PHYSICAL REVIEW B 72, 075307 共2005兲

Electron spin dynamics in impure quantum wells for arbitrary spin-orbit coupling C. Grimaldi* Ecole Polytechnique Fédérale de Lausanne, LPM, Station 17, CH-1015 Lausanne, Switzerland and DPMC, Université de Genève, 24 Quai Ernest-Ansermet, CH-1211 Genève 4, Switzerland 共Received 9 January 2005; revised manuscript received 27 March 2005; published 2 August 2005兲 The time evolution of the electron spin polarization of a disordered two-dimensional electron gas is calculated within the Boltzmann formalism for arbitrary couplings to a Rashba spin-orbit field. It is shown that the classical Dyakonov-Perel mechanism of spin relaxation gets deeply modified for sufficiently strong Rashba fields, in which case new regimes of spin decay are identified. DOI: 10.1103/PhysRevB.72.075307

PACS number共s兲: 72.25.Rb, 71.70.Ej, 72.25.Dc

The physics of transport of electron spins in lowdimensional systems has become a popular theme of research due to the possible impact in future electronic applications.1,2 Key subjects of studies concern the problem of controlling the electron spin polarization through the tailoring of the spin-orbit 共SO兲 interaction3,4 and the knowledge of the physical parameters governing the spin relaxation time ␶s. Main sources of SO coupling are the Rashba interaction arising from structural inversion asymmetries of lowdimensional structures such as quantum wells or wires5 and the Dresselhaus interaction present in bulk crystals lacking symmetry of inversion.6 In the presence of Rashba and/or Dresselhaus interactions, ␶s basically arises from the randomization of the SO 共pseudo兲magnetic field induced by momentum scattering with impurities or other interactions 共Dyakonov-Perel mechanism7兲. Of great interest for spintronic applications are materials with strong SO interaction since they are more effective spin-manipulators2 and spin-current generators through the spin-Hall effect.8 In this respect, systems of interest may be, for example, HgTe quantum wells whose Rashba interaction leads to SO band splitting ⌬so as large as ⬃30 meV,9 or surface states of metals and semimetals which display giant SO band splittings of the order of 0.2 eV or larger,10,11 or even the heavy Fermion superconductor CePt3Si,12 whose lack of bulk inversion symmetry leads to ⌬so ⯝ 50– 200 meV.13 For all these systems, the SO splitting ⌬so is no longer the smallest energy scale, as in most semiconductors, and can be comparable to the Fermi energy ⑀F or even larger. Despite that the time evolution of the spin polarization S has been thoroughly studied in the weak SO limit ⌬so / ⑀F = 0,7,14–16 only few partial studies have been devoted to the strong SO case ⌬so / ⑀F ⫽ 0.17–19 Of particular interest is the problem of assessing the role of ⌬so / ⑀F on the spin relaxation mechanism and of clarifying to which extent the classical Dyakonov-Perel 共DP兲 description7 gets modified by finite values of ⌬so / ⑀F. In this report, the z-component of S is solved explicitly for any value of ⌬so / ⑀F for a two-dimensional electron gas confined in the x-y plane subjected to the Rashba interaction. It is found that for a sufficiently strong Rashba interaction the DP relaxation mechanism gets modified, with the spin polarization displaying a slow 共power law兲 decay with time. Furthermore, in the extreme ⌬so / ⑀F Ⰷ 1 limit, a fast exponen1098-0121/2005/72共7兲/075307共4兲/$23.00

tial decay is obtained with ␶s proportional to the momentum scattering time ␶ p, i.e., spin relaxation is enhanced by disorder. Let us consider an electron gas whose noninteracting Hamiltonian H0 is † cks + H0 = 兺 ⑀kcks k,s

ប 兺 ⍀k · ␴ˆ ss⬘cks† cks⬘ , 2 k,ss

共1兲

⬘

† 共cks兲 is the creation 共annihilation兲 operator for an where cks ˆ is the electron with momentum k and spin index s = ↑ , ↓ , ␴ spin-vector operator with components given by the Pauli matrices, and ⍀k is a k dependent SO pseudopotential vector whose explicit form is not essential for the moment. In the above expression, ⑀k is the electron dispersion in the absence of SO coupling. Let us consider as a momentum-relaxation mechanism a coupling Vkk⬘ to spin-conserving impurities located at random positions Ri. Assuming that Vkk⬘ is switched on at time t = t0, the time evolution of S共t兲 for t ⬎ t0 can be obtained from the equation of motion of the density matrix ␳共t兲:20

