Transport Equations and Spin-Charge Propagating Mode in the Two Dimensional Hole Gas

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by  Taylor L. Hughes

Transport Equations and Spin-Charge Propagating Mode
in the Two Dimensional Hole Gas
Taylor L. Hughes,1,2 B. Andrei Bernevig,1,2 and Yaroslaw B. Bazaliy2,3
1Department of Physics, Stanford University, Stanford, CA 94305
6 2IBM-Stanford Spintronic Science and Applications Center, 650 Harry Rd. San Jose, CA 95120
0 3IBM Almaden Research Center, 650 Harry Rd. San Jose, CA 95120
0
We find that the spin-charge motion in a strongly confined two-dimensional hole gas (2DHG)
2
supports a propagating mode of cubic dispersion apart from the diffusive mode due to momentum
n scattering. Propagating modes seem to be a generic property of systems with spin-orbit coupling.
a ThrougharigorousKeldyshapproach,weobtainthetransportequationsforthe2DHG,weanalyze
J
thebehavioroftheholespinrelaxation time, thediffusion coefficients, andthespin-chargecoupled

6 motion.
1

] The correct description of the motion of charge and the recentexperimentalobservations[7]attributedto the
l
l spinpacketsinmaterialssuchassemiconductorsrequires spin-coulombdrag. The Dyakonov-Perelrelaxationtime
a
h appropriate transport equations. While transport equa- for the heavy holes’ spin is calculated as well. We dis-
- tionsforthe Fermiliquidusuallydescribeseparatediffu- cuss the relation of the previously calculated spin Hall
s
e sionprocessesforspinandchargedensities[1],spin-orbit conductivity in the ballistic regime [8, 9] and this for-
m (SO) coupling alters their form dramatically. For exam- malism. In addition, the transport equations support
. ple, the presence of coupling between spin and charge, a a spin-charge propagating mode, analogous to the one
t
a direct result of SO coupling, leads to the possibility of predicted in Ref. [3]. However, its dispersion relation is
m controlling the spin-orientationby electric fields [2]. An- changed because of the different form of the SO inter-
- otherexampleisthe recentlyproposedspin-chargemode action, and is cubic rather than linear in momentum.
d
[3] propagating with a velocity equal to the Rashba SO The effects of Coulombrepulsion onthe spectrum of the
n
coupling. This mode has no counterpart in usual trans- propagating mode are also accounted for in the present
o
c port and is intimately related to a non-equilibrium ver- treatment.
[ sionofthespin-currentintheground-stateofthesystem The most general single electron Hamiltonian with

[4]. spin-orbit coupling for spin 1/2 systems is:
1
v k2
3 Inthecaseofthetwo-dimensionalelectrongas(2DEG) H = +λ (k)σ , i=1,2,3 (1)
i i
5 with Rashba spin-orbit interaction, diffusive transport 2m
3 equations are known [5, 6]. A similar analysis for the with λ (~k) = λ ( ~k) due to time reversal invariance.
01 two-dimensional hole gas (2DHG), where the spin-orbit The eniergy ei−geniv−alues are E±(k) = k2/2m λ(k)
6 couplingismuchstronger,hassofarbeenlacking. Inthis where λ = √λiλi. The retarded and advanced±Gr|een’s|
0 letter we obtain the transport equations for very thin p- functio|ns|of the clean 2DHG have the form :
/ doped quantum wells in which only the heavy-holeband
t 1 1 sλ σ /λ
a is populated. We start with a general Keldysh equation GR,A(k,ǫ)= − i i | | (2)
m 2 ǫ˜+sλ i/(2τ)
for spin-1/2 systems with generic SO coupling, and then s=±1 ±
X
- particularize to the case of the heavy hole 2DHG. The
