Electron spin relaxation in p-type GaAs quantum wells

Y Zhou¹, J H Jiang² and M W Wu^1,2,3

¹ Hefei National Laboratory for Physical Sciences at Microscale, University of Science and Technology of China, Hefei, Anhui 230026, People's Republic of China
² Department of Physics, University of Science and Technology of China, Hefei, Anhui 230026, People's Republic of China

³ Author to whom any correspondence should be addressed.

E-mail: mwwu@ustc.edu.cn.

Received 8 July 2009
Published 20 November 2009

Contents

• 1. Introduction
• 2. Kinetic spin Bloch equations
• 3. Results and discussions
  • 3.1. Comparison of the BAP and DP mechanisms
  • 3.2. Multi-hole-subband effect
• 4. Conclusion and discussion
• Acknowledgments
• Appendix. Comparison with experiment
• References

1. Introduction

In recent years, much attention has been devoted to semiconductor spintronics both theoretically and experimentally due to the potential application of spin-based devices [1]–[3]. In order to manipulate the spin relaxation such that the information is well preserved before required operations are completed, it is crucial to gain a thorough understanding of spin relaxations. In p-doped III–V semiconductors, the main electron spin relaxation mechanisms have been recognized as [1] the Bir–Aronov–Pikus (BAP) mechanism [4] and the D'yakonov–Perel' (DP) mechanism [5]. In the DP mechanism, electron spins decay due to their precessions around the momentum-dependent spin–orbit fields (inhomogeneous broadening) [6] during the free flight between adjacent scattering events. In the BAP mechanism, spin relaxes due to spin flip caused by exchange interaction with holes. It was believed that in p-doped bulk samples, the BAP mechanism dominates the spin relaxation process at high doping density and low temperature, whereas the DP mechanism is more important at low doping density and high temperature [1], [7]–[10]. In a two-dimensional system, Maialle [11] calculated the spin relaxation time (SRT) due to these two mechanisms at zero temperature using the single-particle approach and showed that these two SRTs have nearly the same order of magnitude. However, as pointed out by Zhou and Wu recently [12], there are some common problems in the published literature: the SRT due to the BAP mechanism was calculated based on the elastic scattering approximation, which is invalid at low temperature due to the omission of Pauli blocking. Also, the investigation of the SRT due to the DP mechanism was also inadequate because the Coulomb scattering is not included in the framework of single-particle theory.

Zhou and Wu applied the fully microscopic kinetic spin Bloch equation (KSBE) approach [6, 13] to investigate the spin relaxation in p-type GaAs quantum wells [12]. The KSBE approach has achieved good success in the study of spin dynamics in semiconductors, where not only the results are in good agreement with the previous experiments, but also many predictions have been confirmed by the latest experiments [6], [12]–[30]. Via this approach, they explicitly included all the relevant scatterings and obtained the accurate SRT due to these two mechanisms. It was found that the BAP mechanism is always less efficient than the DP mechanism for moderate and high excitation densities where N_ex0.1N_h (N_ex (N_h) is the excitation (hole) density), in contrast to the common belief in the literature [1], [7]–[10]. This claim has very recently been confirmed experimentally by Yang et al  [29]. A similar conclusion was also obtained in bulk GaAs later [19].

However, for very low excitation density where the Pauli blocking of electrons is negligible, for high hole density where the contribution from the high subbands or different hole bands becomes significant and/or for high impurity density where the spin relaxation due to the DP mechanism is suppressed, whether the BAP mechanism can be more efficient is still questionable. In the present work, we extend the KBSEs to include both the lowest subband of light hole (LH) and the lowest two subbands of heavy hole (HH), and compare the relative importance of the DP and BAP mechanisms in wider ranges of temperature, hole density, excitation density and impurity density. We present a `phase-diagram-like' picture indicating the dominant spin relaxation mechanism. In the case with no impurity and high excitation density, our results show that the BAP mechanism is unimportant at low temperature, in agreement with [12]. Nevertheless, since more hole subbands and bands are included in our model, we are able to discuss the case with higher hole density. We find that the BAP mechanism can surpass the DP mechanism at high temperature for sufficiently high hole density. In the case with no impurity and low excitation density, the BAP mechanism can surpass the DP mechanism for wider hole density and temperature ranges. Moreover, we also find that in both cases above, the regime where the BAP mechanism is more efficient is always around the hole Fermi temperature for high hole density, regardless of excitation density. However, in the high impurity density case with the impurity density being identical to the hole density, the behavior is very different from the impurity-free case: the regime of hole density where the BAP mechanism is more efficient becomes larger, and the regime of temperature becomes wider, ranging roughly from the electron Fermi temperature to the hole Fermi temperature. We also show that the multi-hole-subband effect leads to a very intriguing hole-density dependence of SRT at low temperature.

This paper is organized as follows. In section 2, we set up the KSBEs. In section 3, we compare the relative importance of the BAP and DP mechanisms in different parameter regimes and investigate the multi-hole-subband effect. We conclude in section 4. A comparison of the calculation from the KSBEs with the experimental data of a p-type GaAs quantum well is given in the appendix.

2. Kinetic spin Bloch equations

We investigate a p-type (001) GaAs quantum well of width a with its growth direction along the z-axis. The width is assumed to be small enough so that only the lowest subband of electron, the lowest two subbands of HH and the lowest subband of LH are relevant for the electron and hole densities we discuss. The envelope functions of the relevant subbands are calculated under the finite-well-depth assumption [12, 17]. The barrier layer is chosen to be Al_0.4Ga_0.6As where the barrier heights of electron and hole are 328 and 177meV, respectively [31]. We focus on the metallic regime where most of the carriers are in extended states. Since the hole spins relax very rapidly (only several picoseconds), we assume that the hole system is always in equilibrium.

Via the nonequilibrium Green function method [32], we construct the KSBEs as follows [12, 13]:

with representing the electron single-particle density matrix with a two-dimensional momentum k = (k_x, k_y), whose diagonal and off-diagonal elements describe the electron distribution function and spin coherence, respectively. The coherent term can be written as (≡1 throughout this paper)

in which [A,B]≡AB–BA is the commutator. h(k) represents the spin–orbit coupling of electrons composed of the Dresselhaus [33] and Rashba [34] terms. For GaAs quantum wells, the Dresselhaus term is dominant [35] and

where k_z² stands for the average of the operator –(∂/∂z)² over the state of the lowest subband of electron, and γ_D = 8.6 eV Å³ denotes the Dresselhaus spin–orbit coupling coefficient [25, 36]. is the effective magnetic field from the Coulomb Hartree–Fock contribution [14]. For the screened Coulomb potential, the screening from electrons and holes is calculated under the random phase approximation [12, 37]. The scattering term consists of electron–impurity, electron–phonon, electron–electron Coulomb, electron–hole Coulomb and electron–hole exchange scatterings. The expressions of these scatterings are given in detail in [12]. Here we just extend the electron–hole Coulomb and exchange scatterings to the multi-hole-subband case. The expression of electron–hole Coulomb scattering is still similar to that in [12]. The complete electron–hole exchange scattering term is written as

Here and are the electron density matrices; are the electron spin ladder operators. λ = HH⁽ⁿ⁾, LH⁽ⁿ⁾ with the superscript being the subband index of hole. f^h_{k, λ} is the hole distribution on the λth hole band. The matrix comes from the long-range term of the electron–hole exchange interaction Hamiltonian and can be written as [38]^Note4 , where ΔE_LT is the longitudinal–transverse splitting in bulk, |_3D(0)|² = 1/(πa₀³) with a₀ being the exciton Bohr radius, and and ) are operators in hole spin space. The matrix is given by [38] (in the order of , , , , , )

where K = k + k '–q is the center-of-mass momentum of the electron–hole pair with K_± = K_x±iK_y. The form factors can be written as

with

In equation (5), it is seen that most of the nonzero elements in matrix contain K_±² or K_±, and thus the magnitudes of these terms increase with increasing K. The only exception is M^–_{–1/2, 1/2}, where the K dependence is only from the form factor . Consequently, the magnitude of M^–_{–1/2, 1/2} decreases with K. This K dependence contributes to an intriguing hole-density dependence of spin relaxation to be addressed in this work.

3. Results and discussions

By numerically solving the KSBEs with all the scatterings explicitly included, one is able to obtain the SRT from the temporal evolution of the electron spin polarization along the z-axis. We choose initial spin polarization P = 4%^Note5  and well width a = 10 nm; the external magnetic field B = 0 unless otherwise specified. The other material parameters are listed in table 1 [11, 39, 40].

3.1. Comparison of the BAP and DP mechanisms

We first examine the relative importance of the BAP and DP mechanisms for different parameters in p-type GaAs quantum wells. In figure 1, the ratio of the SRT due to the BAP mechanism to that due to the DP mechanism is plotted as a function of temperature and hole density in the cases with no/high impurity and low/high excitation densities. From this figure, one can recognize the parameter regime where the DP or BAP mechanism is more important. It is also shown that the multi-hole-subband effect becomes significant for high temperature and/or high hole density (the regime above the yellow solid curve). Here and hereafter, the multi-hole-subband refers to either the high HH subband or the LH subband. Although the multi-hole-subband effect has an important effect on electron spin relaxation in the relevant regime, the main physics is still the same as that in the single-hole-subband model. Therefore, in this subsection, we first discuss the general behavior about how the relative importance of the BAP and DP mechanisms is influenced by the temperature, hole density, excitation density and impurity density, which is analogous in both the multi-hole-subband and single-hole-subband models. We then investigate the special features from the contribution of high hole subbands in the next subsection.

In the case with no impurity and high excitation density (figure 1(a)), our results are consistent with [12]: the BAP mechanism is unimportant at low temperature, which is in stark contrast with the common belief in the literature [1], [7]–[10]. Moreover, since we extend the scope of our investigation to higher hole density by including more hole subbands in our model, it is discovered that the BAP mechanism can surpass the DP mechanism in the regime with high temperature and sufficiently high hole density (the regime embraced by the black dashed curve).

In the case with no impurity and low excitation density (figure 1(b)), one can see that the regime where the BAP mechanism surpasses the DP mechanism becomes larger. The underlying physics is shown in figure 2(a). It is seen that the SRTs due to the BAP and DP mechanisms both decrease with increasing excitation density (N_ex = N_e), but the amplitude of the latter is much larger than the former. The decrease of τ_DP comes from the increase of the inhomogeneous broadening |h_k|²N_ex [14]^Note6 , and the decrease of τ_BAP is mainly from the increase of the average electron velocity v_kN_ex^0.5 [11]. Moreover, the increase of the Pauli blocking of electrons can partially compensate the effect of the increase of v_k [12]. Consequently, τ_BAP decreases with N_ex much more slowly than τ_DP and the relative importance of τ_BAP is enhanced for lower excitation density. It is also noted that when the electron system is in the nondegenerate regime (T > T_F^e = E_F^e/k_B), the inhomogeneous broadening and v_k are not sensitive to N_ex. Thus the ratio τ_BAP/τ_DP changes little with the excitation density.

Figure 2. SRTs due to the DP and BAP mechanisms, the total SRT together with the ratio τBAP/τDP versus temperature T for Nex = 109 cm–2 (curves with •), 3×1010 cm–2 (curves with ) and 1011 cm–2 (curves with ) with hole-density Nh = 5×1011 cm–2 and impurity densities (a) Ni = 0 and (b) Ni = Nh. The electron Fermi temperatures for those excitation densities are TFe = 0.41, 12.4 and 41.5 K, respectively. The hole Fermi temperature is TFh = 124 K. Note that the scale of τBAP/τDP is on the right-hand side of the frame. The multi-hole-subband effect is taken into account in the calculation.

By comparing figures 1(a) and (b), it is seen that the regimes where the BAP mechanism is more efficient in both cases are around the hole Fermi temperature T_F^h = E_F^h/k_B for high hole density. Here E_F^h represents the Fermi energy of the hole at zero temperature, calculated with the HH⁽¹⁾, LH⁽¹⁾ and HH⁽²⁾ subbands included. A typical case is shown in figure 2(a) for N_h = 5×10¹¹ cm^–2.^Note7  It is shown that the ratio τ_BAP/τ_DP first decreases and then increases with increasing T.^Note8  The minimum is around T_F^h = 124 K, regardless of excitation density. The underlying physics is as follows. On the one hand, the SRT due to the DP mechanism first increases and then decreases with T and the peak appears around the hole Fermi temperature. This is because the electron–hole Coulomb scattering, which dominates the momentum scattering, increases with increasing temperature in the degenerate regime (T < T_F^h) and decreases with T in the nondegenerate regime (T > T_F^h), similar to the electron–electron Coulomb scattering [17, 41, 42]. On the other hand, the SRT due to the BAP mechanism first decreases rapidly and then slowly with T. The decrease of τ_BAP is mainly from the decrease of the Pauli blocking of holes and the increase of the matrix elements in equation (5) [12]. In the high-temperature (nondegenerate) regime, Pauli blocking becomes very weak, and thus τ_BAP decreases slowly with T. Under the combined effect of these two mechanisms, a valley in the ratio τ_BAP/τ_DP appears around T_F^h.

Moreover, we also show that in the regime where the DP mechanism is dominant at all temperatures, e.g. the high excitation density case (the curves with squares in figure 2(a)), the total SRT shows a peak around the hole Fermi temperature. This temperature dependence is similar to the peak first predicted theoretically and then confirmed experimentally in n-type samples [17, 21, 43]. The only difference is that the peak in the previous work comes from the electron–electron Coulomb scattering and hence appears around the electron Fermi temperature, whereas the peak here originates from the electron–hole Coulomb scattering and thus appears around the hole Fermi temperature.

Then we turn to the case of high impurity density with N_i = N_h (figures 1(c) and (d)). In this case, the regime where the BAP mechanism is more important becomes larger than that in the impurity-free case. The scenario is that the higher impurity density strengthens the electron–impurity scattering and suppresses the DP mechanism, consequently enhancing the relative importance of the BAP mechanism. Interestingly, it is also seen that the temperature regime where the BAP mechanism surpasses the DP mechanism in this case is very different from that in the impurity-free case. This regime is roughly from the electron Fermi temperature to the hole Fermi temperature for high hole density. To explore the underlying physics, we plot the SRTs due to the BAP and DP mechanisms in figure 2(b) for N_h = 5×10¹¹ cm^–2. It is seen that the SRT due to the DP mechanism first decreases slowly and then rapidly with temperature. This is because the electron–impurity scattering, which dominates the momentum scattering, has a very weak temperature dependence. Thus the temperature dependence of τ_DP is mainly determined by the inhomogeneous broadening from the spin–orbit coupling. It is also noted that the inhomogeneous broadening changes little with temperature when T < T_F^e, hence τ_DP varies with T very mildly at low temperature. In contrast, as mentioned above, the SRT due to the BAP mechanism first decreases rapidly and then slowly with temperature. As a result, the temperature dependence of τ_BAP/τ_DP can be easily understood. When T < T_F^e, τ_DP decreases with T slower than τ_BAP; thus the ratio decreases with T. In the case with T > T_F^h, τ_DP decreases with T faster than τ_BAP; hence the ratio increases with T. The ratio τ_BAP/τ_DP varies mildly when temperature varies from T_F^e to T_F^h. Consequently, when hole density is high enough, the BAP mechanism can surpass the DP mechanism in the temperature regime between these two temperatures. In particular, in the case with high impurity and very low electron excitation densities (figure 1(d)), the electron Fermi temperature (0.41 K) is much lower than the lowest temperature (5 K) of our computation and the hole Fermi temperature is close to the highest temperature (300 K) of our computation. As a result, the BAP mechanism dominates the spin relaxation in the whole temperature regime of our investigation.

