= Solution
A <solar cell> uses the <photovoltaic effect> to turn <photon> <energy> into electrical work. In a <semiconductor>, above-gap <photon absorption> generates an <Electron> in the <conduction band> and an <electron hole> in the <valence band>. A <p-n junction>, together with selective electrical contacts, collects the two carrier species at opposite terminals. Carrier <diffusion> and the built-in <electric field> help separate them before <electron-hole recombination>. Illumination maintains different <quasi-Fermi levels> for <Electrons> and <electron holes>; their splitting, divided by $e$, sets the available terminal voltage in an ideal device. The equilibrium <built-in potential> alone cannot continuously deliver work.
For an ideal junction, the illuminated output <current density> is
$$
J(V)=J_{\rm sc}-J_0\left(e^{eV/(k_BT)}-1\right),\qquad
V_{\rm oc}=\frac{k_BT}{e}\log\left(1+\frac{J_{\rm sc}}{J_0}\right).
$$
Here $J_{\rm sc}$ is the photogenerated short-circuit <current density> and $J_0$ the <diode saturation current>. The useful operating point maximizes $VJ$, rather than either current or voltage separately. With <solar-cell fill factor> $\mathrm{FF}=P_{\max}/(V_{\rm oc}I_{\rm sc})$,
$$
\eta=\frac{P_{\max}}{P_{\rm in}}
=\frac{V_{\rm oc}I_{\rm sc}\mathrm{FF}}{P_{\rm in}}.
$$
Series resistance, shunts, incomplete <photon absorption> and <nonradiative recombination> reduce the attainable efficiency.
\Image[/past-exam-of-the-quantum-condensed-matter-course-of-the-university-of-cambridge/2012-solar-cell-output.png]
{title=Ideal illuminated <solar cell>: current, delivered power and the maximum-power point}
The <Shockley-Queisser limit> removes these avoidable losses but retains the following physical constraints. There is one <semiconductor> junction with one <band gap>; each absorbed above-gap <photon> yields at most one collected <electron-hole pair>; carriers thermalize to the cell <temperature>; and the inverse optical process produces unavoidable <radiative recombination>. If $\Phi_{\odot}(E)$ is the incident <photon> flux per unit <energy>, ideal <photon absorption> gives
$$
J_{\rm sc}=e\int_{E_g}^{\infty}\Phi_{\odot}(E)\,dE.
$$
The same ability to absorb <light> requires the junction to emit it. At a forward voltage $V$, the emitted <photon> population has a <chemical potential> $eV$; in the ordinary nondegenerate limit the resulting recombination current is the exponential diode term above. Maximizing $VJ$ with this radiative $J_0$, and then choosing $E_g$, gives the limiting efficiency.
A small <band gap> collects many <photons> but gives a low voltage and large thermalization losses. A large <band gap> permits a larger voltage but transmits too much of the incident spectrum. Below-gap <photons> are lost to transmission, excess <photon> <energy> above the gap becomes heat, and radiative emission and its associated entropy loss keep $eV_{\rm oc}$ below $E_g$. \b[The optimum single-junction conversion is roughly thirty percent under ordinary unconcentrated sunlight; the precise ceiling depends on spectrum, cell <temperature> and optical geometry.] The original detailed-balance calculation gave about $30\%$ at $E_g\simeq1.1\,\mathrm{eV}$ under its specified assumptions.https://doi.org/10.1063/1.1736034[Detailed Balance Limit of Efficiency of p-n Junction Solar Cells, Journal of Applied Physics 32 (1961), 510–519; abstract and calculation]{ref} Multiple junctions and optical concentration change the assumptions and consequently change the bound.
Back to article page
