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We present an open-system quantum-mechanical 3D real-space study of theconduction band structure and conductive properties of two semiconductorsystems, interesting for their beyond-Moore and quantum computing applications:phosphorus $\delta$-layers in silicon and the corresponding $\delta$-layertunnel junctions. In order to evaluate size quantization effects on theconductivity, we consider two principal cases: nanoscale finite-widthstructures, used in transistors, and infinitely-wide structures, electricalproperties of which are typically known experimentally. For devices widths$W<10$~nm, quantization effects are strong and it is shown that the number ofpropagating modes determines not only the conductivity, but the distinctivespatial distribution of the current-carrying electron states. For $W>10$~nm,the quantization effects practically vanish and the conductivity tends to theinfinitely-wide device values. For tunnel junctions, two distinct conductivityregimes are predicted due to the strong conduction band quantization.

Conductivity and size quantization effects in semiconductor $\delta$-layer systems