# Advances in High-Speed, High-Power Photodiodes: From Fundamentals to Applications
## 2 Overall Design Considerations for Photodiodes
### 3-dB Bandwidth Analysis and Discussion  
Response time is a physical quantity that characterizes how quickly a PD responds to an incident light signal, that is, it represents the total time it takes for a PD to convert an optical signal into an electrical signal. The total response time of PD generally consists of three parts: (a) diffusion time of electrons in the absorption layer, \(\tau_{a}\), (b) circuit time constant, \(\tau_{\text{RC}}\), determined by junction capacitance and load resistance, and (c) drift time of electrons in the collection layer, \(\tau_{\text{c}}\)[44]. Figure 3: (a) The cutoff wavelength–related refractive index and permittivity for In\({}_{1-x}\)Ga\({}_{x}\)As\({}_{y}\)P\({}_{1-y}\) with the As–composition changed from 0 to 1. (b) Solutions for suppressing the bandgap discontinuity by compositional graded quaternary material and dipole–doping regimes. PIN-PD is usually made up of parts (a) and (b), while (c) belongs to UTC-PD [44; 53; 54]. Therefore, the total response time for general UTC-PD can be given as  
\[\tau=\sqrt{\tau_{\rm a}^{2}+\tau_{\rm c}^{2}+\tau_{\rm RC}^{2}}, \tag{1}\]  
while the PIN-PD can be dealt with as a special case, just having a single intrinsic/depletion layer, \(\rm W_{a}\). The RC-delay time-limited bandwidth, \(\rm f_{RC}\), and transit-time-limited bandwidth, \(\rm f_{RC}\), usually consist of the main 3-dB bandwidth of a PD, which here the diffusion is neglected at the high-frequency response. We assume that a PD has one single intrinsic layer (Figure 4) with the i-layer thickness of d and the area of A, so the \(\rm f_{RC}\) can be expressed as  
\[\rm f_{RC}=\frac{1}{2\pi RC}=\frac{1}{2\pi R}\frac{d}{\varepsilon A}, \tag{2}\]  
where the junction capacitance \(\rm C=\varepsilon A/d\) approximately equals PD capacitance (in ideal cases), the \(\varepsilon\) is the relative permittivity of i-layer, and R is the resistance and equals the sum of diode series resistance, \(\rm R_{S}\), and the load resistance, \(\rm R_{L}\), has a typical value of 50 \(\Omega\). It needs to be noted that Equation (2) will be changed if a PD has multiple dielectric/depletion layers and the corresponding values will be replaced by the relative permittivity, \(\varepsilon_{\rm req}\), and the total depletion thickness, \(\rm d_{T}\), respectively, which are shown as follows [55]:  
\[\varepsilon_{\rm req}=\left[\sum_{\rm m=1}^{n}\frac{d_{\rm m}}{d_{\rm T} \varepsilon_{\rm m}}\right]^{-1}, \tag{3}\]  
and  
\[\rm d_{T}=d_{1}+d_{2}+\cdots+d_{n}. \tag{4}\]  
Therefore, the total capacitance, \(\rm C_{T}\), and the RC-limited 3-dB bandwidth, \(\rm f_{3 dB,RC}\), can be rewritten as  
\[\rm C_{T}=\frac{\varepsilon_{0}\varepsilon_{\rm req}A}{d_{T}}=8.854\times 10^ {-6}\frac{\varepsilon_{\rm req}A}{d_{T}}\ \ \ (\rm pF), \tag{5}\]  
and  
\[\rm f_{3 dB,RC}=\frac{1}{2\pi RC_{T}}=\frac{1}{2\pi R}\frac{d_{T}}{\varepsilon_ {0}\varepsilon_{\rm req}A}, \tag{6}\]  
where \(\varepsilon_{0}\) is the vacuum permittivity, with a value of \(8.854\times 10^{-12}\rm F/m\). For a UTC-PD with the same thickness of the absorber and collector, assuming a pure diffusion case without quasi-field, the 3-dB bandwidth can be given as [44; 54]  
\[\rm f_{3 dB,tr}\cong\frac{1}{2\pi\tau_{a}}=\frac{1}{2\pi\Big{(}W_{a}^{2}/3D_{e }+W_{a}/\
u_{th}\Big{)}}, \tag{7}\]  
where \(\rm D_{e}\) is the electron diffusion constant, \(\
u_{th}\) is the thermionic velocity and the related equations and values are shown in Table 1.  
Figure 4: Cross–sectional diagram of PIN–PD and PD symbol. In addition, for the electron drift/delay time-limited bandwidth in the collector of UTC-PD, \(f_{\rm 3 dB,co}\), can be considered as  
\[\tau=\sqrt{\tau_{\rm a}^{2}+\tau_{\rm c}^{2}+\tau_{\rm RC}^{2}}, \tag{1}\]0  
where the electron overshoot velocity in the InP collector is \(v_{\rm e,os}=4\times 10^{7}\) cm/s, while the corresponding electron drift time \(\tau_{\rm c}=W_{\rm c}/v_{\rm e,os}\) which is around 0.5\(\sim\)1 ps for a collector with a thickness of 200-400 nm.