Thermodynamic properties of a two-dimensional electron gas in the presence of a weak spatially modulated periodic potential

Abstract

A systematic investigation on the influence of an additional periodic modulation potential which is weak, either electric or magnetic in nature, and spatially modulated along one dimension, on the equilibrium thermodynamic properties of a two-dimensional electron gas in an externally applied magnetic field is presented. The application of such an additional modulation potential results in a broadening of the Landau level energy spectrum into bands whose widths oscillate as a function of the externally applied magnetic field. Such oscillations are found to reflect the commensurability of the two different length scales present in the system, namely the cyclotron diameter at the Fermi level and the period of the modulation. We show that such commensurability effects are also to be found in all thermodynamic quantities of the system. They appear at low magnetic fields as an amplitude modulation of the well-known de Haas-van Alphen-type oscillations, familiar from the homogeneous two-dimensional electron gas system in an external magnetic field, which may or may not be resolved depending on temperature and are only weakly dependent on temperature. Their origin lies in the oscillations occurring in the bandwidths and they are consequently completely different in origin from the usual de Haas-van Alphen-type oscillations. In particular, we show that commensurability oscillations are to be found in the chemical potential, Helmholtz free energy, internal energy, electronic entropy, electronic specific heat, orbital magnetization and orbital magnetic susceptibility of such weakly modulated systems. We find that the resulting commensurability oscillations in each thermodynamic function exhibit well-defined phase relations between the electric and magnetic modulations except in the case of the orbital magnetization and the orbital magnetic susceptibility. Explicit asymptotic expressions for the chemical potential, Helmholtz free energy and orbital magnetization, in the quasi-classical limit of small magnetic fields and small but finite temperatures, are also given.