An Approach to the Quantum Hall Effect in Three- Dimensional Electron Systems

Abstract

We present a theoretical framework to describe the integer quantum Hall effect (IQHE) in three-dimensional (3D) electron systems. This extends our previous single-electron approach, which was successfully applied to two-dimensional (2D) systems such as semiconductor quantum wells and graphene, where insights provided into both the IQHE and the fractional quantum Hall effect (FQHE). Starting from the graphene model, where the unconventional sequence of Hall plateaux, 2(2n+1), naturally emerges from Landau quantization, we generalize the formulation to 3D semimetals with low carrier density and high mobility, where recent experiments have reported signatures of the QHE. For 3D systems, the model accounts for strong band anisotropy by introducing an effective correction to the cyclotron frequency, and, also by considering large effective gyromagnetic factor, as observed in semimetallic materials. From the calculated density of states and carrier concentration, we derive semiclassical expressions for the diagonal and Hall conductivities. The resulting Hall conductivity exhibits quantized values in agreement with theoretical predictions and experimental observations of 3D quantum Hall states. Simulations reproduce both Hall plateaux and Shubnikov de Haas oscillations under realistic parameter sets. Our results demonstrate that the IQHE in 3D semimetals can be understood as a natural extension of the single-electron Landau quantization framework originally developed for 2D systems. This provides a unified picture of quantum magnetotransport across dimensions, highlighting the crucial role of low carrier density and high mobility. The model further suggests new avenues for analyzing thermodynamic and transport properties in 3D systems under quantum Hall conditions.