Einstein relation for multiple degenerate energy-band semiconductors

What is it about?

A generalized Einstein relation for electron gases in multiple energy-band degenerate semiconductors is derived under near-equilibrium conditions based on the semiclassical drift-diffusion equation. In addition, a more generalized Einstein relation extended to three nonparabolic energy band structures is also derived to be used to approximately describe the Einstein relation for hole gases in degenerate semiconductors, with the application of an isotropic approximation to each of the three valence band structures.

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Photo by Eugene Golovesov on Unsplash

Photo by Eugene Golovesov on Unsplash

Why is it important?

The derivation shows that the Einstein relation for carrier gases in multiple energy bands generally depends on their mobilities unless the energy band structures are parabolic. In particular, it is shown that highly degenerate GaAs exhibits the unusual property that the ratio of the diffusion coefficient to mobility for conduction electrons saturates under two different conditions of electron concentration (or Fermi energy) with increasing electron concentration (or Fermi energy). It is also shown that for GaSb, the effect of the upper conduction band is so substantial that the use of any formula obtained for carriers in a single conduction band is generally unsuitable for describing the Einstein relation for conduction electrons in this semiconductor.

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