Eur. Phys. J. B73, 229–242 (2010) DOI:10.1140/epjb/e2009-00437-3
Regular Article
T
HE
E
UROPEAN
P
HYSICAL
J
OURNAL
B
Quantum kinetic equation for spin relaxation and spin Hall effect
in GaAs
H.C. Lee1,a and C.-Y. Mou2,3
1 Institute of Physics, National Chiao Tung University, Hsinchu, Taiwan 2 Department of Physics, National Tsing Hua University, Hsinchu, Taiwan
3 Physics Division, National Center for Theoretical Sciences, P.O. Box. 2-131, Taiwan
Received 30 July 2009 / Received in final form 22 October 2009
Published online 24 December 2009 – c EDP Sciences, Societ`a Italiana di Fisica, Springer-Verlag 2009 Abstract. We present a general quantum kinetic theory of spin transport based on the Kadanoff- Baym equation (KBE), which we use to study dynamical spin processes in semiconductors right down to femtosec-ond and nanometer scales. In our application of KBE we describe the evolution of the non-equilibrium 2×2 matrix Green function for carrier spin, averaged over the thermal bath. Spin relaxation effects are treated within the Kadanoff-Baym Ansatz (KBA), while carrier interactions are treated within the random-phase model of screening. We track the detailed oscillation of the spin- polarized carrier state within the coher-ence time. Our general kinetic approach also allows description of the spin Hall effect when both impurity scattering and the Fr¨ohlich interaction are included in the KBE collision term. We find that the level of spin current is very sensitive to the density of impurities, and that the Fr¨ohlich interaction can generate a considerable spin current. Significantly, the Fr¨ohlich term leads to a unique type of oscillatory behaviour in the spin current that is independent of impurity scattering effects.
PACS. 05.30.-d Quantum statistical mechanics – 05.60.Gg Quantum transport – 72.25.Dc Spin polarized transport in semiconductors – 73.63.Hs Quantum wells
1 Introduction
Manipulation of the spin degree of freedom in semicon-ductors has attracted considerable attention over the last decade [1–4] due to potential applications such as in quan-tum computation [1,2], magnetic random access mem-ory [3] and spin transistors [4]. Recently, a very fast spin relaxation in GaAs was reported [5]. The time as short as 110 fs± 10% for heavy holes was experimentally dis-covered. Such a time scale is likely too short for indus-trial applications, but is significant in quantum kinetic theory since energy non-conserving events [6] and mem-ory effects [7,8] are active in this time scale. In a previous report [7], memory effects are shown to be appreciable in the time evolution of non-equilibrium carriers, imply-ing that the memory effect on carrier- carrier scatterimply-ing (CCS) resembles Rabi oscillation. That result [7] moti-vates us to study whether the carrier also oscillates be-tween distinct spin-polarized states due to the memory effect prior to spin relaxation. In this work, starting with the Pauli equation, a 2×2 spin-dependent non-equilibrium Green-function matrix was utilized to construct the KBE, which was then applied to spin-dependent non-equilibrium
a e-mail: hclee@mail.nctu.edu.tw
CCS in the presence of a D’yakonov Perel’ (DP) mag-netic field [9]. The quantum kinetic oscillation between distinct spin-polarized states due to the memory (non-Markovian) CCS is demonstrated. Additionally, another oscillation that is the spin precession caused by spin-orbit coupling (SOC) term, Δij, at the frequency of−1|ImΔ12| is described.
The spin Hall effect (SHE) [10,11], namely, the ap-pearance of spin transverse transport driven by a longi-tudinal electric field, was predicted by D’yakonov Perel’ over 30 years ago. Recent experimental verification of the SHE [12] has prompted considerable discussion [13–15]. Since reversely spin-polarized particles have distinct di-rections of spin current (SC), such spin-particle separa-tion without using magnetic fields, magnetic materials or magnetic dopants makes spin-based device fabrica-tion compatible with convenfabrica-tional semiconductor process technology and is likely to be important in spintronic applications [1–4]. Theoretical interests of the SHE in-volve intrinsic (or extrinsic) scatterings [16,17] and bal-listic (or diffusive) transport [18,19]. Another concern is that, once spintronic devices are manufactured, the device model based on Boltzmann theory [20,21] is no longer ap-propriate as the device shrinks rapidly down to the scale in
length with momentum uncertainty compared to carrier’s momentum. Thus a quantum transport description [22,23] for the SHE is needed. In contrast to Kubo formula [24,25] and Keldysh formalism under quasiclassical approxima-tion [26], we base on an ab initio method to construct the spin-dependent KBE by using the non-equilibrium Green functions, into which the retarded Green functions by solving the spin-dependent Dyson equation are input. This work derives the KBE for the SHE, which incor-porates intrinsic, extrinsic effects, impurity and Fr¨ohlich interactions. Within the delta interaction approximation, a concise formula for the first-order SC with respect to an electric field can be obtained, indicating that only a quantum well (QW), not bulk, can have a non-zero SC due to the presence of both bulk (BIA) and surface in-version asymmetries (SIA). Furthermore, our numerical results show that the SC is very sensitive to the impurity density and that Fr¨ohlich interaction can generate a re-markable SC. Significantly, Fr¨ohlich interaction also leads a unique oscillatory behaviour in the SC, probably one kind of quantum kinetic oscillation, which does not occur in the impurity-induced SC.
The remainder of this paper is organized as follows. In Section 2, a general spin-dependent KBE is derived. In Section 3, the general KBE is applied to spin relaxation. Section 4 is devoted to derive and solve numerically a spatially-independent KBE for the SHE. By solving a long standing problem regarding spatially-entangled collision integrals, a spatially-dependent KBE for spin accumula-tion on lateral sides due to the SHE [27,28] is presented in Section 5 Conclusions are finally drawn in Section 6.
2 Derivation of spin-dependent KBE
The Pauli Hamiltonian is considered first. The field oper-ator ψ(r) is defined askuk(r)ck, where the annihilation operator ckis for Fermions and uk(r) is the single-particle state with wave vector k. The field operator is anti-commute. The Heisenberg equation of motion for ψ↑ and ψ↓[29–32] can then be written as ∂tψ↑=−i−1([ψ↑, H11] +[ψ↓, H12]) and ∂tψ↓ =−i−1([ψ↑, H21] + [ψ↓, H22]), re-spectively, where the diagonal and off-diagonal elements of Pauli Hamiltonian H11H12 H21H22 are−2m2∇2+U (r, t)+Δii and Δij(i=j), respectively; [ , ] denotes the commutator; U (r, t) is the potential energy; and Δ is the SOC term (see Appendix A). To keep the article concise, a compact form for either definitions or equations is used. This will greatly reduce the length of mathematical expression. Original ex-pressions for these compact forms are separately shown in Appendix B for better understanding.
The contour Green function is defined as Gc(1, 1) ≡
1
iTC[ψH(1)ψH+(1)], where 1(
)
presents (r1( ), t1()),
stands for the thermal average operator, and contour Tc[ ] follows the conventional path [33]. Via the equa-tion of moequa-tion for ψ↑(↓)and Green-function definition, the
spin-dependent Dyson equation is written as
D(1)Gc(1, 1) = δ C(1− 1)I + Cd2Σ c(1, 2)Gc(2, 1), (1a) [D∗(1)Gc(1, 1)]T = δC(1− 1)I + Cd2G cT (1, 2)ΣcT(2, 1), (1b) where δC(t1 − t1) is a contour delta function [33].
