Hole spin relaxation in $ p $-type (111) GaAs quantum wells

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Mar 18, 2012 - mation potential and vsl being the velocity of the lon- gitudinal sound wave.23 For ...... A. Ritchie, and M. Y. Simmons, Phys. Rev. B 62, 13034.
Hole spin relaxation in p-type (111) GaAs quantum wells L. Wang and M. W. Wu∗

arXiv:1203.3931v1 [cond-mat.mes-hall] 18 Mar 2012

Hefei National Laboratory for Physical Sciences at Microscale and Department of Physics, University of Science and Technology of China, Hefei, Anhui, 230026, China (Dated: March 20, 2012) Hole spin relaxation in p-type (111) GaAs quantum wells is investigated in the case with only the lowest hole subband, which is heavy-hole like in (111) GaAs/AlAs and light-hole like in (111) GaAs/InP quantum wells, being relevant. The subband L¨ owdin perturbation method is applied to obtain the effective Hamiltonian including the Dresselhaus and Rashba spin-orbit couplings. Under a proper gate voltage, the total in-plane effective magnetic field in (111) GaAs/AlAs quantum wells can be strongly suppressed in the whole momentum space, while the one in (111) GaAs/InP quantum wells can be suppressed only on a special momentum circle. The hole spin relaxation due to the D’yakonov-Perel’ and Elliott-Yafet mechanisms is calculated by means of the fully microscopic kinetic spin Bloch equation approach with all the relevant scatterings explicitly included. For (111) GaAs/AlAs quantum wells, extremely long heavy-hole spin relaxation time (upto hundreds of nanoseconds) is predicted. In addition, we predict a pronounced peak in the gate-voltage dependence of the heavy-hole spin relaxation time due to the D’yakonov-Perel’ mechanism. This peak origins from the suppression of the unique inhomogeneous broadening in (111) GaAs/AlAs quantum wells. Moreover, the Elliott-Yafet mechanism influences the spin relaxation only around the peak area due to the small spin mixing between the heavy and light holes in quantum wells with small well width. We also show the anisotropy of the spin relaxation. In (111) GaAs/InP quantum wells, a mild peak, similar to the case for electrons in (111) GaAs quantum wells, is also predicted in the gate-voltage dependence of the light-hole spin relaxation time. The contribution of the Elliott-Yafet mechanism is always negligible in this case. PACS numbers: 72.25.Rb, 71.70.Ej, 71.55.Eq, 73.21.Fg

I.

INTRODUCTION

Spin dynamics in semiconductor quantum wells (QWs) has been intensively investigated in recent years due to the fast development of semiconductor spintronics.1–7 In III-V zinc-blende QW systems, spin relaxation is mainly determined by the D’yakonov-Perel’ (DP) mechanism,8 which results from the momentum scattering together with the momentum-dependent effective magnetic field induced by the Dresselhaus9 and/or Rashba10 spinorbit couplings (SOCs). So far, most studies focus on the (001)11–28 and (110) QWs.29–35 In these works, many efforts have been devoted to obtain a long spin relaxation time (SRT).11,14,15,18–22,24–27,29,33,34 In (001) QWs, when the strengths of the Dresselhaus and Rashba terms are comparable, electron spin relaxation along the [110] or [1¯ 10] direction can be strongly suppressed.11,14,15,18–22,24–27 For (110) symmetric QWs, the effective magnetic field due to the Dresselhaus term is oriented along the growth direction, which leads to an absent DP relaxation for electron spins along this direction.29,33,34 Recently, some attention has been devoted to (111) QWs where electron spin relaxation also shows rich properties.36–40 Cartoix` a et al.36 pointed out that to the first order in the momentum, the effective magnetic field from both the Dresselhaus and Rashba terms Ω1 = (αD +αR )(ky , −kx , 0) (with zk[111], xk[11¯ 2] and yk[¯110]), can be suppressed to zero by setting the Rashba coefficient αR to cancel with the Dresselhaus coefficient αD

and hence the DP spin relaxation becomes absent for all three spin components. However, this strict cancellation can not occur due to the existence√of the cubic Dresselγ haus term Ω3 = 2√ [−k2 ky , k2 kx , 2(3kx2 ky − ky3 )]. Sun 3 et al.38 showed that with this cubic term included, the cancellation occurs only for the in-plane effective magnetic field on a special momentum circle, which may lead to a peak in the density or temperature dependence of the SRT. Experimentally, Balocchi et al.40 measured the gate-voltage dependence of the SRT and a two-order of magnitude increase of the SRT can be observed, reaching values larger than 30 ns by applying an external electric field of 50 kV/cm along the growth direction of the QW. However, to the best of our knowledge, there are still no reports on hole spin relaxation in (111) zinc-blende QWs. In the present work, we investigate the hole spin relaxation in p-type (111) GaAs QWs with only the lowest subband being relevant. In GaAs/AlAs QWs, the lowest hole subband is heavy-hole (HH) like. The Dresselhaus and Rashba SOC terms of the lowest HH subband ΩhD(R) can be written as ΩhD = [βxh (3kx2 ky − ky3 ), βyh (kx3 − 3kx ky2 ), ΩhR

=

βzh (3kx2 ky − ky3 )],

[αhx (3kx2 ky − ky3 ), αhy (kx3 αhz (3kx2 ky − ky3 )]eEz ,



(1)

3kx ky2 ), (2)

respectively, which are obtained by the subband L¨ owdin partitioning method.41,42 The coefficients are given in

2 Appendix A. Here, Ez is the electric field along the growth direction. It is noted that the Rashba term ΩhR has not been reported in the previous literature and interestingly its component ΩhRi (i = x, y, z) can be strictly cancelled by the corresponding Dresselhaus one ΩhDi by setting the gate voltage Ezi = −βih /(eαhi ).