d where ǫ˜ = ǫ + k2/(2m) k2/(2m) and ǫ is measured
n same approach can be applied other types of SO cou- fromthe Fermi enFergyǫ −=k2/2m. The density matrix
o plings,suchasthe3DDresselhausswithlittleornomod- F F
g =g(k,r,t)is a2 2matrixinspinindices,whoseinte-
c ification. The equationsobtained here formthe basis for ×
v: studies of transport and spin-orbit effects in 2DHG sys- graloverthemomentumspaceyieldsthereduceddensity
matrix ρ=ρ(r,t) :
i tems. We find that, unlike inthe caseof 2DEG,the spin
X
orbiteffectsdonotproducefirstorderderivativetermsin ddk
ar the transport equations, but appear only in second and ρ(r,t)= (2π)d+1νg(k,r,t)≡n(r)+Si(r)σi (3)
even third order derivative terms, which now have to be Z
where n(r,t),S (r,t) are the charge and spin densities, d
taken into consideration. As such, the 2DHG does not i
isthespace-dimension,andνisthed-dimensionaldensity
exhibit the uniform spin polarization under an applied
of states. The charge-spin density g satisfies a Keldysh-
electric field, or the spin-galvanic effect that its 2DEG
type equation which takes electron scattering into ac-
counterpart does. The diffusion coefficient becomes a
count through an isotropic momentum relaxation time
tensor with components for spin and charge which de-
τ:
pend on the spin-orbit coupling. Its spin component
turns out to usually be smaller than the charge compo- ∂g g i 1 ∂H ∂g
+ +i[H,g]= (GRρ ρGA) , (4)
nent, which might provide an alternate explanation for ∂t τ τ − −2 ∂k ∂r
(cid:26) i i(cid:27)
2
This quantum Boltzmann equation can be solved iter- continuity equation expanded up to the third order in
atively in the case of small spatial gradients g of the spatial derivatives has the form:
∇
charge-spin density [6]. Fourier transforming with re-
∂Nµ Nµ
spect to time and defining a linear functional K(g) = + (1 δ )=Cµν∂ Nν +Dµν∂ ∂ Nν
∂t τ − µ,0 i i ij i j
i(GRρ ρGA) 1 ∂H, ∂g , one obtains an equation s
τ − − 2 ∂ki ∂ri +Hµν∂ ∂ ∂ Nν (9)
for g(k,r,ω): n o ijk i j k
(2λ2 Ω2)K(g)+2λ λ σ K(g)σ Ωλ [σ ,K(g)] where i,j,k = x,y .We first concentrate on the coeffi-
g =i − i j i j − i i cients Cµν and{Dµν.}In the 2DHG case we find Cµν 0
Ω(4λ2 Ω2) i ij i ≡
− duetothesymmetriesoftheHamiltonian. Thisisincon-
(5)
with Ω ω + i/τ. Expanding in gradients g = g0 + trast with the linear Rashba model of the 2DEG, where
δg(1) +δ≡g(2) +..., a perturbative chain of equations is nonzerolineartermswerecalculatedinRef. [3]. The dif-
fusion coefficient tensor Dµν has only three distinct ele-
obtained up to the desired accuracy. Integrating over ij
ments due to the x⇆ y symmetry of the 2DHG Hamil-
the momentum and taking the DC limit ωτ << 1 one
tonian. As we havealready mentioned the generalforms
obtains the transport equations for ρ. In the case of the
for the transportcoefficients are very cumbersome so we
Rashba coupling λ =αǫ k this was done in Ref. [6].
i ij j
assume small SO coupling (ǫ ǫ ) and expand up to
For2Dsystemswith anisotropicFermisurface,which α F
linearorderin ǫα inallcoefficie≪nts (D’sandH’s)except
includes the Rashba model, the 2DHG model, and the ǫF
for the charge diffusion coefficient whose first correction
2D Dresselhauss model, in the limit of ǫ τ 1, ǫ
F ≪ α ≡ due to SO coupling is at order (ǫα)2. We find
λ(kF) ǫF the continuity equation takes the form: ǫF
| |≪
∂∂ρt = −(4λ2d2−2λk2Ω2)τ(ρ− 21σmρσm) (6) Dx00x = D0 1+ 98(cid:18)ǫǫFα(cid:19)2(cid:18)1+ 1+24ζ2(cid:19)!
