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DOE OSTI2024

Emergence of the isotropic Kitaev honeycomb lattice $α$–RuCl 3 and its magnetic properties

Here, we present a comprehensive investigation of the crystal and magnetic structures of the van der Waals antiferromagnet $α$–RuCl 3 using single crystal x-ray and neutron diffraction. The crystal structure at room temperature is a monoclinic (C2/m). However, with decreasing temperature, a remarkable first-order structural phase transition is observed, leading to the emergence of a rhombohedral ($R\bar3$) structure characterized by three-fold rotational symmetry forming an isotropic honeycomb lattice. On further cooling, a zigzag-type antiferromagnetic order develops below $T_{\textrm{N}} = 6\sim6.6$ K. The critical exponent of the magnetic order parameter was determined to be $\beta = 0.11(1)$, which is close to the two-dimensional Ising model. Additionally, the angular dependence of the magnetic critical field of the zigzag antiferromagnetic order for the polarized ferromagnetic phase reveals a six-fold rotational symmetry within the ab–plane. These findings reflect the symmetry associated with the Ising-like bond-dependent Kitaev spin interactions and underscore the universality of the Kitaev interaction-dominated antiferromagnetic system.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2020

Charge density waves on a half-filled decorated honeycomb lattice

Tight binding models like the Hubbard Hamiltonian are most often explored in the context of uniform intersite hopping $\textit{t}$. The electron-electron interactions, if sufficiently large compared to this translationally invariant $\textit{t}$, can give rise to ordered magnetic phases and Mott insulator transitions, especially at commensurate filling. The more complex situation of nonuniform $\textit{t}$ has been studied within a number of situations, perhaps most prominently in multiband geometries where there is a natural distinction of hopping between orbitals of different degree of overlap. In this paper we explore related questions arising from the interplay of multiple kinetic energy scales and electron-phonon interactions. Specifically, we use determinant quantum Monte Carlo (DQMC) to solve the half-filled Holstein Hamiltonian on a “decorated honeycomb lattice,” consisting of hexagons with internal hopping $\textit{t}$ coupled together by $\textit{t'}$. This modulation of the hopping introduces a gap in the Dirac spectrum and affects the nature of the topological phases. Here, we determine the range of $\textit{t/t'}$ values which support a charge density wave phase about the Dirac point of uniform hopping $\textit{t = t'}$, as well as the critical transition temperature $T_c$. The QMC simulations are compared with the results of mean field theory.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2023

Quantum phases in the honeycomb-lattice J 1 – J 3 ferro-antiferromagnetic model

The significant role of quantum effects in magnets with competing interactions remains at the forefront of condensed matter physics for over 50 years, inspiring a multitude of quests for exotic states, models that can realize them, and real materials that can host them. The elusive spin-liquid states with strong entanglement are but one example; others include valence-bond phases with spatial symmetry breaking, quantum spin nematics that are quantum analogues of liquid crystals, and an especially extensive class of unconventional magnetically ordered phases that do not appear in the classical solutions of the underlying spin models. It is the latter group of phenomena that creates a broader context for the research effort in the present work concerning the search for and an identification of the novel ground states of quantum magnets. In this work, we have studied one of the paradigmatic models in quantum magnetism that is also attracting a significant recent interest because it appears to be providing a tantalizingly close description for many of the newly synthesized materials. We demonstrated that the phase diagram of the quantum ferro-antiferromagnetic model on the honeycomb lattice differs dramatically from the classical one. It hosts the double-zigzag and Ising-z phases as unexpected intermediaries between ferromagnetic and zigzag states that are also extended beyond their classical regions of stability. In broad agreement with quantum order-by-disorder arguments, these collinear phases completely supersede the classical spiral state.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2020

Noncollinear antiferromagnetic order in the buckled honeycomb lattice of magnetoelectric Co 4 Ta 2 O 9 determined by single-crystal neutron diffraction

Co 4 Ta 2 O 9 exhibits a three-dimensional magnetic lattice based on the buckled honeycomb motif. It shows unusual magnetoelectric effects, including the sign change and nonlinearity. These effects cannot be understood without the detailed knowledge of the magnetic structure. Herein, we report neutron diffraction and direction-dependent magnetic susceptibility measurements on Co 4 Ta 2 O 9 single crystals. In this work, below 20.3 K, we find a long-range antiferromagnetic order in the alternating buckled and flat honeycomb layers of Co 2+ ions stacked along the c axis. Within experimental accuracy, the magnetic moments lie in the ab plane. They form a canted antiferromagnetic structure with a tilt angle of ~14° at 15 K in the buckled layers, while the magnetic moments in each flat layer are collinear. This is directly evidenced by a finite (0, 0, 3) magnetic Bragg peak intensity, which would be absent in the collinear magnetic order. The magnetic space group is C2/c'. It is different from the previously reported C2/c' group, also found in the isostructural Co 4 Nb 2 O 9 . The revised magnetic structure successfully explains the major features of the magnetoelectric tensor of Co 4 Ta 2 O 9 within the framework of the spin-flop model.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2020

Impact of disorder on dynamics and ordering in the honeycomb-lattice iridate Na 2 IrO 3

Kitaev's honeycomb spin-liquid model and its proposed realization in materials such as α-RuCl 3 , Li 2 IrO 3 , and Na 2 IrO 3 continue to present open questions about how the dynamics of a spin liquid are modified in the presence of non-Kitaev interactions as well as the presence of inhomogeneities. Here we use 23 Na nuclear magnetic resonance to probe both static and dynamical magnetic properties in single-crystal Na 2 IrO 3 . We find that the NMR shift follows the bulk susceptibility above 30 K but deviates from it below; moreover below T N the spectra show a broad distribution of internal magnetic fields. Both of these results provide evidence for inequivalent magnetic sites at low temperature, suggesting inhomogeneities are important for the magnetism. The spin-lattice relaxation rate is isotropic and diverges at T N , suggesting that the Kitaev cubic axes may control the critical quantum spin fluctuations. Finally, in the ordered state, we observe gapless excitations, which may arise from site substitution, emergent defects from milder disorder, or possibly be associated with nearby quantum paramagnetic states distinct from the Kitaev spin liquid.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2021

Single-ion anisotropy is necessary and appropriate to study the magnetic behavior of ${\mathrm{Tb}}^{3+}$ moments with ${J}_{\text{eff}}=\frac{1}{2}$ on the honeycomb lattice in ${\mathrm{Tb}}_{2}{\mathrm{Ir}}_{3}{\mathrm{Ga}}_{9}$

By developing models of increasing complexity, we show that a model without single-ion anisotropy (SIA) cannot explain the magnetic properties of J eff = 1 / 2 Tb 3+ moments in the orthorhombically distorted, honeycomb material Tb 2 Ir 3 Ga 9 . In four different models for the magnetization of a single honeycomb layer, the only sources of anisotropy are symmetric exchange interactions J nαβ = J nβα along three different bonds n , an anisotropic $\underline{g}$ tensor, and a Dzyalloshinskii-Moriya interaction (asymmetric exchange) that produces the observed canted moment along b . With 21 parameters, the best such model yields χ 2 = 0.065 , which is substantially smaller than χ 2 = 0.112 obtained using a Heisenberg model containing six parameters including easy-axis anisotropy. However, models without SIA fail to reproduce the linear dependence of the magnetization with a field perpendicular to the Ising axis while predicting a saturation magnetization that is far too low. Due to the complex crystal-field environments, we argue that SIA is necessary to study low-symmetry, three-dimensional J eff = 1 / 2 materials containing Tb 3+ ions.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2021

Antiferromagnetic character of the quantum phase transition in the Hubbard model on the honeycomb lattice

