Spin-wave excitations evidencing the Kitaev interaction in single crystalline $\alpha$-RuCl$_3$

TL;DR

This study provides experimental evidence of Kitaev interactions in $ ext{α-RuCl}_3$ through inelastic neutron scattering and theoretical modeling, confirming the material's potential to realize quantum spin liquid physics.

Contribution

The paper combines neutron scattering experiments with first-principles calculations to quantify Kitaev and off-diagonal interactions in $ ext{α-RuCl}_3$, demonstrating real material realization.

Findings

Observation of a ~2 meV spin-wave gap at the M point

Kitaev interaction $K$ is significantly larger than Heisenberg exchange

Effective model fits yield $K=-6.8$ meV and $ ext{Γ}=9.5$ meV

Abstract

Kitaev interactions underlying a quantum spin liquid have been long sought, but experimental data from which their strengths can be determined directly is still lacking. Here, by carrying out inelastic neutron scattering measurements on high-quality single crystals of $\alpha$-RuCl$_3$, we observe spin-wave spectra with a gap of $\sim$2 meV around the M point of the two-dimensional Brillouin zone. We derive an effective-spin model in the strong-coupling limit based on energy bands obtained from first-principle calculations, and find that the anisotropic Kitaev interaction $K$ term and the isotropic antiferromagentic off-diagonal exchange interaction $\Gamma$ term are significantly larger than the Heisenberg exchange coupling $J$ term. Our experimental data can be well fit using an effective-spin model with $K=-6.8$ meV and $\Gamma=9.5$ meV. These results demonstrate explicitly that Kitaev physics is realized in real materials.

Figures

Equations

$$H=\sum _{⟨ij⟩\in \alpha \beta (\gamma )}[K{S}_i^{\gamma }{S}_j^{\gamma }+\Gamma (S_i^{\alpha }{S}_j^{\beta }+S_i^{\beta }{S}_j^{\alpha })],$$

Full text

Quantum spin liquids (QSLs) are an exotic topological state of matter in which strong quantum fluctuations prevent conventional magnetic order from establishing down to zero temperature Anderson (1973). Examples of such states have been proposed in geometrically frustrated quantum magnets having a small spin of $S=1/2$, where the isotropic Heisenberg interaction $J$ cannot be satisfied simultaneously among different sites Balents (2010); Han et al. (2012). In these materials, the degeneracy is so large that the exact state of the system has been challenging to determine. The Kitaev QSL provides an alternative Kitaev (2006); it is realized in the exactly solvable Kitaev spin model, which has a bond-dependent anisotropic exchange interaction $K$ with an intrinsic frustration of the spin on a single site Baskaran et al. (2007); Kitaev (2006). The Kitaev QSL can host topological order and non-Abelian statistics Kitaev (2006); Chaloupka et al. (2010); Singh et al. (2012); Knolle et al. (2014, 2015), the latter property being associated with proposals for fault-tolerant quantum computation Kitaev (2003); Nayak et al. (2008); Kitaev (2006). Therefore, investigating Kitaev spin liquids is of both fundamental and practical importance.

Engineering of Kitaev interactions in real materials was first proposed in iridates Jackeli and Khaliullin (2009). In these materials, iridium and oxygen ions form edge-sharing octahedra Gardner et al. (2010), same as that sketched in Fig. 1(a), but with Ru3+ and Cl*-* replaced by Ir4+ and O2- respectively. As illustrated in Fig. 1(b) Gretarsson et al. (2013); Kim et al. (2009); Birol and Haule (2015); Rau et al. (2016), a strong octahedral crystal field splits the five $d$ orbitals into triply-degenerate $t_{2g}$ and doubly-degenerate $e_{g}$ states. Due to the substantial spin-orbital coupling (SOC) of 5$d$ electrons, the degeneracy of the $t_{2g}$ states is lifted by the formation of $J_{\rm{eff}}=3/2$ and 1/2 bands, with a total angular momentum of 3/2 and 1/2 respectively. Since the $J_{\rm{eff}}=1/2$ band is narrow, a Mott gap opens in the presence of modest electron correlations, and the system is driven to a Mott insulating state with an effective spin of $1/2$. In these systems, the magnetic interaction of the effective spin is intrinsically anisotropic and frustrated due to the spatially anisotropic nature of the $d$ orbitals and the SOC. The delicate bond configuration of the octahedra makes this interaction prominent, naturally fitting Kitaev’s proposal Jackeli and Khaliullin (2009); Chaloupka et al. (2010); Hwan Chun et al. (2015); Chaloupka et al. (2013). More recently, the layered honeycomb-lattice compound $\alpha$-RuCl3 [see Fig. 1(a)] has been suggested to be another material where Kitaev physics may be applicable Kim et al. (2015); Sandilands et al. (2015); Plumb et al. (2014). For $\alpha$-RuCl3 with 4$d$ electrons, the SOC constant $\lambda$ of $\sim$0.13 eV Sandilands et al. (2016) is weaker than that of the iridates with $\lambda\approx 0.4$ eV Rau et al. (2016), but since the band is also narrower due to stronger electron correlations, the same $J_{\textnormal{eff}}=1/2$ state as illustrated in Fig. 1(b) can also be achieved Pollini (1996); Koitzsch et al. (2016). Indeed, $\alpha$-RuCl3 has closer-to-ideal bond configurations so that Kitaev interactions dominate over the isotropic Heisenberg exchange coupling Fletcher et al. (1967); Sizyuk et al. (2016); Chaloupka and Khaliullin (2016). The most powerful approach to identify magnetic interactions is to perform inelastic neutron scattering (INS) measurements on single crystals to explore their dynamical spin response. However, iridium is a strong absorber of neutrons, making neutron scattering experiments challenging Choi et al. (2012); as for $\alpha$-RuCl3, there has been only one report of INS measurements on powder samples Banerjee et al. (2016a).

