## 2 TLS Principle
### Comprehensive Model
#### 2.2.2 The Variation of the Current or Voltage in the Circuit  
When the laser beam is focused on the abnormal spot or the normal spot of the DUT, a temperature variation \(\Delta T\) that meets Equation (12) is formed. The variation of resistance \(\Delta R\) and the Seebeck voltage \(\Delta V_{\text{S}}\) are respectively:  
\[\Delta R=\int\Delta\rho\frac{\text{d}l}{A}=\int\rho_{0}\alpha_{\text{TCR}} \frac{\Delta T}{A}\text{d}l \tag{13}\]  
and:  
\[\Delta V_{\text{S}}=\int S_{1-2}\
abla(\Delta T)\cdot\text{d}\vec{l}_{1-2} \tag{14}\]  
where \(\text{d}l_{1-2}\) is a tiny length in the direction of the current flowing, \(A\) is the cross-sectional area, and \(\
abla(\Delta T)\cdot\text{d}\vec{l}_{1-2}\) is the temperature difference between the two contact materials. At the abnormal spot of the failed device, thermal diffusion is more difficult, so the local temperature is much higher. Therefore, the values of \(\Delta R\) and \(\Delta V_{\text{S}}\) are bigger than those at the normal spot. The difference can be detected by testing the current or voltage.  
As for a device under constant voltage bias \(V_{0}\), \(\Delta R\) and \(\Delta V_{\text{S}}\) will cause a variation of the current. The value of current variation \(\Delta I\) is:  
\[\Delta I=I_{\text{TLS}}-I=\frac{V_{0}+\Delta V_{\text{S}}}{R+\Delta R}-\frac{ V_{0}}{R} \tag{15}\]  
where \(V_{0}\) is the voltage bias, \(l_{0}\) is the current before the laser irradiation starts, and \(R\) is the resistance of the DUT [14]. The laser power used in the experiment is less than 100 mW, resulting in very small \(\Delta R\) and \(\Delta V_{\text{S}}\). Therefore, Equation (15) is approximated to:  
\[\Delta I=\frac{\Delta V_{\text{S}}}{R+\Delta R}+\left(\frac{V_{0}}{R+\Delta R} -\frac{V_{0}}{R}\right)\approx\frac{\Delta V_{\text{S}}}{R}-\frac{V_{0}}{R^{ 2}}\Delta R \tag{16}\]  
There are two parts on the right of Equation (16), the first of which is mainly contributed by the Seebeck effect, and the second is dominated by laser-induced resistance change. Therefore, the OBIRCH and SEI techniques coexist in a test. While \(V_{0}\) is equal to a critical value \(V_{\text{c}}\), \(\Delta I\) will be equal to zero. From Formula (16), the value of \(V_{\text{c}}\) is:  
\[V_{\text{c}}=\frac{\Delta V_{\text{S}}}{\Delta R}R \tag{17}\]  
When \(V_{0}>V_{\text{c}}\), \(\Delta I<0\), that is the OBIRCH technique dominates. Otherwise, the SEI dominates.  
While constant current bias \(I_{0}\) is applied, the TIVA and SEI will coexist, and the voltage variation \(\Delta V\) is:  
\[\Delta V=V_{\text{TLS}}-V=[I_{0}\left(R+\Delta R\right)-\Delta V_{\text{S}}] -I_{0}R=-\Delta V_{\text{S}}+I_{0}\Delta R \tag{18}\]  
The contributions of the two effects are still opposite. Besides, a critical current bias \(I_{\text{c}}\) that causes \(\Delta I=0\) is [14]:  
\[I_{\text{c}}=\frac{\Delta V_{\text{S}}}{\Delta R} \tag{19}\]  
When \(I_{0}>I_{\text{c}}\), \(\Delta V>0\), and the TIVA technique dominates. When \(I_{0}<I_{\text{c}}\), the SEI dominates. Therefore, the SEI technique always requires low bias conditions.  
For an active device, according to [14], an extra weak current \(I_{\text{PN}}\) exists after laser irradiation, which is preliminarily interpreted as a light-current signal. \(I_{\text{PN}}\) has little influence on the current variation \(\Delta I\) or voltage variation \(\Delta V\).