Spin Relaxation in a Two-Dimensional Electron Gas in a Perpendicular Magnetic Field

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by  A. A. Burkov

Spin Relaxation in a Two-Dimensional Electron Gas in a Perpendicular Magnetic
Field
A.A. Burkov and Leon Balents
Department of Physics, University of California, Santa Barbara, CA 93106
(Dated: February 2, 2008)
4
0 We consider the problem of spin relaxation in a two-dimensional electron gas (2DEG) in a per-
0 pendicularmagnetic field. Weassume that thespin relaxation is inducedby theRashba spin-orbit
2 (SO) interaction, which appears due to the inversion asymmetry of the confining potential. Our
n solutionisbasedonamicroscopicevaluationofthespindensityresponsefunctionofthe2DEGwith
a impurities and SO interaction. We derive explicit expressions for the transverse and longitudinal
J spin relaxation rates. Ouranalysis shows, in particular, that thespinrelaxation ratesexhibit mag-
1 netoquantum oscillations,whichareanalogoustotheShubnikov-deHaasoscillationsoftheelectrical
2 resistivity. These oscillations can beobserved, for example, in time-resolved optical experiments.

]
l I. INTRODUCTION and ω τ is large. Our theory is based on a microscopic
l c
a evaluation of the spin density response function of a dis-
h
ordered 2DEG system with the Rashba SO interaction
- The study of spin relaxationin semiconductor 2DEGs
s inaperpendicularmagneticfield. Weobtainexplicitan-
e isanimportantareaoftheemergingfieldofspintronics.1
alytical expressions for the longitudinal and transverse
m The dominant spin relaxation mechanism in these sys-
spin relaxationrates:
tems can typically be associated2 with the Rashba SO
.
at interaction,3whichexistsduetotheinversionasymmetry 1 8λ2mǫFτ/¯h2
m of the confining potential in semiconductor heterostruc- τ = 1+(ω τ)2 ,
z c
ture based 2DEG systems. A simple semiclassical pic-
- 1 1
d ture of the spin relaxation, due to Dyakonov and Perel = ,
n (DP),2,4isthatsincetheRashbainteractionhastheform τ⊥ 2τz
o of a momentum-dependent magnetic field λzˆ·[τ ×p], it whichreducetothewell-knownDyakonov-Kachorovskii2
c
induces precessionofthe spinofamovingelectron. This
[ expressions in the zero-field limit. We show, in particu-
precessionleads to the randomizationofthe spin atlong
lar,thattheapplicationofaperpendicularmagneticfield
1 times. The DP spin relaxation rate is given, to leading leads to quantum oscillations of the spin relaxationrate,
v order in λ, by the simple expression 1/τ ∼(λp /¯h)2τ,
sf F which are analogous to the well-known quantum oscil-
8
where p is the Fermi momentum and τ is the elastic
9 F lations of transport coefficients. These oscillations arise
scattering time. Quasiclassically, λp /¯h is simply the
3 F fromthemagneticfielddependenceofthescatteringtime
1 spin-precession frequency, associated with the Rashba τ, which will be calculated below.
0 field. The distinguishing feature of the DP spin relax- Thepaperisorganizedasfollows. InsectionII wecal-
4 ation is that the relaxation rate is smaller in more dis-
culatethedisorder-averagedGreen’sfunctionofa2DEG
0 ordered systems, since impurity scattering disrupts the
system with the Rashba SO interaction in a perpendicu-
/
t Rashbaspinprecessionbyrandomizingtheelectron’smo- larmagnetic field using the self-consistentBornapproxi-
a
mentum.
m mation (SCBA). In section III the spin-density response
It is well-known that transport processes in 2DEG function of our system is calculated by summing SCBA
-
d systems are strongly influenced by the application of a self-energy and ladder vertex corrections to the bare po-
n perpendicular magnetic field. The perpendicular field larizationdiagram. The transverseandlongitudinalspin
o quantizes electron’s energy spectrum, which manifests relaxationratesarethenextractedfromthe poles ofthis
c spectacularly in Shubnikov-de Haas (SdH) oscillations response function.
:
v of the resistivity and eventually leads to the quantum
i Hall effect. In this paper we address the question of
X
howaperpendicularmagneticfieldinfluencestheRashba II. DENSITY OF STATES AND DISORDER
r SELF-ENERGY
a SO interaction-inducedspin relaxationin semiconductor
2DEG systems. This problemhas been addressedbefore
by several authors, both in the semiclassical5,6,7,8 and The simplest single-particle Hamiltonian describing
in the quantum limits,9,10,11,12,13,14 but a complete and the dynamics of electronsin a semiconductor2DEG in a
rigorous analysis is still lacking. perpendicular magnetic field can be written as:
Inthis paperwewillprovidesuchananalysis. Wewill π2 ∆
mainly be interestedinthe regimeofmoderatemagnetic H = +λzˆ·[τ ×π]− zτz, (1)
0
fields, such that ωcτ <∼ 1 (here ωc is the cyclotron fre- 2m 2
quency), and ǫ ≫ ¯hω . Our analysis can, however, be whereπ =p+(e/c)Aisthe kinetic momentum,τ isthe
F c
generalized to the quantum Hall limit, when ǫ ∼ ¯hω spinoperatorand∆ =gµ BistheZeemanenergy. The
F c z B
2
second term in Eq.(1) describes the Rashba SO interac- Green’s function. This is a very good approximation for
tion. Hamiltonian H can be easily diagonalized15,16,17 the high Landau level filling factors that we are assum-
0
if one notices that the Rashba term mixes only neigh- ing. SCBAequationfortheretardeddisorderself-energy
boring Landau levels with opposite spin directions. The in our case reads:
eigenstates are thus given by:
ΣRna(ǫ)= Γna,n′a′GRn′a′(ǫ), (8)
|n,ai=u |n,↓i+v |n−1,↑i, (2) n′a′
na na X
where
where a=1,2 and |n,σi is the n-th Landau level eigen-
statewithspinσ =↑,↓. Thecorrespondingeigen-energies Γna,n′a′ = hVna,n′a′(k,k′)Vn′a′,na(k′,k)i, (9)
are: k′
X
and
2
¯hω +∆
ǫna =h¯ωcn+(−1)as(cid:18) c2 z(cid:19) +2λ2m¯hωcn, (3) GRna(ǫ)= ǫ−ǫna−1 ΣRna(ǫ), (10)
where ω =eB/mc. Note that n=1,2,... for a=1, but istheretardedSCBAGreen’sfunction. Angularbrackets
c
n = 0,1,... for a = 2. The amplitudes u and v are in Eq.(9) denote disorder average.
na na
given by: Matrix elements of Γ can be easily evaluated for δ-
function impurity potential and are given by:
1 (−1)a(h¯ω +∆ )/2
u = i + c z , ¯h2ω
na s2 (h¯ωc+∆z)2/4+2λ2m¯hωcn Γna,n′a′ = 2πτc |una|2|un′a′|2+|vna|2|vn′a′|2 , (11)
0
v = (−1)a+1p1 + (−1)a+1(h¯ωc+∆z)/2 . where τ0 is the ela(cid:0)stic impurity scattering tim(cid:1)e in the
na s2 (h¯ωc+∆z)2/4+2λ2m¯hωcn absence of the magnetic field.
Eq.(8) is too complicated to be solved analytically
p (4) without approximations. Fortunately, only very minor
simplification is needed to make this equation solvable.
Usingtheabovebasisofsingle-particleeigenstates,the
The main complication is the dependence of the matrix
Hamiltonian of the 2DEG system in the presence of im-
Γonthe Landaulevelindexn. Thisdependence is,how-
purity scattering potential can be written as:
ever,inessential,sinceitappearsentirelyduetotheterm
H = ǫnac†nakcnak+ Vna,n′a′(k,k′)c†nakcn′a′k′, 2caλn2mb¯heωscanfeilny Eigqn.o(4re).d,TshinecdeeptheendReanscheboanSnOininttheirsactteiromn
nak nak,n′a′k′
X X can be assumed to be a weak perturbation, in the sense
(5)
that λp ≪ǫ . In this case we can make a replacement
where index k denotes orbit-center quantum numbers in F F
the Landau gauge and n→ǫF/¯hωcintheRashbaterminEqs.(3)and(4),which
makes the amplitudes u and v and, consequently, Γ,
na na
independent of n. This simple approximation allows us
Vna,n′a′(k,k′)= drΨ∗nak(rσ)V(r)Ψn′a′k′(rσ), (6) to solve Eq.(8) analytically.