In this Letter, by performing INS measurements on high-quality single crystals of $\alpha$-RuCl3, we map out the low-energy spin-wave excitation spectra. By using results obtained from first-principle calculations and fitting the experimental data with a minimal effective-spin model, we determine the two leading interactions, including a ferromagnetic Kitaev $K$ term of $-6.8$ meV, and an antiferromagnetic symmetric off-diagonal $\Gamma$ term of $9.5$ meV. Our results clearly indicate that the anisotropic Kitaev coupling plays a key role in the exotic physics of $\alpha$-RuCl3.

Single crystals of $\alpha$-RuCl3 were grown by the chemical-vapor-transport method. The crystals are plate like, as shown in Fig. 1(c), with a typical mass of 20 mg for each piece. Specific heat and susceptibility measurements were conducted in a Physical Property Measurement System (PPMS) from Quantum Design. Neutron scattering measurements were performed on two triple-axis spectrometers, TAIPAN at ANSTO, and PUMA at MLZ Sobolev and Park (2015), both using a fixed-final-energy mode with $E_{f}=14.7$ meV. Measurements on both instruments were performed under double-focusing conditions for both the monochromator and analyzer. To maximize neutron flux, no collimator was used in the measurements. On TAIPAN, we put a pyrolitic graphite (PG) filter after the sample to reduce higher-order neutrons; on PUMA, we put two PG filters after the sample for the same purpose, and one sapphire filter before the monochromator to suppress epithermal neutrons. Under these conditions, the energy resolution (half width at half maximum) for both instruments was $\sim$1 meV at the Bragg position. For the measurements on TAIPAN, we co-aligned 50 single crystals weighing about 1 g in total using a Laue X-ray diffractometer. Crystals were glued to both faces of the aluminum plates using hydrogen-free cytop grease. Some of the plates before assembling are shown in Fig. 1(b). These crystals were well aligned such that the overall mosaic was $\sim$1.4∘ as determined from the rocking scan through the (300) peak. The samples on the plates were oriented such that the scattering plane was ($H0L$), where we obtained the neutron diffraction data. As for the measurements on PUMA, we used 80 single-crystal pieces weighing $\sim$1.5 g with an overall mosaic of $\sim$4.7∘ measured at (300). The measurements on PUMA were conducted in the $(HK0)$ plane, where we obtained the INS data. The data were analyzed and plotted with the Interactive Data Language programming environment. There have been some discrepancies on whether the low-temperature structure belongs to the monoclinic C2/m Johnson et al. (2015); Cao et al. (2016), or trigonal P3112 group Stroganov and Ovchinnikov (1957); Kim and Kee (2016). Recently, it has been reported that a phase transition from monoclinic to trigonal structure occurs at $150$ K upon cooling Kubota et al. (2015). Throughout the paper, we use P3112 notation with $a=5.96$ Å and $c=17.2$ Å. Such a structure has been commonly adopted for neutron measurements Sears et al. (2015); Plumb et al. (2014); Banerjee et al. (2016a). The wave vector $\bm{Q}$ is expressed as ($HKL$) reciprocal lattice unit (r.l.u.) of $(a^{*},\,b^{*},\,c^{*})=(4\pi/\sqrt{3}a,\,4\pi/\sqrt{3}b,\,2\pi/c)$.

