quantum hall effect macroscopic and mesoscopic electron transport

8/2/2019 Quantum Hall Effect Macroscopic and Mesoscopic Electron Transport

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Quantum Hall Effect:

Macroscopic and Mesoscopic Electron Transport

D.C. Glattli

CEA Saclay, Gif-sur-Yvette, France

Abstract. We give here an introduction to the Integer and the Fractional QuantumHall Effect. Some of the fascinating transport properties of macroscopic samples inthis regime are reviewed. Then, the mesoscopic electronic transport at the edge ofa quantum Hall sample is described. In particular, the Luttinger liquid physics of

fractional edges is discussed.

1 Introduction

The quantum Hall effect is perhaps one of the most beautiful macroscopic man-ifestation of quantum mechanics in condensed matter after superconductivityand superfluidity. The phenomenon is observed in a two-dimensional electrongas (2DEG) at low temperature in a high perpendicular magnetic field. The

physics of the quantum Hall effect is that of a flat macroscopic atom made ofup to 1011 electrons. Here, the magic quantum numbers are replaced by magicvalues p/q of filling factor = ns/n which measures the electron density ns inunits of the flux quantum density n = B/0 where 0 = h/e. Sweeping thisparameter, by varying the magnetic field or the density, reveals a series of newquantum fluids signaled by plateaus in the Hall resistance. The plateaus take theremarkable values qp

he2 each time crosses remarkable values of p/q: an integer

[1] (q = 1) or a fraction with odd denominator [2,3] (q = 2s + 1). For integervalues the physical origin of the phenomenon is the Fermi statistics and the

cyclotron motion quantization in 2D, while for fractional values an additionalingredient is needed: the Coulomb interaction. These ingredients are the simplestone can imagine. No interaction with the host material is needed as in the case ofsuperconductivity. However, a little amount of disorder is required to reveal theHall plateaus: it is a rare and unusual case where imperfections help to reveal afundamental quantum effect! It is remarkable that such a simple system has com-pletely renewed our knowledge of quantum excitations. Topological fractionallycharged excitations [3] with fractional anyonic or exclusonic quantum statistics[4], composite fermions [5] or composite bosons [25], skyrmions [7,8], etc., ... are

the natural elementary excitations required to understand the quantum Hall ef-fect. The quantum Hall effects have made real some delicate concepts inventedfor the purpose of particle physics theories or used in mathematical physics forquantum integrable systems [9]. I should emphasize that none of these conceptsare complicated or difficult to understand. They all suggest how rich and fasci-nating is the physics of the quantum Hall effect.

C. Berthier, L.P. Levy, and G. Martinez (Eds.): LNP 595, pp. 146, 2002.c Springer-Verlag Berlin Heidelberg 2002


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2 D.C. Glattli

As the quantum Hall effect is a fundamental property of electrons confinedin 2D, and not a property of specific materials, any realization of a clean 2Dmetal should reveal this effect, provided the magnetic field is strong enough toreach a quantum flux density commensurable with the electron density. The

highest static magnetic fields presently available require low electrons densities,typically lower than approximately 1016m2 (a field of 20 Tesla corresponds ton 5.1015m2). Also the experimental discovery of quantum Hall effect wouldnot have been possible without the emergence of high-quality low density electronsystems. These samples are a direct product of the semiconductor technology.From the discovery of the integer quantum Hall effect (IQHE) more than twentyyears ago [1], followed later by the discovery of the fractional quantum Hall effect(FQHE) [2], to the recent observation of new phases at large filling factors, asthose discussed in Foglers chapter in this book, one can say that each three to

five years emerges a new concept associated with the discovery of a new kind ofcollective quantum state. This regular rate of surprising discoveries is driven bythe constant improvement of sample quality. This improvement is illustrated inFig. 1 where the Hall plateaus observed in two state-of-the-art samples, separatedby approximately fifteen years, are displayed.

It is a hard task to give an overview of twenty years of such beautiful physics.There have been already many well documented reviews on this subject writtenby the most prominent specialists in the field [2,13,8]. As the readership of thisbook comes from various areas, I will not include the latest developments whichwould be only appreciated by specialists in the field. Instead, I will try to providethe reader with the knowledge of the basic physics and of the experimental

Fig. 1. Hall and longitudinal resistance versus magnetic field measured in typical highmobililty samples. Left: state of the art GaAs/GaAlAs heterojunctions in 1986 (fromRef. [11]); right: todays state of the art sample (from Ref. [2])


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Quantum Hall Effect: Macroscopic and Mesoscopic Electron Transport 3

manifestations of the quantum Hall effect. I will restrict myself to the transportproperties.

After a presentation of the peculiarities of transport in two-dimensions anda presentation of the quantum Hall effect, I will describe the realization of ex-

perimental 2D electron systems. Then a chapter will be devoted to the physicsof transport in macroscopic samples. The last chapter will address the physicsof edge states which carry the current in mesoscopic quantum Hall samples. Iwill discuss the electronic transport in the light of the Luttinger liquid physics.

1.1 The Hall Effect in 2D

In this section we will introduce some basic definitions using the simple Drudemodel for the conductivity of a 2D electron gas treated classically. In the presence

of a perpendicular magnetic field Bz the Lorenz force ev Bz adds to thelongitudinal electric force e

E. The 2D current density

j = ensv is

j =

jl +

jt = l

E + tz E (1)

The longitudinal conductivity l (often denoted as xx) is associated with dissi-

pation (jl .

E = 0), while the transverse or Hall conductivity t (often denoted

as xy) corresponds to dissipationless transport (jt .

E = 0). Alternatively, the

local longitudinal electric field can be related to the transverse and longitudinal

current density E = ljl tz jt (2)There are useful relations between the s and s:

l =0

1+2c2 =

l2l

+2t(3)

t =ensB lc =

t2l

+2t(4)

In the first relation 0 = nse2(B)/m is the Drude conductivity, c = eB/m

is the cyclotron frequency and is the B dependent collision time. The quantityt is the Hall resistance RH

t =B

ens RH (5)

It has the remarkable property of being independent of material parametersexcept the electron density. This is not the case for t, but Eq. (4) shows thatt H = 1/RH in the limit of vanishing longitudinal conductance. This limitis reached in the conditions of the Quantum Hall Effect as we will see below.

1.2 What Is Special in 2D?

In two dimensions, the conductivity (resistivity) has the dimension of a con-ductance (resistance). In zero magnetic field, the classical Drude conductivityo = nse

2 /m can be rewritten as o = (e2/h)kFl, where kF is the Fermi


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4 D.C. Glattli

wave vector and l is the mean free path. From this, we see that the classicalconductivity can be expressed in terms of quantum units: the universal conduc-tance quantum e2/h times a dimensionless number which carries informationon the microscopic details (here the ratio of the mean free path over the Fermi

wavelength).Similar quantum units can be used for the classical Hall conductance in

2D. Indeed, instead of expressing the magnetic field B using conventional units(Tesla, Gauss, ...), we can use quantum units for the magnetic field. B can beuniquely defined by the number n of flux quanta 0 = h/e per unit area:B = (h/e)n. The classical Hall conductance becomes:

H =1

RH=

e2

h

nsn

(6)

In these units, H is just the ratio of the number of electrons over the numberof flux quanta in the plane.

Another interesting property in two dimensions is the small sensitivity of theHall resistance to disorder. Fig. 2 displays schematically a strip of a 2D conductorconnected to contacts without (a) and with (b) disorder. We will treat electronsclassically. In the limit of l 0, the Hall angle defined as tan(H) = t/l is90 and t = H. The current lines are parallel to the equipotential lines. Ina Hall measurement, the Hall voltage VH is defined as the potential difference

between the upper and lower edge. If I is the current flowing through the sam-ple, the Hall resistance is given by: I = (1/RH)VH = tVH. More generally, forany point A and B belonging respectively to the lower and upper boundary, onehave I =

AB(j dl).z = H

AB E.dl. For a uniform sample, the path used

to calculate the potential difference is not important. In particular the voltagedrop V across the contact is always V = VH because of the lack of dissipation,assuming no contact resistance. Now lets introduce some defects inside the sam-ple. The defects can be holes in the sample, i.e. insulating regions. They willstrongly modify the current lines. However, provided it is possible to find a path

A B connecting the two contacts following an unperturbed region of constant

Fig. 2. Schematic representation of current line without (a) and with (b) disorder in

a classical dissipationless 2D Hall bar. The path A B is arbitrary. In (b) holes,ingrey, have been randomly introduced in the sample. Provided one can find a path withconstant Hall resistance and 90 Hall angle to go from the lower corner of the rightcontact to the upper corner of the left contact, the voltage drop is not affected by thepresence of holes. Note that the current lines correspond here to the net current andnot to microscopic electron flow


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t = H ,with 90Hall angle, the voltage drop remains equal to the Hall voltage.

The same result would be obtained by filling the holes with a conductor havinginfinite longitudinal conductance and negligible Hall conductance. This lack ofsensitivity to disorder is a property of electromagnetism in 2D. It is topological

and survives when quantum mechanics is considered. This is a basic ingredientwhich makes the Hall resistance of the quantum Hall effect insensitive to disor-der on macroscopic scale. Of course, to have a perfectly vanishing longitudinalconductance and to built regions of perfectly constant (quantized) density, i.e.constant H, requires quantum mechanical effects and are unlikely to be observedin classical systems.