D(∗)(1()
) is a 2× 2 matrix, of which the diagonal element
is (−)i∂t
1() − H
(∗)
jj (j = 1 or 2) and the off-diagonal element is−Hjj(∗)(j=j).Σc(1, 1) [Gc(1, 1)] is a 2× 2 spin-dependent contour self- energy [Green-function] matrix, of which the diagonal element is Σssc (1, 1) [Gcss(1, 1)] and the off-diagonal element is Σssc(1, 1)[Gcss(1, 1)], where s= s in this context.
Via equations (1a), (1b) and the Langreth theorem [34,35], two kinds of KBE are obtained.
D(1)G(1, 1) ss( )− D∗(1)G(1, 1) ss() − s=↑,↓ Σss, Gss() + Σss, Gss( ) = 1 2 s=↑,↓ Σss, Gss( ) − Gss(), Σss (2a) D(1)G(1, 1) ss( ) − D∗(1)G(1, 1) ss()= s=↑,↓ ΣssrGss( )+ Σss Gass() −Gr ss( )Σss− Gss()Σssa , (2b)
where the retarded, advanced, lesser and greater Green functions (self energies) follow the definition in refer-ences [22,23]. Notably, Σ(1, 1) = [Σr(1, 1) + Σa(1, 1)]/2 while G(1, 1) = [Gr(1, 1) + Ga(1, 1)]/2, and ΣG and GΣ are abbreviated forms of Cd2Σ(1, 2)G(2, 1)
and Cd2G(1, 2)Σ(2, 1), respectively. Additionally, { , }
stands for the anti-commutator. Equations (2a) and (2b) will be applied to spin relaxation and the SHE (including the subsequent topic for spin accumulation), respectively.
3 KBE for spin relaxation
This section focuses on spin relaxation. Assume a one-band (conduction one-band) model and spatial independence. The equation for the spin-dependent carrier’s distribution matrix fk(t) at the wave vector k on the band struc-ture can be obtained using equation (2a) at an equal-time
limit [22], wherefk(t) equals−iGk(t, t = t). i [D(t)fk(t)− D∗(t)fk(t)]ss( )= 1 2 t −∞dt s=↑,↓ Σss ,k(t, t)Gss(),k(t, t) +Gss(),k(t, t)Σss,k(t, t)− Gss( ),k(t, t)Σss,k(t, t) −Σss,k(t, t)Gss( ),k(t, t) . (3)
By applying the Langreth theorem [34,35], the two-time lesser (greater) CCS self energy within the random phase approximation can be expressed as
i q G,s 1s2,k−q(t, t ) ¯V, s1s2,q(t, t), where ¯Vs,1s2,q(t, t)
is the mean-field screened potential at the exchanged wave vectorq and can be presented as
t −∞dt1
t −∞dt1
V¯qr(t, t1)L,s1s2,q(t1, t1) ¯Vqa(t1, t)
according to the Dyson-like equation [22]. Note that the notation of s1s2 can be spin-polarized or spin-flip, unlike that of ss. The polarization function L,s1s2,q(t1, t1) equals −2ikG,s1s2,k+q(t1, t1)G,s1s2,k(t1, t1). The retarded
(advanced) screened potential ¯Vqr(a)(t, t) is assumed as ¯
Vqδ(t− t), where ¯Vqis the screened Coulomb interaction. Therefore, Σs,1s2,k(t, t) can be written as
22
q,k ¯
Vq(t) ¯Vq(t)G,s1s2,k−q(t, t)G,s1s2,k+q(t, t)
× G,s1s2,k(t, t).
Using the KBA [36]
G,s1s2(t, t) = i[Grs1s2(t, t)G,s1s2(t, t) − G, s1s2(t, t)G a s1s2(t, t )] and the plane-wave approximation, Gr,as1s2,k(t, t) =
∓i
θ(±t ∓ t) exp[(−iek∓ γ)(t − t)/], where ekis kinetic
energy and γ is the damping constant; thus equation (3) can be rewritten as [37] ∂tfss(),k(t)−1 fss(),k(t) Im Δ12, s =↓ Im Δ21, s =↑ = 1 2 q,k ¯ Vq(t) t −∞dt V¯q(t) exp[−γ(t−t)/] cos[δ(t−t)/] × s=↑,↓ fss,k−q(t)fss,k+q(t)[1− fss,k(t)] × [1 − fss(),k(t)]− fss(),k(t)fss,k(t)[1− fss,k−q(t)] × [1 − fss,k+q(t)] , (4)
where δ, which is the non-conserving energy of CCS, equals ek−q+ ek+q− ek− ek. fk−qfk+q(1− fk)(1− fk) and fkfk(1− fk−q)(1− fk+q) are two kinds of Pauli factor.
The physical picture of quantum kinetic oscillation can be captured by drawing the memory integral in equa-tion (4) that is parallel to an atomic system. For ex-ample, take the last term on the right-hand side (RHS) in equation (4), states k and k − q can be regarded as two distinct levels of an atom, while the scattering from k to k+q can be regarded as an applied field at the strength of V (q) exp(−iωt). With the quantum dy-namical derivation [38], a Rabi-oscillation-like equation can be obtained as i ˙fk = fk−qVk,k−q and i ˙fk−q =
fkVk−q,k. Integrating ˙fk−q to time and inputting it into the other equation yields the memory integral of ˙fk = − 0tdt−2|Vk,k−q|2fk. The derived result demonstrates the equivalence between the memory integral and oscilla-tion. Thus the behaviour of the memory integral in equa-tion (4) is essentially equivalent to the carrier-carrier oscil-lation; however, equation (4) is somewhat complex due to Pauli factors and the summation over momentum space in band structure. The oscillation results from the quantum coherence between carriers. As time passes, the quantum coherence collapses and the quantum process reduces to a monotonic scattering process, which no longer oscillates between thek and k − q states, as if an atom stops oscil-lating when the quantum coherence between the photon and atom collapses and eventually emits (or absorbs) a photon. Similarly, equation (4) indicates an initially spin-coherent carrier oscillates between distinct spin-polarized states within the quantum coherence time due to the mem-ory integral. As the quantum coherence collapses, the car-rier stops oscillating and eventually completes spin relax-ation process. This oscillrelax-ation is generic and thus valid for other interactions. The CCS, not Fr¨ohlich interaction, was considered in this section because the exchange energy of CCS can be zero, which favors the oscillation between two distinct spin-polarized states with the same energy.
In addition to the carrier-carrier oscillation, equa-tion (4) shows another oscillaequa-tion (spin precession) that can be understood from the corresponding homogeneous solution. By inputting ∂tfss,k(t) in equation (4) into ∂t equation (4), the following equation is generated,
¨
fss−ImΔ12ImΔ21
2 fss= CT , where CT is the collision
term. As Δ21 = Δ∗12, ImΔ21 equals −ImΔ12. Accord-ingly, the homogeneous solution of equation (4) indicates the spin-polarized distribution function has an oscillating frequency at−1|ImΔ12|.
