Table Of ContentLandau level broadening without disorder, non-integer plateaus without interactions –
an alternative model of the quantum Hall effect
Tobias Kramer1
1 Department of Physics, Harvard University, 17 Oxford Street, Cambridge, MA 02138, USA.∗
6 (Dated: January 25, 2006)
0
I review some aspects of an alternative model of the quantum Hall effect, which is not based
0
on the presence of disorder potentials. Instead, a quantization of the electronic drift current in
2
thepresence of crossed electric and magnetic fields is employed to construct a non-linear transport
n theory. Anotherimportantingredientofthealternativetheoryisthecouplingofthetwo-dimensional
a electron gas to the leads and the applied voltages. By working in a picture, where the external
J
voltagesfixthechemicalpotentialinthe2Dsubsystem,theexperimentallyobservedlinearrelation
1 between the voltage and the location of the quantum Hall plateaus finds an natural explanation.
3 Also, theclassical Hall effect emerges as a natural limit of thequantumHall effect.
For low temperatures (or high currents), a non-integer substructure splits higher Landau levels
] into sublevels. The appearence of substructure and non-integer plateaus in the resistivity is not
l
l linkedtoelectron-electron interactions, butcaused bythepresenceofa(linear) electricfield. Some
a
of theresulting fractions correspond exactly tohalf-integer plateaus.
h
-
s PACSnumbers: 73.43.Cd
e
m
I. THE CLASSICAL HALL EFFECT. oriented perpendicular to both, electric E and magnetic
.
at B fields. Averaging the equation of motions over one
m cyclotronperiod T =2πm/(eB)yields the drift-velocity:
A purely electric field leads to a uniform acceleration
- of a charged particle, whereas a purely magnetic field
d
forcestheparticleonacircularpath. Thecombinationof
n 1 t+T
bothfieldsgivesrisetotheelectrondriftmotion,whichis v = dt′r˙(t′)=(E ×B)/B2. (1)
o d T Z
c t
[
The drift-velocity v is also independent of the initial
d
2 16000 velocity r˙(0).
v In the following I consider the electronic motion in a
m] 14000
1 h 1h two-dimensionalsubsystem. The orientationofthe mag-
2 O 12000 2e2 netic field is shownin Fig.3. The constantdrift-velocity
6 [
y hasimportantconsequencesforthetransportofelectrons
1 x 10000
ρ
0 y 1h
at/06 Resistivit 468000000000 14eh2 3e2 ρxx×1000 hm] 1124000000 52eh2 41eh2
m O 10000
d- 2000 ρ[xy 8000 27eh2 31eh2
con 0 0 2 M a4gnetic 6Field 8[T] 10 12 stivity 6000 NO61eh1221eh2NO15eh229eh2 14eh2 ρxx×1000
: si 4000
v Re
i 2000
X
FIG. 1: Classical Hall line (straight line) vs. quantum Hall
r curve, calculated from [1]. The QHE leads to a quantized 0
a rfoersistthaenvcaeluρexsyi=n b1ireah2ck,eits=):1e,ff2e,c3t,i.v.e..mPasasramm∗et=er0s.1(r,emfeorebnilciteys 3 4 Mag5netic fiel6d [T] 7 8
µ=17m2V−1s−1,effectiveg-factorg∗ =10[2],temperature
T =1K,currentjx =1Am−1,averagecarrierdensityNav =
2.4 × 1015 m−2 (corresponding to a fixed Fermi energy of FIG. 2: Same as Fig. 1, but at T =150 mK. The substruc-
EF =11.6 meV.) tureof theLDOSin higherLandaulevels isvisible(compare
Fig. 5). Half-filled plateaus exist at ρxy =(52,72)eh2, however
not at ρxy = (92,121)eh2. Notice that even for T = 0 K the
substructure remains in place, thus giving rise to subdivided
∗Electronicaddress: [email protected] Landau levels in a non-interacting particle model.
