## 4 DC Characterization and Model Validation
### HiCuM Parameter Extraction Methodology  
InP DHBT DC characterization were performed on several device dimensions (0.5 and 0.7 \(\upmu\)m emitter widths and emitter lengths of 5, 7 and 10 \(\upmu\)m) and compared with the HiCuM compact model simulation. For geometry scalable parameter extraction, we employed a dedicated macro model containing the scaling laws for all HiCuM Level 2.0 (L2) parameters related to emitter dimensions. Before the extraction of the model parameters using HiCuM/L2, some technological parameter values are declared and/or pre-calculated such as, for instance, sheet resistances, zero-bias hole charge, area and perimeter dimensions as well as given design rules and process layout information [12]. Parameters related to low-injection conditions are first extracted [12] since at high-current regime, a strong correlation exists between model parameters. Hence, the next step is the junction capacitances extraction. To extract the junction capacitances, the \"cold \(S\)-parameters\" method has been used, in which the transistor is biased to have negligible DC current flow, rendering forward bias across each junction limited to typically less than 0.5 V. The measured \(S\)-parameters under these conditions are then converted into \(Y\) parameters from which, after proper de-embedding, the depletion capacitances can be derived. The scalable extraction of collector and base current parameters is performed at low injection to prevent model parameter correlation usually in play between high-injection parameters. Considering the collector current split into its internal region and the periphery of the emitter window dimensions:  
\[I_{C}=I_{CA}\ A_{E0}+I_{CP}\ P_{E0} \tag{5}\]  
here \(A_{E0}\) is the effective emitter area (m\({}^{2}\)) and \(P_{E0}\) is the effective perimeter (in m). \(I_{CA}\) (A.m\({}^{-2}\)) and \(I_{CP}\) (in A.m\({}^{-1}\)) are the contributions from the intrinsic region and periphery of the device, respectively. Once the low-injection parameters are extracted, model parameters related to the high current effects are considered. The starting point is the extraction of the transit time at \(V_{BC}=0\) V using the already extracted low-injection parameters. The following step is to take into consideration the high current as well as self-heating effects when extracting the transit time. The first step of the transit time parameter extraction method consists in determining the transit frequency from \(S\)-parameter measurements at various collector current densities and various voltages \(V_{CE}\) or \(V_{BC}\). The transit frequency \(f_{T}\) is determined using the current gain bandwidth product by e.g., choosing a spot frequency in the \(-20\) dB/decade roll-off region of the current gain. The second step is the determination of the low-current transit time \(\tau_{f0}\) and the transit time increase \(\Delta\tau_{f}\) at high current densities. Then, in the high-current region, HiCuM/L2 model parameters are extracted by optimizing the region where \(f_{T}\) reaches the peak along with the Gummel plot high-\(V_{BE}\) region. The extraction of thermal resistance is of particular importance [18] since it is strongly coupled with other model parameters in the high-current regime. To obtain a good estimation of the self-heating effects, we used the intersection method [14] to extract the values of R\({}_{\text{TH}}\) from measurements. Additionally, a more detailed thermal impedance model has been adopted to take into account dynamic self-heating, which is detailed in Section 5. In HiCuM/L2, the temperature dependence of parameters for the collector current and emitter back-injection component of the base current is described through the associated saturation currents [12], while the temperature dependence of the zero bias hole charge is fixed by the previously determined capacitance parameters. In order to extract the model parameters, collector and base current measurements with respect to temperature are employed. Gummel plots at \(V_{BC}\) = 0 V are well suited for this purpose.