Spin Relaxation under Optical Orientation in Semiconductors

CHAPTER 3

Spin Relaxation under Optical Orientation in Semiconductors G.E. PIKUS and A.N. TITKOV A.F. ioffe

Physico-TechnicalInstitute USSR Academy of Sciences 194021

Leningrad USSR

Optical © Elsevier Science Publishers B. V.f 1984

F. Meier and B.P.

73

Orientation Edited by Zakharchenya

Contents 1. Introduction

77

2. Spin relaxation mechanisms for free carriers

78

2.1. Band structure, scattering mechanisms and electron-hole exchange interaction in direct-gap A 3 B 5 compounds

78

2.1.1. Two-band model

78

2.1.2. Scattering of electrons by phonons and impurities

81

2.1.3. Electron-hole exchange interaction

82

2.2. The Elliott-Yafet (EY) mechanism

83

2.2.1. Long-range interaction

84

2.2.2. Short-range interaction

85

2.3. The D'yakonov-Perel'(DP) mechanism

87

2.3.1. General expression for spin relaxation rate

88

2.3.2. Electron spin relaxation in A 3 B 5 compounds

90

2.3.3. Electron spin relaxation in uniaxially deformed crystals

91

2.3.4. Spin relaxation of holes in uniaxially deformed crystals

92

2.3.5. Effect of magnetic field on spin relaxation

93

2.3.6. Spin relaxation of hot electrons

96

2.4. The Bir-Aronov-Pikus (BAP) mechanism

98

2.4.1. Scattering of electrons by free holes

98

2.4.2. Electron spin relaxation under strong hole scattering

101

2.4.3. Spin relaxation due to scattering on bound holes

103

2.4.4. Calculation of effective constant of spin-spin interaction

104

2.4.5. Role of Debye screening

105

2.5. Effect of reabsorption on optical orientation of electrons

105

2.6. Relative efficiencies of the Elliott-Yafet, D'yakonov-Perel' and Bir-Aronov-Pikus mechanisms

107

3. Experimental study of free carrier spin relaxation in A 3 B 5 compounds

108

3.1. Optical orientation method

109

3.2. Spin relaxation in p-type InSb (Elliott-Yafet mechanism)

110

3.3. Spin relaxation in p-type GaAs and GaSb

112

3.3.1. The D'yakonov-Perel' mechanism in GaAs 3.3.2. Suppression of spin relaxation in longitudinal magnetic

74

112 field

114

75 3.3.3. The Bir-Aronov-Pikus mechanism in GaAs and GaSb 3.3.4. Dependence of spin relaxation rate on acceptor concentration 3.4. Spin relaxation of carriers in uniaxially deformed crystals

116 120 123

3.4.1. Free electrons

123

3.4.2. Free holes

125

3.5. Spin relaxation of hot electrons

127

4. Conclusion

129

References

129

7. Introduction Application of the optical orientation method to semiconductors has created the unique opportunity to measure very short spin relaxation times of electrons and holes, down to 1 0 ~ 1 2 s, inaccessible to the traditional ESR technique. The optical orientation method has allowed to study spin relaxation of both free and bound carriers in new conditions, namely, in crystals with different doping levels and in a wide temperature range. As a result, very interesting new experimental data were obtained which could not be explained in the framework of spin relaxation mechanisms known at that time. This situation stimulated further theoretical studies which have brought forth two new spin relaxation mechanisms. (1) The lack of an inversion centre in some semiconductor compounds leads to spin splitting of the conduction band. D'yakonov and Perel' (1971a, b) have proposed a relaxation mechanism due to this splitting. They have analysed this relaxation in terms of motional narrowing. (2) In optical orientation experiments one studies spin relaxation of nonequilibrium carriers, usually photoelectrons in p-type crystals. In the case of high hole concentration another relaxation mechanism, due to the exchange interaction between electrons and holes, can be very efficient. A detailed description of this mechanism has been given by Bir et al. (1975). In some cases the Elliott-Yafet spin relaxation mechanism (Elliott 1954, Yafet 1963) can also be important in optical orientation experiments. In this mechanism spin relaxation of carriers is connected with momentum relaxation through spin-orbit coupling. Under certain conditions in semiconductors an appreciable reabsorption of recombination radiation with simultaneous creation of new electron-hole pairs can take place. This reabsorption process causes a decrease of photoelectron spin orientation and can be treated also as a spin relaxation process. In sect. 2 we shall present the theory of the above-mentioned spin relaxation mechanisms. Special attention will be given to spin relaxation under a uniaxial compressive stress and in a longitudinal magnetic field. We shall denote these relaxation mechanisms by the names of their originators, namely, DP, BAP and EY mechanisms, respectively. To perform a uniform treatment of the relaxation mechanisms we shall restrict ourselves to one group of crystals—cubic direct-gap semiconductors with the zinc-blende structure. Here 77

78

G.E. Pikus and A.N.

Titkov

we take into account that the main experiments on the optical orientation of free carriers have been performed on A 3 B 5 compounds belonging to this group. In sect. 3 we shall discuss the available experimental data on variation of spin relaxation rate with temperature and doping level in different crystals and also analyze the suppression of spin relaxation in a longitudinal magnetic field and the effect of uniaxial compression. From a detailed comparison of experi­ mental data with theoretical predictions the temperature and concentration regions will be determined where the considered mechanisms dominate. In the present survey we shall not consider specific relaxation mechanisms operative in the case of optical orientation and alignment of free and bound excitons. These mechanisms have recently been considered in detail by Pikus and Ivchenko (1982). We shall also not consider relaxation mechanisms for carriers bound on impurities.

2. Spin relaxation mechanisms for free carriers As was mentioned above we shall consider spin relaxation mechanisms on the example of A 3 B 5 compounds with a zinc-blende structure. In the beginning of this section we briefly review some data on the band structure of these compounds, the main scattering mechanisms, electron-phonon coupling, and electron-hole exchange interaction, which are necessary for the consecutive theoretical description of spin relaxation mechanisms. 2.1. Band structure, scattering mechanisms and electron-hole exchange interaction in direct-gap A3B5 compounds Without taking electron spin into account, the conduction band in A 3 B 5 compounds is non-degenerate and the corresponding wave function is of S type. When electron spin is taken into account, the representation Γ χ trans­ forms into the representation Γ 7 with the two corresponding wave functions 1 / 2 , 1 / 2 and 1/2, - 1 / 2 , the total angular momentum s = 1/2 being quan­ tized along the ζ axis with projections sz = ± 1 / 2 . The valence band Γ 1 5 is threefold degenerate, without taking into account the spin, and the correspond­ ing wave functions are X9 Y and Ζ functions. As a result of spin-orbit interaction, the Γ 1 5 band splits into two Γ 8 and Γ 6 valence bands with corresponding total angular momenta . 7 = 3 / 2 (Γ 8 ) and ,7 = 1/2 (Γ 6 ), and projections Jz = ± 3/2, ± 1/2 (Γ 8 ) and Jz = ± 1/2 (Γ 6 ). Two - band model For A 3 B 5 compounds an identical description of the energy spectrum in the conduction Γ 7 as well as the valence Γ 8 , Γ 6 bands near k = 0 can be carried

2.7.7.

Spin relaxation

79

Table 1 Two-band Hamiltonian ν

υ)

i!.i>

υ

υ'

ι;

iU>

é§.-!>

c 0

2m't

/i *ζ

/i*_

0

-/i*

/R

ê_

/'

0

0

í

G'-E,

-Η'

-Η'*

F'-E%

-/Γ ^

c h2k2

0

„ + Cje

0

+

-/i*

+

/i*.

V


-ê*

0

Η'

-/iff'

-ν/ϊ/'

V Η'*

G'-Et

0

/iff'

V

/i*-*

/§>

Ι'*

0

κ*

0

-/i*,*

-/!>:

/i/r*

C -

F'

fi

V


/'•

fir*

-/iff'*

υ'



C -

L

F'

y/ll'

fi

-vTA-*

C -

-]/2Γ*

F'

i(G'-F') -Ε,-Δ

0

0

-Ε%-Α

(1) other components are obtained by cyclic permutation of indices. tf± = /F ( * , ± ί * , ) . F/-/i/A2 +

P-{S\PZ\Z)%

^ (it -3Ar r ) + /

G'= A'k2 - \B'(k2

2

2

-3^)

«c+^(e-3e„),

+

H' = /' = - \f$

αε-\ί(ζ-2>εζζ), D'kz(kx-iky)-J(exz-ieyz)-ai0(uy-\ux),

B'(k*

- k]) + iD'kxky

-\)/3fi(exx

-

eyy)+i*fexy+iJ0uz.

80

G.E. Pikus and A.N. Titkov

out in the framework of the two-band model (Kane 1957a, b). The 8 x 8 matrix describing the energy spectrum in these bands is shown in table 1 (see, for example, Bir and Pikus 1972). This matrix also includes the terms accounting for the relative displacement of sublattices U and for uniaxial deformation, described by the deformation tensor ε. Here m is the free-electron mass, m the conduction-band effective mass, \/m^ the contribution to l / m from all bands other than the Γ band, constants A\ B' and D' are related to the contribution of bands other than the T band, constant m determines an interband term quadratic in k and is related to the interaction of Γ and Γ bands with other bands, E is the band gap, Δ the spin-orbit splitting of the valence band, a,
e

15

x

cv

λ

15

g

0

x

2

2

8

+ ^{2mlE y (K-o)

Hc = ^

6

+ ^ C (o^).

1/2

2

g

(2)

3

Here σ, are the Pauli matrices, κ = k (k^ - k^\ ψ = e k components are obtained by cyclic permutation of indices), ζ

2

-l_ 2 ^ 1 i l _ A \ J_ =

me

m

i

E

i

\

i)

m' ' 1/2

1

"

e

to)~ >

"t-lTt^ -

zx

x

- ek zy

(other

v

A

J

11

ζ

3

E + A> %

= ! * Q , [ 2 V , e ( i - in)] ~

c

x / l

-

In the equations for a and C , and also in further formulae, we neglect the contribution of m'~ to m~ . The Hamiltonian H for the valence Γ band, framed in formula (1) with double lines, does not change its form when the interaction with the c-band is taken into account. However, all the primed symbols F\ G\ H' and / ' should be correspondingly replaced by F G, Η and I which are, in turn, obtained by the substitution of A = A' + A , c

x

3

l

w

8

9

Q

* We wrote formula (1) in the canonnical basis instead of the Luttinger basis, used earlier by Bir and Pikus (1972, see Table 24.2). We have also corrected some misprints which occurred there. Additional terms, arising only in crystals without inversion centre, are included in accordance with Bir and Pikus (1961, see Table 4 in their sect. 2).

Spin relaxation

81

B = B' + A0 and D = D' + \/3 A0 for A', B' and D'. Here 1 (ΛΡ\2_

°

h2

3£,U J ~

6m (l-h)' e

Furthermore, the interaction with the c-band adds to Hw terms cubic in k and linear in k and ε, / / V 2 = a v ^ ( 2 m ^ g ) _ 1 / 2( / - < c ) + Q ( / ' < p ) ,

(3)

where (Andrianov et al. 1981) „ - _ ι ^ £ - _ Ι ^ / , ι 2 "ν~ η " 3mc/A

Γ 3 η)

ι/2 '

r

°4~

2η~

Q^P 3 Eg m '

When relativistic corrections are taken into account, Hv should also include terms linear in k (Kane 1957a, b, Bir and Pikus 1972)

Η =± φ ν ) ,

V ={j {j -J )} . 2

ν3

z

z

etc.

2

symm

(4)

Here Jt are the matrices of the hole angular momentum operator in the basis of Y^2 functions and { AB } s y mm = \(AB + BA). It follows from symmetry considerations that Hc and Hv should also contain, respectively, ( σ · £ ) and ( / · £ ) terms (Bir and Pikus 1961) where ξζ = kz(exx - eyy) etc. However, these terms appear only if we take iitfo account the spin-orbit mixing of Γ 8 states, arising from Γ 1 5 (or Γ 2 5 ) states, and Γ 1 2 states. Numerical coefficients for these terms should be much smaller than C 3 or C 4 , so hereafter we shall not consider these terms. In the following we are also interested in the energy spectrum of the upper valence sub-band split off the valence band by compressive uniaxial stress along the (001) axis. The Hamiltonian for holes in this sub-band, for which Jz = ± 1 / 2 , has the form Hy = -{A

+ B)k]-{A-\B)(kl

+ K^ (2m £j3

e

1 / 2

+ k*)

[2(a-K)-a cj4-v^/(aA-aA)z i

(5)

The hole g-factor in this sub-band is anisotropic: g± = 2g | (. Here it has been taken into account that, turning from the Γ 8 electron spectrum to the spectrum of holes, we should replace Η by - Η and Jt by - Jr 2.1.2. Scattering of electrons by phonons and impurities Formula (1) describes the influence on the energy spectrum of both the static stress and the short-range (deformation) interaction with acoustic and optical phonons causing the displacement of sublattices U. In crystals with T d symme­ try, in addition to the short-range interaction there exists also long-range

82

G.E. Pikus and A.N. Titkov

interaction, which is usually predominating. Polar long-range interaction with optical phonons is described by the Frohlich Hamiltonian

aft-icte^i\ . .. r

(6)

9k k

Here I is the unit matrix, C = e
1

0

0

2 lM M ]

l

2

Here U and U are the displacements of the sublattices formed by atoms with masses M and M 2 , respectively. In formula (1) U is defined in the same way. Long-range electron-phonon piezoelectric interaction in crystals with T symmetry is described by the Hamiltonian x

2

x

d

@l>,k =

'^^ti^yz

+ q *zx + ,

0)

l

y

where β is the piezoelectrical modulus. Interaction with ionized impurities is described by the Hamiltonian 0

4ne2

1

where L D e b is the Debye screening length. 2.1.3. Electron - hole exchange interaction In addition to the usual Coulomb interaction defined by the Hamiltonian (8), in crystals there exists also electron-hole exchange interaction. The significant property of this interaction is its dependence on the relative orientation of electron and hole spins. The Hamiltonian for electron-hole exchange interac­ tion can be written in the form (Pikus and Bir 1971, Bir and Pikus 1972) HEX = ™\m(r)SK,K,.

(9)

Here r = r e - r h is the position of the electron relative to the hole, hK and hK' are the total quasi-momenta of the interacting electron and hole in the initial and final states, respectively. The factor πα\, where a is the exciton Bohr radius, is introduced in eq. (9) to express the operator 2 in energy units. There are two different contributions to 3). The first one is connected with short-range h

Spin relaxation

83

electron-hole exchange interaction at the same site. For A 3 B 5 compounds this contribution can be written in the form

S „ = }*„(/· σ),

(10)

where ^ e x is the exchange splitting of the Γ 7 χ Γ 8 exciton states with total angular momenta J = l ( Γ 1 5 ) and J=2. It should be mentioned here that there is an additional anisotropic contribution to 2 which is proportional to Σ,^Λτ, and leads to the splitting of the exciton state with / = 2 into two levels, Γ 1 2 and Γ 2 5 . However, we have neglected this contribution in eq. (10), since the constant in front of the sum Σ , ^ σ , is usually much smaller than 4 e x . The second contribution to 2 comes from long-range annihilation interac­ tion ™\2™(K)

-

4vh2e2 Γ ' i m1eO0E2

(Ρ0ί·Κ)(Ρ0ί·Κ)* V > K1

á ú)

where P0i is the matrix element of the momentum operator between the initial state (the ground state of the crystal) and the final state, represented by the electron-hole pair with momentum hK in the state /. If the final states of electron and hole are, respectively, m and n9 P0i can be written as P0i = (X'n\Pi\m), where J f is the time inversion operator. This long-range interac­ tion leads to the longitudinal-transverse splitting of the electric-dipole-active exciton state. The Hamiltonian (11), as well as the Hamiltonian (10), permits spin-flip scattering of electrons by holes due to spin-orbit splitting of the valence band. It is appropriate to mention here, that the possibility of optical orientation of electrons is caused exclusively by the spin-orbit splitting of the Γ 1 5 band (see, for example, the review by D'yakonov and Perel' in this volume). Below, we shall consider the specific spin relaxation mechanisms in detail. 2.2. The Elliott-Yafet

(EY) mechanism

As was first pointed out by Elliott (1954) and later by Yafet (1963), spin-orbit interaction in crystals causes the mixing of electron wave functions with opposite spin vector orientations. As a result, in the process of momentum scattering the disorientation of electron spin also becomes possible. After­ wards, the spin relaxation rate for this mechanism has been calculated in detail by a number of authors (Pavlov and Firsov 1965, Pavlov 1966, Abakumov and Yassievich 1971, Chazalviel 1975). In these calculations different momentum relaxation processes have been taken into account. In the following subsections we shall, in short, present calculations of the spin relaxation rate for the cases of electron scattering by acoustic and optical phonons and by impurities.

84

G.E. Pikus and A.N.

Titkov

2.2.1. Long-range interaction Taking into account the mixing of wave functions of the Γ band with wave functions of the Γ and Γ bands, we can note the spin-flip matrix element for transitions between m and m' states in the conduction band as follows: 7

8

6

MMI

zjk TJ

^m'k' ,s'k,ris'k'

_

' '- ~t,

Hm k

^sk^sk,

mk

{E -E ){E ,-E )

mk

s

m

s

(ΛΊ\

•

m

[

U

)

Here H* = H * are the terms linear in k of formula (1), responsible for the mixing of wave functions, and H™* are the matrix elements of the operator H which determines the electron interaction with phonons or impurities. The indices m and s refer to the states in conduction and valence bands, respec­ tively. According to eqs. (6)-(8) for the long-range interaction, m

m

int

J/INT

i/INT n

s'k\sk

R n k\k'Os,s'

~

so using the formula (1) we can rewrite the expression for H , as: k

„ „, i*(a[*'*])i,(i-jii)

k

2

in

M

~ Hk\k

k\k

ι

ΤΓΓΛ

*

^

For elastic ' £ [ * scattering, * ] ) = 4 ^when s i n ^k' , = k, 2

is the electron kinetic energy, θ is the scattering angle. where E = h k /2m Since we are interested here in the spin relaxation rate of electrons isotropically distributed in k space, the transition probability should be averaged over all directions of ft. Then it follows for the spin relaxation rate that 2

1

7

=

2

2

e

k

2π

Λ

fdtiL

τ / ^ Σ ΐ ^ .

Ι β ( ^ - ^ )

(14)

2

Λ

Λ

k'

whereas the momentum relaxation rate is

i = χ D ^ < , J ( l - c o s d ) 8 ( E , - E ). 2

m

k

k

Then (15)

Spin relaxation

85

where Φ=

ίι_ι(\-μ2)ο(μ)άμ Ιι_ι(ΐ-μ)σ(μ)άμ

Here μ = cos0, and σ(μ) is the scattering cross section. For scattering by polar optical phonons, if the electron kinetic energy Ek is much higher than the LO phonon energy Λ ω ί ο , we can obtain σ(μ) ~ (1 - μ) and Φ = 1. For scattering by ionized impurities σ(μ) ~ [(1 - μ)+2/β]~2, where β = (2A:L D e b) 2 and 2(2 + /?) ln(l + j8)-2j8/(2 + j9) β \η(\ + β)-β/(1 + β) · After averaging over the Maxwell distribution of electrons the general expres­ sion for the rate of spin relaxation due to the EY mechanism can be written in the form

with _16φ(Ε*τρ-ιχΕΛτρ)

_ 32

Γ

2 7

(Ek)(kBT)2

^E2kr^){Ekfp) (kBTf

'

where the angle brackets denote the averaging over the Maxwell distribution. Hereafter it is taken into account accour that rp, which determines the electron mobility jti e = e T p / m e , is given by (fpEk)/(Ek)

=

2(r E )/3k T. p

k

B

In the case of scattering by polar optical phonons, r equals 2.0 and for the 2 2 scattering by ionized impurities r = 3.0Φ(βλ) where βι = 4A:BrL Debme/ft . For scattering by piezoelectrical phonons the accurate evaluation of 1 / T s and 1 /rp is rather difficult due to the anisotropy of the 3>pk in eq. (7). However, if one takes for 2p k its value averaged over all directions of q = k - k\ r will have the same value as in the case of scattering by polar optical phonons, since for both cases o(q) ~l/q2 (see, for example, Anselm 1978). 2.2.2. Short - range interaction In the case of short-range interaction it follows from formula (1) that for scattering by optical phonons it is sufficient to take into account the admixture of wave functions of the Γ 8 and Γ 6 bands to only one of the wave functions of the ΤΊ band, since for this interaction scattering of electrons is possible from the Γ 8 or Γ 6 band to the Γ 7 band. Therefore, taking account of the interband

86

G.E. Pikus and A.N. Titkov

scattering we obtain: TJ m'k\mk~

__

n

irk

Hmt

4- f / i nt

m11fkf,sk,1Isk',mk~I1mfk',skrisk,mk L· T? — T7

V*

k %k

1 T)h
-

_

( Ë Ð \ V 1' /

*

Correspondingly, for the operator H , , spin-flip scattering, it follows that

H

Hk

which gives the probability of

+ k'j\) (18)

Τ Τ ΰ] Τ Γ - ·

J ~ ~ F 7 ~ ~ Z

[2m £ (l-h)] e

g

In contrast to the case of long-range interaction, scattering due to the shortrange potential occurs not only on longitudinal but also on transverse phonons. For the scattering on polar optical phonons, neglecting the difference in the energies of longitudinal and transverse phonons, the following relation between l/f and l/f can be found: s

f,

p

21fpEtE0l-h'

>

where E = C2h2/4 E . The estimate shows that, for example, for GaAs this inequality holds if 40eV, a being the lattice constant. Thus, short-range interaction with optical phonons does not alter the temperature dependence of f . A similar situation takes place for the interband deforma­ tion-potential scattering by acoustic phonons when the main contribution to l/f is given by piezoelectrical interaction. However, if the value of 1/τ is determined also by the deformation-potential interaction and interband contri­ bution still prevails, calculation according to eq. (17) gives 0

2

0

g

0

s

p

ρ

1_

7

l(C2\

2

Ek

1

Hence: 1

1 ICA2

,knT

(21)

When the intraband contribution is dominating, calculation according to eq.

Kiy

Spin relaxation (12) leads to the following expression for H

k

87

,\

k

f ( - * ι » Μ * [ * * ] ) - ^Σσ,[Α'Λ],(ε,. - * )

= ^

2

e

^ Σ^, [Μ'],|,

(22)

7

where a, 6 and are the deformation-potential constants for the valence band. Calculation according to eq. (14) with the substitution of eq. (22) for leads to a relation for l/f similar to that of eq. (15), with H ,, m k

mk

s

where * =* + 2

2

— (^ (2—η) l

+ y -\fSM)+

(2t+yf3J).

2

2

1

5(2

V)

In this case 1 / T is given by eq. (16) with r = 2 ( # / C ) . As will be shown below, the constant C can be determined from the measurement of spin relaxation times in uniaxially deformed crystals. For GaAs its value, as it is shown in subsection 3.4.1, constitutes approximately 3 eV. Comparison of eqs. (16), (23) and (21) shows that for a «10 eV and at T> 80 Κ the intraband contribution given by eqs. (16) and (23) prevails over the interband contribution given by eq. (21). The smallness of C in compari­ son with a is explained by the fact that the only reason for a non-zero value of C is the absence of inversion symmetry in the crystal. 2

s

1

2

2

2

2.3. The D yakonov - Perel (DP) mechanism y

9

It follows from eq. (3)-(5) that in crystals without inversion centre the degeneracy of the conduction band is completely lifted for k Φ 0. Electrons with the same wave vector k but with opposite spin orientations have different energies. The degeneracy holds only for k directions along the principal axes (100) and (111). Spin splitting of the conduction band states is equivalent to the presence in the crystal of an effective internal magnetic field with magni­ tude and orientation dependent on magnitude and orientation of k. For Ω being the Larmor frequency in this effective magnetic field, the Ω>1/τ , transverse component of the electron spin vanishes before the first momentum the angular scattering occurs. In the opposite limiting case, i.e., at Ω<1/τ rotation of electron spin over time r is small and spin relaxation occurs as a result of a number of accidental small rotations, which appreciably lowers the ρ

ρ9

p

G.E. Pikus and A.N.

88

Titkov

spin relaxation rate. In A B compounds for thermalized electrons the latter condition, Ω τ ^ ^ Ι , is usually valid. In the next subsection, following the original works by D'yakonov and Perel' (1971a, b), we shall represent the derivation of the spin relaxation rate 1 / T s in this case. 3

5

2.3.1. General expression for spin relaxation rate The electron density matrix p(k) with components p -(A), /, j being the indices of electron spin states, is defined by /7

^

+ j[H {kMk)]

+ ZW Mk)-p(k'))-G,

1

(24)

k

k'

where G is the generation matrix, τ is the lifetime, W > is the probability of scattering, supposed to be diagonal with spin, H (k) is the spin splitting operator. p(k) can be rewritten in the form ρ(Λ) = ρ + ρ (Λ), where ρ is the result of averaging p(k) over all directions of k. The following equation for ρ can be obtained from eq. (24): kk

x

1

£ + jr[H (k), (k)] 1

= G.

Pl

(25)

Here the overbar denotes averaging over all directions of k. The equation for Pi(k) can also be found from eq. (24) with account taken of eq. (25):

+ Σ ^ ( Ρ Ι ( ë ) - Ρ Ι ( ^

/

) ) - ^ ( ^ ) + ^ = Ο .

k'

For Ωτ
x

x

\{H (k),p]+f^o(e)( {k)-p (k')) 1

Pl

= 0.

l

(26)

We shall search for a solution of eq. (26) in the form

Ρι(*) = - - χ [ * ι ( * ) / ρ ] ,

(27)

where τ* is a parameter which should be determined from eq. (26). In order to find the solution we have to expand H (k) in eq. (27) over the spherical functions x

Spin relaxation

89

Here it is taken into account that the power index i is the same for all terms. Now we can substitute eq. (27) into eq. (26), making use of the well-known relation for spherical functions (see, for example, Varshalovich et al. 1975) ΙΥ^',Ψ')ο{θ)^

=

Υ^,ψ)ζο{θ)Ρ^Θ)ύηθάθ

Here # and φ are the angular coordinates of ft, and φ' are the analogous coordinates for k\ θ is the angle between k and k\ and i^(cos0) is the Legendre polynomial. Then from eq. (26) we get — = ί + 1σ(μ)(ΐτ* •'-ι

Ρ;(μ))άμ

w h e ^ = cos0.

(28)

Bearing in mind that momentum scattering time is given by ^ = /

+

1

σ(μ)(ΐ-Λ(μ))<1μ

the expression for l / τ * can be rewritten in the form 1

1

— = y/zT "

/ + 1σ ( μ ) ( ΐ - / > , ( μ ) ) < 1 μ where γ, = — 1 t . / σ(μ)(1-μ)(1/ι

(29)

Substituting eq. (27) into eq. (25) we obtain the following equation for p(k): ^ + Y[H (k)[H (k),p}] 1

l

=G.

(30)

The second term in eq. (30) describes the spin relaxation due to the DP mechanism,

(tL~£l

f l

'<*>["-<*»]l

< > 31

For spin s =1/2 the matrix Hx{k) can be written as // 1 (Λ) = ^ [ σ · ί 2 ( Λ ) ] ,

(32)

and ρ in the form Ρ = Κ*·«0. (33) where σ, are the Pauli matrices and ^ is the average of electron spin projections on the jc, direction: si = ^Sp(p*a,). Then, substituting eqs. (32) and (33) into eq. (30) and setting equal the coefficients at the same components σ, we obtain

G.E. Pikus and A.N. Titkov

90

where

(VF}-jO (k)Qj(k)^. t

The other relations are obtained by cyclic permutation of indices. It follows from eq. (34) that the electron spin relaxation time is generally a tensor, with components

^- = y f (W-^),

^-γ,τ,(ϊζϊΖ;)

f p

(35)

Here

2.3.2. Electron spin relaxation in A3B5 compounds According to eq. (2) the Hamiltonian H (k) for conduction electrons has the form of eq. (32) where x

Q=

ach2(2mlEg)~l/2K.

Bearing in mind that K2=zmk6 and (ic~ic~) = 0 (/'Φ j % we conclude, in accor­ dance with general symmetry considerations, that for conduction electrons 1 /f is scalar and, as follows from eq. (35), is defined by s

After averaging over the Maxwell distribution we obtain for the spin relaxation rate 1

(k2 BT)3

Λ

7-G A dhr-. T

(37)

2i

where 0

=

^2_

(r El)(E ) p

k

(f E )(k TY p

k

B

_ 16

«

(fpE'k)

(r E ){k Tf p

k

B

The parameter Q depends on the operative scattering process. For r ~ E£: p

β = # γ 3 ( ' + *)(" + !)· (38) For scattering by polar optical phonons, when σ(μ) = (1 —/χ)""1, substitution of Ρ (μ) = ιμ(5μ2 - 3) in eq. (29) gives γ = 6/41. In this case ν = 0.5 and it 3

3

Spin relaxation

91

follows from eq. (38) that Q = 0.8. For scattering by ionized impurities, as is seen from eq. (8), the main contribution to l / τ * is given by the scattering for small angle Θ. Then l - M < i 3 - 1 = ( 2 / c L D e b) 2 and σ(μ) does not depend on μ. For small scattering angles, 1 - Ρ (μ) - 6(1 μ) and we obtain γ = 6. In this case ν = 3/2, therefore Q = 32/21 «1.5. For scattering by acoustic phonons through the deformation-potential inter­ action, σ(μ) = const, and γ = 1 . With ρ = — 1 / 2 we obtain β = 2.7. For scattering by piezoelectric phonons we can replace @p, by its average value, as done in subsection 2.2.1. Then we find that o(q)~\/q2 and (> = 0.8, the same as for scattering by polar optical phonons. 3

3

3

k

2.3.3. Electron spin relaxation in uniaxially deformed crystals According to eq. (2), in uniaxially deformed crystals the electron Hamiltonian Hx{k) in addition to terms cubic in k contains also terms linear in k. These additional terms are also defined by eq. (32) in which, however, Λ β is replaced by with

etc. Deformation splitting of the conduction band should cause an additional contribution to the electron spin relaxation rate. In this case

and other components are obtained by cyclic permutation of coordinates x, y, z. According to eq. (35) for compressive stress along the (111) axis, when e xy xz yz είιι/3> ^ corresponding contribution to l/f is defined by =

£

=

£

=

e

s

1

4

*U

*S.,

9 7

3

h

2

\ 3

(»)

(<*».

Here it is taken into account that according to eq. (29) y = 1. The parameter ε[ in eq. (39) is the uniaxial strain along the (111) axis: l

η

ε

ίιι

=

ιιι ~ ιϊο

ε

ε

=

544All/2>

where P and S are the applied stress and compliance coefficient, respec­ tively. The principal axis of the tensor l/f lies in the (111) direction. In this coordinate system, according to eq. (39) lu

u

s

T

s,N

s T, ±

1 1

h Ε l

(Ι-^η)

G.E. Pikus and A.N. Titkov

92

Here we have substituted for C as expressed by eq. (2). Then, after averaging over the Maxwell distribution we obtain 3

1

2

16

kT

η

R

2

,

,

^ " ^ • τ ^ ( ϊ γ ϊ ϊ ) ^ » ) · τ

For compression along the (110) axis, when e equal to

= e'llQ/2,

xy

ε

110

=

1ε1 0

and ε = ε νζ

F

S,22

=

ε 1ΪΟ

=

the strain ε[ being ιο

2^44Λΐ0'

= 0, we get

χζ

1

_

,

( 4 1 )

_2_ F

S , ^

=

2

=

S , XFV

2

3

Ë

(Qe,v)2.

2

(42)

In this case, the principal axis of the tensor l/f lies in the (001) direction which is perpendicular to the direction of stress, and s

*,«

τ,.,

^

Μ

^

Γ

2

°J'

(43)

U

The averaged value is defined by T

m

U

3 2 Λ

£ g

( l

_

h )

(C2e

1 1 0

).

(44)

2. J . 4 Sp/« relaxation of holes in uniaxially deformed crystals In strain-free crystals the spin relaxation time for holes in the Γ valence band is in order of magnitude close to the hole momentum relaxation time as a result of the strong spin-orbit coupling. In uniaxially stressed crystals the deformation splitting of valence sub-bands
e

Α

sh

ρ

Ε

Spin relaxation

93

scattering cross section will not be taken into account. According to eq. (5) for terms cubic in k the values of Ω, in eq. (32) are defined by the relations ΗΩΖ = a v ( 2 m ^ g )

/cz,

ΗΩΧ v = 2av{lm\E^j

1/2

1/ 2

Kxy.

a n Bearing in mind that icf^mk6 d (κ~κ^) = 0, from eq. (35) we obtain the following expression for the hole spin relaxation rate:

ίι,.ι > ' Λ . Ι 105 " "H'E.\m, The principal axis of the tensor l / f is directed along the (001) axis. After averaging over the Maxwell distribution we obtain sh

-2(kBT)

(

m

h

5 Vβ,Λ^ϊ7" ^ Γ· =

τ

(46)

Λ βpiezoelectrical phonons ρ equals 3.2, for For scattering by polar optical and ionized impurity scattering Q = 6.1, and for deformation-potential scattering by acoustic phonons Q = 11.0. The contribution of terms linear in k is defined by eq. (32), with ΗΩΧ = 2]f3 / i k x , ΗΩγ = 2v/3 t k y and ΗΩΖ = 0. Then, from eq. (35) follows 1 ητ= -T T sh,||

2

sh,±

=

16τρ^ρΕχ ft

,

E^mJVh2.

(47)

After averaging over the Maxwell distribution we obtain T T

L

sh,||

= -

1 T

- = 24^Brel.

sh,±

hl

(48)

If we accept for & its estimate of 1 0 ~ 1 0 eV cm for CdS, the value 1 0 " 5 eV can be obtained for Ev Then comparison of eqs. (46) and (48) shows that the contribution of terms cubic in k, defined by eq. (46), should prevail at T> 20-50 K. Here we took Κ| = Κ|/2η-0.1. 2.3.5. Effect of magnetic field on spin relaxation Experiments on the optical orientation of electron spins have shown that an external longitudinal magnetic field, i.e., a magnetic field parallel to the direction of light propagation, can appreciably suppress the electron spin relaxation in the internal random magnetic field. The effect, as follows from symmetry considerations, should take place for any orientation of the external magnetic field. However, in optical orientation experiments the longitudinal

94

G.E. Pikus and A.N. Titkov

geometry is the most favourable for its observation, since a transverse magnetic field also causes rapid disorientation of electron spins due to the Hanle effect. The effect of a longitudinal magnetic field has been firstly established for the case of spin relaxation of localized electrons (Parsons 1971, Berkovits et al. 1973, Hermann 1977). These authors have found that spin relaxation of electrons localized in potential wells is mainly caused by hyperfine interaction of electron spins with magnetic momenta of lattice nuclei, the hyperfine magnetic field being randomly changed due to the hopping migration of electrons over localized states in the crystal*. In p-type crystals a random magnetic field can also be caused by the exchange interaction of electrons with localized holes (Ivchenko and Takunov 1976). The first explanation for the effect of an external magnetic field on electron spin relaxation has been proposed by D'yakonov and Perel' (1973). They noticed that the Larmor precession of electron spins about a strong enough longitudinal magnetic field should suppress the precession about the internal random magnetic field, thus preserving the initial orientation of electron spins. This effect becomes noticeable when the Larmor precession frequency fiL = gpBB/h exceeds the rate of change of the internal magnetic field l / r . Obviously, the same magnetic field effect should take place in the case of free electron spin relaxation due to the DP mechanism. For the DP mechanism change of internal magnetic field occurs simultaneously with electron momen­ tum scattering. In this case T is determined by the momentum scattering time τ , and the effect of a longitudinal magnetic field should be noticeable at UL > τ~ι. This inequality has a simple explanation. The electron spin rotation Δφ about the internal magnetic field over the time rp equals 0 ( A ) τ , where β (A) is the frequency of spin precession about the internal magnetic field. Then the mean-square angular rotation over a time t » τρ can be written as i n t

I N T

(Δφ)2 = (Ω(Ιί)τρ)2ί/τρ,

where Ω 2(Α) is the angular average of Ω 2(Α) over all

possible directions of A. Setting (Δφ)2 equjd to unity we obtain a simple estimate of the spin relaxation rate, T s - 1( 0 ) = β 2 (Α)τ / 7. From eq. (35) it follows does not influence the spin that an external magnetic field with Q <^r~ relaxation rate. In a strong magnetic field with fiL > r p l the angular rotation over the time rp does not exceed ^φ(2?) = fi(A)/i2L, since i2(A)/fi L is the l

L

*In general, the rate of electron spin relaxation due to hyperfine interaction of electron and nuclear spins is proportional to the interaction time T I N. T For localized electrons this time equals the time of electron transition to a neighbouring potential well. In the case of free electrons, T I NT has a much smaller value of the order of h/Ek. Therefore, for free electrons hyperfine interaction with lattice nuclei is unimportant. It should be also be noted here that due to hyperfine interaction of electrons with lattice nuclei the optical orientation of electron spins can be transferred to nuclear spins. This phenomenon is discussed in detail in the reviews chs. 2, 5 and 9 of the present volume.

Spin relaxation

95

maximum declination of precession axes from the external magnetic field direction. Then an estimate of the spin relaxation rate 1 / T S ( Z ? ) in strong magnetic field can be derived from the condition ( C 2 ( * ) / 0 > ) ( τ 5 ( Β ) / τ , ) = τ5(Β)Ω2(^τρ/Ω^

=1.

From this the following relation can be obtained to define the decrease of spin relaxation rate 1/Ts(B) with magnetic field B: 1

1

r,(B)

1 T s (0) 1 +

(VlTP)2'

Parallel to the Larmor precession there is another process causing weakening of the DP mechanism in an external magnetic field, namely, the electron orbital motion in a magnetic field. The role of this process has been theoreti­ cally studied in the limits of both classical and quantizing magnetic field (see, respectively, Ivchenko 1973 and Zakharchenya et al. 1976). As is well known, an external magnetic field Β causes a precession of electron momentum hk about the Β direction, with the cyclotron frequency i2c = eB/mcc. In a strong magnetic field, for which Ω0τρ»19 the average values of momentum components perpendicular to the direction of Β reduce to zero and only the momentum component parallel to the direction of Β does not vary. In this case, for a magnetic field Β directed along the (100) or (111) axes, an averaging of all components of the vector κ(Λ) in eq. (2) over all k directions k ± Β leads to the zero values, which means a complete suppression of the DP mechanism. For arbitrary orientations of the magnetic field B, the averaged values of the components of the vector fc(ft), in general, do not vanish. Nevertheless, the DP mechanism is still appreciably weakened. In A 3 B 5 compounds the ratio Ωι^/Ω(: = ^ηι&/2ηι\\^ usually rather small. In the two-band model C L / O c = [ η / ( 3 - t j ) ] - / w c / / w , a nd only at t j - > 1 and w i e « w the value of S L / i 2 c approaches 1/2. So orbital motion seems to be the main process responsible for suppression of the DP mechanism in an external magnetic field. In the general case, the rate of electron spin relaxation due to the DP mechanism should begin to decrease at Ω τ > 1, then saturate at Ω τ »1 and finally decrease to zero at Q T »l. A detailed calculation of 1 / T s decrease in a longitudinal magnetic field due to electron orbital motion has been performed in the limit of a classical magnetic field (Ivchenko 1973), i.e., at the condition that #i2 c < kBT. This author solved the kinetic equation for px(k)9 (26), with an additional term — ([kQc] V Pi(k)) included. The operator Hx{k) in eq. (26) was expanded over the spherical functions in the coordinate system fixed to the vector Β and a solution for px(k) was also searched for in the form of an expansion over spherical functions. This procedure was greatly simplified by the fact that in 0 ρ

L

£ ρ

k

p

96

G.E. Pikus and A.N. Titkov

strong magnetic field, when Ω τ > 1, the inequality i2 f » 1 is also satisfied, since f s / f » $ c / i 2 L . In this case, due to the Hanle effect all electron spin components perpendicular to Β decrease to zero and there remain only the spin components parallel to B. Then it follows from eq. (26) € ρ

1 %{B)

L s

— Χ 5(1-4Γ,+457;) 177,-1177; 1 1 1 15(7;-9Γ2)- — + -2 f s(0) 8 l + (2i2 c x*) 2 1 + ( G CT * )

+

3(1-4Γ!+9Γ2)

(49)

l + (3fi c r*) 2

Here l/f s (0) is defined by eq. (36), τ* by eqs. (28) and (29) with { = 3, Tx = 1/B\B2B2 + B2B2 + B2B2), T2 =l/B6(B2B2B2). In the case of a weak magnetic field, for which Ω τ* « : 1, the formula (49) can be simplified to 0

1

1 l-(4-^r )(fi T*) \ f s(0) 2

1

%(B)

C

(50)

Averaging of eq. (50) over the Maxwell distribution gives 1 Tt(*)

- { i - e ( i - t f 7 i ) ( o cT , ) 2} .

