Is Herbertsmithite far from an ideal antiferromagnet? Ab-initio answer including in-plane Dzyaloshinskii-Moriya interactions and coupling with extra-plane impurities

TL;DR

Is Herbertsmithite far from an ideal antiferromagnet? The paper provides a quantitative, ab initio assessment of the spin Hamiltonian for Herbertsmithite, including both in-plane and out-of-plane Dzyaloshinskii–Moriya interactions and exchange with inter-plane Cu impurities. By combining an embedded cluster approach (DFT) and relativistic wave-function theory (CASSCF/SO-SI/CASPT2) with effective Hamiltonian mapping, the authors extract $J_1 \\approx 181.3$ K (WFT) and $J_1 \\approx 239$ K (DFT) for the dominant isotropic exchange, very small second-neighbor couplings, and considerable ferromagnetic impurity couplings with $J_5' \\approx -48$ K and $J_5'' \\approx -73$ K. They also find a strong, predominantly in-plane DM component with $|\boldsymbol{d}^{\\parallel}_{ij}| \\sim 4.7$ K and $|\boldsymbol{d}^{\\perp}_{ij}| \\sim 1.7$ K, while the symmetric anisotropy parameters $D$ and $E$ remain below 1 K. With an estimated impurity fraction of about $15\%$, these results indicate that 2D magnetic models only describe part of the physics and that 3D coupling and disorder must be incorporated for a faithful description.

Abstract

Herbertsmithite is known as the archetype of a S=1/2 nearest-neighbor Heisenberg antiferromagnet on the Kagomé lattice, theoretically presumed to be a quantum gapless spin liquid. However, more and more experiments reveal that the model suffers from deviations from the ideal one, evidenced at very low temperatures. This detailed ab initio study focuses on two such deviations that have never been quantitatively calculated: the anisotropic exchange interactions and the Heisenberg exchange with extra-plane magnetic impurities. The Dzyaloshinskii-Moriya interaction is found to have an in-plane component almost three times larger than the out-of-plane component, but typically obviated in theoretical studies. Moreover, it is shown that the extra-plane magnetic impurities have a strong ferromagnetic interaction (minus half the main exchange $J_1$ ) with the Kagomé magnetic sites. Combined with an estimated occurrence of these magnetic impurities of $\sim15\%$, the present results indicate that two-dimensional magnetic models only describe part of the physics.

Figures (6)

• Figure 1: Clusters considered in the DFT calculations: (a) 3-copper cluster, (b) 13-copper cluster, (c) 12-copper cluster. (a), (b), (c) consider the in-plane copper only. (d) 7-copper cluster with an inter-plane Cu$^{2+}$ at the position of the Zn$^{2+}$ ion.
• Figure 2: Scheme of the magnetic couplings for clusters (a), (b), (c) and (d) of Fig. \ref{['fig:1']}. The $J_1"$ and $J_5"$ refer to pairs of Cu$^{2+}$ ions bridged by the oxygens that moved away from the impurity while for $J_1'$ and $J_5'$, the bridging oxygens have moved closer to the impurity.
• Figure 3: CAS(4,3)SCF active orbital calculated on a fragment involving two in-plane copper ions and the optimized oxygen orbital (left) and the two magnetic orbitals (right).
• Figure 4: CAS(4,4)SCF active orbitals calculated for a tetra-nuclear fragment constituted of three in-plane (below) and an inter-plane (above) copper ions.
• Figure 5: Two views of the proper magnetic axes of the symmetric tensor of anisotropy calculated for a bi-nuclear fragment. The $Z$ axis points toward the bridging oxygen while $X$ is the inter-nuclear axis.