α-RuCl₃ and other Kitaev materials

Quantum spin liquid (QSL) is a state of matter where spins are strongly entangled without magnetic order down to absolute zero temperature.^1–6 This new quantum state of matter has attracted much theoretical attention ever since Anderson’s resonating valence bond state proposal in 1973.⁷ Until recently, experimental research on quantum spin liquids had focused on materials with geometrical frustration, such as spin systems on a Kagome lattice or a pyrochlore lattice.² However, in recent years, the search for quantum spin liquid materials expanded to include so-called Kitaev materials, which have become one of the most popular topics in magnetism research these days.^8–12 

Kitaev introduced his eponymous model in 2006.¹³ What is remarkable about this model is that it is an exactly solvable model with a quantum spin liquid ground state with Majorana fermions as its elementary excitation, that is, this model is tailor-made for topological quantum computing.¹⁴ However, the required bond-dependent Ising interactions, called Kitaev interactions, were not considered realistic. Then, Jackeli and Khaliullin¹⁵ showed that, unlike spins, spatial information is encoded in the orbital angular momentum, and thus, Kitaev interactions can be realized in materials with large spin–orbit coupling (SOC). This material challenge was quickly met by Singh and Gegenwart¹⁶ when they reported synthesizing Na₂IrO₃ single crystals, which opened the door to a decade-long Kitaev materials research that remains active until today. Many other Kitaev materials have been reported following this.¹¹ The ingredient seems to be d⁵ iridates on a honeycomb lattice with edge-sharing octahedra. For example, α-Li₂IrO₃ was synthesized in a powder form,¹⁷ with single crystals followed.¹⁸ In addition, β- and γ-polytypes of Li₂IrO₃ and Cu₂IrO₃ have been synthesized.^19–21 Another promising direction is using a topotactic replacement of lithium in α-Li₂IrO₃ with hydrogen or silver ions to obtain H₃LiIr₂O₆ and Ag₃LiIr₂O₆.^22,23

Due to the large neutron absorption cross section of Ir, it is difficult to study iridates using neutron scattering. To circumvent this problem, Plumb et al. at the University of Toronto proposed a 4d⁵ material, α-RuCl₃, as a candidate material to study Kitaev physics.²⁴ They showed that even though the bare SOC in Ru is smaller than Ir (about 1/3), its close-to-ideal octahedral environment makes the effect of SOC important compared to other competing energy scales, such as a trigonal crystal field, which was supported with x-ray absorption spectroscopy, optical spectroscopy, and density functional theory studies.²⁴ Subsequent Raman scattering²⁵ and inelastic neutron scattering^26,27 studies have reported unusual broad low energy magnetic excitations, suggestive of fractional magnetic excitations found in a quantum spin liquid. Despite many studies reporting unusual and interesting magnetism, α-RuCl₃ has a magnetically ordered ground state and is clearly not a quantum spin liquid. It was, however, recognized that a moderate in-plane magnetic field could suppress the zigzag magnetic order of α-RuCl₃.^28–34 Although initial thermal conductivity³⁰ and NMR studies³¹ suggested gapped and gapless QSL, respectively, the nature of the field-induced paramagnetic state has been intensely debated and still remains far from being settled.

In 2018, Kasahara et al. at the University of Kyoto reported their measurement of thermal Hall conductivity within the ab-plane when the magnetic order is suppressed with the magnetic field.³⁵ They found that the thermal Hall conductivity (κ_xy) is quantized at half the expected value of the integer quantum Hall effect, indicating that the heat carrier in this phase may be Majorana fermions.³⁵ This amazing result has generated huge interest since, if confirmed, this is the first unequivocal experimental evidence of a quantum spin liquid phase. However, this result remains highly controversial since no other reports confirming the observed quantized thermal Hall effect have been published, except for the work by Bruin et al. in 2021, using the sample from the same batch used by the Kyoto group.³⁶ In the meantime, Czajka et al. at Princeton reported that they were not able to observe the quantized thermal Hall effect.³⁷ Instead, they observed quantum oscillations in their thermal conductivity (κ_xx) measurements.³⁷ Lefrançois et al. at Sherbrooke have also recently reported that they were not able to observe the quantized thermal Hall effect.³⁸ These results point to the sample dependence of the observed thermal Hall effect and highlight the crucial importance to have a clean sample.^39,40

