Giant spin-orbit torque and spin current generation in carriers at oxide interfaces

Interfacing polar insulating oxides may result in the formation of interface-confined, high density carriers with a number of emergent properties. A local coupling of carriers to a helimagnetic oxide layer is shown to lead to an effective spin-orbit interaction with a band splitting determined by the local s-d exchange interaction. As a result a bias-induced, non-equilibrium long-lived spin density normal to the spiral plane is predicted. We find that the spin accumulation is strongly enhanced when only one of the two bands is at the Fermi level and the chemical potential is close to the band edge. As a consequence we predict a large spin-transfer torque and spin current emission that can be utilized for oxide-based spintronic applications.


Introduction
Non-magnetic control of spins is essential for a number of spintronic devices with novel functionalities and lowenergy consumptions. The driving mechanisms are diverse [1,2], such as the spin-Hall effect [3][4][5], currentinduced spin torques [6,7], the spin-Seebeck effect [8], or the magnetoelectric effect [9][10][11][12][13]. Generally, a key common element is the presence of a spin orbital interaction (SOI) which can lead to a substantial currentdriven torque or spin accumulation. This in turn can be utilized to steer the magnetization of a ferromagnetic (FM) layer [7] for example. The largest efficiency of spin-transfer torques so far is observed in the topological insulator (TI) bismuth selenide (Bi 2 Se 3 ) with the Rashba SOI [14,15] and magnetically doped TI hetrostructures [16]. Here we show that the spin-torque ratio (i.e., the strength of the torque per unit charge density) can be further enhanced by orders of magnitude when utilizing an emergent spin-polarization of a high-density twodimensional electron gas (2DEG) or liquid formed at the interface of oxide heterostructures that involve helical magnetic order.
Such 2DEGs with carrier density~10 14 cm −2 and mobility~10 4 cm 2 V −1 s −1 were realized for LaAlO 3 / SrTiO 3 (LAO/STO) or RTiO 3 /STO polar interfaces (where R is a trivalent rare Earth ion) [17]. Moreover, the STO-layer retains the bulk character even down to a thickness of 2 unit cells [18]. This sheet of 2DEG can be excellently confined in STO/1-monolayer-RO/STO heterostructures [19] and STO/SrTi 0.8 Nb 0.2 O 3 quantumwell structures when the thickness of the SrTi 0.8 Nb 0.2 O 3 layer becomes less than 1.56 nm [20]. Experiments on the gate-control of the Rashba-type spin-orbit coupling in LAO/STO heterostructures have been reported [21,22]. Furthermore, experiments [23] evidenced anisotropic magnetoresistance in (STO) quantum wells (<1 nm, with carrier density7 10 14 cm −2 ) epitaxially embedded in ferrimagnetic GdTiO 3 (GTO) or antiferromagnetic SmTiO 3 . What happens if a helimagnet film is involved, e.g. if RMnO 3 (R=Tb, Dy, Gd) [24,25] replaces one of the GTO layer or is deposited atop of highly confined STO quantum wells [20,26]. Experimentally, the helical spin structure of TbMnO 3 (TMO), for instance, is found to persist down to 6nm films on YAlO 3 [001] substrates [27]. In this case, due to electronic correlations the 2DEG carriers couple to the local magnetic moments whose spiral structure leads to an (gauge) effective SOI [28]. Acting on traversing carriers this SOI results in an emergent spin polarization of 2DEG, as shown below. Remarkably this effective SOI resembles the semiconductor case for which the Rashba SOI and Dresselhaus SOI have equal strengths Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. [29,30]. Therefore, only a weak decay of spin polarization coherence occurs during a (nonmagnetic) momentum-dependent intra-band scattering. At low temperatures, the inter-band scattering is suppressed because of the band separation stemming from the local s-d exchange interaction. Thus, a large and long-lived spin accumulation emerges in 2DEG as only the lower subband is occupied with electrons. Such a generated spin accumulation is very useful for oxide-based spintronics, e.g., as highly efficient generators of spin transfer torque steering the magnetization of an attached FM layer (see figure 1(a)), or for coherent spin-current injection (see figure 1(b)). Further functionalities are expected with the RMnO 3 thin films grown on YAlO 3 [27] or STO [31] substrates. For example, some helimagnets (TMO) possess spin-current driven ferroelectric polarization and hence the SOI can be tuned by a moderate electric field [32].