d␳ˆ kk⬘共t兲 dt

i i = − 关Eˆk␳ˆ kk⬘共t兲 − ␳ˆ kk⬘共t兲Eˆk⬘兴 − ⌫ˆ kk⬘共t兲, ប ប

共2兲

where ␳ˆ kk⬘共t兲 is a 2 ⫻ 2 matrix with elements ␳ks,k⬘s⬘共t兲 ˆ , and = 具ks兩␳共t兲兩k⬘s⬘典, Eˆk = ⑀k + ប / 2⍀k · ␴ ⌫ˆ kk⬘共t兲 = V 兺 关eiRi·共k−p兲␳ˆ pk⬘共t兲 − eiRi·共p−k⬘兲␳ˆ kp共t兲兴,

共3兲

pi

where it has been assumed for simplicity that the momentum dependence of the impurity potential can be neglected 共Vkk⬘ = V兲. Equation 共2兲 can be formally integrated and, after the usual adiabatic 共t0 → −⬁兲 and Markov approximations,20,21 ␳ˆ kk⬘共t兲 reduces to

␳ˆ kk⬘共t兲 ⯝ −

i ប

冕

⬁

0

dt⬘e−␦t⬘e−iEkt⬘/ប⌫ˆ kk⬘共t兲eiEk⬘t⬘/ប , ˆ

ˆ

共4兲

where the limit ␦ → 0+ must be taken after the integration. By using the anticommutation property of the Pauli matrices the exponential operators in Eq. 共4兲 can be put in a form more suitable for integration over t⬘:

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©2005 The American Physical Society

PHYSICAL REVIEW B 72, 075307 共2005兲

C. GRIMALDI

兺冉 k

冊

d2Ske dAk − = 0, dt2 dt

共7兲

where Ak = ⍀k ⫻ Sko . By taking the odd part of Eq. 共6兲, the equation of motion of Ak becomes

冋

2␲nV2 k dAk Ak + ␣␤ Ap = ⍀k ⫻ 共⍀k ⫻ Ske 兲 − 兺 p dt 4ប p␣␤

FIG. 1. 共a兲 Rashba SO split electron dispersions ប2 / 2m共k ± kR兲2 in units of ⑀R = m / 8␥2R. The horizontal dashed line indicates the Fermi level. 共b兲 Density of states for the ⫾ bands. 共c兲 Plots of ⌫k, Eq. 共13兲, and ␹k, Eq. 共14兲.

e±iEkt⬘/ប = ˆ

1 兺 e±iEk␣t⬘/ប共1 + ␣⍀ˆ k · ␴ˆ 兲, 2 ␣

k

obtained by integrating over t⬘. By retaining only the scattering contributions 共Boltzmann approximation兲,19 the final result is therefore 1 2␲nV2 dSk ˆ ·⍀ ˆ 兲兴 = ⍀k ⫻ Sk − 兵关1 − ␣␤共⍀ 兺 兺 k p 4 dt ប p ␣␤

dSke 2␲nV2 = Ak − 兺 共Se − Spe + ប␣⍀ˆ k f ko dt 4ប p␣␤ k ˆ f o 兲␦共E − E 兲, − ប␤⍀ p p k␣ p␤

⫻␦共Ek␣ − Ep␤兲,

where is the odd part of f k. By using Eq. 共8兲 the term dAk / dt in Eq. 共7兲 can be eliminated in favor of Ak and Ske . Next, by using Eq. 共9兲, also the terms containing Ak can be eliminated and Eq. 共7兲 reduces to an equation of motion of the component Ske only that is sufficient to find the time evolution of S = 兺kSke . Let us consider the z component of S, Sz, for which the presence of f ko in Eq. 共9兲 has no effect since these terms have zero component in the z direction. In this way, Eq. 共7兲 reduces to

