Approximate steady-state solutions of the continuity equation containing effects of diffusion, drift, and recombination are obtained for the carrier diffusion region of a forward-biased $p-n$ junction over a wide range of injection. These are used to predict the voltage ($V$) dependence and the spatial distribution of electroluminescence (EL) originating from a deep level. Regions where ln (EL) versus $\frac{\mathrm{eV}}{\mathrm{kT}}$ has slope 1, ½, $\frac{\mathrm{}(m+1\mathrm{})\mathrm{}}{m}$, and ½ are found ($m$ is defined by the current-voltage characteristic: $I=I_0\exp \frac{\mathrm{eV}}{\mathrm{mkT}}$). These predictions hold for either Shockley-Read-Hall or donor-acceptor-pair recombination. The spatial distribution of electroluminescence is a simple exponential in the first region. For the next two regions for each recombination mechanism, it contains a saturated layer of constant brightness adjacent to the junction which expands rapidly with voltage. The latter behavior persists in the fourth region for donor-acceptor-pair recombination, but for Shockley-Read-Hall recombination a further increase in brightness near the junction occurs. Comparison of these predictions with experimental results on gallium phosphide electro-luminescent diodes is made in the following paper.

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