Dynamical current-induced ferromagnetic and antiferromagnetic resonances
DDynamical current-induced ferromagnetic and antiferromagnetic resonances
F. S. M. Guimar˜aes , S. Lounis , A. T. Costa , and R. B. Muniz Peter Gr¨unberg Institut and Institute for Advanced Simulation,Forschungszentrum J¨ulich & JARA, D-52428 J¨ulich, Germany Instituto de F´ısica, Universidade Federal Fluminense, Niter´oi, Brazil (Dated: November 5, 2018)We demonstrate that ferromagnetic and antiferromagnetic excitations can be triggered by thedynamical spin accumulations induced by the bulk and surface contributions of the spin Hall effect.Due to the spin-orbit interaction, a time-dependent spin density is generated by an oscillatory elec-tric field applied parallel to the atomic planes of Fe/W(110) multilayers. For symmetric trilayers ofFe/W/Fe in which the Fe layers are ferromagnetically coupled, we demonstrate that only the col-lective out-of-phase precession mode is excited, while the uniform (in-phase) mode remains silent.When they are antiferromagnetically coupled, the oscillatory electric field sets the Fe magnetizationsinto elliptical precession motions with opposite angular velocities. The manipulation of different col-lective spin-wave dynamical modes through the engineering of the multilayers and their thicknessesmay be used to develop ultrafast spintronics devices. Our work provides a general framework thatprobes the realistic responses of materials in the time or frequency domain.
The interplay between charge, spin and orbital angu-lar momentum in nano-structured systems is significantlywidening the prospects of future technologies [1, 2]. Spin-orbit coupling (SOC) is responsible for a variety of fas-cinating phenomena in condensed matter physics. Forexample, the lack of inversion symmetry activates theDzyaloshinskii-Moriya interaction which favors the oc-currence of non collinear ground-state magnetic configu-rations [3–5]. Combined with time-reversal symmetry, itleads to protected conducting states in the so-called topo-logical insulators [6], where spin injection and spin-to-charge conversion were recently demonstrated with thespin-pumping technique [7]. In fact, the generation ofspin currents and spin accumulations by an electric cur-rent, in particular, has been a subject of much inter-est and research recently [8–16]. Several groups showedthat these non-equilibrium quantities can be used to seta magnetization into precessional motion in metallic sys-tems [17–19], including antiferromagnets [20]. Two re-cent reviews of the major experimental and theoreticalresults concerning the charge-to-spin conversion are out-lined in Refs. [21, 22], for both metal and semiconductordevices.So far, theoretical approaches to current-induced spincurrents, accumulations and torques in systems withmore elaborate electronic structures are restricted to thecase in which the applied electric field is static [23–27].Here, we take it one step further, and investigate thedynamic magnetic response which is driven by a time-dependent electric field, as realized in the original ex-periments reported in Refs. [17–19, 28]. One advantageof such an electronic-structure-based method is that itnaturally includes all surfaces, interfaces, and bulk con-tributions [10, 29, 30] to the spin Hall effect, includingthe coupling between local moments and the current-induced spin accumulation of conduction electrons [31],the transparency through the interface [17], and the spin- dependent scattering by the surfaces and interfaces [25].Our framework is general enough to describe all kinds ofdynamical Hall effects (which may be called ac Hall ef-fects) and their reciprocal counterparts. We focus here,however, on the intrinsic (band-related) contributions tothe ac spin Hall effect only.In this Rapid Communication, we shall develop a mi-croscopic theory for the current-induced magnetic re-sponse based on the premise that the amplitude of theexternal electric field is sufficiently weak to allow usto explore its effects within linear response theory. Inthis framework, we demonstrate—in ultrathin films ofFe and W(110)—that ferromagnetic resonances can beinduced by ac electric fields owing to the spin-orbit in-teraction, and distinct modes can be excited dependingon the type of magnetic interaction between the mag-netic layers (ferromagnetic or antiferromagnetic). Im-plicitly, the excitation of the spin-wave modes indicatesthe presence of spin-orbit torques that are dynamical innature. The studied phenomena are the reciprocal of theac spin pumping and inverse spin Hall effect, which areone order of magnitude larger than their dc counterpart[32]—which adds up to the importance of a dynamicaldescription. The considered applied electric field couplesto the charge density, and we are able to calculate theinduced spin disturbances and spin currents along thetransverse directions of the external field, up to first or-der in the field intensity. We show that these quantitiescan be expressed in terms of generalized susceptibilitiesthat may be calculated with the use of the random-phaseapproximation (RPA) of many-body theory. The addi-tional complexity that arises when the RPA decouplingscheme is carried out in the presence of the spin-orbitinteraction is the appearance of four coupled equationsinvolving four distinct response functions that must besolved simultaneously [33].Here we are mainly interested in systems based on a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b transition metals where Coulomb interactions play animportant role. Thus, to accomplish this task explic-itly, we consider that the electronic structure is de-scribed quite generally by a Hamiltonian ˆ H = ˆ H +ˆ H int + ˆ H so , where ˆ H symbolizes the electronic kineticenergy plus a spin-independent local potential, ˆ H int de-notes the electron-electron interaction, and ˆ H so standsfor the spin-orbit interaction term. We choose an atomicbasis set to represent these operators, which then ac-quire the following forms, ˆ H = (cid:80) ijσ (cid:80) µν t µνij c † iµσ c jνσ ,where c † iµσ creates an electron of spin σ in atomic or-bital µ on the site at R i , and the transfer integrals t µνij are parametrized following the standard Slater-Kostertight-binding formalism [34]. We assume that the ef-fective electron-electron interaction U is of short range,and keep only on-site interactions in ˆ H int . Hence,ˆ H int = (cid:80) iµν (cid:80) µ (cid:48) ν (cid:48) (cid:80) σσ (cid:48) U i ; µν,µ (cid:48) ν (cid:48) c † iµσ c † iνσ (cid:48) c iν (cid:48) σ (cid:48) c iµ (cid:48) σ ,where U i ; µν,µ (cid:48) ν (cid:48) is a matrix element of the effective elec-tron interaction between orbitals, all centered on thesame site i . In the spin-orbit term we also take intoaccount intra-atomic interactions only, and write ˆ H so = (cid:80) iµν (cid:80) σσ (cid:48) ξ i (cid:104) iµσ | L · S | iνσ (cid:48) (cid:105) c † iµσ c iνσ (cid:48) , where ξ i denotesthe spin-orbit coupling constant on site i , and L and S are the orbital angular momentum and spin operators,respectively.In order to calculate the desired spin responses in thepresence of the spin-orbit interaction, it is useful to in-troduce the generalized spin susceptibilities χ σ σ σ σ µνγξijk(cid:96) ( t ) = − i (cid:126) Θ( t ) (cid:104) [ c † iµσ ( t ) c jνσ ( t ) , c † kγσ c (cid:96)ξσ ] (cid:105) , (1)where each σ i symbolizes either ↑ or ↓ spin directions.We may represent them as a 4 × σσ (cid:48) = ↑↓ , ↑↑ , ↓↓ , ↓↑ (+ , ↑ , ↓ , − ). Withinthe RPA it is possible to express all elements in termsof the noninteracting spin susceptibilities χ (0) , that aregenerated by evaluating the commutators which enterinto Eq. (1) in the noninteracting ground state. In ma-trix form the relation is schematically given by [ χ ( ω )] = (cid:2) χ (0) ( ω ) (cid:3) − (cid:2) χ (0) ( ω ) (cid:3) [ U ] [ χ ( ω )], where χ (0) ijk(cid:96) ( ω ) = (cid:126) (cid:90) dω (cid:48) f ( ω (cid:48) ) { g jk ( ω (cid:48) + ω ) (cid:61) [ g (cid:96)i ( ω (cid:48) )]+ g − (cid:96)i ( ω (cid:48) − ω ) (cid:61) [ g jk ( ω (cid:48) )] (cid:9) . (2)Here, to simplify the notation, we have omitted the spinand orbital indices, assuming that they are included inthe site indices. We define (cid:61) [ g ] = i π [ g − g − ], where g and g − represent the retarded and advanced one-electronpropagators, respectively, and f ( ω ) is the usual Fermidistribution function. We remark that at this stage weare ignoring long-range Coulomb interactions which arerelevant to ensure charge conservation, especially in thestatic limit of homogeneous fields. Edwards [35] has re-cently shown that for bulk systems this may not be so significant for relatively small SOC.We begin by examining an ultrathin film of W(110)with atomic planes stacked along the ˆz direction, choos-ing the ˆx and ˆy Cartesian axes parallel to the layers,in the [1¯10] and [001] directions, respectively. Assuming U = 1 eV and ξ = 0 .
