Download Integer and fractional quantum Hall effects

Eur. J. Phys. 21 (2000) 535–548. Printed in the UK
PII: S0143-0807(00)12908-3
Integer and fractional quantum Hall
effects
Karol I Wysokiński
Institute of Physics, M Curie-Skłodowska University, PL-20 031 Lublin, Poland
Received 28 March 2000
Abstract. The quantum Hall effect (QHE) is a set of phenomena observed at low temperature
in a two-dimensional electron gas subject to a strong perpendicular magnetic field. It manifests
itself as a quantization of the non-diagonal element ρxy of the resistivity tensor, accompanied by
the simultaneous vanishing of ρxx for ranges of the magnetic field. For the integer quantum Hall
effect (IQHE), ρxy = h/νe2 , where h is the Planck constant, e is the charge of an electron and ν
is an integer, while for the fractional quantum Hall effect (FQHE), ν is a simple fraction. In spite
of the similar phenomenology deep and profound differences between the two effects exist.
In the article I discuss the precision of the Hall quantization in IQHE and the new types of
quantum fluids observed in FQHE. Some recent theoretical and experimental discoveries connected
with quantum Hall liquids are briefly mentioned.
1. Introduction: the Hall effect
It was recognized long ago that measurements of transport properties of conducting materials
give a lot of useful information about their internal properties. The interpretation of most
experiments is possible on the basis of the classical Drude theory but a real understanding
of them can be achieved with full recognition of the quantum mechanical nature of charge
carriers, their dynamics and interactions [1].
It has been known since 1879 that the application of the magnetic field B to the thin
conducting slab, along which the current flows, produces a voltage VH across the sample
transverse to the current flow (see figure 1 for typical setup). The appearance of the voltage,
Figure 1. The standard Hall bar geometry used in studies of the quantum Hall effect to measure
ρxx and ρxy .
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© 2000 IOP Publishing Ltd
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K I Wysokiński
VH , is known as the Hall effect. It is due to a Lorentz force acting on charges moving in the
presence of a magnetic field. In equilibrium the Lorentz force FL = qvD B is balanced by the
electric force qVH /Ly , so VH = vD BLy . Here, vD is the average drift velocity of carriers, q
is their charge and Ly is the width of the sample. Noting that the current I can be expressed
as the product of the drift velocity vD , the charge density n and the cross-sectional area of
the sample S = Ly d (here d is the thickness of the slab; not shown in figure 1), we find the
perpendicular resistivity RH = VH /I to be
RH =
B
B
=
qnd
qNS
(1)
where n is the number of carriers per unit volume and NS = nd is the number of carriers per
unit surface area.
The measurement of the Hall resistance RH (or Hall constant RH /B) gives information
about the density NS and the sign (q = ±e) of charge carriers in metals and semiconductors.
It does not depend on the mass of the carriers or on other parameters of the material different
from carrier density. Owing to this independence the Hall effect has become a standard tool
of material characterization. The direct proportionality of the Hall resistivity to the local
magnetic field B is used to measure the magnetic field and its distribution. Recently, devices
called scanning Hall probe microscopes have been constructed which are so precise as to allow
inter alia detailed determination of the magnetic field distribution near the vortices in type II
superconductors [2].
So far we have been talking about bulk three-dimensional samples. What about twodimensional systems? At first sight nothing special can be inferred from equation (1). The
calculations [3] of transport properties of the two-dimensional electron gas have only shown
that departures from the linear dependence of Hall resistivity on the magnetic field can be
expected. On the other hand, we know [4] that in systems with reduced dimensionality the
quantum effects become important. Both integer (IQHE) and fractional (FQHE) quantum
Hall effects are macroscopic manifestations of the quantum behaviour of two-dimensional
electrons.
Before presenting the results of measurements let us briefly discuss the ways of obtaining
a two-dimensional electron gas.
Advances in technology allow us to produce semiconductor structures in which the
motion of carriers (electrons or holes) is limited to two dimensions. There are two such
systems in which a two-dimensional electron gas (2DEG) can be generated and studied. The
first comprises silicon metal-oxide-semiconductor devices such as, for example, MOS fieldeffect transistors (Si-MOSFETs), and the second GaAs/AlGaAs heterostructures [5]. The
electron density can be easily controlled in Si-MOSFET by changing the gate voltage [5].
