Current-induced spin polarization of a magnetized two-dimensional electron gas with Rashba spin-orbit interaction
A. Dyrdal J. Barnas V. K. Dugaev







^1Faculty of Physics, Adam Mickiewicz University,
ul. Umultowska 85, 61-614 Poznań, Poland 

^2  Institute of Molecular Physics, Polish Academy of Sciences,
ul. M. Smoluchowskiego 17, 60-179 Poznań, Poland

^3 Department of Physics and Medical Engineering, Rzeszów University of Technology, al. Powstańców Warszawy 6, 35-959 Rzeszów, Poland



Current-induced spin polarization in a two-dimensional electron gas with Rashba spin-orbit interaction is considered theoretically in terms of the Matsubara Green functions. This formalism allows to describe temperature dependence of the induced spin polarization. The electron gas is assumed to be coupled to a magnetic substrate via exchange interaction. Analytical and numerical results on the temperature dependence of spin polarization have been obtained in the linear response regime. The spin polarization has been presented as a sum of two terms – one proportional to the relaxation time and the other related to the Berry phase corresponding to the electronic bands of the magnetized Rashba gas. The spin-orbit torque due to Rashba interaction is also discussed. Such a torque appears as a result of the exchange coupling between the non-equilibrium spin polarization and magnetic moment of the underlayer.

71.70.Ej,  75.76.+j, 85.75.-d, 72.25.Mk

Current-induced spin polarization of a magnetized two-dimensional electron gas with Rashba spin-orbit interaction
    A. Dyrdał^1, J. Barnaś^1,2 and V. K. Dugaev^3
    December 30, 2023
=================================================================================================================







§ INTRODUCTION



Spin-orbit interaction leads, in general, to a number of  interesting transport phenomena, that enable generation and control of spin currents in a pure electrical manner. Two of the most prominent examples are the spin Hall and spin Nernst effects. The former (latter) effect consists in generation of pure spin current flowing perpendicularly to an external electric field (temperature gradient) applied to the system. These effects play currently an essential role in the processes of electrical generation and detection of spin currents<cit.>. For instance, the spin current can be used as origin of spin torque exerted on magnetic moments of a ferromagnet in a bilayer system consisting of a magnetic layer attached to a nonmagnetic one with strong spin-orbit coupling. This torque, in turn,  may induce magnetic dynamics and  even can reverse magnetic moment of the magnetic layer when the spin current exceeds some critical value.


Another consequence of the spin-orbit interaction in a system with mobile electrons is the current-induced  nonequilibrium spin polarization of conduction electrons. This effect was predicted theoretically in the '70s <cit.> for a two-dimensional electron gas (2DEG) with Rashba spin-orbit interaction, and then it was studied in various systems exhibiting spin-orbit interaction <cit.>. The current-induced spin polarization was also observed experimentally <cit.>, and currently it is attracting attention of many researchers <cit.>.

The current-induced spin polarization can also arise in a magnetic system, when it includes spin-orbit coupling. In such a case the induced non-equilibrium spin polarization interacts with the local magnetization via exchange coupling and creates a torque exerted on the magnetic moment <cit.>.

Moreover, it has been also shown that not only external electric field, but also a temperature gradient may lead to spin-orbit driven spin polarization <cit.>. These observations initiated a wide interest  in the field- and thermally-induced spin-orbit torques and new ways of magnetization switching, that could be alternative to the switching induced by spin transfer torques <cit.>.


In this paper we present theoretical results on the current-induced spin polarization of a magnetic 2DEG with Rashba spin-orbit interaction. Such a system is a basic model of various magnetic semiconductor heterostructures. The system consists of a 2DEG deposited on a magnetic substrate and interacting with the substrate via exchange interaction (see also Fig.1). To calculate the current-induced spin polarization we use the Matsubara Green function formalism which enables description of the temperature variation of the induced spin polarization. We derive some general formulas for the polarization and also present numerical results. The induced spin polarization is shown to include generally a term due to Berry curvature of the corresponding electron bands. Similar terms also appear in the spin-orbit torques following from exchange interaction of the electrons and magnetic underlayer.

The paper is organized as follows. In section 2 we describe the model system and also present the theoretical formalism and derive general formulas for the current-induced spin polarization.  In Section 3 we present analytical and numerical results in some specific situations; first, we consider the nonequilibrium spin polarization in the absence of  exchange field (Section 3 A), then we present results for  exchange field oriented perpendicularly to the plane of 2DEG (Section 3 B) and for exchange field  oriented in plane of 2DEG and  collinear (perpendicular) to the electric current, Section 3C (Section 3 D). In Section 4 we discuss the spin polarization in a general case of  arbitrarily oriented  exchange field. In Section 5, in turn, we consider relation of the nonequilibrium spin polarization with the Berry curvature of the corresponding electronic bands. The induced spin-orbit torque is briefly discussed in Section 6, while summary and final conclusions are in Section 7.





§ THEORETICAL OUTLINE
 


We consider a magnetized 2DEG with Rashba spin-orbit interaction, as shown schematically in Fig.1. The 2DEG is assumed to be deposited on a ferromagnetic substrate which creates an effective exchange field acting on the electron gas.




 §.§ Model

The single-particle Hamiltonian describing such a system can be written in the following form:

    Ĥ = ħ^2 k^2/2 mσ_0 + α (k_yσ_x - k_xσ_y) + 𝐇· ,

where  σ_0 and σ_n (for  n = {x, y, z}) are the unit and Pauli matrices   defined in the spin space, the parameter α in the second term of the Hamiltonian describes strength of the Rashba spin-orbit interaction, while k_x and k_y are the in-plane wavevector components. The third term of the above Hamiltonian describes  the effect of exchange field due to a magnetic substrate. This exchange field can be written as 𝐇 =J𝐌, with J standing for the  exchange parameter (J>0 for a ferromagnetic coupling between the 2DEG and magnetic substrate). Note, the exchange field 𝐇 is measured here in energy units. In spherical coordinates (see Fig.1),  components of the exchange field,  𝐇 = (H_x, H_y, H_z), can be written as

	
    H_x    = JM_x = JMsin (θ ) cos (ξ ),
    
    	H_y    = JM_y = JMsin (θ )sin (ξ ),
    
    	H_z    = JM_z = JMcos (θ ),


where M=|𝐌|, while θ and ξ are the polar and azimuthal angles, as defined in Fig. 1. In general, we take into account the temperature dependence of the magnetization  M(T), and assume it obeys the Bloch's law
M=M(T) = M_0[1 - (T/T_c)^3/2], where T_c is the Curie temperature of the magnetic substrate, and M_0 is the corresponding zero-temperature magnetization.

