A re-examination of experimental evidence on the spectral dependence of the optical transition matrix element associated with thin-film silicon

In light of the importance of experimental error to this analysis, two sections will be devoted to this particular aspect of the analysis.

While the experimental results reported on in reference [39] are now over a quarter of a century old, no other more recently acquired collection of comparable experimental results is available. Accordingly, the experimental results presented in reference [39] are drawn upon for the purposes of this study.

The material that reference [39] examined, a-Si:H prepared through conventional plasma enhanced chemical vapor deposition, has served as the workhorse for the large area electronics field for many years. All “device quality” forms of a-Si:H, prepared through similar approaches, should exhibit very similar optical response characteristics, as has been established by the prominent researchers in the field. So we focus our analysis on optical response data corresponding to a-Si:H for three important reasons: (1) it provides a benchmark thin-film silicon material (all other forms of thin-film silicon may be viewed as being deviations from this reference material), (2) there is no other thin-film silicon material that has the extensive reservoir of experimental data on its optical response as a-Si:H, and (3) there is no other data set available corresponding to another form of thin-film silicon that is comparable to that of reference [39] for the case of a-Si:H.

The photo-thermal defection experimental measurements were performed using CCl\(_{4}\) as the deflecting medium. The influence of substrate absorption on the results was monitored throughout the experiment and shown to be negligible. The sub-gap optical response results found through photo-thermal defection spectroscopy were found to be within a factor of “2” equal to those determined using the photoconductive approach. For the purposes of their quantitative analysis, reference [39] draw upon their photoconductive derived results.

Reference [39] were forced to adopt a number of assumptions in order to acquire actual numerical values for their experimental results. For example, they had to estimate the DOS values at the valence band and conduction band mobility edges, values that they only could estimate to within a factor of “2”. Secondly, they had to arithmetically average the low photon energy ellipsometry results in order to remove the interference fringes that inevitably arise from the constructive and destructive interference that occurs. Third, they had to make an assumption related to the form of the tail states as DOS results acquired through the use of time-of-flight measurements are only available from 0.15 eV (and beyond) into the mobility gap. This leaves part of these DOS functions not experimentally accessible. So any determination of the spectral dependence of \(\mathcal{R}^{2} \left( E_{{\text{ph}}} \right) \) will be subject to uncertainty. That is why an evaluation of the limits over which a correct experimental resolution of \(\mathcal{R}^{2} \left( E_{{\text{ph}}} \right) \) could be reasonably expected is a topic that is certainly worthy of investigation within the framework of this study.

Experimental instruments, such as those employed for the measurements of reference [39], have a limited dynamic range. This, of course, imposes fundamental limitations on the resolution of the measured quantities.

In order to simplify matters, we set \(\mathcal{R}^{2} \left( E_{{\text{ph}}} \right) \) to 10 Å\(^{2}\) for the purposes of this analysis, noting that spectral variations in \(\mathcal{R}^{2} \left( E_{{\text{ph}}} \right) \) are unlikely to significantly shape this result. The errors, \(\Delta \epsilon _{2}\) and \(\Delta J\), are set to \(10^{-4}\) and \(10^{38}\) cm\(^{-6}\)eV\(^{-1}\), respectively, while \(\epsilon _{2} \left( E_{{\text{ph}}} \right) \) and \(J \left( E_{{\text{ph}}} \right) \) are set to the spectral evaluations found by reference [39]. Eq. (4) is used in order to perform this error evaluation.

The quantification of how the probability of correctly resolving an experimental quantity is related to the corresponding signal-to-noise ratio represents a problem of considerable importance to experimental science. Our approach, while admittedly simplistic, captures the essence of this problem.

Many authors (see, for example, Nunomura et al. [47]) merely assume that the initial spectral range prescribed by reference [39] for the constant spectral dependence of \(\mathcal{R}^{2} \left( E_{{\text{ph}}} \right) \) for the case of a-Si:H, i.e., 0.6 to 3.4 eV, is valid, and this has an impact on their experimental interpretations. Some authors (see, for example, [48, 49]) have found results for the spectral dependence of \(\mathcal{R}^{2} \left( E_{{\text{ph}}} \right) \) that are quite distinct from those of reference [39]. We believe that the results of the current analysis would have been helpful to these authors. Hopefully, our article, when published, will be helpful to those that follow.