Applied Science and Convergence Technology 2021; 30(6): 191-194
Published online November 30, 2021
Copyright © The Korean Vacuum Society.
Department of Physics and Institute for Accelerator Science, Kangwon National University, Chuncheon 24341, Republic of Korea
Correspondence to:E-mail: firstname.lastname@example.org
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Recently, α-RuCl3 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 4d- 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-RuBr3 and α-RuI3, which are expected to host the same spin-orbit-entangled orbitals and Kitaev exchange interactions with α-RuCl3. This study investigated electronic structures of α-RuCl3, α-RuBr3, and α-RuI3 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-RuBr3 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
Kitaev’s finding of an exactly solvable magnetic model on a twodimensional honeycomb lattice , 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 , 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-RuCl3 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-RuCl3 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 , and completely removing them is challenging owing to the multi-orbital nature of candidate systems and the resulting various hopping processes . 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-RuBr3 [18–20] and α-Rul3 [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
Thus, this study focused on the electronic structure of α-RuBr3 and α-Rul3, comparisons with α-RuCl3, and the formation of the spinorbital- entangled
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
Figure 1 shows the model crystal structure we chose for α-RuX3 (X = Cl, Br, I) in this study (Space group: P3112). There have been reports of different space group symmetries and stacking patterns for these compounds [20,22], but our choice of the P3112 stacking is not a serious concern because interlayer interactions are fairly small in all three compounds . Moreover, while the zigzag-type antiferromagnetic order (which requires larger in-plane periodicity) has been reported as the true ground state for α-RuCl3, 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-RuCl3 and α-Rul3 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.
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
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
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
where α, β, γ denote bond directions and the relevant spin components (
The hopping terms
Because we obtained the hopping parameters from ab-initio calculations, the magnitudes of
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
In this study preliminary first-principles density functional theory calculations were performed for α-RuCl3, α-RuBr3, and α-Rul3, to assess their viability for realizing Kitaev magnetism. The results indicated that α-RuCl3 and α-RuBr3 may host the spin-orbit-entangled
Note that some possible effects of minor lattice distortions, for example the effect of trigonal crystal fields, were not discussed in this study . 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-RuBr3 can be another interesting system to study Kitaev physics in addition to α-RuCl3. Lastly, since α-Rul3 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 .
We thank K.-Y. Choi for useful discussions, and acknowledge the support from the Korea Research Fellow (KRF) Program and the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education (NRF-2019H1D3A1A01- 102984 and NRF- 2020R1C1C1005900). We also thank the support of computational resources including technical assistance from the National Supercomputing Center of Korea (Grant No. KSC- 2020-CRE- 0156)
The authors declare no conflicts of interest.