(3)

The cancellation gate voltages Ezx (about 20 kV/cm) and Ezy (about 22 kV/cm) are close to each other, indicating that the in-plane components of the effective magnetic field can be strongly suppressed. This cancellation leads to a strong suppression of the in-plane inhomogeneous broadening43 and hence an extremely long SRT for spins along the growth direction. It is further noted that although the out-of-plane component of the effective magnetic field of the lowest HH subband can also be suppressed to zero, the cancellation gate voltage Ezz (about 103 kV/cm) is too large to realize. Therefore, a large anisotropy between the in-plane and out-of-plane spin relaxations is expected. We also investigate the hole spin relaxation in GaAs/InP QWs where the lowest hole subband is lighthole (LH) like. The Dresselhaus and Rashba SOC terms are then given by ΩlD = [(βxl1 k2 + βxl2 hkz2 i)ky , −(βxl1 k2 + βxl2 hkz2 i)kx , ΩlR

=

βzl (3kx2 ky − ky3 )],

[αlx k2 ky , −αlx k2 kx , αlz (3kx2 ky

This paper is organized as follows. In Sec. II, the effective Hamiltonian of the lowest hole subband in (111) GaAs/AlAs and GaAs/InP QWs is derived. The KSBEs are also constructed in this section. Then in Sec. III, we investigate the hole spin relaxations in GaAs/AlAs and GaAs/InP QWs respectively. We summarize in Sec. IV.

(4)



ky3 )]eEz ,

(5)

respectively. One finds that similar to the case of electrons in (111) QWs,38 the in-plane components of these two terms can be suppressed to zero only on a special momentum circle. The out-of-plane component can also be suppressed to zero under an unrealistic gate voltage. We calculate the hole spin relaxation by the kinetic spin Bloch equation (KSBE) approach,5,12 with all the relevant scatterings explicitly included. As the ElliottYafet44 (EY) mechanism is the leading spin relaxation mechanism in hole spin relaxation in bulk,45 we calculate the SRT due to both the DP and EY mechanisms in the present work. For (111) GaAs/AlAs QWs, we find that the EY mechanism is negligible unless the DP mechanism can be greatly suppressed. By tuning the gate voltage, we predict a pronounced peak in the SRT where the SRT can be extremely long (upto hundreds of nanoseconds). This property is favorable for spin manipulation, storage and device design. This peak results from the suppression of the unique inhomogeneous broadening43 in (111) QWs. The effects of hole density, temperature and impurity density on the suppression of the spin relaxation are investigated. We also analyze the anisotropy of the spin relaxation. For (111) GaAs/InP QWs, a peak is also predicted in the gate-voltage dependence of the SRT. However, the peak becomes less pronounced compared with the case in (111) GaAs/AlAs QWs. The contribution of the EY mechanism is demonstrated to be negligible to the predicted properties.

II.

EFFECTIVE HAMILTONIAN AND KSBES

We start our investigation from the p-type (111) GaAs/AlAs and GaAs/InP QWs. The HH and LH can be described by the 4 × 4 Hamiltonian b H4×4 = HL + H8v8v + Hǫ + V (z)I4 + HE I4 ,

(6)

since the contribution from the conduction and splitoff bands is marginal in the present calculation. Here, (0) (k) HL = HL + HL is the Luttinger Hamiltonian46 with (0) b HL corresponding to the part with kx,y = 0, H8v8v 9 is the Dresselhaus SOC of the HH and LH bands and Hǫ stands for the strain Hamiltonian based on the theory of Bir and Pikus.42,47 Their expressions are given in Appendix B. V (z) is the confinement applied along the growth direction within the infinite-depth well potential approximation and I4 is a 4×4 unit matrix. HE = −eEz z represents the electric field term, which is the source of the Rashba SOC.10 (0) By solving the Schr¨odinger equation of H0 = HL + V (z)I4 + Hǫ , the subband energies for the HH and LH are given by = Enh,l z

nz 2 π 2 ~2 2 2mh,l z a

∓ ES1 − ES2 ,

(7)

where nz stands for the subband index along the growth 0 direction, a is the well width and mh,l = γ1 m z ∓2γ3 represents the effective masses of the HH and LH along the z-axis. γ1,3 and m0 stand for the Kohn-Luttinger parameters and the free electron mass, respectively. The contributions of the strain ES1 and ES2 are given in Eqs. (B4) and (B5) in Appendix B. It is noted that ES2 is just an energy shift for all the HH and LH subbands and can be neglected in the present calculation. For GaAs/AlAs QWs, the contribution of the strain is neglected due to the extremely small lattice mismatch and hence the lowest hole subband is HH like. However, for GaAs/InP QWs, due to the existence of strain, the lowest hole subband is LH like. In our calculation, we focus on the case with only the lowest hole subband being relevant while other subbands have much higher energies. We first construct the effective Hamiltonian of the lowest hole subband in (111) GaAs QWs by the subband L¨ owdin partitioning method41,42 upto the third order of the in-plane momentum. The effective Hamiltonian of the lowest hole subband in (111) GaAs/AlAs QWs can be written as σ h (8) Heff = εhk + (ΩhD + ΩhR ) · . 2

