Pis'ma v ZhETF, vol.88, iss. 12, pp.934-938
© 2008 December 25
Spin relaxation in the impurity band of a semiconductor in the
external magnetic field
I. S. Lyubinskiy
A.F. Ioffe Physical Technical Institute RAS, 194021 St. Petersburg, Russia Submitted 10 November 2008
Spin relaxation in the impurity band of a 2D semiconductor with spin-split spectrum and hyperfine interaction in the external magnetic field is considered. Two contributions to the spin relaxation are shown to be relevant: the one given by optimal impurity configurations with the hop-waiting time inversely proportional to the external magnetic field and another one related to electron motion over large distances. The average spin relaxation rate is calculated.
PACS: 71.55.Jv, 71.70.Ej, 72.25.Rb, 85.75.^d
Spin dynamics in semiconductors has attracted much attention in the last decades [1, 2]. In particular, a number of experimental [3-9] and theoretical [10-15] works are devoted to the investigation of spin relaxation in the impurity band of a semiconductor. An increasing interest to this problem is motivated by experimental observation of up-to-microsecond spin lifetimes in n-doped bulk GaAs and GaAs/AlGaAs heterostructures, which makes them good candidates for the use in possible spin-tronics applications.
Spin relaxation in the impurity band is usually driven by hyperfine interaction or spin-orbit coupling. Since the nuclear spin relaxation time is very long (tjv ~ ~ 0.1 ms), hyperfine interaction can be treated as a ran-dom-in-space static magnetic field with the associated spin precession frequency u>n = A/t/N, where A is the hyperfine coupling constant and N is the number of nuclei within the volume occupied by the wave function [16] (the directions of the random magnetic field for electrons located on different impurities are not correlated). For spin-orbit coupling, the associated spin precession frequency wp is a power function of the electron momentum p [17-19] (in the 2D case, uip is linear in p [19]). As a result, spin-orbit coupling leads to spin rotation in the process of phonon-assisted hops from one impurity to another by the angle (j) « wPoAr/i;o, where Ar is the distance between impurities and po = mv0 is the under-the-barrier momentum.
There can be several mechanisms of spin relaxation in the impurity band. Like in quantum dots (QDs), spin relaxation might be driven by phonon-assisted transitions between Zeeman sublevels of the ground state of an impurity. This mechanism of spin relaxation is well studied in QDs [20, 21]. For isolated shallow donor or
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small QD, it gives spin relaxation times of the order of Is at fields B « IT (Zeeman energy £z « 0.3K) [22, 23]. Other mechanisms of spin relaxation involve electron hops from one donor to another. For such mechanisms, the spin relaxation rate can be roughly estimated as [7, 10, 24]:
1/ts = w2nthc, 1/ts = 4>2/The, (1)
for the case of hyperfine interaction and spin-orbit coupling respectively (here The is the characteristic hop waiting time). These equations are based on the classical picture of the angular spin diffusion in a random magnetic field (in the case of hyperfine interaction the direction of spin precession changes randomly after each hop; in the case of spin-orbit coupling the spin rotates in a random direction in the process of a hop). However, this picture does not account for the exponential variation of the hop waiting times:
rhl = r0 exp (2Ar/o), (2)
rh2 = r0 exp (2Ar/o + A£/T) (3)
for phonon emission and absorbtion respectively (here Ar is the distance between impurities, A£ is the distance between their energy levels, a = eH2/2me2 is the Bohr radius, and T is the temperature). The main consequence of such inhomogeneity is that it is impossible to introduce an universal time scale for the system under consideration. This fact is confirmed by about ten-fold decrease of the experimentally measured spin correlation time in the bulk GaAs at the crossover from hyperfine-interaction-induced to spin-orbit-induced spin relaxation (see Fig.3 in Ref. [7]). The effects of the inhomogeneity on spin relaxation in the absence of the external magnetic field were considered in Refs. [13, 14] for the systems with spin-split spectrum. In particular,
Spin relaxation in the impurity band of a semiconductor in the external magnetic field
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it was found that there are two essentially different contributions to spin relaxation: the one related to electron hops over the pairs of impurities with the size of the order of the Bohr radius and another one related to the motion over large distances. In Ref. [9], the dependence of the spin relaxation rate on the external magnetic field was measured experimentally in the impurity band of the bulk GaAs. It was found that the relaxation time first increases, then decreases, then increases again as a function of the magnetic field (see Fig.3 in Ref. [9]).
