Theoretical study of a quantum-point-contact model.

Abstract

The effects of a short-range impurity potential V of radius ${\mathit{a}}_{0}$ on a model two-dimensional (2D) quantum-point contact (QPC) are studied. In this model the 2D electron gas is confined between two symmetrical hyperbolas. \ensuremath{\Vert}V\ifmmode \tilde{}\else \~{}\fi{}\ensuremath{\Vert}=2${\mathit{ma}}_{0}^{2}$\ensuremath{\Vert}V\ensuremath{\Vert}/${\mathrm{\ensuremath{\Elzxh}}}^{2}$ is assumed to be small but \ensuremath{\Vert}V\ensuremath{\Vert} can be high on the scale of Fermi energy. A bound state at an attractive short-range impurity exists, at finite widths w of the constriction, only if the potential or w exceed some threshold values. The dependence of the binding energy on the width of the constriction and on the parameters of the impurity is found. An explicit expression for the conductance G(E) of the QPC with an impurity (attractive or repulsive) as a function of the Fermi energy E is derived. At small \ensuremath{\Vert}V\ifmmode \tilde{}\else \~{}\fi{}\ensuremath{\Vert} the sign of the impurity effect on the conductance, \ensuremath{\Delta}G, is opposite to the sign of the potential. As the depth of the attractive potential increases, \ensuremath{\Delta}G(E) becomes negative in some regions of Fermi energy. For stronger potentials \ensuremath{\Delta}G is negative for all Fermi energies both for attractive and repulsive impurities. For attractive impurities the dependence of \ensuremath{\Delta}G(E) on V is nonmonotonic: the \ensuremath{\Vert}\ensuremath{\Delta}G(E)\ensuremath{\Vert} peaks and minima are especially high, and the total conductance G(E) is a nonmonotonic function of Fermi energy, when the depth of the potential is close to the threshold value at which a bound state appears. At even higher \ensuremath{\Vert}V\ensuremath{\Vert} the \ensuremath{\Vert}\ensuremath{\Delta}G(E)\ensuremath{\Vert} peaks and minima decrease but are on the order of conductance quantum ${\mathit{e}}^{2}$/2\ensuremath{\pi}\ensuremath{\Elzxh}, and G(E) is monotonic.The \ensuremath{\Vert}\ensuremath{\Delta}G(E)\ensuremath{\Vert} peaks and minima correspond, respectively, to the conductance steps, i.e., regions of transition from one plateau to the next one, and to the feet of these steps. The modulation resistance noise generated by random abrupt changes of the impurity potential is discussed. At high-impurity potentials the deviation of a conductance plateau from its quantized value grows with its number, i.e., the plateaus become progressively less perfect and more noisy. The results are compared with those found earlier for quasi-one-dimensional QPC models.