4 KBE for the spin Hall effect
This section describes how the KBE is applied to the SHE. The driven potential energy, U (r1, t1) =−qcr1· E, is con-sidered, where qcis the charge andE is an electric field. By applying Wigner transformation to equation (2b), where
r = r1 − r1, τ = t1 − t1, R = (r1 +r1)/2 and
equation (2b) in Fourier domain after transformation un-der the scalar potential gauge (φ =−qcE · R) and vector potential gauge (A = −qcTE) [22], respectively, can be represented as i ∂T+ qcE · ∇k+ k m∗∂ω ˜ Gss( )(k, ω, T ) − 2i ˜Gss()(k, ω, T ) Im Δ 12, s =↓ Im Δ21, s =↑ = dτ dτexp(iωτ ) s=↑,↓ ˆ Pss(k1,2, τ1,2, T1,2) × ˆQss()(k1,2, τ1,2, T1,2), (5)
where m∗is the effective mass. Additionally,
ˆ Pss(k1,2, τ1,2, T1,2) ˆQss(k1,2, τ1,2, T1,2) ≡ [ ˆΣssr(k1, τ1, T1) ˆGss(k2, τ2, T2) + ˆΣss(k1, τ1, T1) × ˆGass(k2, τ2, T2)− ˆGrss(k1, τ1, T1) ˆΣss (k2, τ2, T2) − ˆGss(k1, τ1, T1) ˆΣssa(k2, τ2, T2)], k1,2 ≡ k +2q E(τ±τ 2), τ1,2 ≡ τ 2 ∓ τ , τ≡ t 2− T
and T1,2 ≡ T ± τ2,1. Average-space (R) dependence is omitted and considered in the next section. By solving the Dyson equation, the retarded Green function can be obtained. ˜ Grss(k, ω) ≈ 1 ω − ek− Re σssr − iIm σrss − q2c2E2 4m∗(ω − ek− Re σssr − iIm σssr)4, (6a) ˜ Grss(k, ω) ≈ − q 2 c2E2 m∗(ω − ek− Re σsrs− iIm σrss)5 × Re Δ21, s=↑ Re Δ12, s=↓, (6b)
where E is |E| and σrss denotes the equilibrium retarded self energy. The detailed derivation is shown in Ap-pendix C.
By applying Taylor expansion to the electric field in equation (5) and approximating ˜G(k, ω) as ˜g(k, ω) + E ˜G(1)(k, ω) + E2G˜(2)(k, ω), the first-order KBE for the SHE can then be obtained based on the perturbation method. qc 2˜a2ss∂ωnFˆε · ˜ ξss∇kγ˜ss− ˜γss∇kξ˜ss + 2i ˜G(1)ss(k, ω) × Im Δ12, s=↓ Im Δ21, s=↑ = i[˜γssG˜(1)ss (k, ω) − ˜assΣ˜ss(1)(k, ω)], (7a) ˜ G(1)ss(k, ω) 2Im Δ12, s =↓ 2Im Δ21, s =↑ = ˜ γssG˜(1)ss (k, ω) − ˜assΣ˜ss(1) (k, ω), (7b)
where nF is Fermi-Dirac distribution. Additionally, ˆε ≡
E/E, ˜ξss(k, ω) ≡ ω − ek − ˜σss(k, ω), ˜σss(k, ω) ≡ [˜σrss(k, ω) + ˜σssa (k, ω)]/2, ˜ass(k, ω) ≡ i[˜gssr(k, ω) − ˜
gass(k, ω)], and ˜γss(k, ω) ≡ i[˜σssr(k, ω) − ˜σssa(k, ω)]. Note that (non-)equilibrium Green function or (non-) equilib-rium self energy is written in a (capital) lower letter in this paper. The second-order KBE for the SHE is shown in Appendix D, where an algorithm for solving the KBE up to the 2nd order solution is provided. In this report, only the 1st order solution is demonstrated.
For an impurity interaction, the lowest-order self en-ergy is given by NiΩq[|V (q)|2G˜,rs1s2(k − q, ω)], where
Ni is the impurity density, Ω is sample volume and V (q) is the Coulomb interaction. Thus ˜σss(e−imp)r (k, ω) can be rearranged asq NiΩ|V (q)|2
ω−ek−q+iτss(e−imp)−1 , where τss(e−imp)is
the spin-electron impurity scattering time.
For an electron-phonon interaction, the lowest-order lesser (retarded) self energy is given by
i 2π q dω|Mq|2[G()s1s2(k − q, ω − ω)D(r)s1s2(q, ω) + G(r)s1s2(k − q, ω − ω)D()s1s2(q, ω) + G(r)s1s2(k − q, ω − ω)D(r)s1s2(q, ω)],
where|Mq|2 and Ds1s2(q, ω) are the electron-phonon
in-teraction strength and phonon’s propagator, respectively. Then, ˜Σs
1s2(e−ph)(k, ω) can be rearranged as
q,± |Mq|2 Nph(q) +1 2 ± 1 2 ˜ Gs1s2(k − q, ω ± ωq) , where Nph(q) is the phonon number, and ˜σss(e−ph)r (k, ω) is rearranged asq,±|Mq|2 (ω±ωNphq(q)±1+n)−e F(ek−q)
k−q+iτss(e−ph)−1 , where
τss(e−ph)is the spin-electron phonon scattering time. To simplify the calculation of self energy, the inter-action strength in momentum space, |Mq|2, is assumed as M20δ(q) called a delta interaction approximation later, where Mq
2
for the electron-phonon and impurity inter-actions is |Mq|2 and|V (q)|2, respectively. Under the ap-proximation, equations (7a) and (7b) can be rewritten as
− qc 2˜a 2 ↑↑∂ωnF ˜ ξ↑↑∂kyγ˜↑↑− ˜γ↑↑∂kyξ˜↑↑ + 2Im Δ12(k)Im ˜G(1)↓↑ (k) = ˜γ↑↑Im ˜G(1)↑↑ (k) −A M20Nα 4π2 a˜↑↑Im ˜G (1) ↑↑ (k), (8a) 2Im Δ21(k)Im ˜G(1)↑↑ (k) = ˜ γ↓↓Im ˜G(1)↓↑ (k) −A M20Nα 4π2 ˜a↑↑Im ˜G (1) ↓↑ (k), (8b)
Im ˜G(1)↑↑ (k) = −qc 2˜a2↑↑∂ωnF ˜ ξ↑↑∂ky˜γ↑↑− ˜γ↑↑∂kyξ˜↑↑ ˜ γ↓↓−A M20Nα 4π2 ˜a↑↑ ˜ γ↑↑−A M20Nα 4π2 ˜a↑↑ ˜ γ↓↓−A M20Nα 4π2 ˜a↑↑ − 4Im Δ12(k)ImΔ21(k) . (9)
x
y (current direction)
(electric field direction)
z
L
zL
xL
yO
In plane
R
//(lateral cross section)
Fig. 1. Electrically-biased sample’s coordinate and geometry for the SHE.
where the sample’s coordinate is defined in Figure 1, and A denotes sample’s lateral area (LxLz) with respect to the electric field. Notably, Nα is equal to NiΩ and 2Nph(0) + 1 for impurity interactions and electron-phonon interactions, respectively.
Equations (8a) and (8b) can easily derive Im ˜G(1)↑↑ (k), which is shown as
See equation (9) above.
