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Theory of the inverse spin galvanic effect in quantum wells

The understanding of the fundamentals of spin and charge densities and currents interconversion by spin-orbit coupling can enable efficient applications beyond the possibilities offered by conventional electronics. For this purpose we consider various forms of the frequency-dependent inverse spin galvanic effect (ISGE) in semiconductor quantum wells and epilayers taking into account the cubic in the electron momentum spin-orbit coupling in the Rashba and Dresselhaus forms, concentrating on the current-induced spin polarization (CISP). We find that including the cubic terms qualitatively explains recent findings of the CISP in InGaAs epilayers being the strongest if the internal spin-orbit coupling field is the smallest and vice versa (Norman et . 2014, Luengo et al. 2017), in contrast to the common understanding. Our results provide a promising framework for the control of spin transport in future spintronics devices.

The understanding of the fundamentals of spin and charge densities and currents interconversion by spin-orbit coupling can enable efficient applications beyond the possibilities offered by conventional electronics. For this purpose we consider various forms of the frequency-dependent inverse spin galvanic effect (ISGE) in semiconductor quantum wells and epilayers taking into account the cubic in the electron momentum spin-orbit coupling in the Rashba and Dresselhaus forms, concentrating on the current-induced spin polarization (CISP). We find that including the cubic terms qualitatively explains recent findings of the CISP in InGaAs epilayers being the strongest if the internal spin-orbit coupling field is the smallest and vice versa [1,2], in contrast to the common understanding. Our results provide a promising framework for the control of spin transport in future spintronics devices.

I. INTRODUCTION
The spin galvanic effect (SGE) and its Onsager reciprocal effect are currently the focus of an intense investigation in a large variety of physical systems including metals, semiconductors, van der Waals heterostructures and topological insulators [3][4][5][6][7]. The effect allows the "spinto-charge interconversion", where a non-equilibrium spin polarization yields an electrical current (SGE) and, conversely, an applied electrical current is able to orient the electron spin producing the ISGE. In the latter case one speaks also of CISP. In the literature different names refer to the same effect, often depending on the context where the phenomenon is being investigated. A discussion about the nomenclature can be found in Ref. [8]. On symmetry grounds the SGE arises when, due to restricted symmetry conditions as in gyrotropic media [9,10], specific components of polar and axial vectors transform according to the same representation. On a microscopic level, instead, the lack of inversion symmetry lifts the spin degeneracy leading to a momentum dependent spin splitting, which acts as an internal effective magnetic field. As a consequence Bloch electron states have their spin quantization axis dependent on the momentum direction. This aspect gives rise to a well defined spin texture around the Fermi surface, which can be experimentally measured, for instance, by the spin-pumping technique [6,[11][12][13] and by pump-probe techniques as in semiconducting epilayers [1,2,14]. After pumping polarized radiation into the electron system, one can observe a degree of precession of the induced spin polarization in the internal magnetic field. An essential ingredient is the external electric field, which unbalances the occupation of momentum states, yielding a net internal field. In a semiconducting epilayer the spin-orbit coupling (SOC) acts via two microscopic mechanisms. At the bulk level, the lack of inversion symmetry of the lattice as in GaAs het-erostructures is responsible for the Dresselhaus term [15], which depends on the third power of the electron momentum. However, when the electron system is confined in one direction, say along the z axis, the Dresselhaus spinorbit coupling (DSOC) becomes linear in momentum. On the other hand the lack of inversion symmetry with respect to the growth direction of the epilayer yields the Rashba term [16], which is linear in the momentum and in the spin operators. A combination of the linear DSOC and the Rashba spin-orbit coupling (RSOC) produces a characteristic spin texture, where maximum and minimum values of the internal field align along the [1,1] and [1,-1] crystallographic axes, depending on the strength of the two types of SOC. The pump-probe technique used in Refs. [1,2] is capable of reconstructing the texture of the internal magnetic field by varying the direction of the applied electric field, thus allowing the measurement of the DSOC and RSOC.