Eigenvalues of the Hamiltonian (<ref>) take the form

    E_±
    = ε_k±λ_𝐤 ,

where ε_k = ħ^2 k^2/2 m (with k^2 = k_x^2 + k_y^2), while  λ_𝐤 = [H^2 + α^2 k^2 -
2α (H_yk_x -H_xk_y)]^1/2.

Below we present the theoretical method based on the Matsubara-Green function formalism, and also derive a general formula for the nonequilibrium spin polarization induced by an external electric field.



 §.§ Method and general solution for current-induced spin polarization


To describe spin polarization induced by an external electric field we introduce
a time-dependent external electromagnetic field of frequency ω/ħ (note, here ω is energy) described by the vector potential
𝐀(t)=𝐀(ω)exp (-iω t/ħ).
The  electric field is  related to 𝐀 via the formula 𝐀(ω)=(ħ /iω )𝐄(ω).  Hamiltonian H_𝐀^E describing interaction of the system with the external field (treated as a perturbation) takes the form

    Ĥ_𝐀^E(t) =  - 𝐣̂^el·𝐀(t).

Here, the operator of the electric current density is defined as 𝐣̂^el = e 𝐯̂; with  e being the charge of electron (e < 0), and   𝐯̂=(1/ħ )∂Ĥ/∂𝐤 being the electron velocity operator. The  x and y components of the velocity operator have the following explicit form:

    v̂_x = ħ k/mcos(ϕ) σ_0 - α/ħσ_y,
    v̂_y = ħ k/msin(ϕ) σ_0 + α/ħσ_x,

where ϕ is the angle between the wavevector k and the axis x, i.e. k_x=kcos (ϕ) and k_y=ksin (ϕ), while the last terms in Eq. (<ref>) and  Eq. (<ref>) represent components of the anomalous velocity that  originates from the Rashba spin-orbit interaction.

Without loss of generality, we assume in this paper that the external electric field is oriented along the x-axis. Thus, the α-th (α =x, y, z) component of the quantum-mechanical average value of  spin polarization
induced by the external electric field can be found in the Matsubara-Green functions formalism  from the following formula:


    S_α(i ω_m)= 1/β∑_𝐤, nTr{ŝ_α G_𝐤(i ε_n + i ω_m) Ĥ_𝐀^E (i ω_m)G_𝐤(i ε_n) },

where ŝ_α = ħσ_α/2 is the operator of the α's spin component,  β =1/k_BT (with T and k_B denoting the temperature and Boltzmann constant, respectively), ε_n=(2n+1)iπ k_BT and ω_m=2miπ k_BT are the Matsubara energies, while  G_𝐤(i ε_n) are the Matsubara Green functions (in the 2× 2 matrix form).
Note, the  perturbation term takes now the form Ĥ_𝐀^E(i ω_m)=-ev̂_xA_x(i ω_m), with the amplitude of the vector potential A_x(i ω_m) determined by the amplitude E_x(i ω_m) of electric field through the relation A_x(i ω_m) = E_x(i ω_m) ħ/i (i ω_m).

Taking into account the explicit form of Ĥ_𝐀^E(i ω_m),
one can rewrite Eq.(<ref>)  in the form


    S_α (i ω_m)= - 1/βe E_x(iω_m) ħ/i (i ω_m)
    ×∑_𝐤, nTr{ŝ_α G_𝐤(i ε_n + i ω_m) v̂_x G_𝐤(i ε_n) }.


The sum over Matsubara energies in the above expression can be
calculated by the method of contour integration, <cit.>

    1/β∑_nŝ_α G_𝐤(i ε_n + i ω_m) v̂_x G_𝐤(i ε_n) 
    
    = - ∫_𝒞dz/2 π i f(z) ŝ_α G_𝐤(z + i ω_m) v̂_x G_𝐤(z) ,

where 𝒞 denotes the appropriate contour of integration and f(z) is a meromorphic function of the form (e^β z + 1)^-1, that has simple poles at the odd integers n, z = i ε_n (for details see Refs [abrikosov,mahan]).

Upon analytical continuation one obtains

    S_α(ω) = 
    
    - e ħ/ω E_xTr∑_𝐤∫d ε/2 π f(ε) ŝ_α( G_𝐤^R(ε + ω) v̂_x [G_𝐤^R(ε) - G_𝐤^A(ε)]. 
    
    + . [G_𝐤^R(ε) - G_𝐤^A(ε)] v̂_x G_𝐤^A(ε - ω)).

Here, f(ε ) is the Fermi-Dirac distribution function and  G_𝐤^R/A(ε)  is the impurity-averaged retarded/advanced Green function corresponding to the Hamiltonian (1). The Green functions take the following explicit form:

    G_𝐤^R/A(ε) = G_𝐤  0^R/A(ε) σ_0
     + G_𝐤  x^R/A(ε) σ_x + G_𝐤  y^R/A (ε) σ_y + G_𝐤  z^R/A(ε) σ_z,

where

	
    G_𝐤  0^R/A(ε)    = 1/2 [G_+(ε) + G_-(ε)], 
    
    	G_𝐤  x^R/A(ε)    = 1/2 λ_𝐤 (α k_y + H_x)[G_+(ε) -G_-(ε)],
    
    	G_𝐤  y^R/A(ε)    = 1/2 λ_𝐤 (-α k_x + H_y)[G_+(ε)-G_-(ε)],
    
    	G_𝐤  z^R/A(ε)    =  1/2 λ_𝐤 H_z [G_+(ε) -G_-(ε)],


with G_±^R(ε) = [ε + μ - E_±  + iΓ]^-1 and G_±^A(ε) = [ε + μ - E_±  - iΓ]^-1. Note, we assumed  Γ = ħ/2τ, with equal effective relaxation time τ in the two subbands.

 Using equation (<ref>) as a starting point and performing integration over ε we get finally  the following formula for the three components of the current-induced spin polarization:
 
    S_x = - e E_xħ
    ×∫d^2𝐤/(2π)^2{1/2 Γħ^2 k_x/2 m λ_𝐤 (α k_y + H_x) [f'(E_+) - f'(E_-)] . 
    
