Spin-Hall Effect in Two-Dimensional Electron Systems with Rashba Spin-Orbit Coupling and Disorder

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by  L. Sheng

Spin-Hall Effect in Two-Dimensional Electron Systems with Rashba Spin-Orbit
Coupling and Disorder
L. Sheng1, D. N. Sheng2 and C. S. Ting1
5 1Department of Physics and Texas Center for Superconductivity, University of Houston, Houston, Texas 77204
0 2Department of Physics and Astronomy, California State University, Northridge, California 91330
0
2 Using the four-terminal Landauer-Bu¨ttiker formula and Green’s function approach, we calculate
numerically thespin-Hallconductanceinatwo-dimensional junction system with theRashbaspin-
n
orbit (SO) coupling and disorder. We find that the spin-Hall conductance can be much greater or
a
J smallerthantheuniversalvaluee/8π,dependingonthemagnitudeoftheSOcoupling,theelectron
Fermienergyandthedisorderstrength. Thespin-Hallconductancedoesnotvanishwithincreasing
8
samplesize for a widerange of disorderstrength. Ournumericalcalculation reveals that anonzero

SO coupling can induce electron delocalization for disorder strength smaller than a critical value,
]
l and thenonvanishingspin-Hall effect appears mainly in themetallic regime.
l
a
h PACSnumbers: 72.10.-d,72.15.Gd,71.70.Ej,72.15.Rn
-
s
e
m The emerging field of spintronics,[1, 2] which is aimed In this Letter, the spin-Hall conductance (SHC) in a
at exquisite control over the transport of electron spins 2DESjunctionwiththeRashbaSOcouplingisstudiedby
.
t insolid-statesystems,hasattractedmuchrecentinterest. using the four-terminal Landauer-Bu¨ttiker(LB) formula
a
m One central issue in the field is how to effectively gener- with the aid of the Green’s functions. We find that the
ate spin-polarizedcurrents in paramagneticsemiconduc- SHC does not take the universal value, and it depends
-
d tors. In the past severalyears,many works [1, 2, 3, 4, 5] critically on the magnitude of the SO coupling, the elec-
n have been devoted to the study of injection of spin- tronFermienergy,andthe disorderstrength. Forawide
o
polarized charge flows into the nonmagnetic semicon- range disorder strength, we show that the SHC does not
c
ductors from ferromagnetic metals. Recent discovery of decrease with sample size and extrapolates to nonzero
[
intrinsic spin-Hall effect in p-doped semiconductors by values in the limit of large system. The numerical cal-
3
Murakami et al. [6] and in Rashba spin-orbit (SO) cou- culation of electron localization length based upon the
v
pled two-dimensional electron system (2DES) by Sinova transfer matrix method also reveals that the Rashba SO
8
3 et al. [7] may possibly lead to a new solution to the is- coupling can induce a metallic phase, and the spin-Hall
0 sue. For the Rashba SO coupling model, the spin-Hall effect is mainly confined in the metallic regime. The ori-
9 conductivity is found to have a universalvalue e/8π in a ginofthenonuniversalSHC inthe 2DESjunctionisalso
0
clean bulk sample when the two Rashba bands are both discussed.
4
occupied, being insensitive to the SO coupling strength Let us consider a two-dimensional junction consisting
0
/ and electron Fermi energy [7]. of an impure square sample of side L connected with
t
a four ideal leads, as illustrated in the inset of Fig. 1. The
While the spin-Hall effect has generated much inter-
m leads are connected to four electron reservoirs at chem-
est in the research community, [8, 9, 10, 11, 12, 13, 14,
d- 15, 16, 17] theoretical works remain highly controversial ical potentials µ0, µ1, µ2 and µ3. In the tight-binding
representation,theHamiltonianforthesystemincluding
n regarding its fate in the presence of disorder. Within a
the sample and the leads can be written as [18, 19]
o semiclassical treatment of disorder scattering, Burkov et
c
v: aHla.l[l1e0ff]eacntdonSlcyhlsiuemrvaivnensaantdwLeaokssd[i1s1o]rdshero.wOedntthhaetostphiner- H = −t X c†i,σcj,σ+Xεic†iσciσ
hijiσ iσ
i
X hand, Inoue et al. [14] pointed out that the spin-Hall ef-
r fect vanishes even for weak disorder taking into account + VSOXh(cid:16)c†i,↑ci+δx,↓−c†i,↓ci+δx,↑(cid:17)
a the vertex corrections. Mishchenko et al. [15] further i
showed that the dc spin-Hall current vanishes in an im- − i c† c +c† c +H.c. . (1)
(cid:16) i,↑ i+δy,↓ i,↓ i+δy,↑(cid:17) i
pure bulk sample, but may exist near the boundary of a
finitesystem. Nomuraetal.[16]evaluatedtheKubofor- Here,V istheSOcouplingstrength,ε ≡0intheleads
SO i
mulabycalculatingthesingle-particleeigenstatesinmo- and are uniformly distributed between [−W/2,W/2] in
mentum space with finite momentum cutoff, and found the sample, which accounts for nonmagnetic disorder.