Theoretical investigations of both linear DSOC and RSOC in a two-dimensional electron gas (2DEG) [17,18] concluded that the induced spin polarization is proportional to the internal magnetic field and, hence, the former aligns with the latter. However, the experimental results of Refs. [1,2] showed the opposite behavior: the maximum spin polarization occurs in correspondence of the minimum value of the internal field and vice versa. Based on the model developed in Refs. [19,20], a possible explanation has been proposed in Ref. [2], by allowing SOC also from random impurities (see also Refs. [21,22]). The latter have a two-fold effect. On the one hand, they introduce a second channel for spin relaxation, referred to as the Elliott-Yafet mechanism, in addition to the Dyakonov-Perel (DP) one associated with the linear RSOC and DSOC. On the other hand, as found in Refs. [19,20], the interplay of linear RSOC and DSOC with the impurity SOC yields a negative SGE, which tends to decrease the induced spin polarization described in Refs. arXiv:1804.03607v1 [cond-mat.mes-hall] 10 Apr 2018 [17,18].
The aim of this paper is to study theoretically the experimentally relevant regimes of the ISGE in semiconductor structures, including the dependence of the spin polarization on the frequency of the driving electric field. The understanding of the frequency-dependent response allows one to set the limits on the timescale of the spin control by the electric field. We extend our studies beyond the conventional diffusive regime, that is to the case when the spin precession rate due to the spin-orbit coupling is of the order of the impurity-determined scattering rate, as can be achieved in modern high-mobility structures (see Ref. [23] as an example). To correspond to the experimental realizations, in addition to the linearin-the electron momentum spin-orbit coupling, we include the cubic terms in the Hamiltonian. These SOC terms are important for weak antilocalization in quantum wells [24] and for the persistent spin helix dynamics [25,26]. We demonstrate that the unusual experimental results of Refs. [1,2], can be explained by taking into account this cubic SOC. Indeed, the steady current-induced spin density is controlled by the balancing of the spingeneration and spin-relaxation torques. In the absence of cubic SOC, the linear RSOC and the DSOC contribute to both torques and, as a result, one obtains the alignment of the spin polarization along the internal SOC field. Our results demonstrate in detail that the cubic SOC, by itself, can only affect the spin-relaxation torques without inducing a spin-generation torque. When both linear and cubic SOCs are present, the generation-and relaxationtorques are affected differently and the spin polarization is no longer bound to align along the internal field, corresponding to the results of Refs. [1,2].
The layout of the paper is as follows. In Section II we introduce the formalism based on the Eilenberger equation for the quasiclassical Green function. In Section III we apply this formalism to the evaluation of the ISGE in the case of linear RSOC and DSOC and study its frequency dependence within and beyond the diffusive regime. Section IV demonstrates the absence of the ISGE when only the cubic SOC is present. In Section V we consider the interplay between linear and cubic RSOC and in Section VI we analyze the ISGE in a system where both linear RSOC and DSOC are present together with cubic DSOC. The Section VII presents the conclusions and relation to the experiment. Some details of the calculations are provided in the Appendices.

II. THE EILENBERGER EQUATION
We consider electrons confined in a two-dimensional (xy) plane subject to impurity scattering and in the presence of SOC. The Hamiltonian of the model in the pres-ence of a generic intrinsic SOC has the form where V (r) and p = (p x , p y ) represent the impurity potential and the vector of the momentum, respectively. The random potential has zero average and V (r)V (r ) = δ(r − r )n i v 2 0 , with v 0 being the singleimpurity scattering amplitude and n i being the impurity concentration. In the following, we choose units such that = 1 for the sake of simplicity. The vector b can be defined as the effective magnetic field due to the Rashba-Dresselhaus SOC. In Ref. [27], the Eilenberger equation for the quasiclassical Green function was derived in the presence of a SOC of the type shown in the Hamiltonian (1). To present a consistent analysis, we first recall the key steps of the derivation. The starting point is leftright subtracted Dyson equation for the Keldysh Green functionǦ, which has the form [28] ∂ tǦ + 1 2 where the self-energyΣ includes disorder effects and the curly brackets denote