver and protective coating layers. Then, permalloy (Py) with a thickness d = 40 nm is deposited on the prepared and precut DVD substrates using e-beam evaporation. The thickness of the film is independently tested with a profilometer by measuring films simultaneously deposited on glass substrates. Atomic force microscopy (AFM) confirms the profile modulation parameters, the periodicity d = 740 nm and modulation height h = 60-80 nm. As main experiments and numerical simulations are performed with Py/DVD structures, for comparison purposes we also prepared permalloy structures with different profile-modulation parameters using substrates derived from Blu-ray or CD discs, with d = 320 nm and h = 25-30 nm in Py/BR structures, and d = 1600 nm and h = 120-140 nm (Py/CD), see Fig. 1 (a) for the schematics. Ferromagnetic resonance curves are recorded using the Bruker EPR Spectrometer at 10 GHz microwave frequency (X Band). The sample is placed inside the microwave cavity with the sample orientation corresponding to the external magnetic field, H in plane of the film, Fig 1(b). The direction of the grooves makes an angle q with H. The signal (the derivative of the microwave absorption vs H) is recorded for various q.  \nFig. 1. (a) Profile schematics of Py/DVD, Py/BR and Py/CD  structures (as indicated). (b) Orientation of the sample in FMR experiments. 074014802220d =740 nm60 -80 nmnm\n032064096025 -30 nm320 nmnm08001600240032001600nm120-140 nmnm\nPermalloy film,\nwith thickness \nd\n=\n40 nm\nPolycarbonate\nsubstratePy/DVDPy/BRPy/CD\nqH(b)(a)h Typical ferromagnetic resonance spectra observed in Py/DVD are shown in Fig.2 (a). (In this plot, the field is shown in Oersted as is in the original recordings.) The signal position depends on the orientation angle, shifting from the lowest to highest field when q changes from 0 to 90 deg, corresponding, respectively, to the parallel and perpendicular orientations of the grooves in respect to the external field.  \n The position of the main FMR peak (determined from fitting with the Lorentzian lines) is plotted as the function of q,  Fig. 2 (b). The dependence closely follows cos 2q  function. An additional component can be clearly distinguished in some orientations, with the position strongly dependent on the angle as well. The relative strength of this component slightly varies from sample to sample while the angular behavior (Fig. 2 (c), open circles) follows the same cos 2q dependence.  We explored the FMR in the other structures as well. A single peak in the Py/CD structures demonstrates the same angular dependence (Fig. 2(c), stars) but with a smaller amplitude. The behavior of Py/BR is different. In a broad range of angles (q> 40o), a single FMR peak is observed with practically no dependence on the orientation. Splitting into two peaks and a strong angular dependence is observed for low angles. Note, that in the current work we concentrate on the Py/DVD system; the other systems will be studied in detail elsewhere.  Discussion and Modeling The angular dependence of the main FMR signal in Py/DVD is strongly reminiscent of that in ferromagnetic films with the growth-induced in-plane uniaxial anisotropy [28,29]. This can be expected, taking into account that the structural geometry of our gratings is comparable with that of crystalline films [28] but with a submicron-size of the features instead of interatomic distances. Note that films with this type of the magnetic anisotropy exhibit a sharp reorientation of the magnetization upon a small increment of the magnetic field [30, 31], and are of particular interest for optically induced magnetization switching via angular momentum transfer from light to matter [32, 33].  Assuming a relatively small magnetic anisotropy  Hp < H < M, the FMR condition in our excitation geometry reads [34], \nFig. 2. (a) Typical FMR spectra and (b) position of the main peak as the function of q in Py/DVD.  Dashed trace is fitting with Eq. 1. (c) Peak positions in  Py/CD (stars) and Py/DVD (circles), main peak (closed symbols), additional component (open symbols). (d) Positions of two peaks in Py/BR vs angle.   (b)\n0.1240.1280.1320.1360.140.144\n090180270360Peak position [T]\nq[deg]0.090.110.130.15-100-402080140200260Peak Position [T]q[deg] (c)(d)0.080.10.120.140.16-303090150210270330Peak Position [T]q[deg] -1800-1300-800-30020070012001700\n90011001300150017001900Amplitude [arb.u.]H[ Oe]90 deg60 deg(a)\n45 deg26 deg!w\t\"!g\"#=\t$𝐻+𝑀+\t𝐻$cos#𝜃,\t$𝐻+𝐻$cos2𝜃,,     (1).    predicting the periodic dependence with the period of 180o,  minimum at 0o and maximum at 90o as has been observed in the experiment. Here w is the FMR frequency, g is the gyromagnetic ratio, M is the magnetization, and Hp is the anisotropy field [35]. Fitting the experimental data with Eq. 1, the effective anisotropy fields are estimated as 𝜇%Hp =  7.6 mT in the Py/DVD and 3.7 mT in Py/CD, assuming the saturation magnetization of permalloy, M = MS = 6.4 *105 A/m.  This is in agreement with the literature [12-15] as well, where patterned magnetic structures such as films deposited on the grating-like substrates [13] exhibit a uniaxial anisotropy with the easy axis parallel to the direction of grooves. However, in our Py/DVD systems, additional features are clearly seen, which could not be described with this simple approximation.  In order to better understand magnetic behavior of the Py/DVD structure, we perform numerical simulations, considering a meander-like structure with the following parameters: saturation magnetization of permalloy M = MS = 6*105 A/m, periodicity d = 740 nm, modulation height h = 80 nm, thickness of  permalloy, d = 50 nm at horizontal stages,  and various thicknesses w ≤ d at vertical walls. Since the deposition of metal on the top of the substrate can produce vertical walls with reduced thickness, additional simulations have been performed to explore the role of reduced w.  As we found, small variations in the range of possible thicknesses  0 < w ≤ d  do not significantly affect the results. The detailed study of FMR behavior in patterned systems with different parameters including variations in the periodicity, film thickness and the shape of the profile modulation (sine-wave or rectangular) is the subject of a separate study. In the numerical simulations, w e apply an approach discussed in detail in [19] and perform micromagnetic modeling of our structure using the MuMax3 software [36].  The calculation is based on the Landau-Lifshits-Gilbert equation with damping parameter of 0.007. We consider the same geometry as in the experiment: the external magnetic field lies in plane with the structure and the angle, q, between the direction of grooves and field varies from 00 to 90. The microwave field with the magnitude of 10-4 T is applied along y axis (perpendicular to the H direction).  First we estimate local distributions of the static magnetization M and effective internal fields Hi (resulting from external and demagnetization fields). In Figure 3, distributions of Hi (absolute values) are shown at various rotation angles: when the external magnetic field is perpendicular to the direction of grooves (Fig. 3(a)); makes an angle of 45o (Fig. 3 (b)) or parallel to the grooves (Fig, 3(c)). As one can see, at q = 900 and 45o, the magnetic response of the material is strongly modulated in space with maxima of Hi observed in the middle of horizontal stages and \nFig. 3. Distribution of the internal magnetic field at different rotation angles: (a) q  = 90o, (b)  q  = 45o (c) q  =  00. µ0H0 = 0.2 T. \nµ0H [T] µ0H [T] µ0H [T] minima observed at the corners. The magnitude of this spatial modulation decreases with the decrease in the angle, vanishing at the parallel orientation of the grooves and external field, Fig. 3 (c). Absorption at the 10 GHz frequency is calculated following the approach [19] as the function of the external magnetic field. In a flat film, a single resonance peak is expected at µ0H0  = 0.142 T, while in the profile- modulated structures, several peaks can be resolved with the positions dependent on the orientation.  Three major peaks predicted in the Py/DVD structure (with the period d= 740 nm and wall thickness w= 12.5 nm) are shown in Fig. 4 (a). With an increase in q, these peaks shift toward H0, Fig. 4 (b). The angular dependences for the Peak 1 and Peak 2 positions fairly well correspond to those of the main and additional peaks observed in experiment, Fig. 4(c).  \nLet us assume that the peaks observed below H0 both in simulations and experiments correspond to the resonance modes formed by the magnetic surface spin waves (MSSW [17-19, 35]). (The Peak 4 in Fig. 4 (a) with the position above H0 is likely related to the volume modes and is not discussed here).  When the field H and the magnetization, M, are directed along the grooves (q = 0),  MSSW waves are excited in the perpendicular direction, see Fig.5(a) which illustrates the amplitude of the precessing transverse magnetization m(t) under the resonance conditions.  Since the spatial modulation (defined by the grating period) has the period of d in this direction, the resonance condition is expected at k = pN/d,      (2) where N = 1,2,3.. is an integer.  If the magnetization is directed under an angle q in respect to the groove direction (Fig 5 (b)), the period of the modulation becomes d/cosq, and the k vector of the resonance mode can be found from the condition, k = 𝑁&' cosq.          (3)   \nFig. 4. (a) Simulated microwave (10 GHz)  absorption vs field in Py/DVD .  (b)  Peak positions vs orientation angle, (c) Comparison with experiment. Experimental data are shown with symbols, results of numerical simulations are solid traces. Numbers 1, 2, 3 indicate corresponding peaks. \nLet us replot the peak position estimated from the numerical calculations (Fig. 4 (b)) as the function of the k-vector estimated from the Eq. 3 with N =1 for the dataset 1, and N= 2 and N = 3 for the datasets 2 and 3 correspondingly. All the data fits the single curve (Fig. 5 (c)) confirming our assumptions above. Note, that, alternatively, similar speculations can be applied assuming a twice smaller period d/2 of magnetic modulation (which corresponds to the width of each horizontal segment). In this case, the graph (c) will be stretched twice along the x axis. As discussed below, our data rather indicate that d is the period of modulation. In frames of the same numerical approach, instantaneous values of the dynamic (transverse) magnetization component mx are calculated under the resonance conditions of Peaks 1, 2 and 3 at q = 0, see the panels on the left sides of Fig. 6 (a-c) and their analysis on the right. These patterns are rather complicated, in particular, inside the vertical walls.  However, the presence of standing waves is evident, with the k vectors of () , 2() and 3()  corresponding to the resonances 1, 2 and 3 respectively.   We put together theory and experiment in Figure 6 (d). Points are the experimental results with the abscissas calculated as following: Red circles and triangles: Py/DVD (dDVD = 740 nm). The k vectors are calculated as k =  &'\"#\"N cosq ,  N = 1  for the main peak (circles), and N = 2 for the second peak (triangles). Green stars: Py/CD (dCD = 1600 nm). The k vectors are calculated at the same manner assuming N = 1,  k = &'$\" cosq.  We make an attempt to add the results obtained in Py/BR ( dBR = 320 nm) as well. Points (blue squares) fit the general tendency if we use the second (smaller) peak (which shows strong angular dependence) and estimate k  = &'%& cosq. The dashed curve is obtained from the numerical simulations ( figure 5(c)). The solid curve is shown for comparison; it is obtained from Eq. 4 which describes the dispersion curve of the MSSW for the flat film [37] assuming the film thickness d = 50 nm,  !w\t\"!g\"#=\t𝐻(𝐻+𝑀)+*'+(\t1−𝑒,#-.\t) .        (4) In\tprinciple,\tassuming\ta\tcertain\trelationship\tbetween\tthe\tfield\tand\tthe\tresonance\tfrequency\tone\tcan\tderive\tthe\tdispersion\tcurve\tof\tthe\tspin\twaves\tin\tour\tstructures\tfrom\tthe\tdata\tplotted\tin\tFig.\t6.\tIt\tis\talready\t\nFig, 5. (a, b) Standing modes at (a) perpendicular orientation and  (b) an arbitrary angle; (c) Resonance field vs k, simulations. The numbers 1, 2, 3 indicate corresponding datasets of Fig. 4. (c)\n00.020.040.060.080.10.120.140.16\n0481216Peak position [T]k [rad/mm]123(a)\nk0=pN/dM\n-k0\nHN = 1N = 2N = 3H\nx\nzqkk(b)seen\tthat\tat\thigher\tk,\t\tthe\tdispersion\tcurve\tis\tclose\tto\tthat\tin\ta\tflat\tfilm.\t\tThis\tis\texpected\tsince\tthe\tspin\twaves\twith\tsmaller\twavelengths\tare\tless\taffected\tby\tthe\tprofile\tgeometry.\t\t\n\tIn\t conclusion,\t1D\t profile-modulated\t permalloy\t films\t (continuous\t gratings)\tare\tstudied\twith\tthe\tferromagnetic\tresonance\tmethod.\tThe\tposition\tof\tthe\tmain\tresonance\tis\tfound\tto\tbe\tstrongly\tdependent\ton\tthe\tangle\tbetween\tthe\tdirection\tof\tthe\tgrooves\tand\tthe\texternal\tmagnetic\tfield\t(which\tis\tkept\tin-plane).\tThe\tangular\tdependence\tis\tvery\tsimilar\tto\tthat\tobserved\tin\tcrystalline\tfilms\twith\tuniaxial\tmagnetic\tanisotropy\tand\tin-plane\teasy\taxis.\tAdditional\tresonances\tof\ta\tsmaller\tmagnitude\tare\tobserved\tat\tlower\tfields,\twhich\tshow\ta\tsimilar\tangular\tdependence\tas\twell.\tThe\tfindings\tare\tdiscussed\tin\tterms\tof\tMSSW\tresonances\tand\tconfirmed\tby\tmicromagnetic\tsimulations.\t\tAcknowledgements  N. Noginova, M. Shahabuddin and S. Nesbit would like to acknowledge financial support from National Science Foundation (NSF) (1830886), Air Force Office of Scientific Research(AFOSR) (FA9550-18-1-0417) and Department of Defence (DoD) (W911NF1810472). The work of authors from Russian Federation is carried out within the framework of the state task and partially was supported by Russian Foundation for Basic Research, projects No. 18-57-16001. \tReferences 1. Fundamentals and Applications of Nanophotonics, Ed.: J. Haus, 2016 Elsevier Ltd. 2016, pp 253-307. 2. Tutorials in Metamaterials, Eds.: M.A. Noginov, V. A. Podolskiy,  CRC Press, 2011, 308 p. \n-0.00150.00050370740\n-0.0050.00302004006000.0030.0060370740\nmx/M0mx/M0 mx/M0(a)(b)(c)0.040.060.080.10.120.140.16\n-16-80816B [T]\nk [rad/mm](d)\n Fig. 6. (a-c)  Distributions of dynamic magnetization mx (left panels) and profiles of mx along the half-depth of the lower horizontal segment (plots on the right) at the resonance conditions of Peaks 1, 2 and 3 at q = 0, (a) µ0H =0.126T,  (b) µ0H  = 0.0835 Tm and (c) µ0H =0.052T. Red dashed traces are sine waves with the periodicity of (a) 2d,  (b) d,  and (c) 2d/3. (d) Experiment (points) and theory (traces). Py/DVD,  main peak (circles), additional peak b: (triangles); Py/CD (stars), Py/BR (squares). Numerical simulations (dashed trace) and predictions for a flat film, Eq (4) (solid trace).  3. A. L. Adenot-Engelvin, C. Dudek, P. Toneguzzo, O. Acher. “Microwave properties of ferromagnetic composites and metamaterials”. J. of Eur. Cer. Soc., 27,  1029-1033 (2007). 4. Y. Poo, R. Wu, G. He, P. Chen, J. Xu, R. Chen, “Experimental verification of a tunable left-handed material by bias magnetic fields”, Appl. Phy. Lett. 96, 161902 (2010). 5. C. Morales, J. Dewdney, S. Pal, S. Skidmore, K. Stojak, H. Srikanth, T. Weller, and J. Wang, \"Tunable magneto-dielectric polymer nanocomposites for microwave applications\" IEEE Transactions on Microwave Theory and Techniques 59, 302-309 (2011). 6. N. Noginova, Q. L. Williams, P. Dallas and E. P. Giannelis, \"Magnetic nanoparticles for tunable microwave metamaterials\", Proc. SPIE 8455, Metamaterials: Fundamentals and Applications V, 845531 (2012). 7.  M. Mruczkiewicz, M. Krawczyk, R. V. Mikhaylovskiy, and V. V. Kruglyak. Towards high-frequency negative permeability using magnonic crystals in meta- material design. Phys. Rev. B, 86(2):024425, (2012 ) 8.  B. Bates, N. Greene, N. Noginova, “Characterization of ferromagnetic/dielectric systems for metamaterials applications”, Proc. SPIE, 91602R-91602R-4  (2014). 9. Choy, Tuck C. (1999). Effective Medium Theory. Oxford: Clarendon Press. ISBN 978-0-19-851892-1. 10. J. Fischbacher, A. Kovacs, H. Oezelt, M. Gusenbauer, D. Suess,  and T. Schrefl, “Effective uniaxial anisotropy in easy-plane materials through nanostructuring,”  Appl. Phys. Lett. 111, 192407 (2017). 11. N. Noginova, Yu. Barnakov, A. Radocea, V.A. Atsarkin, \"Role of dipole-dipole interactions in half-field quantum transitions in magnetic nanoparticles \", J. Magn. Magn. Mater. 323, 2264 (2011).  12. M. O. Liedke, M. Körner, K. Lenz, M. Fritzsche, M. Ranjan, A. Keller, E. Čižmár, S. A. Zvyagin, S. Facsko, K. Potzger, J. Lindner, and J. Fassbender, “Crossover in the surface anisotropy contributions of ferromagnetic films on rippled Si surfaces, “Phys. Rev. B 87, 024424 (2013). 13. J. Berendt, J. M. Teixeira, A. García-García, M. Raposo, P. A. Ribeiro, J. Dubowik, G. N. Kakazei, and D. S. Schmool, “Tunable magnetic anisotropy in permalloy thin films grown on holographic relief gratings,” Appl. Phys. Lett. 104, 082408 (2014). 14. R. Moroni, D. Sekiba, F. Buatier de Mongeot, G. Gonella, C. Boragno, L. Mattera, and U. Valbus, “Uniaxial Magnetic Anisotropy in Nanostructured C o / C u ( 001 ) : From Surface Ripples to Nanowires,” Phys. Rev. Lett. 91, 167207 (2003) 15. S. A. Mollick,  R. Singh, M. Kumar,  S.n Bhattacharyya, and T. Som, “Strong uniaxial magnetic anisotropy in Co films on highly ordered grating-like nanopatterned Ge surfaces,” Nanotechnology 29 125302, (2018). 16. V V Kruglyak, S O; Demokritov, D Grundler, \"Magnonics,\".J. of Phys. D: Appl. Phys.. 43, 264001 (2010). 17. SL Vysotskii, SA Nikitov, and Yu A Filimonov, “Magnetostatic spin waves in two-dimensional periodic structures (magnetophoton crystals),” J. Exp. Theor. Phys. 101, 547–553 (2005). 18. M.L. Sokolovskyy and M. Krawczyk. The magnetostatic modes in planar one-dimensional magnonic crystals with nanoscale sizes. J. Nanopar. Res. 13, 6085–6091 (2011). 19. E. N. Beginin, A.V. Sadovnikov, V.K. Sakharov, A.I. Stognij, Y.V. Khivintsev, S.A. Nikitov. “Collective and localized modes in 3D magnonic crystals,” J. Magn. Magn. Mat. 492, 165647 (2019) 20. J. O. Vasseur, L. Dobrzynski, B. Djafari-Rouhani, and H. Puszkarski. Magnon band structure of periodic composites. Phys. Rev. B 54 1043, (1996). 21. H. Puszkarski and M. Krawczyk. \"Magnonic crystals – the magnetic counterpart of photonic crystals,\"  Solid State Phenom., 94:125 (2003). 22. G. Gubbiotti,  L. L. Xiong,  F. Montoncello, and A. O. Adeyeye, “Collective spin waves in arrays of permalloy nanowires with single-side periodically modulated width,” Appl. Phys. Lett. 111, 192403 (2017) 23. P. Malagò, L. Giovannini, R. Zivieri, P. Gruszecki, and M. Krawczyk,” Spin-wave dynamics in permalloy/cobalt magnonic crystals in the presence of a nonmagnetic spacer,”  Phys. Rev. B 92, 064416 (2015). 24. J. D Joannopoulos, P. R Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature, 386, 143–149, (1997). 25. K Ohtaka, \"Energy band of photons and low-energy photon diffraction\". Phys. Rev.B. 19,  5057–67 (1979). 26. M. Shahabuddin, N. Noginova, “Plasmonic System with In-Plane Magnetic Anisotropy for Plasmon Based Magnetic Switching,” OSA Techn. Digests, CLEO: QELS, 2019, paper # JW2A. 27. M. Shahabuddin, D. Keene, M. Durach, V. S. Posvyanskii, V. A. Atsarkin N. Noginova, “Magnetically dependent plasmon drag in permalloy structures,” arXiv preprint ArXiv 1909.02675 (2020). 28. C. Tang, M. Aldosary, Z. Jiang, H. Chang,  B. Madon, K. Chan,  M. Wu, J. E. Garay, and J. Shi, “ Exquisite growth control and magnetic properties of yttrium iron garnet thin films,” Appl. Phys. Lett. 108, 102403 (2016). 29. Yu-J. Zhang, L. Wu, J. Ma, Q.-H. Zhang, A. Fujimori, J. Ma, Yu-H. Lin, C.-W. Nan & N.-X. Sun, “Interfacial orbital preferential occupation induced controllable uniaxial magnetic anisotropy observed in Ni/NiO(110) heterostructures”, npj Quantum Materials 2, 17 (2017) 30. V. A. Atsarkin, V. V. Demidov, A. E. Mefed, V. Yu. Nagorkin, “Magnetic pseudoresonance in manganite thin films”, Appl.Magn. Res. 45, 809-816 (2014). 31. T. M. Vasilevskaya, S. A. Volodin, D. I. Sementsov, “Ferromagnetic resonance and bistability field in a uniaxial magnetic film”, Tech. Phys. 56, 1373 (2011). 32. M. Durach, N .Noginova,”Spin angular momentum transfer and plasmogalvanic phenomena”, Phys. Rev. B 96, 195411 (2017).  33. K.Y. Bliokh, A.Y. Bekshaev, F. Nori, “Optical momentum and angular momentum in complex media: from the Abraham–Minkowski debate to unusual properties of surface plasmon-polaritons,” New J. Phys. 19, 123014 (2017). 34. V. V. Demidov, I V. Borisenko,  A A Klimov, G A Ovsyannikov, A M Petrzhik and S A Nikitov, “Magnetic anisotropy in strained manganite films and bicrystal junctions,”  JETP 112, 825 (2011) 35. A. G. Gurevich and G. A. Melkov, Magnetization Oscillation and Waves, New York: CRC, 1996.  36. A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F. Garcia-Sanchez, & B. Van Waeyenberge,  \"The design and verification of MuMax3,\". AIP Advances 4, 107133 (2014).. 37. D. D Stancil. Theory of magnetostatic waves. Springer Science & Business Media, 2012. "    },    {        "title": "1707.08784v2.Electron_spin_resonance_for_the_detection_of_long_range_spin_nematic_order.pdf",        "content": "Electron spin resonance for the detection of long-range spin nematic order\nShunsuke C. Furuya1and Tsutomu Momoi1, 2\n1Condensed Matter Theory Laboratory, RIKEN, Wako, Saitama 351-0198, Japan\n2RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama, 351-0198, Japan\n(Dated: August 31, 2021)\nSpin nematic phase is a quantum magnetic phase characterized by a quadrupolar order parameter.\nSince the quadrupole operators are directly coupled to neither the magnetic \feld nor the neutron,\ncurrently, it is an important issue to develop a method for detecting the long-range spin nematic\norder. In this paper we propose that electron spin resonance (ESR) measurements enable us to\ndetect the long-range spin nematic order. We show that the frequency of the paramagnetic resonance\npeak in the ESR spectrum is shifted by the ferroquadrupolar order parameter together with other\nquantities. The ferroquadrupolar order parameter is extractable from the angular dependence of\nthe frequency shift. In contrast, the antiferroquadrupolar order parameter is usually invisible in the\nfrequency shift. Instead, the long-range antiferroquadrupolar order yields a characteristic resonance\npeak in the ESR spectrum, which we call a magnon-pair resonance peak. This resonance corresponds\nto the excitation of the bound magnon pair at the wave vector k=0. Re\recting the condensation of\nbound magnon pairs, the \feld dependence of the magnon-pair resonance frequency shows a singular\nupturn at the saturation \feld. Moreover, the intensity of the magnon-pair resonance peak shows\na characteristic angular dependence and it vanishes when the magnetic \feld is parallel to one of\nthe axes that diagonalize the weak anisotropic interactions. We con\frm these general properties\nof the magnon-pair resonance peak in the spin nematic phase by studying an S= 1 bilinear-\nbiquadratic model on the square lattice in the linear \ravor-wave approximation. In addition, we\nargue applications to the S= 1=2 frustrated ferromagnets and also the S= 1=2 orthogonal dimer\nspin system SrCu 2(BO 3)2, both of which are candidate materials of spin nematics. Our theory\nfor the antiferroquadrupolar ordered phase is consistent with many features of the magnon-pair\nresonance peak experimentally observed in the low-magnetization regime of SrCu 2(BO 3)2.\nI. INTRODUCTION\nSpin nematic phase is a hidden ordered phase of quan-\ntum magnets where the spin rotation symmetry is spon-\ntaneously broken but, di\u000berent from the ferromagnetic\nand the antiferromagnetic phases, the time reversal sym-\nmetry is kept intact. The spin nematic phase is char-\nacterized by a quadrupolar order parameter made of a\nsymmetric pair of electron spins [1{3]. The emergence\nof the spin nematic phase requires the absence of the\nspontaneous dipolar orders, implying interplay and frus-\ntration of spin-spin interactions behind the order. Until\ntoday, much e\u000bort has been made to explore the spin\nnematic phase in S= 1=2 frustrated ferromagnets both\ntheoretically [4{21] and experimentally [22{27], in the\nS= 1=2 orthogonal dimer spin system [28, 29], and also\ninS\u00151 spin systems with biquadratic interactions [30{\n35]. Recently, the research \feld of the spin nematic order\nexpands to the \feld of an iron pnictide superconductor,\nFeSe [36{38].\nIn the current situation surrounding researches of the\nspin nematic phase, one of the most important problems\nis to develop a method for detecting the spin nematic\norder in an experimentally feasible way. The di\u000eculty\nin the problem is that the quadrupole operators are di-\nrectly coupled to neither the magnetic \feld nor the neu-\ntron. Several theoretical proposals were recently made\nby studying theories of the nuclear magnetic resonance\n(NMR) [39{41], inelastic neutron scattering [42{44], in-\nelastic light scattering [45], resonant inelastic x-ray scat-tering [46], and electron spin resonance (ESR) [47]. As\npointed out in Ref. [47], various quantities related to ESR\nare naturally coupled to quadrupole operators and thus\nESR is a promising way for detecting the hidden spin\nnematic order. In ESR experiments, bound triplon-pair\nmodes were observed in the spin gapped phase of the\northogonal dimer spin system SrCu 2(BO 3)2[29] and a\nbound magnon-pair mode was also in the fully polarized\nphase of Sr 2CoGe 2O7[48]. Bound magnon pairs can give\nthe instability to the spin nematic ordering when they\nclose the energy gap [5]. The behavior of these bound pair\nmodes in ESR spectrum is however not yet understood\ninside the spin nematic phase from both the theoretical\nand experimental sides. In view of the current situations,\nan ESR theory for identi\fcation of the quadrupolar order\nin the spin nematic phase is called for.\nIn this paper, we theoretically study ESR in the spin\nnematic phase to elucidate how to identify the quadrupo-\nlar order from the ESR spectrum. We propose two meth-\nods of detecting spin nematic orders. In the \frst part\nof the paper, we demonstrate for quite a generic model\nthat ESR measurements enable us to extract the ferro-\nquadrupolar (FQ) order parameter (i.e. the spin nematic\norder parameter developed at the wave vector k=0)\nfrom the frequency shift of the electron paramagnetic res-\nonance (EPR) peak in the ESR spectrum, as one of the\nauthors brie\ry mentioned in Ref. [47]. Incidentally the\nantiferroquadrupolar (AFQ) order parameter is not de-\ntectable in this method. In the latter part of the paper,\nwe propose a complementary ESR measurement suitable\nfor identi\fcation of the AFQ order, taking an example ofarXiv:1707.08784v2  [cond-mat.str-el]  21 Mar 20182\nbound magnon pairEPRunpaired magnon at k MFrequency\n05101520\nMagnetic ﬁeld H/J 110 5 10 15\nFIG. 1. Three characteristic resonance frequencies of ESR in\nthe antiferroquadrupolar (AFQ) phase of an S= 1 bilinear-\nbiquadratic model (4.5) on the square lattice. We used the\nparameters J11= 1,J12= 0:1, andJ22= 2 following Ref. [43].\nThe saturation \feld is given by HAFQ\nc = 12. The system is in\nthe AFQ phase in the \feld range 0 \u0014H <HAFQ\nc and in the\nfully polarized phase in the range HAFQ\nc<H. The magnon-\npair resonance (4.80) (the solid curve) and the unpaired-\nmagnon resonance (4.81) at the wave vector kM= (\u0019;\u0019)\n(the dot-dashed line) are found in addition to the electron\nparamagnetic resonance (4.79) (the dashed line).\nanS= 1 spin model. In this method, we focus on res-\nonance of bound magnon-pair excitations, which charac-\nteristically appear in quadrupolar order phases signaling\nthe condensation of bound magnon pairs. The corre-\nsponding resonance peak, which we call the magnon-pair\nresonance peak, appears at a \fnite frequency in the ESR\nspectrum in the AFQ phase. This resonance peak can be\nclearly distinguished from others since the peak intensity\nhas a characteristic angular dependence and the peak fre-\nquency also has a distinctive \feld dependence. A typical\n\feld dependence of the magnon-pair resonance frequency\nis shown as the solid curve in Fig. 1 for the AFQ phase\nin anS= 1 bilinear-biquadratic model. Having the two\nmethods of ESR for the FQ and the AFQ orders, we can\ncharacterize the FQ and the AFQ phases clearly. We can\napply our theory to S= 1=2 candidate materials of spin\nnematics, such as S= 1=2 frustrated ferromagnets and\nS= 1=2 orthogonal dimer spin compound SrCu 2(BO 3)2.\nOur theory for the AFQ ordered phase is consistent with\nmany features of the experimentally observed magnon-\npair resonance peak [29] in SrCu 2(BO 3)2, which suggests\nthe existence of a spin nematic order.\nThis paper is organized as follows. In Sec. II, we review\nbrie\ry an important identity [Eq. (2.5)] that we rely on\nin this paper. Using the identity, we show in Sec. III that\nthe EPR frequency is shifted by the FQ order parameter\ndepending on the direction of the sample. We use a little\ntrick in order to extract the FQ order parameter from\nthe frequency shift. Since the EPR frequency is usually\ninsensitive to the AFQ order parameter, in Sec. IV, we\nfocus on a low-energy bosonic excitation that quali\fes\nas evidence of the presence of the long-range AFQ or-der. In Sec. IV A, we brie\ry explain the concept of how\nto detect the boson in ESR experiments in general. To\n\resh out the general discussion, we take an example of\nanS= 1 bilinear-biquadratic model on the square lat-\ntice (Secs. IV B and IV C) and study its ESR with the aid\nof the linear \ravor-wave theory. The linear \ravor-wave\ntheory in the fully polarized phase (Sec. IV D) and in the\nAFQ phase (Sec. IV F) shows that the boson correspond-\ning to the bound magnon-pair excitation yields a reso-\nnance peak in the ESR spectrum in addition to the EPR\none. The magnon-pair resonance is closely investigated\nin the fully polarized phase in Sec. IV E. Section IV G is\ndevoted to the investigation of additional resonances in\nthe AFQ phase, where we \fnd the magnon-pair resonance\nand another unpaired magnon resonance. Some related\ndiscussions are addressed in Sec. V. Section V B contains\napplications to the S= 1=2 frustrated ferromagnets and\ntheS= 1=2 orthogonal dimer spin system. Finally, we\nsummarize the paper in Sec. VI.\nII. FRAMEWORK\nHere we describe a generic model that we deal with\nin the paper and an important identity that we rely on\nthroughout our discussions. We consider quantum spin\nsystems described by the following Hamiltonian:\nH=HSU(2)\u0000HSz+H0; (2.1)\nwhereHSU(2) denotes interactions that respect the SU(2)\nrotation symmetry of the spin, the second term is the\nZeeman energy, Sz=P\niSz\niis thezcomponent of the\ntotal spin S=P\niSi, andH0is an anisotropic interaction\nthat breaks weakly the SU(2) symmetry of HSU(2) . For\nsimplicity, we take ~=kB=a0= 1 hereafter, where a0\nis the lattice constant.\nWe regardH0as a perturbation to the Hamiltonian\nH0=HSU(2)\u0000HSz: (2.2)\nWe emphasize that, in this paper, we consider the cases\nwhere the spin nematic phase emerges in the unperturbed\nsystem with the Hamiltonian (2.2) and the perturbation\nH0a\u000bects neither the existence of the quadrupolar order\nnor the magnitude of the quadrupolar order parameter.\nOn the other hand, when we consider ESR of quantum\nspin systems, however small H0may be, we must take\nthe anisotropic interaction H0into account.\nThe ESR absorption spectrum is given by the imag-\ninary part of the retarded Green's function of the to-\ntal spin. The ESR spectrum I(!) in the Faraday\ncon\fguration is related to GR\nS+S\u0000(!), whereS\u0006=\nSx\u0006iSyare the ladder operators of the total spin,\nGR\nO1O2(!) is the retarded Green's function GR\nO1O2(!) =\n\u0000iR1\n0dtei!th[O1(t);O2(0)]i.\nIn fact, when the applied electromagnetic wave is cir-\ncularly polarized, the ESR spectrum I(!) is given by [49]\nI(!) =!H2\nR\n8\u0002\n\u0000ImGR\nS+S\u0000(!)\u0003\n(2.3)3\nwithin the linear response theory, where HRis the am-\nplitude of the external oscillating magnetic \feld. On the\nother hand, when the applied electromagnetic wave is\ncompletely unpolarized, the ESR spectrum is given by\n(appendix A)\nI(!) =!H2\nR\n8\u0002\n\u0000ImGR\nS+S\u0000(!)\u0000ImGR\nS\u0000S+(!)\u0003\n:(2.4)\nThe result (2.4) is invariant under any rotation around\nthezaxis, the direction along which the electro-\nmagnetic wave propagates. Since Im GR\nS\u0000S+(!) =\n\u0000ImGR\nS+S\u0000(\u0000!), the ESR spectrum (2.4) is fully de-\ntermined from the retarded Green's function, GR\nS+S\u0000(!).\nHere, we note that the frequency !is positive since it\nrepresents the energy of the photon.\nIn the remainder of the paper, we deal with the ESR\nspectrum of Eq. (2.3) because inclusion of GR\nS\u0000S+(!) has\nno impact on the conclusions of this paper. The Green's\nfunctionGR\nS+S\u0000(!) satis\fes the identity [47, 50]\nGR\nS+S\u0000(!) =2hSzi\n!\u0000H\u0000h[A;S\u0000]i\n(!\u0000H)2\n+1\n(!\u0000H)2GR\nAAy(!); (2.5)\nwhere!\u0000His shorthand for !\u0000H+i0 andAis an\noperator de\fned by\nA= [H0;S+]: (2.6)\nEquation (2.5) is an exact relation that simply results\nfrom the equation of motion of S\u0006[47, 50]. Now it is\nclear that if there was no anisotropy, i.e. H0= 0, the\nESR spectrum I(!) would contain only the single EPR\npeak,\nI(!) =IEPR(!) =\u0019H2\nRH\n4hSzi\u000e(!\u0000H): (2.7)\nThe anisotropic interaction H0, however small it may be,\nis always present in the materials, giving rise to the \fnite\nlinewidth to the EPR peak (2.7). It also yields other\nresonance peaks.\nTheAoperator determines qualitative and quanti-\ntative properties of the ESR spectrum. In particular,\nEq. (2.5) shows that if there is an additional resonance\npeak well isolated from the EPR one, it must come from\nthe retarded Green's function GR\nAAy(!). In fact, many in-\nteresting resonance peaks are found in ESR experiments\nand explained on the basis of the identity (2.5) [51, 52].\nLet us split the ESR spectrum (2.3) into two parts:\nthe EPR peak IEPR(!) and the other peaks I0(!) well\nisolated from the former, if they exist,\nI(!) =IEPR(!) +I0(!): (2.8)\nIn the leading order of the perturbation, the additional\npeakI0(!) is given by\nI0(!) =!H2\nR\n8(!\u0000H)2\u0002\n\u0000ImGR\nAAy(!)\u0003\n; (2.9)where the full retarded Green's function GR\nAAy(!) in\nEq. (2.5) is replaced to the unperturbed retarded Green's\nfunctionGR\nAAy(!) of the unperturbed system (2.2). As\nfar asI0(!) is concerned, measuring ESR is equivalent\nto measuring the resonance of the complex operator A\ngiven in Eq. (2.6). We emphasize that the resonance\npeak position of I0(!) is determined fully in the unper-\nturbed system. While the intensity of I0(!) is propor-\ntional to the square of the small coupling constant of H0,\nthe frequency dependence is fully determined from the\nunperturbed Green's function.\nIII. FREQUENCY SHIFT BY THE FQ ORDER\nIn this section, we discuss e\u000bects of the FQ order on\nthe EPR absorption peak, showing that the FQ order\nparameter can be extracted experimentally from the EPR\nfrequency shift.\nA. Generic case\nWe start with a general discussion. Let us consider\na perturbative expansion of the identity (2.5) up to the\n\frst order of the perturbation,\nGR\nS+S\u0000(!)'2hSzi\n!\u0000H\u0012\n1\u0000h[A;S\u0000]i0\n2hSzi01\n!\u0000H\u0013\n;(3.1)\nwherehOi0denotes the average taken in the unperturbed\nsystem (2.2) [53]. The approximation (3.1) implies that\nthe resonance frequency of the EPR peak is shifted from\n!r=Hto\n!r'H\u0000h[A;S\u0000]i0\n2hSzi0: (3.2)\nThe formula (3.2) is valid as long as the anisotropic\ninteraction is perturbative, the paramagnetic resonance\npeak is notsplit into plural peaks by the perturbation\nH0[54], andh[A;S\u0000]i0is real. An example of the split-\nting is found in one-dimensional quantum antiferromag-\nnets with a uniform Dzyaloshinskii-Moriya (DM) interac-\ntion [47, 55]. While it is nontrivial whether h[A;S\u0000]i0is\nreal in the presence of the FQ order, we show in Sec. III B\nthat it is indeed real on the basis of a speci\fc model.\nIn this paper we denote the frequency shift by \u000e!r=\n!r\u0000H. The formula (3.2) is rephrased as\n\u000e!r=\u0000h[A;S\u0000]i0\n2hSzi0: (3.3)\nA key observation is to notice a fact that a quadratic\ninteractionH0usually makes the commutator [ A;S\u0000]\nquadratic in the formula (3.3). To see it, we take a generic\nexample,\nH0=X\np=x;y;zX\nhi;jin\u000e(n)\npSp\niSp\nj; (3.4)4\nwhereSp\nj(p=x;y;z ) denotes the p-component of spin\nSoperators on jth site,hi;jinrepresents a pair of the\nnth neighbor sites iandj. In particular,hi;ji0means\ni=j. The interaction (3.4) covers quite a wide range\nof anisotropic spin-spin interactions found in quantum\nmagnets. In the following discussions, we choose the\nmost dominant anisotropic interaction. For quantum\nspin systems with S= 1=2, the most likely anisotropic\ninteraction is the anisotropic exchange interaction on the\nnearest-neighbor bond ( n= 1). For spin systems with\nS\u00151, the most likely one is the single-ion anisotropy\n(n= 0). In principle, the anisotropic exchange interac-\ntion on the nearest-neighbor bond can be the dominant\none even for S\u00151. Because these two types of anisotropy\ngive rise to resemblant frequency shifts, it is di\u000ecult to\njudge whether the dominant anisotropic interaction lives\nin single-spin sites or in bonds connecting two spin sites.\nThe resemblance was demonstrated in the S= 1 Heisen-\nberg antiferromagnetic chain [56].\nWe assume the following inequalities\nj\u000e(n)\nzj>j\u000e(n)\nxj\u0015j\u000e(n)\nyj= 0; (3.5)\nwithout loss of generality, for the most dominant\nanisotropy. From Eq. (3.3), the anisotropic interaction\n(3.4) leads to the frequency shift\n\u000e!r=\u00002\u000e(n)\nz\u0000\u000e(n)\nx\n2hSzi0X\nhi;jinh(3Sz\niSz\nj\u0000Si\u0001Sj)i0\u0000\u000e(n)\nx\n2hSzi0X\nhi;jinhS\u0000\niS\u0000\nji0: (3.6)\nThis equation contains the operator S\u0000\niS\u0000\nj, which\ncreates or annihilates a pair of magnons. Since\nbound magnon pairs condense in the spin nematic\nphase,P\nhi;jinhS\u0000\niS\u0000\nji0is proportional to the condensed\namount of bound magnon pairs at the wave vector k=0,\nthat is, the FQ order parameter.\nThe averageh(3Sz\niSz\nj\u0000Si\u0001Sj)i0on the \frst line of\nEq. (3.6) is also nonzero simply because of the magnetic\n\feld along the zdirection. In fact, the frequency shift\nin several one-dimensional quantum magnets, where the\nlong-range spin nematic order is absent, was explained\non the basis of Eq. (3.6) without the second line [56{58].\nLet us introduce a little trick in order to get rid of the\ncontribution unrelated to the spin nematic order. Here,\nwe rotate the material around the yaxis by an angle\n\u0012. The rotation only a\u000bects the form of the anisotropic\ninteraction (3.4) as\nH0=X\nhi;jin\u0002\n(\u000e(n)\nzcos2\u0012+\u000e(n)\nxsin2\u0012)Sz\niSz\nj\n+ (\u000e(n)\nzsin2\u0012+\u000e(n)\nxcos2\u0012)Sx\niSx\nj\n\u0000(\u000e(n)\nz\u0000\u000e(n)\nx) sin\u0012cos\u0012(Sz\niSx\nj+Sx\niSz\nj)\u0003\n:(3.7)\nThe interactions in the unperturbed system H0is invari-\nant under the rotation. The rotated anisotropic interac-\ntion leads to the frequency shift,\n\u000e!r(\u0012) =(\u000e(n)\nz\u0000\u000e(n)\nx)(3 cos2\u0012\u00001) +\u000e(n)\nx\n2hSzi0X\nhi;jinh(3Sz\niSz\nj\u0000Si\u0001Sj)i0+(\u000e(n)\nz\u0000\u000e(n)\nx) sin2\u0012+\u000e(n)\nx\n2hSzi0X\nhi;jinhS\u0000\niS\u0000\nji0\n\u0000(\u000e(n)\nz\u0000\u000e(n)\nx) sin\u0012cos\u0012\n2hSzi0X\nhi;jinhf2(Sz\niS+\nj+S+\niSz\nj) + 3(Sz\niS\u0000\nj+S\u0000\niSz\nj)gi0; (3.8)\nwhere we added the argument to \u000e!ron the left hand\nside to clarify that the frequency shift is a function of\n\u0012. If the rotation does nota\u000bect the direction where the\nFQ order grows, the angular dependence comes only out\nof the coe\u000ecients of those averages. The validity of the\nassumption is to be con\frmed in the next subsection for\na speci\fc example.\nThe EPR frequency shift (3.8) consists of\nthree parts: (i) the uniaxial part coupled to\nthe averageh(3Sz\niSz\nj\u0000Si\u0001Sj)i0, (ii) the FQ or-\nder part coupled to the FQ order parameterhS\u0000\niS\u0000\nji0, and (iii) the o\u000b-diagonal part coupled to\nhf2(Sz\niS+\nj+S+\niSz\nj) + 3(Sz\niS\u0000\nj+S\u0000\niSz\nj)gi0. Note again\nthat the operator 3 Sz\niSz\nj\u0000Si\u0001Sjis not an FQ order\nparameter although it is a quadrupole operator. That\noperator has a nonzero expectation value when the z\naxis is inequivalent to the xyplane by applying the\nmagnetic \feld along the zaxis. Looking at the angular\ndependence of the frequency shift (3.8), we can separate\nthe FQ order parameter from the other terms as follows.\nLet us focus on \u000e!r(0) and\u000e!r(\u0019=2) because the sec-\nond line of Eq. (3.8) vanishes at \u0012= 0 mod\u0019=2. Those\nfrequency shifts are rewritten as5\n\u000e!r(0) +\u000e!r(\u0019=2) =\u000e(n)\nz+\u000e(n)\nx\n2hSzi0\u0012X\nhi;jinh(3Sz\niSz\nj\u0000Si\u0001Sj)i0+X\nhi;jinhS\u0000\niS\u0000\nji0\u0013\n; (3.9)\n\u000e!r(0)\u0000\u000e!r(\u0019=2) =\u000e(n)\nz\u0000\u000e(n)\nx\n2hSzi0\u0012\n3X\nhi;jinh(3Sz\niSz\nj\u0000Si\u0001Sj)i0\u0000X\nhi;jinhS\u0000\niS\u0000\nji0\u0013\n: (3.10)\nSince\u000e(n)\np(p=x;z) are known parameters from ESR\nmeasurements at high enough temperatures out of the\nFQ phase and the magnetization hSzi0is also known in-\ndependently of the ESR experiments. Thus, combining\n\u000e!r(0) and\u000e!r(\u0019=2), we can obtain the FQ order pa-\nrameterP\nhi;jinhS\u0000\niS\u0000\nji0. In the uniaxially anisotropic\ncase of\u000e(n)\nz6= 0 and\u000e(n)\nx=\u000e(n)\ny= 0, the procedure is\nsimpli\fed thanks to the following simple relation,\n1\n2\u000e!r(0) +\u000e!r(\u0019=2) =\u000e(n)\nz\n2hSzi0X\njhS\u0000\niS\u0000\nji0:(3.11)\nB. Speci\fc case\nThe important remaining tasks in this section are to\ncon\frm that the frequency shift (3.3) is real and that the\naverages in Eq. (3.8) is invariant under the rotation. To\ndo so, we take a speci\fc example of an S= 1 bilinear-\nbiquadratic model with the single-ion anisotropy,\nH=X\nn=1;2X\nhi;jin\u0002\nJnSi\u0001Sj+Kn(Si\u0001Sj)2\u0003\n\u0000H(Szcos\u0012+Sxsin\u0012)\n+DX\nj(Sz\nj)2+EX\nj\b\n(Sx\nj)2\u0000(Sy\nj)2\t\n;(3.12)\non the square lattice. Rotating the system about y-axis\nby angle\u0012, we can rede\fne the Hamiltonian as\nH=X\nn=1;2X\nhi;jin\u0002\nJnSi\u0001Sj+Kn(Si\u0001Sj)2\u0003\n\u0000HSz\n+X\nj\u0002\n(Dcos2\u0012+Esin2\u0012)(Sz\nj)2\n+ (Dsin2\u0012+Ecos2\u0012)(Sx\nj)2\u0000E(Sy\nj)2\n\u0000(D\u0000E) sin\u0012cos\u0012(Sz\njSx\nj+Sx\njSz\nj)\u0003\n:(3.13)\nThe latter representation is easier to handle. Here we\nassume, without loss of generality, that both DandE\nhave the same sign. Note that the form of the anisotropic\ninteraction of Eq. (3.13) is a special case of Eq. (3.7). In\nthe language of \u000e(n)\npin Eq. (3.7), the parameters Dand\nEcorresponds to them as\n\u000e(0)\nz=D+E; \u000e(0)\nx= 2E: (3.14)Note that we also imposed the condition (3.5). We as-\nsume thatJn>0 andKn<0 forn= 1;2 and that the\nsingle-ion anisotropy can be seen as a perturbation.\nIt was shown at the mean-\feld level that the ground\nstate of the unperturbed model is in the FQ phase when\nbothjK1j=J1andjK2j=J2are large enough [38]. The\nmean-\feld FQ ground state j 0iis represented as a prod-\nuct state,\nj 0i=Y\njj\u001e0(\u0012H;')ij; (3.15)\nof the local state j\u001e0(\u0012H;')ij,\nj\u001e0(\u0012H;')ij=i(ei'cos\u0012Hj1ij\u0000e\u0000i'sin\u0012Hj\u00001ij);\n(3.16)\nwherejmijis the eigenstate of Sz\njwith the eigenvalue m.\nThe angles \u0012Hand', which are real, are determined so\nthat the ground-state energy is minimized. The ground\nstate has the FQ order\nX\njh 0j(S\u0000\nj)2j 0i=\u0000Nei2'sin 2\u0012H; (3.17)\nwhich is in general complex. Here, Nis the number of\nspins. In the following, we show that ei2'is real in the\npresence of the single-ion anisotropy as long as it is per-\nturbative.\nTo discuss the ground state energy of the mean-\feld\nFQ state (3.15), we introduce the quadrupole operators\nQj=0\nBBBBB@Qx2\u0000y2\nj\nQ3z2\u0000r2\nj\nQxy\nj\nQyz\nj\nQzx\nj1\nCCCCCA=0\nBBBB@(Sx\nj)2\u0000(Sy\nj)2\n[2(Sz\nj)2\u0000(Sx\nj)2\u0000(Sy\nj)2]=p\n3\nSx\njSy\nj+Sy\njSx\nj\nSy\njSz\nj+Sz\njSy\nj\nSz\njSx\nj+Sx\njSz\nj1\nCCCCA\n(3.18)\nand rewrite the Hamiltonian (3.13) as\nH=X\nhi;jin\u0014Jn\n2(Si\u0001Sj+Qi\u0001Qj)\n+Jn\u0000Kn\n2(Si\u0001Sj\u0000Qi\u0001Qj)\u0015\n\u0000HSz\n+2ND\n3+D(3 cos2\u0012\u00001) +Esin2\u0012\n2p\n3Q3z2\u0000r26\n+Dsin2\u0012+E(cos2\u0012+ 1)\n2Qx2\u0000y2\n\u0000(D\u0000E) sin\u0012cos\u0012Qzx(3.19)with Q\u0011P\njQj. Writing the local state as\nj\u001e(\u0012H;')ij=e1j1ij+e0j0ij+e\u00001j\u00001ij, we can rep-\nresent the ground-state energy EFQ=h 0jHj 0ias [43]\nEFQ\nN= 2(J1+J2)je\u0001\u0016ej2\u00002(J1+J2\u0000K1\u0000K2)j2e1\u0016e\u00001\u0000(e0)2j2\u0000H(je1j2\u0000je\u00001j2)\n+2D\n3+D(3 cos2\u0012\u00001) +Esin2\u0012\n6(je1j2+je\u00001j2\u00002je0j2)\n+Dsin2\u0012+E(cos2\u0012+ 1)\n2p\n2(e\u00001\u0016e1+e1\u0016e\u00001)\u0000(D\u0000E) sin\u0012cos\u0012(e1\u0016e0+e0\u0016e1\u0000e0\u0016e\u00001\u0000e\u00001\u0016e0);(3.20)\nwhere \u0016eais the complex conjugate of ea(a= 1;0;\u00001).\nPlugginge1=iei'cos\u0012H,e0= 0, and e\u00001=\n\u0000ie\u0000i'sin\u0012Hinto Eq. (3.20), we obtain\nEFQ\nN= 2(J1+J2)\u00002(J1+J2\u0000K1\u0000K2) sin22\u0012H\n\u0000Hcos 2\u0012H+3D(cos2\u0012+ 1) +Esin2\u0012\n6\n+Dsin2\u0012+E(cos2\u0012+ 1)p\n2cos 2'sin 2\u0012H:\n(3.21)\nLet us determine \u0012Hand'in the spirit of the perturba-\ntion theory. First, when D=E= 0, the ground-state\nenergy becomes\nE0\nFQ\nN= 2(K1+K2)\u0000H2\n2Hsat\n+ 2(J1+J2\u0000K1\u0000K2)\u0012\ncos 2\u0012H\u0000H\nHsat\u00132\n(3.22)\nwith the saturation \feld Hsat= 4(J1+J2\u0000K1\u0000K2).\nSinceKn<0< Jnfor alln= 1;2, the angle \u0012His\ndetermined to be\n\u00120\nH=1\n2cos\u00001\u0012H\nHsat\u0013\n(3.23)\nin 0\u0014\u00120\nH\u0014\u0019=2 when 0\u0014H <H satand\u00120\nH= 0 when\nHsat<H. We restrict ourselves to the former case where\nthe system is in the FQ phase. The ground state j 0ihas\nthe FQ order,\nX\njh 0j(S\u0000\nj)2j 0i=\u0000Nei2's\n1\u0000\u0012H\nHsat\u00132\n;(3.24)\nwhere the angle 'determines the direction of quadrupo-\nlar directors. Next, we treat the single-ion anisotropy\nperturbatively. Weak anisotropies DandEdo not mod-\nify the solution \u0012H=\u00120\nHat the lowest order of the pertur-\nbation. The anisotropy, however small it may be, serve asa symmetry breaking \feld and determines '. From the\nenergy in Eq. (3.21) the angle 'is chosen, for arbitrary\nangle\u0012, as cos 2'= 1 when both DandEare negative,\nand cos 2'=\u00001 when both are positive. In any case,\nsince sin 2'= 0, we thus \fnd ei2'= cos 2'2Rand that\nthe FQ order parameterP\njh(S\u0000\nj)2iis real, satisfying the\nrelation\nX\njh(S\u0000\nj)2i0=X\njhQx2\u0000y2\nji0; (3.25)\nfor the model (3.13) at low enough temperatures. The\nFQ order parameter is thus given by\nhQx2\u0000y2i0\u0011X\njhQx2\u0000y2\nji0\n=\u0000Ncos 2's\n1\u0000\u0012H\nHsat\u00132\n: (3.26)\nAt the same time, we can also conclude that the ob-\ntained solutions of 'and\u0012Hare independent of the\nangle\u0012of the rotation. That is, the FQ order pa-\nrameter is invariant under the rotation around the y\naxis. Also, the averagesP\njh(3(Sz\nj)2\u00002)i0=NandP\njh2(Sz\njS+\nj+S+\njSz\nj) + 3(Sz\njS\u0000\nj+S\u0000\njSz\nj)i0= 0 turn out\nto be independent of \u0012. In particular, the latter average\nvanishes because it involves the creation and the annihi-\nlation of the gapped unpaired magnon. The assumption\nmade in deriving Eq. (3.8) in the previous subsection is\nthus justi\fed.\nFinally, the frequency shift (3.8) of the model (3.13) at\nzero temperature becomes\n\u000e!r'(D\u0000E)(3 cos2\u0012\u00001) + 2E\n2hSzi0=N\n\u0000(D\u0000E) sin2\u0012+ 2E\n2hSzi0hQx2\u0000y2i0:(3.27)\nWe emphasize that the value of the FQ order parameter\nis determined independent of perturbative anisotropies\nDandE. Those anisotropies merely \fx the angle ',7\nwhich was spontaneously determined in the unperturbed\nsystem.\nAlthough we do not show that the frequency shift is\nreal for generic cases, it is reasonable to expect that\nthe result holds true generally as long as the mean\n\feld approximation is applicable and the perturbative\nanisotropic interaction is given by either the single-ion\nanisotropy or the anisotropic exchange interaction (3.4).\nC. Experimental applications\nFor the analysis of experimental data, we comment\non the comparison between the frequency shift and the\nlinewidth of the EPR peak. Should the linewidth be\nlarger than the magnitude of the EPR frequency shift,\nthe frequency shift would be undetectable. However, this\nis not the case as long as the anisotropic interaction is\nsmall enough compared to the isotropic ones. Accord-\ning to the generic perturbation theory (3.3), the EPR\nfrequency shift is of the \frst order of the perturbative\nanisotropic interaction, whereas the linewidth is of the\nsecond order because it is proportional to Im GR\nAAy(!=\nH) [50, 59]. Therefore, given the long-range FQ order\nis well developed, the linewidth is small enough not to\nmask the frequency shift (3.3).\nWe conclude the section, referring to chromium spinel\noxides. Chromium spinels ACr 2O4(A = Zn;Cd;Hg) are\nconsidered as an S= 3=2 pyrochlore Heisenberg antifer-\nromagnet with biquadratic interactions [60]. Recently it\nwas pointed out that those chromium spinels can have\nthe FQ phase next to the fully polarized phase [35]. In-\ndeed, high-\feld measurements discovered the presence of\na classically unexpected phase just below the fully polar-\nized phase [61{63]. We emphasize that the result (3.8)\nis applicable to those interesting compounds. In fact, in\nderiving Eq. (3.8), we only speci\fed the anisotropic inter-\nactionH0and speci\fed neither the spin quantum number\nnor the form of the SU(2)-symmetric interaction in the\nHamiltonian (2.1).\nIV. EMERGENCE OF ANOTHER PEAK BY\nTHE AFQ ORDER: MAGNON-PAIR\nRESONANCE\nThe frequency shift (3.8) is insensitive to the AFQ or-\nder since the AFQ order parameter has alternating sign\ndepending on the position. In this section, we develop\nan alternative way for detecting the AFQ order, studying\nthe additional absorption I0(!) in ESR spectrum (2.8).\nHere, the point is that the Aoperator creates a magnon\npair excitation.A. Concept\nLet us explain the concept of our method, taking the\nfollowing example of the anisotropic interaction:\nH0=\u000e0X\nhi;ji1(Sx\niSz\nj+Sz\niSx\nj): (4.1)\nHere, we do not specify the precise form of HSU(2) . We\njust assume that the unperturbed system (2.2) has the\nAFQ ground state. The Aoperator for Eq. (4.1) is given\nby\nA=\u000e0X\nhi;ji1(S+\niS+\nj\u00003Sz\niSz\nj+Si\u0001Sj): (4.2)\nThe \frst term of Eq. (4.2) creates or annihilates the\nmagnon pair formed on the nearest-neighbor bond. This\nshows that dynamics of the bound magnon pair is di-\nrectly observable in ESR experiments throungh the rela-\ntion (2.9).\nIn the spin nematic phase with the AFQ order, the\nbound magnon pair can acquire an excitation gap, say \u0001,\natk=0, though it becomes gapless at the wave vector\nof the AFQ ordered ground state. If so, the ESR spec-\ntrum in the spin nematic ordered phase contains a sharp\nresonance peak whose frequency corresponds to the exci-\ntation gap of the bound magnon pair at k=0. As shown\nin Eq. (2.9), the ESR spectrum (2.3) contains a contribu-\ntion of theAoperator through the imaginary part of the\nretarded Green's function of A. SinceAinvolves the cre-\nation and annihilation operators of the bound magnon\npair, the ESR spectrum will have a resonance peak at\n!= \u0001, i.e.I0(!)/A MPR\u000e(!\u0000\u0001). We call this peak the\nmagnon-pair resonance peak. In Secs. IV D and IV F, we\ncon\frm that the ESR spectrum indeed yields the sharp\nmagnon-pair resonance by taking an example of an S= 1\nspin model on the square lattice.\nThe interaction (4.1) results from the rotation of a spin\nanisotropy, e.g. \u000e(z)P\nhi;ji1Sz\niSz\nj, around the yaxis as\nwe did in Eq. (3.7). For a general rotation angle \u0012, the\nresultant magnon pairing operator of the rotated inter-\naction (3.7) shows the angular dependence of sin \u0012cos\u0012.\nThis gives a characteristic angle dependence in the in-\ntensityAMPR of the magnon-pair resonance peak. [See\nEq. (4.88).] This angular dependence was a key to char-\nacterize qualitatively the quadrupolar liquid state by us-\ning ESR [47].\nB. Model\nTo \resh out the general discussion described in\nSec. IV A, we take an example of an S= 1 spin model\non the square lattice given by the following unperturbed\nHamiltonian\nH0=X\nhi;ji1J11\u0002\nSi\u0001Sj+ (Si\u0001Sj)2\u00038\n\u0000X\nhi;ji2\b\nJ12\u0002\nSi\u0001Sj+ (Si\u0001Sj)2\u0003\n+J22(Si\u0001Sj)2\t\n\u0000HSz(4.3)\nand the perturbative single-ion anisotropy\nH0=X\ni\u0002\n(Dcos2\u0012+Esin2\u0012)(Sz\ni)2\n+ (Dsin2\u0012+Ecos2\u0012)(Sx\ni)2\u0000E(Sy\ni)2\n\u0000(D\u0000E) sin\u0012cos\u0012(Sz\niSx\ni+Sx\niSz\ni)\u0003\n: (4.4)\nThis form of the anisotropy is the same as in Eq. (3.13),\nwhich is obtained by rotating the common form D(Sz\ni)2+\nEf(Sx\ni)2\u0000(Sy\ni)2gabouty-axis by angle \u0012. We assume\nthat the couplings J11,J12, andJ22are all positive.\nThe bilinear-biquadratic interaction in Eq. (4.3) has the\nSU(3) symmetry when J22= 0 and its low-energy be-\nhavior was closely investigated in Ref. [43]. It is easy to\ncon\frm that the unperturbed Hamiltonian (4.3) is writ-\nten as\nH0=J11\n2X\nhi;ji1(Qi\u0001Qj+Si\u0001Sj)\n\u0000X\nhi;ji2\u0014J12\n2(Qi\u0001Qj+Si\u0001Sj)\n+J22\n2(Qi\u0001Qj\u0000Si\u0001Sj)\u0015\n\u0000HSz; (4.5)\nby using the quadrupole operators Qjof Eq. (3.18).\nWe employ the model (4.3) for the following reasons.\nFirst, the model (4.3) does not su\u000ber from a known tech-\nnical problem of the linear \ravor-wave theory, that is,\nviolation [33, 64] of the frequency sum rule [65] of the\ndynamical structure factorR1\n0d!!S\u000b\u000b(k=0;!) = 0,\nwhich holds true for the unperturbed Hamiltonian (2.2)\nat zero magnetic \feld H= 0. Here, S\u000b\u000b(k=0;!) =\n\u0000ImGR\nS\u000bS\u000b(!)=N. The linear \ravor-wave theory does\nnot always satisfy the exact sum rule but can be recovered\nby including three- and four-particle interactions [64]. It\nis the great advantage of the model (4.3) that we can\nomit such a complicated procedure. This sum rule at\nzero magnetic \feld is related to the exact result of the\nEPR frequency !=Hin the ESR spectrum (2.7). We\nwill brie\ry comment about an example of the violation\nof the exact result (2.7) in the linear \ravor-wave theory\nin the last part of Sec. IV F. Second, the model (4.3) ex-\nhibits the AFQ phase in quite a wide \feld range up to\nthe saturation \feld [ HAFQ\nc in Eq. (4.11)]. We will come\nback to those points in Secs. IV C and IV G. Last but not\nleast, theS= 1 model is closely related to an S= 1=2\nfrustrated ferromagnetic model on a square lattice [43].\nWe emphasize that the results about the AFQ phase ob-\ntained in the present paper also hold true for the S= 1=2\nmodel. Here, the single-ion anisotropy (4.4) in the S= 1\nmodel is translated into anisotropic exchange interactions\nin theS= 1=2 model. See Sec. V B for further discus-\nsions.Following the general discussion in Sec. II, we study the\nadditional peak I0(!) given by Eq. (2.9). Here we only\nneed to derive the unperturbed retarded Green's function\nGR\nAAy(!) of the operatorA= [H0;S+]. For that purpose,\nwe use the linear \ravor-wave theory [30, 43, 66] for the\nunperturbed system (4.5) at zero temperature.\nC. Mean-\feld ground state\nAs well as the spin-wave theory, the \ravor-wave theory\nis developed by taking into account quantum \ructuations\naround the ordered state. We need to start with de-\nriving the AFQ ground state of the bilinear-biquadratic\nmodel (4.5) in a site-decoupled semi-classical approxima-\ntion. Let us denote its mean-\feld ground state by j 0i.\nUnder a strong enough magnetic \feld, j 0iis in the fully\npolarized phase and exactly given by\nj 0i=Y\njij1ij: (4.6)\nAs the magnetic \feld is decreased, the ground state of the\nmodel (4.5) enters into a partially polarized phase. The\nmean-\feld ground state of the partially polarized phase\nhas AFQ order, which is given in the form\nj 0i=Y\njj\u001e0(eikM\u0001rj\u0012H;')ij(4.7)\nwith\nj\u001e0(\u0012H;')ij=i(ei'cos\u0012Hj1ij\u0000e\u0000i'sin\u0012Hj\u00001ij)\n(4.8)\nfor 0\u0014\u0012H\u0014\u0019=2 and 0\u0014' < \u0019 . Here kM= (\u0019;\u0019) is\nthe wave vector where the AFQ order grows, rjspeci\fes\nthe location of the spin Sj, andeikM\u0001rj=\u00061 gives the\nstaggered sign depending on the sublattice. The factor\nsin\u0012Hrepresents the fraction of the condensed bound\nmagnon pair. The state j\u001e0(\u0012H;')ijis also given by an\nSU(3) rotation of the polarized state,\nj\u001e0(\u0012H;')ij=iexp(i'Sz\nj) exp(i\u0012HQxy\nj)j1ij:(4.9)\nAs we did in Sec. III B, we determine \u0012Hfor the unper-\nturbed system and then study 'dependence perturba-\ntively.\nFor the unperturbed Hamiltonian H0, the mean-\feld\nground-state energy of the AFQ state is given by [43]\nE0\nAFQ\nN=\u00002(J12+J22) + 2(J11+J22) cos22\u0012H\n\u0000Hcos 2\u0012H; (4.10)\nwhereNis the number of sites. This energy is minimized\nat\u0012H= 0 forH\u0015HAFQ\nc, whereHAFQ\nc denotes the\nsaturation \feld\nHAFQ\nc = 4(J11+J22); (4.11)9\nand at\n\u0012H=1\n2cos\u00001\u0012H\nHAFQ\nc\u0013\n(4.12)\nforH <HAFQ\nc.\nIn the AFQ phase, the saturation \feld HAFQ\nc is the\ncritical \feld where bound magnon pairs start condensing\nwhen the \feld is decreased. Another mean-\feld solution\nis an antiferromagnetically ordered state given by a stag-\ngered SU(2) rotation of the polarized state\nj\u001e0(eikM\u0001rj\u0012H;')ij=iexp(i'Sz\nj) exp(ieikM\u0001rj\u0012HSy\nj)j1ij:\n(4.13)\nIn this solution, the saturation \feld is\nHAFM\nc = 2(6J11+J22); (4.14)\nwhere the single magnon closes the energy gap. When\nHis decreased, if HAFQ\nc> HAFM\nc, bound magnon\npairs condense below the saturation \feld before the un-\npaired magnon does. The comparison between HAFQ\nc\nandHAFM\nc shows that the AFQ phase is realized for\n4J11<J22 (4.15)\nat least near the saturation. In Sec. IV D, we see that in-\nclusion of the quantum \ructuation relaxes the condition\n(4.15) to\nJ11<J22: (4.16)\nThe anisotropic perturbation H0[Eq. (4.4)] can make\nthe ground-state energy depend on '. In the mean-\n\feld calculation, the \frst-order perturbation changes\nE0\nAFQ=Nto\nEAFQ\nN=\u00002(J12+J22) + 2(J11+J22) cos22\u0012H\n\u0000Hcos 2\u0012H+3D(cos2\u0012+ 1) +Esin2\u0012\n6:(4.17)\nIn contrast to the case of the FQ state (3.21), the mean-\n\feld energy (4.17) of the AFQ state does not depend on\nthe angle'despite the equivalent form of the anisotropic\ninteraction in these two systems. This 'independence\nshows that the angle 'is undetermined in the AFQ state\nat the mean-\feld level. To determine ', we need to go\nbeyond the mean-\feld level. We leave it undetermined\nsince'is insigni\fcant for our calculations below.D. Linear \ravor-wave theory in the fully polarized\nphase\nIn this subsection we discuss the \ravor-wave theory\n[64] in the fully polarized phase above the saturation \feld\nHAFQ\nc of the AFQ phase. Later in Sec. IV F we discuss\nthe \ravor-wave theory in the AFQ phase below HAFQ\nc.\nThe \ravor-wave theory is formulated in terms of\nSchwinger bosons. Let us denote the creation and annihi-\nlation operators of the Schwinger bosons at the jth site by\nby\nj;mandbj;m, respectively. The \ravor index m= 1;0;\u00001\ncorresponds to the eigenvalue of Sz\nj. Using these bosons,\nan operator Ojat thejth site is written as\nOj=X\nm;m0=1;0;\u00001by\nj;m~Omm0\njbj;m0 (4.18)\nwhere ~Ojdenotes a 3\u00023 matrix whose element is given by\n~Omm0\nj=jhmjOjjm0ij. The Schwinger bosons are subject\nto the constraint\nX\nm=1;0;\u00001by\nj;mbj;m= 1 (4.19)\nfor every site j.\nThe 3\u00023 matrix ~Ojis easily found by writing j1i=\n(1 0 0)T,j0i= (0 1 0)T, andj\u00001i= (0 0 1)T. For exam-\nple, the matrices for the spin operators S\u000b\nj(\u000b=x;y;z )\nare given by\n~Sx\nj=1p\n20\n@0 1 0\n1 0 1\n0 1 01\nA;~Sy\nj=ip\n20\n@0\u00001 0\n1 0\u00001\n0 1 01\nA;\n~Sz\nj=0\n@1 0 0\n0 0 0\n0 0\u000011\nA: (4.20)\nThe matrix representation of Qiis easily obtained by\ncombining the matrices of Si(4.20).\nThe fully polarized phase can be seen as a condensation\nphase of the bj;1bosons. We can replace the operators\nbj;1andby\nj;1by ac-number, the fraction of the condensed\nboson,\nbj;1=by\nj;1=q\n1\u0000ay\njaj\u0000by\njbj: (4.21)\nHere, we rewrote boson operators as\nbj;0=bj; bj;\u00001=aj; (4.22)\nto simplify the notation. We thus end up with the\nSchwinger boson representation of the spin operator,\nSx\nj=1p\n2\u0012q\n1\u0000ay\njaj\u0000by\njbjbj+by\njq\n1\u0000ay\njaj\u0000by\njbj+ay\njbj+by\njaj\u0013\n; (4.23)10\nSy\nj=\u0000ip\n2\u0012q\n1\u0000ay\njaj\u0000by\njbjbj\u0000by\njq\n1\u0000ay\njaj\u0000by\njbj+by\njaj\u0000ay\njbj\u0013\n; (4.24)\nSz\nj= 1\u00002ay\njaj\u0000by\njbj: (4.25)\nAs it is expected, the ladder operator S\u0000\nj=Sx\nj\u0000iSy\njinvolves the creation operator by\nj. Likewise, ( S\u0000\nj)2=Qx2\u0000y2\nj\u0000iQxy\nj\ninvolves the creation operator ay\nj. In fact, the Schwinger boson representation of Qjis\nQx2\u0000y2\nj =q\n1\u0000ay\njay\nj\u0000by\njbjaj+ay\njq\n1\u0000ay\njaj\u0000by\njbj; (4.26)\nQ3z2\u0000r2\nj =1p\n3(1\u00003by\njbj); (4.27)\nQxy\nj=i\u0012\nay\njq\n1\u0000ay\njaj\u0000by\njbj\u0000q\n1\u0000ay\njaj\u0000by\njbjaj\u0013\n; (4.28)\nQyz\nj=\u0000ip\n2\u0012q\n1\u0000ay\njaj\u0000by\njbjbj\u0000by\njq\n1\u0000ay\njaj\u0000by\njbj+ay\njbj\u0000by\njaj\u0013\n; (4.29)\nQzx\nj=\u00001p\n2\u0014\n\u0000\u0012q\n1\u0000ay\njaj\u0000by\njbjbj+by\njq\n1\u0000ay\njaj\u0000by\njbj\u0013\n+ay\njbj+by\njaj\u0015\n: (4.30)\nUp to the linear order of creation and annihilation op-\nerators, only Qx2\u0000y2\nj'ay\nj+ajandQxy\nj'i(ay\nj\u0000aj)\ncan create the \\ a\" boson that corresponds to the bound\nmagnon pair.\nUp to the quadratic order of the creation and annihila-\ntion operators, the unperturbed Hamiltonian (4.5) turns\ne\u000bectively into\nH0=X\nk\u0002\n!a(k)ay\nkak+!b(k)by\nkbk\u0003\n; (4.31)\nwhereakandbkare the Fourier transforms of ajandbj,\nrespectively, and !a;b(k) are given by\n!a(k) =\u00004\u0002\nJ11(1\u0000\r(1)\nk)\u0000(J12+J22)(1\u0000\r(2)\nk)\u0003\n\u00008J22+ 2H; (4.32)\n!b(k) =\u00004J11(1\u0000\r(1)\nk) + 4J12(1\u0000\r(2)\nk) +H;(4.33)\nwith\n\r(1)\nk=1\n2(coskx+ cosky); (4.34)\n\r(2)\nk= coskxcosky: (4.35)\nIn the parameter range\nJ22>J11; (4.36)the single a-boson state at k=kMhas the lowest\neigenenergy when the magnetic \feld is close to the satu-\nration \feld, while the aboson has the larger excitation\nenergy than the bboson at extremely strong \felds.\nTwo kinds of bosons have the following excitation gaps:\n!a(kM) = 2(H\u0000HAFQ\nc); (4.37)\n!b(kM) = 4(J22\u0000J11) +H\u0000HAFQ\nc: (4.38)\nThus, the bound magnon pair ( aboson) condenses at\nH=HAFQ\ncwhile the unpaired magnon ( bboson) remains\ngapped when the condition (4.36) is satis\fed. In contrast,\nthey both have excitation gaps at k=0, which we denote\nby \u0001aand \u0001b,\n\u0001a\u0011!a(0) = 8J11+ 2(H\u0000HAFQ\nc); (4.39)\n\u0001b\u0011!b(0) =H: (4.40)\nWithin the framework of the linear \ravor-wave theory,\nthe EPR (2.7) of the unperturbed system (4.5) at tem-\nperaturesT\u001cHis understood as the excitation of the\nbboson from the ground state.\nE. Magnon-pair resonance in the fully polarized\nphase\nHere we study the additional ESR spectrum I0(!),\ngiven by Eq. (2.9), in the fully polarized phase. Using the\nlinear \ravor-wave theory (4.31), we evaluate the retarded\nGreen's function GR\nAAy(!). TheAoperator determined\nfrom the rotated single-ion anisotropy (4.4) is\nA=X\nj\u0002\n(Dcos2\u0012+Esin2\u0012)(Qzx\nj+iQyz\nj)\u0000(Dsin2\u0012+Ecos2\u0012)Qzx\nj+iEQyz\nj11\n\u0000(D\u0000E) sin\u0012cos\u0012(Qx2\u0000y2\nj +iQxy\nj\u0000p\n3Q3z2\u0000r2\nj )\u0003\n: (4.41)\nUp to the linear order of the creation and the annihilation operators, it is approximated as\nAp\nN'p\n2(Dcos2\u0012+Esin2\u0012)by\nk=0\u00001p\n2(Dsin2\u0012+Ecos2\u0012)(bk=0+by\nk=0) +Ep\n2(bk=0\u0000by\nk=0)\n\u00002(D\u0000E) sin\u0012cos\u0012ak=0: (4.42)\nAll the terms in the \frst line of Eq. (4.42) yield the\nEPR peak. The term in the second line, containing the\naboson operator, yields the delta-function magnon-pair\nresonance peak \u000e(!\u0000\u0001a) at!= \u0001a. According to\nEq. (4.39), the slope of the resonance frequency != \u0001a\nas a function of His double of that of the EPR one (4.40)\nbecause the \\ a\" boson creates the magnon pair and the\n\\b\" boson creates the single unpaired magnon.\nEquation (4.42) also indicates that the intensity of the\nmagnon-pair resonance peak shows the angular depen-\ndence of sin2\u0012cos2\u0012:\nI0(!)'N(D\u0000E)2H2\nR\n2\u0001a\n(\u0001a\u0000H)2sin2\u0012cos2\u0012\u000e(!\u0000\u0001a):\n(4.43)\nF. Linear \ravor-wave theory in the AFQ phase\nWe move on to the discussion of the linear \ravor-wave\ntheory in the AFQ phase [64]. Since the angle 'of the\nAFQ directors is not pinned by the anisotropy in the\nmean-\fled approximation, we consider the AFQ state for\nthe general '. We note that this degeneracy is not lifted\neven by the \frst order perturbation of the anisotropy\nin the linear \ravor-wave approximation as shown in Ap-\npendix B. We leave it as an open question to determine '\nbecause the determination of 'has little impact on our\nconclusions in this paper, as shown in Sec IV G.\nIn the fully polarized phase, we took into account low-\nenergy excitations from the fully polarized state by re-\nplacing a local base j1iwith eitherj0iorj\u00001i. In the\nAFQ phase, the mean-\feld ground state [Eq. (4.7)] isobtained from the fully polarized state by performing an\nalternate SU(3) rotation (4.9),\nj 0i=Y\nji~R(eikM\u0001rj\u0012H;')j1ij; (4.44)\nwhere the matrix ~R(eikM\u0001rj\u0012H;') is expressed as\n~R(eikM\u0001rj\u0012H;') = exp(i'~Sz\nj) exp(ieikM\u0001rj\u0012H~Qxy\nj)\n(4.45)\nwith\nexp(i'~Sz\nj) =0\n@ei'0 0\n0 1 0\n0 0e\u0000i'1\nA; (4.46)\nexp(ieikM\u0001rj\u0012H~Qxy\nj) =0\n@cos\u0012H 0eikM\u0001rjsin\u0012H\n0 1 0\n\u0000eikM\u0001rjsin\u0012H0 cos\u0012H1\nA:\n(4.47)\nIn this representation, excitations above the AFQ state\nare formally described by local replacements of j1ito\neitherj0iorj\u00001i, similar to the case of the fully polarized\nphase. As well as the ground state (4.44), the Schwinger\nboson representation of an operator Ojis given by the\nSU(3) rotation of Eq. (4.18),\nOj=X\nm;m0by\ni;m[~Ry(eikM\u0001rj\u0012H)~Oj~R(eikM\u0001rj\u0012H)]mm0bi;m0:\n(4.48)\nFor'= 0, the spin operator Sjand the quadrupole\noperator Qjin the AFQ phase are related to the\nSchwinger boson representation in the FP phase, given\nin Eqs. (4.23){(4.30), as follows:\n0\n@Sx\nj\nQzx\nj1\nA=1p\n20\n@cos\u0012H\u0000eikM\u0001rjsin\u0012H\neikM\u0001rjsin\u0012H cos\u0012H1\nA0\n@q\n1\u0000ay\njaj\u0000by\njbjbj+by\njq\n1\u0000ay\njaj\u0000by\njbj+by\njaj+ay\njbjq\n1\u0000ay\njaj\u0000by\njbjbj+by\njq\n1\u0000ay\njaj\u0000by\njbj\u0000by\njaj\u0000ay\njbj1\nA;\n(4.49)\n0\n@Sy\nj\nQyz\nj1\nA=1p\n2i0\n@cos\u0012HeikM\u0001rjsin\u0012H\n\u0000eikM\u0001rjsin\u0012H cos\u0012H1\nA0\n@q\n1\u0000ay\njaj\u0000by\njbjbj\u0000by\njq\n1\u0000ay\njaj\u0000by\njbj+by\njaj\u0000ay\njbjq\n1\u0000ay\njaj\u0000by\njbjbj\u0000by\njq\n1\u0000ay\njaj\u0000by\njbj\u0000by\njaj+ay\njbj1\nA;\n(4.50)12\n0\n@Sz\nj\nQx2\u0000y2\nj1\nA=0\n@cos 2\u0012HeikM\u0001rjsin 2\u0012H\n\u0000eikM\u0001rjsin 2\u0012H cos 2\u0012H1\nA0\n@1\u00002ay\njaj\u0000by\njbjq\n1\u0000ay\njaj\u0000by\njbjaj+ay\njq\n1\u0000ay\njaj\u0000by\njbj1\nA; (4.51)\n0\n@Qxy\nj\nQ3z2\u0000r2\nj1\nA=0\n@\u0000iq\n1\u0000ay\njaj\u0000by\njbjaj+iay\njq\n1\u0000ay\njaj\u0000by\njbj\n1p\n3(1\u00003by\njbj)1\nA: (4.52)\nFor'6= 0, they are rotated as\n \nSx\nj(')\nSy\nj(')!\n= \ncos'sin'\n\u0000sin'cos'! \nSx\nj(0)\nSy\nj(0)!\n; (4.53)\n \nSz\nj(')\nQ3z2\u0000r2\nj (')!\n= \nSz\nj(0)\nQ3z2\u0000r2\nj (0)!\n; (4.54)\n \nQx2\u0000y2\nj (')\nQxy\nj(')!\n= \ncos 2'sin 2'\n\u0000sin 2'cos 2'! \nQx2\u0000y2\nj (0)\nQxy\nj(0)!\n;\n(4.55)\n \nQzx\nj(')\nQyz\nj(')!\n= \ncos'sin'\n\u0000sin'cos'! \nQzx\nj(0)\nQyz\nj(0)!\n:(4.56)\nUp to the quadratic terms, the unperturbed Hamilto-\nnian (4.5) is split into two parts,\nH0'Ha\n0+Hb\n0; (4.57)\nwhere\nHa\n0=X\nk\u0014\nAkay\nkak+Bk\n2(ay\nkay\n\u0000k+aka\u0000k)\u0015\n;(4.58)\nHb\n0=X\nk\u0014\nCkby\nkbk+Dk\n4(by\nk+kMby\n\u0000k+bk+kMb\u0000k)\u0015\n:\n(4.59)\nThe parameters Ak,Bk,Ck, andDkare given by\nAk= 4J11sin22\u0012H\u00004J11(1\u0000\r(1)\nk) cos22\u0012H\n+ 4J12(1\u0000\r(2)\nk) + 4J22sin22\u0012H\n\u00004J22(1 +\r(2)\nk) cos22\u0012H+ 2Hcos 2\u0012H;(4.60)\nBk=\u00004J11\r(1)\nksin22\u0012H+ 4J22\r(2)\nksin22\u0012H;(4.61)\nCk=\u00004J11cos22\u0012H+ 4J11\r(1)\nkcos 2\u0012H\n+ 4J12(1\u0000\r(2)\nk) + 4J22sin22\u0012H\n+Hcos 2\u0012H; (4.62)\nDk=\u00004J22\r(2)\nksin 2\u0012H: (4.63)\nTo diagonalize the Hamiltonians (4.58) and (4.59), we\nperform the following Bogoliubov transformations,\n\u0012ak\nay\n\u0000k\u0013\n=\u0012\ncosh \u0002a\nk\u0000sinh \u0002a\nk\n\u0000sinh \u0002a\nkcosh \u0002a\nk\u0013\u0012\u000bk\n\u000by\n\u0000k\u0013\n;(4.64)\u0012bk+kM\nby\n\u0000k\u0013\n=\u0012\ncosh \u0002b\nk\u0000sinh \u0002b\nk\n\u0000sinh \u0002b\nkcosh \u0002b\nk\u0013\u0012\fk+kM\n\fy\n\u0000k\u0013\n:\n(4.65)\nThe parameters \u0002a\nkand \u0002b\nkare determined in order to\neliminate the o\u000b-diagonal terms:\n\u0002a\nk=1\n2tanh\u00001\u0012Bk\nAk\u0013\n; (4.66)\n\u0002b\nk=1\n2tanh\u00001\u00122Dk\nCk+Ck+kM\u0013\n: (4.67)\nThe Bogoliubov transformations diagonalize the Hamil-\ntonian (4.57) to\nH0=X\nk\u0002\n!a(k)\u000by\nk\u000bk+!b(k)\fy\nk\fk\u0003\n(4.68)\nwith the following dispersion relations,\n!a(k) =q\nA2\nk\u0000B2\nk; (4.69)\n!b(k) =Ck\u0000Ck+kM\n2+s\u0012Ck+Ck+kM\n2\u00132\n\u0000D2\nk:\n(4.70)\nInheriting the terminology in the fully polarized phase,\nwe call the bosons created by \u000by\nkand\fy\nkas the \\a\" bo-\nson and the \\ b\" boson, respectively, also in the AFQ\nphase. At k=kM, the \\a\" boson corresponding to the\nbound magnon pair is gapless, whereas the \\ b\" boson\ncorresponding to the unpaired magnon is gapped:\n!a(kM) = 0; (4.71)\n!b(kM) =J22\u0000J11\nJ22+J11H: (4.72)\nThe \\a\" boson is the characteristic Nambu-Goldstone bo-\nson that accompanies the AFQ ordered ground state. At\nk=0, both excitations are gapped:\n\u0001a=!a(0) = 8J11\u0014\n1 +J22\u0000J11\nJ11\u001a\n1\u0000\u0012H\nHAFQ\nc\u00132\u001b\u00151=2\n;\n(4.73)\n\u0001b=!b(0) =H: (4.74)\nThe linear \ravor-wave theory reproduces the exact\nEPR frequency !=H[Eq. (2.7)] of the unperturbed13\nsystem (4.5) both in the fully polarized phase and in the\nAFQ phase. The reproduction of the exact result is an\nimportant criterion of appropriateness of the low-energy\ne\u000bective theory. The criterion is akin to the sum rule\nmentioned in Ref. [64]. If we include an SU(2)-symmetric\nbut SU(3)-asymmetric interaction,\nHa=\u0000J21\n2X\nhi;ji1\u0000\nQi\u0001Qj\u0000Si\u0001Sj\u0001\n(4.75)\nto the unperturbed Hamiltonian (4.5), the linear \ravor-\nwave theory fails to reproduce the exact EPR frequency\n!=Hbecause \u0001 bin the AFQ phase is modi\fed to\n\u0001b= 4J11cos 2\u0012H+ 4J22s\n1\u0000\u0012J22\u0000J21\nJ22\u00132\nsin22\u0012H:\n(4.76)This technical problem is an artifact of the linear \ravor-\nwave theory and not essential to our purpose of demon-\nstrating the general properties of the magnon-pair reso-\nnance in the AFQ phase. Thus, we put aside this prob-\nably complicated discussion, restricting ourselves to the\nmodel with J21= 0.\nG. Magnon-pair resonance in the AFQ phase\nHere we study the additional ESR spectrum I0(!),\ngiven by Eq. (2.9), in the AFQ phase. We approximate\nthe operatorAup to the quadratic order of the spin op-\nerators, that is, the quadratic order of the \foperators\nand the linear order of the \u000boperators. TheAoperator\n(2.6) is represented as\nAp\nN'3(D\u0000E) cos 2\u0012+D+ 3E\n2p\n2e\u0000i'\b\n(cos\u0012Hcosh \u0002b\n0\u0000sin\u0012Hsinh \u0002b\n0)\f0+ (\u0000cos\u0012Hsinh \u0002b\n0+ sin\u0012Hcosh \u0002b\n0)\fy\nkM\t\n\u0000(D\u0000E) cos 2\u0012\u0000(D+ 3E)\n2p\n2ei'\b\n(cos\u0012Hcosh \u0002b\n0\u0000sin\u0012Hsinh \u0002b\n0)\fy\n0+ (\u0000cos\u0012Hsinh \u0002b\n0+ sin\u0012Hcosh \u0002b\n0)\fkM\t\n\u00002(D\u0000E) sin\u0012cos\u0012e\u0000i2'\b\ncos2\u0012H(\u000b0cosh \u0002a\n0\u0000\u000by\n0sinh \u0002a\n0)\u0000sin2\u0012H(\u000by\n0cosh \u0002a\n0\u0000\u000b0sinh \u0002a\n0)\t\n\u0000(D\u0000E) sin\u0012cos\u0012X\nk\b\ne\u0000i2'sin 2\u0012H(\fy\nk+kMcosh \u0002b\nk\u0000\f\u0000ksinh \u0002b\nk)(\fkcosh \u0002b\nk\u0000\fy\n\u0000k+kMsinh \u0002b\nk)\n+ 3(\fy\nkcosh \u0002b\nk\u0000\f\u0000k+kMsinh \u0002b\nk)(\fkcosh \u0002b\nk\u0000\fy\n\u0000k+kMsinh \u0002b\nk)\t\n: (4.77)\nThe term containing the \u000boperators creates the abo-\nson at k=0, whereas the linear terms of the \foper-\nators create the bboson at either k=0ork=kM.\nThe quadratic term of the \foperators contributes to the\ntwo-magnon continuum made of two scattering bbosons.\nWhile thebboson excitation at k=0is involved in the\nEPR peakIEPR(!) with the resonance frequency !=H,\ntheaboson excitation at k=0gives rise to the magnon-\npair resonance peak IMPR(!). Thebboson excitation\natk=kMand two-unpaired-magnon excitation result\nin unpaired magnon resonance peak IkM(!) and broad\ntwo-magnon continuum I2-mag (!), respectively. In total,\nthe ESR spectrum I(!) =IEPR(!)+I0(!) contains three\nsharp peaks and a broad continuum:\nI(!) =IEPR(!)+IMPR(!)+IkM(!)+I2-mag (!);(4.78)which are given at the leading order of the perturbation\nby\nIEPR(!)'\u0019H\n4hSzi0\u000e(!\u0000H); (4.79)\nIMPR(!)'AMPR\u000e(!\u0000\u0001a); (4.80)\nIkM(!)'AkM\u000e\u0000\n!\u0000!b(kM)\u0001\n; (4.81)\nI2-mag (!)'X\nkF(k)\u0002\nsin22\u0012H\b\n\u000e(!\u0000!b(\u0000k)\u0000!b(k))\n+\u000e(!\u0000!b(\u0000k) +!b(\u0000k+kM))\t\n+ 18\u000e(!\u0000!b(\u0000k+kM)\u0000!b(k))\u0003\n:\n(4.82)\nThe intensities AMPR andAkMare given by\nAMPR\nN=\u0019\n8(D\u0000E)2sin2\u0012cos2\u0012\u0001a\n(\u0001a\u0000H)2(cos2\u0012Hcosh \u0002a\n0+ sin2\u0012Hsinh \u0002a\n0)2; (4.83)\nAkM\nN=\u0019\n16!b(kM)\n2(!b(kM)\u0000H)2\u0012(D\u0000E) cos 2\u0012\u0000D\u00003E\n2\u00132\n(sin\u0012Hcosh \u0002b\n0\u0000cos\u0012Hsinh \u0002b\n0)2; (4.84)14\nand the factor F(k) is\nF(k)\nN=1\n8(D\u0000E)2sin2\u0012cos2\u0012\u0012sinh 2\u0002b\nk\n2\u00132\n(4.85)\nwithin the lowest-order perturbation theory. Note that these are independent of the angle 'of the quadrupolar order.\nThis'independence comes as a consequence of the linear \ravor-wave approximation and the \frst-order perturbation.\nHigher-order processes can induce 'dependent corrections to the above results.\nThe intensities AMPR andAkMcan be rephrased as\nAMPR\nN=\u0019\n8(D\u0000E)2sin2\u0012cos2\u0012\u0001a\n(\u0001a\u0000H)2\u001a2H\nHAFQ\nc+\u0014\u0012\n1 +\u0012H\nHAFQ\nc\u00132\u0013\u0012J22+J11\n2J11\u0000J22\u0000J11\n2J11\u0012H\nHAFQ\nc\u00132\u0013\n+4(J22\u0000J11)\n\u0001a\u0012\n1\u0000\u0012H\nHAFQ\nc\u00132\u00132\u0015\u001b\n; (4.86)\nAkM\nN=\u0019\n16\u0012(D\u0000E) cos 2\u0012\u0000D\u00003E\n2\u00132J2\n22\u0000J2\n11\n2J2\n11H\u0012HAFQ\nc\nH\u0000H\nHAFQ\nc\u0013\n: (4.87)\nThe presence of the magnon-pair resonance peak\nIMPR(!) is a direct consequence of the quadrupolar or-\nder in the ground state. We found that the magnon-pair\nresonance peak appears at the \fnite frequency != \u0001ain\n(a) ✓=⇡/4\n(b) ✓=⇡/2bound magnon pairunpaired magnon at kMIntensity02468\nMagnetic ﬁeld H/J11051015bound magnon pairunpaired magnon at kMIntensity00.51.01.52.0\nMagnetic ﬁeld H/J11051015\nFIG. 2. Field dependence of the peak intensities of magnon-\npair resonance (4.86) and unpaired magnon resonance (4.87)\nin the antiferroquadrupolar phase in the S= 1 spin model\nused in Fig. 1. The saturation \feld is given by HAFQ\nc = 12.\nWe assumed E= 0 for simplicity. The vertical axis is given\nin the unit of D2. The angle \u0012between the magnetic \feld and\nthe sample is taken to be (a) \u0012=\u0019=4 and (b)\u0012=\u0019=2.the AFQ phase, which is continuously connected to the\nmagnon-pair resonance peak found in the fully polarized\nphase. Re\recting the condensation of bound magnon\npairs, the \feld dependence of the resonance frequency\n!= \u0001ashows a singular upturn at the saturation \feld\nH=HAFQ\nc (Fig. 1). In addition, there is another peak\nIkM(!) [Eq. (4.81)] which is absent in the fully polarized\nphase. This peak IkM(!) corresponds to creation of the\nsingle \\b\" boson at k=kM. Although ESR usually in-\nvolves excitations at k=0only as Eq. (2.3) shows, the\nAFQ order with the wave vector kMmakes the resonance\natk=kMpossible.\nIn general, the magnon-pair resonance peak IMPR(!)\ncould be masked by the broad two-magnon continuum.\nHowever there is a better chance to observe this reso-\nnance peak near the saturation \feld. The lowest en-\nergy of the continuum takes the highest value 2 !b(kM) =\n2(J22\u0000J11)HAFQ\nc=(J22+J11) at the saturation \feld. For\nthe parameter range J22>1:702J11, this lower edge of\nthe continuum is well above the magnon-pair resonance\nfrequency \u0001 a= 8J11at the saturation. It is also worth\nmentioning the \feld dependence of the intensities. The\nintensity of magnon-pair resonance peak in IMPR(!) re-\nmains \fnite the near the saturation \feld. In contrast,\nthe intensities of the unpaired magnon peak in IkM(!)\nand the two-magnon continuum I2-mag (!), both of which\noriginate from the unpaired magnon excitations, vanish\nnear saturation as\nAkM/HAFQ\nc\u0000H\nF(k)/HAFQ\nc\u0000H\nforH < HAFQ\nc. Hence the continuum I2-mag (!) disap-\npears around the saturation \feld HAFQ\nc. Therefore, our\nmethod is more e\u000bective under the high magnetic \feld.\nThe peak intensity AMPR of the magnon-pair reso-\nnance has a strong \feld dependence, showing a diver-\ngence at a certain \feld H\u0003belowHAFQ\nc [see Fig. 2(a)].15\nThis divergence occurs when the magnon-pair resonance\npeak merges into the EPR one at H'H\u0003. The diver-\ngence comes from the factor \u0001 a=(\u0001a\u0000H)2in Eq. (4.83);\nas Eq. (4.73) and Fig. 1 show, the excitation gap \u0001 aof\nthe bound magnon pair equals to HatH\u0003=J22=[J22\u0000\nJ11+J11(Hc=8J11)2]. The intensity AkM[Eq. (4.87)]\nof the unpaired magnon resonance also shows the diver-\ngence atH= 0 (Fig. 2), because of merging of the peak\ninto the EPR one at H'0.\nWe note that the EPR and the MPR peaks can be\nmixed under certain interactions. The EPR peak and\nthe MPR peak are generated by application of \fy\nk=0and\n\u000by\nk=0to a given eigenstate, respectively. To mix those\nresonances, an anisotropic interaction including a term\nsuch as\fk\u000by\nk0or\fy\nk\u000bk0is necessary. For example, a uni-\nform DM interaction with Dvector parallel to the xaxis\ncan generate e\u000bectively such an interaction,P\nhi;ji1D\u0001\nSi\u0002Sj=P\nhi;ji1jDj[(bi\u0000by\ni)(aj+ay\nj)\u0000(ai+ay\ni)(bj\u0000by\nj)].\nIf an anisotropic interaction allows the mixing, it will\nbe di\u000ecut to distinguish the MPR peak from the EPR\none when their resonance frequencies are close, because\nthe EPR peak has a \fnite linewidth in the presence of\nanisotropic interactions. When they are apart, the mix-\ning is not important as long as the anisotropic interactins\nare perturbative.\nAlthough the intensities su\u000ber from the insigni\fcant\ndivergences, the resonance frequencies are free from any\nsingular behavior in the linear \ravor-wave approxima-\ntion except for the singular bent point at the saturation\n\feldHAFQ\nc due to physically reasonable characteristic\nupturn for H > HAFQ\nc (Fig. 1). Using the magnon-pair\nresonance frequency and the unpaired-magnon resonance\nfrequency at kM, we can identify the AFQ order phase\nin quite a wide \feld range.\nAngular dependence of the intensity of the magnon-\npair resonance (4.83) enables another method of identi-\n\fcation free from the technical problems. The magnon-\npair resonance peak shows a characteristic angular de-\npendence\nAMPR/sin2\u0012cos2\u0012; (4.88)\nwhich makes the magnon-pair resonance peak vanish\nwhen the magnetic \feld is parallel to the x,y, orzaxes\n[as shown in Fig. 2(b)]. The angular dependence (4.88)\nholds true independent of the model and the theoretical\ntechnique, as we pointed out in Sec. IV A. This angu-\nlar dependence re\rects the fact that the operators Qyz\nj\nandQzx\njneither create nor annihilate the bound magnon\npair, di\u000berent from Qx2\u0000y2\nj andQxy\nj. Thus, the charac-\nteristic angular dependence of AMPR (4.88) quali\fes as\nan evidence of the formation of the bound magnon pair\nin the system (4.5) with the single-ion anisotropy.\nWe note that the linewidth of the EPR peak of\nS= 1=2 frustrated ferromagnetic chain compounds\nis also expected to show the angular dependence of\nsin2\u0012cos2\u0012[47]. That angular dependence of thelinewidth comes from the same root as the intensity of\nthe magnon-pair resonance peak.\nV. DISCUSSIONS\nHere, we discuss some issues related to the magnon-\npair resonance peak in the ESR spectrum shown in\nSec. IV. We also discuss applications of our theory to\nS= 1=2 spin systems.\nA. E\u000bects of other anisotropic interactions\nIn Sec. IV, we assumed the single-ion anisotropy (4.4)\nas an example of the perturbative anisotropic interaction.\nSince ESR depends crucially on the form of the perturba-\ntive anisotropic interaction H0, it is necessary to con\frm\nthat our results obtained in Sec. IV is robust against in-\nclusion of other kinds of anisotropic interactions.\nThe anisotropic exchange interaction (3.4) on the nth\nneighbor bond leads to the same result because the\nbound magnon pair is not localized at a single site but\nspread around bonds [35]. If the Aoperator (2.6) con-\ntains some of operators that have the same symmetry as\nthe wavefunction of two-magnon bound state, it gener-\nates the magnon-pair resonance to the ESR spectrum\nthrough the formula (2.9). In contrast, the DM in-\nteractionH0\nDM=P\nhi;jinDij\u0001Si\u0002Sjis irrelevant to\nthe magnon-pair resonance (4.80) and to the unpaired\nmagnon resonance (4.81) because of the symmetry; the\nDM interaction neither create nor annihilate the bound\nmagnon pair on the bond because it is antisymmetric\nwith respect to the bond-centered inversion, whereas the\nwavefunction of the bound magnon pair on the bond is\nsymmetric.\nB. Applications to the spin nematic order in\nS= 1=2spin systems\nIn Sec. IV, we studied the S= 1 spin model to demon-\nstrate the magnon-pair resonance. We can apply this\nresult to the spin nematic order in S= 1=2 spin sys-\ntems performing an appropriate mapping of low-energy\ndegrees of freedoms.\n1.S= 1=2frustrated ferromagnets\nIn the case of spin-1/2 frustrated ferromagnets, the\nspin nematic order parameter, de\fned on the nearest\nneighbor bonds, has k=0wave vector, in which the\nsign of it alternates inside the unit cell of the crystal\nstructure [5, 16, 17, 19{21]. For example, on the square\nlattice, the two quadrupolar directors on di\u000berent bonds\nalong two unit vectors e1ande2are orthogonal to each\nother [5, 9]. Because of this sign change, the S= 1=216\nspin nematic order parameter is not captured into the\nfrequency shift discussed in Sec. III, even though it has\nk=0wave vector.\nLow-energy degrees of freedom in the S= 1=2 spin ne-\nmatic systems are given by S= 1 spin degrees of freedom\nformed on the nearest neighbor bonds [5, 42]. These ex-\ncitation modes are e\u000bectively related to the excitations\nof theS= 1 AFQ state through a mapping between\nbond degrees of freedom in S= 1=2 spin systems and\non-site spin degrees of freedom in S= 1 spins [43], where\ntheS= 1 spins are assigned on the center points of\nthe nearest-neighbor bonds of S= 1=2 spins. On the\nsquare lattice, the gapless excitations with k=kMin\ntheS= 1 AFQ state correspond to the excitations with\nk=0wave vector and B1irreducible representation of\nthe space group C4vin theS= 1=2 spin nematic states.\nAbove the saturation H >HAFQ\nc, this mode is the low-\nest excitation which closes the gap at the saturation \feld.\nHowever this mode is inaccessible in ESR measurements,\nsince ESR is directly accessible only to the k=0wave\nvector modes with the A1(trivial) irreducible representa-\ntion. Only the gapped excitation modes with k=0can\nbe observed among the bound magnon pair excitations\nas same as in the S= 1 AFQ state discussed in Sec. IV.\n2.S= 1=2orthogonal dimer spin system\nThe spin nematic phase can also appear in spin-gapped\nsystems when bound magon (triplon) pairs close the en-\nergy gap in an applied \feld [28]. In the S= 1=2 Heisen-\nberg model on the Shastry-Sutherland lattice [67], which\nis also called an orthogonal dimer spin model, it was the-\noretically demonstrated that the ground state is an exact\ndimer state with a \fnite energy gap [67, 68] and bound\ntwo-triplon excited states [28, 69, 70] are stabilized at\nzero \feld by the correlated hopping process [71]. The-\noretical calculations [28] pointed out that a two-triplon\nbound state with Sz= 2 can have a lower energy than\ntwo triplon continuum above the gapped ground state\nand that the energy-gap closing in an applied magnetic\n\feld leads to the condensation of bound triplon pairs.\nSince the lowest energy state of the bound pair in the\nSz= 2 sector has the wave vector k=kM, the \feld-\ninduced condensed phase becomes an AFQ phase [72].\nIn this system, anisotropic interactions between two\northogonal dimers cause the operator S+\niS+\njon the inter-\ndimer bonds in the Aoperator. This operator creates a\ntriplon pair on a nearest-neighbor pair of dimers, which\ngives rise to a triplon-pair resonance peak in the ESR\nspectrum. In the spin gap phase, the resonance frequency\nbehaves as\n!= \u0001a+ 2(Hc\u0000H); (5.1)\nwhereHcdenotes the onset-\feld of the magnetization\nprocess and \u0001 athe energy gap of the bound triplon pair\natk=0at the critical \feld H=Hc. We note that thebound pair closes the gap at k= (\u0019;\u0019) and this excita-\ntion is well dispersive, i.e. \u0001 a>0 [28]. In the magnetic\nphase above Hc, we expect that this peak continuously\nconnects to the triplon pair resonance peak in the AFQ\nphase showing a singularity in the \feld dependence of the\nfrequency at the critical \feld.\nIn an ESR study [29], bound triplon-pair resonance\npeaks were indeed observed in the S= 1=2 orthogonal\ndimer spin compound SrCu 2(BO 3)2. The lowest-energy\nresonance peak of the bound triplon pairs shows the\n\feld dependence (5.1), having a strong intensity around\nH=Hc. Even after the peak frequency changes the\nslope as a function of a \feld around H=Hc, the reso-\nnance peak remains with strong intensity inside the mag-\nnetic phase between the spin gapped and the 1/8-plateau\nphases when the magnetic \feld is parallel to aaxis. The\nimplication of this resonance peak inside the magnetic\nphase has not been properly understood until now. Our\nresearch elucidates that this ESR result has already sug-\ngested the appearance of a spin nematic order in the\nground state of the \feld-induced magnetic phase below\nthe 1/8-plateau. This system deserves further investiga-\ntions.\nTo compare the \feld dependence of the resonance peak\nwith observed results in real compounds, we need to\ninclude mixing between the ground state and excited\nstates. For example, DM interaction induces a mix-\ning between the singlet ground state and triplon excited\nstates [73]. In the case of the bound two-triplon excited\nstate, anisotropic interactions can induce mixing with the\nsinglet ground state. This can be easily seen by consid-\nering an anisotropy on the inter-dimer bonds\n2\u000e(Sx\niSx\nj\u0000Sy\niSy\nj) =\u000e(S+\niS+\nj+S\u0000\niS\u0000\nj); (5.2)\nwhich mixes the bound two-triplon state with the singlet\ndimer state. This e\u000bect might smear the singularity in\nthe \feld dependence of the resonance frequency at H=\nHc.\nC. Magnon-pair resonance in the case of\nferroquadrupolar order\nLastly we comment on the additional peaks in the ESR\nspectrum in the FQ phase.\nIn the FQ phase, only the EPR peak will be found in\nthe ESR spectrum. The bound magnon-pair excitation\nin the FQ phase is gapless at k=0, i.e. \u0001a= 0, but\nthe resonance at != 0 is invisible in the ESR spectrum\nfor the factor !in Eq. (2.3). Note that the unpaired\nmagnon resonance peak at k=kMin the AFQ phase\ncorresponds to the EPR peak in the FQ phase since the\nFQ order grows at k=0.\nIf the magnetic \feld is above the saturation \feld HFQ\nc,\nthe bound magnon pair excitation opens a gap, showing\na characteristic \feld dependence \u0001 a= 2(H\u0000HFQ\nc). This\npeak is observable in the ESR measurements as it comes\nfrom the k=0modes. This gives a clear di\u000berence from17\nthe case of the AFQ order; the lowest excitation which\ncloses the gap as 2( H\u0000HAFQ\nc) above the AFQ phase has\nthek=kMwave vector [Eq. (4.37)] and it cannot be\nobserved in ESR measurements. Thus, the appearance\nof this peak in the ESR spectrum above the saturation\n\feldHFQ\ncand the disappearance below HFQ\ncsignal the\nemergence of the FQ phase below HFQ\nc.\nVI. SUMMARY\nIn this paper we showed that the FQ and the AFQ\norders are distinguishable in ESR experiments. We stud-\nied both the frequency shift of EPR resonance and the\nfrequencies of the additional resonance peaks in the ESR\nspectrum.\nFor the generic spin model (2.1), the FQ order param-\neter turned out to shift the resonance frequency of the\nEPR peak in the ESR spectrum. The EPR frequency\nshift shows a characteristic angular dependence [as shown\nin Eq. (3.8)] on a rotation of the material around the y\naxis keeping the magnetic \feld parallel to the zaxis. Here\nwe determined the yandzaxes so that the anisotropic\nspin interactions in these spin components are, respec-\ntively, weakest and strongest. Thus the FQ order pa-\nrameter can be extracted from the frequency shift. For\nexample, as Eqs. (3.9) and (3.10) show, the frequency\nshifts at\u0012= 0 and\u0019=2 enable us to determine the FQ\norder parameter hS\u0000\niS\u0000\nji0experimentally because only\ntwo quantitiesh3Sz\niSz\nj\u0000Si\u0001Sji0andhS\u0000\niS\u0000\nji0are the\nundetermined variables in these equations. In particular,\nwhen the perturbative anisotropic interaction is uniax-\nial, the FQ order parameter is simply derived from the\nsingle equation (3.11). The unexplained high \feld phase\nof chromium spinels is an interesting target to which this\nmethod is applicable.\nIn the case of the AFQ order, though the EPR fre-\nquency shift is usually insensitive to the order parameter,\n\fngerprints of the AFQ order appear in the additional\nresonance peaks other than the EPR peak in the ESR\nspectrum. As far as the resonance peaks well isolated\nfrom the EPR one are concerned, the ESR spectrum, as\nshown in Eq. (2.9), is derived from the spectrum of the re-\ntarded Green function of the operator A= [H0;S+] given\nby the small anisotropic interaction H0. The operatorA\nis usually quadratic containing the magnon pair creation\noperator [see Eqs. (4.2) and (4.41)]. This is one of the\nmost interesting properties of the ESR spectrum. Except\nfor the vicinity of the EPR peak at !=H, measuring\nthe ESR spectrum is e\u000bectively equivalent to measuring\nthe spectrum of the operator A. We note that, in our\npertubative analysis, the anisotropic interaction plays no\nrole of yielding the spin nematic phase and of making\nthe magnitude of the quadrupolar order parameter grow.\nThose are fully determined in the unperturbed system.\nThe long-range AFQ order yields two additional sharp\nresonance peaks in the ESR spectrum. One is at-\ntributed to the resonance of the bound magnon-pair exci-tation, which we called the magnon-pair resonance. The\nmagnon-pair resonance is also found in the fully polar-\nized phase adjacent to the AFQ phase, where the reso-\nnance frequency shows the linear \feld dependence whose\nslope is double of that of the EPR frequency, as was\nexperimentally observed in Ref. [48]. (A similar triplon-\npair resonance peak was also experimentally observed in\nRef. [29].) With decreasing the magnetic \feld, the sys-\ntem enters into the AFQ phase, where the magnon-pair\nresonance frequency shows the singular upturn as a func-\ntion of the magnetic \feld, re\recting the condensation of\nbound magnon pairs (Fig. 1). The other resonance peak\nis attributed to the excitation of the unpaired magnon\nat the wave vector kM= (\u0019;\u0019). Usually, ESR detects\nexcitations at the wave vector k=0. In the AFQ phase,\nsince the ground state structure has the wave vector kM,\nthe excitation gap of the magnon at kMbecomes visible\nin the ESR spectrum as an independent resonance peak.\nOur results on the FQ and the AFQ orders are valid\nas long as (1) the anisotropic interaction is small enough\nto be seen as a perturbation to the system and (2) the\nanisotropic interaction is governed mainly by the single-\nion anisotropy or the anisotropic exchange interaction.\nThe weak DM interaction has no impact on the result\nobtained in this paper because the DM interaction is an-\ntisymmetric with respect to the bond-centered inversion\nunlike the spin nematic order parameter. Though we re-\nstricted ourselves to the cases of weak anisotropic inter-\nactions in this paper, it is also interesting to investigate\ncases governed by a large anisotropic interaction such as\nthe case of Ref. [48]. While the formula of the frequency\nshift [Eq. (3.3)] is invalid in such cases, the discussion of\nthe sharp magnon-pair resonance peak isolated from the\nEPR one will be qualitatively valid even in the case of\nlarge anisotropic interactions though we need to derive\nthe full Green's function GR\nAAy(!) instead.\nACKNOWLEDGMENTS\nWe thank Akira Furusaki, Masayuki Hagiwara, Hi-\nroyuki Nojiri, Nic Shannon, and Shintaro Takayoshi for\nhelpful discussions. The present work is supported by\nJSPS KAKENHI Grant Nos. 16J04731 and 16K05425.\nAppendix A: Polarization independence of the ESR\nspectrum\nIn this Appendix, we derive the ESR spectrum of the\nunpolarized microwave. To discuss it, we review the lin-\near response theory of the ESR absorption spectrum I(!)\nin a generic spin system described by\n~H(t) =H\u0000Z\u0019\n\u0000\u0019d\u0012Z\u0019\n\u0000\u0019d\u001eA(\u0012)X(t;\u0012;\u001e ); (A1)18\nwhereHis the Hamiltonian of the spin system of our\ninterest,A(\u0012) is a total spin operator\nA(\u0012) =Sxcos\u0012+Sysin\u0012; (A2)\nandX(t;\u0012;\u001e ) is the oscillating magnetic \feld,\nX(t;\u0012;\u001e ) =hR(\u0012;\u001e) cos(!t+\u001e): (A3)\nThe amplitude hR(\u0012;\u001e) follows a distribution which can\nbe random or not. For example, the Hamiltonian under\na circularly polarized magnetic \feld\n~H(t) =H\u0000HR\n2(S+ei!t+S\u0000e\u0000i!t) (A4)\nis a special case of Eq. (A1) with\nhR(\u0012;\u001e) =HR\n2\u0002\n\u000e(\u0012)\u000e(\u001e) +\u000e(\u0012\u0000\u0019\n2)\u000e(\u001e\u0000\u0019\n2)\u0003\n:(A5)\nThe ESR absorption spectrum I(!) is give by the en-ergy absorption rate per a period of the oscillating \feld,\nI(!) =!\n2\u0019Z2\u0019=!\n0dtd\ndtTr[\u001a(t)~H(t)]; (A6)\nwhere\u001a(t) is the density matrix in the Heisenberg pic-\nture,\n\u001a(t) =U(t)exp(\u0000~H(t)=T)\nTr[exp(\u0000~H(t)=T)]Uy(t) (A7)\nwith\nU(t) = exp\u0012\niZt\n0dt0~H(t0)\u0013\n: (A8)\nIfhR(\u0012;\u001e) follows a random distribution, we replace\nEq. (A6) with\nI(!) =!\n2\u0019Z2\u0019=!\n0dtd\ndtTr[\u001a(t)~H(t)]; (A9)\nwhereOdenotes the average of the quantity Owith re-\nspect to the random distribution.\nLet us \frst consider the case that hR(\u0012;\u001e) is uniquely\ndetermined such as Eq. (A5). The energy absorption rate\n(A6) is written as\nI(!) =\u0000!\n2\u0019Z2\u0019=!\n0dtZ\u0019\n\u0000\u0019d\u0012d\u001e Tr[\u001a(t)A(\u0012)]@X(t;\u0012;\u001e )\n@t: (A10)\nWithin the linear response, the trace Tr[ \u001a(t)A(\u0012)] is approximated as\nTr[\u001a(t)A(\u0012)]'hA(\u0012)i+iZ1\n0dt0Z\u0019\n\u0000\u0019d\u00120d\u001e0X(t\u0000t0;\u00120;\u001e0)h[A(t0;\u0012);A(0;\u00120)]i: (A11)\nHere, the average h\u0001iis taken with respect to the Hamiltonian HandA(t;\u0012) is de\fned as\nA(t0;\u0012) =eit0HA(\u0012)e\u0000it0H: (A12)\nTaking these relations into account, we \fnd that the ESR energy absorption rate (A10) is given by\nI(!)'\u0000i!\n2\u0019Z2\u0019=!\n0dtZ1\n0dt0Z\u0019\n\u0000\u0019d\u0012d\u001ed\u00120d\u001e0@X(t;\u0012;\u001e )\n@tX(t\u0000t0;\u00120;\u001e0)h[A(t0;\u0012);A(0;\u00120)]i\n=i!\n8Z1\n0dt0Z\u0019\n\u0000\u0019d\u0012d\u001ed\u00120d\u001e0hR(\u0012;\u001e)hR(\u00120;\u001e0) sin(!t0+\u001e\u0000\u001e0)\b\nh[S+(t0);S\u0000(0)]ie\u0000i(\u0012\u0000\u00120)\n+h[S\u0000(t0);S+(0)]iei(\u0012\u0000\u00120)+h[S+(t0);S+(0)]ie\u0000i(\u0012+\u00120)+h[S\u0000(t0);S\u0000(0)]iei(\u0012+\u00120)\t\n:(A13)\nWhenhR(\u0012;\u001e) is given by Eq. (A5), the ESR absorption rate (A13) becomes\nI(!) =!H2\nR\n8\u0002\n\u0000ImGR\nS+S\u0000(!)\u0003\n; (A14)\nwhich reproduces Eq. (2.3).\nWe next consider the case that hR(\u0012;\u001e) follows a random distribution. Applying the linear response theory (A11)\nto the random averaged energy absorption rate (A9), we \fnd\nI(!) =\u0000!\n2\u0019Z2\u0019=!\n0dtZ\u0019\n\u0000\u0019d\u0012d\u001e Tr[\u001a(t)A(\u0012)]@X(t;\u0012;\u001e )\n@t19\n'i!\n8Z1\n0dt0Z\u0019\n\u0000\u0019d\u0012d\u001ed\u00120d\u001e0hR(\u0012;\u001e)hR(\u00120;\u001e0) sin(!t0+\u001e\u0000\u001e0)\b\nh[S+(t0);S\u0000(0)]ie\u0000i(\u0012\u0000\u00120)\n+h[S\u0000(t0);S+(0)]iei(\u0012\u0000\u00120)+h[S+(t0);S+(0)]ie\u0000i(\u0012+\u00120)+h[S\u0000(t0);S\u0000(0)]iei(\u0012+\u00120)\t\n:(A15)\nIf the applied electromagnetic wave is \\white\", that is, if\nthe random average hR(\u0012;\u001e)hR(\u00120;\u001e0) satis\fes\nhR(\u0012;\u001e)hR(\u00120;\u001e0) =H2\nR\n(2\u0019)2\u000e(\u0012\u0000\u00120)\u000e(\u001e\u0000\u001e0);(A16)\nthe ESR energy absorption rate (A15) is simpli\fed as\nI(!) =!H2\nR\n8\u0002\n\u0000ImGR\nS+S\u0000(!)\u0000ImGR\nS\u0000S+(!)\u0003\n:(A17)\nAppendix B: Perturbations of the ground state\nenergy in the AFQ phase\nThis appendix is devoted to estimation of the ground\nstate energy in the AFQ phase of the S= 1 model on\nthe square lattice described by the Hamiltonian\nH=H0+H0; (B1)\nwhereH0is the unperturbed Hamiltonian (4.5). We take\nthe single-ion anisotropy (4.4) as the perturbation H0.As we discussed in Sec. IV C, the angle 'that speci-\n\fes the direction of the AFQ order growing is not deter-\nmined at the mean-\feld level. Here, we estimate the '\ndependence of the ground-state energy using the linear\n\ravor-wave theory explained in Sec. IV F. The perturba-\ntive single-ion anisotropy shifts the ground state energy\nfrom its unperturbed value by an amount\n\u000eE0=hGSjH0jGSi; (B2)\nup to the \frst order of the perturbation. Here, jGSiis\nthe ground state in the AFQ phase of the unperturbed\nHamiltonian, or the vacuum annihilated by \u000bkand\fk\nfor all k:\n\u000bkjGSi=\fkjGSi= 0: (B3)\nThe energy shift (B2) in the Schwinger boson language\nis given by\n\u000eE0= const:\u0000\u0012\nD\u00003(D\u0000E)\n2sin2\u0012\u0013X\nkhGSjby\nkbkjGSi\n+1\n2\b\nDsin2\u0012+E(1 + cos2\u0012)\t\ncos 2'sin 2\u0012HX\nkhGSj(2ay\nkak+kM+by\nkbk+kM)jGSi: (B4)\nUsing the Bogoliubov transformations (4.64) and (4.65)\nand the property (B3) in the ground state, we can further\nreduce\u000eE0to\n\u000eE0= const:\u0000\u0012\nD\u00003(D\u0000E)\n2sin2\u0012\u0013X\nksinh2\u0002b\nk:\n(B5)The shift (B5) is independent of the angle 'and hence\nleaves'undetermined.\n[1] M. Blume and Y. Y. Hsieh, J. Appl. Phys. 40, 1249\n(1969).\n[2] H. H. Chen and P. M. Levy, Phys. Rev. Lett. 27, 1383\n(1971).\n[3] A. Andreev and I. Grishchuk, Zh. Eksp. Teor. Fiz. 87,\n467 (1984) [Sov. Phys. JETP 60, 267 (1984)].\n[4] A. V. Chubukov, Phys. Rev. B 44, 4693 (1991).\n[5] N. Shannon, T. Momoi, and P. Sindzingre, Phys. Rev.\nLett. 96, 027213 (2006).\n[6] T. Momoi, P. Sindzingre, and N. Shannon, Phys. Rev.Lett. 97, 257204 (2006).\n[7] H. T. Ueda and K. Totsuka, Phys. Rev. B 76, 214428\n(2007).\n[8] R. Shindou and T. Momoi, Phys. Rev. B 80, 064410\n(2009).\n[9] R. Shindou, S. Yunoki, and T. Momoi, Phys. Rev. B 84,\n134414 (2011).\n[10] F. Heidrich-Meisner, A. Honecker, and T. Vekua, Phys.\nRev. B 74, 020403 (2006).\n[11] T. Vekua, A. Honecker, H.-J. Mikeska, and F. Heidrich-20\nMeisner, Phys. Rev. B 76, 174420 (2007).\n[12] L. Kecke, T. Momoi, and A. Furusaki, Phys. Rev. B 76,\n060407 (2007).\n[13] T. Hikihara, L. Kecke, T. Momoi, and A. Furusaki, Phys.\nRev. B 78, 144404 (2008).\n[14] J. Sudan, A. L uscher, and A. M. L auchli, Phys. Rev. B\n80, 140402 (2009).\n[15] H. T. Ueda and K. Totsuka, Phys. Rev. B 80, 014417\n(2009).\n[16] M. E. Zhitomirsky and H. Tsunetsugu, Euro. Phys. Lett.\n92, 37001 (2010).\n[17] M. Sato, T. Hikihara, and T. Momoi, Phys. Rev. Lett.\n110, 077206 (2013).\n[18] O. A. Starykh and L. Balents, Phys. Rev. B 89, 104407\n(2014).\n[19] T. Momoi, P. Sindzingre, and K. Kubo, Phys. Rev. Lett.\n108, 057206 (2012).\n[20] H. T. Ueda and T. Momoi, Phys. Rev. B 87, 144417\n(2013).\n[21] O. Janson, S. Furukawa, T. Momoi, P. Sindzingre,\nJ. Richter, and K. Held, Phys. Rev. Lett. 117, 037206\n(2016).\n[22] K. Nawa, Y. Okamoto, A. Matsuo, K. Kindo, Y. Kita-\nhara, S. Yoshida, S. Ikeda, S. Hara, T. Sakurai, S. Okubo,\nH. Ohta, and Z. Hiroi, J. Phys. Soc. Jpn. 83, 103702\n(2014).\n[23] N. B uttgen, K. Nawa, T. Fujita, M. Hagiwara, P. Kuhns,\nA. Proko\fev, A. P. Reyes, L. E. Svistov, K. Yoshimura,\nand M. Takigawa, Phys. Rev. B 90, 134401 (2014).\n[24] K. Nawa, M. Takigawa, S. Kr amer, M. Horvati\u0013 c,\nC. Berthier, M. Yoshida, and K. Yoshimura, Phys. Rev.\nB96, 134423 (2017).\n[25] H.-J. Grafe, S. Nishimoto, M. Iakovleva, E. Vavilova,\nA. Alfonsov, M.-I. Sturza, S. Wurmehl, H. Nojiri, H. Ros-\nner, J. Richter, et al. , Sci. Rep. 7, 6720 (2017).\n[26] A. Orlova, E. L. Green, J. M. Law, D. I. Gorbunov,\nG. Chanda, S. Kr amer, M. Horvati\u0013 c, R. K. Kremer,\nJ. Wosnitza, and G. L. J. A. Rikken, Phys. Rev. Lett.\n118, 247201 (2017).\n[27] M. Yoshida, K. Nawa, H. Ishikawa, M. Takigawa,\nM. Jeong, S. Kr amer, M. Horvati\u0013 c, C. Berthier, K. Mat-\nsui, T. Goto, S. Kimura, T. Sasaki, J. Yamaura,\nH. Yoshida, Y. Okamoto, and Z. Hiroi, Phys. Rev. B\n96, 180413 (2017).\n[28] T. Momoi and K. Totsuka, Phys. Rev. B 62, 15067\n(2000).\n[29] H. Nojiri, H. Kageyama, Y. Ueda, and M. Motokawa, J.\nPhys. Soc. Jpn. 72, 3243 (2003).\n[30] N. Papanicolaou, Nucl. Phys. B 305, 367 (1988).\n[31] K. Tanaka, A. Tanaka, and T. Idogaki, J. Phys. A: Math.\nGen.34, 8767 (2001).\n[32] K. Harada and N. Kawashima, Phys. Rev. B 65, 052403\n(2002).\n[33] H. Tsunetsugu and M. Arikawa, J. Phys. Soc. Jpn. 75,\n083701 (2006).\n[34] A. L auchli, F. Mila, and K. Penc, Phys. Rev. Lett. 97,\n087205 (2006).\n[35] E. Takata, T. Momoi, and M. Oshikawa,\narXiv:1510.02373.\n[36] F. Wang, S. A. Kivelson, and D.-H. Lee, Nat. Phys. 11,\n959 (2015).\n[37] R. Yu and Q. Si, Phys. Rev. Lett. 115, 116401 (2015).\n[38] Z. Wang, W.-J. Hu, and A. H. Nevidomskyy, Phys. Rev.\nLett. 116, 247203 (2016).[39] M. Sato, T. Momoi, and A. Furusaki, Phys. Rev. B 79,\n060406 (2009).\n[40] M. Sato, T. Hikihara, and T. Momoi, Phys. Rev. B 83,\n064405 (2011).\n[41] A. Smerald and N. Shannon, Phys. Rev. B 93, 184419\n(2016).\n[42] R. Shindou, S. Yunoki, and T. Momoi, Phys. Rev. B 87,\n054429 (2013).\n[43] A. Smerald, H. T. Ueda, and N. Shannon, Phys. Rev. B\n91, 174402 (2015).\n[44] H. Onishi, J. Phys. Soc. Jpn. 84, 083702 (2015).\n[45] F. Michaud, F. Vernay, and F. Mila, Phys. Rev. B 84,\n184424 (2011).\n[46] L. Savary and T. Senthil, arXiv:1506.04752.\n[47] S. C. Furuya, Phys. Rev. B 95, 014416 (2017).\n[48] M. Akaki, D. Yoshizawa, A. Okutani, T. Kida,\nJ. Romh\u0013 anyi, K. Penc, and M. Hagiwara, Phys. Rev.\nB96, 214406 (2017).\n[49] R. Kubo and K. Tomita, J. Phys. Soc. Jpn. 9, 888 (1954).\n[50] M. Oshikawa and I. A\u000feck, Phys. Rev. B 65, 134410\n(2002).\n[51] M. Ozerov, M. Maksymenko, J. Wosnitza, A. Honecker,\nC. P. Landee, M. M. Turnbull, S. C. Furuya, T. Gia-\nmarchi, and S. A. Zvyagin, Phys. Rev. B 92, 241113\n(2015).\n[52] S. C. Furuya and M. Oshikawa, Phys. Rev. Lett. 109,\n247603 (2012).\n[53] We note that, when a certain symmetry is broken spon-\ntaneously in the unperturbed system and it is also broken\nby the anisotropy in the full Hamiltonian H, one must\nchoose a proper direction of the in\fnitesimal auxiliary\nsymmetry breaking \feld in the unperturbed system so\nthat the ground state is smoothly deformed by imposing\nweak perturbation.\n[54] Y. Maeda and M. Oshikawa, J. Phys. Soc. Jpn. 74, 283\n(2005).\n[55] K. Y. Povarov, A. I. Smirnov, O. A. Starykh, S. V.\nPetrov, and A. Y. Shapiro, Phys. Rev. Lett. 107, 037204\n(2011).\n[56] S. C. Furuya, T. Suzuki, S. Takayoshi, Y. Maeda, and\nM. Oshikawa, Phys. Rev. B 84, 180410 (2011).\n[57] Y. Maeda, K. Sakai, and M. Oshikawa, Phys. Rev. Lett.\n95, 037602 (2005).\n[58] S. C. Furuya, P. Bouillot, C. Kollath, M. Oshikawa, and\nT. Giamarchi, Phys. Rev. Lett. 108, 037204 (2012).\n[59] H. Mori and K. Kawasaki, Prog. Theor. Phys. 27, 529\n(1962).\n[60] K. Penc, N. Shannon, and H. Shiba, Phys. Rev. Lett.\n93, 197203 (2004).\n[61] S. Kimura, M. Hagiwara, T. Takeuchi, H. Yamaguchi,\nH. Ueda, Y. Ueda, and K. Kindo, Phys. Rev. B 83,\n214401 (2011).\n[62] A. Miyata, H. Ueda, and S. Takeyama, arXiv:1302.3664.\n[63] A. Miyata, H. Ueda, Y. Ueda, Y. Motome, N. Shannon,\nK. Penc, and S. Takeyama, J. Phys. Soc. Jpn. 81, 114701\n(2012).\n[64] A. Smerald and N. Shannon, Phys. Rev. B 88, 184430\n(2013).\n[65] P. C. Hohenberg and W. F. Brinkman, Phys. Rev. B 10,\n128 (1974).\n[66] N. Papanicolaou, Nucl. Phys. B 240, 281 (1984).\n[67] B. S. Shastry and B. Sutherland, Physica B+C 108, 1069\n(1981).\n[68] S. Miyahara and K. Ueda, Phys. Rev. Lett. 82, 370121\n(1999).\n[69] C. Knetter, A. B uhler, E. M uller-Hartmann, and G. S.\nUhrig, Phys. Rev. Lett. 85, 3958 (2000).\n[70] K. Totsuka, S. Miyahara, and K. Ueda, Phys. Rev. Lett.\n86, 520 (2001).\n[71] T. Momoi and K. Totsuka, Phys. Rev. B 61, 3231 (2000).\n[72] The \feld-induced antiferroquadrupolar (AFQ) phase in\nthe spin-gapped systems is also described as a low-densitycondensate of bound magnon pairs, as same as the AFQ\nphase near the saturation \feld. The vacuum is a singlet\nground state and the bosonic particles are bound triplon\npairs in the former case, whereas the vacuum is the fully\npolarized state and the bosons are bound magnon pairs\nin the latter.\n[73] J. Romh\u0013 anyi, K. Totsuka, and K. Penc, Phys. Rev. B\n83, 024413 (2011)."    },    {        "title": "1912.04447v2.Josephson_current_through_a_ferromagnetic_bilayer__Beyond_the_quasiclassical_approximation.pdf",        "content": "arXiv:1912.04447v2  [cond-mat.supr-con]  28 Dec 2019Josephson current through a ferromagnetic bilayer: Beyond the quasiclassical\napproximation\nHao Meng,1,2Yajie Ren,1Javier E. Villegas,3and A. I. Buzdin2,4,∗\n1School of Physics and Telecommunication Engineering,\nShaanxi University of Technology, Hanzhong 723001, China\n2University Bordeaux, LOMA UMR-CNRS 5798, F-33405 Talence C edex, France\n3Unit´ e Mixte de Physique CNRS/Thales, Universit´ e Paris-S ud,\nUniversit´ e Paris Saclay, 1 Avenue A. Fresnel, 91767 Palais eau, France\n4Sechenov First Moscow State Medical University, Moscow, 11 9991, Russia\n(Dated: January 1, 2020)\nBased on the Bogoliubov-de Gennes equations, we provide an e xact numerical solution for the\ncritical current of Josephson junctions with a composite fe rromagnetic bilayer. We demonstrate that\nfor the antiparallel orientation of the magnetic moments of the bilayer, the presence of a potential\nbarrier at the bilayer interface results in large oscillati ons of the critical current as a function of\nferromagnet thickness and/or exchange ﬁeld. Because of thi s, and remarkably, in the range of small\nexchange ﬁeld and thicknesses, the magnetism leads to the in crease of the critical current. This\neﬀect is well pronounced at low temperature but disappears n earTc. If the potential barrier is\nreplaced by a spin-active barrier at the bilayer interface t he conventional 0- πtransition, similar\nto the case of an uniform ferromagnetic Josephson junction, is observed. Strikingly, for a parallel\norientation of the magnetic moments of the bilayer, the pres ence of the spin-active barrier restores\nthe anomalous behavior—potential barrier in the antiparal lel case. These behaviors result from\nthe resonant tunneling of Cooper pairs across the composite barrier—an eﬀect related to the spin-\ndependent Fermi vector in the presence of the ferromagnets’ exchange ﬁeld.\nI. INTRODUCTION\nInrecentyearsthesuperconductor(S)-ferromagnet(F)\nsystems attracted a lot of attention due to the possibil-\nity to fabricate the new devices based on the supercon-\nducting spintronics1–5. The properties of diﬀerent S/F\nsystems may be rather well qualitatively understood in\nthe framework of quasiclassical Eilenberger6and Usadel7\napproaches. However, the applicability of these meth-\nods assumes that the exchange ﬁeld hin the ferromagnet\nshould be much smaller than the Fermi energy h≪EF\nand the use of the Usadel equations implies even more\nrestrictive conditions hτ≪1, whereτis the electrons\nscattering time. These circumstances lead to the fact\nthat some subtle qualitative eﬀects may be missed by the\nquasiclassical approach, see, for example8–10. Moreover,\na lot of experimental activities with the S/F heterostruc-\ntures deal with the strong ferromagnets (or even half-\nmetals11–13) for which the quasiclassical approximation\ncannot provide an adequate quantitative description.\nThe alternative approach for the analysis of proximity\neﬀects in strongferromagnetsis the use of the microscop-\nical approach on the basis of the Bogoliubov-de Gennes\n(BdG) equations14. The exact numerical solutions of\nthese equations may provide additional information to\nthe quasiclassical approach and this method was used\nin15–24and references cited therein. Recently the inter-\nesting experimental results were obtained for the Joseph-\nson junctions containing a ferromagnetic spin valve25–31.\nTaking in mind these experiments in the present work\nwe study the SFS junctions with composite F layer con-\nsisting of two parallel or antiparallel ferromagnetic layers\nseparated by either a potential or a spin-active barrier.Note that previously the Josephson junctions with fer-\nromagnetic bilayers were studied theoretically by diﬀer-\nent methods15–19,32–39. However, most of the theoretical\nanalysis was made in the framework of the quasiclassi-\ncal approach, while in the present work we discuss some\neﬀects which cannot be found by this approach and has\nnot been discussed before. We have calculated the criti-\ncal current of the Josephson junctions with a composite\n(spin-valve) F 1F2interlayer and studied the role of the\npotential and spin-active barrier at F 1/F2interface. The\nobtained results show an anomalous behavior of the crit-\nical current as a function of the exchange ﬁeld and/or\nF layer thickness which is very sensitive to the type of\nbarrier at the F 1/F2interface.\nII. MODEL AND FORMULA\nThe consideredSF 1F2S Josephsonjunction with a cen-\ntral potential or spin-active barrier is shown schemati-\ncally in Fig. 1. The xaxis is chosen to be perpendicu-\nlar to the layer interfaces with the origin located at the\ncentral F 1/F2interface. The BCS mean-ﬁeld eﬀective\nHamiltonian is2,14\nHeﬀ=/summationdisplay\nα,β/integraldisplay\ndr/braceleftBig\nˆψ†\nα(r)[He−(hzˆσz)αα]ˆψα(r)\n+1\n2/bracketleftBig\n(iˆσy)αβ∆(r)ˆψ†\nα(r)ˆψ†\nβ(r)+H.c./bracketrightBig\n+ˆψ†\nα(r)(Uˆσ0−/vector ρ·/vector σ)αβˆψβ(r)/bracerightBig\n, (1)\nwhereHe=−/planckover2pi12∇2\n2m−EF, andˆψ†\nα(r) andˆψα(r) repre-\nsent creation and annihilation operators with spin α.σ02\nSz\ny\n0x\nd2−d1F1\nSF2\nFIG. 1. The sketch of SF 1F2S Josephson junction with a po-\ntential or spin-active barrier at F 1/F2interface. The lengths\nof F1and F 2are denoted by d1andd2, respectively.\ndenotes a 2 ×2 unit matrix, and /vector σ= (ˆσx,ˆσy,ˆσz) is the\nvector of Pauli matrices. Here mdenotes the eﬀective\nmass of the quasiparticles in both the superconductors\nand the ferromagnets and EFis the Fermi energy. We\nassume equal Fermi energies in the diﬀerent regions of\nthe junction. The superconducting gap is supposed to\nbe constant in the superconducting leads and absent in-\nside the ferromagnetic region:\n∆(r) =\n\n∆eiφ/2, x< −d1\n0, −d1<x<d 2\n∆e−iφ/2, x>d 2,(2)\nwhere ∆ is the magnitude of the gap, and φis the phase\ndiﬀerence between the two superconducting leads. This\napproximation is justiﬁed when, for example, the width\nof the superconducting layers is much larger than the\nwidth of F layers. We model the central F 1/F2interface\nby aδfunction potential barrier which consists of a spin-\nindependent part U=V0δ(x) and a spin-active part /vector ρ=\n(ρx,ρy,ρz)δ(x). The exchange ﬁeld in two ferromagnetic\nlayers is parallel or antiparallel to the zaxis. It has the\nform\nhz=/braceleftbigg\nh1ˆz,−d1<x<0\n±h2ˆz,0<x<d 2,(3)\nwhere ˆzis the unit vector along the zaxis.\nTo diagonalize the eﬀective Hamiltonian, we use the\nBogoliubov transformation ˆψα(r) =/summationtext\nn[unα(r)ˆγn+\nv∗\nnα(r)ˆγ†\nn] and take into account the anticommutation re-\nlations of the quasiparticle annihilation operator ˆ γnand\ncreation operator ˆ γ†\nn. Using the presentation unα(r) =\nuα\npeipx,vnα(r) =vα\npeipx, the resulting Bogoliubov–de\nGennes (BdG) equations can be expressed as14\n/parenleftbiggˆH0+ˆVδ(x)iˆσy∆(x)\n−iˆσy∆∗(x)−ˆH0−ˆV∗δ(x)/parenrightbigg/parenleftbigg\nˆu(x)\nˆv(x)/parenrightbigg\n=ǫ/parenleftbigg\nˆu(x)\nˆv(x)/parenrightbigg\n,\n(4)\nwhere\nˆH0=/parenleftbigg\nξp−hz0\n0ξp+hz/parenrightbigg\n,and\nˆV=/parenleftbigg\nV0−ρz−(ρx−iρy)\n−(ρx+iρy)V0+ρz/parenrightbigg\n.\nHereξp=/planckover2pi12p2\n2m−EF, and ˆu(x) = [u↑\np(x),u↓\np(x)]Tand\nˆv(x) = [v↑\np(x),v↓\np(x)]Tare quasiparticle and quasihole\nwave functions, respectively.\nThe BdG equation (4) can be solved for each super-\nconducting electrode and each ferromagnetic layer, re-\nspectively. For a given energy ǫin the superconducting\ngap, we ﬁnd the following plane-wavesolutions in the left\nsuperconducting electrode:\nψS\nL(x) =C1ˆζ1e−ik+\nSx+C2ˆζ2eik−\nSx(5)\n+C3ˆζ3e−ik+\nSx+C2ˆζ4eik−\nSx,\nwherek±\nS=kF/radicalBig\n1±i√\n∆2−ǫ2/EF−(k/ba∇dbl/kF)2are the\nlongitudinalcomponentsofthewavevectorsforquasipar-\nticles in both superconductors. ˆζ1= [1,0,0,R1e−iφ/2]T,\nˆζ2= [1,0,0,R2e−iφ/2]T,ˆζ3= [0,1,−R1e−iφ/2,0]T, and\nˆζ4= [0,1,−R2e−iφ/2,0]Tare the four basis wave func-\ntions of the left superconductor, in which R1(2)= (ǫ∓\ni√\n∆2−ǫ2)/∆. The corresponding wave function in the\nright superconducting electrode can be described by\nψS\nR(x) =D1ˆη1eik+\nSx+D2ˆη2e−ik−\nSx(6)\n+D3ˆη3eik+\nSx+D4ˆη4e−ik−\nSx,\nwhere ˆη1= [1,0,0,R1eiφ/2]T, ˆη2= [1,0,0,R2eiφ/2]T,\nˆη3= [0,1,−R1eiφ/2,0]T, and ˆη4= [0,1,−R2eiφ/2,0]T.\nThe wave function in the F 1layer is\nψF1(x) = (M1eik1x+M′\n1e−ik1x)ˆe1+(M2eik2x+M′\n2e−ik2x)ˆe2\n+(M3eik3x+M′\n3e−ik3x)ˆe3+(M4eik4x+M′\n4e−ik4x)ˆe4,\n(7)\nwhere ˆe1= (1 0 0 0)T, ˆe2= (0 1 0 0)T,\nˆe3= (0 0 1 0)T, and ˆe4= (0 0 0 1)Tare ba-\nsis wave functions in the ferromagnetic region, and\nk1(2)=kF/radicalBig\n1+(ǫ±h1)/EF−/parenleftbig\nk/ba∇dbl/kF/parenrightbig2andk3(4)=\nkF/radicalBig\n1−(ǫ∓h1)/EF−/parenleftbig\nk/ba∇dbl/kF/parenrightbig2are the longitudinal\ncomponents of the wave vectors for the quasiparticles in\nthe F1layer. The corresponding wave function ψF2(x)\nin the F 2layer can be obtained from Eq. (7) by replace-\nmenth1→h2. It is worthy to note that the parallel\ncomponent k/ba∇dblis conserved in transport processes of the\nquasiparticles.\nThe wave functions [ ψS\nL(x),ψF1(x),ψF2(x), and\nψS\nR(x)] and their ﬁrst derivatives should satisfy the\nboundary conditions at the S/F 1, F1/F2, and F 2/S in-3\nterfaces,\nψS\nL(−d1) =ψF1(−d1),\n∂ψS\nL\n∂x|x=−d1=∂ψF1\n∂x|x=−d1, (8)\nψF1(0) =ψF2(0),\ndψF2\ndx|x=0+−dψF1\ndx|x=0−=kF/parenleftbiggˆW0\n0ˆW∗/parenrightbigg\nψ(0),(9)\nψF2(d2) =ψS\nR(d2),\n∂ψF2\n∂x|x=d2=∂ψS\nR\n∂x|x=d2, (10)\nwhere\nˆW=/parenleftbigg\nZ−Pz−(Px−iPy)\n−(Px+iPy)Z+Pz/parenrightbigg\n.(11)\nWe deﬁne the dimensionless spin-independent parameter\nZ= 2mV0/(/planckover2pi12kF) measuring the strength of the poten-\ntial barrierand the dimensionless spin-dependent param-\neterPx,y,z= 2mρx,y,z/(/planckover2pi12kF) describing the spin-active\nbarrier at the F 1/F2interface. For simplicity, we just\nconsider the eﬀect of y-component Pyand ignore the role\nofx- andy-components ( PxandPz).\nFromtheseboundaryconditionswecansetup24linear\nequations in the following form:\nˆAX=ˆB, (12)\nwhereXcontains 24 scattering coeﬃcients and ˆAis a\n24×24 matrix. The solution of the characteristic equa-\ntion\ndetˆA= 0 (13)\nallows one to identify two Andreev bound-state solutions\nfor energies EAω(ω=1, 2). Below we will consider the\ncase of the short Josephson junction with a thickness\nmuch smaller than the superconducting coherence length\nξ. In such a case the main contribution to the Josephson\ncurrent is provided by the Andreev bound states (see,\ne.g.,40,41). In a one-dimensional (1D) SF 1F2S junction,\nthe Josephson current can be calculated by the general\nformula\nI1d(φ) =2e\n/planckover2pi1∂Ω\n∂φ, (14)\nwhere Ω is the phase-dependent thermodynamic poten-\ntial. This potential can be obtained from the excitation\nspectrum by using the formula42,43\nΩ =−2T/summationdisplay\nωln/bracketleftbigg\n2coshEAω(φ)\n2T/bracketrightbigg\n,(15)\nwhere ∆,h1,h2,ZandPyare assumed to be the equi-\nlibrium values, which minimize the free energy of the\nSF1F2S structure and depend on microscopic parame-\nters44. The summation in (15) is taken over all positive(b) (a)\nFIG. 2. The dependence of the 3D critical current I3d\nc\non the ferromagnetic thickness kFdfor the exchange ﬁeld\nh/EF= 0.1 (a) and on the exchange ﬁeld h/EFfor the ferro-\nmagnetic thickness kFd= 20 (b) when the potential barrier Z\ntakes several diﬀerent values. Here we consider an antipara llel\norientation of the exchange ﬁelds.\nAndreev energies [0 < EAω(φ)<∆]. For each value\nofφ, we solve Eq. (13) numerically to obtain the two\nspin-polarized Andreev levels. Since the Andreev energy\nspectra are doubled as they include the Bogoliubov re-\ndundancy, and only half of the energy states should be\ntaken into account, we can ﬁnd the 1D Josephsoncurrent\nvia Eqs. (14) and (15).\nIn a three-dimensional (3D) case, the Josephson cur-\nrent can be expressed as\nI3d(φ) =S\n4π22e∆\n/planckover2pi1/integraldisplaykF\n0I1d(k/ba∇dbl)2πk/ba∇dbldk/ba∇dbl\n=4π∆\neRN/integraldisplay1\n0I1d(˜k/ba∇dbl)˜k/ba∇dbld˜k/ba∇dbl, (16)\nwhereR−1\nN=e2k2\nFS/(4π2/planckover2pi1) isthe Sharvinresistanceand\n˜k/ba∇dbl=k/ba∇dbl/kFis the normalized wave vector. The 3D crit-\nical current can be derived from I3d\nc=maxφ|I3d(φ)|.\nIII. RESULTS AND DISCUSSIONS\nIn our calculations we use the superconducting gap ∆\nas a unit of energy and take the Fermi energy EF=\n1000∆. All lengths and the exchange ﬁeld strengths\nare measured in units of the inverse Fermi wave vec-\ntorkFand the Fermi energy EF, respectively. Note\nthat the approximation of the short Josephson junc-\ntion (kFd1,kFd2≪1000) is fully satisﬁed in the pre-\nsented calculations. The normalized unit of current is\nI0= 2π∆/(eRN) in the 3D case.4\n(a)\n(b)\nFIG. 3. The dependence of the 3D critical current I3d\ncon the\nferromagnetic thickness kFd(a) and on the exchange ﬁeld\nh/EF(b) in the case of an antiparallel orientation. Here the\npotential barrier is Z= 2.\nWe study the SF 1F2S structures with a potential bar-\nrierZor a spin-active barrier Pyat the F 1/F2interface.\nWe present the results for h1=h2=hfor parallel ex-\nchange ﬁeld and h1=−h2=hfor antiparallel exchange\nﬁeld, and deﬁne the ferromgnetic thickness d1=d2=d.\nWe draw in Fig. 2 the dependence of the critical cur-\nrentI3d\ncon the ferromagnetic thickness kFdand the ex-\nchange ﬁeld h/EFfor an antiparallel alignment of the\nmagnetic moment h1=−h2=hwhen the potential bar-\nrierZtakes several diﬀerent values. It is shown that\nthe critical current decreases monotonically with the in-\ncreasing exchange ﬁeld for the transparent F 1/F2inter-\nfaceZ= 0, while it reveals the oscillating behavior for\nZ >0. By increasing Z, the amplitude of the critical\ncurrent decreases as a whole, but the oscillation behavior\nstill remains. The critical currentshowsthe same charac-\nteristic if one increases the ferromagnetic thickness kFd.\nThese features indicate that the oscillation of the criti-\ncal current originates from the resonant tunneling of the\nCooper pairs occurring between the F 1and F2layers. In\nfact the spin-dependent wave vector of the pairing elec-(a) (b)\nFIG. 4. The dependence of the 3D critical current I3d\nc\non the ferromagnetic thickness kFdfor the exchange ﬁeld\nh/EF= 0.1 (a) and on the exchange ﬁeld h/EFfor the ferro-\nmagnetic thickness kFd= 20 (b) in the case of an antiparallel\norientation. Here the temperature T/∆ varies from 0 to 0.9\nwith a step 0.1, which corresponds to the curves from top to\nbottom. The potential barrier is Z= 2.\ntrons will change when the Cooper pairs pass through\nthe F1and F2layer, and therefore the phase evolution of\nthe Cooper pairs leads to the resonances occurring in F 1\nand F2. Therefore, the oscillation period depends on the\nexchange ﬁeld and/or thickness of the ferromagnets, not\non the properties of the central insulating barrier.\nIf one changes the exchange ﬁeld h/EFand thickness\nkFdof the ferromagnetic layers, the oscillation period of\nthe critical current changes accordingly. The calculation\nresults are illustrated in Fig. 3. The observed oscillations\nremind us of the oscillations observed previously in19for\nthe 1D model of the junction with noncollinear magne-\ntization and attributed to the geometrical resonances.\nThe interesting consequence of the presence of barrier is\nthe counterintuitive increase of the critical current with\nincreasing exchange ﬁeld [up to hmax/EF∼0.12 when\nkFd= 10, see Fig. 3(b)] or ferromagnetic layer thickness\n[up tokFdmax∼24 whenh/EF= 0.05, see Fig. 3(a)].\nNote that the similar increase of the current with the ex-\nchange ﬁeld was obtained in the models of S/F tunnel\nstructures34–36. The key diﬀerence between our results\nand Refs.34–36is that the initial increase wasnot followed\nby the oscillatory behavior of the critical current with\nexchange ﬁeld and/or ferromagnetic layer thickness. We\nﬁndalsothatthegrowthrangeofthecriticalcurrentwith\nthe exchange ﬁeld strongly depends on the ferromagnetic\nlayer thicknesses. Moreover, note that in presence of\na potential barrier the critical current slightly increases\nwith normal-metalthicknesswhen both ferromagnetsbe-\ncome the normal-metal ( h/EF= 0) [see Fig. 3(a)]. This5\n(a)\n(b)\nFIG. 5. The dependence of the 3D critical current I3d\ncon the\nferromagnetic thickness kFd(a) and on the exchange ﬁeld\nh/EF(b) in the case of a parallel orientation. Here the spin-\nactive barrier is taken as Py= 2.\ncircumstance reﬂects the presence of some resonance ef-\nfects in this case too. It should be noted that the os-\ncillatory eﬀect mentioned above is revealed only at low\ntemperatures. The dependence of the critical current on\nthe temperature is illustrated in Fig. 4. We see that the\noscillations of the critical current I3d\ncwill decrease as the\ntemperature T/∆ increases, which should be related to\nthe smearing of the resonance tunneling from the lowest\nAndreev levels. When T/∆ reaches 0.9, the oscillation\ncompletely disappears and I3d\ncdecreases monotonously.\nIn Fig. 5, we show the variation characteristics of the\ncritical current in SF 1F2S structure with the magnetic\nmoment in F 1and F 2being parallel and with a spin-\nactive barrier at the F 1/F2interface. It is found that the\ncritical current also displays an oscillating behavior. The\nbehavior is similar to the cases in which the magnetic\nmoments are antiparallel and there is a potential barrier\nat the F 1/F2interface. So we can say that the spin-\nactive barrier at the F 1/F2interface can play two roles:\n(i) It creates a spin-ﬂip eﬀect to ﬂip the spin of the con-\nduction electrons crossing the F 1/F2interface. The two(b)(a)\nFIG. 6. The dependence of the 3D critical current I3d\ncon the\nferromagnetic thickness kFd(a) and on the exchange ﬁeld\nh/EF(b) in the case of an antiparallel orientation. Here the\nspin-active barrier is taken as Py= 2. The dips at the curves\nsignal the transitions between 0 and πstates.\nferromagnets have the same energy band because of the\nparallel polarized direction of the magnetic moments. In\nsuch a case, spin- ↑(↓) electrons will be transformed into\nspin-↓(↑) electronswhen they passfrom the F 1layerinto\nthe F2layer. The same electron will occupy the opposite\nspin band in the F 1and F2layers. This situation is sim-\nilar to the antiparallel ferromagnets without the spin-ﬂip\nin the central interface. (ii) It acts as a potential barrier,\nwhich hinders electron tunneling and reduces the trans-\nmission of the F 1/F2interface. Therefore, we can still\nsee the oscillating phenomenon of the critical current in\nthis structure.\nSimilarly, the abovetworolescausedbythe spin-active\nbarrier can also present in the antiparallel SF 1F2S junc-\ntion. If one only considers the role of the spin-ﬂip eﬀect,\nthe antiparallel SF 1F2S junction with a central spin-ﬂip\nis equivalent to a homogenous SFS junction. In this\ncase, the 0- πtransition will resume. For example, at\nh/EF= 0.05 the inversion of the current sign takes place\natkFd≈10 andkFd≈40 (see Fig. 6(a) and Fig. 16\n(b)(a)\nFIG. 7. The dependence of the 3D critical current I3d\ncon the\nferromagnetic thickness kFd(a) and on the exchange ﬁeld\nh/EF(b) in the case of a parallel orientation. Here the po-\ntential barrier is taken as Z= 2. The dips at the curves signal\nthe transitions between 0 and πstates.\nin Supplemental Material45). In other words, the junc-\ntion is inπstate forkFd≺10 andkFd≻40, as well\nas it will become 0 state in the region 10 ≺kFd≺40.\nMoreover, the insulating property of the spin-active bar-\nrier causes a resonant tunneling of electrons. This re-\nsults in the largest peaks that appear periodically in the\ncurrentI3d\nc. For example, if one looks at the curve for\nh/EF= 0.10 in Fig 6(a), the resonance produces the\npeaks atkFd≈10 andkFd≈40, which appear in simi-\nlar positions in Fig. 3(a).\nTo further demonstrate the coexistence of resonant\ntunneling and 0- πtransition, we calculated the current\nin the parallel SF 1F2S junction with a central potential\nbarrier. It is known that, in a uniform SFS junction,\nthe critical current decays with increasing ferromagnetic\nthickness (or exchange ﬁeld) and also reveals oscillations\ncaused by the 0- πtransition. If the potential barrier\nis introduced at the center of the ferromagnet, the am-\nplitude of the critical current will be suppressed overall\nbecause the potential barrier reduces the transmission of(b)(a)\nFIG. 8. The dependence of the 3D critical current I3d\ncon\nthe ferromagnetic thickness kFd1(a) and on the exchange\nﬁeldh1/EF(b) for the SF 1S conﬁguration ( d2= 0) with the\npotential barrier Z= 2 at the right F 1/S interface. The dips\nat the curves signal the transitions between 0 and πstates.\nthe conduction electrons. Meanwhile, the resonant tun-\nneling of the conduction electrons between F 1and F 2\nlayers induces the periodic peaks in the critical current.\nAs a result, the critical current shows singular features\nin Fig. 7 and Fig. 2 of the Supplemental Material45.\nFinally, in order to illustrate the previous conjecture\nregarding resonant tunneling, we discuss the current in\nthe SF 1S junction ( d2= 0) with the potential or spin-\nactive barriers at the F 1/S interface. As shown in Figs. 8\nand 9, the critical current displays a damped oscilla-\ntionwith increasingthickness kFd1and/orexchangeﬁeld\nh1/EF. These current oscillations can be attributed to\nthe 0-πtransition (see Figs. 3 and 4 in the Supplemental\nMaterial45) but not to the periodic peaks induced by the\nresonanttunnelingbetweentheF 1andF2layers,because\nthe resonant tunneling cannot exist in these structures.\nInaddition, we ﬁnd thatthe criticalcurrentatthe transi-\ntionbetweenthe0and πstatesisclosetozerointheSF 1S\njunction with the potential barrier Z= 2 at the F 1/S in-\nterface, when the thickness kFd1and/or exchange ﬁeld7\n(a)\n(b)\nFIG. 9. The dependence of the 3D critical current I3d\ncon the\nferromagnetic thickness kFd1(a) and on the exchange ﬁeld\nh1/EF(b) for the SF 1S conﬁguration ( d2= 0) with the spin-\nactive barrier Py= 2 at the right F 1/S interface. The dips at\nthe curves signal the transitions between 0 and πstates.\nh1/EFtake larger values. However, this current is much\nlarger when the F 1/S interface has a spin-active barrier\nPy= 2 (see Fig. 9). This may be related to the impor-\ntant contribution from the second harmonic current inthe presence of spin-active interface structure34,46–48.\nIV. CONCLUSION\nOn the basis of the exact numerical solution of the\nBogoliubov-de Gennes equations, we have studied the\nJosephson current in the SF 1F2S junctions containing a\npotential or spin-active barrier at F 1/F2interface. We\nshow that at low temperature the potential barrier may\nresultin largeoscillationsofthecriticalcurrentasafunc-\ntion of the ferromagnetic layer thickness and exchange\nﬁeld even for the antiparallel orientation of the mag-\nnetic moment in the F 1and F 2layers. Such behavior\nis related to the interference eﬀects of the electrons wave\nfunctions and may be considered as some form of the ge-\nometrical resonance phenomena. Speciﬁcally, comparing\nto the normal-metal junction ( h= 0 in our model), the\nexchange ﬁeld ( h >0) can enhance the critical current\nfor the antiparallel conﬁguration. In contrast, the spin-\nactive barrier in this antiparallel conﬁguration leads to\nthe 0-πtransitions, which is similar to the case of uni-\nformSFS junction. The spin-activebarrierinthe parallel\nconﬁguration can also cause the oscillations of the criti-\ncal current. The obtained results may be useful for the\ninterpretation of the experimental data on the Josephson\njunctions with composite ferromagnetic barrier.\nACKNOWLEDGMENTS\nThe authors thank A. Melnikov and S. Mironov for\nuseful discussions and suggestions. This work was\nsupported by French ANR projects SUPERTRONICS\nand OPTOFLUXONICS, EU Network COST CA16218\n(NANOCOHYBRI), ANR-DFG grant “Fermi-NESt”,\nand ERC 647100 “SUSPINTRONICS”. H. Meng was\nsupported by the National Natural Science Foundation\nof China (Grant No.11604195 and No.11447112) and the\nYouth Hundred TalentsProgrammeofShaanxiProvince.\n∗alexandre.bouzdine@u-bordeaux.fr\n1A. A. Golubov, M. Yu. Kupriyanov, and E. Ilichev, The\ncurrent-phase relation in Josephson junctions, Rev. Mod.\nPhys.76, 411 (2004).\n2A. I. Buzdin, Proximity eﬀects in superconductor-\nferromagnet heterostructures, Rev. Mod. Phys. 77, 935\n(2005).\n3F. S. Bergeret, A. F. Volkov, and K. B. Efetov,\nOdd triplet superconductivity and related phenomena in\nsuperconductor-ferromagnet structures, Rev. Mod. Phys.\n77, 1321-1373 (2005).\n4J. Linder and J. W. A. Robinson, Superconducting spin-\ntronics, Nat. Phys. 11, 307 (2015).5M. Eschrig, Spin-polarized supercurrents for spintronics :\na review of current progress, Rep. Prog. Phys. 78, 104501\n(2015).\n6G. Eilenberger, Transformation of Gorkov’s equation for\ntype II superconductors into transport-like equations, Z.\nPhys.214, 195-213 (1968).\n7K. D. Usadel, Generalized Diﬀusion Equation for Super-\nconducting Alloys, Phys. Rev. Lett. 25, 507 (1970).\n8C. R. Reeg and D. L. Maslov, Proximity-induced triplet\nsuperconductivity in Rashba materials, Phys. Rev. B 92,\n134512 (2015).\n9M. A. Silaev, I. V. Tokatly, and F. S. Bergeret, Anoma-\nlouscurrentindiﬀusiveferromagnetic Josephsonjunction s,\nPhys. Rev. B 95, 184508 (2017).8\n10Hao Meng, A. V. Samokhvalov, and A. I. Buzdin, Nonuni-\nform superconductivity and Josephson eﬀect in a conical\nferromagnet, Phys. Rev. B 99, 024503 (2019).\n11C. Visani, Z. Sefrioui, J. Tornos, C. Leon, J.\nBriatico, M. Bibes, A. Barth´ el´ emy, J. Santamar´ ıa,\nand Javier E. Villegas,Equal-spin Andreev reﬂection\nand long-range coherent transport in high-temperature\nsuperconductor/half-metallic ferromagnet junctions, Na -\nture Physics 8, 539-543 (2012).\n12C. Visani, F. Cuellar, A. P´ erez-Mu˜ noz, Z. Sefrioui,\nC. Le´ on, J. Santamar´ ıa, and Javier E. Villegas,\nMagnetic ﬁeld inﬂuence on the proximity eﬀect at\nYBa2Cu3O7/La2/3Ca1/3MnO3superconductor/half-metal\ninterfaces, Phys. Rev. B 92, 014519 (2015).\n13M. Egilmez, J. W. A. Robinson, Judith L. MacManus-\nDriscoll, L. Chen, H. Wang and M. G. Blamire, Supercur-\nrents in half-metallic ferromagnetic La 0.7Ca0.3MnO3, Eu-\nrophys. Lett. 106, 37003 (2014).\n14P. G. de Gennes, Superconductivity of Metals and Alloys,\nBenjamin, New York, 1966 (Chap.5).\n15Z. Radovi´ c, N. Lazarides, and N. Flytzanis, Josephson\neﬀect in double-barrier superconductor-ferromagnet junc -\ntions, Phys. Rev. B 68, 014501 (2003).\n16Klaus Halterman and Oriol T. Valls, Layered ferromagnet-\nsuperconductor structures: The πstate and proximity ef-\nfects, Phys. Rev. B 69, 014517 (2004).\n17Paul H.Barsic, Oriol T. Valls, andKlaus Halterman, Ther-\nmodynamics and phase diagrams of layered superconduc-\ntor/ferromagnet nanostructures, Phys. Rev. B 75, 104502\n(2007).\n18Klaus Halterman, Oriol T. Valls, and Chien-Te Wu,\nCharge and spin currents in ferromagnetic Josephson junc-\ntions, Phys. Rev. B 92, 174516 (2015).\n19Z. Pajovi´ c, M. Boˇ zovi´ c, Z. Radovi´ c, J. Cayssol, and A.\nBuzdin, Josephson coupling through ferromagnetic hetero-\njunctions with noncollinear magnetizations, Phys. Rev. B\n74, 184509 (2006).\n20Klaus Halterman and Mohammad Alidoust, Half-metallic\nsuperconducting triplet spin valve, Phys. Rev. B 94,\n064503 (2016).\n21Klaus Halterman and Mohammad Alidoust, Josephson\ncurrents and spin-transfer torques in ballistic SFSFS nano -\njunctions, Supercond. Sci. Technol. 29, 055007 (2016).\n22Mohammad Alidoust and Klaus Halterman, Half-metallic\nsuperconducting triplet spin multivalves, Phys. Rev. B 97,\n064517 (2018).\n23Klaus Halterman and Mohammad Alidoust, Induced en-\nergy gap in ﬁnite-sized superconductor/ferromagnet hy-\nbrids, Phys. Rev. B 98, 134510 (2018).\n24Chien-Te WuandKlaus Halterman, Spintransport in half-\nmetallic ferromagnet-superconductor junctions, Phys.Re v.\nB98, 054518 (2018).\n25C. Bell, G. Burnell, C. W. Leung, E. J. Tarte,D.-J.\nKang, and M. G. Blamire, Controllable Josephson current\nthrough a pseudospin-valve structure, Appl. Phys. Lett.\n84, 1153-1155 (2004).\n26J. W. A. Robinson, Gabor B. Halasz, A. I. Buzdin, and\nM. G. Blamire, Enhanced Supercurrents in Josephson\nJunctions ContainingNonparallel Ferromagnetic Domains,\nPhys. Rev. Lett. 104, 207001 (2010).\n27B. Baek, W. H. Rippard, S. P. Benz, S. E. Russek, and P.\nD. Dresselhaus, Hybrid superconducting-magnetic mem-\nory device using competing order parameters, Nat. Com-\nmun.5, 3888 (2014).28M.A.E.Qader, R.K.Singh, S.N.Galvi, L.Yu,J.M.Row-\nell, and N. Newman, Switching at small magnetic ﬁelds in\nJosephson junctions fabricated with ferromagnetic barrie r\nlayers, Appl. Phys. Lett. 104, 022602 (2014).\n29B. Baek, W. H. Rippard, M. R. Pufall, S. P. Benz, S. E.\nRussek, H. Rogalla, and P. D. Dresselhaus, Spin-Transfer\nTorqueSwitchinginNanopillar Superconducting-Magnetic\nHybridJosephson Junctions, Phys. Rev.Applied 3, 011001\n(2015).\n30E. C. Gingrich, Bethany M. Niedzielski, Joseph A. Glick,\nYixing Wang, D. L. Miller, Reza Loloee, W. P. Pratt Jr,\nand Norman O. Birge, Controllable 0- πJosephson junc-\ntions containing a ferromagnetic spin valve, Nat. Phys. 12,\n564 (2016).\n31Bethany M. Niedzielski, T. J. Bertus, Joseph A. Glick, R.\nLoloee, W. P. Pratt, Jr., and Norman O. Birge, Spin-valve\nJosephson junctions for cryogenic memory, Phys. Rev. B\n97, 024517 (2018).\n32A. V.Samokhvalov, R.I. Shekhter, andA. I.Buzdin, Stim-\nulation of a singlet superconductivity in SFS weak links by\nspin-exchange scattering of Cooper pairs, Sci. Rep. 4, 5671\n(2014).\n33Iryna Kulagina and Jacob Linder, Spin supercurrent, mag-\nnetization dynamics, and ϕ-state in spin-textured Joseph-\nson junctions, Phys. Rev. B 90, 054504 (2014).\n34F. S. Bergeret, A. F. Volkov, and K. B. Efetov, Enhance-\nment of the Josephson Current by an Exchange Field in\nSuperconductor-Ferromagnet Structures, Phys. Rev. Lett.\n86, 3140 (2001).\n35V. N. Krivoruchko and E. A. Koshina, From inversion to\nenhancement of the dc Josephson current in S/F-I-F/S\ntunnel structures, Phys. Rev. B 64, 172511 (2001).\n36Elena Koshina and Vladimir Krivoruchko, Spin polariza-\ntion and π-phase state of the Josephson contact: Critical\ncurrent of mesoscopic SFIFS and SFIS junctions, Phys.\nRev. B63, 224515 (2001).\n37A. A. Golubov, M. Yu. Kupriyanov, and Ya. V. Fominov,\nCritical currentinSFIFSjunctions, Pis’maZh. Eksp.Teor.\nFiz.75, 223 (2002) [JETP Lett. 75, 190 (2002)].\n38Ya. M. Blanter and F. W. J. Hekking, Supercurrent in\nlong SFFS junctions with antiparallel domain conﬁgura-\ntion, Phys. Rev. B 69, 024525 (2004).\n39B. Crouzy, S. Tollis, and D. A. Ivanov, Josephson cur-\nrent in a superconductor-ferromagnet junction with two\nnoncollinear magnetic domains, Phys. Rev. B 75, 054503\n(2007).\n40Philip F. Bagwell, Suppression of the Josephson current\nthrough a narrow, mesoscopic, semiconductor channel by\na single impurity, Phys. Rev. B 46, 12573 (1992).\n41C. W. J. Beenakker, Universal limit of critical-current ﬂuc -\ntuations in mesoscopic Josephson junctions, Phys. Rev.\nLett.67, 3836 (1991).\n42J. Bardeen, R. K¨ umel, A. E. Jacobs, and L. Tewordt,\nStructure of Vortex Lines in Pure Superconductors, Phys.\nRev.187, 556 (1969).\n43J. Cayssol and G. Montambaux, Exchange-induced or-\ndinary reﬂection in a single-channel superconductor-\nferromagnet-superconductor junction, Phys. Rev. B 70,\n224520 (2004).\n44L. N. Bulaevskii, A. I. Buzdin, M. L. Kuli´ c, and S.\nV. Panyukov, Coexistence of superconductivity and mag-\nnetism theoretical predictions and experimental results,\nAdv. Phys. 34, 175-261 (1985).9\n45See Supplemental Material for comparison between the ab-\nsolute value of critical current |I3d\nc|and the original critical\ncurrentI3d\nc.\n46Luka Trifunovic, Long-Range Superharmonic Josephson\nCurrent, Phys. Rev. Lett. 107, 047001 (2011).47Caroline Richard, Manuel Houzet, and Julia S. Meyer, Su-\nperharmonic Long-Range Triplet Current in a Diﬀusive\nJosephson Junction, Phys. Rev. Lett. 110, 217004 (2013).\n48A. S. Melnikov, A. V. Samokhvalov, S. M. Kuznetsova,\nandA.I.Buzdin, InterferencePhenomenaandLong-Range\nProximity Eﬀect in Clean Superconductor-Ferromagnet\nSystems, Phys. Rev. Lett. 109, 237006 (2012).arXiv:1912.04447v2  [cond-mat.supr-con]  28 Dec 2019Supplementary material for “Josephson current through a fe rromagnetic bilayer:\nbeyond the quasiclassical approximation”\nHao Meng,1,2Yajie Ren,1Javier E. Villegas,3and A. I. Buzdin2,4,∗\n1School of Physics and Telecommunication Engineering,\nShaanxi University of Technology, Hanzhong 723001, China\n2University Bordeaux, LOMA UMR-CNRS 5798, F-33405 Talence C edex, France\n3Unit´ e Mixte de Physique CNRS/Thales, Universit´ e Paris-S ud,\nUniversit´ e Paris Saclay, 1 Avenue A. Fresnel, 91767 Palais eau, France\n4Sechenov First Moscow State Medical University, Moscow, 11 9991, Russia\n(Dated: January 1, 2020)\nIn our article, the critical current is deﬁned by formula\nI3d\nc=maxφ|I3d(φ)|, so one cannot see the inversion of\nthe current sign, which correspondsto the 0- πtransition.\nIn order to clearly show the 0- πtransition, we plot the\nabsolute value of critical current |I3d\nc|and the original\ncritical current I3d\ncin the following ﬁgures.\nFigure 1 shows the critical current in the antiparal-\nlel SF1F2S junction with the central spin-active barrier\nPy= 2. Meanwhile, in Fig. 2 we plot the critical cur-\nrent in the parallel SF 1F2S junction with the central po-tential barrier Z= 2. From the right columns in the\nabove two ﬁgures, we can see that the critical current\nI3d\ncchanges between negative and positive values. This\nfeature demonstrates that the crossover between 0 and π\nstates takes place.\nIn addition, we calculate the critical current in the\nSF1S junction ( d2= 0) with potential barrier Z= 0 and\nspin-activebarrier Py= 2atF 1/Sinterface. Correspond-\ning results are illustrated in Figs. 3 and 4, respectively.\nThe 0-πtransition can also be seen in these two cases.2\n(a)\n(b) (B)(A)\nFIG. 1. The dependence of the absolute value of critical curr ent|I3d\nc|(left column) and the original critical current I3d\nc(right\ncolumn) on the ferromagnetic thickness kFd[(a) and (A)] and on the exchange ﬁeld h/EF[(b) and (B)] for the antiparallel\nSF1F2S junction with the central spin-active barrier Py= 2. Figures in left column correspond to Fig.6 in our article .3\n(a)\n(b)(A)\n(B)\nFIG. 2. The dependence of the absolute value of critical curr ent|I3d\nc|(left column) and the original critical current I3d\nc(right\ncolumn) on the ferromagnetic thickness kFd[(a) and (A)] and on the exchange ﬁeld h/EF[(b) and (B)] for the parallel SF 1F2S\njunction with the central potential barrier Z= 2. Figures in left column correspond to Fig.7 in our article .4\n(a) (A)\n(B) (b)\nFIG. 3. The dependence of the absolute value of critical curr ent|I3d\nc|(left column) and the original critical current I3d\nc(right\ncolumn) on the ferromagnetic thickness kFd1[(a) and (A)] and on the exchange ﬁeld h1/EF[(b) and (B)] for SF 1S junction\nwith potential barrier Z= 2 at F 1/S interface. Figures in left column correspond to Fig.8 in o ur article.5\n(A)\n(B)(a)\n(b)\nFIG. 4. The dependence of the absolute value of critical curr ent|I3d\nc|(left column) and the original critical current I3d\nc(right\ncolumn) on the ferromagnetic thickness kFd1[(a) and (A)] and on the exchange ﬁeld h1/EF[(b) and (B)] for SF 1S junction\nwith spin-active barrier Py= 2 at F 1/S interface. Figures in left column correspond to Fig.9 in o ur article."    },    {        "title": "1003.3769v1.Dynamics_of_magnetization_on_the_topological_surface.pdf",        "content": "arXiv:1003.3769v1  [cond-mat.mes-hall]  19 Mar 2010Dynamics of magnetization on the topological surface\nTakehito Yokoyama1, Jiadong Zang2,3, and Naoto Nagaosa2,4\n1Department of Physics, Tokyo Institute of Technology, Toky o 152-8551, Japan\n2Department of Applied Physics, University of Tokyo, Tokyo 1 13-8656, Japan\n3Department of Physics, Fudan University, Shanghai 200433, China\n4Cross Correlated Materials Research Group (CMRG), ASI, RIK EN, WAKO 351-0198, Japan\n(Dated: October 17, 2018)\nWe investigate theoretically the dynamics of magnetizatio n coupled to the surface Dirac fermions\nof athree dimensional topological insulator, byderiving t heLandau-Lifshitz-Gilbert (LLG) equation\nin the presence of charge current. Both the inverse spin-Gal vanic eﬀect and the Gilbert damping\ncoeﬃcient αare related to the two-dimensional diagonal conductivity σxxof the Dirac fermion,\nwhile the Berry phase of the ferromagnetic moment to the Hall conductivity σxy. The spin transfer\ntorque and the so-called β-terms are shown to be negligibly small. Anomalous behavior s in various\nphenomena including the ferromagnetic resonance are predi cted in terms of this LLG equation.\nPACS numbers: 73.43.Nq, 72.25.Dc, 85.75.-d\nTopologicalinsulator(TI) providesa new state of mat-\nter topologically distinct from the conventional band in-\nsulator[1]. In particular,the edge channelsorthe surface\nstates are described by Dirac fermions and protected by\nthe band gap in the bulk states, and backward scatter-\ning is forbidden by the time-reversal symmetry. From\nthe viewpoint of the spintronics, it oﬀers a unique op-\nportunity to pursue novel functions since the relativistic\nspin-orbit interaction plays an essential role there. Actu-\nally, several proposals have been made such as the quan-\ntized magneto-electric eﬀect [2], giant spin rotation [3],\nmagneto-transport phenomena [4], and superconducting\nproximity eﬀect including Majorana fermions [5–7].\nAlso, a recent study focuses on the inverse spin-\nGalvanic eﬀect in a TI/ferromagnet interface, predicting\nthe current-induced magnetization reversal due to the\nHall current on the TI [8]. In Ref. [8], the Fermi energy is\nassumed to be in the gap of the Dirac dispersion opened\nby the exchange coupling. In this case, the quantized\nHall liquid is realized, and there occurs no dissipation\ncoming from the surface Dirac fermions.\nHowever, in realistic systems, it is rather diﬃcult to\ntune the Fermi energy in the gap since the proximity-\ninducedexchangeﬁeldisexpectedtobearound5-50meV.\nTherefore, it is important to consider the generic case\nwhere the Fermi energy is at the ﬁnite density of states\nof Dirac fermions, where the diagonal conductivity is\nmuch larger than the transverse one, and the damping of\nthe magnetizationbecomes appreciable. Related systems\nare semiconductors and metals with Rashba spin-orbit\ninteraction, where the spin-Galvanic eﬀect and current\ninduced magnetization reversal have been predicted [9]\nand experimentally observed [10, 11]. Compared with\nthese systems where the Rashba coupling constant is a\nkey parameter, the spin and momentum in TI is tightly\nrelated to each other corresponding to the strong cou-\npling limit of spin-orbit interaction, and hence the gigan-\ntic spin-Galvanic eﬀect is expected.\n\u0001\u0000 \u0002 \u0003\u0004 \u0005 \u0006\n\u0007 \b \t\n\n\u000b \f \r \u000e\u000f\n\u0010\u0011 \u0012 \u0013 \u0014 \u0015\u0016 \u0017\u0018\u0019\u001a \u001b\u001c \u001d\n\u001e\u001f\nFIG. 1: (Color online) (a) Illustration of the Dirac dispers ion\non top of TI. The Fermi level εFis far above the surface\ngap opened by magnetization in the ferromagnetic layer. (b)\nSketch of FMR experiment in the soft magnetic layer. The\nsubstrate in the ﬁgure is TI, which is capped by a layer of\nsoft ferromagnet. The magnetization precesses around the\nexternal magnetic ﬁeld Heﬀ.\nIn this letter, we study the dynamics of the magnetiza-\ntion coupled to the surface Dirac fermion of TI. Landau-\nLifshitz-Gilbert (LLG) equationin the presenceofcharge\ncurrent is derived microscopically, and (i) inverse spin-\nGalvanic eﬀect, (ii) Gilbert damping coeﬃcient α, (iii)\ntheso-called β-terms, and(iv)thecorrectiontotheBerry\nphase, are derived in a uniﬁed fashion. It is found that\nthese are expressed by relatively small number of param-\neters, i.e., the velocity vF, Fermi wave number kF, ex-\nchange coupling M, and the transport lifetime τof the\nDirac fermions. It is also clariﬁed that the terms re-\nlated to the spatial gradient are negligibly small when\nthe surface state is a good metal. With this LLG equa-\ntion, we propose a ferromagnetic resonance (FMR) ex-\nperiment, wheremodiﬁcationsoftheresonancefrequency\nand Gilbert damping are predicted. Combined with the\ntransport measurement of the Hall conductivity, FMR\nprovide several tests of our theory.2\nDerivation of LLG equation. — By attaching a ferro-\nmagnet on the TI as shown in Fig. 1, we can consider a\ntopological surface state where conducting electrons in-\nteract with localized spins, S, through the exchange ﬁeld\nHex=−M/integraldisplay\ndrn(r)·ˆσ(r). (1)\nHere, we set S=Snwith a unit vector npointing in the\ndirection of spin, ˆσ(r) =c†(r)σc(r) represents (twice)\nthe electronspindensity, with c†= (c†\n↑,c†\n↓) beingelectron\ncreation operators, σthe Pauli spin-matrix vector, and\nMbeing the exchange coupling energy. The total Hamil-\ntonian of the system is given by Htot=HS+Hel+Hex,\nwhereHSandHelare those for localized spins and con-\nducting electrons, respectively.\nThe dynamics of magnetization can be described by\nthe LLG equation\n˙n=γ0Heﬀ×n+α0˙n×n+t′\nel, (2)\nwhereγ0Heﬀandα0are an eﬀective ﬁeld and a Gilbert\ndamping constant, respectively, both coming from HS.\nEﬀects of conducting electrons are contained in the spin\ntorque\ntel(r)≡s0t′\nel(r) =Mn(r)×∝angbracketleftˆσ(r)∝angbracketrightne,(3)\nwhich arises from Hex. Here, s0≡S/a2is the local-\nized spin per area a2. In the following, we thus calculate\nspin polarization of conducting electrons perpendicular\nton,∝angbracketleftˆσ⊥(r)∝angbracketrightne, in such nonequilibrium states with cur-\nrent ﬂow and spatially varying magnetization to derive\ntheβ-term, or with time-dependent magnetization for\nGilbert damping. Here and hereafter, ∝angbracketleft···∝angbracketrightnerepresents\nstatistical average in such nonequilibrium states.\nFollowing Refs. [12–14] we consider a small transverse\nﬂuctuation, u= (ux,uy,0),|u| ≪1, around a uniformly\nmagnetized state, n= ˆz, such that n= ˆz+u. In the\n‘unperturbed’ state, n= ˆz, the electrons are described\nby the Hamiltonian\nH0=/summationdisplay\nkvF(kyσx−kxσy)−Mσz−εF+Vimp(4)\nwhereVimpis the impurity potential given by Vimp=\nu/summationtext\niδ(r−Ri) in the ﬁrst-quantization form. We take\na quenched average for the impurity positions Ri. The\nelectron damping rate is then given by γ= 1/(2τ) =\nπniu2νFin the ﬁrst Born approximation. Here, niis the\nconcentration of impurities, and νF=εF/(2πv2\nF) is the\ndensity of states at εF. We assume that γ≪vFkF=/radicalbig\nε2\nF−M2,M, and calculate spin transfer torque in the\nlowest non-trivial order.\nIn the presence of u(r,t) =u(q,ω)ei(q·r−ωt), the con-\nducting electrons feel a perturbation (note that Hel+\nHex=H0+H1)\nH1=−M/summationdisplay\nkσc†\nk+qσck·u(q,ω)e−iωt,(5)and acquires a transverse component\n∝angbracketleftˆσ′α\n⊥(q,ω)∝angbracketrightne=Mχαβ\n⊥(q,ω+i0)uβ(q,ω) (6)\nin the ﬁrst order in uin the momentum and frequency\nrepresentation. Here, χαβ\n⊥is the transverse spin suscep-\ntibility in a uniformly magnetized state with α,β=x,y,\nand summing over βis implied.\nNow, we study the ω-linear terms in the uniform ( q=\n0) part of the transverse spin susceptibility, χαβ\n⊥(q=\n0,ω+i0). We make the following transformation of the\noperator:\nc=U˜c=1/radicalbig\n2ε(ε+M)/parenleftbigg\nvF(ky+ikx)\nε+M/parenrightbigg\n˜c(7)\nwithε=/radicalbig\n(vFk)2+M2. Note U†U= 1,U†σxU=\nvFky/ε,andU†σyU=−vFkx/ε. This transformation\nmaps two component operator cinto one component op-\nerator on the upper Dirac cone ˜ c. With this new op-\nerator, we calculate the transverse spin susceptibility in\nMatsubara form\nχαβ\n⊥(0,iωλ) =/integraldisplayβ\n0dτeiωλτ/angbracketleftbig\nTτσα(0,τ)σβ(0,0)/angbracketrightbig\n=−T/summationdisplay\nk,nU†σαU˜G(k,iεn+iωλ)U†σβU˜G(k,iεn) (8)\nwith˜G(k,iεn) = (iεn−ε+εF+iγsgn(εn))−1. By sym-\nmetry consideration of the integrand in k-integral, we\nﬁndχαβ\n⊥(0,iωλ)∝δαβ. After some calculations, we ob-\ntain the torque stemming from the time evolution:\ntα\nel=M2iω\n2π1\n2v2\nF/parenleftbiggvFkF\nεF/parenrightbigg2\nεFτn×u (9)\n=1\n2/parenleftbiggMvFkF\nεF/parenrightbigg2\nνFτ˙ n×n.(10)\nThis result ﬁts the conventional Gilbert damping with\nα=1\n2/parenleftbiggMvFkF\nεF/parenrightbigg2\nνFτa2\n¯hS. (11)\nWe next examine the case of ﬁnite current by applying\na d.c. electric ﬁeld E, and calculate a linear response of\nσα\n⊥toE, i.e.,< σα\n⊥(q)>ne=Kα\ni(q)Ei. First, it is clear\nthatKα\ni(q=0) =−εiασxx/(evF) where εiαandσxx\nare the anti-symmetric tensor and diagonal conductivity,\nrespectively, because electron’s spin is ”attached” to its\nmomentum. This representsthe inversespin-Galvanicef-\nfect, i.e., chargecurrentinduces magneticmoment. Since\nwe assume that Fermi level is far away from the surface\ngap,σxx≫σxywhereσxyis the Hall conductivity. The\ndominant term in χis thusχxy∝σxx. This is quite\ndiﬀerent from the case studied in Ref. [8], where Fermi\nlevel lies inside the surface gap and therefore σxxis van-\nishing. Hence, the only contribution to the inverse spin-\nGalvanic eﬀect is χxx∝σxy, which is much smaller than3\nthe eﬀect proposed in this letter. Compared with the in-\nverse spin-Galvanic eﬀect in Rashba system [9–11], this\neﬀect is much stronger since the small Rashba coupling\nconstant, i.e., the small factor αRkF/EFin Eq. (16) of\nRef. [9], does not appear in the present case. Taking into\naccount the realistic numbers with α= 10−11eVmandvF= 3×105m/s, oneﬁndsthat theinversespin-Galvanic\neﬀect in the present system is ∼50times largerthan that\nin Rashba systems.\nThe next leading order terms of the expansion in uβ\nandqjcan be obtained by considering the four-point ver-\ntices [12] as\n∝angbracketleftˆσα\n⊥(q)∝angbracketrightne=−eMπ\n45i\n8πε2\nFεikεjl[δαβδkl+δαkδβl+δαlδβk]qjuβEi (12)\n=−eM5i\n32ε2\nF[q·Euα−q·(u×ˆz)(E×ˆz)α+u·(E×ˆz)(q×ˆz)α]. (13)\nTherefore, the spin torque steming from the spatial gradient has the form:\ntβ\nel=−β1\n2e[n×(j·∇)n−(j−(j·n)ˆz)∇·(n×ˆz)+(∇−(n·∇)ˆz)n·(j×ˆz)] (14)\nwherej=σCEwith charge current jand conductivity\nσC=e2\n4π/parenleftBig\nvFkF\nεF/parenrightBig2\nεFτ. and\nβ=5π\n4εFτ/parenleftbiggM\nvFkF/parenrightbigg2\n. (15)\nFrom Eq.(14), one can ﬁnd the followings: (i) The spin\ntransfer torque of the form ( j·∇)nis missing since\nwe consider the upper Dirac cone only. (ii) The β-\nterm has a form essentially diﬀerent from that in the\nconventioal one.[12, 15, 16] In contrast to the conven-\ntionalferromagnet,[12] thisconstantcomesfromthe non-\nmagnetic impurity. Considering vFkF∼=εF, we get\nα/β∼(εFτ)2from Eqs. (11) and (15). Therefore,\ntheβ-terms are negligible for a good surface metal, i.e.,\nεFτ≫1.\nUp to now, we consider only one branch of the band\nwhere the Fermi energy is sitting. When we consider the\n2-band structure, i.e., the 2 ×2 matrix Hamiltonian H=\nvF[(ky+Mnx\nvF)σx−(kx−Mny\nvF)σy], we have the correctionto the Berry phase term. In analogy with the minimal\ncoupling of electromagnetic ﬁeld, A=−M\nevF(−ny,nx)\nplays the same role as the U(1) gauge. By integrating\nthe fermions out, one can get a Chern-Simons term in\ntermsofthemagnetization LCS=σxyǫµνρAµ∂νAρwhere\nµ,ν,ρ=t,x,y. When the gradient of magnetization van-\nishes, it can be rewritten as\nLCS=σxy(M\nevF)2(nx˙ny−ny˙nx).(16)\nThis additional term can be interpreted as an additional\nBerryphase for the magnetization. In fact, as nzremains\nconstant in the present case, we have [ nx,ny] =inz.\nTherefore, nxandnybecome conjugate variables up\nto a factor, which naturally leads to a Berry phase:\nnx˙ny−ny˙nx. This term is exactly equivalent to the\nChern-Simons term.\nIncluding all the terms derived above, we ﬁnally arrive\nat a modiﬁed LLG equation:\n˙n−2σxy(M\nevF)2˙n/(s0N) =γ0Heﬀ×n+/parenleftbiggM\nevFs0N/parenrightbigg\n(−j+(n·j)ˆz)+(α0+α/N)˙n×n+tβ\nel/(s0N) (17)\nwhereNisthenumberofferromagneticlayers. Notethat\nα-,β- andBerryphaseterms originatefromthe interplay\nbetween Dirac fermions and local magnetization which\npersists over a few layers of the ferromagnet. Therefore,\nthe overall coeﬃcients are divided by the number of fer-\nromagnetic layers N.Ferromagnetic resonance. —Observingthe smallvalue\nofβ, the spatial gradient of magnetization can be ne-\nglected for the time being. Only one uniform domain in\nthe absence of current is taken into account for simplic-\nity. Without loss of generality, assume that an external\nmagnetic ﬁeld is applied along zdirection, and consider4\nthe ferromagnet precession around that ﬁeld. ˙ nz= 0\nis kept in the ﬁrst order approximation, namely nzis a\nconstant in the time evolution. By inserting the ansatz\nnx(y)(t) =nx(y)e−iωtinto the modiﬁed LLG equation,\none obtains\nℜω=ξ\nξ2+η2ω0,ℑω=−η\nξ2+η2ω0(18)\nwhereη= (α0+α/N),ω0=γ0Heffandξ= 1−\n2σxy(M\nevF)2/(s0N). Expanding up to the ﬁrst order in\nσxyandη, one gets ℜω=ω0+ 2σxy(M\nevF)2ω0/(s0N)\nandℑω=ηω0. Therefore, the precession frequency ac-\nquires a shift proportional to σxyin the presence of in-\nterplay between Dirac fermions and the ferromagnetic\nlayer. The relative shift of ℜωis 2σxy(M\nevF)2ω0/(s0N) =\n1\nπSNM\nεF(Ma\nvF)2∼1\nN(M\nεF)3[17]. By tuning the Fermi level,\nthis shift can be accessible experimentally.\nMeanwhile, the Gilbert damping constant αcan be\nmeasured directly without referring to the theoretical\nexpression in Eq. (11). One can investigate the fer-\nromagnetic layer thickness dependence of FMR line-\nwidth. While increasing the thickness Nof ferromagnet,\nthe Gilbert damping constant stemming from the Dirac\nfermions decreases inversely proportional to the thick-\nness. Taking into account the realistic estimation with\nεFτ∼100 and M/εF∼0.3, one has α/s0∼1, while\nα0∼0.001 usually. Therefore, even for a hundred of lay-\ners of ferromagnet, the contribution from the proximity\neﬀect is still signiﬁcant compared to the one coming from\ntheferromagnetitself. Observingthattheimaginarypart\nof resonance frequency in Eq. (18) is proportional to η,\none may plot the relation between the FMR peak broad-\nening, namely ℑω, and 1/N. The broadening is a linear\nfunction of 1 /N, and approaches the value of the ferro-\nmagnet at large thickness limit. We can ﬁnd the value of\nαfrom the slope of the plot.\nOn the other hand, the real part of FMR frequency\nprovides rich physics as well. Since in the presence of\nadditional Berry phase, the frequency shift is propor-\ntional to the Hall conductivity on the surface of TI, it\nleads to a new method to measure the Hall conductiv-\nity without four-terminal probe. In an ideal case when\nthe Fermi level lies inside the surface gap, this quantity\nis quantized as σ0\nxy=e2\n2h. However, in realistic case,\nFermi level is away from the surface gap, and therefore\nthe Hall conductivity is reduced to σxy=e2\n2hMnz\nεF[17]. As\na result, the shift of resonance frequency is proportional\nton2\nz∝cos2θ, and the FMR isotropy is broken. Here,\nθis the angle between eﬀective magnetic ﬁeld and the\nnormal to the surface of TI. One can perform an angle\nresolved FMR measurement. The signal proportional to\ncos2θcomes from additional Berry phase.\nSince parameters αandβdepend on Mandτ, it\nis quite important to measure these quantities directly.Molecular-beam epitaxy method can be applied to grow\nTI coated by a thin layer of soft ferromagnet. As is re-\nquired in the above calculation, Fermi level of TI should\nlie inside the bulk band gap. Also, the soft ferromag-\nnet should be an insulator or a metal with proper work\nfunction. One may employ angular resolved photoemis-\nsion spectroscopy(ARPES) or scanning tunneling micro-\nscope techniques to measure the surface gap ∆ opened\nby the ferromagnet, which is given by ∆ = Mnz. As the\neasy axis nzcan be found experimentally, Mcan be ﬁxed\nas well. On the other hand, the lifetime τis indirectly\ndetermined by measuring the diagonal conductivity σxx\nviaσxx=e2\n4π/parenleftBig\nvFkF\nεF/parenrightBig2\nεFτ. Finally, Fermi surface can\nbe determined by ARPES, and all parameters in LLG\nequation Eq.(17) can be obtained.\nIn summary, we have investigated theoretically the\ndynamics of magnetization on the surface of a three\ndimensional topological insulator. We have derived\nthe Landau-Lifshitz-Gilbert equation in the presence of\ncharge current, and analyzed the inverse spin-Galvanic\neﬀect and ferromegnetic resonance predicting anomalous\nfeatures of these phenomena.\nThis work is supported by Grant-in-Aid for Scientiﬁc\nResearch (Grants No. 17071007, 17071005, 19048008\n19048015, and 21244053) from the Ministry of Educa-\ntion, Culture, Sports, Science and Technology of Japan.\n[1] M. Z. Hasan and C. L. Kane, arXiv:1002.3895; X. L.\nQi and S. C. Zhang, Physics Today, 63, 33 (2010) and\nreferences therein.\n[2] X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Phys. Rev. B\n78, 195424 (2008).\n[3] T. Yokoyama, Y. Tanaka, and N. Nagaosa, Phys. Rev.\nLett.102, 166801 (2009).\n[4] T. Yokoyama, Y. Tanaka, and N. Nagaosa, Phys. Rev. B\n81, 121401(R) (2010).\n[5] L. Fu and C. L. Kane, Phys. Rev. Lett. 100, 096407\n(2008).\n[6] Y. Tanaka, T. Yokoyama, and N. Nagaosa, Phys. Rev.\nLett.103, 107002 (2009).\n[7] J. Linder, Y. Tanaka, T. Yokoyama, A. Sudbø, and N.\nNagaosa, Phys. Rev. Lett. 104, 067001 (2010).\n[8] I. Garate and M. Franz, arXiv:0911.0106.\n[9] A. Manchon and S. Zhang, Phys. Rev. B 78, 212405\n(2008).\n[10] A. Chernyshov et al., Nature Phys. 5, 656 (2009).\n[11] I.M. Miron et al., Nature Materials 9, 230 (2010).\n[12] H. Kohno, G. Tatara, and J. Shibata, J. Phys. Soc. Jpn.\n75, 113706 (2006).\n[13] Y.Tserkovnyak,H.J.Skadsem, A.Brataas, andG.E.W.\nBauer, Phys. Rev. B 74, 144405 (2006); Y. Tserkovnyak,\nA. Brataas, and G. E. Bauer, J. Magn. Magn. Mater.\n320, 1282 (2008).\n[14] Y. Tserkovnyak, G.A. Fiete and B.I. Halperin, Appl.\nPhys. Lett. 84, 5234 (2004).\n[15] Clement H. Wong and Y. Tserkovnyak, Phys. Rev. B 80,5\n184411 (2009).\n[16] S. Zhang and Steven S.-L. Zhang, Phys. Rev. Lett. 102,\n086601 (2009).[17] J. Zang, and N. Nagaosa, arXiv:1001.1578"    },    {        "title": "1906.02730v2.Current_noise_geometrically_generated_by_a_driven_magnet.pdf",        "content": "Current noise geometrically generated by a driven magnet\nTim Ludwig1;2, Igor S. Burmistrov2;3;1;4, Yuval Gefen5, Alexander Shnirman1;4\n1Institut f ur Theorie der Kondensierten Materie,\nKarlsruhe Institute of Technology, 76128 Karlsruhe, Germany\n2L.D. Landau Institute for Theoretical Physics RAS, Kosygina street 2, 119334 Moscow, Russia\n3Laboratory for Condensed Matter Physics, National Research University Higher School of Economics, 101000 Moscow, Russia\n4Institute of Nanotechnology, Karlsruhe Institute of Technology, 76021 Karlsruhe, Germany and\n5Department of Condensed Matter Physics, Weizmann Institute of Science, 76100 Rehovot, Israel\nWe consider a non-equilibrium cross-response phenomenon, whereby a driven magnetization gives\nrise to electric shot noise (but no d.c. current). This e\u000bect is realized on a nano-scale, with\na small metallic ferromagnet which is tunnel-coupled to two normal metal leads. The driving\ngives rise to a precessing magnetization. The geometrically generated noise is related to a non-\nequilibrium distribution in the ferromagnet. Our protocol provides a new channel for detecting and\ncharacterizing ferromagnetic resonance.\nO\u000b-diagonal (cross-) response phenomena, e.g. the\nthermoelectric e\u000bect, are ubiquitous in physics. In spin-\ntronic systems, by applying an electric charge current one\ncan drive magnetization dynamics and vice versa [1{7].\nThis usually requires magnetic contacts which allow for\na conversion between spin and charge currents; see how-\never [8]. In this Letter we report a higher order strongly\nnon-equilibrium cross-response e\u000bect. Namely, we show\nthat by driving magnetization dynamics one can gener-\nate electric shot noise [9, 10] without generating charge\ncurrent. Strikingly, no magnetic leads are needed and the\nleads can be at equilibrium with each other.\nWe consider a small metallic ferromagnet with mag-\nnetization driven to precess. The ferromagnet is tunnel-\ncoupled to two normal metal leads; see Fig. 1. The\nprecessing magnetization drives the electrons of the ferro-\nmagnet into a strongly non-equilibrium state. This e\u000bect\nis most pronounced if the ferromagnet is small enough\nsuch that internal relaxation is negligible compared to\nthe relaxation due to the coupling to the leads. The pre-\nFIG. 1: A small metallic ferromagnet with precessing magne-\ntization is tunnel-coupled to two normal metal leads, which\nare at equilibrium with each other. The precessing magneti-\nzation pumps a spin-current from the small ferromagnet into\nthe leads [4{6]. The average charge current vanishes by sym-\nmetry. Thus, the current of spin-up electrons and spin-down\nelectrons balance each other on average and in each junction\nseparately; in the ferromagnet, the precessing magnetization\nmixes spin-up and spin-down electrons. All four spin-resolved\nelectron currents are \ructuating. These \ructuations combine\nto give rise to the noise of left to right (transport) charge\ncurrent.cessing magnetization, in turn, induces non-equilibrium\nshot noise of the electric current. The non-equilibrium\ndistribution responsible for the shot noise is governed by\nthe geometric Berry phase due to precessing magnetiza-\ntion, branding the shot noise geometric. This shot noise\nexists even when both leads are in equilibrium with each\nother, although the average charge current vanishes then.\nShot noise is particularly interesting in spintronics be-\ncause it gives insights into the magnetic con\fguration\nand its dynamics which may be hard to obtain otherwise\n[11{17].\nResults. |In order to describe dynamics of the mag-\nnetization of a small ferromagnet we use the macrospin\napproximation, i.e., the magnetization is given by a sin-\ngle vectorM=M(sin\u0012cos\u001e;sin\u0012sin\u001e;cos\u0012). We as-\nsume a steady state precession of the magnetization at\na constant polar angle \u0012and with a constant precession\nfrequency _'. Under this assumptions, we found that the\ncharge current vanishes on average, I= 0, but the cur-\nrent noise remains \fnite:\nS= 4gtT+gtsin2\u0012\u0010\n_\u001ecoth_\u001e\n2T\u00002T\u0011\n: (1)\nHeregt= 2(\u001a\"+\u001a#)\u0000l\u0000r=(\u0000l+ \u0000r) is the total conduc-\ntance of the double tunnel-junction with spin-dependent\ndensity of states of the small ferromagnet \u001a\u001b. The rates\n\u0000land \u0000rcharacterize the spin-conserved tunneling to\nleft and right leads respectively. The precessing mag-\nnetization pumps a spin-current into the adjacent leads\n[4{6], which drives the electron system into a strong non-\nequilibrium state [18, 19]; see Fig. 3. At high tempera-\ntures (T\u001d_\u001e), the noise is dominated by the \frst term\nS\u00194gtT, which is the standard thermal noise. At low\ntemperatures ( T\u001c_\u001esin2\u0012), however, the noise is dom-\ninated by the second term S\u0019gtsin2\u0012j_\u001ej. The time-\ndependence of the magnetization is the source of driving\nfor the electron system. Therefore, the precession fre-\nquency _\u001eacts like a voltage bias for standard shot noise.\nApplication to FMR-driven magnet. |Now let us con-\nsider our setup under conditions of a ferromagnetic res-\nonance (FMR). The dynamics of the magnetization isarXiv:1906.02730v2  [cond-mat.mes-hall]  13 Jun 20192\nphenomenologically described by the Landau-Lifshitz-\nGilbert equation _m=m\u0002B\u0000\u000bm\u0002 _m, where\nm=M=Mis the direction of the magnetization and \u000bis\nthe Gilbert-damping coe\u000ecient. For the FMR-setup, we\nchoose the magnetic \feld B= (\n cos!dt;\n sin!dt;B0)\nwith a \fxed component B0inz\u0000direction and, perpen-\ndicular to it, a small driving \feld with strength \n and fre-\nquency!d. For negligible internal relaxation, the damp-\ning is dominated by the coupling to the leads. Without\ndriving, the Gilbert-damping would relax the magnetiza-\ntion towards \u0012= 0. With driving (\n 6= 0), however, the\nmagnetization can be brought into a steady state pre-\ncession. That is, after the decay of transient e\u000bects, the\nmagnetization precesses at the frequency of the driving\n\feld _\u001e=!dand the polar angle \u0012is determined by the\ncompetition between Gilbert-damping and FMR-driving.\nExplicitly,\u0012is determined by\nsin2\u0012=(\n++ \n\u0000)2\n\n2\n++ \n2\n\u0000+ 2\u00012+ 2q\n(\u00012+ \n2\n+)(\u00012+ \n2\n\u0000);\n(2)\nwith \n\u0006= \n\u0006\u000b!dand the detuning parameter \u0001 =\n!d+B0. The dependence of sin2\u0012on precession fre-\nquency!dhas a resonant character with a maximum at\n!d=\u0000B0. This ferromagnetic resonance of the magne-\ntization's steady state precession directly translates into\na resonance in the current noise; see Fig. 2. At low\ntemperatures, T\u001c!dsin2\u0012, the form of the resonance\nin the current noise resembles the FMR structure of the\nstationary precession angle. At higher temperatures, the\nresonance in the current noise can be visible on top of the\nconstant thermal noise. Now we explain how our results\nwere derived.\nThe e\u000bective action. |Because the dynamics of the\nmagnetization creates non-equilibrium conditions, we ap-\nply Keldysh formalism [20{22]. The Keldysh generating\nfunction isZ=R\nD[\u0016\t;\t] exp (iS) with the action,\nS=I\nKdt\u0016\t (i@t\u0000hs\u0000^\u0006) \t; (3)\nwhere the integral is along the Keldysh contour and\n\t;\u0016\t denote the fermionic \felds of the small ferromag-\nnet. The self-energy operator ^\u0006 is de\fned by [ ^\u0006\t](t) =H\ndt0\u0006(t\u0000t0)\t(t0), where \u0006 = \u0006 l+ \u0006ris the self-energy\narising from the tunnel-coupling to the left lead \u0006 land\nright lead \u0006 r. The self-energy contains the essential in-\nformation about the tunnel-coupling to the leads: \frst,\nthe retarded and advanced part contain the tunneling-\nrates \u0006R=A\nl=r(!) =\u0007i\u0000l=r; second, the Keldysh part con-\ntains the distribution functions of the leads \u0006K\nl=r(!) =\n\u00002i\u0000l=rFl=r(!), whereFl=r(!) = 1\u00002fl=r(!) with the\nFermi-distributions fl=r(!) = 1=[exp[(!\u0000\u0016)=T]+1]. We\nemphasize that the ferromagnet's distribution function\nfs, respectively Fs, is not yet known explicitly but it\nis implicitly determined by the action, Eq. (3). This\ndistribution function is governed by the coupling to the\nFIG. 2: When the steady state precession of the magnetiza-\ntion is maintained by driving with a FMR-setup, the polar\nangle\u0012depends on driving frequency _\u001e=!d. The peak of\nsin2\u0012at!d=\u0000B0(\u0001 = 0) is a typical FMR-peak. We\nshow the zero-frequency noise of charge current that is gener-\nated by the precessing magnetization; we subtract the ther-\nmal contribution and normalize onto the value of the total\nconductance, that is, we show ( S\u00004gtT)=gt. The generated\nnoise of charge current clearly re\rects the peak structure of\nsin2\u0012in the FMR-setup. Parameters in \fgure: \u000b= 0:04,\n\n=(\u000bB0) = 0:63.\nleads andthe dynamics of the magnetization which en-\nters through the e\u000bective single-particle Hamiltonian,\nhs=h0\u0000M\u001b=2 (4)\nwhere\u001bis the vector of Pauli-matrices and h0is a spin-\ndegenerate single-particle Hamiltonian of the small fer-\nromagnet. For the derivation of the charge noise, the\nmagnetization is considered to be a classical \feld with\ngiven dynamics (steady state precession).\nThe charge current and its noise are determined with\nhelp of a counting \feld \u0015, which is introduced into\nthe self-energy related to the left lead \u0006 l!\u0006l(\u0015).\nWe follow Ref. [16], and introduce \u0015such that the\ncharge transported through the left junction is deter-\nmined ashQli=i@\u0015Z(\u0015)j\u0015=0with the corresponding\nnoise\nQ2\nl\u000b\n= (i@\u0015)2Z(\u0015)j\u0015=0; details are provided in\nsupplementary material (SM). We can now integrate out\nthe fermions to obtain Z(\u0015) = exp[iS(\u0015)] with the action\niS(\u0015) = tr ln\u0002\ni@t\u0000h0+M\u001b=2\u0000\u0006(\u0015)\u0003\n: (5)\nThe magnetization's time-dependence makes it compli-\ncated to proceed. It is, thus, very convenient to trans-\nform to a frame of reference in which the magnetization\nis time-independent.\nRotating frame. |The magnetization is rotated onto\nthez-axis at all times,\nRyM\u001bR=M\u001bz; (6)\nwith a time-dependent rotation in spin-space R. While\nsimplifying the magnetic part, this rotation also comes\nat a cost: because of its time-dependence, it gives rise to3\na new term iRy_Runder the tr ln, see Eq. (5), and also\nrotates the self-energy Ry\u0006R. After rotation, the action\nbecomes\niS(\u0015) = tr ln\u0002\ni@t\u0000h0+M\u001bz=2 +iRy_R\u0000Ry\u0006(\u0015)R| {z }\n=:~G\u00001\n\u0015\u0003\n;\n(7)\nwhere ~G\u00001\n\u0015de\fned the rotating-frame Green's function\n~G\u0015. Following Ref. [23], we choose the Euler-angle rep-\nresentation R=e\u0000i\u001e\n2\u001bze\u0000i\u0012\n2\u001byei\u001e\u0000\u001f\n2\u001bz, where\u001e;\u0012are the\nangles characterizing the magnetization and the gauge-\nfreedom\u001fis \fxed by _\u001f=_\u001e(1\u0000cos\u0012). This choice elim-\ninates the spin-diagonal part of iRy_Rwhich contains in-\nformation about the Berry phase. However, the Berry\nphase is not eliminated; instead it is shifted to the ro-\ntated self-energy.\nRotating-frame distribution functions. |Because re-\ntarded and advanced parts of the self-energy are trivial\nin spin-space and local in time, the rotation only a\u000bects\nthe Keldysh part. While the Keldysh part \u0006K(t\u0000t0) =\n\u00002i[\u0000lFl(t\u0000t0) + \u0000rFr(t\u0000t0)] is also trivial in spin-\nspace, it is non-local in time because of the distribution\nfunctionsFl=r(t\u0000t0). It follows, Ry(t)\u0006K(t\u0000t0)R(t0) =\n\u00002i[\u0000l~Fl(t;t0)+\u0000r~Fr(t;t0)] with the rotating-frame distri-\nbution functions ~Fl=r(t;t0) =Ry(t)Fl=r(t\u0000t0)R(t0). For\nthe following, it is convenient to change to the Wigner\ntime-coordinates \u0016t= (t+t0)=2, \u0001t=t\u0000t0and to per-\nform a Fourier-transformation \u0001 t!!. For steady state\nprecessions (with \u0012and _\u001econstant), the spin-diagonal\nparts of the rotating-frame distribution functions are\ngiven by ~F\u001b\nl=r(!) = [ ~Fl=r(!)]\u001b\u001b= cos2\u0012\n2Fl=r(!+\u001b!\u0000) +\nsin2\u0012\n2Fl=r(!+ \u0016\u001b!+). These distributions are governed\nby the magnetization dynamics via the Berry-phase in\n!\u0006=_\u001e(1\u0006cos\u0012)=2.\nAdiabatic approximation. |In order to proceed, we\nhave to determine the rotating-frame Green's function\n~G\u0015for vanishing counting \feld \u0015= 0. In principle this\nposes a complicated problem, since the spin-o\u000b-diagonal\nelements of its inverse ~G\u00001\n0depend on time. However,\nwe assume the magnetization Mto be the largest rele-\nvant energy scale in the small ferromagnet. This allows\nus to disregard the spin-o\u000b-diagonal elements of ~G\u00001\n0for\nthe determination of ~G0In particular, we disregard spin-\no\u000b-diagonal elements of iRy_Rwhich are related to tran-\nsitions between spin-up and spin-down states; this cor-\nresponds to an adiabatic approximation [23]. Further-\nmore, we disregard spin-o\u000b-diagonal elements of the ro-\ntated self-energy. It is, now, straightforward to obtain\nthe rotating-frame Green's function,\n~GR=A\n0;a\u001b(!) =1\n!\u0000\u0018a\u001b\u0006i\u0000\u0006;\n~GK\n0;a\u001b(!) =\u00002i\u0000\u0006~F\u001b\ns(!)\n(!\u0000\u0018a\u001b)2+ \u00002\n\u0006;(8)\nwith the total level broadening \u0000 \u0006= \u0000l+ \u0000r. The spin-\ndependent single-particle energy is \u0018a\u001b=\u000fa\u0000M\u001b= 2,\nFIG. 3: (a) The spin-diagonal part of the small ferromag-\nnet's rotating-frame distribution function is shown for spin-up\n(red solid) and spin-down (blue dashed). The areas shaded in\nblue and red are all equal in size: sin2\u0012j_\u001ej=4, which means\nthat the electrons are redistributed in energy space for each\nspin-polarization separately. (b) The noise of charge current,\nEq. (11), is determined by the integral over 1 \u0000~F\u001b\ns(!)~F\u001b\nl(!),\nwhich itself is governed by the distribution function ~f\u001b\ns(!).\nThe contribution to this noise is identical for both spin-\npolarizations, as the shaded areas are equal in size (red for\nspin-up; blue for spin-down). Parameters in \fgures: \u0012=\u0019=3\nand!d<0.\nwhere\u000faare the eigenenergies of h0with corresponding\neigenstates a. The rotating-frame distribution function\nof the small ferromagnet,\n~F\u001b\ns(!) =\u0002\n\u0000l~F\u001b\nl(!) + \u0000r~F\u001b\nr(!)\u0003\n=\u0000\u0006; (9)\nis a superposition of the leads' rotating-frame distribu-\ntion functions. In absence of bias, the rotating-frame\ndistribution functions are exactly the same in all three\nsystems ~F\u001b\ns(!) =~F\u001b\nl(!) =~F\u001b\nr(!); see Fig. 3.\nIt is worthwhile to emphasize that the transformation\ninto the rotating frame is a crucial step that allows us\nto solve the problem. The reason is as follows. As we\ndiscussed above it is enough to \fnd the spin-diagonal\ncomponents of the rotating-frame distribution function.\nHowever, as one can check [24], the knowledge of the\nspin-diagonal components of the rotating-frame distribu-\ntion function is not enough in order to determine the\ndistribution function in the laboratory frame.\nCharge current and its noise. |The zero-frequency\ncharge current Ilis de\fned via the transported charge\nhQli=R\ndtIl. Di\u000berentiating the generating func-\ntion, the transported charge is determined to hQli=\n\u0000itr[~G0~\u00060\nl], where ~\u00060\nl=@\u0015~\u0006l(\u0015)j\u0015=0is the derivative\nof the rotated self-energy ~\u0006(\u0015) =Ry\u0006(\u0015)R. For the cur-4\nrent, we \fnd [24]\nIl=X\n\u001b\u001a\u001b\u0000lZ\nd![~F\u001b\nl(!)\u0000~F\u001b\ns(!)] = 0; (10)\nwhere we de\fned the spin-dependent density of states,\n\u001a\u001b(!) =P\na1\n\u0019\u0000\u0006\n(!\u0000\u0018a\u001b)2+\u00002\n\u0006. We assumed it to be ap-\nproximately constant \u001a\u001b(!) =\u001a\u001bon all scales smaller\nthanM. The resulting formula for the charge current is\nthe Landauer formula [25] with rotating-frame distribu-\ntion functions. This re\rects the fact that the amount of\ntransported charge is an observable which has to be in-\ndependent of the frame of reference. Explicitly, however,\nthe current vanishes, since no bias is applied.\nSimilar to the average current, the zero-frequency\nnoise [26] of charge current Slis de\fned viahhQ2\nlii=R\ndtSl=2. Di\u000berentiating the generating function, the\nnoise of transported charge is determined to hhQ2\nlii=\ntr[~G0~\u000600\nl] + tr[ ~G0\n0~\u00060\nl], where ~\u000600\nl=@2\n\u0015~\u0006l(\u0015)j\u0015=0is the\nsecond derivative of the rotated self-energy and ~G0\n0=\n@\u0015~G\u0015\f\f\n\u0015=0=~G0~\u00060\nl~G0is the derivative of the rotating-\nframe Green's function. For the noise, we \fnd [24]\nSl=X\n\u001bg\u001bZ\nd!n\u0002\n1\u0000~F\u001b\ns(!)~F\u001b\nl(!)\u0003\n+\n+\u0000l\n\u0000r~F\u001b\ns(!)\u0002~F\u001b\nl(!)\u0000~F\u001b\ns(!)\u0003o\n;\n(11)\nwhereg\u001b= 2\u001a\u001b\u0000l\u0000r=(\u0000l+\u0000r) is the spin-dependent con-\nductance of the double tunnel-junction. After the inte-\ngration over frequency, we obtain Eq. (1) as result for\nthe shot noise.\nDiscussion. | In our relatively simple model which ex-\ncludes internal relaxation, we were able to properly derive\nthe non-equilibrium distribution function ~F\u001b\ns(!) given by\nEq. (9). This, in particular, guarantees that the charge\nconservation laws are satis\fed. Indeed, since the small\nferromagnet cannot store additional charges for an in\f-\nnite time, charge conservation requires Il=\u0000Ir=:I\nandSl=Sr=:Sat zero frequency. For the right junc-\ntion, current Irand noiseSrcan be obtained from eqs.\n(10) and (11) by exchanging \u0000 l$\u0000rand substituting\n~F\u001b\nl(!)!~F\u001b\nr(!). As expected, we \fnd Il=\u0000Irand\nSl=Sr.\nIn the presence of internal relaxation one might be\ntempted to impose a physically motivated distributionfunction in the small ferromagnet as a shortcut of a full\ncalculation. We emphasize, however, that the charge\nconservation condition Sl=Srputs a strong restriction\nonto possible distribution functions. In particular, charge\nconservation would be violated if ~F\u001b\ns(!) is just replaced\nby an equilibrium distribution function with an adjusted\nelectrochemical potential. Thus, a straightforward appli-\ncation of the results of Ref. [16] obtained for a single\ntunnel junction to the double tunnel-junction considered\nhere is not possible.\nWe expect the e\u000bects of internal relaxation to be three-\nfold: (i) the Gilbert-damping coe\u000ecient \u000bcan be in-\ncreased (spin-orbit coupling) and, thereby, the polar an-\ngle\u0012of steady state precessions is changed; (ii) internal\nrelaxation tends to bring the magnet's rotating-frame dis-\ntribution function ~F\u001b\ns(!) towards equilibrium; (iii) the\nformal result for the noise, Eq. (11), has to be changed\nin order not to violate charge conservation when the dis-\ntribution function changes. However, for weak internal\nrelaxation, these e\u000bects might be taken into account per-\nturbatively and, therefore, we expect our results to be\nrobust against \fnite but small internal relaxation.\nConclusion. | We have found a higher order non-\nequilibrium o\u000b-diagonal response e\u000bect. Namely, we have\nshown that zero-frequency shot noise of charge current is\ngenerated by a precessing magnetization of a small fer-\nromagnet which is tunnel-coupled to two normal metal\nleads. This noise, Eq. (11), crucially depends on the\nelectronic distribution function which is in turn geomet-\nrically governed by the magnetization dynamics; see Fig.\n3. Thus, the noise of the charge current, Eq. (1), is gen-\nerated by the precession of the magnetization. For the\nFMR-setup, Fig. 2, this e\u000bect can be used to detect the\nmagnetization dynamics in spite of the vanishing average\ncurrent.\nAcknowledgements. |We thank G. E. W. Bauer, M.\nKe\u0019ler, and W. Wulfhekel for fruitful discussions. This\nwork was supported by DFG Research Grant No. SH\n81/3-1. The research of T.L. is partially supported by\nthe Russian Foundation for Basic Research under the\nGrant No. 19-32-50005. Furthermore, T.L. acknowl-\nedges KHYS of KIT and the Feinberg Graduate school\nof WIS for supporting a stay at WIS; I.S.B. acknowl-\nedges RAS Program Topical problems in low tempera-\nture physics, the Alexander von Humboldt Foundation,\nand the Basic research program of HSE; Y.G. acknowl-\nedges the DFG Research Grant RO 2247/11-1 and the\nItalia-Israel QUANTRA.\n[1] J. C. Slonczewski, Journal of Magnetism and Magnetic\nMaterials 159, L1 (1996).\n[2] L. Berger, Phys. Rev. B 54, 9353 (1996), URL https:\n//link.aps.org/doi/10.1103/PhysRevB.54.9353 .\n[3] L. Berger, Phys. Rev. B 59, 11465 (1999), URL https:\n//link.aps.org/doi/10.1103/PhysRevB.59.11465 .\n[4] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.Rev. Lett. 88, 117601 (2002), URL https://link.aps.\norg/doi/10.1103/PhysRevLett.88.117601 .\n[5] A. Brataas, Y. Tserkovnyak, G. E. W. Bauer, and B. I.\nHalperin, Phys. Rev. B 66, 060404 (2002), URL https:\n//link.aps.org/doi/10.1103/PhysRevB.66.060404 .\n[6] Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I.\nHalperin, Rev. Mod. Phys. 77, 1375 (2005), URL https:5\n//link.aps.org/doi/10.1103/RevModPhys.77.1375 .\n[7] Y. Tserkovnyak, T. Moriyama, and J. Q. Xiao, Phys.\nRev. B 78, 020401 (2008), URL https://link.aps.org/\ndoi/10.1103/PhysRevB.78.020401 .\n[8] M. Jonson, R. Shekhter, O. Entin-Wohlman, A. Aharony,\nH. Park, and D. Radi\u0013 c, arXiv preprint arXiv:1903.03321\n(2019).\n[9] R. Landauer, Nature 392, 658 (1998).\n[10] Y. M. Blanter and M. B uttiker, Physics reports 336, 1\n(2000).\n[11] J. Foros, A. Brataas, G. E. W. Bauer, and\nY. Tserkovnyak, Phys. Rev. B 79, 214407 (2009),\nURL https://link.aps.org/doi/10.1103/PhysRevB.\n79.214407 .\n[12] T. Arakawa, J. Shiogai, M. Ciorga, M. Utz,\nD. Schuh, M. Kohda, J. Nitta, D. Bougeard,\nD. Weiss, T. Ono, et al., Phys. Rev. Lett. 114,\n016601 (2015), URL https://link.aps.org/doi/10.\n1103/PhysRevLett.114.016601 .\n[13] A. Kamra and W. Belzig, Phys. Rev. B 94,\n014419 (2016), URL https://link.aps.org/doi/10.\n1103/PhysRevB.94.014419 .\n[14] A. Kamra and W. Belzig, Phys. Rev. Lett. 116,\n146601 (2016), URL https://link.aps.org/doi/10.\n1103/PhysRevLett.116.146601 .\n[15] J. P. Cascales, D. Herranz, U. Ebels, J. A. Katine, and\nF. G. Aliev, Applied Physics Letters 107, 052401 (2015).\n[16] P. Virtanen and T. T. Heikkil a, Phys. Rev. Lett. 118,\n237701 (2017), URL https://link.aps.org/doi/10.\n1103/PhysRevLett.118.237701 .\n[17] B. Horovitz and A. Golub, arxiv [cond-mat] p.\n1904.11221v1 (2019).\n[18] T. Ludwig, I. S. Burmistrov, Y. Gefen, and A. Shnirman,\nPhys. Rev. B 95, 075425 (2017), URL https://link.\naps.org/doi/10.1103/PhysRevB.95.075425 .\n[19] T. Ludwig, I. S. Burmistrov, Y. Gefen, and A. Shnirman,\nPhys. Rev. B 99, 045429 (2019), URL https://link.\naps.org/doi/10.1103/PhysRevB.99.045429 .\n[20] A. Kamenev, Field Theory of Non-Equilibrium Systems\n(Cambridge University Press, 2011).\n[21] A. Kamenev and A. Levchenko, Ad-\nvances in Physics 58, 197 (2009),\nhttps://doi.org/10.1080/00018730902850504, URL\nhttps://doi.org/10.1080/00018730902850504 .\n[22] A. Altland and B. D. Simons, Condensed matter \feld\ntheory (Cambridge University Press, 2010).\n[23] A. Shnirman, Y. Gefen, A. Saha, I. S. Burmistrov,\nM. N. Kiselev, and A. Altland, Phys. Rev. Lett. 114,\n176806 (2015), URL https://link.aps.org/doi/10.\n1103/PhysRevLett.114.176806 .\n[24] See Supplemental Material.\n[25] R. Landauer, IBM Journal of Research and Development\n1, 223 (1957).\n[26] Strictly speaking, it should be called low-frequency noise,\nas \ructuations of the magnetization become important\nfor measurement times tmlonger that the typical re-\nlaxation time \u001c\u0012of the\u0012coordinate [16]. This relax-\nation time, however, scales with the magnetization length\n\u001c\u0012\u0018Mwhich we assumed to be very large.ONLINE SUPPLEMENTAL MATERIAL\nCurrent noise geometrically generated by a driven magnet\nTim Ludwig1,2, Igor S. Burmistrov2,3,1,4, Yuval Gefen5, Alexander Shnirman1,4\n1Institut f¨ ur Theorie der Kondensierten Materie,\nKarlsruhe Institute of Technology, 76128 Karlsruhe, Germany\n2L.D. Landau Institute for Theoretical Physics RAS, Kosygina street 2, 119334 Moscow, Russia\n3Laboratory for Condensed Matter Physics, National Research University Higher School of Economics, 101000 Moscow, Russia\n4Institute of Nanotechnology, Karlsruhe Institute of Technology, 76021 Karlsruhe, Germany and\n5Department of Condensed Matter Physics, Weizmann Institute of Science, 76100 Rehovot, Israel\nIn this supplemental material, we present (i) details of the introduction of counting ﬁelds, (ii)\ndetails of the derivation of results for the charge current and its noise, and (iii) a detailed explanation\nwhy the knowledge of spin-diagonal components of the rotating-frame distribution function is not\nsuﬃcient to determine the laboratory-frame distribution function.\nS.I. DETAILS OF THE DERIVATION FOR CHARGE CURRENT AND ITS NOISE\nA. Derivation of eﬀective action\nBefore leads are integrated out, the full Hamiltonian is H=Hs+Hl+HrwhereHsdescribes the small ferromagnet\nandHl,Hrdescribe the left and right lead respectively and include the tunnel-coupling to the small ferromagnet.\nExplicitly, Hs=/summationtext\nασσ/prime[hs]aσσ/primec†\naσcaσ/prime, wherec†\naσcreates (caσannihilates) a particle in the orbital state awith\nspinσ. The single particle Hamiltonian is hs=h0−M(t)σ\n2, whereM(t) is the time dependent magnetization\nof the small ferromagnet, σis the vector of Pauli-matrices, and h0=/summationtext\naσ/epsilon1ac†\naσcaσis the spin-degenerate single-\nparticle Hamiltonian. The leads are assumed to be non-interacting and the tunnel-coupling is assumed to be spin-\nconserving. The (coupling to the) left lead is described by Hl=/summationtext\nγσ/epsilon1γc†\nγσcγσ+/summationtext\naγσ(tl,anc†\naσcγσ+h.c.), where\nγ= (n,k) is a collective index for momentum kand transport channel n. Respectively, c†\nγσ,cγσare creation and\nannihilation operators for electrons in state γwith spinσand/epsilon1γis the corresponding energy. The tunneling through\nthe left junction is described by the tunneling-matrix tl. Analogously, Hr=/summationtext\n˜γσ/epsilon1˜γc†\n˜γσc˜γσ+/summationtext\na˜γσ(tr,a˜nc†\naσc˜γσ+h.c.)\ndescribes (the coupling to) the right lead. Now, applying Keldysh formalism leads to the Keldysh partition function\nZ=/integraltext\nD[¯Ψ,Ψ]eiSwith the actionS=/contintegraltext\nKdt[¯Ψi∂tΨ−H(¯Ψ,Ψ)], where ¯Ψ,Ψ denote all fermionic ﬁelds including\nthose of the leads. Would we directly integrate out the fermionic ﬁelds of the leads, we would obtain the action\nS=/contintegraltext\nKdt¯Ψ(i∂t−hs−ˆΣ)Ψ, where ¯Ψ,Ψ denote only fermionic ﬁelds of the small ferromagnet and the self-energy\noperator ˆΣ is deﬁned by [ ˆΣΨ](t) =/contintegraltext\ndt/primeΣ(t−t/prime)Ψ(t/prime); compare Eq. (3) of the main text. The self-energy Σ = Σ l+ Σr\narises from tunnel-coupling to left lead Σ l=tlGlt†\nland right lead Σ r=trGrt†\nrrespectively, where G−1\nl,γ=i∂t−/epsilon1γ\nandG−1\nr,γ=i∂t−/epsilon1˜γ. However, before integrating out the leads, we should introduce the counting ﬁeld.\nB. Introduction of counting ﬁelds\nFormulas for the charge current and its noise can be conveniently derived via the introduction of a counting ﬁeld.\nFollowing Ref. [1], we introduce a counting ﬁeld λ(t) for the charge transported into the left lead Qlby adding\nSc=−/contintegraltext\nKdt˙λ(t)/summationtext\nγσ¯ΨγσΨγσto the action, that is, S→S +Sc. This newly added term is eliminated by a gauge\ntransformation for the fermionic ﬁelds of the left lead: Ψ γσ→e−iλ(t)Ψγσand ¯Ψγσ→¯Ψγσeiλ(t). While this gauge\ntransformation eliminates Sc, it modiﬁes the tunneling-matrix as\ntl→tle−iλ(t)andt†\nl→t†\nle+iλ(t). (S1)\nWhen the leads are integrated out, the counting ﬁeld is transferred to the self-energy of the left lead Σ l(t−t/prime)→\ne−iλ(t)Σl(t−t/prime)e+iλ(t/prime). For compact notation, we write Σ l(λ) =e−iλ(t)Σl(t−t/prime)e+iλ(t/prime). We assume the counting ﬁeld\nto have only a quantum component λ±(t) =±λq(t)/2. For simplicity, we assume the counting ﬁeld to be constant\nλq(t) =λ, which is possible as we are interested in the zero-frequency current and noise [2].\nNow, we integrate out the leads as indicated above. Afterwards, we also integrate out the fermionic ﬁelds of the\nsmall ferromagnet to obtain the Keldysh partition function\nZ(λ) =eiS(λ)(S2)2\nwith the action\nS(λ) =−itr ln/bracketleftbig\ni∂t−h0+Mσ/2−Σ(λ)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nG−1\nλ/bracketrightbig\n, (S3)\nwhere the counting ﬁeld is contained in the self-energy Σ = Σ l(λ) + Σr. The counting ﬁeld was introduced in such a\nway that the charge transported through the left junction is determined as\n/angbracketleftQl/angbracketright=i∂λZ(λ)|λ=0and/angbracketleftbig\nQ2\nl/angbracketrightbig\n= (i∂λ)2Z(λ)|λ=0(S4)\nand analog for higher moments. Would we directly take the derivative with respect to the counting ﬁeld λ, we would\nobtain for the charge /angbracketleftQl/angbracketright=−itr[G0Σ/prime\nl] with Σ/prime\nl=∂λΣl(λ)|λ=0and for its noise/angbracketleft/angbracketleftQ2\nl/angbracketright/angbracketright= tr[G0Σ/prime/prime\nl] + tr[G/prime\n0Σ/prime\nl] with\nΣ/prime/prime\nl=∂2\nλΣl(λ)|λ=0andG/prime\n0=∂λGλ/vextendsingle/vextendsingle\nλ=0=G0Σ/prime\nlG0. The problem is, however, that we do not know the laboratory-\nframe Green’s function G0. This is the reason for making the transformation to the rotating frame, where we can\ndetermine the Green’s function ˜G0.\nC. Derivation of results for charge current and its noise.\nThe counting ﬁeld is chosen to have a quantum component only and to be constant in time, that is λ±(t) =±λq\n2.\nTo keep the notation simple, we drop the quantum-index λq→λin the following. After changing to the rotating\nframe with R†MσR=Mσz, the generating function is given by Z(λ) =eiS(λ)with the new action\nS(λ) =−itr ln/bracketleftbig\ni∂t−/epsilon1α+Mσz/2−R†Σ(λ)R/bracehtipupleft/bracehtipdownright/bracehtipdownleft /bracehtipupright\n˜G−1\nλ/bracketrightbig\n. (S5)\nDue to the time-dependence of the rotations, a new term iR†˙Rarises. However, as in the main text, the spin-diagonal\npart ofiR†˙Ris eliminated by the choice of gauge ˙ χ=˙φ(1−cosθ) and the spin-oﬀ-diagonal part of iR†˙Ris disregarded\nin an adiabatic approximation.\nUsing equation (S4), a straightforward diﬀerentiation with respect to the counting ﬁeld leads to the ﬁrst moment\n/angbracketleftQl/angbracketright=−itr/bracketleftbig˜G0˜Σ/prime\nl/bracketrightbig\n, where ˜Σ/prime\nl=∂λR†Σl(λ)R/vextendsingle/vextendsingle\nλ=0. From the main text, we know the spin-diagonal parts of the rotated\nself-energy ˜ΣR/A\nσσ(ω) =∓iΓland˜ΣK\nσσ(ω) =−2iΓl˜Fσ\nl(ω) and also the Green’s function ˜GR/A\n0(ω)≈1/(ω−ξaσ±iΓΣ) and\n˜GK\n0(ω)≈−2iΓΣ˜Fσ\ns(ω)/[(ω−ξaσ)2+ Γ2\nΣ]. The spin-oﬀ-diagonal parts can be disregarded due the large magnetization\nM. This leads to\n/angbracketleftQl/angbracketright=/integraldisplay\ndt/integraldisplay\ndω/summationdisplay\nσρσΓl/bracketleftbig˜Fσ\nl(ω)−˜Fσ\ns(ω)/bracketrightbig\n, (S6)\nwhere we assumed the density of states ρσ(ω) =/summationtext\naΓΣ\n(ω−ξaσ)2+Γ2\nΣto be approximately constant on all scales smaller\nthanMaround the leads’ electrochemical potentials µ, i.e.,ρσ(µ+ω) =ρσ. With/angbracketleftQl/angbracketright=/integraltext\ndtIl, we can immediately\nread of the result for the current Il=/summationtext\nσρσΓl/integraltext\ndω[˜Fσ\nl(ω)−˜Fσ\ns(ω)] which is the Landauer formula with rotating-\nframe distribution functions. The results for the right junction can be obtained analogously by introducing a counting\nﬁeld for the right lead. The formal result for Iris analog to Ilbut with the replacements Γ l→Γrand ˜Fσ\nl(ω)→˜Fσ\nr(ω).\nHowever, charge conservation demands Il+Ir= 0 at zero frequency. Due to the absence of bias, we explicitly ﬁnd\nIl= 0 andIr= 0 which could have been expected from symmetry.\nFor the second moment follows /angbracketleftQ2\nl/angbracketright=N0\nl+N1\nl+N2\nlwithN0\nl=−/parenleftbig\ntr/bracketleftbig˜G0˜Σ/prime\nl/bracketrightbig/parenrightbig2=/angbracketleftQl/angbracketright2, such that we obtain the\ncumulant\n/angbracketleft/angbracketleftQ2\nl/angbracketright/angbracketright=/angbracketleftQ2\nl/angbracketright−/angbracketleftQl/angbracketright2=N1\nl+N2\nl. (S7)\nFormally,N1\nl= tr/bracketleftbig˜G0˜Σ/prime/prime\nl/bracketrightbig\nandN2\nl= tr/bracketleftbig˜G0˜Σ/prime\nl˜G0˜Σ/prime\nl/bracketrightbig\n, where ˜Σ/prime/prime\nl=∂2\nλR†Σl(λ)R/vextendsingle/vextendsingle\nλ=0and inN2\nlwe used∂λ˜Gλ/vextendsingle/vextendsingle\nλ=0=\n˜G0˜Σ/prime\nl˜G0. The term N2\nlarises from the dependence of the Green’s function on the counting ﬁeld. Thus, this term can\nbe interpreted as a reaction of the distribution function of the small ferromagnet to tunneling of electrons. One might\nwounder, if it is important to take those reactions of the distribution function into account. The answer is a clear\nyes. Would we approximate /angbracketleft/angbracketleftQ2\nl/angbracketright/angbracketright≈N1\nland proceeding analogously for the right contact /angbracketleft/angbracketleftQ2\nr/angbracketright/angbracketright≈N1\nr, we would3\nﬁnd/angbracketleft/angbracketleftQ2\nl/angbracketright/angbracketright/negationslash=/angbracketleft/angbracketleftQ2\nr/angbracketright/angbracketrightwhich violates charge conservation, as we consider zero-frequency. More explicitly, we obtain\nN1\nl=/integraldisplay\ndt/integraldisplay\ndω/summationdisplay\nσρσ(ω) Γl/bracketleftbig\n1−˜Fσ\ns(ω)˜Fσ\nl(ω)/bracketrightbig\n(S8)\nN2\nl=/integraldisplay\ndt/integraldisplay\ndω/summationdisplay\nσΓ2\nl\nΓΣ/bracketleftBig\n2¯ρσ(ω)˜Fσ\ns(ω)˜Fσ\nl(ω)−¯ρσ(ω)˜Fσ\ns(ω)˜Fσ\ns(ω)−ρσ(ω)−2[˜Fσ\nl(ω)]2(¯ρσ(ω)−ρσ(ω))/bracketrightBig\n, (S9)\nwith two diﬀerently broadened densities of states ρσ(ω) =/summationtext\na1\nπΓΣ\n(ω−ξaσ)2+Γ2\nΣand ¯ρσ(ω) =/summationtext\na1\nπ2Γ3\nΣ\n((ω−ξaσ)2+Γ2\nΣ)2. We\nassume this diﬀerence in broadenings to be insigniﬁcant, that is, we approximate ¯ ρσ(ω)≈ρσ(ω). As for the current,\nwe assume the density of states to be approximately constant ρσ(ω)≈ρσ. It follows,\n/angbracketleft/angbracketleftQ2\nl/angbracketright/angbracketright=/integraldisplay\ndt/integraldisplay\ndω/summationdisplay\nσρσΓrΓl\nΓΣ/braceleftBig/bracketleftbig\n1−˜Fσ\ns(ω)˜Fσ\nl(ω)/bracketrightbig\n+Γl\nΓr˜Fσ\ns(ω)/bracketleftbig˜Fσ\nl(ω)−˜Fσ\ns(ω)/bracketrightbig/bracerightBig\n. (S10)\nFrom this result, we can read oﬀ the noise of charge current Sl, which is deﬁned by /angbracketleft/angbracketleftQ2/angbracketright/angbracketright=/integraltext\ndtSl/2. And indeed,\nwithN1\nlandN2\nlboth taken into account, charge conservation is satisﬁed /angbracketleft/angbracketleftQ2\nl/angbracketright/angbracketright=N1\nl+N2\nl=N1\nr+N2\nr=/angbracketleft/angbracketleftQ2\nr/angbracketright/angbracketright,\nwhen we proceed analogously for the right contact.\nS.II. LABORATORY-FRAME DISTRIBUTION FUNCTION WOULD BE HARD TO DETERMINE\nThe rotating-frame distribution functions of the leads ˜Fl/r(t,t/prime) have been found by the rotation of the laboratory-\nframe distribution functions Fl/r(t−t/prime), that is, ˜Fl/r(t,t/prime) =R†(t)Fl/r(t−t/prime)R(t/prime). We could also perform the\ninverse rotation to obtain Fl/r(t−t/prime) =R(t)˜Fl/r(t,t/prime)R†(t/prime), which determines the leads’ laboratory-frame distribution\nfunctions in terms of their rotating-frame distribution functions. Analogously, the laboratory-frame distribution\nfunction of the small ferromagnet is determined by Fs(t,t/prime) =R(t)˜Fs(t,t/prime)R†(t/prime), where ˜Fs(t,t/prime) is the rotating-frame\ndistribution function. With spin-indices written explicitly follows\nFσσ/prime/prime/prime\ns(t,t/prime) =/summationdisplay\nσ/primeσ/prime/primeRσσ/prime(t)˜Fσ/primeσ/prime/prime\ns(t,t/prime)[R†(t/prime)]σ/prime/primeσ/prime/prime/prime. (S11)\nUsing theθ=const. and ˙φ=const. , the spin-diagonal contributions of the laboratory-frame distribution function\nbecome\nF↑↑\ns(¯t,ω) = cos2θ\n2˜F↑↑\ns(ω−ω−) + sin2θ\n2˜F↓↓\ns(ω−ω+)−sinθ\n2/bracketleftBig\n˜F↓↑\ns(ω−˙φ/2)e−i˙φcosθ¯t+˜F↑↓\ns(ω−˙φ/2)ei˙φcosθ¯t/bracketrightBig\n,\n(S12)\nF↓↓\ns(¯t,ω) = cos2θ\n2˜F↓↓\ns(ω+ω−) + sin2θ\n2˜F↑↑\ns(ω+ω+) +sinθ\n2/bracketleftBig\n˜F↓↑\ns(ω+˙φ/2)e−i˙φcosθ¯t+˜F↑↓\ns(ω+˙φ/2)ei˙φcosθ¯t/bracketrightBig\n,\n(S13)\nwhere ¯t= (t+t/prime)/2. However, despite this formal result, it is hard to make further progress. While we know the spin-\ndiagonal parts of the rotating-frame distribution function, ˜F↑↑\ns(ω) and ˜F↓↓\ns(ω), we do not know the spin-oﬀ-diagonal\nparts ˜F↓↑\ns(ω) and ˜F↑↓\ns(ω). The spin-diagonal parts are hard to determine, because of the time-dependence of the spin-\noﬀ-diagonal contributions of the rotating-frame self-energy and the retarded and advanced part of the rotating-frame\nGreen’s functions. So, we cannot determine the spin-diagonal parts of the laboratory-frame distribution functions.\nIn strong contrast to the distribution function Fs, it is straightforward to determine the laboratory-frame Keldysh\nGreen’s function GKfrom its rotating-frame version ˜GK. The inverse rotation GK(t,t/prime) =R(t)˜GK(t,t/prime)R†(t/prime) is\nanalog to to the distribution function Fs. With spin-indices written out explicitly we obtain,\nGK\nσσ/prime/prime/prime(t,t/prime) =/summationdisplay\nσ/primeσ/prime/primeRσσ/prime(t)˜GK\nσ/primeσ/prime/prime(t,t/prime)[R†(t/prime)]σ/prime/primeσ/prime/prime/prime. (S14)\nHowever, in strong contrast to the distribution function, we know that the spin-oﬀ-diagonal contributions to the\nrotating-frame Keldysh Green’s function ˜GK\nσ¯σ=˜GR\nσ˜ΣK\nσ¯σ˜GA\n¯σare suppressed by the large value of the magnetization M.\nTherefore, it is suﬃcient to take into account only the spin-diagonal elements. A straightforward calculation yields,\nGK\nσσ(ω) = cos2θ\n2˜GK\nσσ(ω−σω−) + sin2θ\n2˜GK\n¯σ¯σ(ω−σω+), (S15)\nGK\nσ¯σ(¯t,ω) =sinθ\n2/bracketleftbig˜GK\n↑↑(ω+ cosθ˙φ/2)−˜GK\n↓↓(ω−cosθ˙φ/2)/bracketrightbig\ne−iσ˙φ¯t, (S16)4\nwith the spin-diagonal part of the rotating-frame Keldysh Green’s function ˜GK\n0(ω)≈−2iΓΣ˜Fσ\ns(ω)/[(ω−ξaσ)2+ Γ2\nΣ].\n[1] P. Virtanen and T. T. Heikkil¨ a, Phys. Rev. Lett. 118, 237701 (2017), URL https://link.aps.org/doi/10.1103/\nPhysRevLett.118.237701 .\n[2] Formally, we should consider a ﬁnite time interval ∆ t, i.e. we should choose λq(t) =λΘ(t+ ∆t/2)Θ(∆t/2−t), and then\ntake the limit of ∆ t→∞ to obtain the zero-frequency results for the transported charge and its noise. For simplicity of\nnotation, we skip this and choose a constant counting ﬁeld."    },    {        "title": "2402.06792v1.Resonances_involving_integer_magnons_and_spin_1_2_excitations_in_a_magnetism_modulated_two_dimensional_electron_gas.pdf",        "content": "arXiv:2402.06792v1  [cond-mat.mes-hall]  9 Feb 2024Resonances involving integer magnons and spin-1/2 excitat ions in\na magnetism modulated two-dimensional electron gas\nLin Zhang,1,2Yi Wang,2,3Shu-Yu Zheng,4Li Lu,4and Chi Zhang2,3,5,∗\n1Department of Mathematics and Physics,\nNorth China Electric Power University, Beijing 102206, Chi na\n2State Key Laboratory of Superlattices and Microstructures (SKLSM),\nInstitute of Semiconductors, Chinese Academy of Science,\nP.O. Box 912, Beijing 100083, China\n3College of Material Science and Opto-Electronic Technolog y,\nUniversity of Chinese Academy of Sciences, Beijing 100190, China\n4Beijing National Laboratory for Condensed Matter Physics,\nInstitute of Physics, Chinese Academy of Sciences, Beijing 100190, China\n5CAS Center for Excellence in Topological Quantum Computati on,\nUniversity of Chinese Academy of Sciences, Beijing 100190, China\nAbstract\nWe conduct an experimental study of high-mobility two-dime nsional electron gas (2DEG) in\nGaAs/AlGaAs quantum wells modulated by strong magnetism at an in-plane magnetic ﬁeld ( B).\nThe modulated B-ﬁelds are performed via the single stripe and gratings whic h are made of the\nheavy rare earth metal Terbium (Tb) thin ﬁlms on the sample su rface. The robust ferromagnetic\nresonances(FMRs) persisttothetemperatureof50-70 Kinbo ththestripeandgratingsamples, for\nthe ferromagnetism (FM) phase of Tb exists at above 100 K. The high-order (with integer numbers\nofj≡¯hω/gµ BB= 1,2,...) magnetic resonances can also be observed in the stripes tructure via the\nmicrowave (MW) photovoltaic detection and magnetoresista nce under microwave irradiation. In\naddition, the resonance features around j= 1/2 are robust in the single-stripe modulated sample,\nwhich suggests the spinons with spin-1/2 collective excita tions in a 1D Heisenberg model.\n1Introduction.—\nIn a magnetic ﬁeld modulated 2DEG, ambient physics phenomena have been revealed,\nand the spatially modulated magnetic ﬁeld plays an important role in elec tric driven spin\nresonance [1], magnetically modulated quantum Hall edge modes [2], an d helical magnetic\nmodes dependent on Chern numbers [3, 4]. Recent studies have disc overed the topological\nsurface states in topological insulators [5] and nontrivial states in topological semimetals\n[6]. In addition, theoretical and experimental studies of magnons a re in prosperous in two-\ndimensional materials (e.g. single layer and twisted bilayer graphene u nder a perpendicular\nB) [7–9], andferromagneticorantiferromagneticmatters [10–13]. T hepotentialapplication\nof magnons has been developed in both theories and models [14–16].\nThe FMRs under MW and transport are revealed in quantum wires mod ulated 2DEG\n[17]. In the further study of 2DEG with the modulated prism magnets of small sizes (at\nabout 100 nm scales) on the surface, magnetic dipolar interaction c an be detected in photo-\nvoltage (PV) measurements [17, 18]. In this paper, we report the t ransport of the strongly\nmagnetically-modulated 2DEG at a small in-plane Bwith MW manipulations. The single\nwire and grating structures that are made of rare earth element T b are deposited on the\nsurface of embedded 2DEG. The transport and MW photovoltaic me asurements of our FM-\nmodulated 2DEG are aﬀected by spin wave transmission in proximity. W e demonstrate that\n2DEG as a sensor can detect a series of magnetic resonances unde r an in-plane magnetic\nexcitation, which involves an integer number of magnons in transpor t. This ﬁnding accom-\nplishes the manipulation of magnons in the electron system. In additio n, in Tb single wire\nmodulated sample, we observe the strong resonance features at twice of the eﬀective mag-\nnetic ﬁelds, which suggests the spinon excitations in 1D. Due to the h igh Curie temperature\nof Tb, the observed magnetic resonances persist until around 50 to 70 K.\nMethod.— Our 2DEG wafer has a high electron mobility µ= 3.0×106cm2/Vs and a\ndensityns= 1.6×1011cm−2in MBE grown GaAs/AlGaAs heterostructure, which is about\n130 nm below the surface. The thin ferromagnetic metal structur es are fabricated on (20\nµm long and 10 µm wide) Hallbars. The magnetic modulation generates a stray magnet ic\nﬁeld, the perpendicular component of which deﬂects the ballistic tra jectory in the 2DEG,\ncoupling the electrical properties of the 2DEG with the magnetic pro perties of the grating\n[17]. We conduct the measurements in two types of structures: on e had a magnetic strip\nwith a width of dA= 650 nm on the surface of the Hallbar (sample A), and the other had\n2nine magnetic gratings with a width of single wire dB= 500 nm fabricated on the Hallbar\nwith a pitch period of a= 1000 nm (sample B). To avoid the edge eﬀects, we fabricate the\nwire or the stripes with the length of l= 60µm which is much longer than the length of\nmesa in Hallbar (20 µm). The single stripe structure on the surface of sample A is shown\nin Fig. 1, and the gratings on sample B are shown in Fig. 4. The mesas of the Hallbar\nare fabricated by UV lithography, and the single wire and stripes of t he Tb thin ﬁlm are\nevaporated via the magnetron sputtering.\nThe magnetism of the rare earth elements (or rare earth metals) s tems from the RKKY-\nlike exchange between the 4f conduction electrons and the local sp ins [19]. The magnetic\nmoment of the Tb atom is very large ( mTb∼9.0) [20]. The electron group state of Tb is\nlabeled as (6H15/2) with a spin angular momentum quantum number (q.n.) S= 5/2, the\norbital angular momentum q.n.of L= 5, and the total angular momentum q.n. J= 15/2.\nIn our experiments, the electron channel is coupled to the Tb thin ﬁ lm on the surface while\nmodulated by the spin wave stimulated by the MW [17], and the resonan ces related to spin\nmodes can be detected in the PV and transport study.\nOur experiment is performed in a VTI with a base temperature of 1.5 K and a vector\nmagnet of 8 T/ 2 T. The PV measurements arecarried out by choppin g theMW radiationat\n870 Hz with a lock-in ampliﬁer without excitation current [21, 22]. The high-resolution MW\nphotovoltaic eﬀect a series of discrete magnetic resonances at sm allB-ﬁelds. An external\nin-planBﬁeld (BxorBy) is applied to magnetize the Tb stripe (as shown in the inset of Fig.\n1(a)). The PV is detected via the modulated MW source by means of lo ck-in ampliﬁers.\nExperimental Results.—\nFerromagnetic resonance is an eﬃcient tool to probe spin waves wh ich involves magnons.\nThe motion of magnetization can be simpliﬁed by the Landau-Lifshitz- Gilbert equation:\n∂− →M\n∂t=−γ(− →M×− − →Heff)+G\nγM2\ns[− →M×∂− →M\n∂t] (1)\nwhich includes two sections: the ﬁrst term represents the preces sion, and the second term\nexpresses the viscous damping (with a Gilbert constant G) which is used in the description\nof relaxations. In various measurements, the resonance resemb les a Lorentzian lineshape,\nand the linewidth is related to the relaxation process. The resonant ﬁeld is dependent on\nvarious factors, e.g. the anisotropy, g-factor, and magnetization.\nIn experiments, the FMR features can be detected via the magnet ic susceptibility mea-\n3surements in resonators or the photovoltaic probes. The power is absorbed by the processing\nmagnetization of the material. In a FM phase, the frequency of the spin precession induced\nby the magnetization is given by the relation in [23]:\nω=γµ0/radicalBig\n[He\ndc+(Nx−Ny)Mdc][He\ndc+(Nz−Ny)Mdc] (2)\n, whereγ=gµB/¯his the gyromagnetic ratio, Mdcrepresents the magnetization induced by\na staticB-ﬁeld, and He\ndcrepresents an eﬀective magnetic ﬁeld caused by the dc component\nof the applied in-plane B[18]. The demagnetization factors Nx,Ny, andNzare solely\ndependent on the geometry of the sample [24, 25]. Suppose the re lationM=χH, Eq. (2)\ncan be simpliﬁed as:\nω=γµ0He\ndc/radicalBig\n(1−Nyχ)(1−Nyχ+Nzχ) (3)\n. InoursampleswithFMmetallicrectangularprisms, thegeometricpa rameters Nx= 0, and\nNyandNzcan be calculated for prototype geometries [26]. FMR occurs in our s amples and\nthe MW frequency is proportional to the external B-ﬁeld. In a FM phase the relation B=\nµ0His eﬀective at a low B-ﬁeld. So Eq.(3) can be further simpliﬁed as ¯ hω= ¯hγ(µ0H)Ag≈\ngµBB, where the geometry factor is deﬁned as: Ag≡/radicalBig\n(1−Nyχ)(1−Nyχ+Nzχ), which\nis very close to 1 for the rectangular prism geometry.\nMoreover, the slope of sample A (∆ µ0He\ndc)/∆fis about ∼0.519 kG/GHz, and that of\nsample B is ∼0.522 kG/GHz. Based on Eq. (3), the magneto-crystalline anisotrop y results\nin an internal magnetic ﬁeld, leading to an Hhincluded in Ha[23]. In our analysis, the\ncrystal-ﬁeld anisotropy of Tb exhibits a shift ﬁeld of He\na=Ha−Hh.\nWe conduct PV measurements on 2DES with the Tb wire (or gratings) structure on the\nsurface. As the Tb wire (or gratings) is magnetized along the y-axis, magnetic poles form\non the edges of the stripe(s), generating a spatially varying B-ﬁeld with two components.\nThez-component of the B-ﬁeld that induces eddy currents transfers magnetization to the\n2DEG via the high-frequency electromagnetic wave, thus forms a s eries of dips in the PV.\nIn our PV detection (PV vs. ( B−µ0HC)) in sample A, the FMR features at T= 1.5 K\nare robust at MW frequencies from 1 to 18 GHz which is linear to the oc curring in-plan B\nalong the y-axis. Figure 1(a) display the experimental data of PV which display a series\nresonance features superposed with a background linear to B-ﬁeld. As shown in the PV\ntrace extracted the background (∆PV) in Fig. 1(b), FMR occurs a t (By−µ0HC) around\n5.1 kG at f= 10 GHz MW. The resonance absorptions in PV detection are robust near\n4By−µ0HC= 5.1 kG and 10.2 kG. To our surprise, the amplitude at By−µ0HC= 10.2\nkG that corresponds to j= 1/2 is much stronger than that of the FMR j= 1. Because\nthe FMR is accompanied by the magnons whose (bosonic) excitation e xists in both 2D\nand 3D. In the theory of 1D ferromagnetic chain of Heisenberg mod el [16], the term in\nHamiltonian gµBB < s z>provides a plausible explanation: the spinon transition occurrs\naccompanied with the spin-1/2 collective excitations (spinons) which is neither the fermions\nnor the bosons in 1D conﬁnement [27].\nIn addition, a series of magnetic resonances in PV, i.e., near By−µ0HC= 2.55 and\n1.61 kG, are observable, which correspond to the resonances at j= 2 and 3 respectively.\nMoreover, minor resonances are detected at very low B-ﬁelds. The small dips at low B\n(below 2 kG) suggest that the spin wave modes oscillate within the mag netic stripe well\n[18]. The spin wave modes that propagate transversely inside the st ripe become conﬁned\nwithin the region of the lower internal ﬁeld [28]. In our sample with a larg e width ( w) of\na single Tb wire, the local spin wave mode (i.e., the dipolar electron spin w ave (DESW))\nthat stems from the geometry conﬁnement [18] is weakened to som e degree in the PV\nexperiments. So the magnetic resonances that involve the integer magnons dominate the\ndipolar spin wave modes at very low in-plane B.\nIn our MW f-dependent PV detection with a frequency range of ∼4−16 GHz, a series of\nmagneticresonancesoccuratinteger j, whichisdeﬁnedas j≡¯hω/(gµBBAg)≈¯hω/(gµBB),\nwhereAgis the geometry factor expressed in Eq.(3) and Ag∼1. The PV signals (of sample\nA) atf= 4,10,14 GHz and at a sweeping down Byare shown in Fig. 2(a), 2(b), and 2(c),\nrespectively. At 4 and 10 GHz resonances occur at j= 1 and 1/2, which are marked by the\nblack and red arrows. But in the PV curve at 14 GHz in Fig. 2(c), the s olid black arrows\nand dashed blue arrows highlight the resonances at j= 1 and 2 respectively, for the location\nofj= 1/2 is beyond the limit of in-plane Bof the magnet in the VTI. A prototype FMR (at\nj= 1) occurs at a higher magnetic ﬁeld which is proportional (or linear) to the excited MW\nfrequency (e.g., By−µ0HC= 2.13 kG at 4 GHz, and By−µ0HC= 7.31 kG at 14 GHz). The\ndata points of By−µ0HCvs. frequency illustrate a distinct slope of µ0∆He\ndc/(∆f) = 0.507\nkG/GHz for sample A, and µ0¯Hh= 0.26 kG, as shown in Fig. 2(d). And the eﬀective\ng-factor can be obtained as approximately g∼1.9, which is very similar to those of Co and\nDy [18]. The magnon transmission is perpendicular to the applied B, e.g., the transmission\nof magnons along the x-direction with an external Balong the y-axis.\n5Thef-dependent PV results at a Balong the x-direction of sample A are shown in Fig.\n2(e) and 2(f), the FMRs (at j= 1) are still observable ( Bx−µ0HCis around 4.7 kG at\nf= 9 GHz, and is around 7 kG at 14 GHz). It is distinct from the experime ntal results\nof Dy modulated 2DEG [29], which is explained by the invalid magnetic modu lation in\nthe conﬁguration: due to the shape of the slim rectangular prism wit hl≫d, the magnon\ntransmission along the length ( l) leads to much weaker scattering than that along the width\n(d) direction [16]. But our observation of FMR at Balong the x-axis does not support\nthe analysis. One plausible reason is the diﬀusion length (or the corre lation length) of the\nmagnons is longer than that of adjacent Hallbar arms and the width o f Hallbar.\nAmong all heavy rare earth elements, Tb has one of the highest Cur ie temperature\nTC∼220 K [20], which is valuable in the potential application of the hybrid dev ices. At\nbase-Taround 1.5 K, the magnetic resonances remain robust with the Tb st ripe magnetized\nalong the y-direction, as shown in Fig. 3(a). At the high Trange of 50 -70 K, the magnetic\nresonances (at various integer j) are robust at 50-70 K at ( By−µ0HC)∼5.2 and 10.0 kG.\nMoreover, the prime magnetic resonance (FMR at j= 1) persists until around 100 K with\nthe magnetization along the y-axis, and ﬁnally disappears at 150 K.\nFor comparison, we detect the T-dependent resonances at a Balong the x-direction, as\nshown in Fig. 3(b). The FMRs at f= 9 GHz remain robust until up to 30 K, and the\nfeatures become very weak at T∼50 K. And the transitional temperatures of the FM\ndisappearance in 2DEG are much lower than those at the transitiona l point of Terbium.\nBut the high-order magnetic resonances j= 2,3 cannot be observed in our T-dependent PV\ndetection at Bx.\nMagnetic resonances in 2DEG with an FM grating structure exhibit mo re complex [18]\nmagnetic excitations than the single stripe structure, which is resu lted from the dipolar\nmagnetization waves (DMW) propagating between the nearest neig hboring stripes [18].\nBut atf= 9,11,15 GHz MW (in Fig. 4(a), 4(b), and 4(c), respectively), no feature s\nof DMW are detectable in our device (sample B). A plausible reason is th at the distance\nbetween the nearest neighboring Tb wires is not small enough, so th at the coupling between\nthe neighboring wires is relatively weak and the DMW mechanism is domina ted by the\nmagnetic resonance features.\nFigure 4(d) shows the PV and resistance ( Rxx) measurement results (of sample B) at a\nB-ﬁeld along the y-direction under f= 16 GHz MW irradiation. The magnetic resonance\n6/s45/s49/s48 /s48 /s49/s48/s45/s53/s48/s48/s48\n/s80/s86/s32/s40 /s86/s41\n/s66\n/s121/s45\n/s48/s72\n/s67/s32/s40/s107/s71/s41/s97/s41\n/s48 /s53 /s49/s48/s45/s49/s48/s48/s45/s53/s48/s48/s53/s48\n/s51/s50/s106/s32/s61/s32/s49\n/s98/s41/s80/s86/s32/s40 /s86/s41\n/s66\n/s121/s45\n/s48/s72\n/s67/s32/s40/s107/s71/s41/s49/s47/s50\nFIG. 1: (Color online) Panel (a): Photovoltage (at 1.5 K) of s ample A under 10 GHz MW irra-\ndiation (black curve). The inset display the MW PV detection of Tb-stripe-2DEG via a lock-in\nampliﬁer. The yellow bar with a width of 650 nm shows the 70 nm t hick Tb ﬁlm on the Hallbar.\n(b): For clarify the resonance features we removed the backg round of curve and obtained the ∆PV\ntrace, where the robust resonance at j∼1,2 and 1/2 are marked by the arrows.\nfeatures at j= 1,2 can be observed at the shoulders in PV and at around the peaks in\nresistance, which are highlighted by the black color arrows.\nDiscussion.—\n(1) We observe robust FMRs in both single wire or grating patterned samples. The\nlinewidth of the resonances is directly relevant to the relaxation pro cessions. In our ob-\nservation, the FWHM of FMR in the 1D single FM wire sample does not exh ibit obvious\nB-dependence at low T. But the FWHM of FMR increases with the occurring B-ﬁeld (as\nshown in Fig.4). Because the width of FMR increases with the dipolar ma gnetic ﬁelds which\n7/s48 /s53 /s49/s48 /s49/s53/s48/s52/s56\n/s45/s56 /s48 /s56/s53/s48/s49/s48/s48\n/s45/s56 /s48 /s56/s48/s50/s48/s48\n/s45/s56 /s48 /s56/s45/s51/s48/s51/s45/s54 /s48 /s54/s45/s54/s48/s48/s48\n/s45/s56 /s48 /s56/s48/s52/s48/s100/s41/s66 /s32/s40/s107/s71/s41\n/s102/s32 /s32/s40/s71/s72/s122/s41/s97/s41/s32\n/s98/s41\n/s99/s41/s80/s86/s32/s40 /s86/s41/s102/s32/s32/s61/s32/s52/s32/s71/s72/s122\n/s49/s48/s32/s71/s72/s122\n/s66\n/s121/s45 /s72\n/s67/s32/s40/s107/s71/s41/s49/s52/s32/s71/s72/s122/s102/s41/s101/s41/s80/s86/s32/s40 /s86/s41/s80/s86/s32/s40 /s86/s41/s57/s32/s71/s72/s122\n/s32/s66\n/s120/s45 /s72\n/s67/s32/s32/s32/s40/s107/s71/s41/s49/s52/s32/s71/s72/s122\nFIG. 2: (Color online) Panel (a), (b), (c): FMRs (at 1.5 K) of s ample A at f= 4,10,14 GHz\nMW and at an in-plane Balong y-axis ( B/bardbly), which are highlighted by the arrows. (d): MW\nfrequency ( f) versus ( By−µ0HC) in FMR ( j= 1). (e), (f): FMRs at 9 and 14 GHz and at B/bardblx\n(PV vs. ( Bx−µ0HC)), thej= 1 features are highlighted by the arrows.\ninvolves the dipolar mode between the adjacent rectangular prisms (i.e. the DMW) [18].\n(2) In the 2DEG sample patterned with single wire or grating structu res, the local spin\nwavemodeincludesDESWandDMWwhichoriginatefromthedipolarinter action(oreﬀect)\nwithin the single wire or between the nearest neighboring wires [18]. In our sample with\na large width of wire or gap between the adjacent stripes, DESW and DMW are weakened\nto a great extent. And the magnetic resonances that involve integ er magnons dominate the\nlocal spin wave modes.\n(3) The comparison of our measurements between single wire and gr atings structures\ndisplay diﬀerences in the excitations around j= 1/2. The robust excitations of S= 1/2\n8/s45/s49/s48 /s48 /s49/s48/s45/s53/s48/s48/s48/s80/s86/s32/s40 /s118/s41\n/s66\n/s121/s45 /s72\n/s67/s32/s40/s107/s71/s41/s32/s53/s32/s75\n/s32/s53/s48\n/s32/s55/s48\n/s49/s48/s32/s71/s72/s122/s97/s41\n/s45/s49/s48 /s45/s53 /s48 /s53 /s49/s48/s45/s53/s48/s48/s48/s98/s41/s80/s86/s32 /s40 /s86 /s41\n/s66\n/s120/s45\n/s48/s72\n/s67/s32/s40/s107/s71/s41/s32/s49/s46/s53/s32/s75/s32/s32 /s32/s49/s48\n/s32/s51/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s51/s48\n/s32/s53/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s53/s48\n/s57/s32/s71/s72/s122\nFIG. 3: (Color online) Panel (a): T-dependent PV of sample A at f = 10 GHz MW with applied\nBlongy-axis (B/bardbly). The FMR persistent at 70 K. (b) T-dependent PV of sample A at 9 GHz\nMW and Balongx-direction ( B/bardblx).\ncan be detected only in the 1D single wire-modulated 2DEG, but canno t be observed in\nthe 2DEG sample with gratings structures. So in the 2DES with 1D FM- modulation, the\noccurrence of excitations at j= 1/2 suggests the occurrence of spinons with S= 1/2. As\nwe know, the magnetic resonance transitions involve magnons, for instance, the spin wave\nof the FMR in the microwave PV measurements in the 2D or 3D models re quire the integer\ntransition of < Sz>, because magnon is a typical boson. So in the 2DEG with gratings\nstructure, the 1D characteristics of the FM-modulation is substa ntially weakened. However,\nin the real 1D models, the rules of the boson or fermion absorption is invalid, and the\nexcitation of spinons is a very plausible explanation.\n(4) Ideally, the Tb thin ﬁlm appears to be a ferromagnetic phase belo wT= 220 K. And\nwe observe distinct resonance features at j= 1/2, which correspond to the transition with\na collective excitation of S= 1/2. In comparison, a recent experimental study revealed\nthe excitation with spin-1/2 in the antiferromagnetic (AFM) chain of 1D Heisenberg model\n[27], which diﬀers from our study in ferromagnetic chain. It seems like ly that the spin-1/2\nexcitations occur in AFM or ferrimagnetic chain in 1D models. So the oc currence of the\n9/s45/s49/s48 /s48 /s49/s48/s45/s56/s48/s48/s45/s52/s48/s48/s97 /s41/s49/s53/s32/s71/s72/s122\n/s45/s49/s48 /s48 /s49/s48/s45/s49/s48/s48/s48/s45/s53/s48/s48\n/s32/s49/s49/s32/s71/s72/s122\n/s45/s49/s48 /s48 /s49/s48/s45/s49/s50/s48/s48/s45/s49/s49/s48/s48\n/s99/s41/s80/s86/s32/s40 /s86/s41\n/s57/s32/s71/s72/s122\n/s66\n/s121/s45\n/s48/s72\n/s67/s32/s40/s107/s71/s41/s98/s41\n/s45/s49/s48 /s48 /s49/s48/s45/s53/s48/s48/s106/s32/s61/s49\n/s50/s100/s41\n/s80/s86/s32/s40 /s86/s41\n/s66\n/s121/s45\n/s48/s72\n/s67/s32/s40/s107/s71/s41/s49/s54/s32/s71/s72/s122/s106/s32/s61/s49\n/s50\n/s48/s53/s48/s48\n/s82\n/s120/s120/s32/s40 /s41\nFIG. 4: (Color online) Panel (a), (b), (c): FMRs (at 1.5 K) of t he (gratings) sample B in the PV\ndetection at B/bardbly), with the frequencies of 15, 11, 9 GHz, respectively. (d): T hef= 16 GHz\nPV and the resistance under MW irradiation (at f= 16 GHz), at a B/bardbly. Inset: the schematic\ndiagram of the grating structures on sample B. The yellow bar s on the Hallbar displays the 70 nm\nthick Tb gratings.\npolarized FM chain at a B-ﬁeld with spin-1/2 excitation is reasonable, although the details\nof the mechanism remain to be explored.\nIn addition, the resonances at j= 1/2 can be explained in the frame of spin-1/2 in\nspinon liquids [30]. However, in our T-dependent study the resonance features at j= 1/2\npersist until around 50 K, which is much higher than the Fermi energ y in the channel. Thus\nthe collective behavior (such as the FMR in the electron channel) is th e only reasonable\nexplanation, and the ESR mechanism can be ruled out in our study.\nConclusion.— In summary, we observed the resonance phenomena that involve a ninteger\nnumber of magnons in the transport and the photovoltage measur ements of a magnetism-\nmodulated2DES.First ofall, wedetect aseries ofmagneticresonan ces in1DFM-modulated\nhigh mobility 2DEG, including the FMR (at j= 1) and resonance at j= 1/2 which persist\nat highTuntil about 50 K. At base- Tof 1.5 K, the robust features at j= 1/2 suggest\n10that the spinon exciations with spin-1/2 are eﬀective in the 1D Heisen berg model. Secondly,\nthe magnetic resonances with an integer j(j= 2,3) are also observable in both single wire\nand gratings patterned samples. Moreover, in the 2DEG with Tb-gr atings structures, the\nresonances that involve an integer number of magnons ( j= 1,2,...) remain robust in both\nPV and resistance under MW irradiations. Based on our research, t he transmission and\nmanipulation of a ﬁnite number of magnons can be utilized to develop po tential devices for\nmagnon-based quantum computation. And the observation of spin -1/2 resonances indicates\nthe collective excitations of spinons in the 1D ferromagnetic Heisenb erg model, which will\nstimulate further discovery in both theory and experiment.\nThis project is supported by the National Science Foundation of Ch ina (Grant\nNo.11974339), and by the Strategic Priority Research Program of the Chinese Academy\nof Science (Grant No. XDB 0460000). L.Z., Y.W. and C.Z. performed t he experiments;\nL.Z. and C.Z. analyzed the data; L.Z draft and C.Z. wrote the paper; L.Z. and Y.W. carried\nout the cleanroom work; S.Z. and L.L. grew the high-quality semicond uctor wafers; C.Z.\nconceived and supervised the project.\n∗Electronic address: zhangchi@semi.ac.cn\n[1] A. Nogaret and F. M. Peeters, Electrically induced spin r esonance ﬂuorescence. I. Theory,\nPhys. Rev. B 76, 075311 (2007).\n[2] A. Nogaret, P. Mondal, A. Kumar, S. Ghosh, H. Beere, and D. Ritchie, Crossover between\nmagnetic and electric edges in quantum Hall systems, Phys. Rev. B 96, 081302(R) (2017).\n[3] F. D. M. Haldane, and D. P. Arovas, Quantized spin current s in two-dimensional chiral mag-\nnets,Phys. Rev. B 52, 4223 (1995).\n[4] A. F. Young, J. D. Sanchez-Yamagishi, B. Hunt, S. H. Choi, K. Watanabe, T. Taniguchi, R.\nC. Ashoori, and P. Jarillo Herrero, Tunable symmetry breaki ng and helical edge transport in\na graphene quantum spin Hall state, Nature505, 528 (2014).\n[5] M. Z. Hasan, and C. L. Kane, Colloquium: Topological insu latorsRev. Mod. Phys. 82, 3045\n(2010).\n[6] S. Y. Xu, et al., Discovery of a Weyl fermion semimetal and topological Ferm i arcs,Science\n349, 613 (2015).\n11[7] Nemin Wei, Chunli Huang, and Allan H. MacDonald, Scatter ing of Magnons at Graphene\nQuantum-Hall-Magnet Junctions, Phys. Rev. Lett. 126, 117203 (2021).\n[8] Yahya Alavirad and Jay Sau, Ferromagnetism and its stabi lity from the one-magnon spectrum\nin twisted bilayer graphene, Phys. Rev. B 102, 235123 (2020).\n[9] Haoxin Zhou, Chunli Huang, Nemin Wei, Takashi Taniguchi , Kenji Watanabe, Michael P.\nZaletel, Zlatko Papic, Allan H. MacDonald, and Andrea F. You ng, Strong-Magnetic-Field\nMagnon Transport in Monolayer Graphene, Phys. Rev. X 12, 021060 (2022).\n[10] J. H. E. Griﬃths, Anomalous high-frequency resistance of ferromagnetic metals, Nature158\n(4019), 670C671 (1946).\n[11] G. P. Rodrigue; H. Meyer; R. V. Jones, Resonance Measure ments in Magnetic Garnets, J.\nAppl. Phys. 31, S376CS382 (1960).\n[12] S. Klingler, A. V. Chumak, T. Mewes, B. Khodadadi, C. Mew es, C. Dubs, O. Surzhenko, B.\nHillebrands, A. Conca, Measurements of the exchange stiﬀnes s of YIG ﬁlms by microwave\nresonance techniques, Phys. D: Appl. Phys. 48, 015001 (2015).\n[13] S. Lee, S. Grudichak, J. Sklenar, C. C. Tsai, M. Jang, Q. Y ang, H. Zhang, John B. Ketterson,\nFerromagnetic resonance of a YIG ﬁlm in the low frequency reg ime,J. Appl. Phys. 120,\n033905 (2016).\n[14] Bin Wei, Jia-Ji Zhu, Yun Song, and Kai Chang, Renormaliz ation of gapped magnon excitation\nin monolayer MnBi 2Te4by magnon-magnon interaction Phys. Rev. B 104, 174436 (2021).\n[15] Shasha Ke, Yun-Mei Li, Wen-Kai Lou, and Kai Chang, Pure m agnon valley currents in a\npatterned ferromagnetic thin ﬁlm, Phys. Rev. B 107, 104426, (2023).\n[16] Florian Meier and Daniel Loss, Magnetization Transpor t and Quantized Spin Conductance,\nPhys. Rev. Lett. 90, 167204 (2003).\n[17] A. Nogaret, Electron dynamics in inhomogeneous magnet ic ﬁelds, J. Phys.: Condens. Matter\n22, 253201 (2010).\n[18] P. Saraiva, A. Nogaret, J. C. Portal, H. E. Beere, and D. A . Ritchie, Dipolar spin waves of\nlateral magnetic superlattices, Phys. Rev. B 82, 224417 (2010).\n[19] J. F. Cooke, Role of electron-electron interactions in the RKKY theory of magnetism, J. Appl.\nPhys.50, 1782 (1979).\n[20] D. R. Lide, and R. David, CRC Handbook of Chemistry and Ph ysics, 90th Ed., Taylor &\nFrancis (2009).\n12[21] JianMi, JianliWang, L.N.Pfeiﬀer, K.W.West, K.W.Baldw in, andChiZhang,Photovoltage\ndetection of edgemagnetoplasmon oscillations andgiant ma gnetoplasmon resonances ina two-\ndimensional hole system, Phys. Rev. B 94, 125411 (2016).\n[22] Jian-Feng Zhang, Xiao-Wei Zhang, Xiaohu Zheng, Haiwen Liu, Jian Mi, ZhujunYuan, Shuang\nJia, X. C. Xie, and ChiZhang, Microwave-enhanced photovolt aic quantumoscillations of Weyl\nfermions, Phys. Rev. B 102, 205413 (2020).\n[23] C. Kittel, On the Theory of Ferromagnetic Resonance Abs orption,Phys. Rev. 73, 155 (1948).\n[24] C. Kittel, D. F. Holcomb, Introduction to Solid State Ph ysics, 8th Ed., John Wiley & Sons\n(2004).\n[25] A. G. Gurevich and G. A, Melkov, Magnetization: Oscilla tions and Waves, 1st Ed., CRC\nPress (1996).\n[26] D.-X. Chen, E. Pardo, and A. Sanchez, Demagnetizing fac tors for rectangular prisms, IEEE\nTransactions on Magnetics ,41(6), 2077-2088 (2005).\n[27] A. K. Bera, B. Lake, F. H. L. Essler, L. Vanderstraeten, C . Hubig, U. Schollw?ck, A. T. M.\nN. Islam, A. Schneidewind, and D. L. Quintero-Castro, Spino n conﬁnement in a quasi-one-\ndimensional anisotropic Heisenberg magnet, Phys. Rev. B 96, 054423 (2017).\n[28] C. Bayer, J. Jorzick, S. O. Demokritov, A. N. Slavin, K. Y . Guslienko, D. V. Berkov, N. L.\nGorn, M. P. Kostylev and B. Hillebrands, Spin-wave excitati ons in ﬁnite rectangular elements,\nSpin dynamics in conﬁned magnetic structures III: 57-103 (2 006).\n[29] A. Nogaret, F. Nasirpouri, J.-C. Portal, H. E. Beere, D. A. Ritchie, A. T. Hindmarch, and C.\nH. Marrows, Double spin resonance in a spatially periodic ma gnetic ﬁeld with zero average,\nEuro. Phys. Lett. 94, 28001 (2011).\n[30] Kirill Yu. Povarov, Timofei A. Soldatov, Ren-Bo Wang, A ndrey Zheludev, Alexander I.\nSmirnov, and Oleg A. Starykh, Electron Spin Resonance of the Interacting Spinon Liquid,\nPhys. Rev. Lett. 128, 187202 (2022).\n13"    },    {        "title": "1012.2397v1.Spin_orbit_driven_ferromagnetic_resonance__A_nanoscale_magnetic_characterisation_technique.pdf",        "content": "arXiv:1012.2397v1  [cond-mat.mes-hall]  10 Dec 2010Spin-orbit driven ferromagnetic resonance: A nanoscale ma gnetic\ncharacterisation technique\nD. Fang,1H. Kurebayashi,1J. Wunderlich,2,3K. V´ yborn´ y,3L. P. Zˆ arbo,3\nR. P. Campion,4A. Casiraghi,4B. L. Gallagher,4T. Jungwirth,3,4and A. J. Ferguson1,∗\n1Microelectronics Group, Cavendish Laboratory, Universit y of Cambridge,\nJJ Thomson Avenue, Cambridge CB3 0HE, UK\n2Hitachi Cambridge Laboratory, Cambridge CB3 0HE, UK\n3Institute of Physics ASCR, v.v.i., Cukrovarnick´ a 10, 162 5 3 Praha 6, Czech Republic\n4School of Physics and Astronomy,\nUniversity of Nottingham, Nottingham NG7 2RD, UK\nAbstract\nWe demonstrate a scalable new ferromagnetic resonance (FMR ) technique based on the spin-\norbitinteraction. AnalternatingcurrentdrivesFMRinuni formferromagneticstructurespatterned\nfrom the dilute magnetic semiconductors (Ga,Mn)As and (Ga, Mn)(As,P). This allows the direct\nmeasurement of magnetic anisotropy coeﬃcients and damping parameters for individual nano-\nbars. By analysing the ferromagnetic resonance lineshape, we perform vector magnetometry on\nthe current-induced driving ﬁeld, observing contribution s with symmetries of both the Dresselhaus\nand Rashba spin-orbit interactions.\n1Ferromagnetic resonance (FMR) is the most common technique for exploring spin-\ndynamics phenomena and for the magnetic characterisation of fer romagnets.1However,\npreviously developed FMR techniques, based on exciting the magnet ic system by an ex-\nternal alternating magnetic ﬁeld from a resonant cavity2–4or a micro-waveguide,5–8struggle\nto simultaneously achieve scalability of the technique to nano-size ob jects, uniformity of the\nexcitation ﬁeld, and the range of available excitation frequencies. W e introduce an FMR\ntechnique applicable to individual nanomagnets in which the FMR driving ﬁeld is generated\nin the probed magnet itself. The excitation is driven by the eﬀective ﬁ eld generated by\nan alternating electrical current passing through the ferromagn et, which results from the\ncombined eﬀect of the spin-orbit (SO) coupling and exchange intera ction.9–11Our SO-FMR\ncan be operated at tuneable frequencies and we demonstrate its s ensitivity and scalabil-\nity by measuring the variation of micromagnetic parameters of lithog raphically patterned\n(Ga,Mn)As and (Ga,Mn)(As,P) nano-bars.\nFMR induced by driving an alternating current directly through the p robed sample has\nbeen previously demonstrated for speciﬁc non-uniform magnetic n ano-devices such as spin-\nvalves.12,13The experiments utilised the spin-transfer torque in which spin-pola rised electri-\ncal current acts on spatially varying magnetisation14and can be viewed as a macroscopic\nangular momentum transfer eﬀect. Our SO-FMR (Figure 1a) does n ot require the speciﬁc\nsamples with a non-collinear magnetisation proﬁle. The method can be applied to a broad\nrange of systems including uniformly polarised nanomagnets. This is b ecause the eﬀective\nﬁeld utilised in the SO-FMR does not rely on the spatial variation of the magnetisation vec-\ntor but on a microscopic non-collinearity of individual electron spins d ue to their relativistic,\nSO-coupled band structure. Speciﬁcally, when an electrical curre nt traverses through the\nuniformly magnetised material, the resulting non-equilibrium distribut ion of occupied states\nin the SO-coupled carrier bands yields a non-equilibrium spin polarisatio n.15–17The polari-\nsation produces a transverse component of the internal exchan ge ﬁeld (can be viewed as an\neﬀective magneticﬁeld) andatorqueis appliedto themagnetisationv ector.9,18This current-\ninduced eﬀective ﬁeld is generic to ferromagnets with SO-coupling an d inversion asymmetry\nin their band structure. Previously it has been utilised for magnetisa tion switching in the\nferromagnetic semiconductor (Ga,Mn)As10and for domain nucleation in a Pt/Co/AlO x\nstack.11\nThemicro andnano-barsemployed inour SO-FMRstudy arepattern ed by electron beam\n2lithography on 25 nm-thick ﬁlms of (Ga 0.94,Mn0.06)As and (Ga 0.94,Mn0.06)(As0.9,P0.1), grown\nby low-temperature molecular beam epitaxy. The (III,Mn)V ferrom agnetic semiconductors\nused in our study are particularly favourable systems for observin g and exploring SO-FMR\nbecause of the compatibility of the material with advanced semicond uctor nanofabrication\ntechniques, because the carrier bands have strong SO-coupling, and the (III,Mn)V nanos-\ntructures have a rich phenomenology in their micromagnetic parame ters. In the following\ntext we demonstrate our scalable SO-FMR technique in lithographica lly patterned bars of\nwidth ranging from several µm’s to 80 nm (Figure 1b).\nIn order to drive SO-FMR we pass a microwave-frequency current through the nano-bar.\nThis is achieved by wire-bonding the sample between an open-circuit c oplanar transmis-\nsion line and a low-frequency connection which also provides a microwa ve ground (Fig-\nure 1c). Since the microwave excitation ﬁeld originates from the mat erial properties, only\na 2-terminal nano-bar (a resistor) is required in our experiment, e nabling simple and rapid\nsample fabrication. For detection of FMR we utilise a frequency mixing eﬀect based on\nthe anisotropic magnetoresistance (AMR).2–7When the magnetisation precession is driven,\na time-dependent change ∆ R(t) in the longitudinal resistance from the equilibrium value\nRoccurs (due to the AMR). The resistance oscillates with the same fr equency as the mi-\ncrowave current, therefore causing frequency mixing and a direc tly measurable dc voltage\nVdcis generated across the nano-bar. This voltage is our observable p roviding a direct probe\nof the amplitude and phase of the magnetisation precession with res pect to the microwave\ncurrent.\nWe ﬁrst show measurements on a 80 nm-wide nano-bar patterned in the [1¯10] direction\nfrom the (Ga,Mn)(As,P) epilayer. The magnetic ﬁeld dependence of Vdcis measured at\ndiﬀerent microwave frequencies and taken at a temperature of 6 K . The frequency of the\nincident current is ﬁxed while an external dc magnetic ﬁeld H0is swept and a well-deﬁned\nresonance peak appears (Figure 2a). The peak is well-ﬁtted by the solution of the Landau-\nLifshitz-Gilbert (LLG) equation, which describes the dynamics of pr ecessional motion of the\nmagnetisation:\nVdc=Vsym∆H2\n(H0−Hres)2+∆H2+Vasy∆H(H0−∆H)\n(H0−Hres)2+∆H2(1)\nHereHresis the ﬁeld at which resonance occurs and ∆ His the linewidth (half width at\nhalf maximum) of the FMR peak. The resonance lineshape is a combinat ion of symmetric\n3and anti-symmetric Lorentzian functions with amplitudes VsymandVasy, respectively. Their\nrelative contributions are determined by the phase of the driving ﬁe ld with respect to the\ncurrent, and the direction of the driving ﬁeld (see Equation 3 & 4).\nFigure 2b plots the frequency-dependence of the resonance ﬁeld Hres. It is described by\nthe equation for ferromagnetic resonance:19\n/parenleftbiggω\nγ/parenrightbigg2\n=µ2\n0(Hres+H′\nani)(Hres+H′′\nani) (2)\nwhereH′\naniandH′′\naniare terms containing the demagnetisation and anisotropy energies of\nthe ferromagnet (see Methods). A gyromagnetic constant γcharacteristic for Mn2+spins\nof 176 GHz/T (g-factor 2) is used for the ﬁtting. This, together w ith the good agreement\nbetween the observed peaks and the ﬁtted results from the LLG e quation, conﬁrms that we\nobserve the coherent precession of Mn spins.\nTheFMRlinewidth(∆ H= ∆Hinhomo+αω/γ)describesthedampingintheferromagnetic\nsystem. The broadband nature of our setup allows us to determine the inhomogeneous\n(2.5mT)andfrequency-dependent contributionstothedamping( Figure2c)thatcorrespond\ntoGilbert-dampingconstant α=0.023. UsingavectorﬁeldcryostatwealsoperformtheSO-\nFMR measurements for diﬀerent orientations of the external mag netic ﬁeld. In Figure 2d we\npresent the data from an in-plane scan of the magnetic ﬁeld showing that there is a strong\nuniaxial anisotropy perpendicular to the bar direction. By analysing the peak positions\n(Figure 2e) using Equation 2 we quantify the anisotropy ﬁelds and ﬁn dµ0H2/bardbl=−180 mT\n(uniaxial) and µ0H4/bardbl= 68 mT (biaxial).\nWenow demonstrate that SO-FMR can beapplied to comparative inve stigations of nano-\nbarswheretheanisotropiesdiﬀerfrombulkvalues. Theeﬀectofst rain-relaxation, duetothe\nlithographic patterning, on the magnetic anisotropy of (Ga,Mn)As n ano-bars has previously\nbeen studied by electrical transport20–22and optically-detected FMR.8We ﬁrst compare\nthe eﬀect of strain-relaxation between 500 nm bars under compre ssive ((Ga,Mn)As) and\ntensile ((Ga,Mn)(As,P)) growth strain. The in-plane anisotropies ar e studied; although\n(Ga,Mn)(As,P) is out-of-plane magnetised23, the applied ﬁeld H0brings the magnetisation\ninto plane. In (Ga,Mn)As we observe an additional uniaxial contribut ion to the anisotropy\n(µ0HU= 32 mT) along the bar (Figure 3a & c) with a similar magnitude to previou s\nreports.8,20,22By contrast in the (Ga,Mn)(As,P) nano-bar (Figure 3b & c) the sign o f the\nuniaxial anisotropy ( µ0HU=−50.1 mT) has reversed andthe easy axis is now perpendicular\n4to the bar. This can be understood in terms of the sign of the strain relaxation: these\nmaterials become magnetically easier in the direction of most compres sive (least tensile)\nstrain. So when the tensile strain of the (Ga,Mn)(As,P) nano-bar re laxes, it introduces an\neasy axis perpendicular to the bar (Figure 3d). Furthermore we me asure (Ga,Mn)(As,P)\nbars of diﬀerent widths and observe a decrease in the strain-relax ation induced anisotropy\nfrom the 80 nm bar ( µ0HU=−270 mT) to the 500 nm bar ( µ0HU=−50.1 mT), and almost\nno eﬀect of strain-relaxation in the 4 µm bar (µ0HU=−10.5 mT).\nAs well as being able to determine the patterning-induced change in a nisotropy, we also\ncompare the damping among the nano-bars of diﬀerent sizes. The f requency-dependent\nterm (related to damping) increases for decreasing bar width: α= 0.004 (4µm-wide), 0.006\n(500 nm) and 0.023 (80 nm). The signiﬁcantly higher value of Gilbert da mping at 80 nm\ncompared with the 500 nm and 4 µm bars may be due to damage during the etching process.\nThe frequency-independent term is relevant in the case of strain r elaxation as it indicates\nthe inhomogeneity of anisotropy ﬁelds within the bar itself. The inter mediate case of 500 nm\nshows greater inhomogeneity ∆ Hinhomo= 9.9 mT than the 4 µm bar ∆ Hinhomo= 5.4 mT,\nexplained by the increased variation in local anisotropy. By contras t, for 80 nm bar reduces\nto ∆Hinhomo= 2.5 mT, indicative of a high degree of strain-relaxation.\nTo characterise SO-FMR we must understand the direction and amp litude of the eﬀective\nﬁeldheﬀthat drives magnetisation precession. Similar to the experiments on STT-FMR in\nspin-valves12,13we are able to perform vector magnetometry on the driving ﬁeld fro m the\nangle dependence of the amplitude of the FMR peak. For a vector dr iving ﬁeld heﬀ(t) =\n(hx,hy,hz)eiωtin-phase with the microwave current I(t) = (Ix,0,0)eiωt, the amplitudes of\nthe two components of the FMR peak are:\nVsym(θ) =I∆R\n2Asymsin(2θ)hz (3)\nVasy(θ) =I∆R\n2Aasysin(2θ)(hxsinθ+hycosθ) (4)\nwhere ∆Ris the AMR coeﬃcient of the ferromagnetic sample, θis the angle between the\napplied ﬁeld H0and the current I, andAsym(asy)are constants determined by the magnetic\nanisotropies. Hence by decomposing the resonance lineshape into VsymandVasy, and by\nmeasurements of the AMR and magnetic anisotropies we are able to d educe the components\nofheﬀ.\nNo component of Vsymis seen to behave as sin(2 θ), indicating that the driving ﬁeld heﬀ\n5is predominantly in-plane. Figure 4a shows the angle-dependence of Vasyfor a 500 nm-\nwide (Ga,Mn)As bar patterned in the [1 ¯10] direction. We see that Vasy(θ) comprises a\n−sin(2θ)cos(θ) term, indicating that the driving ﬁeld is perpendicular to I. In a [110]\ndevice (Figure 3a) the amplitude of Vasyhas the opposite sign, indicating that the driving\nﬁeld has reversed. For nano-bars along [100] and [010] (Figure 3b) , theVasycurve is a\nsuperposition of sin(2 θ)sin(θ) and sin(2 θ)cos(θ) functions, showing that the driving ﬁeld\nconsists of components both parallel and perpendicular to I.\nThese data are most clearly seen by plotting the dependence of the magnitude and di-\nrection of the eﬀective ﬁeld on the current (nano-bar) orientatio n (Figure 3c). Two con-\ntributions to the driving ﬁeld are observed with diﬀerent symmetry, heﬀ=hR+hD. The\nﬁeldshRandhDhave angular dependence on Ireminiscent of the angular dependence of\nRashba and Dresselhaus SO ﬁelds in the momentum space, respectiv ely.24,25The ﬁeld with\nDresselhaus symmetry, as previously observed in magnetisation sw itching experiments,10\nis due to the diagonal elements in the strain tensor (due to the lattic e mismatch between\nGaAs substrate and (Ga,Mn)As). Therefore hDchanges sign between the (Ga,Mn)As and\n(Ga,Mn)(As,P) materials (comparing Figure 4c and 4d). The Rashba s ymmetry ﬁeld hR\ncan be modelled by oﬀ-diagonal elements in the strain tensor. This st rain is not physically\npresent in the crystal structure of (Ga,Mn)As epilayers. It has b een introduced, however,\nin previous studies to model the in-plane uniaxial anisotropy presen t in (Ga,Mn)As and the\nﬁtted values of this eﬀective oﬀ-diagonal strain are typically sever al times smaller than the\ndiagonal, growth-induced strain.26This is consistent with the observed smaller magnitude\nofhR= 6.5µT thanhD= 18µT (values given at j= 105Acm−2). BothhDandhRare\nmeasured to be linear in current density (Figure 4e & f). We observe a larger magnitude\nofhDat a given current density in the (Ga,Mn)(As,P) nano-bars. This is ex plained by\nthe larger magnitude of the growth strain and larger resistivity (lar gerEat given j) of\n(Ga,Mn)(As,P) as compared to the (Ga,Mn)As ﬁlm.23\nIn conclusion, we perform variable-frequency FMR experiments on individual micro and\nnano-bars of uniform ferromagnetic semiconductors (Ga,Mn)As a nd (Ga,Mn)(As,P). The\nFMR is driven by a torque at microwave frequencies whose origin lies in t he internal eﬀective\nﬁeld (due to the SO-coupling and exchange interaction) of the prob ed ferromagnet. We have\ndemonstrated the utility of our SO-FMR technique by determining th e rich characteristics of\nmagnetic anisotropy ﬁelds and damping coeﬃcients in the studied nan oscale ferromagnetic\n6semiconductor samples. Inaddition, we have performed vector ma gnetometry onthe driving\nﬁeld allowing us to measure a previously unobserved contribution to t he current-induced\nﬁeld in the studied ferromagnets with symmetry of the Rashba SO-in teraction. Our work\ndemonstrates a new scalable FMR technique which provides an unpre cedented method to\nperform magnetic characterisation of uniform ferromagnetic nan ostructures and to study\nthe nature of the current-induced eﬀective magnetic ﬁeld in SO-co upled ferromagnets.\nWe acknowledge fruitful discussions with Ion Garate, Allan H. MacDo nald and Leonid\nRokhinson and support from EU Grants FP7-214499 NAMASTE, FP7 -215368 SemiSpin-\nNet, ERC Advanced Grant, from Czech Republic Grants AV0Z10100 521, KAN400100652,\nLC510, KJB100100802 and Preamium Academiae, DF acknowledges s upport from Cam-\nbridgeOverseas Trusts andHitachi Cambridge Laboratory, A.J.F. acknowledges thesupport\nof a Hitachi research fellowship.\n∗Electronic address: ajf1006@cam.ac.uk\n1Vonsovski ˇi, S. V.Ferromagnetic Resonance (Pergamon, Oxford, 1966).\n2Goennenwein, S. T. B. et al.Electrically detected ferromagnetic resonance. Appl. Phys. Lett.\n90(2007).\n3Mecking, N., Gui, Y. S. & Hu, C.-M. Microwave photovoltage an d photoresistance eﬀects in\nferromagnetic microstrips. Phys. Rev. B 76(2007).\n4Hui, X.et al.Electric detection of ferromagnetic resonance in single cr ystal iron ﬁlm. Appl.\nPhys. Lett. 93(2008).\n5Costache, M. V., Watts, S. M., Sladkov, M., van der Wal, C. H. & van Wees, B. J. Large cone\nangle magnetization precession of an individual nanopatte rned ferromagnet with dc electrical\ndetection. Appl. Phys. Lett. 89(2006).\n6Costache, M. V., Sladkov, M., van der Wal, C. H. & van Wees, B. J . On-chip detection of\nferromagnetic resonance of a single submicron Permalloy st rip.Appl. Phys. Lett. 89(2006).\n7Yamaguchi, A. et al.Broadband ferromagnetic resonance of Ni 81Fe19wires using a rectifying\neﬀect.Phys. Rev. B 78(2008).\n8Hoﬀmann, F. et al.Mapping the magnetic anisotropy in (Ga,Mn)As nanostructur es.Phys.\nRev. B80(2009).\n79Manchon, A. & Zhang, S. Theory of spin torque due to spin-orbi t coupling. Phys. Rev. B 79\n(2009).\n10Chernyshov, A. et al.Evidence for reversible control of magnetization in a ferro magnetic ma-\nterial by means of spin-orbit magnetic ﬁeld. Nature Phys. 5, 656–659 (2009).\n11Miron, I. M. et al.Current-driven spin torque induced by the Rashba eﬀect in a fe rromagnetic\nmetal layer. Nature Mater. 9, 230–234 (2010).\n12Tulapurkar, A. A. et al.Spin-torque diode eﬀect in magnetic tunnel junctions. Nature438,\n339–342 (2005).\n13Sankey, J. C. et al.Spin-transfer-driven ferromagnetic resonance of individ ual nanomagnets.\nPhys. Rev. Lett. 96(2006).\n14Myers, E. B., Ralph, D. C., Katine, J. A., Louie, R. N. & Buhrma n, R. A. Current-induced\nswitching of domains in magnetic multilayer devices. Science285, 867–870 (1999).\n15Edelstein, V. Spin polarization of conduction electrons in duced by electric current in two-\ndimensional asymmetric electron systems. Solid State Commun. 73, 233–235 (1990).\n16Inoue, J., Bauer, G. E. W. & Molenkamp, L. W. Diﬀuse transport a nd spin accumulation in a\nRashba two-dimensional electron gas. Phys. Rev. B 67(2003).\n17Silov, A. Y. et al.Current-induced spin polarization at a single heterojunct ion.Appl. Phys.\nLett.85, 5929–5931 (2004).\n18Garate, I. & MacDonald, A. H. Inﬂuence of a transport current on magnetic anisotropy in\ngyrotropic ferromagnets. Phys. Rev. B 88(2009).\n19Liu, X. & Furdyna, J. K. Ferromagnetic resonance in Ga 1−xMnxAs dilute magnetic semicon-\nductors. J. Phys.: Cond. Matt. 18, R245– R279 (2006).\n20H¨ umpfner, S. et al.Lithographic engineering of anisotropies in (Ga,Mn)As. Appl. Phys. Lett.\n90(2007).\n21Wunderlich, J. et al.Local control of magnetocrystalline anisotropy in(Ga,Mn) As microdevices:\nDemonstration in current-induced switching. Phys. Rev. B 76(2007).\n22Wenisch, J. et al.Control of magnetic anisotropy in (Ga,Mn)As by lithography -induced strain\nrelaxation. Phys. Rev. Lett. 99(2007).\n23Rushforth, A. W. et al.Molecular beam epitaxy grown (Ga,Mn)(As,P) with perpendic ular to\nplane magnetic easy axis. J. Appl. Phys. 104(2008).\n24Dresselhaus, G. Spin-orbit coupling eﬀects in zinc blende st ructures. Phys. Rev. 100, 580–586\n8(1955).\n25Bychkov, Y. A. & Rashba, E. I. Oscillatory eﬀects and the magne tic susceptibility of carriers\nin inversion layers. J. Phys. C: Solid State Phys. 17, 6039–6045 (1984).\n26Zemen, J., Kuˇ cera, J., Olejn´ ık, K. & Jungwirth, T. Magneto crystalline anisotropies in\n(Ga,Mn)As: Systematic theoretical study and comparison wi th experiment. Phys. Rev. B 80\n(2009).\n9Figure 1, Principle of the experiment and its setup. a, Precession of the mag-\nnetisation vector Maround the total magnetic ﬁeld Htot.Mis subject to a damping torque\nταdue to energy dissipation, which causes the magnetic motion to relax towardsHtot. The\ndriving torque τSOdue to current-induced eﬀective ﬁeld counters the eﬀect of damp ing, and\nleads to steady-state motion ∂M/∂t=−γM×Htot.b,SEM image of a 80 nm-wide bar,\npatterned from the (Ga,Mn)(As,P) wafer. c,Schematic of the experimental setup.\n10Figure 2, Spin-orbit driven ferromagnetic resonance. a, Vdcmeasured at 8, 10\nand 12 GHz (circles) on the 80 nm-wide device. The resonance peaks are clearly observed\nand can be well-described by Equation 1 (solid lines are the ﬁtted resu lts). The diﬀerence\nin the signal level at diﬀerent ωis caused by the frequency-dependent attenuation of the\nmicrowave circuit. b,The resonance ﬁeld Hresas a function of the microwave frequency\n(black triangles). The red solid line is the ﬁtted results to Equation 2. c,Frequency-\ndependence of the FMR linewidth ∆ H(black squares). The data are ﬁtted to a straight line\nto extract information on ∆ Hinhomoandα.d,Vdcmeasured from in-plane rotational scans\nof the external ﬁeld H0. The colour scale represents the magnitude of the voltage. ϕis the\nanglebetween themagnetisationvector Mandthe[100]crystalline axis. e,Angle-plotofthe\nresonance ﬁeld Hres, which is extracted by ﬁtting to each FMR peak using Equation 1 (blac k\ncircles). The red line is a ﬁtting curve to Equation 2 to calculate the ma gnetic anisotropy.