Detailed Balance in Single-Junction Photovoltaics

The Shockley–Queisser limit is the detailed-balance efficiency limit of an ideal single-junction photovoltaic absorber. Its derivation requires only a few assumptions: photons with energy above the bandgap are absorbed, each absorbed photon generates one electron–hole pair, excess carrier energy thermalizes to the band edges, and recombination is purely radiative. The solar cell can then be treated as a photon converter coupled to two radiation fields: sunlight incident from a hot, highly directional source and thermal emission from the cell into its colder surroundings.

The electrical current is determined by the difference between absorbed and emitted photon fluxes:

$$J(V)=q\left[\Phi_{\mathrm{abs}}-\Phi_{\mathrm{em}}(V)\right].$$

The central theoretical point is that the emitted radiation has a non-zero photon chemical potential. Under illumination, the electron and hole populations develop separate quasi-Fermi levels, and their splitting satisfies

$$\mu_\gamma=\Delta E_F=qV.$$

The generalized Planck distribution therefore becomes

$$\phi(E,V)\propto \frac{E^2} {\exp\left[(E-qV)/kT\right]-1}.$$

Voltage changes the thermodynamic state of the emitted photon field. As $V$ increases, radiative emission rises exponentially. At open circuit, no net current is extracted, so absorbed and emitted photon fluxes balance:

$$\Phi_{\mathrm{abs}}=\Phi_{\mathrm{em}}(V_{\mathrm{OC}}).$$

This condition fixes the maximum radiative open-circuit voltage. Even an ideal absorber cannot generally reach

$$qV_{\mathrm{OC}}=E_g,$$

because the absorbed solar photon flux is finite while the cell can emit into a much larger optical phase space. The resulting voltage deficit therefore exists even in the complete absence of non-radiative recombination.

This loss has a thermodynamic interpretation. Sunlight arrives from the small solid angle subtended by the Sun, whereas an ordinary planar solar cell emits over a much wider range of angles. The outgoing photon field has greater optical entropy than the incoming field. That increase in entropy reduces the maximum extractable free energy and therefore lowers $V_{\mathrm{OC}}$. Optical concentration increases the incident étendue and partially suppresses this angular entropy penalty, which is why the detailed-balance efficiency limit rises under concentration.

The bandgap introduces two additional fundamental losses. Photons with

$$E<E_g$$

cannot generate carriers and are lost by transmission. For photons with

$$E>E_g,$$

the excess energy

$$E-E_g$$

is rapidly dissipated during carrier thermalization. A smaller $E_g$ increases the number of absorbed photons but lowers the achievable voltage and increases thermalization loss. A larger $E_g$ preserves more energy per absorbed photon but rejects a larger fraction of the solar spectrum. The Shockley–Queisser optimum emerges from this competition between photon harvesting, thermalization, and radiative free-energy loss.

The maximum efficiency can be written as

$$\max_V \frac{J(V)V}{P_{\mathrm{in}}}.$$

For an unconcentrated single-junction cell under the standard terrestrial solar spectrum, this maximum is approximately 33% for a bandgap near$1.3$–$1.4,\mathrm{eV}$.

The deeper significance of the SQ limit is that it identifies the thermodynamic cost of converting a broadband photon distribution into electrical free energy through a single thermalized electronic transition. Any route beyond this limit must alter at least one element of that conversion process, for example by dividing the spectrum among multiple bandgaps, extracting carriers before thermalization, generating multiple electronic excitations from one photon, or modifying the optical phase space through concentration. The SQ limit is therefore a detailed-balance boundary associated with a specific class of photovoltaic energy conversion.