# The Sliding-Aperture Transform and Its Applicability to Deep-Level Transient Spectroscopy
## 3 Results  
We present a mathematical \"sliding-aperture\" transform (or SLAP) to obtain defect parameters from capacitance transients measured at constant temperature. Each capacitance transient is an integer sum over \(k\) defect species of exponential transients, Equation (2), each with its own set of parameters, \(C_{i}\), and \(R_{i}\), where the subscript \"\(i\)\" enumerates the defects. For any fixed temperature, there is a single constant offset, \(B\), corresponding to the quiescent reverse bias capacitance value at infinite time when all defects have been ionized. The SLAP transform is plotted as a function of the rate variable \(\rho=1/t\) and produces a defect signature composed of a set of peaks whose locations identify the rates \(R_{i}\), and whose peak values define the peak capacitances \(C_{i}\). Spectra taken at different fixed temperature will have defect signatures shifted to different positions on the \(\rho\) axis, according to Equation (1). Analysis of the temperature dependence of the obtained \(R_{i}\) values then gives the desired defect parameters \(E_{i}\) and \(P_{i}\). Like traditional DLTS, the background, \(B\), subtracts out and needs to be determined by other means to accurately determine defect concentration from the transient maximum. The SLAP transform is defined as  
\[SLAP(\rho)\equiv C\left(\frac{N}{\rho}\right)-C\left(\frac{1}{\rho}\right), \tag{5}\]  
where all quantities are evaluated at the same constant temperature and the SLAP function is expressed in terms of the rate variable \(\rho=1/t\). The aperture \(N\) is defined as  
\[N=\frac{t_{B}}{t_{A}}, \tag{6}\]  
where \(t_{A}\) represents the first sampled time after the removal of the fill pulse. In the simplest case, for a transient sampled at a fixed sampling period, the first few ordered pairs of the SLAP transform for the \(N=2\) case will be:  
\[\left(\frac{1}{t_{A}},\ C(2t_{A})-C(t_{A})\right)\]  
\[\left(\frac{1}{2t_{A}},\ C(4t_{A})-C(2t_{A})\right)\]  
\[\left(\frac{1}{3t_{A}},\ C(6t_{A})-C(3t_{A})\right)\]  
\[\vdots\]  
\[etc.\]  
Any waveform, sampled at a fixed sampling rate, can be SLAP-transformed using the above algorithm. This work concentrates on the special case, where the original function is of the form of Equation (2), namely a decaying exponential plus a constant baseline. This leads to the following SLAP function:  
\[SLAP(\rho)\equiv\pm C\ \bigg{[}\exp\left(-N\ \frac{R}{\rho}\right)-\exp\left(- \frac{R}{\rho}\right)\bigg{]}. \tag{7}\]  
Transforming a digital waveform, as described above, will lead to a waveform that can be curve-fit with the SLAP function of Equation (7). Alternatively, the SLAP function of Equation (7) can be used as an equivalent representation of the decaying exponentials of Figure 1 with the constant background, \(B(T)\), eliminated, as shown in Figure 2. Here, a positive and a negative SLAP peak are plotted in the inverse time, or rate domain \(\rho\), on axis, with the axis scaled by factors notated as \(\sigma_{x}\) and \(\sigma_{y}\). These scale factors are obtained by first noting that the SLAP peak will be located at \(R\), where,  
\[R=\frac{\ln(N)}{N-1}\ \rho_{peak}=\sigma_{x}\rho_{peak}. \tag{8}\]with direct substitution of Equation (8) into (7) leading to  
\[N=\frac{t_{B}}{t_{A}}, \tag{6}\]0  
which defines both the \(\sigma_{x}\) and \(\sigma_{y}\) scaling factors. In general, each defect species is characterized by the physical attributes associated with Equation (1), namely emission rate and capture parameter. This leads to the SLAP transformation for multiple emitters as an integer sum of the following form:  
\[N=\frac{t_{B}}{t_{A}}, \tag{6}\]1  
which produces \(k\) peaks, each of which defines a pair of defect parameters (\(R_{i}\), \(C_{i}\)). The strength of the SLAP approach to transient analysis is the ability to use Equation (10) and a simple curve-fitting algorithm to curve-fit a multipeak spectrum based on an integer number of defects.