Contributing scientists: J. Göres, I. Kukushkin, J. Smet
Introduction
Strongly
interacting many-body problems are pervasive in physics. For
instance, in a cubic centimetre of condensed matter typically 1023
electrons repel each other and interact with a comparable number of
positively charged nuclei. The motion of one electron elicits a
response from all others. No wonder that the search for an exact
solution is usually a desperate undertaking. Ingenious recipes have
been developed, based on quantum field theory, to tackle these
problems systematically: they identify fictitious entities -
quasi-particles - that recast the system of strongly interacting
real bodies into a simpler one, composed of weakly or even
non-interacting bodies, while still capturing the essential physics.
Landau's quasi-electrons, bare electrons dressed with a cloud of
positive charge, are a celebrated example which successfully
describes the behaviour of metals. More recently, composite fermions
- electrons in a different guise - have emerged to account in
single-particle terms for the fractional quantum Hall effect, an
electron-electron correlation phenomenon par excellence.
Quantum Hall Effects
Quantum
Hall effects arise when electrons are constrained to move in a plane
and are exposed to a perpendicular magnetic field. They are quantum
mechanical descendants of a classical effect discovered by Edwin Hall
more than a century ago. He observed that a current-carrying
conductor in the presence of a field develops a voltage perpendicular
to both the current flow and the field. Ever since, a measurement of
the Hall voltage has been a valuable characterization method in solid
state physics, because it reveals the number as well as the sign of
current-carrying charges. Its application to clean, near-perfect
two-dimensional conductors at low temperatures brought a new twist.
Here, the Hall voltage does not simply rise linearly with applied
field. Instead, it shows plateaux as if the Hall voltage is frozen
near specific field values. An example of a Hall measurement is shown
in Fig. 1. Across the plateaux, the voltage drop in the direction of
the current flow vanishes; this is the second hallmark of the quantum
Hall effects.
Fig.
1: The integer and fractional quantum Hall effects. The longitudinal
resistivity measured along the direction of the current flow vanishes
whenever an integer number of Landau levels are completely filled (ν
= 1,2,3,...). This integer quantum Hall effect is accompanied by
plateaux in the Hall resistivity and can be understood in terms of
single particle physics. The fractional quantum Hall effect occurs
mainly when only the lowest Landau level is occupied at higher
magnetic fields for rational values of the filling ν
of the form p/q, where p and q are mutual primes. Despite the
phenomenological similarity to the integer quantum effect, the
fractional quantum Hall effect has a very different microscopic
origin. It can only be accounted for by considering the Coulomb
interaction among the electrons. By introducing suitable
quasi-particles, referred to as composite fermions, it is possible to
describe the fractional quantum Hall effect in single particle terms.
The successive depopulation of composite fermion Landau levels
produces an integer quantum Hall effect of composite fermions and is
equivalent to the fractional quantum Hall effect of the original
electrons.
The
magnetic field sends the electrons into circular orbits. Classically,
any radius is allowed. Quantum mechanics, however, dictates discrete
values of the radius, much as it imposes distinct Bohr orbits on an
atom. It is an outcome of the discrete character of magnetic flux:
the applied field derives from a large collection of flux quanta,
each contributing the smallest unit of magnetic flux to the total
value. According to the laws of quantum mechanics, only electron
orbits that enclose exactly one such quantum or multiple quanta of
magnetic flux are legitimate. Like the Bohr orbits, each of these
orbits has a discrete energy associated with it - a Landau level.
At fixed field, the larger the radius of an orbit, the higher its
energy. The electrons are distributed among the orbits or Landau
levels so as to minimize the total energy, keeping in mind that each
orbit only fits one electron. However, many orbits of equal size (and
thus energy) are spread throughout the sample. To be precise, each
Landau level can accommodate as many electrons as the total number of
flux quanta that thread the sample. As the field is raised and the
number of flux quanta increased, Landau levels can take up ever more
electrons and levels with higher energy are successively depopulated.
The filling factor ν
denotes the number of filled Landau levels. When ν takes
on an integer value, the system will initially resist the addition of
an extra electron, as it has to broach a new Landau level with higher
energy. The system is said to be incompressible and this
incompressibility is at the heart of the integer quantum Hall effect.
Its
cousin, the fractional quantum Hall effect, ensues mainly at higher
fields when one Landau level is partially occupied and the filling
takes on a fraction that can be expressed as a ratio of integers, ν
= p/q, with p and q integers. Despite its
experimental resemblance, it cannot be accounted for directly in the
above picture, considering only the motion of a single electron.
After all, if the one occupied level is only partially filled; why
the incompressibility? Early on, it was recognized that all electrons
must participate to bring about this effect. At many of these
fractional fillings, electrons apparently succeed in becoming
arranged within the Landau level so as to significantly reduce their
mutual repulsion. As we are typically dealing with as many as 1010
electrons, all of which interact with each other, developing an
intuitive understanding of this effect seems hopeless.