the anticommutator. In the Wigner coordinates, the Green function is described aš G =Ǧ(p, , x, t), where p and are the Fourier transform of the relative coordinates x 1 − x 2 , t 1 − t 2 and x = (x 1 + x 2 )/2, t = (t 1 + t 2 )/2 are coordinates of the center of mass. Whenever it is not strictly necessary, we drop the explicit dependenceǦ(p, , x, t) for simplicity's sake. The quasiclassical Green function is defined aš where ξ = p 2 /2m−µ is the energy measured with respect to the chemical potential µ in the absence of SOC. For the Green function, following [27] we make the ansatž with G R 0 and G A 0 being, respectively, the retarded and advanced Green functions in the absence of external perturbations with the self-energy Σ R(A) (derived later) due to the impurity potential and σ 0 the identity matrix. According to the ansatz (4), in the equilibrium one obtainš Since the main contribution to the ξ-integral comes from the domain |ξ| µ, it is sufficient to expand b around the small values of ξ. In the limit of |b| small compared to the Fermi energy, we have where the subscript "0" denotes the values taken at the Fermi surface and the p ± refers to the Fermi momentum in the ±-band. It is useful to introduce the projection operators for the two spin subbands as As a result, the semiclassical Green functionǧ, defined in Eq. (3), can be written aš from which we finď where ∂ ξ is the partial derivative taken with respect to ξ. By means of (10), one can show thať and, moreover, for any function of momentum one obtains: Eqs. (3) and (13), by integrating over the energy ξ and retaining terms up to the first order in |b|/ F , allow to derive the Eilenberger equation in the form [27,29,30] ν=± ∂ tǧν + 1 2 The self-energyΣ appears in the collision integral on the right hand side and describes the spin-independent scattering by disorder. The standard self-energy in the limit of the self-consistent Born approximation has the form [ where n 0 = m/2π is the density of states in the absence of SOC (with m being the electron effective mass). The brackets · · · denote the angular average over the momentum directions. Finally, τ is the elastic scattering time at the Fermi level. Notice thatg R andg A do not depend on the SOC and, thus, have no spin structure, i.e.g R = σ 0 and g A = −σ 0 solve the retarded and advanced components, respectively, of Eq. (14). Then, by using Eq. (10), we show that g R = σ 0 − ∂ ξ (b 0 · σ). Hence in the equilibrium we have which defines g eq . The Keldysh (K) component of the collision integral can be presented in the form Then the Keldysh components of the linearized Eilenberger equation according to Eq. (14) can be written as [27] ( where, by definingp = p/|p|, In the presence of SOC, g K can be written as a system of four equations according to the spin structure of the quasiclassical Keldysh Green function, i.e.
The internal magnetic fields b = b (N ) , due to intrinsic RSOC and DSOC can be classified by the power N of their momentum dependence [32]. Notice that we use the notation (N ) for the superscript to emphasize the label character of the symbol N and to avoid confusion with the power function. In the above equation,b does not depend on the modulus of the momentum. Hence, the retarded component of the Green function according to Eqs. (12)-(16) reads where N = 1 (N = 3) for the linear (cubic) SOC. In the presence of both SOCs b = b (1) + b (3) .
We now consider the Eilenberger equation in the presence of an external electric field. In order to study an infinite system under a uniform time-dependent electric field, we use the minimal substitution where |e| and E are the absolute values of the electron charge and the applied electric field, respectively, and E ≡ (Ê x ,Ê y ) = (cos φ, sin φ) with φ being the angle of the field with respect to the x-axis. Hence, we can go back to Eq. (18) and solve it for the system under the influence of a uniform time-dependent electric field E = EÊ as from which one obtains Notice that g K in Eqs. (26)(27) represents a column vector, whose components are defined in Eq. (23). By taking the angular average of Eq. (27), one obtains a closed equation for g K in terms of which the physical observables, such as the spin polarizations S i are calculated with [33] By using the Pauli matrices expansion of Eq. (23) in Eq. (19), one can write the Eilenberger equation (26) and (27) we can neglect N 1 to leading order in b 0 / F , thus decoupling the spin sector from the charge one. Furthermore, by using the expressions of M 0 , N 0 and S E from the Eqs.(A1, A2, A3) one can show that after angular average g z decouples from g x and g y . As a result the in-plane spin dynamics is reduced to problems described by 2 × 2 matrices.

