Quantum Hall effect in a high-mobility two-dimensional electron gas on the surface of a cylinder

0.7 MB·English

by  K. -J. Friedland

Quantum Hall effect in a high-mobility two-dimensional electron gas
on the surface of a cylinder
K. -J. Friedland1,∗ A. Siddiki2,3, R. Hey1, H. Kostial1, A. Riedel1, and D. K. Maude4
1 Paul-Drude-Institut fu¨r Festk¨orperelektronik, Hausvogteiplatz 5–7, 10117 Berlin, Germany
2Physics Department, Arnold Sommerfeld Center for Theoretical Physics, and Center for NanoScience,
Ludwig-Maximilans-Universit¨at, Theresienstrasse 37, 80333 Munich, Germany
3Physics Department, Faculty of Arts and Sciences, 48170-Kotekli, Mugla, Turkey and
4Grenoble High Magnetic Field Laboratory,Centre National de la
9 Recherche Scientifique, 25 avenue des Martyrs, 38042 Grenoble, France
0 (Dated: January 13, 2009)
0
2 The quantum Hall effect is investigated in a high-mobility two-dimensional electron gas on the
surfaceofacylinder. Thenoveltopology leadstoaspatially varyingfillingfactoralongthecurrent
n
path. The resulting inhomogeneous current-density distribution gives rise to additional features
a
in the magneto-transport, such as resistance asymmetry and modified longitudinal resistances. We
J
experimentallydemonstratethattheasymmetryrelationssatisfiedintheintegerfillingfactorregime
2
arevalidalsointhetransitionregimetonon-integerfillingfactors,therebysuggestingamoregeneral
1
form of these asymmetry relations. A model is developed based on the screening theory of the

integer quantum Hall effect that allows the self-consistent calculation of the local electron density
]
l andtherebythelocalcurrentdensityincludingthecurrentalongincompressiblestripes. Themodel,
l
a which also includes the so-called ‘static skin effect’ to account for the current density distribution
h in the compressible regions, is capable of explaining the main experimental observations. Due to
- the existence of an incompressible-compressible transition in the bulk, the system behaves always
s
metal-like incontrasttotheconventionalLandauer-Bu¨ttikerdescription,inwhichthebulkremains
e
m completely insulating throughout thequantized Hall plateau regime.
t. PACSnumbers: 73.23.Ad,73.43.Fj,73.43.-f
a
m
- I. INTRODUCTION spect to the substrate facets9, demonstrated that the
d spatial current-density distribution is modified, thereby
n
creating striking lateral electric field asymmetries. Simi-
o The self-rolling of thin pseudomorphically strained
larly, in wave guides on cylindrical surfaces the chemical
c semiconductorbilayersystemsbasedonepitaxialhetero-
[ potential differences measured along opposite edges of
1 jpurnocptoiosends bgyroPwrninbzyanmdolceocwuloarrk-ebresa1malleopwitsaxtoyi(nMveBstEig)aates the Hall bar and with opposite magnetic field directions
wasshowntodiffer byafactorof1000oreventoreverse
v physical properties of systems with nontrivial topology.
