Up to this point, trapping has been defined for transitions from one band state in the substrate into one trap state or vice versa. Each of the possible transitions is associated with one rate entering the trapping dynamics. In order to reduce the set of rate equations, compact analytical expressions for the overall trapping rates into one defect are required. In this section, a derivation of the sought expressions will be presented for the case of elastic as well as inelastic tunneling.

The derivation below follows the approach of Tewksbury [23] with some slight modifications. For a proper description of charge trapping, Fermi’s golden rule is taken as a starting point. This fundamental law (see Appendix A.1) gives the rate for a transition between a certain initial and final state. In its most general form it reads:

The subscripts and denote a trap or a band state, respectively, and represents the tunneling matrix element. The -function indicates that the electron energy before and after the transition must be conserved as it is required for elastic tunneling. In semiconductor devices, the charge carriers captured in the dielectric can originate from several different energy levels of the substrate valence or conduction band. In order to account for their contribution to the whole tunneling rate, the sum over all states has to be taken. Here, and is the trap level and the electron energy in the substrate, respectively. The subscript in equation (2.33) has been omitted since only tunneling into and out of a certain trap is considered. The matrix element involves the trap wavefunction, whose exact form is in general unknown. Often, the calculation of the matrix element is simplified to a one-dimensional -type trap potential. Its solution [23], consists of the factorsIn this derivation, the calculation of the matrix element has been reduced to a one-dimensional problem in favor of a compact and analytical expression. In order to correct for this approximation, the term must be introduced following Freeman’s approach [103].

Keep in mind that these equations describe elastic tunneling, meaning that a trap can only exchange charge carriers with those bound states, whose energy coincides with the trap level — even though can have different components . In this derivation, the cross-sections correspond to fitting parameters but can be estimated by analytical expressions presented in [23]. Since the values obtained by these expressions range around and are subject to small variations, the cross-sections are assumed to be constant throughout this thesis.

Compact expressions for inelastic transitions are provided by the framework of the Shockley-Read-Hall (SRH) theory [127]. It has been developed to describe recombination centers in bulk but has also been extended for the case of electron or hole trapping into dielectrics [128]. The following derivation of the SRH rates is generalized to NMP transitions, presuming the case of hole trapping, and starts from equation (2.27).

The trap and the band state involved in the NMP transition are labeled by the subscripts and , respectively. denotes the thermodynamic trap level and an arbitrary state in the valence band. Since the NMP theory assumes thermal transitions, the energies of the states and may differ, meaning that the trap can in principle exchange charge carriers with the entire valence band. In order to account for this fact, a sum over all band states has to be carried out.Making use of

with and being the trap level in the flat band case and the effective valence band weights, respectively, the rates (2.69) and (2.70) can be rewritten as with The corresponding expression for electron trapping from the conduction band can be derived in an analogous manner and reads