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Kitaev interactions, which are bond-related anisotropic interactions induced by spin-orbit coupling (SOC), may produce quantum spin liquid states in two-dimensional (2D) magnetic hexagonal lattices such as RuCl3. Generally, the strong SOCs in these materials come from heavy metal elements such as Ru in RuCl3. In recent years, some related studies have shown the presence of Kitaev effects in some 2D monolayers of ortho-octahedral structures containing heavy ligand elements, such as CrGeTe3 and CrSiTe3. However, there are relatively few reports on the Kitaev interactions in 2D monolayer 1T structures. In this paper, we calculate and analyse the atomic and electronic structures of 1T-CoI2 and the Kitaev interactions contained therein by the first-principles calculation program VASP. The structure of 1T-CoI2 is a triangular lattice with an emphasis on the coordinating element I. The energy dispersion relation
$ {E}_{{\mathrm{S}}}\left(\boldsymbol{q}\right)={E}_{{\mathrm{N}}+{\mathrm{S}}}\left(\boldsymbol{q}\right)-{E}_{{\mathrm{N}}}\left(\boldsymbol{q}\right) $ for the contained Kitaev action is isolated by calculating the energy dispersion relation$ {E}_{{\mathrm{N}}}\left(\boldsymbol{q}\right) $ for the spin-spiral of monolayer CoI2 without SOC and the energy dispersion relation$ {E}_{{\mathrm{N}}+{\mathrm{S}}}\left(\boldsymbol{q}\right) $ considering SOC by using the generalized Bloch condition combined with the spin-spiral method. The parameters of the Heisenberg exchange interaction induced by the SOC are obtained by fitting the dispersion law of the$ {E}_{{\mathrm{S}}}\left(\boldsymbol{q}\right) $ to the Kitaev exchange interaction with the parameters of the Kitaev exchange interaction. The fitted curves obtained with the fitted parameters are in good agreement with the calculated values, indicating the accuracy of our calculations. Calculated fits show that the monolayer CoI2 is dominated by Heisenberg action, with the third nearest neighbour having the largest absolute value of J at –1.81 meV. In addition to this, there are strong Kitaev interactions in the monolayer CoI2, where Γ1 reaches 1.09 meV. We predict that the Kitaev interactions are universally applicable to transition metal triangular lattices with 1T structure. It is shown that the CoI2 can be used as an alternative material for Kitaev and lays a theoretical foundation for exploring Kitaev interactions in other 2D magnetic materials.-
Keywords:
- monolayer 1T-CoI2 /
- first principle /
- two-dimensional magnetic materials /
- Kitaev interaction
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图 1 (a)单层1T-CoI2的基矢图, 其中a1和a2为基矢, b1和b2为倒格矢, G, M和H为第一布里渊区的高对称k点; (b)是单层1T-CoI2加上真空层后的原子结构侧视图, 蓝色球、紫色球分别表示Co和I原子; (c)是扩展后5$ \times $5超胞俯视图, 用来描述Co原子间HBI和Kitaev相互作用, 以中间的Co原子为中心, 红色数字表示Co原子的近邻位置. 设置两个坐标系, [XYZ]表示的是Co—Co键, 三角晶格上的第1近邻, 第2近邻和第3近邻Co—Co键都标记在图中, 绿色、蓝色和黄色分别表示X, Y和Z键. [xyz]表示相互垂直的3个Co—I键坐标系, 其中x, y和z所表示的Co—I键分别垂直于X, Y和Z的Co—Co键所在平面
Figure 1. (a) Base vector diagram of 1T-CoI2, where a1 and a2 are the base vectors, b1 and b2 are the reciprocal lattice vectors, G, M and H are highly symmetric k-points of the first Brillouin zone; (b) a side view of the atomic structure of 1T-CoI2 with a vacuum layer. The blue and purple balls represent Co and I atoms, respectively; (c) the top view of the expanded 5$ \times $5 supercell, which is used to describe the HBI and Kitaev interactions between Co atoms, centered on the Co atom in the middle, and the red numbers indicate the neighboring positions of the Co atoms. Two coordinate systems are set: [XYZ] represents the Co—Co bond, the first, second and third neighbor Co—Co bond on the triangular lattice are marked in the figure. Green, blue and yellow bonds indicate the X, Y and Z bond, respectively. [xyz] represents three Co—I bond coordinate systems perpendicular to each other, where the Co—I bonds represented by x, y and z are perpendicular to the plane where the Co—Co bonds of X, Y and Z are located, respectively.
