Magnetoelectric effect and magnetic phase diagram of a polar ferrimagnet ${\mathrm{CaBaFe}}_4{\mathrm{O}}_7$

The magnetic phase diagram of a polar ferrimagnet ${\mathrm{CaBaFe}}_4{\mathrm{O}}_7$ with a magnetic easy axis has been investigated by measurements of magnetization, specific heat, and magnetoelectricity. A ferrimagnetic transition takes place at $T_{\mathrm{C}1}=275$ K within the orthorhombic phase followed by a second magnetic transition at $T_{\mathrm{C}2}=211$ K. Below $T_{\mathrm{C}2}$, successive metamagnetic transitions occur for magnetic fields applied perpendicular to the easy axis, implying a sequential emergence of magnetic states which are neither collinear nor coplanar. The observation of the static magnetoelectric effect was limited to temperatures below 120 K due to the conducting nature of the crystals at higher temperatures. The magnitude of the ferroelectric polarization shows large changes between the different field-induced magnetic phases. The low-field state is characterized by a large linear magnetoelectric coefficient of ${\alpha }_{cc}=39$ ps/m, while a gigantic polarization change of $\mathrm{\Delta }P=850 \mu \mathrm{C}/{\mathrm{m}}^2$ is observed for ${\mu }_{o}H=14$ T applied along the easy axis.

Figure 1

(a) Orthorhombic unit cell of ${\mathrm{CaBaFe}}_4{\mathrm{O}}_7$ composed of 4 formula units. ${\mathrm{FeO}}_4$ tetrahedra belonging to the alternating kagome and triangular layers are indicated by brown and green colors, respectively [23, 43]. The divalent Ca and Ba cations are located on top of each other in the voids of the stacking layers. (b) One kagome and one triangular layer as viewed from the crystallographic $c$ direction. The asymmetric shape of the tetrahedra, their misorientation relative to the main crystallographic directions of the regular triangular and kagome lattices, and the relative displacement of the Ca and Ba ions in the $ab$ plane are manifestations of the orthorhombic distortion.

Figure 2

Magnetic-field dependence of the magnetization measured in the low-temperature phase of ${\mathrm{CaBaFe}}_4{\mathrm{O}}_7$ for magnetic fields parallel (black and green) and perpendicular (red and blue) to the $c$ direction up to ${\mu }_{o}H=32$ T. The borders of the metamagnetic transitions are labeled by the ${\mu }_o{H}_{\mathrm{C}3}$ lower critical and the ${\mu }_o{H}_{\mathrm{C}4}$ upper critical fields, showing up for $\mathbf{H}\perp c$. Both are characterized by finite hysteresis.

Figure 3

Temperature dependence of the magnetization in field cooling (FC) with ${\mu }_{o}H=1$ T applied along and perpendicular to the $c$ direction, and the specific heat of ${\mathrm{CaBaFe}}_4{\mathrm{O}}_7$ in zero field in a warming run. Three phase transitions are observed in the specific heat among which the one at $T_{\mathrm{S}}=380$ K is likely to be of structural origin. The other two transitions are strongly manifested in the magnetization as well. $T_{\mathrm{C}1}=275$ K is the Curie temperature of the ferrimagnetic ordering, and $T_{\mathrm{C}2}=211$ K is a second magnetic transition, where magnetization along the $c$ axis decreases slightly. The inset shows a magnified view of the magnetization in the field-cooling run around $T_{\mathrm{C}1}$ and $T_{\mathrm{C}2}$ with $\mathbf{H}\perp c$.

Figure 4

Curie plot of the high-temperature magnetic susceptibility of ${\mathrm{CaBaFe}}_4{\mathrm{O}}_7$ with magnetic fields applied along and perpendicular to the $c$ axis, in field-cooling (FC) and field-warming (FW) runs. Vertical dashed lines indicate the structural transition temperature ($T_{\mathrm{S}}$) as determined from the zero-field specific-heat measurements and the ferrimagnetic transition temperature ($T_{\mathrm{C}1}$).

