Electron spin relaxation in $p$-type GaAs quantum wells

We investigate electron spin relaxation in $p$-type GaAs quantum wells from a fully microscopic kinetic spin Bloch equation approach, with all the relevant scatterings, such as the electron-impurity, electron-phonon, electron-electron Coulomb, electron-hole Coulomb and electron-hole exchange (the Bir-Aronov-Pikus mechanism) scatterings explicitly included. From this approach, we examine the relative importance of the D'yakonov-Perel' and Bir-Aronov-Pikus mechanisms in wide ranges of temperature, hole density, excitation density and impurity density, and present a phase-diagram--like picture showing the parameter regime where the D'yakonov-Perel' or Bir-Aronov-Pikus mechanism is more important. By including more hole subbands and bands in our model, we are able to study spin dynamics at high hole density. It is shown that the Bir-Aronov-Pikus mechanism can surpass the D'yakonov-Perel' mechanism in some temperature regime with sufficiently high hole density for various impurity and excitation densities. We also discover that in the impurity-free case the temperature regime where the Bir-Aronov-Pikus mechanism is more efficient than the D'yakonov-Perel' one is 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, this regime is roughly from the electron Fermi temperature to the hole Fermi temperature. Particularly, the Bir-Aronov-Pikus mechanism can dominate the spin relaxation in the {\em whole} temperature regime of the investigation (from 5 K to 300 K) in the case with high impurity and very low excitation densities, since the electron (hole) Fermi temperature is... (The remaining is omitted due to the limit of the space)


I. 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. 1,2,3 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 pdoped 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. 1,7,8,9,10 In two-dimensional system, Maialle 11 calculated the spin relaxation time (SRT) due to these two mechanisms at zero temperature by 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 lately, 12 there are some common problems in the previous 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 the 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 frame of the 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 the 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. 6,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30 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 ex 0.1N h [N ex (N h ) is the excitation (hole) density], in contrast to the common belief in the previous literature. 1,7,8,9,10 This claim has very recently been confirmed experimentally by Yang et al.. 29 Moreover, 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 consistence with Ref. 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 Sec. II, we set up the KSBEs. In Sec. III, 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 Sec. IV. A comparison of the calculation from the KSBEs with the experimental data of a p-type GaAs quantum well is given in the Appendix.

II. 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 finitewell-depth assumption. 12,17 The barrier layer is chosen to be Al 0.4 Ga 0.6 As where the barrier heights of electron and hole are 328 and 177 meV, 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 sys-tem is always in the equilibrium.
Via the nonequilibrium Green function method, 32 we construct the KSBEs as follows: 12,13 withρ k 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) where k 2 z stands for the average of the operator −(∂/∂z) 2 over the state of the lowest subband of electron, and γ D = 8.6 eV·Å 3 denotes the Dresselhaus spin-orbit coupling coefficient. 25,36Σ HF (k) 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 ∂ tρk | scat consists of the electron-impurity, electron-phonon, electronelectron Coulomb, electron-hole Coulomb, and electronhole exchange scatterings. The expressions of these scatterings are given in detail in Ref. 12. Here we just extend the electron-hole Coulomb and exchange scatterings to the multi-hole-subband case. The expression of electronhole Coulomb scattering is still similar to that in Ref. 12. The complete electron-hole exchange scattering term is written as Hereρ > k = 1 −ρ k andρ < k =ρ k are the electron density matrices;ŝ ± =ŝ x ± iŝ y 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 T ± comes from the long-range term of the electronhole exchange interaction Hamiltonian and can be written asT ± = (1) , is the center-of-mass momentum of the electron-hole pair with K ± = K x ± iK y . The form factors can be written as In Eq. (5), it is seen that most of the nonzero elements in matrixM − contain K 2 ± or K ± , and thus the magnitudes of these terms increase with increasing K. The only exception is M − − decreases with K. This K dependence contributes to an intriguing hole density dependence of spin relaxation to be addressed in this work.

III. 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 % 40 and well width a = 10 nm, external magnetic field B = 0 unless otherwise specified. The other material parameters are listed in Table I FIG. 1: (Color online) Ratio of the SRT due to the BAP mechanism to that due to the DP mechanism, τBAP/τDP, as function of temperature and hole density with (a) Ni = 0, The black dashed curves indicate the cases satisfying τBAP/τDP = 1. Note the smaller the ratio τBAP/τDP is, the more important the BAP mechanism becomes. The yellow solid curves indicate the cases satisfying ∂µ h [N LH (1) In the regime above the yellow curve the multi-hole-subband effect becomes significant.

A. 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 Fig. 1, the ratio of the SRT due to the BAP mechanism to that due to the DP mechanism is plotted as 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-holesubband 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 important effect on electron spin relaxation in the relevant regime, the main physics is still the same as that in the single-holesubband 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 multihole-subband and single-hole-subband models. We then investigate the special features from the contribution of high hole subbands in next subsection.
In the case with no impurity and high excitation density [ Fig. 1   i.e., the BAP mechanism is unimportant at low temperature, which is in stark contrast with the common belief in the literature. 1,7,8,9,10 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 [ Fig. 1(b)], one can see that the regime where the BAP mechanism surpasses the DP mechanism becomes larger. The underlying physics is shown in Fig. 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 | 2 ∝ N ex , 14,43 and the decrease of τ BAP is mainly from the increase of the average electron velocity v k ∝ N 0.5 ex . 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 e F = E e F /k B ), the inhomogeneous broadening and v k is not sensitive to N ex . Thus the ratio τ BAP /τ DP changes little with the excitation density.
By comparing Fig. 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 h Here E h F represents the Fermi energy of hole at zero temperature calculated with the HH (1) , LH (1) and HH (2) subbands included. A typical case is shown in Fig. 2 44 It is shown that the ratio τ BAP /τ DP first decreases and then increases with increasing T . 45 The minimum is around T h F = 124 K, regardless of excitation density. The underlying physics is as follows. On 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 h F ) and decreases with T in the nondegenerate regime (T > T h F ), similar to the electron-electron Coulomb scattering. 17,46,47 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 Eq. (5). 12 In high temperature (nondegenerate) regime, the Pauli blocking becomes very weak, and thus τ BAP decreases slowly with T . Under the combined effect of these two mechanisms, the valley in the ratio τ BAP /τ DP appears around T h F . 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 Fig. 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. 17,21,48 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 [ Fig. 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 enhances 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 Fig. 2(b) for N h = 5 × 10 11 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 e F , hence τ DP varies with T very mildly at low temperature. On the contrary, 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 e F , τ DP decreases with T slower than τ BAP , thus the ratio decreases with T . In the case with T > T h F , 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 e F to T h F . 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 [ Fig. 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 impurityfree and high impurity density cases originate from the different dominant momentum scatterings: the electronhole 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 ratio τ BAP /τ DP . In the case with moderate impurity density, these two scatterings both 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.

B. Multi-hole-subband effect
Now we investigate the multi-hole-subband effect on the 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 ∂ µ h N λ /∂ µ h N h as the criterion of the contribution from λ hole subband. We further show the regime where the contribution from high hole subbands becomes significant in Fig. 1 (the regime above the yellow curve), where ∂ µ h (N LH (1) + N HH (2) )/∂ µ h N 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 Eq. (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-ofmass 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 the spin relaxation. The SRTs due to the DP and BAP mechanisms as well as the ratio τ BAP /τ DP are plotted in Fig. 3 as function of temperature for a typical case with N i = 0, N h = 5 × 10 11 cm −2 and N ex = 10 11 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 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-holesubband effect becomes more pronounced at high temperature. This is because the occupation of the high hole subbands becomes larger when temperature increases. From Fig. 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 minimum becomes closer to the hole Fermi temperature. The underlying physics is as follows. In the degenerate regime (T < T h F ), it is seen that compared with those in the single-holesubband 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 h F . 19,20 Nevertheless, in the nondegenerate regime (T > T h F ), 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 accelerating 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 electronhole 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 Eq. (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 − − decreases, while the contributions from the other matrix elements increases. These two trends counteract with each other and make τ BAP decrease with increasing T very slowly at high temperature. Consequently, when T > T h F , the ratio τ BAP /τ DP shows a steeper increase with the rising temperature in the multi-hole-subband calculation. Therefore, both trends when the temperature is below and above T h F make the minimum of τ BAP /τ DP appear at the temperature closer to T h F in the multi-hole-subband

calculation.
We also investigate the multi-hole-subband effect on the hole-density dependence of the spin relaxation. In Fig. 4, the SRTs due to various mechanisms, the total SRT together with the ratio τ BAP /τ DP are plotted as function of hole density. It is interesting to see from Fig. 4(a) that at low temperature, the 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 doublepeak structure is solely from the DP mechanism. The first peak can be understood as follows. The electronhole Coulomb scattering increases with N h in the nondegenerate regime (E h F < k B T ) from the increase of the hole distribution, but decreases with N h in the degenerate regime (E h F > k B T ) 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 h F = k B T . 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 e F = k B T . It is also seen that the second peak only appears in the multi-hole-subband calculation, but becomes 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 × 10 11 cm −2 , the contribution from the LH (1) subband becomes important. 49 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 (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 h F − ∆E LH (1) = k B T . 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 > 9×10 11 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 h F − ∆E LH (1) = k B T . 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 [ Fig. 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 the 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.

IV. CONCLUSION AND DISCUSSION
In conclusion, we have performed a comprehensive investigation of electron spin relaxation in p-type GaAs quantum wells from a fully microscopic KSBE approach. All relevant scatterings, such as, the electron-impurity, electron-phonon, electron-electron Coulomb, electronhole 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 Ref. 12: i.e., the BAP mechanism is unimportant at low temperature, which is in stark contrast with the common belief in the literature. 1,7,8,9,10 However, since we extend the scope of our investigation to higher hole density by including more hole subbands in the model, it is discovered that the BAP mechanism can surpass the DP mechanism in the regime with high temperature and sufficiently high hole density. In the 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 the 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 the 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 electronhole 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 In the high impurity density case with identical impurity and hole densities, there are 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 the 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 the 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 electronhole 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, 1,50 which may be shorter than the spin relaxation time in some parameter regime such as the high impurity density case. However, it has been demonstrated that spin relaxation time much longer than the lifetime can be measured via the Hanle or other timeresolved measurements. 1,51 Hence the calculated long spin relaxation times 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.

APPENDIX: COMPARISON WITH EXPERIMENT
We compare the calculation via the KSBE approach with the experimental results in Ref. 50. In Fig. 5, we plot the temperature dependence of the SRTs from our computation and from the experiment in p-type GaAs quantum well. Here a = 3 nm, N h = 3 × 10 11 cm −2 and N ex = 2.6 × 10 9 cm −2 as indicated in the experiment. 50 N i = 0.3N h is obtained by fitting the mobility µ ≃ 4000 cm 2 V −1 s −1 given in Ref. 50. We take the Dresselhaus spin-orbit coupling parameter as γ D = 18.3 eV·Å 3 . As the range of γ D in bulk GaAs calculated and measured via various methods is from 6.4 to 25.5 eV·Å 3 , 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 likely to be from the fact that the electron temperature T e is higher than the lattice temperature T in 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.