Field-induced intermediate ordered phase and anisotropic interlayer interactions in $\alpha \text{−}{\mathrm{RuCl}}_3$

In $\alpha \text{−}{\mathrm{RuCl}}_3$, an external magnetic field applied within the honeycomb plane can induce a transition from a magnetically ordered state to a disordered state that is potentially related to the Kitaev quantum spin liquid. In zero field, single crystals with minimal stacking faults display a low-temperature state with in-plane zigzag antiferromagnetic order and a three-layer periodicity in the direction perpendicular to the honeycomb planes. Here, we present angle-dependent magnetization, ac susceptibility, and thermal transport data that demonstrate the presence of an additional intermediate-field ordered state at fields below the transition to the disordered phase. Neutron-diffraction results show that the magnetic structure in this phase is characterized by a six-layer periodicity in the direction perpendicular to the honeycomb planes. Theoretically, the intermediate ordered phase can be accounted for by including spin-anisotropic couplings between the layers in a three-dimensional spin model. Together, this demonstrates the importance of interlayer exchange interactions in $\alpha \text{−}{\mathrm{RuCl}}_3$.

Figure 1

(a) Temperature–magnetic-field phase diagram of $\alpha \text{−}{\mathrm{RuCl}}_3$ for in-plane fields perpendicular to Ru-Ru bonds (see Fig. 2), as constructed from ac susceptibility measurements. The Néel temperature as extracted from dc susceptibility measurements [10] is shown as black squares for comparison. The dome-shaped intermediate ordered phase zz2 occurs between the low-field ordered phase zz1 and the disordered phase at higher fields. (b) Phase diagram of $\alpha \text{−}{\mathrm{RuCl}}_3$ at $T=2\mathrm{K}$ as function of in-plane angle and magnetic field from ac and dc susceptibility measurements. The lines are a guide to the eye.

Figure 2

Definition of the in-plane angle $\phi$ within the first Brillouin zone of the two-dimensional (2D) reciprocal lattice (gray hexagon) and with respect to the real-space orientation of the Ru-Ru bonds (red/blue/green). Directions that are equivalent to (1,0,0) (black) and (1,1,0) (red) correspond to angles $\phi \equiv 0^{\circ }mod{60}^{\circ }$ and $\phi \equiv {30}^{\circ }mod{60}^{\circ }$, respectively. ${\mathbf{a}}^{*}$ and ${\mathbf{b}}^{*}$ denote the reciprocal-lattice vectors in the $R\overline{3}$ structure.

Figure 3

(a) Field-dependent magnetization for different in-plane field directions $\phi$ at $T=2\mathrm{K}$. Arrows indicate kinks in the magnetization. (b) Real part of the ac susceptibility ${\chi }_{\mathrm{ac}}^{\prime }$ as a function of dc magnetic field for in-plane field directions $\mathbf{B}\parallel \{1,0,0\}$ ($\phi \equiv 0^{\circ }mod{60}^{\circ }$) (upper panel) and $\mathbf{B}\parallel \{1,1,0\}$ ($\phi \equiv {30}^{\circ }mod{60}^{\circ }$) (lower panel) at $T=2\mathrm{K}$. The frequency of the 1 mT ac field is 1 kHz. (c) Polar plot of the angular dependence of the dc magnetization ($M/B$) plotted for various field strengths at $T=2\mathrm{K}$. A sixfold oscillation as a function of $\phi$ is visible. The maxima and minima are reversed around $6\mathrm{T}$. (d) The angle dependence of the magnetization above $T_{\mathrm{N}}$ at $T=10\mathrm{K}$. (e) ${\chi }_{\mathrm{ac}}^{\prime }$ at various fixed temperatures showing two anomalies as a function of $\{1,1,0\}$ magnetic field strength for $T\lesssim 4\mathrm{K}$. (f) Thermal conductivity at various fixed temperatures as a function of $\{1,1,0\}$ magnetic field strength. The curves in (e) and (f) are offset for clarity. (g) Linear plot of the angle dependence of the magnetization ($M/B$) at various fields, for $T=2\mathrm{K}$. The data are the same as those plotted in (c). The exchange of the minima and maxima between 5 and 6 T is clearly visible in this plot.

Figure 4

Top row: Neutron diffraction intensities at $\left|L\right|=1$ for (a) $B=3.5\mathrm{T}$ and (b) $B=6.6\mathrm{T}$. Magnetic Bragg peaks are visible at the $\mathbf{M}$ points $(1/2,0,1)$, $(-1/2,0,1)$, $(0,-1/2,1)$, and $(0,1/2,1)$. Bottom row: Intensities at $\left|L\right|=0.5$ for (c) $B=3.5\mathrm{T}$ and (d) $B=6.6\mathrm{T}$. Additional weak magnetic Bragg peaks appear at the $\mathbf{M}$ points in the 6.6 T data. The perpendicular integration range in is $\mathrm{\Delta }L=\pm 0.025\mathrm{r}.\mathrm{l}.\mathrm{u}.$ and the data have been averaged over positive and negative $L$. The first Brillouin zone is indicated by the dashed hexagon, and the arrows in (d) point to the weak intensity observed at the $\mathbf{M}$ points. The stripy intensity appearing in the upper corners is spurious and caused by imperfect background subtraction for larger wave vectors.

Figure 5

(a) $\mathbf{M}$-point $(1/2,0,L)$ neutron-diffraction intensities as a function of the out-of-plane wave-vector component $L$ for $B=3.5\mathrm{T}$ (blue) and $B=6.6\mathrm{T}$ (red). The intensities have been averaged over the four different $\mathbf{M}$ points shown in Fig. 4. (b) $\mathit{\Gamma }$-point neutron-diffraction intensities. The data in (a,b) are integrated in $\mathrm{\Delta }H=\pm 0.03\mathrm{r}.\mathrm{l}.\mathrm{u}.$ and $\mathrm{\Delta }K=\pm 0.03\mathrm{r}.\mathrm{l}.\mathrm{u}.$. (c) Magnetic structure factor modeling for the $\mathbf{M}$ point for the zz1 (blue) and zz2 (red) phases as discussed in the text. Note that the intensities in the zz2 phase at integer $L$ are from the 3f-zz structure and the ones at half-integer $L$ are from the 6f-zz structure; they coexist at intermediate fields. (d) Magnetic structure factor modeling for the $\mathit{\Gamma }$ point (see the text).

Figure 6

Intensities of three different magnetic Bragg peaks at 2 K as a function of field strength for $\mathbf{B}\parallel \{1,1,0\}$, as obtained from Gaussian fits of 1D cuts through the data at each field strength. The lines are a guide to the eye. Error bars represent one standard deviation of the fitted intensity.

Figure 7

$R\overline{3}$ crystal structure and interlayer couplings. The three inequivalent honeycomb layers are shown in black, red, and blue. The dashed rhombus indicates the crystallographic unit cell, consisting of two spins per layer. (a) Top view, with a viewpoint along the crystallographic $\mathbf{c}$ direction. (b) Front view, with a viewpoint along an in-plane direction perpendicular to a Ru-Ru bond. The interlayer couplings $J_{\perp }$ and $K_{\perp }$ are depicted in dashed green and dotted purple lines, respectively.

Figure 8

Two different stackings of in-plane zigzag configurations. The view is along the a+b direction as in Fig. 7 (perpendicular to a Ru-Ru bond). Spins that point in the same direction are represented by the same (filled or open) symbol; the different colors denote crystallographically inequivalent layers. The magnetic unit cells are indicated by dashed rectangles. (a) Threefold zigzag stacking (3f-zz). (b) Sixfold zigzag stacking (6f-zz). Note that in (a) spins directly above each other point in different directions (antiferromagnetic alignment), while they point in the same direction (ferromagnetic alignment) in (b).

Figure 9

Classical phase diagram of a microscopic spin model as a function of magnetic field $B$ and in-plane angle $\phi$ in the limit of small interlayer couplings, $|K_{\perp }|,|J_{\perp }|\ll A$, with $(-K_{\perp })/J_{\perp }=1.14$. 3f-zz and 6f-zz denote ordered zigzag configurations with threefold and sixfold layer periodicity, respectively. The experimental phases zz1 and zz2 in Fig. 1 are described by 3f-zz and a coexistence of 3f-zz and 6f-zz, respectively. $B_{\mathrm{c}1}$ (blue) denotes the first-order transition between different zigzag stackings; $B_{\mathrm{c}2}$ (black) denotes the transition to the disordered phase.

Figure 10

Classical magnetization $(M/B)$ at various fields in the microscopic spin model as a function of in-plane angle $\phi$ in the limit of small interlayer couplings, $|K_{\perp }|,|J_{\perp }|\ll A$, where $A$ sets the overall energy scale.

Figure 11

1D cuts through the neutron-diffraction data in the honeycomb plane at 3.5 and 6.6 T. Panels (a)–(d) show cuts with $\left|L\right|=1$, and (e) and (f) show cuts with $\left|L\right|=0.5$. The data are integrated over the perpendicular directions by $\pm 0.03$ r.l.u. in the honeycomb plane, and by $\pm 0.025$ r.l.u. along $L$. In addition, the data are averaged over positive and negative $L$. The inset of panel (a) shows the direction of the cuts in the honeycomb plane. Note the difference in $y$-axes scaling between panels (a)–(d) and (e)–(h).