\n11Figure 3, SO-FMR on devices patterned from diﬀerent materia ls and with\nvarious sizes. a, Hres(ϕ) measured from an in-plane rotational scan on a 500 nm-wide\n(Ga,Mn 0.06)As bar (patterned along the [010] axis). The circles are measurem ent data, and\nthe solid line is the ﬁtted results to Equation 2. The black arrow marks the long axis of the\nnano-bar. b,Hres(ϕ) measured on a (Ga,Mn 0.06)(As,P 0.1) device with identical shape and\norientation. c,Comparison of the in-plane anisotropy ﬁelds Hibetween the two samples.\nd,Schematic of the strain relaxation in the compressively-strained (G a,Mn)As and and\ntensile-strained (Ga,Mn)(As,P) nanostructures. e,Comparison of the magnetic anisotropy\n(in terms of the proﬁles of Hres) among 80, 500 and 4000 nm-wide (Ga,Mn)(As,P) bars. f,\nThe linewidth ∆ Hof the FMR signals measured on the three devices.\n12Figure 4, Characterisation of the driving ﬁeld in both (Ga,M n)As and\n(Ga,Mn)(As,P) devices. a–b, Amplitudes of the anti-symmetric part of the FMR sig-\nnalVasy, measured on a group of 500 nm-wide (Ga,Mn)As bars (circles), pat terned along\ndiﬀerent crystalline directions. The solid lines are ﬁtted results to Eq uation 4. c,Plot\nof the magnitude and direction of the current-induced eﬀective ﬁe ldheﬀmeasured on the\n(Ga,Mn)As nano-bars, scaled for a current density j= 105A/cm2.d,Similar plot for heﬀ\nmeasured on the (Ga,Mn)(As,P) devices. e–f,Current density dependence of hDandhRin\nboth (Ga,Mn)As and (Ga,Mn)(As,P) nano-bars. A second horizonta l scale is included for\nthe electric ﬁeld, calculated from the device resistance (values give n in Methods).\n13"    },    {        "title": "1408.5842v1.Electrical_detection_of_ferromagnetic_resonance_in_ferromagnet_n_GaAs_heterostructures_by_tunneling_anisotropic_magnetoresistance.pdf",        "content": "arXiv:1408.5842v1  [cond-mat.mtrl-sci]  25 Aug 2014Electrical detection of ferromagnetic resonance in ferrom agnet/ n-GaAs\nheterostructures by tunneling anisotropic magnetoresist ance\nC. Liu,1Y. Boyko,1,a)C. C. Geppert,1K. D. Christie,1G. Stecklein,1S. J. Patel,2\nC. J. Palmstrøm,2, 3and P. A. Crowell1,b)\n1)School of Physics and Astronomy, University of Minnesota, M inneapolis,\nMinnesota 55455, USA\n2)Department of Materials, University of California, Santa B arbara,\nCalifornia 93106, USA\n3)Department of Electrical and Computer Engineering, Univer sity of California,\nSanta Barbara, California 93106, USA\nWe observe a dc voltage peak at ferromagnetic resonance (FMR ) in samples consisting\nof a single ferromagnetic (FM) layer grown epitaxially on th en−GaAs (001) surface.\nThe FMR peak is detected as an interfacial voltage with a symm etric line shape\nand is present in samples based on various FM/ n-GaAs hetrostructures, including\nCo2MnSi/ n-GaAs, Co 2FeSi/ n-GaAs and Fe/ n-GaAs. We show that the interface bias\nvoltage dependence of the FMR signal is identical to that of t he tunneling anisotropic\nmagnetoresistance (TAMR) over most of the bias range. Furth ermore, we show how\nthe precessing magnetization yields a dc FMR signal through the TAMR eﬀect and\nhow the TAMR phenomenon can be used to predict the angular dep endence of the\nFMR signal. This TAMR-induced FMR peak can be observed under conditions\nwhere no spin accumulation is present and no spin-polarized current ﬂows in the\nsemiconductor.\na)Current address: Department of Physics, University of Mary land, College Park, MD 20742, USA\nb)Electronic mail: crowell@physics.umn.edu\n1One of the goals of spintronics research is to develop tools f or manipulating electron spins\nin semiconductors.1Although many approaches are based on spin-polarized charg e currents,\na separate class of eﬀects is based on the phenomenon of spin p umping, in which a non-\nequilibrium spin population is generated by ferromagnetic resonance (FMR).2In the case\nof metals and semiconductors, a common method of detecting t his eﬀect is to measure the\ndc voltage generated by the pumped spin current through the i nverse spin Hall eﬀect.3–5To\ncorrectly interpret these measurements, it is essential to understand all of the mechanisms by\nwhich the FMR can contribute to the generation of dc voltages . Among these are anisotropic\nmagnetoresistance (AMR)6and the planar Hall eﬀect.7,8\nIn this Letter, we report on electrically detected FMR in epi taxial ferromagnet (FM)/ n-\nGaAs (001) heterostructures. The FM/GaAs interfaces in eac h of these devices are Schottky\ntunnel barriers. We ﬁnd that the dominant contribution to th e electrically detected FMR\nsignal under reverse and small forward bias current is tunne ling anisotropic magnetoresis-\ntance (TAMR). The measured TAMR signal is used to predict the bias dependence of the\nFMR signal as well as its dependence on the magnetic ﬁeld orie ntation. The agreement with\nthe predictions of our model, in which spin transport in the s emiconductor plays no role, is\nexcellent.\nThe FM/ n-GaAs heterostructures investigated in this experiment we re grown by molec-\nular beam epitaxy on GaAs (001) substrates. The growth start ed with a 500 nm undoped\nGaAs buﬀer layer, followed by 2500 nm of Si-doped n-GaAs ( n= 3−5×1016cm−3). The\njunction region consists of a 15 nm n→n+-GaAs transition layer followed by 15-18 nm\nn+(5×1018cm−3) GaAs.9The 5 nm thick FM ﬁlm is then deposited epitaxially, followed by\n10 nm thick Al and Au capping layers. The FM ﬁlms studied are Co 2MnSi, Co 2FeSi, and Fe,\nwith deposition temperatures of 220◦C, 270◦C, and room temperature, respectively. The ﬁrst\ntwo materials are Heusler alloys that are promising candida tes for spintronics research.10–13\nDevices fabricated from these heterostructures all show no n-local spin valve and Hanle sig-\nnals in traditional electrical spin injection/detection m easurements at low temperatures.9\nThe FMR signals discussed in this paper are not strongly temp erature dependent, so only\nroom temperature measurements will be presented.\nFigure 1(a) depicts the measurement geometry for our experi ment, where φis the in-\nplane angle relative to the crystal axis [1 ¯10], which is the in-plane magnetic hard axis, and\nθis the out-of-plane angle, measured relative to the (001) pl ane. For FMR measurements,\n2Figure 1. (color online) (a) Schematic of the measurement ge ometry. The middle contact, which\nhas a lateral size of 5 ×50µm, is ferromagnetic. On top of the ferromagnet is a 100 nm thick\ngold wire. The other two contacts are fabricated from CuGe. HandM(t) represent the applied\nmagnetic ﬁeld and the time-dependent magnetization respec tively. hrfrepresents the in-plane\nOersted ﬁeld generated by the microwave current ﬂowing in th e gold wire. (b) The dc voltage\npeak measured on a Co 2MnSi sample at 8 GHz under a bias voltage of -0.5V (reverse bia s) as the\nﬁeld is swept through the resonance. The solid line is a Loren tzian ﬁt. The inset of (b) shows the\ncurrent-voltage characteristic of the Co 2MnSi/ n-GaAs interface. (c) Experimentally determined\nFMR frequency as a function of the magnetic ﬁeld (solid squar es) applied along the [1 ¯10] direction,\nwhich is the in-plane magnetic hard axis. The FMR frequency c alculated from the Kittel formula\nis shown using the solid line.\nthe magnetic ﬁeld is applied in the (001) plane. A dc bias curr ent is combined with the\nmicrowave excitation signal using a bias-T and coupled into the sample using a coaxial cable.\nThe microwave current passes through a 100 nm thick gold laye r deposited on top of the FM\n3contact, generating an in-plane Oersted ﬁeld hrfalong the [110] direction. We use a lock-in\nampliﬁer to measure the voltage of FM contact with respect to a CuGe (non-magnetic)\ncounter electrode.14As the magnetic ﬁeld is swept through the resonance, a dc volt age peak\nis measured. Figure 1(b) shows the resonance peak for a Co 2MnSi sample. Similar peaks, all\nwith a symmetric lineshape, are observed in the other two het erostructures. By varying the\nexcitation frequency, the FMR frequency can be measured as a function of magnetic ﬁeld, as\nshown using solid squares in Fig. 1(c). The frequency calcul ated from the Kittel formula15\nis shown by the solid curve in Fig. 1(c). In applying the Kitte l formula, the saturation\nmagnetization Msand uniaxial anisotropy Kuwere determined from measurements of the\nsaturation ﬁeld along [001] and [1 ¯10] directions.\nBecause the FMR measurement uses the 3-terminal conﬁgurati on,16the observed FMR\npeak corresponds to a change in the voltage across the FM/ n-GaAs interface. Careful\ncharacterization of the FM/ n-GaAs interface allows us to identify the mechanism respons ible\nfor this FMR peak. In these epitaxally grown samples, a Schot tky tunnel barrier exists at\nthe FM/ n-GaAs interface.17Spin-orbit interactions due to the Rashba ﬁeld at the interf ace\nas well as the Dresselhaus ﬁeld in the tunnel barrier lead to a dependence of the tunneling\nresistance on the orientation of the magnetization with res pect to the crystal axes.18,19This\nphenomenon is called tunneling anisotropic magnetoresist ance (TAMR) and is present in all\nof our samples.\nThe TAMR eﬀect is shown for a Co 2MnSi device in Figs. 2(a) and (b). The in-plane\nTAMR [ R(φ)−R(0)]/R(0), where R(φ) is the interfacial resistance when the magnetization\nvector is oriented along φ, is shown as a function of φin Fig. 2(a), and the out-of-plane\nTAMR is shown as a function of the out-of-plane ﬁeld in Fig. 2( b). A linear background\ndue to a slight misalignment of the sample has been subtracte d. The observed TAMR eﬀect\nin our heterostructures is similar in magnitude to the resul ts from other studies of FM-\nGaAs interfaces.18,20The solid line in Fig. 2(a) is ﬁt using a sin2φfunction, from which we\nobtain the magnitude ∆ Riof the in-plane TAMR. Given the ordinary shape anisotropy of\na thin ﬁlm, the out-of-plane TAMR should depend quadratical ly on magnetic ﬁeld below\nsaturation. A ﬁt is shown using the solid curve in Fig. 2(b). T he out-of-plane TAMR ∆ Ro\nis the diﬀerence between the resistances measured at zero ﬁe ld, for which the magnetization\nlies along [110], and at saturation, for which it lies along [ 001]. The full angular dependence\n4/s45/s49/s56/s48 /s45/s57/s48 /s48 /s57/s48 /s49/s56/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52\n/s79/s117/s116/s45/s111/s102/s45/s112/s108/s97/s110/s101/s32/s84/s65/s77/s82/s32/s40/s37/s41/s32/s32/s73/s110/s45/s112/s108/s97/s110/s101/s32/s84/s65/s77/s82/s32/s40/s37/s41\n/s32/s40/s100/s101/s103/s41/s45/s50/s48 /s45/s49/s48 /s48 /s49/s48 /s50/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52\n/s40/s98/s41\n/s40/s100/s41/s40/s97/s41\n/s32/s32\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s40/s107/s79/s101/s41\n/s45/s50/s55/s48 /s45/s49/s56/s48 /s45/s57/s48 /s48 /s57/s48/s48/s50/s52/s40/s99/s41/s73/s110/s45/s112/s108/s97/s110/s101/s32/s84/s65/s77/s82/s32/s118/s111/s108/s116/s97/s103/s101/s32/s40/s109/s86/s41/s45/s32/s49/s46/s49/s51/s32/s86\n/s45/s32/s48/s46/s54/s56/s32/s86\n/s45/s32/s48/s46/s50/s53/s32/s86\n/s45/s49/s46/s52 /s45/s49/s46/s50 /s45/s49/s46/s48 /s45/s48/s46/s56/s48/s51/s48/s54/s48\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100 /s32/s40/s107/s79/s101/s41/s40/s100/s101/s103/s41\n/s86/s111/s108/s116/s97/s103/s101/s32/s40 /s86/s41\n/s45/s32/s50/s46/s49/s32/s86\n/s45/s32/s48/s46/s56/s54/s32/s86\n/s45/s32/s48/s46/s49/s32/s86\nFigure 2. (color online) (a), (b) The in-plane and out-of-pl ane TAMR measured on a Co 2MnSi\nsample at bias voltages of -0.5 V and -2.1 V respectively. In ( a) a magnetic ﬁeld much larger\nthan the in-plane hard-axis saturation ﬁeld was used to rota te the magnetization in the ﬁlm plane.\nIn (b) a magnetic ﬁeld normal to the ﬁlm was applied to gradual ly align the magnetization into\nthe out-of-plane direction. (c) The interface voltage as a f unction of the in-plane magnetization\ndirection for diﬀerent reverse bias voltages. (d) The ﬁeld- swept FMR peak measured at diﬀerent\nreverse bias voltages.\nof the TAMR can be written as\nR(φ, θ) =R(0,0)−∆Ricos2θsin2φ+ ∆Rh\nosin2θ, (1)\nwhere ∆ Rh\no=R(0, π/2)−R(0,0) = ∆ Ro−∆Riis the out-of-plane TAMR measured relative\nto the [1 ¯10] direction. Similar angular dependencies of the TAMR are observed for any bias\nvoltage, as shown in Fig. 2(c) for the in-plane case. There is a marked similarity in the\nbias-dependence of the magnitude of the FMR peak, which is sh own in Fig. 2(d).\nWe now show that the FMR signals in Fig. 1 and Fig. 2(d) are due t o TAMR. The\nprimary evidence comes from a comparison of the bias voltage dependence of the magnitude\nof the FMR peak and the in-plane TAMR voltage ∆ Vi. These are shown in Fig. 3 for all\nthree FM materials. In each case, the FMR and TAMR signals und er reverse bias ( V < 0)\nare directly proportional to each other. For clarity, the y-scale for forward bias ( V > 0) is\nmagniﬁed by the factors shown for each sample. A similar scal ing between the TAMR and\nFMR is observed for small forward bias voltages, although th e proportionality breaks down\nas the forward bias voltage increases. This breakdown of sca ling between FMR and TAMR\n5/s48/s50/s53/s53/s48\n/s40/s99/s41/s40/s98/s41/s67/s111\n/s50/s77/s110/s83/s105/s32/s32\n/s32/s70/s77/s82/s32/s112/s101 /s97 /s107\n/s32/s84/s65/s77/s82/s32/s101 /s102/s102/s101 /s99/s116/s40/s97/s41\n/s48/s50/s52\n/s32\n/s48/s50/s53/s53/s48\n/s67/s111\n/s50/s70/s101/s83/s105/s70/s77/s82/s32/s112/s101/s97/s107/s32/s40 /s86/s41\n/s48/s49/s50/s51/s88/s32/s52\n/s84/s65/s77/s82 /s32/s73 /s82\n/s105/s32/s40/s109/s86/s41\n/s88/s32/s50\n/s45/s50/s46/s48 /s45/s49/s46/s53 /s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53/s48/s51/s54\n/s70/s101\n/s73/s110/s116/s101/s114/s102 /s97/s99/s101/s32/s98/s105/s97/s115/s32/s118 /s111/s108/s116/s97/s103/s101/s32/s40/s86/s41/s48/s50/s52\nFigure 3. The magnitudes of the dc voltage peak at FMR (square s, left axis) and the TAMR\nvoltage I∆Ri(triangles, right axis) for the three diﬀerent FM-GaAs hete rostructures at room\ntemperature. Positive voltages correspond to forward bias (ﬂow of electrons into the metal) across\nthe Schottky barrier. In (a) and (b), the plots on the forward bias side are magniﬁed to see the\nsign diﬀerence of the FMR peak and TAMR eﬀect between the two s amples.\nis due to the existence of a spin accumulation, the consequen ces of which will be discussed\nin a future publication.\nIn a similar experiment carried out in a waveguide, in which t he sample orientation can\nbe changed with respect to the microwave ﬁeld, we ﬁnd that the peak shape is insensitive\nto the direction of the microwave electric ﬁeld, in contrast to the case of rectiﬁcation of\nordinary AMR.21This observation implies that the dc voltage generated by th e FMR is\nsensitive only to the precessing magnetization and is indep endent of the microwave current\nﬂowing in the FM. The relevant mechanism is illustrated in Fi g. 4(a), which shows the\nTAMR voltage as a function of φandθ. When the FM contact is driven on resonance, the\nmagnetization follows an elliptical trajectory in ( φ, θ) space. The resonant trajectory can be\ncalculated from the known anisotropy surface, and the examp le for the case ( φ= 0, θ= 0)\nis shown as the solid red curve in Fig. 4(a). On average, the TA MR voltage in the presence\nof a precessing magnetization increases relative to its equilibrium value at ( φ= 0, θ= 0).\nThis eﬀect is proportional to the local curvature of the TAMR surface and the square of the\n6angular amplitude of precession.\nTo explore this eﬀect more quantitatively, we investigate t he dependence of the FMR peak\non the in-plane orientation of the magnetization at a ﬁxed re verse bias. This measurement\nis carried out in a waveguide, and the orientation of the micr owave magnetic ﬁeld is the\nsame as in Fig. 1(a). Figure 4(b) shows the FMR peak magnitude observed from a Co 2MnSi\nsample as a function of φ. The FMR peak is largest at φ= 0◦and undergoes a sign change\nbefore approaching zero as φ→90◦.\nFrom the above discussion, we can derive an expression for th e magnitude of the FMR\nvoltage peak as a function of the in-plane angle of the magnet ization. We expand the\ninterface voltage to second order in small deviations δφandδθabout their equilibrium\nvalues. We retain only those terms that will not vanish after taking a time average:\nV=IR(φ, θ) =IR(φ,0) +1\n2I∂2R(φ,θ)\n∂φ2/vextendsingle/vextendsingle/vextendsingle\nθ=0(δφcosωt)2\n+1\n2I∂2R(φ,θ)\n∂θ2/vextendsingle/vextendsingle/vextendsingle\nθ=0(δθsinωt)2, (2)\nwhere Iis the interface bias current, δφandδθare the in-plane and out-of-plane precession\ncone angles respectively, and ωis the resonance frequency. With the substitution of the\nmeasured R(φ, θ) from Eq. 1 into Eq. 2 and taking of the time average, we obtain :\n/angbracketleftV/angbracketright=IR(φ,0)−1\n2I∆Ricos 2φ(δφ)2\n+1\n2(I∆Risin2φ+I∆Rh\no)(δθ)2. (3)\nIn Eq. 3, the sum of the last two terms, which depend on the prec essional cone angles,\nis the voltage of the FMR peak. The precessional cone angles a reδφ=δφ0cosφand\nδθ=δθ0cosφ, where δφ0andδθ0are the in-plane and out-of-plane angular amplitudes\natφ= 0◦. The factor cos φaccounts for the change in the component of the microwave\nmagnetic ﬁeld perpendicular to the magnetization. Finally we obtain:\nVF MR(φ) =−1\n2I∆Ricos 2φcos2φ(δφ0)2\n+1\n2(I∆Risin2φ+I∆Rh\no) cos2φ(δθ0)2. (4)\nTo calculate VF MR from Eq. 4, I∆RiandI∆Rh\noare obtained from the TAMR mea-\nsurement. Because of the shape anisotropy of the thin ﬁlm, th e second term involving the\nout-of-plane cone angle in Eq. 4 is signiﬁcantly smaller tha n the ﬁrst term. The magnitude\nof the FMR peak should therefore be proportional to I∆Ri, as observed in Fig. 3. The\n7Figure 4. (color online) (a) The interface voltage as a funct ion of the orientation ( φ, θ) of the\nmagnetization due to the TAMR eﬀect. On resonance the magnet ization traces a trajectory on the\n3D surface. (b) The measured FMR peak size (open circles) as a function of the in-plane angle of\nthe applied ﬁeld. The solid line is a ﬁt to Eq. 4.\nquadratic dependence on δφandδθimplies that the FMR peak should be symmetric, in\nagreement with experiment. In Fig. 4(b) the solid curve is a ﬁ t of the in-plane angle de-\npendence of the FMR signal using Eq. 4. The angular amplitude sδφ0andδθ0are the only\nﬁtting parameters. We ﬁnd δφ0= 8.4◦±0.3◦andδθ0= 3.7◦±0.2◦for the in-plane and\nout-of-plane cone angles respectively. We calculated the d ynamical susceptibility for this\nsample using the measured saturation magnetization and ani sotropy, from which we ﬁnd\nthe ratio δφ0/δθ 0≈2.2, in reasonable agreement with the value of 2.3 obtained fro m the ﬁt\nof the angle dependence data in Fig. 4.\nWe emphasize that the mechanism discussed in this paper is es sentially a modulation\nof the tunneling current due to the precession of the magneti zation. This is distinct from\nspin pumping, in which a spin current is generated directly b y the precessing magnetization.\nBecause of the signiﬁcant Schottky tunnel barrier present i n these devices, we expect spin\npumping eﬀects to be small. In fact, we have not been able to ob serve any inverse spin Hall\neﬀect on resonance at zero bias, in spite of the fact that devi ces fabricated from the same\n8heterostructures do function as non-local spin valves. On t he other hand, the tunnel barrier\nin these samples enhances the TAMR eﬀect. As noted above, we d o observe a signiﬁcant\ndeviation of the FMR signal from the TAMR under forward bias v oltages, as can be seen\nin Figs. 3(a) and (b). In determining the extent to which thes e deviations are due to spin\naccumulation, a reliable means for separating the TAMR comp onent, as described here, is\nessential.\nIn summary, we have performed electrically detected FMR exp eriments on epitaxial\nFM/n-GaAs heterostructures. We observe a strong dc voltage peak at the FM/ n-GaAs inter-\nface at resonance in a variety of heterostructures with diﬀe rent ferromagnets. In each case,\nthe predominant origin of the FMR peak is the tunneling aniso tropic magnetoresistance.\nThis contribution must be considered in any measurement in w hich the FM/semiconductor\ninterface is biased.\nThis work was supported by NSF under DMR-1104951, the MRSEC p rogram of NSF\nunder DMR 08-19885, and C-SPIN, one of the six centers of STAR net, a SRC program\nsponsored by MARCO and DARPA.\nREFERENCES\n1I. Žutić, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004).\n2Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. Let t.88, 117601 (2002).\n3E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Appl. Phys. Le tt.88, 182509 (2006).\n4K. Ando, S. Takahashi, J. Ieda, H. Kurebayashi, T. Trypiniot is, C. H. W. Barnes,\nS. Maekawa, and E. Saitoh, Nat. Mater. 10, 655 (2011).\n5H. Kurebayashi, O. Dzyapko, V. E. Demidov, D. Fang, A. J. Ferg uson, and S. O. Demokri-\ntov, Nat. Mater. 10, 660 (2011).\n6N. Mecking, Y. S. Gui, and C.-M. Hu, Phys. Rev. B 76, 224430 (2007).\n7L. Chen, F. Matsukura, and H. Ohno, Nat. Commun. 4, (2013).\n8L. Chen, S. Ikeda, F. Matsukura, and H. Ohno, Appl. Phys. Expr ess7, 013002 (2014).\n9X. Lou, C. Adelmann, S. A. Crooker, E. S. Garlid, J. Zhang, K. S . M. Reddy, S. D. Flexner,\nC. J. Palmstrøm, and P. A. Crowell, Nat. Phys. 3, 197 (2007).\n10I. Galanakis, P. H. Dederichs, and N. Papanikolaou, Phys. Re v. B 66, 134428 (2002).\n11W. H. Wang, M. Przybylski, W. Kuch, L. I. Chelaru, J. Wang, Y. F . Lu, J. Barthel, H. L.\n9Meyerheim, and J. Kirschner, Phys. Rev. B 71, 144416 (2005).\n12Z. Gercsi, A. Rajanikanth, Y. K. Takahashi, K. Hono, M. Kikuc hi, N. Tezuka, and\nK. Inomata, Appl. Phys. Lett. 89, 082512 (2006).\n13R. Farshchi and M. Ramsteiner, J. Appl. Phys. 113, 191101 (2013).\n14M. O. Aboelfotoh, C. L. Lin, and J. M. Woodall, Appl. Phys. Let t.65, 3245 (1994).\n15C. Kittel, Phys. Rev. 73, 155 (1948).\n16X. Lou, C. Adelmann, M. Furis, S. A. Crooker, C. J. Palmstrøm, and P. A. Crowell, Phys.\nRev. Lett. 96, 176603 (2006).\n17G. Schmidt, J. Phys. D: Appl. Phys. 38, R107 (2005).\n18J. Moser, A. Matos-Abiague, D. Schuh, W. Wegscheider, J. Fab ian, and D. Weiss, Phys.\nRev. Lett. 99, 056601 (2007).\n19A. Matos-Abiague, M. Gmitra, and J. Fabian, Phys. Rev. B 80, 045312 (2009).\n20T. Uemura, M. Harada, T. Akiho, K.-i. Matsuda, and M. Yamamot o, Appl. Phys. Lett.\n98, 102503 (2011).\n21M. Harder, Z. X. Cao, Y. S. Gui, X. L. Fan, and C.-M. Hu, Phys. Re v. B 84, 054423\n(2011).\n10"    },    {        "title": "1605.02829v1.Photoenhanced_spin_valley_polarization_and_tunneling_magnetoresistance_in_ferromagnetic_normal_ferromagnetic_silicene_junction.pdf",        "content": "Photoenhanced spin /valley polarization and tunneling magnetoresistance in\nferromagnetic-normal-ferromagnetic silicene junction\nLe Bin Ho1, 2,\u0003and Tran Nguyen Lan1, 3,y\n1Ho Chi Minh City Institute of Physics, VAST, Ho Chi Minh City, Vietnam\n2Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan\n3Department of Physics, University of Michigan, Ann Arbor, Michigan, 48109, USA\n(Dated: October 28, 2018)\nWe theoretically demonstrate a simple way to signiﬁcantly enhance the spin /valley polarizations and tunnel-\ning magnetoresistnace (TMR) in a ferromagnetic-normal-ferromagnetic (FNF) silicene junction by applying a\ncircularly polarized light in o \u000b-resonant regime to the second ferromagnetic (FM) region. We show that the\nfully spin-polarized current can be realized in certain ranges of light intensity. Increasing the incident energy in\nthe presence of light will induce a transition of perfect spin polarization from positive to negative or vice versa\ndepending on magnetic conﬁguration (parallel or anti-parallel) of FNF junction. Additionally, under a circularly\npolarized light, valley polarization is very sensitive to electric ﬁeld and the perfect valley polarization can be\nachieved even when staggered electric ﬁeld is much smaller than exchange ﬁeld. The most important result we\nwould like to emphasize in this paper is that the perfect spin polarization and 100% TMR induced by a circularly\npolarized light are completely independent of barrier height in normal region. Furthermore, the sign reversal of\nTMR can be observed when the polarized direction of light is changed. A condition for observing the 100%\nTMR is also reported. Our results are expected to be informative for real applications of FNF silicene junction,\nespecially in spintronics.\nI. INTRODUCTION\nSilicene, a two-dimensional allotrope of Si, has attracted\na great attention in both theory and experiment. This ma-\nterial not only shares all intriguing electronic properties of\ngraphene, but it has some superior advantages compared to\ngraphene, such as strong spin-orbit coupling and buckled hon-\neycomb structure. While the former enables us to realize\nthe quantum spin Hall e \u000bect1, the latter allows us to control\nthe bulk band gap of silicene by applying an external electric\nﬁeld2.\nRecently, Ezawa reported multiple topological phase tran-\nsitions of silicene in the presence of electric and exchange\nﬁelds3–6. Inﬂuences of these ﬁelds on ballistic transport of\nsingle and multiple barrier junctions of silicene have been also\nextensively investigated7–13. These studies have found many\ninteresting and novel transport phenomena, which is not anal-\nogous in graphene, for examples, ﬁeld-dependent spin- and\nvalley-polarized currents or ﬁeld-dependent transport gap.\nAlthough a circularly polarized light in o \u000b-resonant regime\ncan also produce a topological phase transition in silicene14,\nits e\u000bect on ballistic transport has not received a deserving\nattention and only done by a few recent works. Tahir and\nSchwingenschl ¨ogl15studied the Hall and longitudinal conduc-\ntivities of silicene and germanene in the presence of a per-\npendicular electric and magnetic ﬁelds taking into account the\ne\u000bects of o \u000b-resonant light. Meantime, Niu and Dong16re-\nported e \u000bects of o \u000b-resonant light in combination with stag-\ngered electric ﬁeld or gate voltage on spin and valley polar-\nizations of a normal silicene junction. These authors also con-\nsidered the variation of TMR with the gate voltage applied\nto third region of a FNF silicene junction when FM regions\nwere exposed to a ﬁxed circularly polarized light. Neverthe-\nless, a detailed investigation on the inﬂuence of circularly po-\nlarized light itself on spin and valley polarizations as well as\nTMR of a FNF silicene junction has not been fully establishedyet. Thus, some important e \u000bects still remain unexplored. On\none hand, the exchange ﬁeld in FM regions strongly break\nthe spin degeneracy of band structure. On the other hand,\nthe circularly polarized light in the o \u000b-resonant regime will\ndi\u000berently open band gaps of di \u000berent spin channels, which\nis unlike in graphene because of strong spin-orbit coupling\nin silicene. Consequently, it is expected to obtain intriguing\nspin-polarized transport phenomena under the interplay be-\ntween exchange ﬁeld and circularly polarized light in a FNF\nsilicene junction.\nRecent studies have shown that spin and valley polariza-\ntions of a FNF silicene junction strongly depend on the bar-\nrier height in normal region17,18. Furthermore, TMR has an\noscillatory behavior with respect to incident energy and bar-\nrier height19. It is therefore non-trivial to obtain a robust po-\nlarized current and large TMR value in practice. A technique\nto enhance the spin and valley polarizations as well as TMR\nof a FNF silicene junction is then highly desirable. Very re-\ncently, Saxena et al.20theoretically showed that by applying\nan appropriate external electric ﬁeld to FM regions, the 100%\npositive TMR regardless of barrier Ucan be found.\nIn this paper, we systematically investigate the ballistic\ntransport in a FNF silicene junction in the presence of a circu-\nlarly polarized light in o \u000b-resonant regime applied to the third\nregion. We show that in the presence of light, the sign of per-\nfect spin polarization can be switched by increasing incident\nenergy. The fully valley-polarized current is very sensitive to\nthe staggered electric potential even when it is much smaller\nthan exchange ﬁeld. The most importantly, the perfect spin\npolarization and 100% TMR irrespective of barrier height in\nNM region can be obtained. Moreover, the sign of TMR can\nbe also fully controlled by the polarized direction of light. The\ncondition to realize 100% TMR is also provided.arXiv:1605.02829v1  [cond-mat.mes-hall]  10 May 20162\nII. THEORY\nA. Model Hamiltonian\nLet us now shortly introduce the circularly polarized light,\nA(t)=A(sin(\nt);cos(\nt)), which is perpendicularly applied\nto the silicene junction. \nis the frequency of light, \n>0 for\nright polarization and \n<0 for left polarization. In this work,\nwe only focus on the o\u000b-resonant regime. Such regime occurs\nwhen ~\n\u001dt0, where t0=1:6 eV is the nearest-neighbor\nhopping in silicene. In the limit of eAv F=~\n\u001c1:0 (with vF\nas the Fermi velocity), the o \u000b-resonant light is presented viaa\nstatic e \u000bective Hamiltonian\nHe f f=\u0011\u0015\n\u001cz; (1)\nwhere\u0015\n=(eAv F)2=~\n,\u0011stands for valley index, and\n\u001ci(with i=x;y;z) are the Pauli matrices of the sublattice\npseudospin. A detailed derivation of He f fcan be found in\nRefs.14,21.\nFIG. 1. Schematic picture of a FNF silicene junction, in which the\nthird region (FM2) is exposed to a circularly polarized light. Lis the\nlength of normal region (NM).\nThe low-energy e \u000bective Hamiltonian of a FNF silicene\njunction depicted in Fig. 1 is given by\nH=~vF\u0010\n\u0011kx\u001cx+ky\u001cy\u0011\n+ \u0001\u0011\u001b\u001cz+U\u0000\u001bhr; (2)\nwith vF\u00195:5\u0002105m/s for silicene, \u0001\u0011\u001b=\u0011\u001b\u0015 S O\u0000\u0001z+\n\u0011\u0015\n,k2=k2\nx+k2\ny, and indices r=1;2;3 stand for regions\nI (FM1), II (NM), and III (FM2), respectively. \u0015S O=3:9\nmeV is the spin-orbit coupling constant in silicene and \u0001zis\nthe staggered electric potential between AandBsublattices,\nwhich is induced by an external electric ﬁeld perpendicular to\nthe plane of silicene. \u0011=\u00061 corresponds to the KandK0\npoints, and \u001b=\u00061 respectively denotes spin up and down. U\nis the barrier potential induced by a gate voltage applied to the\nnormal silicene region. hris the exchange ﬁeld in FM regions\n(h2=0), where h1=h3=handh1=\u0000h3=hcorrespond to\nP and AP conﬁgurations. Note that because the Hamiltonian\npresented circularly polarized light in o \u000b-resonant regime is\nvalley degenerate, the circularly polarized light itself cannot\ngenerate a valley-polarized current. However, thanks to the\nvalley degeneracy, it is expected to provide signiﬁcant e \u000bects\non spin-polarized transport properties. Furthermore, it is easy\nto recognize that the valley polarization induced by staggered\nelectric ﬁeld will be signiﬁcantly enhanced if the junction is\nadditionally exposed to a circularly polarized light. In thiswork, the electric ﬁeld and circularly polarized light are only\napplied to the third region.\nB. Transport calculation\nThe general eigenfunctions of Hamiltonian (2) are of\nthe form (x;y)=(uA;uB)Teikxxeikyy. Writing down 2-\ncomponent Dirac equation using Hamiltonian (2) and wave-\nfunction (x;y) for each region, we obtain\n \nEr\u0000mr ~vF(\u0011kx\u0000iky)\n~vF(\u0011kx+iky) Er+mr! \nuA\nuB!\n=0; (3)\nwhere, E1=E+\u001bh1;E2=E\u0000U;E3=E+\u001bh3;m1=\nm2=\u0011\u001b\u0015 S O;m3=\u0011\u001b\u0015 S O\u0000\u0001z+\u0011\u0015\n. Here, Eis an incident\nenergy.\nWe then deﬁne transfer matrices Mras\nMr=0BBBBBBB@eiKr\nX\u0018e\u0000iKr\nX\u0018\n\u0011Kr\nX\u0000iKY\nEr+mreiKr\nX\u0018\u0000\u0011Kr\nX+iKY\nEr+mre\u0000iKr\nX\u00181CCCCCCCA; (4)\nwith dimensionless quantity \u0018=x=Land\nKY=Lky;\nKr\nX=sign( Er+mr)q\u0000E2r\u0000m2r\u0001\u0000K2\nY:\nNote that the translational invariance along the yaxis has been\nused. These transfer matrices Mrwill lead to the relation\n \nuA\nuB!\n=Mr \nar\nbr!\n; (5)\nwhere ( ar;br) are unknown wavefunction coe \u000ecients of re-\ngion r.\nLetr\u001b\u0011andt\u001b\u0011be reﬂection coe \u000ecient in region I and\ntransmission coe \u000ecient in region III, respectively, we have\n(a1;b1)=(1;r\u001b\u0011) and ( a3;b3)=(t\u001b\u0011;0). We ﬁnally obtain the\nfollowing equation\n \n1\nr\u001b\u0011!\n=M\u00001\n1M2M\u00001\n2M3 \nt\u001b\u0011\n0!\n: (6)\nThe transmission probability is then evaluated according to\nthe formula T\u001b\u0011=jt\u001b\u0011j2. The normalized spin-valley depen-\ndent conductance at zero temperature is evaluated according\nto Landauer-B ¨uttiker formalism as\nG\u001b\u0011=1\n2Z\u0019=2\n\u0000\u0019=2T\u001b\u0011(E;\u001e) cos(\u001e)d\u001e: (7)\nThe spin(valley) polarization Ps(v)andT MR (in %) are then\ndeﬁned as follows\nPs(v)=G\"(K)\u0000G#(K0)\nG; (8)\nT MR =GP\u0000GAP\nGP+GAP\u0002100%; (9)\nwhere the spin(valley)-resolved and total conductances are\nin turn given by G\u001b(\u0011)=\u0010\nG\u001bK(\"\u0011)+G\u001bK0(#\u0011)\u0011\n=2 and G=\nG\"(K)+G#(K0).GPandGAPare total conductances of P and\nAP conﬁgurations, respectively.3\nIII. NUMERICAL RESULTS\nEzawa showed that the lowest frequency to satisfy the o \u000b-\nresonant condition for silicene is 1000 THz14. In this work,\nwe have assumed that the junction is exposed to a circularly\npolarized light in the soft x-ray regime with high frequency\nof 4000 THz. With the maximum value of \u0015\nup to 6.0\u0015S O\nused herein, the limit eAv F=~\n\u001c1:0 still holds if a strong\nintensity I=(eA\n)2=(8\u0019~\u000b)\u00190:4\u00021012Wcm\u00002(with\u000b=\n1=137) is used, i.e. eAv F=~\n\u00190:09.\nA. Spin and valley polarizations\nFirst, we will analyze the spin polarizations of both P and\nAP conﬁgurations under the presence of circularly polarized\nlight. The staggered electric potential will not be included\n(\u0001z=0) to fully explore the role of circularly polarized light.\nFig. 2 displays the spin-resolved conductances as functions\nof\u0015\n. Generally, spin-up and spin-down conductances reach\nmaxima when \u0015\nis small and decrease when \u0015\nis further\nenhanced. Particularly, away from \u0015\n=\u0015S O=1:0 the spin-\ndown conductance of P conﬁguration vanishes rapidly, while\nthe spin-up conductance remains nonzero. In the AP conﬁg-\nuration, the decay of spin-up conductance is much faster than\nthat of spin-down channel when \u0015\n=\u0015S Ois away from –1.0.\nThese leads to fully spin-polarized currents in certain ranges\nof\u0015\nas marked by gray regions on the ﬁgure. In particular,\nP and AP conﬁgurations yield the positive and negative spin\npolarizations, respectively.\n 0 0.4 0.8 1.2 1.6 2\n−6 −4 −2  0  2  4  6P Conductance\nλΩ/λSOG↑\nG↓\n−6 −4 −2  0  2  4  6AP \nλΩ/λSO\nFIG. 2. Spin-resolved conductances as functions of \u0015\nfor P and AP\nconﬁgurations. Parameters used are: E=\u0015S O=2:0;h=\u0015S O=4:0;and\nU=\u0015S O=20:0.\nThe variations of spin-resolved conductances with \u0015\ncan\nbe clearly understood from low-energy band structures as\nshown in Fig. 3. For P conﬁguration (upper row), the spin-\ndown band gap is closed at the value \u0015\n=\u0015S O=1:0, therefore,\nconductance of this channel reaches maximum. Away from\nthis value, say \u0015\n=\u0015S O=\u00064:0, the spin-down band gap is en-\nlarged and the incident energy E=\u0015S O=2:0 (dotted black line)\nfalls within the gap, while it crosses one spin-up band, leading\nto the full positive spin polarization as observed. The explana-\ntion for AP conﬁguration is similar to that of P conﬁguration.\nThe spin-up band gap of AP conﬁguration is, however, closed\n 0 5 10−4.0E/λSO\n 0 5 101.0\n 0 5 104.0\n 0 5 10\n−4 −2  0  2  4−4.0E/λSO\nk 0 5 10\n−4 −2  0  2  4−1.0\nk 0 5 10\n−4 −2  0  2  44.0\nkFIG. 3. Low-energy band structures of third regions at \u0015\n=\u0015S O=\n1:0;\u00064:0 for P conﬁguration (upper row) and at \u0015\n=\u0015S O=\n\u00001:0;\u00064:0 for AP conﬁguration (lower row). The solid red indicates\nthe spin-up bands and the dash blue indicates spin-down bands. hand\nUare the same as in Fig 2. The dot black line refers to E=\u0015S O=2:0;\nandE=\u0015S O2[0;1] is marked by the green region.\nat\u0015\n=\u0015S O=\u00001:0.\nObviously, the incident energy plays an important role to\nobtain the fully spin-polarized current. In order to examine\nthe interplay between the incident energy Eand\u0015\n, we plot\nthe spin polarizations of both P and AP conﬁgurations as func-\ntions of Eand\u0015\non upper panel of Fig. 4. For P conﬁg-\nuration, the enhancement of incident energy under the right\npolarized light is resulted in the change of spin polarization\nfrom negative to positive. When a left polarized light is ap-\nplied, positive spin polarization irrespective of incident energy\ncan be obtained. For AP conﬁguration, in contrast, the right\npolarized light yields negative spin polarization irrespective\nof incident energy, while the positive to negative transition\noccurs in the case of left polarized light. The transition of\nspin polarization from negative to positive and vice versa can\nbe demonstrated from low-energy band-structure as shown in\nFig. 3. We only consider the band structure of P conﬁguration\n(upper row) at \u0015\n=\u0015S O=4:0 and a similar explanation can be\nused for the AP conﬁguration (lower row) at \u0015\n=\u0015S O=\u00004:0.\nWe can see that in the range [0 ;1] marked by the green re-\ngion, the incident energy will cross the spin-down band while\nit is within the gap of spin-up band, the negative spin polar-\nization is then obtained. On the other hand, when the incident\nenergy increases out of the range, the spin-up density of state\nis nonzero while there is no density of states for spin-down\nchannel, so that the spin polarization is positive. In general,\nby changing circularly polarized light together with incident\nenergy, a desired spin polarization can be easily realized in\nFNF silicene junction.\nTo further explore the enhancement of spin polarization un-\nder circularly polarized light, we plot, for example, the spin\npolarization of P conﬁguration as functions of ferromagnetic\nexchange ﬁeld hand barrier height Uin lower panel of Fig.\n4. In the absence of light, the spin polarization is only occur\nwhen the exchange ﬁeld his small. Moreover, the spin po-4\nlarization strongly oscillates with respect to barrier height U.\nInterestingly, when the light is switched on, the full positive\nspin polarizations regardless of barrier height Uare observed\nin a large range of ferromagnetic exchange ﬁeld h.\nFIG. 4. Upper panel: spin polarization Psas functions of \u0015\nand\ninicident energy Efor P (left) and AP (right) conﬁgurations with\nh=\u0015S O=4:0 and U=\u0015S O=20:0. Lower panel: spin polarization as\nfunctions of exchange ﬁeld hand barrier Uat\u0015\n=\u0015S O=0 (left) and\n4:0 (right) for the P conﬁguration with E=\u0015S O=2:0.\nLet us now brieﬂy discuss on the valley polarization. The\nelectric ﬁeld applied on the third region is switched on to gen-\nerate the valley dependent current. Fig. 5 shows the valley\npolarization Pvas functions of staggered electric potential \u0001z\nand exchange ﬁeld h. Herein, we only consider the P con-\nﬁguration as an example, while the AP conﬁguration can be\ndone similarly. Formally, fully valley-polarized current in FM\nsilicene junction can be only realized when the staggered elec-\ntric potential is larger than the exchange ﬁeld, i.e. \u0001z=h>120.\nThis argument is conﬁrmed by the middle panel of the ﬁg-\nure, where \u0015\n=\u0015S O=0. Once the light is switched on, the\nvalue of \u0001zto generate perfect valley polarization is substan-\ntially reduced, especially for the right polarized light. Fur-\nthermore, the valley polarization is an odd function of \u0015\n. All\nthese situations are easily understood from the second term of\nHamiltonian (2). Since the circularly polarized light in the o \u000b-\nresonant regime is linearly dependent on the valley index \u0011,\nthe valley-degenerate breaking under staggered electric ﬁeld\nwill be signiﬁcantly enhanced and the sign of valley polariza-\ntion will vary accordingly to the direction of light. Generally,\nthe valley polarization in FNF silicene junction generated by\nstaggered electric ﬁeld will become more sensitive under a\ncircularly polarized light.\nB. Tunneling magnetoresistance\nFinally, we consider the e \u000bect of circularly polarized light\non TMR of the FNF silicene junction. To eliminate the contri-\nbution from staggered potential to TMR, the external electric\nFIG. 5. Valley polarization Pvof P conﬁguration as functions of h\nand\u0001zat di\u000berent values of \u0015\n. The other parameters are: E=\u0015S O=\n2:0 and U=\u0015S O=20:0.\nﬁeld is again turned o \u000b(\u0001z=0). Left panel Fig. 6 shows\nTMR in % as functions of incident energy Eand\u0015\n. 100%\nTMR can be achieved when j\u0015\n=\u0015S Oj>4:0. Changing po-\nlarized direction of light from left to right gives rise to the\ntransition from positive to negative TMR as previously ob-\nserved in Ref.16. In contrast to Ref.19, in which TMR of FNF\nsilicene junction strongly oscillates with incident energy, we\nfound that in the presence of circularly polarized light TMR\nis constant within large ranges of incident energy.\nFIG. 6. 2D contour plots of T MR (\u0015\n;E) (left) and T MR (\u0015\n;h)\n(right). The parameters used for left panel: h=\u0015S O=4:0;U=\u0015S O=\n20:0; and for right panel: E=\u0015S O=1:0;U=\u0015S O=20:0.\n−7−3.5 0 3.5 7λΩ/λSO =−5.0E/λSOλΩ/λSO = 0.0 λΩ/λSO = 5.0\n−7−3.5 0 3.5 7\n−4 −2  0  2  4E/λSO\nk−4 −2  0  2  4\nk−4 −2  0  2  4\nk\nFIG. 7. Low-energy band structures at \u0015\n=\u0015S O=\u00005:0 (ﬁrst column),\n\u0015\n=\u0015S O=0:0 (second column), and \u0015\n=\u0015S O=5:0 (last column) for\nP (upper row) and AP (lower row) conﬁgurations. The dot black line\nrefers to E=\u0015S O=1:0.5\nFIG. 8. TMR (in %) as functions of exchange ﬁeld hand barrier U\nat di\u000berent values of \u0015\n. The incident energy is E=\u0015S O=1:0.\nAnalyzing low-energy band structure of P and AP conﬁgu-\nrations as shown in Fig. 7 can provide the physical origin of\ngiant TMR. At \u0015\n=\u0015S O=5:0, the incident energy is within the\nband gap for the P conﬁguration, while it crosses spin down\nband for AP conﬁguration, leading to 100% negative TMR.\nAt\u0015\n=\u0015S O=\u00005:0, the incident energy is within the band gap\nfor the AP conﬁguration and crosses the spin up band for P\nconﬁguration, resulted in the 100% positive TMR. In general,\nthere are two requirements to realize the 100% TMR. On one\nhand, the incident energy has to be located in the band gap\nfor one conﬁguration. On the other hand, it has to cross one\nenergy band for the other conﬁguration. These conditions can\nbe easily derived from the eigenvalue of e \u000bective Hamiltonian\n(2) in third region, E=\u0006p\n(~vFk)2+(\u001b\u0015S O+\u0015\n)2\u0000\u001bh3with\njh3j=h, as\n(h\u0000\u0015S O)+E<j\u0015\nj<(h+\u0015S O)+E: (10)\nThe condition (10) is fulﬁlled by the right panel of Fig. 6, in\nwhich we plot TMR as functions of \u0015\nandhatE=\u0015S O=1.\nIt is common that TMR will strongly oscilate with respect\ntoU19,20and a signiﬁcant TMR value can be only obtained\nwhen the exchange ﬁeld his smaller than the barrier U20.Consequently, it is not easy to get a desired TMR value in\npractice. It is therefore interesting and informative to examine\nthe variation of TMR with the exchange ﬁeld hand barrier U\nin presence of circularly polarized light. The results are sum-\nmarized in Fig. 8. As expected, in the absence of circularly\npolarized light, TMR strongly oscilates with respect to Uand\nonly has the signiﬁcant value at small h. Nevertheless, when\nthe junction is exposed to a left (right) circularly polarized\nlight, 100% positive (negative) TMR can be achieved regard-\nless of U. More importantly, the junction can provide 100%\nTMR in a large range of heven when Uvanishes.\nIV . CONCLUSION\nIn summary, the inﬂuence of a circularly polarized light in\no\u000b-resonant regime on ballistic transport through a FNF sil-\nicene junction has been systematically investigated. We have\nshown that the spin /valley polarizations and TMR are signif-\nicantly enhanced under a circularly polarized light. Particu-\nlarly, tuning incident energy in the presence of a circularly\npolarized light leads to the transition of spin polarization from\npositive to negative and vice versa . Thanks to the valley de-\npendence of o \u000b-resonant circularly polarized light, the per-\nfect valley polarization can be achieved even when staggered\nelectric ﬁeld is much smaller than exchange ﬁeld. The most\nimportantly, the perfect spin polarization and 100% TMR ir-\nrespective of barrier Ucan be realized when the junction is\nexposed to a circularly polarized light. We have also gained a\ncondition for observing the 100% TMR.\nACKNOWLEDGMENT\nThis work is supported by Vietnamese National Founda-\ntion of Science and Technology Development (NAFOSTED)\nunder Grant No. 103.01-2015.14.\n\u0003binho@qi.mp.es.osaka-u.ac.jp\nylatran@umich.edu\n1C.-C. Liu, W. Feng, and Y . Yao, Phys. Rev. Lett. 107, 076802\n(2011).\n2N. D. Drummond, V . Z ´olyomi, and V . I. Fal’ko, Phys. Rev. B 85,\n075423 (2012).\n3M. Ezawa, Phys. Rev. Lett. 109, 055502 (2012).\n4M. Ezawa, New Journal of Physics 14, 033003 (2012).\n5M. Ezawa, Phys. Rev. B 87, 155415 (2013).\n6M. Ezawa, The European Physical Journal B 85, 1 (2012).\n7T. Yokoyama, Phys. Rev. B 87, 241409 (2013).\n8Y . Wang, Applied Physics Letters 104, 032105 (2014).\n9B. Soodchomshom, Journal of Applied Physics 115, 023706\n(2014).\n10N. Missault, P. Vasilopoulos, F. M. Peeters, and B. Van Duppen,\nPhys. Rev. B 93, 125425 (2016).\n11N. Missault, P. Vasilopoulos, V . Vargiamidis, F. M. Peeters, and\nB. Van Duppen, Phys. Rev. B 92, 195423 (2015).\n12V . Vargiamidis and P. Vasilopoulos, Applied Physics Letters 105,\n223105 (2014).13V . Vargiamidis and P. Vasilopoulos, Journal of Applied Physics\n117, 094305 (2015).\n14M. Ezawa, Phys. Rev. Lett. 110, 026603 (2013).\n15M. Tahir and U. Schwingenschl ¨ogl, The European Physical Jour-\nnal B 88, 1 (2015).\n16Z. P. Niu and S. Dong, EPL (Europhysics Letters) 111, 37007\n(2015).\n17X. J. Qiu, Y . F. Cheng, Z. Z. Cao, and J. M. Lei, Journal of Physics\nD: Applied Physics 48, 465105 (2015).\n18Y . Hajati and Z. Rashidian, Superlattices and Microstructures 92,\n264 (2016).\n19Y . Wang and Y . Lou, Journal of Applied Physics 114, 183712\n(2013).\n20R. Saxena, A. Saha, and S. Rao, Phys. Rev. B 92, 245412 (2015).\n21T. Kitagawa, T. Oka, A. Brataas, L. Fu, and E. Demler, Phys. Rev.\nB84, 235108 (2011)."    },    {        "title": "1603.00694v1.Ferromagnetic_resonance_in_submicron_permalloy_stripes.pdf",        "content": " 1 Ferromagnetic resonance in submicron permalloy stripes  \nE. V. Skorohodov *1,2, R. V. Gorev1, R. R. Yakubov2, E. S. Demidov2, Yu. V. Khivintsev3,  \nYu. A. Filimonov3, and V.  L. Mironov1,2* \n1Institute for Physics of Microstructures RAS, 603950, Nyzhny Novgorod,  Russia  \n2Lobachevsky State University, 603950, Nyzhny Novgorod, Russia  \n3Kotel’nikov Institute of Radio -engineering and Electronics RAS, Saratov branch, 410019, Saratov, Russia  \n*Corresponding author: evgeny@ipmras.ru  \nAbstract  \nWe present the results of systematic experimental investigations and micromagnetic simulations for the ferromagnetic \nresonance in rectangular permalloy microstripes. It is shown that the resonant magnetization oscillations have a \ncomplex spatial  structure including a quasi -homogeneous precession, lateral spin -wave resonances and localized edge \nmodes, which strongly depend on sample orientation in an external magnetic field.\n1. Introduction  \nThe microwave properties of the \nferromagnetic planar patt erned structures \n(PPS) is the subject of the recent intensive \ninvestigations motivated by perspective \napplications in magnetoelectronics, \nspintronics and data processing [1,2]. In \nparticular, the PPS consisting of \nferromagnetic elements with different shap e \nand spatial arrangement are considered as the \neffective tuned filters for the ultra high \nfrequency (UHF) electromagnetic radiation \n[3,4]. The ferromagnetic resonance (FMR) \nand spin wave resonances (SWR) in PPS \nstrongly depend on the internal fields \nconne cted with the shape anisotropy, \nexchange and magnetostatic interaction. As a \nresult the absorption spectrum of PPS is \ndefined by the different geometric factors \nsuch as shape, size and spatial arrangement of \nelements in PPS. This opens the wide \nopportuniti es for tuning of UHF absorption by \nchanging the architecture of PPS using the \nnanolithography methods. In this regard, the \nspecial researches are focused on the study of \nferromagnetic resonance in rectangular \nmicrostripes, considered as one of the main \nstructural elements of planar UHF \nmicrosystems. Due to the high shape \nanisotropy the microstripes have a uniform \nmagnetic state and can be used in the \nmicrowave devices without external \nmagnetizing. In  particular, the PPS with strong magnetostatic interaction  enables the \nrealization of tuned filters with different \nabsorption spectra, which can be switched by \nthe external magnetic field [5]. Partially the \ninvestigations of the magnetization \noscillations spectra depending on the size and \nshape of microstripes we re reported in [6 -11]. \nThe special attention was paid to the analysis \nof uniform precession and localized edge \nmodes arising from the non -homogeneity of \ninternal magnetostatic field near the \nboundaries of microstrip [9 -12]. However, the \nsystematic FMR stud ies for different \nmicrostrip orientations in an external \nmagnetic field have not been performed.  \nThe dynamic properties of PPS systems are \nwidely studied by micromagnetic modeling \nbased on the numerical solution of the \nLandau -Lifshitz equation. In general,  authors \nsimulate the spectra of microstrip \neigenfrequencies based on Fourier analysis of \nrelaxation oscillations of the magnetization \n[12]. However, experimentally the FMR is \nstudied often by the analysis of the \nmicrowave absorption at a given frequency i n \ndependence on the magnitude and direction  of \nsweeping external dc magnetic field. \nAppropriate modeling of the spectra and \nspatial distributions of steady -state \nmagnetization oscillations in external \nmagnetic fields was not discussed.   2 In current paper we present the results of \nexperimental measurements and \nmicromagnetic simulations of magnetization oscillations in planar rectangular microstrip \nfor different sample orientations in an \nexternal magnetic field. The main attention is \ngiven to the analysis of m ode structure and \nspatial d istribution of spin oscillation  \n2. Experiments and m ethods  \nThe array of permalloy (Ni 80Fe20) \nrectangular stripes 3000×500×30 nm3 (the \nseparation between stripe ’s centers is about  6 \nμm) were fabricated by electron -beam \nlithography and lift -off process. At the first \nstage the 150 nm resist was deposited on Si \nsubstrate by centrifuging. Then initial mask in \nthe form of rectangular stripes array was \nformed in negative electron resist usi ng \nscanning electron microscope SUPRA 50VP \nwith lithographic facility ELPHY PLUS. \nAfter that the irradiated areas in resist were \nremoved in a selective organic solvent. At the \nnext step the sample was covered by \npermalloy layer (30 nm thick) using \nmagnetro n sputtering. At the final stage the \nareas of permalloy layer except the strip array \nwas removed in the lift -off process. The \ntypical SEM image of strip array is presented \nin Fig. 1(a).  \n \nFig. 1.  (a) The SEM image of permalloy \nrectangular stripes 3000×500×30 nm3. The \nseparation between stripes is 3 μm. (b) The \nassociated coordinate system, connected with \nthe sample.  \nThe selected distance between the \nmicrostrips realizes the weak magnetostatic \ninteraction between neighboring elemen ts in \nthe array, so that the sample FMR spectrum \ncorresponds to a FMR spectrum of separate \nmicrostrip. Note that all microstrip \ndimensions are less than the path length of \nspin waves in permalloy that allows us to observe the spin -wave resonances (the \nform ation of standing spin waves).  \nThe FMR measurements were performed \nwith Bruker EMX Plus -10/12 electron \nparamagnetic resonance spectrometer using a \nTE011 mode of cylindrical resonant cavity and \ndc magnetic field ( H) up to 2 T. The polarized \nmicrowave magnet ic field  \nh~ with freq uency \n9.8 GHz was perpendicular to the H. The \nsamples were driven through the resonance by \nsweeping the H.  \nTheoretically, the microwave absorption in \nmicrostrip was studied by computer \nmicromagnetic modeling based  on numerical \nsolution of Landau -Lifshitz -Gilbert equation \nusing the standard object oriented \nmicromagnetic framework (OOMMF) code \n[13]. We inve stigated the dependencies of \nmicrovawe  power absorption on the \nmagnitude of swept H. Algorithm of \nnumerical simu lation includes follow stages . \n1) The e xternal magnetic field is applied . 2) \nThe s tanding spin waves is excited by \nmicrowave magnetic field . 3) The dependence \nof averaged variable magnetization  amplitude \nfrom the magnitude of external magnetic field  \nis gra phed . This dependence is a curve  \nmicrovawe power absorption on the \nmagnitude of swept H because absorbed \npower is proportional to the sample averaged \nvariable magnetization amplitude. In addition, \nto visualize the mode composition of \noscillations we calcu lated the time \ndependencies of spatial distributions for \ndifferent magnetization components. The \ncalculations were performed for the typical \npermalloy parameters: the saturation \nmagnetization was MS = 8 × 105 A/m, the \nexchange stiffness was A = 8.4 × 10−12 J/m, \nand the damping constant was 0.01 (see [14]). \nWe omitted magnetocrystalline anisotropy, \nassuming a polycrystalline structure of our \nsamples, the perpendicular uniaxial \nanisotropy K =3,9× 104 J/m3. The microwave \nfrequency was 9.8 GHz. In calculations the \nmicrostrip 3000×500×30 nm3 was discretized \ninto rectangular parallelepipeds with a square \nd Y Z \nX \nl w \n(a) (b) 3 μm  3 base of size δ = 10 nm in the x,y plane and \nheight h = 30 nm. The choice of geometrical \ndimensions and aspect ratio (the ratio of \nthickness, width and length) of microstripes \nwas motivated by convenience for the \nexperimental and theoretical study of the \neigenmodes of m agnetization oscillations. In \ncase of high aspect ratio the magnetostatic \ninteraction would give rise to noticeable \nshape anisotropy and substantial \ninhomogeneity of internal magnetic field that \nallows studying the influence of these effects \non FMR spectru m.  \n3. Results and discussion  \nWe investigated the FMR response in \ndependence on sample orientation relatively  \nH\n. In all experiments the exciting magnetic \nfield \nh was directed in the sample plane as \nthe oscillatio n excitation efficiency in this \ncase is maximal.  At the first step we \ninvestigated the FMR in geometry when the  \nexternal magnetic field  was applied along the \nlong side of microstripe  (X axis  in fig. 1b). \nFig. 2a shows experimental ( dashed  red \ncurve) and nu merically calculated ( solid  blue \ncurve) FMR spectrum for this geometry. It is \nseen that there is a good agreement between \nthe experiment and simulation. As is seen \nfrom fig. 2b the resonant oscillations \ncorresponding to resonance fields  1 - 3 have \ncomplex spatial distributions, which are \nsuperposition of different modes of \nlongitudinal spin -wave resonance along the \nmicrostrip.  It is seen that there is a \nsuperposition of long-wave and short -wave \nmodes . The l ong-wave mode is defined by \nmagnetodipole interacti on, short -wave mode \nis defined by exchange interaction.  \nTo explain the observed mode composition \nwe calculated the spectra of spin waves for \nthe microstripe in an external magnetic field \ndirected along the X axis. The theory of the \ndipole -exchange wave’s spectra was \ndeveloped in [15 -17]. Following the proposed \napproach we assumed that the magnetic moments on the surfaces are not fixed and the \nlower modes do not depend on the z \ncoordinate. It is known that the demagnetizing \nfield of a rectangular parallelep iped is \nsubstantially nonuniform that leads to the \ncomplicated dependence of the variable \ncomponent magnetization on coordinates x, y. \nHowever, approximately the distribution of \ntransverse component of the variable \nmagnetization can be written as follows [ 17]: \n ,( , ) cos cosy z s xn ymm x y M k x k y\n,      (1) \nwhere  \nsM  is magnetization in saturation ; \nkxn=(n+1)/l   and kym=(m+1)/w  ; l and w \nare the length  and width of the  microstrip \nrespectively. The deviation of the variable \ncomponent of the magne tization from the \ncosine function is \n/dl  (where \nd  is the \nmicrostrip thickness). On the other hand the \namendment to the spectrum is \n2/dl , which \nis not substantial for the stripes with high \naspect ratio. Thus, the spectrum of \nmagnetization oscillations can be represented \nas follows [16,17]:  \n2 2 2( )( )nm nm\nnm s H M nm H M nm M mn M J k J k F         \n, (2) \nwhere γ is gyromagnetic ratio; \n4Ms M ; \nnm Mnm\nH N H\n; Nnm are demagnetizing \nfactors for different modes i n the microstrip; \nFmn are the matrix element of the dipole -\ndipole interaction; J  is the exchange stiffness \nconstant. Using (2) and taking into account \nthe frequency of the microwave oscillations \n0\n= 9.8 GHz, it is possible to find  the \nresonant magnetic fields for each FMR mode. \nAs can be seen from fig. 2b, in the resonances \nwe observe a superposition of two modes with \nlow and high spatial frequencies (wave \nnumbers). It is connected with the excitation \nof both dipole spin waves with  small \nxnk  as \nwell as exchange waves with large kxn. The \ndispersion relation in this case is non -\nmonotonic and there is the degeneracy in the \nspectrum.   4 \n1 \n2 33 \n(a)               Experiment  \n              Simulation   \n \n \n \n \n \n \n \n \n \n \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. 2. (a) The experimental ( dashed red curve) and simulated ( solid  blue curve) microstrip e FMR \nspectra  (external magnetic field applied along X axis) . (b) The spatial distributions of the \nmagnetization oscillations (amplitude of the z -component) in different resonant fields. (1) is for \nH = 660 Oe; (2) is for H =770 Oe; (3) is for H = 870 Oe.  \nCalculations shows  that the resonances are \nrealized at H = 645 Oe for the modes with \nn = 0 and  n = 84 (resonance No 1); at H = 740 \nOe for the modes with n = 2 and  n = 82 \n(resonance No 2); at H = 850 Oe for the \nmodes with n = 4 and  n = 78 (resonance No 3) \nrespectively. For this resonance spin waves \nthe m ode number m=0. These values of the \nresonance fields and numbers of longitudinal \nSWR modes are in good agreement with the \nresults of micromagnetic simul ation s (fig. 2a). The edge roughness has a big influence for \nconfiguration when external magnetic field \napplied along short side of sample  (Y axis in \nfig 1b) . At the first we considered microstripe \nwithout edge roughness.  As is seen from fig \n3a there is a good agreement between the \nexperiment and the simulation. But intensities \nof edge mode s are different.  For this \nexperimental configuration analytical \ncalculation was performed  also. The \ndegeneracy of resonance magnetic field from \nmode number is observed  for modes with (3) (2) (1)\n) (b) 1 \n2 3               Experiment  \n              Simulation    5 wave numbers ( n = 2; m = 0) and ( n = 2; \nm = 12) for resonance No 1; for modes with \nwave numbers (n = 0; m = 0) and (n = 0; \nm = 10) for resonance No 2; for modes with \nwave numbers ( n = 4; m = 4) and ( n = 4; \nm = 8) for resonance No 3. For other modes \ndegeneracy of resonance magnetic field from \nmode number is not observed . The r esonances  \nNo 4 and No 5 correspond to edge modes. \nThe e dge modes corresponding resonances  \nNo 4 and No 5 have different numbers of \nhalf-waves. The r esonance  No 4 has three half-waves, resonance  No 5 has one half -\nwave  (fig. 3b).  \nTo decrease intensity of edge  mode we  take \ninto account edge roughness with standard \ndeviation 15 nm.  In this case edge mode s are \nsuppressed  (fig. 4a). The visualization  of the \nspatial distributions o f the  variable  \nmagnetization (fig. 4b) shows that edge \nroughness does weak changing of mode \nstructure .  \n \n \n \n \n \n \n \n \n \n \n \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. (a) The experimental ( dashed red curve) and simulated ( solid blue curve) microstripe FMR \nspectra (external magnetic field applied along Y axis) . (b) the spatial distributions of the \nmagnetization oscillations (amplitude of the z -component) in different resonant fields. (1) is for \nH = 1493 Oe; (2) is for H = 1650 Oe; (3) is for H = 1870 Oe; (4) is for H = 1940 Oe; (5) is for \nH = 2140 Oe.                Experiment  \n              Simulation   \n(1) (2) (3) (4) (5) (b) (a) 1 2 3 4 5  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 \nFig. 4 (a) The experimental ( dashed red curve) and simulated ( solid blue curve) microstripe FMR \nspectra  for (external magnetic field applied along Y axis) . (b) the spatial distributions of the \nmagnetization oscillations (amplitude of the z -component) in different resonant fields. (1) is for \nH = 1493 Oe; (2) is for H = 1650 Oe; (3) is for H = 1870 Oe  \nFig 5a shows experimental and simulated \nmicrostripe FMR spectra when external \nmagnetic field ap plied perpendicular to \nsample plane (Z  axis on fig 1(b)) . For this \ngeometry main resonance  corresponding main \nmode and spin -wave resonances locating left \nfrom main resonance  is observed. Also \nvisualization of variable magnetization spatial \ndistribution was  made.  For resonances  No 1-3 \nthe amplitude  of magnetization oscillation  is \nchanged along a long side only,  while f or the resonance s No 4 and No 5 the amplitude s of \nmagnetization oscillation is changed along \nlong and short sides.  In this geometry \ndegeneracy  of resonance magnetic field from \nmode number is absent but the superposition \nof long - and short -wave modes  is observed . \nFor H = 10850 Oe resonance modes are ( n=0; \nm=2) and ( n=18; m=0). For H = 10320 Oe \nresonance modes are (n=0;  m=4) and ( n=34; \nm=0).  \n(1) (2) (3) (b) (a) 1 2 3               Experiment  \n              Simulation   \n              Experiment  \n              Simulation   \n1 2 3  7  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 5. (a) The experimental ( dashed red curve) and simulated ( solid blue curve) microstripe FMR \n(external magnetic field applied along Z axis) . (b) the spatial distributions of the magnetization \noscillations (amplitude of the z -component) in different resonant fields. ( 1) is for H = 11660  Oe; ( 2) \nis for H = 11580 Oe; ( 3) is for H = 11480 Oe; (4) is for H = 10850  Oe; (5) is for H = 10320 Oe.\n \nConclusion  \nThus , the experimental results and \nmicromagnetic modeling have shown that in \nthe planar high aspect ratio microstrip e the \nseries  of FMR resonances are observed. The \nstructure of FMR spectra strongly depends on \nthe sample orientation relative the \nmagnetizing  dc magnetic field. The simulated \nspatial distributions of resonant modes and \nthe spin -wave spectra calculations show n that \nresonant oscillations have the  comp licated  \nstructure corresponding to the superposition of long-wave and short -wave modes,  \nlocalized oscillations at the edges of \nmicrostrip.  The transverse resonances are \nespecial ly sensitive to the state of the strip \nedges (edge roughness, oxidation).  \nAcknowledgements  \nThe authors are very thankful to \nA.A. Fraerman for the useful discussions and \nS.N. Vdovichev for the help in sample \nfabrication . We also thank M. J. Donahue and \nR. D. McMicheael (NIST, USA) for helpful \ncouncils . This work was supported by the \nRussian Foundation for Basic Research \n(project  No. 15-02-04462 ) 2 3 \n(1) (2) (3) (4) (b) \n(5) 1 \n(a) 5 4 3 2 1               Experiment  \n              Simulation   \n              Experiment  \n              Simulation   \n5 4 3 2 1  8 References  \n1. H. Zhang, A . Hoffmann, R . Divan, and P . \nWang , Appl. Phys. Lett., 95, 232503 \n(2009) . \n2. R. Adam, Yu . Khivi ntsev , R. Hertel, C.  M. \nSchneider, A. Hutchison, R. Camley, and \nZ. Celinski , J. Appl. Phys., 101, 09F516 \n(2007).  \n3. B. K. Kuanr, R . Lopusnil, L . M. Malkinski , \nM. Wenger, M. Yu, D. Scherer II, R. \nCamley, and Z. Celinski , J. Appl. Phys., \n103, 07C508 (2008).  \n4. L. Malkinski , M. Yu, A.Y. Vovk , D. \nScherer  II, L. Spinu , W. Zhou , S. \nWhittenburg , Z. Davis , and J.-S. Jung , J. \nAppl . Phys ., 101, 09J110 (2007).  \n5. M. P. Wismayer, B.  W. Southern, X. L. \nFan, Y. S. Gui, R. E. Camley and C. -M. \nHu, Phys. Rev . B 85, 064411 (2012) . \n6. J. Jorzick , S. O. Demokritov , M. Bailleul , \nC. Fermon , B. Hillebrands , K. Guslienko , \nA. Slavin , D. Berkov , and N. Gorn , Phys . \nRev. Lett., 88, 047204  (2002). \n7. P. H. Bryant , Phys . Rev. B, 47, 11255 \n(1993).  \n8. N. Yu. Grigoryeva , D. A. Popov , and B. A. \nKalinikos , Physics of Solid  State , 56(9), \n1806 (2014).   \n9. M. Bailleul, D.  Olligs,  and C. Fermon, \nPhys. Rev. Lett., 91, 137204  (2003).  \n10. R. D. McMichael and B.  B. Maranville , \nPhys. Rev. B, 74, 024424 (2006).  \n11. C. Wen -Bing, H. Man -Gui, Z. Hao, O. Yu, \nand D. Long -Jiang , Chin. Phys. B, 19(8), \n087502 (2010).  \n12. M. Buess, T.  Haug , M. Scheinfein, and \nC.Bac , Phys. Rev. Lett., 94, 127205 \n(2005) . \n13. M. J. Donahue and D.  G. Porter, \nInteragency Report No. NISTIR 6376, \nNational Institute of Standards and \nTechnology, Gaithersburg,  \n(http://math.nist.gov/oommf/ ). \n14. Yu. Yin, F. Pan, M.Ahlberg , M. Ranjbar , \nA. Mojtaba, Ph. Durrenfeld,  M.Haidar, Ya \nZhai, J. Akkerman, A. Delin,L. Bergqvist, \nand R. Dumas, Phys. Rev. B 92, 024427 \n(2015)  \n15. B. A. Kalinikos an d A. N. Slavin, J. Phys. \nC, 19(35), 7013 (1986).  \n16. K. Yu. Guslienko and A. N. Slavin, J. \nAppl. Phys. , 87, 6337  (2000) . \n17. K. Y u. Guslienko, R. W. Chantrell and A. \nN. Slavin , Phys. Rev. B, 68, 024422 \n(2003) . \n "    },    {        "title": "2207.03829v3.Measuring_the_magnon_photon_coupling_in_shaped_ferromagnets__tuning_of_the_resonance_frequency.pdf",        "content": "Measuring the magnon-photon coupling in shaped ferromagnets: tuning of the\nresonance frequency\nSergio Mart\u0013 \u0010nez-Losa del Rinc\u0013 on,1, 2Ignacio Gimeno,1, 2Jorge P\u0013 erez-Bail\u0013 on,1, 2Victor\nRollano,1, 2Fernando Luis,1, 2David Zueco,1, 2and Mar\u0013 \u0010a Jos\u0013 e Mart\u0013 \u0010nez-P\u0013 erez1, 2,\u0003\n1Instituto de Nanociencia y Materiales de Arag\u0013 on (INMA), CSIC-Universidad de Zaragoza, Zaragoza, Spain\n2Departamento de F\u0013 \u0010sica de la Materia Condensada, Universidad de Zaragoza\nCavity photons and ferromagnetic spins excitations can exchange information coherently in hy-\nbrid architectures, at speeds set by their mutual coupling strength. Speed enhancement is usually\nachieved by optimizing the geometry of the electromagnetic cavity. Here we show that the ge-\nometry of the ferromagnet plays also an important role, by setting the fundamental frequency of\nthe magnonic resonator. Using focused ion beam patterning, we vary the aspect ratio of di\u000ber-\nent Permalloy samples reaching operation frequencies above 10 GHz while working at low external\nmagnetic \felds. Additionally, we perform broad band ferromagnetic resonance measurements and\ncavity experiments that demonstrate that the light-matter coupling strength can be estimated us-\ning either open transmission lines or resonant cavities, yielding very good agreement. Finally, we\ndescribe a simple theoretical framework based on electromagnetic and micromagnetic simulations\nthat successfully accounts for the experimental results. This approach can be used to design hybrid\nquantum systems exploiting whatsoever magnetostatic mode excited in ferromagnets of arbitrary\nsize and shape and to tune their operation conditions.\nI. INTRODUCTION\nIn recent years, quantum manipulation and read-out\nprotocols based on circuit quantum electrodynamics [1]\nhave been successfully applied to the \feld of quantum\nmagnonics [2{4]. In these studies, a coherent coupling be-\ntween photons in an electromagnetic cavity and magnons\nin a ferromagnet is achieved in the strong [5, 6], ultra-\nstrong [7{9] and, even, close to the deep-strong regimes\n[10]. This has opened the way to the observation of\ninteresting phenomena with strong potential for appli-\ncations like quantum transduction between optical and\nmicrowave photons [11, 12], dispersive coupling between\nquantum systems [13{15] or non-reciprocal transmission\nof rf signals [16{19].\nElectromagnetic cavities and ferromagnets are di\u000ber-\nent versions of an harmonic oscillator, whose bosonic\nquanta of excitations correspond to photons and\nmagnons, respectively. The resonant frequency of such\ncavities can be tuned by di\u000berent means. This is rel-\natively easy in the case of photons. For instance, size\ndetermines the fundamental modes in three dimensional\nmicrowave cavities and superconducting coplanar waveg-\nuide (CPW) resonators [20]. Superconducting thin-\flms\nare also used for the implementation of lumped element\nLC-resonators with fundamental frequencies given by\n!p= 1=p\nLC, which is, in essence, a geometrical fac-\ntor as well [20]. In the case of soft magnets, spins precess\naround the local e\u000bective magnetic \feld. This \feld re-\nsults from the contribution of the externally applied mag-\nnetic \feldBextand the (much more di\u000ecult to calculate)\ndemagnetizing \feld [21]. Again, the latter is determined\nby the geometry of the sample. In isotropic magnets, e.g.\n\u0003pemar@unizar.esspheres, shape e\u000bects cancel out and the resulting reso-\nnance frequency depends linearly on the external mag-\nnetic \feld as !m=\reBext[22] with\re=2\u0019= 28 GHz/T\nthe electron gyromagnetic ratio.\nIn most experiments performed so far, yttrium iron\ngarnet (YIG) spheres strongly coupled to either three di-\nmensional cavities [5, 9] or superconducting CPW res-\nonators [13] have been the preferred choice. These\nmagnets are commercial and exhibit record low Gilbert\ndamping constant ( \u000b\u001810\u00005), which is bene\fcial for\nthe observation of long lived coherent exchange of infor-\nmation between the photonic and magnonic excitations.\nHowever, YIG spheres do not couple optimally to thin\n\flm superconducting circuits that are pivotal in quantum\napplications [1, 6]. YIG thin \flms are not easily compat-\nible with conventional lithography processes [23] nor for\ncryogenic operation as the optimum substrate material\nfor YIG growth becomes lossy with decreasing temper-\nature [24]. For these reasons, other C-MOS compatible\nmaterials like Permalloy (Py, with a moderate \u000b\u001810\u00002)\n[7, 25{27] or iron-cobalt alloys (Fe 75Co25, with a very\npromising\u000b\u001810\u00003) are drawing attention [28, 29].\nYet, custom tuning of the resonance frequency of such\nmagnetic cavities is still to be demonstrated. This is rel-\nevant since experimental conditions usually impose cer-\ntain constraints due to the use of, e.g., circulators or am-\npli\fers with narrow frequency bandwidth. Additionally,\noperating devices at high frequencies is interesting since\nthe coupling strength gbetween photons and magnons\nincreases linearly with the frequency. The reason is that\ngis given by the Zeeman coupling between the magnon\ndipole moment and the oscillating vacuum \feld in the\ncavity. The latter is:\nbrms=r\n\u00160~!p\n2Ve\u000b; (1)\nwhere, the sub-index denotes the root mean square\n1arXiv:2207.03829v3  [cond-mat.mes-hall]  30 Nov 20222\n(r.m.s.) and Ve\u000bis the e\u000bective volume of the electro-\nmagnetic mode. Ve\u000bdepends on the size of the elec-\ntromagnetic wavelength (thus on the inverse frequency)\nyieldingg/!p.\nIncreasing gis usually accomplished by a geomet-\nrical optimization of the resonator i.e., patterning of\nnanoconstrictions [30{32] and/or reducing the impedance\nof lumpedLC-resonators which is equivalent to reducing\nVe\u000b[8, 33, 34]. These approaches serve to con\fne brms\nwithin the (typically) small volume of the ferromagnet,\nyielding a square-root increase of the coupling. A larger\n(linear) enhancement of gshould be achieved by working\nwith high frequency superconducting resonators. How-\never, the experimental observation of such high frequency\ne\u000bect in isotropic ferromagnets requires polarizing the\nmagnetization with strong magnetic \felds that are detri-\nmental for the operation of superconducting circuits.\nHere we show a direct way to tune the resonance fre-\nquency of suitably shaped micron-sized Py samples while\nkeeping the external magnetic \feld below few tens of mT.\nUsing superconducting CPW transmission lines we \frst\nshow that, depending on the aspect ratio of the ferro-\nmagnet, the broadband ferromagnetic resonance (FMR)\nspectra can be modi\fed at will. We then convert the\ntransmission lines into cavities by opening gap capaci-\ntors. We demonstrate that both approaches can be used\nto experimentally estimate the coupling factors. Our\nexperiments are complemented with a general-purpose\ntheoretical model including realistic electromagnetic and\nmicromagnetic simulations that account for the non-\nhomogeneous spin precession along the volume of the fer-\nromagnet.\nII. RESULTS\nA. Sample fabrication\nWe use four CPW superconducting transmission lines\nfabricated by optical lithography on 150 nm thick Nio-\nbium \flms deposited by sputtering onto single crystalline\nsapphire substrates (see Fig. 1 atop). A constriction of\nwidthwis opened in the central part of two transmission\nlines by focused ion beam (FIB) milling using moderate\ncurrents which result in lateral resolutions below 10 nm\n(see Fig. 1 abottom). These constrictions, with dimen-\nsions matching those of the di\u000berent samples, serve to\nlocally focus brms, yielding enhanced coupling with the\nsmallest ferromagnets.\nOn the other hand, Py thin-\flms of various thicknesses\ntare e-beam evaporated onto 500 nm-thick Si 3N4mem-\nbranes. The ferromagnets are patterned by FIB milling\nto produce four di\u000berent samples labelled #1to#4\nwhose dimensions are summarized in Fig. 2 a. Finally,\nan Omniprobe needle inside the FIB microscope is used\nto transport a micrometric sized Si 3N4palette containing\nthe Py sample to precise positions on top of the super-\nconducting circuit reaching optimum coupling [35]. For\n20 μm\nConstrictionPy\ny\nxz\na\nd\nbrmsBextS21\nbrmsBextS21bc\nT= 4.2 K T< 4.2 K10 μm\nh h\nw wFIG. 1. a: A constriction of width wis FIB patterned on\nthe central part of a conventional Nb CPW transmission line.\nThe Omniprobe needle is used to transport the Si 3N4palette\nwith the Py ferromagnet towards the constriction. b: Broad-\nband FMR experiments are performed by measuring the S21\ntransmission parameter with a VNA while sweeping the exter-\nnal magnetic \feld Bextapplied along the ^ xdirection so that\nthe microwave \feld brmsproduced by the transmission line is\nperpendicular to it. In this way, a quasi homogeneous Kittel\nmagnon mode is excited in the ferromagnet (blue rectangle\nwith colored arrows) lying at distance hfrom the supercon-\nductor. c: Enlarged view of the meander region (left panel)\nwhere \fnger capacitors are opened by FIB (right panel). d:\nThe resulting CPW resonator, with characteristic frequency\n!p, and its coupling to the ferromagnetic sample are charac-\nterized using the VNA.\nthis purpose, it is important to minimize the distance h\nbetween the ferromagnet and the superconducting line\n(Fig. 1 b). This is di\u000ecult in case of sample #2due to\nits large size that causes the Si 3N4palette to bend. We\nestimateh\u00184\u0016m for sample #2andh\u0018500 nm for\nthe rest. Samples #1and#2are located onto the as-\nfabricated transmission lines, sample #3goes onto the\nw= 1\u0016m wide constriction and, \fnally, sample #4lies\non top of the smallest w= 240 nm wide constriction (see\nFig. 2 a.\nB. Broadband FMR\nSamples are cooled down to 4.2 K in liquid helium in\na cryostat containing a superconducting vector magnet.\nBroadband FMR experiments are performed by measur-\ning the transmission coe\u000ecient S21through the trans-\nmission line using a vector network analyzer (VNA) as\nschematized in Fig. 1 b. The external magnetic \feld\nBextis used to polarize the ferromagnet parallel to the\ntransmission line (^ xdirection) so that the microwave \feld\nproduced by the superconductor (along the ^ ydirection)\ncan excite the homogeneous Kittel mode with frequency3\n4008 0 120\nBext(mT)\n4008 0 120\nBext(mT)2468101214Frequency (GHz)#3 #4\n10μm600 nmt= 120 nm\n15μm3μmt= 60 nm\nNx: 0.006\nNy: 0.025\nNz: 0.969Nx: 0.054\nNy: 0.203\nNz: 0.743\n0\ncb#3 #4\n4008 0 120\nBext(mT)#1\nNx: 0.003\nNy: 0.005\nNz: 0.992\n15μm\n15μmt= 20 nm #1a\nDerivative T\n5 6 7\nBext=29 mT\n7 8 9\nFrequency (GHz)57 mT6 7 8\n44 mT\n7 8 9\nFrequency (GHz)61 mT#2\n4008 0 120\nBext(mT)\n25μm\n25μmt= 120 nm\n#2\nNx: 0.009\nNy: 0.013\nNz: 0.978#2\nFIG. 2. a: SEM pictures of samples #1,#2,#3and#4, highlighted with yellow dashed polygons, placed onto the central\nline of CPW superconducting transmission lines. b: Broadband FMR curves of the same samples in derivative mode (see\nMethods ). Fitting the magnon resonance frequency !mto the general Kittel formula (Eq. 2, dashed) yields the corresponding\ndemagnetizing factors (legend). Color scale in all panels goes from \u00000:1 to 0:1. The solid dots in panels #3and#4correspond\nto the experimental data points obtained from cavity experiments (see Fig. 3 b).c: Field derivative of the experimental\ntransmission parameter (see Methods ) corresponding to sample #2(dashed blue) measured at di\u000berent values of Bextxand\n\fts based on Eq. 3 (solid red). Vertical scale goes from \u00001\u000210\u00003to 1\u000210\u00003in all panels.\ngiven by [21]:\n!2\nm=\r2\neh\nBext+(Ny\u0000Nx)\u00160Msi\n\u0002\nh\nBext+ (Nz\u0000Nx)\u00160Msi\n:(2)\nHere,\u00160= 4\u0019\u000210\u00007Tm/A and Ms= 0:86\u0002106\nA/m is the saturation magnetization of Py measured in\na nominally identical substrate. Finally, Nx,NyandNz\nare the magnetometric demagnetizing factors that sat-\nisfyNx+Ny+Nz= 1. These factors can be analytically\nestimated for each sample based on geometrical consid-\nerations [36]. Results are given in Table I.\nExperimental curves corresponding to samples #1,\n#2,#3and#4are shown in Fig. 2 b. The Kittel\nresonance frequency depends strongly on the aspect ra-\ntio of the Py sample. As a matter of fact, !mincreasesCalculated Fitted\nNxNyNzNxNyNz\n#1 0.003 0.003 0.994 0.003 0.005 0.992\n#2 0.009 0.009 0.982 0.009 0.013 0.978\n#3 0.006 0.033 0.960 0.006 0.025 0.969\n#4 0.011 0.195 0.795 0.054 0.203 0.743\nTABLE I. Demagnetizing factors calculated from geometrical\nconsiderations [36] and \fts based on Eq. 2.\nfrom 2 GHz up to more than 10 GHz at Bext\u00180 mT.\nEq. 2 accounts well for all experimental curves, allowing\nto \ft the demagnetizing factors (see insets of Fig. 2 b\nand Table I). In case of samples #1and#2, symmetry\nconsiderations would yield Nx=Ny. However, this is4\nnot the case due to unavoidable imperfections that are\nespecially dramatic in case of sample #2stemming from\nthe above mentioned bending. In case of sample #3\nand, especially, #4,Nyincreases considerably, indicat-\ning that the ^ ydirection becomes progressively harder. As\nexpected,Nxincreases slightly with sample thickness. In\ncase of sample #4we foundNxalmost \fve times larger\nthan the calculated value. This is probably due to a\nsmall missalignment of the external magnetic \feld with\nrespect to the sample's easy axis. This e\u000bect is much\nmore critical in samples with very large aspect ratio as\n#4.\nFMR data allow us to estimate the frequency-\ndependent magnon-photon coupling. From input-output\ntheory, we can express the transmission through an open\nline as [37]:\nT= 1\u0000\u0000\n\u0000 +\r+ i(!m\u0000!): (3)\nWe recall that !mdepends on the magnetic \feld through\nEq. 2. In the above formula, \r= 2\u000b!mis the linewidth\nof the ferromagnetic resonance and \u0000 = g2\u0019=! m.\nWe use Eq. 3 to \ft the experimental data (in derivative\nmode, as explained in the Methods section). A few rep-\nresentative curves corresponding to sample #2are shown\nin Fig. 2 c. From these \fts we estimate \u000b\u00180:01, i.e.,\nthe Py damping factor at cryogenic temperatures. This\nvalue is close to the Gilbert damping parameter mea-\nsured at low and room temperatures [38]. On the other\nhand, we can estimate the frequency-dependent coupling\ngFMR where the sub-index is used to indicate that these\nvalues are obtained from FMR experiments. Table II\ngives coupling values determined at some \fxed frequen-\ncies!m=!p.\nC. Cavity experiments\nWe next transform each transmission line into a res-\nonator by opening gap capacitors using FIB milling (see\nFig. 1 c). In doing so, we chose the resulting cavity\nlength to \ft the \u0015=2 mode of the desired frequency !p\nsummarized in Table II. Also, the capacitors are de-\nsigned to yield resonators with moderate quality factor\nQ\u00181\u00002\u0002103which are easier to measure while keeping\n\u0014<\r [39]. Samples are again immersed in liquid helium\nexcept for sample #4that is cooled down to 10 mK using\na dilution refrigerator. A VNA is used to probe the S21\nfactor as a function of frequency while sweeping Bextas\ndescribed in the transmission line experiments (see Fig.\n1d).\nMeasurements performed on #1and#2evidence that\nthese samples are in the weak coupling regime. On the\nother hand, two avoided crossings can be distinguished in\nthe experimental data obtained with samples #3and#4\n(see Fig. 3 a). These anticrossings appear at positive and\nnegative magnetic \felds, at the intersect with the FMR\ncurves (dashed lines), which is a proof of magnon-photon!p=2\u0019 g FMR=2\u0019 g res=2\u0019 g theo=2\u0019\n(GHz) (MHz) (MHz) (MHz)\n#1 2.8 7 - 6.6\n#2 3.5 16 - 17.3\n#3 4.2 36 36 45.0\n#4 10.5 66 72 72.4\nTABLE II. Photon resonance frequency ( !p); photon-\nmagnon coupling determined from FMR measurements\n(gFMR), from cavity experiments ( gres) and from theory\n(gtheo). The latter is calculated for h= 4\u0016m in case of\nsample #2andh= 500 nm for the rest.\n#3\n-20 -10 0 10 20024\n  \nBext (mT)-60 -30 0 30 6004812\n  \nBext (mT)(-)/2(MHz) ˜ba\n#3 #44.21\n10.4210.4610.50\n4.194.20Frequency (GHz)\n10 -20 -10 0 20\n 30 -60 -30 0 60#4\n-71 -59 -47S21(dBm)\n-92 -42 -67S21(dBm)\nBext(mT) Bext(mT)\nFIG. 3. a: Experimental microwave transmission through\nsuperconducting resonators containing samples #3(!p=2\u0019=\n4:2 GHz,T= 4:2 K) and #4(!p=2\u0019= 10:5 GHz,T= 10\nmK). The dashed lines are the broadband FMR curves shown\nin Fig. 2 b.b: Field dependence of the resonance width \u0014\nfor the same resonators and least-squares \fts (dashed lines)\nto Eq. 4. Data corresponding to sample #3are shifted 2\nmT towards negative \felds due to a small remnant \feld in\nthe superconducting coils. Maxima of \u0014vs.Bextprovide two\nadditional experimental points of FMR at Bext=\u00003:9 and\n0:4 mT for sample #3andBext=\u000623 mT for sample #4\n(see Fig. 2 b).\ncoupling. In case of sample #4, we observe also a number\nof small successive anticrossings at positive \felds. The\norigin of such features is unknown but they might arise\nfrom the interaction between the fundamental mode of\nthe resonator and spurious modes related with the mag-\nnetic behavior of the superconducting cavity itself, i.e.,\nthey are not related with the Permalloy sample.\nCavity experiments can be also used to give an alter-\nnative estimate of the magnon-photon coupling. For this\npurpose we analyze the cavity resonance linewidth \u0014as\na function of external magnetic \feld. The linewidth is\nevaluated by \ftting the experimental transmission to a\nFano-like resonance (see Methods ). The latter accounts5\nbetter for asymmetries in our experimental data that\nlikely arise from interference between the resonator and\na small continuous background signal [40]. Depending\non the coupling strength, the resonance broadens around\nthe intersection \feld according to [41]:\n\u0014= ~\u0014+\rg2\n(!p\u0000!m)2+\r2; (4)\nwhere ~\u0014is the linewidth of the bare cavity that can be\nmeasured far from resonance. Again, we recall that !m\ndepends on the external \feld through Eq. 2. Experi-\nmental\u0014vs.Bextcurves exhibit two clear maxima at\npositive and negative \felds, where level repulsion occurs\n(Fig. 3 b). Fitting these data to Eq. 4 gives a comple-\nmentary estimate of the coupling (labelled gres) that we\nsummarize in Table II.\nD. Theoretical model\nWe \fnish by comparing our experimental results with\nnumerical estimations of the coupling factors. A gener-\nalization of formulas given in Refs. 42 and 43, where\nthe \feld was considered homogeneous in the relevant re-\ngion, follows. Each ferromagnetic sample is discretized\ninto cells. The Hamiltonian interaction is given by\nHI=vcellP\njmj\u0001brms(rj). Here,vcellandmjare the\nvolume and magnetization of the cell, respectively, while\nbrms(rj) is the rms vacuum \feld at the cell's position rj.\nIn each cell, the number of spins is large enough to per-\nform the Holstein-Primako\u000b transformation, so the quan-\ntization of the magnetization can be written in terms of\nbosonic operators as mj= \u0001Mj(ay\nj+aj) where \u0001 Mjis\nthe position-dependent amplitude of the magnetization\nmodulation. The latter will be always valid as we are\nworking in the low photon or low power limit. Once this\nis done, we de\fne the collective mode [44]:\na=1qP\nj(\u0001Mj\u0001brms(rj))2X\nj\u0001Mj\u0001brms(rj)aj\nFinally, taking the continuum limit, the magnon-\nresonator coupling reads:\ng2=\rvcell\n4~X\njbrms(rj)\u0001\u0001Mj; (5)\nwhich is valid for anykind of magnonic mode excited in\nthe ferromagnet, including the Kittel mode studied here.\nWe numerically calculate the spatial distribution of\nbrms(rj) for the di\u000berent resonators used in this work\n(seeMethods ). Figure 4 ashows the resulting in-plane\n(y) component of brmsfor the di\u000berent samples obtained\nat the position of the ferromagnet. We highlight that\nbrmsincreases by a factor \u001830 from sample #1to#4\nstemming from the smallness of the patterned constric-\ntion.\n11 0 11 0 11 0 11 010100\n    \nh (m)   g/2 (MHz)\n  c\n#3 #4 #2 #1\n2.8 GHz 3.5 GHz 4.2 GHz 10.5 GHz\n#3\n#4#2#1\n050600.10.20.3\n#3\n#4#2#1\n20247.88.300.40.600.40.8My(A/m) brms,y(nT) a b\ny\nx00.10.20.3\nh(m)g/2(MHz)cFIG. 4. Spatial distribution of the in-plane ( y) compo-\nnent ofbrms(a) and \u0001M(b) calculated at the position of\nthe ferromagnet. c: Solid lines are the numerical gtheo(at\n\fxed frequency !pgiven above each panel) vs. distance\nh. Data points are the experimental gFMR (dots) and gres\n(stars). From left to right, our experimental data roughly\nfollowg/!p.\nOn the other hand, we need to estimate \u0001 Mjwhich\ndepends on the spatial distribution of the excitation \feld,\ni.e.,brms(rj), and also on the non-homogeneous magnetic\nsusceptibility of the ferromagnet. Figure 4 bshows the\nnumerically calculated ycomponent of \u0001 Mfor the dif-\nferent samples (see Methods ). In general, \u0001 Myis max-\nimum in the inner part of the ferromagnets but decreases\nconsiderably when approaching the edges. This e\u000bect is\nless pronounced in case of sample #1, for which the sus-\nceptibility is substantially more homogeneous. This is\ndue to its small thickness that constrains the modulation\nof the magnetization to the plane.\nFinally, we use Eq. 5 to calculate the theoretical cou-\npling factors gtheoat \fxed frequency !pfor varying dis-\ntance between the ferromagnet and the superconducting\nline, i.e., distance hin Fig. 1 bandd. Results are plotted\nin Fig. 4 c(solid lines) together with the experimental g\nvalues obtained at the same frequencies (scatter).6\nIII. DISCUSSION\nWe start by highlighting that, in case of samples #3\nand#4,gFMR\u0019gres. This demonstrates that the two\nexperimental approaches, i.e., transmission and cavity\ncon\fguration, are equivalent and valid to estimate the\ncoupling. This result is not limited to the use of ferro-\nmagnetic samples but applies to whatsoever hybrid light-\nmatter architecture based on, e.g., paramagnetic spins or\nsuperconducting qubits.\nWe can calculate the e\u000bective coupling per spin for\nsamples #3 and#4 that amounts to 70 and 280\nHz, respectively. The latter approaches the largest\nspin-photon coupling strength achieved so far in opti-\nmized superconductor-peramalloy-insulator hybrid sam-\nples, i.e., 350 Hz [10] and exceeds the largest spin-photon\ncoupling measured with paramagnetic spins under sim-\nilar conditions [30]. As a matter of fact, the gvalues\nreported here are similar to those obtained using opti-\nmizedLC-resonators [25, 33].\nWe next focus our attention on the experimental gFMR\nandgresand compare them to the gtheocalculated with\nour theory (Table II). The main source of error in gtheois\nthe distance h. As shown in Fig. 4 c, experiment and the-\nory agree to a very good extent for the estimated values\nofh, con\frming the validity of our model.\nEstimating the coupling strength in hybrid magnon-\nphoton architectures is essential for designing reliable\nexperiments. Here, we exploit the shape of thin-\flm\nferromagnets as tuning knob to modify their resonance\nfrequency. This requires an additional account for the\nlargely non-uniform demagnetizing \feld in rectangular\nprisms. However, little attention has been given till\nnow to the distribution of magnetization precession along\nthe volume of real ferromagnets [45, 46]. In contrast,\nour approach involves the calculation of both the three-\ndimensional distribution of vacuum \feld created by the\ncavity and the magnetic susceptibility of the ferromag-\nnet. This is important even for Kittel modes in ellipsoids\nof revolution but becomes critical when dealing with real-\nistic thin-\flms for which magnetization precession might\nbe highly elliptical. Examples of the latter include spin\nmodes in rectangular prisms but also skyrmions or \rux-\nclosure magnetic textures like vortices [42, 43]. As we\nhave demonstrated, Eq. 5 can be used for calculating\nthe coupling between superconducting circuits of what-\never geometry and magnetostatic modes of any kind ex-\ncited in ferromagnetic samples of arbitrary shape as long\nas this mode can be e\u000eciently reproduced by micromag-\nnetic simulations.\nFinally, our experimental data serve to demonstrate\nthe direct relation between the coupling strength and\nthe cavity frequency (see Fig. 4 c). Deviations from the\nexpected linear dependence arise from the di\u000berent ge-\nometries of the resonators and the particular volumes\nand demagnetizing factors of each sample. The latter\na\u000bects the magnetic susceptibility of the corresponding\nmagnetostatic (Kittel) mode. Besides these factors, ourexperimental data roughly follow g/!p. This is partic-\nularly evident for samples #3and#4. Increasing the\ncavity operation frequency from !p=2\u0019= 4:2 GHz ( #3)\nup to!p=2\u0019= 10:5 GHz ( #4) yields twice the coupling\nstrength.\nWe stress that patterning nanoconstrictions or reduc-\ning the impedance yields a square-root increase of gwhile\nkeeping the losses in the ferromagnet unchanged, mean-\ning these approaches serve to increase the cooperativity.\nOn the other hand, increasing the operation frequency\nyields a linear increase of g, at the cost of also increasing\n\r= 2\u000b!p. That is to say, increasing !pdoes not ease\nentering into the strong coupling regime but does have\nthe e\u000bect of linearly increasing the rate of information\nexchange between photon and magnon excitations.\nIV. CONCLUSIONS\nWe demonstrate the possibility of tuning the operation\nfrequency in cavity magnonics by controlling the shape\nof the ferromagnet. Varying the aspect ratio of di\u000berent\nPy samples we reach operation frequencies up to 10 :5\nGHz at\u001820 mT. Working at such high frequencies with\nisotropic spherical ferromagnets would have required an\nexternal \feld close to 0 :4 T. High frequency operation\ntogether with the use of patterned nanoconstrictions in\nsuperconducting transmission lines enhances the net cou-\npling between magnetic materials and cavity photons al-\nlowing us to reach an average single spin coupling of 280\nHz.\nOur experimental results demonstrate that the light-\nmatter coupling strength can be reliably estimated from\nboth open transmission or cavity experiments. Be-\nsides, this has allowed combining broadband spectro-\nscopic measurements, which provide the full picture of\nthe ferromagnetic excitations, with experiments on su-\nperconducting resonators, which show how these modes\ncouple to cavity photons.\nFinally, we provide a simple recipe to theoretically cal-\nculategin hybrid systems combining superconducting\ncircuits and realistic ferromagnetic materials. Our ap-\nproach can be used to design quantum magnonic experi-\nments exploiting not only the homogeneous Kittel mode\nbut any higher order magnonic mode in ferromagnets of\narbitrary size and shape and to tune the operation condi-\ntions, frequency and magnetic \feld, to \ft those required\nin a particular experiment or application.\nV. METHODS\nA. Data \ftting\nOpen transmission data is analyzed by performing the\npseudo-derivative of the transmission parameter. For this7\npurpose, we calculate\nDerivativeTexp=Texp(Bi+1\next)\u0000Texp(Bi\next)\nTexp(Bi+1\next);\nwhereTexpis the linear amplitude of the transmission\nTexp= 10S21=20and super-indexes iandi+ 1 refer to\nsuccessive \feld values. Resulting curves vs. frequency\nand \feld are ploted in Fig. 2 bandc. We then calcu-\nlate the pseudo-derivative of the absolute value of Eq. 3,\nwhich is given by:\njTj=s\n\r2+ (!m\u0000!)2\n\u00002+ 2\u0000\r+\r2+ (!m\u0000!)2;\nwhere!mand\r= 2\u000b!mare evaluated at each Bi\nextusing\nEq. 2. We \fnally \ft the pseudo-derivative of jTjto the\nexperimental data as shown in Fig. 2 cto obtain the\ncoupling strenght g= (\u0000!m=\u0019)1=2.\nCavity experiments are analized by \ftting the experi-\nmental resonance to a Fano curve:\nT=y0+A\u0000!\u0000!0\n\r=2+q\u00012\n1 +\u0000!\u0000!0\n\r=2\u000121\n1 +q2;\nwithy0an o\u000bset,Athe amplitude and qthe Fano param-\neter that de\fnes the lineshape of the resonance. Finally,\n1=(1+q2) is a normalization factor. Fano resonances can\nbe observed when the resonators is not perfectly matched\nto the input output lines, yielding a parasitic path be-\ntween connectors. q= 0 yields an antiresonance whereas\nforq!\u00061 the Lorentzian resonance with linewidth \r\nis recovered.\nB. simulations\nThe spatial distribution of brms(rj) is calculated using\nthe \fnite-element software 3D-MLSI [47] that solves Lon-\ndon equations in a user-de\fned superconducting circuitwith \rowing current irms. The latter is the rms zero-point\ncurrent that can be obtained as [48]:\nirms=!pr\n~\u0019\n4Z0; (6)\nwithZ0= 50 \n the impedance of the circuit. Inserting\nthe experimental frequencies !pinto Eq. 6, we calculate\nirmsfor each resonator. From here, 3D-MLSI allows cal-\nculating the spatial distribution of supercurrents and the\nresulting brms(rj) at each cell jinside the volume of the\nferromagnet.\nWe \fnally calculate the spatial distribution of \u0001 Mj\nresulting from the spatial-dependent brms(rj) using the\nmicromagnetic solver MUMAX3[49]. For this purpose,\nwe excite the ferromagnet with a time- and spatial-\ndependent \feld bexcit=brms(rj) cos(!mt). As a result,\nwe obtain the time evolution of the magnetization at each\ncelljand \ft the resulting curve to an equation of the\nform \u0001 Mjsin(!mt+\u000ej). The coupling is \fnally calcu-\nlated applying Eq. 5.\nVI. ACKNOWLEDGMENTS\nThis work was partly funded and supported by\nthe spanish MCIN/ AEI /10.13039/501100011033/ and\nFEDER ( Una manera de hacer Europa ) through grant\nRTI2018-096075-B-C21, the BBVA Foundation through\n\"Beca Leonardo a Investigadores y Creadores Cultur-\nales 2019\", the Arag\u0013 on Regional Government through\nproject E09 20R ( Construyendo Europa desde Arag\u0013 on ),\nCSIC through project 202160I034, the CSIC program\nfor the Spanish Recovery, Transformation and Resilience\nPlan funded by the Recovery and Resilience Facility of\nthe European Union, established by the Regulation (EU)\n2020/2094, the EU through FET-OPEN (862893 FAT-\nMOLS) and the European Research Council (ERC) un-\nder the European Union's Horizon 2020 research and in-\nnovation programme (948986 QFaST). S. M.-L. R. ac-\nknowledges a FPI grant from the spanish MCIN. Authors\nwould like to acknowledge the use of Servicio General de\nApoyo a la Investigaci\u0013 on-SAI, Universidad de Zaragoza.\n[1] A. Blais, J. Gambetta, A. Wallra\u000b, D. I. Schuster, S. M.\nGirvin, M. H. Devoret, and R. J. Schoelkopf, Physical\nReview A 75, 032329 (2007).\n[2] M. Harder, B. M. Yao, Y. S. Gui, and C.-M. Hu, Journal\nof Applied Physics 129, 201101 (2021).\n[3] Y. Li, W. Zhang, V. Tyberkevych, W.-K. Kwok, A. Ho\u000b-\nmann, and V. Novosad, Journal of Applied Physics 128,\n130902 (2020).\n[4] D. Lachance-Quirion, Y. Tabuchi, A. Gloppe, K. Usami,\nand Y. Nakamura, Applied Physics Express 12, 070101\n(2019).\n[5] Y. Tabuchi, S. Ishino, T. Ishikawa, R. Yamazaki, K. Us-\nami, and Y. Nakamura, Physical Review Letters 113,\n083603 (2014).[6] H. Huebl, C. W. Zollitsch, J. Lotze, F. Hocke, M. Greifen-\nstein, A. Marx, R. Gross, and S. T. B. Goennenwein,\nPhysical Review Letters 111, 127003 (2013).\n[7] I. A. Golovchanskiy, N. N. Abramov, V. S. Stolyarov,\nM. Weides, V. V. Ryazanov, A. A. Golubov, A. V. Usti-\nnov, and M. Y. Kupriyanov, Science Advances 7(2021).\n[8] G. Flower, M. Goryachev, J. Bourhill, and M. E. Tobar,\nNew Journal of Physics 21, 095004 (2019).\n[9] X. Zhang, C.-L. Zou, L. Jiang, and H. X. Tang, Physical\nReview Letters 113, 156401 (2014).\n[10] I. Golovchanskiy, N. Abramov, V. Stolyarov, A. Golubov,\nM. Y. Kupriyanov, V. Ryazanov, and A. Ustinov, Phys-\nical Review Applied 16, 034029 (2021).8\n[11] N. J. Lambert, A. Rueda, F. Sedlmeir, and H. G. L.\nSchwefel, Advanced Quantum Technologies 3, 1900077\n(2020).\n[12] R. Hisatomi, A. Osada, Y. Tabuchi, T. Ishikawa,\nA. Noguchi, R. Yamazaki, K. Usami, and Y. Nakamura,\nPhysical Review B 93, 174427 (2016).\n[13] Y. Li, V. G. Yefremenko, M. Lisovenko, C. Trevillian,\nT. Polakovic, T. W. Cecil, P. S. Barry, J. Pearson, R. Di-\nvan, V. Tyberkevych, C. L. Chang, U. Welp, W.-K.\nKwok, and V. Novosad, Physical Review Letters 128,\n047701 (2022).\n[14] D. Lachance-Quirion, S. P. Wolski, Y. Tabuchi, S. Kono,\nK. Usami, and Y. Nakamura, Science 367, 425 (2020).\n[15] S. Wolski, D. Lachance-Quirion, Y. Tabuchi, S. Kono,\nA. Noguchi, K. Usami, and Y. Nakamura, Physical Re-\nview Letters 125, 117701 (2020).\n[16] Y.-Y. Wang, S. van Geldern, T. Connolly, Y.-X. Wang,\nA. Shilcusky, A. McDonald, A. A. Clerk, and C. Wang,\nPhysical Review Applied 16, 064066 (2021).\n[17] N. Zhu, X. Han, C.-L. Zou, M. Xu, and H. X. Tang,\nPhysical Review A 101, 043842 (2020).\n[18] X. Zhang, A. Galda, X. Han, D. Jin, and V. M. Vinokur,\nPhysical Review Applied 13, 044039 (2020).\n[19] Y.-P. Wang, J. Rao, Y. Yang, P.-C. Xu, Y. Gui, B. Yao,\nJ. You, and C.-M. Hu, Physical Review Letters 123,\n127202 (2019).\n[20] D. M. Pozar, Microwave engineering (Wiley, 2011).\n[21] C. Kittel and D. F. Holcomb, American Journal of\nPhysics 35, 547 (1967).\n[22] L. R. Walker, Physical Review 105, 390 (1957).\n[23] P. G. Baity, D. A. Bozhko, R. Mac^ edo, W. Smith, R. C.\nHolland, S. Danilin, V. Seferai, J. Barbosa, R. R. Peroor,\nS. Goldman, U. Nasti, J. Paul, R. H. Had\feld, S. McVi-\ntie, and M. Weides, Applied Physics Letters 119, 033502\n(2021).\n[24] L. Mihalceanu, V. I. Vasyuchka, D. A. Bozhko,\nT. Langner, A. Y. Nechiporuk, V. F. Romanyuk, B. Hille-\nbrands, and A. A. Serga, Physical Review B 97, 214405\n(2018).\n[25] J. T. Hou and L. Liu, Physical Review Letters 123,\n107702 (2019).\n[26] Y. Li, T. Polakovic, Y.-L. Wang, J. Xu, S. Lendinez,\nZ. Zhang, J. Ding, T. Khaire, H. Saglam, R. Divan,\nJ. Pearson, W.-K. Kwok, Z. Xiao, V. Novosad, A. Ho\u000b-\nmann, and W. Zhang, Physical Review Letters 123,\n107701 (2019).\n[27] I. A. Golovchanskiy, N. N. Abramov, V. S. Stolyarov,\nI. V. Shchetinin, P. S. Dzhumaev, A. S. Averkin, S. N.\nKozlov, A. A. Golubov, V. V. Ryazanov, and A. V. Usti-\nnov, Journal of Applied Physics 123, 173904 (2018).\n[28] I. Haygood, M. Pufall, E. Edwards, J. M. Shaw, and\nW. Rippard, Physical Review Applied 15, 054021 (2021).[29] M. A. W. Schoen, D. Thonig, M. L. Schneider, T. J.\nSilva, H. T. Nembach, O. Eriksson, O. Karis, and J. M.\nShaw, Nature Physics 12, 839 (2016).\n[30] I. Gimeno, W. Kersten, M. C. Pallar\u0013 es, P. Hermosilla,\nM. J. Mart\u0013 \u0010nez-P\u0013 erez, M. D. Jenkins, A. Angerer,\nC. S\u0013 anchez-Azqueta, D. Zueco, J. Majer, A. Lostao, and\nF. Luis, ACS Nano 14, 8707 (2020).\n[31] P. Haikka, Y. Kubo, A. Bienfait, P. Bertet, and\nK. M\u001clmer, Physical Review A 95, 022306 (2017).\n[32] M. D. Jenkins, U. Naether, M. Ciria, J. Ses\u0013 e, J. Atkinson,\nC. S\u0013 anchez-Azqueta, E. del Barco, J. Majer, D. Zueco,\nand F. Luis, Applied Physics Letters 105, 162601 (2014).\n[33] S. Probst, A. Bienfait, P. Campagne-Ibarcq, J. J. Pla,\nB. Albanese, J. F. D. S. Barbosa, T. Schenkel, D. Vion,\nD. Esteve, K. M\u001clmer, J. J. L. Morton, R. Heeres, and\nP. Bertet, Applied Physics Letters 111, 202604 (2017).\n[34] C. Eichler, A. Sigillito, S. Lyon, and J. Petta, Physical\nReview Letters 118, 037701 (2017).\n[35] M. J. Mart\u0013 \u0010nez-P\u0013 erez, J. Pablo-Navarro, B. M uller,\nR. Kleiner, C. Mag\u0013 en, D. Koelle, J. M. de Teresa, and\nJ. Sese, Nano Letters 18, 7674 (2018).\n[36] A. Aharoni, Journal of Applied Physics 83, 3432 (1998).\n[37] D. Roy, C. Wilson, and O. Firstenberg, Reviews of Mod-\nern Physics 89, 021001 (2017).\n[38] Y. Zhao, Q. Song, S.-H. Yang, T. Su, W. Yuan, S. S. P.\nParkin, J. Shi, and W. Han, Scienti\fc Reports 6(2016).\n[39] M. G oppl, A. Fragner, M. Baur, R. Bianchetti, S. Filipp,\nJ. M. Fink, P. J. Leek, G. Puebla, L. Ste\u000ben, and A. Wall-\nra\u000b, Journal of Applied Physics 104, 113904 (2008).\n[40] U. Fano, Physical Review 124, 1866 (1961).\n[41] P. Bushev, A. K. Feofanov, H. Rotzinger, I. Protopopov,\nJ. H. Cole, C. M. Wilson, G. Fischer, A. Lukashenko, and\nA. V. Ustinov, Physical Review B 84, 060501 (2011).\n[42] M. J. Mart\u0013 \u0010nez-P\u0013 erez and D. Zueco, New Journal of\nPhysics 21, 115002 (2019).\n[43] M. J. Mart\u0013 \u0010nez-P\u0013 erez and D. Zueco, ACS Photonics 6,\n360 (2018).\n[44] T. H ummer, G. M. Reuther, P. H anggi, and D. Zueco,\nPhysical Review A 85, 052320 (2012).\n[45] R. Mac^ edo, R. C. Holland, P. G. Baity, L. J. McLellan,\nK. L. Livesey, R. L. Stamps, M. P. Weides, and D. A.\nBozhko, Physical Review Applied 15, 024065 (2021).\n[46] J. Bourhill, V. Castel, A. Manchec, and G. Cochet, Jour-\nnal of Applied Physics 128, 073904 (2020).\n[47] M. M. Khapaev, M. Y. Kupriyanov, E. Goldobin, and\nM. Siegel, Superconductor Science and Technology 16,\n24 (2002).\n[48] M. Jenkins, T. H ~A¼mmer, M. J. Mart\u0013 \u0010nez-P\u0013 erez,\nJ. Garc\u0013 \u0010a-Ripoll, D. Zueco, and F. Luis, New Journal\nof Physics 15, 095007 (2013).\n[49] A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen,\nF. Garcia-Sanchez, and B. V. Waeyenberge, AIP Ad-\nvances 4, 107133 (2014)."    },    {        "title": "2103.00653v1.Spin_and_charge_interconversion_in_Dirac_semimetal_thin_films.pdf",        "content": "arXiv:2103.00653v1  [cond-mat.mtrl-sci]  28 Feb 2021Spin and charge interconversion in Dirac semimetal thin ﬁlm s\nWilson Yanez,1Yongxi Ou,1Run Xiao,1Jahyun Koo,2Jacob T. Held,3\nSupriya Ghosh,3Jeﬀrey Rable,1Timothy Pillsbury,1Enrique Gonz´ alez\nDelgado,4Kezhou Yang,5Juan Chamorro,6Alexander J. Grutter,7Patrick\nQuarterman,7Anthony Richardella,1Abhronil Sengupta,8Tyrel McQueen,6\nJulie A. Borchers,7K. Andre Mkhoyan,3Binghai Yan,2and Nitin Samarth1,∗\n1Department of Physics, Pennsylvania State University,\nUniversity Park, Pennsylvania 16802, USA\n2Department of Condensed Matter Physics,\nWeizmann Institute of Science, Rehovot 7610001, Israel\n3Department of Chemical Engineering and Materials Science,\nUniversity of Minnesota, Minneapolis, Minnesota 55455, US A\n4Department of Physics and Electronics,\nHumacao Campus of the University of Puerto Rico,\n100 Tejas Avenue, Humacao, Puerto Rico 00791-4300\n5Department of Materials Science and Engineering,\nPennsylvania State University, University Park, Pennsylv ania 16802, USA\n6Department of Chemistry, Johns Hopkins University, Baltim ore, MD USA\n7NIST Center for Neutron Research,\nNational Institute of Standards and Technology,\nGaithersburg, Maryland 20899, USA\n8School of Electrical Engineering and Computer Science,\nPennsylvania State University, University Park, Pennsylv ania 16802, USA\n(Dated: March 2, 2021)\n1Abstract\nWe report spin-to-charge and charge-to-spin conversion at room temperature in heterostruc-\nture devices that interface an archetypal Dirac semimetal, Cd3As2, with a metallic ferromagnet,\nNi0.80Fe0.20(permalloy). The spin-charge interconversion is detected by both spin torque ferro-\nmagnetic resonance and ferromagnetic resonance driven spi n pumping. Analysis of the symmetric\nand anti-symmetric components of the mixing voltage in spin torque ferromagnetic resonance and\nthe frequency and power dependence of the spin pumping signa l show that the behavior of these\nprocesses is consistent with previously reported spin-cha rge interconversion mechanisms in heavy\nmetals, topological insulators, and Weyl semimetals. We ﬁn d that the eﬃciency of spin-charge\ninterconversion in Cd 3As2/permalloy bilayers can be comparable to that in heavy metal s. We\ndiscuss the underlying mechanisms by comparing our results with ﬁrst principles calculations.\nI. INTRODUCTION\nSpin-to-charge conversion and its Onsager reciprocal charge-t o-spin conversion have been\nextensively studied in several materials with either strong spin-orb it coupling (such as heavy\nmetals, HMs) or with spin momentum “locking” (as in the surface stat es of topological\ninsulators) [ 1–12]. The latter class of materials motivated the emergence of “topolog ical\nspintronics,” broadly aimed at exploiting the strong spin-momentum c orrelation in helical\ntopological states for spintronic devices. Theoretical studies pr ovide strong motivation for\nextending such studies to topological semimetals. For example, in We yl semimetals, calcu-\nlations predict a large spin Hall conductivity, the Rashba-Edelstein e ﬀect, and eﬃcient spin-\nto-charge conversion in magnetic Weyl semimetal-normal metal he terostructures [ 13–15].\nRecent experiments have reported current-induced spin-orbit t orques in the Weyl semimetal\nWTe2with eﬃciencies comparable to HMs [ 16,17]. In addition, ﬁeld-free current-induced\nmagnetization switching has been demonstrated at room temperat ure in WTe 2/ferromagnet\nheterostructures with current densities lower than HMs and topo logical insulators [ 18]. This\ncontext motivates the exploration of spin-charge interconversio n in a diﬀerent class of topo-\nlogical semimetals of contemporary interest, the Dirac semimetal ( DSM), where only a few\nstudies of spin transport have been reported [ 19–21]. We might anticipate that the inter-\n∗nsamarth@psu.edu\n2play between a bulk three-dimensional linear dispersion with non-triv ial topology and the\nspin polarization of surface states could give rise to eﬃcient spin-ch arge interconversion and\nthus elicit interest for spintronics [ 14,18,22–25]. Unlike the intuitive appeal of topological\ninsulators and Weyl semimetals for topological spintronics, howeve r, the spin degeneracy\ninherent in the topological band structure of a DSM raises a fundam ental question: what is\nthe spin Hall conductivity (SHC) of a DSM and could it be relevant for s pintronics?\nIn this paper, we report measurements of spin-charge interconv ersion at room temper-\nature in bilayers of the archetypal DSM Cd 3As2and a conventional metallic ferromagnet,\nNi0.80Fe0.20(permalloy, Py) via spin torque ferromagnetic resonance (ST-FMR ) and ferro-\nmagnetic resonance driven spin pumping (SP). We also compare our e xperimental results\nwith ﬁrst principles calculations of the SHC in Cd 3As2. In bilayers with an imperfect (oxi-\ndized) interface, we ﬁnda SHCthat ismuch largerthanpredicted by theoryat theestimated\nchemical potential and comparable to that of HMs. We attribute th e dominant contribu-\ntions to extrinsic eﬀects. Measurements of bilayers with a clean inte rface show a smaller\nSHC, consistent with our theoretical predictions for intrinsic cont ributions to spin-charge\ninterconversion.\nII. SAMPLE SYNTHESIS AND CHARACTERIZATION\nWe ﬁrst discuss the synthesis, as well as structural and interfac ial characterization, of\nCd3As2/Py heterostructures. We used molecular beam epitaxy (MBE) to gr ow Cd 3As2thin\nﬁlms under ultrahigh vacuum conditions (pressure P <10−7Pa) in a Veeco 930 system. We\nﬁrst grew an intrinsic, relaxed GaSb (111) buﬀer layer (100 nm thick ) on a semi-insulating\nGaAs(111)B substrate using elemental Ga (5N) and Sb (5N) sourc e materials evaporated\nfrom standard eﬀusion cells. The growth conditions used were stan dard for the III-V semi-\nconductor. The Cd 3As2layer was then deposited at a substrate temperature Ts∼200◦C.\nFor most of the samples discussed in this paper, the source materia ls are elemental Cd (5N\npurity) and As (5N purity) evaporated from conventional eﬀusion cells; the beam equivalent\nﬂux ratio of Cd:As was 3:2. Some samples were also grown by evaporat ing a compound\nCd3As2source, also from an eﬀusion cell. Typical growth rates were 0.5 nm/ min. During\nMBE growth, we obtain an unreconstructed streaky reﬂection hig h energy electron diﬀrac-\ntion (RHEED) pattern (Figs. 1 (a) and 1 (b)) which shows the expec tedC3symmetry of\n3the Cd 3As2crystal. X-ray diﬀraction (Fig. 1c) shows peaks corresponding to Cd3As2grown\nin the∝angbracketleft112∝angbracketrightorientation as well as the peaks of the GaSb (111) buﬀer layer and t he GaAs\n(111) substrate. Typical double crystal rocking curves for diﬀr action peaks from Cd 3As2\nhave widths of ∼0.11◦. Atomic force microscopy (AFM) (Fig. 1(d)) shows large domains\n(500 nm lateral islands) with 0.5 nm high steps. These are all signatur es of epitaxial growth\nof reasonable quality Cd 3As2thin ﬁlms, similar to those reported in prior literature [ 26].\nThe measurement of spin-charge interconversion in Cd 3As2requires the synthesis of het-\nerostructures that interface a well-characterized ferromagne t with the DSM. To this end, we\ntransferred the MBE-grown Cd 3As2ﬁlms to a diﬀerent vacuum chamber for the deposition\nof Py thin ﬁlms using an e-beam evaporator. All the spin-charge inte rconversion data in the\nmain manuscript is taken on samples that involved brief (10-15 minute s) exposure of the\nCd3As2ﬁlm to ambient atmosphere before the subsequent deposition of a P y thin ﬁlm, fol-\nlowed by 1 nm to 3 nm of either Ta or Al as a capping layer to prevent ox idation. Although\ntechnical limitations currently constrain our capability for the rout ine deposition of Py on\nCd3As2without breaking vacuum, we do include some spin-charge interconv ersion data on\na sample grown with a pristine interface (see Appendix D).\nWe further characterized the Cd 3As2/Py heterostructures using atomic-resolution high-\nangle annular dark-ﬁeld (HAADF) scanning transmission electron mic roscopy (STEM). An\nexampleofsuchdataforaheterostructurewithbriefambient exp osurebetweenMBEgrowth\nof Cd3As2and Py deposition is shown in Figs. 1 (e) and 1 (f): the former reveals coherent\ngrowth of the entire heterostructure, albeit with the presence o f a 1 nm cadmium oxide layer\nat the Cd 3As2/Py interface as conﬁrmed by energy dispersive x-ray (EDX) spect roscopy.\nMore detailed STEM andEDXcharacterization is availablein Appendix D. Since proximity-\ninduced magnetism at the interface between the DSM and metallic fer romagnet might play\na role in spin-charge interconversion, we also used polarized neutro n reﬂectometry (PNR)\nmeasurements to characterize the magnetic behavior at the Cd 3As2/Py interface. We do not\nobserve any evidence for proximity-induced magnetism in the Cd 3As2layer (see Appendix\nE).\nTo conﬁrm the presence of a DSM state in the MBE-grown Cd 3As2thin ﬁlms, we carried\nout angle resolved photoemission spectroscopy (ARPES) measure ments on a few samples\nthat were transferred fromthe MBEchamber to alocal ARPES mea surement chamber while\nmaintaining a vacuum environment with pressure 1 .3/lessorsimilarµPa. We used the 21 eV helium\n4(d)\n500 nm \nPermalloy \nCd/g925As/g924\nGaSb \nGaAs 100 nm (e)/s49/s48/s48/s32/s32/s49/s48/s51/s32/s32/s49/s48/s54/s32/s32/s49/s48/s57/s76/s111/s103/s32/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s91/s97/s46/s117/s46/s93 /s49/s48/s48 /s56/s48 /s54/s48 /s52/s48 /s50/s48\n/s50/s32/s32/s32/s91/s68/s101/s103/s93/s32\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s67/s100 /s51/s65/s115/s50/s32/s40/s49/s49/s50/s41\n/s32/s71/s97/s83/s98/s32/s40/s49/s49/s49/s41\n/s32/s71/s97/s65/s115/s32/s40/s49/s49/s49/s41\n/g84\n(c)(d)\n(c (c (c (c (c (c (c (c (c (c (c (c (c (c (c (c (c (c (c (c (c (c (c (c (c ))))))))))))))))))))))(d (d (d (d (d (d (d (d (d (d (d (d (d (d (d (d (d (d (d (d (d )))))))))))))))))\n10 nm (f) Cd/g925As/g924(c)\n(d)\n10 nm \n8\n6\n4\n2\n0(a) (b)\n[111] [110]\nFIG. 1. RHEED pattern of the Cd 3As2ﬁlm along the [ ¯1¯11] (a) and [1 ¯10] (b) crystal directions.\n(c) X-ray diﬀraction 2 θscan of Cd 3As2ﬁlms. (d) AFM image of the Cd 3As2surface. High-angle\nannular dark-ﬁeld TEM image showing the diﬀerent layers in a c omplete device stack (e) and just\nthe Cd 3As2layer (f).\nIαspectral line from a helium plasma lamp isolated via a monochromator as the photon\nsource. Theemitted photoelectronswere imagedby aScientaOmicr on DA30Lanalyzer with\namaximumspectralresolutionof6meV.Figures2(a)and2(b)show ARPESmeasurements\nfrom a Cd 3As2thin ﬁlm at room temperature. The spectra are similar to those repo rted\nin the prior literature on cleaved bulk crystals of Cd 3As2[27–29]. The data shows the\ncharacteristic linear dispersion in the vicinity of the charge neutral point expected for a\nDSM. The ARPES data also indicate that the charge neutral point is lo cated 0.2 eV below\nthe chemical potential. This is consistent among several samples wit h diﬀerent thicknesses.\n5It is important to note that ARPES cannot probe the surface stat e of these Cd 3As2ﬁlms\nonce the surface has been capped with Py. Thus, we do not have an y direct knowledge of\nthe nature of the electronic states at the Cd 3As2/Py interface. This is a limitation in all the\npublished studies on topological spintronics thus far.\nWe carried out electrical magneto-transport measurements on A s-capped Cd 3As2thin\nﬁlms at room temperature, revealing a Hall resistance with non-linea r dependence on mag-\nnetic ﬁeld, indicative of coexisting electron and hole type of carriers as expected for a chemi-\ncalpotentialclosetothechargeneutralpoint[ 30–32]. Inthelowﬁeldlimit(therangeneeded\nfor ST-FMR and SP experiments), the electrical transport is mainly electron type with a\ncarrier density andmobility ofaround n= 1018−1019cm−3andµ= 4000−6000 cm2V−1s−1.\nexp Ef\ncal Ef\n(a) (b)\n(d) (c)\nFIG. 2. (a) Electronic band structureof Cd 3As2thin ﬁlms obtained by ARPES. Themeasurements\nare taken at T= 300 K along the M−Γ−Mdirection, projected on the (112) surface. (b) Second\nderivative of the spectrum shown in (a). (c) Calculated band structure and spin Hall conductivity\n(SHC) of bulk Cd 3As2The color in the band structure shows the spin Berry curvatur e (Ωz\nxy) with\nzalong the [001] direction. (d) Three independent component s of SHC with zalong the [112]\ndirection as a function of chemical potential.\n6III. THEORETICAL CALCULATION OF SPIN HALL CONDUCTIVITY\nTo obtain a theoretical understanding of the spin-charge interco nversion in Cd 3As2, we\nperformed density-functional theory (DFT) calculations in the fr amework of the generalized\ngradient approximation[ 33] with a full-potential local-orbital minimum-basis code (FPLO)\n[34]. Spin-orbit coupling was included. The crystal structure of Cd 3As2belongs to space\ngroupI41/acd(No.142) and has inversion symmetry [ 35]. From DFT calculations, we pro-\njectedtheBlochwave functionintoatomic-likeWannierfunctionsof Cd-5sorbitalandAs-4 p\norbitals. Then we build an accurate tight-binding Hamiltonian( ˆH) to calculate the intrinsic\nspin Hall conductivity components (Figs. 2(c) and 2(d)) [ 36]. Full details of the calculation\nare given in Appendix A.\nIn addition, our calculation reveals an out-of-plane spin polarization in the spin current\ngenerated by the intrinsic SHC σz\nzx= 16 S·cm−1(/planckover2pi1/e), where spin current direction zrefers\nto the [112] direction. The non-zero σz\nzxis induced by the symmetry breaking along the [112]\naxis. If the intrinsic SHE prevails under certain experimental condit ions, we expect that\nthe out-of-plane polarization may play a signiﬁcant role in the spin-or bit torque. We also\nﬁnd that the SHC has several local maxima around the charge neut ral point. This could\ntentatively explain the variation of the spin-charge interconversio n eﬃciency with changes\nof chemical potential seen in Ref. [ 19,20]. It also oﬀers an opportunity to enhance the\nintrinsic SHC in Cd 3As2thin ﬁlms via chemical doping or electrostatic gating.\nIV. SPIN TORQUE FERROMAGNETIC RESONANCE\nWe now discuss the spin-charge interconversion in Cd 3As2/Py bilayers via ST-FMR. We\nhave measured a total of 12 devices all showing the interconversio n phenomenon. In all these\ndevices except for one, the interface between Cd 3As2and Py has some oxidation due to brief\nambient exposure. In ST-FMR experiments, we apply a radio freque ncy (rf) charge current\nin devices with lateral device dimensions 50 µm×10µm patterned from bilayers of 12 nm\nthick Cd 3As2and 4 and 6 nm thick Py in the presence of an in-plane external magne tic\nﬁeld. This process excites the magnetization dynamics of the ferro magnetic layer by means\nof Oersted ﬁeld and spin current at the interface. The latter resu lts from the rf charge\ncurrent creating a ﬂow of angular momentum perpendicular to the in terface (charge-to-spin\n7conversion) in the DSM (Fig 3(a)). The magnetization dynamics are t hen probed using the\nanisotropic magnetoresistance of the ferromagnetic layer which, along with the rf current,\nproduces a DC mixing voltage ( Vmix) in the device. The in-plane and out-of-plane torques\ncan then be extracted by ﬁtting the Vmixwith a symmetric and antisymmetric Lorentzian\nfunction:\nVmix=S∆2\nCd3As2+A∆Cd3As2(H−HRes)\n∆2\nCd3As2+(H−HRes)2. (1)\nHere, ∆ Cd3As2istheline width oftheabsorptionspectrum, His theapplied external ﬁeld,\nHResis the ﬁeld at which we see the resonance and S(A) is the magnitude of the symmetric\n(antisymmetric) component. The frequency dependence of the a bsorption spectrum is well\nunderstood for thin ﬁlms using the Kittel equation f=γ\n2π/radicalbig\nHRes(HRes+MEff). This can\nbe ﬁtted to our data using the known gyromagnetic ratio of the elec tron (γ), giving eﬀective\nmagnetization values of around MEff= 560 kA /m to 640 kA /m which is similar to values\nreported by FMR studies in Py ﬁlms [ 37].\nIn addition, we use the ratio of the symmetric and antisymmetric com ponents to extract\nthe spin torque eﬃciency deﬁned as the ratio of the rf spin current (Js) generated in the\ndevice and the charge current ( Jc) applied by an external source [ 38,39]:\nξ=2e\n/planckover2pi1Js\nJc=S\nAeµ0MStCd3As2tNiFe\n/planckover2pi1/bracketleftbigg\n1+/parenleftbiggMEff\nHRes/parenrightbigg/bracketrightbigg1/2\n. (2)\nHere,eis the charge of the electron, /planckover2pi1is the reduced Planck constant, µ0is the perme-\nability of free space, MS= 560 kA /m is the saturation magnetization of Py (measured in\na Quantum Design superconducting quantum interference device m agnetometer), tCd3As2is\nthe thickness of the Cd 3As2layer, and tNiFeis the thickness of the ferromagnet.\nFigure 3 (b) shows the mixing voltage of a device at diﬀerent frequen cies ranging from\n3.5 GHz to 5 GHz. The spin torque eﬃciency ξis extracted from the measured signal by\nseparating it into a symmetric and an antisymmetric Lorentzian comp onent, as shown in\nFig. 3 (c). If we assume that the spin current has a bulk origin, we ob tain a spin torque\neﬃciency ξ= 0.27 (0.22) with 6 (4) nm of Py, comparable to values in HM and providing\na lower bound on the spin Hall angle due to the less than ideal spin tran sparency [ 39].\nThe angle dependence (Fig. 3(d)) also shows the expected symmet ry of the in-plane and\nout-of-plane torques from, for example, the bulk spin Hall eﬀect ( SHE) or Rashba-Edelstein\n8eﬀect [10,38,40,41]. It also shows no spin polarization in the out-of-plane or current\nﬂow directions (see Appendix B). Moreover, the symmetric compon ent of the torque shows\nthe same sign as in Pt and opposite to that in W. The spin torque eﬃcien cy can then be\ncombined with the Cd 3As2conductivity ( σxx≈1250 Scm−1) to extract the SHC:\nσSH=/planckover2pi1\n2eθSHσxx/greaterorsimilar140/planckover2pi1\neS\ncm(3)\nAs a ﬁrst step to better understand the intrinsic contributions to the SHC of Cd 3As2,\nwe have studied ST-FMR in a device fabricated from Cd 3As2/Py bilayers grown using a\nfullin vacuo transfer procedure. STEM characterization of such a sample sho wed a clean\nCd3As2/Py interface. The ST-FMR signal is however weaker than in the devic es fabricated\nfrom bilayers with an oxidized interface. Analysis of the ST-FMR data shows that the\nspin torque eﬃciency ( ξ= 0.10) and the SHC ( σSH= 63/planckover2pi1\neS\ncm) are closer to the intrinsic\ntheoretical value (see Appendix D), indicating the possibility that th e interfacial cadmium\noxide layer enhances the extrinsic contributions to the SHC.\nV. SPIN PUMPING\nTocomplement theST-FMRmeasurements ofspin-chargeintercon version, wealsocarried\nout SP measurements on Cd 3As2/Py heterostructures. All the data presented here was\nmeasured using samples with a partially oxidized interface. In these e xperiments, we placed\nthe Cd 3As2/Py heterostructure in a grounded microstrip transmission line. We s tudied the\nFMR of the Py ferromagnet by applying a ﬁxed rf signal in the transm ission line while\nsweeping an external magnetic ﬁeld (Fig. 4(a)) [ 1,2,8,42]. The external magnetic ﬁeld\nexcites the dynamics of the ferromagnet, generating precession of its magnetization which\ncan then be measured as a change in the power absorbed under res onance conditions (Fig.\n4(c)) [1,8,12]. The frequency dependence of this FMR phenomenon is again chara cterized\nby the Kittel equation. As we performed the FMR experiment, we sim ultaneously measured\nthe voltage generated in the Cd 3As2ﬁlm under resonance conditions (Fig 4(b)). This signal\ncan be decomposed into two contributions. The dominant contribut ion is linear in power\n(Fig. 4(d))andchangessignwithmagneticﬁelddirection( VSP). Thesecondarycontribution\nis less than 1 .5µVand does not change sign under ﬁeld reversal (see Appendix C)[ 43]. The\nbehavior of VSP, which we attribute to spin to charge conversion due to spin pumping from\n9/s45/s54/s48/s45/s53/s48/s45/s52/s48/s45/s51/s48/s45/s50/s48/s45/s49/s48/s48/s32/s86/s32/s91/s181/s86/s93\n/s45/s52/s48 /s45/s51/s48 /s45/s50/s48\n/s32/s72/s32/s91/s109/s84/s93/s32/s68/s97/s116/s97\n/s32/s70/s105/s116\n/s32/s86/s115\n/s32/s86/s97\n/s32\n/s32/s45/s49/s48/s48/s45/s53/s48/s48/s53/s48/s49/s48/s48/s32/s86/s32/s91/s181/s86/s93\n/s50/s55/s48 /s49/s56/s48 /s57/s48 /s48 /s45/s57/s48\n/s32    /s91/s100/s101/s103/s93/s32/s83/s121/s109/s109/s101/s116/s114/s105/s99\n/s32/s65/s110/s116/s105/s115/s121/s109/s109/s101/s116/s114/s105/s99\n/s32 /s32/s32/s70/s105/s116(b)\n(c) (d)\n/g77\n50 /g80m\n/g77 T = 300K (a)\nFIG.3. (a)Illustrationofcharge-to-spin conversionduet oST-FMR.(b)MeasuredST-FMRmixing\nvoltage signal of a Py(4nm)/Cd 3As2(12nm) heterostructure with a microwave power of 20 dBm at\ndiﬀerent frequencies (voltage oﬀset has been removed for clar ity). Inset: Optical microscope image\nof the ST-FMR device. (c) ST-FMR spectra of the same device at 4 GHz and 20 dBm showing\nthe measured data, and the ﬁt needed to extract the symmetric (Vs) and antisymmetric (Va)\ncomponents of the torque. (d) Angle dependence of the symmet ric and antisymmetric components\nof the torque ﬁtted with sin(φ)cos(φ)2\nthe ferromagnet is similar to the inverse spin Hall eﬀect (ISHE) in HMs and the inverse\nRashba-Edelstein eﬀect (IREE) in topological insulators and 2D elec tron systems in which\nan electric ﬁeld ( /vectorE) is generated in a direction perpendicular to the spin current ( JS) and\nits spin polarization ( σ), i.e:/vectorE∝/vectorJs×/vector σ[1,8,12,37]. Again, the sign of the voltage signal\nmatched that of Pt, consistent with our ST-FMR results.\nWehavemeasuredSPinseveral heterostructures withCd 3As2ﬁlmthickness rangingfrom\n12 nm to 200 nm (measured by STEM and x-ray reﬂectivity) and a Py t hickness of 30 nm.\n10Additionally, as a control experiment, we performed the same SP me asurement in a sample\nwith just Py and the GaSb buﬀer layer; we did not detect a voltage sig nal under resonance\nconditions. Thus, we are sure that the spin-to-charge conversio n is due to the presence of\nthe Cd 3As2layer. Finally, we use the broadening of the linewidth of the absorptio n spectra\nin the Cd 3As2samples (∆ HCd3As2) and the GaSb samples (∆ HGaSb) to compare the spin\nmixing conductance ( g↑↓) in the two cases [ 1,8,44]:\ng↑↓=2π√\n3MSγtNiFe\ngµBw(∆HCd3As2−∆HGaSb). (4)\nHere,gis the Land´ e factor, µBis the Bohr magneton, and wis the width of our samples\n(ranging between 5 mm and 12 mm). We obtain a spin mixing conductanc e in the range\nbetween 1 .0×1018m−2to 3.5×1018m−2. The diﬀerence in spin mixing conductance\namongst samples might be due to variations in the Py /Cd3As2interface during fabrication.\nWecanfurther compute the spincurrent pumped into the Cd 3As2layer using equation [ 1,8]:\nJS=2e\n/planckover2pi1g↑↓h2\nRF/planckover2pi1ω2/bracketleftig\nγ4πMS+/radicalbig\n(γ4πMS)2+4ω2/bracketrightig\n2π(∆HCd3As2)2[(γ4πMS)2+4ω2]. (5)\nHere,ωis the resonance frequency and hrfis the magnetic ﬁeld generated by the trans-\nmission line which has been calibrated using the ISHE of Pt and the IREE of Bi2Se3. We\nobtain values of JSthat range between 0 .5×104Am−2to 3.4×104Am−2.\nFinally, we compute the eﬃciency of the spin-to-charge conversion by calculating the\ncharge current generated in the ﬁlm by normalizing the measured vo ltage signal by the\nsample width (w) and its resistance ( R). Since DSMs can have both surface and bulk states\n[23], we tentatively describe the eﬃciency of the conversion in terms of bulk spin Hall angle\n(θSH) with spin diﬀusion length ( λSD) and surface inverse Rashba-Edelstein eﬀect with\neﬀective length ( λIREE) [1,8,9].\nVsp\nwR=−/bracketleftbigg\nθSHλSDtanh/parenleftbiggtCd3As2\n2λSD/parenrightbigg\n+λIREE/bracketrightbigg\nJS (6)\nLarge variations in sample fabrication (diﬀerences in roughness, ca rrier density, and\nPy/Cd3As2interface) prevent reliable ﬁts to equation 6. Nevertheless, if we assume just\na bulk contribution with a spin diﬀusion length in Cd 3As2that is smaller than the thickness\nof our thicker samples (200 nm), we can approximate θSHλSD≈0.1 nm−1.2 nm. This\nlarge range might be due to diﬀerences in the interface among sample s.\n11/s52/s48\n/s51/s48\n/s50/s48\n/s49/s48\n/s48/s32/s86/s115/s112/s32/s91/s181/s86/s93\n/s50/s48 /s49/s48\n/s32/s72/s32/s91/s109/s84/s93\n/s32/s100/s73/s47/s100/s72/s32/s91/s97/s46/s117/s46/s93/s32/s86/s115/s112\n/s32/s100/s73/s47/s100/s72/s51/s48\n/s50/s53\n/s50/s48\n/s49/s53\n/s49/s48\n/s53/s32/s74/s67/s32/s91/s110/s65/s32/s109/s45/s49/s93\n/s50/s48/s48 /s49/s53/s48 /s49/s48/s48 /s53/s48\n/s32/s80/s32/s91/s109/s87/s93/s32/s67/s100/s51/s65/s115/s50\n/s32/s80/s116(c) (d)(b)\nT = 300K(a)\nFIG. 4. (a) Illustration of spin-to-charge conversion due t o SP. (b) Voltage signal measured in the\nCd3As2(40 nm)/Py(30 nm) heterostructure at room temperature with an applied microwave power\nof 20 dBm. (c) SP voltage signal ( VSP) and FMR absorption spectra ( dI/dH) measured at 3 GHz\nand 23 dBm. (d) Charge current ( JC) as a function of power for Cd 3As2(40 nm)/Py(30 nm) and\nPy(30 nm)/Pt(20 nm) heterostructures showing linear behav ior.\nVI. DISCUSSION\nOur measurements of spin-charge interconversion at room tempe rature in Py /Cd3As2\nheterostructures reveals ξ≈0.22−0.27,σSH/greaterorsimilar140h/e Scm−1andθSHλSD≈0.1 nm−\n1.2 nm. The spin torque eﬃciency is comparable to the reported values of HMs such as\nβ-W and Pt [ 1,7,12,39]. Diﬀerences in the interface during fabrication makes it diﬃcult to\nreliably compare the results in both experiments. Nevertheless, th is seems to indicate short\n(/lessorsimilar5.5 nm) spin diﬀusion lengths in Cd 3As2. We expect that further improvements will\nallow us to do a systematic thickness dependence study of this phen omenon to determine\n12the SHC and the spin diﬀusion length more accurately.\nThe SHE can be theoretically classiﬁed into intrinsic and extrinsic cont ributions. As\ndiscussed earlier, the intrinsic SHC, which is determined by the spin Be rry curvature in the\nwave function, can be calculated from the ﬁrst-principles band str ucture (see Appendix A).\nConsider x,yin the (112) crystallographic plane and the spin current along z, as in the\nexperimental setup. We obtain the related SHC σy\nzx= 10 S·cm−1(/planckover2pi1/e). This is about one\norder of magnitude smaller than the experimental value ∼140 S·cm−1(/planckover2pi1/e) measured in\nsamples with an imperfect oxidized interface. On the other hand, pr eliminary measurements\nof a clean interface between the DSM and FM reveal a SHC more cons istent with theory.\nThis implies that the SHE in the samples with an imperfect interface may be dominated by\nextrinsic eﬀects. These extrinsic contributions canhave bothsur face andbulkorigins. There\nare trivial surface states andDSM Fermi arcstates. They can ge nerate spin polarizationand\ninduce spin diﬀusion into the NiFe side when current ﬂows on the surfa ce. Because Fermi\narcs originate from the quantum spin Hall edge states, their SHE co ntribution is equivalent\nto the intrinsic bulk SHC due to the bulk-boundary correspondence . The extrinsic bulk\ncontribution is related to the spin-scattering and strong spin-orb it coupling in the sample.\nIn summary, we have carried out systematic measurements of spin -charge interconversion\nin a topological semimetal, Cd 3As2, revealing the importance of large extrinsic contributions\ntothespinHallconductivity thatcanariseatanimperfectinterfac e. Theseresultsshowthat\nimperfect interfaces can sometimes be useful for enhancing spin- charge conversion eﬃciency.\nOur results also raise caution about the interpretation of spin tran sport measurements in\nCd3As2that attribute observations to surface Fermi arcs.\nACKNOWLEDGMENTS\nThe principal support for this project was provided by SMART, one of seven centers of\nnCORE, a Semiconductor Research Corporation program, sponso red by the National In-\nstitute of Standards and Technology (NIST). This supported the synthesis and standard\ncharacterization of Py /Cd3As2heterostructures as well as spin-charge interconversion mea-\nsurements (WY,NS) and their characterization using STEM (JH, SG, AM). Additional sup-\nport for materials synthesis was provided by the Institute for Qua ntum Matter under DOE\nEFRC grant DE-SC0019331 (RX, JC, NS, TM). The Penn State Two- Dimensional Crys-\n13tal Consortium-Materials Innovation Platform (2DCC-MIP) under NSF Grant No. DMR-\n1539916provided support for ARPES measurements (YO, TP, AR, NS). The magnetometry\nmeasurements were carried out by JR, supported by a grant from the University of Chicago.\nPart of this work was carried out in the College of Science and Enginee ring Characteriza-\ntion Facility, University of Minnesota, which has received capital equ ipment funding from\nthe National Science Foundation through the UMN MRSEC under Awa rd Number DMR-\n2011401 (JH, SG, AM). EG acknowledges support for an undergra duate summer internship\nfrom the Oﬃce of Graduate Educational Equity Programs and Eber ly College of Science at\nthe Pennsylvania State University. Certain commercial equipment, instruments, or materi-\nals (or suppliers, or software, ...) are identiﬁed in this paper to foste r understanding. Such\nidentiﬁcation does not imply recommendation or endorsement by the National Institute of\nStandards and Technology, nor does it imply that the materials or eq uipment identiﬁed are\nnecessarily the best available for the purpose.\n14Appendix A: Intrinsic spin Hall eﬀect calculations\nThe spin current JSk\nigenerated by electric ﬁeld− →Evia spin Hall conductivity (JSk\ni=\nσk\nijEj), where JSk\niﬂows along idirection when spin polarization along kdirection. We\nevaluate the spin Hall conductivity (SHC) ( σk\nij) by the Kubo-formula approach in the linear\nresponse scheme[ 36],\nσk\nij=e\n/planckover2pi1/summationdisplay\nn/integraldisplay\nBZd3k\n(2π)3fn(k)Ωk\nn,ij(k) (A1)\nΩk\nn,ij(k) = 2i/planckover2pi12/summationdisplay\nm/negationslash=n∝angbracketleftun(k)|ˆjk\ni|um(k)∝angbracketright∝angbracketleftum(k)|ˆvj|un(k)∝angbracketright\n(En(k)−Em(k))2(A2)\nHere,ǫnis the eigenvalue of the |n∝angbracketrighteigenstate, and ˆ vi=dˆH\n/planckover2pi1dki(i=x,y,z) is the velocity\noperator, fnis the Fermi-Dirac distribution. ˆjj\niis the spin current operator which is related\nto the velocity operator(ˆ vi) and spin operator(ˆ sj) asˆjj\ni= [ˆvi,ˆsj] Ak-point of grid of 200 ×\n200×200 is used for the numerical integration in Equation A1.\nWe set the x,y,zaxes as the crystallographic a,b,caxes, respectively. According to the\nspacegroupsymmetryandtime-reversalsymmetry, the σk\nijtensorhasonlythreeindependent\nmatrix elements, σz\nxy,σy\nzxandσx\nyz, as shown in Table 1. SHC depends sensitively on the\nchemical potential, as shown in Fig. 5.\nThe DFT band structure is shown in Fig. 5. The Dirac point appears at kz=±kD\nbetween Γ and Z, well consistent with previous work. The Dirac bands contribute ma inly\nto theσz\nxycomponent. This is consistent with the fact that the kz= 0 plane between two\nDirac points is a quantum spin Hall insulator. If we approximate all kzplanes between two\nDirac points are quantum spin Hall states, then we can estimate the 3D SHC by,\nσz\nxy=kD\nkZ(1\nc/2)2e2\nh(/planckover2pi1/2\ne) (A3)\nwherekZis the Γ−Zdistance in the Brillouin zone,2e2\nhis the conductance quantum, c\nis the lattice parameter of the tetragonal unitcell, and/planckover2pi1/2\neconverts the dimension from the\ncharge current to the spin current. By takingkD\nkZ= 14 % from the band structure and lattice\nparameter c= 25.4˚A, we obtain σz\nxy≈40 S·cm−1(/planckover2pi1/e). This is actually very close to the\nnumerical value in Table 1.\nWhen rotating the zaxis to the [112] crystallographic axis, which is approximately the\n[111] direction in the Cartesian’s coordinate. We can transfrom the SHC tensor by the\n15unitary rotation [ 45],\nσ[112]k\nij=/summationdisplay\nl,m,nDilDjmDknσ[001]n\nlm(A4)\nwhereDlmis the rotationmatrix element that rotates the zaxis to the [112] crystallographic\naxis. The transformed SHC matrix is shown in Tables 1 and 2.\nWhenx,yalign in the (112) crystallographic plane, the new x,y,zaxes are not high-\nsymmetry lines as those before. Therefore, new matrix elements e merge in the SHC. When\ncharge ﬂows in the x,yplane and spin ﬂows along zas the experiment setup, σy\nzx,σz\nzx,σx\nzy\nare the relevant SHC matrix elements. Here, σy\nzx= 10 S·cm−1(/planckover2pi1/e) is about one order of\nmagnitude smaller than the experimental SHC, 140 S ·cm−1(/planckover2pi1/e). This suggests that the\nmain contributions to the observed SHE are related to extrinsic eﬀe cts. In addition, σz\nzx\nindicates the existence of an out-of-plane spin component in the sp in current. Suppose the\nintrinsic SHE prevails under certain experimental condition, we expe ct that the out-of-plane\npolarization may play a signiﬁcant role in the spin-orbit torque.\nFig. 5. Band structure, spin Berry curvature and spin Hall conductivity a long the (001)\ndirection. Calculated band structure and spin hall conductivity of b ulk Cd 3As2. The color\nin the band structure shows the spin Berry curvature (Ωz\nxy). The right panel presents three\nindependent values of spin Hall conductivity calculated in the ∝angbracketleft001∝angbracketrightdirection as a function\nof chemical potential.\n16TABLE I. Calculated spin Hall conductivity of Cd 3As2at the charge neutral point. The SHC is\nin the unit of ( /planckover2pi1/e)(S/cm).\nσxσyσz\nMagnetic Laue group\n4/mmm z[001]\n0 0 0\n0 0−σy\nxz\n0−σy\nzx0\n\n0 0σy\nxz\n0 0 0\nσy\nzx0 0\n\n0σz\nxy0\n−σz\nxy0 0\n0 0 0\n\nCalculation\nz[001]\n0 0 0\n0 0−1\n0 8 0\n\n0 0 1\n0 0 0\n−8 0 0\n\n0 40 0\n−40 0 0\n0 0 0\n\nCalculation\nz[112]\n0 0 0\n0−3−6\n0 3 3\n\n0−19−26\n23 0 0\n24 0 0\n\n0 13 19\n−8 0 0\n−23 0 0\n\nTABLE II. Calculated spin Hall conductivity of Cd 3As2at the experimental Fermi level (0.2 eV\nabove the charge neutral point). The SHC is in the unit of ( /planckover2pi1/e)(S/cm).\nσxσyσz\nMagnetic Laue group\n4/mmm z[001]\n0 0 0\n0 0−σy\nxz\n0−σy\nzx0\n\n0 0σy\nxz\n0 0 0\nσy\nzx0 0\n\n0σz\nxy0\n−σz\nxy0 0\n0 0 0\n\nCalculation\nz[001]\n0 0 0\n0 0 12\n0−32 0\n\n0 0−12\n0 0 0\n32 0 0\n\n0−1 0\n1 0 0\n0 0 0\n\nCalculation\nz[112]\n0 0 0\n0 10 25\n0−19−10\n\n0 6−3\n−15 0 0\n10 0 0\n\n0 8−6\n−20 0 0\n16 0 0\n\n17Appendix B: Analysis of spin polarization in ST-FMR measurements\nFollowing the procedure sketched in [ 41], we can study the direction of the spin polariza-\ntionofthe spincurrent in Cd 3As2byperforming anangledependent ST-FMR measurement.\nDiﬀerent polarizations will contribute to diﬀerent functional forms of the spin current (Fig.\n6 (a)):\nVSx∝sin(φ)sin(2φ), VSy∝cos(φ)sin(2φ),VSz∝sin(2φ) (B1)\nWe ﬁt the symmetric and antisymmetric ST-FMR data with a spin polariz ationalong the\ny axis and along all three axes (Fig. 6 (c) and (d)). The two ﬁts do no t show any signiﬁcant\ndiﬀerences. In the latter case, the amplitudes of the voltage signa l with spin polarization\nalong the xandzaxis (VSxandVSz) are less than 5% the value with spin polarization along\ntheyaxis (VSy). Thus, we are conﬁdent that the direction of the spin polarization of the\nmeasured torque is in the plane of the sample and transverse to the current direction.\n-1.0-0.50.00.51.0Amplitude [A.U.]\n350300250200150100500\nAngle [deg] Sx\n Sy\n Sz\n-100x10-6-50050100VAsym [V]\n350300250200150100500\nAngle [deg] Asym\n fitSy\n fitSySxSz-60x10-6-40-2002040Vsym [V]\n350300250200150100500\nAngle [deg] Sym\n fitSy\n fitSySxSz\n50 /g80m\nXY\nZ(a) (b)\n(c)\n(d)\nFig. 6. (a) Theoretical angle dependent functional form of the ST-FMR s ignal with spin\npolarization along diﬀerent axis. (b) Image of one of our samples sho wing the coordinate\n18axis. Angle dependence of the symmetric (b) and antisymmetric com ponents of the torque\nﬁtted with spin polarization along the y axis (red) and all axes (black) .\nAppendix C: Symmetric and antisymmetric components of the spin pumping signal\nAs mentioned in the main text, the voltage signal measured in our sam ples (Fig. 7(a))\ncan be decomposed in an anstisymmetric signal that changes sign un der ﬁeld reversal ( Vasym\nshown in Fig. 7(b)) and a symmetric one that does not ( Vsymshown in Fig. 7(c)). When\nwe increased the power in the transmission line, we found that Vasymincreases linearly\nas expected from ferromagnetic driven spin pumping into the Cd 3As2layer from the NiFe\nferromagnet. The symmetric signal has an amplitude of less than 4% of the asymmetric one\nand is non-linear with power. This is consistent with a possible Seebeck eﬀect induced by\nthe microstrip not being perfectly centered over the sample [ 43].\n/s45/s52/s48/s45/s50/s48/s48/s50/s48/s52/s48/s32/s32/s86/s32/s91/s181/s86/s93\n/s50/s48 /s49/s48 /s48 /s45/s49/s48 /s45/s50/s48\n/s32/s72/s32/s91/s109/s84/s93/s32/s50/s51/s32/s100/s66/s109\n/s32/s50/s49/s32/s100/s66/s109\n/s32/s49/s57/s32/s100/s66/s109\n/s32/s49/s55/s32/s100/s66/s109\n/s32/s49/s53/s32/s100/s66/s109\n/s45/s49/s46/s50/s45/s48/s46/s56/s45/s48/s46/s52/s48/s46/s48/s32/s86/s115/s121/s109/s32/s91/s181/s86/s93\n/s50/s48/s48 /s49/s53/s48 /s49/s48/s48 /s53/s48\n/s32/s80/s32/s91/s109/s87/s93/s52/s48\n/s51/s48\n/s50/s48\n/s49/s48/s32/s86/s97/s115/s121/s109 /s32/s91/s181/s86/s93\n/s50/s48/s48 /s49/s53/s48 /s49/s48/s48 /s53/s48\n/s32/s80/s32/s91/s109/s87/s93(a)\n(b) (c)\nFig. 7.(a) Voltage signal measured in a Py(30 nm) /Cd3As2(40 nm) heterostructure as we\nsweepthemagneticﬁeld. Magnitudeoftheantisymmetric(b)andsy mmetric(c)components\nof the voltage signal at resonance condition as a function of power .\nAppendix D: Study of the Cd3As2/Py interface\nIn the samples presented in the main text, atomic resolution high-an gle annular dark-\nﬁeld (HAADF) scanning transmission electron microscopy (STEM) an d energy dispersive\n19X-ray spectroscopy (EDX) of the Cd 3As2/Py heterostructures show a 1 nm to 1.5 nm thick\namorphous CdOx layer at the interface (Fig. 8). This probably form s during the ambient\ncondition transfer of samples from the MBE chamber to the metal e vaporation chamber.\nFig. 8.STEM-EDX elemental map of the Cd 3As2/Py interface. (a) HAADF-STEM image\nand elemental maps of Ni, Fe, O, Cd, and As. Scale bar is 10 nm. (b) Co ncentration of\nthe elements in (a) across the interface. Note the ∼1 nm amorphous layer of CdO xat the\ninterface\nTo avoid the oxidation eﬀect at the Cd 3As2/Py interface, we have fabricated ST-FMR\ndevices using a full in vacuum transfer procedure with a vacuum suit case. Diﬀerences in\nsample holder conﬁgurations that allowed the compatibility between c hambers seemed to\nincrease the roughness of the Cd 3As2layer which increased the noise level in our measure-\nments. Nevertheless as seen in Fig. 9, we have successfully been ab le to remove the oxide\nlayer at the interface. Moreover, we have been able to measure an ST-FMR signal that\nclearly shows resonance up to 8 GHz (Fig. 10). This allowed us to comp ute the spin torque\neﬃciency ξ= 0.10 and SHC σSH/greaterorsimilar63/planckover2pi1\neS\ncmof this device. This experimental value is closer\nto the intrinsic theoretical one and seems to indicate the possibility t hat the CdO xlayer en-\nhances the extrinsic spin Hall eﬀect. Furthermore, the data corr oborates that the measured\nsignal is due to charge to spin conversion in the Cd 3As2/Py interface as presented in the\nmain text.\n20Fig. 9. (a) HAADF-STEM image of the top layers of the device. The scale bar is 10\nnm. (b) Concentration of the elements in (a) across the device inte rface layers shown. (c)\nElemental maps of O, Al, Ni, Fe, As, Cd, Ga and Sb. Scale bar is 10 nm.\n0.7\n0.6\n0.5\n0.4\n0.3\n0.2\n0.1\n0.0Vmix [mV]\n150 100 50 0 -50 -100 -150\nField [mT]3 GHz\n4 GHz\n5 GHz\n6 GHz\n7 GHz\n8 GHz\nFig. 10. ST-FMR mixing voltage measured at diﬀerent frequencies in a complet elyinvacuo\n21grown heterostructure.\nAppendix E: PNR measurements in Cd3As2/Pyheterostructures\nTo examine the possibility of proximity-induced magnetism in the Cd 3As2layer caused\nby the adjacent ferromagnetic Py layer, we performed room-tem perature polarized neutron\nreﬂectometry (PNR) measurements at the PBR instrument at the NIST Center for Neu-\ntron Research (NCNR). The incident neutrons ( λ= 4.75˚A−1) were spin polarized parallel\nor antiparallel to the 2 T applied in-plane magnetic ﬁeld, and the specu lar reﬂectivity was\nmeasured as a function of Q, the momentum transfer vector along the ﬁlm normal direction.\nThe spin-dependent neutron reﬂectivity is sensitive to the nuclear and magnetic scattering\nlength density (SLD) depth proﬁles, so that depth-resolved infor mation on the structure and\nmagnetization may be extracted through ﬁtting. Since the 2 T in-pla ne ﬁeld is more than\nsuﬃcient to saturate the Py magnetization, no in-plane component of the magnetization\nis expected to be perpendicular to the applied ﬁeld. Therefore, only the non-spin ﬂip re-\nﬂectivity cross sections, sensitive to the in-plane magnetization pa rallel to the applied ﬁeld,\nwere collected while the spin-ﬂip cross sections, sensitive only to the in-plane magnetiza-\ntion perpendicular to the applied ﬁeld, were not. We reduced and ana lyzed the data using\nthe Reductus and Reﬂ1D software programs, respectively [ 46,47]. Parameter uncertainties\nwere estimated using a Markov chain Monte Carlo (MCMC) method as im plemented in the\nDREAM algorithm of the BUMPS python package.\nThe best ﬁt to the data, shown in Figure 11, was generated using a m odel which allowed\nfor a proximity-induced magnetization of varying thickness in the Cd 3As2at the interface\nwith the Py layer. However, the best ﬁt to the data yields no induced magnetization in the\nCd3As2at the interface, indicating a lack of proximity-induced magnetizatio n within mea-\nsurement sensitivity. Since the thickness and magnetization of a pr oximity-magnetized layer\nare often highly coupled parameters, uncertainty analysis was per formed using a modiﬁed\nmodel in which the proximity-magnetized layer thickness was ﬁxed at a typical value of 2\nnm. Inthis case, DREAManalysis indicates an extremely small upper lim it (95%conﬁdence\ninterval) of 10 kA/m. The model shown does not incorporate the Cd Oxlayer observed in\nSTEM imaging, as modeling which incorporated this layer did not yield a no tably diﬀerent\nﬁt from those that did not. This will be understood given that the ex tremely thin 1 nm\n22layer yields oscillations with a periodicity of approximately 6 .28 nm−1in Q, far outside the\nmeasured range. Further, the expected CdO xthickness is the same order of magnitude as\nthe ﬁtted roughness at the Cd 3As2/Py interface. Lastly, we note that the theoretical nuclear\nSLD (4.065×10−4nm−2) of CdO xis between that of Cd 3As2and Py. Taken together,\nthese factors suggest that the structural features associat ed with the CdO xlayer are likely\nto be extremely muted. Rather, we propose that the CdO xlayer will blend into the Cd 3As2\ninterface and will likely instead manifest in the SLD proﬁle as a small mag netically dead\nlayer inthePyattheCd 3As2interface. Indeed, incorporatingsuch amagneticallydeadlayer\nin Py near the Cd 3As2interface into the model yields a thickness of 1.04 nm ±0.07 nm,\nin excellent agreement with the STEM and EELS. PNR measurements t herefore indicate a\nlack of proximity-induced magnetization at room-temperature and conﬁrm the presence of\na nonmagnetic oxide-like layer at the interface.\n0.2 0.4 0.6 0.8 \nQ  (nm−1)10−1100101102R/RGaAs\n(a) ↑↑\n↓↓\n0.2 0.4 0.6 0.8 \nQ  (nm−1)−0.5 0.0 0.5 1.0 Spin Asymmetry (b) \n0 25 50 75 100 125 150 \nZ (nm) 02468SLD (10−4 nm−2)\n(c)\nGaAs GaSb \nCd3As2\nNiFe \nAlOXSLDNuclear\nSLDMagnetic\n0500 1000 1500 2000 2500 3000 \nMagnetization (kA/m) (a) \n(b) (c) \nFig. 11 (a) Spin-dependent PNR data from a Cd 3As2/Py bilayer at room-temperature in\nan applied ﬁeld of 2 T alongside a theoretical ﬁt. We show the Fresnel reﬂectivity, where the\nmeasured reﬂectivity is normalized by the Q-dependent theoretica l reﬂectivity of the GaAs\nsubstrate. (b) Measured spin asymmetry, deﬁned as (R↑↑- R↓↓)/(R↑↑+ R↓↓) alongside\ntheoretical ﬁt. (c) Best ﬁt nuclear and magnetic scattering lengt h density (SLD) proﬁles\nused to generate the ﬁts shown. The right axis shows the magnetiz ation which is equivalent\nto a given magnetic SLD value.\n23Appendix F: Summary of the samples measured in this study\nAs mentioned in the main text, we have measured spin and charge inte rconversion in 12\ndevices. Table III shows the list of the samples used in this study and the measurements\nperformed.\nTABLE III. Samples used in this study indicating the nominal thickness of the Cd 3As2and Py\nlayers and the measurement performedto study thespin-char ge interconversion. Samples with an *\nwere used to calibrate the deposition rate via transmision e lectron microscopy and x-ray reﬂectivity\n(XRR).\nSample\nnumberCd3As2\nthickness (nm)Py\nthickness (nm)Measurement\nperformed\n190130A 40 30 SP\n190509A* 50 30 SP, TEM, XRR\n190510A* 125 30 SP, TEM, XRR\n190612A 50 30 SP\n190703A 75 30 SP\n190718A 150 30 SP\n190719A 200 30 SP\n190814A 80 4 ST-FMR\n190929A 20 4 ST-FMR,\n191002B* 19 4 ST-FMR, XRR\n191127B* 12 4, 5, 6 SP, ST-FMR, XRR\n201125A* 8 7 ST-FMR, TEM\n24[1] O. Mosendz, V. Vlaminck, J. E. Pearson, F. Y. Fradin, G. E. W. Bauer, S. D. Bader, and\nA. Hoﬀmann, Detection and quantiﬁcation of inverse spin Hall eﬀect from spin pumping in\npermalloy/normal metal bilayers ,Phys. Rev. B 82, 214403 (2010) .\n[2] V. Vlaminck, J. E. Pearson, S. D. Bader, and A. Hoﬀmann, Dependence of spin-\npumping spin Hall eﬀect measurements on layer thicknesses a nd stacking order ,\nPhys. Rev. B 88, 064414 (2013) .\n[3] I. Mihai Miron, G. Gaudin, S. Auﬀret, B. Rodmacq, A. Schuhl , S. Pizzini, J. Vogel, and\nP. Gambardella, Current-driven spin torque induced by the Rashba eﬀect in a fe rromagnetic\nmetal layer ,Nature Materials 9, 230 (2010) .\n[4] L.Liu, C.-F. Pai, Y.Li, H.W.Tseng, D.C.Ralph,andR.A. B uhrman, Spin-Torque Switching\nwith the Giant Spin Hall Eﬀect of Tantalum ,Science336, 555 (2012) .\n[5] Y. Ou, D. C. Ralph, and R. A. Buhrman, Strong Enhancement of the Spin Hall Eﬀect by Spin\nFluctuations near the Curie Point of FexPt1−xAlloys,Phys. Rev. Lett. 120, 097203 (2018) .\n[6] A. Hoﬀmann, Spin Hall Eﬀects in Metals ,IEEE Transactions on Magnetics 49, 5172 (2013) .\n[7] C.-F. Pai, L. Liu, Y. Li, H. W. Tseng, D. C. Ralph, and R. A. B uhrman,\nSpin transfer torque devices utilizing the giant spin Hall e ﬀect of tungsten ,\nApplied Physics Letters 101, 122404 (2012) .\n[8] H. Wang, J. Kally, J. S. Lee, T. Liu, H. Chang, D. R. Hickey, K. A.\nMkhoyan, M. Wu, A. Richardella, and N. Samarth, Surface-State-Dominated Spin-\nCharge Current Conversion in Topological-Insulator–Ferromag netic-Insulator Heterostruc-\ntures,Phys. Rev. Lett. 117, 076601 (2016) .\n[9] H. Wang, J. Kally, C. S ¸ahin, T. Liu, W. Yanez, E. J. Kamp, A . Richardella, M. Wu, M. E.\nFlatt´ e, and N. Samarth, Fermi level dependent spin pumping from a magnetic insulato r into\na topological insulator ,Phys. Rev. Research 1, 012014 (2019) .\n[10] A. R. Mellnik, J. S. Lee, A. Richardella, J. L. Grab, P. J. Mintun, M. H. Fischer, A. Vaezi,\nA. Manchon, E.-A. Kim, N. Samarth, and D. C. Ralph, Spin-transfer torque generated by a\ntopological insulator ,Nature511, 449 (2014) .\n[11] K.Kondou, R.Yoshimi, A.Tsukazaki, Y.Fukuma, J.Matsu no, K.S.Takahashi, M.Kawasaki,\nY. Tokura, and Y. Otani, Fermi-level-dependent charge-to-spin current conversio n by Dirac\n25surface states of topological insulators ,Nature Physics 12, 1027 (2016) .\n[12] Y. Wang, R. Ramaswamy, and H. Yang, FMR-related phenomena in spintronic devices ,\nJournal of Physics D: Applied Physics 51, 273002 (2018) .\n[13] Y. Sun, Y. Zhang, C. Felser, and B. Yan, Strong intrinsic spin Hall eﬀect in the TaAs family\nof Weyl semimetals ,Phys. Rev. Lett. 117, 146403 (2016) .\n[14] A. Johansson, J. Henk, and I. Mertig, Edelstein eﬀect in Weyl semimetals ,\nPhys. Rev. B 97, 085417 (2018) .\n[15] S. S.-L. Zhang, A. A. Burkov, I. Martin, and O. G. Heinone n,Spin-to-Charge Conversion in\nMagnetic Weyl Semimetals ,Phys. Rev. Lett. 123, 187201 (2019) .\n[16] D. MacNeill, G. M. Stiehl, M. H. D. Guimaraes, R. A. Buhrm an, J. Park, and D. C.\nRalph,Control of spin-orbit torques through crystal symmetry in WTe2/ferromagnet bilay-\ners,Nature Physics 13, 300 (2016) .\n[17] D. MacNeill, G. M. Stiehl, M. H. D. Guimar˜ aes, N. D. Reyn olds, R. A. Buhrman,\nand D. C. Ralph, Thickness dependence of spin-orbit torques generated by WTe2,\nPhys. Rev. B 96, 054450 (2017) .\n[18] S. Shi, S. Liang, Z. Zhu, K. Cai, S. D. Pollard, Y. Wang, J. Wang, Q. Wang,\nP. He, J. Yu, G. Eda, G. Liang, and H. Yang, All-electric magnetization\nswitching and Dzyaloshinskii-Moriya interaction in WTe2/ ferromagnet heterostructures ,\nNature Nanotechnology 14, 945 (2019) .\n[19] B.-C. Lin, S. Wang, A.-Q. Wang, Y. Li, R.-R. Li, K. Xia, D. Yu, and Z.-M. Liao, Elec-\ntric Control of Fermi Arc Spin Transport in Individual Topolog ical Semimetal Nanowires ,\nPhys. Rev. Lett. 124, 116802 (2020) .\n[20] A.-Q. Wang, P.-Z. Xiang, X.-G. Ye, W.-Z. Zheng, D. Yu, an d Z.-M. Liao, Room-Temperature\nManipulation of Spin Texture in a Dirac Semimetal ,Phys. Rev. Applied 14, 054044 (2020) .\n[21] H. Xu, J. Wei, H. Zhou, J. Feng, T. Xu, H. Du, C. He, Y. Huang , J. Zhang, Y. Liu,\nH.-C. Wu, C. Guo, X. Wang, Y. Guang, H. Wei, Y. Peng, W. Jiang, G . Yu, and\nX. Han, High Spin Hall Conductivity in Large-Area Type-II Dirac Semim etal PtTe2 ,\nAdvanced Materials 32, 2000513 (2020) ,.\n[22] B.-C. Lin, S. Wang, A.-Q. Wang, Y. Li, R.-R. Li, K. Xia, D. Yu, and Z.-M. Liao, Elec-\ntric Control of Fermi Arc Spin Transport in Individual Topolog ical Semimetal Nanowires ,\nPhys. Rev. Lett. 124, 116802 (2020) .\n26[23] S.-Y. Xu, C. Liu, S. K. Kushwaha, R. Sankar, J. W. Krizan, I. Belopolski, M. Neupan,\nG. Bian, N. Alidoust, T.-R. Chang, H.-T. Jeng, C.-Y. Huang, W .-F. Tsai, H. Lin, P. P.\nShibayev, F. Chou, R. J. Cava, and M. Z. Hasan, Observation of Fermi arc surface states in\na topological metal ,Science347, 294 (2015) .\n[24] S. Shi, Y. Ou, S. V. Aradhya, D. C. Ralph, and R. A. Buhrman ,Fast Low-Current Spin-\nOrbit-Torque Switching of Magnetic Tunnel Junctions throu gh Atomic Modiﬁcations of the\nFree-Layer Interfaces ,Phys. Rev. Applied 9, 011002 (2018) .\n[25] S. Fukami and H. Ohno, Magnetization switching schemes for nanoscale three-term inal spin-\ntronics devices ,Japanese Journal of Applied Physics 56, 0802A1 (2017) .\n[26] T. Schumann, M. Goyal, H. Kim, and S. Stemmer, Molecular beam epitaxy of Cd3As2 on a\nIII-V substrate ,APL Materials 4, 126110 (2016) .\n[27] Z. K. Liu, J. Jiang, B. Zhou, Z. J. Wang, Y. Zhang, H. M. Wen g, D. Prabhakaran, S.-K.\nMo, H. Peng, P. Dudin, T. Kim, M. Hoesch, Z. Fang, X. Dai, Z. X. S hen, D. L. Feng,\nZ. Hussain, and Y. L. Chen, A stable three-dimensional topological Dirac semimetal Cd3 As2,\nNature Materials 13, 677 (2014) .\n[28] S. Borisenko, Q. Gibson, D. Evtushinsky, V. Zabolotnyy , B. B¨ uchner, and\nR. J. Cava, Experimental Realization of a Three-Dimensional Dirac Semi metal,\nPhys. Rev. Lett. 113, 027603 (2014) .\n[29] H. Yi, Z. Wang, C. Chen, Y. Shi, Y. Feng, A. Liang, Z. Xie, S . He, J. He, Y. Peng,\nX. Liu, Y. Liu, L. Zhao, G. Liu, X. Dong, J. Zhang, M. Nakatake, M. Arita, K. Shi-\nmada, H. Namatame, M. Taniguchi, Z. Xu, C. Chen, X. Dai, Z. Fan g, and X. J.\nZhou,Evidence of Topological Surface State in Three-Dimensional Dirac Semimetal Cd3As2 ,\nScientiﬁc Reports 4, 6106 (2014) .\n[30] Y. Liu, C. Zhang, X. Yuan, T. Lei, C. Wang, D. Di Sante, A. N arayan, L. He, S. Picozzi,\nS. Sanvito, R. Che, and F. Xiu, Gate-tunable quantum oscillations in ambipolar Cd3As2 thin\nﬁlms,NPG Asia Materials 7, e221 (2015) .\n[31] P. Cheng, C. Zhang, Y. Liu, X. Yuan, F. Song, Q. Sun, P. Zho u, D. W.\nZhang, and F. Xiu, Thickness-dependent quantum oscillations in Cd3As2thin ﬁlm s,\nNew Journal of Physics 18, 083003 (2016) .\n[32] C.-Z. Li, J.-G. Li, L.-X. Wang, L. Zhang, J.-M. Zhang, D. Yu, and Z.-M. Liao, Two-Carrier\nTransport Induced Hall Anomaly and Large Tunable Magnetores istance in Dirac Semimetal\n27Cd3As2 Nanoplates ,ACS Nano 10, 6020 (2016) .\n[33] J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized Gradient Approximation Made Simple ,\nPhys. Rev. Lett. 77, 3865 (1996) .\n[34] K. Koepernik and H. Eschrig, Full-potential nonorthogonal local-orbital minimum-bas is band-\nstructure scheme ,Physical Review B 59, 1743 (1999) .\n[35] M. N. Ali, Q. Gibson, S. Jeon, B. B. Zhou, A. Yazdani, and R . J. Cava, The crystal\nand electronic structures of Cd3As2, the three-dimensional electronic analogue of graphene ,\nInorganic chemistry 53, 4062 (2014) .\n[36] J. Sinova, S. O. Valenzuela, J. Wunderlich, C. Back, and T. Jungwirth, Spin hall eﬀects ,\nReviews of Modern Physics 87, 1213 (2015) .\n[37] E. Lesne, Y. Fu, S. Oyarzun, J. C. Rojas-S´ anchez, D. C. V az, H. Naganuma, G. Sicoli, J.-P.\nAttan´ e, M. Jamet, E. Jacquet, J.-M. George, A. Barth´ el´ em y, H. Jaﬀr` es, A. Fert, M. Bibes,\nand L. Vila, Highly eﬃcient and tunable spin-to-charge conversion thro ugh Rashba coupling at\noxide?interfaces ,Nature Materials 15, 1261 (2016) .\n[38] L.Liu, T.Moriyama, D. C.Ralph,andR.A.Buhrman, Spin-Torque Ferromagnetic Resonance\nInduced by the Spin Hall Eﬀect ,Phys. Rev. Lett. 106, 036601 (2011) .\n[39] C.-F. Pai, Y. Ou, L. H. Vilela-Le˜ ao, D. C. Ralph, and R. A . Buhrman, Dependence of\nthe eﬃciency of spin Hall torque on the transparency of Pt/fe rromagnetic layer interfaces ,\nPhys. Rev. B 92, 064426 (2015) .\n[40] Y. Ou, C.-F. Pai, S.Shi, D. C.Ralph, andR. A.Buhrman, Origin of ﬁeldlike spin-orbit torques\nin heavy metal/ferromagnet/oxide thin ﬁlm heterostructur es,Phys. Rev. B 94, 140414 (2016) .\n[41] Y. Ou, Z. Wang, C. S. Chang, H. P. Nair, H. Paik, N. Reynold s, D. C. Ralph, D. A. Muller,\nD. G. Schlom, andR. A. Buhrman, Exceptionally High, Strongly Temperature Dependent, Spin\nHall Conductivity of SrRuO3 ,Nano Letters 19, 3663 (2019) .\n[42] M. Jamali, J. S. Lee, J. S. Jeong, F. Mahfouzi, Y. Lv, Z. Zh ao, B. K. Nikolic, K. A.\nMkhoyan, N. Samarth, and J.-P. Wang, Giant Spin Pumping and Inverse Spin Hall Eﬀect\nin the Presence of Surface and Bulk Spin-Orbit Coupling of Top ological Insulator Bi2Se3 ,\nNano Letters 15, 7126 (2015) .\n[43] Y. Shiomi, K. Nomura, Y. Kajiwara, K. Eto, M. Novak, K. Se gawa, Y. Ando, and\nE. Saitoh, Spin-Electricity Conversion Induced by Spin Injection into To pological Insulators ,\nPhys. Rev. Lett. 113, 196601 (2014) .\n28[44] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Enhanced Gilbert Damping in Thin Ferro-\nmagnetic Films ,Phys. Rev. Lett. 88, 117601 (2002) .\n[45] M. Seemann, D. Koedderitzsch, S. Wimmer, and H. Ebert, Symmetry-imposed shape of linear\nresponse tensors ,Phys. Rev. B 92, 155138 (2015) .\n[46] B. Maranville, W. Ratcliﬀ II, and P. Kienzle, Reductus: a stateless Python data reduction\nservice with a browser front end ,Journal of Applied Crystallography 51, 1500 (2018) .\n[47] B. Kirby, P. Kienzle, B. Maranville, N. Berk, J. Krycka, F. Heinrich, and C. Majkrzak, Phase-\nsensitive specular neutron reﬂectometry for imaging the na nometer scale composition depth\nproﬁle of thin-ﬁlm materials ,Current Opinion in Colloid and Interface Science 17, 44 (2012) .\n29"    },    {        "title": "1505.02467v1.An_electromechanical_Ising_machine.pdf",        "content": "An electromechanical Ising machine\nI. Mahboob\u0003and H. Yamaguchi\nNTT Basic Research Laboratories, NTT Corporation, Atsugi-shi, Kanagawa 243-0198, Japan\nSolving intractable mathematical problems in simulators composed of atoms, ions, photons or elec-\ntrons has recently emerged as a subject of intense interest. Here we extend this concept to phonons\nthat are localised in spectrally pure resonances in an electromechanical system which enables their\ninteractions to be exquisitely fashioned via electrical means. We harness this platform to emulate\nthe Ising Hamiltonian whose spin 1/2 particles are replicated by the phase bistable vibrations from\na parametric resonance where multiple resonances play the role of a spin bath. The coupling be-\ntween the mechanical pseudo spins is created by generating thermomechanical two-mode squeezed\nstates which impart correlations between resonances that can imitate a ferromagnetic, random or\nan anti-ferromagnetic state on demand. These results suggest an electromechanical simulator for\nthe Ising Hamiltonian could be built with a large number of spins with multiple degrees of coupling,\na task that would overwhelm a conventional computer.\nElectro/optomechanical systems consisting of a high\nquality mechanical resonance embedded in an electri-\ncal/optical transduction circuit [1{4] have emerged as\nversatile platform in which sensors with unprecedented\nsensitivities can be developed [5{7], in which mechan-\nical nonlinearities can be dynamically engineered and\nharvested [8{10] and even quantum mechanics within a\nmacroscopic context can be studied [11{13]. However\nthis unprecedented success has had the unfortunate con-\nsequence of sti\ring further advancement of mechanical\nsystems beyond these extensively explored concepts.\nMeanwhile quantum simulators have been proposed\nas devices which could be used to solve problems in\nquantum physics that are beyond the capacity of clas-\nsical computers [14]. This notion can also be applied to\nmathematical problems for example the Ising Hamilto-\nnian used to describe a spin glass which is characterised\nas an NP-hard problem that cannot be solved via conven-\ntional computing protocols [15]. However in spite of this\nformidable challenge, the ground state spin con\fguration\nof the Ising Hamiltonian could yield solutions to optimi-\nsation problems that can be mapped on to its spin lattice\nand thus its e\u000ecient extraction is highly desired [16]. One\napproach to rapidly determining the ground state of the\nIsing Hamiltonian is to employ quantum annealing where\nquantum tunnelling is harnessed to search the energy\nlandscape corresponding to a given problem programmed\ninto the underlying spin con\fguration [17{19]. Recently\nan alternative and apparently classical approach to this\nproblem has emerged where a time multiplexed optical\nparametric resonator network is programmed with an\nIsing Hamiltonian composed of four spins. The ground\nstate is then determined by simply activating the net-\nwork which preferentially resonates in its global poten-\ntial minima [20, 21]. Here this concept is developed in a\nfrequency multiplexed electromechanical parametric res-\nonator and we show that this platform can readily be\nextended to multiple electromechanical parametric res-\nonators with arbitrary degrees of coupling namely all the\nnecessary prerequisites to solving the Ising Hamiltonian\nFIG. 1: The electromechanical system. a, A false colour\nelectron micrograph of the coupled mechanical resonators sus-\ntaining the symmetric and asymmetric vibration modes. The\nmeasurements were performed at room temperature and in\na high vacuum and the mechanical vibrations were detected\nvia a laser interferometer which was demodulated either in\na spectrum analyser (SA) or in a phase sensitive detector\nutilising the heterodyne mixing setup detailed in the circuit\nschematic. Five signal generators were employed with two\npiezoelectrically activating the parametric resonances at 2 !n,\ntwo generating the reference signals for the phase sensitive de-\ntectors at!nand the \ffth creating the coupling between the\nparametric resonances via non-degenerate parametric down-\nconversion when activated at !P.\nin a regime that has no analytical solution and where its\nnumerical solution would demand exorbitant computa-\ntional resources.\nThe Ising model conceived using the tools of statis-\ntical physics describes ferromagnetism and in the ab-\nsence of a magnetic \feld its Hamiltonian is given by\n\u0000NX\nikJik\u001bi\u001bkwithNparticles each with two spin statesarXiv:1505.02467v1  [cond-mat.mes-hall]  11 May 20152\n\u001b=\u00061 and the coupling between the ithandkthparti-\ncles is parametrised by Jik. In the \frst steps to building\nan electromechanical Ising machine, the fundamentals of\nthe Ising Hamiltonian need to be replicated in the elec-\ntromechanical domain namely a mechanical pseudo spin\nto encode\u001b, multiple mechanical spins playing the role of\nanNparticle bath and \fnally coupling between the me-\nchanical spins Jikwhose magnitude and polarity can be\ncontrolled and has the potential to be extended beyond\nnearest neighbour spins.\nThe prototype electromechanical system in which these\nconcepts are investigated is shown in Fig. 1 and it con-\nsists of two strongly coupled mechanical beams which\nsustain a symmetric (S) and an asymmetric (A) mode\nat!S=2\u0019\u0019245 kHz and !A=2\u0019\u0019260 kHz with qual-\nity factors (Q) of 1300 and 2300 respectively (see Fig.\nS1 in Supplementary Information (SI)). The mechani-\ncal elements are integrated with piezoelectric transduc-\ners which enable the underlying harmonic potential of\nboth modes to be parametrically modulated leading to a\nsystem Hamiltonian given by\nH=AX\nn=S\u0010\np2\nn=2 +!2\nnq2\nn=2\u0000\n1\u00002\u0000ncos(2!nt)\n+\fnq2\nn=2\u0001\u0011\n+ \u0003qSqAcos(!Pt+\u001e) (1)\nwhere the summation expresses the kinetic and poten-\ntial energies from both modes in terms of their position\nqnand momentum pn[9, 22, 23]. The potential en-\nergy term contains three contributions, the second arising\nfrom the periodic modulation of the mechanical spring\nconstant with amplitude \u0000 nat twice the natural mode\nfrequency which yields degenerate parametric ampli\fca-\ntion and parametric resonance [24, 25] and the third due\nto the well-known anharmonicity from the Du\u000eng non-\nlinearity\fnwhich appears at large displacements [26].\nA unique feature of the parametric resonance is that it\ncan only vibrate with two phases and thus it provides the\nideal construct within which a classical spin can be har-\nboured [9, 20]. The last term describes non-degenerate\nparametric down-conversion when the periodic modula-\ntion is activated with a pump ( P) amplitude \u0003 at the\nsum frequency of both modes ( !P=!S+!A) which re-\nsults in the symmetric and asymmetric modes becoming\nampli\fed and correlated yielding a two-mode squeezed\nstate [22].\nThis Hamiltonian can be transformed in the rotating\nframe approximation with the introduction of a dimen-\nsionless canonical position coordinate Qnand conjugate\nmomentum Pn, as detailed in SI, which yields\nH=\u0000AX\nn=S\u0000\n!n\u00002\nn=3\f2\nn\u0001\n+JSA\u001bS\u001bA (2)\n0.0 0.1 0.2 0.3-1.0-0.8-0.6-0.4-0.20.0\n QS:PA\n QA:PS\nVP(ωP) = 200 mVrms\n  Correlation coefficient\nVS(2ωS) and VA(2ωA) (Vrms)-2 -1 0 1 2-2-1012\n  PS (pm)\nQA (pm)\n-2 -1 0 1 2-2-1012\n  PS (pm)\nQA (pm)0.0 0.1 0.2 0.3 0.4 0.5-1.0-0.8-0.6-0.4-0.20.0\nQS:PA\nQA:PS\n  Correlation coefficient\nVP(ωP) (Vrms)-3 -2 -1 0 1 2 3-50-2502550\n  \nQA (nm)PA (pm)\n-3 -2 -1 0 1 2 3-60-40-200204060\n  \nQS (nm)PS (pm)a b \nc d \ne f \nσA=1 \nσA=-1 \nσS=1 \nσS=-1 LOW HIGH \nLOW HIGH FIG. 2: Mechanical spins and spin coupling. a andb,\nThe occupation probabilities of the parametric resonances in\nphase space are measured with the parametric actuation pe-\nriodically activated with VS(2!S) = 0.7 V rmsandVA(2!A)\n= 0.6 V rmsrespectively. The resultant phase portraits in-\ndicate that when a parametric resonance is activated, each\nmode evolves from its stationary state at the origin, which is\nbroadened by thermomechanical noise \ructuations, to one of\nthe two available oscillating states with a \u0019phase separation\nthat can be exploited to host classical spin information. c,\nThe phase portrait reconstructed from the cross-quadratures\nof the symmetric and asymmetric mode as function of pump\nintensity. d,The corresponding correlation coe\u000ecient reveals\nthat the two modes become indistinguishable as the pump am-\nplitude is increased where the colours indicate the amplitudes\nin the previous \fgure. e,A two-mode squeezed state gener-\nated withVP(2!P) = 0.2 V rmswhich is then enhanced via\ndegenerate parametric ampli\fcation of both modes. f,The\ncorresponding correlation coe\u000ecient con\frms that the initial\nweakly correlated state can be enhanced by the degenerate\nparametric ampli\fcation where the colour coding indicates\nthe amplitudes in the previous \fgure.\nwhere the \frst term describes the energy correspond-\ning to the two stable oscillation phases of a paramet-\nric resonance at Pn= 0 and Qn=\u001bnp\n4\u0000n=3\fn3\n[23, 27] and the second term quanti\fes the coupling\nJSA= \u0003 cos(\u001e)p\n\u0000S\u0000A=9!S!A\fS\fAbetween the modes\nwith their two phases encoding \u001bn=\u00061. Thus this elec-\ntromechanical system can be formally mapped onto the\nIsing model composed of two classical spins arising from\nthe phase bistable parametric resonances of both modes\nthat can be coupled from the non-degenerate parametric\ndown-conversion. Pleasingly, this analysis also reveals\nthat the magnitude of the coupling between the mechan-\nical spins created by \u0003 can be enhanced by the degenerate\nparametric resonances that are activated when \u0000 n\u0015\rn,\nwhere the damping rate \rn=!n=Qn[9], and its polarity\ni.e. the sign of JSAcan be selected by simply tuning the\npump phase \u001e[28].\nTo experimentally con\frm these expectations, the\nspring constant of either mode is modulated at Vn(2!n)\nvia the stress induced by the piezoelectric transduc-\ners namely \u0000 nin the above equations is activated [9].\nThis results in a given modes thermomechanical \ructu-\nations initially being enhanced corresponding to degen-\nerate parametric ampli\fcation [24] and once the rate at\nwhich the phonons are generated begins to exceed their\nrate of decay namely \u0000 n\u0015\rnit results in in\fnite am-\npli\fcation corresponding to a parametric resonance as\ndepicted in Figs. S2 [9, 25]. The parametric resonance,\nunder periodic driving, is then projected into phase space\nby mixing with a local oscillator locked onto its resonance\nfrequency and demodulated in a phase sensitive detec-\ntor (see circuit schematic in Fig. 1). This measurement\nyields two time series for the in-phase Qnand quadra-\nturePncomponents that enables a phase portrait to be\nreconstructed as shown in Figs. 2a and 2b which prob-\nabilistically unveils the parametric resonances from both\nmodes being able to vibrate with only two phases sep-\narated by\u0019radians as described above [9]. This phase\nbistable vibration enables a classical spin to be mimicked\nin the mechanical domain where hence forth the positive\nin-phase component is de\fned as spin up i.e. \u001bn= 1 and\nvice versa.\nIn order to create coupling between the pseudo spins\nencapsulated by the symmetric and asymmetric modes, a\nmechanical two-mode squeezed state is created by pump-\ning the spring constant of both modes at VP(!P) corre-\nsponding to \u0003 in the above equations [22]. This results in\nthe thermomechanical \ructuations of both modes being\nsimultaneously ampli\fed from non-degenerate paramet-\nric down-conversion as shown in Fig. S2 and projecting\nthis output in phase space reveals it to be phase insen-\nsitive as shown Fig. S4 [29]. In contrast, the degenerate\nparametric ampli\fcation detailed above is phase sensi-\ntive with only the in-phase component being ampli\fed\nwhilst the quadrature component is squeezed as shown\nin Fig. S3 [24]. However, the simultaneous generation\nof phonons in both modes also results in their vibrations\nbecoming correlated which can be identi\fed by recon-\nstructing their cross-quadratures in phase space namelythe in-phase component of the asymmetric mode versus\nthe quadrature component of the symmetric mode (or\nvice versa) as shown in Fig. 2c (Fig. S4c). The resul-\ntant squashed distribution implies that the motion from\nboth modes has become intertwined which can statisti-\ncally be con\frmed by evaluating their correlation coe\u000e-\ncientcov(QAPS)=\u0010QA\u0010PSwhere the numerator describes\nthe covariance and \u0010is the standard deviation. The re-\nsults of this analysis (also including QS:PAwhose phase\nportraits are shown in Fig. S4) as a function of pump in-\ntensity are shown in Fig. 2d which reveals that in this\nprocess the correlation coe\u000ecient tends to -1 indicating a\nperfect anti-correlation that is phonons pairs are simulta-\nneously generated in both modes and with an increasing\nrate which renders their vibrations classically indistin-\nguishable.\nTo transfer this correlation into the parametric res-\nonances, a weakly correlated state is \frst created at\nVP(!P)=200 mV rmsas shown in Figs. 2e and 2f. Next\ndegenerate parametric ampli\fcation is continuously ac-\ntivated in both modes and the resultant correlation co-\ne\u000ecient is monitored as function of both VS(2!S) and\nVA(2!A). As expected from equation 2, the correlations\nbetween the two modes can be enhanced from the degen-\nerate parametric ampli\fcation which can readily be seen\nby the narrower squashed distribution in phase space as\nshown in Fig. 2e and quantitatively con\frmed via the\ncorrelation coe\u000ecient in Fig. 2f. Indeed once maximum\ndegenerate parametric ampli\fcation has been achieved\nand both modes start to parametrically resonate, a per-\nfectly correlated state is created even with weak pump\nintensity. Although experimentally the evolution of the\ncorrelation coe\u000ecient across the transition from degen-\nerate parametric ampli\fcation to parametric resonance\ncannot be seamlessly measured as the phase sensitive\ndetectors became overloaded, this transition is readily\ncon\frmed both numerically and analytically by solving\nequation 1 in this con\fguration as shown in Figs. S5 and\nS6.\nBased on the above results, the essential features of\nthe Ising Hamiltonian between two spin 1/2 particles in\nthe electromechanical domain can now be demonstrated\nusing the pulse sequence depicted in Fig. 3a. First a\ntwo-mode squeezed state is created which is undetectable\nvia thermomechanical \ructuations namely VP(!P)!0.\nNext, the symmetric mode is activated o\u000b-resonance i.e.\n2(!S+ \u0001) which results in no vibration. This is then\nfollowed by the asymmetric mode being parametrically\nactivated at 2 !Awhere the tension created by its mo-\ntion modi\fes the spring constant and shifts the natu-\nral frequency of the symmetric mode by \u0001 resulting in\nits parametric resonance being simultaneously activated\nas shown in Fig. 3b. In this formation, periodically\ntriggering the asymmetric mode results in the symmet-\nric mode becoming activating with the same period and\nmonitoring the in-phase component from both modes on4\n   \n \n0.3 Hz  \n0100 200 300 400 500 600 7000.00.20.40.60.81.0\n  Correlation coefficient\nVP(ωP) (µVrms)\n-0.8-0.40.00.40.8QS (nm)\n0 10 20 30 40 50 60 70 80 90 100-2-1012 QA (nm)\nTime (s)-0.8-0.40.00.40.8\n QS (nm)\n0 10 20 30 40 50 60 70 80 90 100-2-1012 QA (nm)\nTime (s)\n-0.8-0.40.00.40.8\n QS (nm)\n0 10 20 30 40 50 60 70 80 90 100-2-1012 QA (nm)\nTime (s)-0.8-0.40.00.40.8\n QS (nm)\n0 10 20 30 40 50 60 70 80 90 100-2-1012 QA (nm)\nTime (s)\n247000 248000 261000 26200010-1410-1310-1210-11\n  SX0.5 (mHz-0.5)\nFrequency (Hz)-60-45-30-15 015304560-1.0-0.8-0.6-0.4-0.20.00.20.40.60.81.0\n  Correlation coefficient\nφ -φ0 (degrees)\nor \n or \n or \nTc \nor \nor \n or \n or a \nor \nor \n∆ c \nd \ne f \ng \nh b \nFIG. 3: Emulating the Ising model in the mechanical domain. a, The pulse sequence used to demonstrate the funda-\nmentals of the Ising Hamiltonian in an electromechanical system. b,The spectral response of the symmetric and asymmetric\nmodes in response to the above pulse sequence. c,The temporal response of the mechanical spins to the above pulse sequence\nwhich are readout via the in-phase component of their motion with VS(2(!S+ \u0001)) = 0.7 V rms,VA(2!A) = 0.7 V rmsand\nVP(!P)=600\u0016Vrmswhich reveals ferromagnetic ordering with the two spins always in parallel alignment where the spin orien-\ntations are visualised by the arrows with red (green) indicating the symmetric (asymmetric) mode. All the temporal outputs are\nmeasured over 1000 seconds to enable statistically reliable analysis. d,The correlation coe\u000ecient extracted from the switching\noutputs as a function of pump intensity. At the largest pump amplitude the mechanical spins exhibit ferromagnetic ordering\nbut as the pump is reduced to zero, the spins become disordered undergoing an e\u000bective second order phase transition with\nthe pump playing the role of temperature and a critical temperature (T c) of 500\u0016Vrms.e,A segment of the temporal output\nused to extract the above correlation coe\u000ecient with the pump deactivated which clearly reveals that the mechanical spins\nhave become decoupled yielding uncorrelated switching events. g,The correlation coe\u000ecient extracted from the mechanical\nsystem in response to the above pulse sequence as a function of the pump phase with VP(!P)=600\u0016Vrms. As the pump\nphase is adjusted, the mechanical spins exhibiting ferromagnetic ordering at \u001e\u0000\u001e0=67\u000edetailed in the temporal output in f\ntransition through a disordered state with random spin alignment to \fnally an anti-correlated state at \u001e\u0000\u001e0=-63\u000ewith the\nspins exhibiting anti-ferromagnetic ordering as detailed in the temporal output in h. Note that the value of \u001e0is determined\nby the spectral position of the readout probe with respect to the resonances of the two modes.\nresonance enables identi\fcation of their spin orientation\n(see Figs. 2a and 2b). Correlations from the two-mode\nsqueezing can then be statistically quanti\fed by moni-\ntoring the resultant switching outputs from both modes.\nFor instance, activating the asymmetric mode will result\nin either a spin up or a spin down the choice of which\nis determined by its thermal \ructuations. However, if a\ntwo-mode squeezed state is created then the switching\noutput observed from the symmetric mode will be corre-\nlated with that from the asymmetric mode.\nImplementing this protocol with a weak pump yields\nthe output shown in Fig. 3c where an up spin from the\nasymmetric mode coincides with an up spin in the sym-metric mode and vice versa. However if the pump is de-\nactivated, the two modes become uncoupled and switch\nrandomly as shown in Fig. 3e. The correlation coe\u000ecient\ncan then be extracted from these switching outputs as a\nfunction of pump intensity as depicted in Fig. 3d. This\nreveals the mechanical pseudo spins can achieve parallel\nalignment when the pump is activated and the corre-\nsponding correlation coe\u000ecient tends to unity. This par-\nallel con\fguration can be controllably deactivated via the\npump and the correlation coe\u000ecient smoothly transitions\nto zero mimicking an e\u000bective second order ferromagnetic\nphase transition. Consequently VP(!P) can convincingly\nplay the role of Jikin the Ising Hamiltonian.5\nNow in order to control the sign of the coupling, the\npump phase is adjusted as depicted in Fig. 1 and detailed\nin equation 2. Indeed measuring the two-mode squeezing\nvia thermomechanical noise \ructuations in phase-space\nas a function of \u001eclearly reveals that the anti-correlated\nstate depicted in Fig. 2c can be tuned to a correlated\nstate as shown in Fig. S7 [28]. Repeating the protocol\noutlined in Fig. 3a but now as function of \u001ewith a suf-\n\fciently intense pump so that correlations are generated\nwith perfect \fdelity yields the result shown in Fig. 3g.\nHere the parallel spin alignment from the two modes as\ncaptured in Fig. 3f can be continuously tuned to an anti-\nparallel alignment via \u001eas shown in Fig. 3h correspond-\ning to an e\u000bective phase transition from ferromagnetism\nto anti-ferromagnetism or in other words Jik!\u0000Jikin\nthe Ising model.\nIndeed the pulse sequence shown in Fig. 3a can also\nbe executed with the o\u000b resonant excitation of the asym-\nmetric mode in the second step and triggering via the\nsymmetric mode as detailed in Fig. S8. Interestingly, the\ndata shown in Fig. 3c (and Fig. S8c) indicates that corre-\nlations from the two-mode squeezing can be observed via\nthe parametric resonances with a pump amplitude that\nis three orders of magnitude smaller than the equivalent\nmeasurement utilising thermomechanical noise \ructua-\ntions as shown in Fig. 2d. This observation suggests an\nalternative approach to identifying mechanical vacuum\nsqueezed states via the large signal from their parametric\nresonance instead of their vacuum noise [30] which could\nprove pivotal in detecting a macroscopic all-mechanical\nentangled state [22].\nThe results detailed in Fig. 3 con\frm that the requisite\nfeatures of the Ising Hamiltonian can be reproduced in\nthe electromechanical domain. However its implementa-\ntion with two mechanical modes in this instance is trivial\nand therefore it is instructive to examine the prospects of\nan electromechanical Ising machine composed of a large\nnumber of spin particles with multiple degrees of cou-\npling.\nBased on the above demonstrations, an electromechan-\nical Ising machine visualised in Fig. 4 would seem feasi-\nble. Here multiple mechanical elements with di\u000berent fre-\nquencies are weakly mechanically coupled to their neigh-\nbours. The elements encode spin information via the\nbistable phase of their piezoelectrically activated para-\nmetric resonance in their fundamental mode, from the\nright clamping point, where this information can also\nbe pre-programmed [9]. The key di\u000berence here is that\nspin information is stored in each mechanical element\nrather than modes composed of more than one element\nas demonstrated above. On the left clamping point a\ncoupling gate electrode is de\fned which interconnects the\nmechanical spins via the piezoelectrically activated para-\nmetric down-conversion pump at the sum frequency of\ntwo elements for instance !i+!i+1as shown in Fig. 4.\nNaturally, the reduced mechanical coupling between the\n2(𝜔𝑖+3) \n2(𝜔𝑖+2) \n2(𝜔𝑖+1) \n2(𝜔𝑖) \n2(𝜔𝑖−1) \n2(𝜔𝑖−2) \n𝝎𝒊+𝝎𝒊−𝟏  𝝎𝒊+𝝎𝒊+𝟏  𝝎𝒊+𝝎𝒊+𝟐  \n𝝎𝒊+𝝎𝒊−𝟐  \n𝝎𝒊+𝝎𝒊−𝟑  𝝎𝒊+𝝎𝒊+𝟑  FIG. 4: The electromechanical Ising machine. A con-\nceptual image of an electromechanical Ising machine based on\nan array of mechanical elements each parametrically resonat-\ning to encode classical spin information (red corresponds to\nspin up and green to spin down) where two-mode squeezing\nis used to create coupling between elements. The parametric\nresonances are piezoelectrically activated and readout via the\nindividual gate electrodes (orange) on each mechanical ele-\nment where the ithelement has a natural frequency !i. The\nglobal gate electrode (orange) located on the left clamping\npoint of all the mechanical elements can enable coupling be-\ntween any pair of mechanical spins with the application of a\nsum frequency pump modulation. Utilising FDM, the pump\ncan execute multiple degrees of coupling between a large num-\nber of spins potentially enabling the Ising Hamiltonian to be\nsolved in a con\fguration that is beyond the reach of conven-\ntional computers.\nelements will require a stronger pump amplitude which\nwould be feasible as even weak pumping can e\u000eciently\ncouple the mechanical spins as demonstrated in Fig. 3d.\nEach mechanical spin could then be readily coupled to all\nits neighbours by employing a frequency division multi-\nplexed (FDM) pump composed of multiple sum frequen-\ncies where Fig. 4 explicitly depicts the FDM pump input\nnecessary to couple the ithelement to its \frst six near-\nest neighbours. The compact form of this coupling is in\nstark contrast to the optical Ising machine which requires\ndelay lines that increase both in number for more spins\nand length for higher order couplings [21]. The FDM\npiezoelectric pump in principle enables a large number\nof mechanical spins to sustain multiple degrees of cou-\npling permitting the electromechanical Ising machine to\nbe programmed with problems that are unsolvable with\nconventional computers. Although this platform could\ntackle NP-hard problems [20, 21], it does not o\u000ber speed\nup as it employs classical annealing where thermome-\nchanical \ructuations of the mechanical elements drive the\nsearch in the underlying potential energy landscape for\nthe ground state. However tantalisingly if all the mechan-\nical elements are operated in their ground state [11{13]\nthen quantum tunnelling could be harnessed to increase6\nthe speed of search thus making an inexorable case for\ndevelopment of such an electromechanical Ising machine.\n\u0003Electronic address: imran.mahboob@lab.ntt.co.jp\n[1] Roukes, M. Nanoelectromechanical systems face the fu-\nture. Phys. World 14, 25{31 (2001).\n[2] Ekinci, K. L. & Roukes, M. L. Nanoelectromechanical\nsystems. Rev. Sci. Instrum. 76, 061101 (2005).\n[3] Marquardt, F. & Girvin, S. M. Optomechanics. Physics\n2, 40 (2009).\n[4] Aspelmeyer, M., Kippenberg, T. J. & Marquardt, F.\nCavity optomechanics. Rev. Mod. Phys. 86, 1391 (2014).\n[5] Rugar, D., Budakian, R., Mamin, H. J. & Chui, B. W.\nSingle spin detection by magnetic resonance force mi-\ncroscopy. Nature 430, 329 (2004).\n[6] Arcizet, O. et al. A single nitrogen-vacancy defect cou-\npled to a nanomechanical oscillator. Nature Phys. 7, 879{\n883 (2011).\n[7] Chaste, J. et al. A nanomechanical mass sensor with yoc-\ntogram resolution. Nature Nanotech. 7, 301{304 (2012).\n[8] Badzey, R. L. & Mohanty, P. Coherent signal ampli\fca-\ntion in bistable nanomechanical oscillators by stochastic\nresonance. Nature 437, 995 (2005).\n[9] Mahboob, I. & Yamaguchi, H. Bit storage and bit \rip op-\nerations in an electromechanical oscillator. Nature Nan-\notech. 3, 275{279 (2008).\n[10] Eichler, A. et al. Nonlinear damping in mechanical res-\nonators made from carbon nanotubes and graphene. Na-\nture Nanotech. 6, 339{342 (2011).\n[11] O'Connell, A. D. et al. Quantum ground state and single-\nphonon control of a mechanical resonator. Nature 464,\n697{703 (2010).\n[12] Teufel, J. D. et al. Sideband cooling of micromechanical\nmotion to the quantum ground state. Nature 475, 359{\n363 (2011).\n[13] Chan, J. et al. Laser cooling of a nanomechanical oscil-\nlator into its quantum ground state. Nature 478, 89{92\n(2011).\n[14] Buluta, I. & Nori, F. Quantum simulators. Science 326,\n108{111 (2009).\n[15] Barahona, F. On the computational complexity of ising\nspin glass models. J. Phys. A: Math. Gen. 15, 3241\n(1982).\n[16] Kirkpatrick, S., Jr, C. D. G. & Vecchi, M. P. Opti-\nmization by simulated annealing. Science 220, 671{680\n(1983).\n[17] Finnila, A. B., Gomez, M. A., Sebenik, C., Stenson, C. &\nDoll, J. D. Quantum annealing: A new method for min-\nimizing multidimensional functions. Chem. Phys. Lett.\n219, 343{348 (1994).\n[18] Santoro, G. E., Marto^ n\u0013 ak, R., Tosatti, E. & Car, R. The-\nory of quantum annealing of an ising spin glass. Science\n292, 472{475 (2002).[19] Johnson, M. W. et al. Quantum annealing with manu-\nfactured spins. Nature 473, 194{198 (2011).\n[20] Wang, Z., Marandi, A., Wen, K., Byer, R. L. & Ya-\nmamoto, Y. Coherent ising machine based on degenerate\noptical parametric oscillators. Phys. Rev. A 88, 063853\n(2013).\n[21] Marandi, A., Wang, Z., Takata, K., Byer, R. L. & Ya-\nmamoto, Y. Network of time-multiplexed optical para-\nmetric oscillators as a coherent ising machine. Nature\nPhoton. 8, 937{942 (2014).\n[22] Mahboob, I., Okamoto, H., Onomitsu, K. & Yamaguchi,\nH. Two-mode thermal-noise squeezing in an electrome-\nchanical resonator. Phys. Rev. Lett. 113, 167203 (2014).\n[23] Ryvkine, D. & Dykman, M. I. Resonant symmetry lifting\nin a parametrically modulated oscillator. Phys. Rev. E\n74, 061118 (2006).\n[24] Rugar, D. & Gr utter, P. Mechanical parametric ampli\f-\ncation and thermomechanical noise squeezing. Phys. Rev.\nLett.67, 699{702 (1991).\n[25] Turner, K. L. et al. Five parametric resonances in a\nmicroelectromechanical system. Nature 396, 149 (1998).\n[26] Aldridge, J. S. & Cleland, A. N. Noise-enabled pre-\ncision measurements of a du\u000eng nanomechanical res-\nonator. Phys. Rev. Lett. 94, 156403 (2005).\n[27] Mahboob, I., Froitier, C. & Yamaguchi, H. A symmetry-\nbreaking electromechanical detector. Appl. Phys. Lett.\n96, 213103 (2010).\n[28] Flurin, E., Roch, N., Mallet, F., Devoret, M. H. & Huard,\nB. Generating entangled microwave radiation over two\ntransmission lines. Phys. Rev. Lett. 109, 183901 (2012).\n[29] Bergeal, N. et al. Phase-preserving ampli\fcation near the\nquantum limit with a josephson ring modulator. Nature\n465, 64{68 (2010).\n[30] Lin, Z. R. et al. Josephson parametric phase-locked os-\ncillator and its application to dispersive readout of su-\nperconducting qubits. Nature Commun. 5, 4480 (2014).\nAcknowledgements The authors are grateful to H.\nOkamoto, N. Kitajima for fabricating the device to Y.\nIshikawa, K. Onomitsu for growing the heterostructure and\nto N. Lambert, F. Nori for discussions and comments. This\nwork was partly supported by JSPS KAKENHI Grant No.\n23241046.\nAuthor contributions H.Y. and I.M. conceived the idea,\nI.M. performed the measurements, analysed the data, con-\nducted the numerical modelling and wrote the paper. H.Y.\ndeveloped the analytical model and planned the project.\nSupplementary Information is linked to the online version\nof the paper.\nCompeting \fnancial interests The authors declare that\nthey have no competing \fnancial interests."    },    {        "title": "2006.13582v1.Ferromagnetic_induced_Kondo_effect_in_graphene_with_a_magnetic_impurity.pdf",        "content": "arXiv:2006.13582v1  [cond-mat.str-el]  24 Jun 2020Ferromagnetic induced Kondo eﬀect in graphene with a magnet ic impurity\nGao-Yang Li,1Tie-Feng Fang,1,∗Ai-Min Guo,2and Qing-Feng Sun3,4,5\n1School of Physical Science and Technology, Lanzhou Univers ity, Lanzhou 730000, China\n2Hunan Key Laboratory for Super-microstructure and Ultrafa st Process,\nSchool of Physics and Electronics, Central South Universit y, Changsha 410083, China\n3International Center for Quantum Materials, School of Phys ics, Peking University, Beijing 100871, China\n4Collaborative Innovation Center of Quantum Matter, Beijin g 100871, China\n5CAS Center for Excellence in Topological Quantum Computati on,\nUniversity of Chinese Academy of Sciences, Beijing 100190, China\n(Dated: June 25, 2020)\nWe investigate the many-body eﬀects of a magnetic adatom in f erromagnetic graphene by using\nthe numerical renormalization group method. The nontrivia l band dispersion of ferromagnetic\ngraphene gives rise to interesting Kondo physics diﬀerent f rom that in conventional ferromagnetic\nmaterials. For a half-ﬁlled impurity in undoped graphene, t he presence of ferromagnetism can\nbring forth Kondo correlations, yielding two kink structur es in the local spectral function near the\nFermi energy. When the spin splitting of local occupations i s compensated by an external magnetic\nﬁeld, the two Kondo kinks merge into a full Kondo resonance ch aracterizing the fully screened\nground state. Strikingly, we ﬁnd the resulting Kondo temper ature monotonically increases with\nthe spin polarization of Dirac electrons, which violates th e common sense that ferromagnetic bands\nare usually detrimental to Kondo correlations. Doped ferro magnetic graphene can behave as half\nmetals, where its density of states at the Fermi energy linea rly vanishes for one spin direction but\nkeeps ﬁnite for the opposite direction. In this regime, we de monstrate an abnormal Kondo resonance\nthat occurs in the ﬁrst spin direction, while completely abs ent in the other one.\nI. INTRODUCTION\nThe Kondo eﬀect [1] describes the screening of a local\nspin by conduction electrons, representing a paradigm\nof many-body correlations in condensed matter physics.\nIts original solution by Wilson’s renormalization group\nmethod has invoked some of the most profound con-\ncepts in theoretical physics [2]. While the eﬀect has al-\nready been studied in bulk materials for some 60 years\n[3], recent two decades have seen a resurgent interest in\nexploring Kondo physics in artiﬁcial nanostructures [4].\nThanks to the great tunability of these nanoscale devices\nand their ﬂexibility for integration with various exotic\nmaterials, many fascinating Kondo phenomena absent in\nbulk systemshavenowbeen revealedwith unprecedented\ncontrol. Of particular interest are the competition of\nKondo correlations with other many-body eﬀects (e.g.,\nferromagnetism [5] and superconductivity [6–8]) and the\nscreening mechanism in unconventional electronic envi-\nronments [9].\nThe inﬂuence of itinerant electron ferromagnetism on\nKondo correlations has been intensively studied in quan-\ntum dots attached to ferromagneticelectrodes [10–12]. It\nwas shown that in the presence of particle-hole symme-\ntry the Kondo peak in the density of states of the dot re-\nmains unsplit even at ﬁnite lead polarizations[13]. When\ncharge ﬂuctuations are allowed in the asymmetric case,\ntheleadferromagnetismcausesaneﬀectiveexchangeﬁeld\nacting on the dot [14–18], leading to splitting and sup-\npression of the Kondo resonance [19–27]. Interestingly,\nthis exchange ﬁeld can be fully compensated by an ap-\npropriately tuned external magnetic ﬁeld, thus restoringthe full Kondo resonance [5, 16, 17, 19, 21]. It should be\nemphasized that although the unsplit Kondo resonance\nshows up at the compensated ﬁeld or in the particle-hole\nsymmetric case, the lead ferromagnetism always plays a\ndestructive role, which reduces the Kondo temperature\n[14,16–19]andhencesuppressesKondocorrelations. No-\ntably, these destructive inﬂuence of ferromagnetism were\ndemonstrated in published studies only considering con-\nventional ferromagnetic electrodes with smooth and fea-\ntureless (if not completely ﬂat) density of states around\nthe Fermi energy. Detailed information on the interplay\nof Kondo correlations and ferromagnetism in unconven-\ntionalmaterialswithexoticbandstructureisstilllacking.\nGraphene [28–30], with exotic Dirac-like electronic ex-\ncitations, is one of such materials. It exhibits a good deal\nof remarkable correlated phenomena, including the frac-\ntionalquantumHalleﬀect [31,32], unconventionalsuper-\nconductivity[33], andcorrelatedinsulatingstates[34]. In\nparticular, graphene provides a perfect realization of the\npseudogap Kondo problem [35–37], when local magnetic\nmoments are created in graphene either by adatom de-\nposition [38–40] or via point defects [41–45]. The result-\ning Kondo physics has attracted much research interest\nin the last decade [46–61]. Graphene can even realize\nsalient Kondo models with multiple screening channels\n[62–64], having the SU(4) symmetry [65], and exhibit-\ning the super-Ohmic dissipation [66]. While these stud-\nies have demonstrated the carrier doping, explicit impu-\nrity positions, and local orbital properties being crucial\nfor characterizing the graphene Kondo system, the in-\nﬂuence of ferromagnetism has not yet been addressed.\nIn fact, there are several approaches [67–69] to induce2\nlong-range ferromagnetic order in graphene without sac-\nriﬁcing its excellent conductivity, e.g., placing graphene\non an insulating magnetic substrate [68]. Such ferromag-\nneticgrapheneprovidesaﬂexibleplatformforprobingin-\ntriguing Kondo correlations arising from the interaction\nof a local moment with spin-polarized Dirac fermions.\nIn this paper, we shall study the Kondo physics of a\nmagnetic adatom on ferromagnetic graphene using the\nnumerical renormalization group (NRG) method [2, 70–\n75]. We demonstrate that unlike conventional ferromag-\nnetic materials, the graphene ferromagnetism has con-\nstructive inﬂuence on the Kondo eﬀect due to its exotic\nband dispersion. Although a particle-hole symmetric im-\npurity in undoped graphene always keeps an unscreened\nlocal moment, Kondo correlations can be induced by the\nlong-range ferromagnetic order present in graphene, giv-\ning rise to two kink structures in the local spectral den-\nsity near the Fermi energy. By an appropriately tuned\nexternalmagneticﬁeld, thetwoKondokinksmergeintoa\nfull Kondo resonance, signaling the fully screened ground\nstate. The resulting Kondo temperature increases with\nthe ferromagnetic exchange ﬁeld and Kondo correlations\nare thus enhanced. Remarkably, ferromagnetic graphene\ncan be driven into the half-metallic regime, when it is\ndoped such that the Fermi level aligns with the Dirac\npoint of one spin component while the hybridization of\nthe other spin component is ﬁnite at the Fermi energy.\nIn this regime, we demonstrate an abnormal Kondo res-\nonance that occurs in the ﬁrst spin component, but com-\npletely absent in the other one. These Kondophenomena\nshould be accessible by scanning tunneling spectroscopy\nmeasurements.\nThe remainder of the paper is organized as follows.\nSec.II introduces the model Hamiltonian and provides\nsome necessary details of the NRG method. Numerical\nresults and discussion are presented in Sec.III, followed\nby a conclusion in Sec. IV.\nII. MODEL AND METHOD\nThe system under consideration consists of a magnetic\nimpurity adsorbed on the ferromagnetic graphene, which\nis modeled by the following Hamiltonian\nH=Himp+Hg+Hhyb. (1)\nHereHimpdescribes the isolated impurity,\nHimp=/summationdisplay\nσεdd†\nσdσ+Ud†\n↑d↑d†\n↓d↓, (2)\nwheredσannihilates an electron with spin σ=↑,↓and\nenergyεdin the localized magnetic orbital. Uis the\nCoulomb repulsion when the orbital is doubly occupied.\nIn the tight-binding representation, the Hamiltonian Hgof the ferromagnetic graphene reads [76]\nHg=/summationdisplay\ni,σ(ε0−µ+σh)(a†\niσaiσ+b†\niσbiσ)\n−/summationdisplay\n/angbracketleftij/angbracketright,σt(a†\niσbjσ+H.c.), (3)\nwhereaiσ(biσ) annihilates an electron on site iin sub-\nlatticeA(B) of the graphene honeycomb lattice, ε0is\nthe Dirac-point energy of nonmagnetic graphene, tis the\nnearest-neighbor hopping energy, and the chemical po-\ntentialµcan be tuned by a gate voltage. For ferromag-\nnetic graphene, a nonzero exchange ﬁeld harises from\nthe ferromagnetic interaction due to the proximity cou-\npling with a magnetic insulator [68]. The explicit form of\nthe hybridization Hamiltonian Hhybcould be very com-\nplicated, depending on the symmetry of the localized im-\npurity orbital and its position relative to the honeycomb\nlattice. We consider in this work the simplest case where\nthe impurity atom is adsorbed on the top of a carbon\natom, say on site i= 0 in sublattice A. In this case, the\nadatom can only hybridizes with this carbon atom and\nthespeciﬁcsymmetryofthelocalizedorbitalisirrelevant,\nyielding\nHhyb=/summationdisplay\nσV(a†\n0σdσ+H.c.), (4)\nwithVrepresenting the hybridization amplitude.\nIn the momentum space, we introduce the operators of\nDirac fermions [66],\nCkσα=1√\n2/parenleftbigg\nakσ−αφk\n|φk|bkσ/parenrightbigg\n, (5)\nto diagonalize the graphene Hamiltonian\nHg=/summationdisplay\nk,σ,αεkσαC†\nkσαCkσα. (6)\nHereα=±labels the conduction and valence bands,\nφk=/summationtext3\nj=1eik·δjwithδjthe three vectors connecting\none site with its nearest neighbors in the honeycomb lat-\ntice. Note that the presence of ferromagnetism lifts the\nspin degeneracy of the dispersion\nεkσα=αt|φk|+ε0−µ+σh. (7)\nIn this basis, the hybridization becomes\nHhyb=/summationdisplay\nk,σ,αV√\n2N(C†\nkσαdσ+H.c.),(8)\nwithNthe number of unit cells in the graphene.\nEquations (1), (2), and (6)-(8) constitute the standard\nsingle-impurity Anderson model. Its impurity properties\nare fully determined by the so-called hybridization func-\ntion\nΓσ(ω) =πV2\n2N/summationdisplay\nk,αδ(ω−εkσα). (9)3\n/s45/s48/s46/s50 /s45/s48/s46/s49 /s48/s46/s48 /s48/s46/s49 /s48/s46/s50/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51\n/s45/s48/s46/s51 /s45/s48/s46/s50 /s45/s48/s46/s49 /s48/s46/s48 /s48/s46/s49 /s48/s46/s50/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51\n/s119 /s32/s47 /s32/s68/s71\n/s115/s40/s119 /s41/s32 /s32/s32/s32/s32/s71\n/s48\n/s119 /s32/s47 /s32/s68/s47/s69\n/s70\n/s32/s32/s32/s40/s97/s41/s40/s98/s41 /s69\n/s70/s47/s71\n/s115/s40/s119 /s41/s32 /s32/s32/s32/s32/s71\n/s48\nFIG. 1: Spin-resolved hybridization function Γ σ(ω) atµ=ε0\n(a) andµ=ε0+h(b). Solid and dashed thin lines represent\nthe hybridization of a top-site impurity with ferromagneti c\n(h/D= 0.1) and nonmagnetic ( h/D= 0) graphene, respec-\ntively. The thick gray lines indicate the Fermi energy EF.\nTypical characteristics of Γ σ(ω) include the presence of\nvan Hove singularities at high energies and the low-\nenergy linear behavior near the Dirac point. Since high-\nenergy structures of Γ σ(ω) are irrelevant to the Kondo\nphysics, it is suﬃcient to only concentrate on its low-\nenergy structure. Speciﬁcally, close to the Dirac point,\nthe graphene dispersion and hence the hybridization\nfunction are linear,\nεkσα≃α/planckover2pi1vF|k|+ε0−µ+σh, (10)\nΓσ(ω)≃V2Ω0\nN/planckover2pi12v2\nF|ω−ε0+µ−σh|,(11)\nmeasuring kwith respect to the Dirac-point momen-\ntum. Here vFis the Fermi velocity and Ω 0is the\narea of graphene unit cell. For notation simplicity, a\nhybridization-function prefactor, Γ 0≡V2Ω0D\nN/planckover2pi12v2\nFwithD\nthe half bandwidth of Dirac fermions, is used hereafter.\nIt is apparent that in ferromagnetic graphene, the mass-\nless Dirac fermions, characterized by the linear disper-\nsion of Eq.(10), are spin polarized due to the long-range\nferromagnetic ordering. Consequently, the hybridization\nfunction Eq.(11), which fully accounts for the inﬂuence\noftheDirac-fermionbathonthe impurity, isalsospinpo-\nlarized. Itisouraiminthispapertoexplorethegraphene\nKondo physics subject to this spin polarization of Dirac\nfermions.\nThe nontrivial spin splitting of the hybridization func-\ntion, as shown in Fig.1, suggests that the Kondo physics\nwill exhibit interesting modulations under variation of\nthe ferromagnetic exchange ﬁeld hand/or the chemical\npotential µ. It iswellestablished[51] that inthe presence\nof particle-hole symmetry, no Kondo screening is possi-\nble for impurities in neutral ( µ=ε0) and nonmagnetic\n(h= 0) graphene for which Γ σ(ω) vanishes as |ω|at the\nFermi energy EF= 0 [the dashed line in Fig.1(a)]. As\nsoon as the ferromagnetism is turned on, the hybridiza-\ntion function at the Fermi energy acquires a ﬁnite value\nΓ0|h|/Dfor both spin species [the intersection of the two\nsolid lines in Fig.1(a)], in favourofdeveloping the Kondoscreening as in normal metals. On the other hand, the\nspin-splittingofthehybridizationfunction awayfromthe\nFermi energy is obviously unfavourable to the Kondo ef-\nfect. We expect intriguingKondo phenomena arisingdue\nto these two conﬂicting consequences of the ferromag-\nnetism in graphene. Remarkably, when the chemical po-\ntentialµis tuned to the spin-dependent Dirac-point en-\nergyε0+σh, sayµ=ε0+h, the ferromagnetic graphene\nbehaves like a Dirac half metal: the Fermi level aligns\nwith the spin- ↑Dirac point while the spin- ↓hybridiza-\ntion function isﬁnite atthe Fermienergy[Fig.1(b)]. Can\nDiracfermionsinthisexotichalf-metalregimestillscreen\nthe local spin? If can, what is the manifestation of the\nresultant Kondo eﬀect?\nWeshalladdresstheseissuesbyusingtheNRGmethod\n[2, 70, 71] based on the full density matrix algorithm [72–\n75]. The full density-matrix NRG [74] is one of the most\npowerful methods for quantum impurity systems. It it-\neratively diagonalizes the Hamiltonian and yields an ap-\nproximatebut complete set ofeigenstates. This complete\nbasis set [72] can be used to calculate dynamical proper-\nties [73] such as the impurity spectral function, as well as\nthermodynamic properties [75] such as the impurity en-\ntropy. What followsarethe numericalresults obtained in\nthe units of D=kB=/planckover2pi1= 1. NRG calculations are per-\nformed by using a discretization parameter Λ = 1 .8 for\ndynamical properties and Λ = 1 .8∼2.5 for thermody-\nnamic quantities, and retaining MK= 512∼1024 states\nper iteration. Discretespectral datais smoothened based\non the log-Gaussian kernel proposed in Ref.[74] with a\nbroadening parameter α= 0.4∼0.5. Results are zav-\neraged over Nz= 1∼4 calculations. Throughout this\nwork, we ﬁx the Dirac-point energy ε0= 0 and the local\nCoulomb interaction U= 0.2. The temperature Tis set\nto zero for all results, unless indicated otherwise.\nIII. NUMERICAL RESULTS AND DISCUSSION\nA. Charge-neutral nonmagnetic graphene\nWe ﬁrst revisit the Kondo physics of a magnetic\nadatom on the nonmagnetic ( h= 0) graphene at charge\nneutrality ( µ= 0). Figure 2(a) displays the resulting\nphase diagram in the plane spanned by the impurity\nparticle-hole asymmetry δ=εd+U/2 and the hybridiza-\ntion Γ 0. The phase boundary is determined according to\nthe zero-temperature values of the impurity contribution\nto entropy [71], Simp(T) =S(T)−S0(T). Here\nS(T) =β/angbracketleftH/angbracketright+ln[Tr(e−βH)] (12)\nand\nS0(T) =β/angbracketleftHg/angbracketright+ln[Tr(e−βHg)], (13)\nwithβ= 1/T, are the entropy of the system with\nand without the impurity, respectively. The central4\n/s49/s48/s45/s52\n/s49/s48/s45/s50\n/s49/s48/s48\n/s45/s48/s46/s49 /s48/s46/s48 /s48/s46/s49/s49/s48/s45/s50/s49/s48/s48/s49/s48/s50/s45/s48/s46/s52 /s45/s48/s46/s50 /s48/s46/s48 /s48/s46/s50 /s48/s46/s52/s48/s50/s52/s54\n/s83/s105/s110/s103/s108/s101/s116 /s83/s105/s110/s103/s108/s101/s116\n/s32/s32/s71\n/s48/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s85\n/s100 /s32/s32/s47 /s32/s32/s85/s47/s68/s111/s117/s98/s108/s101/s116/s40/s97/s41\n/s40/s98/s41\n/s48/s108/s110/s50/s108/s110/s52\n/s32/s83\n/s105/s109/s112\n/s84 /s32/s47 /s32/s32/s68/s40/s99/s41\n/s119 /s32/s32/s47 /s32/s32/s32/s68/s71\n/s48/s32/s32/s32/s65\n/s115/s40/s119 /s41\nFIG. 2: (a) Impurity phase diagram for charge-neutral non-\nmagnetic graphene in the plane spanned by the impurity\nasymmetry δand the hybridization Γ 0. (b) Impurity entropy\nSimp(T) and (c) spectral function Aσ(ω), for the parameter\nset{δ/U,Γ0/U}={−0.4,2}(/squaresolid),{0,2}(•), and{0.4,2}(/trianglesolid)\nmarked in (a).\n(green) region of Fig.2(a) represents the local moment\nphase, characterized by a residual entropy Simp(0) = ln2\n[Fig.2(b)]. In this doublet phase, the impurity moment\nis asymptotically decoupled from the graphene and be-\nhaves like a free local moment. On the other hand, in\nthe two side (orange) regions of Fig.2(a), the impurity\nentropy goes to zero, Simp(0) = 0 [Fig.2(b)], indicating\na fully screened singlet ground state that is described by\nthe asymmetric strong-coupling ﬁxed point. The sepa-\nratrix between the singlet and doublet phases is actu-\nally a many-body level crossing, which can also be deter-\nmined by the discontinuities of the ground-state energy\nas a function of δand Γ0. In the screened singlet phase,\nKondo correlations can develop when charge ﬂuctuations\nare suppressed. But there is no sharp boundary between\nthe Kondo and mixed-valence regimes. It is instead a\nsmooth crossover as the impurity level, εdorεd+U,\napproaches the Fermi level.\nA striking feature of the Kondo eﬀect in neutral non-\nmagnetic graphene is that the impurity spectral function[71],Aσ(ω) =−1\nπImGret\nσ(ω) with\nGret\nσ(ω) =−i/integraldisplay∞\n−∞dt eiωtθ(t)/angbracketleftbig/braceleftbig\ndσ(t), d†\nσ/bracerightbig/angbracketrightbig\n,(14)\nis not peaked at the Fermi energy, exhibiting no charac-\nteristic Kondo resonance. In fact, as shown in Fig.2(c),\nboth in the Kondo and local-moment regimes, the impu-\nrity spectral density vanishes linearly at the Fermi en-\nergy, but featuring broad peaks away from EF. It seems\nthat thereisnoclear-cutspectralsignaturetodistinguish\nthese two phases. All the features discussed in this sub-\nsection are more or less understood in the literature (see\nRef.[51], for a review).\nB. Ferromagnetic graphene at charge neutrality\nNow we turn to investigate the eﬀect of ferromagnetic\ngraphene on the impurity atom. In Sec.III B, we ﬁx the\nchemical potential µ= 0 to obtain results for undoped\ngraphene, while Sec.III C presents results in the Dirac\nhalf-metal regime reached by tuning µ. In both subsec-\ntions, wealwaysconsidertheimpuritybeingparticle-hole\nsymmetric ( δ= 0).\nStarting from the local moment regime at zero ex-\nchange ﬁeld ( h= 0), Fig.3(a) shows the impurity spec-\ntral feature A(ε) =A↑(ω) +A↓(ω) with increasing h.\nBesides the broadening of the Hubbard bands and the\nincreasing of spectral density at the Fermi energy [in-\nset of Fig.3(b)], the most striking phenomenon induced\nby the exchange ﬁeld is the appearance of two kink (or\nshoulder) structures near the Fermi energy. Note that\nashincreases, the two kinks move away from each other\nbut seem more evident. We attribute them to the Kondo\neﬀect arising from many-body correlations between the\nimpurity and spin-polarized Dirac electrons in graphene,\nas demonstrated in the following.\nThe corresponding evolution of the impurity entropy\nSimp(T) is shown in Fig.3(b), which illustrates a sup-\npression of Simp(T) by ﬁnite hat low temperatures. Un-\nlike in nonmagnetic metals, this suppression of the im-\npurity entropy is not an unambiguous signature of the\nfully screened Kondo state. The latter in fact cannot de-\nvelop in the presence of spin asymmetry caused by the\nexchange ﬁeld. In other words, the impurity entropy is\nnot a good quantity for characterizing Kondo physics in\nferromagnetic graphene.\nIn order to elaborate the Kondo nature of the kink\nstructure found in Fig.3(a), we investigate in detail the\nkink position as a function of the exchange ﬁled h. Since\nit is diﬃcult to precisely determine the kink position, the\nfollowing strategy is adopted. We assume that the kink\nstructure is formed by the superposition\nF(ω) =F1(ω)+F2(ω) (15)5\n/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53\n/s48 /s49/s48/s46/s48/s48/s46/s49/s49/s46/s52/s40/s116/s111/s112/s41 /s32 /s32 /s32 /s32 /s49/s46/s48/s44/s32/s49/s46/s50/s44 /s32 /s32 /s32 /s32 /s48/s46/s54/s44/s32/s48/s46/s56/s44 /s32 /s32 /s32 /s32 /s48/s46/s50/s44/s32/s48/s46/s52/s44 /s32 /s32 /s32 /s32 /s104 /s32/s32/s32/s32/s32/s47 /s32/s32/s32/s32/s32/s32/s32/s32/s71\n/s48/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s71\n/s48/s32/s32/s65/s40/s48/s41\n/s32/s32/s71\n/s48/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s65 /s40/s119 /s41\n/s119 /s32 /s32 /s32 /s32 /s32 /s32 /s47 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s71\n/s48\n/s104 /s32 /s32 /s47/s71\n/s48/s61 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s48 /s32 /s32 /s32 /s32 /s40/s98/s116/s109/s41/s44\n/s40/s97/s41\n/s40/s98/s41\n/s48/s46/s54/s48/s46/s50/s48/s46/s48/s56/s48/s46/s48/s51\n/s32/s32/s32\n/s32/s48/s108/s110/s52/s83\n/s105/s109/s112\n/s84 /s32 /s32 /s32 /s32 /s32 /s32 /s47 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s68/s108/s110/s50\n/s49/s48/s45/s51\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 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s49/s48/s45/s50\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 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s49/s48/s45/s49\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 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32/s32/s49/s48/s48/s104 /s32 /s32 /s32 /s32 /s32 /s47 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s71\n/s48/s32 /s32 /s32 /s32 /s32 /s61 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s32 /s48\nFIG. 3: (a) Impurity spectral density A(ω) and (b) impurity\nentropySimp(T) for diﬀerent ferromagnetic exchange ﬁeld h.\nParameters used are µ= 0, Γ 0= 0.1, andδ= 0. The curves\nin (a) are oﬀset for clarity, and the dashed lines guide the\nevolution of kink structure in A(ω). The inset of (b) shows\nthe spectral density at the Fermi energy as a function of h.\nof the Kondo resonance peak [77]\nF1(ω) =c1Re/bracketleftbigg/parenleftbiggic3\nω−c2+ic3/parenrightbigg1\n2/bracketrightbigg\n(16)\nand a linear spectrum F2(ω) =c4+c5|ω|. Here param-\netersc1∼c5are determined by using Eq.(15) to ﬁt the\nspin-resolved NRG spectral data [Figs.4(a) and 4(b)]. In\nparticular, we take the ﬁtting parameter c2, which gives\nthe position of the Kondo resonance peak, as the kink\nposition. The resulting dependence of the kink position\non the ﬁeld his given in Fig.4(c). On the other hand,\nperturbative scaling analysis [15, 17] indicates that the\nferromagnetism of Dirac electrons can induce a local ex-\nchange ﬁeld acting on the impurity, which in turn leads\nto a spin-dependent renormalization /tildewideεdσand a spin split-\nting ∆ = |/tildewideεd↑−/tildewideεd↓|of the impurity level εd:\n/tildewideεdσ=εd−1\nπ/integraldisplay\ndω/braceleftbiggΓσ(ω)[1−f(ω)]\nω−εd+Γ¯σ(ω)f(ω)\nεd+U−ω/bracerightbigg\n,\n(17)\nwheref(ω) is the Fermi distribution function. As shown\nin Fig.4(c), the local spin splitting ±∆ calculated from/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54/s45/s48/s46/s50/s45/s48/s46/s49/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51/s48/s46/s48/s48/s48/s46/s48/s54/s48/s46/s49/s50\n/s45/s48/s46/s51 /s45/s48/s46/s50 /s45/s48/s46/s49 /s48/s46/s48/s48/s46/s48/s48/s48/s46/s48/s54/s48/s46/s49/s50\n/s32/s32/s32\n/s40/s99/s41/s71\n/s48/s32/s32/s65 /s40/s119 /s41\n/s71\n/s48/s32/s32/s65 /s40/s119 /s41/s107/s105/s110/s107/s32/s112/s111/s115/s105/s116/s105/s111/s110/s32/s78/s82/s71\n/s32\n/s104 /s32/s32/s47 /s32/s71\n/s48/s119 /s47/s71\n/s48/s119 /s47/s71\n/s48\n/s177 /s32/s32/s32/s78/s82/s71\n/s32/s102/s105/s116/s116/s105/s110/s103\n/s32/s32\n/s32/s32\n/s40/s98/s41\n/s32/s78/s82/s71\n/s32/s102/s105/s116/s116/s105/s110/s103\n/s32/s32\n/s32/s32\n/s40/s97/s41\nFIG. 4: (a) and (b) NRG results for the Kondo shoulder in\nthe spin-resolved spectral function Aσ(ω) ath/Γ0= 0.32, ﬁt-\nted by Eq.(15) in the main text. (c) NRG results for the\nposition of spin-resolved Kondo shoulder as a function of th e\nferromagnetic exchange ﬁeld h, in comparison with the eﬀec-\ntive local-level splitting ±∆ [from Eq.(17) in the main text].\nParameters used are µ= 0, Γ 0= 0.1, andδ= 0.\nEq.(17) are in good agreement with the kink position\ncalculated by the NRG method. This implies that the\ntwokinkstructuresappearinginthespectralfunctionare\nactually two split Kondo peaks distorted by graphene’s\nlinear dispersion.\nAnother feature that reveals the Kondo nature of the\nkink structure is its magnetic ﬁeld dependence. We ap-\nply a local magnetic ﬁeld Bon the impurity atom. This\nadds the Zeeman energy, B(n↑−n↓) withnσ=d†\nσdσ, to\nthe Hamiltonian (1). Figures 5(a) and 5(b) present the\nevolution of Kondo kink with the magnetic ﬁeld. While\nthe magnetic ﬁeld Bparallel to the exchange ﬁeld hsim-\nply suppresses the Kondo kinks [Fig.5(b)], the situation\nis more interesting when Bis antiparallel to h[Fig.5(a)].\nAs the antiparallelmagneticﬁeld increases, the twokinks\nﬁrst approach to each other and deform into two peaks.\nThe two peaks then merge into a single Kondo reso-\nnance at a particular ﬁeld B=Bc[see the red curve in\nFig.5(a)]. Increasing further the magnetic ﬁeld, the reso-\nnance peak splits again and eventually fades away. Since\natB=Bcthe spin asymmetry in the local occupancies\nalso vanishes [see the point n↑=n↓= 0.5 in Fig.5(c)],\nwe actually obtain a fully-screened ground state and the\nstrong-coupling Kondo limit is reached at Bc. This com-\npensationeﬀectofthe externalmagneticﬁeldhasalready\nbeen observed for Kondo eﬀect in conventionalferromag-\nnetic materials [5, 16, 17, 19, 21]. Unlike the previous6\n/s45/s48/s46/s52 /s45/s48/s46/s50 /s48/s46/s48 /s48/s46/s50 /s48/s46/s52/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48\n/s45/s48/s46/s52 /s48/s46/s48 /s48/s46/s52/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51\n/s45/s48/s46/s48/s55/s50/s52/s55/s52 /s45/s48/s46/s48/s55/s50/s52/s55/s49/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s45/s48/s46/s51/s45/s48/s46/s48/s55/s50/s52/s55/s50/s53/s45/s48/s46/s48/s54/s32\n/s32/s48 /s32/s32/s65/s40 /s41/s32\n/s32\n/s47 /s32/s40/s97/s41\n/s66/s47 /s32\n/s48/s61 /s32/s32/s48\n/s48/s46/s48/s56/s40/s98/s116/s109/s41/s66/s47\n/s48/s61 /s32/s32/s48/s40/s116/s111/s112/s41/s44/s32/s48/s46/s48/s50/s44/s32\n/s40/s98/s41/s48/s32/s32/s65/s40 /s41\n/s48/s46/s48/s52/s44/s32/s48/s46/s48/s54/s44\n/s32/s32\n/s40/s99/s41/s110/s32/s110\n/s32/s110\n/s66/s47 /s32\nFIG. 5: (a),(b) Evolution of the Kondo shoulder in the spec-\ntral function A(ω) with the variation of external magnetic\nﬁeldB. (c) Spin-resolved local occupation nσas a function\nofB. Note that the magnetic ﬁeld at which n↑=n↓= 0.5 in\n(c) is exactly the same magnetic ﬁeld where the two Kondo\nshoulders merge into a single Kondo peak [the red curve in\n(a)]. Parameters used are h/Γ0= 0.5,µ= 0, Γ 0= 0.1, and\nδ= 0.\ncases, the compensation ﬁeld in our case needs to be ex-\ntremely ﬁnely tuned because the step in the occupancy\nnσ(B) as a function of Bis very steep [Fig.5(c)] due to\nthe exotic linear spectrum of ferromagnetic graphene.\nIn the presence of external magnetic ﬁeld B, the renor-\nmalization Eq.(17) of the impurity level within the per-\nturbative scaling theory [15, 17] should be modiﬁed in\nthe following way: in the right hand side of Eq.(17), the\nﬁrst two εdbeing replaced with εd+σB, while the third\none being replaced with εd+ ¯σB. Then the compen-\nsation ﬁeld Bccan be determined under the condition\n∆ =|/tildewideεd↑−/tildewideεd↓|= 0 which means the external ﬁeld fully\ncompensates the local spin splitting induced by the fer-\nromagnetic Dirac electrons. Figure 6(a) shows that the\ncompensation ﬁeld Bc(h) as a function of hcalculated by\nthe scaling theory is in good agreementwith the NRG re-\nsults calculated via the criterion n↑(Bc) =n↓(Bc).\nFixing the magnetic ﬁeld at Bc(h) for any nonzero\nh, a full Kondo resonance always develops in the local\nspectral density and its width at half maximum gives\nthe Kondo temperature TK. We ﬁnd that upon in-\ncreasing h, the Kondo resonance broadens and lowers\n[Fig.6(b)], resulting in an increasing Kondo temperature\n[Fig.6(c)]. This feature is strikingly diﬀerent from the\nKondo physics in conventional ferromagnetic materials.\nIn conventional materials, intensive studies [14, 16–19]\nhave demonstrated the destructive inﬂuence of ferromag-\nnetism, which reduces TKand suppresses Kondo corre-\nlations, even though the compensation ﬁeld is already/s48/s53/s49/s48/s49/s53/s48/s46/s48 /s48/s46/s52 /s48/s46/s56 /s49/s46/s50 /s49/s46/s54\n/s45/s48/s46/s50/s53 /s48/s46/s48/s48 /s48/s46/s50/s53/s45/s49/s46/s52/s48/s46/s48/s49/s46/s52\n/s104 /s32/s47 /s32/s71\n/s48/s66 /s32/s47 /s32/s71\n/s48/s104 /s32/s47 /s32/s71\n/s48/s32/s32\n/s32/s48 /s46 /s48 /s48 /s48 /s46 /s48 /s48 /s49 /s53 /s49 /s48 /s46 /s48 /s48 /s51 /s48 /s50 /s48 /s46 /s48 /s48 /s52 /s53 /s51 /s48 /s46 /s48 /s48 /s54 /s48 /s52 /s48 /s46 /s48 /s48 /s55 /s53 /s53 /s48 /s46 /s48 /s48 /s57 /s48 /s54 /s48 /s46 /s48 /s49 /s48 /s54 /s48 /s46 /s48 /s49 /s50 /s49 /s48 /s46 /s48 /s49 /s51 /s54 /s48 /s46 /s48 /s49 /s53 /s49 /s48 /s46 /s48 /s49 /s54 /s54 /s48 /s46 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/s48/s46/s55/s56\n/s40/s97/s41/s71\n/s48/s32/s32/s65/s40/s119/s41/s68/s47/s71\n/s48\n/s45/s49/s48/s45/s57\n/s45/s49/s48/s45/s54\n/s45/s49/s48/s45/s51/s66/s61/s66\n/s99/s40/s98/s41/s32\n/s32/s45/s49/s48/s48/s32/s48\n/s32/s48/s46/s52\n/s32/s48/s46/s53\n/s32/s48/s46/s55\n/s32/s49\n/s32/s49/s46/s53\n/s49/s48/s48\n/s49/s48/s45/s51\n/s49/s48/s45/s54\n/s49/s48/s45/s57/s104/s47 /s71\n/s48/s32\n/s32/s49/s48/s45/s57/s49/s48/s45/s54/s40/s99/s41/s32\n/s119 /s47 /s32/s71\n/s48/s32\n/s32/s84\n/s75/s49/s48/s45/s51\nFIG. 6: (a) Eﬀective spin splitting ∆ of the impurity level as\na function of the exchange ﬁeld hin ferromagnetic graphene\nand the local external magnetic ﬁeld B, calculated by the per-\nturbative scaling theory. Solid circles are the NRG results of\nthe compensation magnetic ﬁeld Bc(h). (b) Impurity spectral\ndensityA(ω) atB=Bc(h) for various h. (c) Kondo tempera-\nture as a function of h. Parameters used are µ= 0, Γ 0= 0.1,\nandδ= 0.\napplied. On the contrary, we demonstrate here that\nalthough the Kondo eﬀect is absent for a particle-hole\nsymmetric adatom in undoped graphene, the presence of\nferromagnetism can induce this eﬀect by enhancing the\nKondo temperature and Kondo correlations. This con-\nstructive inﬂuence of ferromagnetism on the graphene\nKondo physics is one of our main ﬁndings. An intu-\nitive understanding of this behavior can be achieved by\nexamining the variation of graphene density of states\nwith the ferromagnetic exchange ﬁeld h. As shown in\nFig.1(a) and Eq.(11), the hybridization at the Fermi\nenergy is a monotonically increasing function of h, i.e.,\nΓσ(EF) = Γ0|h|/Dforµ=ε0= 0, so is the graphene\ndensity of states. Therefore, as hincreases, more Dirac\nelectrons can gather near the Fermi energy, favouring the\nformation of a Kondo singlet at large energy scales.\nC. Kondo physics in the Dirac half-metal regime\nIn this subsection, we ﬁx the ferromagnetic exchange\nﬁeldh/Γ0= 0.5 in graphene and investigate the eﬀect of\ncarrier doping by tuning the chemical potential µ. Note\nthat we only present results of electron doping ( µ >0)\nbecause for a symmetric ( δ= 0) impurity the physics\narising from hole doping ( µ <0) is exactly the same\nas in the electron doping case. At zero magnetic ﬁeld,\nFig.7(a) shows the evolution of Kondo shoulders in the\nspectral function A(ω) =A↑(ω)+A↓(ω) with increasing\nµ >0. While the shoulder at ω >0, which is of the spin-7\n↑species, monotonically rises as µincreases, the shoul-\nder atω <0, which is of the spin- ↓species, goes down\nﬁrst and then rises again. Particularly, in the half-metal\nregime of µ=hwhere the Fermi level aligns with the\nspin-↑Dirac point but Γ ↓(ω) is ﬁnite at the Fermi energy\n[Fig.1(b)], the spin- ↓shoulder is completely suppressed,\nleaving only the spin- ↑shoulder [see the red curve in\nFig.7(a)]. Staying in this half-metal regime, we proceed\nour study by applying the magnetic ﬁeld Bto adjust\nthe local spectral feature. As Bincreases to compen-\nsate the spin asymmetry of local occupations, the spin- ↑\nKondo shoulder gradually develops into the full Kondo\nresonance at the Fermi energy [Figs.7(b) and 7(c)]. On\nthe other hand, the spin- ↓spectral function A↓(ω) is al-\nways featureless near the Fermi energy, although its two\nHubbard bands become more symmetric [Fig.7(d)]. De-\nspite the absence of Kondo resonance in A↓(ω), the local\nspin is still fully screened by Dirac electrons in the half-\nmetal regime, since n↑=n↓can always be achieved at\nBc. We ﬁnd again that the compensation ﬁeld Bc(µ) as\na function of µ, determined by the NRG method via the\ncriterion n↑(Bc) =n↓(Bc), is in good agreementwith the\ncorresponding results of the perturbative scaling theory\n[Fig.7(e)].\nThe Kondo physics revealed here for the Dirac half\nmetals seems peculiar: the Kondo resonance shows up\nonly for the spin- ↑component, while Γ ↑(EF) vanishes\nlinearly and Γ ↓(EF) is ﬁnite. Nonetheless, this feature\nis somewhat consistent with previous results obtained in\nconventional ferromagnetic materials having an energy-\nindependent hybridization Γ σ. In such impurity systems,\nthe Friedel sum rule [17–19] can be used to relate the\nspectralfunction Aσ(EF) atthe Fermienergytothelocal\noccupation nσ,\nAσ(EF) =sin2(πnσ)\nπΓσ. (18)\nWhen equal occupation n↑=n↓is achieved by the exter-\nnal compensation ﬁeld Bc, Eq.(18) implies that a small\nhybridization in one spin direction results in a large am-\nplitude of the Kondo resonance in the same spin direc-\ntion, and vice versa. Although our ﬁnding is somewhat\nunderstandable from the above analysis, to the best of\nour knowledge, such kind of fully polarized Kondo res-\nonance appearing only in one spin direction has never\nbeendiscoveredbefore. Apparently,theexceptionalband\nstructure offerromagneticgrapheneis crucial in inducing\nthis unusual phenomenon.\nIV. CONCLUSION\nWe have studied the constructive inﬂuence of ferro-\nmagnetism on the Kondo physics of an impurity atom\nadsorbed on ferromagnetic graphene. It is demonstrated\nthat for a symmetric impurity in undoped graphene,/s45/s48/s46/s52 /s48/s46/s48 /s48/s46/s52/s48/s46/s48/s48/s46/s52/s48/s46/s56\n/s45/s49 /s48 /s49/s48/s46/s48/s48/s46/s53/s49/s46/s48/s48/s50/s52/s45/s48/s46/s51 /s48/s46/s48 /s48/s46/s51/s48/s46/s48/s48/s46/s51/s48/s46/s54\n/s48/s46/s53/s44/s32/s48/s46/s56/s44/s32/s49/s71\n/s48/s32/s32/s65 /s40/s119 /s41/s40/s97/s41\n/s119 /s32/s32/s47 /s32/s71\n/s48\n/s71\n/s48/s32/s32/s65 /s40/s119 /s41/s71\n/s48/s32/s32/s32/s65 /s40/s119 /s41/s71\n/s48/s32/s32/s65/s40 /s119 /s41\n/s32/s32\n/s109 /s32/s47 /s32/s71\n/s48/s32/s32/s61 /s32/s32/s48/s44/s32/s48/s46/s51/s44\n/s32/s32\n/s119 /s32/s32/s47 /s32/s71\n/s48/s40/s100/s41/s32\n/s32/s109 /s32/s61 /s32/s48/s44/s32/s57/s46/s56/s49/s44/s32/s49/s48/s46/s50/s44\n/s45/s48/s46/s50 /s48/s46/s48 /s48/s46/s50/s45/s50/s48/s50\n/s32/s109 /s32/s32/s47 /s32/s71\n/s48\n/s66 /s32/s47 /s32/s32/s71\n/s48/s48/s46/s48/s48/s48 /s57/s46/s56/s48/s49/s69/s45/s48/s52 /s48/s46/s48/s48/s49/s57/s54/s48 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/s48/s46/s53/s55/s50/s52 /s48/s46 /s53/s55/s51/s51 /s48/s46/s53/s55/s52/s51 /s48/s46/s53/s55/s53/s51 /s48/s46/s53/s55/s54/s51 /s48/s46/s53/s55/s55/s51 /s48/s46/s53/s55/s56/s50 /s48/s46/s53/s55/s57/s50 /s48/s46/s53/s56/s48/s50 /s48/s46/s53/s56/s49/s50 /s48/s46/s53/s56/s50/s50 /s48/s46/s53/s56/s51/s49 /s48/s46/s53/s56/s52/s49 /s48/s46/s53/s56/s53/s49 /s48/s46/s53/s56/s54/s49 /s48/s46/s53/s56/s55/s49 /s48/s46/s53/s56/s56/s48 /s48/s46/s53/s56/s57/s48 /s48/s46/s53/s57/s48/s48/s68/s47/s71\n/s40/s101/s41/s32 /s32 /s32 /s32 /s32\n/s32 /s32/s48/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s48/s46/s53/s57\n/s48/s32/s109/s61/s56/s46/s53/s56\n/s32/s109/s61/s57/s46/s56/s49\n/s32/s109/s61\n/s119 /s32/s32/s47 /s32/s71\n/s48/s45/s49/s48/s45/s53\n/s45/s49/s48/s45/s49/s48/s32 /s32/s40/s99/s41/s32\n/s32/s32\n/s45/s49/s48/s48/s66 /s32/s61 /s32/s40/s49/s45/s49/s48/s45/s109\n/s41/s66\n/s99\n/s49/s48/s48\n/s49/s48/s45/s53\n/s49/s48/s45/s49/s48\n/s32/s32\n/s32/s119 /s32/s32/s47 /s32/s71\n/s48/s40/s98/s41\n/s32/s32\n/s66/s61/s48/s44/s32/s45/s48/s46/s48/s48/s51/s44/s32/s45/s48/s46/s48/s48/s53/s44/s32\n/s45/s48/s46/s48/s48/s54/s44/s32/s66\n/s99\nFIG. 7: (a) Local spectral density A(ω) for various chemi-\ncal potentials µat zero magnetic ﬁeld. (b)-(d) Spin-resolved\nspectral density Aσ(ω) for various magnetic ﬁelds Bin the\nhalf-metal regime ( µ=h). Here the compensation ﬁeld\nBc/Γ0≃ −0.0646688 and the exponent mis introduced for\nthe convenience of signifying magnetic ﬁelds very close to Bc.\nIn (a), (b), and (d), arrows indicate the evolution of spectr al\nfunctions with the parameters listed in the ﬁgures. (e) Eﬀec -\ntive spin splitting ∆ of the impurity level as a function of µ\nandB, calculated by the perturbative scaling theory. Solid\ncircles are theNRGresults ofthe compensation magnetic ﬁel d\nBc(µ). Parameters used are h/Γ0= 0.5, Γ0= 0.1, andδ= 0.\nKondo correlations can emerge only in the presence of\nferromagnetism in graphene and the spin polarization\nof Dirac electrons strikingly enhances the Kondo tem-\nperature. Driving ferromagnetic graphene into the half-\nmetallic regime by carrier doping, we predict an abnor-\nmal Kondo resonance that develops in one spin direction\nbut is absent in the opposite direction. These intrigu-\ning features can be locally probed by scanning tunnel-\ning microscopy, and are in principle also accessible in\nbulk transport measurements. Our results predicted in\nthispaper,e.g.,thefullyspin-polarizedKondoresonance,\nmay have potential applications to spintronics based on\nferromagnetic graphene.\nACKNOWLEDGEMENTS\nThis work is ﬁnancially supported by NSF-China\n(Grants No. 11574007, No. 11504066, No. 11874428, and8\nNo. 11874187),National Key Research and Development\nProgram of China (Grant No. 2017YFA0303301), Bei-\njingMunicipal Science&TechnologyCommission(Grant\nNo. Z181100004218001), and Innovation-Driven Project\nof Central South University (Grant No. 2018CX044).\n∗Corresponding author: fangtiefeng@lzu.edu.cn\n[1] J. Kondo, Prog. Theor. Phys. 32, 37 (1964).\n[2] K.G. Wilson, Rev. Mod. Phys. 47, 773 (1975).\n[3] A.C. Hewson, The Kondo Problem to Heavy Fermions\n(Cambridge University Press, Cambridge, UK, 1993).\n[4] L.P. Kouwenhoven and L.I. Glazman, Phys. World 14,\n33 (2001).\n[5] J. R. Hauptmann, J. Paaske, and P. E. Lindelof, Nat.\nPhys.4, 373 (2008).\n[6] K.J. Franke, G. Schulze, and J.I. Pascual, Science 332,\n940 (2011).\n[7] T.F. Fang, A.M. Guo, H.T. Lu, H.G. Luo, and Q.F.\nSun, Phys. Rev. B 96, 085131 (2017).\n[8] T.F. Fang, A.M. Guo, and Q.F. Sun, Phys. Rev. B 97,\n235115 (2018).\n[9] R. Wang, W. Su, J.X. Zhu, C.S. Ting, H. Li, C.F.\nChen, B.G. Wang, and X.Q. Wang, Phys. Rev. Lett.\n122, 087001 (2019).\n[10] N. Sergueev, Q.F. Sun, H. Guo, B.G. Wang, and J.\nWang, Phys. Rev. B 65, 165303 (2002).\n[11] P. Zhang, Q.-K. Xue, Y. Wang, and X. C. Xie, Phys.\nRev. Lett. 89, 286803 (2002).\n[12] R. L´ opez and D. S´ anchez, Phys. Rev. Lett. 90, 116602\n(2003).\n[13] M.-S. Choi, D. S´ anchez, and R. L´ opez, Phys. Rev. Lett.\n92, 056601 (2004).\n[14] J. Martinek, Y. Utsumi, H. Imamura, J. Barna´ s, S.\nMaekawa, J. K¨ onig, and G. Sch¨ on, Phys. Rev. Lett. 91,\n127203 (2003).\n[15] J. Martinek, M. Sindel, L. Borda, J. Barna´ s, R. Bulla,\nJ. K¨ onig, G. Sch¨ on, S. Maekawa, and J. von Delft, Phys.\nRev. B72, 121302(R) (2005).\n[16] P. Simon, P. S. Cornaglia, D. Feinberg, and C. A. Bal-\nseiro, Phys. Rev. B 75, 045310 (2007).\n[17] M. Sindel, L. Borda, J. Martinek, R. Bulla, J. K¨ onig, G.\nSch¨ on, S. Maekawa, and J. von Delft, Phys. Rev. B 76,\n045321 (2007).\n[18] D. Matsubayashi and M. Eto, Phys. Rev. B 75, 165319\n(2007).\n[19] J. Martinek, M. Sindel, L. Borda, J. Barna´ s, J. K¨ onig,\nG. Sch¨ on, and J. von Delft, Phys. Rev. Lett. 91, 247202\n(2003).\n[20] A. N. Pasupathy, R. C. Bialczak, J. Martinek, J. E.\nGrose, L. A. K. Donev, P. L. McEuen, and D. C. Ralph,\nScience306, 86 (2004).\n[21] K. Hamaya, M. Kitabatake, K. Shibata, M. Jung, M.\nKawamura, K. Hirakawa, T. Machida, T. Taniyama, S.\nIshida, and Y. Arakawa, Appl. Phys. Lett. 91, 232105\n(2007).\n[22] L. Hofstetter, A. Geresdi, M. Aagesen, J. Nyg˚ ard, C.\nSch¨ onenberger, and S. Csonka, Phys. Rev. Lett. 104,\n246804 (2010).\n[23] L. Li, Y.Y. Ni, T.F. Fang, and H.G. Luo, Phys. Rev. B\n84, 235405 (2011).[24] R.ˇZitko, J.S. Lim, R. L´ opez, J. Martinek, and P. Simon,\nPhys. Rev. Lett. 108, 166605 (2012).\n[25] K.P. W´ ojcik and I. Weymann, Phys. Rev. B 91, 134422\n(2015).\n[26] I. Weymann and K.P. W´ ojcik, Phys. Rev. B 95, 155427\n(2017).\n[27] I. Weymann, R. Chirla, P. Trocha, and C.P. Moca, Phys.\nRev. B97, 085404 (2018).\n[28] P. R. Wallace, Phys. Rev. 71, 622 (1947).\n[29] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang,\nY. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A.\nFirsov, Science 306, 666 (2004).\n[30] A. H. C. Neto, F. Guinea, N. M. R. Peres, K. S.\nNovoselov, and A. K. Geim, Rev. Mod. Phys. 81, 109\n(2009).\n[31] X. Du, I.Skachko, F. Duerr, A.Luican, andE. Y.Andrei,\nNature462, 192 (2009).\n[32] K. I. Bolotin, F. Ghahari, M. D. Shulman, H. L. Stormer,\nand P. Kim, Nature 462, 196 (2009).\n[33] Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi,\nE. Kaxiras, and P. Jarillo-Herrero, Nature 556, 43\n(2018).\n[34] Y. Cao, V. Fatemi, A. Demir, S. Fang, S. L. Tomarken,\nJ. Y. Luo, J. D. Sanchez-Yamagishi, K. Watanabe, T.\nTaniguchi, E. Kaxiras, R. C. Ashoori, and P. Jarillo-\nHerrero, Nature 556, 80 (2018).\n[35] D. Withoﬀ and E. Fradkin, Phys. Rev. Lett. 64, 1835\n(1990).\n[36] K. Ingersent, Phys. Rev. B 54, 11936 (1996).\n[37] C. Gonzalez-Buxton and K. Ingersent, Phys. Rev. B 57,\n14254 (1998).\n[38] T. Eelbo, M. Wa´ sniowska, P. Thakur, M. Gyamﬁ, B.\nSachs, T. O. Wehling, S. Forti, U. Starke, C. Tieg, A.\nI. Lichtenstein, and R. Wiesendanger, Phys. Rev. Lett.\n110, 136804 (2013).\n[39] F. Donati, Q. Dubout, G. Aut` es, F. Patthey, F. Calleja,\nP. Gambardella, O. V. Yazyev, and H. Brune, Phys. Rev.\nLett.111, 236801 (2013).\n[40] F. Donati, L. Gragnaniello, A. Cavallin, F. D. Natterer ,\nQ. Dubout, M. Pivetta, F. Patthey, J. Dreiser, C. Pi-\namonteze, S. Rusponi, and H. Brune, Phys. Rev. Lett.\n113, 177201 (2014).\n[41] O. V. Yazyev and L. Helm, Phys. Rev. B 75, 125408\n(2007).\n[42] Y. Wang, Y. Huang, Y. Song, X. Zhang, Y. Ma, J. Liang,\nand Y. Chen, Nano Lett. 9, 220 (2009).\n[43] R. R. Nair, M. Sepioni, I.-L. Tsai, O. Lehtinen, J.\nKeinonen, A. V. Krasheninnikov, T. Thomson, A. K.\nGeim, and I. V. Grigorieva, Nat. Phys. 8, 199 (2012).\n[44] M.Ziatdinov, S.Fujii, K.Kusakabe, M.Kiguchi, T.Mori ,\nand T. Enoki, Phys. Rev. B 89, 155405 (2014).\n[45] E. Zaminpayma, M. E. Razavi, and P. Nayebi, Applied\nSurface Science 414, 101 (2017).\n[46] P.S. Cornaglia, G. Usaj, and C.A. Balseiro, Phys. Rev.\nLett.102, 046801 (2009).\n[47] H.B. Zhuang, Q.F. Sun, and X.C. Xie, EPL (Europhys.\nLett.)86, 58004 (2009).\n[48] D. Jacob andG.Kotliar, Phys.Rev.B 82, 085423 (2010).\n[49] M. Vojta, L. Fritz, and R. Bulla, EPL (Europhys. Lett.)\n90, 27006 (2010).\n[50] J.-H. Chen, L. Li, W.G. Cullen, E.D. Williams, and\nM.S. Fuhrer, Nat. Phys. 7, 535 (2011).\n[51] L. Fritz and M. Vojta, Rep. Prog. Phys. 76, 032501\n(2013); M. Vojta and L. Fritz, Phys. Rev. B 70, 0945029\n(2004); L. Fritz and M. Vojta, Phys. Rev. B 70, 214427\n(2004).\n[52] V.V. Mkhitaryan and E.G. Mishchenko, Phys. Rev.\nLett.110, 086805 (2013).\n[53] D. Mastrogiuseppe, A. Wong, K. Ingersent, S.E. Ulloa,\nand N. Sandler, Phys. Rev. B 89, 081101(R) (2014);\nPhys. Rev. B 90, 035426 (2014).\n[54] P.W. Lo, G.Y. Guo, and F.B. Anders, Phys. Rev. B 89,\n195424 (2014).\n[55] L. Craco, S.S. Carara, T.A. da Silva Pereira, and M.V.\nMiloˇ sevi´ c, Phys. Rev. B 93, 155417 (2016).\n[56] D.A. Ruiz-Tijerina and L.G.G.V. Dias da Silva, Phys.\nRev. B95, 115408 (2017).\n[57] A. Allerdt, A.E. Feiguin, and S. Das Sarma, Phys. Rev.\nB95, 104402 (2017).\n[58] J. Hwang, K. Kim, H. Ryu, J. Kim, J.E Lee, S. Kim,\nM. Kang, B.G. Park, A. Lanzara, J. Chung, S.K. Mo, J.\nDenlinger, B.I. Min, and C. Hwang, Nano Lett. 18, 3661\n(2018).\n[59] Y. Jiang, P.W. Lo, D. May, G. Li, G.Y. Guo, F.B. An-\nders, T. Taniguchi, K. Watanabe, J. Mao, and E.Y. An-\ndrei, Nat. Commun. 9, 2349 (2018).\n[60] D. May, P.W. Lo, K. Deltenre, A. Henke, J. Mao, Y.\nJiang, G. Li, E.Y. Andrei, G.Y. Guo, and F.B. Anders,\nPhys. Rev. B 97, 155419 (2018).\n[61] D. Zhai, K. Ingersent, S.E. Ulloa, and N. Sandler, Phys.\nRev. B99, 195410 (2019).\n[62] Z.G. Zhu, K.H. Ding, and J. Berakdar, EPL (Europhys.\nLett.)90, 67001 (2010).\n[63] M. Kharitonov and G. Kotliar, Phys. Rev. B 88,\n201103(R) (2013).[64] T.H. Lee, K.Y.J. Zhang, C.H. Chung, and S. Kirchner,\nPhys. Rev. B 88, 085431 (2013).\n[65] T.O. Wehling, A.V. Balatsky, M.I. Katsnelson, A.I.\nLichtenstein, and A. Rosch, Phys. Rev. B 81, 115427\n(2010).\n[66] B. Uchoa, T.G. Rappoport, and A.H.C. Neto, Phys.\nRev. Lett. 106, 016801 (2011).\n[67] H. Haugen, D. Huertas-Hernando, and A. Brataas, Phys.\nRev. B77, 115406 (2008).\n[68] Z. Wang, C. Tang, R. Sachs, Y. Barlas, and J. Shi, Phys.\nRev. Lett. 114, 016603 (2015).\n[69] P. Wei, S, Lee, F. Lamaitre, L. Pinel, D. Cutaia, W. Cha,\nF. Katmis, Y. Zhu, D. Haiman, J. Hone, J.S. Moodera,\nand C.T. Chen, Nat. Mater. 15, 711 (2016).\n[70] H.R. Krishna-murthy, J.W. Wilkins, and K.G. Wilson,\nPhys. Rev. B 21, 1003 (1980); 21, 1044 (1980).\n[71] R. Bulla, T.A. Costi, and T. Pruschke, Rev. Mod. Phys.\n80, 395 (2008).\n[72] F.B. Anders and A. Schiller, Phys. Rev. Lett. 95, 196801\n(2005); Phys. Rev. B 74, 245113 (2006).\n[73] R. Peters, T. Pruschke, and F. B. Anders, Phys. Rev. B\n74, 245114 (2006).\n[74] A. Weichselbaum and J. von Delft, Phys. Rev. Lett. 99,\n076402 (2007).\n[75] T.F. Fang, N.H. Tong, Z. Cao, Q.F. Sun, and H.G. Luo,\nPhys. Rev. B 92, 155129 (2015).\n[76] Q.F. Sun and X.C. Xie, Phys. Rev. Lett. 104, 066805\n(2010).\n[77] H.O. Frota, Phys. Rev. B 45, 1096 (1992)."    },    {        "title": "1403.4002v2.Spin_resonance_without_spin_splitting.pdf",        "content": "arXiv:1403.4002v2  [cond-mat.mes-hall]  3 Apr 2016Spin resonance without spin splitting\nM. Hell(1,2), B. Sothmann(3), M. Leijnse(4), M. R. Wegewijs(1,2,5), and J. K¨ onig(6)\n(1) Peter Gr¨ unberg Institut, Forschungszentrum J¨ ulich, 52425 J¨ ulich, Germany\n(2) JARA- Fundamentals of Future Information Technology\n(3) D´ epartement de Physique Th´ eorique, Universit´ e de Ge n` eve, CH-1211 Gen` eve 4, Switzerland\n(4) Solid State Physics and Nanometer Structure Consortium (nmC@LU), Lund University, 221 00 Lund, Sweden\n(5) Institute for Theory of Statistical Physics, RWTH Aache n, 52056 Aachen, Germany\n(6) Theoretische Physik and CENIDE, Universit¨ at Duisburg -Essen, 47048 Duisburg, Germany\n(Dated: August 8, 2018)\nWe predict that a single-level quantum dot without discerni ble splitting of its spin states develops\na spin-precession resonance in charge transport when embed ded into a spin valve. The resonance\noccurs in the generic situation of Coulomb blockaded transp ort with ferromagnetic leads whose\npolarizations deviate from perfect antiparallel alignmen t. The resonance appears when electrically\ntuning the interaction-induced exchange ﬁeld perpendicul ar to one of the polarizations – a simple\ncondition relying on vectors in contrast to usual resonance conditions associated with energy split-\ntings. The spin resonance can be detected by stationary dI/dVspectroscopy and by oscillations in\nthe time-averaged current using a gate-pulsing scheme. The generic noncollinearity of the ferromag-\nnets and junction asymmetry allow for an all-electric deter mination of the spin-injection asymmetry,\nthe anisotropy of spin relaxation, and the magnitude of the e xchange ﬁeld. We also investigate the\nimpact of a nearby superconductor on the resonance position . Our simplistic model turns out to be\ngeneric for a broad class of coherent few-level quantum syst ems.\nPACS numbers: 85.75.-d, 73.63.Kv, 85.35.-p\nI. INTRODUCTION\nGaining fast, coherent control over a few spins or even\na single spin is at the heart of current experimental ef-\nforts in both spintronics [1–3] and solid-state quantum\ncomputing [4–7]. Single-molecule magnets in gateable\nnanojunctions [8–11] or adatoms and molecules manipu-\nlated by STM [1, 12–14] provide a bottom-up approach\nto achieve this goal. Promising top-down routes combine\nconventional spin valves [15–19] with nanoscale quantum\ndot (QD) devices [20–26]. Such coherent quantum sys-\ntems are typically manipulated through resonance tech-\nniques, e.g., by electromagnetic pulses [5, 6]. In gen-\neral, this requiresthat the frequencyofthe applied pulses\nmatches the splitting of, e.g., a two-level system. In this\npaper we predict that quite generically resonances can\nappear in systems with quasi-degenerate levels that do\nnot involve such a matching to a splitting. Instead, a\ncondition involving vectorshas to be satisﬁed.\nWe illustrate this for a QD embedded in a noncollinear\nspin valve, a speciﬁc example relevant for spintronics and\nspin-based quantum computation. It leads to an unex-\npected, strongly gate-voltage dependent feature in the\nstationary nonlinear conductance ( dI/dV b) extending all\nacross the Coulomb blockade regime. It arises under\nnonequilibrium conditions but disappears upon revers-\ning the bias voltage. Strikingly, it can appear at voltages\nmuch larger or smaller than any of the naively expected\nenergy scales, showing that it does not ﬁt into the usual\nclassiﬁcationofresonances. All these featuresdistinguish\nthis resonance from known eﬀects in the Coulomb block-\nade regime [24, 27, 28], including those due to inelastic\ncotunneling resolving excitations [29–31], the Kondo ef-\nfect [32–35], and another zero-bias anomaly speciﬁc toQD spin valves [24, 28].\nThe anomalous resonance we predict here relies on the\ncoherent precession of a single spin that is driven by the\nCoulombinteraction-induced exchange ﬁeld [35–41]. The\nexchange ﬁeld is a generic renormalization eﬀect [31, 42–\n45] arising from quantum ﬂuctuations of QD electrons\ninto the attached ferromagnets. This leads to a spin-\ndependent level shift, i.e., an eﬀective magnetic ﬁeld, be-\ncause the tunneling rates into the ferromagnets are spin-\ndependent. While this exchange ﬁeld has been measured\nfor strong tunnel coupling Γ as an induced level splitting\nfor collinear polarizations [32–35], the sharp resonance\nthat we predict here appears for moderate tunnel cou-\nplings when this splitting cannotbe resolved. In this\ncase, the exchange ﬁeld can still have an impact under\nthe additional requirement that the rotational symmetry\nis broken completely by a noncollinear magnetic conﬁgu-\nrationofthespinvalve. Here, eachferromagnetinducesa\ncontribution to the exchange ﬁeld along its polarization,\nwhich strongly depends on the applied voltages. This\nadds a tunable component to the exchange ﬁeld that is\nperpendicular to the injected spins. This induces a spin\nprecession that results in measurable consequences for\nthe stationary conductance [36, 40, 46] and the noise\nspectrum [47, 48], also for hybrid setups with a super-\nconductor [49].\nHowever, the features discussed so far change on large\nvoltage scales in contrast to the sharp resonance pre-\nsentedinthis work. Thisrelatestothelimitation ofthese\nprior works to the sequential tunneling regime where the\nelectron dwell times 1 /Γ are too small for single spins\nto precess by a large angle. To ﬁnd a sharp resonance\none needs a suppression of the spin decoherence, which\nis achieved in our case by an exponentially small leading-2\norder Γ contribution due to the Coulomb interaction U.\nOur spin resonance thus appears in the Coulomb block-\nade regime of a QD spin valve where the spin decoher-\nence is limited by higher-order contributions ∝Γ2/U,\nwhile the spin-precession period is still dominated by the\nleading-order Γ exchange ﬁeld [50].\nOnly few studies address spin-precession eﬀects in the\nCoulomb-blockade regime [50, 51]. What has been over-\nlooked in those works is that a simple QD spin valve al-\nready has built-in capabilities for single-spin operations\nthrough the gate-voltage control over the exchange ﬁeld\ndirection in the ﬁxed, nearly antiparallel conﬁguration.\nWe show that the resulting spin resonance can be ex-\nploited in a gate-pulsing scheme to provide single-spin\ncontrolforquantum-gateoperations. Time-averagedcur-\nrent measurements directly probe the underdamped spin\nprecession.\nThe paper is organized as follows: In Sec. II, we ﬁrst\nintroduce the QD spin-valve model under consideration\nand discuss our quantum master equation approach to\ndescribe the dynamics of the QD system. Based on the\nsolution of these equations, we compute the stationary\nconductance that exhibits the above-mentioned spin res-\nonance. We substantiate the simple resonance condition\nin Sec. III and identify relevant parameter combinations\nthat characterize the resonance features (position and\nwidth). We further suggest procedures to extract these\nparameters from experimental data in order to charac-\nterize QD spin valves. Next, we propose in Sec. IV a\nsimple gate-pulsing scheme, which is shown to reveal the\nunderdampedspin precessionoccurringnearthe spin res-\nonance. Finally, we summarize our ﬁndings in Sec. V\nand argue that the resonance mechanism described here\nis generic for a broad class of coherently evolving quan-\ntum systems renormalized through their environment.\nII. MODEL AND KINETIC EQUATIONS\nThe spin resonance appears in the simplest QD spin-\nvalve model one can think of, which is introduced in Sec.\nIIA.Thisisremarkablesincethismodelofaninteracting,\nsingle, spin-degenerate orbital level, which is tunnel cou-\npled to two noncollinearly polarized ferromagnetic leads,\nhas been studied quite intensively. In Sec. IIB, we dis-\ncuss the quantum master equation for the QD density\noperatorρ, which is required to address this spintronic\neﬀect. Basedonthesolutionoftheseequations,wederive\nthe current that exhibits the spin-resonance feature.\nA. Complete breaking of rotational symmetry in\nquantum-dot spin valves\nThe system under study, see Fig. 1, consists of a QD,\nwhich is tunnel coupled to two ferromagnetic leads r, la-\nbeled withr=s(ource),d(rain). The Hamiltonian reads:Htot=H+/summationdisplay\nr=s,dHr+HT. (1)\nThe QD is modeled by a single, spin-degenerate, inter-\nacting orbital level,\nH=/summationdisplay\nσεd†\nσdσ+Ud†\n↑d↑d†\n↓d↓, (2)\nwhered†\nσ(dσ) are fermionic ﬁeld operators that create\n(annihilate) electrons with spin σin the QD. The QD\nHamiltonian (2) is spin isotropic , that is,\n[H,ˆSi] = 0, (3)\nwhere\nˆSi=/summationdisplay\nσσ′1\n2(σi)σσ′d†\nσdσ′ (4)\nis theith Cartesian component ( i=x,y,z) of the spin\nvector operator and the σidenote the Pauli matrices.\nThe spin isotropy of the QD model implies that the two\nspin states are degenerate.\nBy contrast, the spin symmetry is broken in the ferro-\nmagnets, held at equal temperature Tand diﬀerent elec-\ntrochemical potentials µs(d)=±Vb/2, with Hamiltonian\nHr=/summationdisplay\nkσεrkσc†\nrkσcrkσ, (5)\nwherec†\nrkσ(crkσ) are fermionic ﬁeld operators that cre-\nate (annihilate) electronsin single-particlestates |rkσ∝an}brack⌉tri}ht=\n|rk∝an}brack⌉tri}ht ⊗ |σ∝an}brack⌉tri}htr=c†\nrkσ|0∝an}brack⌉tri}htof leadr. The spin-quantization\naxis is chosen for each ferromagnet along its polarization\nvectornr. The spin-dependent band structure of the fer-\nromagnets is described by the spin-dependent density of\nstates (DOS), here limiting ourselves to a ﬂat band with\nνrσ(ω) =/summationdisplay\nkδ(ω−εrkσ) = ¯νr(1+σnr),(6)\nfor|ω|< W(half-bandwidth) and zero otherwise. In\nEq. (6), we introduced the spin-averaged DOS ¯ νr=\n(νr,↑+νr,↓)/2 and the polarization nr=|nr|= (νr,↑−\nνr,↓)/(νr,↑+νr,↓), which do not depend on the frequency\nω.\nThe breaking of the spin symmetry in the ferromagnets\nis expressed by\n[Hr,ˆnr,⊥·ˆSr]∝n⌉}ationslash= 0. (7)\nHere,ˆnr,⊥·ˆSris a component of the spin operator\nˆSr=/summationtext\nkσσ′r∝an}brack⌉tl⌉{tσ|ˆs|σ′∝an}brack⌉tri}htrc†\nrkσcrkσ′of ferromagnet ralong\na unit vector ˆnr,⊥that is perpendicular to ˆnr. Note that\nfor each ferromagnet, the axial symmetry along its spon-\ntaneous magnetization direction, given by ˆnr, remains\nintact: [Hr,ˆnr·ˆSr] = 0. Importantly, the spin resonance3\nFIG. 1: Schematic setup of a QD spin valve, indicating the\nspin-precession resonance mechanism.\nrelies on a complete breaking of the spin symmetry by\nthe ferromagnets, which means the full Hamiltonian does\nnot commute with anycomponent i=x,y,zof the total\nspin operator ˆStot=ˆS+/summationtext\nrˆSr, that is,\n[H,ˆStot,i]∝n⌉}ationslash= 0. (8)\nThisisachievedfornoncollinearlypolarizedferromagnets\nwith polarizations nsandndatanangle θ=π−α∝n⌉}ationslash= 0,π.\nFinally, the tunnel coupling Hamiltonian reads\nHT=/summationdisplay\nrkσtr,σσ′d†\nσcrkσ′+H.c.. (9)\nHere, the tunneling amplitudes are assumed to be k- and\ntherefore energy independent as well as spin conserving ,\nthat is,\n[HT,ˆStot] = 0. (10)\nHowever, since d†\nσandcrkσmay refer to diﬀerent spin\nquantization axes, the tunneling amplitudes\ntr,σσ′=∝an}brack⌉tl⌉{tσ|σ′∝an}brack⌉tri}htrtr, (11)\nincorporate an overlap factor of the spin states while the\nbaretunnelingamplitudes trarespin-independent. They\nset the spin-averaged tunneling rates by Γ r= 2π¯νr|tr|2.\nB. Kinetic equations and charge current for\ninﬁnite interaction energy\nThe transport signatures of the QD spin valve are gov-\nerned by the nonequilibrium dynamics on the QD, de-\nscribed by its reduced density operator ρ= Trres(ρtot).\nOur reduced density operator approach starts, as usual,\nfrom the von Neumann equation ˙ ρtot=−i[H,ρtot] for\nthe density operator of the full system, ρtot. Eliminating\nthe reservoir degrees of freedom results in the following\nkinetic equation for the reduced density operator of the\nQD:\n˙ρ(t) =−i[H,ρ(t)]+Wρ(t). (12)\nHere, wehavemade anadditionalMarkovapproximation\nsince we are interested either in the stationary current\nobtained from the stationary state satisfying ˙ ρst= 0 (forwhich it is irrelevant) or the time-dependent current for\nwhich non-Markovian corrections are of subordinate im-\nportance in our case as discussed in Appendix B5. Thus,\nthe eﬀects due to the coupling to the leads are incorpo-\nrated through the zero-frequency kernel W.