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Electron Spin Relaxation in a Semiconductor Quantum Well

A fully microscopic theory of electron spin relaxation by the D'yakonov-Perel' type spin-orbit coupling is developed for a semiconductor quantum well with a magnetic field applied in the growth direction of the well. We derive the Bloch equations for an electron spin in the well and define microscopic expressions for the spin relaxation times. The dependencies of the electron spin relaxation rate on the lowest quantum well subband energy, magnetic field and temperature are analyzed.


I. INTRODUCTION
Spin phenomenology in semiconductor structures has been at the focal point of research interest over the past few years in connection with proposals for spin-based quantum devices [1]. Spin implementation of quantum computation, optical switches, magnetic memory cells, etc., calls for a precise knowledge of spin dynamics and, in particular, the spin relaxation rates. The basic mechanisms responsible for spin relaxation are those of D'yakonov-Perel' (DP) [2,3], Elliott-Yafet [4], and Bir-Aronov-Pikus [5]. It was shown [6] that for III-V and II-VI compounds, which are the most promising materials for device purposes, the DP mechanism dominates at moderate temperatures and low hole concentrations. The spin relaxation time due to the DP mechanism has generally been expressed in the following semiphenomenological form [2,3]: for bulk semiconductors, and for quantum well structures, where α describes conduction band spin splitting due to lack of inversion symmetry (α = 0.07 for GaAs), ε g is the band gap, and T is the Kelvin temperature (k B = 1). The numerical coefficient Q depends on the orbital scattering mechanism, and p 2 z is the average square of momentum in the quantum well growth direction. It should be noted that these formulas involve the average momentum relaxation time, τ p , as a phenomenological parameter.
In the present paper we develop a fully microscopic theory of spin dynamics and apply it to a quantum well structure in the presence of an external magnetic field, directed along the well growth direction, taking account of various scattering mechanisms. Our theory facilitates the microscopic determination of τ p and its temperature and magnetic field dependencies in terms of the fundamental material parameters.
Our analysis involves explicit recognition of the two stages in the relaxation process corresponding to the relaxation time hierarchies involved in (a) electron thermalization due to interaction with phonons, and (b) spin relaxation. In the first stage of solution, (a), we determine the relaxation rates and fluctuation characteristics of electron orbital motion due to coupling to the phonon bath. Spin relaxation dynamics (the slowest process in the system) can be neglected in this stage.
The second stage, (b), proceeds with analysis of the spin relaxation process due to spinorbit interaction, wherein the orbital degrees of freedom are considered as an effective heat bath, having the characteristics determined in the first stage. A standard set of Bloch equations with two distinct relaxation times (longitudinal relaxation time, T 1 , responsible for spin magnetic moment relaxation, and transverse relaxation time, T 2 , responsible for decoherence) is derived in this second stage. In both stages of our analyses we employ the method proposed in Ref. [7] and developed in Ref. [8,9].

