Conductance quantization of an ideal Sharvin contact

Thorough calculations of the conductance of an idealized quantum point contact are presented. The point contact is modelled as a configuration in which two conductive half-spaces are separated by a non-conductive planar screen with a circular window of a given radius a. The screen is considered as opaque for the electrons with the exception of the window. From the viewpoint of the electrical conduction, the half-spaces are interpreted as leads which are perfectly coalesced with one another across the window. We assume that the leads are reservoirs of a strongly degenerate gas of electrons. If the leads are made from an n-type degenerate semiconductor, the Fermi wavelength λF = 2π/kF = 2π/(3π2n)1/3 may be tens of nanometers. The theory of the conductance ΓS of such a point contact is reconsidered. Within the framework of the approximation that we use assuming that T = 0, we show that the dependence of ΓS on the variable kFa manifests curved steps terminated by singular spikes. Owing to stochastic influences, the spikes should be seen as maxima in measured spectra. We predict that these maxima should demarcate the edges of the steps in the ΓS vs. kFa plot. This prediction is in agreement with the STM observations reported recently by Nagaoka et al. [K. Nagaoka, S. Yaginuma, T. Nagao, T. Nakayama, Phys. Rev. B. 74 (2006) 033310].