Via equation (9), one can understand why and when the SHE occurs. A conventional definition of the SC as
−kkx
m∗Im ˜G↑↑(k)Eyis used, where the SC arises unless the Fermi sphere of spin species shifts toward the direction perpendicular to the applied field, i.e. Im ˜G(1)↑↑ (k) is an asymmetric function of kx. Since all equilibrium functions and their derivatives with respect to kyin equation (9) are kxsymmetric, SOC term Δij(k) becomes the unique fac-tor in generating the SC. From Appendix A, Im ˜G(1)↑↑ (k) remains kxsymmetric when either BIA or SIA is included in the SOC term except the fact that both BIA and SIA are considered. Accordingly, equation (9) indicates that the SC can only appear in a QW and vanishes in bulk. The prediction is against the observation of the SHE in bulk [39]. This is probably due to the delta interaction approximation used in our calculation. If the interaction strength has a slight broadening in momentum space, a non-zero SC may appear in bulk. Otherwise, an earlier theoretical report based on the Keldysh formalism [40] shows a vanishing SC in bulk, which is the same as our prediction. As for the QW, the SC has a nonzero value even under the delta interaction approximation. The SHE of two-dimensional (2D) electrons [41] and 2D holes [42] in GaAs has been experimentally demonstrated.
In addition to the SOC requirement, the SC between two spin species must be distinct either in direction or
magnitude, or their spin fluxes cannot be distinguished experimentally. Thus the differential SC (DSC), which is defined as Jx,↑↑s − Jx,↓↓s , is useful. Whether the DSC ex-ists depends on the electron-impurity (-phonon) scattering rate. If the scattering rate varies with the spin species, the DSC exists; otherwise, the DSC goes to zero. In the latter case, although the DSC vanishes, the sum of the SC as Jx,↑↑s + Jx,↓↓s survives, implying that a transverse electric current likely exists. According to equation (9), the flux of electrons is in the direction opposite that of holes and the electrical current caused by the opposite charges will not cancel each other. Therefore, the derived result reveals the existence of Hall current without any external magnetic field except for the DP field, which is self-induced due to the BIA and SIA. In contrast, using the Keldysh formalism in the quasiclassical approximation [26], the Hall current due to the SOC can occur only when the ferromagnetic contact is present.
The work now calculates the SC in a 10 nm-wide GaAs/Al0.3Ga0.7 As QW at sheet densities (n2D) of 1010cm−2and 1011cm−2 at room temperature, and com-pares the SC of the impurity and Fr¨ohlich interactions. The extrinsic (M0 = 0) and intrinsic (
M0 = 0) SCs are evaluated for each case. The extrinsic SC stands for the calculation with both non-equilibrium and equilibrium self energies present (diffusive regime), whereas the intrinsic SC stands for the calculation with equilibrium self energy only (ballistic regime). For impurity interactions, com-plete ionization (i.e., n2D = NiL) and Brooks- Herring approximation for calculating the electron-impurity scat-tering time are assumed, where
τss(e−imp)= 24.5πε2GaAs√m∗e1.5k q4cZNi ln [1+8m∗ek/(2qs2)]−[1+(2q2s/(8m∗ek))]−1 with qs being 2m∗qc2nF(k = 0)/εGaAs2, εGaAs being the GaAs dielectric constant, Z = 14 for silicon impu-rity, and L being the width of QW. For Fr¨ohlich inter-action (polar optical phonon abbreviated as POP), two symmetric interface phonon modes and the lowest con-fined phonon mode based on the dielectric continuum model [43–45] are considered. The average electron-POP scattering time τss(e−ph)of 165 fs was used. The SC shown in following figures is multiplied by an electron’s charge.
Figures2a and2b show the extrinsic and intrinsic SCs caused by impurity, respectively. The extrinsic and intrin-sic SCs are in the reverse direction and the intrinintrin-sic SC is significantly higher than the extrinsic SC. At a carrier density of 1011 (1010) cm−2, the intrinsic SC is even ten-(several) thousand times higher than the extrinsic SC. In
0 20 40 60 80 100 0 2 4 6 8 10 extrinsic SHE (impurity case) (a) Applied voltage (mV)
Spin charge curr
ent density ( μ A/m 2 ) n2D=1011cm-2 n2D=1010cm-2 0 15 30 45 60 75
Spin charge current density
(mA/m 2 ) 0 20 40 60 80 100 -80 -60 -40 -20 0 (b) intrinsic SHE (impurity case) Applied voltage (mV) Spin charge cu rrent density (mA/m 2 ) n2D=1011cm-2 n2D=1010cm-2 -6 -5 -4 -3 -2 -1 0
Spin charge current density
(kA/m
2
)
Fig. 2. The first-order impurity-induced spin charge current density for (a) the extrinsic SHE and (b) the intrinsic SHE in a 10 nm-wide GaAs/Al0.3Ga0.7As QW at room temperature within the complete ionization and Brooks-Herring approxi-mations.
clean limit the spin Hall conductance is due to the intrin-sic SC. Once the impurity exists, the intrinintrin-sic SC replaced with the extrinsic SC therefore results in a significant re-duction of the SC by defect scattering as described in the Inoue’s report [24]. Furthermore, the SC, especially the intrinsic SC, is extremely sensitive to the carrier density. While the extrinsic SC at a carrier density of 1011cm−2is 6000 times higher than the extrinsic SC at 1010cm−2, the intrinsic SC at a carrier density of 1011 cm−2 is several 10 000 times higher than the intrinsic SC at a density of 1010 cm−2. The considerable density dependence results from Im ˜σss(e−imp)r , which is proportional to Niτss(e−imp)−1 when ω − ek τss(e−imp)−1 . Due to τss(e−imp)−1 ∝ Ni, Im ˜σrss(e−imp) has the Ni2 dependence. ˜γss and ˜ass re-lates to Im ˜σrss. ˜γss =−2Im ˜σssr, and ˜ass ∝ Im ˜σrss when ω − ek− Re σssr Im σssr. Thus ˜γss and ˜ass also has the Ni2 dependence. By canceling Ni factors between the nu-merator and denominator in equation (9), Im ˜G(1)↑↑ (k) for the intrinsic and extrinsic cases can be shown to depend on ˜a2↑↑∂ωnF ∝ Ni5and ˜a2↑↑∝ Ni4, respectively. (Note that n2D = Ni.) Thus the max. ratio of the SC between 1010 and 1011 cm−2 can reach to several 10 000 times.
0 20 40 60 80 100 0.00 0.05 0.10 0.15 0.20 0.25 extrinsic SHE (phonon case) (a) Applied voltage (mV)
Spin charge current density
(A/m 2 ) n2D=1011cm-2 n2D=1010cm-2 0 2 4 6 8 10 12 14
Spin charge current density
(A/m 2 ) 0 20 40 60 80 100 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 (b) intrinsic SHE (phonon case) Applied voltage (mV)
Spin charge current density
(A/m 2 ) n2D=1011cm-2 n2D=1010cm-2 -1.5 -1.2 -0.9 -0.6 -0.3 0.0
Spin charge current density
(kA/m
2
)
Fig. 3. The first-order POP-induced spin charge current den-sity for (a) the extrinsic SHE and (b) the intrinsic SHE in a 10 nm-wide GaAs/Al0.3Ga0.7As QW at room temperature based on the dielectric continuum model, where the two sym-metric interface phonon mode and the lowest confined phonon mode are included.
