4.2 Capture cross-sections
4.2.1 The conductance method
The principle of the conductance method is based on the measurement of the MIS parallel conductance Gp, which is a measure of the energy loss due to non- equilibrium charge transfer between interface states and the semiconductor. Such
4.2. Capture cross-sections 47
energy loss occurs when interface states are in partial equilibrium with Fermi level variations in the semiconductor induced by the applied AC voltage signal. Charge transfer between the semiconductor and states above or below the Fermi level results in an energy loss that is measured as a conductance. When measured as a function of angular frequency ω, the normalised interface state conductance
hGpi/ωforms a peak centred on the frequency of maximum energy loss. The peak frequency is related to the capture cross-section of the interface states, while the area under the peak is proportional to the density of these states.
The conductance method is particularly powerful because the measured quan- tity, the MIS parallel conductance Gp, generally relates directly and solely to the interface states. This is not the case for capacitance-based methods, where the measured capacitance contains contributions from the semiconductor and dielec- tric which must be separated from that of the interface states. As a result, the conductance method is capable of accurately measuring extremely small interface state densities, on the order of 109eV−1cm−2 [8].
A significant source of uncertainty in the conductance method is that the surface carrier concentration is not measured directly, but must be calculated from the surface potential determined from a C–V measurement or some other method. Because the dependence of the surface concentration on the surface potential is exponential, the extracted capture cross-sections are very sensitive to error in the surface potential. As a result, such measurements can never be more accurate than about a factor of two.
In analysing conductance measurement data, it is usual to make the simpli- fying assumption that minority carrier capture and emission processes may be neglected. This allows a considerable simplification of the analysis, which is why this approach has long been the conventional one [8]. However, it restricts the energy range over which the method is valid to that in which majority carrier processes are dominant. In practice, this means that σp can be determined in the lower part of the bandgap, and σn in the upper part of the bandgap, from measurements of p-type and n-type samples respectively. Furthermore, neither
σp nor σn may be determined accurately close to midgap, where minority and majority carrier concentrations are comparable.
In order to use the measured capture cross-sections to model surface recombi- nation, it is therefore necessary to make some assumption concerning the values of σp and σn over the energy range in which they were not measured (i.e. the greater part of the bandgap). This is typically done by extrapolating the mea- sured values in some way [10], [11], [47], [145]–[147]. Such an approach leads to
large uncertainties, being capable of producing vastly different results depending on the functional dependence chosen for the extrapolated values.
Still more crude is an approach taken by some authors of extrapolating the measured σp and σn to midgap, and treating the resulting capture cross-section ratio as characteristic of recombination at the interface in question. It ought to be remembered that under typical levels of illumination, and especially when the surface charge is significant, interface states across the greater part of the bandgap contribute equally to recombination, not merely those at midgap, and the value of both capture cross-sections must therefore be known over the same range.
A major problem for approaches based on measuring only majority carrier capture cross-sections is that interface states tend to be donor-like in the lower half of the bandgap and acceptor-like in the upper half. Consequently, the minority carrier capture cross-sections tend to be larger than those for majority carriers in both halves of the bandgap. Since recombination will be dominated by the states with the largest capture cross-sections, it will usually be these (unmeasured) minority carrier capture cross-sections that determine the recombination rate.
The limitations imposed by neglecting minority carrier capture can be over- come by considering the complete small-signal MIS equivalent circuit, including minority carrier processes. This approach was first demonstrated by Cooper and Schwartz [148] for the Si–SiO2 interface in 1974, but apart from the work pre- sented here, some of which was previously published in [33], it appears only to have been applied by [149] (to the Ge–SiO2 interface). The likely reason for this is that the resulting equations are significantly more complex, and must be solved numerically, which complicates analysis. However, modern computing power makes this a feasible task even for large datasets. The ensuing advantages are considerable. With this approach it is possible to determine both minority and majority capture cross-sections at the same energy over an energy range from near flatbands, through depletion, midgap, and weak inversion, from mea- surements on a single sample. Consequently, this is the approach adopted in this work. The relevant theory and equations are described in Appendix B.