On narrow-gap electron spectra of semiconductors two-band model with electron-phonon interaction

A two-band model of a narrow-gap semiconductor with interband electron-phonon and $\mathbf{k\cdot P}$-interactions is studied. For the functions that describe the contribution of the electron-phonon interaction to the interband mixing, an equation of the type obtained in superconductivity theory is derived. The equation has a nontrivial solution if a certain condition imposed on the parameters of the semiconductor is satisfied. The renormalization of the energies of the electron bands by the interband electron-phonon interaction is taken into account. The behavior of the corrections to the electron gap as a function of the temperature is estimated and their relationship to experiments is discussed.