In this work we provide a unified, comprehensive treatment of all operators that contribute to the antiferromagnetic, ferromagnetic, and charge-density-wave structure factors and order parameters of the hexagonal Hubbard Model. We use the Hybrid Monte Carlo algorithm to perform a systematic, carefully controlled analysis in the temporal Trotter error and of the thermodynamic limit. We expect our findings to improve the consistency of Monte Carlo determinations of critical exponents. We perform a data collapse analysis and determine the critical exponent β = 0.898 (37) for the semimetal-Mott insulator transition in the hexagonal Hubbard Model. Our methods are applicable to a wide range of lattice theories of strongly correlated electrons.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2026

Decorated Honeycomb Lattice and Successive Magnetic Transitions of the Delafossite Derivative K 4 Ni 5 Te 3 O 16

Here, we report the synthesis and characterization of the delafossite derivative K 4 Ni 5 Te 3 O 16 , which hosts a decorated honeycomb Ni 2+ lattice. Single-crystal X-ray diffraction, electron microscopy, and neutron diffraction establish a noncentrosymmetric Pmm2 crystal structure that combines regular honeycomb units with elongated motifs formed by Ni 2+ trimers along extended edges. Magnetic susceptibility reveals strong antiferromagnetic interactions, while heat capacity measurements identify two successive magnetic transitions at 30 and 12 K. Neutron diffraction shows that these transitions correspond to a progression from partial to full long-range magnetic order on distinct Ni sites, reflecting competition among the various exchange interactions. The resulting magnetic point group, mm2.1′, inherits the lattice polarity and permits high-order magnetoelectric coupling. Consistent with this symmetry, a quadratic magnetic-field-induced polarization is experimentally observed below the magnetic ordering temperature, likely arising from exchange-driven magnetostriction coupled to the polar lattice. These results establish K 4 Ni 5 Te 3 O 16 as an interesting platform for engineering B-site ordering in delafossite derivatives to realize complex quantum magnetism and functional responses.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2020

Magnon damping in the zigzag phase of the Kitaev-Heisenberg- Γ model on a honeycomb lattice

We calculate dispersions and damping of the magnetic excitations in the model that is relevant to a description of the magnetic properties of iridium oxides α-Li 2 IrO 3 and Na 2 IrO 3 , and Ru-based materials such as α-RuCl 3 . Focusing on the parameter regime with the zigzag spin pattern in the ground state that is consistent with experiments, we are able to obtain explicit analytical expressions for magnon energies and eigenstates and go beyond the standard linear spin-wave theory approximation by calculating magnon damping and demonstrating its role in the dynamical structure factor. Here, we show that the magnon damping effects in both Born and self-consistent approximations are very significant, underscoring the importance of nonlinear magnon coupling in interpreting broad features in the neutron-scattering spectra.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2023

K 2 Co 2 TeO 6 : A layered magnet with a S = 1 2 Co 2 + honeycomb lattice

Recent observations of Kitaev interactions in d 7 electron configurations have generated research interest in Co 2+ based honeycomb magnets. Here, we report the synthesis of a K 2 Co 2 TeO 6 single crystal. X-ray diffraction and electron microscopy experiments show that K 2 Co 2 TeO 6 crystallizes in a $P$6 3 /$mcm$ space group with undistorted honeycomb layers of Co 2+ ions. Anisotropic magnetic and thermodynamic characterizations demonstrate an effective S = $\frac{1}{2}$ state of Co 2+ at low temperature, competing magnetic phases, and weak magnetic transitions that can be suppressed by applying fields. Moreover, the interlayer K disorder level is tunable by thermal treatments, and it affects the magnetic features obviously. In conclusion, our findings provide a new and promising platform for studying Kitaev physics.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2021

Local Spin Ice Order Induced Planar Hall Effect in Nd–Sn Artificial Honeycomb Lattice

Geometrically frustrated materials, such as spin ice or kagome lattice, are known to exhibit exotic Hall effect phenomena due to spin chirality. For the first time, Hall effect mechanism is explored in an artificial honeycomb spin ice of Nd–Sn element using Hall probe and polarized neutron reflectivity measurements. In an interesting observation, a strong enhancement in Hall signal at relatively higher temperature of T ~20 K is detected. The effect is attributed to the planar Hall effect due to magnetic moment configuration in spin ice state in low field application. In the antiferromagnetic state of neodymium at low temperature, applied field induced coupling between atomic Nd moments and conduction electrons in underlying lattice causes distinct increment in Hall resistivity at very modest field of H ~ 0.015 T. The experimental findings suggest the development of a new research vista to study the planar and the field induced Hall effects in artificial spin ice.

