Suppression of compressible edge channels and spatial spin polarization in the integer quantum Hall regime

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by  S. Ihnatsenka

Suppression of compressible edge channels and spatial spin polarization in the integer
quantum Hall regime
S. Ihnatsenka∗ and I. V. Zozoulenko
Solid State Electronics, Department of Science and Technology (ITN),
Link¨oping University, 60174 Norrk¨oping, Sweden
6
(Dated: February 6, 2008)
0
0 We perform systematic numerical studies of the structure of spin-resolved compressible strips in
2 split-gate quantum wires taking into account the exchange and correlation interactions within the
n densityfunctionaltheoryinthelocalspin-densityapproximation. Wefindthatforrealisticparame-
a tersofthewiretheexchangeinteractioncancompletelysuppresstheformation ofthecompressible
J strips. Asthedepletionlengthormagneticfieldareincreased,thecompressiblestripsstartstoform
5 firstforthespin-downandthenforspin-upedgechannels. Wedemonstratethatthewidthsofthese
2 stripsplusthespatialseparation betweenthemcausedbytheexchangeinteractionareequaltothe
width of the compressible strip calculated in the Hartree approximation for spinless electrons. We

] also discuss the effect of electron density on the suppression of the compressible strips in quantum
ll wires.
a
h PACSnumbers: 73.21.Hb,73.43.-f,73.63.Nm,73.23.Ad
-
s
e
m Quantitative analytical description of the edge states tum dots. We find that for realistic parameters of the
in terms of compressible and incompressible strips given wire the exchange interaction can completely suppress
.
t in seminal papers of Chklovskii et al.1 has been the ba- the formation of the compressible strips. The exchange
a
m sis for understanding of various features of the mag- interactioncausesaspatialseparationbetweenthestates
netotransport phenomena in the quantum Hall regime. of opposite spins, with the separation distance being
d- PurelyelectrostaticdescriptionofChklovskiiet al.1 does equalto the widthofthe compressiblestripcomputedin
n not however take into account the exchange interaction theHartreeapproximationofspinlesselectrons(whichis
o between the electrons. This interactions is known to well-described by Chklovskii et al theory1). As the de-
c dramatically affect the edge channel structure bringing pletion length or magnetic field are increased, the com-
[
about qualitatively new features absent in a model of pressiblestrips startsto formfirstforthe spin-downand

1 spinless electrons. These features include, for exam- thenforspin-upstates. Wedemonstratethatinthiscase
v ple, spatial separation of the edge states belonging to the widths of these compressible strips plus the spatial
6 different spin species2,3, and pronounced spin polariza- spinseparationbetweenthem due to the exchangeinter-
9
tion of the electron density in quantum wires exhibiting action are equal to the width of the compressible strip
5
1/B-periodicity related to the subband depopulation4. for spinless electrons calculated in the Hartree approxi-
1
0 A detailed knowledge of the structure of the compress- mation.
6 ible strips inthe presenceofexchangeinteractionas well We consider a quantum wire in a perpendicular mag-
0 as information about spatial separation of spin-up and netic fieldB described by the HamiltonianH = Hσ,
σ
t/ spin-down edge channels are necessary for the under- P
a standing and interpretation of a variety magnetotrans-
m Hσ =H +V (y)+Vσ (y)+gµ Bσ, (1)
portphenomenasuchastunnellingthroughquantumdot 0 conf eff b
- and antidot structures in the edge state regime5,6, un- where H is the kinetic energy in the Landau gauge,
d 0
on uansutiadloptesrtiroudcictuitryeso7fatnhde Amhaanryonootvh-eBr8o.hmTohsecimllaatiinonpsuirn- H0 = −2~m2∗ (cid:26)(cid:16)∂∂x − ei~By(cid:17)2+ ∂∂y22(cid:27), σ = ±21 describes
c pose of the present paper is to provide a quantitative spin-up and spin-down states, ↑ , ↓, and m∗ = 0.067m
e
: description of the structure of spin-resolved compress-
v is the GaAs effective mass. The last term in Eq. (1) ac-
i ible strips in split-gated quantum wires taking into ac- countsforZeemanenergywhereµ =e~/2m istheBohr
X count the exchange interaction between the electrons. b e
magneton,andthebulkgfactorofGaAsisg =−0.44.In
r To this end we solve numerically the Schr¨odinger equa- the split-gate geometry the confining potential V (y)
a conf
tion with exchange and correlationinteractions included
due to the gates, donor layers and the Schottky barrier
within the density functional theory (DFT) in the lo- is well approximated by the parabolic confinement3,4,
cal spin-density approximation4. The choice of DFT V (y) = V + m∗ (ω y)2, where V defines the bot-
conf 0 2 0 0
for the description of many-electroneffects is motivated,
tom of the potential (we set the Fermi energy E = 0).
F
on one hand, by its efficiency in practical implementa- The effective potential, Vσ (y) within the framework of
tion within a standard Kohn-Sham formalism9, and, on eff
the Kohn-Sham density functional theory reads9,
the other hand, by the excellent agreement between the
DFTandexactdiagonalization10 andvariationalMonte- Vσ (r)=V (y)+Vσ(y), (2)
eff H xc
Carlo calculations11,12 performed for few-electron quan-
where V (r) is the Hartree potential due to the electron
H
2
define the width of the compressible strips wH within
comp
the energy window |E − E | < 2πkT ; the depletion
F
length l is extracted from the calculated self-consistent
density distribution by fitting to the Chklovskii et al.
1/2
dependence1 n(y) = n(0) y−l , see Fig. 4(a)). The
y+l
(cid:16) (cid:17)
correspondencebetweentheChklovskiietal. predictions
and Hartree calculations for wH is good, being better
comp
for wires with smaller depletion length l. This is related
tothefactthatthe parabolicconfinementpotentialused
inpresentcalculationsisnotfullyequivalenttothesplit-
gateChklovskiiet al. model1, especiallyforsmoothcon-
finement(largerl). Wefinallynotethatintheconsidered
magnetic field interval the effect of Zeeman term on the
FIG. 1: (Color online) Width of the compressible strips in
quantumwires withdifferentdepletionlengthsl=20nm(a) subbandstructureisnegligible,suchthatwecanreferto
and100nm(b). Parametersofthewires: ~ω0=3meV(a),1 the Hartree results as to the case of spinless electrons.
meV(b);V0 =−0.2eV.Thecorrespondingelectrondensities
n(0)≈3·1015m−2 and 2.5·1015m−2. Temperature T =1K. Having confirmed that the Hartree calculations are
well-described by Chklovskii et al. model1, we turn to
the effect of the exchange interaction on the structure
densityn(y)= nσ(y)(includingthemirrorcharges),4 of the compressible strips. Figure 2 (a) shows the elec-
σ
eP2 +∞ (y−y′)2 tron density profiles ν(y) = n(y)/nB calculated in the
V (y)=− dy′n(y′)ln , (3) Hartree and DFT approaches for a representative fill-
H 4πε0εr Z−∞ (y−y′)2+4b2 ing factor ν(0) ≈ 2.2. As expected, the density pro-
files are practically the same, whereas the correspond-
with b being the distance from the electron gas to the ing subband structures are strikingly different4. In what
interface (we choose b = 60 nm). For the exchange and
follows we shall concentrate on the outermost spin-up
correlationpotentialV (y)weutilizethewidelyusedpa-
xc andspin-downedgestatescorrespondingtothesubbands
rameterizationofTanatarandCerperly13(seeRef.[4]for
N = 1,2. (All the conclusions reported below hold also
explicit expressions for Vσ(y)) . This parameterization
xc for higher subbands). In the Hartree approximation the
is valid for magnetic fields corresponding to the filling
spin-degenerate N = 1,2 subbands form a compressible
factor ν > 1, which sets the limit of applicability of our
strip of the width wH , see Fig. 2 (d). Figure 2 (g)
results. The spin-resolved electron density reads comp
shows corresponding subband structure in the DFT ap-
1 ∞ proximation, where exchange interaction lifts the spin
nσ(y)=− ℑ dEGσ(y,y,E)fFD(E−EF), (4) degeneracy by pushing the spin-up and spin-down sub-
π Z
−∞
bands respectively below and above Fermi energy. This
whereGσ(y,y,E)isthe retardedGreen’sfunctioncorre- occurs because the exchange potential for spin-up elec-
spondingtotheHamiltonian(1)andf (E−E )isthe trons depends on the density of spin-down electrons and
FD F
Fermi-Dirac distribution function. The Green’s function vice versa4,13. In the compressible region the subbands
of the wire, the electron and current densities are cal- areonly partiallyfilled(becausef <1inthe the win-
FD
culatedself-consistentlyusing the technique describedin dow|E−E |.2πkT),and,therefore,the populationof
F
detail in Ref. [4]. spin-upandspin-downsubbands canbe different. Inthe