冕 冋 ⬁

0

共6兲

where n is the impurity concentration and f k = 21 Tr共¯␳ˆ k兲 is the electron occupation distribution function. Equation 共6兲 is very general, and holds true for both bulk and lowdimensional systems with no restrictions on the particular form of ⍀k. In this paper, Eq. 共6兲 is used to find the time evolution of the electron spin polarization S = 兺kSk for a twodimensional electron gas confined in the x-y plane and subjected to a Rashba SO interaction. In such a case, the effective SO field is ⍀k = 共␥Rky , −␥Rkx , 0兲 and the corresponding SO split bands depend solely on the magnitude k of the wave vector and can be written as Ek␣ = Ek␣ = ប2 / 2m共k + ␣kR兲2, where m is the electron mass and kR = m / 2ប␥R.22 In the following, instead of using ⌬so = ប␥RkF, the SO splitting will be parametrized by the Rashba energy ⑀R = ប2kR2 / 2m = m / 8␥R2 , that is the minimum interband excitation energy for an electron sitting at the bottom of the lower band 关see Fig. 1共a兲兴. Let us consider the even, Ske , and odd, Sko , parts with respect to k of Sk. It is then clear that S = 兺kSke and, from Eq. 共6兲, dS / dt = 兺k⍀k ⫻ Sk = 兺k⍀k ⫻ Sko , where ⍀−k = −⍀k has been used. Of course, dS / dt is also equal to 兺kdSke / dt, so that, after a further derivative with respect to time, the equation of motion can be recast in the following form:

共9兲

f ko = 21 共f k − f −k兲

ˆ 关⍀ ˆ · 共S − S 兲兴 + ␣␤⍀ ˆ ⫻共Sk − Sp兲 + ␣␤⍀ k p k p p ˆ · 共S − S 兲兴 + ប共␣⍀ ˆ + ␤⍀ ˆ 兲共f − f 兲其 ⫻关⍀ k k p k p k p

共8兲

where the summation over momenta has canceled all terms ˆ 兲共⍀ ⫻ So 兲 − 共⍀ ⫻ ⍀ ˆ 兲 ˆ ·⍀ odd in p and the identity 共⍀ k p k k p p o ˆ ˆ ˆ ⫻共⍀k · Sp兲 = 关Ap − 共⍀k · Ap兲⍀k兴共k / p兲 has been used. From Eq. 共6兲, the equation of motion of Ske is instead given by

共5兲

ˆ = ⍀ / 兩⍀ 兩, and E = ⑀ ± ប / 2兩⍀ 兩 are the where ␣ = ± 1, ⍀ k k k k± k k eigenvalues of H0 关Eq. 共1兲兴. At this point, averaging over the impurity positions Ri restores the translational invariance: 具␳ˆ kk⬘共t兲典imp = ␦k,k⬘¯␳ˆ k共t兲, and, finally, the equation of motion of ˆ ¯␳ˆ 兲, is electron spins for a given wave vector k, S = ប / 2Tr共␴ k

册

k ˆ ˆ − ␣␤ 共⍀ k · Ap兲⍀k ␦共Ek␣ − E p␤兲, p

冉

册

冊

e e dkk d2Szk ␹k e ⌫k dSzk 2 2 + ␥ k + S + = 0, 共10兲 zk R 2␲ dt2 ␶ p dt 4␶2p

e e where Szk = 兰20␲d␾Szk / 2␲, with ␾ being the angle between 2 the directions of k and the x axis, ␶−1 p = 2␲ / បnV N0 is the momentum relaxation rate for a two-dimensional electron gas with zero SO splitting and density of states 共DOS兲 N0 = m / 2␲ប2, and

⌫k =

1 兺 4N0 ␣␤

␹k =

1 兺 N0 ␣␤

冕 冉 ⬁

0

冕

⬁

0

冊

dpp p 1 + ␣␤ ␦共Ek␣ − E p␤兲, 2␲ k

共11兲

dpp 共⌫k − ⌫ p兲␦共Ek␣ − E p␤兲. 2␲

共12兲

A solution to Eq. 共10兲 is obtained by equating to zero the expression within square brackets, which leads to a homogee 共t兲. In neous differential equation of the second order for Szk this way the functions ⌫k and ␹k assume, respectively, the meaning of renormalization of the damping term and of a shift of the 共bare兲 precessional frequency ␥Rk. It can be easily realized from Eqs. 共11兲 and 共12兲 that in the weak SO limit ⑀F / ⑀R → ⬁, for which Ek± → ប2k2 / 2m, both the damping renormalization and the frequency shift are absent 共⌫k = 1 and ␹k = 0兲, indicating that these quantities stem from additional intra- and inter-band scattering channels opened when ⑀F / ⑀R is finite. Let us take a closer look at ⌫k and ␹k by performing the integration over p in Eqs. 共11兲 and 共12兲

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ELECTRON SPIN DYNAMICS IN IMPURE QUANTUM…

⌫k =

␹k =

冦

冦

kR , kR − k 1+

0 艋 k 艋 kR

共2kR − k兲kR , kR 艋 k 艋 2kR 共k − kR兲k 2kR 艋 k,

1,

2k2 + 3kRk kR2 − k2 共2kR − k兲k k2 − kR2

冧

,

0 艋 k 艋 kR

,

kR 艋 k 艋 2kR

−

共k − 2kR兲2 , 2kR 艋 k 艋 3kR 共k − kR兲共3kR − k兲

−

共4kR − k兲kR , 3kR 艋 k 艋 4kR 共k − kR兲共k − 3kR兲 4kR 艋 k.

