We obtain the most general forms of rank-2 and rank-3 tensors allowed by the crystal symmetries of the honeycomb lattice of edge-sharing octahedra for crystals belonging to different crystallographic point groups, including the monoclinic point group $2/m$ and the trigonal (or rhombohedral) point group $\bar{3}$. Our results are relevant for two-dimensional materials, such as $α$-RuCl$_3$, CrI$_3$, and the honeycomb iridates. We focus on the magnetic-field-dependent thermal conductivity tensor $κ_{ij}(\mathbf{H})$, which describes a system's longitudinal and thermal Hall responses, for the cases when the magnetic field is applied along high-symmetry directions, perpendicular to the plane and in the plane. We highlight some unexpected results, such as the equality of fully-longitudinal components to partially-transverse components in rank-3 tensors for systems with three-fold rotational symmetry, and make testable predictions for the thermal conductivity tensor.
We report the crystal structure and magnetic behavior of the $4d^3$ spin-$\frac32$ silicophosphate MoP$_3$SiO$_{11}$ studied by high-resolution synchrotron x-ray diffraction, neutron diffraction, thermodynamic measurements, and ab initio band-structure calculations. Our data revise the crystallographic symmetry of this compound and establish its rhombohedral space group ($R\bar 3c$) along with the geometrically perfect honeycomb lattice of the Mo$^{3+}$ ions residing in disconnected MoO$_6$ octahedra. Long-range antiferromagnetic order with the propagation vector $\mathbf k=0$ observed below $…
We report the structures and physical properties of two new iridates, NaIrO$_3$ and Sr$_3$CaIr$_2$O$_9$, both of which contain continuous two-dimensional honeycomb connectivity. NaIrO3 is produced by room temperature oxidative deintercalation of sodium from Na$_2$IrO$_3$, and contains edge-sharing IrO6 octahedra that form a planar honeycomb lattice. Sr$_3$CaIr$_2$O$_9$, produced via conventional solid-state synthesis, hosts a buckled honeycomb lattice with novel corner-sharing connectivity between IrO6 octahedra. Both of these new compounds are comprised of Ir$^{5+}$ (5$d^4$) and exhibit negli…
We report $\textit{ab-initio}$ density functional theory calculation and Raman scattering results to explore the electronic structure of Ba$_5$CuIr$_3$O$_{12}$ single crystals. This insulating iridate, consisting of face-sharing IrO$_6$ octahedra forming quasi-one-dimensional chains, cannot be described by the local $j_{\rm eff}$=1/2 moment picture commonly adopted for discussing electronic and magnetic properties of iridate compounds with IrO$_6$ octahedra. The shorter Ir-Ir distance in the face-sharing geometry, compared to corner- or edge-sharing structures, leads to strong covalency betwee…
Theoretical studies have predicted the existence of topological magnons in honeycomb compounds with zig-zag antiferromagnetic (AFM) order. Here we report the discovery of zig-zag AFM order in the layered and non-centrosymmetric honeycomb nickelate Ni$_2$Mo$_3$O$_8$ through a combination of magnetization, specific heat, x-ray and neutron diffraction and electron paramagnetic resonance measurements. It is the first example of such order in an integer-spin non-centrosymmetric structure ($P$$_6$3$mc$). Further, each of the two distinct sites of the bipartite honeycomb lattice has a unique crystal …
We report magnetic properties of a 4f-honeycomb iodide SmI3 made up of edge-shared network of SmI6 octahedra. High temperature magnetic susceptibility indicates Γ7 Kramers doublet ground state of Sm3+ (4f5) ions stabilized by the spin-orbit coupling and octahedral crystal electric field, which interact with Sm-I-Sm bond angle nearly 90 degree. Magnetization measurements down to 0.1 K detected antiferromagnetic correlations and an anomaly in the magnetization curve before saturation without a sign of long-range order. Relevance between SmI3 and the antiferromagnetic Kitaev material proposed in …
Transition metal oxides whose lattice structure has edge-sharing network of octahedra constitute a diverse group of intriguing materials besides compounds with corner-sharing octahedra such as perovskites. We present a theoretical investigation of the interplay among spin, orbital and lattice degrees of freedom in these materials. We focus on $t_{2g}$ electron systems where a keen competition among those degrees of freedom is expected to emerge under a relatively weak Jahn-Teller coupling. We study the interplay between spin and orbital degrees of freedom in vanadium spinels and titanium pyrox…
Using ab initio wave-function-based calculations, we provide valuable insights with regard to the magnetic exchange in 5$d$ and 4$d$ oxides with face-sharing ligand octahedra, BaIrO$_3$ and BaRhO$_3$. Surprisingly strong antiferromagnetic Heisenberg interactions as large as 400 meV are computed for idealized iridate structures with 90$^{\circ}$ Ir-O-Ir bond angles and in the range of 125 meV for angles of 80$^{\circ}$ as measured experimentally in BaIrO$_3$. These estimates exceed the values derived so far for corner-sharing and edge-sharing systems and motivate more detailed experimental inve…
Specific features of orbital and spin structure of transition metal compounds in the case of the face-sharing MO$_6$ octahedra are analyzed. In this geometry, we consider the form of the spin--orbital Hamiltonian for transition metal ions with double ($e_g^σ$) or triple ($t_{2g}$) orbital degeneracy. Trigonal distortions typical of the structures with face-sharing octahedra lead to splitting of $t_{2g}$ orbitals into an $a_{1g}$ singlet and $e_g^π$ doublet. For both doublets ($e_g^σ$ and $e_g^π$), in the case of one electron or hole per site, we arrive at a symmetric model with the orbital and…
We investigated the magnetic properties of the ilmenite-type iridate CdIrO$_3$ with a honeycomb lattice formed by Ir$^{4+}$ ions prepared via a solid-state metathesis. The magnetization measurements with using the powder sample reveal a large effective magnetic moment and a fairly strong antiferromagnetic interaction, indicating a deviation from the Kitaev model. Considering the relationship between magnetism and crystal structure in CdIrO$_3$ with comparing with the other ilmenite-type iridates ZnIrO$_3$ and MgIrO$_3$, we conclude that insulating CdIrO$_3$ cannot be describe as a $J_{\rm eff}…
We consider the structure of anisotropic exchange interactions in ytterbium-based insulating rare-earth magnets built from edge-sharing octahedra. We argue the features of trivalent ytterbium and this structural configuration allow for a qualitative determination of the different anisotropic exchange regimes that may manifest in such compounds. The validity of such super-exchange calculations is tested through comparison to the well-characterized breathing pyrochlore compound Ba$_3$Yb$_2$Zn$_5$O$_{11}$. With this in hand, we then consider application to three-dimensional pyrochlore spinels as …
Ferromagnetism in the two-dimensional CrI$_3$ has generated a lot of excitement, and it was recently proposed that the spin-orbit coupling (SOC) in Iodine may generate bond-dependent spin interactions leading to magnetic anisotropy. Here we derive a microscopic spin model of S=3/2 on transition metals surrounded by heavy ligands in honeycomb Mott insulators using a strong-coupling perturbation theory. For ideal octahedra we find Heisenberg and Kitaev interactions, which favor the magnetic moment along the cubic axis via quantum fluctuations. When a slight trigonal distortion of the octahedra i…
We investigated the static and dynamic magnetic properties of the polar ferrimagnet Mn$_2$Mo$_3$O$_8$ in three magnetically ordered phases via magnetization, magnetic torque, and THz absorption spectroscopy measurements. The observed magnetic field dependence of the spin-wave resonances, including Brillouin zone-center and zone-boundary excitations, magnetization, and torque, are well described by an extended two-sublattice antiferromagnetic classical mean-field model. In this orbitally quenched system, the competing weak easy-plane and easy-axis single-ion anisotropies of the two crystallogra…
Novel magnetic ordering on the honeycomb lattice due to emergent weak anisotropic interactions generated by the mixing between the $J=1/2$ sector and the magnetically inactive 3/2 sector is investigated in a three-orbital interacting electron model in the absence of Hund's coupling. Self-consistent determination of magnetic order yields anisotropic Néel and zigzag orders for different parameter regimes, highlighting the effect of the emergent single-ion anisotropy. Study of magnon excitations shows extremely small magnon energy scale compared to the hopping energy scale, and enhancement of ani…
