In the case of the solar cell just discussed, the primary goal is to convert as much optical power as possible into electrical power. When the photodiode is used as a light detector, however, it is generally more important that the detector output be linear with the incident light power. In this section, we examine the linearity of photodiode detector circuits using the two types of biasing modes.

Photovoltaic Mode

Consider first the photovoltaic bias mode shown in Fig. 14-2a. When R is very large (open-circuit condition), the load line is nearly horizontal, and the operating point is close to the V_daxis where i 0. In that case, Eq. (14-3) becomes

Figure 14-4 Variation of solar cell efficiency with load resistance for Example 14-1. Area of the cell is 4 cm². Optimum efficiency is higher and occurs at a higher load resistance when the diode ideality factor ( ) is 2 rather than 1.

with i given by Eq. (14-2). Solving for the diode voltage gives

If the induced photocurrent is much greater than the dark current ( > i₀), this becomes

where

The diode voltage, therefore, varies logarithmically with the incident power for i i₀. For i i₀, however,

Defining the quantity

the diode voltage can be written as

This result suggests that the photodiode can be modeled as an ideal current source connected in parallel with a resistor R_sh, as depicted in Fig. 14-5. Since i = 0 for an open circuit,

Figure 14-5 When i i₀, the photodiode can be modeled as an ideal current source in parallel with a shunt resistance R_sh.

the diode voltage becomes V_d = i R_sh, in agreement with Eq. (14-11). Since R_shappears in parallel with the current source, it is termed a shunt resistance. Higher values of R_share generally desirable, because the detector is then more sensitive to weak light signals (V_d large for small i ).

Values of shunt resistance vary widely, and are higher for wider bandgap materials, for which i₀is smaller. R_shis also higher at lower temperature (less thermal generation of electron-hole pairs) and for smaller junction area (since i₀ = J₀A). For a typical room-temperature silicon photodiode with 1 cm² area, R_sh 10 M .

The response of the open-circuit photodiode to varying optical powers can be summarized as follows. At low incident power levels the response is linear with power, whereas at high power levels the response becomes logarithmic. The transition between these two regimes corresponds to i ~ i₀, which is equivalent to i R_sh ~ V_T. This deviation from linearity at high optical powers is referred to as saturation of the output signal, and is generally to be avoided.

To increase the range of optical powers over which the photodiode response is linear, the load resistance R_Lcan be made small. This makes the load line in Fig. 14-3 nearly vertical, intersecting the diode curves close to the current axis (V_d 0). Since the diode curves are approximately evenly spaced for V_d 0, the operating point moves downward along the -i axis in proportion to the optical power. The voltage across the resistor, V_R = iR_L, is therefore linear with the optical power, as desired.

This conclusion can also be arrived at analytically. If V_dV_T, the approximation e^x 1 + x allows Eq. (14-3) to be written as

or

Using V_d = -iR_Lfor the photovoltaic mode (Fig. 14-2a) and solving for i gives

Since i P_in, we conclude that V_R = iR_L in, as desired.

The equations above can be understood in terms of the equivalent circuit shown in Fig. 14-6. As before, the photodiode is represented as an ideal current source shunted by the resistance R_sh. The load resistor R_Lis now connected in parallel with both of these. Defining the current i_shthrough the shunt resistance to be positive in the downward direction, we have by the junction rule

This is equivalent to Eq. (14-12), using i_sh = V_d / R_sh. It is left as an exercise to show that Eq. (14-13) can be derived from this equivalent circuit model.

When R_L R_sh, Eq. (14-13) becomes i -i , and the diode voltage is

Figure 14-6 Equivalent circuit for a photodiode biased with load resistor R_L. This model is valid when V_d V_T

Defining the output of the detector as , we then have

where Eq. (14-2) has been used.

This result shows that under the two specified conditions the detector voltage V_dis linear not only with the incident optical power but also with the load resistance. In practice, it is easier to measure a larger voltage, so a larger R_Lis desirable. However, as R_Lis increased, one of these two conditions will eventually break down, and V_dwill no longer increase with R_L. One possibility is that R_L R_shbreaks down, while the condition still holds. In the limit where R_L R_sh, Eq. (14-13) gives

which becomes independent of R_L. This is the same result obtained in Eq. (14-11) for the open-circuit condition R_L . Although the detector output V_out is no longer linear with R_Lin this case, it is still linear with P_in.

The other possibility as R_Lincreases is that the condition , we then have breaks down first, in which case the exact expression in Eq. (14-3) must be used. Under these conditions, the output V_out is no longer linear with either R_Lor P_in. According to Eq. (14-14), this saturation will occur at a certain value of the product P_inRL. For higher R_L, saturation occurs at a lower P_in, and at higher P_in, saturation occurs at a lower R_L. There is, therefore, a trade-off between sensitivity (large output for small input) and dynamic range (range of inputs for which response is linear). The saturation with incident power for different values of load resistance is illustrated in Fig. 14-7.