D'yakonov-Perel' spin relaxation under electron-electron collisions in QWs

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by  M. M. Glazov

D’yakonov-Perel’ spin relaxation under electron-electron collisions in QWs
M.M. Glazov and E.L. Ivchenko
A.F. Ioffe Physico-Technical Institute, Russian Academy of Sciences, 194021 St. Petersburg, Russia
The D’yakonov-Perel’ mechanism of spin relaxation is connected with the spin splitting of the
3 electron dispersion curve in crystals lacking a center of symmetry. In a two-dimensional noncen-
0 trosymmetric system, e.g. quantum well or heterojunction, the spin splitting is a linear function
0 of k, at least for small values of k. We demonstrate that the spin relaxation time τs due to the
2 spin splitting iscontrolled not only bymomentum relaxation processes as widely accepted butalso
n by electron-electron collisions which make no effect on the electron mobility. In order to calculate
a thetimeτs takingintoaccounttheelectron-electron scattering wehavesolvedthetwo-dimensional
J kinetic equation for the electron spin density matrix. We show how the theory can be extended to
7 allow for degenerate distribution of thespin-polarized two-dimensional electron gas.
2

] I. INTRODUCTION
l
l
a
h Inrecentyearsthe electronspinrelaxationprocesseshavereceivedmuchattentioninconnectionwith variousspin-
- tronics applications. For free electrons in quantum well structures the following four mechanisms of spin decoherence
s
aremostimportant(seeRefs.1,2andreferencestherein): theElliot-Yafet,D’yakonov-Perel’,Bir-Aronov-Pikusmech-
e
m anisms and electron spin-flip scattering by paramagnetic centers. This paper is devoted to the D’yakonov-Perel’spin
relaxation (DPSR) in which case the spin relaxation time, τ , is related to the spin splitting and given by Ref. 3
. s
t
a τ−1 Ω2 τ . (1)
m s ∝h ki
- Here Ωk is the effective Larmor frequency describing the splitting of the electron spin dispersion branches, the angle
d
brackets mean averaging over the electron energy distribution and τ is a microscopic electron relaxation time. In a
n two-dimensional (2D) system laking a center of a symmetry, e.g. quantum well or heterojunction, the frequency Ω
o k
is linear in k. In this case the time τ was usually identified with the momentum relaxation time that determines the
c
[ electron mobility Refs. 4,5,6,7,8. In Ref. 9 we have shown that the inverse time τ−1 is determined not only by the

momentumscatteringratebutcontainsalsoanindependentcontributionfromelectron-electroncollisionswhichmake
1 noeffect onthe electronmobility. Really,electron-electroncollisionschangethe directionofk andΩ and,therefore,
v k
they control the Dyakonov-Perel’spin relaxation exactly in the same way as any other scattering processes do.
9
1 Thepaperisorganizedasfollows. Section2containsthediscussionofmechanismsofthespinsplittingoftheelectron
5 subbands inquantumwellsandelectronspindecoherencedueto spinsplitting, Sec. 3describes the D’yakonov-Perel’
1 spin relaxation mechanism in terms of the kinetic theory, in Sec. 4 we analyze the electron-electron collision integral
0 for the spin-polarized electron gas, and Sec. 5 describes the solution of the kinetic equation with the account of
3
electron-electron collisions.
0
/
t
a
II. SPIN SPLITTING OF ELECTRON SUBBANDS
m
-
d In the parabolic approximation the effective electron Hamiltonian in the zinc-blende-based (001)-grown quantum
n well (QW) can be written as
o
c ¯h2k2
v: H= 2m +β1(σyky −σxkx)+β2(σxky −σykx), (2)
i
X where x [100], y [010], β are constants, m is the in-plane effective mass, σ are the Pauli matrices and k are
1,2 l n
k k
r the components of the 2D electron wavevector, k. The terms with higher powers in k are not considered here. In
a the symmetrical QWs (the D point group) the linear-k spin-dependent term proportional to β is allowed only,
2d 1
it is called the bulk-inversion asymmetry (BIA) term. In asymmetrical QW structures characterized by the C
2v
point-group symmetry, there exists another spin-dependent contribution proportionalto β and called the structure-
2
inversion asymmetry (SIA) term or the Rashba term Refs. 10,11 (see also Ref. 12 and references therein). The
structure asymmetry can be related with non-equivalent normal and inverted interfaces, external or built-in electric
fields, compositionally stepped QWs etc.