\nTo facilitate the analytical discussion of the QD spin\ndynamics, we express Eq. (12) in terms of coupled\nequations for the occupation probabilities pnfor each\nof the charge states n= 0,1,2, and the average spin\nS= TrQD(ˆSρ). The equivalence of these two represen-\ntations is shown in Appendix A. To keep all analytic\nexpressions as simple as possible, we focus ﬁrst on the\nlimit ofU→ ∞, for which double occupancy of the QD\nis suppressed (which implies p2= 0). In this case, the\nkinetic equations read\n˙p0=−2Γ0p0+Γ1p1+2GpS·S,\n˙S= +G0\nSpp0−1\n2G1\nSpp1−RS·S−B×S,(13)\nwith ˙p0=−˙p1due to probability conservation: p0+p1=\n1. Equation (13) is the most general form of any time-\nlocal quantum master equation for the QD system we\nstudy. It extends common master equation approaches\nfor the occupation probabilities pnby including their in-\ntense coupling (through the vectors G0\nSp,G1\nSp,GpS) to\nthe coherences [52, 53] of the degenerate spin states, con-\ntained in the spin vector S. Furthermore, the spin is sub-\nject to a torque corresponding to an eﬀective exchange\nﬁeldB. This eﬀective ﬁeld arises from quantum ﬂuctua-\ntions of QD electrons into the attached ferromagnets and\nisthekeyfactoringeneratingthespinresonance. Finally,\nthe spin Sis subject to a spin decay, which is described\nby the symmetric tensor RS. The spin-decay tensor can\nbecome signiﬁcantly anisotropic in the Coulomb block-\nade regime due to cotunneling processes. This aﬀects\nthe width of the spin resonance, which we discuss in Sec.\nIIIF. Extending usual master equations in the way de-\nscribed above is a necessity for noncollinear spin valves,\ni.e., when the rotational symmetry is completely broken\n(see Sec. IIA).\nTo compute all coeﬃcients in Eq. (13), we systematically\nexpand the kernel in the tunneling rates Γ r= 2π¯νr|tr|2\nusing the real-time diagrammatic technique [52, 54] and\nwe include all leading-order Γ and next-to-leading order\nΓ2-terms. This has been done analytically, starting from\na general Liouville-space formulation of the real-time di-\nagrammatic approach [54]. The resulting expressions for\nthe rates in the above quantum master equations (13)\nare given in Appendix B1. Our results extend previous\nworks in that we account for bothrenormalization eﬀects\ndue to the dynamics of coherences andnext-to-leading\norder Γ2corrections. This enables us to make reliable\npredictions about the spin resonance in the Coulomb\nblockade regime. This development was motivated by\nRef. 26 where the spin resonance was found at the ﬂank\nof the single-electron tunneling peak but could not be\ntracked into the Coulomb blockade regime because the\nquantum master equation used there included only lead-\ning order Γ processes. The interested reader may ﬁnd4\ndetails on our technical advances and how they extend\nprevious works in Apps. B1 and B2.\nAfter solving Eq. (13) for the occupation probabilities\nand the average spin, one can compute the average cur-\nrent from lead rinto the QD by\nIr= 2Γr,0p0−Γr,1p1−2Gr,pS·S,(14)\nwhere the rates are given in Appendix B1. In Appendix\nB3, we explain how to solve Eq. (13), which is actually a\nnontrivial task in the Coulomb blockade regime because\nO(Γ) contributions can become smaller in magnitude\nthanO(Γ2) contributions. With the analytical results\nobtained in this way, we are able to gain a physical\nunderstanding of the QD dynamics governing the spin\nresonance and they are, moreover, used to derive ap-\nproximation formulas for the current near the resonance.\nHowever, our analytical results are restricted to the\nlimitU→ ∞. To study the case of ﬁnitecharging\nenergyU, we use a computer code to evaluate the kernel\nWnumerically. The code is based on the formulas\ngiven in Ref. 52, which we extended to account for\ncouplings between diagonal and nondiagonal density\noperator matrix elements to O(Γ2). To ensure that our\nperturbative approach is valid, we set for all plots Γ <T\nand we further checked that the numerically computed\nfeatures scale at least as O(Γ2) when the tunnel coupling\nis lowered (see also comments in Appendix B3). Thus,\nthe predicted spin resonance in the Coulomb blockade\nregime can appear in an experiment also at elevated\ntemperatures in the sense T >Γ, for which the Kondo\neﬀect is not present.\nIII. STATIONARY-CONDUCTANCE\nRESONANCE: CHARACTERIZATION OF\nQUANTUM-DOT SPIN VALVES\nWith our model and technique established, we ﬁrst\nturn to the discussion of the spin resonance in the sta-\ntionary conductance. Here, we study the generalization\nof Eq. (13) to ﬁnite Uand solve it numerically unless\nstated otherwise. We ﬁrst present the most important\nfeatures in the stationary conductance in Sec. IIIA be-\nfore we scrutinize the parameter dependence of the res-\nonance position and width in detail in the following sec-\ntions. We further expound that the nontrivial parameter\ndependence can be used to characterize QD spin valves\nin an alternative way.\nA. Spin resonance in stability diagram\nAmainresultofthispaperispresentedinFig. 2,which\nshows the stationary conductance for the setup sketched\nin Fig. 2(a) obtained from the extension ofEqs. (13) and\n(14) to ﬁnite U. We ﬁnd a sharp wigglein the nonlinear\nconductance dI/dV b, i.e., apeakinthecurrentplottedvs.Vb, which extends through the entire Coulomb-blockade\nregion. Notably, the resonance starts at the Coulomb di-\namond edge, then bends towards the particle-hole sym-\nmetry point at ( Vb= 0, ε=−U/2), where its magnitude\nvanishes, and then continues point-symmetrically. We\ntherefore focus our discussion ﬁrst on the Vg<U/2 part\nof Fig. 2 and chose the labels “source” and “drain” such\nthat the lead with the larger spin-injection rate Γ rnris\nthe source for Vb>0.\nFIG. 2: Diﬀerential conductance dI/dV bfor the setup shown\nin Fig. 1 for the current from the source into the QD for\nΓs= 2Γ d= 0.01U,T= 0.05U,W= 50U,ns=nd=\n0.99,α= 0.01π. The white dashed curve follows from the\nresonance condition (15). Signatures in the conductance ca n\nalready be found for ns,nd/greaterorsimilar0.6, andα <0.4πas discussed\nin Sec. IIIB; here we use larger polarizations and smaller α\nfor illustrational purposes.\nTo understand the origin of the spin resonance, we note\nthat the current through the QD is largely suppressed for\nantiparallel polarizations by the spin-valve eﬀect: Elec-\ntrons of spin-majority type coming from the source get\nstuck in the QD because they are of spin minority type\nfor the drain. Thus, the tunneling rate for these elec-\ntrons from the QD into the drain is small. However,\nif the polarizations of the electrodes are merely slightly\nnoncollinear , the spin resonance appears in Fig. 2. The\nreason for this sharp resonance is that the drain contri-\nbution to the exchange ﬁeld, B=Bsˆns+Bdˆnd, adds a\ncomponent Bd,⊥=Bdsinαthat is perpendicular to the\nsource polarization ns, i.e.,B= (Bs+Bd,/bardbl)ˆns+Bd,⊥ˆn⊥\nwithBd,/bardbl=Bdcosα[cf. Eq. (16) below]. The seem-\ningly innocuous component Bd,⊥causes a precession of\nthe spin injected along nstowardsnd. Consequently, the\nelectron can easily leave the QD to the drain, prevent-\ning an accumulation of spin antiparallel to the drain as\nexpected from prior works [36, 40]. We note that such a\ntransverse component appears only because of the non-\ncollinearity of the ferromagnets’ polarizations, i.e., the\ncomplete breaking of the rotational symmetry. If the\npolarizations are collinear, the exchange ﬁeld is aligned\nalongthe commonpolarizationaxisand thereforenospin\nprecession is possible.\nThe spin-precession feature shown in Fig. 2 is unexpect-5\nedly sharp since the spin-valve eﬀect is lifted onlyfor a\nspeciﬁc bias voltage V∗\nb. The reason is that the spin ro-\ntation is eﬀective only if the opening angle of the spin\nprecession is large [cf. Fig. 3(c)(ii)]. Hence, the reso-\nnance appears when the total exchange ﬁeld component\nparallel to the source polarization nsvanishes, i.e., when\nthe following scalar condition is satisﬁed:\nB·ˆns=Bs+Bd,/bardbl= 0. (15)\nIn contrast to usual resonance conditions, it incorporates\ntwovectors.\nThe resonance position can be predicted from the O(Γ)\napproximation for the exchange ﬁeld [40],\nB=/summationdisplay\nrΓrnr[φr(ε)−φr(ε+U)],(16)\nwith spin-polarization vector nrpointing in the polariza-\ntion direction of the ferromagnet. Equation (16) includes\nthe renormalization function\nφr(ε) =/integraldisplay+W\n−Wdω\nπf[(ω−µr)/T]\nω−ε(17)\n=1\nπ/bracketleftbigg\n−Reψ/parenleftbigg1\n2+iε−µr\n2πT/parenrightbigg\n+log/parenleftbiggW\n2πT/parenrightbigg/bracketrightbigg\n,\nincorporating the digamma function ψ, the Fermi func-\ntionf(x) = 1/(ex+ 1), and electrochemical potentials\nµs,d=±Vb/2. Inserting Eq. (16) into the resonance\ncondition (15) and solving for the resonant bias V∗\nbas a\nfunction of Vgyields the white dashed curve in Fig. 2.\nThis simple physical idea thus nicely ties in with the re-\nsults of our full numerical calculations as we further work\nout in Sec. IIIC and with our analytical results based on\nthe kinetic equations (13) for U→ ∞in Sec. IIIF. The\nfull theory is, however, still needed for understanding the\nresonance peak height and shape.\nRemarkably, for a given gate voltage Vg, the condition\n(15) is fulﬁlled only for one bias polarity when the elec-\ntrodes are asymmetrically coupled to the QD. This is one\nfeaturethat canbe usedtoruleout othereﬀectsin exper-\nimental data, for example, those due to inelastic cotun-\nneling, which typically show signatures for both bias po-\nlarities. Otherdistinguishingfeaturesarethepeakheight\nand width as discussed in Sec. IIIF.\nHere, we ﬁrst focus on the explanation of the strong cur-\nrent rectiﬁcation, which can be attributed to the elec-\ntrical tunability of the exchange ﬁeld direction : In Fig.\n3(a), we plot Bs,Bd,/bardbl, and their sum B||=Bs+Bd,||as\nfunction of the bias Vb. For electrode rthe magnitude Br\nis maximal when µr=εorµr=ε+Uand vanishes mid-\nway atµr=ε+U/2 [marked in Fig. 3(b) by (i) for r=d\nand in (iii) for r=s]. In the vicinity of these points, the\nexchange ﬁeld Bcomes from only one electrode, point-\ning along nsornd, see Fig. 3(c)(i) and (iii), respectively.\nHere, the spin precesses with a small opening angle and\nthe spin transport stays blocked. However, when tun-\ning the bias between these two cancellation points, theexchange ﬁeld rotates [see Fig. 3(c)(ii)] and the sum\nB/bardblvanishes for a speciﬁc bias voltage V∗\nband polarity.\nThis electric tunability illustrates that renormalization-\ninduced eﬀective ﬁelds can intervene with the coherent\nevolution of two-level systems in a controlled way to pro-\nduce unexpected resonances as shown in Fig. 2.\nFIG. 3: Main panel (a) and sketched zoom-in (b): Ex-\nchange ﬁeld component along nsfrom the source electrode\n(Bs, green), the drain electrode ( Bd,/bardbl, blue), and their sum\n(Bs+Bd,/bardbl, red) as a function of VbforVg= 0.375U, with\nother parameters as in Fig. 2. (c) Illustration of the spin\nprecession (gray) for diﬀerent directions of the exchange ﬁ eld\n(red), taken for diﬀerent Vbas indicated in (b). The opening\nangle is maximal for (ii) at Vb=V∗\nbwhere Eq. (15) holds.\nFigure 2 further clearly shows that the bias scale V∗\nbdoes\nnot match any obvious energy scale of the problem, at-\ntesting to its nonspectral, vectorial nature. Depending\non the gate voltage, it may exceed Γ, T, and even ap-\nproach a sizable fraction of U[cf. Fig. 7(a)]. As we show\nin Sec. IIIC, the eﬀect may be exploited to characterize\nQD spin valves in situ.\nSimilarly, additionally attaching a superconductor to the\nQD, see Sec. IIID, the spin-resonance position remains\ndistinct from the energy scales set by the Andreev bound\nstates formed on the QD [55]. The eﬀect of the Andreev\nboundstatesistomodify theexchangeﬁeld B[49], which\nshifts the resonance position in the full calculation no-\ntably as accurately predicted by the resonance condition\n(15) when inserting the modiﬁed exchange ﬁeld B. This\nis explained further in Sec. IIID. The above conﬁrms\nthat Eq. (15) truly captures the essence of the spin res-\nonance under various situations and identiﬁes a mecha-\nnism of a highly voltage-dependent loss of magnetoresis-\ntance for QD spin valves that is active already for small\nnoncollinearity angles.6\nB. Experimental feasibility: Polarization vectors\nIn the above section, we used large polarizations ns=\nnd=n= 0.99forillustrationalpurposes. Achievingsuch\nhigh values is a central goal of spintronics, yet currently\npresents a challenge. However, this large value was only\nused to make the resonance as clear as possible in Fig.\n2 but to observe our predicted feature, this is actually\nnot needed. In Fig. 4(a), we show the nonlinear con-\nductancedI/dV bin the stationary state for lower values\nof the polarization n=ns=nd. Clearly, already for\npolarizations n/greaterorsimilar0.6 a discernible modiﬁcation of the\nconductance can be seen. In situpolarizations as large\nasn∼0.8 have already been achieved with half-metallic\nelectrodes in experiments [56].\nWe also note that the small noncollinearity angle α=\n0.01πused in Fig. 2 shows that the assumption of per-\nfect collinearity often made in theoretical analyses of\nspin-valve devices can lead to highly nongeneric results.\nHowever, the spin resonance is not limited to small non-\ncollinearity angles: Figure 4(b) shows the conductance in\nthe vicinity of the resonance for diﬀerent noncollinearity\nanglesαand one ﬁnds a region of negative diﬀerential\nconductance even for αas large as 0 .4π. We conclude\nthat it is not essential to have a noncollinearity angle\nvery precisely close to α= 0 and extraordinary large\npolarizations n≈1 to see a resonance feature in the sta-\nbility diagram. Large polarizations of n/greaterorsimilar0.8 as aimed\nat by eﬀorts in spintronics and angles α/lessorsimilar0.2πshould\nbe suﬃcient to observe features of the spin resonance.\nMoreover, Fig. 4(b) shows that the resonance position\nchanges as a function of the angle α. This can be ex-\nploited to measure the angle αas we discuss in the next\nsection.\nFIG. 4: Diﬀerential conductance dI/dV bas a function of bias\nvoltageVbfor gate voltage Vg= 7.5T, varying (a) the polar-\nization magnitude ns=nd=nas indicated for ﬁxed non-\ncollinearity angle α= 0.01πand (b) varying the angle αas\nindicated for ﬁxed ns=nd= 0.99. All other parameters are\nas in Fig. 2(a).C. Extracting the spin-injection asymmetry from\nresonance position\nTo investigate the parameter dependence of the reso-\nnance position systematically, we introduce the energy\nlevel detuning from the symmetry point ε=−U/2,\nδ=U+2ε, (18)\nwherethespin-resonancebiaspositiongoesthroughzero.\nAs shown in Appendix C using particle-hole symmetry,\nit is suﬃcient to discuss only the case δ >0 andVb>0\nsince the results obtained are easily related to those for\nnegative values. We thus limit our discussion here to the\nleft half ofthe Coulomb diamond of the stability diagram\nin Fig. 2. We recast the resonance condition (15) as\na\nq= 1, (19)\nwith the asymmetry ratio of the spin-injection rates,\na:=Γsns\nΓdndcos(α), (20)\nand electrically tunable ratio\nq:=φd(ε)−φd(ε+U)\nφs(ε)−φs(ε+U). (21)\nThe above condition a/q= 1 has been used to gener-\nate the perfectly matching white dashed curve in Fig.\n2(a) by solving it for the resonant bias V∗\nbas function\nofVg. Thus, we ﬁnd on a numerical basis that the\nO(Γ) approximation for the exchange ﬁeld is suﬃcient\nto reliably predict the resonance position for the full nu-\nmerical calculation up to O(Γ2). Deep in the Coulomb\nblockade regime when the distance of the electrochem-\nical potentials from one of the level positions is large,\nminr=s,d[|ε−µr|,|ε+U−µr|]≫T, the real part of the\ndigamma function can be approximated by a logarithm,\nthat is, ReΨ(1 /2+ix)≈ln|x|. This leads to\nq≈ln|(1+˜δ+˜Vb)/(1−˜δ−˜Vb)|\nln|(1+˜δ−˜Vb)/(1−˜δ+˜Vb)|.(22)\nThus, the factor qbecomes independent of temperature\nand it exclusively depends on the electrical parameters\nsuch as bias through the ratio ˜Vb=Vb/U/greaterorequalslant0 and the\ngate voltage through the ratio ˜δ= 1+2ε/U. As a conse-\nquence, the resonance feature is just rescaled inside the\nCoulomb diamond when the latter is made larger by in-\ncreasing the interaction energy U, cf. Fig. 5(d) below.\nWe emphasize that the nontrivial voltage dependence of\nthe resonance position derives from the drastic changes\nin thedirection of the exchange ﬁeld vector B, rather\nthan its magnitude.\nTo substantiate the simple condition (19) further, we\nnext show in Fig. 5 full numerical results for the res-\nonance when changing various parameters in the setup7\nFIG. 5: Numerically computed diﬀerential conductance\ndI/dV bas a function of bias voltage Vbwhen modifying\nseveral parameters but keeping the spin-injection asymme-\ntry (20) ﬁxed. (a) The tunnel couplings are varied as\nΓd= Γs/(2cos(α)) = 10−3T ...10−1Tin four equidistant\nsteps, keeping ns=nd= 0.99 andα= 0.01πﬁxed. The\ncurves are vertically oﬀset by 10−2with respect to each\nother. (b) The polarization magnitudes are varied as nd=\nns/cos(α) = 0.6...0.99 in four equidistant steps, keeping\nΓd= Γs/2 = 0.1Tandα= 0.01πconstant. (c) The non-\ncollinearity angle is varied as α= 0.85π...0.97πin four\nequidistant steps, adjusting Γ d= Γs/(2/radicalbig\ncos(α)) = 0.1Tand\nnd=ns//radicalbig\ncos(α) = 0.99. The parameters in (a)–(c) are cho-\nsen such that a= 2, the other parameters are U= 20T,\nVg= 7T, andW= 1000T. (d) The interaction energy\nis varied as U= 40T ...100Tin four equidistant steps for\nns=nd= 0.99, Γd= Γs/2 = 0.1T,α= 0.01π,Vg= 0.45U,\nandW= 1000T.\nsuch that the asymmetry aremains constant. According\nto our prediction from Eq. (19), this leaves the resonant\nbiasV∗\nbunchanged, which is conﬁrmed byFig. 5. For ex-\nample, when changing the tunnel couplings in Fig. 5(a)\nand the polarization in Fig. 5(b), the resonance width\nand height are aﬀected, but the resonance bias position\nindeed stays unaltered. In Fig. 5(c), we also change the\nnoncollinearityangle αwhile adaptingboth polarizations\nand tunnel couplings to keep aﬁxed. Finally, we in-\ncreasein Fig. 5(d) the interactionenergy Uand ﬁnd that\nthe resonance condition (19) depends only on the ratios\n˜Vb=Vb/Uand˜δ= 1+2ε/Uof the voltages and the in-\nteractionenergyforstrongCoulombblockade conditions.\nBy contrast, the width of the resonance changes signiﬁ-\ncantly because Uaﬀects the spin-decoherence rates, see\nSec. IIIF.\nWe next outline a simple procedure for determining the\nasymmetry afrom an experimentally measured stability\ndiagram. Here, we use that the resonance condition can\nbedrasticallysimpliﬁedinthevicinityoftheparticle-hole\nsymmetry point. For ˜δ≪1, the condition a/q= 1 im-\nplies that the resonant bias also satisﬁes ˜V∗\nb≪1. Thentheresonancepositioncanbe foundbyalinearexpansion\nof the logarithms in Eq. (22), which results in a linear\ndependence of the resonant bias on the detuning,\n˜V∗\nb=κ(α)˜δ, (23)\nwith slope\nκ(α) =a(α)−1\na(α)+1. (24)\nThe slope (24) becomes minimal in the limit α→0 (for\nα= 0 the spin resonance vanishes and the slope cannot\nbe measured). The slope increases quadratically with α\nas a simple expansion of Eq. (24) for small αshows and\nreachesκ= 1 forα=π/2. Measuring the slope of the\nresonanceposition near the particle-holesymmetry point\nin Fig. 2 allows one to directly extract the spin-injection\nasymmetry a(0) = Γ sns/Γdndand to measure the angle\nα. This can be achieved in two ways: First, if one has\nexperimental access to this slope for a single, accurately\ndetermined angle α, one can directly determine a(0).\nAlternatively, if one has continuous control over αbut\nthe values for αare not known, one can experimentally\nrecord pairs [ αi,κ(αi)] and use Eq. (24) by inserting\nEq. (20) as a ﬁtting formula with the single parameter\na(0). After these two possible “calibration” procedures,\none can conversely extract the angle αby measuring the\nslope. All this illustrates the usefulness of the novel spin\nresonance as alternative and simple route for (partially)\ncharacterizing QD spin-valve setups in-situ.\nD. Impact of proximal superconductor on\nresonance position\nTo illustrate the broad applicability of our resonance\nconcept, we study a modiﬁcation of model (1) by adding\na superconducting terminal at electrochemical potential\nµsup= 0, tunnel coupled to the QD with rate Γ sup, as\nsketched in Fig. 6(a). In the limit of inﬁnite supercon-\nducting gap, ∆ → ∞, the eﬀect of the superconductor\ncan be incorporated by adding a pairing term\nHP=−1\n2Γsup(d†\n↑d†\n↓+d↓d↑) (25)\nto the QD Hamiltonian (2) [57].\nIn the presence of a superconductor, the dependence of\nthe leading-order exchange ﬁeld on the electric parame-\nters, contained in the ratio q[Eq. (21)], is modiﬁed: One\nhas to replace Eq. (18) by [49]\nφr(ε) =/summationdisplay\nγγ′=±γ′\n2π/parenleftBig\n1+γδ\n2εA/parenrightBig\nReΨ/parenleftbig1\n2+iεr,γ′γ\n2πT/parenrightbig\n,(26)\nwith the modiﬁed energies εr,γ′γ=γ′U\n2+γεA−µrdue\nto Andreev reﬂection processes, incorporating the An-\ndreev bound state energies εA=1\n2/radicalBig\nδ2+Γ2supfor detun-\ningδ=U+ 2ε. In the limit of Γ sup→0, Eq. (26)8\nFIG. 6: (a) Modiﬁcation of the quantum-dot spin valve de-\npicted in Fig. 1(a) including a superconducting terminal. ( b)\nDiﬀerential conductance dI/dV bfor setup (a) for the cur-\nrent from the source into the QD for Γ s= 2Γd= 0.01U,\nΓsup= 0.75U,T= 0.025U,W= 50U,ns=nd= 0.99,\nα= 0.01π. The white (black) dashed curve follows from the\nresonance condition (15) including (excluding) the eﬀect o f\nthe Andreev bound states. We excluded cotunneling from\nthe calculations for (b). We comment on this in Appendix\nB4.\nreduces to Eq. (18). Solving the condition a/q= 1 with\nqmodiﬁed through Eq. (26) for nonzero(zero) Γ supgives\nthe white (black) dashed curve in Fig. 6(b). Clearly, the\npresence of the superconductor leads to a signiﬁcant shift\nof the resonance position.\nAgain approximating the real part of the digamma func-\ntion deep in the Coulomb blockade regime by a logarith-\nmic expression, we ﬁnd\nq=/summationtext\nγ=±/parenleftBig\n1+γ˜δ\n2˜εA/parenrightBig\nln/vextendsingle/vextendsingle/vextendsingle1+2γ˜εA+˜Vb\n1−2γ˜εA−˜Vb/vextendsingle/vextendsingle/vextendsingle\n/summationtext\nγ=±/parenleftBig\n1+γ˜δ\n2˜εA/parenrightBig\nln/vextendsingle/vextendsingle/vextendsingle1+2γ˜εA−˜Vb\n1−2γ˜εA+˜Vb/vextendsingle/vextendsingle/vextendsingle,(27)\nwith ˜εA=εA/U. The slope ˜ κof the linear resonance\ncondition ˜V∗\nb= ˜κ˜δ, which is valid near the particle-hole\nsymmetry point, reads in this case:\n˜κ(α) =κ(α)ln/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1+˜Γsup\n1−˜Γsup/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−˜Γ2\nsup\n2˜Γsup\n=κ(α)(1−˜Γ2\nsup)+O(˜Γ3\nsup),(28)\nwith˜Γsup= Γsup/Uandκgiven by Eq. (24). Hence,\ntuning the tunnel coupling of a proximal superconduc-\ntor does not only shift the single-electron tunneling reso-\nnance positions in the stability diagram, but also sup-\npresses the slope of the spin resonance. This can be\nexploited to extract the coupling to the superconductor˜Γsupin an alternative way from the stability diagram: If\nthe tunnel coupling Γ supcan be eﬀectively suppressed,\nwhich leads from Fig. 6(b) to Fig. 2, one can obtain\n˜Γsupfrom Eq. (28) by inserting the experimentally mea-\nsured values for κ(α) and ˜κ(α). This may be advanta-\ngeous since the broadening of the spin resonance can be\nmuch smaller than that of the single-electron tunneling\nresonances, as demonstrated by Fig. 6(b). If one has ad-\nditional control over the angle α, the broadening of the\nspin resonance due to cotunneling processes, which are\nnot included in Fig. 6(b), can be compensated by reduc-\ningα, see Sec. IIIF).\nWe ﬁnally comment on our assumption of an inﬁnite su-\nperconducting gap ∆. In experiments, the gap ∆ can\nbe∼1 meV and therefore on the order of typical charg-\ning energies [58]. Hybrid superconductor-ferromagnetic\nstructures havealso been realized with somewhat smaller\ngaps of∼100µeV [59]. However, as long as the bias Vb\nis smaller than ∆ and the Andreev bound state energies,\nreal tunneling processes due to the superconductor are\nstrongly suppressed and renormalization eﬀects due to\nquantum ﬂuctuations dominate. This expectation is un-\nderpinned by a recent theoretical study [60], which con-\nsiders corrections to the inﬁnite-gap approximation by\nexpanding in 1 /∆. It turns out that the main eﬀect of\nthe ﬁnite gap is to shift the Andreev bound state ener-\ngies rather than leading to modiﬁcations of the current.\nTherefore, we expect that the form of our resonance con-\ndition (15) should be valid for ﬁnite ∆ when tuning close\nto the particle-hole symmetry point where the resonance\nappears for small bias.\nOur study thus illustrates a new, fruitful and experimen-\ntally relevant interplay of superconductivity and spin-\ntronics. Exploring the situation of a ﬁnite superconduct-\ning gap ∆ when Vb∼∆ is an interesting open question\nthat presents additional technical challenges beyond the\nscope of this paper. For the rest of this paper, we return\nto the case when no superconducting leads are present,\ni.e., Γsup= 0.\nE. “Half-sided Coulomb diamond” and zero-bias\npeak\nAs just illustrated by the superconducting hybrid\nsetup, the spin resonance position sensitively reacts to\nmodiﬁcations of the exchange ﬁeld through the ratio q\nregulating the dependence on voltages. However, the\nresonance position can also be changed by the other fac-\ntor in Eq. (19), the spin-injection asymmetry a. We\nillustrate this for the two extreme cases leading to trans-\nport stability diagrams which would be puzzling if one\nwere to experimentally obtain them without having fur-\nther microscopic information: For very large asymme-\ntries,a≫1, the resonance becomes parallel to Coulomb\nedges, forminga“half-sidedCoulombdiamond,”whereas\nfor negligible asymmetry, a= 1, the resonance appears\nasazero-biasconductancepeak. EventhoughtheKondo9\nresonance and the zero-bias anomaly of Refs. 24 and 28\nalso appear at zero bias, our spin resonanceis clearly dis-\ntinguished from these features as we explain below.\nFIG. 7: Diﬀerential conductance dI/dV bas a function of gate\nvoltageVgand bias voltage Vb. In (a), the spin-injection ratio\nisa= 10 with Γ s= 0.1//radicalbig\ncos(α) and Γ d= 0.01//radicalbig\ncos(α)\nand in (b) the spin-injection ratio is a= 1 with Γ s= Γd=\n0.01/radicalbig\ncos(α)T. All other parameters are as in Fig. 2.\nWe ﬁrst note that the resonance position can appear in\nthe entire voltage range by changing κthrough the tun-\nneling rates, polarizations, and the angle α, limited only\nby the condition\n0/lessorequalslant˜V∗\nb/lessorequalslant˜δ,(˜δ >0) (29)\nif the electrode with the larger spin injection rate be-\ncomes the source for Vb>0 and ifα < π/2 (which is\nneeded for a sharp feature). The restriction (29) is read-\nily proved from Eq. (19): Since the asymmetry param-\netera= Γsns/[Γdndcos(α)]/greaterorequalslant1, it follows that q/greaterorequalslant1.\nThe parameter qhas a magnitude larger than 1 if the\nnumerator in Eq. (19) is larger than that of the denom-\ninator, which implies ˜V∗\nb/greaterorequalslant0. Forqto be positive, one\nadditionally has to demand ˜V∗\nb/lessorequalslant˜δsince˜δ/greaterorequalslant0. The\nanalogous constraint in the other half of the Coulomb-\nblockade region,\n˜δ/lessorequalslant˜V∗\nb/lessorequalslant0,(˜δ<0), (30)\nfollows by similar arguments (see Appendix C). The\nabove inequalities turn into an equality for the two ex-treme cases illustrated in Fig. 7.\nFIG. 8: Diﬀerential conductance dI/dV bas a function of bias\nvoltageVbfor Γs= Γd= 0.5T,ns=nd= 0.99,α= 0.01π,\nU= 40T, andW= 1000T. For the gate voltage that re-\nstores the particle-hole symmetry, ˜δ= 0, the spin resonance\nis absent and the broad zero-bias anomaly of Refs. 24 and 28\nis visible. For ˜δ= 0.025, when the particle-hole symmetry\nis absent, the conductance proﬁle for small bias is by con-\ntrast completely dominated by the spin resonance. We chose\nhereratherlarger tunnelcouplingstocomparewiththeabov e-\ncited references. We note that the conductance is shown ther e\nfor strictly antiparallel polarizations, α= 0,which has negligi-\nble impact on the zero-bias anomaly as compared to the case\nα= 0.01πconsidered here. Note that the zero-bias anomaly\ndoes not appear in Fig. 7 because the tunnel couplings are\nsmaller there, yet our spin resonance persists for these par am-\neter values.\nFirst,weshowtheresonanceinFig. 7(a)forstrongasym-\nmetrya≫1. Here, the resonance position is at ˜V∗\nb=˜δ,\ni.e., parallel to the Coulomb diamond edges. Strikingly,\nthe resonance is much sharper than the temperature-\nbroadened single-electron tunneling resonances because\nthe width is not simply limited by temperature Tand\ntunnel coupling Γ (see Sec. IIIF). If one were to mea-\nsure such a signature and would have no further micro-\nscopic information one would thus wonder why this fea-\nture is not thermally broadened, whereas the others can\nbe demonstrated to change with temperature.\nSecond, in Fig. 7(b) we show the resonance for perfect\nsymmetry,a= 1, in which case it appears at ˜V∗\nb= 0 and\nonly for an odd number of electrons on the QD. This sig-\nnature could in fact be mistaken for features due to the\nKondoeﬀect forelectrodeswith negligiblepolarizationor\nthe zero-biasanomalydiscussedinRefs. 24and28, which\nare otherwise very dissimilar. One should note that the\nKondoeﬀectrequiresstrongtunnelcouplings(Γ /T≫1),\nwhereas the spin resonance also appears in the interme-\ndiate coupling regime (still Γ /T <1). Moreover, the\nspin resonance disappears at the particle-hole symmetry\npoint, while the Kondo eﬀect can remain at this point.10\nIt can even appear onlyat this point for strong, parallel\nspin polarizations of the electrodes [32, 34, 37, 61] since\nthe exchange ﬁeld B=0there (For the case of strong\nmagnetizations of the electrodes this is no longer true\n[35]). For strong, antiparallel polarizations – the con-\nﬁguration close to where the spin resonance occurs – it\ndepends on the asymmetries of the spin-injection rates\nwhether the Kondo eﬀect emerges or not.\nFurthermore, for symmetric spin-injection rates, one\nshould also not mistake the spin resonance for the zero-\nbias anomaly studied in Refs. 24 and 28, caused by the\ninterplay of the voltage dependence of the cotunneling\nspin-ﬂip rates with the spin-valve eﬀect. Both eﬀects can\nin fact appear together and, as we demonstrate in Fig. 8,\nthe spin resonance may be even much larger and sharper\nthan the zero-bias anomaly. However, it depends on the\nchoice of the parameters which of the two is more pro-\nnounced: For example, while the width of the zero-bias\nanomalyissetbytemperature, thewidthofthespinreso-\nnance is independent of Tand determined instead by the\nangleαand a combination of the spin-decay rates and\nthe exchange ﬁeld, which depends strongly on the ap-\nplied gate voltage (see Sec. IIIF). Moreover, in contrast\nto the spin resonance, the zero-bias anomaly persists at\nthe particle-holesymmetry point and for strictly antipar-\nallel lead polarizations.\nThe above illustrates that our spin resonance is really\nan independent conductance feature, distinct from other\nfeatures and can moreover be identiﬁed unambiguously\nin an experiment.\nF. Extracting the anisotropy of the spin-decay\ntensor from the resonance shape\nBesides the resonance position we have focused on\nso far, the resonance shape provides additional valu-\nable information about the QD spin-valve: In particular,\none can extract information about the anisotropy of the\nspin-decay rates, that is, the spin-relaxation rate Γ ||=\nˆns·RS·ˆnsand the spin-dephasingrate Γ ⊥=ˆn⊥·RS·ˆn⊥,\nwhereˆn⊥isaunitvectorperpendicularto ˆns. Incontrast\nto the position, the shape is signiﬁcantly inﬂuenced by\ncotunneling corrections and crucially relies on the tech-\nnical developments we report.\nTo illustrate this, we now restrict our attention to volt-\nages near the resonance (such that B/bardbl/B⊥/lessorsimilar1) in\nthe limit of strong Coulomb blockade ( Vb/2ε≪1 and\nU→ ∞), small noncollinearity angles ( α≪1), symmet-\nric polarization magnitudes ns=nd=n, and small spin\ninjection asymmetry ( κ≪1). In this case, the station-\nary current (14), I=Is=−Id, ﬂowing through the QD\ncan be approximated by\nIappr=I0[1−A(1−M)], (31)with\nI0=/summationtext\nr,τr(−1)τΓr,τΓ¯r,¯τ\n2Γ0+Γ1, (32)\nA= 2/summationtext\nr,τr(−1)τ(Gr,pS·ˆns)/parenleftbig\nGτ\nSp·ˆns/parenrightbig\nΓ¯r,¯τ/summationtext\nr,τ(−1)τΓ||Γr,τΓ¯r,¯τ,(33)\nM=M0\n1+[(a/q−1)/H]2, (34)\nwhere all rates are deﬁned in Appendix B1. Here, τ\ntakes the values 0 and 1, ¯ τ:= 1−τ, and the factor r\nin the above sums takes the value r= + (r=−) for\nr=s(r=d) and ¯r=−r. Finally, we introduced the\nabbreviations\nM0=1\n1+(Γ /bardblΓ⊥)/B2\n⊥, (35)\nH=α/radicalBigg\nΓ⊥\nΓ/bardblM0. (36)\nEquation (31) can be interpreted as follows: The value\nI0is the current obtained when ignoring the spin accu-\nmulation, that is, forcing “by hand” S= 0 in the kinetic\nequations (13). Note that I0does not coincide with the\ncurrent for zero polarization since the charge-relaxation\nrates (B2) also depend on the polarizations. The actual\nnonzero spin accumulation S∝n⌉}ationslash= 0 on the QD acts back\non the charge dynamics, thus suppressing the current to\na fraction 1 −A<1 of the current I0. However, for any\nnonzeroαthe exchange-ﬁeld induced spin precession can\nin turn suppress this spin-valve eﬀect. This is captured\nby the factor 1 −M, whereMis a Lorentzian function in\nthe parameter a/q−1 with intensity M0and widthH.\nThe current becomes maximal at a/q= 1, which is the\nresonance condition (19).\nThe peak value of the current resonance depends on two\ncompeting inﬂuences of the cotunneling contributions to\nthe current: On the one hand, they increase the maxi-\nmally achievable current I0by providing additional tun-\nneling processes, but on the other hand they enhance\nthe spin decay, which limits the eﬀectiveness of the spin\nprecession by suppressing M0and thereby M. The de-\ncisive parameter that controls the current peak value is\nthe ratio\nb:=|B⊥|//radicalBig\nΓ⊥Γ|| (37)\nof the perpendicular exchange ﬁeld component and the\nspin-decay rates. Notably, the spin resonance appears\nboth for (i) the strongly underdamped case b≫1 and\nfor (ii) the critically damped case b∼1, while it disap-\npears for (iii) the strongly overdamped case when b≪1,\nwhereM0→0 and therefore Mhas negligible impact on\nthe current. The “optimal” value for a maximal current\nenhancement is given for b≈1. However, even for b<1\nbut not yet b≪1, the spin precession can still signiﬁ-\ncantly enhance the current to produce a sharp feature in11\nthe conductance as in Fig. 2. Therefore, the occurrence\nofthe spin resonancein the stationary conductanceis not\nyet evidence ofan underdamped spin precession. By con-\ntrast, the pulsing scheme discussed below in Sec. IVA is\nable to unambiguously demonstrate underdamped spin\nprecession.\nBefore discussing the time-dependent results, we ﬁrst\ncompare the stationary features in the cases (i) and (ii)\nand moreover explain how they can be exploited to ex-\ntract the spin-decay properties from electron transport\nmeasurements. A salient ﬁnding of this scheme is that\nthe electrical tunability of the exchange ﬁeld allows for\nanall-electric probing of the anisotropic spin-decay ten-\nsorRSin-situ. This scheme resembles that of Ref. 46\nwhere the interplay of the exchange ﬁeld with an exter-\nnal perpendicular magnetic ﬁeld was used to extract the\nspin-dephasing rate. Here, one utilizes the built-in ex-\nchange ﬁeld instead.\nFIG. 9: Stationary current as a function of bias voltage Vbup\ntoO(Γ) and O(Γ2) as indicated. (a) Strongly underdamped\ncase (b≫1) in the large- Ulimit. The current is computed\nﬁrst numerically from the extension ofEq. (13)tothe ﬁnite- U\ncase forU= 1000T/3 (red, denoted by I) and approximated\nby formula (31) in the limit of U→ ∞(blue, denoted by\nIappr). We also show the current (32) for zero spin accumu-\nlation (dashed black, denoted by I0). The other parameters\nareVg= 75T/3, Γs= 2Γd= 0.2T/3,ns=nd= 0.99,\nα= 0.01π, andW= 1000T/3. This choice of parameters\nimpliesb/greaterorsimilar5 [given by Eq. (37)] for both O(Γ) and O(Γ2)\nat the resonance. The approximated current and the numer-\nically computed current match well but not perfectly. The\nmain reason for the deviation is that the resonance does not\nappear here under strong Coulomb blockade conditions, as re -\nquired for Eq. (31) to be strictly valid. These conditions ar e\nmet in Fig. 10(b) below, where approximation and numerical\nsolution match perfectly. However, if we go deeper into the\nCoulomb blockade regime here, the resonance disappears in\nO(Γ), cf. Ref. 26. Therefore, to make a comparison between\ntheO(Γ) and O(Γ2) current, we considered the resonance\ncloser to the single-electron tunneling regime. (b) Critic ally\ndamped case ( b≈1) in the ﬁnite- Ucase: Stationary current\nup toO(Γ) and O(Γ2) as a function of bias voltage Vbnu-\nmerically calculated from Eq. (13) for Vg= 5Twith all other\nparameters as in Fig. 2, implying b≈0.5 forO(Γ),b≈0.2\nforO(Γ2). Note that our approximation formula (31) cannot\nbe applied for the ﬁnite- Ucase employed here.\n(i)Underdamped regime (b≫1). In this regime, the cur-\nrent is restoredto the full value I0at resonance( a/q= 1)\nsinceM0≈1. This is illustrated in Fig. 9(a), in whichwe plot the current numerically obtained from Eq. (13)\nextended to ﬁnite Uand the approximationformula(31).\nBoth are close to the value of I0(black dashed line) at\nthe resonance. Both agree well, but not perfectly, as we\nexplain further in the caption of Fig. 9. The resonance\nwidth,\nH≈α/radicalBigg\nΓ⊥\nΓ/bardbl, (38)\ndirectly yields the anisotropy of the spin-decay tensor,\nΓ⊥/Γ/bardbl, when the angle αis known. To extract Γ ⊥/Γ/bardbl\nfrom experimental data, one ﬁrst determines the spin-\ninjection asymmetry afrom the resonance position, as\ndescribed in Sec. IIIC. One then ﬁts Eq. (31) to gate\nor bias traces of the current peak, expressing a/q−1\nwith the help of Eq. (22) as a function of bias and gate\nvoltage. In the resulting expression, the functions I0,\nH, andAappear. For ﬁtting to experimental data,\nwe suggest to treat these slowly varying functions as\nconstant ﬁtting parameters near the resonance.\n(ii)Critical damping (b≈1). When the spin-decay rate\nis comparable to the spin-precession rate, the current\npeak value is not completely restored to I0asM0reaches\nonly a fraction of 1. Here, the spin decay limits the\nmaximally achievable rotation angle for the QD spin\nbefore it decays or tunnels out. This is visible in Fig.\n9(b), where the peak current may become smaller in\nO(Γ2) as comparedto that in O(Γ), where the spin decay\nis much slower. Furthermore, cotunneling corrections\naﬀect the width Hmore strongly than in the strongly\nunderdamped regime: Here, the width is not exclusively\ndetermined by the ratio Γ ⊥/Γ||but also incorporates\nb, which diﬀers depending on whether cotunneling\ncorrections are included or not. This illustrates that –\nin contrast to the resonance position – for the accurate\nprediction of the resonance shape the next-to-leading\norder corrections are indispensable. The pronounced\nsensitivity of the resonance to cotunneling processes in\nthe critically damped limit b≈1 is also interesting for\nthe characterization of the QD spin valve: Once B⊥is\ndetermined, e.g., from the pulsing scheme (see Sec. IV),\nwe may again use Eq. (31) as ﬁtting formula, taking M0\nnow as an additional ﬁtting parameter. One may then\nextract the spin relaxation rate Γ /bardbland the dephasing\nrate Γ ⊥individually by combining the results for Hand\nM0.\nIV. GATE-PULSING SCHEME: ALL-ELECTRIC\nSINGLE-SPIN OPERATIONS\nIn principle, the transport-induced spin decoherence\ntime∼U/Γ2can be made comparable or longer than ex-\nperimentallymeasuredspin-dephasingtimes dueto other\nmechanisms (see Sec. IVC) by reducing the tunneling12\nrates. Hence, multiple revolutions of an individual QD\nspin are feasible. Probing this underdamped spin preces-\nsion requires time-resolved measurements. At ﬁrst sight,\nit may seem challenging to utilize our transport setup for\nspin detection: Many spin-to-charge conversion readout\nschemes rely on a large energy splitting B≫Tbetween\nthe two spin states allowing the QD electron to leave into\nanattachedelectrodeonlyifithasonetypeofspin. How-\never, as there is no discernible spin splitting in our case,\nsuch an energy-selective readout scheme [62] is not appli-\ncable here. We therefore suggest to employ a tunneling-\nrate selective readout [62], which is naturally provided\nby the strongly spin-polarized ferromagnets in our setup.\nAs we predict in Sec. IVA, this only requires the adapta-\ntion of an experimentally well-developed pulsing scheme\n[63]. Using this scheme, underdamped oscillations in the\ntime-averaged current can be probed as a function of the\npulsing duration. To optimize the contrast in the aver-\nage current oscillations, the pulsing durations have to be\nchosen appropriately as we explain in Sec. IVB.\nIn contrast to other transport transport features in the\nCoulomb blockade regime such as the Kondo eﬀect or the\nzero-bias anomaly discussed in Refs. 24 and 28, the spin\nresonance does not destroy the coherence of the QD spin.\nThis is an advantage as it allows all-electric spin control\nto be accomplished even without the need of an external\nmagnetic ﬁeld or spin-orbit interaction. Only the basic\ntool of spintronics is required: large polarizations of the\nferromagnets.\nA. Probing underdamped spin precession from\naverage current\nThe procedureofthe simple pulsingschemeis sketched\nin Fig. 10(a): At ﬁxed bias voltage Vb, one repeatedly\napplies a rectangular voltage pulse to the gate electrode,\nswitching from V0\ngtoVgfor a time duration τ, and then\nback toV0\ngfor a time duration τ0. Figure 10(b) shows\nthe stationary current as function of Vg, exhibiting the\nspin resonance. We suggest to probe the time-averaged\ncurrent over many pulses,\n¯I=/integraldisplayt\n0dt′\ntIs(t′) (t≫τ,τ0), (39)\nvarying the time duration τ. Figures 10(c)–10(e)\nillustrate that the time-averaged current oscillates as\na function of τwith a period given by 2 π/|B|, which\ncoincides with the period of the plotted spin oscillations.\nThus, one can extract the magnitude of the exchange\nﬁeld|B|at (Vb,Vg). The oscillations can be physically\nunderstood as follows: By switching from the spin-valve\nblocked reference voltage V0\ng[with ﬁeld B0nearly\ncollinear with ns, cf. panel (i) in Fig. 10(a)] to a voltage\nVgwhere the exchange ﬁeld Bprecesses the injected\nspin, the electron is more probable to escape upon return\ntoV0\ngprovided the duration τmatches a half-integer\nmultiple of the precession time τP= 2π/|B|.\nFIG. 10: (a) Schematics of the pulsing scheme. (b) Station-\nary current as function of Vg, obtained by solving Eq. (13)\nexactly ( Ist, green), by neglecting the spin accumulation, i.e.,\nforcingS= 0 (I0, dashed black), and by taking the approxi-\nmation (31) near resonance ( Iappr\nst, red), see Sec. IIIF. (c)–(e)\nAverage current ¯I=/integraltextt\n0(dt′/t)Is(t′) (t≫τ,τ0) (green curves)\nas a function of τfor three diﬀerent Vgas indicated and for\nﬁxedτ0= 2·103/T= 0.46τP, andV0\ng= 30T. The times τ0\nandτare given in units of the precession period at resonance,\nτP≈4.7·103/T. The current is oﬀset by ¯Ist, the current that\nwould ﬂow if the QD were in the stationary state at each in-\nstant of time. Also plotted is the spin component along the\ndrain polarization S·ˆnd(blue curves) computed from Eq.\n(13) for initial condition S=ˆns/2 andp1= 1−p0= 1.\nThroughout we used ns=nd= 0.99 (see caption of Fig. 2),\nα= 0.005π,Vb= 50T,W= 500T, Γs= 0.15T, Γd= 0.1T.\nThe plots are obtained by numerically solving the analytica lly\nderived kinetic equations (13) in the limit U→ ∞using the\nscheme discussed in Appendix B3. To make use of analyt-\nical results, we need a tiny angle αhere. For ﬁnite U, this\nrestriction is unnecessary.\nWe compute the average current shown in Figs. 10(c)–\n10(e) as follows: Taking the stationary state at V0\ngas\ninitial state ρ(0), we obtain the time-dependent solution\nforρ(t) by solving the kinetic equations (13). This yields\nthe time-dependent particle current Is(t) from Eq. (14).\nFor both the current and the kinetic equations the rates\nare time-dependently switched by changing the gate\nvoltageV0\ng↔Vgin the respective expressions according\nto the pulsing scheme. To ensure that Eq. (39) really\ngives the current measured in a circuit, we checked\nthat ˙p1(t)≪ |Is(t)|,|Id(t)|, i.e., the magnitudes of the\ncurrents ﬂowing out of the source, |Is(t)|, and into the\ndrain,|Id(t)|, are nearly the same. Under this condition\ndisplacement currents can be neglected, as explained,\nfor example, in Ref. 64. We comment in Appendix B5\non the importance of non-Markovian corrections that we13\nneglect here.\nThe key feature of the current oscillations shown in\nFigs. 10(c)–10(e) is that the visibility strongly depends\non the voltage Vgcontrolling the opening angle of the\nprecession. The visibility becomes maximal at the reso-\nnance in Fig. 10(d). To prove our claim that the current\noscillations are correlated with a spin precession, we\ncompare in Figs. 10(c)–10(e) the time-averaged current\nwith time- dependent spin-projection curves, which are\nobtained as follows: We take the initial state ρ(0) to\nbe the maximally polarized state with spin S=ˆns/2\nand corresponding occupation probabilities p1= 1 and\np0= 0, i.e., we do not start from the stationary state\nat gate voltage Vg. We then solve the kinetic equations\n(13) time-dependently, keeping the gate voltage ﬁxed\natVg. The resulting spin vector S(t) is then projected\non the drain polarization direction ˆnd, which yields\nthe diﬀerent spin projection curves S(t)·ˆnsin Figs.\n10(c)–10(e) with τ=t. This comparison shows that the\ncurrent actually oscillates with the same frequency with\nwhich a spin would precess in a QD held at gate voltage\nVg.\nFinally, by going slightly oﬀ-resonance the precession\naxiscan be fully tuned within the plane of polarizations\nwhile maintaining full control over the precession angle\nthroughτ. This allowsall single-spin operationsrequired\nfor quantum algorithms to be implemented.\nB. Optimizing the pulsing scheme\nTo set up an experiment that probes the underdamped\nspin precession, we provide here some additional infor-\nmation under which conditions the contrast of the signal\nobtained by the pulsing scheme is maximized. For this\npurpose, we discuss the ratio ¯I/¯Istof the time-averaged\ncurrent (39) from the pulsing scheme, ¯I, to the station-\nary current ¯Ist. The latter is obtained by replacing the\ntime-dependent current Is(t) =Ist(Vg(t)) in Eq. (39) by\nthe stationary current, switching only the gate voltage\nVgas a parameter time-dependently.\nFirst, underdamped precession cycles of a single spin are\nfeasible only if the spin-decay rate is much smaller than\nthe spin-precession rate at the resonance (see Sec. IIIF),\nthat is, if\nb=|B⊥|//radicalBig\nΓ⊥Γ||≫1. (40)\nThis condition is diﬀerent from the condition that max-\nimizes the stationary current , cf. Sec. IIIF. There, we\nfound a ratio b∼1 to be optimal because then roughly\none revolution takes place within the average electron\ndwell time on the QD. If the tunneling rate allows for\nmultiple precession cycles, the stationary resonant cur-\nrent is suppressed again because tunneling happens in-\nfrequently, even if its spin has optimal overlap with the\ndrainpolarization. Thiscurrentsuppressionnearthe res-\nonance does not appear for the gate-pulsing scheme sinceone returns to a gate voltage V0\ngcloser to the single-\nelectron tunneling resonance where the tunneling rate is\nlarger and the electron can leave the QD quickly after it\nhas been precessed at gate voltage Vg. Thus, the larger\nthe ratiob, the clearer the current oscillations are and\nthe longer they persist.\nThe second important set of parameters that has to be\noptimized are the dwell times τ0andτat voltageV0\ngand\nVg. A ﬁrst requirement is that\nτ0/lessorsimilarτ (41)\nbecause ifτ0≫τthe system is most of the time not at\nresonance and the average current is determined by the\ndynamics at gate voltage V0\ng. Condition (41) is fulﬁlled\nfor most values of τshown in Fig. 10. However, there is\nanother condition that is equally important: We ﬁnd on\na numerical basis that τ0is chosen optimally as\nτ0≈0.1τ0\nT≈0.1/I0\nst, (42)\nwith the electron dwell time τ0\nTat gate voltage V0\ng, which\ncan be estimated by the inverse of the stationary particle\ncurrentI0\nstat voltageV0\ng. Ifτ0/greaterorsimilarτ0\nT, the averagecurrent\nis mostly determined by the large stationary currentI0\nst,\ni.e., the precession-induced initial modiﬁcation of the\ncurrent atV0\ngis rather insigniﬁcant. This is illustrated\nin Fig. 11(a), in which we plot the ratio of the average\ncurrent ¯Iobtained from the pulsing scheme and the\naverage current ¯Istthat is obtained if the QD was in\nthe stationary state all the time (but switching between\nthe diﬀerent levels at the two gate voltages): Clearly,\nfor small (but not very small) times τ0, the current is\ndrastically enhanced over the stationary current due\nto the gate pulsing, while the ratio decreases if τ0\napproaches τ0\nT. In Fig. 10, we use a value τ0/τ0\nT∼1,\nwhich already yields a sizable enhancement.\nHowever, if τ0≪τ0\nT, the ratio ¯I/Istbecomes drastically\nsuppressed as Fig. 11(a) also shows. In this case, the\nQD electron does not have enough time to tunnel out\nof the QD when the gate voltage is switched to V0\ng.\nThe average current is then mostly determined by the\ntime-averaged current at resonance. This is illustrated\nin Fig. 11(b), which shows the time-dependent current\nIs(t) (blue) besides its average current ¯I(green), which\nloses contrast after roughly two cycles.\nWe conclude that for setting up and optimizing the puls-\ning scheme in an experiment, the initial characterization\nof the spin valve is of the utmost importance. Once the\ntime scales are known our above discussion should be a\nguide to choose the pulsing times properly.\nC. Experimental feasibility: spin decoherence\nFinally, we provide rough estimates for the spin-decay\ntimes and spin-precession periods for experimentally\nachievable parameters, demonstrating the feasibility of14\nFIG. 11: (a) Ratio ¯I/¯Istas a function of the duration τ0in\nunits of the electron dwell time τ0\nT= 1/I0\nst≈4.7·104/T.\nHere,¯Iis the average current (39) for the pulsing scheme\nand¯Istis the stationary current obtained if the QD was in\nthe stationary state all the time. The gate voltage Vg=\nV∗\ng= 59.8Tis tuned to the resonance and τ= 2500/ T≈\n0.53τPso that a nearly maximal enhancement of the current\noccurs after the precession. (b) Average current ¯Iand time-\ndependent current Is(t) as function of the pulse time τwith\nτP≈4.7·103/Tandτ0= 10/T= 2.1·10−3τP≪τ. We\nsubtract the stationary current Istﬂowing at gate voltage\nVg=V∗\ng. The results shown in (a) and (b) are computed for\na single pulse, N= 1 [t=τ0+τin Eq. (39)], and all other\nparameters are the same as in Fig. 10.\nunderdamped spin precession cycles in the Coulomb\nblockade regime. Typical spin-dephasing times of\n∼10−30ns have been measured in GaAs QDs [4, 7, 65]\nand are also compatible with measurements involving\ncarbonnanotubes(CNTs)[56]. Inourcase,thecotunnel-\ningcurrentthroughtheQDleadstoadditionaldephasing\nwith time constant ∼U/Γ2∼10/µeV∼30ns for typical\nvalues of Γ ∼0.01meV and U∼5meV feasible both\nfor semiconductor QDs and CNT QDs. The energy\nscale related to the exchange ﬁeld may be estimated as\nµBB > µ BBd,⊥≈µB|log(1/2)|Γdndsinα/π∼0.7µeV\nfornd∼0.5 andα∼0.2π. This translates into a\nmaximal precession period of T∼2π/0.7µeV∼6ns at\nthe resonance and even smaller periods away from it.\nThus, indeed, the spin precession period can be made\nsmaller than the spin-decay time.\nOne may wonder whether the spin resonance could\nalso be observed in the strong-coupling regime Γ ≫T.\nThis regime has been under intense experimental in-\nvestigation (using collinear polarizations so far) since\nthe exchange ﬁeld can be probed there by the strong\nspin splitting it induces, aﬀecting the Kondo resonance\n[33–35]. Increasing the tunnel coupling Γ, however,\nenhances the spin-decoherence rate more strongly than\nthe spin-precession rate. Moreover, spin-ﬂip processes\ndriving the Kondo eﬀect for small bias voltage also\ndestroy the QD spin coherence. Thus, the spin preces-\nsion may not be underdamped any more in the strong\ncoupling regime. In addition, the smaller spin-precession\nperiods, which are approximately ∼100 ps as extracted\nfrom exchange-ﬁeld magnitudes in Ref. 34, makes it\nmore challenging to apply the pulsing scheme described\nabove. By contrast, spin-resonance features are more\nlikely to be seen in the stationary current, which requiresthe spin-precession rate only to be comparable to the\nspin-decoherence rate. The reader should note that the\nwidth of the spin resonance can be much smaller than\nthe spin-decoherence rate as our analysis in Sec. IIIF\nshows. Thus, it is worth investigating the spin resonance\nin the strong-coupling regime further. We ﬁnally note\nthat in the meantime of revising the manuscript of this\npaper, signatures of the spin resonance have also been\nfound by one of the authors in waiting-time distributions\n[66].\nV. SUMMARY AND OUTLOOK\nWe have identiﬁed a spin resonance that, unlike usual\nresonances, does not appear when scalar energies of the\nlocal quantum system and reservoir match. Instead, the\ncondition (15) based on vectors, B·ˆns= 0, needs to\nbe satisﬁed. The resonance emerges in the simplest QD\nspin-valve setup one can think of: an interacting spin-\ndegenerate single level which is tunnel-coupled to two\nferromagnets for almost(but not exactly) antiparallel\npolarizations nsandnd. Forthis magnetic conﬁguration,\nthe direction of the exchange ﬁeld Bstrongly depends on\nthe applied voltages, which generates a sharp feature all\nacross the Coulomb diamond of the transport stability\ndiagram.\nThe resonance is clearly distinguished from other\nfeatures in the stability diagram: First, it emerges\nonly for nonzero noncollinearity angle αand responds\nsensitively to changes in α. Second, it depends strongly\non the asymmetries of the spin-injection rates. For small\nasymmetries, it exhibits a strong current rectiﬁcation\neﬀect while for symmetric spin-injection rates it lies\nat zero bias. Third, when these parameters cannot be\ncontrolled in an experiment, one can use the peculiar\nvoltage-dependent line shape of the spin resonance to\ntell it apart from other features. For example, the\nspin resonance vanishes at the particle-hole symmetry\npoint. Furthermore, its width is not given by a simple\ncombination of the tunnel couplings, temperature, or\nany other natural energy scale. Instead, it depends on\nthe ratio of the gate-voltage dependent exchange ﬁeld\nand the spin-decay rates. The latter features contrast\nparticularly with those from the Kondo eﬀect or the\nzero-bias anomaly predicted earlier in Refs. 24 and 28 .\nWhile the resonance position is entirely dictated by\nthe exchange ﬁeld direction , the shape of the resonance\n(peak value, width) is strongly inﬂuenced by the QD spin\ndecay. We have identiﬁed the ratio b=|B⊥|//radicalbigΓ⊥Γ||to\nbe the relevant parameter that determines the resonance\nshape. This ratio involves the perpendicular exchange\nﬁeld component B⊥=B·ˆn⊥, the spin-relaxation\nrate Γ ||=ˆns· RS·ˆns, and the spin-dephasing rate\nΓ⊥=ˆn⊥· RS·ˆn⊥. The resonance appears for b/greaterorsimilar0.1\nin the stationary transport, which is satisﬁed in the\nCoulomb-blockade regime. There, the spin decay is15\nlimited by next-to-leading order processes (cotunneling)\nwith rate Γ ⊥,||∼Γ2/U, which can indeed be made\nsmaller by reducing the tunnel couplings than the\nspin-precession frequency B⊥∼Γ at resonance. The\nstrongest contrast in the stability diagram is expected\nforb≈1 when roughly half a spin revolution happens\nwithin the electron dwell time in the QD. By contrast,\nin the limit b≫1, the spin coherence lasts much\nlonger than one spin revolution and underdamped spin-\nprecession cycles becomes feasible. The underdamped\nspin precession leads to no qualitative modiﬁcations\nof the resonance in the stationary conductance, but\ncan nevertheless be probed experimentally by a simple\ngate-pulsing scheme. In the latter, the precession axis is\ncontrolled by electrical means and the rotation angle by\nthe duration of the pulses. This allows one to determine\nthe magnitude of the exchange ﬁeld and in combination\nwith stationary-conductance measurements to determine\nthe anisotropy of the spin decay all-electrically. This\ncan even be used to realize every single-spin qubit gate\noperation in an all-electric way.\nBesides opening new avenues for spintronics and single-\nspin control, the spin resonance studied here provides an\nillustration of a generic concept in the simplest conceiv-\nable setting: such an anomalous resonance can appear in\nany open quantum system with quasi-degenerate states\nwhose coherence is described by a Bloch vector. For a\ntwo-level system, it is required that (i) the Bloch vector\nsuﬀers only from little decoherence, (ii) the coherent\nevolution is dominated by a renormalization-induced\nﬁeld vector – because of level degeneracy – (iii) which\nis induced by an environment that breaks symmetries\n(which are often present in idealized models). When\nsuch a system is tuned by experimentally accessible pa-\nrameters, a resonance unrelated to any energy splitting\ncan appear when the ﬁeld vector becomes perpendicular\nto the Bloch vector.\nThis can be extended to N-fold degenerate multiplets,\ndescribed by a generalized Bloch vector and an associ-\nated renormalization ﬁeld vector. Interestingly, in this\ncase the precession takes place in a higher dimensional\nspace and is expected to be overlooked even more easily\nas compared to the simple case studied here. Scenarios\ncan be envisaged in nuclear spin systems, [67] double\nQDs [68], or vibrating molecular devices [53, 69, 70].\nOur simple example shows that, interestingly, Coulomb\ninteractions realize both requirements (i) and (ii) while\nnoncollinear spin valves naturally provide (iii).\nAcknowledgements\nWe acknowledge discussions with M. Baumg¨ artel, S.\nDas, S. Herzog, M. Misiorny, and F. Reckermann, and ﬁ-\nnancialsupportoftheSwissNationalScienceFoundation\nvia the NCCR QSIT [B. S.], and the Swedish Research\nCouncil (VR) [M. L.].Appendix A: Quantum master equations and Pauli\nsuperbasis\nIn this Appendix, we outline how our coupled diﬀeren-\ntial equations for operator averages (13) can be obtained\nfrom the general kinetic equation (12) for the reduced\noperator of the QD. For this purpose, we use a Liouville-\nspace notation whose key elements we brieﬂy review.\nApplying the real-time diagrammatic technique [52, 54],\none can express the kinetic equation of the reduced den-\nsity operator ρ(t) of the QD as\n|˙ρ(t)) =−iL|ρ(t))+/integraldisplayt\n−∞dt′W(t−t′)|ρ(t′)).(A1)\nHere, we introduced a bra-ket notation for linear opera-\ntors|A) :H → Hacting on a Hilbert space H, which al-\ntogether form the Liouville space L. Furthermore, Land\nWdenotesuperoperators,whichareoperators S:L → L\nmapping a Liouville-space element on another Liouville-\nspaceelement. In particular, L•= [H,•]mediatesthein-\nternalevolutionofthedensityoperatorbytheQDHamil-\ntonianH, where the dot “ •” denotes the operator that L\nacts on. Furthermore, W(t−t′) is the real-time diagram-\nmatic kernel that incorporates the eﬀect of the environ-\nment on the evolution of the reduced system. We next\ncarry out a Markov approximation, that is, we replace\n|ρ)(t′)≈ |ρ)(t) in Eq. (A1) and obtain\n|˙ρ(t)) = [−iL+W]|ρ(t)). (A2)\nHere,W=/integraltext∞\n0dteiztW(t)|z=i0is the zero-frequency\ncomponent of the kernel. One can prove [52, 54] that the\nstationary state calculated from Eq. (A2) is the exact\nstationary solution of Eq. (A1). For actual calculations,\nEqs. (A1) and (A2) are expressed in terms of matrix ele-\nments. To achieve this goal, one introduces the following\nscalar product in Liouville space: [71]\n(A|B) := Tr(A†B). (A3)\nAn orthonormal superbasis is a set of superstates {|A)}\nthat is orthonormal with respect to this scalar product,\n(A|B) =δAB, (A4)\nand satisﬁes the completeness relation:\nI=/summationdisplay\nA|A)(A|. (A5)\nHere,Idenotes the superidentity I|A) =|A) forany |A).\nAs a consequence, any superstate |O) can be expanded\ninto such an orthonormal basis by |O) =/summationtext\nAOA|A)\nwith coeﬃcients OA= (A|O) and any superoperator\ncan be expressed as S=/summationtext\nA,BSAB|A)(B|withSAB=\n(A|[S|B)].\nUsually, Eq. (A2) is expressed in terms of matrix ele-\nments for the superbasis |a,b) :=|a∝an}brack⌉tri}ht∝an}brack⌉tl⌉{tb|, which yields\n˙ρab−iLab\na′b′ρa′b′+Wab\na′b′ρa′b′, (A6)16\nwithρab= (a,b|ρ) = Tr([ |a∝an}brack⌉tri}ht∝an}brack⌉tl⌉{tb|]†ρ) andSab\na′b′=\n(a,b|S|a′,b′) = Tr([|a∝an}brack⌉tri}ht∝an}brack⌉tl⌉{tb|]†[S|a′∝an}brack⌉tri}ht∝an}brack⌉tl⌉{tb′|]) forS=L,W. In\nthe Keldysh-contour formulation of real-time diagram-\nmatics, [72, 73] diagram rules are given for the kernel\nmatrix elements Wab\na′b′. The diagonal matrix elements\nρaaare interpreted as occupation probabilities and the\noﬀ-diagonal elements ρabas coherences. For a diﬀerent\nchoice of the basis states, however, the coherences in the\nformer basis contribute to the occupation probabilities in\nthe newbasis. Thus, the interpretationas“probabilities”\nand “coherences” is meaningful only if a speciﬁc basis is\nsingled out by the symmetry of the problem. For the\nsingle-level Anderson model we consider here, this would\nbe the case for nonmagnetic electrodes. In this case, one\ncan start from the Hilbert space basis {|0∝an}brack⌉tri}ht,| ↑ ∝an}brack⌉tri}ht,| ↓ ∝an}brack⌉tri}ht,\n|2∝an}brack⌉tri}ht}with a ﬁxed quantization axis for the spin. All co-\nherences between spin states are zero in the stationary\nlimit.\nFor noncollinear lead polarizations, such a spin quantiza-\ntion axis does not exist, c.f., Sec. IIA. Thus, it is helpful\nto expand Eq. (A2) in terms of diﬀerent supermatrix el-\nements such that all expressions are independent of the\nquantization axis. For this purpose, we chose a superba-\nsis{|A)}consisting of observables . The reduced density\nmatrix can then be expanded as\n|ρ) =/summationdisplay\nAA|A), (A7)\nwhere\nA= (A|ρ) = Tr(Aρ) (A8)\nis the expectation value of observable A(=A†) - an ob-\nject with an intuitive physical interpretation in contrast\nto the matrix elements ρab.\nFor the single-level Anderson model, a suitable superba-\nsis is the Pauli superbasis . We focus here on the charge-\nconserving setup without superconductor (see Fig. 1).\nFor the nondegenerate subspaces with zero ( n= 0) and\ntwo (n= 2) electrons, these are simply the projectors\n|ˇr0\n0) :=ˆP0=|0∝an}brack⌉tri}ht∝an}brack⌉tl⌉{t0| (A9)\nand\n|ˇr2\n0) :=ˆP2=|2∝an}brack⌉tri}ht∝an}brack⌉tl⌉{t2|. (A10)\nFor the subspace of charge state n= 1, we introduce\n|ˇr1\nµ) :=/summationdisplay\nσσ′(ˇrµ)σσ′|σ∝an}brack⌉tri}ht∝an}brack⌉tl⌉{tσ′|, (A11)\nwhere,µ= 0,1,2,3, (ˇr0)σσ′=δσσ′/√\n2, and (ˇrµ=i)σσ′=\n(σi)σσ′/√\n2 involving the Pauli matrices σifori= 1,2,3.\nThe element |ˇr1\n0) =ˆP1/√\n2 is proportional to the scalar\nprojection operator on charge state 1 and the elements\n|ˇr1\ni) =√\n2ˆSiareproportionaltothe vectorcomponentsofthe spin operator. Altogether, these six superstates pro-\nvide an orthonormalbasis for the subspace of the charge-\ndiagonal QD operators,\n(ˇrn\nµ|ˇrn′\nµ′) =δnn′δµµ′, (A12)\nIC=/summationdisplay\nnµ|ˇrn\nµ)(ˇrn\nµ|, (A13)\nwhereICdenotes the identity operator in the subspace\nof charge-diagonal operators. The factors 1/√\n2 are in-\ntroduced in the deﬁnition of (ˇ rµ)σσ′to avoid additional\nfactors in Eqs. (A12) and (A13). The Pauli superbasis is\nsuﬃcient to expand the density operator |ρ) [78] which\nreads by applying Eq. (A7):\n|ρ) =1√\n2/summationdisplay\nnpn|ˇrn\n0)+√\n2S·|ˇr1),(A14)\nwherep1=√\n2Tr(ˇr1\n0ρ),p0/2= Tr(ˇr0/2\n0ρ) are the occupa-\ntion probabilities of charge state nandS= Tr/parenleftbigˇr1ρ/parenrightbig\n/√\n2\nis the average spin operator (4). Importantly, Eq. (A14)\niscovariant , i.e., form-invariant under a change of the\nspin-quantization axis or the real-space coordinate sys-\ntem. This also illustrates that working in Liouville space\ndoes not only give more compact expressions but it also\nresults in a physically more transparent description of\nthe QD state and its dynamics.\nAppendix B: Kinetic equations and current for\nquantum-dot spin valve\n1. Rates for kinetic equations and current\nIn this Appendix, we give all expressions for the rates\nin the kinetic equations (13), which read:\n˙p0=−2Γ0p0+Γ1p1+2GpS·S,\n˙S= +G0\nSpp0−1\n2G1\nSpp1−RS·S−B×S.(B1)\nThe charge-relaxation rates are given by\nΓ0/1= Γ±\n0±Im(K+\n00+1\n2/summationdisplay\nρK−\nρρ),(B2)\nwhere Greek indices take the values ρ= 0,1,2,3 and\nLatin indices take the values i= 1,2,3. Furthermore,\nthe vectorial spin-to-charge conversion rates are given by\n(GpS)i= Γ−\ni−Im/parenleftbig\nK+\ni0+1\n2K−\ni0+1\n2K−\n0i/parenrightbig\n−1\n2/summationdisplay\njkεijkRe(K−\njk), (B3)\nthe vectorial charge-to-spin conversion rates are given by\n(G0/1\nSp)i= Γ±\ni±Im/parenleftbig\nK+\ni0+1\n2K−\ni0+1\n2K−\n0i/parenrightbig\n∓1\n2/summationdisplay\njkεijkRe(K−\njk), (B4)17\nthe symmetric spin-decay tensor is deﬁned by\n(RS)ij=δijΓ−\n0+δijIm/parenleftBigg\n−1\n2K−\n00+1\n2/summationdisplay\niK−\nii−D−+\n00/parenrightBigg\n−1\n2Im(K−\nij+K−\nji+X+−\nij+X+−\nji),(B5)\nand, ﬁnally, the vectorial exchange ﬁeld reads\nBi=βi+Re/parenleftbig1\n2K−\ni0−1\n2K−\n0i+D−+\ni0/parenrightbig\n.(B6)\nThe above rates ﬁrst contain terms of O(Γ), Γχ\nρ(ε) =/summationtext\nrΓχ\nr,ρ(ε) andβρ(ε) =/summationtext\nrβr,ρ(ε), with\nΓχ\nr,ρ=0(ε) = Γχ\nr(ε) = 2π|tr|2¯νrf(χ(ε−µr)/T),(B7)\nΓχ\nr,ρ=i(ε) = Γχ\nr(ε)nr,i(i= 1,2,3), (B8)\nβr,ρ(ε) =P/integraldisplay+W\n−Wdω\nπΓ+\nr,ρ(ω)\nε−ω, (B9)\nwithPdenoting the principal value integral and the\nFermi function f(x) = 1/(ex+1). Here, the spatial com-\nponentsρ= 1,2,3 point along by the polarization vector\nnrof leadr. Furthermore, the O(Γ2) contributions in-\ncorporate two diﬀerent tensors, namely\nXχ2χ1\nρ2ρ1=/integraldisplay+W\n−W/integraldisplay+W\n−Wdω1\nπdω2\nπΓχ2\nρ2(ω2)Γχ1\nρ1(ω1)\n1\ni0+ω2−ε1\ni0+ω2−ω11\ni0−ω1+ε,(B10)\nandDχ2χ1ρ2ρ1given by the same expression when replacing\nthe right-most denominator in the above expression by\n1/[i0+ω2−ε]. In contrastto previousworks, we evaluate\nthe full complex integral to completely capture the dy-\nnamics of the spin coherences in the Coulomb blockade\nregime to order Γ2. Adding the X- andD-integrals, we\nobtain the simpler function\nKχ1\nρ2ρ1= ¯χ2(Xχ2χ1\nρ2ρ1+Dχ2χ1\nρ2ρ1) (B11)\n= [χ1β′\nρ2βρ1+Γ′\nρ2Γχ1\nρ1]+i[χ1Γ′\nρ2βρ1−β′\nρ2Γχ1\nρ1],\nwhere Γ′\nρ=dΓ+\nρ/dεandβ′\nρ=dβρ/dε.\nWe note that the leading-order Γ contribution to the\nspin-relaxation tensor (B5) is isotropic while the next-\nto-leading order Γ2contribution renders the spin decay\nanisotropic. Sincethe leading-ordertermis suppressedin\nthe Coulomb-blockade regime, the spin decay can indeed\nbecome signiﬁcantly anisotropic. In contrast to the de-\ncay rates, the ﬁrst-order Γ contribution to the exchange\nﬁeld (B9) is only logarithmically suppressed.\nThe expression for the average current from lead rinto\nthe QD reads\nIr= 2Γr,0p0−Γr,1p1−2Gr,pS·S(B12)with\nΓr,0/1= Γ±\nr±Im(K+\nr,00)+1\n2/summationdisplay\nρIm(K−\nr,ρρ),(B13)\n(Gr,pS)i= Γ−\nr,i−Im/parenleftbig\nK+\nr,i0+1\n2K−\nr,i0+1\n2K−\nr,0i/parenrightbig\n+Im(X+−\nr,0i−X−+\nr,i0)\n−1\n2/summationdisplay\nρ2ρ1εiρ2ρ1Re(K−\nr,ρ2ρ1), (B14)\nwhereXχ2χ1r,ρ2ρ1is obtained from Eq. (B10) by replacing\nΓχ2ρ2(ω2)→Γχ2r,ρ2(ω2) andKχ\nr,ρ2ρ1is obtained from Eq.\n(B12) by replacing β′\nρ2→(βr,ρ2)′and Γ′\nρ2→(Γr,ρ2)′,\nrespectively.\nTheX-type integrals, Eq. (B10), and the corresponding\nD-type integrals are computed numerically as we explain\nnext. We convert the double frequency integral into a\ndouble summation over Matsubara frequencies by ﬁrst\nsubstituting x2=ω2/Tandx1=−ω2/Tand splitting the\nFermi functions f(xT) =g+(x) +g−(x) into their sym-\nmetric part g+(x) = 1/2 and their antisymmetric part\ng−(x) =−tanh(x/2)/2, respectively. We then integrate\noverx1andx2using complex integration, closing the in-\ntegration contour in the upper half of the complex plane.\nBy virtue of the residue theorem, one can derive the fol-\nlowing relation for the generic type of integrals occurring\nafter these manipulations\n/integraldisplay+R\n−Rdx1/integraldisplay+R\n−Rdx2gq1(x1)gq2(x2)\n1\nxj−λ2+i01\nx1+x2+i01\nx1−λ1+i0\n=−4π2δq1,−δq2,−kR/summationdisplay\nk1,k21\nzkj−λ21\nzk1+zk21\nzk1−λ1\n+2πiδq1,−δj,1kR/summationdisplay\nk11\nzk1−λ21\nzk1−λ1kR/summationdisplay\nk2Mq2\nk2\n+O/parenleftbigg1\nR/parenrightbigg\n, (B15)\nwherej= 1,2,q1,q2=±,zk1,2=iπ(2k1,2+ 1) are the\nMatsubara frequencies, and 0 ≤k≤kR=⌈R\n2π−1\n2⌉with\n⌈x⌉denoting the smallest integer that is not less than x.\nWe additionally used the abbreviation\nMq2\nk2=1\n2/bracketleftbigg\nq2ln/parenleftbiggzk2+iR\nzk2+R/parenrightbigg\n+ln/parenleftbiggzk2−R\nzk2+iR/parenrightbigg/bracketrightbigg\n.(B16)\nThe above double Matsubara sums are then evaluated\nnumerically.\n2. Extension of former studies\nIn this Appendix, we compare our kinetic equations\n(13) to those of prior studies of QD spin valves and re-\nsults from other approaches: In fact, our theoretical ap-\nproach presents a technical step forward relative to the18\nprevious works, which is a reason why the spin resonance\nhas been overlooked for a long time.\nQuamtum master equations. First, the lowest-order Γ\ncontribution to our equations complies with the results\ngiven in Refs. 36 and 40 taking the limit U→ ∞. How-\never, a lowest-orderexpansion in Γ is not suﬃcient in the\nCoulomb blockade regime since these terms are exponen-\ntially suppressed with the distance |ε−µr|/Tfrom the\nFermi levels. By contrast, some O(Γ2)-terms are only\nalgebraically suppressed and therefore dominate there.\nIn particular, this is associated with a spin decay due\nto cotunneling that could obliterate the coherent spin-\nprecession features. This was noted in Ref. 26 where the\nspin resonancewasreported to emergeon the ﬂank ofthe\nCoulomb diamond using an O(Γ) kinetic equation, but\ncould not be reliably followed into the Coulomb block-\nade regime. However, the sharp resonance feature we\nﬁnd here even when O(Γ2) cotunneling corrections are\nincluded shows that spin precession eﬀects can still be\ndominant – as anticipated in the introduction from time-\nscale estimations.\nNext-to-leading order corrections ∼Γ2have been in-\ncluded in other studies of the same model, for example,\nin Refs. 24, 28, and 74; however these works address\nonlycollinearly polarized ferromagnets. Here, the spin\nprecession cannot occur since the spin accumulation and\nthe exchange ﬁeld are collinear (cf. the expressionsof the\nrates in Appendix B1). In Ref. 75, also the noncollinear\nmagneticconﬁgurationisstudied, but the QDis assumed\nto be deposited on a ferromagnetic substrate causing a\nlarge splitting ε↑−ε↓≫Γ of the two spin states as com-\nparedtothetunnelcoupling,sothatthespincomponents\ntransverse to this splitting ﬁeld have negligible impact.\nThe diﬃcult case of degenerate QD spin states, non-\ncollinear polarizations, and cotunneling corrections has\nto our knowledge been addressed only in Ref. 50. While\nthe kinetic equations given there include all the terms\nthat correspond to the imaginary parts up to O(Γ2) and\nthe real parts up to O(Γ) in the rates (B2) - (B6), our\nequations additionally include the O(Γ2) corrections to\nthe real parts of these rates. This is done via the Mat-\nsubara double summation (B15), which is implemented\nnumerically. For othermodels with higherdegree ofsym-\nmetry, which only require the imaginary part of these\nintegrals, this can be avoided (see Ref. 52). Thus, we in-\nclude, for example, in the exchange ﬁeld (B6) all renor-\nmalization eﬀects up toO(Γ2). Our results actually con-\nﬁrm that the O(Γ2) corrections to the exchange ﬁeld are\nnot important near the particle-hole symmetry point, at\nleast for an accurateprediction ofthe resonanceposition.\nHowever, this is not clear from the start and required a\ncareful numerical examination. Furthermore, our kinetic\nequations (13) are compactly expressed in equations for\nphysically meaningful observable averages.\nOther methods. Several other works dealing with non-\ncollinearly polarized leads employ completely diﬀerent\ntechniques, such as a Green’s function approach in the\nnoninteracting approximation [25], in a Hartree-Fock ap-proximation [21, 22, 51], or restricted to zero bias [23].\nAs these works do not employ kinetic equations, a di-\nrect comparison of the results is more diﬃcult. Some of\nthese studies address diﬀerent exchange ﬁeld eﬀects also\nfor noncollinear polarizations; [23, 51] yet, a sharp reso-\nnance has not been reported there.\nThus, even though we investigate in this paper the well-\nstudied Anderson QD model with noncollinear ferromag-\nnets, our technically advanced analysis gives us access to\na parameter regime for which reliable predictions were\nhardly possible before. This allows us to go beyond pre-\nvious works. The reason that our spin resonance without\nspinsplittinghasbeenoverlookedsofaristhatit requires\nthe careful treatment of the combination of (i) slow deco-\nherence of the spin in the Coulomb blockade regime, (ii)\nthe degeneracy of the spin levels allowing the coherent\nevolutionto be dominated by the exchangeﬁeld, and (iii)\ncompleterotationalsymmetrybreakingbynoncollinearly\npolarized ferromagnet.\n3. Solving the quantum master equation in the\nCoulomb blockade regime\nAs explained in Sec. IIB, in the Coulomb blockade\nregime the next-to-leading order Γ2contributions can\ndominate over the leading-order Γ contributions. This\nalso requires careful consideration when solving Eq. (13)\nfor the occupation probabilities pnand the average spin\nS: To solve the kinetic equations one could perform a\nsystematic perturbation expansion not only for the ker-\nnels but also for the probabilities pn=p(0)\nn+p(1)\nn+...\nand the spin S=S(0)+S(1)+...in orders of Γ, and\nsolve Eq. (12) then order by order in Γ. This has the\nadvantage that the current is evaluated consistently to\na given order in Γ. This procedure works well as long\nas lowest-order Γ tunneling processes (sequential tunnel-\ning) arepresent but fails in the Coulomb blockade regime\nwhere sequential tunneling is exponentially suppressed\nand cotunneling dominates. This is particularly impor-\ntant for the results obtained for inﬁnite U, shown in Fig.\n10. Therefore, we use an alternative procedure in which\nonly the kernels (but not the probabilities and the spin)\nare expanded in powers of Γ. It is, thus, the kernelsthat\nareto be consistently evaluated to agiven orderin Γ: not\nthe density operator or observables such as the current.\nThis issue has been thoroughly discussed elsewhere for\nour model [24] but also in a more general context [52] in-\ncluding, e.g., vibrational degrees of freedom on the QD.\nAlthough the current we obtain may comprise terms of\norder Γ3, we checked that the spin resonance is clearly\nnot an artifact of those terms. By varying Γ, the res-\nonance current is found to scale maximally as Γ2or a\nlower power but not as Γ3.19\n4. Incorporation of superconducting terminal\nWe comment here on the results that we show in Fig.\n6 when adding a proximal superconductor to the setup.\nTo simplify the analyticalcalculations(which arealready\nquite involved without a superconductor), we included\nhere only the leading-order Γ contribution in the tunnel-\ning rates. Consistent with this, the charging energy U\nhas been chosen of moderate size there. There are sev-\neral reasons why this simpliﬁcation does not aﬀect the\nconclusions we draw from Fig. 6 that concern only the\nresonance position. First, we note that the eﬀect of the\nsuperconductor is clearly visible when moving into the\nCoulomb diamond but still within the thermal broaden-\ning window of 4 Taround the single-electron tunneling\nresonances in Fig. 6. Here, a leading-order Γ calculation\ngives reliable predictions without any question. Second,\nwe note that this regime covers quite a large part of Fig.\n6 since the presence of the superconductor reduces the\nsize of the eﬀective Coulomb diamond in the stability di-\nagram, as one sees from comparing Figs. 2 and 6. The\nexponential suppression of the O(Γ) rates is thus attenu-\nated, butitmaystillbestrongneartheparticle-holesym-\nmetry point. Here, one should in principle include O(Γ2)\ncorrections. However – and this is our third point – by\ncomparing results of O(Γ) [not shown here], and O(Γ2)\n[Fig. 2]forthe sameparameterswithout superconductor,\nwe know that the resonance is not diminished, as clearly\ndemonstrated by Fig. 2, but only slightly broadened due\nto the additional spin decay introduced by cotunneling\n[c.f. Sec. IIIF]. Once the resonance appears, its position\nis determined by the ﬁrst-order exchange ﬁeld [cf. Sec.\nIIIA], modiﬁed by the proximal superconductor [cf. Sec.\nIIID], the eﬀect we wished to illustrate here. The cotun-\nneling corrections are not needed to draw a conclusion\nabout the resonance position.\n5. Non-Markovian corrections\nFinally, we comment on the validity of the Markovian\napproximation underlying our kinetic equations (13) for\nour study of the time-dependent pulsing scheme. To\nstudy time-dependent problems in the Coulomb blockade\nregime, onemust in principle alsoinclude non-Markovian\ncorrectionsinto the kernel [76]. However, non-Markovian\ncorrections appear only as modiﬁcations of the next-to-\nleading order contributions. Thus, non-Markovian cor-\nrections do not aﬀect the exchange ﬁeld, which is domi-\nnatedbyleading-ordertermsanddeterminesthe position\nof the spin resonance and the frequency of the spin pre-\ncession. On the contrary, the corrections do alter the\nspin-decay tensor RSand thereby the time constant of\nthe damped spin oscillations. In spite of this, the lat-\nter will still be of O(Γ2/U) in the Coulomb blockade\nregime, which we have identiﬁed as the crucial require-\nment for the underdamped spin precession. Including\nnon-Markovian corrections is, hence, required only for aquantitative analysis but not to demonstrate the viabil-\nity of an underdamped spin precession, which is our aim\nhere. It should be noted that if such accuracy is desir-\nable, other spin-decay mechanisms should also be taken\ninto account (see Sec. IVC), which is clearly beyond the\nscope of this work.\nAppendix C: Particle-hole symmetry\nIn Sec. IIIC, our discussion of the resonance position\napplied only to the left half of the Coulomb diamond,\ni.e., for gate voltages δ=U+2ε >0; cf. Fig. 2. Here,\nwe show that the resonance extends point-symmetrically\nwithrespecttotheparticle-holesymmetrypoint( δ,Vb) =\n(0,0). In the region to the right of this point, where\nδ <0, the resonance condition requires the exchange\nﬁeld to be perpendicular to the drainpolarization:\nB·ˆnd= 0(δ <0). (C1)\nThis condition is fulﬁlled for negative resonant bias V∗\nb<\n0. Thus, the drain refers here to the same physical elec-\ntrode as the source in Eq. (15) because changing the sign\nof the bias exchanges the role of source and drain.\nEquation (C1) can be understood physically as follows:\nForδ <0, the electrochemical potential of the leads is\ncloser to that for the doubly occupied QD and therefore\nthe current predominantly involves the doubly occupied\nQDstate. Consequently, whenanelectronleavestheQD,\nit leaves behind a holepolarized along ˆnd. However, an\naccumulation of hole spins can be eﬃciently prevented\nby the exchange ﬁeld Bif the latter is directed perpen-\ndicular to ˆnd, that is, if condition (C1) is fulﬁlled.\nIn analogy to Eq. (19), condition (C1) can be recast as\na′\nq′= 1, (C2)\nwith\na′=Γsnscos(α)\nΓdnd, (C3)\nand\nq′:=φd(ε)−φd(ε+U)\nφs(ε)−φs(ε+U)(δ<0),(C4)\nwhere the spin injection asymmetry ratio a′is deﬁned\ndiﬀerently as in Eq. (19). Equation (C2) complies with\nEq. (19): Mapping ( δ,Vb)→(−δ,−Vb), we have to\nreplaceq→1/q′anda→1/a′since the roles of source\nand drain are interchanged. It is therefore suﬃcient to\ndiscuss only the case δ >0 as we did in Sec. IIIC as all\nresults hold for δ <0 accordingly by reversing the signs\nofδandVb.20\n[1] S. Loth, K. von Bergmann, M. Ternes, A. F. Otte,\nC. Lutz, and A. Heinrich, Nature Phys. 6, 340 (2010).\n[2] L. Besombes, C. L. Cao, S. Jamet, H. Boukari, and\nJ. Fern´ andez-Rossier, Phys. Rev. B 86, 165306 (2012).\n[3] A. A. Khajetoorians, B. Baxevanis, C. H¨ ubner,\nT.Schlenk,S.Krause, T.O.Wehling, S.Lounis, A.Licht-\nenstein, D. Pfannkuche, J. Wiebe, et al., Science 339, 55\n(2013).\n[4] J. R. Petta, A. C. Johnson, J. M. Taylor, E. A. Laird,\nA. Yacoby, M. D. Lukin, C. M. Marcus, M. P. Hanson,\nand A. C. Gossard, Science 309, 2180 (2005).\n[5] F. H. L. Koppens, C. Buizert, K. J. Tielrooij, I. T. Vink,\nK. C. Nowack, T. Meunier, L. P. Kouwenhoven, and\nL. M. K. Vandersypen, Nature 442, 766 (2006).\n[6] K. C. Nowack, F. H. L. Koppens, Y. V. Nazarov, and\nL. M. K. Vandersypen, Science 318, 1430 (2007).\n[7] H. Bluhm, S. Foletti, I. Neder, M. Rudner, D. Mahalu,\nV. Umansky, and A. Yacoby, Nature Phys. 7, 109 (2010).\n[8] J. Fern´ andez-Rossier and R. Aguado, Phys. Rev. Lett.\n98, 106805 (2007).\n[9] L. Bogani and W. Wernsdorfer, Nature Mat. 7, 179\n(2008).\n[10] A. S. Zyazin, J. W. van den Berg, E. A. Osorio, H. S.\nvan der Zant, N. P. Konstantinidis, F. May, M. Leijnse,\nW. Hofstetter, M. R. Wegewijs, C. Danieli, et al., Nano\nLett.10, 3307 (2010).\n[11] M. Urdampilleta, S. Klyatskaya, J.-P. Cleuziou,\nM.Ruben, and W. Wernsdorfer, Nature Mater. 10, 502\n(2011).\n[12] A. J. Heinrich, J. A. Gupta, C. P. Lutz, and D. M. Eigler,\nScience306, 466 (2004).\n[13] S. Kahle, Z. Deng, N. Malinowski, C. Tonnoir,\nA. Forment-Aliaga, N. Thontasen, G. Rinke, D. Le,\nV. Turkowski, T. S. Rahman, et al., Nano Lett. 12, 518\n(2012).\n[14] B. W. Heinrich, L. Braun, J. I. Pascual, andK. J. Franke,\nNature Phys. 9, 765 (2013).\n[15] M. Julliere, Phys. Lett. A 54, 225 (1975).\n[16] M. N. Baibich, J. M. Broto, A. Fert, F. Nguyen Van Dau,\nF. Petroﬀ, P. Etienne, G. Creuzet, A. Friederich, and\nJ. Chazelas, Phys. Rev. Lett. 61, 2472 (1988).\n[17] G. Binasch, P. Gr¨ unberg, F. Saurenbach, and W. Zinn,\nPhys. Rev. B 39, 4828 (1989).\n[18] R. E. Camley and J. Barna´ s, Phys. Rev. Lett. 63, 664\n(1989).\n[19] J. C. Slonczewski, Phys. Rev. B 39, 6995 (1989).\n[20] P. Seneor, A. Bernand-Mantel, and F. Petroﬀ, J. Phys.\nCondens. Matter 19, 165222 (2007).\n[21] N. Sergueev, Q. F. Sun, H. Guo, B. G. Wang, and\nJ. Wang, Phys. Rev. B 65, 165303 (2002).\n[22] M. Wilczy´ nski, R. ´Swirkowicz, W. Rudzi´ nski, J. Barna´ s,\nand V. Dugaev, J. Magn. Magn. Mater. 290–291 , 209\n(2005).\n[23] J. Fransson, Eur. Phys. Lett. 70, 796 (2005).\n[24] I. Weymann, J. K¨ onig, J. Martinek, J. Barna´ s, and\nG. Sch¨ on, Phys. Rev. B 72, 115334 (2005).\n[25] J. N. Pedersen, J. Q. Thomassen, and K. Flensberg, prb\n72, 045341 (2005).\n[26] M. M. E. Baumg¨ artel, M. Hell, S. Das, and M. R.\nWegewijs, Phys. Rev. Lett. 107, 087202 (2011).\n[27] G. Usaj and H. U. Baranger, Phys. Rev. B 63, 184418(2001).\n[28] I. Weymann, J. Barna´ s, J. K¨ onig, J. Martinek, and\nG. Sch¨ on, Phys. Rev. B 72, 113301 (2005).\n[29] D. M. Zumb¨ uhl, C. M. Marcus, M. P. Hanson, and A. C.\nGossard, Phys. Rev. Lett. 93, 256801 (2004).\n[30] J. Paaske, A. Rosch, P. W¨ olﬂe, N. Mason, C. M. Marcus,\nand J. Nyg˚ ard, Nature Phys. 2, 460 (2006).\n[31] G. Kirˇ sanskas, J. Paaske, and K. Flensberg, Phys. Rev.\nB86, 075452 (2012).\n[32] A. N. Pasupathy, R. C. Bialczak, J. Martinek, J. E.\nGrose, L. A. K. Donev, P. L. McEuen, and D. C. Ralph,\nScience306, 86 (2004).\n[33] K. Hamaya, M. Kitabatake, K. Shibata, M. Jung,\nM. Kawamura, K. Hirakawa, T. Machida, T. Taniyama,\nS. Ishida, and Y. Arakawa, Appl. Phys. Lett. 91, 232105\n(2007).\n[34] J. R. Hauptmann, J. Paaske, and P. E. Lindelof, Nature\nPhys.4, 373 (2008).\n[35] M. Gaass, A. K. H¨ uttel, K. Kang, I. Weymann, J. von\nDelft, and C. Strunk, Phys. Rev. Lett. 107, 176808\n(2011).\n[36] J. K¨ onig and J. Martinek, Phys. Rev. Lett. 90, 166602\n(2003).\n[37] J. Martinek, M. Sindel, L. Borda, J. Barna´ s, J. K¨ onig,\nG. Sch¨ on, and J. von Delft, Phys. Rev. Lett. 91, 247202\n(2003).\n[38] J. K¨ onig, J. Martinek, J. Barna´ s, and G. Sch¨ on, in\nCFN Lectures on Functional Nanostructures , edited by\nK. Busch et al. (Springer, Berlin, 2005), Lecture Notes\nin Physics, chap. Quantum dots attached to ferromag-\nnetic leads: exchange ﬁeld, spin precession, and Kondo\neﬀect, p. 145.\n[39] M.-S. Choi, D. S´ anchez, and R. L´ opez, Physical Review\nLetters92, 056601 (2004).\n[40] M. Braun, J. K¨ onig, and J. Martinek, Phys. Rev. B 70,\n195345 (2004).\n[41] J. Martinek, M. Sindel, L. Borda, J. Barna´ s, R. Bulla,\nJ. K¨ onig, G. Sch¨ on, S. Maekawa, and J. von Delft, Phys.\nRev. B72, 121302 (2005).\n[42] J. V. Holm, H. I. Jørgensen, K. Grove-Rasmussen,\nJ. Paaske, K. Flensberg, and P. E. Lindelof, Phys. Rev.\nB77, 161406 (2008).\n[43] J. Splettstoesser, M. Governale, andJ. K¨ onig, Phys. R ev.\nB86, 035432 (2012).\n[44] S. Sobczyk, A. Donarini, and M. Grifoni, Phys. Rev. B\n85, 205408 (2012).\n[45] M. Misiorny, M. Hell, and M. R. Wegewijs, Nature Phys.\n9, 801 (2013).\n[46] M. Braun, J. K¨ onig, and J. Martinek, EPL 72, 294\n(2005).\n[47] M. Braun, J. K¨ onig, and J. Martinek, Phys. Rev. B 74,\n075328 (2006).\n[48] B. Sothmann and J. K¨ onig, Phys. Rev. B 82, 245319\n(2010).\n[49] B. Sothmann, D. Futterer, M. Governale, and J. K¨ onig,\nPhys. Rev. B 82, 094514 (2010).\n[50] I. Weymann and J. Barna´ s, Phys. Rev. B 75, 155308\n(2007).\n[51] W. Rudzi´ nski, J. Barna´ s, R. ´Swirkowicz, and\nM. Wilczy´ nski, Phys. Rev. B 71, 205307 (2005).\n[52] M. Leijnse and M. R. Wegewijs, Phys. Rev. B 78, 23542421\n(2008).\n[53] S. Koller, M. Grifoni, M. Leijnse, and M. R. Wegewijs,\nPhys. Rev. B 82, 235307 (2010).\n[54] H. Schoeller, Eur. Phys. J. Special Topics 168, 179\n(2009).\n[55] A. Martin-Rodero and A. Levy Yeyati, Adv. Phys. 60,\n899 (2011).\n[56] L. E. Hueso, J. M. Pruneda, V. Ferrari, G. Burnell, J. P.\nValds-Herrera, B. D. Simons, P. B. Littlewood, E. Arta-\ncho, A. Fert, and N. D. Mathur, Nature 445, 410 (2007).\n[57] A. V. Rozhkov and D. P. Arovas, Phys. Rev. B 62, 6687\n(2000).\n[58] S. De Franceschi, L. Kouwenhoven, C. Sch¨ onenberger,\nand W. Wernsdorfer, Nature Nanotechnology 5, 703\n(2010).\n[59] L. Hofstetter, A. Geresdi, M. Aagesen, J. Nyg˚ ard,\nC. Sch¨ onenberger, and S. Csonka, Phys. Rev. Lett. 104,\n246804 (2010).\n[60] D. Futterer, J. Swiebodzinski, M. Governale, and\nJ. K¨ onig, Phys. Rev. B 87, 014509 (2013).\n[61] J. Martinek, Y. Utsumi, H. Imamura, J. Barna´ s,\nS. Maekawa, J. Konig, and G. Sch¨ on, Phys. Rev. Lett.\n91, 127203 (2003).\n[62] R. Hanson, L. P. Kouwenhoven, J. R. Petta, S. Tarucha,\nand L. M. Vandersypen, Rev. Mod. Phys. 79, 1217\n(2007).\n[63] T. Fujisawa, D.G. Austing, Y.Tokura, Y.Hirayama, and\nS. Tarucha, J. Phys. Condens. Matter 15, R1395 (2003).\n[64] H. Schoeller, Mesoscopic Electron Transport (Kluwer,\nAmsterdam, 1997), chap. Transport through interacting\nquantum dots, p. 291.\n[65] F. H. L. Koppens, K. C. Nowack, and L. M. K. Vander-\nsypen, Phys. Rev. Lett. 100, 236802 (2008).[66] B. Sothmann, Phys. Rev. B 90, 155315 (2014).\n[67] C. P. Slichter, Principles of Magnetic Resonance , vol. 1\n(Springer, Berlin, 1990).\n[68] B. Wunsch, M. Braun, J. K¨ onig, and D. Pfannkuche,\nPhys. Rev. B 72, 205319 (2005).\n[69] A. Donarini, M. Grifoni, and K. Richter, Phys.Rev. Lett .\n97, 166801 (2006).\n[70] M. G. Schultz, Phys. Rev. B 82, 155408 (2010).\n[71] H.-P. Breuer and F. Petruccione, The Theory of Open\nQuantum Systems (Oxford University Press, Oxford,\n2002).\n[72] J. K¨ onig, J. Schmid, H. Schoeller, and G. Sch¨ on, Phys.\nRev. B54, 16820 (1996).\n[73] J. K¨ onig, H. Schoeller, and G. Sch¨ on, Phys. Rev. Lett.\n76, 1715 (1996).\n[74] S. Koller, M. Grifoni, and J. Paaske, Phys. Rev. B 85,\n045313 (2012).\n[75] I. Weymann and J. Barna´ s, Eur. Phys. J. B 46, 289\n(2005).\n[76] J. Splettstoesser, M. Governale, J. K¨ onig, and\nM. B¨ uttiker, Phys. Rev. B 81, 165318 (2010).\n[77] R. B. Saptsov and M. R. Wegewijs, Phys. Rev. B 90,\n045407 (2014).\n[78] In principle, ten more superbasis elements have to be\nadded to form a complete set of all superstates of the\nLiouville space of the QD, which has dimension 42= 16,\nbecause of the four-dimensional Hilbert space considered\nhere. However, for expanding any observable or the den-\nsity operator, eight of these superbasis elements are ob-\nsolete since they do not conserve the fermion parity [77].\nFurthermore, we restrict our considerations here to the\ncharge-conserving model without a superconductor."    },    {        "title": "1011.3663v1.Negative_refraction_in_natural_ferromagnetic_metals.pdf",        "content": "arXiv:1011.3663v1  [cond-mat.mtrl-sci]  16 Nov 2010Negative refraction in natural ferromagnetic metals\nS. Engelbrecht,1A. M. Shuvaev,1Y. Luo,2V. Moshnyaga,2and A. Pimenov1\n1Experimentelle Physik IV, Universit¨ at W¨ urzburg, 97074 W ¨ urzburg, Germany\n2I. Physikalisches Institut, Universit¨ at G¨ ottingen, 370 73 G¨ ottingen, Germany\n(Dated: September 4, 2018)\nAbstract\nIt is generally believed that Veselago’s criterion for nega tive refraction cannot be fulﬁlled in\nnatural materials. However, considering imaginary parts o f the permittivity ( ε) and permeability\n(µ) and for metals at not too high frequencies the general condi tion for negative refraction becomes\nextremely simple: Re(µ)<0⇒Re(n)<0. Here we demonstrate experimentally that in such\nnatural metals as pure Co and FeCo alloy the negative values o f the refractive index are achieved\nclose to the frequency of the ferromagnetic resonance. Larg e values of the negative refraction can\nbe obtained at room temperature and they can easily be tuned i n moderate magnetic ﬁelds.\n1The topic of negative refraction [1] has attracted much research interest in the last years\n[2, 3]. Various possible realizations of a negative index material has be en proposed [4–10].\nExperimental demonstrations of negative refraction in metamate rials [11, 12] and multilay-\ners [7] arebased onthe classical criterion by Veselago which require s simultaneous negativity\nof electric permittivity εand magnetic permeability µ[1]. However, natural materials with\nthese properties most probably do not exist. An alternative way to achieve negative re-\nfraction in natural material may try to use the extended criterion [13, 14] which takes into\naccount imaginary parts of the permittivity and permeability:\n(ε1+|ε∗|)(µ1+|µ∗|)< ε2µ2. (1)\nThis equation is straightforwardly derived from the inequality Re(√ε∗µ∗)<0. Here\nε∗=ε1+iε2denotes the complex dielectric permittivity and µ∗=µ1+iµ2the complex\nmagnetic permeability, respectively. Eq. (1) has an interesting con sequence if we consider\nthe electrodynamics of metals [15]. For metals at frequencies far be low the scattering rate\nthe imaginary part of the permittivity dominates: ε1<< ε2and, therefore, |ε∗| ≈ε2. In\nthis case Eq. (1) can be transformed to a simple ﬁnal condition for t he negative refraction:\nµ1<0⇒Re(n∗)<0 . This condition should be fulﬁlled in ferromagnetic metals close to\nthe ferromagnetic resonance if the strength of the mode is high en ough. First experimental\ndemonstration of negative refraction in a ferromagnetic metal [9] utilized the ferromagnetic\nresonance in a Ca doped LaMnO 3. Although a suﬃciency of the simple condition µ1<0\nfor metals has been proven, the existence of natural materials wit h negative refraction still\nhad to be demonstrated.\nIn this Letter we present the results of the refractive index expe riments in real natural\nmetals. As examples of natural ferromagnetic metals we have chos en pure Cobalt and Fe/Co\nalloy. Using polarization and phase controlled experiments in the millimet er wave range we\ndemonstrate that close to the frequency of the ferromagnetic r esonance the refractive index\nof Co and FeCo alloy indeed goes deep into the negative regime.\nPureCobalt andFe 0.5Co0.5alloywere prepared as polycrystalline thinﬁlms by magnetron\nsputtering technique. The thickness of both samples was 150 ±30nm. As substrate MgO\nwas used, whose millimeter wave properties have been determined in a separate experiment\nas nMgO= (3.09±0.01)+1.4·10−9(T[K])3. Here the temperature Tis given in Kelvin. The\nimaginary part of the refractive index of the MgO substrate in the f requency range of the\n2FIG. 1: a) transmittance and b) phase shift of FeCo alloy for t wo diﬀerent excitation geometries\nat T=170K. Within the geometry ˜h⊥B0the ferromagnetic resonance can be excited, whereas for\n˜h||B0no excitation occurs. The inset sketches the magnetically a ctive geometry ( ˜h⊥B0). Here¯k\ndenotes the wave vector, ˜hand ˜eare the ac magnetic and electric ﬁelds of the radiation, and B 0is\nthe static external magnetic ﬁeld.\npresent experiment was below 1 ·10−4at all temperatures.\nThe experiments in the millimeter frequency range were carried out in a Mach-Zehnder\ninterferometer setup [16, 17] using backward wave oscillators as r adiation source. This\nspectrometer enables to measure both the transmittance and ph ase shift as function of\nfrequency, temperature or external applied magnetic ﬁeld within c ontrolled polarization\ngeometries. Magnetic ﬁeld experiments were carried out using a split coil magnet with\n3polypropylene windows.\nThe experimental spectra obtained were analyzed using the Fresn el formulas for the\ncomplex transmission coeﬃcient of the substrate-ﬁlm system [16– 18]. A non-trivial problem\nof the experiment is the separation of dielectric and magnetic prope rties of the sample. In\nthis case four independent experimental values are necessary. I n present experiments, we\nobtainedthetransmittanceandphaseshiftofthesamplewithintwo diﬀerent polarizationsof\nthe incident radiation, ˜h||B0and˜h⊥B0. Here˜his theac magnetic ﬁeld of theradiationand\nB0istheexternal magneticﬁeld. FromthesolutionoftheBloch’sequat ions forthemagnetic\nmoments in external magnetic ﬁelds it follows that nonzero magnetic susceptibility can only\nbe obtained in the geometry ˜h⊥B0[19]. The inset of Fig. 1 shows the geometry in which\nthe ferromagnetic resonance is excited. To determine the parame ters of the ferromagnetic\nresonance a Lorentz line shape was used, i.e. the magnetic permeab ility was taken as\nµ∗(B) = 1+∆µBB0\nB2−B2\n0−iB0Γ. (2)\nHere Γ is the width of the resonance, B0the resonance ﬁeld and B the external applied\nmagnetic ﬁeld.\nFig. 1 shows the transmittance and phase shift of the FeCo alloy at T =170K and at a\nfrequency of 140GHz in both excitation geometries. As can be clear ly seen in the geometry\n˜h||B0no excitation of the ferromagnetic resonance occurs ( µ∗=1) and, therefore, we can\nuse this geometry to determine the dielectric properties of the met al ﬁlms. These properties\nare obtained from the absolute transmittance spectra in zero ﬁeld and are well described by\nconventional Drude conductivity of a metal in the millimeter range (T able I).\nFurthermore, in the nonmagnetic geometry no magnetoresistanc e occurs within our ex-\nperimental resolution neither in the FeCo alloy nor in pure Co. Theref ore we can assume a\nmagnetic ﬁeld independent behavior of the permittivity. In contras t, in the geometry ˜h⊥B\nthe resonance can be excited which allows to independently determin e the permeability.\nThe excitation frequency is dependent on the resonance ﬁeld and f ollows the relationship\nωres=γ/radicalBig\n(H0+4πM0)H0due to the geometry of the experiment (Voigt geometry) [20].\nHereωresdenotes the excitation frequency, γis the gyromagnetic ratio and M0is the static\nmagnetization. Using the dielectric permittivity obtained from the ma gnetically inactive\ngeometry we are able to calculate the magnetic permeability which is sh own in Fig. 2 for\nthe Co sample at T=190K.\n4FIG.2: Complexmagneticpermeabilityobtainedfromtransm issionandphaseshiftdata(symbols)\nandﬁttedusingEq. 2(solid lines). Closeto theresonanceth ereal partofthepermeability becomes\nnegative.\nTable I summarizes the ﬁlm properties for FeCo and Co at two diﬀeren t excitation fre-\nquencies. For both samples we get quite similar electrodynamic param eters. Weak temper-\nature dependence of the resonance ﬁelds is due to the decreasing static magnetization with\nincreasing temperature. These eﬀects are quite small, since the Cu rie temperatures of both\nCo and FeCo are far above room temperature (Co: T C=1390K [21], FeCo: T C=1253K\n[22]).\nWith the dielectric permittivity and the magnetic permeability determin ed, we can now\ncalculate the refractive index by n =√ε∗µ∗. Here we note that due to a continuous variation\nof the refractive index as function of external magnetic ﬁelds the sign of the square root is\nobtained automatically on crossing the imaginary axis. Fig. 3 a) and b) show the magnetic\nﬁeld dependence of the complex refractive index of the FeCo and Co sample, respectively.\nThe shaded area denotes the range in which the refractive index be comes negative. Here we\n5TABLE I: Electrodynamic parameters for FeCo and Co samples a t two diﬀerent excitation fre-\nquencies and temperatures.\nsample FeCo Co\nfrequency[GHz] 140 182 140 180\nT = 10K\nB0[T] 3.73 ±0.01 5.12 ±0.01 3.87 ±0.01 5.20 ±0.01\n∆µ0.32±0.05 0.16 ±0.05 0.20 ±0.05 0.13 ±0.05\nΓ [T] 0.07 ±0.02 0.08 ±0.02 0.08 ±0.02 0.11 ±0.02\nσ[106Ω−1m−1] 8 ±2 12 ±2\nT = 300K\nB0[T] 3.76 ±0.01 5.17 ±0.01 3.89 ±0.01 5.21 ±0.01\n∆µ0.29±0.05 0.27 ±0.05 0.38 ±0.05 0.21 ±0.05\nΓ [T] 0.08 ±0.02 0.1±0.02 0.07 ±0.02 0.11 ±0.02\nσ[106Ω−1m−1] 6 ±1 6 ±1\nsee that even at room temperature these metals show a negative in dex of refraction, which is\nalso highly tunable by the applied magnetic ﬁeld. We note that the seco nd weak resonance\nwhich can be seen in these plots (at ∼3.7T for FeCo and ∼3.8T for Co) most probably\narises due to inhomogeneities in the ﬁlm magnetization [23]. The estimat e of the best value\nof the ﬁgure of merit for our samples gives |n|/κ∼0.8. This value is already close to the\ntheoretical limit |n|/κ= 1 for ferromagnetic metals.\nA remaining problem with the present materials is the high absorption c oeﬃcient in the\nregion of negative refraction. High absorption values leads to small transmissions of the\norder of 10−4and have to be overcome for possible applications. A possible solution to\nthis problem could be the usage of either superconductors or very clean metals. In these\ncases and in the millimeter range the conductivity will be purely imaginar y, i.e. no internal\nabsorption will be present. Unfortunately, at current stage of r esearch both solution would\nbe diﬃcult, as e.g. magnetism and superconductivity tend to exclude each other.\nIn conclusion, we have shown that the refractive index in natural f erromagnetic metals\nbecomes negative close to the frequency of the ferromagnetic re sonance. We have investi-\n6FIG. 3: Magnetic ﬁeld dependent refractive index for the a) F eCo alloy and b) metallic Co at\nT=300Kandat140GHz. Theshadedareadenotestherangeinwhi chthereal partof therefractive\nindex becomes negative. Symbols - experimental data, solid lines are ﬁts using Eq. (2)\ngated pure Co and FeCo alloy both revealing negative values of the re fractive index over\na large temperature range and at diﬀerent excitation frequencies . Both metals show large\nnegative index of refraction at room temperature which is highly tun able by the external\nmagnetic ﬁeld.\n7[1] V. G. Veselago, Sov. Phys. Usp. 10, 509 (1968).\n[2] J. B. Pendry and D. R. Smith, Phys. Today 57, 37 (2004).\n[3] D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, Science 305, 788 (2004).\n[4] D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, Phys. Rev. Lett.\n84, 4184 (2000).\n[5] E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M . Soukoulis, Nature 423, 604\n(2003).\n[6] J. B. Pendry, Science 306, 1353 (2004).\n[7] A. Pimenov, A. Loidl, P. Przyslupski, and B. Dabrowski, P hys. Rev. Lett. 95, 247009 (2005).\n[8] V. A. Podolskiy and E. E. Narimanov, Phys. Rev. B 71, 201101 (2005).\n[9] A. Pimenov, A. Loidl, K. Gehrke, V. Moshnyaga, and K. Samw er, Phys. Rev. Lett. 98, 197401\n(2007).\n[10] J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A . Genov, G. Bartal, and X. Zhang,\nNature455, 376 (2008).\n[11] R. A. Shelby, D. R. Smith, and S. Schultz, Science 292, 77 (2001).\n[12] W. J. Padilla, D. N. Basov, and D. R. Smith, Mater Today 9, 28 (2006).\n[13] M. W. McCall, A. Lakhtakia, and W. S. Weiglhofer, Eur. J. Phys.23, 353 (2002).\n[14] A. D. Boardman, N. King, and L. Velasco, Electromagneti cs25, 365 (2005).\n[15] M.DresselandG.Gr¨ uner, Electrodynamics of Solids: Optical Properties of Electron s in Matter\n(Cambridge University Press, Cambridge, 2002), 1st ed.\n[16] A. Pimenov, S. Tachos, T. Rudolf, A. Loidl, D. Schrupp, M . Sing, and R. Claessen, Physical\nReview B 72, 035131 (2005).\n[17] G. V. Kozlov and A. A. Volkov, in Millimeter and Submillimeter Wave Spectroscopy of Solids ,\nedited by G. Gr¨ uner (Springer, Berlin, 1998), p. 51.\n[18] M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 2006).\n[19] A. Abragam and B. Bleaney, Electron Paramagnetic Resonance of Transition Ions (Dover,\nNew York, 1970).\n[20] A. K. Zvezdin and V. A. Kotov, Modern Magnetooptics and Magnetooptical Materials (Insti-\ntute of Physics Publ., Bristol, 1997).\n8[21] P. Mohn and E. P. Wohlfarth, J. Phys. F: Met. Phys. 17, 2421 (1987).\n[22] A. M. James and M. P. Lord, Macmillan’s Chemical and Physical Data (The Macmillan Press\nLtd, 1992).\n[23] A. V. Semeno, V. V. Glushkov, A. V. Bogach, N. E. Sluchank o, A. V. Dukhnenko, V. B.\nFillippov, N. Y. Shitsevalova, and S. V. Demishev, Physical Review B 79, 014423 (2009).\n9"    },    {        "title": "1902.00077v2.Fundamental_Spin_Interactions_Underlying_the_Magnetic_Anisotropy_in_the_Kitaev_Ferromagnet_CrI__3_.pdf",        "content": "Fundamental Spin Interactions Underlying the Magnetic Anisotropy\nin the Kitaev Ferromagnet CrI 3\nInhee Lee,1,∗Franz G. Utermohlen,1,†Daniel Weber,2Kyusung Hwang,1, 3Chi Zhang,1\nJohan van Tol,4Joshua E. Goldberger,2Nandini Trivedi,1and P. Chris Hammel1,‡\n1Department of Physics, The Ohio State University, Columbus, OH 43210, USA\n2Department of Chemistry and Biochemistry, The Ohio State University, Columbus, OH 43210, USA\n3School of Physics, Korea Institute for Advanced Study, Seoul, 130-722, Korea\n4National High Magnetic Field Laboratory, Florida State University, Tallahassee, FL 32310, USA\n(Dated: November 20, 2019)\nWe lay the foundation for determining the microscopic spin interactions in two-dimensional (2D)\nferromagnets by combining angle-dependent ferromagnetic resonance (FMR) experiments on high\nquality CrI 3single crystals with theoretical modeling based on symmetries. We discover that the\nKitaev interaction is the strongest in this material with K∼ −5.2 meV, 25 times larger than the\nHeisenberg exchange J∼ −0.2 meV, and responsible for opening the ∼5 meV gap at the Dirac\npoints in the spin-wave dispersion. Furthermore, we ﬁnd that the symmetric oﬀ-diagonal anisotropy\nΓ∼ −67.5µeV, though small, is crucial for opening a ∼0.3 meV gap in the magnon spectrum at the\nzone center and stabilizing ferromagnetism in the 2D limit. The high resolution of the FMR data\nfurther reveals a µeV-scale quadrupolar contribution to the S= 3/2 magnetism. Our identiﬁcation\nof the underlying exchange anisotropies opens paths toward 2D ferromagnets with higher TCas well\nas magnetically frustrated quantum spin liquids based on Kitaev physics.\nTwo-dimensional (2D) van der Waals (vdW) ferro-\nmagnets [1, 2] have recently emerged as an exciting\nplatform for the development of 2D spintronic applica-\ntions [3, 4] and novel 2D spin order [5, 6]. These 2D\nferromagnets must have magnetic anisotropy, since the\nMermin–Wagner theorem forbids 2D materials with a\ncontinuous spin-rotation symmetry from spontaneously\nmagnetizing at ﬁnite temperature [7]. Understanding\n2D ferromagnets thus requires a thorough knowledge of\nthis anisotropy. However, it remains an open question\nwhich fundamental magnetic interactions correctly de-\nscribe these materials and generate this anisotropy.\nIn this Letter we answer this question for CrI 3, one\nof the most robust 2D ferromagnets with a TCof 45 K\nfor the monolayer [1]. We ﬁrst construct a general\nHamiltonian based on its crystal symmetries containing\nanisotropic Kitaev Kand symmetric oﬀ-diagonal Γ inter-\nactions in addition to the Heisenberg Jinteractions. We\ndetermine the strength of these interactions using ferro-\nmagnetic resonance (FMR).\nFMR provides spectroscopically precise measurements\nof magnetic anisotropy, magnetization, spin-wave modes,\nand damping [8–10]. The structure of the magnetic\nanisotropy of a given material can be obtained from\nangle-dependent FMR by measuring the change in the\nresonance ﬁeld as the direction of the external ﬁeld H0\nis varied [8]. At 2 K, CrI 3single crystals have a ∼3 T\nanisotropy ﬁeld Haoriented normal to the layer plane\n[11, 12]. This large Haresults in a resonance frequency\nof at least ω/2π∼100 GHz in an out-of-plane ﬁeld.\nWe performed angle-dependent FMR using a heterodyne\nquasi-optical electron spin resonance spectrometer [13].\nThe measurement was implemented at ω/2π= 120 and\n240 GHz and at T= 5–80 K. The angle θHbetween H0and thee3-axis normal to the sample plane (see Fig. 1(d))\nis varied by rotating the thin CrI 3single crystal plate\nabout the axis indicated by the orange line in Fig. 1(a).\nA representative example of the FMR spectra for diﬀer-\nentθHat 240 GHz and 5 K is shown in Fig. 2(a).\nThe resonance ﬁeld Hres(θH,ω,T ), plotted in\nFig. 2(b)–(g), shows two distinct anisotropy features as\nθHis varied, which we label ∆ HAand ∆HBin Fig. 2(a):\n∆HAis the shift in Hresfrom the free ion contribution\nω/γCr, whereγCris the gyromagnetic ratio of Cr3+, and\n∆HBis the diﬀerence in HresbetweenθHand 180◦−θH.\nThese anisotropy features are crucial to understanding\nthe magnetic behavior of CrI 3and are central to our\nsymmetry-based theoretical analysis.\nIn order to analyze the anisotropies measured in FMR\nand determine the microscopic exchange interactions, we\nbegin by writing the most general Hamiltonian allowed\nby the symmetries of a monolayer with undistorted CrI 6\noctahedra: the crystal lattice is globally invariant under\n(i) time reversal, (ii) 120◦rotations about the e3-axis\nat each Cr3+ion, (iii) Cr–Cr-bond-centered spatial in-\nversion, (iv) 180◦rotations about the Cr–Cr bonds, and\n(v) locally invariant under 180◦rotations about the axis\nperpendicular to a Cr–Cr bond’s superexchange plane.\nBased on these symmetries, we obtain the general\nHamiltonian:\nH=HS+HQ−gµBH0·/summationdisplay\niSi, (1)arXiv:1902.00077v2  [cond-mat.mes-hall]  18 Nov 20192\nFIG. 1. (a) Optical image of the CrI 3single crystal for the FMR experiment (axis of rotation shown in orange). The internal\nangles of the cleaved edges are multiples of 30◦. The sample thickness is ∼35µm. (b) Schematic of the honeycomb lattice of\nthe Cr3+ions (dark blue) inside the iodine octahedron (upper: violet, lower: pink). Octahedral coordinate axes x,y,z (black),\nFMR coordinate axes e1,e2,e3, and Kitaev bonds x(red),y(green),z(blue) are indicated. (c) Pair of neighboring edge-sharing\noctahedra highlighting the local symmetries and the superexchange plane (blue). (d) FMR coordinate system.\nwhere\nHS=/summationdisplay\n/angbracketleftij/angbracketright∈λµ(ν)[JSi·Sj+KSν\niSν\nj+ Γ(Sλ\niSµ\nj+Sµ\niSλ\nj)]\n+/summationdisplay\n/angbracketleftij/angbracketright∈interlayerJ⊥Si·Sj (2)\ndescribes the spin–spin interactions, HQdescribes the\nquadrupole–quadrupole interactions (see Supplement),\nSiis the spin-3/2 operator for the Cr3+ion at site i,\n−gµBH0·/summationtext\niSiis the Zeeman coupling, gis the g-factor\nof Cr3+,µBis the Bohr magneton, and J⊥is the inter-\nlayer Heisenberg coupling [14]. /angbracketleftij/angbracketright∈λµ(ν) denotes that\nthe Cr3+ions at the neighboring sites i,jare interacting\nvia aν-bond, where λ,µ,ν∈{x,y,z}.\nWe next determine the spin interaction parameters in\nthe Hamiltonian. From the resonance ﬁeld Hres(θH,ω,T )\nwe determine the value of J+K/3 =−1.94 meV, which\nappears as a combination in mean ﬁeld theory (MFT)\nand determines how quickly ∆ HAand ∆HBshrink with\nincreasing temperature; and Γ = −67.5µeV, which de-\ntermines the size of ∆ HAat low temperatures. The de-\ntailed ﬁtting procedure is described in the Supplement.\nFrom the switching ﬁeld ∼0.6 T in bilayer CrI 3[1, 3, 15]\nwe estimate|J⊥|∼0.03 meV, which is negligible com-\npared toJ+K/3. Remarkably, the high spectroscopic\nprecision of FMR also enables us to estimate the µeV-\nscale quadrupole interaction constants (listed in Table 1),\nwhich give rise to ∆ HBin Fig. 2(a). The calculated Hres\nandMs(T) are in reasonable agreement with the data at\nall temperatures and frequencies (Fig. 2(b)–(g)). From\nthe known TC= 61 K of bulk CrI 3, we then determine\nthe value of K=−5.2 meV, which automatically ﬁxes\nthe value of J=−0.2 meV (see Fig. 3(a)).\nA key ﬁnding of our analysis is that the Kitaev interac-\ntion is the dominant interaction in CrI 3, almost 25 times\nstronger than the Heisenberg interaction. A strong sig-\nnature of this Kitaev interaction in CrI 3is the∼5 meV\nDirac gap (∆ K) at ˜Kin the spin-wave dispersion, asshown in Fig. 3(d), which is corroborated by a recent in-\nelastic neutron scattering experiment [18]. Furthermore,\nin the absence of the Kitaev interaction, TCis incorrectly\nestimated to be 100 K (Fig. 3(a)).\nIt is important to note that Kitaev anisotropic ex-\nchange interactions arise naturally for 2D honeycomb\nnetworks of edge-sharing octahedrally-coordinated tran-\nsition metals, as found in CrI 3and discussed previously in\nA2IrO3(A= Na,Li) [19, 20] and α-RuCl 3[21]. Electrons\nfrom a transition metal (TM) cation can hop to a neigh-\nboring TM cation via their shared ligands Xalong two\npathways (see Fig. 1(c)) [22–24]. In the presence of strong\nspin–orbit coupling (SOC) on either the cation, ligand,\nor both, the destructive interference between compet-\ning exchange pathways produce Kitaev interactions and\nweaken the Heisenberg interaction [25]. Even though the\nKitaev interaction leads to frustration, the spin moments\nin CrI 3are large (S= 3/2), so quantum ﬂuctuations\nTABLE I. Values of the spin and quadrupole interaction con-\nstants in the Hamiltonian for CrI 3bulk crystals (Eq. (1))\nand the angle dependence of the anisotropies they generate\nin terms of the direction cosines α,β,γ (compare to Fig. 4(c)).\nThe constants with a subscript Qare the quadrupole inter-\naction constants described in the Supplement. The values\nare determined experimentally (with uncertainties of ∼0.1%)\nthrough angle-dependent FMR and the known TC= 61 K.\nCoupling constant Value ( µeV) Angle dependence\nJ -212 1\nK -5190 1\nΓ -67.5 αβ+βγ+γα\nJQ+KQ/3 2.40 α2β2+β2γ2+γ2α2\nΓQ -2.69α2β2+β2γ2+γ2α2,\nαβγ(α+β+γ)\nΓ/prime\nQ -0.372αβ+βγ+γα,\nα2β2+β2γ2+γ2α2,\nαβγ(α+β+γ)\nK/prime\nQ -0.170 α2β2+β2γ2+γ2α23\nFIG. 2. (a) Evolution of the FMR spectrum as θHis var-\nied, measured at 240 GHz and 5 K. Each spectrum is oﬀset\nand scaled moderately for clarity. The same oﬀset is applied\nforθHand 180◦−θH. ∆HAand ∆HBare two anisotropy\nfeatures in Hres.ω/γCrdenotes the corresponding Hresfor a\nfree ion spin. (b) Hresvs.θHobtained from (a). The marker\nsize indicates the signal peak area in the Lorentzian ﬁts of the\nFMR spectrum. The red (blue) markers and labels indicates\nthe range of angles from 0◦to 90◦(90◦to 180◦). The solid\nand dashed black lines are ﬁts calculated from Landau the-\nory (Eq. (3)) and MFT of our model Hamiltonian (Eq. (1)),\nrespectively. Similarly, (c)–(g) show Hresvs.θHfor various\nfrequencies and temperatures.\nare not strong enough to produce a quantum spin liquid\nstate.\nWe next construct a Landau free energy functional\n(FEF) to map out the various magnetic anisotropies in\nCrI3and further connect the coeﬃcients of the Landau\nFIG. 3. (a) Dependence of TCon the spin interaction param-\netersJ,K, Γ under the experimental constraint J+K/3≡\nE0=−1.94 meV. (b) Dependence of TConJandKfor ﬁxed\nΓ =−67.5µeV. In (a) and (b), ( J0,K0,Γ0) (ﬁlled red cir-\ncles) are the values of J,K, Γ (listed in Table 1) that ﬁt the\nFMR data and the known TC= 61 K of bulk CrI 3; the ma-\ngenta and green lines are contour lines for TC= 61 K (bulk)\nandTC= 45 K (monolayer). (c) Dependence of TCon the\nanisotropy ﬁeld Ha(and on Γ) for Cr X3(X= Cl,Br,I) bulk\ncrystals [16, 17]. The values of Haused are for temperatures\nmostly below 5 K. (d) Spin-wave dispersion calculation along\nthe momentum-space path ˜K–˜Γ–˜M–˜K. The blue and red\nplots correspond to ( J,K, Γ) = (E0,0,Γ0) and (J0,K0,Γ0),\nrespectively. Note that the Kitaev interaction is responsible\nfor opening the gap ∆ Kbetween the bands at the Dirac point\n˜K. We zoom in on the area in the dashed black box to show\nthe gap ∆ Γ=−3SΓ at the zero-momentum point ˜Γ, where\nS= 3/2 is the spin of the Cr3+ions.\nFEF to the exchange interaction constants. The Landau\nFEF based on the underlying symmetries up to sixth or-\nder in the direction cosines α,β,γ (the components of the\nsaturation magnetization Msalong thex,y,z directions)\n(Fig. 1(b)) is given by [26–28]:\nFL= 2πM2\nscos2θ+K21(αβ+βγ+γα)\n+K41(α2β2+β2γ2+γ2α2) +K42αβγ(α+β+γ)\n+K61α2β2γ2+K62(α3β3+β3γ3+γ3α3)\n+K63αβγ(α3+β3+γ3)−Ms·H0, (3)4\nwhere 2πM2\nscos2θis the shape anisotropy, θis the angle\nbetween Msand thee3-axis (Fig. 1(d)), and Kpq(ω,T)\nare the coeﬃcients associated with the magnetocrys-\ntalline anisotropies plotted in Fig. 4(c). The FEF de-\ntermines the resonance condition Eq. (S4) of ωand\nHres(θH,ω,T ) (see Supplement). The values of the\nKpq(ω,T) that ﬁt the data are shown in Fig. 4(a), and\nthe corresponding ﬁts are shown in Fig. 2(b)–(g).\nWe map out the total Landau FEF FLshown in\nFig. 4(d) using the Kpqobtained at 5 K for 240 GHz. We\nﬁnd that the uniaxial term FL,21=K21(αβ+βγ+γα)\nis the dominant anisotropy in CrI 3, havingFL,21(θ=\n90◦)−FL,21(θ= 0◦)∼220µeV/Cr (corresponding to\nHa∼2.5 T), which primarily accounts for the large ∆ HA\nin Fig. 2(a). The higher-order anisotropy terms ( K4q,\nK6q) in Fig. 4(c) account for the small shift ∆ HBsince\nthey are not symmetric about the ﬁlm plane.\nBy combining the microscopic spin interaction and\nLandau theory approaches, we can provide insight into\nthe magnetic anisotropy produced by each interaction in\nthe Hamiltonian (Eq. (1)). For example, for the Γ inter-\naction we look at the free energy diﬀerence\n∆FΓ=FH(J,K, Γ,JQ,...)−FH(J,K, 0,JQ,...),(4)\nplotted in Fig. 4(e), and compare its angular structure\nto that of the anisotropies associated with the Kpqco-\neﬃcients in the Landau FEF (plotted in Fig. 4(c)). We\nﬁnd that Γ is mainly responsible for the large uniaxial\nanisotropy in CrI 3associated with K21underlying the\n∆HA. It also plays the crucial role of stabilizing ferro-\nmagnetism in a CrI 3monolayer by opening a ∼0.3 meV\ngap (∆ Γ) at the zone center ˜Γ in the spin-wave spectrum\n(see Fig. 3(d)). The much smaller quadrupole terms gen-\nerate the higher-order anisotropy terms associated with\nK4qandK6qunderlying the ∆ HB. Even though JandK\ngenerate no magnetic anisotropy, from the MFT estimate\nkBTMFT\nC =−5\n4(3J+K+ 2Γ) we see that they determine\nthe scale for TCsince they are much larger than Γ.\nOur model also describes the relation between the\nanisotropy ﬁeld HaandTCfor the chromium trihalides\n(X= Cl,Br,I). By inferring their values of Γ using\nthe low-temperature relation Ha/similarequal−3S2Γ/(MsVCr) ob-\ntained from MFT, where VCris the volume per Cr3+ion\nin CrI 3, we can compare the predicted TCvs. Γ rela-\ntion using the values of JandKobtained for bulk CrI 3\nto the known values of TCandHafor bulk Cr X3(see\nFig. 3(c)) [16, 17]. We note that although the prediction\ncurve agrees closely with the data for CrCl 3and CrBr 3,\nthis does not imply that they have the same JandKas\nCrI3; in fact, we expect Kto be much weaker in CrCl 3\nand CrBr 3since Cl−and Br−have weaker SOC than I−.\nGiven that CrI 3has aTCof 61 K for bulk crystals and\n45 K for a monolayer, we can speculate on the changes\nin the values of the spin interaction constants J,K, and\nΓ that might occur upon exfoliation. A reduction in the\nstrength of one of these interactions by a factor of 2–3\nFIG. 4. (a) Temperature dependence of the coeﬃcients Kpq\nassociated with the basic anisotropy structures shown in (c)\nfor 120 and 240 GHz. (b) Saturation magnetization Ms(T)\nobtained from SQUID magnetometry (out-of-plane (OP) and\nin-plane (IP)), a MFT analysis of the FMR data, and a zero-\nﬁeld spin-wave theory (SWT) analysis using the values of the\nspin interaction constants found (listed in Table 1). In (a)\nand (b), the lines connecting the markers are guides to the\neye. (c) Basic anisotropy structure in terms of the direction\ncosinesα,β,γ (the projections of the magnetization onto the\nx,y,z directions). The sizes are rescaled relative to that for\nαβ+βγ+γαwith the indicated magniﬁcations. Red (blue)\ndenotes positive (negative) values. (d) Total anisotropy FEF\nFLfor 240 GHz and 5 K constructed from Eq. (3). Orange\n(cyan) represents positive (negative) values. (e) Contribution\nof Γ to the FEF, ∆ FΓ, at 5 K. (c)–(e) are plotted with the\ncoordinate axes e1,e2,e3.\nor of several interactions by a smaller amount, perhaps\nas a result of crystal distortions, would lower TCby the\nappropriate amount (see Fig. 3(a) and (b)). FMR studies\non monolayer CrI 3are needed to explore this further.\nIn conclusion, our symmetry-based theoretical analysis\nof angle-dependent FMR measurements of single crystal\nCrI3has revealed strong Kitaev interactions in honey-\ncomb CrI 3, almost 25 times larger than the standard\nHeisenberg exchange, that open a ∼5 meV gap at the\nDirac points in the magnon dispersion, our prediction\nthat was recently corroborated by an inelastic neutron\nscattering study of CrI 3[18]. Such Kitaev interactions\narise naturally in edge-sharing octahedra due to SOC and\nthe interference of exchange pathways. We also found5\na small anisotropic Γ exchange that generates the large\nmagnetic anisotropy in CrI 3, opens a gap at the zone\ncenter, and stabilizes ferromagnetic long-range order in\n2D. This is in contrast to previous studies, which have\nused Ising anisotropy [4, 6, 29–32] or single-ion anisotropy\n[2, 14, 18, 33] to explain this large magnetic anisotropy;\nhowever, the former is not allowed by the crystal sym-\nmetries of CrI 3, whereas the latter is estimated to be too\nsmall [29] due to the weak SOC on the Cr3+ion. Our\nwork also provides insight needed to devise new 2D ma-\nterials with properties ranging from high- TCmagnetism\nto quantum spin liquid states.\nAngle-dependent FMR and our symmetry-based anal-\nysis can readily be applied to other 2D materials in order\nto correctly characterize their magnetic interactions. In\nparticular, we propose performing these FMR measure-\nments on the S= 1/2 Kitaev material α-RuCl 3, which\nlike CrI 3has Kitaev, Heisenberg, and Γ interactions, but\nwhose interaction constants are still hotly debated [34].\nWe thank W. Zhang for helpful discussions. This work\nwas supported by the Center for Emergent Materials, an\nNSF-funded MRSEC under Award No. DMR-1420451.\nJ.E.G. acknowledges the Camille and Henry Dreyfus\nFoundation for partial support. D.W. gratefully acknowl-\nedges the ﬁnancial support by the German Science Foun-\ndation (DFG) under the fellowship number WE6480/1.\nPart of this work was performed at the National High\nMagnetic Field Laboratory, which is supported by NSF\nCooperative Agreements No. DMR-1157490 and DMR-\n1644779 and the State of Florida.\n∗lee.2338@osu.edu\n†I.L. and F.G.U. contributed equally to this work.\n‡hammel@physics.osu.edu\n[1] B. Huang, G. Clark, E. Navarro-Moratalla, D. R. Klein,\nR. Cheng, K. L. Seyler, D. Zhong, E. Schmidgall, M. A.\nMcGuire, D. H. Cobden, W. Yao, D. Xiao, P. Jarillo-\nHerrero, and X. Xu, Nature 546, 270 (2017).\n[2] C. Gong, L. Li, Z. Li, H. Ji, A. Stern, Y. Xia, T. Cao,\nW. Bao, C. Wang, Y. Wang, Z. Q. Qiu, R. J. Cava, S. G.\nLouie, J. Xia, and X. Zhang, Nature 546, 265 (2017).\n[3] S. Jiang, L. Li, Z. Wang, K. F. Mak, and J. Shan, Nature\nNanotechnology 13, 549 (2018).\n[4] D. R. Klein, D. MacNeill, J. L. Lado, D. Soriano,\nE. Navarro-Moratalla, K. Watanabe, T. Taniguchi,\nS. Manni, P. Canﬁeld, J. Fern´ andez-Rossier, and\nP. Jarillo-Herrero, Science (2018), 10.1126/sci-\nence.aar3617.\n[5] S. S. Pershoguba, S. Banerjee, J. C. Lashley, J. Park,\nH.˚Agren, G. Aeppli, and A. V. Balatsky, Phys. Rev. X\n8, 011010 (2018).\n[6] J. Liu, M. Shi, P. Mo, and J. Lu, AIP Advances 8, 055316\n(2018).\n[7] N. D. Mermin and H. Wagner, Phys. Rev. Lett. 17, 1133(1966).\n[8] M. Farle, Reports on Progress in Physics 61, 755 (1998).\n[9] R. D. McMichael, D. J. Twisselmann, and A. Kunz,\nPhys. Rev. Lett. 90, 227601 (2003).\n[10] I. Lee, Y. Obukhov, G. Xiang, A. Hauser, F. Yang,\nP. Banerjee, D. V. Pelekhov, and P. C. Hammel, Na-\nture466, 845 (2010).\n[11] M. A. McGuire, H. Dixit, V. R. Cooper, and B. C. Sales,\nChemistry of Materials 27, 612 (2015).\n[12] J. F. Dillon and C. E. Olson, Journal of Applied Physics\n36, 1259 (1965).\n[13] J. van Tol, L.-C. Brunel, and R. J. Wylde, Review of\nScientiﬁc Instruments 76, 074101 (2005).\n[14] A. Narath, Phys. Rev. 140, A854 (1965).\n[15] S. Jiang, J. Shan, and K. F. Mak, Nature Materials 17,\n406 (2018).\n[16] M. McGuire, Crystals 7, 121 (2017).\n[17] B. Kuhlow, physica status solidi (a) 72, 161 (1982).\n[18] L. Chen, J.-H. Chung, B. Gao, T. Chen, M. B. Stone,\nA. I. Kolesnikov, Q. Huang, and P. Dai, Phys. Rev. X\n8, 041028 (2018).\n[19] Y. Singh, S. Manni, J. Reuther, T. Berlijn, R. Thomale,\nW. Ku, S. Trebst, and P. Gegenwart, Phys. Rev. Lett.\n108, 127203 (2012).\n[20] H. Gretarsson, J. P. Clancy, X. Liu, J. P. Hill, E. Bozin,\nY. Singh, S. Manni, P. Gegenwart, J. Kim, A. H. Said,\nD. Casa, T. Gog, M. H. Upton, H.-S. Kim, J. Yu, V. M.\nKatukuri, L. Hozoi, J. van den Brink, and Y.-J. Kim,\nPhys. Rev. Lett. 110, 076402 (2013).\n[21] A. Banerjee, C. A. Bridges, J.-Q. Yan, A. A. Aczel, L. Li,\nM. B. Stone, G. E. Granroth, M. D. Lumsden, Y. Yiu,\nJ. Knolle, S. Bhattacharjee, D. L. Kovrizhin, R. Moess-\nner, D. A. Tennant, D. G. Mandrus, and S. E. Nagler,\nNature Materials 15, 733 (2016).\n[22] G. Jackeli and G. Khaliullin, Phys. Rev. Lett. 102,\n017205 (2009).\n[23] J. G. Rau, E. K.-H. Lee, and H.-Y. Kee, Phys. Rev. Lett.\n112, 077204 (2014).\n[24] H.-S. Kim, V. S. V., A. Catuneanu, and H.-Y. Kee, Phys.\nRev. B 91, 241110 (2015).\n[25] P. P. Stavropoulos, D. Pereira, and H.-Y. Kee, Phys.\nRev. Lett. 123, 037203 (2019).\n[26] J. H. van Vleck, Phys. Rev. 52, 1178 (1937).\n[27] C. Zener, Phys. Rev. 96, 1335 (1954).\n[28] E. Ascher, Helv. Phys. Acta 39, 466 (1966).\n[29] J. L. Lado and J. Fern´ andez-Rossier, 2D Materials 4,\n035002 (2017).\n[30] J. Liu, M. Shi, J. Lu, and M. P. Anantram, Phys. Rev.\nB97, 054416 (2018).\n[31] F. Zheng, J. Zhao, Z. Liu, M. Li, M. Zhou, S. Zhang,\nand P. Zhang, Nanoscale 10, 14298 (2018).\n[32] W. Jin, H. H. Kim, Z. Ye, S. Li, P. Rezaie, F. Diaz,\nS. Siddiq, E. Wauer, B. Yang, C. Li, S. Tian, K. Sun,\nH. Lei, A. W. Tsen, L. Zhao, and R. He, Nature Com-\nmunications 9, 5122 (2018).\n[33] C. Xu, J. Feng, H. Xiang, and L. Bellaiche, npj Compu-\ntational Materials 4, 57 (2018).\n[34] S. M. Winter, A. A. Tsirlin, M. Daghofer, J. van den\nBrink, Y. Singh, P. Gegenwart, and R. Valent´ ı, Journal\nof Physics: Condensed Matter 29, 493002 (2017).Supplementary Materials for\nFundamental Spin Interactions Underlying the Magnetic Anisotropy in the Kitaev\nFerromagnet CrI 3\nInhee Lee,1,∗Franz G. Utermohlen,1,†Daniel Weber,2Kyusung Hwang,1, 3Chi Zhang,1\nJohan van Tol,4Joshua E. Goldberger,2Nandini Trivedi,1and P. Chris Hammel1,‡\n1Department of Physics, The Ohio State University, Columbus, OH 43210, USA\n2Department of Chemistry and Biochemistry, The Ohio State University, Columbus, OH 43210, USA\n3School of Physics, Korea Institute for Advanced Study, Seoul, 130-722, Korea\n4National High Magnetic Field Laboratory, Florida State University, Tallahassee, FL 32310, USA\n(Dated: November 20, 2019)\nCONTENTS\nI. Sample growth, structural characterization and magnetometry 1\nII. Determination of sample rotation axis in the crystal structure 2\nIII. Quasi-optical spectrometer measurements 2\nIV. Symmetries 2\nV. Quadrupole–quadrupole interactions 2\nVI. Mean ﬁeld theory (MFT) analysis 3\nVII. Calculation of the FMR frequency from the free energy functional (FEF) 4\nVIII. Estimating the saturation magnetization from FMR and ﬁtting the angle-dependent FMR data 4\nIX. Low-temperature relation between Haand Γ 5\nX. Linear spin-wave theory (SWT) analysis 5\nXI. Supplementary ﬁgures 6\nReferences 8\nI. SAMPLE GROWTH, STRUCTURAL CHARACTERIZATION AND MAGNETOMETRY\nChromium chunks (purity 99.996%) and recrystallized iodine (purity 99.9985%) were purchased from Alfa Aesar.\nStoichiometric amounts of Cr (102.1 mg, 1.964 mmol) and I 2(747.9 mg, 2.946 mmol) were sealed in an evacuated\nquartz ampoule (270 mm length, 15 mm inner diameter, 1 mm wall strength). The ampoule was heated in a three-zone\nfurnace for 18–72 h, depending on the desired crystallite size. The feed zone, which contained the reactant mixture,\nwas kept at 650◦C, while the crystals grew in the growth zone at 550◦C. The resulting crystalline platelets had a\nblack-red luster and were characterized by PXRD and SQUID magnetometry as reported in previous work [1]. At\nroom temperature, the platelets crystallized in the C2/mspace group, with the lattice parameters a= 6.9041(9) ˚A,\nb= 11.8991(10) ˚A,c= 7.0080(2) ˚A andβ= 108.735(5)◦similar to previous reports [2]. SQUID magnetometry\nshowed a ferromagnetic transition at TC= 61 K in the bulk crystals, as well as a small peak in the log χvs.Tplot\n∗lee.2338@osu.edu\n†I.L. and F.G.U. contributed equally to this work.\n‡hammel@physics.osu.eduarXiv:1902.00077v2  [cond-mat.mes-hall]  18 Nov 20192\natT= 210–220 K due to the structural phase transition from the high-temperature structure with the space group\nC2/mto the low-temperature modiﬁcation in the R¯3 space group.\nII. DETERMINATION OF SAMPLE ROTATION AXIS IN THE CRYSTAL STRUCTURE\nThe cleaved edges of our sample used in the experiment fall at angles separated by 30◦, as shown in Fig. 1(a). In\nterms of the microscopic crystal structure, this implies that the rotation axis corresponds to either the solid or dashed\norange line in Fig. S1. The ﬁts obtained by considering rotation about the solid line correctly reproduce the ∆ HB\nfeature (see Fig. 2(a)) in the angle-dependent ferromagnetic resonance (FMR) data (ﬁts are shown in Fig. 2(b)–2(g)),\nindicating that the experimental rotation axis is the solid line.\nIII. QUASI-OPTICAL SPECTROMETER MEASUREMENTS\nIn the heterodyne quasi-optical spectrometer, two colors of polarized microwave with ∆ ω/2π= 5 GHz were used\nas one for interacting with the sample and the other for reference. The FMR signal is probed from the change\nof microwave polarization such as Faraday rotation after interacting with the samples in resonance condition. We\nused two microwave frequencies 120 and 240 GHz which determine the main operating optical components such as\nmicrowave sources/detectors, horns and corrugated waveguides. The variable temperature dynamic ﬂow cryostat was\nused to measure at 5–80 K. The ﬁeld modulated signal was measured using a lock-in ampliﬁer. For angle-dependent\nFMR, we performed the manual in-situ sample holder rotation via an externally controlled worm drive.\nIV. SYMMETRIES\nEach pair of neighboring Cr3+ions interacts via superexchange mediated by the two I−ions it shares. Deﬁning x-,\ny-, andz-axes along the three Cr–I bond directions, as shown in Fig. 1(b), these four ions lie on a plane parallel to\nthexy-,yz-, orzx-plane; we refer to these Cr–Cr superexchange interactions as z-,x-, andy-bonds, respectively (a\nz-bond is shown in Fig. 1(c)). A CrI 3monolayer is globally invariant under time-reversal ( T), 120◦rotations about the\ne3-axis at each site ( Ce3\n3), and Cr–Cr-bond-centered spatial inversion ( i). Note that the CrI 6octahedra are actually\nslightly compressed along the e3-axis (the Cr–I–Cr angles are approximately 93◦, instead of 90◦) [3]. However, this\ndistortion is small enough that we do not consider it in our model for simplicity. Accordingly, single-ion magnetic\nanisotropy for Cr3+is excluded due to the quenched orbital angular momentum for the electron conﬁguration 3d3.\nWe therefore treat the system with another global symmetry, namely invariance under 180◦rotations about each of\nthe three Cr–Cr bond axes ( C/bardbl\n2). Furthermore, assuming interactions only between neighboring Cr3+ions, the system\nwill be locally invariant under 180◦rotations about the axis perpendicular to the bond’s superexchange plane (i.e.,\nthe plane containing both Cr3+ions and the two I−ions they share) passing through the center of the bond ( C⊥\n2).\nUnder these transformations, the S= 3/2 spin operator Sicorresponding to a Cr3+ion at siteiinteracting with a\nneighboring Cr3+ion at sitejvia az-bond transforms as\nT: Si→−Si\ni: Si→Sj\nC/bardbl\n2: (Sx\ni,Sy\ni,Sz\ni)→(−Sy\ni,−Sx\ni,−Sz\ni)\nC⊥\n2: (Sx\ni,Sy\ni,Sz\ni)→(−Sx\nj,−Sy\nj,Sz\nj), (S1)\nand similarly for x- andy-bonds byCe3\n3symmetry.\nV. QUADRUPOLE–QUADRUPOLE INTERACTIONS\nThe quadrupole–quadrupole interaction term HQin our model Hamiltonian (Eq. (2)) is\nHQ=/summationdisplay\n/angbracketleftij/angbracketright∈λµ(ν)/bracketleftbig\nJQ(Qxy\niQxy\nj+Qyz\niQyz\nj+Qzx\niQzx\nj) +KQQλµ\niQλµ\nj+K/prime\nQQλ2−µ2\niQλ2−µ2\nj +K/prime/prime\nQQν2\niQν2\nj\n+ ΓQ(Qµν\niQνλ\nj+Qνλ\niQµν\nj) + Γ/prime\nQ(Qλµ\niQν2\nj+Qν2\niQλµ\nj)/bracketrightbig\n, (S2)3\nwhereQλµ\ni,Qλ2−µ2\ni,Qν2\niare quadrupole operators for site iand are given by\nQλµ\ni=Sλ\niSµ\ni+Sµ\niSλ\ni,\nQλ2−µ2\ni = (Sλ\ni)2−(Sµ\ni)2,\nQν2\ni=1√\n3[3(Sν\ni)2−S2\ni] =1√\n3[2(Sν\ni)2−(Sλ\ni)2−(Sµ\ni)2]. (S3)\nWe can expressHQmore compactly as\nHQ=/summationdisplay\n/angbracketleftij/angbracketright∈λµ(ν)(Qν\ni)TIQQν\nj, (S4)\nwhere Qν\ni= (Qµν\ni,Qνλ\ni,Qλµ\ni,Qλ2−µ2\ni,Qν2\ni) and\nIQ=\nJQΓQ 0 0 0\nΓQJQ 0 0 0\n0 0JQ+KQ0 Γ/prime\nQ\n0 0 0 K/prime\nQ0\n0 0 Γ/prime\nQ 0K/prime/prime\nQ\n. (S5)\nVI. MEAN FIELD THEORY (MFT) ANALYSIS\nWe perform a mean ﬁeld analysis of the spin–spin interaction terms Sλ\niSλ/prime\nj(λ,λ/prime∈{x,y,z}) and quadrupole–\nquadrupole interaction terms Qη\niQη/prime\nj(η,η/prime∈{λµ,λ2−µ2,ν2}andλ,µ,ν∈{x,y,z}) in the Hamiltonian, where i/negationslash=j,\nusing the approximations\nSλ\niSλ/prime\nj≈/angbracketleftSλ/prime\nj/angbracketrightSλ\ni+/angbracketleftSλ\ni/angbracketrightSλ/prime\nj−/angbracketleftSλ\ni/angbracketright/angbracketleftSλ/prime\nj/angbracketright,\nQη\niQη/prime\nj≈/angbracketleftQη/prime\nj/angbracketrightQη\ni+/angbracketleftQη\ni/angbracketrightQη/prime\nj−/angbracketleftQη\ni/angbracketright/angbracketleftQη/prime\nj/angbracketright. (S6)\nAssuming\n/angbracketleftSi/angbracketright=1\nγCrmand/angbracketleftQν\ni/angbracketright=1\nγ2\nCrqν(S7)\nfor all sites i, whereγCris the gyromagnetic ratio of a Cr3+ion,m= (mx,my,mz) is the magnetic moment of a Cr3+\nion, and\nqν=\n2mµmν\n2mνmλ\n2mλmµ\nm2\nµ−m2\nν\n1√\n3[m2\nν−S(S+ 1)/planckover2pi12γ2\nCr]\n. (S8)\nThe mean ﬁeld Hamiltonian is then given by\nHMF=N/summationdisplay\ni=1/bracketleftbig\nE0+HS·Si+Hx\nQ·Qx\ni+Hy\nQ·Qy\ni+Hz\nQ·Qz\ni/bracketrightbig\n, (S9)4\nwhere\nE0≡ES,0+EQ,0,\nES,0≡−1\n2γ2\nCrmT/bracketleftbig\nIx\nS+Iy\nS+Iz\nS/bracketrightbig\nm,\nEQ,0≡−1\n2γ4\nCr/bracketleftbig\n(qx)TIQqx+ (qy)TIQqy+ (qz)TIQqz/bracketrightbig\n,\nHS≡1\nγCrmT/bracketleftbig\nIx\nS+Iy\nS+Iz\nS/bracketrightbig\n−γCrH,\nHν\nQ≡1\nγ2\nCr(qν)TIQ,\nIx\nS≡\nJ+K0 0\n0JΓ\n0 ΓJ\n,Iy\nS≡\nJ 0 Γ\n0J+K0\nΓ 0J\n,Iz\nS≡\nJΓ 0\nΓJ 0\n0 0J+K\n. (S10)\nFrom the mean ﬁeld Hamiltonian we can then obtain the system’s mean ﬁeld free energy\nFMF=−kBTlnZMF, (S11)\nwhere\nZMF= Tre−βHMF(S12)\nis the mean ﬁeld partition function, β= 1/(kBT) is the inverse temperature, and Tr denotes the trace.\nVII. CALCULATION OF THE FMR FREQUENCY FROM THE FREE ENERGY FUNCTIONAL (FEF)\nWe obtain the resonance condition, which relates the resonance frequency ωto the resonance ﬁeld Hres, from the\nFEF using the Smit–Beljers–Suhl equation [4],\nω=γCr\nMssinθ0/radicalBig\nF0\nθθF0\nφφ−(F0\nθφ)2, (S13)\nwhereF0\nρσ≡∂2F/∂ρ∂σ/vextendsingle/vextendsingle\nM=Ms(ρ,σ∈{θ,φ}), and Msis the system’s equilibrium saturation magnetization vector.\nVIII. ESTIMATING THE SATURATION MAGNETIZATION FROM FMR AND FITTING THE\nANGLE-DEPENDENT FMR DATA\nIn this section we outline our iterative procedure to estimate the saturation magnetization Ms(T) from FMR (green\nline in Fig. 3(b)) and ﬁt the angle-dependent FMR data using a model Hamiltonian (dashed lines in Fig. 2(b)–2(g))\n(the procedure for ﬁtting the data from Landau theory is very similar):\n1. Provide initial guess of the values of the coupling constants in the Hamiltonian (this does not have to be accurate;\nwe will reﬁne the guess through further iterations)\n2. For each temperature Tmeasured:\n(a) Provide initial guess of the value of Ms(T) (reﬁned through further iterations)\n(b) For each value of the resonance frequency ωand resonance ﬁeld angle θHmeasured:\ni. Provide initial guess of the magnitude of the resonance ﬁeld Hres,ﬁt (reﬁned through iterations)\nii. Compute the mean ﬁeld FEF FH(θ,φ) =−kBTlnZMF(whereZMF= Tre−HMF/kBTis the mean ﬁeld\npartition function and HMFis the mean ﬁeld Hamiltonian)\niii. Find the magnetization direction ( θ0,φ0) that minimizes FH(θ,φ)\niv. Calculate the ﬁt value of the resonance frequency ωﬁtusing Eq. (S13)\nv. Ifωﬁtdiﬀers signiﬁcantly from ω, adjust the estimate of Hres5\nvi. Repeat steps i–v until ωﬁtconverges to ω\n(c) Adjust the estimate of Ms(T)\n(d) Repeat steps (b) and (c) until Hres,ﬁt(ω,T) agrees as closely as possible to Hres(ω,T)\n3. Adjust the values of the coupling constants in the Hamiltonian\n4. Repeat steps 2 and 3 until Hres,ﬁt(ω,T) agrees as closely as possible to Hres(ω,T)\nNote that even though MFT inherently overestimates the tendency of a system to order and therefore overestimates\nmagnetization (and thus TCtoo), the values of Ms(T) obtained this way are experimentally accurate since they are\nobtained from ﬁtting the FMR data, and not from minimizing the mean ﬁeld FEF as a function of magnetization\n(see Fig. S2 for a comparison between the Ms(T) obtained from ﬁtting the FMR data, shown in green, vs. from\nminimizing the mean ﬁeld FEF, shown in blue).\nIX. LOW-TEMPERATURE RELATION BETWEEN HaAND Γ\nThe uniaxial anisotropy ﬁeld Hais given by\nHa=2[F(θ= 90◦)−F(θ= 0◦)]\nMsVCr(S14)\nwhereFis the FEF and VCris the volume occupied by a single Cr3+ion in CrI 3. Within MFT, Hais approximately\nHa∼=−3S2\nMsVCrΓ (S15)\nforT/lessorsimilar20 K.\nX. LINEAR SPIN-WAVE THEORY (SWT) ANALYSIS\nWe perform a spin-wave analysis of the spin–spin interactions in the Hamiltonian using the linearized Holstein–\nPrimakoﬀ representation of the S= 3/2 spin operators in order to calculate the magnetization and spin-wave dis-\npersion. Using i(j) to index the spins on sublattice A(B) and assuming the system is in a ferromagnetic state\nmagnetized in the e3-direction, the spin operators for sublattice Amap to\nS+\ni≈√\n2Sai, S−\ni≈√\n2Sa†\ni, Se3\ni=S−a†\niai (S16)\nin this representation, and similarly for sublattice B, wherea†\ni,ai(b†\nj,bj) are bosonic creation and annihilation\noperators for sublattice A(B). By substituting these expressions, the JKΓ Hamiltonian maps to\nHSW\nJKΓ=/summationdisplay\nk/braceleftBig\nd(a†\nkak+b†\nkbk) +/bracketleftbig\np(k)akb†\nk+q(k)akb−k+ H.c./bracketrightbig/bracerightBig\n+Eg, (S17)\nwhere\nd≡−S(3J+K+ 2Γ), (S18)\np(k)≡S\n3(3J+K−Γ)(1 +eik·a1+eik·a2), (S19)\nq(k)≡S\n3(K+ 2Γ)/parenleftBigg\n1−i√\n3\n2eik·a1+1 +i√\n3\n2eik·a2−1/parenrightBigg\n, (S20)\na1,a2are the basis vectors of the honeycomb lattice, and Egis the ground state energy. From the spin-wave excitation\nspectrumEkobtained from Eq. (S17) we calculate the magnetization per Cr3+ion using\nM(T) =gµB/parenleftBigg\nS−1\n2N/summationdisplay\nk1\neEk/(kBT)−1/parenrightBigg\n, (S21)\nwhere the sum is over the ﬁrst Brillouin zone, g≈2 is the g-factor of the Cr3+ions,Nis the number of unit cells (or\nequivalently, the number of points in k-space), and the factor of 1/2 in front of the sum is due to having two Cr3+\nions per unit cell.6\nXI. SUPPLEMENTARY FIGURES\nFIG. S1. CrI 3Sample rotation axis for angle-dependent FMR. The sample is rotated around the axis indicated by the\nsolid orange line in our angle-dependent FMR experiment. From the internal angles of the cleaved edges of the CrI 3crystal in\nFig. 1(a) we determine that the sample rotation axis corresponds to either the solid or dashed orange line. The ﬁts obtained\nby considering rotation about the solid line correctly reproduce the angle-dependent FMR data (unlike the ﬁts for the dashed\nline), indicating that the experimental rotation axis is the solid line.7\n0 20 40 60 80T(K) 0.00.51.01.52.02.53.0Ms(μB/Cr)\nMinimizes mean field FEF\nSQUID\nFMR fit\nFIG. S2. Comparison of the saturation magnetization Ms(T)obtained through diﬀerent methods. The estimates\nofMs(T) shown are for the CrI 3crystal in the resonance ﬁelds Hresapplied in the out-of-plane direction for the 120 GHz FMR\nexperiment (see Fig. 2(e)–2(g)). The blue, red, and green markers and lines correspond to the values obtained by minimizing\nthe mean ﬁeld FEF, using SQUID magnetometry, and ﬁtting the FMR data (described in Section VIII), respectively. The\nlines connecting the markers are guides to the eye. Note that the values of Ms(T) obtained by minimizing the mean ﬁeld\nFEF decrease more slowly with temperature than for the other two methods because MFT underestimates the role of thermal\nﬂuctuations and thus overestimates the tendency of a system to order.8\n[1] D. Shcherbakov, P. Stepanov, D. Weber, Y. Wang, J. Hu, Y. Zhu, K. Watanabe, T. Taniguchi, Z. Mao, W. Windl,\nJ. Goldberger, M. Bockrath, and C. N. Lau, Nano Letters 18, 4214 (2018), pMID: 29863369.\n[2] W.-B. Zhang, Q. Qu, P. Zhu, and C.-H. Lam, J. Mater. Chem. C 3, 12457 (2015).\n[3] M. A. McGuire, H. Dixit, V. R. Cooper, and B. C. Sales, Chemistry of Materials 27, 612 (2015).\n[4] J. Smit and H. G. Beljers, Phil. Res. Rep. 10, 113 (1955)."    },    {        "title": "1301.7186v1.Influence_of_MgO_tunnel_barrier_thickness_on_spin_transfer_ferromagnetic_resonance_and_torque_in_magnetic_tunnel_junctions.pdf",        "content": "arXiv:1301.7186v1  [cond-mat.mes-hall]  30 Jan 2013Inﬂuence of MgO tunnel barrier thickness on spin-transfer f erromagnetic resonance\nand torque in magnetic tunnel junctions\nWitold Skowro´ nski,∗Maciej Czapkiewicz, Marek Frankowski, Jerzy Wrona, and Tomasz S tobiecki\nDepartment of Electronics, AGH University of Science and Te chnology, Al. Mickiewicza 30, 30-059 Krak´ ow, Poland\nG¨ unter Reiss\nThin Films and Physics of Nanostructures, Bielefeld Univer sity, 33615 Bielefeld, Germany\nKhattiya Chalapat and Gheorghe S. Paraoanu\nLow temperature laboratory, Aalto University, P.O.Box 151 00, FI-02015 Aalto, Finland\nSebastiaan van Dijken\nNanoSpin, Department of Applied Physics, Aalto University School of Science, P.O.Box 15100, FI-00076 Aalto, Finland\n(Dated: March 27, 2022)\nSpin-transfer ferromagnetic resonance (ST-FMR) in symmet ric magnetic tunnel junctions (MTJs)\nwith a varied thickness of the MgO tunnel barrier (0.75 nm < tMgO<1.05 nm) is studied using the\nspin-torque diode eﬀect. The application of an RF current in to nanosized MTJs generates a DC\nmixing voltage across the device when the frequency is in res onance with the resistance oscillations\narising from the spin transfer torque. Magnetization prece ssion in the free and reference layers of the\nMTJs is analyzed by comparing ST-FMR signals with macrospin and micromagnetic simulations.\nFrom ST-FMR spectra at diﬀerent DC bias voltage, the in-plan e and perpendicular torkances are\nderived. The experiments and free-electron model calculat ions show that the absolute torque values\nare independent of tunnel barrier thickness. The inﬂuence o f coupling between the free and reference\nlayer of the MTJs on the ST-FMR signals and the derived torkan ces are discussed.\nI. INTRODUCTION\nHigh density magnetic random access memories can\nbe implemented using current-induced magnetization\nswitching (CIMS) [1] which is caused by interactions be-\ntween spin-polarized current and the magnetization of\nthe free layer (FL) in magnetic tunnel junction (MTJ)\ncells. This phenomenon is called the spin-transfer-torque\n(STT) eﬀect [2, 3]. Moreover, STT is utilized in MTJ\nnano-oscillators that generate signals in the GHz fre-\nquency range [4–6]. In order to optimize MTJ param-\neters, so that they can compete with existing memory\nand microwave technologies, it is necessary to fully un-\nderstand STT. The spin-torque diode eﬀect enables quan-\ntitative measurements of STT parameters [7–9]. In this\nwork, we use the spin-torque diode eﬀect to investi-\ngate the dependence of in-plane and perpendicular spin\ntorques on MgO tunnel barrier thickness. The tunnel\nbarrier determines the transport properties of the device,\nas it aﬀects the tunneling magnetoresistance (TMR) ra-\ntio, the resistance area (RA) product and the coupling\nbetween the FL and the reference layer (RL). We show\nthat the spin-torque ferromagnetic resonance (ST-FMR)\nspectra contain a double resonance mode for very thin\nMgO barriers due to strong ferromagnetic interlayer cou-\npling. Moreover, the in-plane and perpendicular spin-\ntorques do not depend on MgO barrier thickness, in\n∗Electronic address: skowron@agh.edu.plagreement with free electron models [10].\nII. EXPERIMENTAL\nThe MTJ stack with a MgO wedge tunnel barrier\nwas deposited in a Singulus Timaris cluster tool sys-\ntem. The multilayer structure consisted of the follow-\ning materials (thickness in nm): Ta(5) / CuN(50) /\nTa(3) / CuN(50) / Ta(3) / PtMn (16) / Co 70Fe30(2)\n/ Ru(0.9) / Co 40Fe40B20(2.3) / wedge MgO(0.7 - 1.1)\n/ Co40Fe40B20(2.3) / Ta(10) / CuN(30) / Ru(7). The\nslope of the MgO wedge barrier was approximately 0.017\nnm/cm. The deposition process was similar to the one\nused in our previous studies [11, 12]. After thin-ﬁlm de-\nposition, three diﬀerent parts of the sample were selected\nfor patterning into nanometer size pillars (later in the pa-\nper referred to as S1, S2 and S3, see Table I for details).\nUsing a three-steps electron beam lithography process,\nwhich included ion beam milling, lift-oﬀ and oxide and\nconducting layers deposition steps, nanopillars with an\nelliptical cross-section of 250 ×150 nm were fabricated.\nThe pillars were etched to the PtMn layer. The elec-\ntric leads to each MTJ nanopillar consisted of coplanar\nwaveguides which were designed to match an impedance\nof 50 Ohms. To ensure good RF performance, the over-\nlap between the top and bottom leads was about 4 µm2,\nwhich resulted in a capacitance of less than 1 ×10−14\nF. Each set of MTJs with a constant MgO tunnel barrier\nconsisted of 10 - 15 nanopillars.\nST-FMR measurement were conducted in a frequency\nrange from 2 to 12 GHz. In these experiments, the appli-2\nTABLEI: Summaryofstatic parameters ofthepreparedMTJ\nnanopillars.\nSample No. MgO thickness TMR RA product Hs\n(nm) (%) (Ω µm2) (Oe)\nS1 1.01 170 9.6 -21.7\nS2 0.95 165 6.24 -3.7\nS3 0.76 110 2.86 47\ncation of an RF current to an MTJ generated a DC volt-\nage (also called mixing voltage Vmix) across the device,\nwhen the current frequency was brought into resonance\nwith the resistance oscillations arising from the STT. The\nMTJs were placed in an in-plane magnetic ﬁeld at an an-\ngle ofβ= 70◦with respect to the easy magnetization\naxis (except for the case presented in Fig. 3(b)), so\nthat a large variety of angles θbetween the junction’s\nFL and RL could be obtained. We estimated θfrom the\nassumption, that the resistance Rof the MTJ changes as\nfollows:\ncos(θ) =/parenleftbiggRAP+RP\n2−R/parenrightbigg/parenleftbigg2\nRAP−RP/parenrightbigg\n(1)\nwhereRAPandRPare the resistance of the MTJ for\nan antiparallel and parallel alignment of the FL and RL\nmagnetization, respectively. In order to obtain the clear-\nest STT results [13], the strength and angle of the exter-\nnal magnetic ﬁeld was adjusted so that magnetization of\nthe FL is perpendicular to the magnetization of the RL\n(θ= 90◦). The magnitude of the RF input signal, con-\nnected to the MTJ through the capacitive lead of a bias\ntee, was ﬁxed to -15 dBm. This resulted in a RF current\n(IRF) between 5 µA and 25 µA, depending on the sample\nresistance. IRFwas calculated on the basis of the non-\nresonant background signal, using a model proposed in\nRef. [8]. The bias voltage was fed through the inductive\nlead of the bias tee. Vmixwas measured using a AC cou-\npled lock-in ampliﬁer, which was synchronized with the\namplitude modulated signal from the RF generator. In\nthis paper, positive bias voltage indicates electron trans-\nport from the bottom RL to the top FL.\nIII. RESULTS AND DISCUSSION\nTable I summarizes the TMR, the RA product and the\nstatic oﬀset magnetic ﬁeld ( HS) for three sets of MTJs\nwith diﬀerent MgO tunnel barrier thickness. The rep-\nresentative TMR vs. magnetic ﬁeld loops are presented\nin Fig. 1. The high TMR ratio of 170% for a 1.01 nm\nthick barrier and the exponential decrease in RA product\nwith decreasing MgO thickness conﬁrm good tunnel bar-\nrier quality [11]. Similar TMR ratios and RA products\nwere measured on full wafers using a current in-plane\ntunnelling (CIPT) technique before patterning [12]. The\noverall oﬀset ﬁeld ( HS) is shifted approximately 30 - 40/s45/s50/s48/s48 /s48 /s50/s48/s48/s48/s53/s48/s49/s48/s48/s49/s53/s48\n/s32/s84/s77/s82/s32/s40/s37/s41\n/s70/s105/s101/s108/s100/s32/s40/s79/s101/s41/s32/s83/s49\n/s32/s83/s50\n/s32/s83/s51\nFIG. 1: TMR vs. magnetic ﬁeld loops of samples S1-S3.\n/s48/s50/s52/s54/s56/s49/s48/s49/s50\n/s48/s49/s48/s50/s48/s51/s48/s52/s48/s53/s48\n/s51 /s52 /s53 /s54 /s55/s48/s53/s49/s48/s49/s53/s50/s48/s32/s32/s54/s48/s48\n/s53/s53/s48\n/s53/s48/s48\n/s52/s53/s48\n/s52/s48/s48\n/s51/s53/s48\n/s51/s48/s48\n/s50/s53/s48\n/s50/s48/s48\n/s49/s53/s48\n/s32/s97/s41/s32/s83/s49\n/s98/s41/s32/s83/s50\n/s52/s53/s48\n/s52/s48/s48\n/s51/s53/s48\n/s51/s48/s48\n/s50/s53/s48/s86\n/s109 /s105/s120 /s32/s40 /s86/s41/s69/s120/s116/s101/s114/s110/s97/s108/s32/s109/s97/s103/s110/s101/s116/s105/s99/s32/s102/s105/s101/s108/s100/s32/s40/s79/s101/s41\n/s55/s48/s48\n/s54/s53/s48\n/s54/s48/s48\n/s53/s53/s48\n/s53/s48/s48\n/s32\n/s70/s114/s101/s113/s117/s101/s99/s121 /s32/s40/s71/s72/s122/s41/s99/s41/s32/s83/s51\n/s32/s51/s48/s48/s32\n/s32/s50/s53/s48/s32\n/s32/s50/s48/s48/s32\n/s32/s49/s53/s48/s32\n/s32/s49/s48/s48/s32/s53/s53/s48/s32\n/s53/s48/s48/s32\n/s52/s53/s48/s32\n/s52/s48/s48/s32\n/s51/s53/s48/s32\nFIG. 2: ST-FMR spectra of samples S1 (a), S2 (b) and S3 (c)\nmeasured with various magnetic ﬁeld applied at an angle of\nβ= 70◦with respect to the easy magnetization axis. Only\nthe RF signal (without DC bias voltage) was supplied to the\nMTJ. For sample S3 (c) two closely spaced peaks are visible.3\n/s48/s53/s49/s48/s49/s53/s50/s48\n/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s53/s49/s48/s49/s53/s50/s48/s72\n/s69/s88/s84/s77\n/s82/s69/s70\n/s32\n/s32/s97/s41/s32 /s32/s61/s32/s55 /s48 /s176 /s32/s32/s70 /s76 /s49 /s32/s101 /s120 /s112 \n/s32/s70 /s76 /s50 /s32/s101 /s120 /s112 \n/s32/s70 /s76 /s32/s115/s105/s109\n/s32/s82 /s76 /s32/s115/s105/s109/s32/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s71/s72/s122/s41\n/s32/s98/s41/s32 /s32/s61/s32/s51 /s48 /s176 /s32/s32/s70 /s76 /s32/s101 /s120 /s112 \n/s32/s82 /s76 /s32/s101 /s120 /s112 \n/s32/s70 /s76 /s32/s115/s105/s109\n/s32/s82 /s76 /s32/s115/s105/s109/s32\n/s70/s105/s101/s108/s100/s32/s40/s107 /s79/s101/s41/s72\n/s69/s88/s84/s77\n/s82/s69/s70\nFIG. 3: The dispersion relation of sample S3 measured with\nthe magnetic ﬁeld applied at an angle of β= 70◦(a) andβ\n= 30◦(b) with respect to the easy magnetization axis. The\nsolid and dashed lines represent macrospin simulations of t he\nFL and RL, respectively. (a) At an angle of β= 70◦, the\nresonance frequency of two slightly separated FL modes in-\ncrease with increasing magnetic ﬁeld. (b) At an angle of β\n= 30◦, magnetization precessions in both the FL and RL are\nmeasured.\nOe with respect to the wafer-level measurements due to\ndipolar magnetostatic stray-ﬁeld coupling in the nanopil-\nlar junctions. For the MTJ with a 1.01 nm thick tunnel\nbarrier, antiferromagnetic stray-ﬁeld coupling dominates\nthe interaction between FL and RL ( HS= -21.7 Oe).\nA reduction of the barrier thickness to 0.76 nm reverses\nthe sign of the oﬀset ﬁeld ( HS= 47 Oe). In this case,\nthe FL and RL couple ferromagnetically due to direct\ninteractions across the thin MgO tunnel barrier.\nA. ST-FMR\nTypical ST-FMR signals (without DC bias voltage) for\nsamples S1 - S3 are presented in Fig. 2. We note that\na single symmetric peak is measured for sample S2 in\na wide magnetic ﬁeld range. For this sample, the cou-\npling between FL and RL is negligible. Moreover, the\nmonotonic increase of the resonance frequency with ap-\nplied magnetic ﬁeld indicates that the FMR signal orig-inates from magnetization precession in the FL [14]. A\nsimilar behavior is observed for sample S1, wherein the\neﬀective coupling between FL and RL is weakly antiferro-\nmagnetic. However, for sample S3, which is characterized\nby strong ferromagnetic coupling between FL and RL, an\nadditional peak is measured. The origin of this double\nresonance mode is not entirely clear. In previous publica-\ntions, it has been attributed to domain formation in the\nFL [15], higher-order spin wave excitations [16] and mag-\nnetization precession in other layers of spin-valve MTJs\n[17]. To analyze the double resonance mode in sample S3\nin more detail, we performed macrospin simulations us-\ning the model presented in Ref. [18]. This model, based\non the Stoner-Wolfarth approach, assumes coherent ro-\ntation of the FL and RL magnetization. By minimizing\nthe system energy we ﬁnd the angle of the FL and RL\nmagnetizations with respect to the easy axis and on this\nbasis, we calculate the dispersion relation. The simulated\ndispersion relations that are obtained for β= 70◦and for\nβ= 30◦are presented in Fig. 3 together with the mea-\nsured ST-FMR spectra. For β= 30◦, the experimental\nand simulated FMR modes of the FL and RL are in good\nquantitative agreement. We note that the FMR signal of\nthe RL is only measured when a large positive magnetic\nﬁeld is applied to the nanopillar junctions. The reso-\nnance frequency of the RL decreases with increasing ﬁeld\nstrength in this ﬁeld range. The frequency of the double\nresonance peak in the spectra for β= 70◦(Fig. 3(a)),\non the other hand, increase with applied ﬁeld strength.\nThe experimental dispersion relations now closely match\nsimulated curve. Based on this analysis, we attribute the\ndouble resonance mode to inhomogeneous magnetization\nprecession in the FL rather than FMR in the RL or any\nother magnetic layer of the MTJ stack.\nTo further elucidate the origin of double-mode FL spec-\ntra, we simulated the resonance characteristics of MTJ\nnanopillars using oommf software [19] with an additional\nextension enabling calculations of TMR and STT eﬀects\n[20]. In these micromagnetic simulations, elliptical mul-\ntilayer systems with a 2 nm thick FL, a 1 nm thick MgO\ntunnel barrier, a 2 nm thick high-anisotropy RL, antifer-\nromagnetically coupled to a 2 nm thick exchange-biased\npinned layer (PL), were used. The area of the junction\nwas identical to the experimental structures. The inter-\nlayer exchange coupling and anisotropy energies were ex-\nperimentally determined by magnetic and magnetotrans-\nport measurements. Variation of the ferromagnetic inter-\nlayer exchange coupling from 0 to 19 µJ/m2in the sim-\nulations yielded results comparable to the experimental\ndata. We note that dipolar coupling between the FL and\nRL is intrinsically calculated and taken into account in\noommf . Thus depending on the strength of the interlayer\nexchange coupling (input parameter), the eﬀective cou-\npling between FL and RL varies from antiferromagnetic\nto ferromagnetic in accordance with the experimental re-\nsults on samples S1 - S3.\nThe dynamic simulations were conducted in the fol-\nlowing way: ﬁrst, an external magnetic ﬁeld was applied4\n/s48/s49/s50\n/s51 /s52 /s53 /s54 /s55/s48/s49/s50/s51/s65/s110/s105/s115/s111/s116/s114/s111/s112/s121 /s32/s99/s111/s110/s115/s116/s46\n/s32/s49/s48/s32/s107/s74/s47/s109/s51\n/s32/s49/s56/s32/s107/s74/s47/s109/s51/s65/s110/s105/s115/s111/s116/s114/s111/s112/s121 /s32/s99/s111/s110/s115/s116/s46\n/s32/s53/s32/s107/s74/s47/s109/s51\n/s32/s49/s56/s32/s107/s74/s47/s109/s51\n/s32/s32\n/s32/s70/s114/s101/s113/s117/s101/s110/s99/s121 /s32/s40/s71/s72/s122/s41/s97/s41/s32/s83/s50/s44/s32/s74/s32/s61/s32/s54/s32 /s74/s47/s109/s50\n/s72/s32/s61/s32/s52/s53/s48/s32/s79/s101\n/s98/s41/s32/s83/s51/s44/s32/s74/s32/s61/s32/s49/s57/s32 /s74/s47/s109/s50\n/s72/s32/s61/s32/s52/s53/s48/s32/s79/s101/s86\n/s65 /s67 /s32/s40/s97/s46/s117/s46/s41\nFIG. 4: Simulated ST-FMR curves for weak (a) and strong\n(b) ferromagnetic interlayer exchange coupling. In oommf\nsimulations, a voltage step was used to excite magnetizatio n\nprecession in the FL of a MTJ structure. The dimensions of\nthe simulated and experimental junctions are identical. Th e\nexistence of closely-spaced double-peak ST-FMR signal for\nstrong coupling is independent of the anisotropy constant.\nat an angle with respect to the magnetic easy axis. Af-\nter relaxation, a voltage step was applied to exert a STT\non the FL. The voltage step amplitude was adjusted, so\nthat the FL magnetization oscillations changed the MTJ\nresistance by a few Ohms. The used values correspond\nto an AC current of a few µA, which closely mimic the\nexperimental conditions and ensures that the magneti-\nzation oscillations are within the linear regime. Finally,\nthe resonance spectra were obtained by Fourier transfor-\nmation of the time-derivative damped oscillation of the\nsimulated tunneling magnetoresistance.\nFigure 4 presents the simulated ST-FMR spectra for\ntwo MTJ nanopillars that closely resemble experimental\nsamples S2 and S3. The simulations conﬁrm that the\nmagnetization of the RL does not precess under these\nconditions ( β= 70◦) in the investigated frequency range.\nFor a weak interlayer exchange coupling energy of J=\n6µJ/m2(sample S2), a single resonance peak is sim-\nulated for diﬀerent FL anisotropy energies - Fig. 4a -\nand diﬀerent magnetic ﬁeld strength (not shown), which\nfulﬁlls the Kittel dispersion relation. For a larger ferro-\nmagnetic coupling energy of J= 19µJ/m2(sample S3),\nan additional broad resonance peak was resolved in the\nsimulations (Fig. 4(b)), regardless of the FL magnetic\nanisotropy. This behavior is reminiscent to the experi-mental behavior of sample S3 with a 0.76 nm thin MgO\ntunnel barrier. The simulations thus conﬁrm that the\nthe double resonance mode originates from inhomoge-\nneous magnetization precession in the FL of the MTJ\nnanopillar stack due to strong interlayer exchange cou-\npling between FL and RL.\nB. Torques and torkances\nIn order to obtain the STT components, i.e., in-plane\ntorqueτ/bardbland perpendicular torque τ⊥, from the ST-\nFMR measurements, we used the model presented in Ref.\n[13]. Here, we assume a simpliﬁed formula for Vmix:\nVmix=1\n4∂2V\n∂I2I2\nRF (2a)\n+1\n2∂2V\n∂I∂θ¯hγsinθ\n4eMSVolσI2\nRF[ξ/bardblS(ω)−ξ⊥ΩA(ω)],\n(2b)\nwhere ¯his the reduced Planck’s constant, γis the gyro-\nmagnetic ratio, eis the electron charge, Volis the volume\nof the FL, MSis the saturation magnetization of the FL,\nσis the linewidth, ξ/bardbl= 2(e/¯hsinθ)(dV/dI)dτ/bardbl/dV and\nξ⊥= 2(e/¯hsinθ)(dV/dI)dτ⊥/dV are the magnitudes of\nthe symmetric S(ω)=[1+(ω-ωm)2/σ2]−1and asymmet-\nricA(ω)=[(ω-ωm)/σ]S(ω) lorentzians components, and\nΩ⊥=γNxMeﬀ/ωm,Nx=4π+(Hz-Hasin2β)/Meﬀ, where\nωmis the resonant frequency, Hzis the sum of the applied\nexternal magnetic ﬁeld and the oﬀset ﬁeld acting on the\nprecessing FL, Hais the in-plane anisotropy ﬁeld of the\nFL and 4 πMeﬀis the eﬀective out-of-plane anisotropy of\nthe FL. We neglected the terms (2c) - (2g) of Ref. [13]\nbecause in our case θ≈90◦.\nFigure 5a presents a comparison of the in-plane\ntorkance in samples S1, S2, and S3. The absolute value\nof the in-plane torkance increases with decreasing barrier\nthickness and it only weakly depends on DC bias voltage.\nAccording to Slonczewski’s free electron model for elas-\ntic tunneling in symmetric MTJs, the in-plane torkance\nis proportional to the diﬀerential conductance measured\nfor parallel alignment of FL and RL [21]:\ndτ/bardbl\ndV=¯h\n2e2p\n1 +p2/parenleftbiggdI\ndV/parenrightbigg\n/bardbl(3)\nBy using Jullieres model to derive the spin polariza-\ntion of the tunneling current pat V = 0 V, we found a\ngood match between our experimental data and theoret-\nical calculations based on Eq.3 (Fig. 5(a)). The absolute\ntorque values in Fig. 5(b) were obtained by numerical\nintegration of the data in Fig. 5(a). Obviously, the in-\nplane torque varies linearly with DC bias current and it\nis independent of MgO tunnel barrier thickness. These\nresults are in good agreement with previously published\nexperimental data in Refs [8, 9, 13, 22] and calculations\nbased on an ab initio approach [23, 24].5\n/s48/s53/s49/s48/s49/s53\n/s45/s50/s45/s49/s48/s49/s50\n/s45/s48/s46/s52 /s45/s48/s46/s50 /s48/s46/s48 /s48/s46/s50 /s48/s46/s52/s45/s50/s45/s49/s48/s49/s50\n/s45/s49 /s48 /s49/s45/s48/s46/s50/s45/s48/s46/s49/s48/s46/s48\n/s32/s83/s49\n/s32/s83/s50\n/s32/s83/s51/s32/s99/s111/s109 /s112/s46\n/s32/s83/s51/s32/s114/s97/s119/s32/s32/s100 /s47/s100/s86 /s32/s40/s49/s48/s45/s49/s57\n/s32/s67/s41\n/s32/s32\n/s32\n/s124/s124/s32/s40/s49/s48/s45/s49/s57\n/s32/s78/s109 /s41/s32/s32/s100 /s47/s100/s86/s32/s40/s49/s48/s45/s49/s57\n/s32/s67/s41\n/s86/s111/s108/s116/s97/s103/s101/s32/s40/s86/s41/s100/s41/s99/s41/s98/s41\n/s32/s40/s49/s48/s45/s49/s57\n/s32/s78/s109/s41/s32\n/s67/s117/s114/s114/s101/s110/s116/s32/s40/s109/s65/s41/s97/s41\nFIG. 5: Bias dependence of the in-plane torkance (a), in-\nplane torque (b), perpendicular torkance (c) and perpendic -\nular torque (d) for MTJs with diﬀerent MgO barrier thick-\nness. The solid lines in (a) represent calculations based on\nEq. 3. The torque values are numerically integrated from\nexperimentally determined torkances. τ⊥for sample S3 was\ncompensated for an error originating from asymmetric ST-\nFMR resonances.\nExperimental data on the perpendicular torkance are\nsummarized in Fig. 5(c). For samples S1 and S2, the\ntorkance decreases with DC bias voltage and dτ⊥/dV =\n0 for zero DC bias voltage as predicted by theoretical cal-\nculations. However, a discrepancy is observed for sample\nS3. In this sample, strong ferromagnetic coupling be-\ntween the FL and RL of the MTJs results in asymmet-\nrical double resonance modes in the ST-FMR spectra.\nThe ﬁtting procedure based on Eq. 2 therefore intro-\nduces an error in the experimental torkance values for\nthis sample. A good match with theoretical calculations\nis obtained when this artifact is compensated by subtrac-\ntion of a constant torkance value. Figure 5(d) illustrates\nthat the absolute perpendicular torque varies quadrati-\ncally with DC bias current. Moreover, τ⊥is similar for all\nsamples. We note that diﬀerent torque versus bias depen-\ndencies have been measured recently. Especially, it has\nbeen shown that the shape of τ⊥(V) curves can change\nfrom quadratic to linear [25, 26]. However, such eﬀectswere only measured in asymmetric MTJs with diﬀerent\nFL and RL electrodes. In our junctions, the composition\nand thickness of the CoFeB electrodes are the same\nIV. SUMMARY\nIn summary, we have investigated MTJ nanopillars\nwith varied MgO tunnel barrier thickness using the spin-\ntorque diode eﬀect. We measured a symmetric ST-FMR\nsignal for samples with tMgO>0.9 nm. In this case,\nthe coupling between FL and RL is weakly antiferro-\nmagnetic. Contrary, double and closely-spaced resonance\nmodes were obtained for MTJs with a 0.76 nm thick\ntunnel barrier. Macrospin and micromagnetic simula-\ntions indicate that the asymmetric double-peaks orig-\ninate from inhomogeneous magnetization precession in\nthe FL caused by ferromagnetic coupling to the RL. The\nin-plane and perpendicular torques scale with DC bias\ncurrent and they are independent of MgO tunnel barrier\nthickness.\nAcknowledgement\nWe would like to thank Singulus Technologies AG\nfor consultation and technical help with MgO wedge\nMTJs preparation. We also acknowledge help of Micha/suppress l\nWilczy´ nski, Piotr Ogrodnik and Renata ´Swirkowicz for\na fruitful discussion regarding STT models. Project\nsupported by the Polish National Science Center grant\n515544538, Polish Ministry of Science and Higher Ed-\nucation Diamond Grant DI2011001541 and Swiss Con-\ntribution by NANOSPIN PSPB-045/2010 grant. W.S\nand T.S. acknowledges the Foundation for Polish Sci-\nence MPD Programme co-ﬁnanced by the EU European\nRegional Development Fund. G.R. acknowledges sup-\nport from the DFG (contract RE 1052/21-1). G.S.P\nand K.C. acknowledge support from the Commission of\nHigher Education of Thailand and Academy of Finland\n(nos. 129896, 118122, and 135135). S.v.D. acknowledges\nﬁnancial support from the Academy of Finland (grant\nno. 127731).\n[1] Huai, Y., Albert, F., Nguyen, P., Pakala, M., and Valet,\nT.Appl. Phys. Lett. 84, 3118 (2004).\n[2] Berger, L. Phys. Rev. B 54, 9353 (1996).\n[3] Slonczewski, J. C. J. Magn. Magn. Mater. 159, L1\n(1996).\n[4] Petit, S., Baraduc, C., Thirion, C., Ebels, U., Liu, Y., L i,\nM., Wang, P., and Dieny, B. Phys. Rev. Lett. 98, 184420\n(2007).\n[5] Deac, A. M., Fukushima, A., Kubota, H., Maehara, H.,\nSuzuki, Y., Yuasa, S., Nagamine, Y., Tsunekawa, K.,\nDjayaprawira, D. D., and Watanabe, N. Nature Phys. 4,803 (2008).\n[6] Skowro´ nski, W., Stobiecki, T., Wrona, J., Reiss, G., va n\nDijken, S. Appl. Phys. Express 5, 063005 (2012).\n[7] Tulapurkar, A. A., Suzuki, Y., Fukushima, A., Kubota,\nH., Maehara, H., Tsunekawa, K., Djayaprawira, D. D.,\nWatanabe, N., and Yuasa, S. Nature438, 339 (2005).\n[8] Sankey, J. C., Cui, Y., Sun, J. Z., Slonczewski, J. C.,\nBuhrman, R. A., and Ralph, D. C. Nature Phys. 4, 67\n(2007).\n[9] Kubota, H., Fukushima, A., Yakushiji, K., Nagahama,\nT., Yuasa, S., Ando, K., Maehara, H., Nagamine, Y.,6\nTsunekawa, K., Djayaprawira, D. D., Watanabe, N., and\nSuzuki, Y. Nature Phys. 4, 37(2007).\n[10] Wilczy´ nski, M., Barna´ s, J., and ´Swirkowicz, R. Phys.\nRev. B77, 054434 (2008).\n[11] Skowro´ nski, W., Stobiecki, T., Wrona, J., Rott, K.,\nThomas, A., Reiss, G., and van Dijken, S. J. Appl. Phys.\n107, 093917 (2010).\n[12] Wrona, J., Langer, J., Ocker, B., Maass, W., Kanak, J.,\nStobiecki, T., and Powronik, W. In J. Phys.: Conf. Ser.\n200, 052032, (2010).\n[13] Wang, C., Cui, Y., Sun, J., Katine, J., Buhrman, R., and\nRalph, D. Phys. Rev. B 79, 224416 (2009).\n[14] Cornelissen, S., Bianchini, L., Devolder, T., Kim, J., Roy,\nW. V., Lagae, L., and Chappert, C. Phys. Rev. B 81,\n144408 (2010).\n[15] Lee, K., Deac, A., Redon, O., Nozi´ eres, J., and Dieny, B .\nNature Mater. 3, 877 (2004).\n[16] Helmer, A., Cornelissen, S., Devolder, T., Kim, J. V.,\nvan Roy, W., Lagae, L., and Chappert, C. Phys. Rev. B\n81, 94416 (2010).\n[17] Sankey, J., Braganca, P., Garcia, A., Krivorotov, I.,\nBuhrman, R., and Ralph, D. Phys. Rev. Lett. 96, 227601\n(2006).\n[18] Serrano-Guisan, S., Skowro´ nski, W., Wrona, J., Liebi ng,\nN., Czapkiewicz, M., Stobiecki, T., Reiss, G., and Schu-macher, H. W. J. Appl. Phys. 110, 023906 (2011).\n[19] Donahue, M. J. and Porter, D. G. Interagency Report\nNISTIR 6376NIST, USA (1999).\n[20] Frankowski, M., Czapkiewicz, M.\nhttp://www.layer.uci.agh.edu.pl/M.Frankowski\n[21] Slonczewski, J. C. and Sun, J. Z. J. Magn. Magn. Mater.\n310, 169 (2007).\n[22] Wang, Chen and Cui, Yong-Tao and Katine, Jordan A.\nand Buhrman, Robert A. and Ralph, Daniel C. Nature\nPhys.7, 496 (2011).\n[23] Heiliger, C. and Stiles, M. Phys. Rev. Lett. 100, 186805\n(2008).\n[24] Jia, X., Xia, K., Ke, Y., and Guo, H. Phys. Rev. B 84,\n014401 (2011).\n[25] Chanthbouala, A., Matsumoto, R., Grollier, J., Cros, V .,\nAnane, A., Fert, A., Khvalkovskiy, A. V., Zvezdin, K. A.,\nNishimura, K., Nagamine, Y., Maehara, H., Tsunekawa,\nK., Fukushima, A., and Yuasa, S. Nature Phys. 7, 626\n(2011).\n[26] Matsumoto, R., Chanthbouala, A., Grollier, J., Cros, V .,\nFert, A., Nishimura, K., Nagamine, Y., Maehara, H.,\nTsunekawa, K., Fukushima, A., and Yuasa, S. Appl.\nPhys. Express 4, 063001 (2011)."    },    {        "title": "2201.06060v2.Ferromagnetic_resonance_modulation_in__d__wave_superconductor_ferromagnetic_insulator_bilayer_systems.pdf",        "content": "Ferromagnetic resonance modulation in d-wave superconductor/ferromagnetic\ninsulator bilayer systems\nYuya Ominato,1Ai Yamakage,2Takeo Kato,3and Mamoru Matsuo1, 4, 5, 6\n1Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China.\n2Department of Physics, Nagoya University, Nagoya 464-8602, Japan\n3Institute for Solid State Physics, The University of Tokyo, Kashiwa 277-8581, Japan\n4CAS Center for Excellence in Topological Quantum Computation,\nUniversity of Chinese Academy of Sciences, Beijing 100190, China\n5Advanced Science Research Center, Japan Atomic Energy Agency, Tokai 319-1195, Japan\n6RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan\n(Dated: May 6, 2022)\nWe investigate ferromagnetic resonance (FMR) modulation in d-wave superconductor\n(SC)/ferromagnetic insulator (FI) bilayer systems theoretically. The modulation of the Gilbert\ndamping in these systems re\rects the existence of nodes in the d-wave SC and shows power-law\ndecay characteristics within the low-temperature and low-frequency limit. Our results indicate the\ne\u000bectiveness of use of spin pumping as a probe technique to determine the symmetry of unconven-\ntional SCs with high sensitivity for nanoscale thin \flms.\nI. INTRODUCTION\nSpin pumping (SP)1,2is a versatile method that can\nbe used to generate spin currents at magnetic junctions.\nWhile SP has been used for spin accumulation in vari-\nous materials in the \feld of spintronics3,4, it has recently\nbeen recognized that SP can also be used to detect spin\nexcitation in nanostructured materials5, including mag-\nnetic thin \flms6, two-dimensional electron systems7{9,\nand magnetic impurities on metal surfaces10. Notably,\nspin excitation detection using SP is sensitive even for\nsuch nanoscale thin \flms for which detection by con-\nventional bulk measurement techniques such as nuclear\nmagnetic resonance and neutron scattering experiment is\ndi\u000ecult.\nRecently, spin injection into s-wave superconductors\n(SCs) has been a subject of intensive study both theoret-\nically11{20and experimentally21{34. While the research\ninto spin transport in s-wave SC/magnet junctions is ex-\npected to see rapid development, expansion of the devel-\nopment targets toward unconventional SCs represents a\nfascinating research direction. Nevertheless, SP into un-\nconventional SCs has only been considered in a few recent\nworks35,36. In particular, SP into a d-wave SC, which is\none of the simplest unconventional SCs that can be real-\nized in cuprate SCs37, has not been studied theoretically\nto the best of our knowledge, although experimental SP\nin ad-wave SC has been reported recently38.\nIn this work, we investigate SP theoretically in a bi-\nlayer magnetic junction composed of a d-wave SC and\na ferromagnetic insulator (FI), as shown in Fig. 1. We\napply a static magnetic \feld along the xdirection and\nconsider the ferromagnetic resonance (FMR) experiment\nof the FI induced by microwave irradiation. In this setup,\nthe FMR linewidth is determined by the sum of the in-\ntrinsic contribution made by the Gilbert damping of the\nbulk FI and the interface contribution, which originates\nfrom the spin transfer caused by exchange coupling be-\nMicrowavex yz\nSpin current\nFerromagnetic resonanceInteractionFIG. 1. Schematic of the d-wave SC/FI bilayer system. The\ntwo-dimensional d-wave SC is placed on the FI. Precessional\nmotion of the magnetization is induced by microwave irradia-\ntion. The spins are injected and the magnetization dynamics\nare modulated because of the interface magnetic interaction.\ntween thed-wave SC and the FI. We then calculate the\ninterface contribution to the FMR linewidth, which is\ncalled the modulation of the Gilbert damping hereafter,\nusing microscopic theory based on the second-order per-\nturbation39{41. We show that the temperature depen-\ndence of the modulation of the Gilbert damping exhibits\na coherent peak below the transition temperature that\nis weaker than that of s-wave SCs11,13{15. We also show\nthat because of the existence of nodes in the d-wave SCs,\nthe FMR linewidth enhancement due to SP remains even\nat zero temperature.\nThe paper is organized as follows. In Sec. II, we in-\ntroduce the model Hamiltonian of the SC/FI bilayer sys-\ntem. In Sec. III, we present the formalism to calculate\nthe modulation of the Gilbert damping. In Sec. IV, we\npresent the numerical results and explain the detailed\nbehavior of the modulation of the Gilbert damping. In\nSec. V, we brie\ry discuss the relation to other SC sym-\nmetries, the proximity e\u000bect, and the di\u000berence between\nd-wave SC/FI junctions and d-wave SC/ferromagnetic\nmetal junctions. We also discuss the e\u000bect of an e\u000bectivearXiv:2201.06060v2  [cond-mat.mes-hall]  5 May 20222\nZeeman \feld due to the exchange coupling. In Sec. VI,\nwe present our conclusion and future perspectives.\nII. MODEL\nThe model Hamiltonian of the SC/FI bilayer system\nHis given by\nH=HFI+HdSC+HT: (1)\nThe \frst term HFIis the ferromagnetic Heisenberg\nmodel, which is given by\nHFI=\u0000JX\nhi;jiSi\u0001Sj\u0000~\rhdcX\njSx\nj; (2)\nwhereJ>0 is the exchange coupling constant, hi;ji\nrepresents summation over all the nearest-neighbor sites,\nSjis the localized spin at site jin the FI,\ris the gy-\nromagnetic ratio, and hdcis the static magnetic \feld.\nThe localized spin Sjis described as shown using the\nbosonic operators bjandby\njof the Holstein-Primako\u000b\ntransformation42\nS+\nj=Sy\nj+iSz\nj=\u0010\n2S\u0000by\njbj\u00111=2\nbj; (3)\nS\u0000\nj=Sy\nj\u0000iSz\nj=by\nj\u0010\n2S\u0000by\njbj\u00111=2\n; (4)\nSx\nj=S\u0000by\njbj; (5)\nwhere we require [ bi;by\nj] =\u000ei;jto ensure that S+\nj,S\u0000\nj,\nandSx\njsatisfy the commutation relation of angular mo-\nmentum. The deviation of Sx\njfrom its maximum value S\nis quanti\fed using the boson particle number. It is conve-\nnient to represent the bosonic operators in the reciprocal\nspace as follows\nbk=1p\nNX\nje\u0000ik\u0001rjbj; by\nk=1p\nNX\njeik\u0001rjby\nj;(6)\nwhereNis the number of sites. The magnon opera-\ntors with wave vector k= (kx;ky;kz) satisfy [bk;by\nk0] =\n\u000ek;k0. Assuming that the deviation is small, i.e., that\nhby\njbji=S\u001c1, the ladder operators S\u0006\njcan be approx-\nimated asS+\nj\u0019(2S)1=2bjandS\u0000\nj\u0019(2S)1=2by\nj, which\nis called the spin-wave approximation. The Hamiltonian\nHFIis then written as\nHFI\u0019X\nk~!kby\nkbk; (7)\nwhere we assume a parabolic dispersion ~!k=Dk2+\n~\rhdcwith a spin sti\u000bness constant Dand the constant\nterms are omitted.\nThe second term HdSCis the mean-\feld Hamiltonian\nfor the two-dimensional d-wave SC, and is given by\nHdSC=X\nk(cy\nk\";c\u0000k#)\u0012\n\u0018k \u0001k\n\u0001k\u0000\u0018k\u0013\u0012ck\"\ncy\n\u0000k#\u0013\n;(8)wherecy\nk\u001bandck\u001bdenote the creation and annihilation\noperators, respectively, of the electrons with the wave\nvectork= (kx;ky) and thexcomponent of the spin\n\u001b=\";#, and\u0018k=~2k2=2m\u0000\u0016is the energy of conduc-\ntion electrons measured from the chemical potential \u0016.\nWe assume that the d-wave pair potential has the form\n\u0001k= \u0001 cos 2\u001ekwith the phenomenological temperature\ndependence\n\u0001 = 1:76kBTctanh \n1:74r\nTc\nT\u00001!\n; (9)\nwhere\u001ek= arctan(ky=kx) denotes the azimuth angle of\nk. Using the Bogoliubov transformation given by\n\u0012ck\"\ncy\n\u0000k#\u0013\n=\u0012\nuk\u0000vk\nvkuk\u0013\u0012\rk\"\n\ry\n\u0000k#\u0013\n; (10)\nwhere\ry\nk\u001band\rk\u001bdenote the creation and annihilation\noperators of the Bogoliubov quasiparticles, respectively,\nandukandvkare given by\nuk=r\nEk+\u0018k\n2Ek; vk=r\nEk\u0000\u0018k\n2Ek; (11)\nwith the quasiparticle energy Ek=p\n\u00182\nk+ \u00012\nk, the mean-\n\feld Hamiltonian can be diagonalized as\nHdSC=X\nk(\ry\nk\";\r\u0000k#)\u0012\nEk 0\n0\u0000Ek\u0013\u0012\rk\"\n\ry\n\u0000k#\u0013\n:(12)\nThe density of states of the d-wave SC is given by43\nD(E)=Dn= Re\u00142\n\u0019K\u0012\u00012\nE2\u0013\u0015\n; (13)\nwhereDn=Am= 2\u0019~2is the density of states per spin of\nthe normal state, Ais the system area, and K(x) is the\ncomplete elliptic integral of the \frst kind in terms of the\nparameterx, where\nK(x) =Z\u0019=2\n0d\u001ep\n1\u0000xcos2\u001e: (14)\nD(E) diverges at E=\u0001 = 1 and decreases linearly when\nE=\u0001\u001c1 because of the nodal structure of \u0001 k. The\ndensity of states for an s-wave SC, in contrast, has a\ngap forjEj<\u0001. This di\u000berence leads to distinct FMR\nmodulation behaviors, as shown below.\nThe third term HTdescribes the spin transfer between\nthe SC and the FI at the interface\nHT=X\nq;k\u0000\nJq;k\u001b+\nqS\u0000\nk+J\u0003\nq;k\u001b\u0000\n\u0000qS+\n\u0000k\u0001\n; (15)\nwhereJq;kis the matrix element of the spin transfer pro-\ncesses, and \u001b\u0006\nq= (\u001by\nq\u0006i\u001bz\nq)=2 andS\u0006\nk=Sy\nk\u0006iSz\nkare3\n(a) Spin transfer process (b) Self-energy\nJq,kJ*q,k\np/uni2191p+q/uni2193\np/uni2191p+q/uni2193\n−k −k\n/uni03A3R\nk(/uni03C9)=\nFIG. 2. (a) Diagrams of the bare vertices of the spin transfer\nprocesses at the interface. (b) Self-energy within the second-\norder perturbation.\nthe Fourier components of the ladder operators and are\ngiven by\n\u001b+\nq=X\npcy\np\"cp+q#; \u001b\u0000\n\u0000q=X\npcy\np+q#cp\"; (16)\nS\u0000\n\u0000k\u0019(2S)1=2by\nk; S+\nk\u0019(2S)1=2bk: (17)\nUsing the expressions above, HTcan be written as\nHT\u0019p\n2SX\np;q;k\u0010\nJq;kcy\np\"cp+q#by\n\u0000k+J\u0003\nq;kcy\np+q#cp\"b\u0000k\u0011\n:\n(18)\nThe \frst (second) term describes a magnon emission\n(absorption) process accompanying an electron spin-\rip\nfrom down to up (from up to down). A diagrammatic\nrepresentation of the interface interactions is shown in\nFig. 2 (a).\nIn this work, we drop a diagonal exchange coupling at\nthe interface, whose Hamiltonian is given as\nHZ=X\nq;kJq;k\u001bx\nqSx\nk: (19)\nThis term does not change the number of magnons in\nthe FI and induces an e\u000bective Zeeman \feld on electrons\nin the two-dimensional d-wave SC. We expect that this\nterm does not a\u000bect our main result because the coupling\nstrength is expected to be much smaller than the super-\nconducting gap and the microwave photon energy. We\nwill discuss this e\u000bect in Sec. V brie\ry.\nIII. FORMULATION\nThe coupling between the localized spin and the mi-\ncrowave is given by\nV(t) =\u0000~\rhacX\ni(Sy\nicos!t\u0000Sz\nisin!t); (20)wherehacis the amplitude of the transverse oscillating\nmagnetic \feld with frequency !. The microwave irra-\ndiation induces the precessional motion of the localized\nspin. The Gilbert damping constant can be read from\nthe retarded magnon propagator de\fned by\nGR\nk(t) =1\ni~\u0012(t)h[S+\nk(t);S\u0000\n\u0000k(0)]i; (21)\nwhere\u0012(t) is a step function. Second-order perturbation\ncalculation of the magnon propagator with respect to the\ninterface interaction was performed and the expression of\nthe self-energy was derived in the study of SP39{41. Fol-\nlowing calculation of the second-order perturbation with\nrespect to Jq;k, the Fourier transform of the retarded\nmagnon propagator is given by\nGR\nk(!) =2S=~\n!\u0000!k+i\u000b!\u0000(2S=~)\u0006R\nk(!); (22)\nwhere\u000bis the intrinsic Gilbert damping constant that\nwas introduced phenomenologically44{46. The diagram\nof the self-energy \u0006R\nk(!) is shown in Fig. 2 (b). From the\nexpressions given above, the modulation of the Gilbert\ndamping constant is given by\n\u000e\u000b=\u00002SIm \u0006R\nk=0(!)\n~!: (23)\nWithin the second-order perturbation, the self-energy\nis given by\n\u0006R\nk(!) =\u0000X\nqjJq;kj2\u001fR\nq(!); (24)\nwhere\u001fR\nq(!) represents the dynamic spin susceptibility\nof thed-wave SC de\fned by\n\u001fR\nq(!) =\u00001\ni~Z\ndtei(!+i0)t\u0012(t)h[\u001b+\nq(t);\u001b\u0000\n\u0000q(0)]i:(25)\nSubstituting the ladder operators in terms of the Bogoli-\nubov quasiparticle operators into the above expression\nand performing a straightforward calculation, we then\nobtain434\n\u001fR\nq(!) =\u0000X\npX\n\u0015=\u00061X\n\u00150=\u00061\u0012(\u0018p+\u0015Ep)(\u0018p+q+\u00150Ep+q) + \u0001 p\u0001p+q\n4\u0015Ep\u00150Ep+q\u0013f(\u0015Ep)\u0000f(\u00150Ep+q)\n\u0015Ep\u0000\u00150Ep+q+~!+i0; (26)\nwheref(E) = 1=(eE=kBT+ 1) is the Fermi distribution\nfunction.\nIn this paper, we focus on a rough interface modeled in\nterms of the mean J1and variance J22of the distribution\nofJq;k(see Appendix A for detail). The con\fgurationally\naveraged coupling constant is given by\njJq;k=0j2=J12\u000eq;0+J22: (27)\nIn this case, \u000e\u000bis written as\n\u000e\u000b=2SJ12\n~!Im\u001fR\nq=0(!) +2SJ22\n~!X\nqIm\u001fR\nq(!):(28)\nThe \frst term represents the momentum-conserved spin-\ntransfer processes, which vanish as directly veri\fed from\nEq. (26). This vanishment always occurs in spin-singlet\nSCs, including sandd-wave SCs, since the spin is\nconserved43. Consequently, the enhanced Gilbert damp-\ning is contributed from spin-transfer processes induced\nby the roughness proportional to the variance J22\n\u000e\u000b=2SJ22\n~!X\nqIm\u001fR\nq(!): (29)\nThe wave number summation can be replaced as\nX\nq(\u0001\u0001\u0001)!Dn\n2\u0019Z1\n\u00001d\u0018Z2\u0019\n0d\u001e(\u0001\u0001\u0001): (30)\nChanging the integral variable from \u0018toEand substi-\ntuting Eq. (26) into Eq. (29), we \fnally obtain\n\u000e\u000b=2\u0019SJ 22D2\nn\n~!Z1\n\u00001dE[f(E)\u0000f(E+~!)]\n\u0002Re\u00142\n\u0019K\u0012\u00012\nE2\u0013\u0015\nRe\u00142\n\u0019K\u0012\u00012\n(E+~!)2\u0013\u0015\n:\n(31)\nNote that the coherence factor vanishes in the above ex-\npression by performing the angular integral. The en-\nhanced Gilbert damping in the normal state is given by\n\u000e\u000bn= 2\u0019SJ 22D2\nn; (32)\nfor the lowest order of !. This expression means that \u000e\u000b\nis proportional to the product of the spin-up and spin-\ndown densities of states at the Fermi level7.IV. GILBERT DAMPING MODULATION\nFigure 3 shows the enhanced Gilbert damping constant\n\u000e\u000bas a function of temperature for several FMR frequen-\ncies, where \u000e\u000bis normalized with respect to its value in\nthe normal state. We compare \u000e\u000bin thed-wave SC shown\nin Figs. 3 (a) and (c) to that in the s-wave SC shown in\nFigs. 3 (b) and (d). The enhanced Gilbert damping for\nthes-wave SC is given by13\n\u000e\u000b=2\u0019SJ 22D2\nn\n~!Z1\n\u00001dE[f(E)\u0000f(E+~!)]\n\u0002\u0012\n1 +\u00012\nE(E+~!)\u0013\n\u0002Re\u0014jEjp\nE2\u0000\u00012\u0015\nRe\"\njE+~!jp\n(E+~!)2\u0000\u00012#\n;\n(33)\nwhere the temperature dependence of \u0001 is the same as\nthat for the d-wave SC, given by Eq. (9). Note that\nthe BCS theory we are based on, which is valid when\nthe Fermi energy is much larger than \u0001, is described by\nonly some universal parameters, including Tc, and inde-\npendent of the detail of the system in the normal state.\nWhen ~!=k BTc= 0:1,\u000e\u000bshows a coherence peak just\nbelow the transition temperature Tc. However, the co-\nherence peak of the d-wave SC is smaller than that of\nthes-wave SC. Within the low temperature limit, \u000e\u000bin\nthed-wave SC shows power-law decay behavior described\nby\u000e\u000b/T2. This is in contrast to \u000e\u000bin thes-wave SC,\nwhich shows exponential decay. The di\u000berence in the low\ntemperature region originates from the densities of states\nin thed-wave ands-wave SCs, which have gapless and full\ngap structures, respectively. When the FMR frequency\nincreases, the coherence peak is suppressed, and \u000e\u000bde-\ncays monotonically with decreasing temperature. \u000e\u000bhas\na kink structure at ~!= 2\u0001, where the FMR frequency\ncorresponds to the superconducting gap.\nFigure 4 shows \u000e\u000batT= 0 as a function of !. In\nthed-wave SC,\u000e\u000bgrows from zero with increasing !as\n\u000e\u000b/!2. When the value of \u000e\u000bbecomes comparable to\nthe normal state value, the increase in \u000e\u000bis suppressed,\nand\u000e\u000bthen approaches the value in the normal state.\nIn contrast, \u000e\u000bin thes-wave SC vanishes as long as the\ncondition that ~! < 2\u0001 is satis\fed. When ~!exceeds\n2\u0001,\u000e\u000bthen increases with increasing !and approaches\nthe normal state value. This di\u000berence also originates\nfrom the distinct spectral functions of the d-wave and\ns-wave SCs. Under the low temperature condition that\nT= 0:1Tc, the frequency dependence of \u000e\u000bdoes not5\n0.1 5.0\nT/Tc0.0 1.2 0.2 0.4 0.6 0.8 1.00.01.2\n0.20.40.60.81.0/uni03B4/uni03B1//uni03B4/uni03B1n0.1\n0.5\n1.0\n1.5\n2.03.04.05.0/uni210F/uni03C9/kBTc\nT/Tc0.0 1.2 0.2 0.4 0.6 0.8 1.00.02.0\n0.51.01.5/uni03B4/uni03B1//uni03B4/uni03B1n(a)  d-wave (b)  s-wave\n(c)  d-wave (d)  s-wave\nT/Tc0.0 1.2 0.2 0.4 0.6 0.8 1.00.01.2\n0.20.40.60.81.0/uni03B4/uni03B1//uni03B4/uni03B1nT/Tc0.0 1.2 0.2 0.4 0.6 0.8 1.00.02.0\n0.51.01.5/uni03B4/uni03B1//uni03B4/uni03B1n0.1\n0.5\n1.0\n1.5\n2.03.04.05.0\nFIG. 3. Enhanced Gilbert damping \u000e\u000bas a function of tem-\nperatureT. The left panels (a) and (c) show \u000e\u000bin thed-\nwave SC in the low and high frequency cases, respectively.\nThe right panels (b) and (d) show \u000e\u000bin thes-wave SC in the\nlow and high frequency cases, respectively. \u000e\u000bnis the normal\nstate value.\nchange for the s-wave SC, and it only changes in the\nlow-frequency region where ~!.kBTfor thed-wave SC\n(see the inset in Fig. 4).\nV. DISCUSSION\nWe discuss the modulation of the Gilbert damping\nin SCs with nodes other than the d-wave SC consid-\nered in this work. Other SCs with nodes are expected\nto exhibit the power-law decay behavior within the low-\ntemperature and low-frequency limit as the d-wave SCs.\nHowever, the exponent of the power can di\u000ber due to\nthe di\u000berence of the quasiparticle density of states. Fur-\nthermore, in the p-wave states, two signi\fcant di\u000berences\narise due to spin-triplet Cooper pairs. First, the uni-\nform spin susceptibility \u001fR\nq=0(!) can be \fnite in the spin-\ntriplet SCs because the spin is not conserved. Second,\nthe enhanced Gilbert damping exhibits anisotropy and\nthe value changes by changing the relative angle between\nthe Cooper pair spin and localized spin35.\nIn our work, proximity e\u000bect between FIs and SCs\nwas not taken into account because the FMR modula-\ntion was calculated by second-order perturbation based\non the tunnel Hamiltonian. Reduction of superconduct-\n/uni210F/uni03C9=2/uni0394(T =0)\ns-waved-wave\n00.2\n2 4 6 81.2\n0.60.81.0/uni03B4/uni03B1//uni03B4/uni03B1n\n/uni210F/uni03C9/kBTc0.4\n0.0\n100 10.05\n0.000.1T/Tc=0.0FIG. 4. Enhanced Gilbert damping \u000e\u000bas a function of fre-\nquency!. The vertical dotted line indicates the resonance\nfrequency ~!= 2\u0001(T= 0). The inset shows an enlarged\nview in the low-frequency region.\ning gap due to the proximity e\u000bect15and e\u000bect of the\nsubgap Andreev bound states that appear in the ab-axis\njunction47would also be an important problem left for\nfuture works.\nPhysics of the FMR modulation for d-wave\nSC/ferromagnetic metal junctions is rather di\u000ber-\nent from that for d-wave SC/FI junctions. For d-wave\nSC/ferromagnetic metal junctions, spin transport is\ndescribed by electron hopping across a junction and\nthe FMR modulation is determined by the product\nof the density of states of electrons for a d-wave SC\nand a ferromagnetic metal. (We note that the FMR\nmodulation is determined by a spin susceptibility of\nd-wave SC, which in general includes di\u000berent informa-\ntion from the density of states of electrons.) While the\nFMR modulation is expected to be reduced below a SC\ntransition temperature due to opening an energy gap, its\ntemperature dependence would be di\u000berent from results\nobtained in our work.\nFinally, let us discuss e\u000bect of the diagonal exchange\ncoupling given in Eq. (19) (see also the last part of\nSec. II). This term causes an exchange bias, i.e., an e\u000bec-\ntive Zeeman \feld on conduction electrons in the d-wave\nSC, which is derived as follows. First, the x-component\nof the localized spin is approximated as hSx\nji \u0019S,\nwhich gives Sx\nk\u0019Sp\nN\u000ek;0. Next, the matrix element\nJq;k=0is replaced by the con\fgurationally averaged value\nJq;k=0=J1\u000eq;0. Consequently, the e\u000bective Zeeman\n\feld term is given by\nHZ\u0019EZX\np(cy\np\"cp\"\u0000cy\np#cp#); (34)\nwhere we introduced a Zeeman energy as EZ=J1Sp\nN.\nThis term induces spin splitting of conduction electrons6\nin thed-wave SC and changes the spin susceptibility of\nthe SC. The spin-splitting e\u000bect causes a spin excitation\ngap and modi\fes the frequency dependence in Fig. 4, that\nwill provide additional information on the exchange cou-\npling at the interface. In actual experimental setup for\nthed-wave SC, however, the Zeeman energy, that is less\nthan the exchange bias between a magnetic insulator and\na metal, is estimated to be of the order of 0 :1 erg=cm2.\nThis leads to the exchange coupling that is much less\nthanJ\u00180:1 meV for YIG48. Therefore, we expect that\nthe interfacial exchange coupling is much smaller than\nthe superconducting gap and the microwave photon en-\nergy though it has not been measured so far. A detailed\nanalysis for this spin-splitting e\u000bect is left for a future\nproblem.\nVI. CONCLUSION\nIn this work, we have investigated Gilbert damping\nmodulation in the d-wave SC/FI bilayer system. The\nenhanced Gilbert damping constant in this case is pro-\nportional to the imaginary part of the dynamic spin sus-\nceptibility of the d-wave SC. We found that the Gilbert\ndamping modulation re\rects the gapless excitation that\nis inherent in d-wave SCs. The coherence peak is sup-\npressed in the d-wave SC when compared with that in\nthes-wave SC. In addition, the di\u000berences in the spec-\ntral functions for the d-wave ands-wave SCs with gap-\nless and full-gap structures lead to power-law and ex-\nponential decays within the low-temperature limit, re-\nspectively. Within the low-temperature limit, \u000e\u000bin the\nd-wave SC increases with increasing !, while\u000e\u000bin the\ns-wave SC remains almost zero as long as the excitation\nenergy ~!remains smaller than the superconducting gap\n2\u0001.\nOur results illustrate the usefulness of measurement of\nthe FMR modulation of unconventional SCs for determi-\nnation of their symmetry through spin excitation. We\nhope that this fascinating feature will be veri\fed exper-\nimentally in d-wave SC/FI junctions in the near future.\nTo date, one interesting result of FMR modulation in\nd-wave SC/ferromagnetic metal structures has been re-\nported38. This modulation can be dependent on metallic\nstates, which are outside the scope of the theory pre-\nsented here. The FMR modulation caused by ferromag-\nnetic metals is another subject that will have to be clar-\ni\fed theoretically in future work.\nFurthermore, our work provides the most fundamental\nbasis for application to analysis of junctions with vari-\nous anisotropic SCs. For example, some anisotropic SCs\nare topological and have an intrinsic gapless surface state.\nSP can be accessible and can control the spin excitation of\nthe surface states because of its interface sensitivity. The\nextension of SP to anisotropic and topological supercon-\nductivity represents one of the most attractive directions\nfor further development of superconducting spintronics.\nAcknowledgments.| This work is partially supportedby the Priority Program of Chinese Academy of Sciences,\nGrant No. XDB28000000. We acknowledge JSPS KAK-\nENHI for Grants (No. JP20H01863, No. JP20K03835,\nNo. JP20K03831, No. JP20H04635, and No.21H04565).\nAppendix A: Magnon self-energy induced by a\nrough interface\nThe roughness of the interface is taken into account\nas an uncorrelated (white noise) distribution of the ex-\nchange couplings35, as shown below. We start with an\nexchange model in the real space\nHex=X\njZ\nd2rJ(r;rj)\u001b(r)\u0001Sj\n=X\nq;kJq;k\u001bq\u0001Sk: (A1)\nThe spin density \u001b(r) in the SC and the spin Sjin the\nFI are represented in the momentum space as\n\u001b(r) =1\nAX\nqeiq\u0001r\u001bq; (A2)\nSj=1p\nNX\nkeik\u0001rjSk; (A3)\nwhereAdenotes the area of the system and Nis the\nnumber of sites. The exchange coupling constant is also\nobtained to be\nJq;k=1\nAp\nNX\njZ\nd2rei(q\u0001r+k\u0001rj)J(r;rj): (A4)\nThe exchange model Hexis decomposed into the spin\ntransfer term HTand the e\u000bective Zeeman \feld term HZ\nasHex=HT+HZ.\nNow we consider the roughness e\u000bect of the interface.\nUncorrelated roughness is expressed by the mean J1and\nvarianceJ22as\n1p\nNX\njJ(r;rj) =J1; (A5)\n1\nNX\njj0J(r;rj)J(r0;rj0)\u0000J12=J22A\u000e2(r\u0000r0);(A6)\nwhereOis the con\fgurational average of Oover the\nroughness. The above expressions lead to the con\fgu-\nrationally averaged self-energy\n\u0006R\nk=0(!) =\u0000X\nqjJq;k=0j2\u001fR\nq(!)\n=\u0000J12\u001fR\nq=0(!)\u0000J22X\nq\u001fR\nq(!); (A7)\nwhich coincides with the model Eq. (27) in the main text.\nThis model provides a smooth connection between the7\nspecular (J12\u001fR\nq=0) and di\u000buse ( J22P\nq\u001fR\nq) limits. The\nuncorrelated roughness case introduced above is a simplelinear interpolation of the two. Extensions to correlated\nroughness can be made straightforwardly.\n1Y. Tserkovnyak, A. Brataas, and G. E. Bauer, Phys. Rev.\nLett. 88, 117601 (2002).\n2F. Hellman, A. Ho\u000bmann, Y. Tserkovnyak, G. S. D. Beach,\nE. E. Fullerton, C. Leighton, A. H. MacDonald, D. C.\nRalph, D. A. Arena, H. A. D urr, P. Fischer, J. Grollier,\nJ. P. Heremans, T. Jungwirth, A. V. Kimel, B. Koop-\nmans, I. N. Krivorotov, S. J. May, A. K. Petford-Long,\nJ. M. Rondinelli, N. Samarth, I. K. Schuller, A. N. Slavin,\nM. D. Stiles, O. Tchernyshyov, A. Thiaville, and B. L.\nZink, Rev. Mod. Phys. 89, 025006 (2017).\n3I. Zutic, J. Fabian, and S. Das Sarma, Rev. Mod. Phys.\n76, 323 (2004).\n4E. Y. Tsymbal and I. Zuti\u0013 c, eds., Spintronics Handbook,\nSecond Edition: Spin Transport and Magnetism (CRC\nPress, 2019).\n5W. Han, S. Maekawa, and X.-C. Xie, Nat. Mater. 19, 139\n(2020).\n6Z. Qiu, J. Li, D. Hou, E. Arenholz, A. T. N'Diaye, A. Tan,\nK.-i. Uchida, K. Sato, S. Okamoto, Y. Tserkovnyak, Z. Q.\nQiu, and E. Saitoh, Nat. Commun. 7, 12670 (2016).\n7Y. Ominato and M. Matsuo, J. Phys. Soc. Jpn. 89, 053704\n(2020).\n8Y. Ominato, J. Fujimoto, and M. Matsuo, Phys. Rev.\nLett. 124, 166803 (2020).\n9M. Yama, M. Tatsuno, T. Kato, and M. Matsuo, Phys.\nRev. B 104, 054410 (2021).\n10T. Yamamoto, T. Kato, and M. Matsuo, Phys. Rev. B\n104, L121401 (2021).\n11M. Inoue, M. Ichioka, and H. Adachi, Phys. Rev. B 96,\n024414 (2017).\n12T. Taira, M. Ichioka, S. Takei, and H. Adachi, Physical\nReview B 98, 214437 (2018).\n13T. Kato, Y. Ohnuma, M. Matsuo, J. Rech, T. Jonckheere,\nand T. Martin, Phys. Rev. B 99, 144411 (2019).\n14M. A. Silaev, Phys. Rev. B 102, 144521 (2020).\n15M. A. Silaev, Phys. Rev. B 102, 180502 (2020).\n16V. Vargas and A. Moura, Journal of Magnetism and Mag-\nnetic Materials 494, 165813 (2020).\n17V. Vargas and A. Moura, Phys. Rev. B 102, 024412 (2020).\n18R. Ojaj arvi, J. Manninen, T. T. Heikkil a, and P. Virtanen,\nPhys. Rev. B 101, 115406 (2020).\n19H. T. Simensen, L. G. Johnsen, J. Linder, and A. Brataas,\nPhys. Rev. B 103, 024524 (2021).\n20E. H. Fyhn and J. Linder, Phys. Rev. B 103, 134508\n(2021).\n21C. Bell, S. Milikisyants, M. Huber, and J. Aarts, Phys.\nRev. Lett. 100, 047002 (2008).\n22T. Wakamura, H. Akaike, Y. Omori, Y. Niimi, S. Taka-\nhashi, A. Fujimaki, S. Maekawa, and Y. Otani, Nature\nmaterials 14, 675 (2015).\n23K.-R. Jeon, C. Ciccarelli, A. J. Ferguson, H. Kurebayashi,\nL. F. Cohen, X. Montiel, M. Eschrig, J. W. A. Robinson,\nand M. G. Blamire, Nat. Mater. 17, 499 (2018).\n24Y. Yao, Q. Song, Y. Takamura, J. P. Cascales, W. Yuan,\nY. Ma, Y. Yun, X. C. Xie, J. S. Moodera, and W. Han,\nPhys. Rev. B 97, 224414 (2018).\n25L.-L. Li, Y.-L. Zhao, X.-X. Zhang, and Y. Sun, Chin.Phys. Lett. 35, 077401 (2018).\n26M. Umeda, Y. Shiomi, T. Kikkawa, T. Niizeki,\nJ. Lustikova, S. Takahashi, and E. Saitoh, Applied Physics\nLetters 112, 232601 (2018).\n27K.-R. Jeon, C. Ciccarelli, H. Kurebayashi, L. F. Cohen,\nX. Montiel, M. Eschrig, T. Wagner, S. Komori, A. Srivas-\ntava, J. W. Robinson, and M. G. Blamire, Phys. Rev.\nAppl. 11, 014061 (2019).\n28K.-R. Jeon, C. Ciccarelli, H. Kurebayashi, L. F. Cohen,\nS. Komori, J. W. A. Robinson, and M. G. Blamire, Phys.\nRev. B 99, 144503 (2019).\n29K.-R. Jeon, C. Ciccarelli, H. Kurebayashi, L. F. Cohen,\nX. Montiel, M. Eschrig, S. Komori, J. W. Robinson, and\nM. G. Blamire, Physical Review B 99, 024507 (2019).\n30K. Rogdakis, A. Sud, M. Amado, C. M. Lee, L. McKenzie-\nSell, K. R. Jeon, M. Cubukcu, M. G. Blamire, J. W. A.\nRobinson, L. F. Cohen, and H. Kurebayashi, Phys. Rev.\nMater. 3, 014406 (2019).\n31I. Golovchanskiy, N. Abramov, V. Stolyarov, V. Chichkov,\nM. Silaev, I. Shchetinin, A. Golubov, V. Ryazanov,\nA. Ustinov, and M. Kupriyanov, Phys. Rev. Appl. 14,\n024086 (2020).\n32Y. Zhao, Y. Yuan, K. Fan, and Y. Zhou, Appl. Phys.\nExpress 13, 033002 (2020).\n33M. M uller, L. Liensberger, L. Flacke, H. Huebl, A. Kamra,\nW. Belzig, R. Gross, M. Weiler, and M. Althammer, Phys.\nRev. Lett. 126, 087201 (2021).\n34Y. Yao, R. Cai, T. Yu, Y. Ma, W. Xing, Y. Ji, X.-C. Xie,\nS.-H. Yang, and W. Han, Sci. Adv. 7, eabh3686 (2021).\n35Y. Ominato, A. Yamakage, and M. Matsuo, arXiv preprint\narXiv:2103.05871 (2021).\n36L. G. Johnsen, H. T. Simensen, A. Brataas, and J. Linder,\nPhys. Rev. Lett. 127, 207001 (2021).\n37C. C. Tsuei and J. R. Kirtley, Rev. Mod. Phys. 72, 969\n(2000).\n38S. J. Carreira, D. Sanchez-Manzano, M.-W. Yoo,\nK. Seurre, V. Rouco, A. Sander, J. Santamar\u0013 \u0010a, A. Anane,\nand J. E. Villegas, Phys. Rev. B 104, 144428 (2021).\n39Y. Ohnuma, H. Adachi, E. Saitoh, and S. Maekawa, Phys.\nRev. B 89, 174417 (2014).\n40Y. Ohnuma, M. Matsuo, and S. Maekawa, Phys. Rev. B\n96, 134412 (2017).\n41M. Matsuo, Y. Ohnuma, T. Kato, and S. Maekawa, Phys.\nRev. Lett. 120, 037201 (2018).\n42T. Holstein and H. Primako\u000b, Physical Review 58, 1098\n(1940).\n43P. Coleman, Introduction to Many-Body Physics (Cam-\nbridge University Press, 2015).\n44T. Kasuya and R. C. LeCraw, Phys. Rev. Lett. 6, 223\n(1961).\n45V. Cherepanov, I. Kolokolov, and V. L'vov, Phys. Rep.\n229, 81 (1993).\n46L. Jin, Y. Wang, G. Lu, J. Li, Y. He, Z. Zhong, and\nH. Zhang, AIP Advances 9, 025301 (2019).\n47Y. Tanaka and S. Kashiwaya, Phys. Rev. Lett. 74, 3451\n(1995).\n48J. Nogu\u0013 es and I. K. Schuller, Journal of Magnetism and8\nMagnetic Materials 192, 203 (1999)."    },    {        "title": "1408.1594v1.Ferromagnetic_resonance_of_a_magnetic_dimer_with_dipolar_coupling.pdf",        "content": "Ferromagnetic resonance of a magnetic dimer with dipolar coupling\nA. F. Franco, J. L. Déjardin, and H. Kachkachi\nLaboratoire PROMES CNRS UPR 8521, Université de Perpignan Via Domitia,\nRambla de la thermodynamique - Tecnosud, F-66100 Perpignan Cedex, France\nWe develop a general formalism for analyzing the ferromagnetic resonance characteristics of\na magnetic dimer consisting of two magnetic elements (in a horizontal or vertical conﬁguration)\ncoupled by dipolar interaction, taking account of their ﬁnite-size and aspect ratio. We study the\neffect on the resonance frequency and resonance ﬁeld of the applied magnetic ﬁeld (in amplitude\nand direction), the inter-element coupling, and the uniaxial anisotropy in various conﬁgurations.\nWe obtain analytical expressions for the resonance frequency in various regimes of the interlayer\ncoupling. We (numerically) investigate the behavior of the resonance ﬁeld in the corresponding\nregimes. The critical value of the applied magnetic ﬁeld at which the resonance frequency van-\nishes may be an increasing or a decreasing function of the dimer’s coupling, depending on the\nanisotropy conﬁguration. It is also a function of the nanomagnets aspect ratio in the case of\nin-plane anisotropy. This and several other results of this work, when compared with experi-\nments using the standard ferromagnetic resonance with ﬁxed frequency, or the network analyzer\nwith varying frequency and applied magnetic ﬁeld, provide a useful means for characterizing the\neffective anisotropy and coupling within systems of stacked or assembled nanomagnets.\nI. INTRODUCTION\nToday magnetic multilayers beneﬁt from a renewed interest owing to the plethora of potential applications1–6\nthey offer and the still challenging issues they raise for fundamental research7,8. Among the latter, interlayer\ncoupling is the focus of most of the investigations and debates as it constitutes one of the key physical pa-\nrameters that determine the overall behavior and physical properties of the multilayer structures. For this\nreason, many theoretical and experimental investigations are carried out towards a better control of this pa-\nrameter and a better understanding of its effect on the magnetization dynamics in magnetic multilayers. In\nparticular, the nature of this coupling is of special interest and its characterization is still a target of intense\ninvestigation1,3,4,9–24. Depending on the intrinsic properties of the constituting layers (underlying material,\nthickness, roughness, energy parameters, etc.) and the mutual interaction (chemical and physical), the inter-\nlayer coupling may change in nature and strength, leading to a variety of physical phenomena and applications.\nOn the other hand, assemblies of magnetic nanoparticles, deposited on a substrate or embedded in a ma-\ntrix, provide another playground for various investigations, experimental and theoretical, with stimulating\nchallenges both for fundamental research and practical applications25–29. Here, the coupling between pairs of\nnanoparticles is of paramount importance in the understanding of the dynamics of the system. During the last\ndecades several approaches have been developed in order to fathom the role of inter-particle interactions in\nthe onset of the macroscopic behavior of the assembly. Indeed, these interactions have a strong bearing on the\ndistribution of the energy barriers, the relaxation rates, and the related dynamical observables such as the ac\nsusceptibility30–42and hysteresis.\nFor the experimental investigation of either magnetic multilayers or assemblies of nanoparticles, there\nare nowadays many well-established techniques for precise measurements, such as the ferromagnetic reso-\nnance (FMR)20,43–46, Brillouin Light Scattering (BLS)47,48, and the ever developing optical and magneto-optical\ntechniques49–53. The technique of FMR is one of the well established and easiest techniques20,43–46that al-\nlows for a fairly precise probe of the magnetization dynamics, specially in magnetic hetero-structures. In the\ncase of dipolar interacting magnetic elements, a comparison between theory and FMR spectrum provides us\nwith a useful means to characterize the inter-elements coupling and help identify its origin and estimate its\nmagnitude.\nIn this work, we accordingly investigate the FMR characteristics (resonance frequency and resonance ﬁeld) of\na magnetic dimer composed of two (identical) nanomagnets coupled by dipolar interactions (DI). The two mag-\nnetic elements are set up in either i) a vertical conﬁguration thus modeling a multilayer with two magnetic lay-\ners separated by a nonmagnetic spacer, representing the generic situation that is relevant in spintronics54, or ii)\na horizontal conﬁguration where the two magnetic elements form a pair of nanodisks or circular nanopillars55,\nor still a pair of nanoparticles deposited on a substrate or embedded in a nonmagnetic matrix. We compute\nthe resonance frequencies and the resonance ﬁelds upon varying the coupling strength and the direction and\namplitude of the applied magnetic ﬁeld, for various orientations of the anisotropy axes of the two magnetic\nelements. In this work, we investigate the effect of the dipolar coupling between the magnetic elements of ﬁnitearXiv:1408.1594v1  [cond-mat.mtrl-sci]  7 Aug 20142\nFigure 1: Setup of the magnetic dimer.\nsize, thus going beyond the standard dipole-dipole approximation for the magnetostatic energy.\nThe present work is organized as follows. In the next section we state the problem at hand and introduce\nthe model for the magnetic dimer studied. Then, we establish the energy expression for the vertical and hori-\nzontal setup that takes account of the ﬁnite size and shape of the magnetic elements. In section III we derive\nthe various expressions for the resonance frequency of the magnetic dimer (vertical or horizontal) for various\nconﬁgurations of the anisotropy axes (longitudinal, transverse and mixed). We plot the resonance frequency\nagainst the dipolar coupling or the applied ﬁeld magnitude and the resonance ﬁeld as a function of the applied\nﬁeld polar angle. We end this work with a conclusion of our main results and discuss further extensions thereof.\nII. SYSTEM SETUP AND ENERGY\nWe consider a magnetic dimer composed of two identical ferromagnets, either two rectangular slabs or two\nthin cylinders. In the case of slabs we will consider the situation where they are arranged in a trilayer vertical\nstack with a nonmagnetic spacer. In this case, the centers of the three layers are along the zaxis and the\ncenter-to-center distance between the two magnetic layers is henceforth denoted by D, see Fig. 1 a). This is\na model for a multilayer system of two ferromagnetic thin ﬁlms coupled via a nonmagnetic layer of a variable\nthickness. In Fig. 1 b) and c) we consider a pair of disk-shaped ferromagnets with centers either along the z\naxis or on the same plane z= 0, separated by a center-to-center distance D. The two disks (actually cylinders)\nhave diameter 2Rand thickness 2t. The system of two disks mimics either multilayer samples as those used\nin spintronics54or a pair of nanodisks or circular nanopillars55. The magnetic elements are ferromagnets with\nuniform magnetization, with in-plane or out-of-plane (effective) anisotropy, depending on their aspect ratio and\nunderlying material.\nIn the case of a vertical setup (a or b) the two magnets are coupled via a nonmagnetic layer by a dipolar\ninteraction (DI). For the horizontal set up 1 c) the two magnets are also coupled by the long-ranged dipolar\ninteraction through, e.g.a nonmagnetic matrix or just air in the case of nanopillars. For the magnetostatic\nenergy corresponding to the dipolar interaction, the induced magnetic state of the magnetic dimer depends on\nthe orientation of the vector connecting the two magnets, the magnetic dimer bond, which is either along the z\naxis as in cases a) and b) or in the xyplane as in the case c).\nIn the sequel we will use spherical coordinates for all vectors involved. Hence, for the magnetic moments\nwe write mi=Misi, withMi=M0Vi,ksik= 1andsi(\u0012i;'i);i= 1;2. Since the two magnetic elements are\nsupposed to be identical, M1=M2=Mwill denote the saturation magnetization of each element. The applied\nﬁeld is H= (\u00160H)eh, withkehk= 1andeh(\u0012h;'h), and the anisotropy axes are denoted by ei(\u0012a\ni;'a\ni);with\nkeik= 1. The polar and azimuthal angles \u0012and'are deﬁned in Fig. 1.\nWe now write the energy of the magnetic dimer. Regarding the magnetic layers as being constituted by\ndifferential elements of volume dVcarrying each the magnetic moment dmi=M0dVsi;i= 1;2, the energy of\nthe elementary magnetic moment dmimay be written as3\ndEi=X\ni=1;2dE(free)\ni +dE(int)\nwheredE(free)\ni is the energy of each individual (non-interacting) magnet that comprises the Zeeman and\nanisotropy contributions\ndE(free)=\u0000X\ni=1;2h\nM0dV \u0016 0H(si\u0001eh) +KdV (si\u0001ei)2i\nwhereKis the anisotropy constant.\nAs mentioned earlier, for each magnet the (effective) anisotropy axis is assumed to be unique (the same for\nall elements) and the magnetization is uniform. Therefore, integrating over the surface of each magnet leads\nto the total (intrinsic) energy (in S.I. unit) of the magnetic dimer\nE(free)=\u0000X\ni=1;2h\nM\u00160Hsi\u0001eh+KV(si\u0001ei)2i\n: (1)\nIt is convenient to measure this energy in terms of the anisotropy energy KVand accordingly introduce the\n(dimensionless) energy of the (free) magnetic dimer\nE(free)\u0011E(free)\nKV=\u0000X\ni=1;2h\n2hsi\u0001eh+k(si\u0001ei)2i\n(2)\nwhere we have introduced the dimensionless ﬁeld parameter\nh\u0011H\nHa(3)\nwith\nHa=2KV\nM\nbeing the anisotropy ﬁeld.\nIn Eq. (2)kis a simple “ﬂag” merely introduced to keep track of the anisotropy contribution in later develop-\nments; it assumes the value 0(1) in the absence (presence) of anisotropy, respectively.\nThe interaction contribution dE(Int)for the elements dmi, considered here as point dipoles, is given by the\nstandard dipole-dipole approximation\ndE(Int)=\u0010\u00160\n4\u0019\u0011\u001a2dm1\u0001dm2\u00003 (dm1\u0001\u001a) (dm2\u0001\u001a)\n\u001a5;\n=\u0010\u00160\n4\u0019\u0011\ndm1\u0001D12dm2 (4)\nwhere\u001a=r1\u0000r2is the vector connecting the centers of the two magnets and D12the2nd-rank tensor with\nD\u000b\f\n12\u00111\n\u001a5\u0000\n\u001a2\u000e\u000b\f\u00003\u001a\u000b\u001a\f\u0001\n=1\n\u001a3\u0010\n\u000e\u000b\f\u00003e\u000b\n12e\f\n12\u0011\ne12\u0011\u001a=\u001a:\nFor ﬁnite-size magnets the magnetostatic interaction energy involves the shape of the magnets with the help\nof appropriate shape functions56,57. In the case of a vertical setup, either with oblong slabs or disks, it is well\nknown that the dipolar interaction between two ﬂat atomic planes of inﬁnite lateral dimensions vanishes. In\nfact, this interaction would appear in principle only in the presence of roughness. However, because of boundary\nand ﬁnite-size effects the DI does exist between ﬁnite and ﬂat planes (slabs and discs) and its intensity depends\non the shape, the lateral size, and the distance between the two planes12,56,57.4\nA. Rectangular slabs\nIn the case of two thin ﬁlms, modeled here as two atomic planes of lateral dimension Land a distance Dapart\n[Fig. 1a], the anisotropy ﬁeld Hain Eq. (3) may be written as Ha= 2Ks=\u001bwhere\u001bis the magnetic moment per\nunit area. For cobalt, for instance, \u001b= 1:7\u0016B=\u0000\n0:5a2\u0001\n, using the fact that the area a2contains two Co atoms,\na\u00183:55o\nAbeing the lattice step. The surface anisotropy constant Kscan be obtained from experiments on\ncobalt thin ﬁlms8,58,59for which it was evaluated to Ks'0:5 erg=cm2= 5\u000210\u00004J=m2. So,Ha\u00184 T.\nThe energy of the DI is obtained by integrating Eq. (4) over the two planes leading to (the subscript “VS”\nstands for vertical slabs)\nE(Int)\nVS=\u0015\u0002\ncos ('1\u0000'2)s?\n1s?\n2\u00002sz\n1sz\n2\u0003\n: (5)\nwith the coefﬁcient\n\u0015=\u0010\u00160\n4\u0019\u0011M2\nD3\u0002Iv\ns(\u000e): (6)\nIn Eq. (5), we have used the notation si(\u0012i;'i) =\u0000\ns?\nicos'i;s?\nisin'i;sz\ni\u0001\n.Iv\ns(\u000e)is a (double) surface integral12\nwhose analytic expression reads ( \u000e\u0011D\nL)\nIv\ns(\u000e) = 4\u000e3h\n\u000e\u00002p\n1 +\u000e2+p\n2 +\u000e2+1\n2log\u0000\n1 +1\n\u000e2\u0001\n+ log\u0010\n1+p\n1+\u000e2\n1+p\n2+\u000e2\u0011i\n:\nFor small values of \u000ethe integralIv\ns(\u000e)increases with \u000eas4\u000e3\u0014p\n2\u00002 + log\u0012\n2\n\u000e(1+p\n2)\u0013\u0015\nwhile for large \u000eit\ndoes so as 1\u0000\u000e\u00002+ 17\u000e\u00004=16. We see that as the distance between the two magnets becomes very small, i.e.\n\u000e!0, the integralIv\ns(\u000e)and thereby the DI vanishes as it should. On the other hand, as the two magnets are\nvery far apart, i.e.\u000ebecomes very large, Iv\ns(\u000e)!1and thereby the dipolar interaction reaches the limit of the\ndipole-dipole approximation of point dipoles. This behavior is clearly seen in Fig. 2.\nEq. (5) then gives the energy of interaction between two identical planes uniformly magnetized in arbitrary\ndirections. In the particular case of in-plane magnetization ( s?\n1=s?\n2= 1and'1='2= 0), dealt with in Ref.\n12, the energy E(DDI)in Eq. (5) reduces to\nE=\u0000\u0015s1\u0001D12\u0001s2: (7)\nIn fact, we may obtain a ferromagnetic or an antiferromagnetic in-plane ordering with s1\u0001D12\u0001s2=s1\u0001s2\u0000\n3 (s1\u0001e12)\u0001(s2\u0001e12) =\u00061, or an out-of-plane ordering along the bond e12with the energy s1\u0001D12\u0001s2=\u00072.\nB. Disks\nFor the setup in Fig. 1 (b and c) with two disk-shaped magnets of radius Rand (center-to-center) dis-\ntanceDapart, the DI energy was computed in Refs. [56,57] upon including the shape function to account\nfor the fact that the dipole-dipole approximation is no longer valid for close enough magnets of arbitrary\nshape. Each of these two cases was considered with both in-plane and out-of-plane magnetization. In or-\nder to write the corresponding expression of the DI energy in a compact form we introduce the notation\n\u001a= (rcos'\u001a;rsin'\u001a;z); \u001a2=r2+z2. We also introduce the vector pfor bookkeeping the shape parame-\nters of the disks, i.e.p= (R;\u001c)with\u001c=t=R,2tbeing the disk thickness. For arbitrary orientations of the two\nmagnetic moments, the energy of the system is expressed in terms of the integrals Sn(r;z;p);n= 1;2;3given\nin Eqs. (39, 40, 41) of Ref. 57. In general, they can only be computed numerically.\nIn the case of the vertical setup [Fig. 1b], i.e.(z=D;r= 0;'\u001a= 0), with arbitrary orientations of the two\nmagnetic moments, we ﬁnd that the energy can be written as (the subscript “VD” stands for vertical disks)\nE(Int)\nVD=\u0015\u0002\ncos ('1\u0000'2)s?\n1s?\n2\u00002sz\n1sz\n2\u0003\n(8)\nwhere the coefﬁcient \u0015is now given by5\n\u0015=\u0010\u00160\n4\u0019\u0011M2\nD3\u0002Iv\nd(\u0010;\u001c): (9)\nHere the shape integral (for vertical disks) Iv\nd(\u0010;\u001c)reduces to57\nIv\nd(\u0010;\u001c) = 8\u00103\u001c1\u0002\n0dq\nq2J2\n1(q)e\u00002q\u0010\u001c[cosh (2q\u001c)\u00001] (10)\nwhere we have introduced the dimensionless parameters\n\u0010\u0011D\n2t; \u001c =t\nR\nandJ1(x)is the Bessel function of the ﬁrst kind.\n0 20 40 60 80 10000.20.40.60.81\nSquares\nDisks\nζI  d(ζ,τ),v\nτ = 0.1τ = 0.2\nτ = 0.05I  v(τζ)s\nFigure 2: Shape integrals Iv\ns(\u001c\u0010)andIv\nd(\u0010;\u001c)for the vertical setup of rectangular slabs or disks, as functions of the\n(dimensionless) distance \u0010.\nThe two DI coefﬁcients (6) and (9) for slabs and disks, respectively, can be compared upon noting that \u0010=\u000e=\u001c.\nSo for a given \u001cwe compare the corresponding shape integrals Iv\ns(\u001c\u0010)andIv\nd(\u0010;\u001c), as functions of \u0010. This is\nshown in Fig. 2. Note that Iv\nd(\u0010;\u001c)is a special case of the general integrals Sn(r;z;p);n= 1;2;3introduced\nin Ref. 57. For elements of ﬁnite size, all integrals Sn(r;z;p);n= 1;2;3approach 1when the interaction\napproaches the pure dipolar interaction or the magnetic elements are replaced by point dipoles. In this case,\n\u0015!\u0000\u00160\n4\u0019\u0001\nM2=D3which is indeed the coefﬁcient of the dipole-dipole interaction [see Eq. (4)]. In Fig. 2 it is\nseen thatIv\ns(\u001c\u0010)andIv\nd(\u0010;\u001c)do go to 1as\u0010goes to inﬁnity, i.e.for a large distance between the two disks. On\nthe other hand, for small \u0010, or very large radius R, the integralsIv\ns(\u001c\u0010)andIv\nd(\u0010;\u001c)tend to zero as expected\nsince then the DI vanishes between inﬁnite thin layers. Finally, we see that as the aspect ratio \u001cincreases, the\ntwo integrals increase and this implies that, for a ﬁxed distance D, the interaction is stronger between thicker\nmagnets.\nFor the horizontal setup in the xyplane with an arbitrary angle '\u001abetween the dimer’s bond and the xaxis,\ni.e.(z= 0;r=D;'\u001a), we have the energy for the dipolar interaction\nE(Int)\nHD=\u0015= [cos ('1\u0000'2)\u0000(2 + \b) cos ( '1\u0000'\u001a) cos ('2\u0000'\u001a)]s?\n1s?\n2+ \bsz\n1sz\n2: (11)6\nwith\n\u0015=\u0010\u00160\n4\u0019\u0011M2\nD3\u0002Ih\nd(\u0010;\u001c) (12)\nand\b (\u0010;\u001c)\u0011Jh\nd(\u0010;\u001c)=Ih\nd(\u0010;\u001c).Ih\nd(\u0010;\u001c)andJh\nd(\u0010;\u001c)are two shape integrals given by57\nIh\nd(\u0010;\u001c) = 16\u00102\u001c1\u0002\n0dq\nq2J2\n1(q)J1(2\u0010\u001cq)\"\n1\u0000\u0000\n1\u0000e\u00002q\u001c\u0001\n2q\u001c#\n;\nJh\nd(\u0010;\u001c) = 16\u00103\u001c1\u0002\n0dq\nq2J2\n1(q)J0(2\u0010\u001cq)\u0000\n1\u0000e\u00002\u001cq\u0001\n: (13)\nNote that the energy (11), especially with '\u001a=\u0019=2, is of the same form as (5) and (8) but with different\ncoefﬁcients for the transverse and longitudinal components of the magnetic moments. This reﬂects the fact\nthat in the horizontal setup, the dimer’s bond is not along the axis of the rotational symmetry of the disks.\nThe horizontal setup in Fig. 1c corresponds to the conﬁguration (z= 0;r=D;'\u001a=\u0019=2)with the dimer’s bond\nalong theyaxis. In section III B, we investigate the FMR characteristics for the horizontal dimer with a bond\nalong thexaxis, i.e.with'\u001a= 0ande12kex. This choice is motivated by the relative ease in analyzing the\nvarious stationary points of the total energy and the consequent simplicity of the derivation of the corresponding\nanalytical expressions for the eigenfrequencies.\nC. Total energy\nCollecting all contributions (and dividing by KV) we obtain the total (dimensionless) energy of the magnetic\ndimer\nE=E(free)+E(Int)\nwhereE(free)is given in Eq. (2).\nFor two magnets uniformly magnetized in arbitrary directions, E(Int)is given by Eq. (5) for the vertical stack\nof rectangular slabs (upon dividing by KV), by Eq. (8) for the two disks along the zaxis and by Eq. (11) for\nthe two disks on the same xyplane. In fact, the three cases can be encompassed in the following compact form,\nwhich is a generalization of the dipole-dipole interaction (4)\nE(Int)=\u0018[s1\u0001Js2\u00003 (s1\u0001e12) (s2\u0001e12)]: (14)\nThe diagonal “exchange matrix” Jand the coefﬁcient  are given by\n8\n>><\n>>:J=I;  = 1;vertical setup ;\nJ=0\n@1 0 0\n0 1 0\n0 0 \b1\nA;  =2+\b\n3horizontal setup :(15)\nFor convenience, we have also deﬁned the (dimensionless) coupling constant \u0018\u0011\u0015=(KV)which explicitly\nreads\n\u0018=\u0010\u00160\n4\u0019\u0011M2=D3\nKV\u00028\n>>><\n>>>:Iv\ns(\u000e); vertical slabs ;\nIv\nd(\u0010;\u001c);vertical disks ;\nIh\nd(\u0010;\u001c);horizontal disks :\nTherefore, the magnetic state of the dipolar-coupled dimer is obtained by minimizing, with respect to the\nangles\u0012i;'i, the total energy given by the combined equations (2, 14).7\nFigure 3: Magnetic dimer setup with transverse anisotropy on the left and longitudinal anisotropy on the right. The\nnonmagnetic spacer is not shown.\nIII. FMR CHARACTERISTICS\nWe consider three situations regarding the orientation of the (effective) anisotropy easy axes within the\nmagnetic elements, with respect to both the applied ﬁeld and the dimer’s bond e12. The magnetic ﬁeld will be\nvaried both in magnitude and direction. Then, the two anisotropy easy axes e1;e2will be either (for the vertical\ndimer)\n\u000fparallel to each other but perpendicular to the applied ﬁeld, which is parallel to the dimer’s bond, and\nthis setup will be referred to as the transverse anisotropy (TA). This is the easy-axis geometry [see Fig. 3\n(left)].\n\u000fparallel to each other and to the applied ﬁeld and thus perpendicular to the dimer’s bond e12, a situation\nthat will be referred to as the longitudinal anisotropy (LA). The easy axes lie in the plane of the magnetic\nelements. This is the easy-plane geometry [see Fig. 3 (right)].\nLongitudinal and transverse here refer to the orientation with respect to the applied ﬁeld [see Fig. 3]. We will\nalso consider the case of mixed anisotropy (MA), i.e.with the easy axis of one of the magnetic layers along the\nﬁeld and the other perpendicular to it.\nFor the horizontal setup we also consider the cases of longitudinal and transverse anisotropy, together with\na situation usually studied experimentally where the direction of the applied ﬁeld and that of the anisotropy\neasy axes are interchanged.\nIn order to compute the FMR characteristics (resonance frequency and resonance ﬁeld), for a given conﬁgura-\ntion of the system under study, we use the standard method for a system of many degrees of freedom, namely we\nﬁrst determine the absolute minimum of the total energy for the given setup of the system. Then, we linearize\nthe Landau-Lifshitz equation near this minimum leading to an eigenvalue problem. We solve the latter for a\nﬁxed (in direction and amplitude) magnetic ﬁeld to obtain the eigenfrequencies, i.e.the resonance frequencies\nwhich are functions of the applied ﬁeld and all other physical parameters. In addition, the results obtained\nin this work have been recovered using the general theory of magnetic oscillations in antiferromagnets and\nferrimagnets [see e.g.chapter 3 of the textbook 60] which also proceeds by linearizing the equations of motion\nupon writing the magnetic moments and the effective ﬁelds as sums of steady and alternating components,\nassuming the latter to be small as compared to the former. Then setting to zero the determinant of the ensuing\n(eigenvalue) scalar equations, one establishes the characteristic equation for the frequencies of free oscillations.\nFor zero damping the solutions of this equation yield the eigenfrequencies. On the other hand, the eigenvec-\ntors of this eigenvalue problem correspond to the eigenmodes of the system. Thus, for the particular case of a\nmagnetic dimer, one obtains the so-called binding andanti-binding mode s.\nFor a ﬁxed frequency the eigenvalue problem can be solved for the magnetic ﬁeld and this yields the resonance\nﬁeld as a function of the direction of the applied ﬁeld in addition to the other materials parameters. Depending\non the situation, the eigenvalue problem may be solved analytically thus rendering analytical expressions for\nthe resonance frequencies. This is the case for LA and TA. However, for MA and, in general, the problem can\nonly be solved numerically. In this case, some care is necessary when determining the absolute minimum of the\nwhole system. In our work, we combine a Metropolis algorithm to (roughly) ﬁnd the magnetic moment direction\nthat corresponds to the global minimum and then use the Landau-Lifshitz equation with small damping to zero\nin on the absolute minimum, i.e.we do a down-hill search of the minimum61.8\nFor the task at hand in this work, namely the calculation of FMR spectra of magnetic dimers with different\nconﬁgurations, we now give a few orders of magnitude of the various parameters involved. These are the\nmagnetic ﬁeld, the (effective) uniaxial anisotropy, and the inter-layer DI coupling.\n\u000fslab (or rectangular) ﬁlms: these may represent two thin Co or Fe layers separated by a nonmagnetic\nspacer a few nanometers thick. The (intra-layer) anisotropy constant (per unit area) is Ks\u00183:56\u0002\n10\u00005J=m2and the isotropic exchange coupling is J\u00182\u000210\u00002J=m2. Then, the DI coupling evaluates to62\n\u0015\u00183\u000210\u00004J=m2.\n\u000fDisks (or thin cylinders): we consider the system studied in Refs. 55,63. The authors study a system of\nthree pairs of twin disks of FeV of diameter 2Rwith a center-to-center distance Dplaced on the same plane\n[Fig 1 c]. The disks are perpendicularly magnetized (along the zaxis) by an external ﬁeld \u00160H= 1:72T\nand the saturation magnetization is M0'1:45\u0002106A/m. The disks order in a bcc crystal structure63\nand their anisotropy constant and exchange coupling were estimated to be K'4:1\u0002104J=m3,J=Aa0'\n8:6\u000210\u000020joules . The disks are of radius R= 300 nm and thickness 2 t= 26:7 nm, and are separated\nby a center-to-center distance D= 800;1000;1200 nm. Then, for the shortest distance D= 800 nm the\ncoefﬁcient in Eq. (11) evaluates to \u0015=\u0000\u00160\n4\u0019\u0001\nM2=D3'2:36\u000210\u000017joules and thereby \u0018'0:076. For\nD= 1000 nm;1200 nm we respectively have: \u0018'0:039;0:023.\nA word is in order regarding the normalization of the frequency. Starting from the Landau-Lifshitz equation\nwe may divide by the anisotropy ﬁeld Haand then deﬁne the dimensionless time t1\u0011t=tswheretsis the\ncharacteristic time of the underlying material given by ts= (\rHa)\u00001, with\r'1:76\u00021011(Ts)\u00001being the\ngyromagnetic factor. For the FeV disks, for instance, ts'10\u000010s. Therefore, we later use the notation ~!\u0011\n!\u0002ts=!=(\rHa)for the dimensionless frequency and for a frequency \u0017(in GHz) we write \u0017'10 GHz\u0002~!=(2\u0019).\nA. Vertical dimer\nWe compute the resonance frequency as a function of the magnitude of the magnetic ﬁeld and the resonance\nﬁeld and the dimer’s coupling as a function of the applied ﬁeld polar angle \u0012hwith 0\u0014\u0012h\u0014\u0019for a ﬁxed\nazimuthal angle 'h= 0. So the ﬁeld is rotated in the xzplane.\n1. Transverse anisotropy (TA)\nThe magnetic ﬁeld is along the xaxis (\u0012h=\u0019=2) and the anisotropy axes are parallel to the MD bond e12\n[see Fig. 3 left]. By minimizing the total energy comprising the two contributions in Eq. (2) and Eq. (14) with\nrespect to the two polar angles \u0012i;i= 1;2, we obtain the energy global minimum\n\u001asin\u00121= sin\u00122=2h\n2k+3\u0018=2; h\u0014hc;\n\u00121=\u00122=\u0019\n2; h>h c(16)\nwhere the critical magnetic ﬁeld hcis given by\nhc=k+3\n2\u0018 (17)\nor in S.I. units Hc=Ha+ 3\u0015=M .\nForh=hcthere is a change of regime, namely from a regime where the magnetic ﬁeld is dominating thus\nforcing the two magnetic moments to lie along its direction ( x), and the regime where there is a competition\nbetween, on one hand, the magnetic ﬁeld along the xaxis and, on the other, the effective ﬁeld along the zaxis\ncomprising the anisotropy and DI contributions.\nThe resonance frequencies for TA are\n~!+\nres\u0011\u0012!\n\rHa\u0013+\nres=8\n>><\n>>:q\n(2k+ 3\u0018)2\u0000(2h)2; h\u0014hc;\nq\n(2h)2\u00002h(2k+ 3\u0018); h>hc:(18)\nfor the ﬁrst mode and9\n0 0.5 100.511.522.5\nh = H/H hc = k + 3   /2(ω/γ     )\nξ = 0.1ξ = 0.2\nξ = 0\naH ares\nξTransverse anisotropy(Anti)binding mode\nFigure 4: Resonance frequency of the binding mode (continuous lines) and the anti-binding mode (dashed lines) as a function\nof the magnetic ﬁeld and varying dipolar interlayer coupling, with transverse anisotropy.\n~!\u0000\nres\u0011\u0012!\n\rHa\u0013\u0000\nres=8\n>>><\n>>>:s\n(2h)2\n3\u0014\n1\u00004\u0010\n2k\n2k+3\u0018\u00112\u0015\n+ (2k+ 3\u0018)2; h\u0014hc;\np\n(2h\u00002\u0018) (2h\u00002k+\u0018); h>h c:(19)\nfor the second mode.\nLet us now brieﬂy analyze the corresponding eigen-oscillations. When the two magnetic moments are brought\ntogether, the mutual interaction of their (degenerate) resonances or modes can induce a splitting of the modes\ninto pairs characterized by the so-called binding/anti-binding states. To see this, one writes the magnetic\nmomentMias a sum of the equilibrium component M(0)\niand an alternating component mi(assumed to be\nsmall compared to the former), i.e.Mi'M(0)\ni+mi; i= 1;2. Then, upon solving the linearized (coupled)\nequations of motion for the two vectors mi; i= 1;2one obtains the eigen-frequencies and the corresponding\neigen-vectors (modes) of the system. In the present situation, for the mode with frequency ~!+\nreswe obtain\nmy\n1=my\n2andmz\n1=mz\n2,i.e.the two vectors m1andm2are identical and as such they precess together. This\nis the uniform mode or the binding mode . On the other hand, at the frequency !\u0000\nresone getsmy\n1=\u0000my\n2and\nmz\n1=\u0000mz\n2, which implies that m2=\u0000m1, and this corresponds to the anti-binding mode . It is clear that in the\nabsence of coupling ( \u0018= 0) the two frequencies become degenerate (equal). For instance, setting \u0018= 0in the\nﬁrst line of Eqs. (18, 19) renders the (doubly degenerate) resonance frequency of a single magnetic moment\n~!\u0006\nres(\u0018= 0) = 2p\nk2\u0000h2: (20)\nSimilarly, in zero ﬁeld ( h= 0) the effective ﬁeld is along the MD bond. Indeed, the DI energy is minimized\nwhen the two magnetic moments are parallel to each other and pointing along the vector e12, which is also the\ndirection of the MD effective anisotropy axis. In this case, the resonance frequency reads [from Eqs. (18, 19)]\n~!\u0006\nres(h= 0) = 2k+ 3\u0018:\nLet us now discuss the behavior of the resonance frequency as we vary the applied ﬁeld hand DI intensity\n\u0018. We do so only for the rectangular slabs since the change in the case of disks is only of a little quantitative\nimpact.\nIn Fig. 4 we plot the frequency ~!+\nresin continuous curves and ~!\u0000\nresin dashed curves, against the ﬁeld magni-\ntude, for different values of the DI interlayer coupling. ~!\u0000\nres(\u0018= 0) is not plotted as it coincides with ~!+\nres(\u0018= 0).10\n00.511.522.53\n0.2 0 0.2 0.4 0.6 0.8 1 1.2(H/Ha)res\nθh/πξ= 0.4ξ= 0.6ξ= 0.75ξ= 0Transverse anisotropy\nFigure 5: Resonance ﬁeld as a function of the magnetic ﬁeld direction for the frequency ~!= 2:8(\u001828 GHz) and varying\ndipolar interlayer coupling, for the magnetic trilayer with transverse anisotropy.\nWe ﬁrst discuss the behavior of the frequency ~!+\nres. The results are similar to the case of uniaxial anisotropy60,61\nin a transverse ﬁeld but the curves are now shifted by the DI contribution that brings an additional anisotropy.\nAs explained above, this is due to the fact that for the present MD setup, the DI and uniaxial anisotropy have\nadditive effects leading to a (larger) effective anisotropy ﬁeld perpendicular to the magnetic ﬁeld. Similarly to\nthe case of uniaxial anisotropy the resonance frequency goes to zero at some critical value hcof the applied ﬁeld.\nThis is usually used to determine the anisotropy ﬁeld from experiments. In the present case, this could be used\nto determine the strength of the interlayer coupling between two magnetic layers of known anisotropy. For a\nmagnetic ﬁeld above the saturation value, Eq. (18) yields ~!+\nres\u0000!2h, which corresponds to the straight line\nseen at high ﬁelds in Fig. 4. In this binding mode, the two magnetic moments remain parallel to each and the\nmagnetic dimer behaves like a single magnetic moment but with larger stiffness due to the DI coupling. This\nis why the curve ~!res(h)is similar to that of a single magnetic moment but with an increasing hcas a function\nof the dimer’s coupling.\nTurning now to the frequency ~!\u0000\nreswe have two main observations, in comparison with the frequency ~!+\nres.\nFirst, the anti-binding mode frequency is higher than that of the binding mode, as is usually the case. Indeed,\nthe anti-binding mode is an excitation of the system that is higher in energy than the binding mode, considered\nas the ground state. In this mode we have m2=\u0000m1and thereby the two magnetic moments Mi; i= 1;2\ndo not remain parallel to each other, thus leading to a “negative” contribution to the effective anisotropy. This\nin turn induces a decrease of the critical point hcwith increasing coupling, as can be seen by extrapolating to\nlower ﬁelds the dashed straight lines in Fig. 4. Moreover, from Eq. (19) we can see that under high ﬁelds the\nresonance frequency ~!\u0000\nresbecomes independent of the coupling.\nThe issue of binding and anti-binding modes and their comparison is rather involved and requires a thorough\nanalysis. For example, a more precise investigation of these modes and their dynamics can be achieved by\ncomputing (both analytically and numerically) the time evolution of the two magnetic moments on- and off-\nresonance upon varying the applied ﬁeld and the coupling. In order not to make the present article too bulky,\nwith the risk of drowning its main message, in the sequel we restrict our discussion to the binding modes and\ntheir characteristics, leaving the analysis of anti-binding modes for a separate work.11\nIn Fig. 5 we show the results for the resonance ﬁeld for the same setup as in Fig. 4, for a ﬁxed frequency\n~!. The valley and the peak correspond to the easy and hard directions, respectively. We see that as the DI\nincreases the effect of the easy direction is enhanced leading to a wider valley and a lower resonance ﬁeld. The\neffect of the hard direction is also enhanced but affects a narrower range of ﬁeld directions around \u0012=\u0019=2. In\nsummary, as the DI increases the easy axis range widens whereas that of the hard axis shrinks. The fact that\nthe amplitude of the resonance ﬁeld globally decreases when the DI increases is simply due to the fact that\nwith a stronger effective ﬁeld, a smaller magnetic ﬁeld is needed to satisfy the resonance condition.\n2. Longitudinal anisotropy (LA)\nThe magnetic ﬁeld is still along the xaxis (\u0012h=\u0019=2) but now the anisotropy axes are parallel to it but\nperpendicular to the MD bond e12[see Fig. 3 right]. In this case, the DI and uniaxial anisotropy contributions\ninduce different preferred directions for the two magnetic moments and thereby they are in competition with\neach other. Therefore, the two magnetic moments may be parallel or anti-parallel to each other and lie in the\nxzplane. Denoting by \u0012the polar angle they make to the zaxis, the MD energy then reads E=\u00004hsin\u0012\u0000\nsin2\u0012+\u0018\u0000\n1\u00003 cos2\u0012\u0001\nand upon minimizing it with respect to \u0012we obtain the following minima: \u0012=\u0006\u0019=2or\nsin\u0012= 2h=(3\u0018\u00002k). Hence, we have the following three distinct states with their respective energies\n\u00121=\u0019\n2=\u00122;E1=\u00002h\u00002k+\u0018; (21a)\n\u00121=\u0019\n2=\u0000\u00122;E2=\u00002k\u0000\u0018; (21b)\nsin\u00121=2h\n3\u0018\u00002k= sin\u00122;E3=\u0000(2h)2\n3\u0018\u00002k\u00002\u0018: (21c)\nNote that for the last state to exist \u0018must satisfy \u0018\u00152\n3(k+h)\u0011\u0018min. The antiferromagnetic state (21b),\nwithM1=\u0000M2, results from the fact that the DI tends to align the magnetic moments in an anti-parallel\nconﬁguration when they are normal to the DI bond. Hence, the transition from the ferromagnetic state (21a)\nalong the ﬁeld direction to the antiferromagnetic state (21b), still along the ﬁeld direction, occurs when \u0018reaches\nthe valueh. Next, as the DI increases the system undergoes the transition from the antiferromagnetic state\n(21b) to the (oblique) ferromagnetic state (21c), tilted towards the MD bond, when \u0018crosses the value\n\u0018tilt\u00112\n3h\n(2k) +p\nk2\u00003h2i\n; (22)\nobtained by setting E2=E3.\nThe system then selects one minimum or the other according to the strength of the magnetic ﬁeld as compared\nto the other two energy contributions. More precisely, we have two ﬁeld regimes separated by the saturation\nﬁeldhs=k=2, which is obtained by solving E1=E2for\u0018=\u0018min.\nTherefore, we have the following cases:\n1. For a weak ﬁeld, h\u0014hs, a comparison of DI to both the ﬁeld and the anisotropy contributions, yields three\nregimes:\n(a) for weak coupling \u0018 < h , the ﬁeld is strong enough to drive the two magnetic moments along its\ndirection (xaxis). Hence the ferromagnetic state in Eq. (21a) is selected. The resonance frequency in\nthis case is given by\n~!res=p\n(2k+ 2h) (2k+ 2h\u00003\u0018): (23)\n(b) for intermediate coupling h < \u0018\u0014\u0018tilt, the anisotropy is still dominant but the DI contribution\nis stronger than the magnetic ﬁeld contribution, thus leading to the antiferromagnetic state (21b)\nalong the anisotropy axis. In this case, The resonance frequency reads\n~!res=q\nA\u0000p\nB (24)\nwithA\u0011(2h)2+ (2k+\u0018)2+ 2\u00182andB\u00119\u00182(2k+\u0018)2+ (2h)2(4k+\u0018) (4k+ 3\u0018).12\n(c) for a strong coupling, i.e.\u0018 >\u0018 tilt, the system orders in the oblique ferromagnetic state (21c) and the\nresonance frequency is then given by\n~!res=p\u0018\n3\u0018\u00002k\u0002q\n(2h)2(2k+ 3\u0018)\u0000(2k\u0000\u0018) (3\u0018\u00002k)2: (25)\n2. For strong ﬁelds h>hsthere are only two regimes for the DI coupling. Indeed, since the latter competes\nwith the magnetic ﬁeld and thereby only the transition from state (21a) to the state (21c) takes place and\nthis occurs when the DI coupling reaches the value \u0018=\u0018min. Hence, for \u0018\u0014\u0018minthe system minimum is\ngiven by (21a) and the resonance frequency is the same as in Eq. (23). For \u0018 >\u0018 minthe system orders in\nthe global state (21c) and the corresponding resonance frequency is that given in Eq. (25).\nNote that in standard FMR measurements one has to apply a sufﬁciently strong ﬁeld, i.e.h > hs, in order\nto saturate the magnetic system which then occupies an energy minimum. This would mean in the present\nwork that the regime with h\u0014hsfound above, and included here for completeness, might not be relevant\nto all systems studied by the FMR technique. For instance, in Ref. 55, the applied ﬁeld is H= 1:72 T while\nHa'0:056 T .\nIn Fig. 6 we plot the FMR frequency against the DI coupling \u0018. The upper panel is for the case h= 0,\nincluded as a reference. In this particular case, there is a competition between the DI and the (effective)\nuniaxial anisotropy so that when \u0018increases there occurs a transition from the state (21b) to the state (21c) at\n\u0018= 2k. Therefore, for \u0018\u00142kthe minimum is at \u00121=\u0019\n2=\u0000\u00122and the resonance frequency reads ~!res=p2k\u0000\u0018\nwhereas for \u0018>2kthe minimum is at \u00121=\u00122= 0;\u0019and the resonance frequency becomes ~!res=p\n\u0018(\u0018\u00002k). In\nthe caseh<hs, the middle panel exhibits the three regimes found above. In the ﬁrst regime the two magnetic\nmoments are in a ferromagnetic state and the FMR frequency decreases as \u0018increases. In the second regime,\nthe two magnetic moments are antiferromagnetic and precess in opposite directions. The FMR frequency again\ndecreases when \u0018increases. Finally, in the third regime the two magnetic moments are again parallel to each\nother and the FMR frequency increases with \u0018towards the asymptote \u0018=h, similarly to the case of a single\nmagnetic moment. On the other hand, for h>hs(lower panel) the FMR plots exhibit two distinct regimes with\na particular value of \u0018for which!resgoes to zero.\nIn Fig. (7) we plot the resonance ﬁeld for ~!= 2:8against the ﬁeld polar angle \u0012h. This presents the evolution\nof the competition between the DI and uniaxial anisotropy as the former is increased. For low values of \u0018the\nuniaxial anisotropy dominates and thereby imposes the direction of the system’s effective easy axis ( \u0012h=\u0019=2).\nAs the interaction further increases the DI axis becomes easier while the anisotropy axis becomes harder. This\ninduces a “cone-like” behavior of the easy axis for high values of DI [see the curve with \u0018= 1, where the easy\naxis now comprises a wider range of \u0012has compared to the curve with \u0018= 0]. The difference in the behavior of\nthe hard axis for \u0018= 0:5is due to the different nature of the DI contribution to the effective anisotropy, when\ncompared to that of the uniaxial anisotropy. The DI turns out to have a stronger effect along the direction\nperpendicular to the MD bond, inducing a hard axis at a much faster rate than the one it induces for the easy\naxis in the parallel direction.\n3. Mixed anisotropy\nHere we deal with the situation where one of the magnetic layers has an easy axis anisotropy and the magne-\ntization then points normal to the layer’s plane, whereas the second magnetic layer has an easy-plane magne-\ntization. More precisely, we set e1kezande2kex. Whereas for the longitudinal and transverse conﬁgurations\nanalytical expressions have been obtained for the resonance frequency, in the present situation the analytical\nexpressions for the stationary points are too cumbersome for practical use. For this reason, they have been\ncomputed numerically. Note that the different orientations of the easy axes may be a consequence of different\nthicknesses. This means that the two magnetic layers are not identical, as it has been assumed hitherto. A\ndifference in the amplitude of the two magnetic moments will have a quantitative impact on the results, e.g.\nshifts in critical values. However, in the sequel we ignore this difference and concentrate on the qualitative\nbehavior of the dimer.\nThe DI magnetic dimer with mixed anisotropy is somewhat similar to that with longitudinal anisotropy,\ninsofar as the DI again competes with a longitudinal uniaxial anisotropy, but now it does so against only the\ncontribution (to anisotropy) of one of the layers, thus decreasing (for h= 0) the critical value of \u0018from 2kto\n2k=p\n3.13\n0 1 2 300.511.52(ω/γΗ )res\nξk = 1, h = 0Longitudinal anisotropya\n0 1 2 300.511.522.5ares\nξk = 1, h = 0.25Longitudinal anisotropy\n2h2/3 [2 k + (k2  - 3 h2)1/2](ω/γΗ  )\n0 1 2 3 401234(ω/γΗ  ) res\nξk = 1, h = 1.0Longitudinal anisotropy\n2/3 (k + h)a\nFigure 6: Resonance frequency against the interlayer DI coupling \u0018for the values of the applied ﬁeld hmarking the three\nregimes discussed in the text.14\n00.511.522.5\n0 0.2 0.4 0.6 0.8 1 1.2(H/Ha)res\nθh/πξ= 0\nξ= 0.2\nξ= 0.5\nξ= 1Longitudinal anisotropy\nFigure 7: Resonance ﬁeld as a function of the magnetic ﬁeld direction in the xzplane for the frequency ~!= 2:8and varying\nDI interlayer coupling and longitudinal anisotropy.\nIn Fig. 8, we see that in the absence of the interlayer coupling ( \u0018= 0), the effective easy axis is located\nsomewhere between the two (layers) anisotropy axes (the individual anisotropy axes are seen as hard axes\nbecause of the competition between them). As the DI becomes stronger the precession around the direction\n\u0012h=\u0019=2becomes unfavorable and the precession angle decreases, while the precession around the MD bond\nbecomes easier with a wider range of values for the ﬁeld direction. The easy axis effectively starts to shift\ntowards the MD bond until the interaction is strong enough to completely overcome the effect of the transverse\ncomponent of the anisotropy. For \u0018= 0:75the interaction-anisotropy competition leads to a wide easy cone\naround the zaxis, where Hresis almost constant. We again observe the enhanced hard axis behavior induced\nby the DI along the transverse direction. By further increasing the interaction ( \u0018= 1) we again observe the\npeak at\u0012h=\u0019=2and a clear easy axis along the MD bond. This implies that the longitudinal component of the\nanisotropy is no longer the most relevant contribution to the system and the dynamics is governed only by DI,\nand to a lesser extent, by the transverse anisotropy.\nFig. 9 shows a comparison of the FMR frequency as a function of the dipolar interaction parameter \u0018for the\nthree anisotropy conﬁgurations, in the absence of magnetic ﬁeld (h= 0). We see that for \u0018= 0the frequencies\nstart from the same value which indicates that the contribution of the uniaxial anisotropy is the same for\nthe three conﬁgurations of the anisotropy axes. This changes as \u0018increases since the effect of DI depends on\nthe anisotropy conﬁguration. For a magnetic dimer with TA, ~!resis a monotonically increasing function of \u0018\nwhile the LA and MA conﬁgurations clearly exhibit the competition between the uniaxial anisotropy and DI.\nThe critical value of \u0018(denoted her by \u0018FMR\nc) at which ~!res= 0, and that depends on the conﬁguration of the\nsystem, represents the value of the DI at which the competing anisotropy and interaction ﬁelds compensate\nfor each other. The competing ﬁelds for each anisotropy conﬁguration are: i) for a DI magnetic dimer with LA,\nthe two (uniaxial) anisotropy ﬁelds against the interaction ﬁeld, and ii) for a DI magnetic dimer with MA, the\nlongitudinal anisotropy ﬁeld on one hand against the transverse anisotropy and interaction ﬁeld, on the other.\nThis explains why the critical value of \u0018assumes a higher value for LA than for MA, as the interaction has to\novercome a stronger competing ﬁeld due to the anisotropy of both magnetic layers. When \u0018exceeds the critical\nvalue\u0018FMR\nc the system enters the strong coupling regime, where the FMR resonance of the three anisotropy15\n00.511.522.53\n0 0.2 0.4 0.6 0.8 1 1.2(H/Ha)res\nθh/πξ= 0\nξ= 0.2\nξ= 0.5\nξ= 1ξ= 0.75Mixed anisotropy\nFigure 8: Resonance ﬁeld vs ﬁeld direction in the xzplane, with ~!= 2:8and different values of the dipolar interaction in\nthe MA conﬁguration.\nconﬁgurations behaves in a similar way again, differing only by an additive factor that depends on the nature\nof the competition between the anisotropy and the DI.\nf(GHz) TA MA LA\n\u0018= 1 13:53:976:36\n\u0018= 3 22:513:77:79\nTable I: Resonance frequency for \u0018= 1and3for the three anisotropy conﬁgurations.\nTable I gives the resonance frequency of cobalt layers for the three anisotropy conﬁgurations and two values\nof DI.\u0018= 1< \u0018FMR\nc and\u0018= 3> \u0018FMR\nc hold for TA, LA and MA. We clearly see that TA always leads to the\nfastest precession. LA precesses slightly faster than MA when \u0018 <\u0018FMR\nc, and the MA precession is faster than\nthat of the LA otherwise.\nB. Horizontal dimer\nNow, we deal with the horizontal setup of the magnetic dimer which, for convenience, we take here along\nthexaxis, i.e.e12kex. This means that in Eq. (11) we set '\u001a= 0. As discussed earlier, this choice makes\nit relatively easier to analyze the stationary points of the energy and to derive analytical expressions for the\neigenfrequencies. Note that the energy expression (11) applies only to the case of a horizontal setup in the xy\nplane.\nWe only consider the case of two coupled disks. We treat two anisotropy conﬁgurations, either with the easy\naxes along the ﬁeld direction xor perpendicular to the ﬁeld, along the zdirection. We also deal with the case\nof a ﬁeld applied along the zaxis and the two anisotropy axes along the xaxis, as in the experimental study of16\n0123456\n0 1 2 3 4(ω/γHa)res\nξTA\nMA\nLA\nFigure 9: Resonance frequency for transverse, mixed, and longitudinal anisotropy conﬁgurations of the DI magnetic dimer.\nRef. 55 performed on FeV disks with aspect ratio \u001c=t=R'0:0445 and\u0010=D=2t'30:For such parameters,\n\b'0:58.\n1. Easy axes parallel to the ﬁeld (LA)\nThe minimum in this case is simply given by\n\u00121=\u0019\n2=\u00122\nbecause all the contributions to the effective ﬁeld are along the xaxis. The resonance frequency is then given\nby\n~!res=p\n(2h+ 2k+\u0018) (2h+ 2k+ \b\u0018) (26)\nAn obvious consequence is the additive nature of the contributions from the applied ﬁeld, the DI, and the\nanisotropy of this particular setup. Fig. 10 shows this additive effect through the fact that the resonance\nfrequency is a monotonously increasing function of the applied ﬁeld. Moreover, the effect of various values\nof the DI and the shape factor is merely to shift upwards the resonance frequency. This is incidentally in\nagreement with the results discussed earlier for the vertical dimer in that the DI induces a faster precession as\nit increases.17\n23456\n0 0.5 1 1.5 2(ω/γHa)res\nH/Haξ= 0ξ= 0.4Φ = 0.58Longitudinal anisotropy\n23456\n0 0.5 1 1.5 2(ω/γHa)res\nH/HaΦ = 0\nΦ = 1ξ= 0.4Longitudinal anisotropy\nFigure 10: Resonance frequency for different values of the dipolar interaction \u0018(left) and the shape factor \b(right). The\nanisotropy easy axes are parallel to the applied ﬁeld.\n2. Easy axes perpendicular to the ﬁeld (TA)\nIn this case, the minima of the system are\n8\n><\n>:sin\u00121= sin\u00122=2h\n2k\u0000\u0018; cos\u00121=\u0000cos\u00122; h\u0014hc;\n\u00121=\u00122=\u0019\n2; h>h c:(27)\nSimilarly to the vertical dimer, here again there is a minimal value \u0018min= 2k\u00002hof the DI coupling \u0018in order\nfor the ﬁrst state to exist. We may also consider this condition as leading to a critical value of the magnetic ﬁeld\ngiven byhc=k\u0000\u0018=2.\nThe corresponding resonance frequencies are (for the binding mode)\n~!res=8\n>><\n>>:1p2k\u0000\u0018rh\n(2k\u0000\u0018)2\u0000(2h)2i\n[2k\u0000(1\u0000\b)\u0018]; h\u0014hc;\np\n(2h+\u0018\u00002k) (2h+ \b\u0018); h>h c:(28)\nOne should notice the additional dependence of the resonance frequency on the shape function \b. However,\nthe latter does not enter the energy minima of the system in Eq. (27).\nFig. 11 shows the resonance frequency as a function of the applied ﬁeld for different values of \u0018and \b. It\ncan be seen that increasing the DI decreases the critical value of the ﬁeld at which the minimum of the system\nchanges, since hc=k\u0000\u0018=2. This is due to the fact that the minimum sin\u0012i= 2h=(2k\u0000\u0018)results from the\ncompetition between the anisotropy and the combined effect of the applied ﬁeld and the DI. Thus, if the DI\nis stronger, a weaker ﬁeld will be necessary to overcome completely the effect of the anisotropy. This is to be\ncompared with Fig. 4 where the critical value hc=k+ 3\u0018=2increases with \u0018. Therefore, if the anisotropy ﬁeld\nof a material measured by FMR comes out smaller than that of the individual layers, the results above hints to\nthe possibility of a non negligible DI acting at the interface.\n3. Horizontal dimer with vertical magnetic ﬁeld\nA situation that is also of interest and which can be easily set up experimentally is the one where the ori-\nentations of the magnetic ﬁeld and easy axes are swapped with respect to the previous case, i.e.ehkezand18\n00.511.522.533.5\n0 0.5 1 1.5 2 2.5(ω/γHa)res\nH/Haξ= 0ξ= 0.2ξ= 0.4\nΦ = 0.58Transverse anisotropy\n00.511.522.533.5\n0 0.5 1 1.5 2(ω/γHa)res\nH/HaΦ = 0Φ = 0.5Φ = 1ξ= 0.4Transverse anisotropy\nFigure 11: Resonance frequency for different values of the dipolar interaction \u0018and the shape factor \b. The anisotropy easy\naxes are perpendicular to the applied ﬁeld.\neikex;i= 1;2. This was considered, for example, in the case of two coupled vertical disks of FeV54,55. In this\nsituation the energy minima of the magnetic dimer are given by\n8\n><\n>:cos\u00121= cos\u00122=2h\n2k+\u0018(1+2\b); h\u0014hTA\nc;\n\u00121=\u00122= 0; h>hTA\nc(29)\nwhere now the critical value of the magnetic ﬁeld is hTA\nc=k+\u0018(1 + 2\b)=2.\nUpon comparing the energy minima in Eqs. (27) and (29) we realize that a swap of the ﬁeld direction and that\nof the anisotropy easy axes leads to a qualitatively different result. The reason is that, owing to the additional\nanisotropy induced by the dipolar interaction, having the dimer’s bond parallel to the anisotropy easy axes\nleads to a stronger effective anisotropy than when the dimer’s bond is perpendicular to them. Obviously, the\ncorresponding resonance frequencies are also rather different.\nForh<hTA\nc, the corresponding eigenvalues that yield the resonance frequencies read\n~!1\nres=1\n2k+\u0018(1 + 2\b)rh\n[2k+\u0018(1 + 2\b)]2(2k+\u0018)\u0000(2h)2[2k\u0000\u0018(1 + 2\b)]i\n(2k+\u0018\b) (30)\nand\n~!2\nres=1p\n2k+\u0018(1 + 2\b)rh\n[2k+\u0018(1 + 2\b)]2\u0000(2h)2i\n[2k+\u0018(2 + \b)]: (31)\nFor\u0018= 0 we have ~!1\nres= ~!2\nres. However, for \u00186= 0 the eigenvalue (30) is the resonance frequency of the\nfundamental mode and the eigenvalue (31) is the resonance frequency of the higher-order precession mode.\nNote that in general, the DI increases the resonance frequency of the higher-order mode while reducing that of\nthe fundamental mode. However, in a horizontal magnetic dimer with TA this is not the case, and there is a\nmode crossing (or the modes are swapped) at a ﬁeld\u0000\nhTA\nc\u0001swapgiven by\n\u0000\nhTA\nc\u0001swap=hTA\nc\u0002s\n2k+\u0018(1 + \b)\n4k+\u0018(1 + 2\b)(32)\nobtained by setting ~!1\nres= ~!2\nres. Note that\u0000\nhTA\nc\u0001swap<hTA\ncsince \b>1=2and usually \u0018<2k. This indicates that\nthe eigenfrequency ~!1\nresthat corresponds to the fundamental mode for an applied ﬁeld h <\u0000\nhTA\nc\u0001swapbecomes\nthat of the higher-order mode for higher values of the ﬁeld, where the frequency of the fundamental mode is\nthen given by ~!2\nres. We may summarize the different regimes as follows19\n~!res=8\n>>>>><\n>>>>>:~!1\nres; h<\u0000\nhTA\nc\u0001swap;\n~!2\nres;\u0000\nhTA\nc\u0001swap<h<hTA\nc;\np\n[2h\u00002k\u0000\u0018(1 + 2\b)] [2 h+\u0018(1\u0000\b)]: h>hTA\nc:(33)\nNote that for k6= 0andh<hTA\ncthe frequency of the non-interacting dimer ( \u0018= 0) given by Eq. (20) is easily\nrecovered from the expressions above.