We have carefully characterized the crystals by energy-dispersive X-ray spectroscopy (confirming the composition), a Laue X-ray diffractometer (confirming the single crystallinity), and a PPMS. Heat capacity and susceptibility measured in the PPMS are presented in Fig. 1(d), which show that there are two transitions in the samples. The transition temperatures of $\sim$8 K ($T_{\rm{N1}}$) and 14 K ($T_{\rm{N2}}$) almost coincide with the onsets of the intensities for the two magnetic Bragg peaks (0.5, 0, 1) and (0.5, 0, 1.5), respectively [shown in Fig. 1(e)]. The two transitions have been reported in a number of works Sears et al. (2015); Plumb et al. (2014); Kubota et al. (2015), and have been interpreted as a result of ABC- and AB-type stacking arrangements of the honeycomb layers Banerjee et al. (2016a). For samples with the ABC-type stacking structure alone, only the low-temperature transition has been observed Cao et al. (2016). The observed magnetic order with an in-plane wave vector of (0.5, 0) instead of (0.5, 0.5) is consistent with the zigzag order within the honeycomb layer, as shown in Fig. 1(a).

In $\alpha$-RuCl3, magnetic layers are only weakly bonded with the van der Waals force along the $c$ axis, resulting in a quasi-two-dimensional nature of the magnetic interactions Sears et al. (2015); Plumb et al. (2014); Banerjee et al. (2016a). We therefore investigate the spin dynamics within the $a$-$b$ plane. There is a broad continuum as shown in Fig. 2(a), (b), (c) and (f), on top of which features around the M point at temperatures below $T_{\rm{N1}}$ are evident, as shown in Fig. 2(a) and (b). By subtracting the 25-K data (above $T_{\rm{N2}}$) from the 3-K data (below $T_{\rm{N1}}$), the differences become obvious, as shown in Fig. 2(d), (e), (g) and (h). The resulting peaks, with positions indicated by the dashed lines, are dispersive. By plotting the differences of the energy scans at various $\bm{Q}$ values on the $\Gamma^{\prime}$-$\Gamma$ path, we have obtained the dispersion along the [100] direction as shown in Fig. 3(a); by plotting the differences of various $\bm{Q}$ scans at $E=3$ and 2.5 meV, we have obtained constant-energy cuts as shown in Fig. 3(c) and (d).

In the dispersion shown in Fig. 3(a), there is a gap of $\sim$2 meV at the M point, similar to INS results on powder samples Banerjee et al. (2016a); however, it shows maximum intensity near the M point with an energy of $\sim$2.5 meV, and concave curvature around this point, both of which are different from the powder results Banerjee et al. (2016a). The energy cuts in Fig. 3(c) and (d) show that the intensities are concentrated around the M point, with that of the 2.5-meV cut in Fig. 3(d) centering at ($\pm 0.55$, 0, 0). We suspect that this off-M-point behavior is an artifact resulting from the data subtraction, as the scattering is more pronounced at low-$\bm{Q}$ positions before the subtraction. From the dispersion shown in Fig. 3(a), the center of $H$ is determined to be about -0.53 r.l.u., which is the same as that obtained by fitting the $H$ scan shown in Fig. 2(h). Comparing Fig. 3(c) and (d), it is obvious that the 3-meV data are broader along both directions (parallel and perpendicular to $[H00]$) than the 2.5-meV data. A width of 0.20 and 0.14 r.l.u. along the $[H00]$ direction is determined from Fig. 3(c) and (d), respectively. These values are close to the results of 0.22 and 0.16 r.l.u. for $E=3$ and 2.5 meV respectively, extracted from the dispersion shown in Fig. 3(a). As discussed above, this material has a zigzag magnetic order with an in-plane wave vector of (0.5, 0), $i.e.$, the M point. All of the behavior of this dispersion point to the conclusion that the magnetic excitations are spin waves arising from the zigzag magnetic order phase.

The electronic structure of $\alpha$-RuCl3 can be described by the five-orbital Hubbard model with an intermediate spin-orbital interaction Winter et al. (2016); Kim et al. (2015). For the Ru3+ $4d^{5}$ electrons, the combined effects of the octahedral crystal field, SOC, and electron correlations lead to an effective picture of a hole residing on the $J_{\rm eff}=1/2$ Kramers doublet, as illustrated in Fig. 1(b) Pollini (1996); Koitzsch et al. (2016); Kim et al. (2015). In the large Hubbard $U$ limit, the effective Hamiltonian for the pseudospin is obtained by projecting the corresponding spin-orbital model onto the Kramers doublet Chaloupka et al. (2013); Jackeli and Khaliullin (2009). In this context, we determine the exchange interactions in the effective-spin model by using tight-binding parameters obtained from first-principle calculations Wang et al. (2016), and the results are summarized in Fig. 4. In the parameter ranges relevant to this system Plumb et al. (2014); Sandilands et al. (2016); Pollini (1996); Kim et al. (2015); Winter et al. (2016), we find that the Kitaev interaction $K$ term is ferromagnetic, and the off-diagonal exchange $\Gamma$ term is antiferromagnetic, both of which are significantly larger than the Heisenberg $J$ term, as shown in Fig. 4. The small $J$ value is likely to be a result of the virtual intraband hopping between the $J_{\rm eff}=1/2$ states, and the interband hopping between the $J_{\rm eff}=1/2$ and $3/2$ states, both of which contribute to the $J$ term Winter et al. (2016), nearly cancelling out Wang et al. (2016). We thus consider the following $K$-$\Gamma$ effective-spin model:

[TABLE]

where $\alpha\beta(\gamma)$ labels one bond with $\alpha$, $\beta$ and $\gamma$ being the spin directions.

