Ensembles of Electron States in Cubic Lattices Quantized in the Presence of a Magnetic Field

Abstract

Quantization of the Landau levels and their density versus energy in cubic crystal lattices are examined for the case of weak and strong magnetic fields. Bands of the tightly-bound s-electron states are chosen as examples of the calculations. With the assumption of a constant phase taken in the quantization process, the density of the Landau levels calculated for weak magnetic fields is found practically identical with the planar density of the Bloch states derived in each lattice case. Simultaneously, the frequency of the electron gyration in the lattice due to the action of the external magnetic field becomes proportional to the reciprocal value of the density of the Landau states. For strong magnetic fields the Roth phase and its connection with the Berry phase are also examined. Repercussions of the applied theory to the quanta of the weak-field Hall conductivity are demonstrated.