Heterostructures utilizing topological insulators exhibit a remarkable
spin-torque efficiency. However, the exact origin of the strong torque, in
particular whether it stems from the spin-momentum locking of the topological
surface states or rather from spin-Hall physics of the topological-insulator
bulk remains unclear. Here, we explore a mechanism of spin-torque generation
purely based on the topological surface states. We consider
topological-insulator-based bilayers involving ferromagnetic metal (TI/FM) and
magnetically doped topological insulators (TI/mdTI), respectively. By ascribing
the key theoretical differences between the two setups to location and number
of active surface states, we describe both setups within the same framework of
spin diffusion of the non-equilibrium spin density of the topological surface
states. For the TI/FM bilayer, we find large spin-torque efficiencies of
roughly equal magnitude for both in-plane and out-of-plane spin torques. For
the TI/mdTI bilayer, we elucidate the dominance of the spin-transfer-like
torque. However, we cannot explain the orders of magnitude enhancement
reported. Nevertheless, our model gives an intuitive picture of spin-torque
generation in topological-insulator-based bilayers and provides theoretical
constraints on spin-torque generation due to topological surface states.Comment: updated version including a discussion of TI/FM as well as TI/
  magnetically doped TI heterostructure

Fischer, Mark H

Kim, Eun-Ah

Manchon, Aurelien

Vaezi, Abolhassan

English

arXiv

Heterostructures utilizing topological insulators exhibit a remarkable spin-torque efficiency. However, the exact origin of the strong torque, in particular whether it stems from the spin-momentum locking of the topological surface states or rather from spin-Hall physics of the topological-insulator bulk, remains unclear. Here, we explore a mechanism of spin-torque generation purely based on the topological surface states. We consider topological-insulator-based bilayers involving ferromagnetic metal (TI/FM) and magnetically doped topological insulators (TI/mdTI), respectively. By ascribing the key theoretical differences between the two setups to location and number of active surface states, we describe both setups within the same framework of spin diffusion of the nonequilibrium spin density of the topological surface states. For the TI/FM bilayer, we find large spin-torque efficiencies of roughly equal magnitude for both in-plane and out-of-plane spin torques. For the TI/mdTI bilayer, we elucidate the dominance of the spin-transfer-like torque. However, we cannot explain the orders of magnitude enhancement reported. Nevertheless, our model gives an intuitive picture of spin-torque generation in topological-insulator-based bilayers and provides theoretical constraints on spin-torque generation due to topological surface states.The authors are grateful to A. Mellnik and D. Ralph for
helpful discussions.M.H.F. and E.-A.K. acknowledge support
from NSF Grant No. DMR-0955822 and from NSF Grant No.
DMR-1120296 to the Cornell Center for Materials Research.
M.H.F. further acknowledges the Swiss Society of Friends
of the Weizmann Institute of Science. A.M. was supported
by the King Abdullah University of Science and Technology
(KAUST)

Fischer, Mark H.

KAUST Repository (King Abdullah University of Science and Technology)

  Spin-torque generation in topological insulator basedheterostructures  KAUSTRepositoryItem type ArticleAuthors Fischer, Mark H.; Vaezi, Abolhassan; Manchon, Aurelien;Kim, Eun-AhCitation Spin-torque generation in topological insulator basedheterostructures 2016, 93 (12) Physical Review BEprint version Publisher's Version/PDFDOI 10.1103/PhysRevB.93.125303Publisher American Physical Society (APS)Journal Physical Review BRights Archived with thanks to Physical Review BDownloaded 6-Dec-2017 19:33:51Link to item http://hdl.handle.net/10754/603505PHYSICAL REVIEW B 93, 125303 (2016)Spin-torque generation in topological insulator based heterostructuresMark H. Fischer,1 Abolhassan Vaezi,2,3 Aurelien Manchon,4 and Eun-Ah Kim31Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 7610001, Israel2Department of Physics, Stanford University, Stanford, California 94305, USA3Department of Physics, Cornell University, Ithaca, New York 14853, USA4King Abdullah University of Science and Technology (KAUST), Physical Sciences and Engineering Division,Thuwal 23955-6900, Saudi Arabia(Received 21 May 2013; published 11 March 2016)Heterostructures utilizing topological insulators exhibit a remarkable spin-torque efficiency. However, the exactorigin of the strong torque, in particular whether it stems from the spin-momentum locking of the topologicalsurface states or rather from spin-Hall physics of the topological-insulator bulk, remains unclear. Here, weexplore a mechanism of spin-torque generation purely based on the topological surface states. We considertopological-insulator-based bilayers involving ferromagnetic metal (TI/FM) and magnetically doped topologicalinsulators (TI/mdTI), respectively. By ascribing the key theoretical differences between the two setups to locationand number of active surface states, we describe both setups within the same framework of spin diffusion of thenonequilibrium spin density of the topological surface states. For the TI/FM bilayer, we find large spin-torqueefficiencies of roughly equal magnitude for both in-plane and out-of-plane spin torques. For the TI/mdTIbilayer, we elucidate the dominance of the spin-transfer-like torque. However, we cannot explain the orders ofmagnitude enhancement reported. Nevertheless, our model gives an intuitive picture of spin-torque generationin topological-insulator-based bilayers and provides theoretical constraints on spin-torque generation due totopological surface states.DOI: 10.1103/PhysRevB.93.125303I. INTRODUCTIONHarnessing the spin-momentum locking of the surfacestates of topological insulators holds great promise forspintronics applications. Indeed, recent experiments ontopological insulator/ferromagnetic metal (TI/FM) [1,2] andtopological insulator/magnetically doped topological insulator(TI/mdTI) heterostructures [3] observed a large spin-torqueefficiency, the figure of merit for their application. The torquemeasured in these two sets of experiments, however, differsquite significantly. While the TI/FM experiments exhibitspin-transfer- and fieldlike torques of comparable magnitude,the TI/mdTI has predominantly spin-transfer-like torque, andthus resembles the spin-Hall setup of heavy metal (HM)/FMbilayers [4–6]. Its efficiency, however, exceeds the HM/FMbilayers’ by several orders of magnitude.Devices consisting of topological insulators and ferromag-netic metals have so far mainly been the focus of theoreticalstudies in the context of magnetotransport, where the FMaffects the transport properties of the topological surface states[7–9]. Most theoretical investigations of torque generationusing topological insulators, however, have focused on (ideal)TI/ferromagnetic insulator (FI) hybrid structures [10–13].There, a current through the topological surface state mainlyresults in a nonequilibrium spin density due to the surfacestates’ helical spin structure (inverse spin-galvanic effect).Adding to the Oersted field, this acts as a magnetic field onthe ferromagnetic moments [10,11]. This effect can clearly notaccount for either of the two setups.In this work, we investigate TI/FM and TI/mdTI bilayersassuming that in both setups the spin torque originates inthe spin-momentum locking of the topological surface states.After a short description of our approach based on spindiffusion into the ferromagnetic layer [1], we discuss firstthe TI/FM bilayer with an in-plane magnetization, assuming atopological state at the interface; see Fig. 1(a). While it is not apriori clear that a TI next to a FM hosts a topological interfacestate, such a state is supported by density functional theorycalculations [14]. Then, we investigate the TI/mdTI structure.To describe this setup within the same scheme, we assume thatboth sides of the structure are “metallic,” i.e., have bulk states.Furthermore, we do not expect topological interface statesbetween the two TIs, but topological surface states on eachside of the total structure; [15] see Fig. 1(b). Note that whilea current in the bulk may lead to additional contributions tothe spin torque due to the spin Hall effect [4–6], we focus hereentirely on the role of the topological surface states. Finally,we discuss our findings and propose ways to disentangle thevarious contributions to the spin torque.II. METHODThe states at the surface of a topological insulator canexert a torque on an adjacent ferromagnet, which for in-planemagnetization is purely field like [16]. This fieldlike torque canintuitively be understood looking at the surface states describedby the Dirac HamiltonianHk = vF(zˆ × σ ) · k − μ (1)with σ the Pauli matrices acting in spin space and zˆ theunit vector in the z direction. Further, μ = 0 is the chemicalpotential away from the charge neutrality point. The velocityoperator v = ∂kHk is directly proportional to the spin operatorS = (/2)σ and readsv = 2vF(zˆ × S). (2)While the TI has a vanishing equilibrium spin expectation,a finite current density jx = en〈vx〉neq [Figs. 1(a) and 1(b)],where e is the electron’s charge and n is the electron density,2469-9950/2016/93(12)/125303(4) 125303-1 ©2016 American Physical SocietyFISCHER, VAEZI, MANCHON, AND KIM PHYSICAL REVIEW B 93, 125303 (2016)Mj(a)FMTId45ºxyz(b)jmdTITIdd21j21FIG. 1. The heterostructures we consider in this work: (a) TI /FM bilayer [1,2] with a topological surface state at the interface and(b) TI/magnetically doped TI bilayer [3] with surface states at the twoopposite surfaces (indicated in red). The current in both cases runsin x direction and the in-plane magnetization M = M m is along thein-plane diagonal.yields a spin density〈Sy〉neq = − 2evF jx. (3)It is important to note that in a steady-state situation ofa translationally invariant system (see [17] for effects ofscattering on FM boundaries), which is the situation we areinterested in, there is no transfer of momentum between thetopological surface state and the adjacent ferromagnet. Hence,there is also no net transfer of spin from the surface states to theferromagnet as is the case in the situation of the spin Hall effect.However, the magnetic moments of the ferromagnetic layercouple to the surface-state spins through Hex = −ex m · Swith m the magnetization direction in the ferromagnet [10,11].Thus, the spin polarization on the TI surface leads to afieldlike torque of the form T = ex m × 〈S〉neq, which foran in-plane magnetization is out of plane. We show in thefollowing how for an FM layer thicker than the diffusionlength, spin diffusion leads to an additional in-plane torque(Slonczewski-like torque), in a way similar to the spin-currentinjection in HM/FM bilayers [4–6].Given the spin polarization at the TI surface, Eq. (3), as aninput, we consider the diffusion of (itinerant) spins into theferromagnetic metal and the torque they thereby exert. Thediffusion (in the z direction) leads to a steady-state transversespin density through [18]0 = −∇ · Ji − 1τJ(S × m)i − 1τφ[ m × (S × m)]i − Siτsf,(4)where the spin current (for the ith spin component) is given byJi = −D ∇Si (5)with D the diffusion coefficient. The second term in Eq. (4)describes the precession of the spins around the moments of theFM with τJ the spin precession time. The third term capturesthe relaxation of the spin component perpendicular to m withτφ the spin decoherence time, and the last term is the spindiffusion with time scale τsf . In the following, we use λsf =5 nm [19] and values for λJ and λφ of order 1 nm (λ2i = Dτi).III. TI/FM BILAYERFor the setup of Refs. [1,2], Fig. 