III. INVERSE SPIN-GALVANIC EFFECT: BEYOND THE DIFFUSIVE REGIME
In this section we will evaluate the ISGE in the presence of the linear RSOC and DSOC. The evaluation will not be restricted to the diffusive approximation bτ 1, where the SOC is small compared to the disorder broadening. Hence, we will extend the previous results obtained in the diffusive regime [19,20,[34][35][36][37][38][39][40][41][42]. This case will also serve as an example of the way our formalism works. In a 2DEG the effective magnetic field due to the combination of the linear RSOC and DSOC reads [43] b (1) where α 1 and β 1 are the magnitudes of the linear RSOC and DSOC, respectively. The terms S E proportional to the uniform electric field are derived by using Eq. (A3) withẼ = −|e|Eτ v F ∂ g eq and s 11 = α 1 sin 2φ + β 1 cos 2φ, By performing the angular average of Eq. (27) for a uniform system, one getŝ includes both the spin relaxation and the frequency dependence effects. To solve the above equation, we have to perform several integrals with respect to the momentum direction, as listed in Appendix B. Under the uniform time-dependent electric field, we have where a = 2τ p F and L = 1 − iτ Ω, with Ω the variable associated to the Fourier transform with respect to time t (see the term with the time derivative in Eq. (19)) . The matrix elements appearing in Eq.(34) read with For the two dimensionless quantities aα 1 and aβ 1 we may consider two different regimes. As we assumed at the beginning, the SO splitting and the disorder broadening are much smaller than the Fermi energy F . For instance, in the Rashba model we have We can rewrite aα 1 in terms of the two small parameters α 1 /v F and 1/ F τ as According to Eq. (38) and the relation between α 1 /v F and 1/ F τ one can define two different regimes depending on which one dominates [44,45]. The first one is the diffusive regime, corresponding to a high impurity concentration, i.e. aα 1 1 and the Dyakonov-Perel spin relaxation. The second regime which occurs at aα 1 1, goes beyond the diffusive limit and describes the opposite situation of a relatively low concentration of impurities, where the spin relaxation time is close to τ [46]. To analyze these two regimes and a crossover between them in a simple form, we focus in this Section on a model with the linear RSOC and DSOC only. In the diffusive limit (C → 0) we can neglect the terms with higher order of the Rashba-Dresselhaus SOC since In such a case, the second term in Eq. (35) and the first one in Eq. (36) vanish. One may notice also that C = 0 when either α 1 = 0 or β 1 = 0. Finally, by using Eq. (28) we can write a generalized Bloch equation for the spin density asΓ where the matrixω describing the spin generation torque in the right hand side of the above equation is given bŷ with S 0 = −|e|τ n 0 E and and Correspondingly,Γ has the phenomenological meaning of a spin relaxation torque matrix, and the resulting spin density S is obtained as a result of the balance between the generation and the relaxation torques. Clearly, in the diffusive regime, when C 1, δ is very small, and one recovers the standard DP spin relaxation. Equation (41) is one of the main results of the paper. The contributions to the spin torque, dependent on δ and C, appear in Eq. (41) only when the interplay of the Rashba-Dresselhaus SOC is considered beyond the conventional diffusive approximation of Refs. [19,20,38]. Furthermore, the powers of L take into account terms relevant at high frequencies.
We also notice that at α 2 1 = β 2 1 , the contributions of the RSOC and the DSOC cancel each other, which leads to a pure gauge configuration [47], where the CISP does not appear.

A. Inverse spin galvanic effect in the linear Rashba model
In this subsection, we will solve the generalized Bloch equations (41) numerically for different RSOC magnitudes. After setting β 1 = 0 in Eq. (41), the Bloch equations in the 2DEG Rashba model read with S α 0 = −|e|n 0 τ α 1 . In the static limit when the frequency is zero, i.e. L = 1, the spin polarization becomes which is the Edelstein result [18]. In the following equation, we consider the frequency-dependent ISGE by inserting L = 1 − iΩτ in Eq. (46). In this case, the real and imaginary components of the spin density become zero, respectively, when When the imaginary part of the ISGE vanishes, the real part dominates, and vice versa, leading to a dependence of the ISGE on the field frequency. We define the frequency-dependent spin-galvanic (SG) conductivity, which can be found from Eq. (46), as Notice that both the charge current and spin density are odd under time reversal. Hence the Onsager relations imply that the SG and inverse SG conductivities are equal.
In the numerics we use the normalized conductivities to evaluate their frequency behavior, Ω max being the frequency at the maximum χ ij SG . In Fig. 1, we plot the real and imaginary parts as well as the absolute value of the normalized conductivity χ yx as a function of the frequency in units of S α 0 for the different magnitudes of RSOC. At sufficiently high frequencies, the conductivity vanishes, according to Eqs. (48)- (49), and a significant conductivity oscillation appears at Ω ∼ α 1 p F if one goes beyond the diffusive regime, that is the condition α 1 p F τ 1 is satisfied.