Generalities and proposed setup
The proposed structures are sketched in figures 1(a)-(b). The essential building block is composed of the oxidebased 2DEG and a helimagnet such as TMO (e.g. GTO/STO/TMO or STO/LAO/TMO). Below = T 27 C K, TMO exhibits a spin-driven multiferroicity and a helical spin structure in the bc (or ab) plane [24,32]. In the following xz-plane is taken as the spiral plane, i.e.,  . The coupling of 2DEG to the helical spin order is captured by the s-d Hamiltonian It should be noted that the vector potential associated with the internal (real) magnetic field of the magnetic moments is negligible [28]. =  P k is the momentum operator, m is the effective electron mass, U is the local exchange coupling strength, and s is a vector with Pauli matrices components. Performing the unitary local transformation , the quantization axis becomes oriented along the vector m (r) at each point (s y does not change, as H then transforms to (a brief summary of mathematical details is included in the appendix A) The k-dependence of the effective SOI is analogous to the Dresselhaus [110] model and the balanced Rashba and Dresselhaus spin-orbit couplings [29,30,37,38], in which case the system possesses an exact SU(2) symmetry that renders spin lifetime infinite. In our case, given the large electron density and the high mobility of oxide electron gases [17], the electron-phonon and electron-impurity scattering are usually weak. Scattering is dominated by Coulomb interactions. However, due to the relatively large Fermi energy (∼0.5 eV), a moderate exchange interaction U implies long-lived spin coherence in 2DEG with inhomogeneous direction of spin polarization [39]. Furthermore, the degeneracy of each subband may result in screening of the Coulomb interaction at charged defects [40]. Together with the suppression of the kinetic energy due to the large effective mass, we expect in general large spin-orbit effects in oxide heterostructures.

Induced spin density
By mixing orbital and spin degrees of freedom, the effective SOI suggests a current-induced spin density in 2DEG [7]. Generally, such a non-equilibrium spin density influenced by applied electric fields and temperature gradients can be written as [41] w x y z , , and are spatial subscripts. s, q, and e denote respectively the spin, heat, and charge degrees of freedom. Formally, the conductivities k n ij s (here n = e or q) is introducible via the linear response function of the homogeneous spin density S i to a weak vector potential n A j . Given that the vector potential n A j is related linearly to a current density n J j , the response can be denoted by the Kubo product n ŵŜ J ; i j   [42].
. From equation (5) follows that only the conductivities k n yx s are finite. This means a steady non-equilibrium spin density S y (normal to the spiral plane) is generated by a steady current n J x in response to electric field E x and/or temperature gradient - T T x along the spiral wave vector, similar to the Aronov-Lyanda-Geller-Edelstein (ALGE) effect (also known as the Edelstein effect) [44][45][46]. No spin-Hall conductivity are expected due to our special form of SOI as a result of the coplanar spiral magnetic ordering [28,41]. Furthermore, we consider (the experimentally relevant) situation where the applied perturbations E x and - T T x are small such that a linear response approach is viable. Thus, the effect of the induced current n J x on the local magnetic moments is weak and can be neglected. Performing a Matsubara sum over e n and for w  0 the static spin density reads is the Fermi distribution function (its energy derivative). t k is the momentum-relaxation time. Obviously the induced spin density vanishes for a collinear spin order in which case a = 0. In the limit of U=0, the velocityv and the current density n J j come to be spin-independent and we have k = n 0 ij s , leading to a vanishing spin density as well. For strong s-d exchange interaction a U k F  with k F being the Fermi wave vector, the above equations evidence a linear q m -dependence of S y (see the dash lines in figure 2). The sign of q m and hence of S y is invertible by a transverse E field.
For low temperature e ¢ ( ) n F is strongly peaked which yields meV μm −1 , which are two orders larger than the values of GaAs-based quantum wells with the Rashba SOI in [43]. In figure 1, numerical results of spin density, given by equations (8) and (9) with or without the approximation a U k F  , are shown as functions of μ and U and T=20 K which is below the multiferroic transition temperature of TMO [24,32]. The low-temperature approximate solutions, equations (10) and (11) explain well the spin polarization behavior. The spin density is strongly enhanced only if the lower subband ( -E k ) is occupied with electrons as m < | | U (see figure 1(d)) and achieves its extremal value at the band edge. The small discrepancies stem from deviation from a U k F  when SOI coefficient a is large (see figure 2). Here we emphasize that this spin density survives even for weak s-d exchange interaction as long as the thermal excitation energy is lower than the band splitting (i.e., kT U  ) and m » U 1. For m > | | U both subbands are occupied with electrons or nearly empty. The numerical results show a non-zero spin density (see figure 2), which is two orders of magnitude lower than the values with only the -E k being occupied, however it is still the same amplitudes as that in semiconductor quantum wells with Rashba SOI [43]. Whereas, the linear q mdependence of the spin polarization n S y holds for all cases with weak effective SOI. Experimentally, such large and long-lived spin density at complex oxide interfaces is advantageous for highly efficient spin-orbit effects, as discussed below.

Giant spin transfer torques
Different from the usual case of a pure Hall-effect geometry, in which a spin toque would be due to a spin current injected into a FM layer, here the diffusion of the spin accumulation from the 2DEG results in a giant spin transfer torques, e.g., on a FM top electrode, for instance, a Co electrode (see figure 1(a) and [47]). Let the thickness of the FM layer be d FM and the z axis has its origin at the 2DEG/FM interface. In the steady-state regime the boundary conditions of itinerant spin density, ( ) z S FM in the FM are [14]: with  0 being the diffusion coefficient. From the equations governing the spin dynamics in the FM layer [48], the diffusion leads to a steady-state (itinerant) spin distribution [14] 1 with t J , t f , and t sf being the spin precession time, the spin decoherence time, and the spin diffusion-time in the FM electrode, respectively. S FM is perpendicular to the FM magnetization. The initial (three-dimensional) spin density S 0 is assumed to be aligned with the spin accumulation in 2DEG due to the exchange interaction, c = n d S S 0 E G where d EG is the thickness of the electron gas. χ describes the interface effect and is of the order 1 [14]. The spin torque on the FM moments is defined as the spatial change of the spin current compensated by the spin relaxation [14,48] In the limit of a thick FM layer, we find  [32], and l = 40 nm m for Co [49], the spin torque ratio is

Discussions
Our analysis assumes that the magnetic order is exactly coplanar in the spiral plane but it usually has a small deviation in experiment. This would result in an another (random) weak effective SOI in the spiral plane and introduces a spin dephasing in 2DEG. On the other hand, the parity symmetry breaking at the RMnO 3 interfaces due to ferroelectric polarizations in cycloidal RMnO 3 gives rise to a conventional Rashba-type SOI. Comparing the saturated polarization in RMnO 3 (~´--5 10 C m 4 2 [32]) with that in SrTiO 3 interfaces (~-0.1C m 2 [34]), the strength of the Rashba SOI at the RMnO 3 interface is expected to be however much smaller than 10 −2 eV Å at SrTiO 3 interfaces [21], which is already less than 10% of the gauge effective SOI considered here. Therefore, the average spin decoherence induced by these corrections is small. Furthermore, the electric field [32] and magnetic field [31] tuning the spiral spin structure (i.e. q m ) offer new ways to manipulate spin torques. Very recently, extreme mobility enhancement of 2DEG is found at LAO/STO interface by inserting a single-unit-cell insulating layer of La 1−x Sr x MnO 3 (LSMO) [56]. Mn 2+ sites with strong spin-orbit coupling are mostly trapped in the LSMO buffer layer, whereas the 2DEG carriers stay on the Ti site. In principle, the Dzyaloshinskii-Moriya interaction would arise from spin-orbit scattering of electrons in an inversion-asymmetric crystal field [57], we have then helical magnetic order and polar LSMO buffer layer [58], which could be another promising candidate of helimagnetic ultra-thin film in our considerations. q r q r sin , 0, cos , , 0, 0 m m is the spiral vector. At low temperature, the magnetization dynamics is much slower than that of the 2DEG carriers, so we can treat the local magnetic moments as classical and static. By interfacing 2DEG and the helimagnet, the carriers are subject to the local s-d exchange interaction and the dynamics of the 2DEG is governed by the Hamiltonian equation (1). Upon the gauge transformation , the quantization axis becomes oriented along the vector m (r) at each point. It simplifies the last spatially inhomegeneous term in equation(1) as Such a transformation introduces however an additional (spinor) gauge potential . Clearly, the gauge effective SOI is determined purely by the nontrivial topology of the spiral magnetic texture, and its strength depends linearly on the carriers wave vector k and on the magnetic spiral helicity q m . For the collinear magnetic phase (  q 0 m ) this effective SOI vanishes. By diagonalizing the HamiltonianH , we have two carrier subbands shown in figure 3(  Obviously, s á ñ y is odd in k x , and hence it would vanish upon the summation over a symmetric k space. However, in the presence of an electric field and/or a temperature gradient along the x direction, the Fermi surface is displaced. We find thus á ñ ¹ k 0 x and consequently a finite value of the induced spin density s = á ñ  S 2 y y . The other two components, s á ñ x and s á ñ z are periodic in the original space and averaged so to zero.