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1. Introduction
The study of the conductance of electrons belongs to the very
heart of condensed matter physics. Traditionally the quantum theory of
electronic conductivity was built on the picture of an electron being
multiply scattered by impurities and diffusing through the solid. A
cardinal concept in the description of the diffusion of the electron is
the mean free path, the average length the electron travels before it
suffers a collision. The electron executes a zigzag motion and the mean
free path is the average zigzag length. The appearance of strong
multiple scattering correlates with a very short mean free path. The
Ohmic electronic conductivity is direcly proportional to the mean free
path and the wave number at the Fermi surface.
2. Localization transition
Will any increase of the degree of disorder lead to just a decrease of
the mean free path and thus of the conductivity, or is it possible that
some unexpected anomaly could develop on the way, with perhaps a
mimimal attainable conductivity? This issue was raised and solved in an
absolutely brilliant way by Philip Anderson (in 1958, at that time at
Bell Labs). He conceived the idea of electron localization: Beyond a
critical amount of impurity scattering the diffusive motion of the
electron will come to a halt. Later work by Mott, Ioffe and Regel
predicted the transition to occur when the mean free path becomes
smaller than the wavelength. There is not much to wave anymore for a
wave when its mean free path has become shorter than its De Broglie
wave length. This stopping or localization has dramatic consequences
for the conductivity - the material turns into an insulator, exactly
the opposite to what happens in superconductivity. Anderson
localization has become an industry in solid state physics. These days
the number of papers on electron localization that have appeared -
mainly theoretical - runs into the thousands. At the same time,
experimental observations are sparse and covered with disputes and
controversies. The omnipresence of electron-electron interactions makes
it very difficult to observe the effect in Rheinkultur, since repulsive
interactions lead to another kind of a localization, called Mott
localization. The best experimental support for electron localization
is perhaps the observation of variable range hopping - a very special
temperature dependence of the conductivity explained by thermally
activated localized electrons - and the observation of the quantum Hall
effect.
3. Minimum conductivity
The concept of a mimimum conductivity, devised by Nevill Mott in the
sixties and supported by him until his death, was an idea that was
explicitly mentioned by the Nobel Committee in 1977 when sharing the
Nobel Prize with Philip Anderson and John van Vleck. Yet it conflicts
with the celebrated scaling theory of localization published by the
'gang of four' (Abrahams, Licciardello, Ramakrishnan and Anderson
himself) at the end of the seventies. The concept of a mimimum
conductivity is now widely believed to be incorrect: the Anderson
metal-insulator transition (MIT) is believed to be continuous. The main
support of this idea comes from exact numercial solutions of the
Anderson tight binding model (Schreiber, MacKinnon, Kramer).
4. Interactions
In the localization problem beyond the single-electron picture,
there are two main issues. First, one can study the effect of interactions
on the single-particle states. Here the most striking effect is the Coulomb
gap - a drastic suppression of the single-particle density of states near
the Fermi level, predicted by Efros and Shklovskii in (1975).
Further developments in this direction led to the concept of a Coulomb
glass. Whether electron-electron
interaction in the absence phonons is still able to produce analogous hopping
conduction, remained a puzzle for some time. On the one hand, Fleishman
and Anderson (1980) have shown that Mott's arguments
could not be applied at sufficiently low temperatures unless the interaction
is sufficiently long-range. On the other, Altshuler, Aronov and Khmelnitskii
have shown that at sufficiently high temperatures
electron-electron interactions lead to dephasing which completely destroyes
localization of the single-particle wave functions. Recent work by
Basko, Aleiner and Altshuler
analyzed the perturbation theory in electron-electron interaction to all orders
and showed that the electron-electron interaction alone is unable to produce
any hopping conduction unless the temperature exceeds some critical value.
At this critical temperature a metal-insulator transition occurs, which may also
be viewed as an Anderson transition between localization and delocalization
of many-electron states in the many-body Fock space.
In a 1D lattice with repulsive interactions small enough to have no gap,
another issue exists. It is known that small
disorder localizes the ground state.
An issue first raised by Shepelyansky is whether interactions
can lead to a localization length of the two-particles states that is much
larger than the one for the single-particle states.
Numerical work by Schreiber does not seem to confirm this.
5. Mathematical definition
So what exactly is localization? Many refer to the 'vanishing
of diffusion' put forward by Anderson in his 1958 article, and thus to
the vanishing of conductivity'. For those who have really read this
article in detail know that Anderson localization is much more than
that. For Anderson localization to occur almost all single electronic
states must become localized by the disorder, and one should think in
terms of bound states, rather than the familiar extended Bloch states,
to describe electron propagation. Clearly any statement about the
underlying Hamiltonian is a much stronger statement than one addressing
a macroscopic observable such as the diffusion constant. In an
unbounded localized medium a finite number of bound states per unite
volume exists, and the spectrum of the Hamiltonian no longer takes the
familiar, continuous spectrum associated with plane or Bloch waves, but
rather becomes dense pointlike. The Hamiltonian has eigenvalues that
are infinitely close to each other, and with exponentially localized
eigenfunctions.
6. Thouless criterion
The above mathematical definition is undoubtedly the best definition of Anderson
localization, but it is not of much use to an experiment. Clearly,
unbounded media do not exist, and eigenvalues or even eigenfunctions
are rarely measured in condensed matter. What thus happens in finite
open media, where one can measure transmission and conductance? This
issue was raised by D.J. Thouless in a work that is by most colleagues
considered as the best contribution to localization after its
discovery. In a finite open medium, eigenvalues are repelled by the
quantization conditions. At the same time they achieve a finite width
due to leakage through the boundaries. Thouless reasoned that Anderson
localization sets in when the level width is smaller than the level
separation. The dimensionless ratio of both energies - universal
notation g - is now known as the dimensionless Thouless conductance. It
can be shown that it is directly equal to (e^2/h times) the Ohmic
conductance. This is the desired link between spectrum and transport.
The Thouless conductance features in the scaling theory of localization
that predicts how it is to vary with the size of the medium. It is now
known that the dimensionless conductance governs all aspects of
localization, including statistical fluctuations. The inequality 'g
< 1' is the celebrated Thouless criterion for Anderson localization.
Quite surprisingly, in one and two dimensional random media this
criterion can be reached for any degree of disorder by just increasing
the size of the medium. Only in higher dimensions a real critical point
exists associated with a continuous phase transction.
7. Progress in theory
New advances were also made on the theoretical side. Random matrix
theory (Carlo Beenakker, Pier Mello, Narendra Kumar..) now provides a
complete, nonperturbative describtion of localization in quasi
one-dimensional, open random media, including statistical fluctuations.
One of the most successful theories developed for electron localization
(Dieter Vollhardt and Peter Wölfle) was adapted to deal with open
boundaries and finite systems (Sergey Skipetrov and Bart van Tiggelen).
In this theory the diffusion constant becomes a function of position.
This is more precise than the paradigm that near localization the
transport properties become scale dependent. The theory is seen to be
consistent with the time-dependent diffusion observed with microwaves,
light and ultrasound.
8. Classical waves
The idea that localization could be observed with classical
waves started to emerge in the beginning of the 1980's, basically as a
result of two important developments in electron localization: (1)
Anderson localization was phrased in a different language: it was more
seen as an interference effect in multiple scattering, and (2) for
electrons a new phenomenon was observed, which was called weak
localization. Sajeev John (at that time at Princeton) had already
published a paper on localization of light, and an unpublished paper by
Philip Anderson (referred to as the theory of white paint) had already
been circulating among some theoreticians. Unfortunately, it is quite
difficult to find media in which one can get short mean free paths for
classical waves. Nature puts some serious constraints on the
experimentalist. One cannot just make the volume fraction of scatterers
larger and larger, in order to increase the amount of scattering.
Beyond 50 % the waves start to scatter from the interstices rather than
from the matter and this is ineffective. Furthermore there are some
fundamental constraints on the cross-section. Unlike for electrons,
classical waves have a small cross-section at small frequencies. As for
electrons the scattering cross-section can never exceed the wavelength
squared. Finally classical waves suffer from absorption which we want
to avoid. An important parameter under control in an experiment is the
contrast in index of refraction. Titaniumdioxide has an index of
refraction of 2.7 (in the visible) but this contrast was seen not to be
enough to reach localization.
9. Experiments
It was realized by Sajeev John that the advantages that
electrons have to localize at small energies - that is close to a band
edge - also show up for photons travelling in disordered photonic
crystals. Since then the quest for three dimensional photonic crystals
was often justfied to observe Anderson localization. Diederik Wiersma
et al. reported experiments in the near-infrared in GaAs powders. This
frequency is in the electronic gap of the semiconductor which implies
small optical absorption. The index of refraction is there 3.5, and
apparently enough to achieve localization. Near the Anderson transition
the light transmission should vary with the inverse of the square of
the sample thickness and in the localized regime the transmission
should fall exponentially with length. Again, the observation was
contested. Georg Maret (now at Konstanz, Germany) and coworkers argued
that the results on GaAs could well be interpreted on the basis of
classical diffusion theory if small absorption would be included. Azi
Genack (CUNY) has been the pioneer in the field of localization of
microwaves. One of the advantages of microwaves over optical
experiments is the fact that microwave detectors can routinely be used
to detect both amplitude and phase of the electric field. In optics
this requires sophisticated hetero-dyne techniques. Genack
emphasized that localization experiments should focuss on statistical
fluctuations, in addition to the more conventional transport
properties. These fluctuations would make it easier to discriminate
between localization and absorption. In the localized regime,
fluctuations are nongaussian and large, inversely proportional to the
Thouless conductance. The localization of microwaves also revealed a
time dependent diffusion constant if one insists on modeling the
transport with familiar diffusion theory. Anomalous time-dependent
diffusion in transmission is also seen in recent experiments with
visible light carried out in the group of Georg Maret, and with elastic
waves in the group of John Page (Winnipeg). In the localized regime one
expects the diffusion constant to be inversely proportional to time.
The Haifa group ( Schwartz et al) recently made the first report
transverse localization.This phenomenon was predicted by the Amsterdam
group (De Vries, Lagendijk) some 15 years ago in three-dimensional
media with one homogeneous dimension along which the propagation is
ballistic. In the remaining two dimensions the waves are seen to
localize in space.
10. Cold atoms
Final recent developments come from the cold atom community.
Since the discovery of Bose-Einstein in 1995 cold atoms constitute a
new tool to study hot topics in condensed matter physics, among which
clearly Anderson localization. After turning off the trap in which the
atoms were cooled, the atoms see potential barriers proportional to the
light intensity. Two Nature papers - one from the group of Orsay (Alain
Aspect and Philippe Bouyer), and one from the Florence group of Massimo
Inguscio - report on Anderson localization of cold atoms in one
dimension. A number of issues of fundamental importance is and will be
raised by these studies. First, the role of interactions can be
addressed, which can be fine-tuned externally when working with cold
atoms. The influence of interactions on localization is still not fully
understood and conflicting theoretical predictions exist. Secondly, the
speckle potential has long-range correlations and theoretical studies
suggest that even in one-dimension a mobility edge might exist. Of
course, the three-dimensional localization of cold atoms is high on the
agenda of several teams. One challenge is to measure the critical
exponents of the Anderson transition.
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