3 2

2

Here, εhk = ~2mkh , with the in-plane effective mass mht t given in Appendix A, and σ are the Pauli matrices. The Dresselhaus and Rashba SOC terms ΩhD(R) are given in Eqs. (1) and (2). To facilitate the understanding of the effective magnetic field from these SOC terms, we plot the in-plane components around an arbitrary equal-energy surface under the cancellation gate voltage Ezy [Eq. (3)] in Fig. 1. It is shown that the Dresselhaus term almost completely cancels with the Rashba one, leading to a strong suppression of the total in-plane effective magnetic field.

h

ΩD h ΩR h



(×10)

ky kx

FIG. 1: (Color online) Schematic of the in-plane effective magnetic fields of the lowest HH subband in (111) GaAs/AlAs QWs around an arbitrary equal-energy surface. Red (blue) arrows stand for in-plane components of the Dresselhaus (Rashba) term ΩhD(R) and green arrows represent the sum of the Dresselhaus and Rashba terms Ωh , with its amplitude enlarged by 10 for clarity at the cancellation gate voltage Ezy .

Similarly, the effective Hamiltonian of the lowest hole subband in (111) GaAs/InP QWs reads l Heff = εlk + (ΩlD + ΩlR ) · 2

σ , 2

(9)

2

in which εlk = ~2mkl and mlt stands for the in-plane eft fective mass with its form given in Appendix A. The Dresselhaus and Rashba SOC terms ΩlD(R) of the LH are given in Eqs. (4) and (5). We then construct the microscopic KSBEs5,12 to study the hole spin relaxations in GaAs/AlAs and GaAs/InP QWs, respectively. The KSBEs read5,12 ρ˙ k,σσ′ = ρ˙ k,σσ′ |coh + ρ˙ k,σσ′ |scat ,

(10)

where ρk,σσ′ represent the density matrix elements of holes with the diagonal ones ρk,σσ ≡ fkσ (σ = ±1/2) describing the distribution functions and the off-diagonal ones ρk,1/2 −1/2 = ρ∗k,−1/2 1/2 ≡ ρk denoting the spin coherence. ρ˙ k,σσ′ |coh are the coherent terms describing the coherent spin precession due to the effective magnetic field from both the Dresselhaus and Rashba terms while the Hartree-Fock Coulomb interaction is neglected due to

the small spin polarization.5 Their expressions are given by ∂fk,σ = −2σ[Ωx (k)Imρk + Ωy (k)Reρk ], (11) ∂t coh 1 ∂ρk = [iΩx (k) + Ωy (k)](fk1/2 − fk−1/2 ) ∂t coh 2 −iΩz (k)ρk . (12) Here, Ωi (i = x, y, z) are the three components of the effective magnetic field from both the Dresselhaus and Rashba terms. ρ˙ k,σσ′ |scat are the scattering terms including the spin conserving ones, i.e., the hole-hole (Vq2 ), holeimpurity (Uq2 ), hole-acoustic-phonon (|MQ,AC |2 ), polar polar 2 hole-longitudinal-optical-phonon (|MQ,LO | ) and nonnonp 2 polar hole-optical-phonon (|MQ,λ | ) scatterings with λ = LO/TO denoting longitudinal-optical/transverseoptical phonon and the spin-flip ones due to the EY mechanism.44 Their detailed expressions can be found in Ref. 48 (Note the scatterings in Ref. 48 are given in bulk and one needs to transform them to twodimensional cases). The scattering matrix elements Vq2 = P P vQ |I(iqz )|2 2 2 2 ( qz ǫ(q) ) and Uq2 = qz [Zi vQ /ǫ(q)] |I(iqz )| , with Zi being the charge number of the impurity (assumed to be 1 in our calculation).23 ǫ(q) = 1 − P P fk+qσ −fkσ 2 k,σ εk+q −εk stands for the screening qz vQ |I(iqz )| under random phase approximation;49 the bare Coulomb 2 2 2 2 potential vQ = 4πe Q2 with Q = q + qz ; and the form 2 factor |I(iqz )|2 = π 4 sin y/[y 2 (y 2 − π 2 )2 ] with y = qz a/2. polar 2 −1 2 |MQ,LO | = [2πe2 ωLO /(q2 +qz2 )](κ−1 ∞ −κ0 )|I(iqz )| with ωLO denoting the LO phonon energy and κ0 (κ∞ ) benonp 2 ing the static (optical) dielectric constant.16 |MQ,λ | = 3~d20 |I(iqz )|2 2dωλ a2GaAs

for the nonpolar HH-optical-phonon η~d2

nonp 2 scatterings and |MQ,λ | = 2dωλ a20 |I(iqz )|2 for the GaAs nonpolar LH-optical-phonon ones with d0 , d, aGaAs , and ωTO representing the optical deformation potential,50 the mass density,23 the lattice constant of GaAs,42 and the TO phonon energy,45 respectively. η = 1(5) for λ = LO (TO). The matrix element for the hole-acoustic phonon scattering due to the deformation potential reads 2 Q def 2 |MQ,AC |2 = ~Ξ 2dvsl |I(iqz )| with Ξ denoting the deformation potential and vsl being the velocity of the longitudinal sound wave.23 For the hole-acoustic phonon pl scattering due to the piezoelectric coupling, |MQ,AC |2 = ′



