An important characteristic of any photodetector is its response time the time it takes for the detector output to change in response to changes in the input light intensity. It was noted in Chapter 13 that the response time of photoconductive-type detectors is quite poor because of the electron replenishment process, which keeps the induced photocurrent flowing for the duration of the electron's lifetime. In photodiode detectors, this replenishment process is suppressed by the p-n junction, which presents a barrier to the movement of majority carriers. The response time is thereby significantly improved, since the time taken for charge carriers to move through the high-field region of the junction (the carrier transit time) can be much shorter than the carrier lifetime. This improved time response is countered in part, however, by the capacitance of the p-n junction and associated RC time constant. In this section, we consider the implications and relative importance of transit time and capacitance in determining the photodiode response time.

Junction Capacitance

The capacitance of a p-n junction can be evaluated by determining how the charge on either side of the junction changes in response to a changing diode voltage. We will assume for simplicity a highly doped p region and weakly doped n region, so N_A N_Das in Fig. 10-11. In this case, most of the charge depletion region is on the n side (d d_n d_p), and the junction width d is given by Eq. (10-20). When an external voltage V is applied to the diode, the internal potential V₀is replaced by V₀ - V, giving

where V is positive for forward bias and negative for reverse bias. A change V in external voltage causes a change in junction width of

as can be verified by taking the differential of both sides of Eq. (14-20). Most of this change in width occurs on the n side of the junction, since d_n d_p. Writing the charge of the uncovered ion cores in the n region as Q = eN_D(Ad_n), the change in this charge is

There is an equal but opposite change in the charge on the p side of the junction. Combining Eqs. (14-21) and (14-22), the junction capacitance is then

which is the familiar expression for the capacitance of a parallel-plate capacitor of plate area A and spacing d, separated by a medium with dielectric constant . Using Eq. (14-20) for d gives the result

For reverse-bias voltage V=-V_B , this becomes

It is seen that the capacitance of the p-n junction is not constant, but rather decreases with increasing V_B . This is a consequence of the junction width d increasing with V_B .

The junction capacitance can be considered to be in parallel with the diodes in Fig. 14-2, which leads to a first-order RC circuit time response. If the incident power is suddenly switched from zero to some constant value at t = 0, the output voltage increases exponentially in time according to

which is shown graphically in Fig. 14-9. The product RC is known as the time constant of the circuit, and has units of seconds with R in ohms and C in farads. The time constant measures how quickly the output responds to a changing input. For example, at time t = RC the output has risen to 63.2% of the final steady-state value, and at time t = 2RC it has risen to 86.5% of this value.

Figure 14-9 In an exponential rise, the time constant RC gives the time taken for the output to reach 63.2% of the final steady-state value. The rise time t_ris the time taken for the output to go from 10% to 90% of the final value.

An alternative measure of the time response is the rise time, defined as the time taken to rise from 10% to 90% of the final value. For the RC circuit, it is straightforward to show (see Problem 14.6) that

The rise time is a more general definition for time response than the time constant, because it applies equally well to an exponential or nonexponential time dependence. When the time response is nonexponential, Eq. (14-27) can be interpreted as defining an effective time constant in terms of the rise time.

The rise time or RC time constant characterizes the detector time response to a step-function intensity modulation, in which the incident light intensity is suddenly changed from one value to another. If instead the incident light intensity is sinusoidally modulated, then the output will be sinusoidally modulated as well. These two different types of modulation were discussed in connection with the LED time response (see Fig. 11-3). The output amplitude is approximately independent of frequency up to a limiting upper value, the bandwidth, above which the amplitude becomes smaller. Denoting the 3 dB electrical bandwidth as B, we have

where Eq. (11-13) has been used with = RC and B =f_e. The bandwidth can be written in terms of the rise time as

using Eq. (14-27). This last expression can be taken as defining the 3 dB bandwidth in the case of nonexponential time response.

The above results show that a smaller capacitance leads to a faster time response and larger bandwidth, which is usually desirable for a photodetector. According to Eq. (14-25), there are several parameters that can be adjusted to reduce the capacitance. For example, the reverse-bias voltage V_Bcan be increased. This makes the photocon-ductive mode (V_B > 0) inherently faster than the photovoltaic mode (V_B = 0). Indeed, this is another characteristic advantage of the photoconductive mode, in addition to the increased dynamic range that was discussed earlier. There is a practical limit to V_B, however, due to electrical breakdown in the junction. Typical reverse-bias voltages are 5-10 V.

Another way to reduce the capacitance is to decrease the junction area A. Smaller detector areas, therefore, give a faster time response. The downside of this approach is that it may not be possible to direct all the available light onto the semiconductor material if A is too small. The ability to focus light onto a small detector area is governed by the brightness of the light source, as discussed in Appendix A (see also Chapter 15). High-brightness sources such as a laser can be focused to a very small area (~ ²), which allows a small detector area to be used without loss of efficiency. Low-brightness sources such as an LED or incandenscent filament, however, can be imaged only onto a much larger area. If a photodiode with small A is used to detect light of low brightness, much of the light to be measured will not strike the detector's active area, making the detector less sensitive. This results in a sensitivity-speed trade-off, which must be optimized for best performance in a particular application.

The other parameter in Eq. (14-25) that can be adjusted to reduce the capacitance is N_D, the density of donors on the weakly doped n side of the junction. The capacitance is reduced when N_Dis made smaller, because the junction width d then increases. This is one reason that in most photodiodes, one side of the junction is very weakly doped. There are other reasons for this also, as we will soon see.

Reducing the capacitance is one way to reduce the RC time constant, but it is not the only way. Reducing the load resistance R_Lhas the same effect, although this decreases the detector sensitivity, as seen in Eq. (14-19) and Fig. 14-8. Large R_Lis best for high sensitivity, and small R_Lis best for high speed, resulting in another sensitivity-speed trade-off. The dynamic range is also better for small R_L, as previously discussed. These various trade-offs are summarized in Fig. 14-10.