Monoclinic crystal structure of $\alpha -{\mathrm{RuCl}}_3$ and the zigzag antiferromagnetic ground state

The layered honeycomb magnet $\alpha -{\mathrm{RuCl}}_3$ has been proposed as a candidate to realize a Kitaev spin model with strongly frustrated, bond-dependent, anisotropic interactions between spin-orbit entangled $j_{\mathrm{eff}}=\frac{1}{2}{\mathrm{Ru}}^{3+}$ magnetic moments. Here, we report a detailed study of the three-dimensional crystal structure using x-ray diffraction on untwinned crystals combined with structural relaxation calculations. We consider several models for the stacking of honeycomb layers and find evidence for a parent crystal structure with a monoclinic unit cell corresponding to a stacking of layers with a unidirectional in-plane offset, with occasional in-plane sliding stacking faults, in contrast with the currently assumed trigonal three-layer stacking periodicity. We report electronic band-structure calculations for the monoclinic structure, which find support for the applicability of the $j_{\mathrm{eff}}=\frac{1}{2}$ picture once spin-orbit coupling and electron correlations are included. Of the three nearest-neighbor Ru-Ru bonds that comprise the honeycomb lattice, the monoclinic structure makes the bond parallel to the $b$ axis nonequivalent to the other two, and we propose that the resulting differences in the magnitude of the anisotropic exchange along these bonds could provide a natural mechanism to explain the previously reported spin gap in powder inelastic neutron scattering measurements, in contrast to spin models based on the three-fold symmetric trigonal structure, which predict a gapless spectrum within linear spin wave theory. Our susceptibility measurements on both powders and stacked crystals, as well as magnetic neutron powder diffraction, show a single magnetic transition upon cooling below $T_{\mathrm{N}}\approx 13$ K. The analysis of our neutron powder diffraction data provides evidence for zigzag magnetic order in the honeycomb layers with an antiferromagnetic stacking between layers. Magnetization measurements on stacked single crystals in pulsed field up to 60 T show a single transition around 8 T for in-plane fields followed by a gradual, asymptotic approach to magnetization saturation, as characteristic of strongly anisotropic exchange interactions.

Figure 1

Observed x-ray diffraction patterns (log intensity scale) for an untwinned crystal of $\alpha -{\mathrm{RuCl}}_3$ at 80 K (a)–(c), 300 K (d)–(f), shown for three different planes, compared with calculations (g)–(i) for the monoclinic $C2/m$ structural model (Fig. 2) and the trigonal $P{3}_{1}12$ model (j)–(l). All wave vectors are labeled in r.l.u. units of the monoclinic cell and ${\mathbf{a}}_h^{*}$, ${\mathbf{b}}_h^{*}$, and ${\mathbf{c}}_h^{*}$ denote reciprocal lattice vectors of the hexagonal primitive cell of the trigonal structure (for the relation between the hexagonal and monoclinic axes, see Sec. 3c). Note the sharp peaks in the data are in good agreement with the monoclinic model [compare (d)–(f) with (g)–(i)], whereas the “supercell” peaks expected in the case of the trigonal model (k)–(l) at fractional positions $l=n+1/3,n+2/3$ ($n$ integer) are clearly absent from the data, instead only diffuse scattering is found in those places.

Figure 2

Monoclinic crystal structure of $\alpha -{\mathrm{RuCl}}_3$, showing the unit cell as a black outline, Ru as gray balls, and Cl as green. (a) Projection onto the $ac$ plane. (b) Basal layer projected onto the $ab$ plane.

Figure 3

Observed structure factor squared values of all three families of diffraction peaks compared to those calculated by fitting the $C2/m$ monoclinic structural model with fixed theoretical atomic fractional coordinates to data measured at (a) room temperature, and (b) 80 K.

Figure 4

(a) Magnetic susceptibility as a function of temperature for a stack of single crystals (red circles, $\mathbf{H}\perp {\mathbf{c}}^{*}$) representative of those used in the x-ray diffraction experiments described in Sec. 3, and the powder sample used in the neutron diffraction experiments discussed in Sec. 4b (black triangles), in a magnetic field $H=1000$ Oe. (b) Temperature dependence of the integrated intensity of the two magnetic reflections observed in neutron powder diffraction pattern in Fig. 7, normalized to an average of unity at low temperatures. The dashed line is a guide to the eye.

Figure 5

(a) Magnetic phase diagram for single-crystal $\alpha -{\mathrm{RuCl}}_3$ in magnetic field $\mathbf{H}\perp {\mathbf{c}}^{*}$. Solid points mark the maxima in the differential susceptibility $dM/dH$ derived from data shown in panel (b) (upper traces). Open symbols mark the maximum in $M\left(T\right)$ VSM temperature sweeps, as shown in the pane inset for constant magnetic field values close to the phase boundary. The dashed line is a guide to the eye phase boundary between the zigzag antiferromagnetic phase (yellow shading) and paramagnetic (PM, blue shading). (b) $M(H,T)$ data recorded in the rising part of 15 T field pulses at a series of constant temperatures. At lower temperatures, the steep rise in $M\left(H\right)$ is strongly suggestive of a field-induced phase transition near 8 T. (c) $M(H,T)$ data recorded in the rising part of 60 T field pulses in both the antiferromagnetic and paramagnetic phases.

Figure 6

The zigzag magnetic structure of $\alpha -{\mathrm{RuCl}}_3$. The magnetic moments of ruthenium atoms colored red and blue are aligned antiparallel and oriented within the $ac$ plane. Ru-Ru connections are drawn in thick black lines to illustrate the honeycomb layers, and the $C2/m$ monoclinic unit cell is drawn in thin gray lines.

Figure 7

Neutron powder diffraction data measured at 6 K, with the 20 K paramagnetic data subtracted. The diffraction patterns for both zigzag (black solid line) and stripy (brown dashed line) models are calculated and plotted for a moment oriented along ${\mathbf{c}}^{*}$, a similar level of agreement for the zigzag structure could be obtained for a general moment direction in the $ac$ plane. Inset: fit to the $\left(001\right)$ nuclear Bragg reflection, unaffected by stacking faults, used for calibrating the magnetic diffraction intensities.

Figure 8

GGA density of states projected onto the quasimolecular orbital basis of (a) $\alpha -{\mathrm{RuCl}}_3$ in the $C2/m$ structure, (b) $\alpha -{\mathrm{RuCl}}_3$ in the $P{3}_{1}12$ structure [27], and (c) ${\mathrm{Na}}_2{\mathrm{IrO}}_3$.

Figure 9

Band structure and density of states of $\alpha -{\mathrm{RuCl}}_3$ in the $C2/m$ structure obtained within (a) GGA, (b) GGA + SO, and (c) GGA + $\mathrm{SO}+U$ ($U_{\mathrm{eff}}=3$ eV). The right panel shows the projected nonmagnetic GGA and GGA + SO density of states onto the quasimolecular orbital basis [44, 45] and the GGA + $\mathrm{SO}+U$ density of states onto the relativistic $j_{\mathrm{eff}}$ basis.