In GaAs/AlGaAs heterostructure, the charge density is essentially constant. Because of the
difference in the position of the bottom of the conduction band in GaAs and AlGaAs the
electrons from ionized Si donors move towards the GaAs/AlGaAs interface (see figure 2). The
attraction of electrons by positive Si ions makes them reside near the GaAs/AlGaAs interface.
The electrons experience a triangular effective potential, which leads to the formation of the
so-called electric subbands. The energy gaps between subbands are of the order of tens of
meV. For concentrations n in the range of 2 × 1011 cm−2 , only the lowest electric subband
is occupied. Under these conditions the motion of electrons is confined to the plane of the
interface.
The long-range nature of Coulomb forces from ionized Si donors is a source of potential
fluctuations in the region of 2DEG. At low temperature this is a main source of electron
scattering. To diminish scattering, the extra layer (thickness up to 1000 Å) of undoped AlGaAs
is placed between Si doped AlGaAs and GaAs to spatially separate strong scattering centres
from 2DEG. As a result, the scattering of electrons off potential fluctuations is substantially
reduced and the mobility of electrons increased. The mobility µ tells us how easily the
electrons are moving through a sample in response to the applied voltage. With this technique,
Integer and fractional quantum Hall effects
537
Figure 2. Modulation doped GaAs/Alx Ga1−x As heterostructure. The energy gaps of GaAs and
AlGaAs are denoted by Eg1 and Eg2 , respectively. E is the first electric subband in the ‘triangular’
well formed in GaAs, EF is the position of the Fermi level.
called modulation doping [6], mobilities of 2DEG in GaAs/AlGaAs heterostructures have been
obtained in excess of 107 cm2 V−1 s−1 at low temperatures. In such good quality materials the
distance travelled by an electron between two collisions amounts to a 0.2–0.3 mm.
In spite of large mobilities, 2DEG in GaAs/AlGaAs heterostructures cannot be considered
as an ideal gas. The amount of impurities is still large. In particular, the randomly placed Al
atoms in Alx Ga1−x As alloy are the effective scattering centres. In fact, as we shall see, a fair
amount of impurities is necessary for the observation of quantum Hall effects.
2. The integer quantum Hall effect
The IQHE was discovered in 1980 by von Klitzing et al [7]. It manifests itself as a series
of plateaux in the Hall resistance of the two-dimensional electron gas (see figure 3). The
plateaux extend over a range of charge densities in a constant magnetic field (Si-MOSFETs)
or a range of magnetic fields under the condition of constant electron density (GaAs/AlGaAs
heterostructure). Remarkably, the value of the Hall resistivity on the plateau is to high accuracy
538
K I Wysokiński
Figure 3. Integer quantum Hall effect measured [11] at temperature 8 mK in GaAs/AlGaAs
heterostructure characterized by electron density NS = 4.2 × 1011 cm−2 and mobility µ =
1.8 × 105 cm2 V−1 s−1 .
given by
h
25 812.807
=
(2)
νe2
ν
where h is the Planck constant, e is the electron charge and ν is an integer. In the original
paper [7] a quantization precision of 10−5 was observed. Soon after the discovery it turned out
that the precision of quantization of RH is a few parts in 108 or better [8]. As a result, the Hall
resistance has been adopted as an international standard of resistance [9]. For a discussion of
the practical realization of quantum Hall resistance standards see the recent review [10].
To understand the high precision of the measurement it is important to realize that in
two dimensions Hall resistance RH and Hall resistivity ρxy coincide (notation as in figure 1,
Vy = VH )
VH /Ly
Ey
VH
=
=
= ρxy .
(3)
RH =
I
I /Ly
jx
RH =
This means that single and potentially very accurate ‘electric measurement’ of RH suffices
for determination of the resistivity ρxy . To obtain information on the other component of the
tensor, i.e. the longitudinal resistivity ρxx , a knowledge of both R and the sample dimensions
Lx , Ly is necessary and
Lx
Ex Lx
V
= ρxx .