We report on a member of the spin-disordered honeycomb lattice antiferromagnet in a quasi-one-dimensional cobaltate Ba_3Co_2O_6(CO_3)_0.7. Resistivity exhibits as semimetallic along the face-sharing CoO6 chains. Magnetic susceptibility shows strongly anisotropic Ising-spin character with the easy axis along the chain due to significant spin-orbit coupling and a trigonal crystal field. Nevertheless, ^135Ba NMR detects no indication of the long-range magnetic order down to 0.48 K. Marginally itinerant electrons possess large entropy and low-lying excitations with a Wilson ratio R_W = 116, which …
We have combined single crystal neutron and x-ray diffractions to investigate the magnetic and crystal structures of the honeycomb lattice $\rm Na_2IrO_3$. The system orders magnetically below $18.1(2)$ K with Ir$^{4+}$ ions forming zigzag spin chains within the layered honeycomb network with ordered moment of $\rm 0.22(1) μ_B$/Ir site. Such a configuration sharply contrasts the N{é}el or stripe states proposed in the Kitaev-Heisenberg model. The structure refinement reveals that the Ir atoms form nearly ideal 2D honeycomb lattice while the $\rm IrO_6$ octahedra experience a trigonal distortio…
The Kitaev honeycomb model has been suggested as a useful starting point to describe the honeycomb iridates. However, the zigzag ordering seen in Na2IrO3 and the magnetic transition in Li2IrO3 have raised questions to their connection to the Kitaev model and to the microscopic origin of these magnetic states, given their structural similarities. Here we study how the magnetic phases near the Kitaev spin liquid are affected by the inclusion of trigonal distortion of the oxygen octahedra within a purely nearest neighbour model. Using a combination classical analysis and exact diagonalization we …
We report the revised crystal structure, static and dynamic magnetic properties of quasi-two dimensional honeycomb-lattice silver delafossite Ag3Co2SbO6. The magnetic susceptibility and specific heat data are consistent with the onset of antiferromagnetic long range order at low temperatures with Néel temperature TN ~ 21.2 K. In addition, the magnetization curves revealed a field-induced (spin-flop type) transition below TN in moderate magnetic fields. The GGA+U calculations show the importance of the orbital degrees of freedom, which maintain a hierarchy of exchange interaction in the system.…
Ba4NbMn3O12 is reported, synthesized by a solid state method in air. The crystal structure, determined by performing refinements on room temperature powder X-ray diffraction data by the Rietveld method, consists of Mn3O12 trimers in the configuration of three face-sharing MnO6 octahedra, with the trimers arranged in triangular planes. An effective moment of 4.82 μB/f.u is observed and competing antiferromagnetic and ferromagnetic interactions between Mn ions are inferred from the Weiss temperature of -4 K and the ferrimagnetic ordering transition of approximately 42 K. Ba4NbMn3O12 is a semicon…
We theoretically propose possible magnetism-induced negative thermal expansion in honeycomb-lattice antiferromagnets with edge-sharing networks of $MX_6$ octahedra where $M$ and $X$ are transition-metal and ligand ions, respectively. In this crystal structure, the nearest-neighbor exchange interaction is composed of two competing contributions, i.e., the antiferromagnetic contribution from a direct 180$^\circ$ $M$-$M$ bond and the ferromagnetic contribution from 90$^\circ$ $M$-$X$-$M$ bonds, amplitudes of which have different bond-length dependence. Numerical analysis of the spin-lattice model…
VI$_{3}$ is a ferromagnet with planar honeycomb sheets of bonded V$^{3+}$ ions held together by van der Waals forces. We apply neutron spectroscopy to measure the two dimensional ($J/J_{c} \approx 17$) magnetic excitations in the ferromagnetic phase, finding two energetically gapped ($Δ\approx k_{B} T_{c} \approx$ 55 K) and dispersive excitations. We apply a multi-level spin wave formalism to describe the spectra in terms of two coexisting domains hosting differing V$^{3+}$ orbital ground states built from contrasting distorted octahedral environments. This analysis fits a common nearest neigh…