We stress that the different behaviors in the impurity-free and high impurity density cases originate from the different dominant momentum scatterings: the electron–hole Coulomb scattering in the impurity-free case and the electron–impurity scattering in the high impurity density case. The different dominant scatterings lead to the different temperature dependences of τ_DP, and hence the different behaviors of the τ_BAP/τ_DP ratio. In the case with moderate impurity density, these two scatterings contribute to the DP spin relaxation, thus the trend of the temperature dependence of τ_DP is between those in the impurity-free and high impurity density cases. As a result, the temperature regime where the BAP mechanism is more efficient than the DP mechanism is from some temperature between the electron and hole Fermi temperatures to the hole Fermi temperature.

3.2. Multi-hole-subband effect

Now we investigate the multi-hole-subband effect on spin relaxation. In our model, besides the first HH subband, we also consider the contribution from the first LH subband and the second HH subband. Since only the hole states around the Fermi surface can contribute to the electron–hole Coulomb or exchange scattering, we choose ∂_μhN_λ/∂_μhN_h as the criterion of the contribution from the λ hole subband. We further show the regime where the contribution from high hole subbands becomes significant in figure 1 (the regime above the yellow curve), where ∂_μh(N_LH⁽¹⁾ + N_HH⁽²⁾)/∂_μhN_h > 0.1. It is noted that we only discuss the combined effect of the DP spin relaxation from the LH⁽¹⁾ and HH⁽²⁾ subbands in the following, as the effects on the electron–hole Coulomb scattering from these two subbands are analogous. Moreover, the matrix elements in equation (5) relevant to the HH⁽²⁾ subband are one order of magnitude smaller than those relevant to the LH⁽¹⁾ subband for the relevant range of center-of-mass momentum K in the following cases. Therefore, we only discuss the effect on the BAP spin relaxation from the LH⁽¹⁾ subband.

We first show how the multi-hole-subband effect influences the temperature dependence of spin relaxation. SRTs due to the DP and BAP mechanisms as well as the τ_BAP/τ_DP ratio are plotted in figure 3 as a function of temperature for a typical case with N_i = 0, N_h = 5×10¹¹ cm^–2 and N_ex = 10¹¹ cm^–2. It is seen that after considering the contribution from high hole subbands, τ_BAP decreases but τ_DP increases, and hence the importance of the BAP mechanism is enhanced. The underlying physics is as follows. The states in high hole subbands provide an additional scattering channel, and the electron–hole Coulomb and exchange scatterings are both enhanced. The former suppresses the DP mechanism in the strong scattering limit, and the latter leads to an enhancement of the BAP mechanism. Both make the BAP mechanism become more important compared with the DP mechanism. It is also seen that the multi-hole-subband effect becomes more pronounced at high temperature. This is because the occupation of the high hole subbands becomes larger when temperature increases.

From figure 3, one also finds that the multi-hole-subband effect does not significantly affect the trend of the temperature dependence of the SRT. The main change after the inclusion of the high hole subbands is that the temperature at which τ_BAP/τ_DP reaches a minimum becomes closer to the hole Fermi temperature. The underlying physics is as follows. In the degenerate regime (T < T_F^h), it is seen that compared with those in the single-hole-subband model, τ_DP (τ_BAP) in the multi-hole-subband model increases (decreases) faster with increasing temperature, both originate from the increase in the occupation of the high hole subbands and hence the increase of the electron–hole Coulomb and exchange scatterings. This leads to a faster decrease of τ_BAP/τ_DP with increasing temperature when T < T_F^h [19, 20]. Nevertheless, in the nondegenerate regime (T > T_F^h), it is seen that τ_DP (τ_BAP) in the multi-hole-subband model decreases faster (slower) than that in the single-hole-subband model. The acceleration in the decrease of τ_DP can be understood as follows. In the nondegenerate regime, the electron–hole Coulomb scattering decreases with temperature. With the contribution from high hole subbands, the electron–hole Coulomb scattering becomes stronger and thus the decrease rate also becomes larger. Therefore, τ_DP decreases faster in the multi-hole-subband calculation [19, 20]. The slowdown in the decrease of τ_BAP originates from the anomalous K dependence of the matrix element M^–_{–1/2, 1/2} in equation (5), which is relevant to the LH⁽¹⁾ subband. As discussed above, in the nondegenerate regime, the temperature dependence of τ_BAP is mainly from the matrix elements. It is also noted that the magnitude of M^–_{–1/2, 1/2} decreases with K, whereas the magnitudes of the other matrix elements increase with K. With the increase of temperature, more holes and electrons are distributed at larger momentums, the contribution from M^–_{–1/2, 1/2} decreases and the contributions from the other matrix elements increase. These two trends counteract with each other and make τ_BAP decrease with increasing T very slowly at high temperature. Consequently, when T > T_F^h, the ratio τ_BAP/τ_DP shows a steeper increase with rising temperature in the multi-hole-subband calculation. Therefore, both trends when the temperature is below and above T_F^h make the minimum of τ_BAP/τ_DP appear at the temperature closer to T_F^h in the multi-hole-subband calculation.