0,

共13兲

冧

共14兲

␦␮ 兺 ␣ f 0共Ek␣ − ␮兲, ␥ Rk ␣

共15兲

where f 0共x兲 = 关exp共x / T兲 + 1兴−1 is the Fermi distribution function and ␮ = 21 共␮↑ + ␮↓兲.24 Furthermore, by imposing that limt→⬁兩Szk共t兲兩 ⬍ ⬁ and by arbitrarily choosing dSz共0兲 / dt = 0 for ␶−1 p = 0, at zero temperature 共␮ = ⑀F兲 Sz共t兲 is readily found to be Sz共t兲 = −

␦␮ 2␲␥R

冕 冋 k⬎

k⬍

冉 冑 冊 冉 冊 冉 冑 冊册

dk ␪共⌺k − ⌫2k 兲exp −

+ ␪共⌫2k − ⌺k兲exp −

⌫k + ⌺k t 2␶ p

⌺k ⌫k t cosh t 2␶ p 2␶ p

冉 冊 冉冑

Sz共t兲 = Sz共0兲exp −

For k ⬎ 4kR, ⌫k and ␹k are the same as in the zero SO limit, while for lower momenta they acquire a strong k dependence 关plotted in Fig. 1共c兲兴 arising from the combined effect of the reduced dimensionality 共D = 2兲 and the SO interaction. In particular, the divergence of ⌫k at k = kR and those of ␹k at k = kR and k = 3kR are due to scattering processes probing the SO induced van Hove singularity of the DOS of the lower subband which diverges as N−共E兲 ⬀ 冑⑀R / E 关see Fig. 1共b兲兴. As it is shown below, such strong k dependence has important consequences on the spin-polarization dynamics. Let us now turn to evaluate the explicit time dependence e 共t兲 is given by of Sz. From Eq. 共10兲, a general solution for Szk 冑 a linear combination of exp兵−关共⌫k ± ⌺k兲 / 2␶ p兴t其, where ⌺k = ⌫2k − ␹k − 共2␶ p␥Rk兲2, whose coefficients are fixed by imposing some initial conditions. If at t = 0 electrons have been prepared with a nonequilibrium z-spin occupation but equilibrium distribution for each spin branch,23 then the initial density matrix can be approximated by ␳共0兲 = Z−1 exp关−共H0 − ␮↑N↑ − ␮↓N↓兲 / T兴, where T is the temperature, Z = Tr ⫻exp关−共H0 − ␮↑N↑ − ␮↓N↓兲 / T兴, and ␮↑共↓兲 and N↑共↓兲 are the initial chemical potential and electron number for spin up 共down兲. Hence at the lowest order in the initial weak spin e imbalance ␦␮ = 21 共␮↑ − ␮↓兲, Szk 共0兲 is then e Szk 共0兲 = −

⑀F / ⑀R ⬎ 1 and k⭵ = kR共1 ± 冑⑀F / ⑀R兲 for ⑀F / ⑀R ⬍ 1 关see Fig. 1共a兲兴. For ⑀F / ⑀R → ⬁, k⭵ → kF and Eq. 共16兲 reduces to the classical formula25

,

共16兲

where ␪ is the unit step function, k⭵ = kR共冑⑀F / ⑀R ± 1兲 for

冊

t 1 − 共2␶ p⍀R兲2 t cosh , 共17兲 2␶ p 2␶ p

where Sz共0兲 = −ប␦␮N0 and ⍀R = ␥RkF is the Rashba frequency which characterizes the 共damped兲 spin precession behavior Sz共t兲 ⬇ Sz共0兲exp共−t / 2␶ p兲cos共⍀Rt兲 for 2␶ p⍀R Ⰷ 1 and the DP relaxational decay Sz共t兲 ⬇ Sz共0兲exp共−␶ p⍀R2 t兲 for 2␶ p⍀R Ⰶ 1.2 For finite values of ⑀F / ⑀R, Eq. 共16兲 starts to deviate from the classical regime 共17兲. Let us first consider ⑀F / ⑀R 艌 25. In this case, the integration over k in Eq. 共16兲 spans values necessarily larger than 4kR, where, according to Eqs. 共13兲 and 共14兲, ⌫k = 1 and ␹k = 0 关see also Fig. 1共c兲兴. For weak impurity scattering 共2␶ p␥Rk⬍ Ⰷ 1兲 ⌺k is negative and Eq. 共16兲 reduces to