2
2DEG integer quantum Hall effect, the resistivity ρ is quan-
MagneticFieldB xy
J Econ tized:
E1 E(3D)
Uy W eVg EE0F EvFal ρxy = ihe2, i=1,2,3,... (4)
The conditions for the observation of the quantum Hall
Ux E(3D(cid:0)(cid:1))(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1) effect are low temperatures and very clean samples.
F(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
x Metal SiO2 p-Si Interestingly, no standard theory of the integer quan-
y
tum Hall effect is available. While there exist several
z-direction
models which lead to a quantized resistivity, basic ques-
tionsremainunanswered: forexample,thebreakdownof
the quantizedresistivity abovea criticalcurrentis docu-
FIG.3: Left panel: Schematicviewof aHall bar. A current
mentedexperimentally,butremainsachallengeformost
J is flowing through a two-dimensional electron gas (2DEG)
in thex−y–plane, which is oriented perpendicularto an ex- theories.
ternalmagneticfieldB. Thedeflectedelectronsatthesample In the case of the electric field, the difficulty comes
edges produce a Hall voltage Uy over the sample width W, fromthefactthatcurrenttheoriesofthequantumversion
which is measured along with the longitudinal voltage drop of the Hall effect are not based on the Hamiltonian of
Ux. Rightpanel(adaptedfrom[3],Fig.1): Schematicpicture crossed electric and magnetic fields, but rather on the
of a Metal-Oxide-Semiconductor (MOS) device. The two- additionofadisorderpotentialtoapurelymagneticfield:
dimensionalelectrongas(2DEG)attheinterfacebetweenthe
oxide and the silicon is controlled by applying a gate voltage
Vg. The gate voltage changes the Fermi energy EF(3D) of the Hlattice,disorder = p− eA(r) 2/(2m)+VLD(r), (5)
semiconductor,whichinturncouplestotheFermi-energyEF h c i
ofthe2DEG.IfEF <(E1−E0)holds,theelectronsonlypop- where V (r) denotes a periodic lattice potential and
LD
ulate theground state of the1D quantumwell in z-direction
possiblyuncorrelateddisorderpotentials(whichareoften
thathastheeigenenergyE ,whichlinksbothFermienergies
0 assumed to disappear on the average: drV (r) = 0).
via EF =EF(3D)−E0. This Hamiltonian differs from the clasRsical HLDall Hamil-
tonian by the omission of the electric Hall field. The
disorder potential becomes an essential part of the de-
in a solid which is placed in a magnetic field. In a clas-
scription and the appearance of a quantized conductiv-
sicalHallexperiment deflected electronsform anelectric
ity is linked to the presence of a fluctuating potential-
fieldalongtheedgesofametal. Theconductingelectrons
landscape V [6, 7]. Also it cannot sustain an electric
propagateinthepresenceofthiselectricHallfield,which LD
field, which would require that the potential landscape
can be used to determine the carrier-density in the sam-
is not averagedto zero. Thus for most previous theories
ple [4]. Completely neglecting scattering events, one can
of the quantum Hall effect, the electric Hall-field brake
extractthe basic relationbetweenthe classicalcurrentJ
is disregarded. In contrast to the classical Hall effect,
disorder forms an essential part of the model.
J=Nev , (2)
d
II. QUANTIZED SLOPES IN THE QUANTUM
(N denotes the electron density, e the electronic charge)
HALL EFFECT.
and the resistivity tensor ρ (or it’s inverse, the conduc-
tivity tensor σ) from Ohm’s law:
In this section I explore the connection between the
density of states (DOS), the Fermi energy,andthe num-
Ne 0 −1
J=ρ−1·E ⇒ ρ−1 =σ = . (3) ber ofcurrentcarriers. In principle,the number (orden-
B (cid:18)1 0 (cid:19)
sity of carriers) is obtained by a convolution integral of
two independent quantities: the DOS and the probabil-
Theresistivityρ =B/(Ne)isproportionaltothemag-
xy ity of occupation of a quantum state, which is given by
netic field. Notice that the classical Hall effect does in
the Fermi-Dirac distribution:
principle not depend on the presence of disorder or scat-
∞
tering processes. The “electric Hall-field brake” ensures
N(E ,T) = n(E)f(E,E ,T)dE (6)
F F
a constant drift velocity. Z
−∞
−1
f(E,E ,T) = e(E−EF)/(kT)+1 . (7)