We also investigate the multi-hole-subband effect on the hole-density dependence of spin relaxation. In figure 4, SRTs due to various mechanisms, the total SRT together with the ratio τ_BAP/τ_DP are plotted as a function of hole density. It is interesting to see from figure 4(a) that at low temperature, spin relaxation has a very intriguing hole-density dependence. In the impurity-free case, the hole-density dependence of the total SRT shows a double-peak structure, i.e. it first increases slightly, then decreases, again increases rapidly and finally decreases with increasing hole density. In the high impurity density case with N_i = N_h, the total SRT shows first a peak and then a valley as a function of hole density. We first discuss the impurity-free case, where the BAP mechanism is negligible and the double-peak structure is solely from the DP mechanism. The first peak can be understood as follows. The electron–hole Coulomb scattering increases with N_h in the nondegenerate regime (E_F^h < k_BT) from the increase of hole distribution, but decreases with N_h in the degenerate regime (E_F^h > k_BT) due to the increase of the Pauli blocking of holes [19, 20]. Hence τ_DP first increases and then decreases with N_h, with the peak appearing around the hole density satisfying E_F^h = k_BT. This behavior is similar to the peak predicted in n-type samples [19], where the peak originates from the electron–electron Coulomb scattering and hence appears around the electron density satisfying E_F^e = k_BT. It is also seen that the second peak only appears in the multi-hole-subband calculation, but is absent in the single-hole-subband calculation (the green dashed curve with circles), which indicates that this peak comes from the contribution of the electron–hole Coulomb scattering from high hole subbands. In fact, the scenario is similar to the first one. When N_h > 6×10¹¹ cm^–2, the contribution from the LH⁽¹⁾ subband becomes important.^Note9  Since the holes in the LH⁽¹⁾ subband are still in the nondegenerate regime, the electron–hole Coulomb scattering increases with increasing hole density. Thus τ_DP increases with N_h. When N_h > 9×10¹¹ cm^–2, i.e. E_F^h–ΔE_LH⁽¹⁾ > k_BT with ΔE_LH⁽¹⁾ representing the energy splitting between the LH⁽¹⁾ and HH⁽¹⁾ subbands, the holes in the LH⁽¹⁾ subband are also in the degenerate regime, and thus the effect of the Pauli blocking becomes significant. Consequently τ_DP decreases with N_h. The second peak appears around the hole density satisfying E_F^h–ΔE_LH⁽¹⁾ = k_BT.

Figure 4. SRTs due to the DP and BAP mechanisms, the total SRT together with the ratio τBAP/τDP versus hole-density Nh with impurity densities Ni = 0 () and Ni = Nh () at (a) T = 10 K and (b) 100 K (note that the scale of τBAP/τDP is on the right-hand side of the frame). The excitation density Nex = 109 cm–2. The solid curves are the results calculated with the lowest two subbands of HH and the lowest subband of LH. The dashed curves are those calculated with only the lowest subband of HH. The two purple dotted vertical lines indicate the hole densities satisfying EFh = kBT and EFh–ΔELH(1) = kBT, respectively (note that the second dotted line in (b) is very close to the right frame). Here ΔELH(1) represents the energy splitting between the LH(1) and HH(1) subbands.

Then we turn to the case of high impurity density with the impurity density being identical to the hole density. The scenario of the peak is as follows. The SRT due to the DP mechanism increases monotonically with hole density, since the electron–impurity scattering increases with N_h ( = N_i). Moreover, the SRT due to the BAP mechanism decreases with N_h due to the increase of the hole distribution of the HH⁽¹⁾ and LH⁽¹⁾ subbands, similar to the electron–hole Coulomb scattering. As a result, the peak appears around the hole density where the BAP mechanism surpasses the DP mechanism [20]. It is also seen that τ_tot increases with hole density when N_h > 9×10¹¹ cm^–2. The underlying physics is as follows. Similar to the previous discussion of the temperature dependence, with the increase of hole density, the decrease of the matrix element M^–_{–1/2, 1/2} counteracts the increase of the other matrix elements. Thus the dependence of the hole density from the matrix elements is weak. Consequently, when the holes in the HH⁽¹⁾ and LH⁽¹⁾ subbands are both in the degenerate regime, the effect of the increase of the Pauli blocking is dominant, and τ_BAP increases with N_h. The valley appears around the hole density satisfying E_F^h–ΔE_LH⁽¹⁾ = k_BT. It is also noted that the increase of τ_BAP only appears in the multi-hole-subband calculation. In the single-hole-subband calculation, τ_BAP does not increase with N_h but remains almost a constant for high hole density (the blue dashed curve), since the increase of the Pauli blocking is counteracted by the increase of the matrix elements relevant to the HH⁽¹⁾ subband.

The hole-density dependence of the spin relaxation at high temperature is also investigated (figure 4(b)). Differing from the behavior at low temperature, it is seen that there is only one peak in both the impurity-free and high impurity density cases. We further show that the peaks in both cases come from the competition of the DP and BAP mechanisms, which is similar to the peak in the case with high impurity density and low temperature. The absence of the peak from the electron–hole Coulomb scattering is due to the multi-hole-subband effect. As discussed above, when the hole density is high enough so that the holes in the lowest subband are in the degenerate regime, the contribution of the electron–hole Coulomb scattering from the lowest hole subband decreases with increasing hole density due to the increase of Pauli blocking. However, at high temperature the contribution from high hole subbands is also important in this hole-density regime. It is further noted that the holes in the high subbands are still in the nondegenerate regime; thus the contribution from the high hole subbands increases rapidly with N_h and surpasses the effect from the lowest hole subband. Consequently, τ_DP continues to increase with N_h and the Coulomb peak disappears.

4. Conclusion and discussion

In conclusion, we have performed a comprehensive investigation on electron spin relaxation in p-type GaAs quantum wells from a fully microscopic KSBE approach. All relevant scatterings, such as, electron–impurity, electron–phonon, electron–electron Coulomb, electron–hole Coulomb and electron–hole exchange (the BAP mechanism) scatterings, are explicitly included.

We present a phase-diagram-like picture showing the parameter regime where the DP or BAP mechanism is more important. In the case with no impurity and high excitation density, our results are consistent with those in [12]: the BAP mechanism is unimportant at low temperature, which is in stark contrast with the common belief in the literature [1], [7]–[10]. However, by extending the scope of investigation to higher hole density with more hole subbands included, we discover that the BAP mechanism can surpass the DP mechanism in the regime with high temperature and sufficiently high hole density. In cases with low excitation density and/or high impurity density, the regime where the BAP mechanism surpasses the DP mechanism becomes larger. We also show that the temperature regime where the BAP mechanism is more efficient than the DP mechanism is very different in the impurity-free and high impurity density cases. In the impurity-free case, this regime is around the hole Fermi temperature for high hole density, regardless of excitation density. However, in the high impurity density case with identical hole and impurity densities, this regime is roughly from the electron Fermi temperature to the hole Fermi temperature. This is because the dominant scatterings in these two cases are the electron–hole Coulomb scattering and the electron–impurity scattering, respectively. The different dominant scatterings lead to the different temperature dependences of τ_DP, and hence the different behaviors of the ratio τ_BAP/τ_DP. In particular, in the case with high impurity and very low electron excitation densities, the electron (hole) Fermi temperature is much lower than (close to) the lowest (highest) temperature of our investigation. Consequently, the BAP mechanism can dominate the spin relaxation in the whole temperature regime. Moreover, we predict that for the impurity-free case, in the regime where the DP mechanism dominates the spin relaxation, e.g. the cases with high excitation or low hole density, the total SRT presents a peak around the hole Fermi temperature, which is from the nonmonotonic temperature dependence of the electron–hole Coulomb scattering.

The multi-hole-subband effect on spin relaxation is also revealed. It is shown that the multi-hole-subband effect enhances the relative importance of the BAP mechanism significantly for high temperature and/or high hole density. We also predict that at low temperature the spin relaxation has a very intriguing hole-density dependence, thanks to the contribution from high hole subbands. In the impurity-free case, the total SRT shows a double-peak structure. Both peaks originate from the nonmonotonic hole-density dependence of the electron–hole Coulomb scattering. The only difference is that the first (second) peak comes from the contribution from the HH⁽¹⁾ (LH⁽¹⁾) subband, and hence appears around the hole density satisfying E_F^h = k_BT (E_F^h–ΔE_LH⁽¹⁾ = k_BT). In the high impurity density case with identical impurity and hole densities, there is first a peak and then a valley. The peak is formed as the DP and BAP mechanisms compete with each other: the SRT due to the DP (BAP) mechanism increases (decreases) with T as the DP (BAP) mechanism dominates at low (high) temperature. Moreover, since the decrease of the matrix element M^–_{–1/2, 1/2} counteracts the increase of the other matrix elements of BAP scattering, the hole-density dependence from the matrix elements is weak. Consequently, when the holes in the HH⁽¹⁾ and LH⁽¹⁾ subbands are both in the degenerate regime, the effect of the increase of Pauli blocking is dominant, and τ_BAP increases with N_h. Therefore the valley is formed. However, at high temperature, we show that the peak from the electron–hole Coulomb scattering disappears and only the peak from the competition of the BAP and DP mechanisms remains.