Significant sample dependence of the magnetic properties of α-RuCl₃ was recognized from early on. Earlier powder studies reported T_N ∼ 14 K,⁴¹ but later, single-crystal neutron diffraction studies showed T_N ∼ 7 K.^42,43 It was found that T_N ∼ 7 K is usually found for crystals with fewer stacking faults, while a lower-quality sample with a large number of stacking faults is usually associated with T_N ∼ 14 K.⁴³ Many earlier studies have been carried out for samples with both transitions, but the sample quality has improved, and the most recent studies seem to use only T_N ∼ 7 K samples. Most importantly, the aforementioned thermal Hall studies all report that the samples used in their studies show T_N ∼ 7 K.^35–38 Therefore, it is still unclear what distinguishes the samples showing quantized κ_xy from the other clean samples.

Another unresolved but related issue is the low-temperature structure. The structure of α-RuCl₃ was reinvestigated by Cao et al.⁴³ and Johnson et al.,⁴⁴ which showed that the room temperature structure is monoclinic with the space group C2/m, instead of the hexagonal structure (space group P3₁12) identified earlier.⁴⁵ This means that there is a unique a-direction, breaking the C₃ symmetry,^43,44 which is important as recent studies have illustrated in-plane anisotropy of thermodynamic and magnetic properties.^46–49 It is also well known that α-RuCl₃ goes through a first-order structural phase transition around $∼150$ K.⁵⁰ However, there is still no consensus as to the low temperature structure.⁵⁰ The main difficulty with the structure determination is the strong diffuse scattering that sets in below the structural transition temperature. Such diffuse scattering is observed even for a clean sample without diffuse scattering at room temperature, when it is cooled below the structural transition temperature.

This Perspective is not meant to be a comprehensive review article, and only selective topics and references will be discussed. There are many excellent review articles on quantum spin liquids in general^1–6 and Kitaev QSL.^8–12 There are also recent review articles on the materials aspect of quantum spin liquids.^51,52 In this Perspective, we will critically examine α-RuCl₃ with a focus on the crystal growth, quality, and structure. We will also discuss future directions for Kitaev materials beyond α-RuCl₃ in Sec. IV.

In 2006, Kitaev published a remarkable paper¹³ titled “Anyons in an exactly solved model and beyond.” It should be noted that the word “solved” is used instead of the usual “solvable” since he not only proposed an exactly solvable model, but he also showed how to solve this model by representing spins with Majorana operators. This model is now commonly known as the Kitaev model or Kitaev’s honeycomb model in the condensed matter community. There are many excellent reviews on this model^8–11,53,54 and on a more general quantum compass model.⁵⁵ The readers are referred to these for further insights. Despite the remarkable properties of the Kitaev model, this did not receive much attention from the experimental community initially. The concept of QSL grew out of Anderson’s resonating valence bond⁷ and high Tc cuprates,⁵⁶ and insulating S = 1/2 compounds in the geometrically frustrating lattice was the dominant story of QSL at the time.^1,2,57,58 After all, the bond-dependent anisotropic interaction (Kitaev interaction) required to realize the Kitaev model seemed unrealistic. As mentioned above, the paper of Jackeli and Khaliullin published in 2009 changed everything.¹⁵ It was proposed that a d-electron system in an octahedral crystal field could have effective low energy physics governed by Kitaev interactions as long as the SOC is strong enough to entangle the spin and orbital degrees of freedom, giving rise to the local magnetic moment described using Kramer’s doublet of isospin states (or often called J_eff = 1/2 states following the work of Kim et al.⁵⁹). The original paper by Jackeli and Khaliullin focused on the mechanism to suppress isotropic superexchange interaction J, which occurs when the Ir–O–Ir bond forms a 90° bond.¹⁵ Suppressing this J interaction is important to reveal the relatively weak Kitaev interaction, K, in these materials.

