Giant edge spin accumulation in a symmetric quantum well with two subbands

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by  Alexander Khaetskii

Giant edge spin accumulation in a symmetric quantum well with two subbands
Alexander Khaetskii1 and J. Carlos Egues2
1Department of Physics, University at Buffalo, SUNY, Buffalo, NY 14260-1500
2Instituto de F´ısica de S˜ao Carlos, Universidade de S˜ao Paulo, 13560-970, S˜ao Carlos, S˜ao Paulo, Brazil
(Dated: February 2, 2016)
We have studied the edge spin accumulation in a high mobility two-dimensional electron gas
formed in a symmetric well with two subbands. This study is strongly motivated by the recent
6 experiment of Hernandez et al. [Phys. Rev. B 88, 161305(R) (2013)] who demonstrated the spin
1
accumulation near the edges of a bilayer symmetric GaAs structure in contrast to no effect in a
0
single-layer configuration. The intrinsic mechanism of the spin-orbit interaction we consider arises
2
fromthecouplingbetweentwosubbandstatesofoppositeparities. Weobtainaparametricallylarge

n magnitude of the edge spin density for the two-subband sample as compared to the usual single-
a subbandstructure. Weshowthatthepresenceofagap inthesystem,i.e.,theenergyseparation ∆
J
between the two subband bottoms, changes drastically the picture of the edge spin accumulation.

9 Thusonecaneasily proceedfrom theregimeofweakspinaccumulationtotheregimeofstrongone
2 byvaryingtheFermienergy(electrondensity)and/or∆. Weestimatethatbychangingthegap ∆
from zero up to 1÷2 K, the magnitude of the effect changes by three orders of magnitude. This

] opens upthe possibility for thedesign of new spintronicdevices.
l
l
a
PACSnumbers: 72.25.-b,73.23.-b,73.50.Bk
h
-
s
e Spincurrentsandspinaccumulation[1,2]aretopicsof ing and intriguing in many aspects. It demonstrated the
m
great current interest both experimentally and theoret- spinaccumulationnearthe edgesofahighmobilitytwo-
. ically. These phenomena are due to the spin-orbit cou- dimensionalelectronsysteminabilayersymmetricGaAs
t
a pling andareimportantfor the future ofspinelectronics structure in contrast to no effect in a single-layerconfig-
m
[3]. There are two distinct spin-orbit mechanisms, the uration. The observed effect is quite big, i.e., it seems
- extrinsic one due to the Mott asymmetry in the electron bigger than in the experiment by Kato et al. [10], where
d
n scatteringoffimpurities[4,5],andtheintrinsicone[6,7] foraGaAssampletheresultwasexplainedbytheextrin-
o due to spin-orbit induced splitting of the electron spec- sic interaction with impurities. Note that the structure
c trum. Thelatterhasrecentlyrenewedtheinterestinthis studied in [22] has inversionsymmetry and therefore the
[
field, which has existed for quite a long time [4, 5, 8, 9]. usual Rashba term is absent. On the other hand, the

1 Theedgespin-densityaccumulation,relatedtoeitherthe linear-in-momentum Dresselhaus term is known not to
v Mottasymmetrybyimpurities [10] (2Delectrons)orthe lead to a spin current in the bulk. In addition, it cannot
6
intrinsic mechanism (2D holes) [11, 12], has been exper- create a smooth, within the scale of the spin-precession
2
imentally observed. length, edge spin density upon boundary scattering for
0
0 Itis known[2, 13–15]that inthe diffusive regimewith any reasonably short-ranged impurity scattering in the
0 a weak spin orbit interaction, i.e., when the spin diffu- bulk, as is shown in Ref. [21]. There is clearly a signif-
2. sion length is much larger than the mean free path, the icant difference between the observed edge spin density
0 edge spin density is entirely due to the spin flux coming in two-subband vs. the one-subband cases (i.e., strong
6 from the bulk. In contrast, the physics of the edge spin- edge accumulation effect vs no effect). This motivated
1 density accumulation for the intrinsic mechanism in the us to look for the explanation of this phenomenon us-
:
v opposite case of strong spin-orbit splitting [16] only re- ing the inter-subband Rashba-like Hamiltonian arising
i cently has been understood [17–21]. This includes both in two-subband wells [24],[25].