The high temperature magnetic order in SrRu$_2$O$_6$ was studied by measuring magnetization and neutron powder diffraction with both polarized and unpolarized neutrons. SrRu$_2$O$_6$ crystallizes into the hexagonal lead antimonate (PbSb$_2$O$_6$, space group \textit{P}$\overline{3}$1\textit{m}) structure with layers of edge-sharing RuO$_6$ octahedra separated by Sr$^{2+}$ ions. SrRu$_2$O$_6$ orders at $T_N$=565\,K with Ru moments coupled antiferromagnetically both in-plane and out-of-plane. The magnetic moment is 1.30(2) $μ_\mathrm{B}$/Ru at room temperature and is along the crystallographic \…
Co$^{2+}$ ions in an octahedral crystal field, stabilise a j$_{eff}$ = 1/2 ground state with an orbital degree of freedom and have been recently put forward for realising Kitaev interactions, a prediction we have tested by investigating spin dynamics in two cobalt honeycomb lattice compounds, Na$_2$Co$_2$TeO$_6$ and Na$_3$Co$_2$SbO$_6$, using inelastic neutron scattering. We used linear spin wave theory to show that the magnetic spectra can be reproduced with a spin Hamiltonian including a dominant Kitaev nearest-neighbour interaction, weaker Heisenberg interactions up to the third neighbour a…
Antiferromagnetic and ferro/ferrimagnetic orders are typically exclusive in nature, thus, their co-existence in atomic-scale proximity is expected only in heterostructures. Breaking this paradigm and broadening the range of unconventional magnetic states, we report here on an atomic-scale hybrid spin state, which is stabilized in three-dimensional crystals of the polar antiferromagnet Co$_2$Mo$_3$O$_8$ by magnetic fields applied perpendicular to the \emph{Co} honeycomb layers and possesses a spontaneous in-plane ferromagnetic moment. Our microscopic spin model, capturing the observed field dep…
Strong spin-orbital-coupling magnetic systems with the honeycomb structure can provide bond-directional interactions which may result in Kitaev quantum spin liquids and exotic anyonic excitations. However, one of the key ingredients in real materials$-$disorders$-$has been much less studied in Kitaev systems. Here we synthesized a trigonal SrIr$_2$O$_{6-δ}$ with $δ\approx 0.25$, which consists of two-dimensional honeycomb Ir planes with edge-sharing IrO$_6$ octahedra. First-principles computation and experimental measurements suggest that the electronic system is gapped, and there should be no…
The crystal and magnetic structures and underlying magnetic interactions of Bi4Fe5O13F, a model system for studying the physics of the Cairo pentagonal spin lattice, are investigated by transmission electron microscopy, low-temperature synchrotron x-ray and neutron powder diffraction, thermodynamic measurements, and density functional band-structure calculations. The crystal structure of Bi4Fe5O13F contains infinite rutile-like chains of edge-sharing FeO6 octahedra interconnected by the Fe2O7 groups of two corner-sharing FeO4 tetrahedra. The cavities between the chains are filled with the fluo…
The Kitaev-Heisenberg (KH) model has been proposed to capture magnetic interactions in iridate Mott insulators on the honeycomb lattice. We show that analogous interactions arise in many other geometries built from edge-sharing IrO_6 octahedra, including the pyrochlore and hyperkagome lattices relevant to Ir2O4 and Na4Ir3O8 respectively. The Kitaev spin liquid exact solution does not generalize to these lattices. However, a different exactly soluble point of the honeycomb lattice KH model, obtained by a four-sublattice transformation to a ferromagnet, generalizes to all these lattices. A Klein…
We have investigated the thermodynamic and local magnetic properties of the Mott insulating system Ag$_{3}$LiRu$_{2}$O$_{6}$ containing Ru$^{4+}$ (4$d$$^{4}$) for novel magnetism. The material crystallizes in a monoclinic $C2/m$ structure with RuO$_{6}$ octahedra forming an edge-shared two-dimensional honeycomb lattice with limited stacking order along the $c$-direction. The large negative Curie-Weiss temperature ($θ_{CW}$ = -57 K) suggests antiferromagnetic interactions among Ru$^{4+}$ ions though magnetic susceptibility and heat capacity show no indication of magnetic long-range order down t…
