Open circuit voltage and band gap

Question

I try to estimate the open-circuit voltage of an ideal solar cell (no non-radiative loss) with perfect absorption, illuminated by a black body at temperature $T_{\odot}$ and at full concentration. I found myself facing incoherent results.

I first tried to use the standard Boltzman approximation to estimate the absorbed and radiated currents

$$J_{abs} =\frac{\pi}{4\pi\hbar^{3}c^{2}}\stackrel{\infty}{\underset{E_{g}}{\int}}dE\,E^{2}\exp\left(\frac{E}{k_{B}T_{\odot}}\right)$$

$$J_{em} =\frac{\pi}{4\pi\hbar^{3}c^{2}}\left(\stackrel{\infty}{\underset{E_{g}}{\int}}dE\,E^{2}\exp\left(\frac{E}{k_{B}T}\right)\right)\exp\left(\frac{qV}{k_{B}T}\right)$$

and the open circuit voltage is given by

$$\exp\left(\frac{qV_{OC}}{k_{B}T}\right) =\frac{\stackrel{\infty}{\underset{E_{g}}{\int}}dE\,E^{2}\exp\left(-\frac{E}{k_{B}T_{\odot}}\right)}{\stackrel{\infty}{\underset{E_{g}}{\int}}dE\,E^{2}\exp\left(-\frac{E}{k_{B}T}\right)}$$

For $E_{g}=1$ eV, the open volage is larger than the gap $V_{OC}=1.05\,E_{gap}$, which makes no sense. This probably shows that Botlzman approximation is not valid in this case. So I tried to carry out the calculation with the generic expression

$$J_{abs} =\frac{\Omega_{\odot}}{4\pi\hbar^{3}c^{2}}\stackrel{\infty}{\underset{E_{g}}{\int}}dE\,\frac{E^{2}}{\exp\left(\frac{E}{k_{B}T_{\odot}}\right)-1}$$ $$J_{em} =\frac{\Omega_{em}}{4\pi\hbar^{3}c^{2}}\stackrel{\infty}{\underset{E_{g}}{\int}}dE\,\frac{E^{2}}{\exp\left(\frac{E-\Delta\mu}{k_{B}T}\right)-1}$$

Here, $J_{em}$ presents an upper bound $\frac{\pi}{4\pi\hbar^{3}c^{2}}\stackrel{\infty}{\underset{E_{g}}{\int}}dE\,\frac{E^{2}}{\exp\left(\frac{E-E_{g}}{k_{B}T}\right)-1}$ reached for $\Delta\mu=E_{g}$. If the source is hot enough (for instance $T_{\odot}=24\,000$ K), this upper limit is lower than the absorbed current $J_{abs}$, which means that the cell can not be at equilibrium.

I am doing a stupid mistake somewhere ? I tend to blame the perfect absorption, which is certainly disturbed at some point by the large population changes...

Answer 1

My mistake : $J_{em}$ is not bounded because the integral diverges as $1/x$ when $\Delta \mu \rightarrow E_g$, so there is always a quasi-fermi level splitting smaller than the gap that insures equilibrium.

I had in mind the finite value of the integral in the Bose-Einstein condensation. The situation is different here because the density of state is not vanishing at the smallest contributing energy. In the BEC case, this compensates the divergence of the numerator ; here it doesn't.