---
title: 'Co₂/₃Mg₁/₃I₂: Kitaev Spin-Liquid on Triangular Lattice'
url: https://www.emergentmind.com/topics/co-_-2-3-mg-_-1-3-i-_2
type: topic
---

# Co₂/₃Mg₁/₃I₂: Kitaev Spin-Liquid on Triangular Lattice

Co$_{2/3}$Mg$_{1/3}$I$_2$ is a van der Waals layered transition-metal iodide that has emerged as a leading candidate for realizing the Kitaev quantum spin-liquid (QSL) physics on a triangular lattice. This compound, generated by partial non-magnetic Mg$^{2+}$ substitution in CoI$_2$, is distinguished by its near-ideal combination of strong bond-dependent (Kitaev) exchange, suppressed geometric frustration, and minimal distortion of the magnetic ion environment. These features position Co$_{2/3}$Mg$_{1/3}$I$_2$ at the forefront of materials research targeting the realization of Kitaev spin liquids in 3$d$-ion, two-dimensional platforms [2512.22453].

## 1. Crystal Structure and Chemical Composition

Co$_{2/3}$Mg$_{1/3}$I$_2$ crystallizes in the trigonal space group P$\overline{3}$m1 as a van der Waals (vdW) layered material. The structure consists of layers of edge-sharing CoI$_6$ octahedra, where the magnetic Co$^{2+}$ (3$d^7$) and non-magnetic Mg$^{2+}$ ions are distributed on the vertices of a triangular network. The partial substitution involves one third of Co sites replaced by Mg, resulting in the formal composition.

The Mg$^{2+}$ sites (ionic radius $\approx$ 0.72 Å, close to that of Co$^{2+}$ at 0.745 Å) act primarily as non-magnetic "blockers" for third-nearest-neighbor (3NN) Co–Co exchanges. Structural metrics remain effectively unchanged by substitution: the in-plane lattice parameters retain $a = b \approx 3.87$ Å, and the interlayer spacing is $c \approx 7.06$ Å. The CoI$_6$ octahedra thus maintain their nearly regular geometry, and the local ligand field acting on Co centers is only minimally perturbed [2512.22453].

## 2. Spin–Orbital Ground State and Single-Ion Physics

Each Co$^{2+}$ ion in Co$_{2/3}$Mg$_{1/3}$I$_2$ resides in an octahedral iodine environment, resulting in a substantial $t_{2g}$–$e_g$ crystal field splitting $\Delta \approx 1.0$ eV. This splitting stabilizes a high-spin $S=3/2$ configuration, as the low-spin alternative would require a crossover threshold $2J_H \approx 1.8$ eV.

The $t_{2g}$ shell is not fully filled, yielding an effective orbital angular momentum $L_{\text{eff}} = 1$. Strong spin–orbit coupling (SOC) with $\lambda \approx 26.6$ meV splits the $L_{\text{eff}}=1$, $S=3/2$ manifold into $J_{\text{eff}} = 1/2, 3/2, 5/2$ multiplets. In Co$_{2/3}$Mg$_{1/3}$I$_2$, the trigonal crystal field distortion is weak ($\delta_{\text{tri}} \approx 4.5$ meV), ensuring that the ground doublet remains a nearly pure $J_{\text{eff}}=1/2$ state, as confirmed by exact diagonalization (approximately 98% purity by weight). The $g$-factor exhibits moderate anisotropy ($g_\parallel \approx 4.6$ in-plane, $g_\perp \approx 3.4$ out-of-plane), but the $J_{\text{eff}}=1/2$ character is robust under these perturbations [2512.22453].

## 3. Minimal Effective Hamiltonian and Exchange Mechanisms

Projection onto the $J_{\text{eff}}=1/2$ lowest-energy Kramers doublets yields a minimal effective spin Hamiltonian of the “Kitaev–Heisenberg” type on a triangular lattice:
$$
H = \sum_{\langle ij \rangle_1} \Big[ J_1\,\mathbf{S}_i \cdot \mathbf{S}_j + K_1\,S_i^\gamma S_j^\gamma \Big] + J_3 \sum_{\langle\langle\langle ij\rangle\rangle\rangle_3} \mathbf{S}_i \cdot \mathbf{S}_j
$$
where:
- $\langle ij \rangle_1$ runs over 1st-nearest-neighbor (1NN) Co–Co bonds,
- $\gamma \in \{x, y, z\}$ marks the three symmetry-inequivalent Kitaev bond types,
- $J_1$ is the isotropic 1NN Heisenberg exchange (AFM),
- $K_1$ is the bond-dependent 1NN Kitaev interaction (FM),
- $J_3$ is the 3NN Heisenberg exchange, responsible for geometric frustration.