1.3 The Surprising Quantizations of the Hall Resistance

Although, there have been already calculations showing that the Shubnikov-deHaas oscillations of the longitudinal resistance and the Hall resistance in 2Dwas highly non trivial [14], the plateaus of the Hall resistance discovered by K.v. Klitzing, G. Dorda and M. Pepper [1] was not anticipated, nor the accurate(metrological) quantization. The series of striking values of the plateaus whichcharacterizes the Integer quantum Hall effect:

RH = (h/e2)1/p (7)

where p is an integer, tells us that, for some range of magnetic field, the electronsparticipating to the Hall resistance have a density pinned to the flux quantumdensity:

ns = pn (8)

as suggested by Eq. (6). The pinning of the density expresses the incompressibil-ity of the 2DEG in the quantum Hall regime. The origin of the incompressibilityis the opening of a gap in the excitations. The existence of a gap directly man-ifests by a vanishing longitudinal resistance on a Hall plateau. Fig. 3 shows

typical transport measurements in this regime.The phenomenon called Integer Quantum Hall Effect can be understood as

resulting from the combination of the Landau level formation and of the Fermi-Dirac statistics. We will see that the fact that it is observable on macroscopicscale is due to a zero temperature quantum phase transition: Localization. Noelectron-electron interactions are required at this point, although in real samples(specially very clean) interactions are strongly present.

Interactions have been necessary to explain a second surprising phenomenonobserved first at higher magnetic fields: the occurrence of new Hall plateaus for

fractional values [2], p being replaced by p/q, with q odd integer, see Fig. 1. Thisnew effect, called the fractional quantum Hall effect comes from the combinationof electron-electron interactions and of quantum statistics within a Landau level.New gaps open at fractional values p/q of the ratio ns/n. Although the physicalorigin is different, the experimental manifestations are very similar to thoseobserved in the IQHE regime, as far as transport properties are concerned.


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to modulate the doping during the growth. The good atomic interface is due tothe very weak lattice mismatch of GaAs and Ga(Al)As.

In both systems, the conduction band is bent toward the potential wall inorder to realize a triangular potential confining the charges close to the inter-

face. Ionization of donors, randomly distributed in a region preferentially wellaway from the interface, is the source for the conduction band electrons whichwill be trapped at the interface to realize the 2DEG. In MOSFET, the electricfield which bends the conduction band is provided by a positively polarized gate.The gate is deposited on the few nm thick SiO2 insulator while the bulk semi-conductor is p-doped (2D hole gas can also be formed with negative bias andn-doped material). In GaAs/Ga(Al)As heterojunctions, electrons are trappedin the GaAs side, the Ga(Al)As region with larger gap provides the potentialwall. The electric field is usually not provided by a gate on the sample surface.

Instead, it is provided by donors situated in the Ga(Al)As region. Such config-uration is possible because the MBE technique allows to modulate the dopingat will. Some of the electrons provided by the donors reach the potential well toform the mobile 2DEG, while most electrons are usually trapped at the surfaceof the sample by surface states and do not participate to the conduction. Fig. 4shows schematically the two systems and the band structure.

The Integer QHE is well observed in Si-MOSFET samples, while the Frac-tional QHE is better observed in GaAs/Ga(Al)As heterojunctions. This is dueto fundamental differences between the two materials. The effective mass to be

considered in Si is 0.19 while in GaAs it is 0.068. For a similar concentrationthis make GaAs electrons more quantum. The typical length scale and size ofthe random potential experienced by the 2D electrons is also different. In a Si-

Fig. 4. Two possible realizations of a 2DEG: silicon MOSFET and GaAs/GaAlAsheterojonction


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8 D.C. Glattli

MOSFET, the SiO2/Si interface roughness provides a wide spectrum of spatialfluctuations with a large intensity. In GaAs/Ga(Al)As the interface is atomi-cally flat and disorder is long range and of small amplitude. Indeed, an undopedspacer layer of thickness d separates the randomly ionized donors from the 2D

electrons. This filters the spatial frequencies of the random potential which arelarger than d1. Also this provides spatial averaging of the potential fluctuationsover a disk of diameter d, thus reducing their amplitude. As a result, zero fieldelectron mobility hardly approaches 500000 cm2V1s1 in silicon while mobilityof 107 can be obtained in GaAs for similar density.

The Hall plateaus of the integer QHE are wider in case of larger disorder(provided the disorder strength does not overcome the Landau level separationenergy). This is why IQHE is well observed in silicon. The fractional QHE re-quires small integer plateaus (in order to reveal the fractions in between). This

implies small disorder with potential fluctuations much smaller than the FQHEgap (which is one order smaller than the Landau level separation as we willsee below). Clean GaAs/Ga(Al)As heterojunctions are thus favored (but someFQHE fractions can be observed in specially grown silicon samples).

The zero field mobility at 4.2 K is a good qualitative measure of disorder. Thehighest mobilities have been obtained in GaAs. The improvement came in twosteps: the modulation doping and, at the end of the eighties, the planar doping[11]. Further continuous improvements are now mostly due to the increase of thepurity during MBE (better vacuum). Indeed, for ultra high mobility samples,residual acceptor impurities remain the dominant scattering mechanism in theGaAs side of the heterojunction, where electrons are confined.

I would not end this paragraph without talking about two new systems.The first one is MBE grown Si/SiGe heterojunctions. Interest for this mate-rial appeared recently and the growth technique improves regularly. The samplequality can reach the level of good quality GaAs/AlGaAs heterojunctions. Witha different effective mass, a different valley degeneracy and a different dielec-tric constant Si/SiGe heterojunctions provide a set of parameters different from

GaAs/Ga(Al)As heterojunctions to study the QHE [17]. The second system ismore recent. The 2D electrons gas is realized in epitaxially grown molecularcrystals such as Tetracene or Pentacene [18]. To confine the electrons a schemesimilar to Si-MOSFET samples is used. A metallic gate, separated from the sur-face of the molecular crystal by an oxide layer, is used to create the 2D electrons(or the 2D holes because the molecular crystals are ambipolar). Holes seem togive the best mobility, typically 105cm2V1s1. According to the large effectivemass m = 1.5me, this corresponds to scattering times 1010s comparable tothe best GaAs/Ga(Al)As heterojunctions. The Fractional QHE has been indeed

observed in these new molecular systems.How far do we know the physical parameters of a 2DEG in a real sample?

Indeed the basic ingredients behind the physics of the QHE are well known: theCoulomb interaction, the Fermi statistics and the quantization of the kineticenergy in magnetic field. Also the effective mass, the dielectric constant and theg factor are well known for each materials. A direct comparison between experi-


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Fig. 5. Triangular well potential confining the electrons at the semiconductor interface

ments and theory should be possible, free of adjustable parameters. In fact this isnot completely true. A first adjustable parameter is the strength of the disordercoming from ionized donors or interface roughness. However, it can be reducedto a very small amount in the purest structures. A second parameter comes fromthe width of the wavefunction in the

z direction which weakens the Coulomb

interaction. Because of this, electrons are not point like but become rods. As-

suming a triangular potential, as shown in Fig. 5 one can estimate the spreadingz of the wavefunction. A crude estimation is obtained by minimizing the sumof the z kinetic energy and of the potential energy 2/2m z2 + eEz z. Thisyields z 2/meEz1/3, where Ez is the electric field bending the conduc-tion band. Typically Ez ens/0 which yields z (a2.aB)1/3 = a/(rs)1/3.The length a = (ns)

1/2 measures the typical distance between electrons andrs = a/aB is the correlation parameter, as in 3D metals (quantitative estima-tions of z can be found in [15]). In GaAs heterojunctions, the Bohr radius isa

B 10nm and r

sis a few units (as in ordinary 3D metals). Thus the

z

exten-sion is of the order of a and produces a significant reduction of the correlationenergy. Its knowledge is important for comparison with theoretical estimationsof the FQHE gaps as discussed in Shankars chapter.

3 Quantum Hall Effect: Bulk Macroscopic Transport

The transport properties of the bulk 2DEG has been studied for a long time.They are many good reviews on this subject. In the following sections I will

only present some important basic aspects of the Integer and of the FractionalQuantum Hall regime. Some of them will be illustrated by experimental resultsborrowed from the work of leading groups in the field. I apology for any im-portant work which will not be quoted. Because of the broad spectrum of thepresent readership the aim is mostly to give a feeling for what the Quantum HallEffect is.


