As a result of multiple scattering from impurities, electron wavefunctions in disordered wires are localized, i.e., they are centered around one spot and decay with exponential tails. Usually, in order to model the electronic wavefunction in a lattice model, the effect of impurities is mimicked by a random on-site potential. With a random on-site potential, all states in the wire are localized, irrespective of their energy. For a different kind of disorder, random hopping amplitudes instead of random on-site energies, the randomness leads to a quite different behavior: At the band center, i.e. at zero energy, the wavefunction may become delocalized. But it does not necessarily have to be so. Considering the anomalous localization properties in the presence of random hopping amplitudes, one may consider the following questions: What are the differences with the conventional case at zero energy? What happens if the energy moves away from zero? Are the anomalous localization properties reflected in an anomalous behavior of the density of states? Can we make predictions for two-dimensional systems? Models with random hopping matrix elements are related to a wide variety of other models in statistical mechanics and condensed matter physics, e.g. glasses, spin models (XY and Ising), narrow-gap semiconductors, conjugated polymers, and the random flux model. For more information, see our group's publications on this subject, which can be found here.

Wavefunctions and energy levels in a metal nanoparticle

For a piece of metal with a finite size, energy levels are a finite distance apart, just as one finds for the "particle in a box" problem in quantum mechanics. Once the spacing between the levels becomes bigger than the temperature or inelastic relaxation rates, these single-particle levels can be distinguished. Experiments measuring the positions of single-particle levels in nanometer-size metal grains are performed in the group of Prof. Ralph.
Such a metal grain is a classic example of a mesoscopic system: The details of the wavefunctions and the energy levels depend sensitively on the shape of the grain and the locations of eventual defects and impurities. As these microscopic details are not known in detail a priori for a grain used in an experiment, it makes no sense to make a theory that explains the measurements on a specific sample. Instead, the only way one can compare theory and experiment is through statistical distributions. Such a distribution can be obtained by considering an ensemble of metal grains (which is still difficult experimentally), or by scanning one grain over the available spectrum of energies.

One property that we look at is that of the distribution of g-factors. These describe how an energy level splits in a magnetic field: Without a magnetic field (and for a non-magnetic grain), all energy levels are two-fold degenerate. (This is known as Kramers' degeneracy.) In a magnetic field B, the degeneracy is lifted. Levels form a doublet, with a separation proportional to B. This is shown in the figure above [taken from Salinas et al, Phys. Rev. B 60, 6137 (1999), who measured energy levels versus B using tunneling spectroscopy]. Apart from fundamental constants, the proportionality constant is the so-called g-factor. For free electrons, the g-factor is two. For electrons in a metal grain, g can be different from two if there is spin-orbit coupling since, in that case, eigenstates are no longer pure spin-up or spin-down states, but a superposition of spin-up and spin-down. The g-factors are different from level to level, so one can ask what their distribution is. Also, we can ask how the g-factor depends on the direction of the magnetic field, and whether there are correlations between the g-factors of neighboring levels. You can find more about this problem in these publications.

Transport properties of open quantum dots

A quantum dot is a small semiconductor (or metal) region that is confined by gates. A quantum dot is called open if it is coupled to source and drain reservoirs via contacts that have a conductance of at least a conductance quantum 2e²/h, because in that case the coupling to the outside world strongly perturbs the wavefunctions inside the quantum dot. The effects of interactions on the transport properties (notably the conductance) of an open quantum dot are twofold: First, interactions give rise to dephasing, the breaking of phase coherence, driving the system more towards a classical behavior and suppressing quantum corrections to the conductance. Second, interactions give rise to the "Coulomb blockade", resulting from the fact that a particle that enters the dot charges it, at the cost of a charging energy. In open dots and in closed dots (quantum dots without leads), these two effects are different. Relevant questions are: is dephasing different in open dots and closed dots? What is the dephasing time for open dots? What are the effects of Coulomb blockade at low temperatures? These questions and others are addressed in these publications.
 

The left part of the figure shows a "theorist vision" of a quantum dot. The conducting part of the two-dimensional electron gas is shown yellow, and the metal gates that define the shape and size of the dot are shown hatched. The right part shows a real quantum dot as used in the group of Prof. Marcus (Harvard). In this picture, the two-dimensional electron gas is shown dark grey, while the metal gates are shown lighter. The conducting region of the two-dimensional electron gas is slightly smaller than the dark area; its actual size depends on the voltages on the gates. By varying the voltages on the two most right gates, the shape of the quantum dot can be changed during an experiment. In this way, an ensemble of macroscopically similar dots, but microscopically different dots can be taken. It is one of the main goals of "mesoscopic physics" to study the fluctuations between members of such an ensemble.

In addition to the conductance, quantum dots have been used to 'pump' electrons. In this case, two gate voltages that govern the shape of the dot are varied periodically in time. It has been predicted, and experimentally verified, that such a parametric variation results in a dc current through the quantum dot. From the theoretical point of view, pumping raises several questions, not only for quantum dots, but for electron pumps in general. What are the necessary conditions for pumping? When is it reversible? How important is quantum mechanical phase coherence for the pumping? What is the effect of electron-electron interactions? You can find more about this in our group's publication list.

Quantum transport and its classical limit

The interference of multiply scattered quantum mechanical matter waves is the origin of small but noticeable corrections to the electrical conductance of a metal at low temperatures. Being an interference phenomenon, these corrections are statistical in nature, highly dependent on the precise location of impurities in the metal. Historically, one separates the interference correction to the electrical conductance into 'weak localization', a small negative correction to the conductance averaged over an ensemble of conductors with different impurity configurations, and the 'conductance fluctuations', the sample-to-sample fluctuations measured with respect to the ensemble average.

What is the fate of quantum interference corrections in the limit that the wavelength of the electrons becomes small in comparison to all other relevant length scales of the device? This limit is a 'classical limit', similar to the transition from wave optics to ray optics that occurs once the typical size of optical elements becomes much larger than the wavelength of light. In electronic transport, this situation can be realized in a two-dimensional electron gas, in which the electronic motion is ballistic up to distances of several micrometers. In a two-dimensional electron gas, a nontrivial device geometry can be achieved from the placement of artificial macroscopic scatterers ('antidots'), or from the additional confinement of the electrons using metal gates (e.g., to confine electrons to a 'quantum dot', see above).

A new regime of quantum interference is entered once the so-called Ehrenfest time t_E becomes comparable to the 'dwell time' t_D, the time the electrons spend inside the conductor. The Ehrenfest time combines knowledge of the classical mechanics in the conductor with Planck's constant h: it is the time during which two points in the (classical) phase space, initially a quantum distance ~ h apart diverge under the influence of the chaotic classical dynamics and reach a separation of classical magnitude. The Ehrenfest time is proportional to ln(h), so that it diverges in the classical limit described in the previous paragraph. The dwell time is a classical time scale, however. Hence, the classical limit of quantum transport corresponds to the regime t_E >> t_D.

Whereas some quantum corrections (most notably the weak localization correction to the conductance) disappear in the classical limit, several others survive, albeit in a modified form. Why this happens and how can be found in our group's publications on this subject.