## 6 High Frequency Characterization and Modelling
### TCAD and Monte Carlo Simulations  
To strengthen the physical basis of the developed compact models, numerical simulations can be used to unravel the principles of additional effects. For example, the high-frequency \(S\)-parameters show a deviation from the expected behavior especially at very high \(V_{CE}\) and beyond 200 GHz, in comparison with the measurements obtained at low to medium \(V_{CE}\) (around 1V). The observed discrepancies between the experimental measurements and the HiCuM model at high \(V_{\text{CE}}\) have been investigated using a calibrated multi-scale TCAD model [26]. By combining a full-band, atomistic, and ballistic quantum transport (QT) solver [27] with the hydrodynamic (HD) model of the Sentaurus Device (S-Device) tool [28], qualitatively and quantitatively reliable agreements between the simulated and the experimental data can be produced. Utilizing the configured simulation framework, the influence of a high electric field in the collector region is studied. Figure 20 shows the bulk bandstructure of In\({}_{0.53}\)Ga\({}_{0.47}\)As, as calculated with tight-binding after calibrating the model with the recommended band gap energies of Ref. [29]. The position of the different valleys of In\({}_{x}\)Ga\({}_{1-x}\)As are plotted as a function of the Indium composition  
Figure 19: Measured (symbol) and simulated (line) normalized frequency response of UTC-PD with geometries of 5 \(\times\) 15 \(\upmu\)m\({}^{2}\) (red symbol), 5 \(\times\) 30 \(\upmu\)m\({}^{2}\) (green symbol) and 7 \(\times\) 25 \(\upmu\)m\({}^{2}\) (blue symbol). in Figure 20b. Note that to achieve a balanced agreement among all targeted parameters, slight discrepancies exist in the calculated and recommended bandgaps, especially in \(E_{X}\). However, the bowing parameters obtained from our tight-binding approach remains in the recommended range. After calibrating the bandstructure model, the conduction band diagrams of the considered DHBT have been simulated at different \(V_{CE}\) with S-Device, as shown in Figure 21a. As \(V_{CE}\) goes from 1 to 2 V, the conduction bands in the collector region are pushed downwards and the electric field at the base-collector interface increases. These potential profiles have then been inserted into a QT solver to compute the electron distribution inside the active E-B-C domain. Results are summarized in Figure 21b. For \(V_{CE}>1.6\) V, a significant increase in the electron concentration inside the collector region is observed. The origin of this phenomenon has been investigated utilizing the energy and spatially resolved electronic distribution inside the simulated structure, as depicted in Figure 21c,d. At low \(V_{CE}\) (Figure 21c), most of the electrons are confined in the emitter region, while the base and collector electron concentrations are almost constant and relatively low. However, as the electric field increases with \(V_{CE}\), electrons situated in the \(\Gamma\) valley start to be transferred to two of the six \(X\) valleys, those that can be reached without scattering, i.e., those whose principle axis is aligned with the electron transport direction. This can be directly observed in Figure 21d. In contrast to Figure 21c, the region with the highest electron density in Figure 21d is located in both the emitter region and the \(X\) valley of the collector. The transfer of electrons from the \(\Gamma\) to the \(X\) valleys is commonly observed in III-V materials, where the electronic occupancy probability of the \(X\) valleys increases with electric field [29]. It should be noticed that the transition of electrons from the \(\Gamma\) to the \(L\) valleys could also contribute to the observed charge increase, but this would require the absorption or emission of a phonon to compensate for the momentum mismatch. The probability for such events to occur is much lower than the elastic transfer of electrons from \(\Gamma\) to \(X\). A discrepancy between the experimental and theoretical onset of this effect can nevertheless be observed.