26 eV for W, and adjusting the cen-ter of its d bands to reproduce the electronic occupationsobtained by density functional theory (DFT) calculations[36] for each atomic plane, one finds that the ground stateof the W film is nonmagnetic, as expected. Let us thensuppose that a spatially uniform harmonic electric field E = E cos( ωt ) ˆu E is applied parallel to the layers in anarbitrary direction ˆu E . In this case, the time-dependentperturbing Hamiltonian is given byˆ V ( t ) = eE (cid:126) ω N (cid:88) k (cid:107) ,σ (cid:88) (cid:96)(cid:96) (cid:48) µν ∇ k (cid:107) t µν(cid:96)(cid:96) (cid:48) ( k (cid:107) ) · ˆu E sin( ωt ) × c † (cid:96)µσ ( k (cid:107) , t ) c (cid:96) (cid:48) νσ ( k (cid:107) , t ) , (3)where (cid:96) and (cid:96) (cid:48) identify atomic planes, and k (cid:107) is awave vector parallel to the layer, belonging to the two-dimensional Brillouin zone. With the use of linear re-sponse theory we may calculate the components of thelocal spin disturbance per atom in plane (cid:96) , induced bythe ac applied electric field by virtue of the SOC. Theyare given by δ (cid:104) ˆ S m(cid:96) ( t ) (cid:105) = A m(cid:96) ( ω ) sin (cid:0) ωt − φ m(cid:96) ( ω ) (cid:1) , (4)where A m(cid:96) ( ω ) = eE (cid:126) ω |D m(cid:96) ( ω ) | represents the amplitude ofthe local spin disturbance, and φ m(cid:96) ( ω ) is the frequency-dependent phase of the complex number D m(cid:96) ( ω ) = (cid:88) k (cid:107) σ (cid:88) (cid:96)(cid:96) (cid:48) µγξ χ mσ µµγξ(cid:96) (cid:96) (cid:96)(cid:96) (cid:48) ( k (cid:107) , ω ) ∇ k (cid:107) t γξ(cid:96)(cid:96) (cid:48) ( k (cid:107) ) · ˆu E . (5)Here, m = x, y, z labels the corresponding spin compo-nents, χ xσ = [ χ ↑↓ σσ + χ ↓↑ σσ ] / χ yσ = [ χ ↑↓ σσ − χ ↓↑ σσ ] / i ,and χ zσ = χ ↑↑ σσ − χ ↓↓ σσ .Due to the presence of SOC, an ac electric field appliedalong the [1¯10] ( ˆx ) direction should produce an ac spinaccumulation (cid:104) ˆ S y(cid:96) (cid:105) (cid:54) = 0 in the W(110) atomic planes as aresult of the bulk spin currents generated by the dynamicspin Hall effect and also from the spin-orbit fields origi-nated in the spin-split surface states. It also gives rise toa bulk pure ac spin current with spin polarization ˆ z thatflows parallel to the layer along the [001] ( ˆy ) direction,but leads to no spin accumulation due to the translationsymmetry of the layers. Similarly, if the field is appliedalong the [001] direction, the W(110) atomic planes areexpected to acquire an ac spin accumulation (cid:104) ˆ S x(cid:96) (cid:105) (cid:54) = 0.In this case, the electric field also generates an ac spincurrent with spin polarization ˆ z that flows along the [1¯10]direction, causing no spin accumulation. This is preciselywhat we have found in our calculations of the spin dis-turbances and currents induced in a free-standing slab ofW(110). The results for the amplitudes and phases of δ (cid:104) ˆ S m(cid:96) ( t ) (cid:105) calculated as functions of the energy E = (cid:126) ω are shown in Fig. 1 for electric fields applied in two per-pendicular directions. Owing to the spatial anisotropyof the (110) two-dimensional lattice, the amplitudes ofthe spin accumulation in the W surface differ consider-ably for electric fields applied along the [1¯10] and [001]directions. One can also appreciate the importance ofthe Coulomb exchange interaction within the W layer bycomparing the amplitudes of the induced magnetic mo-ments obtained with U = 1 eV and U = 0, which aredepicted by the solid and dashed lines, respectively, inFig. 1. The overall increase for U (cid:54) = 0 suggests that theseeffects possibly may be used to excite spin fluctuations(paramagnons) in ultrathin films of nearly ferromagneticmetals such as Pd and Pt, which exhibit relatively largeStoner enhancement factors. The inset illustrates thecorresponding phases φ m(cid:96) ( E ) of the spin disturbances in-duced in the four W atomic planes by an electric fieldapplied along [1¯10] with U = 1 eV. For low values of ω we identify a current-induced staggered spin disturbanceprofile on the W(110) atomic planes. The same featureappears when the field is applied along the [001] directionfor both values of U . This is compatible with the chargecurrent leading to spin accumulations of inverse sign onthe opposite W surfaces, and the spin polarization in-duced by this spin imbalance in each surface decreases asone moves into the W film along the stacking directionin an oscillatory manner with a period of approximatelytwo inter-planar distances, thus favoring the antiferro-magnetic alignment.We shall now discuss the use of the ac charge currentas a way of exciting spin-wave modes in an Fe layer ad-sorbed to a thin film of W(110), consisting of five atomicplanes in total. The ground-state magnetization of theFe layer in this case sets down in-plane along the [1¯10]direction, which is the easy axis. The uniform spin-wavemode observed in a ferromagnetic resonance (FMR) ab-sorption spectrum is revealed as a resonance in the trans-verse dynamical spin susceptibility, which represents theresponse of the system to a time-dependent oscillatorytransverse magnetic field. This is clearly shown in Fig.2(a), which depicts the local transverse spin susceptibil-ity χ + − ( q (cid:107) = 0 , E ) calculated as a function of energy E = (cid:126) ω in the Fe surface layer. The peak position inIm χ + − ( E ) is the anisotropy energy due to the spin-orbit interaction, and the linewidth of the resonance isinversely proportional to the spin-wave lifetime. If in-stead of a transverse magnetic field we apply an oscilla-tory electric field along the easy-axis direction, for exam-ple, we may also calculate the current-induced spin dis-turbances in the Fe layer δ (cid:104) ˆ S m ( t ) (cid:105) within our approach,and their calculated amplitudes A m ( E ) are illustrated inFig. 2(b). They clearly show that both transverse com-ponents of the induced spin disturbances in the Fe layerexhibit a peak precisely at the ferromagnetic resonance Energy (eV)
Energy (meV) y ` ⇡⇡ A m · E / e E ( ˚A ) ˆ x [1¯10] ˆ y [001] FIG. 1. (Color online) Amplitudes of the surface spin distur-bances A y ( E ) (thin black lines) and A x ( E ) (thick red lines)induced in a free-standing slab of W(110) by ac electric fieldsapplied along the [1¯10] and [001] directions, respectively. Theslab comprises four atomic planes which are labeled sequen-tially by (cid:96) = 1–4, starting from one of the W surfaces. Solidlines represent results calculated for U = 1 eV and and dashedlines for U = 0. The inset shows the corresponding phases φ y(cid:96) ( E ) calculated with U = 1 eV for (cid:96) = 1 (black thin solidline) (cid:96) = 2 (red thin dashed line), (cid:96) = 3 (green thick dashedline), and (cid:96) = 4 (blue thin solid line), as a result of an electricfield applied along [1¯10]. energy, demonstrating that the oscillatory electric field isexciting the uniform spin-wave mode by means of the dy-namical spin-orbit torque. We see the appearance of anoscillatory spin disturbance δ (cid:104) ˆ S z ( t ) (cid:105) , with polarizationperpendicular to the Fe surface layer, which is dephasedby approximately π/ δ (cid:104) ˆ S y ( t ) (cid:105) , revealing that themagnetization of the Fe layer is set into an elliptic pre-cessional motion around the easy axis. We note that the y ( z ) component is even (odd) with respect to magneti-zation inversion ( M → − M ), as discussed in Ref. [23].We have also calculated the change in orbital angularmomentum induced in the Fe surface layer by the sameelectric field. Both amplitudes of δ (cid:104) ˆ L y ( t ) (cid:105) and δ (cid:104) ˆ L z ( t ) (cid:105) display well-defined maxima at the same ferromagneticresonance energy, but they are approximately one orderof magnitude smaller than the corresponding values for A m ( E ).We now turn our attention to Fe/W(110)/Fe multilay-ers. We consider two different thicknesses for the tung-sten spacer layer, starting with two atomic planes of Wwhere the magnetizations of the Fe layers are ferromag-netically coupled along the long axis. In this situation,the FMR absorption spectrum exhibits two precessionmodes corresponding to the cases in which those mag-netizations oscillate in phase (acoustic mode) and out ofphase (optical mode), respectively. This is clearly visiblein Fig. 3(a) which shows the local transverse spin sus- -5050 1 2 3 4 5 6 7051015 Energy (meV)a)b) + A m · E / e E ( ˚A ) FIG. 2. (Color online) (a) Real (black dashed line) and imag-inary (red solid line) parts of the local transverse spin sus-ceptibility calculated (in arbitrary units) for a monolayer ofFe/W(110) as functions of energy. (b) Amplitudes of the lo-cal induced spin disturbances A y ( E ) (green solid line) and A z ( E ) (blue dashed line) calculated in the Fe surface layer. ceptibility calculated as a function of energy for one ofthe Fe surface layers. The energy difference between thetwo peaks in Im χ + − ( E ) is a measure of the exchangecoupling between the Fe magnetizations. In Fig. 3(b)we present our calculated results for the amplitudes ofthe transverse spin components induced in the same Fesurface by an oscillatory electric field applied along the[1¯10] direction. They show that only the out-of-phaseprecession mode is excited by the electric field, while theuniform (in-phase) precession mode remains silent. Thisis reasonable for a perfectly symmetric configuration suchas the one we are considering, since the oscillatory spinaccumulations that drive the magnetizations of the op-posite Fe layers into precession are 180 ◦ out of phase.Indeed, the phase differences φ y,z ( ω ) − φ y,z ( ω ) betweenthe spin disturbances induced in the Fe surfaces are bothequal to π for all values of ω . This contrasts with tra-ditional FMR experiments, driven by a time-dependenthomogeneous transverse magnetic field, where the opti-cal mode would not be observed, unless the individualFM layers have different resonance frequencies. Deposi-tion of the layered structure on substrates introduces anasymmetry between the ferromagnetic layers that mayprevent complete cancellation of the torques, enhancingthe acoustic-mode signal. However, this can be tuned bya suitable choice of substrate.By increasing the thickness of the W spacer layer tothree atomic planes, we find that the magnetizations ofthe Fe layers become antiferromagnetically coupled. Welabel the two Fe surfaces in this trilayer by 1 and 5, re-spectively. In fact, assuming that in the ground state theFe magnetizations are ferromagnetically aligned, a cal-culation of Im χ + − ( E = (cid:126) ω ) displays two resonant spin- -1010 10 2000.30.6 a)b) Energy (meV) + A m · E / e E ( ˚A ) FIG. 3. (Color online) Same as in Fig 2 for a Fe/W(110)/Fetrilayer. wave modes—one at a positive angular frequency andanother at a negative value of ω –proving that the Fe lay-ers are indeed antiferromagnetically coupled in this case.However, one may also calculate the local transversespin susceptibilities from the antiferromagnetic (ground)state. The results for the imaginary parts of χ + − and χ + − , calculated as functions of energy, are shown in Fig.4(a). Each shows two extrema with different intensitiesat ± ω , which is consistent with the antiferromagneticcoupling between the Fe layers in the presence of theanisotropy field due to the SOC. In Fig. 4(b) we presentresults for the amplitudes of the local spin disturbances A y ( E ) and A z ( E ) in one of the Fe surface layers. Wealso found the phase differences between the spin dis-turbances induced in the two Fe surface layers to be φ y ( ω ) − φ y ( ω ) = π , and φ z ( ω ) − φ z ( ω ) = 0, for all val-ues of ω . This is consistent with the two magnetizationsbeing set into elliptic precessional motions around theirequilibrium directions, however, with opposite angularvelocities.To estimate the charge-to-spin conversion we define acoefficient γ ( E ) = |A ( E ) | / | j C ( E ) | [25], given by the ra-tio between the amplitudes of the surface-induced spinaccumulation and of the charge current density | j C ( E ) | .In the energy range of interest, | j C ( E ) | E/E is approx-imately constant and, as a result, the curves represent-ing A ( E ) E/eE are basically the same as γ ( E ), exceptfor a constant multiplicative factor. It follows that thecharge-to-spin conversion at the resonance frequency islargely enhanced with respect to its values at very lowfrequencies.To summarize, we investigated dynamical transportproperties in the context of charge-to-spin conversion.For instance, we evaluate spin and orbital angular mo-mentum accumulation induced by an ac charge currentmediated by the spin-orbit interaction. We demonstrate -2-1012 -20 -10 0 10 20051015 a)b) Energy (meV) I m + `` A m · E / e E ( ˚A ) ⌦ FIG. 4. (Color online) (a) Im χ + − ( E ) (red solid line), andIm χ + − ( E ) (black dashed line), calculated (in arbitrary units)as functions of energy for the Fe surface layers of the antiferro-magnetically coupled Fe/W(110)/Fe trilayer. The W spacerlayer has three atomic planes. (b) Amplitudes of the inducedlocal-spin-disturbance components A y ( E ) (green solid line)and A z ( E ) (blue dashed line) calculated for one of the Fesurface layers. that specific spin-precession modes can be excited in thinfilms depending on the magnetic nature of the nanostruc-tures, which may assist their switching and offer a po-tentially useful tool for ac spintronic developments andnanotechnologies. In fact, it was recently shown that thespin-wave excitation is directly related to the switchingrate. [37] Our framework allows for the inspection ofadditional phenomena, such as the whole family of dy-namical Hall effects and all their reciprocal counterparts.We are grateful to A. B. Klautau for providing tight-binding parameters, M. dos Santos Dias and S. Bl¨ugelfor discussions, and computing time on JUROPA su-percomputer at J¨ulich Supercomputing Centre. 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