Spin torque due to non-uniform Rashba spin orbit effect

Following the early theoretical descriptions of the spin-orbit-induced spin torque [S.G. Tan et al., arXiv:0705.3502 (2007); S.G.Tan et al., Ann. Phys.326, 207 (2011)], the first experimental observation of such effect was reported by L. M. Miron et al., Nature Mater, 9, 230 (2010). We present in this article three additional spin torque terms that arise from the non-uniformity in magnetization space of the Rashba spin-orbit effect. We propose a simple Rashba gradient device which could potentially lower switching current by n orders of magnitude, where large n measures a small magnetization change.


Introduction
Manipulating the magnetization of ferromagnetic (FM) materials by means of the spin transfer torque (STT) has been theoretically predicted in 1996 [1,2]. This prediction has since triggered a series of theoretical and experimental studies in both fundamental and application fields. In the applied physics community, the STT phenomenon can be observed in common nanoscale devices like the spin valve sensors, the magnetic tunnel junction storage cell, as well as in magnetic domain walls [3][4][5]. In fact, STT is now widely used for a magnetization switching process known as the current-induced magnetization switching (CIMS). The adoption of CIMS in the switching operation of memory cells has enhanced the packing density of cross-bar memory, and has significantly moved forward the commercialization schedule of these memories.
Recently it was found that spin orbit coupling (SOC) effect in FM materials could yet be another source of spin torque. The best known example is the Rashba SOC, originating from the structural inversion asymmetry (SIA), which has been found in semiconductor heterostructure [6][7][8] or at the surface of non-magnetic or magnetic metal multilayers [9,10] [14,15]. The physics of the SOC spin torque is different from the STT in that spin injection would not be needed in the case of SOC spin torque. It suffices to pass charge current directly into the FM material with SOC, the effect of spin polarization and spin transfer takes place within the single FM layer. By contrast, STT system requires a spin polarizing layer (normally a FM layer with hard magnetization) to spin polarize electrical current, and an efficient transport mechanism to inject spin current into a relatively soft FM layer where spin transfer torque could occur within.
SOC spin torque in a uniform FM system with Rashba effect has been well understood. In the event that Rashba constant varies spatially, a similar distribution of the spin torque would follow straightforwardly. In this paper, we will discuss spin torque effects in a magnetic system with non-uniform spin orbit constant in the space of the magnetization. In this system, Rashba constant is a function of the magnetization i.e.
, as well as its spatial gradient . We consider a Hamiltonian, where care has been taken to ensure its Hermiticity, of a FM structure with Rashba SOC where is the effective electron mass, is the conduction band electron momentum, is a spatially dependent Rashba SOC parameter, ( ) are the Pauli spin matrices, is the s-d exchange integral and is the local magnetization due to the localized d-orbitals in the magnetic materials. The Hamiltonian can be written in the following way (2) where ( ) is a gauge field due solely to the Rashba SOC, which can also be treated as a spin-dependent magnetic vector potential, and has found applications in spintronics based on non-Abelian gauge theory [16][17][18]. In the presence of local magnetization, it will be instructive to perform a separate local gauge transformation which essentially paves the way for the description of the adiabatic alignment of electron spin along the local magnetization. This transformation process aligns reference spin quantization axis to the local magnetization, resulting in a Hamiltonian in the rotated frame of where the unitary rotation matrix ( ⁄ ⁄ ⁄ ⁄ ) has been used, and are respectively, the polar and azimuth orientations of the local magnetization. Now, is the Pauli spin matrix of the new reference frame, | | √ , and are, respectively, the gauge fields due to the Rashba SOC in the rotated frame, and the spatially varying magnetization (domain wall). For better physical clarity, one can regard the gauge field (vector potential) as representing an effective spin particle generated by the combined presence of Rashba SOC and local magnetization.
Now consider a current propagating through the FM structure with spin orbit coupling. The interaction between the fermion field (electron) and the gauge field (effective spin particle) is given by where . Since in the adiabatic system where , spin is constantly aligned to the local magnetization, and there is no probability of the spin assuming its other eigenstate, one has ( ), and arrives at   6 The dynamics of can be described by the Landau-Lifshitz-Gilbert (LLG) equation. The total energy of this magnetic system also includes the exchange energy, magnetostatic energy, anisotropy energy, as well as the electron-spinparticle interaction energy which gives rise to . In the low-damping limit, the local magnetization will precess about a total effective field which can be obtained from the total energy functional, thus . Thus, the effective anisotropy field due to the Rashba SOC can be treated as an externally   1. (a) Schematic of a nanoscale device with current fixed along , and the Rashba effective electric field along , perpendicular to the material surface. (b) a Rashba gradient device comprising two devices of type (a). Technology that ensures a maximum , and a minimum , would deliver a maximum reduction of switching current density.
In the context of Fig.1(a), the first Rashba anisotropy field term of can generate motion of or [11][12]14]; one can verify the above by inspecting with . The second anisotropy fields of is related to the gradient in magnetization space. These fields can exist in all directions depending on the orientation of (see Table 1). The reasonable to suggest that these new anisotropy fields could potentially be useful for setting off the dynamics or triggering the switching of local moment with high saturation. We summarize the effects of the Rashba anisotropy fields for device of Fig.1(a). with in-plane as follows: One now considers a FM nanoscale structure with an anisotropy field along a particular axis and a current density flowing along the axis (Fig.1 propose to realize the effect of by having two devices of Fig.1(a) bordering each other as shown in Fig.1(b) In summary, we have proposed in this paper three additional spin orbit gradient spin torques and discussed the implementation of the Rashba gradient spin torque.
Rasbha gradient with local moment effect has the potential to lower switching current significantly with proper implementation. The domain wall effects are less clear in terms of how they can be harnessed, but we believe the dimension has important physical significance here, in the same way it has been used to lower the switching current with respect to the Rashba gradient with local moment. In general, we remain optimistic that the challenges related to the domain wall can be overcome in the near future and the estimate we provided above will make instant practical sense.