On the thermal broadening of zero-phonon impurity lines in absorption and fluorescence spectra

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Abstract
We examine the problem of the absorption zero‐phonon line shape for dilute impurities in crystals. We consider the usual two level electronic model, where both the ground and excited state Born–Oppenheimer surfaces are harmonic in the phonon coordinates. The difference between the two surfaces (the electron–phonon interaction) has terms which are both linear and quadratic in the phonon coordinates. In contrast to the usual perturbative theories, we calculate the zero–phonon line broadening and shift to all orders in the electron–phonon interaction. We find that only the quadratic term is responsible for line broadening, and that at T=0 K this contribution vanishes. Our results are presented as integrals, which can be performed analytically or numerically, involving the weighted phonon density of states. We also show that within the model, the zero‐phonon lines in the absorption and fluorescence spectra coincide exactly for all temperatures. Our results resolve the theoretical controversy produced by the two previous attempts to solve the line shape problem for strongly coupled electron–phonon systems. The work by Osad’ko is shown to be correct.