F
h i
A. The quantum Hall effect.
For very low temperatures, the Fermi-Dirac distribution
becomes a step-function:
In contrast to the classical Hall effect, the quantum
HalleffectobservedbyvonKlitzing[5]showsanon-linear EF
N(E ) = n(E)dE (8)
variationoftheresistivitywiththemagneticfield. Inthe F Z
−∞
3
B=16 T p=3 p=4 • On the other hand, a system which is part of an
p=2
p=1 electriccircuitcanundergofluctuationsinthenum-
Magneticfield[Tesla]1111165343 p=1 2.5 Gp=a2teVolat3a.0gep=3Vg[Volt] 3p=.45 Magnetic Field [T] 2 p=1 2.5 Gp=a2te Vol t3agpe=3 [V] 3.5p=4 Tpothnirceaetduldairtebseitfioenrfnooeacorsclfutttphhphaeeieerotdcreiiucneslretserraseogt,nfyewt,t-hohpcfeearrortQerbhayHleseinEmstgyhstseretayleeermnictsetoerrtougfornysoesbmr.aeenmtHidhaNoeeiwnn=astebivcficseaxeorlne,ndtcsioe.tf
of available states in the gap between two Landau lev-
els. Another mechanism is needed to “pin” the Fermi-
FIG. 4: Grayscale plot of the conductance σxx as a func- energy in between two Landau levels. A commonly used
tion of the gate voltage Vg and the magnetic field in a sili- approach is the addition of another kind of density of
con MOSFET. Left panel: A schematic representation of the
states, which does not support a current but only pro-
experimental data obtained by Cobden et al., published in
vides a non-zero density of states in the gap. This other
[8], Fig. 2(a). Right panel: theoretical prediction using [1],
with the following parameters (references for the values in kind of electrons act as a reservoirand should buffer the
brackets): transverse effective mass m∗ = 0.19 [9], mobility otherwise oscillatory Fermi energy.
µ = 0.19 m2V−1s−1 [8], effective g-factor g∗ = 5 [2], val- ThealternativemodeloftheQHE[1]followsadifferent
ley splitting in silicon E = 1.3 meV [10], temperature approach: Instead of adding electrons from a reservoir,
valley
T = 1.0 K [8], C/e = 8.6×1015 m−2V−1 [8], Voff = 2.3 V I propose to treat the QHE system as a system which is
[8], jy = 0.1 Am−1 [assumed]. The location of the plateaus part of an electric circuit and is therefore working at a
(enumerated by p) follows quantized slopes. In the transi-
fixedvoltage(orFermienergy)inthree dimensions. The
tion region between two p’s, the theory shows less structure
two-dimensionalsubsystemhasafixedvoltagedifference
compared to theexperimental result.
to the 3D system and therefore has to adjust its number
of carriers in order to fulfill the energy conditions of the
complete system (see also the discussion in [12]).
In the absence of external fields, the DOS of a free, non-
In this picture, the QHE can be seen as caused by
interactingtwo-dimensionalelectrongas(2DEG)isinde-
coupling a system with a fixed number of channels to a
pendent of the energy of the state
larger system. The direct coupling of the Fermi-energy
of the complete system and the subsystem to external
m 0 E <0
n(2D)(E)=Θ(E) , Θ(E)= (9) voltages provides a good way to test this picture.
