The open-circuit voltage, V_{OC}, is the maximum voltage available from a solar cell, and this occurs at zero current. The open-circuit voltage corresponds to the amount of forward bias on the solar cell due to the bias of the solar cell junction with the light-generated current. The open-circuit voltage is shown on the IV curve below.

An equation for V_{oc} is found by setting the net current equal to zero in the solar cell equation to give:

$$V_{OC}=\frac{n k T}{q} \ln \left(\frac{I_{L}}{I_{0}}+1\right)$$

A casual inspection of the above equation might indicate that V_{OC} goes up linearly with temperature. However, this is not the case as I_{0} increases rapidly with temperature primarily due to changes in the intrinsic carrier concentration n_{i}. The effect of temperature is complicated and varies with cell technology. See the page “Effect of Temperature” for more details

V_{OC} **decreases **with temperature. If temperature changes, I_{0} also changes.

The above equation shows that V_{oc} depends on the saturation current of the solar cell and the light-generated current. While I_{sc} typically has a small variation, the key effect is the saturation current, since this may vary by orders of magnitude. The saturation current, I_{0} depends on recombination in the solar cell. Open-circuit voltage is then a measure of the amount of recombination in the device. Silicon solar cells on high quality single crystalline material have open-circuit voltages of up to 764 mV under one sun and AM1.5 conditions1, while commercial devices on multicrystalline silicon typically have open-circuit voltages around 600 mV.

The V_{OC} can also be determined from the carrier concentration 2:

$$V_{OC}=\frac{k T}{q} \ln \left[\frac{\left(N_{A}+\Delta n\right) \Delta n}{n_{i}^{2}}\right]$$

where kT/q is the thermal voltage, N_{A} is the doping concentration, Δn is the excess carrier concentration and n_{i } is the intrinsic carrier concentration. The determination of V_{OC} from the carrier concentration is also termed Implied V_{OC}.

### Voc as a Function of Bandgap, E_{G}

Where the short-circuit current (I_{SC}) decreases with increasing bandgap, the open-circuit voltage increases as the band gap increases. In an ideal device the V_{OC} is limited by radiative recombination and the analysis uses the principle of detailed balance to determine the minimum possible value for J_{0}.

The minimum value of the diode saturation current is given by 3:

$$J_{0}=\frac{q}{k} \frac{15 \sigma}{\pi^{4}} T^{3} \int_{u}^{\infty} \frac{x^{2}}{e^{x}-1} d x$$,

where q is the electronic charge, σ is the Stefan–Boltzmann constant, k is Boltzmann constant, T is the temperature and

$$u=\frac{E_{G}}{k T}$$

Evaluating the integral in the above equation is quite complex. The graph below uses the method outlined in 4

The J_{0} calculated above can be directly plugged into the standard solar cell equation given at the top of the page to determine the V_{OC} so long as the voltage is less than the band gap, as is the case under one sun illumination.

- 1. , “Analysis of the recombination mechanisms of a silicon solar cell with low bandgap-voltage offset”, Journal of Applied Physics, vol. 121, no. 20, p. 205704, 2017.
- 2. , “Contactless determination of current–voltage characteristics and minority-carrier lifetimes in semiconductors from quasi-steady-state photoconductance data”, Applied Physics Letters, vol. 69, pp. 2510-2512, 1996.
- 3. , “On some thermodynamic aspects of photovoltaic solar energy conversion”, Solar Energy Materials and Solar Cells, vol. 36, pp. 201-222, 1995.
- 4. , “Rapid and precise calculations of energy and particle flux for detailed-balance photovoltaic applications”, Solid-State Electronics, vol. 50, pp. 1400-1405, 2006.

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