## 5 Thermal Characterization
### Dynamic Self-Heating and Thermal Impedance Modeling in InP DHBTs
#### 5.1.2 Geometry Scalable Thermal Impedance Model  
Two scenarios of the pyramidal downward heat flow through the device have been schematically illustrated in Figure 9a,b. The heat diffusion angle, \(\theta\), dictates the temperature distribution along the vertical cross section of the device. For an angle \(\theta\)\(<\)\(\theta_{C}\) (where \(\theta_{C}\) is the critical angle at which the heat flow changes from scenario (a) to (b), with \(\theta_{C}\) = tan\({}^{-1}\)[(\(W_{COL}-W_{E}\))/2\(D_{\alpha\alpha}\)]), with \(W_{COL}\) and \(W_{E}\) being the lateral widths of the collector and the emitter, respectively, and \(D_{\alpha\alpha}\) is the height of the passivation layer, the heat flow is not stopped by the isolation surrounding the collector and the heat diffusion is pyramidal along the depth of the transistor (Figure 9a). On the contrary, for an angle \(\theta\)\(>\)\(\theta_{C}\), the heat flow is first blocked by the mesa edge, thus leading to a uniform heat diffusion in one subsection (see Figure 9b), followed by a pyramidal diffusion pattern toward the substrate. Based on previous works [14,19], the heat diffusion angle is found to be always lower than 65\({}^{\circ}\), leading to the heat-flow scenario depicted in Figure 9a. Hence, for the rest of the thermal modeling approach, scenario (a) was considered. Following our previous modeling approach [14] to obtain an optimal representation of the behavior of dynamic self-heating in the DHBTs, we considered three major vertical sections of the heat diffusion, which included the collector region beneath the heat source located at the B-C junction, the sub collector and the substrate. The upward heat flow through the emitter metal layers is represented only by a thermal resistance, R\({}_{\text{THM}}\), in parallel with the remaining electrical part, as shown in Figure 9c. The emitter-metal layer thermal resistance is calculated in a similar manner as the intrinsic device, considering three different thermal conductivities of the emitter layers: InP, InGaAs (indium gallium arsenide) and TiPdAu [20]. A modified Foster-like thermal network was used to represent heat diffusion through these three sections, as illustrated in Figure 9d. Considering the average thermal conductivity for  
Figure 8: \(Y_{12}\) parameter versus frequency showing the zone where dynamic self-heating activates. each section, one can write the following for the thermal resistances of each subsections [14]:  
\[ \begin{array}{c}R_{THF,1}=\frac{1}{2 tan\theta(L_{E}-W_{E})\kappa_{arg1}}ln\left[ \frac{L_{E}(W_{E}+2h_{1}tan\theta)}{W_{E}(L_{E}+2h_{1}tan\theta)}\right]\\ R_{THF,2}=\frac{1}{2 tan\theta(L_{E}-W_{E})\kappa_{arg2}}ln\left[\frac{(L_{E}+2h _{1}tan\theta)(W_{E}+2h_{1}+h_{2}tan\theta)}{W_{E}(L_{E}+2h_{1}tan\theta)} \right]\\ R_{THF,3}=\frac{1}{2 tan\theta(L_{E}-W_{E})\kappa_{arg3}}ln\left[\frac{(L_{E}+2 h_{1}+h_{2})tan\theta)(W_{E}+2h_{1}tan\theta)}{(W_{E}+2(h_{1}+h_{2})tan\theta)(L_{E}+2h _{1}tan\theta)}\right]\end{array}  \tag{7}\]  
here \(h_{1}\), \(h_{2}\) and \(h_{3}\) denote the depths of the three regions calculated depending on the geometry and the heat flow angle. \(W_{E}\) and \(L_{E}\) are the effective emitter width and length, respectively, and \(\kappa_{arg1}\), \(\kappa_{arg2}\) and \(\kappa_{arg3}\) are the average thermal conductivities of the three regions calculated for InP following the approach in [21]. Similarly, the expressions of the thermal capacitances of the three regions are as follows:  
\[ \begin{array}{c}C_{THF,1}=\frac{\kappa_{arg1}}{D_{th}}\left[\frac{4}{3}h_{1} ^{3}tan^{2}\theta+(L_{E}+W_{E})h_{1}^{2}tan\theta+L_{E}W_{E}h_{1}\right]\\ C_{THF,2}=\frac{\kappa_{arg2}}{D_{th}}\left[\frac{4}{3}h_{2}^{3}tan^{2}\theta+ (L_{E}+W_{E}+4h_{1}tan\theta)h_{2}^{2}tan\theta\right]\\ \hskip 14.226378 pt+(L_{E}+2h_{1}tan\theta)(W_{E}+2h_{1}tan\theta)h_{2}\\ \hskip 14.226378 pt\frac{4}{3}h_{3}^{3}tan^{2}\theta+(L_{E}+W_{E}+4h_{2}tan \theta)h_{3}^{2}tan\theta\\ \hskip 14.226378 pt+(L_{E}+2h_{2}tan\theta)(W_{E}+2h_{2}tan\theta)h_{3}\end{array}