Exploring the correlation between the spin-state configuration and the magnetic order in Co-substituted BiFeO3

Co-substituted ${\mathrm{BiFeO}}_3$ systems exhibit both ferroelectric and antiferromagnetic (AFM) orders with a considerably high electric polarization $\mathbf{P}$ and a modestly canted magnetization $\mathbf{M}$ and a cross coupling between them at room temperature. In the present study, employing first-principles calculations and Monte Carlo simulations, we unravel the microscopic interactions that contribute to stabilize this particular canted AFM order with $\mathbf{M}$. Our results propose a correlation between the spin-state configuration of the trivalent ${\mathrm{Co}}^{3+}$ ions and the energetics associated with the magnetic orders. Because of the complex interplay between the Hund's coupling $\left(J_H\right)$ and crystal field splitting $\left({\mathrm{\Delta }}_{\mathrm{CF}}\right)$, the trivalent ${\mathrm{Co}}^{3+}$ ions are known to exhibit three spin $\left(S\right)$ configurations, namely low $(t_{2g}^6\to S=0)$ spin (LS), intermediate $(t_{2g}^5{e}_g^1\to S=1)$ spin (IS), and high $(t_{2g}^4{e}_g^2\to S=2)$ spin (HS). We observe electron correlation induced LS $\to$ HS state transition, which implies a close competition among these spin states in the $R3c$ polar structure. Moreover, our detailed analysis points towards the formation of heterogeneous spin-state configuration at room temperature. While the LS state configuration was not observed to exhibit any particular tendency to stabilize the canted AFM phase, the HS state tends to strongly favor the formation of this desired magnetic order. On the other hand, interestingly, the formation of finite fraction of the IS state can contribute to enhance the magnitude of $\mathbf{M}$. Our investigation is expected to initiate further quest for appropriate magnetic substitutes to ensure enhanced functionalities of the system.