An equivalent circuit model and biasing effects over the gain and bandwidth of waveguide avalanche photodetectors (WG-APDs)

Abstract

Waveguide photodetectors (WG-PDs) are a promising candidate to overcome the trade-off between the quantum efficiency and bandwidth in conventional photodetectors. In this paper, complete physics and circuit models of waveguide-separated absorption grading charge multiplication-avalanche photodetector (WG-SAGCM-APD) are presented. In this model, the effects of the biasing, doping and the thicknesses of the different layers of the photodetectors, the parasitic of the photodetector and the velocities of photogenerated carriers over the multiplication gain and the bandwidth of the photodetectors are studied and taken into consideration. The results obtained the presented model of WG-SAGCM-APD are compared with published experimental results and a better agreement than with a previous model is obtained. The presented circuit model is generic and can be applied to different structures of WG-APDs.

Appendix

The Generation $G_{0}$ in the absorption layer is:

$$\begin{aligned} G_{0}= & {} \frac{P_{i}}{h\nu }.\eta \end{aligned}$$

(19)

$$\begin{aligned} \eta= & {} K(1-R)(1-\exp (-\alpha \Gamma L) \end{aligned}$$

(20)

where $\eta$ is the quantum efficiency as expressed in Alping (1989) that depend on K (coupling efficiency), R (reflectivity of the absorption layer), $\Gamma$(confinement factor of the absorption layer), and the L (length of the WGPD). The time response of the primary electrons $N_{P}(t)$:

$$\begin{aligned} N_{P}(t)= & {} G_{0}\left( u(t)-u\left( t-\frac{X_{g2}}{V_{g2n}}-\frac{X_{c}}{V_{cn}}-\frac{X_{m}}{V_{mn}}\right) \right) \\&+G_{0}\left( \frac{X_{a}+V_{an}\left( \frac{X_{g2}}{V_{g2n}}-\frac{X_{c}}{V_{cn}}-\frac{X_{m}}{V_{mn}}\right) -Van\mathrm {.}t}{Xa}\right) \\&\quad \cdot \left[ u\left( t-\frac{X_{g2}}{V_{g2n}}-\frac{X_{c}}{V_{cn}}-\frac{X_{m}}{V_{mn}}\right) \right. \\&\quad -\left. u\left( t-\frac{X_{g2}}{V_{g2n}}-\frac{X_{c}}{V_{cn}}-\frac{X_{m}}{V_{mn}}-\frac{X_{a}}{V_{an}}\right) \right] \end{aligned}$$

(21)

similarly, the time response of the the primary holes $P_{P}(t)$ can be calculated.

The time response of the secondary electrons $N_{s}(t)$ in the first case of the $X_{a}>\left( X_{m}-W_{d}\right)$

$$\begin{aligned} N_{s}(t)= & {} G_{0}(M_{e}-1)\cdot \left( \frac{\exp \left( -\frac{t}{(M_{e}-1)\tau _{n})}\right) }{(M_{e}-1)\tau _{n})}\right) \\&\quad \cdot \left\{ \left[ \left( \frac{V_{an}t-V_{an}\left( \frac{X_{g2}}{V_{g2n}}+\frac{X_{c}}{V_{cn}}+\frac{W_{d}}{V_{mn}}\right) }{Xa}\right) \right. \right. \\&\quad \cdot \left( u\left( t-\frac{X_{g2}}{V_{g2n}}-\frac{X_{c}}{V_{cn}}-\frac{W_{d}}{V_{mn}}\right) \right. \\&-\left. \left. u\left( t-\frac{X_{g2}}{V_{g2n}}-\frac{X_{c}}{V_{cn}}-\frac{X_{m}}{V_{mn}}\right) \right) \right] \\&+\frac{X_{m}-W_{d}}{X_{a}}.\left[ u\left( t-\frac{X_{g2}}{V_{g2n}}-\frac{X_{c}}{V_{cn}}-\frac{X_{m}}{V_{mn}}\right) \right. \\&-\left. u\left( t-\frac{X_{g2}}{V_{g2n}}-\frac{X_{c}}{V_{cn}}-\frac{X_{a}}{V_{an}}-\frac{W_{d}}{V_{mn}}\right) \right] \\&+\left[ \left( \frac{X_{a}+V_{an}\left( \frac{X_{g2}}{V_{g2n}}+\frac{X_{c}}{V_{cn}}+\frac{X_{m}}{V_{mn}}\right) -Van\mathrm {.}t}{Xa}\right) \right. \\&\quad \cdot \left( u\left( t-\frac{X_{g2}}{V_{g2n}}-\frac{X_{c}}{V_{cn}}-\frac{W_{d}}{V_{mn}}-\frac{X_{a}}{V_{an}}\right) \right. \\&-\left. \left. \left. u\left( t-\frac{X_{g2}}{V_{g2n}}-\frac{X_{c}}{V_{cn}}-\frac{X_{m}}{V_{mn}}-\frac{X_{a}}{V_{an}}\right) \right) \right] \right\} \end{aligned}$$

(22)

similarly, the time response of the secondary holes $P_{s}(t)$ can be calculated. Where $V_{jn}$ and $V_{jp}$ are the velocity of the electrons and the holes in layer j, respectively. Similarly, the secondary electrons $N_{s}(t)$ can be calculated in the second case of $X_{a}<\left( X_{m}-W_{d}\right)$, while the secondary holes equation$P_{s}(t)$ is still the same in the first case.