Mg substitution selectively disrupts the 3NN Co paths, resulting in a pronounced reduction in $J_3$ without appreciable impact on $J_1$ or $K_1$, thereby isolating the bond-dependent frustration intrinsic to the Kitaev limit [2512.22453].

## 4. Numerical Exchange Parameters and Frustration Hierarchy

State-of-the-art density-functional theory (GGA) calculations, combined with maximally-localized Wannier orbital construction and exact diagonalization of multi-orbital Hubbard models, provide the following exchange couplings for Co$_{2/3}$Mg$_{1/3}$I$_2$ (in meV):

| Compound              | $J_1$ | $K_1$ | $J_3$ | $|K_1/J_1|$ | $|K_1/J_3|$ |
|-----------------------|-------|-------|-------|-------------|-------------|
| CoI$_2$               | +0.63 | –4.17 | +2.16 |     6.6     |     1.9     |
| Co$_{2/3}$Mg$_{1/3}$I$_2$ | +0.89 | –3.15 | +0.53 |     3.5     |     5.9     |

In Co$_{2/3}$Mg$_{1/3}$I$_2$, the ratio $|K_1/J_3| \approx 5.9$ highlights a regime where bond-dependent interactions far outweigh both the nearest-neighbor Heisenberg and geometric frustration terms, an arrangement not achieved in pristine CoI$_2$. The $|K_1/J_1|$ value remains large, furnishing a strong Kitaev (FM) exchange domination on the triangular network [2512.22453].

## 5. Computational Methodology

The theoretical investigation employs a combination of first-principles electronic structure calculations, exact diagonalization (ED), and density-matrix renormalization group (DMRG) simulations:

- DFT (GGA) and maximally localized Wannier functions are used to extract the crystal field environments and hopping matrices up to 3NN.
- ED of multi-orbital Hubbard clusters, with explicit Kanamori $U$, Hund’s coupling $J_H$, and SOC, is performed to validate the $J_{\text{eff}}=1/2$ ground state and quantify two-site exchange interactions (via Löwdin projection).
- DMRG simulations on $6 \times 6$ triangular clusters are used to explore the phase diagram of the reduced $J_1$–$K_1$–$J_3$ model, clarifying conditions leading to helical versus spin-liquid magnetic regimes.

These techniques establish a robust, multi-scale characterization of the local, effective, and collective quantum properties of Co$_{2/3}$Mg$_{1/3}$I$_2$ [2512.22453].

## 6. Relevance to Kitaev Quantum Spin-Liquid Physics

Co$_{2/3}$Mg$_{1/3}$I$_2$ is notable for combining (i) a robust ferromagnetic Kitaev interaction ($K_1$), (ii) a weak 1NN Heisenberg term ($J_1$), and (iii) a strongly suppressed long-range (3NN) Heisenberg exchange ($J_3$). This interaction hierarchy ($|K_1| \gg |J_1|, |J_3|$) positions the compound in close proximity to the pure Kitaev limit for a triangular-lattice system.

Magnetic structure-factor calculations for the Mg-substituted system display only diffuse correlations, with the absence of sharp peaks, consistent with a proximate QSL phase. The application of moderate external magnetic fields ($\sim$2–4 T) or pressure could induce a transition into a bona fide Kitaev QSL. Thus, Co$_{2/3}$Mg$_{1/3}$I$_2$ emerges as a vdW-layered, $3d$-ion Kitaev candidate with minimal geometric frustration and maximal manifestation of bond-dependent frustration, making it an optimal platform for studies of Kitaev spin-liquid phenomena [2512.22453].

Source: https://www.emergentmind.com/topics/co-_-2-3-mg-_-1-3-i-_2