With respect to luminescence characterization of silicon solar cells, two general excitation scenarios exist: (a) the EL case where carriers are injected across the pn-junction into the solar cell base in the dark by applying a forward voltage and (b) the PL case where the solar cell is illuminated and held at a specific electrical working point (wp). In order to obtain the minority charge carrier profile Dn(z) for the different excitation scenarios, we solve the differential equation given in Eq. (5.12) for the boundary conditions
defined in Section 3.3.2. Details about the mathematical derivation are given in Hinken et al. (2009a).
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For the EL case, where _geh(z, l) = 0, we have to solve a linear differential equation to obtain the homogeneous solution
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For the PL case, instead, the complete inhomogeneous differential equation has to be solved. The solution is the sum of the homogeneous solutions given by the EL case and the particular solution satisfying the boundary conditions An(0) = 0, which exactly holds under short-circuit conditions. Thus, the general solution for the PL-wp case is
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assuming that
holds.
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Under open-circuit conditions the solution of Eq. (5.12) reads
with Leff being the effective diffusion length as defined in Eq. (5.19), K defined as – її Seff cos 01
1 ^ D a
1 + Sf Leff and again using the approximations ofEq. (A.4). Note that for Seff! 1 the minority charge carrier density at the junction (z = 0) becomes zero, An(0) = 0, and the open-circuit case becomes equal to the short-circuit case.
Under open-circuit conditions, the minority charge carrier density at z = 0 is
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Inserting Eq. (A.7) into Eq. (5.15) yields the open-circuit voltage of the solar cell under test for the specific optical excitation intensity Fimp0.
