For the level shift, it has been implicitly presumed that vibronic
transitions always take place at equilibrium configurations of the
defects. Thereby, it has been ruled out that the defects are thermally
excited^{2}
up to the intersection points of their adiabatic potentials (see Fig. 2.4). However,
this case has been accounted for in the nonradiative multi-phonon theory
(NMP) [107, 115]. This mechanism has already been suggested for random
telegraph noise and noise in microelectronic devices [56], where only a
simplified description of NMP process has been employed. Furthermore, it is
also encountered in the context of phonon-assisted tunneling ionization of
deep centers [116, 117, 118] and discussed on various levels of theoretical
sophistication [119, 120, 55, 121, 122] including additional second-order
effects, such as the Coulomb energy [123, 124, 125] and field-enhancement
factors [116, 117].

In NMP theory, the equation (2.21) is generalized to account for all possible thermal excitations. Then the equation (2.21) must be rewritten as

where is referred to as the lineshape function. ‘’ denotes the thermal average over all initial vibrational states and accounts for the thermal excitations using a sum over weighted Boltzmann factors. The lineshape function eventually depends on the Franck-Condon factor and thus on the complicated shape of the adiabatic potentials of the defects. It is noted that these potentials are not assessable via experiments but can also not be calculated using first-principles calculations (see Section 3.3), which would by far exceed the current computational capabilities. However, they can be reasonably approximated using the harmonic approximation when only small displacements from the equilibrium configuration of the defects are considered. In this approximation, the adiabatic potentials are represented as a Taylor expansion whose linear term vanishes close to the equilibrium configuration. As a result, these potentials become parabolic and therefore, describe harmonic oscillators frequently used in solid state physics. A corresponding configuration coordinate diagram for the vibronic transitions of a defect is depicted in Fig. 2.4. The total energies and in Fig. 2.4 include the contributions from the defect atoms along with its immediate surrounding (and the channel region) and therefore correspond to the adiabatic potentials . For this reason, their corresponding adiabatic potentials differ only in the location of electron involved in the trapping process. In the case of , the electron resides in the channel while, for , it is located at the defect site. In the harmonic approximation, the adiabatic potentials can be written as:The NMP mechanism was suggested several decades ago but has been disregarded in the context of NBTI so far. Nevertheless, this mechanism should be considered as a possible description of charge trapping in NBTI. The underlying theory relies on the complicated quantization effects of the nuclei system and is therefore quite complex in its original variant. However, several convenient and accurate approximations, including the version presented in this section, have been developed over the years and allow for theoretical investigations on a device level.