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

Abstract

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) Comment: 10 pages, 5 figures, to be published in New J. Phys

Full text

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arXiv:0905.2790v2 [cond-mat.mtrl-sci] 3 Nov 2009
Electron spin relaxation in p-type GaAs quantum wells
Y. Zhou,1J. H. Jiang,2and M. W. Wu1, 2, ∗
1Hefei National Laboratory for Physical Sciences at Microscale,
University of Science and Technology of China, Hefei, Anhui, 230026, China
2Department of Physics, University of Science and Technology of China, Hefei, Anhui, 230026, China
(Dated: November 3, 2009)
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. Via 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. It is discovered 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. Moreover, we predict
that for the impurity-free case, in the regime where the D’yakonov-Perel’ mechanism dominates
the spin relaxation at all temperatures, the temperature dependence of the spin relaxation time
presents a peak around the hole Fermi temperature, which originates from the electron-hole Coulomb
scattering. We also predict that at low temperature, the hole-density dependence of the electron
spin relaxation time exhibits a double-peak structure in the impurity-free case, whereas first a peak
and then a valley in the case of identical impurity and hole densities. These intriguing behaviors
are due to the contribution from holes in high subbands.
PACS numbers: 72.25.Rb, 67.30.hj, 71.10.-w
I.INTRODUCTION
In recent years, much attention has been devoted to
semiconductor spintronics both theoretically and exper-
imentally due to the potential application of spin-based
devices.1,2,3In order to manipulate the spin relaxation
such that the information is well preserved before re-
quired operations are completed, it is crucial to gain
a thorough understanding of spin relaxations.
doped III-V semiconductors, the main electron spin re-
laxation mechanisms have been recognized as:1the Bir-
Aronov-Pikus (BAP) mechanism4and the D’yakonov-
Perel’ (DP) mechanism.5In the DP mechanism, elec-
tron spins decay due to their precessions around the
momentum-dependent spin-orbit fields (inhomogeneous
broadening)6during the free flight between adjacent scat-
tering 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.1,7,8,9,10In two-dimensional sys-
tem, Maialle11calculated the spin relaxation time (SRT)
due to these two mechanisms at zero temperature by us-
ing the single-particle approach and showed that these
two SRTs have nearly the same order of magnitude. How-
ever, as pointed out by Zhou and Wu lately,12there
are some common problems in the previous literature:
The SRT due to the BAP mechanism was calculated
In p-
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 ki-
netic spin Bloch equation (KSBE) approach6,13to
investigate the spin relaxation in p-type GaAs quan-
tum wells.12The KSBE approach has achieved good
success in the study of the spin dynamics in semi-
conductors, 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 rel-
evant scatterings and obtained the accurate SRT due
to these two mechanisms.
BAP mechanism is always less efficient than the DP
mechanism for moderate and high excitation densities
where Nex ? 0.1Nh [Nex (Nh) is the excitation (hole)
density], in contrast to the common belief in the previous
literature.1,7,8,9,10This claim has very recently been
confirmed experimentally by Yang et al..29Moreover,
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 den-
sity where the contribution from the high subbands or
different hole bands becomes significant and/or for high
It was found that the

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2
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 tem-
perature, hole density, excitation density and impurity
density. We present a “phase-diagram–like” picture indi-
cating 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. Nev-
ertheless, since more hole subbands and bands are in-
cluded 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 im-
purity 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 tem-
perature 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 temper-
ature 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 im-
portance of the BAP and DP mechanisms in different pa-
rameter regimes and investigate the multi-hole-subband
effect. We conclude in Sec. IV. A comparison of the cal-
culation 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 low-
est 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.12,17The barrier layer is chosen
to be Al0.4Ga0.6As where the barrier heights of electron
and hole are 328 and 177 meV, respectively.31We 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,32we
construct the KSBEs as follows:12,13
∂tˆ ρk= ∂tˆ ρk|coh+ ∂tˆ ρk|scat, (1)
with ˆ ρkrepresenting the electron single-particle density
matrix with a two-dimensional momentum k = (kx,ky),
whose diagonal and off-diagonal elements describe the
electron distribution function and spin coherence respec-
tively. The coherent term can be written as (? ≡ 1
throughout this paper)
∂tˆ ρk|coh= −i
?
h(k) ·ˆ σ
2+ˆΣHF(k), ˆ ρk
?
, (2)
in which [A,B] ≡ AB−BA is the commutator. h(k) rep-
resents the spin-orbit coupling of electrons composed of
the Dresselhaus33and Rashba34terms. For GaAs quan-
tum wells, the Dresselhaus term is dominant35and
h(k) = 2γD
?
kx(k2
y− ?k2
z?), ky(?k2
z? − k2
x), 0
?
, (3)
where ?k2
−(∂/∂z)2over the state of the lowest subband of electron,
and γD= 8.6 eV·˚ A3denotes the Dresselhaus spin-orbit
coupling coefficient.25,36 ˆΣHF(k) is the effective magnetic
field from the Coulomb Hartree-Fock contribution.14For
the screened Coulomb potential, the screening from elec-
trons and holes is calculated under the random phase
approximation.12,37The scattering term ∂tˆ ρk|scat con-
sists of the electron-impurity, electron-phonon, electron-
electron Coulomb, electron-hole Coulomb, and electron-
hole exchange scatterings. The expressions of these scat-
terings 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 electron-
hole Coulomb scattering is still similar to that in Ref. 12.
The complete electron-hole exchange scattering term is
written as
z? stands for the average of the operator
∂tˆ ρk|BAP= −π
?
η=±
k′qλλ′
δ(ǫk−q− ǫk+ ǫh
k′,λ− ǫh
k′−q,λ′)
× |Tη
λλ′(k + k′− q)|2?
− ˆ sηˆ ρ<
ˆ sηˆ ρ>
k−qˆ s−ηˆ ρ<
k(1 − fh
k′,λ)fh
k′−q,λ′
k−qˆ s−ηˆ ρ>
kfh
k′,λ(1 − fh
k′−q,λ′)
?
+ h.c. .(4)
Here ˆ ρ>
density matrices; ˆ s± = ˆ sx± iˆ sy are the electron spin
ladder operators.λ = HH(n),LH(n)with the super-
script being the subband index of hole.
hole distribution on the λth hole band.
ˆT±comes from the long-range term of the electron-
hole exchange interaction Hamiltonian and can be writ-
ten asˆT±=
8
M±,38,39where ∆ELT is the
longitudinal-transverse splitting in bulk; |φ3D(0)|2=
1/(πa3
k
= 1 − ˆ ρk and ˆ ρ<
k
= ˆ ρk are the electron
fh
k,λis the
The matrix
3
∆ELT
|φ3D(0)|2ˆ
0) with a0being the exciton Bohr radius;ˆ M−and

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3
ˆ
M+(= (ˆ
matrixˆ
|3
M−)†) are operators in hole spin space. The
M−is given by38(in the order of |3
2?(2), | −3
2?(1), |−3
2?(1),
2?(2), |1
2?(1), | −1
2?(1))
ˆ
M−(K) =



000
0
0
0
0
0
0
0
0
0
F0
h1h1K2
0
0
−F0
√3K20
0
+0
−F0
√3K2
0
0
F0
l1l1
3K2
0
h1l1
0
0 F0
h2h2K2
+0
−2F1
√3
h2l1
K+
l1h1
000
−
0
2F1
√3K+0
l1h2
−4F2
l1l1
3