T s (0)

(51)

At Evk the spin relaxation rate in the absence of the magnetic field, 1/T s (0), is defined by eq. (37), and Θ=

9ττ 1 γ3

2

(3^ + 7/2)! (^+7/2)![(^-f3/2)!]:

When scattering by polar optical phonons dominates, ν = 1 / 2 , γ 3 = 6/41 and Θ = 5 7 τ ( χ ) 2 ; for scattering by ionized impurities ρ = 3/2, γ 3 = 6 and Θ = 777/12; for scattering by acoustic phonons (deformation-potential interaction) ν = 1/2, γ 3 = 1 and Θ = \m. It follows from eq. (49) that in a strong magnetic field, in accordance with the above qualitative consideration, the spin relaxation rate l/f s reduces to zero for field orientations along the (100) or (111) axes and remains finite for other orientations. For example, at Λ||(110), the rate 1/TS(B)B^O0 equals 15/32f s (0). 2.3.6. Spin relaxation of hot electrons Due to the energy dependence of the conduction band spin-splitting hQ(Ek) - El/2, the efficiency of the DP mechanism rapidly increases with electron energy. Therefore, the initial spin orientation of hot electrons, created far above the bottom of the conduction band, may be appreciably lost already during thermalization. In pure crystals, for which the DP mechanism has the

Spin relaxation

97

highest efficiency, thermalization of hot electrons down to the energy E « Λ ω ί 0 usually occurs via emission of LO phonons. In the case when &{E )T >\, an accurate consideration of spin relaxation at that stage of thermalization should take into account the loss of spin orientation at each cascade step in a similar way as has been done for hot excitons (Ivchenko et al. 1978, Pikus and Ivchenko 1982). However, the relaxation time r is usually short and, as a rule, Ω(ΕΜ)τρΧΟ<1. Further thermalization down to the energy E ~ k T occurs via emission of acoustic phonons, each phonon taking away a small portion of electron energy. At this stage of thermalization the decrease of electron energy is described by 0

K

PXO

p L O

x

B

(52)

at

TE'

where T is the energy relaxation time. Then the loss of spin orientation after a time dt = — rE-dEk/Ek is defined by E

d S — ? d r - S ^ d - £ *T

S

ktL

S

T

From here 5(£1) = 5 ( £ 0 ) e x p { - / ^ ^ } .

(53)

Further spin disorientation (relaxation) is defined by the well-known relation S = S(Ex)r/(T + T s ), where τ is the electron lifetime. Finally, the spin orienta­ tion of electrons near the bottom of the conduction band, taking into account the losses of orientation during thermalization, is defined by

When 3

+J

and W ( £ * )

=

(E0)\E0

we obtain S - S l B

t

)

Έ(ΕΟ)

7

^ - ^ R \ .

where 3+v-n

3kDT ' 2E

" 0

(55)

G.E. Pikus and A.N.

98

Titkov

and, at E0 » kBT, R = (3+v- n)~l. When the electron energy and momen­ tum relaxation are governed by short-range (deformation) interaction, η = 0.5 (Dienys and Pozhela 1971) and, as was pointed out in subsection 2.3.2, ν = - 0 . 5 , i.e., R - 0.5. When piezoelectric interaction predominates, η = - 0 . 5 and ν = 0.5, i.e., R - 0.25. If electron momentum relaxation is determined by scattering on impurities, ν equals 1.5. Then, for electron energy relaxation due to deformation interaction R = 0.25, and R = 0.2 if piezoelectric interaction is more important. In heavily doped p-type crystals an additional energy relaxation mechanism arises due to electron scattering on heavy holes with simultaneous excitation of holes from the heavy to the light valence sub-bands (D'yakonov et al. 1977). Shortening of the electron energy relaxation time τΕ in doped crystals leads to the decrease of spin orientation losses during thermalization. 2.4.

The Bir - Aronov - Pikus (BAP) mechanism

As was pointed out in the Introduction, electron scattering on holes can lead to spin-flip transitions due to exchange and annihilation interaction. Naturally, the EY process also can cause spin-flip transitions in this case. However, estimates show (Bir, Aronov and Pikus 1975) that for electron-hole scattering the EY mechanism is much less efficient than the BAP mechanism caused by exchange interaction. Exchange interaction may also dominate in the case of electron scattering on paramagnetic impurities. Further following the original work by Bir et al. (1975) we shall reproduce the main expressions for the rate of electron spin relaxation due to the BAP mechanism. 2.4.1. Scattering of electrons by free holes In accordance with eq. (9) the electron spin relaxation time due to scattering on holes in the Bohr approximation is defined by

^ - = 4 V E^(0)|V , (p+ ,p)|V (l-/, ,) w

B

mV

ma

9

/>

+

pq

Χβ(£Λ+

(56)

Here m and m' are the spin indices of an electron and a, a' are those of a hole in the initial and final states, respectively, ρ and $ p are the hole wave vector and energy, fp(Sp) is the hole distribution function, hq is the momentum transferred in the course of scattering, ψ(τ) is the wave function of electron-hole relative motion, r = re - r h being the position of the electron relative to the hole. In A 3 B 5 compounds as a rule me
Spin relaxation

99

hK, is practically equal to hp, and operator 2(p,p +q) depends only on the direction of p. Let us consider now the most characteristic cases. A. Holes are non-degenerate. In this case at me
= *

p + q

-£

the transferred energy

p

is smaller than Ek. Then, taking into account that Σρ/ρ = Np9 Np being the hole concentration, eq. (56) for the spin relaxation rate can be reduced to the form

()

7?V.-

1

0

57

V

B

where

/•di2„ ΑΑ'

B. Holes are degenerate with Fermi energy Ek, kBT. Then ω = qvFcos # + ^

,

# being the angle between ρ and

and /zco

/ d^/,(^)(l-/,(^ + M ) = '

1 - exp

-

In this case 1

3 1

h3

Npo\

e

xf^8[hv-hqv¥cosu

rd3q

r+

hwahw 1-exp

kBT

+ j^jS(hu>-Ek

+ Ek_q).

(59)

In the derivation of eq. (59) the operator 3>2{p) was averaged over all directions of ρ and then replaced by The main contribution to 1 / T s is determined by scattering on heavy holes, since their concentration consider-

100

G.E. Pikus and A.N. Titkov

ably exceeds that of light holes. Therefore, in the following we shall understand by mh the mass of heavy holes and under vF their Fermi velocity. In this case Np = ρΙ/3π

=

2

(2ητ^Ργ/2/3π2Η\

Now we shall consider the case when Ek > SF X m e / m h , the electron energy Ek being, as before, smaller than <$F. This inequality is satisfied at hole concentrations just above the critical concentration N0 of hole degeneration. This concentration range can be called "the range of fast electrons", since under this condition the electron velocity exceeds that of holes at the Fermi surface. In that range Ηω - hqvF - hkvF - EkvF/vk because at Ek > kBT the same time h2q2 1 2mh ηω

«: Ek,

the transferred momentum hq does not exceed 2hk. At

k ρ

Λ

For thermalized and not very hot electrons, with Ek
. JUL E ^-— I h

JP

*>k

(mh/mc)(kBT)2/S)F:

r

*mhJ

There Λω/(1 - e x p ( - hu/kT) = kBT. After integration over Λω and Ωρ in eq. (59) it is convenient to replace the integration over q by integration over k' = k-q, neglecting hu in the argument of the last δ-function. Then for the "fast" electron spin relaxation rate we obtain 1

3 vk

kBT

the transferred energy For hot electrons, with Ek » (mh/mc)(kBT)2/S>F, satisfies the relation hu - EkvF/vk » kBT. Then we can assume that (0 /

hu \

at/iw<0, at > 0.

and correspondingly substitute 0 for the lower limit of the integration over hco in eq. (59). Furthermore, taking into account that j

r+1 dμδ(hω-hqvFμ)=l _ 1

(ι—— *9*f \ 0

a\.hu hqvF,

it is convenient to substitute hqvF for the upper limit of the integration over Λ ω and then perform the integration over Λ ω and q, neglecting, as before, hoi

Spin relaxation

101

in the argument of the last δ-function in eq. (59). As a result, for hot electrons we shall obtain 7 - 7 ? £ ν Ts

τ0

vB

6

3

F

Β ·

( « )

?

At high hole concentrations, when E m /m , the hole velocity ex­ ceeds that of electrons. In this concentration range of "fast" holes q « k and Ηω - E
k

h

e

k

ρ

in the argument of the δ-function, Here we have also neglected h q /2m since in the case under consideration this term is smaller than h ω by a factor of m / m . Further, the integration over ω can be performed taking into account the last δ-function in eq. (59), after that it is again convenient to replace the integration over^R by that over k'. Finally, the following relation for the rate of electron spin relaxation in the case of "fast" holes can be obtained: 2

2

h

e

h

,

4

ί # ϋ ,

Λ

(62)

where

I

M

16

+ 8 Z

i

0

l-exp[-z(l-^)]-

At ζ <^1 / 1( z ) = TV 7 T 2+ i z .

At ζ » 1 7 1(z) = TV 7 T 2 + 0 . 1 z 2 .

Therefore 1 / T S - (kBT/£F)2

at E < k T, and l / r - {Ek/SF)2 k

B

s

at E » Α Γ. k

Β

Electron spin relaxation under strong hole scattering In the previous subsection the relations for the electron spin relaxation rate were obtained under the assumption that during the time T that an electron interaction with a hole the hole changes neither its direction of motion nor its spin orientation, i.e., T r , T , where r and T are, respectively, the hole momentum and spin relaxation times. The interaction time T can be esti­ mated from the relation T = λ / V , λ and ν being, respectively, the effective 2.4.2.

I N T

I N T

ph

S H

ph

SH

I N T

i nt

102

G.E. Pikus and A.N. Titkov

length of interaction and the velocity of electron-hole relative motion. For a short-range potential λ = l/q ~ l / / c , therefore in the case of "fast" electrons, when v = ve, ^m = h/Ek, and in the case of "fast" holes, when v = vv, r*t=l/kvF. In the latter case the inequality τ ^ « τ ρ [ ι reduces to A 7 h » l , /h = v¥rph being the hole free path length. From eqs. (60) and (62) it is seen that 1 / T S ~ T I N . T Indeed, the ratio of spin relaxation rates according to eqs. (60) and (62) is proportional to vF/vk, that is, to τ ^ , / τ ^ . r motion of holes through the area of interaction l/k has a For p nti diffusive character and the effective interaction time is given by the relation 2 2 Tj„ t = \ /3D = 1 /3Dk , where D = \v l is the coefficient of hole diffusion. In the case when T^ t > τ £ ρ that is, for 6meD > h, or more precisely for Ekrp/h < (lhk)2, and (lhk)2 < 1 we have «: Th.s Then the electron spin relaxation rate l / r s decreases in comparison with eq. (60) by a factor of η ^ / τ ^ ( = h /6meD and is given by

T>

λ-

F h

1 _ 2 h Ekpk Ts τ 0 meD <£> B

h

( Ek ττψ \k2 BT

.

(63)

where

+1 exp(zy)-l at z » l .

/T+7-1

/ 2 ( Ζ ) = 0.25/ΤΓ.

