Magnetoelastic coupling anisotropy in the Kitaev material $\alpha \text{−}\mathrm{Ru}{\mathrm{Cl}}_3$

The Kitaev material $\alpha \text{−}{\mathrm{RuCl}}_3$ is among the most prominent candidates to host a quantum spin-liquid state endowed with fractionalized excitations. Recent experimental and theoretical investigations have separately revealed the importance of both the magnetoelastic coupling and the magnetic anisotropy in dependence of the applied magnetic-field direction. In this combined theoretical and experimental research, we investigate the anisotropic magnetic and magnetoelastic properties for magnetic fields applied along the main crystallographic axes as well as for fields canted out of the honeycomb plane. We found that the magnetostriction anisotropy is unusually large compared to the anisotropy of the magnetization, which is related to the strong magnetoelastic $\widetilde{{\mathrm{\Gamma }}^{\prime }}$-type coupling in our ab-initio derived model. We observed large, nonsymmetric magnetic anisotropy for magnetic fields canted out of the honeycomb $ab$ plane in opposite directions, namely, toward the $+c^{*}$ or $-c^{*}$ axes, respectively. The observed directional anisotropy is explained by considering the relative orientation of the magnetic field with respect to the coaligned ${\mathrm{RuCl}}_6$ octahedra. Magnetostriction measurements in canted fields support this nonsymmetric magnetic anisotropy; however, these experiments are affected by magnetic torque effects. Comparison of theoretical predictions with experimental findings allow us to recognize the significant contribution of torque effects in experimental setups where $\alpha \text{−}{\mathrm{RuCl}}_3$ is placed in canted magnetic fields.

Figure 1

(a) Single honeycomb layer of $\alpha \text{−}{\mathrm{RuCl}}_3$ and the crystallographic axes ($a, b$, and $c^{*}$). We highlight two directions within the $ab$ plane; the $a$ axis is perpendicular ($\perp bond$) and the $b$ axis is parallel ($\parallel bond$) to one of the Ru-Ru bonds, respectively. Red and blue arrows at the honeycomb sites indicate the zigzag domain with ordering wave-vector $\mathbf{Q}\parallel b$. The dashed line indicates a ${\mathrm{C}}_2$ rotation symmetry around an axis parallel to the $b$ axis. (b) During the magnetization and magnetostriction measurements, the magnetic-field ($\mathbf{H}$) was canted out of the $ab$ plane by an angle of $\vartheta$, while the planar projection of the applied field was either along the $a$ or $b$ axis. (c) Temperature dependence of the magnetization for fields along the main crystallographic axes in the field cooling runs (${\mu }_{0}H=1$ T). Note, that the data for $\mathbf{H}\parallel c^{*}$ are multiplied by a factor of 5 for better visibility. (d, e) Angular dependence of the magnetization at $T=2$ K for fields rotated within the $ab, a{c}^{*}$, and $b{c}^{*}$ planes, respectively. Measurement data are plotted with symbols (full circles) for ${\mu }_{0}H=1$ T and 5 T. In case of $\mathbf{H}\in ab$, the measurements are plotted for ${\mu }_{0}H=2$ T. Dashed curves in panels (e) and (f) correspond to the theoretical calculations.

Figure 2

Magnetic-field dependence of the ${\lambda }_{c^{*}}$ linear magnetostriction coefficient at selected temperatures. The $\mathrm{\Delta }L$ length change was measured along the $c^{*}$ axis and $\mathbf{H}\parallel a$ magnetic field was applied, perpendicular to one of the Ru-Ru bonds.

Figure 3

(a) Field dependence of the linear magnetostriction coefficient ${\lambda }_{c^{*}}$ at $T=3$ K. The magnetic field was applied along the main crystallographic axes, as well as canted out of the $ab$ plane with projection perpendicular to the bond ($\mathbf{H}\in a{c}^{*}$). The $\vartheta =0$ deg and 90 deg angles correspond to the $a$ and $c^{*}$ axes, respectively. The two peaks in the ${\lambda }_{c^{*}}$ data correspond to (${\mu }_0{H}_1$) a phase transition between different zigzag interplane orderings and (${\mu }_0{H}_2$) a transition into the field-induced quantum paramagnetic phase. The ${\mu }_0{H}_1$ and ${\mu }_0{H}_2$ phase boundaries are indicated by triangles, respectively. (b) Angular dependence of the ${\mu }_0{H}_1$ and ${\mu }_0{H}_2$ phase-transition fields. The dashed lines indicate the $\sim 1/cos\vartheta$ field dependence. (c) The field dependence of the ${\lambda }_{c^{*}}/{\kappa }_{c^{*}}$ magnetostriction calculated for $H$ field canted out of the $ab$ plane with $\vartheta$ angle, $\mathbf{H}\in a{c}^{*}$. The inset defines the field angle $\vartheta$. Note that panels (a) and (c) are shown for different field scales. (d) Magnetic-field dependence of the calculated $\tau$ magnetic torque for selected $\vartheta$ canting angles, $\mathbf{H}\in a{c}^{*}$. (e) The effect of magnetic torque on the field dependence of the magnetostriction is modeled with the ${\lambda }_{c^{*}}/{\kappa }_{c^{*}}+A·\left|\tau \right|$ relation with the same $A=2.2·{10}^5$ $\mathrm{Pa}·\mathrm{Rad}$ / (${\mathrm{T}}^2·{\mu }_B$/f.u.) parameter fitted for each curve.

Figure 4

Dissection of the largest contributions to theoretical magnetostriction through Eq. (5). Each curve corresponds to a component of Eq. (5) with different $\mathcal{J}$. The notation $K, J, \mathrm{\Gamma }, {\mathrm{\Gamma }}^{\prime }$, and $\mathbb{G}$ corresponds to the summand components of $\frac{1}{V}\tilde{K}\left(\frac{\partial M}{\partial K}\right), \frac{1}{V}\tilde{J}\left(\frac{\partial M}{\partial J}\right), \frac{1}{V}\tilde{\mathrm{\Gamma }}\left(\frac{\partial M}{\partial \mathrm{\Gamma }}\right), \frac{1}{V}{\tilde{\mathrm{\Gamma }}}^{\prime }\left(\frac{\partial M}{\partial {\mathrm{\Gamma }}^{\prime }}\right)$, and $\frac{1}{V}\tilde{\mathbb{G}}\left(\frac{\partial M}{\partial \mathbb{G}}\right)$, respectively. The line with $\mathcal{J}=\mathbb{G}$ corresponds to magnetoelastic coupling with the $g$ tensor. (a) In-plane field $\mathbf{H}\parallel a$ ($\vartheta =0\mathrm{deg}$); (b) out-of-plane field $\mathbf{H}\parallel c^{*}$ ($\vartheta =90\mathrm{deg}$).

Figure 5

(a) Magnetic-field dependence of the magnetization at $T=2$ K for $\mathbf{H}\in a{c}^{*}$ with $\vartheta =\pm 45$ deg canting out of the $ab$ plane. (b) Magnetic-field dependence of the field derivative and (c) the field dependence of the $\mathrm{\Delta }M$ magnetization difference for the $\vartheta =+45$ deg and $-45$ deg measurements ($\mathrm{\Delta }M=M_{+45\mathrm{deg}}-M_{-45\mathrm{deg}}$). The phase transitions for $\mathbf{H}\in a{c}^{*}$ are indicated by triangles as peaks in $dM/dH$. (d–f) Magnetic-field dependence of the magnetization, field derivative, and $\mathrm{\Delta }M$ at $T=2$ K for $\mathbf{H}\in b{c}^{*}, \vartheta =\pm 45$ deg. (g) Magnetic-field dependence of the ${\lambda }_{c^{*}}$ magnetostriction coefficient at $T=3$ K for $\mathbf{H}\in a{c}^{*}$ and $\mathbf{H}\in b{c}^{*}$ measured in the field-increasing runs with $\vartheta =\pm 45$ deg canting angles. Three ${\lambda }_{c^{*}}\text{−}H$ curves are shown for each configuration with numerals indicating the order of the measurements (complete list is shown in the Supplemental Material [75], Fig. S1). The experimental conditions are illustrated as an inset in panels (a) and (d). While panels (a, b, d, e, and g) show measurements for the field-increasing runs, panels (c, f, and h) show the measurements for the field-increasing and -decreasing runs.

Figure 6

(a) Schematic illustration of the measurement configurations for each experiment. For $\mathbf{H}\in a{c}^{*}, \vartheta =+45$ deg, and $\vartheta =-45$ deg, the net magnetization points along the vertexes and the edges of the ${\mathrm{Cl}}_6$ octahedra, respectively. In both cases of $\mathbf{H}\in b{c}^{*}, \vartheta =\pm 45$ deg, the net magnetization points along the side of the ${\mathrm{RuCl}}_6$ octahedra, which configurations are connected by the ${\mathrm{C}}_2$ rotation in the honeycomb plane. (b) Magnetic-field dependence of the magnetization and (c) $\mathrm{\Delta }M$ magnetization difference calculated for $\mathbf{H}\in a{c}^{*}$ and $\mathbf{H}\in b{c}^{*}$ with $\vartheta =\pm 45$ deg canting out of the $ab$ plane. (d) magnetic-field dependence of the $\tau$ magnetic torque for $\mathbf{H}\in a{c}^{*}, \vartheta =\pm 45$ deg. (e) Magnetic-field dependence of ${\lambda }_{c^{*}}/{\kappa }_{c^{*}}$ magnetostriction and (f) $\mathrm{\Delta }{\lambda }_{c^{*}}/{\kappa }_{c^{*}}$ magnetostriction difference, calculated for $\mathbf{H}\in a{c}^{*}$ and $\mathbf{H}\in b{c}^{*}, \vartheta =\pm 45$ deg. The effect of the magnetic torque on the $\mathrm{\Delta }{\lambda }_{c^{*}}/{\kappa }_{c^{*}}$ magnetostriction difference is illustrated in panel (f); the torque is scaled to the unit of ${\lambda }_{c^{*}}/{\kappa }_{c^{*}}$ with $A=2.2·{10}^5$ $\mathrm{Pa}·\mathrm{Rad}$ / (${\mathrm{T}}^2·{\mu }_B$/f.u.).