Composite Fermions in a nutshell
Much
later after the discovery this effect, it was however suggested that there may be no need to track all electrons to understand this
phenomenon. At high fields, compound particles come onto the scene,
each assembled from an electron and two flux quanta (or
more generally, an even number of them) as depicted in a cartoon-like
fashion in Fig. 1. This bond between electrons and flux quanta turns
out to be a natural way for electrons to avoid each other and the
resulting quasi-particles, named composite fermions, may for many
purposes be viewed as non-interacting. They too are forced by a field
onto circular orbits, which must obey the laws of quantum mechanics.
But unlike electrons, they experience only an effective field,
greatly reduced from the applied field by an amount equal to the
field produced by all the flux quanta of their fellow composite
fermions. So, the effective field is zero when two flux quanta thread
the sample per electron in the two-dimensional electron system (see
the effective field axis in Fig. 1). At non-zero effective field, the
discrete orbits again have Landau levels associated with them.
Filling these gives the integer quantum Hall effect for composite
fermions. Sure enough, this integer quantum Hall effect of composite
fermions occurs precisely at those external applied fields where the
fractional quantum Hall effect is routinely observed. This
appealing analogy between electrons and composite fermions is
summarized in Fig. 2.
Fig.2:
Analogy between electrons and composite fermions. At zero magnetic
field, electrons form a metallic state and hence the excitation
spectrum is gapless. The Fermi surface is circular and the Fermi
wavenumber kF is determined by the electron density n.
Each state in k-space is doubly degenerate due to the spin degree of
freedom. In a non-zero field, electrons execute circular orbits with
a radius Rc. This radius is determined by kF
i.e. the density and shrinks with increasing field. Analogously a
metallic state of composite fermions with a well-defined Fermi
surface ensues when the lowest Landau level is half filled ( ν
equals ½). The Fermi wavenumber of this metallic state is
identical to that of the electron metallic state at zero external
field apart from a factor √2 ,
since the spin degeneracy at the large external fields where
composite fermions are brought alive has been lifted. Composite
fermions do not experience the external field, but the drastically
reduced effective field. When moving away from half filling, the
effective field becomes non-zero and composite fermions are send into
circular orbits much the same way as electrons but the radius of the
orbit is controlled by the effective field instead of the external
applied field.
Experimental proof for the existence of composite fermions
This
analogy has served as a catalyst for experimentalists to dream up
experiments seeking to confirm the existence of composite fermions.
Examples of such experiments are schematically illustrated in Fig. 3.
Geometries previously explored for the study of ballistic transport
phenomena of electrons at small external magnetic fields were
deployed in pursuit of composite fermions. Instead of launching
electrons, composite fermions are ejected from an orifice and bent
into an adjacent slit arranged along a common boundary in the hope to
uncover replica of the transverse magnetic focusing signals for
electrons near zero external magnetic field in the vicinity of the
field where composite fermions emerge. The discovery of
commensurability oscillations of composite fermions in periodic
systems would also strengthen the case for composite fermions and
would support the claim that the relevant length scale of motion is
set by the composite fermion cyclotron radius and not the electron
cyclotron radius, which is at least an order of magnitude smaller in
this magnetic field regime.
Fig.
3: Examples of experimental geometries to proof the existence of
composite fermions. Electrons and composite fermions both move on
circular orbits. The size of the orbits is however determined by
different fields. At high fields, where composite fermions appear,
the electron orbits typically have shrunk to a size of less than 10
nm. The bottom left panel illustrates a magnetic focusing experiment.
Electrons or composite fermions are injected from a quantum point
contact. For the proper value and sign of the external field or the
effective magnetic field, electrons or composite fermions should
enter the second opening placed at a distance of a micron or less.
This current flow can be detected and is turned off when moving to
different fields. The composite fermion theory predicts that current
flow occurs at the same values of the effective field (apart from a
√2
due to the lifted spin degeneracy) as those external applied fields
where one observes magnetic focusing of electrons. The right panel
illustrates a commensurability experiment. It too as well as
variations thereof have served as valuable geometries to proof the
existence of composite fermions. The basic principle is explained in
the text.
Such
a commensurability experiment is illustrated in the right panel of
Fig. 3 and in Fig. 4. A periodic lattice of voids (or so-called
antidots), where the two-dimensional electron system is entirely
depleted, is patterned on top of the sample with electron beam
lithography. A constant current is passed through the sample form the
source to the drain contact. According to a simple Drude model, the
resistivity of the two-dimensional electron system measured along the
direction of the current flow (ρ)
is inversely proportional to the number of charge carriers. When we
turn on the magnetic field, the charge carriers start to move on
circular orbits. When the orbit fits precisely around one antidot or
a set of antidots, a number of charge carriers will be trapped. These
carriers no longer contribute to the current flow. The number of
current carrying particles is reduced and hence the resistance
develops a peak as a larger voltage has to be applied to drive the
same current through the sample. The resulting resistance peaks at
small fields are illustrated in the right panel of Fig. 4. If
composite fermions exist, we anticipate replica of such resistance
peaks near filling factor ν = ½ .
Such replica are indeed observed in experiment. From their position,
one may verify the prediction that the Fermi wavenumber of composite
fermions is a √2
larger than that of the electron Fermi sea. To this
date, it is not possible to account for these resistance features at
high fields without invoking composite fermions.