\n00.511.522.53\n0 0.5 1 1.5 2 2.5(ω/γHa)res\nH/Haξ= 0\nξ= 0.4Φ = 0.58\nξ= 0.2\n00.511.522.53\n0 0.5 1 1.5 2 2.5(ω/γHa)res\nH/HaΦ = 1Φ = 0.5Φ = 0 ξ= 0.4\nFigure 12: Resonance frequency for different values of the dipolar interaction \u0018and the shape factor \b.\nFig. 12 shows the resonance frequency as a function of the applied ﬁeld for different values of \u0018and\b. The\nabrupt change in the slope of the curve observed for low values of the ﬁeld marks the mode swap. It can be\nseen that with increasing DI, the resonance frequency increases for low values and decreases for high values\nof the applied ﬁeld. At the same time, it increases the ﬁeld critical value since hTA\nc=k+\u0018(1 + 2\b)=2. The\nlatter reduces to the critical value (17) for the vertical dimer with the same anisotropy conﬁguration. Indeed,\nas we saw earlier for the vertical setup instead of the two integrals in Eq. (13) we only have the integral (10),\nwhich means that \b!1. The main difference between the two case, i.e.the vertical dimer with TA and the\npresent case of the horizontal dimer is that in the former one has an out-of-plane anisotropy while in the latter\nthe anisotropy is in plane and this explains the appearance of the shape factor \bin the latter situation.\nFurthermore, the plots in Fig. 12 (right) show that as \bincreases the frequency increases for low values and\ndecreases for high values of the applied ﬁeld, while increasing the ﬁeld critical value. The effect of the shape on\nthe resonance frequency is much more noticeable in this conﬁguration than in the other ones dealt with earlier.\nFig. 13 shows the resonance ﬁeld for the present conﬁguration where it is clearly seen that the stronger is\nthe DI contribution the ﬂatter is the minimum corresponding to the direction along the dimer’s bond and the\nlower is the resonance ﬁeld. The reason, as was explained earlier, is the fact that the stronger is the dipolar\ninteraction the smaller is the ﬁeld necessary for the resonance condition.\nIn Ref. 55, for instance, the applied ﬁeld is H= 1:72 T and this implies that the resonance frequency should\nbe given by the last line in Eq. (33) and thereby the increment of the resonance frequency due to DI reads\n\u0001~!DI=p\n[2h\u00002k\u0000\u0018(1 + 2\b)] [2 h+\u0018(1\u0000\b)]\u0000p\n2h(2h\u00002k):\nThen, for the experimentally observed55shift in frequency (induced by the dipolar interactions) of \u0001\u0017'\n50 MHz ,\u0001~!DI=2\u0019= \u0001\u0017=10'5\u000210\u00003or\u0001~!DI'10\u0019\u000210\u00003. The geometrical factors of the system studied\nyield \b'0:58leading to the reduced parameter \u0018'0:082or the DI parameter \u0015=\u0018\u0002KV'5:56\u000210\u000016J, for\nthe inter-elements distance D= 800 nm. This value of \u0018is larger by the factor Ih\nd(30;0:0445)'1:1378 than the20\n00.511.522.5\n0 0.2 0.4 0.6 0.8 1 1.2(H/Ha)res\nθh/πξ= 0.5ξ= 0.75\nξ= 0\nξ= 1Φ = 0.58\nFigure 13: Resonance ﬁeld for different values of the dipolar interaction \u0018.\nvalue that one would obtain within the dipole-dipole approximation where the nanomagnets are considered as\npoint dipoles (for which Ih\nd(\u0010;\u001c)!1). This means that in such a magnetic dimer, the inter-elements distance,\nor more precisely the parameter \u0010is large enough for the dipole-dipole approximation to apply. However, for\narbitrary shapes one has to take account of the aspect ratio and the more general formulas given here for the\nenergy, its minima and the corresponding resonance frequencies.\nIV. CONCLUSION\nIn this work we have developed a uniﬁed formalism for analyzing the ferromagnetic spectrum of a magnetic\ndimer, representing either a trilayer with a nonmagnetic spacer or a pair of nanomagnets (platelets or nanopar-\nticles) coupled by dipolar interaction. This formalism goes beyond the dipole-dipole approximation of point\ndipoles since it fully takes account of the ﬁnite size and aspect ratio of the magnetic nano-elements.\nWe have provided analytical expressions for the ferromagnetic resonance frequency in various regimes of the\napplied ﬁeld and inter-element dipolar coupling, in various conﬁgurations of the anisotropy of the two magnetic\nelements. Among the results that can be easily inferred from these analytical developments is that when the\nﬁeld is applied normal to the dimer’s bond and to the anisotropy easy axes, in either a vertical or a horizon-\ntal setup, the critical value of the magnetic ﬁeld at which the resonance frequency vanishes is an increasing\nfunction of the coupling parameter; and it depends on the shape factor in the case of in-plane anisotropy in\nthe horizontal setup. In the horizontal setup with transverse anisotropy, this critical value decreases for more\nstrongly coupled dimers. On the other hand, in the case of longitudinal anisotropy there is no critical value\nat all and the resonance frequency is a monotonously increasing function of the applied ﬁeld. These features\nprovide an unambiguous means for characterizing the anisotropy and the inter-element coupling in magnetic\ndimers.\nWe have also studied numerically the effects of these physical parameters on the resonance ﬁeld in the various\nregimes. In the vertical setup, a variation of the dimer’s dipolar coupling, which can be achieved by changing\nthe material and thickness of the spacer, can lead to a change of the direction of the effective anisotropy. This is\nseen through a drastic change in the curves of the resonance ﬁeld versus the direction of the applied magnetic\nﬁeld as the inter-layer coupling is increased. These curves are of course routinely obtained in standard FMR\nmeasurements and could help characterize the effective anisotropy and coupling in such magnetic dimers.\nComparison of the expressions and the other results obtained here with experiments using the standard FMR\nwith ﬁxed frequency, or a network-analyzer with varying frequency and magnetic ﬁeld, yields a valuable means\nto characterize the dipolar coupling in systems of two magnetic moments. In addition, an interacting dimer is\nthe building block for a multi-element system such as magnetic multilayers or assemblies of magnetic nanopar-\nticles. For example, in order to investigate the FMR characteristics of an assembly of magnetic nanoparticles\n(spatially organized or not) deposited on a nonmagnetic substrate or embedded in a nonmagnetic insulating\nmatrix, one could reinstate in all the analytical expressions given here the particle’s index iand its volume21\nVitogether with the inter-particle (center-to-center) distance rij. Then, upon summing over the distributions\nof the volume and the lattice distances rij, one could derive new expressions for the resonance frequencies for\nthe assembly. The latter could then be evaluated upon computing the ensuing lattice sums or expanded into\napproximate practical expressions in the case of dilute assemblies.\n1P. Grünberg, R. Schreiber, Y. Pang, M. B. Brodsky, and H. Sowers, Phys. Rev. Lett. 57, 2442 (1986).\n2E. F. Kneller and R. H. Hawig, IEEE Trans. Magn. 27, 3588 (1991).\n3A. Fert, P. Grünberg, A. Barthélémy, F. Petroff, and W. Zinns, J. Magn. Magn. Mater. 1, 140 (1995).\n4R. H. Victora, X. Shen, IEEE Trans. Magn. 41, 537 (2005).\n5D. Suess, T. Schreﬂ, R. Dittrich, M. Kirschner, F. Dorfbauer, G. Hrkac, J. Fidler, J. Magn. Magn. Mater. 290, 551 (2005).\n6G. Woltersdorf, O. Mosendz, B. Heinrich, and C. H. Back, Phys. Rev. Lett. 99, 246603 (2007).\n7B. Hillebrands and K. E. Ounadjela, Spin Dynamics in Conﬁned Magnetic Structures I (Springer, 2002).\n8P. Bruno, Ph.D. thesis, Université de Paris Sud (1989).\n9E. W. Hill, S. L. Tomlinson, and J.-P. Li, J. Appl. Phys. 73, 5978 (1993).\n10P. Bruno, Phys. Rev. B 52, 411 (1995).\n11K. Xia, W. Zhang, M. Lu, and H. Zhai, Phys. Rev. B 55, 12561 (1997).\n12P. Vargas and D. Altbir, Phys. Rev. B 62, 6337 (2000).\n13B. Heinrich, Y. Tserkovnyak, G. Woltersdorf, A. Brataas, R. Urban, and G. E. W. Bauer, Phys. Rev. Lett. 90, 187601\n(2003).\n14P. Wang, W. K. Shen, J. M. Bai, R. H. Victora, J. H. Judy, and W. L. Song, Appl. Phys. Lett. 86, 142504 (2005).\n15D. Suess, T. Schreﬂ, S. Faehler, M. Kirschner, G. Hrkac, F. Dorfbauer, and J. Fidler, Appl. Phys. Lett. 87, 012504 (2005).\n16D. Suess, T. Schreﬂ, M. Kirschner, G. Hrkac, F. Dorfbauer, O. Ertl, and J. Fidler, Appl. Phys. Lett. 41, 3166 (2005).\n17G. N. Kakazei, Yu. G. Pogorelov, M. D. Costa, V. O. Golub, J. B. Sousa, P. P. Freitas, S. Gardoso, and P. E. Wigen, J. Appl.\nPhys. 97, 10A723 (2005).\n18B. Skubic, E. Holmström, D. Iu¸ san, O. Bengone, O. Eriksson, R. Brucas, B. Hjörvarsson, V. Stanciu, and P. Nordblad,\nPhys. Rev. Lett. 96, 057205 (2006).\n19K. C. Schuermann, J. D. Dutson, S. Z. Wu, S. D. Harkness, B. Valcu, H. J. Richter, R. W. Chantrell, and K. O’Grady, J.\nAppl. Phys. 99, 08Q904 (2006).\n20G. Woltersdorf and Ch. H. Back, Phys. Rev. Lett. 99, 227207 (2007).\n21A. Berger, N. Supper, Y. Ikeda, B. Lengsﬁeld, A. Moser, and E. E. Fullerton, Appl. Phys. Lett. 93, 122502 (2008).\n22M. Madami, S. Tacchi, G. Gubbiotti, G. Carlotti, H. Pandana, R. D. Gomez, H. Tanigawa, and T. Ono, J. Appl. Phys. 104,\n063510 (2008).\n23L. Sun, P. Wong, W. Zhang, X. Zou, L. Luo, Y. Zhai, J. Wu, Y. Xu, and H. Zhai, J. Appl. Phys. 109, 033913 (2011).\n24A. F. Franco, and H. Kachkachi, J. Phys.: Condens. Mat. 25, 316003 (2013).\n25M.P. Sharrock, IEEE Trans. Magn. 26, 193 (1990).\n26K.E. Johnson, J. Appl. Phys. 69, 4932 (1991).\n27J.-L. Dormann, D. Fiorani, and E. Tronc, Adv. Chem. Phys. 98, 283 (1997).\n28X. Batlle and A. Labarta, J. Phys. D: Appl. Phys. 35, R15 (2002).\n29P. Tartaj, P. Morales, S. Veintemillas-Verdaguer,T. Gonzalez-Carreño, and C. Serna, J. Phys. D: Appl. Phys. 36, R182\n(2003).\n30J.L. Dormann, L. Bessais, and D. Fiorani, J. Phys. C 21, 2015 (1988).\n31S. Morup and E. Tronc, Phys. Rev. Lett. 72, 3278 (1994).\n32H. Mamiya and I. Nakatani, J. Magn. Magn. Mater. 177, 966 (1998).\n33J.-L. Dormann, and D. Fiorani, J. Magn. Magn. Mater. 202, 251 (1999).\n34D. V. Berkov and N. L. Gorn, J. Phys.: Condens. Matter 13, 9369 (2001).\n35P.E. Jonsson and J.L. Garcia-Palacios, Phys. Rev. B 64, 174416 (2001).\n36P. Allia, M. Coisson, P. Tiberto, F. Vinai, M. Knobel, M. A. Novak, and W. C. Nunes, Phys. Rev. B 64, 144420 (2001).\n37H. Mamiya, I. Nakatani, and T. Furubayashi, Phys. Rev. Lett. 88, 067202 (2002).\n38O. Iglesias and A. Labarta, Phys. Rev. B 70, 144401 (2004).\n39O. Chubykalo-Fesenko and R.W. Chantrell, J. Appl. Phys. 97, 10J315 (2005).\n40S. H. Masunaga, R. F. Jardim., P. F. P. Fichtner, and J.Rivas, Phys. Rev. B 80, 184428 (2009).\n41D. Ledue and R. Patte and H. Kachkachi, J. Nanoscience and Nanotechnology 12, 4953 (2012).\n42Z. Sabsabi, F. Vernay, O. Iglesias, H. Kachkachi, Phys. Rev. B 169, 323 (2013).\n43K.B. Urquhart, B. Heinrich, J.F. Cochran, A.S. Arrott, and Myrtle, J. Appl. Phys. 64, 5334 (1988).\n44B. Heinrich and J.F.C. Cochran, Adv. Phys 42, 523 (1993).\n45B. Heinrich, in Ultrathin magnetic structures II , edited by B. Heinrich and J. Bland (Springer-Verlag, Berlin, 1994), p.\n195.\n46M. Farle, Rep. Prog. Phys. 61, 755 (1998).\n47J.F. Cochran, in Ultrathin magnetic structures II , edited by B. Heinrich and J. Bland (Springer-Verlag, Berlin, 1994), p.\n222.22\n48C. Mathieu, J. Jorzick, A. Frank, S. O. Demokritov, A. N. Slavin, B. Hillebrands, B. Bartenlian, C. Chappert, D. Decanini,\nF. Rousseaux, et al., Phys. Rev. Lett. 81, 3968 (1998).\n49E. Beaurepaire, J.-C. Merle, A. Daunois, and J.-Y. Bigot, Phys. Rev. Lett. 76, 4250 (1996).\n50B. Koopmans, M. van Kampen, J. T. Kohlhepp, and W. J. M. de Jonge, Phys. Rev. Lett. 85, 844 (2000).\n51M. van Kampen, C. Jozsa, J. T. Kohlhepp, P. LeClair, W. J. M. de Jonge, and B. Koopmans, Phys. Rev. Lett. 88, 227201\n(2002).\n52J. Stöer and H. C. Siegmann, Magnetism: from fundamentals to nanoscale dynamics (Springer, Berlin, 2006).\n53Y. Zhu, Modern techniques for characterizing magnetic materials (Kluwer Academic Publishers, Boston, 2005).\n54V.V. Naletov, G. de Loubens, G. Albuquerque, S. Borlenghi, V. Cros, G. Faini, J. Grollier, H. Herdequint, Phys. Rev. B 84,\n224423 (2011).\n55B. Pigeau, C. Hahn, G. de Loubens, V.V. Naletov, O. Klein, K. Mitsuzuka, D. Lacour, M. Hen, S. Andrieu, and F. Montaigne,\nPhys. Rev. Lett. 109, 247602 (2012).\n56S. Tandon, M. Beleggia Y. Zhu, and M. De Graef, J. Magn. Magn. Mater. 271, 9 (2004).\n57M. Beleggia S. Tandon, Y. Zhu, and M. De Graef, J. Magn. Magn. Mater. 278, 270 (2004).\n58P. Bruno, J. Magn. Soc. Jpn 15, 15 (1991).\n59P. Bruno and J.-P. Renard, J. Magn. Magn. Mater. 135, 343 (1994).\n60A.G. Gurevich and G.A. Melkov, Magnetization oscillations and waves (CSC Press, Florida, 1996).\n61H. Kachkachi and D. S. Schmool, Eur. Phys. J. B 56, 27 (2007).\n62D. Altbir, M. Kiwi, R. Ramírez, and I. K. Schuller, J. Magn. Magn. Mater. 149, L246 (1995).\n63K. Mitsuzuka, D. Lacour, M. Hehn, S. Andrieu, and F. Montaigne, Appl. Phys. Lett. 100, 192406 (2012)."    },    {        "title": "1204.4401v1.Ferromagnetism_in_graphene_nanoribbons__split_versus_oxidative_unzipped_ribbons.pdf",        "content": " 1\nFerromagnetism in graphene  nanoribbons: split versus oxidative unzipped \nribbons \nS. S. Rao$,*, S. Narayana Jammalamadaka&,*, A. Stesmans$ and V. V. Moshchalkov& \n$INPAC – Institute for Nanoscale Physics and Chem istry, Semiconductor Physics Laboratory,  \nK.U. Leuven, Celestijnenlaan 200D, B–3001 Leuven, Belgium \n&INPAC – Institute for Nanoscale Physics and Chemistry, K.U. Leuven, Celestijnenlaan \n200D, B–3001 Leuven, Belgium  \nJ. van Tola \naNational High Magnetic Field Laboratory, Centre  for Interdisciplinary Magnetic Resonance, \nFlorida State University,1800 E. Paul Dirac Drive, Tallahassee, Florida 32310, USA \nD. V. Kosynkin1, A. Higginbotham1 and J. M. Tour1,2,3 \n1Department of Chemistry, 2Department of Mechanical Engineering and Materials Science, \n3Smalley Institute for Nanoscale Science and Technology, Rice University, MS-222, 6100  \nMain Street, Houston, Texas 77005, USA.  \n* These authors contributed equally to this work \n  Two types of graphene nanoribbons: (a) pota ssium-split graphene nanoribbons (GNRs), and \n(b) oxidative unzipped and chemically converted graphene nanoribbons (CCGNRs) were \ninvestigated for their magnetic properties usi ng the combination of static magnetization and \nelectron spin resonance measurements. The two types of ribbons possess remarkably different \nmagnetic properties. While the low temperatur e ferromagnet-like feature is observed in both \ntypes of ribbons, such room temperature feature persists only in potassium-split ribbons. The \nGNRs show negative exchange bias, but the CCGNRs exhibit a ‘positive exchange bias’. \nElectron spin resonance measurements infer th at the carbon related defects may responsible \nfor the observed magnetic behaviour in both types of ribbons. Furthermore, proton hyperfine \ncoupling strength has been obtained from hy perfine sublevel correlation experiments \nperformed on the GNRs. Electron spin resonan ce provides no indications for the presence of \npotassium (cluster) related signals, emphasizi ng the intrinsic magnetic nature of the ribbons. \nOur combined experimental results may infer the coexistence of ferromagnetic clusters with anti-ferromagnetic regions leading to disordered magnetic phase. We discuss the origin of the \nobserved contrast in the magnetic behaviours of these two types of ribbons.  \nKey words: Magnetism, graphene nanor ibbons, exchange bias, ESR, HYSCORE \n*Corresponding authors   \nsrinivasasingamaneni@boisestate.edu\n , Surya.Jammalamadaka@fys.kuleuven.be  \n \nAricle reference : Nano Lett. , 2012, 12 (3), pp 1210–1217  2\n \nGraphene is a promising material for spintronic applications due to its high carrier mobility \nat room temperature and its long spin life time resulting from low spin-orbit coupling (~ 10-4 \neV) and low hyperfine interaction [1,2]. Grap hene is a zero band gap semiconductor, and it \nhas been recognized that the graphene nanoribbons (GNRs) may be the potential candidates in future MOSFET devices with excellent ON/OFF ra tios over graphene as the band gap of the \nGNRs can be tuned as a function of ribbon width, quantum confinement, edge geometry, \ndoping and edge functionalization. GNRs present long and reactive edges prone to \nlocalization of electronic edge states and covalent attachment of chemical groups can \nsignificantly alter their electronic and magnetic properties [3].  \n       It has been predicted that zigzag-edged graphene or semi-hydrogenated graphene sheets \nwould exhibit ferromagnetic and spintronic properties [1]. Peculiar magnetic properties such \nas flat band ferromagnetism (FM) in zigzag GNRs, and the presence of spin waves and \ndynamical edge state magnetism have also b een theoretically predicted [4,5]. Recent \ntheoretical studies have indicated that the magnetic properties of zigzag GNRs can be \ncontrolled by an external electric field and predicted that edge states are strongly spin \npolarized [6]. In addition to that, the occurrence of carrier induced FM in GNRs has also been \nenvisaged [7]. Theoretical work suggests the presence of strong ferromagnetic interaction \nranging several 103 K between the edge-state spins in the zigzag edge that can create carbon \nonly FM [8]. The most recent and outstanding works on these materials showed that the magnetic properties are not exclus ively related to the presence of extrinsic magnetic ions but \nstrongly determined by the defects. For example, in pristine graphite, proton irradiation has \nbeen found to be an effective way to produce bulk FM [9]. While bulk graphene is known to \nbe non-magnetic, its derivatives such as graphene oxides, few-layer hydrogenated epitaxial graphene, and hydrogenated carbon nanotubes (CNT s) are reported to show room temperature  3\nFM (RTFM) [10-12]. Interestingly,  a recent scanning tunneling microscopy/spectroscopy of \nGNRs with ultra-smooth edge states with charact eristic splitting in the dI/dV spectra has been \nreported – an unambiguous indication of magnetic ordering [13]. \n        However, the direct assignment and attribution of the observed RTFM has been \nhampered in bulk static magnetization measurem ents as they provide cumulative magnetic \nresponse. One of the main supporting arguments for the observation of intrinsic FM in \ncarbon-derived materials was obtained from X - ray magnetic circular dichroism (XMCD) \nand magnetic force microscopy (MFM) as intr insic defects [14]. In addition to that, the \nimportance of single atom defects such as carbon (defect) contribution and the role of \nadatoms such as hydrogen to the observed FM are still being investigated and widely \ndiscussed [1,11,12,15]. Despite several interes ting magnetic properties of GNRs have been \npredicted theoretically as outlined above, their experimental realization has been at the nascent stage [16,17]. Therefore, to unde rstand the graphene-based magnetism more \nthoroughly, further sensitive experi mental verification is required. Particularly, to use GNRs \nin spin-based applications, it is an utmost important to investigate their true magnetic nature \nand the species (spins) that cause the magnetic order.  \n         For the present work, we have c hosen two sets of GNRs, characterized by long and \nstraight edges. With the advent of recent chemical approaches [18,19], gram-scale GNRs could be produced, thereby facilitating the investigation of their magnetic properties. To \nprobe the magnetic behaviour of the ribbo ns, we use classical static magnetization \nmeasurements coupled with defect-type-specific  electron spin resonance (ESR) technique and \nhyperfine sublevel correlation (HYSCORE) spectroscopy. In this paper, we report on important findings pertaining to the magnetic nature of two types of GNRs, namely those derived by potassium splitting of multi walled carbon nanotubes (MWCNTs) to produce \nGNRs and oxidative unzipped MWCNTs that produce graphene oxide nanoribbons which  4\nwere further reduced by hydrazine, named as chemically converted graphene nanoribbons \n(CCGNRs). Though both the ribbons exhibit low temperature ferromagnetlike (FM-like) behaviour, such room temperature feature persists only in the GNRs. The isothermal magnetization behaviour could be well described by the Langevin function in both the \nribbons, typically applicable for disorder magnet ic materials such as superparamagnets. The \nGNRs show negative exchange bias while the CCGNRs ribbons show positive exchange bias, the later behaviour is unusual. From conventional ESR and HYSCORE data, we infer that carbon-defect related clusters may cause the observed magnetic behaviour in the GNRs. The \npresent results deserve special attention as  the GNRs exhibit FM-like behaviour without \napplying external perturbations such as doping, hydrogenation, applying external electric \nfield, ion implantation or high energy bombardment. Particularly, the present results are linked to technological interest in spintronic ba sed applications with the RTFM-like properties \ncoupled with their semiconducting properties.  \n      The synthesis of K-split GNRs (GNRs) involves [18] heating potassium metal and \nMWCNTs with outside diameter of 40-80 nm a nd approximately 15-20 inner nanotube layers \nare sealed in a glass tube and heated in a furnace at 250 \noC for 14 h, followed by quenching to \neffect the longitudinal splitting process. The splitting process might be further assisted by the generation of H\n2 upon the ethanolic quench. The split MWCNTs are further exfoliated to form \nGNRs upon sonication in chlorosulfonic acid . This procedure pr oduces GNRs free from \noxidative damage with conductivities paralle ling the properties of the best samples of \nmechanically exfoliated graphene. Scanning electron microscopy (SEM) and atomic force \nmicroscopy (AFM) show that the GNRs have  widths of 130-250 nm and a length of 1-5 μm. \nSeveral techniques were used to fully characterize GNRs and to test their electronic \nproperties, as reported elsewhere [18]. CCGNRs were prepared by longitudinal unzipping of \nMWCNTs [19]. Briefly, this method involves the treatment of MWCNTs, consisting of 15-20  5\nconcentric cylinders and 40-80 nm diameter with concentrated H 2SO 4 followed by KMnO 4 \n(an oxidizing agent) at room temperature, consequently reduced by N 2H4, named as \nchemically converted graphene nanoribbons (CCG NRs).  From very preliminary studies, both \nribbon types have mainly zigzag edges with some short domains of armchair edges.  \n     From a broader context, the ribbons made  from potassium are far more conductive as their \nplanes remain pristine throughout the chemistry.  For example, these ribbons, as we have \npublished [18], are as conductive as exfoliated graphene from graphite.  They are almost \nnever monolayered as they remain foliated in stacks of about 5-8 thick since the planes are \npristine.  The edge anions from the reaction were quenched with a proton source, hence \nhydrogen atoms likely edge the ribbons—though precise confirmation of  such a claim is \nalmost impossible using any analytical method available today.  \n      The ribbons made from oxidative unzipping produce graphene oxide nanoribbons \n(GONRs) that are heavily oxidized both on the planes and at the edges due to the strongly \noxidative conditions.  Hence, they are poorly conductive but very well exfoliated in the \naqueous system in which they are prepared, thereby often being monolayered.  For the studies \nhere they were subsequently hydrazine reduced .  Hydrazine will remove about 95% of the \noxygen functionalities [19].  This restores much of the conductivity, but they are still higher in \nresistivity than the potassium-derived ribbons noted above.  Again, the I-V curves of the two \ntypes are well covered in the former papers  [18,19].  Since these are now chemically \nconverted graphene (CCG) nanoribbons, they have holes and some remaining oxygen \nfunctionalities, predominantly on the edges.  Th e increased and remaining edge structures in \nthe oxidative unzipped structures can indeed affect the measurements here.  \n       Static magnetization measuremen ts were performed using vibrating sample \nmagnetometer (VSM) equipped with Oxford cryostat on tightly packed GNRs and CCGNRs  6\nin non-magnetic straw. Conventional first deriva tive CW ESR experiments were carried out in \nthe temperature (T) range of 4.2 – 120 K using a home built K-band ( ≈ 20.6 GHz) ESR \nspectrometer operated under conditions of adia batic slow passage. Conventional low power \nfirst-derivative-absorption d Pµr/dB ( Pµr being the reflected microwave power) spectra were \ndetected through applying sinusoidal modulation (~ 100 kHz, amplitude Bm ~ 0.42 × 10-4 T) \nof the externally applied magnetic field BG\n, with incident microwave power Pµ as well as Bm  \ncautiously reduced to avoid signal distortion.  The defect spin density was quantified by \ndouble numerical integration of the K-band derivative absorption spectra by making use of a \nco-mounted calibrated Si:P intensity marker, also serving as g marker: g (4.2 K = 1.99869) \n[20]. X - band (~ 9.7 GHz) HYSCORE spectroscopy was carried out at T = 12 K with a \nrepetition time of 1 ms using the sequence π/2-τ-π/2-t 1-π-t2-π/2-τ-echo. The mw pulse lengths \nwere t π/2 = 16 ns and t π = 28 ns, starting times t 10 = t 20 = 260 ns, a τ = 164 and a time \nincrement of ∆t = 8 ns (data matrix 256 × 256). Special attention has been paid to make sure \nthat no external impurity was introduced while sample handling during the measurements.               \n           We begin our discussion with the dc magnetization measurements which are \nsummarized in Figs.1-3. Fig. 1a displays the isothermal magnetization data measured on \nGNRs at 5, 10, 50, 100, 200 and 300 K with the magnetic field varied from -5 to + 5 T. A \nclear, s-shape saturated open hysteresis loops  (M (H) curve) were observed at all the \ntemperatures measured including at 300 K, with the coerceivity ( HC) of ~ 0.02 T and the \nsaturated magnetization of ( MS) ~ 0.25 emu/g. These numbers are comparable to those \nreported for some of the carbon based materials [9-12]. From the main panel of Fig.1(a), it \ncan be noted that the saturation magnetization Mo decreases as the temperature increases from \n5 to 300 K, a typical signature of nominal FM-like materials. The top left inset of Fig. 1(a) describes the enlarged view of M-H curves measured on GNRs. The isothermal magnetization data at 5 K could be satisfactorily fitted with a Langevin function L(x) valid for classical spins  7\nof S = ½, typically describes the disordered magnetic behaviour such as superparamagnetism, \ngiven as   \n1\n00\n0 .................... coth (1) ..BB\nBBHHMMkT kTμμ μμ−⎡⎤⎛⎞ ⎛⎞⎢⎥=− ⎜⎟ ⎜⎟⎢⎥⎝⎠ ⎝⎠⎣⎦\nWhere M is the magnetization, Mo is the saturation magnetization in emu/g, µB = 9.274·10-24 \nJ/T is the Bohr magneton, kB = 1.381·10-23 J/K is the Boltzmann constant, T = 5 K is the \ntemperature and H is the magnetic field in Tesla.  As shown in the lower right inset of \nFig.1.(a), a fit to the experimental data leads to a magnetic moment of 34.27 (5) μB suggesting \na clustering of the S = ½ spins. The FM-like phase is found to be highly stable over a period \nof one year as inferred from our isothermal M-H data (not shown). Such sp-electron magnetic \nclustering has been predicted [21] theoretically  from graphene-based materials, also inferred \nfrom other carbon-based materials such as carbon nanofoam [22]. In a recent theoretical work \n[23], such carbon adatom cluster/agglomeration induced FM-like phase has been \ndemonstrated in graphene. Fig.1.(b) shows isothermal M (H) plot s recorded from CCGNRs \nmeasured at various temperatures 1.8, 5, 10 and 15 K. The typical signature of FM-like phase can be inferred from CCGNRs at 1.8 K. However,  as the temperature increases the FM-like \nphase seems to disappear. This particular observation can be seen more clearly from the \nzoomed spectrum, as displayed in the top left inset of Fig. 1(b). These ribbons show prominent FM-like features (open saturated MH hysteresis loop with well-defined coerceivity) at 1.8 K and gradually the paramagnetic feature (more linear MH) takes over as the temperature rises to 10 K. Similar to the case of GNRs, the isothermal magnetization data \nof CCGNRs at 1.8 K could be satisfacto rily fitted with a Langevin function L(x) (c.f. equation \n(1)) valid for classical spins of S = 1/2 as shown in the lower right inset of Fig.1(b), leading to \na magnetic moment of 7. 04 (6) μ\nB suggesting a clustering of S = ½ spins. As one can notice, \nthe effective magnetic moment in the CCGNRs is almost 4.8 times smaller than GNRs which  8\nmight explain the apparent room temperature FM-like feature of GNRs. Such disordered \nmagnetic feature from graphene based materials has been discussed widely in the literature \n[2,9-12,24]. The apparent absence of such room temperature FM-like feature in CCGNRs can be explained as the majority of the edges are reconstructed, passivated or closed [25] upon the \noxidative unzipping process. Only a few remaining magnetic edges would give rise to the \nweak paramagnetic contribution at room temperature [25]. \n         The results of temperature-dependent magnetization measurements for GNRs and \nCCGNRs recorded at 0.1 T are shown in Figs.2 (a,b). First measurement was taken after zero-field cooling (ZFC) to lowest temperature possible (4.4 K) and in the second run the \nmeasurements were taken under field-cooled (FC) conditions. From these plots, several \nsalient features can be inferred : low temperature susceptibility (cf. Fig.2a) of the former is about 3 times higher than the latter (cf. Fig.2b).GNRs show pronounced FC dependence: χ (T) \nis larger under FC conditions when compared to values measured under ZFC conditions (cf. \nFig. 2a). While FC dependence typically shows monotonic increase of magnetic susceptibility \nwith decreasing temperature, the ZFC data (cf. Fig.2a) displays much more complex \nbehaviour. The ZFC data exhibits a broad maximum at around ~ 70 K and low temperature sharp upturn at around ~ 14 K upon progressive cooling. The existence of maxima in ZFC data can be a consequence of competing inte ractions and/or low dimensionality of the \nmagnetic system. The presence of small magnetic domains or small regions with a super \nmagnetic behaviour may also give rise to such a peak. However, the ZFC stationary points are \nabsent in CCGNRs (cf. Fig.2b). Moreover, while cooling from room temperature, both ZFC and FC data follow similar trend, i.e., slow increase of susceptibility untill ~ 30 K followed by a sharp rise. We note here a striking similarity between the ZFC susceptibility measured in \nGNRs with that of modified graphite and carbon nanofoam [26,27].  9\n       To probe the magnetic behaviour of the ribbons further, isothermal M-H measurements \nwere performed on GNRs and CCGNRs at 5 K under ZFC and FC conditions. M (H) loops are plotted in Fig. 3a for GNRs measured under ZFC and FC (FC: 0.1, 0.3, 0.5 and 1 T) conditions. Upon closer inspection, as displayed in the top left inset for clarity, upon field \ncooling, M (H) loops shift towards negative magnetic field axis – a phenomenon known as \n‘negative exchange bias’ (NEB) – an usual phenomenon that has been commonly observed in systems exhibiting co-existing magnetic ph ases such as ferromagnetism (FM) and anti-\nferromagnetism (AFM). Interestingly, in the ca se of CCGNRs, as shown in Fig.3b, the FC \n(FC: 0.1, 0.2, 0.4, 0.6 T) causes M (H) curves to shift towards positive field axis -so called ‘positive exchange bias’ (PEB), as clearly displayed in the top left inset of the Fig.3b. Such \nPEB is unusual for metal free systems such as CCGNRs though it has been reported earlier in bilayer metallic systems such as NiFe/IrMn and FeF\n2-Fe bilayers [28,29], and in \nLa0.67Sr0.33MnO 3/SrRuO 3 [30].  \n         This particular part of our work probes the interactions develop at the magnetic \ninterfaces in two types of graphene nanoribbon samples. Probing such interactions is \nimportant to understand the intrinsic magnetic behaviour of the ribbons. Analogous to the \ncase of intrinsic interface exchange coupling between FM and AFM phases in phase separated \ncharge ordered manganites [31], we studied the cooling field ( HFC) dependence of exchange \nbias ( HEB) on GNRs and CCGNRs.  Now we define the exchange bias field HEB = -\n(Hright+Hleft)/2, Hright and Hleft being the points where the loop intersects the field axis. Let us \nanalyze the field-cooling dependence of the loop parameters Mr, HEB and HC at T = 5 K for \nboth GNRs and CCGNRs.  \n        We did not find significant vertical loop shift (variation of M r) as a function of field \ncooling (data not shown) for both ribbons. The variation of HEB obtained at 5 K as a function \nof cooling field for both GNRs and CCGNRs is pl otted in the right bottom insets of the Figs.  10\n(3a,b), respectively, which deserves special attention. In both GNRs and CCGNRs, the H EB \nincreases strongly and peaks at ~ 0.4 T and ~ 0.2 T, respectively, and then falls as a function of further increase in cooling field. Similar obs ervations were reported earlier (cf. Fig.4b of \n[32] ) for a granualar system of Fe nanoparticles embedded in an Fe oxide matrix. The authors \nin [32] have explained their observation by  invoking the complex interplay of AFM-FM \n(super)-exchange coupling and partic le-particle dipolar interactions, which both can be FM or \nAFM in sign. We shall explain this interesting observation more quantitatively in forthcoming \nparagraph.  \n          Figure 4 describes the variation of coercivity (H\nC) as a function of H FC for (a) GNRs (b) \nCCGNRs measured at 5 K. As it can be noticed, the increase in H c with field cooling up to ~ \n0.4 T (in GNRs) and ~ 0.2 T (in CCGNRs) is coherent with the increase in H EB (cf. Right \nbottom insets of Figs.3(a,b)): the field-cooli ng induces a preferential direction along which \nthe magnetic moments tend to freeze at 5 K and thus the effect of averaging of anisotropy is reduced.  As the cooling field increases up to ~ 0.4 T (in GNRs) and ~ 0.2 T (in CCGNRs) , \nthe exchange anisotropy at the interface between the FM-AFM regions sets in when the \nribbons are cooled down to 5 K and hence a large increase in H\nEB is observed. At higher \ncooling fields, the effective magnetic coupling (Zeeman coupling) between the field and the AFM moments increases which further orients AFM moments along the field direction. At \nthat stage, the Zeeman coupling overcomes the exchange coupling at the interface of FM-\nAFM, thus leading to decrease in H\nEB. A similar explanation has been proposed to account for \nthe positive exchange bias observed in FeF 2 (AFM)-Fe(FM) bilayers [29]. \n        Analogous to phase separated perovskite manganites, the variation of H EB as a function \nof cooling field (H FC) can be expressed by the following relation [31]   11\n0\n2.............. (2)(...)iF C\nEB i FC\nBB fJHHJ L Hgk Tμμ\nμ⎡⎤ ⎛⎞\n−∝ + ⎢⎥ ⎜⎟⎜⎟⎢⎥ ⎝⎠ ⎣⎦    \nJi is the interface exchange coupling constant, g is the gyromagnetic factor, L(x) is the \nLangevin function, x = μHFC/kBTf, µB = 9.274·10-24 J/T is the Bohr magneton, kB = 1.381·10-23 \nJ/K is the Boltzmann constant and μ is the magnetic moment of the clusters, HFC is cooling \nfield and Tf = 5 K. From the bottom right insets of Fig. 3(a,b) it can be seen that the \nexperimental data could be well described by above model as shown by solid curve. The \nobtained values for the Ji for both the GNRs and CCGNRs are ~ - 95 meV and -8.4 meV, \nrespectively. We also estimated the Curie temperatures T C using the mean-field expression T C \n= JS(S+1)/3k B [33] for S = ½, and obtained as T C = 276 K and 24 K, for GNRs and CCGNRs, \nrespectively. These numbers are in close agreement with our experimental observations of ~ 300 K and 10 K for GNRs and CCGNRs, respectively.   \n       According to the literature reports [ 28-30], for PEB systems, i.e) similar to CCGNRs, the \nsign of J i is found to be negative inferring the anti-ferromagnetic exchange coupling exists \nbetween FM and AFM phases. However, in the ca se of systems similar to GNRs, which show \nNEB, it was reported that the sign of J i can be a positive or negative. The observed –ve J i for \nGNRs implies the presence of anti-ferromagne tic exchange coupling between co-existing FM \nand AFM phases. The observation of PEB in CCGNRs could be attributed to their rough \nmicro-structure of the ribbons resulted from heavy oxidative damage due to harsh acidic \ntreatment during the unzipping process, as directly evidenced from the transmission electron \nmicroscopic image [19]. Such rough structure is absent in GNRs [18] which may result in \nsmooth edges facilitating NEB. \n  Fig. 5 shows representative low-power (2.5 nW) K-band ESR spectra measured on  \nGNRs (a) and CCGNRs (b) at 4.2 K. As show n, only one ESR signal is observed at zero- 12\ncrossing g value ( gc) = 2.0025 and 2.0032 , respectively. These numbers are close to free \nelectron g value, 2.0023 inferring that the observed signal does not originate from transition \nmetals. This value falls within the reported [34] carbon ESR signal range ( g = 2.0022 - \n2.0035), indicating the signal may be ascribed to C related dangling bonds of spin S = ½. For \nthe GNRs, the signal deviates from a Lorentzian of peak-to-peak width of ΔBPP ~ 8 × 10-4  T \nwith the spin density of ~ 6 × 1019g-1. For the CCGNRs, the signal closely follows the \nLorentzian, having ΔBPP ~ 6 × 10-4  T  with the spin density of ~ 1.5 × 1017g-1. One can notice \nthat the number of spins in the former case is more than two orders of magnitude higher than \nin the latter. Despite intense signal averaging over broad field ranges under various extreme \nand optimized spectrometer parameter setti ngs, no other signals could be observed over a \nbroad magnetic field sweep range up to 0.9 T. Though intensely searched for, neither any \ncorrelated additional signal structure could be  traced nor any sign of hyperfine structure \npossible ensuing from 1H, 14N, 19K, 16S, or 55Mn nuclei. Aforementioned data and discussed \nexperimental results refute suggestions that  ferromagnetism in GNRs arises only from the \nprecipitation of metals. Instead, the data presented in Figs 1 - 5 offer clear evidence for the importance of factors other than the presence of transition-metals in governing the magnetism \nof these ribbons.  \n Information on possible magnetic nuclei (if any) coupled to the paramagnetic centers can \nbe obtained with hyper ﬁne sublevel correlation (HYSCORE) spectroscopy carried out at 12 K \nand at 9.7 GHz on the GNRs. HYSCORE is a high-resolution 2-dimensional pulsed ESR \ntechnique used here to measure small hyper ﬁne (hf) couplings not resolved in the CW ESR \nspectrum [35]. The HYSCORE spectrum shown in Fig.6 allows to identify two nuclear isotopes coupled to the electron spins. There is a strong peak at 14 MHz  which is the NMR \nfrequency of protons, which corresponds to weak ly coupled protons, and a weaker peak at \naround 3.5 MHz , which corresponds to the NMR frequency of \n13C. Figure 6 displays ridges  13\nalong the proton diagonal line with proton hyperfine couplings in the range of 20-28 MHz  \n(~0.890 mT), implying that some radicals ar e coupled to protons. The nonzero isotropic \ncomponent of the hyper ﬁne coupling results from a Fermi c ontact interaction, which indicates \nan electronic connection between the unpaired electron and the proton(s), rather than a pure \nthrough space dipole-dipole interaction. A single carbon centered radical with bonded \nhydrogen atoms like a methyl radical can be ruled out since in this case typically A(1H) ~ 67 \nMHz  (2.4 mT), a hyper ﬁne coupling much bigger than observed in the HYSCORE spectrum \nand would result in an observable splitting in the GNRs CW ESR spectra (line width 0.6 mT), \ncontrary to the present observation. HYSCORE data from Fig.6 reveals proton hyper ﬁne \ncouplings around 25 MHz , a value comparable to those of conjugated aliphatic radicals [36]. \nIt is likely that these hyperfine couplings  are related to protons on the edges of the \nnanoribbons, in line with the characterization data [18] which show the termination of edges with protons as a result of quenching of aryl potassium edges with ethanol.  \n           Similar to the analysis reported [37] on hydrogen absorption in ball milled graphite,  if \nwe assume the ESR signals result from π radicals, then we can use the proton hyperfine \ncouplings to estimate the electron spin density ρ\nπ at the adjacent carbon atoms with the \nMcConnell equation [38], a H(α) =  QHρπ, where QH ~ 2.7 mT, is an intrinsic coupling for unit \ndensity and a H(α) is the proton hyperfine coupling in mT. The hyperfine coupling of ~ 0.89 \nmT from the HYSCORE, thus corresponds to a carbon electron spin density about 33%. \nInterestingly, this is only slightly higher as the predicted [39-41] spin densities of localized edge states in zigzag graphene nanoribbons.       In addition to the proton hyperfine interactions, the HYSCORE \n13C signals (in natural \nabundance) at low frequencies in Fig.6 indicate 13C hyper ﬁne couplings in the range |A(13C)| \n< 4 MHz  (0.14 mT). These small couplings are clearly not in line with the above mentioned \nspin density of 0.33, which would result in much larger 13C couplings [42]. However, these  14\nlarge couplings with large anisotropy coupled with the low 13C natural abundance will be \ndifficult to detect in a HYSCORE experiment on a non-oriented sample. The observed 13C \nhyperfine splitting can be due to 13C nuclei closer to the center of the nanoribbons with \nsignificantly lower spin density [40,42]. From the results of HYSCORE experiments, it is \npossible that the larger width (~8×10-4T) of the ESR signal from GNRs in comparison to that \n(~ 6×10-4T) of the CCGNRs can be due to the protons present in the GNRs. \n        Hence the hyper ﬁne couplings derived from the HYSCORE data show that the \n(unpaired) spin density, and thus wave function of the radical is spread over small graphitic \nmoieties. Most likely the magnetic states are created from the cleavage of graphene sheets of MWCNTs, during the splitting process, resulting in so-called zigzag and armchair edges. \nThere could also be a population of radicals associated with aliphatic structures. These structural types are thought to create nonbonding π-electron states, which accommodate a \nhigh unpaired spin concentration. The large spatia l but still localized wave functions of the \nradicals allow exchange interactions of varying strengths suf ﬁcient to induce ferromagnetic \norder. From our experimental results, we sugg est that the clustering of magnetic fragments \ncomprises of carbon defects coupled with hydrog en could be one of the possible sources of \nthe ferromagnetism. From our study, it is clearly evident that not all graphene-based materials exhibit FM-like features at room temperature.  It depends upon the preparation process and \nedge atoms. In other words, one may in fer that hydrogen termination facilitates \nferromagnetism [1] where as the oxygen termination quenches magnetism [43].           Despite the clear phenomenological conclusions derived from the data in Figs.1 - 6, several important open questions remain. One unresolved issue is the chemical identity of the clusters involving carbon defects coupled with hydrogen. A related question is whether these \nclusters located at the edges of the ribbons or at the centers of the ribbons which cause \nferromagnetism. A third outstanding question pertains to the relationship among T\nC, cluster  15\nsize, distance between two adatoms and defect concentration that would be anticipated from \nmany models [32,23]. Experiments to address th ese important questions are underway.    \n              To conclude, we have investigated the magnetism of GNRs and CCGNRs using the \ncombination of magnetization and electron spin resonance measurements. Our classical MH \nmeasurements show the occurrence of ferromagnetlike features at low temperature for GNRs \nas well as for CCGNRs. However, such room temperature feature persists only for GNRs. \nWhile probing for in-depth understanding on the observed magnetism, our field cooled MH \ndata infer the presence of anti-ferromagnetic  regions mixed with ferromagnetic phase as \nevidenced from prominent exchange bias detected in both types of ribbons. Anti-\nferromagnetic exchange coupling is found to exist between the FM and AFM phases in both \ntypes of ribbons, though GNRs show NEB whereas CCGNRs show PEB. Our electron spin \nresonance data infer that the carbon related localized states may be responsible for the \nobserved magnetism. Furthermore, hyperfine sublevel correlation experiments performed on \nthe former ribbons infer the presence of proton (H1), may play a role in the observed \nmagnetism, though the exact mechanism is far from clearly understood. ESR provides no \nindications for the presence of potassium (clu ster) related signals, emphasizing the intrinsic \nmagnetic nature of the ribbons. From our combin ed experimental findings, it may be inferred \nthat the coexistence of ferromagnetic clusters with anti-ferromagnetic regions leading to \ndisorder magnetism in both types of ribbons . Taking into account the numerous advantages of \nan organic carbon-based ferromagnetic-like magnetism together with semiconducting \nproperties, such materials could be used for spin-based and magnet-electronic applications. \nFurther experimental and theoretical efforts will not only bring us new insights into such \nnovel materials but also pave the way to the new generation of molecule-based magnets. \nAcknowledgements   16\nSSR and SNJ would like to thank KU Leuven, for research fellowship. This work is supported \nby the K.U. Leuven Excellence financing (I NPAC), by the Flemish Methusalem financing \nand by the IAP network of the Belgian Government. The National High Magnetic Field Laboratory is supported by NSF Cooperative Agreement No. DMR-0654118, and by the State \nof Florida.  \nReferences: \n1. D. Soriano, F. Muñoz-Rojas, J. Fernández- Rossier and J. J. Palacios, Phys. Rev.   \n       B 81, 165409 (2010). \n2.O.V. Yazyev, Phys. Rev. Lett. 101, 037203 (2008). \n \n3. L. R. Radovic and B. Bockrath, J. Am. Chem. Soc. 127, 5917 (2005). \n \n4. F. J. Culchac, A. Latgé and A. T. Costa, New Journal of Physics 13, 033028 (2011). \n\n5. H. Feldner, Z. Y. Meng, T. C. Lang, F. F. Assaad, S. Wessel, and A. Honecker, Phys. Rev.    \n      Lett. 106, 226401 (2008). \n6. Y.W Son, M. L. Cohen and S. G. Louie, Nature 444, 347 (2006) \n7. K. Sawada, F. Ishii, M. Saito, S. Okada, and T. Kawai, Nano Lett.  9, 269 (2009). \n \n8. K. Wakabayashi, M. Fujita, H. Ajiki, and M. Sigrist, Phys. Rev.B 59, 8271 (1999); K.   \n     Kusakabe and M. Maruyama, Phys. Rev. B 67, 092406 (2003). \n9. J. \n ervenka, M. I. Katsnelson and C. F. J. Flipse, Nature Physics 5, 840 (2009).  \n10. P. Esquinazi, R. H€ohne, K.-H. Han, A. Setzer, D. Spemann, T. Butz, Carbon 42, 1213   \n       (2004). \n11. Y. Wang, Y. Huang, Y. Song, X. Zhang, Y. Ma, J. Liang, Y.Chen, Nano Lett.  9, 220      \n        (2009). 12. L. Xie, X. Wang, J. Lu, Z. Ni, Z. Luo, H. Mao, R. Wang, Y. Wang, H. Huang, D. Qi, R.  \n      Liu, T. Yu, Z. Shen, T.Wu, H. Peng, B. Özyilmaz, K. Loh, A. T. S. Wee, Ariando and W.   \n      Chen, Appl. Phys. Lett., 98, 193113 (2011); A. L. Friedman, H. Chun, Y. J. Jung, D.   \n       Heiman, E. R. Glaser, and L. Menon, Phys.Rev. B 81, 115461 (2010)  17\n13. C. Tao, L. Jiao, O. V. Yazyev, Y-C Chen, J. Feng, X. Zhang, R. B. Capaz, J. M. Tour, A.   \n       Zettl, S. G. Louie, H.Dai and M. F. Crommie, Nature Physics 7,616 (2011).   14. H. Ohldag, T. Tyliszczak, R. Ho¨hne, D. Spemann, P. Esquinazi, M. Ungureanu, and T.   \n      Butz, Phys. Rev. Lett. 98, 187204 (2007); H Ohldag, P Esquinazi, E Arenholz, D    \n      Spemann, M Rothermel, A Setzer and T Butz, New Journal of Physics 12, 123012 (2010);    \n      J. Berashevich and T. Chakraborty, Nanotechnology 21, 355201 (2010). 15. N. Leconte, D. Soriano, S. Roche, P. Ordejon, J. C. Charlier and J. J. Palacios, ACS Nano.    \n       5, 3987 (2011). \n16. V. L. J. Joly, M. Kiguchi, S. J. Hao, K. Takai, T. Enoki, R. Sumii, K. Amemiya, H.  \n      Muramatsu, T. Hayashi, Y. A. Kim, M. Endo, J. C. Delgado, F. L. Urías, A. B. Méndez,   \n      H. Terrones, M. Terrones and M. S. Dresselhaus, Phys. Rev. B 81, 245428 (2010) 17. S. S. Rao, A. Stesmans, K. Keunen, D. V. Kosynkin, A. Higginbotham, J. M. Tour, Appl.   \n      Phys. Lett. 98, 083116 (2011). \n18. D. V. Kosynkin, W. Lu, A. Sinitskii, G. Pera, Z. Sun, and J. M. Tour, ACS Nano. 5, 968    \n       (2011). \n19. D. V. Kosynkin, A. L. Higginbotham, A. Sinitskii, J. R. Lomeda, A. Dimiev,       B. K. Price, and J. M. Tour, Nature (London) 458, 872 (2009); T. Shimizu, J. Haruyama,  \n     D. C. Marcano, D. V. Kosinkin, J. M. Tour, K. Hirose and K. Suenaga, Nature Nanotech.        249, 1 (2010); A. Sinitskii, A. A. Fursina, D. V. Kosynkin, A. L. Higginbotham, D.           Natelson and J. M.Tour, Appl. Phys. Lett. 95, 253108 (2009). \n20. A. Stesmans, Phys. Rev. B 48, 2418 (1993). 21. D.W. Boukhvalov and M. I. Katsnelson, ACS Nano. 4, 2440 (2011), and references   \n       therein. \n22.  A. V. Rode, E. G. Gamaly, A. G. Christy, J. G. F Gerald, S. T. Hyde, R. G. Elliman, B.    \n         Luther-Davies, A. I. Veinger, J. Androulakis, and J. Giapintzakis, Phys. Rev. B 70,     \n        054407 (2004). \n23. I. C. Gerber, A. V. Krasheninnikov, A. S Foster and R. M. Nieminen, New Journal of   \n       Physics 12, 113021 (2010). 24. Y. G. Semenov, J. M. Zavada, and K. W. Kim, Phys. Rev. B 84, 165435 (2011); M.   \n      Sepioni, R. R. Nair, S. Rablen, J. Narayanan, F. Tuna, R. Winpenny, A. K. Geim and   \n      I. V. Grigorieva, Phys. Rev. Lett., 105, 207205 (2010). \n25.  J Kunstmann, C. O¨ zdog ˘an, A. Quandt and H. Fehske, Phys. Rev. B 83, 045414 (2011).  \n26. A. W. Mombrú, H. Pardo, R. Faccio, O. F. de Lima, E. R. Leite, G. Zanelatto, A. J. C.    18\n       Lanfredi, C. A. Cardoso and F. M. Araújo-Moreira, Phys. Rev. B 71, 100404 (R)   \n(2005), and the references therein. \n27. D. Ar čon, Z. Jagli čič, A. Zorko, A. V. Rode, A. G. Christy, N. R. Madsen, E. G.    \n            Gamaly, and B. Luther-Davies, Phys. Rev. B 74, 014438 (2006). \n28. S. K. Mishra, F. Radu, H. A. Du¨rr, and W. Eberhardt, Phys. Rev. Lett., 102, 177208  \n       (2009) \n29. J. Nogues, D. Lederman, T. J. Moran and I.  K. Schuller, Phys. Rev. Lett., 76, 4624 (1996) \n30. X. Ke, M. S. Rzchowski, L. J. Belenky and C. B. Eom, Appl. Phys. Lett., 84, 5458 (2004);   \n      J. Appl. Phys. 97, 10K115 (2005). \n31. D. Niebieskikwiat and M. B. Salamon, Phys.Rev.B 72, 174422 (2005) 32. L. D.Bianco, D. Fiorani, A. M. Testa,\nE. Bonetti and L. Signorini, Phys. Rev. B 70,  \n         052401 (2004). 33. L. Pisani, B. Montanari and N. M. Harrison\n, New Journal of Physics 10, 033002 (2008). \n34. R.C.Barklie, Diamond and Related Materials 10,174 (2001). \n35. P. Hofer, A. Grupp, H. Nebenfuhr, M. Mehring, Chem. Phys. Lett.  132, 279 (1986). \n \n36. F. Gerson and W. Huber,  Electron Spin Reso nance Spectroscopy of Organic Radicals;    \n        Wiley-VCH: New York, 2003. \n37. C. I. Smith, H. Miyaoka, T. Ichikawa, M. O. Jones, J. Harmer, W. Ishida, P. P.    \n       Edwards, Y. Kojima and H. Fuji, J. Phys. Chem. C 113, 5409 (2009). \n38. H.M. McConnell. J. Chem. Phys . 24, 632 (1956). \n39. O.V. Yazyev, and M.I. Katsnelson, Phys. Rev. Lett., 100, 047209 (2008). \n40. M. Fujita,  K. Wakabayashi, K. Nakada, and K. Kusukabe, J. Phys. Soc. Jap 65,  \n       1920 (1996). \n41. Y.W Son, M. L. Cohen and S. G. Louie, Phys. Rev. Lett.,97, 216803 (2006). \n42. O.V. Yazyev. Nano Lett 8, 1011 (2011). 43. R. G. A. Veiga, R. H. Miwa, and G. P. Srivastava, J. Chem. Phys. 128, \n           201101 (2008). \n \nFigure Captions:  19\nFig. 1: (Color online) Isothermal magneti zation (M-H) observed on GNRs collected from H \nof -5 to + 5T at various temperatures 5, 10, 50, 100, 200 and 300 K (a). The inset of this figure presents the enlarged version of M-H curve. The lower right inset of this figure \nrepresents the Langevin fit (solid curve) to the experimental data using equation (1). \nIsothermal magnetization (M-H) observed on CCGNRs collected from H of -5 to + 5 T at \nvarious temperatures 1.8, 5, 10, and 15 K (b). The top left inset of this figure shows the enlarged version of M-H curve. The lower right inset of this figure represents the Langevin fit (solid curve) to the experimental data using eq uation (1), see text fo r more details. In the \nformer ribbons, a clear ferromagnetic character can be seen even at room temperature (300 K). However, in the later case, a clear M-H open loop is visible at 1.8 K.  \nFig. 2: (Color online) A temperature dependence of zero-field cooled and field-cooled dc \nmagnetic susceptibilities for (a) GNRs (b) CCGNRs. In both cases, for field cooled measurements, the applied magnetic field was 0.1 T.   \nFig. 3\n:(Color online) Isothermal M-H curves collecte d from GNRs (a) and CCGNRs (b) \nunder zero field and field cooled conditions (0.1, 0.3, 0.5 and 1 T; 0.1, 0.2, 0.4, 0.6 T), \nrespectively.  The top left insets are the corresponding enlarged versions inferring the \nnegative exchange bias in the former case (a), and the positive exchange bias in the later case (b). The solid curves in the bottom right insets are resulted from the optimized computer \nfittings using equation (2), see text for more details. \nFig. 4: (Color online)\nVariation of coercive field (H c) as a function of cooling magnetic field \n(HFC) measured at 5 K for (a) GNRs (b) CCGNRs. The curves are the guide to the eye.  \nFig. 5: First derivative CW K-band ESR spectra measured at 4.2 K on (a) GNRs (b) \nCCGNRs, using Bm= 0.42 × 10-4 T and Pµ = 2.5 nW. The signal at g  ≈ 1.99869 stems from a \nco-mounted Si:P marker sample.  20\nFig. 6: (Color online)  An X-band 2D plot of HYSCORE spectrum collected at 12 K on GNRs \nin frequency coordinates \n\n\n\n\n 21\n\n\nFig. 1: (Color online) Isothermal magneti zation (M-H) observed on GNRs collected from H \nof -5 to + 5T at various temperatures 5, 10, 50, 100, 200 and 300 K (a). The inset of this figure shows the enlarged version of M-H curve. The lower right inset of this figure \nrepresents the Langevin fit (solid curve) to the experimental data using equation (1). \nIsothermal magnetization (M-H) observed on CCGNRs collected from H of -5 to + 5 T at various temperatures 1.8, 5, 10, and 15 K (b). The top left inset of this figure shows the enlarged version of M-H curve. The lower right inset of this figure represents the Langevin fit (solid curve) to the experimental data using eq uation (1), see text fo r more details. In the \nformer ribbons, a clear ferromagnetic character can be seen even at room temperature (300 \nK). However, in the later case, a clear M-H open loop is visible at 1.8 K.   22\n \n\nFig. 2: (Color online)  A temperature dependence of zero-field cooled and field-cooled dc \nmagnetic susceptibilities for (a) GNRs (b) CCGNRs. In both cases, for field cooled measurements, the applied magnetic field was 0.1 T.   \n\n 23\n \n\nFig. 3:(Color online) Isothermal M-H curves collected from GNRs (a) and CCGNRs (b) \nunder zero field and field cooled conditions (0.1, 0.3, 0.5 and 1 T; 0.1, 0.2, 0.4, 0.6 T), \nrespectively.  The top left insets are the corresponding enlarged versions inferring the \nnegative exchange bias in the former case (a), and the positive exchange bias in the later case (b). The solid curves in the bottom right insets are resulted from the optimized computer fittings using equation (2), see text for more details.  24\n\n\n\n \nFig. 4: (Color online) Variation of coercive field (H c) as a function of cooling magnetic field \n(HFC) measured at 5 K for (a) GNRs (b) CCGNRs. The curves are guide to the eye.  \n \n \n  25\n \n \nFig. 5: First derivative CW K-band ESR spectra measured at 4.2 K on (a) GNRs (b) \nCCGNRs, using Bm= 0.42 × 10-4 T and Pµ = 2.5 nW. The signal at g  ≈ 1.99869 stems from a \nco-mounted Si:P marker sample. \n         26\n \nFig. 6: (Color online)  An X-band 2D plot of HYSCORE spectrum collected at 12 K on GNRs \nin frequency coordinates \n\n\n\n"    },    {        "title": "1505.07791v1.Driving_and_detecting_ferromagnetic_resonance_in_insulators_with_the_spin_Hall_effect.pdf",        "content": "APS/123-QED\nDriving and detecting ferromagnetic resonance in insulators with the spin\nHall e \u000bect\nJoseph Sklenar,1, 2Wei Zhang,1Matthias B. Jungﬂeisch,1Wanjun Jiang,1Houchen\nChang,3John E. Pearson,1Mingzhong Wu,3John B. Ketterson,2and Axel Ho \u000bmann1\n1Materials Science Division, Argonne National Laboratory, Argonne IL 60439, USA\n2Department of Physics and Astronomy,\nNorthwestern University, Evanston IL 60208, USA\n3Department of Physics, Colorado State University, Fort Collins CO 80523, USA\n(Dated: November 12, 2021)\nAbstract\nWe demonstrate the generation and detection of spin-torque ferromagnetic resonance in Pt /Y3Fe5O12\n(YIG) bilayers. A unique attribute of this system is that the spin Hall e \u000bect lies at the heart of both the\ngeneration and detection processes and no charge current is passing through the insulating magnetic layer.\nWhen the YIG undergoes resonance, a dc voltage is detected longitudinally along the Pt that can be de-\nscribed by two components. One is the mixing of the spin Hall magnetoresistance with the microwave\ncurrent. The other results from spin pumping into the Pt being converted to a dc current through the inverse\nspin Hall e \u000bect. The voltage is measured with applied magnetic ﬁeld directions that range in-plane to nearly\nperpendicular. We ﬁnd that for magnetic ﬁelds that are mostly out-of-plane, an imaginary component of the\nspin mixing conductance is required to model our data.\n1arXiv:1505.07791v1  [cond-mat.mes-hall]  28 May 2015Magnetic insulators such as Y 3Fe5O12(YIG) with extremely low magnetic damping serve as\npromising platforms for low power data transmission [1–4]. In YIG /Pt bilayers the groundbreak-\ning discovery of magnetization dynamics generated by spin orbit torques of Pt contacts [5] opens\nup new opportunities for device concepts combining electronic, spintronic, and magnonic ap-\nproaches. The spin orbit torques in heavy metals arise from the spin Hall e \u000bect (SHE) [6, 7],\nwhich converts a charge current, Jc, to a spin current, Js, with a conversion e \u000eciency dictated by\na materials speciﬁc parameter, i.e., the spin Hall angle, \u0002S H[8]. The resultant spin current can\ndrive spin-torque ferromagnetic resonance (ST-FMR) in bilayer thin ﬁlms made from metallic fer-\nromagnets and nonmagnetic metals [9]. In such experiments, FMR is driven by the simultaneous\nOersted ﬁeld and oscillating transverse spin current (spin-torque) transformed by SHE from the\nalternating charge current. Electrical detection is made possible via the spin-torque diode e \u000bect\n[11], i.e., the rectiﬁcation of the time dependent bilayer resistance arising from the anisotropic\nmagnetoresistance (AMR) of the ferromagnet [13, 14]. However, such a detection scenario is not\npossible in magnetic insulators due to missing free electrons coupling to magnetic moments and,\nthus, the absence of AMR.\nIn this Letter, we show experimentally that the SHE of a paramagnetic metal can be used for\nboth excitation and detection of ST-FMR for magnetic insulators. We demonstrate magnetiza-\ntion dynamics of a thin YIG layer induced by spin-torque from an adjacent Pt layer, as well as\nsubsequent detection of a dc voltage via the spin-torque diode e \u000bect generated by the anisotropic\nspin Hall magnetoresistance (SMR) of the Pt [12, 14–17]. It bears mentioning that the anisotropic\nresistance of metal ﬁlms on top of ferromagnetic insulators, and interface e \u000bects in general [18],\nare a very active topic and other mechanisms independent of the SHE such as interface proximity\ne\u000bects [19] and interfacial Rashba e \u000bects [20] are being explored as contributors. In this work,\nSMR refers to the dependence of the electrical resistance of the metal on the magnetization direc-\ntion of an adjacent magnetic insulator and is a result of a simultaneous operation of the SHE and\nits inverse (ISHE) as a nonequilibrium phenomenon. Microscopically, this anisotropic behavior\norginates from the dependence of the spin accumulations of conduction electrons at the YIG /Pt\ninterface on the static YIG magnetization. For example, if the static magnetization is aligned with\nthe spin current’s polarization at the interface there is a large backﬂow [21] spin current; on the\nother hand, if the magnetization is orthogonal to the polarization a spin current is absorbed at\ninterface, and consequently the interfacial spin accumulation is reduced.\nModels of spin transport at the YIG /Pt interface that exclude proximity e \u000bects introduce the\n2spin mixing conductance, G\"#, to describe both the magnitude and phase of the interface spin\ncurrent [22]. This concept has been probed in a comprehensive study [23] involving a suite exper-\niments such as spin pumping [24, 25], spin Seebeck detection [26], and SMR measurements. It has\nalso been shown that the value of G\"#for a YIG /Pt interface is heavily dependent on sample fab-\nrication and processing [27]. In these works the spin mixing conductance is typically described as\nbeing purely real. However, for YIG /Pt bilayers it has been theoretically suggested that a non-zero\nvalue of Im( G\"#) should be considered [28]. Furthermore, very recent experiments investigating\nan anamolous spin Hall e \u000bect in Pt have provided evidence for a non-zero Im( G\"#) at the YIG /Pt\ninterface [29]. Here, we will present evidence that for ST-FMR experiments where the magnetic\nﬁeld is tipped out-of-plane (OOP) a non-zero Im( G\"#) is required and evolves as a function of the\nOOP angle.\nWe fabricated YIG(40 nm) /Pt(6 nm) bilayers by in-situ magnetron sputtering on single crystal\ngadolinium gallium garnet (GGG, Gd 3Ga5O12) substrates of 500 \u0016m thickness with [111] orien-\ntation under high-purity argon atomsphere [3, 30]. The bilayers were subsequently patterned into\nmicrostripes in the shape of 500 \u0016m\u0002100\u0016m by photolithography and liquid nitrogen cooled ion\nmilling to remove all the YIG /Pt materials except for the bar structure. In a last fabrication step,\nsquare contact pads made of Ti /Au (3 nm /120 nm) are patterned on top each end of the YIG /Pt\nstripe via photolithography and lift-o \u000b. We conﬁgured our set-up into a ST-FMR scheme that is\nillustrated in Fig. 1 (a). A bias-tee is utilized to allow for simultaneous transmission of microwaves\nas well as dc voltage detection across the Pt. We modulate the amplitude of the microwave current\nat 4 kHz so that the ST-FMR dc signal is detected via a lock-in ampliﬁer to improve signal to\nnoise.\nThe coordinate system that we will reference throughout this work is shown in Fig. 1 (a). The\nangle\u001eis in-plane and lies between the x and y axis. For our experiments \u001ewas always set to 45\u000e.\nThe polar angle \u0012describes the applied magnetic ﬁeld direction OOP, while the polar angle  is\nthe calculated OOP component of the magnetization. Due to geometrical demagnetization ﬁelds\n >\u0012 ; for a given \u0012and applied magnetic ﬁeld  is determined from the following expression:\n2\u0019Me f fsin 2 csc( \u0000\u0012)\u0000Hex=0; (1)\nwhere Me f fis the e \u000bective magnetization of the YIG and Hexis the externally applied magnetic\nﬁeld.\nTo induce ST-FMR in the YIG we passed a ﬁxed 5.5 GHz signal through the Pt while sweeping\n3θ\nΨ\nΦ\nxyz\nH\nM\nV\nYIGPt\n11.52.02.53.03.54.0Field (kG)\nθ (deg.) 10 30 50 70 90\n0 -15 15\nΔH (Oe)4πMeff= 1633 G(a)\n(c)(b)\nθ = 90o\nθ = 5oFIG. 1. A schematic of the bilayer and ST-FMR set-up is shown in (a). In the diagram Hindicates an\nexperimentally applied ﬁeld, and Mindicates the magnetization vector. \u0012describes the tipping of Hfrom\nthe z-axis (thickness direction) and  describes the tipping of Min the same manner. \u001eis an in-plane angle\nbetween the x and y axis; in all our experiments \u001e=45\u000e. (b) ST-FMR traces measured over a range of \u0012\nthat spans from 90\u000e- 5\u000ein 5\u000esteps. In order to show every resonance we plot each resonance centered on\nzero ﬁeld. (c) shows the \u0012dependence of the ST-FMR experiments ﬁt to Eq. (4). 4 \u0019Me f fis extracted from\nthis data set to be 1633 G.\nHexat a ﬁxed\u001eand\u0012. The nominal microwave power level was set to be 10 dBm. The dynamic\nresponse of the system is governed by a modifed LLG equation of motion [28]:\ndˆM\ndt=\u0000j\rjˆM\u0002He f f+\u000b\u000eˆM\u0002dˆM\ndt+j\rj~Js\n2eM sdF; (2)\nwhere He f fincludes the Oersted ﬁeld, Hac, demagnetization ﬁelds, and the applied external dc\nﬁeld Hex. Additional quantities of importance are the intrinsic damping, \u000b\u000eand the spin current at\nthe interface,\nJs=Re(G\"#)\neˆM\u0002(ˆM\u0002\u0016s)+Im(G\"#)\neˆM\u0002\u0016s (3)\nthat originates from the SHE in Pt. Here G\"#is the spin mixing conductance and \u0016sis the spin\naccumulation at the YIG /Pt interface. The oscillatory torque terms that drive the magnetization\nare the ﬁeld from the microwave current in H e f fand the spin torque term that includes Js. The\nangular range that \u0012covered over the course of our experiment was 5\u000e- 90\u000ein steps of 5\u000e. Figure 1\n4(b) plots every trace that was observed over the measureable angular range of \u0012. The OOP ﬁeld\ndependence of the resonances shown in (b) is plotted in Fig. 1 (c). In order to extract the e \u000bective\nsaturation magnetization of our YIG we ﬁt (Fig. 1 (c)) the out-of-plane angular dependence to the\ngeneralized Kittel equation that is given by:\nf=j\rj\n2\u0019q\nH2+4\u0019Me f f(H(sin\u0012sin \u00002 cos\u0012cos )+4\u0019Me f fcos2 ); (4)\nwhere\ris the gyromagnetic ratio taken as 2.8 GHz /kOe. The extracted e \u000bective magnetization is\n4\u0019Me f f=1633 G. We note that this Kittel-like analysis does not account for magnetocrystalline\nanisotropy or exchange energy. For comparison, in a separate work involving the study of spin\nwaves in other thin YIG ﬁlms we measured 4 \u0019Me f f=1553 G [31].\nTo explain our experimental observations, we employ a theory developed by Chiba et. al.\n[28, 32]. Qualitatively, this model desribes a dc voltage that develops longitudinally along the Pt\nﬁlm when a microwave charge current ﬂowing through the Pt induces ferromagnetic resonance in\nthe YIG. There are two di \u000berent contributions to the observed voltage: ﬁrst, there is an analog to\nwhat is observed for Py /Pt bilayers where AMR of the Py mixes with the microwaves to generate\na dc voltage at and near the FMR condition [9]. For YIG /Pt the magnetoresistance resides in the\nPt and is the SMR [12, 15, 16]. Additionally, spin pumping at the YIG /Pt interface can inject a\nspin current into the Pt that can be converted to a dc charge current via the ISHE.\nThe theoretical model [28, 32] predicts that the voltage generated by spin pumping has a purely\nsymmetric lineshape about the resonance condition, and that the voltage induced by SMR also has\na symmetric contribution. Furthermore, the SMR contribution has an antisymmetric contribution\nto the lineshape as well. This model [32] was recently expanded to include a non-zero imaginary\npart of G\"#, a phase shift parameter, \u000e, between the charge current JcandHac, and an OOP applied\ndc Oersted ﬁeld [28]. \u000eshould be considered to be a property of a given device and, for a ﬁxed\nexcitation frequency, should be constant. The addition of the non-zero imaginary part of G\"#along\nwith the phase shift parameter \u000eallows for additional tunability in the net amplitude of both the\nantisymmetric as well as the symmetric contribution to the lineshape.\nAccording to theory, the lineshapes of a ST-FMR experiment for a YIG /Pt bilayer have the\n5following functional forms [28]:\nVS MR=[S1FS(Hex)+A1FA(Hex)] cos\u001esin 2\u001esin\u0012\n\u0000[S2FS(Hex)+A2FA(Hex)] sin3\u001ecos\u0012sin 2\u0012\n+A3sin\u001esin 2\u001esin 2\u0012(5)\nVS P=S3cos\u001esin 2\u001esin\u0012+S4sin3\u001ecos\u0012sin 2\u0012\n+S5sin\u001esin 2\u001esin 2\u0012;(6)\nwhere VS MRarises from SMR and VS Pis from spin pumping. FS(Hex) is the ﬁeld dependent sym-\nmetric lineshape that is given by \u00012=[(Hex\u0000HFMR)2cos2(\u0012\u0000 )+\u00012].FA(Hex) is an antisymmetric\nlineshape that is given by FS(Hex) cos(\u0012\u0000 )(Hex\u0000HFMR)=\u0001. In these equations \u0001is the linewidth\nof the lineshape and HFMRis the ﬁeld under which FMR occurs, which can be obtained from in-\nverting Eq. (3). S1\u0000S2, and A1\u0000A3are coe \u000ecients that rely on the mixing of the oscillatory SMR\nwith the charge current, and all end up being proportional to J2\nc; the other relevant parameters such\nas\u0002S H, G\"#,\u000e,Me f f,dN, and dF, are imbedded within these coe \u000eencients [28]. Two other param-\neters not yet mentioned are contained within these coe \u000ecients; they are the Pt resistivity \u001a, and\nthe spin di \u000busion length \u0015. In our analysis we use \u0015=1.2 nm; this value was determined for Pt by\nspin pumping experiments in Py /Pt bilayers [34]. S3\u0000S5are spin pumping coe \u000ecients that are\nsimilarily proportional to J2\ncand depend on the same quantities listed above for the SMR terms.\nComplete expressions for these coe \u000ecients can be found elsewhere [28].\nIn our analysis there are three ﬁtting parameters assumed to be independent of \u0012:\u0002S H,Jc, and\n\u000e. We did not directly assume that the magnitude or complex composition of G\"#was independent\nof\u0012. Because we have previously measured the \u0002S Hof Pt to be 0.09 we analyze our data with this\nvalue in mind [34]. In other ST-FMR experiments the paramater \u000ehas been assumed to be zero,\ntherefore we will begin our discussion by following this example [9, 10]. This leaves us with ﬁxing\nthe magnitude of Jc. Because the magnitude of G\"#is free we found various values of Jccould be\nused with reasonable G\"#counterparts. In fact, these two parameters are strongly anti-correlated.\nHowever, we found that a given Jcdoes not ensure that the magnitude of G\"#remains constant\nover all\u0012. We typically see an increase in the magnitude of G\"#as the ﬁeld is tipped OOP. The\nvalue of Jc(9\u0002108A=m2) chosen here minimized the variation of G\"#over\u0012which then stays\nwithin 10% of a mean value of 2.44 \u00021014\n\u00001m2.\n610203040506070809011.522.533.5|G  |\nδ = 0o\nδ = 52o\n(I) (II)\nRe(G  )\nIm(G  )\n3\n10 20 30 40 50 60 70 80 90\n10 20 30 40 50 60 70 80 9000.71.42.12.83.5\n00.71.42.12.83.5\nRe(G  )\nIm(G  )θ (deg.) θ (deg.)\nθ (deg.)\nG  G  (a) (b)\n(c)x1014x1014\nx1014(Ω-1m2)\n(Ω-1m2)(Ω-1m2)FIG. 2. The results of the \u0012dependence on both the real and imaginary components of the spin mixing\nconductance are shown above. In (a) jG\"#jis plotted as a function of \u0012for two di \u000berent assumed values of\n\u000e. The circles represent \u000e=0\u000eand the squares represent \u000e=52\u000e. In (b) the real and imaginary components\nofG\"#are plotted as a function of \u0012for\u000e=0\u000e. In (c) the real and imaginary components are plotted for \u000e=\n52\u000e.\nWith \u0002S H,Jc, and\u000eﬁxed we proceeded to investigate the magnitude and complex behavior\nofG\"#as a function of \u0012. Fig. 2 (a) shows the \u0012dependence for our ﬁrst set of assumptions as\ncircles. The complex behavior of G\"#is plotted in Fig 2 (b) where the Re( G\"#) is indicated as\nsquares and the Im( G\"#) is shown as circles. Here, one sees that the composition of G\"#is purely\nimaginary from \u0012=35\u000e- 90\u000e. This region is indicated as IIin the plot. For small values of \u0012(\n<35\u000e) the composition begins to ﬂucuate. This region is indicated with a Iand is shaded blue in\nFig. 2. As seen in Fig. 2 (b), for the smallest values of \u0012,G\"#settles on having real and imaginary\ncomponents with similar magnitude.\nPreviously reported experiments, where the applied magnetic ﬁeld is in-plane, report that G\"#\nis mainly real, which is not consistent with our analysis. A possible explanation may involve the\nparameter\u000e. In fact,\u000ehas been used in a similar ST-FMR experiment where the in-plane ﬁeld\nconﬁguration and a near out-of-plane measurement was performed while G\"#was assumed to be\nreal [33]. If we allow \u000eto vary we ﬁnd that for a value of \u000e=52\u000ewe had a local maximum in\n7the ratio of Re( G\"#)/jG\"#j, at\u0012=90\u000e, as a function of \u000e. Therefore, we believe that a large phase\nshift between the microwave current and the microwave ﬁeld exists making the analysis with a\nnon-zero\u000emore appropriate. With this new value of \u000e, and with the same value of Jcand\u0002S Has\nbefore, we performed again the \u0012dependent analysis. The dependence that G\"#has on\u0012with this\nnon-zero\u000eis shown in ﬁg. 2 (b) plotted as squares. Fig. 2 (c) shows the complex composition of\nG\"#for this non-zero \u000e. In contrast to before, for region II,G\"#is mostly real with little ﬂucuation\nin the angular range \u0012=35\u000e- 90\u000e. However this behavior does not persist; we again we see that in\nregion I, where the ﬁeld approaches a OOP conﬁguration, both the real and imaginary part of G\"#\nbecome appreciably non-zero.\nOne conclusion from the above discussion is that the parameter space used in ﬁtting ST-FMR\nlineshapes in a YIG-Pt bilayer is not well enough constrained. To illustrate this point we show the\nmodel’s ﬂexibility in Fig. 3. Here, we have plotted the model predictions directly on top of the\ndata for both the zero and non-zero \u000eanalysis and we have also chosen representative traces from\nboth region Iand region II. What does emerge is that independent of the assumptions used, for\u0012\n<35\u000eboth a real and imaginary component of G\"#are needed to ﬁt that data. Before summarizing\nwe note that we analyzed our data under di \u000berent assumed values of \u0002S H(not shown). Smaller\nassumed values of \u0002S Hrequire smaller values of \u000eto make G\"#mainly real at \u0012=90\u000e. Near \u0002S H=\n0.06 no\u000eis required. Regardless, we see the same ﬂucuating behavior of the complex composition\nofG\"#for small values of \u0012.\nThe ST-FMR paradigm has been studied with great intensity for spin Hall metal /ferromagnetic\nbilayers where the ferromagnet is a conductor. The present work shows that it can be successfully\nextended to insulating FM materials. Furthermore, it is clear that in addition to an Oersted mi-\ncrowave ﬁeld torque from the Pt strip line, an additional spin torque from spin accumulation at the\nPt/YIG drives the dynamics as well. This particular conclusion is bolstered by a good agreement\nwith theory that includes such spin torques. A very interesting property of bilayers with ferromag-\nnetic insulators such as YIG is that the longitudinal voltage generated along the Pt when ST-FMR\nis taking place is created by e \u000bects that all trace their origin back to the SHE. These detection\nmechanisms set this work apart from metallic ferromagnets where mixing of the microwave cur-\nrent with the AMR of the ferromagnet itself leads to a measurable voltage. In this work we have\nalso have realized a recently proposed model [28] that describes ST-FMR voltages in YIG /Pt bi-\nlayers. We highlight that in order to adequately model our data over the full angular range, the\nvalue of Im( G\"#) was found to be an appreciable quantity for applied magnetic ﬁelds where the\n81.29 1.3 1.31 1.32 1.33\n2.79 2.8 2.81 2.82 2.83 2.84 2.85 2.86\n1.29 1.3 1.31 1.32 1.33\n2.79 2.8 2.81 2.82 2.83 2.84 2.85 2.86Field (kOe) Field (kOe)\nField (kOe) Field (kOe)Signal (nV)\nSP\nSMR\nSP+SMR\nExperiment\nθ = 90o\nδ = 0oθ = 90o\nδ = 52o\nθ = 20o\nδ = 0oθ = 20o\nδ = 52o(a) (b)\n(c) (d)020406080100120\n-20\nSignal (nV)\n-80-4004080120160\n-20020406080100\n020406080100\n-20120\n-40Signal (nV)Signal (nV)\n1.29 1.3 1.31 1.32 1.33\n2.79 2.8 2.81 2.82 2.83 2.84 2.85 2.86\n1.29 1.3 1.31 1.32 1.33\n2.79 2.8 2.81 2.82 2.83 2.84 2.85 2.86Field (kOe) Field (kOe)\nField (kOe) Field (kOe)Signal (nV)\nSP\nSMR\nSP+SMR\nExperiment\nθ = 90o\nδ = 0oθ = 90o\nδ = 52o\nθ = 20o\nδ = 0oθ = 20o\nδ = 52o(a) (b)\n(c) (d)020406080100120\n-20\nSignal (nV)\n-80-4004080120160\n-20020406080100\n020406080100\n-20120\n-40Signal (nV)Signal (nV)\nFIG. 3. Representative ﬁts of the ST-FMR data for both zero and non-zero values of \u000e. Additionally, we\nshow ﬁts to the data for two di \u000berent angles, \u0012=90\u000eand\u0012=20\u000e. These two angles each represent data\nacquired from regions IandIIin ﬁg. 2. The black data points are densely packed together. The total\ntheoretical ﬁt is plotted in red, while the two contributions to the total, spin pumping and SMR, are plotted\nin blue and green respectively.\nmagnetization is sizably tipped OOP.\nWe acknowledge Stephen Wu for assistance with ion-milling used for sample prepartation. The\nwork at Argonne was supported by the U.S. Department of Energy, O \u000ece of Science, Materials\nScience and Engineering Division. Lithography was carried out at the Center for Nanoscale Ma-\nterials, which is supported by DOE, O \u000ece of Science, Basic Energy Science under Contract No.\nDE-AC02-06CH11357. Work at Northwestern utilized facilities maintained by the NSF supported\nNorthwestern Materials Research Center under contract number DMR-1121262. The work at Col-\norado State University was supported by the U. S. Army Research O \u000ece (W911NF-14-1-0501),\nthe U. S. National Science Foundation (ECCS-1231598), C-SPIN (one of the SRC STARnet Cen-\nters sponsored by MARCO and DARPA), and the U. S. Department of Energy (DE-SC0012670).\n[1]Recent Advances in Magnetic Insulators - From Spintronics to Microwave Applications , edited by M.\nWu and A. Ho \u000bmann, Solid State Physics 64, (Academic Press, 2013).\n9[2] H. L. Wang, C. H. Du, P. C. Hammel, and F. Y . Yang, Phys. Rev. B 89, 134404 (2014).\n[3] H. Chang, P. Li, W. Zhang, T. Liu, A. Ho \u000bmann, A. Deng, and M. Wu, IEEE Magn. Lett. 5, 6700104\n(2014).\n[4] P. Pirro, T. Br ¨acher, A.V . Chumak, B. L ¨agel, C. Dubs, O. Surzhenko, P. G ¨ornert, B. Leven and B.\nHillebrands, Appl. Phys. Lett. 104, 012402 (2014).\n[5] Y . Kajiwara, et al., Nature 464, 262 (2010).\n[6] M. I. D‘yakonov and V . I. Perel, Sov. Phys. JETP Lett. 13, 467 (1971).\n[7] J. E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999).\n[8] A. Ho \u000bmann, IEEE Trans. Magn. 49, 5172 (2013).\n[9] L. Liu, T. Moriyama, D. C. Ralph, and R. A. Buhrman, Phys. Rev. Lett. 106, 036601 (2011).\n[10] A. R. Mellnik et al., Nature 511, 449 (2014).\n[11] J. C. Sankey, P. M. Braganca, A. G. F. Garcia, I. N. Krivorotov, R. A. Buhrman, and D. C. Ralph,\nPhys. Rev. Lett. 96, 227601 (2006).\n[12] H. Nakayama, et al., Phys. Rev. Lett. 110, 206601 (2013).\n[13] H. J. Juretschke, J. Appl. Phys. 31, 1401 (1960).\n[14] M. I. Dyakonov, Phys. Rev. Lett. 99, 126601 (2007).\n[15] Y . -T. Chen, S. Takahashi, H. Nakayama, M. Althammer, S. T. B. Goennenwein, E. Saitoh, and G. E.\nW. Bauer, Phys. Rev. B 87, 144411 (2013).\n[16] M. Althammer, et al., Phys. Rev. B 87, 224401 (2013).\n[17] C. Hahn, G. de Loubens, O. Klein, M. Viret, V . V . Naletov, and J. Ben Youssef, Phys. Rev. B 87,\n174417 (2013).\n[18] J. C. Rojas-Sanchez, et al., Phys Rev. Lett. 112, 106602 (2014).\n[19] B. F. Miao, S. Y . Huang, D. Qu, and C. L. Chien, Phys. Rev. Lett. 112, 236601 (2014).\n[20] V . L. Grigoryan, W. Guo, G. E. W. Bauer, and J. Xiao, Phys. Rev. B 90, 161412 (2014).\n[21] H. Jiao and G. E. W. Bauer, Rev. Lett. 110, 217602 (2013).\n[22] Y . Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. Lett. 88, 117601 (2002).\n[23] M. Weiler, M. Althammer, M. Schreier, J. Lotze, M. Pernpeintner, S. Meyer, H. Huebl, R. Gross, A.\nKamra, J. Xiao, Y . T. Chen, H. J. Jiao, G. E. W. Bauer, and S.T.B. Goennenwein, Phys. Rev. Lett. 111,\n176601 (2013).\n[24] O. Mosendz, V . Vlaminck, J. E. Pearson, F. Y . Fradin, G. E. W. Bauer, S. D. Bader, and A. Ho \u000bmann,\nPhys. Rev. B 82, 214403 (2010).\n10[25] L. Bai, P. Hyde, Y . S. Gui, C.-M. Hu, V . Vlaminck, J. E. Pearson, S. D. Bader, and A. Ho \u000bmann, Phys.\nRev. Lett. 111, 217602 (2013).\n[26] T. Kikkawa, et al., Phys. Rev. Lett. 110, 067207 (2013).\n[27] M. B. Jungﬂeisch, V . Lauer, R. Neb, A. V . Chumak, and B. Hillebrands, Appl. Phys. Lett. 103, 022411\n(2013).\n[28] T. Chiba, M. Schreier, G. E. W. Bauer, and S. Takahashi, ArXiv e-prints (2014), arXiv:1409.8406\n[cond-mat.meshall].\n[29] S. Meyer, R. Schlitz, S. Gepr ¨ags, M. Opel, H. Huebl, R. Gross, and S. T. B. Goennenwein, Appl. Phys.\nLett.106, 132402 (2015).\n[30] T. Liu, H. Chang, V . Vlaminck, Y . Sun, M. Kabatek, A. Ho \u000bmann, L. Deng, and M. Wu, J. Appl.\nPhys. 115, 17A501 (2014).\n[31] M. B. Jungﬂeisch, et al., J. Appl. Phys. 117, 17D128 (2015).\n[32] T. Chiba, G. E. W. Bauer, and S. Takahashi, Phys. Rev. Applied 2, 034003 (2014).\n[33] M. Schreier, T. Chiba, A. Niedermayr, J. Lotze, H. Huebl, S. Gepr ¨ags, S. Takahashi, G. E. W. Bauer,\nR. Gross, and S. T. B. Goennenwein, ArXiv e-prints (2014), arXiv:1412.7460 [cond-mat.mes-hall].\n[34] W. Zhang, V . Vlaminck, J. E. Pearson, R. Divan, S. D. Bader and A. Ho \u000bmann, Appl. Phys. Lett. 103,\n242414 (2013).\n11"    },    {        "title": "2204.08500v1.Influence_of_the_presence_of_multiple_resonances_on_material_parameter_determination_using_broadband_ferromagnetic_resonance_spectroscopy.pdf",        "content": "In\ruence of the presence of multiple resonances on material\nparameter determination using broadband ferromagnetic\nresonance spectroscopy\nPrabandha Nakarmi1and Tim Mewes1,\u0003\n1Department of Physics and Astronomy,\nThe University of Alabama, Tuscaloosa, Alabama 35487 , USA\n(Dated: April 20, 2022)\nAbstract\nThe in\ruence of the presence of multiple resonances in ferromagnetic resonance spectra on ex-\ntracted material parameters is investigated using numerical simulations. Our results show that\nthe systematic error of assuming an incorrect number of resonances for a material can lead to\nthe extraction of material parameters that signi\fcantly deviate from any of the true material pa-\nrameters. When noise is present in experimental ferromagnetic resonance spectra increasing the\nfrequency range of the broadband characterization can signi\fcantly reduce the error-margins when\nthe data is analyzed assuming the correct number of resonances present in the material. For the\ncases investigated in this study it was found that analyzing the data using a single resonance results\nin extracted gyromagnetic ratios and e\u000bective magnetization parameters that are consistent with\nthe weighted average of the true material parameters. We further provide a cautionary example\nregarding the extraction of the inhomogeneous linewidth broadening and damping parameters of\nmaterials that contain an unknown number of resonances.\n1arXiv:2204.08500v1  [cond-mat.mtrl-sci]  18 Apr 2022I. INTRODUCTION\nFerromagnetic resonance spectroscopy (FMR) is a well established characterization\nmethod for magnetic materials1{4. While early experimental techniques were based on res-\nonant cavities operated at a single frequency, recent advancements have led to broadband\ncapabilities based on the use of coplanar waveguides5{10, vector network analyzers7,11{17, and\nelectrically detected FMR18{24. Independent of the experimental details used to carry out\nthese measurements ferromagnetic resonance spectroscopy can provide important informa-\ntion about magnetic material properties including the gyromagnetic ratio \r0and the e\u000bective\nmagnetization Me\u000bof the sample under investigation. This is based on the condition for\nferromagnetic resonance in a magnetic material that links the resonance frequency fwith\nthe free energy density eof the system, which according to Smit and Beljers25can be written\nas:25,26:\u0012f\n\r0\n0\u00132\n=1\nM2\nssin2\u0012\"\n@2e\n@\u00122@2e\n@\u001e2\u0000\u0012@2e\n@\u0012@\u001e\u00132#\n; (1)\nwhere\r0\n0=\u00160j\r0jis the reduced gyromagnetic ratio \r0=\r\n2\u0019rescaled by the vacuum perme-\nability\u00160,Msis the saturation magnetization, \u0012the polar angle, and \u001ethe azimuthal angle\nof the magnetization. The general expression for the resonance condition given by equation\n(1) can be simpli\fed when the sample is saturated along a high symmetry direction. In\nthe case of a thin \flm one can show that if the sample is saturated along the out-of-plane\ndirection that the resonance condition is given by27,28:\nf=\r0\n0(H\u0000Me\u000b): (2)\nHere we have introduced the e\u000bective magnetization Me\u000b=Ms\u00002Ku\n\u00160MSthat takes into ac-\ncount the potential presence of a perpendicular anisotropy Ku. The perpendicular anisotropy\nhas the same functional dependence on the polar angle \u0012as the shape anisotropy and there-\nfore cannot be distinguished from it using only FMR. Equation (2) shows that in this\nparticular measurement geometry the resonance frequency of the system depends linearly\non the applied \feld with the slope given by the gyromagnetic ratio, just like in electron spin\nresonance (ESR)29. The only modi\fcation compared to ESR is the presence of the internal\n\feld characterized by Me\u000b. Furthermore, equation (2) clearly shows that in this situation\nfor a given microwave frequency fthe resonance condition is met for exactly one \feld value,\nthe resonance \feld H=Hres.\n2Figure. 1. Experimental ferromagnetic resonance spectra for an M-type hexaferrite41measured with the\nmagnetic \feld applied along the out-of-plane direction over a microwave frequency range from from\n59 [GHz ] to 64 [GHz ] with spectra recorded in 0 :5 [GHz ] intervals.\nHowever, throughout the history of ferromagnetic resonance spectroscopy there have been\nnumerous examples in the literature showing multiple resonances30{40. The reasons given\nin the literature for the existence of multiple resonances are as diverse as the sam-\nples that have been characterized using FMR. They include for example compositional\nvariations30{32, di\u000berent anisotropies33, anisotropy variation across the \flm thickness34,35,\nunsaturated samples36, edge modes37,38, vortex modes38, coupling of multilayers39, and spin\nwave resonances40.\nTherefore when faced with multiple resonances in FMR experiments it can be challenging\nto pinpoint the origin of those resonances or to predict how many resonances are expected\nfor a particular sample. Figure 1 shows an example of experimental spectra featuring multi-\nple resonances in an M-type hexaferrite sample41measured using broadband ferromagnetic\nresonance spectroscopy4.\nIn order to obtain insights regarding the in\ruence of assumptions made during the data\n3analysis process of spectra featuring an unknown number of resonances, we will focus in this\narticle on numerical simulations. For the simulations we assume the presence of multiple\nresonances in a material with a strong perpendicular anisotropy. All resonances are assumed\nto ful\fll equation (2) albeit with slightly di\u000berent material parameters. The simulated data\nis subsequently analyzed just like one would analyze experimental ferromagnetic resonance\nspectroscopy data. As the number of resonances that contribute to an experimental res-\nonance spectrum is not known a priori, we investigate in detail the in\ruence of assuming\ndi\u000berent numbers of resonances on the results of the data analysis.\nII. NUMERICAL TECHNIQUES\nOur simulations are aimed at reproducing data as it would be obtained experimentally\nin coplanar waveguide based broadband ferromagnetic resonance spectroscopy4,42. In our\nsimulations we assume that the material contains four di\u000berent constituents with slightly\ndi\u000berent material properties. For each constituent k, withk=f1;:::;4g, we de\fne a separate\ngyromagnetic ratio \r0\nk, e\u000bective magnetization Me\u000b,k, Gilbert damping parameter \u000bk, and\ninhomogeneous linewidth broadening \u0001 H0;k. For many broadband ferromagnetic resonance\nspectroscopy experimental realizations it is common to record spectra at a \fxed frequency\nwhile sweeping the \feld. In addition, \feld modulation with lock-in detection is frequently\nused to improve the signal-to-noise ratio43,44. We therefore simulate \feld-swept spectra at\ndi\u000berent microwave frequencies f. The simulations assume a Lorentzian lineshape of the\nimaginary part of the magnetic susceptibility \u001f00\nkfor all four constituents of the material.\nFor su\u000eciently small \feld modulation the lock-in detection results in a recorded FMR signal\nproportional to the \frst derivative @\u001f00\nk=@H of the imaginary part of the susceptibility with\nrespect to the \feld H. Generally the \feld dependent FMR signal SFMR,k (H) contribution\nfrom constituent kmeasured in a spectrometer can be written as follows29,45:\nSFMR,k (H) =Ak\u0010\nHres;k\u0000H\n\u0001Hk=2\u0011\n+ 9Bk\u00003Bk\u0010\nHres;k\u0000H\n\u0001Hk=2\u00112\n\u0014\n3 +\u0010\nHres;k\u0000H\n\u0001Hk=2\u00112\u00152; (3)\nwhereAkandBkare the amplitudes of the absorption and dispersion signal contributions\nto the measured signal. While ideally a spectrometer would only measure the absorption\npart of the signal, it is common to observe a mixture of both contributions experimentally45.\n4However, for simplicity we assume for the simulations that Bk= 0 fork=f1;:::;4g, i.e. we\nassume that there is no dispersion contribution to the FMR signals. The other parameter\nthat enters equation (3) is the frequency dependent peak-to-peak linewidth \u0001 Hk, which we\nassume can also be di\u000berent for each constituent k. In our simulations we assume that the\npeak-to-peak linewidth is caused by a Gilbert-like damping term in the Landau-Lifshitz-\nGilbert equation of motion. In this case the frequency dependence of the linewidth is given\nby:\n\u0001Hk(f) = \u0001H0;k+2p\n3\u000bk\n\r0\n0;kf: (4)\nThis enables us to calculate the total signal SFMR(H) for any choice of microwave frequency\nfby summing the responses of the four di\u000berent constituents:\nSFMR(H) =4X\nk=1SFMR,k (H): (5)\nFigure 2 shows an example of simulated FMR spectra using this methodology.\nThis simulated broadband FMR data is then subjected to the same methodology one would\ntypically use to \ft experimental FMR data, see for example44,46,47. As with experimental\ndata the number of resonances or constituents that are present in the spectra is assumed\nto be unknown a priori. We therefore vary the number of resonances Nthat we use when\nattempting to \ft the simulated data. The \ft function is thus the sum of Nresonances:\nSFit\nFMR(H) =NX\nk=1SFMR,k (H); (6)\nwhere theSFMR,k (H) are again given by equation (3), but the resonance \feld, linewidth,\nabsorption amplitude, and dispersion amplitude are now \ft-parameters. We note here that,\njust like for experimental spectra, we do not assume for the \fts that the dispersion amplitude\nis zero but instead keep this a free parameter of the \ft. To distinguish the \ft parameters\nfrom the original parameters used to simulate the spectra we will use a superscript, thus\nthese parameters are labeled HFit\nres;k, \u0001HFit\nk,AFit\nk, andBFit\nkrespectively.\nFor each frequency fthe algorithm \frst estimates initial parameters to \ft the spectrum\nusing a single resonance. Once the \ftting algorithm has converged to a solution, the residual\nbetween the data and the \ftted curve is used to estimate the additional \ft parameters for the\n\ft with two resonances. Once this \ft with two resonances converges, the process is repeated\nby including one more resonance, until a \ft is obtained for the desired number of resonances\n5Figure. 2. Simulated \feld modulated FMR signal SFMR(H) for a material that consists of four di\u000berent\nconstituents. Here \r0\nk= 28 [GHz\nT],\u00160Me\u000b,k=f\u00001:9;\u00001:7;\u00001:5;\u00001:21g[T],\u000bk= 0:01,\n\u00160\u0001H0;k=f200;100;75;150g[mT],Ak=f0:3;1:0;0:2;0:1g, andBk= 0 fork=f1;:::;4g. The spectra are\ncalculated at the frequencies indicated in the legend using 200 \feld points each.\nNof the \ft. Our \ftting algorithm takes advantage of global optimization tools available in\nMatlab48,49. This includes the option of using global search, multi start algorithms50,51, and\nparticle swarm optimization52{55. However, empirically we have found that in most cases all\nthree algorithms tend to lead to very similar results. Therefore we limit our analysis in this\nmanuscript to using particle swarm optimization to search for the global minimum of the\nsum of squares S=P\nir2\niof the residuals ri= (SFMR(Hi)\u0000SFit\nFMR(Hi)).\nOnce the \ft has converged for the desired Nresonances present in the \ft function we compute\nR2and the adjusted- R2values56. Because we are interested in comparing \ft attempts that\nuse a di\u000berent number of resonances Nthe degrees of freedom in our \ft functions will di\u000ber\nsigni\fcantly. Therefore, we will limit our discussion to the adjusted- R2values of the \fts.\nTo estimate the error-margins of the \ft-parameters HFit\nres;k, \u0001HFit\nk,AFit\nk, andBFit\nkwe use\nbootstrapping57. The resonance \felds HFit\nres;kand the associated error-margins \u001bHFit\nres;kwith\n6k=f1;:::;Ngthen serve as input for a linear \ft using equation (2) to determine the\ngyromagnetic ratio \rFit\nk0, e\u000bective magnetization MFit\ne\u000b,kand their respective error-margins.\nIII. RESULTS AND DISCUSSION\nIn the following we will discuss three di\u000berent cases of materials that all contain four\nconstituents with di\u000berent material properties. In section III A we analyze the case of a\nmaterial with constituents that share the same gyromagnetic ratio but di\u000ber with respect to\ntheir e\u000bective magnetizations. This section also provides detailed examples of the methodol-\nogy we used. In section III B we analyze a material with constituents that di\u000ber with respect\nto their gyromagnetic ratios but share the same e\u000bective magnetization. Finally in section\nIII C we revisit the case of a material with a shared gyromagnetic ratio and di\u000berent e\u000bec-\ntive magnetization but now analyze the in\ruence of noise on the results. We conclude this\nsection with a cautionary example regarding the extraction of damping related parameters\nusing equation (4).\nA. Constituents with di\u000berent e\u000bective magnetization\nFor the simulations in this section we assumed that all four constituents of the\nmaterial shared the same gyromagnetic ratio \r0\nk= 28 [GHz\nT], withk=f1;:::;4g. The\nshift between the individual resonance \felds is therefore only caused by the di\u000berence\nin the e\u000bective magnetization of the di\u000berent constituents. We have chosen \u00160Me\u000b,k=\nf\u00001:9;\u00001:7;\u00001:5;\u00001:21g[T]. In addition, we also assumed that all constituents share the\nsame damping parameter \u000bk= 0:01. However, the inhomogeneous broadening was assumed\nto be di\u000berent: \u00160\u0001H0;k=f200;100;75;150g[mT]. The simulated resonances only contain\nan absorptive part and the amplitudes were Ak=f0:3;1:0;0:2;0:1g. For the initial analysis\nwe assumed that the frequency range for the microwave frequency franged from 60 [ GHz ]\nto 64 [GHz ] with spectra recorded in 0 :5 [GHz ] intervals. The results of the simulations are\nshown in \fgure 2.\nAs described above, all spectra are \ftted using di\u000berent numbers of resonances Nin the\n\ftting function. The spectra and results of the \fts are shown in \fgure 3 exemplary for\nspectra with a microwave frequency f= 62 [GHz ]. For \fts that use less than the four\n7resonances that are present in the spectra it is clear from the residuals (shown as insets)\nthat the \ft does not fully describe the data. However, noise present in experimental data\nwill often make this less obvious, for more details on this see the later discussion regarding\nthe in\ruence of noise in section III C. As can be expected, when over \ftting the data using\nN= 5 resonances, the residual does only change slightly compared to N= 4. The oscilla-\ntions of the residuals for N= 4 andN= 5 are thus a result of numerical errors during the\n\ftting process and the condition used to terminate the \ftting algorithm.\nAs shown in \fgure 4 for all \fts the adjusted- R2values are very close to 1. We therefore\nopted to plot the deviation of the adjusted- R2from 1 in this \fgure to make the deviations\nmore easily accessible. As can be expected when over \ftting the data by using N= 5\nresonances the adjusted- R2does not change signi\fcantly and hence we omit this data in the\n\fgure.\nThe resonance \felds HFit\nres;k and their error margins \u001bHFit\nres;kextracted from the \fts of the\nraw data can now be used to determine the gyromagnetic ratio \rFit\nk0and the e\u000bective mag-\nnetizationMFit\ne\u000b,kas well as their error margins, using equation (2). The results are shown\nin \fgure 5 for N=f1;:::;5g. In all cases except for N= 5 the \fts using equation (2)\ndescribe the relationship between microwave frequency and extracted resonance \felds very\nwell. Even close inspection of the \fts provides no evidence that for N=f1;::;3gthe data is\nmissing one or more resonances present in the material. For the over\ftted data using N= 5\nthe four resonances that are present in the material are captured accurately. However, the\nadditional \fctional resonance features a sudden shift and is therefore not well described by\nequation (2). This together with for example the lack of an improvement of the adjusted- R2\ncompared to the N= 4 \ft should provide clear indications that the additional resonance\nof this \ft is an artifact of over\ftting. We will therefore exclude this data from subsequent\ndiscussions.\nThe results of the \fts shown in the Kittel plots in \fgure 5 are summarized in \fgure 6 as\nblue symbols. This \fgure also contains results using an extended frequency range where\nwe used the same methodology but simulated spectra over a microwave frequency range\nfrom 60 [GHz ] to 68 [GHz ] with spectra recorded in 1 [ GHz ] intervals. The statistical\nerror-margins in both graphs of this \fgure are smaller than the size of the symbols. For\ncomparison the values for the e\u000bective magnetization and gyromagnetic ratio used to simu-\nlate the spectra are shown as black dashed lines.\n8(a)N= 1\n (b)N= 2\n(c)N= 3\n (d)N= 4\n(e)N= 5\n(f)\nFigure. 3. Exemplary spectra (black) and \fts (red) for f= 62 [GHz ] and their corresponding residuals\n(insets). The \fts (a)-(e) use an increasing number of resonances N=f1;:::;5gto \ft the simulated\nspectrum that contains k= 4 resonances.\n9(a)N= 1\n (b)N= 2\n(c)N= 3\n (d)N= 4\nFigure. 4. The deviation (1 \u0000adjusted-R2) for the \fts of the spectra is shown for di\u000berent number of\nresonances N=f1;:::;4gused to \ft the simulated spectrum that contains k= 4 resonances.\nOne of the key observations is that when choosing fewer resonances to \ft the data than\nare present in the spectra, the \ftted values for both the e\u000bective magnetization and the\ngyromagnetic ratio deviate signi\fcantly from their true values. However, the corresponding\nKittel plots and the adjusted- R2values give very little indication that these values may not\nbe trustworthy. If we consider for example the case where the \ft used N= 3 resonances\nthe \ft lines in the corresponding Kittel plot (see \fgure 5 (c)) describe the data very well\nand the adjusted- R2values of the \ftted spectra are extremely close to 1 (see \fgure 4).\nBecause we have not added noise to the simulated spectra one can detect the presence of\nthe additional resonance by examining the simulated spectrum and the corresponding \ft\n(see \fgure 3 (c)) or the residual (inset of the same \fgure). Furthermore, for the N= 3\ncase the \ftted values for the e\u000bective magnetization do not agree with any of the true\nvalues of the constituent materials (see \fgure 6). One of the N= 3 e\u000bective magnetization\nvalues is entirely out of the range of values used in the simulation. It is noteworthy that\n10(a)N= 1\n (b)N= 2\n(c)N= 3\n (d)N= 4\n(e)N= 5\nFigure. 5. (a)-(e) Kittel plots based on the resonance \feld extracted from the simulated data using an\nincreasing number of resonances N=f1;:::;5g. The data in each plot was \ftted using equation (2) and\nthe results are indicated in each \fgure.\n11in this case the corresponding gyromagnetic ratio also signi\fcantly underestimates the true\ngyromagnetic ratio of the material.\nThe limiting case where the spectra were \ftted with a single resonance ( N= 1) leads to a\n\ftted gyromagnetic ratio of \rFit0= 27:96\u00060:01[GHz\nT] that at least comes close to the true\nvalue\r0\nk= 28 [GHz\nT] used for all resonances. However, the error-margins obtained from the\n\ft are still smaller than the observed deviation. For the e\u000bective magnetization in this case\none can compare the \ftted value \u00160MFit\ne\u000b=\u00001:692\u00060:001 [T] with the weighted mean of\nthe e\u000bective magnetization Me\u000bof the four resonances present in the spectra:\nMe\u000b=4P\nk=1AkMe\u000b,k\n4P\nk=1Ak: (7)\nFor the simulated spectra one has \u00160Me\u000b=\u00001:6819 [T] which is shown as a dashed green\nline in \fgure 6 (b). One observes that the \ftted value is close to the weighted mean but its\nerror-margins are again smaller than the deviation.\nWe conclude this section by noting that doubling the frequency range of the simulations\ncauses small di\u000berences in the extracted values obtained but it does not fundamentally\nchange them (see \fgure 6).\nB. Constituents with di\u000berent gyromagnetic ratio\nFor the simulations in this section we assumed that all four constituents of the material\nshared the same e\u000bective magnetization \u00160Me\u000b,k=\u00001:7 [T], withk=f1;:::;4g. The shift\nbetween the individual resonance \felds is therefore now only caused by the di\u000berences in the\ngyromagnetic ratio of the di\u000berent constituents. We have chosen \r0\nk=f22;25;27;28g[GHz\nT].\nWe assumed again that all constituents share the same damping parameter \u000bk= 0:01\nand for the inhomogeneous broadening we used \u00160\u0001H0;k=f200;100;75;150g[mT]. As\nbefore the simulated resonances only contain an absorptive part and the amplitudes were\nAk=f0:3;1:0;0:2;0:1g. For the analysis we assumed that the frequency range for the\nmicrowave frequency franged from 60 [ GHz ] to 64 [GHz ] with spectra recorded in 0 :5 [GHz ]\nintervals.\nThe data analysis follows the same methodology as described in the previous section. The\n\fnal results are summarized in \fgure 7.\n12(a)\n (b)\nFigure. 6. (a) E\u000bective magnetization MFit\ne\u000b,kand (b) gyromagnetic ratio \rFit\nk0fork=f1;:::;Ngas a\nfunction of the number of resonances N=f1;:::;4gused to \ft the simulated data using equation (2). The\nerror-margins for all data points are smaller than the symbol size and are thus omitted. Blue symbols use\nspectra covering a frequency range from 60 \u000064 [GHz ] (see \fgure 5) whereas red symbols use an extended\nfrequency range covering 60 \u000068 [GHz ] (see text for details). The black dashed lines in both graphs\nrepresent the values of the four resonances used to simulate the data. The green dashed line represents the\nweighted average of the e\u000bective magnetization.\nThe statistical error-margins in both graphs of this \fgure are again smaller than the size of\nthe symbols and for comparison the values for the e\u000bective magnetization and gyromagnetic\nratio used to simulate the spectra are shown as black dashed lines. The observations are very\nsimilar to those discussed in section III A. When choosing more than one but fewer resonances\nthan are present in the spectra to \ft the data, the \ftted values for both the e\u000bective\nmagnetization and the gyromagnetic ratio deviate signi\fcantly from their true values. The\nresults for a single resonance \ft are again close to the true value of the e\u000bective magnetization\nand the weighted mean of the gyromagnetic ratio. As before, they deviate slightly more\nthan their statistical error-margins from the true value of the e\u000bective magnetization and\nthe weighted mean of the gyromagnetic ratio. While this deviation is particularly obvious\nfor the e\u000bective magnetization as shown in \fgure 7 (a), we note that the y-axis in this plot\ncovers a rather small range of values and the relative deviation from the true value remains\nsmall.\n13(a)\n (b)\nFigure. 7. (a) E\u000bective magnetization MFit\ne\u000b,kand (b) gyromagnetic ratio \rFit\nk0fork=f1;:::;Ngas a\nfunction of the number of resonances N=f1;:::;5gused to \ft the simulated data using equation (2). The\nerror-margins for all data points are smaller than the symbol size and are thus omitted. Blue symbols use\nspectra covering a frequency range from 60 \u000064 [GHz ]. The black dashed lines in both graphs represent\nthe values of the four resonances used to simulate the data. The green dashed line represents the weighted\naverage of the gyromagnetic ratio.\nC. In\ruence of noise\nTo investigate the in\ruence of noise on the results for parameter values obtained after\ndata analysis, we assumed in this section that the constituents of the material di\u000bered re-\ngarding their e\u000bective magnetizations but shared the same gyromagnetic ratio. We used the\nsame parameters for the simulations as we have used in this case earlier, see section III A.\nTo simulate noise, we added normally distributed random noise with zero mean, i.e. \u0016= 0,\nand di\u000berent standard deviations \u001bN=f5\u000110\u00004;1\u000110\u00003;2\u000110\u00003gto the signal. The data\nanalysis was done using exactly the same methodology as described before. One important\ndi\u000berence regarding the \fts of the spectra compared to the case shown in \fgure 3 is that\nwhen \ftting the data with N= 4 the residuals show no systematic variation with \feld but\nsolely random noise. However, for \fts with Nless than the actual number of resonances\nthe residuals still reveal the presence of an additional resonance. This is shown exemplary\nin \fgure 8 for a \ft using N= 3 resonances.\nWith increasing noise level of the spectra the \ftting algorithm has increasing di\u000eculty to\n14(a)N= 3 \ft,\u001bN= 2\u000110\u00003\n(b)N= 3 residual, \u001bN= 2\u000110\u00003\nFigure. 8. Exemplary spectrum (black) and \fts (red) for f= 62 [GHz ] (a) and the corresponding residual\n(b). The simulation uses a standard deviation \u001bN= 2\u000110\u00003for the normal distribution of the noise added\nto the signal.\nproperly \ft all resonances. The error-margins of the \ft parameters extracted from the spec-\ntra therefore also increase. These errors then propagate to the e\u000bective magnetization MFit\ne\u000b,k\nand gyromagnetic ratio \rFit\nk0derived by \ftting equation 2 to the broadband data. Figure 9\nsummarizes the in\ruence of noise on these parameters.\nAs can be expected with increasing noise, the error-margins of the extracted values for the\ne\u000bective magnetization and the gyromagnetic ratio increase. For \fts using a single resonance\nN= 1 the results agree well with the results obtained without any noise and are close to the\nweighted mean of the e\u000bective magnetization and the true value of the gyromagnetic ratio.\nBecause the amplitude Akof the strongest resonance is more than three times larger than\nthe one with the next largest amplitude adding noise to the data does not fundamentally\nchange the behavior of the single resonance \ft of the individual spectra. Hence one can\nexpect the single resonance \ft to be relatively insensitive to noise as long as the strongest\nresonance remains clearly observable.\nConsequently it is not surprising that for N > 1 the resonances with smaller amplitudes\neither are missed entirely when \ftting the individual spectra and/or lead to larger error\nmargins in the values for the e\u000bective magnetization and the gyromagnetic ratio. For the\ne\u000bective magnetization and gyromagnetic ratio the behavior for N= 2 andN= 3 mimics\nthat observed for spectra without noise. The parameter values are comparable to those\n15(a)\n (b)\nFigure. 9. (a) E\u000bective magnetization MFit\ne\u000b,kand (b) gyromagnetic ratio \rFit\nk0fork=f1;:::;Ngas a\nfunction of the number of resonances N=f1;:::;4gused to \ft the simulated data using equation (2). The\nblack dashed lines in both graphs represent the values of the four resonances used to simulate the data.\nThe green dashed line represents the weighted average of the e\u000bective magnetization. The standard\ndeviations\u001bNused for the normal distributed noise added to the spectra is indicated in the legend. To\nimprove readability of the graphs the data sets are slightly shifted to the right with increasing noise level.\nAll but one data set shown here use a frequency range for the spectra from 60 [ GHz ] to 64 [GHz ]. The\nexception is the data shown as red triangles, which uses a broader frequency range covering 60 [ GHz ] to\n68 [GHz ].\nobtained without noise but are more scattered due to the noise in the spectra.\nThe behavior of the \ftted values for the gyromagnetic ratio for N= 4 is interesting. In the\ncase of noiseless data the \ft exactly reproduces the gyromagnetic ratio used in the simula-\ntion. However, for noisy data one observes a relatively large spread. This is noticeable even\nfor the lowest levels of noise and is accompanied with corresponding large error-margins, see\n\fgure 9 (b). The deviations from the true value and the uncertainty of the gyromagnetic\nratio also lead to deviations and increased uncertainty of the e\u000bective magnetization, see\n\fgure 9 (a). This is where an increased frequency range of broadband ferromagnetic reso-\nnance spectroscopy can make a di\u000berence. To illustrate this we have included a simulation\nwith a standard deviation of the noise \u001bN= 5\u000110\u00004and otherwise identical parameters, but\nusing a frequency range for ffrom 60 [GHz ] to 68 [GHz ] with spectra recorded in 1 [ GHz ]\n16intervals. Note that we are thus using the same number of spectra as in the cases covering a\nfrequency range from 60 [ GHz ] to 64 [GHz ], they are simply more spread out in frequency.\nThis data is shown as red triangles in \fgure 9. The increased frequency range results in\na signi\fcantly smaller spread of the gyromagnetic ratio and consequently also leads to val-\nues for the e\u000bective magnetization that are closer to the true values. This observation is\nconsistent with earlier work for materials with a single resonance that used an asymptotic\napproach to further improve the accuracy for the extraction of the gyromagnetic ratio from\nbroadband FMR measurements58.\nFor this example we also investigated the in\ruence noise and model assumptions on the\nextraction of the Gilbert damping parameter \u000bkand the inhomogeneous linewidth broad-\nening \u0001H0;k. In order to extract these parameter from the simulated data one can plot the\nlinewidth \u0001 Hkas a function of the microwave frequency fand \ft this data using equation\n(4). This is shown exemplary in \fgure 10 for a few cases.\nIn part (a) of this \fgure we have used simulation data without noise and a single resonance\n\ftN= 1 to extract the linewidth. While the data is well described by equation (4) neither\nthe extracted damping parameter nor the inhomogeneous broadening re\rects the properties\nof any of the constituents or their weighted means. This becomes even more obvious when\nusing three resonances to \ft the noiseless data as shown in \fgure 10 (b). Here one would\nobtain an unphysical negative inhomogeneous linewidth contribution for one of the reso-\nnances. For this data the careful observer may realize that the data shows some curvature\nthat is not captured by the linear \ft. However, this can easily missed in particular if noise\nis present in the data. For another resonance in the same \fgure the damping parameter\nwould be negative and thus unphysical. Figure 10 (c) shows a plot of the linewidth using\nthe correct number of resonances N= 4 for spectra with noise with a standard deviation\n\u001bN= 5\u000110\u00004. Despite using the simulation data covering the broader frequency range one\ncan still encounter \fts using equation (4) that might suggest a negative damping (red data).\nWe would like to point out that given the relatively small damping parameter and the large\ninhomogeneous broadening we assumed for the simulations they represent somewhat of a\nworst case scenario and one would therefore need a much larger frequency range to accurately\ndetermine the damping parameters and the inhomogeneous linewidth broadening. However,\nthis example clearly demonstrates that extra care has to be taken when attempting to ex-\ntract damping related parameters using equation (4) from spectra that contain a known or\n17(a)\n (b)\n(c)\nFigure. 10. Linewidth as a function of microwave frequency extracted from spectra using (a) a single\nresonanceN= 1, (b) three resonances N= 3, and (c) four resonances N= 4. The simulated data in (a) &\n(b) had no noise added and covered a frequency range from 60 [ GHz ] to 64 [GHz ], the data in (c) had noise\nwith a standard deviations \u001bN= 5\u000110\u00004added and covered a frequency range from 60 [ GHz ] to 68 [GHz ].\nThe dashed lines in all graphs represent the frequency dependence of the linewidth used for the simulations.\nunknown number of resonances. After all, whether the damping parameter is large or small\nis not known a priori. Minimizing the noise in the data and maximizing the frequency range\nare key to minimizing the error margins of the extracted parameters. For completeness we\nshow in \fgure 11 a summary of the results of the linewidth analysis.\nThe most important result from the graphs in this \fgure is that despite using the correct\nnumber of resonances that produced reliable results for the e\u000bective magnetization and the\ngyromagnetic ratio the same data analysis can lead to signi\fcant deviations of the inhomo-\n18(a)\n (b)\nFigure. 11. (a) Inhomogeneous linewidth broadening \u0001 H0;kand (b) damping parameter \u000bkfor\nk=f1;:::;Ngas a function of the number of resonances N=f1;:::;4gused to \ft the simulated linewidth\ndata using equation (4). The black dashed lines in both graphs represent the values of the four resonances\nused to simulate the data. The green dashed line represents the weighted average of the inhomogeneous\nlinewidth broadening. The standard deviations \u001bNused for the normal distributed noise added to the\nspectra is indicated in the legend. To improve readability of the graphs the data sets are slightly shifted to\nthe right with increasing noise level. All but one data set shown here use a frequency range for the spectra\nfrom 60 [GHz ] to 64 [GHz ]. The exception is the data shown as red triangles, which uses a broader\nfrequency range covering 60 [ GHz ] to 68 [GHz ].\ngeneous linewidth and damping parameter of the \ftted values from the true values of the\nmaterial. Furthermore, if an incorrect number of resonances is used to \ft the spectra wide\nvariations of the parameters extracted from the frequency dependence of the linewidth can\nbe expected. In this case the extracted parameters have no discernible relationship with the\ntrue values of the material even without noise present in the simulated data.\nIV. CONCLUSION\nIn summary, we have highlighted some of the pitfalls that one can encounter when using\nbroadband ferromagnetic resonance spectroscopy to characterize materials that have more\nthan a single constituent and therefore exhibit multiple possibly overlapping resonances.\nOur results show that it is desirable to have independent knowledge regarding the number\n19of resonances that are expected in a material to avoid analyzing the data using an incorrect\nnumber of resonances. Our simulations have shown, even in the absence of noise, that as-\nsuming an incorrect number of resonances for the \ft can result in the extraction of material\nparameters that are not consistent with any of the constituents present in the material.\nIn these cases one observes deviations that far exceed the statistical error-margins associ-\nated with \ftting the data. That is, the systematic error of choosing an incorrect number\nof resonances to describe the observed spectra by far exceeds the statistical errors present.\nThis observation applies to constituents that are characterized by di\u000berent e\u000bective mag-\nnetizations and/or by di\u000berent gyromagnetic ratios. We have also shown that adjusted- R2\nvalues only provide limited guidance regarding the correct number of resonances to use when\n\ftting spectra. However, careful inspection of the residuals of the \fts of the spectra can\nprovide important clues to identify the presence of additional resonances. In the absence of\nindependent knowledge about the nature and number of resonances present in a material it\nis therefore advisable to pay close attention to the residuals of the \ftted spectra.\nOur investigations regarding the in\ruence of noise present in ferromagnetic resonance spec-\ntra on the accuracy of the extracted parameters show very similar behavior as observed in\nthe analysis of noiseless data. Not surprisingly noise acts to increase the variability of the\nextracted parameters and their error-margins. However, if the correct number of resonances\nare used to model the experimental spectra then extending the frequency range of the ob-\nservations can signi\fcantly reduce the error-margins of both the extracted gyromagnetic\nratio and the e\u000bective magnetization. We have also found that using a single resonance\nto \ft spectra that clearly contain multiple resonances is surprisingly robust regarding noise\npresent in the data. While there will certainly be exceptions to this, for the cases we inves-\ntigated in this study, we found that the extracted values are in reasonable agreement with\nthe weighted mean of the true values of the material.\nHowever, our example for the extraction of inhomogeneous linewidth broadening and damp-\ning parameters from the frequency dependence of the resonance linewidth shows that the\nsame cannot be said for these parameters. We see large deviations of those parameters from\nthe true material parameters and their weighted means for any analysis that does not use\nthe correct number of resonances present in the material. Even when the correct number of\nresonances is used for the analysis the results remain very sensitive to noise contamination\nof the data. It is therefore advisable to be careful when attempting to extract inhomoge-\n20neous linewidth broadening and damping parameters for materials that contain multiple\nresonances. As expected, minimizing the noise of the data while maximizing the frequency\nrange over which data is collected is also advisable.\nACKNOWLEDGEMENTS\nWe would like to thank Prof. John Xiao for helpful discussions and would like to thank\nhim and Qorvo for providing the M-type hexaferrite sample. We would also like to thank\nDr. Claudia Mewes for helpful discussions. The authors would like to acknowledge funding\nfor this research under the NASA Grant NASA CAN80NSSC18M0023.\nDECLARATION OF COMPETING INTEREST\nThe authors declare that they have no known competing \fnancial interests or personal\nrelationships that could have appeared to in\ruence the work reported in this paper.\nREFERENCES\n\u0003Corresponding author:tmewes@ua.edu\n1V. Arkad'Yev, Zh. Russk. Fiz.-Khim. Obshchestva, Otdel Fiz 44, 165 (1912).\n2S. Von Sovskii, Ferromagnetic Resonance: The Phenomenon of Resonant Absorption of a High-\nFrequency Magnetic Field in Ferromagnetic Substances (Elsevier Science, 2016).\n3M. Farle, Reports on Progress in Physics 61, 755 (1998).\n4T. Mewes and C. K. A. Mewes, \\Ferromagnetic resonance,\" in Magnetic Measurement Tech-\nniques for Materials Characterization , edited by V. Franco and B. Dodrill (Springer Interna-\ntional Publishing, Cham, 2021) pp. 431{452.\n5Y. Sugiyama, K. Kagawa, H. Ohmori, K. Hayashi, M. Hayakawa, and K. Aso, IEEE Transac-\ntions on Magnetics 29, 3867 (1993).\n6S. Zhang, J. B. Sokolo\u000b, and C. Vittoria, Journal of Applied Physics 81, 5076 (1997).\n7C. Bilzer, Microwave susceptibility of thin ferromagnetic \flms: metrology and insight into mag-\nnetization dynamics , Phd, Universit\u0013 e Paris Sud - Paris XI (2007).\n218C. Bilzer, T. Devolder, P. Crozat, C. Chappert, S. Cardoso, and P. P. Freitas, Journal of\nApplied Physics 101, 074505 (2007).\n9H. Lee, Y.-H. A. Wang, C. K. A. Mewes, W. H. Butler, T. Mewes, S. Maat, B. York, M. J.\nCarey, and J. R. Childress, Applied Physics Letters 95, 082502 (2009).\n10S. Sch afer, N. Pachauri, C. K. A. Mewes, T. Mewes, C. Kaiser, Q. Leng, and M. Pakala, Applied\nPhysics Letters 100, 032402 (2012).\n11W. B. Weir, Proceedings of the IEEE 62, 33 (1974).\n12G. Counil, J.-V. Kim, T. Devolder, C. Chappert, K. Shigeto, and Y. Otani, Journal of Applied\nPhysics 95, 5646 (2004).\n13Y. Ding, T. J. Klemmer, and T. M. Crawford, Journal of Applied Physics 96, 2969 (2004).\n14B. Kuanr, R. Camley, and Z. Celinski, Applied Physics Letters 87, 012502 (2005).\n15I. Neudecker, G. Woltersdorf, B. Heinrich, T. Okuno, G. Gubbiotti, and C. Back, Journal of\nMagnetism and Magnetic Materials 307, 148 (2006).\n16S. Kalarickal, P. Krivosik, M. Wu, C. Patton, M. Schneider, P. Kabos, T. Silva, and J. Nibarger,\nJournal of Applied Physics 99, 093909 (2006).\n17J. F. Godsell, S. Kulkarni, T. O'Donnell, and S. Roy, Journal of Applied Physics 107, 033907\n(2010).\n18M. Tsoi, A. Jansen, J. Bass, W.-C. Chiang, V. Tsoi, and P. Wyder, Nature 406, 46 (2000).\n19S. I. Kiselev, J. Sankey, I. Krivorotov, N. Emley, R. Schoelkopf, R. Buhrman, and D. Ralph,\nnature 425, 380 (2003).\n20A. Tulapurkar, Y. Suzuki, A. Fukushima, H. Kubota, H. Maehara, K. Tsunekawa,\nD. Djayaprawira, N. Watanabe, and S. Yuasa, Nature 438, 339 (2005).\n21Y. S. Gui, S. Holland, N. Mecking, and C. M. Hu, Phys. Rev. Lett. 95, 056807 (2005).\n22J. 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).\n23M. Harder, Z. X. Cao, Y. S. Gui, X. L. Fan, and C.-M. Hu, Phys. Rev. B 84, 054423 (2011).\n24Y. Wang, R. Ramaswamy, and H. Yang, Journal of Physics D: Applied Physics 51, 273002\n(2018).\n25J. Smit and H. Beljers, Philips Res. Rep 10, 113 (1955).\n26L. Baselgia, M. Warden, F. Waldner, S. L. Hutton, J. E. Drumheller, Y. Q. He, P. E. Wigen,\nand M. Mary\u0014 sko, Phys. Rev. B 38, 2237 (1988).\n2227C. Kittel, Phys. Rev. 71, 270 (1947).\n28C. Kittel, Phys. Rev. 73, 155 (1948).\n29C. P. Poole, Electron spin resonance: a comprehensive treatise on experimental techniques (In-\nterscience, 1967).\n30S. C. Shin, M. L. Co\feld, and R. H. D. Nuttall, Journal of Applied Physics 61, 4326 (1987),\nhttps://doi.org/10.1063/1.338462.\n31S. H. Talisa, K. C. Yoo, M. Abe, and T. Itoh, Journal of Applied Physics 64, 5819 (1988),\nhttps://doi.org/10.1063/1.342217.\n32S. Aslam, A. Das, M. Khanna, and B. K. Kuanr, Journal of Alloys and Compounds 838,\n155566 (2020).\n33J. O. Artman, D. J. DeSmet, X. Shao, J. C. Cates, C. Alexander, M. R. Parker, E. T.\nLacey, D. G. Lord, and P. J. Grundy, Journal of Applied Physics 70, 6038 (1991),\nhttps://doi.org/10.1063/1.350064.\n34J. P. Omaggio and P. E. Wigen, Journal of Applied Physics 50, 2264 (1979),\nhttps://doi.org/10.1063/1.327020.\n35M. Co\feld, C. Brucker, J. Gau, and J. Gerard, Journal of Magnetism and Magnetic Materials\n54-57 , 1689 (1986).\n36N. Vukadinovic, M. Labrune, J. B. Youssef, A. Marty, J. C. Toussaint, and H. Le Gall, Phys.\nRev. B 65, 054403 (2001).\n37M. J. Pechan, C. Yu, R. L. Compton, J. P. Park, and P. A. Crowell, Journal of Applied Physics\n97, 10J903 (2005), https://doi.org/10.1063/1.1857412.\n38M. J. Pechan, C. Yu, D. Owen, J. Katine, L. Folks, and M. Carey, Journal of Applied Physics\n99, 08C702 (2006), https://doi.org/10.1063/1.2158391.\n39Z. Zhang, P. E. Wigen, and S. S. P. Parkin, Journal of Applied Physics 69, 5649 (1991),\nhttps://doi.org/10.1063/1.347924.\n40S. T. B. Goennenwein, T. Graf, T. Wassner, M. S. Brandt, M. Stutzmann, J. B. Philipp,\nR. Gross, M. Krieger, K. Z urn, P. Ziemann, A. Koeder, S. Frank, W. Schoch, and A. Waag,\nApplied Physics Letters 82, 730 (2003), https://doi.org/10.1063/1.1539550.\n41M-type hexaferrite sample provided by Qorvo, https://www.qorvo.com/ (2022).\n42C. Sterwerf, S. Paul, B. Khodadadi, M. Meinert, J.-M. Schmalhorst, M. Buchmeier,\nC. K. A. Mewes, T. Mewes, and G. Reiss, Journal of Applied Physics 120, 083904 (2016),\n23https://doi.org/10.1063/1.4960705.\n43A. Rai, A. Sapkota, A. Pokhrel, M. Li, M. De Graef, C. Mewes, V. Sokalski, and T. Mewes,\nPhys. Rev. B 102, 174421 (2020).\n44S. Klingler, A. V. Chumak, T. Mewes, B. Khodadadi, C. Mewes, C. Dubs, O. Surzhenko,\nB. Hillebrands, and A. Conca, Journal of Physics D: Applied Physics 48, 015001 (2014).\n45C. J. Oates, F. Y. Ogrin, S. L. Lee, P. C. Riedi, G. M. Smith, and T. Thomson, Journal of\nApplied Physics 91, 1417 (2002), https://doi.org/10.1063/1.1428804.\n46B. Khodadadi, J. B. Mohammadi, J. M. Jones, A. Srivastava, C. Mewes, T. Mewes, and\nC. Kaiser, Phys. Rev. Applied 8, 014024 (2017).\n47S. Wu, K. Abe, T. Nakano, T. Mewes, C. Mewes, G. J. Mankey, and T. Suzuki, Phys. Rev. B\n99, 144416 (2019).\n48MATLAB, R2020a (The MathWorks Inc., Natick, Massachusetts, 2020).\n49MATLAB Global Optimization Toolbox, version 4.3 (The MathWorks Inc., Natick, Mas-\nsachusetts, 2020).\n50Matlab Online Documentation, https://www.mathworks.com/help/gads/\nhow-globalsearch-and-multistart-work.html (2022), accessed: 2022-03-30.\n51Z. Ugray, L. Lasdon, J. Plummer, F. Glover, J. Kelly, and R. Mart\u0013 \u0010, INFORMS Journal on\nComputing 19, 328 (2007), https://doi.org/10.1287/ijoc.1060.0175.\n52Matlab Online Documentation, https://www.mathworks.com/help/gads/\nparticle-swarm-optimization-algorithm.html (2022), accessed: 2022-03-30.\n53J. Kennedy and R. Eberhart, in Proceedings of ICNN'95-international conference on neural\nnetworks , Vol. 4 (IEEE, 1995) pp. 1942{1948.\n54M. E. H. Pedersen, Hvass Lab., Copenhagen, Denmark, Tech. Rep. HL1001 , 1551 (2010).\n55E. Mezura-Montes and C. A. C. Coello, Swarm and Evolutionary Computation 1, 173 (2011).\n56T. O. Kvalseth, The American Statistician 39, 279 (1985).\n57T. J. DiCiccio and B. Efron, Statistical science 11, 189 (1996).\n58J. M. Shaw, H. T. Nembach, T. J. Silva, and C. T. Boone, Journal of Applied Physics 114,\n243906 (2013), https://doi.org/10.1063/1.4852415.\n24"    },    {        "title": "1611.02865v1.Effect_of_nanostructure_layout_on_spin_pumping_phenomena_in_antiferromagnet__nonmagnetic_metal__ferromagnet_multilayered_stacks.pdf",        "content": "E\u000bect of nanostructure layout on spin pumping phenomena in antiferromagnet/\nnonmagnetic metal/ ferromagnet multilayered stacks\nA. F. Kravets,1, 2,a)Olena V. Gomonay,3, 4D. M. Polishchuk,2, 1Yu. O.\nTykhonenko-Polishchuk,2, 1T. I. Polek,1A. I. Tovstolytkin,1and V. Korenivski2\n1)Institute of Magnetism, National Academy of Sciences of Ukraine, 03680 Kyiv,\nUkraine\n2)Nanostructure Physics, Royal Institute of Technology, 10691 Stockholm,\nSweden\n3)Institut f ur Physik, Johannes Gutenberg Universit at Mainz, D-55099 Mainz,\nGermany\n4)National Technical University of Ukraine \\KPI\", 03056, Kyiv,\nUkraine\n(Dated: 18 August 2021)\nIn this work we focus on magnetic relaxation in Mn 80Ir20(12 nm)/ Cu(6 nm)/ Py( dF)\nantiferromagnet/Cu/ferromagnet (AFM/Cu/FM) multilayers with di\u000berent thick-\nness of the ferromagnetic permalloy layer. An e\u000bective FM-AFM interaction medi-\nated via the conduction electrons in the nonmagnetic Cu spacer { the spin-pumping\ne\u000bect { is detected as an increase in the linewidth of the ferromagnetic resonance\n(FMR) spectra and a shift of the resonant magnetic \feld. We further \fnd exper-\nimentally that the spin-pumping-induced contribution to the linewidth is inversely\nproportional to the thickness of the Py layer. We show that this thickness depen-\ndence likely originates from the dissipative dynamics of the free and localized spins\nin the AFM layer. The results obtained could be used for tailoring the dissipative\nproperties of spintronic devices incorporating antiferromagnetic layers.\na)Electronic mail: anatolii@kth.se\n1arXiv:1611.02865v1  [cond-mat.mtrl-sci]  9 Nov 2016Antiferromagnets (AFMs) are attractive materials for spintronic applications. They op-\nerate at high frequencies and thus have the potential to functionally \fll the \\terahertz gap\"\nin electronics. Due to their lack of a macroscopic magnetic moment, AFMs produce no stray\n\felds and therefore potentially can provide higher scalability for magnetic memory devices.\nHigh typical values of the spin-\rop \felds prevent AFMs from spontaneous thermally-induced\nswitching and increase the data retention times. In addition, recent experimental1and the-\noretical investigations2have shown that AFMs are sensitive to spin-polarized currents and\ncan be used as active elements in spintronic devices.\nDirect observation of spintronic e\u000bects in AFMs is challenging due precisely to the same\nreasons that make AFMs competitive with their ferromagnetic counterparts: the magnetore-\nsistance in AFM-based devices is low due to the absence of net magnetization in AFM, and\nthe dynamics require very high excitation frequencies, beyond the capabilities of microwave\ncircuits. An alternative technique to detect the spin dynamics of AFM \flms was recently\nimplemented by a number of groups.3{7This technique is based on the spin pumping e\u000bect,\nwhich is reciprocal to the spin-transfer torque e\u000bect.8,9A metallic ferromagnetic layer (FM)\nis excited at its resonance frequency (FMR) and pumps spin current into a neighbouring non-\nmagnetic layer interfaced with an antiferromagnetic \flm (AFM) at the other surface. The\nlinewidth of the FMR spectrum increases due to the presence of the AFM layer and thereby\nprovides information about the interaction of the nonequilibrium conduction-electron spins\nand the localized AFM moments.\nThe interpretation of such experiments is not quite straightforward, however, as di\u000berent\nprocesses contribute to the e\u000bective damping in a multilayered sample: spin-dependent\nscattering at the interfaces10and in the bulk, energy exchange between the free and\nlocalised spins, spin-di\u000busion, etc. An e\u000ecient theoretical approach to this problem,\nbased on nonequilibrium thermodynamics, was proposed in Ref. 11 for ferromagnetic\n(FM)/nonmagnetic (NM) bilayers, and was further generalized for FM/NM/FM systems.12\nSpin-pumping from an AFM layer was recently predicted in Refs. 13 and 14.\nIn this paper we focus on the dissipative response, expressed via the FMR linewidth,\nof MnIr/Cu/Py multilayers with di\u000berent thickness of the Py layer. We generalize the\nOnsager formalism for the case of the discrete system AFM/NM/FM and calculate the\ne\u000bective Gilbert damping of the FM layer, taking into account the spin-pumping and spin-\naccumulation e\u000bects in both the FM and AFM layers. While the previous experiments3,4\n2have studied the damping dependence vs thickness of the AFM layer, we focus on the\nproperties of the FM layer and especially the FM/NM interface. Our experiments reveal\nan inverse dependence of the additional, AFM-induced damping on the thickness of the\nFM layer, in agreement with our theoretical predictions. Our results should be useful for\ntailoring dissipation in spintronic devices.\nFor the experiments we use multilayers Substrate/Ta(5)/Py(3)/Mn 80Ir20(12)/Cu(6)/\nPy(dF)/Al(4), hereinafter AFM/Cu/FM( dF), with the FM layer of variable thickness, dF=\n3, 6, 9, 12, 15 nm. The numbers in parenthesis denote thickness in nanometers of the corre-\nsponding layers; Py = Ni 80Fe20. In these multilayers, Mn 80Ir20(12), Cu(6) and Py( dF) form\nthe functional combination of the AFM/NM/FM stack, while the other layers are auxiliary.\nThe top Al layer is a protective capping layer. The bottom layers facilitate the formation\nof the optimal crystalline and magnetic structure of Mn 80Ir20(12). We also fabricated a set\nof reference samples with identical structure but without Py(3)/Mn 80Ir20(12) layers.\nThe multilayers were deposited at room temperature (295 K) on thermally oxidized silicon\nsubstrates using magnetron sputtering in an AJA Orion 8-target system.15The base pressure\nin the deposition chamber was 5 \u0002108Torr and the Ar pressure used during deposition was\n3 mTorr. The exchange pinning between Py(3) and Mn 80Ir20(12) layers was set in during\nthe deposition of the multilayers using an in-plane magnetic \feld of 1 kOe.\nWe use an X-band ELEXSYS E500 spectrometer equipped with an automatic goniometer\nto measure the out-of-plane and in-plane angular dependencies of the FMR spectra. The\noperating frequency is 9.85 GHz, the temperature is 295 K. The spectra show no signal\nfrom the Py(3) bu\u000ber layer, while the signal from Py( dF) is clearly visible. We record the\nmagnetic-\feld derivative of the microwave absorption and \ft each spectrum by a Lorentzian\nfunction to obtain the resonance \feld Hrand the linewidth \u0001 in the in-plane and the out-of\nplane geometries [Fig. 1(b)]. Typical FMR spectra measured for the in-plane orientation\nare shown in the inset to Fig. 2(a).\nWhen FMR is excited, a moving magnetization in the FM pumps a spin current into\nthe NM and AFM layers.16The spin current is proportional to the e\u000bective \feld HF, which\ndetermines the magnetic dynamics in the FM layer. The spin current can induce exchange\nof angular momentum between the di\u000berent subsystems of the conduction and localized\nelectrons in the NM and AFM layers. Moreover, it can stimulate additional spin pumping\nfrom the AFM layer induced by the dynamic magnetization MAF, which follows the motion\n3of the localized AFM moments.13,17,18In addition, free conduction-electron spins in our\nmetallic AFM can interact with the dynamic magnetization MAFand also accumulate,\nsimilar to that in the NM layer. While the spin polarization in FM is so strong that\nspin accumulation in it can be neglected, in the metallic AFM spin accumulation and spin\npolarization by the localized moments are comparable. Therefore, the transport of spins\nthrough the AFM/NM/FM system and the corresponding dissipative phenomena within\nthe trilayer depend upon the balance between the free and localized spins within all three\nlayers of the structure.\nTreating the AFM/NM/FM as a discrete system, one can distinguish between \fve sub-\nsystems, shown schematically in Fig. 1(a): three reservoirs of free spins in FM (spin density\nsF), NM (spin density sN), and AFM (spin density sAF), and localized FM (macroscopic\nmagnetization MF\u0011MFmF) and AFM moments (characterized with the N\u0013 eel order pa-\nrameter L=MAFland macroscopic magnetization MAF\u0011MAFmAF). Here we introduce\nthe saturation magnetizations MFandMAFof the FM and AFM layers, respectively. In\nequilibrium, free spins in the FM are mostly parallel to the FM magnetization, sFkMF. In\nthe NM and AFM layers, the population of free spin-up and spin-down electrons is balanced,\nsN=sAF= 0, since MAF= 0.\nIn the framework of linear nonequilibrium thermodynamics, spin densities sF,sN,sAF,\nand magnetizations mF,mAFcan be treated as thermodynamic variables aj,j= 1:::5.\nThe conjugated thermodynamic forces are calculated as the derivatives of free energy:19\nXj=@F=@a j(we assume that the temperature is constant). The thermodynamic forces for\nthe free spins coincide with the spin accumulation potentials \u0016(s)\nF(in FM), \u0016(s)\nN(in NM),\nand\u0016(s)\nAF(in AFM). For the localized moments the corresponding forces are proportional to\nthe e\u000bective \felds MFVFHF(in FM) and MAFVAFHAF(in AFM).\nThermodynamic currents Jj\u0011_ajare related to the thermodynamic forces via the Onsager\ncoe\u000ecients ^L:\n(_mAF;_mF;_sAF=e;_sF=e;_sN=e)T=^L\u0010\nMAFVAFHAF;MFVFHF;\u0016(s)\nAF;\u0016(s)\nF;\u0016(s)\nN\u0011T\n;(1)\nwhereeis electron charge.\nUsing the Onsager reciprocity principle and the symmetry considerations, one can reduce\n4relations (1) to the following form:\n_mAF=\r\u000bAFHAF\u0000\r~\ne2MAFVAFl\u0002\u0010\nGAF\nb\u0016(s)\nAF+GAF\nS\u0016(s)\nFmF\u0011\n\u0002l;\n_mF=\r\u000bFHF\u0000\r~GF\nS\ne2MFVFmF\u0002\u0016(s)\nAF\u0002mF; (2)\n_sAF=\u0000\r~\neGAF\nbHAF+1\neGAF\n0\u0016(s)\nAF;_sF=\u0000\r~\neGF\nbHF+1\neGF\n0\u0016(s)\nFmF;\n_sN=\r~\neGAF\nSHAF\u0000\r~\neGF\nSHF+1\neGN(\u0016(s)\nF\u0000\u0016(s)\nAF);\nwhere\ris the gyromagnetic ratio and ~is the Plank constant. We neglect spin accumulations\nin the NM layer, since the spin-di\u000busion length in the NM layer is relatively long. We\nalso set \u0016(s)\nN= 0 and take into account strong spin polarization in the FM layer, so, that\n\u0016(s)\nF=\u0016(s)\nFmF. In the second equation of (2) we use Landau-Lifshitz representation of\nthe magnetic damping in FM ( /\u000bFHF), as it is consistent with the Onsager's concept of\nconjugated currents ( _mF) and forces ( HF). Conversion to the standard Gilbert form can be\nobtained from equations of motions for FM as HF=mF\u0002_mF=\r.\nThe interpretation of the coe\u000ecients in Eq. (2) is schematically shown in Fig. 1(a). Diag-\nonal coe\u000ecientsLjjfor the localized spins are related with the internal damping in the FM\n(damping parameter \u000bF) and AFM (damping parameter \u000bAF) layers. Diagonal coe\u000ecients\nLjjfor the free spins are proportional to the corresponding conductances, GF\n0andGAF\n0. The\nnondiagonal coe\u000ecients responsible for the cross-coupling e\u000bects between the AFM and FM\nlayers, are of two types. First, the spin-mixing conductances GF\nSandGAF\nSoriginate from the\ndephasing of the free electrons at the FM/NM and NM/AFM interfaces,8and are responsible\nfor the spin-pumping phenomena. The free electrons in NM re\recting from the FM/NM and\nNM/AFM interfaces acquire additional nonequilibrium spin polarization, which is related\nto the dynamic magnetization of the FM and AFM \flms. Second, the bulk conductivities,\nGAF\nbandGF\nbdescribe the exchange of angular momentum between the subsystems of the\nlocalized and free spins in the FM and AFM layers. In our case of strong polarization inside\nthe FM layer, the term with GF\nbcan be neglected. Lastly, GNis the spin conductivity in the\nNM layer.\nThe \frst of Eqs. (2) reproduces the well-known result of AFM spintronics:2,20the spin-\ntorque induced by a spin-polarized current (last term in the r.h.s.). It is clear from Eq. (2)\nthat this torque originates not only from the current polarized by the FM layer, but also\nfrom the spin accumulation inside the AFM layer.\n5FIG. 1. (a) Schematic view of the energy and spin exchange within a trilayer system FM/NM/AFM.\nThe magnetic layers (FM and AFM) are symbolically separated into subsystems of localized\n(coloured area) and free (white area) spins. Wide arrows show the \ruxes that originate from\ndi\u000berent mechanisms. Vertical arrows correspond to spin exchange between the localized and free\nspins inside the FM and AFM layers. (b) Schematic view of the FMR experiment, where HkxOy\nis the in-plane geometry and HkxOz is the out-of-plane geometry.\nThe second of Eqs. (2), for the FM magnetization is similar to the corresponding equation\nfor FM/NM bilayers,4,9,21,22with the only di\u000berence that spin accumulation \u0016(s)\nAFtakes place\nin the AFM layer. This fact re\rects the \\duality\" of the metallic AFM, which manifests the\nproperties of FM (non-zero magnetization of the localized spins) as well as NM (has free spins\nthat can accumulate). There is one principal di\u000berence, however, between spin accumulation\n\u0016(s)\nFin FM/NM systems, and \u0016(s)\nAFin AFM/NM/FM trilayers. The FM magnetization is\nlarge and fully de\fnes the orientation of the spin accumulation in a FM/NM bilayer. In the\nAFM/NM/FM system, the spin accumulation \u0016(s)\nAFis de\fned by the interplay between the\nmagnetic dynamics in the AFM and the spin \row between the FM, NM, and AFM layers,\nwith the result that its spin orientation is de\fned by a non-trivial interplay of a number of\nfactors and point essentially in any direction.\nTo describe the magnetic dynamics of a AFM/NM/FM trilayer one must start from the\nbalance equations for the localized moments in the FM and AFM layers, and take into\naccount the spin \rows through the interfaces and the dissipative terms given by Eq. (2). In\n6particular, the equation for the FM moments can be written as\n_mF=\u0000\rmF\u0002(HF+H)\u00001\neMFVFmF\u0002_sN\u0002mF\n+\r\u000bFHF\u0000\r~GF\nS\ne2MFVFmF\u0002\u0016(s)\nAF\u0002mF; (3)\nThe \frst term in Eq. (3) corresponds to the standard Landau-Lifshits dynamics in the\npresence of external magnetic \feld H. The second term describes a spin \rux through the\ninterface, which coincides with the spin current, \u0000_sN, from the adjacent NM layer. Cross\nproducts with mFre\rect the fact that only transverse (with respect to mF) spin component\n\rows out of FM. Last two terms correspond to the Onsager forces, according to Eq. (2).\nFor the AFM/Cu/FM( dF) system used in our FMR-induced spin pumping experiment,\nwe can set \u0016(s)\nF= 0 as no electric voltage is applied across the structure. We further assume\nno spin accumulation inside the AFM layer, \u0016(s)\nAF= 0, since the spin-di\u000busion length in AFM\n(0.3 nm for Cu/IrMn23) is much shorter than the AFM thickness. Then, from Eq. (1) and\n(3) we obtain the e\u000bective dynamic equation for the FM layer:\n_mF=\u0000\rmF\u0002(HF+H) +\r\u0012\n\u000bF+\r~GF\nS\ne2MFVF\u0013\nHF+\r2~2GAF\nS\ne2MFVFmF\u0002HAF\u0002mF:(4)\nThe second term in the r.h.s. of Eq. (4) points to an increase of the e\u000bective damping\ndue to the presence of the FM/NM interface, which leads to a corresponding increase in the\nFMR linewidth \u0001. In addition, the last term in Eq. (4) predicts a \feld-like contribution to\nthe FM dynamics, which results exclusively from the spin pumping by the AFM layer, as\nthe direct exchange between the FM and AFM is fully suppressed by the Cu spacer. This\n\feld,/HAF\u0002mF, can contribute to the value of the resonant \feld Hr, and the contribution\ncan be estimated as follows. The typical AFMR frequencies are much larger than the FMR\nfrequency of the FM layer, so the dynamics of the AFM is driven solely by the FM, and\nHAF/HF. The additional \feld is then /GAF\nSHF\u0002mF=MFVF.\nAccording to Eq. (4), both spin-pumping-induced corrections to the linewidth and the\nresonant \feld are inversely proportional to MFVF/MFdF. Fig. 2(a) illustrates this tendency\nof \u0001(dF) andHr(dF) measured for our samples.\nTo con\frm the thickness dependence of the e\u000bective damping predicted by Eq. (4), we\ncalculated the incremental change in the AFM-induced linewidth as \u0001 sp= \u0001\u0000\u0001inhom\u0000\u0001ref,\nwhere \u0001 refis the linewidth of the FMR of the reference sample. Contribution \u0001 inhom,\nwhich originates from a possible inhomogeneuity of the sample is calculated according to\n7FIG. 2. (a) In-plane resonance \feld Hr(triangles) and linewidth \u0001 (circles) vs thickness dFof the\nPy layer for AFM/Cu/FM( dF) multilayers (bold symbols, solid lines) and for reference samples\n(open symbols, dashed lines). Inset shows typical FMR spectra for AFM/Cu/FM( dF) samples\nwithdF= 3 and 15 nm. (b) Py-thickness dependence of \u000bspfor AFM/Cu/FM( dF). Inset shows\n\u000bspMFproduct as a function of d\u00001\nF. The solid line is guide to the eye.\nthe procedure described in Refs. 24 and 25. This contribution is below 2 Oe for the samples\nwithdF\u00146 nm and equals to 8 Oe for dF= 3 nm. It should be noted that for the multilayer\nwith the thickest Py layer ( dF= 15 nm), the FMR linewidth ( \u001855 Oe) well agrees with the\nvalues reported by other research groups for well-characterized high-quality Py \flms.3,26\nFigure 2(b) shows the thickness dependence of the spin-pumping-induced contribution to\nGilbert damping obtained from \u0001 sp. In agreement with the theory, Eq. (4), \u000bspMFgrows\nlinearly with d\u00001\nF. We believe that the observed thickness dependence of the damping param-\neter points to the important role of free spins in the magnetic dynamics of the AFM/Cu/FM\ntrilayer. We also conclude that the observed Hr(dF) dependence indicates that the local-\nized AFM moments a\u000bect the dynamics of the FM layer through the dynamic exchange via\nconduction electrons in the system. However, this contribution from the localized moments\ncan be partially masked by the exchanges bias due to the second Py layer and thus requires\nfurther analysis.\nIn summary, we observe spin-pumping e\u000bect in AFM/NM/FM multilayers as an increase\nin the linewidth of the FMR and shift of the resonant magnetic \feld. Basing on Onsager\nformalism, we calculate additional damping and \feld-like torque on FM moments due to\nthe presence of AFM layer. The inverse dependence of damping and resonant \feld vs\n8the thickness of FM layer supports the hypothesis of AFM in\ruence on FM dynamics.\nThe contribution from the spin-pumping e\u000bect to the FMR linewidth is separated and\nshown be a\u000bected by the changes in the thickness of the ferromagnetic layer. The physical\nmechanisms of the observed \u0001 spvs.dFbehaviour are analyzed and show a rich interplay\nof the conduction-vs-lattice spins in the \fve e\u000bective sub-systems of the structure. These\nresults provide a deeper understanding of the spintronic e\u000bects in nanostructures containing\nantiferromagnets and can prove useful for designing future spintronic devices.\nACKNOWLEDGMENTS\nSupport from the Swedish Stiftelse Olle Engkvist Byggm astare, the Swedish Research\nCouncil (VR grant 2014-4548), and the National Academy of Sciences of Ukraine (project\n0115U00974) are gratefully acknowledged. OG acknowledges support of the Fundamental\nresearch program of the National Academy of Sciences of Ukraine \\Fundamental problems\nof creation of new nanomaterials and nanotechnologies\", the ERC Synergy Grant SC2 (No.\n610115), and the Transregional Collaborative Research Center (SFB/TRR) 173 SPIN+X.\nREFERENCES\n1A. H. MacDonald and M. Tsoi, Phil. Trans. R. Soc. A 369, 3098 (2011).\n2E. V. Gomonay and V. M. Loktev, Low Temp. Phys. 40, 17 (2014).\n3P. Merodio, A. Ghosh, C. Lemonias, E. Gautier, U. Ebels, M. Chshiev, H. B\u0013 ea, V. Baltz,\nand W. E. Bailey, Appl. Phys. Lett. 104, 032406 (2014).\n4L. Frangou, S. Oyarz\u0013 un, S. Au\u000bret, L. Vila, S. Gambarelli, and V. Baltz, Phys. Rev. Lett.\n116, 077203 (2016).\n5H. Wang, C. Du, P. C. Hammel, and F. Yang, Phys. Rev. Lett. 113, 097202 (2014).\n6C. Hahn, G. de Loubens, V. V. Naletov, J. Ben Youssef, O. Klein, and M. Viret, Europhys.\nLett. 108, 57005 (2014).\n7Z. Qiu, J. Li, D. Hou, E. Arenholz, A. T. N\u0013Diaye, A. Tan, K.-i. Uchida, K. Sato,\nS. Okamoto, Y. Tserkovnyak, Z. Q. Qiu, and E. Saitoh, Nat. Commun. 7, 12670 (2016).\n8Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. B 66, 224403 (2002).\n9A. Brataas, Y. Tserkovnyak, G. E. W. Bauer, and P. J. Kelly, in Spin Current , edited\n9by S. Maekawa, S. O. Valenzuela, E. Saitoh, and T. Kimura (Oxford University Pess,\nOxford, 2012) p. 87.\n10S. M. Rezende, M. A. Lucena, A. Azevedo, F. M. de Aguiar, J. R. Fermin, and S. S. P.\nParkin, J. Appl. Phys. 93, 7717 (2003).\n11A. Brataas, G. E. W. Bauer, and K. P. J., Phys. Rep. 427, 157 (2006).\n12Phys. Rev. B 90, 214407 (2014).\n13R. Cheng, J. Xiao, Q. Niu, and A. Brataas, Phys. Rev. Lett. 113, 057601 (2014).\n14W. Zhang, M. B. Jung\reisch, W. Jiang, J. E. Pearson, A. Ho\u000bmann, F. Freimuth, and\nY. Mokrousov, Phys. Rev. Lett. 113, 196602 (2014).\n15A. F. Kravets, A. N. Timoshevskii, B. Z. Yanchitsky, M. A. Bergmann, J. Buhler, S. An-\ndersson, and V. Korenivski, Phys. Rev. B 86, 214413 (2012).\n16Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I. Halperin, Rev. Mod. Phys. 77,\n1375 (2005).\n17I. V. Baryakhtar and B. A. Ivanov, Sov. J. Low Temp. Phys. 5, 361 (1979).\n18H. Gomonay and V. Loktev, J. Magn. Soc. Jpn. 32, 535 (2008).\n19H. B. Callen, Phys. Rev. 73, 1349 (1948).\n20R. Cheng and Q. Niu, Phys. Rev. B 89, 081105 (2014).\n21A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer, Phys. Rev. B 84, 054416 (2011).\n22Y. Ohnuma, H. Adachi, E. Saitoh, and S. Maekawa, Phys. Rev. B 89, 174417 (2014).\n23R. Acharyya, H. Y. T. Nguyen, W. P. Pratt, and J. Bass, J. Appl. Phys. 109, 07C503\n(2011).\n24S. Mizukami, Y. Ando, and T. Miyazaki, Jpn. J. Appl. Phys. 40, 580.\n25A. F. Kravets, D. M. Polishchuk, Y. I. Dzhezherya, A. I. Tovstolytkin, V. O. Golub, and\nV. Korenivski, Phys. Rev. B 94, 064429 (2016).\n26E. Montoya, P. Omelchenko, C. Coutts, N. R. Lee-Hone, R. H ubner, D. Broun, B. Heinrich,\nand E. Girt, Phys. Rev. B 94, 054416 (2016).\n10"    },    {        "title": "1503.08408v2.Low_non_linearity_spin_torque_oscillations_driven_by_ferromagnetic_nanocontacts.pdf",        "content": "Low non-linearity spin-torque oscillations driven by ferromagnetic\nnanocontacts\nMuftah Al-Mahdawi,\u0003Yusuke Toda, Yohei Shiokawa, and Masashi Sahashi\nDepartment of Electronic Engineering,\nTohoku University, Sendai 980-8579, Japan\n(Dated: July 24, 2021)\nAbstract\nSpin-torque oscillators are strong candidates as nano-scale microwave generators and detectors.\nHowever, because of large amplitude-phase coupling (non-linearity), phase noise is enhanced over\nother linear auto-oscillators. One way to reduce nonlinearity is to use ferromagnetic layers as a\nresonator and excite them at localized spots, making a resonator-excitor pair. We investigated\nthe excitation of oscillations in dipole-coupled ferromagnetic layers, driven by localized current at\nferromagnetic nano-contacts. Oscillations possessed properties of optical-mode spin-waves and at\nlow \feld (\u0019200 Oe) had high frequency (15 GHz), a moderate precession amplitude (2{3\u000e), and a\nnarrow spectral linewidth ( <3 MHz) due to localized excitation at nano-contacts. Micromagnetic\nsimulation showed emission of resonator's characteristic optical-mode spin-waves from disturbances\ngenerated by domain-wall oscillations at nano-contacts.\nPACS numbers: 75.30.Ds,75.40.Gb,75.78.Cd,85.75.-d\n1arXiv:1503.08408v2  [cond-mat.mes-hall]  11 Jan 2016I. INTRODUCTION\nTransfer of angular momentum from a dc spin-polarized current to a nano-scale fer-\nromagnet (FM) exerts an anti-damping torque, Spin-Transfer Torque (STT), that can\ncompensate intrinsic damping torque and induce stable precession of magnetization.1When\ncombined with a magnetoresistance e\u000bect, like giant magnetoresistance (GMR) or tunnel-\ning magnetoresistance (TMR), high-frequency voltage oscillations are emitted, producing\nSpin-Torque Oscillations (STO).2The same structures can also rectify injected ac voltage\nat resonance,3,4and o\u000b-resonance5. Such microwave nano-oscillators/detectors are sought\nafter for applications like inter-/intra-chip communication, imaging6and non-destructive\ntesting, and lab-on-chip sensors. STOs are suitable candidates for such a role. However,\nlow output power, the trade-o\u000b between power and frequency, and weak coherence of os-\ncillations hinder their applications compared with their semiconductor counterparts. The\nlarge nonlinear phase-amplitude coupling7produces a dilemma between large precession\namplitude and small linewidth. Also, frequency locking of STOs to external reference signal\nbecomes hindered.8Such a dilemma can be overcome by the separation of STOs into an\nexcitation source and a resonating element, so that precession frequency and amplitude will\nbe set by the resonator design, not by the driving STT. Then the linewidth will be reduced\nconsiderably.\nIt was shown that non-uniform current density in TMR-STO resulted in an increase the\namplitude of generated precession,9{12and reduction in linewidth.9,13{15However, the origin\nis still not clear, and the fabrication process is not well understood or easily reproducible.\nOn the other hand, the Ion-Assisted Oxidation (IAO) of ultra-thin aluminum reproducibly\nwas used to fabricate a 1-nm-thin alumina Nano-Oxide Layer (NOL) with direct 2-nm\nNano-Contacts (NCs) between FM layers.16{20In NOL-based STOs, moderate power and\nnarrow linewidths were reported,21{24with oscillation behavior similar to low-TMR-STO.25\nThere is evidence for the presence of FM metallic NCs by magnetoresistance and trans-\nport properties,17transmission-electron micrographs,18,20,26and conductive atomic force\nmicroscopy.18,19,26However, the measured Nano-Contacts Magnetoresistance (NCMR) ratios\nare far below expectation compared with scattering from con\fned Domain-Walls (DW),27{29\nmostly due to the presence of oxygen and non-magnetic impurities.19,26,30\nIn this paper, we propose that the localized precession of DWs at NCs work as excitors\n2of spin-waves in FM layers. This makes frequency completely determined by resonator's\ndesigned eigen-frequency, regardless of the mechanism of excitors. The loss of frequency\ntunability reduces non-linearity and linewidth considerably. After the experimental and\nsimulation description, we present the excited modes in the chosen resonator, then we dis-\ncuss the reduced non-linearity of NCMR-STO. For the resonator, we used a nano-pillar with\ntwo free FM layers, where magnetostatic dipolar \feld provides inter-layer coupling with two\ncoupled-oscillations characteristic modes.31The dynamics of coupled free FM layers are of\npractical interest for high-frequency emission at low applied \feld ( <500 Oe), linewidth nar-\nrowing, and doubling of magnetization precession frequency in resistance oscillations.32{35\nMost importantly, the loss of current-tunability of frequency can be compensated by the\ncontrollability of inter-layer relative angle.\nII. EXPERIMENT AND SIMULATION\nThe \flm stack with designed thickness in nm was: thermally-oxidized silicon sub-\nstrate/electrode layer(Ta 5/Cu 200/Ta 40/chemical-mechanical polishing)/milling 5/Ta\n3/Ru 2/Fe 50Co505/Al 1.3/IAO 20 seconds exposure time/Al 0.3/Fe 50Co505/capping (Cu\n10/Ru 10). The \flm was deposited by magnetron and ion-beam sputtering in the chambers\ndescribed before.19Subsequently, \flms were vacuum-annealed at 270\u000eC and 400\u000eC for 1.5\nhours each with 10-kOe magnetic \feld. The choice of capping material and additional metal\nAl insertion were optimized with annealing process for lower Resistance-Area (RA) product\nand enhanced MR ratio, based on previous work.19,30Current-Perpendicular-to-Plane (CPP)\npillars of elliptical cross-section were micro-fabricated by Ar+ion-milling and electron-beam\nlithography. RA was found from the slope of 4-probe dc resistance vs. area inverse (R-1/A)\nline, and compared with Current-In-Plane-Tunneling (CIPT) measurement of unpatterned\n\flms.\nThe STO microwave emissions were measured by a 26-GHz spectrum analyzer under bias-\ning from a bias-T, and scattering at the measurement probe was measured by a network\nanalyzer. Geometry, angles and coordinates de\fnitions are summarized in Fig. 1(a). The\npositive current was de\fned to be electrons \rowing up. We are presenting the detailed\nmeasurements of a 320 nm \u0002160 nm pillar at \u0018= 60\u000e, although results presented later were\n3qualitatively similar among samples.\nMicromagnetic simulation was done by Nmag, a \fnite-element-method simulator.36Calcula-\ntion geometry consisted of two ellipses same as experimental design separated by 1 nm, and\nNCs were included as 1-nm-radius cylindrical contacts between the two layers. Inter-layer\ndipolar \feld was included through demagnetization \feld calculation. Material parameters\nare: sti\u000bness constant of 2 :3\u000210\u000010erg/cm3, saturation magnetization of 1930 emu/cm3,37\nwith Gilbert damping constant of 0.02, and an unphysical spin polarization of 100%. Mesh\nsize away from NCs was set to 5 nm, and it changed to 0.7 nm inside NCs. For hysteresis\nloops,\u0018-dependence was calculated for 20 randomly-placed NCs. For qualitative understand-\ning of STO dynamics, we compared the cases of zero and four NCs, under the experimental\nconditions of 250-Oe \feld applied at \u0018= 60\u000e. The current pro\fle was approximated to be\ncon\fned in NCs with con\fnement extending 1 nm away from middle of NC into FM layers,\nas most of the voltage drop will be on this region,38although more accurate representation\nis needed.39The total current was +17.5 mA and the current distribution was calculated by\nassuming that a single NC and tunnel barrier resistances are 600 \n and 500 \n, respectively.18\nIII. RESULTS AND DISCUSSION\nRA found from the slope of 4-probe R-1/A line, and CIPT measurements were 0.2 and\n0.3 \n\u0001\u0016m2, respectively. The bias dependence of 4-probe di\u000berential resistance at parallel\nmagnetization state (inset in Fig. 1(b)) was relatively \rat. The resistance temperature\ndependence of similarly conditioned \flms also showed metallic-transport character. This\nindicates that conduction is dominated by transport through NCs and not by tunneling\nthrough oxide barrier. Previously, high-temperature annealing ( >380\u000eC) was hindered\nby manganese di\u000busion from pinning antiferromagnet towards NOL.19Better NOL barrier\nquality and purer NCs were obtained in this report by using manganese-free structure for\nhigher annealing temperature, in addition to insertion and capping layers optimizations.30\nFigure 1(b) shows the two-probe resistance vs. magnetic \feld (R-H) applied at \u0018= 0\u000e\nand 60\u000emeasured at the same position as STO measurements. From the switching \felds\nof easy-axis R-H, interlayer dipolar coupling \feld ( Hic) is estimated at 400 Oe. It is in\nagreement with the estimation of cross-demagnetization,40Hic= 4\u0019\u001a12Ms= 433 Oe, where\n4Ms=1750 emu/cm3is the measured saturation magnetization, and \u001a12= 0:0197 is the cross-\ndemagnetization factor. This dipole inter-layer coupling can be considered equivalent to\nthe usual bilinear coupling through a metal spacer ( J=\u00002dM sHic=\u00000:7 erg/cm2,41\nde\fned negative for anti-parallel coupling). The contribution from coupling through NCs\nand spacer roughness to magnetostatic energy can be neglected. Ferromagnetic coupling\nwas found to be small from free-layer magnetization-loop shift of unpatterned spin-valve\n\flms (IrMn/FeCo(Pinned)/NOL-NCs/FeCo(Free)), with JNCs= 0:01\u00000:02 erg/cm2.19\nMicromagnetic simulation reproduced static R-H, Hicestimation, and the reduction of AP-\nto-P plateau width by NCs (Fig. 1(c)). We chose for oscillation measurements the pillar\nthat had the closest R-H curve to micromagnetic simulation, which had 11.1% MR ratio\nand 0.17 \n\u0016m2RA product. Due to large pillar size, uniform rotational switching was not\nreproducible for \feld applied along easy axis ( \u0018= 0\u000e). At tilted angles, the magnetization\nrotated as a single domain. Thus we are presenting tilted angles results of oscillations\n(Fig. 2).\nLargest power microwave oscillations were observed for \u0018= 60\u000eat\u001915 GHz when applying\nhigh currents. Sample power spectrum with a Lorentzian peak \ftting is shown in Fig. 2(a).\nThere is a drop in resistance at I dc= 14.7 mA accompanied with a jump in oscillation\nfrequency,fosc, a narrowing in full-width-at-half-maximum, \u0001 f, and increase in integrated\npower, P int, indicating a change into auto-oscillation mode,42with a mechanism similar\nto STOs based on pin-hole tunnel junctions (Fig. 2(c)).9Linear \fts to normalized inverse\npower, 1=p, at sub-threshold gave a threshold current, I thof 14.74 mA. The presense of two\nfrequency branches at sub-threshold and high-current regions can be ascribed to edge and\ncenter modes in elliptical geometries.43The highest oscillation power is 0.4 nW (1.6 nW if\ncorrected for impedance mismatch) giving a precession amplitude ( \u0012p) of 2{3\u000e, whereas the\nlowest \u0001fis 3 MHz corresponding to a quality factor of 5000.\nRegarding the excited oscillations, the possible coupled oscillations or spin-waves in two\nlayers of free spins are the optical (anti-phase) mode (OM) and the acoustic (in-phase)\nmode (AM).31Modes frequencies can be found from the solution to coupled Bloch equations\nof the two layers with e\u000bective \feld determined from the free energy.41,44,45We considered\nonly the main contributions of Zeeman energy, \flm demagnetization, and interlayer dipolar\ncoupling. The optical and acoustic eigen-frequencies of an in-plane magnetization precession\ncan be simpli\fed to:\n5\u0012fac\n\r=2\u0019\u00132\n= (Hcos + 4\u0019M s\u00002Hic(cos \u0001\u0012+ 1)) (Hcos ) + 8H2\niccos \u0001\u0012; (1a)\n\u0012fop\n\r=2\u0019\u00132\n= (Hcos + 4\u0019M s\u00002Hic(cos \u0001\u0012+ 1)) (Hcos \u00004Hiccos \u0001\u0012); (1b)\nwhere\r=2\u0019= 2:8 MHz/Oe is the gyromagnetic ratio, and other symbols are de\fned in\nFig. 1(a). We con\frmed the presence of the weaker-amplitude AM (Fig. 2(b)). Using Hic\n= 400 Oe and \u0001 \u0012= 130{150\u000efrom measured R-H and micromagnetic simulation results\ninfopof 14.0{16.1 GHz and facof 3.9{3.5 GHz, which agrees with the observed spectrum.\nThe frequency of OM depends mostly on the coupling strength and relative angle between\nthe layers (the last term on right in Eq. 1b). The weak dependence of foscagainst I dcand\nH (\u0019\u00001:3 MHz/Oe not shown) supports that foscis determined mainly by excitation of\nan OM spin-wave. The maximum fosc(H= 0;\u0001\u0012= 180\u000e) from other devices was 17.8 GHz\nwhich corresponds to Hic= 460 Oe, in agreement with the estimation from the correspond-\ning R-H curve. To increase the oscillation frequency, we reduced the size of elliptical pillars\nto 160 nm\u000280 nm. At I dc= 2.8 mA, H = 185 Oe, and \u0018= 50\u000e, the resulting oscillations\nhadfosc,df=dI dc, and \u0001fof 23.3 GHz, <4 MHz/mA, and 1.3-MHz linewidth, respectively.\nThe corresponding quality factor is more than 17,000.\nAlthough the presented frequency of OM is higher than AM, the measured and simulated\npeak intensities of OM are much larger (Figs. 2(b) and 3(a,c)). This is due to two reasons.\nIn OM, anti-phase dynamics maximize dynamic MR change.33,34In comparison, OM am-\nplitude as measured by Brillouin light scattering is reduced due to canceling contributions\nto light scattering cross-section.41Secondly, the energy required to excite OM is smaller in\nanti-coupled ( J <0) harmonic oscillators. The average energy di\u000berence between AM and\nOM with equal precession amplitudes is: hEaci\u0000hEopi=\u0000J(\u000em1\u0001\u000em2), where\u000em1\u0001\u000em2\nis the characteristic-mode's dimensionless power. For J < 0, excitation of AM requires\nhigher energy than same-amplitude OM.\nIt should be noted that Eqs. 1 were derived for in\fnite-wavelength limit ( i.e.wave-vector-\nthickness product qd\u00190). Quantitative corrections due to dynamic dipolar coupling\nbetween propagating spin-waves cannot be ignored because qd= 1:74 from simulation pre-\nsented later.41,44,45But due to experimental uncertainty in determining Hiccos \u0001\u0012, exact\nquantitative comparison becomes di\u000ecult, and the main conclusions are not changed.\nThe linewidth broadening of STOs compared to linear auto-oscillators is understood to\n6be due to amplitude-phase coupling,7which is expressed by the nonlinearity parameter\n(\u0017).46The nonlinearity of presented results is \u0017= (Idc=\u0000g)(df=dI dc)\u0019\u00000:16. The natu-\nral FMR linewidth (\u0000 g= 934 MHz) is obtained from linear extrapolation of \u0001 fto zero\ncurrent at sub-threshold,47and the agility of oscillation frequency in current ( df=dI dc) was\n-9.6 MHz/mA. This non-linearity is one order of magnitude smaller than other reported\nvalues.15,47,48Because the nonlinearity is very small, the sudden change of oscillation into a\nsingle mode at threshold hinders the applicability of determining \u0017from \u0001f-1=pplot.15The\n\u0001f-1=pslopes were 3.8 and 19.4 MHz =(mA2\u0001\u0016W) for above-threshold and below-threshold\nregimes, respectively.\nNCMR-STO usually showed relatively small agility (16{18 MHz/mA)21,23,25compared with\nother TMR-STOs, leading to smaller nonlinearity and narrow linewidth. Possible reasons for\nlowered agility and nonlinearity in this report can be the optimized fabrication process with\npurer NCs,19,30the coupled oscillations of two layers,33,35,49and the tilted magnetization\nangle away from easy axis.50However, the loss of agility and small linewidths were obtained\nfor various angles and frequencies, which indicates that the improved purity of NCs is the\nmain factor.\nWe used micromagnetic simulation to show how nano-magnets with NCs worked as a\nresonator-excitor pair. Results shown in Fig. 3 agree reasonably with experimental data.\nIn the case of no NCs (upper part in Figs. 3(a{c)) oscillation frequency is similar to the\ncalculated and measured ones, with the optical mode being the dominant component. But\nprecession amplitude ( \u0012p= 0:05\u000e) is very small compared with the experimental value (2{\n3\u000e). With the insertion of four NCs, optical spin-waves were emitted from NCs (Fig. 3(d)),\nand increased \u0012pto 2\u000e(lower part in Fig. 3(a{c)). The origin of perturbation near NCs\nis of similar origin to previous reports.51,52The domain-wall is pushed outside NC-region\ninto the ferromagnetic layer and starts to oscillate at high frequency between N\u0013 eel and\nBloch walls (250 GHz for the chosen geometry and current density)(Fig. 3(e)). These very\nhigh frequency oscillations were localized up to 5{10 nm away from NC (Fig. 3(f)). The\nlocalized precession acts as a point source that generates spin-waves propagating radially at\nthe characteristic mode of the system, which is an optical spin-wave.\nThe implication on the nature of current-induced dynamics is that magnetization preces-\nsion is not induced by STT directly. In Fig. 2(c), at \frst going above I th, localized STT\nincreases amplitude, compensates damping around NCs and increases local precession am-\n7plitude. When local precession amplitude saturates, at I = 15.6 mA, increase of current\nand STT will not change \u0012p, leading to loss of agility and narrowing of linewidth. At this\nstable regime, magnetic layers act as a resonator that is excited by the energy coming\nfrom the point sources at NCs. This makes the presented oscillator similar to a classical\nauto-oscillator, and results in a considerable reduction in linewidth.\nIV. CONCLUSION\nIn conclusion, we presented the measured spin-torque-driven oscillations of a spin-torque\noscillator with nano-contacts between two free ferromagnetic layers coupled antiferrmagneti-\ncally with dipolar-\feld. Resulting oscillation character agrees with propagating optical-mode\nspin-waves. In micromagnetic simulation, inclusion of NCs with localized current density\nshowed that NCs work as point sources, and optical-mode spin-waves were excited in the\nferromagnetic layers. The ferromagnetic layers act as a resonator that is decoupled from\nmechanism of excitation point sources. Since oscillation is not driven by spin-transfer torque,\nlinewidth decreased.\nAlthough we studied dual ferromagnetic layers in this report, same e\u000bect should be the origin\nfor the small non-linearity and linewidth narrowing common in NCMR and pin-hole-TMR\nSTOs.9,13,21,23,25So, by utilizing NCs and using a magnetic resonator that has a large os-\ncillation amplitude, e.g.tilted-anisotropy ferromagnets,53we expect to increase both power\nand quality factor for applications.\nACKNOWLEDGMENTS\nAuthors would like to thank Dr. Andrei Slavin for his insightful comments and discussion.\nThis work was supported by SCOPE program (contract No. 0159-0058) from Ministry of\nInternal A\u000bairs and Communications.\n\u0003mahdawi@ecei.tohoku.ac.jp\n81J. C. Slonczewski, \\Electronic device using magnetic components,\" U.S. Patent No. 5,695,864\n(1997).\n2S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley, R. J. Schoelkopf, R. A. Buhrman,\nand D. C. Ralph, Nature 425, 380{383 (2003).\n3A. A. Tulapurkar, Y. Suzuki, A. Fukushima, H. Kubota, H. Maehara, K. Tsunekawa, D. D.\nDjayaprawira, N. Watanabe, and S. Yuasa, Nature 438, 339 (2005).\n4S. Miwa, S. Ishibashi, H. Tomita, T. Nozaki, E. Tamura, K. Ando, N. Mizuochi, T. Saruya,\nH. Kubota, K. Yakushiji, T. Taniguchi, H. Imamura, A. Fukushima, S. Yuasa, and Y. Suzuki,\nNature Materials 13, 50 (2014).\n5L. Fu, Z. X. Cao, S. Hemour, K. Wu, D. Houssameddine, W. Lu, S. Pistorius, Y. S. Gui, and\nC. Hu, Applied Physics Letters 101, 232406 (2012).\n6L. Fu, W. Lu, D. R. Herrera, D. F. Tapia, Y. S. Gui, S. Pistorius, and C. Hu, Applied Physics\nLetters 105, 122406 (2014).\n7J. Kim, V. Tiberkevich, and A. N. Slavin, Physical Review Letters 100, 017207 (2008).\n8B. Georges, J. Grollier, M. Darques, V. Cros, C. Deranlot, B. Marcilhac, G. Faini, and A. Fert,\nPhysical Review Letters 101, 017201 (2008).\n9D. Houssameddine, S. H. Florez, J. A. Katine, J. Michel, U. Ebels, D. Mauri, O. Ozatay,\nB. Delaet, B. Viala, L. Folks, B. D. Terris, and M. Cyrille, Applied Physics Letters 93, 022505\n(2008).\n10H. Maehara, \\Oscillation element and method for manufacturing oscillation element,\" U.S.\nPatent No. 8,836,438 (2014).\n11H. Maehara, H. Kubota, Y. Suzuki, T. Seki, K. Nishimura, Y. Nagamine, K. Tsunekawa,\nA. Fukushima, A. M. Deac, K. Ando, and S. Yuasa, Applied Physics Express 6, 113005 (2013).\n12H. Maehara, H. Kubota, Y. Suzuki, T. Seki, K. Nishimura, Y. Nagamine, K. Tsunekawa,\nA. Fukushima, H. Arai, T. Taniguchi, H. Imamura, K. Ando, and S. Yuasa, Applied Physics\nExpress 7, 023003 (2014).\n13D. Houssameddine, U. Ebels, B. Dieny, K. Garello, J. Michel, B. Delaet, B. Viala, M. Cyrille,\nJ. A. Katine, and D. Mauri, Physical Review Letters 102, 257202 (2009).\n14T. Devolder, L. Bianchini, J. Kim, P. Crozat, C. Chappert, S. Cornelissen, M. O. d. Beeck, and\nL. Lagae, Journal of Applied Physics 106, 103921 (2009).\n915K. Kudo, T. Nagasawa, R. Sato, and K. Mizushima, Journal of Applied Physics 105, 07D105\n(2009).\n16H. Fukuzawa, H. Yuasa, S. Hashimoto, K. Koi, H. Iwasaki, M. Takagishi, Y. Tanaka, and\nM. Sahashi, IEEE Transactions on Magnetics 40, 2236 (2004).\n17H. N. Fuke, S. Hashimoto, M. Takagishi, H. Iwasaki, S. Kawasaki, K. Miyake, and M. Sahashi,\nIEEE Transactions on Magnetics 43, 2848 (2007).\n18M. Takagishi, H. N. Fuke, S. Hashimoto, H. Iwasaki, S. Kawasaki, R. Shiozaki, and M. Sahashi,\nJournal of Applied Physics 105, 07B725 (2009).\n19Y. Shiokawa, M. Shiota, Y. Watanabe, T. Otsuka, M. Doi, and M. Sahashi, Magnetics, IEEE\nTransactions on 47, 3470 (2011).\n20H. Yuasa, M. Hara, Y. Fuji, and H. Fukuzawa, EPL (Europhysics Letters) 101, 47005 (2013).\n21H. Endo, T. Tanaka, M. Doi, S. Hashimoto, H. N. Fuke, H. Iwasaki, and M. Sahashi, IEEE\nTransactions on Magnetics 45, 3418 (2009).\n22H. Suzuki, H. Endo, T. Nakamura, T. Tanaka, M. Doi, S. Hashimoto, H. N. Fuke, M. Takagishi,\nH. Iwasaki, and M. Sahashi, Journal of Applied Physics 105, 07D124 (2009).\n23H. Suzuki, T. Nakamura, H. Endo, M. Doi, H. Tsukahara, H. Imamura, H. N. Fuke,\nS. Hashimoto, H. Iwasaki, and M. Sahashi, Applied Physics Letters 99, 092507 (2011).\n24M. Doi, H. Endo, K. Shirafuji, S. Kawasaki, M. Sahashi, H. N. Fuke, H. Iwasaki, and H. Ima-\nmura, Journal of Physics D: Applied Physics 44, 092001 (2011).\n25M. Al-Mahdawi, M. Doi, S. Hashimoto, H. N. Fuke, H. Iwasaki, and M. Sahashi, IEEE Trans-\nactions on Magnetics 47, 3380 (2011).\n26M. Al-Mahdawi and M. Sahashi, Applied Physics Letters 104, 032405 (2014).\n27L. R. Tagirov, B. P. Vodopyanov, and K. B. Efetov, Physical Review B 63, 104428 (2001).\n28J. Sato, K. Matsushita, and H. Imamura, Journal of Applied Physics 105, 07D101 (2009).\n29H. Imamura and J. Sato, Journal of Physics: Conference Series 266, 012090 (2011).\n30Y. Shiokawa, J. Jung, T. Otsuka, and M. Sahashi, Journal of Applied Physics 118, 053909\n(2015).\n31P. Gr unberg, Journal of Applied Physics 51, 4338 (1980).\n32T. Seki, H. Tomita, T. Shinjo, and Y. Suzuki, Applied Physics Letters 97, 162508 (2010).\n33T. Moriyama, G. Finocchio, M. Carpentieri, B. Azzerboni, D. C. Ralph, and R. A. Buhrman,\nPhysical Review B 86, 060411 (2012).\n1034P. M. Braganca, K. Pi, R. Zakai, J. R. Childress, and B. A. Gurney, Applied Physics Letters\n103, 232407 (2013).\n35T. Nagasawa, K. Kudo, H. Suto, K. Mizushima, and R. Sato, Applied Physics Letters 105,\n182406 (2014).\n36T. Fischbacher, M. Franchin, G. Bordignon, and H. Fangohr, Magnetics, IEEE Transactions\non43, 2896 (2007).\n37K. Miyake, Y. Okutomi, H. Tsukahara, H. Imamura, and M. Sahashi, Applied Physics Express\n6, 113001 (2013).\n38Y. G. Naidyuk and I. K. Yanson, Point-Contact Spectroscopy (Springer, 2005).\n39N. Strelkov, A. Vedyayev, N. Ryzhanova, D. Gusakova, L. D. Buda-Prejbeanu, M. Chshiev,\nS. Amara, N. de Mestier, C. Baraduc, and B. Dieny, Physical Review B 84, 024416 (2011).\n40O. Dmytriiev, T. Meitzler, E. Bankowski, A. Slavin, and V. Tiberkevich, Journal of Physics:\nCondensed Matter 22, 136001 (2010).\n41M. Grimsditch, S. Kumar, and E. E. Fullerton, Physical Review B 54, 3385 (1996).\n42A. Slavin and V. Tiberkevich, Magnetics, IEEE Transactions on 45, 1875 (2009).\n43A. M. Deac, A. Fukushima, H. Kubota, H. Maehara, Y. Suzuki, S. Yuasa, Y. Nagamine,\nK. Tsunekawa, D. D. Djayaprawira, and N. Watanabe, Nature Physics 4, 803 (2008).\n44J. F. Cochran, J. Rudd, W. B. Muir, B. Heinrich, and Z. Celinski, Physical Review B 42, 508\n(1990).\n45R. Zivieri, L. Giovannini, and F. Nizzoli, Physical Review B 62, 14950 (2000).\n46V. S. Tiberkevich, A. N. Slavin, and J. Kim, Physical Review B 78, 092401 (2008).\n47B. Georges, J. Grollier, V. Cros, A. Fert, A. Fukushima, H. Kubota, K. Yakushijin, S. Yuasa,\nand K. Ando, Physical Review B 80, 060404 (2009).\n48L. Bianchini, S. Cornelissen, J. Kim, T. Devolder, W. v. Roy, L. Lagae, and C. Chappert,\nApplied Physics Letters 97, 032502 (2010).\n49D. Gusakova, M. Quinsat, J. F. Sierra, U. Ebels, B. Dieny, L. D. Buda-Prejbeanu, M. Cyrille,\nV. Tiberkevich, and A. N. Slavin, Applied Physics Letters 99, 052501 (2011).\n50K. Mizushima, T. Nagasawa, K. Kudo, Y. Saito, and R. Sato, Applied Physics Letters 94,\n152501 (2009).\n51H. Arai, H. Tsukahara, and H. Imamura, Applied Physics Letters 101, 092405 (2012).\n1152K. Matsushita, J. Sato, H. Imamura, and M. Sasaki, Journal of Physics: Conference Series\n200, 042016 (2010).\n53T. N. Anh Nguyen, N. Benatmane, V. Fallahi, Y. Fang, S. M. Mohseni, R. K. Dumas, and\nJ.\u0017Akerman, Journal of Magnetism and Magnetic Materials 324, 3929 (2012).\n12FIG. 1. (a) A schematic of the elliptical pillar geometry and de\fnitions of coordinates and angles.\n(b) Two-probe resistance vs. \feld applied at \u0018= 0\u000eand 60\u000e. Inset shows the bias dependence of\ndi\u000berential resistance in parallel magnetizations state. (c) Normalized MR found from micromag-\nnetics simulation, for the cases of no NCs (dashed lines), and with 20 NCs (solid lines).\n13FIG. 2. (a) Representative oscillation power spectrum at \u0018= 60\u000ewith Lorentzian peak \ftting\n(solid line). (b) The acoustic and optical modes of coupled oscillations were observed near 3 GHz\nand 15 GHz, respectively. (c) Current dependencies of oscillation characteristics. Same symbols in\nfoscand \u0001fpanels correspond to each other.\n14FIG. 3. (a{c) Comparison of micromagnetic dynamics with (top part) or without (bottom part)\nNCs atH= 250 Oe, \u0018= 60\u000e, Idc= 17:5 mA . (a) Normalized magnetization's y-component\nof the total system ( my). (b) Magnetization angle of top and bottom layers at ( x;y) = (0;0).\n(c) Spectrum of mytransformed from 250-ns (15-ns) duration of dynamics without (with) NCs.\n(d) Time snapshots with the color representing [ my(r;t)\u0000my(r;0)]. Optical-mode spin-waves\nare emitted from localized excitation at NCs. (e) The origin of localized excitation is a con\fned\ndomain-wall that is oscillating at 250-GHz. (f) Power pro\fle of localized precession around the\ntop-right NC.\n15"    },    {        "title": "1811.01168v1.Ferromagnetic_resonance_in_the_complex_permeability_of_an_Fe__3_O__4__nanosuspension_at_radio_and_microwave_frequencies.pdf",        "content": "arXiv:1811.01168v1  [cond-mat.mes-hall]  3 Nov 2018Ferromagnetic resonance in the complex permeability of an F e3O4nanosuspension at\nradio and microwave frequencies\nJ. Dubreuil and J. S. Bobowski∗\nDepartment of Physics, University of British Columbia, Kel owna, British Columbia, V1V 1V7, Canada\n(Dated: November 6, 2018)\nThe complex permeability of an iron-oxide nanosuspension h as been measured as a function of\nmagnetic ﬁeld strength at RF and microwave frequencies usin g a loop-gap resonator. The particles\nwere suspendedin water andhadan8-nmdiameter Fe 3O4core thatwas coated byDextran. The real\npartofthepermeabilityincreased sharplybeyondafrequen cy-dependentthresholdvalueofthestatic\nmagnetic ﬁeld before saturating. Just beyond this threshol d ﬁeld, there was a peak in the imaginary\npart of the permeability. The permeability measurements, w hich exhibited features associated with\nferromagnetic resonance, were used to determine the depend ence of the microwave absorption on\nstatic magnetic ﬁeld strength. Using the absorption data, t heg-factor of the nanosuspension was\nfound to be 1 .86±0.07.\nIntroduction . Since the discovery of the metal-to-\ninsulator transition in magnetite, Fe 3O4has been one of\nthe most widely studied magnetic systems.1,2Upon cool-\ning magnetite below the so-called Verwey transition tem-\nperautre TV≈125K, its resistivity abruptly increases by\nup to two orders of magnitude in the best single crystals.\nAs described in a review by Walz, there are numerous\nother anomalies in the physical properties of Fe 3O4at\nTV, such as steps in the magnetization, initial suscepti-\nbility, and thermal expansion, and a spike in the speciﬁc\nheat. However, despite decades of intense research, there\nis not yet a consensus understanding of the microscopic\nmechanisms governing the Verwey transition3and it is\nstill being actively investigated both experimentally4–11\nand theoretically12–14.\nShortly after the discovery of the Verwey transi-\ntion, Bickfordobservedresonantmicrowaveabsorptionin\nFe3O4single crystals.15Based on the theory of ferromag-\nnetic resoance developed by Kittel,16,17these data were\nused to determine the crystalline anisotropy and g-factor\nof magnetite. More recently, there has been consider-\nable interest in the magnetic properties of Fe 3O4thin\nﬁlms18–20and nanoparticles21,22. Fe3O4nanaoparticles,\nin particular, have attracted a lot of attention because of\ntheir potential use in biomedical,23environmental,24and\nother applications.25\nIn this paper, we describe our measurements of the\ncomplex permeability of a suspension of 8-nm diame-\nter Fe 3O4particles as a function static magnetic ﬁeld\nstrength. The measurements were made using a novel\ncompact resonator whose resonant frequency could be\ntuned between 500 and 1300MHz. Many previous stud-\nies of the microwave properties of Fe 3O4nanoparticles\nhave used less sensitive non-resonant techniques, often\nwith larger diameter partilces (150 to 300nm). Further-\nmore, in these studies the authors were primarily in-\nterested in the microwave absorption properties in zero\nstatic magnetic ﬁeld.21,22Noginov et al.studied the elec-\ntron magnetic resonance (EMR) signal in a suspension\nof 9-nm diameter Fe 3O4nanoparticles and identiﬁed fea-\ntures which marked quantum transitions.26Shankar et\nal.likewise studied magnetite nanoparticles that were 9to 10nm in size. These authors investigated the tem-\nperature dependence of the ﬁeld-cooled and zero-ﬁeld\ncooled ferromagnetic resonance lineshapes at 9 .45GHz\nand identiﬁed a spin glass transition at 46K.27The mea-\nsurements that we present oﬀer a detailed determination\nof the ﬁeld dependence of the complex permeability of\nan Fe3O4nanosuspension over a relatively wide range of\nRF frequencies.\nThe organization of the paper is as follows: We ﬁrst\nbrieﬂy introduce the the loop-gap resonator (LGR) and\nthe measurement principle. The experimental design\nis then described along with a discussion of its advan-\ntageous and limitations. The experimental results are\nthen presented and compared to complementary mea-\nsurements on Fe 3O4bulk single crystals and thin ﬁlms.\nFinally, the key ﬁndings are brieﬂy summarized.\nLop-gap resonator . The measurements presetned in\nthis paper were made using a loop-gap resonator (LGR).\nIn its mostbasic form, the LGR is madeby cutting anar-\nrowslit alongthe lengthofahollowconductingtube.28,29\nThe eﬀective capacitance of the gap and inductance of\nthe bore determine the resontant frequency f0(typically\nbetween 100MHz and 2GHz). The quality factor or Q\nof the resonance is determined by electromagnetic skin\ndepth and radiative losses.30The radiative losses asso-\nciated with conventional cylindrical LGRs can be sup-\npressed using a toroidal geometry which completely con-\nﬁnesthe magneticﬂux within the boreofthe resonator.31\nThe advantagesofthe LGR include resonatordimensions\nthat are many times smaller than the free-space wave-\nlength of the resonant frequency and good isolation be-\ntween RF electric and magnetic ﬁelds.\nInserting an insulatorinto the gap of a LGR allows one\nto determine the material’s complex permittivity. The\nshift in the resonant frequency is determined by the real\npart of the permittivity while the imaginary part lowers\nthe quality factor of the resonance.30,32In an analogous\nway, the complex permeability of a magnetic material\nloaded in the bore of a LGR can likewise be determined\nfrom the changes in f0andQ.33,34This paper describes\nthe use of a toroidal LGR to measure the change in the\ncomplex permeability of an Fe 3O4nanosuspension as the2\n(a)\n (b)\nFIG. 1. (a) Photograph of the two halves of the toroidal LGR wi th an asymmetrically-placed gap. A coupling loop passes\nthrough a small hole in the top half of the resonator. The outs ide diameter of the LGR is 3 .8cm (1.5inches). (b) Schematic\ndiagram of the cross-section of the toroidal LGR. The resona nt frequency is set by a low-loss dielectric in the gap of the\nresonator. The magnetic nanosupension partially ﬁlls the b ore of the resonator. A uniform static magnetic ﬁeld B0is applied\nperpendicularly to the RF magnetic ﬁeld B.\nstrength of an applied static magnetic ﬁeld is varied.\nExperimental method . Because a new experimental\nmethod was designed speciﬁcally for this measurement,\nwe brieﬂy outline some of the details. Figure 1 shows a\nphotograph of the aluminum toroidal LGR used in our\nexperiments along with a schematic diagram of its cross-\nsection. An ac magnetic ﬂux is introduced into the bore\nof the LGR using a coupling loop at the end of a short\nsection of coaxial transmission line. The opposite end of\nthe transmission line is connected to one port of a vector\nnetwork analyzer(VNA). The VNA supplies a signal and\nmeasures what fraction of the incident signal is reﬂected\nback as a function of frequency. A full equivalent-circuit\nmodel of the experimental setup in shown in Fig. 2.35\nAs shown in Fig. 1(b), the Fe 3O4nanosuspension only\npartially ﬁlls the bore of the LGR. By the symmetry of\nthe toroidal geometry, the current density on the inner\nbore wall is uniform. That is, the current density is the\nsame in the ﬁlled and unﬁlled sections of the bore. On\nthe other hand, the magnetic ﬂux within cross-sections\nof the ﬁlled and unﬁlled regions of the bore are certainly\ndiﬀerent. These observations suggest that the eﬀective\ninductance of the LGR bore can be modeled as the se-\nriescombination Leﬀ=L1+µrL2whereL1= (1−η)L0\nis the inductance of the unﬁlled region, L2=µrηL0is\nthe inductance of the ﬁlled region, ηis the ﬁlling frac-\ntion,L0is the inductance of the bore when η= 0, and\nµr=µ′−jµ′′is the complex relative permeability of the\nnanosuspension ( j=√−1).34Therefore, the eﬀective in-\nductance shown Fig. 2 is complex and can be expressed\nas\nLeﬀ={[1+η(µ′−1)]−jηµ′′}L0.(1)\nAs a result, the impedance jωLeﬀhas a resitive term\ngivenby ηµ′′ωL0that combines in serieswith Reﬀ. Thus,\na non-zero value of µ′′enhances the net resistance and\nsuppresses the resonator Q.Figure 1(b) shows that the gap of the LGR has been\nplaced asymmetrically towards to the top of the res-\nonator. This was done to provide more sample space\nfor the nanosuspension and also to limit, as much as pos-\nsible, fringing electric ﬁelds emanating from the gap from\nreaching the sample. The ﬁgure also shows a low-loss di-\nelectric ﬁlling the gap of the resonator. The dielectric\nis used to adjust the resonance frequency of the LGR.\nChanging the dielectric allowed us to tune the resonant\nfrequency to values between 500 and 1300MHz with-\nout signiﬁcantly reducing Q. The dielectrics used were\nAl2O3, TiO2([100] and [001] substrates), and SrTiO 2.36\nTheCeﬀparameterin the circuit model is predominantly\nset by the dimensions of the slit in the LGR and the di-\nelectric ﬁlling the gap. However, fringing electric ﬁelds\nthat reach the sample also make a small contribution to\nCeﬀsuch that Ceﬀ=C0+εrCfwhereC0+Cfis the ca-\npacitance when the gap is ﬁlled with a low-loss dielectric\nand the resonator bore is empty and εr=ε′−jε′′is the\ncomplex permittivity of the material that partially ﬁlls\nthe resonator bore.37,38\nThe remaining circuit-model parameters are\nReﬀ≈R0/radicalbig\nf/f0whereR0is the eﬀective resistance at\nf0,31L1which is the inductance of the coupling loop,\nandZ0= 50Ω which is both the output impedance\nof the VNA and the characteristic impedance of the\ntransmission line.\nThe circuit in Fig. 2 can be analyzed to determine\nthe impedance Zof the inductively-coupled LGR.34,35\nThe magnitude of reﬂection coeﬃcient at the calibration\nplane of the VNA is then given by\n|S11|=/vextendsingle/vextendsingle/vextendsingle/vextendsingleZ−Z0\nZ+Z0/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (2)\nA total of four successive measurements are required to\ndetermine all of the circuit-model parameters:34(1) A\nmeasurement of S11when the LGR is not in place de-3\nFIG. 2. Complete equivalent circuit of the toroidal LGR coup led to a length of transmission line terminated by a coupling loop.\nA VNA is used to measure the magnitude of the reﬂection coeﬃci ent|S11|at a calibration plane established at the opposite\nend of the transmission line.\ntermines L1. (2) A measurement of |S11|when the bore\nof the resonator is empty determines f0,Q, andM2\n0/R0\nwhereM0is the mutual inductance between L1andL0.\n(3) A measurement of |S11|when the bore is ﬁlled with\ndeionized water, assuming a known ﬁlling fraction ηand\nεrof water, is used to determine the ratio Cf/C0. (4) Fi-\nnally, a measurement of |S11|when the bore is partially\nﬁlledwiththeFe 3O4nanosuspensionisusedtodetermine\nµ′andµ′′.\nThe ﬁnal measurement assumes that the complex per-\nmittivity of the nanosuspension is known. Because we\nhave not precisely determined εrof the nanosuspension,\nour measurements cannot be used to extract the abso-\nlute complex permeability of our sample. However, the\nchange in |S11|in the presence of an applied static mag-\nnetic ﬁeld is very insensitive to the values of both εrand\nCf/C0. Therefore, our measurements can precisely de-\ntermine the dependence of the nanosuspension’s complex\npermeability( µ′−jµ′′)onthestrengthofthestaticﬁeld.\nIn our experiments, the static magnetic ﬁeld was gener-\nated using a solenoid wound using 18-AWG copper wire.\nThe solenoidhad a bore diameter of 7 .6cm and produced\na magnetic ﬁeld of 50mT when supplied with 4A of cur-\nrent.\nNanosuspension . The nanosuspension used in our ex-\nperiments, called Molday ION, is commercially available\nfrom BioPhysics Assay Laboratory, Inc.39It is a suspen-\nsion of Dextran-coated Fe 3O4nanoparticles in distilled\nwater. The mean diameter of the Fe 3O4core is 8nm\nand Molday ION contains 10mg of Fe per milli-liter of\nsuspension.40The Dextran coating is used to suppress\namalagamation of the Fe 3O4nanoparticles and to in-\ncrease the biocompatibility of the suspension for use in\nbiomedical applications. Molday ION is primarily used\nas an MRI contrastagent (see, for example, Refs. 41–43).\nAll of our permeability measurements were made using\n2mL of the Molday ION suspension.\nExperimental results . Figure 3 shows the measured|S11|whenthe resonatorborewasempty, ﬁlled with 2mL\nof water, and ﬁlled with 2mL of the Fe 3O4nanosuspen-\nsion. For these measurements, the LGR gap was ﬁlled\nwith a single-crystal Al 2O3plate. The LGR was also\nequipped with a platinum resistance thermometer. All\n|S11|measurements reported in this paper were taken at\na LGR temperature of 25 .0±0.1◦C.\nThere are eight nearly-indistinguishable plots of |S11|\nmeasured while the LGR bore was empty. Between each\nmeasurement, the two halves of the LGR (see Fig 1(a))\nwere completely separated. By adjusting the torque ap-\nplied to the bolts used to assemble the LGR while watch-\ning the VNA display, it was possible to very reproducibly\nset the empty-bore resonant frequency.\nThe solid lines in the left-hand plot show |S11|after\nadding 2mL of deionized water to the resonator bore.\nAdding the water to the bore reduced the resonant fre-\nquency by about 17%. This eﬀect is due to fringing elec-\ntric ﬁelds from the gap of the LGR that extend into\nthe water. Water has a relatively large dielectric con-\nstant (εr≈78 at 25◦C and low frequencies) such that\nthe contribution from εrCfbecomes non-negligible. Four\nmeasurements of |S11|for the water-ﬁlled resonator are\nshown. Between each measurement, the water was com-\npletely drained from the bore of the resonator and the\nresonator was completely disassembled. The data show\nthat the |S11|measurements are reproducible.\nThe dashed lines in Fig. 3 show |S11|measured when\n2mLoftheFe 3O4isaddedtotheLGR bore. Onceagain,\nrepeated measurements gave consistent results. Rela-\ntiveto the watermeasurements, the nanosuspensiondata\nshow a slight increase in the resonant frequency. This\nshift in frequency is not due to a magnetic eﬀect. Rather,\nit is caused by a nanosuspension dielectric constant that\nis slightly less than that of water. The volume of water\ndisplaced by the nanoparticles results in a lower eﬀective\ndielectric constant of the suspension and shifts the reso-\nnance to a higher frequency. This eﬀect completes with,4\n1.15 1.20 1.250.880.900.920.940.960.981.00\nFrequency (GHz)|S11,ﬁlled|\nwater\nUSPIO\n1.43 1.45 1.470.00.20.40.60.81.0\nFrequency (GHz)|S11,empty|\nFIG. 3. The right-hand plot shows the resonance of the\ntoroidal LGR when its bore was empty. There are eight\nnearly-identicalmeasurementsof |S11,empty|shownintheplot.\nThe left-had plot shows measured reﬂection coeﬃcient when\nthe bore of the resonator was ﬁlled with 2mL of water (solid\nlines) and 2mL of the Fe 3O4nanosuspension (dashed lines).\nThere are four measurements each for a water-ﬁlled and\nnanosuspension-ﬁlled LGR. All of these data were obtained\nwith an Al 2O3plate in the gap of the resonator.\nand dominates, the expected paramagnetic permeability\nof the nanosuspension which would tend to lower the res-\nonant frequency relative to water. For this reason, the\nabsolute permeability of the nanosuspension cannot be\ndetermined without precise knowledge of the permittiv-\nity ofboth waterandthe nanosuspensionat the measure-\nment frequency. However, in an applied magnetic ﬁeld,\nonly changes in nanosuspension’s permeability can result\nin a change to |S11|. Therefore, the observed magnetic\nﬁeld dependence of |S11|can be used to determine how\nµ′andµ′′deviate from their zero-ﬁeld values.\nFigure 4 shows the evolution |S11|as the static mag-\nnetic ﬁeld strength applied to the nanosuspension-ﬁlled\nLGR is varied. The data show the full range of magnetic\nﬁeld strengths explored, but only a subset of the values\ntested. The solid lines correspond |S11|measured while\nsweeping from high ﬁeld to low ﬁeld and the dashed lines\ncorrespond to a low-to-high ﬁeld sweep. The overlap of\nthe solid and dashed lines conﬁrms that there was no\nappreciable temperature drift of the LGR/sample during\nthe magnetic ﬁeld sweep. We did identical magnetic ﬁeld\nsweeps when the bore of the resonator was empty and\nﬁlled with 2mL of water. In both cases, |S11|did not de-\nviate from its zero-ﬁeld response at any of the magnetic\nﬁeld strengths used in our experiments.\nBeforediscussingthequantitativeresults,weﬁrstqual-1.16 1 .18 1 .20 1 .22 1 .24 0.90 0.92 0.94 0.96 0.98 1.00 \nFrequency (GHz) |S11 |\n46 .4mT \n40.6 \n34.8 \n29.0 \n23.2 \n17.4 \n11.6 \n5.8 \n0\nFIG. 4. Measurements of |S11|with the bore of the LGR par-\ntially ﬁlled with the nanosuspension at diﬀerent static mag -\nnetic ﬁeld strengths. The solid curves were obtained while r e-\nducing the ﬁeld strength from 50mT to zero and the dashed\ncurves were taken while increasing the strength of the stati c\nﬁeld. For these measurements the gap of the LGR was ﬁlled\nwith an Al 2O3dielectric.\nitatively interpret the |S11|data of Fig. 4. At the lowest\nstatic ﬁeld strengths, the resonant frequency is approxi-\nmatelyconstantwhichimpliesaconstant µ′. Afterreach-\ning a threshold ﬁeld of about 30mT, the |S11|resonant\nfrequency starts to decrease which corresponds to an in-\ncreasing µ′. The depth of the |S11|dip can be used to\ntrackthe changesin Q–the greaterthe depth, the higher\nthe quality factor. Figure 4, shows that Qinitially de-\ncreases with increasing ﬁeld before reaching a minimum\nand then increasing. These data suggest a µ′′that in-\ncreases, peaks, and then decreases as a function of the\napplied static ﬁeld strength. The data imply that µ′′\npeaks at static ﬁeld strength near 40mT.\nThe|S11|data of Fig. 4 were ﬁt to the equivalent cir-\ncuit model and the extracted changes in µ′andµ′′ver-\nsusB0are shown using triangular data points in Fig. 5.\nAs anticipated, ∆ µ′≡µ′(B0)−µ′(0) is initially ﬂat and\nthen increases sharply after exceeding a threshold value\nofB0. After the sharp increase, the ∆ µ′data plateau.\n∆µ′′≡µ′′(B0)−µ′′(0) goes through a broad peak as a\nfunction of magnetic ﬁeld with the peak of ∆ µ′′located\nat the predicted value of B0.\nOur measurements of ∆ µ′and ∆µ′′were done at a\ntotal of ﬁve diﬀerent frequencies. The LGR resonant\nfrequency was tuned by changing the low-loss dielectric\nin the gap. Figure 5 shows all ﬁve data sets for both\n∆µ′and ∆µ′′. The real part of the permeability re-\nmains constant up until a frequency-dependent thresh-\nold magnetic ﬁeld after which ∆ µ′rapidly increases. For\nthe three lowest measurement frequencies, ∆ µ′saturates5\n(a)0 10 20 30 40 0.000 0.002 0.004 0.006 0.008 \nMagnetic Field (mT) ∆µ′=µ′(B0)−µ′(0) 1288MHz \n1197 \n827 \n595 \n493 \n10 \nMagnetic Field (mT) Magnetic Field (mT) (b)0 10 20 30 40 −0.002 0.000 0.002 0.004 0.006 0.008 0.010 \nMagnetic Field (mT) ∆µ′′ =µ′′ (B0)−µ′′ (0) \nFIG. 5. The real and imaginary parts of the change in permeabi lity of the Fe 3O4nanosuspension verses static magnetic ﬁeld\nstrength. (a) The real part of the permeability, initially c onstant, rises sharply beyond a threshold value of the stati c ﬁeld\nstrength and then saturates. The gray markers highlight min ima in ∆ µ′that occur just before the sharp increase. (b) The\nimaginary part of the permeability peaks at a frequency-dep endent value of the magnetic ﬁeld strength.\nafter the sharp rise. The saturation region in the two\nhighest-frequency measurements was not observed be-\ncause we could not apply suﬃcient magnetic ﬁelds us-\ning our solenoid. The ∆ µ′′datasets all initially increase\nbefore reachinga peak value and then falling oﬀ as B0in-\ncreases. At the highest magnetic ﬁelds, we ﬁnd ∆ µ′′<0\nin the three lowest-frequency datasets . Extrapolating\nthe ∆µ′′tails out to very high ﬁeld strengths, where one\nexpectsµ′′→0, would be one way of estimating the\nzero-ﬁeld values of µ′′.\nThe ∆µrdata have been measured with reasonably\nhigh resolution. Given that the Fe content in the\nnanosuspension is relatively dilute, the absolute perme-\nability of the sample is expected to be close to one.\nTherefore, we are measuring changes in the permeability\nwith a precision that is greater than 0 .01%. The error\nbars in the ∆ µ′measurements are approximately equal\nto the size of the data points and they are even smaller\nin the ∆µ′′measurements. This precision has allowed us\nto identify a small but distinct feature in the ∆ µ′data.\nJustbeforethe sharpincrease,∆ µ′passesthroughamin-\nimum. These minima are clearly visible in all but the\n1197MHz data and have be identiﬁed using rectangular\nmarkers below the magnetic ﬁeld axis in Fig. 5(a). Al-\nthough wehavenot identiﬁed the sourceofthe minima in\n∆µ′, they could be related to the minimum Bickford ob-\nserved in his original microwave absorption experiments\non bulk Fe 3O4crystals.15,44\nEnergy dissipation, or absorption, in a ferromagnetic\nmaterial is proportional to44\nEd∝/bracketleftbigg/radicalBig\n(µ′)2+(µ′′)2+µ′′/bracketrightbigg−1/2\n. (3)For the relatively dilute Fe 3O4nanosuspension, the\nreal and imaginary parts of the zero-ﬁeld permeability\ncan be written as µ′(0) = 1+ δ1andµ′′(0) =δ2, where\nδ1,δ2≪1. Therefore, the components of permeability\nin a static magnetic ﬁeld are given by µ′= 1+δ1+∆µ′\nandµ′′=δ2+∆µ′′such that, to within a very good ap-\nproximation\nEd∝1+δ1+δ2\n2+∆µ′+∆µ′′\n2. (4)\nThe static ﬁeld dependence of the energy dissipation is,\ntherefore, simply determined from the sum ∆ µ′+∆µ′′.\nA plot of ∆ µ′+∆µ′′as a function of static magnetic\nﬁeld strength is shown in Fig. 6(a). A broad peak in\nthe microwave absorption is clearly visible in all but the\n1288MHz dataset. Rectangular markers in the ﬁgure\nidentify the values of the static ﬁeld B0,pat which the\nabsorption peaks. The plot in Fig. 6(b) shows that B0,p\nvaries linearly with measurement frequency f1. Here,f1\nrepresents the resonant frequency of the LGR when its\nbore is ﬁlled with 2mL of the nanosuspension. Assuming\ngµBB0,p=hf1, whereµBis the Bohr magneton and h\nis Planck’s constant, the data have been ﬁt to a straight\nlinewithzerointerceptsoastoextractabest-ﬁt valuefor\ntheg-factor. Theslopeofthebest-ﬁtline is0 .539±0.019\nwhich corresponds to g= 1.86±0.07. Microwave exper-\niments on thin ﬁlms of Fe 3O4have also reported values\nofgthat are less than two,20whereas experiments on\nbulk single-crystals of magnetite ﬁnd g= 2.12 at room\ntemperature.15\nSummary . There has been growing interest in probing\nthe magnetic properties of thin ﬁlms, nanoparticles, and\nexotic nanostructures.18–22,45–50Using a LGR, we have\ndeveloped a sensitive experimental technique to make6\n(a)0 10 20 30 40 0.000 0.004 0.008 0.012 0.016 \nMagnetic Field (mT) ∆µ′+∆ µ′′ 1288MHz \n1197 \n827 \n595 \n493 \n(b)0 1 2 3 4 5 60.00.51.01.52.02.53.0\nhf 1( ➭eV) µBB0,p( ➭eV) \ng= 2 \nFIG. 6. (a) Plot of the magnetic ﬁeld dependence of the microw ave absorption by the Fe 3O4nanosuspension at the ﬁve\nmeasurement frequencies. (b) Plot of µBB0,pas a function of hf1.B0,pis the static ﬁeld strength at which ∆ µ′+∆µ′′peaks\nandf1is the resonance frequency of the LGR when partially ﬁlled wi th the nanosuspension. The data were ﬁt to a straight\nline with zero intercept. The dashed line has a slope of 0 .5 and corresponds to g= 2.\nprecision measurements of the permeability of a dilute\nFe3O4nanosuspension as the strength of a static mag-\nnetic ﬁeld was varied. The LGR dimensions can be many\ntimes smaller than the free-space wavelength of the reso-\nnant frequency which allows one to work at low frequen-\ncies with relatively small sample volumes. By placing\nlow-loss dielectrics in the gap of the resonator, the mea-\nsurement frequency was tuned from 500 to 1300MHz.\nThe realpartofthe nanosuspension’spermeabilitywas\nobserved to increase sharply after exceeding a frequency-\ndependent threshold magnetic ﬁeld. ∆ µ′then reached\nsaturation at higher magnetic ﬁelds. At four out of the\nﬁve measurement frequencies, a small dip in ∆ µ′was ob-\nserved just before the onset of the sharp rise. The imag-\ninary part of the permeability exhibited a broad peak as\na function of the applied static magnetic ﬁeld strength.The ∆µ′and ∆µ′′measurements were used to extract\nthe magnetic ﬁeld dependence of the nanosuspension’s\nmicrowave absorption. The magnetic ﬁeld B0,pat the\nabsorption peak was plotted as a function of the mea-\nsurement frequency. The resulting linear relationship,\ncharacteristic of a ferromagnetic resonance, yielded a g-\nfactor of 1 .86±0.07.\nACKNOWLEDGMENTS\nWe wish to acknowledge the support of Thomas John-\nsonwhogenerouslyprovidedaccesstoanAgilentE5061A\nVNA.\n∗jake.bobowski@ubc.ca;\nhttps://people.ok.ubc.ca/jbobowsk\n1T. Okamura, Sci. Rep. Tohoku Imp. Univ. 21, 231 (1932).\n2E. J. W. Verwey, Nature 144, 327 (1939).\n3F. Wlaz, J. Phys.: Condens. Matter 14, R285 (2002).\n4N. C. Bartelt, S. Nie, E. Starodub, I. Bernal-Villamil,\nS. Gallego, L. Vergara, K. F. McCarty, and J. de la\nFiguera, Phys. Rev. B 88, 235436 (2013).\n5R. Ramos, M. H. Aguirre, A. Anad´ on, J. Blasco, I. Lucas,\nK. Uchida, P. A. Algarabel, L. Morell´ on, E. Saitoh, and\nM. R. Ibarra, Phys. Rev. B 90, 054422 (2014).\n6M. Taguchi, A. Chainani, S. Ueda, M. Matsunami,\nY. Ishida, R. Eguchi, S. Tsuda, Y. Takata, M. Yabashi,\nK. Tamasaku, Y. Nishino, T. Ishikawa, H. Daimon,S. Todo, H. Tanaka, M. Oura, Y. Senba, H. Ohashi, and\nS. Shin, Phys. Rev. Lett. 115, 256405 (2015).\n7M. S. Senn, J. P. Wright, J. Cumby, and J. P. Attﬁeld,\nPhys. Rev. B 92, 024104 (2015).\n8F. Randi, I. Vergara, F. Novelli, M. Esposito,\nM. Dell’Angela, V. A. M. Brabers, P. Metcalf, R. Kukreja,\nH. A. D¨ urr, D. Fausti, M. Gr¨ uninger, and F. Parmigiani,\nPhys. Rev. B 93, 054305 (2016).\n9L. Mart´ ın-Garc´ ıa, A. Mascaraque, B. M. Pab´ on, R. Bliem,\nG. S. Parkinson, G. Chen, A. K. Schmid, and J. de la\nFiguera, Phys. Rev. B 93, 134419 (2016).\n10T. P. Ma, Y. Yang, Z. Ding, Z. H. Chen, H. B. Zhao,\nP. Werner, S. S. P. Parkin, and Y. Z. Wu, Phys. Rev. B\n95, 014103 (2017).7\n11S. Borroni, E. Baldini, V. M. Katukuri, A. Mann, K. Par-\nlinski, D. Legut, C. Arrell, F. van Mourik, J. Teyssier,\nA. Kozlowski, P. Piekarz, O. V. Yazyev, A. M. Ole´ s,\nJ. Lorenzana, and F. Carbone, Phys. Rev. B 96, 104308\n(2017).\n12H. Seo, M. Ogata, and H. Fukuyama, Phys. Rev. B 65,\n085107 (2002).\n13H.-T. Jeng, G. Y. Guo, and D. J. Huang, Phys. Rev. Lett.\n93, 156403 (2004).\n14L. Craco, M. S. Laad, and E. M¨ uller-Hartmann, Phys.\nRev. B74, 064425 (2006).\n15L. R. Bickford, Jr., Phys. Rev. 78, 449 (1950).\n16C. Kittel, Phys. Rev. 73, 155 (1948).\n17C. Kittel, Phys. Rev. 76, 743 (1949).\n18L. McGuigan, R. C. Barklie, R. G. S. Soﬁn, S. K. Arora,\nand I. V. Shvets, Phys. Rev. B 77, 174424 (2008).\n19M. Monti, B. Santos, A. Mascaraque, O. Rodr´ ıguez de la\nFuente, M. A. Ni˜ no, T. O. Mentes, A. Locatelli, K. F.\nMcCarty, J. F. Marco, and J. de la Figuera, Phys. Rev. B\n85, 020404(R) (2012).\n20M. Nagata, K. Tanabe, T. Moriyama, D. Chiba, J.-I.\nOhe, M. Myoka, T. Niizeki, H. Yanagihara, E. Kita, and\nT. Ono, IEEE Trans. Magn. 50, 1400203 (2014).\n21S. Ni, S. Lin, Q. Pan, F. Yang, K. Huang, and D. He, J.\nPhys. D: Appl. Phys. 42, 055004 (2009).\n22K. Jia, R. Zhao, J. Zhong, and X. Liu, J. Magn. Magn.\nMater.322, 2167 (2010).\n23R. Dinali, A. Ebrahiminezhad, M. Manley-Harris,\nY. Ghasemi, and A. Berenjian, Crit. Rev. Microbiol. 43,\n493 (2017).\n24P. Xu, G. M. Zeng, D. L. Huang, C. L. Feng, S. Hu, M. H.\nZhao, C. Lai, Z. Wei, C. Huang, G. X. Xie, and Z. F. Liu,\nSci. Total Environ 424, 1 (2012).\n25S. F. Hasany, A. Rehman, R. Jose, and I. Ahmed, AIP\nConf. Proc. 1502, 298 (2012).\n26M. M. Noginov, N. Noginova, O. Amponsah, R. Bah,\nR. Rakhimov, and V. A. Atsarkin, J. Magn. Magn. Mater.\n320, 2228 (2008).\n27A. Shankar, M. Chand, G. A. Basheed, S. Thakur, and\nR. P. Pant, J. Magn. Magn. Mater. 374, 696 (2015).\n28W. N. Hardy and L. A. Whitehead, Rev. Sci. Instrum. 52,\n213 (1981).\n29W. Froncisz and J. S. Hyde, J. Magn. Reson. (1969-1992)47, 515 (1982).\n30J. S. Bobowski, Am. J. Phys. 81, 899 (2013).\n31J. S. Bobowski and H. Nakahara, Rev. Sci. Instrum. 87,\n024701 (2016).\n32J. S. Bobowski and A. P. Clements, IEEE Trans. Microw.\nTheory Tech. PP, 1 (2017).\n33J. S. Bobowski, in 2015 Conference on Laboratory Instruc-\ntion Beyond the First Year of College , BFY Conference\n(College Park, MD, 2015) pp. 20–23.\n34J. S. Bobowski, Can. J. Phys. 96, 878 (2018).\n35G. A. Rinard, R. W. Quine, S. S. Eaton, and G. R. Eaton,\nJ. Magn. Reson. Ser A 105, 137 (1993).\n36All dielectric substrates were purchased from MTI Corpo-\nration (Richmond, CA).\n37M. A. Stuchly and S. S. Stuchly, IEEE Trans. Instrum.\nMeas.IM-29, 176 (1980).\n38J. S. Bobowski and T. Johnson, Prog. Electromagn. Res.\nB40, 159 (2012).\n39BioPhysics Assay Laboratory, Inc., http://www.biopal.\ncom/(2018), online; accessed 25 July 2018.\n40Product Data Sheet , Molday ION. BioPhysics Assay Lab-\noratory, Inc.\n41F. Hyodo, G. V. R. Chandramouli, S. Matsumoto, K.-I.\nMatsumoto, J. B. Mitchell, M. C. Krishna, and J. P. Mu-\nnasinghe, Int. J. Oncol. 35, 797 (2009).\n42T.-A. Perles-Barbacaru, D. Procissi, A. V. Demyanenko,\nand R. E. Jocobs, NMR Biomed. 25, 498 (2012).\n43T. K. Hitchens, L. Lui, L. M. Foley, V. Simplaceanu, E. T.\nAhrens, and C. Ho, Magn. Reson. Med. 73, 367 (2015).\n44W. A. Yager, Phys. Rev. 75, 316 (1949).\n45D. D. Awschalom, D. P. DiVincenzo, and J. F. Smyth,\nScience258, 414 (1992).\n46S. Wirth, M. Field, D. D. Awschalom, and S. von Moln´ ar,\nPhys. Rev. B 57, R14028(R) (1998).\n47X. Liu, S. W. Or, C. M. Leung, and S. L. Ho, J. Appl.\nPhys.113, 17B307 (2013).\n48D. Zhang, J. Cheng, X. Yang, B. Zhao, and M. Cao, J.\nMater. Sci. 49, 7221 (2014).\n49Y. Liu, T. Cui, T. Wu, Y. Li, and G. Tong, Nanotechnol-\nogy27, 165707 (2016).\n50K. Zhang, X. Gao, Q. Zhang, H. Chen, and X. Chen, J.\nMagn. Magn. Mater. 452, 55 (2018)."    },    {        "title": "1204.1491v1.Andreev_current_induced_by_ferromagnetic_resonance.pdf",        "content": "arXiv:1204.1491v1  [cond-mat.supr-con]  6 Apr 2012Andreev current induced by ferromagnetic resonance\nCaroline Richard,1Manuel Houzet,1and Julia S. Meyer1\n1SPSMS, UMR-E 9001 CEA/UJF-Grenoble 1, INAC, Grenoble, F-38 054, France\n(Dated: November 30, 2018)\nWe study charge transport through a metallic dot coupled to a superconducting and a ferromag-\nnetic lead with a precessing magnetization due to ferromagn etic resonance. Using the quasiclassical\ntheory, we ﬁnd that the magnetization precession induces a d c current in the subgap regime even\nin the absence of a bias voltage. This eﬀect is due to the recti ﬁcation of the ac spin currents at the\ninterface with the ferromagnet; it exists in the absence of s pin current in the superconductor. When\nthe dot is strongly coupled to the superconductor, we ﬁnd a st rong enhancement in a wide range of\nparameters as compared to the induced current in the normal s tate.\nPACS numbers: 74.45.+c, 75.76.+j, 72.25.-b\nSpin-transfertorqueallowsonetomanipulatethemag-\nnetization of a ferromagnetic (F) layer by means of a\nspin-polarized current [1, 2]. Random-access memories\nusing this eﬀect in order to induce magnetization rever-\nsal of the active elements are on their way to commer-\ncialization. The reverse eﬀect, namely the generation of\na spin current in a normal metal (N) by means of a dy-\nnamically precessing ferromagnetic metal, has also been\npredicted [3]. In the absence of direct spin probes, this\neﬀect may be measured by using a second ferromagnet as\nan analyzer that converts the spin current into a charge\ncurrent. However,it waspointedout theoretically[4] and\nmeasuredexperimentally[5,6]thatasingleF/Njunction\nis enough to both generate and detect the spin current\nthrough the generation of a dc voltage at ferromagnetic\nresonance (FMR) in an open-circuit geometry. At the\norigin of this phenomenon is the spin accumulation on\nthe normal side of the junction – due to the precession-\ninduced spin current – which is typically diﬀerent from\nthat on the ferromagnetic side. If transmissions for the\nmajority and minority electron species through the junc-\ntionarediﬀerent, thediﬀerenceinspinaccumulationgen-\nerates a net charge current which must be compensated\nby a diﬀerence in electrochemical potentials such that\nno charge accumulation occurs. Spin relaxation inhibits\nspin accumulation and, thus, suppresses the eﬀect.\nThe aim of our work is to explore how this eﬀect is\nmodiﬁed in a ferromagnet/superconductorjunction. The\ncombination of ferromagnetic and superconducting (S)\nmaterials has been shown to lead to a variety of inter-\nesting spin phenomena [7, 8]. However, the study of the\ninterplaybetweenmagnetizationdynamicsandsupercon-\nductivity isarelativelynewtopic. Experimentally, anar-\nrowing of the FMR width at the superconducting transi-\ntion was observed in an F/S bilayer [9]. Theoretically, it\nwas proposed that a dynamically precessing ferromagnet\nmay generate a long-range proximity eﬀect [10]. This\neﬀect would manifest itself in the enhancement of the\ncritical current in a phase-biased ferromagnetic Joseph-\nson junction under FMR conditions. Related signatures\nin the tunneling density of states of the F layer have alsodot\n{Gr} {Gl,Gm}\nFIG. 1: Setup of the junction. A metallic dot is coupled\nto a ferromagnetic lead with precessing magnetization m(t)\non the left and to a normal or superconducting lead on the\nright. The left barrier is characterized by the conductance s\nGlandGm, deﬁned above Eq. (3), whereas the right barrier\nis characterized by the conductance Gr.\nbeen investigated [11]. However, these works disregard\ninterface eﬀects and, therefore, do not take into account\nthe possible generation of an FMR-induced dc voltage.\nFinally, let us note that the ﬁrst experiments on voltage\ngeneration by FMR of Refs. [5, 6] were performed with\nAl as the normal metal, which becomes superconducting\nat low temperatures.\nAs the FMR-generated charge current in an F/N junc-\ntion is typically associated with a spin current, one may\nwonder what happens in an F/S junction in the sub-\ngapregime, where transportis mediated by Andreevpro-\ncesses [12]. We show that the generation of charge cur-\nrent in the absence of a spin current in a conventional\nsinglet superconductor is possible. In fact, the absence\nof spin currents in the superconductor may even lead to\na strong enhancement of the induced charge current as\ncompared to the normal state.\nAs in the normal state, the two main ingredients nec-\nessary to generate the eﬀect are spin-dependent trans-\nmissions through the junction and a spin accumulation\nregion [19]. The simplest setup meeting these require-\nments is a metallic dot coupled through tunnel barriers\nto a ferromagnet and to a superconductor (see Fig. 1).\nCoulomb blockade eﬀects are neglected, assuming that\nthe conductances of the barriers largely exceed the con-\nductance quantum.\nThe magnetization precession in the ferromagnetic2\nlead is described by a time-dependent exchange ﬁeld,\nJ(t) =Jm(t) with\nm(t) = (sinθcosΩt,sinθsinΩt,cosθ),(1)\nacting on the spin of the conduction electrons. Here the\nprecession frequency, Ω, and the tilt angle, θ, are both\ntunable with external dc and rf ﬁelds under standard\nFMRconditions[13]. Wewillconsiderthemasexternally\nﬁxed parameters.\nThe precession of the magnetization drives the system\nout of equilibrium and, thus, may generate a current. To\ndescribe the system, we use the quasiclassical Keldysh\ntheory [14]. In particular, the current through the junc-\ntion can be expressed in terms of the quasiclassicalGreen\nfunction ˇ gof the dot. Here ˇ gis a matrix in Keldysh,\nNambu, and spin space. In Keldysh space, it has a trian-\ngular structure with retarded (ˆ gR), advanced (ˆ gA), and\nKeldysh (ˆ gK) components. Furthermore, it satisﬁes the\nnormalization condition ˇ g2= 1.\nThe equations determining the Green functions are\nmost conveniently written in the rotational frame for\nthe magnetization precession, where the problem is sta-\ntionary [10]. Assuming that the conductance of the dot\nlargely exceeds the conductances of the junctions with\nthe leads, the equation determining ˇ gmay be cast in the\nform\n−i2πGQ\nδ/bracketleftbigg/parenleftbigg\nE+Ω\n2σz/parenrightbigg\nτz,ˇg/bracketrightbigg\n+ˇIl+ˇIr= 0.(2)\nHereσiandτiare Pauli matrices in Nambu and spin\nspace, respectively ( i=x,y,z). Furthermore, GQ=\ne2/πis the conductance quantum (in units where /planckover2pi1= 1),\nandδis the mean level spacing in the dot. The spin-\ndependent energy shift ±Ω/2 is a spin-resolved chemical\npotential induced by the transformation from the labo-\nratory to the rotational frame. The boundary conditions\nwith the ferromagnetic ( l= left) and superconducting\n(r= right) leads are represented by the matrix currents\nˇIl/rand depend on the Green functions ˇ gl/rdescribing\nthe non-equilibrium state in the leads due to the magne-\ntization precession.\nTunneling through an F/N interface is generally spin-\ndependent. The relevant processes can be characterized\nby the total conductance of the junction, Gl, and the\ndiﬀerence between the conductances for the majorityand\nminority electrons, Gm[20]. The matrix current at the\ntunnel interface between the dot and the ferromagnet\nthen takes the form [15]\nˇIl=Gl\n2[ˇgl,ˇg]+Gm\n4[{m·στz,ˇgl},ˇg],(3)\nwherem≡m(0). Within the quasiclassical approxima-\ntion, we assume |Gm| ≪Gl. Thus, Gmcan be treated\nperturbatively.The Green function in the F lead, ˇ gl, is determined by\n/bracketleftbigg/parenleftbigg\nE+Ω\n2σz+Jm·σ/parenrightbigg\nτz−ˇΣ,ˇgl/bracketrightbigg\n= 0,(4)\nwhere the self-energy ˇΣ =−iΓˇgN(E+(Ω/2)σz) accounts\nfor inelastic scattering in the relaxation time approxi-\nmation. Here, 1 /Γ is the inelastic scattering time and\nˇgNis the equilibrium Green function in a normal metal.\nNamely, ˆ gR(A)\nN(E) =±τzand ˆgK\nN(E) = 2τzf(E), where\nf(E) = tanh( E/2T) is related to the Fermi distribution\nat temperature T. For a large exchange ﬁeld, J≫Ω,Γ,\nthe solution of Eq. (4) takes the form ˆ gR(A)\nl=±τz\nand ˆgK\nl= 2τz(f++f−cosθm·σ), where f±(E) =\n[f(E+Ω/2)±f(E−Ω/2)]/2.\nThe matrix current at the dot-superconductor tunnel\ninterface is given as\nˇIr=Gr\n2[ˇgr,ˇg], (5)\nwhereGris the conductance of the junction. The Green\nfunction in the S lead reads ˇ gr= ˇgS(E+(Ω/2)σz), where\nˇgSis theequilibrium Greenfunction in asuperconductor.\nNamely, ˆ gR(A)\nS(E) = (−iEτz+∆τx)//radicalbig\n∆2−(E±i0+)2\nand ˆgK\nS(E) = [ˆgR\nS(E)−ˆgA\nS(E)]f(E), where ∆ is the su-\nperconducting order parameter (taken to be real).\nNow we have all the ingredients necessary to deter-\nmine the Green function in the dot and subsequently the\nspin and charge currents at both interfaces. The charge\ncurrents are given by\nIl/r=1\n16e/integraldisplay\ndETr[τzˆIK\nl/r]. (6)\nCurrent conservation ensures that I≡Il=−Ir.\nThe spin currents in the rotational frame are given by\nIl/r=−1\n32e2/integraldisplay\ndETr[σˆIK\nl/r]. (7)\nIn the laboratoryframe, they decompose into adc contri-\nbutionalongtheprecessionaxis, Iα,z, andaccomponents\nin the perpendicular plane, Iα,x/y(t) =Iα,x/ycosΩt∓\nIα,y/xsinΩt. Contrarily to the charge current, the spin\ncurrents do not need to be conserved. Eq. (2) yields\nIl+Ir−Ω\n16δ/integraldisplay\ndETr[(ˆz×σ)τzˆgK] = 0.(8)\nThus, only the dc spin current along ˆzis conserved.\nWhile our main interest are the FMR-induced cur-\nrents in the subgap regime of an F-dot-S junction, we\nﬁrst study the simpler case of an F-dot-N junction for\ncomparison. For better readibility, in the following, we\nwill normalize conductances by GΣ=Gl+Grand en-\nergies by the Thouless energy Eg=GΣδ/(4πGQ). In\nparticular, we introduce the dimensionless conductances3\nγα=Gα/GΣ(α=l,r,m) as well as the dimensionless\nenergies ǫ=E/Egandω= Ω/(2Eg).\nA normal lead is described by setting ∆ = 0 in the\nabove equations for ˇ gr. In the absence of superconduc-\ntivity, the retarded and advanced Green functions in the\ndot are trivial, ˆ gR(A)=±τz. The Keldysh component is\nobtained with the help of Eqs. (2), (3), and (5). There is\na remarkablerelation between the spin current at the left\ncontact with the F lead and the charge current to lowest\norder in γm,\nI=2eγmγr\nγlm(t)·Il(t) =2eγmγr\nγl(Il,xsinθ+Il,zcosθ),\n(9)\nnamely the charge current is proportional to the pro-\njection of the spin current onto the instantaneous mag-\nnetization axis of the barrier due to the spin-dependent\nconductance Gm. That is, the charge current originates\nfrom two eﬀects: (i) the rectiﬁcation of the ac in-plane\nspin current pumped from the ferromagnet, and (ii) the\nconversion of the dc spin current along the ˆz-axis into\na charge current. It turns out that the two eﬀects have\nopposite sign, and that the former dominates over the\nlatter. Namely, we ﬁnd\nIl,x=GlEg\n2e2ω(γr+ω2)\n1+ω2sinθcosθ,(10a)\nIl,z=−GlEg\n2e2ωγrsin2θ. (10b)\nNote that, in the limit γr≪γl, the spin current along\ntheˆz-axis is negligible. By contrast, in the limit γl≪\nγr, the two components are of comparable magnitude\nand almost complete cancellation between the competing\neﬀects takes place.\nThe charge current reads\nI=GmEg\neγrγlω3sin2θcosθ\n1+ω2. (11)\nAt large precession frequency, ω≫1, the\ncurrent scales linearly with frequency, I≃\n(GlGrGm/2eG2\nΣ)Ωsin2θcosθ. In particular, in an\nopen-circuit geometry, this would correspond to an\nFMR-induced dc voltage eV= (Gm/2GΣ)Ωsin2θcosθ\nin accordance with Refs. [4–6]. At ω≪1, spin-\nrelaxation mechanisms induced by the tunnel coupling\nof the dot to the leads tend to suppress the eﬀect.\nWe now turn to the F-dot-S junction. In the subgap\nregime, the spin current at the interface with the super-\nconductor vanishes, Ir= 0. Thus, Iz,l= 0. However, an\nac spin current is present at the interface with the fer-\nromagnet. Then, the Andreev charge current originates\nentirely from the rectiﬁcation of this ac spin current.\nRestricting ourselves to energy scales much smaller\nthan ∆, the Green function in the superconducting lead\ntakes the simple form ˆ gR(A)\nr=τxand ˆgK\nr= 0. Takingγmas a small parameter, we search for a perturbative\nsolution of equation (2) in the form ˇ g= ˇg0+γmˇg1+...\nDue to the proximity eﬀect, now the retarded and ad-\nvanced Green functions of the dot are modiﬁed as well.\nTo zeroth order in γm, an explicit solution is given by\nˆgR(A)\n0=γrτx+[−i(ǫ+ωσz)±γl]τz/radicalbig\nγ2r−(ǫ+ωσz±iγl)2.(12)\nHere,γris the eﬀective minigap due to the coupling with\nthe S lead[16], ωan eﬀective exchangeﬁeld, and γlyields\nabroadeningoftheenergylevelsduetothecouplingwith\nthe F lead. The Keldysh Green function can be cast in\nthe form ˆ gK\n0= ˆgRˆϕ−ˆϕˆgAwith\nˆϕ=f++f−cosθ/bracketleftbiggγlsinθ\nω2+γ2\nl(γlσx−ωσy)+cosθσz/bracketrightbigg\n.(13)\nThe function ˆ ϕcan be interpreted as a matrix-\ndistribution function. Note that it does not depend on\nγras subgap electrons only thermalize with the F lead.\nTo ﬁrst order in γm, a solution which satisﬁes the nor-\nmalization condition, ˇ g2= 1, is obtained in the form\nˇg1= ˇg0ˇX−ˇXˇg0. For the advanced and retarded compo-\nnents, one ﬁnds ˆXR(A)=∓(sinθ/2ω)[iγr/(ǫ±iγl)τx+\nτz]σy. The Keldysh component can be decomposed as\nˆXK=XK\nxτx+XK\nzτz, whereXK\nxandXK\nzsolve the cou-\npled equations\n2γlXK\nz−iω[σz,XK\nz] (14a)\n= 2sinθ[cosθsinθf−+f+(σx−γl\nωσy)],\n2ǫXK\nx−2iγrXK\nz+ω{σz,XK\nx} (14b)\n= 2iγlγrsinθ\nω(γ2\nl+ǫ2)[γlf+σy−ǫf−cosθ(cosθσx−sinθσz)].\nEvaluating the current at the right interface, Eq. (6)\nyieldsI=−iγmGr/(16e)/integraltext\ndETr[τyˆgK\n1]. Inserting\nthe solution for ˆ gK\n1and using the property ˆ gR\n0(−ǫ) =\n−σxτzˆgA\n0(ǫ)σxτz, we obtain the current\nI=1\n2I0γ2\nrω\nγ2\nl+ω2/integraldisplay\ndǫǫf−\n(γ2\nl+ǫ2)(ǫ+ω)(15)\n×/summationdisplay\n±−γl(ǫ+ω)±i(γ2\nl−ǫω)/radicalbig\nγ2r−(ǫ+ω±iγl)2,\nwhereI0= (GmEg/e)sin2θcosθ. The current as a func-\ntion of frequency for diﬀerent values of γl= 1−γris\nshowninFig.2. Simpleanalyticexpressionscanbefound\nin diﬀerent asymptotic regimes.\nIn particular, at temperature T= 0 and low frequency\n|ω| ≪γ, whereγ= (γ2\nl+γ2\nr)1/2, the FMR-induced cur-\nrent is given as\nI≃10\n3I0γ2\nrγl\nγ7ω5. (16)4\n0.0 0.5 1.0 1.50.00.20.40.60.81.01.21.4\n/CapOmeΓa/Slash1/LParen12 Eg/RParen1I/Slash1I0Γl/Equal0.6Γl/Equal0.4Γl/Equal0.2Γl/Equal0.01\nFIG. 2: Andreevcurrentinduced byferromagnetic resonance\nas a function of precession frequency for diﬀerent values of\nγl=Gl/(Gr+Gl). HereI0= (GmEg/e)sin2θcosθ.\nThe large power ω5indicates the strong suppression of\nthe eﬀect.\nAt large frequencies |ω| ≫γ, the current saturates.\nThe frequency-independent value is given by\nI≃π\n2I0sign(ω)×/braceleftbigg\nγ2\nr, γr≪γl,\n1, γ l≪γr.(17)\nThe saturation can be understood as Andreev processes\nbecome ineﬃcient at energies larger than the minigap\n[17].\nDepending whether thedot ismorestronglycoupled to\nthe ferromagnet or to the superconductor, the crossover\nbetween these asymptotic regimes is diﬀerent. If the dot\nis weakly coupled to the superconductor, γr≪γl, a\nsmooth crossover happens at ω∼1≫γrwith a typi-\ncal current I/I0∼γ2\nr. By contrast, if the dot is weakly\ncoupled to the ferromagnet, γl≪γr, the crossoverin the\nregionω∼1/2 is described by\nI≃I0×/braceleftbiggγl/(2√\n−δω),−1≪δω≪ −γl,\n2√\nδω, γ l≪δω≪1,(18)\nwhereδω≡ω−1/2, with a typical current I/I0∼√γl\natω= 1/2.\nWhile in the asymptotic regimes of ωvery small or\nvery large, the current is suppressed as compared to the\nnormal state, there is in fact a wide intermediate regime\nwhereit maybe stronglyenhanced. ComparingEqs.(11)\nand (17), one notices that, if the dot is strongly coupled\nto the superconductor, in the regime 1 /2< ω < γ−1\nl\nthe induced current in the superconducting state exceeds\nthe induced current in the normal state. This eﬀect may\nbe understood due to the absence of a dc spin current\nalong along the ˆz-axis which leads to a strong suppres-\nsion of the eﬀect in the normal state. Fig. 3 shows the\ncurrent in the superconducting and normal state as well\nthe contribution due to rectiﬁcation only in the normal\nstate. The ratio between the current in the supercon-\nducting state and the latter contribution in the normal0.00.51.01.52.02.53.03.54.00.00.20.40.60.8\n/CapOmeΓa/Slash1/LParen12 Eg/RParen1I/Slash1I0\nFIG. 3: Induced current in the superconducting (dotted\nline) and normal state for γl= 0.4. The thin line shows the\ncontribution to the normal state current due to rectiﬁcatio n\nonly.\nstatereﬂectsthe ratiobetweenAndreevand normalstate\nconductances in an N-dot-S junction.\nSo far we assumed that the magnetization in the ferro-\nmagnet is uniform. However, boundary eﬀects may lead\nto a suppression of the magnetization in the vicinity of\nthe F/N interface. This would result in a diﬀerent reso-\nnance frequency at the barrier than in the ferromagnetic\nreservoir and, consequently, in a tilt angle θB/ne}ationslash=θat\nFMR. The eﬀect can be accounted for by replacing m\nwithmB= (sinθB,0,cosθB) in Eq. (3). In particular,\natθB= 0, the spin dependent conductance Gmrefers to\nthe constant axis ˆz.\nIn the normal case, the relation between spin\nand charge currents, Eq. (9), now reads I=\n(2eγmγr/γl)mB(t)·Il(t). While at θB=θthe recti-\nﬁcation of the in-plane ac spin currents always domi-\nnates over the conversion of the dc spin current along\nˆzinto a charge current, this eﬀect is completely sup-\npressed at θB= 0. As a consequence, at θB= 0, the\ncharge current, I=−(GmEg/e)ωγ2\nrsin2θ, has the op-\nposite sign compared to Eq. (11). In general, both ef-\nfects are important. The sign reversaloccurs at tan θB=\n[1−γlω2/(γr+ω2)]tanθ.\nIn the superconducting case, the dc spin current along\nˆzis always zero, and the charge current is due entirely\nto the rectiﬁcation of the in-plane ac spin currents. As a\nconsequence, we ﬁnd that the charge current vanishes at\nθB= 0. The general result is obtained from Eq. (15) by\nreplacing I0withIB\n0= (GmEg/e)sinθBsinθcosθ.\nInsummary, we demonstratethat asubgapchargecur-\nrent in an F/S junction may be induced by ferromagnetic\nresonance. The eﬀect is due to the rectiﬁcation of ac\nspin currents generated by the precessing magnetization\nin the ferromagnet. In the normal case, a competing\neﬀect of conversion of a dc spin current into a charge\ncurrent exists. This eﬀect is absent in an F/S junction as\nthe superconductor cannot carry a subgap spin current.\nAs a consequence, the induced current in the supercon-5\nducting state may be strongly enhanced as compared to\nthe normal state. Interesting non-equilibrium phenom-\nena should be expected in ferromagnetic Josephson junc-\ntions under ferromagnetic resonance conditions.\nWe acknowledge funding through an ANR grant\n(ANR-11-JS04-003-01)andanEU-FP7MarieCurieIRG.\n[1] J. C. Slonczewski, Journal of Magnetism and Magnetic\nMaterials, 159, L1 (1996).\n[2] L. Berger, Phys. Rev. B, 54, 9353 (1996).\n[3] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. Lett., 88, 117601 (2002).\n[4] X. Wang, G. E. W. Bauer, B. J. van Wees, A. Brataas,\nandY.Tserkovnyak,Phys. Rev.Lett., 97, 216602 (2006).\n[5] M. V. Costache, M. Sladkov, S. M. Watts, C. H. van der\nWal, and B. J. van Wees, Phys. Rev. Lett., 97, 216603\n(2006).\n[6] T. Moriyama, R. Cao, X. Fan, G. Xuan, B. K. Nikoli´ c,\nY. Tserkovnyak, J. Kolodzey, and J. Q. Xiao, Phys. Rev.\nLett.,100, 067602 (2008).\n[7] A. I. Buzdin, Rev. Mod. Phys., 77, 935 (2005).\n[8] F. S. Bergeret, A. F. Volkov, and K. B. Efetov, Rev.\nMod. Phys., 77, 1321 (2005).\n[9] C. Bell, S. Milikisyants, M. Huber, and J. Aarts, Phys.Rev. Lett., 100, 047002 (2008).\n[10] M. Houzet, Phys. Rev. Lett., 101, 057009 (2008).\n[11] T. Yokoyama and Y. Tserkovnyak, Phys. Rev. B, 80,\n104416 (2009).\n[12] G. E. Blonder, M. Tinkham, and T. M. Klapwijk, Phys.\nRev. B,25, 4515 (1982).\n[13] C. Kittel, Phys. Rev., 73, 155 (1948).\n[14] Y. V. Nazarov and Y. M. Blanter, Quantum Trans-\nport, Introduction to Nanoscience (Cambridge University\nPress, 2009).\n[15] D. Huertas-Hernando, Y. V. Nazarov, and W. Belzig,\nPhys. Rev. Lett., 88, 047003 (2002).\n[16] C.W.J.Beenakker,Lecture Notes in Physics, 667, 333 (2005).\n[17] A. Volkov, A. Zaitsev, and T. Klapwijk, Physica C: Su-\nperconductivity, 210, 21 (1993).\n[18] H. J. Skadsem, A. Brataas, J. Martinek, and\nY. Tserkovnyak, Phys. Rev. B, 84, 104420 (2011).\n[19] The eﬀect is absent in the theory of Ref. [18], which did\nnot include such a region.\n[20] In addition one may take intoaccount the imaginary part\nof the spin-mixing conductance, Gφ, which accounts for a\nspin-dependent phase shift upon reﬂection. In good F/N\ncontacts with large Fermi velocity mismatch, Gφis typi-\ncally small and therefore neglected. It may, however, be\nlarge in tunnel junctions. As Gφleads to spin relaxation,\nit would suppress the eﬀect studied in this work."    },    {        "title": "1401.1924v1.FMR_Study_of_Co_Ti_Bilayer_Thin_Films.pdf",        "content": " \nFMR Study  of Co/Ti  Bilayer Thin  Films \n \n \nM. Erkovan1,2*, S. Tokdemir Öztürk2, D. Taşkın  Gazioğlu2, R. Topkaya2, \n O. Öztürk2 \n \n1Gebze Institu te of Technology, Department of Physics , Gebze, Kocaeli, Turkey .  \n2Sakarya University, Department of Metallurgy and M aterials Engineering, Sakarya, 54687, Turkey  \n \n \n \nAbstract . We focused on the interaction between two ferromagnetic cobalt layers through  a non-magnetic titanium layer . \nThe magnetic properties o f the structure were characterized by ferromagnetic resonance te chnique (FMR). The data were \ncollected as a function of non -magnetic titanium layer thickness.  Co/Ti multilayer (Ti (50 Å)/ Co(45  Å)/Ti(2 -40 Å)/Co(40 \nÅ)/Ti(100 Å))films were grown  onto naturally oxidized p -type single crystal  Si (100) substrate at UHV cond ition with \nmagnetron sputtering system at room temperature.  The thickness of Ti spacer layer ranges from 2 to 40 Å with 2 Å steps. \nWe did not observe usual optic and acoustic modes ; instead we had two broad overlapped peaks for the films ranged from 6  \nÅ to 40  Å. One interesting result was the high anisotropic resonance field values for these films.   Exchange coupling \nbetween ferromagnetic layers causes shift on resonance field values but these shifts in our samples were much large r than \nexpected.  This larg e anisotropic behavior  is not clear at the moment. Our theoretical model was not able to determine a value \nfor the exchange coupling parameter. One reason can be the close thickness values for Co sublayers. The other reason can be \nthe Ti non -magnetic layer . If titanium did not grow layer by layer on cobalt, the cobalt ferromagnetic layers may behave as a \nsingle layer. As a result one cannot observe exchange interaction between ferromagnetic layers through non -magnetic \nspacer.  \n \nKeyword:  Magnetic Multilayer,  Ferromagnetic Materials , Ferromagnetic Resonance.   \nPACS: 75.70.Cn, 75.50.Cc, 76.50.+g  \nIntroduction  \nMagnetic multilayer films have interesting magnetic properties compare d to single layer magnetic films such as \ngiant magneto resistance effect (GMR), [1,2] tunneling magneto resistance effect (TMR). [3,4] Magnetic \nmultilayer systems are composed of alternating ferromagnetic layers and non -magnetic layers. [5] If the non-\nmagnetic material is metal, GMR effect may be observed. In order to observe TMR effect , the non-magnetic  \nlayer has to be insulating material. Another important magnetic property is observing oscillating interlayer \nexchange coupling which changes with magnetic and non -magnetic layer compositions and thicknesses. [6,7] \nThe interlayer exchange coupl ing between two ferromagnetic layers is ferromagnetic (antiferromagnetic) when \nthe magnetization vectors of magnetic layers are parallel ( anti -parallel). The magnetic multilayer films are \ngrown  by many different deposition techniques [8, 9]. But the films  prepared with the magnetic sputtering -\ndeposition process in Ultra High Vacuum (UHV) conditions have advantages  such as growing conditions lead to \nthe formation of more complete thin -film layers and consequently periodic behavior of exchange coupling \nparam eter is clearly observable.  The non -magnetic layer homogeneity is crucially  important to identify the \nexchange coupling species. The quality of the homogeneity of films is very high  [10] when the films are \nprepared with this technique.  There are a lot of magnetic characterization techniques to investigate magnetic \nproperties of magnetic multilayer films such as FMR, VSM, SQUID, MOKE  [10, 11] etc. Especially  \nferromagnetic resonance is a very suitable technique to identify interaction species between ferroma gnetic \nlayers. T wo different modes are generally observed  in these structures depending on magnetic layer thickness, \none of the modes is called optic mode and the other is called acoustic mode and their relative intensities and \npositions change with exchan ge interaction. If the exchange interaction is ferromagnetic (anti -ferromagnetic) the \noptic (acoustic) mode has lower resonance field value than the acoustic (optic) mode’s value  [12, 13]. There are \nsome studies on Co/Ti multila yer structures in the litera ture [ 14-18]. One study focused on the structural \ntransformation of cobalt from the polycrystalline to the soft nanocrystalline structure in Co/Ti multilayer system. \nFMR was used to determine the change in effective anisotropy field for t he critical thickn ess of Cobalt [ 14].  P. \nWu et all.  investigated the magnetization and interface structures of the Co/Ti multilayer films. The saturation \nmagnetization of the film is found to decrease linearly with 1/d Co and also decrease with 1/d Ti [15].  M. Schmidt M\nM\nH\nM\nHx y z \nHet all. used the Co/Ti films for pseudo spin  valve structures with Cu layer [ 16]. These studies do not give any \ninformation about interlayer exchange interaction between two cobalt layers. On the other hand , Smardz used \nVSM technique in his study  [17] to invest igate the interlayer exchange interaction. Results showed some \nconditions to observe exchange coupling between two ferromagnetic cobalt layers. According to the author , Co \nsublayers are ferromagnetically coupled up to Ti spacer if thickness is about 19  Å.  Furthermore, a weak \nantiferromagnetic coupling of the Co sublayers was observed for a Ti thickness range between 19 and 27 Å.In \nthis study, Cobalt was chosen as ferromagnetic layer and Titanium was chosen for nonmagnetic transition metal \nas a spacer. The interlayer exchange coupling between ferromagnetic cobalt layers was investigated as a \nfunction of titanium thickness with ferromagnetic resonance (FMR) technique.  \nSample  Preparation  \n \n Co/Ti multilayer (Ti (50 Å)/ Co(45  Å)/Ti(2 -40 Å)/Co(40 Å)/Ti(100 Å))films were grown  onto naturally \noxidized p -type single crystal  Si (100) substrate at UHV condition with magnetron sputtering system at room \ntemperature.  The thickness of Ti spacer layer ranges from 2 to 40 Å with 2 Å steps.  RF power supply was used \nfor sputt ering the cobalt magnetic layers, and DC power supply was used for sputtering the titanium layers. Both \ncobalt and titanium deposition ratio were determined by X Ray Photoelectron Spectroscopy (XPS) and Quartz \nCrystal Monitoring (QCM). All details about de position ratio determination wi th XPS are given in reference 1 8.  \nBefore the substrates were loaded to the ultra high vacuum chamber, they were cleaned with ethanol and \nmethanol for  ten minutes by ultrasonic cleaner. In order to remove surface roughness, t he substrates were heated \nat UHV conditions by PBN heater and hold at 600 °C for 20 minutes. Although  the system base pressure was \nabout 5x10-8 mbar, the deposition pressure was about 1.2 -1.5x10-3 mbar. The water -cooled target with 3 in. in \ndiameter provid es a good homogeneity in thickness. High purity argon gas (6N) was used for sputtering.  \nFrom earlier experiences  [13] it was found that  measurable exchange coupling could be observed between \nferromagnetic layers through a non magnetic spacer when the thicknesses of bottom and upper ferromagnetic \nlayers were different but close to each other. For that purpose the thicknesses of bottom and upper Co layers \nwere chosen as 45 Å and 40 Å, respectively. The buffer titanium layer was 50 Å to remove substrate effe cts and \nthe cap layer titanium was 100 Å to protect films from atmospheric effects.  \nExperimental Results  \n \nA Bruker EMX model X -band electron spin resonance spectrometer was used for FMR measurements. \nThe microwave frequency was 9.5 GHz. Two measurements w ere taken as a function of the angle of the \nexternal dc field with respect to the film normal at room temperature. The sample sketch, relative orientation of \nthe equilibrium magnetization vector M, the applied dc magnetic field vector H, and the experiment al \ncoordinate system are shown in Fig. 1(a). When the magnetic field is parallel to the plane of the film we call this \nposition as in plane geometry (IPG) and  when the magnetic field is perpendicular to the plane of the film we call \nit out of plane  geometr y (OPG). The picture of the prepared trilayer structure is shown in Fig. 1(b).  The \nmagnetic field component of microwave is always kept perpendicular to the dc field during the sample rotation. \nThe applie d microwave field always remains  in sample plane for  conventional geometry and power is kept small \nenough to avoid saturation, as well. A small modulation field of 100 kHz was applied in parallel to the dc \nmagnetic field in order to record the field derivative of absorption power.  \n (b) \n \n \nFIGURE 1 .( a) Repre sentative picture of Ti (50 Å)/ Co (45  Å)/ Ti (2 -28 Å)/ Co (40 Å)/ Ti (100 Å) multilayer structures. (b) \nRelative orientations of the external dc magnetic field and magnetization vectors with respect to sample plane.   \nFigure 2 shows the FMR spectra for di fferent Co thickness when the external magnetic field is parallel to the \nfilm plane. Only one FMR mode with narrow line width was observed for the thinnest spacers up to 6 Å.  The \nsignals get broader as the spacer thickness increases and single FMR signal becomes doublet. Especially 30  Å \nand 32  Å films show this feature clearly. Magnetic resonance positions are nearly  the same  for all films; \nhowever resonance positions shift to the lower magnetic field values for 10  Å, 20 Å, and 30  Å.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFIGURE  2. Experimental FMR spectra of the cobalt single layer films with different spacer thickness when the external \nfield is parallel to the sample  plane.  \n \nThe FMR spectra for the perpendicular position are shown in Fi gure 3 (a). A s imilar s ituation was observed for  \nthis case too. For thin spacers only one narrow FMR mode was observed up to 6 Å. As spacer (Ti) increases two (a) \n0.0 0.5 1.0 1.5 2.030Å27Å\n36Å34Å32Å25Å\n28Å16Å\n17Å\n18Å\n26Å21Å\n24Å23Å12Å\n38Å7Å4Å3Å\n22Å11Å2Å\n6Å\n10Å\n20Å\n  \nMagnetic Field (kOe)Intensity (arb. units)\n40Ånot well -resolved FMR modes were observed. However in perpendicular position, the spacer thickness plays an \nimportant ro le in resonance positions of these modes.  The dependence of the resonance field on the spacer \nthickness is much stronger for the optical mode. Figure 3 (b) shows how sensitive the resonance positions are to \nthe thickness of the spacer. One can see an oscil lation behavior . Roughly peaks occur for 10  Å, 20 Å, and 30  Å \nfilms .  The d istances between two modes do not  depend on the spacer thickness.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFIGURE  3.  E xperimental FMR spectra (a) and resonance positions (b) of the cobalt  single l ayer films with different spacer \nthickness when the external field is perpendicular to the sample  plane.  \n \nSimulation of The FMR Data  \n \nThe collected FMR data was tried to be analyzed with the theoretical model which was explained in \ndetails in reference 12.   The model is suitable  for a system consisting of N magnetic layers with saturation \nmagnetization Ms and layer thickness. However the computer program developed for this purpose   was not able \nto fit the experimental spectra to the theoretical model.  The p osition of the resonance peaks for each ultrathin \nCo(45  Å)Ti (2 -28 Å) Co(40  Å) trilayers were not able to  be determined by any suitable  saturation \nmagnetization Ms, effective anisotropy and exchange coupling of the magnetic layers.  \n(a) \n(b) \n12 13 14 15 16 1740Å38Å30Å\n34Å\n36Å16Å\n22Å\n24Å\n25Å\n32Å28Å27Å26Å23Å20Å\n21Å11Å3Å 4Å\n10Å\n17Å\n18Å12Å7Å6Å\n \n  \nMagnetic Field (kOe)Intensity (arb. units)2Å\n(b) The doublet peaks for particular Ti thickness were expected to represent the optical and acoustic \nmodes. We know that if ferromagnetic layers are magnetically equivalent, a single resonance peak is observed \ndue to simultaneous excitations of precession of magnetization in all l ayers. On the other hand , if layers are \nmagnetically  nonequivalent  two resonance modes are observed in FMR curves for OPG case as a result of \nexchange coupling of magnetically nonequivalent neighboring layers. If magnetic properties of different layers \nare very close to each other, two modes come close to each other and additionally if  the damping param eter is \nrelatively larger,  these two peaks overlap and give a distorted single line. Choice of 40 Å and 45 Å Co layers \ncould be the case for inaccuracy of ex change parameter.  \nHowever in this study, the anisotropy of the resonance field values as a function of Ti thickness is very \nlarge and periodic (nearly 10  Å). The reason for this large shift is not obvious. Even though there  is small \nexchange interaction be tween different layers; this cannot be responsible for such  a large shift.  \n \nConclusion  \n \nThere is a long-range oscillatory indirect magnetic exchange coupling between two ferromagnetic layers \nseparated by thin layers of the nonmagnetic transition metals . Parkin [22] showed that this spectacular \nphenomenon occurs with almost any metal as the spacer material.  As a result of this indirect exchange \ninteraction two different resonance modes (optical and acoustic) are observed on the FMR spectra. Their relative \nresonance field values and intensities help to characterize the magnetic interaction between ferromagnetic \nlayers. In this study, we did not observe two well resolved modes for all samples; instead we had two broad \noverlapped peaks. Unfortunately , these broa d peaks were not able to be identified by the theoretical model \nwhich is suitable for thin multilayer system s. Smardz  [19,20,22] observed that Co sublayers are very weakly \nexchange coupled  or decoupled for dTi > 27 Å in the previous  hysteresis measurements  for 170 Å Co-dTi-170 \nÅ Co trilayer.  Also he didn’t find any indication for the antiferromagnetic  coupling between Co sublayers. In \nour experiment, FMR spectra support that Co sublayers are very weakly exchange coupled for dTi > 6 Å. One \ninteresting thing is that the thick ness of Ti spacer does not  affect the magnitude of the exchange coupling but \nrather affects the magnetic resonance positions in a periodic way.  \nThere are two possible reasons for not clearly observing exchange coupling interaction between ferromagnetic \nlayers. The f irst reason is the choice of two magnetic layer thicknesses. In our experiment thickness differences \nbetween Co sublayers were only 5 Å.  It seems this choice may not be the right  one. However , pure Co 40 Å and \nCo 45 Å films gave  resonance peaks at different magne tic field positions, so we do not  think that sublayers \nthicknesses is the issue here.  The second reason may  be the choice of non -magnetic sp acer. If titanium did not \ngrow layer by  layer it did not  behave as spacer betwee n magnetic layers. As a result, two Co magnetic layers \nbehaved such as one layer and simultaneous excitations of precession of magnetization occurred in all layers.  \nAcknowledgments  \n \nThis work was partly supported by TUBITAK (Project No :TBAG -106T576 ) and by  State Planning \nOrganization of Turkey (DPT -Project No: 2007K120900 ). We gratefully acknowledge that all samples used in \nthis study were grown at the Nanotechnology Center of Gebze Institute of Technology.  \nReferences  \n1. M. N. Baibich, J. M. Broto, A. Fert, F. Nguyen Van Dau, F. Petroff, Phys. Rev . 61  21 (1988) . \n2. G. Binasch, P.Grünberg, F. Saurenbach, W. Zinn, Phys. Rev. B  39 7 (1989) .  \n3. T. Miyazaki, N. Tezuka, J. Magn. Magn. Mater . 139, 231 (1995) . \n4. J. S. Moodera, Lisa R. Kinder, Terrilyn M. Wong, and  R. Meservey, Phys. Rev. Lett . 74 3273  (1995) . \n5. M. Pardavi -Horvath, Magnetic Multilayers Book , Editors L.H.Bennett, R.E.Watson, Singapore  World Scientific,  1994 . \n6. P. Grünberg, R. Schreiber, Y.Pang, M.B. Brodsky and H.Sowers, Phys. Rev. Lett . 57 2442  (1986). \n7. S.S.Parkin, N. More and K.P.Roche, Phys. Rev. Lett . 64 2304  (1990) . \n8. N.D. Telling, M.J. Bonder, G.A. Jones, C.A. Faunce, P.J Grundy, D.G. Lord,  D.E. Joyce , Magnetics, IEEE Transactions  \n37(4) 2308  (2001) . \n9. MP Dariel, D. Dayan, A. Languille, Magnetic Multila yers Book , Editors L.H.Bennett, R.E.Watson, Singapore , World \nScientific 1994, pp.355 -373.  \n10. M T Johnson, P J H Bloemen, F J A den Broeder and J J de Vries ,  Rep. Prog. Phys . 59 1409  (1996) . \n11. J.J. Krebs, P.Lubitz, A. Chaiken and G.A.Prinz , Phys. Rev. Lett. 63, 1645 (1989); J.  Appl. Phys . 67 5920  (1990) . \n12. R.Topkaya, M. Erkovan, A. Öztürk, O. Öztürk, M. Özdemir, B.Aktaş , J. App. Phys . 108 23910  (2010) . \n13. M. Erkovan , S.T. Öztürk, R. Topkaya, M. Özdemir, B.Aktaş, O. Öztürk ,  J. App. Phys . 110 (2011) . \n14. K. Chrzumnicka , J. Dubowik, R. Gontarz, I. Gogciafiska, H. Ratajczak, L. Smardz F. Stobiecki, B. Szymafisk, J.Magn. \nMagn. Mater.  160 349 (1996) . 16. Ping Wu, E.Y. Jiang, C.D. Wang, J. Magn. Magn. Mater . 168 43 (1997) . \n17. M. Schmidt ,  F. Stobiecki, and B. Szyma ński, Phys. Stat. Sol. (a)  196(1) , 56 (2003) . \n18. M. Erkovan, Azerbaijan Journal Of Physics Fizika , Vol XVI(2),   342 (2010) . \n19. L. Smardz,  K. Smardz, H. Niedoba , Jorn. Magn. Magn. Mater . 220, 175 (2000) . \n20. L. Smardz, Czechoslovak Journal of Physics  52, 209 (2002) . \n21. S. S. P. Parkin, Phys. Rev. Lett . 67 3598  (1991) . \n22. L. Smardz , Solid State Communications  112 693 (1999) . \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n "    },    {        "title": "0912.3409v1.Quantum_level_control_in_a_III_V_based_ferromagnetic_semiconductor_heterostructure_with_a_GaMnAs_quantum_well_and_double_barriers.pdf",        "content": "c 1cQuantum-level control in a III-V-based ferromagneti c-semiconductor heterostructure \nwith a GaMnAs quantum well and double barriers \nc\nShinobucOhya a) c\nDepartment of Electrical Engineering and Information Systems, The University of Tokyo, \n7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan, and P RESTO Japan Science and \nTechnology Agency, 4-1-8 Honcho, Kawaguchi, Saitama  332-0012, Japan \nc\nIriyacMunetacandcMasaakicTanaka b) c\nDepartment of Electrical Engineering and Information Systems, The University of Tokyo, \n7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan \nc\nc Wec investigatec thec spin@dependentc tunnelingc proper tiesc inc ac three@terminalc \nIII@V@basedc ferromagnetic@semiconductorc heterostruc turec withc ac 2.5@nm@thickc GaMnAsc \nquantumcwellc(QW)candcdoublecbarriers.c c Wecsuccessf ullyccontrolcthecquantumclevelscandc \nmodulatecthecspin@dependentccurrentcwithcvaryingcth ecvoltagecofcthecelectrodecconnectedctoc \nthecGaMnAscQW.c c Ourcresultscwillcopencupcacnewcposs ibilitycforcrealizingcthree@terminalc \nspincresonant@tunnelingcdevices.c\nc\na)cElectroniccmail:cohya@cryst.t.u@tokyo.ac.jpc\nb)cElectroniccmail:cmasaaki@ee.t.u@tokyo.ac.jpcc 2cIII@V@basedcferromagnetic@semiconductorcGaMnAschete rostructurescareconecofcthec \nmostcidealcsystemscforcfuturecsemiconductor@basedcs pintroniccdevices.c c Especially,cthechighc\ncoherencycofcthecvalence@bandcholescincthesecstruct uresciscacverycpromisingcfeaturecforcdevicesc \nusingcthecquantum@sizeceffect.c c Recently,cwechavecc learlycobservedcthecresonantctunnelingc \neffectcandcthecincreasecofctunnelcmagnetoresistance c(TMR)cinducedcbycresonantctunnelingcinc \nthecGaMnAscquantum@wellc(QW)cdouble@barrierc(DB)che terostructuresc(QWDBs).1c c Onecofc \nthecnextcimportantcgoalsciscthecrealizationcofc“thr ee@terminal”cGaMnAscQWDBcdevicescsuchc \nasc spinc resonant@tunnelingc transistors,2c wherec thec spin@dependentc quantumc levelsc canc bec \nexternallyccontrolledcbycapplyingcthecvoltagectocth ecelectrodecconnectedctocthecGaMnAscQW.c c\nInccomparisoncwithcthecmetal@basedcmagneticctunnelc transistors, 3,4c weccancmoreceffectivelyc \ndesignc thec heterostructurec forc controllingc thec spin @polarizedc coherentc holesc utilizingc thec \nwell@establishedcband@engineeringctechniquecofcsemi conductors.c c Furthermore,cthecmetallicc \nfeaturecofcGaMnAscallowscusctocmakecacgoodccontactc tocthecGaMnAscQWclayer,cthuscmorec \nefficientccontrolcofcthecquantumclevelsciscpossible cthancthatcincusualcsemiconductor@basedc \nthree@terminalcquantumcheterostructuresccontainingc acQWcelectrode.5c c Here,cwecdemonstratec \nthecquantum@levelccontrolcincacthree@terminalcGaMnA scQWDBcheterostructurecandcshowcthec \nsuccessfulcmodulationcofcthecspin@dependentccurrent cincthiscdevice.c\nFigurec1cillustratescthecschematicccross@sectionalc structurecofcthecdevicecinvestigatedc \nhere.c c ThecGaMnAscQWDBcdevicecwascgrowncbycmolecula rcbeamcepitaxycandcconsistscof,c\nfromcthecsurfacecside,cGa 0.94 Mn 0.06 As(10cnm)/cGaAsc(1cnm)/cAl 0.94 Mn 0.06 As(4cnm)/cGaAs(2c \nnm)/c Ga 0.94 Mn 0.06 Asc QW(2.5c nm)/c GaAs(1c nm)/c AlAs(4c nm)/c GaAs:Bec (100 c nm)c onc ac\np+GaAs(001)csubstrate.c c ThecBecconcentrationcofcthecB e@dopedcGaAsc(GaAs:Be)clayercwasc\n2µ10 18c cm @3.c c Thec thinc GaAsc spacerc layersc werec insertedc toc smo othc thec surface.c c Thec \nGaAs:Be,cAlAscandctheclowestcGaAscspacerclayerscwer ecgrowncatchighctemperaturescofc600,c \n550,c andc 600c ºC,c respectively.c c Thec GaMnAsc QW,c GaAs /c AlMnAs/c GaAs,c andc thec topc c 3cGaMnAsclayerscwerecgrowncatclowctemperaturescofc240 ,c200,c225ºC,crespectively.c c Incthisc \nstructure,cwecusedcAlMnAscascthecupperctunnelcbarri er.c c WecconfirmedcthatcAlMnAscactscascac \nparamagneticctunnelcbarriercagainstcGaMnAscwithcacb arriercheightcofc110cmeV.6c c ThecCuriec \ntemperaturec ( TC)c ofc thec GaMnAsc layersc wasc estimatedc toc bec ~60c Kc by c magnetizationc \nmeasurements.c c Aftercthecgrowth,cwecfabricatedcacri ng@shapedcelectrodec(namedcQW)cwhosec \nareacisc36@timesclargercthancthatcofctheccentralcel ectrodec(namedcTOP)concthecsample,candcthec \nregioncsandwichedcbetweencthesecelectrodescwasccare fullycetchedctocthecdepthcofcthecAlMnAsc \nbarrierclayer.c c Duectoctheclargecdifferencecincarea csizecbetweencthesecelectrodes,cthecenergyc \npotentialcofcthecGaMnAscQWccancbeceffectivelyccontr olledcbycthecvoltagecofcQWc( VQW ).c c Inc \nthecfollowingcspin@dependentctransportcmeasurements ,cthecTOPcelectrodeciscgrounded,candcwec \nfocusconctheccurrentc IcthroughcTOPctocthecgroundcwhencchangingc VQW candcthecvoltagec Vcofcthec \nsubstratec (namedc SUB).  