its sign.4,5
0 Using a specific heterojunction, where the high-mobility
9 This large resistance anisotropy, which even persists
two-dimensional electron gas (2DEG) in a 13nm-wide
4 at higher magnetic fields, was intuitively explained by
GaAs single quantum well could be effectively protected
1 the so-called bending away of one-dimensional Landau-
from charged surface states, the electron mobility in
. states(1DLS)fromtheedgesintothebulk4,11,asdemon-
1
the quantum well remains high even after fabrication
0 of freestanding layers2 and particularly in semiconduc- stratedinFig.1(b). Figure1 showsschematicallya Hall
9 bar structure oriented along the periphery of a cylin-
tor tubes.3,4 Implementing this new design, the low-
0 der as used for our investigation. A current I is im-
: temperature mean free path of electrons lS can be kept posed between the current leads E−A, whichEtAherefore
v
long, comparable to the curvature radius r of the tube,
i flows parallel to the gradient k = δB⊥/δy and imposes
X opening the way to investigate curvature-related adia- the chemical potentials µ at terminals i. By adopt-
baticmotionofelectronsonacylindricalsurface,suchas i
r ing the Landauer-Bu¨ttiker formalism the longitudinal
a ‘trochoid’- or ‘snake’-like trajectories.3,5
resistances can be calculated for integer filling factors
Placing a tube with a high mobility 2DEG in a static ν =hn(2eB)−1 =1,2,3... as follows:
and homogeneous magnetic field B , the fundamental
0
dominantmodificationis the gradualchangeofthe com- µ −µ h 1 1
ponentofthemagneticfieldperpendiculartothesurface RL = D C = ( − )=RH −RH
DC I 2e2 ν ν 0 DF
B⊥ along the periphery of the tube, which is equivalent EA 0 DF (1)
µ −µ h 1 1
to a gradualchangeofthe filling factor ν. This is anim- RL = F G = ( − )=RH −RH
portant modification for the quantum Hall effect, which FG IEA 2e2 ν0 νCG 0 CG
has recently stimulated notable theoretical interest.6,7 Here, the position y⊥ at which the magnetic field B0
Earlier investigations of the magneto-transport with is directed along the normal to the surface n, is located
spatially varying magnetic fields, created by a density between the leads F −G and D−C. ν , ν are filling
0 ij
gradient8 or by magnetic field barriers inclined with re- factorsatthepositionsy⊥ andoftheHallleadpairsi−j,
2
respectively. h denotes Planck’s constant and e the elec- gin of this effect is the gradual change of the Hall field
tronic charge. For clarity, we use the superscripts L and alongthe Hallbarwhichactsonthelongitudinalelectric
H for the longitudinal and Hall resistances,respectively. fieldsothatitbecomesdifferentonbothsidesoftheHall
The arrows in Fig. 1 indicate the chirality of the 1DLS bar. Microscopically, the SSE is a result of an exponen-
anddeterminethoseHallleads,fromwhichthepotential tial current-squeezing towards one of the Hall bar edges
is induced into the opposite longitudinal lead pair for a and is characterized by the skin length L = (kµ)−1,
skin
given direction of the magnetic field. For the situation where µ is the carrier mobility. Asymptotically, for high
in Fig. 1, the Hall resistance RH induces a finite RL , magneticfieldstheSSEisdescribedbythesameEq.(2),
DF DC
while the Hall voltage RH do so for RL , etc. in the form of: RL =0, RL =RH −RH .
CG FG CB BH CG BH
The longitudinal resistances for pairs of leads outside Despite this similarity, both mechanisms differ antag-
the position y⊥ read: onistically in their microscopically origin. For the ex-
planation of the SSE it is assumed that a current flows
µ −µ
RL = C B =0 exclusively at one edge of the Hall bar which changes to
CB I
EA theoppositeonebyinvertingthemagneticfielddirection.
RL = µG−µH = h ( 1 − 1 )=RH −RH . Incontrast,theapplicationoftheLandauer-Bu¨ttikerfor-
GH I 2e2 ν ν CG BH malism for the 1DLS states presupposes current flow at
EA CG BH
(2) both edges of the Hall bar. In the quantum Hall regime,
for the situation presented in Fig. 1, the longitudinal re-
Reversingthedirectionofthemagneticfieldresultsinan sistance RL with leads, which are still bound by the
interchange of RDLC ⇀↽RFLG and RCLB ⇀↽RGLH . outermostCeBdge channels, remains zero at all times. In
The resistance anisotropy in Hall bars with magnetic
contrast, the bending of the innermost 1DLS channels
field gradient along the current direction is also well
into the opposite leads causes the nonzero longitudinal
knownfromclassical(metal-like)electrontransportstud- resistanceRL thatcompensatesthechangeofthetrans-
GH
ies at low magnetic fields. The anisotropy was also pre-
verse Hall-voltages.
dicted by Chaplik, and is referred to as the ‘static skin
Inthispaper,wepresentquantumHalleffectmeasure-
effect’ (SSE).10,11 An experimental demonstration was
ments of a high- mobility 2DEG on a cylinder surface
reportedby Mendach andcoworkers.12 The physical ori-
and show that a significant part of the results cannot be
explained by the simplified 1DLS-approach. We observe
clearindicationsthattheactualcurrent-densitydistribu-
(a)
tion in the Hall bar should be reconsidered and propose

anewmodelwhichtakesintoaccountmorepreciselythe

sequential current flow along incompressible stripes and
metal-like compressible regions, for which a current dis-
tribution according to the SSE should be considered.

II. EXPERIMENTAL

(b) The layer stack, with an overall thickness of 192 nm
including the high mobility 2DEG, was grown on top of
a 20nm-thick In Ga As stressor layer, an essential
0.15 0.85
component of the strained multi-layered films (SMLF).
An additional 50nm-thick AlAs sacrificial layer is intro-
duced below the SMLF in order to separate the SMLF
from the substrate.
Forthefabricationofcurved2DEGs,wefirstfabricate
conventionalHallbarstructuresintheplanarheterojunc-
tion along the [100] crystal direction. The two 20 µm-
wide Hall bar arms and three opposite 4µm-narrow lead
pairs, separated by 10 µm, are connected to Ohmic con-
tact pads outside of the rolling area in a similar man-
ner as the recently developed technology to fabricate
FIG. 1: (a) Sketchof aHall-bar on theperipheryof a cylin-
der. (b) Schematicof such aHall barindicating thegradient laterally structured and rolled up 2DEGs with Ohmic
ofthemagneticfieldk,theimposedcurrentIEA andimposed contacts.13,14 Subsequently,theSMLFincludingtheHall
chemical potentials µi at leads i. The magnetic field is per- bar was released by selective etching away of the sacrifi-
pendicular at the position y⊥ (y = 0 is defined to be at the cialAlAslayerwitha5%HFacid/watersolutionat4◦C
centerHallleadsC−G). The1DLSareshownschematically. starting from a [010] edge. In order to relax the strain,
The arrows indicate their chirality . the SMLF rolls up along the [100] direction forming a
3

6 1.1 1.1
-d B -d B
^ ^
(a)
4 D C B 1.0 1.0
E A )H
2 BSd 0.9 0.9R
F G H e DF
60 (2 0.8 0.8en
/
n
(b) u.) h
b. 0.7 D-F C-G 0.7
) 4 ar
O (
WWWWR (k 2 SdH 0.6 -10 -5 0 5 0.6

-0,6 -0,9 B-1- 1,2 -1,5 position y m(mmm m)
60 ^ ^^^
(c)
FIG.3: FundamentalfrequencyShubnikovdeHaasoscillation
4 BS−d1Hinunitsof2e(hn)−1calculatedfromRDLC(closedcircles)
L H
and RFG (closed squares). The low field Hall resistance RDF
2 in units of (en)−1 is shown by is small open circles.
0
-30 -20 -10 0 10 20 30 as B⊥ =B0cos(ϕδy), where ϕδy = arcsin(δy/r). Accord-
ingly, the magnetic field gradient can be calculated as
B (T) ∼
0 k =B0δy/r2. When we consider the given mobility and
thefieldvalueB =0.66T,wecanestimateaskinlength
∼ 0
FIG. 2: Longitudinal resistances with a magnetic field B0 Lskin =670nmatthepositionsofthenextleftandright
normal to the surface at the center Hall-leads y⊥ = 0. (a) pairsoftheHallleads. Asthedirectionofcurrentsqueez-
L L
RDC,theinsetshowstheorientationoftheHall-bar,(b)RCB, ingisdeterminedbythesignofthefieldgradient,wefind
L
theinsetshowsthesecondderivativeofRCB,asafunctionof that for positive magnetic field values, that the current
L
thereciprocal of themagnetic field, (c) RDB. isconcentratedexponentiallyclosetothe upper Hallbar
edge between the D−C leads, while the current is con-
centrated exponentially close to the lower Hall bar edge
completetubewitharadiusrofabout20µm. Wereport between the G−H leads. Inverting the magnetic field
on specific structures which are described in Ref. 3 and directionresultsinachangeoftheHallbaredgesforthe
whichhaveacarrierdensityofn∼=(6.8−7.2)×1015m−2 current flow.
and a mobility of up to 90 m2(Vs)−1 along the [100] Incontrast,ascanbeseeninFig.2(c),thelongitudinal
crystal direction before and after rolling-up. All pre- resistances measured between leads D and B is nearly
sented measurements were carried out at a temperature symmetric,despitethefactthatRL resultsfromcurrent
DB
T= 100 mK. flow in different spatial areas.
III. RESULTS AND DISCUSSION B. Shubnikov de Haas oscillations
A. Asymmetry of the longitudinal resistances We observe a complex structure of the Shubnikov de
Haasoscillations(SdHO).Inparticular,aclearbeatingin
The strong asymmetry of the longitudinal resistances theSdHOresultsinnodesinthesecondderivativeofthe
for the current parallel to the magnetic field gradient k longitudinal resistances with respect to the inverse mag-
is demonstrated in Fig. 2. The magnetic field is perpen- netic field as seen, for example, in the inset of Fig. 2(b).
dicular to the surface around the center Hall lead pair As a result, the low-field SdHO are composed of at least
C −G the position of which we define as y⊥ = 0. The two fundamental SdHO frequencies BS−d1H, as calculated
longitudinalresistancesRL - onthe rightsideandRL by a Fourier transform analysis.
CB DC
- on the left side of this position differ strongly for a Wehaveanalyzedthe two-frequencySdHO patternby
given magnetic field and are asymmetric with respect to rotating the tube around the cylinder axis through an
the direction of the magnetic field. For example, at a angle ϕ, thereby shifting the position y⊥ away from the
magnetic field of B0 = 0.66 T, where RCLB shows a min- center pair of Hall leads C −G. For y⊥ values between
imum, the ratio RL /RL exceeds 300. With the de- the longitudinal voltage leads D −C, Fig. 3 shows the
DC CB
viation δy towards either side of the perpendicular field dimensionless value hn/(2eBSdH) as a function of y⊥. In
position, the component of the magnetic field decreases thesamefigure,wepresentalsothe dataforthe classical
4

Halleffect RH en,which correspondsnicely to the lower 25
frequency SdDHFO branch. Therefore, we conclude that (a) R H
CG
this brancharises from the B⊥ values at the pair of Hall ) 20
leads D−F, which induce a voltage at the leads D−C. WWWWk
Theupperbranch,closetohn/(2eBSdH)=1,reflectsthe e ( 15
SdHOforB0atthepositionsy⊥. Weconclude,therefore, c
n
that the two-frequency SdHO pattern is in accordance n
a
with Eq. (1) in the form of RDLC = R0H −RDHF,RFLG = st 10 1 H
Rre0Hsis−taRnCHcGesarsetflheectSsdHthOeofiflltihnegcofarcrteosprsonvdailnugesloνngiatunddinνal esi n RBH
at y⊥ and the corresponding pair of Hall lea0ds i − jij, ll r 5 2
a
respectively.
H
0
12
(b) RL
C. Quantum Hall effect 10 GH
The quantum Hall effect can be observed for a wide ) 8 H H
R -R
rangeofmagneticfieldgradients. Figure4showstheHall WWWW(k n n CG BH
resistancesRBHH andRCHG andthelongitudinalresistance R 6 n 2-n1
RGLH for y⊥ = −9.4 µm (close to the pair of Hall leads 2 1
D−F), which represents a large gradient case. The fill- 4
ing factors differ substantially for subsequent Hall leads.
