## 2 Compact Model Development
### Model Formulation  
The first compact model developed in [11] serves as the starting point of the modeling framework presented in this work. In order to make the model versatile across different technologies and geometries, improvements in model equations and implementations have been made. In the following sections, the details of the principal improvements are presented. Firstly, the direct current contribution, \(I_{F}\), has been slightly modified to take into account the two operating regimes: low and high injections [15]. This formulation, conventionally used for SPICE modeling of PN and PIN diodes, accounts for the change in the slope of the current-voltage characteristic at the knee current (\(I_{K}\)), as the type of injected charge carrier changes with the increasing forward bias. Thus, the direct current equation becomes the following:  
\[I_{F}=\frac{I_{D}}{1+\sqrt{\frac{I_{D}}{I_{K}}}} \tag{1}\]  
Here, \(I_{D}\) is the classic diode equation [16] and \(I_{K}\) is the knee current parameter that signifies the transition between the medium and high injection regimes. Among other improvements, we incorporate a more detailed description of the of the series resistances through decorrelation of the resistive contributions as functions of the applied bias and device geometry. In the first version of the model, only one series resistance, independent of the bias voltage, was considered. It reflects the resistance of the contacts and the spreading resistance. However, the thickness and the doping level of the collector varies from one technology to another (usually \(N^{-}\) doped). When the collector is not fully depleted, an additional bias-dependent resistance appears which, when the collector is fully depleted, becomes negligible. In our new model, therefore, these two contributions are separated. The dependence of the resistance of the active region, \(R_{N^{-}}\), on the applied voltage, \(V_{d}\), can be expressed as follows:  
\[R_{N^{-}}(V_{d})=R_{N^{-}}(0)\frac{w_{T}-w_{scr}(V_{d})}{w_{T}-w_{SCR0}} \tag{2}\]  
where \(R_{N^{-}}(0)\) is the resistance of the undepleted collector at _zero_ bias, \(w_{T}\) is the width of the collector and \(w_{SCR0}\) is the width of the space charge region at _zero_ bias. \(w_{scr}(V_{d})\) and \(R_{N^{-}}(0)\) can be expressed with the following set of equations [17]:  
\[w_{scr}(V_{d})=w_{scr0}\Bigg{(}1-\frac{V_{d}}{V_{j}}\Bigg{)}^{Mmr} \tag{3}\]  
Figure 2: (**a**) SEM micrograph of a GaInAsSb/InP UTC-PD; (**b**) Schematic of the intrinsic mesa UTC-PD showing the location of the equivalent circuit elements; (**c**) the Verilog-A topology of the UTC-PD photocurrent transfer function; (**d**) Top-view SEM micrograph of a GaInAsSb/InP UTC-PD connected to the GSG pads. \[R_{N^{-}}(0)=\frac{w_{T}-w_{SCR0}}{q.\mu.N^{-}.A} \tag{4}\]  
where \(V_{j}\) is the junction potential and \(M_{mr}\) is the grading coefficient at medium reverse bias of the \(P-N^{-}\) junction, \(q\) is the electronic charge, \(\mu\) is the electron carrier mobility in the collector, \(N^{-}\) is the doping concentration in the collector and \(A\) is the UTC-PD active area. The classical model of the bias-dependence of the junction capacitance has evolved into a more mature form in the HiCuM/L2 model. We refer here mainly to the formulation for the base-collector junction. Considering different thicknesses, the collector may not be completely depleted at equilibrium. In this case, the junction capacitance follows the conventional model [17] until the collector is completely depleted, the voltage drop becomes small due to the high doping level of the waveguide (or n-contact).