free 2π¯h2 (cid:26)1 E >0
If the Fermi energy is directly determined by a gate
voltage V (minus a constant offset voltage V ) via
whereasforcrossedelectricandmagneticfields,theDOS g o
becomesasumofshiftedoscillatordensities[11],eq.(20): E =α(V −V ), (13)
F g o
∞
itispossibletoobtaintheintersectionpointsofthe(clas-
nE×B(E) = nk,E×B(E), (10) sical)HallresistivitywiththequantizedHallgraph. The
kX=0 intersection points are obtained by equating both resis-
nk,E×B(E) = 2[kH+k1(kE!πk3//Γ2)l2]2Γe−Ek2/Γ2, (11) tivities for the same Fermi energy EF
B B
Rcl = =! Rqm = ,
where Hk(x) denotes the Hermite polynomial. The level xy eNav xy e 0EF nE×B,↑↓(E,E,B)dE
width parameter Γ and the energies E are given by
k EF R m∗
(2D)
N = dE2n (E)= E , (14)
Γ = eEy ¯h/(eB) av Z0 free π¯h2 F
eBp
ω = (12) where n(2D)(E) is given by (9) multiplied by two to
ELk = E2m−Γ2/(4h¯ωL)−(2k+1)h¯ωL. bacycoeuqn.t(f1rfe0oe)r wthiteh stphienaddedgiteinoenraocfyaanspdinn-sEp×liBtt(iEng,E(,sBee)
Sec. 5.5.2 in Ref. [13]). Note that the intersection point
The question how the two-dimensional (quantized) is not necessarily exactly in the middle of a plateau (see
subsystem is coupled to the contacts in an experiments Fig. 1).
is important for a model of the QHE. In principle, one At the plateaus Rqm = h , i=1,2,3,... holds and
xy e2i
can think of two possibilities: simultaneously one reaches the intersection point (14)
with the classical Hall line Rqm = Rcl . Therefore the
• Onecanviewthe subsystemascompletelyisolated xy xy
magnetic field values at the intersection points with the
and filled with a fixed number of particles. In this
quantized resistivity are given by
N =constpicture,achangeintheunderlyingDOS
(i.e. by a change in the magnetic field), yields in h B h
= ⇒B = N . (15)
principle a change of the energy in the system. e2i eN i ei av
av
4
6 B=5T,E⊥=4000V/m 6 B=5T,E⊥=12000V/m generation of the Hall field is not included in theories of
5 5 the QHE. Since the steady-state crossed-fields configu-
4 NE×B(E) 4 NE×B(E) ration is reached on a short time-scale and the electric
3 nE×B(E) 3 field remains present, the field has to be included in the
2 2 nE×B(E) propagation of successive electrons. Interestingly, basic
1 1 quantities like the local density of states are changed in
0 0 1 2 3 4 5 6 7 8 9 10 0 0 1 2 3 4 5 6 7 8 9 10 the presence of an electric field [11, 13, 15].
EnergyEinmultiplesofh¯ωL EnergyEinmultiplesofh¯ωL The presence of the electric Hall field does in general
not destroy the gaps between two purely magnetic Lan-
dau levels, but broadens the Landau levels and imprints
FIG. 5: Local density of states (LDOS) nE×B(E) in crossed
electricandmagneticfields,see[11,15]. NE×B(E)denotesthe a different substructure on each level. These properties
carrierdensityobtainedfromNE×B(E)= −E∞nE×B(E′)dE′. are reflected in a non-trivial form of the local density of
NotethesubstructurewithinLandaulevelRsandthebroaden- states (see Fig. 5) and show the divergence of the quan-
ing dependenton theelectric field value. tum Hall effect from a classical electron drift picture:
• For emission from a localized contact, the drift
An example for such an intersection point is B = 10 T depends not only on the field ratio, but also on
2
in Fig. 1. Now it is possible to derive how the magnetic the kinetic energy of the electrons: for certain en-
fieldvalueoftheintersectionpointschangesasafunction ergyranges,localizedcurrentsareformedwithzero
of the Fermi energy and therefore of the average parti- macroscopic flux and the electron propagation is
cle number. Using Eq. (15), I obtain for the slopes in blocked. This is in stark contrast to the classical
GaAs/AlGaAs heterostructures case, where every electron can participate in the
driftmotion,independentofits initial(kinetic)en-
∂B h ergy.
