Theory of spin-polarized bipolar transport in magnetic $p-n$ junctions

The interplay between spin and charge transport in electrically and magnetically inhomogeneous semiconductor systems is investigated theoretically. In particular, the theory of spin-polarized bipolar transport in magnetic $p-n$ junctions is formulated, generalizing the classic Shockley model. The theory assumes that in the depletion layer the nonequilibrium chemical potentials of spin-up and spin-down carriers are constant and carrier recombination and spin relaxation are inhibited. Under the general conditions of an applied bias and externally injected (source) spin, the model formulates analytically carrier and spin transport in magnetic $p-n$ junctions at low bias. The evaluation of the carrier and spin densities at the depletion layer establishes the necessary boundary conditions for solving the diffusive transport equations in the bulk regions separately, thus greatly simplifying the problem. The carrier and spin density and current profiles in the bulk regions are calculated and the $I-V$ characteristics of the junction are obtained. It is demonstrated that spin injection through the depletion layer of a magnetic $p-n$ junction is not possible unless nonequilibrium spin accumulates in the bulk regions—either by external spin injection or by the application of a large bias. Implications of the theory for majority spin injection across the depletion layer, minority spin pumping and spin amplification, giant magnetoresistance, spin-voltaic effect, biasing electrode spin injection, and magnetic drift in the bulk regions are discussed in details, and illustrated using the example of a GaAs based magnetic $p-n$ junction.