1(a), we solve Eqs. (4) and(5) requiring no spin current through the outer boundary of theFM, J (d) = 0, where d is the thickness of the ferromagneticlayer. For the TI/FM interface, we assume that, due to theexchange interaction, the itinerant spins of the FM right at theinterface align with the spin density of the TI interface, i.e.,S(0) = γ 〈S〉neq with γ of order 1 [20]. With these boundaryconditions, the spin distribution in the z direction is given byˆS(z) = S⊥(z) + iSz(z) = S0 cosh[ˆk(z − d)]cosh( ˆkd) (6)withˆk =√λ−2‖ − iλ−2J , (7)and λ−2‖ = λ−2sf + λ−2φ . S⊥(z) is the in-plane spin densityand S0 = |S(0) × m| is the initial spin density (z = 0), bothperpendicular to m. Figure 2 shows the in-plane spin densityS⊥ perpendicular to the magnetization (solid line) and Sz alongthe z axis (dashed line) for d = 8 nm. Note that this thicknessd ≈ 8 nm  1/k′ with ˆk = k′ + ik′′. Using Eq. (6), we canthus approximateˆS(z) ≈ S0e− ˆkz = S0 cos k′′ze−k′z − iS0 sin k′′ze−k′z, (8)i.e., both components oscillate and decrease exponentially; seeFig. 2.Figure 3 shows the integrated torque as a function of theFM layer thickness d. Assuming the spin angular momentumto be a good quantum number, the torque is given by the spatialchange of the spin current compensated by the spin relaxation,ˆT =∫ d0dz[− ∂z ˆJ (z) − 1τsfˆS(z)], (9)FIG. 2. Spin accumulation in the ferromagnet (d = 8 nm) as afunction of distance z from the TI/FM boundary, where the solid(dashed) line denotes S⊥ (Sz). For these plots, we used a spindecoherence length of λφ = 1 nm and the spin diffusion lengthof Permalloy λsf = 5 nm [19]. Green (red) curves correspond to aspin-precession length λJ = 1 nm (λJ = 0.5 nm).125303-2SPIN-TORQUE GENERATION IN TOPOLOGICAL . . . PHYSICAL REVIEW B 93, 125303 (2016)FIG. 3. Integrated torque as a function of the FM thickness d . Weagain set λsf = 5 nm, and the solid (dashed) lines denote the in-plane(out-of-plane) torque.where we again use the short forms ˆT = T⊥ + iTz and ˆJ =J⊥ + iJz. Given the spin distribution in the z direction ofEq. (6), we findˆT = S0(1λ2φ− iλ2J)Dˆksinh( ˆkd)cosh( ˆkd) (10)→ S0Dˆk(1λ2φ− iλ2J). (11)For the limit in the last line, we used d → ∞. As expectedfrom the fast decay of the spin density in Fig. 2, the torqueis “deposited” within only a few nanometers. The total torqueexerted on the ferromagnet as a function of the thickness dthus stays constant with layer thickness.For the geometry described in Fig. 1(a), the spin polariza-tion perpendicular to the magnetization of the FM is√2/2of the total polarization 〈Sy〉neq, and we find for the thick-FMlimit (d  1/k′)ˆT = −2Dˆk(1λ2φ− iλ2J)√22jxevF. (12)In analogy to the spin Hall angle θSH = (2eJS)/(JC), whichdescribes the spin Hall current per charge current, we definethe spin-torque efficiencyˆθ =ˆTjx2e= −√22DvF ˆk(1λ2φ− iλ2J). (13)For λJ ∼ λφ  λsf , the out-of-plane and in-plane spin-torqueefficiencies are of comparable magnitude. Using λJ = λφ =1 nm, λsf = 5 nm, vF = 5 × 105 ms−1, and a typical diffusioncoefficient D = 1 − 10 cm2 s−1, we find for the in-plane andout-of-plane-torque efficiency |θ⊥| = 0.15 − 1.5 and |θz| =0.065–0.65.IV. TI/mdTI BILAYERWe apply the same scheme now to investigate the setup ofRef. [3], Fig. 1(b), namely a bilayer of a TI (thickness d1) anda Cr-doped TI (thickness d2). At sufficiently low temperature,the doped TI exhibits ferromagnetism due to the magneticmoments introduced by Cr doping [21]. Within our approach,FIG. 4. The two torque components as a function of the TIthickness d1 for S1 = −S2 for λJ = λφ = 1 nm (in the mdTI) andλsf = 5 nm (on both sides) and d2 = 6 nm. The solid (dashed) linedenotes the in-plane (out-of-plane) torque. Panel (b) shows the twocomponents for fixed d1 = 3 nm [gray bar in (a)] as a function of theratio | S1|/| S2| for | S1| + |S2| fixed.the key difference between the TI/mdTI bilayer setup and theTI/FM setup is then the spatial location of the topologicalsurface states. Assuming no topological distinction betweenTI and mdTI, we do not anticipate a topological state at theinterface. Instead, we expect two surface states, one on eachnaked surface [see Fig. 1(b)]. These two surfaces carry thecurrent j1 and j2 with associated spin-polarization S1 and S2.Now the boundary conditions