II. FORMULATION
The Hamiltonian of part (a) is given by where spin and its interactions are neglected in the first stage, and is the Hamiltonian of a two-dimensional electron in a quantum well with harmonic confinement in z-direction having frequency ω 0 . We assume the temperature to be low enough, so that only the lowest energy subband of the quantum well is occupied. In this case the motion of an electron can be described by means of two electron in-plane velocity components, which in the presence of a constant, uniform magnetic field directed along the z-axis B = (0, 0, B) are given by where ω k is the phonon energy and b + k and b k are creation and annihilation operators, respectively. The term U(r ⊥ , t) in Eq.(3) describes the electron coupling to phonons and impurities.
The electron-phonon interaction is given by where is the phonon heat bath variable (ζ is the strength of the electron-phonon coupling), and the electron variable conjugate to this operator is defined as Here, L 3 is the volume of the crystal, r ⊥ = (x, y), and f (k z ) = exp − k 2 z 4mω 0 is the electron confinement form factor. The response function of uncoupled phonon heat bath variables, ϕ k (t; t 1 ), and their correlation function, M k (t; t 1 ), are given by and where Q (0) k (t) are the unperturbed phonon variables, Θ(t − t 1 ) is the Heavyside unit step function, and [..., ...] + denotes an anticommutator.
The electron-impurity interaction is described by where U k are the spatial Fourier components of the impurity potential with correlation In summary, III. ORBITAL DYNAMICS Employing the Hamiltonian of Eq.(3), we obtain Langevin-like operator equations of motion determining in-plane electron dynamics, as given by (the derivation is presented in with fluctuation sources ξ x,y (t) and collision terms G x [V x (t); V y (t)] given by Eqs.(A8) and (A9). It should be emphasized that the expressions for fluctuation sources (Eq.(A8)) are obtained from microscopic analysis, and it is possible to calculate their correlation functions of any order. In particular, the correlation function of the fluctuation sources (in the case of weak coupling or Gaussian statistics of the unperturbed phonon variables) is given by We can separate the electron velocity operator into its average and fluctuation parts, Due to relaxation processes only the fluctuating part is nonzero, V x,y (t) = 0, and the equations of motion for the fluctuating components take the forms To further simplify Eqs. (18), we have to calculate the (anti)commutators involved in Eqs. (16), (A8), and (A9). This procedure is presented in Appendix B resulting in the following simplified equations: where the damping rate, γ 0 = γ x = γ y , and the frequency shift, δ = δ x = δ y , are given by and Here, τ = t − t 1 and τ 2 c (k ⊥ ) is given by Eq.(B11). It follows from Eq. (19) that the Fourier transforms of velocity correlation functions are given by where we neglected terms of the of order δ/ω c << 1. The correlation function of the fluctuation forces is defined as To carry out the τ − and k−integrations in Eqs. (20) and (25), we have to employ explicit expressions for the response and correlation functions for a particular scattering mechanism.
In the present paper we consider polar optical phonons, deformational acoustic phonons and charged impurities as possible scattering mechanisms. Their corresponding response and correlation functions are listed below.
Deformational acoustic phonons: where D is the deformation potential , ρ is the crystal density, and u is the sound velocity.
Static charged impurities have only correlation function, given by where r 0 is the screening radius, n * t = α n α Z 2 a , n α is the impurity concentration for species α, and Z α is their charge number. We neglect the cross correlations of scattering processes and, consequently, they are additive in both the damping rate and in the correlation function of the fluctuation sources: Substituting the corresponding response and correlation functions in Eqs. (20) and (25), we obtain the contributions of polar optic phonons as and the contributions of deformational acoustic phonons as and and, finally, the contributions of the charged impurities as and In all these formulae we use the notation k =

IV. SPIN DYNAMICS
In the second stage we analyze spin relaxation due to DP interaction between spin and electron orbital motion. The corresponding spin interaction Hamiltonian is given by [2,3] where µ B − → σ · − → B is the Zeeman splitting Hamiltonian term, µ B is the Bohr magneton, with For a quantum well strongly confined in < 001 > direction, H DP can be simplified as [3] H where and with p 2 z = m ω 0 /2 if the confinement is parabolic.
Considering orbital motion to play the role of an effective heat bath with correlation functions given by Eqs. (23) and (24), and employing a second application of the method of Refs. [7][8][9], we obtain a set of Bloch equations for the average spin projections as where σ x , σ y , σ z are the Pauli matrices and σ 0 z = − tanh ( ω B /2T ) is the equilibrium zcomponent of spin. ω B = gµ B B/ , µ B = |e| /2m 0 c is the Bohr magneton, and the g-factor is −0.44 for GaAs. It should be emphasized that we determine the relaxation times T 1 and T 2 microscopically as (44) While this result is reminiscent of a formula obtained by Ivchenko for bulk semiconductors [10], that differs from the present quantum well case under consideration here in that the (ii) via temperature dependence of the momentum relaxation rate, γ 0 = γ 0 (T ).
In the absence of a magnetic field (ω c = ω B = 0, V 2 x = K(0)/2γ 0 = T /m), we recover Eq. (2) as the zero field limit: wherein our microscopic analysis yields the phenomenological constant τ p of Eqs. (1,2) as Figs. 1(a,b) exhibit the dependence of the relaxation rate on the energy of the lowest electronic subband of the quantum well (which is given by E 0 = ω 0 /2 for the case of harmonic confinement) for various temperatures and magnetic field strengths. It is evident from these figures that at low temperature the applied magnetic field suppresses spin relaxation, whereas at higher temperatures the momentum relaxation rate dominates in the denominator of Eq. (44) and there is negligible magnetic field dependence in this case.
It should be emphasized that our microscopic calculations yield quantitative agreement with the experimental results of Ref. [6]. The full magnetic field dependence (up to 1000 Gs) of the spin relaxation rate is presented in Fig. 2 for ω 0 = 0.01 eV . This dependence has the well-known [2] Lorentzian shape, evident in Eq. (44), and for a reasonable range of magnetic field strength it is not affected by the magnetic field dependence of the velocity fluctuations, which are embedded in the function K(ω B ). (However, for the case of dilute magnetic semiconductors having a large electron g-factor, to which our general analysis is also applicable, the magnetic field dependence of the function K(ω B ) can be of crucial importance.) The magnetic field dependence for the case of bulk semiconductors (represented by a sum of several Lorentzians) was obtained in Ref. [10] and is also reconfirmed by our microscopic analysis [11].
The temperature dependence of the spin relaxation rate is shown in Fig. 3. The nonmonotonic behavior is due to the dominance of different scattering mechanisms in different temperature ranges. At low temperatures (before the peak), the momentum scattering rate is determined by impurities and is almost independent of temperature, and, consequently, we have almost linear growth of the spin relaxation rate with increasing temperature, as predicted by Eq. (2). However, as temperature further increases, optical phonons become the dominant scattering mechanism and, with this, the momentum scattering time becomes temperature dependent, resulting in suppression of the relaxation rate. It is important to note that in the temperature range in which phonons dominate, the relaxation rate is almost independent of the magnetic field, whereas at low temperatures the magnetic field shifts the peak and gives rise to deviations from linear behavior.

