Abstract
A generalized semiclassical quantization condition for cyclotron orbits was recently proposed by Gao and Niu , that goes beyond the Onsager relation . In addition to the integrated density of states, it formally involves magnetic response functions of all orders in the magnetic field. In particular, up to second order, it requires the knowledge of the spontaneous magnetization and the magnetic susceptibility, as was early anticipated by Roth . We study three applications of this relation focusing on two-dimensional electrons. First, we obtain magnetic response functions from Landau levels. Second we obtain Landau levels from response functions. Third we study magnetic oscillations in metals and propose a proper way to analyze Landau plots (i.e. the oscillation index nn as a function of the inverse magnetic field 1/B1/B) in order to extract quantities such as a zero-field phase-shift. Whereas the frequency of 1/B1/B-oscillations depends on the zero-field energy spectrum, the zero-field phase-shift depends on the geometry of the cell-periodic Bloch states via two contributions: the Berry phase and the average orbital magnetic moment on the Fermi surface. We also quantify deviations from linearity in Landau plots (i.e. aperiodic magnetic oscillations), as recently measured in surface states of three-dimensional topological insulators and emphasized by Wright and McKenzie .
Highlights
The quantization of closed cyclotron orbits for Bloch electrons in the presence of a magnetic field leads to the formation of Landau levels (LLs) [5]
When describing surface states of topological insulators, the Rashba model is only valid for small wave-vectors and only the inner circle should be considered (in Figure 4(a) or (b), it means that only the full line should be considered and not the dashed one)
The quantization condition recently proposed by Gao and Niu is a powerful generalization of the Onsager relation
Summary
The quantization of closed cyclotron orbits for Bloch electrons in the presence of a magnetic field leads to the formation of Landau levels (LLs) [5]. In a recent insightful paper, Gao and Niu [1] proposed a further extension by systematically including higher-order corrections in the magnetic field in a compact and thermodynamic manner Their equation generalizes Onsager’s relation [2]. 3) The phase of magnetic oscillations (related to γ(ε, B)), such as Shubnikov-de Haas oscillations in the longitudinal resistance or de Haas-van Alphen oscillations in the magnetization, can be derived from the Roth-Gao-Niu relation This helps to analyze Landau plots (i.e. index n of oscillations as a function of the inverse magnetic field 1/B). We first review the Roth-Gao-Niu quantization condition and its validity (section 2), present the three type of consequences: from LLs to response functions.
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