Adiabatic electron transport in a quantum point cosh-contact

Abstract

Quantum point contacts (QPCs) are quantum-dimensional constrictions between two electrically conductive regions. Their width is comparable to the de Broglie wavelength of an electron. They are of interest to those studying fundamental mesoscopic phenomena, particularly quantization of ballistic conductivity. Furthermore, QPCs have promising applications as sensitive charge sensors and elements of quantum computing devices. In this paper, the author investigates electron transport in a two-dimensional QPC whose shape is described by a hyperbolic cosine (cosh-contact). To analyze electron behavior in the narrow channel, adiabatic approximation was used. This method offers an approximate solution to complex quantum mechanical problems. It involves a two-stage approach. First, the behavior of the fast subsystem is analyzed while the slow subsystem remains in a fixed state. Then it is studied how the parameters of the slow subsystem affect the entire quantum system. Using the method of adiabatic separation of variables, an effective Hamiltonian for the slow subsystem was obtained. This Hamiltonian describes the longitudinal motion of electrons in the quantum cosh-contact. The energy transmission coefficient of electrons was calculated using an analytical solution of the stationary Schrödinger equation derived for the effective Hamiltonian of the slow subsystem. The conductivity of the cosh-contact was computed for both zero and finite temperatures. The analytical expression obtained for the conductivity demonstrates its stepwise character. The paper offers a qualitative interpretation of this result: each step corresponds to the opening of new quantum channels for electron transport. Graphical results showing how temperature affects the conductivity of the quantum point cosh-contact are presented.