Evolution of the persistent spin helix in the presence of Hartree-Fock fields

1.6 MB·English

by  Matthias C. Lüffe

Evolution of the persistent spin helix in the presence of Hartree-Fock fields
Matthias C. Lu¨ffe,1 Jeroen Danon,1,2 and Tamara S. Nunner1
1Dahlem Center for Complex Quantum Systems and Fachbereich Physik, Freie Universita¨t Berlin, 14195 Berlin, Germany
2Niels Bohr International Academy, Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen, Denmark
(Dated: January 9, 2013)
Wederiveaspindiffusionequationforaspin-orbitcoupledtwo-dimensionalelectrongasincluding
the Hartree-Fock field resulting from 1st order electron-electron interactions. We find that the life-
timeofthepersistentspinhelix,whichemergesforequallinearRashba-andDresselhausspin-orbit
interactions, can be enhanced considerably for large initial spin polarizations due to the Hartree-
Fockfield. Thereasonisareductionofthesymmetry-breakingcubicDresselhausscatteringrateby
3 the Hartree-Fock field. Also higher harmonics are generated and the polarization of the persistent
1 spin helix rotates out of the (Sy,Sz)-plane acquiring a finite Sx-component. This effect becomes
0 more pronounced, when the cubic Dresselhaus spin-orbit interaction is large.
2

n
I. INTRODUCTION electron-electron collisions, which arise in 2nd order in
a
J the electron-electron interaction but we have neglected
the Hartree-Fock term, which arises in 1st order. This
8 In recent years the spin dynamics in semiconductors
hasattractedgreatinterest, becauseintegratingthespin approximationisvalid,whentheinitialspinpolarization
] is small and seems to be appropriate for the present ex-
ll degree of freedom into semiconducting devices opens up perimental observations of the PSH state.4 On the other
a newpossibilitiesofrealizingspintronicsorquantumcom-
hand, it is experimentally also possible to realize large
h putationaldevices.1 Forthesepotentialapplicationslong
initialspinpolarizations,12 wheretheHartree-Fockterm,
- spinlifetimesarehighlydesirable. Inthisrespecttheper-
s which acts like an effective magnetic field pointing along
e sistentspinhelix(PSH),2,3aspin-densitywavestatewith
the local spin polarization, is known to enhance the spin
m ideally infinite lifetime, is of great interest. This state
lifetime considerably.12–14 Therefore it is the purpose of
. exists in two-dimensional electron systems with Rashba
t the present paper to analyze the effect of the Hartree-
a and linear Dresselhaus spin orbit interactions of equal
Fock term on the PSH in the diffusive D’yakonov Perel’
m magnitude due to a SU(2) symmetry of the underlying
regime. In particular we want to answer the following
Hamiltonian.3 On a less abstract level, the PSH state
- questions: (i) what is the effect the Hartree-Fock term
d emerges because for Rashba and linear Dresselhaus spin
n on the lifetime of the PSH-state and (ii) does it modify
orbit interactions of equal magnitude the resulting spin
o the pattern of the spin-polarization of the PSH-state?
orbit interaction points into the same direction for all
c This article is organized as follows: In Sec. III we
[ electronicmomentumstates. Themagnitudeofthespin-
orbitinteractiongrowslinearlywiththeprojectionofthe presentthespindiffusionequationforatwo-dimensional

1 electron gas with Rashba- and Dresselhaus spin-orbit in-
momentum on the PSH wave-vector, such, that all spins
v teractions including the Hartree-Fock field, which arises
rotate by the same angles when they diffuse along the
2 from 1st order electron-electron interaction. In Sec. IV
PSH direction.
6
we show how the presence of the Hartree-Fock term en-
6 InrealsystemsthelifetimeofthePSHisnolongerin-
hances the lifetime of the PSH state. Furthermore we
1 finiteduetothepresenceofadditionaltermswhichbreak
demonstrate that the presence of the Hartree-Fock term
.
1 theSU(2)symmetryoftheHamiltonian. Ithasbeensug- also generates higher harmonics in the spin polarization.
0 gested, that the presence of cubic Dresselhaus spin orbit
In particular, we show that the 2nd harmonic corre-
3 interaction is the dominant symmetry-breaking mecha-
sponds to an S -component and thus rotates the spin-
1 nism leading to a finite lifetime of the PSH state.4,9,10 x
: polarization out of the (Sy,Sz)-plane. The appendices
v Experimentally, the PSH state has been observed in
contain details about the derivation of the Hartree-Fock
i GaAs/AlGaAs quantum wells,4–7 where the cubic Dres-
X term (appendix A) and of the spin diffusion equation
selhaus spin orbit interaction is quite small, as well as in
(appendix B).
r (In,Ga)As quantum wells,8 where the cubic Dresselhaus
a
spin orbit interaction is much larger.
