Localization and interaction effects in two dimensional electron systems
Such properties of normal metal as conductivity, spin susceptibility are strongly affected by disorder and electron-electron interactions. The most interesting case is realized in two dimensions. In the absence of spin-orbit interaction and electron-electron interactions the standard theory indicates the absence of metal-insulator transition in two-dimensions such that all electronic states are localized at zero temperature. If the electron-electron interaction is presented only in the singlet channel the theory predicts the absence of transition as well. In the presence of electron-electron interaction in the triplet channel the situation becomes more complicated and there are theoretical estimates in the favor of metal-insulator transition. Moreover, since 1994 there have been found a number of experimental results made on two-dimensional electron system in various types of heterostructures which are in favor of the existence of metal-insulator transition. However, upto now the existence of metal-insulator transition in two-dimensions is under debates.
Literature:
A.M. Finkel’stein, in Electron Liquid in Disordered Conductors, Soviet Scientific Reviews, Vol. 14, edited by I.M. Khalatnikov (Harwood, London, 1990)
Coulomb blockade phenomena in single-electron devices
Over the last two decades theoretical and experimental interest has been devoted to the Coulomb blockade problem which is one of the manifestations of the electron-electron interactions in the mesoscopic systems. One among other possibilities for experimental observation of the Coulomb blockade is a metallic island connected by a tunneling contact to a metallic lead and coupled capacitively to a gate (single electron transistor). At low temperatures it becomes important that only interger number of electrons can appear on the island. The number of electrons is controlled by the gate voltage. At low temperatures the current through single electron transistor is strongly enhanced then the gate voltage is such that the Coulomb energy of the state with N and N+1 electrons on the island are exactly the same. If value of the gate voltage does not satisfy this condition then the current is suppressed. This phenomenon is called as Coulomb blockade. The conductance of a single electron transistor demonstrates periodic dependence of the conductance on the gate voltage.
Literature:
G. Schon and A.D. Zaikin, Phys. Rep. 198, 237 (1990).
Single Charge Tunneling, ed. by H. Grabert and M.H. Devoret (Plenum, New York, 1992).
I. L. Aleiner, P.W. Brouwer, and L. I. Glazman, Phys. Rep. 358, 309 (2002).
Non-equilibrium phenomena in disordered interacting electron systems
The study of quantum electron kinetics in metallic conductors and, in particular, problems related to inelastic electron scattering represent the basic directions of mesoscopics. Most theoretical and experimental research on the quantum transport in mesoscopic systems involves an estimate of the electron inelastic scattering time [or corresponding length scale]. Its comparison with the physical parameters of the system allows us to understand when the concept of electrons with the well-defined phase and energy is relevant for the electron transport description, and when the nonequilibrium distribution function can be approximated by the Fermi-Dirac distribution with an effective electron temperature. The possibility to manipulate the inelastic scattering rates with the external fields, e.g., by a magnetic field, is one of the fundamental questions of electron kinetics. The typical main building block of a mesoscopic device is a diffusive normal metal or a heavily doped semiconductor wire connected to massive electrodes acting as reservoirs. Electrons in the device interact with each other. In addition, electrons are coupled to phonons, electromagnetic environment, and so on. Understanding of transport phenomena in mesoscopic devices is based on the ability to solve the quantum kinetic equation for the electron density matrix (distribution function) in which interactions are taken into account via scattering integrals. The derivation of the scattering integrals is another fundamental problem of electron kinetics. Electron-electron interaction usually provides the strongest mechanism for energy relaxation in metallic conductors at low temperatures which are typical for experiments.
Literature:
E. M. Lifshitz and L. P. Pitaevskii, Course in Theoretical Physics (Pergamon, Oxford, 1981), Vol. 10.
B. L. Altshuler and A. G. Aronov, in Electron-Electron Interactions in Disordered Conductors, edited by A. J. Efros and M. Pollack (Elsevier, Amsterdam, 1985).
Y. Imry, Introduction to Mesoscopic Physics (Oxford University Press, New York, 1997).
F. Giazotto et al., Rev. Mod. Phys. 78, 217 (2006).
Integer and fractional quantum Hall effects
The integral and fractional quantum Hall effects are remarkable and richly complex phenomena of nature with a level of significance that is comparable to that of superconductivity and superfluidity. The robust quantization of the Hall conductance is observed in experiments on two dimensional electron systems at low temperatures (50mK- 4K) and strong perpendicular magnetic fields (1 - 20T). This quantization phenomenon originally came as a complete surprise in Physics and appeared to be in fundamental conflict with the prevailing ideas on electron transport in metals, especially the semiclassical theory that explains the ordinary Hall effect. Although discovered relatively recently the quantum Hall effect has already led to two Nobel prizes in Physics, one in 1985 for the discovery of the integral quantum Hall effect (K. von Klitzing) and one in 1998 for the discovery of the fractional quantum Hall effect (R. Laughlin, H. Stormer, D. Tsui). In spite of the huge number of theoretical and experimental papers that have appeared over the years, and the dramatic progress that has been made, our microscopic understanding of the quantum Hall effect is still far from being complete. From the experimental side, the list of observed but unexplained transport phenomena is still growing and the subject matter, as it now stands, goes well beyond any of the theoretical scenario's that were originally proposed to explain the quantum Hall plateaus. The main reason why a microscopic theory of the quantum Hall effect is complicated is the fact the problem generally lacks a “small parameter''. Any attempt to force the phenomenon in the mold of “mean-field'' theory is doomed to fail. A satisfactory theory of the quantum Hall effect must treat the various different microscopic factors on an equal footing. These factors are i) strong magnetic fields, ii) a random impurity potential and iii) the effects of electron-electron interactions.
Literature:
The Quantum Hall effect, edited by R.E. Prange and S.M. Girvin, (Springer-Verlag, Berlin, 1987)
T. Chakraborty and P. Pietilainen, The Fractional Quantum Hall Effect, (Springer-Verlag, Berlin, 1988).
I.S. Burmistrov Landau Institute for Theoretical Physics 142432 Russia Moscow region Chernogolovka Prosp. acad. Semenova 1a tel. +74957029317 fax +74957029317 e-mail: burmi AT itp.ac.ru http://burmi.itp.ac.ru