# Spin-Orbit-Entangled Nature of Magnetic Moments and Kitaev Magnetism in Layered Halides
α-RuX_{3} (X = Cl, Br, I)

Heung-Sik Kim

## Abstract

Recently, α-RuCl_{3} has been extensively studied because of potential bond-dependent Kitaev magnetic
exchange interactions and the resulting quantum spin liquid phase that can be realized
therein. The covalency between Ru 4*d*- and Cl *p*-orbitals is crucial for inducing large Kitaev interactions in this compound, therefore
replacing Cl with heavier halogen elements such as Br or I appears to be a promising
method for further promoting the Kitaev interaction. There have been several reports
on synthesis of α-RuBr_{3} and α-RuI_{3}, which are expected to host the same spin-orbit-entangled orbitals and Kitaev exchange
interactions with α-RuCl_{3}. This study investigated electronic structures of α-RuCl_{3}, α-RuBr_{3}, and α-RuI_{3} via comparisons, focusing on the cooperation of the spin-orbit coupling and on-site
Coulomb repulsions to realize the spin-orbit-entangled pseudospin-1/2 at Ru sites.
In addition, magnetic exchange interactions of all three compounds were estimated,
thereby demonstrating that α-RuBr_{3} can be promising candidates for realizing Kitaev spin liquid phases in solid-state
systems.

**Keywords:**Kitaev magnetism, Layered transition metal halides, Spin-orbit coupling, Strongly correlated electron systems, Frustrated magnetism, Density functional theory

## 1. Introduction

Kitaev’s finding of an exactly solvable magnetic model on a twodimensional honeycomb
lattice [1], which promises a fault-tolerant quantum computation via realization of Majorana
fermions with non- Abelian statistics, ushered in a plethora of theoretical and experimental
studies that aimed at realizing the Kitaev physics within condensed matter systems.
A general guiding principle was reported by Jackeli and Khaliullin [2], followed by a number of theoretical and experimental studies that attempted to
determine and evaluate the viability of the Jackeli-Khaliullin mechanism in realistic
systems [3, 4]. Currently, α-RuCl_{3} is considered the most promising candidate [5–15], wherein half-quantized thermal Hall conductivity, considered as the essential evidence
for the presence of the Kitaev spin liquid (KSL) state [1, 13, 15], is reported.

However, attaining the KSL state in α-RuCl_{3} is hindered by the presence of magnetic exchange interactions, in addition to Kitaev
interactions, which induce static magnetism and disrupt the KSL phase. Specifically,
the presence of the first- and third-nearest-neighbor Heisenberg interaction is detrimental
for the Kitaev magnetism [16], and completely removing them is challenging owing to the multi-orbital nature of
candidate systems and the resulting various hopping processes [17]. Therefore, selective enhancement of relevant electron hopping channels that contribute
to the Kitaev’s exchange interaction are essential for the realization of the KSL
phase in realistic situations.

Recently, certain theoretical and experimental reports on α-RuBr_{3} [1820] and α-Rul_{3} [19, 21, 22] have been presented, focusing on the possibility of promoting and realizing the
KSL phase. As discussed in further detail later, the heavier ligand ion results in
stronger hybridization between Ru 4*d*- and ligand (Cl, Br, I) *p*-orbitals, which may lead to stronger Kitaev interactions. However, it may also induce
unwanted enhancement of further-neighbor and even inter-layer Heisenberg interactions,
which are not favorable to realizing the KSL phases. Moreover, enhanced bandwidth
due to larger *d-p* hybridization can also disturb the formation of the local spin-orbital-entangled
*j _{eff}* = 1/2 moments, a critical element of the Kitaev’s exchange interactions. Therefore,
evaluation of such quantities via first-principles electronic structure calculations
is crucial during the early stage of studies regarding α-RuBr

_{3}and α-Rul

_{3}.

Thus, this study focused on the electronic structure of α-RuBr_{3} and α-Rul_{3}, comparisons with α-RuCl_{3}, and the formation of the spinorbital- entangled *j _{eff}* = 1/2 moments in the compounds. Furthermore, the magnitudes of magnetic exchange
interactions were estimated via employing first-principles Wannierization of Ru

*t*bands, showing α-RuBr

_{2g}_{3}to be a promising platform to further investigate the Kitaev magnetism in condensed matter systems.

## 2. Computational details

For the optimizations of cell parameters and internal coordinates of all three compounds,
the Vienna ab-initio Simulation Package (VASP), which uses the projector-augmented
wave (PAW) basis set [23,24] was employed. 500 eV of plane-wave energy curoff and 7×7×3 Γ-centered *k*-grid sampling were adopted. Further, effects of electron correlations in structural
optimizations were considered via the Strongly Constrained and Appropriately Normed
(SCAN) semi-local functional [25], which is parameter-free, combined with a revised Perdew-Burke-Ernzerhof (PBEsol)
[26] exchange-correlation functional. Following the structural optimizations, effects
of atomic spin-orbit coupling (SOC) and on-site Coulomb repulsions were studied employing
a linear-combination- of-pseudo-atomic-orbital basis code OPENMX [27, 28], where projections onto the spin-orbit-entangled *j _{eff}* = 1/2 and 3/2 states and the Wannierization of Ru

*t*bands were performed [2931]. Double zeta plus polarization (DZP) bases and 300 Ry of energy cut-off for real space integrations were employed in the OPENMX calculations, and the effects of on-site Coulomb repulsions were incorporated via a simplified variant of the rotationally-invariant DFT+U formalism [32] combined with the local density approximation [33, 34]. The value of

_{2g}*U*was chosen as 2 eV, which has been considered to be a reasonable value in previous studies [6, 10]. Electron hopping integrals were computed using Wannier orbital method in the nonmagnetic phase without including SOC and on-site Coulomb interactions, thereby eliminating the double-counting of such effects.

_{eff}≡ U - J## 3. Results and discussion

### 3.1. Crystal structures

Figure 1 shows the model crystal structure we chose for α-RuX_{3} (X = Cl, Br, I) in this study (Space group: P3_{1}12). There have been reports of different space group symmetries and stacking patterns
for these compounds [20,22], but our choice of the P3_{1}12 stacking is not a serious concern because interlayer interactions are fairly small
in all three compounds [10]. Moreover, while the zigzag-type antiferromagnetic order (which requires larger
in-plane periodicity) has been reported as the true ground state for α-RuCl_{3}, the effect of different antiferromagnetism within the RuCl3 layer on structural
and electronics properties is insignificant. Therefore, a Néel-type antiferromagnetism
was employed in our structural optimizations.

Table I presents PBEsol+SCAN-optimized lattice parameters of all three compounds in the presence
of SOC. Our result predicted lattice parameters of α-RuCl_{3} and α-Rul_{3} to be approximately 1% larger than experimentally reported values [22,35]; however, the tendency toward cell expansion with the ligand ion becoming heavier
was well-captured within our results.

### 3.2. Electronic structure and spin-orbital-entangled states

Figure 2 summarizes our calculation results for all three compounds, showing the band structures
and projected densities of states (PDOS) onto the spin-orbit-entangled *j _{eff}* =1/2 and 3/2 states. The Ru

*j*states, eigenstates of the SOC Hamiltonian within the Ru

_{eff}*t*orbital, are defined as

_{2g}On comparing the left and right columns in Fig. 2, it is evident that the inclusion of the on-site Coulomb interaction enhanced the
splitting between the *j _{eff}* = 1/2 and 3/2 states. The SOC being effectively enhanced by

*U*, has been discussed in previous studies regarding α-RuCl

_{eff}_{3}[6, 22]. Further, the inclusion of magnetism and the opening of the band gap further enhances the splitting (comparing the middle and right columns in Fig. 2), consequently driving α-RuCl

_{3}and α-Rul

_{3}good Kitaev magnet candidates. However, in contrast to the results obtained in our study [Fig. 2(a)] α-Rul

_{3}has been reported to be metallic [21]. The discrepancy may originate from technical details (e.g., the choice of local atomic projectors employed in the DFT+

*U*methodology). We comment that, the results of this study indicate that the band gap size of α-Rul

_{eff}_{3}is much smaller than that of α-RuCl

_{3}and α-RuBr

_{3}[comparing Figs. 2(c), 2(f), and 2(i)], which is consistent with the fact that larger

*d-p*hybridization in α-Rul

_{3}induces larger bandwidth and weaker electron correlation effects. In addition, the enhancement of the

*j*= 1/2 − 3/2 splitting introduced by

_{eff}*U*is not significant [Figs. 2(g) and 2(h)] compared to other compounds, possibly because of the larger

_{eff}*d-p*hybridization in α-Rul

_{3}.

### 3.3. Electron hopping channels and magnetic exchange interactions

Having established the spin-orbit-coupled nature of our systems, we next switch to
magnetic exchange interactions. Figure 3 represents major nearest-neighbor hopping channels between Ru *t _{2g}* orbitals. Among these the first t-term, which is mediated via an intermediate anion
p-orbital, is critical to inducing the Kitaev interaction, whereas the remaining terms
result in conventional Heisenberg or symmetric anisotropy interactions. Specifically,
the general magnetic Hamiltonian in α-RuX

_{3}is approximately expressed as [17],

where α, β, γ denote bond directions and the relevant spin components (*x, y, z*) involved in the Kitaev interactions, respectively, and ⟨*ij*⟩ and ⟨⟨⟨*ij*⟩⟩⟩ indicate the nearest-neighbor and third- nearest-neighbor *i-j* sites, respectively. The *J*, *K*, and Γ are explicitly obtained via perturbation theory as follows [17],

where *U* and *J _{H}* are the strength of the on-site Coulomb repulsion (not necessarily the same as

*U*) and Hund’s coupling within the Ru

_{eff}*t*orbital, respectively.

_{2g}The hopping terms *t*, *t'*, and *t''* can be obtained using the Wannier function method [29, 30], which are tabulated in Table II The magnitude of the *d-p-d* hopping term *t* decreased slightly when changing from α-RuCl_{3} to α-Rul_{3}. In addition, the magnitudes of *t'* and *t''*, which introduce Heisenberg *J*, decreased when Cl was replaced with Br or I. The origin of this behavior is unclear
at this point and deserves further study.

Because we obtained the hopping parameters from ab-initio calculations, the magnitudes
of *J*, *K*, and Γ could be estimated as a function of *U* and *JH* . We set *U = U _{eff}*=2 eV, and computed exchange interactions as a function of

*JH*/

*U*. Figure 4 shows the results, and owing to the diminishing

*t'*and

*t''*in α-RuBr

_{3}and α-Rul

_{3}, the size of

*J*and Γ reduced when changing from α-RuCl

_{3}to α-Rul

_{3}. As α-Rul

_{3}has been reported to be metal, α-RuBr

_{3}appears to be the best candidate to host Kitaev magnetism in this series of compounds.

We conclude this section by mentioning third-nearest-neighbor and inter-layer hopping
elements. Table II indicates that both terms increased as the ligand anion became heavier, as expected
in the beginning. However, the enhancement is not that significant compared to the
changes in *t'* and *t''*. Although many different channels can constructively contribute to further neighbor
and interlayer hopping channels, as discussed in Winter et al. [16], and Catuneanu et al. [36], it can be speculated that the enhancement in further neighbor and interlayer hopping
channels may not cause substantial changes in α-RuBr_{3} and α-Rul_{3}, thereby rendering α-RuBr_{3} a promising candidate for realizing Kitaev magnetism in this family.

## 4. Conclusions

In this study preliminary first-principles density functional theory calculations
were performed for α-RuCl_{3}, α-RuBr_{3}, and α-Rul_{3}, to assess their viability for realizing Kitaev magnetism. The results indicated
that α-RuCl_{3} and α-RuBr_{3} may host the spin-orbit-entangled *j _{eff}* = 1/2 moment, and that α-RuBr

_{3}shows even stronger Kitaev magnetism compared to α-RuCl

_{3}.

Note that some possible effects of minor lattice distortions, for example the effect
of trigonal crystal fields, were not discussed in this study [37]. Estimation of next- and third-nearest-neighbor Heisenberg interactions that induce
long-range magnetic orders needs to be done as well for further studies on magnetic
properties of these systems. Hence continuing theoretical and experimental studies
are needed at this moment, but we believe that α-RuBr_{3} can be another interesting system to study Kitaev physics in addition to α-RuCl_{3}. Lastly, since α-Rul_{3} has been reported to be nonmagnetic and metallic, it has potentials to host topological
band insulating phases with weakto- intermediate electron correlations as previously
suggested [36].

## Article information

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