X
the case of a ballistic structure in which the mean free Here we follow the method proposed in [21] to calcu-
r
a path is the largest lengthscale in the problem, and the late the edge spin density which appears due to bound-
case of a diffusive sample but with large spin-orbit split- ary scattering [Fig. 1(a)] in the quasi-ballistic regime for
ting of the spectrum so that the spin-precession length a Rashba-like Hamiltonian with inter-subband coupling
is smaller than the mean free path. This latter case we [24],[25]describingthetwo-subbandwell[Fig.1(b)]inthe
termthequasi-ballisticregime. Inparticular,ithasbeen experimentofRef. [22]. Inthis quasi-ballisticregimethe
shown [21] that in this regime there is no direct relation characteristic length of the spin accumulation near the
between edge spin-density and bulk spin current. For boundaryissmallerthanthemeanfreepath. Infact,the
example, in the case of 2D holes in the quasi-ballistic mean free path in the experiment is around 30 µm [22],
regime the edge spin-density, which is due to the spin whichindeedexceedsallthe characteristiclengthsofour
currentfromthe bulk, isparametricallysmallerthanthe theory. We believe that we have explained the experi-
density generated upon the boundary scattering. mental results, in particular, the large magnitude of the
The experiment of Hernandez et al. [22] is interest- edge spin density for the two-subband sample compared
2
!
!
a) p2 p1 ε b()p) ∆co.rrIensptohnisdrsetgoimtheethceontdwiotiosunbsboafnedxspaerriemoecnctup[2i3ed].,Mwhoirceh-
over,thespin-orbitinteractionisweakandtherefore”co-
ϕ
herent” and spin-precession length scales are large com-
1
ϕ pared to the Fermi wave length, L ,L λ . Our
2 εF calculation shows that the characteri∆sticssp≫atialFscale of
!"
the edge spin density is Λ=L L / L2 +L2.
∆ s ∆ s
ε Model Hamiltonian. — The Hamiltonian of a sym-
o ∆ p
metric quantum well with two subbands and inter-
ε
subband-induced SO interaction resembles that of the
e
p p
2 1 ordinaryRashbamodel. Incontrasttothe latter,the in-
tersubband SO interaction is nonzero even in symmetric
structures. As derived in Refs.[24, 25], the 4 4 Hamil-
×
tonian describing a (symmetric) two-subband well is
Figure 1. Schematics of the boundary specular scattering in
the presence of spin-orbit coupling. Plus and minus modes H =( p2 +ε )1 1 ε τ 1+(η/~)τ (p σ p σ(1)),
are shown for the same energy and the same wave vectors 2m + ⊗ − − z ⊗ x⊗ x y − y x
along theboundary.
where m is the effective mass, ε = (ε ε )/2, ε and
± o e e
±
ε are quantized energies of the lowest (even) and first
o
excited (odd) subbands, respectively, measuredfrom the
totheusualsingle-bandstructurewitheithertheRashba
bottom of the quantum well, τ denote the Pauli ma-
or Dresselhaus interactions. x,y,z
trices describing the subband (or pseudospin) degree of
Interestingly, we have found that the presence of the
freedom, and σ are Pauli matrices referring to the
x,y,z
gap∆betweentwosub-bandedges[Fig.1(b)]drastically
electron spin. The inter-subband SO coupling η is ex-
changes the scenario of the edge spin accumulation, so
pressed[24]intermsofthe gradientsoftheHartree-type
that the physics is now determined by the value of the
contribution to the electron potential, the external gate
parameter ξ = 2ηk /∆ L /L [Fig. 2], where L =
F ∆ s ∆ and doping potentials, and the structural quantum-well
~v /∆, L = ~2/2mη ar≡e the ‘coherence” and the spin-
F s potentialprofile. Notethatthegapis∆=ε ε =2ε .
