Effective medium theory based analytical models for the potential and field distributions in arrays of nanoscale junctions

Recently, we developed an Effective Medium Theory (EMT) for the Space-Charge Region electrostatics of Schottky and p-n junctions in arrays of nanofilms (NFs), nanowires, and nanotubes in a dielectric ambient and gave formulas for their junction depletion width and screening length characterizing the space-charge tail. In the present work, we develop this EMT further and derive simple formulas for the potential and field distributions in the semiconductor and dielectric media of the array. The formulas derived are validated with numerical simulations. It is shown that the potential and field distributions perpendicular to the junction plane in the array correspond to those in a bulk junction with an effective semiconductor medium, whose permittivity and doping are their weighted averages over the cross-sectional areas of the semiconductor and dielectric; the shapes of the cross-sections are immaterial. We also analyze a single NF junction, treating it as a limiting case of an array, and obtain the following key results. For negligible film thickness, the depletion width depends linearly on applied voltage and inverse of doping; the peak electric field depends linearly on doping and inverse of ambient permittivity and varies very gradually with applied voltage. These features of a thin film junction are remarkably different from the bulk junction, wherein the depletion width and peak field have a square-root dependence on applied voltage.