′ 2

32π 2 ~e2 e214 (3qx qy qz ) 2 dvsl Q7 |I(iqz )| for the longitudinal phonon κ20 32π 2 ~e2 e214 pt 1 ′2 ′2 ′2 ′2 and |MQ,AC |2 = dvst Q5 [qx qy + qy qz + κ20 ′ ′ ′ 2 (3qx qy qz ) qz′ 2 qx′ 2 − ]|I(iqz )|2 for the transverse phonon, Q2 1 ′ where qx = √6 qx − √12 qy + √13 qz , qy′ = √16 qx + √12 qy + √13 qz and qz′ = − √26 qx + √13 qz . vst represents the velocity of the transverse sound wave and e14 is the piezoelectric ˆ k,k′ for the lowest HH and constant.23 The spin mixing Λ

4 TABLE I: Parameters used in the calculation (from Ref. 42 unless otherwise specified) 6.85 −81.93 eV˚ A3 ˚ −0.98 eVA3 5.66 ˚ A 5.32 48 eVa 1.41×109 V/mc 5.29×103 m/sc

γ1 b41 b52 aAlAs C12 d0 e14 vsl

γ2 b42 Du′ aInP C44 ωLO κ∞ vst

2.10 1.47 eV˚ A3 4.67 eV 5.87 ˚ A 5.94 35.4 meVb 10.8c 2.48×103 m/sc

γ3 b51 aGaAs C11 d ωTO Ξ κ0

2.90 0.49 eV˚ A3 ˚ 5.65 A 11.81 5.31 g/cm3 33.2 meVb 8.5 eVc 12.9c

a

Reference 50 Reference 16 c Reference 23 b

LH subbands in the spin-flip scattering terms are detailed in Appendix C.

III.

NUMERICAL RESULTS

We present our results obtained by numerically solving the KSBEs. All the parameters used in our calculation are listed in Table I. The initial spin polarization is along the QW growth direction except otherwise specified. We set the initial spin polarization as 2.5 %. Note that the SRT is insensitive to the small spin polarization.16 It is noted that the system is always in the strong scattering limit in the present investigation.

10

1

-1

10

-1

10

-3

10

-3

5

10

15 20 25 Ez (kV/cm)

30

35

2 -1

10

10

0

HH spin relaxation in GaAs/AlAs QWs

Before a full investigation by numerically solving the KSBEs, we start from an analytical analysis of the DP spin relaxation of the lowest hole subband, which is HH like, in (111) GaAs/AlAs QWs. In the strong scattering limit, the evolution of spin polarization along the z-axis can be approximated by51 S˙z = −τp∗ [Sz hΩ2x + Ω2y i − Sx hΩx Ωz i − Sy hΩy Ωz i], (13) with τp∗ standing for the effective momentum scattering time contributed by the hole-hole, hole-impurity and hole-phonon scatterings.16,43 The evolutions of the inplane components Sx and Sy can be obtained by index permutation. It is noted that unlike (001) III-V zincblende QWs where Ωz is absent and the inhomogeneous broadening of the spin polarization along the z-axis is determined by hΩ2⊥ i = hΩ2x + Ω2y i,5 for (111) QWs, Ωz and hΩx Ωz i are nonzero under the effective magnetic field from both the Dresselhaus and Rashba terms given in Eqs. (1) and (2). This makes the evolutions more complex compared with (001) QWs, i.e., S˙z = −τp∗ [Sz hΩ2x + Ω2y i − Sx hΩx Ωz i], S˙x = −τp∗ [Sx hΩ2y + Ω2z i − Sz hΩx Ωz i], S˙y = −τ ∗ [Sy hΩ2 + Ω2 i]. p

x

z

40

FIG. 2: (Color online) HH SRT along the z-axis τsz and the −1 inverse of the inhomogeneous broadening hΩ2y i of the lowest hole subband in (111) GaAs/AlAs QWs against the gate voltage Ez . Red solid (blue dashed) curve represents the SRT due to the DP (both DP and EY) mechanism. The temperature T = 30 K, the hole density Nh = 3 × 1011 cm−2 and the −1 impurity density Ni = 0. Note the scale of hΩ2y i is on the right-hand side of the frame.

(14) (15) (16)

The exact solutions are given in Appendix D. Specially by setting the initial spin polarization along the z-axis as Sz (0), we obtain Sz = Sz (0)e−t/τsz approximately by considering that hΩ2x i, hΩ2y i and hΩx Ωz i are much smaller than hΩ2z i in our investigation. The SRT along the z-axis reads τsz = τp∗ −1 hΩ2y i

T=30 K 11 -2 Nh=3×10 cm Ni=0

1

3

z

τs (ns)

10

3

(arb. unit)

10

A.