冉 冊

Sz共t兲 ⬇ Sz共0兲exp −

t sin共⍀⬎t兲 − sin共⍀⬍t兲 , 2␶ p 共⍀⬎ − ⍀⬍兲t

共18兲

where ⍀⭵ = ␥Rk⭵. The main feature of Eq. 共18兲 is that Sz共t兲 oscillates with two different Rashba frequencies ⍀⭵ associated with the two SO splitted Fermi surfaces, Fig. 1共a兲. Two distinct Rashba frequencies characterize also the relaxation regime obtained when the scattering with impurities is strong enough that 2␶ p␥Rk⬎ Ⰶ 1 共and so ⌺k ⬎ 0兲 holds true. Also in this case the integration of Eq. 共16兲 is elementary and Sz共t兲 ⬇ Sz共0兲

冑␲ erf共冑␶p⍀⬎2 t兲 − erf共冑␶p⍀⬍2 t兲 , 2 冑␶p共⍀⬎ − ⍀⬍兲2t

共19兲

where erf is the error function. The spin precession and relaxation regimes of Eqs. 共18兲 and 共19兲 are governed solely by the enhanced momentum phase space settled by finite values of ⑀F / ⑀R. Instead, for ⑀F / ⑀R ⬍ 25, also the momentum dependence of ⌫k and ␹k becomes relevant, leading to important anomalous features of the spin dynamics. One of these is particularly striking and it is found when 1 艋 ⑀F / ⑀R 艋 9. In this case, the integration over k in Eq. 共16兲 crosses the point k = 2kR where ␹k changes sign 关see Fig. 1共c兲兴. Hence if 4␶ p␥RkR Ⰶ 1, 冑⌺k can be expanded as ⌫k − 共2kR − k兲 / 3kR for k ⬍ 2kR, while when k ⬎ 2kR ␹k becomes negative leading to exponentially small contributions to Sz共t兲 for sufficiently long times. Equation 共16兲 then can be approximated to Sz共t兲 ⬇ −

␦␮ 2␲␥R

冕

2kR

k⬍

dk −关共2k −k兲/6␶ k 兴t 3Sz共0兲 ␶ p R p R ⬇ e . 2 t 2 共20兲

The surprising result of Eq. 共20兲 provides the rather interesting prediction that, for sufficiently strong SO interaction and momentum scattering, the spin polarization decays as a power law rather than exponentially. In this case therefore the memory of the initial spin polarization can be much longer lived than in the DP regime. This is shown in Fig. 2共a兲 where Eq. 共16兲 is solved for ⑀R = 10 meV, ⑀F = 80 meV, and

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C. GRIMALDI

⑀F / ⑀R Ⰶ 1 limit in which the integration of Eq. 共16兲 becomes restricted to a narrow region around k = kR where both ⌫k and ␹k diverge as 1 / 兩k − kR兩, so that Eq. 共16兲 becomes

冉 冊

Sz共t兲 ⬀ exp −

FIG. 2. 共a兲 Spin-polarization time evolution from Eq. 共16兲 共solid line兲 for ⑀R = 10 meV and ⑀F = 80 meV corresponding to the band parameters of surface states in Bi共111兲 共Ref. 11兲. The momentum lifetime has been chosen to be ␶ p = 20 fs, in agreement with the angle-resolved photoemission results of Ref. 26. The dotted line is the power law decay of Eq. 共20兲 while the gray solid line is Eq. 共17兲. 共b兲 Crossover from the power law decay of Eq. 共20兲 and the fast relaxation decay of Eq. 共21兲 obtained for ␶ p⑀F / ប = 1 / 40 and several values of ⑀F / ⑀R.