σ Z
X It is convenient to introduce the following new nota-
are the matrix elements of the impurity potential V(r) tion. Let
in the basis (2). The wavefunctions Ψ (rσ) are given
nak 2
¯hω +∆
by: ∆=2 c z +2λ2mǫ . (12)
F
s 2
(cid:18) (cid:19)
Ψ (r↑) = v φ (r),
nak na n−1,k Then ǫ =h¯ω n+(−1)a∆/2. Also define:
Ψ (r↓) = u φ (r), (7) na c
nak na n,k
u =icos(ϑ), v =sin(ϑ),
1 1
where φ (r) are the Landau-leveleigenfunctions in the
n,k u =isin(ϑ), v =−cos(ϑ), (13)
2 2
Landau gauge.
Our analysis of the spin relaxation in a system, de- where
scribed by Eq.(5), is based on a calculation of the spin
1 ¯hω +∆
densityresponsefunction,whichissimilartoAndo’scal- cos(ϑ)= − c z. (14)
2 2∆
culationoftheconductivityofa2DEGinaperpendicular r
field.18 Our calculation is technically more complicated Using this notation, the matrix elements of Γ are given
than Ando’s due to the presence of the SO interactions by:
and also due to the fact that vertex corrections to the
¯h2ω 1
polarizationbubble, which vanish in the case of the con- Γ = Γ = c 1− sin2(2ϑ) ,
11 22
2πτ 2
ductivity calculation, are crucial in our case. 0 (cid:18) (cid:19)
Following Ando, we use the self-consistent Born ¯h2ω
Γ = Γ = c sin2(2ϑ). (15)
approximation (SCBA) to find the disorder-averaged 12 21 πτ
0
3
Letus alsomaketheusualassumptionthattherealpart different from the usual expression, which is periodic in
ofthedisorderself-energycanbeabsorbedintothechem- ¯hω ,duetothepresenceoftheRashbaandZeemansplit-
c
ical potential. Then SCBA equation can be written as: tings. In what follows we will ignore this modification of
the density of states, putting λ = 0 and ∆ = 0 in the
z
oscillatory term. This does not mean, of course, that we
ImΣRa(ǫ)= Γaa′ImGRna′(ǫ)=−2π2ℓ2 Γaa′̺a′(ǫ), are ignoring SO interactions altogether, since they still
n,a′ a′ enter in the matrix elements of Γ. Then the density of
X X
(16) states becomes independent of the index a and is given
where by the following simple expression:
1
̺ (ǫ)=− ImGR (ǫ), (17) ∞
a 2π2ℓ2 na ̺(ǫ)=̺ 1+2 e−πp/ωcτ0cos 2πpǫ −πp , (22)
n 0
X " p=1 (cid:18) ¯hωc (cid:19)#
X
is the density of states and ℓ= ¯hc/eB is the magnetic
length. Writing out Eq.(17) explicitly, we have: where ̺ (ǫ) = ̺ (ǫ) = ̺(ǫ)/2 and ̺ = m/¯h2π is the
p 1 2 0
zero-field density of states. If the magnetic field is not
1
̺a(ǫ)=−2π2ℓ2 too large, i.e. if ωcτ <∼ 1, only the first term in the sum
over p in Eq.(22) needs to be retained. Then the oscil-
∞ ImΣR(ǫ) latory term in the density of states is purely sinusoidal.
× a .
[ǫ−¯hω n−(−1)a∆−¯hω δ ]2+[ImΣR(ǫ)]2 Substituting Eq.(22) in Eq.(16) we find that SCBA self-
n=0 c c a,1 a
X energyisalsoindependentoftheindexaandisgivenby:
(18)
ThesumoverLandaulevelindicesintheaboveequation ¯h π¯h3
can be done using Poissonsummation formula:19 ImΣR(ǫ)≡− =− ̺(ǫ). (23)
2τ(ǫ) 2mτ
0
∞ f(0) ∞ ∞
f(n)= + dnf(n)e2πipn. (19) The above disorder self-energy reduces to the usual ex-
2
nX=0 p=X−∞Z0 pression −¯h/2τ0 in the zero-field limit. At finite fields,
the oscillatory term in the density of states in Eq.(23)
Uptothispoint,ourcalculationswereapplicabletomag-
is at the origin of a number of magnetoquantum oscil-
netic fields of general strength. To proceed further, we
lation phenomena, including the SdH oscillations of the
will restrict ourselves to the regime when ǫ τ/¯h ≫ 1
F electrical resistivity. As will be shown later, it also leads
and ǫ ≫¯hω . (The calculation we present can be done
F c tosimilarmagnetoquantumoscillationeffectsinthespin
in other regimes as well, but the actual procedure will
relaxation rate.
be a little different.) This allows us to neglect the term
f(0)/2 in Eq.(19). Then, changing the integration vari-
able to x=h¯ω n−ǫ+(−1)a∆−¯hω δ and extending
c c a,1 III. SPIN DENSITY RESPONSE FUNCTION
the lower limit of integration to −∞, which is justified
by the above assumptions, we obtain:
Wecannowcalculatespindensityresponsefunctionof
1 ∞ ∞ dx a disordered 2DEG system with the Rashba SO interac-
̺a(ǫ)=−2π2ℓ2 ¯hω tions in a perpendicular magnetic field. Transverse and
p=−∞Z−∞ c longitudinal spin relaxation rates can be found from the
X
x+ǫ−(−1)a∆−¯hω δ ImΣR(ǫ) poles of the corresponding spin density response func-
× exp 2πip c a,1 a .
(cid:18) ¯hωc (cid:19)x2+[ΣRa(ǫ)]2 tciuolnast.ionThtheactaclcaunlabtieondoinsesiamtilzaerrotofiethlde.2a0nWaloegowuilsl cuasle-
(20)
¯h=1unitsinthe restofthissection,exceptforthefinal
results.
Calculating the above integral, we obtain:
We start from the general density matrix response
∞ function:
m
̺ (ǫ) = 1+2 e−πp/ωcτ0
a 2π¯h2 " Xp=1 χσ1σ2,σ3σ4(r−r′,t−t′)=
× cos 2πpǫ−(−1)a∆−¯hωcδa,1 , − iθ(t−t′)h ̺ˆ†σ1σ2(r,t),̺ˆσ3σ4(r′,t′) i, (24)
¯hω
(cid:18) c (cid:19)(cid:21) (cid:2) (cid:3)
(21) where ̺ˆσ1σ2(r,t) = Ψ†σ2(r,t)Ψσ1(r,t) is the generalized
density operator, whose expectation value is the density
where we have replaced ImΣR(ǫ), which appears in the matrix. As usual, the calculation is most conveniently
a
exponentafterintegration,byitszero-fieldvalue−¯h/2τ . done by analytical continuation of the imaginary time
0
The oscillatory term in the above expression is slightly responsefunction.21 Fouriertransformedimaginarytime
4
At low frequencies and low temperatures Eq.(30) can be
(a)
= + simplified to:
iΩ ̺(ǫ )
χ(q,Ω)= P(q,ǫ −iη,ǫ +Ω+iη)+ F . (31)
F F
2π 2
Thus, the problem reduces to calculating the matrix
P0(q,ǫ −iη,ǫ +Ω+iη). We will in fact be interested
F F
(b) = + only in the relaxation of uniform spin polarization and
thus will put q = 0 henceforth. The calculation of P0
inourcasecloselyresemblestheanalogouscalculationin
FIG.1: (a)SCBAequationforthedisorder-averagedGreen’s the zero-field problem.20
function; (b) Equation for the vertex function D. Dashed We start from the definition of P0 Eq.(27), using the
lines denote impurity potential correlator hV(r)V(r′)i. disorder-averagedGreen’sfunctionscalculatedinsection
II:
response function can be written as: P0 (ǫ −iη,ǫ +Ω++iη)
σ1σ2,σ3σ4 F F
1
1 = drdr′GA (r′,r,ǫ )GR (r,r′,ǫ +Ω)
χσ1σ2,σ3σ4(q,iΩ)= β Pσ1σ2,σ3σ4(q,iω,iω+iΩ), L2 Z σ3σ1 F σ2σ4 F
Xiω (25) = 1 drdr′ Ψnak(r′,σ3)Ψ∗nak(r,σ1)
where L2 Z nak,n′a′k′ ǫF −ǫna−i/2τ
X
Ψ (r,σ )Ψ∗ (r′,σ )
hPGσ1σ2,σ(3rσ′,4r(r,i−ω)rG′,iω,(irω,r+′,iiΩω)+=iΩ)i (26) × nǫ′aF′k+′ Ω−2ǫn′na′′a+′k′i/2τ 4 , (32)
σ3σ1 σ2σ4
where τ ≡ τ(ǫ ) and the wavefunctions are given by
is the polarization bubble diagram and G are imaginary F
Eq.(7). Thespatialintegralsintheaboveequationcanbe
time Green’s functions. P can be calculated by sum-
easily evaluated using the orthonormality of the Landau
ming all the SCBA diagrams for the disorder self-energy
gauge eigenfuctions and are given by:
and all the ladder vertex corrections to the polarization
bubble,whichconstitutesaconservingapproximationfor
the density matrix response function. The result can be drΨ∗nak(r,↑)Ψn′a′k′(r,↑) = va∗va′δn,n′δk,k′,
written in matrix notation as P = P0D, where P0 is Z
the “bare” polarization bubble with only the self-energy drΨ∗nak(r,↓)Ψn′a′k′(r,↓) = u∗aua′δn,n′δk,k′,
corrections included:
Z
P0 (r−r′,iω,iω+iΩ)= drΨ∗nak(r,↑)Ψn′a′k′(r,↓) = va∗ua′δn,n′+1δk,k′,
σ1σ2,σ3σ4
Z
G (r′,r,iω)G (r,r′,iω+iΩ), (27)
σ3σ1 σ2σ4 drΨ∗nak(r,↓)Ψn′a′k′(r,↑) = u∗ava′δn,n′−1δk,k′.(33)
with G here being the disorder-averaged Green’s func- Z
tion. The vertex part, or, as it is often called, diffusion The sums over Landau level indices in Eq.(32) can be
propagator,D satisfies the following matrix equation: done as follows. We first decouple the product of two
Green’s functions appearing in (32) as:
D =1+γP0D, (28)
1
where γ ≡hV2(r)i. The solution of this equation is
(ǫF −ǫna−i/2τ)(ǫF +Ω−ǫn′a′ +i/2τ)
D = 1−γP0 −1. (29) = 1
Ω+(−1)a∆−(−1)a′∆+i/τ
We can now calculate(cid:2)the retar(cid:3)ded real time response 1 1
× − .
function by analytically continuing Eq.(25) to real fre- (cid:18)ǫF −ǫna−i/2τ ǫF +Ω−ǫn′a′ +i/2τ(cid:19)
quencies. The result can be written as an integral along
(34)
the branch cut of the SCBA Green’s function:
∞ dǫ The real parts of the decoupled sums over Landau level
χ(q,Ω)= nF(ǫ)[P(q,ǫ+iη,ǫ+Ω+iη) indices can then be neglected by our assumption that
2πi
Z−∞ ǫ ≫¯hω ,λp ,¯hΩ. Theimaginarypartsaresimplypro-
F c F
− P(q,ǫ−iη,ǫ+Ω+iη)+P(q,ǫ−Ω−iη,ǫ+iη) portional to the density of states at Fermi energy ̺(ǫ ).
F
− P(q,ǫ−Ω−iη,ǫ−iη)]. (30) Thus we obtain the following expressions for the matrix
5
elements of P0: rates are given by:
γP↑0↑,↑↑ = γP↓0↓,↓↓ =(cid:18)1− 21sin2(2ϑ)(cid:19)f0(Ω) τ1z = 8λ12+m(ǫωFcττ/)¯h22,
sin2(2ϑ) 1 1
+ [f+(Ω)+f−(Ω)], = . (40)
4 τ 2τ
⊥ z
sin2(2ϑ)
γP0 = γP0 = [2f (Ω)−f (Ω)−f (Ω)],
↑↑,↓↓ ↓↓,↑↑ 4 0 + − Eq.(40) is our main result. The form of these ex-
sin2(2ϑ) pressions is rather similar to the well-known Dyakonov-
γP0 = f (Ω+ω )+sin4(ϑ)f (Ω+ω ) Kachorovskiiexpressions2forthespinrelaxationratesin
↑↓,↑↓ 2 0 c + c the zero field case, except for the factor 1/[1+(ω τ)2],
+ cos4(ϑ)f (Ω+ω ), c
− c whichdescribesthe suppressionofspinrelaxationbythe
sin2(2ϑ) perpendicular field. The mechanism of this suppression
γP0 = f (Ω−ω )+sin4(ϑ)f (Ω−ω )
↓↑,↓↑ 2 0 c − c can be understood in quasiclassical terms by imagining
+ cos4(ϑ)f (Ω−ω ), (35) electronsmoving alongclassicalcyclotronorbits andun-
+ c
dergoing impurity scattering and the Rashba spin pre-
cession. Orbital motion affects the spin precession,since
where functions f and f are given by:
0 ±
the direction of the Rashba field is always transverse
1 to the direction of electron’s velocity, and thus changes
f (Ω) = ,
0 as the electron moves along a circular cyclotron orbit.