In order to outline the effect of the exchange interac- DFTcalculation,thispopulationdifference(triggeredby
tion on the structure of the compressible strips we first Zeeman splitting) is strongly enhanced by the exchange
perform calculations in the Hartree approximation (set- interaction leading to different effective potentials for
ting Vσ(y)=0) and, following Suzuki and Ando15 com- spin-upandspin-downelectrons. Thiscausesthesepara-
xc
parethem withthe predictionsofChklovskiiet al.1. Ac- tionofthesubbandswhichmagnitudecanbecomparable
cording to the Chklovskii et al. theory1 the width and to the Landau level spacing ~ω. As the result, the com-
positionofcompressibleandincompressiblestripsarede- pressibleregion(presentintheHartreeapproximation)is
termined by the filling factor ν(0) in the bulk (i.e in the suppressedandthespin-upandspin-downstatesbecome
middle of the wire, y = 0), and the depletion length spatially separated by the distance d ≈ wH . This
sep comp
l. Figure 1 shows the magnetic field dependence of the is illustrated in Fig. 2 (j) which shows the current den-
width of the compressible strips, w , for two repre- sities for the outermostspin-up and spin-downchannels,
comp
sentative quantum wires of different depletion lengths l. peaked at the positions where the corresponding spin-
The corresponding electron density profiles (local filling up and spin-down subbands intersect the Fermi energy.
factors) ν(y)= n(y)/n (n = eB/h) and magnetosub- (The current densities in the Hartree approximation are
B B
band structures are illustrated in Figs. 2 and 3 for some practicallydegeneratedanddelocalizedwithinthe whole
representativevalues of the depletionlengths l andmag- extension of the compressible strip). Outside this region
neticfieldrespectively. (FollowingSuzukiandAndo15we the subbands remains degenerate because they are fully
3
FIG. 2: (Color online) (a)-(c) The electron density profile ν(y)=n(y)/nB, (d)-(i) the magnetosubband structure in Hartree
andDFTapproximationsand(j)-(l)theDFTcurrentdensitiesinwireswithdifferentdepletionlengthsl=20nm,40nm,100
nm (first, second and third columns respectively) for the filling factor ν(0) ≈ 2.2. Fat solid and dashed lines show the total
confining potentials for spin-up and spin-down electrons respectively. Parameters of thewires: V0 =−0.2 eV; ~ω0= 3 meV, 2
meV, 1 meV (first, second and third columns respectively). The corresponding magnetic fields are B ≈5.9T, 5.4T, 4.9T; and
theelectron density n(0)≈(3,2.75,2.5)·1015m−2. T =1K, EF =0
occupiedwhenE .E −2πkT. Astheresult,thecorre- pletely suppresses the compressible strip and the spin-
F
sponding spin-up and spin-down densities are the same, up and spin-down spin channels become spatially sepa-
hence the exchange and correlation potentials for spin- rated with d ≈ wH as described above. For larger
sep comp
up and spin-down electrons are equal, Vx↑c(y) = Vx↓c(y). l the compressible strip of the width wc↓oDmFpT starts to
(Note thattowardsthecenterofthe wirethe degeneracy form for the spin-down subband (Fig. 2 (h)), such that
ofN =1,2subbandsisliftedagainbecauseofthe differ- d +w↓DFT ≈wH , see Fig. 2, second column. With
sep comp comp
ence of spin-up and spin-down densities for electrons in further increase of the depletion length l the compress-
the subbands N =3,4.) ible strip of the width w↑DFT starts to form also for the
comp
Evolution of the subband structure in quantum wires spin-up edge channel, see Fig. 2, third column). Note
as the depletion length l is increased is shown in Fig. 2. that decrease of the spatial separation between the edge
The described above scenario of the suppression of the channels by steeping the confining potential was experi-
compressible strips holds also for larger l and B, with mentally demonstrated by Mu¨ller et al17.
one new important feature. According to the electro-
static description of Chklovskii et al.1, the compressible Qualitatively similar evolution of the magnetosub-
strips are more easily formed in a structure with larger bands takes place when the magnetic field is increased.
l, which is confirmed experimentally16. This feature is This is illustrated in Fig. 3 for two representative filling
clearly manifested in Hartree calculations, Figs. 2 (d)- factors ν(0), where the exchange interaction completely
(f), where the width of the compressible strip grows as suppress the compressible strip for ν(0)=3, whereas for
l is increased (see also Fig. 1). The same effect holds ν(0) = 2 the compressible strip starts to form for the
true in the presence of the exchange interaction. When spin-down channel. The effect of the exchange interac-
l is small as in Figs. 2(g), the exchange interaction com- tion on the structure of the compressible strips is sum-
4
fect of the exchange interaction will be reduced in wires
with larger densities. Figure 4 shows the density profiles
FIG. 3: (Color online) (a),(b) The electron density profile
ν(y) = n(y)/nB and (c)-(f) the magnetosubband structure
in Hartree and DFT approximations in the wire with l =
20nm for different bulk filling factors ν(0) = 3 (left column)
and ν(0) = 2 (right column) (corresponding to B ≈ 4.4T
and 6.4T respectively). Fat solid and dashed lines show the
totalconfiningpotentialsforspin-upandspin-downelectrons
respectively. Parameters of the wire are ~ω0 =3 meV, V0 =
−0.2 eV, corresponding to n(0) ≈ 3·1015m−2. T = 1K,
EF =0. FIG.4: (Coloronline)(a)Theelectrondensityprofileν(y)=
n(y)/nB, and (b)-(e) magnetosubband structure in Hartree
and DFT approximations for two different quantum wires
marized in Fig. 1: the spatial separation between spin-up that have the same depletion length l = 50nm for the fill-
and spin down edge channels plus the widths of the cor- ing factor ν(0) ≈ 2.1. Fat solid and dashed lines show the
totalconfiningpotentialsforspin-upandspin-downelectrons
responding compressible strips are equal to the width of
the compressible strip calculated in the Hartree approxi- respectively. Parametersofthewires, ~ω0 andV0,are(b),(d)
1.5 meV, -0.1 eV, and (c),(e) 2.6 meV, -0.3 eV. T = 1K,
mation, d +w↑DFT +w↓DFT ≈wH .
sep comp comp comp EF =0.
Itshouldbealsonotedthatrelativelyweakspatialspin
polarizationdue to the exchange interaction occurs even
for the fields when the width of the compressible strip is
negligible, wH ≈0. This can be seen in Fig. 1 (a) for and magnetosubband structure for two different quan-
comp tum wires characterized by the same depletion length l.
B .4T.
The corresponding Hartree density profiles (as well as
Let us now discuss the effect of the electron density
wH ) are practically the same, whereas the DFT den-
on suppression of the compressible strips. According to comp
Chklovskiiet al. theory1, at the givenfilling factor ν(0), sity profiles show some difference. In accordance to the
expectation,the widthofthe compressiblestripwDFT is
the depletion length l represents the only relevant scale comp
largerfor higher electrondensity, cf. Figs. 4 (d) and (e).
determining the electron density profile. This means
that widths of the compressible strips wH in differ- Toconclude,weperformadetailednumericalstudy of
comp
ent wires with different confinement strengths ~ω and the effect of the exchange interaction on the structure of
electron densities n(0) are expected to be the same as compressible edge channels, and we find that this inter-
soon as their depletion lengths are the same. On the action can completely or partly suppress the formation
other hand, the exchange interaction is density depen- of the compressible strips.
dent, favoring stronger spin polarization for lower elec- S. I. acknowledges financial support from the Royal
tron densities13,18. It is expected therefore that the ef- Swedish Academy of Sciences and the Swedish Institute.
5
∗ Permanent address: Centre of Nanoelectronics, Depart- Ford,M.Y.Simmons,andD.A.Ritchie,Phys.Rev.B68
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