## 2 TLS Principle  
### Classical Model  
Because of the reflection of the front-side metal layer, it is difficult for the laser to reach deep levels from the front side of the device. Backside laser irradiation has proven to be an effective method [8,11]. In TLS, a continuous infrared laser (wavelength of 1.3 \(\upmu\)m) is used to scan the back-side silicon substrate for two reasons,  
* The photon's energy for the 1.3 \(\upmu\)m infrared laser (\(\frac{hc}{\lambda}\sim\) 0.95 eV) is lower than the silicon band gap energy (1.12 eV), so the laser hardly generates a photo-current [10];
* Silicon is almost transparent at such a wavelength, so the laser beam can reach the active area through the back-side substrate at low loss [11].  
It is well known that thermal energy leads to an increasing temperature \(\Delta T\), causing a variation of resistivity:  
\[\Delta\rho=\rho_{0}\left(\alpha_{\text{TCR}}-2\delta_{r}\right)\Delta T \tag{1}\]  
where \(\rho_{0}\), \(\alpha_{\text{TCR}}\), and \(\delta_{r}\) are respectively the electrical resistivity, the Thermal Coefficient of Resistivity (TCR), and the thermal expansion coefficient of the material. \(\delta_{r}\) is so small relative to \(\alpha_{\text{TCR}}\) that it is negligible in most cases [15]. The resistivity and temperature coefficients of some metal materials are shown in Table 1.  
In the OBIRCH technique, a constant voltage bias \(V_{0}\) is applied to the Device Under Test (DUT) [12]. Due to the changed resistance, the current \(\Delta I\) varies as:  
\[\Delta I=-\frac{\Delta R}{R}I=-\frac{\Delta R}{R^{2}}V_{0} \tag{2}\]  
where \(\Delta R\) is the resistance variation and \(I\) is the current of the DUT before laser irradiation. In the TIVA technique, however, a constant current bias \(I_{0}\) is used [12]. The variation of the voltage \(\Delta V\) is:  
\[\Delta V\equiv\frac{\Delta R}{R}V=I_{0}\Delta R \tag{3}\]  
where \(V\) is the voltage of the DUT before laser irradiation.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Material** & \(\rho_{0}\)**(20 \({}^{\circ}\)C) (\(\Omega\cdot\mathbf{m}\))** & \(\mathbf{\kappa_{\text{TCR}}}\) (\({}^{\circ}\)C\({}^{-1}\))** \\ \hline Al & 2.9 \(\times\) 10\({}^{-8}\) & 4.3 \(\times\) 10\({}^{-3}\) \\ Cu & 1.7 \(\times\) 10\({}^{-8}\) & 3.9 \(\times\) 10\({}^{-3}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Resistivity and temperature coefficients of some metal materials [16].  
The SEI technique, which typically uses zero-bias or constant current bias, is based on the so-called Seebeck effect. While two contiguous materials have different temperatures (see Figure 1) [7], a Seebeck voltage is generated according to the following equation:  
\[\Delta V_{S}\equiv\left(S_{1}-S_{2}\right)\Delta T=S_{1-2}\Delta T \tag{4}\]  
where \(\Delta T\) is the temperature difference between two materials and \(S_{1-2}=\frac{\mathrm{d}V}{\mathrm{d}T}\) is the Seebeck coefficient. Seebeck coefficients for several materials or junctions are presented in Table 2. It can be positive or negative in different contacted materials. The Seebeck voltage is independent of the circuit resistance, and it has an effect on the voltage or current of the circuit.