Shape effects of the conductance of a quantum ballistic constriction in a two-dimensional electron gas

A hyperbolic model is used to describe the shape effect of the quantized conductance of a microscopic constriction in a two-dimensional electron gas. The constriction is given by two hyperbolas of β = β0 and π - β0 in elliptic coordinates (α, β). The conductance G of the constriction is calculated by using Mathieuʹs functions which satisfy Schrödingerʹs equation and the hyperbolic boundary conditions. It is found that the number of channels NC depends not only on the constriction width W but also the slope-related coordinate β0. It is also found that tunnelling is the important factor to determine the shape of the quantized G graph and depends on β0. Quantization of G is more possible for the large-β0 constriction (i.e. the long and smooth shape) than for the small-β0 constriction (i.e. the short and steep shape). For comparison, we calculate G of the hyperbolic constrictions using the adiabatic approximation that the shape is smooth enough for quantization. All adiabatic results for every β0 have almost the same G curve, which is similar to one obtained by the first scheme for smoothly changing shape, say β0 = 45°.