It is convenientto introduce the Cartesiancoordinatesx′ [1¯10], y′ [110],z [001]which allowto write a sum of the
k k k
BIA and SIA terms in the form
c1(k)=(β−σx′ky′ +β+σy′kx′)/2, (3)
H
2
where β± =2(β2 β1). The effective Larmor frequency Ωk is defined by c1 =(h¯/2)Ωk σ, its components are
± H ·
Ωk,x′ =β−ky′/¯h, Ωk,y′ =β+kx′/¯h, Ωk,z =0. (4)
The spin splitting at k = (kx′,ky′) is ¯hΩk = β+2kx2′ +β−2ky2′. If only one of the linear-k terms, BIA or SIA, is
q
nonzero then β2 =β2 β2 and the splitting h¯Ω =βk is angular independent.
− + ≡ k
Consider an electron gas occupying the lowest conduction subband e1 and assume that, at the moment t =0, the
electrons are spin-polarized in the same direction along, say, the growth axis z. Owing to the spin-splitting of the
electron subband the electron spin in the state with the wavevector k precesses around the axis Ω which lies in the
k
interface plane, see Eq.(4). In case of the largesplitting, Ω 1/τ, the spin of electronin the state k will oscillate
k
| |≫
with time as s (t) = s (0)cosΩ t. In the case of Ω being isotropic in the QW plane, the spin polarized electrons
z z k k
whichoccupythecircleofthefixedradiusinthek spaceshowthesimilaroscillatorybehaviorfortheirs component.
z
However,ifβ2 =β2 and/orthe electrons occupystates with differentvalues of k,the scatterin Ω results ina fast
− 6 + | | k
non-exponential decay of s .
z
D’yakonov and Perel’ Ref. 3 were the first to show that the processes of electron-momentum scattering result in
the slowing off the spin decoherence caused by the spin splitting. In the collision-dominated (“motional-narrowing”)
limit Ω 1/τ,this resultsinanexponentialdecayofthe averagespin: s¯ (t)=s¯ (0)exp( t/τ ),whereτ isgiven
k z z s s
| |≪ −
by Eq. (1). The dimensionless coefficient in Eq. (1) can be obtained from the solution of kinetic equation Refs. 3,4.
This approachis valid as long as the energy relaxation time τ is short as compared to τ .
ε s
III. DPSR IN TERMS OF THE KINETIC THEORY
In the frame of kinetic theory, the electron distribution in the wavevector and spin spaces is described by a 2 2
×
spin-density matrix
ρˆ =f +s σ. (5)
k k k
·
Here f = Tr(ρˆ /2) is the average occupation of the two spin states with the wavevector k, or distribution function
k k
of electrons in the k-space, and the average spin in the k state is s = Tr[ρˆ (σ/2)]. If we neglect the spin splitting
k k
then, for arbitrary degeneracy of an electron gas with non-equilibrium spin-state occupation but equilibrium energy
distribution within each spin branch, the electron spin-density matrix can be presented as
E µ¯ µ˜(σo ) −1
ρˆ0 = exp k− − s +1 , (6)
k (cid:26) (cid:20) k T (cid:21) (cid:27)
B
whereE =h¯2k2/2m,k istheBoltzmannconstant,T isthetemperature,o istheunitvectorinthespinpolarization
k B s
direction, µ± =µ¯ µ˜ are the effective Fermi energies for electrons with the spin component 1/2 or 1/2 along os so
± −
that the energy distribution functions of electrons with the spin 1/2 are given respectively by
±
−1
Ek µ±
fk,± = exp − +1 .
