### 7.1 Transition Rates according to the NMP Theory

The transition rates of charge transfer reactions must be discussed on the basis of configuration coordinate diagrams. One such a diagram is depicted in Fig. 7.1 for the case of hole trapping. The adiabatic potentials and in the configuration coordinate diagrams are approximated by the parabolas around their respective minima and assuming the harmonic approximation:

and denote the vibrational frequency of the oscillator potential when the defect is in the charge state and , respectively. It is stressed that these oscillator potentials of both charge states are assumed to have different curvatures in this derivation. The transition barriers and differ by the energy , which can be expressed as
using the relations

corresponds to the separation of the trap level from the valence band edge and gives the kinetic energy of the substrate hole. Making use of expression (7.3), the difference between the transition barriers can be expressed as:

Using the identity (2.62) and equation (7.6), one obtains the following relation:
Analogously to the equation (2.59), the trapping dynamics are governed by the rate equation
with
Making use of (7.6), the above rate equation can be simplified to
In order to evaluate the integral in the above equation, analytic expressions for and are required. The former is defined as the energy difference between and the intersection point IP in the configuration coordinate diagram. The position of of this point can be derived from the condition
with
Inserting the expression (7.13) into equation (7.1), one obtains
where the Huang Rhys factor is defined by the equation
A second order expansion of equation (7.15) delivers
According to equation (7.14), the oscillator frequencies and differ and thus the quantity deviates from unity. Since enters the above expression for the barrier height, the oscillator frequencies have a strong impact on the transition rates. When the kinetic energy of the substrate hole is taken into account, must be replaced by and equation (7.17) can be rewritten as
In the case of strong electron-phonon coupling, holds and the third term can be neglected. Assuming parabolic bands (see Appendix A.4), the valence band density of states can be expressed as
with
Using (7.19), the integral in equation (7.11) can be evaluated as
and simplifies to
with
Here, denotes the Gamma function, which is defined by
With (7.22), the compact form of the rate equation (7.11) can be rewritten as
From this, it follows that
Just as in the standard SRH theory (see Section 2.5), the right-hand side of equation (7.26) can be simplified using the definition
with
Analogously to Section 2.5, thermal equilibrium can be assumed so that the trap occupation follows Fermi-Dirac statistics and detailed balance applies for the rate equation (7.25). Then equals and the hole capture and emission time constant reads
It is emphasized that the barrier heights are correctly calculated by determining the crossing point of two parabolas. Thereby, one avoids the artificial differentiation, whether is located above or below , as it has been the case in equation (6.10) of the TSM. Additionally, the NMP barriers have not been assumed to be independent of the energy of the hole in contrast to the Kirton model and the TSM.