Quantum theory of the electrical conductivity of semiconductors with a non‐standard energy band

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Abstract
The theory developed by Adams and Holstein is generalized to the case of an isotropic but non‐parabolic energy band, such as the conduction band in InSb. The problem of the electron spectrum in crossed electric and magnetic fields is solved taking into account all the interactions of the conduction band with the valence bands. It is shown that in this case, as for a parabolic band, an electric field appears in the spectrum within the linear approximation. The solution of the equation of motion for the density matrix shows that the non‐parabolicity enters into the dissipative current only through the energy conservation law or the scattering process. The general formula for the electrical conductivity tensor is applied to degenerate semiconductors; Shubnikov‐de‐Haas oscillation conditions are obtained when spin splitting of the Landau levels is considered and the positions of the oscillation maxima are calculated. Inelastic electron scattering by optical phonons is also considered. The conditions for magnetophonon resonance in semiconductors with non‐parabolic energy bands are derived. It is demonstrated that all the maxima of Gurevich‐Firsov oscillations except the first have the natural width.

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