# Weak ferromagnetism in hexagonal Mn3Z (Z=Sn, Ge, Ga) alloys

## TL;DR

This study combines theoretical models and first-principles calculations to understand the origin and characteristics of weak ferromagnetism in hexagonal Mn3Z alloys, aligning with experimental observations.

## Contribution

It introduces a comprehensive approach integrating spin models, group theory, and density functional theory to analyze weak ferromagnetism in Mn3Z compounds, providing new insights into their magnetic ground states.

## Key findings

- Weak ferromagnetic states are explained by lattice symmetry and spin-orbit coupling.
- Analytical expressions for weak ferromagnetic distortions are derived.
- Results align well with experimental data and previous calculations.

## Abstract

We present combined spin model and first principles electronic structure calculations to study the weak ferromagnetism in bulk Mn$_3$Z (Z=Sn, Ge, Ga) compounds. The spin model parameters were determined from a spin-cluster expansion technique based on the relativistic disordered local moment formalism implemented in the screened Korringa--Kohn--Rostoker method. We describe the magnetic ground state of the system within a three-sublattice model and investigate the formation of the weak ferromagnetic states in terms of the relevant model parameters. First, we give a group-theoretical argument how the point-group symmetry of the lattice leads to the formation of weak ferromagnetic states. Then we study the ground states of the classical spin model and derive analytical expressions for the weak ferromagnetic distortions by recovering the main results of the group-theoretical analysis. As a third approach we obtain the weak ferromagnetic ground states from self-consistent density functional calculations and compare our results with previous first principles calculations and with available experimental data. In particular, we demonstrate that the orbital moments follow a decomposition predicted by group theory. For a deeper understanding of the formation of weak ferromagnetism we selectively trace the effect of the spin-orbit coupling at the Mn and Z sites. In addition, for the case of Mn$_3$Ga, we gain information on the role of the induced moment of Ga from constrained local density functional calculations.