B. Inverse spin-galvanic effect in the linear Rashba-Dresselhaus SOC
As shown in the previous subsection, the ISGE shows a different behavior with respect to the dimensionless parameter aα 1 . In this subsection we consider the ISGE in the presence of both the RSOC and DSOC. In the diffusive regime we assume aα 1 1 and aβ 1 1 for high impurity concentration. In this limit, we can neglect the higher order terms in the Rashba-Dresselhaus SOC and in Ωτ . Hence, the generalized Bloch equation has the same form as Eq. (41) with theΓ andω given bŷ Notice that in the diffusive approximation, keeping just the first order in Ωτ yields the standard form of the Bloch equation, which coincides with the result of Refs. [19,20,38], when the extrinsic effect is not considered. For β 1 = 0, the above equations reproduce the results for the Rashba model presented in Eq. (46) for the diffusive limit. In the limit of the spin helix regime, where RSOC and DSOC are close to each other, we can write the spin polarization of Eq. (41) as where ∆ = α 1 − β 1 and |∆| |α 1 |, Ωτ 1. We notice that there is no effect for ∆ = 0, as expected, and the typical frequency scale is Ω ∼ a 2 ∆ 2 /2τ . This can be appreciated explicitly by the numerical evaluation of the SG conductivities when the RSOC and DSOC are present. In Fig. 2(a-b), we plot the normalized conductivities, χ xx and χ xy , as function of the frequency for different values of α 1 and β 1 in the diffusive regime. The different scale in the frequency behavior from the top to the bottom plots is related to the difference between the two RSOC and DSOC as shown in Eq. (54). In the diffusive regime, there is no finite-frequency peak in the conductivity, independent of the spin-orbit coupling details.
To solve numerically the generalized Bloch equations beyond the diffusive approximation, we have to keep all the orders of the spin-orbit coupling field b and frequencies Ωτ . As we demonstrated in Eq. (41), in this regime several new terms in the spin relaxation and the spin generation torques contribute to the Bloch equation. Figures 2(c-d) show the numerically obtained absolute value of the SG conductivity as a function of frequency beyond the diffusive regime. In contrast to the diffusive regime, we find a finite-Ω SG conductivity. The latter increases with the difference of the magnitudes of RSOC and DSOC.
By using Eqs. (A1)-(A3) one obtains for the generalized Bloch equation: The above equations show that, in the cubic RSOC and DSOC model, the spin generation torqueωE vanishes, although the spin relaxation rateΓ is nonzero. Notice that this result can also be derived in the diagrammatic approach, where it appears as a consequence of the vanishing of the vertex corrections. The latter contain the first harmonics of φ and, hence, the b field with the third harmonics does not contribute. This was first noticed by Murakami in the theory of the spin Hall effect [50].

V. INVERSE SPIN GALVANIC EFFECT IN THE LINEAR AND CUBIC RASHBA MODEL
As we have seen in the previous section, in the cubic SOC case, the ISGE does not exist. Here we evaluate the ISGE in the presence of both the linear and cubic RSOCs. In this case, the internal magnetic field reads where α 1 and α 3 are the magnitudes of the linear and cubic Rashba SOC, respectively. With the field b R in Eq. (A3), the source S E in Eq. (31) becomes: By using Eqs. (A1) and (A2), the matrix M 0 (N 0 + N 1 ) can be written as with a 1 = aα 1 and a 3 = ap 2 F α 3 and M 11 = (L 2 + 1 2 (a 2 1 + a 2 3 ))A 0 (66) and .
The diffusive regime occurs when a 1 1 and a 3 1, and for this regime one has D 1. In such case, all the integrals except the first one in Eqs. (66) and (67) can be neglected. Moreover D = 0 when either a 1 = 0 or a 3 = 0.
Finally, by using Eqs. (A1) and (A3) the matrixω appearing on the right hand side of Eq. (41) can be presented as:ω with and where the formulas for A 0 , . . . , A 6 are provided in Appendix B. Notice that when the cubic Rashba SOC goes to zero (α 3 = 0), Eqs. (70, 71, 72) reproduce the result derived in Eq. (47). Furthermore, Eqs. (71, 72) become zero when α 1 = 0, irrespective of α 3 . As a result we found that when the linear and cubic RSOC are present, the ISGE is strongly modified by several new terms in the spin relaxation and the spin generation torques.

VI. THE EFFECTS OF THE LINEAR RSOC AND DSOC WITH THE CUBIC DSOC
In this section we will evaluate the ISGE in the presence of the linear Rashba-Dresselhaus SOC combined with the cubic Dresselhaus SOC. To make the comparison with the experiments easier, we limit ourselves to the diffusive regime. For the given SOC, the effective SO field b is defined as with α 1 , β 1 and β 3 being the above introduced magnitudes of the linear (Rashba and Dresselhaus) and cubic (Dresselhaus) SOC. According to Eq. (A3) and using the form of Eq. (41) we can show To evaluate the ISGE in the diffusive regime, we need to expand in Eq. (27) where the DP spin relaxation for the linear RSOC and DSOC (Γ 1 ) and the cubic DSOC (Γ 3 ) are given bŷ Since the cubic SOC does not produce the spin generation torque by itself, one can expect that this torque contains the terms originating from the linear coupling and its interplay with the cubic one. Hence, the static limit (Ω = 0) of the Bloch equations in Eq. (41) can be rewritten as where the superscript indices correspond to the linear (1) and cubic parts (3) of the SOC and to their interplay (1,3). By using Eq. (A1) and Eq. (A3) the matricesω 1 and δω 1,3 appearing in the spin generation torque in the right hand side of Eq. (79) are given bŷ To obtain the spin polarizations, we take the inverse of the matrixΓ and multiply it by the spin generation torqueω. In the zero-frequency limit,Γ −1 is given bŷ Then the spin polarization is defined by where In Eq. (85), the first term corresponds to the linear Rashba-Dresselhaus SOC, whereas the second one is produced by the presence of the linear and cubic SOCs and represents their joint action. To compare our results with Ref. [19], we begin by considering the simple case of β 3 = 0. Thus, Eq. (85) is equivalent to Eqs. (52) and (53) and the spin polarization is given by where B int is the spin-orbit field induced by the electric current.
To analyze the frequency behavior of the CISP, we consider its real and imaginary components. The imaginary part originates from L = 1−iΩτ in the matrixΓ, whereas the spin generation torque is frequency independent in the diffusive regime. The normalized conductivities, χ xx and χ yx , are shown in Fig. 3 as a function of frequency for different values of the cubic DSOC (2β 3 τ p 3 F ) and fixed values 2α 1 τ p F = 0.1 and 2β 1 τ p F = 0.3. We have shown that in the presence of the linear RSOC-DSOC and cubic DSOC the conductivity is the result of the interplay of these two mechanisms. Besides, the anisotropy of the spin polarization can be controlled by the strength of the cubic DSOC in addition to the present linear RSOC and DSOC.
To analyze such anisotropy of the ISGE, in Fig. 4 we present the vector plot of the spin polarization as a function of the electric field direction for different values of the cubic DSOC. In the two upper diagrams the ISGE is shown without (β 3 = 0, top) and with a weak cubic DSOC (2β 3 τ p 3 F = 0.1, middle). One can see that the largest magnitude of the ISGE occurs for the electric field and the current along the crystallographic direction [1,1] (effective linear SOC α 1 + β 1 ) while the smallest effect occurs for the field along the [1,-1] direction (effec-  tive linear SOC α 1 − β 1 ). In addition, the bottom plot in Fig. 4 shows that the increase in β 3 (2β 3 τ p 3 F = 0.2) considerably modifies the anisotropy of the ISGE, i.e. the strongest polarization is produced now for the spin along the [1,-1] direction and the smallest one corresponds to the [1,1] direction. This picture is consistent with the experimental result [1,2].

VII. CONCLUSIONS
In this work, we have studied theoretically the currentinduced spin orientation in quantum wells by applying the approach based on the quasiclassical Green functions. The theory has been developed for the systems where both the linear and the cubic in the electron momentum spin-orbit couplings are present, and the corresponding Eilenberger equation have been derived. From this equation we obtained the generalized Bloch equations governing the spin dynamics of the carriers, permitting us to study a strong spin-orbit coupling sufficient to place the spin dynamics beyond the diffusive regime. Compared with previous studies, in the case of a sufficiently strong coupling, we found several new terms arising from the interplay of spin-orbit coupling symmetries. For the linear in the momentum coupling, we calculated numerically the current-induced spin polarization as a function of frequency of the driving electric field. We obtained that this polarization can be increased by using the highfrequency fields. Since the linear coupling contributes to both the spin generation and spin relaxation torques, in the static limit the spin polarization always aligns along the internal spin-orbit coupling "magnetic" field. We noticed that the purely cubic SOC has only effect on the spin relaxation torque, without inducing a spin generation torque. When both the linear and the cubic coupling are present, the spin generation and spin relaxation are affected differently. As a result, the spin polarization is no longer parallel to the spin-orbit field, depending on the relative strength of the linear and cubic couplings. This feature agrees with recent experiments [1,2]. In general, the approach developed in this paper can both lead to a better understanding of the spin transport in semiconductors and to a finding of efficient operational regimes of spintronics devices. The matrix forms of the linear operators appearing in the expressions (19) and (27) are