Rotomagnetic coupling in fine-grained multiferroic $\mathrm{BiFe}{\mathrm{O}}_3$: Theory and experiment

Using Landau-Ginzburg-Devonshire (LGD) theory for $\mathrm{BiFe}{\mathrm{O}}_3$ dense fine-grained ceramics with quasispherical grains and nanosized intergrain spaces enriched by elastic defects, we calculated a surprisingly strong size-induced increase in the antiferromagnetic transition temperature caused by the joint action of rotomagnetic and magnetostrictive coupling. Notably, all parameters included in the LGD functional have been extracted from experiments, not assumed. Complementarily, we performed experiments for dense $\mathrm{BiFe}{\mathrm{O}}_3$ ceramics, which revealed that the shift of the antiferromagnetic transition is to $T_N\sim 690\mathrm{K}$ instead of $T_N\sim 645\mathrm{K}$ for a single crystal. To explain the result theoretically, we consider the possibility of controlling the antiferromagnetic state of multiferroic $\mathrm{BiFe}{\mathrm{O}}_3$ via biquadratic antiferrodistortive rotomagnetic, rotoelectric, magnetoelectric, and magnetostrictive couplings. According to our calculations, the highest contribution is the rotostriction contribution, while the magnetostrictive and electrostriction contributions appear smaller.

Figure 1

(a) Schematics of the spherical grain with radius $R$ covered by a shell of thickness $R_0$, where the stresses are accumulated. (b) Cross section of the eight densely packed spherical grains of radius $R$ placed inside the cube with edge $4R$.

Figure 2

(a) Dependences of the AFM transition temperature $T_{\mathrm{AFM}}$ vs the grain radius $R$ calculated from Eq. (8) for several shell thicknesses $R_0=50\mathrm{nm}$ (curves 1), $R_0=25\mathrm{nm}$ (curves 2), $R_0=10\mathrm{nm}$ (curves 3), and $R_0=5\mathrm{nm}$ (curves 4). The total Vegard coefficient $W={\sum }_m{W}_m$ is equal to $-20{\AA }^3$ for solid curves and $+20{\AA }^3$ for dashed curves. (b) Separate contributions (rotomagnetic, magnetostriction, and rotoelectric) to the $T_{\mathrm{AFM}}$. Surface tension coefficient $\mu =5\mathrm{N}/\mathrm{m}$ and total defect concentration in the shell ${\sum }_m\delta N_m={10}^{27}{\mathrm{m}}^{-3}$. Other parameters are taken from Table 1.

Figure 3

Color map of the AFM transition temperature $T_{\mathrm{AFM}}$ in coordinates “grain radius $R$–shell thickness $R_0$” calculated from Eq. (9) for the same parameters as in Fig. 2 and Table 1.

Figure 4

(a,b) Temperature dependences of magnetization measured in the BFO ceramics under ZFC (lower curve) and FC condition (magnetic field of 1 kOe, upper curve). Plot (b) is the part of plot (a) in the vicinity of the AFM transition temperature. Error bars are shown by red lines. XRD data shown in the inset (c) illustrate the phase purity of the studied BFO ceramics (a tiny amount of the $\mathrm{F}{\mathrm{e}}_2{\mathrm{O}}_3$ impurity phase is marked by an asterisk). (d)–(f) SEM images of the dense BFO ceramics for different magnifications.

Figure 5

The dependence of the averaged transition temperature $\langle T_{\mathrm{AFM}}\rangle$ on the average grain radius $\langle R\rangle$ calculated from Eq. (10) for minimal grain radius $R_{min}\approx 50\mathrm{nm}$, shell thickness $R_0\approx 45\mathrm{nm}$, and Vegard coefficient $W=-20{\AA }^3$. Other parameters are the same as in Fig. 2 and Table 1.