Magnetic Phase Diagrams and Spin Hamiltonian of Monoclinic $α$-RuCl$_3$ from Angle-Dependent Torque Studies

Abstract

The layered honeycomb $α$-RuCl$_3$ has been much explored as a candidate to display unconventional cooperative magnetism between spin-orbit entangled $j_{\rm eff}=1/2$ Ru$^{3+}$ magnetic moments stabilized by strongly-frustrated, Kitaev and other bond-dependent anisotropic interactions. Most studies so far have focused on relatively large single crystals that undergo a structural phase transition from monoclinic to rhombohedral upon cooling below 150 K and a magnetic transition to zigzag order around 7 K. Here we study the magnetism of high-quality, very small (sub 100 $μ$m diameter) single crystals, which remain monoclinic upon cooling to low temperatures and display a sharp magnetic transition at 14 K. Using highly-sensitive piezo-cantilever magnetic torque measurements in fields up to 16 T, we report a comprehensive study of the magnetic phase diagrams for magnetic field rotated in three orthogonal crystallographic planes. Our extensive torque data collected upon varying almost continuously the angular orientation of the field as well as the field magnitude, allows clear detection of phase transitions via anomalies in the raw torque data and its higher order derivatives with respect to both angle and field magnitude. The obtained magnetic phase diagrams show many differences compared to rhombohedral samples, displaying reduced rotational symmetry, substantially higher fields required to suppress the spontaneous magnetic order, and distinct field-induced phases. We provide a direct comparison of the torque data with mean-field calculations for magnetic Hamiltonians appropriate for $α$-RuCl$_3$ allowing for the monoclinic lattice symmetry and propose minimal magnetic structure models for all explored regions of the magnetic phase diagrams.