Figures3a and3b show the extrinsic and intrinsic SCs caused by the POP, respectively. Like the impurity case, the direction of the extrinsic SC is opposite that of the intrinsic SC, and the intrinsic SC is higher than the ex-trinsic SC; however, the ratio between the inex-trinsic SC and extrinsic SC is not so considerable. At a carrier density of 1011 (1010) cm−2, the intrinsic SC is approximately 130 (2) times higher than the extrinsic SC. The density de-pendence of SC is strong, but also not as strong as that of impurity interaction. While the extrinsic SC at a carrier density of 1011 cm−2 is roughly 40 times higher than the extrinsic SC at 1010 cm−2, the intrinsic SC at a carrier density of 1011 cm−2 is almost 2500 times higher than the intrinsic SC at 1010 cm−2. This is because τss(e−ph) does not have the n2D dependence. Thus Im ˜σrss(e−ph) as well as ˜γss and ˜ass is only proportional to n2D. Fur-thermore, unlike the impurity case, Nα is independent on the n2D. Therefore, by canceling n2D factors in equa-tion (9), Im ˜G(1)↑↑ (k) for both intrinsic and extrinsic cases shows the dependence of ˜a2↑↑∂ωnF ∝ n32D. As a result, the maximum ratio of the SC between 1010 and 1011cm−2 is about 1000 times.
0.0 0.2 0.4 0.6 0.8 1.0 -0.02 0.00 0.02 0.04 0.06 0.08 0.10 Spi n c h ar ge c u rr e n t density ( A /m 2) Scaling factor β n2D=10 9 cm-2 (a) 0.0 0.2 0.4 0.6 0.8 1.0 -0.4 -0.2 0.0 0.2 0.4 0.6 Spi n c h ar ge c u rr e n t density ( A /m 2 ) Scaling factor β n2D=3x10 9 cm-2 (b) 0.0 0.2 0.4 0.6 0.8 1.0 -3 -2 -1 0 1 2 n2D=5x109cm-2 Spi n char g e c u rr ent densi ty ( A/m 2) Scaling factor β (c) 0.0 0.2 0.4 0.6 0.8 1.0 -10 -8 -6 -4 -2 0 2 4 Spin char g e c u rr ent density ( A /m 2 ) Scaling factor β n2D=7x109cm-2 (d)
Fig. 4. Spin charge current density caused by Fr¨ohlich interaction as a function of scaling factor at sheet carrier densities of (a) 109 (b) 3× 109 (c) 5× 109 (d) 7× 109 cm−2.
Generally, the SC caused by the POP is much higher than the SC caused by impurity. For example, the ratio of the extrinsic POP-induced SC to the extrinsic impurity-induced SC is close to 25 000(160) at a carrier density of 1010 (1011) cm−2. The exception is that the intrin-sic POP-induced SC becomes one-fourth of the intrinintrin-sic impurity-induced SC when the density is at 1011 cm−2. Excluding the case, the POP- induced SC is always dom-inant; however, the importance of Fr¨ohlich interaction on the SHE has not been described. If the spin effect on the electron-POP scattering rate τss(e−ph)−1 is also strong, a considerable POP-induced DSC exists, further increasing the importance of Fr¨ohlich interaction on the application of devices based on the SHE.
Additionally, Fr¨ohlich interaction leads to a unique os-cillatory behaviour in the SC that does not appear with the impurity-induced SC. By introducing the scaling fac-tor β, i.e.,M20 ≡ βM20 (0, intrinsic SC; 1, extrinsic SC), Figures 4a–4d show the POP-induced SC as a function of the scaling factor at carrier densities of 109, 3× 109, 5×109and 7×109cm−2, respectively, where oscillation as a function of the scaling factor is shown. The scaling factor can result from Coulomb screening effect. As the carrier density increases, the oscillation weakens and eventually disappears. Notably, the oscillatory behaviour is depen-dent on the scaling factor. While the oscillatory amplitude shows a modulation as a quantum beat, oscillatory fre-quency gets lower as the scaling factor increases. Interest-ingly, the phonon number (Nα) has the same mathemati-cal role in equation (9) as the smathemati-caling factor. Therefore, the oscillatory behaviour can also appear in the SC-voltage
curve because of the temperature dependence of phonon excitation number, which is associated with an applied voltage.
5 KBE for spin accumulation
This section is to derive the spatially dependent KBE for spin accumulation, where the in-plane (R) dependence is considered. With the Neumann boundary condition, the lesser Green function ˜G(k, ω, R) can be expressed as √ 1 LxLz KG˜ (k, ω, K ) exp[iK· (R− L/2)], where K = π nx Lx,nzLz
with nx,z being an integer and L be-ing (Lx, Lz). With the Fourier expansion, the LHS of equation (2b), i.e., the driving term (DT) of spatially-dependent KBE, after Wigner transformation and Fourier transform under the scalar gauge can be written as
DTss() = 1 √ LxLz K exp[iK· (R− L/2)] × i∂T− 2 m∗k · K+iqcE · ∇k+ k m∗∂ω × ˜Gss( )(k, ω, K, T )− 2i ˜Gss()(k, ω, K, T ) × Im Δ12, s=↓ Im Δ21, s=↑ . (10)
The RHS of equation (2b), i.e. the collision term (CT), becomes difficult to transform to the (k, ω) domain when the average spaceR dependence involves due to the spa-tial entanglement in dr2dt2{Σ[r1− r2,12(r1+r2), t1, t2]
G[r2− r1,12(r2+r1), t2, t1]}. However, the Fourier ex-pansion for the average space R is efficient to elimi-nate the entanglement. Hence, the CT can be successfully transformed to the (k, ω) domain.
CTss()(k, ω, R, T ) = 1 LxLz K K exp[iK·(R−L/2)] × dτ dτexp(iωτ ) s=↑,↓ ˆ Pssk1,2, τ1,2,K1,2, T1,2 × ˆQss() k 1,2, τ1,2,K1,2, T1,2, (11) where k1,2 =k1,2± 12K2,1. K1,2 = 12(K± K). The detailed derivation is shown in Appendix E.
Hence, the first and the second order R-dependent KBEs can be determined and shown as, respectively,
qcˆε · ∇k+mk∗∂ω ˜ gss δss()+i 2 m∗k · KG˜ (1) ss( )(k, ω, K) − 2 ˜G(1)ss()(k, ω, K) Im Δ12, s=↓ Im Δ21, s=↑ − qc 2ˆε · δss( ) √ LxLz × K ∂ωP˜eqss∇kQ˜eqss− ∇kP˜eqss∂ωQ˜eqss =√−i LxLz × K s=↑,↓ ˜ Pss(k1,2, ω,K1,2) × ˜Qss()(k1,2, ω,K1,2) 1, (12a) qcˆε · ∇k+ k m∗∂ω ˜ G(1) ss( )(k, ω, K) +i 2 m∗k · KG˜ (2) ss( )(k, ω, K) − 2 ˜G(2) ss()(k, ω, K) Im Δ12, s =↓ Im Δ21, s =↑ − iqc2 82 δss()ε·ˆ √ LxLz × K ∂ω2P˜eqss∂k2Q˜eqss− ∂k2P˜eqss∂ω2Q˜eqss = −i √ LxLz K s=↑,↓ ˜ Pss(k1,2, ω,K1,2) × ˜Qss()(k1,2, ω,K1,2) 2, (12b)
where the subscript (1,2) in the middle bracket denotes the order expansion with respect to the electric field. The retarded Green function (retarded self energy) shown in equations (C.8a) and (C.8b) can be input into the CT on the RHS ofR-dependent KBEs.
The R-dependent KBE is important to the study of the SHE because detection of spin flux density is still a major restriction for current measurements; however, spin accumulation can be verified experimentally using Kerr spectroscopy [12]. Since the KBE is a very gen-eral approach, the issue of spin accumulation in a ballis-tic regime [18,19] or a nanometer scale where Boltzmann
theory no longer fits can be governed by the KBE. In addition to spin applications, the analytic KBE is es-pecially important for spatial quantum kinetic effects, which have been still less understood because the ana-lytic spin-independent, spatially-dependent KBE has not been derived before this work. Based on equations (12a) and (12b), spatial quantum kinetic effects such as momen-tum non-conservation and spatial coherence [46,47] can be studied and compared with temporal quantum kinetic ef-fects involving the energy non-conservation [6] and mem-ory effect [7,8].
