Triangular Lattice System Research Articles

Competing higher order topological superconducting phases in triangular lattice magnet-superconductor hybrid systems

Competing higher order topological superconducting phases in triangular lattice magnet-superconductor hybrid systems

Signature of elementary excitations in Se-substituted layered triangular-lattice antiferromagnet CuCrS2

Geometrical spin frustration in a layered triangular-lattice system prohibits the long-range magnetic order and could give rise to unusual spin-disordered states of matter in which spins are strongly correlated and highly entangled with low-energy excitations. Here we investigate the low-temperature magnetic and specific heat properties of the layered triangular-lattice system CuCr(S1-xSex)2 within the entire Se-for-S substitution range. In magnetization measurements, the x = 0 phase exhibits a sharp cusp-like antiferromagnetic transition at 38 K, while for all the Se-substituted samples a broad maximum along with a spin-glass-like freezing at low temperatures is seen. The Curie-Weiss temperature systematically increases from −122 K for x = 0 to + 15 K for x = 1. These observations suggest that the magnetic ground state is determined by the competition between antiferromagnetic direct exchange interactions and ferromagnetic super-exchange interactions. For x = 0, the magnetic contribution to heat capacity exhibits a sharp jump at the magnetic transition temperature indicative of 3D long-range magnetic order, while for the x > 0 samples it increases smoothly across the magnetic transition range following a power-law (∼T2.1) behavior. All these results are in line with an existence of unusual gapless spin-liquid like excitations originating from the spin frustrations and competing nearest-neighbor interactions in CuCr(S,Se)2.

Thermodynamic properties of the Mott insulator-metal transition in a triangular lattice system without magnetic order

The organic system, $\kappa$-[(BEDT-TTF)$_{1-x}$(BEDT-STF)$_x$]$_2$Cu$_2$(CN)$_3$, showing the Mott transition between a nonmagnetic Mott insulating (NMI) state and a Fermi liquid (FL), is systematically studied by calorimetric measurements. An increase of the electronic heat capacity at the transition from the NMI state to the FL state which keeps the triangular dimer lattice demonstrates that the charge sector lost in the Mott insulating state is recovered in the FL state. We observed that the remaining low-energy spin excitations in the Mott insulating state show unique temperature dependence, and that the NMI state has a larger lattice entropy originating from the frustrated lattice, which leads to the Pomeranchuk-like effect on the electron localization. Near the Mott boundary, an unexpected enhancement and magnetic-field dependence of heat capacity are observed. This anomalous heat capacity is different from the behavior in the typical first-order Mott transition and shows similarities with quantum critical behavior. To reconcile our results with previously reported scenarios about a spin gap and the first-order Mott transition, further studies are desired.

Synthesis, crystal structure, and physical properties of a triangular-lattice magnet K2Co(HPO4)(C2O4)(OH2)⋅H2O

A previously unreported geometrically frustrating cobalt oxalatophosphate, K2Co(HPO4)(C2O4)(OH2)⋅H2O (KCoCP), is solvothermally synthesized and characterized. KCoCP shows a typical two-dimensional (2D) layered structure of [Co(HPO4)(C2O4)(OH2)]2− separated by K+ and H2O, which is isostructural to the previously reported Fe analogue. Specifically, the magnetic Co2+ ions sit on a distorted triangular lattice (TL) and introduce geometrical frustration into the system. The mainstay of the 2D layer [Co(HPO4)(C2O4)(OH2)]2− is cobalt oxalate skeleton with HPO4 and H2O groups staggering on both sides. The Bravais–Friedel–Donnay–Harker (BFDH) theory, which is used to predict the polyhedral morphology and crystal growth faces, indicates a chamfer polyhedron morphology of KCoCP crystal. UV–vis diffuse reflection and electronic structure calculation suggest KCoCP being a semiconductor with an indirect band gap (0.214 ​eV). The absence of long-range magnetic ordering for KCoCP is proved via a combination of magnetic susceptibility and specific-heat measurements down to 1.8 ​K. Thus, KCoCP shows potential to investigate unconventional low temperature physical properties in TL system.

Dimensional reduction and incommensurate dynamic correlations in the S=frac{1}{2} triangular-lattice antiferromagnet Ca3ReO5Cl2

The observation of spinon excitations in the S=frac{1}{2} triangular antiferromagnet Ca3ReO5Cl2 reveals a quasi-one-dimensional (1D) nature of magnetic correlations, in spite of the nominally 2D magnetic structure. This phenomenon is known as frustration-induced dimensional reduction. Here, we present high-field electron spin resonance spectroscopy and magnetization studies of Ca3ReO5Cl2, allowing us not only to refine spin-Hamiltonian parameters, but also to investigate peculiarities of its low-energy spin dynamics. We argue that the presence of the uniform Dzyaloshinskii-Moriya interaction (DMI) shifts the spinon continuum in momentum space and, as a result, opens a zero-field gap at the Γ point. We observed this gap directly. The shift is found to be consistent with the structural modulation in the ordered state, suggesting this material as a perfect model triangular-lattice system, where a pure DMI-spiral ground state can be realized.

