Constrained DFT+$U$ approach for understanding the magnetic behaviour of ACr$_{2}$O$_{4}$ (A=Zn, Mg, Cd and Hg) compounds

TL;DR

This study uses constrained DFT+$U$ calculations to accurately model the magnetic interactions in ACr₂O₄ spinels, resolving previous inconsistencies and highlighting the importance of constraints for matching experimental data.

Contribution

It demonstrates that constrained DFT+$U$ calculations provide a better understanding of magnetic behavior in ACr₂O₄ compounds than unconstrained methods.

Findings

Constrained calculations align with experimental exchange coupling constants.

Unconstrained calculations show inconsistent behavior with experimental data.

Curie-Weiss temperatures follow the experimental trend when using constrained methods.

Abstract

In this work, we try to understand the inconsistency reported by [Yaresko, Phys. Rev. B. {\bf 77}, 115106 (2008)] in the theoretically estimated nature and the variation of magnitude of nearest neighbour exchange coupling constant ($\arrowvert${\it J$_{1}$}$\arrowvert$) with increasing $U$ in ACr$_{2}$O$_{4}$ (A=Zn, Cd, Mg and Hg) compounds by using density functional theory. In unconstrained calculations, the nature and variation of $\arrowvert${\it J$_{1}$}$\arrowvert$ as a function of $U$ in the present study are not consistent with the experimental data and not according to the relation, {\it J$_{1}$}$\propto$$\frac{t^{2}}{U}$ especially for CdCr$_{2}$O$_{4}$ and HgCr$_{2}$O$_{4}$ for $U >$3 eV and U=2-6 eV, respectively. Such an inconsistent behavior of $\arrowvert${\it J$_{1}$}$\arrowvert$ is almost similar to that of Yaresko for these two compounds for $U$=2-4 eV. For ZnCr$_{2}$O$_{4}$ and MgCr$_{2}$O$_{4}$, the nature and the variation of $\arrowvert${\it J$_{1}$}$\arrowvert$ in the present work are in accordance with the experimental data and above mentioned relation for $U$=2-6 eV and are similar to that of Yaresko for ZnCr$_{2}$O$_{4}$ for $U$=2-4 eV. However, in constrained calculations the nature and variation of $\arrowvert${\it J$_{1}$}$_{c}$$\arrowvert$ in the present work are according to experimental data and above above mentioned relation for all four compounds. Hence, the present study shows the importance of constrained calculations in understanding the magnetic behaviour of these spinels. The values of magnitude of Curie-Weiss temperature [$\arrowvert$($\varTheta$$_{CW}$)$_{c}$$\arrowvert$] for ZnCr$_{2}$O$_{4}$$>$MgCr$_{2}$O$_{4}$$>$CdCr$_{2}$O$_{4}$$>$HgCr$_{2}$O$_{4}$ for $U$=2-5 eV, which are according to the order of experimentally observed values for these spinels.