6 Conclusion
This work presents a general quantum kinetic theory of spin dynamics, in which the KBE is applied to spin re-laxation and the SHE. First, the equation governing the time evolution of spin relaxation via the DP magnetic field among non-equilibrium CCS was constructed. Quantum kinetic oscillation between distinct spin-polarized states within the quantum coherence time, as well as SOC-induced oscillation, was identified. Second, the quantum transport equation of the SHE in the presence of impu-rity and Fr¨ohlich interactions was formulated. The equa-tion can interpret why the SHE exists and when the SC is no longer zero. Furthermore, the numerical results in-dicate that the SC is very sensitive to impurity density, while Fr¨ohlich interaction can result in a considerable SC and lead to a unique oscillation in the SC. Finally, the
R-dependent KBE for spin accumulation was derived, and is especially useful for exploring spatial quantum ki-netic effects.
The author Lee would like to thank Prof. S.A. Lyon’s enlight-enment for the work. The work was financially supported by the project of National Science Council under grant number NSC 95-2112-M-007-013.
Appendix A: SOC terms in DP
mechanism [48]
The DP Hamiltonian is given by HDP =σ · Ω(k), where σ is the Pauli matrix,
ΩBIA(k) = γ ⎛ ⎝kx(k 2 y− kz2) ky(k2z− k2x) kz(kx2− ky2) ⎞ ⎠ and ΩSIA(k) =α ⎛ ⎝k−ky x 0 ⎞ ⎠
are the effective magnetic field for the bulk inversion asym-metry (BIA) and the surface inversion asymasym-metry (SIA), respectively. In GaAs, the BIA coefficient γ equals 27 e˚A3 independent of carrier density and the Rashba coefficient
α decreases linearly from−10−4 eV ˚Aat the sheet density of 1011 cm−2to −5 × 10−3 eV ˚Aat n2Dof 8× 1011cm−2. The SOC term in bulk can therefore be represented as Δ11= Δ22= 0 and Δ12= Δ∗21= γkx(ky2−kz2)−iγky(kz2− k2x). In a QW, Ω = γ ⎛ ⎝kx(k 2 x− kz2) ky(kz2 − kx2) 0 ⎞ ⎠ + ΩSIA(k), where k2 z = 14 16.5π m∗n2De2 2ε∞ 2/3
and ε is the high-frequency dielectric constant. The SOC diagonal elements in a QW remain zero while the off-diagonal can be represented as Δ12 = Δ∗21 = γkx(ky2−
k2
z) − iγky(k2z − k2x) + αky+ iαkx.
Appendix B: Recovery of compact definitions
and equations
For clarity, the definitions in Section 2 are presented in an original form. H ≡ H11H12 H21H22 = ⎛ ⎝− 2 2m∇2+U (r, t)+Δ11 Δ12 Δ21 −2m2∇2+U (r, t)+Δ22 ⎞ ⎠ , D(1) ≡ ⎛ ⎝D11(1) D12(1) D21(1) D22(1) ⎞ ⎠ = ⎛ ⎝i∂t1− H11(1) −H12(1) −H21(1) i∂t1− H22(1) ⎞ ⎠ , D∗(1)≡ ⎛ ⎝D ∗ 11(1) D∗12(1) D∗21(1) D∗22(1) ⎞ ⎠ = ⎛ ⎝−i∂t1 − H11∗(1) −H12∗ (1) −H∗ 21(1) −i∂t1 − H22∗(1) ⎞ ⎠ , G(1, 1)≡ ⎛ ⎝G↑↑(1, 1 ) G ↑↓(1, 1) G↓↑(1, 1) G↓↓(1, 1) ⎞ ⎠ , Σ(1, 1)≡ ⎛ ⎝Σ↑↑(1, 1 ) Σ ↑↓(1, 1) Σ↓↑(1, 1) Σ↓↓(1, 1) ⎞ ⎠ . (B.1)
where 1 and 1 present (r1, t1) and (r1, t1), respectively.
The original form of Dyson equation shown in equa-tions (1a) and (1b) is given by
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ D11(1)G↑↑(1, 1) + D12(1)G↓↑(1, 1) = δC(1− 1) + Cd2 [Σ↑↑(1, 2)G↑↑(2, 1 ) + Σ ↑↓(1, 2)G↓↑(2, 1)], (B.2a) D∗11(1)G↑↑(1, 1) + D12∗ (1)G↓↑(1, 1) = δC(1− 1) + Cd2 [G↑↑(1, 2)Σ↑↑(2, 1 ) + G ↓↑(1, 2)Σ↑↓(2, 1)] , (B.2a)’ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ D11(1)G↑↓(1, 1) + D12(1)G↓↓(1, 1) = Cd2 [Σ↑↑(1, 2)G↑↓(2, 1 ) + Σ ↑↓(1, 2)G↓↓(2, 1)] , (B.2b) D∗11(1)G↑↓(1, 1) + D12∗ (1)G↓↓(1, 1) = Cd2 [G↑↓(1, 2)Σ↑↑(2, 1 ) + G ↓↓(1, 2)Σ↑↓(2, 1)] , (B.2b)’ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ D21(1)G↑↑(1, 1) + D22(1)G↓↑(1, 1) = Cd2 [Σ↓↑(1, 2)G↑↑(2, 1 ) + Σ ↓↓(1, 2)G↓↑(2, 1)] , (B.2c) D∗21(1)G↑↑(1, 1) + D∗22(1)G↓↑(1, 1) = Cd2 [G↑↑(1, 2)Σ↓↑(2, 1 ) + G ↓↑(1, 2)Σ↓↓(2, 1)] , (B.2c)’ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ D21(1)G↑↓(1, 1) + D22(1)G↓↓(1, 1) = δC(1− 1) + C d2 [Σ↓↑(1, 2)G↑↓(2, 1) + Σ↓↓(1, 2)G↓↓(2, 1)] , (B.2d) D∗21(1)G↑↓(1, 1) + D∗22(1)G↓↓(1, 1) = δC(1− 1) + Cd2 [G↑↓(1, 2)Σ↓↑(2, 1 ) + G ↓↓(1, 2)Σ↓↓(2, 1)] , (B.2d)’ Given the Langreth theorem [34,35], if Z(t1, t1) = Cdτ
X(t1, τ )Y (τ, t1), then Z(t1, t1) = C dτ [Xr(t1, τ )Y(τ, t1) + X(t1, τ )Ya(τ, t1)]. (B.3) Applying (B.3) to (B.2), two kinds of KBE shown in equa-tions (2a) and (2b) can then be obtained. To clarify the spin notation, the original form of the two kinds of KBE is shown. For the first kind of KBE in equation (2a), the
detailed expression is given by [D11(1)− D∗11(1)]G↑↑(1, 1) + [D12(1)− D∗12(1)]G↓↑(1, 1) − [Σ↑↑, G↑↑]− [Σ↑↑ , G↑↑]− [Σ↑↓, G↓↑]− [Σ↑↓ , G↓↑] = 1 2{Σ ↑↑, G↑↑}− 1 2{G ↑↑, Σ↑↑ }+ 1 2{Σ ↑↓, G↓↑}− 1 2{G ↓↑, Σ↑↓ }, (B.4a) [D11(1)− D∗11(1)]G↑↓(1, 1) + [D12(1)− D∗12(1)]G↓↓(1, 1) − [Σ↑↑, G↑↓]− [Σ↑↑ , G↑↓]− [Σ↑↓, G↓↓]− [Σ↑↓ , G↓↓] = 1 2{Σ ↑↑, G↑↓}− 1 2{G ↑↓, Σ↑↑ }+ 1 2{Σ ↑↓, G↓↓}− 1 2{G ↓↓, Σ↑↓ }, (B.4b) [D21(1)− D∗21(1)]G↑↑(1, 1) + [D22(1)− D∗22(1)]G↓↑(1, 1) − [Σ↓↑, G↑↑]− [Σ↓↑ , G↑↑]− [Σ↓↓, G↓↑]− [Σ↓↓ , G↓↑] = 1 2{Σ ↓↑, G↑↑}− 1 2{G ↑↑, Σ↓↑ }+ 1 2{Σ ↓↓, G↓↑}− 1 2{G ↓↑, Σ↓↓ }, (B.4c) [D21(1)− D∗21(1)]G↑↓(1, 1) + [D22(1)− D∗22(1)]G↓↓(1, 1) − [Σ↓↑, G↑↓]− [Σ↓↑ , G↑↓]− [Σ↓↓, G↓↓]− [Σ↓↓ , G↓↓] = 1 2{Σ ↓↑, G↑↓}− 1 2{G ↑↓, Σ↓↑ }+ 1 2{Σ ↓↓, G↓↓}− 1 2{G ↓↓, Σ↓↓ }, (B.4d) where reminding that ΣG and GΣ are abbreviated forms
of Cd2Σ(1, 2)G(2, 1) and Cd2G(1, 2)Σ(2, 1),
respec-tively. Additionally, [ , ] and{ , } stand for the commuta-tor and anti-commutacommuta-tor, respectively.
