On Energy References when a Magnetic Field is Applied. A Practical Guide to Reference Energies with and without a Magnetic Field

Abstract

We present a simple analysis of the energy references and of the magnetic-field-dependent and system-dependent features of the Landau energy levels. The energy reference used in describing Landau quantization is usually taken as given, with the Fermi levels of the electron system and the coupled reservoir aligned in thermal equilibrium. Nevertheless, the dependence of the Landau-level spectrum on magnetic field can make the relation between these energy references less transparent. In this work, we show that when a magnetic field is applied and the confined Landau levels $E_n$ are established, both the Fermi level $E_F$ and the bottom energy $E_0$ remain fixed when the magnetic field varies. We show that the bottom energy $E_0$ is independent of $B$, and determined by a system-dependent field $B_1$, defined by the condition that the first Landau level equals the Fermi energy. In contrast, the minimum of the parabolic well depends explicitly on the magnetic field. The resulting picture provides a simple way of displaying the relative motion of the Landau levels and the fixed Fermi level without introducing a magnetic-field-dependent energy zero.