Exact Analytical Formula for the Excess Noise Factor for Mixed Carrier Injection Avalanche Photodiodes

The well-known analytical formula for the excess noise factor associated with avalanche photodiodes (APDs), developed by R. J. McIntyre in 1966, assumes the injection of either an electron or a hole at the edge of the APD's avalanche region. This formula is based on the statistics of the probabilities of carriers gaining and losing energy subject to high electric fields. However, this analytical formula, is not applicable in cases when photons are absorbed inside the avalanche region (even though the physics of the high field transport remains the same), and its use may severely underestimate or overestimate the actual excess noise factor depending on the absorption profile and the hole-to-electron ionization coefficient ratio, k. Here, an easy-to-use exact analytical formula is derived for the excess noise factor of APDs while taking into account a mixed-carrier initiated avalanche multiplication process, which is triggered by a parent electron-hole pair at an arbitrarily specified location within the multiplication region. The derivation relies on analytically solving a special case of a previously reported recursive integral equations [Hayat et al., IEEE Trans. Electron Devices, vol. 39, no. 3, pp. 546-552, Mar. 1992.], and the result matches the formula reported by McIntyre in 1999 using a different and limited technique. In addition, an expression for the excess noise factor is presented in the case when the location of the parent electron-hole pair within the multiplication region obeys an arbitrary exponential distribution. The results show that in contrast to the case of edge parent-electron injection, when mixed injection is allowed even a small level of hole ionization (e.g., small k ~ 0.0001) causes the excess noise factor to increase dramatically, depending on the absorption profile as it ranges from narrow to flat within the multiplication region. The theoretical results are validated against experimental results for Si APDs.