2.5 Effective Rates into Single Traps

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.

2.5.1 Elastic Electron 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 factors
and
with
Note that electron and hole tunneling proceed through an energy barrier formed by the oxide conduction and valence band, respectively. The matrix elements for these cases are labeled by ‘’ and ‘’, accordingly. The second factor of the matrix element in equation (2.35) arises from the exponential decay of the electron wavefunction, which can be derived using the WKB approximation. Since this factor shows a dependence on the carrier kinetic energy perpendicular to the interface, the summation over all band states must be split into a one-dimensional and a two-dimensional part (see Appendix A.4):
In a factor of two has been introduced in order to account for spin degeneracy. Note that the DOS has been derived based on the parabolic band approximation, however, the potential well in the inversion layer gives rise to the formation of subbands (see Section 2.1) and in consequence a different DOS. This means that the equation (2.40) neglects quantization effects in the inversion layer. Nevertheless, inserting the expression (2.40) into equation (2.34) yields
which can be simplified to
The factor enters the above equation as a square, which corresponds to a transmission coefficient of an electron through an energy barrier [126], and will be referred to as the ‘WKB factor’ in the following.

In 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.
Employing to the parabolic band approximation (see Section A.45), the sum over all band states can be approximated by an integral over the valence band DOS , where can be identified with the energy of the valence band state (see Appendix A.4).
Using the expression (2.49), the transition rate can be rewritten as
For a hole capture process, one must account for the joint probability that the trap must be occupied by an electron while the band state with an energy is empty. The first condition is considered by the trap occupancy . The second one can be expressed by
with being the Fermi-Dirac distribution
With this joint probability, one obtains the rate equation for hole capture,
where the ‘minus’ sign on the right-hand side in the above equation reflects the fact that the trap occupancy is decreased by a hole capture event. An analogous argumentation for hole emission yields
where equals the Fermi-Dirac distribution . Combining the rate equations for hole capture (2.54) and emission (2.55) and using the shorthands
one obtains
If thermal equilibrium is assumed [127], the trap level is occupied according to the Fermi-Dirac statistics
and detailed balance
must be employed. Using the identity
one obtains
Inserting this result back into equation (2.59) yields
The hole capture time constant can be expressed as
where the first term on the right-hand side of the above equation is defined as the product of a cross section and the thermal velocity . The definition of the hole emission time constant follows from equation (2.64).
For the ratio of the time constants, one obtains the well-known relation
which is frequently invoked in the context of charge trapping — in particular for NMP models [5512156124125]. Note that the quantity in equation (2.65) contains the matrix element and the Frank-Condon factor . The former is associated with an electron tunneling process and thus often approximated by a WKB factor. The latter is strongly determined by the barrier height of an NMP transition (see Section 2.4). This suggests that also the cross section in the SRH theory should somehow reflect this barrier dependence. For a hole capture process with (cf. Fig. 2.5), the barrier height can be split into two components, namely and . The former is defined as
where and are the trap level and the valence band edge in the absence of an electric field. The value of corresponds to minimal energy required for a transition. The latter, that is , represents only the remainder to the overall barrier . For the , the component vanishes and only remains. Using the above definitions of the barriers, the cross section can be written as
where and denote a temperature-independent cross section and the characteristic tunneling length. It is remarked at this point that the transition barriers has been assumed to be independent of the energy of the holes.

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
with
Finally, it should be mentioned that the conventional SRH theory as established in [127] does not account for charge carrier tunneling and the possible presence of thermal barriers.