␶ p = 20 fs. These values correspond to the band parameters of surface states of Bi共111兲 extracted from Ref. 11 and from the relaxation time estimated in Ref. 26. Another striking feature is that obtained in the extreme

*Electronic address: [email protected] Prinz, Phys. Today 48 共4兲, 58 共1995兲. 2 I. Žutić, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 共2004兲. 3 S. Datta and B. Das, Appl. Phys. Lett. 56, 665 共1990兲. 4 N. S. Averkiev and L. E. Golub, Phys. Rev. B 60, 15582 共1999兲; J. Schliemann, J. C. Egues, and D. Loss, Phys. Rev. Lett. 90, 146801 共2003兲. 5 E. I. Rashba, Sov. Phys. Solid State 2, 1224 共1960兲. 6 G. Dresselhaus, Phys. Rev. 100, 580 共1955兲. 7 M. I. Dyakonov and V. I. Perel, Fiz. Tverd. Tela 共Leningrad兲 13, 3581 共1971兲 关Sov. Phys. Solid State 13, 3023 共1971兲兴. 8 J. Sinova, D. Culcer, Q. Niu, N. A. Sinitsyn, T. Jungwirth, and A. H. MacDonald, Phys. Rev. Lett. 92, 126603 共2004兲. 9 Y. S. Gui et al., Phys. Rev. B 70, 115328 共2004兲. 10 E. Rotenberg, J. W. Chung, and S. D. Kevan, Phys. Rev. Lett. 82, 4066 共1999兲. 11 Yu. M. Koroteev, G. Bihlmayer, J. E. Gayone, E. V. Chulkov, S. Blugel, P. M. Echenique, and P. Hofmann, Phys. Rev. Lett. 93, 046403 共2004兲. 12 E. Bauer, G. Hilseler, H. Michor, C. Paul, E. W. Scheidt, A. Gribanov, Y. Seropegin, H. Noel, M. Sigrist, and P. Rogl, Phys. Rev. Lett. 92, 027003 共2004兲. 13 K. V. Samokhin, E. S. Zijlstra, and S. K. Bose, Phys. Rev. B 69, 1 G.

t , 8␶ p

共21兲

indicating that for extremely strong SO interaction, momentum scattering increases the spin-polarization decay. The power decay of Eq. 共20兲 and the fast relaxation regime of Eq. 共21兲 are plotted in Fig. 2共b兲 from a numerical integration of Eq. 共16兲 for different values of ⑀F / ⑀R. To conclude, the kinetic equations describing the time evolution of the spin polarization have been formulated for arbitrary strength of the SO interaction. Explicit solutions for quantum wells with Rashba-like SO interactions predict the failure of the DP relaxation formula for sufficiently strong SO couplings. In particular, the memory of the initial spin state can be strongly enhanced or reduced depending on ⑀F / ⑀R due to the strong momentum dependence of the renormalized damping and precession frequency. The tailoring of such features suggests an alternative route for spin manipulation in strong SO systems for spintronic applications. I thank E. Cappelluti for fruitful discussions.

094514 共2004兲. M. W. Wu, J. Phys. Soc. Jpn. 70, 2195 共2001兲. 15 M. M. Glazov and E. L. Ivchenko, J. Supercond. 16, 735 共2003兲. 16 F. X. Bronold, A. Saxena, and D. L. Smith, Phys. Rev. B 70, 245210 共2004兲. 17 E. L. Ivchenko, Yu. B. Lyanda-Geller, and G. E. Pikus, Zh. Eksp. Teor. Fiz. 98, 989 共1990兲 关Sov. Phys. JETP 71, 550 共1990兲兴. 18 A. A. Burkov, A. S. Núñez, and A. H. MacDonald, Phys. Rev. B 70, 155308 共2004兲. 19 C. Lechner and U. Rössler, Phys. Rev. B 72, 045311 共2005兲. 20 K. Blum, Density Matrix Theory and Applications 共Plenum, New York, 1981兲. 21 T. Kuhn and F. Rossi, Phys. Rev. B 46, 7496 共1992兲. 22 The following results are equally valid for quantum wells with zero Rashba interaction but with a Dresselhaus coupling of the form ⍀k = 共␥Dkx , −␥Dky , 0兲. 23 This initial configuration can be obtained by ultrashort pump pulses as explained, for example, in Ref. 14. 24 At the lowest order in ␦␮ the diagonal part of the initial density 1 matrix is simply f k共0兲 = 2 兺␣ f 0共Ek␣ − ␮兲. 25 V. N. Gridnev, JETP Lett. 74, 380 共2001兲. 26 C. Kirkegaard, T. K. Kim, and Ph. Hofmann, New J. Phys. 7, 99 共2005兲. 14

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