For the second kind of KBE in equation (2b), the de-tailed expression is given by
D11(1)G↑↑(1, 1)− D11∗ (1)G↑↑(1, 1) + D12(1)G↓↑(1, 1) − D∗ 12(1)G↓↑(1, 1) = Σ↑↑r G↑↑ + Σ↑↑Ga↑↑− Gr↑↑Σ↑↑ − G↑↑Σa↑↑+ Σ↑↓r G↓↑+ Σ↑↓ Ga↓↑− Gr↓↑Σ↑↓ − G↓↑Σ↑↓a , (B.5a) D11(1)G↑↓(1, 1)− D11∗ (1)G↑↓(1, 1) + D12(1)G↓↓(1, 1) − D∗ 12(1)G↓↓(1, 1) = Σ↑↑r G↑↓ + Σ↑↑Ga↑↓− Gr↑↓Σ↑↑ − G↑↓Σa↑↑+ Σ↑↓r G↓↓+ Σ↑↓ Ga↓↓− Gr↓↓Σ↑↓ − G↓↓Σ↑↓a , (B.5b) D21(1)G↑↑(1, 1)− D21∗ (1)G↑↑(1, 1) + D22(1)G↓↑(1, 1) − D∗ 22(1)G↓↑(1, 1) = Σ↓↑r G↑↑ + Σ↓↑Ga↑↑− Gr↑↑Σ↓↑ − G↑↑Σa↓↑+ Σ↓↓r G↓↑+ Σ↓↓ Ga↓↑− Gr↓↑Σ↓↓ − G↓↑Σ↓↓a , (B.5c) D21(1)G↑↓(1, 1)− D∗21(1)G↑↓(1, 1) + D22(1)G↓↓(1, 1) − D∗ 22(1)G↓↓(1, 1) = Σ↓↑r G↑↓ + Σ↓↑Ga↑↓− Gr↑↓Σ↓↑ − G↑↓Σa↓↑+ Σr↓↓G↓↓+ Σ↓↓ Ga↓↓− Gr↓↓Σ↓↓− G↓↓Σ↓↓a . (B.5d)
Appendix C: Seeking retarded Green function
for the spin Hall effect
According to Langreth theorem, Zr(t1, t1) = Cdτ
Xr(t1, τ )Yr(τ, t1). With the relation, (B2a) + (B2a)’ and
(B2c) + (B2c)’ can be presented as, respectively
[D11(1)+D∗11(1)]Gr↑↑(1, 1)+[D12(1)+D∗12(1)]Gr↓↑(1, 1) = 2δ(1− 1) + Σr↑↑Gr↑↑+ Gr↑↑Σ↑↑r + Σ↑↓r Gr↓↑+ Gr↓↑Σ↑↓r ,
(C.1a)
[D21(1)+D∗21(1)]Gr↑↑(1, 1)+[D22(1)+D∗22(1)]Gr↓↑(1, 1) = Σ↓↑r Gr↑↑+ Gr↑↑Σ↓↑r + Σr↓↓Gr↓↑+ Gr↓↑Σ↓↓r , (C.1b) which is another form of spin-dependent Dyson equation and where Gr↑↑(1, 1) and Gr↓↑(1, 1) can be found out to input into the KBE for the SHE shown in equation (5).
(C.1a) and (C.1b) after applying the Wigner transfor-mation and Fourier transform under the scalar gauge, i.e. dτ dr exp[i(ω − qcE · R/)τ − ik · r]G(r, τ, R, T ), become ω − ek+q 2 cE2 8m∗∂ 2 ω ˜ Gr↑↑(k, ω) − Re Δ12G˜r↓↑(k, ω) = 1, (C.2a) −Re Δ21G˜r↑↑(k, ω) + ω − ek+q 2 cE2 8m∗∂ 2 ω ˜ Gr↓↑(k, ω) = 0, (C.2b) where the retarded Green function is assumed to be in-dependent of R (spatially homogeneous) and T (station-ary); the collision term of (C.1) and (C.2) is set to be zero first and considered later. The retarded Green function can be obtained in the τ domain. Applying the transform
ˆ
Gr(k, τ) ≡ −∞∞ dω exp(−iωτ) ˜Gr(k, ω), note that this is under neither the scalar nor the vector potential gauge, (C.2a) and (C.2b) become
i∂τ− ek−q 2 cE2 8m∗τ 2Gˆr ↑↑(k, τ) − Re Δ12Gˆr↓↑(k, τ) = δ(τ), (C.3a) −Re Δ21Gˆr↑↑(k, τ)+ i∂τ− ek−q 2 cE2 8m∗τ 2Gˆr ↓↑(k, τ) = 0, (C.3b)
where the two retarded Green functions can be found out by using an iterative method. Set ˆGr↓↑(k, τ) = 0 in (C.3a), then ˆ Gr↑↑(k, τ) = −i θ(τ ) exp −i ek τ + qc2E2 24m∗τ 3,
where θ(τ ) is a step function. Hence, ˜Gr↑↑(k, ω) ≈ ω−e1
k−
q2
c2E2
4m∗(ω−ek)4, where the first order Taylor expansion for the
exponential function was made. Considering the equilib-rium self energy σssr as the collision term in (C.1a) then yields ˜ Gr↑↑(k, ω) ≈ 1 ω − ek− Re σ↑↑r − iIm σr↑↑ − q2c2E2 4m∗ ω − ek− Re σ↑↑r − iIm σr↑↑ 4. (C.4a) Inputting ˆGr↑↑(k, τ) into (C.3b) can find out ˆGr↓↑(k, τ) equal to−ReΔ21 2 τ θ(τ ) exp −i ek τ + q 2 cE2 24m∗τ3 , Chang-ing it to the frequency domain then yields
˜ Gr↓↑(k, ω) ≈ − q 2 c2Re Δ21E2 m∗ ω − ek− Re σ↑↑r − iIm σ↑↑r 5, (C.4b) where the equilibrium spin-flip term is set zero.