To obtain parameters which fit the INS data, we calculate the dynamical spin-spin correlation function utilizing linear spin-wave theory Coldea et al. (2003); Muniz et al. (2014). In the calculations, we have summed the intensities over the three 120∘-twinned magnetic domains, taking into account polarization factors and the instrumental resolution. Numerical results for dispersions along high-symmetry paths are presented in Fig. 3(b). To compare with experimental data, we plot the calculated results along the $\Gamma^{\prime}$-$\Gamma$ path as a solid line in Fig. 3(a). With a ferromagnetic Kitaev interaction $K=-6.8$ meV, and an antiferromagnetic $\Gamma=9.5$ meV, the calculated dispersion agrees well with the data. Due to weak intensities, we have observed only one branch of the spin waves. The calculation shows an intensity maximum near the M point, in line with the experimental results. The spin-wave velocity can be increased slightly to match the data with $K=-7.2$ meV and $\Gamma=9.5$ meV. However, this combination reduces the gap size to 1.5 meV. One can increase $\Gamma$ as well to keep $|K/\Gamma|$ constant, and thus maintain the gap of 2 meV. We find that $|K/\Gamma|$ in the range of 0.65 to 0.8 gives gap sizes of about 2 meV. However, a noticeable change in $\Gamma$ requires significant changes of both $U$ and $\lambda$, as indicated in Fig. 4(a) and (b). Overall, the choice of the above parameter set is reasonable since the resulting fit well captures the maximum intensities, gap value, and intensities above 4 meV. Energy cuts at 3 and 2.5 meV for the calculated dispersions are shown in Fig. 3(e) and (f) respectively, which agree with the experimental data shown in Fig. 3(c) and (d) qualitatively.

To describe the anisotropic magnetic interactions in iridates and $\alpha$-RuCl3, a Heisenberg-Kitaev ($J$-$K$) model is commonly adopted, where $K$ plays a dominant role Chaloupka et al. (2010, 2013); Chaloupka and Khaliullin (2016). However, it has been found that this model has some difficulties in reproducing the zigzag magnetic order Foyevtsova et al. (2013), and a couple of alternatives have been proposed, including modified $J$-$K$ models with long-range terms Kimchi and You (2011); Choi et al. (2012), or even a completely different quasimolecular-orbital model Mazin et al. (2012). Models with additional terms such as the $\Gamma$ term have also been put forward Winter et al. (2016); Rau et al. (2014); Sizyuk et al. (2014). As discussed above, to mimic the gap of 2 meV, a finite $\Gamma$ (comparable to $K$) is necessary (see also Ref. Banerjee et al. (2016a)). Furthermore, in Ref. Rau et al. (2014), it has been shown theoretically that for the zigzag ordered state, a major portion of the parameter space is filled by a large ferromagnetic $K$ and antiferromagnetic $\Gamma$ term, but nearly zero $J$ term, supporting our conclusions.

In summary, we have presented neutron scattering results on single crystals of $\alpha$-RuCl3 showing that the excited states associated with the zigzag magnetic order can be well described by a minimal $K$-$\Gamma$ effective-spin model. These results make a strong case that $\alpha$-RuCl3 is a prime candidate in realizing Kitaev physics. As for a Kitaev QSL, the only two known material systems by now, iridates and $\alpha$-RuCl3, indeed have some obvious drawbacks—they both show magnetic order, albeit weakly Plumb et al. (2014); Ye et al. (2012); Dhital et al. (2013); Ye et al. (2013). Recently, it has been shown that in $\alpha$-RuCl3 the magnetic order can be fully suppressed by an external magnetic field applied parallel to the plane Majumder et al. (2015); Johnson et al. (2015); Kubota et al. (2015). It should be intriguing to study the material’s spin dynamics under such a field.

KJR, JHW, and WW contributed equally to the work. Work at Nanjing University was supported by NSFC Nos. 11374143, 11674157, 11190023, 11374138 and 11204125, and National Key Projects for Research & Development of China with grant No. 2016YFA0300401. XR was supported in part by NSFC Nos. 11374024 and 11522429. We thank Yuan Li for providing access to his lab facilities at Peking University. We thank J. A. Schneeloch for proof reading.

Note added: Upon writing up this manuscript, we became aware of an INS work on single crystalline $\alpha$-RuCl3 Banerjee et al. (2016b).