The above inequality > implies that equation (63) corresponds to the case of "fast" holes. Then from comparison of eqs. (62) and (63) it follows that the change of hole motion character from free to diffusive leads to an increase of the electron spin relaxation rate 1 / T s by a factor of τ - ^ / τ ^ = (/h/r)~\ i.e., proportionally to the increase of interaction time due to the strong hole momentum scattering. the interaction In the region Ek < h/rsh and r®t > T s h (or (lhk)2 < rph/rsh) time is controlled by the hole spin relaxation time T S H, and 1 / T s in comparison Ekrsh/h, with eq. (60) decreases by a factor of

1

_ 2 El

»k

( Ek \

where / ϊ ( ζ )

" 2 ϊ ) _ χ. eρx p( (ζzνy )) --l 1 ^

;

1 β

at z » l , 7 3 (z) = 0.4. Figure 1 shows the regions of validity of eqs. (60)-(64). The solid line ve = vF separates the region of "fast" electrons (a and b), and that of "fast"

Spin relaxation

103

B

me

eF

ft

1

U

a

c

Tph

d e ο Fig. 1. The regions of validity of eqs. (60)-(64): (a) "fast" electron region, eq. (60); (b) hot electron region, eq. (61); (c) "fast" hole region, eq. (62); (d) region of diffusive motion of holes, eq. (63); (e) region of rapid hole spin relaxation, eq. (64).

holes (c). In region (a), where the electrons are thermalized, the electron spin relaxation rate 1 / T s is defined by eq. (60). In the hot-electron region (b) 1 / r s is defined by eq. (61). In the "fast"-hole region (c) l / r s is given by eq. (62). The dashed lines 6meD = A, (lhk)2 = 1 and (lhk)2 = r / T border the region of diffusive motion (d), in which case 1 / T s is defined by eq. (63). The dashed lines a n c T T T T E/Jph/h = /7h/ sh * ('ιΛ) 2 = />h/ sh border the region of rapid hole spin relaxation (e), in which eq. (64) is valid. P H

SH

2.4.3. Spin relaxation due to scattering on bound holes In calculation of the matrix element of electron scattering by bound holes it is convenient to write the hole wave function in p-representation. Then the main contribution to the expansion of ΨΥϊ over e ' is given by the term ρ - ^ g j , where # B h is the hole Bohr radius on an acceptor. At me
(65)

104

G.E. Pikus and A.N.

Titkov

where NA is the acceptor concentration. The factor 1 / τ 0 differs from that in eq. (58) by substitution of (66) for ^ s 2 , g being the degeneracy of the ground acceptor state. 2.4.4. Calculation of effective constant of spin-spin interaction In cubic crystals, when D = / 3 Β (spherical approximation) and the constants A, Β and C have the same sign, the Bloch wave functions of the degenerate Γ 8 valence band transform like the spherical functions Y3/^ with L = + 3/2 for heavy holes and like the functions Y^/2 with / , = ± 1/2 for light holes, the ζ axis being parallel to /?. In the assumed spherical limit, averaging over all possible directions of ρ is equivalent to averaging over directions of /. As a result, the exchange contribution to the operator defined by eq. (58), proves to be the same for both heavy and light holes and equals (67) For holes bound on acceptors the exchange contribution to S s 2 is larger and, in accordance with eq. (66), equals (68) Annihilation interaction in the spherical limit gives a contribution to 3)1 only for the light holes, which is £2

^s,ann

=1Λ2

(69)

41η„,

where 4 a n n is the longitudinal-transverse splitting of the optically active 7 = 1 level of the Γ 8 Χ Γ 7 exciton. When a deviation from the spherical symmetry is taken into account, there arises also an annihilation contribution to 3)^ for the heavy holes. Its value can be estimated by the inequality D

A4

For a hole bound on an acceptor the annihilation contribution to <@s2 equals (70) Usually the annihilation contribution is much smaller than the exchange one. If these contributions are comparable, the cross-terms proportional to Aex X 4 a n n should be also taken into account in <®s2.

Spin relaxation 2.4.5. Role of Debye screening When screening is negligible and ve» defined by the Sommerfeld factor, |ø (0)|2 =

í 1_βιρ

105

vh, the modulus squared |ψ(0)| 2 is

(-τ).

(71)

This formula is valid when the Debye screening where κ = kaB = (Ek/EB)l/2. length L D e b exceeds both aB and k~l. With increasing hole concentration Np the Debye length L D e b decreases, which results in the progressive reduction of I ψ (0)| 2 down to unity. However, when screening by holes becomes important, i.e. at LOcbk ~l and L D e b « aB, accurate calculation of |ψ(0)| 2 is complicated by the fact that the inequality L3OcbNp » 1 does not hold any more and, therefore, the screening cannot be considered static. Theoretical estimates of I ψ (0)| 2 in this case have been attempted by Bir et al. (1974). It is important to note that both the theoretical estimates and the experimental data (Rogachev and Sablina 1968) show that |ψ(0)| 2 reaches unity only at rather high hole concentrations far exceeding the critical value N0 at which the holes become degenerate. In the case of "fast" holes, i.e., at (mh/me)kBT, |ψ(0)| 2 = 1 . When an electron is scattered by a hole localized on an acceptor, |ψ(0)| 2 also equals unity, since in this case the acceptor, as a whole, is neutral and kaBh
106

G.E. Pikus and A.N.

Titkov

doped crystals this limitation does not exist and reabsorption may be im­ portant, which, indeed, has been found experimentally (Alferov et al. 1974). Calculation of the rate of spin relaxation due to reabsorption has been attempted by Kleinman and Miller (1981). However, the results obtained by these authors practically do not differ from the well known expression for radiative recombination time r in doped crystals, i.e., for the electron lifetime determined by electron-hole recombination with emission of photons. In crystals where the holes are strongly degenerate, i.e., < f F » k h T , the radiative lifetime of thermalized electrons is given by r ad

1 ^ T

e2P2Egn

8

3

RAD

(72)

m2hV

Here η is the refractive index, c is the velocity of light, Ρ is the constant in formula (1) defined by the relation Ρ = (S\PZ\Z). Equation (72) takes into account transitions to both heavy-hole and light-hole valence sub-bands which give equal contributions to l A r a d. In crystals where the holes are non-degener­ ate, 1 / T decreases by a factor of fp(#p) of which the value at δ ~ kT(mc/mh) practically equals fp(0) = \Np{2irh2/mhkBTY/2. The expressions for 1 / T s obtained by Kleinman and Miller (1981) differ from corresponding relations for 1 / T only by a numerical factor of 11/12 for degenerate and of 11/24 for non-degenerate holes. At the same time, as it was mentioned above, in degenerate crystals the reabsorption and, correspondingly, the related losses of electron spin orientation should be unimportant. The simplified method used by Kleinman and Miller (1981) for calculation of 1 / T s does not take into account also the variations of both electron orientation and recombination radiation polarization with the distance from the crystal surface, when reabsorption is effective. It is obvious that reabsorp­ tion of recombination radiation is more important in the volume of the crystal than at its surface. Therefore the degree of both electron orientation and recombination radiation polarization should decrease inside the crystal. In these conditions, for calculation of the polarization degree of recombination radiation emerging out of the crystal one should firstly find the distributions of recombination radiation and its polarization at different distances from the crystal surface. In the case of optical orientation of free electrons this problem has not been solved yet. A similar problem has been studied earlier by Ivchenko et al. (1980) for the case of resonant excitation of excitons experiencing elastic scattering. It was proved that using the Chandrasekhar (1950) method, modified by taking account of light reflection from internal crystal surfaces, it is possible to find an expression for the polarization of emerging recombination radiation without a precise definition of its distribution in the crystal. These calculations show that at low values of quantum yield ώ 0 = τ 0 / ( τ 0 + T R A ) D, τ 0 being the non-radia­ tive recombination time, the effective spin relaxation time exceeds r . On the R A D

R A D

rad

Spin relaxation

107

other hand, when
2.6. Relative efficiencies of the Elliott -Yafet, D'yakonov-Perel9 Bir-Aronov-Pikus mechanisms

and

Here we would like to discuss the main differences distinguishing the EY, DP and BAP spin relaxation mechanisms. The EY and DP mechanisms principally differ by their opposite dependences on τ ρ : for the former mechanism 1 / T S ~ T p _ 1, whereas for the latter one 1 / τ ~ τ . Therefore, when scattering on impurities predominates, the efficiency of the EY mechanism increases and that of the DP mechanism decreases with increasing impurity concentration. The efficiency of the BAP mechanism, as in the case of the EY mechanism, also increases with acceptor concentration. However, above the critical con­ centration TVQ, leading to metalization of acceptors, the increase of 1 / T s slows down mainly due to the screening of |ψ(0)| 2 by the delocalized holes. At heavy doping, when the holes are strongly degenerate and |ψ(0)| 2 « 1 , in the case of "fast" electrons, according to eq. (60) the rate 1 / T s increases proportionally to N . In the contrary case of "fast" holes, according to eq. (62), the value of 1 / T s does not depend on Np at all. For electron scattering by phonons the spin relaxation rate 1 / T s for both the EY and DP mechanisms does not depend on the impurity concentration and rapidly increases with temperature. As a rule, the dependences of 1 / T s on temperature for these mechanisms are different which gives an additional possibility for the EY and DP mechanisms to be distinguished. For the BAP mechanism the temperature dependence of 1 / T s in moderately doped crystals is mainly determined by the variation with temperature of |ψ(0)| 4 and by the redistribution of holes between acceptors and valence band states. Both the EY and DP mechanisms are related to the spin-orbit splitting of the valence band. It is interesting to compare their efficiencies in the same conditions. For example, let us consider the case when scattering on polar optical or piezoelectrical phonons predominates. Then, according to eqs. (16) and (37) 8

ρ

l/3

p

(73)

G.E. Pikus and A.N. Titkov

108

Here we have substituted for a c as expressed by eq. (2). Since h/τρ < kBT <^ £ g , it follows from eq. (73) that 1 / T s E Y can exceed 1 / T s d p only in small-gap semiconductors with w i e « w i c v . Let us also compare the efficiencies of the EY and BAP mechanisms in doped crystals when scattering on ionized acceptors and delocalized holes predominates. For non-degenerate holes, assuming the number of scattering centers to be equal to 2N A , in accordance with eqs. (15), (57) and (58) we get 2

's,BAP T

s,EY

F 128 27 ®,Ea

Ιø(ο)Γ * (/0, 2

where Φ2(β)

(74)

=2

2+ β β

2β 2+ β

ln(H-iB)-

For degenerate holes, which do not give any appreciable contribution to momentum scattering, the relation T S B A/ TP s E Y according to eq. (60) differs from eq. (74) by a factor of S /3k T. Then it follows that the EY mechanism can be more efficient than the BAP mechanism in materials with large spin-orbit splitting Δ and small energy gap Eg. An increase of hole concentra­ tion favours the relative reinforcement of the EY mechanism and, in general, may cause the replacement of the BAP mechanism by the EY mechanism, which should manifest itself in a change of the dependence of 1 / T s on Np. >

F

B

3. Experimental study of free carrier spin relaxation in compounds

AB 3

5

In recent years a number of studies on optical orientation has been devoted to the problem of free carrier spin relaxation in A 3 B 5 compounds. Benoit a la Guillaume et al. (1974) have reported the observation of the EY mechanism in the case of spin relaxation of degenerate electrons in InSb crystals in which, according to the theoretical estimates, the EY mechanism should have the maximum efficiency. Manifestations of the DP mechanism for spin relaxation of electrons in moderately doped GaAlAs and GaAs crystals at high tempera­ tures has been observed by Clark et al. (1975, 1976, respectively). These authors, however, could not explain the temperature variation of the spin relaxation time at low temperatures where, as was shown later by Safarov and Titkov (1980), relaxation via the BAP mechanism predominates. The BAP and DP mechanisms have been studied in detail by Aronov et al. (1979, 1983), Sakharov et al. (1981), and Maruschak et al. (1983b), where the electron spin relaxation time in GaAs and GaSb crystals was measured in wide ranges of temperature and doping. The efficiencies of the EY, DP and BAP mechanisms

Spin relaxation

109

in some other A 3 B 5 compounds have been estimated by Fishman and Lampel (1977). The dominant role of the DP mechanism in the spin relaxation of hot electrons has been demonstrated in GaAs crystals by Ekimov and Safarov (1971) and Dzjioev et al. (1971), and in GaSb crystals by Zakharchenya et al. (1976). Spin relaxation in uniaxially stressed GaAs crystals of n- and p-type was studied by Titkov et al. (1978) and D'yakonov et al. (1982), respectively. Prior to setting forth the main results of these experiments we shall give an outline of the optical orientation method of measuring the time T s and τ. 3.1. Optical orientation method In optical orientation experiments the degree of circular polarization of luminescence ρ is given by

p = p (l + T / T r \ 0

(75)

s

where p 0 is the maximum possible degree of polarization determined, in the absence of spin relaxation, by the selection rules for interband optical transi­ tions. For Γ 8 -* ΓΊ transitions p 0 equals 0.25. A magnetic field Β transverse to the direction of light propagation causes depolarization of the luminescence, because of the precession of electron spins in the magnetic field (the Hanle effect). The decrease in the degree of polarization is expressed by

ρ(Β) = ρ{ΐ + ΩΐΤ η-\

(76)

$

where i2 L is the Larmor frequency in the transverse field B, T' = T s + τ - 1 . Seen from eqs. (75) and (76) is the possibility to determine T s and τ indepen­ dently by measuring the degree of luminescence polarization ρ and observing its decrease in the transverse magnetic field p(B). The time Ts is usually determined from the field value Bl/2 for which the initial polarization is halved: p(2? 1 / 2) = p(0)/2. The electron spin relaxation time T s and lifetime τ are then 1

Po p0-p(0) Po P(0)

-1

(77) ft

(78)

For the first time this method was used in the study of times T s and τ as functions of temperature in moderately doped GaAlAs crystals (Garbuzov et al. 1971). The data obtained in this work are presented in fig. 2a. Figure 2b shows the variation of T s and τ with doping level in GaSb crystals (Aronov et al. 1979, Titkov et al. 1981). The data presented in fig. 2 demonstrate that the optical orientation method is a universal tool for determining small values of T S

G.E. Pikus and A.N.

