Stacking faults in $\alpha \text{−}{\mathrm{RuCl}}_3$ revealed by local electric polarization

We present out-of-plane dielectric and magnetodielectric measurements of single-crystalline $\alpha \text{−}{\mathrm{RuCl}}_3$ with various degrees of stacking faults. A frequency-dependent, but field-independent, dielectric anomaly appears at $T_A(f=100\mathrm{kHz})\sim 4$ K once both magnetic transitions at $T_{N1}\sim 7$ K and $T_{N2}\sim 14$ K set in. The observed dielectric anomaly is attributed to the emergence of possible local electric polarizations whose inversion symmetry is broken by inhomogeneously distributed stacking faults. A field-induced intermediate phase is only observed when a magnetic field is applied perpendicular to the Ru-Ru bonds for samples with minimal stacking faults. Less pronounced in-plane anisotropy is found in samples with a sizable contribution from stacking imperfections. Our findings suggest that dielectric measurement is a sensitive probe in detecting the structural and magnetic properties, which may be a promising tool, especially in studying $\alpha \text{−}{\mathrm{RuCl}}_3$ thin-film devices. Moreover, the stacking details of ${\mathrm{RuCl}}_3$ layers strongly affect the ground state both in the magnetic and electric channels. Such a fragile ground state against stacking faults needs to be overcome for realistic applications utilizing the magnetic and/or electric properties of Kitaev-based physics in $\alpha \text{−}{\mathrm{RuCl}}_3$.

Figure 1

(a)–(c) Temperature dependence of the normalized dielectric constant, $\varepsilon \left(T\right)/{\varepsilon }_{300\mathrm{K}}$, measured at zero field for three typical samples. The electric field was applied perpendicular to the $ab$ plane ($\mathbf{E}\left|\right|c\perp ab$ in the $R\overline{3}$ representation). A sample-dependent hysteretic structural transition is observed in all three samples using a temperature ramping rate of 1 K/min ($T_{sc}$: transition temperature for cooling; $T_{sw}$: transition temperature upon warming). (d)–(f) Temperature evolution of specific heat near the magnetic transitions for the three corresponding samples. Vertical dashed lines in (d)–(f) mark the magnetic transitions.

Figure 2

(a)–(c) Temperature evolution of $\varepsilon /{\varepsilon }_{20\mathrm{K}}$, recorded at $f=100$ kHz. In-plane magnetic fields were applied along the reciprocal space $(1\text{,}0,0)$ direction [parallel to the real-space Ru-Ru bonds; see the inset in (a)]. Curves have been shifted vertically for clarity. (d) The enlarged view of $\varepsilon /{\varepsilon }_{10\mathrm{K}}$ near $T_{N1}$. The open squares are experimental raw data and the lines are guides to the eyes. The inset of (d) shows $d\varepsilon /dT$. (e) and (f) The dissipation ($D$) as a function of temperature measured in fixed fields of samples 2 and 3, respectively. Inset in (a): An illustration of the in-plane reciprocal space (inner purple hexagon) and real space (outer blue hexagon) representations. Solid triangles in (a)–(c) track the variation of $T_{N1}$ with respect to in-plane magnetic fields. Vertical dashed lines in (b) and (c) mark the dielectric anomaly and the transition at $T_{N2}$ [determined by the specific heat data in Figs. 1 and 1].

Figure 3

Frequency dependence of (a), (c), (e) $\varepsilon /{\varepsilon }_{20\mathrm{K}}$ and (b), (d), (f) the corresponding dissipation ($D$) of all samples measured in zero field. The dielectric anomaly depends strongly on frequency and $T_A$ shifts gradually to higher temperatures for increasing frequency. (g) A schematic illustration of inversion symmetry breaking caused by nonuniformly distributed stacking faults. The black (red) cross marked out by a dashed rectangle labels the inversion center of each $ABC$ ($AB$) unit. The inversion counterpart is missing for the bottom $ABC$ segment when an $AB$ unit is inserted asymmetrically. The inset in (e) and (f) is a fitting of the frequency dependence of $T_A$ according to the Vogel-Fulcher law $f=f_{0}exp\left[-E/k_B(T_A-T_0)\right]$ using the Boltzmann constant $k_B$, a temperature constant $T_0=-4$ K, an activation energy $E=143$ K, and a characteristic frequency $f_0=7.6\times {10}^{12}$ Hz.

Figure 4

Magnetodielectric effect measured at fixed temperatures and constant frequency $f=100$ kHz. In (a) and (b), magnetic fields were along and perpendicular to the Ru-Ru bonds with $\mathbf{H}\parallel (1,0,0)$ and $\mathbf{H}\parallel (1,\overline{2},0)$, respectively, for sample 1. (c), (d) and (e), (f) Same measurements with (a) and (b) for samples 2 and 3, respectively. Curves are shifted vertically for clarity.

Figure 5

$T\text{−}H$ phase diagrams of the studied three samples derived from Figs. 2 and 4. (a) and (b) $T\text{−}H$ phase diagrams of sample 1 with fields applied along $\mathbf{H}\parallel (1,0,0)$ and $\mathbf{H}\parallel (1,\overline{2},0)$, respectively. (c), (d) and (e), (f) Same representations to those of (a) and (b) for samples 2 and 3, respectively. The symbols ZZ3 and ZZ2 represent three and two domains in the zigzag phase, respectively. The field-induced intermediate state is labeled as the $X$ phase. The paramagnetic state is labeled as PM and the zigzag AF order formed below $T_{N2}$ is represented as AFM 2. Green squares were obtained in specific heat experiments.