Fig.
4: Antidot experiment. Right panel: At certain magnetic fields the
electron cyclotron orbit fits exactly around one or a number of
antidots. Many carriers are trapped around these antidots and hence
no longer contribute to current flow between the source and the drain
contacts. As a result the resistance of the sample develops a maxima
at these magnetic fields. The same experiment can be repeated at high
magnetic field values, where composite fermions are expected to come
onto the scene. Indeed similar features are observed. Composite
fermions are more fragile and their mean free path is much smaller
than that of electrons and so only the fundamental peak for
encircling a single antidot is observed in experiment.
Other properties of composite fermions
The
previous experiments confirm that composite fermions precess, like
electrons, along circular cyclotron orbits, with a diameter
determined by the effective field, rather
than the external applied field. These experiments owe much of their
persuasiveness to their independence on the poorly known composite
fermion dispersion. They only rely on the existence of a Fermi
surface with the predicted Fermi wavenumber. Unraveling the energy
dispersion of composite fermions turned out far more challenging.
Indeed, what is the mass of composite fermions? With what frequency
do they execute the cyclotron orbit? For electrons this cyclotron
frequency is determined by the well-known conduction band mass of the
underlying GaAs crystal, in which the two-dimensional electron system
forms. It is also relatively straightforward to detect the cyclotron
frequency. When microwave radiation with the same frequency is
incident on the sample, electrons are accelerated and eventually give
up the acquired energy by heating up the sample. This resonant
heating can be detected in absorption experiments.
The
effective mass of composite fermions is no longer related to the band
mass of the original electrons, but is entirely generated from
electron-electron interactions and, hence, difficult to predict. The
search of the composite fermion cyclotron resonance requires
substantial sophistication over conventional methods, used to detect
the electron cyclotron resonance, since Kohn's theorem must be
outwitted. This theorem states that in a translationally invariant
system, homogeneous radiation can only couple to the center-of-mass
coordinate and can not excite other internal degrees of freedom.
Phenomena originating from electron-electron interactions will thus
not be reflected in the absorption spectrum. An elegant way to bypass
this theorem is to impose a periodic density modulation to break
translational invariance. The non-zero wavevectors defined by the
appropriately chosen modulation may then offer access to the
cyclotron transitions of composite fermions, even though they are
likely to remain very weak. Therefore, the development of a detection
scheme, that boosts the sensitivity to resonant microwave absorption
by several orders of magnitude in comparison with traditional
techniques, is a prerequisite for these studies.
Fig.
5:Optical setup for the detection of the cyclotron resonance of
composite fermions. The photoluminescence is detected in the presence
of quasi-monochromatic microwave radiation with the help of a
spectrometer and CCD camera. The spectrum under microwave radiation
is compared with the spectrum if no high frequency radiation is
incident on the sample. Resonant and non-resonant heating processes
produces differences between both spectra, that are best brought out
be integrating the absolute value of difference spectrum over the
entire optical wavelength range. The resulting integrated value is
referred to as the absorption intensity and is plotted as a function
of magnetic field for two different electron densities on the right.
Near the integer and fractional quantum Hall states ν = 1 and ν = 1/3 non-resonant heating takes place and produces peaks that are
largely insensitive to the frequency of the microwave radiation. They
are not of interest here. In the vicinity of filling factor n = ½,
two symmetrically arranged peaks appear, whose separation does scale
with the microwave frequency. We assert that these originate from the
cyclotron resonance of composite fermions. The cyclotron resonance of
electrons is also apparent in these graphs at very low field.
Fig.
5 illustrates results from an optical technique that delivers the
required sensitivity. Apart from the strong cyclotron resonance
signal of electrons at low B-field, several peaks, that scale
with a variation of the density, emerge near filling 1, 1/2 and 1/3.
Those peak positions associated with ν
= 1 and 1/3 remain fixed when tuning the microwave frequency and are
ascribed to heating induced by non-resonant absorption of microwave
power. In contrast, the weak maxima surrounding filling 1/2 readily
respond to a change in frequency as illustrated in Fig. 6. We assert
that this resonance is the long searched for cyclotron resonance of
composite fermions. By repeating the same experiment at a number of
different frequencies, we can extract the mass of composite fermions.
It is far heavier than that of the original electrons and also
depends on the carrier density.
Fig. 6:The absorption intensity near filling factor ν
= 1/2 measured in photoluminescence exeriments for three different
frequencies. The locations of the composite fermion cyclotron
resonance features scale linearly with microwave frequency and
suggest a composite fermion effective mass more than 15 times larger
than the electron effective mass in GaAs for the chosen density of
1.1 x 1015/m2
Outlook
Even
though, we have come a long way in deciphering the properties of
composite fermions, some important questions remain. So far, the main
thrust has been on ballistic transport experiments exploring the
particle like behavior of composite fermions. What about the particle
wave duality. What is the phase coherence length of composite
fermions and is it possible to carry out an Aharonov Bohm like
experiment for composite fermions? These are some intriguing
questions that we wish to address in the future.
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