The $t_{2g}$ orbitals of an edge-shared transition-metal oxide with a honeycomb lattice structure form dispersionless electronic bands when only hopping mediated by the edge-sharing oxygens is accessible. This is due to the formation of isolated quasimolecular orbitals (QMOs) in each hexagon, introduced recently by Mazin et al. [Phys. Rev. Lett. 109, 197201 (2012)], which stabilizes a band insulating phase for $t_{2g}^5$ systems. However, with help of the exact diagonalization method to treat the electron kinetics and correlations on an equal footing, we find that the QMOs are fragile against …
We report discovery of a new iridate family K$_x$Ir$_y$O$_2$ with depleted triangular lattice planes made up of edge sharing IrO$_6$ octahedra separated by K planes. Such a material interpolates between the triangular and honeycomb lattices and is a new playground for Kitaev physics. The materials are Mott insulators with $y = 1 - x/4$. Physical property measurements for the $x \approx 0.85$ material are reported. Using magnetic susceptibility $χ$ versus temperature $T$ measurements we find $S_{eff} = 1/2$ moments interacting strongly with a Weiss temperature $θ\approx - 180$~K and no magnetic…
We report a high-resolution neutron diffraction study on the orbitally-degenerate spin-1/2 hexagonal antiferromagnet AgNiO2. A structural transition to a tripled unit cell with expanded and contracted NiO6 octahedra indicates root(3) x root(3) charge order on the Ni triangular lattice. This suggests charge order as a possible mechanism of lifting the orbital degeneracy in the presence of charge fluctuations, as an alternative to Jahn-Teller distortions. A novel magnetic ground state is observed at base temperatures with the electron-rich S = 1 Ni sites arranged in alternating ferromagnetic row…
The structural and magnetic properties of the hexagonal four-layer form of SrMnO$_3$ have been investigated by combining magnetization measurements, electron diffraction and high-resolution synchrotron X-ray and neutron powder diffraction. Below 350K, there is subtle structural phase transition from hexagonal symmetry (space group $P6_3/mmc$) to orthorhombic symmetry (space group $C222_1$) where the hexagonal metric is preserved. The second-order phase transition involves a slight tilting of the corner-sharing Mn$_{2}$O$_{9}$ units composed of 2 face-sharing MnO$_6$ octahedra and the associate…
We show how to compute the exact partition function for lattice statistical-mechanical models whose Boltzmann weights obey a special "crossing" symmetry. The crossing symmetry equates partition functions on different trivalent graphs, allowing a transformation to a graph where the partition function is easily computed. The simplest example is counting the number of nets without ends on the honeycomb lattice, including a weight per branching. Other examples include an Ising model on the Kagome' lattice with three-spin interactions, dimers on any graph of corner-sharing triangles, and non-crossi…
We investigated the paramagnetic resonance in single crystals of LiCuVO$_4$ with special attention to the angular variation of the absorption spectrum. To explain the large resonance linewidth of the order of 1 kOe, we analyzed the anisotropic exchange interaction in the chains of edge-sharing CuO$_6$ octahedra, taking into account the ring-exchange geometry of the nearest-neighbor coupling via two symmetric rectangular Cu-O bonds. The exchange parameters, which can be estimated from theoretical considerations, nicely agree with the parameters obtained from the angular dependence of the linewi…
Two new transition-metal tellurium oxy-chlorides of the general formula Ni_5(TeO_3)_4X_2 (X=Cl, Br) have been isolated during the investigation of the ternary phase diagram NiO-NiCl2-TeO2. They crystallize in the monoclinic system, space group C2/c, and for the case of Ni_5(TeO_3)_4Cl_2 the unit cell is a=19.5674(2) A, b=5.2457(1) A, c=16.3084(1) A, with beta=125.289(1). The structure is layered and built up of corner connected [Ni_5O_17X_2] entities, made of five nickel(II) octahedra associated by edge and face sharing. The tellurium(IV) atoms are fixed upon the nickel layers. Their lone pair…
