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Before considering measurements of the capture cross-sections of the Si–Al₂O₃ interface states, it will be useful to make a distinction between real and apparent capture cross-sections, and to illustrate some of the ways in which the latter may differ from the former in misleading ways.

In extracting capture cross-section values from measured conductance or DLTS data it is usually implicitly assumed that a) all states at a given energy have the same σn and σp (i.e. only a single species of interface states is present), and b)

Dit, σn, and σp are approximately independent of energy over the range probed by the conductance measurement AC signal or by the DLTS bias pulse (where this range is broadened by the Fermi occupation function at finite temperatures and by surface potential fluctuations). These assumptions significantly simplify

2_{Note that Tanaka and Iwauchi [22] used the tunnelling model of [150] to account for fre-} quency dispersion, rather than the surface potential fluctuation model of [144] used here and in most recent works, resulting in a systematically larger value forσn. The value ofσn corre- sponding to the measurements of [22] is reassessed in Fig. 4.7

3_{Each of the preceding authors used different values of v}

th in extractingσp and σn, either 2×107_{cm s}−1 _{[22], 5×10}7_{cm s}−1 _{[54], or 1×10}7_{cm s}−1 _{[47]. The values mentioned in the} text have been normalised tovth= 2×107cm s−1.

the analysis and therefore are routinely employed in practice. We will call the values of σn and σp extracted under such assumptions the “apparent” values.

Unfortunately, these assumptions are often invalid. Interface defect states are not in general distributed uniformly with energy across the semiconductor bandgap, but exist as statistically broadened energy distributions centred around certain characteristic peak energies [9], [151]. Furthermore, the generally observed existence of a continuum of interface states throughout the bandgap implies that these separate distributions (due to different interface state species) are likely to overlap over at least some part of the energy range. The apparent values of

σn and σp will deviate systematically from the true values whenever the energy dependence of Dit,σn, or σp is significant, or when interface states with different

σn or σp are present in the energy range probed by the measurement. Because the relevant measurement techniques in fact probe states spanning a significant range of energies (on the order of several kT /q to either side of the mean surface potential), the resulting errors can be quite significant.

The equations presented in Appendix B allow hGpi/ω to be calculated as a function of frequency f and average surface potential for arbitrary combinations of defect states with arbitrary Dit(E), σp(E), and σn(E) dependences. Surface potential fluctuations are described by the normalised standard deviation of sur- face potential σs. The corresponding apparentσp andσn may then be calculated by fitting such modelled hGpi/ω vs f curves with the same equations, assuming a single defect type at each energy, with energy independent Dit, σp, and σn.

Fig. 4.2 shows the divergence between the apparent and real values ofDit(E),

σp(E), andσn(E), due to energy dependence inDit(E) only, for a single Gaussian interface state distribution with energy-independent capture cross-sections. The apparent values of σp(E), σn(E), and σs(E) only approach their input values in the energy range where Dit(E) is a slowly varying function of energy. Outside of this range, the apparent values deviate systematically from the inputs due to the energy dependence of Dit(E). The apparent Dit(E) is simply broadened relative to the input function by the effects of thermal broadening and surface potential fluctuations, which limit the ability of the technique to resolve fine features in the Dit(E) distribution.

At some threshold energy in inversion the deviation ofσp, σn becomes expo- nential with respect to energy. This occurs when the minority carrier capture resistance becomes small enough that the states are effectively coupled together via the inversion layer and respond as an ensemble with a single time constant. At this point the shape of the conductance peak becomes independent of σp, σn,

4.2. Capture cross-sections 51 ıS ıQ 'LW ıS ıQ ' LW e0 e 3 ıQ 9e ıS e0 3 (ee(Le03 7 7 7 7

Figure 4.2: Apparentσp,σn, andDitvs energy for a single Gaussian distribution of interface states with energy-independent capture cross-sections. The input values of σp,σn, and Dit are shown as dashed lines. σs= 3.

ıS ıQ 'LW ıQ% ıS% ıS$ ıQ$ ' LW 2 . ) .7.. .7. .7. .7. ıQ 4 ıS 2 ) .7. .7. .7. .7. .7. .7. ( (L 2) 79 79 7 79 79

Figure 4.3: Apparentσp,σn, andDitvs energy for two overlapping Gaussian dis- tributions of interface states (labelled A and B) with different energy-independent capture cross-sections. The input values of σp, σn, and Dit for each distribution are shown as dashed lines. σs= 3.

and σs, while the peak frequency depends on both σp and σs. There is therefore too little information to determine any parameter except Dit. The energy at which this threshold occurs shifts to higher energies as σn decreases.

The dependence of the apparent interface state properties on energy becomes more complex when a second distribution of interface states with different prop- erties is added to the model. Fig. 4.3 shows the apparent properties that result from the addition of a second distribution of interface states with different capture cross-sections from the first, where the properties of the first defect are unchanged from the previous example except for its location in the bandgap. In this case the apparent capture cross-sections only approach the true values in the range where one or other of the distributions is dominant. In the range where the distribu- tions overlap, the apparent capture cross-sections exhibit a pronounced energy dependence as they transition between those of the two states.

Note that even in this transition region, excellent fits to the simulatedhGpi/ω vs f data are achieved with a model that includes only a single type of interface state. This is because the contributions of the two states frequently merge into a single peak that could have been produced by a single type of state. Conse- quently in this and many other cases it is difficult or impossible to distinguish the presence of more than one type of interface state merely from the measured

hGpi/ω vs f characteristics at a given voltage. It is therefore rather from the energy dependence of the apparent properties that we must infer the presence of multiple types of states.

The distinction between real and apparent interface state properties is often not appreciated by experimenters. It is common practice to accept the apparent values ofσnandσp at face value, even when this results in obvious inconsistencies between the assumptions of the analysis and its results, because the apparent

Dit, σn, or σp are, in fact, strongly varying functions of energy. The preceding discussion and simulation results make clear that a strong energy dependence of the apparent σn and σp does not necessarily imply any energy dependence of the real values, but may result from the energy dependence of Dit, or from a change in the relative concentrations of two or more interface state species with different energy-independent capture cross-sections. This may create a highly misleading picture of the energy dependence of interface properties, which is exacerbated when these spurious energy dependences are used to extrapolate σn and σp outside the measured energy range (for examples of this practice see [11], [145]–[147]).

A final important distinction needs to be made between “apparent” and “ef- fective” capture cross-sections. It should be stressed that the apparent capture cross-sections are not “effective” values from the point of view of recombination. That is, the recombination rate calculated from the apparent values does not

4.2. Capture cross-sections 53

in general equal the recombination rate that would be calculated from the true values (in fact it may differ by orders of magnitude). The direct use of apparent capture cross-sections to calculate recombination may therefore be a source of significant error.