(5)
where K = k + k′− q is the center-of-mass momentum
of the electron-hole pair with K±= Kx±iKy. The form
factors can be written as
Fp
λλ′(K) =
?
dqz
2π
qp
z
K2+ q2
z
fλλ′(qz)(6)
with
fλλ′(qz) =
?
dzdz′ξe(z′)ζλ′
h(z′)eiqz(z−z′)ζλ
h(z)ξe(z).
(7)
In Eq. (5), it is seen that most of the nonzero elements
in matrix ˆM−contain K2
nitudes of these terms increase with increasing K. The
only exception is M−
−1
from the form factor F0
l1l1. Consequently the magnitude
of M−
−1
tributes to an intriguing hole density dependence of spin
relaxation to be addressed in this work.
±or K±, and thus the mag-
2,1
2, where the K dependence is only
2,1
2decreases with K. This K dependence con-
TABLE I: Material parameters used in the calculation
ge
m∗
m∗
κ0
D
vst
Ξ
∆SO
∆ELT 0.08 meV
−0.44m∗
m∗
m∗
κ∞
e
0.067m0
HH,?0.112m0
LH,?0.211m0
12.9
5.31 × 103kg/m3e14
2.48 × 103m/s
8.5 eV
0.341 eV
HH,⊥0.377m0
LH,⊥0.091m0
10.8
1.41 × 109V/m
vsl
5.29 × 103m/s
ωLO
35.4 meV
Eg
1.55 eV
a0
146.1˚ A
III.RESULTS AND DISCUSSIONS
By numerically solving the KSBEs with all the scat-
terings explicitly included, one is able to obtain the SRT
from the temporal evolution of the electron spin polariza-
tion along the z-axis. We choose initial spin polarization
P = 4 %40and well width a = 10 nm, external mag-
netic field B = 0 unless otherwise specified. The other
material parameters are listed in Table I.11,41,42
10-2
10-2
10-2
10-2
10-1
10-1
10-1
10-1
100
100
100
100
101
101
101
101
102
102
102
102
300
(a)
10
100
(b)
Nh ( 1011 cm-2 )
T ( K )
τBAP / τDP
300
(c)
1 10
10
100
(d)
1 10
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,
Nex = 1011cm−2; (b) Ni = 0, Nex = 109cm−2; (c) Ni = Nh,
Nex = 1011cm−2; (d) Ni = Nh, Nex = 109cm−2. 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 indi-
cate the cases satisfying ∂µh[NLH(1) + NHH(2)]/∂µhNh = 0.1.
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 excita-
tion 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 important ef-
fect 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 im-
portance 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 next subsection.
In the case with no impurity and high excitation den-
sity [Fig. 1(a)], our results are consistent with Ref. 12:

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4
101
102
103
104
105
100
101
102
103
104
τ ( ps )
τBAP / τDP
(a)
Ni = 0
τtot
τDP
τBAP
τBAP / τDP
103
104
105
10 100
100
101
T ( K )
τ ( ps )
τBAP / τDP
(b)
Ni = Nh
τtot
τDP
τBAP
τBAP / τDP
FIG. 2: (Color online) SRTs due to the DP and BAP mech-
anisms, the total SRT together with the ratio τBAP/τDP
vs.
temperature T for Nex = 109cm−2(curves with •),
3 × 1010cm−2(curves with ?) and 1011cm−2(curves with
?) with hole density Nh= 5 × 1011cm−2and impurity den-
sities (a) Ni = 0 and (b) Ni = Nh.
temperatures for those excitation densities are Te
12.4 and 41.5 K, respectively. The hole Fermi temperature is
Th
side of the frame. The multi-hole-subband effect is taken into
account in the calculation.
The electron Fermi
F = 0.41,
F= 124 K. Note the scale of τBAP/τDP is on the right-hand
i.e., the BAP mechanism is unimportant at low temper-
ature, which is in stark contrast with the common belief
in the literature.1,7,8,9,10Moreover, since we extend the
scope of our investigation to higher hole density by in-
cluding 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 (Nex=Ne),
but the amplitude of the latter is much larger than the
former. The decrease of τDPcomes from the increase of
the inhomogeneous broadening ?|hk|2? ∝ Nex,14,43and
the decrease of τBAPis mainly from the increase of the
average electron velocity ?vk? ∝ N0.5
increase of the Pauli blocking of electrons can partially
compensate the effect of the increase of ?vk?.12Con-
sequently, τBAP decreases with Nex much more slowly
than τDP and the relative importance of τBAP is en-
hanced for lower excitation density. It is also noted that
when the electron system is in the nondegenerate regime
(T > Te
F/kB), the inhomogeneous broadening and
?vk? is not sensitive to Nex. Thus the ratio τBAP/τDP
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 Th
Eh
F/kB for high hole density.
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(a) for Nh= 5×1011cm−2.44It is
shown that the ratio τBAP/τDPfirst decreases and then
increases with increasing T.45The minimum is around
Th
F= 124 K, regardless of excitation density. The under-
lying 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 < Th
F) and decreases with T in the nondegen-
erate regime (T > Th
F), similar to the electron-electron
Coulomb scattering.17,46,47On the other hand, the SRT
due to the BAP mechanism first decreases rapidly and
then slowly with T. The decrease of τBAPis mainly from
the decrease of the Pauli blocking of holes and the in-
crease of the matrix elements in Eq. (5).12In high tem-
perature (nondegenerate) regime, the Pauli blocking be-
comes very weak, and thus τBAP decreases slowly with
T. Under the combined effect of these two mechanisms,
the valley in the ratio τBAP/τDPappears around Th
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 sim-
ilar to the peak first predicted theoretically and then con-
firmed experimentally in n-type samples.17,21,48The 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 tem-
perature.
Then we turn to the case of high impurity density
with Ni = Nh [Fig. 1(c) and (d)].
regime where the BAP mechanism is more important be-
comes larger than that in the impurity-free case. The
scenario is that the higher impurity density strengthens
ex.11Moreover, the
F= Ee
F=
Here Eh
Frepresents the
F.
In this case, the

