Charge trapping models found in literature usually assume elastic electron tunneling, which ignores the atomic configuration of the defects. In the Franck-Condon theory [107, 108, 109, 110], however, changes in the defect configuration play a crucial role in the charge trapping process. For instance, this theory predicts that the defect levels can be subject to a shift [111, 112, 113, 114] (discussed in Section 2.2). In order to support the physical understanding of such trapping processes, one has to deal with the basics of microscopic theories, which provide the most complete description of a physical process, such as charge trapping.

Therefore, the Schrödinger equation [92, 91] for a system involving electrons and nuclei is taken as a starting point. Its Hamiltonian can be formulated as

where the subscripts and refer to nuclei and electrons, respectively. and are the coordinates of a certain electron or a certain nucleus , respectively, whereas the curly braces stand for the whole set of coordinates, such as , , , for . The Hamiltonian contains all contributions from the electrons () and nuclei () kinetic energies as well as from the Coulombic electron-electron , nucleus-nucleus , and electron-nucleus interactions. Then, the Schrödinger equation reads where the energy of the whole system of the electrons and the nuclei is denoted as . It is important to note at this point that the wavefunctions depend on all positions of the electrons as well as the nuclei and thus the equation (2.14) cannot be solved due to its mathematical complexity.In order to obtain an approximate solution, one usually employs the Born-Oppenheimer approximation, also known as the adiabatic approximation [92, 91], which states that the electrons move much faster than the nuclei. In this picture, the electrons instantaneously adapt to each atomic configuration and the motion of the nuclei can be neglected for the solution of the electron system. Based on this assumption the wavefunction can be split into an electron () and a nuclei () part.

According to the Franck-Condon principle [107, 108, 109, 110], these vibronic transitions involve a change in the electronic as well as in the vibrational state. The corresponding rate can be calculated using Fermi’s golden rule (see Appendix A.1)

with being the perturbation operator. and denote the initial and the final vibronic state, respectively, where the latin and greek symbols refer to the electronic and vibrational states. According to the Franck-Condon approximation, the matrix element can be separated into two factors: