Kitaev interaction and possible spin liquid state in CoI2 and Co2/3Mg1/3I2

Abstract

Kitaev materials are of great interest due to their potential in realizing quantum spin liquid (QSL) states and applications in topological quantum computing. In the pursuit of realizing Kitaev QSL, a Mott insulator with strong bond-dependent frustration and weak geometric frustration is highly desirable. Here we explore Kitaev physics in the van der Waals triangular antiferromagnet (AF) CoI$_2$, through the spin-orbital states and Wannier function analyses, exact diagonalization and density matrix renormalization group study of the electronic structure and magnetic properties. We find that the high-spin Co$^{2+}$ ion is in the $J_\mathrm{eff}=1/2$ state because of strong spin-orbit coupling, and the weak trigonal elongation and crystal field contribute to the observed weak in-plane magnetic anisotropy. The strong $t_{2g}$-$e_g$ hopping via the strong Co 3$d$-I 5$p$ hybridization gives rise to a strong Kitaev interaction ($K_1$) at the first nearest neighbors (1NN), and the long Co-Co distance and the weak $t_{2g}$-$t_{2g}$ hoppings determine a weak Heisenberg interaction $J_1$. The resultant $|K_1/J_1|$ = 6.63 confirms a strong bond-dependent frustration, while the geometric frustration due to the 3NN Heisenberg interaction $J_3$ gets involved, and they all together result in the experimental helical AF order in CoI$_2$. We then propose to suppress the $J_3$ using a partial Mg substitution for Co, and indeed we find that Co$_{2/3}$Mg$_{1/3}$I$_2$ has the much reduced geometric frustration but hosts the robust bond-dependent frustration, and thus it would be a promising Kitaev material being so far closest to the QSL state.

Figures (18)

• Figure 1: (a) The $c$-axis view of crystal structure and local basis of CoI$_2$. The cyan and yellow balls correspond to Co and I atoms. The local X, Y and Z-axis and the corresponding Kitaev bonds of each axis are indicated by blue, green and red color. (b) Energy splitting of $d^7$ high spin state under SOC and trigonal crystal field. The excited energy levels are predicted by ED calculation. (c) The occupation number of ground state under different trigonal crystal field. The inset shows out-of-plane (blue curve) and in-plane (yellow curve) magnetic moments under different trigonal crystal field. The gray line denotes trigonal crystal field strength $\delta$ of CoI$_2$ in our calculation.
• Figure 2: The schematic diagram of (a) the dominant $t_{2g}$-$e_g$ hopping: indirect hopping between $d_{xy}$ and $d_{3z^2-r^2}$ orbitals and (c) the dominant $t_{2g}$-$t_{2g}$ hopping: direct hopping between $d_{xy}$ orbitals. The corresponding Wannier functions are plotted in (b) and (d).
• Figure 3: The DMRG results and phase diagrams of Kitaev-Heisenberg model on triangular lattice (a) without $J_3$ and (b) with $J_3$. The ground state energy $E_\mathrm{GS}/A$ per site and its second derivative $d^2E_\mathrm{GS}/Ad\varphi^2$ are indicated by blue and red line respectively.
• Figure 4: (a) The crystal structure of Co$_{2/3}$Mg$_{1/3}$I$_2$. The cyan, red and yellow balls represent Co, Mg and I atoms respectively. (b) Magnetic structure factor $S(q)$ of Co$_{2/3}$Mg$_{1/3}$I$_2$ with $30\%$$J_3$. The colormap is limited to the range of 0 to the maximum value $S(q)_\mathrm{max}$ of zigzag and stripe order as shown in Fig. S11 of SM SM. (c) Phase diagram for FM Kitaev interaction $K<0$ by ED. The exchange parameters of Na$_2$IrO$_3$ and RuCl$_3$ are from Ref. Winter2016.
• Figure S1: (a) Schematic diagram of local CoI$_6$ octahedron. (b) Splitting of energy levels of Co $d$ orbitals under local octahedral crystal field and global trigonal crystal field. Under octahedral crystal field, the degenerate $d$ orbitals split into $t_{2g}$ orbitals and $e_g$ orbitals with energy difference $\Delta$. The trigonal distortion along [111] direction further split $t_{2g}$ orbitals into $e_{g}^\pi$ and $a_{1g}$ orbitals with energy difference $3\delta$. (c) Structure and coordinate of two-site model of Z bond.