Quantum statistical theory of semiconductor junctions in thermal equilibrium

By means of a quantum-mechanical phase-space distribution function and its corresponding Boltzmann equation, the free-carrier and electric-field distributions of one-dimensional semiconductor junctions ($n-p$, $p-p^{+}$, etc.) are evaluated. It is shown that quantum and exchange corrections, which have been neglected in the past, play an important role in the determination of the built-in electric field within the transition region, the region in which the doping concentration changes rapidly (from $n$-type to $p$-type material for instance). This is particularly true in cases of high doping concentrations, i.e., when carrier densities become degenerate. Exact expressions will be given for the maximum built-in electric field in case of abrupt junctions. It is also shown that the exchange effect induces a slight change in the position of the band edges which persists through the homogeneous (neutral) part of the junction far away from the transition region. A numerical example is given and the quantitative differences between heavily doped (degenerate) and nondegenerate (classical) junction characteristics (maximum electric field, built-in voltage and carrier concentration within the transition region) are determined. The theory is briefly generalized to encompass high-low junctions.