The observed temperature dependence of the life-
time of the PSH state4 suggests, that also electron- II. MODEL
electron interactions strongly affect the lifetime of the
PSHstate.10,11 Inourpreviouswork10 wefoundthatthe We describe conduction band electrons in a (001)
considerationofelectron-electroninteractionincombina- grownquantumwellwithRashba-andDresselhausspin-
tion with cubic Dresselhaus spin orbit interaction as the orbit interactions, potential impurities and electron-
symmetry breaking mechanism can fairly well account electron interactions. For simplicity we focus on the
fortheobservedtemperaturedependenceofthePSHlife- limit of zero temperature, where the 2nd order electron-
time. Inthisprevioustreatmentwehaveonlyconsidered electron scattering vanishes but the 1st order contribu-
2
tionduetoelectron-electroninteraction,i.e.,theHartree- Finally,theHamiltonian(1)containselectron-electron
Fock term, is finite. Although our treatment is strictly interactions,
valid only at zero temperature, we expect, that qualita-
1 (cid:88)
tively similar results hold also for small but finite tem- H = V ψ† ψ† ψ ψ (8)
peratures, because the temperature dependence of the e-e 2V k3,k4,k1,k2 k4s2 k3s1 k1s1 k2s2
k1...k4
Hartree-Fock term is weak and the presence of 2nd or- s1,s2
der electron-electron interactions does not qualitatively with a Thomas-Fermi screened Coulomb potential of the
affect the PSH state.10 form V ≈ v˜(|k − k |)δ where
k3,k4,k1,k2 3 1 k1+k2−k3−k4,0
theWfeollcoownisnidgeursteh(cid:126)e=fol1lo):wing Hamiltonian (here, and in v˜(q) = εqm2πqa∗ and εq ≈ 1 + q2a∗ with a∗ = 4πmεe02εr
being the effective Bohr radius.
Following the procedure of our previous treatment of
H =H +H +H , (1)
0 imp e-e thePSH-state10weusetheNonequilibriumstatisticalop-
eratormethod16toderivekineticequationsforthecharge
where the first term represents a two-dimensional elec-
and spin components of the Wigner-transformed density
tron gas (2DEG) with a quadratic dispersion and intrin-
matrix
sic spin-orbit interaction
ρˆ (x,t)=n (x,t)+s (x,t)·σ. (9)
(cid:88) k k k
H = ψ† H ψ (2)
0 ks(cid:48) 0s(cid:48)s ks
Since, to zeroth order in b/E , the kinetic equations for
s,s(cid:48);k F
spin and charge decouple we only need to consider the
and the 2×2 matrix in spin space spin component and find for the spin density s :
k
H0 =(cid:15)k+b(k)·σ. (3) ∂tsk+v·∂xsk+2sk×b(k)=Jikmp+Jeke(1) (10)
with v = k /m, where the index i = x,y labels the in-
The ψk†s(ψks) are creation (annihilation) operators for plane sipatiail directions. For simplicity, we focus here
electrons with momentum k and spin projection s. on the limit of zero temperature, where the 2nd or-
Within the standard envelope function approximation15 der electron-electron collision integral vanishes and we
one finds (cid:15)k = 2km2 where m is the effective mass. The keep only the first order electron-electron collision inte-
vectorofPaulimatricesisdenotedbyσ andthein-plane gralJee(1). AsweshowinappendixA,Jee(1) canbecast
k k
spin-orbit field intoaneffectivemagneticfield,i.e.,aHartree-Fockterm,
arisingfromthemeanpolarizationofthePSH-stateitself
b(k)=b (k)+b (k)+b (k) (4)
R D cD
(cid:90) dq
Jee(1) =−2s × v˜(q)s . (11)
contains Rashba spin-orbit interaction k k (2π)2 k+q
 k  The impurity collision integral in Eq. (10) is given by:
y
bR(k)=αvF −kx (5) Jimp =−(cid:88)W δ((cid:15) −(cid:15) )(s −s ), (12)
0 k kk(cid:48) k k(cid:48) k k(cid:48)
k(cid:48)
as well as linear and cubic Dresselhaus spin-orbit inter- where the transition rate is given by W =
kk(cid:48)
actions 2πn |v(k(cid:48)−k)|2 with the impurity concentration n .
i i
   
k sin3θ
y k3
bD(k)=βvF kx, bcD(k)=γvF 4 −cos3θ. III. SPIN-DIFFUSION EQUATION INCLUDING
0 0 HARTREE-FOCK FIELDS
(6)
Here, vF is the Fermi velocity, the angle θ gives the di- In the diffusive strong scattering D’yakonov-Perel’
rection of k with respect to the x axis, which we choose regime b(k )τ (cid:28) 1 (where τ is the momentum relax-
F
to point along the (110)-crystal direction. ation time) one can derive a diffusion equation for the
Furthermore, we have included in Eq. (1) scalar spin density of the form (see appendix B for details)
electron-impurity interactions,
∂ S=DˆS+H +H . (13)
t 1 3
1 (cid:88)
H = ψ† U ψ , (7) WefocusforsimplicityonequalRashbaandDresselhaus
imp V k(cid:48)s k(cid:48)k ks
s;k,k(cid:48) spin-orbitinteractionsα=β,i.e.,theusualconditionfor
the PSH state. In this case we find for the matrix Dˆ
arising from scattering of electrons at potential impuri-
tpioetsenVtiimalp(oxf)ea=ch(cid:80)sinigvl(exim−pRurii)t,ywahnedre{Rvi(}x)ardeetnhoeteims pthue- Dˆ =D˜∂y20−γ˜cdD˜(∂y2−0q02)−γ˜cd 2q00D˜∂y  (14)
rity positions, eventually to be averaged over. 0 −2q D˜∂ D˜(∂2−q2)−2γ˜
0 y y 0 cd
3
where q = 4mv α is the PSH wave-vector. This corre- with B →0) reads
0 F 1,3
sponds to the usual spin diffusion matrix in the absence
S =0 (21)
of Hartree-Fock fields but with a renormalized diffusion x
constant D˜ and a renormalized cubic Dresselhaus scat- S =−S sin(q y)e−t/τPSH
y 0 0
tering rate γ˜cd, given by (cid:32)(cid:114) γ2 γ (cid:33)
S =S 1+ cd − cd cos(q y)e−t/τPSH
D γ z 0 Γ2 Γ 0
D˜ = , γ˜ = cd , (15)
1+(2B τ)2 cd 1+(2B τ)2 (cid:16) γ (cid:17)
1 3 ≈S 1− cd cos(q y)e−t/τPSH,
0 Γ 0
where D = 1v2τ is the ordinary diffusion constant and
2 F where the lifetime τ is given by
γ = 1v2γ2k6τ isthecubicDresselhausscatteringrate. PSH
cd 8 F F
We also introduced the effective Hartree-Fock fields 1 1 3 1(cid:113) 3
= Γ+ γ − Γ2+γ2 ≈ γ . (22)
τ 2 2 cd 2 cd 2 cd
PSH
B =χ S, (16)
1,3 1,3
Here, we have denoted the D’yakonov-Perel’ relaxation
whicharetheHartree-Fockfieldsforthewindingnumber rate by Γ = 4q2D and the approximate expressions
0
one (s±1) and winding number three (s±3) components for Sz and τPSH are valid to lowest order in γcd/Γ ∝
of the spin density [bcD(kF)/bR(kF)]2, which is basically the square of the
ratio of the cubic Dresselhaus spin orbit interaction over
s = (cid:88) s (k)eilθ. (17) the linear spin orbit interactions.
