Rhombohedral distortion-driven ferroelectric order and exchange bias effect in geometrically frustrated ${\mathrm{ZnFe}}_2{\mathrm{O}}_4$

The rhombohedral distortion-driven occurrence of the spontaneous electric polarization at a reasonably high temperature $\left(T\right)$ and an exchange bias (EB) effect below the antiferromagnetic (AFM) Néel temperature $\left(T_N\right)$ are revealed in ${\mathrm{ZnFe}}_2{\mathrm{O}}_4$. We observe the magnetic memory effect, suggesting a cooperative glassy magnetic state below $T_N$. The phase separation between the long-range AFM and the glassy magnetic component leads to the EB effect below $T_N$. The synchrotron diffraction studies confirm a structural transition to a polar $R3m$ structure from the cubic spinel structure, involving a strong rhombohedral distortion. This distortion correlates with the occurrence of the spontaneous electric polarization $\left(P\right)$ below $\sim 110$ K $\left(T_{\text{FE}1}\right)$, which is accompanied by a short-range order (SRO). The $P$-value enhances further due to an additional rhombohedral distortion below $T_N$. A considerable magnetoelectric (ME) coupling is detected below $T_{\text{FE}1}$. The increase of $P$ is $\sim 7.2$% for 5 T at the liquid nitrogen temperature. There has been a fundamental interest among the community in the unique result of the occurrence of ferroelectric (FE) order coexisting with SRO and having a significant ME coupling, which is accompanied by a strong rhombohedral structural distortion analogous to that observed in ${\mathrm{BiFeO}}_3$.

Figure 1

(a) A part of the pyrochlore structure formed by the Fe atoms crystallized in $Fd\overline{3}m$ structure within the unit cell. (b) Antiferromagnetically coupled tetrahedron unit, with examples of the magnetic frustrations denoted by the “?” marks.

Figure 2

(a) Zn $2p$ and (b) Fe $2p$ core level XPS of ZFO. (c) ${}^{57}\mathrm{Fe}$ Mössbauer spectrum recorded in the transmission mode at 300 K. The fits in (a), (b), and (c) are shown by the solid curves. (d) SEM image highlighting the particles.

Figure 3

(a) The $T$ variations of ZFC-FC magnetization curves recorded at 7 mT. The arrow in the inset highlights $T_N$, below which ZFC and FC magnetization curves deviate from each other. (b) The $dM/dT$ with $T$ recorded in ZFC mode, highlighting $T_{\text{FE}1}$. (c) The $T$ variation of the ZFC curve recorded at 7 mT, defined as a reference curve and the memory curve, as indicated by the solid curve. The arrow indicates the aging temperature at 10 K. (d) Difference between these curves shown by $\mathrm{\Delta }M$ plotted with $T$, highlighting a sharp “dip” close to 10 K. (e) The $t$ dependence of $M$ at 12 K, 9 K, and again at 12 K for $t1$, $t2$, and $t3$, respectively, with 7 mT after cooling in ZFC mode. (f) The $t$ dependence of $M$ at 12 K for $t1+t3$ recorded at 7 mT, following a single stretched exponential function. (g) The $t$ dependence of $M$ at 12 K, 9 K, and again at 12 K for $t1$, $t2$, and $t3$, respectively in zero-field after cooling in FC mode with a field of 7 mT. (f) $t$ dependence of $M$ at 12 K for $t1+t3$ in zero-field, following a single stretched exponential function. The $T$ vs $t$ plots for the experiments in ZFC and FC modes are shown at the bottom of panels (e) and (g), respectively.

Figure 4

(a) Hysteresis loop recorded up to maximum field of $\pm 5$ T at 2 K for a cooling field of 1 T. The inset highlights that the loop closes around 1.4 T, as indicated by an arrow. (b) Negative and (c) positive shifts are shown for a cooling field of $\pm 1$ T. Cooling field dependence of (d) EB field and (e) coercive field at 2 K. Temperature variation of (f) EB field for a cooling field of 2 T. The inset of (d) illustrates a representative illustration of the occurrence of a pinned layer at the interface between AFM and glassy magnetic phases.

Figure 5

The $T$ variations of (a) $I_p$ recorded with a constant $T$-sweep rate of 5 K/min at different poling temperatures ($T_{\text{pole}}$), as described in the text and for a poling field ($E$) of 500 kV/m; (b) $P$ for $E=+300,\pm 500$ kV/m, and 500 kV/m with a magnetic field of 5 T; (c) $I_p$ for $E=\pm 500$ kV/m and the same with a magnetic field of 5 T for $T_{\text{pole}}$ at 15 K; and (d) $P$, as calculated from the data given in (c). The inset of (a) highlights another low-$T$ peak and (c) shows the dc bias current ($I_{\text{DC}}$) with $T$ with a “dip” at $\sim 10.5$ K. The $T$ variations of (e) ${\epsilon }^{\prime }$ at $f=2$ kHz and (f) MD(%), showing $T_{\text{FE}2}$ and $T_N$ by the arrows. Plot of (g) $M^2$ with MD(%) at 10 K for $f=2$ kHz.

Figure 6

(a) High-$2\theta$ peaks shown at different $T$ around $T_{\text{FE}1}$ at 110 K. (b) Integrated intensity ($I_{\mathrm{area}}$) of (600) peak with $T$, showing arrows at $T_{\text{FE}1}$ and $T_N$. Comparison of (c) the refinements (red solid curve) of the selected $2\theta$ peaks at 90 K using $Fd\overline{3}m$ and $R3m$ space groups. Rietveld refinement of the (d) diffraction pattern at 90 K using $R3m$ structure. The inset amplifies the result on the quality of refinement in a selected high-$2\theta$ region. The $T$ variations of (e) refined lattice constants, $a$ and $c$ in range of 8.7–250 K, (f) $c$, and (g) $a$ at low-$T$ with $R3m$ structure, highlighting $T_{\text{FE}2}$ and $T_N$ by the arrows, (h) $V$ in the entire $T$ range, and (i) $V$ at low-$T$, showing $T_{\text{FE}2}$ and $T_N$.

Figure 7

(a) Connecting cubic unit cell along the [111] direction formed by Fe1 (0,0,0) atoms at the corner of the unit cell of $Fd\overline{3}m$ structure and a rhombohedral structure shown by the thick red colored lines. Axes of the rhombohedral structure are defined as [100]${}_r$, [010]${}_r$, and [001]${}_r$. (b) Hexagonal structure formed by Fe1 (0,0,0) atoms crystallized in $R3m$ space group and formation of a rhombohedral structure. The $T$ variations of (c) lattice constants, $a_r$, and (d) $\alpha$ of the isohedral rhombohedron, showing $T_{\text{FE}1},T_{\text{FE}2}$, and $T_N$ by the arrows.