that the spin-Hall effect does not decrease with sample The lattice constant is taken to be unity, and δ and
x
size at rather weak disorder. Therefore, further inves- δ are unit vectors along the x and y directions. In the
y
tigations of disorder effect in the SO coupled 2DES are verticalleads2 and3, V is assumedto be zeroin order
SO
highly desirable. to avoid spin-flip effect, so that a probability-conserved
2

spin current can be detected in the leads. 1.2
The electrical current outgoing through lead l W = 0
0.8 W = 1t
can be calculated from the LB formula [20] I =
l W = 2t
i(se2t/hhe)tPotla′6=lleTlelc,lt′r(oUnl′t−raUnls)m, iwsshieornecUole=fficµiel/n(t−freo)manldeaTdl,ll′′ ) 0.4
to leadl. A number of symmetry relations for the trans- p(e/4 0.0
mission coefficients result from the time-reversal and in- G sH 3
versioninvarianceofthesystemafteraverageofdisorder L
-0.4
configurations,useofwhichwillbeimplied. Weconsider
0L 1
that a currentI is driventhroughleads 0 and1, andad- y
-0.8
just U ’s to make I = −I = I and I = I = 0. Since 2
l 1 0 3 2
x
in the present system the off-diagonal conductance G
xy -1.2
vanishesbysymmetry,U −U equalstothelongitudinal -4 -2 0 2 4
0 1 E/t
voltagedropcausedbythecurrentflowI. Inthevertical
leads 2 and 3, where V = 0, the electrical currents are
SO FIG. 1: Spin-Hall conductance GsH for some disorder
separable for the two spin subbands I =I +I with ↑
l l↑ l↓ strengthsasafunctionofelectronFermienergyE. Here,the
and↓forspinsparallelandantiparalleltothez-axis. The sample size L = 40 and the spin-orbit coupling VSO = 0.5t.
spin current is given by I(l) = [h¯/2(−e)](I −I ). By Inset is a schematic view of the four-terminal junction.
sH l↑ l↓
use of the LB formula, it is straightforwardto obtain for

thetransversespincurrentIs(H3) =−Is(H2) =GsH(U0−U1). 1.0
Here, the proportional coefficient 1.0 L = 60 0.8G
e 0 .8 L = 40 00..46sH (e/4p W = t
GsH =−4π(T3↑,0−T3↓,0) , (2) ) 0.2) W = 2t
pe/40.6 0 1 2W3/t4 5 60.0 WW == 34tt
(
is the SHC, where T3σ,0 is the electron transmis- G sH WW == 56tt
sion coefficient from lead 0 to spin-σ subband in lead 0.4
3. Equation (2) can be calculated in terms of the
0.2
nonequilibrium Green’s functions [21, 22, 23] G =
sH
−(e/4π)Tr(Γ ηGrΓ Ga). Here, η = 1 and −1 in the
3 0 0.0
spin-↑ and spin-↓ subspaces, respectively, and Γl = 20 40 60 80 100
L
i[Σ − (Σ )†] with Σ the retarded electron self-energy
l l l
inthesampleduetoelectronhoppingcouplingwithlead
FIG.2: Spin-Hallconductanceasafunctionofsamplesize L
l. The retarded Green’s function Gr is given by for different disorder strengths at E = −2t and VSO = 0.5t.