Figure 5

Magnetic-field dependence of the polarization change and corresponding magnetization curves for magnetic fields applied along the $c$ axis [panels (a) and (c)] and within the $ab$ plane [panels (b) and (d)] at $T=80$ K. The largest change was observed in the $c$ component of the polarization irrespective of the orientation of the magnetic field. The inset to (a) shows the symmetric butterfly loop in $\mathrm{\Delta }\mathbf{P}\parallel c$ in the $-1$ T $\le {\mu }_{o}H\le$ 1 T region. The inset to (b) shows the polarization changes perpendicular to the $c$ axis plotted in an enlarged scale. In the inset of (c), $c$-axis magnetization is magnified to show the hysteresis corresponding to that of the polarization change.

Figure 6

Magnetic-field dependence of polarization change $\mathrm{\Delta }P$ at different temperatures, for (a) $\mathrm{\Delta }\mathbf{P}\parallel c$ and $\mathbf{H}\parallel c$ axis, (b) $\mathrm{\Delta }\mathbf{P}\parallel c$ and $\mathbf{H}\perp c$ axis, and (c) $\mathrm{\Delta }\mathbf{P}\parallel \mathbf{H}$ and $\mathbf{H}\perp c$ axis. In panels (a), (b), and (c), each polarization curve is shifted vertically by 200 $\mu \mathrm{C}/{\mathrm{m}}^2$, 200 $\mu \mathrm{C}/{\mathrm{m}}^2$, and 100 $\mu \mathrm{C}/{\mathrm{m}}^2$, respectively, for the purpose of clarity. $\mathrm{\Delta }P$ values marked by cross symbols are plotted in Fig. 7 against temperature. (d) Field dependence of the magnetization at various temperatures with magnetic fields applied along the $c$ axis. The inset shows a magnified view of the hysteresis region. (e) Field-dependent magnetization curves at different temperatures when the field is applied perpendicular to $c$. Each curve is shifted vertically by $0.5{\mu }_B/\mathrm{f}.\mathrm{u}.$ for better visibility. Data collected in increasing and decreasing magnetic fields are plotted with solid and dashed lines, respectively. The magnetic phase transitions ${\mu }_o{H}_{\mathrm{C}1},{\mu }_o{H}_{\mathrm{C}3}$, and ${\mu }_o{H}_{\mathrm{C}4}$ are indicated by solid up and empty down triangles for the field increasing and decreasing runs, respectively. These phase transitions are determined as local maximum of the $\frac{dM}{dH}$ curves, and are plotted in the ${\mu }_{o}H-T$ phase diagram by symbols of the same colors (Fig. 8). For more details about the determination of the phase boundaries see the Supplemental Material [52].

Figure 7

(a) Temperature dependence of the linear magnetoelectric coefficients ${\alpha }_{cc}=\frac{d{P}_c}{d{H}_c}$ and ${\alpha }_{aa}=\frac{d{P}_a}{d{H}_a}$ measured in the $\mathrm{\Delta }\mathbf{P}\parallel \mathbf{H}$ configurations. (b) Temperature evolution of the polarization changes in representative magnetic fields in three measurement configurations. The corresponding values are indicated by crosses in Figs. 6–6.

Figure 8

Temperature versus magnetic field phase diagram for fields applied (a) parallel and (b) perpendicular to the $c$ axis. Phase boundaries based on magnetization measurements are represented by full or empty triangles, while those obtained from specific-heat measurements are denoted by diamonds. Direction of the triangles illustrates the following measurement conditions: triangles pointing up and down correspond to isothermal magnetization measurements with increasing and decreasing fields, respectively, while triangles pointing left and right indicate magnetization measurements at fixed external fields measured with decreasing and increasing temperatures, respectively. The phase boundaries are represented by dashed lines as a guide to the eyes. The finite-field crossover region where the spin polarization rapidly develops in the paramagnetic state is indicated by color gradation. For a detailed discussion of the different phases and the determination of the phase boundaries see the main text and the Supplemental Material [52].