Notec thatc thec signc ofc Vc isc thec oppositec toc thatc definedc inc ourc \npreviouscpapers,1,7c socthecresonantclevelscarecdetectedcincthec V>0cregioncwhenc VQW ciscnotc \napplied.c\nFigurec2(a)cshowscthec dI/dV @Vccurvescincparallelcmagnetizationcwhenc VQW ciscvariedc \nfromc@0.05ctoc+0.05cVc(fromcbottomctoctop)cwithcthe cvoltagecstepcofc0.01cVcatc3.6cK.c c Inc \neveryc curve,c oscillationsc duec toc resonantc tunneling c ofc thec heavy@holec firstc statec (HH1),c \nlight@holecfirstcstatec(LH1),candcheavy@holecsecond cstatec(HH2)carecobserved.c c Thecvalleyscofc \nthesecoscillationsclinearlycshiftctocthechighercvol tagescwithcincreasingc VQW cascshowncbycthec\nblackcdottedclines.c c Figurec2(b)cshowscthecschemati ccvalence@bandcdiagramscofcourcGaMnAsc \nQWDBcheterostructurecinctermscofcholecenergycwhenc VQW ciscpositivec(uppercgraph),czeroc \n(middlecgraph),candcnegativec(bottomcgraph). 8c c Black,credcandcblueclinescarecthecvalencecbandc \natcthecΓcpointc( Ev),cresonantclevels,candcthecchemicalcpotentialc µcofcthecGaAs:Becelectrode.c c \nAsccancbecseencincthesecpictures,cwhenc VQW ciscincreasedcfromcnegativectocpositive,c Vcforc\ndetectingcthesecresonantclevelsciscincreased.c c Thus ,cthecresonantcpeakscarecshiftedctochigherc c 4cvoltagesc withc increasingc VQW ,c whichc indicatesc thatc thec quantumc levelsc arec succe ssfullyc \ncontrolledcbycchangingc VQW .c c IncFig.c2(a),cweccancalsocseecothercoscillations cindicatedcbyc\nlinescAcandcB,cwhichcwillcbecdiscussedclater.c\nFigurec3cshowscthecmappingscofc(a)c dI/dV candc(b)c d2I/dV 2cascfunctionscofc VQW candc Vc\nincparallelcmagnetizationcatc3.6cK.c c Here,cthecreso nantclevelscofcHH1,cLH1,candcHH2carec \ntracedcbycthecblackcdottedccurves.c c Thecshiftcinc Vcofcthesecresonantclevelsciscalmostcsaturatedc\nwhenc |VQW |c getsc largerc thanc ~0.2c V.c c Thisc reasonc isc explaine dc asc follows.c c VQW ,c\ncorrespondingctocthecvoltagecdropcbetweencTOPcandcQ Wcthroughctheccurrentcpathcindicatedcbyc \nI’’cincFig.c1,ciscmainlycconsumedcatcthecAlMnAscbarrie rcbeneathcthecTOPcelectrodecandcincthec\nGaMnAscQWcplanecbetweencthecTOPcandcQWcelectrodes.9c c Withcincreasingc| VQW |,cthectunnelc \nresistancecofcAlMnAscbecomescmuchcsmallercduectocit sclowcbarriercheightc(~110cmeV),6cwhilec \nthecresistancecofcthecGaMnAscQWcplanecdoescnotcdepe ndconc VQW .c c Ascacresult,c VQW cisc \nconsumedcmostlycincthecGaMnAscQWcplanecwhenc| VQW |cisclarge,cthuscthecshiftcofcthecresonantc \npeaksciscsaturated.c c Wecnotecthatcthecresonantcleve lscarecdetectedcwhencthecenergycregionc \noccupiedcbycholescincthecGaAs:Becelectrodeccrossesc thecresonantclevelscincthecGaMnAscQWc\nregardlesscofctheccurrentcdirection.c c Thus,cthecres onantclevelscarecdetectedcbothcinc V>0candc \nV<0cregions.c c Incourcstudiescincludingcothercthree@t erminalcGaMnAscQWDBcdevicesc(notc \nshown),cwecfoundcthatcthecshiftcofcthecresonantclev elsctendsctocbeclargercwhencthec TCcofcthec \nGaMnAsc QWc isc higher,c whichc meansc thatc thec metallicc naturec ofc thec GaMnAsc QWc isc \nimportantcforcimprovingctheccontrollabilitycofcthec quantumclevels.c\nIncFig.c3(a),cthereciscacdeepcvalleyctracedcwithcth ecwhitecdottedccurvecAcpassingc \nthroughcthecoriginc( V=VQW =0).c c ThiscvalleyccorrespondsctoctheclinecAcincFig. 2c(a)candcthec \ncurvecAcincFig.c3(b).c c Thiscvalleyciscconsideredcto cbecthecbiascconditioncwherecthecchemicalc \npotentialscofcTOPcandcQWcarecthecsame.c c Ifcthecpote ntialcofcthecGaMnAscQWcwerecperfectlyc\ncontrolledcbyc VQW ,cthesecvalleyscwouldcnotcbecobserved.c c Thiscresult cmeanscthatcthecpotentialc c 5cofcthecGaMnAscQWciscinfluencedcalsocbyc V ascwellcascbyc VQW .c c Weccancseecacsmallcvalleyc \ntracedcwithctheccurvecBcincFig.c3(a),cwhichccancbec morecclearlycseencincFig.c2(a)c(linecB)candc \nFig.c3(b)c(whitecdottedccurvecB).c c Theyccorrespondc tocthecbiascconditioncwherecthecchemicalc \npotentialscbetweencQWcandcSUBcarecthecsame.c\nIncFig.c3(a),ctherecarecvalleysctracedcwithcthecwhi teclinescCcmovingcalmostclinearlyc\nwithcchangingc VQW ,cwhichcoverlapcthecabove@mentionedcresonantclevels cwhenc|VQW |ciscsmallerc\nthanc~0.05cV.c c Thesecvalleysccorrespondctocthecreso nantclevelscformedcincthecareacofcthec \nGaMnAscQWcplanecbeneathcthecring@shapedcQWcelectrod ec(SeecFig.c1),cwherecthecpotentialcisc \nalmostc ideallyc controlledc byc VQW c duec toc itsc proximityc toc thec QWc electrode.c c Sincec t hec \nGaMnAscQWciscmetallic,c(probablycnon@ballisticccomp onentcof)ctheccurrentc Iccollectedcatcthec \nTOPcelectrodeciscexpectedctocbecwidelycspreadcincth ecGaMnAscQWcplane.c c Thus,cacpartcofc \nthisccurrentciscdetectedcincthecmeasurementscofc I.c c Thesecvalleyscwerecmorecclearlycseencinc \nthec dIQW /dV cmappingcascfunctionscofc VQW candc V (notcshown),cwherec IQW cisctheccurrentcgoingc \noutcthroughcthecQWcelectrode.c\nFigurec4cshowscthec Vcdependencecofcthecmagneto@currentc(MC)cratiocdefin edcbyc\n(IP@IAP )/ IAP cwithcvariousc VQWc fromc0ctoc@0.14cVcatc3.6cK,cwherec IP (IAP )crepresentsc Icincparallelc \n(anti@parallel)cmagnetization.c c (Forcthecmeasuremen tscofcthecspin@dependentccurrent,cseecourc \npreviousc paper. 1)c c Toc controlc thec magnetizationc alignmentc ofc thec Ga MnAsc layers,c thec \nmagneticc fieldc wasc appliedc alongc thec [100]c axisc inc plane.c c Thesec MCc vs.c Vc datac werec \nmathematicallycderivedcfromcthec I-Vcdatacincparallelcandcanti@parallelcmagnetizationsc atczeroc \nmagneticcfield.c c Withcchangingc VQW cfromc0ctoc@0.14cV,cthecMCcpeakcatcLH1ciscdecreased ,c \nwhilecthecMCcpeakcatcHH2ciscincreased.c c Thiscopposi tecbehaviorciscduectocthecincreasingc \ncurrentcfromcTOPctocQWcidentifiedcbyc I’’cincFig.c1cwithcchangingc VQW cfromc0ctoc@0.14cV.c c \nHere,cwecdefinecthisccurrentcasc I’’Pc( I’’AP )candcthecdirectccurrentctransferredcbetweencTOPcan dc \nSUBcidentifiedcbyc I’cincFig.c1casc I’Pc( I’AP )cincparallelc(anti@parallel)cmagnetization,cthencM Ccisc c 6cexpressedc asc [( I’P -I’’P)@(I’AP @I’’AP )]/( I’AP @I’’AP ).c c Here,c wec takec thec MC@ Vc curvec withc\nVQW =@0.03c(lightcblueccurve)cascancexamplecforcexplain ingcthecMC@VcbehaviorcshowncincFig.c \n4.c c Whenc V<<0,cMCciscdominatedcbyc I’Pcandc I’AP ,cthuscthectypicalc VcdependencecofcMCc\nappears,cwherecMCciscmonotonicallycdecreasedcwithci ncreasingc|V|.c c Whenc Vciscaroundczero,c \nI’Pcandc I’AP careccomparablectoc I’’Pcandc I’’AP ,crespectively.c c Thus,cthecdenominatorcincthec \ndefinitioncofcMCcbecomescclosectoczero,candcMCcbeco mescinfinity.c c Whenc Vciscincreasedc \nmore,cthecsigncofcthecdenominatorciscchanged,cthusc MCcbecomescnegativecinfinity.c c Incthec \nV>>0cregion,cI’Pcandc I’AP carecdominantcagain,cthuscMCcbecomescpositivecandcm onotonicallyc \ndecreasesctoczero.c c AlthoughcthecLH1cpeakcnearcthec originciscmoreclargelycaffectedcbyc I’’Pcandc \nI’’AP candcMCciscdecreasedcwhenc VQW ciscchangedcfromc0ctoc@0.04cV,cthecMCcpeakcatcHH2ci sc \nincreasedcduectocthecsmallercinfluencecofc I’’Pcandc I’’AP .c c ThecMCcincreasecatcHH2cincthecrangec \nofc VQW cfromc0ctoc@0.1cVciscmainlycbecausecthecenergycofcH H2cbecomescclosectocthecchemicalc \npotentialcofcTOPcwherecachighcspincpolarizationcisc expected.c c Incthisc VQW crange,cthecbiasc \nvoltagecofcthecMCcpeakcatcHH2cmovesctowardcthecnega tivecdirectioncwithcchangingc VQW cfromc \n0ctoc@0.1cV,cfollowingcthecHH2cpeak’scshiftcascshow ncincFig.c2(a),c3(a),candc(b).c c Thiscmeansc \nthatc wec successfullyc controlledc thec spin@dependentc currentc byc electricallyc modulatingc thec \nquantumclevelscincthecGaMnAscQW.c c When  V QW ciscchangedcfromc@0.1ctoc@0.14cV,cthecMCc\npeakcatcHH2cmovesctowardcthecpositivecdirection.c c I ncFig.c3(a)candc(b),cweccancseecthatcthec \nresonantclevelcofcHH2ciscmergedcwithcthecvalleycAcw hencVQW cisc~@0.1cVcandcthecvalleycAc\nbecomescdominantcwhenc VQW c<@0.1cV.c c Thus,cthecTMRcincreasecobservedcwhenc VQW c<c@0.1c \nVciscattributedctocthecvalleycA.c\nIncthecpresentcdevice,cthecpowercgainchascnotcbeenc obtainedcduectoctheclargectunnelingc \ncurrentc I’’cbetweencTOPcandcQW.c c Ifcweccancinsertcacthickcbarr iercwithcaclowcbarriercheightc \nbetweencTOPcandcQWcwithoutcsuppressingcMC,cthecspin @transistorcoperationcwithcacpowerc \ngainc isc expectedc toc bec obtained,c whichc willc leadc to c quantumc spinc devices,c suchc asc spinc c 7cresonant@tunnelingctransistorscandcmagneticcmonosta ble@bistablectransitionclogiccelements.10c\nIncsummary,cwechavecfabricatedcthecthree@terminalcG aMnAscQWDBcdevice.c c Wec\ncontrolledc thec quantumc levelsc ofc thec 2.5@nmc thickc G aMnAsc QWc andc modulatedc thec \nspin@dependentccurrentcwithcvaryingc VQW ,cwhichcisclargelycattributedctocthecmetalliccfeatu recofc \nthecGaMnAscQW.c c ThecMCcratiocwascincreasedcbycdecre asingcthecpotentialcofcthecGaMnAsc \nQWcinctermscofcholecenergycbycapplyingcthecnegative cbiascvoltagectocthecQWcelectrode.c\nc\nThisc workc wasc partlyc supportedc byc Grant@in@Aidsc for c Scientificc Researchc No.c \n18106007,c No.c 19048018,c andc No.c 20686002,c thec Speci alc Coordinationc Programsc forc \nPromotingcSciencecandcTechnology,cR&DcforcNext@gene rationcInformationcTechnologycbyc \nMEXT,candcPRESTOcofcJST.  c 8cReferences \nc\n1c S.cOhya,cP.cN.cHai,cY.cMizuno,candcM.cTanaka,cPhys .cRev.cBc 75 ,c155328c(2007).c\n2c Y.cMizuno,cS.cOhya,cP.cN.cHai,candcM.cTanaka,cAppl .cPhys.cLett.c 90 ,c162505c(2007).c c\n3c K.cMizushima,cT.cKinno,cT.cYamauchi,candcK.cTanaka ,cIEEEcTrans.cMagn.c 33 ,c3500c(1997).c\n4c S.cvancDijken,cX.cJiang,candcS.cS.cP.cParkin,cAppl .cPhys.cLett.c 83 ,c951c(2003).c c\n5c Lepsa,cM.I.,cvancdecRoer,cTh.G.,cKwaspen,cJ.J.M.,c Smalbrugge,cE.,cvancdercVleuten,cW.,c \nKaufmann,cL.M.F.c1997cInternationalcSemiconductorcC onferencec20thcEdition.cCAS'97c \nProceedingsc(Cat.cNo.c97TH8261)  1 ,c139c(1997).c\n6c S.cOhya,cI.cMuneta,cP.cN.cHai,candcM.cTanaka,cAppl .cphys.cLett.c 95 ,c242503c(2009),c \narXiv:0912.3045.c\n7c S.cOhya,cI.cMuneta,cP.cN.cHai,candcM.cTanaka,csubm itted.c\n8c IncRef.c7,cwecpreciselycestimatedcthecenergycdiffe rencecbetweenc EvcandcthecFermiclevelcandc \nfoundcthatcitcdependsconcthecGaMnAscQWcthickness.c c Here,cwecneglectcthisceffectcforc \nsimplicity.c\n9c Thereciscacsmallccurrentcpathcthroughcthecunderlyi ngcGaAs:BeclayercandcthecAlAscbarriercasc \nindicatedcbycdottedcpathcincFig.c1,cbutcwecneglecte dcthiscpath.c c Wecalsocneglectedcthec \nresistancecofcthecAlMnAscbarriercjustcbeneathcthecr ing@shapedcQWcelectrode,cbecausecitcisc \nmuchcsmallercthancthatcbeneathcthecTOPcelectrodecdu ectoctheclargecdifferencecincareacsizec \nbetweencthecTOPcandcQWcelectrodes.c\n10 c C.cErtlercandcJ.cFabian,cPhys.cRev.cBc 75 ,c195323c(2007).cc 9cc\nc\nc\nc\nc\nFIG.c1.c c Schematicccross@sectionalcstructurecofcthe cthree@terminalcdevicecinvestigatedchere.c c \nThecGaMnAscQWDBcheterostructureccomprises,cfromcthe csurfacecside,cGa 0.94 Mn 0.06 As(10c\nnm)/cGaAsc(1cnm)/cAl 0.94 Mn 0.06 As(4cnm)/cGaAs(2cnm)/cGa 0.94 Mn 0.06 AscQW(2.5cnm)/cGaAs(1c\nnm)/cAlAs(4cnm)/cGaAs:Bec(100cnm)concacp +GaAs(001)csubstrate.c c I’ representsctheccurrentc \npathc directlyc flowingc fromc SUBc toc TOP.c c I’’c isc thec currentc pathc flowingc fromc thec TOPc \nelectrodectocthecQWcelectrodecthroughcthecGaMnAscQW cplane.c c Dottedcpathcisctheconec \nthroughcthecGaAs:Beclayer.c c Notecthatcthereciscalso caccurrentcpathcdirectlycfromcSUBctocQW.cc 10 cc\nc\nc\nc\nc\nc\nc\nFIG.c2.c(a)c dI/dV @Vccurvescincparallelcmagnetizationcwhenc VQW ciscvariedcfromc@0.05ctoc+0.05cVc \n(fromcbottomctoctop)cwithcthecvoltagecstepcofc0.01c Vcatc3.6cK.c c (b)cSchematiccvalence@bandc \ndiagramsc ofc ourc GaMnAsc QWDBc heterostructurec inc term sc ofc holec energyc whenc VQW c isc \npositivec(uppercgraph),czeroc(middlecgraph),candcne gativec(bottom).c c Black,credcandcbluec \nlinescarecthecvalencecbandcatcthecΓcpointc( EV),cresonantclevels,candcthecchemicalcpotentialc µcofc \nthecGaAs:Becelectrode.cc 11 cc\nc\nc\nc\nc\nc\nc\nc\nFIG.c 3.c (a)c dI/dV c andc (b)c d2I/dV 2c mappingsc asc functionsc ofc VQW c andc Vc inc parallelc \nmagnetizationcatc3.6cK.c c Here,cthecresonantclevelsc ofcHH1,cLH1,candcHH2carectracedcbycthec \nblackcdottedccurves.c c Forcthecexplanationcofcotherc curvescA@C,cseecthectext.cc 12 cc\nc\nc\nc\nc\nc\nFIG.c4.cVcdependencecofcMCcdefinedcbyc( IP@IAP )/ IAP catc3.6cKcwithcvariousc VQWc fromc0ctoc@0.14c \nV,cwherec IP (IAP )crepresentsc Icincparallelc(anti@parallel)cmagnetization.c\nc"    },    {        "title": "1401.3543v1.Structural_and_magnetic_properties_of_irradiated_SiC.pdf",        "content": "Page | 1 \n Structural and magnetic properties of irradiated  \nSiC  \nYutian Wang1, 4, Xuliang Chen2, Lin Li1, 3, Artem Shalimov1, Wei Tong5, Slawomir  \nPrucnal1, Frans Munnik1, Zhaorong Yang2, Wolfgang Skorupa1, Manfred Helm1,4, \nShengqiang Zhou1 \n1. Institute of Ion Beam Physi cs and Materials Research, Helmholtz -Zentrum \nDresden -Rossendorf (HZDR), P.O. Box 510119, 01314 Dresden, Germany  \n2. Key Laboratory of Materials Physics, Institute of Solid State Physics, Chinese \nAcademy of Sciences, Hefei 230031, People's Republic of China  \n3. Department of Physics and Electronics, School of Science, Beijing University \nof Chemical Technology, Beijing 100029, China  \n4. Technische Universität Dresden, 01062 Dresden, Germany  \n5. High Magnetic Field Laboratory, Hefei Institutes of Physical Science , Chinese \nAcademy of Sciences, Hefei 2300 31, People's Republic of China  \n \nAbstract:  \nWe present a comprehensive structural characterization of ferromagnetic SiC single \ncrystals  induced by Ne ion irradiation . The ferromagnetism has been confirmed by \nelectron spin resonanc e and possible transition metal impurities can be excluded to \nbe the origin of the observed ferromagnetism. Using X -ray diffraction and Rutherford \nbackscattering/channeling spectroscopy, we estimate the damage to the crystallinity \nof SiC which mutually  influences the ferromagnetism in SiC.  Page | 2 \n  \nI. Introduction:  \nDefect -induced ferromagnetism  in materials which do  not contain  partially filled 3d or \n4f electrons  has recently entered the focus  of condensed matter research [1]. It \nchallenges the mJ paradigm for magn etism, where m refers the local moment and J \nstands for the interaction between the local moments. Experimentally,  defect -induced \nferromagnetism was observed in many materials, including graphite  [2-5] and  various \noxides [6-12]. SiC single crystal s are emerging as another candidate for this \ninvestigation and ha ve been shown to be ferromagnetic after particle irradiation  [13, \n14] or after aluminum doping  [15]. The magnetization in SiC is found to sensitively \ndepend on the fluence of ion or neutron irradiatio n. Compared with other materials , \nSiC is commercially available at large scale with the microelectronic quality grade  \n[16]. In this paper, we present a systematic structural investigation in correlation with \nthe magnetic properties of SiC prepared by Ne io n irradiation. A possible Fe, Co or Ni \ncontamination in SiC is excluded by particle  induced X -ray emission (PIXE) and by \nAuger electron spectroscopy (AES) measurements . The appearance of \nferromagnetism has been confirmed by electron spin  resonance  (ESR) spectroscopy . \nMeasurements of X -ray diffraction  (XRD ) and Rutherford backscattering/channeling \nspectroscopy  (RBS/C)  reveal the damage to the crystallinity of SiC which can \nextinguish  the ferromagnetism in SiC.  \nII. Experimental methods:  \nThe 6H-SiC(0001 ) wafer  purchased from KMT corporation (Hefei, China)  is one -side \npolished and semi -insulating.  The same wafer was cut into smaller pieces which \nwere  implanted with Ne ions at an energy of 140 keV. The ion fluences were 5×1013, Page | 3 \n 1×1014, 5×1014, 1×1015 cm-2, and conseq uently the samples will be named  as 5E13, \n1E14, 5E14 and 1E15, respectively. Magnetometry was performed using a MPMS -XL \nmagnetometer from Quantum Design. The ferromagnetic resonance was measured \nat 9.46 GHz by an electron paramagnetic resonance spectromete r (Bruker ELEXSYS \nE500 ).  PIXE  was performed using 3 MeV protons with a broad beam  of 1 mm2, while  \nAES was done by  a scanning Auger electron spectrometer Microlab 310F (Fisons \nInstruments). In order  to evaluate the crystalline variation after ion irradiati on, \nsynchrotron radiation XRD  was carried out  at the Rossendorf beamline (BM20) at the \nESRF with an X-ray wavelength of 0.1078 nm.  As a complementary method, RBS/C \nwith 1.7 MeV He+ was used to  quantitatively determine the ion -implantation -induced \natomic d isorder of the Si sublattice in 6H -SiC.  \nIII. Results and discussion:  \n-2500 0 2500-2-1012@300 K\n  \n  \nField (Oe)Magnetization (emu/g)  Virgin\n  5E13\n  1E14\n  5E14\n  1E15(a)\n-4000 -2000 02000 4000\n  \n   Virgin\n 5E13\n0 1600 3200Intensity (arb. u.)\n  \n100 K300 K(b) sample 1E14\nField (Oe)\n \nFig. 1: (a) Magnetization vs. field measured at 300 K for all samples: the diamagnetic background from \nthe substrate has been subtracted.  The magnetization was calculated by assum ing a thin layer of 460 \nnm thickness. The inset shows the magnetization measured for the virgin sample (black line) and Page | 4 \n sample 5E13 without subtracting the background. (b) ESR spectra of sample 1E14 at temperatures  \nbetween 100 and 300 K : a broad resonance peak appears indicating ferromagnetic resonance . The \nblack line  indicate s the resonance centers which slightly change with temperature.  \n \nA. Magnetic properties  \nThe magnetic properties of Ne irradiated samples at di fferent temperature s have \nbeen reported in our previous paper  [13]. The inset of Fig. 1(a) shows the \nmagnetization at 300 K measured for  sample 5E13 and the virgin SiC from the same \nwafer. The virgin SiC  only shows diamagnetism, while sample 5E13 presents a \nferromagnetic hysteresis. Fig. 1 (a) shows the magnetization at 300 K for all samples \nafter subtracting the diamagnetic background. The saturation magnetization shows a \nnonlinear dependence with increasing Ne fluence (defect  concentration ) and reach es \nits maximum for sample 1E14.  \nThe appearance of the ferromagnetism in ion irradiated SiC is further confirmed by \nESR. We have measured virgin SiC, sample 1E14 and one paramagnetic SiC in a \nbroad field range. Only for sample 1E14, a broad ESR peak, and a shift  of the center  \nof the resonance at lower temperatures are clearly observed (Fig . 1b). Even at 300 K, \nthe ESR signal shows a very large shift from the free -electron position  (around 3 300 \nOe), show ing that th e sample is ferromagnetic and  has Tc above 300 K. The g-factor \nincreases slightly with decreasing temperatures.    \nB. Scrutinizing possible magnetic contaminations  \nFor the investigation of defect -induced ferromagnetism, the crucial issue is to \ncarefully examine if the ferromagnetic contamination is actuall y responsible for the Page | 5 \n observed ferromagnetism. The magnetization [Fig. 1(a)] shows a clear dependence \non the ion fluence, however the magnetic contamination s are expected to be random \nor to increase with ion fluence  if they are due to ion implantation . We further applied \nPIXE and AES to scrutinize Fe/Co/Ni elements in our studied SiC specimens.  Figure \n2 show s the PIXE spectra for virgin SiC and 5E13.  In the spectrum, the narrow peak \nis from Si K -line X-ray emission. The broad peak is due to the Bremsstrahlu ng \nbackground.  No transition metal impurities were detected within  the detection limit  of \naround  1 µg/g.   \n0 2 4 6 8 10 12 14100101102103104\n  Yield (counts)\nEnergy (keV) Virgin 6H-SiC\n 5E13Si K-edge\nFe/Co/Ni K-edge\n \nFig. 2 PIXE spectrum for the virgin SiC wafer and a ferromagnetic sample. Within the detection limit, \nwe do  not observe a ny Fe, Co or Ni contamination.  \n \nTo exclude contamination introduced during implantation and handling  of the \nsamples, we applied the surface sensitive AES method to the ferromagnetic pieces. \nFor most elements, the detection limit of AES is in the range of 0.01-0.1 at %, which is \naround 1012-1013 atom/cm2 by assuming a detection depth of 2 nm. We measured \nboth the front and back sides for virgin and ferrom agnetic SiC after Ne irradiatio n and \ndid not observe any signal from  Fe/Co/Ni  within the detection limit  (not shown ).  Page | 6 \n C. Structural properties  \nIn order to correlat e magnetization with structure , we performed XRD and RBS/C to \ncheck the crystalline deformation and defects.  \nFigure 3 depicts the θ–2θ scans at SiC(000  12) recorded for Ne ion irradiated 6H -\nSiC. A  virgin sample is also shown for comparison.  All curves show a main sharp \ndiffraction peak at the diffraction angle θ = θ Bragg coming from the sample volum \nbelow  the irradiated part. The diffraction peaks  at the low angle side of the main \nBragg peak, (θ<θ Bragg) are the characteristic s of an expansion gradient of the lattice \nalong SiC[0001] . The elastic strain in the near surface region (Δd/d) can be deduced \nfrom the position of this satellite peak. Except for the largest fluence applied here \n(1E15), a fring e pattern is clearly visible. This feature is due to the presence of the \nnon-homogeneous lattice expansion  along the depth and arises from interferences \nbetween x -ray beams diffracted from regions with the same lattice spacing . The \nmaximum value of the nor mal strain (Δd/d) M is estimated to be roughly proportional to \nthe fluence.  Furthermore, we also performed the corresponding reciprocal spacing \nmapping  (RSM) around  SiC(000  12) (see Figure 4). Only a vertical streak coming \nfrom the irradiated region of the  crystals is observed. No broadening in the horizontal \ndirection (equivalent to a rocking curve) is measured as compared with that obtained \nfrom virgin crystals. These observations indicate that the defects are randomly \ndistributed point defects or small d efect clusters  [18]. \nFor the sample irradiated with the largest fluence of 1E15 cm-2, the absence of the \nsatellite peak shows that the near -surface region of this sample is heavily damaged, \ni.e. no more crystalline. Such a signal is consistent with a stron gly defective \ncrystalline lattice characterized by the presence of extended defects : amorphous Page | 7 \n clusters or layer [ 18]. The conclusions from XRD observation are  fully consistent with \nthe observation by positron annihilation spectroscopy, which reveals the c lustering of \nvacancies  [13].  \n0.75 0.76 0.77 0.78 0.79 0.80 0.8110-310-210-1100101102103104105106107\n  Intensity (a.u.)\n1/d (1/ Å)Resolution=0.0177 °\n Virgin\n 5E13\n 1E14\n 5E14\n 1E15\n \nFig. 3 XRD 2θ/θ scans of the virgin and Ne irradiated 6H -SiC recorded in the vicinity of the (000  12) \nreflection.     \n Page | 8 \n \n \nFig. 4 Reciprocal space mapping of SiC(000  12). There is no indication of broadening along q x. The \nstreak labeled CTR is the crystal truncation ro d. The streak labeled PSD is due to the transmittance \nfunction of the position sensitive detector.  \n \nThe RBS/C spectra alo ng the SiC[0001] axis are shown in Fig. 5 for virgin and for Ne \nirradiated samples. The random spectrum is assumed equivalent to a comp letely \namorphized SiC, while the virgin spectrum corresponds to an essentially damage -\nfree crystal. The minimum yield (χmin), the ratio of the channeling  spectrum to the \nrandom spectrum, is 2.3% for the virgin SiC crystals at the surface region. For Page | 9 \n sample 5E13 (not shown) , the channeling spectrum is only slightly higher than the \nvirgin sample. A peak in the damage depth around 150 nm is becoming visible for \nfluences of 5×1014 and 1×1015 /cm2. Note that for sample  1E15  χmin is 93%, indicating \nthe almost full amorphization  at the damage peak. This is in agreement with the \nthreshold displacement per atom ( DPA) values  for amorphization  reported  in \nliterature  [19]. DPA presents the irradiation effect considering both ion fluence and \nenergy [20].  \n200 400 600 800 1000\n  \n Yield (arb. u.)a  Random\nb  Virgin\nc  1E14\nd  5E14\ne  1E15\nEnergy (keV)SiCa\ne\nd\nc\nbDepth (nm)0 100 200\n \nFig. 5: RBS/C spectra for 6H-SiC implanted with Ne ions. A random spectrum and a channeling \nspectrum from a virgin sample are also included for comparison. The dashed  (dotted)  line indicates \nthe position of the damage peaks  for Si (carbon) . A rough depth scale is given.  \n024\n    (d/d0) (%)\n04080\n(c) Magnetization\n     at 5 K(b) Disorder\n min (%)(a) Strain\n0.01 0.1 1012\n M (emu/g)\nDPA\n Page | 10 \n Fig. 6: Evolution of the saturation magnetization , the Si -sublattice disorder and the maximum  strain  \nwith DPA. The gray area indicates the DPA range for optimizing the ferromagnetism.  \nIV. Summary  \nCombining  both structural and magnetic properties  of ion irradiated SiC , we are able \nto conclude  a mutual  relation between defect concentration  and magnetization. As \nshown in Fig. 6, there is a narrow DPA window, within which the sample is  \nferromagneti c. The initial introduction of str uctural disorder  (mainly point defects)  \nleads to pronounced magnetization. Further in creasing disorder induces \nagglomeration of point defects and finally amorphizes the host SiC crystal , \nconsequently  the ferromagnetism nearly drops to zero .  \nAcknowledgemen t \nThe work was financially  supported by the Helmholtz Association  (VH-NG-713).   Y. \nWang thanks the China Scholarship Council for supporting his stay at HZDR.   \n \nReference  \n1. J. M. D. Coey, P . Stamenov, R . D. Gunning, M . Venkatesan, K Paul, New J. Phys. 12, 0 53025 \n(2010).  \n2. P. Esquinazi, D. Spemann, R. Höhne, A. Setzer, K. -H. Han, and T. Butz, Phys. Rev. Lett. 91, \n227201  (2003).  \n3. Z. He, X. Yang, H. Xia, X. Zhou, M. Zhao, Y. Song, and T. Wang, Carbon 49, 1931 (2011) . . \n4. X. Yang , Z. He, W . Li, H . Xia, Y . Song, X . Zhou, X . Liu, M . Zhao, T . Wang, and K . Hou, J. Appl. \nPhys. 109, 083933 (2011).  \n5. J. Barzola -Quiquia, W. Böhlmann, P. Esquinazi, A. Schadewitz, A. Ballestar, S. Dusari, L. \nSchultze -Nobre, and B. K erstin g, Appl. Phys. Lett. 98, 192511 (2011) . \n6. M. Venkatesan, C. B. Fitzgerald, and J. M. D. Coey, Nature 430, 630 (2004).  Page | 11 \n 7. Q. Xu, H . Schmidt, S . Zhou, K . Potzger, M . Helm, H . Hochmuth, M . Lorenz, A . Setzer, P . \nEsquinazi, C . Meinecke, and M . Grundmann, Appl.  Phys. Lett. 92, 082508 (2008).  \n8. M. Khalid, M. Ziese, A. Setzer, P. Esquinazi, M. Lorenz, H. Hochmuth, M. Grundmann, D. \nSpemann, T. Butz, G. Brauer, W. Anwand, G. Fischer, W. A. Adeagbo, W. Hergert, and A. \nErnst, Phys. Rev. B 80, 035331 (2009 ).  \n9. X. Zuo , S. Yoon, A . Yang, W . Duan, C . Vittoria, and V . Harris, J. Appl. Phys. 105, 07C508 (2009).  \n10. C. Araujo, M . Kapilashrami, X . Jun, O. D. Jayakumar, S . Nagar, Y . Wu, C . Århammar, B . \nJohansson, L . Belova, R . Ahuja, G. Gehring, and K. V. Rao, Appl. Phys. Lett. 96, 2 32505 (2010).  \n11. H. Thakur, P. Thakur, R . Kumar, N. B. Brookes, K. K. Sharma , A. P. Singh , Y. Kumar, S. Gautam , \nand K. H. Chae , Appl. Phys. Lett. 98, 192512 (2011).  \n12. C. Gómez -Polo, S. Larumbe, and J. M. Pastor, J. Appl. Phys. 113, 17B511 (2013).  \n13. L. Li, S. Pr ucnal, S. D. Yao, K. Potzger, W. Anwand, A. Wagner, and S. Zhou , Appl. Phys. Lett. \n98, 222508 (2011).  \n14. Y. Liu, G. Wang, S. Wang, J. Yang, L. Chen, X. Qin, B. Song, B. Wang, and X. Chen, Phys . Rev. \nLett. 106, 087205 (2011).  \n15. B. Song, H. Bao, H. Li, M. Lei, T.  Peng, J. Jian, J. Liu, W. Wang, W. Wang, and X. Chen, J. Am. \nChem. Soc. 131, 1376 (2009).   \n16. J. R. Jenny, D. P. Malta, S. G. Müller, A. R. Powell, V. F. Tsvetkov, H. M. D. Hobgood, R. C. \nGlass, and C. H. Carter: J. Electron. Mater. 32 , 432 (2003).   \n17. P. Esqu inazi, J. Barzola -Quiquia, D. Spemann, M. Rothermel, H. Ohldag, N. Garc´ıa, A. Setzer, \nT. Butz, J. Magn. Magn. Mater. 322 , 1156  (2010) . \n18. A. Debelle, L. Thome,  D. Dompoint, A. Boulle, F. Garrido, J. Jagielski and D. Chaussende , J. \nPhys. D: Appl. Phys. 43 , 455408  (2010) . \n19. W.J. Weber, L.M. Wang, N. Yu, and N.J. Hess, Mater. Sci. Eng. A 253, 62 (1998).  \n20. J. Ziegler, J. Biersack, and U. Littmark, The Stopping and Range of Ions in Matter  (Pergamon, \nNew York, 1985).  Page | 12 \n Fig. captions  \nFig. 1: (a) Magnetization vs. field m easured at 300 K for all samples: the diamagnetic background from \nthe substrate has been subtracted. The magnetization was calculated by assum ing a thin layer of 460 \nnm thickness. The inset shows the magnetization measured for the virgin sample (black line ) and \nsample 5E13 without subtracting the background. (b) ESR spectra of sample 1E14 at temperatures \nbetween 100 and 300 K: a broad resonance peak appears indicating ferromagnetic resonance. The \nblack line indicates the resonance centers which slightly cha nge with temperature.  \nFig. 2 PIXE spectrum for the virgin SiC wafer and a ferromagnetic sample. Within the detection limit, \nwe do  not observe any Fe, Co or Ni contamination.  \nFig. 3 XRD 2θ/θ scans of the virgin and Ne irradiated 6H -SiC recorded in the vici nity of the (000  12) \nreflection.     \nFig. 4 Reciprocal space mapping of SiC(000  12). There is no indication of broadening along q x. The \nstreak labeled CTR is the crystal truncation rod. The streak labeled PSD is due to the transmittance \nfunction of the pos ition sensitive detector.  \nFig. 5: RBS/C spectra for 6H -SiC implanted with Ne ions. A random spectrum and a channeling \nspectrum from a virgin sample are also included for comparison. The dashed  (dotted)  line indicates \nthe position of the damage peaks  for Si  (carbon) . A rough depth scale is given.  \nFig. 6: Evolution of the saturation magnetization, the Si -sublattice disorder and the maximum strain \nwith DPA. The gray area indicates the DPA range for optimizing the ferromagnetism.  \n "    },    {        "title": "1311.7620v1.Magnon_radiation_by_moving_Abrikosov_vortices_in_ferromagnetic_superconductors_and_superconductor_ferromagnet_multilayers.pdf",        "content": "arXiv:1311.7620v1  [cond-mat.supr-con]  29 Nov 2013Magnon radiation by moving Abrikosov vortices in ferromagn etic superconductors\nand superconductor-ferromagnet multilayers\nA. A. Bespalov,1,2A. S. Mel’nikov,1,3and A. I. Buzdin2\n1Institute for Physics of Microstructures, Russian Academy of Sciences, GSP-105, 603950, Nizhny Novgorod, Russia\n2Universit´ e Bordeaux, CNRS, LOMA, UMR 5798, F-33400 Talenc e, France\n3Nizhny Novgorod State University, 22 Gagarin av., 603950, N izhny Novgorod, Russia\nIn systems combining type-II superconductivity and magnet ism the non-stationary magnetic ﬁeld\nof moving Abrikosov vortices may excite spin waves, or magno ns. This eﬀect leads to the appear-\nance of an additional damping force acting on the vortices. B y solving the London and Landau-\nLifshitz-Gilbert equations we calculate the magnetic mome nt induced force acting on vortices in\nferromagnetic superconductors and superconductor/ferro magnet superlattices. If the vortices are\ndrivenby a dc force, magnon generation due to the Cherenkov r esonance starts as the vortexvelocity\nexceeds some threshold value. For an ideal vortex lattice th is leads to an anisotropic contribution\nto the resistivity and to the appearance of resonance peaks o n the current voltage characteristics.\nFor a disordered vortex array the current will exhibit a step -like increase at some critical voltage.\nIf the vortices are driven by an ac force with a frequency ω, the interaction with magnetic moments\nwill lead to a frequency-dependent magnetic contribution ηMto the vortex viscosity. If ωis below\nthe ferromagnetic resonance frequency ωF, vortices acquire additional inertia. For ω > ω Fdissipa-\ntion is enhanced due to magnon generation. The viscosity ηMcan be extracted from the surface\nimpedance of the ferromagnetic superconductor. Estimates of the magnetic force acting on vortices\nfor the U-based ferromagnetic superconductors and cuprate /manganite superlattices are given.\nPACS numbers: 74.25.Uv, 75.30.Ds, 74.25.nn\nI. INTRODUCTION\nWithin the last 13 years a number of fascinating com-\npounds has been discovered, revealing the coexistence\nof ferromagnetism and superconductivity in the bulk.1–4\nThese compounds are U-based ferromagnets which be-\ncome superconducting at temperatures ∼1K under ap-\nplied pressure, or even at atmospheric pressure. Experi-\nmental investigation of magnetic properties of these ma-\nterials in the superconducting state is hampered by the\nMeissner eﬀect, making static measurements ineﬃcient.\nHowever, important parameters can be extracted from\ndynamical measurements of the spin-wave (magnon)\nspectrum, which can be determined, e. g., by microwave\nprobing5,6or using Abrikosov vortex motion.7,8\nA number of papers has been devoted to theoreti-\ncal investigation of the magnon spectrum in magnetic\nsuperconductors.5,6,9–13Buzdin9determined the magnon\nspectrum in a superconducting antiferromagnet with an\neasy-axis anisotropy. Diﬀerent types of spin waves in fer-\nromagnetic superconductors in the Meissner state have\nbeen studied by Braude, Sonin and Logoboy,5,6,10,11in-\ncluding surface waves and domain wall waves.\nExperimental measurements of the ac magnetic sus-\nceptibility ofsuperconductingferromagnetsrevealedthat\nthe screening of the magnetic ﬁeld created by mag-\nnetic moments in these materials is incomplete.2,3This\nindicates that the superconducting transition in the\nU-compounds occurs in the spontaneous vortex state.\nOnly two papers so far have addressed the inﬂuence of\nAbrikosov vortices on the magnon spectrum in ferromag-\nnetic superconductors. In Ref. 12 coupled magnetic\nmoment-vortex dynamics has been studied in the limitof long wavelength λw≫a, whereais the inter-vortex\ndistance. Later,13this analysis has been extended to the\ncaseλw/lessorsimilara. It has been demonstrated that in the pres-\nence of a vortex lattice the magnon spectrum acquires a\nBloch-like band structure.\nTo study the spin wave spectrum experimentally two\nsimple procedures have been proposed. The ﬁrst method\nconsistsindirectexcitationofmagnonsbyanelectromag-\nneticwaveincidentatthesample.5,6,13Then, information\nabout the spin wave spectrum can be extracted from the\nfrequency dependent surface impedance Z. Note that\nthis procedure can be applied also to ordinary ferromag-\nnets, but inallcasesthehigh qualitycrystallinesurfaceis\nrequired. The surface impedance has been calculated for\na ferromagnetic superconductor in the Meissner state5,6\nand in the mixed state13for a static vortex lattice.\nThe second method is based on the indirect magnon\nexcitation: an external source of current sets in motion\nthe Abrikosov vortices, which start to radiate magnons\nwhen the Cherenkov resonance condition is satisﬁed.7,8\nHere, the current-voltage characteristics yield informa-\ntion about the magnon spectrum. Since this method in-\nvolves Abrikosov vortices, it is speciﬁc for superconduct-\ning materials. Diﬀerent phenomena arising from vortex-\nmagnetic moment interaction in magnetic superconduc-\ntors have been studied by Bulaevskii et al.7,8,14–17In\nRefs. 7 and 14 the dissipation power due to magnon\ngeneration by a moving with a constant velocity vortex\nlattice in a superconducting antiferromagnet has been\ncalculated. In Ref. 8 this result has been generalized for\nthe case of a vortex lattice driven by a superposition of\nac and dc currents. In Refs. 15 and 16 a polaronic mech-\nanism of self-induced vortex pinning in magnetic super-2\nconductors is discussed. The motion of the vortex lattice\nunder the action of dc15and ac currents16has been stud-\nied. Finally, in Ref. 17it has been predicted that the ﬂux\nﬂow should lead to the creation of domain walls in sys-\ntems with slow relaxation of the magnetic moments.\nIn the present paper, by solving the phenomenological\nLondon and Landau-Lifshitz equations, we analyze the\nproblem of magnon generation by moving Abrikosovvor-\ntices in ferromagnetic superconductors and superconduc-\ntor/ferromagnet (SF) multilayers. Theoretical investiga-\ntion of the latter systems is relevant in view of the recent\nsuccess in fabrication and characterization of cuprate-\nmanganite superlattices.18–20Also, recently an experi-\nmental study of the ﬂux-ﬂow resistivity in Nb/PdNi/Nb\ntrilayers has been reported.21Our consideration of bulk\nferromagnetic superconductors, on the other hand, is rel-\nevant to the U-based compounds mentioned above. In\nthis aspect, the present work complements the preceding\npapers,7,8,14–17which concetrated mainly on antiferro-\nmagnetic materials. As we show, the presence of ferro-\nmagnetism introduces its own speciﬁcs, as the magnon\nspectrum in ferromagnets diﬀers from the antiferromag-\nnetic spectrum. Our results also include the comparison\nof the cases of a disordered and regular vortex lattice.\nThe outline of the paper is as follows. In Sec. II we\ngiveamodeloftheferromagneticsuperconductorandde-\nrive a general equation for the magnetic moment induced\nforcefMacting on vortices in ferromagnetic supercon-\nductors. In Sec. III this force is calculated for a vortex\nlattice and disordered vortex array moving under the ac-\ntionofadctransportcurrent. Here, thediﬀerencesinthe\ndependence of fMvs. vortex velocity for ferromagnetic\nand antiferromagnetic materials are discussed. Section\nIV is devoted to vortex motion under the action of an ac\ndriving force. The magnetic contributions to the vortex\nviscosity and vortex mass are determined. In Sec. V it\nis shown how the force fMcan be estimated experimen-\ntally by measuringthe surfaceimpedance. In Sec. VI the\ngeneralization of our calculations for the SF multilayers\nis discussed. In the conclusion a summary of our results\nis given.\nII. THE INTERACTION FORCE BETWEEN\nVORTICES AND MAGNETIC MOMENTS:\nGENERAL EQUATIONS\nIn the London approximation the free energy of the\nferromagnetic superconductor in the mixed state can be\ntaken in the form\nF=/integraltext/bracketleftBig\n1\n8πλ2/parenleftbig\nA+Φ0\n2π∇θS/parenrightbig2+(rotA−4πM)2\n8π\n+α\n2/parenleftBig\n∂M\n∂xi∂M\n∂xi/parenrightBig\n+KM2\n⊥\n2−BHe\n4π/bracketrightBig\nd3r.(1)\nHereλis the London penetration depth, Ais the vector\npotential, Φ 0is the ﬂux quantum (Φ 0=π/planckover2pi1c/|e|>0),\nθSis the superconducting order parameter phase, Mis\nthe magnetization, and αis a constant characterizingCompound UGe2UCoGe URhGe\nλ, nm 10001200 3450\nL, nm 13,6 45 900\nHan, T∼100∼10∼10\nµU1,4µB0,07µB0,3µB\nωF, Hz∼1013∼1010∼1011\nVth, cm/s ∼107∼105∼107\nK=Han/M∼104∼104∼103−104\nTABLE I. Parameters of some ferromagnetic superconduc-\ntors.L=/radicalbig\nα/Kis the eﬀective domain wall width, Hanis\nthe anisotropy ﬁeld, µUis the magnetic moment per U atom,\nµBis the Bohr magneton, ωFis the ferromagnetic resonance\nfrequency, and Vthis the critical vortex velocity for magnon\nradiation (see Sec. III). The data have been taken from Refs.\n1, 22–24.\nthe exchangeinteraction. The U-based ferromagnetic su-\nperconductors, listed in Table I, have a strong easy-axis\nmagnetocrystalline anisotropy, which is accounted for by\nthe term KM2\n⊥/2, where Kis an anisotropy constant,\nM⊥=M−(e·M)e, andeis a unit vector along the\nanisotropy axis. The term BHe/4πin Eq. (1) accounts\nfor a uniform external ﬁeld He. All terms in the right-\nhand side of Eq. (1) are integrated over the whole space,\nexceptfortheﬁrstterm, containing λ, whichisintegrated\nover the sample volume. Certainly, outside the sample\nM= 0. In the sample the magnetization modulus is\nconstant.\nFirst, we determine the equilibrium state by minimiz-\ningFwith respect to M, and then with respect to A\nandθS. We note that anisotropy ﬁeld Han=KMis\ntypically very large (see Table I): it is comparable to or\ngreater than the upper critical ﬁeld. This means that the\ninequality B≪Hanholds for any internal ﬁeld Bthat\ndoes not suppress superconductivity. Then the trans-\nverse component of the magnetization M⊥can be esti-\nmated as M⊥/lessorsimilarB/K≪M. SinceK≫1, in a zero\napproximation with respect to K−1we can neglect the\ntransverse magnetization (even in the anisotropy energy,\nwhich appears to be proportional to K−1). Then\nF(A,θS)≈/integraltext/bracketleftBig\n1\n8πλ2/parenleftbig\nA+Φ0\n2π∇θS/parenrightbig2\n+B2\n8π−BM0−BHe\n4π+2πM2/bracketrightBig\nd3r,(2)\nwhereM0=Me. Foranarbitraryshapedsamplefurther\nminimization can not be performed analytically. Here\nwe assume the ferromagnetic superconductor to be an\nellipsoid. The results derived below should be also valid\nin the extreme cases of slabs and long cylinders. It is\nreasonable to assume that the average internal magnetic\nﬁeldB0inanellipsoidalsamplewillbe uniform(compare\nwith a dielectric ellipsoid in a uniform external ﬁeld - see\nRef. 25). Denoting the superconductor volume as V, we3\ncan rewrite the free energy as\nF=V/parenleftbig\nfS(B0)−M0B0−B0He\n4π/parenrightbig\n+/integraltext\nr/∈V/bracketleftBig\nB2\n8π−BHe\n4π/bracketrightBig\nd3r+const, (3)\nwhere the constant does not depend on the magnetic in-\nductionB, andfSis the free energy density of the vortex\nlattice:\nfS(B0) =/angbracketleftBigg\n1\n8πλ2/parenleftbigg\nA+Φ0\n2π∇θS/parenrightbigg2\n+B2\n8π/angbracketrightBigg\n.(4)\nAveragingis performed overa volumethat is much larger\nthan the inter-vortex distance. The function fS(B0) can\nbe determined explicitly by solving the London equation\n(11) with a given vortex lattice density, corresponding\nto the average ﬁeld B0. To transform the integral in\nEq. (3) we introduce several quantities: the self-ﬁeld of\nthe sample BS=B−He, the magnetization MSdue\nto supercurrents, the eﬀective full magnetization Meﬀ=\nM0+MS, and the eﬀective H-ﬁeld Heﬀ=BS−4πMeﬀ.\nThen the integral can be transformed as\n/integraltext\nr/∈V/bracketleftBig\nB2\n8π−BHe\n4π/bracketrightBig\nd3r=/integraltext\nr/∈VB2\nS\n8πd3r−/integraltext\nr/∈VH2\ne\n8πd3r\n=/integraltextH2\neff\n8πd3r−/integraltext\nr∈VH2\neff\n8πd3r+const\n=V\n2Meﬀ/parenleftBig\nˆN−ˆN2\n4π/parenrightBig\nMeﬀ+const.\nHereˆNis the demagnetizing tensor, connecting the eﬀec-\ntive magnetization and eﬀective ﬁeld inside the sample:\nHeﬀ=−ˆNMeﬀ. Analytical and numerical values of ˆN\ncan be found in Ref. 26. Finally, if we eliminate Meﬀ\nusing the relation\nMeﬀ= (4π−ˆN)−1(B0−He),\nwe obtain\nF\nV=fS(B0)−M0B0−B0He\n4π\n+1\n8π(B0−He)ˆN(4π−ˆN)−1(B0−He)+const .(5)\nHere, the only variable is the internal ﬁeld B0, which\nshould be determined from the equation\n∂F\n∂B0= 0. (6)\nEquations (5) and (6) completely deﬁne the equilibrium\nstate of the ferromagnetic superconductor.\nNow we proceed from statics to coupled vortex and\nmagnetization dynamics. We focus on two systems, for\nwhich the derivation of the force acting on vortices is\nalmost identical: a bulk ferromagnetic superconductor,\nand an SF multilayer (see Fig. 1), where S is an or-\ndinary type-II superconductor, and F is a ferromagnet\nwith a strong easy-axis anisotropy: K≫1. For the\nmultilayer system the same expression (1) for the free\nenergy is used with M= 0 in the superconductor and\nFIG. 1. A scheme of the SF multilayer system. The dashed\nlines denote vortices.\nλ=∞in the ferromagnet. We neglect the Joseph-\nson coupling between neighboring S layers. This is jus-\ntiﬁed for /greaterorsimilar10 nm thick ferromagnets: in the case of\nan ordinary (non-triplet) proximity eﬀect, superconduct-\ning correlations decay exponentially on a scale of several\nnanometers in the ferromagnet.27\nLet the vortices be aligned along the z-axis (which is\nperpendicular to the S/F interface in the multilayer sys-\ntem). They may form a regular or disordered lattice.\nWhen the vortices are set in motion by a dc or ac trans-\nport current, their time-dependent positions are given by\nthe vectorfunctions Ri(z,t), lyingin the xy-plane, where\ni= 1..Nv, andNvis the number of vortices. In our cal-\nculations we will assume the vortices to be straight, i. e.\nRidoes not depend on z.\nAs vortices move, the magnetic moments start to ﬂuc-\ntuate. We describe the magnetization dynamics using\nthe Landau-Lifshitz-Gilbert equation28\n∂M\n∂t=γ/parenleftbigg\nM×δF\nδM/parenrightbigg\n+ν\nM2/parenleftbigg\nM×∂M\n∂t/parenrightbigg\n,(7)\nwhereγis the gyromagnetic ratio, νis a dissipation con-\nstant, and the free energy Fis given by Eq. (1).\nThe force acting on a single vortex per unit length of\nthe vortex equals\nfi=−1\nLv∂F\n∂Ri. (8)\nwhereLvis the vortex length. Averaging fiover all vor-\ntices, we obtain the average force\nf=−1\nLvNv/summationdisplay\ni∂F\n∂Ri, (9)\nWhen we considered the equilibrium state, the mag-\nnetization component perpendicular to the easy axis e\nhas been neglected. Now we have to abandon this ap-\nproximation, as it would lead to a vanishing force acting4\non the vortices from the side of the magnetic moments.\nWe putM=M0+m, wherem≈M⊥,|m| ≪M, and\nlinearize Eq. (7) with respect to m:\n∂m\n∂t=−γM0×/parenleftbig\nα∇2m−Km+B/parenrightbig\n+ν\nM2M0×∂m\n∂t.\n(10)\nFrom this equation it is evident that magnetization ﬂuc-\ntuations are excited if the vortex ﬁeld is not parallel to\nthe magnetization easy-axis. In a ferromagnetic super-\nconductor this may be achieved by applying an exter-\nnal ﬁeld at an angle to the magnetization easy axis or\nby choosing an appropriate sample geometry (for exam-\nple, anellipsoidalsamplewiththemagnetizationdirected\nalong neither of the principal axes).\nThe magnetic induction inside the superconductor\nshould be determined from the London equation\nδF\nδA= 0,or\n−∇2B+B\nλ2=Φ0\nλ2z0/summationtext\niδ(2)(ρ−Ri)+4π\ncrotrotm,(11)\nwherez0is a unit vector along the z-axis. In the case of\nthe multilayer system (see Fig. 1), Maxwell’s equations\ninside the F-layers read\nrotB= 4πrotM,divB= 0.(12)\nOn the SF-interface appropriate boundary conditions\nmust be imposed:\nBz/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nF=Bz/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nS, H x,y/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nF=Hx,y/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nS(13)\n∂m\n∂z/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nF= 0. (14)\nThe last condition follows directly from Eq. (7), if no\nsurface term is present in the free energy (1).\nWe present the magnetic ﬁeld as the sum of the vortex\nﬁeldhand the magnetization ﬁeld bMdeﬁned by\n−∇2h+h\nλ2=Φ0\nλ2z0/summationdisplay\niδ(2)(ρ−Ri),(15)\n−∇2bM+bM\nλ2= 4πrotrotm (16)\ninside the superconductor, and\nroth= 0,divh= 0, (17)\nrotbM= 4πrotm,divbM= 0 (18)\nin the ferromagnetic layers.\nIn Eqs. (11) - (18) we neglected the magnetic ﬁeld in-\nduced by normal currents. These are given by j=σE,\nwhereσis the normal conductivity, and Eis the elet-\nric ﬁeld. We will ﬁrst estimate the contribution of the\nnormal currents ﬂowing in the F-layers of the multilayersystemtothe magneticﬁeld. UsingbothMaxwell’sequa-\ntions for rot Band rotE, we obtain\nrotrotB=4πσF\ncrotE=−4πσF\nc2∂B\n∂t,\nor\n∂2B\n∂z2+∇2\nρB−4πσF\nc2∂B\n∂t= 0, (19)\nwhereσFis the conductivity in the magnetic layers. As-\nsuming\n∂B\n∂t≈ −(VL∇ρ)B,\nwhereVLis the ﬂux velocity, we can see that the in-\nﬂuence of the normal currents on the magnetic ﬁeld is\nnegligible, if the inequality\n4πσF\nc2lVL≪1\nholds, where lis the characteristic in-plane length scale\nof the problem. Similar arguments can be applied to the\nS-layers. Then, for the multilayer system we ﬁnd the\nfollowing constraint on the vortex velocity:\nVL≪c2\n4πmax(σn,σF)l, (20)\nwhereσnis the normal-state conductivity of the super-\nconductor. In the case of a bulk ferromagnetic supercon-\nductor, we have to demand\nVL≪c2\n4πσnl.(21)\nAs we will see, the main length scales of the problem are\nthe inter-vortex distance aand the length L=/radicalbig\nα/K,\nwhich is of the order of the domain wall width of the\nferromagnet (or ferromagnetic superconductor). Further\non we assume that Eqs. (20) and (21) with l= min(a,L)\nare satisﬁed. Then, we may not take into account the\nnormal currents in Eqs. (11) - (18).\nIn the free energy (1) the interaction of the vortices\nwith magnetizationoriginatesfrom the Zeeman-liketerm\n(a brief explanation is given in Appendix A)\nFZ=−/integraldisplay\nMhd3r.\nThen, the magnetic moment induced force fMacting on\nthe vortices is\nfM=−1\nLvNv/integraldisplay\nmz∇hzd3r. (22)\nHere it has been assumed that the perpendicular to the\nz-axis component of the ﬁeld his negligible. In the case5\nof SF multilayers, this is true for a suﬃciently small pe-\nriod of the structure. To draw a parallel with preced-\ning works,7,8,14–17where the susceptibility formalism has\nbeen used, we note that fMcan be written in the form\nfM=−1\nLvNv/integraldisplay\n(ˆχzzhz)∇hzd3r,\nwhere ˆχzzis the susceptibility operator. If desired, the\nexplicit form of ˆ χzzcan be easily derived from Eq. (29),\ngiven below.\nAll further calculationsin this section andSecs. III, IV\nandVarecarriedoutforaferromagneticsuperconductor.\nIn Section VI we discuss how our results can be extended\nto the case of the multilayer system.\nIn the Fourier representation Eq. (22) reads\nfM=4π2\nNvi/integraldisplay\nqmqzh∗\nqzd2q, (23)\nwhere for any function X(ρ) its Fourier transform is de-\nﬁned as\nXq=1\n(2π)2/integraldisplay\nX(ρ)e−iqρd2ρ.\nBy Fourier transforming Eqs. (10), (15), and (16), as-\nsuming that all quantities do not depend on z, we obtain\n∂mq\n∂t=−γM0×/parenleftbig\n−(K+αq2)mq+bMq+hq/parenrightbig\n+ν\nM2M0×∂mq\n∂t, (24)\nhqz=Φ0\n4π2(1+λ2q2)/summationdisplay\nie−iqRi(t).(25)\nbMq=−4πq×(q×mq)\nq2+λ−2. (26)\nIt can be seen that the absolute value of the term bMq\nin Eq. (24) is much smaller than |Kmq|. Further on we\nwill neglect the magnetization ﬁeld bMq.\nEquation (24) is an inhomogeneous linear diﬀerential\nequationwithconstantcoeﬃcientswithrespectto mq. It\ncan be solved using standard methods. We are interested\nin thez-component of the magnetization, which equals\nmqz=γMi\n2sin2θ/integraltextt\n−∞hqz(t′)/braceleftBig/parenleftbig\n1+iν\nM/parenrightbig−1\n×exp/bracketleftBig\n−/parenleftbig\n1+iν\nM/parenrightbig−1iω(q)(t−t′)/bracketrightBig\n−/parenleftbig\n1−iν\nM/parenrightbig−1exp/bracketleftBig/parenleftbig\n1−iν\nM/parenrightbig−1iω(q)(t−t′)/bracketrightBig/bracerightBig\ndt′,(27)\nwhereθis the angle between eandz0, and\nω(q) =γM(K+αq2) =ωF(1+L2q2) (28)\ngives the magnon dispersion law in an ordinary ferro-\nmagnet, if the dipole-dipole interaction is not taken intoaccount (see Ref. 29). Here, ωF=γMKis the ferro-\nmagnetic resonance frequency. In the small dissipation\nlimit,ν≪M, we have\nmqz=γMi\n2sin2θ/integraltextt\n−∞hqz(t′)/braceleftbig\nexp/bracketleftbig/parenleftbig\n−i−ν\nM/parenrightbig\nω(q)(t−t′)/bracketrightbig\n×/parenleftbig\n1−iν\nM/parenrightbig\n−exp/bracketleftbig/parenleftbig\ni−ν\nM/parenrightbig\nω(q)(t−t′)/bracketrightbig/parenleftbig\n1+iν\nM/parenrightbig/bracerightbig\ndt′.\n(29)\nThen, the force fMtakes the form\nfM=2π2γM\nNvsin2θ/integraltext\nd2q/integraltextt\n−∞hqz(t′)h∗\nqz(t)\n/braceleftbig\nexp/bracketleftbig/parenleftbig\ni−ν\nM/parenrightbig\nω(q)(t−t′)/bracketrightbig/parenleftbig\n1+iν\nM/parenrightbig\n−exp/bracketleftbig/parenleftbig\n−i−ν\nM/parenrightbig\nω(q)(t−t′)/bracketrightbig/parenleftbig\n1−iν\nM/parenrightbig/bracerightbig\nqdt′.(30)\nIII. MAGNON RADIATION BY VORTICES\nMOVING WITH A CONSTANT VELOCITY\nLet us consider the motion of vortices under the action\nof a constant external force (e. g., spacially uniform and\ntime-independent transport current). Then the positions\nof individual vortices are given by\nRi(t) =Ri0+VLt+∆Ri(t). (31)\nHere the vectors Ri0denote the vortexpositions in a reg-\nular lattice, VLis the average ﬂux velocity, and ∆ Ri(t)\nis responsible for ﬂuctuations of vortices due to interac-\ntions with pinning cites ( /angb∇acketleft∆Ri(t)/angb∇acket∇ight= 0). It should be\nstressed here that we do not take into account the inﬂu-\nence of pinning on the ﬂux velocity. The eﬀect that is\nimportant for us is the vortex lattice distortion caused\nby impurities, which strongly inﬂuences the eﬃciency of\nmagnon generation.\nThe product of magnetic ﬁelds under the integral in\nEq. (30) is\nhqz(t′)h∗\nqz(t) =/parenleftbiggΦ0\n4π2(1+λ2q2)/parenrightbigg2\neiqVL(t−t′)K,(32)\nK=/summationtext\ni,jexp[iq(Rj0−Ri0)+iq(∆Rj(t)−∆Ri(t′))].\nBelow we consider the cases of a perfect vortex lattice\nand a disordered vortex array.\nA. A perfect vortex lattice.\nThe approximation used in this section is valid for suf-\nﬁciently weak pinning, when we can put\n/angbracketleftBig\neiq(∆Rj(t)−∆Ri(t′))/angbracketrightBig\n≈1,\nwhere the averaging is over i. To ensure the fulﬁllment\nof this condition it is suﬃcient to demand\n∆Rq≪1, (33)6\nwhere ∆ Ris the characteristic displacement of vortices\nfrom their positions in a perfect lattice. The inequality\n(33)mustholdforall qgivingaconsiderablecontribution\nto the integral in Eq. (30). In the end of Sec. IIIB it\nwill be shown that this leads to the condition\n∆R≪L. (34)\nWhen (34) holds, we have\nK=4π2NvB0\nΦ0/summationdisplay\nGδ(q−G), (35)\nwhereGare the vectors of the lattice, reciprocal to the\nvortex lattice. After integration over qandt′the mag-\nnetic force takes the form\nfM= Φ0B0γMsin2θ/summationtext\nGG\n(1+λ2G2)2\n×iω(G)+ν\nMGVL\nω2(G)−(VLG)2−2iν\nMVLGω(G).\nWhen the terms corresponding to Gand−Gare com-\nbined, this can be written as\nfM=−γνB0Φ0sin2θ/summationtext\nGG(GVL)\n(1+λ2G2)2\n×(GVL)2+ω2(G)\n[ω2(G)−(GVL)2]2+4ν2\nM2(GVL)2ω2(G),(36)\nwhere small terms of the order of ν/Min the numerator\nhave been droped. From this it follows that the force has\nlocal maxima when for some G=G0the condition\nω(G0)≈VLG0 (37)\nis satisﬁed. This relation presents the well-known\nCherenkov resonance condition. When Eq. (37) holds,\nmagnons with the wave vector G0are eﬀectively gener-\nated. When the vortex velocity is close to a resonance\nvalue, in the sum in Eq. (36) we can drop all terms ex-\ncept the two resonant terms corresponding to G0and\n−G0. Then\nfM≈ −γνB0Φ0sin2θG0\n(1+λ2G2\n0)2\n×ω(G0)\n(ω(G0)−VLG0)2+ν2\nM2ω2(G0).(38)\nIt can be seen that the fMvs.VLdependence for a given\nvortex velocity direction exhibits a Lorentzian-like peak\nwith the width\n∆VL=ν\nMω(G0)VL\nG0VL.\nThe maximum value of fMis\n|fM|max=γM2B0Φ0G0sin2θ\n(1+λ2G2\n0)2νω(G0). (39)\nAnother remarkeable feature is that the force is directed\natsomeangletothevelocityofthevortices: fMisparallel\ntoG0, and not VL. The angle between fMandVL\nmay range from 0◦to 90◦. This eﬀect also follows fromEquation (3) in Ref. 15, though the authors did not\nmention it, because it has been assumed that VLandfM\nare always parallel.\nLet us discuss how the Cherenkov resonances inﬂuence\nthecurrent-voltagecharacteristics. Abrikosovvortexmo-\ntion in a superconductor is governed by the equation\nΦ0\ncj×z0=−f (40)\nThe term in the left-hand side represents the Lorentz\nforce with jbeing the macroscopic supercurrent density.\nAll other forces are represented by the term f. We take\ninto account two contributions to f: the viscous drag\nforce−ηVLandfM. Here,ηis the viscosity due to order\nparameter relaxation processes and normal current ﬂow-\ning through the vortex core.30Taking the cross product\nof Eq. (40) and z0, we obtain the expression for the\ncurrent\nj=−cη\nΦ0VL×z0+c\nΦ0fM(VL)×z0.(41)\nThe relation between jandEis the established via\nE=−1\nc(VL×B), (42)\nwhichfollowsfromFaraday’slaw. Accordingto Eq. (41),\nthe vortex-magnetic moment interaction leads to an in-\ncrease ∆ jof the current density at a given electric ﬁeld\nE:\n∆j=c\nΦ0fM/parenleftbiggc\nB0E×z0/parenrightbigg\n×z0.\nAccording to Eq. (38), near the Cherenkov resonance we\nhave\n∆j=γνB0csin2θz0×G0\n(1+λ2G2\n0)2\n×ω(G0)/bracketleftBig\nω(G0)−c\nB0(z0×G0)E/bracketrightBig2\n+ν2\nM2ω2(G0).(43)\nThisrelationindicatesthat theI-Vcurveexhibitsaseries\nof peaks corresponding to the resonance electric ﬁelds\ngiven by\nω(G)−c\nB0(z0×G)E≈0 (44)\nMoreover, close to the resonance the additional current\n∆jis directed along the vector z0×G0and notE. The\nangle between ∆ jandEmay range from 0◦to 90◦.\nThus, locally the resistance is anisotropic. Considering\nmacroscopic ferrromagnetic superconductors and multi-\nlayer systems, care should be taken when applying Eq.\n(43) to the whole sample: it is known that even a small\nconcentration of pinning cites destroysthe long-rangeor-\nder in the vortex lattice.31In fact, vortex lattice domains\nare formed in large superconducting samples – see Ref.\n32 and references therein. The vortex nearest-neighbor\ndirections are typically linked to crystal axes. Hence, in7\nmonocrystalline samples there are only few energetically\nfavorable orientations of the vortex lattices. This fact\nallows us to put forward a qualitative argument. Let us\ndenote as Gthe set of all reciprocal lattice vectors for all\nvortex lattice domains. Since there are only few possible\norientations of the domains, the set Gconsists of isolated\npoints. We claim that when the applied electric ﬁeld sat-\nisﬁes Eq. (44) for some G∈ G, the enhancement of the\ncurrent should be observable. Hence, even if there are\nseveral vortex lattice domains, the peaks on the current-\nvoltage characteristics are present. The measurement of\nthe peak voltages at diﬀerent applied magnetic ﬁelds al-\nlows to probe the magnon spectrum ω(q).\nB. A disordered vortex array\nIn this section we analyze the opposite extreme case of\nchaotically placed vortices. This situation may be real-\nized in weak magnetic ﬁelds, B0/lessorsimilarΦ0/λ2, when vortex-\nvortex interaction is weak and the lattice is easily de-\nstroyed by defects and thermal ﬂuctuations.\nWe assume\n/angbracketleftBig\neiq(∆Rj(t)−∆Ri(t′))/angbracketrightBig\n≈0. (45)\nwhereqis not too small, and the averaging is over i/negationslash=j.\nNote that the behavior of Kat small qalmost does not\ninﬂuence the force fM, sinceKenters the integral in Eq.\n(30) with a factor q. Fori=jwe have\n/angbracketleftBig\neiq(∆Ri(t)−∆Ri(t′))/angbracketrightBig\n= 1\nwhent=t′, and\n/angbracketleftBig\neiq(∆Ri(t)−∆Ri(t′))/angbracketrightBig\n=/vextendsingle/vextendsingle/vextendsingle/angbracketleftBig\neiq∆Ri(t)/angbracketrightBig/vextendsingle/vextendsingle/vextendsingle2\n≪1\nwhen|t−t′| → ∞. To derive some qualitative results,\nwe make the following assumption:\n/angbracketleftBig\neiq(∆Ri(t)−∆Ri(t′))/angbracketrightBig\n=e−|t−t′|/τ(q),\nwhere the the time τ(q) is chosen so that\n/angb∇acketleft|q(∆Ri(t)−∆Ri(t′))|/angb∇acket∇ight ∼1 att−t′=τ(q). Then\nK=Nve−|t−t′|/τ(q), (46)\nand after integration over t′Eq. (30) yields\nfM=γMΦ2\n0sin2θ\n8π2/integraltextqd2q\n(1+λ2q2)2/bracketleftBig\n1+iν\nM\nτ−1(q)+ν\nMω(q)−iω(q)−iqVL\n−1−iν\nM\nτ−1(q)+ν\nMω(q)+iω(q)−iqVL/bracketrightBig\n. (47)\nIt can be seen here that in the case of fast vortex ﬂuc-\ntuations, τ−1(q)≫ω(q), magnon generation is strongly\nsuppressed, as the integral is proportional to τ(q). Wewillanalyzeindetailtheoppositelimitingcase, τ−1(q)≪\nω(q). Then\nfM≈γMΦ2\n0sin2θ\n4π2/integraltextqd2q\n(1+λ2q2)2\n×iω(q)\nω2(q)−(qVL)2−2i(qVL)τ−1\n1(q), (48)\nwhereτ−1\n1(q) =τ−1(q)+νω(q)/M, and in the numerator\nterms proportional to ν/Mhave been droped. The main\ncontribution to the integral comes from qlying in the\nvicinity of two circles in the q-plane, given by ω(q) =\n±qVL(this equation speciﬁes the Cherenkov resonance\ncondition). Near the circle ω(q) =qVLwe make the\nfollowing transformation:\nω2(q)−(qVL)2−2i(qVL)τ−1\n1(q)\n≈2ω(q)(ω(q)−qVL−iτ−1\n1(q)).\nForthecircle ω(q) =−qVLthetransformationsareanal-\nogous. Then\nfM≈γMΦ2\n0sin2θ\n4π2/integraldisplayqd2q\n(1+λ2q2)2ℜi\nω(q)−qVL−iτ−1\n1(q).\n(49)\nThe last fraction in the right-hand side resembles the\nexpression\nℜi\nf(x)−iǫ,\nwhich reduces to δ(f(x)) when ǫ→+0. Hence, the last\nfactor in Eq. (49) also can be replaced by a δ-function,\nwhenτ−1\n1(q) issuﬃcientlysmall. Toderivethelimitation\nonτ−1\n1(q) we direct the qx-axis along VLand rewrite the\ndenominator of the large fraction in Eq. (49) as follows:\nωF(1+L2q2)−qxVL−iτ−1\n1(q)\n=ωF/parenleftBig\n1−V2\nL\nV2\nth/parenrightBig\n−iτ−1\n1(q)+ωFL2/bracketleftbigg/parenleftBig\nqx−VL\n2L2ωF/parenrightBig2\n+q2\ny/bracketrightbigg\n,\nwhereVth= 2ωFL. Now it is evident that the δ-function\ncan be introduced in Eq. (49) when\nτ−1\n1(q)≪ωF/vextendsingle/vextendsingle/vextendsingle/vextendsingleV2\nL\nV2\nth−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle.\nThen\nfM≈ −γMΦ2\n0sin2θ\n4π/integraldisplayqd2q\n(1+λ2q2)2δ(ω(q)−qVL).(50)\nHere, two points should be noted: (i) the expression for\nfMdoes not depend on the dissipation rate and on the\nartiﬁcially introduced time τ(q); (ii) Equation (50) can\nbe derived from Eq. (36) in the limit of an extremely\nsparse vortex lattice, when summation can be replaced\nby integration.\nTechnical details of integration in Eq. (50) are given\nin Appendix B. The ﬁnal result is\nfM=−γMΦ2\n0sin2θ\n8λ4ω2\nF/parenleftBigg\n1+/parenleftbiggVL\nλωF/parenrightbigg2/parenrightBigg−3/2\nΘ(VL−Vth)VL\n(51)8\nforλ≫L. Equation (51) asserts that the quantity Vthis\nthe magnon generation threshold velocity. The maximal\nvalue offMis reached at VL=λωF√\n2≫Vth:\n|fM|max=Φ2\n0γMsin2θ\n8√\n2λ3ωF33/2. (52)\nThe inﬂuence of the magnetic force fMon the current-\nvoltage characteristics in general has been discussed in\nthe previous section. According to Eqs. (41) and (42),\nat the electric ﬁeld E=VthB0/cthe average current\ndensity should exhibit a stepwise increase by\n∆j=c\nΦ0fM(Vth) =γMΦ0csin2θ\n8λ4ω2\nF/parenleftBigg\n1+/parenleftbiggVth\nλωF/parenrightbigg2/parenrightBigg−3/2\nVth.\nThe maximum enhancement of the current density due\ntovortex-magneticmomentinteractionisreachedat E=√\n2λωFB0/cand equals\n∆jmax=Φ0cγMsin2θ\n8√\n2λ3ωF33/2.\nIn Ref. 14 it has been predicted that in antiferromag-\nnetic superconductors in the sparse lattice limit the cur-\nrent enhancement ∆ jis proportional to√VL−Vc(at\nVL> Vc), where Vcis some critical velocity. This result\nis in contrast with ours: we found that ∆ j∼Θ(VL−Vth)\nnear the magnon generation threshold. This diﬀerence\nis due to diﬀerent magnon spectra in ferromagnets and\nantiferromagnets – see Fig. 2. In an antiferromagnet\nω(q) =/radicalbig\nω2\n0+s2q2, whereω0is a gap frequency, and sis\nthe short-wavelength magnon velocity. As the vortex ve-\nlocity is increased, the resonance condition ω(q) =VLq\nisﬁrstsatisﬁedatinﬁnitely large q. However,at q≫ξ−1,\nwhereξis the coherence length, the Fourier components\nhqzareexponentially small. Magnongenerationbecomes\neﬃcient at q∼ξ, which is reached at a critical velocity\nthat roughly equals Vc=/radicalbig\nω2\n0ξ2+s2. In short, the gen-\neration threshold in antiferromagnetic superconductors\ncorresponds to an intersection of the curves ω=ω(q)\nandω=VLqatq∼ξ−1(see Fig. 2a), yielding a ∆ j∼√VL−Vcdependence. On the contrary, in ferromag-\nnetic superconductors at VL=Vththe curves ω=ω(q)\nandω=VLqtouch each other at q=L−1< ξ−1(see\nFig. 2b). This fact leads to a stepwise increase of the\ncurrent at the threshold vortex velocity.\nFinally, we need to make a remark concerning the con-\ndition (33), providing that the ideal lattice approxima-\ntion can be used. It follows from Fig. 2b that near\nthe generation threshold magnons with wave numbers\nq≈L−1are generated. This means that for VL/greaterorsimilarVth\nthe main contribution to the integral in Eq. (30) comes\nfromq∼L−1. Thus, the condition (34) should be im-\nposed to ensure the applicability of the perfect lattice\napproximation./s113\n/s45/s49/s48\n/s45/s49/s32/s40/s50\n/s48/s43 /s115/s50\n/s113/s50\n/s41/s49/s47/s50\n/s32/s115/s113\n/s32/s86\n/s76/s113\n/s76/s45/s49/s113/s98\n/s70/s32\n/s70/s40/s49/s43 /s76/s50\n/s113/s50\n/s41\n/s32/s86\n/s76/s113/s97\nFIG. 2. The magnon spectra in an (a) antiferromagnet and\n(b) ferromagnet. The dash-dotted line is given by ω=VLq,\nwhereVLis the vortex velocity at which magnon generation\nbecomes eﬃcient ( VL=Vcfor antiferromagnetic supercon-\nductors and VL=Vthfor ferromagnetic superconducors).\nC. Estimates of the threshold vortex velocity in\nferromagnetic superconductors and SF multilayers.\nLet us check whether it is possible to observe the\nfeatures connected with the Cherenkov resonances on\nthe current-voltage characteristics of ferromagnetic su-\nperconductors and SF multilayers. To satisfy the condi-\ntion (37) suﬃciently large vortex velocities VL> Vthare\nrequired. Estimates of the theshold velocity for known\nferromagnetic superconductors are given in Table I. One\ncanseethatthe valuesof Vthareverylarge. Thequestion\narises if such velocities are compatible with superconduc-\ntivity in the U-based superconductors. To investigate\nthis question we will estimate the supercurrent density\njthwhich is suﬃcient to accelerate the vortices up to the\nvelocityVth. Equation (41) yields\njth≈cη\nΦ0Vth (53)\nFor the viscosity ηwe use the Bardeen and Stephen\nexpression33(which is a good estimate for relatively slow9\nprocesses)34\nη= Φ0Hc2σn/c2, (54)\nwhereHc2= Φ0/(2πξ2) is the upper critical ﬁeld. For\nthe normal state conductivity we use Drude’s estimate\nσn∼e2nℓ\nmVF.\nHerenis the concentration of charge carriers, mis their\nmass,ℓis the mean free path, and VFis the Fermi veloc-\nity. Then\njth∼e2nℓHc2Vth\nmcVF. (55)\nThis value should be compared with the depairing cur-\nrent density which is given within the BCS theory by\njcr∼en∆\nmVF,\nwhere ∆ is the superconducting gap. We demand jth≪\njcr. Using the relation ∆ ∼/planckover2pi1VF/ξ(valid for clean super-\nconductors) we can rewrite the inequality above as\nVth≪ξ\nℓVF. (56)\nIn the U-based compounds the coexistence of supercon-\nductivity and ferromagnetism appears in clean samples\nwithℓ/greaterorsimilarξ. The Fermi velocities are of the order of\n108cm/sin UGe 2and 105cm/sin UCoGe and URhGe –\nsee Refs. 35–37. Thus, the inequality (56) is satisﬁed in\nneither of these compounds, and our model breaks down\nat vortex velocities below Vth. This is a consequence\nof the high magnetic anisotropy and large quasiparticle\nmass in the U-compounds.\nThe situation seems to be more optimistic in SF su-\nperlattices. Certainly, we should consider if Eq. (54) is\nvalid for multilayers. A study of the vortex viscosity in\nsuperconductor/normal metal multilayers is presented in\nRefs. 38 and 39. It has been shown the Bardeen-Stephen\nviscosity (54) may be signiﬁcantly modiﬁed for vortices\ninclined with respect to the z-axis, or for strongly con-\nducting normal metal layers. Still, in our case Eq. (54)\nis a good order-of magnitude estimate for dS∼dFand\nσF/lessorsimilarσn, where dSanddFare the thicknesses of the\nsuperconducting and ferromagnetic layers (see Fig. 1),\nrespectively.\nRecently, a number of experimental papers18–20\nhave reported successful fabrication of high-quality\nYBa2Cu3O7/La2/3Ca1/3MnO3superlattices. In Ref. 40\nthe value Han= 1200Oefor La 0.7Ca0.3MnO3is given,\nthough it is noted that the anisotropy is signiﬁcantly in-\nﬂuenced by strain. The measured domain wall width in\nthe same compound, denoted as δin Ref. 41, is 12 nm.\nAssuming γ∼µB//planckover2pi1, whereµBis the Bohr magneton,\nwe obtain the following estimate for the vortex threshold\nvelocity:\nVth= 2γHanL∼104cm/s. (57)The Fermi velocity in YBa 2Cu3O7is of the order of or\ngreater than 107cm/s.42Thus, the condition (56) can\nsurely be satisﬁed in the cuprate/manganite superlat-\ntices.\nIV. MAGNON RADIATION BY A\nHARMONICALLY OSCILLATING VORTEX\nLATTICE\nAs it has been shown in Sec. IIIC, magnon genera-\ntion in U-based ferromagnetic superconductors by a vor-\ntex array moving with constant velocity seems problem-\natic due to the extremely high required vortex veloci-\nties. In this section we study a more feasible approach\nto magnon generation in magnetic superconductors, an-\nalyzing the case of a harmonic external current acting on\nthe vortices. Experimentally, the oscillating current in\nthe superconductor can be created using the microwave\ntechnique (for example, see Ref. 21). Then, the surface\nimpedance yields information about the high-frequency\nproperties of the sample – see Sec. V.\nSubjected to the action of a harmonic force, in the\nlinear regime the vortices oscillate harmonically:\nRi(t) =R′\ni0+Re−iωt+R∗e−iωt.(58)\nHereR′\ni0are the equilibrium positions of the vortices,\nwhich are deﬁned by vortex-vortex interaction as well as\npinning. The vectors R′\ni0do not necessarily form a reg-\nular lattice, unlike Ri0.Ris the amplitude of vortex\noscillations. We will consider frequencies of the order of\nthe ferromagnetic resonance frequency in ferromagnetic\nsuperconductors, ωF∼100GHz. This frequency is sev-\neral orders of magnitude larger than the typical depin-\nning frequency.43This fact allowsto neglect the inﬂuence\nof the pinning force on vortex motion and to assume that\nthe oscillation amplitudes of all vortices are equal to R.\nThe product of the magnetic ﬁelds in Eq. (30) equals\nhqz(t′)h∗\nqz(t) =/parenleftBig\nΦ0\n4π2(1+λ2q2)/parenrightBig2\nK′eiq(Ri(t)−Ri(t′))\n≈/parenleftBig\nΦ0\n4π2(1+λ2q2)/parenrightBig2\nK′[1+iq(Ri(t)−Ri(t′))],(59)\nwhere\nK′=/summationdisplay\ni,je−iqR′\ni0+iqR′\nj0=Nv/angbracketleftBigg/summationdisplay\nje−iqR′\ni0+iqR′\nj0/angbracketrightBigg\n.\n(60)\nHere, the averaging is over i. The linear with respect to\nRcontribution to the force takes the form\nfM=γMΦ2\n0\n8π2Nvsin2θ/integraltext\nd2q/integraltextt\n−∞iK′qR\n(1+λ2q2)2(e−iωt−e−iωt′)\n×/braceleftBig\ne[iω(q)−ν\nMω(q)](t−t′)−e[−iω(q)−ν\nMω(q)](t−t′)/bracerightBig\nqdt′+c.c.\n≈γMΦ2\n0\n4π2Nvsin2θe−iωt/integraltextd2qK′(q)qR\n(1+λ2q2)2\n×/bracketleftBig\nω(q)\nω2(q)−ω2−2iν\nMωω(q)−ω−1(q)/bracketrightBig\nq+c.c.(61)10\n/s48/s44/s48/s48 /s48/s44/s48/s49 /s48/s44/s48/s50 /s48/s44/s48/s51/s45/s49/s44/s48/s45/s48/s44/s56/s45/s48/s44/s54/s45/s48/s44/s52/s45/s48/s44/s50/s48/s44/s48\n/s32/s32/s40\n/s77/s41\n/s66\n/s48/s76/s50\n/s47\n/s48/s32 /s48/s46/s53\n/s70\n/s32 /s48/s46/s54\n/s70\n/s32 /s48/s46/s55\n/s70\n/s32 /s48/s46/s56\n/s70\n/s32 /s48/s46/s57\n/s70\nFIG. 3. The ℑ(ηM) vs. magnetic ﬁeld dependence at fre-\nquencies below the ferromagnetic resonance frequency for\nan ideal triangular vortex lattice (see Eq. (66)). η0=\nγMΦ2\n0sin2θ/(2λ4ω2\nF).\nHerec.c.denotes the complex conjugate. Like before, we\nneglected small terms of the order of ν/M.\nTo proceed further, the explicit form of K′(q) is re-\nquired. Again, we will consider the cases of a pefect\nvortex lattice and a disordered array.\nA. A perfect vortex lattice.\nLet us assume that pinning is suﬃciently weak, so that\nq∆R≪1, (62)\nwhere ∆ R∼ |R′\ni0−Ri0|is the characteristic deviation\nof the vortices from their positions in a perfect lattice.\nThe inequality (62) should hold for all qgiving a con-\nsiderable contribution to the integral in Eq. (61). The\ncharacteristic value of qwill be estimated below.\nForq∆R≪1 we have\nK′=4π2NvB0\nΦ0/summationdisplay\nGδ(q−G). (63)\nSubstituting Eq. (63) into Eq. (61), assuming that the\nvortex lattice is either square or regular triangular, we\nobtain\nfM=iωηMRe−iωt+c.c., (64)\nηM=−iγMΦ0B0\n2ωsin2θ/summationtext\nGG2\n(1+λ2G2)2\n×/bracketleftBig\nω(G)\nω2(G)−ω2−2iν\nMωω(G)−ω−1(G)/bracketrightBig\n.(65)\nHere, wehaveintroducedthe complexquantity ηM, play-\ning the role of a generalized vortex viscosity. Indeed,\nwhenηMis purely real, the magnetic force is simply\nfM=−ηMdRi/dt. In our system there is a phase shiftbetween the vortex velocity and fM, and the more gen-\neral expression (64) is valid. Further on we will call ηM\nthe magnetic viscosity.\nThe ideal vortex lattice is likely to form when vortex-\nvortex interaction is suﬃciently strong, or the inter-\nvortex distance is suﬃently small. Let this distance be\nmuch smaller than the London penetration depth, which\nmeansB0≫Φ0/λ2. ThenλG≫1 for all G/negationslash= 0, and\nηM≈ −iγMΦ0B0ω\n2λ4sin2θ/summationtext\nG/negationslash=0G−2ω−1(G)\n×/bracketleftbig\nω2(G)−ω2−2iν\nMωω(G)/bracketrightbig−1.(66)\nNow we consider the behavior of ηMin diﬀerent fre-\nquency ranges. First, let the frequency be below the\nferromagnetic resonance frequency ( ω < ω M). Then\nmagnon generation is ineﬃcient. However, if we put\nν= 0, the force fMwill not vanish below the generation\nthreshold, unlike in the case of constant vortex velocity.\nInstead, the magnetic viscosity will be purely imaginary,\nsignifying that there are no magnetic losses. In Fig. 3\nwe plot the imaginary part of ηvs. magnetic ﬁeld B0de-\npendencies for diﬀerent frequencies (below ωF) and for a\nﬁxed angle θ.\nReturning to the condition (62), one can see that for\nω < ω Fthe characterisitic values of Gin Eq. (66) are\nof the order of L−1. Hence, ∆ R≪Lis required for Eq.\n(66) to be valid.\nAt frequencies above the ferromagnetic resonance fre-\nquency magnetic dissipation can not be neglected, and\nthe real part of ηMbecomes signiﬁcant. In Fig. 4 we\nplot theηMvs.B0dependencies for diﬀerent frequencies\nand for a ﬁxed angle θand dissipation rate ν/M= 0.02.\nThegraphsexhibit asequenceofLorentzian-like( ℜ(ηM))\nandN-shaped( ℑ(ηM))features,locatedatsomeresonant\nﬁeld values, BR, which are determined from the relation\nω(G) =ω. (67)\nFor small ﬁelds these features may overlap, but the reso-\nnance correspondingto the highest ﬁeld remains well dis-\ntinguishable. For a triangular vortex lattice the largest\nresonance ﬁeld equals\nBR=√\n3\n8π2Φ0\nL2/parenleftbiggω\nωF−1/parenrightbigg\n,\nand for a square lattice\nBR=1\n4π2Φ0\nL2/parenleftbiggω\nωF−1/parenrightbigg\n.\nSolving Eq. (67) with respect to G, we obtain\nG=L−1/radicalbiggω−ωF\nωF. (68)\nHence, the peaks on the ηMvs.B0dependences must\nbe observable if the characteristic deviation ∆ Rof the\nvortices from their positions in an ideal lattice satisﬁes\n∆R≪G−1=L/radicalbiggωF\nω−ωF. (69)11\n/s45/s49/s44/s48/s45/s48/s44/s53/s48/s44/s48/s48/s44/s53/s49/s44/s48/s49/s44/s53/s45/s49/s48/s49/s50/s51\n/s48/s44/s48/s48 /s48/s44/s48/s49 /s48/s44/s48/s50 /s48/s44/s48/s51/s45/s48/s44/s53/s48/s44/s48/s48/s44/s53/s32/s61/s49/s46/s53\n/s70/s32 /s40\n/s77/s41\n/s32 /s40\n/s77/s41\n/s32/s32/s32\n/s32/s32/s47\n/s48\n/s32/s32 /s40\n/s77/s41\n/s32 /s40\n/s77/s41\n/s32\n/s32/s32/s61/s49/s46/s49\n/s70\n/s32/s61/s50/s46/s48\n/s70/s32 /s32/s32\n/s66\n/s48/s76/s50\n/s47\n/s48/s32 /s40\n/s77/s41\n/s32 /s40\n/s77/s41\nFIG. 4. The ηMvs. magnetic ﬁeld dependencies for fre-\nquencies above the ferromagnetic resonance frequency (see\nEq. (66)). η0=γMΦ2\n0sin2θ/(2λ4ω2\nF). The vortices form an\nideal triangular lattice.\nNote that for frequencies close to ωFthis condition is\nweaker than ∆ R≪L.\nWe conclude this section by giving a numeric estimate\nof the magnetic viscosity. When the resonance condition\n(67) is satisﬁed, we obtain from Eq. (66)\nηM∼γMΦ0B0\nλ4G2ω2M\nν.\nSinceB0G−2∼Φ0, and the lowest allowable value of ω\nisωF=γMK, we have\nηM/lessorsimilarΦ2\n0\nKλ4ωFM\nν. (70)\nThen, according to Eq. (54), the ratio of ηMtoηis\nηM/η/lessorsimilarM\nνξ2c2\nKλ4ωFσn. (71)\nWe will make the numeric estimate for UCoGe, the ferro-\nmagnetic superconductor with the lowest ferromagnetic\nresonancefrequency. In Ref. 3 we ﬁnd the value 12 µΩcm\nfor the normal resistivity, and the maximal value 200 ˚A/s48/s44/s48 /s48/s44/s53 /s49/s44/s48 /s49/s44/s53 /s50/s44/s48/s45/s49/s48/s45/s53/s48/s53/s49/s48/s48\n/s70/s32 /s40\n/s77/s41\n/s32 /s40\n/s77/s41\nFIG. 5. The frequency dependence of the magnetic vis-\ncosity,ηM, for a disordered vortex array – see Eq. (76).\nThe value ln( λ/L) = 4.3 of UGe 2has been used. η0=\nγMΦ2\n0sin2θ/(2λ4ω2\nF).\nfor the coherence length. Using Table I, we obtain\nηM/η∼M\nν3×10−5. (72)\nData on the ratio M/νare not available yet. The small\nfactor 10−5in Eq. (72) appears due to the large magne-\ntocrystalline anisotropy of UCoGe: it can be seen from\nEq. (71) that ηM/ηis proportional to K−2, sinceωF=\nγMK. Hence, to increase the ratio of ηMtoη, com-\npounds (or multilayer systems) with a lower anisotropy\nare preferable.\nB. A disordered vortex array.\nNow let us assume that due to relatively strong pin-\nning the vortices are placed chaotically, i. e., there is no\ncorrelation between R′\ni0andR′\nj0fori/negationslash=j. Then\nK′=Nv/parenleftBig/integraltextB0\nΦ0e−iqR′\ni0+iqR′\nj0d2R′\nj0+1/parenrightBig\n=Nv/parenleftBig\n4π2B0\nΦ0δ(q)+1/parenrightBig\n. (73)\nIn a real vortex lattice there is a short-rangeorder, which\nleads to the smearing of the delta-function on a scale of\nthe order of the inverse inter-vortex distance. However,\nthe behavior of K′at small qis not important, since K′\nenters the integral in Eq. (61) with a factor q2. For\nclarity, we stress here that the product hqz(t′)h∗\nqz(t) does\nnot decay with increasing t−t′, unlike in the case of a\nconstant driving force – see Eqs. (32) and (46). This is\nexplained by the fact that the vortices oscillate close to\ntheir equilibrium positions and do not travel from one\npinning cite to another. Thus, the positions Ri(t) and\nRi(t′) of a single vortex are always well correlated, cor-\nresponding to an inﬁnite correlation time τ(q).12\nWithK′givenby Eq. (73) the magneticviscositytakes\nthe form\nηM=−iγMΦ2\n0\n4πωsin2θ/integraltext∞\n0q3dq\n(1+q2λ2)2\n×/bracketleftBig\nω(q)\nω2(q)−ω2−iǫ−ω−1(q)/bracketrightBig\n. (74)\nHere, like in Sec. IIIB, we assume that the imaginary\nterm−iǫ(ǫ >0) in the denominator is an inﬁnitesimal.\nTo simplify the expression in the right-hand side of Eq.\n(74), we note that the contribution to the integral from\nsmallq(q/lessorsimilarλ−1) can be neglected in the λ≫Llimit.\nThen we can put 1+ λ2q2≈λ2q2, and cut the integral\noﬀ atq=λ−1:\nηM=−iγMΦ2\n0\n4πωλ4/integraltext∞\nλ−1dq\nq\n×/bracketleftBig\nω(q)\n(ω+ω(q))(ω(q)−ω−iǫ)−ω−1(q)/bracketrightBig\n.(75)\nFurther integration should not present diﬃculties. For\nλ≫Lwe obtain\nηM=γMΦ2\n0sin2θ\n8πωωFλ4/braceleftBig\nπωF\n2(ω−ωF)Θ(ω−ωF)−i/bracketleftBig\n2ω2\nω2\nF−ω2lnλ\nL\n+ωF\n2(ω+ωF)lnω+ωF\nωF+ωF\n2(ωF−ω)ln/vextendsingle/vextendsingle/vextendsingleωF−ω\nωF/vextendsingle/vextendsingle/vextendsingle/bracketrightBig/bracerightBig\n.(76)\nLike in the previous section, below the ferromagnetic res-\nonance frequency the magnetic viscosity is purely imagi-\nnary. However, unlike in the case of a perfect fortex lat-\ntice, now the viscosity does not depend on the magnetic\nﬁeld. It should be also noted that in the limit B0→0\nEq. (65) after summation transforms into (76), i. e., the\ncases of isolated vortices and chaotically placed vortices\nare equivalent, like in Sec. III. The ηMvs.ωdependence\nis depicted in Fig. 5.\nC. Vortex mass.\nAs we have seen, at ω < ω Fthe magnetic viscosity is\nimaginary. Moreover, at ω≪ωFthe viscosity is propor-\ntional to ω. This signiﬁes that the vortex can be ascribed\na mass per unit length, Mv, so that the equation of mo-\ntion becomes\nMvd2Ri\ndt2=fext, (77)\nwherefextincludes all forces, except for the force fM.\nThe mass is deﬁned by\nMv=iηM\nω/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nω=0. (78)\nBefore we give explicit expressions for Mv, we should\ncomment on the connection between the vortex mass en-\nhancement and the self-induced polaronic pinning mech-\nanism, studied in Refs. 15 and 16. In the mentioned\npapers it has been assumed that the magnetization dy-\nnamics is purely dissipative, i. e., ν/M≫1, which is\nin contrast to our case. In fact, ν/M≫1 is a necessarycondition for the formationof polaronlikevortices. Thus,\nthe polaronic pinning mechanism contributes rather to\nthe real part of ηMthan to its imaginary part, an it is\nnot related to the vortex mass enhancement discussed\nhere.\nUsing Eq. (66), we ﬁnd that the magnetic contribution\nto the vortex mass for a perfect lattice is\nMv=γMΦ0B0\n2λ4sin2θ/summationdisplay\nG/negationslash=0G−2ω−3(G) (79)\nwhenB0≫Φ0/λ2. For a disordered array we obtain\nfrom Eq. (74)\nMv=γMΦ2\n0\n4πω3\nFsin2θ/integraltext∞\n0q3dq\n(1+q2λ2)2(1+L2q2)3\n≈γMΦ2\n0sin2θ\n16πω3\nFλ4/parenleftbig\n4lnλ\nL−5/parenrightbig\n(λ≫L).(80)\nLet us estimate the characteristic magnetic contribution\nMvto the vortexmass andcompareit with the electronic\ncontribution (see, for example, Ref. 44), which is present\nin any superconductor:\nMe=2\nπ3m2VF\n/planckover2pi1. (81)\nWe give estimates for the ferromagnetic superconductor\nURhGe. The valuesof ωF=γMKandλcanbe found in\nTable I. The electron mass and Fermi velocity for one of\nthe Fermi surface pockets of URhGe have been measured\nin Ref. 37. The values given there are m= 22meand\nVF= 4.4×105cm/s, wheremeis the free electron mass.\nThen\nMv∼γMΦ2\n0\n16πω3\nFλ4≈10−24g/cm, M e∼10−20g/cm.\nIt can be seen that the magnetic contribution to the vor-\ntex mass is negligible for URhGe. Estimates for UGe 2\nand UCoGe yield the same result. This happens due to\nthe very large ferromagnetic resonance frequency ωFin\nthese compounds: note that the right-hand side of Eq.\n(80) contains ω−3\nF. The situation is the same as for the\nmagnetic viscosity – see Eqs. (71) and (72). Thus, the\nmagneticmass Mvshould be detectablein materialswith\na smaller ferromagnetic resonance frequency.\nV. DISCUSSION OF THE MAGNETIC\nVISCOSITY MEASUREMENT\nAsimple experimentalmethod to studyvortexdynam-\nics in type-II superconductors consists in the measure-\nment of the surface impedance. A possible geometry for\nsuch experiment is depicted in Fig. 6. We consider the\nsimplestsituation, whenthevorticesareperpendicularto\nthesamplesurface,andtheprobingelectromagneticwave\nwith the amplitude heis normally incident on this sur-\nface. Then, foranon-magneticsuperconductor( M0= 0)13\nheHe\nB0\nM0Lx\nLy\nLzx\nyzFerromagnetic\nsuperconductor\nFIG. 6. The geometry for the measurement of the surface\nimpedance of a ferromagnetic superconductor. The dashed\nlines denote vortices.\ntheory30,45predicts that in a wide range of parameters\nthe surface impedance Z(ω) equals\nZ(ω) =/parenleftbigg−iωµρf\n4π/parenrightbigg1/2\n, (82)\nwhereµis the static diﬀerential magnetic permeability,\nµ=dB0z\ndHez,\nandρfis the ﬂux-ﬂow resistance,\nρf=BΦ0\nc2η.\nThus, the experimental value of the surface impedance\nprovides information about the viscosity coeﬃcient η.\nWe will prove that for a ferromagnetic superconductor\na range of parameters exists, where Eq. (82) can be ap-\nplied, if the magnetic viscosity is taken into account: η\nshould be replaced by η+ηM.\nFirst, we outline the applicability conditions of Eq.\n(82) for an ordinary superconductor. Within the con-\ntinuous medium approximation used in Ref. 45 an alter-\nnating external ﬁeld heexcites a long-wavelength and a\nshort-wavelength mode in the superconductor. For con-\nvenience, we will call these modes type-1 and type-2, and\ndenote the z-projections of their wave vectors as k1and\nk2, respectively. These quantities are explicitly deﬁned\nbyEquation(24)inRef. 45. Tousethesimpleexpression\n(82) fortheimpedance, threeconditionsmustbefulﬁlled:\n(i)|k1|λ≪1, (ii)|k1| ≪ |k2|, and (iii) |k2|Lz≫1 (Lz\nis the sample thickness – see Fig. 6). According to Ref.\n45, the conditions (i) and (ii) are satisﬁed, if\nω≪ωC=Φ0C∗\n44\nB0λ2η, (83)whereC∗\n44is an elastic modulus of the vortex lattice.\nThis inequality presents a limitation on the frequency.\nWe would like to note that in the limit Hc1≪B0≪Hc2,\nwhereHc1is the lower critical ﬁeld, the condition (83)\ncan be weakened, namely\nω≪ωB=Φ0B0\n4πλ2η(ωB≫ωC).(84)\nThis follows directly from Equation (22) in Ref. 45.\nLetusturn tothecaseofaferromagneticsuperconduc-\ntor. We assume the sample is a slab with dimensions Lx,\nLyandLz, whereLz≪Lx,Ly– see Fig. 6. The x-axis,\nparallel to the large surface of the sample, is the magne-\ntization easy-axis. In fact, the slab geometry is not a key\npoint for us, but the equilibrium magnetization must be\nparallel to one of the sample surfaces. By applying an\nexternal ﬁeld we can provide that the internal ﬁeld B0is\nparallel to the z-axis. In the slab geometry the compo-\nnents of the demagnetizing tensor are Nxx≈0,Nyy≈0,\nNzz≈4π. Then, according to Eq. (5), if the external\nﬁeld isHe= (−4πM,0,Hez), the internal ﬁeld equals\nB0= (0,0,Hez).\nNow we discuss the surface impedance of a ferromag-\nnetic superconductor. Compared to the case of a conven-\ntional superconductor, an additional complication arises\ndue to the presence of new degrees of freedom. These\nare connected with magnetization dynamics and lead to\nthe appearance of new magnon-like modes. Such modes\ncan be directly excited by an electromagnetic wave even\nin the absence of vortices,5,6and they may signiﬁcantly\ninﬂuence the surface impedance. However, in our geome-\ntry the excitation of these modes can be avoided, as will\nbe demonstrated below.\nIf the frequency is not too close to the ferromag-\nnetic resonance frequency ( |ω−ωF|/ωF≫K−1) we\ncan neglect the magnetostatic interaction in the Landau-\nLifshitz equation when analyzing the additional magnon-\nlike modes, as we have done in Sec. II (where the term\nbMqhas been dropped). Then, in the limit of small dis-\nsipation, Eq. (10) takes the form\n∂m\n∂t=−γM0×/parenleftbigg\nα∂2m\n∂z2−Km/parenrightbigg\n.(85)\nThis yields two modes, which we label as type-3 and 4:\nm= (z0∓iy0)m3,4eik3,4z,\nk3=L−1/radicalBig\nω\nωF−1, k 4=iL−1/radicalBig\nω\nωF+1,(86)\nwherem3andm4are scalar amplitudes. Now suppose\nthat the magnetic ﬁeld hein the probing electromagnetic\nwave oscillates along the x-axis, i. e., along the equilib-\nrium magnetization (see Fig. 6). We assume that inside\nthe sample the alternating magnetic induction /angb∇acketleftb/angb∇acket∇ight, aver-\naged over the xy-plane, is also parallel to the x-axis. It\nwill be shown that this statement is self-consistent. In-\ndeed, for /angb∇acketleftb/angb∇acket∇ightparallel to M0we see from Eqs. (10) and\n(14) that ∂/angb∇acketleftm/angb∇acket∇ight/∂t= 0. This means that the magnon-\nlike type-3 and 4 modes are not excited. In the type-114\nand 2 modes /angb∇acketleftm/angb∇acket∇ight= 0, but /angb∇acketleftb/angb∇acket∇ight /negationslash= 0. Hence, these modes\ndiﬀer from their analoguesin non-magneticsuperconduc-\ntors only by the presence of the magnetic contribution to\ntheviscosity, ηM, whichisduetotheFourier-components\nmqwithq/negationslash= 0. Then, according to Ref. 45, the internal\nﬁeld/angb∇acketleftb/angb∇acket∇ightwill be parallel to the probing ﬁeld he(which\nfollows from the London equation (11), if the deforma-\ntion of the vortex lattice is taken into account). Thus,\nwe have proved the validity of our assumption, having\nshown in addition that only the type-1 and 2 modes are\nexcited.\nStrictly speaking, the eﬀective viscosity for the long-\nwavelength type-1 mode diﬀers from η+ηM, because the\nvortices are not straight. However, since |k1| ≪λ−1, the\nradius of curvature of the vortices is suﬃciently large to\nmake this diﬀerence negligible.\nAn electromagnetic wave polarized in the y-direction\n(he=heyy0) requires separate treatement, which is out-\nsidethescopeofthispaper. Here,themagnon-likemodes\nof type 3 and 4 must be taken into account. For a study\nof the surface impedance in the case he⊥M0(in a diﬀer-\nent geometry) see Ref. 13.\nVI. MAGNON EXCITATION IN SF\nMULTILAYERS.\nIn this section it is shown how our results can be ex-\ntended to the case of SF multilayers with S and F be-\ning an ordinary type-II superconductor and ordinary fer-\nromagnet, respectively. We consider structures with a\nsuﬃciently small period d(see below) and with vortices\noriented perpendicular to the layer surfaces – see Fig. 1.\nThen, the generalization of the results from Secs. II - IV\nis straightforward, if two points are taken into account:\n(i) Since the magnetic moments now occupy only a frac-\ntion of the sample, the force fMis reduced by a factor of\nd/d′\nF, whered′\nF≤dFis the eﬀective thickness of the fer-\nromagnetic layer. Formally, all expressions for fM, start-\ning with Eq. (23), should be multiplied by d′\nF/d. The\nquantities d′\nFanddFcoincide, if the mutual inﬂuence\nof the superconducting and magnetic orders is negligi-\nble. However, this is not the case for cuprate/manganite\nsuperlattices. Experimental papers report giant super-\nconductivity induced modulation of the magnetization19\nand the suppression of magnetic order in the mangan-\nite layer close to the SF interface.20In the latter case,\nd′\nF< dF, but both quantities are of the same order of\nmagnitude.\n(ii) Due to the fact that the structure is only partially\nsuperconducting, the in-plane London penetration depth\nnow equals λeﬀ=λ(d/dS)1/2– see Ref. 46, for example.\nThe expression for the single vortex ﬁeld\nhqz≈Φ0\n4π2(1+q2λ2\neﬀ)(87)\ncan be used if the period dof the structure is muchsmaller than the characteristic in-plane length scale of\nthe problem. To apply our results for the case of a\nconstant driving force, we have to demand d≪L, ac-\ncording to Sec. III. The constraint is somewhat weaker\nin the case of the harmonic driving current. Indeed,\nforω > ω Fthe main contribution to fMcomes from\nq≈L−1/radicalbig\nω/ωF−1, hence, the limitation on the period\nof the structure is\nd≪L/radicalbiggωF\nω−ωF.\nThus, for ( ω−ωF)/ωF≪1 the thickness dmay be of\nthe order of or larger than the domain wall width.\nFinally, we will discuss brieﬂy a recent paper by\nTorokhtii et al.,21where the ﬂux-ﬂow resistivity in\nNb/PdNi/Nb trilayers has been measured. It has been\nreported that in the presence of the magnetic PdNi\nlayer the ﬂux-ﬂow resistivity in Nb exceeds the Bardeen-\nStephen estimate,33as if the vortex viscosity is reduced\nbythe interactionwith magneticmoments. At ﬁrst sight,\nthis seems to contradict our prediction. However, this\nexperiment can not be interpreted in the framework of\nthe model used here, since the ferromagnetic alloy PdNi\ndoes not posess a well-deﬁned magnetic anisotropy, and\nthe magnon modes can not be characterized by a wave\nvectorqduetothe lackoftranslationalsymmetry. More-\nover, the dependence of the critical temperature ofNb on\nthe PdNi layer thickness signiﬁes strong inﬂuence of the\nmagnetic order on superconductivity. We suppose that\nthe explanation of the viscosity reduction in the men-\ntioned experiment requires a more complicated micro-\nscopic treatment.\nVII. CONCLUSION\nWe have calculated the magnetic moment in-\nduced force fMacting on moving Abrikosov vor-\ntices in ferromagnetic superconductors and supercon-\nductor/ferromagnet multilayers. When the vortices are\ndriven by a dc transport current, magnons are eﬃciently\ngenerated when the vortex velocity exceeds the value\nVth= 2ωFL. As a result, narrow peaks appear on the\ncurrent-voltage characteristics of the superconductor, if\nthe vortices form a regular lattice. Within a vortex lat-\ntice domain the current may be not parallel to the elec-\ntric ﬁeld. For a disordered vortex array a step-like fea-\ntureshouldappearonthe current-voltagecharacteristics.\nThis behavior is in contrast with antiferromagnetic su-\nperconductors, where the increase of the current at the\nmagnongenerationthresholdisproportionalto√U−Uc,\nwhereUis the voltage, and Ucis some threshold value.14\nAccording to our estimates, the transport current re-\nquired to reach the vortex velocity Vthin the U-based\nferromagnetic superconductors is of the order the depair-\ning current due to the large magnetic anisotropy of these\ncompounds. On the other hand, in cuprate/manganite15\nmultilayers18–20the requiredcurrent is well below the de-\npairingcurrent,sothementionedfeaturesmaybeobserv-\nable on the current-voltage characteristics of such sys-\ntems.\nIf the vortices are driven by an ac current, the interac-\ntion with magnetic moments results in the appearance of\nacomplexmagneticcontribution ηMtothevortexviscos-\nity. We determined this quantity for the cases of an ideal\nvortex lattice and a disordered vortex array. For low fre-\nquencies, ω≪ωF, the magnetic contribution to the vor-\ntex mass has been estimated. From the ηMvs. magnetic\nﬁeld and frequency dependencies the magnon spectrum\nin the ferromagnetic superconductor can be extracted.\nExperimentally, ηMcan be determined by measuring the\nsurface impedance of the sample in the geometry, where\nthe equlibrium magnetization isparallelto the oscillating\nexternal magnetic ﬁeld.\nACKNOWLEDGEMENTS\nWe are grateful to L. Bulaevskii for useful discussions\nand valuable comments. This work was supported in\npart by the Russian Foundation for Basic Research, Eu-\nropean IRSES program SIMTECH (contract n.246937),\nthe French ANR program ”electroVortex” and LabEx\n”Amadeus” program.\nAppendix A\nIn this appendix we will prove that the magnetic mo-\nment induced force acting on vortices can be written as\n(22). Wehavetocalculatethe variationofthefreeenergy\nwhen all vortices are shifted by an equal vector, and the\nmagnetization is kept ﬁxed. To simplify the calculations\nwe use the fact that the free energy acquires the same\nvariation if the vortices are kept ﬁxed, and the magneti-\nzation is shifted in the opposite direction. Then\nδF=/integraldisplay/parenleftbiggδF\nδAδA+δF\nδMδM/parenrightbigg\nd3r.\nAccording to the London equation δF/δA= 0 the ﬁrst\nterm in the right-hand side vanishes. Also, the terms in\nEq. (1) which depend only on M(e. g., the exchange en-\nergy) are not aﬀected by the magnetization shift. Hence,\nonly the term\nδF=−/integraldisplay\nBδMd3r, (A1)\nremains, and the force acting on a vortex is\n(fM)xi=1\nNvLv/integraldisplay\nB∂M\n∂xid3r=−1\nNvLv/integraldisplay∂B\n∂xiMd3r\nPresenting the magnetic ﬁeld as B=h+bM, we have\nfM=fM1+fM2. (A2)(fM1)xi=−1\nNvLv/integraltext∂bM\n∂xiMd3r,\n(fM2)xi=−1\nNvLv/integraltext∂h\n∂xiMd3r.\nNote that the term fM1does not depend on the vortex\npositions. Hence, to calculate this term we can place the\nvortices anywhere in the superconductor. Let us posi-\ntion the vortices in an area with uniform magnetization\n(M= const). Then, fM2vanishes, and fM=fM1. On\nthe other hand, in the area with homogenous magneti-\nzationbM= 0 inside the superconductor (in the fer-\nromagnetic superconductor this happens due to London\nscreening, and in the SF multilayer system the ﬁeld bM\nis simply conﬁned to the ferromagnetic layers). Hence,\nthe magnetization has no inﬂuence on the magnetic ﬁeld\nand supercurrent in the vortex region, and the force fM\nvanishes. Then, fM1= 0, and for any vortex positions\nfM=fM2. From this follows Eq. (22).\nAppendix B\nIn this appendix we show how the integral in Eq. (50)\ncan be evaluated. We introduce the dimensionless quan-\ntitiesl=L/λ,lv=VL/(ωFλ) andg=λq, and direct\nthegx-axis along VL. ThenfMy= 0, and\nfMx=−γMΦ2\n0sin2θ\n4πλ3ωF/integraltextgxd2g\n(1+g2)2δ(1+l2g2−lvgx)\n=−γMΦ2\n0sin2θ\n4πλ2VL/integraltext(1+l2g2)δ(1+l2g2−lvgx)\n(1+g2)2d2g\n=−γMΦ2\n0sin2θ\n4πλ2VL/bracketleftBig\nl2/integraltextδ(1+l2g2−lvgx)\n1+g2d2g\n+(1−l2)/integraltextδ(1+l2g2−lvgx)\n(1+g2)2d2g/bracketrightBig\n. (B1)\nNow we make a coordinate shift, redesignating gx−\nlv/(2l2) bygx:\nfMx=−γMΦ2\n0sin2θ\n4πλ2VL/braceleftbigg/integraltextδ(g2−g2\n0)d2g\n1+g2y+(gx+lv\n2l2)2\n+(l−2−1)/integraltextδ(g2−g2\n0)d2g/bracketleftBig\n1+g2y+(gx+lv\n2l2)2/bracketrightBig2/bracerightBigg\n,(B2)\nwhere\ng2\n0=l−2/parenleftbiggl2\nv\n4l2−1/parenrightbigg\n.\nFurther we assume that VL> Vth, so that g2\n0>0 (at\nVL< VthfM= 0). Integration over the modulus of gis\nnow straightforward. Then\nfMx=−γMΦ2\n0sin2θ\n8πλ2VL/bracketleftbigg/integraltext2π\n0dϕ\n1+g2\n0+l2v\n4l4+lv\nl2g0cosϕ\n+/integraltext2π\n0(l−2−1)dϕ/parenleftbigg\n1+g2\n0+l2v\n4l4+lv\nl2g0cosϕ/parenrightbigg2\n, (B3)\nwhereϕis the polar angle in the g-plane. Integration\ncan be completed using standard methods or a table of16\nintegrals. The result is\nfMx=−γMΦ2\n0sin2θ\n8λ2VL(l−2+1)l2\nv\nl4/bracketleftbigg\n(1−l−2)2+l2\nv\nl4/bracketrightbigg−3/2\n.\n(B4)If we return to dimensional variables and recall that L≪\nλ, we obtain Eq. (51).\n1S. S. Saxena et al., Nature 406, 587 (2000).\n2Dai Aoki et al., Nature 413, 613 (2001).\n3N. T. Huy et al., Phys. Rev. Lett. 99, 067006 (2007).\n4J. Flouquet and A. Buzdin, Physics World 15, 41 (2002).\n5V. Braude and E. B. Sonin, Phys. Rev. Lett. 93, 117001\n(2004).\n6V. Braude, Phys. Rev. B 74, 054515 (2006).\n7L. N. Bulaevskii, M. Hru˘ ska, and M. P. Maley, Phys Rev.\nLett.95, 207002 (2005).\n8Shi-Zeng Lin and Lev N. Bulaevskii, Phys. Rev. B 85,\n134508 (2012).\n9A. I. Buzdin, JETP Lett. 40, 956 (1985) [Pis’ma v ZhETF\n40, 193-196 (1984)].\n10V.BraudeandE.B. Sonin, Europhys.Lett. 72, 124(2005).\n11N. A. Logoboy and E. B. Sonin, Phys. Rev. B, 75, 153206\n(2007).\n12T. K.NgandC. M. Varma, Phys.Rev.B 58, 11624 (1998).\n13A. A. Bespalov and A. I. Buzdin, Phys. Rev. B 87, 094509\n(2013).\n14A. Shekhter, L. N. Bulaevskii, and C. D. Batista, Phys\nRev. Lett. 106, 037001 (2011).\n15Lev N. Bulaevskii and Shi-Zeng Lin, Phys. Rev. Lett. 109,\n027001 (2012).\n16Lev N. Bulaevskii and Shi-Zeng Lin, Phys. Rev. B 86,\n224513 (2012).\n17S.-Z. Lin, L. N. Bulaevskii, and C. D. Batista, Phys. Rev.\nB86, 180506(R) (2012).\n18H.-U.Habermeier, G.Cristiani, R.K.Kremer, O.Lebedev,\nand G. Van Tendeloo, Physica C 364-365 , 298 (2001);\nZ. Sefrioui et al., Phys. Rev. B 67, 214511 (2003); Todd\nHoldenet al., Phys. Rev. B 69, 064505 (2004); V. Pe˜ na et\nal., Phys. Rev. Lett. 94, 057002 (2005); V. K. Malik et al.,\nPhys. Rev. B 85, 054514 (2012).\n19J. Hoppler et al., Nature Materials 8, 315 (2009).\n20J. Stahn et al., Phys. Rev. B 71, 140509(R) (2005); A.\nHoﬀmann et al., Phys. Rev. B 72, 140407(R) (2005); D. K.\nSatapathy et al., Phys. Rev. Lett. 108, 197201 (2012); M.\nA. Uribe-Laverde et al., Phys. Rev. B 87, 115105 (2013).\n21K. Torokhtii, N. Pompeo, C. Meneghini, C. Attanasio, C.\nCirillo, E. A. Ilyina, S. Sarti, and E. Silva, J. Supercond.\nNov. Magn. 26, 571 (2013).\n22Vu Hung Dao, Sebastien Burdin, and Alexandre Buzdin,\nPhys. Rev. B 84, 134503 (2011).\n23A. B. Shick, Phys. Rev. B 65, 180509 (2002).\n24N. T. Huy, D. E. de Nijs, Y. K. Huang, and A. de Visser,Phys. Rev. Lett. 100, 077002 (2008).\n25L.D. Landau, E.M. Lifshitz, L.P. Pitaevskii (1984). Elec-\ntrodynamics of Continuous Media . Vol. 8 (2nd ed.).\nButterworth-Heinemann.\n26J. A. Osborn, Phys. Rev. 67, 351 (1945).\n27A. I. Buzdin, Rev. Mod. Phys. 77, 935 (2005).\n28T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004).\n29L. P. Pitaevskii, E.M. Lifshitz (1980). Statistical Physic s,\nPart 2. Vol. 9 (1st ed.). Butterworth-Heinemann.\n30L. P. Gorkov and N. B. Kopnin, Usp. Fiz. Nauk 116, 413\n(1975) [Sov. Phys. Usp. 18, 496 (1975)].\n31A. I. Larkin, Zh. Eksp. Teor. Fiz. 58, 1466 (1970) [Sov.\nPhys. JETP 31, 784 (1970)].\n32M. Iavarone et al., Phys. Rev. B 78, 174518 (2008); M.\nLaveret. al., Phys. Rev. B 79, 014518 (2009); M. R. Es-\nkildsenet al., Phys. Rev. B 79, 100501(R) (2009); H. A.\nHansonet al., Phys.Rev.B 84, 014506 (2011); H.Kawano-\nFurukawa et. al, Phys, Rev B 84, 024507.\n33I. Bardeen and M. I. Stephen, Phys. Rev. 140, A1197\n(1965).\n34M. Yu. Kupriyanov and K. K. Likharev, Sov. Phys. JETP\n41, 755 (1976) [Zh. Eksp. Teor. Fiz. 68, 1506 (1975)].\n35E. D. Bauer, R. P. Dickey, V. S. Zapf and M. B. Maple, J.\nPhys.: Condens. Matter 13, L759-L770 (2001).\n36Dai Aoki, Ilya Sheikin, Tatsuma D. Matsuda, Valentin\nTaufour, Georg Knebel, and Jacques Flouquet, J. Phys.\nSoc. Jpn. 80, 013705 (2011).\n37E. A. Yelland, J. M. Barraclough,W. Wang, K. V.\nKamenev and A. D. Huxley, Nature Physics 7, 890 (2011).\n38A. S. Mel’nikov, Phys. Rev. B 53, 449 (1996).\n39A. S. Mel’nikov, Phys. Rev. Lett. 77, 2786 (1996).\n40N. D. Mathur, M.-H. Jo, J. E. Evetts, and M. G. Blamire,\nJ. Appl. Phys. 89, 3388 (2001).\n41S.J.Lloyd, N.D.Mathur, J.C.Loudon, andP.A.Midgley,\nPhys. Rev. B 64, 172407 (2001).\n42N. Hass, D. Ilzycer, G. Deutscher, G. Desgardin, I. Monot,\nand M. Weger, J. Sup. Nov. Magn 5, 191 (1992).\n43Y. Shapira and L. J. Neuringer, Phys. Rev. 154, 375\n(1967).\n44G. Blatter, M. Y. Feigel’man, Y. B. Geshkenbein, A. I.\nLarkin, and V. M. Vinokur, Rev. Mod. Phys. 66, 1125\n(1994).\n45E. B. Sonin, K. B. Traito, Phys. Rev. B 50, 13547 (1994).\n46R. A. Klemm, Layered Superconductors (Oxford University\nPress, Oxford, England, 2012), Vol. 1."    },    {        "title": "2403.03518v2.Tracing_Dirac_points_of_topological_surface_states_by_ferromagnetic_resonance.pdf",        "content": "Tracing Dirac points of topological surface states by ferromagnetic resonance\nLaura Pietanesi1,a, Magdalena Marganska2, Thomas Mayer2, Michael Barth2, Lin Chen1, Ji Zou3,\nAdrian Weindl2, Alexander Liebig2, Rebeca D´ ıaz-Pardo1, Dhavala Suri1,5, Florian Schmid2, Franz\nJ. Gießibl2, Klaus Richter2, Yaroslav Tserkovnayk3, Matthias Kronseder2,a, Christian H. Back1,4,b\n1School of Natural Sciences, Department of Physics,\nTechnical University of Munich,\n85748 Garching, Germany\n2Institute for Experimental and Applied Physics,\nUniversity of Regensburg, 93040 Regensburg, Germany\n3Department of Physics and Astronomy,\nUniversity of California,\nLos Angeles, California 90095, USA\n4Center for Quantum Engineering (ZQE),\nTechnical University Munich, 85748 Garching, Germany\n5Center for Nanoscience and Engineering,\nIndian Institute of Science, Bengaluru 560 012, India\naThese authors have contributed equally\nbchristian.back@tum.de\nFerromagnetic resonance is used to reveal features of the buried electronic band structure at\ninterfaces between ferromagnetic metals and topological insulators. By monitoring the evolution\nof magnetic damping, the application of this method to a hybrid structure consisting of a ferro-\nmagnetic layer and a 3D topological insulator reveals a clear fingerprint of the Dirac point and\nexhibits additional features of the interfacial band structure not otherwise observable. The under-\nlying spin-pumping mechanism is discussed in the framework of dissipation of angular momentum\nby topological surface states (TSSs). Tuning of the Fermi level within the TSS was verified both by\nvarying the stoichiometry of the topological insulator layer and by electrostatic backgating and the\ndamping values obtained in both cases show a remarkable agreement. The high energy resolution of\nthis method additionally allows us to resolve the energetic shift of the local Dirac points generated\nby local variations of the electrostatic potential. Calculations based on the chiral tunneling process\nnaturally occurring in TSS agree well with the experimental results.\nINTRODUCTION\nThree dimensional topological insulators (TIs) possess\nan insulating bulk and metallic topological surface states\n(TSS) that exhibit spin-momentum locking [1, 2]: the\ncarrier spins are orthogonal to their momentum. Around\nthis feature revolves the concept that TSSs could pro-\nvide a very efficient way to convert spin currents into\ncharge currents and vice versa [3–5]. However, since\n3D TIs contain heavy elements, the bulk states also ex-\nhibit strong spin-orbit coupling and can therefore con-\ntribute to spin-charge interconversion (SCI) phenomena\nthrough the conventional spin Hall effect [6–9]. Angle re-\nsolved photoemission electron spectroscopy, ideally with\nspin analysis, as well as scanning tunneling spectroscopy\n(STS) are the most popular techniques for the investi-\ngation of the electronic band structure with high energy\nresolution, however, they cannot be applied to study SCI\nor the buried interface band structure. Methods to study\nSCI typically require ferromagnet (FM)/TI hybrid struc-\ntures where the magnetization of the FM can be manip-\nulated via spin-orbit torques generated by the TI or by\nobserving the loss of angular momentum in ferromagnetic\nresonance (FMR) type experiments.\nTo realize high-quality hybrid structures with clean in-terfaces and easily tunable band structure, the method\nof choice is molecular beam epitaxy (MBE), enabling\nTI/FM growth without disrupting the vacuum. While\nMBE growth offers the advantage of well-prepared in-\nterfaces, it is associated with significant problems aris-\ning from inadvertent doping by crystal defects leading to\nbulk conducting samples. Therefore, disentangling the\ncontribution of TSSs and bulk conducting states involved\nin SCI remains an important task. This problem has pre-\nviously been addressed using concentration doping and\nthe electric field effect to study SCI as well as the spin\nSeebeck effect [7, 9–12], however, the conversion mecha-\nnisms, arising from the bulk or TSS, remain unsettled.\nIn this work, we combine MBE growth of hybrid TI/FM\nfilms with high quality interfaces on substrates suitable\nfor electric backgating with FMR-based spin-pumping\nexperiments which we use as a tool to probe the effective\ninterface band structure of the TI in contact to a FM\nlayer. We also propose a theoretical model to describe\nspin-pumping from FM primarily into TSSs. Electro-\nstatic potential fluctuations arising from crystal defects\nlead to an imperfect Dirac point alignment throughout\nthe sample surface and give rise to additional features\nin the spectrum which can be accounted for within our\ntheoretical framework.arXiv:2403.03518v2  [cond-mat.mes-hall]  7 Mar 20242\nHextGNDVBG\nGND\n0 5 10 15 20 25010203040\n60 100 140 Py\n BS/BSTS/Py (0 V)\n BS/BSTS/Py (9 V) \nf (GHz)Ampl. (arb.u.)\n0 Hext \n(mT)0DH (mT) μμ(a) (b)\nFigure 1. (a) Schematic of the spin-pumping FMR setup comprising a coplanar-waveguide (CPW) to externally drive a\nferromagnetic system while an external magnetic field H extis applied to the system. The temperature of the CPW and sample,\nwhich lies face down on the CPW, can be varied between 8-300K. Application of an electric field between the backside and the Py\nlayer in the sample stack SrTiO 3(111)/(1)BS/(10)BSTS/Py/AlO xis used to gate the system leading to a change of the position\nof the Fermi level of BSTS. (b) Linewidth dependence on frequency for Py (black curve) at 9 K and (1)BS/(10)BSTS/Py with\nx= 0.67 at 10 K and 0 V (red curve) as well as 9 V backgate (blue curve) voltages. The inset shows a typical FMR absorption\ncurve and the corresponding fit (derivative of a Lorentzian line).\nEXPERIMENTAL\nThe investigated TI heterostructures consist of a\nsingle quintuple layer (QL) Bi 2Se3(BS) and 10 QL\n(Bi1−xSbx)2(Te1−ySey)3(BSTS) bilayers grown by MBE\non SrTiO 3(111) (STO) substrates. With the n-p het-\nerostructure approach established in [13, 14], we are able\nto strongly suppress the unwanted carrier contribution\ndue to bulk doping and to determine the position of the\nFermi level in the upper TSS [15]. The single, intrinsi-\ncally n-type BS QL is used as an epitaxial and electro-\nstatic seed layer significantly improving the crystal qual-\nity [13]. BSTS can be stoichiometrically tuned to p-type:\nthe n-p heterostructure inherently leads to a band bend-\ning along the growth direction that allows the Fermi level\nto be placed well inside the band gap at the top surface\n[13, 15]. For details concerning the electrical transport\ncharacterization of the complete sample series we refer to\nsection I of the Supplemental Material [16]. From trans-\nport characterization experiments we conclude that in\nthe (1 QL)Bi 2Se3/ (10 QL)(Bi 1−xSbx)2(Te1−ySey)3se-\nries, samples with yfixed to ≈0.9 show maximized bulk\nband gap [13, 17–19], and x≈0.7 places the Fermi en-\nergy in the vicinity of the Dirac point at the top sur-\nface, as will be explained further below. Since spin-\npumping is most sensitive to the interface with the fer-\nromagnet, these compositions are ideal for the ferromag-\nnetic resonance/spin-pumping experiments described in\nthe following.\nIn order to enable FMR measurements, 10 nm thick\npermalloy (Ni 80Fe20, Py) layers were grown on top of the\nBSTS layers in a separate MBE chamber connected via\nan ultra high vacuum tunnel. 7 nm thick AlO xcapping\nlayers were grown on top of all samples (with or without\nthe Py layer) to protect the structures from oxidation.Full film samples were placed on top of a coplanar waveg-\nuide, as shown in Fig. 1(a), and FMR measurements were\nperformed in a temperature range from room tempera-\nture to 8 K at fixed frequencies sweeping the external\nmagnetic field. Resonance spectra, as shown in the inset\nof Fig. 1(b) were recorded for frequencies between 5 - 25\nGHz, and fitted using the derivative of a Lorentzian pro-\nfile to extract the resonance field and linewidth. From\nthe frequency dependence of the linewidth, the Gilbert\ndamping parameter α, which is a measure for the loss of\nangular momentum, can be extracted as the slope of a\nlinear fit [20, 21].\nRESULTS\nIn TI/FM bilayer structures, spins are pumped from\nFM into the TI when the magnetization is driven by mi-\ncrowave magnetic fields. Thus, by measuring αand sub-\ntracting the intrinsic damping of FM, it is possible to\ndetermine the TI’s efficiency for absorbing angular mo-\nmentum near the Fermi energy. The precessing magneti-\nzation of the FM layer thereby acts as source of angular\nmomentum and simultaneously as a detector for the loss\nof angular momentum. By variation of the Sb-content\nin the BSTS layer or by variation of the backgate (BG)\nvoltage, characteristics of the local band structure and\ndensity of states (DOS) in the TI layer can be recon-\nstructed as will be discussed further below.\nFigure 1(b) shows linewidth vs. frequency obtained\nfrom FMR measurements for a Py sample (black curve)\nand a BS/BSTS/Py sample with x= 0.67 at 0 V (red\ncurve) and 9 V (blue curve) backgate voltages performed\nat 10 K. From the linear dependence of the linewidth\non the frequency, see Eq. (S1) in the Supplemental Ma-3\nterial, we get the following values for α: 0.006(9) for\npure Py and 0.022(3) for BS/BSTS/Py at 0 V. ∆ α(T) =\nαTI/FM(T)−αFM(T), indicates the TI’s efficiency for ab-\nsorbing angular momentum as a function of temperature;\nnote that the temperature dependence of αfor a pure Py\nsample grown on STO is almost negligible, see Supple-\nmental Material Fig. S1(c).\nTo investigate ∆ αas a function of the position of the\nFermi level, backgated FMR/spin-pumping experiments\nwere performed. Efficient backgating is enabled by build-\ning a capacitor with a conducting layer at the back side\nof the STO substrate and using Py as a front electrode.\nThe choice of STO as substrate allows efficient gating,\nthanks to its structural phase transition at 105 K [22]:\nbelow this temperature the substrate’s dielectric constant\nincreases dramatically to around 7000 at ∼10K (see Sup-\nplemental Material Fig. S4). A schematic band diagram\nas shown in Fig. 2(a) middle panel can be drawn for the\nn-p heterostructure: when the BSTS is p-type and thick\nenough, the band bending generated by the n-p junction\nenables a precise control of the Fermi level position in\nthe bulk band gap at the top surface and drastically re-\nduces the metallic contribution of the 1QL BS [13, 15].\nApplying an electric field via backgating allows shifting\nthe Fermi level primarily at the interface to the STO\nsubstrate. Since the band diagram is anchored at the\nmetallic Py interface, the band diagram in growth direc-\ntion can be viewed as a lever arm when applying elec-\ntric fields. Consequently, a small non-zero shift of the\nband diagram in the top layers will result when backgat-\ning the heterostructure, see Fig. 2(a). This enables ex-\ntremely precise control of the position of the Dirac point\nof the top-TSS with respect to the Fermi level with a\nshift ofdEDP\ndVBG=−3meV\nVas verified in STM/STS exper-\niments (see Supplemental Material Fig. S5). Note that\nsince the energy of the precessing magnetization is in the\nrange of a few tens of µeV for the frequency range used\nin our experiments, FMR/spin-pumping has the neces-\nsary high energy resolution to resolve features within the\nband gap. To determine the optimum stoichiometry for\nthe spin-pumping investigations, the temperature depen-\ndence of ∆ αhas been measured for the sample series\nwith yfixed to ≈0.90 and variable Sb concentration x,\nsee Supplemental Material Fig. S1. We would like to em-\nphasize here that spin-pumping is mostly sensitive to the\nFM/TI interface. In contrast, transport experiments re-\nflect the whole conducting layer consisting of bottom-,\ntop-TSS and bulk channels [23, 24]. Thus, for choosing\nthe most suitable FM/TI-hybrid either temperature de-\npendent spin-pumping experiments or other methods like\nthe photogalvanic effect [15] can be used. For the former\napproach, α(T) reflects the temperature dependence of\nthe top-TSS and parts of the bulk systems, see Fig. S2,\nand can be summarized as follows: when the Fermi en-\nergy in the topmost TI-layers is within the bulk band gap,\nsee Fig. 2(a) middle panel and Fig. 2(c), the TSS gov-erns the efficiency of spin-pumping at low temperatures\nand consequently ∆ αstarts to increase below ≈70 K. At\nelevated temperatures, thermally populated bulk states\ncounteract this effect [25, 26], leading to reduced dissipa-\ntion of angular momentum into the TSS. For the latter\nmethod, it was shown in [15] for the same TI-structure\nthat the Fermi energy is indeed close to the Dirac point in\nthe top-TSS for x≈0.7. Hence, both studies confirmed\nthe choice of 0 .67< x < 0.73 as the ideal concentration\nrange even when replacing the AlO xcapping layer with\nPy. Please note, the Fermi level/Dirac point positions de-\ntermined by ARPES [13] and STM (see section IV of the\nSupplemental Material) on uncapped films most likely\ndiffer from the positions in the Py/AlO xcapped samples\nmeasured in the FMR experiments. Removal of the Se\ncapping layers leaves sample surfaces that are likely to\nbe n-type due to residues and Se vacancies, as they are\ngenerally n-type dopants for Bi-based TIs.\nSubsequently, in a first overview experiment, ∆ αwas\nmeasured as a function of backgate voltage in the range of\n±60 V, see Fig. 2(b), for a sample with x= 0.67 at 10 K.\nWe observe that αcan be significantly manipulated by\nbackgating; as shown exemplarily in Fig. 1(b), the value\nof ∆α(V) increases from 0.015(3) at 0 V to 0.024(3) for\na backgate voltage of +9 V. The results summarized in\nFig. 2(b) can be divided into three regions: two satura-\ntion regimes for large backgate voltages (shaded in green)\nand a region around zero backgate voltage (shaded in\nred). In the red shaded region a minimum of ∆ αis ob-\nserved at -5 V. Away from the Dirac point, but still in\nthe red region, ∆ αincreases, followed by a decrease un-\ntil it reaches saturation in the green shaded region. At\nlarge positive or negative gate voltages, the Fermi en-\nergy in the layers close to the STO substrate touches the\nbulk conduction or valence bands, respectively, see Fig.\n2(a). This means that the bottom layers become highly\nconductive and screen any further changes to the electric\nbackgate; consequently saturation of ∆ αis reached. It is\nimportant to note that due to this effect and the uncer-\ntainty in the electrostatic lever arm from bottom to top\nlayer in the np-heterostructure, the linear gate voltage\nscale does not translate into a linear energy scale for the\nposition of the Fermi level in the top-TSS.\nWe were able to show a reciprocity between the two\ndifferent experiments performed: varying the Sb concen-\ntration and measuring ∆ αon different samples at low T\nagrees well with the backgated measurements of ∆ αon\nsingle samples, as shown in Fig. 3. Note that in these\nexperiments the relative position of the Fermi energy in\nthe top-TSS was varied in two ways, by changing the\nstoichiometry via Sb-content xand by application of a\nbackgate voltage VBG. Both dependencies, α(EF(x)) and\nα(EF(VBG)), show very similar behavior and are juxta-\nposed in Fig. 3. This is a first hint that the DOS of the\nTSS governs spin-charge conversion and therefore the loss\nof angular momentum measured with ∆ α. The agree-4\nσxx,TSS \nσxx,b\nξSP\nE µ(0V)E\nkx < 0.4\nx~0.70DOSbulkDOSTSS0 << V BGV<< 0BGE\nSrTiO3BS\nBSTSPy\nV ~ 0BGECBEVB\nEF\n-60 -40 -20 0 20 40 600.0120.0160.0200.028\nVBG (V)Δα\nE\nz\nE(a) (b)\n(c) (d)\nFigure 2. (a) Schematics of the band diagrams along the growth direction of the heterostructure of BS/BSTS with a Py\nferromagnet on top grown on the substrate SrTiO 3. At VBG∼0 the intentional internal band bending sets the Fermi energy\nin the top-TSS close to the Dirac point. Application of BG voltages distorts the conduction and valence bands changing\nalso the energetic alignment of the top-Dirac cone. (b) Gilbert damping versus applied backgate voltage for a sample with\n(1)BS/(10)BSTS/Py with x= 0.67 measured at 10 K. The damping for STO/Py is subtracted: ∆ α=αTI/FM−αFM. (c)\nSimplified sketch of the band structure considering bulk states and the Dirac cone at the top surface; the Fermi level position\nfor different Sb concentrations is shown on the left and a sketch of the density of states for TSS and bulk at T=0 is shown\non the right. (d) Schematic representation of different contributions of the bulk, σxx,b(green line), and of the TSSs, σxx,TSS\n(yellow line), to the longitudinal conductivity σxxresulting in a spin-pumping efficiency ξSP(red dashed-dotted line).\nment between the two measurements is a further confir-\nmation that the electrostatic backgate is an efficient way\nof varying the Fermi energy. Note that while a variation\nof SCI within the band gap disagrees with the prediction\nof being approximately constant as suggested in [11], it\nis fully in line with the results shown in [10] and the\nfollowing theoretical considerations.\nTHEORETICAL BACKGROUND\nIn order to understand how the spin-pumping signal is\nrelated to the band structure and density of states of the\nTI, the theory of spin-pumping from FM into a TI has\nto be examined. Let us assume the injection layer Py to\nbe magnetized along the y-direction, with the Cartesian\ndirections schematically indicated in Fig. 4(c). Then, an-\ngular momentum transfer is related to the transformation\nof a spin current directed along the z-direction into an in-\nplane charge current in the TI. The TI/FM heterostruc-\nture can be described by a minimal Hamiltonian [27]\nH=vF(p−eA)·z×ˆσ+J mzˆσzincluding the kinetic en-\nergy of Dirac electrons and a term describing the couplingJof the Dirac electron spin ˆ σto the spin dynamics m(t)\nof the ferromagnet. The in-plane components of the cou-\npling between the FM and the TSS are absorbed into the\nvector potential A:A=J\nevFm×z=J\nevF(my,−mx,0).\nIf the static in-plane magnetization points along the y-\ndirection, the external drive with frequency ωwill lead\nto an elliptical precessional motion with amplitude δm\nin the x-direction and a z-component of the precession\nwhich is much reduced due to the demagnetizing field. At\nferromagnetic resonance the vector potential gives rise to\nan in-plane electric field E(t) =−∂tA=−Jδm ω\nevFsinωty.\nThis in-plane electric field along the y-direction eventu-\nally drives an electric current and dissipates power ac-\ncording to\nP≡j·E=σE2\ny=σ\u0012Jδm ω\nevF\u00132\n, (1)\nwhere Pis the power density and jthe current density.\nIf we assume an isotropic in-plane longitudinal conduc-\ntivity σ≡σxx=σyy, this equation is valid for different\norientations of the FM. On the other hand, the rate of\nenergy dissipation per unit volume, associated with the5\n0.8 0.7 0.6 0.5 0.4 0.30.0060.0080.010\nx-10 0 10 20 30 40 50 60 70\n0.0150.0200.025VBG (V)(a)\n(b)\nΔα ΔαVBG (V)\n-10 0 10 20 30 40 50 60\n0.0150.0200.025\n0.7 0.6 0.5 0.4 0.30.0060.0080.010\nx\nFigure 3. To enable a direct comparison, a part of the\ndata presented already in Fig. 2(b) is replotted in (a) for the\nsample with x∼0.67 measured at 10 K. At VBG∼-5V the\nFermi energy is at the Dirac point. (b) ∆ αdependence on\nSb concentration measured at 10 K at VBG= 0. The Fermi\nenergy shifts towards the conduction band with increasing\nVBGor decreasing xas shown in the sketch of the Dirac cone\nbetween the panels. The measurements shown in (a) have\nbeen recorded on the same sample as shown in (b) at x=0.67\nwith 12 months separation.\npredominantly in-plane spin dynamics, can be evaluated\nto\nP=αs\n2˙m2\nx=αs δm2ω2\n2, (2)\nwhere αis the Gilbert damping parameter and sis the\nlocal spin density. Combining (1) and (2) leads to\nα=2J2\nse2v2\nFσ . (3)\nHence, for magnetic coupling to 2D-TSS, damping is pro-\nportional to the longitudinal conductivity of the Dirac\nelectrons: α∝σxx,TSS[27–29], assuming predominantly\nin-plane spin fluctuations. Note that this dependence is\nrooted in the form of the kinetic term in the Hamiltonian\nleading to the electron velocity operator ˆv=vFz×ˆσ,\nwhich is essentially given by the spin.\nAlthough 180◦-backscattering is strictly forbidden for\nTSS, all other angles are generally allowed for momen-\ntum scattering processes. In real systems any source\nof disorder leads to scattering and the type of disorder\nsets, in general, a finite scale for the allowed momentumtransfer ( |∆k|< k0, with k0being a cutoff for the max-\nimum momentum transfer). Away from the Dirac point\n(kF> k0), the allowed scattering angles are limited by\nthis cutoff which increases the conductivity. Close to the\nDirac point, scattering covers the whole Fermi surface\n(kF< k0), albeit anisotropically, due to the Dirac sup-\npression of the backscattering. This leads to a reduced\nconductivity at low doping. In sec. VI of the Supplemen-\ntal Material, the influence of non-isotropic scattering on\nthe conductivity is further investigated.\nTranslating the longitudinal conductivity to magnetic\ndamping leads to the following scenario: within the band\ngap, αis fully governed by the TSS, see Supplemen-\ntal Material Fig. S2. It is thus necessarily low at the\nDirac point and increases away from the Dirac point as\nshown in Fig. 2(b). When the band structure is richer,\ni.e. when additionally considering bulk states outside\nthe band gap (with conductivity σxx,b), or trivial in-gap\nstates (e.g. impurity levels or interface states with con-\nductivity σxx,i),αis dictated by the interplay of differ-\nent conductivities. Both, σxx,bandσxx,ihave the effect\nof lowering the conductivity σxx,TSSdue to hybridization\nand consequently enhanced scattering probabilities. This\ntranslates into a simple recipe for the spin-pumping effi-\nciency: the larger the conductivities of trivial states, the\nlower the spin-pumping efficiency. A simple version of\nthese arguments considering only TSS and bulk states is\nsketched in Fig. 2(d), with σxx,TSSgoverning the over-\nall spin-pumping efficiency ξSPandσxx,blowering ξSP.\nAs the valence band edge is closer and the valence band\nconductivity is larger than the conduction band edge and\nconductivity [7, 11] the reduction of αsets in earlier and\nis stronger towards the valence band ( E <0) than in the\ncase of the conduction band (0 < E), meaning that the\ntotal damping remains at a larger value when the Fermi\nenergy moves towards the conduction band compared to\nthe valence band direction. The red curve in Fig. 2(d),\nrepresenting the expected spin-pumping efficiency, agrees\nwith the measured data shown in Fig. 2(b) for x= 0.67.\nHIGH-RESOLUTION SCANS\nSo far, in the theoretical description of the spin-\npumping mechanism, we assumed a homogeneous sys-\ntem, i.e. a system where the band structure is indepen-\ndent of the lateral position. However, in real extended\nsystems the energetic position of the Dirac point with re-\nspect to EFvaries as a function of lateral position [30–32]\nresulting in p- and n-type regions when EFis tuned to\nthe vicinity of the Dirac points, see Fig. 4(b),(c). As a\nconsequence, the spin-pumping efficiency must also vary\nlocally, i.e. ξ=ξ(r), with rbeing a lateral position in\nthe 2D-TSS. To detect these variations experimentally,\nwe performed high-resolution backgate voltage scans near\nthe energetic position of the Dirac point. This resulted in6\n 0 0.01 0.02 0.03 0.04 0.05 0.06\n-10  0  10  20  30T (a.u.)\nE (meV) 0 0.5 1\n-0.01  0  0.01  0.02  0.03T (E),   ρ(E) ⋅T(E) (a.u.)\nE (eV)unwei ghtedDOS-wei ghted, rescaled\n 0.001 0.01\n 0 0.2 0.4 0.6 0.8\n-0.01 0 0.01 0.03\n-π/2 0 π/2E (eV)\nφ\nnpnnnn\nnpn\nnnna\nb 0 4 8 12\nD  = 330nm\nkxky\n-0.01 0 0.01 0.03\n-π/2 0 π/2E (eV)\nφ 0.001 0.01\ne\nfkxkyD V \n 0 1 2 3 4 5 6 7 8\nD  = 160nm\nx (µm)V (meV) 4\n 0 8 12\nD  = 3.6nm\nxzy\n0.015\nxp-region\nEDPE(x) view\nExEDPE\nn- n-regionEF\nxED \nV \nkxkyd\nxEDPE\nnnnD \nV e\nnpnnnnvivr\nvttop view\nxy\nreﬂection \nand \ntransmission\nprobability\nat \npn-boundaryneeds to be updated\nneeds to be updated 0 0.02 0.04 0.06 0.08\n-0.04  0  0.04  0.08  0.122G (e /h)\nE (eV)-0.04  0  0.04  0.08  0.12T(E)⋅ ρ(E) (arb.u.)\nE (eV)TSS\ndouble TSS \nzero gap SC\n0°90°\n270°0°90°\n270°1\n1\ntrans-\nmissionreﬂection TSS\ndouble TSS (x 8)\nzero gap SC  (x 200)\nTSS\ndouble TSS \nzero gap SC0°90°\n270°1 0 0.02 0.04 0.08\n-0.04  0  0.04  0.08  0.122G (e /h)\nE (eV)TSS\n(a)\n(b)(c)\n(d)\n(e)(f)\n(g)-8 -6 -4 -2 0 2 40.0090.0120.019\nVBG (V)Δα\n0.015\nFigure 4. (a) High-resolution scan of ∆ αaround VBG∼0. Both measurements (red and black dots) have been performed\non the same sample with (1)BS/(10)BSTS/Py and x= 0.7, measured at T=10K. (b) Band alignment for a spatially extended\npotential barrier in a single TSS. (c) Corrugated potential energy landscape with local n-p junctions for EFbetween the Dirac\npoints (purple dashed line in (b)). A right moving electron with spin up which crosses the n-p boundary has to become a\nright moving hole with spin up ((b)), undergoing the chiral tunneling and reflection process as shown in (d). In chiral systems\nsome incidence angles lead to enhanced transmission ((e) left), for a non-chiral zero-gap semiconducting system transmission is\nexponentially suppressed (right). (f) Analytically calculated quantum mechanical transmission for three systems (single TSS,\ndouble TSS and zero gap semiconductor) with a potential barrier (as in (b)) of height V= 60 meV and width D= 60 nm,\nweighted by the DOS. The chiral systems, single and double TSS, show two dips at the Dirac point energies with a pronounced\nmaximum between them. Removing the chiral nature of the system results in zero transmission for energies at which n-p or\np-n junctions occur as in this case the transmission is mediated by evanescent waves. (g) Numerical transport calculations for a\nsingle potential step in the same systems as in (f) at T= 10 K reveal the same qualitative features as the quantum mechanical\napproach in (f). The results in (f) and (g) are rescaled for visibility (rescaling factor in brackets in the legend, see also the\nSupplemental Material).\nthe appearance of two distinct minima in the dependence\nof magnetic damping on backgate voltage, which will be\nexplained below. Indeed, two distinct minima could be\nreproducibly measured as shown in Fig. 4(a).\nSpin-pumping integrates over the potential landscape\nof the entire sample near the interface between FM and\nthe TI, and thus two scenarios can be drawn if local vari-\nations are taken into account. In the first scenario, which\nis in the limit of a TSS with low overall conductivity, the\nspin-pumping efficiency is fully determined by the instant\nprocess of spin-charge interconversion. In this limit the\nactual charge transport within the TSS plays a minor\nrole. As an example, if we assume a corrugated energy\nlandscape with two distinct potential levels randomly dis-\ntributed across the surface, the integrated spin-pumping\nefficiency will have two minima at the two Dirac point\nenergies. The relative depth of the two minima reflects\nthe ratio of the areas occupied by the two different po-\ntential levels. Away from the Dirac points the potentialvariations become negligible. In [32] Bi Se-antisite defects\nlead to the occurrence of two energetically distinct defect\nstates depending on the position of the defect in the lat-\ntice. As a result, potential fluctuations in the Dirac point\nenergy centered around two main energies were observed\nin [32] and are in line with our observation of two main\ndips in damping.\nIn the second scenario the overall TSS conductivity\nis assumed to be large and charge transport within the\nTSS after the SCI process does play an important role.\nConsidering again a corrugated energy landscape with a\nglobal Fermi energy in the vicinity of the Dirac points,\nas in Fig. 4(b) with n- and p-type regions, an additional\ntransport mechanism occurs, that of chiral tunneling.\nThe chiral nature of massless Dirac fermion systems,\nlike graphene [33–35] and TIs, is the basis for this pos-\nitive contribution to the conductivity between the two\nDirac point levels and is closely related to Klein tunnel-\ning. The original Klein tunneling describes the angle-7\ndependent tunneling process of a Dirac fermion propa-\ngating across a very high electrostatic barrier which leads\nin non-relativistic systems to an exponential decay of the\nstate, but in relativistic cases the transmission probabil-\nity reaches unity for normal incidence, independent of\nthe barrier height and width [36–41]. In chiral systems,\nsuch as the spin-momentum locked TSS, charge carri-\ners propagate through the barriers not via evanescent\nwaves as usual, but via travelling waves belonging to the\nother branch of states. In the n-p-n tunneling depicted\nin Fig. 4(b), the incoming electrons from the positive\nDirac cone (n) are transmitted across the central barrier\nthrough the extended eigenstates from the negative Dirac\ncone (p). The overall transport is determined by the in-\nterplay between the momentum selection rules, spinorial\ncompatibility between incoming and transmitted states,\nand the density of states of the incoming carriers. The\nDirac cone of the TSS has two special features – it is\nchiral and its density of states (DOS) vanishes at the\nDirac point. In order to gauge the importance of each of\nthese features, we compare the TSS to two other systems.\nOne of them consists of two coupled TSSs, analogous\nto bilayer graphene, and is chiral but always has finite\nDOS. The second is a zero-gap semiconductor, which is\nnot chiral, but has constant DOS. Both these artificial\nsystems are discussed purely for the sake of illustration.\nNote that in the following description and calculation\nbulk contributions are neglected. Now let us consider\nthe transmission through a 1D barrier extending in the\nydirection, such as shown in Fig. 4(b). For a general\nangle, the electrons are partly reflected and partly trans-\nmitted (see Fig. 4(e)). At normal incidence, transmission\nreaches unity for the single TSS, while for the double TSS\nit is strictly zero and the charge is transmitted only for\noblique incidence angles. For the zero-gap semiconduc-\ntor transmission is exponentially suppressed, as shown\nschematically in Fig. 4(e). When we integrate the quan-\ntum mechanical transmission over the incidence angles,\nand multiply it by the density of states of the incoming\nelectrons, for both chiral systems we find an enhanced\ntransmission in the energy range between two minima\ncorresponding to the top and bottom of the barrier. In\nthe same energy range, transmission for the non-chiral\nzero-gap semiconductor is fully suppressed as can be seen\nin Fig. 4(f). Similar conclusions can be drawn from a nu-\nmerical calculation of transport in the three systems with\nthe same potential landscape (cf. Fig. 4(g)), where for\nthe single and double TSS we see again a conductance\npeak within the barrier, flanked by two dips, while for\na zero gap semiconductor the conductance in this en-\nergy range is zero. Qualitatively, the same results are\nobtained numerically for a potential landscape with 2D\npotential islands, shown in Sec. VIII of the Supplemental\nMaterial. The central conductance peak flanked by two\ndips persists also for 2D potential disorder. The source of\nthe discrepancy between the shapes of the calculated andmeasured curves is the highly nonlinear coupling between\nthe gate voltage and the position of EFof the top-TSS\nin our devices, hence the true shape of the conductance,\nG(EF), inferred from spin-pumping, may be different.\nFor a more detailed discussion of chiral tunneling, mod-\nelling of the three systems and transport calculations we\nrefer the reader to Sec. VII and VIII of the Supplemental\nMaterial.\nSummarizing the second scenario in which after the oc-\ncurrence of SCI further charge transport through a corru-\ngated energy landscape plays a role: Analytical and nu-\nmerical calculations confirm the suppression of the con-\nductivity exactly at the Dirac points, leading to a simi-\nlar result as in the first scenario with the main difference\nthat in this case the levels of the minima are independent\nof the area ratio as long as the percolation limit is not\nreached. Furthermore, charge transport requires passing\nboth np- or pn-boundary at least once within the mean\nfree path. All calculations confirm that the formation of\na conductivity peak (and hence a peak in the spin pump-\ning efficiency) between two distinct dips is caused by the\nchiral nature of the system.\nFinally, for the sake of completeness, we would like\nto mention a third possible scenario which, however, we\nbelieve is rather unlikely. It is based on the possibil-\nity of electron hopping between the FM and the TSS\n[42]. In [42] it has been shown that due to the hy-\nbridization between the electronic bands of the TSS and\na mono-crystalline FM, for strong enough coupling and\na precise matching of the bands of the FM and the TI\navoided crossings can occur in the band structure, open-\ning smaller gaps within the main gap of the TI. This\nin turn may lead to the occurrence of several minima\nin the conductivity and thus in spin-pumping efficiency.\nOur polycrystalline permalloy films contain grains with\nmany lattice orientations, each of them potentially hy-\nbridizing with different parts of the TSS Dirac cone. In\nconsequence, we would not expect to observe well-defined\ndips/gaps, but rather a background uniform in energy.\nCONCLUSION\nIn conclusion, we have shown that spin-pumping in\ncombination with backgating can be used as a high res-\nolution tool to reveal details of the buried energy land-\nscape of topological insulators. The sensitivity of the\nspin-pumping mechanism to features in the TI band\nstructure was theoretically explained by the dependence\nof the damping parameter on the longitudinal conduc-\ntivity. We also measured the dependence of ∆ αon the\nFermi energy in two equivalent ways: changing the stoi-\nchiometry and applying a backgate. Both measurements\nare in good agreement and show the exceptional control\nof the position of the Fermi energy in the top-TSS of the\nn-p-heterostructure. Further, application of this method8\nwith high resolution shows a rich structure of the spin-\npumping efficiency within the bandgap which can be re-\nlated to the corrugated energy landscape within the TI.\nAmong the three possible explanations for these extra\nfeatures, the chiral tunneling effect seems to be most\nlikely since it does not rely on low conductivity of the\nTSS or a specific matching of the lattice and electronic\nband structure of FM and TI. In addition, only the chi-\nral tunneling effect can lead to the formation of a peak\nbetween conduction dips. On the contrary, in non-chiral\nsystems the conductivity would be fully suppressed for\nenergies within the range of potential fluctuations.\nACKNOWLEDGEMENT\nWe acknowledge the financial support of the\nDeutsche Forschungsgemeinschaft through Project ID\n422 314695032-SFB1277 (Subprojects A01, A07, A08 &\nB04). The work at UCLA (JZ and YT) was supported\nby the US Department of Energy, Office of Basic Energy\nSciences under Grant No. DE-SC0012190.\n[1] M. Z. Hasan and C. L. Kane, Colloquium: Topological\ninsulators, Rev. Mod. Phys. 82, 3045 (2010).\n[2] J. E. Moore, The birth of topological insulators, Nature\n464, 194 (2010).\n[3] S. Zhang and A. Fert, Conversion between spin and\ncharge currents with topological insulators, Phys. Rev.\nB94, 184423 (2016).\n[4] Y. Shiomi, K. Nomura, Y. Kajiwara, K. Eto, M. No-\nvak, K. Segawa, Y. Ando, and E. Saitoh, Spin-electricity\nconversion induced by spin injection into topological in-\nsulators, Phys. Rev. Lett. 113, 196601 (2014).\n[5] P. Deorani, J. Son, K. Banerjee, N. Koirala, M. Brahlek,\nS. Oh, and H. Yang, Observation of inverse spin Hall ef-\nfect in bismuth selenide, Phys. Rev. B 90, 094403 (2014).\n[6] J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back,\nand T. Jungwirth, Spin Hall effects, Rev. Mod. Phys. 87,\n1213 (2015).\n[7] A. Mellnik, J. Lee, A. Richardella, J. Grab, P. Mintun,\nM. H. Fischer, A. Vaezi, A. Manchon, E.-A. Kim,\nN. Samarth, et al. , Spin-transfer torque generated by a\ntopological insulator, Nature 511, 449 (2014).\n[8] F. Yang, S. Ghatak, A. A. Taskin, K. Segawa, Y. Ando,\nM. Shiraishi, Y. Kanai, K. Matsumoto, A. Rosch, and\nY. Ando, Switching of charge-current-induced spin po-\nlarization in the topological insulator BiSbTeSe 2, Phys.\nRev. B 94, 075304 (2016).\n[9] D. Zhu, Y. Wang, S. Shi, K.-L. Teo, Y. Wu, and\nH. Yang, Highly efficient charge-to-spin conversion from\nin situ Bi 2Se3/Fe heterostructures, Applied Physics Let-\nters118, 062403 (2021).\n[10] K. Kondou, R. Yoshimi, A. Tsukazaki, Y. Fukuma,\nJ. Matsuno, K. Takahashi, M. Kawasaki, Y. Tokura, and\nY. Otani, Fermi-level-dependent charge-to-spin currentconversion by Dirac surface states of topological insula-\ntors, Nature Physics 12, 1027 (2016).\n[11] H. Wang, J. Kally, C. S ¸ahin, T. Liu, W. Yanez, E. J.\nKamp, A. Richardella, M. Wu, M. E. Flatt´ e, and\nN. Samarth, Fermi level dependent spin pumping from\na magnetic insulator into a topological insulator, Phys.\nRev. Research 1, 012014 (2019).\n[12] Z. Jiang, C.-Z. Chang, M. R. Masir, C. Tang, Y. Xu,\nJ. S. Moodera, A. H. MacDonald, and J. Shi, Enhanced\nspin Seebeck effect signal due to spin-momentum locked\ntopological surface states, Nature communications 7, 1\n(2016).\n[13] T. Mayer, H. Werner, F. Schmid, R. Diaz-Pardo, J. Fujii,\nI. Vobornik, C. H. Back, M. Kronseder, and D. Bougeard,\nTransport properties of band engineered p-n heterostruc-\ntures of epitaxial (Bi 1−xSbx)2(Te1−ySey)3topological in-\nsulators, Phys. Rev. Mater. 5, 014202 (2021).\n[14] M. Eschbach, E. M ly´ nczak, J. Kellner, J. Kampmeier,\nM. Lanius, E. Neumann, C. Weyrich, M. Gehlmann,\nP. Gospodariˇ c, S. D¨ oring, G. Mussler, N. Demarina,\nM. Luysberg, G. Bihlmayer, T. Sch¨ apers, L. Plucin-\nski, S. Bl¨ ugel, M. Morgenstern, C. M. Schneider, and\nD. Gr¨ utzmacher, Realization of a vertical topological\np–n junction in epitaxial Sb 2Te3/Bi2Te3heterostruc-\ntures, Nature Communications 6, 8816 (2015).\n[15] S. N. Danilov, L. E. Golub, T. Mayer, A. Beer,\nS. Binder, E. M¨ onch, J. Min´ ar, M. Kronseder, C. H.\nBack, D. Bougeard, and S. D. Ganichev, Superlinear Pho-\ntogalvanic Effects in (Bi 0.3Sb0.7)2(Te0.1Se0.9)3: Probing\nThree-Dimensional Topological Insulator Surface States\nat Room Temperature, Phys. Rev. Appl. 16, 064030\n(2021).\n[16] The Supplemental Material is available via the following\nlink: https://doi.org/10.1103/PhysRevB.109.064424 ,\nThe Supplemental Material covers, first, the results of\ndifferent experimental methods to characterize the sam-\nple series used in the main manuscript. On the other\nhand, the theoretical framework is laid out on which our\nquantum mechanical calculations are based on. Results\nfrom these quantum mechanical calculations for one- and\ntwo-dimensional transport with potential fluctuations are\nshown complementary to the results shown in the main\nmanuscript. The Supplemental Material also contains\nRefs. [18, 43–62].\n[17] Z. Ren, A. A. Taskin, S. Sasaki, K. Segawa, and Y. Ando,\nOptimizing Bi 2−xSbxTe3−ySeysolid solutions to ap-\nproach the intrinsic topological insulator regime, Phys.\nRev. B 84, 165311 (2011).\n[18] T. Arakane, T. Sato, S. Souma, K. Kosaka, K. Nakayama,\nM. Komatsu, T. Takahashi, Z. Ren, K. Segawa, and\nY. Ando, Tunable Dirac cone in the topological insulator\nBi2−xSbxTe3−ySey, Nat Commun 3, 636 (2012).\n[19] B. Xia, P. Ren, A. Sulaev, P. Liu, S.-Q. Shen, and\nL. Wang, Indications of surface-dominated transport in\nsingle crystalline nanoflake devices of topological insula-\ntor Bi 1.5Sb0.5Te1.8Se1.2, Phys. Rev. B 87, 085442 (2013).\n[20] Z. Celinski and B. Heinrich, Ferromagnetic resonance\nlinewidth of Fe ultrathin films grown on a bcc Cu sub-\nstrate, Journal of Applied Physics 70, 5935 (1991).\n[21] Z. Celinski, K. Urquhart, and B. Heinrich, Using ferro-\nmagnetic resonance to measure the magnetic moments\nof ultrathin films, Journal of Magnetism and Magnetic\nMaterials 166, 6 (1997).9\n[22] H. Weaver, Dielectric properties of single crystals of\nSrTiO 3at low temperatures, Journal of Physics and\nChemistry of Solids 11, 274 (1959).\n[23] S. Just, F. L¨ upke, V. Cherepanov, F. S. Tautz, and\nB. Voigtl¨ ander, Parasitic conduction channels in topo-\nlogical insulator thin films, Phys. Rev. B 101, 245413\n(2020).\n[24] D. Backes, D. Huang, R. Mansell, M. Lanius, J. Kamp-\nmeier, D. Ritchie, G. Mussler, G. Gumbs, D. Gr¨ utz-\nmacher, and V. Narayan, Disentangling surface and bulk\ntransport in topological-insulator p-n junctions, Phys.\nRev. B 96, 125125 (2017).\n[25] B. Skinner, T. Chen, and B. I. Shklovskii, Why Is the\nBulk Resistivity of Topological Insulators So Small?,\nPhys. Rev. Lett. 109, 176801 (2012).\n[26] B. Skinner, T. Chen, and B. I. Shklovskii, Effects of bulk\ncharged impurities on the bulk and surface transport in\nthree-dimensional topological insulators, Journal of Ex-\nperimental and Theoretical Physics 117, 579 (2013).\n[27] Y. Tserkovnyak and D. Loss, Thin-film magnetization\ndynamics on the surface of a topological insulator, Phys.\nRev. Lett. 108, 187201 (2012).\n[28] A. Sakai and H. Kohno, Spin torques and charge trans-\nport on the surface of topological insulator, Phys. Rev.\nB89, 165307 (2014).\n[29] T. Chiba, S. Takahashi, and G. E. W. Bauer, Magnetic-\nproximity-induced magnetoresistance on topological in-\nsulators, Phys. Rev. B 95, 094428 (2017).\n[30] T. Knispel, W. Jolie, N. Borgwardt, J. Lux, Z. Wang,\nY. Ando, A. Rosch, T. Michely, and M. Gr¨ uninger,\nCharge puddles in the bulk and on the surface of the\ntopological insulator BiSbTeSe 2studied by scanning tun-\nneling microscopy and optical spectroscopy, Phys. Rev.\nB96, 195135 (2017).\n[31] T. B¨ omerich, J. Lux, Q. T. Feng, and A. Rosch, Length\nscale of puddle formation in compensation-doped semi-\nconductors and topological insulators, Phys. Rev. B 96,\n075204 (2017).\n[32] C. Mann, D. West, I. Miotkowski, Y. P. Chen, S. Zhang,\nand C.-K. Shih, Mapping the 3D surface potential in\nBi2Se3, Nature communications 4, 2277 (2013).\n[33] A. F. Young and P. Kim, Quantum interference and Klein\ntunnelling in graphene heterojunctions, Nature Physics\n5, 222 (2009).\n[34] N. Stander, B. Huard, and D. Goldhaber-Gordon, Evi-\ndence for Klein Tunneling in Graphene p−nJunctions,\nPhys. Rev. Lett. 102, 026807 (2009).\n[35] J. Martin, N. Akerman, G. Ulbricht, T. Lohmann, J. H.\nSmet, K. von Klitzing, and A. Yacoby, Observation\nof electron–hole puddles in graphene using a scanning\nsingle-electron transistor, Nature Physics 4, 144 (2008).\n[36] O. Klein and Y. Nishina, ¨Uber die Streuung von\nStrahlung durch freie Elektronen nach der neuen rela-\ntivistischen Quantendynamik von Dirac, Zeitschrift f¨ ur\nPhysik 52, 853 (1929).\n[37] N. H. Tu, Y. Tanabe, Y. Satake, K. K. Huynh,\nand K. Tanigaki, In-plane topological p-n junc-\ntion in the three-dimensional topological insulator\nBi2−xSbxTe3−ySey, Nature Communications 7, 13763\n(2016).\n[38] K. M. M. Habib, R. N. Sajjad, and A. W. Ghosh, Chi-\nral Tunneling of Topological States: Towards the Effi-\ncient Generation of Spin Current Using Spin-MomentumLocking, Phys. Rev. Lett. 114, 176801 (2015).\n[39] Y. Xie, Y. Tan, and A. W. Ghosh, Spintronic signatures\nof Klein tunneling in topological insulators, Phys. Rev.\nB96, 205151 (2017).\n[40] M.-H. Liu, J. Bundesmann, and K. Richter, Spin-\ndependent Klein tunneling in graphene: Role of Rashba\nspin-orbit coupling, Phys. Rev. B 85, 085406 (2012).\n[41] M. I. Katsnelson, K. S. Novoselov, and A. K. Geim, Chi-\nral tunnelling and the Klein paradox in graphene, Nature\nPhysics 2, 620 (2006).\n[42] K. Arimoto, T. Koretsune, and K. Nomura, Quantum\nanomalous Hall effect in a three-dimensional topological-\ninsulator–thin-film-ferromagnetic-metal heterostructure,\nPhys. Rev. B 103, 235315 (2021).\n[43] A. Roy, S. Guchhait, S. Sonde, R. Dey, T. Pramanik,\nA. Rai, H. C. P. Movva, L. Colombo, and S. K. Banerjee,\nTwo-dimensional weak anti-localization in Bi 2Te3thin\nfilm grown on Si(111)-7 x7 surface by molecular beam epi-\ntaxy, Applied Physics Letters 102, 163118 (2013).\n[44] W. Ko, J. Park, I. Jeon, H. W. Kim, H. Kwon, Y. Oh,\nJ. S. Kim, H. Suh, S. W. Hwang, and C. Chung, Lo-\ncal potential fluctuation of topological surface states\nin Bi 1.5Sb0.5Te1.7Se1.3observed by Landau level spec-\ntroscopy, Applied Physics Letters 108, 083109 (2016).\n[45] C. A. Mack, Generating random rough edges, surfaces,\nand volumes, Appl. Opt. 52, 1472 (2013).\n[46] T. Trivedi, S. Sonde, H. C. P. Movva, and S. K. Banerjee,\nWeak antilocalization and universal conductance fluctu-\nations in bismuth telluro-sulfide topological insulators,\nJournal of Applied Physics 119, 055706 (2016).\n[47] H. Beidenkopf, P. Roushan, J. Seo, L. Gorman, I. Droz-\ndov, Y. S. Hor, R. J. Cava, and A. Yazdani, Spatial fluc-\ntuations of helical Dirac fermions on the surface of topo-\nlogical insulators, Nature Physics 7, 939 (2011).\n[48] C. W. Groth, M. Wimmer, A. R. Akhmerov, and\nX. Waintal, Kwant: a software package for quantum\ntransport, New Journal of Physics 16, 063065 (2014).\n[49] H. Nielsen and M. Ninomiya, A no-go theorem for reg-\nularizing chiral fermions, Physics Letters B 105, 219\n(1981).\n[50] D. S. Fisher and P. A. Lee, Relation between conductivity\nand transmission matrix, Phys. Rev. B 23, 6851 (1981).\n[51] F. Guinea, C. Tejedor, F. Flores, and E. Louis, Effective\ntwo-dimensional hamiltonian at surfaces, Phys. Rev. B\n28, 4397 (1983).\n[52] M. Liu, C.-Z. Chang, Z. Zhang, Y. Zhang, W. Ruan,\nK. He, L.-l. Wang, X. Chen, J.-F. Jia, S.-C. Zhang, Q.-\nK. Xue, X. Ma, and Y. Wang, Electron interaction-driven\ninsulating ground state in Bi 2Se3topological insulators\nin the two-dimensional limit, Phys. Rev. B 83, 165440\n(2011).\n[53] J. Chen, X. Y. He, K. H. Wu, Z. Q. Ji, L. Lu, J. R. Shi,\nJ. H. Smet, and Y. Q. Li, Tunable surface conductivity\nin Bi 2Se3revealed in diffusive electron transport, Phys.\nRev. B 83, 241304 (2011).\n[54] J. Wang, A. M. DaSilva, C.-Z. Chang, K. He, J. K. Jain,\nN. Samarth, X.-C. Ma, Q.-K. Xue, and M. H. W. Chan,\nEvidence for electron-electron interaction in topological\ninsulator thin films, Phys. Rev. B 83, 245438 (2011).\n[55] Y. Takagaki, B. Jenichen, U. Jahn, M. Ramsteiner,\nand K.-J. Friedland, Weak antilocalization and electron-\nelectron interaction effects in Cu-doped Bi 2Se3films,\nPhys. Rev. B 85, 115314 (2012).10\n[56] S.-P. Chiu and J.-J. Lin, Weak antilocalization in topo-\nlogical insulator Bi 2Te3microflakes, Phys. Rev. B 87,\n035122 (2013).\n[57] N. I. Fedotov and S. V. Zaitsev-Zotov, Experimental\nsearch for one-dimensional edge states at surface steps of\nthe topological insulator Bi 2Se3: Distinguishing between\neffects and artifacts, Phys. Rev. B 95, 155403 (2017).\n[58] J. Dufouleur, E. Xypakis, B. B¨ uchner, R. Giraud, and\nJ. H. Bardarson, Suppression of scattering in quantum\nconfined 2D helical Dirac systems, Phys. Rev. B 97,\n075401 (2018).[59] L. Susskind, Lattice fermions, Phys. Rev. D 16, 3031\n(1977).\n[60] R. Stacey, Eliminating lattice fermion doubling, Phys.\nRev. D 26, 468 (1982).\n[61] H.-Z. Lu and S.-Q. Shen, Finite-temperature conductiv-\nity and magnetoconductivity of topological insulators,\nPhys. Rev. Lett. 112, 146601 (2014).\n[62] E. McCann and V. I. Fal’ko, Landau-Level Degeneracy\nand Quantum Hall Effect in a Graphite Bilayer, Phys.\nRev. Lett. 96, 086805 (2006)."    },    {        "title": "2311.12846v1.Electronic_structure_of_CrO__2__probed_by_NMR_and_DFT.pdf",        "content": "Electronic structure of CrO 2probed by NMR and DFT\nVojtˇ ech Chlan,1,∗Anna A. Shmyreva,1, 2and Helena ˇStˇ ep´ ankov´ a1\n1Charles University, Faculty of Mathematics and Physics,\nDepartment of Low Temperature Physics, V Holeˇ soviˇ ck´ ach 2, Prague 18000, Czech Republic\n2St. Petersburg State University, Center for Magnetic Resonance,\nUniversitetskiy pr. 26, St. Petersburg 198504, Russia\nElectronic structure of ferromagnetic half-metal CrO 2was studied by means of53Cr nuclear mag-\nnetic resonance (NMR) spectroscopy and density functional theory (DFT). The measured NMR\nspectrum consists of three distinct spectral lines and is interpreted as a triplet arising due to electric\nquadrupole interaction. The observed NMR parameters agree well with those obtained from elec-\ntronic structure calculations, corresponding to the presence of Cr4+with fully occupied localized dxy\nsinglet and partially occupied degenerated dxzanddyzstates, as required by ferromagnetic double\nexchange mechanism. With high accuracy the orbital occupations and valence states of all Cr atoms\nwithin the CrO 2structure are found uniform.\nChromium dioxide (CrO 2) is a promising material for\napplications in spintronics owing to being a ferromagnetic\nhalf-metal.1,2The majority-spin electrons are metallic\nwhile there is a gap in the minority spin channel, leading\nto spin polarization close to 100 % at the Fermi level.3–5\nCrO 2crystallizes in a tetragonal rutile structure6\n(space group P4 2/mnm ) where the two Cr atoms occupy\noctahedral 2 asites at (0,0,0) and ( ½,½,½). Chromium 3d\norbitals are split into lower-lying t2gtriplet, occupied by\ntwo electrons, and excited egdoublet. The tetragonal dis-\ntortion of the octahedron leads to a localized dxysinglet\nand degenerated dxzanddyzstates hosting the remain-\ning electron.7,8The microscopic mechanism of ferromag-\nnetism in CrO 2is double-exchange (DE), analogous to\nDE in manganites with a mixed valence state. However,\nin CrO 2the localized and the itinerant t2gelectrons at\nboth Cr atoms are involved and in contrast to mangan-\nites this specific DE does not require the Cr atoms to be\nin different oxidation states.9,10The metallic behavior\nof CrO 2arises due to the dispersed chromium 3d states\nstrongly hybridized with oxygen 2p states and stretching\nacross the Fermi level.9\nThese basic features of the electronic structure are well\nreproduced by density functional theory (DFT) calcu-\nlations already within the framework of the local-spin-\ndensity approximation (LSDA) or the generalized gradi-\nent approximation (GGA).7,8,11,12Improved description\nof the orbital character, Cr orbital moments and other,\nmore subtle properties require to include some electron\ncorrelation effects by the application of the LDA+U\napproach,9,13–15or by the dynamical mean field theory\n– expectably a more adequate treatment of the half-\nmetallic nature of CrO 2.16,17\nThe (half)metallic character of CrO 2was confirmed\ndirectly by numerous experiments,3–5,18,19however, the\nevidence of the specific orbital character of Cr, essen-\ntial for the DE, is less straightforward to obtain experi-\nmentally. Optical measurements20–22as well as measure-\nments of the x-ray absorption spectroscopy (XAS) and\nthe x-ray magnetic circular dichroism (XMCD)15,23–25were interpreted in line with the picture of the band\nstructure obtained by the calculations, however, other\nstudies suggested different mechanism of DE, involv-\ning Cr atoms with mixed valence, by using XAS and\nXMCD,26x-ray photoelectron spectroscopy,15or neutron\npowder diffraction.27Further such conclusion28–30has\nbeen drawn from the nuclear magnetic resonance spec-\ntroscopy (NMR), which is particularly sensitive to the\nvalence state and the local orbital arrangement of atoms\nin magnetic materials.\nThe first NMR measurements of CrO 2by Nishihara\net al.31in 1972 were interpreted as originating from two\ndifferent Cr species in the structure, because their53Cr\nNMR spectra at 4.2 K clearly showed two distinct reso-\nnance lines at 26.3 and 36.7 MHz. Since then, it has been\nnaturally assumed that there are two non-equivalent Cr\nspecies with markedly different hyperfine fields. Nishi-\nhara et al. assigned the line at 36.7 MHz to Cr in un-\nperturbed CrO 2structure while the second line was at-\ntributed to the presence of vacancies or similar struc-\ntural imperfections. Later53Cr NMR study28assigned\nboth53Cr spectral lines as pertaining to the proper CrO 2\nstructure and the large difference in frequencies of the\nspectral lines was explained as a consequence of two dif-\nferent valence states Cr(4±δ)+; in analogy to DE mecha-\nnism in manganites, where two Mn species with different\nvalence states can be observed in55Mn NMR spectra.32,33\nHowever, such an interpretation requires the difference\nδin valence of the two Cr atoms to be relatively high\n(δ∼0.4), which was not confirmed by other experimental\nmethods or recognized in the calculations. Therefore, the\nspectrum was interpreted differently in later53Cr NMR\nstudies29,30assuming that the pair of Cr atoms has much\nsmaller difference in the valence states ( δ∼0.03), but\nrather differs in orbital occupations, allowing to explain\nthe observed NMR spectrum by the anisotropic contribu-\ntions to the hyperfine field at53Cr nuclei. In very recent\n53Cr NMR study including detailed analysis of the NMR\nrelaxation times34the authors arrived to a similar conclu-\nsion that the two observed53Cr spectral lines correspondarXiv:2311.12846v1  [cond-mat.mtrl-sci]  13 Oct 20232\nFIG. 1.53Cr NMR spectrum at 4.2 K in zero external mag-\nnetic field. Red lines denote positions of the NMR lines based\non the DFT-calculated Vzzandη. Experimental spectrum\nwas normalized by f2to reflect the linear frequency depen-\ndences of both the NMR probe inductance and the population\ndifferences of nuclear energy levels.\nto Cr crystal sites with different local magnetic fields yet\nat the same time the Cr atoms may possess the same\nvalence.\nIn this work we clear out these seeming contradictions\nby introducing new53Cr NMR experiments where we de-\ntected (besides the two previously observed lines at 37.16\nand 26.40 MHz) an additional spectral line at lower fre-\nquency. This line has been omitted in all previous NMR\nworks on CrO 2, and thus – understandably – the exist-\ning interpretations of the53Cr NMR incorrectly assumed\npresence of two different Cr atoms in CrO 2. In the view of\nthe appearance of the third line, we reinterpret the53Cr\nNMR spectrum in CrO 2as a triplet owing to nuclear\nelectric quadrupole interaction, which is rather strong\nhere. Our NMR experiments thus show unambiguously\nthat all Cr atoms in the CrO 2structure are crystallo-\ngraphically and magnetically equivalent. Moreover, by\nusing density functional theory (DFT) modeling we show\nthat the observed NMR spectra correspond to the par-\nticular orbital arrangement, which is responsible for the\nspecific DE mechanism in CrO 2involving the localized\nand the itinerant t2gelectrons at both Cr atoms in the\nunit cell.9,10\nStudied CrO 2powder sample was supplied by Sigma\nAldrich (MagtrieveTM) and checked by x-ray diffraction\nas CrO 2rutile structure with negligible traces of Cr 2O3.\nFrequency swept53Cr NMR spectrum was acquired in\nzero external magnetic field at temperature of 4.2 K. At\neach frequency step the NMR probe was properly tuned\nand matched, and Carr-Purcell-Meiboom-Gill pulse train\nwas applied. All spin echos in the train were recorded\nFIG. 2. Schematics of the effect of magnetic field Band elec-\ntric field gradient (EFG, parameters Vzzandη) on the energy\nlevels of the53Cr nucleus (spin I=3/2). From left to right,\nthe original degenerate ground state of the nucleus undergoes\nZeeman splitting under the magnetic field Binto four levels\n(labeled by the magnetic quantum number mI), and these are\nfurther shifted by the nuclear electric quadrupole interaction.\nThree transitions labeled with frequencies correspond to the\nthree spectral lines observed in the NMR spectrum. Numeri-\ncal values of B,|Vzz|, and ηwere obtained from the fit of the\nexperimental NMR spectrum. The sign of Vzzwas obtained\nfrom DFT calculations.\nand their sum Fourier transformed. The measured spec-\ntrum (Fig. 1) consists of three intense spectral lines at\n37.16, 26.40, and 15.63 MHz and several two orders of\nmagnitude weaker lines spread over the displayed spec-\ntral region. We assign the intense triplet to the bulk\nCrO 2phase, origin of the weaker signals is unknown, but\nwe assume they arise from regions close to surface of the\nCrO 2particles with less defined stoichiometry and crys-\ntalline arrangement. In this work we shall focus on the\nintense triplet only.\nTwo lines at higher frequencies, 37.16 and 26.40 MHz,\ncorrespond to the lines documented in the previous NMR\nworks,28–31,34whereas the third line at 15.63 MHz is the\none that has not been observed before. Nuclear spin\nnumber of53Cr isotope I=3/2, and thus in a mag-\nnetic field Bthe energy level of the nuclear ground state\nis Zeeman-split onto four equidistant stationary energy\nlevels – a situation that would lead to a single line in\nthe NMR spectrum (gyromagnetic ratio35of53Cr nu-\ncleiγ\n2π=−2.4115 MHz T−1). When additionally to the\nmagnetic field an electric field gradient (EFG) is present\nat nuclei (nuclear quadrupole moment36of53Cr nuclei\nQ=−150(50) milibarn, 1 milibarn = 10−31m2), the\nfour energy levels are further (unevenly) shifted due to\nthe electric quadrupole interaction, which yields three\ndifferent transitions observable in the NMR spectrum3\nFIG. 3. Enhancement of the NMR signal in the double-\nresonance experiment. Population inversion at one transition\ncauses an increase of the measured intensity at another tran-\nsition of the53Cr nuclear multiplet. 100 % corresponds to\nintensity without an inverting pulse.\n(see energy level diagram in Fig. 2). It is customary\nto label the energy levels by the magnetic quantum num-\nbermIof the original Zeeman eigenstates, and so in our\ncase the line at 26.40 MHz arises due to the transition\nbetween levels |−1/2⟩and|1/2⟩(central transition, CT),\nwhile the lines at 37.16 and 15.63 MHz (satellite transi-\ntions, ST) correspond to the transitions |1/2⟩ ↔ |3/2⟩and\n|−3/2⟩ ↔ |−1/2⟩, respectively. In case of53Cr nucleus\nin CrO 2without application of external magnetic field,\nthe magnetic field Bat Cr nuclei results from hyperfine\nmagnetic interaction of the53Cr nuclear spin with the\norbital and spin moments of electrons, mostly with the\nonsite Cr 3d states: directly (dipolar interaction) as well\nas mediated by spins of the s-states via Fermi contact\ninteraction. Nonzero EFG appears at Cr sites owing to\nthe local symmetry being lower than cubic (point group\nof the site symmetry is mmm ).\nWe claim that the three observed spectral lines at\n37.16, 26.40, and 15.63 MHz arise due to splitting of\nthe spectrum of53Cr in equivalent sites because of the\nelectric quadrupole interaction. In such a case the three\ntransitions are realized within one energy-level multiplet\n(Fig. 2), which can be unambiguously demonstrated by a\ndouble-resonance NMR experiment.37Energy level |1/2⟩\nis involved in two transitions: |1/2⟩ ↔ |3/2⟩producing one\nof the satellite lines (37.16 MHz), and |−1/2⟩ ↔ |1/2⟩, cor-\nresponding to the CT at 26.40 MHz. Inducing population\ntransfer between the levels of one transition changes the\npopulation difference for the other transition, which then\naffects the intensity of the corresponding spectral line.\nE.g., applying hard 180◦radiofrequency (rf) pulse at fre-\nquency of 26.40 MHz (CT) inverts the populations of lev-els|−1/2⟩and|1/2⟩and the intensity of subsequently mea-\nsured line at 37.16 MHz (ST) is enhanced in dependence\non the delay ∆ between the inverting 180◦pulse at CT\nand the measuring echo-pulse sequence at ST (Fig. 3).\nIrradiation by the 180◦rf pulse at 15.63 MHz and mea-\nsuring at 26.40 MHz yields analogous result. The in-\nduced enhancement decreases with increasing delay ∆\ndue to the nuclear spin-lattice relaxation process: in\nCrO 2at 4.2 K the nuclear relaxation is relatively fast\nand ∆ ∼0.3 s is sufficient for the inverted populations\nto revert back to the thermal equilibrium. The enhance-\nment due to inversion is given by the Boltzmann dis-\ntribution for the53Cr multiplet displayed in Fig. 2 and\nat temperature T= 4.2 K the maximum enhancement\nequals ∼171 and ∼158 % when applying inverting rf pulse\nat 26.40 and 15.63 MHz, respectively. This theoretical\nlimit is not fully achieved in our experiments, though,\nmost likely due to relatively large linewidth of the spec-\ntral lines compared to limited spectral bandwidth of the\ninverting pulse, and possibly also due to spin diffusion.\nNonetheless, the observed enhancement directly proves\nthat the three observed spectral lines are not three indi-\nvidual53Cr species (with three different local magnetic\nfields), but belong to a triplet due to electric quadrupole\ninteraction and thus originate from one type of Cr atoms\nwith a single value of local magnetic field.\nIn order to confirm interpretation of the NMR spec-\ntrum by another independent method providing infor-\nmation on the electronic structure and hyperfine pa-\nrameters, the electronic structure of CrO 2was modelled\nwithin the DFT using the full-potential augmented plane-\nwave method implemented in WIEN2k.38In fact, our\nDFT calculations predicted the position of the line at\n15.63 MHz prior performing the experiments, which em-\nphasizes the importance of calculations in this field. Lat-\ntice parameters a= 4.4841 ˚A,c= 2.9745 ˚A, and oxy-\ngen parameter u= 0.30168 were fully relaxed within the\nspace group P4 2/mnm . Perdew-Burke-Ernzerhof vari-\nant of the GGA exchange-correlation potential39was\nemployed and the description of the electronic correla-\ntions was improved by the GGA+U approach applied to\nCr 3d states with parameters Ueff=U−J= 3.5 eV\nandJ= 0 eV. Atomic sphere radii were chosen as 2.0\nand 1.5 a0for Cr and O, respectively (Bohr unit a0∼\n0.529˚A). We used computational parameters well con-\nverged with respect to EFG: a basis set of 1072 functions\n(RMTKmax = 8 .0) and 2588 k-points (mesh 15 ×15×23)\nin the irreducible part of the Brillouin zone. Spin-orbit\ninteraction was introduced for the semi-core and valence\nelectrons within the second variational method using the\nscalar-relativistic approximation.40\nThe frequencies of the spectral lines of the53Cr triplet\nare in general determined from eigenvalues of the spin\nHamiltonian of electric quadrupole interaction and Zee-\nmann interaction41for spin I=3/2:4\nH=eQV zz\n4I(2I−1)\u0010\n3ˆI2\nz−ˆI2+η\n2(ˆI2\n++ˆI2\n−)\u0011\n+ (1)\n+γB\n2\u0010\nˆI+e−iφsinϑ+ˆI−eiφsinϑ+ 2ˆIzcosϑ\u0011\nexpressed using nuclear spin operators ( ˆIz,ˆI,ˆI±=\nˆIx±iˆIy) within the principal axis system of the EFG ten-\nsor. The EFG tensor Vis defined by its largest principal\ncomponent Vzz,|Vzz| ≥ |Vyy| ≥ |Vxx|, and the asymme-\ntry factor η=Vxx−Vyy\nVzz(0≤η≤1).Qandγdenote\nquadrupole moment and magnetogyric ratio of the nu-\ncleus in the ground state. Without an external magnetic\nfield, the magnetic field Bat the53Cr nucleus is given\nby the hyperfine magnetic field Bhf. Orientation of Bhf\nwith respect to the main axes of EFG tensor is expressed\nvia spherical angles ϑandφ. For Cr and other 3d ele-\nments, the direction of Bhfis antiparallel to the direc-\ntion of atomic magnetic moment. In CrO 2in zero exter-\nnal magnetic field both the direction of magnetization42\nand the direction of Vzzprincipal axis lie parallel to the\ntetragonal axis c, i.e., ϑ= 0 and the dependence on φis\nremoved from the Hamiltonian (Eq. 1). Parameters Vzz\nandηcan be evaluated from the charge density calcu-\nlated by DFT and are usually in a good agreement with\nexperimental values for various compounds.43–45Values\nofVzzandηfor53Cr nuclei in CrO 2were calculated as\nVzz=−5.95(5)·1021Vm−2andη= 0.31(10) and arise\npredominately due to the Cr 3d states (d-d contribution),\nhowever, a weaker p-p contribution to Vzzfrom the oxy-\ngen 2p states is also present. It is usual in NMR to ex-\npress the strength of the electric quadrupole interaction\nby the quadrupole coupling constant, CQ=eQV zz\nh, which\nhere equals 21.6 MHz.\nThe hyperfine magnetic field Bhfat Cr nuclei arises\nfrom the interaction of the nuclear spin with the or-\nbital and spin moments of electrons surrounding the nu-\ncleus and is also obtainable from the calculations of elec-\ntronic structure, however, for nuclei of transition-metal\nelements the Fermi contact term of the Bhfis usually\nunderestimated by the DFT calculations.46In our case\nthe calculated Bhf= 8.14(10) T at53Cr nuclei is lower\nby about 26 % than the experimentally observed value.\nFrequencies of the spectral lines obtained using the\nvalue of Bfrom experiment and the values of Vzzand\nηfrom the DFT calculation are compared to the exper-\niment in Fig. 1. The calculated Vzzmatches the experi-\nment very well, the calculations underestimate the value\nofη, although the spectral shape is relatively insensitive\ntoηin this case. It should be noted, however, that the\nnuclear quadrupole moment Qof53Cr is known only rela-\ntively inaccurately,36Q= 150(50) milibarn. We presume\nthat the accuracy of our DFT calculation of the EFG pa-\nrameters is significantly higher than the accuracy of Q\nand could be in principle used to refine its value.\nFIG. 4. The dependence of calculated53Cr Vzz on the\nvalence state of Cr atoms (a) and their orbital arrangement\n(b).\nThe DFT calculations can also provide parameters de-\nscribing the anisotropy of the hyperfine magnetic field,\ni.e., its dependence on the direction of magnetization,\nwhich can significantly influence the NMR spectrum in\nmagnetic materials.47Our calculations show that for\nCrO 2this anisotropy contribution is relatively large (it\nchanges from +2 .2 T for [001] direction to −5.0 T for\n[100] direction) and can be used to interpret NMR ex-\nperiments on CrO 2in external magnetic fields.28–31Es-\npecially for the latter work,30where single crystal thin\nfilms were used, the explanation is relatively straight-\nforward, since the frequency of the CT (26.40 MHz) is\npredominately influenced by the change of the local mag-\nnetic field due its anisotropy, whereas the ST at 37.16 is\nadditionally strongly influenced by the change of the ori-\nentation of the local magnetic field (angles ϑandφin\nEq. 1) with respect to the EFG tensor.\nWe have shown that the observed53Cr NMR spectrum\nconsists of a triplet of lines due to electric quadrupole\ninteraction, which corresponds to a single Cr species\npresent in the CrO 2structure. NMR parameters ex-\ntracted from the experiment agree well with the calcu-\nlated ones, indicating that the DFT calculations provide\nrather realistic description of CrO 2electronic structure.\nIn the following analysis we point out that this NMR-\nDFT correspondence is unique by showing that even a\nrelatively low difference in the valence states or the or-\nbital occupations of Cr atoms would lead to a notable\nchange in the NMR spectrum.\nIn order to inspect the connection between Vzzand\nthe electronic state of Cr, we deliberately perturbed the\ncalculated ground state and evaluated the dependence of\nVzzon the valence state of Cr atom and, independently,\nalso on the orbital arrangement of Cr 3d states. The\npositive charges of the two Cr nuclei in the unit cell of\nCrO 2are represented by the Coulomb potentials propor-\ntional to Z= 24. Different valence states can be imposed\non the two Cr atoms by modifying their atomic number\nZ′= 24±δZ. Subsequent DFT calculation with such\nmodified atomic numbers will reach a new ground state\nwith the two Cr atoms possessing adequately differenti-\nated valence states. The dependence of calculated Vzz5\non the value of δZis shown in Fig. 4a and we can esti-\nmate that δZ= 0.01, i.e., difference in the valence states\nof 0.02, would cause a difference of 0 .15·1021Vm−2be-\ntween the values of Vzzof the two Cr atoms, leading\nto a splitting of the ST lines about 2x larger than their\nlinewidth, which would be well noticeable in the NMR\nexperiment.\nA simple illustration of how sensitively the EFG de-\npends on the orbital distribution can be established by\nconsidering the d-d valence contribution to Vzzand its\nproportionality to the ”anisotropy count” of Cr 3d:48,49\nVzz≃∆nd=dxy+dx2−y2−1\n2(dxz+dyz)−dz2(2)\nwhere dxy,dx2−y2,dxz,dyz, and dz2are occupation\nnumbers of the respective Cr d-states. Vzzof Cr in CrO 2\nis dominated by the d-d contribution and according to\nEq. 2 the value of Vzzshould increase with increasing\noccupation of dxyor with decreasing occupation of dxz\nanddyz. We may artificially perturb the occupations\nof Cr d-states by manually adjusting the corresponding\noccupation matrix in the calculations. Then, the applied\norbital potential within the GGA+U framework pushes\nthe occupations towards the desired state. From Fig. 4b\nwe estimate that already a very small change, δorb= 0.01,\nin the occupation of any of the t2gd-states would lead\nto an observable change of Vzzby about 0 .4·1021Vm−2,\nproducing a well visible frequency shift/splitting of the\nsatellite lines in the NMR spectrum of ∼700 kHz. Given\nthe observed widths of the satellite lines in the NMR\nspectrum we may conclude that the occupations of Cr d-\nstates in CrO 2are identical within the accuracy of 0.001.\nIn conclusion, the53Cr NMR spectrum of CrO 2mea-\nsured at 4.2 K was interpreted on the basis of presence\nof strong nuclear electric quadrupole interaction. The\ncalculations of electronic structure fully explain the ob-\nserved NMR spectrum, which shows that the orbital oc-\ncupations and valence states of both Cr sites in the unit\ncell of CrO 2are identical and in line with the picture\nprevalent in the literature, i.e., the localized dxysinglet\nis occupied by one electron and the degenerated dxzand\ndyzstates share the remaining electron of Cr4+.\nWe thank R. Kuˇ zel for the x-ray measurement. Com-\nputational resources were provided by the e-INFRA CZ\nproject (ID:90254), supported by the Ministry of Educa-\ntion, Youth and Sports of the Czech Republic.\n∗vojtech.chlan@mff.cuni.cz\n[1] R. A. de Groot, F. M. Mueller, P. G. van Engen, and\nK. H. J. Buschow, Phys. Rev. Lett. 50, 2024 (1983).\n[2] J. M. D. Coey and M. Venkatesan, J. Appl. Phys. 91,\n8345 (2002).\n[3] Y. Ji, G. J. Strijkers, F. Y. Yang, C. L. Chien, J. M.\nByers, A. Anguelouch, G. Xiao, and A. Gupta, Phys.\nRev. Lett. 86, 5585 (2001).[4] R. J. Soulen, J. M. Byers, M. S. Osofsky, B. Nadgorny,\nT. Ambrose, S. F. Cheng, P. R. Broussard, C. T. Tanaka,\nJ. Nowak, J. S. Moodera, A. Barry, and J. M. D. Coey,\nScience 282, 85 (1998).\n[5] D. J. Huang, L. H. Tjeng, J. Chen, C. F. Chang, W. P.\nWu, A. D. Rata, T. Himba, S. C. Chung, S.-G. Shyu,\nC.-C. Wu, and C. T. Chen, Surf. Rev. Lett. 09, 1007\n(2002).\n[6] P. Porta, M. Marezio, J. Remeika, and P. Dernier, Mater.\nRes. Bull. 7, 157 (1972).\n[7] P. I. Sorantin and K. Schwarz, Inorg. Chem. 31, 567\n(1992).\n[8] S. P. Lewis, P. B. Allen, and T. Sasaki, Phys. Rev. B 55,\n10253 (1997).\n[9] M. A. Korotin, V. I. Anisimov, D. I. Khomskii, and G. A.\nSawatzky, Phys. Rev. Lett. 80, 4305 (1998).\n[10] P. Schlottmann, Phys. Rev. B 67, 174419 (2003).\n[11] K. Schwarz, J. Phys. F: Met. Phys. 16, L211 (1986).\n[12] J. Kuneˇ s, P. Nov´ ak, P. M. Oppeneer, C. K¨ onig,\nM. Fraune, U. R¨ udiger, G. G¨ untherodt, and\nC. Ambrosch-Draxl, Phys. Rev. B 65, 165105 (2002).\n[13] I. I. Mazin, D. J. Singh, and C. Ambrosch-Draxl, Phys.\nRev. B 59, 411 (1999).\n[14] M. S. Laad, L. Craco, and E. M¨ uller-Hartmann, Phys.\nRev. B 64, 214421 (2001).\n[15] D. J. Huang, H.-T. Jeng, C. F. Chang, G. Y. Guo,\nJ. Chen, W. P. Wu, S. C. Chung, S. G. Shyu, C. C.\nWu, H.-J. Lin, and C. T. Chen, 66, 174440 (2002).\n[16] I. V. Solovyev, I. V. Kashin, and V. V. Mazurenko, Phys.\nRev. B 92, 144407 (2015).\n[17] L. Chioncel, H. Allmaier, E. Arrigoni, A. Yamasaki,\nM. Daghofer, M. I. Katsnelson, and A. I. Lichtenstein,\nPhys. Rev. B 75, 140406 (2007).\n[18] K. A. Yates, W. R. Branford, F. Magnus, Y. Miyoshi,\nB. Morris, L. F. Cohen, P. M. Sousa, O. Conde, and\nA. J. Silvestre, Appl. Phys. Lett. 91, 172504 (2007).\n[19] T. Tsujioka, T. Mizokawa, J. Okamoto, A. Fujimori,\nM. Nohara, H. Takagi, K. Yamaura, and M. Takano,\nPhys. Rev. B 56, R15509 (1997).\n[20] E. J. Singley, C. P. Weber, D. N. Basov, A. Barry, and\nJ. M. D. Coey, Phys. Rev. B 60, 4126 (1999).\n[21] H. Huang, K. Seu, A. Reilly, Y. Kadmon, and W. F.\nEgelhoff, J. Appl. Phys. 97, 10C309 (2005).\n[22] M. K. Stewart, K. B. Chetry, B. Chapler, M. M. Qazil-\nbash, A. A. Schafgans, A. Gupta, T. E. Tiwald, and D. N.\nBasov, Phys. Rev. B 79, 144414 (2009).\n[23] C. F. Chang, D. J. Huang, A. Tanaka, G. Y. Guo, S. C.\nChung, S.-T. Kao, S. G. Shyu, and C. T. Chen, Phys.\nRev. B 71, 052407 (2005).\n[24] C. B. Stagarescu, X. Su, D. E. Eastman, K. N. Altmann,\nF. J. Himpsel, and A. Gupta, Phys. Rev. B 61, R9233\n(2000).\n[25] P. Zimmermann, N. Bouldi, M. O. Hunault, M. Sikora,\nJ. M. Ablett, J.-P. Rueff, B. Lebert, P. Sainctavit, F. M.\nde Groot, and A. Juhin, J. Electron Spectrosc. Relat.\nPhenom. 222, 74 (2018).\n[26] S. Seong, E. Lee, H. W. Kim, B. Min, S. Lee, J. Dho,\nY. Kim, J.-Y. Kim, and J.-S. Kang, J. Magn. Magn.\nMater. 452, 447 (2018).\n[27] K. Kodama, K. Ikeda, M. Isobe, H. Takeda, M. Itoh,\nY. Ueda, S. ichi Shamoto, and T. Otomo, J. Phys. Soc.\nJapan 85, 094709 (2016).\n[28] J. H. Shim, S. Lee, J. Dho, and D.-H. Kim, Phys. Rev.\nLett. 99, 057209 (2007).6\n[29] H. Takeda, Y. Shimizu, M. Itoh, M. Isobe, and Y. Ueda,\nJ. Phys. Conf. Ser. 400, 032098 (2012).\n[30] H. Takeda, Y. Shimizu, Y. Kobayashi, M. Itoh, T. Jin-\nno, M. Isobe, Y. Ueda, S. Yoshida, Y. Muraoka, and\nT. Yokoya, Phys. Rev. B 93, 235129 (2016).\n[31] H. Nishihara, T. Tsuda, A. Hirai, and T. Shinjo, J. Phys.\nSoc. Japan 32, 85 (1972).\n[32] M. Savosta and P. Nov´ ak, J. Magn. Magn. Mater. 196-\n197, 518 (1999).\n[33] M. M. Savosta, V. I. Kamenev, V. A. Borodin, P. Nov´ ak,\nM. Maryˇ sko, J. Hejtm´ anek, K. D¨ orr, and M. Sahana,\nPhys. Rev. B 67, 094403 (2003).\n[34] Y. V. Piskunov, A. F. Sadykov, V. V. Ogloblichev, A. G.\nSmolnikov, A. P. Gerashenko, and P. Z. Si, Phys. Rev. B\n106, 094428 (2022).\n[35] N. Stone, At. Data Nucl. Data Tables 90, 75 (2005).\n[36] W. Ertmer, U. Johann, and R. Mosmann, Z. Phys. A\n309, 1 (1982).\n[37] J. Haase, N. J. Curro, R. Stern, and C. P. Slichter, Mol.\nPhys. 95, 891 (1998).\n[38] P. Blaha, K. Schwarz, F. Tran, R. Laskowski, G. K. H.\nMadsen, and L. D. Marks, J. Chem. Phys. 152, 074101\n(2020).[39] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev.\nLett. 77, 3865 (1996).\n[40] A. H. MacDonald, W. E. Picket, and D. D. Koelling, J.\nPhys. C Solid State Phys. 13, 2675 (1980).\n[41] A. Abragam, The Principles of Nuclear Magnetism\n(Clarendon Press, Oxford, 1961).\n[42] F. Y. Yang, C. L. Chien, E. F. Ferrari, X. W. Li, G. Xiao,\nand A. Gupta, Appl. Phys. Lett. 77, 286 (2000).\n[43] S. Cottenier, V. Bellini, M. C ¸akmak, F. Manghi, and\nM. Rots, Phys. Rev. B 70, 155418 (2004).\n[44] Y. Zagorodniy, V. Chlan, H. ˇStˇ ep´ ankov´ a, Y. Fomichov,\nJ. Pejchal, V. Laguta, and M. Nikl, J. Phys. Chem. Solids\n126, 93 (2019).\n[45] K. Choudhary, J. N. Ansari, I. I. Mazin, and K. L. Sauer,\nSci. Data 7, 362 (2020).\n[46] P. Nov´ ak and V. Chlan, Phys. Rev. B 81, 174412 (2010).\n[47] V. Chlan, H. ˇStˇ ep´ ankov´ a, R. ˇRezn´ ıˇ cek, and P. Nov´ ak,\nSolid State Nucl. Magn. Reson. 40, 27 (2011).\n[48] P. Blaha, K. Schwarz, and P. H. Dederichs, Phys. Rev. B\n37, 2792 (1988).\n[49] C. Ambrosch-Draxl, P. Blaha, and K. Schwarz, J. Con-\ndens. Matter Phys. 1, 4491 (1989)."    },    {        "title": "2109.09341v1.Electron_Spin_Resonance_of_the_itinerant_ferromagnets_LaCrGe3__CeCrGe3_and_PrCrGe3.pdf",        "content": "Electron Spin Resonance of the itinerant\nferromagnets LaCrGe 3, CeCrGe 3and PrCrGe 3\nJ org Sichelschmidt1, Thomas Gruner1z\n1Max Planck Institute for Chemical Physics of Solids, D-01187 Dresden,\nGermany\nDebarchan Das2x, Zakir Hossain2;3\n2Department of Physics, Indian Institute of Technology, Kanpur 208016, India\n3Institute of Low Temperature and Structure Research, Ok\u0013 olna 2, 50-422\nWroclaw, Poland\nE-mail: Sichelschmidt@cpfs.mpg.de\nAbstract. We report Electron Spin Resonance of the itinerant ferromagnets\nLaCrGe 3, CeCrGe 3, and PrCrGe 3. These compounds show well de\fned and very\nsimilar spectra of itinerant Cr 3 dspins in the paramagnetic temperature region.\nUpon cooling and crossing the Cr-ferromagnetic ordering (below around 90 K)\nstrong spectral structures start to dominate the resonance spectra in a quite\ndi\u000berent manner in the three compounds. In the Ce- and Pr-compounds the\nresonance is only visible in the paramagnetic region whereas in the La-compound\nthe resonance can be followed far below the ferromagnetic ordering temperature.\nThis behavior will be discussed in terms of the speci\fc interplay between the 4 f\nand 3dmagnetism which appears quite remarkable since CeCrGe 3displays heavy\nfermion behavior even in the magnetically ordered state.\nPACS numbers: 71.27.+a, 75.20.Hr, 76.30.-v\nzPresent address: Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, United\nKingdom\nxPresent address: Laboratory for Muon Spin Spectroscopy, Paul Scherrer Institut, CH-5232 Villigen\nPSI, SwitzerlandarXiv:2109.09341v1  [cond-mat.str-el]  20 Sep 2021ESR of the itinerant ferromagnets RECrGe 3 2\n1. Introduction\nThe interplay between local 4 fmagnetism and itinerant 3 dferromagnetism is an\nimportant topic in heavy fermion systems and superconducting pnictide systems\n[1, 2]. This is also relevant for systems such as CeCrGe 3which show 4 fKondo\nlattice and heavy fermion behavior in the presence of ferromagnetic order of\nitinerant 3dmoments [3, 4]. A comparison of CeCrGe 3with LaCrGe 3revealed a\nremarkably large Sommerfeld coe\u000ecient [3] which characterizes the speci\fc heat in\nthe magnetically ordered state. The roles of spin \ructuations and quantum criticality\nin itinerant ferromagnetic systems were investigated in the systems LaV xCr1\u0000xGe3\nand CeCr 1\u0000xTixGe3where a strong suppression of ferromagnetic order was observed\n[5, 6]. Recently, LaCrGe 3has drawn considerable attention in the search for a pressure\ndriven ferromagnetic quantum critical point [7, 8, 9]. For the investigation of the spin\ndynamics of metals with strong ferromagnetic correlations the spectroscopic method\nof Electron Spin Resonance (ESR) provided important results in 3 dmetals like TiBe 2\n[10], ZrZn 2[11, 12], NbFe 2[13] as well as 4 fheavy fermion metals like YbRh 2Si2[14]\nand CeRuPO [15, 16]. ESR is probing the magnetic ion locally and directly in contrast\nto magnetization measurements which probe the bulk of the sample. Here we report\nthe Cr3+ESR in LaCrGe 3, CeCrGe 3, and PrCrGe 3in which the magnetism is based on\ndi\u000berent interactions between 3 dand 4felectrons, namely pure itinerant 3 dmagnetism\n(LaCrGe 3, 4f0), heavy fermion ferromagnetism with 3 d{4fhybridization (CeCrGe 3,\n4f1), and itinerant 3 d/ local 4fmagnetism (PrCrGe 3, 4f2). Distinct di\u000berences in\nthe ESR properties of these compounds are mainly found in the ferromagnetically\nordered state.\n2. Experimental Details\nPolycrystalline samples of RECrGe 3with RE = La, Ce and Pr were prepared and\ncharacterized as described previously [3]. All magnetic properties were measured using\na Quantum Design superconducting quantum interference device - vibrating-sample\nmagnetometer (QD SQUID VSM). Speci\fc heat and electrical four-point resistivity\nwere measured in a commercial QD Physical Property Measurement System (PPMS)\nequipped with a3He option. The ESR experiments were carried out using a standard\ncontinuous wave spectrometer at X-band frequencies ( \u0017= 9:4 GHz). The temperature\nwas varied between 4 and 295 K with a He-\row cryostat.\nESR probes the absorbed power Pof a transversal magnetic microwave \feld as\na function of a static and external magnetic \feld \u00160H[17]. To improve the signal-to-\nnoise ratio, we used a lock-in technique by modulating the static \feld, which yields\nthe derivative of the resonance signal dP=dH . The measured ESR spectra were \ftted\nwith a Lorentzian function including the in\ruence of the counter-rotating component\nof the linearly polarized microwave \feld [13]. From the \ft, we obtained the resonance\n\feldHres(which determines the ESR g-factorg=h\u0017=\u0016 BHres), and the linewidth\n\u0001H(half-width at half maximum). In metals, \u0001 His a direct measure of the spin\nlattice relaxation time 1 =T1and its temperature and frequency dependences reveal\nthe nature of the participating relaxation mechanisms.ESR of the itinerant ferromagnets RECrGe 3 3\n02040(a)P\nrR ECrGe3/s109\n0H = 0.32 T/s99 (10-6 m3 \nmol-1)L\naRE = Ce0\n24( b)T\n PrC\n(4f )T\n PrC\n(Cr)T LaC\nT CeC\n0  T/s114 (10-6 Ωm)0\n5 01 001 5001002\n040(c)0\n TC\n = 0C (J mol-1 \nK-1)T\n (K)C = 0\nFigure 1. Temperature dependence of (a) magnetic susceptibility \u001f(T), (b)\nelectrical resistivity \u001a(T), and (c) speci\fc heat C(T) for LaCrGe 3, CeCrGe 3\nand PrCrGe 3. At high temperatures ferromagnetic transitions are indicated by\nTLa;Ce;Pr\nCwhich are de\fned as the point of in\rection of \u001f(T). AtTPr\nC(4f)\u001916 K\nthe Pr-compound undergoes another transition. Both \u001a(T) andC(T) show\ndistinct anomalies at the respective ordering temperatures, too. The C(T) data\nof CeCrGe 3and LaCrGe 3are shifted by 20 and 40 J mol\u00001K\u00001, respectively.\n11 01 000.1110T\n Prc\n(Cr)T\n Prc\n(4f )/s1090H = 0 TPrCrGe3C /T (J mol-1 \nK-2)T\n (K)γ + αnT -3\nFigure 2. Temperature dependence of C=T of PrCrGe 3. AtT < 2 K the main\ncontribution is the high temperature tail of the nuclear Schottky peak. The solid\nline is a \ft with Cn=T=\r+\u000bnT\u00003.ESR of the itinerant ferromagnets RECrGe 3 4\n3. Experimental Results\n3.1. Magnetic, transport, and thermal properties\nMeasurements of the electrical transport and thermodynamic properties were carried\nout for PrCrGe 3as was done previously for LaCrGe 3and CeCrGe 3in Ref. [3].\nFurthermore, we performed DC-susceptibility measurements on all RECrGe 3(see\n\fgure 1a) in a \feld of 0.32 T to get the magnetic response close to the ESR resonance\n\feldHres(see \fgure 5). The magnetic properties of LaCrGe 3and CeCrGe 3are\nconsistent with earlier reports [3, 18, 19].\nTable 1. Summary of magnetic susceptibility data analysis of RECrGe 3.\nRE\u0016e\u000b;total\u0016e\u000b(RE)\u0016e\u000b(Cr)T\u001f\nC(Cr)T\u001f\nC(4f) \u0002\n(\u0016B) (\u0016B) (\u0016B) (K) (K) (K)\nLa 2.44 0 2.44 90 - 104\nCe 3.38 2.54 2.2 70 - 72\nPr 4.10 3.58 2.0 102 16 105\nInterestingly, PrCrGe 3exhibits twoanomalies, one at \u0018102 K due to the ordering\nof itinerant Cr- moments, and another at \u001816 K suggesting a low-temperature\ntransition due to the ordering of Pr3+moments. From linear \ftting the inverse\nsusceptibility data with a Curie-Weiss law ( \u001f\u00001= (T\u0000\u0002)=CCW) for the temperature\nrange between 200 K and 400K which is well above the ordering temperature TC\n(de\fned as the in\rection point of \u001f(T)) we obtained the e\u000bective moments ( \u0016e\u000b) and\nWeiss temperatures (\u0002), see summarized values in table 1. The Cr contribution in\nthe e\u000bective moment \u0016e\u000b(Cr) for CeCrGe 3and PrCrGe 3can be estimated assuming\nstable 3+ valence states for the rare earth ions. A similar approach was discussed\npreviously [3, 6]. The resulting moments \u0016e\u000b(Cr) summarized in table 1 are less than\nexpected for free Cr3+ions (3:8\u0016B). These reduced moments hint to an itinerant\ncharacter of the Cr magnetism in all measured RECrGe 3compounds.\nPositive and large values of \u0002 for all cases are indicative of a strong ferromagnetic\nexchange interaction in these systems. The electrical resistivity data, \u001a(T), in \fgure 1b\nexhibit a sharp drop below the magnetic transition temperatures due to the reduction\nof spin disorder resistivity. Furthermore, while LaCrGe 3and PrCrGe 3show metallic\nbehavior,\u001a(T) for CeCrGe 3is similar to that of heavy fermion materials [3]. The\nresistivity anomaly at 102 K in PrCrGe 3is due to magnetic order of Cr-moments\nand the anomaly near 16K is likely due to the ordering of Pr3+moments. The kink\ntemperatures in \u001a(T) are consistent with the TCvalues determined from \u001f(T) (see\n\fgures 1a and b).\nSpeci\fc heat C(T) data are presented in \fgures 1c and 2. The bulk nature\nof the magnetic transitions are con\frmed by pronounced anomalies in C(T) near\nTC. CeCrGe 3exhibits a signi\fcant enhancement of the Sommerfeld coe\u000ecient\n\r= 130 mJ mol\u00001K\u00002(renormalization of \u001845) at low temperatures, making it\na moderate heavy fermion system [3, 6]. PrCrGe 3, on the other hand, exhibits a\nsmaller\rof about 66 mJ mol\u00001K\u00002but shows a large nuclear speci\fc heat contribution\n(\u000bn) at low temperatures. As shown in \fgure 2, below 1 K, C(T)=Tfor PrCrGe 3\nincreases dramatically due to the nuclear Schottky e\u000bect which is caused by the\nstrong interaction between the nuclear magnetic moments with the strong magneticESR of the itinerant ferromagnets RECrGe 3 5\n\feld produced by the 4 felectrons at the nuclear site resulting in the splitting of the\nnuclear hyper\fne levels [20, 21, 22]. Low temperature C(T < 2 K) data of PrCrGe 3\nwere \ftted with C=T =\r+\u000bnT\u00003:Remarkably, \u000bnamounts to\u0018454 mJ K mol\u00001\nwhich is signi\fcantly higher than \u000bnof pure Pr metal ( \u001856 mJ K mol\u00001) at zero\napplied \feld [23] suggesting the presence of a strong hyper\fne interaction in PrCrGe 3.\n3.2. Electron Spin Resonance\nTypical ESR spectra at room temperature, in the paramagnetic state of LaCrGe 3,\nCeCrGe 3, and PrCrGe 3are shown in \fgure 3. For all compounds the spectra could\nbe well described by symmetric Lorentzian line shapes, indicating sample grain sizes\nbeing smaller than the microwave penetration depth. This resulted in the powder-\naveraged parameters g= 2:10\u00060:02 and\u00160\u0001H= 55\u00062 mT. The deviations from the\nLorentzian shapes are due to an anisotropy of the gfactor which, however, could not\nbe reliably resolved by \ftting with powder-averaged Lorentzians due to broad lines\nand broad background structures.\n0.20 .40 .6L\naCrGe3dP/dH (arb. units)C\neCrGe30\n.20 .40 .6T = 290 KPrCrGe3/s109\n0H (T)\nFigure 3. ESR spectra (symbols) at T= 290 K for LaCrGe 3, CeCrGe 3,\nand PrCrGe 3and Lorentzian shape (lines) with g= 2:10\u00060:02 and\u00160\u0001H=\n55\u00062 mT.\nThe obtained gfactor describes the S= 3=2 spins of Cr3+ions which are centered\nin octahedra which form face sharing chains in the hexagonal (P6 3=mmc space group)\ncrystal structure. However, for an insulating environment and in an octahedral \feld,\none expects for local Cr3+(Hund's rule4F3=2state) agvalue slightly below 2 for\nthe lowest lying level which is e\u000bectively an Sstate because of the quenched orbital\nmoment [17]. In order to explain the deviation from the observed gvalue, \u0001g, one\nneeds to take into account that the Cr 3 dstates are delocalized in narrow d-bands,\nleading to the band ferromagnetism in all three compounds [24]. Therefore, the\nresonance may be interpreted as a conduction electron spin resonance of conduction\nspins in a narrow band which is based almost entirely on Cr d-states. Then \u0001 gis\nan e\u000bect of spin-orbit coupling of the conduction electrons and depends on the band\nstructure [25].\nFigure 4 illustrates the temperature dependence of the derivative microwave\nabsorption for the three compounds. For temperatures above the ordering\ntemperatures the wavy features correspond to the paramagnetic resonance signal.\nInterestingly, in LaCrGe 3this feature is also visible in the ordered temperature regionESR of the itinerant ferromagnets RECrGe 3 6\nLaCrGe3dP/dH (arb. units)\nµ0H (T) T (K)0\n0.2\n0.4\n0.6\n0.8\n050100150200250dP/dH (arb. units) dP/dH (arb. units)\n0\n0.2\n0.4\n0.6\n0.8\n0501001502002500\n0.2\n0.4\n0.6\n0.8\n050100150200250\nCeCrGe3\nPrCrGe3\nFigure 4. Temperature dependence of the ESR spectra dP=dH (H) for LaCrGe 3,\nCeCrGe 3and PrCrGe 3.\nindicating a ferromagnetic resonance signal. Close to the magnetic phase transitions\nthe microwave absorption is dominated at low \felds ( <0:1 T) by pronounced\nnon-resonant structures re\recting a strongly \feld dependent magnetization. These\nstructures have been disregarded when \ftting the derivative microwave absorption\nwith a Lorentzian shape.\nThe results of a Lorentzian line \ftting are displayed in \fgure 5. The e\u000bect of\nferromagnetic ordering is to shift the line towards lower resonance \felds which is\nnicely seen for LaCrGe 3. For CeCrGe 3and PrCrGe 3the line becomes undetectable\ndue to extreme broadening and shift. The reason for this quite di\u000berent behavior to\nLaCrGe 3may be the much stronger magnetic anisotropy in CeCrGe 3and PrCrGe 3\nwhich arises from the 4 fstates. In the ordered state of LaCrGe 3demagnetization \felds\ncould be a relevant contribution to the observed resonance \feld. Assuming for the\ninvestigated polycrystalline powder spherically shaped samples with a demagnetizing\nfactor ofN= 1=3 we obtain demagnetization NM(T;\u0016 0H= 320 mT) values below\n0.1 mT in the entire temperature range. Thus, this \\sample-shape anisotropy\\ has a\ntiny e\u000bect, leaving anisotropy \felds as a major source for the observed line shift.ESR of the itinerant ferromagnets RECrGe 3 7\n0100200300(a)L aCrGe3 \n /s1090ΔH (mT)(b)C eCrGe3 \n (\nc)P rCrGe3 \n 0\n1 002 00200250300350(d)L aCrGe3T\n Lac\n/s1090Hres (mT)0\n1 002 00(e)C eCrGe3T\n Cec\nT\n (K)01 002 003 00(f)P rCrGe3T\n Prc\n(Cr) \nFigure 5. Linewidth \u00160\u0001Hand resonance \feld \u00160Hres.TCindicates the\nferromagnetic ordering temperature. TCwas determined by the point of in\rection\nin the\u001f(T) data.\n4. Discussion and Conclusion\nThe results of the ESR study on LaCrGe 3, CeCrGe 3and PrCrGe 3are similar in\nthe paramagnetic region but strongly depend on the presence of 4 felectrons for\ntemperatures close to and below the ferromagnetic ordering temperature TC, see \fgure\n5. The coupling between 4 fand 3dmagnetism obviously in\ruences the spin resonance\nof Cr 3delectrons. Therefore, the e\u000bect of 4 fmagnetism seems to be relevant at\nrelatively high temperatures although 4 fmagnetic ordering occurs well below the\nordering of the 3 dmoments. For example, in PrCrGe 3the Pr ordering is re\rected\nin an additional increase of the magnetic susceptibility below 30 K (see \fgure 1a)\nwhereas 3dmagnetic ordering is at TC= 102 K. Approaching TCin the paramagnetic\nregion, the linewidth increases due to critical \ructuations and a reduction of exchange\nnarrowing processes [26] whereas the resonance \feld decreases because internal \felds\nare built up towards TC. BelowTC, in PrCrGe 3and CeCrGe 3, the anisotropy inherent\nin the 4fmagnetism leads to additional broadening such that the 3 dresonance is\nunobservable below TC(see \fgures 5b, c, e and f).\nIn case of LaCrGe 3(4f0), the pure 3 dmagnetism determines the resonance\nproperties. Besides the clear signatures at ferromagnetic ordering there are also\nanomalies around 35 K and 125 K in the linewidth and resonance \feld (see \fgures 5a\nand d, respectively). Around 35 K, measurements of the thermoelectric power show a\nbroad hump which was related to the ordering of Cr moments [3]. The small maximum\nin the linewidth data around 125 K has no correspondence to other quantities and is\nthe result of a broad, unresolved background line superimposed on the main line.\nUpon decreasing the temperature below 125 K the main line starts to dominate and\nthe \ftted ESR parameters approach the main line properties.\nInterestingly, there is no clear di\u000berence in the temperature dependence of the\nESR parameters between PrCrGe 3and CeCrGe 3compounds. However, one wouldESR of the itinerant ferromagnets RECrGe 3 8\nexpect a di\u000berence since both compounds show a di\u000berent coupling among their 4 f-3d\nelectron systems. PrCrGe 3(4f2) has a stable, classic 4 fmoment [a non-magnetic\nsinglet is possible which, however, is not indicated in the magnetic susceptibility (see\nupturn in\u001f(T) below 30 K, \fgure 1a)]. CeCrGe 3(4f1) is a heavy fermion ferromagnet\nwhere 4fand conduction electron (3 d) states are hybridized leading to Kondo lattice\nbehavior of the electrical resistivity and an enhanced Sommerfeld coe\u000ecient of the\nspeci\fc heat [3]. The Kondo e\u000bect was shown to be essential for the observability of\na Yb3+-based heavy electron spin resonance in YbRh 2Si2[27, 28]. In this respect, for\ntemperatures above TC, it seems remarkable that the Cr 3 dresonance is very similar in\nPrCrGe 3and CeCrGe 3, regardless of the presence or absence of 4 fconduction-electron\nhybridization.\nAcknowledgements\nWe acknowledge valuable discussions with Christoph Geibel. ZH would like to\nacknowledge the Polish National Agency for Academic Exchange (NAWA) for ULAM\nfellowship.\nReferences\n[1] Krellner C, Burkhardt U and Geibel C 2009 Physica B 4043206\n[2] Sarkar R, Jesche A, Krellner C, Baenitz M, Geibel C, Mazumdar C and Poddar A 2010 Phys.\nRev. B 82054423\n[3] Das D, Gruner T, Pfau H, Paramanik U B, Burkhardt U, Geibel C and Hossain Z 2014 J. Phys.\nCondens. Matter 26106001\n[4] Das D, Nandi S, da Silva I, Adroja D T and Hossain Z 2016 Phys. Rev. B 94174415\n[5] Lin X, Taufour V, Bud'ko S L and Can\feld P C 2013 Phys. Rev. B 88(9) 094405\n[6] Das D, Bhattacharyya A, Anand V K, Hillier A D, Taylor J W, Gruner T, Geibel C, Adroja\nD T and Hossain Z 2015 J. Phys. Condens. Matter 27016004\n[7] Rana K, Kotegawa H, Ullah R R, Gati E, Bud'ko S L, Can\feld P C, Tou H, Taufour V and\nFurukawa Y 2021 Phys. Rev. B 103174426\n[8] Kaluarachchi U S, Bud'ko S L, Can\feld P C and Taufour V 2017 Nat. Commun. 8546\n[9] Taufour V, Kaluarachchi U S, Khasanov R, Nguyen M C, Guguchia Z, Biswas P K, Bonf\u0012 a P,\nDe Renzi R, Lin X, Kim S K, Mun E D, Kim H, Furukawa Y, Wang C Z, Ho K M, Bud'ko\nS L and Can\feld P C 2016 Phys. Rev. Lett. 117037207\n[10] Shaltiel D, Ioshpe D, Grayevsky A, Zevin V and Smith J L 1987 Phys. Rev. B 364090\n[11] Walsh W M, Knapp G S, L W Rupp J and Schmidt P H 1970 J. Appl. Phys. 411081\n[12] F orster T, Sichelschmidt J, Gr uner D, Brando M, Kimura N and Steglich F 2010 J. Phys.: Conf.\nSer.200012035\n[13] Rauch D, Kraken M, Litterst F J, S ullow S, Luetkens H, Brando M, F orster T, Sichelschmidt\nJ, Neubauer A, P\reiderer C, Duncan W J and Grosche F M 2015 Phys. Rev. B 91174404\n[14] Sichelschmidt J, Ivanshin V A, Ferstl J, Geibel C and Steglich F 2003 Phys. Rev. Lett. 91156401\n[15] Krellner C, F orster T, Jeevan H, Geibel C and Sichelschmidt J 2008 Phys. Rev. Lett. 100066401\n[16] F orster T, Sichelschmidt J, Krellner C, Geibel C and Steglich F 2010 J. Phys. Condens. Matter\n22435603\n[17] Abragam A and Bleaney B 1970 Electron Paramagnetic Resonance of Transition Ions (Oxford:\nClarendon Press)\n[18] Synoradzki K, Das D, Frackowiak A, Szymanski D, Skokowski P and Kaczorowski D 2019 J.\nAppl. Phys. 126075114\n[19] Yang X, Pan J, Liu S, Yang M, Cao L, Chu D and Sun K 2021 Phys. Rev. B 103104405\n[20] Grivei E, Bayot V, Piraux L and Issi J P 1995 Phys. Rev. B 511301\n[21] Lounasmaa O V and Guenther R A 1962 Phys. Rev. 1261357\n[22] Steppke A, Brando M, Oeschler N, Krellner C, Geibel C and Steglich F 2010 Phys. Status Solidi\nB247737\n[23] Pathak A K, Paudyal D, Mudryk Y, Gschneidner K A and Pecharsky V K 2013 Phys. Rev. Lett.\n110186405ESR of the itinerant ferromagnets RECrGe 3 9\n[24] Bie H, Zelinska O Y, Tkachuk A V and Mar A 2007 Chemistry of Materials 194613\n[25] D\u0013 ora B and Simon F 2009 Phys. Rev. Lett. 102137001\n[26] Anderson P W and Weiss P R 1953 Rev. Mod. Phys. 25269\n[27] Kochelaev B I, Belov S I, Skvortsova A M, Kutuzov A S, Sichelschmidt J, Wykho\u000b J, Geibel C\nand Steglich F 2009 Eur. Phys. J. B 72485{489\n[28] W ol\re P and Abrahams E 2009 Phys. Rev. B 80235112"    },    {        "title": "2103.05743v2.Self_induced_spin_orbit_torques_in_metallic_ferromagnets.pdf",        "content": "Self-induced spin-orbit torques in metallic ferromagnets\nH\u0013 ector Ochoa,1;2Ricardo Zarzuela,1;3and Yaroslav Tserkovnyak1\n1Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA\n2Department of Physics, Columbia University, New York, NY 10027, USA\n3Institut f ur Physik, Johannes Gutenberg Universit at Mainz, D-55099 Mainz, Germany\nWe present a phenomenological theory of spin-orbit torques in a metallic ferromagnet with spin-\nrelaxing boundaries. The model is rooted in the coupled di\u000busion of charge and spin in the bulk\nof the ferromagnet, where we account for the anomalous Hall e\u000bects as well as the anisotropic\nmagnetoresistance in the corresponding constitutive relations for both charge and spin sectors. The\ndi\u000busion equations are supplemented with suitable boundary conditions re\recting the spin-sink\ncapacity of the environment. In inversion-asymmetric heterostructures, the uncompensated spin\naccumulation exerts a dissipative torque on the order parameter, giving rise to a current-dependent\nlinewidth in the ferromagnetic resonance with a characteristic angular dependence. We compare\nour model to recent spin-torque ferromagnetic resonance measurements, illustrating how rich self-\ninduced spin-torque phenomenology can arise even in simple magnetic structures.\nI. INTRODUCTION\nSpin-transfer torques in magnetic devices, i.e., the\ntransfer of angular momentum leveraged by itinerant\nelectrons to the magnetization dynamics,1,2enable the\nelectrical control of the latter3{6and are of interest for\ndiverse technological applications. For example, spin-\ntransfer torques can compensate the action of damping\nforces, sustaining a large-angle precessional motion in\nmagnetic nano-oscillators,7{9a phenomenon of potential\ninterest in the \feld of neuromorphic computing.10Recent\nadvances on this front exploit the torques of relativistic\norigin generated at the interface between a magnet and\na heavy metal when charge \rows in the latter.11These\ntorques can be described in terms of nonequilibrium accu-\nmulations due to the interfacial Edelstein12and/or spin\nHall e\u000bects,13,14which rely on the lack of inversion sym-\nmetry imposed by the geometry of the device.\nSpin accumulations can also be generated by spin-\npolarized currents in metallic ferromagnets without the\nactive intervention of adjacent normal metals. In\nfact, the broken symmetries associated with the spon-\ntaneous magnetic ordering allow for more complex\nspin-current responses,15{18such as the anisotropic\nmagnetoresistance19(AMR, which includes the so-called\nplanar Hall e\u000bect20), leading to di\u000berent mechanisms\nof spin transfer.21{23In this article, we present a min-\nimal model for the spin-orbit torques generated by elec-\ntronic currents in a heterostructure consisting of a ferro-\nmagnetic metal (FM) sandwiched between spin-relaxing\nlayers, such as heavy normal metals (NM). Our theory\nis complementary to recent ab initio studies.24{26The\nmodel relies on a phenomenological description of the\n\rows of charge and longitudinal (to the magnetic or-\nder) spin, accompanied by suitable boundary conditions\nde\fned at the interfaces. We \fnd that, when the het-\nerostructure is inversion asymmetric, the uncompensated\nspin accumulation induced by a current density jcex-\nerts a damping-like torque (normalized by volume) onthe magnetization of the form\n\u001cd=\u0011~\n2e\u0000sL(^z\u0001n) (n\u0002^z\u0002n) (1)\n\u0002[#Hn\u0001(^z\u0002jc) +%MR(^z\u0001n) (jc\u0001n)]:\nHere,nis a unit vector along the collective spin density,\n\u0000eis the electron charge, Lis the thickness of the \flm,\nand \u0000sis a dimensionless number characterizing the spin\nrelaxation rate in the bulk of the ferromagnetic metal.\nThe dimensionless coe\u000ecients #Hand%MRare related to\nthe anomalous Hall and AMR e\u000bects, respectively, while\n\u0011(with units of the inverse volume) characterizes the\ntorque by the out-of-equilibrium longitudinal spins on\nthe order parameter at the interface. This spin torque\ncan be either direct or mediated by magnons,27leading\nto a characteristic temperature dependence in the latter\ncase.28The above expression has been derived under the\nassumption that the magnetization dynamics occur on\nlong timescales (longer than, e.g., the longitudinal spin-\n\rip time,\u001cs), so that the itinerant electrons respond to\na (quasi-)static magnetic background during their trans-\nport.\nThe torque (1) a\u000bects the linewidth of the ferromag-\nnetic resonance as electron charge \rows through the sys-\ntem. When the static component of the magnetization\nlies within the plane de\fned by the normal of the layered\nheterostructure and the charge current (the xzplane in\nFig. 1), the shift in the resonance linewidth follows\n\u0001B(\u0012)\u0019Axz\u0010\nsin 2\u0012+1\n2sin 4\u0012\u0011\n; (2)\nwhere\u0012is the polar angle of the magnetization relative\nto thezaxis andAxzis the single \ftting parameter pro-\nportional to %MR=L, see Eq. (14). If, on the other hand,\nthe order parameter lies within the plane perpendicular\nto the current and the heterostructure (the yzplane in\nFig. 1), the shift in the resonance linewidth reads\n\u0001B(\u0012)\u0019Ayz(sin\u0012+ sin 3\u0012); (3)\nwhere the prefactor Ayznow scales linearly with #H=L.\nThe linewidth shift vanishes for the magnetization withinarXiv:2103.05743v2  [cond-mat.mes-hall]  4 Jun 20212\nxz\nxz\ny\nFIG. 1: a) Schematic representation of the heterostructure\nunder consideration. b) Vertical pro\fle of the spin accumula-\ntion within the ferromagnetic \flm (of thickness L) when both\nnormal metals behave as bad spin sinks. c) The same when\none of the metals is a good spin sink. The uncompensated\nspin accumulation is localized close to the interface with the\nbad spin sink on a length scale set by the spin di\u000busion length\nin the ferromagnet, `s.\nthe \flm's ( xy) plane, according to the torque (1). We\nnote that these dependences for the resonance linewidth\nhave been derived under the simpli\fed assumption of cir-\ncular precession of the magnetization.\nThe manuscript is structured as follows: We present\nour model and derive the main results in Secs. II and III.\nIn Sec. IV, we analyze data from recent spin-torque fer-\nromagnetic resonance (ST-FMR) measurements22per-\nformed in di\u000berent NM/FM heterostructures. We con-\nclude by discussing and summarizing our \fndings in\nSec. V.\nII. PHENOMENOLOGICAL MODEL\nWe consider a two-dimensional stack of a ferromag-\nnetic conductor sandwiched between normal metals, see\nFig. 1(a). The relevant hydrodynamical variables are\nthe charge density, \u001ac, and the longitudinal spin den-\nsity,\u001as\u0011n\u0001\u001as. The conjugate thermodynamic forcesare\u0016c\u0011\u0000e\u000e\u001acFand\u0016s\u0011~\u000e\u001asF, whereFis the free\nenergy of the itinerant magnet. We assume that there\nare no slow variables related to transverse spin dynam-\nics, apart from a coherent Landau-Lifshitz-type preces-\nsion. In particular, any electronic spin dynamics relative\nto the collective order should relax very fast. In essence,\nwe are constructing a phenomenology in which the spin\ndegrees of freedom are coarse-grained down to the direc-\ntional spin-density variable nand its magnitude that is\nparametrized by \u001as(which, while generally consisting of\nboth the electronic and thermally-excited magnonic con-\ntributions, has our focus on the former). In the bulk of\nthe ferromagnetic metal, we have local conservation laws\nof the form\n@t\u001ac+r\u0001jc= 0; (4a)\n@t\u001as+r\u0001js=\u0000\u0000s\u0016s; (4b)\nwhere \u0000s=~\u0017F=2\u001cs. Here,\u001csand\u0017Fare, respectively,\nthe spin-relaxation time and density of states per volume\nat the Fermi level. These continuity equations must be\nsupplemented with constitutive relations of the form:\n\u0012jc\n2e\n~js\u0013\n=\u001b\u0012^\u001bc[n] ^\u001bx[n]\n^\u001bT\nx[\u0000n] ^\u001bs[n]\u0013\u00121\ner\u0016c\n\u00001\n2er\u0016s\u0013\n;(5)\nwhere the o\u000b-diagonal matrix elements are related by the\nOnsager reciprocal relations.\nIn a featureless, isotropic ferromagnet, the normalized\nconductivity tensors have the following general structure:\n[^\u001bc]ij=\u000eij+#\u000fijknk+%ninj; (6a)\n[^\u001bs]ij=\u000eij+#s\u000fijknk+%sninj; (6b)\n[^\u001bx]ij=P\u000eij+#x\u000fijknk+%xninj: (6c)\nHere,\u001bis the total conductivity, which, in the two-\nchannel phenomenology with little spin mixing, is given\nby\u001b\u0019\u001b\"+\u001b#, in terms of the conductivity \u001b\"(\u001b#)\nof the majority (minority) electrons. The dimensionless\nparameterP\u0019(\u001b\"\u0000\u001b#)=(\u001b\"+\u001b#) measures the spin\npolarization of the electrical current. The dimensionless\ncoe\u000ecients #and%parametrize the anomalous Hall and\nAMR e\u000bects, respectively. The coe\u000ecients #sand%s\nparametrize analogous e\u000bects in the spin sector, while\n#xand%xare associated with similar spin-charge cross\nterms. Microscopically, all these phenomenological con-\nstants depend on relativistic interactions and can typi-\ncally be assumed to be small: j#j\u0018j#s;xj;%j\u0018j%s;xj\u001c1.\nWe apply these equations to the device geometry de-\npicted in Fig. 1(a), with the charge \rowing in the xdi-\nrection, jc=j^x. For simplicity, we assume translational\ninvariance along y, and we focus on the charge and spin\naccumulations deep inside the ferromagnet, namely, far\naway from the leads.29For the asymmetric case, con-\nsisting of the top/bottom normal metals being bad/good\nspin sinks, the spin accumulation along the transverse3\ndirection reads (see Appendix A)\n\u0016s(z) =2eE`0\nssinh\u0010\nz\n`0s\u0011\nh\n1 +%sn2z\u0000(P+%xn2z)2\n1+%n2zi\ncosh\u0010\nL\n`0s\u0011 (7)\n\u0002\u0014\n#xny+%xnxnz\u0000P+%xn2\nz\n1 +%n2z(%nxnz+#ny)\u0015\n;\nwhose pro\fle is shown in Fig. 1(c). Here, z >0 is mea-\nsured from the good spin sink and `0\nsdenotes a magnetic-\norder dependent spin di\u000busion length, see Eq. (A4). The\nparameterEis associated with the voltage drop between\nthe leads. In our \fnal expression, we neglect a small\nmisalignment between the applied current and the elec-\ntric \feld associated with the Hall/MR e\u000bects, and write\nsimplyE\u0019j=\u001b.\nIII. TORQUE-INDUCED LINEWIDTH\nNext we focus on the absorption power in a usual ferro-\nmagnetic resonance (FMR) experiment. We assume here-\nafter the low-frequency regime for the ac \feld, namely\n!T1\u001c1; furthermore, we assume that the correspond-\ning wavelength is much larger than the size of the sample\nand, therefore, the itinerant ferromagnet exhibits a uni-\nform dynamic state. The dynamics of the order param-\neter is described by the Landau-Lifshitz-Gilbert (LLG)\nequation,30,31\ns(1 +\u000bn\u0002)_n=n\u0002He\u000b+\u001c; (8)\nwheresis the saturated spin density, \u000bdenotes the\nGilbert damping constant and He\u000b=\u0000\u000eF=\u000enis the\nthermodynamic force conjugate to the order parame-\nter. To simplify the analysis, we will disregard in what\nfollows anisotropy terms. Consequently, the magnetic\nfree energy contains nothing but the Zeeman energy,\nF=\u0000\rsn\u0001(B0+b), with\rbeing the gyromagnetic\nratio andB0,b(t) denoting the strong dc and weak ac\ncomponents of the magnetic \feld, respectively.\nWhen re\rection symmetry along the heterostructure\naxis ( ^z) is broken while retaining the axial ( C1v) sym-\nmetry, the most generic torques to the lowest order in\nthe spin accumulation can be written as27\n\u001c=\u00110\u0016s(^z\u0001n)n\u0002^z+\u0011\u0016s(^z\u0001n)n\u0002^z\u0002n:(9)\nHere\u0011,\u00110are phenomenological constants (with units\nof inverse of volume). One can imagine two possible\nmicroscopic mechanisms for these torques. In one sce-\nnario, the electronic spin accumulation exerts directly\na torque on the order parameter due to the inversion-\nsymmetry breaking-induced spin-orbit coupling at the\ninterface. Another possibility is an inelastic channel me-\ndiated by magnons: the electronic spin accumulation is\n\frst converted into a magnon chemical potential,28and\nthe magnon cloud subsequently exerts a torque on thecoherent spin dynamics.27The damping torque, second\nterm in Eq. (9), reduces to the expression in Eq. (1),\nwhere the prefactor comes from the average spin accu-\nmulation across the \flm thickness, \u0016 \u0016s=1\nLRL\n0dz\u0016s(z).\nThis is only di\u000berent from 0 for asymmetric heterostruc-\ntures; from Eq. (7), to the leading order in relativistic\ne\u000bects (i.e., assuming L\u001d`s), we have\n\u0016\u0016s\u00192e`2\ns\n\u001bL\u0002\n#Hn\u0001(^z\u0002jc) +%MR(^z\u0001n)(jc\u0001n)\u0003\n;(10)\nwhere we have introduced #H=#x\u0000P#and%MR=\n%x\u0000P%. Note that the prefactor can be conveniently\nwritten as\n2e`2\ns\n\u001bL=\u001cs\ne\u0017FL=~\n2e\u0000sL; (11)\nyielding the prefactor in Eq. (1).\nIn the measurements discussed later in Sec. IV, the\nstatic magnetic \feld was in the ballpark of 0.1 T,22which\ntranslates into a Larmor frequency of !L=\rB0'10\nGHz. Just like in Ref. 22, we assume for simplicity that in\nequilibrium the order parameter follows the static compo-\nnent of the magnetic \feld, n0/B0. Whenb(t) =be\u0000i!t\nis switched on, the order parameter acquires a small\ntransverse ac component, n(t) =n0+\u0010(t). The ab-\nsorption power (averaged over an oscillation period) is\nproportional to the imaginary part of the transverse com-\nponent of the susceptibility tensor, which can be obtained\nfrom the solution of the linearized LLG equation for \u0010,\n\u001ft(!) =\r(!L\u0000i\u000b!)\n(!L\u0000i\u000b!)2\u0000\u0002\n!+i\u0011\u0016\u0016s\ns(^z\u0001n0)2\u00032:(12)\nClose to the resonant frequency, !\u0019!L, the absorption\npower goes as\nP(!)/!Im\u001ft(!)\u0019\r!\n2\u0000\n(!\u0000!L)2+ \u00002; (13a)\nwith the resonance linewidth as a function of n0and jc\ngiven by\n\u0000 =\u000b!L+\u0011~\n2es\u0000sL(^z\u0001n0)2[#Hn0\u0001(^z\u0002jc) (13b)\n+%MR(^z\u0001n0) (jc\u0001n0)]:\nIV. COMPARISON TO ST-FMR DATA\nThe current-induced shift in the resonance linewidth\ncorresponds roughly to \u0001 B\u0018\u0000=\rafter subtracting the\nGilbert damping contribution,\n\u0001B\njc\u0019\u0011~\n8e\rs\u0000sLh\n#H(sin\u0012+ sin 3\u0012) sin\u001e (14)\n+%MR\u0000\nsin 2\u0012+1\n2sin 4\u0012\u0001\ncos\u001ei\n;4\nwhere\u001eand\u0012denote the azimuthal and polar angles of\nthe magnetization measured with respect to the direction\nof the current and ^z, respectively.\nFigure 2 depicts the experimental values of the reso-\nnance linewidth shifts reported in Ref. 22 for three asym-\nmetric nanostrips Ta/NM/FM/Ta, where FM denotes a\n[Co/Ni] 2/Co magnetic superlattice and NM is either Au\n[panel (a)], Pd [panel (b)] or Pt [panel (c)]. The corre-\nsponding measurements were taken at room temperature\nand FM layers of thickness L= 5:11 nm were deposited\nduring the fabrication of the nanostrips. In the same\nexperiment, the current-induced shift was dramatically\nreduced for symmetric heterostructures.\nThe data points in Fig. 2 corresponds to measurements\nwith the static magnetic \feld lying within the plane de-\n\fned by ^zand the current ( \u001e= 0), as depicted in the\ngeometry of Fig. 1. Red curves in Fig. 2 correspond to the\nsecond line of Eq. (14) with the overall factor \u0011~=8e\rs\u0000sL\nand%MRcombined in a single \ftting parameter, Axz.\nThe formula reproduces well the angular dependence ob-\nserved in the experiments for Au and Pd. In particu-\nlar, our model captures the extra beating /sin 4\u0012ob-\nserved in the data, which goes beyond the behavior that\nmay be naively expected for a magnetoresistance com-\npatible with the reduced symmetry of the heterostruc-\nture, ( ^z\u0001n)(jc\u0001n)/sin 2\u0012. For Pt, our model does\nnot yield the correct angular dependence for the reso-\nnance linewidth (neither does the \ftting of the formula\n/sin 2\u0012showed in dashed line), which suggests that the\nstrong spin-orbit interaction in Pt modi\fes physics be-\nyond our simple spin-sink boundary condition.\nThe data for the case where the static \feld lies within\nthe plane of the heterostructure ( \u0012=\u0019=2) con\frms this\nscenario (see Fig. 3a in Ref. 22). Our model predicts no\nshift, which is indeed the case for Au within the experi-\nmental error, while for Pt heterostructures there is a siz-\nable shift resulting from a more conventional mechanism\nrooted in the spin Hall e\u000bect (see Appendix B) originat-\ning in the heavy metal,5which appears to be the largest\ncontribution for the totality of the angular dependence\n(/sin\u001ein this con\fguration, with some corrections).\nV. DISCUSSION\nOne important property of our model is that it repro-\nduces well the extra beating in the angular dependence\nof the linewidth shifts of Au and Pd heterostructures.\nSpeci\fcally, the model \fxes the relative strength of the\nsin 2\u0012and sin 4\u0012components in the signal, with only one\nglobal \ftting parameter measuring the overall strength\nof the current-induced shift. The similar values of Axz\nfor both nanostrips ( Axz'1:5\u000110\u00003Oe m/A for Au\nandAxz'1:3\u000110\u00003Oe m/A for Pd) agrees well with\nthe basic ingredient of the model, namely, that the ex-\nact nature of the normal metals play a secondary role\nbeyond de\fning the boundary conditions for the coupled\nspin-charge di\u000busion in the ferromagnetic metal. The in-\n(a)   Au\n(b)   Pd\n(c)   Pt/uni03B8/uni03B8 Oe m/A)  Oe m/A)  (10−3 Oe m/A)  (10−3\n/uni03B8 (10−3\nFIG. 2: Values of the resonance linewidth shift for the\nlowest-frequency mode measured in Ta/Au/FM/Ta [panel\n(a)], Ta/Pd/FM/Ta [panel (b)] and Ta/Pt/FM/Ta [panel (c)]\nat room temperature, extracted from Ref. 22. Red lines rep-\nresent the \ftting of the second line of Eq. (14) to the data\nwithAxz=\u0011~%MR=8e\rs\u0000sL. We obtainAxz'1:5\u000110\u00003\nOe m/A for Au and Axz'1:3\u000110\u00003Oe m/A for Pd, with\ncoe\u000ecients of determination R2\nAu= 0:93 andR2\nPd= 0:95, re-\nspectively. For Pt in panel (c) we obtain Axz'0:74\u000110\u00003\nOe m/A (with R2\nPt= 0:68). The blue dashed line represents\nthe \ftting of sin 2 \u0012to the data, with R2= 0:88 in that case\n(the same \ft to Au and Pd data yields similar values of R2).\nterface with Pt, however, seems to play a more active role\nin the magnetization dynamics. The same trend is con-\n\frmed by the data in the xyplane of Ref. 22. Our model\npredict a nil shift, compatible with the data in Au het-\nerostructures. There is, however, a sizable shift coming\nfrom the adjacent Pt \flm and also, to a smaller extent,\nin the Pd case. We conclude that the spin-orbit e\u000bects in\nPt give rise to the more conventional external torques,11\nwhile Au is dominated by our internal mechanism. Pd5\n(which is electronically similar to Pt, but with a weaker\nspin-orbit interaction) seems to be somewhat intermedi-\nate and displays both mechanisms, with a stronger self-\ninduced torque, as suggested by the good \ft to the xz\nplane data shown in Fig. 2b.\nThe previous analysis and, in particular, the disagree-\nment between our model and the experimental data for\nPt makes clear that the present theory is not the most\ngeneral one. Thus, it is worth discussing our results in\nthe context of a more general phenomenology guided by\nsymmetry, in the spirit of Ref. 32. In order to make\ncontact with our model, in the following construction,\nwe incorporate the separation of time scales between the\ndynamics of the order parameter and the itinerant de-\ngrees of freedom, by considering only symmetry-allowed\ninterfacial torques up to linear order in the current den-\nsity jc. For simplicity, we discuss only asymmetric het-\nerostructures with the principal axis oriented along ^z. As\nin Sec. II, the ferromagnets are assumed to be isotropic,\nwhile the presence of normal metals reduces the sym-\nmetry down to C1v(for the subsequent notation, see\nRef. 33).\nTorques must be orthogonal to the magnetic order\nn, yielding two possibilities at the interface, up to a\n(pseudo)scalar prefactor:\n^z\u0002n; (15a)\nn\u0002^z\u0002n: (15b)\nThe torques must also transform as n, namely, the zcom-\nponent must be a pseudoscalar ( A2representation), and\nthe rest of the components form a vector ( E1represen-\ntation). The two candidates in Eqs. (15) behave accord-\ningly only if multiplied by a pseudoscalar (e.g., nz, which\nleads to Eq. 9). Here, we consider all the possible pseu-\ndoscalars up to linear order in the current density. De-\ncomposing jcin its collinear and orthogonal components\nto the projection of n, we have again two possibilities:\nn\u0001jc; (16a)\n(^z\u0001n) [n\u0001(^z\u0002jc)]: (16b)\nThe combination of these two pseudoscalars with the\ntwo vectors in Eq. (15) generate four groups of mag-\nnetic torques.33In particular, the Hall torque, \frst term\nin Eq. (1), follows directly by combining Eqs. (15b) and\n(16b). Additional ST-FMR measurements with the static\ncomponent of the \feld within the plane perpendicular to\nthe current ( \u001e=\u0019=2 in our expressions, yzplane in\nFig. 1) would directly test this Hall contribution, as de-\nscribed by Eq. (3) in our model.\nIn general, we should include higher powers in ^z\u0001n\n(only even powers are allowed by symmetry) weighted\nby di\u000berent phenomenological constants; for example,\n(n\u0001jc)\u0002\nA+B(^z\u0001n)2+:::\u0003\nn\u0002^z\u0002n: (17)\nThe magnetoresistance torque, second term in Eq. (1),\ncorresponds to the case of A= 0, which is speci\fc to ourtransport/torque model. For a more general expansion as\nin Ref. 32, we should allow also for geometrical factors of\nthe form 1=[1\u0000(^z\u0001n)2], as is the case, for example, for the\nusual spin Hall torque generated by spin accumulation in\nan adjacent normal metal, see Appendix B.\nIn conclusion, we have presented a theory for self-\ninduced torques in NM1/FM/NM2 heterostructures.\nThe model relies on the separation of time scales be-\ntween the magnetization and electron dynamics. The\nlatter is described by di\u000busion equations for the charge\nand longitudinal spin coupled via constitutive relations\nthat include the anomalous Hall and AMR e\u000bects in the\nbulk of the ferromagnet generated by the static compo-\nnent of the magnetization. Both e\u000bects produce steady-\nstate spin accumulations at the interfaces with the nor-\nmal metals which, in turn, exert a net torque on the\norder parameter if uncompensated. The damping-like\ninterfacial torques are manifested through characteristic\nmodel-speci\fc beatings in the angular dependence of the\nST-FMR resonance linewidths. Other signatures are the\ndependence on the thickness of the heterostructure or the\ntemperature dependence contained in \u001csand the coupling\n\u0011.\nAcknowledgments\nThe authors are grateful to Eric Montoya and Ilya\nKrivorotov for sharing their data before publication and\nbringing this problem to our attention. This work was\nsupported by the U.S. Department of Energy, O\u000ece of\nBasic Energy Sciences under Award No. DE-SC0012190.\nAppendix A: Spin accumulation in the steady state\nTranslational invariance along yyields that the elec-\ntrochemical potential \u0016cand the spin accumulation \u0016s\nto be functions of the coordinates xandzonly. Since\nwe have restricted ourselves to a spatial region of the\nmagnet far way from the leads, we also disregard the\ndependence of \u0016son the coordinate xand approximate\n\u0016c(x;z)\u0019eEx+ ~\u0016c(z). Supposing that the noncollinear-\nity between the applied current and the induced electric\n\feld is small, we neglect it when writing Eq. (10), and\napproximate E\u0019j=\u001b.\nThe spin and charge accumulations along zare related\nby the condition ^z\u0001jc= 0, that is\n\u0000\n1 +%n2\nz\u0001\n@z~\u0016c+ (#ny+%nxnz)eE=P+%xn2\nz\n2@z\u0016s:\n(A1)\nBy using this last relation, the spin continuity can be\nrecast as\n\"\n1 +%sn2\nz\u0000\u0000\nP+%xn2\nz\u00012\n1 +%n2z#\n@2\nz\u0016s=\u0016s\n`2s; (A2)6\nwhere`s\u0011p\n~\u001b=4e2\u0000sis the spin di\u000busion length. The\nsolution to this equation reads in general\n\u0016s(z) =\u0016be\u0000z\n`0s+\u0016tez\u0000L\n`0s; (A3)\nwith\n`0\ns=`ss\n1 +%sn2z\u0000(P+%xn2z)2\n1 +%n2z: (A4)\nThe coe\u000ecients \u0016b;tcan be inferred from the boundary\nconditions on the spin \row across the bottom/top inter-\nfaces, placed at z= 0 andz=L, respectively.\na. Both normal metals are poor spin sinks\nIn that case the condition ^z\u0001js= 0 holds at both\ninterfaces. The spin accumulation along the transverse\ndirection reads then\n\u0016s(z) =2eE`0\nssinh\u0010\nz\u0000L=2\n`0s\u0011\nh\n1 +%sn2z\u0000(P+%xn2z)2\n1+%n2zi\ncosh\u0010\nL\n2`0s\u0011 (A5)\n\u0002\u0014\n#xny+%xnxnz\u0000P+%xn2\nz\n1 +%n2z(%nxnz+#ny)\u0015\n;\nwhose pro\fle is depicted in Fig. 1(b).b. Good/poor spin sinks at the bottom/top interfaces\nNow we consider the asymmetric case where the top\n(bottom) normal metal is, as before, a bad (good) spin\nsink; we have then the identity \u0016s(z= 0) = 0, i.e., there\nis no spin accumulation at the interface with the good\nspin sink. The spin accumulation is given in this case by\nEq. (7).\nAppendix B: Spin Hall torque\nAn asymmetric heterostructure should generally yield\nalso the spin Hall torque, which can be generated by a\nheavy normal metal such as Pt:11\n\u001csH= (\u0011sH+#sHn\u0002) (^z\u0002jc)\u0002n: (B1)\nHere\u0011sHis a phenomenological parameter associated, for\nexample, with the Edelstein e\u000bect12and parametrizing\nthe reactive component of the torque; #sHis the spin-Hall\nangle characterizing the dissipative counterpart. Here we\nshow that this torque can be expressed in terms of the\nbasis introduced in Eqs. (15) following the prescription\ndiscussed in the main text.\nFirst observe that we can write ( ^z\u0002jc)\u0002n= (^z\u0001\nn)jc\u0000(n\u0001jc)^z. By decomposing jcin its longitudinal\nand transverse components to n, we have then\n(^z\u0002jc)\u0002n=\u0000(n\u0001jc) (n\u0002^z\u0002n) + ( ^z\u0001n) (n\u0002jc\u0002n): (B2)\nThe \frst term is already of the desired form, namely, a combination of vector (15b) and pseudoscalar (16a). For the\nsecond term, the transverse component of the current can be expanded in the basis of Eqs. (15); speci\fcally, we have\nn\u0002jc\u0002n=n\u0001(^z\u0002jc)\n(^z\u0001n)2\u00001^z\u0002n+(^z\u0001n)(n\u0001jc)\n(^z\u0001n)2\u00001n\u0002^z\u0002n: (B3)\nRegrouping all these terms, we can \fnally write\n(^z\u0002jc)\u0002n=1\n(^z\u0001n)2\u00001[(n\u0001jc) (n\u0002^z\u0002n) + ( ^z\u0001n) [n\u0001(^z\u0002jc)] (^z\u0002n)]: (B4)\nThe terms between brackets are of the desired form: a\ncombination of vector (15b) and pseudoscalar (16a) (\frst\nterm) and the same vector with pseudoscalar (16b) (sec-\nond term). The overall prefactor is a scalar that seems\nto result in a divergence as napproaches ^z. This is just\nan artifact of the basis choice in Eqs. (15), which is ill-\nde\fned in that limit. Note, however, that the overall\nexpression in the right-hand side of this last equation issmooth and gives the correct limit ( ^z\u0002jc)\u0002n!jcwhen\nn!^z.\nFinally, from Eq. (B3), it also directly follows that the\ntwo other basic spin-orbit torques, ( ^z\u0001n)jc\u0002nand\n(^z\u0001n)n\u0002jc\u0002n(which, together with torques (B1),\nconstitute the full basis32) can be similarly expanded in\nterms of Eqs. (15) and (16).\n1L. Berger, J. Appl. Phys. 49, 2156 (1978); Phys. Rev. B\n54, 9353 (1996).2J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996).7\n3K. Ando, S. Takahashi, K. Harii, K. Sasage, J. Ieda, S.\nMaekawa, and E. Saitoh, Phys. Rev. Lett. 101, 036601\n(2008).\n4I. M. Miron, K. Garello, G. Gaudin, P.-J. Zermatten, M. V.\nCostache, S. Au\u000bret, S. Bandiera, B. Rodmacq, A. Schuhl,\nand P. Gambardella, Nature 476, 189 (2011).\n5L. Liu, T. Moriyama, D. C. Ralph, and R. A. Buhrman,\nPhys. Rev. Lett. 106, 036601 (2011).\n6L. Liu, C.-F. Pai, Y. Li, H. W. Tseng, D. C. Ralph, R. A.\nBuhrman, Science 336, 555 (2012).\n7L. Liu, C.-F. Pai, D. C. Ralph, and R. A. Buhrman, Phys.\nRev. Lett. 109, 186602 (2012).\n8V. E. Demidov, S. Urazhdin, H. Ulrichs, V. Tiberkevich,\nA. Slavin, D. Baither, G. Schmitz, and S. O. Demokritov,\nNat. Mater. 11, 1028 (2012).\n9Z. Duan, A. Smith, L. Yang, B. Youngblood, J. Lindner,\nV. E. Demidov, S. O. Demokritov, and I. N. Krivorotov,\nNat. Commun. 5, 5616 (2014).\n10J. Torrejon, M. Riou, F. A. Araujo, S. Tsunegi, G. Khalsa,\nD. Querlioz, P. Bortolotti, V. Cros, K. Yakushiji, A.\nFukushima, H. Kubota, S. Yuasa, M. D. Stiles, and J.\nGrollier, Nature 547, 428 (2017).\n11Y. Tserkovnyak and S. A. Bender, Phys. Rev. B 90, 014428\n(2014).\n12V. M. Edelstein, J. Phys.: Condens. Matter 7, 1 (1995).\n13M. I. D'yakonov, V. I. Perel', J. Exp. Theor. Phys. Lett.\n13, 467 (1971).\n14J. E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999).\n15T. Taniguchi, J. Grollier, and M. D. Stiles, Phys. Rev.\nAppl. 3, 044001 (2015).\n16C. Ortiz Pauyac, M. Chshiev, A. Manchon, and S. A. Niko-\nlaev, Phys. Rev. Lett. 120, 176802 (2018).\n17K.-W. Kim and K.-J. Lee, Phys. Rev. Lett. 125, 207205\n(2020).\n18A. Davidson, V. P. Amin, W. S. Aljuaid, P. M. Haney, andX. Fan, Phys. Lett. A 384, 126228 (2020).\n19H. J. Juretschke, J. Appl. Phys. 31, 1401 (1960).\n20S. Kokado, M. Tsunoda, K. Harigaya, and A. Sakuma, J.\nPhys. Soc. Japan 81, 024705 (2012).\n21A. Ghosh, K. Garello, C. O. Avci, M. Gabureac, and P.\nGambardella, Phys. Rev. Applied 7, 014004 (2017).\n22C. Safranski, E. A. Montoya, and I. N. Krivorotov, Nat.\nNanotech. 14, 27 (2019).\n23M. Haidar, A. A. Awad, M. Dvornik, R. Khymyn, A.\nHoushang, and J. Akerman, Nat. Comm. 10, 2362 (2019).\n24V. P. Amin, Junwen Li, M. D. Stiles, and P. M. Haney,\nPhys. Rev. B 99, 220405(R) (2019).\n25K. D. Belashchenko, A. A. Kovalev, and M. van Schilf-\ngaarde, Phys. Rev. Mater. 3, 011401(R) (2019).\n26D. Go, F. Freimuth, J.-P. Hanke, F. Xue, O. Gomonay,\nK.-J. Lee, S. Bl ugel, P. M. Haney, H.-W. Lee, and Y.\nMokrousov, Phys. Rev. Res. 2, 033401 (2020).\n27B. Flebus, P. Upadhyaya, R. A. Duine, and Y.\nTserkovnyak, Phys. Rev. B 94, 214428 (2016).\n28E. G. Tveten, A. Brataas, and Y. Tserkovnyak, Phys. Rev.\nB92, 180412(R) (2015).\n29Speci\fcally, at distances much longer than the spin di\u000bu-\nsion length, `s.\n30L. D. Landau and E. M. Lifshitz, Phys. Zeitsch. der Sow.\n8, 153 (1935); L. D. Landau, E. M. Lifshitz, and L. P.\nPitaevskii, Statistical Physics, Part 2 , 3rd ed. (Pergamon,\nOxford, 1980).\n31T. L. Gilbert, Phys. Rev. 100, 1243 (1955); IEEE Trans.\nMagn. 40, 3443 (2004).\n32K. Garello, I. M. Miron, C. O. Avci, F. Freimuth, Y.\nMokrousov, S. Bl ugel, S. Au\u000bret, O. Boulle, G. Gaudin,\nand P. Gambardella, Nat. Nanotech. 8, 587 (2013).\n33M. Tinkham, Group Theory and Quantum Mechanics\n(Dover Publications, Mineola, NY, 2003)."    },    {        "title": "1705.09423v1.Static_and_Dynamic_Magnetic_Properties_of_FeMn_Pt_Multilayers.pdf",        "content": "1\n \n \nStatic and \nD\nynamic \nMa\ngnetic \nP\nroperties of FeMn/Pt \nM\nultilayers\n \n \nZiyan \nLuo\n1\n, Y\numeng\n \nYang\n1\n, Y\nanjun\n \nXu\n1,2\n,\n \nMengzhen\n \nZhang\n \n3\n, \nBaoxi Xu\n2\n, \nJingsheng \nChen\n3\n, \nand Y\nihong\n \nWu\n1\n,\n*\n \n1\nDepartment of Electrical and Computer Engineering, National University of Singapore, 4 \nEn\ngineering Drive 3, Singapore 117583, Singapore\n \n2\nData Storage Institute, A*STAR (Agency for Science, Technology and Research), 2 Fusionopolis \nWay, 08\n-\n01 Innovis, Singapore 138634, Singapore\n \n3\nDepartment of Materials Science and Engineering, National Universi\nty of Singapore, 117575\n \nRecently we \nhave \ndemonstrated the presence of spin\n-\norbit toque in \nFeMn/P\nt\n \nmultilayer\ns\n \nwhich\n, in \ncombination with \nthe \nanisotropy field, is able to rotate \nits\n \nmagnetization \nconsecutively from 0\no\n \nto\n \n360\no\n \nwithout any external field. Her\ne, we report on \nan \ninvestigation of static and dynamic magnetic properties \nof FeMn/Pt multilayers using combined techniques of magnetometry, ferromagnetic resonance, inverse \nspin Hall effect and spin Hall magnetoresistance\n \nmeasurements\n. \nThe \nFeMn/Pt multila\nyer \nwas found to \nexhibit ferromagnetic properties\n,\n \nand \nits\n \ntemperature dependence of saturation magnetization can be fitted \nwell using a phenomenological model by including a finite distribution in Curie temperature due to subtle \nthickness variations acros\ns the \nmultilayer \nsample\ns\n. \nThe non\n-\nuniformity in static magnetic properties \nis \nalso manifested in the \nf\nerromagnetic resonance spectr\na, which\n \ntypically exhibit a broad resonance peak\n. \nA damping parameter \nof around \n0.106\n \nis derived from the \nfrequency dependen\nce of ferromagnetic \nresonance \nlinewidth\n, which is \ncomparable to the reported values for other \ntypes of \nPt\n-\nbased multilayers\n. \nClear inverse spin Hall signals and spin Hall magnetoresistance have been observed in all samples below \nthe Curie temperature, whic\nh \ncorroborate\n \nthe \nstrong \nspin\n-\norbit torque effect observed previously. \n   \n \n \n \n \n                                        \n        \n \n*\n \nAuthor to whom correspondence should be addressed: elewuyh@nus.edu.sg\n 2\n \n \nI.\n \nINTRODUCTION\n \nM\nultilayer structures\n \nconsisting of ultrathin nonmagnetic \n(NM) \nlayers, particularly Pt and Pd, and \nferro\nmagnetic\n \n(FM)\n \nlayers such as Co and Fe, have been of both fun\ndamental and \ntechnological interest \nsince late 1980’s.\n1\n \nWhen the thicknesses of both NM and FM layers are controlled withi\nn \na certain \nran\nge, \ntypically less than\n \n1.5 nm, \nthe \nmultilayer \nas a whole \nexhibits ferromagnetic properties with dominant\nly\n \nperpendicular magnetic anisotropy (PMA). \nSome \nof these \nmultilayer\n \nfilms have\n \nalready \nbeen applied\n \nin \nmagneto\n-\noptic recording\n2\n \nand\n \nmore recently \nal\nso \nin magnetic tunnel junctions \nas \npart of the reference \nlayer.\n3,4\n \nStimulated by earlier work on proximity effect at the FeMn and Pt interface,\n5\n \nwe have recently \ncarried out a systematic study of FeMn/Pt multilayers.\n6,7\n \nDespite the fact that FeMn is an antiferromagnet \n(AFM), FeMn/Pt multilayers with ultrathin FeMn \nand Pt layer\ns\n \n(< 1 nm) \nwere found to exhibit \nglo\nbal FM \nordering\n \nwith in\n-\nplane \nmagnetic \nanisotropy. \nA large field\n-\nlike spin\n-\norbit torque (SO\nT) was found to be \npresent \nin the multilayer \nwhen a charge current \nflows through it\n.\n7\n \nQuantification of the SOT strength was \ncarried out by varying the thicknesses of both FeMn and\n \nPt systematically and the results \ncorroborate \nthe \nspin Hall \neffect (SHE) \nscenario, \ni.e.\n, spin current is generated and absorbed by the multilayer\n,\n \nthereby \ngenerating the\n \nSOT.  We have \nfurther demonstrated that \nthe \nSOT is able to rotate the \nmagnetization o\nf \nFeMn/Pt multilayers\n \nby 360\no\n \nwithout any external field\n. \nThese results demonstrate \nclearly \nthe potential of \nFeMn/Pt multilayers in \nmemory\n \nand \nsensor\n \napplications. \n \nIn order to gain further insights into the SOT generation mechanism in FeMn/Pt multilayers,\n \ni\nn this \npaper, we report on ferromagnetic resonance (FMR), inverse spin Hall effect (ISHE) and spin Hall \nmagnetoresistance (SMR) \nstudies of\n \nmultilayer \nsamples which exhibit clear SOT effect. \nBefore \nproceeding to dynamics studies, the static magnetic prope\nrties of the multilayers were characterized using \nmagnetometry at variable temperatures. Special emphasis \nwa\ns placed on the understanding of the \ntemperature dependence of the saturation magnetization. From fitting of the experimental data using \ndifferent m\nodels, it is \nfound\n \nthat the multilayers \nexhibit the characteristic of three\n-\ndimensional Heisenberg \nuniversality class with a finite Curie temperature distribution. This correlates well with the large linewidth 3\n \n \nof resonance peaks observed in both FMR and IS\nHE. A large damping parameter (~\n \n0.106\n) is derived \nfrom the frequency dependence of the FMR, which is comparable to the values reported previously for \nother types of Pt\n-\nbased multilayers.  \nThe observation of both ISHE and SMR suggests the presence of spin \ncurrent generation/absorption processes, corroborating the strong SOT effect observed previously\n. The \nrole of asymmetric FeMn/Pt and Pt/FeMn interfaces in generating the SOT \nis\n \ndiscussed\n \nfor samples with \nrelatively thick FeMn and Pt layers, whereas for sam\nples with ultrathin Pt \nas well as co\n-\nsputtered samples, \nextrinsic SHE/ISHE may play a more important role.\n \n \n \nII.\n \nEXPRIMENTAL\n \n[FeMn\n(\nt\nFeMn\n)\n/Pt\n(\nt\nPt\n)\n]\nn\n \nmultilayers \n(here \nn\n \ndenotes the \nrepeating period\n) \nwith different FeMn and Pt \nthicknesses, \nt\nFeMn\n \nand \nt\nPt\n, \nwere pr\nepared on SiO\n2\n/Si substrates \nby magnetron sputtering \nwith a base \npressure of 2\n \n×\n \n10\n-\n8\n \nTorr \nand working pressure of 3\n \n×\n \n10\n-\n3\n \nTorr, respectively.\n \nThe nominal composition of \nFe:Mn is 50:50.\n \nThe structur\nal properties \nof the multilayers w\nere\n \ncharacterized using\n \nboth \nX\n-\nray \ndiffraction (XRD)\n \nand X\n-\nray reflectivity \n(XRR) \nanalysis\n. Magnetic measurements were carried out using \na Quantum Design vibrating sample magnetometer (VSM) with the samples cut into a size of 2\n \nmm\n×2mm. \nThe FMR measurements were \nperformed\n \nat r\noom temperature via a coplanar waveguide \n(CPW), designed to have \nan \nimpedance \nof 50 \n\n \nwithin a broad frequency range up to 20 GHz. The \nwaveguide, 5\n \nmm \nlong, has a signal line of 150 \n\nm and a \nsignal to ground line spacing of 20 \n\nm. The two \nsignal lines\n \nof t\nhe CPW were connected to \na\n \nVector Network Analyzer (VNA)\n \nvia high\n-\nfrequency probes\n. \nThe FMR spectra were obtained by placing \na 2\n \nmm \n× \n2\n \nmm \nsample directly on \nthe CPW\n \nwith sample \nsurface facing down\n \nand taking readings of the \nS\n21\n \nsignal while \nsweeping a\n \nDC\n \nmagnetic field \nin\n \nthe \nsignal \nline \ndirection. \nFor \nISHE\n \nmeasurements, \nthe samples were patterned into H\nall bar\ns\n \nwith a lateral dimension\n \nof 2000 \n\nm \n×\n \n120 \n\nm\n \nby \ncombined techniques of photolithography, sputtering deposition and lift\n-\noff. \nFollowing \nthe Hall bar fabrication\n, a 100 nm SiO\n2\n \ninsulating layer was deposited to isolate \nelectrical 4\n \n \nconduction between \nthe waveguide \nand \nthe multilayer \nwith\n \nthe \ncontacts to the H\nall bar uncovered \nfor \nsubsequent electrical measurements\n. \nThe last step was to deposit a 150\n-\n\nm \nwide\n \nand 200\n-\nnm thick Cu \ncoplanar waveguide and four 500\n \n\nm \n×\n \n500 \n\nm \ncontact pads\n. Th\ne same \nHall bar\n \nwas used to measure the \nSMR, which was obt\nained by rotating the samples under a constant field of 3\n \nkOe\n \nin the \nxy\n, \nyx\n, and \nzx\n \nplanes, respectively.\n \n \n \nFIG. 1. \nX\n-\nray diffraction\n \npattern\n \nof \nPt(1)/[FeMn(0.6)/Pt(0.4)]\n10\n. \nDotted lines indicate the (111) peak position of Pt and FeMn, \nrespectively.\n \n \nIII.\n \nRESU\nLTS AND DISCUSSION\n \nA. Structural properties\n \nThe \nas\n-\ndeposited multilayers were characterized using both \nhigh\n-\nangle \nXRD and \nsmall\n-\nangle \nXRR. \nFi\ngure\n \n1 \nshows the XRD pattern of \nPt(1)/[FeMn(0.6)/Pt(0.4)]\n10\n, covering the range of bulk fcc Pt (111) \npeak at 39.8° \nand bulk fcc FeMn (111) peak at 43.5°, using the Cu Kα radiation\n \n(\n\n1.541 Å)\n.\n \nHere the \nnumber and symbols inside the parentheses denote the thickness of individual FeMn and Pt layers in \nnm\n. \nIn order to prevent oxidation, all the samples except stated oth\nerwise were all covered by a 1 nm Pt capping \nlayer. \nAs can be seen\n \nfrom the figure\n, t\nhe diffraction pattern\n \nis\n \ndominated by a main peak at 40.\n5\n° \n–\n \n40.\n6\n°, \n37\n40\n43\n46\nFeMn (111)\nIntensity (a.u.)\n2\n\n\n\nDeg\n\n Pt (111)5\n \n \nwhich \nfalls\n \nbetween the bulk Pt (111) and FeMn (111) peaks\n. Th\nis suggests that the multilayer is \n(\n111\n)\n \ntextured and its lattice spacing is the average of those of Pt and FeMn, though it is more dominantly of \nPt \ncharacteristic\n. \nThe FeMn (111) peak is almost at the same level of the baseline, which is presumably \ncaused by the combined effect of ultrathin thi\nckness, interface mixing and small scattering cross sections \nof Fe and Mn as compared to Pt. Similar phenomena\n \nha\nve\n \nalso been \nobserved in Co/Pt multilayers\n, in \nwhich the peak position is near that of Pt and increases with increasing the Co thickness\n.\n8\n-\n10\n \nThe \nsmall\n-\nangle\n \nXRR \nwas measured\n \nwith an \nincident angle\n \nin the range of \n0°\n \n–\n \n10\n° with a s\ntep of 0.\n02\n°\n.\n \nFigure 2 \nshows the XRR of a\n \nmultilayer \nwith structure: \nPt(1)/[FeMn(0.6)/Pt(0.\n6\n)]\n3\n0\n \nand another co\n-\nsputtered \nsamp\nle\n,\n \ni.e.\n, Pt and FeMn were deposited simultaneously using the same \ndeposit\nion\n \ntime \nand power\n. \nThe \nn\n \n= 1 \nBr\nagg maxim\num\n \ncorrespond\ning\n \nto a period of \n1.06 \nnm\n \n(about 20%\n \nsmaller than\n \nthe\n \nnominal\n \nvalues\n) \nis clearly observed in the spectrum for the multilayer sa\nmple\n \n(red solid\n-\nline)\n. \nIn contrast, only thickness \ninduced fringes are observed in the spectrum for the co\n-\nsputtered sample\n \n(blue dotted\n-\nline)\n. The result \ndemonstrate\ns\n \nthat the multilayer has a well\n-\ndefined periodicity.\n \n \n \nFIG. 2. XRR \npatterns \nof\n \nPt(1)/[Fe\nMn(0.6)/Pt(0.6)]\n30\n \nmultilayer sample \n(red \nsolid\n-\nline)\n \nand co\n-\nsputtered sample (blue \ndotted\n-\nline)\n \ndeposited under the same condition. \n  \n \n \nB. Magnetic properties\n \n0\n2\n4\n6\n8\n10\n2\n\n\n\nDeg\n\n \nMultilayer sample\n \nCo-sputtered sample\nIntensity (a.u.)6\n \n \nAll the multilayers with \nt\nFeMn\n \n< 0.8 nm and \nt\nPt\n \n< 1 nm exhibit ferromagnetic properties with in\n-\nplane \nmagnetic \nanisotropy. The Curie temperature \n(\nT\nC\n) \nvaries from 250 K to 380 K, depending on both the total \nand individual layer thicknesses. Figure 3a shows \nthe hysteresis loop of \nPt(1)/[FeMn(0.6)/Pt(0.3)]\n10\n \nat 50 \nK and 300 K, respectively. The coercivi\nty at 50 K is around \n240\n \nOe, but it decreases rapidly \nto \nabout \n1 \nOe \nat 300 K. Such kin\nd of behavior is typical of \nsamples\n \nexhibiting\n \nFM properties above \nroom temperature\n \n(RT)\n.\n \nFig\nure\n \n3\nb shows the saturation magnetization as a function of the temperature \n(\nM\n-\nT\n) \nwith \nthe \nFeMn \nlayer thickness \n(\nt\nFeMn\n) fixed \nat 0.6\n \nnm and Pt layer thickness \n(\nt\nPt\n)\n \nranging \nfrom 0.1 nm to 0.8 nm. \nAs\n \nit was \nfound that a minimum repeating period of 3\n \n–\n \n4 is required for most samples to exhibit ferromagnetic \nproperties above \nRT\n, we fix\ned the repe\ntition \nperiod for all the samples at 10\n. \nAlthough the polarized Pt \nalso contributes to the measured magnetic moment, it is difficult to quantify it for samples with different \nthickness combinations and at different temperature. Therefore, as an \napproximation, we only take the \noverall FeMn volume into consideration when calculating the saturation magnetization. \nAs shown in the \nfigure, the \nM\ns\n \nat \nlow\n-\ntemperature\n \nincreases with \nincreasing \nt\nPt\n, though\n \nthe sample with \nt\nPt\n \n= 0.1 nm \nhas a \nsignificantly s\nmaller magnetization. An opposite trend is observed for \nT\nC\n \nwhich \ndecreases with \nt\nPt\n, \nsaturating\n \nat about \n300 \nK when the adjacent \nFeMn\n \nlayers are completely separated \nmagnetically \nby the Pt \nlayer. \nBoth trends\n \nare in \nqualitatively agreement with\n \nfindings\n \nrep\norted \nfor\n \nCo/Pt multilayers\n,\n11\n \nwhich can \nbe accounted for by the proximity effect at P\nt/FeMn interfaces. Pt is known to be just under the Stoner \nlimit \nthat\n \ncan be readily polarized when it is in direct contact with ferromagnetic materials. \nIn the present\n \ncase, although FeMn is an AFM in bulk phase, \nit shall behave like a \n“\nsuperpara\n-\nAFM\n”\n \nwhe\nn \nit is ultra\nthin, \ni.e.\n, \nt\nFeMn\n \n< 1 nm. This can be inferred\n \nfrom exchange bias studies in FeMn\n-\nbased AFM/FM bilayers, \nwhich have \nrevealed that a minimum thickness of 4\n \n–\n \n5 nm is required for FeMn to establish a measurable \nexchange bias to the FM at \nRT\n.\n12\n \nDespite its superpara\n-\nAFM nature, \nwhen it forms a multilayer with Pt, \nthe mutual interaction at their interfaces promotes FM order in both layers which eventually extends \nthroughout the\n \nmultilayer when both layers are u\nl\ntrathin. \nTherefore, as long as Pt is thin enough to \nallow \ncomplete pol\narization \nby the adjacent FeMn layers the average magnetic moment at low temperature will 7\n \n \nincrease with the Pt th\nickness. On the other hand, the decrease of\n \nT\nC\n \nat increasing\n \nPt thickness \nis \npresumably due to\n \nweakening of exchange cou\npling throughout the mu\nltilayer caused by incomplete \npolarization of the Pt layers\n \nat central regions\n. \nThe anomaly at \nt\nPt\n \n= 0.1 nm \ncan be readily understood by \ntaking into account the effect of interface roughness. At this thickness, \nPt is probably partially \ndiscontinuous, resul\nting in direct coupling of neighboring FeMn layers at certain locations and thereby \nreduc\nes\n \nthe saturation magnetization and \nT\nC\n.  \n \n \n  \n \nFIG. 3. (a) Hysteresis loop \nof \nPt(1)/[FeMn(0.6)/Pt(0.3)]\n10\n \nmeasured at 50 K\n \n(\nsquare\n)\n \nand 300 K\n \n(\ncircle\n)\n, respectively\n. (\nb) \nSaturation magnetization as a function of temperature. The legend (\nt\n1\n,t\n2\n) denotes a multilayer with a FeMn thickness of \nt\n1\n \nand \nPt thickness of \nt\n2\n.\n \nT\nhe \nnumber of \nperiod \nfor all samples \nis \nfixed\n \nat 10\n.\n \n \nIn order to gain more insights on the magnetic prope\nrties of the multilayers, we examine the \nM\n-\nT\n \ncurves using different models.\n \nThe temperature\n-\ndependence of \nmagnetization for a ferromagnet\n \nat low\n-\ntemperature can be calculated \nfrom t\nhe number of \nthermally excited \nmagnons\n \n–\n \nquanta of \nspin\n-\nwave\n. \nAssociated wi\nth each magnon is a magnetic moment \ng\n\nB\n, and therefore the total moment of magnon is \ngiven by\n \n-600\n-300\n0\n300\n600\n-600\n-300\n0\n300\n600\n50\n150\n250\n350\n0\n200\n400\n600\n800\n(b)\nMagnetic Field (Oe)\n \n50 K\n \n300 K\n(a)\nM\ns \n(\nemu/cm\n3\n)\nM\ns \n(\nemu/cm\n3\n)\nTemperature (K)\n \n(0.6,0.1)\n \n(0.6,0.2)\n \n(0.6,0.3)\n \n(0.6,0.4)\n \n(0.6,0.5)\n \n(0.6,0.6)\n \n(0.6,0.8)8\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n(1)\n \nwhere \ng\n \nis the electron g\n-\nfactor, \n\nB\n \nis the Bohr magneton, \nħ\n \nis the reduced Planck’s constant, \n\nk\n \nis the \nmagnon frequency, and \nk\nB\n \nis the Boltzmann’s \nconstant. \nUnder\n \nthe long wavelength limit, the magnon \ndispersion relation may in general be written \nas \n\n\n\nn\nk\nDk\n, where \nD\n \nis the spin\n-\nwave stiffness, and \nn\n \n= \n2 for a ferromagnet and \nn\n \n= 1 for an AFM. Substitute the dispersion relation into\n \nEq.\n \n(1)\n, one obtains\n  \n \n \n \n \n \n\n\n2\n3\n0\n3/\n2\n4\n(2 )\nexp / 1\n1 3 3\n2\nB\nn\nB\nn\nB\nB\ng\nk dk\nN\nDk k T\nk T\ng\nn n n D\n \n\n \n\n\n\n\n \n   \n \n   \n \n   \n \n\n \n \n \n \n \n \n \n \n \n \n \n \n  \n(2)\n \nwhere\n \n\n \nis the Riemann zeta function and \n\n \nis the Gamma function. \nEq.\n \n(\n2) can be used to calculate the \ntemperature dependence of magnetization in FM or stagger order parameter in AFM.\n \nSince the FeMn/Pt \nmultilayers exhibit ferromagnetic properties despite the fac\nt that bulk FeMn is an AFM, in what follows \nwe only focus on FM\n. \nBy substituting \nn\n \n= 2 into Eq. (2), we obtain\n \nthe Bloch \nT\n3/2\n \nlaw, \ni.e.\n,  \n \n \n \n \n \n3/2\n3/2\n( ) (0)(1 )\nM T M B T\n \n \n \n \n \n \n \n \n \n \n \n \n                   \n \n(3)\n \nwhere \nB\n3/2\n \nis a constant proportional to \nD\n-\n3/2\n. Alth\nough the Bloch \nT\n3/2\n \nlaw can satisfactorily explain the \nM\n-\nT\n \ndependence at low temperature, it fails at high temperature because of the neglect of magnon\n-\nmagnon \ninteractions and deviation of the dispersion relation from \n \nat large \nk\n.\n \nFor a Heisenberg \nferromagnet,\n \nthe high\n-\ntemperature effect may\n \nbe included in \nM(T)\n \nby introducing a temperature\n-\ndependent \nD\n, namely, \n5/2\n5/2\n( ) (0)(1 )\nD T D B T\n \n, where \nB\n5/2\n \nis a constant.\n13\n \nAs a result, the \nM(T)\n \nin a wide \ntemperature range can be modelled by\n \n                   \n3/2\n3/2\n5/2\n5/2\n( ) (0) 1\n1\nT\nM T M B\nB T\n \n \n \n \n \n \n\n \n \n.\n \n \n \n     \n \n \n \n \n \n \n \n   \n     \n(4)\n \nWhen \nB\n5/2\n \nis small, \nM(T)\n \ncan be approximated as\n 9\n \n \n    \n3/2 4\n3/2 3/2 5/2\n3\n( ) (0) 1\n2\nM T M B T B B T\n \n  \n \n \n \n \n \n \n \n \n \n \n \n        \n    \n  \n(5)\n \nAlthough \nEq. (5) improves the fitting at higher temperature as compared to the Bloch \nT\n3/2\n \nlaw, it is still \nunable to reproduce the \nM\n-\nT\n \ncurve i\nn the entire temperature\n \nrange\n, and \nthe deviation \nfrom experimental \nvalue \ntends to increase\n \nnear \nT\nC\n \ndue to the critical behavior of ferromagnet.\n \nIn order to \nimprove the fitting near \nT\nC\n \nby taking into account the critical behavior\n, \nwe invoke the \nsemi\n-\nempiri\ncal model developed by M. D. Kuz`min,\n14\n  \nwhich turned out to be very successful in fitting \nthe \nM\n-\nT\n \ncurves of many different types of magn\netic materials. According to this model, the temperature\n-\ndependent magnetization of a ferromagnet is given by:\n \n  \n \n \n \n   \n3/2 5/2\n( ) (0) 1 (1 )\n\n \n   \n   \n \n   \n \n   \n \nC C\nT T\nM T M s s\nT T\n    \n \n \n \n \n \n \n \n \n        \n(\n6\n)\n \nwhere \nM(0)\n \nis the magnetization at zero temperature, \nT\nC\n \nis the Curie temperature, \ns\n \nis \nthe so\n-\ncalled shape \nparameter \nwith a value in\n \nthe range of 0 \n–\n \n2.5\n, and \n\n \nis the critical exponent whose value is determined by \nthe universality class of the material: 0.125 for two\n-\ndimensional Ising, 0.325 for three\n-\ndimensional (3D) \nIsing, 0.346 for 3D XY, 0.365 for 3D Heisenberg, and 0.5 for mean\n-\nfield theory\n15,16\n. On the other hand, \nfor surface magnetism, \n\n \nis in the range of 0.75\n \n–\n \n0.89.\n17,18\n \nT\nhe shape parameter \ns\n \nis determined by the \ndependence of exchange interaction, including its sign, on interatomic distance\n \nin 3D Heisenberg \nmagnets\n.\n19\n \nThis may have implications to multilayer samples as lattice distortion and strain are \nunavoidable at the interfaces due to la\nrge lattice match between and FeMn and Pt.\n \nThe \nM\n-\nT\n \ndependence\n \nshown in Fig.\n \n3b can be fitted reasonably well using Eq. (6) \nwith \n\n \n \n=\n \n1.01\n \n~\n \n2.55\n \nand \ns\n \n= \n-\n0.85\n \n~\n \n-\n0.45\n, except that \nthe \nfitted magnetization drops \nto zero \nmore quickly \nas \ncompared to \nthe \nexperimental\n \ndata.\n \nThe large \n\n \nvalues seem to \nsuggest that the \nM\n-\nT\n \nof FeMn/Pt multilayers follows \nthe surface scaling behavior. \nHowev\ner, a careful examination of the results suggest\ns\n \nthat this may not be \nthe case because \nwe found that \n\n \ndecreases as \nt\nPt\n \nincreases\n.\n \nAn opposite trend would have been observed 10\n \n \nshould it were due to surface mechanism because a thick Pt layer would help enhan\nce the 2D nature of \nferromagnetism at the interfaces. \nThis prompted us to consider other possible factors that may affect the \nshape \nof \nM\n-\nT\n \nof the multilayers \nin a more prominent way\n \nas compared to \nthe case of \na uniform 3D \nferromagnet. \nThe one that came int\no our attention \nis the \nhigh \nsensitivity of \nT\nC\n \nto the Pt\n \nthickness\n \nas \nmanifested in the M\n-\nT curves in Fig. 3\nb; this may lead to finite distribution of \nT\nC\n \nthroughout the multilayer \ndue to thickness variation induced by interface roughness.\n \nWhen this happens\n,\n \nthe magnetization may \ndrop more slowly near \nT\nC\n, \nas\n \nobserved experimentally. To this end, we modified Eq. (6) by including a \nnormal distribution of \nT\nC\n, which leads to\n \n \n3/2 5/2\n2\n0\n2\n0\n1 ( )\n( ) (0) 1 (1 ) exp\n2\n2\n\n\n\n \n     \n\n    \n \n   \n \n\n\n \n     \n \n\nC C\nC\nC C C\nC\nT T T T\nM T M s s dT\nT T T\nT\n  \n \n \n \n \n  \n(\n7\n)\n \nwhere \nT\nC0\n \nis the mean value of \nT\nC\n \nand \n\nT\nC\n \nis its s\ntandard deviation. As shown in Fig.\n4\na\n, all the \nM\n-\nT\n \ncurves \ncan be fitted very well using \nEq. (7)\n \nwith a fixed \n\n \nvalue of 0.365\n, especially near the \nT\nC\n \nregion. \nNote that \n\n \n= 0.365 is the critical exponent for 3D Heisenberg\n \nferromagnet\n. \nFor the sake of clarit\ny, all the curves in \nFig.\n \n4\na\n \nexcept for the one for \nt\nPt\n \n= 0.1 nm are shifted vertically. In the figure, symbols are \nthe \nexperimental \ndata and solid\n-\nlines are fitting results. \nThe fitting values for \nM(0)\n, \nT\nC\n0\n, and \n\nT\nC\n, and \ns\n \nas a function of Pt \nthickness \nar\ne shown in Fig.\n \n4b\n,\n \n4c\n,\n \nand 4d\n \nrespectively. \nExcept for the sample with smallest \nt\nPt\n, the trends \nof \nM(0) \n-\n \nt\nPt\n \nand \nT\nC\n \n-\n \nt\nPt\n \nare opposite with each other, \ni.e.\n, the former increases whereas the latter decreases \nwith \nt\nPt\n. Both are manifestation of the fact t\nhat the global FM ordering in FeMn/Pt multilayers originates \nfrom the proximity effect at FeMn/Pt interfaces, as discussed above. It is interesting to note that \n\nT\nC\n \nalso \nincreases when \nt\nPt\n \ndecreases, and i\nmportantly, the range of \n\nT\nC\n \nfor \nsamples with \nt\nPt\n \n= 0.1 \n–\n \n0.\n8\n \nnm \ncorresponds to the \nrange of \naverage \nT\nC\n \nof \nall samples with \nt\nPt\n \nranging from 0.1 nm to 0.8 nm. \nThese results \nare consistent with\n \nthe \nT\nC\n \nfluctuation \nscenario\n,\n \ni.e.\n, \na larger fluctuation \nin \nT\nC\n \nis expected in samples with \nsmaller \nt\nPt\n \ndue to \ninterface roughness and its\n \nrange should be \ncorresponding to the difference in average \nT\nC\n \nwhen \nt\nP\nt\n \nvaries from 0\n.1\n \nto 0.8 nm or less\n. \nAn\nother\n \nimportant result \nderived\n \nfrom \ncurve fitting is the \nt\nPt\n 11\n \n \n-\n \ndependence of the shape parameter \ns\n. According to M. D. Kuz`min \net al.\n,\n \nfor 3D Heisenberg magnets,\n \ns\n \nis determined by the dependence of exchange interaction\n \non interatomic distance.\n19\n \nIt is generally positive \n \n \nFIG. 4. (a) Experimental \nM\n-\nT \ncurves (open symbols) and fitted results (s\nolid lines). The experimental data are the same as \nthose shown in Fig. 3b, but are shifted for clarity (except for the \nt\nPt\n \n= 0.1 nm sample). (b) M\n0\n, (c) T\nC0\n \n(triangle) and \nΔ\nT\nC\n \n(square), and (d) \ns,\n \nas a function of \nt\nPt\n \nobtained from the fittings. \n \n \nwith a\n \ns\nmall \ns\n \n(< 0.4) \ncorresponding to\n \nmetallic FM\ns\n \nwith long\n-\nrange ferromagnetic ordering\n \nand high \nT\nC\n, \nwhereas\n \na \nlarge \ns\n \n(> 0.8) is indicative of competing exchange interaction\ns and the resultant \nmaterial \ntypically has a \n \nlow \nT\nC\n. \nAs shown in Fig.\n \n4\nd\n, \ns\n \nis small \nand positive for samples with \nt\nPt\n \n= 0.6 nm and 0.8 \nnm, but it turns negative for smaller \nt\nPt\n. When \ns\n \nis negative, the \nT\n3/2\n \nterm of the base of Eq. \n(\n6\n)\n \nbecomes \npositive, or in other words, it contributes positively to \nM(T)\n \nwhen temperature increases. This i\ns \ncounterintuitive for 3D Heisenberg\n \nferromagnet\n. It suggests that, in addition to isotropic exchange \ncoupling, interfacial Dzyaloshinskii\n-\nMoriya interaction (DMI) may \nplay a role, particularly in samples \nwith smaller \nt\nPt\n.\n \nAs DMI \nfavors non\n-\ncollinear align\nment of spins, a weakening of DMI at moderate\nly\n \nelevated temperature may give a relative boost of isotropic exchange coupling\n,\n \nthereby resulting in a \n0\n100\n200\n300\n400\n0\n500\n1000\n1500\n0.0\n0.5\n1.0\n200\n400\n600\n0.0\n0.5\n1.0\n-6\n-3\n0\n0.0\n0.5\n1.0\n250\n350\n450\n\nT\nC\n (K)\nT\nC0\n\n(K)\nM\ns\n \n(\nemu/cm\n3\n)\n0\n50\n100\nPt thickness (nm)\n(d)\n(c)\n(b)\n(0.6,0.8)\n(0.6,0.6)\n(0.6,0.5)\n(0.6,0.4)\n(0.6,0.3)\n(0.6,0.2)\nTemperature (K)\n(0.6,0.1)\n(a)\nM(0) (emu/cm\n3\n)\nPt thickness (nm)\ns\n \nPt thickness (nm)12\n \n \npositive contribution to the magnetic moment \nat intermediate temperature range. This may explain why \ns\n \nis \nnegative, though further studies are required to quantify the effect of DMI on temperature dependence \nof magnetization in these multilayers. \n \n \n \n \nFIG. 5. (a) FMR spectra of \nPt(1)/[FeMn(0.6)/Pt(0.5)]\n80\n \nat fixed frequency ranging from 2 GHz to 4 GHz. (b) Dat\na (\nsquare \nsymbol) and fitting (line) for FMR signal at \nf \n= 3 GHz. (c) Full width at half maximum\n \nof the resonance pea\nk (triangle \nsymbol) are plotted \nagaint\n \nthe \nfrequency.\n \nThe solid line is a linear fit to the data. \n \n \nC. \nFMR measurements and d\namping constan\nt\n \nMagnetic damping plays a\n \nkey\n \nrole in the m\nagnetization dynamics of magnetic materials, which can \nbe treated phenomenologically by including a damping term\n \n( )\nM dM dt\n\n\n \n \nin the Landau\n-\nLifshitz\n-\nGilbert (LLG) equation. Here, \n\n \nis the Gilbert da\nmping constant, which \ncharacterize\ns\n \nthe strength of\n \ndamping.\n \nIt is commonly assumed that \nthe origin of Gilbert damping\n \nis spin\n-\norbit coupling (SO\nC\n), same \nas that of magnetic anisotropy. Since SOC is also the origin of spin\n-\norbit torque, naturally it would \nbe of \n1.0\n2.0\n3.0\n4.0\n5.0\n400\n500\n600\n-2\n-1\n0\n1\n2\n0\n1\n2\n(b)\nFWHM (Oe)\n(a)\n |\nS\n21\n|\n (dB)\n0.04 dB\nField (kOe)\nFrequency (GHz)\n4.0 GHz\n3.5 GHz\n3.0 GHz\n2.5 GHz\nField (kOe)\n2.0 GHz\n |\nS\n21\n|\n (dB)\n(c)13\n \n \ninterest to measure the damping constant of FeMn/Pt multilayers and correlate it with SOT or ISHE\n. \nThe \neffective damping constant, including both intrinsic and extrinsic contributions, can be de\nduced \nfrom the \nFMR line width\n \nas a function of resonance\n \nfrequency\n.\n \nFigure \n5\na\n \nshows \nthe\n \nFMR spectr\na\n \nof\n \na\n \nPt(1)/[FeMn(0.\n6\n)/Pt(0.\n5\n)]\n80\n \nmultilayer\n \nextracted by VNA \nat\n \ndifferent frequencies ranging from 2\n \nGHz to \n \n4\n \nGHz \nwith a\n \nsweeping \nDC\n \nmagnetic field. \nCompared with a homogeneous FM layer, the FMR peak is \nrather b\nroad. This is presumably caused by the variation in \nT\nC\n \nand \nM\ns\n \nthroughout the multilayer as \ndiscussed above. Nevertheless, t\nhe \naverage resonance \nfields at different frequencies \ncan \nstill \nbe described \nby\n \nthe \nKittel equation\n20\n \n \n0\n2 ( )\nFMR FMR s\nf H H M\n  \n \n \n \n \n \n \n \n \n \n \n \n       \n      \n \n(8)\n \n \nwh\nere \nf\n \nis the frequency, \n\n \nis the effective gyromagnetic ratio\n,\n \nM\ns\n \nis the saturation magnetization\n, \nH\nFMR\n \nis \nthe resonance field, and \n\no\n \nis the vacuum permeability\n. The FMR spectra \nnear the resonance region \ncan \nbe \nroughly\n \nfitted by \nthe\n \nsuperposition of \na symmetric and an antisymmet\nric peak\n.\n \nAs an example, Fig.\n \n5b \nshows the f\nitting result \nat \nf\n \n=\n \n3 GHz\n \nfor Pt(1)/[FeMn(0.6)/Pt(0.5)]\n80\n.\n \nThe \nfull\n \nwidth at half maximum \n(\nFWHM\n) of \nthe \nsymmetric \npeak with \nLorentz \nshape\n \nis plotted in Fig. \n5c \n(empty square)\n \nas a function of \nfrequency\n. The soli\nd\n-\nline is the linear fitting \nto\n \nthe\n \nrelation\n21\n \n \n0\n0\n4\n( ) ( )\nH f f H\n\n\n \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n    \n      \n \n(9)\n \nwhe\nre \n\n \nis the\n \neffective\n \ndamping parameter\n \nand \n\nH\n0\n \nis \nzero\n-\nfrequency linewidth\n \ncaused by\n \nmagnetic \ninhomogeneit\ny\n \nof the sample\n. The large \n\nH\n0\n \nvalue is consistent with \ndistribution of \nT\nC\n \ndiscussed in IIIB\n. \nFrom the linear fitting, we obtained an\n \neffective \ndamping \nparameter \nof \n0.\n106 \nfor this specific sample\n, \nwhich is around \none\n \norder \nof magnitude \nlarger than that of permalloy at \nthe\n \nsame thickness\n,\n22\n \nbut\n \nis \n \ncomparable to that of Pt/Co multilayers\n.\n23,24\n \nThis affirms our previous argument of the twofold role of Pt \nin Pt/FeMn multilayer\ns\n7\n,  \ni.e.\n, it \npromot\nes\n \nglobal \nFM ordering via\n \nproxi\nmity effects \nat\n \nPt/FeMn interfaces\n \nand at the same time it functions simultaneously as both a spin current generator and \nan \nabsorber\n. \nIt is\n 14\n \n \npostulated that both the proximity effect and spin\n-\ncurrent absorption contribute to the enhancement of \n\n,\n25, \n26\n \nthou\ngh it is difficult to determine which factor is dominant. \nWhen Pt is magnetized\n, it will be a FM with \nlarge SOC which will lead to large damping. FeMn is known to have a small SOC. However, being \nsandwiched by Pt in the multilayer structure, t\nhe \nprecession\n \nof its \nmagnetization \nunder ferromagnetic \nresonance will pump spin current into the \nneighboring Pt layers\n, \nwhich \nagain \nwill\n \nlead to \nthe\n \nenhancement \nof damping. \nAlthough a large damping constant is undesirable for applications which require the use of \nspin \ntorque transferred from other layers to switch its magnetization, it can be \neffectively exploited for \nSOT\n-\nbased applications, \ni.e.\n, to generate SOT internally by a charge current. This is exactly what we have \nreported in our earlier work, in which we have \ndemonstrated that it is possible to switch the magnetization \nof FeMn/Pt multilayers by SOT without any external field.\n7\n \nIt is worth pointing out that the damping \nparameter extracte\nd above may be overestimated considering the fact that sample inhomogeneity may also \ncontribute to the large FM linewidth. \n \n \nD. Inverse spin Hall effect\n \nIn the aforementioned FMR measurements, we attribute the enhancement of \n\n \npartially to the \nabsorption of spin current by the Pt layers. As we will discuss shortly in the SMR experiments, for \nmultilayers with relatively thick Pt and FeMn, we may treat them as consisting of alternating FM and HM \nlayers. However, if the Pt and Fe\nMn layers are ultrathin, it is more appropriate to treat the multilayer \nequivalently as a single FM layer. We \nconsider\n \nthe multilayer case first. If we focus on a specific FeMn \nlayer inside the multilayer structure, there are two interfaces with the adjace\nnt Pt layers. To differentiate \nthese two interfaces, we call Pt/FeMn the upper interface and FeMn/Pt the lower interface. These two \ninterfaces are not necessarily to be identical due to the large lattice mismatch between Pt and FeMn.\n27\n \nAlthough the FeMn/Pt\n \nmultilayer behaves like a single phase FM, the magnetic moment is presumably \nmainly from the FeMn layer. Under the FMR condition, the precession of magnetization in the FeMn layer 15\n \n \npumps spin current into the adjacent Pt layers, which is subsequently absor\nbed either completely or \npartially depending on the Pt layer thickness. This leads to the enhancement of damping constant as \ndiscussed above. If the two interfaces are symmetrical, there should not be a net spin current following \ninside the multilayer afte\nr we take into account the contributions of all the individual layers. However, if \nthe two interfaces are asymmetrical and have different spin\n-\nmixing conductance, a net spin current will \nbe generated due to broken inversion symmetry. When this happens, a t\nransverse electromotive force \n(EMF) will be generated due to ISHE, which can be detected as a voltage signal under open circuit \ncondition.\n \nIn this context, we have measured the voltage across the two side\n-\ncontacts of the sample \nsimultaneously with the FMR \nmeasurements. \n \n \n \n \n \nFIG. 6. (a) Measurement geometry of ISHE and FMR. (b) ISHE and (c) FMR spectra for Pt(1)/[FeMn(0.6)/Pt(0.5)]\n50\n \nmeasured \nat 3.0 GHz. (d) Voltage signal as a function of positive (circle) and negative (square) magnetic field for \nPt(1)/[Fe\nMn(0.6)/Pt(0.4)]\n10\n \nat 3 GHz. (e) Decomposition of measured voltage signal for Pt(1)/[FeMn(0.6)/Pt(0.4)]\n10 \nat 3GHz \ninto symmetric  and antisymmetric  components. Symbols are raw data as shown in (d). \nDash dotted\n \nand dashed lines show the \nsymmetric and antis\nymmetric components, respectively. The solid\n-\nline shows the combined fitting results. \n \n \n \n-1\n0\n1\n0.0\n0.5\n1.0\n1.5\n0.0\n0.3\n0.6\n0.9\n1 \n\nV\n \nVoltage  (\n\nV)\nH (kOe)\n0.02 dB\n |\nS\n21\n|\n (dB)\n1 \n\nV\n \nH (kOe)\nVoltage \n(\n\nV\n)\n1 \n\nV\n \n(d)\n(c)\n(b)\n(a)\n(e)\nx\ny\nz\nH\nh\nm\nH (kOe)\nVoltage (\n\nV)\n Exp.\n Fitting\n Vasym\n Vsym 16\n \n \nFig. 6a shows the measurement geometry, where \nh\nm\n \nis the \nrf\n \ndriving field and \nH\n \nis the external field. \nThe measurement was firstly performed on multilayer sample Pt(1)/\n[FeMn(0.6)/Pt(0.5)]\n50\n \nwith \nn\n \n= 50. \nThe peak position of the ISHE signal in Fig. 6b and FMR spectrum in Fig. 6c show a good correspondence \nwith each other, suggesting that the ISHE signal might be directly related to FMR absorption. Following \nthat, we carri\ned out the same measurements on Pt(1)/[FeMn(0.6)/Pt(\nt\nPt\n)]\n10\n \nsamples with \nt\nPt\n \nranging from \n0.1 nm to 0.8 nm, respectively. Although the FMR signal of the sample with \nn\n \n= 10 was too weak to be \ndetected due to small absorption, the voltage could still be dete\ncted for samples with relatively large \nM\ns\n \nat \nRT\n \n \nwith \nt\nPt \n=\n \n0.2 \n–\n \n0.5 nm; however, we could not detect any voltage signal for samples with \nt\nPt\n \n= 0.1 \nnm, 0.6 nm and 0.8 nm due to the small \nM\ns\n \nat RT. As an example, Fig. 6d shows the measured voltage for \nPt(1\n)/[FeMn(0.6)/Pt(0.4)]\n10\n \nas a function of external magnetic field at fixed frequency of 3 GHz. As can \nbe seen, the peak contains both symmetric and antisymmetric components with respect to the resonan\nce\n \nfield and its polarity changes when field reverses. Al\nthough the transverse voltage can be readily detected \nunder FMR, analysis of the signal is not straightforward because, in addition to ISHE, it also contains \ncontributions due to non\n-\nISHE related effects such as spin rectification effect (SRE) and anomalou\ns Nernst \neffect (ANE). The ANE is caused by the temperature gradient due to microwave heating and, as reported \nin several studies, is generally smaller than the SRE effect.\n \n28,29\n \nThe SRE signal contains both \nanisotropic \nmagnetoresistance (\nAMR\n)\n \nand \nanomalou\ns Hall effect (AHE) contributions\n \nand exhibits complex \nsymmetry and sign dependence on the applied external field, \nH\n. \nBased on previous FMR studies in \ndifferent measurement geometries,\n30,31\n \nthere are mainly three contributions to the measured voltage signa\nl \nin the present case: (i) symmetric component due to the ISHE, (ii) symmetric component due to AHE, and \n(iii) antisymmetric component due to AMR. Based on this, we firstly decompose the obtained voltage \nsignal into the symmetric and antisymmetric componen\nts. Fig. 6e shows the symmetric and antisymmetric \nvoltage components of the sample \nPt(1)/[FeMn(0.6)/Pt(0.4)]\n10\n. In this specific case, the peak value of the \nsymmetric component is around 0.97 \nμV. Based on its symmetry and polarity, the symmetric component 17\n \n \nshould \ncontain both ISHE and AHE contributions. As our experimental setup does not allow us to perform \naccurate \nangle\n-\ndependent measurement, here we estimate the magnitude of AHE signal using \nknown \nparameters. Following Chen \net al.\n,\n3\n2\n \nthe Lorentzian contribution of AHE is approximately given by \n \n                                  \n, 0\n,\ncos cos\n2 (2 )\nrf s AHE m\nAHE L\nFMR s\nI R h\nV\nH M\n\n\n \n\n\n                                            \n \n      \n    \n      \n(10)\n \nwhere \n, ,0\n/\nrf s rf wg s\nI I R R\n\n \nwith\n,0\nrf\nI\n \nthe magnitude of \nrf \ndriving current and \nwg\nR\n,\ns\nR\nthe resistance of \ncoplanar waveguide and sample, respectively\n,\n \nAHE\nR\n\nis the anomalous Hall resistance, \ns\nM\nis the \nsaturation magnetization, \nFMR\nH\n \nis the resonant magnetic field,\n \nm\nh\n \nis the \nrf\n \nmagnetic field along \nx\n \ndirection, \n0\n\nis the angle between the direction of exte\nrnal magnetic field and coplanar waveguide, and \n\n \nis the phase of \nrf\n \nfield with respect to \nrf\n \ndriving current. In the present case, \n,\n0.23 0.03\nrf s\nI\n \n \nmA \n(calculated from the microwave power assuming maximum delivery effici\nency), \n1.06 0.11\nAHE\nR\n \n \nΩ \n(from static measurement),\n \n262.4 2.8\ns\nM\n \nemu/cm\n3\n, \n548.7 9.2\nFMR\nH\n \n \nOe, \n36.8 4.7\nm\nh\n \n \nOe \n(calculated from \nrf\n \ncurrent),\n0\n0\no\n\n\n \nand\n0.106 0.01\n\n \n. Based on t\nhese parameters, we obtain \n7\n,\n(1.79 0.8) 10\nAHE L\nV\n\n  \nV, which is around one order of magnitude smaller than the measured \nsymmetric voltage component. Since the phase difference between the \nrf\n \nfield and \nrf\n \ncurrent is unknown, \nwe assume \n\n= \n0 in the calculation, which might have led to a slight overestimation of the AHE signal. \nBased on the discussion above, we believe that the symmetric component of the measured voltage signal \nis mainly from the ISHE. Before we end this section, it is\n \nworth pointing out that the above discussion \nbased on asymmetry in upper and lower interfaces may not apply to multilayers with ultrathin FeMn and \nPt as the interfaces are not well defined. This poses a question as to whether the ISHE signal can still be \ndetected in these kind\ns\n \nof samples. As we will discuss in the SMR section, we believe that in this case, we \nstill can detect the ISHE due to extrinsic spin Hall and inverse spin Hall effect. \n \n 18\n \n \nE. SMR measurement\n \n  \n \nFIG. 7. (a) Geometry of angle\n-\ndependent \nMR measurement. (b) Angle\n-\ndependent MR of Pt(1)/[FeMn(0.6)/Pt(0.3)]\n10\n. (c) Data \n(square symbol) and fitting (line) of SMR ratio as a function of \nt\nPt\n \nfor FeMn(3)/Pt(\nt\nPt\n) bilayers. Inset shows the calculated SMR \n(line) for FeMn(0.6)/Pt(\nt\nPt\n) bilayers\n \nat small\n \nPt thickness as well as experimentally obtained SMR ratio (triangle symbol) for \nPt(1)/[FeMn(0.6)/Pt(\nt\nPt\n)]\n10\n \nmultilayers. \n \n \nBoth FMR and ISHE measurements confirm that spin current generation and absorption occur \nsimultaneously in the multilayer. This is e\nxactly the ingredient for generating \nboth SOT and \nSMR\n, which \nthemselves are complementary processes of each other\n.\n33 \nTo confirm this, we have performed SMR \nmeasurements for the same batch of samples used for the ISHE measurements. \nFig\nure\n \n7\na \nshows the \ngeome\ntry \nof the SMR, or \nangle\n-\ndependent magnetoresistance (MR) measurements\n, which\n \nwere\n \ncarried \nout with \nan \napplied field\n \nof 30 kOe\n \nrotating in \nthe \nzy\n, \nzx\n, and \nxy\n \nplanes, respectively. All the multilayer \nsamples exhibit clear\n \nSMR signal. \nAs a typical example, F\nigure \n7\nb \nshows t\nhe \nangle\n-\ndependent MR of\n \nPt(1)/[FeMn(0.\n6\n)/Pt(0.\n3\n)]\n10\n. \nFrom the angle\n-\ndependence, we can see that only AMR \nis observed when the \n0\n90\n180\n270\n360\n2828\n2829\n2830\n2831\n0.0\n0.5\n1.0\n1E-05\n1E-04\n1E-03\n0\n5\n10\n15\n0\n10\n20\nx\ny\nz\nH\nθ\nxy\nx\ny\nz\nH\nθ\nzx\nx\ny\nz\nH\nθ\nzy\nj\n(c)\n(a)\n(b)\n\nxy\n\nzy\n\nzx\n (deg)\nR\nxx\n \n(\n\n)\n \nxy\n \nzy\n \nzx\n\nR/R\nxx  \nPt thickness (nm)\n\nR/R\nxx \n(\n\n10\n-4\n)\nPt thickness (nm)19\n \n \nfield is rotated in the \nz\nx\n \nplane, whereas the signal obtained in the \nzy\n \nplane is dominantly from SMR. When \nthe fi\neld is rotated in the \nxy\n \nplane, both AMR and SMR are detected.\n \nRecently, Manchon developed a \nmodel for SMR in AFM/HM bilayer,\n34\n \nwhich applies to the collinear AFM with well\n-\ndefined Neel order \n1 2\nn m m\n \n  \n, where \n1\nm\n\n, \n2\nm\n\n \nare the unit vector of the two spin sublattices, respectively. According to this \nmodel, the SMR of AFM/HM bilayers is given by\n \n2 1 2\n1 1\n( )\n(1 cosh )\n(1 tanh )(1 tanh )\nN N N\nSH\nN N\nxx N N AF AF N\nN N\nd\nR\nd d\nR d d\n   \n \n\n  \n   \n \n \n\n \n \n\n\n \n\n \n \n \n                  \n(\n1\n1\n)\n \n \nwith\n \n, , , , ,\n1 ( ) tanh( )\nAF AF AF\nAF\nr d\n   \n    \n \n    \n,\n \n1\n, , , ,\n( ) tanh ( )\nAF AF AF\nN N AF\nd\n     \n\n   \n\n   \n, \nAF AF AF\nsf\nD\n \n\n \nand\n(1 1 )\nAF AF AF AF\nsf\nD\n\n  \n \n \n. \nHere, the subscript \n( )\n\n\n \nrefers to the \nconfiguration\n \nwhen the spin \npolarization aligns parallel (transverse) to the Neel order parameter, \n\nSH\n \nis the spin Hall angle, \nD\nAF\n \nis\n \nthe \nelectron diffusion coefficient in \nthe \nAF\nM\n, \nAF\nsf\n\nis \nthe conventional isotropic spin relaxation time, \nAF\n\n\n \nis \nthe spin dephasing time that relaxes only the spin component that \nis transverse to the Neel order parameter, \nr\n \nis \nthe interfacial resistivity\n, \n\nN\n, \n\nN\n \n(\n\nAF\n) \nand \nd\nN\n \n(\nd\nAF\n) are spin diffusion length, conductivity and thickness \nof the HM (AFM) layer, respectively. \nA\ns shown in Fig.\n \n7\nc\n, by varying the Pt \nthickness \nsystematicall\ny, we \nfound that the thickness\n-\ndependence of SMR \nof \nFeMn\n(3)\n/Pt\n(\nt\nPt\n)\n \nbilayers can be fitted well using the \nfollowing parameters\n: \n\nN\n \n \n=\n \n1.05 ± 0.05\n \nnm, \nFeMn\n\n\n \n= \n4.1 ± 0.1 \nnm,\n \nFeMn\n\n\n \n= \n1.70 ± 0.07\n \nnm\n, \n\nSH\n \n= 0.2\n8 ± \n0.03\n,\n \nFeMn\nsf\n\n= \n(4.25 ± 0.25)\n \n×\n \n10\n-\n14\n \ns,\n \nFeMn\n\n\n \n= \n(7.75 ± 0.25) \n×\n \n10\n-\n1\n5\n \ns\n, \n\nN\n \n= 4\n.0\n \n×\n \n10\n6 \nS/m,\n \nFeMn\n\n\n \n= \n1\n.0 \n×\n \n10\n6 \nS/m, and \nFeMn\n\n\n \n= \n1.5\n \n×\n \n10\n6 \nS/m. \nThe inset of Fig.\n \n7\nc shows the regio\nn with small Pt thickness in log\n-\nscale, together with the experimental SMR of \nPt(1)/[FeMn(0.6)/Pt(\nt\nPt\n)]\n10\n \nmultilayers. \nAs can \nbe seen, the \nexperimental SMR values \nfor multilayers \nare significantly larger than the simulated results for bilayers, \nparticularl\ny at very small Pt thickness. \nThe difference \nbecomes smaller when the Pt thickness increases. \nThis suggests that when Pt is thick, the multilayer can be considered as comprising of magnetically 20\n \n \ndecoupled bilayers and therefore the SMR ratio should be the s\name for both types of samples. However, \nwhen \nt\nPt\n \nis very small, the multilayer behaves more like a “single phase” FM; this is the reason wh\ny the \nSMR is different from that of bilayers with small \nt\nPt\n. \nThe observation of large SMR \nin \nthe \nmultilayer\ns\n \nsuggests\n \nthat there is spin current generation/absorption process taking place inside the multilayer, \npresumably \ndue to either intrinsic \n(for samples with thick Pt) \nor extrinsic SHE/ISHE \n(for samples with \nultrathin Pt) \nor the combination of both.\n \nThis is also the \nreason why a large SOT was observed in these \nstructures.\n \n \n \n \n \nFIG. 8. (a) Angle\n-\ndependent MR of co\n-\nsputtered sample; (b) AMR and (c) SMR of co\n-\nsputtered and \nmultilayer samples with same nominal composition and thicknesses.\n \n0\n90\n180\n270\n360\n3666\n3668\n0\n90\n180\n270\n360\n0\n2\n4\n0\n90\n180\n270\n360\n0\n3\n6\n(c)\n(b)\nR\nxx\n \n(\n\n)\n\nxy\n\nzy\n\nzx\n (deg)\n \nxy\n \nzy\n \nzx\n(a)\n\nR/R\nxx \n(\n\n10\n-4\n)\n\nzx\n (deg)\nCo-sputtered\nMultilayer  \n\nR/R\nxx \n(\n\n10\n-4\n)\n\nzy \n(deg)\nCo-sputtered\nMultilayer  21\n \n \nTo shed some light on the origin \nof SMR, particularly, in structures with ultrathin Pt layers, we have \nalso fabricated and measured the SMR of co\n-\nsputtered sample\ns\n. At the same nominal thickness and \ncomposition, the co\n-\nsputtered sample is more resistive than its multilayer counterpart, co\nnsistent with its \nmore disordered structure. Despite the structur\nal difference\n, SMR of similar magnitude of that of \nmultilayers was also observed in co\n-\nsputtered samples. Fig.8 (a) shows the angle\n-\ndependent MR of a co\n-\nsputtered FeMn:Pt  sample with overall\n \nnominal thickness of \nt\nFeMn\n \n= 6 nm and  \nt\nPt\n \n= 3 nm (calculated from \nthe deposition power and duration). Both the AMR and SMR components are present in the angle\n-\ndependent MR. In Fig\ns\n. 8b and 8c, we show the normalized AMR and SMR curve for both the co\n-\nsput\ntered \nand Pt(1)/[FeMn(0.6)/Pt(0.3)]\n10\n \nmultilayer sample. The nominal thickness and composition are the same \nfor the two samples and both samples are capped with a 1 nm Pt. We have confirmed that the SMR for \nPt(1)/[FeMn(0.6)/Pt(0.3)]\n10\n  \nand [FeMn(0.6)/Pt(0.\n3)]\n10\n \nis almost the same, and therefore, the 1 nm Pt \ncapping layer is not responsible for the SMR observed in both cases. Although further studies are required \nto elucidate the SMR mechanism in both co\n-\nsputtered and multilayer samples with ultrathin layers\n, the \nobserved SMR can be qualitatively explained using the drift\n-\ndiffusion model by taking into account both \nthe precession and dephasing of SHE\n-\ngenerated spin inside a single FM with large spin\n-\norbit coupling. \nThe dynamics of the SHE\n-\ngenerated spin accum\nulation\n \nˆ\nS\n \nis governed by the coupled equations:\n35\n  \n \n                       \n2\n2 2 2\n1 1 1\nˆ ˆ ˆ ˆ\nˆ ˆ ˆ\n( )\nS\nS S m m S m S\n \n  \n      \n \n \n \n \n \n \n \n \n \n \n          \n(12a)\n \n                      \nˆ\nˆ\nˆ\n( )\nS\ni i SH i\nJ D S e n\n\n    \n \n \n \n \n \n \n \n \n       \n \n \n                     \n \n(\n12b)\n \nw\nhere\n \nˆ\nS\n \nis the non\n-\nequilibrium spin density generated by SHE, \nˆ\nm\n \nis direction of the local magnetization, \n\n\n, \n\n\nand\n \nS\n\n \nare spin precession, dephas\ning and spin\n-\nflip diffusion length, respectively,\n \nˆ\nS\ni\nJ\n \nis the \ni\nth\n \ncomponent of spin current with polarization in\n \nˆ\nS\n \ndirection, \nD\n \nis the diffusion coefficient, \nn\n \nis the charge \ndensity,\n \nSH\n\n \nis the spin Hall angle, and\n \nˆ\ni\ne\n \nis a unit vector. The angle\n-\ndependence of MR (or simply SMR) 22\n \n \nappears due to the additional electromotive force generated by\n \nˆ\nS\ni\nJ\n \nvia ISHE. Eq. 12a can be better \nunderstood by con\nsidering the special cases: i) \n\n\n, \n\n\n \n≫\n \nS\n\n, \nii) \n\n\n \n≫\n \nS\n\n, \n\n\n, \nand iii)\n \nS\n\n \n \n≫\n \n\n\n \n, \n\n\n. \nIn \ncase i), the first two terms at the r\night\n-\nhand\n-\nside of Eq. 12a can be ignored, which leads to the spin diffusion \nequation \nfor\n \na no\nn\n-\nmagnetic metal. In this case, there will be no SMR\n-\nlike angle\n-\ndependent MR unless \nwhen it is in contact with a ferromagnetic layer. In case ii), the 2\nnd\n \nterm can\n \nbe ignored, which leads to  \n2\n2 2\n1 1\nˆ ˆ ˆ\nˆ\nS\nS S m S\n\n \n   \n. \nThis is similar to the case of Hanle MR (HMR) in HM except that the spin \nprecession in HMR is caused by an external field,\n36\n \nwhereas in the present case it is caused by the exchange \nfield of the FM i\ntself. In the last case, both spin precession and dephasing terms have to be taken into \naccount on equal footing. To estimate the influence of these two terms on the spin density, we consider \ntwo special cases which is related to the transverse and vertica\nl MR, \ni.e.,\n \ni)\n \nˆ\nm\n \n=\n(\n0\n,\n1\n,\n0\n)\n \nand ii)\n \nˆ\nm\n \n=\n(\n0\n,\n0\n,\n1\n)\n. In the thin film geometry, we are mainly concerned about the spin accumulation on the top and \nbottom surfaces which have a spin polarization dominantly in \ny\n-\ndir\nection. In this case, when \nˆ\nm\n \n=\n(\n0\n,\n1\n,\n0\n)\n, \n  \nboth the precession and dephasing terms can be ignored. \nUnder this condition\n, spin accumulation occurs \non both surfaces, resulting in a diffusion spin current reflected back to the sample. T\nhis will lead a smaller \nresistivity due to ISHE effect. On the other hand, when\n \nˆ\nm\n=\n(\n0\n,\n0\n,\n1\n)\n, the dephasing and diffusion term can \nbe combined, leading to\n2\n2 2\n1 1\nˆ ˆ ˆ\nˆ\nS S m S\n\n \n   \n, where\n2 2 2\n1 1 1\nS\n\n  \n \n.\n \nThis expression is \nsimilar to the \ncase of HMR except that the spin diffusion length is replaced by \nan equivalent diffusion\n \nlength. We can \nlet\n4.1\nFeMn\nS\n \n \n\n \nnm\n \nand\n1.7\nFeMn\n \n  \n\n  \n \nnm\n, \nand \nthen\n1.57\n\n\n \nnm. \nSince this equation is similar \nto the case of HMR, we can use the solution given in \nthe \nsupplementary material of S. Vélez \net al.\n36\n \nto \nestimate the SMR\n-\nlike resistance change due to the first term, which is given by \n \n                      \n2\n2\n3\n( )\n4\nSH\nxx\nR\nR d\n\n\n \n\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n      \n(\n13)\n 23\n \n \nwhere \nd\n \nis the sample thickness, \nR\n\n \nis the change in longitudinal resistance and\n \nxx\nR\n \nis the longitudinal \nresistance at zero field.\n \nUsing\n0.1\nSH\n\n\n, \nd\n \n= 10 nm, \n1.7\n\n\n\n \nnm\n \nand\n \n1.57\n\n\n \nnm, \nwe obtain an MR ratio\n0.1%\nxx\nR\nR\n\n\n, \nwhich is on the same order of magnitude of SMR observed \nexperimentally\n. Although the \nexact value depends on the parameters used, we believe it does explain the salient feat\nure of the MR \nresponse observed in both the co\n-\nsputter and multilayer samples with ultrathin Pt and FeMn layer. \nHowever, when the Pt layer is sufficiently thick, the bilayer model seems to be more appropriate as \nmanifested in the agreement \nbetween experime\nnt and theoretical model \nshown in Fig.7.  \n \n \nF. Discussion\n \nIn this study, we \ninvestigated \nthe static and dynamic properties of [FeMn/Pt]\nn\n \nmultilayers by \ncomb\nined techniques of magnetometry, \nFMR, ISHE and SMR\n, and found a g\nood correlation \nin\n \nthe results \nobta\nined by different techniques\n. \nFirst\n, the FMR and \nISHE signals can only be detected in samples \nwith \nsu\nfficient\nly\n \nlarge \nM\ns\n \nat room temperature, which typically happens in \nsamples with \na \nlarge\n \nrepe\ntitio\nn \nperiod\n, and magnetic inhomogeneity due to thickness\n-\nsen\nsitive \nT\nc\n \nvariation is well reflected in the broad \npeak appeared in the FMR and ISHE spectra\n. Second, t\nhe FMR peak positions correspond well with th\nose \nof \nISHE. \nThird, \nSMR with a magnitude comparable to that of FeMn/Pt bilayer was observed\n, supporting \nthe \npresence of large SOT\n. \nAll these results in combination with the fact that the multilayer behaves like \na \n3D Heisenberg ferromagnet and exhibits a large SOT \nseem to \nsuggest that there is a broken inversion \nsymmetry (BIS) inside the multilayers.\n \nThe most\n \nlik\nely origin of the BIS in the multilayer is the crystalline \nasymmetry of the FeMn/Pt and Pt/FeMn interface\n \ncaused by the different atomic size\n. \nAccording to Liu \net al.\n,\n27\n \nthe atom radii of Pt and FeMn are 0.139 nm and 0.127 nm, respectively. When depositing Pt on \nfcc (111) textured FeMn layer, \nthe crystal direction and atom packing will have to change in order to \naccommodate the large Pt atoms as the (111) plane is already\n \nclose\n-\npacked. On the other hand, the situation 24\n \n \nwill be different when smaller Fe and Mn atoms are deposited on fcc (111) textured Pt layer. This will \nlead to local inversion asymmetry in the multilayer. Similar phenomenon has also been reported for \nCo/Pt\n3\n7,38\n \nand Co/\nPd\n39\n \nmultilayers. \nThis explains why a large SOT is generated when a charge current is \napplied to the multilayer, as we demonstrated previously.\n \nHowever, the observation of SMR in co\n-\nsputtered samples with a magnitude comparable to the multilaye\nr sug\ngests the observed phenomena can \nalso be explained by simultaneous actions of extrinsic SHE and ISHE, particularly in multilayers with \nultrathin FeMn and Pt. Further studies are required to \nevaluate the relative contribution of intrinsic and \nextrinsic\n \nSHE and ISHE in FeMn/Pt multilayers\n \nwith different thickness combinations.\n \n \n \nIV.\n \nCONCLUSIONS\n \nThe static and dynamic magnetic properties of FeMn/Pt multilayers have been studied using \ncombined techniques of magnetometry, \nFMR\n, \nISHE\n \nand \nSMR\n. Despite the fact tha\nt FeMn is an AFM in \nthe bulk phase, FeMn/Pt multilayers with ultrathin FeMn (\nt\nFeMn\n \n< 0.8 nm) and\n \nPt\n \n(\nt\nPt\n \n< 1.0 nm) layers \nexhibit ferromagnetic properties with in\n-\nplane\n \nmagnetic\n \nanisotropy. The temperature dependence of \nsaturation magnetization can be fitt\ned well using a phenomenological model developed for 3D Heisenberg \nmagnet by including a finite distribution in \nT\nC\n. The latter is attributed to the high sensitivity of magnetic \nproperties to subtle changes in the individual layer thicknesses. The finite di\nstribution of \nT\nC\n \ncorrelates well \nwith the broad absorption peaks observed in the FMR spectra. A large damping parameter (~\n \n0.106\n) is \nderived from the frequency dependence of \nFMR\n \nlinewidth, which \nis comparable to the values reported for \nCo/Pt\n \nmultilayers\n. C\nlear \nISHE\n \nsignals and \nSMR\n \nhave been observed in all samples below the Curie \ntemperature, which corroborate the strong \nSOT\n \neffect observed previously. \nThe latter is attributed to the \ncrystalline asymmetry between the top FeMn/Pt and bottom Pt/FeMn interface\ns\n \nwhen the Pt layer is \nrelatively thick. However, for samples with ultrathin Pt, extrinsic SHE/ISHE may play a more important \nrole in the phenomena observed.\n \n 25\n \n \nACKNOWLEDGMENTS\n \nY.H.W.\n \nwould like to acknowledge support by the Singapore National Research Found\nation, Prime \nMinister's Office, under its Competitive Research Programme (Grant No. NRF\n-\nCRP10\n-\n2012\n-\n03) and \nMinistry of Education, Singapore under its Tier 2 Grant (Grant No. MOE2013\n-\nT2\n-\n2\n-\n096). Y.H.W. \nand \nJ.S.C. are \nmember\ns\n \nof the Singapore Spintronics Cons\nortium (SG\n-\nSPIN).\n \n \n 26\n \n \nREFERENCES\n \n \n1\nP. Carcia, J. Appl. Phys. \n63,\n \n5066 (1988).\n \n \n2\nY. Ochiai, S. Hashimoto, and K. Aso, \nIEEE Trans. Magn.\n \n25\n,\n \n3755\n(1989).\n \n \n3\nJ.\n-\nH. Park, C. Park, T. Jeong, M. T. Moneck, N. T. Nufer, and J.\n-\nG. Zhu, J. Appl. Phys. \n103,\n \n07A917 \n     \n  \n(2008).\n \n \n4\nH. Sato, S. Ikeda, S. Fukami, H. Honjo, S. Ishikawa, M. Yamanouchi, K. Mizunuma, F. Matsukura, and \nH. Ohno, \nJpn. J. Appl. Phys.\n \n53,\n \n04EM02 (2014).\n \n \n5\nY. Liu, C. Jin, Y. Fu, J. Teng, M. Li, Z. Liu, and G. Yu, \nJ. Phys. D: Appl. Phys.\n \n41,\n \n205006 (2\n008).\n \n \n6\nY. Yang, Y. Xu, X. Zhang, Y. Wang, S. Zhang, R.\n-\nW. Li, M. S. Mirshekarloo, K. Yao, and Y. Wu, Phys. \nRev. B \n93,\n \n094402 (2016).\n \n \n7\nY. Xu, Y. Yang, K. Yao, B. Xu, and Y. Wu, \nSci. Rep.\n \n6\n, 26180\n \n(2016).\n \n \n8\nS. Maat, K. Takano, S. Parkin, and E. E. Fullerto\nn, Phys. Rev. Lett. \n87,\n \n087202 (2001).\n \n \n9\nC.\n-\nJ. Lin, G. Gorman, C. Lee, R. Farrow, E. Marinero, H. Do, H. Notarys, and C. Chien, J. Magn. Magn. \nMater. \n93,\n \n194 (1991).\n \n10\nD. Weller, L. Folks, M. Best, E. E. Fullerton, B. Terris, G. Kusinski, K. Krishnan, and \nG. Thomas,\n \nJ. \nAppl. Phys. \n89,\n \n7525 (2001).\n \n11\nS. Hashimoto, Y. Ochiai, and K. Aso, J. Appl. Phys. \n67,\n \n2136 (1990).\n \n12\nX. Lang, W. Zheng, and Q. Jiang, Nanotechnology \n18,\n \n155701 (2007).\n \n13\nD. C. Mattis, \nThe Theory of Magnetism Made Simple: An Introduction to P\nhysical Concepts and to Some \nUseful Mathematical Methods\n \n(World Scientific, 2006)\n \np.410. \n \n14\nM. Kuz’min, \nPhys. Rev. Lett. \n94,\n \n107204 (2005).\n \n15\nJ. Le Guillou and J. Zinn\n-\nJustin, \nPhys. Rev. Lett.\n \n39,\n \n95 (1977).\n \n16\nJ. J. Binney, N. Dowrick, A. Fisher, and M. Ne\nwman, \nThe theory of critical phenomena: an introduction \nto the renormalization group\n \n(Oxford University Press, Inc., 1992).\n \n17\nS. Alvarado, M. Campagna, and H. Hopster, Phys. Rev. Lett. \n48,\n \n51 (1982).\n 27\n \n \n18\nU. Gradmann, J. Magn. Magn. Mater. \n100,\n \n481 (1991).\n \n19\nM. D. Kuz’min, M. Richter, and A. N. Yaresko, Phys. Rev. B \n73,\n \n100401 (2006).\n \n20\nC. Kittel, \nPhys. Rev. \n73,\n \n155 (1948).\n \n21\nH. T. Nembach, T. J. Silva, J. M. Shaw, M. L. Schneider, M. J. Carey, S. Maat, and J. R. Childress, \nPhys. Rev. B \n84\n, 054424 \n(2011).\n \n22\nY.\n \nTserkovnyak, A. Brataas, and G. E. Bauer, Phys. Rev. Lett. \n88,\n \n117601 (2002).\n \n23\nS. Mizukami, E. Sajitha, D. Watanabe, F. Wu, T. Miyazaki, H. Naganuma, M. Oogane, and Y. Ando, \nAppl. Phys. Lett. \n96,\n \n152502 (2010).\n \n24\nA. Caprile, M. Pasquale, M. Kuepferling, \nM. Coïsson, T. Y. Lee, and S. H. Lim, \nIEEE Magn. Lett.\n \n5,\n \n3000304\n \n(2014).\n \n25\nS. Mizukami, Y. Ando, and T. Miyazaki, Jpn. J. Appl. Phys. \n40\n, 580 (2001).\n \n26\nX. Qiu, W. Legrand, P. He, Y. Wu, J. Yu, R. Ramaswamy, A. Manchon, and H. Yang, Phys. Rev. Lett.  \n117\n, \n217206 (2016). \n \n27\nY. Liu, Y. Fu, C. Jin, and C. Feng, Rare Met. \n29,\n \n473 (2010).\n \n28\nL. Chen, F. Matsukura, and H. Ohno, Nat. Commun. \n4,\n \n2055 (2013). \n \n29\nA. Tsukahara, Y. Ando, Y. Kitamura, H. Emoto, E. Shikoh, M. P. Delmo, T. Shinjo, and M. Shiraishi, \nPhys. R\nev. B\n \n89\n, 235317\n \n(2014).\n \n30\nM. Harder, Y. Gui, and C.\n-\nM. Hu, Phys. Rep. \n661\n, 1 (2016).\n \n31\nO. Mosendz, V. Vlaminck, J. E. Pearson, F. Y. Fradin, G. E. W. Bauer, S. D. Bader, and A. Hoffmann, \nPhys. Rev. B \n82\n, 214403\n \n(2010)\n.\n \n32\nH. Chen, X. Fan, H. Zhou, W. Wang,\n \nY. S. Gui, C. M. Hu, and D. Xue, \nJ. Appl. Phys. \n113,\n \n17C732 \n(2013)\n.\n \n3\n3\nH. Nakayama, M. Althammer, Y.\n-\nT. Chen, K. Uchida, Y. Kajiwara, D. Kikuchi, T. Ohtani, S. Geprägs, \nM. Opel, and S. Takahashi, Phys. Rev. Lett. \n110,\n \n206601 (2013).\n 28\n \n \n34\nA. Manchon, \nPhys. Sta\ntus Solidi RRL\n \n11\n, 1600409\n \n(2017).\n \n35\nM. Jamali, K. Narayanapillai, X. Qiu, L. M. Loong, A. Manchon, and H. Yang, Phys. Rev. Lett. \n111\n, \n246602 (2013)\n.\n \n36\nS. Vélez, V. N. Golovach, A. Bedoya\n-\nPinto, M. Isasa, E. Sagasta, M. Abadia, C. Rogero, L. E. Hueso, \nF. S\n. Bergeret, and F. Casanova, Phys. Rev. Lett. \n116\n, 016603 (2016).\n \n3\n7\nR. Lavrijsen, P. Haazen, E. Mure, J. Franken, J. Kohlhepp, H. Swagten, and B. Koopmans, Appl. Phys. \nLett. \n100,\n \n262408 (2012).\n \n3\n8\nS. Bandiera, R. Sousa, B. Rodmacq, and B. Dieny\n, \nIEEE Magn. \nLett. \n2,\n \n3000504 (2011).\n \n3\n9\nD.\n-\nO. Kim, K. M. Song, Y. Choi, B.\n-\nC. Min, J.\n-\nS. Kim, J. W. Choi, and D. R. Lee, \nSci. Rep. \n6\n,\n \n25391 \n(2016).\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 29\n \n \nFIGURE CAPTIONS\n \n \nFIG. 1. X\n-\nray diffraction pattern of \nPt(1)/[FeMn(0.6)/Pt(0.4)]\n10\n. Dotted lines indicate t\nhe (\n111\n) peak \nposition of Pt and FeMn, respectively.\n \n \nFIG. 2. XRR patterns of \nPt(1)/[FeMn(0.6)/Pt(0.6)]\n30\n \nmultilayer sample \n(red solid\n-\nline)\n \nand co\n-\nsputtered \nsample (blue dotted\n-\nline) deposited under the same condition.   \n \n \nFIG. 3. (a) Hysteresis loop of \nP\nt(1)/[FeMn(0.6)/Pt(0.3)]\n10\n \nmeasured at 50 K (square) and 300 K (circle), \nrespectively. (b) Saturation magnetization as a function of temperature. The legend (\nt\n1\n,t\n2\n) denotes a \nmultilayer with a FeMn thickness of \nt\n1\n \nand Pt thickness of \nt\n2\n. The number of peri\nod for all samples is fixed \nat \n10\n.\n \n \nFIG. 4. (a) Experimental \nM\n-\nT \ncurves (open symbols) and fitted results (solid lines). The experimental data \nare the same as those shown in Fig. 3b, but are shifted for clarity (except for the \nt\nPt\n \n= 0.1 nm sample). (b) \nM\n0\n,\n \n(c) T\nC0\n \n(triangle) and \nΔ\nT\nC\n \n(square), and (d) \ns,\n \nas a function of \nt\nPt\n \nobtained from the fittings. \n \n \nFIG. 5. (a) FMR spectra of \nPt(1)/[FeMn(0.6)/Pt(0.5)]\n80\n \nat fixed frequency ranging from 2 GHz to 4 GHz. \n(b) Data (square symbol) and fitting (line) for FMR s\nignal at \nf \n= 3 GHz. (c) Full width at half maximum \nof the resonance pea\nk (triangle symbol) are plotted versus\n \nthe frequency. The solid line is a linear fit to \nthe data. \n \n 30\n \n \nFIG. 6. (a) Measurement geometry of ISHE and FMR. (b) ISHE and (c) FMR spectra for \nPt\n(1)/[FeMn(0.6)/Pt(0.5)]\n50\n \nmeasured at 3.0 GHz. (d) V oltage signal as a function of positive (circle) and \nnegative (square) magnetic field for Pt(1)/[FeMn(0.6)/Pt(0.4)]\n10\n \nat 3 GHz. (e) Decomposition of measured \nvoltage signal for Pt(1)/[FeMn(0.6)/Pt(0.4)]\n10\n \nat 3GHz into symmetric  and antisymmetric  components. \nSymbols are raw data as shown in (d). Dash\n \ndotted and dashed lines show the symmetric and \nantisymmetric components, respectively. The solid\n-\nline shows the combined fitting results. \n \n \n \nFIG. \n7\n. (a) Geom\netry of angle\n-\ndependent MR measurement. (b) Angle\n-\ndependent MR of \nPt(1)/[FeMn(0.6)/Pt(0.3)]\n10\n. (c) Data (square symbol) and fitting (line) of SMR ratio as a function of \nt\nPt\n \nfor FeMn(3)/Pt(\nt\nPt\n) bilayers. Inset shows the calculated SMR (line) for FeMn(0.6)/P\nt(\nt\nPt\n) bilayers\n \nat small \nPt thickness as well as experimentally obtained SMR ratio (triangle symbol) for Pt(1)/[FeMn(0.6)/Pt(\nt\nPt\n)]\n10\n \nmultilayers. \n \n \nFIG. 8. (a) Angle\n-\ndependent MR of co\n-\nsputtered sample; (b) AMR and (c) SMR of co\n-\nsputtered and \nmultilayer sa\nmples with same nominal composition and thicknesses.\n \n "    },    {        "title": "2001.09732v2.Resonant_thermal_energy_transfer_to_magnons_in_a_ferromagnetic_nanolayer.pdf",        "content": "1Experimentelle Physik 2, Technische Universität Dortmund, Otto -Hahn -Str. 4a, 44227 Dortmund, \nGermany.  2Ioffe Institute, Politechnycheskaya 26, 194021 St. Petersburg, Russia.  3Department of \nTheoretical Physics, V.E. Lashkaryov Institute of Semiconductor Physics, Pr. Nauky 41, 03028 Kyiv, \nUkraine. 4 LAUM, CNRS UMR 6613, Le Mans Universit é, 72085 Le Mans, Fra nce. 5School of Physics and \nAstronomy, University of Nottingham, Nottingham NG7 2RD, United Kingdom.  \n*e-mail: michal.kobecki@tu -dortmund.de ; alexey.shcherbakov@tu -dortmund.de .  \n Resonant thermal energy transfer to magnons  \nin a ferromagnetic nanolayer  \nMichal Kobecki1*, Alexey V. Scherbakov1,2*, Tetiana L. Linnik3, Serhii M. Kukhtaruk1,3, \nVitalyi E. Gusev4, Debi  P. Pattnaik5, Ilya A. Akimov1,2, Andrew W. Rushforth5,   \nAndrey V. Akimov5, and Manfred Bayer1,2 \nEnergy harvesting is a modern concept which makes dissipated heat useful by transferring thermal \nenergy to other excitations. Most of the existing principles for energy harvesting are realized in \nsystems which are heated continuously, for example generating DC voltage in thermoelectric \ndevices. Here we present the concept of high -frequency energy harvesting where the dissipated \nheat in a sample excites resonant  magnons in a 5 -nm thick ferromagnetic metal layer. The sample \nis excited by femtosecond laser pulses with a repetition rate of 10 GHz which results in temperature \nmodulation at the same frequency with amplitude  ~0.1 K. The alternating temperature excites \nmagnons in the ferromagnetic nanolayer which are detected by measuring the net magnetization \nprecession. When the magnon frequency is brought onto resonance with the optical excitation, a \n12-fold increase of the amplitude of precession indicates efficient resonant heat transfer from the \nlattice to coherent magnons. The demonstrated principle may be used for energy harvesting in \nvarious nanodevices operating at GHz and sub -THz frequency ranges.  \nThe transfer of thermal energy to mechanical, electrical or magn etic excitations is of great \ninterest and is widely considered for energy harvesting when waste heat is transferred to more \nusable types of energy [1]. The rational utilization of heat is a critical task for nanoscale \nelectronics, where operations are acco mpanied by extensive production of parasitic heat. \nNanoscale devices for communication and computing operate with digital signals, which are \ngenerated by ultrafast current or optical pulses with high repetition rate. The heat generated  in \nthis case is also  modulated at the same frequency and could be transferred to non -thermal types \nof excitation which become the elements of the energy harvesting process. This process would \nbe most efficient in the case of resonance, when the modulated heat is transferred  \nto a sub -system with the same intrinsic frequency. Resonant heat transfer is widely used  in \nphoto -acoustics and was proposed for the first time by Bell in 1881 [2]. There the modulated \noptical signal is absorbed in a system with acoustic resonance at the f requency of modulation \nand as a result the modulated heat resonantly excites the acoustic wave. Modulation \nfrequencies in photo -acoustics do not exce ed the MHz frequency range [3], while \nthermomodulation with frequencies up to 400 GHz [4] is used in picose cond ultrasonics for \nstudying sub -THz coherent phonon dynamics.  \nHere we propose to convert high frequency (GHz) modulated heat to magnons, which \nare collective spin excitations in magnetically ordered materials. The manipulation of coherent \nhigh -frequency magnons on the nanoscale is one of the most prospective concepts  for \ninformation technologies [5], also in the quantum regime [6]. While the most common way to \nexcite coherent magnons is non -thermal and based on microwave techniques [7], thermal \nmethods ha ve proven successful in ferromagnetic metals. These methods are based on ultrafast \nmodulation of the magnetic anisotropy induced by rapid lattice heating, e.g. due  to the  \n2 \n absorption of optical pulses [8,9]. In ferromagnetic metals, the intrinsic magnon fre quency \ndepends on the external magnetic field and may be varied between ~1 and ~100 GHz [10]. This \nrange covers the clocking frequency for most electronic devices and, thus, magnons are suitable \nto receive dissipated heat modulated at the resonant frequenc y.  \nIn the present paper, we introduce the concept of transferring heat dissipated  in a \nsemiconductor from the relaxation of hot electrons excited by a 10 -GHz laser to coherent \nmagnons. In our experiments the energy harvester is a metallic ferromagnetic fi lm of 5 -nm \nthickness grown on a semiconductor substrate. We demonstrate an efficient heat –magnon \ntransfer by measuring a 12-fold increase of the fundamental magnon mode amplitude  at the \nresonance conditions. The experiments and related theoretical analysis  show that GHz \nmodulation of the temperature on the scale of ~ 0.1 K is sufficient for the excitation of magnons \nwith amplitude reliable for the operation of spintronic devices.  \nResults  \nThe ferromagnetic energy harvester is a 5 -nm layer of Galfenol (Fe 0.81Ga 0.19) grown  by \nmagnetron sputtering on a (001) semi -insulating GaAs substrate and covered by a 2 -nm Cr cap \nlayer to prevent oxidation. The chosen composition of Fe and Ga is characterized by a high Curie \ntemperature ( Tc≈900 K) and large saturation magnet ization M0 [11].  Experiments were carried \nout at ambient conditions at room temperature  with an external magnetic field, B, applied in \nthe layer plane at an angle -/8 from the [100] crystallographic direction, which corresponds to \nthe maximal sensitivity  of the magnetization, M, to the temperature -induced changes of the \nmagnetic anisotropy [12]. In the studied layer the lowest, fundamental mode of the quantized \nmagnon spectrum is well separated from the higher -order modes due to the large exchange \nmode splitting [12]. As a result, the magnon spectrum consists of a narrow single spectral line  of \nLorentzian shape with a width of ≈500 MHz. Example s of the magnon spectrum  for several \nvalues of B measured by monitoring the magnetization precession excited by a single optical \npulse are shown in Fig. 1a (for details see Methods section). The experimentally measured \ndependence of magnon frequency f on B is shown by  the symbols  in the right panel  of Fig. 1a .   \nFig. 1b illustrates a schematic of the pump -probe expe riment for magnon energy \nharvesting from high -frequency modulated heat. A femtosecond pump laser with repetition rate \nf0=10 GHz and  average power up to W=150 mW is used for modulation of the lattice \ntemperature . A second laser with repetition rate of 1 GHz and average power of 18 mW is used \nto probe the response of the magnons to the high -frequency heat modulation by means of the \ntransient polar Kerr rotation effect. The pump and probe beams are focused on the Fe 81Ga 19 \nlayer into overlapping spots of 1 7 m and 14 m diameter, respectively, using different sectors  \nof the same reflective microscope objective (for details see the Method s section) . \nFig. 1c (upper panel) shows the calculated temporal evolution of the lat tice temperature \nT(t) in the Fe 0.81Ga 0.19 layer under 10 -GHz pump optical excitation. The calculations were \nperformed by solving the heat equations taking into account the thermal resistance at the \nFe0.81Ga 0.19/GaAs interface (see Methods section  and Suppl ementary Notes 1 and 2 ). The \ntemperature modulation amplitude 𝛿𝑇~𝑊/(𝐶𝑟2), where C and r are the heat capacity and \nradius of the excitation spot.  It is seen that the lattice temperature oscillates with amplitude T \non a stationary background which exceeds the room temperature T0 by T. Both T and T \nincrease linearly with W: 𝛿𝑇=𝑝𝑊 and Δ𝑇=𝑃𝑊 where p and P are constants which for our \nstructure   equal   to 4.9 K/W  and  510 K/W  respectively.  The   temperature   modulation  is  not    \n3 \n harmonics at the frequencies fn = nf0, where n is an integer. The harmonic amplitude decreases \nwith the increase of n.   \nThe idea of the experiments illustrated in Fig. 1a is to exploit the periodic thermal  \nmodulation for exciting coherent magnons and to monitor the magnetization precession by \ntransient Kerr rotation (KR) measurements of the out -of-plane magnetization projection, 𝛿𝑀𝑧, \nfor various values of B. We expect a resonant increase of the precession  amplitude at the \nresonances  when B=B n, which correspond to f = fn. The expected values  of Bn for the first three \nharmonics are shown in the right panel of Fig. 1a by the vertical arrows. The idea is analogous \nto the excitation of a harmonic oscillator with tunable eigenfrequency f by a periodic force , \nwhich acts on the oscillator with repe tition rate  f0. For magnetization precession,  this force is \ngenerated by the modulated temperature T(t), and the magnon eigenfrequency f is controlled \nby the external magnetic field.  \nFigure 2 shows the transient K R signals measured for W=95 mW and  various values of \nB. The signals measured in the vicinity of the first resonance ( n=1) have  a harmonic shape. At \nthese conditions, we fit the si gnals by a sine function  𝛿𝑀 𝑧(𝑡)=𝐴𝐵sin(𝜔1𝑡+𝜑𝐵) where \nω1=2πf1, and AB and φB are the magnetic field -dependent amplitude and phase of the harmonic \noscillations. Times t=0 and 100 ps correspond  to excitation of the sample by the pump pulses. It \nis seen that the amplitude  of the signal is maximal  when B=B 1=44 mT, which corresponds to the \nfirst resonance f=f1 (see Fig. 1a ). In the vicinity of the resonance the phase changes from  \nφB= -π/2 for B<B 1 to φB=+π/2 for B>B 1. At the  resonance φB≈0. Similar results  are observed for \nthe signals measured in the vicinity of the  second resonance ( n=2). The amplitude of the \noscillations  at B=B2=250 mT are 1.4 times smaller  than for the first resonance and the same \nconclusions  as for the first resonance can be made concer ning the field dependences of the \namplitude  and phase of the signal around B2. Away from  the resonances the signals are periodic \nbut not harmonic. An example of a non -resonant signal measured at B=190 mT is presented in \nthe right panel of Fig. 2. The amplitude of the signal is much smaller than in the case of resonance \nand the temporal shape is impossible to fit with a single sine function.   \n \nFigure 1 .  a Fast Fourier transform spectra showing the fundamental magnon mode for several values \nof magnetic field, B, (left panel) and the field dependence of its frequency, f, (right panel) for the 5 -nm \nthick Fe 0.81Ga0.19 energy harvesting layer. Symbols show f(B) obtained by fast Fourier transformation \nof Kerr rotation signals measured in a single pulse pump -probe experiment. Solid lines are calculated \ndependences for ΔT=0 K (upper curve) and ΔT=200 K (lower curve). The d ashed horizontal lines show \nthe frequencies of the harmonics in the temperature modulation spectrum induced by the 10 -GHz \noptical excitation. The vertical arrows indicate the expected resonances for the fundamental magnon \nmode at these harmonics. b Scheme of the experiment. c Calculated temporal evolution of the \nFe0.81Ga0.19 lattice temperature induced by the 10 -GHz optical excitation of the Fe 0.81Ga0.19/GaAs \nheterostructure at the pump excitation power W=95 mW (upper panel) and its  Fourier spectrum (lower \npanel).  \n0200 400 600102030\nFrequency (GHz) B3 B2 B1n=3\nn=2\nMagnetic field (mT)n=1\n102030400 mT\n35 mT100 mT250 mT Frequency (GHz)B=550 mT\nb\n48.048.549.00.0 0.2 0.4 0.6 0.8 1.0\n Time (ns)\nTT T(t)-T0 (K)\nT\nAmplitudea c\n010 20 30 40 505432\n Frequency (GHz) Amplituden=1 \n4 \n  \nFigure 2. Transient Kerr rotation signals measured in the vicinity of the first (left panel) and second \n(central panel) resonances and at the intermediate magnetic field of 190 mT, corresponding to off -\nresonance (right panel).  The amplification of the precession am plitude observed at B=B1=44 mT and \nB=B2=250 mT is due to thermally -induced resonant driving of the precession M by the oscillating effective \nfield B eff. The torque Q acting on the magnetization (see sketch at the top right) is maximal at the \nresonance conditions: f=nf0. \n \nFigure 3. a  Measured magnetic field dependences of the root mean square (RMS) amplitude of the Kerr \nrotation signal, 𝐴𝐵̃,  for three values of pump excitation power, W. The vertical arrows indicate the \nmagnetic fields Bn at which the resonance condition f=fn is fulfilled; the horizontal bars indicate zero signal \nlevels. b Measured (symbols) and calculated (solid lines) zoom ed fragments of the field dependences of \n𝐴𝐵̃ (upper panel) and the phase (lower panel) in the vicinity of the first resonance ( n=1) for W=95 mW; \nthe dashed horizontal line shows 𝐴𝐵̃  at the intermediate off -resonance field. c Calculated field \ndependences  of the RMS precession amplitude for the background temperatures ΔT obtained from the \nexperimental dependences in a. d Power dependence of the relative precession amplitude (left axis) and \nthe corresponding precession angle (right axis) at the first resona nce (n=1) when f=f0. \n \n-50 0 50 100-2-10123456\n50 mT\n44 mTKR angle ( rad)\nTime (ps)30 mT\nn=1\n-50 0 50 100out of resonance\nTime (ps)190 mT\n-50 0 50 100 260 mT\n250 mT\nTime (ps)240 mT\nn=2Q\nBeffM\n0 100 200 300 400 500n=3n=2\n95 mW\n55 mW\n RMS, AB (arb. units)\n35 mWn=1~\n-20 -10 0 10 20n=1Phase\nB-B1 (mT)\n0 50 100 150 2000.000.050.100.150.20\n  (degree) mz (%)\n W (mW)n=1\n0 100 200 300 400 500RMS, AB (arb. units)Magnetic field (mT)\nMagnetic field (mT)77 \n44 \n28 n=1\nn=2\nn=3\nT (K)\n~n=1RMS, AB (arb. units)W=95 mW\n0.00.20.40.6a b\nc\nd \n5 \n To plot the dependence of the measured signal amplitude as a function of B we present \nthe root -mean -square  (RMS)  amplitude 𝐴̃𝐵 of the measured KR signal. The symbols in Fig. 3a \nshow the field dependences 𝐴̃𝐵(𝐵) for three pump excitation powers W. It is clearly seen that  \nthe dependences have peaks at the resonant values of Bn corresponding to n = 1, 2 and 3 .  \nFig. 3b shows zoomed fragments of the field dependence s of the amplitude and phase, \nrespectively, obtained for the first resonance at W=95 mW. The 1 2-fold increase of 𝐴̃𝐵 at the \nresonance condition is  clearly seen by comparison with the out -of-resonance RMS amplitude \nshown  by the horizontal dashed line  in the  upper panel of Fig. 3b . No peaks in the dependence \nof the precession amplitude on B are detected in the case of single pulse excitation , where  𝐴̃𝐵 \ngradually decreases with the increase of B.   \nThe values of the measured resonance fields Bn shift to slightly higher fields when W \nincreases. We explain this shift by the heat -induced decrease of the magnon frequencies [1 3] \nand respective increase of the magnetic field value required for achieving the resonance \nconditions . We use the values of t his shift to obtain the background temperature Δ T of the \nGalfenol film by comparison with the known  dependence of f(B) on temperature . Two \ndependences f(B) calculated using the known dependen ce of the Galfenol magnetic parameters \non temperature [ 13,14 ] are demonstrated in the right panel of Fig. 1a  by the solid lines (for \ndetails see Supplementary Note 2). The corresponding values  of the background temperature \nobtained from the experimentally measured shift s of the resonances are T=28, 44 and 77 K for \nW=35, 55 and 95 mW, respectively. They are 40% higher than the values calculated theoretically \nfrom the heat equations. We attribute this difference to the additional background heating by  \nthe probe beam  that is not considered in the theoretical modeling.   \nDiscussion  \nIn the analysis, we consider thermal modulation  of the magnetic anisotropy  as the main \nmechanism for magnon excitation in our experiment and do not take into account other \nmechanisms (e.g. thermal strain).  This approach is based on previous experiments and \ntheoretical analysis where various mechanisms of  laser -pulse induced excitation  of magnons in \nGalfenol were considered [14 ]. We also exclude the effect  of thermal gradient s inside a \nferromagnetic film (we esti mate 4% temperature difference across the film) . This gradient in \nspecial  cases can induce ac -spin transfer [15,16] modulated at the laser repetition rate, f0. In our \nexperiment, d ue to the uniform in-plane magnetization in a single thin ferroma gnetic layer, this \ntransfer does not produce a torque on the magnetization  [17]. The dominating role of the \nthermal modulation of the magneto crystalline anisotropy is also confirmed by the dependence \nof the precession amplitude excited by a single laser pu lse on the direction of the external \nmagnetic field . This dependence demonstrates a four -fold in-plane symmetry with a slight \nuniaxial distortion, w hich corresponds to the magneto crystalline anisotropy of the studied layer  \n[12]. \nWe describe t he effect of t hermal modulation on the magnetization, M, by consider ing \nthe magnetization precession motion in a time -dependent effective field Beff as schematically \nshown in Fig. 2  [14]. The magnetization precession is  described by the Landau -Lifshitz -Gilbert  \n(LLG)  equation [7]:  \n         𝑑𝐦\n𝑑𝑡=−𝛾0𝐦×𝐁eff (𝑡)+𝛼0𝐦×𝑑𝐦\n𝑑𝑡,                (1)  \nwhere  𝐦=𝐌/𝑀0 is the normalized magnetization  and 𝛼0 is the Gilbert damping parameter. \nThe effective magnetic field is determined as 𝐁𝐞𝐟𝐟=−∇𝐦𝐹𝐌(𝐦,𝑡),  where FM  \nis the normalized free energy density  [7]:  \n6 \n 𝐹𝑀(𝑚)=−(𝐦∙𝐁)+𝐵𝑑𝑚𝑧2+𝐾1(𝑚𝑥2𝑚𝑦2+𝑚𝑧2𝑚𝑦2+𝑚𝑥2𝑚𝑧2)−𝐾𝑢(𝐦∙𝐬)2,  (2) \nwhere  the first term  is the Zeeman energy, the second term is the demagnetization energy  \n(𝐵𝑑=𝜇0𝑀0\n2) and the following two terms describe the cubic and uniaxial magnetocrystalline \nanisotropy with respective  coefficients K1 and Ku, with  the unit vector s|| [110] along the uniaxial \nanisotropy axis. In the chosen coordinate system, the x, y, and z axes  correspond to the main \ncrystallographic directions [100], [010]  and [001] (normal to the layer plane), respectively. At \nequilibrium, i.e.  without  temperature modulation , the value of Beff determine s the fundamental \nmagnon frequency , while its direction se ts the equilibrium orientation of the magnetization  m. \nThe change of temperature alters Beff through the temperature dependent parameters M0, K1 \nand Ku. Within the studied temperature range their dependence on T is linear  [13]: 𝑋=𝑋𝑅𝑇+\n𝛽𝑋(𝑇−𝑇0), where X = M0, K1 or Ku, the index RT indicates the room -temperature value s and 𝛽𝑋 \nis the corresponding thermal coefficient . The room temperature values  𝑀0𝑅𝑇= 1.96 T, 𝐾1𝑅𝑇= 20 \nmT and 𝐾𝑢𝑅𝑇= 9 mT  for our sample  are found from fitting the experimental dependence f(B) \n(shown in  the right panel of  Fig 1a ) [12,14 ]. The thermal coefficients,  𝛽𝑀0=−0.97 mT/K,  \n𝛽𝐾1=−0.046  mT/K  and 𝛽𝐾𝑢=−0.025 mT/K for Galfenol  are taken from previous  \nstudies [ 13,14]. The Gilbert damping coefficient = 0.00 6 is obtained from the precession \nkinetics excited by a single laser pulse.      \nThe m odulation amplitude  T<<T and for solving Eq. (1)  we assume that  the magnitude  \nof 𝐁𝐞𝐟𝐟 is constant and the temperature modulation affects only its direction. We also consider \nonly the ground magnon mode with zero wave vector , i.e. we assume a uniform precession , due \nto the large diameter ( 17 μm) of the excitation spot, which limits the in -plane wave vector of  \nmagnons  to 𝑞||≤ 4700 cm-1. In the 5 -nm thick  Galfenol layer  the frequency range  for such 𝑞|| \ndoes not exceed 270 MHz [ 18, 19], which is clearly below  the 500-MHz spectral width of the \nfundamental magnon mode. For details of the magnon dispersion and contribution of magnons \nwith finite wave vectors in a 100 -nm thick layer, see Supplementary Note 4 .   \nWith the se assumption s, the LLG equation can be reduced  to a system of equations (see \nthe Methods  section for details ) for the magnetization components, where the thermal \nmodulation of the anisotropy coefficients alters  the direction of Beff and generates a tangential \ndriving torque, Q, acting on the magnetization [1 4] as schematically shown in the in set of Fig. 2.  \nThe vector Beff follows the temporal temperature evolution T(t) shown in Fig. 1c, which can be \nwritten as sum of background heating and a Fourier series expansion for the  small periodic \ntemperature modulation:  \n 𝑇(𝑡)=𝑇0+Δ𝑇+𝛿𝑇∑ 𝑢𝑛sin(𝜔𝑛𝑡+𝜓𝑛)∞\n𝑛=1 .   (3) \nHere the amplitude and phase of the nth harmonic are  𝑢𝑛=2\n𝜋𝜔1𝜏0   \n√1+𝜔𝑛2𝜏02    and  \n𝜓𝑛=arctan (1\n𝜔𝑛𝜏0) , respectively, 𝜔𝑛=2𝜋𝑓𝑛, 𝜏0 is the characteristic cooling time of the \nGalfenol layer, which is determined by the thermal boundary resistance, R, \nat the interface with GaAs [17]. T he solution of the linearized Eq. (1) for the mz- component \nexpanded as a Fourier series has the form:  \n        𝛿𝑚𝑧(𝑡)=∑ 𝐴𝑛(𝐵)sin (𝜔𝑛𝑡+𝜑𝑛(𝐵))∞\n𝑛=1  ,        (4)  \nwhere An(B) and φn(B) are the time -independent amplitude and phase  for the nth harmonic. Like \nin the case of a simple harmonic oscillator, at the resonant conditions f(B) = fn the precession \namplitude reaches  maximum with zero phase -shift relative to the driving force 𝜑𝑛(𝐵)≈0. The \nanalytical expressions for An(B) and φn(B) are  presented in the Methods section.   \n7 \n Figure 3c shows the field dependences of the RMS amplitude of the calculated mz(t) \nfor three values of the background temperature T which correspond to the W indicated near \nthe experimentally measured data in Fig. 3a . The agreement  of the measured and calculated \nspectral shapes  is explicitly demonstrated in the zoomed  dependences of the RMS amplitude \nand phase in Fig. 3b, where the calculated  dependences (solid lines) for n=1 are presented \ntogether with the experimental  values. Good  agreement is observed also for higher n and W.  \nThe measured power dependence of the resonant magnon amplitude for n=1 in the \nstudied sample is shown in Fig. 3d. The precession amplitudes are obtained from the absolute \nvalues of the Kerr rotation angle in the tran sient Kerr signal s [12]. The field independent small \namplitude background, which is due to the response of the electron system and not related to \nthe magnetization [ 21], has been  subtracted from the experimental values. At W=100 mW the \nprecession amplitude reach es 0.1% of the saturation magnetization M0. To get this value,  the \nparameters of the magnetic anisotropy have  to be modulated by the oscillating  temperature \nwith the amplitude T=0.5 4 K. This is in perfect agreement with the values obtained by solving \nthe thermal equations  which for the same power give T=0.49  K. The achieved p recession \namplitude , which correspond s to the precession angle  ≈ 0.25o (with  the ellipticity of 0.2) and \nthe generate d ac- induction Mz ~ 1 mT , is large enough for prospective spintronics applications: \ndetection of spin currents by spin pumping [ 22,23]; manipulating single spin states of NV- \ncenters in nanodiamonds by microwave magnetic fields [2 4]; excitation of propagating sp in \nwaves in magnonic devices [25 ].  \nFor the practical use of resonant energy harvesting in a device excited optically or \nelectrically, it is important to achieve high efficiency of the transformation of the modulated \npart of the thermal  energy into the oscillations of magnetization. This efficiency is defined by \nthe ratio of the magnetization  angle  of precession and the power W injected into  a device. In \nour experiment,  we demonstrate  W = 2.5 °/W which  is only one order of magnitude less than \nthe efficiency of conventional mesoscopic microwave devices  [26] and of the same order as \ndevices utilizing surface acoustic waves for spin pumping [ 27]. At resonance, t he magnetization \nprecession remains  harmonic up to the maxim um used power , and, thus, non-linear \n(anharmonic) effects do not affect  the transformation efficiency. However, the shift of the \nresonance frequency with the increas ing T at higher W affect s the linear power dependence \nof the precession amplitude at fixed magnetic field . This temperature -induced “nonlinearity” is \nknown from  the conventional microwave experiments on ferromagnetic resonance [ 28]. The \neffect of this nonlinear ity is very weak in the studied structure. For instance, for the second \nresonance ( n=2) the 20% -change of the background temperature without chang ing thermal \nmodulation amplitude results in a ~3% decrease of the precession  amplitude . Thus, stability of \nthe background temperature is not critical for the suggested concept.  \nThe dependences of T on the parameters of the used structure and materials define \nthe efficiency of the heat transfer to the magnons. For optical excitation , shorter penet ration \ndepth s and smaller hea t capacit ies of the ferromagnetic layer increase T and correspondingly \n/W. It is convenient to define the dimensionless resonant  harvesting efficiency, =T/T, \nwhich governs the ratio of the useful temperature modulation T relative to the parasitic \nheating, T. Obviously, i t is favorable to have  as large as possible. For instance, faster cooling \nto the bulk of the substrate leads to an increase of T with a simultaneous  decrease of T and \ncorrespondingly to an increase of  .  In our particular experiment, where both the Fe 0.81Ga 0.19 \nnanolayer and the GaAs substrate are excited optically we get ~ 10-2. The value of  increases \nwith the decrease of the thickness of ferromagnetic layer and the increase of substrate  \n8 \n thermoconductivity (see Supplementary Note 3). For instance, for the same ferromagnetic layer \non a silicon substrate increases by a factor of  1.5. An efficient way to increase  and the \nefficiency of the heat -magnon transfer would be to use a multilayer  structure where T is \ngoverned by the rapid escape of heat to the substrate while the high frequency thermal waves \nwhich define T are localized inside or in the vicinity of the harvester layer. To demonstrate that \nthe concept is applicable for electrical  nanodevices we have considered the case of a thin \nconducting layer (e.g. graphene) where Joule heat can be generated as a result of passing GHz \ncurrent pulses  and transferred to the ferromagnetic harvester through a thin spacer layer. For \nhigh efficiency the spacer layer should have high thermal conductivity while the ferromagnetic \nlayer should have low specific heat  capacity . In the example presented in the Supplementary \nNote 3 we  get  410-3. \n In our experiments, we excite resonances up to 30 GHz. To reach higher, sub -THz \nfrequencies, the corresponding bandwidth should be available for both temperature and \nmagnon modulation. For temperature modulation , we refer to the picosecond acoustic \nexperim ent [ 4] where the temperature in a 12-nm Al film was modulated at a frequency up to \n400 GHz. The  thermal properties of ferromagnetic metals do not differ much from normal metals \nand thus s imilar  modulation in thin films is achievable . For optical excitatio n of metals , it is \nessential that optically excited hot electrons pass the ir energy to the lattice in a time less than 1 \nps. Theoretically, for the parameters fixed as above the amplitude of the temperature \noscillations is proportional to the laser power a nd inverse ly proportional to the frequency up to \nsub-THz frequencies . For the magnons  in thin (Fe,Ga) films , narrow -band precession with \nfrequencies up to 100 GHz was demonstrated  at strong magnetic field s  [12].  Moreover, novel \nmetallic ferrimagnetic materials (e.g. Mn -Te compounds) possess narrow magnon resonances \nat frequencies up to hundreds of GHz at low magnetic fields [ 27]. At the lower frequency end, \nthere is no limit for temperature modulation while fo r magnons the lower frequency varies from \nhundreds of MHz (e.g. for garnets) up to several GHz in metallic ferromagnets. The available \nchoice of a proper magnetic material is not limited to the Galfenol  used in our study. This novel \nferromagnet  is actively  studied nowadays [ 30] and presents  a good example for realization of \nthe demonstrated concept due to its pronounced magnetocrystalline anisotropy and narrow \nmagnon resonances (long lifetime of precession) . However, other ferro - and ferrimagnetic \nmaterials  possessing similar properties will also work as resonant harvesters  for transferring \nparasitic heat modulated at high frequenc ies to magnons .  \nTo conclude, we have demonstrated how the heat generated during 10 GHz pulsed \noptical excitation is used for the excitation of resonant magnons. The amplitude of the \nmagnetization precession at the fundamental magnon frequency increases enormously when \nthe repetition rate of the optical pulses is equal to the magnon frequency. The resonances also \noccur at the higher harmonic frequencies, i.e. 20 and 30 GHz. An amplitude of the temperature \nmodulation of order ~ 0.1 K is sufficient to generate AC magnetization  with an amplitude of 1 mT \n[31] which is sufficient  to be exploited in various applications including information and \nquantum technologies. The concept of resonant heat transfer can be considered as a \nprospective method to generate magnons in ferromagnetic  layers deposited on processors and \nother microchips operating at GHz and sub -THz clock frequencies.  \nMethods  \nPump -probe measurements  with 10 -GHz repetition rate : We use d two synchronized Titanium \nsapphire lasers (Taccor x10 and Gigajet 20c from Laser Quan tum) generating 50 fs pulses with \nrepetition rates of 10 GHz (pump) and 1 GHz  (probe)  at center wavelength s of 810 and 780 nm, \nrespectively.  The beams were  focused onto the sample using a reflective microscope objective with  \n9 \n a magnification factor of 15 comprising 4 sectors through which light could enter and exit the \nobjective [ 32]. The incidence angles of the laser beams are 17 °. The spot diameters for the pump \nand probe beam s were set to 17  μm and 14 μm, respectively.  The diameters are defined at the 1/e2-\nlevel of the Gaussian spatial energy distribution s. The coherent response of the magnetization was \nmeasured by monitoring the polar Kerr rotation of the reflected probe pulses using a differential \nscheme based on a Wollaston prism and a balanced optic al receiver with 10 -MHz bandwidth. \nTemporal resolution was achieved by means of an asynchronous optical sampling (ASOPS) technique \n[33], wher e we used an offset fre quency of 20 kHz between the 10 GHz laser and the 10th harmonics \nof the 1-GHz laser in order to resolve the transient signals in 100 ps time window between pump \npulses  with the time resolution of 50 fs.  The sample was mounted between the poles of a dipole \nelectromagnet.   \nMeasurements of the magnon spectrum with single pulse exc itation. The magnon spectrum of the \nstudied sample was obtained by a conventional magneto -optical pump -probe scheme based on two \nmode -locked Erbium -doped ring fiber lasers (FemtoFiber Ultra 780 and FemtoFiber Ultra 1050 from \nToptica). The lasers generate p ulses of 150 -fs duration with  a repetition rate of 80 MHz at \nwavelengths of 1046 nm (pump pulses) and 780 nm (probe pulses). The magnetization precession \nwas excited by the pump pulses with energy of 3 nJ per pulse focused on the Galfenol film surface to \na spot of 17 μm diameter. The linearly polarized probe pulses with energy of 30 pJ per pulse hitting \nat normal incidence to the sample surface were focused to a spot of 1  μm diameter in the center of \nthe pump spot. The coherent response was measured by transient polar Kerr rotation in the same \nway as for the 10-GHz measurements. Temporal resolution was achieved also by the ASOPS \ntechnique w ith a frequency offset of 800 Hz , which in com bination with the 80 -MHz repetition rate \nallowed measurement of the ti me-resolved signals in the time window of 12.5 ns with 150-fs time \nresolution. The magnon spectra and the magnetic field dependence of the central frequency were \nobtained by fast Fourier transforming the corresponding transient Kerr rotation signals.    \nModeling of temperature evolution . In the case of periodic laser pulse excitation, the temperature \noscillates around the background temperature 𝑇0+Δ𝑇  with amplitude δ 𝑇. The amplitude δ𝑇 is \nestimated by solving the standard heat equations for the lattice temperatures of Fe 0.81Ga0.19 and \nGaAs for periodic excitation by laser pulses using Comsol Multiphysics® [34] (see Supplementary Note \n1). We estimate the background heating Δ𝑇 from the s olution of the 3D stationary heat equation for \ncw-laser excitation (see Supplementary Note 2 ). We use the fol lowing parameters for GaAs [35 ]: \nrefractive index 3.7+ i0.09 which results in an absorption length of 740 nm; heat capacity 1.76×106 \nJm-3K-1 and the rmal conductivity 55 Wm-1K-1 [36,37]. We assume that the optical and thermal \nparameters of Fe 0.81Ga0.19 are close to the parameters of Fe: refractive index 2.9+ i3.4 [38 ]; heat \ncapacity 3.8×106 Jm-3K-1 and thermal conductivity 80 Wm-1K-1 [39].  As a result the reflection \ncoefficient of the (Fe,Ga)/GaAs heterostructure is 0.4 and the absorption coefficients for the \nFe0.81Ga0.19 layer and GaAs substrate are 0.1 and 0.49, respectively. The calculated values of T and \nT at the excitation power W=95 mW are 48 K and 0.4 7 K, respectively .  \nIn our calculations, we consider the thermal boundary resistance, R, which determines the \nflux through the Fe 0.81Ga0.19/GaAs interface [17]. In order to estimate R, we measured the reflectivity \nsignal from the studied structure in the case of low -frequency (80 -MHz) laser excitation. Using the \nexperimentally observed decay time of the transient reflectivity 𝜏0 = 200 ps we found the thermal \nboundary resistance R=10-8 m2KW-1. HIM  \nModelling the periodic laser pulse -induced magnetization precession. To calculate the \nmagnetization precession amplitudes An(B) and phases φn(B), we apply the approach developed in \nRef. [ 14] where the detailed theory of magnetization precession due to ultrafast heating of a Galfenol \nfilm was considered. We analyze the precession by rewriting  the LLG equation in a spherical \ncoordinate system with in -plane azimuthal angle, , and polar angle, , (= 0 corresponds to the \nlayer normal, z), which describes the direction of the normalized magnetization, m. Assuming that \nthe changes of and  from  the equilibrium angles 0 and 0 are small and  induced by the  \n10 \n modulation of the cubic and uniaxial anisotropy constants, the LLG equation in linear approximation  \nhas the form : \n𝜕𝛿𝜙\n𝜕𝑡=𝛾0𝐹𝜃𝜃𝛿𝜃+𝛼0𝜕𝛿𝜃\n𝜕𝑡,                                                (5) \n𝜕𝛿𝜃\n𝜕𝑡=𝛾0𝐹𝜙𝜙𝛿𝜙−𝛾0𝐹𝜙𝐾1𝛿𝐾1−𝛾0𝐹𝜙𝐾𝑢𝛿𝐾𝑢(𝑡)−𝛼0𝜕𝛿𝜙\n𝜕𝑡, \nWhere  the 𝐹𝑖,𝑗=𝜕2\n𝜕𝑖𝜕𝑗𝐹𝑀  (𝑖,𝑗=𝜃,𝜙,𝐾1,𝐾𝑢) are calculated for the equilibrium (in-plane ) \norientation  of m, 0(B,T), 0 =/2 and  is the Gilbert damping parameter. At the in-plane \nequilibrium orientation  of m, 𝐹𝜃𝜙=𝐹𝜃𝐾1=𝐹𝜃𝐾𝑢 =0  and there is no torque due to the thermal  \nchange of M0.  \nIn Eq. ( 4) the precession amplitude and phase are determined as 𝐴𝑛(𝐵)=√𝑎𝑛2(𝐵)+𝑏𝑛2(𝐵) \nand 𝜑𝑛(𝐵)=arctan (𝑎𝑛(𝐵)/𝑏𝑛(𝐵)), where 𝑎𝑛(𝐵)=(−𝜔𝑛𝜉𝑄𝑛𝑏+2𝜏𝑀−1𝜔𝑛2𝑄𝑛𝑎)/𝐺 and \n𝑏𝑛(𝐵)=(𝜔𝑛𝜉𝑄𝑛𝑎+2𝜏𝑀−1𝜔𝑛2𝑄𝑛𝑏)/𝐺, with 𝐺=𝜉2+4𝜔𝑛2𝜏𝑀−2, and 𝜉 =𝜔𝑛2−𝜔2. Here 𝜔=2𝜋𝑓(𝐵) \nand 𝜏𝑀−1=𝛼0𝛾0(𝐹𝜃𝜃+𝐹𝜙𝜙)/2 are the field -dependent frequency and decay time of the \nfundamental magnon mode  [12]. The Fourier series expansion coefficients of the  effective driving \nforce  𝑄𝑛𝑎 and 𝑄𝑛𝑏 are determined by the t emporal  modulation of the temperature given by Eq. (2)  \nand are equal to 𝑄𝑛𝑎=2\n𝜋∙𝜔1𝜏0\n1+𝜔𝑛2𝜏02𝑄 and 𝑄𝑛𝑏=2\n𝜋∙𝜔1𝜔𝑛𝜏02 \n1+𝜔𝑛2𝜏02𝑄, where  𝑄=𝛾0(sin(4𝜙0)𝛽𝐾1/2−\n cos (2𝜙0)𝛽𝐾𝑢)𝛿𝑇. \nReferences  \n[1] Benenti, G., Casati, G., Saito, K. & Whitney, R. S., Fundamental aspects of steady -state \nconversion of heat to work at the nanoscale. Phys. Rep . 694, 1–124 (2017).  \n[2] Bell, A. G., Upon the production of sound by radiant energy. Philos. Mag. J. Sci . 11, 510 –528 \n(1881).  \n[3] Manomar, S. & Razansky, D., Photoacoustics: a historical review.  Adv. Opt. Photonics  8, 586-\n617 (2016).  \n[4] Klieber, C., Peronne, E., Katayama, K., Choi, J.,  Yamaguchi, M., Pezeril, T. & Nelson, K. A., \nNarrow -band acoustic attenuation measurements in vitreous silica at frequencies between 20 \nand 400 GHz. Appl. Phys. Lett.  98, 211908 (2011).  \n[5] Chumak, A. V., Vasyuchka, V. I., Serga A. A., & Hillebrands B., Magnon  spintronics. Nature \nPhys. , 11, 453 – 461 (2015).  \n[6] Yi Li et al., Strong Coupling between magnons and microwave photons in on -chip ferromagnet -\nsuperconductor thin -film devices. Phys. Rev. Lett.  123, 107701 (2019).  \n[7] Gurevich, A.G. & Melkov, G.A., Magnetizatio n oscillations and waves (CRC -Press, Boca Raton, \n1996).  \n[8] van Kampen, M., et al. All-optical probe of coherent spin waves, Phys. Rev. Lett.  88, 227201 \n(2002).  \n[9] Vomir, M., Andrade, L. H. F., Guidoni, L., Beaurepaire, E., & Bigot, J. -Y., Real space trajectory \nof the ultrafast magnetization dynamics in ferromagnetic metals. Phys. Rev. Lett.  94, 237601 \n(2005).  \n[10] Frait. Z. & MacFaden, H., Ferromagnetic resonance in metals. Frequency dependence. Phys. \nRev. 139, A1173 – A1181 (1965).  \n[11] Restorff, J.B. et al., Tetragonal  magnetostriction and magnetoelastic coupling in Fe -Al, Fe -Ga, \nFe-Ge, Fe -Si, Fe -Ga-Al, and Fe -Ga-Ge alloys . J. Appl. Phys.  111, 023905 (2012).  \n[12] Scherbakov, A.V. et al., Optical excitation of single - and multimode magnetization precession \nin Fe -Ga nanolayers . Phys. Rev. Appl.  11, 031003 (2019).  \n[13] Clark, A. E. et al., Temperature dependence of the magnetic anisotropy and magnetostriction \nof Fe 100−xGax (x=8.6, 16.6, 28.5). J. of Appl. Phys.  97, 10M316 (2005).  \n[14] Kats, V. N. et al., Ultrafast changes of magnetic ani sotropy driven by laser -generated coherent \nand noncoherent phonons in metallic films, Phys. Rev. B  93, 214422 (2016).   \n11 \n [15] Melnikov, A. et al., Ultrafast Transport of Laser -Excited Spin -Polarized Carriers in \nAu/Fe/MgO(001) . Phys. Rev. Lett. 107, 076601 (2011).  \n[16] Kimling, J. et al., Picosecond spin Seebeck effect, Phys. Rev. Lett.  118, 057207 (2017).  \n[17] Razdolskii, I., et al. Nanoscale interface confinement of ultrafast spin transfer torque driving \nnon-uniform spin dynamics, Nature Commun . 8, 15007 (2017).   \n[18] Kamimaki, A., Iihama, S., Sasaki, Y., Ando, Y., & Mizukami, S. Reciprocal excitation of \npropagating spin waves by a laser pulse and their reciprocal mapping in magnetic metal films. \nPhys. Rev. B  96, 014438 (2017).  \n[19] Khokhlov, N. E. et al., Optical excitation  of propagating magnetostatic waves in an epitaxial \ngalfenol film by ultrafast magnetic anisotropy change. Phys. Rev. Appl. 12, 044044 (2019).  \n[20] Krenzer, B., Hanisch, A., Duvenbeck, A., Rethfeld, B., and Horn -von Hoegen, M. Heat transport \nin nanoscale heter osystems: A Numerical and Analytical Study. J.  Nanomat.  2008 , 590609 \n(2008).  \n[21] Razdolski, I.  et al. , Analysis of the time -resolved magneto -optical Kerr effect for ultrafast \nmagnetization dynamics in ferromagnetic thin films . J. Phys. C  29, 174002 (2017).  \n[22] Ando, K. et al., Inverse spin -Hall effect induced by spin pumping in metallic system. J. Appl. \nPhys . 109, 103913 (2011).  \n[23] Weiler M. et al., Spin pumping with coherent elastic waves. Phys. Rev. Lett . 108, 176601 \n(2012).  \n[24] Fuchs, G. D. et al., Excited -state spectroscopy using single spin manipulation in diamond. Phys. \nRev. Lett.  101, 117601 (2008).  \n[25] Demidov, V.E. et al., Excitation of coherent propagating spin waves by pure spin currents. \nNature Commun.  7, 10446 (2016).  \n[26] Gupta, S. et al., Important role of magnetization precession angle measurement in inverse spin \nHall effect induced by spin pumping. Appl. Phys. Lett.  110, 022404 (2017).  \n[27] Weiler, M. Spin Pumping with Coherent Elastic Waves. Phys. Rev. Lett.  108, 176601 (2012) .  \n[28] Damon, R. W., Thermally‐induced nonlinear effects in the ferromagnetic resonance. Phys. \nStat. Sol.  8, K89 -K93 (1965).  \n[29] Mizukami, S. Long -lived ultrafast spin precession in manganese alloys films with a large \nperpendicular magnetic anisotropy. Phys. Rev.  Lett.  106, 117201 (2011).  \n[30] Parkes, D. E. et al., Magnetostrictive thin films for microwave spintronics. Sci. Reports. 3, 2220 \n(2013).  \n[31] Salasyuk, A.S. et al., Generation of a localized microwave magnetic field by coherent phonons \nin a ferromagnetic nanogra ting. Phys. Rev. B  97, 060404(R) (2018).  \n[32] Kreilkamp, L. E. et al., Terahertz dynamics of lattice vibrations in Au/CdTe plasmonic crystals: \nphotoinduced segregation of Te and enhancement of optical response. Phys. Rev. B 93, \n125404 (2016).  \n[33] Bartels, A., Hudert, F., Janke, C., Dekorsy, T. & Köhler, K., Femtosecond time -resolved optical \npump -probe spectroscopy at kilohertz -scan -rates over nanosecond -time -delays without \nmechanical delay line. Appl. Phys. Lett.  88, 041117 (2006).  \n[34]  COMSOL Multiphysics® v. 5.4. http://www.comsol.com . COMSOL AB, Stockholm, Sweden.  \n[35] Aspnes, D. E., Kelso, S. M., Logan, R. A., and Bhat, R., Optical properties of Al xGa1-xAs. J. App. \nPhys.  60, 754 - 767(1986).  \n[36] Chen, B., Fu, X. Tang, J., Lysevych, M., Tan, H. H., Jagadish, C., Zewail, A. H., Dynamics and \ncontrol of gold -encapped gallium arsenide nanowires imaged by 4D electron microscopy. Proc. \nNat. Acad. Sci.  114, 12876 - 12881, (2017).  \n[37] Carlson, R. O., Slack, G. A., Silve rman, S. J., Thermal conductivity of GaAs and GaAs 1−xPx laser \nsemiconductors. J. App. Phys . 36, 505 – 507 (1965).  \n[38] Johnson, P. B.  and Christy, R. W., Optical constants of transition metals: Ti, V, Cr, Mn, Fe, Co, \nNi, and Pd. Phys. Rev. B  9, 5056 - 5070 (19 74). \n[39] Lide, D.R., CRC Handbook of Chemistry and Physics (CRC Press, Boca Raton, FL, 2003).   \n12 \n Acknowledgements  \nWe are grateful to Ilya Razdolski for fruitful discussions. The work was supported by the \nBundesministerium fur Bildung und Forschung through the pro ject VIP+ \"Nanomagnetron\" and by \nthe Deutsche Forschungsgemeinschaft and the Russian Foundation for Basic Research (Grants No. \n19-52-12065) in the frame of the International Collaborative Research Center TRR 160. The \ncooperation between TU Dortmund, the La shkaryov Institute and the Ioffe Institute was supported \nby the Volkswagen Foundation (grant no. 97758).  \nAuthor contributions  \nM.K. constructed the experimental setup, carried out the experiment and performed the data \nanalysis. A.V.S., I.A.A, and A.V.A. designed the experiment and supervised the experiment and \ntheoretical modeling, T.L.L., S.M.K and V.E.G. performed theoretical an alysis and numerical \nmodeling, A.W.R. and D.P.P. designed and deposited the Galfenol samples, D.P.P. characterized the \nGalfenol samples, M.K., A.V.S., T.L.L, S.M.K, V.E.G., I.A.A, A.W.R., A.V.A and M.B. discussed the results \nand wrote the manuscript.      \nCompeting interests  \nThe authors declare no competing interests.  \nData availability  \nThe data that support s the plots within this paper and other findings of this study are available from \nthe corresponding authors upon reasonable request.  \n \n   \n13 \n Supplementary Information  \nNote 1. Modeling of heat transport for periodically pulsed optical excitation  \nIn order to describe the heat transport in the system of a plane metallic film on a substrate we \nused the one -dimensional two -temperature model for the lattice heating temperatures of the \nfilm, 𝑇𝑓, and the substrate, 𝑇𝑠:  \n                              𝑐𝑓𝜕𝑇𝑓 \n𝜕𝑡−𝜅𝑓𝜕2𝑇𝑓 \n𝜕𝑧2=𝐹𝑝𝐴𝑓𝑔(𝑡)𝑑𝑆\n𝑑𝑧 ,                                                        (S1)  \n                    𝑐𝑠𝜕𝑇𝑠 \n𝜕𝑡−𝜅𝑠𝜕2𝑇𝑠 \n𝜕𝑧2=𝐹𝑝𝐴𝑠𝑔(𝑡)𝑑𝑆\n𝑑𝑧,                                                          (S2)  \nwhere 𝑐𝑗, 𝜅𝑗, and 𝐴𝑗 are the heat capacity, thermal conductivity, and fraction of absorbed optical \nenergy relative to the incident excitation energy (further referred to as absorption coefficients), \nrespectively ( 𝑗=𝑓,𝑠). The pump laser pulse fluence 𝐹𝑝=2𝑊/(𝑓0𝜋𝑟2) , where  𝑊  is the \naverage laser power, 𝑓0=10 GHz is the laser repetition rate, and 𝑟 is the radius of the pump \nspot defined by 1/𝑒2 of the intensity. The function, 𝑔(𝑡)=2\n𝜏𝑝√2\n𝜋 ∑ exp (−2(𝑡−𝑛−1\n𝑓0)2\n𝜏𝑝2)∞\n𝑛=1  , \ndescribes the temporal evolution of the laser pulses with duration, 𝜏𝑝,  and repetition rate, 𝑓0. \nEverywhere throughout the Supplementary Information, we chose the coordinate system with \nthe direction of the z -axis pointing from the substrate to the film.  \nWe found the absorption coefficients from the solution of Maxwell equations for the \nthree -layer (air/metal/substrate) system:  \n 𝐴𝑓=1−|(𝑘𝑎𝑘𝑓−𝑘𝑓𝑘𝑠)cos(𝑘𝑓ℎ)−𝑖(𝑘𝑎𝑘𝑠−𝑘𝑓2)sin(𝑘𝑓ℎ)\n(𝑘𝑎𝑘𝑓+𝑘𝑓𝑘𝑠)cos(𝑘𝑓ℎ)−𝑖(𝑘𝑎𝑘𝑠+𝑘𝑓2)sin(𝑘𝑓ℎ)|2\n−𝐴𝑠,                               (S3) \n 𝐴𝑠=Re(𝑛S)|2𝑘𝑎𝑘𝑓\n(𝑘𝑎𝑘𝑓+𝑘𝑓𝑘𝑠)cos(𝑘𝑓ℎ)−𝑖(𝑘𝑎𝑘𝑠+𝑘𝑓2)sin(𝑘𝑓ℎ)|2\n,                                  (S4) \nwhere 𝑘𝑗=𝑘𝑎𝑛𝑗 is the complex wave vector of light in the air ( j=a), film ( j=f), and substrate( j=s), \n𝑛𝑗 is the complex refractive index, ℎ is the thickness of the film.  \nThe function 𝑑𝑆(𝑧)/𝑑𝑧  describes the spatial distribution of the energy density \ndeposition by the pump light, which penetrates into the medium. More specificall y, 𝑆(𝑧) is \nnormalized to the value of the flux of electromagnetic energy at the metal surface. We used \nCOMSOL Multiphysics® [S1] to calculate 𝑆(𝑧).  \nThe equations (S1) -(S4) must be complemented by the boundary conditions. At the \ninterface between the f ilm and substrate (i.e. at  𝑧=0), the boundary conditions read:  \n𝜅𝑓𝜕𝑇𝑓 \n𝜕𝑧|𝑧=0=𝜅𝑠𝜕𝑇𝑠 \n𝜕𝑧|𝑧=0=(𝑇𝑓−𝑇𝑠)/𝑅,                                                     (S5)  \nwhere R is the thermal boundary resistance. At the interface of air and film,  we assumed a zero \nheat flux boundary condition, i.e.   𝜅𝑓𝜕𝑇𝑓 \n𝜕𝑧|𝑧=ℎ=0. We take the thickness of the substrate large \nenough to minimize the impact of any boundary condition at the back side of the substrate.    \n  \n  \n14 \n The described approach was implemented in CO MSOL Multiphysics®. The example of \nthe average lattice heating temperature of the film 𝑇̅𝑓=∫𝑇𝑓𝑑𝑧\nℎℎ\n0 for 𝑊=95 mW, 𝑟=8.5 μm \nis shown in Fig. S1. One can see that the temperature oscillates with the excitation frequency \nf0=10 GHz around a slowly increas ing background. Here we want to mention that our \ncalculations show that in the discussed one -dimensional (1D) heat transport problem the \nbackground temperature does not reach a stationary value but, diverges with time. However, \nas one can clearly see from Fig. S1, the amplitude of thermal oscillations, 𝛿𝑇, converges very \nquickly. In order to estimate the stationary background temperature Δ𝑇 we solved analytically \nthe stationary 3D problem, see the next Supplementary Note.  \n \nNote 2. Estimation of background temperature Δ T \nTheory  \nIn order to calculate the background temperature rise Δ𝑇, we solved analytically the stationary \n3D heat equation for 𝑇𝑠. Taking into account that the optical excitation energy is absorbed in a \nlayer with a thickness much  less than the radius of excitation at the surface, we may assume \nthat the thermal source is located at the interface ( z=0). Then the equation and boundary \ncondition may be written as:  \n  (𝜕2\n𝜕𝑥2+𝜕2\n𝜕𝑦2+𝜕2\n𝜕𝑧2)𝑇𝑠=0,                                                                   (S6) \n and  \n 𝜅𝑠𝜕𝑇𝑠 \n𝜕𝑧|𝑧=0=(𝐴𝑓+𝐴𝑠) 𝐼 𝑒𝑥𝑝 (−2𝑥2+𝑦2\n𝑟2),                                          (S7)  \nwhere 𝐼 is the laser intensity which is related to the laser power by  \n 𝑊=𝐼∬𝑒𝑥𝑝 (−2𝑥2+𝑦2\n𝑟2)𝑑𝑥𝑑𝑦 =𝐼𝜋𝑟2\n2.∞\n−∞                                                      (S8)  \nFig. S1. The temporal evolution of the average temperature 𝑇̅𝑓(𝑡) for periodic laser \npulse excitation with average power W=95 mW incident on the FeGa film with a \nthickness h=5nm deposited on the GaAs substrate.  \n \n15 \n The analytical solution of these equations at z=0 and at the center of the laser beam ( x=y=0) \nreads  \n   Δ𝑇=(𝐴𝑓+𝐴𝑠)𝑊\n√2𝜋𝜅𝑠𝑟.                                                                          (S9) \nIt is obvious that for optical excitation from the side of the film (absorption length smaller in the \nmetal than in the substrate) we have 𝑇𝑓>𝑇𝑠 because of the boundary resistance.  We estimate \nthe jump of the temperature at the interface, using the boundary condition (S5). In this case, \nthe temperature jump at the interface of the film and substrate is given by  \n  (𝑇𝑓−𝑇𝑠)|𝑧=0= (𝐴𝑓+𝐴𝑠)𝐼𝑅=(𝐴𝑓+𝐴𝑠)𝑊2𝑅\n𝜋𝑟2.                                               (S10) \nFor the p arameters of our experiment, the temperature (S9) is 10 times larger than the \ntemperature jump (S10). Thus, the effect of the temperature jump can be neglected.  \nExperiment  \nWe also derive the background temperature Δ T from the measured dependences of the \nresonance fields B 1, B 2, and B 3 on W. For this we take the derivatives dB/dW  from the \nexperimental data in Fig. 3a and compare them with the calculated derivatives dB/dT  [see \ntheoretical curves f(B) in Fig. 1(a) of the main text]. As a result, we get the va lues for \ndT/dW=(dB/dW)/(dB/dT)  for the three resonant frequencies 10, 20 and 30 GHz. The results are \npresented in the Table S1.   \nTable S1 . The measured values of the resonant magnetic fields B 1, B2, and B 3  \nExcitation \nPower (mW)  Resonant Fields (mT)  \n10 GHz 20 GHz  30 GHz  \n35 34 232 490 \n55 38 234 492 \n95 42 240 498 \n \nTable S2 . Power and temperature derivatives for the three resonant frequencies  \n 10 GHz  20 GHz  30 GHz  \ndB/dW  \n(mT/mW)  0.13  0.13  0.13  \ndB/dT  \n(mT/K)  0.19  0.13  0.16  \ndT/dW  \n(K/mW)  0.7 0.9 0.7 \n \nTable S3 . Values of the background temperature rise.  \nPower (mW)  ΔT (K)  \n Calculated  Extracted from experiment  \n35 18 282 \n55 28 443 \n95 48 775 \n  \n16 \n The average value for dT/dW  = 81050 K/W . The estimated temperature rise ΔT= (dT/dW )W for \nthe three values of W are presented in Table S3 and compared with the theoretical values  The \ndifference between the values calculated and extracted from the experimen ts is less than 40%. \nThis difference may be due to additional heating by the probe which is not included into the \ntheoretical calculations.  \nNote 3. Dependence of =T/T on parameters and design of the device  \nFigure S2 shows the dependencies of the dynamical harvesting efficiency  =T/T on the film \nthickness and thermal conductivity of the substrate. The value of  does not depend on power, \nbecause both 𝛿𝑇 and  Δ𝑇 depend linearly on 𝑊. As one can see from Fig. S2a,  decreases with \nincreasing fi lm thickness. From Fig. S2b one can see that  increases linearly with thermal \nconductivity of the substrate, because T is inversely proportional to 𝜅𝑠 (see (S9)). Interestingly, \nin the considered frequency range, 𝛿𝑇 does not depend on 𝑅, 𝜅𝑙, 𝑐𝑠, as well, and depends only \non 𝑐𝑓.  Thus, the parameter  is controlled by Eq. (S9).  Hence, in the case of a Si substrate, the \nefficiency can be more than two times larger than for GaAs, due to the difference in their heat \nconductivities. The dependence of the dynamical harvesting efficiency on 𝑐𝑓 is shown in Fig. 2c. \nThus, the efficiency can be larger than in Galfenol for materials with smaller 𝑐𝑓.   \nFinally, we discuss the design of a device with a ferromagnetic harvester, which uses the \nheat generated in a conducting layer in the vicinity of the harvester (see Fig. S3). Consider a thin \nconducting layer (e.g., a quantum well or a graphene layer) with thickness, 𝑙, which is separated \nfrom the harvester by a spacer with thickness, 𝑠. The harvester’s thickness is ℎ. We calculate  \nfor such a device using an  approach similar to that described in the Supplementary Notes 1 and \n2. In our example, we take ℎ=𝑙=5 nm, and 𝑠=2.5 nm. For the conducting layer, spacer and  . \n  \n \nFig. S2 . Dependencies of the efficiency parameter,  on ℎ (a), 𝜅𝑠 (b), and 𝑐𝑓 (c). \n \n17 \n  \nsubstrate, we use the same thermal parameters as for GaAs. We consider that the heat is \nperiodically released only in the conducting layer at 𝑓0=10 GHz. In the real device the heat \nsource may be Joule heat emitted as result of an electrical current inject ed on the clock \nfrequency 𝑓0.  For the harvester, we take Galfenol. In this case, the calculated efficiently  \n=0.004 which is only two times smaller than the efficiency of the system used in our \nexperiments.  \n \nNote 4. Magnon dispersion  \nThe frequency of t hermally excited spin waves (magnons) in the studied Galfenol nanolayer is \ndetermined by the external magnetic field, B, and the magnon wave vector, q.  The frequency \nof the ground magnon mode, which corresponds to the case of uniform precession ( q=0), is \ngiven by the well -known expression [S2]:  \n𝑓=𝛾0\n2𝜋√𝐹𝜙𝜙𝐹𝜃𝜃,      (S11)  \nwhere 0 is the gyromagnetic ratio, and 𝐹𝜃𝜃=𝜕2𝐹𝑀\n𝜕𝜃2  and 𝐹𝜙𝜙=𝜕2𝐹𝑀\n𝜕𝜙2  are the second derivatives \nof the free energy density (see Eq.2 in the main text) with respect to the in -plane azimuthal \nangle, , and polar angle, , calculated at the equilibrium orientation of magnetization. The \nangles are counted from the [100] crystallographic direction and from the normal to the layer \nplane, respectively.  Using the free energ y density as in Ref. [3] and considering the case of an \nin-plane external magnetic field we get:   \n𝐹𝜃𝜃=𝐵cos(𝜙𝑀−𝜙𝐵) +𝜇0𝑀0+𝐾1\n2(cos4𝜙𝑀+3)+𝐾𝑢(1+𝑠𝑖𝑛2𝜙𝑀)   (S12)  \n𝐹𝜙𝜙=𝐵cos(𝜙𝑀−𝜙𝐵)+2𝐾1cos4𝜙𝑀+2𝐾𝑢sin2𝜙𝑀    \nwhere K1, Ku and M0 are the temperature dependent cubic and uniaxial anisotropy coefficients, \nand the saturation magnetization, respectively; B is the in -plane angle of the external magnetic \nfield and M is the field -dependent angle of the equilibrium orientation of magneti zation (at \nB>0.3T, M=B). In the experimental temperature range the temperature dependences of K1, Ku \nand M0 may be written as: 𝑋=𝑋𝑅𝑇+𝛽𝑋(𝑇−𝑇0),  where X= K1, Ku or M0 [S4]; XRT are their \nvalues at room temperature and 𝛽𝑋=𝜕𝑋\n∂T is the temperature independent coefficient. The \ndependences f(B) calculated for B=-/8 for room temperature and T=200 K are shown in Fig.  \nFig. S3 . Scheme of the device with a thin conducting layer and a ferromagnetic harvester.  \n \n18 \n 1a of the main text. The parameters used for the calculations are the following [S3,S4]:  \n 𝑀0𝑅𝑇=1.95 T,,𝐾1𝑅𝑇=20 mT, 𝐾𝑢𝑅𝑇=9 mT, 𝛽𝑀=−0.97 mT/K, 𝛽𝐾1=− 0.046 mT/K,  \n 𝛽𝐾𝑢=−0.025 mT/K.  \n  Due to the finite laser spot size, the thermal modulation generates also magnons with \nnon-zero in -plane wave vectors, 𝑞||. The range of in -plane wave vectors is set by the laser spot. \nIt is found from the Fourier transform of the Gaussian distribution of the laser intensity (see Eq. \n(S7)). Therefore, we obtain  0≤𝑞||≤4\n𝑟,  where r is the laser spot radius. In our experiment with \n𝑟=8.5 m the upper limit is 𝑞||≤ 4700 cm-1. This range corres ponds to magneto -static spin \nwaves with a very weak dispersion determined by the magnetic dipole -dipole interaction. This \ndispersion is strongest for the wave vectors perpendicular to the external magnetic field [S5]. In \nthe case of a thin ferromagnetic fi lm of thickness, h, where 𝑞||h\n2𝜋 <<1 it can be written in a \nsimplified form as [S6]:  \n𝑓𝑞||=𝑓+𝛾0\n2𝜋𝜈 𝑞||ℎ,      (S13)  \nwhere 𝜈 is the dispersion coefficient determined by the main parameters of the ferromagnet: \nK1, Ku and M0. For the studied Galfenol layer its value at room temperature is 𝜈≈ 3.8 T [S7] and \nin our experiment the frequencies of the thermally driven magnons with finite in -plane wave \nvector do not exceed  270 MHz. This value is less than the spectral width of th e fundamental \nmagnon mode (0.5 GHz) determined by the precession decay. Thus, the thermally driven \nmagnons with finite in -plane wave vectors may be considered as degenerate.  \n The spectrum of magnons with finite tangential wave vector, q, is quantized due  to \nspatial confinement along the layer normal: 𝑞⊥,𝑛=𝜋𝑛\nℎ , where n=1,2,3… In layers with \nnanometer thickness, the corresponding dispersion is determined by the exchange interaction \nand in a simplified form can be written as:        \n  𝑓𝑛=𝑓+𝛾0\n2𝜋𝜚𝐷𝑞⊥,𝑛2,     (S14)  \nwhere D is the spin stiffness constant and  𝜚 is a field dependent coefficient approaching 1 with \nincreasing B [S8]. In the studied Galfenol layer with h=5 nm, the large penetration depth of the \nlaser pulse in comparison with  the layer t hickness results in  uniform thermal distribution along \nthe layer normal. Thus, the odd magnon modes ( n=1,3,5) cannot be excited. The lowest higher -\norder mode, which can be driven by thermal modulation, corresponds to n=2.  For the Galfenol \nspin stiffness D=1.5×10-17 Tm2 [S8, S9], the frequency of this mode exceeds the ground mode \nfrequency by more than 500 GHz. Due to the extremely low amplitude of such high -frequency \nharmonics in the thermal spectrum, this mode as well as other higher -order even modes are not \nexcited. Thus, in the studied layer of 5 -nm thickness the fundamental magnon mode is solely \ndriven by the modulated heat.   \nThe situation changes drastically in a ferromagnetic layer with thickness ~100 nm, which \npossesses significantly less spectral s plitting of the higher -order magnon modes with finite q. \nDue to the small penetration depth of light in comparison with the layer thickness, both odd and \neven modes can be excited [S8,S10]. This results in a drastic broadening of the magnon spectrum \nand a  corresponding dephasing of the magnetization precession with destructive effects on the \nresonant energy harvesting. This is clearly seen in Fig. S4, which summarizes the experimental \nresults obtained for a Galfenol layer of thickness h=105 nm.  The upper inset shows the magnon   \n   \n19 \n spectrum obtained by fast Fourier transforming the Kerr rotation signal measured in a single \npulse pump -probe experiment at B=200 mT. In addition to the dominating fundamental magnon \nmode, the FFT includes several spectral lines, which correspond to the higher -order magnon \nmodes with n=1…5. Their spectral splitting is described with high accuracy by the dispersion \nrelation [S7].  The result of such a spectral broadening becomes apparent in the transient KR \nsignal measured under resonant conditions when f=f0=10 GHz (see the lower inset). The non -\nharmonic character of the thermally driven oscillations in the KR signal is clearly see n. As a result, \nthe field dependence of the RMS Kerr amplitude shown in the main panel consists of several \nbroad peaks instead of narrow resonances as observed in the 5 -nm Galfenol layer. At B=75 mT, \nwhich corresponds to the maximum amplitude of precession , its absolute values and RMS are \n15-times smaller than in the 5 -nm layer.         \nReferences  \n[S1] COMSOL Multiphysics® v. 5.4. http://www.comsol.com. COMSOL AB, Stockholm, Sweden.  \n[S2] Gurevich, A.G. & Melkov, G.A., Magnetization oscillations and waves (CRC -Press, Boca Raton, \n1996).  \n[S3] Kats, V. N. et al., Ultrafast changes of magnetic anisotropy driven by laser -generated coherent and \nnoncoherent phonons in metallic films, Phys. Rev. B  93, 214422 (2016).  \n[S4] Clark, A. E. et al., Temperature dependence of the magnetic anisotropy and magnetostriction of \nFe100−xGa x (x=8.6, 16.6, 28.5). J. of Appl. Phys.  97, 10M316 (2005).  \n[S5] Damon, R. W. & ESHBACH J. R., J. Phys. Chem. Solids  19, 308 -320 (1961).  \n[S6] Kamimaki, A., Iihama, S., Sasaki, Y., Ando, Y., & Mizukami, S. Reciprocal excit ation of propagating \nspin waves by a laser pulse and their reciprocal mapping in magnetic metal films. Phys. Rev. B  96, \n014438 (2017).  \n[S7] Khokhlov, N. E. et al., Optical excitation of propagating magnetostatic waves in an epitaxial galfenol \nfilm by ultrafast magnetic anisotropy change. Phys. Rev. Appl. 12, 044044 (2019).  \n[S8] Scherbakov, A.V. et al., Optical excitation of single - and multimode m agnetization precession in Fe -\nGa nanolayers . Phys. Rev. Appl.  11, 031003 (2019).  \n[S9] Gopman, D. B., Sampath, V., Ahmad, B. H., Bandyopadhyay, S., & Atulasimha, J., Static and dynamic \nmagnetic properties of sputtered Fe -Ga thin films, IEEE Trans Magn.  53, 6101304 (2017).  \n[S10]  van Kampen, M. et al., All -optical probe of coherent spin waves. Phys. Rev. Lett.  88, 227201 (2002).    \nFig. S4. Magnetic field dependences of the root mean square (RMS) Kerr rotation \namplitude, 𝐴𝐵̃, measured in a Galfenol layer of 105 -nm thickness for the excitation power \nW=140 mW. The upper inset shows the magnon spectrum obtained as the fast Fourier \ntransform of the Kerr rotation signal measured in a single pulse pump -probe experiment \nat B=200 mT. The lower inset shows the transient Kerr rotation signal measured for  \n10-GHz optical excitation at B=75 mT  \n \n \n0 100 200 300 400 5000.00.20.40.60.81.0RMS, AB (arb. units) \nMagnetic field (mT)10 20 30Amplitude \nFerquency (GHz)B=200 mT\nfM\n-50 0 50 100fM=10 GHzKR angle  \nTime (ps)B=75 mT"    },    {        "title": "1212.5794v1.Magnetoelectric_coupling_in_a_ferroelectric_ferromagnetic_chain_revealed_by_ferromagnetic_resonance.pdf",        "content": "arXiv:1212.5794v1  [cond-mat.mtrl-sci]  23 Dec 2012Magnetoelectric coupling in a ferroelectric/ferromagnet ic chain revealed by\nferromagnetic resonance\nA. Sukhov1,2, P.P. Horley2, C.-L. Jia1,3, J. Berakdar1\n1Institut f¨ ur Physik, Martin-Luther Universit¨ at Halle-W ittenberg, 06120 Halle (Saale), Germany\n2Centro de Investigaci´ on en Materiales Avanzados (CIMAV S. C.),\nChihuahua/Monterrey, 31109 Chihuahua, Mexico\n3Key Laboratory for Magnetism and Magnetic Materials of the\nMinistry of Education, Lanzhou University, Lanzhou 730000 , China\nUnderstanding the multiferroic coupling is one of the key is sues in the ﬁeld of multiferroics. As\nshown here theoretically, the ferromagnetic resonance (FM R) renders possible an access to the\nmagnetoelectric coupling coeﬃcient in composite multifer roics. This we evidence by a detailed\nanalysis and numerical calculations of FMR in an unstrained chain of BaTiO 3in the tetragonal\nphase in contact with Fe, including the eﬀect of depolarizin g ﬁeld. The spectra of the absorbed\npower in FMR are found to be sensitive to the orientation of th e interface electric polarization and\nto an applied static electric ﬁeld. Here we propose a method f or measuring the magnetoelectric\ncoupling coeﬃcient by means of FMR.\nPACS numbers: 85.80.Jm,76.50.+g,75.78.-n\nIntroduction.- Materials with multi ferroic (magnetic,\nelectric, and/orelastic) orders, calledmultiferroics(MF),\nhave attracted increased attention again [1–4], mainly\ndue to the discovery that the notoriously small multifer-\nroic coupling in bulk matter may well be increased by\na controlled engineering of low dimensional compounds,\nopening thus the way for the design of qualitatively new\ndevice concepts [5–7]. For instance, magnetoelectricity\nallows for the control of magnetism with an electric ﬁeld\nwhich has a large potential for environmentally friendly\nsensorics and spintronics applications with low-energy\nconsumption. The progress in this ﬁeld depends criti-\ncally on the understanding of the magnetoelectric (ME)\ncoupling and on developing methods to assess its prop-\nerties. This is particularly so, as currently several ME\ncoupling mechanisms are being discussed, e.g. in Refs.\n[6, 8–10]. On the other hand, an established approach\nfor probing the ferromagnetic response is the ferromag-\nnetic resonance [11–14] in which the sample is usually\nsubjected to crossed static and time-dependent magnetic\nﬁelds. Hence, it is natural and timely to envisage a pos-\nsible mapping of the multiferroic dynamics in an FMR\nsetup with the aim to draw conclusions on the nature\nof the ME coupling and relate it to the FMR signal. In\nfact, FMR has been experimentally shown [15, 16] to be\nsensitive to acoustic waves in ferromagnetic-ferroelectric\nstructure. To our knowledge, multiferroic FMR, as sug-\ngested below has not yet been realized experimentally for\nthechoseninterface,thoughseveralstudies areknownfor\nmultiferroic interfaces with other types of ME-coupling\n[17, 18]. To be speciﬁc, we focus on a special class of\nME coupled materials, so-called composite MFs [6, 19],\nthat may be synthesized from a wide range of mate-\nrials that, when composed together yield a strong ME\ncoupling and stable ferroelectric (FE) and ferromagnetic\n(FM) orders at room temperatures. An example that\nhas been studied intensively, theoretically and experi-\nmentally is BaTiO 3(BTO)/Fe [20–30]. The ME couplingin this system is predicted to be an interfacial eﬀect and\nrelatively high [21, 28, 31, 32]. An experimental evidence\nis presented in [20].\nFor a reliable prediction, a realistic modelling is indis-\npensable since the ME coupling is relatively weak in to-\ntal (due to its interfacial nature) and the MF dynamics\nis governed by a series of interrelated eﬀects, as shown\nbelow. The route followed here is based on a combina-\ntion of the Landau-Lifshits-Gilbert and the Ginzburg-\nLandau dynamics using the total MF free energy FΣ\ndensity [24, 33, 34], including the ferroelectric FFE, the\nferromagnetic FFM, and the part EINTthat involves the\nME interface coupling. FFEincludes the energy densi-\nties corresponding to the Ginzburg-Landau-Devonshire\npotential FGLD[35, 36], to the spatial inhomogeneity of\nthe order parameter FGE[37], the depolarizing ﬁeld con-\ntribution FDF[38], the dipole-dipole interaction FFE\nDDI, as\nwell as the applied external electric ﬁeld FAEF.FFM\n[39] incorporates the nearest-neighbor exchange interac-\ntionFEXC, the anisotropy contribution FANI, the FM\ndipole-dipole interaction FFM\nDDIand the energy density of\nthe interaction with an external magnetic ﬁeld FAMF.\nRealistic parameters corresponding to bulk BaTiO 3in\nthe tetragonal phase and bulk Fe are employed. Our\nsimulation show that the FMR spectra of the absorbed\npower are indeed sensitive to the FE order as controlled\nby an applied static electric ﬁeld, evidencing thus the ac-\ncess of FMR to the ME coupling. The key ﬁnding of\nthis study is a practical proposal to estimate the mag-\nnetoelectric coupling coeﬃcient from the FMR spectra.\nWith the reﬁnement in the spatiotemporal resolution of\nthe FMR technique we expect the multiferroic FMR to\nplaya vital role in uncoveringthe details of ME coupling.\nTheoretical model.- Our treatment is based on the\nGinzburg-Landau phenomenology, i.e. we study the dy-\nnamicsofcoarse-grainedorderparameters( /vectorPiand/vectorMj)of\nthe FE/FM chain, that result from an averaging of the\nmicroscopic quantities over an appropriate cell. These2\ncells form in our calculations the sites iorjfor the local\n/vectorPior/vectorMj. At the interface (site i=j= 1) of the FE/FM\ncomposite the mobile spin-polarized electrons in the FM\nrearrange as to screen the electric polarization at the FE\npart [40], leading thus in eﬀect to a local ME coupling of\nthe interface magnetization /vectorM1with the interface polar-\nization/vectorP1that can be expressed [28] as EINT=λ/vectorP1·/vectorM1,\nwhereλis the ME coupling coeﬃcient. The FE polariza-\ntion vector /vectorPidevelops in time according to the Landau-\nKhalatnikov (LKh) equation[41]\nγνd/vectorPi\ndt=−δFΣ\nδ/vectorPi, (1)\nwith the FE relaxation constant γν= 2.5·\n10−5[Vms/C][42]. The magnetization dynamics /vectorMj\nis governed by the Landau-Lifshitz-Gilbert (LLG)\nequation[43, 44]\nd/vectorMj\ndt=−γ\n1+α2\nFM/bracketleftBig\n/vectorMj×/vectorHLEF\nj(t)/bracketrightBig\n(2)\n−αFMγ\n(1+α2\nFM)MS/bracketleftBig\n/vectorMj×/bracketleftBig\n/vectorMj×/vectorHLEF\nj(t)/bracketrightBig/bracketrightBig\n,\nwhereγis the gyromagnetic ratio and αFMis a Gilbert\ndamping constant. The local eﬀective ﬁeld is deﬁned as\n/vectorHLEF\nj(t)≡ −δFΣ\nδ/vectorMj. The FMR power absorbed by the\nchain we infer from (cf. e.g., [45, 46])\nPFMR=−µ0a3\nFM/summationdisplay\nj1\nNTT/integraldisplayNTT\n0/vectorMj(t)·∂/vectorHΣ\n∂tdt,(3)\nwhereµ0isthevacuummagneticconstant, aFMisthecell\nsize,NTandT= 2π/ωare the number and the period\nof the external magnetic ﬁeld cycles, respectively. The\nsum runs over magnetization sites j. The total magnetic\nﬁeld applied to the system is /vectorHΣ(t) =H/vector ez+H0cosωt/vector ey,\nwhereas H0≪H.\nIt the following the imaginary part of the transverse\nmagnetic susceptibility χ′′will be calculated, for which\nχ′′∼PFMR\nµ0MSH0ωholds[13, 14].\nContact of ultrathin FE and FM ( E= 0).-First, we nu-\nmerically model a contact of a single FE site and a single\nFM site in zero E-ﬁeld. Employing the FE free energy\ndensity for the tetragonal phase of BaTiO 3[24]\nFN=1\nFE=αFE\n2P2\nz+βFE\n4P4\nz (4)\nand the FM free energy density for a uniaxial crystal in\nan external magnetic ﬁeld [24]\nFN=1\nFM=−K1\nM2\nS(Mz)2−µ0/vectorM·/vectorHΣ(t).(5)\nwe calculated the spectra shown in Fig. 1. The posi-\ntion of the peak for the case of zero ME coupling follows\nfrom[47]1\nγω|res(λ= 0) =µ0HLEF|res=2K1\nM2\nSMz+µ0Hres\nFIG. 1. FMR spectra for the multiferroic chain of NFE= 1\nandNFM= 1-sites. BaTiO 3has the tetragonal phase with\nthe following coeﬃcients in the free energy density (eq. (4) )\nαFE=−2.77·107[Vm/C][37], βFE= 1.7·108[Vm5/C3][37]\nandPS= 0.265 [C/m2][37]. The parameters of bulk Fe\nassumed in the calculations (eq. (5)) for this ﬁgure are:\nK1= 4.8·104[J/m3][39],MS= 1.71·106[A/m][39] and\nαFM= 0.1. For the time-dependent magnetic ﬁeld it is taken\nµ0H0= 28·10−3[T],ω/(2π) = 4·109[Hz]. Inset shows the\ndependence of resonance peaks on λ.\naccording to which we ﬁnd µ0Hres(λ= 0)≈0.087 [T]\n(Fig. 1) [48]. For a ﬁnite ME coupling constant the reso-\nnance ﬁeld is modiﬁed by the anisotropy induced by the\nscreened polarization\nµ0Hres=1\nγω|res(λ/negationslash= 0)−2K1\nM2\nSMz+λPz.(6)\nThis relation reveals that the resonance condition de-\npends on the orientation of the surface polarization. In\nthe simulations the spectra are calculated at equilibrium\nwith the eﬀect of opposite orientation of the polarization\nand magnetization. For a ﬁnite λthe magnetization is\naligned along zand the polarization Pzis negative. This\nresults in a shift of the resonance peaks towards smaller\nﬁelds (Fig. 1, λ/negationslash= 0). The shift of resonance ﬁelds re-\nmains linear as a function of λ(inset of Fig. 1).\nWe also note, that the ME coupling does not aﬀect the\nintensity and the width of the spectra, similarly to the\neﬀect ofthe uniaxialanisotropyaxisoriented along zaxis\nin macro-spin FM nanoparticles[46].\nContact of ultrathin FE and FM ( E/negationslash= 0).-Considering\nthe MF chain in the presence of an electric ﬁeld Ewhich\nacts directly on the electrically active part of the chain,\nwe expect for E/bardblzthat the resonances shift to weaker\nmagnetic ﬁelds (eq. (6)). To favor the stability of the\nmagnetization along the zdirection, we increase the fre-\nquency of the oscillating magnetic ﬁeld which lowers the\nintensity of χ′′. Similarly to Fig. 1, a ﬁnite λleads to a\nshift of the resonance ﬁeld towards smaller ﬁelds ( λ= 0\nis not shown in Fig. 2), this is also predicted by a very\nrecent study [29]. The nonzero electric ﬁeld applied par-\nallel to the MF chain acts on the FE polarization and3\nFIG. 2. FMR spectra for the multiferroic chain of NFE= 1\nandNFM= 1-sites. All parameters are adopted from Fig.\n1, except the ME coupling parameter λ= 0.2 [s/F] and the\nfrequency of the applied magnetic ﬁeld ω/(2π) = 6·109[Hz].\nindirectly changes the ME coupling energy enlarging the\nresonance ﬁelds (eq. (6) for the ﬁeld /vectorE=E/vector ez).\nLet us deﬁne the shift of the resonance position from eq.\n(6)foraﬁnite electricﬁeldwithareferencetozeroE-ﬁeld\nµ0∆H≡Hres(E/negationslash= 0)−Hres(E= 0) =\nλ∆Pz≡λ[Pz(E/negationslash= 0)−Pz(E= 0)],(7)\nand plot µ0∆Hagainst ∆ Pz(Fig.3, left), we can deter-\nmine from the slope of the obtained dependence the ME\ncoupling parameter λ= 0.202 [s/F], the original value\nof which was set to 0 .2 [s/F]. The analysis of the po-\nlarization states shows that the abrupt increase of the\nresonance ﬁeld for E= 5·106[V/m] (Fig. 2) refers to\nthe reversal of the polarization from the antiparallel to\nthe parallel arrangement with respect to the magnetiza-\ntion orientation in the FM part. Since re-polarization\nof the FE modiﬁes signiﬁcantly the system and creates\nan energy jump at the interface responsible for a parallel\norientation of FE and FM sites, it is natural to expect\nthat the corresponding resonance point will deviate from\nthe linear dependence, which is proved by Fig. 3 (left\npanel). Therefore, a linear ﬁt of the spectral data should\nbeperformedforthepointsobtainedfortheelectricﬁelds\nthat do not result in re-polarization of the ferroelectrics\n(E <5·105[V/m]). If the spectral points measured for\nhigh ﬁelds were included into the ﬁtting data set, the\nresulting values of λcan exceed the actual value con-\nsiderably. Notwithstanding the diﬃculties of FE polar-\nization measurements [49] during the FMR experiment,\nthe proposed procedure gives a transparent method for\nobtaining the magnitude of the ME-coupling coeﬃcient\n[50].\nThin FE/FM contact for E/negationslash= 0.-For a thicker MF\nsystem the total FE energy density additionally includes\ninteractions of the neighboring sites[24] and the dipole-\ndipole interactions FFE\nDDI[34]FN>1\nFE=/summationtext\ni/parenleftBig\nαFE\n2P2\nzi+\nβFE\n4P4\nzi+κFE\n2(Pzi+1−Pzi)2−PziEz/parenrightBig\n+FFE\nDDI. The FM\nFIG. 3. Shifts of resonance positions calculated according to\nexpression (7) for diﬀerent applied electric ﬁelds. Values of\nFE polarizations are calculated for the resonance magnetic\nﬁelds presented in Fig. 2. Resonance curves for the ﬁelds\nE={1.0;3.0;4.0;4.5}·106[V/m] are not shown in Fig. 2.\nenergy density (eq. (5)) is supplemented by the ex-\nchange interaction and the FM dipole-dipole interaction\nFFM\nDDI[34]FN>1\nFM=/summationtext\nj/parenleftBig\n−A\na2\nFMM2\nS/vectorMj·/vectorMj+1−K1\nM2\nS(Mzj)2−\nµ0/vectorMj·/vectorHΣ(t)/parenrightBig\n+FFM\nDDI. The ME coupling in our case is\nlimited to the vicinity of the FE/FM interface. Hence,\nwe expect the inﬂuence of ME coupling on MF dynamics\nto be less pronounced as compared with the single-state\ncase (Fig. 2), which is evidenced by Fig. 4. Never-\ntheless, the spectral lines are clearly distinguished for\nFE switching from antiparallel to parallel orientation re-\ngarding the direction of magnetization vectors in FM\npart. The FE switching occurs for lower ﬁeld values\nin comparison with single-state case because the pres-\nence of interaction between the sites lowers the barrier\nof the Ginzburg-Landau-Devonshire potential. Applying\nthe procedure outlined above, we plotted (inset to Fig.\n4) the variation of peak position versus the averaged fer-\nroelectric polarization. Remarkably, the obtained points\nﬁts well to a straight line for the case of low polariza-\ntion values corresponding to a non-switched ferroelectric\nlayer. The points related to the switched value diverge\nfrom a linear scaling and were not considered. The slope\nof the linear ﬁt obtained from the plot is 0.01376, which\nis lower than the value of λ= 0.2 [s/F]. This eﬀect most\nprobablyappearsbecausethemagnetoelectriccouplingis\nlimited to a single interface and then propagates through\na chain of ﬁve sites at each side of the device, resulting in\na smaller variation of FMR spectra. A straightforward\nstep consisting in a multiplication of the ﬁtted value of\nλby the number of sites yields the value of 0.0688 [s/F]\nthat is three times smaller than λ. We think that it is\npossible to ﬁnd a proper re-normalization constant from\ngeometric considerations, which, however, go beyond the\nscopeofthisletter. Themostimportantresultisthat the\npeak position / polarization variation plots present the\nsame linear dependence as that observed for the single-4\nFIG. 4. FMR spectra for a chain of NFE= 5 and NFM= 5\nsites, cell size: aFE=aFM= 5·10−9[m]. The applied ﬁeld\nfrequency is ω/(2π) = 30·109[Hz], FM exchange stiﬀness is\nA= 2.1·10−11[J/m] [39], and λ= 0.2[s/F]. Otherparameters\nare as in Fig. 1. The FE coupling strength is calculated as\nκFE=G11/a2\nFE, whereG11= 51.·10−11[Jm3/C2][37]. Insets\nshow the peak on an enlarged scale resolving the states with\nnon-switched and switched ferroelectric layer. The linear ﬁt\nof the resonance shifts involves the non-switched polariza tion.\nsite system, which endorse the proposed method for the\nmeasurement of the magnetoelectric coupling for thicker\ncomposite multiferroic systems.\nRemarks and conclusions.- As shown above, the FMR\nspectra of the absorbed power of an unstrained thin com-\npositemutliferroicchaindependcriticallyontheMEcou-\npling. The peaks of resonance absorption are sensitive to\nthe magnitude and the orientation of the FE polariza-\ntion vector (Fig. 1) in the absence of an electric ﬁeld.\nApplying a static electric ﬁeld changes the value of Pzand causes a shift in the peak position according to (eq.\n(6)). Figs. 2 and 3 demonstrate how in our case the ME\ncoupling can be accessed. At ﬁrst, the static electric ﬁeld\nis applied along the direction ofthe FE minima such that\nit shifts sizably the peaks relative to the ﬁeld-free case.\nThe shifts of the ﬁelds µ0∆H(eq. 7) are plotted against\nthe measured FE polarization, yielding the ME coupling\ncoeﬃcient as the slope of the µ0∆H(∆Pz) dependence\n(Fig. 3, left panel). The plot of µ0∆H(E) (Fig. 3, right\npanel) is nonlinear, hence, the ME coupling can only be\ndetermined for known Pz(E) dependence. We note, the\nµ0∆H(E) dependence will be diﬀerent for the rhombo-\nhedral phase[34] of the FE [51].\nBased on the results in (Figs. 3, 4), on the estimates\nmade for the ME coupling coeﬃcient [24, 34], and on the\ncalculations for the elongated MF chains [33] we suggest\nto choose a MF contact consisting of a thin FM layer\n(electrode) and a thick FE. The thin FM layer serves to\navoid additional peaks in the FMR spectra due to spin\nwaves as well as to avoid a decay of the ME coupling\nin long FMs, whereas the thick FE part stabilizes the\nFE polarization for measuring Pz(Fig.3, left). Note, for\nhighlighting the ME dynamics dominated by the interfa-\ncial screening charges, the principal axis of Fe is rotated\nby 45◦with respected to BTO[100], we have then a good\nlattice match ( ∼1.4%) for the epitaxial growth of bcc Fe\non the tetragonal BTO.\nACKNOWLEDGMENTS\nThe authors gratefully acknowledge the support of the\nGerman Research Foundation by the grant SU 690/1-\n1 and SFB 762, the grant of CONACYT as Basic Sci-\nence Project 129269 (Mexico), the National Natural Sci-\nence Foundation of China (Grant No. 11104123)and the\nNational Basic Research Program of China (Grant No.\n2012CB933101).\n[1] N. Spaldin, M. Fiebig, Science 309, 391 (2005).\n[2] M. Fiebig, J. Phys. D: Appl. Phys. 38, R123 (2005).\n[3] W.Eerenstein, N.D. Mathur, J.F. Scott, Nature 442, 759\n(2006).\n[4] R. Ramesh, N.A. Spaldin, Nat. Mater. 6, 21 (2007).\n[5] M. Bibes, A. Barth´ el´ emy, Nat. Mater. 7, 425 (2008).\n[6] C.A.F. Vaz, J. Hoﬀman, C.H. Ahn, R. Ramesh, Adv.\nMater.22, 2900 (2010).\n[7] F. Zavaliche, T. Zhao, H. Zheng, F. Straub, M. P. Cruz,\nP.-L. Yang, D. Hao, R. Ramesh, Nano. Lett. 7(2007)\n1586.\n[8] K.F. Wang, J.-M. Liu, Z.F. Ren, Adv. Phys. 58, 321\n(2009).\n[9] G. Lawes, G. Srinivasan, J. Phys. D: Appl. Phys. 44,\n243001 (2011).\n[10] X.L. Lu, Y.S. Kim, S. Goetze, X. Li, S. Dong, P. Werner,\nM. Alexe, D. Hesse, Nano. Lett. 11, 3202 (2011).\n[11] J.H. Griﬃths, Nature 158, 670 (1946).[12] E.K. Zavoiskii, J. Phys. U.S.S.R. 10, 170 (1946).\n[13]Ferromagnetic Resonance: The Phenomenon of Reso-\nnant Absorption of a High-Frequency Magnetic Field\nin Ferromagnetic Substances. , S.V. Vonsovskii, (Oxford,\nPergamonn, 1966).\n[14] For a recent review see M. Farle, Rep. Prog. Phys. 61,\n755 (1998).\n[15] M. Weiler, L. Dreher, C. Heeg, H. Huebl, R. Gross, M.\nS. Brandt, S.T.B. Goennenwein, Phys. Rev. Lett. 106,\n117601 (2011).\n[16] S. Geprgs, A. Brandlmaier, M. Opel, R. Gross, S.T.B.\nGoennenwein, Appl. Phys. Lett. 96, 142509 (2010).\n[17] M.I. Bichurin, I.A.Kornev, V.M.Petrov, A.S.Tatarenk o,\nYu. V. Kiliba, G. Srinivasan, Phys. Rev. B 64, 094409\n(2001).\n[18] N. Li, M. Liu, Z. Zhou, N.X. Sun, D.V.B. Murthy, G.\nSrinivasan, T.M. Klein, V.M. Petrov, A. Gupta, Appl.\nPhys. Lett. 99, 192502 (2011).5\n[19] C.-W. Nan, M.I. Bichurin, S. Dong, D. Viehland, G.\nSrinivasan, J. Appl. Phys. 103, 031101 (2008).\n[20] H.L. Meyerheim, F. Klimenta, A. Ernst, K. Mohseni,\nS. Ostanin, M. Fechner, S. Parihar, I.V. Maznichenko,\nI. Mertig, J. Kirschner, Phys. Rev. Lett. 106, 087203\n(2011).\n[21] C.-G. Duan, S. S. Jaswal, E. Y. Tsymbal, Phys. Rev.\nLett.97, 047201 (2006).\n[22] S. Sahoo, S. Polisetty, C.-G. Duan, Sitaram S. Jaswal,\nEvgeny Y. Tsymbal, C. Binek, Phys. Rev. B 76, 092108\n(2007).\n[23] C.-G. Duan, J. P. Velev, R. F. Sabirianov, W. N. Mei, S.\nS. Jaswal, E. Y. Tsymbal, Appl. Phys. Lett. 92, 122905\n(2008).\n[24] A. Sukhov, C.-L. Jia, P.P. Horley, J. Berakdar, J. Phys. :\nCond. Matt. 22, 352201 (2010).\n[25] T. Taniyama, K. Akasaka, D.-S. Fu, M. Itoh, J. Appl.\nPhys.105, 07D901 (2009).\n[26] S. Valencia, A. Crassous, L. Bocher, V. Garcia, X. Moya,\nR. O. Cheriﬁ, C. Deranlot, K. Bouzehouane, S. Fusil, A.\nZobelli, A.Gloter, N.D.Mathur, A.Gaupp, R.Abrudan,\nF. Radu, A. Barth´ el´ emy, M. Bibes, Nat. Mater. 10, 753\n(2011).\n[27] G. Venkataiah, Y. Shirahata, I. Suzuki, M. Itoh, T.\nTaniyama, J. Appl. Phys. 111, 033921 (2012).\n[28] T. Cai, S. Ju, J. Lee, N. Sai, A. A. Demkov, Q. Niu, Z.\nLi, J. Shi, E. Wang, Phys. Rev. B 80, 140415(R) (2009).\n[29] V. Gunawan, R.L. Stamps, Phys. Rev. B 85, 104411\n(2012).\n[30] C.-L. Jia, A.Sukhov,P.P. Horley, J. Berakdar, Europhy s.\nLett.99, 17004 (2012).\n[31] M. Fechner, I.V. Maznichenko, S. Ostanin, A. Ernst, J.\nHenk, I. Mertig, Phys. Stat. Solidi B 247, 1600 (2010).\n[32] J.-W. Lee, N. Sai, T. Cai, Q. Niu, A.A. Demkov, Phys.\nRev. B81, 144425 (2010).\n[33] C.-L. Jia, A. Sukhov, P.P. Horley, J. Berakdar, J. Phys. :Confer. Ser. 303, 012061 (2011).\n[34] P.P. Horley, A. Sukhov, C. Jia, E. Mart´ ınez, J. Berakda r,\nPhys. Rev. B 85, 054401 (2012).\n[35] V.L. Ginzburg, Usp. Fiz. Nauk 38, 490 (1949).\n[36] A.F. Devonshire, Philos. Mag. 40, 1040 (1949).\n[37] J. Hlinka, P. M´ arton, Phys. Rev. B 74, 104104 (2006).\n[38]Physics of Ferroelectrics , K. Rabe, Ch. H. Ahn, J.-M.\nTriscone (Eds.), (Springer, Berlin 2007).\n[39]Magnetism and magnetic materials , J.M.D. Coey, (Cam-\nbridge University Press, Cambridge 2010), pp. 385.\n[40] S.F. Zhang, Phys. Rev. Lett. 83, 640 (1999).\n[41] L.D. Landau, I.M. Khalatnikov, Dokl. Akad. Nauk SSSR\n46, 469 (1954).\n[42] P. M´ arton, J. Hlinka, Ferroelectrics 373, 139 (2008).\n[43] L.D. Landau, E.M. Lifshitz, Phys. Z. Sowjet. 8, 153\n(1935).\n[44] T.L. Gilbert, Phys. Rev. 100, 1243 (1955) (abstract\nonly); T.L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004).\n[45] K.D. Usadel, Phys. Rev. B 73, 212405 (2006).\n[46] A. Sukhov, K.D. Usadel, U. Nowak, J. Magn. Magn.\nMater.320, 31 (2008).\n[47] C. Kittel, Phys. Rev. 73, 155 (1948).\n[48] The magnetization is supposed to be aligned by a reso-\nnant magnetic ﬁeld along the zaxis (Mz≈MS).\n[49] C.B. Sawyer, C.H. Tower, Phys. Rev. 35, 269 (1930).\n[50] A straightforward method in an experiment would be to\nplotµ0∆Has a function of the applied electric ﬁeld E.\nTheresonanceshiftsagainst theelectric ﬁeldareshownin\nFig. 3 (right), which has a nonlinear character and can be\nexplainedduetothesolution fortheFEpolarization from\nδFΣ\nδPz= 0, yielding P3\nz+αFE\nβFEPz+1\nβFE(λMS−E) = 0.\nThis eq. gives a strongly nonlinear solution for Pz(E),\nwhich, assuming the relation (7), suggests a nontrivial\ndependence of µ0∆H(E) even for the case of low ﬁelds,\nwhen the FE polarization is not reversed yet.\n[51] A. Sukhov, P.P. Horley, C.-L. Jia, J. Berakdar, unpub-\nlished."    },    {        "title": "1003.5344v1.Giant_magnetic_broadening_of_ferromagnetic_resonance_in_a_GMR_Co_Ag_Co_Gd_quadlayer.pdf",        "content": "arXiv:1003.5344v1  [cond-mat.mtrl-sci]  28 Mar 2010Giant magnetic broadening of ferromagnetic resonance in a G MR Co/Ag/Co/Gd\nquadlayer.\nS. Demirtas and M. B. Salamon\nUniversity of Texas at Dallas\nA. R. Koymen\nUniversity of Texas at Arlington\n(Dated: December 14, 2018)\nBoth magnetic-resonance damping and the giant magnetoresi stance eﬀect have been predicted\nto be strongly aﬀected by the local density of states in thin f erromagnetic ﬁlms. We employ the\nantiferromagnetic coupling between Co and Gd to provide a sp ontaneous change from parallel to an-\ntiparallel alignment of two Co ﬁlms. A sharp increase in magn etic damping accompanies the change\nfrom parallel to antiparallel alignment, analogous to resi stivity changes in giant magnetoresistance.\nThe discovery of giant magnetoresistance (GMR) by\nBaibichetal.[1]hasledtoimportantapplicationsinmag-\nnetic recording and data storage. Nonetheless, a fun-\ndamental understanding of the microscopic mechanism\nremains a subject of continuing research.[2, 3] Early\nwork[4, 5] considered spin-dependent scattering to be the\nprimary mechanism for GMR eﬀects, and indeed such\nscattering can considerably enhance them.[6] However,\nSchep, et al.[7] were the ﬁrst to demonstrate that sig-\nniﬁcant GMR (for currents perpendicular to the mag-\nnetic layers (CPP) at least) is possible in a perfect mag-\nnetic superlattice, a consequence ofs-d hybridization and\nresultant diﬀerential localization of electronic states be-\ntweeenparallel(P) and antiparallel(AP) alignment. The\nsame quantum-well states strongly modify the eﬀective-\nness of scatterersat the interface[3], thereby contributing\nto GMR for in-plane currents (CIP) as well. The aim\nof this paper is to provide independent evidence for sub-\nstantial changes in the local density of states accompa-\nnying a transition from P to AP alignment. Exploiting\nthe strong antiferromagnetic coupling between Co and\nGd, we fabricated a GMR structure that spontaneously\nreverses the relative orientation of two Co layers as the\ntemperature is reduced. Upon reversal from P to AP\nalignment, the width of the ferromagnetic resonance line\nof the free Co layer sharply changes its temperature de-\npendence. We interpret these results in the context of\nthe so-called torque-correlation model of ferromagnetic\ndamping, [8–10] applicable to Co, in which the linewidth\nis directly related to the local density of states; by anal-\nogy, we term the increased broadening Giant Magneto-\nBroadening (GMB).\nWe have prepared a trilayer structure of Co/Ag/Co\nwith an underlying Gd layer; the Ag layer is suﬃ-\nciently thick that there is no exchange coupling of the\ntwo Co layers. Co and Gd are strongly coupled\nantiferromagnetically.[11] Above, and somewhat below,\nthe Curie temperature of Gd, the two Co layers are fer-\nromagnetically aligned in a modest magnetic ﬁeld. As\nthe temperature is reduced, the magnetic moment of Gd/s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s48/s46/s48/s48/s48/s48/s48/s48/s46/s48/s48/s48/s48/s50/s48/s46/s48/s48/s48/s48/s52/s48/s46/s48/s48/s48/s48/s54/s48/s46/s48/s48/s48/s48/s56/s48/s46/s48/s48/s48/s49/s48/s77/s97/s103/s110/s101/s116/s105/s99/s32/s77/s111/s109/s101/s110/s116/s32/s91/s101/s109/s117/s93\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s91/s75/s93/s72/s61/s49/s48/s48/s32/s79/s101\nFIG. 1: Magnetic moment as a function of temperature for\nthe [Co 40 ˚A/Gd 100 ˚A] bilayer. Minimum corresponds to\nTcomp.\nincreases. Below the compensation temperature Tcomp,\nthe Gd moment exceeds that of its adjacent Co layer,\ncausing it to align with the magnetic ﬁeld, producing AP\nalignment of the two Co layers. In Fig. 1, we show the\nlow-ﬁeld magnetization of a Ag(10 nm)/Co(4nm) bilayer\non Gd(10 nm). The minimum in net magnetization at\nTcomp= 170 K reﬂects the point at which the magneti-\nzation of the underlying Gd and its adjacent Co layer are\nequaland opposite, orientedperpendicularto the applied\nﬁeld. The small paramagnetic moment at Tcompresults\nfrom the canting of the opposing moments toward the\napplied ﬁeld direction.\nMultilayer samples were created at room temperature\nusing a dc magnetron sputtering system. The base pres-\nsure of the deposition chamber was 10−9Torr. Ultra\nhigh purity argon gas was used and the deposition pres-2\nsure was 3 mTorr. An in situquartz thickness monitor,\ncalibrated by a stylus proﬁlometer, measures the deposi-\ntion thicknesses. Samples were sputtered from pure Gd,\nCo and Ag targets on Si (100) substrates. Ag layers 200\nAngstrom ( ˚A) thick were used as buﬀer layers in all sam-\nples. The Co(1)/Ag/Co(2)/Gd multilayer was created\nwith a 100 ˚A nonmagnetic Ag spacer between the two 4\nnm-Co layers, thick enough to suppress any long range\nexchange interactions. A 100- ˚A Ag cap layer completed\nthe deposition. The Curie temperature TCof the Gd\nthin ﬁlm is 240 K, somewhat below the bulk value.\nThe absorption spectrum as a function of applied\nmagnetic ﬁeld for the Co(1)/Ag/Co(2)/Gd multilayer is\nshown in Fig. 2 at room temperature. The microwave\nfrequency is 10 GHz and the applied ﬁeld is in the plane\nof the sample. Two Lorentzian derivative ﬁts are also\nshown in Fig. 2 to identify two diﬀerent resonances. Sep-\narationofthe adjacent absorptionpeaks can be made be-\ncause, as shown previously,[12] a proximate layer of Gd\nreduces the ﬁeld for resonance and signiﬁcantly increases\nthe linewidth of Co thin ﬁlms. This leads to the conclu-\nsion that the broader reasonance is associated with the\nCo(2) layer. Fig 3 shows the temperature dependence\nof the linewidth associated with Co(1) and Co(2) reso-\nnances. Above the Curie temperature of Gd ( TC= 240\nK), both resonancelines broadenslightly with decreasing\ntemperature. Below TCthe Co(2) resonanceis no longer\nseen while the Co(1) resonance ﬁrst broadens abruptly\nand then continues to increase with decreasing tempera-\nture to the compensation point, Tcomp= 170 K. Below\nTcomp, the linewidth increases much more strongly with\ndecreasing temperature, exceeding the resonant ﬁeld be-\nlow 100 K.\nFerromagnetic resonance is generally treated phe-\nnomenologically via the Landau-Lifshitz-Gilbert (LLG)\nequation of motion,[13]\nd− →m\ndt=−γ− →m×− →H+α− →m×d− →m\ndt. (1)\nwhere− →mis the reduced magnetization vector, γ,the gy-\nromagnetic ratio and α,the Gilbert damping parameter.\nRelaxation in metallic ferromagnet ﬁlms has convention-\nally been attributed to the transfer of angular momen-\ntum from the precessing magnetization to the spin of\nthe conduction electrons via s-dexchange and the subse-\nquent relaxation of the conduction electron polarization\nvia spin-dependent scattering.[14] More recently, atten-\ntion has been focused on the so-called torque-correlation\nmodel ﬁrst introduced by Kambersky.[15] In this pro-\ncess, the time-dependent magnetization induces charge-\ncurrents in the conduction electrons via the spin-orbit\ninteraction. These, in turn, exert torque on the mag-\nnetization, transferring angular momentum to the lattice\nvia the relaxation of charge currents. The longer the re-\nlaxation time τof these currents, the greater the torque\nand the broader the line. For intraband transitions,/s48 /s50/s48/s48 /s52/s48/s48 /s54/s48/s48 /s56/s48/s48 /s49/s48/s48/s48 /s49/s50/s48/s48 /s49/s52/s48/s48/s45/s49/s50/s48/s48/s48/s45/s54/s48/s48/s48/s48/s54/s48/s48/s48/s49/s50/s48/s48/s48\n/s32/s32/s65/s98/s115/s111/s114/s112/s116/s105/s111/s110/s32/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s91/s97/s46/s117/s46/s93\n/s77/s97/s103/s110/s101/s116/s105/s99/s32/s70/s105/s101/s108/s100/s32/s91/s71/s93/s51/s48/s48/s32/s75\nFIG. 2: FMR absorption spectra for the [Co 40 ˚A/Ag 100 ˚A\n/Co 40˚A/Gd 100 ˚A] ﬁlm at room temperature. Linewidths\nwere found making two Lorentzian ﬁts to the overall absorp-\ntion spectra\nGilmore, et al.[10] have shown that\nα(T) =γτ(T)\n2µ0m/summationdisplay\nnk|Γn(k)|2/parenleftbigg\n−∂f\n∂ε/parenrightbigg\n,(2)\nwhereτis the orbital relaxation time of the conduc-\ntion electron, Γ n(k) is the torque matrix element from\nthe spin-orbit interaction, and ( −∂f/∂ε) is the negative\nderivative of the Fermi function. The sum is over band\nindices. The interplay between the two mechanisms has\nbeen discussed by several authors.[9, 16] By artiﬁcially\nchanging the Fermi energy in their band calculations,\nGilmore et al. demonstrate specﬁcally that the summa-\ntion in Eq. (2) follows the density of states for Co and\nother ferromagnetic metals. Note that the linewidth is\nrelated to the Gilbert parameter by ∆ H= 1.16ωα/γ,\nwhereω/2π= 10 GHz is the applied microwave fre-\nquency.\nAs seen in Fig. 3, the Co(1) linewidth gradually in-\ncreases with decreasing temperature from TCtoTcomp\nand then increases more rapidly below; this is the GMB\neﬀect. A linewidth that increases with decreasing tem-\nperature is indicative [9] that the torque-correlation pro-\ncess dominates over spin damping, evidently becoming\neven more dominant below Tcomp. In the absence of\ntorque-correlation processes, spin-damping, which varies\nτ(T)−1, would require a mechanism that, upon rever-\nsal of the Co(2) magnetization, increases with decreas-\ning temperature at a rate that overcomes the increase\ninτ(T). The band structure of the Co(1) layer, on the\nother hand, will change dramatically upon the transition\nfrom P to AP alignment.[7], thereby changing the den-3\n/s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s50/s53/s48/s51/s48/s48/s51/s53/s48/s52/s48/s48/s52/s53/s48\n/s67/s111/s40/s49/s41/s67/s111/s40/s50/s41/s47/s71/s100\n/s84\n/s99/s111/s109/s112/s67/s111/s40/s49/s41/s47/s65/s103/s47/s67/s111/s40/s50/s41/s47/s71/s100\n/s67/s111/s40/s49/s41/s32/s105/s115/s32/s97/s110/s116/s105/s112/s97/s114/s97/s108/s108/s101/s108/s32/s116/s111/s32/s67/s111/s40/s50/s41\n/s67/s111/s40/s49/s41/s32/s105/s115/s32/s112/s97/s114/s97/s108/s108/s101/s108/s32/s116/s111/s32/s67/s111/s40/s50/s41\n/s32/s32/s76/s105/s110/s101/s119/s105/s100/s116/s104/s32/s91/s79/s101/s93\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s91/s75/s93\nFIG. 3: FMR linewidth as a function of temperature for par-\nallel and antiparallel alignment of Co layers in [Co 40 ˚A/Ag\n100˚A /Co 40 ˚A/Gd 100 ˚A] ﬁlm.\nsity of states in the Co(1) layer. Further, Binder et al.[3]\nshowedthat impuritieslocated within a Colayerin GMR\nstructures exhibit dramatically larger relaxation rates in\nAP vs P alignment, again reﬂecting an increase in the lo-\ncal density of states. Impurities located at the interface\nbetween Co and Cu, in their calculation, are seven times\nmore eﬀective as scatterers in the AP conﬁguration; the\neﬀect is even larger for impurities in the center of the Co\nlayer. Similarly, the torque matrix element Γ n(k), which\ntracks with the density of states, [10] should reﬂect the\nsame increasein localdensity ofstates in the AP conﬁgu-\nration. We attribute the seven-fold increase in the slope\nof ∆H(T) shown in Fig. 3, therefore, to an increase in\nthe summation in Eq.(2) and consequently, to a stronger\ndependence on τ(T). Further, Steiauf and F¨ ahnle[17]\nhave shown, in the context of the torque-correlation ap-\nproach, that band-structure eﬀects in lower-dimensional\nstructures dramatically increase the Gilbert parameter\nof Co relative to the bulk metal. We suggest that, in\nthe single layer considered by Steiauf and F¨ ahnle, both\nspin sub-bands are localized, much as in the case of AP\nalignment, while only one sub-band is localized in the P\nconﬁguration. We conclude that the largeenchancement\nof the temperature dependence of the linewidth in our\nGMR structure–the GMB eﬀect–conﬁrms both the dom-\ninance of the torque-correlation process in spin damping\nand the importance of electron localization in the GMR\neﬀect.\nThere have been, of course, many studies of magnetic\nrelaxation in thin metallic ﬁlms and multilayers. For ex-\nample, an experiment by Urban, et al.[18] found that\nthe relaxation rate for a thin Fe layer was larger when asecond Fe layer, separated by an Au spacer, was added.\nBecause the increase depends on the thickness dof the\nresonating layer, they ascribed it to torques that occur\nat single ferromagnetic-normal metal interfaces, with no\nrole proposed for the thicker Fe layer beyond acting as a\nsinkforspincurrents. Wesuggestthatlocalizationeﬀects\nmay play a role, even though the conduction electrons in\nFe are less polarized than in Co. A very similar exper-\niment [19] showed that when the resonance of the two\nlayers in an Fe/Au/Fe are made to coincide by judicious\nchoice of in-plane ﬁeld angle, the linewidths are equal\nand narrowest. This was interpreted in terms of spin\npumping between the two layers. In that picture, the\noﬀ-resonance ferromagnetic layer acts as a perfect spin\nsink, except when the two layers have a common reso-\nnant ﬁeld. Then spin currents generated in each layer\ncompensate the spin-sink eﬀect of the other.However,the\nresonances coincide when the eﬀective ﬁeld is the same\nin each layer, which may also maximize ferromagnetic\nalignment and minimize localization. As seen in Fig. 2\nin the present experiment, the Co(2) and Co(1) reso-\nnances overlap at room temperature, and therefore each\nmay be narrowed by spin pumping. Below Tc,however,\nthe Co(2) resonance is no longer detected, with the anti-\nferromagnetic coupling to the ferromagnetic moment of\nGd shifting the resonance out of the observed ﬁeld range.\nAs a consequence, we expect dynamical coupling due to\nspin-pumping to disappear below the Gd transition, giv-\ningrisetotheobservedjumpinthelinewidthoftheCo(1)\nresonance.\nTo summarize, we argue that the change in the tem-\nperature dependence of the ferromagnetic linewidth that\noccurs at the transition between P and AP alignment,\nprovides independent conﬁrmation of the role of quan-\ntum conﬁnement in GMR structures. At the same time,\nit provides further evidence that the torque-correlation\nmodel playsasubstantialrolein spin relaxationin metal-\nlic ferromagnets, especially in Co, which is nearly a half-\nmetal. Clearly,similarexperimentsusingFeandpermal-\nloy, where the torque-correlationmodel may be less dom-\ninant, are clearly in order.\nOne of us (ARK) wishes to acknowlege the support of\nthe Welch Foundation through Grant No. Y-1215.\n[1] M. N. Baibich, J. M. Broto, A. Fert, Nguyen Van Dau,\nF. Petroﬀ, P. Etienne, G. Creuzet, A. Friederich and J.\nChazelas, Phys. Rev. Lett. 61, 2472 (1988).\n[2] P. Zahn, J. Binder, I. Mertig, R. Zeller and P.H. Ded-\nerichs, Phys. Rev. Lett. 80, 4309 (1998).\n[3] J. Binder, P. Zahn, and I. Mertig, J. Appl. Phys. 89,\n7107 (2001).\n[4] R. E. Camley and J. Barna´ s, Phys. Rev. Lett. 63, 664\n(1989).\n[5] P. Levy, S. Zhang, and A. Fert, Phys. Rev. Lett. 65, 16434\n(1990).\n[6] P. Zahn, I.Mertig, M. Richter andH. Eschrig, Phys. Rev.\nLett.75, 2996 (1995).\n[7] Kees M. Schep, Paul J. Kelly and Gerrit E. W. Bauer,\nPhys. Rev. Lett. 74, 586 (1995).\n[8] J. Kune˘ s and V. Kambersk´ y, Phys. Rev. B65, 212411\n(2002).\n[9] K. Gilmore, Y. U. Idzerda and M. D. Stiles, Phys. Rev.\nLett.99,027204((2007)..\n[10] K. Gilmore, Y. U. Idzerda and M. D. Stiles, J. Appl.\nPhys.103, 07D303 (2008).\n[11] S. Demirtas, M. R. Hossu, R. E. Camley, H. C. Mireles,\nand A. R. Koymen, Phys. Rev. B 72, 184433 (2005).\n[12] S. Demirtas, R. E. Camley, Z. Celinsky, M. R. Hossu, A.\nR. Koymen, C. Yuand M. J. Pechan ( arXiv:1002.4889v1[cond-mat.mtrl-sci])\n[13] T.L. Gilbert, Phys. Rev. 100, 1243 (1955).\n[14] Y. Tserkovnyak, A. Brataas and G. E. W. Bauer, Phys.\nRev. B66, 224403 (2002).\n[15] V. Kambersky, Can. J. Phys. 48, 2906 (1970).\n[16] Y. Tserkovnyak, G. A. Fiete and B. I. Halperin, Appl.\nPhys. Lett. 54, 5234 (2004).\n[17] D. Staiauf and M. F¨ ahnle, Phys. Rev. B 73, 064450\n(2005).\n[18] K. Urban, G. Woltersdorf and B. Heinrich, Phys. Rev.\nLett.87, 217204 (2001).\n[19] B. Heinrich, Y. Tserkovnyak, G. Woltersdorf, A. Bratas ,\nR. Urban and G. E. W. Bauer, Phys. Rev. Lett. 90,\n187601 (2003)."    },    {        "title": "2005.13850v1.Spin_Pumping_Induced_Non_Linear_Electric_Current_on_the_Surface_of_a_Ferromagnetic_Topological_Insulator.pdf",        "content": "Spin-Pumping-Induced Non-Linear Electric Current on the Surface of a Ferromagnetic Topological\nInsulator\nYusuke Hama1,\u0003and Kentaro Nomura2, 3\n1National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan\n2Institute of Material Research, Tohoku University 2-1-1 Katahira, Aoba-ku, Sendai, 980-8577, Japan\n3Center for Spintronics Research Network, Tohoku University 2-1-1 Katahira, Aoba-ku, Sendai, 980-8577, Japan\n(Dated: May 29, 2020)\nWe investigate the spin-pumping-induced electric current on the surface of a three-dimensional topological\ninsulator hybridized with a ferromagnet, namely, ferromagnetic topological insulator. In order to do this, we\nestablish the microscopic formalism and construct the perturbation theory using a Keldysh Green’s function\napproach. We analyze how this electric current is generated by an exchange interaction and an external ac mag-\nnetic ﬁeld, which is the driving force of ferromagnetic resonance as well as the spin pumping. The mechanism is\nas follows. First, the ferromagnetic resonance is driven and a zero-momentum magnon emerges. It is the ﬂuctu-\nation from the saturation magnetization pointing parallel to the precession axis of the ferromagnetic resonance.\nAfter then, the spin pumping is generated with the zero-momentum magnon being the carrier of spin. The zero-\nmomentum magnon and the topological insulator surface state couples through the exchange interaction and the\nspin carried by the magnon is transferred to it. Owing to the spin-momentum locking, the transferred spin is\nconverted into the momentum of topological insulator surface state leading to the generation of electric current\nﬂowing perpendicular to the precession axis of the ferromagnetic resonance. It is quadratic in the amplitude of\nexternal ac magnetic ﬁeld whereas it is linear to the strength of the exchange interaction. The associated electric\nvoltage is described by the spectrum of zero-momentum magnon. The non-linearity of spin-pumping-induced\nelectric current in the ac magnetic ﬁeld as well as the linearity in the exchange-interaction strength reﬂects that\nthe surface of ferromagnetic topological insulator has a high-performing functionality of generating the electric\ncharge current by magnetic controlling.\nI. INTRODUCTION\nQuantum technologies for hybridizing two or more sub\nquantum systems have been advancing rapidly with many\ntypes of elements ranging from solid-state systems to atomic-\nmolecular and optical systems having been used, for exam-\nple, electrons and nuclei in GaAs semiconductors, nitrogen-\nvacancy centers in diamonds, superconducting qubits, and\natoms and cavities composing cavity quantum electrodynamic\nsystems [1–13]. The functionalities of these hybrid quantum\nsystems are superior to or richer than those of any individual\nsub quantum systems and are characterized in the way they\nare composed of. By selecting sets of sub quantum systems\nwhich are the best choices to engineer the hybrid quantum\nsystem which has the functionality to perform the task you are\naiming, it becomes a powerful tool to execute quantum-state\ncontrolling, quantum information processing, and spintronics.\nThe key issue for spintronics is to perform a high-efﬁcient\nconversion of electric charge and spin degrees of freedom\nor the coherent controlling of electricity and magnetism with\nlowering sufﬁciently an energy consumption (Joule heating).\nIn order to accomplish these tasks, we have to search for ma-\nterials having potentials to create physical processes which\ncan be utilized for them and use these materials to engineer\nquantum devices. Examples include the non-magnetic heavy\nmetals with strong spin-orbit interaction which exhibits (in-\nverse) spin Hall effect like Pt and materials composed of metal\nand oxide possessing Rashba interfaces [14–24]. Recently,\n\u0003current address: Quemix Inc., 2-11-2 Nihombashi, Chuo-ku, Tokyo 103-\n0027, Japantopological insulator (TI) is considered to be a good candi-\ndate for a component of spintronics devices because TI ex-\nhibits bulk state with strong spin-orbit coupling as well as\nsurface state whose spin and momentum are strongly coupled\nwhich is called the spin-momentum locking (high-efﬁcient\nconvertibility of spin and electric charge current) [25–28].\nIn addition, the hybrid quantum system of magnetic materi-\nals and TI, namely, the magnetic TI, has been intensively in-\nvestigated from both theoretical and experimental points of\nview [29–59]. Including the quantum anomalous Hall ef-\nfect, the magnetic TI exhibits rich quantum phenomena ow-\ning to the composition of magnetism and spin-momentum\nlocking (multifunctionality). Because of the multifunction-\nality and the high-efﬁcient convertibility of spin and electric\ncharge current, the magnetic TI is considered to be one of\nthe promising candidate for spintronics devices and a large\nnumber of investigations have been made toward this goal\n[29, 38–45, 47, 51, 52, 54, 55, 57–59]. Although great ef-\nforts have been made for this, we still have not satisfactorily\nachieved the microscopic understanding of physics at the in-\nterface between the magnet and TI. For instance, we have not\nunderstand satisfactorily the way and how efﬁciently the spin\ntransferred from the magnet can be converted into the elec-\ntric current and/or voltage (spin pumping and the associated\nphenomena; inverse spin Hall effect and inverse Edelstein ef-\nfect) whereas the electric current of TI being converted into\nthe magnetization dynamics and/or a spin current (spin-orbit\ntorque, spin Hall effect, and Edelstein effect). Such complex-\nities are arising from the fact that the spin current is not a con-\nserved current in the macroscopic systems and the difﬁculties\nto distinguish whether the contribution to the electric charge\ncurrent under observation is coming from the surface state orarXiv:2005.13850v1  [cond-mat.mes-hall]  28 May 20202\nthe bulk state. It is important and an urgent issue to challenge\nanalyzing these problems in order to achieve a deeper under-\nstanding of the conversion between the electric current (orbital\ndegrees of freedom) and the magnetization dynamics (spin po-\nlarization as well as the spin current) in the magnetic TI, the\nphysics at the interface between magnets and TI surface state\nboth theoretically and experimentally, and further, to realize\nthe coherent controlling of TI surface state and magnetization\ntoward spintronics application.\nIn this paper, we will focus on the physics of TI surface\nstate and construct the microscopic theory for the quantum\ntransport phenomena at the interface between ferromagnet and\nTI. In order to do this, we use a Keldysh (non-equilibrium or\ncontour-time) Green’s function approach. We investigate the\nelectric current of TI surface state as well as the associated\nelectric voltage induced by the spin pumping originating in\nthe ferromagnetic resonance (FMR) driven by an external ac\nmagnetic ﬁeld. We analyze in detail how this electric current\nis created by the ac magnetic ﬁeld and the exchange interac-\ntion between the localized spin in the ferromagnet and the TI\nsurface state. We show that when the spin is carried from the\nzero-momentum magnon, which is created by the FMR, to\nthe TI surface state through the exchange interaction, due to\nthe spin-momentum locking this carried spin is converted into\nthe momentum. Then correspondingly, the electric current is\ninduced in the direction perpendicular to the precession axis\nof FMR, namely, spin-pumping-induced electric current. It\nis the quadratic response to the ac magnetic ﬁeld whereas it\nscales linearly to the strength of exchange coupling. On the\nother side, the associated electric voltage has a structure rep-\nresented by the spectrum of zero-momentum magnon which\nclearly reﬂects that the driving force of this electric voltage is\nthe spin pumping. Our result enables us to understand clearly\nnot only the mechanism of the spin-pumping-induced electric\ncurrent and its characteristic, but it also gives us a qualitative\nexplanation for the experimental results reported previously\n[42, 59].\nThis paper is organized as follows. In Sec. II, we present\nour microscopic model of the composite system of ferromag-\nnet and TI surface state. Then, we construct the formalism for\ndescribing the time evolution of this system using the Keldysh\nGreen’s function approach. Based on it, we present a mathe-\nmatical representation for the electric current of TI surface\nstate at the non-equilibrium steady state. Next, to calculate\nthis electric current we establish the perturbation theory for\nthe Keldysh Green’s function where the external ac magnetic\nﬁeld and the exchange interaction are regarded as perturbative\nterms. In Sec. III, which presents the main result of this pa-\nper, we discuss in detail the generation of electric current of TI\nsurface state induced by the spin pumping as well as the asso-\nciated electric voltage. By analyzing the structure of Feynman\ndiagram for the perturbative Green’s function, we discuss the\nmechanism of the spin-pumping-induced electric current as\nwell as its characteristics. Then, we make a comparison be-\ntween our result and the experimental results [42, 59] through\nthe characteristic of electric voltage. Sec. IV is devoted to the\nconclusion and outlook of this paper.\nFIG. 1. Schematic illustration of the ferromagnetic TI. FM is rep-\nresented by the Hamiltonian HFMwhile the surface state of TI is\nexpressed by HTI. The localized spin in FM couples with the spin\nof TI surface state through the exchange interaction Vexc. The total\nHamiltonian of this system is given by H=HFM+HTI+Vexc:\nII. MICROSCOPIC THEORY\nIn this section, we ﬁrst present our microscopic model for\nthe ferromagnetic TI. Based on it, we establish the formalism\nto describe the time evolution of this system generated by the\nspin pumping. Then we evaluate the electric current of the\nTI surface state at the non-equilibrium steady state using the\nKeldysh Green’s function approach. We do this by construct-\ning the perturbation theory for the Keldysh Green’s function\nso that the ac magnetic ﬁeld and the exchange interaction are\ntreated as perturbative terms.\nA. Modeling and Formalism\nThe ferromagnetic TI is the composite system of a ferro-\nmagnet (FM) and the three-dimensional TI. We take a spa-\ntial cartesian coordinate so that the xyplane is parallel to the\ninterface between the FM and TI whereas the zaxis perpen-\ndicular to it. The surface state of TI appears in the xyplane.\nThe TI surface state and a localized spin in the FM are cou-\npled through the exchange interaction. The illustration of fer-\nromagnetic TI is presented in Fig. 1. Experimentally, this\nsystem is created by doping the magnetic atoms (for instance,\nCr, V , and Mn) to the TI or implementing the heterostructure\nof ferromagnetic materials (e.g., a ferromagnetic insulator as\nwell as metal including EuS, EuO, YIG, and permalloy such\nasNi81Fe19and CoFeB) and the TI [33, 34, 36–40, 42, 45–\n54, 56–59]. The examples of three-dimensional TI include\ntetradymites Bi2Se3andBi2Te3[25–28, 60]. Hereinafter, let\nus focus on the interface between FM and TI and model the\ncomposite system of localized spin at this interface and the\nTI surface state (let us call it the surface of ferromagnetic\nTI). The spin pumping and the associated inverse spin Hall\neffect in the heterostructure systems composed of ferromag-\nnetic metal (or ferromagnetic insulator such as YIG) and Pt\nor NiPd alloy have been modeled in [19]. By referring to it,\nwe model the surface of ferromagnetic TI for describing the\nspin pumping process and the associated electric-current gen-\neration. It is described by the Hamiltonian H=H0+Vexc;3\nwhereH0=HFM+HTIwithHTI=\u0016HTI\n0+Himp. The\nHamiltonian \u0016HTI\n0is the unperturbed Hamiltonian of the TI\nsurface state consisting of the spin-momentum-locking term\nwith the dispersion relation being measured from the chem-\nical potential \u0016TI:\u0016HTI\n0= (HTI\n0\u0000\u0016TINTI). The operator\nNTIis the number operator of TI surface state. Hereinafter,\nlet us take the chemical potential \u0016TIto be equal to the Fermi\nenergy of TI and denote it as \u000fF.Himpis an impurity potential\nterm and assume to be spin independent (non magnetic). HFM\nis the unperturbed Hamiltonian of FM and take its chemical\npotential to be zero (\u0016FM= 0):Vexcis the exchange inter-\naction between the localized spin in FM and TI surface state.\nThe Hamiltonians \u0016HTI\n0andHimpare given by\n\u0016HTI\n0=Z\nd2x y\n\u000b0(x)\u0010\nH(0)\nTI(x)\u0000\u000fF1\u0011\n\u000b0\u000b \u000b(x);(1)\nHimp=Z\nd2x y\n\u000b0(x)Himp\n\u000b0\u000b(x) \u000b(x); (2)\nwhere\u0010\nH(0)\nTI(x)\u0011\n\u000b0\u000b=\u0000i~vF(\u001by@x\u0000\u001bx@y)\u000b0\u000b; (3)\nHimp\n\u000b0\u000b(x) =NimpX\niimp=1Vimp(x\u0000Ximp\niimp)\u00011\u000b0\u000b: (4)\nThe operators  \u000b(x)and y\n\u000b(x)are the annihilation and cre-\nation operators of the TI surface state at the two-dimensional\nspatial coordinate x= (x;y), respectively. The index \u000b=\";#\ndescribes the spin degrees of freedom of TI surface state. The\nsummation is taken for two repeated indices \u000band\u000b0in Eqs.\n(1) and (2).vF\u00185:0\u0002105m/s is the Fermi velocity while \u001bx\nand\u001byare the Pauli matrices. Vimp(x\u0000Ximp\niimp)in Eq. (4) is\nthe impurity potential and the vector Ximp\niimp= (Ximp\niimp;Yimp\niimp)\nis the coordinate of iimp-th impurity. Nimpis the total num-\nber of impurities. 1\u000b0\u000bis the two by two unit matrix. For\nthe details of TI-surface-state ﬁeld operators  \u000band y\n\u000b0, see\nsubSec. A 1 in Appendix A. The Hamiltonian HFMis given\nby\nHFM=\u0000JnxX\nhijiSi\u0001Sj+~\rX\niB0Sy\ni: (5)\nThe three-component vector Si= (Sx\ni;Sy\nj;Sz\ni)represents\nthe localized spin of the FM at the spatial coordinate ri=\n(rx\ni;ry\ni):The indices iandjruns from 1 to NlocwithNloc\ndenoting the total number of localized spin at the inter-\nface between FM and TI. Jnxis the strength of the nearest-\nneighboring exchange interaction. The summationP\nhijiis\ntaken for nearest-neighboring pairs. For any i, the localized\nspinSisatisﬁesS2\ni= (Sx\ni)2+ (Sy\ni)2+ (Sz\ni)2=S0(S0+ 1)\nwithS0its spin magnitude. \ris the gyromagnetic ratio of lo-\ncalized spin. The static magnetic ﬁeld B0is applied to the y\ndirection and the saturation magnetization is created along this\ndirection. Hereinafter we will not include the demagnetizing\ncoefﬁcient for simplicity. The exchange interaction Vexchas\nthe form\nVexc=\u0000JexcX\niX\na=x;y;zsa(ri)Sa\ni; (6)\nFIG. 2. Diagrammatic representation of the time evolution of fer-\nromagnetic TI surface. At far past ( t=\u00001), the ferromagnetic TI\nsurface is in the thermal equilibrium state described by the grand-\ncanonical ensemble \u001aGC(H;\f;\u000f F):After then, at t=t0the exter-\nnal magnetic ﬁeld Hext(t)is applied and the FMR as well as the spin\npumping are driven. The time evolution of this system is represented\nby the density matrix \u001aH(t). At sufﬁciently a long time ( t\u001dt0),\nthe surface of ferromagnetic TI is in the non-equilibrium steady state\nand the associated quantum transport of TI surface state is generated.\nwhereJexcis the strength of the exchange interaction.\nsa(ri) = y\n\u000b0(ri)(\u001ba\n\u000b0\u000b=2) \u000b(ri)is the spin density of TI\nsurface state at the coordinate ri.\nNext, let us discuss the time evolution of this system. At\ninitial time ( t=\u00001), the ferromagnetic TI is in the thermal\nequilibrium state with the temperature T. It is represented by\nthe grand-canonical ensemble with its density matrix\n\u001aGC(H;\f;\u000f F) =exp (\u0000\fH)\nTr (exp (\u0000\fH)); (7)\nwhere\f\u00001=kBTwithkBthe Boltzmann con-\nstant. Note that the TI-Fermi-energy dependence\nis included in the Hamiltonian H:Att=t0, we\napply an ac external magnetic ﬁeld Bext(t) =\nBext\u0000\nsin\u0000\nsgn(B0)\u0001!extt\u0001\n;0;cos\u0000\nsgn(B0)\u0001!extt\u0001\u0001\n,\nwhere sgn(B0) = +1 (\u00001)whenB0>0 (<0). Here we\nhave taken a circular polarized light. Bextand!extare its\namplitude and frequency, respectively. This triggers the ferro-\nmagnetic resonance (FMR). The system at t > t 0is going to\nbe described by the total Hamiltonian H(t) =H+Hext(t)\nwhereHext(t)is given by\nHext(t) =~\rX\na=x;zX\niBext\na(t)Sa\ni: (8)\nFor later convenience, we decompose the total Hamiltonian\nH(t)into the formH(t) =H0+H0(t)withH0(t) =\nVexc+Hext(t):The precession axis of the FMR is along the\nydirection owing to the static magnetic ﬁeld B0. Once the\nFMR is triggered, a spin transfer occurs from FM to the TI4\nsurface state mediated by the exchange interaction Vexc, i.e.,\nspin pumping. As a result, a spin polarization as well as an as-\nsociated non-equilibrium state is generated on the surface of\nTI. Such a physical process (the time evolution of the system\natt>t 0) is represented by the density matrix [62, 63]\n\u001aH(t) =U(t;t0)\u001aC(H;\f;\u000f F)Uy(t;t0); (9)\nwhere the time-evolution operator U(t;t0)is given by\nU(t;t0) =Texp\u0012\n\u0000i\n~Zt\nt0H(t0)dt0\u0013\n; (10)\nwith the symbol Tdenoting the time-ordering product of real\ntime. By using the density matrix in Eq. (9), the expectation\nof a physical operator Aatt>t 0is expressed by\nhA(t)i= Tr [AH(t)\u001aC(H;\f;\u000f F)]; (11)\nwhereAH(t) =Uy(t;t0)AU(t;t0).\nX\u000b\n=\nTr(X\u001aC(H;\f;\u000f F))represents the thermal average taken\nwith respect to the Hamiltonian H. At sufﬁciently a long\ntime (t\u001dt0), the non-equilibrium steady state is realized\nand the quantum transport phenomena of TI surface state is\ngenerated. To summarize the above description, in Fig. 2 we\npresent the diagrammatic structure of time evolution of the\nferromagnetic TI surface.\nSince the microscopic formalism for the time evolution of\nthe ferromagnetic TI surface as well as that for the expectation\nof the physical operators have been established, let us discuss\nthe quantum transport phenomena on the surface of ferromag-\nnetic TI at the non-equilibrium steady state. When the spin\npumping is driven by the FMR, the y-polarized spin is injected\nfrom FM to TI surface. We write the spin current associated\nwith this spin pumping process as Jspin\ny;z. The ﬁrst subscript y\ndenotes the direction of the spin polarization whereas the sec-\nond subscript zdescribes the ﬂowing direction of spin current.\nThrough the exchange interaction Vexc, the spin current ﬂows\ntoward the TI surface. Some portion of Jspin\ny;z is going to be\nconverted into the momentum (the electric current ﬂowing on\nthe surface of TI) due to the spin-momentum locking. Besides\nthat, it might be converted into other types of phenomena, for\ninstance, a dissipation process like a spin relaxation process or\nthe spin current which bounces back to FM. From such a con-\nsideration, the exact evaluation of the spin current Jspin\ny;z and\nhow efﬁciently it is converted into the electric current of TI\nsurface state are very difﬁcult tasks. This is because it is hard\nto mathematically deﬁne the spin current since the spin is not\nthe conserved quantity or the spin current is not the conserved\ncurrent in the macroscopic system like a mili-meter-scale sys-\ntem. On the other hand, what has been observed in the ex-\nperiment is the electric voltage induced by the spin pumping\n[42, 59]. By taking into account of this fact, although there\nare some theoretical approaches which treat mathematically\nthe spin current and calculate the spin-to-charge conversion\nefﬁciency using a concept such as spin-mixing conductance\n[15, 19, 22], we do not take such approaches. Instead, we con-\nsider that the y-polarized spin carried from FM to TI surface\nvia spin-pumping process is going to be mainly converted intothe electric charge current of TI surface state. Therefore, in-\nstead of calculating the spin current Jspin\ny;zdirectly and analyze\nhow efﬁciently it is converted into the electric charge current,\nwe calculate directly the electric charge current of the TI sur-\nface state and analyze how it is created by the ac magnetic\nﬁeld and the exchange interaction. Here we calculate the x-\ncomponent of electric charge current density jx(x). It is given\nbyjx(x) =\u0000evF\u0000\n y\n\u000b0(x)\u001by\n\u000b0\u000b \u000b(x)\u0001\n=\u00002evFsy(x);with\n\u0000e(<0)the electric charge and sy(x)is they-component\nspin density of TI surface state at the coordinate x. Such an\nequivalence of the x-component of electric charge current and\nthey-component spin originates in the spin-momentum lock-\ning. By denoting the annihilation and creation operators of\nTI surface state ﬁeld in the Heisenberg picture with respect to\nH(t)as H\u000b(x;t)and y\nH\u000b(x;t), respectively, from Eq. (11)\nthe expectation of the x-component electric current density at\ntimetis given by\nhjx(x;t)i=\u0000evF\u001by\n\u000b0\u000b\n y\nH\u000b0(x;t) H\u000b(x;t)\u000b\n: (12)\nB. Keldysh Green’s Function and Perturbation Theory\nOur next task is to rewrite the expectation value of electric\ncurrent density in Eq. (12) with the Keldysh Green’s function\nand evaluate it by constructing the perturbation theory where\nthe perturbative term is H0(t) =Hext(t) +Vexc. Then, what\nwe evaluate at the end is the spatial and temporal averaged\nelectric current density at the non-equilibrium steady state. It\nis deﬁned by\n\u0016jx=Zd2x\nVZt0+T\nt0dt\nThjx(x;t)i; (13)\nwhereVthe area of TI surface. The time Tis given by\nT= 2\u0019Ntime=!extwithNtime a positive integer. We as-\nsume it to be very large to describe that we are taking the\nlong-time average (Ntime\u001d1). By analyzing the structure\nof perturbative Keldysh Green’s function, we investigate how\nthe TI-surface-state electric current \u0016jxis generated by the spin\npumping in terms of the ac external magnetic ﬁeld and the ex-\nchange interaction.\nFirst, we rewrite the expectation value of electric current\ndensity in Eq. (12) by the ﬁeld operators in the interaction\npicture. We denote the creation and annihilation operators of\nTI surface state in the interaction picture as  y\nH0\u000band H0\u000b,\nrespectively. The expectation value of x-component electric\ncurrent density at the non-equilibrium steady state becomes\n[61]\n\njx(x;t)\u000b\n=ievF\u001by\n\u000b0\u000blim\nt0!t+\nx0!xG<\n\u000b\u000b0(xt;x0t0); (14)\nwhereG<\n\u000b\u000b0(xt;x0t0)is the lesser component of full real-time\nGreen’s function. t+is the time which is inﬁnitesimally later\nthant:t+=t+\u000f+\ntwith\u000f+\nta positive inﬁnitesimal. The\nlesser Green’s function G<\n\u000b\u000b0(xt;x0t0)is redescribed by the5\nFIG. 3. Schematic for the closed contour C. It consists of two\nsub contours C\u0000andC+:The sub contour C\u0000starts from\u001c=\u00001\nand ends at\u001c= +1whereasC+begins from\u001c= +1and reaches\n\u001c=\u00001. The variables tandt0are the real times which are obtained\nby performing the real-time projection on the contour times \u001cand\n\u001c0, respectively. As shown in the diagram in Fig. 2, the contour C\ndescribes the time evolution of the ferromagnetic TI surface such that\nat the far past ( t!\u00001 ) the thermal-equilibrium state represented\nby\u001aGC(H;\f;\u000f F)was realized, and due to the external ﬁeld Hext(t),\nat sufﬁciently a long time ( t!+1) the non-equilibrium state is\ngenerated.\nKeldysh (contour-time) Green’s function deﬁned by [62, 63]\niGC;\u000b\u000b0(x\u001c;x0\u001c0) =D\nTC\u0002\nUexc\nCUext\nC H0\u000b(x\u001c) y\nH0\u000b0(x0\u001c0)\u0003E\n0;\n(15)\nwhere\nX\u000b\n0= Tr(X\u001aGC(H0;\f;\u000f F))is the thermal average\ntaken with respect to the unperturbed Hamiltonian H0. The\ncontourCis the closed path as shown in Fig. 3 and is rep-\nresented by the time variable called the contour time. Let us\ndenote it as \u001c. The symbol TCrepresents the time-ordering\noperator for contour times belonging to C. For instance, if\n\u001c1< \u001c2we haveTC[A1(\u001c1)A2(\u001c2)] =\u0006A2(\u001c2)A1(\u001c1). We\nobtain the positive sign after we exchanged the order between\nA1(\u001c1)andA2(\u001c2)if this exchange was bosonic (exchanging\neven numbers of fermionic operators) while we get the nega-\ntive sign if the exchange was fermionic (exchanging odd num-\nbers of fermionic operators). The contour Cconsists of two\nsub contours C\u0000andC+:The sub contour C\u0000starts from\n\u001c=\u00001 and reaches \u001c= +1while the sub contour C+\nbegins from \u001c= +1and ends at \u001c=\u00001. Such a struc-\nture represents that the timescale of dynamics we are focusing\non is when the non-equilibrium steady state is realized. It is\nwhen sufﬁciently a long time has passed since we applied the\nexternal ﬁeld Hext(t)(at timet0). In order to describe such\na situation, the limit t0!\u00001 is going to be taken while\nfor timet, which is the time when the non-equilibrium state\nwe are focusing on is realized, we take t!1 . The reason\nwe have the two sub contours C\u0000andC+is because, as de-\nscribed in Eq. (11), the physical operators are sandwiched be-\ntween the two time evolution operators Uy(t;t0)andU(t;t0).\nNote that the temporal structure of contour Cis equivalent\nto the structure of time evolution presented in Fig. 2. Thecontour times \u001cand\u001c0in Eq. (15) belong to C\u0000andC+,\nrespectively. In such a temporal conﬁguration, the Keldysh\nfunctionGC;\u000b\u000b0(x\u001c;x0\u001c0)becomes the lesser Green’s func-\ntion via real-time projection. For more details on the real-time\nprojection of the Keldysh Greens’ function formalism see sub-\nSec. B 1 in appendix B.\nThe operatorsUexc\nCandUext\nCare the time-evolution opera-\ntors along the contour Cgenerated by VexcandHext, respec-\ntively. They are deﬁned by\nUext\nC= exp\u0012\n\u0000i\n~Z\nCd~\u001cHext\nH0(~\u001c)\u0013\n;\nUexc\nC= exp\u0012\n\u0000i\n~Z\nCd\u0014\u001cVexc\nH0(\u0014\u001c)\u0013\n: (16)\nHext\nH0(~\u001c)andVexc\nH0(\u0014\u001c)in the above equation are written by\nthe ﬁeld operators in the interaction picture at the contour\ntime ~\u001cor\u0014\u001c. In order to perform the perturbative calcu-\nlation, we rewrite the Hamiltonians Hext\nH0(~\u001c)andVexc\nH0(\u0014\u001c)\nin the momentum representation and reorganize the unper-\nturbed and perturbed terms. For doing this, let us intro-\nduce the Fourier transform of the spin density for TI sur-\nface state. It is given by sa(x) =V\u00001P\nkeik\u0001xsa(k)\nwherek= (kx;ky)is the two-dimensional wavevector of\nTI surface state and sa(k) =P\nk0 y\n\u000b0(k0)(\u001ba\n\u000b0\u000b=2) \u000b(k0+\nk)witha=x;y;z . The (inverse) Fourier trans-\nform of the ﬁeld operator of TI surface state is given\nby H0\u000b(xt) =\u0000\n1=p\nV\u0001P\nkeik\u0001x H0\u000b(kt); H0\u000b(kt) =\u0000\n1=p\nV\u0001R\nd2xe\u0000ik\u0001x H0\u000b(xt):Besides the TI-surface-state\nﬁeld operator in the momentum representation, we introduce\nthe magnon ﬁeld operators represented in the momentum\nspace by re-expressing the localized spin with them (Holstein-\nPrimakoff tranformation). They are given by\nSy\ni=\u0000sgn(B0)0\n@S0\u00001\nNlocX\npp0ay(p0)a(p)ei(p\u0000p0)\u0001ri1\nA;\nS\u0000\ni=Sz\ni\u0000iSx\ni=8\n<\n:q\n2S0\nNlocP\npe\u0000ip\u0001riay(p);(B0<0)q\n2S0\nNlocP\npeip\u0001ria(p);(B0>0);\nS+\ni=Sz\ni+iSx\ni=8\n<\n:q\n2S0\nNlocP\npeip\u0001ria(p);(B0<0)q\n2S0\nNlocP\npe\u0000ip\u0001riay(p);(B0>0);\n(17)\nwherea(p)(ay(p)) denotes the annihilation (creation) oper-\nator of magnon with momentum p= (px;py). The annihi-\nlation and creation operators of magnon satisfy the commuta-\ntion relation [a(p);ay(q)] =\u000e(p\u0000q)with all others being\nzero. By using the spin density sa(k)and the magnon ﬁeld\noperatorsa(p)anday(p0), the Hamiltonian of the surface of\nferromagnetic TI is re-expressed in the momentum space as6\n\u0016HTI\n0=~vFX\nk \u000b0(k)\u0000\n\u001by(kx+kx\n0)\u0000\u001bxky\u0000\u000fF1\u0001\n\u000b0\u000b \u000b(k); (18)\nHimp=1\nVX\nkq\u000bvimp(q)\u001aimp(q) y\n\u000b(k+q) \u000b(k); (19)\nHFM\n0=X\np\u000fFM\npay(p)a(p);\nVexc=8\n<\n:\u0000q\nS0\n2NlocV2P\nqp\u0010\nJexc\n(qp)s\u0000(q)a(p) +Jsd\u0003\n(qp)s+(\u0000q)ay(p)\u0011\n+Jexc\nVP\npp0ay(p0)a(p)sy(p0\u0000p);(B0<0)\n\u0000q\nS0\n2NlocV2P\nqp\u0010\nJexc\n(qp)s+(q)a(p) +Jsd\u0003\n(qp)s\u0000(\u0000q)ay(p)\u0011\n\u0000Jexc\nVP\npp0ay(p0)a(p)sy(p0\u0000p);(B0>0);\n(20)\nHext(t) =~\rBextr\nNlocS0\n2\u0010\nay(0)e\u0000i!extt+a(0)ei!extt\u0011\n; (21)\nwherevimp(q) =R\nd2xe\u0000iq\u0001xVimp(x)and\u001aimp(q) =PNimp\ni=1e\u0000iq\u0001Xi. We will take vimp(0) = 0 .\u000fFM\np =\nzJnxS0(1\u0000\rp)+~\rejB0jis the dispersion relation of magnon\nwith\rp=z\u00001P\n\u001ae\u0000ip\u0001\u001a: zis the number of nearest-\nneighboring sites for localized spins and \u001arepresents the\nnearest-neighboring-site vector. The quantity Jexc\n(qp)is deﬁned\nbyJexc\n(qp)=P\niJexcei(q+p)\u0001ri:By comparing the Hamilto-\nnian \u0016HTI\n0in Eq. (1) and that in Eq. (18), we see that be-\ncause of the exchange interaction the Dirac point of TI sur-\nface state (the point where the dispersion of TI surface state\nbecomes zero) is shifted to the momentum k0= (kx\n0;0)with\nkx\n0= sgn(B0)(JexcS0n2D\nloc)=(2~vF). Heren2D\nloc=Nloc=Vis\nthe two-dimensional number density of localized spins. For\nthe convenience, we perform the Fourier transformation on\nthe ﬁeld operators of TI surface state  \u000b(x)and y\n\u000b0(x)by\nthe shifted momentum ~k=k+k0:As a result, the formula\nof Hamiltonian HTI\n0in Eq. (18) described by the shifted mo-\nmentum ~kis going to be equivalent to that in Eq. (1) rep-\nresented by the original momentum k. Hereinafter we just\nsimply write the shifted momentum ~kask. Note that with-\nout the impurity effect, the TI surface state exhibits the linear\ndispersion relation \u000fTI\nk=~vFkwithk=p\n(kx)2+ (ky)2.\nConsequently, the surface of ferromagnetic TI is remodeled\nas the hybrid quantum system of magnon and TI surface state\nwith the Hamiltonians in Eqs. (18) - (21).\nNext, in order to construct the perturbation the-\nory for the Green’s function GC;\u000b\u000b0(x\u001c;x0\u001c0)in\nEq. (15) let us perform the Fourier transforma-\ntion with taking the limit x0!x. We have\nGC;\u000b\u000b0(x\u001c;x0\u001c0) =V\u00001P\nkk0ei(k\u0000k0)xGC;\u000b\u000b0(k\u001c;k0\u001c0).\nHereGC;\u000b\u000b0(k\u001c;k0\u001c0)is given by GC;\u000b\u000b0(k\u001c;k0\u001c0) =\n\u0000iD\nTC\u0002\nUexc\nCUext\nC H0\u000b(k\u001c) y\nH0\u000b0(k0\u001c0)\u0003E\n0. Then, we\nperform the perturbative expansion on GC;\u000b\u000b0(k\u001c;k0\u001c0)by\nexpanding the two operators Uext\nCandUexc\nCin Eq. (16) with\nrespect toHext\nH0(\u0014\u001c)andVexc\nH0(~\u001c), respectively. It is going tobe represented in the form\nGC;\u000b\u000b0(k\u001c;k0\u001c0) =1X\nn=01X\nn0=0G(n;n0)\nC;\u000b\u000b0(k\u001c;k0\u001c0): (22)\nWe have used the superscript ( n;n0)in the right-hand side of\nEq. (22) to describe that the perturbative Green’s function\nG(n;n0)\nC;\u000b\u000b0(k\u001c;k0\u001c0)is in then-th order of Hextwhile it is\nin then0-th order of Vexc. Note that the Green’s functionP1\nn0=0G(0;n0)\nC;\u000b\u000b0(k\u001c;k0\u001c0)is the full thermal-equilibrium\nGreen’s function since it does not contain the external-ﬁeld\nHamiltonian Hext. At the non-equilibrium steady state, what\nwe observed in the experiment is the deviation (ﬂuctuation)\nfrom the thermal-averaged value at thermal equilibrium.\nThus, we calculate and show the expectation value of\njx(x;t)\u000b\nin Eq. (14) as well as the spatial and temporal\naveraged electric current \u0016jxin Eq. (13) for n\u00151:\nAs a result, the perturbative Green’s function\nG(n;n0)\nC;\u000b\u000b0(k\u001c;k0\u001c0)is expressed by the unperturbed Keldysh\nGreen’s functions of TI surface state and magnon given by\niG0\nC;\u000b\u000b0(x\u001c;x0\u001c0) =D\nTC\u0002\n H0\u000b(x\u001c) y\nH0\u000b0(x0\u001c0)\u0003E\n0;\n(23)\niD0\nC(q\u001c;q0\u001c0) =D\nTC\u0002\naH0(q\u001c)ay\nH0(q0\u001c0)\u0003E\n0: (24)\nG0\nC;\u000b\u000b0(x\u001c;x0\u001c0)is the unperturbed Keldysh Green’s function\nof the TI surface state while D0\nC(q\u001c;q0\u001c0)is that of magnon.\nThe Fourier transform of G0\nC;\u000b\u000b0(x\u001c;x0\u001c0)is given as\nG0\nC;\u000b\u000b0(x\u001c;x0\u001c0) =V\u00001P\nkk0ei(k\u0000k0)xG0\nC;\u000b\u000b0(k\u001c;k0\u001c0).\nNote that bothG0\nC;\u000b\u000b0(k\u001c;k0\u001c0)andD0\nC(q\u001c;q0\u001c0)are diago-\nnal in momentum: G0\nC;\u000b\u000b0(k\u001c;k0\u001c0) =G0\nC;\u000b\u000b0(k;\u001c;\u001c0)\u000ekk0\nandD0\nC(q\u001c;q0\u001c0) =D0\nC(q;\u001c;\u001c0)\u000eqq0. To obtain the\nphysical observables like the electric current of TI sur-\nface state, we project the contour times onto the real-time\naxis. Then, the Keldysh Green’s functions G0\nC;\u000b\u000b0(x\u001c;x0\u001c0)\nandD0\nC(q\u001c;q0\u001c0)are rewritten by the unperturbed real-time7\nGreen’s functions: G0\nC;\u000b\u000b0(k;\u001c;\u001c0)!\u0016g\u0017\n\u000b\u000b0(k;t\u0000t0)and\nD0\nC(q; ~\u001c;~\u001c0)!\u0016D~\u0017(q;~t\u0000~t0). Here\u0017;~\u0017= t;<;>; ~tde-\nnoting the time-ordered, lesser, greater, and anti-time-ordered\ncomponents, respectively. t;t0;~t, and ~t0are real-time vari-\nables introduced by the real-time projection and correspond\nto\u001c;\u001c0~\u001c, and ~\u001c0, respectively. After the Keldysh Green’sfunctions are transformed into the real-time Green’s functions\nthey are represented by the differences of two real-time vari-\nables. As a result, the perturbative Keldysh Green’s function\nG(n;n0)\nC;\u000b\u000b0(k\u001c;k0\u001c0)is redescribed as products of unperturbed\nreal-time Green’s functions. Its formula can be organized with\nthe retarded and advanced components in the momentum-\nfrequency representation given by\n\u0016gr\n\u000b\u000b0(k;!) =(1+~H0)\u000b\u000b0\n2\u0010\n!+!F\u0000!TI\nk+i\n2\u001crel\nTI\u0011+(1\u0000~H0)\u000b\u000b0\n2\u0010\n!+!F+!TI\nk+i\n2\u001crel\nTI\u0011\n\u0016ga\n\u000b\u000b0(k;!) =(1+~H0)\u000b\u000b0\n2\u0010\n!+!F\u0000!TI\nk\u0000i\n2\u001crel\nTI\u0011+(1\u0000~H0)\u000b\u000b0\n2\u0010\n!+!F+!TI\nk\u0000i\n2\u001crel\nTI\u0011; (25)\n\u0016Dr(0;!) =1\n!\u0000!FM\n0+i\u000b!;\u0016Da(0;!) =1\n!\u0000!FM\n0\u0000i\u000b!; (26)\nwhere 1is the two by two unit matrix and ~H0is given by\n~H0= \n0\u0000i(kx\u0000iky)\nki(kx+iky)\nk0!\n: (27)\nThe Green’s functions \u0016gr(a)\n\u000b\u000b0(k;!)and \u0016Dr(a)(0;!)in Eqs.\n(25) and (26) are the retarded (advanced) components of TI-\nsurface-state and zero-momentum magnon Green’s functions,\nrespectively. The frequencies !TI\nk;!F, and!FM\n0are deﬁned\nby!TI\nk=~\u00001\u000fTI\nk,!F=~\u00001\u000fF, and!FM\n0=~\u00001\u000fFM\n0=\n\rjB0j, respectively. \u001crel\nTIis the relaxation time of the TI sur-\nface state due to the impurity effect Himpwhile the constant \u000b\nappearing in the magnon Green’s function is the Gilbert damp-\ning constant. We put bars on top of these Green’s functions\nto express that we have taken into account the impurity and\ndamping effects.\nThe perturbation theory for the Keldysh Green’s function in\nthe above way enables us to clearly explore how the electric\ncurrent on the surface of TI is induced by the spin pumping in\nterms of the external ac magnetic ﬁeld and the exchange inter-\naction. In Appendix A, we present the details for the real-time\nGreen’s functions of TI surface state as well as the derivation\nof retarded and advanced components of impurity-averaged\nGreen’s functions given in Eq. (25) using the imaginary-time\nGreen’s function formalism. Moreover, we describe the real-\ntime Green’s functions of magnon and then discuss the deriva-\ntion of retarded and advanced Green’s functions in Eq. (26)\nusing the Landau-Lifshitz-Gilbert equation. In Appendix B,\nwe present the detailed description for the Keldysh Green’s\nfunction formalism as well as the relation between Keldysh\nGreen’s function and real-time Green’s function. Further,\nwe show some formulas of Keldysh Green’s function formal-\nism and by applying them we demonstrate the derivation of\nimpurity-averaged Green’s functions of TI surface state for\nthe retarded, advanced, lesser, and greater components.III. SPIN-PUMPING-INDUCED NON-LINEAR ELECTRIC\nCURRENT\nSince we have established the perturbation theory, we now\nevaluate the electric current of the TI surface state induced by\nthe spin pumping. Let us ﬁrst present the diagrammatic rep-\nresentation of our perturbative Green’s function. Based on it,\nwe microscopically analyze how the electric current is gen-\nerated by the external magnetic ﬁeld and the exchange inter-\naction. Then, we show the structure of spin-pumping-induced\nelectric current as well as the associated electric voltage repre-\nsented by the static external magnetic ﬁeld, the amplitude and\nthe frequency of ac magnetic ﬁeld, the exchange-interaction\nstrength, the relaxation time originating in the non-magnetic\nimpurity, and the Gilbert-damping constant. Finally, we com-\npare our result of the electric voltage with the experimental\nresults [42, 59].\nLet us evaluate the right-hand side of Eq. (22). We denote\nthe expectation of spatial and temporal averaged x-component\nelectric current corresponding to the term G(n;n0)\nC;\u000b\u000b0(k\u001c;k0\u001c0)\nas\u0016j(n;n0)\nx:First, we can show that the spatial and temporal\naveraged electric current \u0016j(1;1)\nx is zero. This implies that the\nsurface of ferromagnetic TI does not show the linear response\nin the ac external magnetic ﬁeld. Next, let us present the next-\nleading-order term \u0016j(2;1)\nx. In order to obtain this, we calculate\nthe perturbative Green’s function G(2;1)\nC;\u000b\u000b0(k\u001c;k0\u001c0)given in\nthe right-hand side of Eq. (22). First, we expand the time-\nevolution operators Uext\nCandUexc\nCwith respect to Hext\nH0(~\u001c)\nandVexc\nH0(\u0014\u001c), respectively. With using the Wick’s theorem\nthe perturbative Green’s function G(2;1)\nC;\u000b\u000b0(k\u001c;k0\u001c0)is given\nin terms of the unperturbed Keldysh Green’s functions of TI\nsurface state and magnon as8\nG(2;1)\nC;\u000b\u000b0(k\u001c;k0\u001c0) =isgn(B0)\u0012\n\u0000i\n~\u00133\u0000\n~\rBext\u00012Jexcn2D\nlocS0\n4Z\nCd~\u001c1d~\u001c2d\u0014\u001c1e\u0000i!ext(~\u001c1\u0000~\u001c2)X\npp0k1\n\u0002\u0002\nD0\nC(0; ~\u001c1;~\u001c2)D0\nC(p; 0+)\u000ep;p0+D0\nC(0; ~\u001c2;\u0014\u001c1)D0\nC(0; \u0014\u001c1;~\u001c1)\u000ep;0\u000ep0;0\u0003\n\u0002h\nG0\nC;\u000b\u000b0\n1(k;\u001c;\u0014\u001c1)\u001by\n\u000b0\n1\u000b1G0\nC;\u000b1\u000b0(k0; \u0014\u001c1;\u001c0)\u000ek;k1\u000ek0;k1+p0\u0000p\n\u0000G0\nC;\u000b\u000b0(k; 0+)\u001by\n\u000b0\n1\u000b1G0\nC;\u000b1\u000b0\n1(k1; 0+)\u000ek;k0\u000ek1;k1+p0\u0000pi\n: (28)\nWe note that in the above equation the positive inﬁnitesimal\ntime difference 0+forG0\nC;\u000b\u000b0(k; 0+)is equal tot0\u0000t:Since\n\u001c(=t)2C\u0000while\u001c0(=t0)2C+, the Green’s function\nG0\nC;\u000b\u000b0(k; 0+)is the lesser Green’s function. On the other\nhand, 0+forG0\nC;\u000b1\u000b0\n1(k1; 0+)is equal to\u001c+\n1\u0000\u001c1:The contour\ntimes\u001c+\n1and\u001c1both belong to the same sub contour C\u0016(\u0016=\n\u0000;+):\nSecond, we perform the real-time projection on the con-tour times ~\u001c1;~\u001c2, and \u0014\u001c1and rewrite the right-hand side of\nEq. (28) by the real-time Green’s functions of TI surface state\nand magnon. Then, the perturbative Keldysh Green’s func-\ntionG(2;1)\nC;\u000b\u000b0(k\u001c;k0\u001c0)in the right-hand side of Eq. (28) be-\ncomes the lesser real-time Green’s function which we write\nasG<(2;1)\n\u000b\u000b0(kt;k0t0)witht0=t+(see also Eq. (14)). Let us\ndenote the real-time variables corresponding to ~\u001c1;~\u001c2, and \u0014\u001c1\nas~t1;~t2, and \u0014t1, respectively. Then, by using the ﬁrst formula\nin Eq. (B10), we obtain\nG<(2;1)\n\u000b\u000b0(kt;k0t0) =isgn(B0)\u0012\n\u0000i\n~\u00133\u0000\n~\rBext\u00012Jexcn2D\nlocS0\n4Z\nd~t1d~t2d\u0014t1e\u0000i!ext(~t1\u0000~t2)\u0016Da(0;~t2\u0000\u0014t1)\u0016Dr(0;\u0014t1\u0000~t1)\n\u0002\u0010\n\u0016gr\n\u000b\u000b0\n1(k;t\u0000\u0014t1)\u001by\n\u000b0\n1\u000b1\u0016g<\n\u000b1\u000b0(k;\u0014t1\u0000t) + \u0016g<\n\u000b\u000b0\n1(k;t\u0000\u0014t1)\u001by\n\u000b0\n1\u000b1\u0016ga\n\u000b1\u000b0(k;\u0014t1\u0000t)\u0011\n\u000ekk0; (29)\nwhere we have usedR\nCd~\u001c1d~\u001c2D0\nC(0; ~\u001c1;~\u001c2) =R\nd~t1d~t2\u0010\n\u0016Dt\u0000\u0016D<\u0000\u0016D>+\u0016D~t\u0011\n(0;~t1\u0000~t2) = 0:\nFurther, the termG0\nC;\u000b\u000b0(k; 0+)G0\nC;\u000b1\u000b0\n1(l; 0+)in Eq. (29)\nvanishes since it describes the disconnected diagram: We\ndenote the real time variables t1andt0\n1which are the real-\ntime projection of the contour times \u001c1and\u001c0\n1, respectively.\nThey satisfy \u001c1< \u001c0\n1because in the exchange-interaction\nHamiltonian Vsdthe operator  y\n\u000b0\n1(l;\u001c0\n1)comes to the left\nside of \u000b1(l;\u001c1). When\u001c1;\u001c0\n12C\u0000we havet1< t0\n1and\nG0\nC;\u000b1\u000b0\n1(l;\u001c1;\u001c0\n1) = \u0016g<\n\u000b1\u000b0\n1(l;t1\u0000t0\n1)whereas for\u001c1;\u001c0\n12C+\nwe obtaint1>t0\n1andG0\nC;\u000b1\u000b0\n1(l;\u001c1;\u001c0\n1) = \u0016g<\n\u000b1\u000b0\n1(l;t1\u0000t0\n1).\nHence, we haveR\nCd\u001c1lim\u001c0\n1!\u001c+\n1G0\nC;\u000b1\u000b0\n1(l;\u001c1;\u001c0\n1) =R\ndt1\u0002\n\u0016g<\n\u000b1\u000b0\n1(l;0\u0000)\u0000\u0016g<\n\u000b1\u000b0\n1(l;0+)\u0003\n= 0 . Here 0\u0000is the\nnegative inﬁnitesimal. For the detail treatments on real-time\nprojection as well as the real-time integration see subSec.\nB 1 in Appendix B. Third, what we do is we perform the\nFourier transforms on the above Green’s functions as, for\ninstance, \u0016Da(0;~t1\u0000\u0014t1) =Rd~!\n2\u0019e\u0000i~!(~t1\u0000\u0014t1)\u0016Da(0;~!)and\n\u0016gt\n\u000b\u000b0\n1(k;t\u0000\u0014t1) =Rd!\n2\u0019e\u0000i!(t\u0000\u0014t1)\u0016gt\n\u000b\u000b0\n1(k;!):Then, using Eq. (B28) the right-hand side of Eq. (29) is\nrewritten by the retarded and advanced Green’s functions in\nEqs. (25) and (26). By performing the temporal integralsR\nd~t1d~t2d\u0014t1and from Eq. (13), the x-component averaged\nelectric current density of TI surface state becomes\nFIG. 4. Feynman diagram for the spin-pumping-induced electric cur-\nrent described by Eq. (30). It consists of two TI-surface-state Green’s\nfunctions (solid lines), two magnon Green’s functions (wavy lines),\nand two vertices denoted by crosses. As described by the orange cir-\ncle, the exchange interaction between the zero-momentum magnon\nand the TI surface state occurs at the right vertex leading to the gen-\neration of electric current on the surface of TI.9\n\u0016j(2;1)\nx= (\u0000evFsgn(B0))\u0012\n\u0000i\n~\u00133\u0000\n~\rBext\u00012\u0012JexcS0n2D\nloc\n4\u0013\n\u0016Dr(0;!ext)\u0016Da(0;!ext)\n\u0002Zd2kd!\n(2\u0019)3f(~!)h\u0010\n\u0016ga\n\u000b\u000b0\n1(k;!)\u001by\n\u000b0\n1\u000b1\u0016ga\n\u000b1\u000b0(k;!)\u0000\u0016gr\n\u000b\u000b0\n1(k;!)\u001by\n\u000b0\n1\u000b1\u0016gr\n\u000b1\u000b0(k;!)\u0011\n\u001by\n\u000b0\u000bi\n; (30)\nwheref(~!) =\u0000\n1 +e\f~!\u0001\u00001and we have taken a contin-\nuum limitV\u00001P\nk!R\nd2k=(2\u0019)2. The right-hand side of\nEq. (30) represents the way the electric current of TI sur-\nface state is induced by the spin pumping due to the external\nmagnetic ﬁeld Hextand the exchange interaction Vexc. To\nsee this clearly, let us describe \u0016j(2;1)\nx diagrammatically and\npresent this in Fig. 4. The solid and the wavy lines represent\nthe Green’s function of TI surface state (fermion line) and that\nof magnon (boson line), respectively. The two vertices are de-\nscribed by crosses where the energy and momentum conserve.\nThe gray and orange circles denote the amplitude of ac mag-\nnetic ﬁeldBextand the exchange-interaction strength Jexc,\nrespectively. The Pauli matrix in the left side originates in the\ngenerator of x-component electric current while the right one\nis coming from the y-component exchange interaction. The\nretarded Green’s function \u0016Dr(0;!ext)appearing in this dia-\ngram describes the emission process of magnon with the zero\nmomentum and the energy ~!extgoing from orange to gray\ncircles whereas the advanced Green’s function \u0016Da(0;!ext)\nrepresents the absorption process going from gray to orange.\nIn the diagram in Fig. 4, the energy and momentum of TI\nsurface state remains unchanged. This is because, ﬁrst, the\nemission and absorption processes of magnon of the energy\n~!extoccur with each process occurring once. Second, the\nmagnon does not carry momentum since the ac magnetic ﬁeld\nHextis spatially homogeneous and so does the Fourier trans-\nform of exchange interaction Jexc\n(qp): for they-component it is\ndescribed by the constant Jexc(see Eq. (20)). Since we have\noverlooked at the structure of our diagram, let us now ana-\nlyze the mechanism of the spin-pumping-induced electric cur-\nrent. Initially, the TI surface state is in the thermal equilibrium\n\u001aGC(H;\f;\u000f F)and the origin of Fermi sphere of TI surface\nstate is atk0= (0;0). When the FMR is triggered at t=t0,\nthe localized spin Sistarts to show its dynamics described by\nthe Landau-Lifshitz-Gilbert equation (see Eq. (A31)) and the\nmagnon of zero momentum and frequency !extemerges. It is\nthe ﬂuctuation of the saturation magnetization in the ydirec-\ntion created by the external magnetic ﬁeld B0. After then, the\nspin pumping occurs associating with the spin current Jpump\ny;z\nﬂowing from FM to the surface of TI. The zero-momentum\nmagnon is going to be the carrier of it. In other words, the\nspin current Jpump\ny;z is the ﬂow of zero-momentum magnon.\nThe magnon couples with the TI surface state through the ex-\nchange interaction Vexc. Then owing to the spin-momentum\nlocking, the magnon acts like an additional momentum of TI\nsurface state. This means that effectively TI surface state ex-\nperiences the coupling between the magnon as an electric ﬁeld\nbeing applied and a non-equilibrium state of TI surface stateis driven. Such a situation can be described as the deviation of\nthe TI-surface-state Fermi circle from the origin (see also Fig.\n5 (b)). On the other side, the TI surface state is affected by\nthe impurity potential Himpgiven by Eq. (2). Then as time\ngoes by, the effect of effective electric ﬁeld of magnon and the\nimpurity effect Himpare going to get balanced. As a result,\nthe TI surface state and the magnon both relax to the non-\nequilibrium steady state and the static electric ﬁeld is created\non the surface of TI. Let us call it the spin-pumping-induced\nelectric ﬁeld ESPI\nx. At the non-equilibrium steady state, the\nTI surface state experiences the ESPI\nxand the spin-pumping-\ninduced electric current \u0016j(2;1)\nx ﬂows on the TI surface as the\nresponse to it. To make the relation between \u0016j(2;1)\nx andESPI\nx\nclear, let us rewrite \u0016j(2;1)\nx with using the electrical conductiv-\nity\u001bTI\nxxas\u0016j(2;1)\nx=\u001bTI\nxxESPI\nx. As in the case of Dirac electrons\nin graphene, the electrical conductivity of TI surface state \u001bTI\nxx\ncan be calculated by using the Boltzmann equation [66]. We\nobtain\u001bTI\nxx=\u000fF\u001crel\nTI\n2~\u0001e2\n2\u0019~. On the other side, the formula\nofESPI\nxis obtained by evaluating the right-hand side of Eq.\n(30). For doing this, ﬁrst we remark that the denominator in\nthe right-hand side of Eq. (30) is a function of the dispersion\nof TI surface state ( !TI\nk=~\u00001\u000fTI\nk), and thus, it is the func-\ntion of the absolute k=p\n(kx)2+ (ky)2. Hence, it means\nthat the denominator in the right-hand side of Eq. (30) is an\neven and symmetric function of kxandky. Due to this fact,\nfor the numerator in the right-hand side of Eq. (30) the terms\nproportional to kxkyas well as (kx)2\u0000(ky)2vanish. As a\nresult, the only terms which remain are the products of two\ndiagonal elements of TI-surface state Green’s function, i.e.,\u0000\n\u0016ga(r)\n\"\"(k;!)\u00012and/or\u0000\n\u0016ga(r)\n##(k;!)\u00012. In the following evalu-\nation, we only retain the ﬁrst term of \u0016gr(a)\n\u000b\u000b0(k;!)in Eq. (25)\nsince only the electronic state in the vicinity of Fermi-energy\nlevel contributes to the electric transport. Next, we perform\nkand!integrals with using three types of approximations.\nWe ﬁrst do from the !integral and rewrite the integrand with\nthe derivative term @f(~!)=@(~!). As the ﬁrst approxima-\ntion, we take the low-temperature limit ( \f!1 ) and we ob-\ntain@f(~!)=@(~!) =\u0000\u000e(~!):By performing the !integral,\nthe integrand becomes the function of the relaxation time \u001crel\nTI,\nthe TI-surface-state dispersion !TI\nk, and the Fermi energy !F\n(=~\u00001\u000fF)given as1=2\u001crel\nTI\u0000\n(!TI\nk\u0000!F)2+(1=2\u001crel\nTI)2\u0001. Next, we rewrite\nthekintegral in the following way:d2k\n(2\u0019)2=~N(\u0018k)d\u0018k,\nwhere ~N(\u0018k) =(\u0018k+\u000fF)\n2\u0019(~vF)2is the density of states per volume\nand\u0018kis the energy of the TI surface state measured with re-\nspect to the Fermi energy. It is deﬁned by \u0018k=\u000fTI\nk\u0000\u000fF. As a10\nresult, the integrand becomes ~N(\u0018k)\u0002~=2\u001crel\nTI\n\u00182\nk+(~=2\u001crel\nTI)2. Then as\na second approximation, we regard only the electronic state in\nthe vicinity of Fermi surface contributes to the electric current.\nIn other words, the TI surface state depends weakly on the\ndensity of states ~N(\u0018k). Therefore, we take ~N(\u0018k)\u0019~N(0):\nOn the other side, the lower limit of \u0018k-integral is\u0000\u000fF. We\nconsider that on the surface of area Va huge number of elec-\ntrons are contained. Hence, as a third approximation we takethe number density of TI surface state nTI\n2Dto be sufﬁciently\nlarge. Since the number density nTI\n2Dis related to the Fermi\nenergy\u000fFas\u000fF=~vFq\n4\u0019nTI\n2D;we take\u000fF!1 (see also\nthe argument below Eqs. (A25) and (B19)). By using these\nthree types of approximations and performe the \u0018k-integral,\nwe have\nESPI\nx(!ext;B0) =\u0000sgn(B0)\u0012JexcS0n2D\nloc\n4evF\u001crel\nTI\u0013(\rBext)2\n(!ext\u0000!FM\n0)2+ (\u000b!ext)2; (31)\nwhere!FM\n0=\rjB0j. Consequently, when the FMR occurs\nwith the frequency !ext, the electric current \u0016j(2;1)\nx as well as\nthe electric ﬁeld ESPI\nxare induced by the spin pumping on\nthe surface of TI. It ﬂows perpendicular to the precession axis\n(yaxis) of FMR owing to the spin-momentum locking. It is\nproportional to the square of the ac-magnetic-ﬁeld amplitude\nBextdescribing that it is the non-linear (quadratic) response\nto the external ac magnetic ﬁeld. In other words, it is pro-\nportional to the power of the applied electromagnetic wave\n(microwave). Like the FMR (magnon) spectrum, the electric\ncurrent \u0016j(2;1)\nx (or the electric ﬁeld ESPI\nx) is described by the\nquantities\rBext,!ext,!FM\n0, and the Gilbert-damping con-\nstant\u000b. Indeed, the spectral function of magnon can be ob-\ntained by multiplying the factor \u000b!extto the third factor of\nESPI\nxin Eq. (31): \u000b!ext\u00021\n(!ext\u0000!FM\n0)2+(\u000b!ext)2. In other\nwords, the retarded Green’s function of magnon is equivalent\nto the magnetic susceptibility (see Eq. (A32) and the argument\nbelow it). Physically, this represents the absorption energy\nof localized spin which we need to drive the FMR (see also\nthe argument after Eq. (8) in [15]). The electric ﬁeld ESPI\nx\ndepends on both magnetic quantities including \rBext;\u000b, the\nexchange interaction strength Jexc, the density of localized\nspinn2D\nloc, and those of TI such as Fermi velocity vF, and\nthe relaxation time \u001crel\nTI. This is natural and reasonable be-\ncause the spin-pumping-induced electric ﬁeld ESPI\nxis real-\nized at the non-equilibrium steady state owing to the com-\nmensuration of the effective electric ﬁeld of magnon and the\nimpurity effect Himpmediated by the exchange interaction.\nBased on the Feynman diagram in Fig. 4, we can under-\nstand why the spin-pumping-induced electric current \u0016j(2;1)\nx\nis the quadratic response to the ac magnetic ﬁeld in the fol-\nlowing way. First, the ac magnetic ﬁeld is used to drive the\nFMR and the associated zero-momentum magnon which cou-\nples with the TI surface state through the exchange interac-\ntion. Second, to generate the transport phenomena of TI sur-\nface state we need to drive the magnon with the ac magnetic\nﬁeld once more. As a result, the electric current of TI surface\nstate becomes the quadratic response to the ac magnetic ﬁeld\nsuch that both the emission and absorption processes of zero-\nmomentum magnon occur. Indeed, this naturally reﬂects that\nFIG. 5. (a) Schematic of the generation of the electric ﬁeld EISHE\nx .\nWhen the spin current Jspin\ny;z is injected to the non-magnetic metal,\nowing to the spin-orbit interaction the inverse spin Hall effect is gen-\nerated so that the both electrons of spins polarized in the positive\nand negative ydirections accumulate on the edge of the sample. As\na result, the electric ﬁeld EISHE\nx and the associated electric current\njISHE\nc emerge. (b) Schematic of the generation of the spin-pumping-\ninduced electric current \u0016j(2;1)\nx. When we drive the FMR and the asso-\nciated spin pumping, the zero-momentum magnon couples with the\nTI surface state through the exchange interaction. Due to the spin-\nmomentum locking, the TI surface state effectively experiences the\nzero-momentum magnon as the electric ﬁeld. As a result, the spin-\npumping-induced electric ﬁeld ESPI\nxas well as the spin-pumping-\ninduced electric current \u0016j(2;1)\nx are generated (inverse Edelstein ef-\nfect). It is described as the ﬂow of Fermi circle of TI surface state.\nthe spin-pumping-induced electric current is generated by the\nelectromagnetic wave whose power is quadratic to the ampli-\ntude of ac magnetic ﬁeld. The spin-pumping-induced elec-\ntric current \u0016j(2;1)\nx (or the spin-pumping-induced electric ﬁeld\nESPI\nx) gets larger by raising the ac-magnetic-ﬁeld amplitude\nBext(or the power of electromagnetic wave) and by choosing\nthe ferromagnetic material exhibiting a strong exchange cou-\npling strength. Such a feature is reﬂecting that the surface of\nferromagnetic TI has a high-performing functionality of gen-\nerating the electric charge current by magnetic controlling.\nTo make our understanding on the spin-pumping-induced\nelectric current \u0016j(2;1)\nx better, let us compare it with the elec-\ntric ﬁeld associated with the inverse spin Hall effect by us-\ning the illustrations presented in Fig. 5. The inverse spin11\nHall effect occurs in, for instance, the hybrid system com-\nprise of FM and non-magnetic heavy metal exhibiting strong\nspin-orbit interaction, for example, the heterojunction of NiFe\nand Pt [19, 20, 22, 24]. When we inject the y-polarized spin\ncurrent to the non-magnetic metal ﬂowing in the zdirection,\ndue to the spin-orbit coupling both the electrons whose spins\nare polarized in the positive and negative ydirections ﬂow\nparallel into the xdirection and accumulate to the edge of\nsample. As a result, the electric ﬁeld, namely, EISHE\nx emerges\nin thexdirection. Simultaneously, the associated electric cur-\nrentjISHE\nc ﬂows in the same direction (Fig. 5(a)). This is\nthe phenomenon in a three-dimensional bulk system. In con-\ntrast, our spin-pumping-induced electric current \u0016j(2;1)\nx is the\nphenomenon intrinsic in the two-dimensional surface system.\nThe mechanism of its generation is not due to the accumula-\ntion of electrons on the edge of sample but due to the effective\nelectric ﬁeld of magnon via the spin-momentum locking. As\nillustrated in Fig. 5(b), the spin-pumping-induced electric cur-\nrent can be described as the ﬂow of Fermi circle of TI surface\nstate. It is nothing but the inverse Edelstein effect which is\nalso realized in systems possessing Rashba interfaces [22, 24].\nFinally, let us make a qualitative comparison between\nour result and the experimental results [42, 59]. What has\nbeen measured in these experiments are the electric voltage\nemerged on the surface due to the spin pumping. Therefore,\nwe calculate the spin-pumping-induced electric voltage and\ncompare its characteristic with the experimental results. Be-\nfore we give a detailed argument, we note here that in [42] the\ndirection of static magnetic ﬁeld B0(the precession axis of\nFMR) is taken to be parallel to the yaxis while in [59] it is\ntaken to be in the xaxis. Since the essence of physics does\nnot change, as we did in Sec. II A we take the precession axis\nof FMR to be in the yaxis (thus, the electric current of TI sur-\nface state or electric voltage emerges in the xdirection). To\nmake our argument clear and simple, in the following we in-\ntroduce the effective electric voltage by using ESPI\nx. First, as\nwe see in Eq. (31) the electric ﬁeld ESPI\nxis spatially homoge-\nneous along the xdirection. Thus, by multiplying ESPI\nxwith\nthe length of TI surface in the xdirectionlTI\nx, we obtain the\nelectric voltage in the xdirection and call it as VSPI\nx. Next,\nwe divideVSPI\nxby the factor\u0010\n\u0000JexcS0n2D\nloclTI\nx\n4evF\u001crel\nTIB2\n0\u0011\nbecause essen-\ntially its characteristic is represented by the Gilbert-damping\nconstant\u000b, the external frequency !ext, and the frequency\n!FM\n0=\rjB0j=\r(sgn(B0)B0). In addition, in the experi-\nment the external frequency !extis ﬁxed whereas the mag-\nnetic ﬁeldB0varies from positive to negative values. By\ntaking account of this, we take !extto be the positive con-\nstant and introduce the “spin-pumping-induced electric volt-\nage\u0016VSPI\nx” deﬁned as the function of B0as\n\u0016VSPI\nx(~B0) =\u0000\u0002(~B0)eP\n\u0010\n~B0\u00001\u00112\n+\u000b2\n+ \u0002(\u0000~B0)eP\n\u0010\n~B0+ 1\u00112\n+\u000b2; (32)\nwhere ~B0= (\rB0)=!extis the dimensionless magnetic ﬁeld\nFIG. 6. Plots of spin-pumping-inducedelectric voltage \u0016VSPI\nx de-\nﬁned by Eq. (32). The vertical axis represents the dimensionless\nmagnetic ﬁeld ~B0whereas the horizontal axis represents the spin-\npumping-induced electric voltage \u0016VSPI\nx. The blue, orange, green,\nand red curves are for P= 0:0100;0:0075;0:0050 , and 0:0025 , re-\nspectively. For all four curves, we take the Gilbert damping constant\n\u000bto be equal to 0.15.\nandP= (Bext)2. It is the quantity describing the power\nof electromagnetic ﬁeld which we apply to derive the FMR.\n\u0002(\u0006~B0)is the Heaviside step function. We plot \u0016VSPI\nx(~B0)\nin Fig. 6 by taking the electromagnetic-wave power Pas\na parameter while we ﬁx the Gilbert-damping constant \u000b\nto 0.15. Here we plot \u0016VSPI\nx for four different conditions;\nP= 0:0100;0:0075;0:0050 , and 0:0025 . The full width of\nhalf maximum is equal to the Gilbert damping constant \u000b:\nThe most striking features of \u0016VSPI\nx are (i) the emergence of\ntwo side peaks and (ii) the linear scaling of two peak val-\nues with respect to the power P; the two side peaks locate\nat~B0=\u00061:The values of two peaks have the same abso-\nlute values ( =eP=\u000b2) but the signs are opposite. Let us now\nlook at the experimental data [42, 59]. First, in [42] the (bulk\ninsulating) TIs were chosen as Bi 1:5Sb0:5Te1:7Se1:3naming\nBSTS and Sn-doped Bi 2Te2Se. On the other hand, for the\nferromagnetic material they choose Ni 81Fe19. Let us focus\non Figs. 3(b) or 4(a) and 3(c). Fig. 3(b) is the experimental\ndata of electric voltage for sample BSTS/Ni 81Fe19with the\nsample size of BSTS is 4\u00023\u00020:1mm3for four different\nmicrowave-power conditions; 0.2mW, 0.15mW, 0.10mW, and\n0.05 mW. Fig 4(a) is the result of electric voltage for samples\nBSTS/Ni 81Fe19with three different sample sizes of BSTS;\n4\u00021\u00020:1mm3,4\u00023\u00020:1mm3, and 2\u00021:5\u00020:2\nmm3. They are plotted as functions of static magnetic ﬁeld\nwhich corresponds to our B0(or~B0). Both of them show\ntwo side peaks as discussed in the previous description. Two\npeak spots appear symmetrically with respect to the origin of\nthe magnetic-ﬁeld axis and the two peak values have (almost)\nthe same absolutes with opposite signs. The similar result\nis reported in [59]. In this work, the magnetic TI was engi-\nneered by creating the heterostructure of YIG and Cr-doped\nTI: YIG/Cr 0:08(Bi0:37Sb0:63)1:92Te3. We will focus on Fig.\n2(c) where the electric voltage is plotted as a function of mag-\nnetic ﬁeld. It shows the similar features as the results shown\nin Fig. 3(b) or Fig 4(a) in [42]: the emergence of two side12\npeaks having opposite signs. The difference between the elec-\ntric voltage in Fig. 3(b) or Fig 4(a) in [42] and that in Fig.\n2(c) in [59] is that the signs of two peaks; in Fig. 3(b) the\npeak value at positive magnetic ﬁeld is negative while it is\npositive in Fig. 2(c) it is positive. Such an opposite-sign\nbehavior, however, is not essential for our analysis and we\nwill not refer to its origin. We note that in [51] the measure-\nment of spin-pumping-induced voltage was performed using\nthe bilayer systems of Bi 2Se3(TI) and CoFeB (ferromagnet).\nIn this experiment, it is considered that the dominant contri-\nbution to the spin-pumping-induced voltage is coming from\nthe inverse spin Hall effect (bulk state) rather than the inverse\nEdelstein effect (surface state). Thus, although the measured\nspin-pumping-induced voltage shows similar characteristics\n(Figs. 2 and 3(a) and (b)) with \u0016VSPI\nx, we will not make a\ncomparison with these experimental results. Next, let us take\na look at Fig. 3(c) in [42]. It shows the microwave-power de-\npendence of peak values for the BSTS sample size 4\u00021\u00020:1\nmm3. Both the positive and negative peak values become\nlarger as microwave power increases. To summarize, from the\nabove analysis we see that the physical behavior of our result\n\u0016VSPI\nxrepresented by Eq. (32) and Fig. 6 match qualitatively\nwith these experimental results.\nIV . CONCLUSION\nIn this paper, we have investigated the electric current on\nthe surface of ferromagnetic TI induced by the spin pumping.\nFirst, we have presented the microscopic model of ferromag-\nnetic TI surface and represented its time evolution. We have\nmathematically formulated how the system evolves from the\nthermal equilibrium state realized in the far past to the non-\nequilibrium steady state driven by the spin pumping (FMR).\nThen we have used the Keldysh Green’s function approach\nto analyze the generation of a spin-pumping-induced elec-\ntric current. We have calculated it by regarding the ac exter-\nnal magnetic ﬁeld and the exchange interaction as perturba-\ntive terms. In this way, we could clearly understand the way\nspin-pumping-induced electric current is generated by these\ntwo interactions. The mechanism is as follows. The FMR is\ntriggered by the ac magnetic ﬁeld and the magnon with the\nzero momentum emerges. It is the ﬂuctuation from the satura-\ntion magnetization. After then, the spin pumping is induced,\nand during such a process, the spin current ﬂows from FM to\nTI carried by the zero-momentum magnon. Through the ex-\nchange interaction, the zero-momentum magnon couples with\nthe TI surface state and the spin is exchanged between them.\nThen owing to the spin-momentum locking, it is converted\ninto momentum and effectively the TI surface state experi-\nences this additional momentum as the applied electric ﬁeld.\nOn the other hand, the TI surface state is affected by the non-\nmagnetic impurity. As a result, at the non-equilibrium steady\nstate these two effects commensurate and the static electric\nﬁeld, i.e., the spin-pumping-induced electric ﬁeld is created\nleading to the generation of the spin-pumping-induced elec-\ntric current. It scales quadratically to the ac magnetic ﬁeld\n(linear to the power of electromagnetic ﬁeld) while it is lin-ear to the strength of the exchange interaction. The effective\nelectric voltage \u0016VSPI\nx in Eq. (32) is expressed by the spec-\ntrum of zero-momentum magnon which clearly reﬂects that it\nis created by the spin pumping (FMR). The effective electric\nvoltage \u0016VSPI\nxshows two side peaks. They emerge when the\nabsolute of external frequency of ac magnetic ﬁeld becomes\nequivalent to the Zeeman gap of magnon. The absolutes of\nthese two peak values are the same while they have the oppo-\nsite signs. Further, the absolutes of two peaks are the increas-\ning function of the microwave power. Such characteristics of\nour effective voltage \u0016VSPI\nxshow qualitatively the good match-\ning with the experimental results of electric voltage reported\nin [42, 59]. Consequently, the spin-pumping-induced elec-\ntric current is the quantum phenomena intrinsic in the hybrid\nquantum system of TI surface state and the zero-momentum\nmagnon. It is the non-linear response to the ac magnetic ﬁeld.\nOur microscopic theory based on the Keldysh Green’s func-\ntion approach makes not only the mechanism as well as the\nstructure of spin-pumping-induced electric current (voltage)\nclear. We believe that our theory can be extended in many\nother types of quantum phenomena occurring at the inter-\nface between the magnetic materials and TI. For instance, we\nwould like to apply our Keldysh Green’s function approach\nto analyze the heat current as well as the spin Seebeck effect\nand the spin-orbit torque in the future. In addition, we become\nable to extract more information on magnets and TI. For in-\nstance, by measuring the peak value of electric voltage we can\nestimate the exchange-coupling strength. Another important\nand interesting issue is the Fermi-energy dependence on spin-\npumping-induced electric voltage. It is important to analyze\nwhether the contribution to electric transport quantities (for\ninstance, the electric voltage) is coming from the surface state\nor the bulk state [51, 54, 59]. The transport properties of Dirac\nelectrons in solids are affected by many types of elements. For\ninstance, the Fermi-energy dependence of Dirac-electron con-\nductivity in graphene differs whether the impurity potential\nis short-range (delta-function) type or long-range (Coulomb)\ntype [66]. For the TI surface state the characteristics of its\nconductivity is not only generated by the impurity effect but\nalso by a scattering process due to a magnetic texture such as\nskyrmion [67]. As our future work, we would like to explore\nthe rich transport phenomena on the surface of a magnetic TI\ninduced by the impurity potentials and the magnetic textures\nwith many types and analyze carefully the characteristics of\nelectric voltage as well as the electrical conductivity as func-\ntions of the Fermi energy.\nTo discuss our result from the application point of view,\nthe non-linear response in the magnetic ﬁeld as well as the\nlinear scaling in the exchange-coupling strength of the spin-\npumping-induced electric current clearly indicates that the\nsurface of ferromagnetic TI possesses the high-performing\nfunctionality of creating the electric charge current or volt-\nage by the magnetic controlling. When we think of engineer-\ning spintronics devices, the merit of using the ferromagnetic\nTI comparing to the hybrid system of FM and metal like the\nNiFe/Pt is the lower energy consumption: The joule heating is\nsuppressed for the ferromagnetic TI because the bulk is insu-\nlating while it is unavoidable for the FM/metal hybrid system13\nsince the bulk is metallic. By designing carefully the larger\nhybrid quantum systems based on the magnet and TI, we will\nbecome able to perform the coherent controlling of magnon\ndynamics and the quantum transport of TI surface state at the\ninterface, and consequently, make a high-efﬁcient conversion\nof the spin and the electric charge current (coherent control-\nling of the magnetism and the electricity). Such investigations\nlead to an important progress on the realization of magnetic-\nTI-based spintronics devices.\nACKNOWLEDGMENTS\nY . H thanks Kanta Asakawa for the having the discussion on\nthe basics of FMR experiment, Yuki Shiomi for having fruitful\ndiscussion on Ref. [42], Minoru Kawamura for discussing the\nphysical interpretation on the non-linearity of spin-pumping-\ninduced electric current, Hiroyasu Yamahara for the discus-\nsion on Refs. [42, 51] as well as the basics of FMR experi-\nment. This work was supported in part by the MEXT Grant-\nin-Aid for Scientiﬁc Research on Innovative Areas KAK-\nENHI Grant Number JP15H05870 (Y . H), and JSPS KAK-\nENHI Grants Nos. JP15H0584 and JP17K05485, JST CREST\nGrant No. JPMJCR18T2, and JSPS KAKENHI Grant No.\nJP20H01830 (K. N).\nAppendix A: Field Quantization, Real-Time Green’s Function,\nand Imaginary-time Green’s function\nIn this section, ﬁrst we present the details of ﬁeld quantiza-\ntion for the TI surface state. Then, we introduced the unper-\nturbed real-time Green’s function. Next we demonstrate the\nderivation of impurity-averaged Green’s function of TI sur-\nface state by using the imaginary-timeGreen’s function for-\nmalism. Further, we show the real-time Green’s functions\nof magnon and the retarded and advanced components of\nmagnon Green’s function including the Gilbert-damping ef-\nfect.\n1. Field Quantization and Real-Time Green’s Functions of TI\nSurface State\nThe spin-momentum-locking Hamiltonian of TI surface\nstate in the momentum space is given by (see also Eq. (3))\nHTI\n0;\u000b0\u000b(k) =~vF(kx\u001by\u0000ky\u001bx)\u000b0\u000b; (A1)\nwhere\u000b;\u000b0=\";#. The eigenvalues of the above Hamil-\ntonian are\u0006\u000fTI\nk=\u0006~vFkwithk=p\n(kx)2+ (ky)2:\nWe denote the positive and negative-energy plane-wave solu-\ntions asu(+)\nk(xt) =u(+)(k)ei(k\u0001x\u0000!TI(k)t)andu(\u0000)\nk(xt) =\nu(\u0000)(k)ei(k\u0001x+!TI(k)t), respectively. The eigenfrequency\n!TI(k)is obtained from \u000fTI\nkas!TI(k) = ~\u00001\u000fTI(k).\nThe vectors u(+)(k) = (u\"(k);u#(k))tandu(\u0000)(k) =(u(\u0000)\n\"(k);u(\u0000)\n#(k))tare two-column vectors with “t\" denot-\ning the transpose. We take them as\nu(+)(k) =1p\n2\u00121\nik+\nk\u0013\n; u(\u0000)(k) =1p\n2\u0012ik\u0000\nk\n1\u0013\n;\n(A2)\nwherek\u0006=kx\u0006iky. The two eigenvectors u(+)(k)and\nu(\u0000)(k)satisfy the completeness relations\nX\n\u000b=\";#u(+)y\n\u000b(k)u(+)\n\u000b(k) =X\n\u000b=\";#u(\u0000)y\n\u000b(k)u(\u0000)\n\u000b(k) = 1;\nX\n\u000b=\";#u(+)y\n\u000b(k)u(\u0000)\n\u000b(k) =X\n\u000b=\";#u(\u0000)y\n\u000b(k)u(+)\n\u000b(k) = 0:\n(A3)\nWith using the plane-wave solutions in Eq. (A2) and the com-\npleteness relations in Eq. (A3), we construct the ﬁeld operator\nof TI surface state. By denoting the ﬁeld operator (annihila-\ntion operator) of TI surface state as  \u000b(x), it is given by\n \u000b(x) =1p\nVX\nk;\u0015=\u0006\u0010\nu(\u0015)\n\u000b(k)eik\u0001x\u0011\n\u0001c(\u0015)(k); (A4)\nwherec(+)(k)andc(\u0000)(k)are annihilation operators of TI\nsurface state whose energy and momentum are (\u000fTI(k);k)\nand(\u0000\u000fTI(k);k), respectively. Vis the area of TI surface.\nFor the ground state, we choose the Dirac sea represented by\nj0i=Y\nkc(\u0000)y(k)j~0i; (A5)\nwherej~0iis the Fock state which satisﬁes c(\u0006)(k)j~0i= 0for\nanyk:Correspondingly, we rewrite the ﬁeld operator in Eq.\n(A4) as\n \u000b(x) =1p\nVX\nk\u0000\nu\u000b(k)eik\u0001xa(k) +v\u000b(k)e\u0000ik\u0001xby(k)\u0001\n;\n(A6)\nwherea(k) =c(+)(k);by(\u0000k) =c(\u0000)(k),u\u000b(k) =\nu(+)\n\u000b(k), andv\u000b(k) =u(\u0000)\n\u000b(\u0000k). The operator a(k)is\nannihilation operator of particle (electron) with the energy\n+\u000fTI(k)and momentum kwhileby(k)is creation opera-\ntor of anti-particle (hole) with the energy +~!TI(k)and\nmomentumk. They satisfy the anti-commutation relations\nfa(k);ay(k0)g=fb(k);by(k0)g=\u000e(k\u0000k0), and all the\nothers are zero. From these anti-commutation relations and\nEq. (A3), we have f \u000b(x); y\n\u000b0(x0)g=\u000e(x\u0000x0)and\nf \u000b(x); \u000b0(x0)g=f y\n\u000b(x); y\n\u000b0(x0)g= 0:By using the\noperator \u000b(x)in Eq. (A6) and its Hermitian conjugate the\nfree Hamiltonian, momentum operator, and number operator\nare described as\nHTI\n0=X\nk\u000fTI(k)\u0000\nay(k)a(k) +by(k)b(k)\u0001\n;\nPi=X\nk~ki\u0000\nay(k)a(k) +by(k)b(k)\u0001\nNTI=X\nk\u0000\nay(k)a(k)\u0000by(k)b(k)\u0001\n; (A7)14\nwherei=x;y: We have neglected the constants in HTI\n0and\nNTIwhich are the contribution from the Dirac sea. Here-\ninafter, we just write NTIasN. Based on the previous argu-\nment, next we consider the Hamiltonian \u0016HTI\n0=HTI\n0\u0000\u000fFN\n(see also Eq. (1)) with \u000fF(>0) the Fermi energy of TI.\nWe have taken the chemical potential of TI surface state to\nbe equal to\u000fF:Correspondingly, we re-describe the ﬁeld op-erator \u000b(x)in Eq. (A6) by three operators a(k),b(+)(k),\nandb(\u0000)(k): The operator a(k)is the annihilation operator of\nmomentumkwith its energy higher than the Fermi energy \u000fF.\nOn the other hand, the operator b(+)(k)is the annihilation op-\nerator of momentum kwith a positive energy which is lower\nthan\u000fF:b(\u0000)(k)is the annihilation operator of momentum k\nwith a negative energy [68]. The representation of operator\n \u000b(x)in terms ofa(k),b(+)(k), andb(\u0000)(k)is given as\n \u000b(x) =1p\nVX\nkh\n\u0002(k\u0000kF)ua\n\u000b(k)eik\u0001xa(k) +\u0010\n\u0002(kF\u0000k)vb(+)\n\u000b(k)b(+)y(k) +vb(\u0000)\n\u000b(k)b(\u0000)y(k)\u0011\ne\u0000ik\u0001xi\n; (A8)\nwhere \u0002(k\u0000kF) (\u0002(kF\u0000k))is the step function and\nkF= (~vF)\u00001\u000fF. The operators a(k),b(+)(k), andb(\u0000)(k)\nsatisfy the anti-commutation relation fa(k);ay(k0)g=fb(+)(k);b(+)y(k0)g=fb(\u0000)(k);b(\u0000)y(k0)g=\u000e(k\u0000k0)\nand all the others are zero. The two-column vectors u\u000b(k);\nvb(+)\n\u000b(k);andvb(\u0000)\n\u000b(k)are given by\nua(k) =1p\n2\u00121\nik+\nk\u0013\n; vb(+)(k) =1p\n2\u0012ik\u0000\nk\n1\u0013\n; vb(\u0000)(k) =1p\n2\u0012\n\u0000ik\u0000\nk\n1\u0013\n: (A9)\nFrom Eqs. (A8) and (A9), the Hamiltonian \u0016HTI\n0is expressed by the operators a(k),b(+)(k), andb(\u0000)(k)as\n\u0016HTI\n0=X\nkh\n\u0002(k\u0000kF)\u0018TI\na(k)ay(k)a(k) + \u0002(kF\u0000k)\u0018TI\nb(+)(k)b(+)y(k)b(+)(k) +\u0018TI\nb(\u0000)(k)b(\u0000)y(k)b(\u0000)(k)i\n; (A10)\nwhere\u0018TI\na(k) =\u0000\u0018TI\nb(+)(k) =\u000fTI(k)\u0000\u000fF, and\n\u0018TI\nb(\u0000)(k) =\u000fTI(k) +\u000fF. For deriving Eq. (A10), we have\nused the anti-commutation relation fb(+)(k);b(+)y(k0)g=\nfb(\u0000)(k);b(\u0000)y(k0)g=\u000e(k\u0000k0):Next, we introduce theﬁeld operator of TI surface state in the interaction picture with\nrespect to the Hamiltonian Let us denote it as  H0\u000b(xt)which\nis deﬁned by  H0\u000b(xt) =ei\u0016HTI\n0t=~ \u000b(x)e\u0000i\u0016HTI\n0t=~. It is de-\nscribed bya(k),b(+)(k), andb(\u0000)(k)as\n H0\u000b(xt) =1p\nVX\nk\u0010\n\u0002(k\u0000kF)ua\n\u000b(k)ei(k\u0001x\u0000!\u0018;TI\na(k)t)a(k) + \u0002(kF\u0000k)vb(+)\n\u000b(k)e\u0000i\u0010\nk\u0001x\u0000!\u0018;TI\nb(+)(k)t\u0011\nb(+)y(k)\n+vb(\u0000)\n\u000b(k)e\u0000i\u0010\nk\u0001x\u0000!\u0018;TI\nb(\u0000)(k)t\u0011\nb(\u0000)y(k)\u0011\n; (A11)\nwhere!\u0018;TI\na(k) = ~\u00001\u0018TI\na(k);!\u0018;TI\nb(+)(k) = ~\u00001\u0018TI\nb(+)(k),\nand!\u0018;TI\nb(\u0000)(k) = ~\u00001\u0018TI\nb(\u0000)(k). By using the ﬁeld oper-\nator H0\u000b(xt)in Eq. (A8) and its Hermitian conjugate\n y\nH0\u000b(xt), we introduce two-point real time Green’s func-tion in the interaction picture. Let us denote the time-\nordered, anti-time-ordered, retarded, and advanced Green’s\nfunctions as gt(0)\n\u000b\u000b0(xt;x0t0),g~t(0)\n\u000b\u000b0(xt;x0t0); gr(0)\n\u000b\u000b0(xt;x0t0),\nandga(0)\n\u000b\u000b0(xt;x0t0);respectively. They are given by15\ngt(0)\n\u000b\u000b0(xt;x0t0) =\u0000iTh H0\u000b(xt) y\nH0\u000b0(x0t0)i\u00160=gt(0)\n\u000b\u000b0(x\u0000x0;t\u0000t0) =1\nVX\nkZd!\n2\u0019ei\u0000\nk\u0001(x\u0000x0)\u0000!(t\u0000t0)\u0001\ngt(0)\n\u000b\u000b0(k!);\ngt(0)\n\u000b\u000b0(k!) =(1+~H0(k))\u000b\u000b0\n2\u00141\u0000nf(\u000fTI\nk)\n!+!F\u0000!TI\nk+i\u0011+nf(\u000fTI\nk)\n!+!F\u0000!TI\nk\u0000i\u0011\u0015\n+(1\u0000~H0(k))\u000b\u000b0\n2\u0014\u0016nf(\u000fTI\nk)\n!+!F+!TI\nk+i\u0011+1\u0000\u0016nf(\u000fTI\nk)\n!+!F+!TI\nk\u0000i\u0011\u0015\n;\ng~t(0)\n\u000b\u000b0(xt;x0t0) =\u0000i~Th H0\u000b(xt) y\nH0\u000b0(x0t0)i\u00160=g~t(0)\n\u000b\u000b0(x\u0000x0;t\u0000t0) =1\nVX\nkZd!\n2\u0019ei\u0000\nk\u0001(x\u0000x0)\u0000!(t\u0000t0)\u0001\ng~t(0)\n\u000b\u000b0(k!);\ng~t(0)\n\u000b\u000b0(k!) =(1+~H0(k))\u000b\u000b0\n2\u0014\u0000nf(\u000fTI\nk)\n!+!F\u0000!TI\nk+i\u0011+nf(\u000fTI\nk)\u00001\n!+!F\u0000!TI\nk\u0000i\u0011\u0015\n+(1\u0000~H0(k))\u000b\u000b0\n2\u0014\u0016nf(\u000fTI\nk)\u00001\n!+!F+!TI\nk+i\u0011+\u0000\u0016nf(\u000fTI\nk)\n!+!F+!TI\nk\u0000i\u0011\u0015\n;\ngr(0)\n\u000b\u000b0(xt;x0t0) =\u0000i\u0012(t\u0000t0)\u0001\u001a(0)\n\u000b\u000b0(xt;x0t0) =gr(0)\n\u000b\u000b0(x\u0000x0;t\u0000t0) =1\nVX\nkZd!\n2\u0019ei\u0000\nk\u0001(x\u0000x0)\u0000!(t\u0000t0)\u0001\ngr(0)\n\u000b\u000b0(k!);\nga(0)\n\u000b\u000b0(xt;x0t0) =i\u0012(t0\u0000t)\u0001\u001a(0)\n\u000b\u000b0(xt;x0t0) =ga(0)\n\u000b\u000b0(x\u0000x0;t\u0000t0) =1\nVX\nkZd!\n2\u0019ei\u0000\nk\u0001(x\u0000x0)\u0000!(t\u0000t0)\u0001\nga(0)\n\u000b\u000b0(k!);\ngr(0)\n\u000b\u000b0(k!) =Zd\u0014!\n2\u0019\u001a(0)\n\u000b\u000b0(k\u0014!)\n!\u0000\u0014!+i\u0011ga(0)\n\u000b\u000b0(k!) =Zd\u0014!\n2\u0019\u001a(0)\n\u000b\u000b0(k\u0014!)\n!\u0000\u0014!\u0000i\u0011; (A12)\nwhereh\u0001\u0001\u0001i \u00140denotes the thermal average taken by the den-\nsity matrix\u001aGC(\u0014H0;\f;\u000f F)with \u0014H0=\u0016HTI\n0: \u0011is a positive\ninﬁnitesimal. The symbols Tand~Tare the time-ordering and\nanti-time-ordering operators, respectively. For the step func-\ntion\u0012(t\u0000t0)we have used \u0012(t\u0000t0) =\u0000Rd!\n2\u0019ie\u0000i!(t\u0000t0)\n!+i\u0011.\nThe functions nf(\u000f)and \u0016nf(\u000f)are given by nf(\u000f) =\u0000\n1 +e\f(\u000f\u0000\u000fF)\u0001\u00001and\u0016nf(\u000f) =\u0000\n1 +e\f(\u000f+\u000fF)\u0001\u00001.nf(\u000f)rep-\nresents the Fermi-Dirac distribution function for the energy\n\u000fwith the chemical potential \u0016which we take to be equalto\u000fF:\u0016nf(\u000f)is the one with the chemical potential equal to\n\u0000\u000fF. They are given by the thermal average of TI-surface-\nstate ﬁeld operators as nf(\u000fTI(k)) =hay(k)a(k)i\u00160= 1\u0000\nhb(+)y(k)b(+)(k)i\u00160and \u0016nf(\u000fTI(k)) =hb(\u0000)y(k)b(\u0000)(k)i\u00160:\nThe matrix ~H0is given by Eq. (27) or\n~H0(k) = \n0\u0000i(kx\u0000iky)\nki(kx+iky)\nk0!\n:\nThe spectral functions \u001a(0)\n\u000b\u000b0(xt;x0t0)and\u001a(0)\n\u000b\u000b0(k!)are\n\u001a(0)\n\u000b\u000b0(xt;x0t0) =hf H0\u000b(xt); y\nH0\u000b0(x0t0)gi\u00160=\u001a(0)\n\u000b\u000b0(x\u0000x0;t\u0000t0) =1\nVX\nkZd!\n2\u0019ei\u0000\nk\u0001(x\u0000x0)\u0000!(t\u0000t0)\u0001\n\u001a(0)\n\u000b\u000b0(k!);\n\u001a(0)\n\u000b\u000b0(k!) =\u0019h\n(1+~H0(k))\u000b\u000b0\u000e(!+!F\u0000!TI\nk) + (1\u0000~H0(k))\u000b\u000b0\u000e(!+!F+!TI\nk)i\n; (A13)\nwherefgin the above ﬁrst equation denotes the anticommu-\ntator:fX;Yg=XY+YX.Besides time-ordered, anti-time-ordered, retarded, and ad-\nvanced components, there are lesser and greater components\ndeﬁned by16\ng<(0)\n\u000b\u000b0(xt;x0t0) =ih y\nH0\u000b0(x0t0) H0\u000b(xt)i\u00160=g<(0)\n\u000b\u000b0(x\u0000x0;t\u0000t0) =1\nVX\nkZd!\n2\u0019ei\u0000\nk\u0001(x\u0000x0)\u0000!(t\u0000t0)\u0001\ng<(0)\n\u000b\u000b0(k!);\ng<(0)\n\u000b\u000b0(k!) =i\u0019f(~!)h\n\u000e(!\u0000!TI\nk+!F)(1+~H0(k))\u000b\u000b0+\u000e(!+!TI\nk+!F)(1\u0000~H0(k))\u000b\u000b0i\n;\ng>(0)\n\u000b\u000b0(xt;x0t0) =\u0000ih H0\u000b(xt) y\nH0\u000b0(x0t0)i\u00160=g>(0)\n\u000b\u000b0(x\u0000x0;t\u0000t0) =1\nVX\nkZd!\n2\u0019ei\u0000\nk\u0001(x\u0000x0)\u0000!(t\u0000t0)\u0001\ng>(0)\n\u000b\u000b0(k!);\ng>(0)\n\u000b\u000b0(k!) =\u0000i\u0019\u0000\n1\u0000f(~!)\u0001h\n\u000e(!\u0000!TI\nk+!F)(1+~H0(k))\u000b\u000b0+\u000e(!+!TI\nk+!F)(1\u0000~H0(k))\u000b\u000b0i\n; (A14)\nwheref(~!) =\u0000\n1 +e\f~!\u0001\u00001. The lesser and greater components of Green’s functions are related to the time-\nordered, anti-time-ordered, retarded, and advanced compo-\nnents through the relations [62, 65]\ng<(0)\n\u000b\u000b0(xt;x0t0) =gt(0)\n\u000b\u000b0(xt;x0t0)\u0000gr(0)\n\u000b\u000b0(xt;x0t0) =g~t(0)\n\u000b\u000b0(xt;x0t0) +ga(0)\n\u000b\u000b0(xt;x0t0);\ng>(0)\n\u000b\u000b0(xt;x0t0) =gt(0)\n\u000b\u000b0(xt;x0t0)\u0000ga(0)\n\u000b\u000b0(xt;x0t0) =g~t(0)\n\u000b\u000b0(xt;x0t0) +gr(0)\n\u000b\u000b0(xt;x0t0): (A15)\nThe above relation also holds for the Green’s functions in\nthe momentum-frequency representation. Further, from Eq.\n(A14) and the formula1\nz\u0000z0\u0006i\u0011= P\u0010\n1\nz\u0000z0\u0011\n\u0007i\u0019\u000e(z\u0000z0),\nwe have\ng<(0)\n\u000b\u000b0(k!) =f(~!)(ga(0)\n\u000b\u000b0(k!)\u0000gr(0)\n\u000b\u000b0(k!));\ng>(0)\n\u000b\u000b0(k!) =\u0000(1\u0000f(~!))(ga(0)\n\u000b\u000b0(k!)\u0000gr(0)\n\u000b\u000b0(k!)):\n(A16)\n2. Impurity-Averaged Imaginary-Time Green’s Function\nWe now include the impurity-potential effect and derive the\nimpurity-averaged Green’s function for the TI surface state.Let us denote a function described by the coordinates of im-\npurities asF(X1;:::;XNimp):The impurity average is de-\nﬁned by\nhF(X1;:::;XNimp)iimp\nave=ZNimpY\ni=1dXi\nVF(X1;:::;XNimp):\n(A17)\nTo perform this on the Green’s functions of TI surface state,\nwe derive the Dyson’s equation for the imaginary-time (Mat-\nsubara) Green’s functions with regarding the impurity poten-\ntial as the perturbation which is given by Eq. (4). First, let\nus introduce the unperturbed imaginary-time Green’s function\ndeﬁned by [61, 64, 65]\nG(0)\nM;\u000b\u000b0(x\u001cM;x0\u001c0\nM) =\u0000\nTM H0\u000b(x\u001cM) y\nH0\u000b0(x0\u001c0\nM)\u000b\n0;GC=G(0)\nM;\u000b\u000b0(x\u0000x0;\u001cM\u0000\u001c0\nM);\n=1\n\f~VX\nkmei\u0000\nk\u0001(x\u0000x0)\u0000i!m(\u001cM\u0000\u001c0\nM)\u0001\nG(0)\nM;\u000b\u000b0(k;i!m);\nG(0)\nM;\u000b\u000b0(k;i!m) =Zd!0\n2\u0019\u001a(0)\n\u000b\u000b0(k!0)\ni!m\u0000!0: (A18)\nHere\u001cM;\u001c0\nMare the imaginary times and TMrepresents\nthe imaginary-time ordering. The fermionic imaginary-time\nGreen’s functionG(0)\nM;\u000b\u000b0(x\u0000x0;\u001cM\u0000\u001c0\nM)in the above equa-\ntion is anti-periodic with respect to the the imaginary time:G(0)\nM;\u000b\u000b0(x\u0000x0;\u001cM\u0000\u001c0\nM) =\u0000G(0)\nM;\u000b\u000b0(x\u0000x0;\u001cM\u0000\u001c0\nM\u0006\f~).\nCorrespondingly, the Matsubara frequency !mis given by\n!m= (2m+ 1)\u0019=(\f~)and the Fourier transform of the\nG(0)\nM;\u000b\u000b0(x\u0000x0;\u001cM\u0000\u001c0\nM)for the temporal component be-17\ncomesG(0)\nM;\u000b\u000b0(x\u0000x0;i!m) =R\f~\n0ei!m(\u001cM\u0000\u001c0\nM)G(0)\nM;\u000b\u000b0(x\u0000\nx0;\u001cM\u0000\u001c0\nM)d(\u001cM\u0000\u001c0\nM).\u001a(0)\n\u000b\u000b0(k!)is the spectral function\ndeﬁned in Eq. (A13). By performing the analytic continua-\ntioni!m!!+i\u0011onG(0)\nM;\u000b\u000b0(k!)in Eq. (A18), we obtain\nthe retarded Green’s function gr(0)\n\u000b\u000b0(k!)in Eq. (A12). On theother side, we have the advanced Green’s function ga(0)\n\u000b\u000b0(k!)\nin Eq. (A12) by i!m!!\u0000i\u0011.\nNext, the Dyson equation owing to the impurity potential is\ngiven by [64]\nGM;\u000b\u000b0(x\u001cM;x0\u001c0\nM) =G0\nM;\u000b\u000b0(x\u001cM;x0\u001c0\nM)\n+Z\f~\n0d\u001cM\n1Z\nd2x1G0\nM;\u000b\u000b0\n1(x\u001cM;x1\u001cM\n1)Himp\n\u000b0\n1\u000b1(x1)GM;\u000b\u000b0(x1\u001cM\n1;x0\u001c0\nM): (A19)\nLet us rewrite the right-hand side of Dyson Equation (A19)\nasGM;\u000b\u000b0(x\u001cM;x0\u001c0\nM) =P\nnG(n)\nM;\u000b\u000b0(x\u001cM;x0\u001c0\nM)whereG(n)\nM;\u000b\u000b0(x\u001cM;x0\u001c0\nM)is in then-th order of impurity potential\nHimp:Its form is represented as\nG(n)\nM;\u000b\u000b0(x\u001cM;x0\u001c0\nM) =Z\f~\n0d\u001cM\n1\u0001\u0001\u0001d\u001cM\nnZ\nd2x1\u0001\u0001\u0001d2xnHimp\n\u000b0\n1\u000b1(x1)\u0001\u0001\u0001Himp\n\u000b0n\u000bn(xn)\n\u0002G0\nM;\u000b\u000b0\nn(x\u0000xn;\u001cM\u0000\u001cM\nn)G0\nM;\u000bn\u000b0\nn\u00001(xn\u0000xn\u00001;\u001cM\nn\u0000\u001cM\nn\u00001)\u0001\u0001\u0001G0\nM;\u000b1\u000b0(x1\u0000x0;\u001cM\n1\u0000\u001c0\nM);\n(A20)\nwhere we have used G0\nM;\u000b\u000b0(x\u001cM;x0\u001c0\nM) =G0\nM;\u000b\u000b0(x\u0000\nx0;\u001cM\u0000\u001c0\nM). Let us perform the Fourier transfor-\nmations on the Green’s function G(n)\nM;\u000b\u000b0(x\u001cM;x0\u001c0\nM).\nSince the impurity potential is time independent, theGreen’s function G(n)\nM;\u000b\u000b0(x\u001cM;x0\u001c0\nM)is described as\nG(n)\nM;\u000b\u000b0(x\u001cM;x0\u001c0\nM) =G(n)\nM;\u000b\u000b0(x;x0;\u001cM\u0000\u001c0\nM):Then the\nFourier transformation is given as G(n)\nM;\u000b\u000b0(x;x0;\u001cM\u0000\u001c0\nM) =\n(\f~V)\u00001P\nmkk0G(n)\nM;\u000b\u000b0(k;k0;i!m)ei(k\u0001x\u0000k0\u0001x0\u0000!m(\u001cM\u0000\u001c0\nM)):\nThe formula ofG(n)\nM;\u000b\u000b0(k;k0;i!m)is represented as\nG(n)\nM;\u000b\u000b0(k;k0;i!m) =1\nVnX\nk1;:::;kn\u00001h\nG(0)\nM;\u000b\u000bn(k;i!m)vimp(k\u0000kn\u00001)ih\nG(0)\nM;\u000bn\u000bn\u00001(kn\u00001;i!m)vimp(kn\u00001\u0000kn\u00002)i\n\u0001\u0001\u0001h\nG(0)\nM;\u000b3\u000b2(k2;i!m)vimp(k2\u0000k1)ih\nG(0)\nM;\u000b2\u000b2(k1;i!m)vimp(k1\u0000k0)i\nG(0)\nM;\u000b1\u000b0(k0;i!m)\n\u0002\u001aimp(k\u0000kn\u00001)\u001aimp(kn\u00001\u0000kn\u00002)\u0001\u0001\u0001\u001aimp(k3\u0000k2)\u001aimp(k2\u0000k1)\u001aimp(k1\u0000k0); (A21)\nwhere we have used (\f~)\u00001R\f~\n0d\u001cexp\u0000\ni(!m\u0000!m0)\u001c\u0001\n=\n\u000em;m0. In the above equation, all the Matsubara frequencies of\nn+1unperturbed Green’s functions in the right-hand side are\nequivalent due to time independence of impurity potential. We\nnow perform the impurity average on Eq. (A21) by assuming\nthat the total number of impurities Nimpis sufﬁciently large.We use the formulas, for instance [65],\nh\u001aimp(k\u0000k0)iimp\nave=Nimp\u000ek;k0;\nh\u001aimp(k)\u001aimp(k0)iimp\nave=Nimp\u000ek+k0;0+N2\nimp\u000ek;0\u000ek0;0:\n(A22)\nForn= 0;1;2, we have18\nhG(0)\nM;\u000b\u000b0(k;k0;i!m)iimp\nave=\u000ek;k0G(0)\nM;\u000b\u000b0(k;i!m);\nhG(1)\nM;\u000b\u000b0(k;k0;i!m)iimp\nave= 0;\nhG(2)\nM;\u000b\u000b0(k;k0;i!m)iimp\nave=\u000ek;k0G(0)\nM;\u000b\u000b1(k;i!m)\u0012\nnimpZd2q\n(2\u0019)2jvimp(q\u0000k)j2G(0)\nM;\u000b1\u000b0(q;i!m)\u0013\nG(0)\nM;\u000b0\u000b0(k0;i!m)\n\u0011\u000ek;k0\u0016G(2)\nM;\u000b\u000b0(k;i!m); (A23)\nwherenimp =Nimp=Vis the number density of impu-\nrities. To derive Eq. (A23), we have taken the contin-\nuum limitV\u00001P\nq!R\nd2q=(2\u0019)2. Further, we used\nv\u0003\nimp(q) =vimp(\u0000q)andvimp(0) = 0:The impurity-\naveraged Green’s function hG(n)\nM;\u000b\u000b0(k;k0;i!m)iimp\navein Eq.\n(A23) is diagonal with respect to momentum. Similarly,\nthe impurity-averaged Green’s function is diagonal in mo-\nmentum for n\u00153[64, 65]. As a result, when the impurity\naverage is taken on Green’s function, it restores the trans-\nlational symmetry. Let us express hGM;\u000b\u000b0(k;k0;i!m)iimp\nave\nandhG(n)\nM;\u000b\u000b0(k;k0;i!m)iimp\nave as \u0016GM;\u000b\u000b0(k;i!m)\u000ek;k0and \u0016G(n)\nM;\u000b\u000b0(k;i!m)\u000ek;k0, respectively. Their Fourier\ntransforms are given by hGM;\u000b\u000b0(x;x0;i!m)iimp\nave\u0011\n\u0016GM;\u000b\u000b0(x\u0000x0;i!m) =V\u00001P\nkeik\u0001(x\u0000x0)\u0016GM;\u000b\u000b0(k;i!m)\nandhG(n)\nM;\u000b\u000b0(x;x0;i!m)iimp\nave\u0011\u0016G(n)\nM;\u000b\u000b0(x\u0000x0;i!m) =\nV\u00001P\nkeik\u0001(x\u0000x0)\u0016G(n)\nM;\u000b\u000b0(k;i!m).\nNext, we reorganize the perturbative expansionP\nn\u0016G(n)\nM;\u000b\u000b0(k;i!m)by representing it as the sum of all\nirreducible Feynman diagrams. Let us denote the associated\nself-energy as \u0006imp(k;i!m)\u000b\u000b0:Then, the Dyson equation\nfor the impurity-averaged Green’s function \u0016GM;\u000b\u000b0(k;i!m)\nis expressed as\n\u0016GM;\u000b\u000b0(k;i!m) =G(0)\nM;\u000b\u000b0(k;i!m) +G(0)\nM;\u000b\u000b2(k;i!m)\u0006imp(k;i!m)\u000b2\u000b1\u0016GM;\u000b1\u000b0(k;i!m): (A24)\nTo evaluate the self-energy \u0006imp(k;i!m), we take the impu-\nrity potential as Vimp(x\u0000Ximp\ni) =v0\u000e(x\u0000Ximp\ni)with\nv0a constant and use the ﬁrst-Born approximation. Then,\nwe havevimp(q) =v0:Since the Fermi energy \u000fFis posi-\ntive, the term in the unperturbed Green’s function which con-tributes dominantly to this evaluation is 1=(i!m+!F\u0000!TI\nk).\nTherefore, when we perform the momentum integral for eval-\nuating \u0006imp(k;i!m), we just retain 1=(i!m+!F\u0000!TI\nk).\nLet us write the self-energy in the ﬁrst-Born approximation as\n\u0006imp\n1BA(k;i!m)\u000b\u000b0. It is evaluated as\n\u0006imp\n1BA(k;i!m)\u000b\u000b0=nimp\n~Zd2q\n(2\u0019)2jvimp(q\u0000k)j2G(0)\nM;\u000b\u000b0(k;i!m) =1\n2nimpv2\n0Z1\n\u0000\u000fFd\u0018~N(\u0018)1\ni\u000fm\u0000\u0018\n\u0019\u00001\n2~N(0)nimpv2\n0Z1\n\u00001d\u0018\u0018+i\u000fm\n\u00182+\u000f2m=\u0000i\n2\u0019~N(0)nimpv2\n0sgn(!m)\u0011\u0000i\n2\u001crel\nTIsgn(!m)\u000e\u000b;\u000b0; (A25)\nwhere\u000fm=~!m. The quantity ~NTI(\u0018)is the density of\nstates per volume of the TI surface state measured from the\nFermi energy. It is given by ~NTI(\u0018) = (\u000fF+\u0018)=\u0000\n2\u0019(~vF)2\u0001\n:\nFor going from the ﬁrst to second line of Eq. (A25), we have\nmade an approximation such that the density of state ~NTI(\u0018)\nincluded in the integrand can be set with ~NTI(0). This is\nbecause we can consider that the energy state in the vicin-\nity of Fermi level dominantly contributes to the self energy\n\u0006imp\n1BA(k;i!m)\u000b\u000b0:jqj;jkj'kFand ~N(\u0018)'~N(0). Fur-ther, we have replaced the lower limit \u0000\u000fFwith\u00001 since\nwe consider that the number of electrons included in the sur-\nface with its area Vis large enough and the number density\nof TI surface nTI\n2Dcan be taken as large. The relation between\n\u000fFandnTI\n2Dis given by\u000fF=~vFq\n4\u0019nTI\n2D, and hence, we\ntake\u000fF!1:The time\u001crel\nTIis the relaxation time of TI sur-\nface state owing to the impurity effect. We now derive the\nimpurity-averaged green’s function \u0016GM;\u000b\u000b0(k;i!m). From\nEqs. (A24) and (A25) we obtain19\n\u0016GM;\u000b\u000b0(k;i!m) =\u0014\u0010\nG(0)\nM(k;i!m)\u0011\u00001\n+i\n2\u001crel\nTIsgn(!m)\u00011\u0015\u00001\n\u000b\u000b0; (A26)\nwhere\n\u0010\nG(0)\nM(k;i!m)\u0011\u00001\n\u000b\u000b0= (!m+!F)1\u000b\u000b0+!TI\nk~H0;\u000b\u000b0:\n(A27)\nBy performing the analytic continuation i!m!!+\nisgn(!m)\u0011, consequently, we obtain the retarded and ad-\nvanced components of impurity-averaged Green’s functions\n\u0016gr\n\u000b\u000b0(k;!)and\u0016ga\n\u000b\u000b0(k;!)in Eq. (25). The retarded compo-\nnent is obtained for sgn(!m)>0while we get the advancedcomponent for sgn(!m)<0.\n3. Magnon Green’s Functions\nWe present the real-time magnon Green’s functions. First\nlet us show them without the damping effect. Like given in\nEq. (A12), the time-ordered, anti-time-ordered, retarded, and\nadvanced components of magnon Green’s functions in the in-\nteraction picture are\nDt(pt;p0t0) =\u0000iThaH0(pt)ay\nH0(p0t0)i0=\u000ep;p0Dt(p;t\u0000t0) =Zd!\n2\u0019e\u0000i!(t\u0000t0)\u000ep;p0Dt(p!);\nD~t(pt;p0t0) =\u0000i~ThaH0(pt)ay\nH0(p0t0)i0=\u000ep;p0D~t(p;t\u0000t0) =Zd!\n2\u0019e\u0000i!(t\u0000t0)\u000ep;p0D~t(p!);\nDt(p!) =1 +nb(\u000fFM\np)\n!\u0000!FMp+i\u0011\u0000nb(\u000fFM\np)\n!\u0000!FMp\u0000i\u0011; D~t(p!) =nb(\u000fFM\np)\n!\u0000!FMp+i\u0011\u00001 +nb(\u000fFM\np)\n!\u0000!FMp\u0000i\u0011;\nDr(pt;p0t0) =\u0000i\u0012(t\u0000t0)\u0001\u001a(0)(pt;p0t0) =\u000ep;p0Dr(p;t\u0000t0) =\u000ep;p0Zd!\n2\u0019e\u0000i!(t\u0000t0)Dr(p!);\nDa(pt;p0t0) =i\u0012(t0\u0000t)\u0001\u001a(0)(pt;p0t0) =\u000ep;p0Da(p;t\u0000t0) =\u000ep;p0Zd!\n2\u0019e\u0000i!(t\u0000t0)Da(p!);\n\u001a(0)(pt;p0t0) =\u000ep;p0e\u0000i!FM\np(t\u0000t0); Dr(p!) =1\n!\u0000!FMp+i\u0011; Da(p!) =1\n!\u0000!FMp\u0000i\u0011; (A28)\nwhereaH0(pt) =eiH0t=~a(p)e\u0000iH0t=~anday\nH0(pt) =eiH0t=~ay(p)e\u0000iH0t=~.nb(\u000f) =\u0000\ne\f\u000f\u00001\u0001\u00001is the Bose-\nEinstein distribution function. The lesser and greater Green’s\nfunctions are given by\nD>(pt;p0t0) =\u0000ihaH0(pt)ay\nH0(p0t0)i0=\u000ep;p0D>(p;t\u0000t0) =Zd!\n2\u0019e\u0000i!(t\u0000t0)\u000ep;p0D>(p!);\nD<(pt;p0t0) =\u0000ihay\nH0(p0t0)aH0(pt)i0=\u000ep;p0D<(p;t\u0000t0) =Zd!\n2\u0019e\u0000i!(t\u0000t0)\u000ep;p0D<(p!);\nD>(p!) =\u00002\u0019i\u000e(!\u0000!FM\np)\u0000\n1 +nb(\u000fFM\np)\u0001\n; D<(p!) =\u00002\u0019i\u000e(!\u0000!FM\np)nb(\u000fFM\np): (A29)\nThe magnon Green’s functions presented above satisfy ex-\nactly the same relations given in Eq. (A15). Further, from\nEqs. (A28) and (A29), we can verify that with similar to Eq.(A16) these components satisfy the relations\nD<(p!) =\u0000nb(~!) (Da(p!)\u0000Dr(p!));\nD>(p!) =\u0000\u0000\n1 +nb(~!)\u0001\n(Da(p!)\u0000Dr(p!)):(A30)20\nNext, we show the magnon Green’s functions including the\ndamping effect. This is obtained by solving the Landau-\nLifshitz-Gilbert equation\ndSi\ndt=\r\u0000\nB0+Bext(t)\u0001\n\u0002Si\u0000\u000b\nS0\u0012\nSi\u0002dSi\ndt\u0013\n;\n(A31)\nwhereB0= (0;B0;0)withB0a constant. For\nBext(t), we take the same magnetic-ﬁeld con-\nﬁguration as we did in Sec. II: Bext(t) =\nBext\u0000\nsin\u0000\nsgn(B0)\u0001!extt\u0001\n;0;cos\u0000\nsgn(B0)\u0001!extt\u0001\u0001\n.\nThe solution is given in the form Sy\ni=\u0000S0sgn(B0)and\nS\u0000\ni=Sz\ni\u0000iSx\ni=\u0016S\u0000\nie\u0000isgn(B0)!exttwhere \u0016S\u0000\niis a complex\nconstant. The demagnetizing coefﬁcient is going to be\nexcluded. By introducing Bext;\u0000=Bext;z\u0000iBext;xand\nthe magnetic susceptibility as \u001fmag, for sgn(B0)>0we\nre-expressS\u0000\niasS\u0000\ni=\u001fmagBext;\u0000[15]. As a result, we\nobtain\n\u001fmag=\rS0\n!ext\u0000!FM\n0+i\u000b!ext; (A32)\nwhere!FM\n0=\rjB0j, which is the Zeeman gap of magnon.\nWith eliminating the factor \rS0in Eq. (A32), we identify the\nmagnetic susceptibility \u001fmagwith the retarded Green’s func-\ntion\u0016Dr(0;!)in Eq. (26). On the other hand, for sgn(B0)<0\nwe identify the retarded function with the response (magnetic\nsusceptibility) of S+\ni=Sz\ni+iSx\nitoB+\ni=Bz\ni+iBx\niand\nthis is equal to \u001fmagin Eq. (A32). The advanced component\nis given by the complex conjugate of retarded component.\nAppendix B: Keldysh Green’s Functions\nIn this section, ﬁrst we present the formalism for the\nKeldysh Green’s function. Next, we show how the Keldysh\nGreen’s function is related to the real-time Green’s func-\ntion via the real-time projection. Further, with presenting\nsome useful formulas obtained by the the real-time projection,\nwe demonstrate the derivation of impurity-averaged real-time\nGreen’s functions.\n1. Real-Time Projection\nAs discussed in subsec. II B, our starting point is the full\nlesser Green’s function of TI surface state in Eq. (14). We\nrewrite this with the Keldysh Green’s function given by Eq.\n(15) or\niGC;\u000b\u000b0(x\u001c;x0\u001c0) =D\nTC\u0002\nUexc\nCUext\nC y\nH0\u000b0(x0\u001c0) H0\u000b(x\u001c)\u0003E\n0:\nThe time-evolution operators Uexc\nCandUext\nCin the above\nequation are deﬁned by Eq. (16) or\nUexc\nC= exp\u0012\n\u0000i\n~Z\nCd\u0014\u001cVexc\nH0(\u0014\u001c)\u0013\n;\nUext\nC= exp\u0012\n\u0000i\n~Z\nCd~\u001cHext\nH0(~\u001c)\u0013\n:The perturbative calculation is performed by expanding Uexc\nC\nandUext\nCwith respect to Vexc\nH0(\u0014\u001c)andHext\nH0(~\u001c), respectively.\nThen by taking thermal average on them, these perturbative\nexpansions are described by the unperturbed Keldysh Green’s\nfunctions given by Eqs. (23) and (24) or\niG0\nC;\u000b\u000b0(x\u001c;x0\u001c0) =D\nTC\u0002\n H0\u000b(x\u001c) y\nH0\u000b0(x0\u001c0)\u0003E\n0;\niD0\nC(q\u001c;q0\u001c0) =D\nTC\u0002\naH0(q\u001c)ay\nH0(q0\u001c0)\u0003E\n0:\nWe perform the real-time projection to the above Keldysh\nGreen’s functions in order to calculate the physical observ-\nables. We do this by classifying whether the contour time \u001c\nbelongs to the path C\u0000orC+while\u001c0toC\u0000orC+(see Fig.\n3). We have four different conﬁgurations. To represent this\nsituation clearly, let us introduce a two-by-two-matrix Green’s\nfunction (Schwinger-Keldysh Green’s function) [62]\n^G\u000b\u000b0(x\u001c;x0\u001c0) =\u0012^G\u0000\u0000\n\u000b\u000b0(xt\u0000;x0t0\u0000)^G\u0000+\n\u000b\u000b0(xt\u0000;x0t0+)\n^G+\u0000\n\u000b\u000b0(xt+;x0t0\u0000)^G++\n\u000b\u000b0(xt+;x0t0+)\u0013\n;\n(B1)\nwheret\u0006andt0\u0006are both real times. The compo-\nnent ^G\u0016\u00160\n\u000b\u000b0(xt\u0016;x0t0\u00160) (\u0016;\u00160=\u0007)is representing that\nthe contour time \u001c=t\u0016belongs to the contour C\u0016\nwhile\u001c0=t0\u00160belongs to C\u00160. The components\n^G\u0000\u0000\n\u000b\u000b0(xt\u0000;x0t0\u0000);^G\u0000+\n\u000b\u000b0(xt\u0000;x0t0+);^G+\u0000\n\u000b\u000b0(xt+;x0t0\u0000);and\n^G++\n\u000b\u000b0(xt+;x0t0+)are equivalent to time-ordered, lesser,\ngreater, and anti-time-ordered components, respectively.\nSimilarly, we introduce the Schwinger-Keldysh Green’s\nfunction of magnon given by\n^D(q\u001c;q0\u001c0) =\u0012^D\u0000\u0000(qt\u0000;q0t0\u0000)^D\u0000+(qt\u0000;q0t0+)\n^D+\u0000(qt+;q0t0\u0000)^D++(qt+;q0t0+)\u0013\n;\n(B2)\nwhere the components ^D\u0000\u0000(qt\u0000;q0t0\u0000);^D\u0000+(qt\u0000;q0t0+),\n^D+\u0000(qt+;q0t0\u0000);and ^D++(qt+;q0t0+)are equivalent to\ntime-ordered, lesser, greater, and anti-time-ordered Green’s\nfunctions, respectively.\nIn the following, let us show some examples of calculation\nfor the real-time projection on the Keldysh Green’s functions.\nWe will just write the time arguments of functions and omit\nthe arguments of spatial coordinate or momentum since what\nwe want to demonstrate here is the calculation for real-time\nprojection and integrals of real-time variables. We perform\nthe integral along the contour Cby decomposing it into C\u0000\nandC+and rewrite them by the real-time variables.\nFor practice, ﬁrst let us show the simplest example of inte-\ngral along the contour Cgiven by a single contour-time vari-\nable\u001c1. It has a form\nf(\u001c;\u001c0) =Z\nCd\u001c1g(\u001c;\u001c1)h(\u001c1;\u001c0)\n=Z+1\n\u00001\u001c\u00161\u00161\nzdt\u00161\n1f(t;t\u00161\n1)g(t\u00161\n1;t0); (B3)21\nwhere\n\u001c\u00161\u00160\n1z =\u0012\n\u001c\u0000\u0000\nz\u001c\u0000+\nz\n\u001c+\u0000\nz\u001c++\nz\u0013\n=\u0012\n1 0\n0\u00001\u0013\n; (B4)\nandt\u00161\n1is the real-time variable. Via the real-time projec-\ntion, let us rewrite the function f(\u001c;\u001c0)asf\u0017\u0016\u00160(t;t0). Heretandt0are real-time variables corresponding to the real-time\nprojection of the contour times \u001cand\u001c0, respectively. The\nsuperscript\u0017\u0016\u00160= t;<;>; ~twith\u0016;\u00160=\u0006. It describes\nthe situation such that \u001c2C\u0016while\u001c02C\u00160. For instance,\nwhen\u001c2C\u0000while\u001c02C+the function f(\u001c;\u001c0)becomes\nf<(t;t0). In the following, we list the four cases of f(\u001c;\u001c0)\ngiven as\nft(t;t0) =Z+1\n\u00001dt1\u0000\ngt(t;t1)ht(t1;t0)\u0000g<(t;t1)h>(t1;t0)\u0001\n;\nf<(t;t0) =Z+1\n\u00001dt1\u0000\ngt(t;t1)h<(t1;t0)\u0000g<(t;t1)h~t(t1;t0)\u0001\n;\nf>(t;t0) =Z+1\n\u00001dt1\u0000\ng>(t;t1)ht(t1;t0)\u0000g~t(t;t1)h>(t1;t0)\u0001\n;\nf~t(t;t0) =Z+1\n\u00001dt1\u0000\ng>(t;t1)h<(t1;t0)\u0000g~t(t;t1)h~t(t1;t0)\u0001\n: (B5)\nBy using the relations in Eq. (A15), Eq. (B5) can be re- described by the lesser, greater, retarded, and advanced com-\nponents as\nf<(t;t0) =Z+1\n\u00001dt1\u0000\ng<(t;t1)ha(t1;t0) +gr(t;t1)h<(t1;t0)\u0001\n;\nf>(t;t0) =Z+1\n\u00001dt1\u0000\ng>(t;t1)ha(t1;t0) +gr(t;t1)h>(t1;t0)\u0001\n;\nfr(t;t0) =Z+1\n\u00001dt1gr(t;t1)hr(t1;t0); fa(t;t0) =Z+1\n\u00001dt1ga(t;t1)ha(t1;t0): (B6)\nNext, we present an example of temporal function represented\nby two contour-time variables \u001c1and\u001c2given by\nf(\u001c;\u001c0) =Z\nCd\u001c1d\u001c2g(\u001c;\u001c2)h(\u001c2;\u001c1)l(\u001c1;\u001c0): (B7)Like we did in Eq. (B5), we perform the real-time projection\non\u001c1and\u001c2and rewrite them as t1andt2, respectively. As a\nresult, we have\nf<(t;t0) =Z+1\n\u00001dt1dt2\u0000\ng<(t;t2)ha(t2;t1)la(t1;t0) +gr(t;t2)h<(t2;t1)la(t1;t0) +gr(t;t2)hr(t2;t1)l<(t1;t0)\u0001\n;\nf<(t;t0) =Z+1\n\u00001dt1dt2\u0000\ng>(t;t2)ha(t2;t1)la(t1;t0) +gr(t;t2)h>(t2;t1)la(t1;t0) +gr(t;t2)hr(t2;t1)l>(t1;t0)\u0001\n;\nfr(t;t0) =Z+1\n\u00001dt1dt2(gr(t;t2)hr(t2;t1)lr(t1;t0)); fa(t;t0) =Z+1\n\u00001dt1dt2(ga(t;t2)ha(t2;t1)la(t1;t0)):(B8)\nEqs. (B6) and (B8) are called Langreth rules [62, 63]. As a last example, we demonstrate a calculation represented22\nby three contour-time variables ~\u001c1;~\u001c2;and\u001c1. The integral\nwhich we calculate is\nf(\u001c;\u001c0) =Z\nCd~\u001c1d~\u001c2d\u001c1l(~\u001c1;\u001c1)m(\u001c1;~\u001c2)n(\u001c;\u001c1)o(\u001c1;\u001c0):\n(B9)With using the real-time variables ~t1,~t2, andt1corresponding\nto~\u001c1;~\u001c2;and\u001c1, respectively, the right-hand side of Eq. (B9)\nis rewritten as\nf<(t;t0) =Z+1\n\u00001d~t1d~t2dt1la(~t1;t1)mr(t1;~t2)\u0000\nn<(t;t1)oa(t1;t0) +nr(t;t1)o<(t1;t0)\u0001\n;\nf>(t;t0) =Z+1\n\u00001d~t1d~t2dt1la(~t1;t1)mr(t1;~t2)\u0000\nn>(t;t1)oa(t1;t0) +nr(t;t1)o>(t1;t0)\u0001\n;\nfr(t;t0) =Z+1\n\u00001d~t1d~t2dt1la(~t1;t1)mr(t1;~t2)nr(t;t1)or(t1;t0);\nfa(t;t0) =Z+1\n\u00001d~t1d~t2dt1la(~t1;t1)mr(t1;~t2)na(t;t1)oa(t1;t0): (B10)\n2. Impurity-Averaged Real-Time Green’s Function\nLet us apply the Keldysh Green’s function formalism to de-\nrive the retarded, advanced, lesser, and greater components ofimpurity-averaged real-time Green’s functions.\nThe Dyson equation for the Keldysh Green’s function of TI\nsurface state due to the non-magnetic impurity effect is given\nby [16, 62, 63]\nGC;\u000b\u000b0(x\u001c;x0\u001c0) =G0\nC;\u000b\u000b0(x\u001c;x0\u001c0) +Z\nCd\u001c1Z\nd2x1G0\nC;\u000b\u000b0\n1(x\u001c;x1\u001c1)Himp\n\u000b0\n1\u000b1(x1)GC;\u000b1\u000b0(x1\u001c1;x0\u001c0);\n=G0\nC;\u000b\u000b0(x\u001c;x0\u001c0) +Z\nCd\u001c1Z\nd2x1GC;\u000b\u000b0\n1(x\u001c;x1\u001c1)Himp\n\u000b0\n1\u000b1(x1)G0\nC;\u000b1\u000b0(x1\u001c1;x0\u001c0): (B11)\nWe use the formulas given in Eq. (B6) and perform the real-\ntime projection on the contour times \u001c;\u001c0and\u001c1in Eq. (B11).Then, we obtain the Dyson equations for retarded, advanced,\nlesser, and greater Green’s functions given by23\ngr\n\u000b\u000b0(xt;x0t0) =gr(0)\n\u000b\u000b0(xt;x0t0) +Z\ndt1d2x1Himp\n\u000b0\n1\u000b1(x1)gr(0)\n\u000b\u000b0\n1(xt;x1t1)gr\n\u000b1\u000b0(x1t1;x0t0)\n=gr(0)\n\u000b\u000b0(xt;x0t0) +Z\ndt1d2x1Himp\n\u000b0\n1\u000b1(x1)gr\n\u000b\u000b0\n1(xt;x1t1)gr(0)\n\u000b1\u000b0(x1t1;x0t0)\nga\n\u000b\u000b0(xt;x0t0) =ga(0)\n\u000b\u000b0(xt;x0t0) +Z\ndt1d2x1Himp\n\u000b0\n1\u000b1(x1)ga(0)\n\u000b\u000b0\n1(xt;x1t1)ga\n\u000b1\u000b0(x1t1;x0t0)\n=ga(0)\n\u000b\u000b0(xt;x0t0) +Z\ndt1d2x1Himp\n\u000b0\n1\u000b1(x1)ga\n\u000b\u000b0\n1(xt;x1t1)ga(0)\n\u000b1\u000b0(x1t1;x0t0)\ng<\n\u000b\u000b0(xt;x0t0) =g<(0)\n\u000b\u000b0(xt;x0t0) +Z\ndt1d2x1Himp\n\u000b0\n1\u000b1(x1)\u0010\ng<(0)\n\u000b\u000b0\n1(xt;x1t1)ga\n\u000b1\u000b0(x1t1;x0t0) +gr(0)\n\u000b\u000b0\n1(xt;x1t1)g<\n\u000b1\u000b0(x1t1;x0t0)\u0011\n=g<(0)\n\u000b\u000b0(xt;x0t0) +Z\ndt1d2x1Himp\n\u000b0\n1\u000b1(x1)\u0010\ng<\n\u000b\u000b0\n1(xt;x1t1)ga(0)\n\u000b1\u000b0(x1t1;x0t0) +gr\n\u000b\u000b0\n1(xt;x1t1)g<(0)\n\u000b1\u000b0(x1t1;x0t0)\u0011\ng>\n\u000b\u000b0(xt;x0t0) =g>(0)\n\u000b\u000b0(xt;x0t0) +Z\ndt1d2x1Himp\n\u000b0\n1\u000b1(x1)\u0010\ng>(0)\n\u000b\u000b0\n1(xt;x1t1)ga\n\u000b1\u000b0(x1t1;x0t0) +gr(0)\n\u000b\u000b0\n1(xt;x1t1)g>\n\u000b1\u000b0(x1t1;x0t0)\u0011\n=g>(0)\n\u000b\u000b0(xt;x0t0) +Z\ndt1d2x1Himp\n\u000b0\n1\u000b1(x1)\u0010\ng>\n\u000b\u000b0\n1(xt;x1t1)ga(0)\n\u000b1\u000b0(x1t1;x0t0) +gr\n\u000b\u000b0\n1(xt;x1t1)g>(0)\n\u000b1\u000b0(x1t1;x0t0)\u0011\n;\n(B12)\nwhere we have used the relations gt=gr+g<=ga+g>\nandg~t=\u0000gr+g>=\u0000ga+g<as given in Eq. (A15).\nBased on Eq. (B12), we derive the impurity-averaged Green’s\nfunctions for all of these four components. Basically, we can\nobtain them by adopting the same argument given in subSec.\nA 2. At ﬁrst, let us focus on the retarded component. Weexpressgr\n\u000b\u000b0(xt;x0t0)in the perturbative expansion form as\ngr\n\u000b\u000b0(xt;x0t0) =P\nngr(n)\n\u000b\u000b0(xt;x0t0)wheregr(n)\n\u000b\u000b0(xt;x0t0)is\nin then-th order ofHimp. Then we perform the Fourier trans-\nformations\ngr\n\u000b\u000b0(xt;x0t0) =1\nVX\nkk0Zd!d!0\n(2\u0019)2ei((kx\u0000k0x0)\u0000(!t\u0000!0t0))gr\n\u000b\u000b0(k!;k0!0);\ngr(n)\n\u000b\u000b0(xt;x0t0) =1\nVX\nkk0Zd!d!0\n(2\u0019)2ei((kx\u0000k0x0)\u0000(!t\u0000!0t0))gr(n)\n\u000b\u000b0(k!;k0!0): (B13)\nWe take the impurity average on gr\n\u000b\u000b0(k!;k0!0)as well as\ngr(n)\n\u000b\u000b0(k!;k0!0). Let us denote them as hgr\n\u000b\u000b0(k!;k0!0)iimp\nave\nandhgr(n)\n\u000b\u000b0(k!;k0!0)iimp\nave, respectively. As a result, both of\nthem become diagonal in momentum and frequency repre-sented as\nhgr\n\u000b\u000b0(k!;k0!0)iimp\nave= \u0016gr\n\u000b\u000b0(k!)\u000ek;k0\u0001(2\u0019\u000e(!\u0000!0));\nhgr(n)\n\u000b\u000b0(k!;k0!0)iimp\nave= \u0016gr(n)\n\u000b\u000b0(k!)\u000ek;k0\u0001(2\u0019\u000e(!\u0000!0)):\n(B14)\nConsequently, the impurity average of gr\n\u000b\u000b0(xt;x0t0)\nis represented in the translational invariant form:\nhgr\n\u000b\u000b0(xt;x0t0)iimp\nave= \u0016gr\n\u000b\u000b0(x\u0000x0;t\u0000t0). The Fourier trans-\nform of impurity-averaged Green’s function \u0016gr\n\u000b\u000b0(k!)is ob-\ntained from the two-point Green’s function \u0016gr\n\u000b\u000b0(x\u0000x0;t\u0000t0)\nas\n\u0016gr\n\u000b\u000b0(x\u0000x0;t\u0000t0) =1\nVX\nkZd!\n2\u0019ei(k(x\u0000x0)\u0000!(t\u0000t0))\u0016gr\n\u000b\u000b0(k!): (B15)24\nNext, we re-sum the perturbative expansion \u0016gr\n\u000b\u000b0(k!) =P\nn\u0016gr(n)\n\u000b\u000b0(k!)and express it in terms of the self-energy. Here\nwe take the ﬁrst-Born approximation to evaluate this as we did\nin subSec. A 2. As a result, we obtain\n\u0016gr\n\u000b\u000b0(k!) =gr(0)\n\u000b\u000b0(k!) +gr(0)\n\u000b\u000b2(k!)\u0006r(0)\n\u000b2\u000b1(k!)\u0016gr\n\u000b1\u000b0(k!)\n=gr(0)\n\u000b\u000b0(k!) + \u0016gr\n\u000b\u000b2(k!)\u0006r(0)\n\u000b2\u000b1(k!)gr(0)\n\u000b1\u000b0(k!);\n(B16)\nwhere the self-energy \u0006r(0)\n\u000b2\u000b1(k!)is given by\n\u0006r(0)\n\u000b2\u000b1(k!) =nimp\n~VX\nqjvimp(q\u0000k)j2gr(0)\n\u000b2\u000b1(q!):(B17)Formally, we can solve Eq. (B16) for \u0016gr\n\u000b\u000b0(k!)as\n\u0016gr\n\u000b\u000b0(k!) =\u0014\u0010\ngr(0)(k!)\u0011\u00001\n\u0000\u0006r(0)(k!)\u0015\u00001\n\u000b\u000b1\u0010\ngr(0)(k!)\u0011\u00001\n\u000b\u000b0= (!+!F+i\u0011)1\u000b\u000b0\u0000!TI\nk~H0;\u000b\u000b0:\n(B18)\nNext, let us evaluate the self-energy \u0006r(0)\n\u000b2\u000b1(k!)in Eq. (B17).\nIn order to do this, we only retain the term proportional to\n!+!F\u0000!TI\nq+i\u0011ingr(0)\n\u000b2\u000b1. As a result, we have\n\u0006r(0)\n\u000b2\u000b1(k!) =1\n2nimpv2\n0Z1\n0d\u000fN(\u000f)1\n\u000fF\u0000\u000f+i\u0011=1\n2nimpv2\n0Z1\n\u0000\u000fFd\u0018~N(\u0018)1\n\u0000\u0018+i\u0011\n\u0019\u00001\n2~N(0)nimpv2\n0Z1\n\u00001d\u0018\u0018+i\u0011\n\u00182+\u00112=\u00001\n2\u0019~N(0)nimpv2\n0=\u0000i\n2\u001crel\nTI\u000e\u000b2\u000b1; (B19)\nwhereN(\u000f) =\u000f=\u0000\n2\u0019(~vF)2\u0001\nis the density of states per\nvolume of TI surface state with \u000f=~!.