Conventionally, a material with large SOC (λ) is believed to host two types of anisotropic interactions, namely, an antisymmetric and a symmetric part that goes as ∼λ and ∼λ², respectively.⁶⁰ The symmetric part is often ignored, and only the antisymmetric part, termed Dzylloshinsky–Moriya (DM) interaction, is considered in many magnetic materials. In Kitaev materials of interest, the DM interaction is not allowed by symmetry, but the symmetric part, typically called Γ, survives. It was later realized that the Γ interaction cannot be ignored in α-RuCl₃.^61,62 It is now widely believed that the Γ interaction is an important part of the realistic model describing α-RuCl₃.^11,54 Therefore, the minimal model for α-RuCl₃ seems to be the J–K–Γ model. However, often, two additional smaller terms, the third-nearest-neighbor Heisenberg interaction J₃ or Γ′ interactions arising from the non-zero trigonal crystal fields, are included.

Various experimental investigations seem to converge on the fact that ferromagnetic K and antiferromagnetic Γ interactions are essential for describing the physics of α-RuCl₃^{47,48,63–69} although the size of each term is still somewhat controversial. The irony here is that the appeal of the Kitaev model is its exactly solvable nature, but as is often the case, the real materials require the consideration of other terms, such as Γ and J, throwing this problem back to the realm of “not exactly solvable” models. Recent numerical studies have made significant progress in understanding these models, however.^70,71 We would like to mention that the real “Kitaev material” might host a QSL phase that is not exactly the Kitaev QSL as discussed in a recent paper.⁷² 

Ruthenium trichloride is certainly not a new material. Guthrie and Bourland reported the magnetic susceptibility of RuCl₃ in their 1931 paper.⁷⁷ Hill and Beamish reported that there were two types of RuCl₃ (now known as α and β polytypes) in 1950.⁷⁸ However, it was Fletcher et al.⁴¹ who reported a comprehensive investigation of structural and magnetic properties of α- and β-RuCl₃. The sample used in their study was synthesized using the method reported earlier.⁷⁹ 

There are generally two routes for synthesizing RuCl₃ crystals. Growing crystals in the vapor phase is the most common method. For example, earlier scanning-tunneling microscopy,⁸⁰ Hall effect,⁸¹ and optical spectroscopy⁸² studies were all done using crystals obtained using the vapor synthesis method. In addition, some earlier studies^83,84 reported using commercial powder samples, which are mostly α-RuCl₃, but some impurity β phase could be present in such samples. However, the β phase of RuCl₃ forms at lower temperatures and has distinct crystal morphology (whiskers for β and plates for α) so that it is easy to distinguish the β phase crystals from the α phase crystals.

While most α-RuCl₃ samples used in recent studies have been grown using some form of vapor growth methods, each group used somewhat different growth conditions as described in Table I. Chemical vapor transport (CVT) is the method used most commonly. However, often, the growth occurs without additional transport gas (such as chlorine). Ruthenium chloride powder can dissociate and produce chlorine gas, which acts as a transport agent, which is referred to as “auto-transport” in Table I. The sublimation growth occurs at a much higher temperature, but the growth rate tends to be very slow. The tutorial article by May et al.⁷³ describes crystal growths using both CVT and sublimation in detail. At the University of Toronto, we used an intermediate temperature where the CVT and sublimation growths could be controlled using the temperature gradient. These two growth methods produce two types of crystals with different morphology as discussed in Sec. III A.

Note that Kubota et al. has been growing α-RuCl₃ crystals using the vertical Bridgman method from the melts,²⁹ which is distinctly different from the methods used by other groups. It is worth noting that these crystals grown with the Bridgman method are used by both Kasahara et al.³⁵ and Bruin et al.³⁶ to observe the quantized thermal Hall effect. Both Czajka et al.³⁷ and Lefrançois et al.³⁸ reported that they could not observe similar quantized thermal Hall effects in their studies using the samples grown in the vapor phase by the Mandrus group and the Kim group. Comparisons between the samples grown using the Bridgman method and the vapor phase methods are discussed in Sec. III B.