     + α/Γ(α k_y + H_x) (α k_x - H_y)/(2λ_𝐤)^2 + (2Γ)^2 [f'(E_+) + f'(E_-)] 
    
     - α H_z/(2 λ_𝐤)^2(2 Γ)^2/(2λ_𝐤)^2 + (2Γ)^2 [f'(E_+) + f'(E_-)] 
    
     - . α H_z/4 λ_𝐤^3 [f(E_-) - f(E_+)]},

 
    S_y = e E_xħ
    ×∫d^2𝐤/(2π)^2{1/4Γα/λ_𝐤^2 (α k_x - H_y)^2 [f'(E_+) + f'(E_-)].
    
     +ħ^2/m λ_𝐤 (α k_x - H_y) 1/4Γ [f'(E_+) - f'(E_-)]
    
     + . αΓ/λ_𝐤^2(1 - (α k_x - H_y)^2/λ_𝐤^2) f'(E_+) + f'(E_-)/(E_+ - E_-)^2 + (2Γ)^2},

 
    S_z = - e E_xħ
    ×∫d^2𝐤/(2π)^2{α H_z/Γα k_x - H_y/(2 λ_𝐤)^2 + (2Γ)^2 [f'(E_+) + f'(E_-)] .
    
     + ħ^2 k_x/2mH_z/λ_𝐤1/2Γ [f'(E_+) - f'(E_-)]
    
     - α/4 λ_𝐤^2(2Γ)^2 (H_x + α k_y)/(2 λ_𝐤)^2 + (2Γ)^2 [f'(E_+) + f'(E_-)]
    
     -. α/4 λ_𝐤^3 (H_x + α k_y) [f(E_+) - f(E_-)]}.

 Details on the derivation of the  above equations are presented in the Appendix A.

 Before presenting results on the current-induced spin polarization for an arbitrary orientation of the exchange field, we consider first some special cases.






§ SPECIAL CASES
 





 §.§ Zero exchange field


First, we reconsider the limit of zero exchange field, i.e. the limit of a nonmagnetized 2DEG, when only the y component of spin polarization survives. The general expression for S_y takes then the following form:


    S_y = e E_xħ∫dk k/(2π)^2{απ/4 Γ [f'(E_+) + f'(E_-)] . 
    
    + απΓf'(E_+) + f'(E_-)/(2α k)^2 + (2Γ)^2
    
    + . πħ^2 k/4 m Γ [f'(E_+) - f'(E_-)]  },

where f'=∂ f/∂ E.
Note that in this limit the eigenvalues have the form E_± = ε_k ±α k.

In the low-temperature regime, the above integrals can be evaluated analytically and one arrives at

    S_y = - e E_xħ∫dk k/(2π)^2{απ/4 Γ [δ(E_+-μ) + δ(E_--μ)] . 
    
    + απΓδ(E_+-μ) + δ(E_--μ)/(2α k)^2 + (2Γ)^2
    
    + . πħ^2 k/4 m Γ [δ(E_+-μ) - δ(E_--μ)]  }.

When both subbands are occupied (which corresponds to μ > 0), the Dirac delta functions in the above equation can be written in the form

    δ(E_± - μ) = m/√(2 mμħ^2 + m^2α^2)δ(k - k_±),

and finally one obtains

    S_y = 1/2 e E_xm α/2πħ^2τ
    
    - e E_xħ m α/16 πΓ√(2 m μħ^2 + m^2α^2)
    ×[ k_+/1 + (α k_+/Γ)^2 + k_-/1 + (α k_-/Γ)^2],

with k_± = ∓m α/ħ^2 + 1/ħ^2√(m^2α^2 + 2 m μħ^2).
The first term of Eq.(<ref>) corresponds to the Edelstein expression for the current-induced spin polarization in the so-called bubble approximation,

    S^0_y = 1/2 e E_xm α/2πħ^2τ .

Note, the impurity vertex correction is neglected in our  considerations. Such a correction leads to some renormalization of the spin polarization (for details see e.g. Ref. [edelstein90,BDDIchapt]). The second term in (<ref>) is a correction which originates from the imaginary  term in the nominator of the Green function and  products of two retarded or two advanced Green's functions (omitted in Ref. edelstein90). Note, the second term in Eq.(<ref>) vanishes in the quasi-ballistic limit (low impurities concentration), when Γ→ 0.





In the general case, i.e. for arbitrary T and arbitrary chemical potential μ, one should use the general formula (16). However, one point requires some comment. It is known, that for impurities with short-range (δ-like) potential and μ>0, the parameter Γ is constant, while for negative μ it increases and diverges when
μ approaches the bottom of the lower energy band <cit.>. Thus, at a certain value of μ,  μ =μ_ loc, the Ioffe-Regel localization condition <cit.> is obeyed, and the states become localized below μ_ loc. Accordingly, the results are valid beyond the localization regime, i.e. for μ >μ_ loc.


Now, we present some numerical results. In Fig.<ref>(a) we show the temperature dependence of spin polarization for four different values of chemical potential μ. Here, we should mention that the chemical potential also depends on temperature, thus a fixed value of chemical potential means that the carrier concentration varies. If however the system is gated one can keep chemical potential constant. Apart from this, the relaxation time τ (and thus the parameter Γ) may also depend on temperature T. This dependence, however, is neglected in Fig.<ref>. The spin polarization S_y was obtained from Eq.(16) and is normalized there to the corresponding value of S^0_y (note S^0_y does not depend explicitly on temperature). For the largest value of μ, the S_y component remains almost constant in the temperature range shown in Fig.<ref>(a), and is roughly equal to the corresponding value of S^0_y. For smaller values of μ,  in turn, the spin polarization becomes reduced monotonously with increasing T (see the curves for μ =0.02 eV and μ = 0.005 eV). For still lower values of μ, the temperature dependence is nonmonotonous - it first decreases and then slightly increases with temperature. To understand this behaviour we plot in Fig.<ref>(c) the spin polarization S_y  as a function of chemical potential for several values of temperature and the same Γ as in Fig.<ref>(a).  This figure clearly shows that spin polarization tends to S^0_y with increasing μ. Such a behavior is reasonable as the second term in Eq.(19) decreases with increasing μ (the effective role of finite Γ decreases with increasing E_+ - E_-). For small values of μ, however, the second term in Eq.(19) plays a role and the spin polarization is reduced. The temperature dependence appears when E_+ - E_-  at the Fermi level is of the order or smaller than kT, which takes place in the region of small values of μ. Moreover, this figure also shows that S_y decreases with increasing T, except a narrow region of small values of μ, where the temperature dependence  is nonmonotonous, exactly like in Fig.<ref>(a)
The temperature dependence is also shown in Fig.<ref>(b) for several values of Γ. This figure also shows that the correction due to the second term in Eq.(19) increases with increasing Γ. The latter behavior is shown explicitly in  Fig.<ref>(d) for indicated  values of μ. The decrease of spin polarization with increasing Γ is physically clear as the effective separation of the two Rashba bands becomes reduced with increasing Γ. The second term in Eq.(19) plays then an important role and leads to reduction of spin polarization.






