# Advances in High-Speed, High-Power Photodiodes: From Fundamentals to Applications
## 2 Overall Design Considerations for Photodiodes
### Equivalent Circuit Models  
The time-delay equivalent circuit model for air-bridge type waveguide PIN-PD was thoroughly analyzed by [PERSON] et al. [84; 85; 86], as shown in Figure 8. Fundamentally, a basic air-bridge typed-PD is a current generator, and the corresponding equivalent circuit is composed of four parts, as shown in Figure 9, including the PD itself, the air bridge, the coplanar waveguide (CPW), and the 50 \(\Omega\) load [86; 87]. The photocurrent can be obtained and simulated using a current source, \(I_{\mathrm{ph}}\), which is in parallel with the junction capacitance, \(C_{\mathrm{j}}\), and the large junction shunt resistance (meeophms, usually \(\geq\) 100 k\(\Omega\)), \(R_{\mathrm{j}}\), and in series with a small series resistance, \(R_{\mathrm{s}}\) (or \(R_{\mathrm{c}}\)) (related to \(P^{+}\), \(N^{+}\) layers and ohmic contact resistance of metallic connection), as well as the air-bridge inductance from p-contact to p-electrode pad, \(L_{\mathrm{b}}\) (or \(L_{\mathrm{s}}\)), determined by the bridge geometry. The \(C_{\mathrm{dx}}\) is the capacitance between the air bridge and the \(n\)-electrode pad. The characteristic impedance, \(Z_{\mathrm{c}}\), of the CPW, which can be measured with open and short circuit yielding a p-electrode pad capacitance, \(C_{\mathrm{p}}\) and an \(n\)-electrode pad-induced pad inductance, \(L_{\mathrm{p}}\) (or \(L_{\mathrm{G}}\)), is calculated by \(Z_{\mathrm{c}}=\sqrt{L_{\mathrm{p}}/C_{\mathrm{p}}}\) with a common value of 50 \(\Omega\) (e.g., \(L_{\mathrm{p}}=55\) pH, \(C_{\mathrm{p}}=22\) fF, so \(Z_{\mathrm{c}}=50\)\(\Omega\) in [87]). One thing to note is that some PDs do not use air bridge during the device fabrication process, so the parasitic capacitance between the p-electrode and \(n\)-electrode, \(C_{\mathrm{dx}}\) and air-bridge inductance, \(L_{\mathrm{b}}\) (or \(L_{\mathrm{s}}\)) could not be used in the equivalent circuit model, as shown in Figure 10.  
Figure 7: The calculated electron traveling time in the absorber and collector on 3–dB bandwidth (BW) [82; 44; 83]. (**a**) The electron traveling time versus the absorber and collector thickness (assume \(W_{a}\cong W_{c}\)); (**b**) relationship between the 3–dB bandwidth and absorber/collector thickness; (**c**) comparisons of the calculated electron diffusivity versus p–doping concentration in the absorber [83]; and (**d**) summary of three p–dopings in the absorber with the calculated values. To evaluate the opto-microwave conversion properties of high-speed PDs by using S-parameters (such as reflection coefficients S22 and optoelectronic conversion properties S21), a small-signal radio frequency (RF) equivalent circuit for PD is reported considering synthetically both the analysis of the carrier transit effect [88] and the external parasitic resistance-capacitance (RC) delay-time constant [85]. The small-signal RF equivalent circuit is shown in Figure 11, which uses the voltage (RF voltage, \(\mathrm{V_{ac}}\))-controlled current source (VCCS, \(\mathrm{I_{ac}}=\mathrm{g_{m}V_{ac}}\), \(\mathrm{g_{m}}\) is denoted as transconductance representing optical-to-RF conversion quantum efficiency, i.e., responsivity) to replace of the current source, \(\mathrm{I_{ph}}\) in Figures 9 and 10, while the overall frequency response (O-E 3-dB bandwidth) of PD is determined by both of the transit time and the RC-delay time constant, i.e., \(1/\hat{\mathrm{f}}_{\mathrm{3 dB}}^{2}=1/\hat{\mathrm{f}}_{\mathrm{tr}}^{2}+1/ \hat{\mathrm{f}}_{\mathrm{RC}}^{2}\). The transit-time bandwidth, \(\mathrm{f_{tr}}=1/(2\pi\mathrm{\tau_{tr}})\) of ac-photocurrent, \(\mathrm{I_{ac}}\) is determined by the carrier transit-time effect (\(\mathrm{\tau_{tr}}=\mathrm{R_{c}C_{t}}\)), which is indirectly affected by selecting the values of input circuit elements of \(\mathrm{C_{t}}\), \(\mathrm{R_{t}}\), and \(\mathrm{g_{m}}\). The signal source impedance is \(\mathrm{R_{0}}\). The RC-delay time constant bandwidth can be calculated using Equation (6).