Non-classical spin transfer effects in an antiferromagnet
NNon-classical spin transfer effects in an antiferromagnet
Alexander Mitrofanov and Sergei Urazhdin
Department of Physics, Emory University, Atlanta, GA, USA.
We simulate scattering of a spin-polarized electron by a chain of antiferromagnetically coupledquantum Heisenberg spins, to analyze spin-transfer effects not described by the classical modelsof magnetism. Among the effects elucidated by the simulations are efficient excitation of multiplemagnetic excitation quanta by a single electron, which is not possible in ferromagnets due to angularmomentum conservation, as well as quantum interference of spin wavefunctions, making it possibleto induce magnetization dynamics with amplitudes exceeding the transferred magnetic moment.Our results suggest the possibility to utilize non-classical contributions to spin transfer to achieveefficient spin conversion and electronic control of static and dynamical states in antiferromagnets.
Spin transfer (ST) effect - the transfer of spin fromthe itinerant electrons to the localized spins in magneticsystems - has provided unprecedented insights into nano-magnetism, and enabled a variety of efficient magneto-electronic nanodevices [1, 2]. The speed of ST-baseddevices is limited by the characteristic dynamical mag-netization frequencies, on the order of a nanosecond forferromagnets (Fs), while the efficiency is limited by therequirement that the transferred spin is comparable tothe total spin of the nanomagnet [1]. These limitationscan be substantially reduced in ST-driven nanodevicesbased on antiferromagnets (AFs) [3–9]. Indeed, the van-ishing bulk magnetization of AFs reduces the constraintson device efficiency, while the high characteristic dynam-ical magnetization frequencies in AFs, typically two or-ders of magnitude larger than in Fs, enable fast controlof their states [6, 10]. Furthermore, AFs are immune toperturbations by magnetic fields. These features may en-able resilient AF-based memory devices with picosecondswitching times [5, 11, 12], nanoscale oscillators operatingin the
T Hz frequency range [13–15], and ST-driven AFdomain wall motion with extremely high velocities [6, 16].While there are many similarities between ST effectsin Fs and AFs, substantial differences are also expected.The magnetization of Fs can be well approximated as asemi-classical vector field. In particular, in the groundstate of a Heisenberg F, the local spins are aligned [17].For a simple collinear AF, the equivalent would be theN´eel state, where the spins of two magnetic sublatticesare aligned in the opposite directions. However, thisstate is not an eigenstate of the AF Heisenberg Hamilto-nian [18]. Instead, its ground state can be described as aN´eel state dressed with a large population of sublatticemagnons - spin flips spread out on one of the magneticsublattices [19]. Since the effects of ST on the dynamicalmagnetization states can be viewed as stimulated emis-sion of magnons that occurs at a rate proportional tomagnon populations [20–22], the magnon-dressed statesof AFs may be affected by ST very differently from thepure N´eel state.The dressed N´eel states of AFs originate from the non-commutativity of different components of spin [18], which is not described by the semi-classical approximation forthe magnetic order. The resulting contributions to STmay be similar to the ”quantum ST” demonstrated forFs, which also originates from the non-commutativityof different spin components [21, 22].
Here and be-low, we use the terms ”quantum ST”, or equivalently”non-classical ST”, to refer to the contributions to STthat cannot be described within semiclassical approxima-tion for magnetism, but instead require that the local-ized spins forming the magnetization are described bythe Schrdinger equation.
Since the effects of spin non-commutativity are generally much larger in AFs than inFs, more significant quantum contributions to ST maybe also expected.To analyze the quantum problem of interaction be-tween the spin-polarized current and AF, we consider anitinerant electron initially propagating in a non-magneticmedium, and subsequently scattered by a 1D chain of lo-cal AF-coupled Heisenberg spins-1/2. The system can bedescribed by the tight-binding Hamiltonian [23–26]ˆ H = − (cid:88) i b | i (cid:105)(cid:104) i + 1 |− (cid:88) j ( J sd | j (cid:105)(cid:104) j | ⊗ ˆ S j · ˆ s − J ˆ S j · ˆ S j +1 ) , (1)where i enumerates the tight-binding sites of the entiresystem, j - the sites occupied by the localized spins-1/2representing the AF, ˆ s , ˆ S j are the spin operators of theelectron and the local spins. The first term in Eq. (1) de-scribes the itinerant electron hopping, the second term -exchange interaction between the itinerant electron andthe local spins, and the last term - exchange interac-tion between localized spins. Periodic boundary condi-tions for both the electron and the spin chain are used toavoid reflections at the boundaries. For the experimen-tally accessible magnetic fields, the Zeeman contributionis negligible on the timescales considered in our analysis.The evolution of the system is determined by numeri-cally integrating the time-dependent Schrdinger equationwith the Hamiltonian Eq. (1) [26]. To analyze the evo-lution of observable quantities, we determine the density a r X i v : . [ c ond - m a t . m t r l - s c i ] S e p xyz v s A F s i n g l e t xyz S S v s S S S n F ...... S n . .
