Some analytic results in coherent quantum transport

TL;DR

This paper introduces a new, faster formula for calculating Green's functions and transmission in coherent quantum transport, providing analytical solutions for specific quantum wire configurations and their properties.

Contribution

A novel formula for Green's function and transmission calculation that significantly improves computational efficiency and allows for analytical solutions in certain quantum wire models.

Findings

Derived analytical expressions for Green's function and LDOS.

Provided explicit formulas for transmission coefficients.

Solved the uniform dot problem analytically.

Abstract

A quantum wire of uniform cross section (but with eventual disorder) with three regions: dot, left lead, and right lead, is considered. Assuming that the same unitary transformation diagonalizes all unit cells of this wire, we propose a new formula for the calculation of the Greens function (GF) and the coherent transmission coefficient. This formula allows to calculate these quantitites much faster than the standard methods. In particular, the problem of a uniform dot (simple cubic uniform dot attached to the simple cubic wire), with all onsites equal and all hoppings equal is solved fully analytically. The energy and dot-length dependence of the GF, local density of states (LDOS), the transmission coefficient and bound state energies are also derived.