  The theoretical maximum efficiency of a solar cell is typically characterized
by a detailed balance of optical absorption and emission for a semiconductor in
the limit of unity radiative efficiency and an ideal step-function response for
the density of states and absorbance at the semiconductor band edges, known as
the Shockley-Queisser limit. However, real materials have non-abrupt band
edges, which are typically characterized by an exponential distribution of
states, known as an Urbach tail. We develop here a modified detailed balance
limit of solar cells with imperfect band edges, using optoelectronic
reciprocity relations. We find that for semiconductors whose band edges are
broader than the thermal energy, kT, there is an effective renormalized bandgap
given by the quasi-Fermi level splitting within the solar cell. This
renormalized bandgap creates a Stokes shift between the onset of the absorption
and photoluminescence emission energies, which significantly reduces the
maximum achievable efficiency. The abruptness of the band edge density of
states therefore has important implications for the maximum achievable
photovoltaic efficiency.