(4)
=
R=
I
jx L y
Ly
Integer and fractional quantum Hall effects
539
Measurements of Lx and Ly , however, are never very precise. In two dimensions, the
resistances and resistivities are expressed in the same units, namely ohms. Sometimes it
is assumed that Lx = Ly and then R is said to be measured in ohms per square.
Note also that in the two-dimensional system under consideration the symmetries σyy =
σxx , σxy = −σyx hold and the matrix elements of the resistivity tensor ραβ are proportional to
2
2 −1
the corresponding elements of the conductivity tensor σαβ , for example, ρxx = σxx (σxx
+σxy
) .
As can be seen in figure 3, each plateau in ρxy is accompanied by the vanishing of the
longitudinal resistivity ρxx . The value of the longitudinal resistance R measured in the Hall
plateau is as low as 10−10 −1 . This is a lower value than in any non-superconducting
material. The IQHE has been observed in various systems containing a two-dimensional gas
of carriers. The results do not depend on the material, the sample geometry, etc. IQHE,
however, is a low-temperature effect. With increasing temperature the quantization accuracy
is lowered, the plateaux become narrower and eventually vanish, the longitudinal resistivity
takes on non-zero values.
Note that RH measures the combination of physical constants which also enters into the
definition of the Sommerfeld fine-structure constant α −1 = (2/µ0 c)h/e2 . Since both µ0 and
c are defined physical constants, and QHE measurements can be used for a high-precision
determination of α. The effect was, in fact, announced in the original paper [7] as a ‘new
method for high-accuracy determination of the fine structure constant’ and soon obtained a
precision comparable to that of other methods [8].
Before we embark upon a theoretical explanation of the IQHE we have to note another
aspect connected with the precision of quantization. Assuming typical electron concentration
in 2DEG of the order of 2 × 1011 cm−2 and typical sample dimensions of 260 × 400 µm2 [11]
we find that the number of electrons in a two-dimensional channel is N ≈ 2 × 108 . Thus,
the precision
of quantization is of the order of 1/N instead of the much lower precision
√
of 1/ N ≈ 10−4 expected on statistical grounds, connected with fluctuations of physical
parameters in the many-body system. This observation prevents the application of any type of
‘mean-field’ treatment of the system, and the only approximation allowed is replacement of
the sum over states by an integral, a procedure which can be shown to introduce errors smaller
than 1/N.
It turns out that IQHE can be understood solely in terms of single-particle considerations.
Thus, we start with the ideal 2DEG in perpendicular magnetic B and electric E fields. Consider
a typical Hall sample, as shown in figure 1(a), of area Lx Ly placed in a perpendicular magnetic
field B = (0, 0, B). The Hall voltage VH across the sample is a source of the electric field
E = (0, E, 0). The single-electron Hamiltonian is given by
2
2
∂
1
1
∂
−ih̄
H =
(5)
+ eAx +
−ih̄
+ eAy + eEy + Vc (x, y) + Vimp (x, y)
2m
∂x
2m
∂y
where Vc (x, y) is the confining potential, while Vimp (x, y) represents the electron–impurity
scattering potential. The Schrödinger equation H ψ = Eψ with Vc (x, y) and Vimp (x, y) can
only be solved numerically. Without these two potentials the problem is exactly solvable.
Using the Landau gauge Ax = −By, Ay = 0 and writing the electron wavefunction in
the form
1
ψ(x, y) = √ eikx ϕk (x)
Lx
we obtain the equation
h̄2 ∂ 2
1
2
(h̄k − eBy) −
+ eEy ϕ(x) = ε(k)ϕk (y)
(6)
2m
2m ∂y 2
which is an equation for the displaced one-dimensional harmonic oscillator. Its solutions are
expressed in terms of Hermite polynomials Hn as
y − yk
1
2
2
(7)
ϕnk (y) = √ exp −(y − yk ) /2l Hn
l
2n n! πl
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K I Wysokiński
where n = 1, 2, . . .,
n x2
Hn (x) = (−1) e
d
dx
n
e−x
2
√
is the nth Hermite polynomial and l = h̄/eB denotes magnetic length. The wavefunction
ϕnk (y) is seen to be centred around yc = yk , which in turn depends on the wavevector k.
Explicitly,
eEl 2
(8)
hωc
and the eigenenergies become
1
1 (eEl)2
εn (k) = n +
.