Error bars due to statistical fluctuations, being smaller than
1
Gr = , (3) the symbol size, are drawn inside the open symbols. Inset:
E−HC−P3l=0(Σl) spin-Hall conductance as a function of disorder strength for
L=40 and 60.
and Ga = (Gr)†, where E stands for the electron Fermi
energy, and H is the single-particle Hamiltonian of the
C
centralsquaresample only. The self-energiescanbe first SHC.WithincreasingE fromthebandbottomE ≃−4t,
computed exactly by matching up boundary conditions except for a small oscillation due to the discrete energy
for the Green’s function at the interfaces by using the levels in the finite-size sample, GsH increases continu-
transfer matrices of the leads [24]. The Green’s function ouslyuntilE isveryclosetothebandcenterE =0. Itis
Eq. (3) is then obtained through matrix inversion. In easy to see from Fig. 1 that at weak disorder W <∼t the
ourcalculations,GsH is alwaysaveragedoverupto5000 calculated GsH may be greater than the universalvalue,
disorder realizations, whenever W 6=0. namely, 0.5 in our unit e/4π.
In Fig. 1, the SHC G is plotted as a function of the In order to determine the behavior of the spin-Hall ef-
sH
electron Fermi energy E at fixed size L = 40 for several fectin largesystems,we calculatethe SHC as a function
disorder strengths. The SHC is always an odd function of the sample size from L = 10 up to 100 for different
of electron Fermi energy E, and vanishes at the band strengths of disorder, as shown in Fig. 2. For weak dis-
center E = 0. The antisymmetric energy dependence order W <∼ 3t, the SHC first increases with increasing
of the SHC is similar to that of the Hall conductance sample size, and then tends to saturate. In particular,
in a tight-binding model [25], and originates from the for W <∼ t, we see that the SHC can be several times
particle-hole symmetry of the system. For E < 0 and greater than the universal value e/8π, when the system
E >0 the charge carriers are electron-like and hole-like, becomes large. For a stronger disorder 3t<∼W <∼5t, the
respectively, and so make opposite contributions to the SHC is roughly independent of the sample size, and ex-
3

1.0 8
W = 0 (a) (b)
0.8 W = 1t
W = 2t E = -2.0t E = -2.0t
) 6
p40.6 V = 0.5t V = 0.1t
e/ SO SO
G (sH0.4 L
x/ 4 L = 8
0.2 L = 16
L = 32
0.00.0 0.2 0.4 0.6 0.8 1.0 2 L = 64
V /t
SO W W
c c
0
FIG.3: Spin-Hallconductanceasafunctionofspin-orbitcou- 4 6 8 102 4 6 8
W/t W/t
pling strength for some disorder strengths. Here, the sample
size L=40 and theelectron Fermienergy E =−2t.
FIG. 4: Normalized localization length as a function of dis-
order strength calculated on long bars of length 5×105 and
widths L=8, 16, 32 and 64.
trapolatesto a finite value in the large-sizelimit. There-
fore, it is evident that the SHC will not vanish in large
systems in the presence of moderately strong disorder
localization length ξ/L is plotted as a function of disor-
W <∼ 5t. With further increase of W, the SHC becomes der strengthfor V =0.5t andL=8, 16,32and 64. At
vanishingly small at W >∼ 6t, as seen more clearly from weak disorder, ξ/LSOincreases with L, indicating that the
the inset of Fig. 2, indicating that very strong disorder
localizationlength ξ will diverge as L→∞, correspond-
scattering would eventually destroy the spin-Hall effect.
ing to an electron delocalized metallic phase. With the
WefurtherexaminethedependenceoftheSHConthe increase of W, ξ/L goes down and all the curves cross
strength of the SO coupling. As shown in Fig. 3, over- at a point (fixed point) W = W ≃ 6.3t, where ξ/L be-
c
all, the SHC increases with increasing VSO in the range comes independent of bar width L. For W > Wc, ξ/L
0 ≤ VSO ≤ t. For W = 0 or weak disorder, the SHC decreaseswithL,indicatingthatξ willconvergetofinite
displays an interesting oscillation effect with a period values as L→∞, corresponding to an electron localized
much greater than the average level spacing. According insulator phase. Thus the fixed point W = W is the
c
to Eq. (2), the oscillation of the SHC is a manifestation critical disorder strength for the metal-insulator transi-
of the oscillation of the sideway spin-resolved transmis- tion. Our result is consistentwith the earlier calculation
sion coefficients. For a two-terminal junction with the by Ando [18], where a metallic phase was established at
SO coupling, similar oscillation with finite sample size the band center E =0 for a strong Rashba SO coupling.
haspreviouslybeenobservedforthe spin-resolvedtrans- Here, we also study weak SO coupling. In Fig. 4b, we
missioncoefficients[19],wherethe oscillationperiodwas plot the result for a SO coupling strength much smaller
discussed to be the spin precession length Lsp. If we ap- than the electron hopping integral, i.e., V = 0.1t, and
SO
ply the same condition L = nLsp with n an integer and similar phase transition is also revealed at Wc ≃ 4.6t.
notice Lsp ≃ πt/VSO, [19] we can obtain for the equiv- In general, we have performed calculations in the whole
alent period in the SO coupling δVSO ≃ πt/L. For the range from strong to weak SO coupling (details will be
parameters used in Fig. 3, δVSO ≃ 0.08t, which is very presented elsewhere), and found that electron delocal-
close to the period as seen in the figure. This indicates ization occurs for any nonzero SO coupling strength as
that the oscillation of the SHC is due to a spin preces- the magnitude of the disorder varies. Our result is in
sional effect in finite-size systems. Experimentally, VSO agreement with the perturbative calculation of weak lo-
can be varied over a wide range by tuning a gate volt- calization.[30]As V reduces,the criticalW decreases,
SO c
age[26,27],andsothisoscillationeffectmaypossiblybe and the size-independent critical ξ/L increases (so does
observed directly. the critical longitudinal conductance G [28, 29]). In
xx
Electron delocalization is a crucial issue for under- the limit V → 0, we have W → 0 and all electron
SO c
standing electron transport properties in the 2DES, and states become localized, recovering the known regime of
has already been studied experimentally by use of mag- the two-dimensional Anderson model for electron local-
netoresistance measurements [27]. For this reason, we ization [28]. The fact that the critical ξ/L changes with
investigatenumericallywhether the RashbaSO coupling V indicates that the SO coupled 2DES belongs to the
SO
caninduceauniversalelectrondelocalizationinthepres- universalityclassoftwo-parameterscaling[31]. Compar-
enceofdisorder. Accordingtothewell-establishedtrans- ing W = 6.3t calculated in Fig. 4a for V = 0.5t and
c SO
fermatrixapproach,[28,29]wecalculatetheelectronlo- E =−2t with the SHC shown in Fig. 2 for the same pa-
calization length ξ on a bar of essentially infinite length rameters,weseethatnonvanishingspin-Halleffectexists
(5×105) and finite width L. In Fig. 4a, the normalized mainly in the metallic regime.
4
Our numerical study addresses the spin-Hall effect in tute for Theoretical Physics for hospitality and support
a finite-size junction system with leads. A comparison (through PHY99-07949 from KITP), where part of this
betweenthe spin-Halleffect andthe quantum Halleffect work was done.
(QHE)canshedsomelightonthe nonuniversalSHC ob-
tained. For a QHE system, delocalized states exist at
the centers of the discrete Landau levels, which are sep-
aratedby mobility gaps consisting of localizedstates. In
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Acknowledgment: The authors would like to thank (1981); Z. Phys.B 53, 1 (1983).
A. H. MacDonald, Q. Niu and J. Sinova for stimulating
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discussions. ThisworkissupportedbyACS-PRF41752- Rev. B 56, R7053 (1997); Phys. Rev. Lett. 79, 1710
AC10, Research Corporation Award CC5643, the NSF (1997).
grant/DMR-0307170 (DNS), and also by a grant from [30] M. A.Skvortsov,JETP Letters, 67, 133 (1998).
the Robert A. Welch Foundation (CST). DNS wishes [31] D. N. Sheng and Z. Y. Weng, Phys. Rev. Lett. 80, 580
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to thank the Aspen Center for Physics and Kavli Insti-
5
[32] B. K. Nikoli´c, L. P. Zˆarbo and S. Souma, cond-mat/0409334 (2004).
cond-mat/0408693 (2004); E. M. Hankiewicz, L.
W. Molenkamp, T. Jungwirth, and J. Sinova,