i
= , (16)
∂N ei
av • Landau-levelsarebroadenedbythe electricfieldin
or expressing α in terms of the capacitance C for a Si- a non-trivial way. Each Landau level acquires a
MOSFET [α=C/(n(2D)e)] differentsubstructureandwidth,dependentonthe
free level number and the electric and magnetic field
e ∂B h values (see Fig. 5).
i
= . (17)
C ∂Vg ei • Thebroadeningfollowsapowerlaw,whichleadsto
a critical Hall field for the breakdown
These values reflect exactly the experimentally reported
quantizedslopes ([8],Eq.(1),and[14], p.329). Disorder E ∝B3/2. (18)
crit
was deliberately discarded, although it may be impor-
tant for the observed fine-structure in the experiments.
• HigherLandaulevelsbegintooverlapandtherefore
A comparison of the theoretical prediction with experi-
cannot sustain a quantized transport. Note that
mental results is shown in Fig. 4. The excellent agree-
there is a natural broadening occuring due to the
ment supports the underlying model of a Fermi energy
presence of the Hermite polynomials in eq. (10).
whichisdirectlyproportionaltotheappliedgatevoltage,
whiletheactualnumberofparticlesmayfluctuateabout
Experiments by Kawaji et al. [16, 17, 18], who studied
an average value.
the QHE and its breakdownas a function of the electric
Recentexperimentstracetheevolutionoftheplateaus
Hall field, are in precise agreement with the theoretical
as a simultaneous function of the magnetic field B and
predictions. In fact, the same power law as the theoret-
an applied gate-voltage V . Experiments have been per-
g ically calculated one (see eq. 18) is empirically deduced
formed using GaAs heterostructures [14] (see also [1]) as
from the experimental data in [16]. Also, Kawaji ob-
wellasSiliconMOSFETdevices[8](seeFig.4). Bothex-
tains different critical fields for different Landau levels,
periments confirmthe linearlaw for the plateaulocation
whichisexplainedbytheLandau-leveldependentbroad-
in the V –B-plane.
g ening in the theory [11]. The experimental findings can
be explained within the heuristic theory of the Hall con-
ductivity [1], which goes beyond linear response theories
III. THE ROLE OF THE ELECTRIC FIELD IN
andtheirassumptionofalinearrelationbetweenthecon-
THE QUANTUM HALL EFFECT.
ductivity and the current. Instead a non-linear relation
In principle, a microscopic theory of the QHE could
workwithoutthepresenceoftheelectricfieldinthebasic j=σ(B,E)·E (19)
Hamiltonian, since the electric Hall field is quickly build
upastheresponseofthesystemtoanexternallyapplied isderived. Acomparisonofthe theoryandexperimental
voltage. However,tomyknowledge,thistime-dependent data is shown in Fig. 6.
5
4 7µA 28µA 57µA 7µA28µA57µA76µA particle number (see Fig. 11 in [13]) are largest at the
76µA 4 lowest Landau level, leading to an additional enhance-
ment of interactions at strong magnetic fields.
Ω]3 3
[
Rxx2 Ω) 2
esistance1 R(xx 1 V. CONCLUSIONS.
R
0
0 The heuristic theory reviewed in this article has fea-
4.4 4.5 4.6 4.7 4.8 4.9
4.4 4.5 4.6 4.7 4.8 4.9 tures notcontainedin conventionaltheoriesofthe QHE:
MagneticFieldB[T] B(T)
• It incorporates the electric field in the underlying
FIG. 6: Breakdown of the QHE. Diagonal resistance Rxx ∝ density of states and yields the classicalHall effect
σxx as a function of the magnetic field in the i = 4 plateau inthelimitofstrongcurrents. Itexplainsquantita-
for different currents and therefore electric Hall fields: jx = tivelythebreakdownofthequantizedHallconduc-
σxy(B,E)Ey. Left panel: A schematic representation of the tivity. Other theories do not consider the electric
experimental results obtained by Kawaji, published in [17], Hall field, and are thus unable to explain the (ex-
Fig. 2. Right panel: theoretical prediction using the non-
perimentally observed) dependence of the plateau
linear expression fortheconductivityσxx(EF,Ey,B,T,τ)de- width on the electric Hall field.