i Ω(δg(1)+...) 1+24ζ4+32ζ6
− (2π)2m D11 = D
Z xx 0 (1+4ζ2)3
where δg(1) contains first order spatial derivatives, and (cid:18) (cid:19)
1 12ζ2
subsequent terms contain higher order derivatives. Here D33 = D −
xx 0 (1+4ζ2)3
m = 1,2; m = 3. The first term on the right hand side (cid:18) (cid:19)
has zero comp6 onents in the charge-sectorand represents D03 = D0 3ζ2(4ζ2−1) (10)
spin-relaxation with characteristic time: xy ǫFτ (cid:18) (4ζ2+1)2 (cid:19)
1 = 1 2ζ2 (7) where D = vF2τ, and with symmetries Dµν
τ τ 4ζ2+1 0 2 xx →
s (cid:20) (cid:21) Dµν,D11 D22. Another notable symmetry is Dµν =
whereζ ǫ τ.Equation(6)andthe relaxationtime for- Dyyµν sxoxt→he offyy-diagonal terms in the sum Dµν∂ ∂xyNν
≡ α − yx ij i j
mula (7) are valid for any spin-orbit coupling in a 2D cancel. Theexpressionsabovearevalidinthelimitwhen
system with an isotropic Fermi surface. The general ex- the spin-orbit band splitting and momentum relaxation
pressions for higher order derivative terms tend to be energy are assumed to be smaller than the Fermi energy
exceedingly long. i.e. ǫ , 1 <<ǫ . However, the value of ζ is unrestricted
α τ F
Wenowturntoaparticularcase: thetwo-dimensional and we find for some common physical parameters that
hole gas. The valence band of type III V bulk semi- 0<x<1. The transport equation above with these dif-
−
conductors is a spin 3/2 band, separatedinto heavy and fusion coefficients represents one of the main results of
light-hole bands which touch at the Γ point. In a quan- this paper.
tum well a gap opens at the Γ point so that the light Here we briefly show how an interesting new phe-
hole band is moved below the heavy hole band. A suf- nomenon, the spin Hall effect, emerges from the above
ficiently thin well ensures a gap large enough to place equations. Under an applied electric field along the xˆ-
the light-holebandbelowthe Fermilevel. Thenonlythe axis we expect out-of-plane spins to be transported in
heavy-hole Hamiltonian should be considered. Asymme- the yˆ direction[11, 12]. We take into account the effects
try of the quantum well introduces a spin-orbit term of of an electric field per the substitution ~n ~n eνE~
∇ →∇ −
the form λiσi with: [5]. From the transport equations above we see that Ex
λ =αk (3k2 k2), λ =αk (3k2 k2) (8) generates:
x y x− y y x y− x
and λ = 0[10]. This model has an isotropic Fermi sur- Jz =(D03eν)Ex (11)
z y yx
face with λ =αk3. The limit of validity of our calcula-
tions is ~/|τ,| αk3 k2/2m where m is the heavy- whichisinthepreferredform[11,12]andwherethesym-
holemass. ForaFco≪mpaFctnothahtion,wedehfihnethecharge- metry Dxµyν = −Dyµxν gives us the antisymmetric tensor
spin four-vector Nµ = (n,S~), µ = 0,1,2,3. The generic structureJj =σ ǫ Ex. Inthe lowimpurity limit i.e.
i SH ijx
3
FIG. 1: (color online) (a)Charge diffusion coefficient(b)Spin FIG.2: (coloronline) Thirdordercoefficients plottedvs. SO
relaxation time plotted vs SO coupling α(c) Sz diffusion co- coupling strength αand hole-doping δ.
efficient(c)Sy diffusioncoefficient. All3dfiguresareplotted
vs. SO coupling strength α and hole-doping δ.
For the third-order coefficients we find:
as τ we find a value of 3ζ(3+16ζ2+72ζ4+24ζ6)
→∞ Hx0x2x = H0 2(1+4ζ2)3
3 (cid:18) (cid:19)
Dx03y = 4m. (12) H13 = H (4ǫFτζ)(4ζ2−1) (15)
xxx 0 (1+4ζ2)4
(cid:18) (cid:19)
This gives us: Jz = 3eEx. When we take into account,
like [8] that we ayre us8iπng spin-1/2 matrices to represent where H0 = v2Fmτ and up to this order we have the addi-
a spin-3/2system, we accountfor anadditionalfactorof tional symmetries: Hx0x2x =Hx2x0x and Hx1x3x =−Hx3x1x.