(9)
h̄ωc + eEl 2 k −
2
2 h̄ωc
The spectrum has a linear dispersion. It consists of narrow bands separated by energy gaps
h̄ωc = h̄eB/m∗ , where m∗ is the effective mass of electrons in the system. For typical
values of parameters h̄ωc ≈ 1 meV. Note that for the vanishing electric field (E = 0) the
spectrum consists of degenerate Landau levels. To find the degeneracy gn we assume periodic
boundary conditions in the x-direction; ψ(0, y) = ψ(Lx , y). This limits the allowed values
of k to k = (2π/Lx )r, r = 0, ±1, ±2, . . .. The distance k between neighbouring k states is
(2π/Lx ). Because the centre of the wavefunction with wavevector k located at yk as given by
(8) has to fall inside the sample width, i.e. 0 yk Ly , the total degeneracy gn of the nth
Landau level is found to be
Ly
Lx Ly eB
Lx Ly
φ
gn = g = 2
=
(10)
=
=
2
l k
2πl
h̄
φ0
where φ = Lx Ly B is the magnetic flux through the sample area and φ0 = h/e is the
flux quantum. The ratio between the electron density N/Lx Ly and the degeneracy g, i.e.
ν = N h/eB, is called the filling factor.
Suppose now that a number of Landau levels, say n, are fully occupied. This means that
the electron density NS = gn/Lx Ly = neB/ h. If this is applied to equation (1) then we
obtain
E
Bh
h
RH =
=
(11)
= 2
eNs
eneB
ne
in agreement with (2). Unfortunately, this does not yet explain the QHE, because we have only
shown that for a very special density of electrons the Hall resistance takes on a very special
value. Had one changed the electron density then RH would also have changed linearly with
B in complete disagreement with experiment.
In a real, impure system some states are localized and the number of current-carrying
states is smaller than found above. Moreover all other states are scattered by impurities and
there is no reason for the quantization. If, however, we observe quantum values of ρxy , as
experiments show, this is only because of some lucky compensation which takes place. The
impurity potential which localizes some states, changes other states in such a way that they
carry more current, exactly compensating for those that do not. This point is further discussed
in the next section.
yk = l 2 k −
2.1. The role of impurities
As we have seen, the density of states of clean 2DEG in the perpendicular B field consists
of a set of δ-functions centred at εn = h̄ωc (n + 21 ). The impurities bind some states and their
energy levels appear inside band gaps. If the number of impurities is a finite fraction of the
degeneracy of the Landau level (LL) the band gaps eventually close, as shown in figure 4.
The density of states is a continuous function of energy, with maxima centred around LLs and
Integer and fractional quantum Hall effects
541
Figure 4. Schematic dependence of the density of states on the energy for the two-dimensional
electron gas in the presence of impurities. Shaded regions indicate localized states. Recent studies
suggest that extended states exist at a single energy in each Landau band (marked by the dashed
line).
minima in between, and consists of localized and extended states. The localized states do not
carry a current. Because the number of states of the system at hand does not depend on the
existence of impurities, this means that only a small fraction of all states will carry a current.
The precision of quantization and its independence from the material tell us that the extended
states do exist and carry just enough current.
A very important result in this context has been obtained by Prange [12]. This is the exact
solution of a single δ-function impurity potential inside an otherwise ideal system. Such a
potential (independently of whether it is attractive or repulsive) binds a single localized state
from each Landau level. The localized state does not carry the current. Each of the remaining
extended states, however, carries a little bit more current, exactly compensating the loss of
the ‘companion’. This result can be generalized and the compensation proved for a class of
impurity potentials [13, 14].
To see the effect of compensation in some detail let us consider the Hamiltonian (5) with
the term Vimp (x, y) only (Vc = 0) and look at the changes of eigenstates and eigenenergies
of the pure system induced by the impurity potential. In terms of scattering theory some
states will be localized around impurities, while others, the scattering states, will be slightly
modified. Energies of scattering states εα will differ from those of corresponding states εα0 of
the pure system by a small amount εα0 . Energies of localized states on the other hand will
change appreciably. The important point is that energies of extended states depend on the
542
K I Wysokiński
gauge potential, while those of localized states do not.
In the presence of a magnetic field the current operator
e
e
jˆα = p̂ = (−ih̄∇ + eA)
m
m
can be written as jˆ = δ Ĥ /δA(r ). The Hellman–Feynman theorem states that if the
Hamiltonian Ĥ depends on the arbitrary parameter λ then
∂ Ĥ ∂εn (λ)
(12)
= φn (λ) φ (λ)
∂λ n
∂λ
where Ĥ (λ)|φn (λ) = εn (λ)|φn (λ). In the present case, this means that the current jα is
calculated from a knowledge of the gauge dependence of the energies
1 ∂εα
jα =
.
(13)
B ∂A0
Here, A0 is the additional (constant) vector potential so that now Ax = −B(y + A0 ). The
presence of A0 modifies the eigenenergies of a pure system and equation (9) becomes
1
1 (eEl)2
εn0 (k) = hωc n +
.
(14)
+ eE(l 2 k + A0 ) −
2
2 hωc
0
The current carried by the state |nk is thus jnk
= eE/B = j0 . This agrees with the previous
discussion. In the impure system the change of energies εn (k, A0 ) = −η(εn0 (k))/π , where
η(εn0 (k)) is the phase shift and the current carried by the scattering state |nk+ [13]
1 ∂εn (k)
1 ∂η(εn0 (k))
0
0
jnk = jnk
+
= jnk
−
(15)
B ∂A0
Bπ ∂A0
0
differs from jnk
. The total current carried is the sum over all occupied states. Assuming that
the Fermi level lies in the region of localized states we have
occ.
1 ∂η(εn0 (k))
(16)
I = (N − Nl )j0 −
πB nk
∂A0
where Nl is the number of localized states. Noting that
1 ∂η(εη0 (k))
∂η(εn0 (k))
=
(17)
j0
B ∂A0
∂εn0 (k)
and summing over all occupied states we obtain that the second term in (16) equals the total
change of the phase shift. The Levinson theorem, known from scattering theory [13], states
that this change is exactly given by the number of localized states multiplied by −π , i.e. it
equals −πNl . Thus, the total current is found to be I = (N − Nl )j0 + Nl j0 = Nj0 = I0 , i.e.
the same as that carried by a pure system. This proves the compensation theorem for a general
class of potentials. A somewhat different and, in fact, even more general explanation of the
integer Hall plateaux and their independence from disorder has been proposed by Laughlin [15]
(see also his Nobel Lecture and the references cited therein [16]).
In fact, it is the disorder in the system which breaks translational symmetry and leads
to the quantization of Hall conductance. The change of the number of electrons changes the
position of the chemical potential, but as long as it lies in the region of localized states the Hall
current is constant and the free electron value of the resistivity remains intact ρxy = h/ne2 .
This is consistent with experiment.
Moreover, as long as the chemical potential lies in localized states the longitudinal
resistivity is zero at zero temperature, because electrons from current-carrying states cannot
be scattered across the gap, and there is no voltage drop in the direction of current flow. The
longitudinal resistivity ρxx thus vanishes. This explains the data.
Of particular importance for the precision of quantization is the influence of finitetemperature effects. We shall study these effects in the next subsection.
Integer and fractional quantum Hall effects
543
2.2. The role of temperature
The experiments are carried out at non-zero temperatures and various kinds of process
contribute to dissipation and the non-zero value of σxx , even when the Fermi level lies
in the region of localized states. This in turn affects the exactness of the Hall resistivity
quantization. At low temperatures the activation of electrons to higher extended states, and
phonon-induced hopping of carriers between localized states or variable-range hopping yield
important contributions to the transport coefficients.
Experimentally it has been found [17] that with decreasing temperature the slope of the
Hall resistivity dρxy /dB remained activated down to the lowest temperatures, while the T
dependence of ρxx crossed over from activated behaviour ln ρxx ∝ 1/T to the behaviour
ln ρxx ∝ 1/T 1/2 characteristic for variable-range hopping [18]. Note that variable range
hopping increases the value of ρxx over its activated behaviour at the lowest temperatures.
In the quantum Hall situation the states localized around a short-range impurity located
at r0 have the Gaussian form exp[−(r − r0 )2 /2l 2 ]. At finite temperature the electron–phonon
interaction gives rise to transitions between such states. The probability per unit time for the
hopping of an electron from a localized state i with energy εi below the Fermi level εF to
another localized state j with energy εj above εF is the product of the overlap between these
states exp[−|Ri −Rj |2 /2l 2 ] and the Boltzmann factor exp[−|εi −εj |/2kT ] [19]. Normally the
electron initially localized around an impurity at Ri will jump to the nearest impurity state in
¯ ε /kT ]. At the
space. This process gives rise to the usual activated conductivity σxx ∝ exp[−
lowest temperatures there appears a possibility of optimization. It may be more convenient for
an electron to hop a greater distance |Rij | and to find a site with energy difference smaller than
¯ ε . This requires optimal values of Rij and εij = |εi − εj |. The calculations [18]
the typical show that the value of the longitudinal conductivity on the plateau region σxx and the departures
δσxy of σxy from the ideal value e2 / h due to variable range hopping between Gaussian states
are given by
A
B
σxx = exp[−(T0 /T )1/2 ]
δσxy = exp[−(3/2)(T0 /T )1/2 ].
(18)
T
T
The presence of the factor 3/2 in the exponent makes the variable-range hopping contribution
to δσxy much smaller than that to σxx . This prediction has eventually been confirmed
experimentally [20]. The characteristic temperature T0 below which departures from activated
transport are to be expected has been found (in [18]) to be related to the density of states at
the Fermi level by kT0 = δ/NF l 2 , where δ is a number of order 1. The application of the
percolation ideas valid for slowly varying long-range impurity potentials modifies that result
by making T0 lower [21].
In practical realization of the resistivity standard it is of great importance to know the
thermoelectric power of the system which is a source of thermally induced (error) voltages.
These also lead to departures of ρxy from the ideal quantized value. Both the theory and
experiments agree [22] that thermoelectric power tends to zero in the plateau regions taking
on maximal values in transition regions between plateaux.
2.3. Localization in high magnetic fields
The standard scaling theory of localization predicts that in two dimensions all states are
localized in the thermodynamic limit, no matter how weak the disorder is [23]. These results
are evidently in conflict with the quantum Hall effect, which as we have seen requires the
existence of extended states below the Fermi energy for its explanation. The energy which
separates extended and localized states is called a mobility edge. In analogy with bulk systems
the states in the tails of Landau bands are localized (see figure 4). If the Fermi energy lies in
a mobility gap the system is in the quantum Hall state with ρxx = 0 and ρxy quantized. If it
is between lower Ec1 , and upper Ec2 mobility edges then ρxx takes a non-zero value and ρxy
changes from a given quantized value to the next one.
544
K I Wysokiński
Even from the early experiments it was clear that with decreasing temperature the width
of the ρxx peak (B) decreased and the slope of ρxy increased, indicating that the region of
extended states is very narrow, if not of zero width. More detailed experiments have shown
that B ∼ T κ with κ = 0.42 ± 0.04, signalling that extended states do exist at the singleenergy value. The maximal slope of ρxy , as measured by dρxy /dB, has been found to increase
proportionally with the same exponent κ; dρxy /dB ∼ T −κ . All this is consistent with the idea
that lower and upper mobility edges coincide and the extended states exist at a single energy. At
all other energies the states are localized, i.e. their spatial extension ξ is finite. The parameter
ξ is called the localization length. It depends on energy and for energies close to a mobility
edge is expected to diverge as ξ ∝ (E − Ec )−ν . The measured value of ν ≈ 2.34 ± 0.04 agrees
quite well with theoretical estimates (see [24] for a review of various theoretical approaches
and [25] for a classical calculation for the network model).
Each transition region between two consecutive IQHE steps in fact represents a quantum
phase transition from localized to delocalized states. Quantum phase transitions take place
at T = 0 K. Experimentally they are studied at low but finite temperatures and manifest
themselves as a narrowing down of regions, where ρxx = 0 and a growth of the slopes
(dρxy /dB). Pruisken was the first to apply the scaling theory of phase transitions to the
localization in strong magnetic fields [26].
In the ultra-quantum limit of very high magnetic fields and low electron concentrations
the localization due to interactions may be realized. In this limit one expects the formation
of a Wigner crystal. In this state, due to Coulomb repulsion, electrons form a lattice. The
magnetic field which freezes the kinetic energy of an electron favours the formation of the
Wigner crystal. In dirty systems the physics is probably again dominated by disorder.
Before concluding this discussion we mention recent experiments which show [27] that
the metallic phase can exist in the magnetic-field-free two-dimensional electron gas. If it is
true, this may indicate that the simultaneous effect of interactions and disorder can be a source
of more different phases in the phase diagram of 2DEG than expected.
3. The fractional quantum Hall effect
On phenomenological grounds, FQHE looks like ‘ordinary’ IQHE rationalized by the
equation (2), with ν being a fractional number. Originally, the fractional steps with ν = 13
and 23 were discovered in 1982 [28], during high-field measurements aimed at the observation
of Wigner crystallization of electrons. The original data shown in figure 5 were taken at a
few temperatures. The development of the IQHE plateaux (at low B field) accompanied by
vanishing ρxy is observed with the decrease of temperature. At B fields as high as 15 T the
formation of a deep minimum in ρxx and a flattening off of ρxy is observed. The calculated
filling factor indicates partial (1/3) filling of the lowest Landau level.
A further increase in the electron mobility of 2DEG has resulted in new fractions ν.
Figure 6 presents FQHE obtained in ultra-high mobility 2DEG [29]. The plateaux are seen
to appear not only in the lowest Landau level but also in a higher one. As is easily seen,
most fractions have odd denominators. More detailed analysis of these and other data shows
that some sequences are particularly clearly developed. The sequences with ν = p/(2p ± 1)
terminate at ν = 21 . This is one of the very special fractions with even denominators. At this
filling factor one observes some features in ρxx and none in ρxy .
Even though the FQHE effect appears to be the same as IQHE, in that it shows quantization
of Hall resistivity ρxy accompanied by the vanishing of longitudinal resistivity ρxx , its
explanation is completely different. The first successful theory was published in 1983 by
Laughlin [30]. He proposed that the ground state of N electrons in a partially filled Landau
level is described by the many-body wavefunction
N
1 m
2
ψm (z1 , z2 , . . . , zN ) =
(19)
(zj − zk ) exp − 2
|zj |
4l j
j <k
Integer and fractional quantum Hall effects
545
Figure 5. The first observation of the ν = 13 fractional quantum Hall state [28].
where m = 3 and zj = zj + iyj is the position of the j th particle in the (x, y) plane, expressed
as a complex number.
Some tacit features of the Laughlin wavefunctions are worth mentioning. It was chosen to
keep electrons apart and reduce their Coulomb energy. This is achieved by the factors (zi −zj )m .
For positions zi and zj of electrons close to each other, the wavefunction has a small amplitude.
The motion of electrons is thus highly correlated. The important point is that there exists a
gap between the ground state and excited states. The gap is due to interactions. The effect of
disorder is presumably similar to in IQHE. The disorder will localize the quasi-particles in the
tails of bands allowing for a finite width of the plateaux.
The excitations over the ground state are constructed by locally changing the magnetic
field. This is achieved by the adiabatic insertion, perpendicularly to the plane of 2DEG, of a
thin solenoid, which carries exactly one flux quantum. Such a perturbation creates a quasihole. Similarly, the quasi-particle excitation can be formed by locally removing a flux unit of
magnetic field from the system.
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K I Wysokiński
Figure 6. The contemporary view of quantum Hall results obtained on the very-high-mobility
2DEG [29].
At the time of writing more than 30 fractional Hall plateaux have been discovered, and
the appropriate wavefunctions can be deduced from equation (19), which is the simplest in
the hierarchy [31]. It has been shown that the Laughlin wavefunction is an exact many-body
ground-state wavefunction of a class of Hamiltonians [32]. It describes a new state of matter,
an incompressible quantum liquid. The excitations over its ground state carry a fractional
electric charge. These are extended objects whose properties are not related to the electron
properties but are determined by the interactions between electrons. In particular, the energy of
the creation of the quasi-particle has been estimated to be a fraction of the Coulomb interaction
energy e2 / l of two electrons a magnetic length apart from each other.
There are alternative explanations for the FQHE. In one of these, proposed by Jain [33],
the composite fermions are considered. These are electrons with an even number of flux quanta
bound to them. If the external magnetic field B is applied to a system with N electrons and
each of them is dressed with (2s) flux quanta then the effective magnetic field B ∗ in which
new objects, called composite fermions, move is found from B ∗ S = BS − 2sN φ0 (where
S is the sample area) and the effective filling factor ν ∗ = N φ0 /SB ∗ is related to the filling
factor of electrons ν = N φ0 /SB by ν = |ν ∗ |/(2s|ν ∗ | ± 1). In this picture FQHE of 2DEG
looks like an IQHE of composite fermions. Letting s = 1 and ν ∗ = p, this leads to the
previously mentioned sequence of FQHE plateaux. One of the first successful applications of
the composite fermion theory was the explanation of the ν = 21 features observed in ρxx [34].
Integer and fractional quantum Hall effects
547
3.1. Fractional charge of excitations
The elementary excitations in the FQHE state with ν = 13 have been predicted to carry fractional
charge q = 13 e. Laughlin has also shown that this charge quantization is exact. Thus, the
fractional charge of excitations should show up in measurements other than the FQHE itself.
The experimental search started with the work of Clark et al [35] and others [36]. The most
direct observations of fractional charge excitations have recently been reported from shot-noise
experiments [37]. Note, however, that the fraction of the charge is different from the filling
factor ν, as shown in recent experiments [38] which measure the charge 15 e on the Hall plateaux
with ν = 25 .
The idea behind the shot-noise experiments is simple. Owing to the discrete nature of
charge carriers the current experiences small fluctuations, called shot noise. In experiments
the tunnelling of quasi-particles between edges of the small constriction in quantum Hall
samples generates noise in the circuit. This noise, as measured by the spectral density of
current fluctuations, turns out to be directly proportional to the charge of the carriers. The data
obtained for the ν = 13 plateau unequivocally show that the charge of the carriers is q = 13 e.
An experiment to verify this performed at the integral quantum Hall plateau ν = 4 shows, in
agreement with theory of IQHE, that the charge of the carriers equals the electron charge e.
4. Conclusions
We have briefly discussed the issues connected with Hall effects (classical, quantum integer and
fractional), pointing out their universality aspects. The Hall effect is manifestly a universal
phenomenon. The Hall resistivity RH = B/qNS depends only on the quantized charge of
carriers and their density per unit area. It does not depend on the disorder or the sample shape.
Even if one punches holes in a sample the measured parameters q and Ns are the same. This
is not the case for longitudinal resistivity which depends on the impurities, their density and
other details.
This universal character of the Hall effect has shown its full glory in the precision of the
von Klitzing effect. By virtue of a complete freeze out of the kinetic energy of electrons in
the quantizing magnetic field the quantum character of transport shows up as a quantization of
Hall resistivity. The increase in temperature masks quantum effects and returns the classical
behaviour of resistivity, i.e. its proportionality to the magnetic field.
As we have seen, FQHE is observed in very-high-mobility systems. In addition to the
IQHE plateaux, the additional steps appear in ρxy (B) at magnetic fields corresponding to
fractional occupation of Landau levels. Each of these steps signals the formation of a new
quantum liquid. Some of them are of particular interest since they are the simplest members
of a whole family of related states.
In the highest-mobility samples and very intense magnetic fields the interactions and
disorder start to play a comparable role. In this regime the system is an insulator. The
expectation is that its ground state is a pinned Wigner crystal [40]. The two-dimensional
electrons form a crystal which, however, cannot conduct the current by sliding in response to
the electric field because it is pinned by defects.
For a given density NS of electrons in the sample the Hall effect measures the charge of
carriers (electrons or holes). The same is true for IQHE [39] except that quantum mechanics
enters into the discussion and the precision of this measurement is very high. Similarly, in
FQHE, as argued by Laughlin [16], the fractional charge of elementary excitations is measured.
The fractional charges have been directly observed in shot-noise experiments. These
objects also possess fractional statistics, which is related to the topology of the system [41].
The field theory of fractional charge and statistics is a lovely subject intensively studied on
various levels [42]. Students interested in other aspects of these wonderful effects are advised
to consult the original literature or one of the books on the subject [43].
548
K I Wysokiński
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