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3.1 The Integer Quantum Hall Effect

In presence of a magnetic field B = B

z the kinetic energy K = (p+eA)

2

2m of anelectron moving freely in a plane is equivalent to the problem of an electron

moving in a harmonic potential. The set of eigenvalues for the kinetic energy,called Landau levels, is:

En = (n +1

2)c (9)

where c = eB/m is the cyclotron pulsation. Typically c/2 =400 GHz/Tesla

and c =20 Kelvin/Tesla in GaAs/GaAsAl samples. The Landau levels arealready well separated at liquid Helium temperature in field Tesla. The eigenenergies are function of a single quantum number n. As there are two degrees offreedom (the electron moves in 2D) there is a missing quantum number indicat-

ing a huge degeneracy. Using the cylindrical gauge A = (By/2,Bx/2, 0), themeaning of the degeneracy appears clearly if one represents the conjugate pairsof electron coordinates [x, px] and [y, py] by a new set of conjugate pairs, seeFig.6

(x, y) = (X+ , Y + ) (10)

where

[, ] = [vy/c, vx/c] = i/eB (11)[X, Y] = i/eB (12)

The Hamiltonian becomes H = 12 m2c (2 + 2) and does not depend on (X, Y).The first pair of conjugate coordinates represents the fast cyclotron motion withradius

rn = n| 2 + 2 |n1/2 = (n + 12

)1/2lc (13)

The cyclotron radius increases with the orbital Landau level index n and scales

as lc = (/eB)1/2

, the magnetic length. The second pair is the center of the

cyclotron orbit R = (X, Y). This is precisely the freedom to choose the center

Fig. 6. Decomposition of the electron motion into cyclotron orbit coordinates andcoordinates of the center of the cyclotron orbits (left); (middle) Landau levels; (right)chemical potential versus


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of cyclotron orbits which encode the degeneracy of the Landau levels. The de-generacy is equal to the number of way to put the cyclotron orbit centers in theplane. It is equal to the number N = nS = eBS/h of flux quanta 0 = h/e inthe surface S. This a direct consequence of the non commutation of coordinates

(X, Y). We can also say that the 2D electron plane is analogous to the phasespace [P, Q] of a one-dimensional system. It is known that the area of a quantumstate is h. In our 2D system with quantizing magnetic field the flux quantumh/e stands for the Planck constant h.

Assuming first fully spin polarized electrons, it is easy to understand whythere is a vanishing longitudinal resistivity l and conductivity l =

l2l

+2twhen

the Hall resistance RH is precisely h/pe2, p integer. Because of the Pauli prin-

ciple, when the number of electrons N = nsS is exactly pN = pnS, the

ground state corresponds to the complete filling of the p lowest Landau levels,while the next Landau levels are empty. Thus, there is an energy gap c tocreate an electron-hole pair excitation. The 2DEG becomes a perfect insulatorfor longitudinal currents (l = 0), while the current parallel to equipotentialline is exclusively made of dissipationless transverse currents (l = 0). At finitetemperature, longitudinal transport and dissipation are recovered via thermallyactivated inelastic process with energy c.

The central role played by the filling of the Landau levels has lead to definethe filling factor v = ns/n characterizing the QHE. The IQHE occurs for = p.

Regarding thermodynamics, there is a jump c of the chemical potential forthe smallest change of the filling factor around the integer value p: removing asingle flux quantum by lowering the field or adding just a single electron. Thesystem is incompressible.

In the previous view of a clean ideal 2DEG, the striking QHE features areexpected only for an infinitely small width of the parameter around an in-teger. However experiments show a remarkable and accurately quantized Hallplateau with finite width. In the narrow mesoscopic samples considered later, theexplanation is simple: we will see that a finite plateau can be explained using a

Landauer-Buttiker approach of 1D chiral modes at the boundary of the sample.In macroscopic systems, where boundary effects are negligible, the explanationis different. It requires disorder and the associated zero temperature quantumphase transition called Localization. Note that this is one of the rare cases wheredisorder helps for the observation of a very fundamental phenomenon. This iswhat we are going to discuss now.

To get some physical insight, let us assume for simplicity that the randompotential V(x, y) acting on the electrons is smooth such that no appreciableLandau level mixing occurs. This means no strong variations on the lengthscalelc or |lcV| c. The effect of disorder is therefore to lift the degeneracy ofeach Landau level n separately. If we restrict the Hilbert space to the subspaceof a single Landau level n, the resulting Hamiltonian becomes:

Hn = (n + 1/2)c + V(n)(X, Y) (14)


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where V(n) is the potential V averaged over the fast cyclotron motion (i.e. pro-jected over n). Using [X, Y] = i/eB and it is easy to show that the coordinatesR = (X, Y) of the centre of the cyclotron orbits obey the semiclassical equationof motion:

dR/dt = (1/eB)(z V(n)) (15)Electrons move along the equipotential lines. Their velocity is such that theLorenz force compensates the local electric field. As schematically shown in Fig. 7disorder leads to localized states. Electron running along paths of localized statesdo not participate to the macroscopic longitudinal conductivity and if tunnel-ing between neighboring localized states is negligible l remains zero. Localizedstates are mostly found at energies in the middle between the unperturbed Lan-dau Level energies. When varying the magnetic field the localized states providea way to pin the Fermi level and a quantized Hall plateau associated with zerol occurs on a finite parameter range. This is only when the equipotential linesat the Fermi level percolate through the whole sample that a macroscopic zerofinite conductivity is recovered and the Quantum Hall effect breaks down. Thisoccurs for Fermi level close to the unperturbed Landau level energies. At theseparation between the two regimes, when the chemical potential is equal to theso-called mobility-edge energy, there is an abrupt transition from a conductingto localized insulating regime. Localization is a zero temperature quantum phasetransition and this is the reason why the phenomenon is robust and universal.When the Fermi level is pinned in the localized region, electrons participatingto the transverse current have a density still equal to pn and, as shown in the

Fig. 7. Semiclassical percolation picture of the localized states of the QHE. In theschematic figure, b is below the mobility edge EC of the first Landau level: states arelocalized and l vanishes. a is close to the energy of the Landau level and states aredelocalized


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classical model of the introduction, the Hall resistance is unchanged and equalto RH = (1/p)(h/e

2).

This picture can be extended to larger disorder, with appreciable Landaulevel mixing. One can show that the global result is unchanged. For more details

on the localization approach see Ref. [10].

When spin is included, an even filling factor = 2p corresponds to thesituation described above but with the n p orbital Landau levels doublyoccupied. An odd filling factor = 2p 1 corresponds to n p 1 doublyoccupied Landau level and the last Landau level n = p fully polarized. There isalso a gap, but the origin is now the Zeeman energy gBB. With the g-factor ofthe host material (g = 0.44 for GaAs) the gap is expected much smaller thanthe cyclotron gap. Odd filling factor would be difficult to observe. However wewill see later that interactions via the exchange energy enhance the gap to valuesclose to c and the effective g is much larger than g. This is why Hall plateausat odd filling factor are easily observed. What is important for the occurrenceof the quantum Hall effect is not the origin but the existence of the gap. Stateslocalized by disorder in the gap will pin the Fermi level and prevent long rangetransport in the macroscopic sample.

As pointed out by K.v.Klitzing, G. Dorda, and M. Pepper in their pio-neering paper, the accuracy of the plateaus is metrological. This has lead toa resistance standard based on the fundamental constants h and e. The ex-

actness of the quantization is theoretically supported by the gauge argumentof R. Laughlin [19] and also by the concept of topological invariant developedby Thouless (see theoretical chapters in this book) [20]. Already at the begin-ning of the 80s experimentalists were able to produce and measure sampleswith longitudinal resistivity l as low as 10

10 Ohm. The Hall plateau is foundvery flat with RH(T) RH(0) = s.l, s being an empirical parameter foundaround 0.01-0.05. Experiments show that the resistivity obeys an Arrhenius lawl exp(Eg/2kBT) at moderately low temperature characteristic of a conduc-tion gap Eg. This is followed at lower temperature by a variable range hopping

law l exp((T0/T)1/2

) characteristics of assisted tunneling through localizedstates. In the context of localization Eg is called the mobility gap and measuresthe distance between the critical energies separating conducting from insulatingstates. For even filling factors Eg is thus expected to be smaller than c theseparation between two consecutive Landau levels. In Si-MOSFET Eg has beenfound 85%-95% of the cyclotron gap. In clean GaAs/GaAlGa sample it is found10%-25% higher for = 2 and 4. A value larger than c indicates the effect ofthe interaction not included in a simple description of the Integer QHE. Indeeda correlation energy of order e2/40lc has to be added to the cyclotron gap.

The localization theory has been a key to understand the bulk transportproperties of the QHE. Also, important theoretical and experimental work hasbeen done to test this theory. There have been numerous attempts to determinethe critical exponent of the localization length , the typical size of localizedstates which diverges at the mobility edge. The exponent enters in the temper-ature dependence of the transition width separating two Hall plateaus and in


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14 D.C. Glattli

the temperature dependence of the height and width of the associated resistancepeaks. One expects |E Ec| where Ec is the mobility edge energy. Theexponent, not to be confused with the filling factor, is = 4/3 in a classical per-colation picture [21] and = 7/3 if quantum tunneling between close localized

states is included [22]. The first experiments measured the width B(T) Tand the magnetic field derivative of the conductance peaks dl(T)/dB T[23]. The experiments were not able to give a direct access to the critical ex-ponent because the exponent measured = p/2 involves the exponent pof the temperature dependence of the inelastic length, which is not universal.Later, experiments based on a comparison of with the sample size [24] found = 2.3 compatible with the quantum percolation prediction as shown in Fig. 8.More recently, predictions relating the frequency dependence of the longitudinalconductivity l() = (2/3)0 [25] have been checked experimentally and

provide also a critical exponent 2.3 even for surprisingly short values of [26].Although some refinements have to be made from the theoretical side (inclu-sion of interactions, etc. ...) and from the experimental side, there is a commonagreement to believe that Localization is the key ingredient to observe the QHEat macroscopic scale.