precession lengths. Here p = ~k = mv is the Fermi o− e −
F F F Theoretical approach. To calculate the edge spin den-
momentum, η is the inter-subband spin-orbit coupling
sityinthequasi-ballisticregimewefollowthemethodde-
constant. The parameter ξ can have an arbitrary value
velopedinRefs.[20,21]forthecaseofthesingle-subband
even for a small η since the gap ∆ can be made much
Rashba Hamiltonian. Assuming that the spatial scale of
smaller than the Fermi energy. Thus one can easily go
the edgespinaccumulation(Λ)is muchsmallerthanthe
fromthe regimeofweakspinaccumulation(largeξ 1)
mean free path (l) (quasi-ballistic regime), we solve the
≫
overtotheregimeofstrongspinaccumulation(ξ isorder
edge spin problem by the method of scattering states,
of unity) just by slight increase of the gap value. As we
i.e.,wefindtheexactquantummechanicalsolutionofthe
discuss later, for GaAs structure similar to that used in
electronscatteringbyanimpenetrablestraightboundary
Ref. [22], it is enough to change the gap ∆ from zero up
[Fig. 1(a)] at a given Fermi energy. These solutions are
to 1 2K onlyin orderto increasethe magnitude ofthe
then used in the calculation of the (mean) spin density
÷
effect by the three orders.
profile. Depending on the Fermi level location some of
Therefore despite the problem in question resembling the involved states can be evanescent modes. This hap-
very much the usual Rashba problem (there are two pens, for example, when the Fermi level is within the
copies of them because each state is doubly degenerate), gap between the two subbands (ε < ε < ε ), i.e.,
e F o
the presence of the gap changes physics of the edge spin only one subband is occupied. The populations of the
accumulation completely. The most pronounced change incomingstatesarefoundfromthesolutionofthekinetic
happens at L∆ 0.5Ls, where the edge spin density equation for the spin-density matrix in the bulk (2D) of
≃
Sz(x) is maximized and its order of magnitude is given the sample in the presence of electric field [27]. These
by kE/Ls [Fig. 2], which is parametrically largerthan in distribution functions are used as the input parameters
the single-band case. Here kE = eEτtr/~, e is the mod- for the part of the problem related to the scattering by
ulus of the electron charge, E is the magnitude of the the straightboundary. Besides the quasi-ballistic regime
in-plane (driving) electric field [22] and τtr is the trans- (Λ l), we assume that the Fermi energy εF is much
≪
port scattering time due to the impurities in the bulk of larger than the gap, ε ∆, i.e., both sub-bands are
F
≫
a sample. occupied. We also consider the SO interaction weak so
Weconsiderspecularscattering(i.e.,astraightbound- that ηk ε L ,L λ = 2π/k . The ratio
F F ∆ s F F
≪ ⇒ ≫
ary for simplicity, Fig. 1(a)) and a Fermi energy much L /L can be arbitrary.
∆ s
larger than the gap ∆ between the subbands, i.e., ε The Hamiltonian (1) has 4 eigensolutions Ψ ,
F i,s
≫
3
Ψ , Ψ , Ψ , Ψ with the corresponding energy s = , (Kramers pairs). Upon scattering by the
1,↑ 1,↓ 2,↑ 2,↓
↑ ↓
spectrum straight boundary where energy and momentum p
y
along the boundary are conserved, the states in the
ε (p)= p2 +ε ε2 +η2p2/~2, (2) pair Ψ1,↑(ϕ1,θ1),Ψ2,↓(ϕ2,θ2) mix up and form two
1,2 2m +∓ − scattering states, Eqs. (3),(4) [similarly for the pair
q
Ψ (ϕ ,θ ),Ψ (ϕ ,θ )]. For this pair of scattering
where the subscript i = 1,2 corresponds to the lower 1,↓ 1 1 2,↑ 2 2
states, we have for the scattering by the hard wall lo-
(higher) in energy sub-band. Each sub-band is dou-
cated at x=0 [Fig. 1(a)]
bly degenerate with respect to the ”spin direction”
Ψ˜1,↑(x,y)|x=0 =eikyy[Ψ1,↑(π−ϕ1,θ1)e−ik1x+F11,,↑↑Ψ1,↑(ϕ1,θ1)eik1x+F12,,↑↓Ψ2,↓(ϕ2,θ2)eik2x]|x=0 =0, (3)
Ψ˜2,↓(x,y)|x=0 =eikyy[Ψ2,↓(π−ϕ2,θ2)e−ik2x+F21,,↓↑Ψ1,↑(ϕ1,θ1)eik1x+F22,,↓↓Ψ2,↓(ϕ2,θ2)eik2x]|x=0 =0, (4)
with p2 = ~2(k2 + k2), p2 = ~2(k2 + k2), ε (p ) = Fromthe aboveequationswecanfindthecomponentsof
1 y 1 2 y 2 1 1
ε (p ) = ε. The momenta p , p describe states belong- the unitary scattering matrix Sˆ
2 2 1 2
ingtosubbands1and2foragivenenergyε,seeFig.1(b).