 §.§ Exchange field perpendicular to plane of 2DEG


Consider now a magnetized 2DEG and let us begin with the situation when the exchange field (or equivalently substrate magnetization) is perpendicular to the plane of 2DEG, 𝐇 = (0, 0, H_z).
The eigenvalues of Hamiltonian (1) reduce now to the form E_± = ε_k±ζ, with ζ = √(H^2 + α^2 k^2).



The general expressions describing the two nonzero components of spin polarization take the forms

    S_x =  e E_xħ∫dk k/2 πα H_z/(2 ζ)^2{Γ^2/ζ^2 + Γ^2 [f'(E_+) - f'(E_-)] .
    
    - . 1/ζ [f(E_+) - f(E_-)] },



    S_y = e E_xħ∫dk k/(2π)^2απ{α^2 k^2/4 Γζ^2 [f'(E_+) + f'(E_-)] .
    
    + ħ^2 k^2/4 Γ m ζ [f'(E_+) - f'(E_-)] 
    . + Γ(2 - α^2 k^2/ζ^2) f'(E_+) + f'(E_-)/(2ζ)^2 + (2Γ)^2},


while S_z = 0.
Accordingly, the  electric field generates now spin polarization with both in-plane components nonzero, while the component normal to the plane of 2DEG (along the exchange field) vanishes exactly. Thus, the exchange field generates spin polarization along the electric field and also modifies the spin polarization  along the axis y.

In the  low temperature limit equations (<ref>) and (<ref>) lead to the following analytical expressions:

    S_x = -ħe/8 π E_xH_z/α[ζ_+ - ζ_-/ζ_+ζ_-.
    
    +.  α^2(ν_+/ζ_+^21/1 + (ζ_+/Γ)^2 - ν_-/ζ_-^21/1 + (ζ_-/Γ)^2) ],


    S_y = - ħe/16 π E_xα/Γ[ k_+^2/ζ_+ - k_-^2/ζ_-]
    
    - ħe/16 π E_x[(1-H_z/ζ_+) ν_+/1 + (ζ_+/Γ)^2.
    . + (1-H_z/ζ_-) ν_-/1 + (ζ_-/Γ)^2],

where ν_± =m/ħ^2 (1 ±m α^2/ħ^2ζ_±)^-1 represent  the density
 of states corresponding to the E_± subbands, respectively,  ζ_± = ζ (k = k_±), and  k_± = √(2m)/ħ^2√(m α^2 + μħ^2∓√(m^2α^4 + 2 m α^2ħ^2μ + H_z^2ħ^4)) are the Fermi wavevectors corresponding to the two subbands.

In the ballistic limit (extremely long relaxation time), Eq. (<ref>) takes the form

    S_x =  - e E_xħ∫dk k/2 πα H_z/4 ζ^3
     [f(E_+) - f(E_-)],

which after integration over k leads to

    S_x = - ħe/8 π E_xH_z/αζ_+ - ζ_-/ζ_+ζ_-,

i.e. to the first term in Eq.(<ref>).
The above expression does not depend on the relaxation time and due to the mathematical form of Eq.(<ref>)  it may be identified as the Berry phase related contribution to the spin polarization, that in turn may be responsible for anti-damping spin-torque. For details see Section V.

Numerical results on the current-induced spin polarization of the magnetized 2DEG with the exchange field oriented perpendicularly to the plane are shown in Fig.<ref>.
The dependence of S_x and S_z on the exchange field JM_0  is presented in Figs <ref>(a) and <ref>(b), respectively. Figure <ref>(a) clearly shows that S_x vanishes in the limit of
zero exchange field and then its magnitude grows rather fast with increasing JM_0. Then, it decreases to zero for large exchange fields. Magnitude of S_y, in turn, is nozero for zero exchange field, and increases with increasing JM_0. It reaches a maximum at some value of JM_0, and then decreases with a further increase in JM_0. Such a behavior with JM_0 can be understood since the relative role of Rashba coupling decreases with increasing JM_0. Note, that the S_x component is antisymmetrical with respect to sign reversal of JM_0, while the S_y component is then symmetrical.
Figures <ref>(c) and <ref>(d) present the x and y components of spin polarization as a function of temperature for fixed values of chemical potential and JM_0.  In numerical calculations we have assumed T_c = 150 K and therefore the S_x component vanishes for T ≥ 150 K. In turn, the S_y components is remarkably enhanced below T_c and drops to a weakly temperature dependent value (for fixed chemical potential and the parameter Γ) when T ≥ 150 K.

Variation of spin polarization with the chemical potential is presented in Figs <ref>(e) and <ref>(f) for different magnitudes of the exchange field JM_0. Magnitudes of both components increase monotonously with μ when μ is inside the energy region between the bottom edges of the two subbands. For μ in the vicinity of the bottom of higher energy band, these components reach maximum values and for larger μ they decrease with increasing μ. Note, the component S_x is roughly three orders of magnitude smaller than the S_y component.

In Fig. <ref>(g) and <ref>(h) we show the x and y components of the spin polarization as a function of the Rashba coupling constant. These figures clearly show that the absolute values of both components increase roughly linearly with α. However, some deviations from this linear dependence appear above certain values of α, where the increase is smaller.



 §.§ Exchange field in plane of 2DEG and perpendicular to electric field



In this section we consider the current-induced spin polarization for the magnetization vector (exchange field) oriented along  the y axis, i.e. when the exchange field is in the plane of two-dimensional electron gas and perpendicular to the current. In such a case the x and z components of the current-induced spin polarization vanish exactly, and the only nonzero component is  S_y  - like in the case of zero exchange field. This component, however, is modified by the exchange field.

Numerical results for S_y are shown in Fig.<ref>, where variation of S_y with the exchange field JM_0, Fig.<ref>(a), clearly shows that the spin polarization decreases relatively fast with increasing absolute value of JM_0 and is suppressed when the Zeeman-like term (due to exchange coupling to the substrate) dominates over the Rashba term. The suppression to zero of spin polarization at large JM_0 appears due to strong modification of electronic states by the Zeeman like term, and takes place for all values of chemical potential.