In the previous presentation of the integer QHE, interactions have beencrudely neglected. When samples become cleaner and cleaner this is no longerlegitimate. Recently new features in the bulk resistivity have been observed asan anisotropy related to the possible formation of stripes at larger Landau lev-

els. This is due to the Coulomb interactions which lead to natural instabilitytoward charge density formation. It is beyond the scope of this presentation to

Fig. 8. (left) Transport data in Landau level n = 1 at T=25mK. (a) normalizedlongitudinal resistance and (b) Hall resistance as a function the B Bc around thepercolation critical field Bc; (right) half witdh B(T) obtained for different samplesize. Information of the temperature variation and its saturation with sample size allowsdetermination of the critical exponent (after Ref. [24])


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tell more about this and I refer the reader to the chapter by Fogler in this book.Instead I will describe a regime now much more understood, where interactionsare important and which concerns partially filled Landau levels: the FractionalQuantum Hall Effect.

3.2 The Fractional Quantum Hall Effect

We will first give a brief theoretical picture of the fractional QHE. For simplerpresentation we will assume a spin polarized system and the first orbital Landaulevel partially filled: N < N or < 1. We have seen that a Landau levelis highly degenerate. For partial filling, there is a large freedom to occupy thequantum states, i.e. to fill the plane with electrons. However electrons interactand the Coulomb repulsion will reduce our freedom to distribute electrons in the

plane. Let us first consider the limit of infinite magnetic field when the fillingfactor goes to zero. The Gaussian wavefunction describing the cyclotron motionin the first Landau level shrinks to zero. The N electrons being like point chargesbehave classically (no overlap between quantum states) and minimize their en-ergy to form a crystalline state (analogous to the one observed in dilute 2Dclassical electron systems in zero field ). In the present case, for weaker magneticfield, not too small, the wavefunctions overlap. Electron are not localized to alattice but instead form a correlated quantum liquid. However interactions stillbreak the Landau level degeneracy and, for some magic filling factors, a unique

collective wavefunction minimizes the energy. The magic filling factors are foundto be odd denominator fractions: = 1/3,1/5, 2/3, 2/5, 3/5, 2/7, ... [27]. This isthe Fractional QHE. Historically these remarkable states were not anticipated bythe theory and were accidentally discovered by experimentalists doing transportmeasurements and searching for a crystalline state at higher field.

The Laughlin States: The unique wavefunction corresponds to a ground stateseparated from a continuum of excitations by a gap . For = 1/(2s + 1), sinteger, Laughlin proposed a trial wavefunction for the ground state which was

found very accurate. The wavefunction is built from single particle states in thecylindrical vector potential gauge. Using a representation of electron coordinatesas z = x + iy in unit of magnetic length lc, the single particle states in the firstLandau level are:

m =1

22mm!zm exp( |z|2) (16)

It is instructive to look first at the Slater determinant of electrons at filling factor1 which is a Van der Monde determinant. Its factorization gives the followingwavefunction, up to a normalization constant:

1 =

i


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16 D.C. Glattli

zeros at zi = zj reflects the Pauli principle and their multiplicity 1 the Fermistatistics. At filling factor 1/(2s + 1) the polynomial for each zi should be ofdegree of (2s + 1)(N 1) such that all electrons are also uniformly distributedon the (2s + 1)N states available. A uniform distribution of electrons in the

plane requires a very symmetrical polynomial. Also Laughlin proposed the simplepolynomial form [3]:

1/2s+1 =

i


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the hole. The so called quasi-hole carry a charge e = e/3. The energy cost hcan be approximated by the energy required to create a disc of size 0/B andcharge e/3: (4

2/3)(e/3)2/40lc. Similarly quasi-electron excitations with

charge e/3 are possible and correspond to removing a flux quantum to locally

increase the electronic density with an energy e. Quasi-electron or hole excita-tions with charge e/q can be generalized for other filling factor = p/q with qodd. The excitation gap for quasi-electron quasi-hole pairs has been numericallyestimated and calculations agree with a value 0.092e2/40lc for q = 3and does not depend on p (as far as spin polarized electrons are considered). Aninteresting theoretical issue is the statistics associated with the excitations. Itcan be shown that when moving adiabatically two quasi-holes around each otherand exchanging their positions, the collective wavefunction picks up a Berrysphase factor exp(i(2s + 1)). The excitations are not bosons nor fermions but

obey a so-called anyonic statistics, a concept first introduced in the context ofparticle physics.

The simple picture of the FQHE presented here is valid for spin polarizedelectrons. The chapter of Prof. Shankar in this book gives theoretical detailswhen incomplete polarization plays an important role. Now I will concentrateon some experimental features of the transport properties of the FQHE.

Experimental Determinations of the Gap: Although the underlying physicsis different from the physics of the Integer QHE, the basic experimental mani-

festations are similar. The gap associated with the incompressibility leads to aquantization of the Hall resistance at the magic filling factors and to vanishinglongitudinal resistivity and conductivity.

The accuracy of the Hall plateau quantization can not be tested on a metro-logical level as for the Integer QHE. The reason is the lower gap. Less voltageand so less current can be passed through the sample and the signal to noiseratio is lower. However there is no reason to believe to some deviation from thefundamental resistance quantum.

The magnitude of the gap corresponds to the correlation energy anticipated

by Laughlin. Attempts to measure the gap using the thermally activated depen-dence of the resistivity has been done by various groups. Exact comparison withtheory is not that straightforward however. As for the Integer QHE, disorderleads to a mobility edge. The activation energy is then lower than the funda-mental gap of a pure system. Disorder can also be view as random fluctuationsof the filling factor at large length scale which, on average, reduces the effectivegap. In addition, the extension of the electron wavefunction in the direction per-pendicular to the plane softens the Coulomb interaction. This could be includedin the computation of the gap but the extension of the wavefunction is not known

accurately. Finally calculations have to include the Landau level mixing. Indeed,the gap calculated using the Laughlin wavefunction describe interactions withinthe first Landau level. When the correlation energy e2/(40lc) is comparableor larger than c the projection in the first Landau is no longer valid. Thisis the case in real sample where the correlation parameter rs 1. There is ahuge literature on experimental determination of the FQHE gaps using thermal


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18 D.C. Glattli

Fig. 10. The energy gap deduced from the activation energy of the longitudinal resis-tance is compared with theoretical values including Landau-Level mixing effect. Left:GaAs heterojunction (after Ref. [28]) = 1/3, 2/3, 4/3 and 5/3. Right: 2D hole gasconfined in a Tetracene molecular crystal, = 1/3 and 2/3 (after Ref. [18])

activation energy. Fig. 10 shows a typical attempt to compare data with the-ory on two different samples. The first one is a GaAs/Ga(Al)As heterojunction[28] while the second one is a new system based on a hole inversion layer in amolecular crystal, here Tetracene [18]. In both systems the effect of the finite

thickness has been taken into account. In the first one Landau level mixing hasbeen considered and all corrections give a reasonable agreement. In the molecu-lar system, there is a puzzling good agreement without inclusion of Landau levelmixing. This is surprising regarding the larger effective mass (1.3me) and smallerdielectric constant ( 4) implying rs 1. This shows that our understandingis far from being complete.

The intriguing excitations proposed by Laughlin have not been convincinglyprobed using bulk transport measurement [29]. We will see later that the meso-scopic transport properties of edge states offers a mean for unambiguous determi-

nation of their charge. It is important to remark that the fractional quantizationof the Hall conductance is not a signature of the fractional excitations. It is aproperty of the ground state and a consequence of the fractional filling of thesingle particle states of the correlated electrons. This is not a signature of frac-tionally charge excitations (although fractional filling is probably a necessaryrequirement to get fractional excitations).

Composite Fermions: Following the work of Jain [5], a hierarchy of the frac-tional filling factors can be made using the concept of Composite Fermions as a

guide. This hierarchy followed more pioneering work made by Halperin [30] us-ing a different approach to built higher order fractions from the basic 1/(2s + 1)states. Because this physics is also discussed elsewhere in the book, I will nottalk here about the long theoretical improvements of Jains first proposal whichhave lead to the present theoretical understanding. Instead I will show how theCF concept has been a guide and a source of inspiration for experimentalists and


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how it manifests experimentally. The concept is first based on another conceptwhich is the statistical transmutation of electrons in 2D (or 1D). Topological con-siderations show that purely 2D particles are not necessarily bosons or fermionsbut may have any intermediate statistics (an example is the Laughlin quasi-

particles). For the same reason, it is easy to manipulate the statistics of 3Dparticles such as electrons which are Fermions provided they are forced to livein 2D (or 1D). This can be done by attaching an integer number of fictitiousflux quanta to each electron. The price to pay is a redefinition of the wavefunc-tion and of the Hamiltonian. An even number of flux quanta will transformFermions into Fermions while an odd number Fermions into Bosons. In the firstcase we have Composite Fermions while in the second case Composite Bosons(CB). Both approaches have been used in the FQHE context. Both have theirown merit and a bridge between them is possible. CF are believed to be ap-

propriate for high order fractions and to describe the remarkable non QuantumHall electronic state found at = 1/2. CB make an interesting correspondencebetween the = 1/(2s + 1) states and superfluidity [25].

By attaching 2s flux quanta to each electron with a sign opposite to theexternal magnetic field flux, the resulting CF experiences reduced mean field.A mapping can then be done between FQHE states and IQHE states. As anexample, for s = 1, the mean field attached to the new electrons, the compositeFermions, is equivalent and opposite to the magnetic field B1/2 at = 1/2. Thefield experienced by the CF is thus BCF = B

B1/2. A filling factor = 1/3 for

electrons corresponds to a CF filling factor CF = 1. Similarly = p/(2p + 1)becomes CF = p. This describes a series of fractions observed between 1/2 and1/3. For fields lower than B1/2, = p/(2p 1) also becomes CF = ()p andthis describes fractions from 1 to 1/2. In general attaching 2s flux quanta toelectrons describe the fractions = p/(p.2s 1). The following table shows thecorrespondence for 2s = 2:

1/3 2/5 3/7 ... 1/2 ... 3/5 2/3 1CF 1 2 3 ... ... 3 2 1

The composite fermion picture is supported by experimental observations.Indeed, the symmetric variations of the Shubnikov-de Haas oscillations aroundB1/2 are very similar to that observed around B = 0. I should emphasize thatthis is not a real cancellation of the external field, as the Meissner effect insuperconductivity is, but the phenomenon is a pure orbital effect due to the 2sflux attachment. Convincing experiments have shown that the quasiparticles at 1/2 behave very similarly to the quasiparticles at zero field. Fig 11 comparesthe magnetic focusing experiments realized at zero field and around B1/2 (from

Ref.[32]; see also [5]). The radius of the trajectories of the quasiparticles bentby the magnetic field is R = vF/c = kF/eB around zero field and RCF =kFCF/eBCF, where kF = (2ns)

1/2 , kFCF = (4ns)1/2 assuming spin polarized

CF fermions, and BCF = B B1/2. The results of the experiment displayed hereshow that the ratio of the period in magnetic field of the resistance oscillationsverify the relation BCF =

2B.


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20 D.C. Glattli

Fig. 11. Magnetic focusing for zero magnetic field (bottom trace) and for compositeFermions near = 1/2 (top trace) for 9T. The distance between the injector andcollector point contact is 5.3 m. A peak is obtained each time a multiple integer ofthe cyclotron orbit diameter matches the distance (adapted from Ref.[32])

Surface acoustic waves (SAW) experiments have found also such relation. ASAW couples weakly to a 2DEG via the piezoelectric interaction. The minutechange in the SAW propagation velocity resulting from this interaction is mea-surable and bring information about the electrons. In particular, SAW exper-iments probes the commensurability between the wavevector q = /vs of thesurface wave and the cyclotron radius of the composite Fermions. Here is thefrequency of the SAW and vs the propagation velocity. An unexpected finding

of this experiment was the observation of a very peculiar dependance of thelongitudinal CF conductivity with the wavevector: l(q) q. This is due to thefact that the CF do not form an ordinary 2D Fermi liquid but keep very specialcorrelations resulting from the interactions [32]. Another interesting quantitywhich has been studied experimentally is the effective mass of the CF. If thecorrespondence between and CF is exploited further one can introduce an ef-fective mass mCF(p) such that the FQHE gap for the p/(2p 1) series is definedas p = eBCF/mCF(p). The gap is determined by the activation energy of thelongitudinal resistance. There is an agreement for a value mCF(p ) 0.52at B1/2 and for a 1/p or BCF dependance of mCF 0.52 + 0.074BCF. Also thesame value is found on both side ofB1/2 (for example = 2/5 and 2/3, = 3/5and 3/7,... ) [35].

So far, we have not discussed the effect of spin in the quantum Hall regime. Wehave always considered the case of full polarization by the strong magnetic field.Spin effects lead to a very rich physics. Often neglected spin effects have been


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the subject of a growing interest the last ten years. As these effects are discussedin great details in the chapter of Prof. Shankar and in several contributions inthis book, we will mentioned only a few basic points for the consistency of thisreview.

In GaAs, the gap between spin-split Landau level should be 0.3K/Tesla.However the gap observed at odd filling factors 1, 3, 5,... is much larger andeven comparable to c for =1. The first explanation proposed was the en-hancement due to the exchange interaction. Ando et al. found = gBB +nnn+n

e2/(40lm) [15]. Here (n n)/(n + n) denotes the relative differ-ence of the total population of spin down and spin up electrons. For even fillingfactor this is always zero and the enhancement vanishes. For odd filling factor = 2k + 1, the effect decreases as 1/(2k + 1). Indeed, the enhancement ex-perimentally observed increases with magnetic field. Note that the correlation

energy scale is the same as the one of the FQHE. Because of the spin degree offreedom, the possible excitations are not only charged excitations (electron-holepairs), but also collective spin excitations or spin waves. In the simplest modelfor = 1 the dispersion relation of spin waves is SW(k) = gBB for k 0and SW(k) = gBB +

2 e

2/(40lm) when k 1/lm. The excitations havebeen theoretically reconsidered with the introduction of skyrmions [7][8]. It hasbeen proposed that topological excitations should be more favorable than a sin-gle spin flip. The so-called skyrmion excitation corresponds to a spin texturein which the central flipped spin is accompanied by a gradual spin flip of the

surrounding electrons so decreasing the exchange energy. This excitation carriesa unit charge. Charge and spin transport can thus be coupled.

Transport measurements are not able to provide direct information on thedegree of spin polarization of the 2DEG. Measurements of the transport acti-vation energy have been made to study the competition between the Zeemanenergy and the spontaneous polarization due to interaction[36]. Experiments us-ing pressure to change the g factor have been made and have given indication ofskyrmions [37]. Instead specific measurements are needed such as magneto-optics(see the review by Kukushkin in this book), nuclear specific heat measurements

[38], optically pumped nuclear magnetic resonance [40][41] and nuclear magneticresonance [42]. These measurements have shown that, in general, filling factorswith odd numerators are well polarized while even numerators show less or zerospin polarization depending on Zeeman energy. Regarding skyrmions most ex-periments agree with this picture but there are still many unsettled issues.

4 Quantum Hall Effect: Mesoscopic Transport

In the following we will concentrate on transport properties at the edge of a finitequantum Hall conductor. We will see that transport is mediated by a finitenumber of nearly ideal one-dimensional modes representing electron runningaround the edges. These modes are called edge channels. They are chiral andhave very nice properties which make them a prototype of quantum conductorto which the concept of mesoscopic transport can be applied. In the fractional


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22 D.C. Glattli

Fig. 12. (a) Schematic representation of edge states and of (b) the Landau level bend-ing. (c) reflection of an edge state by a controlled artificial impurity called QuantumPoint Contact. (d) analogy with 1D semiclassical trajectories in the phase space

regime these modes form a set of chiral Luttinger liquids, i.e. their physics issimilar to that of one-dimensional interacting electron systems called Tomonaga-Luttinger liquids.

4.1 Edge Channels in the Integer Quantum Hall Regime

We have seen that the quantum Hall effect is due to the existence of a gap arisingat special filling factors. However, in a finite system, gapless excitations appearat the periphery of the quantum Hall conductor. For better understanding, it iseasier to consider first the case of fully spin polarized non-interacting electrons inthe integer QHE regime where the physics is dominated by Landau level forma-tion and Fermi statistics. We will also neglect coupling between Landau levels,an assumption justified in most experimental situations. Within the nth Landau

level, the dynamics of electrons is described by the projected Hamiltonian:

Hn = (n + 1/2)c + V(n)(X, Y) (19)

where V(n)(X, Y) is the potential which confines electrons at the boundary ofthe 2D conductor. The coordinates (X, Y) describing the center of a cyclotronorbit are conjugated: [X, Y] = i/eB. If for example, the electric field due toconfinement is along the

y direction, electrons drift along the boundary, here

the x direction, with a velocity Vx = (1/eB)V(n)/Y. This defines the so-

called edge sates. When populated by electrons, edge states give rise to a chiralpersistent current concentrated at the edges. Simultaneously the nth Landaulevel is bend by the confining potential, see Fig. 12. In the integer quantumHall regime, for a filling factor = p, there are p Landau levels which are fullyoccupied in the bulk of the sample. Because of bending, the filled Landau levelswill cross the Fermi energy EF and lead to gapless excitations at the edges. This


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gives rise to p lines of gapless excitations parallel to the boundary. They arelocated at the coordinate Y = YF,n, n p, given by:

EF = (n + 1/2)c + V(n)(YF,n) (20)

The line of gapless excitations and the associated persistent currents carriedby edge states form what is called an edge channel. At the opposite boundary, notshown in the figure, there are also p similar gapless excitations. Here electronsdrift with the opposite direction as the confinement electric field is of reversedsign. Thus for a filling factor = p, there are p pairs of opposite edge channelsassociated with the p filled Landau levels in the bulk.