V. CONCLUSIONS
In conclusion, we have developed a fully microscopic theory of electron-spin relaxation in semiconductors by the D'yakonov-Perel' mechanism. We have applied this theory to a quantum well structure with a magnetic field in the growth direction. A set of Bloch equations for a spin system has been derived with microscopically determined longitudinal, T 1 , and transverse, T 2 , relaxation times, which are related as T 1 = T 2 /2 for the quantum well structure grown in < 001 > direction [3]. The well-known semiphenomenological expression for the spin relaxation rate [3] emerges as the zero-magnetic field limit of our result.
Furthermore, we have analyzed the dependencies of the electron-spin relaxation rate on the energy of the lowest quantum well subband, on magnetic field strength, and on temperature obtaining quantitative agreement with the experimental results.

APPENDIX A
The derivation of Langevin-like equations for electron velocity operators, Eq.(15) presented here follows the approach developed in our previous works [7][8][9]. We start from Heisenberg equations of motion for electron position and velocity operators for the Hamiltonian of Eq.(3), as If the coupling between the electron subsystem and heat bath is weak, or in the case of Gaussian statistics of unperturbed heat bath variables, the fully coupled Heisenberg heat bath variable is given by [7] Q k (t) = Q (0) where the response function of the phonon heat bath ϕ k (t; t 1 ) is defined by Eq.(10). Substituting Eq.(A2) into Eq.(A1), we have to take into account the fact that only the fully coupled heat bath variable commutes with the electron variable taken at equal times. Accordingly, we perform a symmetrization of both terms of Eq.(A2) with the electron variables, with the To eliminate the unperturbed phonon/impurity variables we employ the Furutsu-Novikov theorem [12]: where the correlation function M k (t; t 1 ) of free phonon variables is given by Eq.(11) and δ/δQ (0) (t 1 ) is the functional derivative with respect to the uncoupled heat bath variable Q (0) (t 1 ). Eq.(A4) can be derived by an application of the Wick theorem, assuming the operator X k (t) to be a functional of {Q (0) (t 1 )} with t 1 ≤ t (See Ref. [7]). This expression is exact only for Gaussian statistics of the variables Q (0) (t), and it also can be applied in the case of weak coupling. For the case of strong coupling, Eq.(A4) requires modification to include functional derivatives of all orders. The functional derivative on the right-hand side of Eq. (A4) is proportional to the commutator [7] in the form with the following result where The resulting simplified stochastic equations for the fluctuating velocity components take the form given by Eq. (19).