We also note that α-RuCl₃ is one of a large class of materials called van der Waals materials, and thin samples down to a monolayer have been made through mechanical and other exfoliation methods.^85,86 It also means that α-RuCl₃ could be used to build various heterostructures with other 2D materials, such as graphene.⁸⁷ Other types of heterostructures with α-RuCl₃ were also investigated. Grönke et al.⁸⁸ reported growing thin film samples of α-RuCl₃ on YSZ substrates using chemical vapor deposition, while Park et al.⁸⁹ were able to grow topological insulating Bi₂Se₃ directly on an α-RuCl₃ crystal substrate.

Crystals with two distinct morphologies are obtained depending on the growth conditions as described in Table I. As shown in Fig. 1(b), plate-like thin crystals with a thickness less than 100 μm and a relatively large area of 1 cm² are obtained when the temperature gradient is 100 °C (see Table I). The growth, in this case, occurs through CVT (auto-transport). However, much thicker crystals (thickness of up to 2 mm) can be grown when we keep the source temperature at 850 °C and the gradient at 50 °C, which creates a condition for the sublimation growth. The sublimation-grown thick crystals are found to have high quality as evidenced by x-ray diffraction and thermodynamic measurements, which is consistent with the result reported in Ref. 73. Characterization of these samples is shown in Figs. 2 and 3. Figure 2 shows representative x-ray diffraction patterns for a thick crystal and a thin crystal. The x-ray diffraction patterns shown in Fig. 2 illustrate a clear difference between the thick and thin crystals. Unlike the sharp Bragg peaks at the integer values of L observed for thick crystals, diffusive rods were observed in thin crystals. Such diffusive rods were observed in previous structural studies of α-RuCl₃^43,44 and were attributed to defects in the stacking between layers by an occasional shift in ±b/3.

The presence of many stacking faults in thin crystals is also apparent in the specific heat data shown in Fig. 3. Unlike the sharp peak at 7 K for thick crystals, multiple features are observed at around 7, 10.5, 12, and 14 K for thin crystals. In their neutron diffraction studies, Banerjee et al.²⁶ found that samples with different layer stacking sequences give different magnetic transition temperatures. In their sample, the 7 and 14 K transitions correspond to the ABCABC and ABAB stacking patterns, respectively. The presence of specific heat anomalies at 10.5, 12, and 14 K in all thin crystals strongly indicates many stacking faults in these crystals. Based on the sharp diffraction peaks and single magnetic transition temperature, one can surmise that thick crystals are of higher quality. May et al.⁷³ also reported similar observations regarding two types of crystals with different thicknesses. However, we should also emphasize that we found some “poor-quality” pieces among thick crystals (even from the same batch). Therefore, the only way to ensure crystal quality is to characterize each piece used in physical property measurements.

The problem with the stacking disorder and multiple magnetic transition temperatures seems to be common for both the CVT/sublimation method and the Bridgman method. Multiple magnetic transitions at 7, 10, 12, and 14 K were found in the earlier samples grown with the Bridgman method,²⁹ which is remarkably similar to the multiple transitions shown in Fig. 3(b). It is still not well understood why different stacking sequence would lead to different magnetic transition temperatures. However, this is currently used as a diagnostic tool for finding a clean sample.

There have been several studies exploring the origin of the differing experimental results using different samples. Yamashita et al.³⁹ investigated three samples grown using the Bridgman method and found that only the sample with the largest κ_xx values at the low temperature maximum (around 5 K, below T_N) shows the half-integer thermal Hall effect. The phonon thermal conductivity rapidly rises with the decreasing temperature below T_N since the phonon–magnon scattering is suppressed with the magnon gap opening. Since the phonon thermal conductivity in this regime is only limited by the phonon mean free path, $κxxmax$ is a measure of sample quality. Kasahara et al.⁴⁰ have recently extended this comparison, including several other samples, including some samples used in previous studies, such as the work of Bruin et al.³⁶ They also reached a similar conclusion that the samples showing half-integer quantization all exhibit $κxxmax$ larger than about 4 W/Km. The samples with $κxxmax$ below this did not show the half-integer quantization. Although earlier samples, indeed, had smaller $κxxmax$ values,^30,74 this criterion is not sufficient to explain the difference between samples. The $κxxmax$ values for the samples grown at Toronto³⁸ and Oak Ridge³⁷ both exceed the 4 W/Km threshold, but the quantized thermal Hall effect was not observed.

