Fully macroscopic description of electrical conduction in metal-insulator-semiconductor structures

In an earlier paper [Phys. Rev. B 26, 6104 (1980)], a fully macroscopic description of semiconductors was presented which, in addition to the usual diffusion-drift current equations, includes two new boundary conditions resulting from the requirement that the conservation of linear momentum of the electron and hole fluids be satisfied at semiconductor interfaces. In the present work, this description is applied to situations involving semiconductors which abut insulators, e.g., metal-insulator-semiconductor structures, in which macroscopic currents flow. Consistent boundary conditions for the limiting cases of small signals and low-level injection are derived from the general boundary conditions of the earlier work. To illustrate the use of these conditions, they, together with the familiar small signal and low-level differential equations, are employed in the study of two experiments, namely steady-state photoconductivity and metal-oxide-semiconductor (MOS) admittance. The analysis of the former shows that for dc situations the two semiconduction boundary conditions may be approximated by a single "outer" condition to be applied at the outer edge of the space-charge region. For small signal cases, but not for low-level situations, this condition is shown to be well approximated by the often used surface recombination velocity condition ($s$ constant). An expression for the surface recombination velocity in terms of macroscopic surface coefficients is derived, and its predicted variation with bias is in qualitative agreement with known results. For the MOS admittance experiment a similar "outer" condition approach is shown to be inadequate. The purely small signal (linear) treatment given constitutes the field description underlying equivalent circuit representations of the MOS capacitor. Approximate solutions for the admittance are obtained in terms of the macroscopic surface coefficients and are found to be in qualitative agreement with published data.