The frequency response of the primary holes $P_{P}(\omega )$ and primary electrons $N_{P}(\omega )$ are calculated by applying a Fourier transform on their time response:

$$\begin{aligned} P_{P}(\omega )= & {} \frac{G_{0}}{j\omega }+\frac{G_{0}V_{ap}}{x_{a}\omega ^{2}}\left[ \exp \left( \frac{-j\omega X_{g1}}{V_{g1p}}\right) \right. \\&\qquad -\left. \exp \left( -j\omega \left( \frac{X_{g1}}{V_{g1p}}+\frac{X_{a}}{V_{ap}}\right) \right) \right] \end{aligned}$$

(23)

$$\begin{aligned} N_{P}(\omega )= \frac{G_{0}}{j\omega } +\frac{G_{0}V_{an}}{x_{a}\omega ^{2}}\left[ \exp \left( -j\omega \left(\frac{X_{g2}}{V_{g2n}}+\frac{X_{m}}{V_{mn}}+\frac{X_{c}}{V_{cn}}\right) \right) - \exp \left( -j\omega \left( \frac{X_{g2}}{V_{g2n}}+\frac{X_{m}}{V_{mn}}+\frac{X_{c}}{V_{cn}}+\frac{X_{a}}{V_{an}}\right) \right) \right] \end{aligned}$$

(24)

Similarly, the frequency response of the secondary holes $P_{S}\left( \omega \right)$ and electrons $N_{S}\left( \omega \right)$ are calculated by applying a Fourier transform on their time response:

$$\begin{aligned} P_{S}\left( \omega \right)= & {} \frac{G_{0}(M_{e}-1)V_{an}}{\left( 1+j\omega \tau _{n}(M_{e}-1)\right) x_{a}\omega ^{2}} \\&\quad \cdot \left\{ \exp \left[ -j\omega \left( \frac{Xg2}{V_{g2n}}+\frac{Xa}{V_{an}}+\frac{Xc}{V_{cn}}+\frac{W_{d}}{V_{mn}}\right) \right] \right. \\&-\exp \left[ -j\omega \left( \frac{Xg2}{V_{g2n}}+\frac{W_{d}}{V_{mn}}+\frac{Xc}{V_{cn}}\right) \right] \\&+\exp \left[ -j\omega \left( \frac{Xg2}{V_{g2n}}+\frac{W_{d}}{V_{mn}}+\frac{Xc}{V_{cn}}+\frac{W_{d}}{V_{mp}}\right. \right. \\&\quad +\left. \left. \frac{Xc}{V_{cp}}+\frac{Xg2}{V_{g2p}}+\frac{Xa}{V_{ap}}+\frac{Xg1}{V_{g1p}}\right) \right] \\&-\exp \left[ -j\omega \left( \frac{Xg2}{V_{g2n}}+\frac{W_{d}}{V_{mn}}+\frac{Xc}{V_{cn}}+\frac{W_{d}}{V_{mp}}\right. \right. \\&\quad +\left. \left. \left. \frac{Xc}{V_{cp}}+\frac{Xg2}{V_{g2p}}+\frac{Xa}{V_{ap}}+\frac{Xg1}{V_{g1p}}+\frac{Xa}{V_{an}}\right) \right] \right\} \end{aligned}$$

(25)

$$\begin{aligned} N_{S}\left( \omega \right)= & {} \frac{G_{0}(M_{e}-1)}{1+j\omega \tau _{n}(M_{e}-1)} \\&\quad \cdot \left\{ \frac{-V_{an}}{x_{a}\omega ^{2}}\exp \left[ -j\omega \left( \frac{Xg2}{V_{g2n}}+\frac{Xc}{V_{cn}}+\frac{W_{d}}{V_{mn}}\right) \right] \right. \\&\quad +\,\frac{1}{j\omega X_{a}}\left[ 1-\frac{V_{an}}{V_{mn}}\right] \left( X_{m}-W_{d}\right) \left( \exp \left[ -j\omega \left( \frac{Xg2}{V_{g2n}}\right. \right. \right. \\&\quad +\,\left. \left. \frac{Xc}{V_{cn}}+\frac{X_{m}}{V_{mn}}\right) \right] \\&-\left. \exp \left[ -j\omega \left( \frac{Xg2}{V_{g2n}}+\frac{Xc}{V_{cn}}+\frac{Xa}{V_{an}}+\frac{W_{d}}{V_{mn}}\right) \right] \right) \\&+\frac{V_{an}}{x_{a}\omega ^{2}}\left( \exp \left[ -j\omega \left( \frac{Xg2}{V_{g2n}}+\frac{Xc}{V_{cn}}+\frac{X_{m}}{V_{mn}}\right) \right] \right. \\&+\left. \exp \left[ -j\omega \left( \frac{Xg2}{V_{g2n}}+\frac{Xc}{V_{cn}}+\frac{Xa}{V_{an}}+\frac{W_{d}}{V_{mn}}\right) \right] \right) \\&-\left. \frac{V_{an}}{x_{a}\omega ^{2}}\exp \left[ -j\omega \left( \frac{Xg2}{V_{g2n}}+\frac{Xc}{V_{cn}}+\frac{X_{m}}{V_{mn}}+\frac{Xa}{V_{an}}\right) \right] \right\} \end{aligned}$$

(26)

Similarly, in the second case of $X_{a}<\left( X_{m}-W_{d}\right)$, the secondary electrons $N_{s}(\omega )$ can be calculated by applying a Fourier transform on its time response: