  We present a theory of superconducting p-n junctions. We consider a 2-band
model of doped bulk semiconductors with attractive interactions between the
charge carriers and derive the superconducting order parameter, the
quasiparticle density of states and the chemical potential as a function of
semiconductor gap $\Delta_0$ and the doping level $\varepsilon$. We verify
previous results for the quantum phase diagram (QPD) for a system with constant
density of states in the conduction and valence band, which show
BCS-Superconductor to Bose-Einstein-Condensation (BEC) and BEC to Insulator
transitions as function of doping level and band gap. Then, we extend it to a
3D density of states and derive the QPD, finding that a BEC phase can only
exist for small band gaps $\Delta_0 < \Delta_0^*$. For larger band gaps, there
is a direct transition from an insulator to a BCS phase. Next, we apply this
theory to study the properties of superconducting p-n junctions, deriving the
spatial variation of the superconducting order parameter along the p-n
junction. We find a spatial crossover between a BCS and BEC condensate, as the
density of charge carriers changes across the p-n junction. For the 2D system,
we find two regimes, when the bulk is in a BCS phase, a BCS-BEC-BCS junction
with a single BEC layer, and a BCS-BEC-I-BEC-BCS junction with two layers of
BEC condensates separated by an insulating layer. In 3D there can also be a
conventional BCS-I-BCS junction for semiconductors with band gaps exceeding
$\Delta_0^*$. Thus, there can be BEC layers in the well controlled setting of
doped semiconductors, where the doping level can be varied to change the
thickness of BEC layers, making Bose Einstein Condensates possibly accessible
to experimental transport and optical studies in solid state materials.