In this letter, we calculate the average spin relaxation rate for the mechanisms described above in the presence of the external magnetic field B. The use of the averaged relaxation rate is justified when relaxation is slow enough so that an electron can walk over a large distance during the spin relaxation time ts (in the opposite case the spin relaxation is governed by escape from the regions with slow relaxation to the regions with fast relaxation [14]). The corresponding condition is [14] ts tc (here tc = tq exp (C£o) is the hop waiting time for so-called critical bond [25], £„ = M4L2W/a2T, C is the coefficient of the order of unity, W = e2 ¡eL4 is the width of the impurity band, and L4 = n^1^2 is the average distance between impurities). We also assume that spin precession in the external magnetic fields is sufficiently fast O0T5 1 (here O0 is the spin precession frequency in the external magnetic field). In this case, the components of the spin perpendicular to the magnetic field are suppressed due to fast precession, and hereafter they will be neglected. Finally, we assume that the temperature is sufficiently small so that we can neglect activation to the conduction band, assume that -C T, and neglect electron-electron interaction.
Our main point is that over a wide range of parameters the main contribution to the spin relaxation is given by the pairs of impurities with the hop waiting time:
The « 1/O0. (4)
Indeed, a common feature of the relaxation mechanisms based on the angular spin diffusion in a random magnetic field is that they are suppressed by applying a longitudinal magnetic field with the associated spin precession frequency larger than the inverse correlation time of the random magnetic field. In the simplest case of a pair of impurities with the hop waiting times tm = r^2 = r^ (AS -C T), the spin relaxation rate is proportional to A02Tft/ (l + fioTft)> where AO is the spin precession frequency in the random magnetic field (in the case of hyperfine interaction AO « wn] in the case of spin-orbit coupling AO « fio0, as shown below). The contribution of the pairs to the spin relaxation rate increases exponentially with Ar for Th < 1/O0 and decrease for r^ > 1/O0.
Taking into account Eqs. (1) and (4), one can estimate the spin relaxation rate on the pairs of impurities as:
1 ¡TS = fu}%/n0, 1 ¡TS = ^20o, (5)
for the case of hyperfine interaction and spin-orbit coupling respectively (here v « (a/Ld)2 T/W is the share of the optimal pairs). At sufficiently small magnetic fields the relaxation is due to electron motion over large distances.
Let us proceed to the rigorous formulation of the problem. We start with the system with spin-split spectrum. The Hamiltonian of the system is
P2
H0 = — + U (r) + fio-ilo/2 + h<Tap/2mLs, (6) 2 to
where U (r) is the impurity potential, Ls is the length characterizing the strength of the spin-orbit coupling, a is the dimensionless tensor with the components of the order of unity, and a is the vector of Pauli matrices. The last term on the right-hand side is the combination of the Bychkov-Rashba spin-orbit coupling [17] and Dres-selhaus spin-orbit coupling averaged over the electron motion in the direction perpendicular to the quantum well [18, 19]. For the following consideration it is convenient to make a transformation, which cancels spin-orbit coupling to the first order in parameters 1 ¡Ls and O0 [26, 27]:
jji _ eia&T/2Ls jje-icrai/2Ls _ ^
As a result, p2
H'0 = + U (r) + nn0<r/2 + h[rt0 x ar/Ls] tr/2.
ZTfl
(8)
Let us consider spin relaxation on a pair of impurities caused by spin precession in the random magnetic field. From the Hamiltonian (8) one can derive an equation, describing spin dynamics:
dS/dt = [(«o + Afl (t)) x S], (9)
where Ail (t) = [fio x (t) /Ls] and the position of an electron r (t) takes two values: rj and r2 (here rij2 are the positions of the impurities). To find the random field correlator k (t) = (Ail (t) Ail (0)), one needs to calculate the Green function Gij (t) of the kinetic equation for an electron on a pair of impurities:
dni/dt = —dnzldt = «2/77,2 - ni/TM, (10)
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I. S. Lyubinskiy
where ni,2 are the probabilities to find an electron at impurity 1 and 2 respectively. Using the Green function of this kinetic equation, we get:
K(t)= ^(Ti)Gij(Ti)Sl(Tj)nj0 = «,¿=1,2
AO2
= 4cosh2(Ag/2T)eXP(^/Tft)' (11)
where Ail = [fi0 x aAr/Ls], Ar = rj — r2 is the size of the pair, 1/th = 1/tm + 1/77,2, and = = Thi/ (thi + f^) is the equilibrium probability to find an electron at impurity i (i = 1,2). Treating the term proportional to Ail (t) in Eq. (9) as a perturbation and using Eq. (11), we get the following evolution equation for the component of the spin parallel to the external magnetic field:
dSn/dt = - J k (t') cos (0 of) Sy (t - t') dt1. (12) The spin relaxation rate on a pair of impurities is
1/Ts{At,A£) =
AO2
Th
4cosh2(A£/2T)l + 02r2'
(13)
Depending on the strength of the external magnetic field, several regimes can be realized. In the case O0 < 1/r
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