图 2 (a), (b)单层1T-CoI2的能带结构, (a)中红色和黑色分别表示自旋向上和自旋向下的能带, (b)考虑SOC计算的能带图; (c), (d)单层1T-CoI2的态密度图, 其中红色曲线表示Co原子, 黑色曲线表示I原子, (c)为单层1T-CoI2的I原子和Co原子的分态态密度, 纵坐标正值表示上自旋的态密度, 负值表示下自旋态密度, 能量为0处的蓝色虚线是费米面, (d)加SOC计算得到Co和I的总态密度图
Figure 2. (a), (b) Band structure of the monolayer 1T-CoI2. The red and black lines in panel (a) indicate the spin-up and spin-down bands, respectively. (c), (d) Density of states (DOS) maps of monolayer 1T-CoI2, where the red and black curves show the DOS of Co and I atoms, respectively. In panel (c), positive and negative values indicate the DOS of the up and down spin, respectively; the blue dashed line at energy 0 is the Fermi level; panel (d) is the DOS of Co and I calculated with SOC.
图 3 (a)只考虑Γ1 = 1时, $ {E}_{{\varGamma }_{1}}(\boldsymbol{q}) $对应的色散关系图; (b), (c)分别是只考虑Γ2 = 1, Γ3 = 1时对应的$ {E}_{{\varGamma }_{2}}(\boldsymbol{q}) $, $ {E}_{{\varGamma }_{3}}(\boldsymbol{q}) $色散关系图; (d)—(f)分别是K1, K2, K3取1时对应的$ {E}_{{K}_{1}}(\boldsymbol{q}) $, $ {E}_{{K}_{2}}(\boldsymbol{q}) $, $ {E}_{{K}_{3}}(\boldsymbol{q}) $色散关系图
Figure 3. (a) Dispersion relation corresponding to $ {E}_{{\varGamma }_{1}}(\boldsymbol{q}) $ when only Γ1 = 1 is considered; (b), (c) the corresponding $ {E}_{{\varGamma }_{2}}(\boldsymbol{q}) $, $ {E}_{{\varGamma }_{3}}(\boldsymbol{q}) $ dispersion relations when only Γ2 = 1 and Γ3 = 1 are considered, respectively; (d)–(f) plots of $ {E}_{{K}_{1}}(\boldsymbol{q}) $, $ {E}_{{K}_{2}}(\boldsymbol{q}) $, $ {E}_{{K}_{3}}(\boldsymbol{q}) $ dispersion relations corresponding to K1, K2, and K3 taken as 1, respectively.
图 4 (a)离散点分别代表的是计算的单层1T-CoI2体系的自旋螺旋能量与波矢q的色散关系$ E\left(\boldsymbol{q}\right) $, 其中N表示不考虑SOC, S是只有SOC; 黑色方框$ {E}_{{\mathrm{N}}+{\mathrm{S}}} $与红色圆圈$ {E}_{{\mathrm{N}}} $是计算值, 蓝色三角$ {E}_{{\mathrm{S}}} $是两者之差. 黑色曲线、红色曲线和蓝色曲线是对应的拟合曲线; H, G, M是图1(a)中第一布里渊区的特殊k点. (b)单层1T-CoI2中海森伯相互作用第1—第8近邻的J值变化趋势图. (c) SOC作用下第1近邻-第3近邻J, K, Γ参数点的变化趋势图. (d)—(g)表示第一布里渊区中H, G, M点和$ E\left(\boldsymbol{q}\right) $中最低点L的磁矩分布图
Figure 4. (a) Discrete points represent the calculated dispersion relation $ E\left(\boldsymbol{q}\right) $ between the spin spiral energy of the 1T-CoI2 system and the wave vector q. Among them, N means that SOC is not considered, and S means only SOC; the black box $ {E}_{{\mathrm{N}}+{\mathrm{S}}} $ and the red circle $ {E}_{{\mathrm{N}}} $ are calculated values, and the blue triangle $ {E}_{{\mathrm{S}}} $ is the difference between the two. The black, red and blue curves are the corresponding fitting ones; H, G, M are special k points in the first Brillouin zone in Fig. 1(a). (b) The J value of the first to eighth neighbors of the HBI in 1T-CoI2. (c) J, K, Γ parameter points from the first neighbor to the third neighbor with SOC. (d)–(g) Magnetic moment distribution diagrams of points H, G, M in the first Brillouin zone and the lowest point L in $ E\left(\boldsymbol{q}\right) $.
