A zero-thickness limit of multilayer structures: a resonant-tunnelling δ′-potential

A zero-thickness limit for two-terminal and three-terminal devices from the quantum electronics domain is analysed. The study is focused on heterostructures composed of a single barrier with, adjacent, one or two prewells. The point interactions obtained in this limit are shown to be described by a family of 'resonant' diagonal matrices that connect the two-sided boundary conditions at the device origin, which are a subclass of the whole four-parameter family of point interactions. Transmission through such a device is absent almost everywhere, except at a few points, whose number and position can be controlled by a gate voltage applied externally to the barrier subsystem. It is remarkable that the existence of resonances in the zero-thickness limit occurs only if a squeezing sequence is constructed in a 'δ′-like' way. In this case, the δ′-limit describes adequately the resonant behaviour of a barrier–well heterostructure with realistic parameters. Simple analytical expressions obtained for resonance sets are supported by direct numerical calculations of the transmission being in agreement with the results of experiments on semiconductor devices. The zero-thickness δ′-like limiting procedure can be used in the design of nanodevices or contacting quantum wires. (paper)