# Spin dynamics of antiferromagnetically coupled ferromagnetic bilayers -- the case of Cr$_2$WO$_6$ and Cr$_2$MoO$_6$

## TL;DR

This paper studies the spin dynamics of antiferromagnetically coupled ferromagnetic bilayers in Cr$_2$WO$_6$ and Cr$_2$MoO$_6$, revealing magnon dispersion and sublattice magnetization reductions through theoretical analysis and comparison with Cr$_2$TeO$_6$.

## Contribution

It provides a theoretical investigation of the spin dynamics in Cr$_2$WO$_6$ and Cr$_2$MoO$_6$ bilayer systems with ferromagnetic intra-bilayer and antiferromagnetic inter-bilayer couplings, expanding understanding beyond previous work on Cr$_2$TeO$_6$.

## Key findings

- Magnon dispersion relations obtained for Cr$_2$WO$_6$ and Cr$_2$MoO$_6$.
- Significant reduction in sublattice magnetization compared to atomic spin value.
- Comparison shows differences and similarities with Cr$_2$TeO$_6$ results.

## Abstract

Recent inelastic neutron diffraction measurements on Cr$_2$(Te, W, Mo)O$_6$ have revealed that these systems consist of bilayers of spin-3/2 Cr$^{3+}$ ions with strong antiferromagnetic inter-bilayer coupling and tuneable intra-bilayer coupling from ferro (for W and Mo) to antiferro (for Te). These measurements have determined the ground state spin structure and the values of sublattice magnetization, which shows significant reduction of sublattice magnetization from the atomic spin value of 3.0$\mu_B$ for Cr$^{3+}$ atoms. In an earlier paper we theoretically investigated the low temperature spin dynamics of Cr$_2$TeO$_6$ bilayer system where both the intra and inter-bilayer couplings are antiferromagnetic. In this paper we investigate Cr$_2$WO$_6$ and Cr$_2$MoO$_6$ systems where intra-bilayer exchange couplings are ferromagnetic but the inter-bilayer exchange couplings are antiferromagnetic. We obtain the magnon dispersion, sublattice magnetization, two-magnon density of states, longitudinal spin-spin correlation function, and its powder average and compare the results for these systems with results for Cr$_2$TeO$_6$.