v
The angles ϕ1, ϕ2 (between the corresponding momenta S1,↑ =F1,↑, S2,↓ =F2,↓, S2,↓ =S1,↑ =F2,↓ x,2,
and the positive direction of the x-axis) are expressed 1,↑ 1,↑ 2,↓ 2,↓ 1,↑ 2,↓ 1,↑ vx,1
r
as sin(ϕ ) = ~k /p and sin(ϕ ) = ~k /p . The angles (5)
1 y 1 2 y 2
θ ,θ are defined via cosθ = 1/ 1+(ηp /~ε )2. where the group velocities in the subbands i = 1,2 are
1 2 1,2 1,2 −
v =∂ε /∂p and the scattering amplitudes are
x,i i x
p
F1,↑ =[e−iϕ1sin(θ /2)sin(θ /2) eiϕ2cos(θ /2)cos(θ /2)]/D; F2,↓ = sinθ cosϕ /D,
1,↑ 1 2 − 1 2 1,↑ − 1 1
F2,↓ = [eiϕ1sin(θ /2)sin(θ /2) e−iϕ2cos(θ /2)cos(θ /2)]/D; F1,↑ = sinθ cosϕ /D,
2,↓ − 1 2 − 1 2 2,↓ − 2 2
D =eiϕ2cos(θ /2)cos(θ /2)+eiϕ1sin(θ /2)sin(θ /2). (6)
1 2 1 2
Similar equations can be written for the pair Heref (ε,k )is thedistributionfunctionofthe electron
i y
Ψ (ϕ ),Ψ (ϕ ),andthecorrespondingscatteringma- stateinthesub-bandiforagivenenergyandgivenwave
1,↓ 1 2,↑ 2
trix elements S11,,↓↓, S22,,↑↑, S12,,↓↑ =S21,,↑↓ can be determined. vectorky alongtheboundary. Thenwecancalculatethe
The expectation value of the z component of the spin mostimportantpartofthe spin density whichis smooth
as a function of coordinates is given by the following ex- on the scale of the Fermi wave length [26] and involves
pression: the interference of the outgoing waves (two last terms in
Eqs. (3) and (4))
dk dε
y
S (x) = f (ε,k )
h z i (2π)2v i y
i,s Z x,i
X
Ψ˜ (x)Sˆ Ψ˜ (x) (7)
i,s z i,s
×h | | i
dk dε
S (x) =Re y S2,↑ (S1,↓)⋆ Ψ (ϕ )σˆ Ψ (ϕ ) +( ⇆ )
h z i (Z (2π)2√vx,1vx,2 h 2,↑ · 2,↑ ·h 1,↓ 1 | z| 2,↑ 2 i ↑ ↓ i
ei(k1−k2)x[f (ε,k ) f (ε,k )] . (8)
1 y 2 y
× − )
Using the expressions above we obtain
4
dk dε √sinθ sinθ (p−1 sinθ +p−1 sinθ )
S (x) = y k 1 2 1 · 1 2 · 2
h z i −(Z (2π)2√v1v2 (cid:20) ycos2[(θ1−θ2)/2]−sinθ1sinθ2sin2[(ϕ1−ϕ2)/2](cid:21)
sin[(p cosϕ p cosϕ )x/~] [f (ε,k ) f (ε,k )] . (9)
1 1 2 2 1 y 2 y
× − · − )
We can calculate the edge spin density for arbitrary 3Φ(ξ) (a) S(x/L ) (b)
z s
values ofthe parameterξ =L /L usingthatp p
∆ s 1 2
− ≪
p , i.e. the energy separation between two sub-bands
F L =0.1L
∆2+4η2k2 is much smaller than the Fermi energy. ∆ s
F
This means that θ θ θ , and for the same reason
p 1− 2 ≪ 1,2
ϕ ϕ ϕ . To proceed we need also the difference
2 1 1,2
− ≪
of the distribution functions entering Eq. (9). It can be
x/L
calculatedfollowingthe proceduredescribedinRef.[27]. s
dWe hLav,ewdhoenreeidtiasstshuemcinogrrethlaetisoentroafdiinuesqoufatlhiteieismkpF−u1ri≪ty ξ=L∆/Ls LLss==12..18 µµmm
s
po≪tentialinthebulkofthestructure. Thefirstcondition Sz(x/Ls) (c) Sz(x/Ls) (d)
meansthatthescatteringinthebulkisofthesmall-angle
L =0.56L L =2L
type. Both conditions are fulfilled for a high mobility ∆ s ∆ s
GaAs structure. After presenting the general formula
below, we will illustrate the calculation for the case of
a weak spin-orbit coupling, L L . The final result
s ∆
≫
derived from Eq. (9) reads
x/L x/L
s s
3k ξ
E
S (x) = Φ(ξ)J(x/Λ); Φ(ξ)= .