rived in [1], with the following parameters (references for
the values in brackets): Effective mass m∗ = 0.1, scat-
tering time τ = 1 × 10−13 s, effective g-factor g∗ = 12 • The many-body aspect is taken into account by
[2], temperature T = 1.2 K, average number of particles constructing a band model of the QHE, which is
Nav = 4.5×1015 m−2 [17] (corresponding to a fixed Fermi filled according to the density of states (DOS) in
energy of EF = 10.7 meV.) Due to the lack of more experi- the presence of the external magnetic field and the
mental data (i.e. over a wider magnetic field range), the pa- electric Hall field. The DOS features gaps in the
rameters should be viewed as empirically derived. However,
plateau regions.
independent of the exact values, the observed power law for
thecriticalHallfield(18)isalwaysreproducedbythetheory.
• The current is calculated in a purely quantum-
mechanicalway,without using perturbativelinear-
IV. NON-INTEGER PLATEAUS. response theory. The theory shows a sharp con-
trastbetweentheclassicalpropagationofelectrons
incrossedelectricandmagneticfieldsemittedfrom
ThesubdivisionofthedensityofstatesforhigherLan-
a localized contact and their quantum-mechanical
dau levels is a suprising result. It is caused by the pres-
motion [11, 13].
ence of the electric Hall field. Normally, interactions are
invoked to explain a splitting of Landau levels into sub-
levels. In the present case, no interactions (or disorder) • IncontrasttoothertheoriesoftheQHE,thismodel
are needed to get a broadening and simultaneously a allows for fluctuations of the number of carriers
splitting of Landau levels. Of special interest are half- about an averagevalue. The coupling between the
filled Landau levels, which are stable against variations Fermi energy of the two-dimensional electron gas
of the electric field value [11]. In a simple spin-splitting and the device is provided by a gate voltage (see
picture, the appearence of spin doubles the appearence Fig. 3). The number of carriers is calculated as
of each Landau level due to an energy shift of a function of the gate voltage (and therefore the
Fermi energy). Note, that N(E ) will provide the
F
1
∆E =± g∗m¯hω . (20) plateaus, while the average drift velocity is con-
L
2 stant. As a result, N(E ) oscillates as a function
F
of the magnetic field for fixed E . The gaps in the
If I assume spin-splitted Landau levels (see Fig. 2), half- F
integer plateaus are expected at 2/5h and 2/7h, but DOS in perpendicular electric and magnetic fields
e2 e2 cause the observed conductivity quantization.
not at 2/9h and 2/11h (where instead a peak oc-
e2 e2
curs at these values). Interestingly, strong anisotropies
havebeenobservedexperimentallyatthesefractions[19], • Surprisingly,crossedelectric and magnetic fieldin-
which warrant a further examination of these fractions, duce a substructure in a Landau-level which leads
i.e. as a function of electric Hall fields. to plateaulike structures at several fractional and
Notice that the present theory does not predict a sub- nearly fractional values of the conductivity quan-
division of the lowest Landau level (which is not in line tum[11]. Althoughtheirvaluesmatchtheobserved
withexperiments). A possibleexplanationis thatmany- FQHEfractionsonlypartially,itisneverthelessre-
body effects become predominant for low Landau levels markable that a non-interacting particle theory al-
at high magnetic fields [19]. Also the fluctuations of the ready generates a fractional pattern.
6
Acknowledgments sky) and the Instituto de Ciencias Nucleares, U.N.A.M.,
(A. Frank) are gratefully acknowledged. I appreci-
I would like to thank the organizers T. Belyaeva, ate helpful discussions with P. Kramer, M. Kleber,
R. Bijker, and E. Mart´inez Quiroz for the opportunity C. Bracher, and A. Frank. This work is supported by
to present this work at XXIX Symposium on Nuclear the Deutsche Forschungsgemeinschaft (grant KR 2889
Physics in Cocoyoc, M´exico. The invitation and hospi- [EmmyNoetherProgramme])andNSEC[E.Heller,Har-
tality of the Instituto de F´isica, U.N.A.M., (M. Moshin- vard].