−1

,

(17)

with hΩ2y i being the inhomogeneous broadening. This inhomogeneous broadening is unique in (111) QWs and has not been reported in the literature. We then numerically calculate the spin relaxation of the lowest hole subband, which is HH like, in (111) GaAs/AlAs QWs where the well width is chosen to be 8 nm. We first investigate the SRT due to the DP mechanism and plot the gate-voltage dependence of the SRT along the z-axis at T = 30 K with the hole density Nh = 3 × 1011 cm−2 by red solid curve in Fig. 2. It is seen that the SRT can be effectively modulated by the gate voltage and a peak appears at Ez ≈ 22 kV/cm. Under this gate voltage, the SRT is as long as 283 ns which is about four orders of magnitude larger than the one without any bias. This peak results from the suppression of the inhomogeneous broadening hΩ2y i as shown by green dotted curve in Fig. 2. This suppression originates from the cancellation of the y-component of the effective magnetic field Ωy [Eq. (3)]. Since the suppression of the DP spin relaxation is determined by the cancellation of the y-component of the

5 150 2000

(a) T=30 K

11

×5

500

(b) 60

×10

4

4

0

0 0

0.5

1

1.5

Nh (10

2 11

2.5 -2

3

3.5

4

20

60

cm

(d)

-2

(c)

τs x τs

9

τs (ps)

1200

800

×10

80

T (K)

10 11

Nh=3×10

400

40

cm )

T=30 K

1600

z

-2

30 ×10

τs (ns)

cm

Ni=0

90

z

1000

Nh=4×10

τs (ns)

z

τs (ns)

120

Ni=0

1500

y

8

T=30 K Nh=3×1011 cm-2 Ni=0

7

4

×10 0

6 0

0.2

0.4

0.6

Ni (Nh)

0.8

1

15

20

25

30

Ez (kV/cm)

FIG. 3: (Color online) HH SRT along the z-axis τsz of the lowest hole subband in (111) GaAs/AlAs QWs as a function of (a) the hole density with temperature T = 30 K and impurity density Ni = 0; (b) temperature with the hole density Nh = 4 × 1011 cm−2 and Ni = 0; and (c) impurity density with T = 30 K and Nh = 3 × 1011 cm−2 . Red curves with × stand for the SRTs due to the DP mechanism under zero bias. Green curves with  (blue ones with •) represent the SRTs due to the DP (both DP and EY) mechanism at the cancellation gate voltage Ez = 22 kV/cm. (d) HH SRT along the x (y)axis τsx (τsy ) against the gate voltage Ez with T = 30 K, Nh = 3 × 1011 cm−2 and Ni = 0.

effective magnetic field Ωy [Eq. (3)], it is robust against the hole density, temperature and impurity density. In Fig. 3(a)-(c), we plot the hole density, temperature and impurity density dependences of the SRT under zero bias and the cancellation gate voltage Ez = 22 kV/cm as red curves with × and green ones with , respectively. It is shown that the SRT at the cancellation gate voltage is always four orders of magnitude larger than the one under zero bias. In addition, we also observe peaks in the hole density and temperature dependences with the peak locations fixed under different gate voltages. The peak in the density dependence of the SRT has been theoretically predicted by Jiang and Wu48 and experimentally confirmed52–55 recently, which is attributed to the crossover from the nondegenerate to degenerate limit. The peak in the temperature dependence of the SRT was predicted by Zhou et al.23 and experimentally realized by Leyland et al.,56 Ruan et al.,57 and Han et al.,54 which is solely caused by the Coulomb scattering.23 The location of the peak is insensitive to the gate voltage just the same as the case for electrons in (111) QWs.38 It is further noted that the EY mechanism is usually the major mechanism in hole spin relaxation in bulk GaAs.45 In order to rule out the influence of the EY mechanism to the predicted phenomena, we plot in Fig. 2 the gate-voltage dependence of the SRT due to both the DP and EY mechanisms as the blue dashed curve. It

is seen that the EY mechanism can be important only around the cancellation gate voltage where the DP spin relaxation is greatly suppressed. Away from the cancellation gate voltage, the contribution of the EY mechanism is negligible since the spin mixing between the HH and LH is extremely small due to the large energy splitting between the HH and LH subbands in QWs with small well width. Moreover, since the EY mechanism is important at the cancellation gate voltage, we also plot in Fig. 3(a)(c) the hole density, temperature and impurity density dependences of the SRT due to both the EY and DP mechanisms under this gate voltage (blue curves with •). We find that the peaks in the hole density and temperature dependences disappear since the SRT caused by the EY mechanism decreases with the hole density and temperature.3,5,45 Moreover, the SRT due to the EY mechanism also decreases with the impurity density.3,5,45 However, away from the cancellation field, the spin relaxation is determined by the DP mechanism only and the peaks still exist in the corresponding temperature and density dependences. Finally, we address the anisotropy of the spin relaxation with respect to the spin polarization direction. The SRT along the z-axis τsz is given in Eq. (17). Similarly, one obtains the SRTs along the x- and y-axes as τsx = τp∗ −1 hΩ2x + Ω2y + Ω2z i −1

τsy = τp∗ −1 hΩ2x + Ω2z i

−1

,

,

(18) (19)

by setting the initial spin polarizations along the x- and y-axes correspondingly. By comparing Eq. (17) with Eqs. (18) and (19), one finds a strong anisotropy between the out-of- and in-plane spin relaxation, since Ωz is about one order of magnitude larger than the in-plane components Ωx and Ωy in (111) QWs with small well width. In addition, a marginal anisotropy also arises between the in-plane spin relaxations from Eqs. (18) and (19). The numerical result of the SRT along the x (y)-axis τsx (τsy ) is shown in Fig. 3(d) by red curve with × (blue one with ). One finds from the figure that both τsx and τsy are about two orders of magnitude smaller than τsz . B.

LH spin relaxation in GaAs/InP QWs

We also investigate the spin relaxation of the lowest hole subband, which is LH like, in (111) GaAs/InP QWs with the well width a = 20 nm. The gate-voltage dependence of the SRT along the z-axis due to the DP mechanism under different temperatures is shown in Fig. 4 and a peak is also observed. We find that the peak becomes less pronounced than that in GaAs/AlAs QWs since the cancellation only occurs on a special momentum circle in GaAs/InP QWs.38 In addition, when the temperature increases, the peak location changes, similar to the case for electrons in (111) GaAs QWs.38 It is noted that the contribution of the EY mechanism is in the order of 10 ns and is therefore always negligible in this case.

6 suggests that the HH in (111) GaAs/AlAs QWs offers unique properties for the spintronic devices and calls for experimental investigations.

5.5

z

τs (ps)

4.5

Acknowledgments

3.5 T=40 K 60 K 12

2.5

Nh=1×10

cm

This work was supported by the National Basic Research Program of China under Grant No. 2012CB922002 and the National Natural Science Foundation of China under Grant No. 10725417.

-2

Ni=0 1.5 -160

-110

-60 -10 Ez (kV/cm)

40

90

FIG. 4: (Color online) LH SRT along the z-axis τsz of the lowest hole subband in (111) GaAs/InP QWs against the gate voltage Ez . Red curve with × (blue one with ) stands for the case with temperature T = 40 (60) K. The hole density Nh = 1 × 1012 cm−2 and impurity density Ni = 0.

IV.

SUMMARY

In summary, we have investigated the hole spin relaxations in p-type (111) GaAs/AlAs and GaAs/InP QWs with only the lowest hole subband being relevant. The lowest hole subband is HH like in the former case while LH like in the latter one. In both systems, we utilize the subband L¨ owdin perturbation method to obtain the effective Hamiltonian including the Dresselhaus and Rashba SOCs. Under a proper gate voltage, the total in-plane effective magnetic field in (111) GaAs/AlAs QWs can be strongly suppressed in the whole momentum space, while the one in (111) GaAs/InP QWs can only be suppressed to zero on a special momentum circle. Therefore, tuning the gate voltage is an efficient way to modulate the hole spin relaxation in (111) QWs. We calculate the hole spin relaxation due to the DP and EY mechanisms by numerically solving the KSBEs with all the relevant scatterings explicitly included. For (111) GaAs/AlAs QWs, the contribution of the EY mechanism is negligible unless the DP mechanism is strongly suppressed. In addition, we predict a pronounced peak in the gate voltage-dependence of the HH SRT where an extremely long SRT (upto hundreds of nanoseconds) can be reached. This peak comes from the suppression of the unique inhomogeneous broadening in (111) GaAs/AlAs QWs, specifically, the cancellation of the y-component of the effective magnetic field. The suppression of the spin relaxation is robust against the hole density, temperature and impurity density. A strong anisotropy between the out-of- and in-plane spin relaxations is also addressed. In (111) GaAs/InP QWs, a peak is also predicted in the gate-voltage dependence of the LH SRT. However, the peak becomes less pronounced compared with the case in (111) GaAs/AlAs QWs since the cancellation only occurs on a special momentum circle in GaAs/InP QWs. The contribution of the EY mechanism is always negligible in this case. This investigation

Appendix A: EFFECTIVE MASS AND SPIN-ORBIT COUPLING COEFFICIENTS

The in-plane effective mass mht in Eq. (8) and the Dresselhaus (Rashba) strengths βih (αhi ) (i = x, y, z) of the lowest hole subband, which is HH like, in Eqs. (1) and (2) are given by X 2B12 Q21n 1 γ1 + γ3 2C 2 z − = − 2 1lh , h lh 2 m0 mt ~ ∆nz 1 ~ ∆11 n 6=1

(A1)

z

4(C1 B3 + B2 C3 ) b42 βxh = √ − ∆lh 3 11 X 4(B1 C4 + 2C2 B2 )Q21n z , + lh ∆ n 1 z n 6=1

(A2)

z

βyh =

4(C1 B3 − B2 C3 ) −b51 + b52 √ + ∆lh 3 11 X 4(2C2 B2 − B1 C4 )Q21n z + , lh ∆ n 1 z n 6=1

(A3)

z

√ √ 23 6 4(C3 B3 − C1 B2 ) 6 b41 − b42 + 2 24 ∆lh 11 X 4(B1 C2 − 2B2 C4 )Q21n z , (A4) + lh ∆ n 1 z n 6=1

βzh = −

z

αhx =

X 4P1n Q1n z z (B1 B3 − C1 C4 n2z + 2B22 lh ∆hh ∆ n 1 n 1 z z n 6=1 z

− C2 C3 n2z ) +

X 4P1n Q1n 1 1 z z ( lh − hh ) lh ∆ ∆ ∆ 11 nz 1 nz 1 n 6=1 z

× (B1 B3 − C1 C4 + 2B22 − C2 C3 ), αhy =

(A5)