50 10 200 . . . . s z , s z S x s x (a) ∆ S z S p i n p r o j ec ti on s t (fs)F (b) AF (c) S p i n p r o j ec ti on s t (fs)S x s x � ∆ s x s z (d) AFF � ∆ s xS p i n p r o j ec ti on s nS z Figure 1. (Color online) (a) Schematics of the simulated sys-tems that consist of a spin-polarized itinerant electron scat-tered by F (top) and AF (bottom), initially in their groundstates. (b,c) Evolution of the expectation values of x - and z -components of the electron spin s and the total spin S ofthe magnetic system that consists of n = 8 spins-1/2, for F(b), and AF (c). ∆ S z is the variation relative to S z = 4.(d) Dependence of the transferred spin on the number n oflocal spins, for F and AF, as labeled. The simulations wereperformed using b = 1 eV, J sd = 0 . J = − . . k = 5 nm − . matrices ˆ ρ e = T r m ˆ ρ and ˆ ρ m = T r e ˆ ρ for the itinerantelectron and the local spins, respectively, by tracing outthe full density matrix ˆ ρ with respect to the other subsys-tem [23]. The expectation value of observable ˆ A pertain-ing to the electron, such as its spin component or energy,is (cid:68) ˆ A (cid:69) = T r ( ˆ A ˆ ρ e ), while the probability of its value a is P a = (cid:104) ψ a | ˆ ρ e | ψ a (cid:105) , where ψ a is the corresponding eigen-state. The quantities pertaining to AF are determinedsimilarly. Hereafter, we for brevity omit brackets on theexpectation values of spins and energy.We start the analysis of ST by comparing its effectson AF to those on a 1D F modeled using Eq. (1) withthe same parameters, except for the opposite sign of J .Both systems are initialized in their ground states - Fspins aligned with the z-axis, and AF spins forming a spinsinglet [18, 27]. We note that all the components of eachlocal spin vanish in the spin singlet state, so it cannot bedescribed semiclassically. Thus, ST in the singlet state ispurely quantum.The electron is initialized, at time t = 0, as a Gaussianwave packet spin-polarized along the x-axis propagatingin the non-magnetic medium towards the magnetic sys-tem [Fig. 1(a)]. The spins of the electron and of themagnetic system start to significantly vary at t > Initial state2 spinons>2 spinons1 magnon (b) (c)(a)
AFF Δ E / | J | number of spins Δ Ε S Τ / Δ Ε J (eV) J sd = 0.3 eV0.2 eV0.1 eV -3 -2 -1 0 10481216 F AF ) / | J | S z ( E - E AF Figure 2. (Color online) (a) Transfer of energy from electronto the local spins vs the chain length n . (b) Relative energiesof the eigenstates for F (left) and AF (right) with n = 6 vstheir spin projection on the z-axis. Only the states with finiteamplitudes after scattering are shown. Stars: ground state,circles: 1-magnon states (for F), squares: 2-spinon states (forAF), triangles: states with more than two spinons. Colorscale: the probability of the state after scattering. Some sym-bols are slightly shifted for clarity. (c) The ratio of energytransferred to magnetic excitations with finite spin to the to-tal transferred energy vs J , at the labeled values of J sd . Thesimulation parameters and the initial states are the same asin Fig. 1, unless specified otherwise. the wave packet approaches the local spins, Figs. 1(b,c).The variations become negligible again at t >
12 fs, afterthe wave packet is completely scattered [26]. The well-defined transitions among these different regimes allowus to unambiguously quantify the ST effects.In the simulations for F, the x-component of the elec-tron spin orthogonal to the local spins becomes reduced,while the corresponding component for the local spins in-creases by the same amount [Fig. 1(b)], consistent withthe theories of ST [1, 20, 28]. The electron also becomespartially spin-polarized along the z-axis, while the corre-sponding local spin component becomes reduced by thesame amount, due to the quantum ST [22, 23, 25].In case of AF, the electron’s initial spin polarization isalso partially transferred to the local spins [Fig. 1(c)]. Incontrast to F, the z-component of electron spin does notvary, consistent with the spin symmetry of the singletstate. The transferred spin increases with increasing sizeof the magnetic system for both F and AF, as expecteddue to the increasing interaction time with the itinerantelectron [Fig. 1(d)]. The spin transferred to AF alwaysremains smaller than both the x- and the z-componentsof spin transferred to F.ST is likely not the only effect controlling the current-induced dynamical processes in AFs. Indeed, the spinangular momentum of a compensated 3D AF in the Ne´elstate is zero. Thus, according to the angular momentumconservation argument, rotation or reversal of the N´eelorder does not require ST. However, switching betweenstable magnetic configurations requires that the magneticsystem overcomes the energy barrier between them, andtherefore energy transferred to the magnetic system mustplay an important role [25].The energy transferred from the scattered electron tothe magnetic system is larger for AF than F [Fig. 2(a)].We reconcile this result with the weaker ST in AF by an-alyzing the dynamical magnetization states induced bythe electron scattering. We use Bethe ansatz to clas-sify the eigenstates of the magnetic systems in termsof the elementary excitations - magnons for F, andspinons - fractionalized spin-1/2 quasiparticles - for the1D AF [18, 26, 27]. The final state of the magnetic sys-tem is projected onto these eigenstates to determine theprobabilities of their excitation.Figure 2(b) shows the energies of the eignestates withnon-zero amplitudes after electron scattering, plottedversus the z-component of their spin, for n = 6. For F , all 6 of the eigenstates excited by ST are 1-magnonstates, with S z = −
2. This result is expected from angu-lar momentum conservation: each magnon carries spin 1,so a spin-1/2 electron can excite at most one magnon.In contrast to F, a variety of multi-quasiparticle eigen-states are excited in AF. The integer spin of the chain ne-cessitates that spinons are generated in pairs, with possi-ble z spin component of −
1, 0 or 1. All these possibilitiesare realized in the studied system, as shown in Fig. 2(b)by squares. Furthermore, in contrast to F, spin conser-vation does not limit the number of generated quasipar-ticles, as long as their spins add up to 0 or 1. Indeed,11 of the 31 eigenstates of AF excited by ST containmore than 2 spinons, as shown in Fig. 2(b) by triangles.These results are consistent with the possibility of many-spinon excitation by neutron scattering [29, 30], and thepredicted enhancement of electron interaction with mag-netic excitations in AFs [19].The results of Fig. 2(b) explain why energy transfer inAF can be more efficient than in F, even though ST isless efficient. In F, spin conservation limits the accessibledynamical magnetization states, and since each magnoncarries the same spin 1, magnon excitation is directly tiedto ST. For AFs, excitation of many different dynamicalstates is allowed by spin conservation. They can havedifferent spin directions, adding up to smaller net spintransfer. The relative significance of ST can be char-acterized by the ratio ∆ E ST / ∆ E of energy transferredto the states with S z = ±
1, to the total transferred en-ergy [31]. The value of ∆ E ST / ∆ E varies with the systemparameters such as the exchange stiffness or the inter-action between the itinerant electron and the localizedspins [Fig. 2(b)], suggesting that the efficiency of non-STexcitation can be enhanced by optimizing materials and � a � t t S x s x �� S p i n p r o j ec ti on t (fs) ∆ (b) xyz S j ''S j ' s B eff ' B eff '' t t Figure 3. (Color online) ST for an anisotropic AF chain with n = 4, initially in the superposition of two Nel states. (a)Evolution of the expectation values of the x spin componentsof electron (solid curve) and of the local spins (dashed curves),at the labeled values of anisotropy ∆. (b) Schematics of spin-up (labeled S (cid:48) j ) and spin-down (labeled S (cid:48)(cid:48) j ) components of thewavefunction for one of local spins at ∆ = 0 .
4, at times labeled t and t in panel (a). Curved arrows and dashed circlesshow the trajectories of the corresponding spin wavefunctioncomponents during ST (left) and after ST (right). B (cid:48) eff and B (cid:48)(cid:48) eff are the effective anisotropy fields experienced by S (cid:48) j and S (cid:48)(cid:48) j , respectively. For ∆ = 0 .
4, ST is compensated by theanisotropy torques at t . These effects are not shown forclarity. experimental parameters.The latter possibility may enable current-induced exci-tation of magnetization dynamics with much larger am-plitudes, and consequently a higher efficiency of current-induced magnetic switching, than would be achievablewith only ST-mediated excitations. This possibility isdemonstrated in Fig. 3 for an anisotropic AF chain offour spins initialized in the state ( |↑↓↑↓(cid:105) − |↓↑↓↑(cid:105) ) / √ J ∆ (cid:80) j ˆ S zj ˆ S zj +1 to the Hamil-tonian Eq. (1), but other forms of anisotropy should pro-duce similar effects. Figure 3(a) shows the spin evolu-tion for different values of ∆. By symmetry, the sum ofthe y- and z-components of both local spin wavefunctioncomponents remain zero. The dependence s x ( t ) is nearlyidentical for all three shown values of ∆ [solid curve inFig. 3(a)]. In contrast, the evolution of S x is stronglydependent on ∆. For ∆ = 0, it mirrors the evolution ofthe electron’s spin, as expected for the isotropic Hamil-tonian. For ∆ = 0 . . S x first slightly increases,and then starts to oscillate with amplitude significantlylarger than the transferred spin. The period of the oscil-lation is larger for ∆ = 0 .