## 5 Thermal Characterization
### Thermal Characterization and Modeling of UTC Photodiodes  
To take into account the self-heating effect in the UTC photodiodes, a classical thermal network, consisting of a thermal resistance and capacitances, has already been implemented in the compact model, as described in [13]. From the extraction point of view, the thermal resistance, \(R_{th}\), of this network can be estimated using the following equations [22]:  
\[\frac{I_{d}}{V_{d}}-\frac{Nk_{B}T_{j}}{qI_{d}}\ln\left(\frac{I_{d}}{I_{s}(T_{j} )}\right)=R-\frac{Nk_{B}T_{j}}{qI_{d}} \tag{9}\]  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Device Area (\(\upmu\)m\({}^{2}\))** & **0.5 \(\times\) 5** & **0.5 \(\times\) 7** & **0.5 \(\times\) 10** & **0.7 \(\times\) 5** & **0.7 \(\times\) 7** & **0.7 \(\times\) 10** \\ \hline \(R_{THI}\) (K/W) & 3680 & 2910 & 1980 & 2714 & 1865 & 1456 \\ \(R_{TII2}\) (K/W) & 1195 & 1095 & 777 & 1297 & 1108 & 856 \\ \(R_{THS}\) (K/W) & 920 & 768 & 680 & 868 & 825 & 773 \\ \(C_{THI1}\) (p//K) & 12.5 & 17 & 29 & 21 & 28 & 43 \\ \(C_{TII2}\) (p/K) & 140 & 215 & 320 & 165 & 220 & 340 \\ \(C_{THS3}\) (rI/K) & 1.75 & 1.9 & 2.23 & 1.9 & 2.13 & 2.45 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Thermal resistance and capacitance contributions from the different epitaxial layers of the DHBT.  
Figure 11: Magnitudes of the thermal impedance, \(Z_{TH}\), for two emitter width \(W_{E}=0.7\)\(\upmu\)m (**a**) and \(0.5\)\(\upmu\)m (**b**).  
with  
\[T_{j}=T_{amb}+R_{th}.I_{d}.V_{d} \tag{10}\]  
where \(I_{d}\) and \(V_{d}\) are the current (in A) and voltage (in V) across the UTC-PD, \(N\) is diode current ideality factor, \(k_{B}\) is the Boltzmann constant (in eVK\({}^{-1}\)), \(T_{j}\) and \(T_{amb}\) are the junction and ambient temperatures (both in K), respectively, \(q\) is the electronic charge (in C), \(I_{s}\) is the reverse saturation current (in A) and \(R\) is the total resistance (in \(\Omega\)) in series with the photodiode. The parameters in Equation (9) have been extracted under high-current injection (at 1 V) conditions, where self-heating occurs. In order to converge on a solution for \(T_{j}\) from both sides of Equation (9), an iteration of the values of \(R_{th}\) was performed to solve the expression. In order to facilitate optimal convergence, the model parameters \(I_{S}\), \(N\) or \(R\) were slightly varied, within an acceptable tolerance that is summarized in Table 3. Subsequently, the extracted values of the thermal resistances of the UTC-PDs are also summarized in Table 3. Interestingly, the extracted values of the thermal resistances of the UTC-PDs are not significantly different (slightly higher) from what can be predicted for these geometries using the \(R_{TH}\) scaling law implemented for the DHBTs. Since the two technologies have been developed by the same foundry with similar epitaxial compositions and similar substrate thermal conductivity (InP), the results are coherent and promising for achieving homogeneous monolithic integration with the two devices. Despite the reasonable \(R_{TH}\) values obtained for the UTC-PDs, further model improvement and refinement of the extraction methodology lie within the scope of future work.  
As emphasized earlier, unlike the HBT modeling framework, which has been refined and standardized for a long time, the UTC compact modeling approach is under development and still has several aspects to improve. While the current framework is reasonably accurate, further improvements can include the addition of more material parameters, the description of photocurrent saturation and the more in-depth analysis of self-heating.