Theory of electron-phonon interaction in the quantum Hall regime

The quantum Hall effect (QHE) is the most fascinating physical phenomenon in low-dimensional systems, and its discovery resulted in the awarding of two Nobel Prizes. Despite extensive worldwide research in this field there are still many unanswered questions related to both fractional and integer quantum Hall effects. Being an essentially quantum phenomenon, the QHE is observed in two-dimensional systems at extremely low temperatures and high magnetic fields. However, there are several unanswered questions relating to the disappearance of the QHE with increasing temperature, decreasing magnetic field, and increasing width of the quasi two-dimensional electron layer. Under these conditions the interaction between electrons and crystal lattice vibrations (phonons), which is usually suppressed in the quantum Hall regime, becomes significant. This attracted our attention to the electron-phonon interaction in the QHE regime and suggested a set of problems to be addressed.

Firstly, we considered electron-phonon interaction in two-subband quasi-two-dimensional systems in quantising magnetic fields. In such systems it is possible to tune the energy separation between Landau levels corresponding to different size-quantisation subbands. There is a strong enhancement of electron-phonon interaction when the inter-level separation is close to the energy of the acoustic phonon with wave-vector of the order of the inverse magnetic length. We carefully studied this resonance phenomenon and predicted Rabi-like oscillations of electron populations in the presence of non-equilibrium phonons [21]. We have also shown the essence of many-electron effects in the same system, and have created a theory of phonon magnetospectroscopy of many-electron few-subband systems at different filling factors. The opening of many-body gaps at odd filling factors [29] and the prospect of phonon spectroscopy of quantum Hall ferromagnets at even filling factors [24] were both considered.

Another problem we considered was related to phonon-assisted recombination of both inter-band and intra-band magnetoexcitons [26,32]. For inter-band excitons the photoluminescence has a single slightly asymmetric line with the width saturating with increasing temperature. This linewidth saturation results from the suppression of the absorption or emission of phonons with momenta exceeding the inverse magnetic length. For intra-band excitons, we predicted a double peak of the emission line when the electron filling factor is odd, and greater than or equal to three. In the latter case the lowest magnetoexciton dispersion curve has a minimum at non-zero momentum. Then, the higher-energy peak results from the direct optical emission of zero-momentum excitons. The origin of the lower-energy peak is the phonon-assisted transition from the non-zero momentum exciton states. With increasing temperature, the higher-energy peak becomes more pronounced and the lower-energy peak vanishes. With increasing well width the separation between the lines becomes smaller, and the temperature at which the lower-energy peak can be observed is decreased. Finally, we have predicted that direct cyclotron absorption in this system at very low temperature results in the generation of non-equilibrium phonons, which becomes suppressed with increasing temperature.

Phonon absorption spectroscopy has recently been applied to probe magnetoroton dispersion in the fractional quantum Hall regime. Whilst the mechanism of magnetoroton creation by absorption of a non-equilibrium phonon was well understood, the mechanism of magnetoroton dissociation into unbound quasiparticles remained a mystery. A single-phonon process could not dissociate a magnetoroton due to restrictions imposed by the energy and momentum conservation laws. However, dissociation is necessary to produce a signal in the dissipative conductivity, which is measured in phonon absorption spectroscopy experiments. We have resolved this puzzle and calculated the rate of magnetoroton dissociation for different fractional filling factors [27,32]. We have shown that the phonon-pulse-induced dissociation of the magnetoroton occurs as a second-order process in the electron-phonon interaction. Depending on the temperature of the two-dimensional electron gas, the magnetoroton decay can be considered either as a direct two-phonon dissociation or as the result of the two-phonon diffusion of a quasiexciton in momentum space. This part of our research provides the basis for our planned future work on magnetoroton-assisted recombination of few-particle anyon excitons in the fractional quantum Hall regime.

Our work on electron-phonon interaction in the quantum Hall regime was supported by a two-year EPSRC research grant, which resulted in eight journal publications and several conference presentations. A number of other papers are currently in preparation. The projects final report received an excellent response from both referees and the panel. One of the objectives of this project was to explain puzzling temperature dependence of the high-current breakdown of the quantum Hall effect observed at Exeter University by the experimental team of Dr Alan Usher. We found out that this dependence resulted from a non-uniform distribution of induced charges [36,41,43] and had nothing to do with electron-phonon interaction. This work led to a very close and fruitful on-going collaboration between our theoretical team and one of the School's most successful experimental groups.