Edge channels are ideal one dimensional (chiral) conductors: the physicalseparation between pairs of opposite edge channels prevents backscattering andelectrons propagate elastically over huge distances (mm at low temperature)

because only one dimensional forward scattering by phonons is permitted. Theseproperties have make them a convenient tool to test the generalized Landauer-Buttiker relations [43] in the context of the mesoscopic quantum transport. Inorder to do that it is necessary to induce intentionally elastic backscatteringin a controllable way. The tool used is a Quantum Point Contact as shownin Fig. 12. A negative potential applied on a metallic gate evaporated on topof the sample depletes electrons to realize a narrow constriction in the 2DEG.This allows a controllable modification of the boundaries of the sample. Theseparation between opposite pairs of edges channels of a given Landau level can

be made so small that the overlap between wavefunctions lead to backscatteringfrom one edge to the other. Indeed, the QPC creates a saddle shape potentialand when the potential at the saddle point is close but below the value VF,n =EF (n + 12 )c, electrons emitted from the upper left edge channel start to bereflected into the lower edge channel with probability R 1 but are still mostlytransmitted with probability T = 1R. When the saddle point potential is aboveVF,n electrons are mostly reflected and rarely transmitted T 1. Fig. 12 showsthe semi classical analogy between the real space coordinates (Y, X) of the 2DHall conductor and the (P, Q) phase space coordinate of a real 1D conductor, as

suggested by the non-commutation relations [Y, X] = i/ |e| B and [P, Q] = i.The physics of tunneling between opposite edge channels is clearly equivalentto that of the tunneling in a 1D system. However the 2D topology allows usto inject or detect electrons at the four corners of the phase space, somethingimpossible with 1D systems.

Four point conductance measurements in combination with gates or QPCshave been used to probe edge states [44]. The experiments use the Buttiker-Landauers multiterminal formula for quantum transport which relates the cur-rent at contact probe to the chemical potential of contact :

I = eh

(M R) eh=

T (21)

M is the number of (edge) channels emitted from contact , chemical potential, and R the sum of their reflection probabilities. T is the sum of transmis-sion probabilities of the channels emitting electrons from contact to contact


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24 D.C. Glattli

. Fig. 13 shows an example of such measurement in the IQHE regime. If wedenotes R, the resistance measured by injecting current between contacts and while measuring the voltage between contacts and , the followinglongitudinal and diagonal resistances are respectively found:

R13,12 = R13,43 =h

e2(

1

G 1

) (22)

R13,13 =h

e21

G(23)

R13,24 =h

e2(

2

1

G) (24)

Here is the filling factor in the lead and G edge channels are reflected(G edge channels transmitted). The two terminal conductance G = G(e

2/h) isequal to the number of channels fully transmitted. One can also says that thiscorresponds to the filling factor at the center of the QPC. In the example shownin Fig. 13 the filling factor is = 8 in the 2D leads. For G = 8 to 2 one cancheck that plateaus in the longitudinal resistance are found at 0, 1/56, 1/24,3/40, 1/8, 5/24, 3/8, in quantum resistance units, as expected.

So far we have considered the integer quantum Hall regime. Note that theabove picture of edge channels generalizes immediately if we include spin split

Landau levels. Now the number of pairs of opposite edge channels in a givensample is given by the number of occupied spin split Landau level, i.e. the fillingfactor. In fact in Fig. 13 the plateaus observed for G = 7, 5, and 3, do correspondto spin split edge channels. In the data, the odd plateaus are clearly less welldefined because the spin gap, even enhanced by exchange interactions, is lessthan the cyclotron gap.

Fig. 13. Longitudinal resistance versus QPC gate voltage showing the reflection of thesix first edge states for = 8, at T=45 mK. The values of G are indicated on theresistance plateaus


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4.2 An Interacting Picture of IQHE Edge States

At short distance the Coulomb repulsion between electrons is responsible forcorrelations which eventually give rise to exchange effects and to the fractional

QHE. But the Coulomb interaction is very special because it is long range. We arenow going to focus on the long range part of this interaction which is responsiblefor screening. In a first approximation it can be treated using electrostatics andcapacitive models in a Thomas-Fermi approach. This approximation is believedqualitatively correct for the smooth edge confining potentials which exist in realsamples.

Screening means that electrons change their density to build up a charge dis-tribution which in turn creates an induced electric field compensating an externalforce. The gap of the quantum Hall effect is however responsible for incompress-

ibility which prevents screening. The competition between both effects leads toa strong modification of the edge channels as schematically shown in Fig. 14. Inthis example going from the bulk to the edge, one starts with an electron densitypinned at ns = 2n ( = 2). The confining potential is unscreened because ofthe energy gap 12c to remove one electron. Here a change of the local densityis impossible. Thus, the Landau levels bend and rise. When the second Landaulevel crosses the Fermi energy, the energy gap vanishes. Now, it costs no energyto depopulate the second Landau level and screening is as effective as in a perfectmetal. The total potential is constant and the Landau level energies are locally

flat. When the second Landau level is empty, ns = n ( = 1) the density isagain pinned until the first Landau level reaches the Fermi energy, and so on.The edges can be viewed as a series of incompressible quantized region of welldefined filling factors, the quantum Hall regions, which alternate with metallic

Fig. 14. Interacting picture of edge states in the Thomas-Fermi picture


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26 D.C. Glattli

strip of compressible regions, the edge channels. From simple electrostatics, thewidth of the incompressible strips is found of the order of (aBohrd)

1/2 where d isthe lengthscale of the confining potential [45].

Regarding quantum transport, the longitudinal and diagonal resistance are

expected to be the same as in the non interacting picture of edge states. Trans-port can not distinguish alone between the two pictures. The rules for calculatingthe current are the following: 1) the compressible strips form equipotential equalto the electrochemical potential of the contact emitting electrons; 2) an in-compressible strip with filling factor i between two compressible strips attachedto contact and carries a excess Hall current (ie

2/h)( ). Applyingthese rules give the same expressions as those given in Eqs. (22) to (24).

This picture can be extended to the fractional quantum Hall regime. Now thegap is that of FQHE and the is can be fractional [46]. Fractional edge states

can be detected using the reflection induced by a QPCs in a similar manner asinteger edge states as shown in Fig. 15 . Here the filling factor in the leads is = 2/3 while the filling factor at the QPC G is varied from 2/3 to 1/3.The longitudinal resistance (h/e2)( 1G 1 ) shows accurate quantization withplateaus at 1 and 3/2. From these values, we can infer the existence of the 2/5and 1/3 fractional edge states respectively.

The picture described here is expected to apply to smooth edge, as it is thecase in ordinary samples. Another approach has been proposed for hard wallconfinement in Ref.[47].

4.3 Luttinger Liquid Properties of Fractional Edge Channels

Wens Approach of Fractional Edge States: Wen [49] has first shown thedeep connection between fractional edge channels and the concept of Tomonaga-Luttinger liquids [48]. We will here repeat the phenomenological hydrodynamicapproach of Wen in the simple case of a Laughlin state in the bulk, filling factor = 1/(2s + 1). We will start with a classical approach and keep only incom-pressibility as a quantum ingredient.

Fig. 15. Fractional edge channel reflection observed for = 2/3. The longitudinalresistance quantization indicates G = 2/5 and 1/3 fractional channels


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The only possible excitations are periphery deformations of the 2D quantumHall conductor which preserves the total area (like a 2D droplet of an ordinaryliquid) as shown schematically in Fig. 16 . If we denote y+(X, t) and y(X, t) thedeformation of the upper and lower boundaries which are respectively located at

the positions Y = YF and Y = YF, the time varying electron density is givenby:

n(X , Y , t) = ns(Y YF y+(X, t)) ns(Y + YF + y(X, t)) (25)

where ns = eB/h and (x) is the Heaviside function. We wish to find theequations of motion for y. To do that we have to remind that, within the firstLandau level, the single particle motion is given by the reduced Hamiltionian

H1 = (1/2)c + V

(1)(X, Y). X and Y being conjugate, the classical Poissons

bracket is {X, Y} = h/eB. Using the equation of motion for the 2D densityn/t + {H1, n} = 0 and assuming translation invariance along the X axis forV(1), leads to the two decoupled equations describing the chiral propagation ofthe shape deformations y+ and y along each boundary Y = YF and Y = YFrespectively:

y+/t + vDy+/X = 0 (26)

y/t

vDy/X = 0 (27)

where vD =1

eB

V(1)/YY=YF is the drift or local Fermi velocity. For sim-plicity we have assumed a similar confining potential at Y = YF leading toequal absolute value of the drift velocity. Lets now concentrate on the upperedge. The potential energy associated with the deformation is

U+ =1

2

dXnsy

2+

V

Y=vD

dX(nsy+)

2 (28)

Fig. 16. Left: Wens approach for the Chiral Luttinger liquid picture of a fractionaledge channel at = 1(2s+1). Right: a similar picture can be used for the (zero magneticfield) 1D interacting electron system called Luttinger liquids. Here, the semiclassicalphase space (p, q) replaces the 2D real space (Y, X)


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28 D.C. Glattli

If we define the charge variation integrated on the upper edge in units of inthe following way

+ = X

nsy+dXornsy+ = +(X) = 1 +X

the action

S =

dXdt

+X

+

t+ vD

+X

(29)

gives the correct propagation equation for y+ and leads to the HamiltonianH+ = U+. We finally arrive to the total hamiltonian for the two decoupledupper and lower edge channels:

H+ = vD dX+X 2 + X 2 (30)= hvD2

dX

+2 + 2 (31)So far the model is purely classical. We are going now to quantize the col-

lective modes found above. From the Lagrangian, one can define momentum

conjugate of

as

=

/X and quantizes the bosonic field:

(X), (X) = i (X X) (32)The physics of these bosonic modes describing the periphery deformations

would not be interesting unless one have to consider the transfer of a bare electronor a Laughlin quasiparticle form one edge to the other. Indeed, the non trivialphysics arises from the fact that adding an electron on a given edge involves aninfinite number of bosonic modes. This is at the origin of strong non-linearitiesin the transport properties, a property not shared by ordinary Fermi liquids.