3 which yields: It was found in Ref.[3] that in 2DEG systems the
SO interaction can produce a mode of coupled spin and
9e charge oscillations (fractionalized spin packets mode, or
σ = (13)
SH
8π spin-galvanic mode) which can be weakly damped even
in the diffusive regime. It was later found that the
in agreement with [8, 9]. This value for the conductiv-
Coulomb interactions of the charge component do not
ity matches the clean-limit and supports the previously
suppress this mode [13]. Here we study the analogue
knownfactthatvertexcorrectionsvanishforthis model.
of such mode in the 2DHG system using the ansatz
[9]
δn(x,t)=δnexp[iqx iωt], S (x,t)=S exp[iqx iωt],
y y
Nontrivial dynamics, in the absence of applied fields, − −
S ,S =0 tofind asolutionofEqs.(9)inthe absenceof
appears in the third order derivative terms of the trans- x z
externalfields. FollowingtheprocedureofRef.[13]toac-
port equations. The sum Hµν∂ ∂ ∂ Nν only depends
ijk i j k count for Coulomb interactions, we obtain the spectrum
on coefficients Hµν , Hµν , and symmetrized combina-
xxx yyy of the mode
tions: H˜µν = Hµν +Hµν +Hµν , and H˜µν = Hµν +
x xxy xyx yxx y yxy
Hµν +Hµν . The number of distinct coefficients is fur- i(φ +φ ) 4γ2φ (φ φ )2
thyeyrxreducxeydyto just four by the exact symmetries of the ω±(q)= − 1 2 ± 2τ 3− 1− 2 (16)
p s
Hamiltonian:
with
1 1
Hx0x2x = Hy0y1y =−3H˜y02 =−3H˜x01 φ = Dx00xκ2R, φ = Dx22xκ2+1 , φ =κ6R ,
1 1 1 D 2 D 3
H20 = H10 = H˜20 = H˜10 0 0
xxx yyy −3 y −3 x
Hx1x3x = −Hy2y3y =−13H˜y13 = 31H˜x23 wRh=er1e+κ2=πκqBl/s,κlasn=dκ√BD=0τνse2islstchheasrpacinte-driizffeutshioenstlreennggtthh,
Hx3x1x = −Hy3y2y =−13H˜y31 = 31H˜x32. (14) othfeth2DeECGouclaosme,bthinetsehraacpteioonf,thaendspγect=ruHmx0ix2sxcτosn/tls3r.ollAedsbiny
4
w calculation shows that the components of the diffusion
coefficienttensordependonthemagnitudeoftheSOcou-
w t >1 plingandthatthespindiffusionisgenerallysmallerthan
p the charge diffusion, thus offering an alternate explana-
tion to spin-coulomb drag for this already experimen-
tallyobserved[7]phenomenon. We findthat,incontrast
to the 2DEG case, the spin-charge coupling enters into
the transport equations starting with the second order
ql >1 derivative terms. Our diffusive equations reproduce the

valueofthespinHallcoefficient,previouslycalculatedfor
1/t
s this systemin the ballistic regime [8, 9], thus supporting
k(cid:13) k(cid:13) k(cid:13) k the conclusionaboutthe absenceofvertexcorrectionsin
- + *
this2DHGsystem. Thefractionalized-spinpacketsprop-
agating mode in the 2DHG is also studied. It is found
FIG. 3: Real (solid line) and imaginary (dashed lines) parts
thatthe Coulombinteractionsuppressesthis mode more
of themode frequenciesω±(k) in an infinite2DHGstructure
with Coulomb repulsion. Two windows of Re(ω) 6= 0 exist. than in the case of electron gas. The spectrum of the
The grey areas are outside thelimits of the diffusiveapprox- mode suggests that its counterpart may exist for 2DHG
imation. systems in the ballistic regime.
TheauthorsagratefultoA.MacDonald,S.Parkin,D.
twodimensionlessparametersγandκ . Theestimatefor Arovas and H. Manoharan for useful discussions. B.A.B
B
γ ǫ τ /k l canbecomparedwiththe2DEGestimate acknowledges support from the SGF. T.L.H. acknowl-
α p F s
γ ∼ ǫ2DEGτ /k l . One observes that the larger edges support from NSF. This work is supported by the
S2ODEspGlit∼tingα ǫ asssocFiasted with 2DHG systems can be NSFundergrantnumbersDMR-0342832andtheUSde-
α
compensated by the small factor τ /τ . The Coulomb partment of Energy, Office of Basic Energy Sciences un-
p s
interactionparameteris large,κ 103 for l =100nm. der contract DE-AC03-76SF00515.
B s
∼
A sketch of the spectrum is shown in Fig. 3. As in
the case of 2DEG, the modes are purely dissipative at
κ = 0, representing the charge diffusion and spin relax-
ation. Spinandchargemotionsgetcoupledatκ>0and
[1] D. Pines and P. Nozieres, The Theory of Quantum Liq-
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6 uids (Addison Wesley, 1994).
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∼
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