z
h i Ls (2ξ2+1) ξ2+1
(10)
p
with the spatial dependence given by the integral
Figure 2. Φ(ξ) vs. ξ a) and the edge spin density hSz(x)i in
x 1 dzz2 x L L units of 106cm−2 for distinct ratios L ∆/Ls and two different
J( )= sin( ), Λ= ∆ s valuesof Ls b)-c),asa function of x/Ls. Note that Φ(ξ) has
Λ Z0 π2 Λ√1−z2 L2∆+L2s a maximum at ξ ∼ 0.56 = L∆/Ls. This non-monotonic be-
(11) havior is reflected in the overall amplitude of hSz(x)i, which
p
For x Λ we have J(x) x/Λ. In opposite limit is maximized for L∆ = 0.56Ls c) and is suppressed as ξ de-
x Λ, w≪e obtain J(x) (Λ/∝x)3/2cos[(x/Λ)+π/4]. viates from this value b), d). Notethat the amplitude of the
≫We see that the phys∝ical picture of the edge spin ac- oscillations isreducedasLs increase(cf. blueandredcurves
in b)-d).)
cumulation is totally different as compared to the case
of usual Rashba system with one sub-band. We recall
that in the latter the edge spin density is strongly sup-
ε (p ) = ε (p ) = ε = ε , Fig. 1(b). Using the condi-
pressed as compared to the value k /L for any reason- 1 1 2 2 F
E s tion k d 1 (small-angle scattering in bulk), we obtain
ably short-ranged impurity scattering in the bulk [21]. F ≫
(τ (p ) τ (p ))/τ 3(p p )/p (3/(k L ). [28]
The suppression factor is d2/L2 1. It is very interest- tr 1 − tr 2 tr ≈ 1− 2 F ≈ F ∆
s ≪ We note that compared to the usual Rashba one-band
ingthatinthecaseL .L suchasuppressiondoesnot
∆ s case the difference of the distribution functions consid-
occur. The reason for this is the presence of the gap ∆
ered here is finite at η = 0, and is of the first order in
in the spectrum.
p p =~/L . Intheone-bandcasethedifferenceofthe
Weak spin-orbit coupling: Ls ≫ L∆. In this case we d1is−tri2bution f∆unctions is of the third order with respect
can calculate the difference of the distribution functions
to p p = ~/L (see Ref. 21). With increasing the
entering Eq. (9) using their expressions at η =0, i.e., 1 − 2 s
gapvalue(∆)fromzero,theinter-subbandspin-orbitin-
f =(eE~k /m)τ (p )∂f /∂ε, (12) teraction decreases and the inter-subband transion rates
1,2 y tr 1,2 0
are suppressed in the collision integral due to impurities
where f is the Fermi function, the electric field E is compared to the intra-subband ones. This leads to the
0
directed along the y-axis, and τ (p) is the momentum- lack of complete cancellation which occurs in the pure
tr
dependent transport scattering time calculated within Rashba model (see Eqs. (14-16) in [21]) and recovering
the Born approximation due to impurity scattering in of the first order effect in p p . This is true for an
1 2
−
the bulk. The values of p ,p are related through arbitrary value of ξ. As for the considered case of the
1 2
5
weak SO coupling, the additional smallness comes in via We acknowledge financial support from FAPESP
the sin2θ L2/L2 1 factor in Eq. (9) , and finally (Funda¸ca˜odeApoio`aPesquisadoEstadodeSa˜oPaulo).