How to define the creation operator for one electron on the upper edge?

By definition, it satisfies:+(X),

(X)

= (X X)(X) (33)

On the other hand, the 1D excess density + is related to the conjugate of +by = v + and we have:

+(X),

+(X

)

= i (X X) (34)

which immediately implies:

exp(i+/) (35) creates a unit charge at X but it is not an electron operator unless it

satisfies Fermi statistics. Exchanging two electrons at position X and X gives


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(X)(X) = exp(i sgn(X X))(X)(X). The requirement that thebare particles are Fermions implies

= 1/(2k + 1) (36)

The beauty of Wens approach is that the Laughlin filling factors appear nat-urally as a consequence of incompressibility and Fermi statistics. To obtain otherfractional filling factors as Jains filling factors, a generalization of this approachis possible. One must introduce additional bosonic modes at the periphery (forexample: p modes for p/(2ps + 1) filling factors).

Finally, one can define similarly the quasiparticle operator which creates acharge 1/(2s + 1) on the edge:

+(X), qp(X) = (X X)qp(X) (37)which writes as

qp exp(i+) (38)

It shows fractional statistics: qp(X)qp(X) = e

(isgn(XX))qp(X)qp(X

).This establishes a direct correspondance between quasi-particles on the edge andLaughlin quasiparticles in the bulk.

The above set of equations for the electron operator and for the bosonicmodes are characteristics of those of a Luttinger liquid. Because of the decou-

pling between upper and lower modes it is called a Chiral Luttinger Liquid. Theconductance e2/h correspond to the conductance ge2/h of a Luttinger Liquidand one usually identifies g = . As for Luttinger liquids there is an algebraicdecay of the correlation functions. One can show that the single particle Greensfunction decreases as (X vDt)1/ . The Tunneling Density of State for anelectron at energy is | EF|(1/1). This means that, for electrons tun-neling between a Fermi liquid and a chiral Luttinger fractional edge channel,the finite temperature conductance G(T) and the zero temperature differentialconductance dI/dV show the following power laws:

G(T) (T /TB) (39)dI

dV (V /VB) (40)

= 1 1 (41)

where TB and VB are related to the coupling energy of the tunnel barrier. Thepower laws characterize a Luttinger liquid and have been experimentally ob-served (see below). The chirality leads to some differences with ordinary Lut-

tinger Liquids for which 1/ is to be replaced by (g1

+ g)/2.

Connection with the 1D Luttinger Liquids: Because some (attendants ofthe School) come from the field of organic conductors and the physics of Lut-tinger liquids is also relevant for these systems for some energy scale, it may beuseful to present the connection, and also the differences, between fractional edge


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30 D.C. Glattli

channels and 1D interacting systems. To do that I will give a phenomenologicalderivation of Luttinger liquids, borrowing Wens approach. I will apply this ap-proach to the semi-classical phase space of the 1D Fermi sea (in zero magneticfield). For those not familiar in this field, I must emphasize that rigorous fully

quantum mechanical microscopic derivations exist (see Ref.[50]). Here, the ideais to avoid heavy mathematics and give some physical intuition of what is goingon.

In the following, the (q, p) coordinates will play the role of (X, Y). Accordingto the Pauli principle, the phase space is incompressible with an electron densityn(p, q) pinned to the value 1/h for pF < p < pF in the ground state. Because itis known that for 1D interacting electrons collective excitations are more relevantthan single particle excitations, the dynamics of electrons will be taken intoaccount by considering deformations p(q, t) of the periphery of the Fermi sea,

as shown in Fig. 16, with:

n(p, q, t) =1

h(p pF p+(q, t)) 1

h(p +pF + p(q, t)) (42)

A collective excitation corresponds to redistribution of the population of sin-gle particles near pF. The Hamiltonian describing the dynamics of the singleparticles occupying the Fermi sea is given by:

H(p, q) = 12m

p2 + V(q) (43)

We will proceed like in the previous section and find the equation of motionof the Fermi sea deformation, using the Poissons bracket n/t + {HH, n} = 0.In the limit p(q, t) pF and using n/p = 1h (p pF) the deformationssatisfy the following equations:

t+ vF

q

p+ +

V

qpF = 0 (44)

t vF

q p V

q pF = 0 (45)

We will assume a short range local interaction between electrons such thatthe potential energy experienced by each single particle occupying the Fermisea is proportional to the linear density

dpn(p, q, t) = 2pFh +

1h(p

+F + p

F ) =

0 + (q, t), i.e. V(q) = (q, t). This gives

t+ (vF + 2

h)

q

p+ + 2

hp = 0 (46)

t (vF + 2 h ) q p 2 h p+ = 0 (47)The solution are collectives modes mixing +pF and pF deformations of the

Fermi sea. The eigenvalues for the propagation velocity are:

u = u with u =

vF(vF + 2

h) (48)


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In standard notations of Luttinger liquid models, interactions between elec-trons moving in the same direction are characterized by an interaction pa-rameter g4 while interactions between electrons moving in opposite directionare characterized by the interaction parameter g2 and the general solution is

u = (vF + g42 )2 ( g22 )2. The interaction chosen here is such that g2 =g4 = 2/h, a common situation. We will use g2 instead of in the following.To make better connection with standard notations, we introduce, in the so-called phase representation, the fields 1 (q) = (q)/0 =

1h0

(p+ + p) and1 (q) = 1h0 (p+ p) which gives

t vF

q= 0 (49)

t + (vF + 2

g2

2 )

q = 0 (50)

From the Lagrangian density in appropriate units 2vF

t

2 u2

q

2,

one can define the momentum (q) = vFt =

q conjugate to the field :

[(q), (q)] = i(q q) (51)and one finally arrives to the well known Luttinger liquid Hamiltonian for thebosonic modes:

H = u2

dq

g

q

2+ g1

q

2(52)

where

g =

vF

vF + 2g22

with u.g = vF andu

g= vF + 2

g22

(53)

The parameter g is called the Luttinger parameter. It plays the role of thefilling factor of the fractional quantum Hall effect. One can show that the

conductance without impurity and for an infinite long system is:

GLL = g.e2

h(54)

For repulsive interactions, g < 1.As in the previous section, using the commutation relations given by (51),

it is found that the operator creating a right (left) moving electron is exp(i( )). One can then calculate various useful correlation functions. Forexample, the tunneling density of state is found to vanish at the Fermi energy as

TDOS | EF|(g+1/g2)/2, a hallmark of Luttinger liquids. Finally in orderto make direct comparison with the Chiral Luttinger Liquid model of fractionaledge channels, let us represent the collective bosonic excitations in term of theeigenmodes solution of (46, 47) = g (55)


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32 D.C. Glattli

associated with the linear densities

=

1

2

q

(56)

The Hamiltonian simply becomes

hu

2g

dq+2 + 2 (57)

and it is similar to the one obtained in Eq. (31) for the case of the fractionalquantum Hall effect.

2D Chiral Luttinger Liquid 1D Luttinger Liquid

g = fractional, g: any values, results frombuilt by interaction in the bulk coupling between pF excitations

GkCLL = .e2

h GLL = g.e2

h

TDOS |EF|(1/1) TDOS | EF|

(g+1/g2)/2

right (left) bosonic modes right (left) bosonic modesdo not mix upper and lower excitations mix pFexcitations

Comparison Between Chiral and non Chiral Luttinger Liquid: The

physics of both systems is very similar. Using the analogy between the 2D spaceand the 1D phase space, the difference in the microscopic origin of the bosonicmodes is clearly established. In some experimental situations, when the 2D elec-tron Hall bar is narrow, width w, and when the energy is such that vDh/ > w,the long range Coulomb interaction can couple the upper and lower modes. Thisintroduces a parameter g2 (here wavevector dependent) similar to that definedfor non chiral Luttinger liquid. The effective Luttinger parameter is no longerg = but g(, g2) = . In particular, in the limit w 0 the qualitative differencebetween both systems becomes very small.

