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Theory of the phonon broadening of impurity spectral lines. Nishikawa, Kyoji

Abstract

The theory of the phonon broadening of impurity spectral lines in homopolar semi-conductors is discussed within the framework of a Kubo-type formulation of the adiabatic dielectric susceptibility and the subsequent calculation of this using the double-time Green's function method. The basic assumption is the smallness of the interaction of the electrons (or holes) bound to impurity sites with the lattice vibrations. This interaction is then treated as a small perturbation of the independent systems of electron and vibrating lattice; the use of the adiabatic approximation is thereby avoided. The so-called decoupling of the infinite hierarchy of equations for the relevant Green's functions is discussed in detail and is given its justification in the present problem. In the case of nondegenerate electronic levels, the line-shape function is obtained explicitly in terms of the matrix elements of the electron-phonon interaction. It is found that the absorption line consists of a sharp peak with a width arising from a finite life-time of the unperturbed states due to the electron-phonon interaction and of a continuous background arising from the multi-phonon processes which accompany the optical absorption. In the degenerate case, a general method of obtaining the line-shape function is discussed and is illustrated in an example. The results are compared with those obtained by previous workers in the field. The general theory is applied to shallow impurity levels in silicon with the use of a modified hydrogenic model and a deformation potential description of the electron-phonon interaction; numerical estimates are made for typical contributions to the widths of the lines in both acceptor and donor cases.

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