110

Titkov

GaAlAs

CO

b

NA = 2x10 1 7CRRR 3

10"

ο° _ ·

μ ο

•

ο ο

•

ο

····

•

o ο

10

ο - Ts 100

GaS b

ο ο 400

ο-ô5 «- ô

(Α)

- Ô=4.2 • ι

m

10 18

Ô, Ê

ι

I

*

10"

Í

Á

(cm" 3

Fig. 2. Variation of spin relaxation time T s and lifetime τ as a function of: (a) temperature in moderately doped GaAlAs (after Garbuzov et al. 1971); (b) doping level in GaSb (after Aronov et al. 1979, and Titkov et al. 1981).

and τ in wide ranges of temperature and doping. It is essential that very short times, down to 1 0 - 1 1 s, are measured in steady-state conditions in a relatively weak applied magnetic field. These field values can be easily estimated. The most unfavourable conditions arise in the crystals with the smallest modulus of electron g-factor. Among A 3 B 5 compounds the GaAs crystals possess the smallest modulus of electron g-factor, |g e | = 0.44 (Weisbuch and Hermann 1977). With this value of the g-factor and at the condition p(0) <^ p 0 , which is usually fulfilled at temperatures and doping levels when T s is small, we get from eq. (77) T S = 2,4X10" Βϊ/ ιι

2

( T s in s when Bl/2

in T).

Thus, to measure T s values in the order 1 0 " 1 1 s we need magnetic fields of several tesla. The situation may be somewhat less favourable in the case of some solid solutions of A 3 B 5 compounds remarkable for their small values of the electron g-factor. 3.2. Spin relaxation in p-type InSb (Elliott-Yafet

mechanism)

Benoit a la Guillaume et al. (1974) have studied spin relaxation of spinoriented photoelectrons in InSb crystals with an acceptor concentration of 2 X 1 0 1 5 c m - 3 at Γ = 1.3 Κ. Excitation with a CO-laser made it possible to

Spin relaxation

111

achieve high concentrations of photoelectrons, n. The electron distribution was found degenerate and could be characterized by two quasi-Fermi levels EF Λ and EF2 for electrons with opposite spin orientations. The degree of electron spin orientation in the experiment was only about 1%. For this reason the energies EFl and EF2 differed little and could be characterized by the average value EF which varied with the change of excitation intensity in the range of 2-7 meV, i.e. corresponded to the increase of photoelectron concentration η from 0.7XlO 1 5 c m - 3 to 5 X 1 0 1 5 c m - 3 . Because of the strong degeneracy of electrons at Γ = 1.3 Κ, only electrons with an energy near EF could experience spin relaxation. Figure 3 shows the dependence of spin relaxation rate 1 / T s on EF obtained by Benoit a la Guillaume et al. (1974). The values of 1 / T s are seen to vary in proportion to EF. It was supposed in this work that at low temperature and relatively small concentration of ionized acceptors electron momentum scattering is mostly due to scattering by photoexcited holes avail­ able in the same concentration as electrons. At kBT
Ο

I

2

0

235

240

E F+ E g ( m e V ) Fig. 3. Square root of spin relaxation rate of degenerate electrons versus electron Fermi level in InSb at Τ = 1.3 Κ (after Benoit a la Guillaume et al. 1974).

112

G.E. Pikus and A.N. Titkov

In these conditions 1/τ ~ NpEk3/2 = const., since the free hole concentra­ tion Np, which equals the electron concentration n, is proportional to EF/2. Because of their large effective mass, the holes remain practically non-degener­ ate. Therefore, from eq. (79) we get 1 / T S ~ EF in agreement with the experi­ mental data. It is worth noting here that the rate of spin relaxation via the DP mechanism at τ ρ = const., according to eq. (36), should be proportional to EF and, thus, that the DP mechanism does not explain the observed dependence of 1 / T s on EF. Making use of the Brooks-Herring formula for scattering of electrons by charged impurities and of the data on electron mobility in η-type InSb crystals doped to similar levels, the following estimate was found for the rate of electron momentum scattering: 1 / T / 7 - 2 X 1 0 1 2 s - 1 . The value of 1 / T s calcu­ lated from eq. (79) [which coincides with eq. (10) in the paper by Chazalviel (1975)] for the above estimate of τρ and at EF = 3 meV equals 10 8 s _ 1 , which is near enough to the experimental value of 4 X 1 0 8 s " 1 at EF = 3 meV (see fig. 3). It can be noted here that the BAP mechanism at Np ~ E\/2 and |ψ(0)| 2 = 1 also gives the relation 1 / T S - EF. From eqs. (57) and (58) in the case of scattering of degenerate electrons by non-degenerate holes at kBT <^ EF and EFΛ — EF 2 ^ EF we obtain 1 \ τ

ΛΑΡ

'F 4

P \£ B

T

ln(l + / ? ) -

β

1 + /?

(80)

In our case l/rp is given by the Brooks-Herring formula. The experimental values of 1 / T s lead to 2% = 3 X 1 0 " 3 meV or Z i ex = 10 2 meV which, however, are somewhat in excess of the estimates for InSb. Moreover, it should be taken into account that, with the BAP mechanism dominating, a considerable contribution to 1 / T s may have come from scattering on holes bound to neutral acceptors which are present in a concentration comparable to photohole concentration. This would have led to a relationship essentially different from the experimentally observed dependence 1 / T S ~ EF. Thus, it should be con­ cluded that spin relaxation of degenerate electrons in InSb is indeed caused by the EY mechanism. 3.3. Spin relaxation in p-type GaAs and GaSb 3.3.1. The D'yakonov-PereT mechanism in GaAs The first experimental observation of the DP mechanism was reported by Clark et al. (1976). The authors studied the spin relaxation of conduction electrons in GaAs crystals with acceptor concentration7V A = 2 X 1 0 1 7 c m - 3 .

Spin relaxation

113

However, the interference of other spin relaxation mechanisms did not allow them to study the new mechanism which revealed itself at high temperatures, above 150 K. This was attained later in the study of purer GaAs crystals with N A = 4 x l 0 1 6 c m - 3 (Maruschak et al. 1983b). The results of this study are presented in fig. 4 which shows the variation of 1 / T s with temperature for two GaAs crystals having almost the same acceptor concentration but different degree of compensation. The compensation of acceptors afforded the possibility of studying crystals with substantially different carrier mobilities. The temperature dependences of the hole mobility in the crystals under study, which reflect variations with temperature of the carrier momentum relaxation time Tp, are also given in fig. 4.

Fig. 4. Variation of spin relaxation rate 1 / T s as a function of temperature for two GaAs samples (1,2) having almost the same acceptor concentration NA = 4 X 1 0 1 6 c m - 3 but different compensation degrees (after Maruschak et al. 1983b). Also shown are the temperature dependences of hole mobility for both samples.

114

G.E. Pikus and A.N. Titkov

The onset of the DP mechanism shows as a sharp increase in the spin relaxation rate at Τ > 30-40 Κ. According to eq. (37), the spin relaxation rate 1 / T s for the DP mechanism should rise in proportion to Τ τ . When the power dependence of τρ on temperature is taken into account, we obtain 1 / T S ~ T3 + n where η varies with temperature. The temperature dependence of η is caused by the variation with temperature of both the relative efficiencies of electron scattering by impurities and phonons, and the number of thermally ionized acceptors. The data in fig. 4, indeed, show that in the temperature range 40 < Τ < 11 Κ, where τρ increases, i.e. η > 0, the slope of the experimental curves exceeds 3, whereas, at Τ > 11 Κ where τρ decreases the slope becomes smaller than 3. The dependence of 1 / T s on the degree of acceptor compensa­ tion provides unambiguous evidence of the domination of the DP mechanism at Τ > 30-40 Κ: the decrease of τρ due to the more intensive electron scattering causes the efficiency of the DP mechanism also to decrease. At Τ < 30-40 Κ the DP mechanism no longer explains the experimental data. The increase of 1 / T s in this temperature range is the consequence of the localization of electrons at low temperatures and of the manifestation in new conditions of other spin relaxation mechanisms. As was mentioned in subsec­ tion 2.3.5, spin relaxation of localized electrons may be caused by hyperfine interaction of electron spins with magnetic momenta of lattice nuclei (Berkovits et al. 1973, D'yakonov and Perel' 1973). Exchange interaction with localized holes may also play an important role (Ivchenko and Takunov, 1976) in a similar way as occurs in the case of excitons (Pikus and Bir, 1974). 3

ρ

3.3.2. Suppression of spin relaxation in longitudinal magnetic field A suppression of the DP mechanism in a longitudinal magnetic field has been found experimentally in both quantizing and classical magnetic fields. The effect of a quantizing magnetic field has been observed by Zakharchenya et al. (1976) in the case of conduction electron spin relaxation in GaSb crystals at Τ = 4.2 Κ. In accordance with the theoretical predictions, the effect of the magnetic field was found to depend on the field orientation relative to the principal crystallographic axes. However, the results obtained had a qualitative character only and no detailed comparison with theory was given. The effect of a classical magnetic field has been studied by Maruschak et al. (1983a) in the case of spin relaxation of thermalized electrons in GaAs crystals at 77 K. The decrease of spin relaxation rate 1 / T s with a magnetic field in samples with different electron mobilities are shown in fig. 5a. In accordance with the theory, the effect of the magnetic field is more pronounced in samples with higher electron mobilities, i.e. with larger values of rp. The authors have also found the dependence of the effect of the magnetic field on its orientation for samples with the same electron mobility. It is evident from fig. 5b that the most rapid decrease of 1 / T s occurs for the magnetic field orientation along the

Spin relaxation

115

 (Ô)

Β (Τ)

Fig. 5. Decrease of spin relaxation rate 1 / T s with increasing magnetic field in GaAs with NA = 4 X 1 0 1 6 c m - 3 at Τ = 11 Κ (after Maruschak et al. 1983a): (a) effect of magnetic field for GaAs samples with different electron mobility; (b) effect of magnetic field for field orientation along different crystallographic axes. Solid curves are theoretical dependences calculated using eq. (81) describing the suppression of the D P mechanism due to electron orbital motion.

(100) axis. The solid curves in figs. 5a and 5b present the theoretical depen­ dences obtained according to the formula T s ( 0)

rs(B)

= \ {15(7\ - 9 Γ 2 ) + 5(1 - 4 7 \ + 8

45T2)F{y)

+ (177\ - \ \ 1 T 2 ) F { 4 y ) + 3(1 - 4Τλ + 9T2)F(9y ) },

(81)

with 11 ί°°Α /·«> *^ x 5×eÑ< xp(-jc) ί ο ρ\ A —' > y = A 3 48 yx The formula (81) has been derived from eq. (49) which was averaged over the Maxwell distribution for the actual case of electron scattering by ionized impurities (fp - Ek/2). The values of rp which permitted to achieve the best fit between experimental data and theory are also presented in figs. 5a and 5b. The good fit of the experimental data by theoretical dependences shows that, as was predicted by Ivchenko (1973), the main cause of the suppression of the DP mechanism in a magnetic field is the cyclotron precession of electron momentum about the magnetic field. Τ7ί \

Ω

τ

2

G.E. Pikus and A.N. Titkov

116

It is worth noting here, that the revealed effect of 1 / T s decreasing in a magnetic field offers a new possibility to measure the momentum relaxation time rp of non-equilibrium electrons in moderately doped p-type crystals. Maruschak et al. (1983b) have used this possibility to find a value of 0.07 for the constant a c with the help of eq. (37) for spin relaxation rate 1/T s (0). 3.3.3. The Bir-Aronov-Pikus mechanism in GaAs and GaSb In GaAs crystals the BAP mechanism reveals itself at the doping level NA > 1 0 1 7 c m ~ 3 , at first in the low-temperature range (above the temperature at which electrons become delocalized). As the acceptor concentration increases the BAP mechanism begins to dominate at progressively higher temperatures.

to ι

GaA s

40

10

10

-

O

-

-

Δ -

-

X

-

/

2

i

>(ST,A )

Ο

° , 1

oX Ο

3(C)

0

Ο

X/

Δ /

^x

/

V

'

y

BAP

/

7

ι 1

/ A -A

AP

/

1

/

1 DP 10c

/ 1

1

40

,

i l l 100

ι

a

ι 400

T.K Fig. 6. Variation of spin relaxation rate 1 / T s as function of temperature: (a) in moderately doped and (1) 7VA = 3 . 5 X 1 0 1/ c m " J (after Clark et al. 1976, GaAs with (2,3) 7 V A = 2 . 2 x l 0 1 Safarov and Titkov 1980, and Aronov et al. 1983); (b) in moderately doped GaSb with N A = 3 x l 0 1 7 c m - 3 (after Aronov et al. 1983). Solid and dashed curves are theoretical dependences for the BAP and D P mechanisms calculated using eqs. (82) and (37), respectively.

Spin relaxation

111

τ,κ Fig. 6b.

Figure 6a shows the variation with temperature of l / r s for two GaAs crystals having ]VA = 2 x l 0 1 7 c m " 3 and iVA = 3, 5 X 1 0 1 7 cm 3, i.e., with the doping level about an order of magnitude higher than in fig. 4 (Clark et al. 1976, Safarov and Titkov 1980, Aronov et al. 1983). The dashed line in fig. 6a depicts the behaviour of 1 / T s for the DP mechanism according to eq. (37). It is evident that this mechanism gives a good description of experimental data only at high temperatures, Γ > 1 5 0 Κ. In the region of lower temperatures an additional mechanism of spin relaxation is seen to exert its influence. This becomes stronger with increasing acceptor concentration. The experimental dependence of 1 / T s in this temperature range is adequately described by the BAP mechanism for electron scattering by both free holes and holes bound on acceptors. In this case, the electron spin relaxation rate is defined by the sum of eqs. (57) and (65). Also taking into account eqs. (67) and (68) we get 4

7 = γ("α4)

ΧΑ

~ Κ

(82)

118

G.E. Pikus and A.N.

Titkov

The curves of 1 / T s according to eq. (82) are presented in fig. 6a as the solid lines. Since in the crystals under study screening plays but a minor role, the values of |ψ(0)| 4 may be calculated from eq. (71). The theoretical curves give a good fit to the experimental data at a ACX value of 4,7 X 1 0 " 5 eV. The observed temperature variation of the spin relaxation rate is a combined effect of three factors: of the increase with temperature of (i) electron velocity vk and (ii) degree of acceptor ionization Np/NA, and (hi) of the decrease with temperature of the Sommerfeld factor |ψ(0)| 2 . A specific feature of the temperature dependence of 1 / T s is its non-monotonic character: in the region of acceptor depletion the rapid decrease of the Sommerfeld factor with rising temperature must lead to a slower spin relaxation and even to a decrease of 1 / T s . The experimental dependences of 1 / T s in fig. 6a exhibit only saturation at Τ > 100 Κ, the expected decrease of 1 / T s at higher temperatures is smeared by the DP mechanism. The BAP mechanism was also identified in the study of spin relaxation in GaSb crystals. Figure 6b shows the temperature variation of 1 / T s for GaSb crystal with NA = 2.8 Χ 1 0 1 7 c m " 3 . As with GaAs crystals, the BAP mechanism was found to predominate in the low temperature range and the DP mecha­ nism predominates at high temperatures. Use of eqs. (37) and (82) for description of the experimental data for GaSb yields the values of 0.10 and 2.4X10" 5 eV for a c and ^ l e x , respectively. The enhancement of the BAP mechanism and simultaneous suppression of the DP mechanism at higher doping results, in GaAs and GaSb crystals with NA > 1 0 1 8 c m " 3 , in the domination of the BAP mechanism in a progressively wider temperature interval up to room temperature. At high acceptor con­ centrations the temperature dependence of 1 / T s for the BAP mechanism changes its character. This is due to the derealization of holes brought about by the metalization of acceptors at J V A « 1 . 2 x l 0 1 8 c m " 3 in GaSb and at NA ^ 4 X 1 0 1 8 c m " 3 in GaAs. As the Fermi level lowers into the valence band the holes become degenerate and, because of the limitation imposed by the Pauli principle, only holes with energy near the Fermi level may cause electron spin relaxation. The temperature variation of the electron spin relaxation rate in degenerate GaAs and GaSb crystals is shown in fig. 7 (Safarov and Titkov 1980, Kleinman and Miller 1981, Aronov et al. 1983). In a wide temperature interval the rate rises with temperature as Γ 3 / 2 . This behaviour is consistent with the theoretical prediction, eq. (60), for the case when electrons relaxing on degenerate holes are "fast", i.e. ve > vh. In GaAs and GaSb even under strong degeneracy, because of substantial difference in the effective masses of electrons and holes, the condition vc > vh is already satisfied at relatively low temperatures, namely, above 10-20 Κ for GaSb crystals and above 20-35 Κ for GaAs crystals. A feature of interest in the 1 / T s curves in fig. 7 is their saturation in the low temperature region. This effect is most pronounced in GaAs crystals, in which

Spin relaxation

I

1

L_I_J

10

1

1

1

119

Ι

1

100

I

Ι

IL

400 T,K

Fig. 7. Temperature dependence of spin relaxation rate 1 / T s in degenerate GaSb with (1) A A = 1 . 8 x l 0 1 8c m - 3 and (2) NA = 6.5 X l O 1 8 c m - 3, and GaAs with (1,3) NA = 5 X l O 1 8 c m " 3 and (2) NA = 4 X l O 1 9 c m - 3. (After Safarov and Titkov 1980 (ST), Kleinman and Miller 1981 (KM), and Aronov et al. 1983 (A).)

saturation starts at 40-60 K. It should be noted that the weak dependence of 1 / T s on Τ at low temperatures cannot be explained by the assumption that at these temperatures the transition to the region of "fast" holes takes place: eq. (62) for spin relaxation rate, which is relevant in this case, contains an even stronger dependence, 1 / T S - T2. We think that the observed behaviour of 1 / T S has its cause in the incomplete electron thermalization in the conduction band during the electron lifetime. The crystals studied had rather short lifetimes of 2 - 3 X l O - 1 0 s for GaAs and 4 - 5 X l O - 1 0 s for GaSb. Besides, the lifetimes were weakly influenced by doping level and temperature. Thermalization of electrons having kinetic energy less than the energy of optical phonons may proceed via scattering on acoustic phonons or on degenerate holes. In the crystals studied the scattering by acoustic phonons cannot provide the effective thermalization channel as the corresponding energy relaxation times τΕ p h of the order 1 0 " 9 - 1 0 " 8 s (Levinson and Levinskii 1976) are far in excess of the

G.E. Pikus and A.N.

120

Titkov

lifetimes. The estimates of energy relaxation times rEh for electron scattering by degenerate holes made by Aronov (1983) show that for electrons with energy equivalent to 10 Κ the times rEh are at least an order of magnitude greater than the lifetimes. Thus, at Τ = 10 Κ the electrons in degenerate GaAs and GaSb crystals are certain not to reach the lattice temperature, so this must be the reason of saturation of 1 / T s at low temperatures. Studies of spin relaxation processes might be used in investigations of thermalization of non-equilibrium electrons. For example, if we compare the time T s determined from the curves in fig. 7 and that calculated for hot electrons according to eq. (61), we shall get an expression for the electron temperature: (83) Here rs(T) is the spin relaxation time for thermalized electrons defined by eq. (60), which can be taken from the experimental curves at T> 60 K, r s (T e ) is the spin relaxation time for hot electrons in the region of the plateau in fig. 7. In deriving eq. (83) it was assumed that in the region of strong screening Iψ(0)| does not depend on Ek. At a lattice temperature of 10 Κ an estimate of the electron temperature from eq. (83) gives Te = 40 Κ for GaAs crystals with NA = 3.8Χ10 1 9 c m " 3 and 7 e = 2 9 K a t iVA = 5 x l 0 1 8 c m " 3 . Another explanation of the same effect has been proposed by Kleinman and Miller (1981) and Miller et al. (1981). These authors have attempted to connect the saturation of the temperature dependence of 1 / T s in degenerate crystals with the decrease of electron orientation degree due to reabsorption of recom­ bination radiation. However, this explanation is not satisfactory as at nearhelium temperatures the reabsorption effect in degenerate crystals should not be of much importance. The lack of an elaborate theoretical description of the proposed relaxation process impairs the possibilities for its experimental identification. It seems, however, that this process is more likely to be effective in non-degenerate crystals where reabsorption effects are stronger and the BAP mechanism is less important. 2

3.3.4. Dependence of spin relaxation rate on acceptor concentration The concentration dependences of the spin relaxation rate 1 / T s in GaAs and GaSb crystals are shown in figs. 8 and 9, respectively. The dependences are given at two different temperatures for each crystal. The temperature variations of 1 / T s considered in the previous subsection demonstrate that in the entire concentration range of 1 0 1 7 - 1 0 2 0 c m " 3 covered in figs. 8 and 9 electron spin relaxation at the lower of two indicated temperatures is caused by the BAP mechanism. At higher temperatures and low doping levels spin relaxation

Spin relaxation

111

proceeds through the DP mechanism which in the range of acceptor degenera­ tion (NA> N0) is replaced again by the BAP mechanism. A point of interest here is a rather weak dependence of the spin relaxation rate 1 / T s on doping level: a rise of acceptor concentration by three orders of magnitude results in an increase of the relaxation rate by less than an order of magnitude. The low-temperature data in figs. 8 and 9 reveal different behaviour of the BAP mechanism in different doping ranges. At concentrations NA < N0 the spin relaxation rate increases in proportion to the acceptor concentration NA, whereas at NA = N0 the rate 1 / T s almost ceases to rise, the latter effect being more pronounced in GaAs crystals. Still higher doping leads to a new increase in 1 / T s which now is proportional to NA/3. At NA < N0 the electron spin relaxation rate in the BAP mechanism is defined by eq. (82). In GaSb crystals at 7 = 15 Κ practically all the holes are localized on acceptors and eq. (82) leads to the experimentally observed dependence 1 / T S ~ NA. The same dependence follows from eq. (82) for GaAs crystals at 77 Κ when the acceptors are partly ionized. This is explained by the fact that the degree of acceptor ionization is almost independent of the doping level: the increase with doping of the number of acceptor states is compensated by the accompanying decrease of the acceptor binding energy. Simultaneously, the increase of free-hole concentration leads to a smaller Sommerfeld factor ΙΨ(0)| 2· The saturation of the dependence of 1 / T s on NA at NA — N0 is explained by the rapid decrease of the Sommerfeld factor |ψ(0)| 2 as a result of screening,

; GaAs - oQ o c ο

ο.

ο ο

/*·

:

10 10

ο °

φ 9 β

9

Ν„

9

71

ο

,

,

, I

10' 8

Ο

= 300Κ = 77Κ 10" 10 ο -Ô ï

-Ô

I

.

é

I -3

I

:

Ν (cm )

Fig. 8. Dependence of spin relaxation rate 1 / T s on acceptor concentration in doped GaAs at Τ =11 and 300 Κ (after Aronov et al. 1983).

G.E. Pikus and A.N. Titkov

122

which almost totally compensates the increase of l / r with rising acceptor concentration. For instance, in GaAs at 77 Κ and N « 5 Χ 1 0 1 7 c m when the screening is not very important, eq. (71) gives |ψ(0)| 4 = 20. At concentra­ tion i V A - 1 0 1 9 c m ~ 3 the factor |ψ(0)| 2 is approximately unity, and, hence, in this concentration range the product |ψ(0)| 4 X iV is practically constant. The smaller extent of the region of constant 1 / T s for GaSb crystals finds an explanation in a reduced significance of the Sommerfeld factor. At Γ = 15 Κ and N < 1 0 c m scattering of electrons occurs predominantly on localized holes, for which | ψ ( 0 ) | 2 = 1 . In the concentration range i V A > 1 0 1 8 c m , because of strong screening by degenerate holes, the factor |ψ(0)| 2 is again close to unity: as follows from the papers by Rogachev and Sablina (1968) and Bir et al. (1974), at Na\ =10, which corresponds to N = 1 0 c m " for GaSb and N ^ 1 0 c m for GaAs, |ψ(0)| 2 lies between 2 and 1. Further decrease of I ψ (0)| 2 with doping proceeds slowly, as the screening length is weakly dependent on the concentration of degenerate holes, L D e b ~ N . When I ψ (0)| 2 is independent of N , eq. (60) for the electron spin relaxation rate for scattering by degenerate holes gives the dependence observed in degenerate crystals, i.e. 1 / T S N . A satisfactory quantitative description of the temperature and concentration dependences of 1 / T in degenerate crystals can be obtained with the same values of A as those found for non-degenerate crystals, namely, 4,5 X l O " 5 s

- 3

A

A

18

- 3

A

3

18

3

p

19

- 3

p

l/6

p

p

1/3

p

s

ex

Fig. 9. Dependence of spin relaxation rate 1 / T s on acceptor concentration in doped GaSb at Τ = 15 and 77 Κ (after Aronov et al. 1983).

Spin relaxation

123

eV for GaAs and 2,4 X l O - 5 eV for GaSb. In this case expression (60) gives a good fit to the experimental data with a choice of the Sommerfeld factor for degenerate crystals of 1,5 for GaAs and 1,7 for GaSb, which agree with the estimates given above.

3.4. Spin relaxation of carriers in uniaxially deformed crystals 3.4.1. Free electrons Recently it has been shown (D'yakonov et al. 1982) that uniaxial compressive stress can intensify spin relaxation of conduction electrons in GaAs crystals. This effect was observed only in moderately doped crystals in which the DP mechanism predominates. In degenerately doped GaAs in which the BAP mechanism becomes more important the uniaxial stress produced no effect. It has been found that in GaAs with acceptor concentration NA - 4 x 10 1 6 c m - 3 a stress Ρ along the (111) and (110) axes causes an increase of electron spin relaxation rate in proportion to the square of the stress, i.e. 1/T S ~ P2. At the same time for compression along the (100) axis the stress produces no change in 1 / T s . Analysis of experimental data has shown that the observed intensifica­ tion of spin relaxation of conduction electrons is caused by the deformation splitting of spin sub-levels of the conduction band, which is described by the last term in eq. (2). The absence of any effect in the case of stress along the (100) axis indicates that, as was assumed in subsection 2.1.1, the correspond­ ing splitting of the conduction band, arising exclusively out of the interaction with more distant bands, is small. D'yakonov et al. (1982) have also established that under stress the halfwidth Bl/2 of the luminescence depolarization curve in a magnetic field transverse to the light propagation direction depends on the field orientation relative to the stress axis: for stress along the (111) axis the halfwidth Bl/2 is larger in the Β LP geometry than at B\\P\ when stress is applied along the (110) axis and the luminescence is observed along the (1Ϊ0) axis, the halfwidth Bl/2 is larger in the B\\P geometry; with stress as before along the (110) axis, but lumines­ cence observed along the (001) axis the halfwidths Bl/2 in the both geometries coincide. At large stress the observed differences in Bl/2 values reach 30-40%. This significant difference in Bl/2 values, which cannot be attributed to anisotropy of the electron g-factor in uniaxially stressed crystals, gives cogent evidence of the dependence of the spin relaxation rate on the orientation of electron spin relative to the stress axis. The anisotropy of the rate 1 / T s in uniaxially stressed crystals is the characteristic feature of spin relaxation due to the deformational splitting of the conduction band, as was theoretically shown in subsection 2.3.3. Consider now the effect of spin relaxation rate anisotropy on the halfwidths Bl/2 of luminescence depolarization curves at various orientations of the

G.E. Pikus and A.N. Titkov

124

magnetic field. In the presence of anisotropy the halfwidth B by

is described

l/2

(84) where

Here ^ = ( ^ 1 - Γ 1 1 ) / Γ 1 1 , V ± ) ~ *7ιί<±)· < l u a n t i t i es / \ \ \ = ani \/\z'z' * V s, ± the longitudinal and transverse components of the tensor 1 / T s , ^ = c o s d / / and / x ^ c o s d , , where and # are the angles between the principal axis Z ' of the tensor 1 / T s and the directions of magnetic field Β and radiation propagation /, respectively. The magnetic field Β is assumed to be directed perpendicular to /. For stress along the (111) axis the principal axis ΖΊ|(111) and, since / ± Z ' , μι = 0. Then =

T

1 +τ

t h e

l

a r e

T

f /

±

=

J_

7

1 +50(ΐ + μ ) + δν 2

//

ο

//

(85)

If Λ||<111>, we find μ = 1 and T^] = Tl if Β -L <111>, μ„ = 0 and T~ \ = (Tl{T±)~1. At large stress 1 / T s » 1 / T , the main contribution to 1 / T s being given by terms linear in ft, accounting for stress. Then the ratio Τ~ \/Τ~ \ tends to 2, since for this case eq. (41) yields 1 / T s ^ = 2 / T s . As a result, the T E S N D ratio ^ i / 2 , ± / ^ i / 2 , i i ( ^ e 7 , ± / ^ e f , i | ) I · agreement with the experimental observations. For stress along the (110) direction the principal axis of the tensor 1 / T s lies along the (001) direction and according to eq. (44) 1 / T S y = 2 / T s , as in the previous case. However, the anisotropy of B is now different, since the principal axis Z ' does not coincide with^ the direction of the stress. When the luminescence is observed along the (1Ϊ0) axis, μ = 0 and as before T is given by eq. (85). Then for B\\P, i.e., Λ||<110>, μ = 0 and T~ \ = (Äà ± whereas for Β 1 Ρ, i.e., Β ± (110), μ is unity and Τ~ \ = T~ . It further follows that B l / 2 ± / B l / 2 l l = (Τ±ι/Τ^ι)1/2 and at large stress this ratio approaches 0.7. When the luminescence is observed along the (001) axis, for both geometries B\\P and Β ±P μ, = 1, μ = 0 and T~ = (Γ„7\ The relationships for halfwidth B obtained for the case P||(110) also correspond to the experimental observations. Making use of B values in different experimental geometries D'yakonov et al. (1982) have determined the spin relaxation rates 1 / T s ^ and 1 / T s for stress along the (111) and (110) axes. The variation of 1 / T s ^ and 1 / T s with 2

9

Η

{

{

{

±

=

1 /2

4 i n

T O

±

l/2

ι

e{

Η

{

2

Η

{

2

Η

(

l/2

l/2

±

±

Spin

0

4

8

12

relaxation

16

Ρ

0

125

4

8

12

16

P (kbar )

2

2

Fig. 10. Variation of spin relaxation rates 1 / T s ,( and 1 / T S

±

2

with stress (a) along the ( 1 1 1 ) axis,

and (b) along the ( 1 1 0 ) and ( 1 0 0 ) axes (after D'yakonov et al. 1982).

Solid lines are calculated

using, respectively, eqs. (41) and (44) with C 2 chosen equal to 2.7 eV.

stress is shown in fig. 10. The solid lines are theoretical dependences according to eqs. (41) and (44). A good fit to the experimental data is obtained when the parameter C3/h is chosen equal to 8 X l O 7 c m s - 1 . At this value of C3/h the quantity C 2 in eq. (2) turns out to be 3.4 eV. 3.4.2. Free holes Optical orientation of free holes in uniaxially stressed crystals has been studied by Titkov et al. (1978) in the case of photoholes in η-type GaAs at Ύ— 2 K. The crystals under study had a doping level of about 10 1 5 c m " 3 . A dye-laser was used to excite holes to just near the top of the valence band, so that their initial energy could not exceed 1 meV. The circular polarization of the luminescence band due to the neutral donor-valence band transition was measured. Without stress the degree of polarization was equal to 4% and Bl/2 amounted to 1.2 T, which leads to l / T = 2.5XlO 1 1 s - 1 . The hole g-factor g h = 2K was taken equal to 2.4. Application of uniaxial stress along the (100) axis caused an increase of the degree of polarization and narrowing of depolarization curves in a transverse magnetic field. The variation of the hole spin relaxation rate with stress obtained from these data is shown in fig. 11. The value of l/rsh is seen to decrease more slowly than follows from theoretical consideration of the EY mechanism which is commonly assumed to be the most important spin relaxation mechanism for free holes in A 3 B 5 compounds. It was indicated in subsection 2.3.4 that the rate 1 / T s in the EY mechanism should decrease with S

H

O

126

G.E. Pikus and A.N.

Titkov

stress in proportion t o P " " with η equal to 2 or 4. However, the experimental value of η is actually only slightly above unity. Moreover, at Ρ > 1.5 kbar the decrease of 1 / T s h saturates at the level of 2 X l O 1 0 s~l. At such stress the valence band splitting amounts approximately 10 meV and, under excitation, the holes are generated only in the upper of the split valence sub-bands. The thermal excitation into the split-off sub-band at 2 Κ can be neglected. In these conditions the saturation of 1 / T s h should be attributed to the manifestation under stress of another spin relaxation mechanism. In rather pure η-type GaAs crystals the most probable mechanism is spin relaxation due to spin splitting of of splitting cubic and the upper valence sub-band. Contributions to l/r linear in k can be found according to eqs. (46) and (48), respectively. At a v = \ac/2t]\ = 0.2, the former contribution amounts to 10 9 s " 1 which is much less than the experimental value. When splitting linear in k is assumed important, the experimental value of 1 / T s h in the saturation region of 2 X l O 1 0 s " 1 leads to an estimate for the constant Ex in eq. (48) of 1.4X10" eV. Here it was also assumed that τρ = 1 /2rsho = 2 X 1 0 " 1 2 s. Such relation of these times occurs for isotropic scattering which is relevant in the actual case of hole scattering by neutral donors. The estimate for constant Ex of 1.4 X l O " 6 eV sh

6

P(kbar ) Fig. 11. Variation of hole spin relaxation rate l/rsh with stress along the ( 1 0 0 ) axis in GaAs (after Titkov et al. 1978).

Spin relaxation

121

implies parameter / to be equal to 1.4X10" eV cm which is about 7 times less than the & value in GdS crystals. In stressed GaAs crystals there was also observed a large difference in Bl/2 values for the B\\P and Β IP geometries. For holes the difference in Bl/2 values is mainly connected with the g-factor anisotropy in the upper valence sub-band, g± = 2g^ = 4κ (Bir and Pikus 1972). In this case according to eq. (85) 11

B

l/2,

±/Β1/2,\\

"= ~ ( £ _L

Τ

± /

Τ

\ \ )

'Α

From eq. (48) it follows that τ~^/τ~]_ = 2. However, the experimental data have shown no detectable anisotropy of the time T e f. The halfwidth Bl/2 at B\\P is approximately twice as much as at Β ± P, in agreement with the ratio g±/g\\3.5. Spin relaxation of hot electrons Figure 12 after Parsons (1971) shows the dependence of the polarization degree ρ of recombination radiation of thermalized electrons in GaSb on the energy of exciting light Α ω. The sharp decrease of polarization degree ρ with the energy of exciting light provides clear evidence for spin relaxation of hot electrons in a passive zone where their energy is already less than the energy of LO phonons, h coLo> &nd further cooling proceeds through emission of a large number of acoustic phonons. When electrons are excited in the conduction band with an initial energy ( A < o - £ g ) w h / ( m h + rae) which is a multiple of /*co L O, they emit LO phonons and fall directly to the bottom of the band retaining their polarization degree practically unchanged. However, the peak amplitude in fig. 12 does not reach the value corresponding to excitation just to the bottom of the conduction band. This is because the condition (Αω - Eg)mh/(mh - mc) = nhcoLO can not be fulfilled simultaneously for all excited electrons, since their initial energies differ in the cases of excitation from light- and heavy-hole valence sub-bands and in the latter case due to the warping of the heavy-hole band. Under excitation with arbitrary energy, after emission of LO phonons electrons get into the passive zone and lose their orientation during subsequent cooling. The loss of electron orientation in the passive zone is described by eq. (54). A similar dependence of the degree of luminescence polarization on the energy of the exciting light was observed in GaAs crystals with NA - 1 0 1 5 c m - 3 (Weisbuch 1977). Also seen in fig. 12 is the gradual decrease of the degree of luminescence polarization with increasing number of emitted LO phonons. This indicates noticeable spin relaxation at each stage of the cascade. We shall not dwell on this subject as the decrease of electron spin orientation by emission of LO phonons is extensively treated by

G.E. Pikus and A.N. Titkov

128

0.81

0.89

0.97

f l£ 0 (eV) Fig. 12. Dependence of degree of circular polarization, p, of recombination radiation on the energy of the exciting light h ω in GaSb at 7 = 5 . 5 K. Also shown is the variation of the total luminescence intensity / (after Parsons 1971).

Mirlin in this volume (ch. 4). It is important to note, however, that the dependence of the degree of luminescence polarization on excitation energy shown in fig. 12 is characteristic only for lightly doped crystals. At acceptor concentrations N > 10 c m the relationships are essentially different (Zakharchenya et al. 1971, Ekimov and Safarov 1971). At high doping levels the electron energy relaxation occurs already mainly through scattering by holes (D'yakonov et al. 1977). This change in energy relaxation mechanism leads to the disappearance of oscilla­ tions on the ρ (Λ ω) curve. At the same time, the decrease of the time τ at higher doping greatly weakens the DP mechanism which is responsible for the relaxation of hot electrons. As a result, the dependence of the degree of polarization on the electron initial energy (Ηω - E )m /(m + m ) becomes weaker and the decrease of the degree of polarization occurs only at electron energies of the order of hundreds of meV. Thus, in GaAs crystals with Λ^ = 10 c m the degree of polarization remains almost unaffected up to electron initial energies of 0.3-0.4 eV. Only at higher energies the action of the DP mechanism leads to a rapid fall of the polarization degree. The energy at which the degree of polarization starts to fall depends on the acceptor concentration N and increases in proportion to N . The relationships between the degree of luminescence polarization and the electron initial energy in heavily doped crystals are presented in the review article by D'yakonov and Perel' in this volume (ch. 2). 18

- 3

A

ρ

g

19

h

h

- 3

Α

/?>

A

A

e

Spin relaxation 10

129

3

BAP 10 10 17

10 18

10 13

10 20

N A( c m - 3 ) Fig. 13. Relative role of the BAP and D P mechanisms in GaAs and GaSb. The curves (1 for GaAs and 2 for GaSb) separate the temperature and concentration regions in which these mechanisms predominate.

4. Conclusion As is evident from the data presented in this review, the most detailed experimental studies of spin relaxation mechanisms have been performed on GaAs and GaSb crystals. These studies have permitted to establish unambiguously the behaviour of the main spin relaxation mechanisms dominating in the conditions of optical orientation experiments. A theoretical description of these mechanisms is given in sect. 2. The estimates performed show that in medium-gap crystals, like GaAs or GaSb, the EY mechanism is much less efficient than the DP and BAP mechanisms. The relative roles of the DP and BAP mechanisms in GaAs and GaSb crystals in different temperature and doping ranges are shown in fig. 13. The curves in fig. 13 separate the regions of the domination of these mechanisms. It should be stressed particularly that quantitative analysis of the spin relaxation rate provides a very effective means of determining a number of principal parameters such as the exchange splitting for excitons, the splittings cubic and linear in k of the conduction and valence bands in an uniaxially stressed and stress-free crystals. Because these splittings are very small, their direct measurements by conventional methods of optical spectroscopy is usually impossible.

References Abakumov, V.N., and I.N. Yassievich, 1971, Zh. Eksp. Teor. Fiz. 6 1 , 2571 [Engl, transl.: 1972, Sov. Phys. JETP 3 4 , 1375].

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