1. Introduction For a 2D system with any HSO(k) (where k lies in the plane of the heterostructure), one can show, similarly to  The degrees of freedom of spin have received a great deal (see also [4–6]), that the spin dynamics of electrons in the of attention throughout the development of semiconductor presence of elastic scattering is described by the following physics. Recently, the spin properties of carries have been equations investigated intensely in low-dimensional semiconductor structures. In electronics, much interest in spin has been ji d(F+ - F-)nAn aroused by recent proposals to construct spin transistors and 1 0 i(t) =- S (t), (1) spin computers based on heterostructures [1,2].
2 2 n=- d(F+ - F-) j The spin-orbit interaction, governing the spin behavior, is 0 much more complex in semiconductor heterostructures than in bulk systems. The bulk spin-orbit terms take a more ji An = Tr H-n, [Hn, j] i.
interesting form in two-dimensional (2D) systems, and, in addition, new terms appear, which are absent in bulk.
It should be noted that this is true only for times longer In  we considered electron spin dunamics in asymmetthan the momentum relaxation time but shorter than the rical heterostructures. A giant anisotropy of spin relaxation spin relaxation times. In Eq. (1), Si are the spin density times caused by interference of different spin-orbit terms has components (i = x, y, z), the integration is performed over been revealed. In this work, we calculate the spin relaxation energy = k2/2m, where m is the electron effective mass, rates in real asymmetrical structures. A heterojunction and F±() are distribution functions of electrons with the spin a triangular quantum well (QW) are considered in detail.
projection equal to ±1/2, i are the Pauli matrices; Hn are The effect of heteropotential asymmetry on spin relaxation the harmonics of the spin-orbit Hamiltonian:
is investigated in a wide range of electron concentrations and temperatures. We show that the giant spin relaxation dk Hn = HSO(k) exp(-ink), (2) anisotropy is governed by external parameters, that opens up new possibilities for spin engineering.
where k is the angular coordinate of k, and the scattering times are given by 2. Theory Let us consider a system with spin-orbit interaction de- = d W(, )(1 - cos n), (3) n scribed by the Hamiltonian HSO(k), where k is a wavevector.
HSO(k) is equivalent to a Zeeman term with effective magwhere W(, ) is the probability of elastic scattering by an netic field dependent on k. In the presence of scattering, the angle for an electron with energy.
wavevector changes and, hence, the effective magnetic field Equation (1) is valid for 2D electrons with any spinchanges too. Therefore, in the case of frequent scattering, orbit interaction HSO(k). Now we consider an asymmetrical the electrons move in a chaotically changing magnetic field.
zinc-blende heterostructure. There are two contributions The spin dynamics in such a system has diffusion character, to HSO(k). The first, the so-called bulk inversion asymwhich leads to loss of any specific spin orientation. This is metry (BIA) term, is due to lack of inversion symmetry called the D’yakonov–Perel’ spin relaxation mechanism , in the bulk material of which the heterostructure is made.
which is the main spin relaxation mechanism is many AIIIBV To calculate this term, one has average the corresponding bulk semiconductors and heterostructures.
bulk expression over the size-quantized motion . We in¶ E-mail: golub@coherent. ioffe. rssi.ru vestigate a heterostructure with the growth direction  7 98 N.S. Averkiev, L.E. Golub, M. Willander coinciding with the z-axis and assume that only the first where elecrton subband is populated. The BIA-term has the form 41 2 z(k) = 2 k2 + 2 k2 - 2 k2 kz z HBIA(k) = x kx (k2 - k2 ) +y ky( k2 -k2), (4) y z z x 1 + 3/where we choose x- and y-directions to be aligned with + 2k6, the principal axes in the heterostructure plane. Here kz is squared the operator (-i/z) averaged over the ground 21 +(k) = - k2 k2 + - k2 kz z state, and is the bulk spin-orbit interaction constant. It is seen that HBIA contains terms both linear and cubic in k.
1 + 3/In asymmetrical heterostructures, there is an additional + 2k6, (10) contribution to the spin-orbit Hamiltonian, which is absent in the bulk. It is caused by structure inversion asymmet 21 (k) = + k2 k2 - + k2 k- z z ry (SIA) and can be written as [7–9] HSIA(k) =(x ky - ykx), (5) 1 + 3/+ 2k6.
where is proportional to the electric field E, acting on an Equations (9), (10) are valid for any electron energy electron:
distribution. If the electron gas is degenerate, then the spin = 0eE. (6) relaxation times are given by Here e is the elementary charge and 0 is a second spinorbit constant determined by both bulk spin-orbit interaction =i(kF), (11) i parameters and properties of heterointerfaces. It should be stressed that, in asymmetrical heterostructures, E is caused where kF is the Fermi wavevector determined by the total 2D mainly by the difference of the wavefunction and band electron concentration N:
parameters at the interfaces, rather than by average electric field .
kF = 2N. (12) HSIA also contains terms linear in k. From Eq. (1) follows that the harmonics with the same n are coupled in the spin In this case, the scattering time 1 in Eqs. (10) coincides with dynamics equations. This leads to interference of linear in the transport relaxation time, tr, which can be determined wavevector BIA- and SIA-terms in spin relaxation . from the electron mobility.
For HSO = HBIA + HSIA, the system has C2v-symmetry. For nondegenerate electrons, the spin relaxation times Therefore Eqs. (1) can be rewritten as follows: are determined, in particular, by the energy dependences of the scattering times 1 and 3. If 1, 3, then Sx ± Sy Sz 3/1 = const and z = -, x ± y = -. (7) z ± 1 4tr 2 2mkBT = 2 k2 + The times z, +, and are the relaxation times of the spin - z 2 z parallel to the axes ,  and , respectively.
+ 2 2mkBT If both spin subsystems come to equilibrium before the - 2 kz onset of spin relaxation, then 1 + 3/1 2mkBT F±() =F0(µ± - ), (8) +( + 2)( + 3) 2, (13) where F0 is the Fermi–Dirac distribution function and µ± are 1 2tr 2mkBT chemical potentials of the electron spin subsystems. If the = ± - k2 z ± spin splitting is small, i. e.
+ 2 2mkBT µ+ - µ µ+, µ - - + (± - k2 ) z then the expressions for the spin relaxation rates 1/i 1 + 3/1 2mkBT (i = z, +, -) have the form +( + 2)( + 3) 2.
Here T is electron temperature and kB is the Boltzmann d(F0/)i(k) 0 constant. In the particular case of short-range scattering, =, (9) i = 0, and 1 = 3 are equal to tr, which is independent d(F0/) of temperature.
Физика и техника полупроводников, 2002, том 36, вып. Spin relaxation in asymmetrical heterostructures Spin relaxation times are very sensitive to the relationship between two spin-orbit interaction strengths, k2 and.
z From Eqs. (11), (13) follows that at low concentration or temperature, 1/z, 1/, and 1/+ are determined by the sum of squared k2 and, by their squared sum, z and squared difference, respectively. This may lead to a considerable difference between the three times, i. e. to a total spin relaxation anisotropy, if k2 and are close in z magnitude.
In real AIIIBV systems, the relations between HBIA and HSIA may be different. HBIA or HSIA may be dominant [11,12], or they may be comparable .
The value of k2 depends on the shape of the heteropoz tential and will be calculated for the given asymmetrical heterostructures below. The constant is known for GaAs from optical orientation experiments . Correct theoretical expressions for and 0 have been derived in terms of the three-band k · p model [13,14]. The heterointerfaces give a contribution to 0 in addition to that from the bulk .
At large wavevectors, 0 starts to depend on k [16,17].
Here we assume concentrations and temperatures to be sufficiently low, allowing us to ignore this effect.
Figure 1. Concentration dependences of the reciprocal spin The spin relaxation rates for two types of asymmetrical relaxation times, 1/z (solid line), 1/ (dashed line), and structures — a heterojunction and a triangilar QW — are 1/+ (dotted line), for a GaAs/AlAs heterostructure at zero calculated below. The scattering is assumed to be shorttemperature. The parameters are given in the text. The inset shows range ( = 0, 3 = 1 = tr). All parameters are chosen to the spin-orbit interaction strengths, k2 (solid line), (dashed), z correspond to GaAs/AlAs heterostructure: = 27 eV · 3, and | k2 -| (dotted), in eV ·, as functions of the electron z m = 0.067m0, where m0 is the free electron mass and concentration N/(1012 cm-2).
0 = 5.33 2. The time tr is taken equal to 0.1 ps and assumed to be independent of the electron concentration.
in (10) is larger for 1/, and the second, for 1/+.
3. Spin relaxation in a heterojunction Therefore, at a certain concentration, the times + and must be equal. From Eqs. (10) (11) follows that this takes In a heterojunction, the extent of the spin-orbit interaction place when is governed by the 2D carrier concentration N; k2 can be z k2 = 4 k2, (16) estimated as follows :
F z 2/which is fulfilled at N = 1.1 · 1013 cm-2 as illustrated in 1 16.5Ne2m k2 =, (14) Fig. 1. At larger concentrations, the spin relaxation is again z 4 totally anisotropic.
where is the dielectric constant. The mean electric Despite that k2 and are close in magnitude over z field acting on an electron can be taken equal to half the a wide range of concentrations (see the inset of Fig. 1), maximum field in the junction: all three spin relaxation rates depend on N monotonically.
This happens because, as the concentration increases, kF in2Ne creases as well, and the terms in HSO which are cubic in the E =. (15) wavevector, become important. The growth of these terms with N dominates the change in ( - k2 )2 in (10), hence z Figure 1 shows the concentration dependence of the the concentration dependence of 1/+ is monotonic.
reciprocal spin relaxation times for degenerate electrons in The situation changes in the case of a Boltzmann gas.
a GaAs/AlAs heterojunction ( = 12.55). The inset shows For non-degenerate electrons, the mean wavevector and the spin-orbit interaction strengths, k2 and, and the z the concentration are independent. For temperatures up absolute value of their difference, as functions of electron to 300 K, the characteristic k2 2mkBT / is much less concentration.
than k2, and the spin relaxation rates are determined by One can see a spin relaxation anisotropy for all three z directions over a wide range of concentrations. 1/+ is the first terms in (13). As a result, all three spin relaxation less than 1/ at small N and greater than 1/ at large times are different up to 300 K at a given concentration. The - concentration. This is due to the fact that the first term results of relevant calculations are presented in Fig. 2.
7 Физика и техника полупроводников, 2002, том 36, вып. 100 N.S. Averkiev, L.E. Golub, M. Willander The times + and are equal to each other at a certain temperature only. According to (13), the corresponding condition is kz T =. (17) mkB(1 + /2) With the GaAs-parameters and = 0 in (14), it can be seen that (17) is satisfied at T 100 K for N = 1011 cm-and at T 290 K for N = 5 · 1011 cm-2, in agreement with Fig. 2.
At a fixed temperature, the spin relaxation rates are governed by the electron concentration. According to Eqs. (13), the dependences of 1/i on N are similar to the curves in the inset of Fig. 1. In particular, form Eqs. (13) follows that 1/z and 1/ must be close in magnitude and both greatly exceed 1/+. In addition, 1/+ depends on concentration non-monotonically. This is confirmed completely by the results presented in Fig. 3. One can see that 1/+ 1/z 1/, and the rate 1/+ has a minimum when plotted as a function of concentration. This Figure 3. Concentration dependences of the reciprocal spin relaxaminimum is at N = 1.4 · 1013 cm-2, when the terms in HSO tion times, 1/+ (solid line), 1/z (dashed line), and 1/ (dotted linear in the wavevector cancel out. The corresponding line), for Boltzmann electron gas in GaAs/AlAs heterostructure at temperatures T, K: 1 — 30, 2 — 77, 3 — 150, 4 — 300.
condition is k2 =. (18) z At this concentration, the spin relaxation time + is very large but remains finite owing to the terms cubic in k.
Therefore the difference in the spin relaxation times is more pronounced at low temperature. At high T, the cubic in the wavevector terms become significant in HSO, and the minimum in 1/+ disappears. However 1/+ is still much less than 1/, i. e. huge spin relaxation anisotropy occurs in the plane of the heterojunction even at room temperature.
4. Spin relaxation in a triangular quantum well In this Section, we investigate spin relaxation in the following asymmetrical system. We consider a structure with infinitely-high barrier at z < 0 and constant electric field E at z > 0.
In the framework of this model, 2/2meE k2 = a, (19) z where dx Ai (x - ) a = 0.78. (20) dx Ai (x - ) Figure 2. Temperature dependences of spin relaxation rates, Here (-) is the first root of the Airy function:
1/ (solid line), 1/z (dashed line) and 1/+ (dotted line), for a GaAs/AlAs, heterostructure at different electron concentrations. Ai (-) =0, 2.338.
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