There are also important differences. A continuous variation of the g = parameter is a priori not expected in the FQHE regime because the magneticfield stabilizes special fractional values of in the bulk (as the Jains series).A generalization of Wens approach for = p/(2sp + 1) shows that one musthave p branches of bosonic modes. These branches interact together and give arelation between the TDOS exponent not simply related as in (41). This is notthe case for 1D systems for which g varies continuously and the relation betweenthe tunneling exponent and g is expected to always hold: = (g + g1 2).For the simplest series of Jains filling factor p/(2p 1) between 1 and 1/3, theexponent is expected to be

=2p + 1

p 1|p| (58)

i.e. constant ( = 2) between filling factors 1/2 and 1/3 and decreasing linearly( = 1/ B) form 2 to 1 between filling factors 1/2 and 1.


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It is also interesting to compare the practical realizations of 1D and fractionalquantum Hall Luttinger liquids. Practically, it is rather difficult to built a cleansingle 1D quantum wire. The pF electronic modes being spatially located at thesame place, any inhomogeneity will produce efficient scattering. The upper and

lower fractional quantum Hall edge channels are usually well separated spatiallyand no unintentional backscattering is possible. This is why fractional quantumHall systems have been the only ideal realization of Luttinger liquids found so far.Also, in 1D systems such as quantum wires [5153], carbon nanotubes [54,55],blue-bronze or quasi-1D organic conductors it is difficult to probe power laws overmore that 1 to 1.5 decades, while in fractional quantum Hall systems the TDOShas been probed over 3 decades giving good determination of the tunnelingexponents and a clear evidence of Luttinger liquids properties [56,57,61].

Experimental Evidence of Chiral Luttinger Liquids: The best evidencefor Luttinger liquid properties is obtained by probing the tunneling density ofstates. To do that, the measurement has to be non-invasive, i.e. a weak tunnelcoupling is required. Indeed, a tunneling experiment measures the TDOS only ifhigher order tunneling processes are negligible, which means small transmissionand small energy (as the current | EF|+1 and, so the effective coupling,will increase with voltage eV or temperature kBT).

Convincing experiments have been performed by the group of A.M. Chang[56,57,61]. The tunnel contact is realized using the cleaved edge overgrowth tech-

nique. By epitaxial growth on the lateral side of a 2DEG, a large tunnel barrieris first defined followed by a metallic contact realized using heavy doped semi-conductor. The advantage is a weak coupling, a barrier strong enough to applylarge voltage without significant change in the (bare) tunnel coupling, and arather well define edge for the 2D electron system. Also, probably important isthe fact that the metallic contact close to the edge provides screening of the longrange Coulomb interaction (short range is needed for having power laws).

Fig. 17 shows example of IVC: a power law of the current with applied voltageI V, = + 1, is well defined over several current decades for = 1/3. Theexponent found is 2.7-2.65 close to the value = 3 predicted by the theory. Forother filling factors similar algebraic variations are also observed. In the samefigure, the tunneling exponent deduced from a series of I-V curves is shown asa function of the magnetic field or 1/. For filling factor 1/2 < < 1/3 theconstant exponent predicted [5860] is not observed and instead the exponentvaries rather linearly with field or 1/ (however, recent experiments made withcleaner samples have shown signs of a plateau in the exponent in a narrow fillingfactor value near 1/3).

Theoretical attempts to explain quantitatively the discrepancies have been

made. Taking into account the long range Coulomb interaction slightly lowersthe exponent. The modified Luttinger liquid theories can also include the finiteconductivity in the bulk for non fractional filling factors. Indeed a finite con-ductivity modifies the dispersion relation of the bosonic chiral modes and sothe exponents. With reasonable parameters these modifications are not yet ableto fully reproduce the data [59,60]. The discrepancy between experiments and


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34 D.C. Glattli

Fig. 17. Left: Log-Log plot of an I-V curve at = 1/3 clearly shows the algebraicvariation of current with voltage which characterizes a Luttinger liquid. Right: theexponent is ploted verus 1/ and compared with theoretical predictions (adaptedfrom Ref[56,57])

predictions may be due to the reconstruction of the edge. Wens model assumesa sharp density variation at the edge of the 2D sample. In real samples, the den-sity decreases smoothly and some additional edge states corresponding to fillingfactor lower than the bulk filling factor may also strongly modify the exponents.

Another possibility to search for chiral Luttinger liquid evidence is to maketransport experiments between two fractional edges. This can be done using anartificial impurity, a Quantum Point Contact. This strategy however is less re-liable than the previous one as it is likely that, when a voltage difference Vdsis applied between both side of the QPC, the microscopic potential which re-sults from the gate of the QPC and the electric field associated with Vds willchange. This will modify the transmission in a trivial manner and will makeidentification of power laws difficult. In the case of a high barrier with tunnelingregime established for all energies and no possibilities to reach the regime of weak

backscattering, the result will be similar to that of A.M. Changs experiments.However, because the dI/dVds characteristics will be proportional to the squareof the TDOS, the exponents for the conductance will be doubled. This approachhas been used by several groups. A difficulty is that sample inhomogeneitiesaround the QPC leads to transmission resonances which are difficult to control.Exploiting Luttinger predictions for tunneling through such a resonant state hasbeen used in [62]. A further difficulty is the high value of the power law for theconductance with temperature or voltage. For = 1/3, = 4 (i.e. 2(1 1)).It has been realized recently, with the help of exact finite temperature calcu-

lations of I-V characteristics, that the exponent 4 can be measured only if theconductance is smaller than 104e2/3h [63]. Up to now, no experimental grouphave tried to do measurements in this limit. For higher conductance instead,an effective exponent, much weaker than 4, will be measured. To understandthis, one have to discuss the physics of Luttinger liquids with an impurity andunderstand how this impurity affects the conductance.


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4.4 Fractional Edge Channels with an Artificial Impurity

What we are going to discuss in the following is also valid for ordinary Luttingerliquids. Indeed we have seen that the chiral Luttinger liquids and ordinary Lut-

tinger liquids are described by the same Hamiltonian. Without loss of generalitywe will take the case of the special filling factor = 1/3 as the physics is betterunderstood and experiments are cleaner.

In the fractional regime, a controllable artificial impurity is easily realizableusing a QPC. The gate potential can be used to change the strength of theimpurity. We will discuss here the case of a weak impurity. Without Luttingerliquid effects such impurity would lead to a weak coupling between upper andlower edges. Practically, the QPC gently pushes the upper edge close to the loweredge to induce a quantum transfer of particles from one edge to the other. Also

the QPC potential is weak enough to do not make appreciable change of thelocal filling factor. This is different from the so-called pinch-off regime where thepotential is so strong that a local electron depletion sets in and a tunnel barrierforms. Nevertheless, the Luttinger liquid theory predicts that in the limit ofsmall energy ( kBT or eVds 0) the strength of the weak artificial impurityis strongly renormalized by the interaction. The system flows to an insulatingstate and the conductance displays the same power law with T or Vds than theone expected for a strong impurity potential (tunnel barrier)

G e2

3h

TB2( 11) 0 for TB (59)

where TB is an energy scale related to the impurity potential. On the other hand,as the impurity is very weak one expects that at high energy a conductance close

to, but smaller than the quantum of conductance e2

3h is recovered. Indeed theLuttinger liquid theory predicts

G =e2

3h GB with GB =

e2

3h TB2(1)

0 for

TB (60)

GB is called the backscattering conductance. Indeed if we call I the forward

current, I0 =e2

3h Vds the current without impurity, the current associated withparticles backscattered by the impurity is IB = I0 I from which one can defineGB = IB/V. The above formulae correspond to strong and weak backscatteringlimits. In the first case there is a weak tunneling of particles between the left andright side, while in the second case there is a weak quantum transfer of particlesbetween the upper edge and the lower edge. There is an interesting duality with

1/.Most experiments at = 1/3 have been performed in an intermediate regime.

First, to reach the strong backscattering regime one needs at least one order ofmagnitude of variation of T with the condition T TB. This implies work-ing with conductances as low as 102(

11)e2/h to obtain conductance values.

Second, if this condition is not respected, theory shows that inclusion of higher


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36 D.C. Glattli

order processes gives lower effective exponents. Indeed the exponent above canbe viewed as the scaling exponent of a renormalization theory. It is valid onlyclose to the critical point (here G = 0).

What are the higher order tunneling terms? The tunneling Hamiltonian cou-

pling the upper and lower edge is

Himp = 12

n

n

exp(in+(0)) exp(in(0)) + exp(in(0)) exp(in+(0))

(61)

where n is the Fourier transform of the potential at 2nkF and the nth term

correspond to the transfer of n particles. Each contribution can be calculatedperturbatively using a renormalization approach. In the two limiting cases strongbackscattering, for energies 0 , it is found [58]

G() = e2

h

n

n |n|2 2(n2

1) (62)

were the coefficient n are non universal. The powers in the series increase ratherrapidly with the order n considered. For = 1/3 for example the second termhas a power 22 while the first one has a power 4. When the energy is below someenergy scale

quantum hall effect macroscopic and mesoscopic electron transport

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