1−iΩτ
However, if ω τ ≪ 1, impurity scattering randomizes
1 c
f±(Ω) = . (36) the direction of electron’s motion and the influence of
1−iΩτ ±i∆τ
the magnetic field on the orbital motion is then negligi-
All other matrix elements of P0 vanish. ble. On the other hand, when ωcτ ≫ 1, electrons can
move aroundcyclotron orbits almost freely and thus the
Tocalculatethetransverseandlongitudinalspinrelax-
Rashba spin precession is averaged out. This picture of
ation rates we need to find the poles of the spin density
the suppression of the DP spin relaxation by a perpen-
response function, or, equivalently, zeroth of the inverse
diffusionpropagatorD−1. Itismosteasilydonebytrans- dicular magnetic field has been discussed before by sev-
forming D−1 to the physical space of charge and spin eralauthors.5,6,7 However,wearenotawareofarigorous
derivation of a simple analytical expression for the spin
density components, which is accomplished by multiply-
relaxation rate, such as Eq.(40).
ing it on both sides with Pauli matrices:
Probably the most noteworthy feature of our result is
1 the oscillatory dependence on the magnetic field, which
D−1 = τα D−1 τβ , (37)
αβ 2 σ1σ2 σ1σ2,σ3σ4 σ4σ3 appearsduetothe oscillatoryterminthe scatteringrate
1/τ in Eqs.(22) and (23). If ω τ < 1, we may neglect
c
whereα,β =c,x,y,z andτc istheidentitymatrix. Thus the (ω τ)2 term in the denominator in Eq.(40) and leave
c
we obtain: only the first term in the sum over p in Eq.(22). In this
case spin relaxation rates will exhibit purely sinusoidal
Dc−c1(Ω) = 1−f0(Ω), SdH-like oscillations. Similar effects have been discussed
sin2(2ϑ) earlier in the context of nuclear spin-lattice relaxation
Dz−z1(Ω) = 1−f0(Ω)+ 2 [2f0(Ω) inquantumHallsystems.22,23,24 However,the possibility
of such magnetoquantum oscillations in the SO-induced
− f (Ω)−f (Ω)],
+ − spin relaxation rates has never been discussed before.
D−1(Ω) = 1− sin2(2ϑ)f (Ω+ω )−sin4(ϑ)f (Ω+ω ) Suchoscillationshaveinfactbeenobservedinrecentex-
+− 2 0 c + c perimentsonspinrelaxationinInGaAsquantumwells.25
− cos4(ϑ)f (Ω+ω ). (38) In conclusion, we have considered the problem of spin
− c
relaxationina2DEGwiththeRashbaSOinteractionsin
Clearly D−1(0) = 0, which is a consequence of charge aperpendicularmagneticfield. Wehavegivenarigorous
cc
conservation. For the spin part of the inverse diffusion microscopicderivationofthetransverseandlongitudinal
propagator,to leading order in λp τ/¯h and Ωτ, and as- spin relaxation rates from the poles of the spin density
F
suming that ∆ ≪¯hω , we obtain: response function in the limit ǫ ≫ ¯hω . It is possi-
z c F c
ble to extend our calculation to other regimes, including
τ−1D−1(Ω) = −iΩ+ 1 , the quantum Hall regime ǫF ∼¯hωc. Our results demon-
zz τ stratethatmagnetoquantumoscillationeffectsshouldbe
z
1 observed in spin relaxation phenomena in 2DEGs. One
τ−1D−1(Ω) = −iΩ+i¯h−1∆ + , (39)
+− z τ ofthe advantagesofourapproachis thatitcanbe easily
⊥
extended to study another very interesting and still un-
where the longitudinal and transverse spin relaxation exploredquestion of what is the effect of Coulombinter-
6
actions on the spin relaxation in semiconductor 2DEGs. Donald and Vanessa Sih. This work was supported
This question will be addressed in a future publication. by DARPA/ONR N00014-99-1-1096, by the NSF under
We would like to acknowledge helpful discussions grantDMR-9985255andbytheSloanandPackardfoun-
with David Awschalom, Yogesh Joglekar, Allan Mac- dations.
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