= Solution
Absorption generates electrons and holes in the perovskite. After relaxation toward the band edges, their nonequilibrium populations are described by separate electron and hole <quasi-Fermi levels>. Selective contacts collect electrons through an electron-transport layer and holes through a hole-transport layer, while suppressing the opposite carriers.
For ideal contacts at open circuit,
$$
\boxed{qV_{\rm OC}\simeq E_{Fn}-E_{Fp}}.
$$
This <quasi-Fermi-level splitting> is not the band gap itself or simply the equilibrium built-in voltage. At equilibrium there is one common Fermi level and no extractable photovoltage. Under illumination at open circuit, generation is balanced by recombination while a nonzero population splitting persists.
\Image[/past-exam-of-the-mphil-in-micro-and-nanotechnology-enterprise-course-of-the-university-of-cambridge/2023/paper-1-additional-solar-cell.png]
{title=Selective extraction, quasi-Fermi splitting and recombination-limited solar-cell voltage}
{description=Original qualitative energy diagram and stipulated illuminated diode curves. Trap-assisted and interface recombination reduce carrier splitting and open-circuit voltage. Band offsets, energy spacings and current scales are illustrative rather than fitted perovskite data.}
Important parasitic paths include trap-assisted <Shockley-Read-Hall recombination> in the bulk, at grain boundaries and at interfaces, and recombination at poorly selective contacts. They lower the steady carrier populations and their quasi-Fermi splitting. <Auger recombination> can matter at high carrier densities but need not dominate an ordinary one-sun device. Radiative recombination is an unavoidable detailed-balance contribution, rather than an avoidable defect loss of the same kind.
An approximate illuminated diode relation is
$$
J(V)=J_{\rm SC}-J_0\left[e^{qV/(n_{\rm id}k_BT)}-1\right],
\qquad
V_{\rm OC}=\frac{n_{\rm id}k_BT}{q}\ln\left(1+\frac{J_{\rm SC}}{J_0}\right).
$$
Increased nonradiative recombination generally increases the dark recombination current and reduces $V_{\rm OC}$; it can also change ideality and fill factor. Shunts add a leakage path. Series resistance mainly reduces voltage under load and fill factor: its simple $IR_s$ drop vanishes at zero terminal current, so it should not be invoked as an automatic direct subtraction from open-circuit voltage.
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