Conductivities of a quantum Hall system with a model disorder potential: Influence of electron-phonon interaction on plateau widths

Abstract

We consider noninteracting electrons in a quasi-two-dimensional strip (extended in the $x$ direction) in the presence of a strong perpendicular magnetic field and a constant electric field in the $y$ direction, and subject to a model disorder potential $V(x,y)$ (``toy model'') which allows us to obtain the exact solutions of the time-dependent Schr\"odinger equation in very good approximation. Further, the electrons are coupled to a heat bath (accoustic phonons) at temperature $T$ that modifies the coherent Schr\"odinger time evolution induced by the electric field ${E}_{y}.$ After elimination of macroscopically unobservable fluctuations by averaging over suitable short time intervals, the time evolution that is relevant for the macroscopic current density can be described by a Boltzmann-type equation, which is solved numerically. The steady-state solution allows us to calculate ${\ensuremath{\sigma}}_{\mathrm{yy}}$ and ${\ensuremath{\sigma}}_{\mathrm{xy}}$ as a function of any physical parameter of the system. Here we give results of ${\ensuremath{\sigma}}_{\mathrm{yy}}$ and of ${\ensuremath{\sigma}}_{\mathrm{xy}}$ for all filling factors and as a function of temperature. Quantized Hall plateaus are obtained at the correct values with high precision. Between two plateaus, the Shubnikov--de Haas peak spreads out and its maximum decreases with increasing temperature in qualitative accordance with typical quantum Hall samples. Further, we obtain the remarkable result that for temperatures below 5 K the plateaus of ${\ensuremath{\sigma}}_{\mathrm{xy}}$ become sensibly larger than those of ${\ensuremath{\sigma}}_{\mathrm{yy}}.$ Our analysis shows that this effect results from electron-phonon interaction. (Such a phenomenon has been known experimentally for a long time, but it has seemed unexplained so far.) Our method of calculation could, in principle, be extended to more complex disorder potentials.