Magnetoelectric responses induced by domain rearrangement and spin structural change in triangular-lattice helimagnets NiI${}_2$ and CoI${}_2$

Dielectric and magnetic properties have been investigated for single crystals of triangular-lattice antiferromagnets NiI${}_2$ and CoI${}_2$. For NiI${}_2$, the proper screw spin order with the magnetic modulation vector $q\sim (0.138,0,1.457)$ induces electric polarization ($P$) along the in-plane direction with respect to the triangular lattice basal plane. The $P$ shows monotonic increase as a function of the poling magnetic field ($H$) along the in-plane direction, suggesting the $H$-induced rearrangement of the multiferroic domain out of six possible domains. For CoI${}_2$, both in-plane and out-of-plane components of $P$ emerge in the helimagnetic ground state, in which two cycloidal magnetic phases with $q_1=(\frac{1}{12},\frac{1}{12},\frac{1}{2})$ and $q_2=(\frac{1}{8},0,\frac{1}{2})$ are supposed to coexist. The application of the in-plane $H$ induces two-step metamagneticlike transitions, which probably goes through another ferroelectric helimagnetic phase as well as a paraelectric spin-collinear phase. Such distinctive magnetoelectric responses in the two simple triangular lattice antiferromagnets demonstrate that even a slight difference in the balance of magnetic interactions leads to a dramatic change of resultant magnetoelectric response in frustrated magnets.

Figure 1

The crystallographic unit cell of (a) CdCl${}_2$ type and (b) CdI${}_2$ type structure. (c) The (001) projection of a triangular-lattice layer of ${\mathit{M}}^{2+}$ ions (filled circles) and two adjacent I${}^{-}$ layers (open circles). Plus (minus) signs indicate that the position of the I atoms is above (below) the ${\mathit{M}}^{2+}$ layer. The symmetry elements at an ${\mathit{M}}^{2+}$ site are also indicated: reflection mirrors ($m$), twofold rotation axes (2), and a threefold rotation axis along the $\left[001\right]$ axis with an inversion center (a triangle with a small circle).

Figure 2

(a) and (b) Schematic illustration of the proper screw spin order with the magnetic modulation vector $q\sim (0.138,0,1.457)$ reported for NiI${}_2$, viewed from the (a) [110] direction or (b) [001] direction. Blue bars in (a) indicate the spin-spiral plane. The compatible symmetry element as well as the allowed electric polarization ($P$) direction is indicated in (b). Closed (open) circles indicate Ni (I) atoms. (c) Six possible multiferroic domains with $P\parallel$$\langle 110\rangle$, (I)$\sim$(VI). Corresponding in-plane magnetic modulation vector $q_{\text{in}}$ and spin chirality (denoted as R or L) are also indicated.

Figure 3

(a)–(c) The temperature dependence of magnetic susceptibility $\chi$, the $\left[110\right]$ component of $\varepsilon$ (${\varepsilon }_{\left[110\right]}$), and electric polarization $P$ measured for NiI${}_2$. The $\left[110\right]$ component of $P$ ($P_{\left[110\right]}$) was measured in the warming process without $E$ and $H$ after the field cooling with $E\parallel \left[110\right]$ and $H\parallel \left[1\overline{1}0\right]$ (denoted as $H_{\text{MEC}}$). (d) and (e) Favorable domain distribution under the electric field ($E$) and the magnetic field ($H_{\text{MEC}}$).

Figure 4

(a) The temperature dependence of $P_{\left[110\right]}$ measured for NiI${}_2$ under various magnitudes of $H$ applied along the $\left[1\overline{1}0\right]$ axis. $P_{\left[110\right]}$ was measured in the warming process without $E$ after the field cooling with $E\parallel \left[110\right]$, while applied $H$ was unchanged during both procedures. The small open triangle indicates the onset of the magnetically-induced hump structure in the $P-T$ curve. (b) Magnetic field dependence of $P_{\left[110\right]}$ value at 10 K, obtained from the $P-T$ scans in Fig. 4a (open squares) and Fig. 3c (triangles). The data points taken with the same magnitude of poling magnetic field $H_{\text{MEC}}$ (=3 T or 14 T) are connected by dashed arrows.

Figure 5

(a) The temperature dependence of $M/H$ taken for NiI${}_2$ under various magnitudes of $H\parallel \left[1\overline{1}0\right]$. The data (except for the one at 1 T) are arbitrarily offset for clarity. Solid (open) circles are data obtained in the cooling (warming) run. Open triangles indicate anomalies observed in the warming runs. The $M/H$ profiles, except for the one with $H=1$ T, are arbitrarily offset for clarity. (b) $T-H$ phase diagram for NiI${}_2$ with $H\parallel \left[1\overline{1}0\right]$, determined by various $T$ scans of $M$ and $P$.

Figure 6

The temperature dependence of $M/H$ and $P_{\left[110\right]}$ near the magnetic transition temperature at $H=14$ T.

Figure 7

(a)–(c) The temperature dependence of $\chi$, the in-plane component of $\varepsilon$ (${\varepsilon }_{\text{in}}$), and $P$ for CoI${}_2$. (d)–(i) Schematic illustrations of possible magnetic structures in CoI${}_2$, viewed from the $\left[001\right]$ direction. Upper (lower) row indicates the ones with $q\parallel$$\langle 110\rangle$ ($q\parallel$$\langle 1\overline{1}0\rangle$). In the magnetic ground state, the spin-spiral plane is confined within the (001) plane [(d) and (g)]. Application of in-plane $H$ is expected to reorient the spin spiral into the $\left(1\overline{1}0\right)$ plane [(e) and (h)] or the $\left(110\right)$ plane [(f) and (i)]. For each configuration, the compatible symmetry element and the possible electric polarization direction are indicated. Red or blue bars in (e), (f), (h), and (i) denote the spin spiral plane perpendicular to the (001) plane.

Figure 8

(a) The temperature dependence of $M/H$, measured for CoI${}_2$ under various magnitudes of $H$ applied perpendicular to the [001] axis. (b) The temperature derivative of magnetization $\text{d}M/\text{d}T$ for the data taken in (a). (c) Magnetic field dependence of $M$ measured with $H\perp \left[001\right]$ at various temperatures. (d) Magnetic field derivative of magnetization $\text{d}M/\text{d}H$ for the data taken in (c). The triangles represent the transition points. In each figure, the data are arbitrarily offset for clarity, except for the ones taken at 1 T [(a) and (b)] or at 2 K [(c) and (d)].

Figure 9

The $T-H$ phase diagram for CoI${}_2$ under $H$ applied perpendicular to the [001] axis. The phase boundaries are determined by various $T$ and $H$ scans of $M$ and $P$. The horizontal broken line represents the presumed phase boundary between paramagnetic (PM) and antiferromagnetic (AF) phases suggested by the neutron diffraction experiment in Ref. 34.

Figure 10

(a) The temperature dependence of the in-plane component of (a) pyroelectric current $I_{\text{p}}$ and (b) electric polarization ($P_{\text{in}}$) measured for CoI${}_2$ in the warming process without $E$, after the cooling with $E$. Here, the in-plane $H$ is kept unchanged during both processes, which is applied normal to $E$. The corresponding data taken with the $E\parallel H$ poling condition is also shown in (c) and (d). In (a) and (c), the $I_{\text{p}}-T$ profiles are arbitrarily offset for clarity, except for the ones at 10 T.

Figure 11

(a) The $H$-direction dependence of $M/H$ measured at 6 T for CoI${}_2$. Here, $H$ is rotated around the [001] axis, and ${\theta }_H$ is defined as the angle between the $H$ direction and an arbitrarily chosen in-plane crystallographic axis. (b) Temperature dependence of the in-plane component of $P$ measured in the warming process without $E$, after the cooling down to 6 K with $E$. During both processes, in-plane $H$ (applied parallel or normal to $E$) was kept unchanged. (c) Temperature dependence of the [001] component of $P$ under the in-plane $H$. The solid (dashed) line represents the procedure that the specimen was cooled with the poling field of $E$ from 15 K to 6 K (2 K); then $E$ was switched off prior to the measurement in the $T$-increasing run. (d) Six possible multiferroic domains with unique in-plane $P$ directions expected for the spin texture shown in Fig. 7h. Corresponding in-plane magnetic modulation vector $q_{\text{in}}$ and spin chirality (denoted as R or L) are indicated. Favorable domain distributions under various combinations of $H$ and $E$ are also indicated in (e) and (f).