A generalized Caroli formula for transmission coefficient with lead-lead coupling

TL;DR

This paper introduces a generalized formula for transmission coefficients in lead-lead systems accounting for lead interactions, enabling analysis of heat and electrical transport in ballistic interfaces with broader applicability.

Contribution

It develops a new generalized transmission coefficient formula that includes lead interactions, extending the classical Caroli formula for ballistic transport analysis.

Findings

Derived a generalized transmission coefficient formula for lead-lead systems.

Established conditions for the formula's validity in ballistic transport.

Provided an explicit formula for one-dimensional interface setups.

Abstract

We present a generalized transmission coefficient formula for the lead-junction-lead system, in which interaction between the leads has been taken into account. Based on it the Caroli formula could be easily recovered and a transmission coefficient formula for interface problem in the ballistic system can be obtained. The condition of validity for the formula is carefully explored. We mainly focus on heat transport. However, the corresponding electrical transport could be similarly dealt with. Also, an illustrative example is given to clarify the precise meaning of the quantities used in the formula, such as the concept of the reduced interacting matrix in different situations. In addition, an explicit transmission coefficient formula for a general one-dimensional interface setup is obtained based on the derived interface formula.