[1] T. Kramer. A heuristic quantum theory of the Quantum Semiclass. Opt., 6:21–27, 2004.
integer quantum hall effect. International Journal [12] A.Y. Lozovoi, A. Alavi, J. Kohanoff, and R.M. Lynden-
of modern physics: B, 2005. In press, preprint: Bell. Ab initio simulation of charged slabs at constant
http://arxiv.org/abs/cond-mat/0509451. chemicalpotential.J.Chem.Phys.,115:1661–1669,2001.
[2] The effectiveg-factor is empirically deduced.I assume a [13] T. Kramer and C. Bracher. Propagation in
constantg∗,whichexplainstheobservedstructuresvery crossed magnetic and electric fields: The quantum
well. Thus a dependency of g∗ on the magnetic field is source approach. In B. Gruber, G. Marmo, and
not needed here. N. Yoshinaga, editors, Symmetries in Science XI,
[3] C.W.J. Beenakker and H. van Houten. Quantum pages 317–353, Dordrecht, 2004. Kluwer. Online:
transport in semiconductor nanostructures. Solid state http://arxiv.org/abs/cond-mat/0309424.
physics, 44:1–228, 1991. [14] S.Ilani,J.Martin,E.Teitelbaum,J.H.Smet,D.Mahalu,
[4] E.H. Hall. On a new action of the magnet on electric V. Umansky,and A. Yacoby. The microscopic nature of
currents. American Journal of Mathematics, 2:287–292, localization inthequantumHalleffect. Nature,427:328–
1879. 332, 2004.
[5] K. v. Klitzing, G. Dorda, and M. Pepper. New [15] T. Kramer. Matter waves from localized sources
method for high-accuracy determination of the fine- in homogeneous force fields. PhD thesis, Tech-
structure constant based on quantized Hall resistance. nische Universit¨at Mu¨nchen, 2003. Online:
Phys. Rev. Lett., 45:494–497, 1980. http://tumb1.biblio.tu-muenchen.de/publ/diss/ph/2003/kramer.ht
[6] J.Hajdu,editor.IntroductiontotheTheoryoftheInteger [16] S. Kawaji, K. Hirakawa, and M. Nagata. Device-width
Quantum Hall Effect. VCH, Weinheim,1994. dependence of plateau width in quantum Hall states.
[7] B.Kramer,S.Kettermann,andT.Ohtsuki.Localization Physica B, 184:17–20, 1993.
inthequantumHallregime.PhysicaE,20:172–187,2003. [17] S. Kawaji. Breakdown of the integer quantum Hall ef-
[8] D.H. Cobden, C.H.W. Barnes, and C.J.B. Ford. Fluc- fect at high currents in GaAs/AlGaAs heterostructures.
tuations and evidence for charging in the quantum Hall Semicond. Sci. Technol., 11:1546–1551, 1996.
effect. Phys. Rev. Lett., 82:4695–4698, 1999. [18] T. Shimada, T. Okamoto, and S. Kawaji. Hall electric
[9] V. Mitin, V. Kochelap, and M. Stroscio. Quantum field-dependent broadening of extended state bands in
Heterostructures: Microelectronics and Optoelectronics. LandaulevelsandbreakdownofthequantumHalleffect.
Cambridge University Press, Cambridge, 1999. Physica B, 249-251:107–110, 1998.
[10] Y. Hirayama, K. Muraki, K. Hashimoto, K. Takashina, [19] M.P. Lilly, K.B. Cooper, J.P. Eisenstein, L.N. Pfeif-
andT.Saku. QuantumHalleffectatLandaulevelcross- fer, and K.W. West. Evidence of an anisotropic
ing. Physica E, 20:133–142, 2003. state of two-dimensional electrons in high landau levels.
[11] T. Kramer, C. Bracher, and M. Kleber. Electron propa- Phys. Rev. Lett., 82:394–397, 1999.
gation incrossed magneticandelectricfields. J. Opt. B:
