Thermally or Magnetically Induced Polarization Reversal in the Multiferroic ${\mathrm{CoCr}}_2{\mathbf{O}}_4$

We report the unexpected evolution, with thermal and magnetic-field ($H$) variations, of the interrelation between the polarization $\mathbf{P}$, magnetization $\mathbf{M}$, and spiral wave vector $\mathbf{Q}$ in ${\mathrm{CoCr}}_2{\mathrm{O}}_4$, which has a ferrimagnetic conical-spiral magnetic order. For example, $\mathbf{P}$ suddenly jumps and changes its sign at the magnetic lock-in transition ($T_L$) with thermal variation, or with isothermal variation of $H$ (without changing its direction) at $T_L$, which surprisingly occurs without change in spiral handedness (i.e., the sign of $\mathbf{Q}$). The presence of multiple spiral sublattices may be behind this unusual behavior.

Figure 1

(a) Crystallographic and low-$T$ magnetic structure of spinel ${\mathrm{CoCr}}_2{\mathrm{O}}_4$. ${\mathrm{Co}}^{2+}$ and ${\mathrm{Cr}}^{3+}$ ions are located at the center of tetrahedral and octahedral ${\mathrm{O}}^{2-}$ cages, respectively. Conical-spiral spins of ${\mathrm{Co}}^{2+}$ and ${\mathrm{Cr}}^{3+}$ ions for only 3 of the 6 sublattices ($A_2$, $B_1$, and $B_2$) are shown for clarity. Cone angles shown are from 12, which, with 13, should be consulted for more details. Also shown are the directions of $\mathbf{M}$, $\mathbf{P}$, and $\mathbf{Q}$. (b) $T$ dependence of magnetization, $\mathbf{M}$, along the $[001]$ direction in $H=0.5\mathrm{T}$ upon cooling, specific heat divided by temperature in $H=0\mathrm{T}$ upon cooling, and dielectric constant in $H=0\mathrm{T}$ at 44 kHz upon warming. The existence of three phase transitions is evident, and $T_C$, $T_S$, and $T_L$ denote temperatures for ferrimagnetic transition, conical spin ordering, and lock-in transition, respectively.

Figure 2

(a) $T$ dependence of electric polarization, $\mathbf{P}$, along the $[\overline{1}10]$ direction, and $\mathbf{M}$ along the $[001]$ direction below 30 K. $\mathbf{P}$ suddenly switches sign when cooling across 14 K without changing the signs of $\mathbf{M}$ and $\mathbf{Q}$. (b) and (c) $H$ dependence of $\mathbf{M}$ and $\mathbf{P}$ at 20 K and 10 K, respectively. The reversal of all of $\mathbf{M}$, $\mathbf{P}$, and $\mathbf{Q}$ is achieved by $H$ reversal.

Figure 3

(a) Contour plots of the Co $L_3$ edge magnetic soft x-ray scattering intensity in the plane defined by ${\mathbf{Q}}_{110}$ and ${\mathbf{Q}}_{1\overline{1}0}$ recorded at temperatures above and below $T_L$ (14 K) with photon energy of 778.4 eV. The $\mathbf{E}$ vector of incident x-rays was parallel to the $[001]$ axis. The contour plots are shown in a logarithmic scale with its order of magnitude expressed by means of color. (b), (d), and (e) Scattering intensity of $\mathbf{Q}$ scans along $[110]$ and $[1\overline{1}0]$ under $+\mathbf{M}$ (solid curves) and $-\mathbf{M}$ (open circles) with circularly polarized x-rays. (c) Depiction of conical-spiral spins where $\pm \mathbf{M}$ go with $\pm \mathbf{Q}$ above and below $T_L$, respectively.

Figure 4

(a) Repeatable polarization switching with $T$ varied linearly with time between 8 K and 20 K. (b) $T$ dependence of polarization around the $T_L$ transition in different applied $H$ (0.1, 0.5, and 3 T), indicating that $T_L$ increases slightly with increasing $H$. (c) $\mathbf{P}$ at 14.0 K vs $H$ after poling the specimen in $H=0.1\mathrm{T}$ and $E\approx 10\mathrm{kV}/\mathrm{cm}$. The initial $(+,+,+)$ state which is defined as [$\mathrm{sgn}(\mathbf{M})$, $\mathrm{sgn}(\mathbf{P})$, $\mathrm{sgn}(\mathbf{Q})$] changes to the $(+,-,+)$ state in $H\approx 3\mathrm{T}$. Negative $H$ scan, reversing $\mathbf{M}$, induces switching of $\mathbf{P}$ and $\mathbf{Q}$, i.e., resulting in the $(-,+,-)$ state.