Various ways to benchmark crystal quality were also discussed by Czajka et al.³⁷ They considered the value of $κxxplateau$ at a high field (13 T) when the spin-fluctuations are gapped due to the magnetic field and κ_xx reaches a plateau. They found $κxxplateau/T=2.2$ W/K²m at 1 K for their best sample. Although no direct comparison is available, the corresponding values for Bridgman crystals seem to show similar $κxxplateau/T$ ranging from 2 to 3 W/K²m.^35,36,90 In other words, there seems to be no clear indicator of the difference in crystal quality between high-quality crystals grown with the CVT/sublimation method and the Bridgman method.

Kasahara et al.⁴⁰ also noted that the difference in the T_N indicates different magnetic Hamiltonians, which could be the reason behind different magnetic field ranges for which the half-integer quantization was observed. They found that the transition temperatures range from 7.2 to 8 K for the samples grown with the Bridgman method for both the “high-quality group” and the “low-quality group” samples. It should be emphasized that this variation of the transition temperature should be distinguished from the several transitions from 7 to 14 K often observed in lower quality samples discussed above [e.g., Fig. 3(b)]. The samples studied in Ref. 40 are all high-quality samples with single transition temperatures ranging from 7.2 to 8 K. Even for the samples grown with the vapor transport method, this type of slight T_N variation has been observed. The transition temperature of the thick crystal shown in Fig. 3(a) is 7.2 K, while that of the thin crystal shown in Fig. 3(b) is 7.5 K (the sharp main peak). Do et al.²⁷ and Sears et al.³² reported that the magnetic ordering temperature is 6.5 K. The specific heat data in Fig. 2 of the work of Cao et al.⁴³ show a shoulder feature around 6.5 K next to the main peak at 7.2 K. We also found that while most thick crystals show T_N = 7.2 K, some other pieces (from the same batch) show a single transition at 6.5 K. Further studies are necessary to understand these different transition temperatures.

We would like to point out that the observed variation in T_N is a strong indication that the magnetic interlayer coupling is a necessary ingredient to explain the magnetism of α-RuCl₃. According to the Mermin–Wagner–Hohenberg theorem, interlayer couplings are required for a quasi-two-dimensional Heisenberg magnet to order at finite temperatures, and it was found that T_N is governed by the interlayer coupling strengths.⁹¹ However, some van der Waals magnetic materials, such as CrI₃, can exhibit finite-temperature magnetic order even in the monolayer limit because of the Ising interaction.⁹² It is interesting to note that Kim and Kee found that Kitaev and Γ interactions are quite sensitive to the stacking structure in α-RuCl₃.⁹³ It is, therefore, reasonable to make a connection between the magnetic properties and different stacking patterns. However, further studies are necessary to understand whether observed T_N variation is due to slight changes in interlayer couplings or bigger modifications in the magnetic Hamiltonian, such as Kitaev interactions.

It will be important to resolve the conflicting experimental reports on the quantized thermal Hall effect in α-RuCl₃, which will require careful sample synthesis and comparison studies. Given the intricate nature of sample preparation, some of the earlier experimental studies could be revisited, which will help clarify the microscopic model that governs this material. Among these, it will be crucial to determine the low-temperature crystal structure using crystals of the highest quality. Current modelings using first-principles calculations rely on the crystal structure parameters obtained at higher temperatures. In addition, a high-quality spin-wave dispersion relation in the zigzag ground state will also help determine the magnetic Hamiltonian unambiguously, which will require single-domain single crystal samples large enough for inelastic neutron scattering.

Honeycomb-lattice iridates, the original Kitaev materials, are still a very active research area, which was reviewed in recent articles^10,11 and will not be further discussed here. However, iridates’ inaccessibility to inelastic neutron scattering may be one of the reasons for slow progress in this area. Resonant inelastic x-ray scattering (RIXS) has been proven to be an excellent alternative that can provide information on the momentum-dependent spin-excitation spectra.⁹⁴ However, the energy resolution of RIXS remains relatively poor, making it difficult to study the relevant low-energy (∼meV) physics of Kitaev quantum spin liquids. The promising results obtained by studying honeycomb iridates and α-RuCl₃ have prompted a broader search for Kitaev physics in other honeycomb magnets with spin–orbit entangled Kramer’s doublets. At the same time, the rising interest in van der Waals magnets has reinvigorated research on all types of honeycomb lattice magnetic materials.^92,95–97

Two routes have been considered to acquire the large spin–orbit coupling necessary for realizing Kitaev materials. One is to explore electron configuration other than d⁵ $(t2g5)$ found in iridates and α-RuCl₃. The other is to rely on ligand ions with large SOC, such as I⁻ or Br⁻, to provide superexchange paths for Kitaev interactions. These two directions are discussed briefly in the following subsections (Secs. IV A and IV B).

