# A microscopic formulation of dynamical spin injection in ferromagnetic-nonmagnetic heterostructures

## TL;DR

This paper presents a detailed microscopic approach to model dynamical spin injection in ferromagnetic-nonmagnetic heterostructures, incorporating effects like spin-orbit coupling and disorder for accurate computation.

## Contribution

It introduces a Green's function-based microscopic formulation for spin pumping, enabling efficient calculations in complex heterostructures with realistic effects.

## Key findings

- Green's function approach effectively models spin injection.
- Inclusion of spin-orbit coupling and disorder enhances realism.
- Recursive methods enable efficient computations.

## Abstract

We develop a microscopic formulation of dynamical spin injection in heterostructure comprising nonmagnetic metals in contact with ferromagnets. The spin pumping current is expressed in terms of Green's function of the nonmagnetic metal attached to the ferromagnet where a precessing magnetization is induced. The formulation allows for the inclusion of spin-orbit coupling and disorder. The Green's functions involved in the expression for the current are expressed in real-space lattice coordinates and can thus be efficiently computed using recursive methods.