X 4P1n Q1n z z (B1 B3 − C1 C4 n2z − 2B22 lh ∆hh ∆ nz 1 nz 1 n 6=1 z

+ C2 C3 n2z ) +

X 4P1n Q1n 1 1 z z ( lh − hh ) lh ∆11 ∆nz 1 ∆nz 1 n 6=1 z

× (B1 B3 − C1 C4 − 2B22 + C2 C3 ),

(A6)

7 αhz =

X 4P1n Q1n z z (B1 B2 − C1 C2 n2z − 2B2 B3 lh ∆hh ∆ n 1 n 1 z z n 6=1 z

+

C3 C4 n2z )

X 4P1n Q1n 1 1 z z ( lh − hh ) + lh ∆11 ∆nz 1 ∆nz 1 n 6=1

δn1 ,n2 ), Qn1 n2 = √ 3 2π 2 1 a2 ( 6 b41

+

7 24 b42

z

× (B1 B2 − C1 C2 − 2B2 B3 + C3 C4 ).

(A7)

The in-plane effective mass mlt in Eq. (9), the Dresselhaus strengths βxl1,l2 , βzl and the Rashba strengths αlx,z of the lowest hole subband, which is LH like, in Eqs. (4) and (5) read X 2(C12 + C32 ) γ1 − γ3 1 2 − = − l hl 2 2 m0 mt ~ ∆11 ~ ∆hl nz 1 n 6=1 z

× βxl1

βxl2 βzl

(B12

+

4B22 )Q21nz ,

4an1 n2 [(−1)n1 +n2 −1] (1 − (n21 −n22 )2 π 2 n1 +n2 2n1 n2 [(−1) −1] (δn1 ,n2 −1), and W1 = (n2 −n2 )a

C4 = 41 (b42 + b51 − b52 ). Pn1 n2 =

(A8)

√ 4 4b41 + 7b42 3 √ − (b51 + b52 ) − hl (B2 C3 = 2 ∆11 4 3 X 4(−B1 C4 + 2C2 B2 )Q21n z (A9) , − C1 B3 ) + hl ∆ nz 1 nz 6=1 √ √ 4 3b41 7 3b42 √ (A10) = − − − 3b52 , √3 √ 3 13 6b42 4(C1 B2 + C3 B3 ) 6b41 − − = − 6 24 ∆hl 11 2 X 4(B1 C2 + 2B2 C4 )Q1n z , (A11) + hl ∆ n 1 z n 6=1 z

1

Appendix B: LUTTINGER HAMILTONIAN FOR (111)-ORIENTED III-V ZINC-BLENDE CRYSTAL

The Luttinger Hamiltonian HL , the Dresselhaus SOC b H8v8v and the strain Hamiltonian Hǫ in Eq. (6) are given in the following. The Luttinger Hamiltonian HL can be written as46   F H I 0 0 I   H∗ G (B1) HL =  ∗ I 0 G −H  ∗ ∗ 0 I −H F in the basis of the eigenstates of Jz with eigenvalues 3 1 1 3 ~2 2 , 2 , − 2 , and − 2 in sequence, where F = 2m0 [(γ1 +

~2 2 2m0 [(γ1 − γ3 )(kx + √ 2 ~ 2 ky2 ) + (γ1 + 2γ3 )kz2 ], H = − 2m + [ 36 (−γ2 + γ3 )k+ 0 √ √ 2 3 ~ 2 3 2 3 (2γ2 + γ3 )k− kz ], and I = − 2m0 [ 3 (γ2 + 2γ3 )k− + √ (0) 2 6 HL = 3 (−γ2 + γ3 )k+ kz ], with k± = kx ± iky . 2 2 ~ kz 2m0 diag(γ1 −2γ3 , γ1 +2γ3 , γ1 +2γ3 , γ1 −2γ3 ) corresponds

γ3 )(kx2 + ky2 ) + (γ1 − 2γ3 )kz2 ], G =

to the part of the Luttinger Hamiltonian HL by setting kx,y = 0. b The Dresselhaus SOC H8v8v is given by42 2

αlx

2

X 4P1n Q1n 1 1 z z + ( hl − ll ) hl ∆ ∆ ∆ 11 nz 1 nz 1 n 6=1 z

αlz

× (B1 B3 − C1 C4 − 2B22 + C2 C3 ), (A12) X 4W1 P1n Q1n X 4W1 P1n Q1n n2 z z z z z = − − hl ll2 ∆hl ∆ 11 ∆nz 1 n 1 n z n 6=1 z

z

X 4P1n Q1n z z − (−B1 B2 + C1 C2 − 2B2 B3 ll ∆hl ∆ nz 1 11 n 6=1 z

+ C3 C4 ) +

X 4P1n Qn 1 z z (B1 B2 − C1 C2 n2z ll ∆hl ∆ nz 1 nz 1 n 6=1 z

+ 2B2 B3 −

C3 C4 n2z ).

