Theory of the isotope shift for zero-phonon optical transitions at traps in semiconductors

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Abstract
An isotope shift has previously been measured for "zero-phonon" optical transitions in semiconductors where the initial or final state involves an electron or hole trapped around an impurity whose mass can be varied by isotopic substitution. The theory of the shift is developed and reasonable agreement found with experimental results for various impurities in GaP. The presence of the trapped carrier reduces the interatomic force constant at and around the impurity, thus softening the normal-mode frequencies of the lattice and reducing the zero-point energy of the oscillators. The isotope shift arises from a cross term in the zero-point energy between the change in impurity mass and the mode softening. The softening of the phonons by trapped or free carriers is also related to the temperature dependence of the band gap of the pure material, from which its magnitude can be estimated. Conversely we learn from the isotope shift that the hole contribution to the temperature dependence of the gap is about three or four times as large as that of the electron.