Magnon dispersion and dynamic spin response in three-dimensional spin models for $\alpha \text{−}{\mathrm{RuCl}}_3$

In the search for experimental realizations of bond-anisotropic Kitaev interactions and resulting spin-liquid phases, the layered magnet $\alpha \text{−}{\mathrm{RuCl}}_3$ is a prime candidate. Its modeling typically involves Heisenberg, Kitaev, and symmetric off-diagonal $\mathrm{\Gamma }$ interactions on the two-dimensional honeycomb lattice. However, recent neutron-scattering experiments point towards a sizable magnetic interlayer coupling. Here we study three-dimensional exchange models for $\alpha \text{−}{\mathrm{RuCl}}_3$, for both possible $R\overline{3}$ and $C2/m$ crystal structures. We discuss the symmetry constraints on the interlayer couplings, construct minimal models, and use them to compute the magnetic mode dispersion and the dynamical spin structure factor, in both the zero-field zigzag phase and the paramagnetic high-field phase. Our predictions for the interlayer mode dispersion shall guide future experiments; they also call for a reevaluation of the quantitative model parameters relevant for $\alpha \text{−}{\mathrm{RuCl}}_3$.

Figure 1

3D crystal structures of $\alpha \text{−}{\mathrm{RuCl}}_3$, showing the stacking of the honeycomb layers and the assumed interlayer interactions of the minimal models employed in this paper. (a) $R\overline{3}$ structure, (b) $C2/m$ structure. In both cases, the conventional trigonal crystallographic unit cell (dashed) contains three ${\mathrm{RuCl}}_3$ layers, and $\mathbf{a},\mathbf{b},\mathbf{c}$ show the directions of the basis vectors of this cell. Each honeycomb layer consists of two sublattices, represented by black and white balls, respectively.

Figure 2

Top view of the 3D zigzag magnetic order with three-layer periodicity along the $c$ axis, showing three layers. (a) $R\overline{3}$ structure, (b) $C2/m$ structure. Red and blue sites correspond to spin directions up and down, respectively. ${\mathbf{a}}^{*},{\mathbf{b}}^{*},{\mathbf{c}}^{*}$ are the reciprocal-lattice vectors in the conventional trigonal basis.

Figure 3

Dynamic spin structure factor $\mathcal{S}(\mathbf{q},\omega )$ (color coded) and mode dispersion (dashed lines) at zero external field, calculated for (a)–(c) the 2D Model A [33] and (d)–(f) the same model augmented by an interlayer coupling $J_{\perp 1}$ of 1 meV in an assumed $R\overline{3}$ structure (Model B). Different columns correspond to different paths in 3D momentum space: (a), (d) in-plane path as shown in panel (a) at $L=0$; (b), (e) vertical out-of-plane path at in-plane momentum (0,0); (c), (f) vertical out-of-plane path at in-plane momentum (0,0.5). Symbols show mode energies extracted from THz spectroscopy (Ref. [41]) and neutron-scattering (INS, Refs. [17, 42]) measurements. All panels involve an averaging over the three symmetry-equivalent zigzag domains.

Figure 4

Dynamic spin structure factor $\mathcal{S}(\mathbf{q},\omega )$ (color coded) and mode dispersion (dashed lines), calculated for Model 1 with $R\overline{3}$ crystal structure. Different rows correspond to different external parameters: (a)–(c) Zero-field zigzag phase; (d)–(f) high-field phase for field along $(1,-2,0)$ (perpendicular to a Ru-Ru bond); (g)-(i) high-field phase for field along (1,0,0) (parallel to a Ru-Ru bond). Different columns correspond to different paths in 3D momentum space: (a), (d), (g) in-plane path as shown in panel (g) at $L=0$; (b), (e), (h) vertical out-of-plane path at in-plane momentum (0,0); (c), (f), (i) vertical out-of-plane path at in-plane momentum (0,0.5). Symbols show experimental mode energies extracted from THz spectroscopy (Ref. [41]), ESR (Ref. [46]), neutron-scattering (INS, Refs. [17, 42]), and thermal transport (Ref. [15]) measurements. Panels (a)–(c) involve an averaging over the three symmetry-equivalent zigzag domains. The agreement with the experimental data is striking.

Figure 5

Same as Fig. 4, but now for Model 2 with $R\overline{3}$ crystal structure. While this parameter set also reproduces the high-field mode energies (see text for details), its does not match quantitatively at zero field.

Figure 6

Same as Fig. 4, but now for Model 3, which has $C2/m$ crystal structure. Here, panels (a)–(c) have been calculated for the energetically favorable zigzag direction and do not involve domain averaging. This model cannot reproduce the qualitative shape of the high-field dispersion; this generically applies to $C2/m$ models, which require a ferromagnetic interlayer coupling in order to stabilize threefold-periodic zigzag order in zero field.