Temperature dependence of S_y is shown in Fig.<ref>(b) for two values of chemical potential and two values of JM_0. For the larger value of  JM_0, the spin polarization S_y vanishes in a broad temperature region and then increases when T approaches the Curie temperature, reaching the magnitude of S_y in the limit of a nonmagnetized  2DEG. This behavior is consistent with that in Fig.4a

In  Fig.<ref>(c) we show S_y as a function of chemical potential.
As follows from this figure, S_y increases monotonously with the chemical potential increasing from the minimum of the lower subband, and then becomes saturated for large values of μ. The rate of this increase as well as the  chemical potential at which the saturation appears depend on JM_0.
Spin polarization as a function of the Rashba parameter α is shown in  Fig.<ref> (d) for indicated values of the exchange field. In general, the y component of spin polarization increases now nonlinearly with the Rashba constant.



 §.§ Exchange field in plane of 2DEG and collinear with electric field





When the exchange field is oriented along the x axis, i.e. it is collinear with the external electric field, the x component of spin polarization vanishes, whereas the y and z components are non-zero. In general, the S_z component of spin polarization is roughly three orders of magnitude smaller than the S_y component. Variation of both components with the exchange field JM_0, temperature, chemical potential, and Rashba constant is presented in Figs<ref> (a-d).


Behavior of the S_y and S_z components with JM_0, T, μ and α is qualitatively similar to the corresponding behavior of the components S_y and S_x in the case with the exchange field normal to the plane of 2DEG, see Fig.<ref>. There are some differences of rather quantitative character, which follow from different electronic bands in these two situations. For instance, the S_z component varies with the chemical potential in a slightly different manner than the S_x component in Fig.3.
Weak difference also appear in the variation of the S_y component with temperature for T below the Curie temperature T_c. Similarly as in Fig. 3, both components behave  almost linearly with the Rashba parameter α.



§ NUMERICAL RESULTS FOR ARBITRARILY ORIENTED EXCHANGE FIELD 


Up to now we have discussed only some specific situations, when the exchange field is oriented along the three main directions: (i) along the electric field, (ii) normal to the electric field and to the plane of 2DEG, and (iii) normal to the electric field and oriented in the plane of 2DEG. Now let us consider a general case, when the exchange field is oriented arbitrarily. This orientation is described by the polar θ and azimuthal ξ  angles, as shown in Fig.1. Generally, all three components of spin polarization (i.e. S_x, S_y and S_z) can be nonzero.
In Fig. <ref> we present these components  as a function of both θ and ξ angles, see left panel in this figure. The right panel, in turn, presents several vertical cross-sections of the corresponding density plots from the left panel. In the specific configurations  considered in the preceding section, the results shown in Fig. 6 reduce to the corresponding ones discussed in Sec.3. This figure shows the regions in the (θ ,ξ) plane, where particular components of the spin polarization are large and where are small or suppressed to zero .




The results in a general case, like those presented in Fig. <ref> are required  when considering magnetic dynamics induced by spin torque due to spin polarization. Magnetic moment (and thus also exchange field) precesses then in space, and this time evolution is associated with time evolution of the spin polarization. In this paper, however,  we do not consider dynamical properties and focus rather on evaluating spin polarization in static situations.



§ RELATION WITH THE BERRY CURVATURE

Recently  H. Kurebayashi et al. <cit.>, based on experimental results, have proposed the  anti-damping spin-orbit torque mediated by the Berry phase <cit.>. In other words, they showed that the Berry curvature gives rise to the spin-orbit torque in systems with broken inversion symmetry.

Our results given by Eqs. (<ref>)-(<ref>) show that when the exchange field is nonzero, the inversion symmetry is broken and the general expressions for the x and z components of the spin polarization  contain terms that do not depend on relaxation rate, but are functions of the Fermi-Dirac distribution function instead of its derivative. Thus,  taking into account  the notation well known in the context of the anomalous Hall effect, we can rewrite Eqs (<ref>)-(<ref>) as follows:

    S_α = S_α^I + S_α^II

where the first term depends on the states in a close vicinity of the Fermi level: S_α^I = S_α[f'(E_±)], while the second term contains information from all electronic states: S_α^II = S_α[f(E_±)].
Now we show that the terms S_α^II are related to the Berry curvature.

To do this let us rewrite the Hamiltonian (<ref>) in the following form:

    H = ε_kσ_0 + 𝐧·

where  𝐧 = (α k_y - H_x, - α k_x - H_y, - H_z), and ε_k = ħ^2 (k_x^2 + k_y^2)/2m. The eigenvectors corresponding to the eigenvalues E_±  can be written as

    | Ψ_+⟩  = (
    [                √(λ_𝐤 + n_z/2 λ_𝐤); -n_x + i n_y/√(2 λ_𝐤 (λ_𝐤 + n_z));                                   ]),
    | Ψ_-⟩  = (
    [ -n_x - i n_y/√(2 λ_𝐤 (λ_𝐤 + n_z));                √(λ_𝐤 + n_z/2 λ_𝐤);                                   ]).

The Berry curvature of the n-th (n=1,2) band, ℬ⃗_n(𝐤), is defined as the rotation of the  Berry connection 𝒜⃗_n(𝐤) = i ⟨Ψ_n| ∇_𝐤|Ψ_n⟩ (for details see Refs Volovik,DiXiaoRevModPhys,NagaosaRevModPhys). Thus one can write

    ℬ_n^z(𝐤) = i [∂/∂ k_x⟨Ψ_n| ∂/∂ k_y|Ψ_n⟩ - ∂/∂ k_y⟨Ψ_n| ∂/∂ k_x|Ψ_n⟩] .

Combining Eqs. (28) to (30) we find for the Berry curvature,

    ℬ_±^z(𝐤) =∓α^2 H_z/2 λ_𝐤^3 .

Taking the expression above into account, the Berry phase related terms in the electrically generated spin polarization can be written as

    S_x^II = 1/2 e E_xħ/α∑_n∫d^2𝐤/(2π)^2 f(E_n) ℬ_n^z(𝐤)
    
    S_y^II = 0
    
    S_z^II = -1/2 e E_xħ∑_n∫d^2𝐤/(2π)^2 f(E_n) α k_y + H_x/α H_zℬ_n^z.

Note, these terms disappear in the absence of exchange field.



§ SPIN-ORBIT TORQUE




Due to exchange interaction, the current induced spin polarization exerts a torque  on the magnetic moment 𝐌.
This torque enters the Landau-Lifshitz-Gilbert equation for magnetic dynamics,

    ∂𝐦 /∂ t = -γ𝐦 ×𝐡_ eff  + α_g   𝐦×∂𝐦 /∂ t  +  ,

where 𝐦 =𝐌 /M is a unit vector along magnetic moment 𝐌, 𝐡_ eff is the effective magnetic field which includes external magnetic field, dipolar field, and anisotropy field, α_g is the Gilbert damping factor, and γ is the giro-magnetic factor.

To find the torque   we write the coupling energy of the magnetic moment and induced spin polarization as  =(2J/ħ )𝐒·𝐌=-𝐌·𝐡_ so, where 𝐡_ so is defined as

    𝐡_ so = -2J/ħ𝐒.

Taking the above into account, one can write
the torque  as a sum of a field-like  torque _f and damping-like torque _d,

    =_f + _d.

These components can be written in terms of the spin-orbit field  𝐡_ so as

    _f = -γ𝐦 ×𝐡_ so

for the field-like term, and

    _d = -α_g  γ𝐦 × (𝐦 ×𝐡_ so)

for the damping-like term.
Since the spin polarization includes terms related to the Berry curvature, the resulting spin-orbit torques include terms related to the Berry curvature as well.





§ SUMMARY AND CONCLUSIONS


Using the Matsubara Green function method we have calculated current-induced spin polarization in a magnetized two-dimensional electron gas with the Rashba spin-orbit interaction. The exchange field is shown to have a significant impact on the spin polarization. First, For some orientations of the exchange field, the component of spin polarization that appears in the absence of exchange field can be enhanced by the exchange field, while for other orientations this component can be suppressed. Second, exchange field  also generates the components of spin polarization which are absent in the limit of vanishing exchange field. We also note, that the states at the band edges may become localized due to disorder and the results may be not valid in the localization regime. 

Analytical and/or numerical results have been presented in some special cases, when exchange field is oriented along current or perpendicular to current (in-plane and perpendicular to the plane of 2DEG in the latter case). Numerical results have been also presented in a general case of arbitrary orientation of exchange field. We have found that the exchange field leads to terms in the spin polarization that can be related to the Berry curvature of the corresponding electron bands. Since the calculated spin polarization generates a torque which may induce dynamics of the magnetic moment, this torque includes terms related to the Berry curvature as well.



	This work was supported by  the Polish Ministry
	of Science and Higher Education through a research
	project Iuventus Plus in years 2015-2017 (project
	No. 0083/IP3/2015/73). A.D. also acknowledges support
	from the Fundation for Polish Science (FNP). V.D. acknowledges support from the National
Science Center in Poland under Grant
No. DEC-2012/06/M/ST3/00042. 






§ DERIVATION OF EQS. (<REF>), (<REF>), (<REF>)


The current induced spin polarizaton is evaluated starting from the equation (<ref>) that we rewrite in the following form:


    S_α = - e E_xħ/ω∫d^2𝐤/(2π)^2(𝒯_S_α^(1) + 𝒯_S_α^(2)),

where:

    𝒯_S_α^(1) = ∫d ε/2 π f(ε) ℐ_S_α^(1)(ε + ω, ε),
    𝒯_S_α^(2) = ∫d ε/2 π f(ε) ℐ_S_α^(2)(ε, ε - ω),

and the following notation has been introduced:

    ℐ_S_α^(1)(ε + ω, ε) = Tr{ŝ_α G_𝐤^R(ε + ω) v̂_x [G_𝐤^R(ε) - G_𝐤^A(ε)]},
    ℐ_S_α^(2)(ε, ε - ω) = Tr{ŝ_α [G_𝐤^R(ε) - G_𝐤^A(ε)] v̂_x G_𝐤^A(ε - ω) },


According to the above notation the S_x component of spin polarization is described by the following expressions:

    ℐ_S_x^(1)(ε + ω, ε) = ħ^2 k_x/2 m λ_𝐤 (α k_y - H_x)
    
    
    [G_𝐤 -^R(ε+ ω) G_𝐤 -^A(ε) - G_𝐤 -^R(ε + ω) G_𝐤 -^R(ε)
    
    
    + G_𝐤 +^R(ε + ω) G_𝐤 +^R (ε) - G_𝐤 +^R(ε + ω) G_𝐤 +^A (ε) ]
    
    - α/2 λ_𝐤^2 (α k_y - H_x) (α k_x + H_y)
    
    
    [ G_𝐤 -^R(ε + ω) G_𝐤 -^A(ε) - G_𝐤 -^R(ε + ω) G_𝐤 +^A(ε)
    
    -G_𝐤 -^R(ε + ω) G_𝐤 -^R(ε) + G_𝐤 -^R(ε + ω) G_𝐤 +^R(ε)
    
    - G_𝐤 +^R(ε + ω) G_𝐤 -^A(ε) + G_𝐤 +^R(ε + ω) G_𝐤 +^A(ε)
    
    + . G_𝐤 +^R(ε + ω) G_𝐤 -^R(ε) - G_𝐤 +^R(ε + ω) G_𝐤 +^R(ε)]
    
    - i α/2λ_𝐤 H_z[ G_𝐤 -^R(ε + ω) G_𝐤 +^A(ε) - G_𝐤 -^R(ε + ω) G_𝐤 +^R(ε)
    
    
     -G_𝐤 +^R(ε + ω) G_𝐤 -^A(ε) + G_𝐤 +^R(ε + ω) G_𝐤 -^R(ε)]



    ℐ_S_x^(2)(ε, ε - ω) = ħ^2 k_x/2 m λ_𝐤 (α k_y - H_x)
    
    
    [ G_𝐤 -^A(ε) G_𝐤 -^A(ε - ω) - G_𝐤 +^A(ε) G_𝐤 +^A(ε - ω)
    
    
    -G_𝐤 -^R(ε) G_𝐤 -^A(ε) + G_𝐤 +^R(ε) G_𝐤 +^A(ε - ω) ]
    
    - α/2 λ_𝐤^2 (α k_x + H_y) (α k_y - H_x)
    
    
    [ G_𝐤 -^A(ε) G_𝐤 -^A(ε - ω) - G_𝐤 +^A(ε) G_𝐤 -^A(ε - ω) .
    
