Optoelectronic reciprocity relations relate properties of a diode under illumination to the photon emission of the same diode under applied voltage. The relations are useful for interpretation of luminescence based measurements of solar cells and modules and for the analysis of recombination losses in solar cells.
Basics Solar cells and light-emitting diodes are both semiconducting diodes that are operated in a different voltage and illumination regime and that serve different purposes. A solar cell is operated under illumination (usually by solar radiation) and is typically kept at the maximum power point where the product of current and voltage are maximized. A light emitting diode is operated at an applied forward bias (without external illumination). While a solar cell converts the energy contained in the electromagnetic waves of the incoming solar radiation into electric power (voltage x current) a light-emitting diode does the inverse, namely converting electrical power into electromagnetic radiation. A solar cell and a light emitting diode are typically made from different materials and optimized for different purposes; however, conceptually every solar cell could be operated as a light emitting diode and vice versa. Given that the operation principles have a high symmetry it is fair to assume that the key figures of merit that are used to characterize photovoltaic and luminescent operation of diodes are related to each other. These relations become particularly simple in a situation, where recombination rates scale linearly with minority carrier density and are explained below.
Reciprocity between the photovoltaic quantum efficiency and the electroluminescence spectrum of a pn-junction diode
The photovoltaic quantum efficiency Q e , P V {\displaystyle Q_{e,PV}} is a spectral quantity that is generally measured as a function of photon energy (or wavelength). The same is true for the electroluminescence spectrum ϕ E L {\displaystyle \phi _{EL}} of a light emitting diode under applied forward voltage V {\displaystyle V} . Under certain conditions specified below, these two properties measured on the same diode are connected via the equation
ϕ E L = Q e , P V ϕ b b [ exp q V k T − 1 ] {\displaystyle \phi _{EL}=Q_{e,PV}\phi _{bb}[\exp {\frac {qV}{kT}}-1]} (1) where ϕ b b {\displaystyle \phi _{bb}} is the black body spectrum emitted by a surface (the diode) into the hemisphere above the diode in units of photons per area, time and electron interval. In this case the black body spectrum is given by
ϕ b b = 2 π h 3 c 2 E 2 exp E / k T − 1 {\displaystyle \phi _{bb}={\frac {2\pi }{h^{3}c^{2}}}{\frac {E^{2}}{\exp {E/kT}-1}}}
where k {\displaystyle k} is the Boltzmann constant, h {\displaystyle h} is the Planck constant, c {\displaystyle c} is the speed of light in vacuum, and T {\displaystyle T} is the temperature of the diode. This simple relation is useful for the analysis of solar cells using luminescence-based characterization methods. Luminescence used for characterization of solar cells is useful because of the ability to image the luminescence of solar cells and modules in short periods of times, while spatially resolved measurements of photovoltaic properties (such as photocurrent or photovoltage) would be very time-consuming and technically difficult. Equation (1) is valid for the practically relevant situation, where the neutral base region of a pn-junction makes up most of the volume of the diode. Typically, the thickness of a crystalline Si solar cell is ~ 200 μm while the thickness of the emitter and space charge region is only on the order of hundreds of nanometers, i.e. three orders of magnitude thinner. In the base of a pn-junction, recombination is typically linear with minority carrier concentration over a large range of injection conditions and charge carrier transport is by diffusion. In this situation, the Donolato theorem. is valid that states that the collection efficiency f c {\displaystyle f_{\text{c}}} is related to the normalized minority carrier concentration δ n ( x ) / δ n ( x = x j ) {\displaystyle \delta n(x)/\delta n(x=x_{j})} via
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