k l
Thepurposeofthepresentpaperistoinvestigatehow
l=0,±1,±3
the pattern and the lifetime τ of the PSH state are
PSH
modified in the presence of the Hartree-Fock field aris-
For χ we find (see appendix A and appendix B)
1,3
ing from first-order electron-electron interactions. In
4π (cid:90) dq (cid:18) qn (cid:19) Sec. IVA we present our results concerning the lifetime,
χ = v˜(|q−k|)f(cid:48)((cid:15) ) 1− cos(nθ ) , based on numerical solution of the diffusion equation
k,n m (2π)2 q kn qk
(13). Then, in Sec. IVB, we present a more analyt-
(18)
ical analysis of Eq. (13). We will simplify the prob-
where θ is the angle between k and q. Assuming a
qk lem and investigate the diffusion equation assuming (i)
Fermi temperature of T = 400K, an effective mass of
F the limit of small cubic Dresselhaus spin-orbit interac-
m = 0.067m and a dielectric constant of ε = 12.9
eff e r tion, γ /Γ (cid:28) 1, and (ii) a time-independent helical
(for GaAs) one finds χ ≡ χ = 34.7cm2/s and χ ≡ cd
1 kF,1 3 magneticfieldinsteadoftheself-consistentHartree-Fock
χ =43.2cm2/s at T =0.
kF,3 field. These assumptions allow for an analytical ap-
The two other terms in (13) read in the limit α=β
proach, making use of the smallness of the parameter
γ /Γ. Although corresponding to a qualitatively differ-
H1 =2D˜(cid:8)B1τ ×(∂y2S)+q02(B1τ ·eˆx)(S×eˆx) (19) encdt situation, this approach provides some insight in the
+q ∂ [B τ ×(S×eˆ )]+q eˆ ×(∂ S×B τ) general structure of the solutions of Eq. (13). Finally, in
0 y 1 x 0 x y 1
Sec. IVC we present full numerical solutions of Eq. (13)
+2q (B τ ·∂ S)(B τ ×eˆ )
0 1 y 1 x
now focusing on the spatial pattern of the PSH for dif-
(cid:9)
+2∂y[B1τ(B1τ ·∂yS)] , ferent initial spin polarizations and for small and large
H = −2γ˜ (B τ,B τ,0)×S. (20) cubic Dresselhaus spin-orbit interaction.
3 cd 3x 3y
Due to the presence of the terms H the spin diffu-
1,3
sionequation(13)becomesanonlinearpartialdifferential IV. MODIFICATIONS OF THE PSH STATE
equation,becausetheHartree-FockfieldsB dependon
1,3
the solution for the spin density S (see Eq. (16)). The A. Lifetime enhancement of the PSH state
exactsolutioncanthusonlybefoundnumerically. Note,
thatforthederivationofthespindiffusionequationonly Our main finding is, similar to previous work,12,13
b(kF)τ (cid:28)1 was required but no restriction applies with that the PSH lifetime is enhanced considerably due to
respecttothemagnitudeoftheHartree-FockfieldsB1,3, the Hartree-Fock field arising from first-order electron-
since those fields have winding number zero. electroninteractions. Theintuitiveexplanationissimply
In our previous treatment10 of the PSH state in the that the Hartree-Fock field is parallel to the spin polar-
presence of cubic Dresselhaus spin-orbit interaction we ization of the helix state and thus strengthens the spin
have found that the PSH state survives in the presence polarization. Since, experimentally this can e.g. be real-
of cubic Dresselhaus spin-orbit interaction but the am- ized by increasing the initial spin polarization, i.e., the
plitudesofthespinpolarizationofthePSH-stateacquire amplitude of the PSH state, we show in Fig. 1 the time
a small ellipticity and a finite lifetime. The PSH eigen- evolutionoftheS spinpolarizationforinitialspinpolar-
z
modeof(14)intheabsenceoftheHartree-Fockfield(i.e. izationsofthePSHstatewith10%(blue)and20%(red)
4
B. Analytical approach for γ /Γ(cid:28)1
cd
Whereas the full nonlinear solution for the spin polar-
without Hartree Fock
10% spin polarization ization in the presence of a self-consistent Hartree-Fock
0.8
20% spin polarization field can only be obtained numerically, we can simplify
the problem and work with a static magnetic field in-
0.6
stead. We take as an ansatz for the PSH state
S
 
0.4 0
SPSH =S0 −sinq0y , (23)
0.2 (1−x)cosq y
0
0 where x parametrizes the ellipticity of the spin polariza-
0 10 20 30 40 50
t/t tion due to the presence of cubic Dresselhaus spin-orbit
PSH
interaction (see Eq. (21)). We assume the fields B to
1,3
be static and proportional to this polarization profile,
FIG. 1. Time dependence of the S -amplitude of the PSH-
z
state for different initial spin polarizations (solid lines). For S
comparison the time evolution without Hartree-Fock field is B1,3 =B1,3|SPSH|. (24)
shown in black. Dashed lines indicate, how the time depen- PSH
denceoftheS -amplitudeofthePSH-statewouldlooklike,if
z If we plug the ansatz (23) and the fields (24) into the
the Hartree-Fock field remained constant at its initial value.
diffusion equation (13), we find that with
γ˜ 1 γ˜ 1
x= cd ≡ cd , (25)
4q2D˜ 1+ 1(B τ)2 Γ˜ 1+ 1(B τ)2
0 2 1 2 1
using the parameters of the quantum well of Ref. 4: a
the instanteneous time-evolution of the PSH state reads
Fermi temperature of T =400K, an effective electronic
F to lowest order in γ /Γ
mass of m = 0.067m , linear Rashba and Dresselhaus cd
eff e
spin orbitcouplings of α=β =0.0013 anda cubic Dres-
S S S
selhaus spin-orbit interaction of γv = 5.0eV˚A3 as well ∂ S =− PSH + 2x + 3yz, (26)
F t PSH τ˜ τ τ
as a momentum relaxation time of τ = 1ps. In the sim- PSH 2x 3yz
ulations we always assume that at time t = 0 the elec-
where
tronic spin polarization corresponds to the PSH state of
Eq.(21). Wethensolveforthetime-dependentpolariza- 1 3 3 γ
= γ˜ = cd , (27)
tionS(t)withtheHartree-Fockfieldswitchedonattime τ˜ 2 cd 21+(2B τ)2
PSH 3
tpo=la0r.izaFtoironcoomfpthaerisPoSnHthsetattiemweiethvoouluttHioanrtorfeeth-FeoSckz fispelidn S S (cid:18) B τ (cid:19)sin(2q0y)
isalsoshown(inblack). Obviously,forrealisticquantum τ2x = 3τ˜ 0 B3τ − 1+ 1(1B τ)2  0 , (28)
well parameters the lifetime of the PSH state can easily 2x PSH 2 1 0
 
be enhanced by factors of 10 to 20 when the initial spin 0
S S (B τ)2
polarization is in the percentage range. τ3yz = 3τ˜ 0 1+ 11(B τ)2 −sin(3q0y). (29)
3yz PSH 2 1 cos(3q0y)
The main cause for the lifetime enhancement of the
PSH-state can be traced back to the reduction of the ef- This equation thus describes two effects: (i) The state
fective cubic Dresselhaus scattering rate γ˜ in Eq. (15) S taken as ansatz tends to relax towards S=0 with
cd PSH
in the presence of a Hartree-Fock field. Since it is the aninstantaneousrelaxationtimeτ˜ ,whichisenhanced
PSH
SU(2)-symmetry breaking cubic Dresselhaus spin-orbit in comparison to the bare τ due to the reduction of
PSH
interaction, that renders the lifetime of the PSH state thecubicDresselhausscatteringrateinthepresenceofa
finite in the first place, a reduction of the effective cu- helical magnetic field. (ii) The polarization starts to de-
bicDresselhausscatteringrateconsiderablyenhancesthe velop higher harmonics (with wave vectors 2q and 3q ),
0 0
lifetimeofthePSHstate. BecausetheHartree-Fockfield including an x-component which was not present before
is proportional to the spin polarization itself, its mag- (cf. Eq. (21)).
nitude and therefore its ability to suppress the effective As we will show below, the magnitudes of the higher
cubic Dresselhaus scattering rate γ˜ decreases with de- harmonics are smaller by factors γ /Γ (see Eq. (33)).
cd cd
creasing spin polarization. As a consequence the spin This means that in the limit of γ /Γ (cid:28) 1 we can ne-
cd
polarization (solid lines) in Fig. 1 decays faster than the glect the contributions of S and S in (26) to first
2x 3yz
exponential decay (dashed lines) expected for a constant approximation, and Eq. (26) becomes valid at all times
field. (at least for constant magnetic fields B ). In this case
1,3
5
one can solve the equation easily to find the simple ex- Sy
ponential decay
S (t)=S e−t/τ˜PSH. (30)
PSH PSH
This exponential solution for constant fields is indicated
by dashed lines in Fig. 1. For the actual time-dependent
Sz
Hartree-Fock field (solid lines in Fig. 1) the lifetime is of
course shorter than τ˜ of Eq. (27): as soon as the spin
PSH
S(y+dy)
polarization of the PSH state starts to decay, the fields
B decreaseproportionally,whichresultsinanincrease S(y)
1,3
S(y−dy)
of the decay rate 1/τ˜ of the PSH state.
PSH
Let us now turn to the generation of higher harmonics
in(26),whichisduetothenonlinearityofthespindiffu-
FIG. 2. For equal amplitudes S(y + dy) = S(y − dy)
sionequation. Inparticularasecondharmonicsin(2q y)-
0 the contribution of the winding number one spins to the S -
term appears in the S component (see Eq. (28)) and a x
x componentdiffusingfromy+dy andfromy−dy tothepoint
third harmonic sin(3q y)-, cos(3q z)-term appears in the
0 0 y would cancel, because they rotate around the same local
Sy- and Sz-components (see Eq. (29)). Of course, this is HartreeFockfieldB(y)butcorrespondto±k-statesandthus
not the end of the story, since, in order to obtainthe full are oriented oppositely. Due to the elliptical profile of the
solution, the second and third harmonics would have to PSHamplitudetheamplitudesS(y+dy)andS(y−dy)differ
be included into the initial expression leading to a suc- and thus a finite Sx-contribution emerges.
cessive generation of all higher harmonics.
The most surprising and at first counterintuitive find-
ingisthegenerationofafiniteSx-componentofthespin to rise from zero to its maximum value within a short
helix, i.e., the observation that a Hartree-Fock field par- switch on period given by 1/Γ˜ (cid:28)τ˜ .
PSH
allel to the local spin polarization rotates the spin po-
Can we understand the physical origin of this sec-
larization of the PSH out of the yz-plane by creating
ond harmonic S -component? Normally, one would ex-
an additional out-of-plane S -component. In order to x
x pectthataparallelmagneticfieldsimplystrengthensthe
estimatethemagnitudeofthisnewsecondharmonicS -
x parallel spin orientation. In the presence of spin-orbit
component of the PSH-state, we modify the ansatz of
fields, however, the spin density consists of parts with
the initial PSH state of Eq. (23) to include the second
different winding numbers, where only winding number
harmonic S -component as
x zero contributes to the local spin polarization while the
  other winding numbers (one and three) average out to
s sin2q y
x 0 zero. Since the anisotropic components of the spin den-
SPSH2x =S0 −sinq0y . (31) sity(windingnumbersoneandthree)arenotnecessarily
(1−x)cosq y
0 paralleltothelocalspindensityandthusnotnecessarily
parallel to the Hartree-Fock field, they can modify the
Pluggingthisansatzintothespindiffusionequation(13)
pattern of the PSH state by rotations around the local
onefindsforthex-componentofS toleadingorder
PSH2x Hartree-Fock field.
in γ /Γ
cd
TheexpressionforthesecondharmonicS -component
x
γ˜ (cid:16) B τ (cid:17) Γ˜ in the presence of a constant Hartree-Fock field (see
∂t(S0sx)=S0 2cd B3τ − 1+ 1(1B τ)2 − 4S0sx. (32) Eq. (33)) suggests that both winding number one and
2 1 windingnumberthreepartsofthespindensitycontribute
to the formation of the S -component. The winding
We see that the second harmonic has a fast relaxation x
rate Γ˜ compared to the time scale 1/γ˜ of the other number one part contributes only via the diffusion term.
cd
Atonefixedpositionyelectronicspinsarriveviadiffusion
terms. Let us solve (32) for s (0) = 0, i.e., starting
x
fromy+δywithamplitudeS(y+δy)andfromy−δywith
from the PSH ansatz of Eq. (23). Assuming that the
amplitude S(y−δy). While diffusing to position y both
spin polarization of the PSH state S remains constant
0
spinsarrivingfromy±δyrotatearoundthelocalHartree-
in time we find
Fock field, which is parallel to S(y) and thus acquire a
sx(t)= γ˜2cΓ˜d(cid:16)B3τ − 1+B1(1Bτ τ)2(cid:17)(cid:16)1−e−14Γ˜t(cid:17). (33) (fiSneieteFSigx.-c2o).mSpionnceentth,esainncgeleSbiestwpeoelanrSiz(eyd+iδnyt)haenydzS-p(yla)nies
2 1
equaltotheanglebetweenS(y)andS(y−δy)theresult-
From this simple consideration we see that the fast re- ing S -components are of opposite sign an would cancel
x
laxation rate of s has two effects: (i) It makes the equi- for equal magnitudes S(y +δy) = S(y −δy), i.e., for a
x
librium amplitude of the S -component very small for PSHstatewithequalS andS amplitudes. However,in
x y z
small cubic Dresselhaus spin-orbit interaction, since it is the presence of cubic Dresselhaus spin orbit interaction
proportional to γ˜ /Γ˜. (ii) It causes the S -component the initial PSH profile is elliptical with the S and S
cd x y z
6
0.004 0
0.002 -0.02
S/Sx0 0 S/Sx0-0.04
-0.002 -0.06
-0.0040 10 20 30 40 50 -0.080 10 20 30 40 50
t/t t/t
PSH PSH
FIG. 3. Time dependence of the S -amplitude, which FIG. 4. Time dependence of the S -amplitude at first max-
x x
arises solely due to the Hartree Fock term, at first maximum imum q y=π/2 (left) and spatial and temporal evolution of
0
q y=π/2(left)andspatialandtemporalevolutionoftheS - the S -amplitude (right) of the PSH state for an initial spin
0 x x
amplitude of the PSH state (right) for an initial spin polar- polarizationof20%andwithlinearspin-orbitinteractionsre-
izationof20%andwithlinearspin-orbitinteractionsreduced duced by a factor of 0.02. (γ /Γ=6.8 and τ =324ps).
cd PSH
by a factor of 0.2. (γ /Γ=0.067 and τ =412ps).
cd PSH
amplitudes differing by approximately γ /Γ. Thus the
cd
amplitudes of the electronic spins arriving from y ±δy
differ due to the elliptical PSH profile leading to the for-
mationofafiniteS -component(seeFig.2). Onlywhen
x
the local spin polarization is parallel to one of symmetry
axes of the ellipse, i.e., for S = 0 or S = 0 the ampli-
y z
tudeoftheelectronicspinsarrivingfromy±δy areequal
and thus S =0 at these points. Since this situation oc-
x FIG.5. SpatialandtemporalevolutionoftheSz-component
cursforq0y =0,π/2,π,3π/2thisexplainstheoscillation (left) and the Sy-component (right) of the PSH state for the
of Sx with double wave vector 2q0. same parameters as used in Fig. 4.