    - G_𝐤 -^R(ε) G_𝐤 -^A(ε - ω) + G_𝐤 +^R(ε) G_𝐤 -^A(ε - ω)
    
    - G_𝐤 -^A(ε) G_𝐤 +^A(ε - ω) + G_𝐤 +^A(ε) G_𝐤 +^A(ε - ω)
    
    + . G_𝐤 -^R(ε) G_𝐤 +^A(ε - ω) - G_𝐤 +^R(ε) G_𝐤 +^A(ε - ω)]
    
    - i α/λ_2𝐤 H_z[ G_𝐤 -^A(ε) G_𝐤 +^A(ε - ω) - G_𝐤 +^A(ε) G_𝐤 -^A(ε - ω) 
    
    -G_𝐤 -^R(ε) G_𝐤 +^A(ε - ω) + G_𝐤 +^R(ε) G_𝐤 -^A(ε - ω) ]

Inserting Eqs. (<ref>), (<ref>) into Eqs. (<ref>) and (<ref>) respectively we get:


    ℜ𝔢[ 𝒯_S_x^(1) + 𝒯_S_x^(2)]
    
    =ωħ^2 k_x/2 m λ_𝐤 (α k_y - H_x) 2 Γ/ω^2 + (2Γ)^2 [f'(E_+) - f'(E_-)]
    
    
    + ωα/2 λ_𝐤^2 (α k_x + H_y) (α k_y - H_x) 2 Γ/ω^2 + (2Γ)^2 [f'(E_+) + f'(E_-)]
    
    - ωΓα/λ_𝐤^2 (α k_x + H_y) (α k_y - H_x)
    
    
    ( f'(E_-)/(E_+ - E_- - ω)^2 + (2Γ)^2 +  f'(E_+)/(E_+ - E_- + ω)^2 + (2Γ)^2)
    
    -ωα/2 λ_𝐤 H_z
    
    ( E_+ - E_- - ω/(E_+ - E_- - ω)^2 + (2Γ)^2f'(E_-) -  E_+ - E_- + ω/(E_+ - E_- + ω)^2 + (2Γ)^2 f'(E_+) )
    
    + ωα/2 λ_𝐤 H_zE_+ - E_-/(E_+ - E_-)^2 - ω^2 [f'(E_+) + f'(E_-)]
    
    - ωα/λ_𝐤 H_zf(E_+) - f(E_-)/(E_+ - E_-)^2 - ω^2+ ω^2α/2 λ_𝐤 H_zf'(E_-) - f'(E_+)/(E_+ - E_-)^2 - ω^2


In the limit of ω→ 0 we find x component of current-induced spin polarization given by Eq.(<ref>).



In turn, the S_y component of spin polarization is expressed by the following functions:

    ℐ_S_y^(1)(ε + ω, ε) = α/2[ G_𝐤 -^R(ε+ω) G_𝐤 -^A(ε) + G_𝐤 +^R(ε+ω) G_𝐤 +^A(ε).
    
    - . G_𝐤 -^R(ε+ω) G_𝐤 -^R(ε) - G_𝐤 +^R(ε+ω) G_𝐤 +^R(ε)]
    
    - ħ^2 k_x/2 m λ_𝐤 (α k_x + H_y)
    
    [ G_𝐤 -^R(ε+ω) G_𝐤 -^A(ε) - G_𝐤 -^R(ε+ω) G_𝐤 -^R(ε).
    
    - . G_𝐤 +^R(ε+ω) G_𝐤 +^A(ε) + G_𝐤 +^R(ε+ω) G_𝐤 +^R(ε)]
    
    - α/2 λ_𝐤^2[ (α k_y - H_x)^2 + H_z^2]
    
    [ G_𝐤 -^R(ε+ω) G_𝐤 -^A(ε) - G_𝐤 +^R(ε+ω) G_𝐤 -^A(ε).
    
    -  G_𝐤 -^R(ε+ω) G_𝐤 -^R(ε) + G_𝐤 -^R(ε+ω) G_𝐤 +^R(ε)
    
    + G_𝐤 -^R(ε+ω) G_𝐤 +^R(ε) - G_𝐤 +^R(ε+ω) G_𝐤 +^R(ε)
    
    - . G_𝐤 -^R(ε+ω) G_𝐤 +^A(ε) + G_𝐤 +^R(ε+ω) G_𝐤 +^A(ε)]



    ℐ_S_y^(2)(ε, ε - ω) = α/2[ G_𝐤 -^A(ε) G_𝐤 -^A(ε - ω) - G_𝐤 -^R(ε) G_𝐤 -^A(ε - ω) .
    
    + . G_𝐤 +^A(ε) G_𝐤 +^A(ε - ω) - G_𝐤 +^R(ε) G_𝐤 +^A(ε - ω) ]
    
    - ħ^2 k_x/2 m λ_𝐤 (α k_x + H_y)
    
    
    [ G_𝐤 -^A(ε) G_𝐤 -^A(ε-ω) - G_𝐤 +^A(ε) G_𝐤 +^A(ε-ω).
    
    - . G_𝐤 -^R(ε) G_𝐤 -^A(ε-ω) + G_𝐤 +^R(ε) G_𝐤 +^A(ε-ω)  ]
    
    - α/2 λ_𝐤^2[ (α k_y - H_x)^2 + H_z^2]
    
    
    [ G_𝐤 -^A(ε) G_𝐤 -^A(ε-ω) - G_𝐤 +^A(ε) G_𝐤 -^A(ε-ω).
    
    - G_𝐤 -^R(ε) G_𝐤 -^A(ε-ω) + G_𝐤 +^R(ε) G_𝐤 -^A(ε-ω)
    
    - G_𝐤 -^A(ε) G_𝐤 +^A(ε-ω)+ G_𝐤 +^A(ε) G_𝐤 +^A(ε-ω)
    
    + . G_𝐤 -^R(ε) G_𝐤 +^A(ε-ω) - G_𝐤 +^R(ε) G_𝐤 +^A(ε-ω)]

After integration over ε  in Eqs. (<ref>) and (<ref>) with integrands given by  (<ref>),(<ref>) we obtain the following expression:


    ℜ𝔢[ 𝒯_S_y^(1) + 𝒯_S_y^(2)]
    =
    - ħ^2 k_x/2 m λ_𝐤 (α k_x + H_y) 2 Γω/ω^2 + (2 Γ)^2 [f'(E_+) - f'(E_-)]
    
    - α/λ_𝐤^2(α k_x + H_y)^2
    2 Γω/ω^2 + (2 Γ)^2 [f'(E_+) - f'(E_-)]
    
    
    - α/λ_𝐤^2Γω[ (α k_y - H_x)^2 + H_z^2]
     
    
    
    (f'(E_-) /(E_+ - E_- - ω)^2 + (2Γ)^2 + f'(E_+) /(E_+ - E_- + ω)^2 + (2Γ)^2)

In the limit ω→ 0 we obtain the formula describing y component of current-induced spin polarization given by Eq.(<ref>).