Binary transition-metal halides have been drawing much attention in recent years. Of course, α-RuCl₃ is one such example, but interest in van der Waals magnets, such as CrI₃, has also been responsible for the explosive growth of the interest in binary transition-metal halides. Crystal and magnetic structures of this family of materials are reviewed in a recent article by McGuire.⁹⁸ Transition-metal dihalides form a triangular lattice, a geometrically frustrated lattice, while trihalides form a honeycomb lattice necessary for the Kitaev interaction. Therefore, both series of materials are being investigated extensively these days.

Stavropoulos et al. found that essential ingredients for a higher spin (S > 1/2) Kitaev model are strong spin–orbit coupling in anions and strong Hund’s coupling in transition metal cations.⁹⁹ Lee and co-workers¹⁰⁰ continued this line of investigation and suggested that the large Dirac point gap in CrI₃, as observed in the inelastic neutron scattering experiments,¹⁰¹ arises from the strong Kitaev interaction due to the anion SOC. However, recent follow-up neutron scattering studies seem to indicate that the next-nearest-neighbor (NNN) DM interaction is the main reason for the Dirac point gap.¹⁰² The latter explanation is consistent with the experimental observation that CrCl₃ remains gapless at the Dirac point.^103–105 Since the spins are ordered in-plane in CrCl₃, the NNN DM interaction is not allowed, unlike CrI₃ and CrBr₃,¹⁰⁶ in which spins are ordered perpendicular to the honeycomb plane. Both CrI₃ and CrBr₃ exhibit large Dirac-point gaps of about 5 and 3.5 meV, respectively.^101,107 However, recent inelastic neutron scattering provides new data, showing that the spin excitation remains gapless at the Dirac point of CrBr₃,¹⁰⁸ which questions the above scenario based on the NNN DM. Further studies are required to determine the microscopic model for this family of important van der Waals magnets.¹⁰⁹ Recently, VI₃, an S = 1 analog of CrI₃, has been drawing much interest as another promising ferromagnetic van der Waals material.^110,111 Interestingly, a recent study reported anomalous thermal Hall effect in VI₃,¹¹² which seems to suggest an exotic magnetic Hamiltonian.

Before turning our attention back to cations, it is worthwhile to mention the recent developments in the study of RuBr₃ and RuI₃.^113–116 Unlike α-RuCl₃, which can exist in both α and β phases, RuBr₃ and RuI₃ have been known to crystallize only in the TiI₃ structure (the same as β-RuCl₃).¹¹⁷ However, recent papers reported high pressure syntheses of α-RuBr₃¹¹⁵ and α-RuI₃.^113,114 What is very exciting is that recent theoretical studies¹¹⁶ predict the ideal pristine crystal of α-RuI₃ to have a nearly vanishing conventional nearest-neighbor Heisenberg interaction and to be a quantum spin liquid candidate of possibly different kind than the Kitaev spin liquid. However, much work seems to be needed for improving the crystal quality of these compounds.

