# The Sliding-Aperture Transform and Its Applicability to Deep-Level Transient Spectroscopy
## 2 Methods  
To investigate defects in semiconducting systems, the ionization or emission rate of trapped carriers \(R(T)\) is typically used, and is given by  
\[R(T)=P\ T^{2}\exp\left[-\frac{E}{k_{b}\ T}\ \right], \tag{1}\]  
where \(T\) is temperature and \(k_{b}\) is Boltzmann's constant. The ionization energy, \(E\), and the capture parameter, \(P\), are unique defect-specific signatures, both required for an a priori determination of emission rate at a specific temperature. If the capture parameter and ionization energy are not known a priori, then the emission rate must be determined as a function of temperature. Once obtained, the capture parameter and ionization energy are extracted using Arrhenius analysis of \(R(T)\)[1, 18]. In practice, a capacitance transient is obtained by collapsing a diode's depletion region for a time, \(t_{fill}\), filling the defect sites with carriers [19]. A reverse bias is then applied at \(t=0\) to expand the depletion region and initiate defect ionization at rates determined by the ionization energy and capture parameter in Equation (1). Changes to diode capacitance occur as the diode depletion width expands or contracts to maintain charge neutrality in the depletion region of the diode as defect ionization occurs. This fill-and-measure sequence produces a capacitance transient, as shown in Figure 1, with the initial value \(\pm\)\(C\), with respect to a steady state value \(B\). The capacitance transient is of the form  
\[C(t,T)=B(T)\pm C\exp[-t\ R(T)], \tag{2}\]  
where \(C\) is proportional to the defect concentration, with a positive sign consistent with minority carrier emission and a negative sign indicating majority carrier emission. The sign of C is the strength of the capacitance-based DLTS technique. Similar current-transient-based techniques cannot distinguish between minority carrier or majority carrier emissions, because each carrier contributes identically to any current measured [20, 21, 22]. The constant value \(B\) is not a function of time, only of temperature, and represents the case of complete defect ionization. In practice, the temperature dependence of the background signal, \(B(T)\), complicates transient analysis because, a priori, it is not known whether the background is a constant or a very slowly ionizing defect site. The SLAP methodology presented below effectively removes this ambiguity. Traditionally, the capacitance transients of DLTS were repeated and averaged using an analog boxcar average [1]. The double-boxcar average technique was configured to sample a repeated input signal at two chosen times, \(t_{A}\) and \(t_{B}\), which define a rate window. This  
Figure 1: DLTS capacitance transients for minority carriers (solid line) and majority carriers (dashed line), following the removal of a fill pulse at time \(t=0\). \(t_{A}\) and \(t_{B}\) represent sampling times for a waveform sampled for a total time, \(t_{\text{max}}\). analog gated measurement was synchronized with the start of defect ionization, producing a derived signal, as follows:  
\[DLTS(t_{A},t_{B},T)\equiv C(t_{B},T)-C(t_{A},T). \tag{3}\]  
Then, for traditional capacitance-based DLTS, temperature is slowly swept, and a peak in the DLTS signal is found at some temperature, \(T_{\text{peak}}\), when the emission rate is given by  
\[R\Big{(}T_{\text{peak}}\Big{)}=\frac{\ln\big{(}\frac{t_{B}}{t_{A}}\big{)}}{t_{B }-t_{A}}. \tag{4}\]  
The double-boxcar average output is then plotted versus temperature to produce a set of \(T_{\text{peak},i}\), each of which defines an emission rate, \(R_{i}\), for the \(i\)th defect, according to Equation (4). Each \(T_{\text{peak},i}\) contributes to the spectral signature unique to the type (donor or acceptor), concentration, capture parameter, and ionization energy of all defects present. Analysis of a DLTS signal deconvolves this signature into an integer number of specific defects.