α-SrCr2O4 has a triangular planar lattice of d3 Cr3+ made from edge sharing CrO6 octahedra; the plane shows a very small orthorhombic distortion from hexagonal symmetry. With a Weiss temperature of -596 K and a three-dimensional magnetic ordering temperature of 43 K, the magnetic system is quasi two-dimensional and frustrated. Neutron powder diffraction shows that the ordered state is an incommensurate helical magnet, with an in-plane propagation vector of k = (0,0.3217(8),0). Temperature dependent synchrotron powder diffraction characterization of the structure shows an increase in the inter-…
We investigated the crystal structure of Rb$_2$Cu$_3$SnF$_{12}$ and its magnetic properties using single crystals. This compound is composed of Kagome layers of corner-sharing CuF$_{6}$ octahedra with a 2a x 2a enlarged cell as compared with the proper Kagome layer. Rb$_2$Cu$_3$SnF$_{12}$ is magnetically described as an $S$=1/2 modified Kagome antiferromagnet with four kinds of neighboring exchange interaction. From magnetic susceptibility and high-field magnetization measurements, it was found that the ground state is a disordered singlet with the spin gap, as predicted from a recent theory. …
We theoretically investigate the electronic band structures and magnetic properties of ilmenites with edge-sharing IrO$_6$ honeycomb layers, $A$IrO$_3$ with $A=$ Mg, Zn, and Mn, in comparison with a collinear antiferromagnet MnTiO$_3$. The compounds with $A=$ Mg and Zn were recently reported in Y.~Haraguchi {\it et al.}, Phys. Rev. Materials {\bf 2}, 054411 (2018), while MnIrO$_3$ has not been synthesized yet but the honeycomb stacking structure was elaborated in a superlattice with MnTiO$_3$ in K.~Miura {\it et al.}, Commun. Mater. {\bf 1}, 55 (2020). We find that, in contrast to MnTiO$_3$, w…
We report a combined experimental and theoretical study of the unusual ferromagnetism in the one-dimensional copper-iridium oxide Sr$_3$CuIrO$_6$. Utilizing Ir $L_3$ edge resonant inelastic x-ray scattering, we reveal a large gap magnetic excitation spectrum. We find that it is caused by an unusual exchange anisotropy generating mechanism, namely, strong ferromagnetic anisotropy arising from antiferromagnetic superexchange, driven by the alternating strong and weak spin-orbit coupling on the $5d$ Ir and 3d Cu magnetic ions, respectively. From symmetry consideration, this novel mechanism is gen…
We used single-crystal x-ray and neutron diffraction to investigate the crystal and magnetic structures of trigonal lattice iridate Ca2Sr2IrO6. The crystal structure is determined to be $R\bar3$ with two distinct Ir sites. The system exhibits long-range antiferromagnetic order below $T_N = 13.1$ K. The magnetic wave vector is identified as $(0,0.5,1)$ with ferromagnetic coupling along the $a$ axis and antiferromagnetic correlation along the $b$ axis. Spins align dominantly within the basal plane along the [1,2,0] direction and tilt 34$^\circ$ towards the $c$ axis. The ordered moment is 0.66(3)…
Quantum spin systems such as magnetic insulators usually show classical magnetic order, but such classical states can give way to quantum liquids with exotic entanglement through two known mechanisms of frustration: geometric frustration in lattices with triangle motifs, and spin-orbit-coupling frustration in the exactly solvable quantum liquid of Kitaev's honeycomb lattice. Here we present the experimental observation of a new kind of frustrated quantum liquid arising in an unlikely place: the magnetic insulator Ba4Ir3O10 where Ir3O12 trimers form an unfrustrated square lattice. Experimentall…
Near itinerancy and slow singlet formation in the triangular lattice NaRuO2
University of California Santa Barbara（加州大学圣塔芭芭拉分校）；Leibniz IFW Dresden（莱布尼茨德累斯顿研究所）；Argonne National Laboratory（阿贡国家实验室）；Helmholtz-Zentrum Dresden-Rossendorf（德累斯顿-罗斯多夫亥姆霍兹中心）；Deutsches Elektronen-Synchrotron DESY（德国电子同步加速器研究中心）
We determined the crystal structure of Cs$_2$LiRuCl$_6$, which was synthesized first in this work, and investigated its magnetic properties. Cs$_2$LiRuCl$_6$ has a hexagonal structure composed of linear chains of face-sharing RuCl$_6$ and LiCl$_6$ octahedra. In two-thirds of the structural chains, Ru$^{3+}$ and Li$^+$ sites are almost ordered, while in the other chains their sites are disordered. This situation is analogous to the ground state of the antiferromagnetic Ising model on a triangular lattice. Using electron paramagnetic resonance, we evaluated the $g$ factors of Ru$^{3+}$ with effe…
