[Paper Review] Understanding the Anomalous Hall effect in Co$_{1/3}$NbS$_{2}$ from crystal and magnetic structures

K. Lu, A. Murzabekova+11 more|arXiv (Cornell University)|Dec 30, 2022

2D Materials and Applications4 citations

TL;DR

This study resolves the paradox of a large anomalous Hall effect (AHE) in non-centrosymmetric Co₁/₃NbS₂ despite its long-assumed collinear antiferromagnetic order by demonstrating via neutron diffraction and DFT that the true magnetic ground state is non-collinear but preserves the same reflection symmetries. The AHE is quantitatively reproduced by DFT using this non-collinear spin structure, validating the 'crystal Hall effect' mechanism driven by chiral lattice symmetry and orbital geometry, independent of net magnetization.

ABSTRACT

A large anomalous Hall effect (AHE) has recently been observed in the intercalated transition metal dichalcogenide (TMDC) Co$_{1/3}$NbS$_{2}$ below a known magnetic phase transition at $T_N$ = 29 K. The spins in this material are widely believed to order in a highly symmetric collinear antiferromagnetic configuration, causing extensive debate about how reports of an AHE can be reconciled with such a state. In this article, we address this controversy by presenting new neutron diffraction data on single crystals of Co$_{1/3}$NbS$_{2}$ and an analysis that implies that moments in this material order into a non-collinear configuration, but one that maintains the same refelction symmetries as the collinear phase. We present new transport and magneto-optic Kerr measurements which show that AHE signatures persist below $T_N$ to temperatures as low as $T$ = 5 K and firmly associate them with the long-range antiferromagnetic order. Finally, we show that these AHE signatures can be quantitatively reproduced by density functional theory (DFT) calculations based on the lattice and spin state determined with neutron diffraction. These combined findings establishes the veracity of the 'crystal Hall effect' picture, which shows how such effects can emerge from the shape of magnetic orbitals in compounds containing chiral lattice symmetry regardless of the symmetry of the ordered spin configuration. These results illuminate a new path for the discovery of anomalous Hall materials and motivate a targeted study of the transport properties of intercalated TMDCs and other compounds containing antiferromagnetic order and chiral lattice symmetry.

Motivation & Objective

• To resolve the longstanding contradiction between the observed large anomalous Hall effect (AHE) and the previously assumed collinear antiferromagnetic order in Co₁/₃NbS₂.
• To determine the true magnetic structure of Co₁/₃NbS₂ using high-resolution neutron diffraction on single crystals.
• To establish a direct link between the measured AHE and long-range antiferromagnetic order below TN = 29 K.
• To validate the origin of the AHE through first-principles density functional theory (DFT) calculations based on the experimentally determined lattice and spin configuration.
• To demonstrate that the AHE arises from the crystal Hall effect, driven by chiral lattice symmetry and orbital geometry, even in the absence of net magnetization.

Proposed method

• Performed single-crystal neutron diffraction using the DEMAND and WAND 2 instruments at the High Flux Isotope Reactor (HFIR), Oak Ridge National Laboratory, to probe both nuclear and magnetic structures.
• Conducted lattice and magnetic structure refinements using the P6₃22 space group and the k = (0.5, 0, 0) propagation vector, testing irreducible representations to identify the correct magnetic symmetry.
• Used the Γ₂ irreducible representation to model the magnetic structure, which best fit the neutron diffraction data, particularly from the WAND 2 instrument with lower error bars.
• Performed DFT calculations with structural and spin parameters derived from neutron data, including full relaxation of the lattice and spin configuration.
• Calculated magneto-optic Kerr effect (MOKE) signals using DFT with varying sulfur positions and magnetic moment signs to test robustness of the AHE signal.
• Assessed k-point convergence of the Kerr signal using 5×10×5 and 4×8×4 Monkhorst-Pack grids to ensure numerical accuracy.

Experimental results

Research questions

• RQ1 Why does Co₁/₃NbS₂ exhibit a large anomalous Hall effect despite being described as a highly symmetric collinear antiferromagnet?
• RQ2 What is the true nature of the magnetic order in Co₁/₃NbS₂, and does it deviate from the previously assumed collinear configuration?
• RQ3 Can the observed AHE be quantitatively reproduced by first-principles calculations based on the experimentally determined magnetic structure?
• RQ4 Is the AHE in this system driven by intrinsic topological effects or by a crystal Hall effect arising from chiral lattice symmetry and orbital anisotropy?
• RQ5 How robust is the AHE signal to variations in sulfur position and magnetic moment orientation, and does it persist under different structural and magnetic configurations?

Key findings

• Neutron diffraction data from both DEMAND and WAND 2 instruments confirm a non-collinear magnetic structure described by the Γ₂ irreducible representation, with Co moments forming a non-collinear spin configuration that preserves the reflection symmetries of the collinear phase.
• The magnetic moments on the four Co sites in the unit cell are non-collinear, with measured values of ±0.965, ±1.671, and ±0.358 μB in x, y, and z directions, respectively, yielding a total moment of 1.962 μB per Co site.
• DFT calculations based on the neutron-determined structure predict a reduced moment of ~1.44 μB per Co site and a slightly larger tilting angle, confirming the stability of the non-collinear ground state.
• The calculated Kerr rotation and ellipticity signals from DFT are of the same order of magnitude across multiple configurations, including reversed magnetic moments and different sulfur positions (0.31, 0.33, 0.66), confirming the robustness of the AHE signal.
• The AHE persists down to 5 K, firmly associating it with long-range antiferromagnetic order rather than short-range fluctuations or weak ferromagnetism.
• The AHE is quantitatively reproduced by DFT using the non-collinear spin structure, providing strong evidence for the 'crystal Hall effect' mechanism in chiral, non-centrosymmetric lattices with non-collinear magnetism.