The and the load resistance RL. Ct is

The speed of response of a photodiode measure the time required for the accumulated charge to become an external current and is generally expressed as the rise time or cut-off frequency. The rise time is the time required for the output signal to change from 10% to 90% of the peak output value and is determined by the some factors. The factors are terminal capacitance Ct and the time constant t1 of load resistance RL, diffusion time t2 of carriers generated outside the depletion layer and carrier transmit time t3 in the depletion layer. Time constant t1 determined by the terminal capacitance Ct of the photodiode and the load resistance RL. Ct is the sum of the package capacitance and the photodiode junction capacitance. t1 is given by:t1 = 2.2 × Ct × RL                                                                                                   Eq 5    To shorten t1, the design must be such that either Ct or RL is made smaller. Cj is nearly proportional to the active area A and inversely proportional to the second to third root of the depletion layer width d. Since the depletion layer width is proportional to the product of the resistivity ? of the substrate material and reverse voltage VR, the following equation is established as:Cj ?  A {(VR + 0.5) × ?} -1/2 to -1/3                                                                      Eq 6Accordingly to shorten t1, a photodiode with small A and large ? should be used with a reverse voltage applied. However, reverse voltage also increases dark current so caution is necessary for use in low-light-level detection. Next the diffusion time t2 of carriers generated outside the depletion layer, carriers may generateoutside the depletion layer when the incident light misses the P-N junction and is absorbed by the substrate section which is below the depletion layer. The time t2 required for these carriers to diffuse may soetimes be greater than several microseconds. Then carrier transmit time t3 in the depletion layer, the transit speed vd at which the carriers travel in the depletion layer is expressed using the traveling rate ? and the electric field E developed in the depletion layer, as in vd = ?E. If the depletion layer witdh be d and the applued voltage be VR, the average electric field E= VR/d, and thus t3 can be approximated as follows:t3 = d /vd = d2 / (? VR)                                                                                           Eq 7To achieve a fast response time for t3, the moving distance of carriers should be short and the reverse voltage larger. The above three factors determine the rise time tr is approximated by the following equation:tr =  ?(?t_1?^2+?t_2?^2+?t_3?^2  )                                                                                             Eq 8PIN photodiodes and avalanche photodiodes are designed such that less carriers are generated outside the depletion layer, Ct is small and the carrier transit time in the depletion layer is short. Therefore, these types are ideally suited for the high speed light detection. The cut-off frequency is the frequency at which the photodiode output decreases by 3 dB from the output at100kHz when the photodiode receives sinewave modulated light from a laser diode. The rise time tr roughly approximates this fc in the formula:tr = 0.35/f_c                                                                                                                      Eq 9