≈ ∆ s ≪
we obtain Helpful discussions with G. Gusev andF. G. G. Hernan-
dez aregreatlyappreciated. A. Khaetskiiis alsograteful
L
∆
S (x) =3k J(x/Λ). (13) to Instituto de F´ısica de Sa˜o Carlos of the University of
h z i E L2
s Sa˜o Paulo for the hospitality.
The calculated edge spin density Eq. (10) is maximal
at L L when it is of the order of k /L . In the op-
∆ s E s
≈
positelimitL L ,whenwithincreasingSOcoupling
s ∆
≪
the spectrumapproachesthe usualRashbatype, S (x)
h z i [1] H.A. Engel, E.I. Rashba, and B.I. Halperin, in Hand-
decreases in magnitude as k L /L2 (see Fig. 2), finally
E s ∆ book of Magnetism and Advanced Magnetic Materials,
approaching the limit calculated in Ref. 21 given by
ed. by H. Kronmuller and S. Parkin, Vol.5 (John Wiley
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s
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× (2005).
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z
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of Ref. [22] have estimated the threshold minimal value 256602 (2005).
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× Rev. Lett.95, 046601 (2005).
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sonofourtheoreticalpredictionsfortheedgespindensity [18] V. A. Zyuzin, P. G. Silvestrov, and E. G. Mishchenko,
with the experimental value of this quantity allows one Phys. Rev.Lett. 99, 106601 (2007).
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[20] A. Khaetskii and E. Sukhorukov, Phys. Rev. B 87,
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[22] F. G. G. Hernandez, L. M. Nunes,G. M. Gusev, and A.
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great difference between the effect in a bilayer structure [23] For the two-subband wide-well sample of Ref. [22] the
compared to the one in a single-layer configuration. We total electron density n = 9.2×1011 cm−2 and we can
show that presence of the gap between two sub-bands estimate the Fermi wave vectors kF,1 and kF,1 by as-
suming that the total electron density is approximately
changes drastically the picture of the edge spin accumu-
lation. Thus, by varying the gap value one can easily ekqFu=alpto2nπ/n2/2w)it=hi1n.7e×ac1h06sucbmb−a1n.dW. HeceanncethkeFn,1d=etekrFm,2in=e
proceed from the regime of strong spin accumulation to the Fermi energy εF =~2kF2/2m=16.4 meV (assuming
theregimeofweakspinaccumulation. Thisopensupthe
m=0.067m0 for aGaAswell). Notethat εF ≫∆=1.4
possibility for the design of new spintronic devices.
6
meV in Ref. [22]. donor layerlocated at distanced from the2D gas. Since
[24] E. Bernardes, J. Schliemann, M. Lee, J. C. Egues, and kFd ≫ 1, the scattering due to impurity potential is of
D.Loss, Phys. Rev.Lett. 99, 076603 (2007). the low-angle type with the characteristic scattering an-
[25] R. S. Calsaverini, E. S. Bernardes, J. C. Egues, and D. gle≃1/kFd≪1.Thereforethetransportscatteringtime
Loss, Phys.Rev.B 78, 155313 (2008). calculated in the Born approximation is proportional to
[26] A fast contribution to the edge spin density which os- the cubic of the momentum τtr(p) = Ap3, where A is
cillates as function of x with 2kF wave vector, gives someconstant.Usingthisfactwecanobtaintheexpres-
parametrically smaller contribution to the total spin sionforthedifferenceofthescatteringtimesgiveninthe
∞
R0 dxhSz(x)i,and we omit it. [21] text.
[27] A.Khaetskii, Phys. Rev.B 73, 115323 (2006). [29] JiyongFuandJ.CarlosEgues,Phys.Rev.B91,075408
[28] We consider a high mobility 2D structure, where scat- (2015).
tering is due to the long-range disorder caused by the