## 2 TLS Principle
### Comprehensive Model
#### 2.2.1 The Evolution of the Temperature Field in the Circuit after Laser Irradiation  
When the laser beam is focused on the active area of the DUT (see Figure 2), a temperature field is generated, following the thermal conduction equation:  
\[\rho c\frac{\partial T}{\partial t}-\kappa\
abla^{2}T=Q(x,y,z,t) \tag{5}\]  
where \(\rho\), \(c\), and \(\kappa\) are respectively the density, heat capacity, and thermal conductivity of the material. \(Q(x,y,z,t)\), as a function of position \((x,y,z)\) and time \(t\), is the heating source produced by the laser beam. Taking the case of a laser beam moving along the positive \(x\) direction as an example, the expression of \(Q\) is:  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Material or Junction** & **S\({}_{1-2}\) (\(\mu\)V/K)** \\ \hline Al & \(-3.5\) * \\ W & \(3.6\) * \\ Al/n + Si & \(287\) ** \\ Al/p + Si & \(-202\) ** \\ \hline \hline \end{tabular}
\end{table}
Table 2: Seebeck coefficients for several materials and junctions [16].  
Figure 1: Seebeck effect principle. A Seebeck voltage is generated due to the temperature difference of the two contiguous materials.  
\[Q(x,y,z,t)=\frac{(1-R)\,P_{0}}{\pi Y\omega^{2}(z)}\exp\left[-\frac{(x-vt)^{2}+y^{2} }{\omega^{2}(z)}-\frac{z}{Y}\right] \tag{6}\]  
where \(P_{0}\), \(\omega^{2}(z)\), \(R\), and \(\gamma\) are respectively the power, spot size, reflectivity, and attenuation length of the laser. \(v\) is the scanning velocity of the laser spot. Equation (6) uses the approximation of the Gaussian intensity profile [15].  
In general, the temperature variation in the xy plane is what we are interested in. Besides, the area of the IC layout is much larger than that of the laser spot. Therefore, the model mentioned above can be transformed into the temperature field distribution in two-dimensional infinite space. Equations (5) and (6) are converted to:  
\[\rho c\frac{\partial T}{\partial t}-\kappa\left(\frac{\partial^{2}T}{\partial x ^{2}}+\frac{\partial^{2}T}{\partial y^{2}}\right)=Q(\vec{r},t) \tag{7}\]  
\[Q(\vec{r},t)=\frac{(1-R)\,P_{0}}{\pi Y\omega^{2}}\exp\left[-\frac{(x-vt)^{2}+y ^{2}}{\omega^{2}}\right] \tag{8}\]  
where \(\vec{r}=(x,y)\). In Equation (8), the laser spot size \(\omega^{2}\) is treated as a constant. Considering that the temperature difference \(\Delta T=T-T_{0}\) is also a special solution to Equation (7), the boundary condition is:  
\[\Delta T(\vec{r},t)\big{|}_{|\vec{r}|\rightarrow\infty}=0,\ \Delta T(\vec{r},t) \big{|}_{t=0}=0 \tag{9}\]  
Equation (7) can be solved by a [PERSON]'s function approach [17] to give:  
\[\Delta T(\vec{r},t)=\int_{0}^{t}\mathrm{d}t^{\prime}\iint_{\Sigma}G(\vec{r},t; \vec{r}^{\prime},t^{\prime})Q(\vec{r}^{\prime},t^{\prime})\mathrm{d}\vec{r}^{\prime} \tag{10}\]  
where \(\Sigma\) is the whole two-dimensional plane and \(G(\vec{r},t;\vec{r}^{\prime},t^{\prime})\) is the Green's function for point excitation at \((\vec{r}^{\prime},t^{\prime})\) given by:  
\[G(\vec{r},t;\vec{r}^{\prime},t^{\prime})=\frac{1}{4\pi k\,(t-t^{\prime})}\exp \left[-\frac{(x-x^{\prime})^{2}+(y-y^{\prime})^{2}}{4k\,(t-t^{\prime})}\right] \eta(t-t^{\prime}) \tag{11}\]  
where \(k=\frac{\kappa}{\rho c}\) and \(\eta(t-t^{\prime})\) is the step (Heaviside) function. Using Equations (8) and (11) in Equation (10), we can obtain:  
\[\Delta T(\vec{r},t)=\frac{(1-R)\,P_{0}}{4\pi\rho cy}\int_{0}^{t}\frac{1}{ \omega^{2}+4k\,(t-t^{\prime})}\exp\left[-\frac{(x-vt^{\prime})^{2}+y^{2}}{ \omega^{2}+4k\,(t-t^{\prime})}\right]\mathrm{d}t^{\prime} \tag{12}\]  
Figure 2: Laser beam focused on the metal layer. The thermal energy will diffuse in all directions.