36 MATERIALS SCIENCE↗
DOE OSTI2020

Spin-orbit exciton in a honeycomb lattice magnet CoTiO 3 : Revealing a link between magnetism in d - and f -electron systems

We carried out inelastic neutron scattering to study the spin-orbital (SO) exciton in a single crystal sample of CoTiO 3 as a function of temperature. CoTiO 3 is a honeycomb magnet with dominant XY-type magnetic interaction and an A-type antiferromagnetic order below T N ≈ 38 K. We found that the SO exciton becomes softer, but acquires a larger bandwidth in the paramagnetic phase, compared to that in the magnetically ordered phase. Moreover, an additional mode is only observed in the intermediate temperature range, as the sample is warmed up above the lowest accessible temperature below T N . Such an unusual temperature dependence observed in this material suggests that its ground states (an S eff = 1 2 doublet) and excited states multiplets are strongly coupled, and therefore cannot be treated independently, as often done in a pseudo-spin model. Furthermore, our observations can be explained by a multi-level theory within random phase approximation that explicitly takes into account both the ground and excited multiplets. The success of our theory, which is originally developed to explain temperature dependence of magnetic excitations in the rare-earth magnets, highlight the similarity between the magnetic excitations in rare-earth systems and those in transition metal systems with strong spin orbit coupling.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2023

Magnetic behavior of manganese chloride on Kagome and honeycomb lattices

The magnetic properties of transparent pink Na 2 Mn 3 Cl 8 and NaMnCl3 are presented. For both phases, the effective magnetic moments are large, while the Curie-Weiss thetas are below 10 K. The compounds crystallize in the trigonal R-3m and R-3 space groups, respectively. A Kagome plane of Mn 2+ octahedrally coordinated to Cl- is found in Na 2 Mn 3 Cl 8 and the material has no apparent magnetic ordering above 1.8 K, while the octahedrally coordinated, honeycomb network of Mn 2+ in NaMnCl 3 orders antiferromagnetically at 6.3 ± 0.2 K. The heat capacity of NaMnCl3 has one transition, consistent with the susceptibility, while Na 2 Mn 3 Cl 8 exhibits a more complex heat capacity with two peaks, at 0.6 K and 1.4 K. A third, broad peak near 6.2 K in the heat capacity of Na 2 Mn 3 Cl 8 is problematic but may be due to the intergrowth of NaMnCl 3 domains.

36 MATERIALS SCIENCE↗
DOE OSTI2022

Two methods to study inelastic neutron-scattering measurements based on ω n (q) versus S(q, ω) applied to the magnetic open honeycomb lattice Tb 2 Ir 3 Ga 9

This work describes two methods to fit the inelastic neutron-scattering spectrum S(q, ω) with wavevector q and frequency ω. The common and well-established method extracts the experimental spin-wave branches ω n (q) from the measured spectra S(q, ω) and then minimizes the difference between the observed and predicted frequencies. When n branches of frequencies are predicted but the measured frequencies overlap to produce only m < n branches, the weighted average of the predicted frequencies must be compared to the observed frequencies. A penalty is then exacted when the width of the predicted frequencies exceeds the width of the observed frequencies. The second method directly compares the measured and predicted intensities S(q, ω) over a grid {q i , ω j } in wavevector and frequency space. After subtracting background noise from the observed intensities, the theoretical intensities are scaled by a simple wavevector-dependent function that reflects the instrumental resolution. Furthermore, the advantages and disadvantages of each approach are demonstrated by studying the open honeycomb material Tb 2 Ir 3 Ga 9 .

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