SHAPE EFFECTS OF THE CONDUCTANCE OF A QUANTUM BALLISTIC CONSTRICTION IN A 2-DIMENSIONAL ELECTRON-GAS

A hyperbolic model is used to describe the shape effect of the quantiz ed conductance of a microscopic constriction in a two-dimensional elec tron gas. The constriction is given by two hyperbolas of beta = beta(0 ) and pi-beta(0) in elliptic coordinates (alpha, beta). The conductanc e G of the constriction is calculated by using Mathieu's functions whi ch satisfy Schrodinger's equation and the hyperbolic boundary conditio ns. It is found that the number of channels N-C depends not only on th e constriction width W but also the slope-related coordinate beta(0). It is also found that tunnelling is the important factor to determine the shape of the quantized G graph and depends on beta(0). Quantizatio n of G is more possible for the large-beta(0) constriction (i.e. the l ong and smooth shape) than for the small-beta(0) constriction (i.e. th e short and steep shape). For comparison, we calculate G of the hyperb olic constrictions using the adiabatic approximation that the shape is smooth enough for quantization. All adiabatic results for every beta( 0), have almost the same G curve, which is similar to one obtained by the first scheme for smoothly changing shape, say beta(0) = 45 degrees .