Line Broadening of Impurity States by Resonant Phonon Interactions

A simplified calculation of the absorption cross section for optical excitation of an impurity is performed. The model considered includes a two-level impurity which may interact strongly with a lattice mode capable of relaxation. The cross section exhibits a strong resonance behavior when the impurity excitation energy is close to the energy of a strongly coupled lattice mode. Maximum broadening occurs on resonance. Off-resonance broadening also occurs with generally asymmetric and energy-shifted line shapes. When the system is far from resonance, the natural line shape is obtained. A sum rule is derived which states that the energy-integrated cross section depends only on the natural linewidth. The case of donors in silicon is discussed, and the predictions of the model calculation are compared qualitatively with the experimental results. The most strongly coupled lattice mode is the TO mode of ${\Sigma }_1$ symmetry with a sharply defined energy corresponding to $q\approx 0.5q_{max}$ along the $〈110〉$ directions.