# Magnetization direction dependent spin Hall effect in 3d ferromagnets

## TL;DR

This study reveals that the spin Hall conductivity in 3d ferromagnets like Fe and Ni varies with magnetization direction, enabling control of spin current magnitude through magnetization orientation.

## Contribution

First-principles calculations demonstrate the anisotropic spin Hall effect in ferromagnets and develop an analytical model explaining this dependence.

## Key findings

- Spin Hall conductivity depends on magnetization direction.
- Electrons with spins orthogonal to magnetization have higher conductivity.
- The effect can be controlled by changing the magnetization orientation.

## Abstract

We have studied the intrinsic spin Hall conductivity of 3d transition metal ferromagnets using first-principles calculations. We find the spin Hall conductivity of bcc Fe and fcc Ni, prototypes of ferromagnetic systems, depends on the direction of magnetization. The spin Hall conductivity of electrons with their spin orientation orthogonal to the magnetization are found to be larger than that when the two are parallel. For example, the former can be more than four times larger than the latter in bcc Fe. Such a difference arises due to the anisotropy of the spin current operator in the spinor space: Its expectation value with the Bloch states depends on the relative angle between the conduction electron spin and the magnetization. A simple analytical form is developed to describe the relation between the spinor states, with respect to the magnetization direction, and the Berry and spin Berry curvatures. The model can account for the characteristics found in the calculations. These results show that ferromagnets can be used to generate spin current and its magnitude can be controlled by the magnetization direction.