for the spin-diffusion equation(4) as stated for the TI/FM bilayer has to change. First, the spindensity on the two sides are S(0) = S1 and S(d1 + d2) = S2. Inaddition, we require that the spin density and the spin currentmatch at the interface, i.e., at z = d1.Figure 4(a) shows the integrated torque of a 6-nm-thickmdTI as a function of d1 for j1 = j2 and thus S1 = −S2, wherewe use again λJ = λφ = 1 nm (in the mdTI) and λsf = 5 nm.For d1 = 0, i.e., no TI next to the mdTI, the contributions fromthe two surface states exactly cancel and upon increasing d1 thetorque grows monotonically with the fieldlike torque alwayssmaller than the transferlike torque. The two currents will ingeneral not be identical, and Fig. 4(b) shows the two torques ford1 = 3 nm and d2 = 6 nm, the dimensions of the experimentalsetup, for different ratios of | S1|/| S2|. As long as | S1| ≈ |S2|,the spin-transfer-like torque dominates, in accordance with theexperimental results of Ref. [3].V. DISCUSSION AND CONCLUSIONSIn this work, we analyzed the spin-torque generation inTI-based heterostructures arising from the spin-momentumlocking of the topological surface states. Considering itinerantspins that diffuse in the ferromagnetic side (either FM or125303-3FISCHER, VAEZI, MANCHON, AND KIM PHYSICAL REVIEW B 93, 125303 (2016)mdTI), we find both an out-of-plane (fieldlike) and an in-plane (Slonczewski-like) torque. For realistic parameters, aspin-torque efficiency of the order of |θ | ≈ 0.1–1 should beexpected. This agrees with the reported values in Refs. [1,2]and is comparable to or larger than the largest value ofspin-torque efficiency observed in HM/FM structures to date[4–6,22]. However, we do not find as large a spin-torqueefficiency as reported in Ref. [3] within our approach.Within our model, both components of the torque stemfrom the combination of the inverse spin-galvanic effect of theTI surface and spin diffusion into the FM. The two torquecomponents not only differ in their direction, but also intheir behavior under M → − M: While the fieldlike torquechanges sign, the Slonczewski-like torque does not. This canhelp distinguish in-plane torque arising from out-of-plane spinpolarization [23] from Slonczewski-like torque. For “metallic”TIs, an additional spin-transfer-like torque arises from the bulkspin Hall effect. As transport is dominated by the surfacestates for thin TIs [24], we still expect the two componentsof the torque to be of comparable magnitude. In the case of theTI/mdTI heterostructure, the fact that the transferlike torqueis more than an order of magnitude larger than the fieldliketorque, however, hints at a dominant contribution from thebulk.In closing, we comment on limits of the applicability of ourapproach to extremely thin FM layers. As the total spin torquestays constant independent of FM layer thickness for d 2 nm, thin FM layers are preferable for device applications.However, our calculation treating the FM layer in the zdirection to be in the diffusive regime relies on a FM layer thatis thicker than its mean free path. For a device with an FM layerthinner than the diffusion length, the device should be modeledusing a semiclassical Boltzmann approach or through quantumtunneling of spins [9,25–27]. Our simple model can alreadyguide ferromagnetic resonance measurements, which do notrequire such thin FM layers, and help distinguish the variouscontributions to the spin torque in TI based heterostructures.ACKNOWLEDGMENTSThe authors are grateful to A. Mellnik and D. Ralph forhelpful discussions. M.H.F. and E.-A.K. acknowledge supportfrom NSF Grant No. DMR-0955822 and from NSF Grant No.DMR-1120296 to the Cornell Center for Materials Research.M.H.F. further acknowledges the Swiss Society of Friendsof the Weizmann Institute of Science. A.M. was supportedby the King Abdullah University of Science and Technology(KAUST).[1] A. R. Mellnik, J. S. Lee, A. Richardella, J. L. Grab, P. J.Mintun, M. H. Fischer, A. Vaezi, A. Manchon, E. A. Kim,N. Samarth, and D. C. Ralph, Nature (London) 511, 449(2014).[2] Y. Wang, P. 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Spin-Torque Generation in Topological-Insulator-Based Heterostructures

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