(cid:20) (cid:18) k T (cid:19) (cid:21)
B
Note that Eq. (6) can be rewritten in the equivalent form Ref. 3
1
ρˆ0k ≡fk0+s0k·σ = 2[fk,++fk,−+(fk,+−fk,−)(σos)] .
The densities n± of 2D electrons with a particular spin can be related with the effective Fermi energies by
m
n± = 2π¯h2 kBT ln(1+eµ±/kBT). (7)
If the spin splitting is non-zero but small compared to h¯/τ, the distribution function Tr[ρ /2] = f0 does not
k k
change, whereas the spin vector obtains a correctionδs =s s0 proportionalto the spin splitting. Therefore, the
k k− k
spin-density matrix may be presented as
ρˆ =ρˆ0 +δs σ. (8)
k k k·
3
The quantum kinetic equation for the spin-density matrix taking into account the electron-electron collisions has
the form
∂ρˆ i
k + [ (k),ρˆ ]+Qˆ ρˆ =0, (9)
c1 k k
∂t ¯h H { }
where [P,R] = PR RP, (k) is the linear-k Hamiltonian (3), and the third term in the left-hand side is the
c1
− H
collisionintegralorthescatteringrate,inthisequationitisa2 2matrix. ItfollowsfromEq.(9)thatthepseudovector
×
kinetic equation for s can be written as
k
ds
k +Ω s +Q s =0, (10)
k k k
dt × { }
where Q s,f = (1/2)Tr(σQˆ ρ ). In the following we ignore spin flips under scattering. Then, say, for the elastic
k k
{ } { }
scattering one has
Qk F = Wk′k(Fk Fk′),
{ } −
Xk′
whereWk′k istheprobabilityratefortheelectrontransitionfromthestatektok′. Thecollisionintegralforelectron-
electronscatteringis consideredin the nextsection. Here it suffices to note that, for the distributionρˆ0, the collision
k
integral vanishes identically. This integral also vanishes after the summation over k which allows, in particular, to
derive from Eq. (10) the following equation of balance for the total averagespin S0 =os(n+ n−):
−
dS
0 + Ω δs =0. (11)
k k
dt ×
Xk
The angular dependence of the non-equilibrium correction δsk is the linear combination of cosΦk = kx′/k and
sinΦk = ky′/k, where Φk is the angle between k and the axis x′. Retaining in the kinetic equation (10) terms
proportional to the first angular harmonics we obtain the equation for δs with the inhomogeneous term linear in
k
S . Then one can substitute the solution in the secondterm of Eq. (11). The final result is that the tensor of inverse
0
spin relaxation times, 1/τs , is diagonal in the coordinate system x′,y′,z and given by Refs. 4,13
αβ
1 = 1 β+ 2 k2τ , 1 = 1 β− 2 k2τ , 1 = 1 + 1 , (12)
τs 2(cid:18) ¯h (cid:19) h pi τs 2(cid:18) ¯h (cid:19) h pi τs τs τs
x′x′ y′y′ zz x′x′ y′y′
where τ is the momentum relaxation time. If among the two contributions, BIA and SIA, to the spin splitting one
p
is dominant and β+ = β− , the spin relaxation times are interconnected by Ref. 4
| | | |
τs =τs =2τs .
x′x′ y′y′ zz
InterplaybetweentheBIAandSIAcontributionscanleadtoagiantspinrelaxationanisotropyRef.13. Inparticular,
if these contributions coincide, β1 =β2, so that β− =0 one has τxs′x′ =τzsz and τys′y′ =∞. In the case β1 =−β2 the
coefficient β =0, the time τs is infinite and τs coincides with τs .
+ x′x′ y′y′ zz
IV. ELECTRON-ELECTRON COLLISIONS IN QWS
Here we will write the electron-electron collision term Qˆ ρˆ in the general case of arbitrary spin-density matrix
k
{ }
ρˆ (inparticular,arbitrarydegeneracyandarbitrarydistribution ofspininthe k-space). Forthis purposewe remind
k
that the matrix element of the Coulomb scattering k,sk + k′,sk′ p,sp + p′,sp′ is given by
→
M(p,sp;p′,sp′|k,sk;k′,sk′)=Vk−pδsp,skδsp′,sk′ −Vk−p′ δsp,sk′δsp′,sk , (13)
where sk,sk′...= 1/2, Vq is a Fourier transform of the 2D Coulomb potential of the electron-electron interaction
±
2πe2
V = , (14)
q
æqΣ
eistheelementarycharge,æisthedielectricconstant,andΣisthesampleareaintheinterfaceplane;inthefollowing
we set Σ=1. Equation (13) takes into account both the direct and exchange Coulomb interaction.
4
In order to present Qˆ ρˆ in a compact form we introduce the 2 2 unit matrix I(1) and Pauli matrices σ(1)
k α
{ } ×
(α =x,y,z) for the spin coordinates s ,s and a similar set of four matrices, I(2) and σ(2), for the spin coordinates
p k α
sp′,sk′. One can check that Eq. (13) allows the following matrix representation
Mˆ =AI(1)I(2)+Bσ(1) σ(2), (15)
·
1 1
A=Vk−p Vk−p′ , B = Vk−p′ . (16)
− 2 −2
Now the collision term for the electron spin-density matrix can be presented as
π
Qˆk ρ = δk+k′,p+p′ δ(Ek+Ek′ Ep Ep′)Tr2G(p,p′;k,k′), (17)
{ } 2h¯ − −
kX′pp′
G(p,p′;k,k′)= (18)
= Mˆ(I(1) ρˆ(1))(I(2) ρˆ(2))Mˆρˆ(1)ρˆ(2)+ρˆ(1)ρˆ(2)Mˆ(I(1) ρˆ(1))(I(2) ρˆ(2))Mˆ
− p − p′ k k′ k k′ − p − p′ −
Mˆρˆ(1)ρˆ(2)Mˆ(I(1) ρˆ(1))(I(2) ρˆ(2)) (I(1) ρˆ(1))(I(2) ρˆ(2))Mˆρˆ(1)ρˆ(2)Mˆ .
− p p′ − k − k′ − − k − k′ p p′
Here the spin-density matrices ρˆ(1)(k)=I(1)fk+σ(1) sk, ρˆ(2)(k′)= I(2)fk′ +σ(2) sk′ etc., the symbol Tr2 means
the trace over the spin variable 2. After the trace is fo·und the index 1 in Tr G(p,p′·;k,k′) can be omitted. In order
2
to derive Eqs. (17), (18) we used the standard diagram technique.
Instead ofequation (9) for the spin-density matrix one can use a scalarequationfor the distributionfunction f in
k
the form
df
k +Q f,s =0 (19)
k
dt { }
and a equation for the spin-distribution vectorfunction as
ds
k +Ω s +Q s,f =0, (20)
k k k
dt × { }
where
1
Q f,s = Tr [Qˆ ρˆ ] (21)
k 1 k
{ } 2 { }
π
= δk+k′,p+p′ δ(Ek+Ek′ Ep Ep′)Tr1Tr2[G(p,p′;k,k′)].
4h¯ − −
kX′pp′
1
Q s,f = Tr [σ(1)Qˆ ρˆ ] (22)
k 1 k
{ } 2 { }
π
= δk+k′,p+p′ δ(Ek+Ek′ Ep Ep′)Tr1Tr2[σ(1)G(p,p′;k,k′)].
4h¯ − −
kX′pp′
For the analysis of the general equations (9), (17), (21), (22) we consider below few particular cases.
Spin-unpolarized electrons. In this case s 0 and the spin-density matrix reduces to a product of the unit 2 2
k
≡ ×
matrix and the distribution function f . Taking into account that
k
Mˆ2 =(A2+3B2)I(1)I(2)+2B(A B)σ(1) σ(2)
− ·
we come to the conventional collision term
2π
Qk f = δk+k′,p+p′ δ(Ek+Ek′ Ep Ep′)(A2+3B2) (23)
{ } ¯h − −
kX′pp′
5
[fkfk′(1 fp)(1 fp′) fpfp′(1 fk)(1 fk′)].
× − − − − −
Note that according to Eq. (16) one has
A2+3B2 =Vk2−p+Vk2−p′ −Vk−pVk−p′
which is one-forth of the function R introduced in Eq. (2.4b) in Ref. 16 and the above collision term agrees with the
equation (2.4a) in the cited paper. It is worth to note that in the sum (23) V2 +V2 can be replaced by 2V2 .
k−p k−p′ k−p
Electrons polarized along the same axis. By using the coordinate system with z parallel to the electron spin
polarization one has s = s 0 and the spin-density matrix is a diagonal matrix with the diagonal components
k,x k,y
f (s = 1/2). It follows then≡that the products ρˆ(1)(k)ρˆ(2)(k′) and [I(1) ρˆ(1)(p)][I(2) ρˆ(2)(p′)] are diagonal as
k,s
± − −
well. We can take into account the spin conservation
M(p,s ;p′,s k,s ;k′,s ) δ
3 4| 1 2 ∝ s3+s4,s1+s2
and use the identity
M(p,s ;p′,s k,s ;k′,s )M(p,s ;p′,s k, s ;k′,s )=0.
3 4 1 2 3 4 1 2
| | −
This allows, in agreement with Ref. 17, rewrite the collision term for f as
k,s
2π
δk+k′,p+p′ δ(Ek+Ek′ Ep Ep′)W(p,s1;p′,s2 k,s;k′,s′)
¯h − − |
kX′pp′sX′s1s2
×[fk,sfk′,s′(1−fp,s1)(1−fp′,s2)−fp,s1fp′,s2(1−fk,s)(1−fk′,s′)].
Here
W(p,s;p′,sk,s;k′,s)=(A+B)2 =(Vk−p Vk−p′)2,
| −
W(p,s;p′, sk,s;k′, s)=(A B)2 =V2 ,
− | − − k−p
W(p, s;p′,sk,s;k′, s)=(2B)2 =V2 ,
− | − k−p′
and other values of W with s +s =s+s′ are zero.
1 2
6
Low electron polarization. If the average electron spin s is small as compared with the occupation probability f
k k
then, in the equation for f , one can ignore the spin polarization at all and use Eq. (23), while, in the equation for
k
s , one can retain in the collision term only the contribution linear in s . The linearized collision term is given by
k k
2π
Qk s,f = δk+k′,p+p′ δ(Ek+Ek′ Ep Ep′) (24)
{ } ¯h − −
k′X,p,p′
× (Vk2−p+Vk2−p′ −Vk−pVk−p′)skF(k′;p,p′)−Vk−pVk−p′sk′F(k;p,p′)
(cid:2)
Vk−p(Vk−p Vk−p′)spF(p′;k,k′) Vk−p′(Vk−p′ Vk−p)sp′F(p;k,k′)] ,
− − − −
where
F(k ;k ,k )=f (1 f )(1 f )+(1 f )f f =f (1 f f )+f f .
1 2 3 k1 − k2 − k3 − k1 k2 k3 k1 − k2 − k3 k2 k3
Equation (24) can be transformed to
2π
Qk s,f = δk+k′,p+p′ δ(Ek+Ek′ Ep Ep′) (25)
{ } ¯h − −
k′X,p,p′
2V2 [s F(k′;p,p′) s F(p′;k,k′)]
× k−p k − p −
(cid:8)
Vk−pVk−p′[skF(k′;p,p′)+sk′F(k′;p,p′) 2spF(p′;k,k′)] .
− − }
6
Here the term proportional to 2V2 is due to the direct Coulomb interaction whereas the term proportional to
k−p
Vk−pVk−p′ comes from the exchange interaction.