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5
the electron-impurity scattering and suppresses the DP
mechanism, consequently enhances the relative impor-
tance of the BAP mechanism. Interestingly, it is also
seen that the temperature regime where the BAP mech-
anism 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 Nh= 5×1011cm−2.
It is seen that the SRT due to the DP mechanism first de-
creases slowly and then rapidly with temperature. This
is because the electron-impurity scattering, which domi-
nates the momentum scattering, has a very weak temper-
ature dependence. Thus the temperature dependence of
τDPis mainly determined by the inhomogeneous broad-
ening from the spin-orbit coupling. It is also noted that
the inhomogeneous broadening changes little with tem-
perature when T < Te
F, hence τDP varies with T very
mildly at low temperature. On the contrary, as men-
tioned above, the SRT due to the BAP mechanism first
decreases rapidly and then slowly with temperature. As
a result, the temperature dependence of τBAP/τDP can
be easily understood. When T < Te
T slower than τBAP, thus the ratio decreases with T.
In the case with T > Th
F, τDP decreases with T faster
than τBAP, hence the ratio increases with T. The ratio
τBAP/τDPvaries mildly when temperature varies from Te
to Th
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.
F, τDPdecreases with
F
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 dif-
ferent temperature dependences of τDP, and hence the
different behaviors of the ratio τBAP/τDP. 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 τDPis between those
in the impurity-free and high impurity density cases. As
a result, the temperature regime where the BAP mech-
anism 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 ∂µhNλ/∂µhNhas the criterion of the contribution
from λ hole subband. We further show the regime where
the contribution from high hole subbands becomes signif-
icant in Fig. 1 (the regime above the yellow curve), where
∂µh(NLH(1) + NHH(2))/∂µhNh> 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 ma-
trix 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-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 in-
fluences the temperature dependence of the spin relax-
ation. The SRTs due to the DP and BAP mechanisms
as well as the ratio τBAP/τDP are plotted in Fig. 3 as
function of temperature for a typical case with Ni= 0,
Nh = 5 × 1011cm−2and Nex = 1011cm−2. It is seen
that after considering the contribution from high hole
subbands, τBAP decreases but τDPincreases, 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-hole-
subband effect becomes more pronounced at high tem-
perature. 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 tem-
perature dependence of the SRT. The main change after
the inclusion of the high hole subbands is that the tem-
perature at which τBAP/τDPreaches minimum becomes
closer to the hole Fermi temperature. The underlying
physics is as follows. In the degenerate regime (T < Th
it is seen that compared with those in the single-hole-
subband model, τDP (τBAP) in the multi-hole-subband
model increases (decreases) faster with increasing tem-
perature, both originate from the increase in the occu-
pation of the high hole subbands and hence the increase
of the electron-hole Coulomb and exchange scatterings.
This leads to a faster decrease of τBAP/τDPwith increas-
ing temperature when T < Th
F),
F.19,20Nevertheless, in the