The frequency response of avalanching photodiodes

The Townsend equations for avalanche breakdown in back biased p-n junctions may be derived from the transport equations for semiconductors. Integral solutions of the time independent equations are well known. An integral solution of the time dependent equations is given for multiplication by one carrier only. An exact solution is given for multiplication by two carriers with equal ionization coefficients in a constant junction field. The Townsend equations are nonlinear because of space charge effects. It is shown, however, that the nonlinearity, which imposes an upper limit on the current multiplication possible, is not important until the total multiplied current approaches the space charge limited current for the junction. Assuming multiplication is due to one carrier, frequency response curves are calculated for constant and linear junction fields and for a generation rate, due to photon absorption, which is either uniform or given by a delta function at the junction boundary. The curves indicate a relatively slight dependence of the frequency response on multiplication. Frequency response curves are also given for multiplication by both carriers with equal ionization coefficients when the junction field is constant. In this case the frequency response decreases continuously as the multiplication is increased. For multiplication by two carriers with unequal ionization coefficients, the frequency response is independent of multiplication until the product of the multiplication and the ratio of the ionization coefficients approaches one. Thereafter the frequency response decreases with multiplication.