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It is well known that the properties of a collection of electrons change significantly when their motion is restricted to fewer than three dimensions. For example, electrons confined to two dimensions in a semiconductor heterostructure exhibit the quantum Hall effects (in which the Hall resistance takes quantized values) upon application of a strong magnetic field perpendicular to their plane. And the conductance of electrons in one dimension, which can be experimentally realized in very narrow wires, is found to be quantized in multiples of 2e²/h.

Collections of other fermions might also be expected to exhibit these effects. This is found to be the case: the integral quantum Hall effect in a two dimensional hole gas (2DHG) was reported three years after the discovery of that effect in electrons. However, holes are useful for more than verifying effects observed in electron gases. This is because holes occupy the valence band of the crystal, and therefore have a different effective mass and angular momentum properties.

The effective mass of a hole is not well defined, because the spin-orbit coupling and degeneracy of the valence band combine to produce non-parabolic bands. Nevertheless, it is still true that the effective mass is about four times larger than that of electrons in GaAs. This property means that the kinetic energy of a hole gas is much smaller than that of an electron gas of the same density. This has important consequences for the many-body properties of hole gases.

The interaction terms of the many-body hamiltonian are just the inter-particle Coulomb interactions, which do not depend upon the particle mass. The importance of the interactions can be expressed as a fraction of kinetic and interaction energies: clearly, interactions are far more important in the hole gas than in the electron gas.

The 2DHGs studied in this group are fabricated in the same way as the two dimensional electron gas (2DEG) - using Molecular Beam Epitaxy to grow GaAs/AlGaAs heterostructures. The holes are introduced into the system by modulaton-doping the crystal with a p-dopant: this is achieved using Si which, when deposited onto an appropriate crystal facet, acts as an acceptor. Research in this group benefits from having the facilities and know-how to produce some of the highest quality hole gases in the world.

Current Experiments

Work to date has focussed on four main areas, and is on-going: (i) The properties of holes in less than two dimensions; (ii) The properties of two-dimensional hole gases in tilted magnetic fields; (iii) Metal-insulator transitions in the 2DHG at zero magnetic field; (iv) Bilayer holes.

In the first category, this group was the first to demonstrate conductance quantization in the one-dimensional hole gas. An example of real data is displayed above (a); it was obtained at 50 mK in a dilution refrigerator and shows that as the width of the wire is reduced, the conductance G drops in steps of 2e²/h. In parallel magnetic fields, we have observed the multiple crossing of one-dimensional hole subbands, and have determined the 1D hole g factors. The crossing is shown above in part (b), as a greyscale. We have also demonstrated the Aharonov-Bohm effect and the focussing of beams of ballistic holes.

In the second category, we have evidence of the importance of the exchange interaction in hole gases in the quantum Hall effect regime. Whilst the formation of Landau levels depends on the component B perpendicular to the plane of the 2DHG, the angular momenta couple to all components. Thus, by tilting the plane of the sample with respect to the field, we can independently vary the Zeeman and cyclotron energies. Very recent experiments reveal many discrepancies between simple theories and the data; it is thought that many-body processes causing phase-transitions might explain these. The figure above shows how the energy gap varies with total B for either 4 (a) or 5 (b) occupied Landau levels in a 2DHG. The characteristic features of the figure are that in (a) the data are roughly linear upto a turning point (TP), which occurs at non-zero energy gap. In contrast, in (b) there is clear curvature in the vicinity of the TP. We argue that in the case of four filled Landau levels, the system rearranges itself (at the TP in the figure) to lower its total energy before the energy gap to single-particle excitations reaches zero.