The winding number three part of the spin distri-
bution function contributes to the formation of an S -
x
component via combined rotations around the local themagnitudeofcubicDresselhausspin-orbitinteraction
Hartree-Fock field and the component of the cubic Dres- (see Eq. (21), where one finds τ−1 ≈ 3γ for γ (cid:28) Γ
PSH 2 cd cd
selhaus spin orbit field parallel to eˆy. Again a finite and τP−S1H ≈ γcd for γcd (cid:29) Γ), this does not significantly
contribution to Sx arises only when the local spin po- affect the absolute lifetime of the PSH state. Alterna-
larization is neither parallel nor perpendicular to eˆy (the tively, the regime of large cubic Dresselhaus spin-orbit
direction of the relevant component of the cubic Dres- interaction could also be reached by using an (In,Ga)As
selhaus spin orbit field) thus leading to an oscillation of quantumwellinstead, orbyincreasingtheFermienergy.
the winding number three contribution to Sx with dou- InFig.3weshowthetemporalandspatialevolutionof
ble wave vector 2q0 similar to the winding number one the Sx-component of the PSH state, which arises solely
case.17 due to the Hartree-Fock term, for only slightly enhanced
cubic Dresselhaus spin orbit interaction. This situation
has been reached by a reduction of the linear spin orbit
C. Modification of PSH pattern interactions by a factor of 0.2, which results in a ratio of
γ /Γ=0.067. AsexpectedfromEq.(33)themagnitude
cd
Since the magnitude of the S -component as well as of the S -component is quite small and its spatial vari-
x x
the other higher harmonics are of higher order in γ /Γ, ation is characterized by a sin(2q y)-dependence. The
cd 0
they will become significant for systems with large cu- initial switch on process now exhibits an oscillatory be-
bic Dresselhaus spin orbit interaction. Therefore it is havior and, contrary to the S - and S -components, the
y z
interesting to investigate how the spatial and temporal magnitudeoftheS -componentsaturatesbeforeitdrops
x
evolution of the PSH state is modified when cubic Dres- to zero quite abruptly.
selhaus spin orbit interaction is large γ ≈Γ. Since the As an example for large cubic Dresselhaus spin orbit
cd
magnitudeofthecubicDresselhausspinorbitinteraction interaction we show in Fig. 4 the spatial and temporal
is given by the crystal symmetry of GaAs, whereas the evolution of the PSH state for linear spin-orbit interac-
magnitudes of the linear Rashba and Dresselhaus spin tions reduced by a factor of 0.02, which results in a ratio
orbit interactions α and β can be varied by changing the of γ /Γ = 6.8. The magnitude of the S -component is
cd x
dopingasymmetryandthewidthofthequantumwell,we nowquitesizable, reachingalmost8%ofthePSHampli-
could access the γ ≈ Γ-regime by reducing the magni- tude at intermediate times. Obviously, the spatial vari-
cd
tudeofthelinearspinorbitinteractionswhilekeepingthe ation is no longer characterized by a simple sin(2q y)-
0
magnitudeofthecubicDresselhausspin-orbitinteraction dependence but even higher harmonics start to play a
fixed. Since the lifetime of the PSH depends mainly on prominent role. For comparison the spatial and tempo-
7
ral evolution of the S - and S -components are shown in 2nd order electron-electron interaction vanishes we con-
y z
Fig. 5. Due to the large ratio of γ /Γ = 6.8 also the sider the kinetic equation only up to first order in V,
cd
amplitude of the S -component is strongly reduced and yielding:
z
the presence of higher harmonics is clearly visisble.
∂ (cid:104)ψ†ψ (cid:105)t−i(cid:104)[H ,ψ†ψ ](cid:105)t =J(1)(t), (A1)
t l(cid:48) l 0 l(cid:48) l ll(cid:48)
with the first order mean field term:
V. CONCLUSIONS
J(1)(t)=i(cid:104)[V,ψ†ψ ](cid:105)t . (A2)
ll(cid:48) l(cid:48) l rel
Basedonaspin-coherentBoltzmanntypeapproachwe
have derived a semiclassical spin diffusion equation for a Weevaluatethemeanfieldterm(A2)fortheelectron-
two-dimensional electron gas with Rashba- and Dressel- electron interactions of Eq. (8). Evaluation of the com-
haus spin-orbit interactions including the Hartree-Fock mutator in Eq. (A2) yields in this notation
field, which arises in first order in the electron-electron
interactions. Inagreementwithprevioustreatments12–14 Jee(1)=i(cid:104)[H ,ψ† ψ ](cid:105) (A3)
11(cid:48) e-e 1(cid:48) 1
we find that the Hartree-Fock term considerably en- 1 (cid:88) (cid:16) (cid:17)
hancesthelifetimeofthepersistentspinhelixstate,when =iV V2(cid:48)3(cid:48)23 δ13(cid:48)(cid:104)ψ1†(cid:48)ψ2†(cid:48)ψ2ψ3(cid:105)−δ1(cid:48)3(cid:104)ψ3†(cid:48)ψ2†(cid:48)ψ2ψ1(cid:105) .
the initial spin polarization is in the percentage range. 2(cid:48)3(cid:48)23
Within our approach we are able to assign the origin of
Upon Wick decomposition of the averages
the lifetime enhancement to the suppression of the ef-
fective cubic Dresselhaus scattering rate by the Hartree (cid:104)ψ†ψ†ψ ψ (cid:105)=−(cid:104)ψ†ψ (cid:105)(cid:104)ψ†ψ (cid:105)+(cid:104)ψ†ψ (cid:105)(cid:104)ψ†ψ (cid:105)
1 2 3 4 1 3 2 4 1 4 2 3
Fock field.
