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A generalized history-dependent recurrence theory for the time-response analysis is derived for avalanche photodiodes with multilayer, heterojunction multiplication regions. The heterojunction multiplication region considered consists of two layers: a high-bandgap Al0.6Ga0.4As energy-buildup layer, which serves to heat up the primary electrons, and a GaAs layer, which serves as the primary avalanching layer. The model is used to optimize the gain-bandwidth product (GBP) by appropriate selection of the width of the energy-buildup layer for a given width of the avalanching layer. The enhanced GBP is a direct consequence of the heating of primary electrons in the energy-buildup layer, which results in a reduced first dead space for the carriers that are injected into the avalanche-active GaAs layer. This effect is akin to the initial-energy effect previously shown to enhance the excess-noise factor characteristics in thin avalanche photodiodes (APDs). Calculations show that the GBP optimization is insensitive to the operational gain and the optimized APD also minimizes the excess-noise factor.