For positive magnetic field values, the longitudinal resis- 2
tances RL and RL are always non zero. As a special
DC GH
case, we indicate in Fig. 4 some of the magnetic field re-
10 15 20 25
gionswherebothHallterminalsareatdifferentbutinte-
gerfillingfactors,thusprovingtheexistenceofquantized B (T)
0
conductanceinthe non-zerolongitudinalresistanceR
GH
inaccordancewithEq.(2). Moreover,inFig.4,itcanbe H H
seen that the equation RGLH = RCHG−RBHH holds for all FreIsGis.ta4n:ce(aR)GLHHalflorreys⊥is=ta-n9ce.4sµRvCsG.Ba0n.dQRuaBnHt,iz(abt)iolnoningiRtuGLdHinaatl
positive magnetic fields values, i.e. also for non-integer ν2ν1/(ν2−ν1)isindicatedbytheshadedrectanglesforthose
filling factors, which is not guarantied by the Landauer- regionswhereRCHG andRBHH areatintegerfillingfactors. We
Bu¨ttiker approach for Eq. (2), but is in agreement with plot thecalculated resistance RCHG - RBHH bya thinblueline.
the local Kirchhoff’s law of voltage distribution in elec-
tronic circuits with current. Therefore, we conclude
that for the large gradient case the equality between the
outer left and outer right expressions in Eqs. (1) and resistanceRH remainsquantizedatcorrespondingmag-
DF
(2) account for the current and voltage distribution in
netic fields, see Fig. 5. In order to explain this effect, we
our system in a more generalfashion than the simplified
will use our model as discussed in the following section.
Landauer-Bu¨ttiker approach for conductance along one-
dimensional channels. We will show that our model can
be used for a more quantitative explanation.
In the case of moderate gradients, i.e. small distances
IV. MODEL
of y⊥ from the corresponding middle pair of Hall leads,
we observe a striking deviation from the set of Eq. (2).
Despite the fact that we should expect RL =0 for any We now discuss our experimental findings in the light
CB
field value, we observe clear resistance maxima, which of self-consistentcalculations of the density distribution.
even increase in height with increasing magnetic field at Weexploittheinherentsimilarityofthefillingfactorgra-
thehighmagneticfieldendofthequantizedHallplateau dient generated by the inhomogeneous magnetic field to
measured for the nearest pair of Hall leads, see Fig. 5. the density gradient and utilize current confinement to
While the maximum values in RL remain an order of one of the Hall bar edges resulting from the SSE. In our
CB
magnitudelowerthenthereverseone,namelyRL ,they model calculations, we assume periodic boundary con-
GH
exceed the background minima due to the SSE at low ditions in two dimensions to describe the Hall bar elec-
magnetic fields by an order of magnitude. We exclude trostatically. Themagneticfieldgradientissimulatedby
that these resistance maxima arise from a certain inac- anelectrondensitygradient,whichessentiallymodelsthe
curacyintheleadfabricationprocess,whichcouldresult filling factor distribution over the Hall bar. The density
in a small cross talk from the voltage inducing Hall lead gradientis generatedbyanexternalpotentialpreserving
pair D −F into the lead C, by ensuring that the Hall the boundary conditions. The total electrostatic poten-
5

20 where κ¯ is an average dielectric constant (= 12.4 for
) ′ ′
(a) RH GaAs) and K(x,y,x,y ) is the solution of the 2D Pois-
WWWWk CG H sonequationsatisfyingthe periodicboundaryconditions
( R
e 15 DF we assume 17 Eqs 3 and 4 form a self-consistent loop,
c
which has to be solved numerically.
n
a In our simulations, we start with a sufficiently high
st RH
i 10 BH temperaturetoassureconvergenceanddecreasethetem-
s
e perature step by step. In the first iteration, we as-
Hall r 5 saunmdecaalchuolmatoegeVnbego(xu,syb)afcrkogmroEunqd. ((5d)onreopr)ladciisntgribnuelt(ixo′n,yn′0)
by this constant distribution. The density gradient is
produced by employing a periodic external potential
V (x,y) = V cos(2πx/L ), where L is the length of
ext 0 y y
the Hall bar and V the amplitude, reproducing also the
80 (b) 0
cosine-like dependence of the perpendicular component
of the magnetic field B⊥, which exactly models the ex-
60 L
R perimental situation represented in the Fig. 2. Here we
) CB
should note that, due to the computational limitations,
WWWWR( 40 we confined our calculations to a rather narrow sample.
Nevertheless, our results are scalable15,16 to larger unit
20 cells, which is, however time consuming.
As it was shown earlier for homogeneous and con-
0 stricted 2DEG systems the calculations reveal that the
waveguide is divided into compressiblebulk regionsand
5 10 15 20 incompressiblestripes19. Figure6presentsthecalculated
spatial distribution of the incompressible stripes (yellow
B (T) areas)forthreecharacteristicvaluesofthemagneticfield
0
as a function of lateral coordinates. Arrows indicate the
FIG.5: (a)Hallresistances RCHG,RDHF andRBHH,(b)longitu- currentdistribution,whichwillbediscussedindetailbe-
L L low. Before proceeding with the discussion of the rela-
dinal resistance RCB for y⊥=-2.1 µm vs. B0. Peaks in RCB
appearatthenearestHallleadpairatthehighmagneticfield tion between incompressible stripes and quantized Hall
end of thequantized Hall plateau. effect, we would like to emphasize the difference in the
distributionofthe incompressiblestripesforthe selected
magnetic fields.
tial energy experienced by a spinless electron is given by
In Fig. 6(a), two incompressible stripes appear along
V (x,y)=V (x,y)+V (x,y)+V (x,y), (3) the edges of the Hall bar, which are slightly curved to-
tot bg ext H
wards the center due to the simulated bending, i.e. the
whereV (x,y)isthebackgroundpotentialgeneratedby
bg external potential V (x,y). The two stripes merge at
the donors, V (x,y) is the external potential resulting ext
ext the center of the Hall bar at a higher magnetic field,
from the gates (which will be used to simulate the filling
ν = 2.1, so that the center becomes completely incom-
factorgradient)andthemutualelectron-electroninterac-
pressible. Whereas, at the highest magnetic field value
tion is described by the Hartree potential V (x,y). We
H considered here the center becomes compressible. In ad-
assume that this total potential varies slowly over the
dition to the difference between the screening proper-
quantum mechanicallength scale,givenby the magnetic
ties of the metal-like compressible (nearly perfect) and
length l = ¯h/mω so that the electron density can
b c insulator-like incompressible regions (very poor),18 their
be calculatedpwithin the Thomas-Fermi approximation
transport properties are also remarkable different. As
in 2D15,16 according to
mentioned before, the compressible regions are metal-
n (x,y)= D(E,x,y)f(E+V (x,y)−µ∗)dE (4) like. Therefore, scattering is finite, and hence resistance
el Z tot is also finite. However,atthe incompressible stripes, the
resistance vanishes somewhat counter intuitively since
where D(E,x,y) is the (local) density of states, f(E) =
∗ the conductance is also zero.19 A simple way of under-
1/[exp(E/k T)+1] the Fermi function, µ the electro-
b standing this phenomenon is to consider the absence of
chemicalpotential(whichisconstantinequilibrium),k
B backscattering within the incompressible stripes. More-
Boltzmann’sconstant,andT thetemperature. Sincethe
over, a simultaneous vanishing of both the longitudinal
Hartreepotentialexplicitly depends onthe electronden-
resistance and conductance is a general feature of two-
sity via
dimensionalsystemssubjectedtoastrongperpendicular
2e2 ′ ′ ′ ′ ′ ′ magnetic field. Based on these arguments, the impor-
V (x,y)= K(x,y,x,y )n (x,y )dxdy , (5)
H κ¯ Z el tantfeaturesoftheintegerquantizedHalleffectandlocal
A
6
D' D C B 10000
0.8
1000
(a) n (x,y)>2 ) 100 RSSE RGLH
0.4
WWWW(

R
n (x,y)<2 10

F G H 1
0.8
0,1
)
m 0 5 10 15 20 25
(b)
B (T)
m(0.4
0
x
n (x,y)=2 L
FIG. 7: The measured longitudinal resistance RGH (black
thick line) vs. B0 and the calculated theory curveRSSE (red
thin line).
0.8
very close to the corresponding values calculated by us-
(c) ing the given mobility, the tube radius, and the width of
0.4 the tube. We see, that the fitted curve follows the ex-
perimentalresultsfairlywell. Inparticular,atlowfields,
SSE
theagreementisnearlyperfectsinceathigherfillingfac-
tors the transition from compressible to incompressible
-0.6 -0.2 0.2 0.6 (in other words metal to insulator) states at the center
y (m m) occurs over a very narrow magnetic field range so that
the bulk remains almost always compressible. However,
at higher fields the measured resistance exhibits oscilla-
FIG.6: Thecalculatedspatialdistributionoflocalfillingfac-
tions around the theoretical curve, which are clear sig-
tors with integer values, ν(x,y)=2 incompressible, (yellow)
natures of a compressible to incompressible transition in
andcompressible,(white)forthreeselectedvaluesofthemag-
the bulk.
neticfieldcorrespondingtoafillingfactoratthey⊥ =0posi-
tion ν =(a) 2.5, (b) 2.1, and(c) 1.9 at 1.6Kelvin . Theunit Now,wecanreconsiderthe currentdistributioninour
cell is chosen to be1×2µm2, spanning48×96 mesh points model. As mentioned above the applied external cur-
in ournumerical simulation. rent is confined to the incompressible stripes, due to
the absence of backscattering. In a conventional Hall
bargeometry,ifanincompressiblestripepercolatesfrom
probe experiments20 can be explained.21,22 source to drain contact, the system is in the quantized
The appearance of a metal-like compressible region Hallregime, i.e. the longitudinalresistancevanishes and
along the current path, see Fig. 6(c) forces us to in- simultaneously the Hall resistance is quantized. Such a
cludeanotherimportantingredientinourmodel,namely situation is observedin Fig. 6(a), where the longitudinal
the SSE. This phenomenon is fundamental. A fixed cur- resistance measured between the leads D −C (or sim-
rent imposed in a bent metal stripe in a magnetic field ilarly F − G, C − B, G − H) vanishes, while at the
becomes confined to one edge of the metal due to the same time the Hall resistance is quantized, according
curvature of the system. The following two-parameter RH = RH = RH = e2/(2h). Similarly, if the center
DF CG BH
expression may be derived using the SSE theory: becomesincompressible,Fig.6(b),theHallresistancere-
mainquantizedetc. Notethatnow,whenthe higherend
B eB/B1 of the quantized Hall plateau is approached, a striking
RSSE =RSSE , (6)
0 B11−eB/B1 effect is observed. When the percolating incompressible
stripe breaks due to the bending of the structure, the
where RSSE is the resistance at B =0, B = r/(µw), bulk becomes metal-like, and therefore the SSE comes
0 0 1
w is the Hall bar width. In Fig. 7, we provide a semi- now into play, Fig. 6(c).
logarithmic plot, fitting the measured longitudinal resis- First, let us discuss the Hall resistance measured be-
tanceRL ofthe highresistancebranchwithRSSE. The tweencontactsD−F: ThequantizedHalleffectremains
GH
fit parameters RSSE=6.03 and B =0.015 T hold for low unchanged, since the bulk is well decoupled from the
0 1
as well as high magnetic fields. In addition, they are edges and the current is flowing from the center incom-
7
pressible region. Such an argument also holds for the ena, which indicate the presence of a specific current-
Hall resistance measured between the contacts B −H. densitydistributionintheHallbar. Themostprominent
Next, if we measure the longitudinal resistance between asymmetryrelationsholdnotonlyforthesimplifiedcase
say D′ −D, we would observe that the resistance van- developedfortheintegerfillingfactors,butalsoinamore
ishes due to the existence of the percolating incompress- general fashion including the transition regions between
ible stripe between these two contacts. However, if we integer filling factors. Indeed, the integer filling factor
measure RH simultaneously, we will see that the quan- caseappearstobe arelativerarecasedue tothe gradual
CG
tization is smeared out since now the bulk behaves like varying filling factor over the current path.
an ordinary metal. At this point, due to the SSE, the
current is diverted towardthe edges of the Hall bar, e.g. We have briefly discussed the screening theory of the
to the upper edge on the left side of the Hall bar and to integer quantum Hall effect and employed this theory to
the lower edge on the right side for the one direction of our system by simulating the filling factor gradient. The
themagneticfieldandviceversafortheoppositefielddi- electron density is obtained self-consistently, while the
rection. Therefore,themeasuredlongitudinalresistances (local) current distribution is derived based on a phe-
RGLH andRDLCwillexhibittheSSEwithsmalldeviations, nomenologicallocalOhm’slaw. Wehaveexplicitlyshown
resulting from the incompressible to compressible tran- that due to the transition from incompressible to com-
sition. This scenario implies also that the current will pressible states in the bulk, the system behaves metal-
flow across the Hall bar at the position y⊥ =0 from one like. Therefore, SSE is observed in our measurements,
edge to the opposite one. We believe that this transition
around the Hall leads RH also accounts for the sharp
CG This model allows us to explain the additional sharp
peak structure of the resistance around the transition
peaks in the resistance near the transition point, which
point in RL and RL , cf. Fig. 5. This effect cannot be
CB FG appear in the otherwise zero-resistance edge of the Hall
explainedbythesimpleLandauerBu¨ttikerapproachand
bar and indicate a peculiar current swing from one edge
indeed it would not simply occur in flat-gated samples.
to the other. Such an effect cannot be explained by the
In the discussion above, we have argued that the SSE
conventional Landauer-Bu¨ttiker formalism, since in this
becomesdominantwhenthecenterofthesystemiscom-
picture the bulk remains completely insulating through-
pressibleandthat sucha transitioncannotbe accounted
out the quantized Hall plateau regime.
for in the 1DLS picture, where the bulk should always
remain incompressible. The other features explained by
the1DLSareequallywellexplainedbythescreeningthe-
ory, naturally, for the case of equilibrium. As an im-
portant point, we should emphasize that the screening
theoryfails tohandle the non-equilibriummeasurements
performedbymanyexperimentalgroups(forareviewsee
Ref. 23), since this theory is based on the assumptionof
a local equilibrium. However, in our case the filling fac-
Acknowledgments
torgradientisNOTgeneratedbythe gates(i.e. creating
non-equilibrium), but by the inhomogeneous perpendic-
ular magnetic field. Therefore, δν(x,y) is adiabatic, and
the system remains in equilibrium. The authors gratefully acknowledge stimulating dis-
cussions with R. R. Gerhardts P. Kleinert and H.T.G.
Grahn. WethankE.WiebickeandM.Ho¨rickefortechni-
V. CONCLUSION calassistance. Oneofus,A.S.,wasfinanciallysupported
byNIMAreaA.TheworkatGHMFLwaspartiallysup-
The quantum Hall effect for a high-mobility 2DEG on ported by the European 6th Framework Program under
acylindersurfaceshowadditionalexperimentalphenom- contract number RITA-CT-3003-505474.
∗ Corresponding author. Electronic address: 4 A. B. Vorob’ev , K.-J. Friedland , H. Kostial, R. Hey, U.
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