First-principles study of role of Kitaev interaction in monolayer 1<i>T</i>-CoI<sub>2</sub>
Authors: Zhu Kai , Huang Can , Cao Bang-Jie , Pan Yan-Fei , Fan Ji-Yu , Ma Chun-Lan , Zhu Yan ,   College of Physics, Nanjing University of Aeronautics and Astronautics, Nanjing 210006, China,   Key Laboratory of Aerospace Information Materials and Physics, Ministry of Industry and Information Technology, Nanjing University of Aeronautics and Astronautics, Nanjing 210006, China,   Jiangsu Key Laboratory of Micro and Nano Heat Fluid Flow Technology and Energy Application, School of Physical Science and Technology, Suzhou University of Science and Technology, Suzhou 215009, China
Published: September 2023

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 RuCl<sub>3</sub>. Generally, the strong SOCs in these materials come from heavy metal elements such as Ru in RuCl<sub>3</sub>. 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 CrGeTe<sub>3</sub> and CrSiTe<sub>3</sub>. However, there are relatively few reports on the Kitaev interactions in 2D monolayer 1<i>T</i> structures. In this paper, we calculate and analyse the atomic and electronic structures of 1<i>T</i>-CoI<sub>2</sub> and the Kitaev interactions contained therein by the first-principles calculation program VASP. The structure of 1<i>T</i>-CoI<sub>2</sub> is a triangular lattice with an emphasis on the coordinating element I. The energy dispersion relation <inline-formula><tex-math id="M2">\begin{document}$ {E}_{{\mathrm{S}}}\left(\boldsymbol{q}\right)={E}_{{\mathrm{N}}+{\mathrm{S}}}\left(\boldsymbol{q}\right)-{E}_{{\mathrm{N}}}\left(\boldsymbol{q}\right) $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="24-20230909_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="24-20230909_M2.png"/></alternatives></inline-formula> for the contained Kitaev action is isolated by calculating the energy dispersion relation <inline-formula><tex-math id="M3">\begin{document}$ {E}_{{\mathrm{N}}}\left(\boldsymbol{q}\right) $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="24-20230909_M3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="24-20230909_M3.png"/></alternatives></inline-formula> for the spin-spiral of monolayer CoI<sub>2</sub> without SOC and the energy dispersion relation <inline-formula><tex-math id="M4">\begin{document}$ {E}_{{\mathrm{N}}+{\mathrm{S}}}\left(\boldsymbol{q}\right) $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="24-20230909_M4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="24-20230909_M4.png"/></alternatives></inline-formula> considering SOC by using the generalized Bloch condition combined with the spin-spiral method. The parameters of the Heisenberg exchange interaction induced by the SOC are obtained by fitting the dispersion law of the <inline-formula><tex-math id="M5">\begin{document}$ {E}_{{\mathrm{S}}}\left(\boldsymbol{q}\right) $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="24-20230909_M5.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="24-20230909_M5.png"/></alternatives></inline-formula> 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 CoI<sub>2</sub> is dominated by Heisenberg action, with the third nearest neighbour having the largest absolute value of <i>J</i> at –1.81 meV. In addition to this, there are strong Kitaev interactions in the monolayer CoI<sub>2</sub>, where <i>Γ</i><sub>1</sub> reaches 1.09 meV. We predict that the Kitaev interactions are universally applicable to transition metal triangular lattices with 1<i>T</i> structure. It is shown that the CoI<sub>2</sub> can be used as an alternative material for Kitaev and lays a theoretical foundation for exploring Kitaev interactions in other 2D magnetic materials.