\u0018=\u000f\u0000\u000fFand\n~N(\u0018) = (\u000fF+\u0018)=\u0000\n2\u0019(~vF)2\u0001\n:The time\u001crel\nTIin the above\nequation is the same quantity appearing in Eq. (A25). As we\nderived Eq. (A25), in the above analysis we considered that\nthe quantum tranport phenomena is induced by the electrons\nin the energy state in the vicinity of Fermi surface and the\nnumber density of TI surface to be sufﬁciently large. Thus,\nwe assumejqj;jkj'kF;\u000f\u001c\u000fFand set ~N(\u0018) = ~N(0)with\ntaking\u000fF!1 for the lower limit in the ﬁrst line of above\nequation. From Eqs. (B16) and (B19), we have the formula\nof\u0016gr\n\u000b\u000b0(k!)and it is the same as the one given in Eq. (25).\nBy applying the similar argument, we can derive the advanced\ncomponents of impurtiy-averaged Green’s function \u0016ga\n\u000b\u000b0(k!).\nLike Eq. (B16), we have\n\u0016ga\n\u000b\u000b0(k!) =ga(0)\n\u000b\u000b0(k!) +ga(0)\n\u000b\u000b2(k!)\u0006a(0)\n\u000b2\u000b1(k!)\u0016ga\n\u000b1\u000b0(k!)\n=ga(0)\n\u000b\u000b0(k!) + \u0016ga\n\u000b\u000b2(k!)\u0006a(0)\n\u000b2\u000b1(k!)ga(0)\n\u000b1\u000b0(k!);\n(B20)\nwhere\n\u0006a(0)\n\u000b2\u000b1(k!) =nimp\n~VX\nqjvimp(q\u0000k)j2ga(0)\n\u000b2\u000b1(q!):(B21)The self-energy \u0006a(0)\n\u000b2\u000b1(k!)in Eq. (B21) can be evaluated\nby the same analysis which we did exactly for deriving Eq.\n(B19). We have \u0006a(0)\n\u000b2\u000b1(k!) =\u0000\u0006r(0)\n\u000b2\u000b1(k!):We solve Eq.\n(B20) for \u0016ga\n\u000b\u000b0(k!)and obtain\n\u0016ga\n\u000b\u000b0(k!) =\u0014\u0010\nga(0)(k!)\u0011\u00001\n\u0000\u0006a(0)(k!)\u0015\u00001\n\u000b\u000b1\u0010\nga(0)(k!)\u0011\u00001\n\u000b\u000b0= (!+!F\u0000i\u0011)1\u000b\u000b0\u0000!TI\nk~H0;\u000b\u000b0;\n(B22)\nwhich is equal to the advanced Green’s function in Eq. (25).\nLastly, we derive the impurity-averaged lesser Green’s func-\ntion referring to the analysis in [16]. By the similar analysis\nused for deriving Eqs. (B16) or (B20), we obtain\n\u0016g<\n\u000b\u000b0(k!) =g<(0)\n\u000b\u000b0(k!) +g<(0)\n\u000b\u000b0\n2(k!)\u0006a(0)\n\u000b0\n2\u000b1(k!)\u0016ga\n\u000b1\u000b0(k!)\n+gr(0)\n\u000b\u000b0\n2(k!)h\n\u0006r(0)\n\u000b0\n2\u000b1(k!)\u0016g<\n\u000b1\u000b0(k!) + \u0006<(0)\n\u000b0\n2\u000b1(k!)\u0016ga\n\u000b1\u000b0(k!)i\n;\n(B23)25\nwhere\n\u0006<(0)\n\u000b2\u000b1(k!) =nimp\n~VX\nqjvimp(q\u0000k)j2g<(0)\n\u000b2\u000b1(q!):(B24)From Eqs. (B18), (B22), and (B24), Eq. (B23) is rewritten as\n\u0016g<\n\u000b\u000b0(k!) = \u0016gr\n\u000b\u000b2(k!)\u0006<(0)\n\u000b2\u000b1(k!)\u0016ga\n\u000b1\u000b0(k!)\n+ \u0016gr\n\u000b\u000b2(k!)\u0014\u0010\ngr(0)(k!)\u0011\u00001\n\u0001g<(0)(k!)\u0001\u0010\nga(0)(k!)\u0011\u00001\u0015\n\u000b2\u000b1\u0016ga\n\u000b1\u000b0(k!): (B25)\nFirst, let us evaluate the second term in Eq. (B25). From Eqs. (A16), (B18), and (B22) we have\n\u0014\u0010\ngr(0)(k!)\u0011\u00001\n\u0001g<(0)(k!)\u0001\u0010\nga(0)(k!)\u0011\u00001\u0015\n\u000b2\u000b1=f(~!)\u0014\u0010\ngr(0)(k!)\u0011\u00001\n\u0000\u0010\nga(0)(k!)\u0011\u00001\u0015\n\u000b2\u000b1\n= 2i\u0011f(~!)1\u000b2\u000b1: (B26)\nBy taking the limit \u0011!0+, we see that the second term in Eq. (B25) vanishes. Next, let us evaluate the ﬁrst term in Eq.\n(B25) using Eqs. (25) and (A16). This becomes\n\u0016gr\n\u000b\u000b2(k!)\u0006<(0)\n\u000b2\u000b1(k!)\u0016ga\n\u000b1\u000b0(k!) =f(~!) (\u0016ga\n\u000b\u000b0(k!)\u0000\u0016gr\n\u000b\u000b0(k!)): (B27)\nBy doing exactly the same analysis for the greater compo-\nnent starting from the Dyson equation (B12), as a result, the\nlesser and greater components of impurity-averaged Green’s\nfunctions are represented by the retarded and advanced com-\nponents as\n\u0016g<\n\u000b\u000b0(k!) =f(~!)(\u0016ga\n\u000b\u000b0(k!)\u0000\u0016gr\n\u000b\u000b0(k!));\n\u0016g>\n\u000b\u000b0(k!) =\u0000(1\u0000f(~!))(\u0016ga\n\u000b\u000b0(k!)\u0000\u0016gr\n\u000b\u000b0(k!)):(B28)Consequently, the impurity-averaged Green’s functions sat-\nisfy exactly the same relations with the ones for non-impurity-\naveraged Green’s functions presented in Eq. (A16).\n[1] M. Wallquist, K. Hammerer, P. Rabl, M. Lukin, and P. Zoller,\nPhys. Scr. T137 014001 (2009).\n[2] Z.-L. Xiang, S. Ashhab, J. Q. You, and F. Nori, Rev. Mod. Phys.\n85, 623 (2013).\n[3] G. Kurizki, P. Bertet, Y . Kubo, K. Mølmer, D. Petrosyan,\nP. Rabl, and J. Schmiedmayer, PNAS 112, 3866 (2015).\n[4] R. Hanson, L. P. Kouwenhoven, J. R. Petta, S. Tarucha,\nL. M. K. Vandersypen, Rev. Mod. Phys. 79, 1217 (2007).\n[5] W. A. Coish and J. Baugh, Phys. Status. Solidi. B 246, 2203\n(2009).\n[6] Y . Hirayama, G. Yusa, K. Hashimoto, N. Kumada, T. Ota, and\nK. Muraki, Semicond. Sci. Technol. 24, 023001 (2009).[7] E. A. Chekhovich, M. N. Kakhonin, A. I. Tartakovskii, A. Ya-\ncoby, H. Bluhm, K. C. Nowack, and L. M. K. Vandersypen,\nNature Mat. 12, 494 (2013).\n[8] R. Schirhagl, K. Chang, M. Loretz, and C. L. Degen, Ann. Rev.\nPhys. Chem. 65, 83 (2014).\n[9] D. Lee, K. W. Lee, J. V . Cady, P. Ovartchaiyapong, and\nA. C. Bleszynski Jayich, J. Opt. 19033001 (2017).\n[10] J. M. Raimond, M. Brune, and S. Haroche, Rev. Mod. Phys. 73,\n565 (2001).\n[11] H. Ritsch, P. Domokos, F. Brennecke, and T. Esslinger, Rev.\nMod. Phys. 85, 553 (2013).\n[12] G. Wendin, Rep. Prog. Phys. 80, 106001 (2017).26\n[13] P. Krantz, M. Kjaergaard, F. Yan, T. P. Orlando, S. Gustavsson,\nand W. D. Oliver, Appl. Phys. Rev. 6, 021318 (2019).\n[14] I. \u0014Zuti\u0013 c, J. Fabian, and S. Das. Sarma, Rev. Mod. Phys. 76, 323\n(2004).\n[15] Y . Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I. Halperin,\nRev. Mod. Phys. 77, 1375 (2005).\n[16] G. Tatara, H. Kohno, and J. Shibata, Phys. Rep. 468, 213\n(2008).\n[17] S. D. Bader, and S. S. P. Parkin, Annu. Rev. Condens. Matter\nPhys. 1, 71 (2010).\n[18] A. Brataas, A. D. Kent, and H. Ohno, Nature Mat. 11, 372\n(2012).\n[19] Y . Ohnuma, H. Adachi, E. Saitoh, and S. Maekawa, Phys. Rev.\nB89, 174417 (2014).\n[20] J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back, and\nT. Jungwirth, Rev. Mod. Phys. 87, 1213 (2015).\n[21] V . K. Josh, Engineering Science and Technology, an Interna-\ntional Journal, 19, 1503 (2016).\n[22] F. Hellman, A. Hoffmann, Y . Tserkovnyak, G. S. D. Beach,\nE. E. Fullerton, C. Leighton, A. H. MacDonald, D. C. Ralph,\nD. A. Arena, H. A. D  urr, P. Fischer, J. Grollier, J. P. Heremans,\nT. Jungwirth, A. V . Kimel, B. Koopmans, I. N. Krivorotov,\nS. J. May, A. K. Petford-Long, J. M. Rondinelli, N. Samarth,\nI. K. Schuller, A. N. Slavin, M. D. Stiles, O. Tchernyshyov,\nA. Thiaville, B. L. Zink, Rev. Mod. Phys. 89, 025006 (2017).\n[23] V . Baltz, A. Manchon, M. Tsoi, T. Moriyama, T. Ono, and\nY . Tserkovnyak, Rev. Mod. Phys. 90, 015005 (2018).\n[24] W. Han, Y . Otani, and S. Maekawa, npj. Quantum Materials.\n(2018) 3:27 ; doi:10.1038/s41535-018-0100-9.\n[25] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010).\n[26] X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011).\n[27] Y . Ando, J. Phys. Soc. Jpn. 82, 102001 (2013).\n[28] M. Franz and L. Molenkamp, Topological Insulators Volume\n6:Contemporary Concepts of Condensed Matter Science (El-\nsevier, Oxford, 2013).\n[29] T. Yokoyama, Y . Tanaka, and N. Nagaosa, Phys. Rev. B 81,\n121401(R) (2010); T. Yokoyama, J. Zang, and N. Nagaosa,\nPhys. Rev. B 81, 241410(R) (2010).\n[30] K. Nomura and N. Nagaosa, Phys. Rev. B 82, 161401(R)\n(2010); Phys. Rev. Lett. 106, 166802 (2011).\n[31] I. Garate and M. Franz, Phys. Rev. Lett. 104, 146802 (2010).\n[32] W.-K. Tse and A. H. MacDonald, Phys. Rev. Lett. 105, 057401\n(2010); Phys. Rev. B 84, 205327 (2011).\n[33] J. G. Checkelsky, J. Ye, Y . Onose, Y . Iwasa, and Y . Tokura, Nat.\nPhys. 8729 (2012).\n[34] J. Henk, M. Flieger, I. V . Maznichenko, I. Mertig, A. Ernst,\nS. V . Eremeev, and E. V . Chulkov, Phys. Rev. Lett. 109, 076801\n(2012).\n[35] Y . Tserkovnyak and D. Loss, Phys. Rev. Lett. 108, 187201\n(2012).\n[36] C.-Z. Chang, J. Zhang, X. Feng, J. Shen, Z. Zhang,1 M. Guo,\nK. Li, Y . Ou, P. Wei, L.-L. Wang, Z.-Q. Ji, Y . Feng, S. Ji,\nX. Chen, J. Jia, X. Dai, Z. Fang, S.-C. Zhang, K. He, Y . Wang,\nL. Lu, X.-C. Ma, Q.-K. Xue, Science 340, 167 (2013).\n[37] P. Wei, F. Katmis, B. A. Assaf, H. Steinberg, P. Jarillo-Herrero,\nD. Heiman, and J. S. Moodera, Phys. Rev. Lett. 110, 186807\n(2013).\n[38] A. R. Mellnik, J. S. Lee, A. Richardella, J. L. Grab, P. J. Mintun,\nM. H. Fischer, A. Vaezi, A. Manchon, E. -A. Kim, N. Samarth,\nand D. C. Ralph, Nature (London) 511, 449 (2014).\n[39] Y . Fan, P. Upadhyaya, X. Kou, M. Lang, S. Takei, Z. Wang,\nJ. Tang, L. He, L.-T. Chang, M. Montazeri, G. Yu, W. Jiang,\nT. Nie, R. N. Schwartz, Y . Tserkovnyak, and K. L. Wang, Nat.\nMater. 13, 699 (2014).[40] P. Deorani, J. Son, K. Banerjee, N. Koirala, M. Brahlek, S. Oh,\nand H. Yang, Phys. Rev. B 90, 094403 (2014).\n[41] F. Mahfouzi, N. Nagaosa, and B. K. Nikoli \u0013c, Phys. Rev. B 90,\n115432 (2014).\n[42] Y . Shiomi, K. Nomura, Y . Kajiwara, K. Eto, M. Novak,\nK. Segawa, Y . Ando, and E. Saitoh, Phys. Rev. Lett. 113,\n196601 (2014).\n[43] A. Sakai and H. Kohno, Phys. Rev. B 89, 165307 (2014).\n[44] K. Taguchi, K. Shintani, and Y . Tanaka, Phys. Rev. B 92,\n035425 (2015).\n[45] A. A. Baker, A. I. Figueroa, L. J. Collins-Mclntyre, G. van der-\nLaan, and T. Hesjedal, Sci. Rep. 5, 7907 (2015).\n[46] I. Lee, C-K. Kima, J. Lee, S. J. L. Billinge, R. Zhong,\nJ. A. Schneeloch, T. Liu, T. Valla, J. M. Tranquada, G. Gu, and\nJ. C. S. Davis, PNAS 112, 1316-1321 (2015).\n[47] X. Kou, Y . Fan, M. Lang, P. Upadhyaya, K. L. Wang, Solid\nState Commun. 215-216 , (2015) 34.\n[48] M. Mogi, R. Yoshimi, A. Tsukazaki, K. Yasuda, Y . Kozuka,\nK. S. Takahashi, M. Kawasaki, and Y . Tokura, Appl. Phys. Lett.\n107, 182401 (2015).\n[49] C.-Z. Chang, W. Zhao, D. Y . Kim, H. Zhang, B. A. As-\nsaf, D. Heiman, S.-C. Zhang, C. Liu, M. H. W. Chan, and\nJ. S. Moodera, Nat. Mater. 14, 473 (2015).\n[50] S. Grauer, S. Schreyeck, M. Winnerlein, K. Brunner, C. Gould,\nand L. W. Molenkamp, Phys. Rev. B 92, 201304(R) (2015).\n[51] M. Jamali, J. S. Lee, J. S. Jeong, F. Mahfouzi, Y . Lv, Z. Zhao,\nB. K. Nikoli \u0013c, K. A. Mkhoyan, N. Samarth, J.-P. Wang, Nano.\nLett. 15, 7126 (2015).\n[52] J.-C. Rojas-S \u0013anchez, S. Oyarz \u0013un, Y . Fu, A. Marty, C. Vergnaud,\nS. Gambarelli, L. Vila, M. Jamet, Y . Ohtsubo, A. Taleb-\nIbrahimi, P. Le F \u0012evre, F. Bertran, N. Reyren, J.-M. George, and\nA. Fert, Phys. Rev. Lett. 116, 096602 (2016).\n[53] C.-Z. Chang and M. Li, J. Phys.: Condens. Matter 28(2016)\n123002.\n[54] K. Kondou, R. Yoshimi, A. Tsukazaki, Y . Fukuma, J. Matsuno,\nK. S. Takahashi, M. Kawasaki, Y . Tokura, and Y . Otani, Nat.\nPhys. 121027 (2016).\n[55] M. H. Fischer, A. Vaezi, A. Manchon, and E.-A. Kim, Phys.\nRev. B 93, 125303 (2016); P. B. Ndiaye, C. A. Akosa,\nM. H. Fischer, A. Vaezi, E.-A. Kim, and A. Manchon, Phys.\nRev. B 96, 014408 (2017).\n[56] M. Mogi, M. Kawamura, A. Tsukazaki, R. Yoshimi, K. S. Taka-\nhashi, M. Kawasaki, and Y . Tokura, Sci. Adv. 3, eeao1667\n(2017).\n[57] N. H. D. Khang, Y . Ueda, and P. N. Hai, Nat. Mater. 17, 808\n(2018).\n[58] C. Tang, Q. Song, C.-Z Chang, Y . Xu, Y . Ohnuma, M. Matsuo,\nY . Liu, W. Yuan, Y . Yao, J. S. Moodera, S. Maekawa, W. Han,\nJ. Shi, Sci. Adv. 4, eeao8660 (2018).\n[59] H. Wang, J. Kally, C. Sahin, T. Liu, W. Yanez, E. J. Kamp,\nA. Richardella, M. Wu, M. E. Flatt \u0013 e, and N. Samarth, Phys.\nRev. Reserch 1, 012014(R) (2019).\n[60] R. J. Cava, H. Ji, M. K. Fuccillo, Q. D. Gibson, and Y . S. Hor,\nJ. Mater. Chem. B, 2013, 1, 3176.\n[61] A. L. Fetter and J. D. Walecka, Quantum Theory of Many-\nParticle Stystems , (Dover Publications, New York, 2003).\n[62] J. Rammer and H. Smith, Rev. Mod. Phys. 58, 323 (1986);\nJ. Rammer, Quantum Field Theory of Non-Equilibrium States ,\n(Cambridge University Press, New York, 2007).\n[63] R. A. Jishi, Feynman Diagram Techniques in Condensed Mat-\nter Physics , (Cambridge University Press, New York, 2013).\n[64] H. Bruus and K. Flensberg, Many-Body Quantum Theory in\nCondensed Matter Physics, An Introduction , (Oxford Univer-\nsity Press, New York, 2004).27\n[65] G. D. Mahan, Many-Particle Physics , (Third Edition, Springer\nScience+Business Media, New York, 2000).\n[66] K. Nomura and A. H. MacDonald, Phys. Rev. Lett. 96, 256602(2006); Phys. Rev. Lett. 98, 076602 (2007).\n[67] Y . Araki and K. Nomura, Phys. Rev. B 96, 165303 (2017).\n[68] V .-H. Nguyen, B.-H. Nguyen and N. D. Dinh, Adv. Nat. Sci.:\nNanosci. Nanotechnol 7(2016) 015013."    },    {        "title": "1501.01171v1.Sensing_magnetic_nanoparticles_using_nano_confined_ferromagnetic_resonances_in_a_magnonic_crystal.pdf",        "content": "Sensing magnetic nanoparticles using nano-conﬁned ferromagnetic\nresonances in a magnonic crystal\nPeter J. Metaxas,\u0003a‡Manu Sushruth,a‡Ryan A. Begley,aJunjia Ding,bxRobert C. Woodward,aIvan\nS. Maksymov,aMaximilian Albert,cWeiwei Wang,cHans Fangohr,cAdekunle O. Adeyeyeband\nMikhail Kostyleva\nWe demonstrate the use of the magnetic-ﬁeld-dependence\nof highly spatially conﬁned, GHz-frequency ferromagnetic\nresonances in a ferromagnetic nanostructure for the detec-\ntion of adsorbed magnetic nanoparticles. This is achieved\nin a large area magnonic crystal consisting of a thin fer-\nromagnetic ﬁlm containing a periodic array of closely\nspaced, nano-scale anti-dots. Stray ﬁelds from nanopar-\nticles within the anti-dots modify resonant dynamic mag-\nnetisation modes in the surrounding magnonic crystal,\ngenerating easily measurable resonance peak shifts. The\nshifts are comparable to the resonance linewidths for high\nanti-dot ﬁlling fractions with their signs and magnitudes\ndependent upon the modes’ localisations (in agreement\nwith micromagnetic simulation results). This is a highly\nencouraging result for the development of frequency-\nbased nanoparticle detectors for high speed nano-scale\nbiosensing.\nMagnetic biosensors, in which biological analytes are tagged\nwith magnetic nanoparticles (MNPs), have excellent potential\nfor solid-state point-of-care medical diagnostics1–3. The tech-\nnique is intrinsically matrix-insesntive1, can compete with\nindustry-standard immunoassays4and can be combined with\nmagnetic separation methods5. The central element of a mag-\nnetic biosensor is a detector for the stray or ‘fringing’ mag-\nnetic ﬁelds generated by magnetised MNPs which are used\nto label, typically in-vitro , analytes of interest within a bi-\nological sample. Previously used sensors include SQuIDs6,\nHall sensors7, ferromagnetic rings8,9and magneto-impedance\ndevices10. However one of the most widely used methods\nis that employing magnetoresistive (MR) magnetic ﬁeld sen-\nsors1–5,11–14which are typically fabricated with at least one\naSchool of Physics, M013, University of Western Australia, 35 Stirling Hwy,\nCrawley WA 6009, Australia. Fax: +61 8 6488 7364; Tel: +61 8 6488 7015;\nE-mail: peter.metaxas@uwa.edu.au\nbInformation Storage Materials Laboratory, Department of Electrical and\nComputer Engineering, National University of Singapore, Singapore-117576,\nSingapore.\ncEngineering and the Environment, University of Southampton, Southamp-\nton, SO17 1BJ, United Kingdom.\nxPresent address: Argonne National Laboratory, 9700 S. Cass Avenue, Ar-\ngonne, IL 60439, USA.\n‡These authors contributed equally to this work.lateral dimension on the order of 10-100 µm. An MNP is de-\ntected when its stray (or ‘fringing’) magnetic ﬁeld modiﬁes\nthe quasi-static magnetic conﬁguration in the ferromagnetic\nMR device. This changes the device’s resistance, enabling\nelectronic MNP detection.\nIt can however be challenging to minimise noise in conven-\ntional MR sensors when reducing the sensor size due to ther-\nmal instabilities of the device’s magnetic conﬁguration15–17.\nA suggested approach to overcome this is to use intrinsically\nhigh frequency detection methods exploiting the strong ﬁeld\ndependence of resonant magnetisation dynamics18–20which\ncan be reliably driven in isolated nanostructures. These dy-\nnamics can be driven electrically in spin torque oscillators\n(e.g.18,21–24and refs. therein) which have been predicted to\nretain high ﬁeld sensing signal to noise ratios at sub-100 nm\ndimensions18,19. Furthermore, real time electrical detection of\nthe dynamics25–27will pave the way for high speed19, nano-\nscale MNP sensing for solid-state ﬂow cytometry28–31.\nIn this work we use a large area32,33magnonic crystal\n(MC)34to macroscopically probe resonant GHz-frequency\nmagnetisation dynamics which are spatially conﬁned to nano-\nmetric regions and experimentally demonstrate their use for\nMNP sensing. In contrast to continuous ferromagnetic layers,\nnanostructured MCs (e.g. Fig. 1a) exhibit a number of distinct\nferromagnetic resonance (FMR) modes with different lateral\nspatial localisations within the crystal’s nano-periodic struc-\nture35–37. A previous study demonstrated that FMR modes\nwithin anti-dot-based MCs are sensitive to magnetic nanos-\ntructures fabricated within the anti-dots, leading to a pro-\nposal for a magnonic biosensor to detect captured MNP-based\ntags38. Here we demonstrate that stray magnetic ﬁelds gen-\nerated by captured MNPs within an anti-dot-based MC do in-\ndeed generate clear resonance peak shifts which are mode-\ndependent and which approach the resonance linewidth when\nthe fraction of MNP-ﬁlled anti-dots is high. Note that reso-\nnances within MNPs can be be detected directly however only\nbroad, relatively weak signals have been observed previously\nfor small collections of MNPs39. Rather, in this work, we\ndetect MNP-induced changes to very well deﬁned resonances\nwithin a periodically nanostructured, high quality ferromag-\nnetic ﬁlm.arXiv:1501.01171v1  [cond-mat.mes-hall]  6 Jan 2015Fig. 1 (a) Scanning electron micrograph of a portion of a magnonic\ncrystal (MC). The scale bar is 1 µm long. (b) Schematic of the\nexperimental setup showing the MC placed face down above a\nmicro-stripline.\nLarge area MCs (4 \u00024 mm2) consisting of arrays of 0.3\nµm wide anti-dots with edge-to-edge spacings of 0.15 µm\nwere fabricated in a 30 nm thick Ni 80Fe20ferromagnetic layer\nusing deep ultraviolet lithography32,33(Fig. 1a). The MCs’\nFMR modes were probed at room temperature using broad-\nband, stripline-based, magnetic ﬁeld modulated FMR spec-\ntroscopy (Fig. 1b). This technique measures the derivative of\nﬁnite width FMR peaks with respect to the external magnetic\nﬁeld, Hext, at a ﬁxed GHz frequency. MC spectra are measured\nboth before and after the addition of cluster-shaped MNPs\n(diameters \u00180:1\u00000:3µm). Two micromagnetic simulation\nmethods have also been used: (i) a time domain (‘ringdown’)\nsimulation using MuMax340which subjects the MC to a ﬁeld\npulse and exploits Fourier analysis on the resulting dynamics\nto extract frequency-resolved information (e.g.41), and; (ii) an\neigenmode method (e.g.42) which directly calculates the sys-\ntem’s resonant modes and mode proﬁles for a given Hext(car-\nried out using FinMag which is the successor to Nmag43). See\nthe supplementary information for additional details.\nAn experimental FMR spectrum for a bare MC (ie. no ad-\nsorbed MNPs) obtained at 11.5 GHz is shown in Fig. 2a in\nwhich two high amplitude resonances near the extremes of\nthe measured Hextrange can be identiﬁed. The frequency\n(f) dependence of the resonance ﬁelds, Hres, of these two\nFMR modes compare well with those predicted from time do-\nmain simulations for the extended (E) and side (S) modes35–37\n(Fig. 2b). In Fig. 2c we show Fourier transformed data from\nthe time domain simulation obtained at µ0Hext=200 mT,\nshowing the E and S modes together with their spatial locali-\nsations inside the MC’s unit cell. The E mode is concentrated\nin bands between rows of anti-dots oriented perpendicular to\nHextwhile the S mode is localised between neighboring anti-\ndots (see also the schematic in Fig. 2d). In both simulation and\nexperiment, a number of lower amplitude modes lie between\nthe side and extended modes, demonstrating good agreement\nin terms of the overall mode structure however these modes\nwill not be discussed in this communication.\nMNPs were adsorbed onto the MCs at Hext=0 by plac-\ning 12 µL of diluted MNP suspension on the MC’s surface\nwhich then was allowed to dry in ambient conditions be-\nFig. 2 (a) Experimental, ﬁeld-resolved FMR trace ( f=11:5 GHz)\nshowing differential absorption peaks corresponding to FMR modes\nin the MC. (b) Comparison of experimental and simulated resonant\nfrequencies versus µ0Hextfor the side (S) and extended (E) modes.\n(c) Frequency-resolved, Fourier transformed time domain simulation\ndata at µ0Hext=200 mT for a single unit cell of the MC with and\nwithout a 150 nm MNP at the centre of the anti-dot. The Fourier\namplitude has been differentiated with respect to fto facilitate\ncomparison with experimental spectra. Insets show the localisation\nof the resonant dynamics for the S and E modes with lighter shading\nindicating a stronger dynamic magnetisation component\nperpendicular to Hext. The anti-dot boundary is shown as a blue\ncircle. (d) Schematic showing the spatial localisation of the E and S\nmodes together with a MNP and a sketch of its stray magnetic ﬁeld.Fig. 3 (a) SEM image of MNPs on the MC with a 300 nm wide\nscale bar. Out of 33 holes counted, 30 contain MNPs. In 7 of these,\nthe ‘captured’ MNP extends outside of the anti-dot. There are 5\nisolated nanoparticles on the MC’s upper surface. FMR traces\nobtained at 11.5 GHz showing the side mode (b) and extended mode\n(c) before (‘bare’) and after the addition of MNPs. Multiple traces\nhave been taken following repeated removal and replacement of the\nMC, a requirement for MNP application, enabling an estimation of\nthe associated experimental uncertainty in ( \u00180:5 mT). Traces have\nbeen vertically scaled and vertically shifted to locally normalize the\ndifferential absorption signals to that obtained with a bare MC.\nfore re-measurement. Scanning electron microscopy of dried\nMNPs on a MC reveals irregularly shaped MNPs with the ma-\njority lying inside the anti-dots (see Fig. 3a and caption). The\nMNPs generate an upward shift in Hresfor the side mode res-\nonance (Fig. 3b) and a downward shift for the extended mode\nresonance (Fig. 3c). Both shifts exceed experimental uncer-\ntainty (see caption of Fig. 3).\nThe observed shifts can be qualitatively understood by con-\nsidering the ﬁeld generated by an idealized magnetized MNP,\nlocated at the center of an anti-dot and magnetized along Hext\n(Fig. 2d). Treating the MNP as a dipole, the y-component\nof its stray magnetic ﬁeld will, to a ﬁrst approximation, rein-\nforce Hextat E mode region but oppose Hextat the S mode re-\ngion. Thus, for a given experimental measurement frequency,\na larger Hextmust be applied to attain the side mode reso-\nnance condition when MNPs are within the anti-dots. Simi-\nlarly, the extended mode will be observed at a lower external\nﬁeld. These predictions are consistent with the experimental\nresults (Fig. 3b,c) as well as analogous numerical results ob-\ntained for anti-ring structures38.\nTo verify these arguments more rigorously, simulations\nwere repeated with a 150nm wide spherical MNP within the\nanti-dot (see supplementary information for further details in-\ncluding the Hext-dependent MNP moment). The time domain\nsimulation result for the MC in the presence of a MNP with\nµ0Hext=200 mT is shown as a red dotted line in Fig. 2c. The\nS mode’s frequency is indeed decreased, consistent with thatpart of the MC experiencing a lower net ﬁeld. Likewise, an\nincreased resonance frequency is predicted for the extended\nmode. To extract numerical values for these shifts, eigenmode\nsimulations were carried out at both µ0Hext=37 mT (to probe\nthe side mode as per Fig. 3b) and at 206 mT (to probe the ex-\ntended mode as per Fig. 3c). This yielded MNP-induced fre-\nquency shifts of +0.311 GHz for the E mode and -0.085 GHz\nfor the S mode, in good agreement with time domain simu-\nlations. The local gradient of the fresvs.Hresdata (Fig. 2b)\nwas then used to convert the frequency shifts into equivalent\nﬁeld shifts. This yielded \u00006:9 mT for the extended mode (45\nGHz/T) and +2:0 mT for the side mode (43 GHz/T). Although\nthe observed shift will depend on the MNP coverage, these\nsimulations which assume one 150 nm wide MNP per anti-\ndot correctly predict the order of magnitude and sign of the\nresonance ﬁeld shift (Fig. 3 where, albeit, not every hole is\nﬁlled and there is a distribution of MNP sizes).\nTo study MNP coverage effects, we measured a second MC\nwith an equivalent anti-dot lattice geometry. We carried out\nconsecutive applications of diluted solutions of MNPs with in-\ncreasing concentration, c, obtaining FMR traces and imaging\nthe MC via SEM before and after the application of each solu-\ntion. Representative SEM images are shown in Figs. 4a-d for\neach concentration. Increases in cvisibly increase the MNP\ncoverage, resulting in an increased peak shift for both the ex-\ntended and side modes. For the lowest coverage, a shift of the\nextended mode is measurable and just higher than the experi-\nmental uncertainty (Fig. 4e). At the highest coverage however\nthe resonance peak shift is much larger and approaches the\npeak-to-peak resonance linewidth. There was some variation\nin the measured shifts for different fhowever no clear, widely\napplicable trends could be determined (except at low ffor the\nside mode, and thus low Hext, where a smaller shift presum-\nably resulted from a lower Hext-induced MNP moment). In\nFigure 4f, we have averaged the S and E modes’ peak shifts\nover the measured frequency range (11 :5\u000016 GHz) and plot-\nted the averaged shifts versus c. Reliably measurable shifts\nof the E mode are observed over the entire frequency range.\nHowever, a higher concentration must be used to register a\nconsistent shift for the side mode. Notably, the simulated\nshifts of +2 mT for the S mode and \u00006:9 mT for the E mode\ncompare well with the shifts observed at c=0:225 µg/µL\nwhere we are close to having every anti-dot ﬁlled with a MNP\n(Fig. 4d)) and thus closest to the simulation condition. The\nMNPs also generate a coverage-dependent linewidth increase\nwith a\u00181.5 times increase on average in the peak-to-peak res-\nonance linewidths at the highest c(Fig. 4g). However, the\nlinewidth broadening does not dominate the observed shifts\nin that the minimum of the differential absorption line consis-\ntently moves in the direction of the peak shift (Fig. 4e). This\nis qualitatively consistent with that expected for a collection\nof differently sized MNPs all acting in unison with the shiftmagnitude depending upon the size of the MNP.\nFig. 4 (a-d) 2.25 µm wide images of the MC following consecutive\napplications of aqueous MNP solutions of increasing concentrations,\nc(µg/µL). (e) Differential absorption peak for the extended mode at\n12 GHz in the bare MC and following adsorption of MNPs for\nincreasing values of c. Extended and side mode peak shifts (f) and\nrelative peak-to-peak linewidth increase (g) as a function of MNP\nsolution concentration. Error bars combine both the uncertainty in\nthe shift measurement at each frequency and the spread of peak\nshifts over the measured frequency range (11.5-16 GHz).\nConclusions\nWe have used an anti-dot based magnonic crystal (MC) to\nexperimentally demonstrate the ability to detect magnetic\nnanoparticles (MNPs) via their inﬂuence on spatially con-\nﬁned, high frequency, ferromagnetic resonance modes. MNPs\nare preferentially captured within the holes which, depend-\ning upon a mode’s spatial localisation, leads to an increased\nor decreased resonant frequency. Resonance shifts are re-\nproduced well by micromagnetic simulations and observable\neven for quite low anti-dot ﬁllings ( \u001815%). A non-dominant\nlinewidth broadening is observed at high MNP coverages. Our\nresults are directly applicable to conﬁned modes in isolatedspintronic18or magnonic44nanostructures. This is encour-\naging for the development of frequency-based spintronic de-\nvices such as spin torque oscillators for nano-scale magnetic\nbiosensing18in applications such as ﬂow cytometry30,31. No-\ntably our observed frequency shifts are signiﬁcantly larger\nthan measured spin torque oscillator linewidths27,45,46how-\never electrical detection will rely on close proximity of the\nMNP to the sensing layer and high GHz/T ﬁeld sensitivities.\nAcknowledgements\nResearch supported by the Australian Research Council’s Dis-\ncovery Early Career Researcher Award (DE120100155) and\nDiscovery Projects scheme (DP110103980), the University\nof Western Australia’s (UWA) RCA, ECRFS, SIRF, UPAIS,\nVacation Scholarship, Teaching Relief and post-doctoral fel-\nlowship schemes, an EPSRC DTC grant (EP/G03690X/1)\nand iVEC through the use of advanced computing resources\nlocated at iVEC@UWA. A.O.A. was supported by the Na-\ntional Research Foundation, Prime Minister’s Ofﬁce, Singa-\npore under its Competitive Research Programme (CRP Award\nNo. NRF-CRP 10-2012-03). The authors thank C. Lu-\neng, D. Turner, A. Suvorova, A. Vansteenkiste, A. Dodd,\nM. House, T.G. St. Pierre, N. Kostylev, C. Bording and J. Izaac\nfor useful discussions, advice and/or assistance. The authors\nacknowledge access to the UWA’s Biomagnetics Wet Labora-\ntory and Magnetic Characterisation Facility as well as the fa-\ncilities, and the scientiﬁc and technical assistance of the Aus-\ntralian Microscopy & Microanalysis Research Facility at the\nCentre for Microscopy, Characterisation & Analysis, The Uni-\nversity of Western Australia, a facility funded by the Univer-\nsity, State and Commonwealth Governments.\nSupplementary information\nMagnonic crystals and magnetic nanoparticles\nThe MCs were composed of 30 thick nm Ni 80Fe20layers\ncovered by an Au capping layer (8 nm thick for the data in\nManuscript Figs. 2 and 3 and 10 nm thick for the data in\nManuscript Fig. 4). They were fabricated on Si substrates\nusing deep ultraviolet lithography with deposition via elec-\ntron beam evaporation followed by lift-off32,33. The anti-\ndot lattice geometries were measured using scanning elec-\ntron microscopy (SEM) with antidot diameters and array pitch\nrounded to the nearest 10 nm. MNPs were nanomag-D iron-\noxide nanoparticles (micromod Partikeltechnologie GmbH)\nwith a colloidally stabilized dextran surface, a quoted MNP\nwidth of 130 nm and a quoted solids content of 25 mg/mL.\nThe latter was used to calculate the concentrations of the di-\nluted MNP solutions (diluted using e-pure water). Magne-\ntometry on freeze dried MNPs was carried out at 300K usinga MPMS3 SQuID magnetometer (Quantum Design Inc.) in\nVSM mode. The magnetic moment per unit volume (Fig. 5)\nwas calculated assuming an iron-oxide density of 5.24 g/cm3.\nSEM was carried out with a Zeiss 1555 VP-FESEM and a FEI\nVerios 460 SEM.\nFig. 5 [Supplementary ﬁgure] Magnetic moment per unit volume\nfor the MNPs at 300K.\nFerromagnetic resonance spectroscopy\nFMR modes were probed at room temperature using broad-\nband, stripline-based, magnetic ﬁeld modulated FMR spec-\ntroscopy (Manuscript Fig. 1b). During measurement, the\nMC’s (01) axis was closely aligned with stripline which it-\nself was aligned with Hext, the latter modulated at 220 Hz.\nThe technique uses a interferometric receiver47and lock-in\nampliﬁer (SR830, Stanford Research Systems) to measure the\nexternal magnetic ﬁeld ( Hext-)derivative of ﬁnite width fer-\nromagnetic resonances in the sample at a set frequency, f,\nand stepped Hext. To obtain the FMR traces, Hextwas in-\ncreased in steps over a range typically spanning \u00180\u0000250\nmT with Hextmeasured at each step using a FH54 Teslame-\nter (Magnet-Physik Dr. Steingroever GmbH). The resultant\nFMR spectra were measured both before and after the addi-\ntion of cluster-shaped MNPs to the MC’s surface. The MCs\nare characterised by a non-zero remanent magnetisation with\nthe stepped Hextsweeps for the measurements carried out for\na single ﬁeld polarity. Thus, the bias ﬁeld and the spatially av-\neraged y-component of the MC’s magnetic moment remained\naligned during the FMR experiments. A microscope cover-\nslip between the MC and the stripline was used in all mea-\nsurements to avoid MNPs, when present, rubbing off onto the\nstripline. A PVC block placed on the stripline board with-\nout contact to the stripline itself was used to ensure consistent\nplacement of the sample with respect to the stripline. This\nenabled excellent reproducibility even with repeated removals\nand replacements of the MC. Reproducibility was conﬁrmedfor each measurement with the associated uncertainty in the\npeak position typically being on the order of 0.5 mT. This can\nbe seen in Manuscript Figs. 3b,c where we show traces ob-\ntained after repeated removals and replacements of the sam-\nple. Upon adding MNPs to the MC, we consistently observed\nan overall decrease in the signal amplitude which was stronger\nthan the MNP-induced linewidth broadening effects. This sig-\nnal amplitude reduction increased with MNP concentration,\nsuggesting a broadband absorption of the microwave power\nby the MNPs. To consistently compare data obtained with and\nwithout MNPs, we vertically scaled the FMR data so that all\ntraces had the same peak-to-peak amplitude. It was also some-\ntimes necessary to introduce a small vertical offset, typically\non the order of a few tens of µV at most.\nMicromagnetic simulations\nTwo micromagnetic simulation methods were used in this\nwork. Both methods simulated a single unit cell of the MC\nwith periodic boundary conditions in the xandydirections, an\n11\u000211 ‘tiled’ macro-geometry48for determining the demag-\nnetizing ﬁeld and the following magnetic parameters: damp-\ninga=0:008, nil intrinsic anisotropy, saturation magnetiza-\ntionMS=8\u0002105A/m, gyromagnetic ratio 2 pg=1:85\u00021011\nrad/T.s and exchange stiffness Aex=13 pJ/m. These values\nofMSandgwere consistent with results from FMR measure-\nments on continuous layers averaged over two reference sam-\nples. The value of gis also close to that determined by Shaw\net al49. MNPs were modeled explicitly assuming a ferromag-\nnetic sphere sitting within the anti-dot with its lower surface\naligned with the lower surface of the MC. For the MNP, we\nused a=0:05,Aexandgequal to that in the MC. MSwas\nread off from Fig. 5 at the value of Hextused in the simulation.\nIPython50, Sumatra51and matplotlib52were used for analy-\nsis, management and visualisation of the simulation data.\nTime domain (ringdown) simulations Time domain or\n‘ringdown’ simulations (e.g. Grimsdith et al.41) were carried\nout using MuMax340version 3.5.3 with cuboid discretisation\ncells (\u00193:52\u00023:52\u00023:75 nm3) wherein the system’s equi-\nlibrium magnetic conﬁguration, m0(r;Hext), at a given Hext,\napplied in the y-direction, is perturbed with a ﬁeld pulse in\nthex-direction (0.5 mT sinc pulse53with a 300 ps offset and\n30 GHz cut-off frequency). Fourier analysis was then applied\nto the time dependent, spatially averaged x-component of the\nmagnetization to extract the characteristic (resonant) frequen-\ncies associated with the resultant excited dynamics. Mode vi-\nsualisations (insets of Manuscript Fig. 2c) were obtained by\ncalculating the spatially resolved Fourier amplitudes for mxat\neach resonant frequency.\nEigenmode simulations The second method was an eigen-\nmode calculation which uses the ﬁnite element micromagneticsimulator, FinMag (successor to Nmag43), to directly deter-\nmine the eigenfrequencies and eigenvectors associated with\na given m0(r;Hext). For relaxation, the system was meshed\nusing a characteristic internode length of lmesh =3:5 nm.\nThe eigenmodes were determined using an algorithm simi-\nlar to that detailed in d’Aquino et al.42on a coarsened mesh\nwith lmesh =7 nm (coarsening was needed due to the algo-\nrithm’s high memory requirements). The nature of the mode\n(e.g. side or extended) was determined via visual inspection of\nthe eigenvectors.\nReferences\n1 R. S. Gaster, D. A. Hall, C. H. Nielsen, S. J. Osterfeld,\nH. Yu, K. E. Mach, R. J. Wilson, B. Murmann, J. C. Liao,\nS. S. Gambhir et al. ,Nat. Med. , 2009, 15, 1327–1332.\n2 B. Srinivasan, Y . Li, Y . Jing, C. Xing, J. Slaton and J.-P.\nWang, Anal. Chem. , 2011, 83, 2996–3002.\n3 P. P. Freitas, F. A. Cardoso, V . C. Martins, S. A. M. Mar-\ntins, J. Loureiro, J. Amaral, R. C. Chaves, S. Cardoso, L. P.\nFonseca, A. M. Sebasti ˜ao and et al., Lab Chip , 2012, 12,\n546–557.\n4 B. Srinivasan, Y . Li, Y . Jing, Y . Xu, X. Yao, C. Xing and\nJ.-P. Wang, Angew. Chem. Int. Ed. , 2009, 48, 2764–2767.\n5 S. J. Osterfeld, H. Yu, R. S. Gaster, S. Caramuta, L. Xu, S.-\nJ. Han, D. A. Hall, R. J. Wilson, S. Sun, R. L. White and\net al., Proc. Natl. Acad. Sci. U.S.A. , 2008, 105, 20637–\n20640.\n6 Y . R. Chemla, H. L. Grossman, Y . Poon, R. McDermott,\nR. Stevens, M. D. Alper and J. Clarke, Proc. Natl. Acad.\nSci. U.S.A. , 2000, 97, 14268–14272.\n7 P.-A. Besse, G. Boero, M. Demierre, V . Pott and\nR. Popovic, Appl. Phys. Lett. , 2002, 80, 4199.\n8 M. M. Miller, G. A. Prinz, S.-F. Cheng and S. Bounnak,\nAppl. Phys. Lett. , 2002, 81, 2211–2213.\n9 J. Llandro, T. J. Hayward, D. Morecroft, J. A. C. Bland,\nF. J. Castano, I. A. Colin and C. A. Ross, Appl. Phys. Lett. ,\n2007, 91, 203904.\n10 J. Devkota, A. Ruiz, P. Mukherjee, H. Srikanth and M.-H.\nPhan, IEEE Trans. Magn. , 2013, 49, 4060–4063.\n11 D. R. Baselt, G. U. Lee, M. Natesan, S. W. Metzger, P. E.\nSheehan and R. J. Colton, Biosens. Bioelectron. , 1998, 13,\n731–739.\n12 D. Hall, R. Gaster, T. Lin, S. Osterfeld, S. Han, B. Mur-\nmann and S. Wang, Biosens. Bioelectron. , 2010, 25, 2051–\n2057.\n13 V . C. Martins, J. Germano, F. A. Cardoso, J. Loureiro,\nS. Cardoso, L. Sousa, M. Piedade, L. P. Fonseca and P. P.\nFreitas, J. Magn. Magn. Mater , 2010, 322, 1655–1663.\n14 Y . Li, Y . Wang, T. Klein and J.-P. Wang, Appl. Phys. Lett. ,2014, 104, 122401.\n15 A. F. Md. Nor, E. W. Hill and M. R. Parker, IEEE Trans.\nMag. , 1998, 34, 1327–1329.\n16 N. Smith, J. Katine, J. R. Childress and M. J. Carey, IEEE\nTrans. Mag. , 2006, 42, 114–119.\n17 Z. Lei, G. Li, W. F. Egelhoff, P. Lai and P. W. Pong, IEEE\nTrans. Mag. , 2011, 47, 602–612.\n18 P. M. Braganca, B. A. Gurney, B. A. Wilson, J. A. Ka-\ntine, S. Maat and J. R. Childress, Nanotechnol. , 2010, 21,\n235202.\n19 K. Mizushima, K. Kudo, T. Nagasawa and R. Sato, J. Appl.\nPhys. , 2010, 107, 063904.\n20 M. Inoue, A. Baryshev, H. Takagi, P. B. Lim, K. Hatafuku,\nJ. Noda and K. Togo, Appl. Phys. Lett. , 2011, 98, 132511.\n21 S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley,\nR. J. Schoelkopf, R. A. Buhrman and D. C. Ralph, Nature ,\n2003, 425, 380383.\n22 W. Rippard, M. Pufall, S. Kaka, S. Russek and T. Silva,\nPhys. Rev. Lett. , 2004, 92, year.\n23 J.-V . Kim, Solid State Phys. , 2012, 63, 217 – 294.\n24 N. Locatelli, V . Cros and J. Grollier, Nat. Mater. , 2013, 13,\n11.\n25 I. N. Krivorotov, N. C. Emley, J. C. Sankey, S. I. Kiselev,\nD. C. Ralph and R. A. Buhrman, Science , 2005, 307, 228–\n231.\n26 H. Suto, T. Nagasawa, K. Kudo, K. Mizushima and\nR. Sato, Appl. Phys. Express , 2010, 4, 013003.\n27 Z. Zeng, P. K. Amiri, I. N. Krivorotov, H. Zhao, G. Finoc-\nchio, J.-P. Wang, J. A. Katine, Y . Huai, J. Langer, K. Galat-\nsiset al. ,ACS Nano , 2012, 6, 6115–6121.\n28 J. Loureiro, R. Ferreira, S. Cardoso, P. P. Freitas, J. Ger-\nmano, C. Fermon, G. Arrias, M. Pannetier-Lecoeur, F. Ri-\nvadulla and J. Rivas, Appl. Phys. Lett. , 2009, 95, 034104.\n29 J. Loureiro, P. Z. Andrade, S. Cardoso, C. L. da Silva,\nJ. M. Cabral and P. P. Freitas, Appl. Phys. Lett. , 2011, 109,\n07B311.\n30 J. Loureiro, P. Z. Andrade, S. Cardoso, C. L. da Silva, J. M.\nCabral and P. P. Freitas, Lab Chip , 2011, 11, 2255–2261.\n31 M. Helou, M. Reisbeck, S. F. Tedde, L. Richter, L. B ¨ar,\nJ. J. Bosch, R. H. Stauber, E. Quandt and O. Hayden, Lab\nChip , 2013, 13, 1035–1038.\n32 N. Singh, S. Goolaup and A. O. Adeyeye, Nanotechnol ,\n2004, 15, 1539.\n33 A. O. Adeyeye and N. Singh, J. Phys. D: Appl. Phys. ,\n2008, 41, 153001.\n34 M. Krawczyk and D. Grundler, J. Phys.: Condens. Matter ,\n2014, 26, 123202.\n35 S. Neusser, B. Botters and D. Grundler, Phys. Rev. B , 2008,\n78, 054406.\n36 S. Tacchi, M. Madami, G. Gubbiotti, G. Carlotti, A. O.Adeyeye, S. Neusser, B. Botters and D. Grundler, IEEE\nTrans. Magn. , 2010, 46, 1440–1443.\n37 R. Zivieri, P. Malag `o, L. Giovannini, S. Tacchi, G. Gub-\nbiotti and A. O. Adeyeye, J. Phys.: Condens. Matter ,\n2013, 25, 336002.\n38 J. Ding, N. Singh, M. Kostylev and A. O. Adeyeye, Phys.\nRev. B , 2013, 88, year.\n39 E. Chatterjee, T. Marr, P. Dhagat and V . T. Remcho, Sens.\nAct. B , 2011, 156, 651–656.\n40 A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen,\nF. Garcia-Sanchez and B. Van Waeyenberg, AIP Adv. ,\n2014, 4, 107133.\n41 M. Grimsditch, G. Leaf, H. Kaper, D. Karpeev and\nR. Camley, Phys. Rev. B , 2004, 69, 174428.\n42 M. d’Aquino, C. Serpico, G. Miano and C. Forestiere, J.\nComp. Phys. , 2009, 228, 6130–6149.\n43 T. Fischbacher, M. Franchin, G. Bordignon and H. Fan-\ngohr, IEEE Trans. Mag. , 2007, 43, 2896–2898.\n44 S. Urazhdin, V . E. Demidov, H. Ulrichs, T. Kendziorczyk,\nT. Kuhn, J. Leuthold, G. Wilde and S. O. Demokritov, Nat.\nNanotechnol. , 2014, 9, 509–513.\n45 V . S. Pribiag, I. N. Krivorotov, G. D. Fuchs, P. M. Bra-\nganca, O. Ozatay, J. C. Sankey, D. C. Ralph and R. A.\nBuhrman, Nat. Phys. , 2007, 3, 498.\n46 R. Lebrun, N. Locatelli, S. Tsunegi, J. Grollier, V . Cros,\nF. Abreu Araujo, H. Kubota, K. Yakushiji, A. Fukushima\nand S. Yuasa, Phys. Rev. App. , 2014, 2, 061001.\n47 E. N. Ivanov and M. Kostylev, Pre-print: arXiv:1402.3459\n[cond-mat.mes-hall] , 2014, (vol.: n/a) , (page: n/a).\n48 H. Fangohr, G. Bordignon, M. Franchin, A. Knittel,\nP. A. J. de Groot and T. Fischbacher, J. Appl. Phys , 2009,\n105, 07D529.\n49 J. M. Shaw, H. T. Nembach, T. J. Silva and C. T. Boone, J.\nAppl. Phys , 2013, 114, 243906.\n50 F. Perez and B. E. Granger, Comput. Sci. Eng. , 2007, 9,\n21–29.\n51 A. P. Davison, Comput. Sci. Eng. , 2012, 14, 48.\n52 J. D. Hunter, Comput. Sci. Eng. , 2007, 9, 90–95.\n53 G. Venkat, D. Kumar, M. Franchin, O. Dmytriiev,\nM. Mruczkiewicz, H. Fangohr, A. Barman, M. Krawczyk\nand A. Prabhakar, IEEE Trans. Magn. , 2013, 49, 524."    },    {        "title": "0810.4020v1.Strong_asymmetry_of_microwave_absorption_by_bi_layer_conducting_ferromagnetic_films_in_the_microstrip_line_based_broadband_ferromagnetic_resonance.pdf",        "content": "arXiv:0810.4020v1  [cond-mat.mtrl-sci]  22 Oct 2008Strong asymmetry of microwave absorption by bi-layer\nconducting ferromagnetic ﬁlms in the microstrip-line base d\nbroadband ferromagnetic resonance\nM. Kostylev∗\nSchool of Physics, The University of Western Australia,\n35 Stirling Highway, Crawley WA 6009, AUSTRALIA\nAbstract\nPeculiarities of ferromagnetic resonance response of cond ucting magnetic bi-layer ﬁlms of nano-\nmetric thicknesses excited by microstrip microwave transd ucers have been studied theoretically.\nStrong asymmetry of the response has been found. Depending o n the order of layers with respect\nto the transducer either the ﬁrst higher-order standing spi n wave mode, or the fundamental mode\nshows the largest response. Film conductivity and lowered s ymmetry of microwave ﬁelds of such\ntransducers are responsible for this behavior. Amplitude o f which mode is larger also depends on\nthe driving frequency. This eﬀect is explained as shielding o f the asymmetric transducer ﬁeld by\neddycurrents in theﬁlms. Thisshieldingremains very eﬃcie nt for ﬁlms withthicknesses well below\nthe microwave skin depth. This eﬀect may be useful for studyin g buried magnetic interfaces and\nshould be accounted for in future development of broadband i nductive ferromagnetic resonance\nmethods.\n∗Electronic address: kostylev@cyllene.uwa.edu.au\n1I. INTRODUCTION\nBroadband inductive microwave setups are emerging as a common to ol with which to\nmeasure dynamical properties of magnetic thin ﬁlms and nano-stru ctures [1, 2, 3, 4]. Their\nmainpart isasection of acoplanar ora microstrip microwave transmis sion line (”‘microwave\ntransducer”) on which top the sample under investigation sits. Abs orption of microwave\npower by the transmission line loaded by the sample is measured by a ne twork analyzer [2]\nor by a combination of a microwave generator and microwave passive components [4]. These\nsetups are useful for accurately measuring spin wave excitations which are used as a probe\nfor characterizing magnetic surfaces and buried interfaces. How ever, consideration needs to\nbe made of the experimental setup when analyzing results [2, 5, 6, 7 , 8, 9]. In particular,\nSchneider et al. [5] mention that measured response is dependent o n the distance of the\nﬁlm from the coplanar surface due to a change in strip line impedance. In another work\nthey report on using a ﬂoating ground plane above the ﬁlm in order to enhance the response\namplitude [10]. The broadband inductive technique was also shown to e xcite not only the\nfundamental ferromagnetic resonance mode, but also higher ord er spin wave resonances [3].\nInthispaperwepresenttheoreticalresultsshowingasymmetricb roadband-FMRresponse\nof multilayered metallic ﬁlms with total thicknesses much smaller than t he microwave skin\ndepth. This eﬀect is one more feature which distinguishes this techn ique from the cavity\nferromagnetic resonance (FMR).\nOn the other hand, recently an interest has emerged to excitation of propagating spin\nwaves in metallic magnetic ﬁlms by narrow (several microns in width) mic rostrip microwave\ntransducers [11, 12, 13, 14]. This interest is largely due to the poss ibility of using spin waves\nto perform logic operations [15, 16, 17, 18, 19, 20]. If necessary, the expressions we obtain\ncan beeasily extended to describe excitation of propagating spin wa ves following suggestions\nin Ref. [21] and Ref. [9].\nII. DYNAMIC EQUATIONS\nTo construct our model we use the system of Maxwell equations wit h no electric-bias\nterm as suggested previously [9, 22, 23]:\n∇×h=4πσ\nce (1)\n2∇·h=−∇·4πm\n∇×e=−iω\nc(h+4πm)\nHeremis the dynamic magnetization, eis an electric ﬁeld, and ωis the eigenfrequency\nof spin waves. To obtain the last of these equations we assumed tha t all dynamic ﬁelds and\nthe dynamic magnetization have the time dependence in the form exp(iωt).\nIn Ref.[9] starting with these equations we constructed a quasi-an alytical theory allowing\nfor inhomogeneous dynamic magnetization in the ﬁlm plane in the form o f spin waves with\na set of in-plane wave numbers kx. This was important in that work, since due to the type\nof symmetry in the coplanar transducers, their microwave ﬁelds do not excite homogeneous\nprecession kx=0 [24]. On the contrary, microstrip transducers show maximum of m icrowave\nabsorption at kx=0 (see e.g. [25] and references therein) due to symmetry of the in -plane\ncomponent of the dynamic magnetic ﬁeld of the transducer hx. For microstrip transducers\nwith a microstrip width much larger than the free spin-wave propaga tion path (about 100\nmicron) in Permalloy (which is the metallic magnetic material with the lowe st spin wave\ndamping) one can consider the dynamic magnetization and all microwa ve ﬁelds being ho-\nmogeneous in the ﬁlm plane (the (x,z) plane). Assuming that the micro wave current of\nthe microstrip and the static magnetic ﬁeld H(which will later enter the magnetic torque\nequation) are along the axis z, these equations can be reduced to a system as follows:\n∂hx/∂y=−4πσ\ncez (2)\n∂ez/∂y=iω\nc(hx+4πmx)\nhy=−my.\nThe last of these equations shows that the out-of-plane compone nt of the dynamic mag-\nnetic ﬁeld is purely of magnetostatic nature and represents the dy namic demagnetizing ﬁeld\nofmy. It remains present when the sample conductivity is set to zero. On the contrary,\nfrom the ﬁrst two equations one sees that the in-plane component of the magnetic ﬁeld is\nof curling nature. Indeed, one can reduce Eqs.(2a) and (2b) to a s econd-order diﬀerential\nequation:\n−iδ2∂2hx/∂y2+hx=−4πmx. (3)\nwhereδ=/radicalBig\nc/(4πσω) is the microwave skin depth. One sees that in the limit mx=0, when\nthe sources of the magnetostatic (potential) ﬁeld vanish this equa tion remains valid. On\n3the other hand, from Eqs.(1) one can ﬁnd that hxvanishes for σ= 0. Thus this component\nis not present in the insulating samples, where all ﬁelds are known to b e magnetostatic, it\nis entirely of curling nature, and is due to the eddy current jzinduced in the conducting\nferromagnetic ﬁlms by the precessing magnetization and the exter nal microwave magnetic\nﬁeld.\nIn Ref.[9] it has been shown how to obtain a particular solution of such inhomogeneous\nequation in terms of Green’s functions. A general solution of the ho mogeneous equation has\nto be added to this solution to obtain the complete solution. This is not diﬃcult to do,\nif magnetic layers composing the multi-layered ﬁlm have the same cond uctivity [26]. But\nif interfaces of magnetic metal layers with diﬀerent values of condu ctivities are present the\nﬁnal analytic expressions become too cumbersome to be used in com putations.\nTherefore, instead of solving Eq.(3) analytically we derive boundary conditions for this\nequation and solve it numerically together with Landau-Lifshitz-Guilb ert equation. From\nthe condition of continuity of hxandezat the interface of two magnetic layers y=L1we\nﬁnd\nσ2∂h(1)\nx/∂y=σ1∂h(2)\nx/∂y (4)\nh(1)\nx=h(2)\nx,\nwhere the indices 1 and 2 denote the layer numbers. The boundary c onditions at the\nouter surfaces of the bilayer ﬁlm are obtained by ﬁrst allowing the ou ter space having ﬁnite\nconductivity σout. Then the magnetic ﬁeld in the outer space is described by Eq.(3) with the\nzero right-hand part. Its solution is obtained requiring vanishing of the microwave magnetic\nﬁeld on both inﬁnities. At one side from the ﬁlm y <0 we have: hout\nx=Aexp(√\niδouty),\nand at the other side from the ﬁlm y > L1+L2we havehout\nx=Bexp(−√\niδouty). Obviously\nthe same boundary conditions (4) are valid at the boundary of a con ducting magnetic ﬁlm\nwith a conducting outer space. Substituting the second of these e xpressions into Eqs.(4) one\nobtains a formula which relates the ﬁeld insidethe ferromagnetic layer to its derivative.\nhx=−1√\niδoutσout\nσ2∂hx\n∂yaty=L1+L2, (5)\nwhereσ2is the conductivity of the second layer of the bi-layer ﬁlm. To derive t he formula\nwhich relates hxto∂hx/∂yat the other ﬁlm boundary we need to include the magnetic ﬁeld\ninduced by the microstrip. We model the microstrip line as a surface c urrent density jzat\n4the ﬁlm surface y= 0. Then following the same procedure one obtains:\nhx=1√\niδoutσout\nσ1∂hx\n∂y−jzaty= 0. (6)\nIn the limit σout<< σ1,σ2Eqs.(5) and (6) reduce to:\nhx=−j0aty= 0, (7)\nhx= 0 at y=L1+L2.\n(In these expressions we use SI units.) The second of these formu las shows that the\ndynamicmagneticﬁeldattheﬁlmsurfacenotfacingthemicrostripline iszero. Thissuggests\nthat total back-reﬂection of the microwave magnetic ﬁeld from th e boundary between two\nmedia with a large diﬀerence in electric conductivity takes place. From the second of Eqs.(4)\none now ﬁnds that the dynamic magnetic ﬁeld outside the ﬁlm y > L1+L2vanishes. Thus\na conducting magnetic layer with a thickness much smaller than δeﬃciently shields the\nmicrowave magnetic ﬁeld. This is in a striking contract with insulating ﬁlm s: the microwave\nmagnetic ﬁeldofa transducer easily penetrates throughmagnetic insulators. It equals −jz/2\neverywhere for y >0 andjz/2 fory <0. (see e.g. Eq.(32) in [25]). (To obtain this result\nfrom our expressions one has to set σ1=σ2=σoutand then take the limit of vanishing\nconductivities.) The latter result allows one to consider the total ﬁe ldhxas a sum of the\nexternal ﬁeld −jz/2 and the ﬁeld of eddy currents. Then one ﬁnds that the ﬁeld of the eddy\ncurrents should grow from −jz/2 aty= 0 tojz/2 aty=L1+L2ensuring no total dynamic\nmagnetic ﬁeld outside the ﬁlm y <0 andy > L1+L2.\nThe obtained boundary conditions (7) are quite asymmetric. This is a big contrast\ncompared with the cavity resonance where a homogeneous extern al microwave magnetic\nﬁeld penetrates the sample through both surfaces and the shieldin g eﬀect is not seen in the\nboundary conditions (Eq.(4.1) in [22]). (We note, that the conditions (7) are in agreement\nwith the theory [22] and for the cavity resonances can be transfo rmed into Eqns. (3.3) or\n(4.1) in the cited work.)\nIn this work we numerically solve the linearized Landau-Lifshitz-Gilber t magnetic torque\nequation [27]\niωm= (8)\n−γ[(ixmx+iymy+Miz)\n×(ix(hx+2A/M2∂2mx\n∂y2) +iy(hy+2A/M2∂2my\n∂y2)+Hciz)],\n5whereMis saturation magnetization of a layer, Ais the exchange constant for the layer,\nHc=H+iαω/γ,αis the Gilbert magnetic damping constant, and iare the unit vectors\nalong the coordinate axes. (The static magnetic ﬁeld Hand the equillibrium magnetization\nare along the z-axis.) The torque equation is solved together with Eqs.(2c),(3), th e electro-\nmagnetic boundary conditions Eqs.(7), and the exchange boundar y conditions at the layer\ninterface. The latter are cast in the form suggested in [28]. Provide d there is no pinning\nof magnetization in both layers at the interface these exchange bo undary conditions at the\ninterface y=L1of two layers 1and2read\n∂m(1)\nx/∂y+A12\nA1m(1)\nx−A12\nA1M1\nM2m(2)\nx= 0 (9)\n∂m(2)\nx/∂y+A12\nA2m(2)\nx+A12\nA2M2\nM1m(1)\nx= 0,\nwhereA12is the inter-layer exchange constant. For the out-of-plane dyna mic magnetization\ncomponents the inter-layer boundary conditions are the same. Th e exchange boundary\nconditions at the outer surfaces of the ﬁlm are, as follows [29]\n∂mx/∂y±dmx= 0, ∂m y/∂y= 0, (10)\nwheredis the pinning parameter at the ﬁlm surface. The positive sign is for th e boundary\ny= 0 and the negative one is for y=L1+L2. To obtain the numerical solution the\nsystem of diﬀerential equations is transformed into a system of ﬁn ite-diﬀerence equations.\nThe latter form a matrix-vector equation with a band matrix of coeﬃ cients. This linear\nalgebraicsystemissolvedusingnumericalmethodsoflinearalgebra. Thewayweincorporate\nboundary conditions between layers and at the outer surfaces of the ﬁlm into the equations\nis shown in the Appendix.\nIII. TRANSDUCER RESPONSE\nThere are two ways to treat the transducer response. One of th em is that of the eﬀective\nmicrowave susceptibility [6, 7, 8]. It assumes that the total microwa ve energy absorbed\nby the material is lost due to magnetic losses by driven precession of magnetization in the\narea of localization of the microwave magnetic ﬁeld of the transduce r. Another method\nwas established several decades ago when magnetostatic spin wav es in monocrystalline fer-\nrimagnetic ﬁlms of yttrium iron garnet (YIG) were found being promis ing for processing\n6microwave signals [30]. It was suggested that linear impedance could b e used to quanti-\ntatively characterize eﬃciency of excitation of propagating spin wa ves by microstrip (see\ne.g. [25, 31, 32, 33, 34] and references therein), and coplanar [24 ] transducers. The linear\nimpedance is calculated as a ratio of the Poynting vector of the ﬂux o f energy of microwave\nﬁeld through the transducer surface to the microwave current in the transducer [25]. This\napproach remains valid also in the case when the transducer width is m uch larger than the\nspin wave free propagation path and the most of the irradiated ene rgy is not carried away by\nspin waves but is lost due to relaxation processes within the reach of the microwave Oersted\nﬁeld of the transducer. In this case it naturally incorporates poss ibility of energy losses due\nto irradiation of propagating spin waves which leave the area of the t ransducer’s microwave\nOersted ﬁeld and carry energy away. This additional loss mechanism for metallic magnetic\nﬁlms was recently discussed in Ref.[2] and experimentally studied in det ail in [11, 12, 13].\nBoth the eﬀective microwave susceptibility and the linear impedance c an be related to the\nscattering coeﬃcient S21 measured in the network-analyzer based broadband FMR [2].\nIn this work we choose the complex linear impedance Zrof a microstrip loaded by the\nferromagnetic ﬁlm as a quantity describing the eﬃciency of microwav e absorption. This is\nbecause in this way, if necessary, our theory can be easily extende d to include the ﬁnite\nwidth of the transducer to describe eﬀects of irradiation of propa gating spin waves with\nnonvanishing wave numbers by narrow transducers which were use d in recent experiments\n[11, 12, 13]. Ref. [21] shows the way the irradiation of propagating w aves can be treated.\nOur calculations are carried out following the suggestion in [25] origina lly made for mag-\nnetostatic spin waves in YIG ﬁlms. We adopt this approach to metallic f erromagnetic ﬁlms.\nNote, that the latter couple to the transducers much more weakly and take much less power\nfrom the transducer. For the in-plane homogeneous surface cur rent and the in-plane homo-\ngeneous microwave electric ﬁeld the expression for ZrEq.(27) from [25] reduces to:\nZr=ez(y= 0)\nj∗zw, (11)\nwherewis the transducer width in the direction x.\nThe obtained values of Zrare then transformed into the value of the scattering coeﬃcient\nS21. We start with the formula for the input impedance Zfof a section of a microstrip line\nloaded by a magnetic ﬁlm. The ﬁlm sample has a length lalongzand sits on top of the\n7transducer. Following Eq.(25) in [25] we have\nZf=Zcz0cosh(γfl)+Zcsinh((γfl))\nz0sinh(γfl)+Zccosh((γfl)). (12)\nIn our case of the broadband FMR z0is the characteristic impedance of the sections of the\nmicrostrip line not covered by the sample (”unloaded microstrip”) an d equals 50 Ohms. Zc\nis the characteristic impedance of the section of the microstrip line lo aded by the sample\n(”loaded microstrip”)\nZc=/radicalBig\n(Z0+Zr)/Y0 (13)\nwithZ0andY0being the complex series resistance and the complex parallel conduc tance of\ntheunloadedmicrostrip, and γfisthecomplexpropagationconstantoftheloadedmicrostrip\nγf=/radicalBig\n(Z0+Zr)Y0. (14)\nThe transmission matrix Tof the loaded microstrip is obtained following Ref.[35]:\nT = [T(1)·T(2)·T(3)], (15)\nwhereT(1)andT(3)are the transmission matrices of junctions of the loaded microstrip with\nthe unloaded microstrip. The former is for the junction at the fron t edge of the sample\nand the latter is for the rear edge. These matrices are deﬁned via t he complex reﬂection\ncoeﬃcient:\nΓ =±Zf−z0\nZf+z0, (16)\nwhere the positive sign is for the front edge and the negative sign is f or the rear edge. The\nelements of these matrices are:\nT(1,3)\n11= T(1,3)\n22= (1−Γ)−1, (17)\nT(1,3)\n12= T(1,3)\n21= Γ(1−Γ)−1.\nThe transmission matrix for the loaded microstip between these two edgesT(2)has only\ndiagonal elements:\nT(2)\n11= 1/T(2)\n22=eγfl. (18)\nThe scattering parameter S21 of the whole loaded microstrip is 1 /T22(Eq.(15)). By multi-\nplying the matrices in Eq.(15) one obtains:\nS21 =Γ2−1\nΓ2eγfl−e−γfl. (19)\n8The magnitude of the linear radiation impedance of magnetostatic sp in waves in YIG is\nof order of z0[24] and one has to use Eqs.(12-14) as they are. But for nanometr ic metallic\nﬁlms and wide microstrip transducers Zr<<50Ω which is clearly seen in experiment as\n|Γ|<<1 [8, 36]. This allows considerable simpliﬁcation of these formulas. From the\ncondition Zr<< z0to the ﬁrst order in Zr/z0from Eqs.(12-20) one obtains\nS21/S210= exp[−Zr\n2z0l], (20)\nwhereS210is the scattering parameter of the transducer with no sample on its top.\nIV. DISCUSSION\nResultsofnumericalcalculationusingthisformalismforthefrequen ciesω/(2π)4,7.5,and\n18 GHz are shown in Fig. 1. We consider a Cobalt (Co)-Permalloy (Py) b i-layer ﬁlm with\nthe parameters shown in the ﬁgure caption. The bilayer has a thick P y layer and a thin Co\nlayer which role is to introduce dynamic magnetization pinning [37] of ma gnetization in the\nPy layer at the layers’ interface while avoiding formation of additiona l resonances localized\nin the Co-layer.\nFrom the ﬁgure one sees that the absorption when the Co-layer fa ces the microstrip is a\nfew times smaller than when the Permalloy layer faces it. One also sees that the amplitude\nof the ﬁrst standing spin wave (the second peak from the right) is v isible only for Co facing\nthe transducer. It grows with frequency and becomes larger the n the fundamental mode\n(the most right-hand peak) at higher frequencies. This result agr ees well with experiment\nwhich will be published elsewhere [36].\nThe next ﬁgure (Fig. 2) explains this strong asymmetry of the ﬁlm re sponse. Its upper\npanel (Fig. 2(a)) shows distribution of dynamic magnetization mx(y) across the bi-layer\nthickness when the surface current is applied from the side of the t hin Cobalt layer. One\nobserves partial dynamic pinning of magnetization at the Py-interf ace with Co which man-\nifests itself as considerably inhomogeneous mx(y)-distribution through the Py-layer for the\nfundamental mode with the minimum at the layer interface. The ﬁrst higher-order mode\nof the stack represents a combination of the fundamental mode o f the Co-layer and the 1st\nSSW of the Py-layer. For the surface current applied at y=L1+L2one obtains the mirror\nimage of this panel.\n9Figure 2(b) shows the hx(y)-dependence for both cases of ﬁlm orientation with respect to\nthe microstrip. First one sees that the distribution satisﬁes the bo undary conditions Eq.(7),\nand in order to satisfy them the magnetic ﬁeld needs to have a large n egative gradient\nthrough the ﬁlm. This ﬁeld may be thought as a combination of the Oer sted ﬁeld of the\nmicrostrip and of the shielding ﬁeld ofthe eddy current jeinthe ﬁlm. The former isconstant\nthrough the ﬁlm and the space y > L 1+L2and equals −j0/2. The latter equals −j0/2\naty≤0, grows through the ﬁlm to reach + j0/2 aty=L1+L2, and remains equal to\nthis value for y > L 1+L2to ensure no dynamic magnetic ﬁeld behind the ﬁlm. From this\nconsideration on can infer that the direction of the eddy current is opposite to j0. If one\nlooks at the plot of the electric ﬁeld (Fig. 2(c)) one ﬁnds that the ed dy current je=4πσ\ncez\nhas a phase of πand thus is indeed anti-aligned to j0.\nFrom Fig. 2(b) one also sees that hx(y) is practically a linear function. Indeed, the\ndeviation from the linearity for the long solid line is less than 10 percent at the maximum\nof deviation. If one looks at Eq.(3) one ﬁnds that without the right- hand part this equation\nhas the solution hc\nxsatisfying the boundary conditions (7) as follows:\nhc\nx=Asinh[√\niδ(y−L1−L2)] (21)\n(here we neglect the inter-layer boundary). In the case case of a thinmonolayer conducting\nﬁlmL1<< δ, L 2= 0 Eq.(22) reduces to a linear dependence hc\nx=A√\niδ(y−L1), where\nA= 2j0/[exp(2√\niδL1)−1]≈j0/(√\niδL1). The line slope is j0which is in agreement with\nour numerical calculations for monolayer ﬁlms. This suggests that t he contribution to the\neddy currents from precessing magnetization (i.e. from the partic ular solution of the inho-\nmogeneous Eq.3) is small and the eddy current in the ferromagnetic ﬁlm is predominantly\ndirectly induced by the microwave magnetic ﬁeld of the transducer. The total hxin the ﬁlm\nconsists then from the transducer Oersted ﬁeld and the ﬁeld of th e directly excited eddy\ncurrent with negligible contribution from precessing magnetization.\nFrom the linearized torque equation (8) one ﬁnds that the dynamic m agnetization is\ndriven by the total ﬁeld hx. If one neglects the small contribution from the precessing\nmagnetization to the total ﬁeld, one can consider (8) as an inhomog eneous equation with\nthe right-hand (driving) term in the form of the linear hx(y) function. This inhomogeneous\nsystem ofdiﬀerential equationscanbeeasily solved analytically, but performing thisisout of\nscope of this paper. Here we just mention that if the resonance mo des are well-resolved, the\n10resonance amplitude of the i-th resonance rishould be proportional to the overlap integral\nIi=/integraltextL1+L2\n0m[i]\nx(y)·hx(y)dy, wherem[i]\nx(y) is the magnetization proﬁle of the i-th eigenmode\nof the ﬁlm. The latter is calculated as an eigenfunction of the operat or which is obtained\nfrom Eq.(8) by setting α=0,hx=0.\nObviously, if the driven resonances are well-resolved (i.e. the reson ance linewidths ∆ H≈\nαω/γare smaller then distances between the neighboring resonances), themxdistributions\nin maxima of resonances (Fig 2(a)) arevery close to the respective m[i]\nx(y)and canbe used to\ndiscussIi. As one sees from this ﬁgure, when the surface current is applied o n the Permalloy\nside of the bi-layer ﬁlm the maximum of the proper distribution of dyna mic magnetization\nfor the fundamental mode m[1]\nx(y) coincides with the maximum of the total driving ﬁeld\nhx(y) and the value of the overlap integral I0is maximized. On the contrary, if the current\nis applied at the Co-surface of the bi-layer the maximum of the driving ﬁeld coincides with\nthe minimum of the dynamic magnetization. I0is noticeably smaller, resulting in a much\nsmaller amplitude of the fundamental mode.\nThis consideration does not apply to the ﬁrst standing spin wave of t he bi-layer. The\nmx(y) distribution for this mode is a quasi-antisymmetric function, but th e proﬁle hx(y)\nremains the same as for the fundamental mode (Fig. 2(b)) and is ch aracterized by a large\nanti-symmetric component. As a result the overlap integral I1does not considerably depend\non the side at which the microwave current is applied, and its value is lar ge. Thus, the\nbehavior of resonance amplitudes in Fig.1 is explained not as increase in excitation eﬃciency\nof the 1st SSW, but as decrease in eﬃciency of excitation of the fun damental mode for the\nspeciﬁc bi-layer ﬁlm orientation with respect to the microwave trans ducer.\nSince eddy currents are involved in this eﬀect, frequency depende nce of the amplitudes\ntakes place. Figure 3 shows S21/S210in the maximum of resonance for diﬀerent modes as\na function of frequency. Figure 3(a)) shows the absolute values, and Fig. 3(b) shows the\nrelative amplitudes of the standing-wave modes with respect to the fundamental mode. One\nsees that the ﬁrst exchange mode for Co facing the microstrip bec omes dominant at 7 GHz\nor so.\nThe model also demonstrates more eﬃcient excitation by the micros trip transducers of\nhigher-order modes in conducting monolayer samples than in insulatin g ﬁlms (Fig. 4). In\nthe case of insulating ﬁlms, if the surface spins are unpinned d= 0 (Eq.(10)) the only mode\nwhichcouplestothetransducer ﬁeldisthefundamental mode[38]. FromFig. 4onesees that\n11in the case of conducting ﬁlms the ﬁrst SSW which is characterized by an anti-symmetric\nproﬁlem[2]\nxmay provide considerable response. As seen in our simulation, this re sponse is\npractically frequency independent which is in contrast to the bi-laye r ﬁlms.\nV. CONCLUSION\nIn this work we theoretically studied peculiarities of ferromagnetic r esonance response\nof conducting magnetic bi-layer ﬁlms of nanometric thicknesses exc ited by wide microstrip\nlines. We found strong asymmetry of the response. Depending on o rdering of layers with\nrespect to the transducer either the ﬁrst higher-order standin g spin wave mode, or the\nfundamental mode showed the largest response. Amplitude of whic h mode is larger also\ndepends on the driving frequency. This theory is in a good agreemen t with an experiment\npublished elsewhere.\nThis eﬀect is explained as shielding by eddy currents induced in the ﬁlm. Our results\nshow that for ﬁlms with thicknesses well below the microwave skin dep th this shielding\nremains very eﬃcient. This ﬁnding may be useful for studying buried magnetic interfaces,\nas it allows more eﬃcient excitation of higher-order standing spin-wa ve modes carrying\ninformation about interface spins.\nVI. ACKNOWLEDGMENT\nTheauthorthanksProf. RobertL.StampsandMr. RhetMagarag giafromtheUniversity\nof Western Australia for fruitful discussions and proof-reading t he manuscript text.\nSupport from the Australian Research Council and the University o f Western Australia\nis gratefully acknowledged.\nVII. APPENDIX: THE DISCRETE MODEL\nHere we show how we construct the discrete model and how we incor porate boundary\nconditions into it. We demonstrate it using the equation (3) for the d ynamic magnetic ﬁeld\nhx. The discrete version of Eq.(8) with boundary conditions (9) and (1 0) has a similar form.\n12We use a three-point formula for discrete diﬀerentiation to obtain t he equation as follows:\n(h(j+1)\nx+h(j−1)\nx−2h(j)\nx)/∆2+iδ−2h(j)\nx+i4πδ−2m(j)\nx= 0, (22)\nwhere ∆ is the mesh step along yandjis the number of a point on the mesh. We locate\nthe points on the mesh such as no point is at the boundary. In partic ular the ﬁrst point on\nthe mesh j= 1 is at y= ∆/2, the last point j=nis aty=L1+L2−∆/2. The points at\nthe interface of two layers have numbers n0andn0+ 1 and are situated at y=L1−∆/2\nandy=L1+∆/2 respectively. Eq.(23) is valid for any values of jexcept for 1, n,n0, and\nn0+1. For these boundary points boundary conditions should be includ ed into the discrete\nsecond derivative in Eq.(23).\nWe assume that the axis ygoes along a horizontal line from the left to the right. Let\nus ﬁrst consider the point j=n0to the left of the interface y=L1which is located\nhalf-way between n0andn0+ 1. We denote an auxiliary point which is located on the\ninterface as j=n0+0.5. The value of magnetic ﬁeld at this point h(n0+0.5)\nxcan be obtained\nby extrapolating hx(y) dependence beyond the point n0from the left using the Taylor\nexpansion. First one calculates the ﬁeld at the point h(n0−0.5)\nxwhich is half way between n0\nandn0−1. Using the Taylor series one obtains:\nh(n0−0.5)\nx (23)\n=h(n0−1)\nx+(∂h(n0−1)\nx/∂y)(∆/2)\n+ (∂2h(n0−1)\nx/∂y2)(∆/2)2\n≈h(n0−1)\nx+(h(n0)\nx−h(n0−2)\nx)/4\n+ (h(n0−2)\nx+h(n0)\nx−2h(n0−1)\nx)/8.\nThis formula is easily simpliﬁed to read\nh(n0−0.5)\nx= 3h(n0)\nx/8+3h(n0−1)\nx/4−h(n0−2)\nx/8. (24)\nSimilarly at the point n0+1.5 which is half way between the two ﬁrst points n0+1n0+2\nto the right from the boundary we obtain\nh(n0+1.5)\nx= 3h(n0+1)\nx/8+3h(n0+2)\nx/4−h(n0+3)\nx/8. (25)\nThe value of magnetic ﬁeld at the point at the boundary j=n0+0.5 ish(b)\nx. The ﬁrst\nderivative ∂hx/∂yenters the upper of boundary conditions (4). To evaluate it at the left\n13from the boundary we use the Taylor expansion again\n∂h(b)\nx/∂y=∂h(n0)\nx/∂y+(∂2h(n0)\nx/∂y2)∆/2 (26)\n≈(3h(b)\nx−4h(n0)\nx+h(n0−0.5)\nx)/∆.\nSimilarly, to the right from the boundary one has\n∂h(b)\nx/∂y= (−3h(b)\nx+4h(n0+1)\nx−h(n0+1.5)\nx)/∆. (27)\nThen substituting Eqs.(27) and (28) into the ﬁrst of the boundary conditions (4) with Eqs.\n(25) and (26) we obtain\nh(b)\nx=(29h(n0+1)\nx−6h(n0+2)\nx+h(n0+3)\nx)σ1\n24(σ1+σ2)(28)\n+(29h(n0)\nx−6h(n0−1)\nx+h(n0−2)\nx)σ2\n24(σ1+σ2).\nThis allows one to evaluate the second derivative at the points n0andn0+1. Thus, instead\nof (23) for the point n0we have\n4(h(b)\nx+h(n0−0.5)\nx−2h(n0)\nx)/∆2+iδ−2h(n0)\nx+i4πδ−2m(n0)\nx= 0, (29)\nwithh(b)\nxandh(n0−0.5)\nxdeﬁned by Eqs.(29) and (25) respectively. A similar expression is\neasily obtained for n0+1. These ﬁnite diﬀerence equations now incorporate the inter-lay er\nelectro-dynamic boundary conditions (4).\nSimilarly, at the outer boundary y=L1+L2from (9) we have h(b)\nx= 0. Then from\nEq.(30) with n0=none derives the ﬁnite diﬀerence equation for j=n\n(13h(n)\nx−6h(n−1)\nx+h(n−2)\nx)/(2∆2)+iδ−2h(n)\nx+i4πδ−2m(n)\nx= 0, (30)\nwhereh(n−0.5)\nxis obtained from Eq.(25) by setting n0=n. In the same way for the point\nj= 1 we have\n(−13h(1)\nx/2+3h(2)\nx−h(3)\nx/2)/(2∆2)+iδ−2h(1)\nx+i4πδ−2m(1)\nx= 4ηj0, (31)\nwhereη= 1/80 is the factor which relates j0measured in Ampere per meter to hxmeasured\nin Oersteds.\nEq.(8) can be discretized in a similar way incorporating exchange boun dary conditions\n(9) and (10) into the exchange operator 2 A/M2∂2m\n∂y2at the points j= 1,n0,n0+ 1,n. In\nparticular, for the point none has\n∂2m(n)\nx\n∂y2≈(d∆−2)(6m(n−1)\nx−m(n−2)\nx)−(13d∆−10)m(n)\nx\n2∆2(3−d∆), (32)\n14wheredis the surface spin pinning parameter from Eq.(10).\nThederivedﬁnite-diﬀerenceequationsformasystemoflinearalgeb raicequations ˆC/vector u=/vectorf,\nwhere/vector uconsists of amplitudes of mx,my, andhxat the points j= 1,2...n. The matrix of\ncoeﬃcients of this system ˆChas a band form. The system is inhomogeneous, with just one\nelement of the vector /vectorfbeing non-zero. This element is the right-hand side of Eq.(32). It\nplays the role of the ”excitation term” for the motion of magnetizat ion. The system ˆC/vector u=/vectorf\ncan be easily solved using numerical methods of linear algebra.\n[1] T. J. Silva, C. S. Lee, T. M. Crawford, and C. T. Rogers, J. A ppl. Phys. 85, 7849 (1999).\n[2] G. Counil, J.-V. Kim, T. Devolder, C. Chappert, K. Shiget o, and Y. Otani, J. Appl. Phys.\n95, 5646 (2004).\n[3] D. Crew, K. Kennewell, M. Lwin, R. Woodward, S. Prasad, an d R. L. Stamps, J. Appl. Phys.,\n97, 10707 (2005).\n[4] M. Kostylev, R. Magaraggia, F. Y. Ogrin, E. Sirotkin, V. F . Mescheryakov, N. Ross, and R.\nL. Stamps, IEEE Trans. On Mag., 44, No. 10 (2008).\n[5] M. L. Schneider, T. Gerrits, A. B. Kos, and T. J. Silva, App l. Phys. Lett. 87, 072509 (2005).\n[6] S. S. Kalarickal, P. Krivosik, M. Wu, C. E. Patton, M. L. Sc hneider, P. Kabos, T. J. Silva,\nand J. P. Nibarger, J. Appl. Phys. 99, 093909 (2006).\n[7] G. Counil, P. Crozat, T. Devolder, C. Chapper, S. Zoll, an d R. Fournel, IEEE Trans. on Mag,\n42, 3321 (2006).\n[8] C. Bilzer, T. Devolder, P. Crozat, C. Chappert, S. Cardos o, and P. P. Freitas, J. Appl. Phys.\n101, 074505 (2007).\n[9] K. J. Kennewell, M. Kostylev, and R. L. Stamps, J. Appl. Ph ys.101, 09D107 1-3 (2007).\n[10] M. L. Schneider, A. B. Kos, and T. J. Silva, Appl. Phys. Le tt.,85, 254 (2004).\n[11] M. Bailleul, D. Olligs, C. Fermon, S.O. Demokritov, Eur ophys. Lett. 56, 741 (2001).\n[12] V. E. Demidov, S. O. Demokritov, K. Rott, P. Krzysteczko , and G. Reiss, Appl. Phys. Lett.,\n91, 252504 (2007).\n[13] V. E. Demidov, S. O. Demokritov, K. Rott, P. Krzysteczko , and G. Reiss, Phys. Rev. B, 77,\n064406 (2008).\n[14] M. Bao, A. Khitun, Y. Wu, J.-Y. Lee, K. L. Wang, and A. P. Ja cob, Appl. Phys. Lett. 93,\n15072509 (2008).\n[15] R. Hertel, W. Wulfhekel, and J. Kirschner, Phys. Rev. Le tt.93, 257202 (2004).\n[16] M. P. Kostylev, A. A. Serga, T. Schneider, B. Leven and B. Hillebrands, Appl. Phys. Lett.,\n87, 153501 (2005).\n[17] T. Schneider, A. A. Serga, B. Leven, and B. Hillebrands, R. L. Stamps, and M. P. Kostylev,\nAppl. Phys. Lett., 92, 022505 (2008).\n[18] T. Schneider, A. Serga, B. Hillebrands, and M. Kostylev , J. of Nanoelectronics and Optoelec-\ntronics,3, No.1, 69-71 (2008).\n[19] A. Khitun and Kang L. Wang, Superlattices and Microstru ctures,38, 184 (2005).\n[20] R. de Sousa and J. E. Moore, Appl. Phys. Lett., 92, 022514 (2008).\n[21] T. Schneider, A. A. Serga, T. Neumann, B. Hillebrands, a nd M. P. Kostylev, Phys. Rev. B\n77, 214411 (2008).\n[22] T. Wolfram and R. E. De Wames, Phys. Rev. B, 4, 3125 (1971).\n[23] N. S. Almeida, and D. L. Mills, Phys. Rev. B, 53, 12232 (1996).\n[24] V. F. Dmitriev, Sov. J. Comm. Technol. 36, 34 (1991).\n[25] V. F. Dmitriev and B. A. Kalinikos, Sov. Phys. J. 31, 432 (1988).\n[26] K. Kennewell, Ph. D. thesis, University of Western Aust ralia, 2008.\n[27] A. G. Gurevich, and G. A. Melkov, Magnetization Oscilla tions and Waves. CRC, 1 edition\n(1996).\n[28] M. Vohl, J. Barnas, and P. Gruenberg, Phys. Rev. B 39, 12003 (1989).\n[29] R. F. Soohoo, Magnetic Thin Films. Harper and Row (1965) .\n[30] W. S. Ishak, Proc. IEEE, 76, 171 (1988).\n[31] A. K. Ganguly, and D. C. Webb, IEEE Trans. Microwave Theo ry Tech., 23, 998 (1975).\n[32] P. R. Emtage, J. Appl. Phys., 53, 5122 (1982).\n[33] G. A. Vugal’ter and V.N. Makhalin, Sov. Phys. Tech. Phys .,30, 296 (1985).\n[34] B. A. Kalinikos and P. A. Kolodin, J. Mag.Mag. Mater., 83, 103 (1990).\n[35] W. Barry, IEEE Trans. Microwave Theory Tech., 34, 80 (1986).\n[36] K. Kennewell, M. Kostylev, R. L. Stamps, M. Ali, D. Gregg , and B. J. Hickey, to bepublished.\n[37] P. E. Wigen, S. F. Kooi, M. R. Shanabarger, and Th. D. Ross ing, Phys. Rev. Lett., 9, 206\n(1962).\n[38] C. Kittel, Phys. Rev. 1101295 (1958).\n16VIII. FIGURE CAPTIONS\nFig. 1. (Color online) Transmission coeﬃcient S21/S210for a bi-layer ferromagnetic\nmetallic ﬁlm. Layer 1: thickness: 10 nm, saturation magnetization 4 πM: 15080 Oe, ex-\nchange constant: 1 .0·106erg/cm, Gilbert damping constant: 0.016, conductivity: 1 .8·107\nSm/m. Layer 2: thickness: 87 nm, saturation magnetization 4 πM: 8042 Oe, exchange con-\nstant: 0.55·106erg/cm, Gilbert damping constant: 0.008, conductivity: 4 .5·106Sm/m.\nInterlayer exchange constant A12= 2·106erg/cm, gyromagnetic constant: γ= 2π·2.92\nrad·MHz/Oe, spins at the outer surfaces of the ﬁlm are unpinned d=0. Microwave trans-\nducer is a microstrip line 1.5 mm in width. (a) microwave frequency is 4 GH z, (b) 7.5 GHz,\n(c) 18 GHz. Thick lines: amplitudes (left axes); thin lines: phase (righ t axes). Solid lines:\nCo layer facing the transducer, dashed lines: Py layer facing the tr ansducer.\nFig. 2 (Color online) Distributions across the ﬁlm thickness. (a) in-pla ne component of\ndynamic magnetization mx. (b) in-plane dynamic magnetic ﬁeld hxfor the fundamental\nmode. (c) microwave electric ﬁeld ezfor the fundamental mode. In all panels thick lines are\nfor the fundamental mode, thin lines are for the ﬁrst standing spin wave mode. Black solid\nlines are for the Co layer facing the transducer. Dashed red line is fo r the Py layer facing\nthe transducer. Blue lines: phase (right axes). All the other para meters are as for Fig. 1.\nFig. 3. (Color online) (a) absolute values of absorption amplitudes S21/S210as a\nfunction of frequency. (b) relative amplitudes of standing spin-wa ve modes with respect\nto the amplitude of the fundamental mode. Bold solid lines: fundamen tal mode, thin blue\nlines: ﬁrst standing spin-wave mode. Other thin lines: higher-order standing spin waves\nseen in Fig. 1(c). Solid black and blue lines: Co layer facing the transdu cer. Dashed red\nlines: Py layer facing the transducer. Parameters of calculation ar e the same as for Fig. 1.\nFig. 4. (Color online) Absolute values (right axis, black lines) of absor ption amplitudes\nS21/S210andtheir ratio(left axis, darkcyanline) asafunctionoffrequency foramonolayer\nPermalloy ﬁlm 97 nm in thickness. Thin line: 1st standing spin-wave mode ; thick line:\nfundamental mode. All the other parameters are as for the Perm alloy layer in Fig. 1.\n17"    },    {        "title": "1909.08293v2.Ferromagnetic_resonance_with_magnetic_phase_selectivity_by_means_of_resonant_elastic_x_ray_scattering_on_a_chiral_magnet.pdf",        "content": "Ferromagnetic resonance with magnetic phase selectivity by\nmeans of resonant elastic x-ray scattering on a chiral magnet\nS. P ollath,1A. Aqeel,2A. Bauer,2C. Luo,3, 2H. Ryll,3F.\nRadu,3C. P\reiderer,2, 4G. Woltersdorf,5and C.H. Back1, 2, 4,\u0003\n1Institut f ur Experimentelle Physik,\nUniversit at Regensburg, D-93040 Regensburg, Germany\n2Physik-Department, Technische Universit at M unchen, D-85748 Garching, Germany\n3Helmholtz-Zentrum Berlin f ur Materialien and Energie, D-12489 Berlin, Germany\n4Munich Center for Quantum Science and Technology\n(MCQST), Schellingstr. 4, D-80799 M unchen\n5Institut f ur Physik, Universit at Halle-Wittenberg, D-06120 Halle (Saale)\n(Dated: September 23, 2019)\nAbstract\nCubic chiral magnets, such as Cu 2OSeO 3, exhibit a variety of non-collinear spin textures, in-\ncluding a trigonal lattice of spin whirls, so-called skyrmions. Using magnetic resonant elastic x-ray\nscattering (REXS) on a crystalline Bragg peak and its magnetic satellites while exciting the sample\nwith magnetic \felds at GHz frequencies, we probe the ferromagnetic resonance modes of these spin\ntextures by means of the scattered intensity. Most notably, the three eigenmodes of the skyrmion\nlattice are detected with large sensitivity. As this novel technique, which we label REXS-FMR,\nis carried out at distinct positions in reciprocal space, it allows to distinguish contributions origi-\nnating from di\u000berent magnetic states, providing information on the precise character, weight and\nmode mixing as a prerequisite of tailored excitations for applications.\n1arXiv:1909.08293v2  [cond-mat.mtrl-sci]  20 Sep 2019Ferromagnetic resonance (FMR) measurements represent a well-established technique for\nthe study of systems with collinear magnetization [1], allowing to extract information on the\nmagnetic energy landscape and material-speci\fc parameters such as the e\u000bective magnetiza-\ntion, the Land\u0013 e gfactor, or the magnetic damping constant \u000b. In systems with non-collinear\nspin textures, measurements of resonant microwave excitations exhibit very complex spectra,\nwhere the identi\fcation of speci\fc modes proves to be prohibitively di\u000ecult. However, in\nview of new technological developments such as antiferromagnetic spintronics or use of quan-\ntum magnetism, the precise identi\fcation of speci\fc modes will be of great importance [2].\nThe cubic chiral magnets MnSi, Fe 1\u0000xCoxSi, and Cu 2OSeO 3represent excellent showcases\nfor the inherent complexity of materials with technological potential. These materials host\nlong-wavelength helimagnetic order including a trigonal lattice of topologically non-trivial\nspin whirls, the so-called skyrmion lattice [3{7]. Their microwave excitations have been\nstudied by means of coplanar waveguides and cavities [8{12]. In the helimagnetic states,\ntwo collective spin-precessional modes, denoted + qand\u0000q, are observed. In the skyrmion\nlattice state three eigenmodes exist, namely a clockwise and a counter-clockwise gyration\nmode as well as a breathing mode [13]. The remarkably detailed understanding of the cubic\nchiral magnet allows to assign these modes in the experimental spectra by comparing their\nresonance frequency and spectral weight, as well as the evolution of the latter as a function\nof temperature and \feld, to results of analytic calculations and micromagnetic simulations\nsolving the Landau{Lifshitz{Gilbert equation taking into account dipolar interactions [12].\nSuch an in-depth understanding, however, may not be available when investigating novel\nmaterials [14{16] or when phenomena such as metastable states, glassy textures, phase\ncoexistence, topological transitions, or pronounced history dependencies play a role [17{25].\nIn this work we present a novel technique, called REXS-FMR, which combines the exci-\ntation of collective modes by means of a coplanar waveguide with the detection by means\nof the scattered intensity in magnetic resonant elastic X-ray scattering (REXS). The inten-\nsity of a crystalline Bragg peak or magnetic satellites may be studied, permitting to clearly\nidentify the hosting magnetic state of a given excitation via the scattering pattern in recip-\nrocal space. As a point of reference, we demonstrate the potential of REXS-FMR using the\ninsulating cubic chiral magnet Cu 2OSeO 3, which was studied previously both by means of\nREXS [26{31] and standard microwave spectroscopy [11, 12, 32{36].\nOur present study was carried out on the beamline PM2 at BESSY II with the VEKMAG\n2end station [37]. The sample was a single-crystal cuboid of Cu 2OSeO 3, cut from an ingot\ngrown by means of chemical vapor transport, with dimensions of 1 :8\u00020:5\u00020:5 mm3and\nedges oriented parallel to [110], [001], and [ \u0016110]. One of the surfaces normal to [001] was\nmechanically polished. The latter facing top, the sample was placed into the gap of a\ncoplanar waveguide with a gap width of 1 mm. The sample slightly protrudes the top surface\nof the waveguide resulting in microwave excitation that comprises both in-plane and out-of-\nplane components. Typical excitation \felds are of the order of 3 \u0016T to 10\u0016T, depending on\nthe excitation frequency. For the REXS measurements, the energy of the circularly polarized\nphotons is tuned to the Cu L 3edge (931 eV). Note that the element speci\fc character of\nREXS is also inherited to REXS-FMR. Further note that only in resonant X-ray scattering\nthe crystallographically forbidden Bragg peak at 2 \u0012\u001996:5\u000eis observed [27, 28, 38, 39].\nThe geometry of the experiment is illustrated in Fig. 1(a). The sample surface is illumi-\nnated by a X-ray spot of 100 \u0016m diameter. The scattered intensity of the structural (001)\nBragg peak and its magnetic satellites is captured using a photo diode. Initially the diode\nangle (2\u0012) is adjusted such that the intensity of the Bragg peak is maximized. Subsequently,\n(2\u0012) remains \fxed and the reciprocal space around the Bragg peak is mapped by varying\nthe sample angle !and the vertical diode position zd. The diode is located behind a pinhole\nwith a diameter of 300 \u0016m that convolutes the measured signal, leading to an elongation of\nthe peaks in linearly scanned zddirection. Despite the rather small sample{detector distance\nof 20 mm, the Bragg peak and its magnetic satellites may be separated clearly. Using the\nsuperconducting vector magnet at VEKMAG, the magnetic \feld is applied either parallel\nto the [110] axis, the [001] axis, or the incoming X-ray beam.\nThe magnetic phase diagram of Cu 2OSeO 3is schematically depicted in Fig. 1(b) [6, 40].\nStarting in the paramagnetic (pm) state at high temperatures and low \felds, long-wavelength\nhelimagnetic order (wavelength \u0015= 620 \u0017A) is observed below the transition temperature\nTc= 58 K. In the helical state, macroscopic domains of helices propagate along one of\nthe easyh001iaxes. In the corresponding reciprocal space map, shown in Fig. 1(c), the\nstructural (001) Bragg peak is surrounded by four magnetic satellite peaks ( \u000e01), ( \u0016\u000e01),\n(0\u000e1), and (0 \u0016\u000e1). Although after zero-\feld cooling the equivalent domains are expected to\nbe populated equally across a bulk sample, the present experiment only maps few domains\ndue to the small X-ray spot size, rendering asymmetric intensity distributions possible.\nUnder applied magnetic \feld, the propagation directions of the helices re-orient into the\n3\feld direction and \fnite net magnetization emerges as the magnetic moments increasingly\ntilt towards the \feld direction. In REXS, this conical state is associated with magnetic\nsatellites along the \feld direction that are observed when the \feld is applied perpendicular\nto the [001] direction, as shown in Fig. 1(d). Further increasing the magnetic \feld to the\ncritical \feld Hc2results in a \feld-polarized state in which the moments are aligned along\nthe \feld and no long-wavelength modulation is observed (not shown).\nAt intermediate magnetic \feld just below Tc, a pocket of skyrmion lattice state is ob-\nserved. The trigonal order of the spin whirls in the plane perpendicular to the \feld translates\nto the characteristic sixfold pattern of magnetic satellites in both small-angle neutron scat-\ntering [41{45] and REXS [26{31], as shown in Fig. 1(e). Under \feld cooling, the skyrmion\nlattice may be frozen-in to lower temperatures as a metastable state [17, 20, 23, 46]. As de-\npicted in Fig. 1(f), the intensity of the magnetic satellites increases by an order of magnitude\ndue to the increase of the magnetic moment with decreasing temperature.\nTypical REXS-FMR data are shown in Fig. 2 for the \feld-polarized state at T= 15 K,\nwhen a \feld larger than \u00160Hc2\u0019125 mT is applied parallel to the beam direction [47].\nDue to the lack of a long-wavelength modulation there are no magnetic satellites around\nthe structural (001) Bragg peak. This peak, however, comprises a magnetic contribution\nthat depends on the magnitude and orientation of the magnetization ~Mwith respect to the\nincident and scattered X-ray wave vector. In turn, the \feld dependence of the magnetization\nmay be inferred by tracking the intensity of the Bragg peak, shown in Fig. 2(a). Clear kinks\nat\u0006Hc2and saturated behavior at larger \felds are observed, when the microwave excitation\nis switched o\u000b (gray curve). Subtle changes in the curve's slope indicate phase transitions\nwhich are discussed in more detail in Refs. [41, 48, 49]. As the magnetic contribution includes\nterms linear and quadratic in ~M, the intensity curve is not symmetric with respect to zero\n\feld. We refer to the Supplementary Material [] and Refs. [50, 51] for information on the\ndetermination of the absolute magnetization values.\nResonant excitation is studied by repeatedly switching on and o\u000b the microwave excitation\nwhile stepping the magnetic \feld, starting in high positive \felds. At each \feld point, the\nREXS intensity under excitation is integrated for 3 s before the excitation is switched o\u000b and\nthe intensity is integrated again for 3 s. For an excitation frequency of 4.5 GHz (red curve),\na minimum of the magnetic intensity contribution emerges in the \feld-polarized state above\nHc2. As shown in Fig. 2(b), the normalized signal di\u000berence, ( Ion\u0000Io\u000b)=(Ion+Io\u000b), at this\n4minimum is of the order of 2%, which translates to a reduction of the magnetization by\nabout 6.5%. Note that the way how the excitation is applied means that the magnetization\nreproducibly switches from its reduced to its regular value at each \feld step. When assuming\nthat precessional motion of the moments causes the reduction, a precession angle of 21\u000eis\nrequired. This value is large but plausible considering the rather low e\u000bective damping\nof\u000b\u001910\u00004observed in the insulator Cu 2OSeO 3[52, 53]. In contrast, a change of the\nmoment of 6.5% by heating e\u000bects requires an increase/decrease of the sample temperature\nby\u001813 K after switching the excitation on/o\u000b in less than the time frame of 1 s resolvable in\nthe present REXS experiment. Such drastic heating e\u000bects, however, would interfere when\nstudying the skyrmion lattice state with its rather narrow temperature width of \u00182 K close\ntoTc, see below, and can be excluded.\nSimultaneously to the REXS measurements, the re\rected microwave power S11of the\ncoplanar waveguide was recorded using a Schottky diode detector. As shown in Fig. 2(c),\nminima in the \feld-polarized state above \u0006Hc2are observed at the same \feld values as in\nREXS. An additional broad signature around zero \feld is attributed to resonant excitations\nin the helimagnetic state, notably of the \u0006q modes. The absence of these resonances in the\nREXS data highlights the potential of REXS-FMR to selectively study individual magnetic\nphases and determine the origin of speci\fc excitations. Figure 2(d) shows that with increas-\ning excitation frequency the resonance \feld in REXS data increases linearly, consistent with\nKittel behavior in the \feld-polarized state. The resonance \feld values in the \feld-polarized\nstate are also in excellent agreement with resonance frequencies inferred from conventional\nmicrowave spectroscopy using a vector network analyzer (spectra not shown).\nAs one of its decisive advantages, REXS-FMR may be carried out not only on structural\nBragg peaks but also on magnetic satellites, allowing to unambiguously make the connection\nbetween the underlying magnetic phase and the resonant mode. In Fig. 3(a), the intensity at\na helical satellite position is shown as a function of \feld for di\u000berent excitation frequencies\natT= 30 K. Similar as before, data are recorded with microwave excitation switched on\nand o\u000b at each magnetic \feld step. The integration time was increased to 10 s. Finite\nintensity arises only around zero \feld, where the helical state is observed in the magnetic\nphase diagram after zero-\feld cooling, cf. Fig. 1(b). Note, however, that the \feld is applied\nalong [001], i.e., an easy axis for the helices in Cu 2OSeO 3, and the measurement starts in\nthe \feld-polarized state. Therefore, when decreasing the \feld to zero through the conical\n5state, the helices may be expected to remain in the helical domain oriented along the \feld\ndirection even in zero \feld [54], resulting in the absence of magnetic satellites in the present\nscattering geometry at temperatures well below Tc.\nThis putative contradiction connects to the recent discovery that Cu 2OSeO 3not only\nhosts a skyrmion lattice at high temperatures, common to all cubic chiral magnets, but\nalso an independent second skyrmion phase at low temperatures [25, 49]. In contrast to\nthe high-temperature phase, the low-temperature phase stabilized by magneto-crystalline\nanisotropies and exists only for \feld values around Hc2applied along [001]. Due to the\ntopological protection inherent to skyrmions and the rather low temperature, the energy\nbarrier of the low-temperature skyrmion phase is comparatively high. As a result, the\nskyrmion state exhibits a rather glassy texture without well-de\fned long-range order. When\nthis state decays at lower \felds by means of coalescence of neighboring skyrmions, a texture\nresembling poorly ordered helices with propagation perpendicular to the \feld forms [55, 56].\nThe weak magnetic satellites associated with such a helical state are detected in our REXS\nexperiment. Perhaps most strikingly, the glassy texture is highly susceptible to changes\ninduced by resonant microwave excitation, as explained in the following.\nAt low and high frequencies, i.e., when the excitation is o\u000b-resonance, data with mi-\ncrowave switched on and o\u000b agree with each other, corroborating that heating e\u000bects are\nnegligible. In resonance, the excitation distinctly reduces the satellite intensity. Consistent\nwith the behavior in the \feld-polarized state, the intensity reproducibly switches between\nits low and high value at each \feld step and the reduction is attributed to a precessional\nmotion of the magnetic moments. Note, however, that data without excitation (gray curves)\nare expected to track each other as long as the same magnetic texture is probed, which is\nclearly not the case.\nThe discrepancy becomes especially pronounced for an excitation frequency of 3.5 GHz,\nfor which the satellite intensity increases by an order of magnitude while the \feld range\nshrinks by a factor of two. This \fnding suggests that the microwave excitation interacts\nwith the glassy magnetic texture described above, improving its long-range order. In reso-\nnance this pumping e\u000bect is particularly e\u000bective and the corresponding magnetic satellite\nin reciprocal space not only increases in intensity but also decisively sharpens.\nWhen also taking into account a small misalignment between sample plane and \feld\ndirection, as a result the scattering condition is only ful\flled in a reduced \feld interval\n6around zero \feld. In Fig. 3(b), the relative reduction of the helical satellite intensity in zero\n\feld is shown as a function of the excitation frequency. Two sharp maxima are observed that\nare unambiguously attributed to the -q and +q collective modes of the helical state. Both\nfrequency values and heights of the maxima are in excellent agreement with the literature [11,\n12].\nFinally, REXS-FMR on the high-temperature skyrmion lattice is presented in Fig. 3(c)\nshowing the intensity at a skyrmion satellite position as a function of \feld at T= 56 K for\ndi\u000berent excitation frequencies. Starting the description in o\u000b-resonance at low excitation\nfrequency (bottom), measurements with and without microwave excitation track each other.\nThe intensity maxima in \fnite \felds around \u000630 mT are attributed to the skyrmion lattice\nstate. In addition, \fnite intensity emerges around zero \feld as the tail of the broad helical\nsatellite close-by in reciprocal space reaches the position of the skyrmion satellite. At higher\nfrequencies, a distinct reduction of the skyrmion satellite intensity is observed under mi-\ncrowave excitation, again attributed to a precessional motion of the magnetic moments. At\nfrequencies above 1.9 GHz, resonance e\u000bects associated with the helical state are observed\naround zero \feld.\nIn order to further analyze the skyrmion resonances, the relative reduction of the satellite\nintensity at 35 mT, i.e., in the skyrmion lattice state, is depicted in Fig. 3(d) as a function\nof the excitation frequency. Three distinct maxima are observed and associated with the\ncounter-clockwise (ccw) gyration, breathing (bre), and clockwise (cw) gyration mode, in\nexcellent agreement with literature [11, 12]. The relative heights of the maxima are also\nperfectly consistent with calculated spectral weight distributions, as the coplanar waveguide\ncombines in-plane excitation, driving the two gyration modes, and out-of-plane excitation,\ndriving the breathing mode [35]. Due to a photon penetration depth of about 30 nm, the\nREXS-FMR measurements probe the recently discovered surface states of the skyrmion\nlattice in Cu 2OSeO 3[57, 58]. Although roughly 25 % of the probed volume are expected\nto contain such surfaces states, no contributions distinguishable from the bulk resonance\nmodes are observed, indicating that the resonance frequencies of bulk and surface states\nare very similar or equal. Note that above and below 2 GHz di\u000berent circulators are used,\nleading to a small o\u000bset in the applied microwave power. Furthermore, the diode position\nwas changed, as indicated in the sketch in Fig. 3(d), resulting in quantitative discrepancies\nin the measured intensity pro\fles at that frequency.\n7In summary, we established a novel X-ray scattering technique, referred to as REXS-\nFMR, that combines microwave excitation by means of a coplanar waveguide with detection\nin reciprocal space by means of magnetic REXS. In the cubic chiral magnet Cu 2OSeO 3, we\nidenti\fed the resonant modes in the \feld-polarized, helimagnetic, and skyrmion lattice state\nby tracking the intensity of the structural (001) Bragg peak and its magnetic satellites under\nmicrowave excitation. The surface sensitive measurement indicates equal magnetic resonance\nfrequencies for skyrmionic surface and bulk states. REXS-FMR also allows for stroboscopic\nmeasurements in order to determine the character of eigenmodes microscopically, e.g., to\ndistinguish between gyration and breathing modes. Also note that the selectivity to cer-\ntain magnetic phases may prove particularly useful in complex magnetic environments for\nwhich the clear identi\fcation of eigenmodes in conventional microwave spectroscopy is chal-\nlenging, such as for multi-domain skyrmion states in lacunar spinels [14, 59, 60], glassy\nskyrmionic textures in Co-Mn-Zn compounds [61{63], or the low-temperature skyrmion\nstate in Cu 2OSeO 3with its concomitant tilted conical state [25, 49, 64].\nWe wish to thank W. Simeth for fruitful discussions and assistance with the experiments.\nC.B. G.W. and F.R. acknowledge funding by the BMBF via the VEKMAG project. S.P.\nG.W. and C.B. acknowledge funding by the German Research Foundation via SPP2137.\nThis project has received funding from the EMPIR programme co-\fnanced by the Partic-\nipating States and from the European Union's Horizon 2020 research and innovation pro-\ngramme. A.B. and C.P. acknowledge \fnancial support through DFG TRR80 (project F7),\nDFG SPP2137, as well as ERC Advanced Grants 291079 (TOPFIT) and 788031 (ExQuiSid).\nThis work has been funded by the Deutsche Forschungsgemeinschaft (DFG, German Re-\nsearch Foundation) under Germanys Excellence Strategy EXC-2111 390814868.\n\u0003e-mail:christian.back@tum.de\n[1] C. Kittel, Phys. Rev. 73, 155 (1948).\n[2] S. Kimura, M. Hagiwara, H. Ueda, Y. Narumi, K. Kindo, H. Yashiro, T. Kashiwagi, and\nH. Takagi, Phys. Rev. Lett. 97, 257202 (2006).\n[3] S. M uhlbauer, B. Binz, F. Jonietz, C. P\reiderer, A. Rosch, A. Neubauer, R. Georgii, and\nP. B oni, Science 323, 915 (2009).\n8[4] X. Z. Yu, Y. Onose, N. Kanazawa, J. H. Park, J. H. Han, Y. Matsui, N. Nagaosa, and\nY. Tokura, Nature 465, 901 (2010).\n[5] X. Z. Yu, N. Kanazawa, Y. Onose, K. Kimoto, W. Z. Zhang, S. Ishiwata, Y. Matsui, and\nY. Tokura, Nat. Mater. 10, 106 (2011).\n[6] S. Seki, X. Z. Yu, S. Ishiwata, and Y. Tokura, Science 336, 198 (2012).\n[7] N. Nagaosa and Y. Tokura, Nat. Nanotechnol. 8, 899 (2013).\n[8] M. Date, K. Okuda, and K. Kadowaki, J. Phys. Soc. Jpn. 42, 1555 (1977).\n[9] M. I. Kobets, K. G. Dergachev, E. N. Khatsko, A. I. Rykova, P. Lemmens, D. Wulferding,\nand H. Berger, Low Temp. Phys. 36, 176 (2010).\n[10] J. D. Koralek, D. Meier, J. P. Hinton, A. Bauer, S. A. Parameswaran, A. Vishwanath,\nR. Ramesh, R. W. Schoenlein, C. P\reiderer, and J. Orenstein, Phys. Rev. Lett. 109, 247204\n(2012).\n[11] Y. Onose, Y. Okamura, S. Seki, S. Ishiwata, and Y. Tokura, Phys. Rev. Lett. 109, 037603\n(2012).\n[12] T. Schwarze, J. Waizner, M. Garst, A. Bauer, I. Stasinopoulos, H. Berger, C. P\reiderer, and\nD. Grundler, Nat. Mater. 14, 478 (2015).\n[13] M. Mochizuki, Phys. Rev. Lett. 108, 017601 (2012).\n[14] I. K\u0013 ezsm\u0013 arki, S. Bord\u0013 acs, P. Milde, E. Neuber, L. M. Eng, J. S. White, H. M. R\u001cnnow, C. D.\nDewhurst, M. Mochizuki, K. Yanai, H. Nakamura, D. Ehlers, V. Tsurkan, and A. Loidl, Nat.\nMater. 14, 1116 (2015).\n[15] A. K. Nayak, V. Kumar, T. Ma, P. Werner, E. Pippel, R. Sahoo, F. Damay, U. K. R o\u0019ler,\nC. Felser, and S. S. P. Parkin, Nature 548, 561 (2017).\n[16] T. Kurumaji, T. Nakajima, V. Ukleev, A. Feoktystov, T. Arima, K. Kakurai, and Y. Tokura,\nPhys. Rev. Lett. 119, 237201 (2017).\n[17] W. M unzer, A. Neubauer, T. Adams, S. M uhlbauer, C. Franz, F. Jonietz, R. Georgii, P. B oni,\nB. Pedersen, M. Schmidt, A. Rosch, and C. P\reiderer, Phys. Rev. B 81, 041203 (2010).\n[18] A. Bauer and C. P\reiderer, Phys. Rev. B 85, 214418 (2012).\n[19] H. Oike, A. Kikkawa, N. Kanazawa, Y. Taguchi, M. Kawasaki, Y. Tokura, and F. Kagawa,\nNat. Phys. 12, 62 (2016).\n[20] Y. Okamura, F. Kagawa, S. Seki, and Y. Tokura, Nat. Commun. 7, 12669 (2016).\n[21] K. Karube, J. S. White, N. Reynolds, J. L. Gavilano, H. Oike, A. Kikkawa, F. Kagawa,\n9Y. Tokunaga, H. M. R\u001cnnow, Y. Tokura, and Y. Taguchi, Nat. Mater. 15, 1237 (2016).\n[22] K. Karube, J. S. White, D. Morikawa, M. Bartkowiak, A. Kikkawa, Y. Tokunaga, T. Arima,\nH. M. R\u001cnnow, Y. Tokura, and Y. Taguchi, Phys. Rev. Mater. 1, 074405 (2017).\n[23] A. Bauer, A. Chacon, M. Halder, and C. P\reiderer, in Topology in Magnetism , edited by\nJ. Zang, V. Cros, and A. Ho\u000bmann (Springer, 2018) p. 151.\n[24] X. Z. Yu, W. Koshibae, Y. Tokunaga, K. Shibata, Y. Taguchi, N. Nagaosa, and Y. Tokura,\nNature 564, 95 (2018).\n[25] A. Chacon, L. Heinen, M. Halder, A. Bauer, W. Simeth, S. M uhlbauer, H. Berger, M. Garst,\nA. Rosch, and C. P\reiderer, Nat. Phys. 14, 936 (2018).\n[26] M. C. Langner, S. Roy, S. K. Mishra, J. C. T. Lee, X. W. Shi, M. A. Hossain, Y.-D. Chuang,\nS. Seki, Y. Tokura, S. D. Kevan, and R. W. Schoenlein, Phys. Rev. Lett. 112, 167202 (2014).\n[27] S. L. Zhang, A. Bauer, H. Berger, C. P\reiderer, G. van der Laan, and T. Hesjedal, Phys.\nRev. B 93, 214420 (2016).\n[28] S. L. Zhang, A. Bauer, D. M. Burn, P. Milde, E. Neuber, L. M. Eng, H. Berger, C. P\reiderer,\nG. van der Laan, and T. Hesjedal, Nano Lett. 16, 3285 (2016).\n[29] S. L. Zhang, A. Bauer, H. Berger, C. P\reiderer, G. van der Laan, and T. Hesjedal, Appl.\nPhys. Lett. 109, 192406 (2016).\n[30] S. L. Zhang, G. van der Laan, and T. Hesjedal, Nat. Commun. 8, 14619 (2017).\n[31] S. Zhang, G. van der Laan, J. M uller, L. Heinen, M. Garst, A. Bauer, H. Berger, C. P\reiderer,\nand T. Hesjedal, Proc. Natl. Acad. Sci. USA 115, 6386 (2018).\n[32] Y. Okamura, F. Kagawa, M. Mochizuki, M. Kubota, S. Seki, S. Ishiwata, M. Kawasaki,\nY. Onose, and Y. Tokura, Nat. Commun. 4, 2391 (2013).\n[33] N. Ogawa, S. Seki, and Y. Tokura, Sci. Rep. 5, 9552 (2015).\n[34] S. Seki, Y. Okamura, K. Kondou, K. Shibata, M. Kubota, R. Takagi, F. Kagawa, M. Kawasaki,\nG. Tatara, Y. Otani, and Y. Tokura, Phys. Rev. B 93, 235131 (2016).\n[35] I. Stasinopoulos, S. Weichselbaumer, A. Bauer, J. Waizner, H. Berger, M. Garst, C. P\reiderer,\nand D. Grundler, Sci. Rep. 7, 7037 (2017).\n[36] M. Weiler, A. Aqeel, M. Mostovoy, A. Leonov, S. Gepr ags, R. Gross, H. Huebl, T. T. M.\nPalstra, and S. T. B. Goennenwein, Phys. Rev. Lett. 119, 237204 (2017).\n[37] T. Noll and F. Radu, in Proc. MEDSI2016, Barcelona, Spain, September 11-16, 2016 , 9\n(JACoW, Geneva, Switzerland, 2017) pp. 370{373.\n10[38] D. H. Templeton and L. K. Templeton, Phys. Rev. B 49, 14850 (1994).\n[39] V. E. Dmitrienko and V. A. Chizhikov, Phys. Rev. Lett. 108, 187203 (2012).\n[40] A. Bauer and C. P\reiderer, Topological Structures in Ferroic Materials: Domain Walls, Vor-\ntices and Skyrmions , edited by J. Seidel (Springer, 2016) p. 1.\n[41] T. Adams, A. Chacon, M. Wagner, A. Bauer, G. Brandl, B. Pedersen, H. Berger, P. Lemmens,\nand C. P\reiderer, Phys. Rev. Lett. 108, 237204 (2012).\n[42] S. Seki, J.-H. Kim, D. S. Inosov, R. Georgii, B. Keimer, S. Ishiwata, and Y. Tokura, Phys.\nRev. B 85, 220406 (2012).\n[43] J. S. White, K. Pr\u0014 sa, P. Huang, A. A. Omrani, I. \u0014Zivkovi\u0013 c, M. Bartkowiak, H. Berger, A. Ma-\ngrez, J. L. Gavilano, G. Nagy, J. Zang, and H. M. R\u001cnnow, Phys. Rev. Lett. 113, 107203\n(2014).\n[44] L. J. Bannenberg, F. Qian, R. M. Dalgliesh, N. Martin, G. Chaboussant, M. Schmidt, D. L.\nSchlagel, T. A. Lograsso, H. Wilhelm, and C. Pappas, Phys. Rev. B 96, 184416 (2017).\n[45] K. Makino, J. D. Reim, D. Higashi, D. Okuyama, T. J. Sato, Y. Nambu, E. P. Gilbert,\nN. Booth, S. Seki, and Y. Tokura, Phys. Rev. B 95, 134412 (2017).\n[46] S. Seki, Y. Okamura, K. Shibata, R. Takagi, N. D. Khanh, F. Kagawa, T. Arima, and\nY. Tokura, Phys. Rev. B 96, 220404 (2017).\n[47] Note that critical \feld values slightly vary with the applied \feld direction in a bar-shaped\nsample due to demagnetization e\u000bects.\n[48] R. B. Versteeg, I. Vergara, S. D. Schfer, D. Bischo\u000b, A. Aqeel, T. T. M. Palstra, M. Grninger,\nand P. H. M. van Loosdrecht, Phys. Rev. B 94, 094409 (2016).\n[49] M. Halder, A. Chacon, A. Bauer, W. Simeth, S. M uhlbauer, H. Berger, L. Heinen, M. Garst,\nA. Rosch, and C. P\reiderer, Phys. Rev. B 98, 144429 (2018).\n[50] G. van der Laan, C. R. Phys. 9, 570 (2008).\n[51] S. L. Zhang, G. van der Laan, and T. Hesjedal, Phys. Rev. B 96, 094401 (2017).\n[52] I. Stasinopoulos, S. Weichselbaumer, A. Bauer, J. Waizner, H. Berger, S. Maendl, M. Garst,\nC. P\reiderer, and D. Grundler, Appl. Phys. Lett. 111, 032408 (2017).\n[53] H. G. Bauer, P. Majchrak, T. Kachel, C. H. Back, and G. Woltersdorf, Nat. Commun. 6,\n8274 (2015).\n[54] A. Bauer, A. Chacon, M. Wagner, M. Halder, R. Georgii, A. Rosch, C. P\reiderer, and\nM. Garst, Phys. Rev. B 95, 024429 (2017).\n11[55] P. Milde, D. K ohler, J. Seidel, L. M. Eng, A. Bauer, A. Chacon, J. Kindervater, S. M uhlbauer,\nC. P\reiderer, S. Buhrandt, C. Sch utte, and A. Rosch, Science 340, 1076 (2013).\n[56] J. Wild, T. N. G. Meier, S. P ollath, M. Kronseder, A. Bauer, A. Chacon, M. Halder,\nM. Schowalter, A. Rosenauer, J. Zweck, J. M uller, A. Rosch, C. P\reiderer, and C. H. Back,\nSci. Adv. 3, e1701704 (2017).\n[57] S. L. Zhang, G. van der Laan, W. W. Wang, A. A. Haghighirad, and T. Hesjedal, Phys. Rev.\nLett. 120, 227202 (2018).\n[58] W. Legrand, J.-Y. Chauleau, D. Maccariello, N. Reyren, S. Collin, K. Bouzehouane, N. Jaouen,\nV. Cros, and A. Fert, Sci Adv 4, eaat0415 (2018).\n[59] D. Ehlers, I. Stasinopoulos, V. Tsurkan, H.-A. Krug von Nidda, T. Feh\u0013 er, A. Leonov,\nI. K\u0013 ezsm\u0013 arki, D. Grundler, and A. Loidl, Phys. Rev. B 94, 014406 (2016).\n[60] Y. Okamura, S. Seki, S. Bord\u0013 acs, A. Butykai, V. Tsurkan, I. K\u0013 ezsm\u0013 arki, and Y. Tokura,\nPhys. Rev. Lett. 122, 057202 (2019).\n[61] Y. Tokunaga, X. Z. Yu, J. S. White, H. M. R\u001cnnow, D. Morikawa, Y. Taguchi, and Y. Tokura,\nNat. Commun. 6, 7638 (2015).\n[62] R. Takagi, D. Morikawa, K. Karube, N. Kanazawa, K. Shibata, G. Tatara, Y. Tokunaga,\nT. Arima, Y. Taguchi, Y. Tokura, and S. Seki, Phys. Rev. B 95, 220406 (2017).\n[63] K. Karube, J. S. White, D. Morikawa, C. D. Dewhurst, R. Cubitt, A. Kikkawa, X. Yu,\nY. Tokunaga, T. Arima, H. M. R\u001cnnow, Y. Tokura, and Y. Taguchi, Sci. Adv. 4, eaar7043\n(2018).\n[64] F. Qian, L. J. Bannenberg, H. Wilhelm, G. Chaboussant, L. M. Debeer-Schmitt, M. P.\nSchmidt, A. Aqeel, T. T. M. Palstra, E. Br uck, A. J. E. Lefering, C. Pappas, M. Mostovoy,\nand A. O. Leonov, Sci. Adv. 4, eaat7323 (2018).\n12b\nbeamline\nPM2sample\ndiode2θω\nH / Hc2\nT (K)zero-field \ncooled\nconicalfp\nhelical pmxy\nx-rays(a) (b) vector magnet\n(c) (d)\n(e) (f)helical      conical     \nskyrmion frozen \nskyrmion Diode height z d (mm)\nSample angle ω (°)\nI (a.u.)[001]\nzd\n60 30 001\n-0.500.5\n0.50\n-0.5(001)(δ01)\n(0δ1) (δ01)(0δ1)\n(001)\n(001) (001)Η || [110]\nΗ || [001]skyrmion\nlattice\nΗ || [001]30 K 70 mT\n30 K 44 mT 56 K 44 mT30 K 70 mTFIG. 1. Setup and typical REXS data. (a) Schematic view of the experimental setup. (b) Schematic\nzero-\feld cooled magnetic phase diagram of Cu 2OSeO 3. (c) Reciprocal space map around the (001)\nBragg peak in the helical state. (d) Reciprocal space map in the conical state. The magnetic \feld\nis applied along the [110] axis, i.e., within the sample plane. (e),(f) Reciprocal space map in the\nskyrmion lattice state. The magnetic \feld is applied along the [001] axis, i.e., perpendicular to\nthe sample plane. Under \feld cooling, the skyrmion lattice may be metastable frozen-in to lower\ntemperatures.\n13(a) (b)\n(d)(c)Bragg intensity I (a.u.)\nμ0H (mT) μ0H (mT)onoff\nIΔ/IΣ (%) S11 (a.u.) Frequency f/fc2\nH || [001]T = 15 K+q\n-qREXS-FMR T = 15 K\nFMR T = 5 Kf = 4.5 GHzFIG. 2. REXS-FMR on the structural (001) peak. (a) Intensity of the Bragg peak as a function\nof magnetic \feld applied parallel to the incoming X-ray beam, tracking the magnetization of the\nsample. Data are recorded at each \feld step with microwave excitation on (red curves) and o\u000b\n(gray curves). (b) Relative change of the intensity under excitation. (c) Re\rected microwave power,\nS11, as a function of \feld. The signature around zero \feld is attributed to the helimagnetic states.\n(d) Resonant modes for magnetic \feld along the [001] axis at low temperature. Frequencies are\nnormalized to their value at Hc2. Resonance \felds inferred from REXS-FMR (solid symbols) are\ncompared to resonance frequencies inferred from microwave spectroscopy (open symbols).\n14-100 0 1000.01.02.03.04.0\n-50 0 500.00.20.40.60.8\nonoff\n0 2 4 6020406080\n020406080-q\n+qccw bre\ncwSatellite intensity I (a.u.)\nμ0H (mT) μ0H (mT)(a)\n(b)(c)\n(d)\nf (GHz) f (GHz)μ0H = 0 mT\nH || [001]\nT = 15 Kμ0H = 33 mT\nH || [001]\nT = 56 K3.003.253.503.754.004.254.505.00f (GHz)\n1.201.401.501.601.701.801.902.002.252.50f (GHz)IΔ / Ioff (%)\n0 2 4 6T=15 K T = 56 KSatellite intensity I (a.u.)excitationIΔ / Ioff (%)FIG. 3. REXS-FMR on the magnetic satellites. (a) Intensity of a helical satellite as a function\nof magnetic \feld for di\u000berent excitation frequencies. Data are recorded at each \feld step with\nmicrowave excitation on (red curves) and o\u000b (gray curves). The large value at 3.5 GHz is attributed\nto changes of the spin texture, see text for details. (b) Relative reduction of the helical satellite\nintensity in zero \feld as a function of frequency. The sketch illustrates the position of the detector\ndiode. Statistical error bars are typically much smaller than the symbol size. (c) Intensity at\nthe reciprocal space position of a skyrmion lattice satellite. Around zero \feld, the tail of the\nbroad helical satellite is observed at the skyrmion satellite position. (d) Relative reduction of the\nskyrmion satellite intensity at a \feld of 35 mT as a function of frequency.\n15The citations in this appended supplement refer to the ones in the main text given above.\nFerromagnetic resonance with magnetic phase selectivity by\nmeans of resonant elastic x-ray scattering on a chiral magnet -\nsupplementary material\nWhen the Bragg-peak intensity is recorded as a function of externally applied magnetic\n\feld a magnetization trace is observed. The observed intensity change is however not directly\nproportional to the change of magnetization due to higher order terms. To extract the actual\nmagnetization trace the theory of van der Laan et al. is applied in the following [50].\nThe scattering cross section I(q) is given by ([50] Equation (15))\nI(q) =Ic(q) +Im(q) +Ii(q) (1)\nwith the scattering vector q=kf\u0000ki, the incident and \fnal photon wave vector kiand\nkf(and their unit vectors ^kiand^kf). The contribution of charge scattering Icis given by\n([50] Equation (16))\nIc(q) =1\n2\f\f\f~f0~\u001a\f\f\f2\u0014\nP\u001b+P\u0019\f\f\f^kf\u0001^ki\f\f\f2\u0015\n(2)\nwith the Fourier transformation of the charge density ~ \u001aand the monopole contribution of\nthe energy dependent resonance amplitude ~f0. Further is P\u001b=P0+P1andP\u0019=P0\u0000P1with\nthe Poincar\u0013 e vector P. For circularly polarized light PT= (1;0;0;\u00061) which is assumed for\nthe following considerations. Equation 2 is independent of the magnetization M(and its\nFourier transformation ~M) of the sample.\nThe contribution of pure magnetic scattering Imfor~Minside the scattering plane is given\nby ([50] Equation (19))\nIm(~Mqk) =1\n2\f\f\f~f1\f\f\f2\u0014\nP\u001b\f\f\f^kf\u0001~M\f\f\f2\n+P\u0019\f\f\f^ki\u0001~M\f\f\f2\u0015\n(3)\nwith the Fourier transformation of the magnetic dipolar contribution to the energy depen-\ndent resonance amplitude ~f1. Finally, the interference term between charge and magnetic\nscatteringIifor circular polarization with ~Minside the scattering plane is given by ([50]\nEquation (21))\n16Ii(~Mqk) =P2Imh\n~f\u0003\n0~\u001a\u0003~f1~\u001a\u0003\u0010\n^ki\u0001~M+ (^kf\u0001~M)(^kf\u0001^ki)\u0011i\n(4)\n+P3Reh\n~f\u0003\n0~\u001a\u0003~f\u0003\n1\u0010\n^ki\u0001~M+ (^kf\u0001~M)(^kf\u0001^ki)\u0011i\nFor the case shown in Fig. 2(a) of the main text, the magnetization is collinear to the\nincident beam. The photon wave vectors kiandkfare \fxed to the (001)-Bragg condition\n(2\u0012= 95:8\u000e). As a function of M(\u00160Hext) the charge scattering Ic=Acis constant. Pis\nassumed for circularly polarized light, meaning P\u001b=P\u0019= 1,P2= 0 andP3=\u00061.\nEquation 1 can then be written as\nI(~Mqk) =Ac+Am\u0012\f\f\f^kf\u0001~M\f\f\f2\n+\f\f\f^ki\u0001~M\f\f\f2\u0013\n\u0006Ai\u0010\n^ki\u0001~M+ (^kf\u0001~M)(^kf\u0001^ki)\u0011\n(5)\nwith the constants Am=1\n2\f\f\f~f1\f\f\f2\nMsandAi= Reh\n~f\u0003\n0~\u001a\u0003~f\u0003\n1i\nMs. The net magnetization\nM(\u00160Hext) along ^kiis assumed to have the following form\nM(\u00160Hext) =8\n><\n>:^kiHext\nHc2Ms;forjHextj<H c2\n^kiMs; otherwise(6)\nwith the second critical \feld Hc2and the saturation magnetization Ms. ForjHextj<H c2\nthis is obviously not true in case of the non-collinear ferromagnet that is investigated, the\nnet magnetization along the \feld however is well-described by this behaviour as can be seen\ne.g. from [41]. The experimental data in Fig. 2(a) of the main text can now be \ftted\nto equation 5 using the parameters Ac,Am,AiandHc2. Fig. 4(a) shows again the data\nfrom Fig. 2(a) of the main text and the resulting \ft for the excitation o\u000b data. The \ft\ngives a ratio of pure magnetic scattering to the interference term ofAi\nAm= 3:2. From the \ft\nparameterHc2the magnetization trace can be plotted using equation 6. Getting Hextfrom\nthe plot where the intensity of the rf o\u000b data (in the non-saturated part) is equal to the\nreduced intensity of the rf on data, one can use Fig. 4(b) to estimate the reduction \u0001 Mof\nMin resonance. A mean value of \u0001 M=Ms\u00190:065 is obtained. The scattering geometry is\nshown in Fig. 4(c). The contributions of charge scattering, pure magnetic and interference\nscattering obtained from the \ft is shown in Fig. 4(d-f).\n17-200 0 200\n0Hext(mT)0.60.811.21.41.61.82I (a.u.)FMR on\nFMR off\nFit off\n-200 0 200\n0Hext(mT)-1-0.8-0.6-0.4-0.200.20.40.60.81M/MsM/M s= 0.92\nM/M s= 0.95\n-1 -0.5 0 0.5 1\nH/Hc200.511.522.5Ii (a.u.)\n-1 -0.5 0 0.5 1\nH/Hc200.050.10.150.20.25Im (a.u.)\n-1-0.500.51Ic (a.u.)ki kf\nM\n-1 -0.5 0 0.5 1\nH/Hc2\nAi = 0.73Ac = 1.19\nAm = 0.23θ=47.9°(a) (b) (c) (d)\n(e) (f)FIG. 4. (a)Magnetization trace obtained by tracking the Bragg peak intensity with applied\nmagnetic \feld (black). The red curve shows the same trace with an applied microwave excitation\nof 4.5 GHz (Compare Fig. 2(a) of the main text). Also included is a \ft to the theoretical formula\n(orange). (b)Magnetization as a function of externally applied magnetic \feld obtained from\n\ft.(c)Scattering geometry present in (a). (d-f) Contribution of charge, pure magnetic and\ninterference scattering to the Bragg peak intensity.\n18"    },    {        "title": "1902.08365v1.A_highly_sensitive_magnetic_sensor_using_a_2D_van_der_Waals_ferromagnetic_material.pdf",        "content": " \n \n1 \n A highly sensitive magnetic  sensor using a 2D van der Waals ferromagnetic material  \nValery Ortiz Jimenez1, Vijaysankar Kalappattil1, Tatiana Eggers1, Manuel Bonilla1, Sadhu \nKolekar1, Pham Thanh Huy2, Matthias Batzill1, and Manh -Huong Phan1,a) \n1 Department of Physics, University of South Florida, Tampa, FL 33620, USA  \n2 Phenikaa Institute for Advanced Study, Phenikaa University, Yen Nghia, Ha -Dong District, \nHanoi 1000, Viet Nam  \n \nTwo-dimensional (2D) van der Waals f erromagnetic materials  are emerging as promis ing \ncandidate s for applications  in ultra-compact spintronic nanodevices, nanosensors, and \ninformation storage . Our recent discovery of the strong room temperature ferromagnetism in \nsingle layers  of VSe 2 grown on graphite or MoS 2 substrate has opened new opportunities to \nexplore these ultrathin  magnets for such applications. In this paper, we present  a new type of \nmagnetic sensor that ultilizes the single layer VSe 2 film as a highly senstive magnetic core. The \nsensor relies in  changes in resonan ce frequency of the LC circuit composed of a soft \nferromagnetic microwire coil that contains the ferro magnetic VSe 2 film subject to applied DC \nmagnetic fields.  The senstivity of the sensor reaches  an extremely high value of 16×106 Hz/Oe, \nmaking it an excellent candidate for a wide range of magnetic sensing applications.   \n \nPACS:  75.50.Kj  \nKeywords: Monolayer  magnets; Magneto -LC resonance; Magnetic sensors  \na) Corresponding author:  phanm@usf.edu  \n \n  \n \n2 \n Since intrinsic long -range ferromagnetic order was realized in 2017 with bulk exfoliated \nmonolayers of Cr 2Ge2Te6 and CrI 3,1,2 the potential of  two-dimensional (2D) van der Waals \nmagnets  has excited  the scientific community .3-11 While these 2D magnets have demonstrated  \ntheir usefulness in magnetoelectric devices, their technological applications are restricted  to low \ntemperature s (< 100 K).6-10 In contrast, recent discoveries  of the room temperature (RT) \nferromagnetism in transition metal dichalcogenide (TMD) monolayers of VSe 2 and MnSe 2 \ngrown by van der Waals epitaxy on various substrates  (graphite, MoS 2, GaSe) may enable \napplications at ambient temperature .3,4,10  \nIn this paper, we present a new type of magnetic sensor that intergrates a single layer \nVSe 2 film within a Co-rich micro wire-based coil. We note that i nduction coil sensors have been \nwidely used due to the simplicity of their construction  and a well known transfer function.12 It is \nestablished that the sensivity of a n induction coil is limited by the number of windings; the \ngreate r the number of turns the hig her the sensitivity . This quickly becomes a problem for \nmodern applications where it is desirable to  limit the size of sensors . Adding a soft ferromangetic \ncore with high relative permeability can largely increase the sensitivity of a coil, and hence allow  \nfor smaller sensor sizes .13 A similar working principle has recently been  used in the design of \nMMC -LC resonator sensors , but cobalt rich magnetic microwires are used instead of non -\nmagnetic conductors such as copper .14 The working principle of the sensor reported in this paper \nis fundamentally different than that of a conventional induction coil sensor . It relies on the \nchanges in resonance frequency caused by external magnetic fields, instead of simply measuring \nthe induction of the coil.  It is also different from the MMC -LC resonator sensor14 that relies in \nchanges in impedance  of the micr owire caused  by external magnetic fields .   \n \n3 \n  A simple model for a coil sensor is a lumped element representation of a non -ideal \ninductor. Winding cylindrical conductors close to each other will introduce parasitic elements \n𝑅𝑝𝑎𝑟 and 𝐶𝑝𝑎𝑟, such that the non -ideal inductor can be represented as a series combination of an \nideal inductor 𝐿 and 𝑅𝑝𝑎𝑟, in parallel with 𝐶𝑝𝑎𝑟.15,16,17 The impedance  of the coil 𝑍𝑐𝑜𝑖𝑙 can then \nbe written as   \n𝑍coil=(𝑍Rpar+𝑍𝐿) || 𝑍Cpar     (1) \n𝑍coil=1\n1\n𝑅par +𝑗𝜔L+1\n1/𝑗𝜔𝐶par      (2) \n𝑍coil=𝑅𝑝𝑎𝑟+𝑗𝜔[L(1−𝜔2LC𝑝𝑎𝑟)−𝐶par𝑅𝑝𝑎𝑟2]\n(1−𝜔2L𝐶𝑝𝑎𝑟)2+(𝜔𝐶𝑝𝑎𝑟𝑅𝑝𝑎𝑟)2,   (3) \nwhere 𝜔 is the angular frequency, and 𝑗 is the imaginary unit . Resonance will occur when the \ninductive reactance (𝑋𝐿) and the capacitive reactance (𝑋𝐶) are equal in magnitude but differ in \nphase by 180 degrees. At this point very little current flows through the wire , the impedance \n𝑍𝐶𝑜𝑖𝑙 becomes very large and self -resonance is achieved.18,19 The resonance frequency is given \nby  \n𝑓0=√1−(Rpar2Cpar/L)\n2𝜋√LCpar,       (4) \nwhere 𝑓0 is the resonance frequency.  We expect a self -resonant behavior in the MMC as well , \nbut the resonance frequency will be different than that of the inductor since the wire is now a \nmagnetic material. We must  also consider the effects of a ferromagnetic core on the sensor. The \ncore will modify the relative permeability in the space within  the coil, and this will in turn \nchange the flux through the coil and hence affect the inductance . Since the permeability is field \ndependent, an external magnetic field will modify the permeability of the core and the resonance  \n \n4 \n frequency with it.  The sensitivity of the sensor is defined as the rate of change of the resonance \nfrequency with respect to the external DC magnetic field , \n                                                                 𝑆𝑒𝑛𝑠𝑖𝑡𝑖𝑣𝑖𝑡𝑦 = 𝑑𝑓0\n𝑑𝐻.                                                      (5) \nThe Q factor is also calculated by measuring the bandwidth , BW, and the resonance frequency \nusing the following relation,  \n 𝑄= 𝑓0\n𝐵𝑊                                                                (6) \nTo make the magnetic sensor, a Co 69.25Fe4.25Si13B12.5Nb1 wire (diameter, ~60 m) was  \nwound into a 15 -turn, 10 mm long coil with a 5 mm internal diameter . The fabrication details \nand material characterization of the microwire can be found elsewhere .20,21 Monolayer  VSe 2 was \nused as the core of the coil . The  single layer films of VSe 2 were grown on  graphite (HOPG) and \nsingle crystal MoS 2 by molecular beam epitaxy (MBE) ; the details of which have been reported \nin our previous work.3 Since both monolayer VSe 2 samples grown on MoS 2 and HOPG show \nsimilar magnetic and sensing properties, in this paper we only report on the properties of \nmonolayer VSe 2 grown on MoS 2. Figure 1(a) shows a typical scanning tunnel microscopy ( STM ) \nimage of the VSe 2 film. It can be se en that the single layer of VSe 2 was epitaxially grown on the \nMoS 2 substrate. The magnetization versus magnetic field ( M-H) curve s, measured by a vibrating \nsample magnetometer (VSM)  on VSe 2 grown on both HOPG and MoS 2, show a soft \nferromagnetic characteristic (small coercive field , small remanent magnetization, and high \nsaturation magnetization) of the single layer VSe 2 film at room temperature  (Fig. 1 (b)), which is \ndesirable for its use as the core of the sensor  that itself operates at room temperature . The core \nwas ~14 mm long and ~0.4 mm wide, allowing us to insert and remove the core with ease. The \nsensor was mounted on a test fixture made of a dielectric material on top of a copper ground  \n \n5 \n plane; SMA connector port s were soldered  to the ground plane  on two ends  and the two leads of \nthe inductor were soldered to the center pin of each connector.   \n One port reflection measurements were performed with an HP 4191A impedance \nanalyzer  over the frequency range 1 – 200 MHz . A simple short -open -load (SOL) calibration \nwas made . The test fixture was connected to the impedance analyzer through a coaxial cable, and \nit was terminated with a 50 ohm cap. T he effects of the cable were removed by the calibration.  \nThe impedance  (Z), resistance  (R), and rea ctance  (X) are calculated  from the reflection \nmeasurement . An external DC magnetic field was generated by a Helmholtz coil . A diagram of \nthe setup is shown in Fig. 1(c,d). The external field was applied transverse to the coil axis  as the \nfrequency was swept.  A frequency sweep was performed for every value of the field which was \nswept from -30 to 30 Oe  in steps of 1 Oe .  \nA set of measurements is performed to characterize the sensor; the resonance frequency \nis found for every value of the field. Resonance is determined by the zero crossing of the \nreactance.  To see the effects of the monolayer core we show in Fig. 2(a)  the reactance of the coil \nwith and without the monolayer core  at zero field . The  presence of the monolayer shifts the  \nresonance frequency of the sensor by a few megahertz.  We then applied the external field and \nobserved how the reactance curve is shifted along with the corresponding resonance frequency as \nshown in Fig. 2(b), for the coil with the monolayer in its core.  The reactance of the coil being \nproportional to the inductance changes drastically with frequency and applied field , as well as \nthe resonance frequency  (see Fig. (2c) ). \nThe resonance frequency is deter mined for every value of the f ield. Figure 3(a) shows the \nchange in resonance frequency as a function of applied field. Unique values of the resonance \nfrequency exist for fields smaller than 5 Oe. At higher fields,  the magnetic wire  saturates , and 𝑓0  \n \n6 \n changes very slowly , rending the sensor inoperable.  A fit of the right branch of the  𝑓0 plot is \nshown in Fig. 3(b)  as well as the sensitivity  of the sensor . Values as large as  16×106 Hz/Oe  are \nachieved  for the sensitivity . The massive value of the sensitivity shows that the present  sensor \ncan be used to accurately determine the magnitude of small magnetic fields as well as subtle \nvariations in them . This brings forth a wide range of applications  such as  bio-detection , where \nmagnetic nanoparticles  are used as bio -markers  for pathogen detection  and quantifying low \nparticle concentrations is neccessay .22 Finally, we have determined the Q factor of the sensor  at \ndifferent values of external DC magnetic field , as shown in Fig. 4 . We have found that for very \nsmall fields, smaller than 1 Oe, the Q factor is 0.6; as the field increases it reaches a minimum \nvalue of 0.36 at 3 Oe.  Most of the losses are associated with the magnetic wire itself ; the core \nslightly decreases the Q factor by less than 0.1. This is consistent with  the soft ferromagnetic \ncharacteristic of monolayer  VSe 2 (Fig. 1(b) ), which predicts minimal magnetic losses.  \nIn summary,  a highly  sensitive magnetic field sensor was built using a magnetic \nmicrowire coil as the sensing element with a single layer  VSe 2 film as  the core.  Sensitivity as \nlarge as 16×106 Hz/Oe for very small fields was obtained.  We successfully  implement ed VSe 2 \nmonolayers in room temperature magnetic field se nsors . Furthermore, core losses were shown to \nbe minimal which is desirable for sensor design.  \nACKNOWLEDGMENTS  \nWork was support ed by  the VISCOSTONE USA under Grant  No. 1253113200  and the National \nScience Foundation under grants DMR -1701390  and ECCS -1608654 . T.E. also acknowledges \nsupport from NASA  Florida Space Grant Consortium (FSGC)  under Award No. 1253 -1124 -00.  \n  \n \n7 \n REFERENCES  \n1 B. Huang, G.  Clark, E. Navarro -Moratalla, D.  R. Klein, R. Cheng, K.  L. Seyler, D. Zhong, E. \nSchmidgall, M.  A McGuire, D.  H Cobden, W. Yao, D.  Xiao, P.  Jarillo -Herrero, and X.  Xu, \nNature  546, 270 (2017).  \n2 C. Gong, L. Li, Z. Li, H. Ji, A. Stern, Y. Xia, T. Cao, W. Bao, C. Wang, Y. Wang, Z. Q. Qiu, \nR. J. Cava, S. G. Louie, J. Xia, and  X. Zhang, Nature  546, 265 (2017).  \n3 M. Bonilla , S. Kolekar, Y.J. Ma, H. Coy Diaz, V. Kalappattil, R. Das, T. Eggers, H.R. \nGutierrez, M. H. Phan, and M. Batzill,  Nature Nanotechnol.  13, 289 (2018) . \n4 Dante J. O’Hara,  T.C. Zhu, A.H. Trout,  A.S. Ahmed,  Y. K. Luo, C.H. Lee, M.R. Brenner,  S. \nRajan,  J.A. Gupta,  D.W. McComb, and  R.K. Kawakami , Nano Lett.  18, 3125 ( 2018 ). \n5 Y.J. Deng, Y .J. Yu, Y .C. Song, J .Z. Zhang, N .Z. Wang, Y .Z. Wu, J .Y. Zhu, J . Wang, X .H. \nChen, and Y .B. Zhang, Nature  563, 94 (2018) . \n6 Jiang, S., Shan, J., Mak, K.F., Electric -field switching of two -dimensional van der Waals \nmagnets.  Nat. Mater.  17, 406-410 (2018 ). \n7 B. Huang, G. Clark, D.R.  Klein, D. MacNeill, E. Navarro -Moratalla, K.L. Seyler, N. Wilson,  \nM.A.  McGuire,  D.H.  Cobden, D. Xiao, W. Yao, P. Jarillo -Herrero, X. Xu, Nat. Nanotechnol.  13, \n544 (2018) . \n8 T.C. Song,  X.H. Cai, M. Wei-Yuan Tu,  X. Zhang,  B. Huang,  N. P. Wilson,  K.L. Seyler,  L. \nZhu, T. Taniguchi,  K. Watanabe,  M.A. McGuire,  D.H. Cobden,  D. Xiao,  W. Yao,  X.D. Xu, \nScience   2018: DOI: 10.1126/science.aar4851  \n9 M. Umar Farooq and J . Hong, npj 2D Materials and Applications (2019) 3:3  \n10 2D magnetism gets hot, Nature Nanotechnology  13, 269 (2018)  \n11 K.S. Burch,  D. Mandrus  and J.G. Park, Magnetism in two -dimensional van der Waals \nmaterials, Nature  563, 47 (2018) .  \n \n8 \n 12 Tumanski, Slawomir. (2007). Induction coil sensors —A review. Measurement Science & \nTechnology - MEAS SCI TECHNOL. 18. 10.1088/0957 -0233/18/3/R01.  \n13 J. Devkota, T. Luong, J. Liu, H. Shen, F. Qin, J. Sun, H. Srikanth, M.H. Phan , J. Appl. Phys.  \n116, 234504  (2014) . \n14 O. Thiabgoh, T. Eggers, and M. H. Phan, Sensors and Actuators A  265, 120 (2017 ) \n15 G. Grandi, M.K. Kazimierczuk, A. Massarini, U. Reggiani, Stray capacitances of single -layer \nsolenoid air -core inductors, IEEE Trans. Ind. Appl.  35, 1162  (1999) . \n16 C. R. Paul, Introduction to electromagnetic compatibility, second ed., John -Wiley, New Jersey, \n2006.  \n17 T.L. Floyd, D. Buchla, Electronics Fundamentals: Circuits, Devices & Applications, eight ed., \nPrentice Hall Press, New Jersey, 2009.  \n18 R.K. Wangsness, Electromagnetic fields, second ed., John -Wiley & Sons, New York, 1986.  \n19 P. Scherz, Practical electronics for inventors, second ed., McGraw -Hill, New York, 2000 . \n20 H. Shen, J. Liu, H. Wang, D. Xing, D. Chen, Y. Liu, J. Sun, Phys. Status.  Solidi.  A 211, 1668  \n(2014) . \n21 H. Wang, D. Xing, X. Wang, J. Sun, Metall. Mater. Trans. A  42A, 1103 -1108  (2010) . \n22 I. Giouroudi and G. Kokkinis, Nanomaterials  7, 171 (2017).  \n \n \n \n \n \n \n \n \n  \n \n9 \n Figure captions  \n \nFIG. 1. (a) A typical STM image of the single layer VSe 2 film grown on single crystal MoS 2; (b) \nthe magnetic hysteresis ( M-H) loop of the film measured at 300 K; (c) A block diagram of the \nmeasurement setup for testing performances of the sensor; (d) the LC-circu it composed of a \nmagnetic microwire with the VSe 2 film inserted in it.  \nFIG. 2. (a) Frequency dependence of the reactance of the sensor with and without the magnetic \nVSe 2 film in the absence of an external magnetic field; (b) Frequency dependence of the \nreactance of the sensor with the magnetic VSe 2 film shows large shifts in the resonant frequency \nwith respect to applied magnetic fields; (c) 2D surface plot shows the magn etic field and \nfrequency dependences of reactance.  \nFIG. 3. (a) The resonant frequency of the sensor ( f0) changes as a function of magnetic field. The \nextremely large change is observed at low fields; (b) A fit of the magnetic field dependent \nresonant freq uency f 0(H) data is displayed, with the inset showing the magnetic field dependence \nof the sensor sensitivity. The extremely large value of the sensor sensitivity indicates the \nusefulness of the sensor for a wide range of sensing applications.  \nFIG. 4. Magnetic field dependence of the Q factor of the sensor. The significant change in Q is \nobserved at low field regime.  \n   \n \n10 \n Figure 1  \n \n \n  \n \n \n11 \n Figure 2 \n \n \n \n \n12 \n Figure 3  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n13 \n Figure 4  \n \n \n \n"    },    {        "title": "0912.4545v1.Microscopic_Coexistence_of_Ferromagnetism_and_Superconductivity_in_Single_Crystal_UCoGe.pdf",        "content": "arXiv:0912.4545v1  [cond-mat.supr-con]  23 Dec 2009Typeset with jpsj2.cls <ver.1.2> Full Paper\nMicroscopic Coexistence of Ferromagnetism and Supercondu ctivity in Single-Crystal\nUCoGe\nTetsuyaOhta1, Taisuke Hattori1∗, KenjiIshida1,2†, Yusuke Nakai1,2, Eisuke Osaki3, Kazuhiko Deguchi3,\nNoriaki K. Sato3, and Isamu Satoh4\n1Department of Physics, Graduate School of Science, Kyoto Un iversity, Kyoto 606-8502, Japan.\n2Transformative Research Project on Iron Pnictides (TRIP), Japan Science and Technology Agency (JST), Tokyo 102-0075,\nJapan.\n3Department of Physics, Graduate School of Science, Nagoya U niversity, Nagoya 464-8602, Japan.\n4Institute for Materials Research, Tohoku University, Send ai 980-8577 Japan.\nUnambiguous evidence for the microscopic coexistence of fe rromagnetism and superconduc-\ntivity in UCoGe ( TCurie∼2.5 K and TSC∼0.6 K) is reported from59Co nuclear quadrupole\nresonance (NQR). The59Co-NQR signal below 1 K indicates ferromagnetism throughou t the\nsample volume, while nuclear spin-lattice relaxation rate 1/T1in the ferromagnetic (FM) phase\ndecreases below TSCdue to the opening of the superconducting(SC) gap. The SC sta te was\nfound to be inhomogeneous, suggestive of a self-induced vor tex state, potentially realizable in\na FM superconductor. In addition, the59Co-NQR spectrum around TCurieshow that the FM\ntransition in UCoGe possesses a ﬁrst-order character, whic h is consistent with the theoreti-\ncal prediction that the low-temperature FM transition in it inerant magnets is generically of\nﬁrst-order.\nKEYWORDS: ferromagnetic superconductor, U-based heavy-f ermion, UCoGe, Nuclear Quadrupole Reso-\nnance\nAfter the discovery of superconductivity in UGe 2\nunder pressure,3the coexistence of superconductivity\nand ferromagnetism becomes one of the major topics\nin condensed-matter physics. This is because ferromag-\nnetism and spin-singlet superconductivity are thought\nto be mutually exclusive.1,2In the presence of a large\nsplitting between the majority and minority spin Fermi\nsurfaces, as in a ferromagnetic (FM) state, more-exotic\nspin-triplet superconductivity is allowed, in which paral-\nlel spins pair within each spin Fermi surface. While FM\nsuperconductors such as UIr4and URhGe5has recently\nbeen demonstrated to occur experimentally, proof that\nthe same charge carriers participate simultaneously in\nboth phenomena has remained elusive.\nIn 2007, new ambient-pressure ferromagnetic (FM)\nsuperconductor UCoGe was discovered by Huy et al.6\nUCoGe is a weak ferromagnet with TCurie= 3 K and the\nordered moments µs= 0.03µB, and shows superconduc-\ntivity at the transition temperature TSC= 0.8 K,6high-\nest within FM superconductors. In order to investigate\nthe correlation between ferromagnetism and supercon-\nductivity,nuclearquadrupoleresonance(NQR)measure-\nments are ideally suited, since they provide microscopic\ninformation about the electronic and magnetic proper-\nties without applying external ﬁelds. In a magnetically\norderedstate,theNQRsignalsplitsorshiftsduetointer-\nnal ﬁelds at the nuclear site, and the nuclear spin-lattice\nrelaxation rate 1 /T1provides site-selective information\nabout the density of states at the Fermi level and thus\nabout the superconducting (SC) gap structure. UCoGe\nis aFM superconductorsuitable forNQR measurements,\nsince it contains an NQR-active element of59Co.\n∗E-mail address: t.hattori@scphys.kyoto-u.ac.jp\n†E-mail address: kishida@scphys.kyoto-u.ac.jpIn the previous letter, we reported59Co-NQR studies\nin a polycrystalline UCoGe with TCurie= 2.5 K and the\nSC onset temperature Tonset\nSC= 0.7 K.11We found in-\nhomogeneous ferromagnetism below TCuriein the poly-\ncrystalline sample, from the observation of the FM and\nnonmagnetic NQR spectra at lowest temperature. In ad-\ndition, the SC anomaly was observed in 1 /T1measured\nin both NQR spectra, suggesting that superconductiv-\nity exists both in the FM and paramagnetic (PM) state.\nHowever, to investigate intrinsic nature of UCoGe,59Co-\nNQR studies on a high-quality single crystal are highly\ndesired.\nAnother interest in UCoGe is its FM transition. The\nthermal FM transition at TCuriein zero ﬁeld is an well-\nknown example of second-order phase transition. How-\never, Belitz et al.theoretically suggested that the suﬃ-\nciently low-temperature phase transition in itinerant fer-\nromagnetism is generically of the ﬁrst order.9Experi-\nmental results supporting this have been reported in a\nfew ferromagnets.7,8In order to investigate the nature of\nthe quantum FM transition occurring at T= 0, UCoGe\nis one of the best ferromagnets, since TCurieof UCoGe is\nas low as 2.5 K.\nFor the abovepurposes, we have performed59Co-NQR\nmeasurements on two samples: one is the polycrystalline\nsample reported in literature11and the other is a 55\nmg single crystal with 1 .65×1.65×1.89 mm3dimen-\nsion, grown by the Czochralskimethod in a tetra-arcfur-\nnace. The sample preparation and characterization will\nbe published elsewhere.13The residual resistivity ratio\nalong the aaxis for the single-crystal samples is 20. The\nFM transition temperature TCurieof the single-crystal\nsample was evaluated to be 2 .45±0.1 K from the Ar-\nrot plots shown in Fig. 1 (a), where magnetization ( M)\n12 J. Phys. Soc. Jpn. Full Paper Author Name\nwas measured in ﬁelds ( H) parallel to the caxis. Ising-\ntype anisotropic behavior was observed with the easy\naxis along c. The magnitude of m0was determined to be\n0.07µB/Co by extraporating at T= 0. The acsuscep-\ntibility, shown in Fig.1 (b), exhibits onset and midpoint\nSC transition temperatures, 0.70 and 0.57 K respectively\nin the single-crystal sample. The FM and SC properties\nof our single crystal are in good agreement with those\nobtained by Huy et al10and D. Aoki et al.18\n0.0 0.2 0.4 0.6 0.8 1.00.00.51.01.52.0\n0.2 0.4 0.6 0.8 1.0(a) \n2.3K \nH/M  (T f.u./ µB)M  2  (10  -3   µB2 / f.u. 2)UCoGe \nsingle crystal\nH // M // c2.1K \n2.2K \n2.4K \n2.5K \n2.6K \n2.7K \nsingle crystal \n(b) δχ ac ∝ -δf   (a. u.) \n  \nT (K)  TSC  ～ 0.57 K\nFig. 1. (Color online)(a) Arrot plots of magnetization isot herms\nmeasured in ﬁelds H/bardblcat temperatures from 2.1 to 2.7 K.\n(b) Temperature dependence of the ac susceptibility χacin zero\nﬁeld. Inset: Crystal structure of UCoGe (orthorhombic TiNi Si\nstructure), drawn in Vesta.21\nFigure 2 (a) shows the59Co-NQR ( I= 7/2) spec-\ntrum observed at 4.2 K in the PM state of the poly-\ncrystalline sample, exhibiting three peaks from the m=\n±1/2↔ ±3/2(ν1),±3/2↔ ±5/2(ν2), and±5/2↔\n±7/2(ν3) transitions. From the resonance peaks, the\nNQR frequency of the principal-axis component of the\nelectric ﬁeld gradient (EFG), |νzz|, is evaluated to be\n2.85 MHz, and the asymmetric parameter ηdeﬁned as\n(νxx−νyy)/νzzis 0.52.The direction ofthe principal axis\nof the EFG is calculated to be tilted by 10◦from the a\naxis in the acplane. The EFG principal axis is parallel\nto the zig-zag chain consisting of the U atoms.12In the\nFM state of the polycrystalline sample, the complicated\n59Co-NQR spectrum shown in Fig. 2 (b) was observed\nat 95 mK. We found signals around 4 and 7 MHz, which\ncould not be observed previously.11When the nuclear4 5 6 7 8 9\n 59 Co-NQR Intensity (arb. units.) \nFrequency (MHz) T < TCurie   ν2ν3\n T > TCurie ν1UCoGe polycrystalline sample \nFig. 2. (color online)59Co-NQR spectra in the polycrystalline\nsample obtained (a) at 4.2 K in the PM state and (b) at 95mK\nin the FM state. The arrows in the FM state are the calculated\nresonance frequencies (see text)\nspin experiences an internal ﬁeld Hint, the Zeeman term\nHZ=−γn/planckover2pi1I·Hint=−γn/planckover2pi1Hint(Izcosθ+Ixsinθ) is\nadded in the electric-quadrupole interaction, where γnis\nthe Co nuclear gyromagnetic ratio, the zcomponent of\nIis along the EFG principal axis, θis the angle between\nthe EFGprincipal axisand Hint, which is in the acplane\nfor the crystallographicalsymmetry. The NQR spectrum\nshown in Fig. 2 (b) is consistently understood by the su-\nperposition of the PM NQR spectra without Hintand\nthe FM NQR spectra with ( Hint,θ) = (910 Oe, 80◦).\nThe internal-ﬁeld direction from the ordered U moments\nis tilted by 80◦from the the EFG principal axis in the ac\nplane, which makes the U ordered moments consistent\nwith the c-axis Ising anisotropy observed.\nIn the single-crystal UCoGe, although three NQR sig-\nnals were observedin the PM state, we could not observe\nsignals associated with ν1andν2in the FM state. The\nundetection of ν1andν2signals would be due to a weak\nsignal/noise ratio and/or short T2eﬀect. In the follow-\ning, focusing onthe NQRsignalaround8.3MHz ( ν3), we\ndiscuss the character of a FM transition in UCoGe and\ndiﬀerences between the polycrystalline and single-crystal\nsamples. Figure 3 (a) shows the temperature variation of\nthe spectrum in the single-crystal sample. The full width\nat half maximum (FWHM) of the 8.3 MHz peak is 70\nkHz at 4.2 K, half that observed in the polycrystalline\nsample, indicating the single-crystal sample has a more\nhomogeneous EFG and higher quality. With decreasing\ntemperature, the intensity of the 8.3 MHz NQR signal\narising from the PM region decreases below ∼3.7 K\nwhile the 8.1 MHz signal originating from the FM re-\ngion appears below 2.7 K. The two NQR signals coexist\nbetween1and2.7K,but the PMsignaldisappearsbelow\n0.9 K. This is in contrast with the temperature variation\nin the polycrystalline sample, where the PM59Co-NQR\nsignal remained even at T= 95 mK shown in Fig. 3 (b).\nThe narrow temperature range of the coexistence in the\nsingle-crystal sample is likely indicative of better sampleJ. Phys. Soc. Jpn. Full Paper Author Name 3\n7.8 8.0 8.2 8.4 8.6 \n7.8 8.0 8.2 8.4 8.6 \nFrequency (MHz) T = 0.1 K (FM) 59 Co-NQR Intensity (arb. units) T = 1.35 K (FM) \nT = 0.9 K (FM)  (a) UCoGe \nSingle CrystalT = 4.2 K      T = 2.0 K (FM) \n  T = 2.5 K   T = 0.095 K  \n  (b) UCoGe \nPoly Crystal\nFig. 3. (a) Temperature dependence of the NQR spectrum from\nthe±5/2⇔ ±7/2 transitions ( ν3) in the single-crystal sample.\nThe blue (red) broken lines represent Gaussian ﬁts to the PM\n(FM) peaks; the solid lines are guides to the eye. (b) NQR spec -\ntrum of the ν3transition in the polycrystalline sample at 95 mK.\nquality. We stress that the single-crystal UCoGe is in the\nhomogeneous FM state throughout the sample below 1\nK from the absence of the PM signal.\nNotably, the temperature dependence of the NQR\nspectruminUCoGethrough TCurieisquitediﬀerentfrom\nthat observed in a second-order transition. For exam-\nple, in the antiferromagnet CeRhIn 5(N` eel temperature\nTN∼3.8 K), the ±5/2↔ ±7/2 transition in115In-NQR\n(I= 9/2) shifts continuously to lower frequencies be-\nlowTN, indicating the continuous development of the\nordered moment.14It is also noteworthy that the reso-\nnance frequency of the FM phase is nearly temperature\nindependent, indicating that the internal ﬁeld remains\nconstant below TCurie. These are in agreement with the\ntheoreticalpredictionthatlow-temperatureitinerantFM\ntransitions are generically of a ﬁrst order.9\nThe temperature dependence of 1 /T1at 8.3 and 8.1\nMHz provides information about spin dynamics related\nto the U moments in the normal state and SC prop-\nerties below TSC. Figures 4 show the recovery curves\nR(t) = 1−m(t)/m(∞)ofthenuclearmagnetization m(t)\nmeasured on the single-crystal sample at 4.2, 0.75, and\n0.14K. Here, m(t) is the nuclear magnetizationat a time\ntafter a saturation pulse. The R(t) data measured at the\nPM 8.3 MHz signal at 4.2 K can be ﬁt consistently by\nthe theoretical function for I= 7/2 with a single T1\ncomponent.?,15Below 2.3 K, 1 /T1was measured at the\nFM 8.1 MHz59Co-NQR peak and is still described by a\nsingle component down to TSC. BelowTSC, slower relax-\nation component was observed in R(t) and this tendency0 1 210 -2 10 -1 10 0\n0 10 20 10 -2 10 -1 10 0\n0 200 400 600 10 -2 10 -1 10 08.3 MHz (PM) \n 4.2 K \n t (ms) \n \n   \nR ( t ) ≡ 1- m ( t )/ m (∞)\n8.1 MHz (FM) \n 750 mK \n t (ms) \n \n   \n \n t (ms)   8.1 MHz (FM) \n 140 mK \nFig. 4. (Color online) Recovery curves R(t) of the nuclear magne-\ntization m(t) at time tafter a saturation pulse with the ﬁts used\nto evaluate 1 /T1.?,15Two relaxation components were clearly\nobserved below TSCin the single-crystal sample.\n0.1 1 10 100 10 -1 10 010 110 210 310 4\n single crystal\nFM (8.1MHz) \n above TSC \n below TSC  \n fast \n slow ~T  3~T 1/ T1 (1/s) \nTemperature (K) UCoGe \n59 Co-NQR \nPM (8.3MHz) \n poly crystal\n single crystalTSC TCurie \nT* \nFig. 5. (Color online) Temperature dependence of59Co 1/T1in\nthe single-crystal sample, along with 1 /T1in the polycrystal be-\ntween 10 and 150 K. 1 /T1was measured at the PM (8.3 MHz)\nfrequency above 2.3 K, shown by closed blue (green) circles f or\nthe single-crystal (polycrystalline) sample. Below 2.3 K, 1/T1\nwas measured at the FM (8.1 MHz) frequency. Two 1 /T1com-\nponents were observed in the SC state, the faster (slower) co m-\nponent denoted by red open (closed) squares; the red broken\ncurve below TSCrepresents the temperature dependence calcu-\nlated assuming line-node gap and ∆ 0/kBTSC= 2.3.4 J. Phys. Soc. Jpn. Full Paper Author Name\nis more pronounced as temperature is lowered. R(t) in\nthe SC state possesses nearly equal amount of the fast\nand slowcomponents, and thus the fast (slow) relaxation\nrate 1/T1was determined by ﬁtting the recovery curve\nof 0.5< R(t)<1(0.01< R(t)<0.5) region, as shown in\nFig. 4 (c)\nFigure 5 shows the temperature dependence of 1 /T1\nin the single-crystal UCoGe down to 70 mK, together\nwith the polycrystalline results between 10 and 150 K.\nThe 1/T1s in the single-crystal and polycrystalline sam-\nples agree well between 10 and 60 K, remaining nearly\nconstant down to T∗∼40 K and gradually decreasing\nbelowT∗. Since the magnetic susceptibility χdeviates\nfrom the Curie-Weiss behavior and the electrical resis-\ntivity along the caxis shows metallic behavior below\naboutT∗,13T∗is regarded as the characteristic temper-\nature below which the U 5 felectrons become itinerant\nwith relatively heavy electron mass. Below 10 K, 1 /T1\nincreases to a remarkable peak at TCurie∼2.5 K due\nto the critical slowing down of U moments. Note that\nχ≡χ(q= 0) also diverges, indicative of FM ordering\nin zero ﬁeld. The strong divergence of 1 /T1implies that\nthe FM transition in UCoGe would be weaklyﬁrst order,\nsince the critical ﬂuctuations are not usually observed in\na ﬁrst-order transition.\nIn the SC state, the fast component of 1 /T1at the FM\nsignal is roughly proportional to T, consistent with Ko-\nrringa behavior characteristic of metals, indicating that\nit originates from non-superconducting regions. In con-\ntrast, the slow component at the FM signal decreases\nrapidly below TSC, roughly as T3, suggestive of line\nnodes in a SC gap. The red broken line in Fig. 5 shows\na ﬁt using the line-node model ∆( θ) = ∆ 0cosθwith\n∆0= 2.3kBTSC. The detection of the SC gap via the FM\nsignal makes this strong unambiguous evidence for mi-\ncroscopic coexistence. Although this is the second piece\nof evidence for microscopic coexistence of itinerant fer-\nromagnetism and superconductivity after73Ge-NQR in\nUGe2under pressure,19the present result on UCoGe is\nmoreunambiguous, since the59Co-NQRsignals from the\nPM and FM states arewell separated(see Fig. 3 (a)) and\nthe NQR measurements were performed on a single crys-\ntal at ambient pressure.\nThe results below TSCprovidesome new insight on the\nnature of the superconductivity in UCoGe. From the re-\nlaxationin the FM signal, nearlyhalfof the sample’s vol-\nume remains non-superconducting even at 70 mK, while\nthe sample is in a homogeneous FM state below TCurie.\nSimilar two-relaxation rate behavior was previously ob-\nserved in the polycrystalline sample.11Since the tem-\nperature where the second component emerges coincides\nwithTSCfor both single-crystaland polycrystalline sam-\nples, the two-relaxation rate behavior would be intrinsic.\nOne possibility is a nonunitary SC state, which is real-\nized in the superﬂuid3He-A1under anexternalmagnetic\nﬁeld.16In such a SC state, although only up-spin pair is\nformed, the SC state is spatially homogeneous. The in-\nhomogeneous SC state in UCoGe appears to be incom-\npatible with the homogeneous nonunitary SC state. Al-\nternative interpretation of this behavior is a self-induced\nvortex (SIV) state.20When superconductivity occurs ina FM state, it has been suggested that a SIV state in\nwhich vortices are generated spontaneously can be sta-\nbleforHc1<4πM < H c2.There,theregionsnearvortex\ncores, where the SC gap is largely suppressed would give\nrise to the fast component of 1 /T1. Our single-crystal re-\nsults suggest that the SC gap is inhomogeneous in a real\nspace as in the polycrystalline sample,11which appears\nto be consistent with the SIV state. The recent obser-\nvation of a slight increase in the muon decay rate below\nTSCmight be related to the SIV state, since the SIV\nwould produce a distribution of internal ﬁelds at the im-\nplanted muon site.17The SIV state has been discussed\ntheoretically, but has never been identiﬁed experimen-\ntally. UCoGe is a promising candidate in which the SIV\nstate may be realized.\nIn conclusion,59Co-NQR measurements on UCoGe\nshow the ﬁrst-order character of the FM transition, and\nthe unambiguous evidence for the microscopic coexis-\ntence of ferromagnetism and superconductivity. The co-\nexistence originates from the same U-5 felectrons, rul-\ning out real-space phase separation. Although ferromag-\nnetism exists homogeneously throughout the sample,\nUCoGe’s superconductivity would be intrinsically inho-\nmogeneous, which might be interpreted in terms of a\nSIV state; further work, particularly low-temperature\nSTM/STS measurements in zero ﬁeld, will be important\nto detect the vortices produced by FM moments in the\nSC state.\nThe authors thank D. C. Peets, S. Yonezawa, H.\nTakatsu, S. Kittaka, and Y. Maeno for experimental sup-\nport and valuable discussions, and H. Harima, H. Ikeda,\nS. Fujimoto, A. de Visser, D. Aoki, and J. Flouquet for\nfruitful discussions.Thisworkwaspartiallysupportedby\nKyoto Univ. LTM center, the ”Heavy Electrons” Grant-\nin-Aid for Scientiﬁc Research on Innovative Areas (No.\n20102006) from MEXT of Japan, a Grant-in-Aid for the\nGlobal COE Program “The Next Generation of Physics,\nSpun from Universality and Emergence” from MEXT of\nJapan, a grant-in-aid for Scientiﬁc Research from JSPS\nand KAKENHI (S) (No. 20224015) from JSPS.\n1) V. L. Ginzburg: Sov. Phys. JETP 4(1957) 153.\n2) D. Fay and J. Appel: Phys. Rev. B 22(1980) 3173.\n3) S. S. Saxena, P. Agrwal, A. Ahilan, F. M. Grosche, R. K. W\nHaselwimmer, M. J. Steiner, E. Pugh, I. R. Walker, S. R. Ju-\nlian, P. Monthoux, G. G. Lonzarich, A. Huxley, I. Sheiken, D.\nBraithwaite, and J. Flouquet: Nature 406(2000) 587.\n4) T. Akazawa, H. Hidaka, T. Fujiwara T. C. Kobayashi, E. Ya-\nmamoto Y. Haga, R. Settai and Y. ¯Onuki: Journal of Phys.\nCondensed Matter 16(2004) L29.\n5) D.Aoki, A.Huxley, E.Ressouche, D.Braithwaite, J.Flouq uet,\nJ-P.Brison, E.Lhotel and C.Paulsen: Nature 413(2001) 613.\n6) N. T. Huy, A. Gasparini, D. E. de Nijs, Y. Huang, J. C.\nP. Klaasse, T. Gortenmulder, A. de Visser, A. Hamann, T.\nG¨ orlach and H. v. L¨ ohneysen: Phys. Rev. Lett. 99(2007)\n067006.\n7) M. Uhlarz, C. Pﬂeiderer and S. M. Hayden: Phys. Rev. Lett.\n93(2004) 256404.\n8) C. Pﬂeiderer, D. Reznik, L. Pintschovlus, H.v. L¨ ohneyse n, M.\nGarst and A. Rosch: Nature 413 (2001) 613.\n9) D. Belitz, T.R.Kirkpatrick and T.Vojta: Phys.Rev. Lett. 82\n(1992) 4707.\n10) N. T. Huy, D. E.de Nijs, Y.K. Huang and A. de Visser: Phys.J. Phys. Soc. Jpn. Full Paper Author Name 5\nRev. Lett. 100(2008) 077002.\n11) T.Ohta, Y.Nakai, Y.Ihara, K.Ishida, K.Deguchi, N.K.Sa to\nand Isamu Satoh: J. Phys. Soc. Jpn. 77(2008) 023707.\n12) H. Harima: unpublished.\n13) N. K. Sato: unpublished.\n14) T. Mito, S. Kawasaki, G. -q. Zheng, Y. Kawasaki, K. Ishida ,\nY. Kitaoka, D. Aoki, Y. Haga and Y. ¯Onuki: Phys. Rev. B 63\n(2001) 220507.\n15) A. Narath: Phys. Rev. 162(1967) 320.\n16) A. J. Leggett: Rev. Mod. Phys. 47(1975) 83.\n17) A. de Visser, N. T. Huy, A. Gasparini, D. E. de Nijs, D. An-\ndreica, C. Baines and A. Amato: Phys. Rev. Lett. 102(2009)167003.\n18) D. Aoki, T. D. Matsuda, V. Taufour, E. Hassinger, G. Knebe l\nand J. Flouquet: J. Phys. Soc. Jpn. 78(2009) 113709.\n19) H. Kotegawa, A. Harada, S. Kawasaki, Y. Kawasaki, Y. Ki-\ntaoka, Y. Haga, E. Yamamoto, Y. ¯Onuki, K. M. Itoh, E. E.\nHaller and H. Harima: J. Phys. Soc. Jpn. 74(2005) 705.\n20) M.Tachiki, H.Matsumoto, T.Koyama and H.Umezawa: Solid\nState Comm. 34(1980) 19.\n21) K. Momma and F. Izumi: J. Appl. Crystallogr. 41(2008) 653.\n22) J. Chepin and J. H. Ross Jr.: J. Phys. Condens. Matters 3\n(1991) 8103."    },    {        "title": "2303.07025v2.Experimental_investigation_of_the_effect_of_topological_insulator_on_the_magnetization_dynamics_of_ferromagnetic_metal___BiSbTe__1_5_Se__1_5___and__Ni__80_Fe__20___heterostructure.pdf",        "content": "Experimental investigation of the effect of topological insulator on the magnetization\ndynamics of ferromagnetic metal: BiSbTe 1.5Se1.5andNi80Fe20heterostructure\nSayani Pal, Soumik Aon, Subhadip Manna, Sambhu G Nath, Kanav Sharma & Chiranjib Mitra∗\nIndian Institute of Science Education and Research Kolkata,\nMohanpur 741246, West Bengal, India\n(Dated: November 27, 2023)\nWe have studied the spin-pumping phenomenon in ferromagnetic metal( Ni80Fe20)/topological\ninsulator( BiSbTe 1.5Se1.5) bilayer system to understand magnetization dynamics of ferromagnetic\nmetal (FM) in contact with a topological insulator (TI). TIs embody a spin-momentum-locked\nsurface state that spans the bulk band gap. Due to this special spin texture of the topological surface\nstate, the spin-charge interconversion efficiency of TI is even higher than that of heavy metals. We\nevaluated the parameters like effective damping coefficient ( αeff), spin-mixing conductance ( g↑↓\neff)\nand spin current density ( j0\nS) to demonstrate an efficient spin transfer in Ni80Fe20/BiSbTe 1.5Se1.5\nheterostructure. To probe the effect of the topological surface state, a systematic low-temperature\nstudy is crucial as the surface state of TI dominates at lower temperatures. The exponential increase\nof ∆Hfor all different thickness combinations of FM/TI bilayers and the enhancement of effective\ndamping coefficient ( αeff) with lowering temperature confirms that the spin chemical potential bias\ngenerated from spin-pumping induces spin current into the TI surface state. Furthermore, low-\ntemperature measurements of effective magnetization (4 πMeff) and magnetic anisotropy field ( Hk)\nshowed anomaly around the same temperature region where the resistivity of TI starts showing\nmetallic behavior due to the dominance of conducting TI surface state. The anomaly in Hkcan\nresult from the emerging exchange coupling between the TI surface state and the local moments of\nthe FM layer at the interface without any long-range ferromagnetic order in TI at the interface.\nINTRODUCTION\nSpintronics is one of the emerging fields that has\nwitnessed remarkable progress on both fundamen-\ntal and technological fronts over the past couple of\ndecades. Phenomena like spin-orbit torque [1], spin\nHall effect[2], giant magnetoresistance [3], tunnelling\nmagnetoresistance [4], domain wall motion [5] provide\nbasics for applications in memory devices[6], storage\ntechnology[7], logic gates [8] and magnetic sensors [9].\nThese devices utilize the spin degrees of freedom of\nelectrons and their interaction with orbital moments\nthrough spin-orbit coupling. Complete knowledge of the\nprocess of generation, manipulation, and detection of\nspin degrees of freedom or the spin current is essential for\nwidespread applications in this field. If one focuses on\nthe currently available spin current generation processes,\nspin-pumping [10, 11] is one of the most efficient methods\nwhere the precessing magnetization in the ferromagnet\n(FM) injects spin current into the adjacent layer by\ntransferring spin angular momentum. This raises a\nneed to study the effect of spin pumping with special\nemphasis on exploring new materials which can give\nrise to significant spin-charge interconversion efficiency.\nTopological insulators (TI) are a new class of materials\nthat have an interesting spin texture of the surface\nstate, owing to spin-momentum locking[14–17]. The\nmomentum direction of the electron in the surface state\nof TI is perpendicularly locked to its spin polarization\n∗Corresponding author:chiranjib@iiserkol.ac.indirection. Thus the spin-charge interconversion for TI\nis even higher than the heavy metals which makes TIs\nsuitable for spintronics application[12]. As surface states\nare robust against deposition of FM layers on top of TI\n[18], the topological surface states remain intact and\ngapless [19]. TI/FM bilayers have been successfully\nused for the spin current generation in spin-pumping\nexperiments[20, 34–37]. The effect of spin pumping can\nbe witnessed in the enhanced damping coefficient ( αeff)\nvalue of the ferromagnet upon excitations of ferromag-\nnetic resonance (FMR) because, in the spin pumping\nprocess, the net transfer of spin angular momentum into\nTI layer brings about an additional damping torque on\nthe precessing magnetization in the FM. It is difficult\nto fabricate a perfect TI thin film where the bulk state\nof TI is completely insulating. Thus for a complete\nunderstanding of the effect of TI surface state on FM\nmagnetization dynamics, the low-temperature study is\nnecessary where the surface states of TI dominate.\nIn this paper, we present the study of the spin-pumping\nphenomenon in ferromagnetic metal (FM)/ topological\ninsulator (TI) bilayer system. We chose Ni80Fe20\nas the FM layer and BiSbTe 1.5Se1.5as the TI layer.\nCurrently, BiSbTe 1.5Se1.5is one of the best 3D TI\nmaterials in which bulk conduction in thin films is\nnegligible even at room temperature and the dominance\nof surface state is very prominent at lower temperatures\n[21–23]. In our low-temperature measurements, we have\nwitnessed exponential enhancement of FMR linewidth\n(∆H) and effective damping coefficient ( αeff) at lower\ntemperatures. It supports the proposal of the spin\nchemical potential bias induced spin current injectionarXiv:2303.07025v2  [cond-mat.mes-hall]  24 Nov 20232\ninto the surface state of TI given by Abdulahad et al. [50].\nFor further investigation of the effect of the TI surface\nstate on the FM magnetization, we have also studied\nlow-temperature variations of effective magnetization\nand anisotropy field. We calculated the interfacial\nmagnetic anisotropy of the bilayer to be in-plane of the\ninterface. At low temperatures, this magnetic anisotropy\nfield shows a hump-like feature concomitant with the\nresistivity behavior of BiSbTe 1.5Se1.5with temperature.\nIt predicts the existence of exchange coupling between\nthe surface states of TI and the local moments of the FM\nlayer which acts perpendicular to the TI/FM interface.\nWe have also evaluated the values of spin-transport\nparameters like spin-mixing conductance, g↑↓\neffand spin\ncurrent density, j0\nsat room temperature to ensure a suc-\ncessful spin injection into the TI layer from the FM layer.\nSAMPLE PREPARATION AND\nCHARACTERIZATION\nFor this particular work, we have prepared dif-\nferent thickness combinations of topological insu-\nlator(TI)/ferromagnet(FM) bilayer heterostructure.\nBiSbTe 1.5Se1.5(BSTS ) has been taken as the TI\nmaterial and Permalloy( Ni80Fe20) has been used as\nthe ferromagnetic material. BSTS thin films were\ngrown on silicon (Si 111) substrate using pulsed laser\ndeposition(PLD) technique [24, 25]. The target material\nwas prepared using 99 .999% pure Bi, Sb, Te, and Se in a\n1:1:1.5:1.5 stoichiometric ratio. The films were deposited\nthrough ablation of the target by a KrF excimer laser\n(248 nm, 25 ns pulse width) at a low repetition rate of\n1Hz and 1 .2Jcm−2laser fluence keeping the substrate\ntemperature fixed at 2500Cand the chamber partial\npressure at 0.5 mbar (base pressure 2 ×10−5mbar)\nwith a continuous flow of Ar gas. After deposition,\nTI films were immediately transferred into the thermal\nevaporation chamber for the deposition of the FM\nlayer. Commercially available 99 .995% pure permalloy\n(Ni80Fe20) pallets were used for deposition. The Py\nfilm was deposited [26] on top of TI film at a rate of\n1.2˚A(crystal monitor: Inficon SQM 160) keeping the\nchamber pressure fixed at 1 ×10−6torr (base pressure\n1×10−7torr). For the characterization of the films\nX-ray diffraction analysis (XRD), field emission scanning\nelectron microscope (FE-SEM) imaging, and atomic\nforce microscopy (AFM) facilities have been used. X-ray\nreflectometry technique has been used for thickness\nmeasurements here. For convenience we are defining the\nBSTS of different thicknesses as follows: 10nm BSTS as\nBSTS1, 21nm BSTS as BSTS2, 28nm BSTS as BSTS3,\nand 37nm BSTS as BSTS4.RESULTS AND DISCUSSION\nFor a systematic study of the FM/TI bilayer system,\nwe have done in-plane FMR measurements in reflection\nmode geometry using a short-circuited CPW as shown\nin fig.1a. We obtained typical FMR signal at different\nmicrowave frequencies for Py(15nm)/BSTS2 sample in\nfig.1b. From the Lorentz formula fitting [53] of the FMR\nsignal we extracted the frequency dependence of the field\nlinewidth (∆ Hvs.f) and the resonance frequency vs.\nresonance field ( fvsH) data as shown in fig.2a and fig.2b\nrespectively. These give us valuable information about\nthe magnetization dynamics in ferromagnet which can\nbe described within the framework proposed by Landau,\nLifshitz, and Gilbert [30],\nd⃗M\ndt=−γ⃗M×⃗Heff+αeff\nMS⃗M×d⃗M\ndt(1)\nwhere, γis the gyromagnetic ratio, ⃗Mis the magneti-\nzation vector, MSis the saturation magnetization, Heff\nis the effective magnetic field which includes the exter-\nnal field, demagnetization and crystalline anisotropy field\nandαeffis the effective damping coefficient of the sys-\ntem.\nFor a given magnetic material at ferromagnetic res-\nonance, the resonance field and frequency follow Kittel\nequation[27] given by,\nf=γ\n2πq\n(H+Hk)(H+Hk+ 4πMeff) (2)\nwhere H,Hk, and 4 πMeffare the externally ap-\nplied field, magnetic anisotropy field, and effective mag-\nnetization respectively. We have obtained Hkand\n4πMefffor different FM/TI bilayer systems by fitting\nthe Kittel equation to the fvs. Hcurve as shown\nin fig.2b. The obtained 4 πMeffvalue contains satura-\ntion magnetization(4 πMs) and other anisotropic contri-\nbutions. We can evaluate 4 πMsvalue by analyzing the\nthickness dependent measurement of 4 πMeffof the FM\nlayer. In the lower thickness region of the ferromagnetic\nthin films, 4 πMeffis inversely proportional to the film\nthickness and follows the equation[28],\n4πMeff= 4πMs−2Ks\nMsd(3)\nwhere Ksis the surface anisotropy constant and dis the\nthickness of the FM film. The slope of the linear fit\ngives the anisotropy field contribution to 4 πMeffand\nthe intercept gives the 4 πMsvalue as shown in fig.2c.\nThe 4 πMeffdoes not depend on the thickness varia-\ntion of BSTS at room temperature but 4 πMefffor Py(t)\nmonolayer samples and for Py(t)/BSTS2 bilayer sam-\nples vary linearly with the inverse Py thickness as shown\nin Fig.2c. From the linear fitting (Eq.3) of 4 πMeff3\n(a)\n (b)\nFIG. 1. (a) In the left diagram, a schematic illustration of the experimental set-up has shown where the FM/TI bilayer is\nplaced upside down on top of a CPW, and in the right diagram, net injected spin current ( Ipump\nS ) due to spin-pumping into\nthe TI layer (BSTS) from the FM layer (Py) has shown, it results faster magnetization relaxation in FM; (b) Ferromagnetic\nResonance spectra of absorption at different frequencies for Py/BSTS bilayer system at room temperature after background\nsubtraction.\n(a)\n (b)\n (c)\nFIG. 2. (a) Field linewidth (∆ H) variation with resonance frequencies ( f) at 300K for Py/BSTS bilayer samples with different\nPy thicknesses. Eq.4 has been used for fitting the curve and to determine the damping coefficient ;(b) Resonance field ( H) vs.\nresonance frequency ( f) for Py(20nm)/BSTS2 system at different temperatures . Eq.2 has been used for fitting the curve and\nto determine the effective magnetization; (c) Effective magnetization (4 πMeff) variation with thickness of Py(t), Py(t)/BSTS2\nand Py(15nm/BSTS(t) at room temperature. Eq3 has been used for fitting the curve and to evaluate saturation magnetization\n(4πMS) and magnetic anisotropy field( Hk).\n(a)\n (b)\nFIG. 3. (a) αeffvariation with Py thickness for Py(t)/BSTS2 heterostructure at room temperature which fits in Eq.5; (b)\nαeffas a function of BSTS thickness for Py(15nm)/BSTS(t) heterostructure at room temperature.4\n(a)\nFIG. 4. Temperature dependence of resistivity of the BSTS\nsample of thickness 21nm deposited on Si(111) substrate.\nvs. 1 /tPydata for the Py(t) and Py(t)/BSTS2 sam-\nples, we evaluated the saturation magnetization, Msof\nthe Py/BSTS bilayer that has been decreased from that\nof the bare Py sample by an amount of 183 emu/cc3.\nIt is a result of the loss of ferromagnetic order in the\nPermalloy layer. Due to the intermixing of the Py and\nBSTS at the interface, a magnetic dead layer could have\nformed at the interface which resulted in the reduction\nof saturation magnetization value as suggested by some\nprevious studies [42–44] also. The Ksvalue has de-\ncreased from 0 .092±0.008erg/cm2in bare Py film to\n0.091±0.015erg/cm2in Py/BSTS2 bilayer. So interfa-\ncial anisotropy constant, Ki(=KPy/TI\ns −KPy\ns) for the\nPy/BSTS2 sample is −0.001erg/cm2. From the nega-\ntive value of Ki, we can ensure an in-plane magnetic\nanisotropy in the Py/BSTS interface at room temper-\nature. A detailed discussion of magnetic anisotropy has\nbeen provided in the last section where the temperature\nvariation of Hkis discussed.\nαeffcan be determined by analysing ∆ Hat different\nfrequencies. ∆ Hcontains both the intrinsic and extrin-\nsic contributions to the damping. Linewidth due to in-\ntrinsic damping is directly proportional to the resonance\nfrequency( f) and follows the equation[29],\n∆H= ∆H0+ (2παeff\nγ)f (4)\nwhere ∆ H0describes inhomogeneous linewidth broad-\nening [38, 39] due to different extrinsic contributions\nlike magnetic inhomogeneities [40, 41], surface roughness,\nand defects in the sample. We have evaluated the αeff\nvalues by fitting the ∆ Hvsfcurve for FM/TI bilayers\nas shown in fig.2a. This αeffconsists of Gilbert damp-\ning in the bulk ferromagnet( αFM) and the enhanced\ndamping( αSP) resulting from spin pumping into the ad-\njacent TI layer [31–33], αeff=αFM+αSP. The αFM\nvalue for bare Py film of thickness 15nm was calculated\nto be 0.0074 and for the FM/TI bilayer system there has\nbeen significant enhancement in the αeffvalue over thebare Py value due to spin pumping, αSP. In this het-\nerostructure, αeffincreases gradually as the thickness of\nPy decreases both for Py(t) and Py(t)/BSTS2 samples as\nshown in fig.3a. From the linear fit of αeffvs. 1/tPydata\nwe have obtained the spin-mixing coefficient, g↑↓\nefffor the\nBSTS/Py interface to be 5 .26×1018±0.71×1018m−2\nby using the equation[34, 37],\nαeff−αFM=gµB\n4πMstFMg↑↓\neff(5)\nwhere, gandµBare the g-factor and Bohr magneton\nrespectively. We have also calculated the spin current\ndensity( j0\ns) for the FM/TI heterostructure using the g↑↓\neff\nvalue in the following equation[20, 36],\nj0\ns=g↑↓\neffγ2h2\nmℏ[4πMsγ+p\n(4πMs)2γ2+ 4ω2]\n8πα2[(4πMs)2γ2+ 4γ2](6)\nwhere γ,hm,ℏ,ω, and αare the gyromagnetic ratio,\nmicrowave magnetic field, Planck’s constant, Larmour\nprecession frequency, and effective damping parameter\nrespectively. Using Eq.6 the j0\nsvalue for Py/BSTS2\nwas obtained to be 0 .901×10−10±0.122×10−10Jm−2\nin our experiment. The g↑↓\neffandj0\nsvalues obtained\nfrom Py thickness-dependent study of αeffare in a\ncomparable range of the previously reported values\nfor other combinations of ferromagnet and TI bilayer\nstructures [34, 35, 37]. This gives evidence of successful\nspin injection into the BSTS layer from the Py layer\ndue to spin pumping [31–33, 50]. We also report the TI\nthickness-dependent study of αeffas shown in fig.3b.\nFor bilayer structures of Py(15nm)/BSTS2(t) there is\na sudden jump in the αeffvalue from that of the bare\nFM film ( αFM = 0.0074) because of spin pumping.\nThen with the thickness variation of TI layer in the\nrange of 10nm to 37nm, αeffincreases slowly from\n0.015 to 0.02. The TI thickness dependence of αeff\nfor Py(15nm)/BSTS(t) bilayer is almost linear which\ncertainly can not be described by the conventional\nspin diffusion theory [48] for FM/NM proposed by\nTserkovnyak et al. [47]. For Py/BSTS heterostructure,\nαeffvs. tBSTS study suggests an efficient spin-sink\nnature of the TI bulk with increasing thickness at\nroom temperature [49]. From the room temperature\nstudy we certainly can not distinguish the TI surface\nstate contribution from the TI bulk state contribution\nbecause growing a BSTS thin film with a perfectly\ninsulating bulk state is still very challenging. Thus it\nwas imperative to study the effect of topological surface\nstate at low-temperature where bulk states of TI get\nsuppressed and surface states of TI starts to dominate.\nIn this section, we have focused on low-temperature\nmeasurements specifically to understand the effect of\ntopological surface states (TSS) on the magnetization\nrelaxation of FM. At higher temperatures, a significant\namount of bulk carriers are available to participate\nin the transport but with the reduction of phonon5\n(a)\n (b)\nFIG. 5. (a)Temperature dependence of the field linewidth (∆ H) for different thickness combinations of Py/BSTS bilayer\nsystems and for a bare Py thin film. The solid lines are the fits in the expression exp(−T/T 0); (b)Temperature dependence of\neffective damping coefficient, αeffof Py(20nm)/BSTS2 and bare Py(20nm) film.\n(a)\n (b)\nFIG. 6. (a)Temperature dependence of effective magnetization of Py(20nm/BSTS2); (b)Temperature dependence of the\nanisotropy field of Py(20nm)/BSTS2.\nscattering, surface carriers dominate at a lower tem-\nperature. From the resistivity vs. temperature data of\nBSTS2 in fig.4, we can see an insulating behavior of\nresistivity due to the enhanced insulating nature of the\nbulk state of TI at higher temperatures and a metallic\nbehavior of resistivity below a certain temperature\nwhere the topological surface states dominate. We\nmeasured temperature variation of FMR linewidth\n(∆H), enhanced damping coefficient ( αeff), anisotropy\nfield ( Hk) and effective magnetization (4 πMeff). For\ndifferent thickness combinations of Py/BSTS bilayer,\nwe obtained the ∆ Hvariation with temperature. It\nincreases exponentially with decreasing temperature\nthat fits the expression, exp(−T/T 0) as shown in fig.5a.\nFor bare Py(15nm) film, we can note that there is no\nsignificant variation in ∆ Hat low temperatures as can\nbe seen from the curve at the bottom of fig.5a. To\ngain further understanding, the temperature variation\nofαeffhas also been studied for Py(20nm)/BSTS2\nas shown in fig.5b and compared with αefffor barePy film. From the enhancement of αeffvalue for\nPy(20nm)/ BSTS2 at room temperature we can ensure a\nsuccessful spin injection due to the spin pumping effect.\nBut the exponential increase of αeffwith decreasing\ntemperature for the bilayer implies a huge increment in\nthe amount of spin angular momentum transfer into the\nTI layer at lower temperatures. We attribute the origin\nof the exponential increase of αeffand ∆ Hat lower\ntemperatures to the spin chemical potential bias induced\nspin current into the surface state of TI as proposed by\nAbdulahad et al. [50]. The induced spin current into\nthe TI surface state at lower temperatures corresponds\nto the rapid relaxation of magnetization precession of\nFM which is reflected in the exponential increase of ∆ H\nandαeffof the ferromagnet.\nTo further investigate the effect of TI surface state on\nthe magnetization of FM, we studied the temperature\nvariation of 4 πMeffandHkfor Py(20nm)/BSTS2. In\nour previous study [26] with bare Py thin films, we have6\nseen that 4 πMeffincreases monotonically as saturation\nmagnetization increases with lowering the temperature.\nBut from fig.6a, we can see that the low-temperature\ndependence of 4 πMefffor Py/BSTS2 bilayer deviates\nfrom the single layer Py film [Supplementary fig.S11(a)].\nThis anomaly in 4 πMeffis related to the change of mag-\nnetic anisotropy energy of the system as well as the other\neffects like spin chemical potential induced current and\nexchange coupling between TSS and FM as mentioned\nby Abdulahad et al. [50]. In a previous section, we\nevaluated the interfacial magnetic anisotropy coefficient\n(Ki=−0.001erg/cm2) to be in-plane of the interface of\nthe Py/BSTS2 bilayer. The anisotropy field associated\nwith the system anisotropy energy shows an interesting\nnature as we lower the temperature. We can see from\nfig.6b that Hkincreases initially with decreasing tem-\nperature until a certain value is reached and then the\nanisotropy field weakens against a further decrease in\ntemperature. Thus we get a hump-like feature of HK\nfor the same temperature region where 4 πMeffshows\nthe anomaly and it is concomitant with the resistivity vs\ntemperature behavior of the BSTS2 sample. The low-\ntemperature behavior of Hkand 4 πMeffcan be justified\nby the argument proposed by Abdulahad et al. [50]. In\ntheir phenomenological model, they propose an existence\nof exchange interaction between the surface states of TI\nand local moments of the ferromagnetic layer. Several\ntheoretical as well as experimental predictions confirm\nthe existence of gapless topological surface states even\nafter transition metal deposition on TI [51, 52]. These\nsurface states can couple with the local moments of the\nFM through exchange interaction without any long-range\nferromagnetic order. This exchange coupling acts per-\npendicular to the TI surface and weakens the in-plane\nanisotropy at lower temperatures where the surface states\nof TI dominate.\nCONCLUSIONS\nIn summary, we have carried out spin-pumping ex-\nperiment in BiSbTe 1.5Se1.5(TI)/ Ni80Fe20(FM) bilayer\nsystem. From the thickness-dependent measurements of\nFM/TI bilayers, we obtained the spin-transport param-\neters like damping coefficient due to spin-pumping, spin\nmixing conductance, and spin current density at room\ntemperature. These results demonstrate a successful spin\ntransfer from the FM layer to the TI layer due to spin-\npumping. We have performed low-temperature measure-\nments to specifically understand the surface state con-\ntribution of TI on the FM magnetization because the\nsurface states of TI are more pronounced at lower tem-\nperatures. We have confirmed the suppression of the\ninsulating bulk state of TI at lower temperatures from\nthe resistivity vs. temperature data of TI. In our low-\ntemperature measurements of FMR linewidth and ef-fective damping coefficient, we have witnessed an expo-\nnential increase in both parameters with the decrease in\ntemperature. It suggests a spin chemical potential bias-\ninduced spin current injection into the surface states of\nTI that gets enhanced at low temperatures [50]. We have\nalso studied temperature variations of the effective mag-\nnetization of the system. It showed a deviation from\nthe bare Py film [26] in the temperature regime where\nTI surface states dominate. This deviation of effective\nmagnetization results from the change in the anisotropy\nenergy of the system. At room temperature, we eval-\nuated the magnetic anisotropy energy coefficient which\nis found to be in-plane of the interface. This in-plane\nanisotropy weakens when conducting surface state of TI\nstarts to dominate. It reflects from the hump-like feature\nin the magnetic anisotropy field vs. temperature data of\nthe bilayer system. The decrease in in-plane magnetic\nanisotropy below a certain temperature can result from\nthe exchange coupling between the surface states of TI\nand the local moments of the FM layer which act per-\npendicular to the interface [50]. Combining the results of\nour low-temperature measurements we can conclude that\nthere exists an exchange coupling between the TI surface\nstate and FM which does not create any long-range ferro-\nmagnetic order in the TI and is unable to alter the overall\nspin texture of the TI surface state at the interface[18].\nHowever, it affects the magnetization dynamics of the\nferromagnetic metal quite significantly. These added fea-\ntures of enhancing the damping coefficients enables an-\nother fast control of magnetization dynamics in the FM\nlayer.\nACKNOWLEDGEMENTS\nThe authors sincerely acknowledge the Ministry\nof Education, Government of India and Science\nand Engineering Research Board (SERB) (grant no:\nEMR/2016/007950), and Department of Science and\nTechnology (grant no. DST/ICPS/Quest/2019/22) for\nfinancial support. S.P. acknowledges the Department\nof Science and Technology(DST)-INSPIRE fellowship In-\ndia, S. A. acknowledges the Ministry of Education of the\nGovernment of India, S.M. acknowledges the Council Of\nScientific and Industrial Research(CSIR), India, S.G.N\nand K.S acknowledges the University Grant Commis-\nsion, India for research fellowship. The authors would\nlike to thank Dr. Partha Mitra of the Department of\nPhysics, Indian Institute of Science Education and Re-\nsearch Kolkata, for providing the lab facilities for sample\ndeposition. The authors would like to acknowledge Prof.\nAnjan Barman and Mr. Pratap Kumar Pal of the De-\npartment of Physics, SN Bose National Centre for Basic\nSciences for helping with thickness measurements using\nthe XRR facility in their institute.7\n[1] Manchon, A. and Zhang, S., 2009. Theory of spin torque\ndue to spin-orbit coupling. Physical Review B, 79(9),\np.094422.\n[2] Liu, L., Moriyama, T., Ralph, D.C. and Buhrman,\nR.A., 2011. Spin-torque ferromagnetic resonance induced\nby the spin Hall effect. Physical review letters, 106(3),\np.036601.\n[3] Xiao, J.Q., Jiang, J.S. and Chien, C.L., 1992. Giant mag-\nnetoresistance in nonmultilayer magnetic systems. Phys-\nical Review Letters, 68(25), p.3749.\n[4] Sharma, M., Wang, S.X. and Nickel, J.H., 1999. Inver-\nsion of spin polarization and tunneling magnetoresistance\nin spin-dependent tunneling junctions. Physical Review\nLetters, 82(3), p.616.\n[5] Ryu, K.S., Thomas, L., Yang, S.H. and Parkin, S., 2013.\nChiral spin torque at magnetic domain walls. Nature nan-\notechnology, 8(7), pp.527-533.\n[6] Sabpreet Bhatti, Rachid Sbiaa, Atsufumi Hiro-\nhata, Hideo Ohno, Shunsuke Fukami, S.N. Pi-\nramanayagam, Spintronics based random access\nmemory: a review, Materials Today, Volume\n20, Issue 9, 2017, Pages 530-548, ISSN 1369-\n7021,https://doi.org/10.1016/j.mattod.2017.07.007\n[7] Yakout, S.M., 2020. Spintronics: future technology for\nnew data storage and communication devices. Journal of\nSuperconductivity and Novel Magnetism, 33(9), pp.2557-\n2580.\n[8] Patil, S., Lyle, A., Harms, J., Lilja, D.J. and Wang, J.P.,\n2010, October. Spintronic logic gates for spintronic data\nusing magnetic tunnel junctions. In 2010 IEEE Inter-\nnational Conference on Computer Design (pp. 125-131).\nIEEE.\n[9] Bertram, H.N., Safonov, V.L. and Jin, Z., 2002. Thermal\nmagnetization noise, damping fundamentals, and mode\nanalysis: application to a thin film GMR sensor. IEEE\ntransactions on magnetics, 38(5), pp.2514-2519.\n[10] Azevedo, A., Vilela-Le˜ ao, L.H., Rodr´ ıguez-Su´ arez, R.L.,\nSantos, A.L. and Rezende, S.M., 2011. Spin pumping and\nanisotropic magnetoresistance voltages in magnetic bilay-\ners: Theory and experiment. Physical review B, 83(14),\np.144402.\n[11] Taniguchi, T. and Imamura, H., 2008. Spin pumping\nin ferromagnetic multilayers. Modern Physics Letters B,\n22(30), pp.2909-2929.\n[12] Vila, M., Hsu, C.H., Garcia, J.H., Ben´ ıtez, L.A., Wain-\ntal, X., Valenzuela, S.O., Pereira, V.M. and Roche,\nS., 2021. Low-symmetry topological materials for large\ncharge-to-spin interconversion: The case of transition\nmetal dichalcogenide monolayers. Physical Review Re-\nsearch, 3(4), p.043230.\n[13] Pesin, D. and MacDonald, A.H., 2012. Spintronics and\npseudospintronics in graphene and topological insulators.\nNature materials, 11(5), pp.409-416.\n[14] Jiang, Z., Chang, C.Z., Masir, M.R., Tang, C., Xu, Y.,\nMoodera, J.S., MacDonald, A.H. and Shi, J., 2016. En-\nhanced spin Seebeck effect signal due to spin-momentum\nlocked topological surface states. Nature communica-\ntions, 7(1), pp.1-7.\n[15] Hasan, M.Z. and Kane, C.L., 2010. Colloquium: topolog-\nical insulators. Reviews of modern physics, 82(4), p.3045.[16] Chen, W., 2020. Spin torque and persistent currents\ncaused by percolation of topological surface states. Phys-\nical Review B, 102(14), p.144442.\n[17] Fischer, M.H., Vaezi, A., Manchon, A. and Kim,\nE.A., 2016. Spin-torque generation in topological insu-\nlator based heterostructures. Physical Review B, 93(12),\np.125303.\n[18] Scholz, M.R., S´ anchez-Barriga, J., Marchenko, D.,\nVarykhalov, A., Volykhov, A., Yashina, L.V. and Rader,\nO., 2012. Tolerance of topological surface states towards\nmagnetic moments: Fe on Bi 2 Se 3. Physical Review\nLetters, 108(25), p.256810.\n[19] Shiomi, Y., Nomura, K., Kajiwara, Y., Eto, K., No-\nvak, M., Segawa, K., Ando, Y. and Saitoh, E., 2014.\nSpin-electricity conversion induced by spin injection into\ntopological insulators. Physical review letters, 113(19),\np.196601.\n[20] Deorani, P., Son, J., Banerjee, K., Koirala, N., Brahlek,\nM., Oh, S. and Yang, H., 2014. Observation of inverse\nspin Hall effect in bismuth selenide. Physical Review B,\n90(9), p.094403.\n[21] Ren, Z., Taskin, A.A., Sasaki, S., Segawa, K. and Ando,\nY., 2011. Optimizing Bi (2-x) Sb x Te (3-y) Se y solid\nsolutions to approach the intrinsic topological insulator\nregime. Physical Review B, 84(16), p.165311.\n[22] Taskin, A.A., Ren, Z., Sasaki, S., Segawa, K. and Ando,\nY., 2011. Observation of Dirac holes and electrons in a\ntopological insulator. Physical Review Letters, 107(1),\np.016801.\n[23] Singh, S., Gopal, R.K., Sarkar, J., Pandey, A., Patel,\nB.G. and Mitra, C., 2017. Linear magnetoresistance and\nsurface to bulk coupling in topological insulator thin\nfilms. Journal of Physics: Condensed Matter, 29(50),\np.505601.1-10.\n[24] Sarkar, J., Singh, S., Gopal, R.K. and Mitra, C., 2017.\nEvidence of robust metallic surface states on the sur-\nface of PLD grown poly-crystalline topological insulator\nthin-film. In Proceedings of the seventeenth international\nconference on thin films: abstracts.\n[25] Pandey, A., Singh, S., Ghosh, B., Manna, S., Gopal, R.K.\nand Mitra, C., 2019. Pulsed laser deposition of highly c-\naxis oriented thin films of BSTS topological insulator.\narXiv preprint arXiv:1910.08100.\n[26] Pal, S., Aon, S., Manna, S. and Mitra, C., 2022. A\nshort-circuited coplanar waveguide for low-temperature\nsingle-port ferromagnetic resonance spectroscopy setup\nto probe the magnetic properties of ferromagnetic thin\nfilms. Review of Scientific Instruments, 93(8), p.083909.\n[27] Kittel C. On the theory of ferromagnetic resonance ab-\nsorption. Physical review. 1948 Jan 15;73(2):155.\n[28] Ruane WT, White SP, Brangham JT, Meng KY,\nPelekhov DV, Yang FY, Hammel PC. Controlling and\npatterning the effective magnetization in Y3Fe5O12 thin\nfilms using ion irradiation. AIP Advances. 2018 May\n8;8(5):056007.\n[29] Zakeri K, Lindner J, Barsukov I, Meckenstock R, Farle\nM, Von H¨ orsten U, Wende H, Keune W, Rocker J, Kalar-\nickal SS, Lenz K. Spin dynamics in ferromagnets: Gilbert\ndamping and two-magnon scattering. Physical Review B.\n2007 Sep 14;76(10):104416.8\n[30] Gilbert, T.L., 2004. A phenomenological theory of damp-\ning in ferromagnetic materials. IEEE transactions on\nmagnetics, 40(6), pp.3443-3449.\n[31] Urban R, Woltersdorf G, Heinrich B. Gilbert damping in\nsingle and multilayer ultrathin films: Role of interfaces\nin nonlocal spin dynamics. Physical review letters. 2001\nNov 2;87(21):217204.\n[32] Azzawi S, Ganguly A, Toka¸ c M, Rowan-Robinson RM,\nSinha J, Hindmarch AT, Barman A, Atkinson D. Evolu-\ntion of damping in ferromagnetic/nonmagnetic thin film\nbilayers as a function of nonmagnetic layer thickness.\nPhysical Review B. 2016 Feb 1;93(5):054402.\n[33] Tserkovnyak, Y., Brataas, A. and Bauer, G.E., 2002.\nEnhanced Gilbert damping in thin ferromagnetic films.\nPhysical review letters, 88(11), p.117601.\n[34] Mendes, J.B.S., Santos, O.A., Holanda, J., Loreto,\nR.P., De Araujo, C.I.L., Chang, C.Z., Moodera, J.S.,\nAzevedo, A. and Rezende, S.M., 2017. Dirac-surface-\nstate-dominated spin to charge current conversion in the\ntopological insulator (Bi 0.22 Sb 0.78) 2 Te 3 films at\nroom temperature. Physical Review B, 96(18), p.180415.\n[35] Deorani, P., Son, J., Banerjee, K., Koirala, N., Brahlek,\nM., Oh, S. and Yang, H., 2014. Observation of inverse\nspin Hall effect in bismuth selenide. Physical Review B,\n90(9), p.094403.\n[36] Ando K, Takahashi S, Ieda J, Kajiwara Y, Nakayama H,\nYoshino T, Harii K, Fujikawa Y, Matsuo M, Maekawa S,\nSaitoh E. Inverse spin-Hall effect induced by spin pump-\ning in metallic system. Journal of applied physics. 2011\nMay 15;109(10):103913.\n[37] Wu CN, Lin YH, Fanchiang YT, Hung HY, Lin HY, Lin\nPH, Lin JG, Lee SF, Hong M, Kwo J. Strongly enhanced\nspin current in topological insulator/ferromagnetic metal\nheterostructures by spin pumping. Journal of Applied\nPhysics. 2015 May 7;117(17):17D148.\n[38] Schneider, M.L., Gerrits, T., Kos, A.B. and Silva, T.J.,\n2007. Experimental determination of the inhomogeneous\ncontribution to linewidth in Permalloy films using a time-\nresolved magneto-optic Kerr effect microprobe. Journal\nof Applied Physics, 102(5).\n[39] Husain, S., Pal, S., Chen, X., Kumar, P., Kumar, A.,\nMondal, A.K., Behera, N., Gupta, N.K., Hait, S., Gupta,\nR. and Brucas, R., 2022. Large Dzyaloshinskii-Moriya\ninteraction and atomic layer thickness dependence in a\nferromagnet-WS 2 heterostructure. Physical Review B,\n105(6), p.064422.\n[40] Schl¨ omann, E., 1969. Inhomogeneous broadening of fer-\nromagnetic resonance lines. Physical review, 182(2),\np.632.\n[41] Jiang, S., Sun, L., Yin, Y., Fu, Y., Luo, C., Zhai, Y.\nand Zhai, H., 2017. Ferromagnetic resonance linewidth\nand two-magnon scattering in Fe1-xGdx thin films. AIP\nAdvances, 7(5).\n[42] Li, J., Wang, Z.Y., Tan, A., Glans, P.A., Arenholz, E.,\nHwang, C., Shi, J. and Qiu, Z.Q., 2012. Magnetic deadlayer at the interface between a Co film and the topo-\nlogical insulator Bi 2 Se 3. Physical Review B, 86(5),\np.054430.\n[43] Walsh, L.A., Smyth, C.M., Barton, A.T., Wang, Q., Che,\nZ., Yue, R., Kim, J., Kim, M.J., Wallace, R.M. and\nHinkle, C.L., 2017. Interface chemistry of contact met-\nals and ferromagnets on the topological insulator Bi2Se3.\nThe Journal of Physical Chemistry C, 121(42), pp.23551-\n23563.\n[44] Ojha, D.K., Chatterjee, R., Lin, Y.L., Wu, Y.H. and\nTseng, Y.C., 2023. Spin-torque efficiency enhanced in\nsputtered topological insulator by interface engineer-\ning. Journal of Magnetism and Magnetic Materials, 572,\np.170638.\n[45] Jamali, Mahdi, et al. ”Giant spin pumping and inverse\nspin Hall effect in the presence of surface and bulk spin-\norbit coupling of topological insulator Bi2Se3.” Nano let-\nters 15.10 (2015): 7126-7132.\n[46] Foros, J., Woltersdorf, G., Heinrich, B. and Brataas, A.,\n2005. Scattering of spin current injected in Pd (001).\nJournal of Applied Physics, 97(10), p.10A714.\n[47] Tserkovnyak, Y., Brataas, A. and Bauer, G.E., 2002.\nSpin pumping and magnetization dynamics in metallic\nmultilayers. Physical Review B, 66(22), p.224403.\n[48] Boone, C.T., Nembach, H.T., Shaw, J.M. and Silva, T.J.,\n2013. Spin transport parameters in metallic multilayers\ndetermined by ferromagnetic resonance measurements\nof spin-pumping. Journal of Applied Physics, 113(15),\np.153906.\n[49] Baker, A.A., Figueroa, A.I., Collins-McIntyre, L.J., Van\nDer Laan, G. and Hesjedal, T., 2015. Spin pump-\ning in ferromagnet-topological insulator-ferromagnet het-\nerostructures. Scientific reports, 5(1), p.7907.\n[50] Abdulahad, F.B., Lin, J.H., Liou, Y., Chiu, W.K.,\nChang, L.J., Kao, M.Y., Liang, J.Z., Hung, D.S. and\nLee, S.F., 2015. Spin chemical potential bias induced sur-\nface current evidenced by spin pumping into the topo-\nlogical insulator B i 2 T e 3. Physical Review B, 92(24),\np.241304.\n[51] Ye, M., Eremeev, S.V., Kuroda, K., Krasovskii, E.E.,\nChulkov, E.V., Takeda, Y., Saitoh, Y., Okamoto, K.,\nZhu, S.Y., Miyamoto, K. and Arita, M., 2012. Quasipar-\nticle interference on the surface of Bi 2 Se 3 induced by\ncobalt adatom in the absence of ferromagnetic ordering.\nPhysical Review B, 85(20), p.205317.\n[52] Scholz, M.R., S´ anchez-Barriga, J., Marchenko, D.,\nVarykhalov, A., Volykhov, A., Yashina, L.V. and Rader,\nO., 2012. Tolerance of topological surface states towards\nmagnetic moments: Fe on Bi 2 Se 3. Physical Review\nLetters, 108(25), p.256810.\n[53] Celinski, Z., Urquhart, K.B. and Heinrich, B., 1997. Us-\ning ferromagnetic resonance to measure the magnetic mo-\nments of ultrathin films. Journal of Magnetism and Mag-\nnetic Materials, 166(1-2), pp.6-26."    },    {        "title": "1905.00771v2.Negligible_thermal_contributions_to_the_spin_pumping_signal_in_ferromagnetic_metal_Platinum_bilayers.pdf",        "content": "1 Negligible  thermal contributions to the spin pumping  signal in \nferromagnetic metal -Platinum bilayers   \nPaul Noël1, Maxen Cosset -Cheneau1, Victor Haspot2, Vincent Maurel3, Christian Lombard3, Manuel \nBibes2, Agnès Barthelemy2, Laurent Vila1, Jean -Philippe Attané1  \n1Univ. Grenoble Alpes, CNRS, CEA, Grenoble INP , IRIG-SPINTEC , F-38000 Grenoble, France  \n2Unité Mixte de Physique, CNRS, Thales, Université Paris -Sud, Université Paris -Saclay, Palaiseau, \nFrance  \n3Univ. Grenoble Alpes, CEA, CNRS, IRIG-SYMMES, F -38000 Grenoble, France  \nAbstract : Spin pumping by ferromagnetic resonance is one of the most common technique to \ndetermine spin hall angles, Edelstein lengths or spin diffusion lengths of a large variety of \nmaterials. In recent years, rising concerns have appeared  regarding the interpretation  of these \nexperiments , underlining that the signal could arise purely from thermoelectric effects, rather than \nfrom coherent spin pumping. Here, we propose a method to evaluate the presence or absence of \nthermal effects in spin  pumping signals, by combining bolometry  and spin pumping by \nferromagnetic resonance measurement s, and comparing their timescale.  Using a cavity  to perform \nthe experiments on Pt \\Permalloy and La0.7Sr0.3MnO 3\\Pt samples , we conclude on the absence of  \nany measurable  thermoelectric contribution  such as the spin Seebeck and a nomalous Nernst \neffects  at resonance . \nI Introduction  \nSpinorbitronics is based on the inter conversion of charge currents into spin currents, by the spin  Hall \nEffect  in bulk materials or by the Edelstein  effect at surface s and interfaces. Determining the spin to \ncharge current conversion efficiency, i.e., the spin Hall Angle or the  inverse  Edelstein length, and the \nspin diffusion length, is a key point to understand e xperimental results and develop applications. \nMaterials with large spin Hall angles are indeed required for a variety of foreseen applications such as \n3-terminal SOT MRAM [1], Magnetoelectric Spinorbit Logic [2] or Terahertz emitters [3]. Since the \nearly 2 000's several techniques have emerged to evaluate this conversion efficiency , using  lateral spin \nvalves [4], Spin Seebeck  measurements  [5], spin pumping FMR  (SP-FMR) [6], magneto -optical Kerr \neffect measurements [7], etc . The spin p umping  FMR technique has been widely used to evaluate the \nspin Hall angles and inverse Edelstein lengths of a large numbers of materials, including heavy metals \n[6,8,9], semiconductors [10], Rashba interfaces [11, 12] and topological insulators [13,14,15,16]. The \nreason of such a wide use relies on t he compatibility of SP -FMR with any multilayer stack  containing a \nferromagnet , and the absence of any complex and costly nanofabrication process . Moreover  \nferromagnetic resonance was an already widely used  spectroscopy technique to evaluate the \ndynamical and non -dynamical properties of ferromagnetic thin films  \nEven for a widely studied material as  Platinum  the estimated values of spin diffusion length and  spin \nHall angle  determined by different techniques  spread  over more than one  order of magnitude ,with \nspin diffusion length ranging from 1.2 nm [17] to 11  nm [18]  and spin Hall angle from 1.2%  [19] to \n38.7 % [20]. This large discrepancy  can be partially expl ained by differences in  Pt resistivity  or \naccount ed for by  interface -related phenomena  such as the spin memory -loss [9,21,22,23,24], but it \nremains  mostly unexplained  [25]. In this particular context , concerns regarding the reliability of the \nSP-FMR technique have been  pointed out . A thermal gradient could indeed arise at the ferromagnetic \nresonance, due to the energy absorption in the ferromagnetic layer [26,27,28,29] and give rise to \nseveral thermoelectric and spin-caloritronics contributions  in particular  the anomalo us Nernst 2 effect (ANE) and the s pin Seebeck effect (SSE)  that would add up to the spin pumping inverse s pin \nHall e ffect (ISHE)  signal.  In this picture, t he signal would thus be due to a combination of the ISHE \nsignal, the spin rectification effects (SRE) and thermal effects  [30].  \nWhile the separation of ISHE from SRE has already been vastly discussed and can be achieved from \nthe angular dependence in differen t measurement geometries [30,31,32,33], disentangling the ISHE \nsignal from thermal effect s remains  an open question . While the ordinary Seebeck e ffect ( OSE) and \nordinary Nernst e ffect ( ONE ) contributions may  be extracted from angular dependences, it is not the \ncase of the ANE and SSE contributions, which possess an angular dependence similar to that of the \nISHE signal. Recent analysis of spin -pumping FMR results h ave even been based on the hypothesis \nthat the observed signals are dominated by the SSE  [27]. In that case the nature of the spin pumping \nsignal would be incoherent  –thermal - and the  extensively used coherent spin pumping model would  \nstrongly misestimate the injected spin current [34]. It is therefore possible that due to a large \ncontribution  of incoherent spin pumping , the estimation of the Spin Hall Angle  by spin pumping FMR  \nin Pt is inaccurate. More generally, a contribution of ordinary thermoelectric effects such as the \nordinary Nernst effect  is also a possible explanation to the discrepancy between the lack of spin \ncharge conversion in Silver Bismuth bilayer measured by means of Longitudinal Spin Seebeck Effect \n(LSSE) [35], and the large one measured by means of spin pumping  [11,12]. Therefore, estimating \npossible thermoelectric or spin -caloritronics contributions is of high importance to obtain an accurate \nevalua tion of the spin charge conversion  efficiency . \nII Experime ntal procedure  \nFigure 1  a) depicts the dynamical spin injection process as  described by Tserkovnyak et al.  [36], which \nis the model used  to analyze  the signal observed in spin pumping FMR measurements . As suggested \nby Yamanoi et al.  [27], the additional dissipation at the FMR could lead to the appearance of a voltage \nalong the x direction. The  absorption at resonance would  lead  to a temperature increase of the \nferromagnet , and thus to a thermal gradient perpendicular to the layers. This thermal gradient would \nlead to the injection of a pure spin current along Z towards the non-magnetic material, converted by \nISHE into an electric  field along X  through a process known as the longitudinal  Spin Seebeck effect  as \nseen in figure 1.b . Owing to the existence of a thermal gradient , the a nomalous Nernst effect  in the \nFM layer  could also appear, creating an electric field along X  as depicted in figure 1 c.   \n \nFigure 1: Schematic representation of the possible spin injection mechanisms at the FMR and of \nthermal gradient related effects. a) Dynamical spin injection. Because of the magnetization \nprecession, the spin current 𝐽𝑆𝑆𝑃 is injected from the FM layer towards the NM layer. An \nelectromotive force 𝐸𝐼𝑆𝐻𝐸  then arises along X, due to ISHE, which can be detected as a voltage in \nopen circuit. b) Spin injection due to the thermal gradient. At the ferromagnetic resonance t he \ntemperature of the FM layer increases creating a thermal gradient 𝛻𝑇 along the z direction and \nthus a thermal spin current injection  𝐽𝑆𝛻𝑇. This spin current is then converted into an \nelectromotive force  𝐸𝐼𝑆𝐻𝐸 . c) Thermal gradient within the f erromagnet could give rise to an \nanomalous Nernst  related electromotive force 𝐸𝐴𝑁𝐸. \n3  \nWe propose to test thes e hypothesis  in two multilayers. The first o ne is a Pt \\Permalloy  bilayer, \narchetypal of spin pumping ISHE experiments  [6, 8, 17, 19], with a large  ANE coefficient in  Permalloy \n(Py) [37], and the second one  is a La0.7Sr0.3MnO 3\\Pt bilayer , in which a large  SSE contribution is \nexpected [38]. The characteristic time scale  of the FMR  spin injection mechanism is  the FMR \nprecession period , which  is of the order of the nanosecond . But the temperature increase  timescale , \nthe time needed to reach a thermal  equilibrium, is of several  seconds [39,40]: thermal and non-\nthermal effect s have  different dynamics. W e thus propose a technique that can be adapted  to any SP-\nFMR  experiment to disentangle  the two mechanisms , by measuring  the time dependence of the spin \npumping signal and of the temperature increase.   \nIII Results and Discussion  \nWe performed SP-FMR  measurement s on a SiO 2\\\\Pt(10) \\Py(20) multilayer  (the numbers in \nparenthes is represent the thickness in nanometers ), on  a 2.40.4 mm² structure positioned in a \nBruker MS5 loop gap cavity.  We performed a FMR measurement at different sweeping rates, at a \npower of 100  mW.  The scheme of the measurement is shown in figure 2 a), and consists in the \nmeasurement of the voltage at the ferromagnetic resonance . The DC magnetic field can be applied in \nthe plane of the sample, and perpendicularly to the electrical contacts. This configuration is the \nparallel configuration. The sample is then turned by 180°, in a position corresponding to the \nantiparallel configuration. As seen in figure 2 b) and c) in both the parallel and antiparallel \nconfiguration s, the signal is  symmetric and  independent of the sweeping time . In both cases we \nsubtracted the offset signal and divided by the square of the radiofrequency  magnetic field h rf to have \ncomparable results. The amplitude of the radiofrequency  field was determined by measuring the Q \nfactor with the sample p laced inside the cavity using the following equation: h rf2 =4PQ/500, with P \nthe microwave power in Watt [33].  \n \nFigure 2: a) Schematic representation of the measurement device used to detect spin pumping ISHE at \nresonance in the Pt(10) \\Py(20) sample (top view). b) Signal obtained in the in -plane par allel configuration for \n4 various sweeping rates, normalized by the square of the rf magnetic field ℎ𝑟𝑓2 for a power of 100mW . The Q \nfactor was 690 and the symmetric voltage of 14.7 µV before normalization.  c) Similar measurement in the in -\nplane antiparallel configuration. The Q factor was 537 and the symmetric voltage of -11.8 µV before \nnormalization  d) Out -of-plane angular dependence of the spin signal fitted using the ISHE angular dependence \nprovided in ref [33] e) Power dependence of the symmetric part of the signal as a function of the microwave \npower in the parallel configuration , the Q factor was 510 . \nRegarding the possibl e contribution of the spin rectifications effects  in Py , the out-of-plane angular \ndependence has also been performed (cf. figure 2 d) . The obtained symmetric signal can be fitted \nwith the ISHE angular dependence model described in reference  [33] as sin(M) with M being the \nmagnetization angle with respect to the out of plane direction.  This also excludes any contribution of \nthe ordinary Seebeck effect , which  would be field  independent , and of the Nernst effect , which  \nwould depend  on the applied field  perpendicular to the the rmal gradient . The signal is also linear \nwith the power (cf. figure 2 e)  indicating a negligible change of magnetization when increasing Power . \nThe signal possesses  the ISHE  angular dependence , and there is  no trace of thermal drift, which \nimplies that if there is  a thermal component to the signal, a steady state with the thermal gradient \nhas to be reached in a characteristic time  well below one second . \n \nFigure 3: a) Schematic representation of the measurement device used to detect a resistance \nchange at resonance in the Pt(10) \\Py(20) sample (top view). b) Change of the two probes \nresistance around the resonance field, for various fiel d sweeping rates (with base resistance \nsubtracted). c) Resistance of the sample as a function of the temperature, measured in a 2 probes \nconfiguration on the same sample. The slope value is 96±2m Ω/K d) Temperature change as a \nfunction of the sweeping rate,  estimated from the increase of resistance at resonance.  \n \nLet us now evaluate  this characteristic time. A temperature increase can occur  at the ferromagnetic \nresonance , due to the increased microwave absorption at resonance  [26,39,40,41]. To evaluate the \ntime dependence of this effect , we adopt the measurement scheme shown in figure 3 a) , where the \nfield is applied out of plane to avoid ISHE or SRE voltage contribution . We used  a fixed DC  current of 1  \nmA, fixed power of 100  mW and measured the change of resistance at resonance,  known as the \nbolometric effect . As can be seen on figure 3 b) we observe an increase  of resistance at resonance ; \n5 more importantly , this increase is highly dependent on the field sweeping rate. The resistance \nincrease s from 7.2±1 mΩ for a sweeping rate of 12  mT/s, to 28.5 ±1 mΩ for a sweeping rate of 0.18  \nmT/s . The result is in stark contrast with fig. 2  where the signal is independent on the sweeping rate . \nThe temperature increase characteristic time is thus of several seconds , as the time spent near \nresonance at a sweeping rat e of 0.18  mT/s is of 20s for a linewidth of 3.5 mT. This timescale is similar \nto what has been observed in previous Electrically Detected FMR experiments  [39,40]. A similar result \nwas obtained with in plane field .  \n \nWe estimate the corresponding temperature increase using the temperature  dependence of the \nresistance of the sample  measured  inside the cavity  (cf. figure 3c ). The linear  behavior  of the \nresistance leads to a  temperature dependence of 96±2 mΩ/K in a range from 100K to Room \ntemperature . The temperature inc rease as a function of the  sweeping rate is shown in figure 3 d. The \nmaximum temperature increase is of 297±10  mK for the slowest sweeping rate , and of 75±10  mK for \nthe fastest . The temperature increase  is thus  found to be strongly dependent on the sweeping time , \nthe temperature stabilization  is not reached  even  after several seconds near resonance . Therefore , \nany effect originating from a temperature change  should vary with the sweeping time. Th e spin \nsignal s measured in the configuration of fig. 2  being totally independent  of the sweeping time, we \ncan conclude that in SiO 2\\\\Pt(10) \\Py(20) the longitudinal s pin Seebeck effect  and anomalous Nernst \neffect  are negligible , and that the observed si gnals are due to  coherent  spin pumping . This \ncomparative measurement can also be performed using a coplanar waveguide setup where \nbolometric effects with a time constant of several seconds were previously observed [40]. Some \nhundreds of mK might appear  to be a small temperature increase when compared to the several \nKelvins usually used in LSSE measurements. Nonetheless due to the large thickness of the su bstrate \ncompared to the Ferromagnetic thin films, the temperature difference between the bottom and the \ntop of the ferromagnetic layer is usually below 1mK  in LSSE measurements  [42]. For the ordinary \nNernst contribution a temperature difference of only a fe w mK could also give rise to signals of some \nµV [43].  \nIn order to verify this lack of thermal contribution in Pt we also performed  a combined  bolometric \nand SP-FMR  measurement s on a  LSAT \\\\La0.7Sr0.3MnO 3(13.8) \\Pt(8.2) sample , measured along the \n[100] direction . La0.7Sr0.3MnO 3 (LSMO ) possess es a high  resistivity compared  to Permalloy  and Plati-\nnum  moreover t he LSMO \\Pt structure is expected to possess a  small er ANE coefficient  but a large r \nSSE contribution than Py\\Pt as demonstrated in Longitudinal Spin Seebeck experiments  [38]. There-\nfore the possible contribution of SSE in this multilayer is expected to be enhanced  compared to \nPt\\Py. In figure s 4b and 4 c similarly to the case of Pt \\Py we can see that  the thermal equilibrium is \nstill not reached even for the slowest sweeping rate , the total temperature increase is of comparable  \namplitude and up to 199±3mK. As can be seen in figure 4e and 4f, the obtained spin signal is inde-\npendent  on the sweeping rate . Here again, t his shows that in this system  the ANE and SSE contribu-\ntions are negligible  compared to the spin pumping ISHE signal . As already observed in LSMO layer, in \nthe out of plane configuration the resonance peak is slightly asymmetric, because of magnetic ho-\nmogeneities  [44], this leads to an asymmetry of the bolometric response as observed in figure 4 b). \nThis is not the case in Permalloy, because the small grain size leads to a str ong exchange narrowing  \n[45] and thus to a small inhomogeneity. This inhomogeneity is not observed in plane and ISHE signal \nin figure 4 e) and f) are consequently symmetric.  \n \nThe NM and FM  stacking order  is inverted in the Pt \\Py sample , which is  why the spin signal is of \nopposite sign when compared to LSMO \\Pt and  to previous results on Co \\Pt [9], as expected for  ISHE  \nsymmetries  [46]. The normalized ISHE signal  is the ISHE voltage divided by the square of the  rf field, \nthe width and the total resistance of the device. The  obtained values are  of 0.78  mV.G-2Ω-1m-1 in 6 LSMO \\Pt and 1.11  mV.G-2Ω-1m-1 in Pt \\Py, similar to the value of 0.85 mV.G-2Ω-1m-1 to 1.13  mV.G-2Ω-1m-\n1 that was previously reported in SiO 2\\\\Co\\Pt of similar thicknesses  at X-band [9] indicating  a similar \ninjected spin currents  in these three structures . \n \nFigure 4: a) Schematic representation of the measurement device used to detect resistance change in the \nLSAT  \\\\La0.3Sr0.7MnO 3(13.8) \\Pt(8.2) sample  (top view).  b) Change of resistance for various sweeping rate . Inset \nshows the  FMR response  out of plane  in LSMO is asymmetric,  with a narrow and a wide peak  c) Temperature \nchange as a function of sweeping rate. Inset shows the  resistance as a function of temperature, slope is of \n303±6  mΩ/K. d) Schematic representation of the measurement device used to detect spin pumping  (top view).  e) \nSignal obtained in the in -plane parallel configuration for various sweeping rates, normalized  by the  squared  rf \nmagnetic field  ℎ𝑟𝑓2. The Q factor was 510 and the symmetric voltage of -18.8  µV before normalization . f) Similar \nmeasurement in the in -plane antiparallel configuration . The Q factor was 269 and the symmetric voltage of 11.8  \nµV before normalization.   \n \nAnother experiment  can be  done to demonstrate the absence of thermal contribution to the spin \npumping signal. The  sample was placed in the parallel configuration and the  external field was  swept \nas fast as possible from  20 mT below the resonance  to the resonance field H res at a fixed rf power of \n100 mW. The sweeping rate in this experiments was limited to 1  mT/s to avoid a large oversho ot of \nthe field when stopping  at the resonance field and thus allowing a fast stabilization of the field \ncomparable to  our time resolution.  \nIn a first step, a 5 mA current is applied in the sample, so that the signal variations correspond mostly \nto resistance variations.  The voltage resulting due to Ohm’s Law is of 5  µV/m Ω using a current of 5mA \nwhile the total spin pumping signal is of around 10  µV at a power of 100  mW. The result s, shown in \nfigure 5, exhibits  a resistance increase when reaching the  resonance field . The time  constant of the \ntemperature increase is of around 10 seconds . In a second step, the same experiment is performed in \nthe open circuit conditions commonly  used for spin pumping experiments. In that case, the maximal \n7 signal is obtained immediately  after reac hing  the resonance field .  This implies t hat the signal \nmeasured in open  circuit conditions is not linked to the slow te mperature increase at resonance but \nto the fast dynamical spin injection mechanism.   \n \nFigure 5:  Change in the measured output voltage as a function of time  during the sweeping of the field from out \nof resonance to resonance  in the SiO2 \\\\Pt(10) \\Py(20) sample for parallel to the plane configuration   with an \ninput current of 5  mA (in black)  and in open circuit conditions  (in red) .  \nIn this study we focused on the possible thermal effects contributions at resonance . As previously \nshown the re is also in our measurements an offset signal  (existing at and out of resonance) , that is \nusually subtracted , and which can also be of thermal origin [29]. The change of temperature due to \nmicrowave absorption out of resonance is orders of magnitude larger than that due to the  resonance \n[40], and it can lead  to larger  thermal gradient s. In order to study that point  we used a technique  very \nsimilar  to that  shown in figure 5 , but out of resonance.We thus compared  the evolution s of the \nresistance  of the sample  and of the offset voltage when increasing  the power entering the cavity . We \nobserved a temperature change more than one order of magnitude larger than the change observed  \nat resonanc e at a similar power , and a similar evolution for both th e resistance and offset voltage  in \nboth samples . These observations confirm that the offset signal has a thermal origin , and that \nactually the temperature increase of the sample  is mostly due to microwave absorption , \nindependently of the  resonance.  Note that this signal contribution, which exists both at and out of \nresonance, does not contribute to the spin signal amplitude.  \nIV Conclusion  \nWe observed in SP- FMR experiments in cavity that the temperature increase at resonance is limited \nto a few hundreds of mK, even at a large  rf power of 100  mW, and is further reduced  to dozens of mK \n8 for faster field sweeping around the resonance . Moreover,  regarding the angular dependencies and \nthe absence of link between the detected signal and the temperature increase at resonance , we can \nconclude that the SSE and ANE are  absent in the signals  at resonance  for both SiO 2\\\\Pt\\Py and \nLSAT \\\\LSMO \\Pt multilayers , and that only dynamical spin injection is involved. The experiment  \npresented  here  can be generalized to any system measured using spin pumping , and in particular to \nRashba interfaces and topological insulators . Indeed , these two groups of materials gather  a large \nnumber of materials of interest for spinorbitronics but possess a very  high thermoelectric figure  of \nmerit  [47]. For example in Bi, Bi 2Se3, or Bi2Te3 it could  give rise to  non-negligible  thermal signals , \nunrelated to the spin-charge interconversion . It might  also be  a useful  way to further study some \nrecently evidenced thermal  effects such as the valley N ernst effect  [48] or to acquire a better \nunderstanding of the nature of the spin pumping signal in  antiferromagnets [49].   \nAcknowledgments:  \nWe acknowledge the financial support by ANR French National Research Agency Toprise  (ANR -16-\nCE24 -0017) , ANR French National Research Agency OISO  (ANR -17-CE24 -0026)  the Laboratoire \nd’excellence LANEF (ANR -10-LABX -51-01) and European Commission via the TOCHA  project H2020 - \nFETPROACT -01-2018 under Grant Agreement 824140 . We are grateful to the EPR facilities available at \nthe national TGE RPE facilities (IR 3443). We thank Jean -François Jacquot and Serge Gambarelli for \ntheir help and advices on the FMR measurem ent setup.  \n \nData availability:  \nThe data that support the findings of this study are available from the corresponding author upon \nreasonable request.  \n                                                           \n1 Prenat, G., Jabeur, K., Di Pendina, G., Boulle, O., Gaudin, G., Spintronics -Based Computing. Springer \nInternational Publishing, 145 –157 (2015)  \n2 Manipatruni, S., Nikonov, D.E., Lin, C. -C., Gosavi, T.A., Liu, H., Prasad, B., Huang, Y . -L., Bonturim, E., Ram esh, R., \nYoung, I.A., Nature 565, 35 (2019)  \n3 Seifert, T., Jaiswal, S., Martens, U., Hannegan, J., Braun, L., Maldonado, P ., Freimuth, F., Kronenberg, A., Henrizi, \nJ., Radu, I., Beaurepaire, E., Mokrousov, Y ., Oppeneer, P .M., Jourdan, M., Jakob, G., Turchi novich, D., Hayden, \nL.M., Wolf, M., Münzenberg, M., Kläui, M., Kampfrath, T., Nature Photonics 10, 483 –488 (2016)  \n4 Vila, L., Kimura, T., Otani, Y . C., Phys. Rev. Lett. 99, 226604 (2007)  \n5 Uchida, K., Takahashi, S., Harii, K., Ieda, J., Koshibae, W., Ando, K., Maekawa, S., Saitoh, E., Nature 455, 778 –\n781 (2008)  \n6 Saitoh, E., Ueda, M., Miyajima, H., Tatara, G., Appl. Phys. Lett. 88, 182509 (2006)  \n7 Kato, Y .K., Myers, R.C., Gossard, A.C., Awschalom, D.D., Science 306, 1910 –1913 (2004)  \n8 Mosendz, O., Pearson, J.E., Fradin, F.Y ., Bauer, G.E.W., Bader, S.D., Hoffmann, A., Phys. Rev. Lett. 104, 046601 \n(2010)  \n9 Rojas -Sanchez,  J.C.,Reyren,  N.,Laczkowski,  P .,Savero,  W.,Attane,  J.P .,Deranlot,  C.,Jamet,  M.,George,  J.M.,Vila,  \nL.,Jaffres,  H., Phys. Rev. Lett. 112, 106602  (2014)  \n10 Ando, K., Saitoh, E., Nature Communications 3, 629 (2012)  \n11 Rojas -Sánchez, J.C., Vila, L., Desfonds, G., Gambarelli, S., Attané, J.P ., De Teresa, J.M., Magén, C., Fert, A., \nNature Communications 4, 2944. (2013)  \n12 Sangiao, S., De Teresa, J.M., Mor ellon, L., Lucas, I., Martinez -Velarte, M.C., Viret, M., Appl. Phys. Lett. 106, \n172403 (2015)  \n13 Jamali, M., Lee, J.S., Jeong, J.S., Mahfouzi, F., Lv, Y ., Zhao, Z., Nikolić, B.K., Mkhoyan, K.A., Samarth, N., Wang, \nJ.-P ., Nano Lett. 15, 7126 –7132 (2015)  \n14 Deorani , P ., Son, J., Banerjee, K., Koirala, N., Brahlek, M., Oh, S., Yang, H., Phys. Rev. B 90, 094403 (2014)  \n15 Shiomi, Y ., Nomura, K., Kajiwara, Y ., Eto, K., Novak, M., Segawa, K., Ando, Y ., Saitoh, E., Phys. Rev. Lett. 113, \n196601 (2014)  \n16 Noel, P .,Thomas , C.,Fu, Y .,Vila, L.,Haas, B. Jouneau, P .H.,Gambarelli, S.,Meunier ,T.,Ballet, P .,Attane, J.P . , Phys. \nRev. Lett. 120, 167201 (2018)  9                                                                                                                                                                                      \n17 Zhang, W., Vlaminck, V., Pearson, J.E., Divan, R., Bader, S.D., Hoffmann, A., Appl. Phys. Lett. 103, 242414 \n(2013)  \n18 Stamm , C., Murer, C., Berritta, M., Feng, J., Gabureac, M., Oppeneer, P .M., Gambardella, P . , Phys. Rev. Lett. \n119, 087203 ( 2017 ) \n19 Feng, Z., Hu, J., Sun, L., You, B., Wu, D., Du, J., Zhang, W., Hu, A., Yang, Y ., Tang, D.M., Zhang, B.S., Ding, H.F., \nPhys. Rev. B  85, 214423 ( 2012 ) \n20 Berger, A.J., Edwards, E.R.J., Nembach, H.T., Kar is, O., Weiler, M., Silva, T.J., Phys. Rev. B 98, 024402 (2018)  \n21 Sagasta, E., Omori, Y ., Isasa, M., Gradhand, M., Hueso, L.E., Niimi, Y ., Otani, Y .C., Casanova, F., Phys. Rev. B 94, \n060412 (2016)  \n22 Wang, Y ., Deorani, P ., Qiu, X., Kwon, J.H., Yang, H.  Appl. Phys. Lett. 105, 152412 (2014)  \n23 Nguyen, M. -H., Ralph, D.C., Buhrman, R.A., Phys. Rev. Lett. 116, 126601 (2016)  \n24 Tao, X., Liu, Q., Miao, B., Yu, R., Feng, Z., Sun, L., You, B., Du, J., Chen, K., Zhang, S., Zhang, L., Yuan, Z., Wu, D., \nDing, H. Science Advances 4, eaat1670 (2018)  \n25 Wesselink, R.J.H., Gupta, K., Yuan, Z., Kelly, P .J., Phys. Rev. B 99, 144409 (2019)  \n26 Yamanoi, K., Yokotani, Y ., Kimura, T., Appl. Phys. Lett. 107, 182 410 (2015)  \n27 Yamanoi, K., Yokotani, Y ., Kimura, T., Phys. Rev. Applied 8, 054031 (2017)  \n28 Lin, W., Chien, C.L., arXiv:1804.01392 [cond -mat] (2018).  \n29 Y . Huo, F.L. Zeng, C. Zhou, and Y . Z. Wu, Phys. Rev. Applied 8, 014022  (2017)  \n30 R. Iguchi and E. Saitoh, J. Phys. Soc. Jpn  86, 011003 (2017)  \n31 Harder, M., Gui, Y ., Hu, C. -M., Physics Reports, 661, 1 –59 (2016)  \n32 Azevedo, A., Vilela -Leao, L.H.,Rodriguez -Suarez, R.L.,LacerdaSantos, A.F., Rezende, S.M., Phys. Rev. B 83, \n144402 (2011)  \n33 Rojas -Sanchez, J. -C., Cubukc u, M., Jain, A., Vergnaud, C., Portemont, C., Ducruet, C., Barski,  A., Marty, A., Vila, \nL., Attane , J.-P ., Augendre, E., Desfonds, G., Gambarelli, S., Jaffrès, H., George, J. -M., Jamet, M., Phys. Rev. B 88, \n064403 (2013)  \n34 Chen, Y . S., Lin, J. G., Huang, S . Y . & Chien, C. L. Phys. Rev. B 99, 220402 (2019).  \n35 Yue,  D., Lin, W., Li, J., Jin, X., Chien , C.L., Phys. Rev. Lett. 121, 037201 (2018)  \n36 Tserkovnyak, Y ., Brataas, A., Bauer, G.E.W., Phys. Rev. B 66, 224403 (2002)  \n37 Holanda , J., Alves Santos, O., Cunha, R.O., Mendes, J.B.S., Rodríguez -Suárez, R.L., Azevedo, A., Rezende, S.M., \nPhys. Rev. B 95, 214421 (2017)  \n38 Wu, B.W., Luo, G.Y ., Lin, J.G., Huang, S.Y ., Phys. Rev. B 96, 060402 (2017)  \n39 Goennenwein, S.T.B., Schink, S.W., Brand lmaier, A., Boger, A., Opel, M., Gross, R., Keizer, R.S., Klapwijk, T.M., \nGupta, A., Huebl, H., Bihler, C., Brandt, M.S., Appl. Phys. Lett. 90, 162507 (2007)  \n40 Gui, Y .S., Mecking, N., Wirthmann, A., Bai, L.H., Hu, C. -M., Appl. Phys. Lett. 91, 082503 (2007)  \n41 Hahn, C., de Loubens, G., Klein, O., Viret, M., Naletov, V.V., Ben Youssef, J., Phys. Rev. B 87, 174417 (2013)  \n42 Kirihara, A., Uchida, K., Kajiwara, Y ., Ishida, M., Nakamura, Y ., Manako, T., Saitoh, E., Yorozu , S., Nature Mater. \n11 686 –9 \n43 Roschewsky,  N., Walker,  E.S.,  Gowtham,  P ., Muschinske,  S., Hellman,  F., Bank,  S.R, Salahuddin , S., \nPhys. Rev. B 99, 195103 (2019)  \n44 Luo, G. Y ., Belmeguenai, M., Roussigné, Y ., Chang, C. R., Lin, J. G., Chérif S. M. , AIP Advances 5, 097148 (2015)  \n45 Hurdequint, H., JMMM 242 –245, (2002)  \n46 Schreier, M., Bauer, G.E.W., Vasyuchka, V.I., Flipse, J., Uchida, K., Lotze, J., Lauer, V., Chumak, A.V., Serga, A.A., \nDaimon, S., Kikkawa, T., Saitoh, E., Wees, B.J. van, Hillebrands, B., Gross, R., Goennenwein, S.T.B., J. Phys. D: \nAppl. Phys. 48, 025001 (2014)  \n47 Goldsmid, H.J., Materials (Basel) 7, 2577 –2592 (2014)  \n48 Dau, M.T., Vergnaud, C., Marty, A., Beigné, C., Gambarelli, S., Maurel, V., Journot, T., Hyot, B., Guillet, T., \nGrévin, B., Okuno, H., Jamet, M., Nature Comm.  10, 5796 (2019)  \n49 Li, J., Wilson, C.B., Cheng, R., Lohmann, M., Kavand, M., Yuan, W., Aldosary, M., Agladze, N., Wei, P ., Sherwin, \nM.S., Shi, J., Nature 578, 70 –74 (2020)  "    },    {        "title": "1511.08664v1.Resonance_spin_charge_phenomena_and_mechanism_of_magnetoresistance_anisotropy_in_manganite_metal_bilayer_structures.pdf",        "content": "1 \n \n \n \n \n \n \nResonance spin -charge phenomena  and mechanism of magnetoresistance anisotropy in \nmanganite /metal  bilayer  structures  \nV.A. Atsarkin, B.V. Sorokin , I.V. Borisenko, V.V. Demidov, and G.A. Ovsyannikov  \nKotel’nikov Institute of Radio Engineering and Electronics of RAS  \n11-7, Mokhovaya Str., 125009 Moscow, Russia  \n  2 \n \n \n \n  \nAbstract  \nThe dc voltage generated under ferromagnetic resonance has been studied in bilayer  structures  \nbased on manganite thin epitaxial films La0.67Sr0.33MnO 3 (LSMO) and non -magnetic metals  (Au, \nPt, and SrRuO 3) in the temperature range up to  the Curie point. The effect is shown to be caused \nby two different phenomena: (1) the resonance dc electromotive force related to aniso tropic \nmagnetoresistance  (AMR)  in the manganite film and (2) pure spin current (spin pumping) \nregistered by means of the inverse spin Hall effect in normal metal. The two phenomena were \nseparated using the angular dependence of the effect, the external mag netic field H 0 being rotated \nin the film plane. It was found that the AMR mechanism in the manganite films differs \nsubstantially from that in traditional ferromagnetic metals being governed by  the colossal \nmagnetoresistance together with the in -plane magne tic anisotropy. The spin pumping effect \nregistered in the bilayers was found to be much lower than that reported for common \nferromagnets; possible reasons are discussed.  \n  3 \n \n \n \n \n \n1. Introduction  \nExcitation of ferromagnetic resonance (FMR) in ferromagnetic metals  (FM) result s in several \ninteresting spin -charge effects, such as a change in electrical resistance  [1-6], appearance of dc \nelectromotive forc e (the resonance e.m.f. effect)  [1, 2, 7 -10], and flowing of pure spin current \nacross the interface of FM with adjacent non -magnetic (“normal”) metal (NM) [ 9, 11 -16]. \nRecently, the resonance spin -charge  phenomena attract heightened attention due to both pure \nphysical interest and prospects of application in spintronics [17, 18 ].  As a rule, the studies in this \nfield have been  performed with common magnetic metals and alloy s; the results w ere explained \nsuccessively in terms of standard ideas on the anisotropic magnetoresistance (AM R) and various \nmanifes tations of the spin Hall effect .  \nMuch  less is known, however, on the resonance spin -charge effects in the doped rare -earth \nmanganites possessing unique magnetic and transport properties prospective for many \napplications (see, for  example, the review articles [ 19-21] and references therein).  The physical \npicture of the spin -charge interplay in manganites is strongly complicated due to s uch features as \ncolossal magnetoresistance (CMR), metal -insulator transition near the Curie point (T C), rich \nphase diagram , tendency to phase segregation , etc. Particularly, it was ascertained recently that \nthe AM R effect in the rare -earth manganites differ s considerably from that in common FMs in its \nmagnitude as well as temperature and angular dependences. Apparently, this points to some \npeculiarities in  the physical mechanism of the effect.  The theory was proposed [ 22] based on \nquantum -mechanical calculation of the spin -orbit interaction with account made for the \ncrystalline field; as a result, some experimental data were explained . However, this approac h is \nlimited to the case of low temperatures ( T<<TC) and still was not supported sufficiently in a wide \nrange of temperatures and manganite compositions.    \nThe so -called pure spin current is one of the most interesting and prospective effects arising \nunder conditions of the FMR excitation [ 11, 12 ]. The pure spin current is not related  with \ntransfer of electric  charge and corresponds to the flow of the non -equilibrium component of the \nspin momentum from the excited ferromagnet to adjacent NM. FI/NM stru ctures were also \napproved for the spin current generation (here FI stands for  ferromagnetic in sulator) [ 23, 24 , 25]. \nIt was shown  [11, 12, 17 ] that the spin-dependent  value of the spin current Is is proportional to 4 \n \n[mdm/dt], where m is the unit vector directed along the ferromagnetic moment M precess ing at \nthe FMR frequency. The  cited  expression is analogous to the Gilbert relaxation term in the \nLandau -Lifshits equation. As a consequence, spin current affects the relaxation width of the \nFMR line which inc reases markedly due to the flowing  out of the transverse magnetization \ncomponent across the interface with NM. This effect can be used for estimation of Is [11]. The \nmost reliable method of detecting and measuring the pure spin curren t is based on registering the \ndc voltage USP (the superscript stands for spin pumping, SP) provided by the inverse spin Hall \neffect (ISHE) in the NM accepting the spin flow  [13]. The magnitude of USP depends on the spin \ncurrent generation rate, the interface transparency for various spin components (the spin \nconduc tance  tensor ), and the ISHE value in the NM. The latter is determined mostly by the spin -\norbit interaction efficiency which increases steep ly in heavy metals. Up to date, the maximum \nvalues of USP (up to several mV) were  obtained in the FI/NM bilayers where the Y3Fe5O12 (YIG) \nepitaxial film was used as the source of the spin current and thin films of Pt, Ta, W, etc.  served \nfor the ISHE registering [ 24, 25].   \nIn the FM/NM structures employing standard ferromagnetic metals such as Fe, Py, etc., the USP \nmagnitude was found to be considerably less [ 9, 15, 16 ]. Besides, in this case one has to separate  \nthe measured dc voltage into two components: UAMR and USP arising in FM and NM, \nrespectively. The reliable method of distinguishing these cont ributions is described in Ref. [9 ]; it \nemploys the specific dependence of UAMR on the angle between the external magnetic field H0 \nand the direction of the microwave current induced in the FM film by the resonance pumping. \nThis technique is widely used; particularly, it was employed in our previous work [ 10] concerned \nwith the resonance e.m.f.  effect in the manganite thin films. In this case, however, a problem \narises because of lack of information about the AMR mechanism in manganites, see above. \nThus, it seems to be reasonable to continue and advance studies of the resonance spin -charge \nphenomena in thin manganite  films and bilayer  structures, including the detection of pure spin \ncurrent and new analysis of the AMR mechanism accounting for the CMR effect present in \nmanganites. These issues are the subject of the present work.  \n2. Experimental  \nThe samples under study were  bilayer structures  LSMO/ Au, LSMO/ Pt and LSMO/S rRuO3, \nbased o n thin epitaxial films of La 0.67Sr0.33MnO 3 (LSMO) . The SrRuO 3 (SRO ) metal is \nparamagnetic above 160 K ; it has good conductivity and allows easy epitaxial growth on \nperovskite substrates [26].  The LSMO films  with the thickness of 50 – 100 nm were epitaxially \ngrown  on the (011) surface of the 550.5 mm NdGaO 3 (NGO ) substrate by laser ablation ; for 5 \n \ndetailed description see Ref s. [27, 28]. To obtain the FM/NM bilayers, Au was deposited over \nthe LSMO film  in situ , whereas  Pt was deposited over LSMO  ex situ . The LSMO/SRO bilayer \nfilms were epitaxially grown  on the same  substrate by RF magnetron sputtering  [29]. It is known  \n[27, 30] that the basic plane of the LSMO films grown on the orthorhombic (110)NGO  substrate \nis (001) ( we use  the pseudocubic notation  for LSMO ), and an additional in -plane  uniaxial \nmagnetic anisotropy  is created  with the easy axis directed along [010] LSMO due to orthorhombic \ndistortions of the LSMO crystal structure . In the samples under study, the substrate plates were \ncut by such a way that the LSMO easy axis  (corresponding to the [11̅0] direction of the NGO \nsubstrate)  coincide d with an edge of the substrate  for LSMO/Au and LSMO/Pt  bilayers  and was \ndeclined from the edge by 4 0 deg. for LSMO/SRO  one (Fig. 1) . \nThe experiments were carried out  in the temperature range of 295 -360 K. The temperature was \nmeasured by use of preliminary  calibrated film resistance  R(T) . Thus, the film under study \nserved as a self -thermometer, allowing to  take into account additional heating produced by \nmeasuring current and microwave pumping. Typical R(T) data for similar LSMO films are \nreported in Ref. [6]. \nThe main set of experiments  was performed with a home -made EPR spectrometer which allowed \nelectrical voltage measuring  and additional irradiation of the sample with high power microwave \nradiation .  The sample was situated horizontally into the central maximum of the microwave \nmagnetic field h of the rectangular TE102 cavity with the loaded quality factor Q ~103 at the \nfrequency /2 ~ 9 GHz . The external magnetic field H0 could be rotated around the vertical \naxis, thus remaining in the film plane. Contact platinum electrodes were sputtered at four corners \nor along opposite sides of the manganite film or covering metal, see Fig. 1.  \n \n \n \n \n \n \n \n 6 \n \n \n \n \n \n  \n \n \n \n \n \nFig. 1.  The sketch of experimental geometry . The easy axis directions are shown at the left panel  \nfor two types of samples  (see the text).  \n \nAs a rule, the voltage U was measured along the sample border parallel to the wider wall of the \ncavity, that is along h (z axis in Fig.1) . Nominally , microwave currents cannot be induced  in this \ndirection . In practice, however, the TE 102 mode is perturbed by the sample and leads, so the z -\ncomponent of microwave current does exist and provides considerable dc effect under study \n[10]. \nThe microwave pumping with the power P up to 250 mW  was supplied by the Gann diodes. To \ndistinguish the effect of resonant pumping, the microwave power was square -wave modulated at \nthe modulation frequency f m = 100 kHz. Correspondingly, t he voltage U was lock -in amplified \nand detected with the reference fr equency f m. To enhance sensitivity, accumulation was used at \nrepeated H0 sweeping across the FMR line. With this technique, the signals down to 0. 01 V \nwere reliably recorded.  The U values given below are converted to dc voltage at the contact \nelectrodes . \nThe external magnet was supplied through  a switch key allowing inversion of the field direction. \nSince  the dc signals  caused  both by AMR  and spin pumping mechanisms change the sign \ntogether with the change of the H0 polarity , hereafter the half -difference of the signals measured  \nz \n \nα \nH0 \nh \nU \nEasy axis (=400) \nEasy axis ( =0) \nNM \nLSMO  \nNGO  \nU 7 \n \nat opposite field directions are presented.  This trick enables one  to avoid any parasitic signals \nwhich do not depen d on the field direction [10 ]. \nThe same setup  could be employed for FMR spectroscopy. When neces sary, more precise \nspectral measurements were performed at the commercial X -range E PR spectrometer Bruker ER \n200. The angular dependences of the FMR spectra were used for determination of the main \nparameters of the LSMO layers , such as saturated magnetization M, uniaxial magnetic \nanisotropy field H u and the FMR line width H (the absorption full width at half maximum ), see \nTable 1. Generally, the H u values at room temperature w ere about hundreds Oe, exceeding \nstrongly the natural crystallographic (cubic) anisotropy of LSMO, so the latter was disregarded . \nNote that the FMR line -width may serve as a sensitive indicator of the sample  quality, including \nthe film homogeneity and possible crystal twinning. In the best LSMO films, the value of H did \nnot exceed 20 Oe at 295 K, and passed through its maximum of about 200 Oe near the Curie \npoint (T C = 330 -350 K).   \nTable 1.  Main parameters and experimental data obtained with the bilayer samples under study.  \nUAMR\nmax and USP\nmax are the peak absolute values  of dc signals caused by the anisotrop ic \nmagne toresistance and spin pumping effects, respectively.   The data shown in six right columns \ncorrespond to the room temperature.  \n \nStructure  tFM/tNM \n(nm/nm)  TC \n(K) M \n(Oe) Hu \n(Oe) ∆H \n(Oe)  \n(deg.)  UAMR\nmax \n(V) USP\nmax \n(V) \nLSMO/Au  65/15 347 285 85 28 0 0.13 0.34 \nLSMO/Pt  80/10  347 302 139 20 0 0.2 0.5 \nLSMO/SRO  60/10  331 271 101 80 40 1.2 0.55 \n \n \n3. Results  \nTypical signals U(H 0) recorded in the samples LSMO /Au  and  LSMO /SRO by passing the \nmagnetic field across FMR conditions are shown in Fig. 2  (insets).  At fixed temperature, the \nsignal magnitudes were found to be proportional to the microwave power P within the available \nrange.  8 \n \n-0,4-0,20,00,20,4\n-90 0 90 180-101LSMO/Au \n Us(V)\n(a)\nLSMO/SRO(b)\n Us(V)\n(deg)1750 20000,00,20,4U (V)\nH0(Oe)\n1500 2000 25000,00,5 U (V)\nH0(Oe)\n \nFig. 2 . Magnitude of symmetric part Us of the resonance dc voltage as a function of the angle   \nbetween the field  direction and the sample edge for (a) LSMO/Au and (b) LSMO/SRO. Filled \ncircles are experimental data. Solid  curves are the best fits to Eq. (1) with =0 (a) and =400 (b). \nDotted and dashed curves correspond to the AMR and SP effects, respectively. Insets: t ypical \nU(H 0) signals recorded u pon FMR pumping .  \n \nIt is expected that these signals are resulted from two different effects: (1) rectification of \nmicrowave signal  due to AMR in the LSMO layer, and (2) spin pumping with detecting the spin \ncurrent by means of ISHE in the NM.  The corresponding contributions will be denoted by UAMR \nand USP. As seen from Fig .2, the U(H 0) signals contain both symmetric a nd antisymmetric \ncomponents . It is known [ 13] that t he shape of USP(H0) is always symmetric and coincides with \nthe FMR absorption line. This is not sufficient, however, for its separation from UAMR, since the \nlatter may contain both symmetric and antisymmetric components, their ratio being dependent on \nthe phase shift between the microwave field and microwave current [9 ]. In the geometry used in \nthis work (Fig.1), the microwave current flowing along U is related to some p erturbations (see \nabove, Section 2), so its phase depends on particular sample and hardly can be determined ab 9 \n \ninitio . As emphasized in Ref. [9], the most reliable way to separate the UAMR and USP signals is \nto analyze  the angular dependence of the effect when the field H0 is rotated in the film plane. \nNote that the antisymmetric signal caused by  the anomalous  Hall effect  [1] is hardly observable \nbecause the out -of-plane magnetization component is small due to the strong demagnetization \nfield [9].  \nThe dependences of the magnitude of the symmetric component Us on the angle  between H0 \nand the direction of U measuring for the two samples are presented in Figs. 2(a, b). As seen in \nthe Figure, t hese data, as well as those for LSMO/Pt (not shown), can be successively \napproximated by the following expression:  \n𝑈𝑠(𝛼)=𝑈0𝐴𝑀𝑅sin2(𝛼−𝛽)∙sin𝛼+𝑈0𝑆𝑃sin3𝛼    (1) \nTwo terms in Eq.(1) are in accordance with the theoretically approved formulae for the AMR \nand SP effects [ 9] and serve as a base for separating them  (note that in our case H 0>>H u, so the \nequilibrium magnetization is directed practically along H0). The common factor sin  in Eq. (1) \nworks in any case, since the FMR precession is excited only by the transverse component of h. \nThe curves corresponding  to each mechanism are also shown in Fig. 2.  Maximum  voltages for  \nthe both effects , as well as the fitting angle s  for various samples are listed in Table 1.  It should \nbe noted that  is zero  for LSMO/Au and LSMO/Pt, but equals to 40 deg. for LSMO/SRO.   \nEvidently, this is related to the fact that the easy axis coincides with the azimuthal origin  (=0) \nin the first two samples, but differ from it just by 4 0 deg. in the latter one.  Thus, our data point to  \nspecific AMR mechanism where the effect depends on  the direction of easy axis rather than the \ndirection of electric current.  \nThis conclusion was supported by additional measurements performed with the LSMO fil m \nsimilar to that used in the LSMO/ Pt sample but without  the NM layer. The UAMR\nij signals were \ndetected using different pairs (i,j) of potential contacts situated at the sample corners , see  Fig. \n3(b). In this case, the registered signals correspond to different directions of the microwave \ncurrents contributing to the measured voltage, whereas the azimuthal angle   is counted from \nthe same origin coinciding with the easy axis . It was found that the UAMR\n12, UAMR\n23 and  UAMR\n34 \nsignals differed in their shape thus pointing to different phases of microwave currents. At the \nsame time, t he dependencies of the signal magnitudes  on  [Fig. 3 (a)] have the same functional \nform, being well described by the first term in Eq. (1) at =0. 10 \n \n \n \n \n \n \n \n \n \nFig. 3. (a) Angular dependences of the potential differences U ij measured between electrical \ncontacts 1,2 ( open triangles), 1,3 ( filled  squares) and 2,3 (circles) in the LSMO film under FMR \npumping. The solid curve corresponds to the first term in Eq. (1) at  = 0. (b) T he sketch of the \nsample.  \n \nTemperature dependences of the symmetric components Us for LSMO/Pt and LSMO/SRO  are \npresented in Fig. 4. The data were taken at different orientations of H 0 (α=β and α=150), thus \nallowing separation of the UAMR(T) and USP(T) functions. It is seen that both effects decrease \nsteeply when approaching T C. \n300 310 320 330 340 3500,00,20,40,6 U (V)\nT (K)\n \nFig. 4. Temperature dependence of the dc voltage caused by spin pumping  (LSMO/Pt, filled  \nsquares; LSMO/SRO,  filled  circles ) and by AMR effect (LSMO/SRO, triangles ). The lines \nconnect the experimental points.   \n \nFinally,  we present the results of some supplementary measurements to be useful in discussion \non the physical mechanism of the resonance e.m.f.  in manganites.  \n-90 -60 -30 0 30 60 90-20246\n  UAMR(V)\n(deg)a \nEasy axis  \nH0, M \n \nh \n4 \n3 \n 1 \n2 \n  U13 \n  U12 \n  U23 \nb 11 \n \nThe anisotropic magnetoresistance was measured directly in the sample with β =  400. The \nmeasuring dc cur rent was directed along the sample edge making an angle β with the easy axis. \nThe dependence of the resistance R on the in-plane H0 direction at H 0 = const . = 1700 Oe and \nT=295 K is shown in Fig. 5 (a).  \n-90 -60 -30 0 30 60 90-0,8-0,40,00,40,8-0,03-0,02-0,010,00\n(b) \n U (V)\nalfa (deg)(a)[R-R(0)]/R(0)  (%)\n \nFig. 5.  (a) Change in the film resistance measured along the sample edge ( =0) as a function of \nH0 direction in the film plane. The arrow indicate s easy axis. Solid curve corresponds to  Eq. (2).  \n(b) Angular dependence of UAMR\n in the same sample. The curve is fitted to  the first term in \nEq.(1) with =400. \n \nFor comparison, the angular dependence of UAMR is presented in Fig .5(b) . As seen from the \nFig.5(a), the AMR data can be  well described by the expression  \n𝑅(𝛼)\n𝑅(0)=1+𝜀𝐴𝑀𝑅cos2(𝛼−𝛽)   (2) 12 \n \nat AMR = (- 2,9  0,1) 10-4. Obviously, this result is consistent with the foregoing  treatment of \nEq. (1).   \nFurther, the differential CMR factor,  \n𝑟𝐶𝑀𝑅 =𝑑𝑅\n𝑑𝐻       (3) \nwas measured in the same sample as a function on temperature, see Fig. 6. Note that the R(H) \nfunction was found to be approximately linear in the field range used in present experiments; the \ndata shown in Fig. 6 were taken around H 0 = 2 kOe.  \n300 320 340 3600,0000,0020,0040,0060,008\nrCMR (Ohm/Oe)\nT (K)050100Hu (Oe)\nTC\n \nFig. 6.  The uniaxial in -plane anisotropy field H u (triangles, left scale) and d ifferential CMR \nfactor rCMR (squares, right  scale) measured in  the LSMO film as a function of temperature . Filled  \ncircles are the normalized rCMR values calculated with the use of Eq. (8), see the text. The curves \nare guides for eyes.  \n \nIn the same Fig. 6, the temperature dependence is plotted  of the uniaxial in -plane anisotropy field \nHu. The H u values were determined from the angular variation of the FMR line position, see Ref. \n[28]. As seen from the plot, the anisotropy field falls approximately linearly upon heating and \ntends to zero near T C. Temperature dependences of electrical resistance R(T), equilibrium 13 \n \nmagnetization M(T), and FMR line width H(T) were obtained as well, but not shown here for \nthe sake of brevity.  \n4. Discussion  \nConsider  firstly physi cal mechanism of the resonance e.m.f.  caused by anisotropic \nmagnetoresistance in LSMO films. Let us return to the experimental scheme shown in Fig.1. In \nthe simple model [ 8], the description is based on the Ohm law, \n𝑈(𝑡)=𝐼(𝑡)∙𝑅(𝑡)       (4) \nwhere I (t) = I0 sin(t) and R (t) are, respectively,  the microwave current and electric resistance in \nthe direction of the potential difference to be measured (z -axis in Fig. 1). Resonance microwave \npumping leads to precession of magnetization ; as a result, the angle (t) between z -axis and M(t) \nis modulated at the FMR frequency. Note that, owing to dema gnetization in a thin film, the \nprecession cone is flattened, so, in the first approximation, one has   \n𝛼(𝑡)=𝛼+𝛿∙sin(𝜔𝑡+𝜑)      (5) \nwhere  is the cone angle in the film  plane and  is the phase shift relative to the microwave \ncurrent.  At this point, the anisotropic magnetoresistance comes into play. Owing to AMR, the \nresistance R depends on the magnetization orientation, and hence on the angle (t). This leads to \ncorresponding modulation of the resistance R(t) and to appearance of the dc  component UAMR = \n<U(t)>. This is the resonance e.m.f. effect under study.  \nNaturally, the UAMR magnitude depends on .  In the traditional approach suitable for standard \nelementary ferromagnetic metals and alloys, the anisotropic part of the magnetoresistance is \nproportional to cos2 [3]. In this case, the angular dependence of UAMR would be represented by  \nthe first term in Eq. (1) with =0. However, our experimen tal data evidence for another behavior. \nIn fact, the effect is not always determined by , but rather by the deflection of the field H0 from \neasy axis. As a result, the angle  is substituted for (-) in Eqs. (1) and (2).  \nAt the first glance , this is consistent with the modern consideration  [25, 31] which predicts that \nthe AMR effect in manganites depends on the field direction relative to the [100] \ncrystallographic axis. This approach, however, is restricted by low temperatures (T<<TC) and \ndoes not take into account the in -plane magnetic anisotropy; thus, it is hardly applicable to our \ndata.  14 \n \nWe propose  another interpretation  taking into account the colossal magnetoresistance effect \n(CMR) which is a characteristic property of manganites,  especially in the temperature range not \nfar from T C. The CMR effect manifests itself in a decrease in resistivity  at H 0 increasing ; this is \ncommonly related to an increase in spin polarization favoring electron jumps between Mn3+ and  \nMn4+ ions [3 2, 19 ]. Evidently, such mechanism can be efficient only at high enough \ntemperatures (not too far from T C), when the maximum  value of saturated magnetization is not \nachieved because of thermal fluctuations. Usually CMR raises with temperature  in ferromagnetic \nphase , reaching its maximum near T C, see Fig. 5. It is reasonable to suppose that the \nmagnetization  magnitude (the length of the M vector)  depends not only on the external field H0, \nbut also on internal fields related to demagnetization factor and magnetic anisotropy. All these \ncontributions are included in an effective field He which can be determined using standard \nmethod s with the use of free energy and equilibrium conditions [ 33].  For the experiments \ndescribed in the present work, He is directed practically along H0 and equal s to \n𝐻𝑒=𝐻0+𝐻𝑢cos2𝛼𝑒     (6) \nwhere e = ( - ) is the angle making by H0 relative to the easy axis. Since CMR reveals  in \nresistance decreasing at field increasing, Eq. (6) predicts that the minimal value of R should be \nreached at H0 directed along easy axis  ([010] in our samples) . Recently such a result was \nobtained experimentally in very thin LSMO films with in -plane un iaxial magnetic anisotropy \n[34].  \nTo check this model quantitatively, let us compare the decrease in resistivity RCMR when  H0 is \nincreased just by the value of H u with the resistivity change RAMR due to H0 rotating by 900 \nfrom the hard to easy axis. The data obtained are as follows:  \n∆𝑅𝐶𝑀𝑅 =𝑟𝐶𝑀𝑅𝐻𝑢=−(0.083 ±0.005 )𝑂ℎ𝑚 \n            (7) \n∆𝑅𝐴𝑀𝑅 =𝜀𝐴𝑀𝑅𝑅=−(0.071 ±0.008 )𝑂ℎ𝑚 \nThe results coincide within the experimental error, evidencing for validity of the model \nsuggested.  \nDiscuss now the temperature dependence of the resonance e.m.f. effect, Fig.4. According to our \nmodel, the voltage  UAMR   should be  represented  by an expression analogous to that used in Refs. \n[8, 9],  but with RCMR substituted for RAMR, see Eq.(7) . As a result, we have:   15 \n \n𝑈𝐴𝑀𝑅(𝑇)=𝐴𝑟𝐶𝑀𝑅 (𝑇)∙𝐻𝑢(𝑇)\n𝑅(𝑇)∙∆𝐻(𝑇)      (8) \nHere the temperature independent factor A includes microwave field parameters, angular \ndependence  and so on. The inverse proportionality of the microwave current to th e resistance \nR(T) is taken into account.  \nTo check the validity of Eq.(8), the measured values of UAMR (Fig. 4) were multiplied by the \nexpression R(T)H(T) /Hu(T), the corresponding data being  obtained in independent \nexperiments. The results ( scaled  relative to the ordinate axis) are plotted together with  the \nexperimental graph of rCMR(T), Fig. 6. It is seen that the calculated and experimental temperature \ndependences of r CMR coincide within the experimental error.  \nNote that substitution  of He for H0 in the  AMR interpretation seems to be quite natural and can \nbe compared with  similar procedure  in the Landau -Lifshits equation [3 3]. More serious grounds \nare needed, however, when considering the resonance e.m.f. effect. Indeed, in terms of the \nsuggested approach, resonance oscillations of e lead to corresponding modulation of the length \nof the M vector which, in its turn, causes oscillations of resistance R(t) in Eq. (4)  owing to the \nCMR mechanism.  Such a model supposes an adiabatic mode of the resonance precession , in the \nsense that equilibrium values of M and R are established at every instant of time. This means \nthat the rate of the “absolute” longitudinal spin relaxation along M is much more than the \nprecession frequency,  \n𝜏1𝑎𝑏𝑠−1≫𝜔~6∙1010𝑠−1    (9) \nAccording to Eq. (9), 1abs-1 exceeds the “ordinary” relaxation rate related to rotating of M \ntoward its equilibrium orientation. At the same time, Eq. (9) is consistent with the  theory [ 35, 36] \nbased on the Landau -Lifshits -Bloch equation with account made for thermal fluctuations. As \nshown in  Ref. [ 36], the rate 1abs-1 in ferromagnetic phase is determined by exchange interaction \nand amounts to about  1013 s-1 at the whole temperature range except the close vicinity of T C, \nwhere a transition occurs to paramagnetic behavior. Note that this theory was successfully \napplied to interpretation of another spin -charge effect, namely, resonance magnetoresistance  in \nLSM O films [ 5, 6]. \nPassing to discussion on the spin pumping, it should be noted that the magnitude of the effect in \nthe LSMO/NM structures studied in the present work was found to be rather low: the values of \nUSP at room temperature amount to fractions of V and decrease steeply when approaching T C. \nAccording to t he theory [15, 16 ], the maximum voltage is det ermined by the expression  16 \n \n  \n                  U𝑚𝑎𝑥𝑆𝑃= − 𝑒𝑆𝐻\n2(𝜎𝑁𝑀𝑡𝑁𝑀+𝜎𝐹𝑀𝑡𝐹𝑀)𝜆𝑆𝐷tanh  (𝑡𝑁𝑀\n2𝜆𝑆𝐷)𝑔⇅𝐿𝑝(ℎ\n∆𝐻)2              (10)   \nwhere e is the electron charge; SH is the spin Hall angle  in NM ; σNM (σFM) and tNM (tFM) are the \nconductivities and thicknesses  of the NM (FM) layers, respectively; g⇅ is the real part of the spin \nmixing conduct ance characterizing the FM/NM interface ; λSD is the spin diffusion length in NM; \nL is the sample length along U; and p ~ 1 is a factor accounting for ellipticity of spin precession \nin the film. From Eq. (10), accounting for  our experimental data and the  metal  parameters cited \nin literature (see, for example, Ref.  [24]), the g⇅ value  can be estimated as being  one or two \norders of magnitude less than typical quantities  reported for various FI/NM and FM/NM \nstructures [ 15, 16, 25 ]. Low USP voltage obtained in our study m ight be related to specific \nfeatures of the LSMO zone diagram, as well as to low spin polarization caused by high enough \ntemperature  (see the temperature dependence of USP in Fig. 4). It seems, however, that  the most \nprobabl e cause of the SP suppression may be defects at  the LSMO/NM interface , such as \nformation of a “dead” layer  at the manganite surface . Further studies are planned . \nIn conclusion, two resonance spin -charge effects, the electromotive force  caused by anisotropic  \nmagnetoresistance and spin pumping (pure spin current ) induced by ferromagnetic resonance \nexcitation have been observed and studied in ferromagnetic manganite epitaxial thin films and \nmanganite/metal bilayers. Temperature dependencies of the e ffects were  traced up to Curie \npoint , where both effects  vanish ed. The two phenomena were separated from each other using \nvariation of the measured voltage as a function on the magnetic field direction. The dc voltage  \ncorresponding to the spin pumping w as found to be considerably lower than th at reported \npreviously for other FM/NM and FI/NM structures.  This may be attributed to some peculiarities \nof the FM/NM interface . Further study of this problem is desirable.  \nIt was also found that the physical mechan ism of e.m.f. caused by anisotropic magnetoresistance  \nin the manganite films , as well as the AMR itself,  differ considerably from similar effects  in \ntraditional ferromagnetic metals. In the manganite films  with in -plane magnetic anisotropy , the \nmain role is played by colossal magnetoresistance  which provides resistivity variations \ndepending on the angle between M and easy axis.  \nAcknowledgments  17 \n \nThe authors are grateful to A.M. Petrzhik, K.I. Constantinian, and Yu.V. Kislinski for their help \nin experiment . The work was supported by the RFBR grant s 14-02-00165 , 14-07-93105 , and 14-\n07-00258 . \nReferences  \n1. Juretschke  H J 1960  J. Appl. Phys.  31 1401  \n2. Egan  W G, Juretschke  H J  1963  J. Appl. Phys.  34 1477  \n3. Gui Y S,  Mecking  N, Wirthmann  A, Bai  L H, Hu C-M 2007  Appl. Phys. Lett.  91 082503  \n4. Hui X, Wirthmann  A, Gui Y S, Tian Y, Jin X F, Chen Z H, Shen S C and Hu C -M 2008  Appl. \nPhys. Lett. 93 232502  \n5. Atsarkin  V A, Demidov  V V, Levkin L V and Petrzhik A M  2010  Phys. Rev.  B 82 144414   \n6. Atsarkin  V A, Demidov  V V 2013  Zh. Eksp. Teor. Fiz.  143 109 [English translation: 2013 JETP  \n116 95] \n7. Gui Y S, Mecking  N, Zhou  X, Williams  G, Hu C-M 2007 Phys. Rev. Lett.  98 107602  \n8. Mecking  N, Gui  Y S, Hu C-M 2007 Phys. Rev.  B 76 224430  \n9. Azevedo  A, Vilela -Leão  L H, Rodríguez -Suárez  R L, Lacerda  Santos  A F, Rezende  S M 2011 \nPhys. Rev.  B 83 144402  \n10. Atsarkin  V A, Sorokin  B V  2014 Zh. Eksp. Teor. Fiz.  146 645 [English translation: 2014 JETP  \n119 567] \n11. Tserkovnyak  Y and Brataas  A 2002 Phys. Rev. Lett.  88 117601  \n12. Brataas  A, Tserkovnyak  Y, Bauer  G E W and Halperin  B I 2002 Phys. Rev.  B 66 060404(R)  \n13. Saitoh  E, Ueda  M and Tatara  G, 2006 Appl. Phys. Lett.  88 182509  \n14. Ando  K, Kajiwara  Y, Takahashi  S and Maekawa  S 2008 Phys. Rev. 78 014413  \n15. Mozendz  O, Vlaminck  V, Pearson  J E, Fradin  F Y, Bauer  G E W, Bader  S D and Hoffmann  A, \n2010 Phys. Rev.  B 82 214403  \n16. Czeschka  F D, Dreher  L, Brandt  M S, Weiler  M, Althammer  M, Imort  I-M, Reiss  G, Thomas  A, \nSchoch  W, Limmer  W, Huebl  H, Gross  R and Goennenwein  S T B 2011 Phys. Rev. Lett.  107 \n046601  18 \n \n17. Tserkovnyak  Y, Brataas  A, Bauer  G E M and Halperin  B I 2005 Rev. Mod. Phys . 77 1375  \n18. Locatelli  N, Cros  V and Grollier  J 2014 Nature Materials  13, 11 \n19. Salamon  M and Jaime  M 2001 Rev. Mod. Phys.  73, 583  \n20. Haghiri -Gosnet  A-M and Renard  J-P 2003 J. Phys. D: Appl. Phys.  36, R127  \n21. Dörr  K, 2006 J. Phys. D: Appl. Phys.  39, R125  \n22. Fuhr  J.D, Granada  M, Steren  L B and Alascio  B  2010 J. Phys.: Condens. Matter  22, 146001  \n23. Heinrich  B, Burrowes  C, Montoya  E, Kardasz  B, Girt  E, Song  Y-Y, Sun  Y and Wu M 2011 \nPhys. Rev. Lett. 107 066604  \n24. Wang  H L, Du C H, Pu Y, Adur  R, Hammel  P C and Yang  F Y 2013 Phys. Rev. B 88 100406(R)  \n25. Wang  H L, Du C H, Pu Y, Adur  R, Hammel  P C and Yang  F Y 2014 Phys. Rev. Lett.  112, \n197201  \n26. Allen  P B., Berger  H, Chauvet  O, Forro  L, Jarlborg  T, Junod  A, Revaz  B and Santi  G 1996 Phys. \nRev. B 53 4393  \n27. Demidov  V V, Borisenko  I V, Klimov  A A, Ovsyannikov  G A, Petrzhik  A M and Nikitov  S A \n2011 Zh. Eksp. Teor. Fiz. 139 943 [English translation: 2011 JETP  112, 825 ].  \n28. Demidov  V V, Ovsyannikov  G A, Petrzhik  A M, Borisenko  I V, Shadrin  A V and Gunnarsson  \nR. 2013 J. Appl. Phys.  113, 163909  \n29. Borisenko  I V, Karpov  M A and Ovsyannikov  G A 2013 Pis’ma v Zhurnal Tekhnicheskoj Fiziki  \n39 1 [English translation: 2013  Technical Physics Letters  39 1027 ] \n30. Boschker  H, Mathews  M, Houwman E P, Nishikawa H, Vailionis A, Koster G, Rijnders G and \nBlank D H A  2009 Phys. Rev.  B 79 214425  \n31. Egilmez  M, Chow  K H and Jung  J A 2011 Mod. Phys. Lett.  B 25 697 \n32. Anderson  P W and Hasegawa  H 1955 Phys. Rev.  100 675 \n33. Gurevich A G and Melkov  G A 1996  Magnetization Oscillations and Waves  (Boca Raton: CRC \nPress ) \n34. Liao Z, Huijben  M, Koster  G and Rijnders  G 2014 APL Materials  2 096112  19 \n \n35. Garanin  D A 1997 Phys. Rev.  B 55 3050  \n36. Chubykalo -Fesenko  O, Nowak  U, Chantrell  R W and Garanin  D 2006 Phys. Rev.  B 74 094436  \n \nCaptions for Figures  \nFig. 1. The sketch of experimental geometry. The easy axis directions are shown at the left panel \nfor two types of samples (see the text).  \n \nFig. 2.  Magnitude of symmetric part Us of the resonance dc voltage as a function of the angle   \nbetween the field direct ion and the sample edge for (a) LSMO/Au and (b) LSMO/SRO. Filled \ncircles are experimental data. Solid curves are the best fits to Eq. (1) with =0 (a) and =400 (b). \nDotted and dashed curves correspond to the AMR and SP effects, respectively. Insets: typic al \nU(H 0) signals recorded upon FMR pumping.  \nFig. 3. (a) Angular dependences of the potential differences U ij measured between electrical \ncontacts 1,2  (open triangles), 1,3 (filled squares) and 2,3 (circles) in the LSMO film under FMR \npumping. The solid curve corresponds to the first term in Eq. (1) at  = 0. (b) The sketch of the \nsample.  \n \nFig. 4.  Temperature dependence of the dc voltage caused by spin pumping (LSMO/Pt, filled \nsquares; LSMO/SRO, filled circles) and by AMR effect (LSMO/SRO, triangles). The lines \nconnect the experimental points.  \n \nFig. 5.  (a) Change in the film resistance measured along the sample edge ( =0) as a function of \nH0 direction i n the film plane. The arrow indicates easy axis. Solid curve corresponds to Eq. (2). \n(b) Angular dependence of UAMR\n in the same sample. The curve is fitted to the first term in \nEq.(1) with =400. \n \nFig. 6.  The uniaxial in -plane anisotropy field H u (triangles, left scale) and differential CMR \nfactor r CMR (squares, right scale) measured in the LSMO film as a function of temperature. Filled \ncircles are the normalized r CMR values calculated with the use of Eq. (8), see the text. The curves \nare guides f or eyes.  \n  20 \n \nTable 1.  Main parameters and experimental data obtained with the bilayer samples under study.  \nUAMR\nmax and USP\nmax are the peak absolute values of dc signals caused by the anisotrop ic \nmagne toresistance and spin pumping effects, respectively.   The data shown in six right columns \ncorrespond to the room temperature.  \n \nStructure  tFM/tNM \n(nm/nm)  TC \n(K) M \n(Oe) Hu \n(Oe) ∆H \n(Oe)  \n(deg.)  UAMR\nmax \n(V) USP\nmax \n(V) \nLSMO/Au  65/15 347 285 85 28 0 0.13 0.34 \nLSMO/Pt  80/10  347 302 139 20 0 0.2 0.5 \nLSMO/SRO  60/10  331 271 101 80 40 1.2 0.55 \n  "    },    {        "title": "1306.4680v1.Asymmetric_Ferromagnetic_Resonance__Universal_Walker_Breakdown__and_Counterflow_Domain_Wall_Motion_in_the_Presence_of_Multiple_Spin_Orbit_Torques.pdf",        "content": "Asymmetric Ferromagnetic Resonance, Universal Walker Breakdown, and Counterﬂow Domain\nWall Motion in the Presence of Multiple Spin-Orbit Torques\nJacob Linder\u0003and Mohammad Alidousty\nDepartment of Physics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway\n(Dated: November 1, 2018)\nWe study the motion of several types of domain wall proﬁles in spin-orbit coupled magnetic nanowires and\nalso the inﬂuence of spin-orbit interaction on the ferromagnetic resonance of uniform magnetic ﬁlms. Whereas\ndomain wall motion in systems without correlations between spin-space and real-space is not sensitive to the\nprecise magnetization texture of the domain wall, spin-orbit interactions break the equivalence between such\ntextures due to the coupling between the momentum and spin of the electrons. In particular, we extend previous\nstudies by fully considering not only the ﬁeld-like contribution from the spin-orbit torque, but also the recently\nderived Slonczewski-like spin-orbit torque. We show that the latter interaction affects both the domain wall\nvelocity and the Walker breakdown threshold non-trivially, which suggests that it should be accounted in exper-\nimental data analysis. We ﬁnd that the presence of multiple spin-orbit torques may render the Walker breakdown\nto be universal in the sense that the threshold is completely independent on the material-dependent Gilbert damp-\ning\u000b, non-adiabaticity \f, and the chirality \u001bof the domain wall. We also ﬁnd that domain wall motion against\nthe current injection is sustained in the presence of multiple spin-orbit torques and that the wall proﬁle will\ndetermine the qualitative inﬂuence of these different types of torques ( e.g.ﬁeld-like and Slonczewski-like). In\naddition, we consider a uniform ferromagnetic layer under a current bias, and ﬁnd that the resonance frequency\nbecomes asymmetric against the current direction in the presence of Slonczewski-like spin-orbit coupling. This\nis in contrast with those cases where such an interaction is absent, where the frequency is found to be symmetric\nwith respect to the current direction. This ﬁnding shows that spin-orbit interactions may offer additional con-\ntrol over pumped and absorbed energy in a ferromagnetic resonance setup by manipulating the injected current\ndirection.\nPACS numbers: 75.78.Fg,75.60.Jk,76.50.+g, 75.76.+j, 85.75.-d\nI. INTRODUCTION\nSpintronics has been a highly fertile research area espe-\ncially over the last two decades1, giving rise to practical devel-\nopments such as read-heads of harddrives, non-volatile mag-\nnetic memory, and other types of magnetic sensors2,3. The\nkey ingredient in this ﬁeld is to utilize the spin-degree of free-\ndom in currents and materials to achieve the desired function-\nality, in particular with an eye to providing a feasible alterna-\ntive to semiconductor technology. One of the main obstacles\nto overcome in this regard is the high energy cost associated\nwith e.g.Joule heating when passing a spin-polarized current\nconsisting of electrons through a device: current-densities of\norder 106A/cm2are needed to perform magnetization switch-\ning via current-induced spin-transfer torque. As an alternative\nmechanism to spin-transfer torque which could circumvent\nthe Joule heating from electrons, magnon-induced magneti-\nzation dynamics has been investigated more recently4–7.\nCurrently, the topic of controllable domain wall motion is\nreceiving much attention (see e.g.Ref. 8 for a very recent re-\nview) due to its potential with regard to the storage and trans-\nfer of information. A domain wall is a topological defect in\na magnetic system where the local magnetic order parame-\nter typically rotates spatially in a fashion that reduces the net\nmagnetic moment of the domain wall area. Owing to their\nsmall size (\u001810 nm) and large velocities ( \u0018100 m/s)9,10,\ncontrollable domain wall motion represents holds real poten-\ntial for tailoring functional devices with fast writing speeds.\nIn addition, there has been several proposals11–14related to\nmagnetic memory functionality due to the non-volatile nature\nof magnetic domains. Walker breakdown15is nevertheless a\n-x+xz\ny\n-x+xz\ny\n +x-x\nxy(a): Bloch(z)\n(b): Bloch(y)(c): Head-to-head\nDomain wall ferromagn\netFIG. 1: (Color online) Schematic setup: a spin-polarized current\nis passed through a domain wall magnetic nanowire with spin-orbit\ncoupling. The spin-orbit interaction may be either intrinsic or in-\nduced via a heavy metal proximate host. We consider several types\nof domain wall conﬁgurations, since the presence of spin-orbit cou-\npling qualitatively distinguishes the domain wall motion with one\ntype of magnetization texture from another. More speciﬁcally, we\nconsider two types of Bloch-domain walls relevant for perpendicular\nmagnetic anisotropy systems in addition to a head-to-head domain\nwall with an in-plane magnetization easy anisotropy.\nlimiting factor in this regard.\nDomain walls can come in several different shapes depend-\ning on the anisotropy energies and dimensionality of the sys-\ntem at hand. In a low-dimensional system such as a magnetic\nnanowire, Bloch walls are one of the most frequent types en-arXiv:1306.4680v1  [cond-mat.mes-hall]  19 Jun 20132\ncountered. However, it is also possible to generate other sorts\nof magnetization textures such as head-to-head domain walls.\nBoth of these wall types are shown in Fig. 1. A key ques-\ntion is whether or not speciﬁc domain wall types are beneﬁ-\ncial with regard to the objectives mentioned above ( e.g. fast\npropagation, low current densities to generate motion). The\nanswer to this question depends on if the spin and position\ndegrees of freedom are correlated in the system, for instance\nvia spin-orbit interaction. In the absence of such spin-orbit in-\nteractions, different types of domain walls behave in the same\nway - the exact magnetization texture has no effect and one\nobtains for instance the same terminal domain wall velocity in\nall cases. The fact changes when spin-orbit coupling is present\nsince the electron transport and spin torque now directly de-\npends on the precise magnetization texture, which warrants a\nspeciﬁc study for how domain wall motion is manifested for\ndifferent types of domain walls. A numerical investigation of\nthis issue was recently put forth in Ref. 16.\nThe inﬂuence of spin-orbit coupling on domain wall mo-\ntion has recently been considered extensively in several theo-\nretical works16–23. On the experimental stage24–27, it has been\ndemonstrated that the presence of spin-orbit coupling indeed\ninﬂuences the domain wall dynamics in a non-trivial way in-\ncluding anomalous behavior such as strongly enhanced do-\nmain wall velocities and induced wall motion in the opposite\ndirection of the electron ﬂow. In order to explain these ﬁnd-\nings, it was shown in Ref. 21 that the presence of spin-orbit\ncoupling would generate not only a ﬁeld-like torque but also\na so-called Slonczewski-like torque28, named such due to its\nformal resemblence to standard current-induced torques in the\nabsence of spin-orbit coupling. Alternatively, these two types\nof spin-orbit torques may be characterized as out-of-plane and\nin-plane components of the total Rashba torque29.\nMotivated by this, we will in this paper derive exact ana-\nlytical expressions for the domain wall velocity and Walker\nbreakdown threshold for several types of domain wall conﬁg-\nurations when including both types of spin-orbit torques in or-\nder to investigate how the Slonczewski-like torque inﬂuences\nthe physics at hand. This way, we expand previous literature17\nwhich has only considered the ﬁeld-like term and show that\nthe inclusion of the Slonczewski-like torque has profound im-\npact on the domain wall velocity and the threshold value of\nWalker breakdown. In fact, we will show that the existence\nof this torque renders the threshold value to be universal in\nthe sense that it is independent on both the Gilbert damping\n\u000b, the non-adiabiticity parameter \f, and the chirality \u001bof the\ndomain wall.\nWe will present a detailed derivation of the equations of\nmotion where possible and show precisely in which manner\nthe spin-orbit coupling inﬂuences both the domain wall ve-\nlocity and the Walker breakdown threshold value. Our analyt-\nical expressions show the precise conditions required to real-\nize domain wall motion against the current ﬂow, as has been\nexperimentally observed recently27, and in particular how the\ndomain wall chirality affects this phenomenon.\nFinally, we investigate how the ferromagnetic resonance\nresponse of a material (or equivalently the dissipation and\npumping of energy) is altered due to the above men-tioned spin-orbit torques. The ferromagnetic resonance ex-\nperiment is an important technique for obtaining informa-\ntion about anisotropy, magnetic damping and magnetization\nreversal41–43,48,50–52. The inﬂuence of spin-polarized current\non Gilbert damping and ferromagnetic resonance have been\nextensively investigated in different situations30,31,44–47,49.\nConsidering a ferromagnetic resonance setup in the pres-\nence of a current bias, we analytically show that the spin-orbit\ninteractions render the resonance frequency to become asym-\nmetric with respect to the direction of current injection. This\nis different from previous works considering a ferromagnetic\nresonance setup in the presence of spin-transfer torques, albeit\nwithout spin-orbit coupling, where the frequency was found to\nbe symmetric with respect to the current direction30,49.\nThis paper is organized as follows. In Sec. II, we\noutline the theoretical framework to be used in our analy-\nsis, namely the Landau-Lifshitz-Gilbert (LLG) equation aug-\nmented to include the role of spin-orbit coupling combined\nwith a collective-coordinate description of the domain wall.\nWe then present our main ﬁndings in Sec. III, in four subsec-\ntions, where the LLG equation is solved in order to obtain both\nthe domain wall velocity, the Walker breakdown threshold and\nthe ferromagnetic resonance frequency. In Subsec. III A we\nconsider Block( z) wall proﬁle, in Subsec III B Block( z) wall\nproﬁle is studied, in Subsec. III C a head-to-head domain wall\nstructure is investigated, and in Subsec. III D we present and\ndiscuss the results of absorbed power by a ferromagnetic ﬁlm\nunder current injection in the presence of Slonczewski-like\nspin-orbit interaction. We ﬁnally summarize our results and\nﬁndings in Sec. IV.\nII. THEORY\nThe starting point of our analysis is the spatio-temporal\nLandau-Lifshitz-Gilbert equation32, augmented to include the\ncontribution from torque terms arising due to the presence of\nspin-orbit coupling. When a current-bias is applied along x\naxis, the full LLG equation takes the form21,33\n@tM =\u0000\rM\u0002(Heff+Hso\u0000\f\nM0M\u0002Hso)\n+\u000b\nM0M\u0002@tM+ \u0000@xM\u0000\f\u0000\nM0M\u0002@xM:(1)\nThe above equation describes the time-dynamics of the local\nmagnetic order parameter M(x;t). The effective ﬁeld Heffis\nformally obtained by a functional derivative of the free energy\nwith respect to the magnetization and will vary depending on\ne.g.the anisotropy conﬁguration of the wire35. The inﬂuence\nof spin-orbit interaction is captured as an effective ﬁeld:\nHso=\u000bRmeS\n~eM0(1 +\f2)^z\u0002j; (2)\nwhere inversion symmetry is broken in the zdirection and\n\u000bRcharacterizes the strength of the spin-orbit coupling. S\nandjis the polarization and density of the injected current,\nwhereasmeandM0is the electron mass and magnitude of\nthe magnetization, respectively. The parameter \fis known as3\nthe non-adiabaticity parameter in the literature, a convention\nwe shall stick to although this terminology is not ideal34.\nThe terms in Eq. (1) have the following physical interpre-\ntation. The effective ﬁeld causes a precession of the magneti-\nzation vector Mand has two extra contributions in terms of\nHsoandM\u0002Hsoin the presence of spin-orbit coupling. The\nformer of these has the exact form of an effective ﬁeld-like\ntorque whereas the latter has the form of a Slonczewksi-like\ntorque. Interestingly, this term was conjectured to exist in the\nexperiment of Miron et al.27in order to explain the results, but\nit was only recently theoretically derived in Refs. 21, 29. A\nkey observation is that the Slonczewski like spin-orbit torque\ndepends on the non-adiabaticity parameter \fwhich also ap-\npears for the conventional non-adiabatic spin-transfer torque\n[last term in Eq. (1)] as is well-known. The term /@xM\nis the adiabatic spin-transfer torque originating from the as-\nsumption that the spin of the conduction electrons follow the\ndomain wall proﬁle perfectly without any loss or spin scatter-\ning and \u0000 =\u0016BP=eM 0(1 +\f2).\nOne of the main goal in this work is to compute the domain\nwall velocity and analyze Walker breakdown for a domain\nwall nanowire with spin-orbit coupling, considering several\ntypes of experimentally relevant domain walls, both with in-\nplane and perpendicular magnetization relative the extension\nof the wire16,25–27. We will take into account both the ﬁeld-\nlike and the Slonczewski-like spin-orbit induced torques. We\nunderline again that the various magnetization textures con-\nsidered in this paper will give qualitatively different behavior\nfor the wall velocity and Walker threshold values precisely\ndue to the spin-orbit interaction which correlates spin- and\nreal-space. For a Bloch( y) domain wall (see Fig. 1), an exact\nanalytical solution for the domain wall velocity vDWis permis-\nsible and we will derive this result in detail. For other types\nof domain walls, a general expression for vDWis not possi-\nble to obtain analytically, thus for completeness, we revert to\na numerical study for these cases. However, it is still possi-\nble to investigate analytically the Walker breakdown thresh-\nold for these domain walls and we show that the chirality of\nthe domain wall conspires with the presence of spin-orbit cou-\npling to qualitatively alter the behavior of Walker breakdown\nin spin-orbit coupled nanowires.\nIII. RESULTS AND DISCUSSION\nWe shall start by investigating domain wall motion in the\npresence of multiple spin-orbit torques and consider three\ntypes of domain wall structures as shown in Fig. 1. For\neach case, we will focus on the domain wall velocity and the\nWalker breakdown threshold value, giving exact analytical re-\nsults where possible. We note that such an exact solution for\nvDWconstitutes the most general analytical expression for the\ndomain wall velocity up to now, including fully the inﬂuence\nof spin-orbit coupling. We then study the ferromagnetic res-\nonance response of a magnetic layer with a Slonczewski-like\nspin-orbit interaction with an injected current into the plane of\nthe layer and using the absorbed power by the ﬁlm, we drive\nthe ferromagnetic resonance expression analytically.A. Bloch(z) wall\nConsider ﬁrst a domain wall proﬁle relevant for magnetic\nnanowires with perpendicular anisotropy25–27(e.g. Co/Ni\nmultilayers), namely a so-called Bloch( z) wall which is\nparametrized as:\nm= (sin\u0012sin\u001e;sin\u0012cos\u001e;\u001bcos\u0012); (3)\nand a corresponding effective ﬁeld:\nHeff=2Aex\nM2\n0r2m\u0000H?mx^x+Hkmz^z+Hext: (4)\nHere,H?andHkare the anisotropy ﬁelds along the hard and\neasy axes of magnetization, respectively, whereas Hextis an\nexternally applied magnetic ﬁeld. The parameter \u001b=\u00061\ncharacterizes the chirality of the domain wall: both signs of\n\u001bgive allowed equilibrium solutions (\u001e= 0) of the LLG-\nequation and describes a spin texture changing from positive\nto negative depending on which direction one is moving in.\nNote that\u001bis also denoted the topological charge of the do-\nmain wall35: the winding direction of the local magnetization\ndictates the effective ”charge” since the sign of \u001bwill deter-\nmine the direction in which an external magnetic ﬁeld moves\nthe domain wall. The components of the magnetization vector\ndepend on both space and time according to15\ncos\u0012= tanh\u0010x\u0000X(t)\n\u0015\u0011\n;\nsin\u0012=sech\u0010x\u0000X(t)\n\u0015\u0011\n: (5)\nEq. (5) is obtained by inserting the magnetization proﬁle m\ninto the LLG equation and solving for \u0012and\u001eunder equilib-\nrium conditions (in which case X(t)is a constant and \u001e= 0).\nThe tilt angle \u001e=\u001e(t)is in general, however, time-dependent\nand causes the domain wall to acquire a ﬁnite component\nalong the hard magnetization axis in an non-equilibrium situ-\nation. A collective-coordinate description of the domain wall\nmotion is obtained if one may identify the time-dependence of\nthe domain-wall center position X(t)and the tilt angle \u001e(t).\nIn general, other modes of deformation can be allowed35.\nHowever, it can be shown that the domain wall may be treated\nas rigid [only depending on X(t)and\u001e(t)] in a collective-\ncoordinate framework when the easy axis anisotropy energy\nKis assumed larger than its hard axis equivalent K?36, i.e.\njKj\u001djK?j.\nIt is useful to write down an explicitly normalized form of\nthe LLG-equation which we will use for all the domain wall\nproﬁles considered in this work. We normalize all quanti-\nties to a dimensionless form as deﬁned by the following LLG\nequation:\n@\u001cm=\u0000m\u0002(Heff+Hso\u0000\fm\u0002Hso)\n+\u000bm\u0002@\u001cm+u@~xm\u0000\fum\u0002@~xm:(6)\nIn the speciﬁc case of a Bloch( z) wall, we then have the nor-\nmalized effective ﬁeld:\nHeff= 2A~r2m\u0000H?mx^x+Hkmz^z;\nHso= ~\u000bRu^y: (7)4\nInserting Eq. (5) into Eq. (6) leads to one pair of equations of\nmotion for the collective coordinates Xand\u001e. These equa-\ntions may be simpliﬁed by using Thiele’s approach37where\none integrates over xand utilizesR1\n\u00001sin2\u0012dx= 2\u0015andR1\n\u00001sin\u0012dx=\u0015\u0019. We then ﬁnd the following dimensionless\nequations:\n\u001a\n\u000b@\u001c\u001e\u0000\u001b@\u001cX=\u001bu\u00001\n2H?sin 2\u001e\u00001\n2~\u000bR\u0019usin\u001e;\n@\u001c\u001e+\u000b\u001b@\u001cX=1\n2\f~\u000bRu\u0019sin\u001e\u0000\fu\u001b:\nHere,X=X=\u0015 is the normalized spatial coordinate of the\ndomain-wall center and ~\u000bRis a dimensionless measure of the\nstrength of the spin-orbit interaction. In the limiting case of\nan absent Slonczewski-like spin-orbit torque where the terms\nproportional to \f\u0002~\u000bRare zero, our results are consistent\nwith Ref. 20. The sin\u001eterms in Eq. (8) makes an exact\nanalytical solution of the equations untractable. As we shall\nsee, a similar situation occurs for the head-to-head domain\nwall case. Nevertheless, it is possible to make further progress\nin the present case with regard to the appearance of so-called\nWalker breakdown15. This phenomenon refers to a threshold\nvalue of the current density for which the domain wall starts\nto rotate with a time-dependent \u001e=\u001e(\u001c)rather than simply\npropagating with a ﬁxed magnetization texture, i.e. constant\n\u001e. In general, it is desirable with as large threshold value as\npossible for Walker breakdown. We note in passing here that\nthe presence of pinning potentials and defects in the sample\nmay also contribute to the threshold value of the current, but\nwe leave this issue for a future work.\nTo investigate the velocity at which breakdown occurs, we\ncombine the equations of motion into a single equation for the\ntilt angle\u001e:\n@\u001c\u001e=1\n1 +\u000b2h1\n2(\f\u0000\u000b)~\u000bRu\u0019sin\u001e\u0000\u001bu(\f\u0000\u000b)\n\u00001\n2\u000bH?sin 2\u001ei\n: (8)\nThere is no Walker breakdown as long as @\u001c\u001e= 0, which\nholds when the tilt angle \u001esatisﬁes the equation:\nsin 2\u001e=(\f\u0000\u000b)u\n\u000bH?(~\u000bR\u0019sin\u001e\u00002\u001b): (9)\nWalker breakdown will occur at a velocity ucsuch that for\nu > ucthere is no stable solution for this equation. Now,\nforj~\u000bR\u0019j<2the right hand side of Eq. (9) will have equal\nsign for its minimum and maximum value as \u001evaries from\n0 to2\u0019. Therefore, Walker breakdown will always occur by\nincreasingu: at some value uc, the minimum value of the\nright hand side of Eq. (9) will be larger than unity and thus\nrender the equation to be void of any solution. However, if\nj~\u000bR\u0019j>2, the minimum and maximum value of the right\nhand side have opposite signs. This means that there must\nbe a crossing of the 0 line at some values of \u001e, and thus an\nintersection with sin 2\u001e. In effect, we can always ﬁnd a stable\nsolution and there will be no Walker breakdown regardless of\nthe velocity uwhen:\n\f\f\f~\u000bR\u0019\n2\f\f\f>1: (10)In other words, for a sufﬁciently large spin-orbit interaction,\nno Walker breakdown occurs. It is interesting to note that this\ncondition is universal in the sense that it is independent on the\ndamping parameter \u000b, the non-adiabiticity parameter \f, and\nthe chirality \u001bof the domain wall. This observation can be\nattributed directly to the presence of the new spin-orbit torque\nproportional to \f. To see this, consider a scenario where only\nthe ﬁeld-like spin-orbit torque /M\u0002Hsois included. All\nterms proportional to \f\u0002~\u000bRare then zero, and we obtain the\nequation\nsin 2\u001e=2\u001bu\nH?\u0010\n1\u0000\f=\u000b\u0000\u001b~\u000bR\u0019\n2sin\u001e\u0001\n; (11)\nwhich must be satisﬁed to prevent Walker breakdown. As\nseen, whether or not the maximum and minimum value of the\nright hand side have equal sign depends on if\n\f\f\f~\u000bR\u0019\n2\f\f\f>j(1\u0000\f=\u000b)j: (12)\nIn this regime, we recover the results of Ref. 20. The effect\nof the Slonczewski-like spin-orbit torque is then to render the\nWalker breakdown universal (independent on \u000b;\f;\u001b ). Let us\nalso consider the implications this torque-term has with regard\nto the magnitude of the threshold value for Walker breakdown.\nComparing Eqs. (10) and (12), we see that the required spin-\norbit interaction ~\u000bRto completely remove the Walker thresh-\nold depends on the ratio \f=\u000b if one does not take into account\nthe Slonczewski-like spin-orbit torque. For \f=\u000b'1, the re-\nquired spin-orbit strength becomes very small. In the more\ngeneral case where the aforementioned torque is included,\nhowever, the required ~\u000bRhas a ﬁxed value. This is shown\nin Fig. 2.\n0 1 2 3 400.511.522.53\nα/βThreshold ˜ αR\n  \nOnly field−like torque\nBoth FL− and SL−torque\nFIG. 2: (Color online) Threshold value for the magnitude of the spin-\norbit coupling above which there is no Walker breakdown. In the\nmore general scenario where both types of spin-orbit torques are ac-\ncounted for, the threshold value for ~\u000bRis constant. When only the\nﬁeld-like torque is considered, the threshold is strongly increased in\nthe regime\u000b=\f < 0:5. In the limit \u000b=\f!1 , the asymptote is\n2=\u0019.5\nWe also give numerical results for the wall velocity for this\nBloch domain wall conﬁguration, using a similar approach as\nin Ref. 38. Let us ﬁrst note that it is possible to infer what\nthe qualitative effect is of the chirality \u001bdirectly from the\nequations of motion Eqs. (8). By making the transformation\n\u001e!\u001b\u001e, it is seen that the equations of motion become in-\ndependent on the chirality \u001b:This means that the domain wall\nvelocity will be the same regardless of the sign of \u001b, whereas\nthe tilt angle \u001eevolves in the opposite direction with time for\nopposite signs of \u001b. In Fig. 3, we therefore present results for\n\u001b= 1without loss of generality and consider two cases with\ndamping\u000blarger or smaller than the non-adiabaticity con-\nstant\fin (a) and (b), respectively. As seen, this qualitatively\naffects the domain wall velocity.\nA particular feature worth noting in (b) is that the abrupt\nchange in wall velocity at a given uis not necessarily syn-\nonymous with the occurrence of Walker breakdown. To see\nthis, consider Fig. 4 where we have plotted the left- and right-\nhand side of the Walker breakdown criterion Eq. (9) in addi-\ntion to the time-evolution of the tilt angle \u001eas an inset. We\nhave set\u000b= 0:005and\f= 0:01and consider two strengths\nof the spin-orbit coupling parameter ~\u000bRin (a) and (b). An\nintersection of the lines in the main panels means that there\nexists a solution to Eq. (9) and that Walker breakdown does\nnot occur. Considering Fig. 4(a) ﬁrst, we see that increasing\nthe current density eventually causes Walker breakdown as the\ndashed and full lines no longer intersect. As a result, \u001eis no\nlonger a constant as seen in the inset and starts to grow with\ntime. We may therefore conclude that the abrupt change in\nwall velocity for ~\u000bR= 0:01seen in Fig. 3(b) does correspond\nto the occurrence of Walker breakdown. However, turning to\nFig. 4(b) it is seen that the dashed and full lines always in-\ntersect even when increasing the current density uabove the\nvalue at which the wall velocity abruptly changes in Fig. 3(b)\nfor~\u000bR= 1 (aroundu= 0:14). What is important to note\nis that their point of intersection changes discontinuously: the\ntilt angle\u001eremains constant so that there is no Walker break-\ndown in the sense of a continuously deforming domain wall.\nInstead, there is an abrupt change in the tilt angle where it\nchanges from one constant value to another.\nB. Bloch(y) wall\nAnother type of domain wall structure which may appear in\nsuch a systems with perpendicular magnetic anisotropy is the\nBloch (y)-wall, having the easy magnetization direction along\ntheyaxis whereas the hard axis remains along the wire direc-\ntion:\nm= (sin\u0012sin\u001e;\u001bcos\u0012;sin\u0012cos\u001e); (13)\nand a corresponding effective ﬁeld:\nHeff=2Aex\nM2\n0r2m\u0000H?mx^x+Hkmy^y+Hext:(14)\nIn this case, the equations of motion for the collective coordi-\nnatesXand\u001etake a different form compared to the Bloch( z)\n0.10.20.30.40.50.60.7−0.6\n−0.5\n−0.4\n−0.3\n−0.2\n−0.1\n0/angbracketleftvDW/angbracketright(a)\n  \n00.10.20.30.40.50.60.7−0.6\n−0.5\n−0.4\n−0.3\n−0.2\n−0.1\n0\nu/angbracketleftvDW/angbracketright(b)\n  ˜αR= 0\n0.001\n0.01\n0.1\n1FIG. 3: (Color online) Domain wall velocity for a Bloch( z) wall\nplotted against the injected current. We have chosen \u001b= 1 without\nloss of generality (see text) and set \f= 0:01andH?=0.5. In (a)\n\u000b > \f (\u000b= 0:02) whereas in (b) \u000b < \f (\u000b= 0:005). Note the\ninverted sign of the yaxis, which simply corresponds to the direction\nof the wall motion.\ncase:\n\u001a\n\u001b@\u001cX+\u000b@\u001c\u001e=\f~\u000bRu\u00001\n2H?sin 2\u001e\u0000u\u001b;\n@\u001c\u001e\u0000\u000b\u001b@\u001cX=\fu\u001b+ ~\u000bRu:\nIn fact, these equations can now be solved analytically in an\nexact manner, using a similar approach as in Ref. 22. Com-\nbining the two above equations yields:\n@\u001c\u001e(1 +\u000b2) =\u0000\u000b\n2H?sin 2\u001e+u[\u001b(\f\u0000\u000b) + ~\u000bR(1 +\u000b\f)]:\n(15)\nConsider Eq. (15) with respect to \u001e=\u001e(\u001c). This is a separa-\nble equation and direct integration gives:\n\u001c=C0\u00001 +\u000b2\np\nA2\u0000\u000b2H2\n?=4atanh\u000bH?=2\u0000Atan\u001ep\nA2\u0000\u000b2H2\n?=4i\n;\n(16)\nwhereC0is an integration constant and we deﬁne:\nA\u0011u[\u001b(\f\u0000\u000b) + ~\u000bR(1 +\u000b\f)]: (17)\nFor brevity of notation, we also introduce B\u0011\u000bH?=2. The\nintegration constant depends on the initial conditions. At\n\u001c= 0, we assume that the domain wall is in its equilibrium\nconﬁguration \u001e= 0, in which case we may write the solution6\n01 2 3 4 5 6 7−2−1.5−1−0.500.51\nφL.h.s and r.h.s of Eq. (10)(b)\n  \n01234567−1.5−1−0.500.51\nφL.h.s and r.h.s of Eq. (10)(a)\n  \n0 5 10\nx 105−400−2000\nτφ(τ)\n  \n0 5 10\nx 105−4−20\nτφ(τ)\n  \nsin(2 φ)\nFIG. 4: (Color online) Plot of left-hand side (dashed line) and right-hand side (full lines) of Eq. (9) in order to illustrate the intersection\npoints. When there is no intersection between the lines, Walker breakdown has occurred. We have set \f= 0:01,\u000b= 0:005and consider (a)\n~\u000bR= 0:01anduranging from 0.24 to 0.30 along the direction of the arrow, in addition to (b) ~\u000bR= 1anduranging from 0.10 to 0.16 along\nthe direction of the arrow. The black arrow between the circles in (b) highlights how the intersection point changes abruptly upon increasing\nu.Insets: Time-evolution of the tilt angle for the same choices of u.\nfor the tilt angle as:\ntan\u001e=B\nA\u0000p\nA2\u0000B2\nAtanh\natan(\u000bB=p\nA2\u0000B2)\n\u0000\u001cp\nA2\u0000B2=(1 +\u000b2)i\n: (18)\nHaving now obtained the full time-dependence of the tilt-\nangle, we insert this back into the original equation of motionin order to ﬁnd the domain wall velocity _X=vDW. The gen-\neral expression for the domain wall velocity is rather large.\nHowever, by utilizing the fact that vDWwill display small-\nscale oscillations it is possible to ﬁnd a simpliﬁed expression\nfor the average domain wall velocity hvDWi. The period of\noscillation is T= (1 +\u000b2)\u0019=p\nA2\u0000B2, which gives us:\nhvDWi=1\nTZT\n0d\u001c\u001b\n\u000b(\nA2\u0000B2\nA(1 +\u000b2)sec2\u0010\natan(\u000bB=p\nA2\u0000B2)\u0000\u001cp\nA2\u0000B2\n1 +\u000b2\u0011\n\u0002h\n1 +\u0010\u000bB\nA\u0000p\nA2\u0000B2Atan[atan(\u000bB=p\nA2\u0000B2)\u0000\u001cp\nA2\u0000B2=(1 +\u000b2)]\u00112i\u00001)\n\u0000u(~\u000bR\u001b+\f)=\u000b: (19)\nThe analytical solution to the above integral and the ﬁnal result is:\nhvDWi=\u001b\n\u000b(1 +\u000b2)sgnfu\u001b(\f\u0000\u000b) +u~\u000bR(1 +\u000b\f)g\u0002Req\n[u\u001b(\f\u0000\u000b) +u~\u000bR(1 +\u000b\f)]2\u0000\u000b2H2\n?=4\u0000u(~\u000bR\u001b+\f)=\u000b;\n(20)\nwhere we have reinstated the original parameters contained in the quantities AandB.\nThe equation forhvDWishows the exact manner in which the\ndomain wall velocity depends on the various torque terms\nsuch as the non-adiabatic contribution \fand the spin-orbit\nterms ~\u000bR, and reveals several important features. It is seen\nthat for this particular domain wall conﬁguration [Bloch (y)],\nthe effect of the Slonczewski-like spin-orbit torque is a smallquantitative correction of order O(\u000b\f), which thus can be ne-\nglected. However, the conventional ﬁeld-like spin-orbit torque\nhas a strong qualitative inﬂuence on the wall dynamics. In\nfact, it is seen that the ~\u000bRterm plays the same role as the non-\nadiabatic conventional torque proportional to \f, but with one\nimportant difference: the spin-orbit torque contribution is chi-7\n0.1 0.2 0.3 0.4 0.5 0.6−0.7\n−0.6\n−0.5\n−0.4\n−0.3\n−0.2\n−0.1\n0/angbracketleftvDW/angbracketright(a)\n  \n0.1 0.2 0.3 0.4 0.5−0.5\n−0.4\n−0.3\n−0.2\n−0.1\n0\n0.1\n0.2(b)\n  \n00.1 0.2 0.3 0.4 0.5 0.6 0.7−0.7\n−0.6\n−0.5\n−0.4\n−0.3\n−0.2\n−0.1\n0\nu/angbracketleftvDW/angbracketright(c)\n  \n0 0.1 0.2 0.3 0.4 0.5−0.6\n−0.4\n−0.2\n0\n0.2\nu(d)\n  \n˜αR= 0\n0.001\n0.01\n0.1\n1\nFIG. 5: (Color on-\nline) The domain\nwall velocityhvDWi\nas a function of the\ncurrent density ufor\nvarious chiralities and\nspin-orbit coupling\nstrengths. (a): Positive\nchirality\u001b= +1 and\n\u000b > \f (\u000b= 0:02).\n(b): Negative chi-\nrality\u001b=\u00001and\n\u000b > \f (\u000b= 0:02).\n(c): Positive chirality\n\u001b= +1 and\u000b < \f\n(\u000b= 0:005). (d):\nNegative chirality\n\u001b=\u00001and\u000b < \f\n(\u000b= 0:005). For\nall plots, we have\nused\f= 0:01and\nH?= 0:5.\nrality dependent, i.e. changes sign with \u001b, whereas the \f-term\ndoes not. As a consequence, the wall may actually propagate\nin opposite direction of the applied current depending on the\nchirality\u001bof the domain wall, as was shown recently in Ref.\n22.\nIt is seen from Eq. (20) that there is either an enhancement\nof the domain wall velocity or a competition between the spin-\norbit induced torque and \f-torque depending on the sign of \u001b.\nWe show this in Fig. 5 where we consider the four possible\ncombinations of wall chirality \u001b(two values, \u001b=\u00061) com-\nbined with whether or not \u000bis larger than \f(two possibilities,\n\u000b > \f or\u000b < \f ). For a positive chirality \u001b= +1 displayed\nin Fig. 5 (a) and (c), the wall moves in the same direction for\nall current densities uas the torque terms in Eq. (20) have the\nsame sign. This is no longer the case for the opposite chirality\n\u001b=\u00001shown in Fig. 5(b) and (d) where the wall velocity\ncan actually change sign as uincreases. This is indicative of\ncounterﬂow domain wall motion where the wall moves in the\nopposite direction of the applied spin current.\nWalker breakdown for the domain wall occurs for veloci-\ntiesu\u0015ucwhere the root in Eq. (20) becomes imaginary,\nnamely:\nuc=\u000bH?\nj2\u001b(\f\u0000\u000b) + 2~\u000bR(1 +\u000b\f)j: (21)\nNote that this is the same as ucthat we would have found us-\ning the arguments in the previous section in order to identify\nthe Walker breakdown from the equations of motion (without\nactually solving them explicitly) and thus serves as a consis-\ntency check for the correctness of Eq. (20). This expression\nis quite generally valid, including the effects of both types of\nspin-orbit torques and both types of conventional spin-transfer\ntorques. As another consistency check, we observe that in\nthe absence of spin-orbit coupling ( ~\u000bR= 0), one ﬁnds that\njucj=\u000bH?=2j\f\u0000\u000bjwhich agrees with Ref. 33. The effectof the spin-orbit interaction is seen to depend explicitly on the\nchirality\u001bof the domain wall. Although Walker breakdown\nis inevitable for the present Bloch( y) domain wall, in contrast\nto the Bloch( z) one, the presence of spin-orbit interactions\n(~\u000bR6= 0) can strongly enhance the threshold velocity due to\nthe competition between the terms \u001b(\f\u0000\u000b)and~\u000bR(1 +\u000b\f)\nin the denominator. When these terms have different sign (ei-\nther for\u001b=\u00001and\f > \u000b or\u001b= 1 and\f < \u000b ), the\nspin-orbit coupling can very strongly enhance the threshold\ncurrent for Walker breakdown. This effect could be used to\ninfer information about the value of \u000band\fprecisely due\nto the non-monotonic behavior of the threshold current as a\nfunction of ~\u000bR.\nWe illustrate this behavior in Fig. 6 where we have cho-\nsen\u001b= +1 . As seen, the threshold velocity decreases in a\nmonotonic fashion with increasing ~\u000bRwhen the damping is\nlow,\u000b < \f . However, when the two terms in the denomina-\ntor differ in sign (which occurs precisely when \u000b > \f ), the\nthreshold velocity uchas a non-monotonic behavior and is in\nfact strongly increases near ~\u000bR=j\f\u0000\u000bj. In this way, one\nmay obtain information regarding the relative size of \u000band\f\nby measuring the threshold velocity.\nC. Head-to-head domain wall\nThe ﬁnal type of domain wall structure we will consider\nappears for in-plane magnetized strips ( e.g.NiFe layer16) and\nis known as a so-called head-to-head domain wall. In this\ncase, the easy axis is parallell with the extension of the wire\nwhereas the hard axis is perpendicular to it:\nm= (\u0000\u001bcos\u0012;sin\u0012cos\u001e;sin\u0012sin\u001e); (22)8\n00.511.522.533.54uc/H⊥\n  \n0 0.005 0.01 0.015 0.02020406080\n˜αRuc/H⊥0.02\n0.015\n0.01\n0.005\n0.035 0.03 α= 0.025α\n(b)(a)\nFIG. 6: (Color online) Critical velocity uc=H?that triggers Walker\nbreakdown. We have chosen \f= 0:02as a representative value\nwhich demonstrates the fundamental behavior of uc. For sufﬁciently\nlow damping \u000b < \f shown in (a), the threshold velocity is lowered\nmonotonically as the spin-orbit interaction ~\u000bRis increased. When\nthe damping becomes stronger such that \u000b > \f ,ucis strongly en-\nhanced in a limited interval of ~\u000bR.\nand a corresponding effective ﬁeld:\nHeff=2Aex\nM2\n0r2m\u0000H?mz^z+Hkmx^x+Hext:(23)\nUsing again Thiele’s approach as described in the previous\nsections, one arrives at exactly the same equations of motion\nas in the Bloch( z) case. The formal reason for this can be\ntraced back to the fact that the effective spin-orbit ﬁeld Hso\nis directed along the yaxis. The magnetization textures of the\nBloch(z) and head-to-head domain walls may be transformed\ninto each other via an SO(3) rotation with an angle \u0019=2ofM\naround theyaxis. Such a rotation leaves Hsoinvariant and\none thus obtains the same equations of motion for both types\nof domain walls. Formally, one can see this by multiplying\nEq. (1) from the left side with:\nU=0\n@0 0\u00001\n0 1 0\n1 0 01\nA; (24)\nand using that\n(Ua)\u0002(Ub) =det(U)(U\u00001)T(a\u0002b): (25)\nSinceU2 SO(3), we have that (U\u00001)T=Uand det (U)=+1.\nBy direct multiplication, one observes that UHso=Hso,\nUMBloch(z)=Mhead-to-head andUHeff\nBloch(z)=Heff\nhead-to-head .\nNote that it is in drastic contrast with the Bloch( y) case whereHsoisnotinvariant under the matrix which rotates MBloch(z)\nintoMBloch(y). The same arguments and results related to the\ndomain wall velocity and Walker breakdown that were dis-\ncussed in Sec. III A then also hold for the present head-to-\nhead domain wall case.\nWe mention here that the equivalence of the Bloch( z) and\nhead-to-head domain wall case found here is contingent on\nthe speciﬁc setup we have considered in Fig. 1. Although\nthis model is the standard one and indeed the most frequently\nemployed setup experimentally, it was recently shown that\nsuch an equivalence does not hold when combining a mag-\nnetic strip/wire with a non-magnetic conductive layer with\nspin-orbit interaction in a non-parallell geometry16. Such a\nmethod actually provides a manner in which the direction of\nthe effective spin-orbit ﬁeld can be changed which could then\nserve as a mean to distinguish between different types of do-\nmain walls, based on their response to an applied current.\nD. Ferromagnetic resonance (FMR) in the presence of\nspin-orbit torques\nWe now turn our attention to another setup where the aim\nis to identify the ferromagnetic resonance response of a ma-\nterial where spin-orbit interactions play a prominent role. To\ndo so, we consider the setup shown in Fig. 7 where a spin-\ncurrent with polarization magnitude and unit vector direction\nS2[0;1]and~S, respectively, is injected into the ferro-\nmagnetic layer where spin-orbit coupling is present. This di-\nrectly inﬂuences the susceptibility tensor and thus both the fer-\nromagnetic resonance frequency/linewidth and the absorbed\npower by the system39.\nTo facilitate the analytical calculations, we will operate\nwith two different coordinate systems. The laboratory (sta-\ntionary) framework xyzis shown in Fig. 7, where the xy\nplane spans the ferromagnetic layer, and xyz denotes a ro-\ntated coordinate system which we will specify the direction\nand purpose of below. A current is injected into the ferromag-\nnetic layer acting with a spin-transfer torque on the magneti-\nzation vector~M. This torque is modiﬁed due to the presence\nof spin-orbit coupling which is taken into account via a ﬁeld\n~Hsoas in the domain-wall treatment. The time-dependent\nLLG motion equation describing the dynamic of ferromag-\nnetic layer magnetization vector then takes the following form\nin this new notation:\n@~M\n@t=\u0000\r~M\u0002~Ht+\u000b\nMS~M\u0002@~M\n@t\n+\r\nMS~M\u0002~M\u0002 (\f~Hso+Ps~S); (26)\n~Hso=\u000bRmeS\n~eMS(1 +\f2)(~n\u0002~Je); Ps=~SJe\n2eMSd:\nHere,\ris the electron gyromagnetic ratio and \u000bis the Gilbert\ndamping constant. Moreover, \fis the non-adiabaticity param-\neter discussed previously, Psis the spin-torque parameter, S\nis the polarization of injected current into the ferromagnetic9\nFIG. 7: (Color online) Schematic setup of the free ferromagnetic\n(FM) layer with a general saturation magnetization direction,~MS,\ndescribed by polar and azimuthal angles \u0012Mand'M, respectively.\nThe thickness of free ferromagnetic layer is denoted by d. The exter-\nnally applied static magnetic ﬁeld~H0, polarization vector of injected\ncharge current~S, spin-orbit coupling torque vector~Hso, and ﬁnally\nnormal unity vector ~nare shown. The ferromagnetic ﬁlm is located\nin the xyplane so that zaxis is normal to the ferromagnetic ﬁlm. The\nspin-orbit coupling is assumed to be induced via a substrate layer into\nthe free ferromagnetic layer. The double dot represents the vector\nquantities in the non-rotated coordinate system (laboratory frame-\nwork).\nlayer, and a normal vector to the plane of ferromagnetic layer\nis represented by ~n(see Fig. 7).\nWe now introduce a rotated coordinate system xyz where\nthe saturation magnetization direction is parallel with the z\naxis. The orientation of the rotated system xyz compared to\nthe stationary one xyzis determined by calculating the equi-\nlibrium orientation of the magnetization order parameter and\nsetting thezaxis to be parallel with it. The details of the cal-\nculations will be discussed in what follows.\nWe deﬁne a transformation matrix which rotates the ﬁxed\ncoordinate system so that its zaxis to be oriented along~MS.\nTherefore, all other vector quantities should be rotated via the\ndeﬁned transformation to be described in this new rotated co-\nordinate system. If we describe~MSby polar and azimuthal\nangles i.e.\u0012Mand'M, in the ﬁxed original coordinate sys-\ntem, a rotation around the zaxis equal to 'Mand then around\nthe rotated yaxis equal to \u0012Mare required for aligning zaxis\nand~MSorientations. Hence, the rotation matrices can be re-\nspectively given by (see Ref. 40 for more details):\nRz(\u0000'M) =0\n@cos'M\u0000sin'M0\nsin'Mcos'M0\n0 0 11\nA;\nRy(\u0012M) =0\n@cos\u0012M0\u0000sin\u0012M\n0 1 0\nsin\u0012M0 cos\u0012M1\nA:\nThe total rotation matrix is thus the multiplication of RyandRzi.e.\nRt=RyRz=0\n@cos\u0012Mcos'M\u0000cos\u0012Msin'M\u0000sin\u0012M\nsin'Mcos'M0\nsin\u0012Mcos'M\u0000sin\u0012Msin'Mcos\u0012M1\nA:(27)\nWe characterize each vector quantity by its polar and az-\nimuthal angle in the ﬁxed original coordinate system shown\nin Fig. 7. Since we assume a homogeneous magnetization\ntexture (macrospin approximation), we have ~r2~M= 0. The\ntotal effective ﬁeld entering the LLG-equation may now be\ndecomposed into the following terms:\n~Ht=~Hdip+~hdip(t) +~Ha+~ha(t) +~Hso\n+b~S+~H0+~hext(t)\n\u0011~H+~h(t): (28)\nAbove,f~Hdip;~hdip(t)gandf~Ha;~ha(t)gare the static and dy-\nnamic parts of the dipole and anisotropy ﬁelds respectively,\n~Hsois the spin-orbit ﬁeld, b~Sis the spin-torque effective ﬁeld\n(which is usually negligible), ~H0is the static externally ap-\nplied ﬁeld, and ﬁnally ~hext(t)is a small rf ﬁeld applied per-\npendicularly to the saturation magnetization direction zin or-\nder to probe the ferromagnetic resonance. To show an exam-\nple of how the quantities in the two coordinate systems are\nrelated, note that the x,y, andzcomponents of the externally\napplied static magnetic ﬁeld ~H0in the rotated coordinate sys-\ntem are given by:\nH0x=H0\b\ncos\u0012Mcos'Msin\u0012H0cos'H0\u0000\ncos\u0012Msin'Msin\u0012H0cos'H0\u0000sin\u0012Mcos\u0012H0\t\n;(29)\nH0y=H0\b\nsin'Msin\u0012H0cos'H0+\n\b\ncos'Msin\u0012H0cos'H0\t\n; (30)\nH0z=H0\b\nsin\u0012Mcos'Msin\u0012H0cos'H0\u0000\nsin\u0012Msin'Msin\u0012H0cos'H0\u0000cos\u0012Mcos\u0012H0\t\n:(31)\nAs mentioned above, the dipole ﬁeld can be divided into static\n~Hdipand dynamic ~hdip(t)parts. In the rotated coordinate\nsystem they may be obtained as31:\n~Hdip=Mcos\u0012M0\n@cos\u0012Msin'M\n\u0000cos'M\nsin\u0012Msin'M1\nA;\n~hdip(t) = 4\u0019my(t) sin\u0012M0\n@cos\u0012Msin'M\n\u0000cos'M\nsin\u0012Msin'M1\nA;\nwhereM\u0019 4\u0019MS\u0000Ha. Assuming a weak rf magnetic\nﬁeld applied transverse to the ^z-direction, we may consider\nthe components of magnetization in the rotated coordinate\nsystem asMz=MS\u001dMx;My. In this case, the fol-\nlowing time-dependent coupled differential equations for the\nprecessing magnetization components are obtained;10\n@Mx\n@t=\u0000\rMyHt\nz+\rMx(\fHso\nz+PsSz)\n+\rMS(Ht\ny\u0000(\fHso\nx+PsSx))\u0000\u000b@My\n@t;\n@My\n@t=\rMxHt\nz+\rMy(\fHso\nz+PsSz)\n\u0000\rMS(Ht\nx+ (\fHso\ny+PsSy)) +\u000b@Mx\n@t;\n@Mz\n@t=@MS\n@t= 0 =\rMx(\u0000Ht\ny+ (\fHso\nx+PsSx))\n+\rMy(Ht\nx+ (\fHso\ny+PsSy)):\nSetting the transverse part of the magnetization and ﬁelds\nequal to zero in the above equations for @tMxand@tMy,\none obtains the equilibrium conditions which specify the ori-\nentation of the zaxis:\n\u001aHx+ (\fsoHso\ny+\fsSy) = 0\nHy\u0000(\fsoHso\nx+\fsSx) = 0: (32)\nThis is consistent with the equation for @tMzand our preas-\nsumption namely; Mz\u001dMx;My. In order to obtain the\nsolution for the transverse components MxandMyto lowest\norder, we now substitute these conditions back into the equa-\ntions of motion for the magnetization components above and\nobtain:\n@Mx\n@t=\u0000\rMyHz+\rMShy(t)\u0000\u000b@My\n@t\n+\rMx(\fHso\nz+PsSz);\n@My\n@t= +\rMxHz\u0000\rMShx(t) +\u000b@Mx\n@t(33)\n+\rMy(\fHso\nz+PsSz):\nIn our calculations we have set the time-dependent ﬁelds suf-\nﬁcienty small so that those terms including higher orders of\ntime-dependent components are negligible. Assuming that\nthe the external time-dependent magnetic ﬁeld induces the\nsame frequency in all time-dependent components of other\nvector quantities (including responses) as itself, \n, we get e.g.\n~hdip(t) =~hdipe\u0000i\nt. By substituting this time-dependency\ninto Eqs. (33) we arrive at ~M(t) =\u001f~hext(t)in which\n~M(t) = (Mx;My)T,~hext(t) = (hext\nx;hext\ny)T, and;\n\u001f=\u0012\n\u001fxx\u001fxy\n\u001fyx\u001fyy\u0013\n: (34)\n\u001fis known as the susceptibility tensor which determines the\nbehavior of magnetization in response to the external time-\ndependent magnetic ﬁeld. The components of the obtained\nsusceptibility tensor in the presence of spin-orbit coupling\nread:\n\u001fxx= +\u0000f\rWy\u0004\u0000\u0001\u000b\n\u0000i(\r\u0001Wy+ \n\u000b\u0004)g;\n\u001fxy=\u0000\u0000f\u0006\u0004 + \u0001\n\u0000i(\u0001\u0006\u0000\n\u0004)g;\n\u001fyx= +\u0000f\u0006\u0004 + \u0001\n\u0000i(\u0001\u0006\u0000\n\u0004)g\n\u001fyy= +\u0000f\rWx\u0004\u0000\u0001\u000b\n\u0000i(\r\u0001Wx+ \n\u000b\u0004)g;where we have deﬁned the following parameters;\n\u0000 =\rMS\n\u00042+ \u00012;\u0006 =\r(\fsoHso\nz+\fsSz);\n\u0004 = \u00072\u0000\n2(1 +\u000b2);\u0007 =q\n\r2WxWy+ \u00062;\n\u0001 = 2\u0006\n\u0000\r\u000b\n(Wx+Wy);\nWx=Hz+Msin\u0012Mcos\u0012Msin'M;\nWy=Hz+Msin\u0012Mcos'M:\nThe susceptibility tensor components may be used to com-\npute physical quantities of interest such as the absorbed\npower (which is experimentally relevant39) by the ferro-\nmagnetic sample with volume Vat frequency \n. In turn,\nthis gives a clear signal of ferromagnetic resonance in the\nabsorption spectrum. This energy dissipation is given by\nPabs\npower =ImfPpowergwherePpower is deﬁned by:\nPpower =\u0000\n2Z\nVdV~hext\u0003\u0001~M=\u0000\n2Z\nVdV~hext\u0003\u0001\u001f~hext\n=\u0000\n2Z\nVdVn\njhext\nxj2\u001fxx+hext\nx\u0003hext\ny\u001fxy+\nhext\ny\u0003hext\nx\u001fyx+jhext\nyj2\u001fyyo\n:\nThis expression simpliﬁes if the rf magnetic ﬁeld only has one\ncomponent, e.g.~h(t) =hext\nx(t), in which case the power\nabsorbed at radio-frequency \ncan be expressed by:\nPabs\npower =\n2Z\nVdV\rMSjhext\nxj2\n\u00042+ \u00012(\r\u0001Wy+ \n\u000b\u0004):\nAlthough the above expressions may be numerically evalu-\nated in our system for a speciﬁc parameter choice, we focus\nbelow on analytical insights that may be gained. In partic-\nular, we are interested in the role played by spin-orbit inter-\nactions and the magnitude/direction of the injected current.\nSo far, our treatment has been general and accounted for sev-\neral terms contributing to the susceptibility tensor. In order\nto identify the role played by current-dependent spin-orbit\ncoupling in the ferromagnetic resonance, we need to derive\nan analytical expression for the ferromagnetic resonance fre-\nquency \nFMR. This is deﬁned as the frequency where the\nPabs\npower has a maximum. In their general form shown above,\nthis cannot be done analytically in an exact manner. How-\never, progress can be made by considering the denominator\nofPabs\npower . This quantity has the following form when all the\nfrequency-dependence is written explicitly:\n\u00042+ \u00012= [\u00072\u0000\n2(1 +\u000b2)]2\n+ \n2[2\u0006\u0000\r\u000b(Wx+Wy)]2: (35)\nFollowing the standard procedure of neglecting the second\nterm above, one may identify the resonance frequency simi-\nlarly to Ref. 31 as \nFMR= \u0007. We have also veriﬁed that this\nholds numerically for a realistic parameter set.\nTo see how the spin-orbit coupling affects \nFMR, one should\nnote in particular its dependence on the current J. It is instruc-\ntive to consider ﬁrst the scenario with zero spin-orbit coupling,11\nin which case the resonance frequency may be written as:\n\nFMR=p\nc1+c2J2; (36)\nwherec1andc2are determined by the quantities in Eq. (35)\nin the limit ~\u000bR!0. Importantly, they are independent on the\ncurrent bias J, which means that the resonance frequency is\ncompletely independent on the direction of the applied current\nas it is only the magnitude J2that enters. Therefore, the cur-\nrent direction cannot alter the \nFMR. Turning on the spin-orbit\ncoupling so that ~\u000bR6= 0, one may in a similar way show from\nthe above equations that the resonance frequency now can be\nwritten as:\n\nFMR=p\n(d1+DJ)(d2+DJ) +d3J2; (37)\nwhere again the coefﬁcients diandDare determined from\nEq. (35). It then follows from Eq. (37) that the resonance\nfrequency will be asymmetric with respect to the applied cur-\nrent direction when spin-orbit coupling is present. In partic-\nular, one obtains different values for \nFMR by reversing the\ncurrentJ!(\u0000J)so that the Z2symmetry in Eq. (36) is\nlost. The main signature of spin-orbit coupling in the current-\nbiased ferromagnetic resonance setup under consideration is\nthen an asymmetric current dependence which should be dis-\ntinguishable from the scenario without spin-orbit interactions.\nIt is interesting to note that the current-dependence on the fer-\nromagnetic resonacne and the linewidth allows one to exert\nsome control over the magnetization dissipation/absorption\nin the system via J. The presence of spin-orbit interactions\nenhances this control since it introduces a directional depen-\ndence which is absent without such interactions.\nIV . SUMMARY\nIn summary, we have considered the inﬂuence of existence\nof spin-orbit interactions on both domain wall motion and fer-romagnetic resonance of a ferromagnetic ﬁlm. Due to the cou-\npling between the momentum and spin of the electrons, the\ndegeneracy between domain wall textures is broken which in\nturn leads to qualitatively different behavior for various wall\nproﬁles, e.g.Bloch vs. Neel domain walls. By taking into ac-\ncount both the ﬁeld- and Slonczewski-like spin-orbit torque,\nwe have derived exact analytical expressions for the wall ve-\nlocity and the onset of Walker breakdown. One of the most\ninteresting consequences of the spin-orbit torques is that they\nrender Walker breakdown to be universal for some wall pro-\nﬁles in the sense that the threshold is completely independent\non the material-dependent damping \u000b, non-adiabaticity \f, and\nthe chirality \u001bof the domain wall. We have also shown that\ndomain wall motion against the current ﬂow is sustained in\nthe presence of multiple spin-orbit torques and that the wall\nproﬁle will determine the qualitative inﬂuence of these differ-\nent types of torques. Finally, we calculated the ferromagnetic\nresonance response of a ferromagnetic material in the pres-\nence of spin-orbit torques, i.e. a setup with a current bias. We\nfound a key signature of the spin-orbit interactions in the res-\nonance frequency, namely that the latter becomes asymmetric\nwith respect to the direction of current injection. This is differ-\nent from usual ferromagnets in the presence of spin-transfer\ntorques in the absence of spin-orbit interactions, where the\nfrequency is found to be symmetric with respect to the current\ndirection.\nAcknowledgments\nM.A. would like to thank P. Landeros for helpful conversa-\ntions and A. Sudbø for valuable discussion.\n\u0003Electronic address: jacob.linder@ntnu.no\nyElectronic address: phymalidoust@gmail.com\n1I. Zutic, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323\n(2004).\n2C. Chappert, A. Fert, and F. Nguyen Van Dau, Nature Mater. 6,\n813 (2007).\n3J. A. Katine anda E. E. Fullerton, J. Magn. Magn. Mater. 320,\n1217 (2008).\n4D. S. Han et al. , Appl. Phys. Lett. 94, 112502 (2009).\n5M. Jamali, H. Yang, and K. J. Lee, Appl. Phys. Lett. 96, 242501\n(2010).\n6P. Yan, X. S. Wang, and X. R. Wang, Phys. Rev. Lett. 107, 177207\n(2011).\n7J. Linder, Phys. Rev. B 86, 054444 (2012).\n8J. Grollier et al. , C. R. Physique 12, 309 (2011).\n9M. Hayashi et al. , Phys. Rev. Lett. 98, 037204 (2007).\n10S. Pizzini et al. , Appl. Phys. Exp. 2, 023003 (2009).\n11A. D. Kent, B. Ozyilmaz, and E. del Barco, Appl. Phys. Lett. 84,3897 (2004).\n12S. Matsunaga et al. , Appl. Phys. Exp. 1, 091301 (2008).\n13S. S. Parkin, M. Hayashi, and L. Thomas,Science 320, 190 (2008).\n14H. Liu et al. , Appl. Phys. Lett. 97, 242510 (2010).\n15N. L. Schryer and L. R. Walker, J. Appl. Phys. 45, 5406 (1974).\n16A. V . Khvalkovsiy et al. , arXiv:1210.3049.\n17K. Obata and G. Tatara, Phys. Rev. B 77, 214429 (2008).\n18A. Manchon and S. Zhang, Phys. Rev. B 79, 094422 (2009).\n19P. M. Haney and M. D. Stiles, Phys. Rev. Lett. 105, 126602\n(2010).\n20J. Ryu, S.-M. Seo, K.-J. Lee, H.-W. Lee, Journal of Mag. Mater.\n324, 1449 (2012).\n21K.-W. Kim, S.-M. Seo, J. Ryu, K.-J. Lee, H.-W. Lee, Phys. Rev.\nB85, 180404(R) (2012).\n22J. Linder, Phys. Rev. B 87, 054434 (2013).\n23E. van der Bijl and R.A. Duine, arXiv:1205.0653.\n24A. Chernyshov et al. , Nature Phys. 5, 656 (2009).\n25T. A. Moore et al. , Appl. Phys. Lett. 95, 179902 (2009).12\n26K.-J. Kim et al. , Appl. Phys. Exp. 3, 083001 (2010).\n27I. M. Miron et al. , Nature. Mat. 10, 419 (2011); I. M. Miron et al. ,\nNature 476, 189 (2011).\n28J.C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996).\n29X. Wang and A. Manchon, arXiv:1111:1216.\n30P. Landeros, R. A. Gallardo, O. Posth, J. Lindner, and D. L. Mills,\nPhys. Rev. B 81, 214434 (2010).\n31P. Landeros, R. E. Arias, and D. L. Mills, Phys. Rev. B 77, 214405\n(2008).\n32L. D. Landau and E. M. Lifshitz, Phys. Z. Sowjetunion 8, 153\n(1935); T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004).\n33A. Thiaville et al. , EPL 69, 990 (2005).\n34J. Xiao, A. Zangwill, and M. D. Stiles, Phys. Rev. B 73, 054428\n(2006)\n35G. Tatara, H. Kohno, J. Shibata, Phys. Rep. 468, 213 (2008).\n36G. Tatara, H. Kohno, J. Shibata, J. Phys. Soc. Jpn. 77, 031003\n(2008).\n37A. A. Thiele, Phys. Rev. Lett. 30, 230 (1973).\n38J. Linder, Phys. Rev. B 84, 094404 (2011).\n39L. Dreher, M. Weiler, M. Pernpeintner, H. Huebl, R. Gross, M.\nS. Brandt, and S. T. B. Goennenwein Phys. Rev. B 86, 134415\n(2012).\n40G. B. Arfken, H. J. Weber, F. Harris, Mathematical Methods for\nPhysicists , ISBN-13: 9780120598250, Academic Press (2000).41M. Farle, Rep. Prog. Phys. 61, 755 (1998).\n42Z. Celinski, K. B. Urquhart, and B. Heinrich, J. Magn. Magn.\nMater. 6, 166 (1997).\n43J. C. Sankey, P. M. Braganca, A. G. F. Garcia, I. N. Krivorotov, R.\nA. Buhrman, and D. C. Ralph, Phys. Rev. Lett. 96, 227601 (2006).\n44R. Urban, G. Woltersdorf, and B. Heinrich, Phys. Rev. Lett. 87,\n217204 (2001).\n45C. Thirion, W. Wernsdorfer, and D. Mailly, Nat. Mater. 2, 524\n(2003).\n46K. Lenz, T. Toli nski, J. Lindner, E. Kosubek, and K. Baberschke,\nPhys. Rev. B 69, 144422 (2004).\n47S. Kaka, M. R. Pufall, W. H. Rippard, T. J. Silva, S. E. Russek,\nand J. A. Katine, Nature (London) 437, 389 (2005).\n48H. K. Lee and Z. Yuan, J. Appl. Phys. 101, 033903 (2007).\n49P.-B. He, Z.-. Li, A.-L. Pan, Q. Wan, Q.-L. Zhang, R.-X. Wang,\nY .-G. Wang, W.-M. Liu, and B.-S. Zou, Phys. Rev. B 78, 054420\n(2008).\n50T. E. Hasty and L. J. Boudreaux, J. Appl. Phys. 32, 1807 (1961).\n51Y .-C. Chen, D.-S. Hung, Y .-D. Yao, S.-F. Lee, H.n-P. Ji, and C.\nYu, J. Appl. Phys. 101, 09C104 (2007).\n52H. C., X. Fan, H. Zhou, W. Wang, Y . S. Gui, C.-M. Hu, and D.\nXue, J. Appl. Phys. 113, 17C732 (2013)."    },    {        "title": "1610.06695v2.Electromotive_forces_generated_in_3d_transition_ferromagnetic_metal_films_themselves_under_their_ferromagnetic_resonance.pdf",        "content": " \n 1Electromotive forces generated in 3d -transition ferromagnetic metal films \nthemselves under their ferromagnetic resonance \n \nKazunari Kanagawa 1, Yoshio Teki 2, and Eiji Shikoh 1,a) \n \n1Graduate School of Engineering, Osaka City Universi ty, 3I3I138 Sugimoto, SumiyoshiIku, \nOsaka 558I8585, Japan \n \n2Graduate School of Science, Osaka City University, 3I3I138 Sugimoto, SumiyoshiIku, Osaka \n558I8585, Japan \n \n(Received ; Accepted ; Published ) \n \nWe report the electromotive force (EMF) properties generated in 3dItransition ferromagnetic \nmetal (FM = Fe, Co, and Ni 80 Fe 20 ) films themselves under their ferromagnetic resona nce (FMR). \nFor Fe and Co films, the EMF due to the anomalous8H all effect is dominantly generated under \ntheir FMR. Meanwhile, for a Ni 80 Fe 20  film, the EMF due to the inverse spin8Hall effect in the \nNi 80 Fe 20  film itself under the FMR is mainly generated. Thi s tendency is qualitatively explained  \n 2with differences of the spin polarization, the spin  Hall conductivity, the anomalous Hall \nconductivity, the magnetization saturation, and the  resistivity of the FM films.  \n \na) E8mail: shikoh@elec.eng.osaka8cu.ac.jp \n  \n 3     In spintronics, recently, the spin8pumping wit h the ferromagnetic resonance (FMR) and \nthe inverse spin8Hall effect (ISHE) have become pow erful techniques to generate a spin current \nand to detect the spin current, respectively. 1812  First, those techniques were used for spin \ninjection from a ferromagnetic metal Ni 81 Fe 19  as a spin injector into a nonmagnetic Pt layer \nworking as a spin detector, by using the Ni 81 Fe 19 /Pt bi8layer structure samples.1 In the spin \ninjection by using the spin8pumping with the FMR, t he conductance mismatch problem between \nthe ferromagnetic material and the target8material,  which causes lowering the spin injection \nefficiency in the case of an electrical spin inject ion method,13,14  is negligible. 4,5,7,8,10,12  Therefore, \nwhile those techniques are very fundamental, those apply to investigate the spin transport \nproperty of various materials, for example, by usin g the ferromagnetic metal (FM) \n/target8material/non8magnetic metal (NM) junction s tructure samples.587,12  Meanwhile, it was \ndiscovered that the electromotive force (EMF) was g enerated in a single layer Ni 80 Fe 20  film \nitself under the FMR.15  The suggested EMF generation mechanism in the sing le8layer Ni 80 Fe 20  \nfilm itself under the FMR is as follows; a spin cur rent is generated due to the magnetic \ninhomogeneity of the Ni 80 Fe 20  film under the FMR condition and converted to a ch arge current \nwith the spin8orbit interaction of Ni 80 Fe 20  film, that is, the ISHE in the Ni 80 Fe 20  film.15  \nPreviously, by using the Y 3Fe 5O12 (YIG)/ferromagnet “bi8layer” structure samples, the  ISHE \ngenerated in 3d8transition ferromagnetic metal film s of Fe, Co, Ni 80 Fe 20  and Ni under the FMR  \n 4of the YIG was observed,9,11  while the EMF generated in a single8layer ferromag netic metal film \nitself under the FMR is not investigated except for  the Ni 80 Fe 20  film.15  If those EMF generation \nphenomena by using the spin8pump with the FMR of a FM material were applied for practical \nuse, the single8layer structure is better than any other multi8layer structures, in terms of the \nmaterial saving and easiness of manufacturing devic es. In this study, the EMF properties \ngenerated in single8layer 3d8transition FM films th emselves under the respective FMR were \ninvestigated. \n \n     Our sample structure and experimental set up a re illustrated in Figure 1. On a \nthermally8oxidized silicon substrate, a ferromagnet ic metal (FM = Fe, Co, and Ni 80 Fe 20 ) was \ndeposited to a thickness of 25 nm by using a conven tional DC magnetron sputtering system. No \nprotection layer was formed on the FM films, as sim ilar to the previous study.15  After forming \nthe FM films, the sample substrates were cut as a r ectangular shape of 4.0 ×1.5 mm 2, to \nmeasure the physical properties. \n     A sample substrate was set into the microwave TE 011 8mode cavity of an electron spin \nresonance (ESR) system (JEOL, JES8TE300) to excite the FMR of the sample. The microwave \nfrequency f to excite the FMR was 9.45 GHz. The EMF property o f the FM sample was \nmeasured by using a nano8voltmeter (Keithley Instru ments, 2182A). Leading wires to detect the  \n 5output voltage properties from a sample were direct ly attached with silver paste at both ends of \nthe film sample. All of the measurements were perfo rmed at RT. \n \n     Figures 2 (a)8(c) show the FMR spectra of samp les at an external magnetic field \norientation angle θ of 0°, with the microwave power PmW  of 200 mW to excite the respective \nFMR. Figs. 2 (a), (b) and (c) are for an Fe sample,  for a Co sample, and for a Ni 80 Fe 20  sample, \nrespectively. As expected, the FMR was observed in all FM films at the respective FMR field \nHFMR  of 1061 Oe for the Fe, 1094 Oe for the Co and 472. 8 Oe for the Ni 80 Fe 20 . The saturation \nmagnetization MS was estimated to be 1061 emu/cc for Fe, 1094 emu/c c for Co and 472.8 \nemu/cc for Ni 80 Fe 20  by using the following equation: 2,3  \n)π4 (S FMR FMR 0 M H H + =γω ,   (1) \nwhere ω0 (= 2 πf) and γ are respectively, the angular frequency of the mic rowave and the \ngyromagnetic ratio of the respective FMs. \n     Figs. 2 (d)8(f) show the output voltage proper ties at the θ of 0° and 180°, with the PmW  of \n200 mW for the excitation of the respective FMR. Fi gs. 2 (d), (e) and (f) are for an Fe sample, \nfor a Co sample, and for a Ni 80 Fe 20  sample, respectively. For the experimental data (o pen \ncircles), components which do not relate to the mag netic field orientation angles are removed by \nusing the eq.(2):   \n 62180 0V VV−=,   (2) \nwhere the V0 and V180  correspond to the EMFs at the θ of 0° and 180°, respectively. Output \nvoltages were observed in all FM films themselves u nder the respective FMR. The polarity of \noutput voltages is inverted in all FM films against  the magnetization reversal of the respective \nFM films. The output voltages increased with the in crease of PmW . These polarity inversion to \nthe magnetization reversal and PmW  dependence of output voltages are similar to previ ous \nstudies using the spin8pump and ISHE.388,10,12,15  To analyze those output voltage properties, the \ndata in Figs. 2 (d)8(f) were fitted by the followin g equation: \nBG 2 2\nFMRFMR\nAHE 2 2\nFMR2\nISHE) () (2\n) ()( VHHHHVHHV HV +Γ+−−Γ−+Γ+−Γ= , (3) \nwhere the first and second terms of the eq. (3) cor respond to the EMF due to the ISHE and the \nEMF due to the extra ordinary Hall effect in FMs, t hat is, the anomalous Hall effect (AHE) in \nFMs, respectively.1,488,10,12,15  The VISHE and VAHE  indicate the magnitude of the EMF due to the \nISHE and that due to the AHE, respectively. Γ is a damping constant in these fittings. The ISHE \nterm is a Lorenz function, in other words, symmetry  to the HFMR , while the AHE term is \nderivative of a Lorenz function, which is anti8symm etry to the HFMR . VBG is background signals \non experiments, which are independent of the extern al magnetic field. The fitting results are \ndrawn with the solid lines in Figs. 2 (d)8(f). The above analysis indicated that the electromotive \nforces generated in the 3d8transition FM metal film s under their FMR were successfully  \n 7observed. The ISHE properties of the FM films were similar to the studies by using “bi8layer” \nstructure samples. 9,11  \n     Table 1 shows the summary of the analysis, whe re the VISHE , VAHE,  the absolute value of \nthe ratio of | VISHE /VAHE |, MS, the ratio to the VISHE  of Co, and the ratio to the VAHE  of Ni 80 Fe 20  are \ndescribed. From the values of the | VISHE /VAHE |, we can say that the AHE is dominant for the Fe \nand Co samples, while the ISHE is dominant for the Ni 80 Fe 20  sample. This tendency has the \nreproducibility and may be due to the difference of  the spin polarization ( PS), spin8Hall \nconductivity ( σSHE ), anomalous Hall conductivity ( σAHE ), MS, and resistivity ( ρ) of the FM films. \nTable 2 shows the data of the PS,16,17  the σSHE ,15,18  the σAHE ,15,18  and the ρ, where only the ρ \nvalues were experimentally obtained in this study. \nIn the ISHE regime, 1 the spin current density 0\nSj is converted to a charge current density \njC, as follows:  3,5  \nσθ×∝0\nS SHE C j j ,   (4) \nwhere the σ and θSHE  are the spin polarization vector of the spin curre nt and the spin8Hall \nangle, which is a kind of a conversion efficiency f rom the 0\nSj to the jC,. Thus, the absolute \nvalue of the jC can be expressed as follows: \n0\nSHE CSj jθ∝ .  (5) \nThe θSHE  is equal to \nCSHE\nσσ, where the σC is the electrical conductivity of a FM film and  \n 8corresponds to 1/ ρ. Thus, the eq. (5) is rewritten as follows; \n0\nSHE CSj jρσ∝ .  (6) \nIn this study, the ISHE is not observed as a charge  current but as an EMF via the sample \nresistance. Under an assumption that the 0\nSj is approximately proportional to the PS in a FM \nfilm, the absolute value of VISHE  is expressed as follows;  \nSPlVSHES2\nISHEσρ∝ ,  (7) \nwhere the l is the length of an FM sample (4 mm in this study) , and the S is the sectional area of \nthe FM sample (1.5 mm × 25 nm). Using the values in  the Table 2, the VISHE  values of Fe, Co, \nand Ni 80 Fe 20  samples to the VISHE  for a Co sample ratio is estimated to be Fe : Co :  Ni 80 Fe 20  = 8 : \n1 : 6, while the experimentally obtained data were Fe : Co : Ni 80 Fe 20  = 36 : 1 : 13 as shown in \nthe Table 1. Those were qualitatively consistent, a lthough the quantitative consistency lacks. \n     The VAHE  in FM films is simply described as follows:19 \nCS AHE IM V∝ ,  (8) \nwhere the IC is a charge current in an FM film. In this study, the IC is generated due to the ISHE \nin FMs. Therefore, the eq. (8) can be rewritten as follows; \nρISHE\nAHEVMVS∝ .    (9) \nSimilarly to the VISHE , using the values in the Table 2 for parameters in  the eq.(9), the VAHE  \nvalues to the VAHE  for a Ni 80 Fe 20  sample ratio is estimated to be Fe : Co : Ni 80 Fe 20  = 5: 1.1 : 1,  \n 9while the values ratio on the experiments is Fe : C o : Ni 80 Fe 20  = 134 : 11 : 1 as shown in the \nTable 1. The tendency of the AHE among the Fe, Co, and Ni 80 Fe 20  samples was also \nqualitatively consistent between the above consider ation and our experiments. The \ndiscrepancies between the above estimation and expe riments for both the VISHE  and VAHE  may \ncome from the difference of the magneto8crystalline  anisotropy of the FM films. In general, it is \nnegligible for Ni 80 Fe 20 , while it strongly affects to the magnetic propert ies for Fe and Co films. \nThe MS values were estimated by using the eq. (1), which is established for a uniform \nmagnetization rotation mode and not considered the strong correlation between the neighbor \nspins. The way how to take the magneto8crystalline anisotropy of FM films into account must \nbe found. Also, the polarities of the respective EM Fs have not considered in this study, yet. \nHowever, the polarities of the output voltages are different among the researches.9,11  Because of \nthe lack of amount of the related studies, the disc ussion about the polarity of output voltages is a \nnext issue. For further investigation, other 3d 8transition FM films with different 3d 8electron \nnumbers are tested. \nFinally, we compared the VISHE  value of the Ni 80 Fe 20  sample in this study with the VISHE  of \nsome Ni8Fe alloy/NM multi8layer samples in previous  studies.183 In this study, the VISHE  was \nestimated to be 14.7 µV for a single layer Ni 80 Fe 20 film sample, while the VISHE  was estimated to \nbe 10 µV ~ 30 µV for “bi8layer” Ni 80 Fe 20 /Pt or Ni 80 Fe 20 /Pd samples.183 Thus, it was indicated  \n 10 that the single FM layer structure under the FMR ge nerates high enough EMF compared to the \nFM/NM multi8layer structure. That is, we successful ly demonstrated a better method to obtain \nthe EMF under the FMR than using any other multi8la yer structures, which has a merit of the \nmaterial saving, and easiness of manufacturing devi ces.  \n \nThe EMF properties generated in Fe, Co, and Ni 80 Fe 20  films themselves under their FMR \nwere investigated. For Fe and Co films, the EMF due  to the AHE was dominantly generated \nunder their FMR, while for Ni 80 Fe 20  films, the EMF due to the ISHE in the Ni 80 Fe 20  film itself \nunder the FMR was mainly generated. This tendency w as qualitatively explained with the PS, \nthe σSHE , the σAHE , the MS, and the ρ of the FM films. \n \n[Acknowledgement] \n     This research was partly supported by a Grant8 in8Aid from the Japan Society for the \nPromotion of Science (JSPS) for Scientific Research  (B) (26286039). \n  \n 11 References \n1E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, App l. Phys. Lett. 88 , 182509 (2006). \n2K. Ando, Y . Kajiwara, S. Takahashi, S. Maekawa, K. Takemoto, M. Takatsu, and E. Saitoh, \nPhys. Rev. B 78 , 014413 (2008). \n3K. Ando, and E. Saitoh, J. Appl. Phys. 108 , 113925 (2010). \n4K. Ando, and E. Saitoh, Nat. Commun. 3, 629 (2012). \n5E. Shikoh, K. Ando, K. Kubo, E. Saitoh, T. Shinjo, and M. Shiraishi, Phys. Rev. Lett. 110 , \n127201 (2013). \n6Y . Kitamura, E. Shikoh, Y . Ando, T. Shinjo, and M. Shiraishi, Sci. Rep. 3, 1739 (2013) \n7Z. Tang, E. Shikoh, H. Ago, K. Kawahara, Y . Ando, T . Shinjo, and M. Shiraishi, Phys. Rev. B \n87 , 140101 (R) (2013). \n8J.C. Rojas8Sánchez, M. Cubukcu, A. Jain, C. Vergnau d, C. Portemont, C. Ducruet, A. Marty, L. \nVila, J.8P. Attané, E. Augendre, G. Desfonds, S. Ga mbarelli, G. Jaffrès, J.8M. George, and M. \nJamet, Phys. Rev. B 88 , 064403 (2013). \n9B.F. Miao, S.Y . Huang, D. Qu, and C.L. Chien, Phys.  Rev. Lett. 111 , 066602 (2013). \n10 Y . Ando, K. Ichiba, S. Yamada, E. Shikoh, T. Shinjo , K. Hamaya, and M. Shiraishi, Phys. Rev. \nB 88 , 140406 (R) (2013). \n11 H. Wang, C. Du, P.C. Hammel, and F. Yang, Appl. Phy s. Lett. 104 , 202405 (2014).  \n 12 12 Y . Tani, Y . Teki, and E. Shikoh, Appl. Phys. Lett. 107 , 242406 (2015). \n13 G. Schmidt, D. Ferrand, L.W. Molenkamp, A.T. Filip,  and B.J. van Wees, Phys. Rev. B 62 , \nR4790 (2000). \n14 A. Fert, and H. Jaffrès, Phys. Rev. B 64 , 184420 (2001). \n15 A. Tsukahara, Y . Ando, Y . Kitamura, H. Emoto, E. Sh ikoh, M. P. Delmo, T. Shinjo, and M. \nShiraishi, Phys. Rev. B 89 , 235317 (2014). \n16P.M. Tedrow, and R. Meservey, Phys. Rev. B 7, 318 (1973). \n17E. Villamor, M. Isasa, L.E. Hueso, and F. Casanova,  Phys. Rev. B 88 , 184411 (2013). \n18T. Naito, D.S. Hirashima, and H. Kontani, Phys. Rev . B 78 , 014413 (2008). \n19R. Karplus, and J. M. Luttinger, Phys. Rev. 95 , 1154 (1954). \n \n  \n 13 Figure captions: \nFig. 1. (Color online) A schematic illustration of our single ferromagnetic metal (FM) layer \nsample and experimental set up. The dimensions of t he FM (Fe, Co and Ni 80 Fe 20 ) layer are 1.5 \nmm × 4.0 mm and the thickness is 25 nm. Two electro des are attached on both ends of the FM \nfilm using silver paste.  \n \nFig. 2. (Color online) (a)8(c) FMR spectra of (a) a n Fe sample, (b) a Co sample, and (c) a \nNi 80 Fe 20  sample under the microwave power of 200 mW. The I is the microwave absorption \nintensity. (d)8(f) Static magnetic field H dependence of the electromotive force (EMF), V, for θ \n= 0° (red line and circles) and 180° (blue line and c ircles). (d), (e) and (f) are for an Fe sample, \nfor a Co sample, and for a Ni 80 Fe 20  sample, respectively.  \n \nTable 1. The analysis results for our Fe, Co, and N i 80 Fe 20  samples. \n \nTable 2. The parameters of Fe, Co, and Ni 80 Fe 20  samples. Only ρ values are obtained in this \nstudy. \n  \n 14 \n \nK. Kanagawa, et al.: FIG. 1.  \n 15 \n \n \nK. Kanagawa, et al.: FIG. 2. \n \n \n \n \n \n \n \n \n  \n 16 K. Kanagawa, et al.: Table 1. \n  Fe Co Ni 80 Fe 20  \nVISHE  (µV)  34.0  80.939  12.4  \nVAHE  (µV) 873.1  85.85  0.544  \nThe ratio to the VISHE  of Co 36  1 13  \nThe ratio to the VAHE  of Ni 80 Fe 20  134  11  1 \n|VISHE /VAHE | 0.466  0.161  22.8  \nMS (emu/cc)  1061  1094  472.8   \n 17  \nK. Kanagawa, et al.: Table 2. \n  Fe  Co  Ni80 Fe 20  \nPS 16,17 0.44  0.34  0.38  \nσSHE  (Ω 81cm 81) 15,18  400  200  133  \nρ (Ωcm) (experimental data) 7.2×10 85 4.2×10 85 12×10 85 \nσAHE  (Ω 81cm 81) 15,18 806  341  73  "    },    {        "title": "2008.03423v2.Two_magnon_frequency_pulling_effect_in_ferromagnetic_resonance.pdf",        "content": "Two-magnon frequency-pulling eﬀect in ferromagnetic resonance\nW. K. Peria,1H. Yu,2S. Lee,2, 3I. Takeuchi,2and P. A. Crowell1,a)\n1)School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455,\nUSA\n2)Department of Materials Science and Engineering, University of Maryland, College Park, Maryland 20742,\nUSA\n3)Department of Physics, Pukyong National University, Busan 48513, South Korea\n(Dated: 27 October 2020)\nWe report the experimental observation in thin ﬁlms of the hybridization of the uniform ferromagnetic resonance mode\nwith nonuniform magnons as a result of the two-magnon scattering mechanism, leading to a frequency-pulling effect\non the ferromagnetic resonance. This effect, when not properly accounted for, leads to a discrepancy in the dependence\nof the ferromagnetic resonance ﬁeld on frequency for different ﬁeld orientations. The frequency-pulling effect is the\ncomplement of the broadening of the ferromagnetic resonance lineshape by two-magnon scattering and can be calcu-\nlated using the same parameters. By accounting for the two-magnon frequency shifts through these means, consistency\nis achieved in ﬁtting data from in-plane and perpendicular-to-plane resonance conditions.\nMagnetization dynamics of ferromagnets are inﬂuenced by\nmany factors, such as magnetocrystalline anisotropy, interfa-\ncial anisotropy, and Gilbert damping. It is of technological\ninterest to study these phenomena in order to understand, for\nexample, the physics governing magnetization switching by\nspin torque.1–3Interfacial anisotropies are particularly rele-\nvant for these applications, where materials with perpendic-\nular anisotropy are sought due to the lower energy cost as-\nsociated with switching of the magnetization.4Anisotropy\narising from broken symmetry at the interfaces in ultrathin\nmagnetic ﬁlms is commonly probed using ferromagnetic res-\nonance (FMR) techniques,5,6usually by measuring the FMR\nﬁelds as a function of driving frequency or applied ﬁeld ori-\nentation.\nTwo-magnon scattering (TMS), an extrinsic relaxation pro-\ncess of uniform magnetization precession in ferromagnets, is\nan important phenomenon that inﬂuences magnetization dy-\nnamics. TMS is commonly observed as a broadening of\nFMR lineshapes, but it may also lead to a shift in the FMR\nfrequency.7–9Although the broadening of the FMR lineshape\ncaused by TMS is often impossible to ignore, the latter effect\nis more subtle and almost universally neglected when FMR\ndata are used to extract static magnetic properties of materi-\nals. Failure to properly account for this effect, however, may\nlead to inaccurate estimates of magnetic anisotropy energies,\npotentially leading to the inference of a substantial interface\nanisotropy where there is none.\nIn this letter, we demonstrate the existence of a frequency-\npulling effect in ferromagnetic resonance induced by two-\nmagnon scattering in polycrystalline Fe 0:7Ga0:3thin ﬁlms for\nin-plane applied magnetic ﬁelds. We arrive at this result\nby calculating the frequency shift based on our analysis of\nthe linewidth broadening caused by two-magnon scattering\n(which is a large effect and is easier to observe). This is pos-\nsible due to the complementary nature of the two phenom-\nena. We show that the observed resonance frequencies are\nonly consistent with the data taken for perpendicular-to-plane\na)Author to whom correspondence should be addressed: crowell@umn.eduﬁelds (i.e., can be ﬁt using the same perpendicular anisotropy\nﬁeld) when they are adjusted to account for the two-magnon\nfrequency shifts. We conclude by demonstrating this effect\nin additional samples of different thicknesses, simultaneously\nshowing that the magnitude of the frequency shifts scales with\nthe magnitude of the two-magnon linewidths as the theory\npredicts.\nThe samples used in this report are 17 nm, 26 nm, and\n33 nm Fe 0:7Ga0:3ﬁlms (thicknesses determined by x-ray re-\nﬂectivity). The 33 nm ﬁlms were deposited on Si/SiO 2sub-\nstrates at room temperature by dc magnetron sputtering of an\nFe0:7Ga0:3target. The base pressure of the deposition cham-\nber was 5\u000210\u00008torr and the working pressure was main-\ntained at 5\u000210\u00003torr by Ar gas (99.999%). The compo-\nsition of the Fe 0:7Ga0:3ﬁlms was quantitatively analyzed by\nenergy dispersive spectroscopy (EDS). The 17 nm and 26 nm\nﬁlms were obtained by etching the 33 nm ﬁlms with an ion\nmill. The lack of magnetic anisotropy in the plane of the ﬁlm\nwas veriﬁed with vibrating sample magnetometry (VSM) and\nFMR for the 33 nm ﬁlm. Grain boundaries were observed\nwith atomic force microscopy (AFM), yielding an average\ngrain diameter of \u001815 nm [see Fig. 1(a)]. This is in good\nagreement with the structural coherence length, which was\nestimated to be 13 nm with XRD. Figure 1(b) shows a two-\ndimensional XRD detector image of the Fe 0:7Ga0:3(110) peak\nfor the 33 nm ﬁlm, where the center of the detector coincides\nwith a scattering vector normal to the ﬁlm plane. The intensity\nof this Bragg peak is approximately constant for ﬁxed values\nof the scattering angle 2 qas the scattering vector is canted\ninto the plane [as evidenced by the “ring” in Fig. 1(b)], indi-\ncating the absence of texture.\nFerromagnetic resonance lineshapes were measured at\nroom temperature using a coplanar waveguide with modula-\ntion of the applied magnetic ﬁeld for lock-in detection as de-\nscribed in Ref. 10. The coplanar waveguide was placed in se-\nries with a rectifying diode that measured the transmitted mi-\ncrowave power. The applied dc magnetic ﬁeld was modulated\nwith a 220 Hz ac magnetic ﬁeld having an amplitude of a few\nOe for lock-in detection of the differential absorption. The mi-\ncrowave frequency was varied up to 52 GHz with power from\n5 to 10 dBm. The lineshapes were measured for both in-planearXiv:2008.03423v2  [cond-mat.mtrl-sci]  26 Oct 20202\n(a)\nTwo-magnon \nscattering(b)\n(c) (d)\n65 60 55 50 45 40 35\n2ș(°)\n0100200300400Intensity\n(arb.units)\nxq\nxyz\nq\nxyz\nFIG. 1. (a) Atomic force microscopy image of the 33 nm Fe 0:7Ga0:3\nﬁlm. Root-mean square roughness is 0.7 nm. (b) Two-dimensional\nXRD detector image of the 33 nm Fe 0:7Ga0:3ﬁlm showing the (110)\nBragg peak (given by the “ring” at 2 q'52\u000e). The center of the de-\ntector corresponds to the symmetric conﬁguration, in which the scat-\ntering vector qis normal to the plane of the ﬁlm ( qx=qy=0). The\nscattering vector is canted into the ﬁlm plane as one moves vertically\nfrom the center of the detector, as indicated by the coordinate axes.\nThe structural coherence length determined from the full-width-at-\nhalf-maximum of the Bragg peak is 13 nm. (c) Thin-ﬁlm magnon\ndispersion for in-plane magnetization and wavevectors qkM, with\nan arrow indicating the two-magnon scattering process. (d) Field-\nswept FMR linewidths of the 33 nm Fe 0:7Ga0:3ﬁlm with in-plane\napplied magnetic ﬁeld overlaid with a ﬁt to a combined two-magnon\nscattering and Gilbert damping model. The Gilbert damping a(a ﬁt\nparameter) and defect correlation length x(ﬁxed) are shown on the\nﬁgure.\n(IP) and perpendicular-to-plane (PP) applied ﬁelds. When the\nmagnetization is IP, there exist magnons degenerate with the\nq=0 FMR magnon [see Fig. 1(c)]. This leads to a possible\nscattering mechanism of the FMR mode, observable through\nits nonlinear effect on the ﬁeld-swept linewidth as a function\nof frequency,7–9shown in Fig. 1(d) for the 33 nm Fe 0:7Ga0:3\nﬁlm. This is the TMS process, and it is allowed as long as\nsome assisting process enables conservation of momentum.\nThe resonance frequency was ﬁt as a function of the applied\nmagnetic ﬁeld H0to the Kittel equation for a thin ﬁlm with no\nin-plane magnetic anisotropy, which reads as\nwFMR =gq\nH0(H0+4pMe f f) (1)\nforH0in the plane and\nwFMR =g(H0\u00004pMe f f) (2)\nforH0perpendicular to the plane,11with gthe gyromagnetic\nratio and 4 pMe f fthe effective demagnetizing ﬁeld. Hence-\n/s48 /s53 /s49 /s48 /s49 /s53 /s50 /s48 /s50 /s53 /s51 /s48 /s51 /s53 /s52 /s48 /s48 /s49 /s48 /s50 /s48 /s51 /s48 /s52 /s48 /s53 /s48 /s72/s32/s80/s80\n/s52 /s77 \n/s101 /s102/s102/s32/s61 /s32/s49 /s49 /s46/s55 /s40/s50 /s41/s32/s107/s79/s101 \n/s103 /s32/s61 /s32/s50 /s46/s49 /s52 /s40/s49 /s41/s52 /s77 \n/s101 /s102/s102/s32/s61 /s32/s49 /s51 /s46/s56 /s50 /s49 /s40/s52 /s41/s32/s107/s79/s101 \n/s103 /s32/s61 /s32/s50 /s46/s48 /s53 /s54 /s52 /s40/s55 /s41/s32/s32/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s71/s72/s122/s41\n/s72\n/s70 /s77 /s82/s32/s40/s107/s79/s101/s41/s72/s32/s73/s80\n/s48 /s53 /s48 /s48 /s49 /s48 /s48 /s48 \n/s51 /s54 /s57 /s72 \n/s70 /s77 /s82 /s32/s40/s79/s101 /s41/s70/s114/s101 /s113 /s117 /s101 /s110 /s99/s121/s32/s40/s71/s72 /s122/s41FIG. 2. Frequency as a function of resonance ﬁeld for the 33 nm ﬁlm\nobtained with IP (red circles) and PP (black squares) orientations of\nthe magnetic ﬁeld, overlaid with ﬁts to Eqs. (1) and (2), respectively.\nFit parameters 4 pMe f fandg-factor are indicated on the ﬁgure for\nboth cases. Inset shows a close-up of the IP orientation at low fre-\nquencies.\nforward we will express the gyromagnetic ratio in terms of\nthe Landé g-factor, viag=gmB=¯h. Figure 2 shows ﬁeld-\ndependent resonance dispersions of the 33 nm ﬁlm for IP and\nPP applied ﬁelds, along with ﬁts to Eqs. (1) and (2). For\nthe PP case we obtain g=2:0564\u00060:0007 and 4 pMe f f=\n13:821\u00060:004 kOe, and for the IP case g=2:14\u00060:01\nand 4 pMe f f=11:7\u00060:2 kOe. The PP value of 4 pMe f f=\n13:821 kOe is higher than previous bulk measurements12but\nis similar to values obtained in thin ﬁlms.13The inset shows a\nclose-up of the IP data at low frequencies, whence it is clear\nthat a discrepancy exists between the IP data and the ﬁt to Eq.\n(1). For the 26 nm and 17 nm thicknesses we observe the same\nqualitative behavior. In the case of the 26 nm ﬁlm we mea-\nsure 4 pMe f f=14:0736\u00060:0006 kOe and g=2:060\u00060:001\nfor PP ﬁelds, compared to 4 pMe f f=12:17\u00060:01 kOe and\ng=2:133\u00060:007 for IP ﬁelds. For the 17 nm ﬁlm we mea-\nsure 4 pMe f f=13:0049\u00060:0005 kOe and g=2:054\u00060:001\nfor PP ﬁelds, compared to 4 pMe f f=11:623\u00060:006 kOe and\ng=2:120\u00060:004 for IP ﬁelds. In addition, the IP data for\nboth the 17 nm and 26 nm ﬁlms cannot be ﬁt well to Eq.\n(1) at low ﬁelds (similar to what is seen for the 33 nm ﬁlm,\nshown in the inset of Fig. 2). There is no in-plane magnetic\nanisotropy, so the discrepancy cannot be attributed to an in-\nplane anisotropy ﬁeld. Furthermore, the parameters yielded\nby the ﬁts in either case are drastically different. In light of\nthese inconsistencies, we proceed to investigate the effect of\nTMS on the IP ﬁeld-dependent resonance dispersion.\nOne of the primary characteristics of TMS is that it can3\nbe suppressed by orienting the magnetization perpendicular-\nto-plane, a result of the disappearance of degeneracies in the\nspin wave dispersion as the magnetization is rotated perpen-\ndicular to the plane.14Later this fact will be used to control for\nTMS, allowing the observation of the noninteracting or “bare”\nproperties when the ﬁlm is perpendicularly magnetized.\nThe breaking of momentum conservation in TMS neces-\nsitates the presence of defects in order to drive the process.\nThere are numerous categories of defects which may cause\nTMS. Among the prominent ones are surface roughness,7,15\ndislocation networks,16and grain boundaries.8,9We will fo-\ncus here on TMS induced by grain boundaries, having con-\nﬁrmed the structural isotropy of the ﬁlms with XRD as wellas observing grains directly with AFM. These characterization\ndata also allow us to constrain the defect lengthscale, which\npartially determines the strength of coupling between q=0\nandq6=0 magnons. In the context of TMS, the grains lead\nto a spatially inhomogeneous and random anisotropy ﬁeld.\nThe inhomogeneous ﬁeld allows for interaction between the\nFMR mode and modes at nonzero wavevector, providing both\nan additional relaxation channel and a change in the effective\nstiffness of the FMR mode. These can be described as imag-\ninary and real shifts in the FMR frequency, respectively. A\nperturbative model of TMS for this system gives the follow-\ning result for the complex frequency shift of the FMR mode\ndue to interactions with modes at nonzero wavevector:8,9,17\nDwTMS =g2x2H02Z\nd2qL0q1\n(1+ (qx)2)3=21\np1\n(w\u0000wFMR)\u0000idw(3)\nwhere H0is the root-mean square inhomogeneity ﬁeld, xis the\ndefect correlation length, L0qis the magnon-magnon coupling\nstrength (see Ref. 17), and dw= (dw=dHjHFMR)(aw=g)is\nthe Gilbert frequency half-width-at-half-maximum linewidth.\nThe imaginary part of Eq. (3) corresponds to the well-known\nTMS contribution to the FMR scattering rate, i. e. linewidth.\nLesser known, however, is the real part, which describes a\nshift of the FMR frequency due to TMS. This effect has been\npreviously reported in ultrathin ﬁlms of Ni 0:5Fe0:5,18but a\nlack of broadband measurements leaves the results open to\ninterpretation (such as the possibility of it having arisen from\ninterface anisotropy). In addition, the strength of two-magnon\nscattering in our system is much greater.\nWe begin our analysis by ﬁtting the ﬁeld-swept linewidths\nto the imaginary part of Eq. (3), including contributions from\nGilbert damping (linear with frequency) and inhomogeneous\nbroadening (constant with frequency, determined from the PP\nmeasurement). We hold the defect correlation length xﬁxed\nto 14 nm based on the structural characterization described\nearlier. The ﬁt parameters are aandH0, which we use to cal-\nculate the FMR frequency shifts from the real part of Eq. (3).\nNotably, the Gilbert damping adetermines both the Gilbert\nlinewidth andthe two-magnon linewidth—the latter is clear\nupon inspection of Eq. (3) and is discussed at length in Ref.\n10.\nThe fractional FMR frequency shifts for the 17 nm, 26 nm,\nand 33 nm ﬁlms are shown in Fig. 3. The solid curves give\nthe predicted fractional frequency shifts based on the ﬁts of\nthe linewidths. The points in Fig. 3 represent the observed\nfractional frequency shifts, determined by taking the differ-\nence between the observed FMR frequencies and the FMR\nfrequencies predicted by Eq. (1) (taking 4 pMe f ffrom the PP\nmeasurement). The strong frequency-pulling effect of TMS\nis evident from the main panel of Fig. 3, with red shifts ap-\nproaching 1 GHz at low frequencies.\nThe two-magnon linewidths for the three ﬁlms are shown in\nthe inset of Fig. 3. The solid curves give the ﬁts to the imag-\ninary part of Eq. (3), while the points are obtained by taking\n/s48 /s49 /s48 /s50 /s48 /s51 /s48 /s52 /s48 /s53 /s48 /s45/s48 /s46/s52 /s45/s48 /s46/s51 /s45/s48 /s46/s50 /s45/s48 /s46/s49 /s48 /s46/s48 \n/s49 /s55 /s32/s110 /s109 /s32\n/s50 /s54 /s32/s110 /s109\n/s51 /s51 /s32/s110 /s109/s70 /s77 /s82/s47\n/s70 /s77 /s82\n/s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s71/s72/s122/s41/s48 /s50 /s48 /s52 /s48 \n/s49 /s48 /s48 /s50 /s48 /s48 /s51 /s48 /s48 /s70/s114/s101 /s113 /s117 /s101 /s110 /s99/s121/s32/s40/s71/s72 /s122/s41/s72\n/s84 /s77 /s83 FIG. 3. Frequency shifts induced by two-magnon scattering for the\n17 nm (red points), 26 nm (blue points), and 33 nm (black points)\nFe0:7Ga0:3ﬁlms. The predicted frequency shifts given by the solid\ncurves are calculated from the real part of Eq. (3) using the ﬁt pa-\nrameters from the ﬁts of the linewidths. Inset shows the two-magnon\nlinewidths for the three ﬁlms along with ﬁts to the imaginary part of\nEq. (3). The two-magnon linewidths are determined by subtracting\nthe Gilbert damping and inhomogeneous linewidths (the inhomoge-\nneous linewidths are determined from the PP measurement).\nthe observed linewidths and subtracting both the Gilbert and\ninhomogeneous broadening (taken from the PP measurement)\ncontributions. A notable observation from Fig. 3 is the corre-\nlation between the magnitudes of the two-magnon linewidths\nand frequency shifts for the three ﬁlms, which is a prediction\nof Eq. 3.\nWe now discuss how the aforementioned inconsistency be-\ntween IP and PP ﬁeld-dependent resonance dispersions, and\nthe inability to obtain a good ﬁt of the IP data to Eq. (1),\ncan be explained by the frequency-pulling effect of TMS.\nThe absence of TMS for PP magnetization is of particular4\n/s48 /s50 /s53 /s48 /s53 /s48 /s48 /s55 /s53 /s48 /s49 /s48 /s48 /s48 /s49 /s50 /s53 /s48 /s48 /s51 /s54 /s57 /s49 /s50 /s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s71/s72/s122/s41\n/s72\n/s70 /s77 /s82/s32/s40/s79/s101/s41/s52 /s77\n/s101 /s102 /s102 /s32/s61/s32/s49 /s51 /s46/s56 /s50 /s49 /s40/s52 /s41/s32/s107/s79/s101 \n/s103 /s32/s61/s32/s50 /s46/s48 /s52 /s53 /s40/s49 /s41\n/s72/s32/s105/s110/s45/s112/s108/s97/s110/s101/s48 /s50 /s52 /s54 /s56 /s49 /s48 /s49 /s50 \n/s50 /s48 /s52 /s48 /s72 \n/s70 /s77 /s82 /s32/s40/s107/s79/s101 /s41/s70/s114/s101 /s113 /s117 /s101 /s110 /s99/s121/s32/s40/s71/s72 /s122/s41/s84 /s77 /s83\nFIG. 4. Ferromagnetic resonance frequencies of the 33 nm ﬁlm at\nlow ﬁelds for the IP conﬁguration. The red data points are blue-\nshifted by amounts given by the red curve in Fig. 3, overlaid with a\nﬁt to Eq. (1). The effective demagnetizing ﬁeld 4 pMe f fis ﬁxed to\n13.821 kOe based on the ﬁt to the PP data (black squares of Fig. 2);\nthe only ﬁt parameter is the Landé g-factor. The blue data points are\nthe observed resonance frequencies, before two-magnon interactions\nare taken into account. The inset shows all of the adjusted resonance\nfrequencies up to high ﬁelds.\nconvenience in our case because it allows determination of\nthe effective demagnetizing ﬁeld 4 pMe f fand the Landé g-\nfactor, which together can in principle be used to predict the\nFMR ﬁeld-dependent dispersion for IP magnetization. A di-\nrect measurement of the dispersion for IP magnetization is not\npossible due to the frequency shifts caused by TMS. To ad-\ndress this problem, the IP FMR frequencies are blue-shifted\nusing the red curve shown in Fig. 3, representing the FMR\nfrequencies in the absence of TMS. We then ﬁt the corrected\nIP FMR frequencies to Eq. (1), ﬁxing 4 pMe f fto the PP value\nof 13.821 kOe (for the 33 nm ﬁlm) and leaving the g-factor\nas a free parameter. (A small amount of surface anisotropy\nmay lead to an anisotropy of the orbital moment of the ﬁlm,\nleading to an anisotropic g-factor.11) The shifted IP FMR fre-\nquencies at low ﬁelds, along with a ﬁt to Eq. (1) for ﬁxed\n4pMe f f, are shown in Fig. 4 for the 33 nm ﬁlm—these values\nrepresent the bare FMR frequencies in the absence of two-\nmagnon scattering. The ﬁt yields g=2:045\u00060:001, which is\nless than 1% lower than the PP value of g=2:0564\u00060:0007.\nAlso shown are the FMR frequencies before being adjusted\nfor two-magnon interactions (blue data points). The inset of\nFig. 4 shows the bare FMR frequencies and ﬁt up to high\nﬁelds. This process was also carried out for the 26 nm and\n17 nm ﬁlms, whereby the agreement with Eq. (1) was signif-\nicantly improved. The ﬁts of the IP data to Eq. (1) yielded\ng=2:039\u00060:002 for the 26 nm ﬁlm and g=2:045\u00060:002\nfor the 17 nm ﬁlm (both <\u00181% lower than the PP values). It is\nclear from Fig. 4 that the frequency-pulling effect of TMS is\nsuccessful at explaining the inconsistencies we have encoun-\ntered.In conclusion, we observe a frequency-pulling effect of the\nferromagnetic resonance in thin ﬁlms of Fe 0:7Ga0:3for mag-\nnetization in the plane of the ﬁlm. It is shown that this ef-\nfect can be explained by the hybridization of the ferromag-\nnetic resonance with nonuniform magnons as a result of the\ntwo-magnon scattering interaction. The frequency shifts can\nbe predicted from the two-magnon induced broadening of the\nlineshapes, whereby a consistency is obtained with the line-\nshapes measured when the magnetization is perpendicular to\nthe plane of the ﬁlm. Our results highlight the importance of\naccounting for two-magnon scattering when using ferromag-\nnetic resonance as a characterization technique, a fact which\nis usually ignored in the determination of static properties.\nThis work was supported by SMART, a center funded\nby nCORE, a Semiconductor Research Corporation program\nsponsored by NIST. Parts of this work were carried out in\nthe Characterization Facility, University of Minnesota, which\nreceives partial support from NSF through the MRSEC pro-\ngram, and the Minnesota Nano Center, which is supported by\nNSF through the National Nano Coordinated Network, Award\nNumber NNCI - 1542202.\nThe data that support the ﬁndings of this study are available\nfrom the corresponding author upon reasonable request.\n1D. C. Ralph and M. D. Stiles, “Spin transfer torques,” J. Magn. Magn.\nMater. 320, 1190 (2008).\n2J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back, and T. Jungwirth,\n“Spin Hall effects,” Rev. Mod. Phys. 87, 1213 (2015).\n3A. Manchon, J. Železný, I. M. Miron, T. Jungwirth, J. Sinova, A. Thiaville,\nK. Garello, and P. Gambardella, “Current-induced spin-orbit torques in fer-\nromagnetic and antiferromagnetic systems,” Rev. Mod. Phys. 91, 035004\n(2019).\n4F. Hellman, A. Hoffmann, Y . Tserkovnyak, G. S. D. Beach, E. E. Fuller-\nton, C. Leighton, A. H. MacDonald, D. C. Ralph, D. A. Arena, H. A. Dürr,\nP. Fischer, J. Grollier, J. P. Heremans, T. Jungwirth, A. V . Kimel, B. Koop-\nmans, I. N. Krivorotov, S. J. May, A. K. Petford-Long, J. M. Rondinelli,\nN. Samarth, I. K. Schuller, A. N. Slavin, M. D. Stiles, O. Tchernyshyov,\nA. Thiaville, and B. L. Zink, “Interface-induced phenomena in magnetism,”\nRev. Mod. Phys. 89, 025006 (2017).\n5J.-M. Beaujour, D. Ravelosona, I. Tudosa, E. E. Fullerton, and A. D. Kent,\n“Ferromagnetic resonance linewidth in ultrathin ﬁlms with perpendicular\nmagnetic anisotropy,” Phys. Rev. B 80, 180415 (2009).\n6L. Chen, S. Mankovsky, S. Wimmer, M. A. W. Schoen, H. S. Körner,\nM. Kronseder, D. Schuh, D. Bougeard, H. Ebert, D. Weiss, and C. H.\nBack, “Emergence of anisotropic Gilbert damping in ultrathin Fe layers\non GaAs(001),” Nat. Phys. 14, 490 (2018).\n7R. Arias and D. L. Mills, “Extrinsic contributions to the ferromagnetic res-\nonance response of ultrathin ﬁlms,” Phys. Rev. B 60, 7395 (1999).\n8R. D. McMichael and P. Krivosik, “Classical Model of Extrinsic Ferromag-\nnetic Resonance Linewidth in Ultrathin Films,” IEEE Trans. Magn. 40, 2\n(2004).\n9P. Krivosik, N. Mo, S. Kalarickal, and C. E. Patton, “Hamiltonian formalism\nfor two magnon scattering microwave relaxation: Theory and applications,”\nJ. Appl. Phys. 101, 083901 (2007).\n10W. K. Peria, T. A. Peterson, A. P. McFadden, T. Qu, C. Liu, C. J. Palmstrøm,\nand P. A. Crowell, “Interplay of large two-magnon ferromagnetic resonance\nlinewidths and low Gilbert damping in Heusler thin ﬁlms,” Phys. Rev. B\n101, 134430 (2020).\n11M. Farle, “Ferromagnetic resonance of ultrathin metallic layers,” Reports\nProg. Phys. 61, 755 (1998).\n12A. E. Clark, M. Wun-Fogle, J. B. Restorff, K. W. Dennis, T. A. Lograsso,\nand R. W. McCallum, “Temperature dependence of the magnetic anisotropy\nand magnetostriction of Fe 100\u0000xGax(x= 8.6, 16.6, 28.5),” J. Appl. Phys. 97,\n10M316 (2005).\n13D. B. Gopman, V . Sampath, H. Ahmad, S. Bandyopadhyay, and J. At-5\nulasimha, “Static and Dynamic Magnetic Properties of Sputtered Fe–Ga\nThin Films,” IEEE Trans. Magn. 53, 1 (2017).\n14M. J. Hurben and C. E. Patton, “Theory of two magnon scattering mi-\ncrowave relaxation and ferromagnetic resonance linewidth in magnetic thin\nﬁlms,” J. Appl. Phys. 83, 4344 (1998).\n15A. Y . Dobin and R. H. Victora, “Surface Roughness Induced Extrinsic\nDamping in Thin Magnetic Films,” Phys. Rev. Lett. 92, 257204 (2004).\n16G. Woltersdorf and B. Heinrich, “Two-magnon scattering in a self-\nassembled nanoscale network of misﬁt dislocations,” Phys. Rev. B 69,184417 (2004).\n17S. S. Kalarickal, P. Krivosik, J. Das, K. S. Kim, and C. E. Patton, “Mi-\ncrowave damping in polycrystalline Fe-Ti-N ﬁlms: Physical mechanisms\nand correlations with composition and structure,” Phys. Rev. B 77, 054427\n(2008).\n18A. Azevedo, A. B. Oliveira, F. M. de Aguiar, and S. M. Rezende, “Extrinsic\ncontributions to spin-wave damping and renormalization in thin Ni 50Fe50\nﬁlms,” Phys. Rev. B 62, 5331 (2000)."    },    {        "title": "1611.02834v1.Theoretical_study_of_the_stripline_ferromagnetic_resonance_response_of_metallic_ferromagnetic_films_based_on_an_analytical_model.pdf",        "content": " 1  Theoretical study of the stripline ferromagnetic resonance  \nresponse of metallic ferromagnetic films based on an \nanalytical model  \n \nR.Hai 1,2  and M. Kostylev 1  \n \n1.School of Physics, University of Western Australia, Crawley, W.A. 6009, Australia  \n2.School of Phy sics, Nanjing University, Nanjing 210093, China  \n \nAbstract:  We develop an advanced analytical model for calculating the broadband stripline \nferromagnetic resonance (FMR) response for metallic ferromagnetic films, taking into account the \nexchange interaction  as well as the exchange boundary conditions at the film surface. This approach \nleads to simple analytical expressions in the Fourier space. As a result, a numerical code which \nimplements inverse Fourier transform of these equations is very quick. This all ows us to explore a wide \nspace of parameters as numerical examples of application of this theory . In particular, we investigate \nthe joint effect of microwave eddy current shielding and magnetisation pinning at the ferromagnetic \nfilm surfaces on the shape o f the stripline FMR response of the film.  \n \n \nI.  INTRODUCTION  \nStripline broadband ferromagnetic resonance (FMR) response has been receiving a significant \nattention in recent years because of its potential for characterisation of magnetic thin films and \nnanostr uctures  [1 - 3]. Also, the stripline geometry is important for various applications of magnetization \ndynamics, such as sensing fields, particles and substances [4 - 6] and in microwave spintronics [7].  \nThe geometry of a stripline ferromagnetic resonance exper iment setup usually contains a \nmicroscopic coplanar (CPW) or microstrip line (MSL) through which a microwave current flows \n(direction z  in Fig. 1). In this work we will be focusing on MSL, and Fig. 1 reflects this geometry . A \nferromagnetic film which is to  be characterized, sits on top of the microstrip line. We will be dealing \nwith metallic films; therefore an insulating spacer separates the film from the microstrip in order to \navoid an electrical contact between them. A static magnetic field H  is applied along the microstrip line. \nPrecession of magnetization in the material is driven by the Oersted field of a microwave current \nflowing through the microstrip line. On resonance, the amplitude of precession increases sharply which \nis seen as an increase in th e microstrip line transmission losses or a decrease in the microstrip line \ntransmission coefficient S21 [1].  \nA number of important peculiarities of the stripline FMR response have been recently discovered. \nFor instance, it has been shown that the linewidth  of the FMR peaks for the samples as measured with \nthis method can be broadened by excitation of travelling spin waves [8] and over - coupling to the \nprobing stripline (“radiation losses”) [9].  \nAlso, it has been demonstrated both experimentally [10 - 13] and theoretically [14 - 16] that the  2  geometry of Fig. 1 breaks the symmetry of the microwave magnetic field incidence on the sample \nsurface. This leads to excitation of microwave eddy currents in a sample under test if it is conducting. \nConsequences of this are especially important for samples with high conductivity of metals -  see Fig. \n21 in [1] for details of the microwave magnetic field configuration in this case.  \nUltimately, the eddy currents lead to strong excitation of standing spin wave modes (SSWM) for  \nt hose ferromagnetic - film samples, for which one would expect much smaller or even vanishing SSWM \namplitudes in the conditions of the conventional cavity FMR [10 - 11].  \nThe first theoretical paper on this subject treated the stripline FMR response only in the limit of a \nvery large microstrip width [15]. Later on, it was found that the width of the microstrip line has a large \neffect on the response [16,17]. In particular, it was shown that the decrease in the microstrip width \ndecreases the FMR peak amplitudes co rresponding to excitation of the standing spin wave modes [16].  \nHowever, the theoretical approaches from those publications have significant drawbacks which did \nnot allow exploring the impact of the finite width of the stripline on the entire space of par ameters for \nthe stripline FMR experiment. The numerical model constructed in [16] is just very slow . The analytical \nsolution from [17] delivers quantitative results almost instantly; however, it was obtained in the \nexchange - free approximation, hence it all ows simulation of only the fundamental FMR peak.  \nIn this work we fill the gap and report an analytical solution for the dipole - exchange case and carry \nout a number of calculations by using the derived model. The obtained results agree well with the fully \nnumerical model from [16]. Furthermore, since a numerical code we developed based on this analytical \nsolution is very fast, we are now able to explore a range of details of the broadband FMR experiment \nwhich have not been studied before. These are the effe cts of the finite width of the stripline and of the \nspacer thickness on the amplitudes of the response of the standing - spin wave modes. Also, we are now \nable to consider radiation losses for the standing wave excitations and the effect of surface anisotrop ies \n(surface spin pinning) on these peaks.  \nThe paper is organized as follows. In Section II we construct the analytical model. In Section III it \nis used to obtain a number of quantitative results which are discussed in detail in that section.  Section \nIV c ontains conclusions.  \n \nII.  Theory  \nWe consider a metallic ferromagnetic film of thickness L  which is homogeneous throughout its volume, \nbut may possess perpendicular surface anisotropy at one or both surfaces. In the model, the \nperpendicular anisotropy is intro duced as surface magnetization pinning [18]. We treat the film as \ninfinitely long along both in - plane axes -  x  and z. The external magnetic field H  is applied in the \npositive z  direction. The film sits on top of a dielectric spacer of a thickness s  separat ing the film from a \nmicrostrip of a width w . We neglect the fact that real microstrips have a small (with respect to w ), but \nfinite thickness and treat the microstrip as infinitely thin in the direction y . The microstrip is supported \nby a dielectric substr ate of thickness d  whose second surface is uniformly metalized. Although the \nmagnetostatic approximation utilized below does not need specifying the dielectric constant for the \nsubstrate, we implicitly assume a specific value for it. It is one which result s in a 50   of characteristic \nimpedance for the microstrip line in the absence of a film on its top.  \nOne standard approach to calculation of the complex impedance for the microstrip lines loaded by \nferromagnetic films is by exploiting the translation invar iance in the z - direction [19]. This corresponds \nto a quasi - static approach for description of microwave transmission lines and results in a \ntwo - dimensional problem which we will be dealing with below . The actual form of the electromagnetic \nfield dependence  on the z  co - ordinate –  in the form of an electromagnetic wave travelling along the \nstripline -  will be accounted for while employing the obtained expression for the complex impedance \nfor calculations of the transmission coefficient S21.   3  In order to descri be the magnetization dynamics in the film we employ the linearized \nLandau - Lifschitz equation (LLE)  \n ( ) eff t        m m H M h ,                               (1)  \nwhere  0 z H  H u ,  0 z M  M u  , and u z  is a u nit vector in the direction z.  In Eq.(1), the dynamic \nmagnetization m has only two components :     , x y m m  m . They are perpendicular to the static \nmagnetization M  whose magnitude 0 M is equal to the s aturation magnetization of the film \n0 S M M   . The dynamic effective field  eff h  consists of two parts: the effective exchange field ex h  is \ngiven by  \n 2 2 \n2 2 ( ) ex x y        h m   ,                               (2)   \nand the dynamic magnetic field h  being solution of  Maxwell equations in the electric - bias free \napproximation  \n    h e  ,                                            ( 3)   \n  h m ,                                          (4)  \n 0 ( ) i     e h m  .                                   (5)   \nHere e  is the microwave electric field,   is electric conductivity,  0  is the permittivi ty of vacuum and   \nis the frequency of the microwave driving field. Dynamic magnetic field h  has two components  \n( , ) x y h h  h  which are perpendicular to the direction along the stripline. Note that the term containing \nelectric permi ttivity is missing on the right hand side of Eq. (3); this is consistent with the standard \nmagnetostatic approximation for ferromagnetic materials [20]. Furthermore, for metals at microwave \nfrequencies, the contribution of the electric bias field to Eq.(3)  is negligible with respect to the \nconductivity one [21,15,17].  \nEven in two dimensions, previous calculations  faced serious difficulties when trying to employ \nreal - space methods [1,14] because of the incompatibility of the length scales for L , w  and d . By  \nexploiting the translational symmetry in the x - direction, that is, by applying the spatial Fourier \ntransformation to both sides of Eqs. (1 - 5) we can significantly simplify the problem. This is because the \nmicrostrip has a infinitely small thickness in the  y - direction and hence can be treated as a boundary \ncondition involving the surface current density [19,22,16,17]. The surface current and its distribution \nalong x  are assumed to be given. Alternatively , it can be calculated self - consistently [17,19], but the \nlatter is out of scope of the present paper.  \nOne more important advantage of the Fourier - space approach is that simple analytical solutions \nexist for the areas above and below the film [16,17]. This allows one to exclude those areas from \nconsideration and consider only the dynamic equations for the film. Exclusion of those areas produces \nspecific boundary conditions for the electromagnetic fields on the film surfaces [17].        \nAccording to the Fourier space approach, we have  \n, exp( ) i t ikx   m h  .                       (6)  \nSubstituting Eq. (6) into Eqs. (3 - 5), we have  \n 0 ( ) z y y ke h m    ,                                    (7)   4   x \ny z h ikh e y      ,                                       (8)  \n y y \nx x h m ikh ikm y y         ,                               (9)    \n 0 (h m ) z \nx x e i y      .                                     (10)  \nIt is easy to verify that, similar to Ref. [21] the ansatz   , qy e h m   solves the sys tem of equations \n(1), (7 - 10). Eqs. (7 - 10) then reduce to  \n   2 2 2 0 x y y kqh iK h i k K m     ,                           (11)  \n 0 x y y x ikh qh qm ikm      ,                               (12)  \nwhere  2 2 \n0 K k i    ,  \nand Eq.( 1) takes the form as follows:  \n   2 2 H \nx x y \nM M h k q m i m                 ,                      (14)  \n   2 2 H \ny y x \nM M h k q m i m                 ,                        (15)  \n where ( ) H H i H      ,  M S M    , and / G H       is the magnetic loss parameter which scales as \nthe Gilbert magnetic damping constant G  .  The magnetic losses have been introduced into Eqs. \n(14,15) phenomenologically, as it is known that the linearized Landau - Lifshitz - Gilb ert Equation \nreduces to the linearised Landau - Lifshitz Equation (1) with a complex value of the applied field given \nby the expression above [20].  Equations (11 - 15) form a homogeneous system of linear algebraic \nequations. On elimination of the h x  and h y  va riables, the system reduces to two equations (A1) for m x  \nand m y  shown in Appendix.  \n Equating the determinant of the system (A1) to zero produces a characteristic equation for the \nproblem.  The above - mentioned approach of first eliminating h x  and h y  from t he system has an \nimportant advantage –  it results in a compact characteristic equation which is bi - cubic with respect to q :  \n3 2 \n0 0 aQ bQ cQ d     ,                            (16)    \nwhere Q = q 2  and the coefficients a , b , c  and d 0  are given in Appendix (E qs.(A2)). The sixth order for the \ncharacteristic equation is in agreement with the earlier treatment of the dipole - exchange spectrum [21] \nand, as we will see below, with the number of available boundary conditions. 1    \nThe cubic equation (16) allows an anal ytic solution.  For completeness, it is also given in the \nappendix, although this is a textbook result. The three roots of (16) are Q 1 , Q 2  and Q 3  (see Eqs.(A3)). \n                                                             \n1 D irect evaluation of the determinant of ( 11 - 14 ) results in an equation of 8 th  order with respect to q , but as the method of elimination of the \nh x  and h y  variab les demonstrates, two of eight roots in total of the 8 th  order equation are meaningless.      5  Accordingly, the 6 roots of the characteristic equation read: 1 1 q Q  , 2 2 q Q  , 3 3 q Q  , \n4 1 q Q  , 5 2 q Q  , 6 3 q Q  . Note that since the coefficients of Eq.(16) are complex numbers, the \nroots are also complex.    \nThe presence of the six roots indicate s that the complete solution for the two components of m  can \nbe expressed in the following form  \n  6 \n1 exp y \ny i i \ni m M q y \n   ,                                    (17)  \n  6 \n1 exp x \nx i i \ni m M q y \n    .                                (18)  \n     In order to determine the co efficients i \nx M  (or i \ny M ), we need six boundary conditions at the film \nsurfaces. Furthermore, in order to obtain a nontrivial solution, at least one of these boundary \nconditions should be inhomogeneous.  \n     The first set of available boundary conditions is the exchange boundary conditions for the \noriginal Landau - Lifshitz equation. They apply to the magnetization vector [18]  \n0 p d y     m m  .                                    (19)  \nWhere d p  is the s urface magnetization pinning constant. The sign in front of p d is nega tive for the \nsurface 0 y  while it is posi tive for the surface y L  . The pinning constant is  non - vanishing if surface \nanisotropies are present at the film surfaces. For instance, in the case of a perpendicular uni - axial \nsurface anisotropy which will be considered in Section III, the m y  component can be pinned but the m x  \ncomponent remains free ( 0, 0 y x \np p d d   ). In the case of an in - plane uni - directional in - plane anisotropy \n(exchange bias), both pinning constants are non - vanishing and depend on the direction of the exchange \nbias field with respect to H [23].   \nAs m  has two components -   x m  and  y m , and this exchange boundary condition applies to both \nsurfaces of the film and to the two m - vector components separately, altogether four independent \nboundary conditions are avail able.  \n     Two more boundary conditions are still needed. These are electromagnetic boundary conditions \nwhich follow from Maxwell Equations. As shown in [17], for the geometry of Fig. 1, they can be cast \nin the form which involves field components inside  the film only . For the surface y L   the respective \nboundary condition reads  \n  0 y y x k h m ih k     .                                      (20)  \nHere we would like to recall that we seek solution of the problem in the Fourier space, ther efore the \nfield and magnetization vector components entering Eq.(20) are actually spatial Fourier components of \nthese quantities.   6  For the surface 0 y  the boundary condition is inhomogeneous because of the presence of a \nmicro wave current I  flowing through the microstrip in the direction z . The linear density of this current \nj ( x ) is assumed to be given. Its spatial Fourier transform reads: 1 ( )exp( ) 2 k j j x ikx dx   \n  .  \nThe inhomogeneous boundary condition for y =0 reads   \n| | sinh(| | ) | | ( )coth(| |( )) sinh(| |( )) y y x k k k d k h m k d s i h i j k k d s k      .                          (21)  \nSubstitution of the solution (17 - 18) together with the relations between the components of m  and h  \n(Eqs. (A4) in Appendix) into the boundary conditions results in a system of 6 algebraic equations \n(Eqs.(A5) in Appendix). The system is non - homogeneous, as one of the equations has a non - vanishing \nright - hand side which follows from the right - hand - side of Eq.(21).  \n   In our work, we solve the system of equations (A5) using numerical methods of linear algebra. \nThe so lution is obtained for the q - values which solve the characteristic equation (16). As the order of \nthe vector - matrix equation (A6) is just 6, the numerical solution is instantaneous, while run on a \npersonal computer.  \nThe obtained numerical solution for m  is  employed to calculate the linear impedance Z r  of the \nmicrostrip loaded by the film, since the latter quantity is an indicator of the microwave magnetic \nabsorption by the film  [15]. The linear impedance can be defined as follows:  \nr U Z I  ,                                              (22)  \nwhere  U  is the linear voltage along the microstip which can be defined as the mean value of the total \nelectric field e z ( x ) induced at the surface of the strip  \n/2 \n/2 1 ( , ) w \nz \nw U e x y s dx w    .                           (23)  \nKeeping in mind that our solution is in the Fourier space it is useful to express U  in terms of the Fourier \ncomponents zk e  of e z ( x ) ( 1 ( )exp( ) 2 zk z e e x ikx dx   \n  ). From the solution of Eqs.(7 - 10) for the area y <0 \n(characterized by   =0, m =0) it follows that  \n \n0 cosh( ) ( 0) ( ) sinh( ) sinh[ ( )] cosh[ ( )] k x \nzk k d j i h y e y s k d k k d s k d s                ,             (24)    \n \nwhere the Fourier transform of the magnetic field at the film surface y =0 reads  \n6 \n1 ( 0) ( ) x \nx hx i i \ni h y C q M \n    .                                (25)  \nAlso, it is convenient to express U  in t erms of the Fourier components of e z ( x ). This expression reads  \nsin 2 \n2 zk kw \nU e dk kw  \n        .                                  (26)    7  This concludes the solution for Z r . Similar to our previous works [16,17], the integral in (26) is \ncomputed numerically in the pr esent work.  \nThus, the complete process of the numerical simulation consists of three steps. The first one (Step 1) \nis to calculate the roots q 1 to q 6  by using the expressions (A3). Because of the analytical character of this \nsolution, the program code imp lementing it delivers those values instantaneously . The next step is to \nsubstitute these q - values into Eqs. (A5) and solve the inhomogeneous system of linear algebraic \nequations numerically . This gives the six amplitudes i \nx M . This comp utation is practically instantaneous, \nas already stated above. Substitution of the obtained i \nx M values into Eq.(25) and the result into Eq.(24) \nconcludes Step 2. Thus, the output of Step 2 is a value of zk e  for some  given value of k .  \nThe two steps have to be repeated numerous times, as we need zk e  values for a large number of \nFourier wave numbers k , in order to implement Step 3 –  carrying out the numerical integration in \nEq.(26).  \nIn our previous  work [16] up to ten hours of computation were needed to produce the final result -  \na Z r  dependence on the applied field H  for a given microwave frequency . The computation was slow \nbecause the model developed in [16] was based on a numerical solution for S tep 1. Half a minute or so \nwas needed to complete this step. As it is repeated numerous times for different k  and H  values, the \ntotal computation time becomes very large.  \nOn the contrary, the present code is very fast. It has been implemented as MatLAB an d MathCAD \nworksheets for a usual personal computer. In both cases a complete Z r ( H ) dependence is obtained \nwithin 10 to 20 seconds. It takes a couple of more seconds to convert the obtained Z r  values into S21 by \nusing Eqs. (18 - 25) from [16].  \n \nIII.  DISCUSSION  \nSev eral numerical tests were run to validate our theory . First we checked the value of Im( Z r ) off \nthe resonance peaks (e.g. for H =0 or a very large H ) and for  =0 (a non - conducting film). We found \nthat Im( Z r ( H =0,  =0)) is in excellent agreement with an analyt ical formula for the linear inductive \nimpedance for a microstrip line not loaded by a ferromagnetic film (see [16,17] for detail).   \nWe also checked the off - resonance distribution of h x  across the film thickness for a highly \nconducting film. For a 50nm - thi ck film, w =1.5mm and s =0, we found that ( 0, 0) ( 0) x h x y j x      \nand ( 0, ) x h x y L    is practically vanishing, as expected. Furthermore, the h x ( y ) dependence is very \nclose to linear between these two points, as also expected. This behavior is cons istent with a strong \nmicrowave shielding effect present for thin metallic films when microwave power is incident on one \nfilm surface only [1,15].  \nAlso, the shape of Z r (H)  dependence as a function of the film thickness was checked. The results \nare shown in Fig. 2.  The first observation from Fig. 2 is that two resonance peaks are present in Panels \n(b) and (c), although no surface pinning was assumed for either of the film surfaces. This is consistent \nwith the calculation in [16] where it was explained as a co nsequence of a perfect microwave shielding \neffect. In the presence of the microwave eddy currents induced in the film by the incident microwave \nmagnetic field of the microstrip line, the magnetization dynamics is driven by the sum of the spatially \n(quasi) - uniform Oersted field of the microwave current in the microstrip and the spatially \nanti - symmetric Oersted field of the eddy current in the film (see Fig. 21 in [1]). The uniform  8  component is responsible for excitation of the fundamental mode of uniform pre cession (the right - hand \npeak in the panels of Fig. 2) and the anti - symmetric component for excitation of the 1 st  standing spin \nwave mode (1 st  SSW , the left - hand peak in the panels). The latter mode is characterized by an \nanti - symmetric distribution of dyna mic magnetization across the film thickness.  \nThe second observation from Fig. 2 is that an increase in L  leads to an upshift in the 1 st  standing \nspin wave mode peak. This behavior and the field positions of both peaks are consistent with Kittel \nEquation fo r the dipole - exchange modes [24]. This is one more confirmation of the validity of our \ntheory . Noteworthy is the absence of the 2 nd  SSW peak in Fig. 2(b), although Kittel Equation predicts \nthat this peak should be located at about 2000 Oe. Hence, the eddy current effect allows one to probe \nthe 1 st  SSW mode only (unless a film possesses asymmetric surface magnetization pinning).  \nNext, we employ the developed numerical code in order to understand the dependence of the \namplitude of the 1 st  SSW peak on the widt h of the microstrip w . In [16] it has been observed that the \namplitude of this peak increases with an increase in w . Now, given the much higher computation speed \nof the present software, we are able to explore this effect in detail.  \nIn order to perform th is study, the calculations are repeated for a number of w  values. The obtained \nZ r ( H ) and S21( H ) traces are fitted with a complex function  \n1 2 \n0 \n1 1 2 2 ( ) ( ) ( ) A A F H A H H i H H H i H           .                (27)  \nHere H 1(2)  is the extracted resonance field for Mode 1 or 2 respectivel y, 1(2) H   is the linewidth of the \nrespective peak. The quantities A 0 , A 1  and A 2  are complex numbers; they are also extracted from the fits. \nIn the following, we use quantities 1 1 / A H   and 2 2 / A H   to chara cterize the resonance peak \nheights.  \nThe fits show that the shape of the Z r ( H ) dependence is in excellent agreement with Eq.(27). Hence, \nthe complex function (27) is the “natural” dependence of Z r  on H . This implies that the shape of the \nS21( H ) dependence may deviate from the one given by Eq.(27), because S21 depends on Z r  in a \ncomplicated and nonlinear way (see Eq.(31) in [17]). Consequently, values of the parameters of Eq.(27) \nextracted from the fits may be different for Z r and S21 traces. This conclusion  is confirmed by our \nnumerical calculations.  \nFig. 3 displays examples of traces obtained for different values of w . One sees that the 1SSW peak \ngrows in amplitude with an increase in w . In [16] this was explained as a more pronounced microwave \nshielding ef fect for wider microstrips. The last panel of Fig. 3 shows the peak amplitude ratio \n2 2 1 1 / / / r A H A H     as a function of w . (Here the index “1” denotes the fundamental mode and “2” \nthe 1 st  SSW .). From this figure one sees that there is a strong depend ence of the amplitude of the 1 st  \nSSW mode on the microstrip width for w  values below 300 micron. For larger w  values the dependence \nsaturates and the effect becomes the same as in the case of normal incidence of a travelling plane \nelectromagnetic wave on t he surface of a ferromagnetic film [25,26]. One also notices that the traces \nare slightly different for Z r  and S21. As discussed above, this is consequence of the fundamental \ndifference in the shapes of Zr ( H ) and S21( H )  dependences. \nSimilar behavior is ob served as a function of the thickness of the dielectric spacer s  (Fig. 4(e).) From \nthe examples of the raw traces in Fig. 4(a - c), one sees that r  increases with an increase in s . This \nhappens because lifting the film with respect to the stripline makes the  microwave Oersted field of the \ncurrent in the stripline more spatially uniform at the position y  where the film is located. The more \nuniform is the field, the more pronounced the microwave shielding is. Hence, the effect of lifting the  9  film is analogous t o increasing w  while keeping s  constant.  \nThis conclusion implies that the r ( s ) dependence should be significant for small w  values only . \nIndeed, the graphs in Fig. 4 were obtained for w =10 microns. For significantly wider striplines ( w >300 \nmicrons) this d ependence is practically vanishing.  \nFrom the raw traces in Fig. 4 one also notices that the increase in s  is accompanied by a decrease in \nthe heights of both peaks. This behavior is caused by a decrease of the strength of coupling of the \nmagnetization dyn amics to the driving magnetic field [9, 17]).   \nThe change in the coupling strength also leads to a dependence of the resonance peak linewidth on \ns . From Fig. 4(e) one sees that the linewidth decreases with a decrease in s . This is consistent with the \neffe ct of radiation losses [9]. Interestingly, the linewidth broadening due to the radiation losses is more \nsignificant for the 1 st  SSW than for the fundamental mode, with broadening being the same for both \nS21 and Z r . This is actually in qualitative agreement  with the experiment in [9], see Fig. 3 (c) in that \npaper. On the contrary, the fundamental - mode linewidth for small values of s  is noticeably larger for \nS21 than for Z r . The latter fact is in agreement with the exchange - free model from [17].  \nLet us now lo ok at the effect of surface magnetization pinning on magnetization dynamics. Fig. 5 \ndisplays a number of S21( H ) traces calculated for different values of the pinning constant. As before, \nwe assume that the films have large conductivity of metals; this lead s to important peculiarities of the \nstripline FMR responses, as we will show below .   \nLet us first discuss the effect of symmetric pinning –  the situation when ( 0) ( ) x x \np p d y d y L    . As seen \nfrom Fig. 5 (a,b,e,f), the main impact of the symmetric pinning  is a shift of the resonance peaks \nupwards or downwards, depending on the sign of the pinning constant. This is in full agreement with \nKittel Equation for spin wave resonance frequencies [24].  No noticeable difference in r  is seen for \nPanels (e) and (f) w ith respect to Panel (a).  \nOn the contrary, asymmetric pinning Fig.5(c,d,g,h) has a strong effect on r . This is because the \nsymmetry is now doubly broken –  on top of the asymmetry of h x ( y ) originating from the single - side \nincidence of the microwave field, t here is also asymmetry in the film’s magnetic parameters in the \ndirection of the film thickness. This makes the value of r  dependent on the film orientation with respect \nto the microstrip line.  \nThis effect was theoretically found in [1] based on considerat ion of a simple model w =  . In that \nwork, it was suggested that it might be useful for experimental determination of the particular film \nsurface (from the two) at which magnetization pinning is present. Fig. 5 confirms this funding with a \nrigorous calculati on for a realistic value of w  and hence the practical importance of this effect.  \n \nIV.  CONCLUSION  \nIn this work, we constructed a new two - dimensional model for calculation of the stripline \nferromagnetic resonance response of metallic ferromagnetic films. Our mo del works much more \nrapidly and is capable of taking exchange interaction and surface magnetisation pinning into account. \nThe acceleration was achieved by analytically solving the initial system of equations describing the \ndynamics.  \nWe also conducted a nu mber of computations with a numerical model which followed from this \ntheory . The numerical code enabled us to explore a large parameter space for the problem. Our \ncomputations confirmed that microwave shielding by eddy currents induced in a ferromagnetic f ilm \nstrongly affects its stripline FMR response if the film has large conductivity . The eddy currents are \nexcited in the film because of the single - side incidence of the microwave field on the film in the \ngeometry of Fig. 1. They lead to excitation of the 1 st  Standing Spin Wave Mode. The amplitude of the \nrespective peak in the raw FMR traces depends on the width of the microstrip – with an increase in the  10  width the microwave magnetic field incident on the film becomes more spatially uniform which leads \nto mo re efficient shielding by the eddy currents.  \nThe developed model also allowed us to investigate radiation losses for the FMR modes. It was \nfound that they are larger for the 1 st  Standing Spin Wave Mode than for the Fundamental one, in \nqualitative agreemen t with an earlier experiment [9].  \nAlso the effect of the surface magnetization pinning on the response was explored in the \nframework of this more rigorous model. It confirmed a conclusion from a previous work that the \nstripline FMR method can be used to e xtract the degree of symmetry of pinning, and if the pinning \nturns to be asymmetric to identify which of film surfaces is characterized by larger (smaller) pinning.  \n \nAppendix  \nBy substituting Eq. (14) and Eq. (15) into Eqs. (11 - 12), we obtain  \n \n  \n  2 2 4 2 2 2 \n4 2 2 2 2 ( ) ( ) 0 \n( ) ( ) ( ) 0 M x M K M k M y \nM K M k M x M y K q kq m i q q K v m \ni q q K m K q kq m      \n                  \n              ,       (A1)  \nw here  2 \n0 i    , 2 \nk H M k       , and 2 \nK k M K      .  \nEquating the determinant of the matrix of this system of equations to zero results in the characteristic \nequation (16). The coefficients of this  equation are as follows:  \nM a   ,  \n2 2 2 ( ) M K M M b k v              ,  \n2 2 2 2 2 2 2 2 2 (3 2 ) 2 ( )( ) ( ) M M M H H H M H c k k v k v k                        ,  \n2 2 2 2 2 2 2 2 2 2 2 \n0 ( ) 2( )( ) ( ) M M M H M H M H d k k v k k v k v                             .        (A2)  \nThe three roots of the cubic equation (16) are as follows.  \n1 Q D R C     \n* \n2 3 3( ) \n2 2 2 R D i R D Q Q B       ,                         (A3)  \nwhere  3 b B a  , 3 c C a   2 C B R D   , and   1/3 3 \n1 /(2 ) 3 / 2 D D B d a BC     ,  and \n1/2 3 2 2 2 \n3 \n1 2 2 3 \n4 2 4 B d B C Bcd d D C a a a            .  Once we have obtained the Q - values, we are able to express m y  \nin terms of m x  with the hel p of Eq.(A1):  \ny my x m C m   ,                        (A4 - 1)  \nw here  4 2 2 \n2 2 ( ) ( ) \n( ) M K M k M \nmy \nM i q q K \nC K q kq    \n                 . A lso, the following relations between the other \nfield components follow from Eqs.(14) - (15):   11  x hx x h C m  ,                       (A4 - 2)  \nwhere  2 2 2 ( )/( ) hx my C K qkC K q    , and  \ny hy x h C m  ,                       (A4 - 3)  \nwhere  ( 1)/ hy my hx C C ik C q    .  \n \nThe system of equations which follows from the boundary value problem is given by:  \n \n6 \n1 ( ) 0 x i \ni p x \ni q d M \n    ,  \n6 \n1 ( ) exp( ) 0 x i \ni p x i \ni q d M q L \n    ,  \n6 \n1 ( ) ( ) 0 y i \ni p my i x \ni q d C q M \n    ,  \n6 \n1 ( ) ( ) exp( ) 0 y i \ni p my i x i \ni q d C q M q L \n    ,                        (A5)  \n  6 \n1 sgn( )sinh( ) ( ) ( ) coth ( ) sgn( ) ( ) \nsinh ( ) i \nmy i hy i hx i x \ni i k k d C q C q k d s i k C q M \nk d s                    ,  \n  6 \n1 ( ) ( ) sgn( ) ( ) exp 0 i \nmy i hy i hx i i x \ni C q C q i k C q q L M \n         .   \nIn this work this system is solved numerically to obtain the six coefficients x \ni M .  \n \nReferences  \n1.  I.S. Maksymov and M. Kostylev, J. Ph ys. E  69 , 253 (2015).  \n2.  J.M. Shaw, H.T . Nembach, T.J. Silva, C.T. Boone, J. Appl. Phys . 114 , 243906 (2013).  \n3.  C. Bilzer, T. Devolder, P . Crozat, C. Chappert, S. Cardoso, and P .P . Freitas, J. Appl. Phys.  101 , \n074505 (2007).  \n4.  M. Inoue, A. Baryshev, H. Takagi, P . B. Lim, K. Hatafuku, J. Noda and K. Togo, Appl. Phys. Lett . \n98 , 132511 (2011).  \n5.  P .J. Metaxas, M. Sushruth, R.A. Begley, J. Ding, R.C. Woodward, I.S. Maksymov, M. Albert, W . \nWang, H. Fangohr, A.O. Adeyeye and M. Kostylev, Appl. Phys. Lett.  106 , 232406 (2015) .  \n6.  C.S. Chang, M. Kostylev, and E. Ivanov, Appl. Phys. Lett.  102 , 142405 (2013).  \n7.  E. Saitoh, M. Ueda, H. Miyajima and G. Tatara, Appl. Phys. Lett . 88 , 182509 (2006).  \n8.  G. Counil, J. - V . Kim, T. Devolder, C. Chappert, K. Shigeto and Y . Otani, J. Appl. Phys.  95 , 5646 \n(2004).  \n9.  M.A. W . Schoen, J.M. Shaw, H.T. Nembach, M. Weiler, and T. J. Silva, Phys. Rev. B  92 , 184417 \n(2015).   12  10.  K.J. Kennewell, M. Kostylev, N. Ross, R. Magaraggia, R.L. Stamps, M. Ali, A.A. Stashkevich, \nD. Greig, and B.J. Hickey, J. Appl. Phys . 108 , 073 917 (2010).  \n11.  M. Kostylev, A.A. Stashkevich, A.O. Adeyeye, C. Shakespeare, N. Kostylev, N. Ross, K. \nKennewell, R. Magaraggia, Y . Roussigné, and R.L. Stamps, J. Appl. Phys . 108 , 103914 (2010).  \n12.  M. Kostylev, J. Ding, E. Ivanov, S. Samarin, and A.O. Adeyeye, J. Appl. Phys.  115 , 173903 \n(2014).  \n13.  M. Kostylev, Z. Yang, I.S. Maksymov, J. Ding, S. Samarin, and A.O. Adeyeye, J. Appl. Phys.  119 , \n103903 (2016).  \n14.  M. Bailleul, Appl. Phys. Lett . 103 , 192405  (2013).  \n15.  M. Kostylev, J. Appl. Phys.  106 , 043903  (2009).  \n16.  Z. Lin and M. Kostylev, J. Appl. Phys . 117 , 053908 (2015).  \n17.  M. Kostylev, J. Appl. Phys . 119 , 013901 (2016).  \n18.  R.F . Soohoo, Phys. Rev.  131 , 594 (1962).  \n19.  V .F . Dmitriev and B. A. Kalinikos, Sov. Phys. J.  31 , 875 (1988).  \n20.  A.G. Gurevich, and G.A. Melkov. Magnetization oscillation s and waves . CRC P ress, 1996.  \n21.  T. Wolfram and R.E. De Wames, Phys. Rev. B  4 , 3125 (1971).  \n22.  P .R. Emtage, J. Appl. Phys . 49 , 4475 (1978).  \n23.  R. Magaraggia, K. Kennewell, M. Kostylev, R.L. Stamps, M. Ali, D. Greig, B.J. Hickey, C.H. \nMarrows, Phys.Rev. B  83 ,  054405  ( 2011).  \n24.  D.D. Stancil and A. Prabhakar, Spin Waves: Theory and Application , Springer, Berlin, 2009.  \n25.  M. Kostylev, J. Appl. Phys . 112 , 093901 (2012).  \n26.  M. Kostylev, J. Appl. Phys.  113 , 053908 (2013).  \n \n \n \n \n  13   \n \n \n \n \nFigure 1 . Sketch of t he modeled geometry . (1) The ground plane of the microstrip \nline. (2) Substrate of the microstrip line of thickness d . (3) Infinitely thin strip of \nwidth w  carrying a microwave current in the direction z . (4) Ferromagnetic film of \nthickness L  in the direct ion y and of width w  in the direction x. The static \nmagnetic field H  is applied along z.   14   \n \nFigure 2 . Calculated FMR traces for different ferromagnetic film thicknesses L . (a) \nL =30nm; (b) L =50nm; (c) L =80nm. Parameters of calculat ion: width of \nmicrostrip line w =350  m; thickness of the line substrate d=300  m; spacer \nthickness s =1  m; microwave frequency is 20GHz; saturation magnetization \n4  M s =10000G; conductivity of the film  =4.5 10 6 S/m; magnetic loss parameter \nfor the ferromagnetic film:  H =53.017 Oe (Gilbert dampin g constant  G =0.008), \nexchange constant A =0.8 10  6  erg/cm. Unpinned surface spins  ( 0 x y \np p d d   )  are \nassumed for both film surfaces.   15  \n \n \n \n30nm \n             (d) \n0 1 2 3 4 5 Ratio of pek amplitudes r \n(unitless)\n0.0 0.1 0.2 0.3 0.4 \nCol 2 vs S21 \nCol 2 vs Zr 85nm \n                (e) \nMicrostrip width (mm) 0 1 2 3 4 5 Ratio of pek amplitudes r \n(unitless)\n0.0 0.1 0.2 0.3 0.4 \nS21 \nZ r \n \nFig. 3. Panels (a) to (c): FMR traces for different microstrip line widths w =10, 100 \nand 350  m resp ectively . Other parameters are the same as for Fig.2(a). (d) and (e): \ndependence of the ratio of the peak amplitudes on w  for two film thicknesses \nL =30nm (d) and L =85nm (e). Parameters of calculation for (d) and (e): L =85 \nmicron, 4  M s =10000G ,  =4 .5 10 6 S/m,  exchange constant A =1.3 10  6 erg/cm,  \nGilbert damping constant 0.008,  w =0.1 mm, s =10micron, spacer dielectric \npermittivity: 1; microstrip substrate thickness d =0.1mm and permittivity 10. \nLength of the film in the direction along the microstrip is 5 mm. Freq uency is 22 \nGHz.   16  \n \n \n \nSpacer thickness (  m) 0 100 200 300 400 500 600 Peak amplitude ratio\n(unitless)\n0.0 0.1 0.2 0.3 0.4 \nZ r \nS21 \nSpacer thickness (  m) 0 100 200 300 400 500 600 Peak linewdth \n(Oe)\n28 32 36 40 44 48 \nfundamental mode, Z r \nfundamental mode, S21 \n1st exchange mode, S21 \n1st exchange mode, Z r (d) \n(e) \n \nFig. 4. Panels (a) to (c) are plotted for spacers of thickness s =10, 100 and 200 \nmic rons respectively . The microstrip line width is 10 micron and other parameters \nare same as in Fig.2(a). (d): dependence of the peak amplitude rati o on s . (e) Peak \nlinewidth dependence on s . Parameters of calculation for (d) and (e): L =85 nm, \nw =10  m, frequency is 10 GHz. The remainder of parameters is the same as for \nFig. 3(d - e).   17  \n \nFigure 5. Examples of FMR traces for different surface magnetization pinning \nconditions. (a) and (e): magnetisation is unpinned 0 y \np d   at both film surfaces. \n(b) and (f): it is equally pinned at both surfaces. The pinning constant 8 1 10 m y \np d    \nfor (b) and 8 1 10 m y \np d   for (f). (c) and (g): single - sided pinning. 0 y \np d   for y = L  \nand 8 1 10 m y \np d    (c) or 8 1 10 m y \np d    (g) for y =0.  (d) and (h): the same, but \nnow 0 y \np d   for y =0, and 8 1 10 m y \np d    (d) or 8 1 10 m y \np d    (h) for y = L .  0 x \np d   \nfor all panels. Width of the microstrip line w =350  m, thickness of the film \nL =50nm; thickness of the spacer s =0; microwave frequency is 19 GHz; saturation \nmagnetization 4  M s =17900G; conductivity of the film  =1.8 10 7 S/m;  Gilbert \ndamping constant is 0.008 . Note that that (a) and ( e) show the same plot; the plot \nis repeated to facilitate its comparison with the other plots.  \n "    },    {        "title": "2002.11694v1.Control_of_spin_dynamics_in_artificial_honeycomb_spin_ice_based_nanodisks.pdf",        "content": "Control of spin dynamics in arti\fcial honeycomb spin-ice-based nanodisks\nMojtaba Taghipour Ka\u000bash,1Wonbae Bang,2, 3, 4Sergi Lendinez,1Axel\nHo\u000bmann,4, 5John B. Ketterson,3and M. Benjamin Jung\reisch1,\u0003\n1Department of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA\n2Institute of Advanced Materials, LG Chem, Daejeon 34122, Korea\n3Department of Physics and Astronomy, Northwestern University, Evanston, IL 60208, USA\n4Materials Science Division, Argonne National Laboratory, Argonne, IL 60439, USA\n5Department of Materials Science and Engineering,\nUniversity of Illinois at Urbana-Champaign, Urbana, IL 61801, USA.\n(Dated: February 27, 2020)\nWe report the experimental and theoretical characterization of the angular-dependent spin dy-\nnamics in arrays of ferromagnetic nanodisks arranged on a honeycomb lattice. The magnetic \feld\nand microwave frequency dependence, measured by broadband ferromagnetic resonance, reveal a\nrich spectrum of modes that is strongly a\u000bected by the microstate of the network. Based on sym-\nmetry arguments with respect to the external \feld, we show that certain parts of the ferromagnetic\nnetwork contribute to the detected signal. A comparison of the experimental data with micromag-\nnetic simulations reveals that di\u000berent subsections of the lattice predominantly contribute to the\nhigh-frequency response of the array. This is con\frmed by optical characterizations using microfo-\ncused Brillouin light scattering. Furthermore, we \fnd indications that nucleation and annihilation\nof vortex-like magnetization con\fgurations in the low-\feld range a\u000bect the dynamics, which is di\u000ber-\nent from clusters of ferromagnetic nanoellipses. Our work opens up new perspectives for designing\nmagnonic devices that combine geometric frustration in gyrotropic vortex crystals at low frequencies\nwith magnonic crystals at high frequencies.\nI. INTRODUCTION\nThe investigation of arti\fcial spin ice (ASI) structures\nhas received increased attention in the magnetism com-\nmunity over the past decade1{7. Beyond the traditional\nstudies on slow dynamics and thermalization e\u000bects in\nASI, there is an e\u000bort to understand the spin dynam-\nics in the microwave frequency regime in those networks,\ne.g., Refs. [8{16] as well as their magnetotransport prop-\nerties, e.g., Refs. [17{21]. Aided by advances in modern\nnanofabrication technologies it is possible to create two-\ndimensional arrays of ferromagnetic nanoscaled elements\nwith a wealth of possible orientations and alignments,\nand to place these elements on a lattice. The original\nintention of creating ASI was to design geometrically\nfrustrated networks that closely mimic the frustration\nin crystalline spin ices, e.g., Refs. [22{25]. Many di\u000ber-\nent lattice structures such as honeycomb, Shakti, Tetris,\nbrickwork, etc. have been investigated since then26{29.\nWhat these lattices have in common is that the ferromag-\nnetic elements consist of elongated singe-domains, also\nreferred to as ` islands ', in which shape anisotropy facil-\nitates an alignment of the magnetic moments to point\nparallel or antiparallel to the easy axis of the island. As\nsuch, these elements essentially have only binary degrees\nof freedom. Recently, new types of non-Ising like lattices\nsuch as XY spin systems30and Potts ASI31,32have been\nexplored and more exotic phase transitions were found.\nFrom a dynamic perspective, recent work by Behncke\net al. showed that arrays of ferromagnetic disks are par-\nticularly interesting as their polarization state can be\ntuned so that geometrical frustration arises through the\npresence and motion of magnetic vortices in the disks33.They showed that frustrated and non-frustrated states\ncan be achieved by changing the frequency of the state\nformation process. However, the individual disks had\nmicrometer diameters { too big to be treated as a XY\nspin system { and only the low-frequency dynamics, cor-\nresponding to the gyrotropic frequencies, of the vortices\nwere studied.\nHere, we describe detailed experimental and theoreti-\ncal characterizations of angular-dependent spin dynam-\nFIG. 1. (a) Schematic view of the experimental FMR setup in-\ncluding SiO 2substrate (purple), coplanar waveguide (CPW)\n(yellow) with two ground lines (G) and one signal line (S),\nand arti\fcial spin ice sample (gray) patterned on top of the\nCPW. The inset shows a scanning electron micrograph of the\nfabricated honeycomb lattice. (b) Schematic view and dimen-\nsions of a vertex of the honeycomb lattice with the direction\nof the microwave magnetic \feld Hrfand the direction of the\nin-plane angle \u0012of the external magnetic \feld Hext.arXiv:2002.11694v1  [cond-mat.mes-hall]  26 Feb 20202\nics observed in arrays of ferromagnetic nanodisks. Our\nnanodisks have a diameter well below one micrometer\nand are closely packed on a honeycomb lattice. The\nbroadband ferromagnetic resonance (1 { 10 GHz) mea-\nsurements are complemented by micromagnetic simula-\ntions using mumax3[34] and micro-focused Brillouin light\nscattering ( \u0016-BLS) spectroscopy. Our results show that\nvortex-like magnetization con\fgurations in the disks af-\nfect the high-frequency dynamics in the honeycomb-disk\nASI networks, which di\u000bers from clusters of ferromag-\nnetic nanoellipses35. Furthermore, we \fnd evidence of\nlocalized magnetization dynamics in our arrays. This\nis the \frst step in combining geometric frustration in\na gyrotropic vortex crystal at low frequencies with a\nmagnonic crystal at high frequencies.\nII. EXPERIMENTAL DETAILS AND\nMICROMAGNETIC APPROACH\nIn the following, we present details on the sample fab-\nrication, ferromagnetic resonance measurements, micro-\nmagnetic simulations, as well as micro-focused Brillouin\nlight scattering spectroscopy.\nA. Sample fabrication\nA multi-step lithography process was used for sam-\nple fabrication. In the \frst step, a coplanar waveguide\n(CPW) was de\fned on a thermally oxidized Si substrate\nusing optical lithography on to which 5-nm Ti and 120-\nnm of Au was deposited by electron-beam evaporation\nfollowed by lift-o\u000b. The width of the signal line is 20\n\u0016m separated by an 8 \u0016m gap between the signal and\nthe 40\u0016m wide ground lines. We have previously shown\nthat the best coupling between the microwave magnetic\n\feld created by the CPW and arrays of nanomagnets is\nobtained by directly patterning the arrays on top of the\nsignal line13. Hence, the ASI structure was written on\nthe signal line by electron-beam lithography. For this\npurpose, a double layer positive resist of MMA (methyl\nmethacrylate)/PMMA (polymethyl methacrylate) was\nused. After exposure and development of the resist stack,\nwe used electron-beam evaporation to deposit 15-nm of\npermalloy (Py, Ni 80Fe20), followed by a lift-o\u000b step. A\nsketch of the fabricated CPW with the magnetic material\nis shown in Fig. 1(a). A scanning electron microscopy\nimage of the honeycomb lattice made of ferromagnetic\nnanodisks on top of the signal line is shown as an inset.\nFigure 1(b) shows the dimensions of the disks as well as\nthe separation of the disks to neighboring disks in the\nhoneycomb lattice. The disks have a diameter of 500 nm\nand are constructed as a pair, where the gap between\nneighboring pairs is 76 nm.B. Broad-band ferromagnetic resonance (FMR)\nmeasurements\nThe patterned devices were characterized by broad-\nband ferromagnetic resonance using a vector network an-\nalyzer (VNA)36. An external magnetic \feld Hextis ap-\nplied in the sample plane, and its angle \u0012with respect\nto the CPW can be controlled by an automated rotating\nmotor, see Fig. 1(b). As shown in Fig. 1(a), a microwave\ncurrent is passed through the CPW, generating an alter-\nnating magnetic \feld H rfin they-axis direction (red ar-\nrows). H rfexcites a collective precessional motion of the\nmagnetic moments in the disks, which can be detected\nusing the VNA.\nThe FMR spectra were recorded as follows: the sam-\nple was \frst saturated by an external in-plane magnetic\n\feld at \u00002100 Oe, followed by recording a reference spec-\ntrum at \u00001500 Oe. Thereafter, the \feld was gradually\nincreased from \u00001000 Oe to 1000 Oe in steps of 10 Oe\nwhile sweeping from 1 GHz to 10 GHz and recording the\ntransmission S21 parameter using the VNA at each \feld\nstep. The reference spectrum is then subtracted from\nthe spectra at each \feld step to account for changes in\nthe transmission characteristics not associated with the\nmagnetic \feld.\nC. Micromagnetic simulations\nAll simulations were conducted using the mumax3mi-\ncromagnetic simulator34. For this purpose, each of the\n15 nm thick 500 nm diameter nanodisks is partitioned\ninto 5 \u00025\u000215 nm3so that the lateral dimensions are\nkept less than the exchange length ( LPy\nex= 5:3 nm).\nThe grid is divided in 512 \u0002512\u00021 cells. Standard\nmagnetic parameters for Py are used: saturation mag-\nnetizationMsat= 800 \u0002103A/m, exchange sti\u000bness\nAex= 13 \u000210\u000012J/m and Gilbert damping parame-\nter\u000b= 0:01. A honeycomb vertex is formed of 6 disks as\nshown in Fig. 1. Two disks build a pair that is arranged\non a honeycomb lattice, where the connecting axis be-\ntween two disks is aligned along the signal line of the\nCPW, while the axes of the two neighboring coupled disks\nare directed at an angle \u0012=\u0006120\u000e, respectively.\nD. Microfocused Brillouin light scattering ( \u0016-BLS)\nThe spatially resolved dynamic response of the reso-\nnant modes in the lattice was measured using \u0016-BLS37.\nFor this purpose, we use a 532 nm single mode laser with\na continuous power of less than 2 mW at the sample posi-\ntion. The external \feld is applied parallel to the direction\nof the signal line of the CPW ( \u0012= 0\u000e). The same sample\nwas used for both the \u0016-BLS and the FMR measure-\nments. Furthermore, we use the same \feld routine as in\nthe FMR measurements and micromagentic simulations:\nFirst, the sample is saturated and then the \feld is swept,3\nFIG. 2. Experimental absorption spectra of our sample, obtained by angular-dependent broadband ferromagnetic resonance.\nThe external magnetic \feld Hextis applied at di\u000berent in-plane angles (a) \u0012= 0\u000e, (b)\u0012= 15\u000e, (c)\u0012= 30\u000ewith respect to the\nsignal line of the CPW ( x-axis). As the magnetic \feld is swept from negative to positive values, the spectra are recorded by a\nvector network analyzer at each \feld step. The white arrow shows the magnetic \feld sweep direction. The red arrow indicates\nthe main mode as discussed in the text.\nwhile changing the excitation frequencies and probing\nthe excited dynamics. A nominal microwave power of\n+22 dBm is used, which is low enough to avoid any non-\nlinearities. Informed by the acquired \feld/frequency re-\nsponse, we choose a particular \feld/frequency combina-\ntion to measure the spatial extent of the spin dynamics.\nIII. RESULTS AND DISCUSSION\nFigures 2(a-c) show the experimental FMR spectra as\nfalse color-coded images for the nanodisks arranged on\na honeycomb lattice for three di\u000berent in-plane angles\n\u0012of 0\u000e, 15\u000eand 30\u000ewith respect to the signal line of\nthe CPW (x-axis). Note that the magnetic \feld is swept\nfrom negative to positive \felds. A dark contrast shows\na strong microwave absorption indicative of an e\u000ecient\nexcitation of spin dynamics in the array, while a brighter\ncolor means a negligible microwave absorption (i.e., an\nabsence of coherent spin dynamics). The choice of these\nthree external \feld angles is based on the symmetry of\nthe lattice and the fact that the angles 45\u000e, 60\u000e, 75\u000e\nand 90\u000eshow similar behavior as the former three. In\nother words, by shape symmetry, one can infer that there\nis a similarity between the following groups of external\napplied \feld angles: i) 0\u000eand 60\u000e; ii) 15\u000e, 45\u000eand 75\u000e;\nand iii) 30\u000eand 90\u000e.\nAs is shown in Fig. 2, we detect a number of di\u000berent\nmodes in the FMR spectra with varying position and in-\ntensity depending on the in-plane magnetic \feld angle \u0012.\nThe spectra can be divided into three di\u000berent regimes:\nOne at high \felds, one at low \felds, and one at interme-\ndiate \felds.\nThe \frst regime ranges from \u00001000 Oe to 20 Oe when\nsweeping the magnetic \feld up, i.e., it starts with a con-\n\fguration in which all magnetic moments in the disks are\nsaturated along \u0000Hextand ends just short of an insta-\nbility.\nIn this region, the mode indicated by the red arrow in\nFig. 2 corresponds to the bulk-like fundamental mode,and is produced by the spins that are mostly aligned\nalong the external magnetic \feld. It shows the strongest\nabsorption for the applied \feld at \u0012= 0\u000e. As the an-\ngle\u0012is increased, the intensity of this mode decreases,\nwhile other lower-lying modes appear in the spectra, see\nFigs. 2(b) and (c). Furthermore, the fundamental mode\nis shifted slightly to higher frequencies as the angle \u0012is\nincreased.\nIn the second regime, ranging from 20 to 120 Oe, we\nobserve a distinct gapin the mode spectrum independent\nof the in-plane \feld angle. Typically, one can observe a\ndecrease of the resonant frequency in this \feld range, fol-\nlowed by the nucleation of magnetic vortices in the disks\naccompanied by the onset of a gyrotropic motion of the\nvortices33. The gyrotropic motion typically occurs at fre-\nquencies well below 1 GHz (depending on the dimensions\nof the disks and speci\fc material parameters). While we\ndo observe the gyrotropic motion in micromagnetic simu-\nlations (as it will be discussed bellow), we did not detect\nany signal below 1 GHz in our experiments. There are\ntwo reasons for this, both of which arise from the smaller\nthickness and lateral dimensions of our sample compared\nto the honeycomb disks studied in Ref. [33]. First, since\nvortex dynamics generally produce a weak signal, less\nthickness can make the signal detection more challenging.\nSecond, as explained in detail below, the simulated mag-\nnetization con\fguration [Fig. 3 (e)] suggests that only\nin one nanodisk (out of six) at each honeycomb vertex is\na vortex formed. Even more importantly, the magnetic\nmoments in the disks curl, and thus the net magnetiza-\ntion in each disk is almost zero. As a result, the overall\nFMR response is minimum as is evident from the gap in\nthe Fig. 2. It is interesting to note that this is indepen-\ndent of the in-plane \feld angle, which is di\u000berent from\nclusters of ferromagnetic nanoellipses35.\nThe third regime ranges from 120 to 300 Oe. In this\n\feld range not all moments have aligned with the exter-\nnal \feld direction. This leads to some minor di\u000berences\nin the resonant dynamics compared with the symmetric\nnegative \felds. These di\u000berences are absent in the high-4\nFIG. 3. Results of micromagnetic simulations. (a) Simulated absorption spectra for an applied in-plane \fled at \u0012= 0\u000e. The\nblack arrow indicates the direction in which the \feld is swept. (b) Corresponding magnetization behavior of the normalized\nx-component of the magnetization ( mx). The insets shows the saturated con\fguration of mxin one vertex for \u00001000 Oe (red)\nand 1000 Oe (blue). (c)-(e) Normalized mx-component value at \felds \u0000800 Oe (green), \u0000200 Oe (red), and 100 Oe (blue),\nrespectively. These \feld are labeled with dashed lines in (a) and (b). (f)-(k) Corresponding 2D FFT intensity pro\fles for\nfrequencies with strong absorption in both experimental and simulation results. (k) Magni\fed vortex dynamics for the 120\u000e\nouter disk in (h) indicated be dashed square. Note that only (k) is plotted in logarithmic scale; all other \fgures are in linear\nscale.\n\feld regime.\nThe last part of the spectra (300 to 1000 Oe) shows\nthe same behavior as the negative \feld regime at high\nmagnetic \feld values. Here, all magnetic domains are\ncompletely aligned with Hextand the resonant spectra\nare similar to the negative \feld range.\nThe experimental FMR spectra are the result of the\ncollective dynamics of the lattice. However, in order to\nget more insight of the two-dimensional pro\fles of the\nmagnetization con\fguration and the spin dynamics we\nperform micromagnetic simulations. For this purpose, a\nprocedure replicating the experimental routine was im-\nplemented in mumax3: First, the magnetization of the\nlattice is saturated by initializing all the spins in \u0000Hext\ndirection. Then, the \feld is swept from \u00001000 Oe to\n1000 Oe [indicated by a black arrow in Fig. 3(a)], with\n10 Oe increment steps. The equilibrium magnetization\nstate at each \feld step is obtained by minimizing the\ntotal energy and then letting the magnetization evolve\nfor 5 ns. This is done to avoid possible saddle points inthe energy landscape. In order to obtain the frequency\nresponse, a perturbative sinc-shaped \feld pulse with an\namplitude of 1 Oe and cut-o\u000b frequency of 50 GHz is\napplied in the y-axis. The time evolution of the magne-\ntization is recorded every 1 ps for a total time of 20 ns.\nA fast-Fourier transform (FFT) is performed on the z-\ncomponent of the time-dependent magnetization to ob-\ntain the characteristic precession frequencies of the sys-\ntem at each \feld step. False-color coded plots of the FFT\nintensity as a function of the external magnetic \feld and\nfrequency can then be obtained, as shown in Fig. 3(a).\nBy comparing the experimental FMR data [Figs. 2(a-\nc)] with their corresponding simulation results [Fig. 3(a)]\nwe \fnd a very good agreement for the di\u000berent \feld\nregimes discussed above. As we will see, that is also\ntrue for the other in-plane \feld angles studied here [Figs.\n4(a) and (e)]. Before we turn to a discussion of the\nspin dynamics, we focus on the static magnetization con-\n\fguration. Figure 3(b) shows the simulated \feld scan\nof the normalized x-component of the static magnetiza-5\nFIG. 4. Results of micromagentic simulations. (a) Simulated absorption spectra for an applied in-plane \feld angle of \u0012= 15\u000e\nin a color-coded image, and (b)-(c) the spatial pro\fles of the simulated spin dynamics at \u0000800 Oe [green dashed line in (a)]\nand frequencies of (b) 7.254 GHz, (c) 8.231 GHz, and (d) 8.797 GHz. (e) Simulated absorption spectra for an applied in-plane\n\feld angle of \u0012= 30\u000e, and (f)-(h) the spatial pro\fles of the spin dynamics at \u0000800 Oe [red dashed line in (e)] and frequencies\nof (f) 7.614 GHz, (g) 8.026 GHz, and (h) 9.106 GHz.\ntion (mx). The vertical dashed lines indicate the same\n\feld values as in Fig. 3(a). They denote \feld values of\n\u0000800 Oe (green), \u0000200 Oe (red), and +100 Oe (blue).\nThe two insets in Fig. 3(b) depict the color code for\nthe normalized magnetization mx, displaying the disks\nof one honeycomb vertex in the saturated state, i.e., at\n\u00001000 Oe (red) and at +1000 Oe (blue). The magnetiza-\ntion con\fguration of one honeycomb vertex is illustrated\nin Figs. 3(c)-(e) at the intermediate \felds as indicated by\nthe dashed lines in (a) and (b) [ \u0000800 Oe (green), \u0000200 Oe\n(red), and +100 Oe (blue)].\nAs is obvious from the simulated magnetization vs. ex-\nternal \feld ( mxvs.Hext) trace shown in Fig. 3(b), the\nmagnetization con\fguration points mostly along the \feld\ndirection in the high-\feld region. As the \feld is approach-\ning zero, the magnetic moments in the nanodisks start to\n\rip as can been seen from the steps in the mxvs.Hext\nplot [Fig. 3(b)], and from the spatially-resolved simula-\ntion results: Fig. 3(c) at \u0000800 Oe, Fig. 3(d) at \u0000200 Oe,\nand Fig. 3(e) at 100 Oe. This \feld range, in which\nthe magnetization con\fguration changes corresponds to\nthe gapregion observed in the dynamics measurements\n[Fig. 2(a)] and simulations [Fig. 3(a)], where no reso-\nnant dynamics are observed. Increasing the magnetic\n\feld even further results in a progressive re-alignment\nof the moments to the positive \feld direction until the\noverall magnetization points in the positive direction.\nIn addition to the FFT intensity spectra, two-\ndimensional pro\fles of the FFT intensity were computedat selected magnetic \felds/frequencies. First, we dis-\ncuss the two-dimensional pro\fles for an in-plane mag-\nnetic \feld angle of 0\u000e. The corresponding 2D maps are\nshown in Figs. 3(f){(k). As mentioned above Figs. 3(c){\n(e) show the magnetization con\fguration, while (f){(k)\ndepict the spatially-resolved dynamics in the elements.\nFor this purpose, the average magnetization as a func-\ntion of time as well as the magnetic con\fguration at each\ntime step were computed. The dynamic pro\fles were ob-\ntained by calculating the FFT of each cell. The following\ndiscussion focuses on three particular magnetic \feld val-\nues,\u0000800 Oe, \u0000200 Oe, and 100 Oe, as indicated by\nthe dashed lines (green, red, and blue, respectively) in\nFigs. 3(a) and (b).\nAt\u0000800 Oe, the \frst fundamental mode [mode A1\nin Fig. 3(a)] with one antinode in an individual disk\n(frequency 8.437 GHz) is localized in the \u0006120\u000eelements\n[Fig. 3(f)]. This mode has the strongest intensity in the\nspectra. On the other hand, the second prominent mode\nindicated in Fig. 3(a) as mode A2 is a higher-order mode\nlocalized in the horizontal nanodisks, see Fig. 3(i). This\nmode is less intense and lies at a higher frequency than\nmode A1. It is characterized by two antinodes.\nAs the \feld is decreased, the resonant frequency grad-\nually decreases. The spatially resolved dynamic pro\fles\nat\u0000200 Oe are shown in Figs. 3(g) and (j). As is visible\nin the 2D maps, higher-order modes with three antinodes\ncontribute to the signal at 4.476 GHz [Fig. 3(g)] and \fve\nantinodes to the signal at 5.695 GHz [Fig. 3(j)] in the6\nlower frequency regime. Note that the di\u000berence in inten-\nsity in each pair of disks is due to the fact that the vertex\nwas simulated without taking into account the neighbor-\ning disks to reduce the computation time. We tested\nsimulations with neighboring disks, i.e. a larger array\n(not shown here), to con\frm that the magnetostatic in-\nteractions from the other disks cause the intensity to be\nequally distributed between the pairs.\nIn a low \feld of 100 Oe the magnetization con\fguration\nfurther creates \rux closure domains to minimize the mag-\nnetostatic energy, see Fig. 3(e). This lowers the internal\ne\u000bective \feld and, hence, the resonance frequency drops\nto MHz frequencies. Moreover, the only disk that experi-\nences the vortex state is the outer 120\u000eelement. As seen\nfrom the 2D pro\fle of the FFT intensity, Fig. 3(h), we\nobserve the vortex gyromotion in the outer disk at 120\u000e\nat 257 MHz; a magni\fed 2D plot of the vortex dynamics\nis shown in Fig. 3(k) in logarithmic scale. Although the\ngeometry of the structure is symmetric with respect to\nthex-axis, the magnetic con\fguration is not, as can be\nseen from Fig. 3(e). This is because in our simulations\nthe external \feld Hextis applied at 1\u000ewith respect to the\nx-axis to break the symmetry and, thus, to avoid possi-\nble saddle points in the simulations. Depending on the\ninitial conditions (e.g. \feld sweep direction, the choice of\na small deviation angle for the magnetic \feld, etc.), the\nvortex state can be created in di\u000berent nanodisks. How-\never, in our simulations it was not possible to create the\nvortex state in all nanodisks of the lattice at the same\ntime. This could also be the reason for us not being able\nto experimentally detect the corresponding mode.\nThe high-\feld regime at positive \felds (i.e. +150 \u0000\n+1000 Oe) resembles the behavior at negative \felds and,\nthus, we omit a detailed discussion here.\nTo demonstrate the ability to turn on the precession\nin di\u000berent parts of the network, we compare the experi-\nmental FMR results at di\u000berent in-plane angles with mi-\ncromagnetic simulations. Figure 4(a) presents the simu-\nlated frequency versus magnetic \feld spectra for \u0012= 15\u000e,\nwhile Fig. 4(b) shows the corresponding simulations for\n\u0012= 30\u000e. We \frst discuss the 15\u000edata. In particular, we\nfocus on the modes labeled as B1, B2 and B3 in Fig. 4(a).\nTheir corresponding spatial pro\fles at \u0000800 Oe [indi-\ncated by a vertical green dashed line in (a)] are shown in\nFigs. 4(b)-(d).\nDue to the applied \feld direction, the axis of symmetry\nfor the modes is at 120\u000e(compared with the 0\u000edirection\nofHextwhen it lies along the 0\u000eaxis). Mode B1, which\nhas the lowest frequency, is localized in the 120\u000edisks.\nAs is obvious from Fig. 4(b), mode B1 arises from a cou-\npling of the resonances in the pair of disks aligned at\n120\u000e. Mode B2 lies between B1 and B3 in Fig. 4(a). In\nthis intermediate frequency range, the resonant response\nis distributed in all nanodisks, while the strongest preces-\nsion is found in the 120\u000edisks [Fig. 4(c)]. The strongest\nmode at \u0000800 Oe is observed in the simulations for mode\nB3, which is in fact a band of modes. The resonance in\nthe spectrum [Fig. 4(a)] mainly stems from the 0\u000eand\nFIG. 5. (a) \u0016-BLS intensity as a function of \feld and exci-\ntation frequency. Two-dimensional \u0016-BLS intensity measure-\nments at a magnetic \feld of \u0000500 Oe and excitation frequen-\ncies of (b) 7.1 GHz and (c) 7.7 GHz. The position of one\nhoneycomb disk vertex is superimposed in gray.\n\u0000120\u000edisks as indicated in Fig. 4(d).\nFigures 4(e)-(h) show the simulation results when the\n\feldHextis applied at 30\u000ewith respect to the x-axis. At\na large \feld [red dashed line for \u0000800 Oe in Fig. 4(e)],\nthe simulated spectrum shows two intense modes (C1 and\nC3) that are well separated by a weaker mode (labeled\nas C2). The corresponding spatial pro\fles are shown in\nFigs. 4(f)-(h). Modes C1 and C2 are localized in the 120\u000e\ndisks. Again, the modes are coupled along the axes of\nsymmetry of the corresponding pairs of disks [Figs. 4(f)\nand (g)], whereas mode C3 is localized in the 0\u000eand\n\u0000120\u000eelements. This result demonstrates the possibil-\nity of controlling the dynamics in di\u000berent parts of the\nnetwork, not only by de\fning the lattice parameters, but\nalso by selecting an appropriate in-plane \feld angle. This\nis well known from antidot lattices, e.g., Refs. [38{41],\nbut has not been shown for arti\fcial spin ices in general,\nand arrays of nanodisks in particular, until now.\nIn the following, we present experimental results ob-\ntained by\u0016-BLS that con\frm the presence of well de\fned\nchannels of spin dynamics in our structures as suggested\nby the simulated 2D dynamics maps. Our \u0016-BLS setup\nallows us to apply an external magnetic \feld only along\nthe CPW guide axis ( \u0012= 0\u000e). A constant microwave7\nsignal of +22 dBm is applied at varying frequencies. In\na \frst step, we record the BLS spectral response as a\nfunction of the external magnetic \feld and excitation\nfrequencies for a \fxed laser position on the sample, see\nFig. 5(a). Although we are not able to resolve the de-\ntailed mode structure we found in FMR, overall the BLS\nspectrum is in agreement with the FMR results shown in\nFig. 2(a). For the 2D BLS imaging, a \fxed magnetic \feld\nof\u0000500 Oe is chosen. At this \fxed magnetic \feld we ob-\nserve two distinct peaks in the BLS spectra for microwave\nexcitation frequencies of 7.1 GHz and 7.7 GHz. In order\nto \fnd the spatial distribution of the precessional modes,\nthe sample was scanned at those frequencies by rastering\nthe laser over the sample. The stage was moved in 100 nm\nsteps, recording the BLS spectrum at each position. The\nresults of the 2D scans are shown in Figs. 5(b) and (c) for\nthe excitation frequencies of 7.1 GHz and 7.7 GHz, re-\nspectively. Note that the spectra were normalized to the\nelastic BLS peak to compensate for any drifts possibly\noccurring during the course of the measurements.\nThe 2D BLS images show a high-intensity signal [rep-\nresented by the red color in Fig. 5] localized in di\u000berent\nnanodisks in the lattice depending on the excitation fre-\nquency. For instance, when the excitation frequency is\n7.1 GHz, the disks showing a higher intensity are the\nones oriented at \u0006120\u000ewith respect to the external mag-\nnetic \feld, Fig. 5(b). However, when the excitation fre-\nquency is 7.7 GHz, a higher BLS intensity is observed in\nthe disks that are oriented along the external magnetic\n\feld direction, Fig. 5(c). This is in very good agree-\nment with our micromagnetic simulations of a single ver-\ntex at\u0012= 0\u000e; compare Fig. 3(f) with Fig. 5(b) and\nFig. 3(i) with Fig. 5(c). In both the simulations and\nin the spatially-resolved BLS measurements, we observe\nthat the lower-frequency precession is localized in the\ndisks oriented at \u0006120\u000e, and that the higher-frequency\nprecession is localized in the disks oriented along the\nmagnetic \feld. However, our BLS measurements are un-\nable to resolve the \fne structure of the magnetization\ndynamics indicated by antinodes observed in the simula-\ntions at higher frequencies [Figs. 3(i) and (j)].\nPrevious wavevector-resolved BLS studies in ASI lat-\ntices, alongside micromagnetic simulations, gave some in-\nsights in the spatial distribution of the precession modes,\nin particular for a square ASI42and an anti-square ASI,\nconsisting of an extended magnetic \flm with empty is-\nlands positioned in a square lattice43. Together with\nother recent results44, our measurements provide now di-\nrect evidence of localized magnetization dynamics in aparticular ASI lattice by using spatially-resolved BLS.\nIV. CONCLUSION\nIn summary, we performed detailed experimental and\ntheoretical characterizations of the angular-dependent\nspin dynamics in a new type of arti\fcial spin-ice lattices,\nan array of ferromagnetic nanodisks arranged on a hon-\neycomb lattice. Using a combination of broadband fer-\nromagnetic resonance spectroscopy and two-dimensional\ndynamic micromagnetic simulations, we showed that the\nmode spectra at di\u000berent in-plane angles are strongly af-\nfected by the microstate of the network. Di\u000berent sub-\nsections of the lattice predominantly contribute to the\nhigh-frequency response of the array and the exact spa-\ntial location of the dynamics can be controlled by the ex-\ncitation frequency, as well as by the in-plane \feld angle.\nFurthermore, we \fnd indications that nucleation and an-\nnihilation of vortex-like magnetization con\fgurations in\nthe low-\feld range a\u000bect the dynamics. This is di\u000ber-\nent from clusters of ferromagnetic nano-ellipses, which\nare typically the building blocks of arti\fcial spin ice.\nOur two-dimensional micromagnetic simulations are fur-\nther con\frmed by optical characterizations using micro-\nfocused Brillouin light scattering, where a good agree-\nment is found. Our work opens up new perspectives for\ndesigning magnonic devices that combine geometric frus-\ntration in a gyrotropic vortex crystal at low frequencies\nwith a magnonic crystal at high frequencies. While be-\nyond the scope of the current work, this could be achieved\nby systematic studies of the dimensions and thickness of\nthe disks, as well as their separation and arrangement in\nthe network.\nACKNOWLEDGEMENTS\nWork at Delaware, including FMR measurements, mi-\ncromagnetic simulations, and data analysis was sup-\nported by the U.S. Department of Energy, O\u000ece of Basic\nEnergy Sciences, Division of Materials Sciences and En-\ngineering under Award DE-SC0020308. Work at North-\nwestern, including experimental design, was supported\nunder NSF Grant No. DMR 1507058. Device fabrica-\ntion and thin \flm deposition were carried out at Argonne\nand supported by the U.S. Department of Energy (DOE),\nO\u000ece of Science, Materials Science and Engineering Di-\nvision. Lithography was carried out at the Center for\nNanoscale Materials, an O\u000ece of Science user facility,\nwhich is supported by DOE, O\u000ece of Science, Basic En-\nergy Science under Contract No. DE-AC02-06CH11357.\n\u0003mbj@udel.edu1C. Nisoli, R. Moessner, and P. Schi\u000ber, Reviews of Modern\nPhysics 85, 1473 (2013).8\n2I. Gilbert, C. Nisoli, and P. Schi\u000ber, Physics Today 69,\n54 (2016).\n3L. J. Heyderman and R. L. Stamps, Journal of Physics:\nCondensed Matter 25, 363201 (2013).\n4S. Lendinez and M. B. Jung\reisch, Journal of Physics:\nCondensed Matter 32, 013001 (2019).\n5E. Iacocca, S. Gliga, and O. Heinonen, \\Tailoring spin\nwave channels in a recon\fgurable arti\fcial spin ice,\"\n(2019), arXiv:1911.05354 [cond-mat.mes-hall].\n6S. H. Skj\u001arv\u001c, C. H. Marrows, R. L. Stamps, and L. J.\nHeyderman, Nature Reviews Physics 2, 13 (2019).\n7A. Ortiz-Ambriz, C. Nisoli, C. Reichhardt, C. J. O. Reich-\nhardt, and P. Tierno, \\Ice rule and emergent frustration in\nparticle ice and beyond,\" (2019), arXiv:1909.13534 [cond-\nmat.soft].\n8X. Zhou, G. L. Chua, N. Singh, and A. O. Adeyeye, Ad-\nvanced Functional Materials 26, 1437 (2016).\n9W. Bang, F. Montoncello, M. B. Jung\reisch, A. Ho\u000bmann,\nL. Giovannini, and J. B. Ketterson, Physical Review B 99,\n014415 (2019).\n10V. S. Bhat and D. Grundler, Physical Review B 98, 174408\n(2018).\n11V. S. Bhat, F. Heimbach, I. Stasinopoulos, and\nD. Grundler, Physical Review B 93, 140401(R) (2016).\n12E. Iacocca, S. Gliga, R. L. Stamps, and O. Heinonen,\nPhysical Review B 93, 134420 (2016).\n13M. B. Jung\reisch, W. Zhang, E. Iacocca, J. Sklenar,\nJ. Ding, W. Jiang, S. Zhang, J. E. Pearson, V. Novosad,\nJ. B. Ketterson, O. Heinonen, and A. Ho\u000bmann, Physical\nReview B 93, 100401(R) (2016).\n14S. Gliga, A. K\u0013 akay, R. Hertel, and O. G. Heinonen, Phys-\nical Review Letters 110, 117205 (2013).\n15T. Dion, D. M. Arroo, K. Yamanoi, T. Kimura, J. C. Gart-\nside, L. F. Cohen, H. Kurebayashi, and W. R. Branford,\nPhys. Rev. B 100, 054433 (2019).\n16D. M. Arroo, J. C. Gartside, and W. R. Branford, Phys.\nRev. B 100, 214425 (2019).\n17M. Tanaka, E. Saitoh, H. Miyajima, T. Yamaoka, and\nY. Iye, Physical Review B 73, 052411 (2006).\n18W. R. Branford, S. Ladak, D. E. Read, K. Zeissler, and\nL. F. Cohen, Science 335, 1597 (2012).\n19M. B. Jung\reisch, J. Sklenar, J. Ding, J. Park, J. E. Pear-\nson, V. Novosad, P. Schi\u000ber, and A. Ho\u000bmann, Physical\nReview Applied 8, 064026 (2017).\n20B. L. Le, J. Park, J. Sklenar, G. W. Chern, C. Nisoli, J. D.\nWatts, M. Manno, D. W. Rench, N. Samarth, C. Leighton,\nand P. Schi\u000ber, Physical Review B 95, 060405(R) (2017).\n21J. Park, B. L. Le, J. Sklenar, G.-W. Chern, J. D. Watts,\nand P. Schi\u000ber, Physical Review B 96, 024436 (2017).\n22R. F. Wang, C. Nisoli, R. S. Freitas, J. Li, W. McConville,\nB. J. Cooley, M. S. Lund, N. Samarth, C. Leighton, V. H.\nCrespi, and P. Schi\u000ber, Nature 439, 303-306 (2006).\n23S. T. Bramwell and M. J. P. Gingras, Science 294, 1495\n(2001).\n24D. J. P. Morris, D. A. Tennant, S. A. Grigera, B. Klemke,\nC. Castelnovo, R. Moessner, C. Czternasty, M. Meiss-\nner, K. C. Rule, J. U. Ho\u000bmann, K. Kiefer, S. Gerischer,\nD. Slobinsky, and R. S. Perry, Science 326, 411 (2009).\n25T. Fennell, P. P. Deen, A. R. Wildes, K. Schmalzl, D. Prab-\nhakaran, A. T. Boothroyd, R. J. Aldus, D. F. McMorrow,and S. T. Bramwell, Science 326, 415 (2009).\n26J. Park, B. L. Le, J. Sklenar, G.-W. Chern, J. D. Watts,\nand P. Schi\u000ber, Physical Review B 96, 024436 (2017).\n27H. Stopfel, E. Ostman, I.-A. Chioar, D. Greving, U. B.\nArnalds, T. P. A. Hase, A. Stein, B. Hj orvarsson, and\nV. Kapaklis, Physical Review B 98, 014435(R) (2018).\n28S. Zhang, I. Gilbert, C. Nisoli, G.-W. Chern, M. J. Erick-\nson, L. Obrien, C. Leighton, P. E. Lammert, V. H. Crespi,\nand P. Schi\u000ber, Nature 500, 553 (2013).\n29I. Gilbert, Y. Lao, I. Carrasquillo, L. OBrien, J. D. Watts,\nM. Manno, C. Leighton, A. Scholl, C. Nisoli, and P. Schif-\nfer, Nature Physics 12, 162 (2016).\n30R. Streubel, N. Kent, S. Dhuey, A. Scholl, S. Kevan,\nand P. Fischer, Nano Letters 18, 7428-7434 (2018),\n10.1021/acs.nanolett.8b01789.\n31D. Louis, D. Lacour, M. Hehn, V. Lomakin, T. Hauet, and\nF. Montaigne, Nature Materials 17, 1076 (2018).\n32J. Sklenar, Y. Lao, A. Albrecht, J. D. Watts, C. Nisoli, G.-\nW. Chern, and P. Schi\u000ber, Nature Physics 15, 191 (2019).\n33C. Behncke, C. F. Adol\u000b, S. Wintz, M. H anze, B. Schulte,\nM. Weigand, S. Finizio, J. Raabe, and G. Meier, Scienti\fc\nReports 8, 186 (2018).\n34A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen,\nF. Garcia-Sanchez, and B. Van Waeyenberge, AIP Ad-\nvances 4, 107133 (2014).\n35W. Bang, F. Montoncello, M. T. Ka\u000bash, A. Ho\u000bmann,\nJ. B. Ketterson, and M. B. Jung\reisch, Journal of Applied\nPhysics 126, 203902 (2019).\n36S. S. Kalarickal, P. Krivosik, M. Wu, C. E. Patton, M. L.\nSchneider, P. Kabos, T. J. Silva, and J. P. Nibarger, Jour-\nnal of Applied Physics 99, 093909 (2006).\n37T. Sebastian, K. Schultheiss, B. Obry, B. Hillebrands, and\nH. Schultheiss, Frontiers in Physics 3, 1589 (2015).\n38S. Neusser, B. Botters, M. Becherer, D. Schmitt-\nLandsiedel, and D. Grundler, Applied Physics Letters 93,\n122501 (2008), https://doi.org/10.1063/1.2988290.\n39J. Ding, D. Tripathy, and A. O. Adeyeye, Jour-\nnal of Applied Physics 109, 07D304 (2011),\nhttps://doi.org/10.1063/1.3535425.\n40S. Tacchi, M. Madami, G. Gubbiotti, G. Carlotti, A. O.\nAdeyeye, S. Neusser, B. Botters, and D. Grundler, IEEE\nTransactions on Magnetics 46, 172 (2010).\n41G. Gubbiotti, F. Montoncello, S. Tacchi, M. Madami,\nG. Carlotti, L. Giovannini, J. Ding, and A. O.\nAdeyeye, Applied Physics Letters 106, 262406 (2015),\nhttps://doi.org/10.1063/1.4923271.\n42Y. Li, G. Gubbiotti, F. Casoli, F. J. T. Gon\u0018 calves, S. A.\nMorley, M. C. Rosamond, E. H. Lin\feld, C. H. Marrows,\nS. McVitie, and R. L. Stamps, Journal of Physics D: Ap-\nplied Physics 50, 015003 (2016).\n43S. Mamica, X. Zhou, A. Adeyeye, M. Krawczyk, and\nG. Gubbiotti, Physical Review B 98, 054405 (2018).\n44V. S. Bhat, S. Watanabe, K. Baumgaertl, and\nD. Grundler, \\Direct observation of magnon modes in\nkagome arti\fcial spin ice with topological defects,\" (2019),\narXiv:1910.00874 [cond-mat.mes-hall]."    },    {        "title": "1009.4108v1.Electron_Spin_Resonance_of_the_ferromagnetic_Kondo_lattice_CeRuPO.pdf",        "content": "arXiv:1009.4108v1  [cond-mat.str-el]  21 Sep 2010Electron Spin Resonance of the ferromagnetic\nKondo lattice CeRuPO\nT F¨ orster, J Sichelschmidt, C Krellner, C Geibel, and F\nSteglich\nMax Planck Institute for Chemical Physics of Solids, D-01187 Dresd en, Germany\nAbstract. The spin dynamics of the ferromagnetic Kondo lattice CeRuPO is\ninvestigated by Electron Spin Resonance (ESR) at microwave frequ encies of 1, 9.4,\nand 34 GHz. The measured resonance can be ascribed to a rarely ob served bulk Ce3+\nresonance in a metallic Ce compound and can be followed below the ferr omagnetic\ntransition temperature TC= 14 K. At T > T Cthe interplay between the RKKY-\nexchange interaction and the crystal electric ﬁeld anisotropy det ermines the ESR\nparameters. Near TCthe spin relaxation rate is inﬂuenced by the critical ﬂuctuations\nof the order parameter.\nPACS numbers: 71.27.+a, 75.20.Hr, 76.30.-v\n1. Introduction\nUntil the discovery of an Electron Spin Resonance (ESR) signal belo w the Kondo\ntemperature TKin YbRh 2Si2[1] it was believed and experimentally well manifested\nthat there should be no ESR signal in heavy fermion systems. This wa s justiﬁed by\nthe strong electronic correlations which originate from the hybridiz ation of the 4 f/5f-\nelectrons and the conduction electrons [2]. Since 2003 only two othe r Kondo lattices\nhave been found which show an ESR Signal with local properties of th eir Kondo-ions\nnamely YbIr 2Si2(I-type) [3] and CeRuPO [4].\nIn CeRuPO a pronounced decrease of the electrical resistivity at t emperatures\nbelow 50 K indicates the onset of coherent Kondo scattering. Ferr omagnetic (FM) order\nappears at TC= 14 K and therefore the system is a rare case of a FM Kondo lattice. In\ncontrast to the Yb-compounds mentioned above, the Ruderman- Kittel-Kasuya-Yosida\n(RKKY) interaction is stronger than the Kondo interaction charac terized by a Kondo\ntemperature TK≈10 K [5]. It was a remarkable observation that in contrast to its\nantiferromagnetic homologue CeOsPO the ferromagnetic CeRuPO s hows a well-deﬁned\nsignal in the regime of coherent Kondo scattering [4]. This fact sugg ests that FM\ncorrelations are important for the observability of a Kondo-ion ESR , which was later on\nsupported by several theoretical investigations [2, 6, 7, 8].\nThispaperreportsdetailedESRinvestigationsofCeRuPOinorderto documentthe\nproperties of the rarely observed Ce3+spin resonance (we are aware of only two others,Electron Spin Resonance of the ferromagnetic Kondo lattice CeRuPO 2\nnamely CeP [9] and CeB 6[10]), and to show the eﬀect of ferromagnetic correlations and\nmagnetic order on the Kondo ion spin resonance.\nThe system reveals a unique magnetic anisotropy: Although CeRuPO is a collinear\nferromagnet with the magnetic moments aligned along the c-axis, the saturation\nmagnetization along this axis is smaller than in the basal plane. This beh aviour\noriginates from diﬀerent anisotropies of the crystal electric ﬁeld ( CEF) and the RKKY\nexchange interaction with respect to the components of the magn etic moment. The\nﬁrst one results in a ground state CEF doublet with a larger saturat ion moment in the\nbasal plane. The latter favours an alignment of the moments along t hec-axis and is\nresponsible for the FM transition [11].\n2. Experimental Details\nThe ESR experiments were carried out using a standard continuous wave spectrometer\ntogether with a He-ﬂow cryostat that allows us to vary the temper ature from 4 to\n300 K. In order to investigate the magnetic ﬁeld dependence we use d three frequencies\nν= 1,9.4, and34GHz(L-, X-, Q-band); fora g-factorof2thiscorresponds toresonance\nﬁelds of 36, 340, and 1200 mT. For the lowest frequency we used a s plit-ring resonator,\nwhich has a lower Q factor than the resonant cavities utilized at highe r frequencies.\nESR probes the absorbed power Pof a transversal magnetic microwave ﬁeld as a\nfunction of a static and external magnetic ﬁeld µ0H. To improve the signal-to-noise\nratio, we used a lock-in technique by modulating the static ﬁeld, which yields the\nderivative of the resonance signal dP/dH. The measured ESR spectra were ﬁtted with a\nmetallic Lorentzian function including the inﬂuence of the counter-r otating component\nof the linearly polarized microwave ﬁeld [12]. From the ﬁt we obtained th e resonance\nﬁeldHres(which determines the ESR g-factorg=hν/µBHres), the ESR intensity IESR,\nand the linewidth ∆ H(half-width at half maximum). In metals ∆ His a direct measure\nof the spin lattice relaxation time 1 /T1and its temperature and frequency dependence\nreveals the nature of the participating relaxation mechanisms.\nFor our measurements we used three CeRuPO single crystals (one f or each\nfrequency) which were grown from the same batch by a Sn-ﬂux met hod [11]. CeRuPO\ncrystallizes inthetetragonalZrCuSiAs-typestructure( P4/nmm)containingalternating\nlayers of OCe 4and RuP 4tetrahedra. The platelet-like single crystals had a surface area\nof about 3 mm2and a thickness of up to 0.08 mm with the crystallographic c-axis\nperpendicular to the platelet plane. Electron-microprobe, X-ray d iﬀraction, and a large\nresidual resistivity ratio ( ρ300K/ρ0= 30) indicated a high sample quality of single-phase\nCeRuPO [11].\n3. Experimental Results and Discussion\nFigure 1 shows the temperature evolution of the ESR spectra for t wo orientations of\nthe crystallographic c-axis: perpendicular ( H⊥c) and parallel ( H∝bardblc) to the quasistaticElectron Spin Resonance of the ferromagnetic Kondo lattice CeRuPO 3\n/s48/s44/s48 /s48/s44/s53 /s49/s44/s48/s72 /s99/s49/s53/s46/s48/s32/s75\n/s32/s32\n/s32/s97/s41\n/s72 /s124/s124 /s99\n/s98/s41/s53/s46/s48/s32/s75/s54/s46/s53/s32/s75/s56/s46/s53/s32/s75/s49/s48/s46/s53/s32/s75/s49/s50/s46/s48/s32/s75/s49/s51/s46/s53/s32/s75/s49/s56/s46/s48/s32/s75\n/s49/s54/s46/s53/s32/s75\n/s32/s32\n/s49/s51/s46/s48/s32/s75/s49/s51/s46/s53/s32/s75/s49/s52/s46/s48/s32/s75/s49/s53/s46/s48/s32/s75/s49/s54/s46/s53/s32/s75/s49/s55/s46/s53/s32/s75/s100/s80/s47/s100/s72 /s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s48/s72 /s32/s40/s84/s41/s50/s48/s46/s48/s32/s75\nFigure 1. Typical ESR spectra at 9.4 GHz of single crystalline CeRuPO at diﬀeren t\ntemperatures and ﬁeld orientations. Dashed lines represent ﬁts w ith a single metallic\nLorentzian. Arrows in a) indicate structures which occur below TC. The structure at\n0.33 T originates from the background.\nmagnetic ﬁeld µ0Hat 9.4 GHz. In the paramagnetic region the ESR signal could be\nobserved in a small temperature range up to ≈23 K, where it is in good agreement with\na single metallic Lorentzian shape for both orientations (dashed lines in ﬁgure 1). In the\nferromagnetically ordered region, the signal displays strongly anis otropic properties: for\nH⊥cadditional structures below the main line appear (see arrows in ﬁgur e 1a) which\nprobably can be related to ferromagnetic resonance modes. The r esonance ﬁeld of the\nmain line increases with lowering the temperature and the signal was v isible down to\nat least 4 K. For H∝bardblcthe ESR line rapidly broadens with decreasing temperature and\nstrongly shifts towards a zero resonance ﬁeld. Therefore, at H∝bardblc, the signal could be\ndetected down to 13 K for X-band and 9 K for Q-band. For the less s ensitive L-band\nsetup the signal could only be observed for H⊥cbetween 12 and 15.5 K. Below we only\ndiscuss the ESR parameters which were obtained by ﬁtting the spec tra with Lorentzian\nlines that describe linewidth and resonance ﬁeld within an error of 15% .\nThe bulk origin of the ESR signal is corroborated by its large signal-to -noise ratio\nand especially by the angle and the temperature dependences (see following sections).\nAlsotheESRintensityfollowsthemagneticsusceptibility χ, andweﬁndalinearrelationElectron Spin Resonance of the ferromagnetic Kondo lattice CeRuPO 4\nIESR∝(χ−χ0) withχ0\n⊥=0.09·10−6m3/mol and χ0\n/bardbl= 1.0·10−6m3/mol (not shown)\nsimilar to what was found in YbRh 2Si2[1]. A quantitative comparison of the ESR\nintensity with a CuSO 4standard conﬁrms that all Ce3+ions within the penetration\ndepth of the microwave contribute to the signal.\nIn our previous publication on polycrystalline CeRuPO we reported a w ell-deﬁned\nESR signal which splits into a low-ﬁeld (LF) and a high-ﬁeld (HF) compon ent near TC\n(note that in the polycrystalline samples TC= 15 K). Both signals showed a pronounced\ntemperature behaviour of their resonance ﬁelds which shift in oppo site directions. We\nsuspected that the components belong to diﬀerent orientations o f the powder grains\ncrystal axes to the resonance ﬁeld [4]. The single crystal data pr esented here conﬁrms\nthis assumption and we could identify the LF and HF component with po wder grains\noriented around a direction parallel to H∝bardblcandH⊥c.\nThestrongshiftoftheresonanceﬁeldbelowtheferromagneticor deringtemperature\nindicates the relevance of demagnetization and magnetic anisotrop y ﬁelds. These\neﬀects strongly depend on the orientation of the external ﬁeld wit h respect to the\nmagnetic- and shape-anisotropy of the sample. The magnetic easy axis coincides with\nthe crystallographic c-axis. Then, in the ferromagnetic phase and H∝bardblcthe resonance\ncondition reads:\nhν=gµB(Hres−NM(H,T)+HA(T)), (1)\nwhereM(H,T) is the magnetization of the sample and HAis the anisotropy\nﬁeld. In order to calculate demagnetizing eﬀects the magnetization M(H,T) was\nmeasured for H∝bardblcin a separate experiment using a commercial Quantum Design\nSQUID magnetometer. The samples are thin platelets, and therefo re, we assume a\ndemagnetizing factor of N= 1 for the magnetization normal to the platelet plane ( H∝bardblc)\nandofN= 0fortheperpendiculardirection( H⊥c). Itturnsoutthatinthetemperature\nregion we discuss here (12-14 K) NM(H,T) is smaller than 1% of the measured\nresonance ﬁeld and, thus, NM(H,T) can safely be neglected. Using equation 1 with\ng/bardbl= 1.18 and the 34 GHz data we ﬁnd for the anisotropy ﬁeld µ0HA≈1.4 T at 10 K\nandµ0HA≈0.65 T at 12 K. Therefore, the temperature dependence of the an isotropy\nﬁeld easily exceeds the estimated 20% temperature variation of the g-factor anisotropy:\ng/bardbl/g⊥= 0.46 at 20 K; g/bardbl/g⊥=M/bardbl/M⊥= 0.36 at 2 K, Ref.[11]. Amore detailed analysis\nofthebehaviour below TCasitwasdoneforFMthinﬁlms inRefs. [13] and[14]doesnot\nlead to reasonable results for CeRuPO. The huge linewidth and the me tallic lineshape\nprevents a suﬃcent accuracy for the determination of the reson ance ﬁeld. Therefore,\nmost of our conclusions are only valid above TC. Although the eﬀect of ferromagnetism\nregarding its anisotropy ﬁelds vanishes at TCcritical ﬂuctuations may still inﬂuence the\ntemperature dependence of the linewidth ∆ H[15].\n3.1. Anisotropy and temperature dependence of the g-factor\nFigure 2 presents the angle dependence of the eﬀective ESR g-fac tor at the diﬀerent\nfrequencies. The crystals were rotated by an angle Θ around an ax is in the basalElectron Spin Resonance of the ferromagnetic Kondo lattice CeRuPO 5\n/s48 /s57/s48 /s49/s56/s48 /s50/s55/s48 /s51/s54/s48/s49/s44/s48/s48/s49/s44/s50/s53/s49/s44/s53/s48/s49/s44/s55/s53/s50/s44/s48/s48/s50/s44/s50/s53\n/s49/s32/s71/s72/s122\n/s49/s52/s32/s75/s57/s46/s52/s32/s71/s72/s122\n/s49/s53/s46/s53/s32/s75/s51/s52/s32/s71/s72/s122\n/s49/s53/s46/s53/s32/s75\n/s32/s32\n/s72 /s124/s124 /s99\n/s72 /s99/s69/s83/s82/s32 /s103/s45/s102/s97/s99/s116/s111/s114\n/s32/s40/s100/s101/s103/s46/s41/s67/s101/s82/s117/s80/s79\nFigure 2. Anisotropy of the eﬀective g-factor at diﬀerent frequencies (1 G Hz 14 K /diamondsolid,\n9.4 GHz 15.5K /trianglesolid, 34 GHz 15.5 K /squaresolid). Θ denotes the angle between the ﬁeld Hand the\nc-axis. Solid lines represent ﬁts a with a uniaxial symmetry (see equat ion (2)). The\n1 GHz data are plotted at a temperature where g(Θ) is phase-shifted by 90◦compared\nto the behaviour above 15 K.\nplane. The data can nicely be ﬁtted with an uniaxial symmetry behavio ur (solid lines\nin ﬁgure 2):\ng(Θ) =/radicalBig\ng2\n/bardblcos2Θ+g2\n⊥sin2Θ. (2)\nThisisexpectedforatetragonalcrystalstructureandcouldfu rthermorebeconﬁrmedby\nrotating the crystal around the c-axis, keeping Θ = 90◦constant: within experimental\naccuracy all ESR parameters were found to be isotropic in this conﬁ guration.\nTheg-factor anisotropy reveals a remarkable temperature dependen ce, namely a\ncrossing of g/bardbl(T) andg⊥(T) resulting in a 90◦-phase-shift of g(Θ) at a temperature\nwhich depends on the magnetic ﬁeld. In ﬁgure 2 the phase shift can b e seen for the\n1 GHz data compared to the data at 9.4 and 34 GHz which are shown fo r temperatures\nwhere the phase shift has not yet occurred.\nWhether or not the observed resonance can be related to Ce3+magnetic moments\ncould be checked by comparing the absolute values of g/bardblandg⊥in the paramagnetic\nregion (see ﬁgure 3) with the g-values expected for Ce3+in a tetragonal CEF\nenvironment. There, the J=5\n2multiplet splits into three Kramers doublets with a\nsingle mixing coeﬃcient η[11]:\nΓ6:/vextendsingle/vextendsingle/vextendsingle/vextendsingle±1\n2/angbracketrightbigg\n, (3)\nΓ(1)\n7:η/vextendsingle/vextendsingle/vextendsingle/vextendsingle±5\n2/angbracketrightbigg\n+/radicalbig\n1−η2/vextendsingle/vextendsingle/vextendsingle/vextendsingle∓3\n2/angbracketrightbigg\n, (4)Electron Spin Resonance of the ferromagnetic Kondo lattice CeRuPO 6\nΓ(2)\n7:/radicalbig\n1−η2/vextendsingle/vextendsingle/vextendsingle/vextendsingle±5\n2/angbracketrightbigg\n−η/vextendsingle/vextendsingle/vextendsingle/vextendsingle∓3\n2/angbracketrightbigg\n. (5)\nAccording to the saturation magnetization and the 4 f-related speciﬁc heat, CeRuPO\nis likely to have a ground-state wave function with Γ 6symmetry and two excited CEF\nlevels at 70 and 320 K with a Γ 7symmetry [11].\nIf a Γ 7ground-state is assumed and the admixture of excited CEF-double ts\nis neglected a calculation of the expected g-factorsgth\n⊥,/bardbl[16] yields values that are\ninconsistent with the experimental data, regardless of which mixing coeﬃcient ηis\nused. In contrast, with gth\n⊥= 2.57 and gth\n/bardbl= 0.86 the Γ 6doublet (3) provides results\nwhich agree well with the experimental values as shown in ﬁgure 3: th e measured g-\nfactors for H⊥candH∝bardblcbecome ﬁeld and temperature independent above T= 18 K,\nmerging to g⊥= 2.58 and g/bardbl= 1.18. Therefore, the observed ESR line can be associated\nwith the resonance of a Γ 6ground-state doublet of Ce3+in CeRuPO. The remaining\ndiﬀerences between theoretical and experimental g-factors could originate from the\ncommonly observed g-shift of paramagnetic ions in a metallic environment [17].\nThe frequency dependence of g/bardblandg⊥displayed in ﬁgure 3 corresponds to ﬁeld-\ndependent internal magnetic ﬁelds which are determined by an inter play between the\nRKKY- and the CEF-interactions of the Ce3+magnetic moments. As mentioned above,\nthe RKKY interaction leads to an FM ordering and alignment of the 4 f-spins along\nthe c-axis [11]. Therefore, in the case H∝bardblc, the internal and external ﬁelds sum up to\nthe resonance ﬁeld, i.e. one needs a much smaller external ﬁeld to re ach the resonance\ncondition which corresponds to an increase of g/bardbl.\nIn the case of H⊥ctheg-factor continuously decreases with lowering the\ntemperature across TC. This decrease shows a frequency dependent slope that is largest\nfor the lowest frequency. This behaviour originates from an eﬀect ive FM coupling of\nthe Ce3+magnetic moments along the c-axis which leads to a reduction of the magnetic\nﬁeld along the basal plane. Therefore a larger external magnetic ﬁ eld is necessary to\nreach the resonance condition and a decrease of g⊥is observed. The fact that this eﬀect\nis largest at small frequencies (ﬁelds) can be related to the magnet ic ﬁeld dependence\nof the isothermal magnetization M(H) of CeRuPO (see inset ﬁgure 3a)): When the\nmagnetic ﬁeld is applied along the crystallographic c-axis (H∝bardblc), one observes a well\ndeﬁned hysteresis and a saturation with a magnetic moment of µc\nsat= 0.43µB. With\nthe magnetic ﬁeld applied perpendicular to the c-axis (H⊥c) no hysteresis is observed.\nInstead one ﬁnds a linear increase up to the critical ﬁeld µ0Hc1= 1 T and a saturated\nmagnetic moment of µab\nsat= 1.2µB[11]. For the ESR measured at 34 GHz, and H⊥c\nthe external ﬁeld is swept between 1 T and 1.6 T which exceeds the sa turation ﬁeld\nHc1. Then, almost all FM moments are aligned along the basal plane, and t he moment\nof the CEF ground state determines the g⊥factor. At frequencies where Hc1is larger\nthanHres(i.e. L- and X-band) the FM moments are not fully rotated towards t he basal\nplane which, thus, results in a reduced g⊥factor.\nA phenomenological approach to describe the temperature depen dence of the g-\nfactor in an anisotropic magnet has been realized by relating the g-factor to the bulk,Electron Spin Resonance of the ferromagnetic Kondo lattice CeRuPO 7\n/s48/s50/s52/s54/s56\n/s45/s50 /s48 /s50/s45/s49/s48/s49\n/s84\n/s67/s49/s32/s71/s72/s122\n/s97/s41\n/s67/s101/s82/s117/s80/s79/s51/s52/s32/s71/s72/s122\n/s32/s103\n/s124/s124/s61/s49/s46/s49/s56/s69/s83/s82/s32 /s103/s45/s102/s97/s99/s116/s111/s114\n/s103 /s61/s50/s46/s53/s56\n/s72 /s99/s72 /s124/s124 /s99\n/s57/s46/s52/s32/s71/s72/s122\n/s49/s48 /s49/s50 /s49/s52 /s49/s54 /s49/s56 /s50/s48/s48/s49/s50\n/s84\n/s67\n/s98/s41/s49/s32/s71/s72/s122/s32 /s32/s84 /s32/s40/s75/s41/s57/s46/s52/s32/s71/s72/s122/s51/s52/s32/s71/s72/s122/s72 /s99/s72\n/s99/s49\n/s72 /s99\n/s84 /s32/s61/s32/s50\n/s32/s75/s32/s77 /s32/s40\n/s66/s47/s67/s101/s41\n/s48/s72 /s32/s40/s84/s41\nFigure 3. Temperature dependence of the eﬀective ESR g-factors for a) H∝bardblcand b)\nH⊥cat 1/diamondsolid, 9.4/trianglesolidand 34 GHz /squaresolid. Dashed lines indicate an extrapolation towards the\n9.4 GHz data. Data points for 34 GHz at 15.5 K H∝bardblc/squareand for 1 GHz at 14 K H∝bardblc\n♦were extrapolated by ﬁtting equation (2) to the angle dependence of theg-factor,\nsee ﬁgure 2. Inset in a) shows the isothermal magnetization as fun ction of the applied\nmagnetic ﬁeld at 2 K for H⊥c◦andH∝bardblc◦, taken from Reference [11].\nstatic susceptibility [18]. This procedure has successfully been applie d, for instance, for\nthe uniaxial ferromagnet CrBr 3and indicates that the ESR in concentrated systems\nprobes a collective mode of the coupled spin system, in contrast to d ilute systems where\nthe resonance ﬁeld is determined entirely by single-ion properties. F or YbRh 2Si2and\nYbIr2Si2the data could be well described by the following relations being valid if t he\nmagnetization is proportional to the applied ﬁeld (“low-ﬁeld limit”) [19 , 20]:\ng⊥(T) =g0\n/bardbl/radicalBigg\nχ⊥(T)\nχ/bardbl(T)and (6)\ng/bardbl(T) =(g0\n⊥)2\ng0\n/bardblχ/bardbl(T)\nχ⊥(T). (7)Electron Spin Resonance of the ferromagnetic Kondo lattice CeRuPO 8\n/s49/s48 /s49/s50 /s49/s52 /s49/s54 /s49/s56 /s50/s48/s50/s52/s54/s56\n/s84\n/s67/s67/s101/s82/s117/s80/s79\n/s57/s46/s52/s32/s71/s72/s122\n/s72 /s99\n/s72 /s124/s124 /s99\n/s32/s32/s69/s83/s82/s32 /s103/s45/s102/s97/s99/s116/s111/s114\n/s84 /s32/s40/s75/s41/s72 /s99/s72 /s124/s124 /s99\nFigure 4. Temperature dependence of the eﬀective ESR g-factor for H⊥c/trianglesolidandH∝bardblc\n△at 9.4 GHz together with calculated g-factors using equations (6), (7) (dashed lines)\nand the susceptibility data measured at 0.1 T [11].\nHere,g0\n/bardbl,⊥belongtothemicroscopic g-tensor, and χ/bardbl,⊥denotesthesusceptibility for H∝bardblc\nandH⊥c. The application of these relations to the ESR of CeRuPO yields reaso nable\nresults for the 9.4 GHz measurements as can be seen by the dashed lines in ﬁgure 4. For\nthe descriptions of these data we used the susceptibility measured at a ﬁeld of 0.1 T [11]\nandg0\n⊥= 2.58,g0\n/bardbl= 1.18 which are the values measured for T >18 K (see ﬁgure 3).\nWe suspect that the deviations of the dashed lines from the data ma inly originate from\nthe fact that a ﬁxed ﬁeld was used for the susceptibility measureme nts, while µ0Hres\nchanges from 0.3 to 1 T for H⊥cand from 0.6 to 0.2 T for H∝bardblc.\n3.2. Temperature and Frequency behaviour of the ESR linewid th\nThe temperature and frequency behaviour of the linewidth of CeRu PO is plotted in\nﬁgure 5. ∆ H/bardbl(T) could only be measured with the 9.4 GHz setup because of the\nlower sensitivity and larger linewidth at the other frequencies. It is w orth noting the\nconsistency with the linewidth data of the polycrystalline samples (se e Fig. 2 in Ref.\n[4]): At temperatures above TCthe linewidth of the polycrystals is dominated by the\nH⊥csignal of the grains. Below TCthe strong shift of the resonance ﬁeld due the FM\ntransition separates the signals, and the linewidth is dominated by th eH∝bardblccomponent.\nIn the paramagnetic regime the general linewidth behaviour resemb les the common\nbehaviour of diluted4f-spins in a metallic environment [17]. There, a Korringa\nmechanism leads to a linear increase of the linewidth, which is then ofte n followed by an\nexponential growth indicating the contribution of excited CEF levels . However, for our\ncase of a high concentration of Ce3+ions in the presence of a Kondo eﬀect the linewidthElectron Spin Resonance of the ferromagnetic Kondo lattice CeRuPO 9\nis determined by a more complicated mechanism. For example many impo rtant features\nof the ESR results of YbRh 2Si2could successfully be explained by the relaxation of a\ncoupled spin mode of conduction electrons and Yb-4 fspins [7].\nIn systems with concentrated magnetic moments the dipolar intera ction can lead\nto a broadening of the linewidth. However, this eﬀect can be neglect ed for CeRuPO\nwhen considering a rough estimate for the high-temperature limit of the linewidth in\npresence of theexchange narrowing processes: ∆ H∞∝∝angbracketleftν2\nDD∝angbracketright\nνex. Here,∝angbracketleftν2\nDD∝angbracketrightisthe second\nmoment of the resonance-frequency distribution due to dipolar br oadening [21], and νex\ndenotestheexchange frequency [22]. Although ∝angbracketleftν2\nDD∝angbracketrightwithinclusionofnearest andnext-\nnearest neighbours results in linewidth values of the order of the ex perimental data, the\nexchange-narrowing process with νex≈/radicalbig\nzS(S+1)I/h≥100 GHz yields a reduction\nby at least two orders of magnitude. The exchange coupling Iwas calculated using the\nWeiss-molecular equation 3 kBΘW=IzS(S+1) (z: number of nearest and next-nearest\nneighbours, Θ W: Weiss temperatures taken from [11] for both ﬁeld orientations).\nIn the temperature region slightly above and below the magnetic ord ering the\nlinewidth data display remarkable dependencies on frequency and ﬁe ld orientation.\nForH⊥cat 9.4 GHz and 34 GHz no anomaly is found around the onset of magnet ic\nordering. This contrasts to the data at 1 GHz ( H⊥c) and at 9.4 GHz ( H∝bardblc) where\nthe linewidth strongly increases upon decreasing the temperature acrossTC. These\ndivergencies resemble the characteristics observed in various fer romagnets like Gd [23],\nNi [24], CrBr 3[25, 26] and CdCr 2Se4[27]. They are referred to as ”critical speeding-up”\n[27] of the spin-relaxation time because of a reduction of the excha nge narrowing of\nmagnetic dipole interactions when TCis approached from above. This eﬀect is present\nunder the condition Hres≪Hex·(a/ξ)5/2which contains the exchange ﬁeld Hex, the\nlattice constant aand the correlation length ξof the spin system [25].\nIn the case of CeRuPO this condition qualitatively may describe the pr esence or\nabsence of divergent linewidth behavior near TC. The ﬁeld dependence in ﬁgure 5b\nshows that Hex·(a/ξ)5/2is larger than the 1 GHz resonance ﬁeld ( ≈60 mT) but similar\nor smaller than the resonance ﬁelds for the data at 9.4 GHz ( ≈300 mT) and 34 GHz\n(≈1100 mT). The orientational dependence in the 9.4 GHz data indicate s that, when\ngoing from H∝bardblctoH⊥c, the critical ﬂuctuations of the spontaneous magnetization\nbecome strongly suppressed, i.e. Hex·(a/ξ)5/2gets reduced compared to Hres.\n4. Conclusion\nWe presented a detailed study of ESR on CeRuPO single crystals at th ree diﬀerent\nfrequency bands. We have shown that the ESR signal displays the lo cal properties of\nCe3+ions with a Γ 6CEF doublet ground-state in a metallic environment. Near TCthe\nESR is inﬂuenced by the critical ﬂuctuations of the spontaneous ma gnetization, and\nat higher temperatures the magneto-crystalline anisotropy of Ce RuPO dominates the\nresonance.\nThe ESR of CeRuPO shares several peculiarities with the ESR of Yb-b ased Kondo-Electron Spin Resonance of the ferromagnetic Kondo lattice CeRuPO 10\n/s49/s50 /s49/s52 /s49/s54 /s49/s56 /s50/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s50/s53/s48/s51/s48/s48/s52/s48/s48/s53/s48/s48/s54/s48/s48\n/s67/s101/s82/s117/s80/s79\n/s49/s32/s71/s72/s122/s57/s46/s52/s32/s71/s72/s122/s51/s52/s32/s71/s72/s122\n/s98/s41\n/s84 /s32/s40/s75/s41/s84\n/s67/s72 /s99\n/s32/s32/s57/s46/s52/s32/s71/s72/s122\n/s97/s41/s72 /s124/s124 /s99\n/s32/s48/s72 /s32/s40/s109/s84/s41/s84\n/s67\nFigure 5. Temperature dependence of the ESR linewidth for a) H∝bardblcand b)H⊥cat\n1/diamondsolid, 9.4/trianglesolidand 34 GHz /squaresolid.\nlattice systems YbRh 2Si2and YbIr 2Si2, respectively: For instance, below the Kondo\ntemperature all rare earth magnetic moments contribute to the r esonance signal and,\ndespite the presence of a dense4flattice, anisotropy and temperature dependence of\nthe resonance show a typical behaviour of diluted4fmoments in a metal.\nAcknowledgements\nWe acknowledge the Volkswagen foundation (I/84689) for ﬁnancia l support.\nReferences\n[1] Sichelschmidt J, Ivanshin V A, Ferstl J, Geibel C and Steglich F 200 3Phys. Rev. Lett. 91156401\n[2] W¨ olﬂe P and Abrahams E 2009 Phys. Rev. B 80235112\n[3] Sichelschmidt J, Wykhoﬀ J, Krug von Nidda H A, Fazlishanov I I, Hos sain Z, Krellner C, Geibel\nC and Steglich F 2007 J. Phys.: Condes. Matter 19016211\n[4] Krellner C, F¨ orster T, Jeevan H, Geibel C and Sichelschmidt J 200 8Phys. Rev. Lett. 100066401Electron Spin Resonance of the ferromagnetic Kondo lattice CeRuPO 11\n[5] Krellner C, Kini N S, Br¨ uning E M, Koch K, Rosner H, Nicklas M, Baen itz M and Geibel C 2007\nPhys. Rev. B 76104418\n[6] Schlottmann P 2009 Phys. Rev. B 79045104\n[7] Kochelaev B I, Belov S I, Skvortsova A M, Kutuzov A S, Sichelschm idt J, Wykhoﬀ J, Geibel C\nand Steglich F 2009 Eur. Phys. J. B 72485–489\n[8] Zvyagin A A, Kataev V and B¨ uchner B 2009 Phys. Rev. B 80024412\n[9] Huang C Y, K S and Cooper B R 1976 Proc. Int. Conf. Cryst. Field Eﬀects Met. Alloys -51–60\n[10] Demishev S V, Semeno A V, Bogach A V, Samarin N A, Ishchenko T V , Filipov V B, Shitsevalova\nN Y and Sluchanko N E 2009 Phys. Rev. B 80245106\n[11] Krellner C and Geibel C 2008 J. Cryst. Growth 3101875–1880\n[12] Wykhoﬀ J, Sichelschmidt J, Lapertot G, Knebel G, Flouquet J, F azlishanov I I, von Nidda H A K,\nKrellner C, Geibel C and Steglich F 2007 Science And Technology Of Advanced Materials 8\n389–392\n[13] AnisimovA N, Farle M, PoulopoulosP, PlatowW, BaberschkeK, Isb ergP, W¨ appling R, Niklasson\nA M N and Eriksson O 1999 Phys. Rev. Lett. 822390–2393\n[14] Farle M, Anisimov A N, K B, J L and H M 2000 Europhys. Lett. 49658–664\n[15] K¨ otzler J and Scheithe W 1975 Phys. Status Solidi B 69389–394\n[16] Abragam A and Bleaney B 1970 Electron Paramagnetic Resonance of Transition Ions (Oxford\nUniversity Press)\n[17] Barnes S E 1981 Adv. Phys. 30801–938\n[18] Huber D L and Seehra M S 1975 J. Phys. Chem. Solids 36723–725\n[19] Huber D L 2009 Journal of Physics: Condensed Matter 21322203\n[20] Gruner T, Wykhoﬀ J, Sichelschmidt J, Krellner C, Geibel C and Ste glich F 2010 Journal of\nPhysics: Condensed Matter 22135602\n[21] Van Vleck J H 1948 Phys. Rev. 741168–1183\n[22] Anderson P W and Weiss P R 1953 Rev. Modern Phys. 25269–276\n[23] Burgardt P and Seehra M S 1977 Phys. Rev. B 161802–1807\n[24] Sp¨ orel F and Biller E 1975 Solid State Commun. 17833 – 835\n[25] Seehra M S and Gupta R P 1974 Phys. Rev. B 9197–202\n[26] Scheithe W, K¨ otzler J and Radhakrishna P 1978 Phys. Lett. A 66419–421\n[27] K¨ otzler J and Philipsborn H V 1978 Phys. Rev. Lett. 40790–793"    },    {        "title": "1102.1805v2.Enhancement_of_the_response_of_non_uniform_resonance_modes_of_a_nanostructure_in_the_Picoprobe_microwave_current_injection_ferromagnetic_resonance.pdf",        "content": " 1 Enhancement of the response of non-uniform resonanc e modes of a \nnanostructure in the Picoprobe microwave-current in jection ferromagnetic \nresonance \n \nC.S.Chang 1, A.O.Adeyeye 2, M.Kostylev 1 and S. Samarin 1 \n \n1School of Physics, University of Western Australia,  Australia ; \n 2Department of Electrical and Computer Engineering, National University of Singapore, \nSingapore  \n \nAbstract: The nonWuniform standing spinWwave modes in thin magnetic films and \nnanostructures provide important information about surfaces and buried interfaces. \nVery often they are lacking in the recorded ferroma gnetic resonance spectra for \nsymmetry reasons. In this work we experimentally de monstrate that by direct injection \nof microwave currents into an array of Permalloy na nostripes using a microscopic \nmicrowave coaxial to coplanar adaptor one can effic iently excite nonWuniform standing \nspin wave modes with odd symmetry. The proposed met hod is quick and allows easy \nspatial mapping of magnetic properties with the res olution down to 100 microns. We \nhave validated this method using an example from a periodical array of nanostripes. \nThe results from direct current injection are compa red to that of microstripWbased FMR \nmeasurements. \n \n \n Standing spin wave modes (SSWMs) are microwave mag netic excitations in confined \ngeometries. The wavelengths of SSWMs are determined  by the size of the sample along the \nconfinement direction and pinning at the surfaces a nd interfaces. It is well known that the \nhomogeneous microwave magnetic field typically used  for ferromagnetic resonance (FMR) cavity \nexperiments does not allow SSWM observation unless pinning [1] of magnetization is present at the \nsample surfaces.  \n The standing wave modes are affected by the inhomo geneous exchange interaction, and thus \ncarry important information about surfaces and buri ed interfaces [1W4], and about the value of \nexchange constant for the material, so the possibil ity of observation of these modes using such a \nsimple tool as FMR is very exciting. Recently, it w as shown that in the microstrip based broadband \nFMR experiment [5W7], it is possible to considerabl y increase the FMR response of the higherWorder \nSSWM modes in conducting ferromagnetic films, due t o injection of eddy currents into the \nsamples. However, in the microstrip FMR the efficie ncy of standingWwave mode excitation is \nstrongly dependent on the driving frequency; in Per malloy (Py) films at the frequencies below 6 \nGHz, the higherWorder standing wave modes show very  small responses, if any. Furthermore, \nmacroscopicWsize coplanar or microstrip transducers  (about 1mm in width, several mm long \nmicrostrip lines) are typically required in order t o excite and observe SSWM. A way to get around \nthis deficiency has recently been shown by Khivints ev et al [18], where the magnetic sample is \nembedded into a microscopic microwave coplanar tran smission line. It has been demonstrated that \nif a magnetic sample is in the form of a stripe of a microscopic  crossWsection is sandwiched between \ntwo highly conducting (copper) layers and the sandw ich forms the signal line of the coplanar \nwaveguide, efficient absorption by the first antiWs ymmetric standing spin wave mode is seen in the  2 transmission (S21) characteristic of the coplanar l ine. The microwave currents flowing in the \nCopper layers create an Oersted field which is anti Wsymmetric across the thickness of the magnetic \nlayer. This forms the necessary conditions for exci tation of this mode.  \n In this letter,  we demonstrate that efficient exc itation of an exchange standing spinWwave \nmode can be achieved in a much simpler way, without  embedding the sample in a characterization \ngear. This gives more flexibility and can be used f or determining the quality of nanomaterials. \nFurthermore, in contrast to Ref.[8], we demonstrate  this effect in patterned nano structured material. \nThe method is based on injection of microwave curre nts directly  into a Permalloy nanostructure \nusing the commercially available Picoprobe®; a micr oscopic microwave coaxial to coplanar \nadaptor. The method is quick and allows easy spatia l mapping of magnetic dynamics with the \nresolution down to 100 microns, which is given by t he minimum lateral size of the tip of the \npicoprobe. Both responses of the fundamental (quasi Wuniform) dipolar modes and of nonWuniform \nexchange modes are seen in this arrangement.  \n The objects of our study are arrays of magnetic na nostripes (MNS). These nanostructures are \npromising for magnonic [9] and magnetoWplasmonic ap plications [10]. In the latter case, covering \nmagnetic material with a thin gold (Au) layer, thus  forming an Au/Py interface, may enhance \nstructure performance [11]. A number of periodic ar rays of parallel Permalloy (Py) nanostripes \nwhich are 100 nm thick, 264 nm wide, 4 mm long and with different interWstripe spacings have been \nfabricated using deep ultraWviolet lithography [12] . SingleWlayer Py samples and samples having a \n10nmWthick Au capping layer have been prepared on S ilicon substrates. A number of reference \ncontinuous films with the same thickness and compos ition have also been fabricated in the same \nprocess. All MNS and films demonstrate similar beha vior, therefore in the following we concentrate \non one singleWlayer MNS sample with an interWstripe  spacing of 150 nm and the respective reference \nfilm.  \n The tip of a picoprobe represents a set of three n eedleWlike tungsten microwave contacts \naligned in a row (see Fig. 1(a)). The central conta ct is connected to the central conductor of the \nfeeding coaxial line and the outer ones are grounde d. The distance from the central contact to the \nexternal ones is called “pitch”. Below we demonstra te results obtained with a picoprobe having a \npitch of 200micron. We saw similar behavior with a picoprobe having the smallest commercially \navailable pitch: 50micron.  \n To drive magnetization precession and to register FMR absorption, contacts of the picoprobe \nare placed on top of the nanostructure or the film such that a microwave current from the contacts is \ndirectly injected into Py. In the case of MNS the s ample is oriented in such a way that the line in th e \nzWdirection connecting the tips of all three contact s of the picoprobe (“tip line”) is parallel to the \napplied static field and to the nanostripes. Thus, a microwave conduction current can flow between \nthe picoprobe contacts. In the case of the continuo us film, care is only taken that the applied field is \noriented along the tip line  such that the field is parallel to the microwave cu rrent and thus \nperpendicular to the microwave magnetic field of th e current. \n The tip of the picoprobe is lowered using a 3D tra nslation stage. We monitor the approach of \nthe picoprobe’s tip to the film surface with a digi tal microscope. Electrical contact is monitored \nusing a dc Ohmmeter. In full electrical contact of the picoprobe with the material, the measured dc \npicoprobe resistance is typically around 6 Ω and 13 0 Ω for the contacts with the continuous film \nand MNS respectively. \n We measure the microwave signal reflected from the  picoprobe as a function of the applied \nmagnetic field Ha for given microwave frequencies. Similar to [13], we modulate the magnetic field \napplied to the sample using two small coils attache d to the poles of an electromagnet. Modulation  3 frequency is 220Hz and the RMS magnetic field produ ced by the coils is 9.5 Oe. The microwave \npower in the frequency range 2W18 GHz is applied to  the picoprobe from a microwave generator \nthrough a circulator. Signal reflected from the pic oprobe is rectified using a microwave detector and \ndetected using a digital lockWin amplifier referenc ed by the same 220 Hz signal. The signal we \nobserve this way represents the first derivative of  the resonance line (Fig. 2). \n In a similar manner, microwave absorption in a mic rostrip FMR is detected. We use a section \nof a microstrip line 0.3 mm wide and 5 mm long (Fig .1 (b)). The sample is placed on top of the \nmicrostrip. MNS are oriented along the microstrip a nd along the applied field (axis z). FMR \nabsorption is detected in the same way as in the pi coprobe experiment, with the exception that a \ncirculator was not used; the measured signal is tha t which is transmitted through the microstrip line.   \n Fig. 2(a) displays the results we obtained on the nanostripe sample. One observes two wellW\nresolved modes, and one partiallyWresolved mode whi ch is visible as an additional small peak on the \nlowerWfield shoulder of the lowerWfield wellWresolv ed mode. The amplitudes of wellWresolved higher \nand lower field modes swap between the picoprobe an d the microstrip experiments which is the \nmain finding in this work. \n In order to explain this effect, one first has to identify these modes. This is easily done using \nthe theory from [14,15]. It states that the resonan ce frequencies of the structured material should \nobey the approximate dispersion relation for spin w aves valid for continuous films. All peculiarities \nof confinement due to nanostructuring is hidden in the coefficients of this equation. We write down \nthis equation in the form \n \n( )( )2( / ) 4 a ex d a ex d H H M H H H H ω γ π = + + − + + , (1) \n \nwhere  Ha is the applied field, Hex  is the effective exchange field, Hd is the effective dynamic dipole \nfield, M is the saturation magnetization of the material, ω  is the spin wave angular frequency, and \nγ/2π =2.82 MHz/Oe is the gyromagnetic coefficient. We f itted the experimental data for ω vs Ha \n(Fig. 3) and extracted the values of M, Hex , and Hd for both modes. For the higher field mode, we \nobtained Hex  =369 Oe and Hd = 0 Oe while for the lower field mode we get  H ex  = 409 Oe and Hd = \n1418 Oe.  \n Based on the large value of Hd, the lower field mode is identified as the fundame ntal (dipole) \nmode of the array. This identification is confirmed  by a numerical simulation using theory from \n[16]. The mode’s resonant field is strongly shifted  down field due to confinement effects resulting \nin strong effective magnetization pinning at the st ripe edges [15] and a large dynamic \ndemagnetizing (dipole) field. Similarly, from the v anishing Hd, the higher field mode is identified as \nthe first exchange mode for MNS. This identificatio n is also confirmed by the numerical simulation. \nThe simulation shows that this mode has a simple (q uasiWuniform) distribution of dynamic \nmagnetization in the array plane but an antiWsymmet ric distribution across the stripe thickness. It \nrepresents the counterpart of the 1 st  exchange SSWM for the continuous film. The dipole field is \nsmall for this mode because of its antiWsymmetric c haracter [17]. Thus the main contribution to the \nmode frequency originates from the exchange energy which depends mainly on the smallest \ndimension of the structure; in our case, on the thi ckness of the nanostripes. Since the MNS \nthickness is the same as the thickness of the refer ence continuous film, one may expect that the \nresonant field for this mode should be close to the  resonant field for the 1 st  SSWM of the film. \n  Fig. 2 (b) demonstrates the results obtained on t he reference continuous sample at 14 \nGHz. One sees that the microstrip data contain one main response which is easily identified as the  4 fundamental (uniform) FMR mode, and a small – barel y observable – mode downWfield from the \nfundamental mode, which has the frequency close to the frequency of the upperWfield mode for \nMNS. The smallWamplitude mode is identified as the first (antiWsymmetric) SSWM. The extremely \nsmall higherWorder response, also when the film is flipped such that film substrate faces the \nmicrostrip transducer [7] (not shown), demonstrates  that the film has nearly perfect unpinned \nsurface spins. The same fact is also confirmed by m easurement with a cavity at 9.45 GHz. The \ngradual enhancement of the absorption amplitude for  first SSWM mode’s at larger frequencies is \nconsistent with the increase in the effect of the e ddy current injection at higher frequencies in the \nmicrostrip FMR [5].  \n This measurement taken on the reference film confi rms that the higher field resonance peak \nfor MNS is the 1 st  exchange mode. Thus, the effect we see in Fig. 2(a ) is enhancement of the signal \nof the first SSWM in the picoprobe experiment with respect to the microstrip one. Interestingly, we \ndo not observe a noticeable effect of this type for  the continuous film. The picoprobe data (Fig.2b) \nshow a small peak at the lower field shoulder of th e fundamental mode consistent with the first \nSSWM. The ratio of the amplitude of the first SSWM mode to the fundamental mode is not \nsignificantly increased. Furthermore, the overall r esonance amplitude obtained using the picoprobe \non the continuous film is about twenty times smalle r compared to the stripline method (Fig. 2b). \n We see that in the confined MNS geometry the direc t injection of the microwave currents \nenhances excitation of the antiWsymmetric thickness  modes with respect to the microstrip FMR, but \nno enhancement is seen for the continuous film. Bel ow we suggest an explanation for this behavior. \n In the microstrip based FMR, magnetization precess ion is excited by the inWplane component \nof the microwave magnetic field which is along x in Fig. 1. This fact is evidenced by rotating the \nmicrostrip by 90 degree with respect to the applied  field, resulting in a significantly diminished \nFMR signal. In the picoprobeWbased experiment, the microwave voltage applied between the tips of \nthe picroprobe induces a microwave current between the tips (along z). The Oersted field (along y) \nof this current drives magnetization precession. If  the current density were uniform across the \nsample thickness (along y), the Oersted field would be an antiWsymmetric fun ction of y with a zero \nat half a sample thickness. Obviously, the current density is not uniform, since for the reference fil m \none observes efficient excitation of the uniform fu ndamental mode and no enhancement of the 1 st  \nantiWsymmetric standing wave mode with respect to t he microstrip FMR.  \n We may explain this similarity to the microstrip F MR result by a similar distribution of the \nmicrowave magnetic field inside the sample. In the microstrip FMR the microwave magnetic skin \neffect leads to the nonWuniformity of the microwave  magnetic field along y [5]. The field is more \nlocalized at the film surface facing the microstrip . The electric  field of the picoprobe is also applied \nto one film surface only. Therefore, one can expect  a similar microwave screening and localization \nof the microwave current and field near the film su rface facing the picoprobe. This asymmetric field \nis able to drive both modes. \n In the MNS geometry the screening effect is weaken ed due to the lateral confinement. One \nmay expect penetration of the microwave electric fi eld inside the stripes through the lateral surfaces  \nof the stripes. For this reason the current density  across the stripe crossWsection is more uniform. \nThis promotes excitation of the antiWsymmetric exch ange mode. However, the fact that the \namplitude of the fundamental mode is still large im plies that the screening effect is not completely \nsuppressed by the presence of the gaps between the stripes. Consequently, the amplitude of the \nmicrowave magnetic field is still larger at the str ucture surface facing the picoprobe than at the one  \nfacing the structure substrate. \n We confirmed this observation by measuring MNS wit h the microstrip setup in the  5 configuration when the microstrip is along the x but the applied field is still along z (Fig. 1c). In this \ngeometry the inWplane component of the microwave ma gnetic field does not contribute to excitation \nof magnetization precession. On the other hand, the  electric field of the microstrip now has a \ncomponent along the stripes, thus one may expect a behavior similar to one seen in the picoprobe \nexperiment. Indeed, in this geometry we also see an  increase in the amplitude of the 1 st  SSWM (Fig. \n2b). However, the increase is not as significant as  in the picoprobe experiment. \n In conclusion, we have demonstrated that by direct  injection of microwave currents using a \nmicroscopic microwave coaxial to coplanar adaptor, one can efficiently excite nonWuniform \nstanding spinWwave modes in magnetic nanostructures  with geometries in which continuity of \nconduction currents is ensured. The proposed method  is quick and allows easy spatial mapping of \nmagnetic properties with the resolution down to 100  microns, which is limited by the minimum \nlateral size of commercially available picoprobes. This suggests applications in express monitoring \nand control of spatial homogeneity of nanostructure d plane magnetic materials.  \n Acknowledgment: Financial support from Australian Research Council and the AustralianW\nIndian Strategic Research fund is acknowledged. \n \nReferences:  \n[1] C. Kittel, Phys. Rev . 110 , 1295 (1958) \n[2] W. Stoecklein, S. S. P. Parkin, and J. C. Scott , Phys.Rev. B  38 , 6847 (1988). \n[3] R. D. McMichael, M. D. Stiles, P. J. Chen, and W. F.Egelhoff, Phys. Rev. B  58 , 8605 (1998). \n[4] R. Magaraggia, M. Kostylev, K. Kennewell, R. L.  Stamps, M. Ali, D. Greig, B. Hickey, and C. \nH. Marrows “Exchange anisotropy pinning of a standi ng spin wave mode”, accepted for publication \nin Phys. Rev. B . \n[5] M. P. Kostylev, J. Appl. Phys . 106 , 043903 (2009). \n[6] . K. J. Kennewell et al., J. Appl. Phys. , 108 , 073917 (2010). \n[7] M. Kostylev et al., J. Appl. Phys . 108 , 103914 (2010). \n[8] Y .V .Khivintsev et al., J.Appl.Phys . 108 , 023907 (2010).  \n[9] V V Kruglyak et al., J. Phys. D: Appl. Phys . 43  264001 (2010). \n[10] A. A. Grunin et al., Appl.Phys.Lett. , 97 , 261908 (2010). \n[11] T.V . Murzina et al., Thin Solid Films  517 , 5918 (2009)  \n[12] A.O. Adeyeye and N.Singh, J.Phys.D:Appl. Phys . 41  153001 (2008). \n[13] M. Belmeguenai et al., Phys. Rev. B  79 , 024419 (2009). \n[14] K. Yu. Guslienko et al., Phys. Rev. B  66 132402 (2002). \n[15] K. Yu. Guslienko and A. N. Slavin, Phys. Rev. B  72 014463 (2002). \n[16] S. Tacchi et al., Phys. Rev. B  82 , 184408 (2010). \n[17] M. P. Kostylev et al., J. Mag. Mag. Mat. , 278 , 397 (2004). \n \nFigure captions: \n \nFig. 1 (Color online) Geometries of experiments. a. ): PicoprobeWbased FMR. b.):Microstrip FMR \nwith the stripline is parallel to the applied field . c.): Microstrip FMR with the stripline perpendicu lar \nto the applied field. Note that the nanostripe arra y is always parallel to the applied field in all ca ses.  \n \nFig. 2. (Color online) FMR traces taken at 14 GHz.;  (a): nanostripe array. (b): continuous film. \nSolid line: picoprobe FMR. Dashed line: stripline F MR with the microstrip parallel to the applied \nfield and the nanostripes. DashWdotted line: stripl ine FMR with the microstrip perpendicular to the  6 applied field and the nanostripe array. The vertica l dashed line shows the position of the 1 st  SSWM \nfor the reference film.  \n \nFig. 3. (Color online) Frequency vs. applied field dependencies for the continuous film (circles: \nfundamental mode, squares: 1 st  SSWM mode) and the nanostripes (diamonds: 1 st  SSWM mode, \ntriangles: fundamental dipole mode). The lines are respective fits with Eq.(1).  \n \n \n \n \nFig. 1 \n \n \n \n \nFig. 2 \n \n  7 \n \n \nFig. 3 "    },    {        "title": "1809.02272v1.Nuclear_spin_pumping_by_pulling_effect.pdf",        "content": "Nuclear spin pumping by pulling e\u000bect\nY. Ohnuma1;2, S. Maekawa2;1, and M. Matsuo1;2\n1Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing, 100190, China.\n2RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan.\n(Dated: September 10, 2018)\nThe nuclear-to-electron spin angular momentum conversion via hyper\fne coupling in a normal\nmetal (NM)/ferromagnet (FM) bilayer system is theoretically investigated by using the nonequi-\nlibrium Green's function method. The spin current generated by the nuclear magnetic resonance\n(NMR) is found to be enhanced by the pulling e\u000bect in the FM when the temperature is lower\nthan NMR resonance frequency. In a Co/Pt bilayer system, we show that the spin current by NMR\nbecomes larger than that of the ferromagnetic resonance (FMR).\nPACS numbers: 72.25.-b, 71.70.Ej, 47.61.Fg\nIntroduction.| Spin current, a \row of electron spins,\nis a key concept in the \feld of spintronics [1]. Gen-\neration of spin current has been demonstrated by us-\ning angular momentum conversion between spin and\nvarious angular momenta in condensed matter such as\nmagnetization[2], photons[3, 4], the orbital motion of\nelectrons, and mechanical angular momentum carried by\nmoving materials[5{7]. In this context, a remaining an-\ngular momentum in condensed matter is nuclear spins.\nBecause a nuclear spin couples to an electron spin via\nthe hyper\fne coupling, the nuclear spin can excite the\nnonequilibrium electron spin dynamics, and then, gener-\nate a spin current in principle. However, the interconver-\nsion of nuclear spin and electron spin has not been ex-\nploited. One of the reasons for this situation is that the\nhyper\fne coupling between electron and nuclear spins is\nrather small compared to the couplings among electron\nspins. In addition, the time scale of nuclear spin is much\nslower than that of electron spin.\nIn order to overcome this di\u000eculty, we consider the am-\npli\fcation of the spin current using the pulling e\u000bect [8].\nAs noted above, the modulation of electron spins by the\nmotion of the nuclear spins is negligible due to the mis-\nmatch of their resonance frequencies. In ferromagnets\nwith a large density of nuclei at low temperature, how-\never, the dynamics of electron spins is modulated by the\nnuclear spins because the coherent motion of high density\nof the nuclear spins is induced. At the nuclear magnetic\nresonance (NMR) condition, the electron spins adiabat-\nically follow the nuclear spins. This e\u000bect is called the\npulling e\u000bect, and in this case, the spin angular momen-\ntum of the nuclei transfers to the electrons with high\ne\u000eciency since the nuclear and electron spins behave as\na coupled system. Hence, the pulling e\u000bect is expected to\namplify the spin current generated by the nuclear spins.\nIn this article, we theoretically investigate the spin-\ncurrent generation due to interconversion of nuclear spins\ninto electron spins via the hyper\fne coupling in a nor-\nmal metal (NM)/ferromagnet (FM) bilayer system. We\nformulate a spin transport theory driven by the nuclear\nspin dynamics in FM using the nonequilibrium Green's\nhNhA\nAhf\nµNµB\nRF hAhAAAAAAFM or FI NM\nFIG. 1. (Color online) Schematic illustration of spin pump-\ning driven by NMR via pulling e\u000bect. Here, \u0016B,hAand\nhNare the Bohr magneton, anisotropic \feld and hyper\fne\n\feld acting on electron spins, respectively, and Ahfand\u0016N\nare coupling constant of hyper\fne interaction and the nuclear\nmagnetic moment, respectively.\nfunction method. It is shown that the spin current gen-\nerated at the interface is enhanced by the pulling e\u000bect\nand takes the maximum value when the temperature is\nlower than NMR frequency. In a Co/Pt bilayer system,\nit is shown that the spin current generated by NMR at\nlow temperature is larger than that of the ferromagnetic\nresonance (FMR). Our theory provides a new method of\ngenerating the spin current using NMR (MHz frequency\nrange) larger than that generated by FMR (GHz fre-\nquency range).arXiv:1809.02272v1  [cond-mat.mes-hall]  7 Sep 20182\nModel.| We consider the spin transport in a bilayer\nsystem, where a normal metal (NM) and a ferromagnet\n(FM) are coupled to each other through the s-dexchange\nat the interface:\nHint=JsdX\ni\u001bi\u0001Si; (1)\nwhere\u001biandSiare conduction spin in NM and localized\nelectron spin in FM on the i-th site at the interface, and\nJsdis the exchange coupling. In addition, localized spins\nin FM are coupled to nuclear spins via hyper\fne coupling:\nHIS=AhfX\njIj\u0001Sj; (2)\nwhere Ijis nuclear spin on the i-th site and Ahfis hy-\nper\fne coupling constant.\nThe spin current generated at the interface is given\nby the rate of change of conduction electron spin in the\nNM,IS:=h^ISi=~P\nih@t\u001bz\nii, where ^IS:=~P\ni@t\u001bz\ni\nis a spin current operator and h\u0001\u0001\u0001i := Tr[^\u001a\u0001\u0001\u0001] denotes\nthe statistical average with the density matrix ^ \u001a. By\nperforming the second-order perturbation with respect\nto the interfacial exchange coupling, the generated spin\ncurrent is given by:\nIS=J2\nsdNint\n~2ReZ\nqk![\u001fR\nqr;!tG<\nkr0;!t+\u001f<\nqr;!tGA\nkr0;!t];(3)\nwhereNintis the number of sites at the interface and\nthe random average is taken over the impurity posi-\ntions at the interface. The lesser (retarded) Green's\nfunction for conduction electron spin, \u001f<(R)\nqr;!t, is de-\n\fned as\u001f<(R)\nqr;!t :=R\nexp[ik\u0001\u000er\u0000i!\u000et]\u001f<(R)(r+\n\u000ert+\u000et;r\u0000\u000ert\u0000\u000et),\u001f<(r1t1;r2t2) =\u0000ih\u001b\u0000\nr2t2\u001b+\nr1t1i,\n\u001fR(r1t1;r2t2) =\u0000i\u0012(t1\u0000t2)h[\u001b+\nr1t1;\u001b\u0000\nr2t2]i, and\u0012(t) is\nthe step function. The lesser (advanced) Green's func-\ntion for localized spin, G<(A)\nkr0;!t, is de\fned by G<(A)\nkr0;!t:=R\nexp[ik\u0001\u000er\u0000i!\u000et]G<(A)(r+\u000ert+\u000et;r\u0000\u000ert\u0000\u000et),\nG<(r1t1;r2t2) =\u0000ihS\u0000\nr2t2S+\nr1t1i, andGA(r1t1;r2t2) =\ni\u0012(t2\u0000t1)h[S+\nr1t1;S\u0000\nr2t2]i.\u001fR\nq!is given by [9] \u001fR\nq!=\n\u001fN\u001csf(1 +\u00152\nNq2+i!\u001csf)\u00001, with\u001fN,\u001csf, and\u0015Nbeing\nthe paramagnetic susceptibility, the spin-\rip relaxation\ntime, and the spin-di\u000busion length in NM, respectively.\nLet us consider the situation, where conduction elec-\ntron spins in the NM are in local thermal equilibrium\nwhereas localized spins in the FM are excited by nuclear\nspins via the hyper\fne coupling, describing by \u000eG<. The\nspin current is reduced to\nIS=J2\nsdNint\n~2Z\nqk!Im\u001fR\nq!Im\u000eG<\nk!: (4)\nSpin pumping by pulling e\u000bect.| From now on, we\ncalculate the lesser function of localized electron spin ex-\ncited by the pulling e\u000bect[8]. A coupled system of local-ized electron and nuclear spins is modeled by the follow-\ning Hamiltonian: HFM=HS+HI+Hac+HIS, where\nHS=JX\nhi;jiSi\u0001Sj+~\reh0X\njSz\nj+D\n2X\nj(Sz\nj)2;(5)\nHI=\u0000~\rNh0X\njIz\nj (6)\nHac=\u0000~\rNhacX\nj(Ix\njcos\u0017t+Iy\njsin\u0017t); (7)\nwhereJis the exchange coupling withP\nhi;jibeing the\nsummation over nearest-neighbor sites, \reis the electron\ngyromagnetic ratio, Dis magnetic anisotropy constant,\n\rNis the nuclear gyromagnetic ratio, h0is a DC exter-\nnal magnetic \feld, and hacand\u0017are the amplitude and\nfrequency of the AC magnetic \feld, respectively.\nThe dynamics of the local electron and nuclear spins\nare given by the Landau-Lifshitz-Gilbert (LLG) equation\nand the Bloch equation:\n_S=\re(S\u0002he) +\u000b\nS0S\u0002_S; (8)\n_Ix;y=\rN(I\u0002hI)x;y\u0000Ix;y\nT2;_Iz=\rN(I\u0002hI)z\u0000Iz\n0\u0000Iz\nT1;(9)\nwhere\u000bis the Gilbert damping constant of the FI, he\nandhIare the magnetic \felds acting on localized spin\nand nuclear spins, and T1andT2are the longitudinal and\ntransverse relaxation times of nuclear spins, respectively.\nThe magnetic \feld acting on the j-site localized spin is\ncalculated by ha\ne;j=@(HI+HS)\n~\re@Sa\njwitha=x;y;z , and thus,\nhx;y\ne;j=Ahf\n~\reIx;y\nj;hz\ne;j=h0+hA+Ahf\n~\reIz\nj;(10)\nwherehA=DhSziis the anisotropic \feld. Here, we\nintroduce the thermal and site averaged z-component of\nelectron spinshSzigiven byhSzi:= (Ne)\u00001PNe\njhSz\nji\nwithNebeing the total number of sites of electron spins.\nBecause the localized spin dynamics is much faster\nthan the nuclear spin dynamics, j\rN=\rej\u001c1, the local-\nized spins adiabatically follow the nuclear spins in FM.\nIn this case, the localized spin can be considered to be\nstatic: _S\u00190. Then, the transverse component of local-\nized magnetic moment is related to the longitudinal one\nasSx;y= (hx;y\ne=hz\ne)Sz, and we obtain\nS\u0006\nj=hN\nh0+hA+hNhIzi=hSziI\u0006\nj: (11)\nHere, we replace approximately the jdependent z com-\nponent of the nuclear spins Iz\njby the thermal and site\naveraged values as hIzigiven byhIzi:= (NI)\u00001PNI\njhIz\nji\nwithNIbeing the total number of sites of nuclear spins.\nIn the above equation S\u0006\njandI\u0006\njare given by S\u0006\nj=\nSx\nj\u0006Sy\njandI\u0006\nj=Ix\nj\u0006Iy\nj, respectively, and the hyper-\n\fne \feldhNis de\fned as hN:=AhfhSi=(~\re)3\nSimilarly, the magnetic \feld hIis calculated by ha\nI;j=\n@(HI+HIS)\n~\rN@Ia\nj:\nh\u0006\nj(t) =hace\u0006i\u0017t\u0000Ahf\n~\rNS\u0006\nj;hz\nj=h0\u0000Ahf\n~\rNSz\nj;(12)\nwhereh\u0006\nj=hx\nj\u0006hy\nj. Using these relations, the Bloch\nequation can be rewritten as\nd\ndtI\u0006\nj=\u0006i(~\u0017NI\u0006\nj+\rNhachIzie\u0006i\u0017t)\u0000I\u0006\nj\nT2;(13)\nwhere ~\u0017Nis the modi\fed NMR frequency given by ~ \u0017N=\n\rNh0+\u0017N(1 +\u0018hIzi=hSzi) with\u0017Nand\u0018being the\nbare NMR frequency and enhancement factor de\fned as\n\u0017N:=AhfhSzi=~and\u0018:=S\u0006\nj=I\u0006\nj, respectively. Insert-\ningI\u0006(t) =R1\n\u00001d\u0017=2\u0019I\u0006\n\u0017ei\u0017tinto Eq. (13), we have\nI\u0006\n\u0017=\u0007hIzi\rNhace\u0006i\u0017t\n\u0017\u0006~\u0017N+iT\u00001\n2: (14)\nUsing Eqs. (11) and (14), the lesser Green's function \u000eG<\nis given by\n\u000eG<\nk!=\u0000i\f\f\f\u0018\rNhachIzi\n\u0017+ ~\u0017N+iT\u00001\n2\f\f\f2\n\u000e(!\u0000\u0017): (15)\nBy inserting this equation into Eq. (4), we obtain the\nspin current generated by nuclear spin dynamics:\nIPull\nS=J2\nsdNint\n~2Z\nqIm\u001fR\nq\u0017(\u0018\rNhachIzi)2\n(\u0017+ ~\u0017N)2+ (1=T2)2:(16)\nAt the resonance condition \u0017=\u0000\rN~hz, Eq. (16) re-\nduces to\nIPull\nS=\u0000Gs(~\u0017N)Aint(hIzi\u0018)2~\u0017N(T2)2(\rNhac)2(17)\nwhereAintis the surface area of the interface expressed\nbyAint=Nintaintwithaintbeing the unit surface\narea of the interface, and Gs(~\u0017N) is given by Gs(~\u0017N) =\n(J2\nsd=~)a\u00001\nintR\nqIm\u001fR\nq~\u0017N=~\u0017N.\nEquation (17) shows that NMR spin pumping is pro-\nportional to the square of the transverse relaxation time\nT2. Because the transverse components of nuclear spins\nrelax to the thermal equilibrium state during T2, the long\nT2leads to the strong non-equilibrium state and enhances\nthe nuclear spin pumping.\nThe temperature dependence of the nuclear spin\npumping is determined mainly by ( hIzi\u0018)2in Eq. (17).\nHere, we calculate the z-component of nuclear spin hIzi\nin mean \feld approximation given by\nhIzi=I0h2I0+ 1\n2I0coth\u00102I0+ 1\n2x\u0011\n\u00001\n2I0cothx\n2i\n;(18)\nwhereI0is nuclear spin value and xis de\fned as x:=\n~~\u0017z=(kBT). When the temperature Tis lower than T\u0003\ngiven byT\u0003:=~~\u0017N=kB, the nuclear spins are fully po-\nlarized andhIzibecomesI0. Because the factor ( hIzi\u0018)2\nFIG. 2. (Color online) (a) Spin current signal IPull\nSplotted\nas a function of Tandh0for a Co/Pt bilayer system. The\nplotted spin current is scaled by FMR spin pumping IFMR\nS =\n1:2\u0002107A/m2. (b) Temperature dependence of IPull\nSat a\n\fxed magnetic \feld h0= 1 mT.\nis an increasing function of hIzi, NMR spin pumping is\nenhanced for T\u001cT\u0003. By contrast, when the temper-\natureTis higher than T\u0003, the temperature and mag-\nnetic \feld dependence of NMR spin current is obtained\nasIPull\nS/(jhNj\u0000h0)3=h02T2.\nAmpli\fcation of NMR spin pumping.| Now we es-\ntimate the spin current (17) for a bilayer system of the\ncobalt and platinum (Co/Pt) where Co and Pt are FM\nand NM layers, respectively. To evaluate Eq. (17), we\ncombine Eq. (17) with the spin current driven by FMR.\nFollowing Ref. 10, we obtain the spin current driven by\nFMR as follows:\nIFMR\nS =Gs(!0)AinthSzi2(\rehac)2\n\u000b2!0; (19)\nwhere\u000bis the Gilbert damping constant given by \u000b=\n\u000b0+\u000e\u000b, with\u000b0and\u000e\u000bbeing the intrinsic and additional\nterms due to the spin pumping, and !0is FMR frequency\nexpressed by !0=\re(h0+hA). We introduce Gs(!0) as\nGs(!0) := (J2\nsd=~)a\u00001\nintR\nqIm\u001fR\nq!0=!0.\nUsing the material parameters in a Co/Pt system[15]\nas\u000b= 0:014,hac= 0:11 mT,\reh0= 2\u0019\u00029:75 GHz and\nIFMR\nS = 1:2\u0002107A/m2, we obtain J2\nsd\u001fN=aint\u00195:2\u0002\n1017eV/m2. Combining these parameters and the pa-\nrameters for59Co in the fcc cobalt, \rN= 6:3015 kHz/Oe,\nhSzi= 0:85[12],I0= 7=2 ,hA= 125 Oe at 3 K[8, 13],\nhN\u001920 T,T2= 20\u000210\u00006s [14],T\u0003\u001910 mK and\nthe spin current generated from NMR at T= 10 mK and\nh0= 1 mT isIPull\nS= 1:5\u00021015\u0002h2\nacA/(T\u0001m2).\nSubstituting hac= 0:11 mT into this result, we show\nthe temperature and magnetic \feld dependence of nu-\nclear spin pumping IPull\nSfor a Co/Pt system in Fig. 2(a),\nwhere the spin current is normalized by FMR spin pump-\ningIFMR\nS. In the region of low temperature or magnetic\n\feld (red colored region), the spin current driven by NMR4\nbecomes larger than that of FMR. In 2(b), we show the\ntemperature dependence of IPull\nSat a \fxed magnetic \feld\nh0= 1 mT. The NMR spin pumping is a decreasing\nfunction of the temperature. We obtain the NMR spin\ncurrent asjIPull\nSj= 1:9\u0002107A/m2atT= 10 mK and\nh0= 1 mT. Comparing jIPull\nSjwith the FMR spin pump-\ning in a Co/Pt system IFMR\nS [15], the NMR spin pumping\natT= 10 mK and h0= 1 mT is ampli\fed to about 1 :5\ntimes larger than that of the FMR spin pumping. Note\nthat FMR spin pumping in a Co/Pt system is almost\nindependent of temperature [16].\nThe enhancement of NMR spin pumping in a Co/Pt\nsystem at low temperature is obtained the competition of\nthe gyromagnetic ratios and relaxation times of the nu-\nclear and electron spins. Comparing Eqs. (17) with (19),\nwe obtain the ratio of IPull\nStoIFMR\nS asIPull\nS=IFMR\nS\u0019\n(\u0018hIzi=hSzi)2(\rN=\re)2\u000b(T2=TFMR), where we introduce\nthe relaxation time of FMR TFMR asTFMR = [(\u000b0+\n\u000e\u000b)!0]\u00001. Considering TFMR calculated as TFMR\u0019\n10\u00008s, we \fndIPull\nS=IFMR\nS\u0019C(\u0018hIzi=hSzi)2withC\nbeing the numerical constant of the order of 1. It is ex-\npected that the ratio of IPull\nStoIFMR\nS is enhanced in FM\nwith large\u000band longT2.\nConclusion.| In this article, we have investigated\nspin-current generation by nuclear spin dynamics via hy-\nper\fne coupling in a normal metal (NM)/ferromagnet\n(FM) bilayer system. We have formulated spin trans-\nport theory using the nonequilibrium Green's function\nmethod. The spin current generated at the interface is\nfound to be enhanced by the pulling e\u000bect and is max-\nimized at the temperature lower than NMR resonance\nfrequency. In a Co/Pt system, we have predicted the\nampli\fcation of the NMR spin current generation. Our\ntheory reveals a new mechanism of angular momentum\nconversion in condensed matter systems, and suggests a\nnew method of generating the spin current using NMR\n(MHz frequency range) which is larger than that gener-\nated by FMR (GHz frequency range).\nAcknowledgements.| The authors thank H. Chudo,\nM. Imai and K. Yamamoto for valuable discussions.This work is \fnancially supported by ERATO-JST\n(JPMJER1402), and KAKENHI (No. 26103005, No.\nJP16H04023, and No. JP26247063) from MEXT, Japan.\n[1] S. Maekawa, S. Valenzuela, E. Saitoh, and T. Kimura ed.,\nSpin Current (Oxford University Press, Oxford, 2012).\n[2] E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Appl.\nPhys. Lett. 88, 182509 (2006).\n[3] Daniel T. Pierce, and Felix Meier, Phys. Rev. B 13, 5484\n(1976).\n[4] K. Ando, M. Morikawa, T. Trypiniotis, Y. Fujikawa, C.\nH. W. Barnes, and E. Saitoh, Appl. Phys. Lett. 96,\n082502 (2010).\n[5] R. Takahashi, M. Matsuo, M. Ono, K. Harii, H. Chudo,\nS. Okayasu. J. Ieda, S. Takahashi, S. Maekawa, and E.\nSaitoh, Nature Physics, 12, 52 (2016).\n[6] M. Matsuo, J. Ieda, K. Harii, E. Saitoh, and S. Maekawa,\nPhys. Rev. B 87, 180402 (2013).\n[7] D. Kobayashi, T. Yoshikawa, M. Matsuo, R. Iguchi, S.\nMaekawa, E. Saitoh, and Y. Nozaki, Phys. Rev. Lett.,\n119, 077202 (2017).\n[8] P. G. de Gennes, P. A. Pincus, F. Hartmann-Boutron,\nand J. M. Winter, Phys. Rev. 129, 1105 (1963).\n[9] P. Fulde and A. Luther, Phys. Rev. 175, 337 (1968).\n[10] Y. Ohnuma, H. Adachi, E. Saitoh, and S. Maekawa,\nPhys. Rev. B 89, 174417 (2014).\n[11] C. P. Slichter, Principles of Magnetic Resonance 2nd ed.\nSpringer-Verlag Berlin Heidelberg GmbH (1978).\n[12] M. Kawakami, T. Hihara, Y. Koi and T. Wakiyama, J.\nPhys. Sot. Japan 331591 (1972).\n[13] H. Br omer and HL. Huber, J. Magn. Mag. Mater 861\n(1978).\n[14] Hirochika Enokiya, J. Phys. Sot. Japan 42796 (1977).\n[15] J.-C. Rojas-S\u0013 anchez, N. Reyren, P. Laczkowski, W.\nSavero, J.-P. Attan\u0013 e, C. Deranlot, M. Jamet, J.-M.\nGeorge, L. Vila, and H. J. Ja\u000br\u0012 es, Phys. Rev. Lett., 112\n106602 (2014).\n[16] T. G. A. Verhagen, H. N. Tinkey, H. C. Overweg, M. van\nSon, M. Huber, J. M. van Ruitenbeek, and J. Aarts, J.\nPhys.: Condens. Matter 28, 056004 (2016)."    }]