Theory of Impurity Light-Absorption Spectrum. The Urbach Rule

Abstract

The line shape of impurity absorption was treated in terms of the general formalism of the adiabatic, Condon, and harmonic approximations. Experimental results of the temperature dependence of Br and I in KCl were presented. The semi-classical methods were discussed in the seminar. It was shown that the quantum-mechanical theory gives a satisfactory explanation of both the Gaussian in the centre and the Urbach rule in the long-wave tail of the absorption band. It is clear from physical considerations that transitions from vibrational levels which correspond to large amplitudes of nuclear vibrations give the main contribution to the edge of the band. Including only terms linear in the electron-phonon interaction may turn out to be insufficient. The generating functions of the optical line shape of impurity absorption were obtained including both displacement of equilibrium positions of normal oscillators and the change of their frequencies. The line shape was investigated by the saddle-point method. It was shown that the frequency effect changes considerably the Gaussian line shape in the centre of the band and the Urbach constants in the long-wave edge. The band edge for forbidden transitions induced by lattice vibrations was also investigated.