(A13)

In above equations, γi are Kohn-Luttinger parameters; b41,42,51,52 stand for the coefficients of the Dresselhaus ll SOC of the valence band; ∆hh n1 n2 (∆n1 n2 ) is the energy difference between n1 -th and n2 -th subbands of HH (LH); ∆hl n1 n2 is the energy difference between n1 -th HH and n2 th LH subbands. B1 = √



3~2 (2γ2 +γ3 ) , 3m0

2 (γ2 +2γ3 ) . C1 and B3 = 3~ 6m 0 √ 9 2 C2 = 4 (b41 + 4 b42 −b51 +b52 ),

= (b41 + C3 =





6~2 (−γ2 +γ3 ) , 6m0 1 π2 9 4 b42 − 2 b52 ) a2 ,

B2 =

3

2

2

2

2

2

− kz′ }Jx′ + cp) − b51 ({kx′ , ky′ + kz′ }{Jx′ , Jy′

z

+

2

b H8v8v = −b41 ({kx′ , ky′ − kz′ }Jx′ + cp) − b42 ({kx′ , ky′

X 4P1n Q1n z z = (B1 B3 − C1 C4 n2z − 2B22 ll ∆hl ∆ n 1 n 1 z z n 6=1 C2 C3 n2z )

2

+ 14 b51 − 41 b52 )( 23 b41 + 67 b42 + 21 b52 ).

2π 2 4a2 (2b42 +b52 ),

and

2

3

2

− Jz′ } + cp) − b52 (kx′ {Jx′ , Jy′ − Jz′ } + cp), (B2) where kx′ = √16 kx − √12 ky + √13 kz , ky′ = √16 kx + √12 ky + √1 kz , k ′ = − √2 kx + √1 kz , J ′ = √1 Jx − √1 Jy + √1 Jz , x z 3 6 3 6 2 3 Jy′ = √16 Jx + √12 Jy + √13 Jz , and Jz′ = − √26 Jx + √13 Jz . Ji (i = x, y, z) represent the spin-3/2 angular momentum matrices and cp denotes the cyclic permutation of the preceding terms. The Bir-Pikus strain Hamiltonian Hǫ reads42,47 Hǫ = diag(−ES1 , ES1 , ES1 , −ES1 ) − ES2 I4 , ES1 = 2Du′ ǫa , ES2 = 3Ddǫb , 1 1 ǫa = − (1 + (111) )ǫk , 3 σ 1 1 ǫb = (2 − (111) )ǫk , 3 σ as − ae ǫk = , ae C11 + 2C12 + 4C44 σ (111) = . 2C11 + 4C12 − 4C44

(B3) (B4) (B5) (B6) (B7) (B8) (B9)

8 Here, Du′ and Dd stand for the deformation potential constants; ǫb and ǫa denote the diagonal and off-diagonal components of the strain tensor; ae and as represent the lattice constants of epilayer (GaAs) and substrate materials (AlAs/InP); and C11 , C12 and C44 are the stiffness constants.42,47 ˆ k,k′ FOR THE Appendix C: SPIN MIXING Λ LOWEST HH AND LH SUBBANDS

ˆ k,k′ in the spin-flip scattering The spin mixing Λ ˆ k,k′ = Iˆ − induced by the EY mechanism reads Λ † † † P (1) (1) (1) (1) (1) (1) 1 − 2Sk Sk′ + Sk′ Sk′ ), with Iˆ being nz (Sk Sk 2 a 2 × 2 unit matrix. For the lowest hole subband, which (1) is HH like, in GaAs/AlAs QWs, Sk is given by

(1)

Sk



 = 

0

H 1nz

− ∆12 hl

0

− ∆42 hl

1nz

H 1nz

− ∆41 hh

1nz

H 1nz

− ∆14 hh

1nz

H 1nz 1nz

H 1nz

− ∆13 hl

1nz

H 1nz

− ∆43 hl

1nz



 . 

1 2

3

4

5 6 7

8 9 10



 (1) Sk =  

H 1nz

0

− ∆23 ll

1nz

H 1nz − ∆32 ll 1nz

0

H 1nz

H 1nz

− ∆21 lh

− ∆24 lh

H 1nz − ∆31 lh 1nz

H 1nz − ∆34 lh 1nz

1nz

1nz



 , 

(C2)

where ∆ll 1nz is the energy difference between the first and (1) nz -th LH subbands. It is noted that Sk′ can be obtained (1) by replacing k in Sk with k′ . Appendix D: EXACT SOLUTIONS OF Eqs. (14) TO (16)

∗ ∗ Sz (0) hΩx Ωz i (−e−λ1 τp t + e−λ2 τp t ) + [(hΩ2x u 2u ∗ ∗ − Ω2z i + u)e−λ1 τp t + (hΩ2z − Ω2x i + u)e−λ2 τp t ], (D1) ∗ Sx (0) = [(hΩ2z − Ω2x i + u)e−λ1 τp t + (hΩ2x − Ω2z i + u) 2u ∗ ∗ ∗ hΩx Ωz i (−e−λ1 τp t + e−λ2 τp t ), × e−λ2 τp t ] + Sz (0) u (D2)

Sz = Sx (0) (C1)

1nz = h1|Hij |nz i in which 1 (nz ) stands for the first Hij (nz -th) subband and Hij represents the i-th row and jth column of the Hamiltonian matrix H4×4 in Eq. (6). ∆hh 1nz is the energy difference between the first and nz -th HH subbands and ∆hl 1nz is the energy difference between the first HH and nz -th LH subbands. For the lowest hole subband, which is LH like, in



(1)

GaAs/InP QWs, Sk is given by

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