Honeycomb cobaltates containing Co²⁺ with an electronic configuration of $t2g5eg2$ have emerged as another class of materials that could potentially realize Kitaev physics.¹¹⁸ It is worth mentioning that a ground doublet with unquenched orbital angular momentum in Co²⁺ has been known for decades¹¹⁹ and has found evidence in many cobalt-based magnets with a magnetic moment of 5–6μ_B, considerably larger than the pure spin value (3.8μ_B for S = 3/2). Traditionally, the pseudo-spin interactions in cobaltates are taken to be a simple XXZ type, which is obtained by projecting a simple Heisenberg interaction onto the ground doublet of a Co²⁺ ion with a trigonal crystal field.¹¹⁹ However, such a simple pseudospin interaction fails to take into account the full spin-orbital exchange processes. Calculations considering realistic exchange processes were reported only recently, suggesting that the Kitaev interaction could be dominant,^118,120 which inspired many experimental works on different cobalt-based honeycomb magnets.¹²¹ We note that Motome et al.⁵⁴ presented a comprehensive examination of potential ingredient ions for Kitaev materials among transition-metal and rare-earth ions. They showed that materials containing high-spin d⁷ ions, such as Co²⁺ and Ni³⁺, and f¹ ions, such as Pr⁴⁺ and Ce³⁺, are good candidates for finding Kitaev physics. Among these materials, Co²⁺ compounds are being investigated intensively in recent years.

One example is Na₂Co₂TeO₆ and Na₂Co₂SbO₆, which exhibits a zigzag order similar to α-RuCl₃ below T_N of $∼30$ K and $∼5$ K, respectively.^122,123 Powder averaged magnetic excitation spectra have been obtained for both materials using inelastic neutron scattering, which were interpreted within a J–K–Γ model with small J₃ (see Sec. II). However, widely different interaction parameters, ranging from a K dominant model with K > 0¹²⁴ and K < 0¹²⁵ to a J dominant model,¹²⁶ appear to fit the data reasonably well within the spin-wave theory. If nontrivial anisotropic interactions such as Γ and K are significant, ordered moment direction could provide useful information for interaction parameters. Indeed, deviations from the zigzag magnetic order were reported in recent studies of Na₂Co₂TeO₆^127,128 although future studies using single-crystal samples will be required to determine the size of the anisotropic interactions. In addition to quantifying the exchange parameters in these materials, the work has also been carried out to suppress the magnetic order in these compounds. Although there has been a thermodynamic signature of a field-induced non-magnetic phase in Na₂Co₂TeO₆,¹²⁶ it is still unclear whether this phase is a spin liquid or a simply polarized phase. Measurement of magnetic excitations in the field induced phase by transport and/or neutron scattering can potentially address this question.

Another well-known example is $BaCo2(AO4)2$ (A = P, As), which has been studied in the past for possible Kosterlitz–Thouless transition due to strong XY exchange anisotropy.¹²⁹ These materials seem to be at the boundary between competing phases, as evidenced by a coexistence of a short-ranged incommensurate helical and commensurate zigzag/stripy order in $BaCo2(PO4)2$¹³⁰ and a field induced transition in $BaCo2(AsO4)2$ at a very low field to a polarized phase.¹³¹ Although the field induced phase was proposed to be a spin-liquid-like phase in Ref. 131, this interpretation is based on a frustrated J₁–J₂–J₃ model in earlier works. However, given the localized nature of 3d orbitals and, therefore, intrinsically small further neighbor interactions, the observed phase competition might be more naturally explained within the frustrated J–K–Γ model with only nearest neighbor coupling. Consideration of Γ and K might also explain the spin canting out of the easy plane observed in the zero field incommensurate phase of $BaCo2(AsO4)2$⁠.¹³² In addition, a highly unusual magnetic excitation was observed in the same material: unlike conventional magnon excitations that emanate from the ordering wave-vectors, a gapped magnon centered at q = 0 was observed at zero field reminiscent of the behavior with a large ferromagnetic Kitaev interaction.¹³³ It might be useful to revisit the magnetic excitations in $BaCo2(AO4)2$ at both the zero field and the polarized phase to quantify the exchange anisotropy beyond a simple XXZ model using modern neutron spectrometers.

The work at the University of Toronto was supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada through Discovery Grant No. RGPIN-2019-06449, the Canada Foundation for Innovation, and the Ontario Research Fund.

The authors have no conflicts to disclose.

Subin Kim: Data curation (lead); Investigation (lead); Visualization (lead); Writing – original draft (supporting); Writing – review & editing (supporting). Bo Yuan: Investigation (supporting); Writing – original draft (supporting). Young-June Kim: Conceptualization (lead); Writing – original draft (lead); Writing – review & editing (lead).

The data that support the findings of this study are available from the corresponding author upon reasonable request.