Non-degenerate 2D electron gas. In this case the function F(k ;k ,k ) reduces to f and the collision terms are
1 2 3 k1
as follows
2π
Qk{f,s}= ¯h δk+k′,p+p′ δ(Ek+Ek′ −Ep−Ep′) (26)
kX′pp′
×[(2Vk2−p−Vk−pVk−p′)(fkfk′ −fpfp′)−Vk−pVk−p′(sk·sk′ −sp·sp′)],
2π
Qk{s,f}= ¯h δk+k′,p+p′ δ(Ek+Ek′ −Ep−Ep′) (27)
kX′pp′
×[2Vk2−p(skfk′ −spfp′)−Vk−pVk−p′(skfk′ +sk′fk−2spfp′)].
Neglecting the exchange interaction given by the term proportionalto the Vk−pVk−p′ the scattering rate Qk f,s is
{ }
independent on the electron spin distribution. Both Q f,s and Q s,f take a simple form
k k
{ } { }
Qk f,s = Wpp′,kk′(fkfk′ fpfp′), Qk s,f0 = Wpp′,kk′(skfk′ spfp′), (28)
{ } − { } −
kX′pp′ kX′pp′
where Wpp′,kk′ is the probability rate for the scattering of a pair of electrons from the k,k′ states to the p,p′ states
2π
Wpp′,kk′ = ¯h δk+k′,p+p′ δ(Ek+Ek′ −Ep−Ep′)2Vk2−p,
an additional factor of 2 takes into account the double degeneracy of the electronic states.
V. ELECTRON-ELECTRON SCATTERING TIME CONTROLLING THE DPSR
Inwhatfollowsweconsideronlyanon-degenerate2Delectrongasinwhichcasethezero-approximationspin-density
matrix (8) can be written as
ρˆ0 =f0(1+2s¯ σ), (29)
k k ·
where fk0 is the Boltzmann distribution function, and s¯is the averagespin per electron, S/(n++n−). The nonequi-
librium correction δs satisfies the equation
k
Ω (2f0s¯ )+Q δs,f0 =0. (30)
k× k 0 k{ }
It should be noted that in the 2D case the collision term (27) does not allow the quasi-elastic and relaxation time
approximations and Eq. (30) must be solved directly.
Theα-componentofthevectorproductΩ s¯ canberepresentedas(Ω s¯ ) =Λ k s¯ ,wherethethird-rank
k 0 k 0 α αβγ β 0γ
tensor Λ in general case of both BIA and SIA×linear-k terms has four nonz×ero components
Λxxz = Λzxx =β+/¯h, Λyyz = Λzyy =β−/¯h.
− −
The function (1/k )Q k F ,f0 is independent of the azimuthal angle Φ (here F is an arbitrary function of
β k β k k k
{ }
k = k)asthe operatorQ δs ,f0 conservesthe angulardistribution inthe k space. Insuchcase the solutionmay
k α
| | { }
be written as follows
k
δs (k)= Λ βs¯ k τ∗ eµ/kBT v(K). (31)
α − αβγ k 0γ T ee
Here we introduced the dimensionless wavevectorK=k/k , k =(2mk T/¯h2)1/2,
T T B
¯hk Tæ2
∗ B
τ = , (32)
ee e4N
7
N =n++n− and v(K) satisfies the equation
Ke−K2 = d2K′ d2P WPP′,KK′ v(K)e−K′2 cosΘv(P)e−P′2 , (33)
Z Z (cid:16) − (cid:17)
f
where Θ is the angle between K and P, P′ =K+K′ P,
−
1
WPP′,KK′ = K P2 δ(K2+K′2−P2−P′2).
| − |
f
Inserting Eq. (31) into Eq. (11) one obtains after summation over k the principal values of the tensor of reciprocal
spin relaxation times
2 2
1 β+kT 1 β−kT 1 1 1
= τ , = τ , = + . (34)
τs (cid:18) ¯h (cid:19) τs (cid:18) ¯h (cid:19) τs τs τs
x′x′ y′y′ zz x′x′ y′y′
The time τ which controls the spin relaxation is given by
∞
1
τ =τ∗I, I = v(K)K2dK. (35)
ee 2Z
0
The parameterτ∗ is presentalsoin the ee-scatteringtime which determines the rate ofenergyexchangebetween 2D
ee
electrons Ref. 15.
Thefunctionv(K)wasexpandedinseriesusingabasissetl (ε)=√2exp( ε)L (2ε),whereL (ε)aretheLaguerre
n n n
polynomials and ε = K2. The expansion was substituted into the right-ha−nd side of Eq. (33), the integration has
been performed by Monte-Carlo method. The problem was reduced to a set of linear inhomogeneous equations for
the expansion coefficients of v(K). The resulting value of I in Eq. (35) was found to be 0.027. Allowance for the
≈
exchange interaction leads to a slight increase of this value to 0.028.
≈
Comparison with ionized impurities scattering. If the ionized impurities of the same concentration N lie inside
the 2D layer then the corresponding transport time is given by τ = (2/π2)τ∗ (see Ref. 16). The spin relaxation
tr ee
time, τii, controlled by scattering by the ionized impurities is given by Eq. (34) where τ is changed by τ /2, thus
s tr
the ratio of the spin relaxation time, τee, governed by electron-electron collisions and the time τii is 3.6, i.e. the
s s ≈
elasticscatteringby impuritiesis lessefficient. If the dopedlayerisseparatedfromthe quantumwellbya spacer,the
influence of the Coulomb potential of ionized impurities on τ is reduced and eventually can be neglected.
s
The aboveresultwas obtainedfor the 2D Coulombpotential U2D(ρ)=e2/æρ,where ρ is the distance between the
electrons in the interface plane. It is this potential that leads to the Fourier transform given by Eq. (14). In order
to analyze the role of the quasi-2D character of the electron wave function confined in a QW of the finite thickness a
one can replace U2D(ρ) by the effective potential obtained by averaging the three-dimensional Coulomb potential as
follows
e2 ϕ2 (z)ϕ2 (z′)
U(ρ)= e1 e1 dzdz′,
æ Z Z ρ2+(z z′)2
−
p
where ϕ (z) is the electron envelope function at the lowest conduction subband e1. The straightforwardcalculation
e1
shows Ref. 18 that for the conduction band offset V ¯h2/ma2 and k a < 1 a value of the time τ = τ∗I increases
c ≫ T ee
with widening the QW as the electron-electron interaction becomes weaker, but an order of magnitude of I and τ
remains the same as in the exact 2D case.
Conclusion. Future Work
Wehaveshownthatelectron-electroncollisionscontroltheD’yakonov-Perel’spinrelaxationinthesamewayasany
scatteringprocessesdo. The electron-electroncollisionintegralhasbeenderivedforanarbitrarydegeneracyandspin
distributionofthe2Delectrongas. Thecalculationshasbeenperformedfornon-degenerate2Dandquasi-2Delectrons
confined in a quantum well. Calculations of the spin relaxationtime controlledby electron-electroncollisions in bulk
semiconductors are in progress. An important next step is the extension of the calculations from non-degenerate to
degenerate spin-polarized electron gas.
In agreement with our theory, the latest optical spin-dynamic measurements in an n-doped
GaAs/AlGaAs QW of high mobility give an experimental evidence of the electron-electron scattering to randomize
the electron spin precession Ref. 19.
8
Note in conclusion that the time τ = τ∗I can be related physically with the momentum relaxation time of an
ee
electron by equilibrium holes of the density N if the electron and hole effective masses are assumed to coincide. In
the electroncollisionswithholesthe directedelectronmomentumis transferredto the hole gasanddecayswithin the
time τ. TheworkwassupportedbyRFBR,andbytheprogramsoftheRussianMinistryofSci. andthePresidium
∼
of the Russian Academy of Sci.
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