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[1] Jin C H, Lin F, Suenaga K, Iijima S 2009 Phys. Rev. Lett. 102 195505
Google Scholar
[2] Liu H, Neal A T, Zhu Z, Luo Z, Xu X, Tománek D, Ye P D 2014 ACS Nano 8 4033
Google Scholar
[3] Min Y, Moon G D, Kim B S, Lim B, Kim J S, Kang C Y, Jeong U 2012 J. Am. Chem. Soc. 134 2872
Google Scholar
[4] Kim D, Sun D, Lu W, Cheng Z, Zhu Y, Le D, Rahman T S, Bartels L 2011 Langmuir 27 11650
Google Scholar
[5] 蒋小红, 秦泗晨, 幸子越, 邹星宇, 邓一帆, 王伟, 王琳 2021 物理学报 70 127801
Google Scholar
Jiang X H, Qin S C, Xing Z Y, Zou X Y, Deng Y F, Wang W, Wang L 2021 Acta. Phys. Sin. 70 127801
Google Scholar
[6] Dai C Y, He P, Luo L X, Zhan P X, Guan B, Deng J 2023 Sci. China Mater. 66 859
Google Scholar
[7] 刘南舒, 王聪, 季威 2022 物理学报 71 127504
Google Scholar
Liu N S, Wang C, Ji W 2022 Acta Phys. Sin. 71 127504
Google Scholar
[8] 吴燕飞, 朱梦媛, 赵瑞杰, 刘心洁, 赵云驰, 魏洪祥, 张静妍, 郑新奇, 申见昕, 黄河, 王守国 2022 物理学报 71 048502
Google Scholar
Wu Y F, Zhu M Y, Zhao R J, Liu X J, Zhao Y C, Wei H X, Zhang J Y, Zheng X Q, Shen J X, Huang H, Wang S G 2022 Acta Phys. Sin. 71 048502
Google Scholar
[9] Magda G Z, Jin X, Hagymási I, Vancsó P, Osváth Z, Nemes-Incze P, Hwang C, Biró L P, Tapasztó L 2014 Nature 514 608
Google Scholar
[10] Amoroso D, Barone P, Picozzi S 2020 Nat. Commun. 11 5784
Google Scholar
[11] Zhang W B, Qu Q, Zhu P, Lam C H 2015 J. Mater. Chem. C 3 12457
Google Scholar
[12] Han H, Zheng H, Wang Q, Yan, Y 2020 Phys. Chem. Chem. Phys. 22 26917
Google Scholar
[13] Cui Q R, Zhu Y M, Ga Y L, Liang J H, Li P, Yu D X, Cui P, Yang H X 2022 Nano Lett. 22 2334
Google Scholar
[14] Li S, Wang S S, Tai B, Wu W, Xiang B, Sheng X L, Yang S A 2021 Phys. Rev. B 103 045114
Google Scholar
[15] Fuh H R, Yan B, Wu S C, Felser C, Chang C R 2016 New J. Phys. 18 113038
Google Scholar
[16] Alsubaie M, Tang C, Wijethunge D, Qi D, Du A 2022 ACS Appl. Electron. Ma. 4 3240
Google Scholar
[17] Kitaev A 2006 Ann. Phys. 321 2
Google Scholar
[18] Jackeli G, Khaliullin G 2009 Phys. Rev. Lett. 102 017205
Google Scholar
[19] Ran K J, Wang J H, Wang W, Dong Z Y, Ren X, Bao S, Li S C, Ma Z, Gan Y, Zhang Y T, Park J Y, Deng G H, Danilkin S, Yu S L, Li J X, Wen J S 2017 Phys. Rev. Lett. 118 107203
Google Scholar