=−δ δ f (k )f (k )
Surprisingly, notonlythelifetimebutalsothepattern k1k3 k2k4 s1s3 1 s2s4 2
+δ δ f (k )f (k ) (A4)
of the spin polarization is modified in the presence of a k1k4 k2k3 s1s4 1 s2s3 2
Hartree Fock field. Due to the nonlinearity of the prob- and by exploiting the Kronecker symbols we obtain a
lem higher harmonics in the spin polarization are gener- mean field term that is diagonal in momentum but re-
ated. The second harmonic corresponds to a spin polar- mains a matrix in spin space
izationinS -directionandthusrotatesthespinpolariza-
x
tion of the persistent spin helix state out of the (Sy,Sz)- Jee(1)(k)= 1 (cid:88)v˜(|k(cid:48)−k|)[f(k(cid:48))f(k)−f(k)f(k(cid:48))] .
plane. These modifications of the spin pattern of the ss(cid:48) iV ss(cid:48)
k(cid:48)
persistent spin helix become significant when the cubic
(A5)
Dresselhausspin-orbitinteractionisofsimilarmagnitude
By decomposing the spin space matrices according to
asthelinearspin-orbitinteractions. Sincethemagnitude
ofthecubicDresselhausspin-orbitinteractionisfixedby fˆ(k)=n +σ·s , Jˆee(1) =Jee(1)+σ·Jee(1), (A6)
crystal symmetry, we have suggested that this regime k k k k k
could be experimentally reached by reducing the magni- one finds that the mean field term entering on the right-
tude of the linear spin-orbit interactions via appropriate hand side of the kinetic equation for the spin density
design of the quantum wells. (Eq. (10)) becomes:
(cid:90) dq
Jee(1) =−2s × v˜(q)s . (A7)
ACKNOWLEDGMENTS k k (2π)2 k+q
We would like to thank P.W. Brouwer for helpful dis-
Appendix B: Spin diffusion equation
cussions. This work was supported by SPP 1285 of the
DFG and by the Alexander von Humboldt Foundation.
We follow the approach used in our previous work10
to set up a diffusion equation for the spin density s
k
validinthediffusiveD’yakonov-Perel’regime,i.e.,inthe
Appendix A: Hartree Fock term
regime b τ (cid:28)1 where due to strong scattering the spin
F
polarization is stabilized. In order to solve equation (10)
ForageneralnonequilibriumproblemwiththeHamil-
for the spin density we expand the spin density into k-
tonian H = H + V containing an exactly solvable
0 space winding numbers
single-particle part H and an interaction V, Zubarev’s
0
Nonequilibrium statistical operator formalism16 allows s = (cid:88) s (k)eilθ, (B1)
k l
to derive a closed set of equations that describe the
l=0,±1,±3
irreversible temporal evolution of the density matrix
(cid:104)ψ†ψ (cid:105)t =Tr[ρ(t)ψ†ψ ]. Herel,l(cid:48) arepossiblycomposed where we keep besides the isotropic component s =
l(cid:48) l l(cid:48) l 0
indices, e.g., for momentum and spin. For details of the −2πf(cid:48)((cid:15) )S anisotropic components with winding num-
m k
derivation, see Ref. 16. Since for zero temperature the bers ±1 and ±3. Other winding numbers would be of
8
higher order in b(k )τ and therefore need not to be con- where B = χ S is the Hartree-Fock field felt by the
F 1 1
sidered in the diffusive regime. winding number one spins and τ is the effective relax-
1
Using this expansion of the spin density, the 1st order ation time for the winding number one part of the spin
electron-electron collision integral (Eq. (A7)) reduces in distribution function. Although in our model τ1 = τ,
the diffusive regime to: thisisnottrueingeneral,e.g. considerationof2ndorder
electron-electronscattering10 willaddanadditionalterm
toτ . Thereforeweusethemoregeneralτ inallourex-
Jeke(1) =−S×(χk,1sk,±1+χk,3sk,±3), (B2) pres1sions in order to make an extension to1including e.g.
2nd order electron-electron interaction straightforward.
where χ and χ have been defined in Eq. (18). Solving Eqs. (B6) one finds
1 3
Plugging the Hartree-Fock precession term (B2) back s =−τ˜ {v∂ s −2b ×s +2τ v(B ×∂ s ) (B7)
c 1 x 0 c 0 1 1 x 0
into the equation for the spin density (10) one finds for −4τ B ×(b ×s )+4τ2vB (B ·∂ s )(cid:9)
the isotropic components of the spin density 1 1 c 0 1 1 1 x 0
s =−τ˜ {v∂ s −2b ×s +2τ v(B ×∂ s )
s 1 y 0 s 0 1 1 y 0
v v −4τ B ×(b ×s )+4τ2vB (B ·∂ s )(cid:9),
∂ s =− ∂ s − ∂ s −s ×b −s ×b −s ×b −s ×b 1 1 s 0 1 1 1 y 0
t 0 2 x c 2 y s c c s s c3 c3 s3 s3 where we have used B1||s0 and
(B3)
τ
with τ˜ = 1 . (B8)
1 1+(2B τ )2
1 1
sc =s1+s−1, sc3 =s3+s−3 (B4) Similarly,thesteadystateofthewindingnumberthree
s =i(s −s ), s =i(s −s ) anisotropiccomponentsofthespindensityisgovernedby
s 1 −1 s3 3 −3
the following equations
and the spin-orbit fields sc3 =2b ×s +2B ×s (B9)
τ c3 0 3 c3
3
s
b =v k(−α+β)eˆ , b =v k(α+β)eˆ , (B5) s3 =2b ×s +2B ×s ,
c F y s F x τ s3 0 3 s3
3
k3 k3
bc3 =−γvF 4 eˆy, bs3 =γvF 4 eˆx. where B3 = χ3S is the Hartree-Fock field felt by the
winding number three spins and τ is the effective relax-
3
ation time for the winding number three part of the spin
In order to obtain a closed equation for s0 one has to distribution function. Although in our model τ3 = τ we
determinesc,ss,sc3 andss3 fromtheanisotropiccompo- keep the more general τ3 in all our expressions based on
nentsoftheBoltzmannequation. Inthediffusiveregime the same reasoning as detailed for τ below Eq. (B6).
1
itissufficienttofindthe(quasi)-equilibriumsolutionsfor Solving the equations for s and s one finds
c3 s3
the anisotropic coefficients, which are obtained by omit-
ting the time derivative of the anisotropic components. sc3 =2τ˜3{bc3×s0+2τ3B3×(bc3×s0)} (B10)
The justification for doing so is that, in order to capture s =2τ˜ {b ×s +2τ B ×(b ×s )},
s3 3 s3 0 3 3 s3 0
the slow precession-diffusion dynamics of the real space
density, we can interpret the time derivative as a coarse- where we have used that B3||s0 and
grained one, i.e. ∂ s → ∆s /∆t with ∆t ≈ b−1 (cid:29) τ.
Then the fast relaxta0tion of t0he anisotropic comFponents τ˜ = τ3 . (B11)
3 1+(2B τ )2
into the steady state at the beginning of each time in- 3 3
terval ∆t contributes only in higher order in b τ to the
F Plugging the solutions for the anisotropic compo-
average over ∆t. Using these approximations one finds
nents (B7) and (B10) back into the equation for the
for the winding number one components
isotropic spin density (B3) and using α = β, i.e., the
condition for the PSH state, one finds for the spin den-
sc =−τ1v∂xs0+2τ1bc×s0+2τ1B1×sc (B6) sity S, which is given by S=(cid:82) (2dπk)2s0 at T =0 the spin
s =−τ v∂ s +2τ b ×s +2τ B ×s , diffusion equation Eq. (13).
s 1 y 0 1 s 0 1 1 s
1 I. Zuti´c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. Lett. 97, 236601 (2006).
76, 323 (2004). 4 J.D. Koralek, C.P. Weber, J. Orenstein, B.A. Bernevig,
2 J. Schliemann, J.C. Egues, and D. Loss, Phys. Rev. Lett. S.-C.Zhang,S.Mack,andD.D.Awschalom,Nature458,
90, 146801 (2003). 610 (2009).
3 B.A. Bernevig, J. Orenstein, and S.-C. Zhang, Phys. Rev. 5 L. Yang, J.D. Koralek, J. Orenstein, D.R. Tibbetts, J.L.
9
Reno and M.P. Lilly, Nature Phys. 8, 153 (2012). 13 M.Q. Weng and M.W. Wu, Phys. Rev. B 68, 075312
6 M.P.Walser,C.Reichl,W.WegscheiderandG.Salis,Na- (2003).
ture Phys. 8, 757 (2012). 14 Y. Takahashi, K. Shizume, and N. Masuhara, Phys. Rev.
7 L. Yang, J.D. Koralek, J. Orenstein, D.R. Tibbetts, J.L. B 60, 4856 (1999).
Reno,andM.P.Lilly,Phys.Rev.Lett.109,246603(2012). 15 R.Winkler, Spin-Orbit Coupling Effects in Two-
8 M. Kohda, V. Lechner, Y. Kunihashi, T. Dollinger, P. Dimensional Electron and Hole Systems, Springer
Olbrich, C. Scho¨nhuber, I. Caspers, V.V. Bel’kov, L.E. Tracts in Modern Physics 191 (2003).
Golub,D.Weiss,K.Richter,J.Nitta,andS.D.Ganichev, 16 D.N. Zubarev, V. Morozov, and G. Ro¨pke, Statistical Me-
Phys. Rev. B 86, 081306(R) (2012). chanics of Nonequilibrium Processes, Vol. 1 (Akademie
9 L. Yang, J. Orenstein, and D.-H. Lee, Phys. Rev. B 82, Verlag, Berlin, 1996).
155324 (2010). 17 This reasoning is based on the equations of appendix B.
10 M. C. Lu¨ffe, J. Kailasvuori and T. S. Nunner, Phys. Rev. Whentheexpressionforthewindingnumberthreepartof
B 84, 075326 (2011). the spin distribution (Eq. (B9)) is plugged into the equa-
11 X. Liu and J. Sinova, Phys. Rev. B 86, 174301 (2012). tionfortheisotropiccomponent(Eq.(B3))onefindsthat
12 D. Stich, J. Zhou, T. Korn, R. Schulz, D. Schuh, theonlyterm,whichgeneratesafiniteS -componentisof
x
W. Wegscheider, M.W. Wu, and C. Schu¨ller, Phys. Rev. the form b ×(s ×(b ×s )) = −(b ×s )(b ·s )
c3 0 c3 0 c3 0 c3 0
Lett. 98, 176401 (2007). where b ∝eˆ .
c3 y