Finally, the S_z component of the nonequilibrium spin polarization is described by following traces:

    ℐ_S_z^(1)(ε + ω, ε) = α^2 k_x/2 λ_𝐤^2 H_z
    
    [ G_𝐤 -^R(ε + ω) G_𝐤 -^A(ε) - G_𝐤 -^R(ε + ω) G_𝐤 +^A(ε) .
    
    - G_𝐤 -^R(ε + ω) G_𝐤 -^R(ε) + G_𝐤 -^R(ε + ω) G_𝐤 +^R(ε)
    
    - G_𝐤 +^R(ε + ω) G_𝐤 -^A(ε) + G_𝐤 +^R(ε + ω) G_𝐤 +^A(ε)
    
    + . G_𝐤 +^R(ε + ω) G_𝐤 -^R(ε) - G_𝐤 +^R(ε + ω) G_𝐤 +^R(ε)]
    
    + ħ^2 k_x/2 m λ_𝐤 H_z
    
    [ G_𝐤 -^R(ε + ω) G_𝐤 -^R(ε) - G_𝐤 -^R(ε + ω) G_𝐤 -^A(ε).
    
    + . G_𝐤 +^R(ε + ω) G_𝐤 +^A(ε) - G_𝐤 +^R(ε + ω) G_𝐤 +^R(ε)]
    
    + α/2 λ_𝐤^2 H_y H_z
    
    [ G_𝐤 -^R(ε + ω) G_𝐤 -^A(ε) - G_𝐤 -^R(ε + ω) G_𝐤 +^A(ε).
    
    - G_𝐤 -^R(ε + ω) G_𝐤 -^R(ε) + G_𝐤 -^R(ε + ω) G_𝐤 +^R(ε)
    
    - G_𝐤 +^R(ε + ω) G_𝐤 -^A(ε) + G_𝐤 +^R(ε + ω) G_𝐤 +^A(ε)
    
    + . G_𝐤 +^R(ε + ω) G_𝐤 -^R (ε) - G_𝐤 +^R(ε + ω) G_𝐤 +^R(ε) ]
    
    + i α/2 λ_𝐤 (H_x - α k_y)
    
    
    [ G_𝐤 -^R(ε + ω) G_𝐤 +^A(ε) - G_𝐤 -^R(ε + ω) G_𝐤 +^R(ε).
    
    - . G_𝐤 +^R(ε + ω) G_𝐤 -^A(ε) + G_𝐤 +^R(ε + ω) G_𝐤 -^R(ε) ]



    ℐ_S_z^(2)(ε, ε - ω) = α^2 k_x/2 λ_𝐤^2 H_z
    
    [ G_𝐤 -^A(ε) G_𝐤 -^A(ε - ω) - G_𝐤 +^A(ε) G_𝐤 -^A(ε - ω).
    
    - G_𝐤 -^R(ε) G_𝐤 -^A(ε - ω) + G_𝐤 +^R(ε) G_𝐤 -^A(ε - ω)
    
    - G_𝐤 -^A(ε) G_𝐤 +^A(ε - ω) + G_𝐤 +^A(ε) G_𝐤 +^A(ε - ω)
    
    + . G_𝐤 -^R(ε) G_𝐤 +^A(ε - ω) - G_𝐤 +^R(ε) G_𝐤 +^A(ε - ω)]
    
    + ħ^2 k_x/2 m λ_𝐤 H_z
    
    [ G_𝐤 +^A(ε) G_𝐤 +^A(ε - ω) - G_𝐤 -^A(ε) G_𝐤 -^A(ε - ω) .
    
    + . G_𝐤 -^R(ε) G_𝐤 -^A(ε - ω) - G_𝐤 +^R(ε) G_𝐤 +^A(ε - ω)]
    
    + α/2 λ_𝐤^2 H_y H_z
    
    [ G_𝐤 -^A(ε) G_𝐤 -^A(ε - ω) - G_𝐤 +^A(ε) G_𝐤 -^A(ε - ω) .
    
    - G_𝐤 -^R(ε) G_𝐤 -^A(ε - ω) + G_𝐤 +^R(ε) G_𝐤 -^A(ε - ω)
    
    - G_𝐤 -^A(ε) G_𝐤 +^A(ε - ω) + G_𝐤 +^A(ε) G_𝐤 +^A(ε - ω)
    
    + . G_𝐤 -^R(ε) G_𝐤 +^A(ε - ω) - G_𝐤 +^R(ε) G_𝐤 +^A(ε - ω)]
    
    + i α/2 λ_𝐤 (H_x - α k_y)
    
    
    [ G_𝐤 -^A(ε) G_𝐤 +^A(ε - ω) - G_𝐤 +^A(ε) G_𝐤 -^A(ε - ω).
    
    - . G_𝐤 -^R(ε) G_𝐤 +^A(ε - ω)  + G_𝐤 +^R(ε) G_𝐤 -^A(ε - ω)]

These two equations combining with Eqs. (<ref>) and (<ref>) lead to the following expression:

    ℜ𝔢[ 𝒯_S_z^I + 𝒯_S_z^II]
    = - ω2 Γ/ω^2 + (2Γ)^2ħ^2 k_x/2 m λ_𝐤 H_z [f'(E_+) - f'(E_-)]
    - ωα/2 λ_𝐤^2 H_z (α k_x + H_y)
    
    
     2 Γ/ω^2 + (2Γ)^2 [f'(E_+) + f'(E_-)]
    
     + ωΓα/λ_𝐤^2 (α k_x + H_y) H_z[f'(E_-)/(E_+ - E_- - ω)^2 + (2Γ)^2 + f'(E_+)/(E_+ - E_- + ω)^2 + (2Γ)^2]
    
     + ωα/2 λ_𝐤 (H_x - α k_y) [E_+ - E_- - ω/(E_+ - E_- - ω)^2 + (2Γ)^2 f'(E_-) + E_+ - E_- + ω/(E_+ - E_- + ω)^2 + (2Γ)^2 f'(E_+)  ]
    
     - ωα (H_x - α k_y) f'(E_+) + f'(E_-)/(E_+ - E_-)^2 - ω^2 - ωα/λ_𝐤 (H_x - α k_y) f(E_-) - f(E_+)/(E_+ - E_-)^2 - ω^2 - ω^2α/2 λ_𝐤 (H_x - α k_y) f'(E_-) - f'(E_+)/(E_+ - E_-)^2 - ω^2

In the dc-limit we get Eq. (<ref>).








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