[20] Sears J A, Chern L E, Kim S, Bereciartua P J, Francoual S, Kim Y B, Kim Y J 2020 Nat. Phys. 16 837
Google Scholar
[21] Kim C, Jeong J, Lin G, Park P, Masuda T, Asai S, Itoh S, Kim H, Zhou H, Ma J 2021 J. Phys. Condens. Mat. 34 045802
Google Scholar
[22] Songvilay M, Robert J, Petit S, Rodriguez-Rivera, J A, Ratcliff W D, Damay F, Balédent V, Jiménez-Ruiz M, Lejay P, Pachoud E, Hadj-Azzem A, Simonet V, Stock C 2020 Phys. Rev. B. 102 224429
Google Scholar
[23] Xu C S, Feng J S, Xiang H J, Laurent B 2018 npj Comput. Mater. 4 57
Google Scholar
[24] Xu C S, Feng J S, Kawamura M, Yamaji Y, Nahas Y, Prokhorenko S, Qi Y, Xiang H J, Bellaiche L 2020 Phys. Rev. Lett. 124 087205
Google Scholar
[25] Kresse G, Joubert D 1999 Phys. Rev. B 59 1758
Google Scholar
[26] Kress G, Furthmuller J 1996 Phys. Rev. B 54 11169
Google Scholar
[27] Jiang Y, Huang C, Zhu Y, Pan Y F, Fan J Y, Ma L C 2022 Sci. China 52 226811
Google Scholar
[28] 李小影, 黄灿, 朱岩, 李晋斌, 樊济宇, 潘燕飞, 施大宁, 马春兰 2018 物理学报 67 137101
Google Scholar
Li X Y, Huang C, Zhu Y, Li J B, Fan J Y, Pan Y F, Shi D N, Ma L C 2018 Acta Phys. Sin. 67 137101
Google Scholar
[29] Perdew J P, Burke K, Ernzerhof M 1996 Phys. Rev. Lett. 77 3865
Google Scholar
[30] Botana A S, Norman M R 2019 Phys. Rev. Mater. 3 044001
Google Scholar
[31] Anisimov V I, Zaanen J, Andersen O K 1991 Phys. Rev. B 44 943
Google Scholar
[32] Dudarev S L, Botton G A, Savrasov S Y, Humphreys C J, Sutton A P 1998 Phys. Rev. B 57 1505
Google Scholar
[33] Melchakova I, Kovaleva E A, Mikhaleva N S, Tomilin F N, Ovchinnikov S G, Kuzubov A A, Avramov P 2020 Int. J. Quantum Chem. 120 26092
Google Scholar
[34] Yang H X, Boulle O, Cros V, Fert A, Chshiev M 2018 Sci. Rep. 8 12356
Google Scholar
[35] Zhu Y, Ma C L, Shi D N, Zhang K C 2014 Phys. Lett. A 378 2234
Google Scholar
[36] Yang H X, Thiaville A, Rohart S, Fert A, Chshiev M 2015 Phys. Rev. Lett. 115 267210
Google Scholar
[37] Pan Y F, Zhu Y, Shi D N, Wei X Y, Ma C L, Zhang K C 2015 J. Alloy. Compd. 644 341
Google Scholar
[38] Kulish V V, Huang W 2017 J. Mater. Chem. C 5 8734
Google Scholar
[39] Kim H S, Catuneanu A, Kee H Y 2015 Phys. Rev. B 91 241110
Google Scholar
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