Electron scattering and bound-state energies in crossed N-chain wires. A comparative study of discrete and continuous models

Abstract

A closed set of equations, which describe electron scattering in a right-angle intersection of two 2D wires, is derived on the basis of the tight-binding Hamiltonian with the interaction between the neighbouring lattice sites. In the continuum limit, N to infinity , Na=constant (N is the number of coupled chains that constitute each wire, a is the lattice constant), the equations are shown to be identical to those known in the matching theory for the continuous version of the model. An analytic solution of the scattering problem obtained yields the scattering amplitude of the single-mode electron-wave propagation and the equations determining the bound-state energies of the system. It is shown that the bound-state to band-state energy gap has a qualitatively different dependence on N (i.e. on wire width) in the cases of even- and odd-parity bound states. The effect reveals itself in ultra-narrow channels, N<or=10. The width dependence of the reflection and transmission probabilities in crossed ultra-narrow wires is also discussed.