Finally, we discuss the relevance to experiments. Since electrons are minority carriers in p-type semiconductors, they have finite lifetime limited by the electron–hole recombination. From experiments, the electron lifetime is found to be on the order of several hundreds of picoseconds [1, 44], which may be shorter than the SRT in some parameter regimes such as the high impurity density case. However, it has been demonstrated that SRT much longer than the lifetime can be measured via the Hanle or other time-resolved measurements [1, 45]. Hence the calculated long SRTs can be observed in reality. In short, our calculation based on realistic parameters provides useful information in a wide range of experimental conditions, which would benefit the understanding on spin relaxation.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant No. 10725417, the National Basic Research Program of China under Grant No. 2006CB922005 and the Knowledge Innovation Project of Chinese Academy of Sciences.

Appendix. Comparison with experiment

We compare the calculation via the KSBE approach with the experimental results in [44]. In figure A.1, we plot the temperature dependence of the SRTs from our computation and from the experiment in a p-type GaAs quantum well. Here a = 3 nm, N_h = 3×10¹¹ cm^–2 and N_ex = 2.6×10⁹ cm^–2, as indicated in the experiment [44]. N_i = 0.3N_h is obtained by fitting the mobility μ≊4000 cm² V^–1 s^–1 given in [44]. We take the Dresselhaus spin–orbit coupling parameter as γ_D = 18.3 eV Å³. As the range of γ_D in bulk GaAs calculated and measured via various methods is from 6.4 to 25.5 eV Å³ [36], our value of γ_D is reasonable. It is seen that our calculation is in good agreement with the experimental data. The deviation at T = 5 K is probably because the electron temperature T_e is higher than the lattice temperature T in the experiment due to photo-excitation. This effect becomes significant at low temperature [1, 9]. It is also noted that the SRT due to the BAP mechanism is over one order of magnitude larger than that from the DP one, i.e. the spin relaxation is totally governed by the DP mechanism in this case. This is consistent with our conclusion that the BAP mechanism is unimportant at low impurity and hole densities.

References

[1]

Meier F and Zakharchenya B P 1984 Optical Orientation  (Amsterdam: North-Holland) 

[2]

Wolf S A, Awschalom D D, Buhrman R A, Daughton J M, von Molnár S, Roukes M L, Chtchelkanova A Y and Treger D M 2001 Science 294 1488 
CrossRefPubMed

[3]

 Awschalom D D, Loss D and Samarth N (ed) 2002 Semiconductor Spintronics and Quantum Computation (Berlin: Springer) 

Žutić I, Fabian J and Das Sarma S 2004 Rev. Mod. Phys. 76 323 
CrossRef
Fabian J, Matos-Abiague A, Ertler C, Stano P and Žutić I 2007 Acta Phys. Slov. 57 565 
CrossRef
 D'yakonov M I (ed) 2008 Spin Physics in Semiconductors (Berlin: Springer) and references therein 
CrossRef

[4]

Bir G L, Aronv A G and Pikus G E 1975 Zh. Eksp. Teor. Fiz. 69 1382 (Sov. Phys. JETP42 705 (1976)) 

[5]

D'yakonov M I and Perel' V I 1971 Zh. Eksp. Teor. Fiz. 60 1954 (Sov. Phys. JETP33, 1053 (1971)) 

D'yakonov M I and Perel' V I 1971 Fiz. Tverd. Tela (Leningrad) 13 3581 (Sov. Phys. Solid State13, 3023 (1972)) 

[6]

Wu M W and Ning C Z 2000 Eur. Phys. J. B 18 373 
CrossRef
Wu M W and Metiu H 2000 Phys. Rev. B 61 2945 
CrossRef
Wu M W 2001 J. Phys. Soc. Japan 70 2195 
CrossRef

[7]

Aronov A G, Pikus G E and Titkov A N 1983 Zh. Eksp. Teor. Fiz. 84 1170 (Sov. Phys. JETP57, 680 (1983)) 

[8]

Fishman G and Lampel G 1977 Phys. Rev. B 16 820 
CrossRef

[9]

Zerrouati K, Fabre F, Bacquet G, Bandet J, Frandon J, Lampel G and Paget D 1988 Phys. Rev. B 37 1334 
CrossRef

[10]

Song P H and Kim K W 2002 Phys. Rev. B 66 035207 
CrossRef

[11]

Maialle M Z 1996 Phys. Rev. B 54 1967 
CrossRef

[12]

Zhou J and Wu M W 2008 Phys. Rev. B 77 075318 
CrossRef

[13]

For a brief review, seeand references therein Wu M W, Weng M Q and Cheng J L 2007 Physics, Chemistry and Application of Nanostructures: Reviews and Short Notes to Nanomeeting 2007 ed V E Borisenko, V S Gurin and S V Gaponenko (Singapore: World Scientific) p 14 For a brief review, seeand references therein 

[14]

Weng M Q and Wu M W 2003 Phys. Rev. B 68 075312 
CrossRef
Weng M Q and Wu M W 2004 Phys. Rev. B 70 195318 
CrossRef

[15]

Weng M Q, Wu M W and Jiang L 2004 Phys. Rev. B 69 245320 
CrossRef

[16]

Jiang L and Wu M W 2005 Phys. Rev. B 72 033311 
CrossRef

[17]

Zhou J, Cheng J L and Wu M W 2007 Phys. Rev. B 75 045305 
CrossRef

[18]

Zhang P, Zhou J and Wu M W 2008 Phys. Rev. B 77 235323 
CrossRef

[19]

