# Application of the Landauer formalism to the calculation of spin current

## TL;DR

This paper applies the Landauer formalism and transfer matrix method to calculate spin currents in magnetic multilayers, providing analytical insights and asymptotic approximations that match numerical results.

## Contribution

It introduces a formalism combining Landauer and transfer matrix approaches for spin current analysis in ballistic quantum regimes, with analytical and asymptotic solutions.

## Key findings

- Closed-form expressions with physical interpretation
- Asymptotic approximations agree with numerical calculations
- Insights into the symmetry and origin of spin current components

## Abstract

In this communication we apply the Landauer method and transfer matrix formalism to the calculation of spin current in magnetic multilayered structures within a ballistic quantum-mechanical regime. The method provides an elegant and intuitive formalism with which to study spin current properties and within which closed-form expressions with a transparent physical interpretation can be obtained. We apply the method to illuminate origin and the symmetry properties of the various spin current components within the parabolic band approximation. We also apply the stationary phase approximation to develop asymptotic approximations to the total spin current within this formalism, and show that these give excellent agreement with full numerical calculations for both barrier and well systems.