## 3 Nondestructive Testing Methods
### Transient Thermal Analysis (TTA)  
TTA on semiconductors is an electrical test method to evaluate the thermal behavior of semiconductors in a thermomechanical set-up. The key parameter is the thermal impedance \(Z_{th}(t)\) defining the temperature change in the semiconductor junction \(\Delta T_{J}(t)\) according to a change in power \(P\). \[Z_{th}(t)=\ \frac{\Delta T_{J}(t)}{P}. \tag{1}\]  
The measurement procedure of TTA for semiconductors is generally defined in [33] and refined for diodes [34], LEDs [35], MOSFETs [36], IGBTs [37; 38], bipolar transistors [39], and integrated circuits (ICs) [40]. However, the measurement principle and measurement sequence are identical for all of them and the measurement sequence is visualized in Figure 3. At the beginning, an internal power loss \(\mathrm{P_{Heat}}\) is applied for the duration \(\mathrm{t_{Heat}}\) to heat up the semiconductor by driving it in some electrical operation condition. Depending on the type of semiconductor, different conditions for heating can be used, e.g., by operating diodes and LEDs in the forward direction with nominal current [41] or operating MOSFETs in the saturation region [42]. After \(\mathrm{t_{Heat}}\), \(\mathrm{P_{Heat}}\) is switched off very quickly to a much lower power level \(\mathrm{P_{Sense}}\), and the semiconductor stays in this condition for the duration \(\mathrm{t_{Sense}}\) and cools down. During \(\mathrm{t_{Sense}}\), the temperature of the semiconductor junction is measured and time-resolved via a temperature-sensitive electrical parameter (TSP) of the semiconductor itself. For different kinds of semiconductors, different TSPs are used, e.g., for diodes and LEDs, the forward voltage \(V_{f}\)[41], and, for MOSFETs, the threshold voltage \(V_{th}\)[42], generally called \(V_{TSP}(t)\) in the following. \(\Delta T_{J}(t)\) is calculated over \(\Delta V_{TSP}(t)\) and the sensitivity (SEN) of the TSP with  
\[\Delta T_{J}(t)=\ \frac{\Delta V_{TSP}(t)}{SEN}=\ \frac{V_{TSP}(t)-V_{TSP}(0)}{SEN}. \tag{2}\]  
It should be mentioned that, in some standards and publications, the k-factor is used instead of SEN, which is basically its reciprocal. \(Z_{th}(t)\) is calculated afterward from \(\Delta T_{J}(t)\) and the power step from \(\mathrm{P_{Heat}}\) to \(\mathrm{P_{Sense}}\) with Equation (1), and it is the basis for any evaluation in TTA. The simplest evaluation method is comparing the thermal resistance \(R_{th}\), being the steady-state termination value \(Z_{th}(\infty)\); however, in that way, only the performance of the whole thermomechanical set-up can be analyzed. If an evaluation of the thermal properties of different material layers inside the thermomechanical set-up is required, \(Z_{th}(t)\) curves have to be compared on a logarithmic timescale as in Figure 4a. The three curves overlap until the point of separation where the thermal paths of the thermomechanical set-ups start to differ. Depending on the position in time of the separation, the defective material layer can be identified. An earlier separation denotes the defective layer being located closer to the heat-generating junction of the semiconductor and vice versa. In addition to the \(Z_{th}(t)\) comparison, there are also deeper mathematical evaluations such as the normalized logarithmic derivative. For this method, the time is substituted by \(z=ln(t)\) and, afterward, the data are derived by \(z\) and logarithmized, resulting in  
\[B(z)=log\bigg{(}\frac{1}{SEN*P}\bigg{)}+log\bigg{(}\frac{dV_{TSP}(z)}{dz}\bigg{)}. \tag{3}\]  
The advantage of this method is that \(V_{TSP}(0)\) is eliminated by the derivation, and \(SEN\) and \(P\) are converted to a linear offset by the logarithm, which is balanced by normalization. A time-intensive determination of these parameters is no longer required. An exemplary evaluation of the data can be seen in Figure 4b with the same dataset as in Figure 4a.