Jiang J H and Wu M W 2009 Phys. Rev. B 79 125206 
CrossRef

[20]

Jiang J H, Zhou Y, Korn T, Schüller C and Wu M W 2009 Phys. Rev. B 79 155201 
CrossRef

[21]

Ruan X Z, Luo H H, Ji Y, Xu Z Y and Umansky V 2008 Phys. Rev. B 77 193307 
CrossRef

[22]

Teng L H, Zhang P, Lai T S and Wu M W 2008 Europhys. Lett. 84 27006 
IOPscience

[23]

Stich D, Zhou J, Korn T, Schulz R, Schuh D, Wegscheider W, Wu M W and Schüller C 2007 Phys. Rev. Lett. 98 176401 
CrossRef
Stich D, Zhou J, Korn T, Schulz R, Schuh D, Wegscheider W, Wu M W and Schüller C 2007 Phys. Rev. B 76 205301 
CrossRef

[24]

Korn T, Stich D, Schulz R, Schuh D, Wegscheider W and Schüller C 2009 Adv. Solid State Phys. 48 143 
CrossRef

[25]

Stich D, Jiang J H, Korn T, Schulz R, Schuh D, Wegscheider W, Wu M W and Schüller C 2007 Phys. Rev. B 76 073309 
CrossRef

[26]

Holleitner A W, Sih V, Myers R C, Gossard A C and Awschalom D D 2007 New J. Phys. 9 342 
IOPscience

[27]

Zhang F, Zheng H Z, Ji Y, Liu J and Li G R 2008 Europhys. Lett. 83 47006 
IOPscience
Zhang F, Zheng H Z, Ji Y, Liu J and Li G R 2008 Europhys. Lett. 83 47007 
IOPscience

[28]

Krauß M, Bratschitsch R, Chen Z, Cundiff S T and Schneider H C arXiv:0902.0270
Preprint

[29]

Yang C, Cui X, Shen S-Q, Xu Z and Ge W 2009 Phys. Rev. B 80 035313 
CrossRef

[30]

Shen K 2009 Chin. Phys. Lett. 26 067201 
IOPscience

[31]

Yu E T, McCaldin J O and McGill T C 1992 Solid State Phys. 46 1 

[32]

Haug H and Jauho A-P 1998 Quantum Kinetics in Transport and Optics of Semiconductors  (Berlin: Springer) 

[33]

Dresselhaus G 1955 Phys. Rev. 100 580 
CrossRef

[34]

Bychkov Y A and Rashba E I 1984 J. Phys. C: Solid State Phys. 17 6039 
IOPscience
Bychkov Y A and Rashba E I 1984 JETP Lett. 39 78 

[35]

Lau W H and Flatté M E 2005 Phys. Rev. B 72 161311(R) 

[36]

Chantis A N, van Schilfgaarde M and Kotani T 2006 Phys. Rev. Lett. 96 086405 
CrossRefPubMed

[37]

Mahan G D 2000 Many Particle Physics  (New York: Kluwer Academic) 

[38]

Maialle M Z, de Andrada e Silva E A and Sham L J 1993 Phys. Rev. B 47 15776 
CrossRef

[39]

 Madelung O. (ed) 1987 Semiconductors Landolt-Börnstein, New Series vol. 17a (Berlin: Springer) 

[40]

Ekardt W, Lösch K and Bimberg D 1979 Phys. Rev. B 20 3303 
CrossRef

[41]

Giulianni G F and Vignale G 2005 Quantum Theory of the Electron Liquid  (Cambridge: Cambridge University Press) 
CrossRef

[42]

Glazov M M and Ivchenko E L 2002 Pis'ma Zh. Eksp. Teor. Fiz. 75 476 (JETP Lett. 75, 403 (2002)) 
CrossRef
Glazov M M and Ivchenko E L 2004 Zh. Eksp. Teor. Fiz. 126 1465 (JETP99, 1279 (2004)) 

[43]

Bronold F X, Saxena A and Smith D L 2004 Phys. Rev. B 70 245210 
CrossRef

[44]

Martin M D, Viña L, Potemski M and Ploog K H 1999 Phys. Status Solidi b 215 229 
CrossRef
Viña L 1999 J. Phys.: Condens. Matter 11 5929 
IOPscience

[45]

Wagner J, Schneider H, Richards D, Fischer A and Ploog K 1993 Phys. Rev. B 47 4786 
CrossRef

Notes

Note4  The contribution from the short-range term is negligible [19].

Note5  Our calculation is not sensitive to the initial spin polarization when it is small.

Note6  Most cases we discuss in this paper are in the strong scattering regime, where τ_DP increases with momentum scattering strength and decreases with inhomogeneous broadening.

Note7  The hole-density N_h consists of both holes from acceptors and those from optical excitation.

Note8  It is seen that in the case with N_ex = 10¹¹ cm^–2, the variation of the SRT due to the DP mechanism becomes very mild when T < 10 K. This is because this case belongs to the intermediate scattering regime, where τ_DP has a weak dependence on the scattering strength.

Note9  It is also noted that the contributiion from the HH⁽²⁾ subband is always negligible in this case.