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IBM Journal of Research and Development  
Volume 32, Number 1, Page 52 (1988)
Basic Concepts in Quantum and Stochastic T...
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Bloch electron in a magnetic field: Mixed dimensionality and the magnetic-field-induced generalized quantum Hall effect

by M. Y. Azbel
The energy spectrum of a Bloch electron in a magnetic field is one-dimensional. This leads to the Peierls instability and the magnetic-field-induced transition to the quantized Hall effect. The wave function is two-dimensional. This decreases the Peierls gap and makes it exponentially vanishing with magnetic field. Disorder lifts the degeneracy and one-dimensionality of the spectrum. High disorder yields a metallic behavior. Intermediate disorder leads to the generalized quantized Hall effect. The latter has a finite magnetoresistance as a semimetal, and Hall plateaus similar to the quantized ones, but they may have any value of the effective charge.
Related Subjects: Carrier transport in small structures; Hall effect; Mesoscopic phenomena; Physics, solid state; Quantum theory and effects