Metal-halide perovskite (MHP) solar cells exhibit long nonradiative lifetimes as a crucial feature enabling high efficiencies. Long nonradiative lifetimes occur if the transfer of electronic into vibrational energy is slow due to, e.g., a low trap density, weak electron-phonon coupling, or the requirement to release many phonons in the electronic transition. Here, we combine known material properties of MHPs with basic models for electron-phonon coupling and multiphonon-transition rates in polar semiconductors. We find that the low phonon energies of MAPbI lead to a strong dependence of recombination rates on trap position, which we deduce from the underlying physical effects determining nonradiative transitions. This is important for nonradiative recombination in MHPs, as it implies that they are rather insensitive to defects that are not at midgap energy, which can lead to long lifetimes. Therefore, the low phonon energies of MHPs are likely an important factor for their optoelectronic performance.

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The concept ofétendue is applied to the propagation of luminescent radiation, and to the transformation of such radiation in absorbing and luminescent media. Central to this analysis is the notion ofétendue as a measure of the number of rays in the beam which permits the definition of entropy and transition to the formalism of statistical mechanics. When considered from the statistical viewpoint,étendue conservation along the path of a beam in clear and transparent media then implies the conservation of entropy. The changes in thermodynamic parameters of a beam upon absorption and re-emission can then be determined in terms of the corresponding changes resulting from the addition or removal of photons from the incident and emitted beam. The thermodynamic theory which follows gives the rate of entropy generation in this process. At moderate light intensities, the results resemble the thermodynamics of a two-dimensional gas. The formalism allows an extension to absorption/emission processes where a high-temperature incident light beam is transformed reversibly into low-temperature luminescent radiation, corresponding to a potential increase in the open-circuit voltage of a solar cell.

This letter presents a thermodynamic analysis of losses in an ideal solar cell. It is shown that the maximum voltage-corresponding to the voltage produced by a hot-carrier solar cell-is equal to the energy of the incident solar photon multiplied by the appropriate Carnot factor. Voltage generated by the usual p-n junction cell is lower on account of entropy generation through kinetic losses, photon cooling, and étendue expansion of the incident beam. Simple expressions can be obtained by an approximation where the energy and entropy changes are modeled by the corresponding expressions for a two-dimensional ideal photon gas. © 2007 American Institute of Physics. ͓DOI: 10.1063/1.2766857͔A solar cell transforms a photon beam at temperature T s Х 6000 K into electrical energy while emitting photons at temperature T o Х 300 K of the solar cell. The photon flux in the beam is given by the usual expressionwhere is the frequency, the integration is over the spectral region ͑ ͒ of luminescence emission of the cell, is the chemical potential ͑in general, nonzero for luminescent radiation͒, and E denotes the étenduewhere dA is an element of the cross sectional area of the beam, d is an element of the solid angle, and is the angle between the normal to dA and the direction of the beam. We assume that the cell represents a perfect, optically thick, emitter of radiation-in other words, it has emissivity equal to unity. If, furthermore, the only recombination processes in the cell are radiative, the detailed balance between the electron current in the external circuit and the absorbed and emitted photon flows gives the I-V characteristic of the cell:where q is the electron charge,V is the voltage and I is the current produced by the solar cell. Equations ͑4͒ assume direct ͑beam͒ illumination of the cell ͑ in =0͒ and take into account the fact that, in addition to sunlight, the cell is exposed to an ambient photon flux at zero chemical potential ͑ amb =0͒ with étendue E out − E in . We note that the étendue E out of the emitted beam must be greater than the étendue of the incident beam E in since, by microscopic reversibility, photons can exit along the same path along which they arrived. For weak to moderate illumination when stimulated emission can be neglected, the cell characteristic ͓Eq. ͑3͔͒ simplifies to the usual form 1where I ᐉ = qJ N ͑in͒ andThis letter gives an alternative thermodynamic description to this, essentially kinetic, view of the conversion process, by considering the energy and entropy balance of photons transferred between the incident and emitted photon beams. If propagating in a clear and transparent medium, these beams are characterized by constant thermodynamic parameters and act as basis states ͑or reservoirs͒; the transfer of photons between them produces work ͑or electrical energy͒ and, in general, a certain amount of waste heat through irreversible entropy generation.Consider a packet of photons traveling with a beam defined by its temperature, étendue, and chemical potential where photons at the ...