Although there are several types of circuits used to measure the photodiode signal current (see Section 14-5), the way that the photodiode is biased falls into one of two fundamental categories. In the photovoltaic mode (Fig. 14-2a), a load resistor R_Lis directly connected across the photodiode, whereas in the photoconductive mode (Fig. 14-2b), the load resistor is connected through a series bias voltage V_B. In either case, the photocurrent generates a voltage V_Racross the load resistor, which constitutes the detector output signal. The photoconductive mode we are discussing here should not be confused with the photoconductive-type detectors discussed in the previous chapter. The distinction is the presence or absence of a p-n junction in the device.

The current i in the circuit depends not only on the incident light intensity, but also on the values of R_Land V_B. To evaluate this current, we add up the potential changes around the circuit loop of Fig. 14-2b, and set the sum equal to zero (voltage loop law):

V_d+V_B+V_R = 0

We have adopted a sign convention for the diode voltage V_dand the resistor voltage V_R such that positive current flows into the positive side of each element. Of course, the actual values of the current or voltages may be either positive or negative. Writing V_R = iR_L and solving for i, we have

Figure 14-1 In a photodiode detector, the motion of electrons and holes across the charge depletion region causes a current in the external circuit. By convention, current i is defined as positive when it enters the p side of the diode. The photocurrent produced is therefore negative.

Equation (14-1) gives a relation between current i and diode voltage V_dthat is imposed by the external circuit. Another relation between i and V_dcomes from the internal constraints of the diode itself. The i-V relation for a semiconductor diode was given earlier in Eq. (10-21) and Fig. (10-14) for the case of no light absorption. When light is absorbed, the electron-hole pairs that are created cause an additional negative current, termed a photocurrent. The magnitude of the photocurrent is given by Eq. (13-7), which can be written here as

Figure 14-2 (a) In the photovoltaic mode, a load resistor is directly connected across the photodiode. (b) In the photoconductive mode, the load resistor in connected in series with a reverse-bias voltage.

where _abs is the fraction of incident photons that are absorbed to create electron-hole pairs. This photocurrent adds to the diode current of Eq. (10-21) to give a total circuit current

According to Eqs. (14-2) and (14-3), the diode i-V curve is shifted downward (along the -i axis) by an amount i , which is proportional to the incident light intensity. A few representative i-Vcurves for a photodiode are shown in Fig. 14-3, for equally spaced values of light intensity. For self-consistency, the diode current i and voltage V_d must satisfy both Eq. (14-1) and the diode i-V relation simultaneously. The solution for i can easily be obtained graphically, by plotting Eq. (14-1) on the same graph as the diode i-V curves. This procedure, shown in Fig. 14-3, is known as a load-line analysis, and Eq. (14-1) is known as the load line. A similar analysis was discussed in Section 11-1 in connection with biasing an LED.

Since the photovoltaic mode is just a special case of the photoconductive mode, with V_B = 0, both circuits can be analyzed in the same fashion using Fig. 14-3. The intersection of the load line and the diode i-V curve corresponds to the operating point of the circuit, which gives the value of both i and V_d. For the photovoltaic mode, the load line passes through the origin, so the operating point is always in the fourth quadrant, with positive V_dand negative i. In the photoconductive mode, the intercept on the V_daxis is at V_d = -V_B, so the operating point can be either in the third or fourth quandrants. The current i is always negative, but V_dcan be either positive or negative.

The photovoltaic and photoconductive modes each have advantages and disadvantages, depending on the application. For low-level light detection, the photovoltaic mode has higher ultimate sensitivity than the photoconductive mode. This is because under dark conditions (no incident light), the photovoltaic operating point is at i = 0, whereas the photoconductive mode is at i = -i₀, the reverse saturation current. This minimum current

Figure 14-3 The operating point in a photodiode circuit is determined by the intersection between the load line and the diode i-V curve. Increasing optical power P_in shifts the i-V curve downward by an amount i P_in, moving the operating point down and to the right.

One important application utilizing the photovoltaic mode is the solar cell, which converts optical power into electrical power. The electrical power supplied to the load resistor is P_elec = i²R_L, where i is determined by Eq. (14-3) with V_d = -iR. Under practical conditions of solar illumination, i i₀, and Eq. (14-3) can be approximated as

with i given by Eq. (14-2). The efficiency

of converting optical power into electrical power can then be calculated by solving Eq. (14-4) for i. Since this is an implicit equation for i, it must be solved numerically or graphically.

An important consideration for the solar cell is the choice of load resistance that maximizes the conversion efficiency. Since the load line in Fig. 14-3 has a slope -1/R_L, the operating point moves close to i = 0 for large R and V_d= 0 for small R. The power iV_d delivered to the resistor will, therefore, have a maximum at some value of R. This optimum value of resistance can be determined graphically, or numerically as in the following example.

EXAMPLE 14-1 A silicon solar cell has an area of 4 cm2, reverse saturation current density 1.5 × 10-8 A/cm2, and diode ideality factor ß = 1. Assume that light of intensity I = 1000 W/m2 and average wavelength 500 nm is incident on the cell, and that 80% of the light is absorbed. Determine the optimum load resistance and power conversion efficiency. Repeat the calculation for ß = 2. Solution: The power striking the cell is (1000 W/m2)(4 × 10-4 m2) = 0.4 W. The photo current is then For room temperature (20°C = 293 K), is the "voltage equivalent of temperature." Putting this in Eq. (14-4) gives (for ß = 1) For a particular value of R, this equation is solved numerically for i, and the efficiency sc = i2R/Pinis calculated. By varying R, the graph shown in Fig. 14-4 is obtained. The optimum efficiency of 8.94% is obtained for R = 2.5 . If ß = 2, Eq. (14-4) becomes and the optimum efficiency increases to 17.9% at R = 5 .

A more detailed model of solar cell efficiency would take into account the variation with wavelength of the optical power from the sun and the fraction of this light that is absorbed by the silicon. In practice, solar cells based on crystalline silicon can have efficiencies as high as 24% in the laboratory, with ~ 15% being typical in commercial devices. Thin films of amorphous silicon (atoms not ordered periodically) are inexpensive to manufacture but have lower efficiencies, typically 13% in the laboratory and 5 7% in commercial devices.