Temperature Dependence of Nonradiative Transitions of an Excited Optical Ion in Solids

We have investigated the temperature dependence of nonradiative transitions of an optical ion in a solid from the excited electronic state to the ground electronic state within the framework of proper adiabatic approximation for a single configuration coordinate, q. The Hamiltonian based on the nonadiabaticity operator has been used to calculate the temperature dependence of nonradiative transition rates. This perturbing Hamiltonian, {H}_na=−{ℏ}²{(\frac{∂}{∂q})}_e{(\frac{∂}{∂q})}_v−\frac{{ℏ}²}{2}{(\frac{{∂}²}{∂{q}²})}_e{(I)}_v consists of two terms. In the literature, the first term is usually used to explain nonradiative transitions of an optical ion in a solid. We have shown that the first term alone leads to a zero of the nonradiative transition rates when s=z for low temperature, where s represents the Huang-Rhys parameter and z relates to the energy gap between the excited state and ground state in terms of vibrational quantum of energy, ℏω. This zero cannot be removed unless the second term is included for calculating the transition rates. We have defined three auxiliary functions, {U}₁, {U}₂ and {U}₃ to describe the temperature dependence of the nonradiative transition rates in a physically meaningful way.