Starting from (B2b) + (B2b)’ and (B2d) + (B2d)’, ˜
Gr↑↓(k, ω) and ˜Gr↓↓(k, ω) can be found out with using the same iterative method.
˜ Gr↓↓(k, ω) ≈ 1 ω − ek− Re σ↓↓r − iIm σ↓↓r − qc22E2 4m∗ ω − ek− Re σ↓↓r − iIm σ↓↓r 4, (C.4a)’ ˜ Gr↑↓(k, ω) ≈ − q 2 c2E2Re Δ12 m∗ ω − ek− Re σ↓↓r − iIm σ↓↓r 5. (C.4b)’ When considering the spatially inhomogeneous case, (C.2a) and (C.2b) becomes
ω − ek+q 2 cE2 8m∗∂ 2 ω ˜ Gr↑↑(k, ω, R) + 2 8m∗ × ∂2 RG˜r↑↑(k, ω, R)− Re Δ12G˜r↓↑(k, ω, R) = 1 (C.5a) − Re Δ21G˜r↑↑(k, ω, R) + ω − ek+q 2 cE2 8m∗∂ 2 ω × ˜Gr↓↑(k, ω, R) + 2 8m∗∂ 2 RG˜r↓↑(k, ω, R) = 0 (C.5b)
With the Neumann boundary condition, the Green func-tion can be written as
˜ Gr(k, ω, R) = √ 1 LxLz K ˜ Gr(k, ω, K) × exp[iKx(x− Lx/2)] exp[iKz(z− Lz/2)], (C.6)
where Kx(z) = nx(z)π/Lx(z) and nx(z) is an integer. The coordinate refers to Figure 1.
Taking (C.6) into (C.5a) and (C.5b), and changing them to the τ domain yields
i∂τ− ek−1 4eK − qc2E2τ2 8m∗ ˆ Gr↑↑(k, τ, K) − Re Δ12Gˆr↓↑(k, τ, K) =−ηδ(τ), (C.7a) − Re Δ21Gˆr↑↑(k, τ, K) + i∂τ− ek−1 4eK− q2cE2τ2 8m∗ × ˆGr↓↑(k, τ, K) = 0, (C.7b) where η = [1− cos(nxπ)][1− cos(nzπ)]/(√LxLzKxKz).
(C.7a) and (C.7b) can then be solved with using the same iterative method as that in the spatially homoge-neous case. Starting from (B2b) + (B2b)’ and (B2d) + (B2d)’, ˜Gr↑↓(k, ω, K) and ˜Gr↓↓(k, ω, K) can also be found out. As a result, the K-dependent retarded Green func-tion can be expressed as
˜ Grss(k, ω, K)≈ −η ω − ek− eK/2− Re σssr − iIm σrss + ηq 2 c2E2 4m∗(ω − ek− eK/2− Re σssr − iIm σssr)4, (C.8a) ˜ Grss(k, ω, K)≈ ηqc22E2 m∗(ω − ek− eK/2− Re σsrs− iIm σrss)5 × Re Δ21, s =↑ Re Δ12, s =↓ . (C.8b)
Appendix D: The 2nd KBE for the SHE
with an algorithm for seeking solutions
Applying Taylor expansion to the electric field in equa-tion (5) and approximating G˜(k, ω) as ˜g(k, ω) + E ˜G(1)(k, ω) + E2G˜(2)(k, ω), the second order KBE for SHE based on the perturbation method can be
expressed as qcˆε · ∇k+mk∗∂ω ˜ G(1)ss (k, ω) − i q c 2 2 ×ˆε· ∂k2g˜ss ∂ω2σ˜ss+∂ω2˜gss∂2kσ˜ss+∂ω2σ˜ss∂k2˜gss+∂k2σ˜ss∂2ωg˜ss − ˜G(2)ss(k, ω) · 2Im Δ12, s=↓ 2Im Δ21, s=↑ =−[˜γssG˜(2)ss (k, ω) − ˜assΣ˜ss(2)(k, ω)] − ˜gss Γ˜ss(e−ph)(2) , (D.1a)
qcˆε · ∇k+mk∗∂ω ˜ G(1)ss (k, ω) − 2 ˜G(2)ss(k, ω) × Im Δ12, s =↓ Im Δ21, s =↑ =− ˜ γssG˜(2)ss (k, ω) − ˜σss A˜(2)ss(k, ω) +˜gssΓ˜ss(2)(k, ω) − ˜assΣ˜ss(2) (k, ω) , (D.1b) where ˜gss(k, ω) ≡ 12[˜gssr(k, ω) + ˜gass(k, ω)]. The sub-script in ˜gss Γ˜ss(e−ph)(2) denotes that the term only exists in the case of an e-ph interaction due to the Langreth theorem [34,35]. ˜A(2)ss( )(k, ω) ≡ i[ ˜Gr(2)ss( )k, ω) − ˜Ga(2)ss( )(k, ω)] and ˜Γss(2)()(k, ω) ≡ i[ ˜Σssr(2)()(k, ω) − ˜Σssa(2)()(k, ω)] can be de-rived using retarded Green functions in equation (6b).
The KBE for the SHE can be solved by using an it-erative algorithm shown below, where the sufficiently ac-curate first-order solutions are iteratively found and then input into the second-order equation to find solutions it-eratively until accuracy is acceptable.
Appendix E: Derivation for collision term
in spatially-dependent KBE
For clarity, we use a simplified notation first. The col-lision term C(r1,r1, t1, t1) = dr2dt2A(r1,r2, t1, t2)× B(r2,r1, t2, t1) after the Wigner transformation becomes C(r, τ, R, T )= dr2dt2A r1−r2, t1−t2,r1+2r2,t1+t2 2 × B r2− r1, t2− t1,r2+r1 2 , t2+ t1 2 , (E.1) where reminding that r = r1− r1, τ = t1− t1, R = (r1+r1)/2 and T = (t1+ t1)/2.
With the spatial inverse Fourier transform under the vector potential gauge, i.e.,
f (r, τ, R, T ) = dk (2π)3exp i k −qcET · r × ˆF (k, τ, R, T ),
Initial solutions within delta interaction approximation
Find (1) ss
G~< from (7b) for e-imp or e-ph interaction
Find (1) s s'
G~< from (7a) for e-imp or e-ph interaction
If 2nd order solution accuracy enough ? Find (2) s s' G~< from (D1a) for e-imp or e-ph interaction
Find (2) ss
G~< from (D1b) for e-imp or e-ph interaction
If 1st order solution accuracy enough ? Finished Yes Yes No No (E.1) becomes C(r, τ, R, T ) = dr2dt2 dk1 (2π)3 dk1 (2π)3 × exp i k1−q2cE(t1+ t2) (r1− r2) × exp i k1−qcE 2 (t2+ t1) (r2− r1) × A k1,r1+2 r2,t1+ t2 2 × B k1,r2+r1 2 , t2+ t1 2 , (E.2)
where the difference time in A and B functions is not marked for simplicity.