Engineering a Skyrmion Crystal in Ferromagnetic/Antiferromagnetic Bilayers Based on Magnetic Frustration Mechanism

We theoretically propose a new stabilization mechanism of a skyrmion crystal (SkX) in a bilayer triangular lattice system consisting of the ferromagnetic and the antiferromagnetic layers. By performing variational calculations and Monte Carlo simulations in a complementary way, we find that a magnetic frustration between the ferromagnetic and antiferromagnetic layers is a source of a finite-Q spiral state and the SkX in the strong interlayer coupling regime. We also show that the degree of frustration is related to the interlayer exchange interaction. The stronger interlayer coupling tends to make the effect of frustration larger, which results in the stabilization of the SkX. The present results not only provide a way of engineering the SkX in the ferromagnetic/antiferromagnetic domain and heterostructure but also imply the possibility of the SkX based on interorbital frustration scenario.

Absence of spin susceptibility decrease in a bulk organic superconductor with triangular lattice

The study of non-$s$-wave unconventional superconductivities in strongly correlated electron systems has been a central issue in condensed matter physics for more than 30 years. In such unconventional superconductivities, $d$-wave Cooper pairing with antiparallel spins has been often observed in various quasi-two-dimensional (quasi-2D) bulk systems. Interestingly, many theories predicted that the triangular lattice causes the $d$-wave pairing to be unstable and may lead to more exotic pairing such as parallel spin (spin-triplet) pairing. Here we focus on a bulk organic triangular-lattice system in which superconductivity emerges near a nonmagnetic Mott insulating phase. We demonstrate, by using low-power nuclear magnetic resonance (NMR) measurements, that the spin susceptibility of the superconducting state retains the normal state value even deep in the superconducting state. This result indicates the possibility that the material exhibits spin-triplet superconductivity. Our finding will bring insights also into understanding the 2D materials with triangular moir\'e superlattices that are considered also to show unconventional superconductivities near Mott-like insulating states.

Imaginary-time hierarchical equations of motion for thermodynamic variables.

The partition function (PF) plays a key role in the calculation of quantum thermodynamic properties of a system that interacts with a heat bath. The imaginary-time hierarchical equations of motion (imHEOM) approach was developed to evaluate in a rigorous manner the PF of a system strongly coupled to a non-Markovian bath. In this paper, we present a numerically efficient scheme to evaluate the imHEOM utilizing the β-differentiated imHEOM (BD-imHEOM) that are obtained by differentiating the elements of the imHEOM with respect to the inverse temperature. This approach allows us to evaluate the system, system-bath interaction, and heat-bath parts of the PF efficiently. Moreover, we employ a polyharmonic decomposition method to construct a concise hierarchical structure with better convergence, thus reducing the cost of numerical integrations. We demonstrate the proposed approach by compute thermodynamic quantities of a spin-boson system and a 2 × 2 antiferromagnetic triangular spin lattice system with an Ohmic spectral distribution.

Non-Fermi liquid behavior below the Néel temperature in the frustrated heavy fermion magnet UAu2

The term Fermi liquid is almost synonymous with the metallic state. The association is known to break down at quantum critical points (QCPs), but these require precise values of tuning parameters, such as pressure and applied magnetic field, to exactly suppress a continuous phase transition temperature to the absolute zero. Three-dimensional non-Fermi liquid states, apart from superconductivity, that are unshackled from a QCP are much rarer and are not currently well understood. Here, we report that the triangular lattice system uranium diauride (UAu2) forms such a state with a non-Fermi liquid low-temperature heat capacity [Formula: see text] and electrical resistivity [Formula: see text] far below its Néel temperature. The magnetic order itself has a novel structure and is accompanied by weak charge modulation that is not simply due to magnetostriction. The charge modulation continues to grow in amplitude with decreasing temperature, suggesting that charge degrees of freedom play an important role in the non-Fermi liquid behavior. In contrast with QCPs, the heat capacity and resistivity we find are unusually resilient in magnetic field. Our results suggest that a combination of magnetic frustration and Kondo physics may result in the emergence of this novel state.

Magnetic properties of the quasi two-dimensional centered honeycomb antiferromagnet GdInO3

The crystal structure and magnetic property of the single crystalline hexagonal rare-earth indium oxides ${\mathrm{GdInO}}_{3}$ have been studied by combing experiments and model calculations. The two inequivalent ${\mathrm{Gd}}^{3+}$ ions form the centered honeycomb lattice, which consists of honeycomb and triangular sublattices. The dc magnetic susceptibility and specific heat measurements suggest two antiferromagnetic phase transitions at ${T}_{\text{N1}}=2.3\phantom{\rule{0.28em}{0ex}}\mathrm{K}$ and ${T}_{\text{N2}}=1.02\phantom{\rule{0.28em}{0ex}}\mathrm{K}$. An inflection point is observed in the isothermal magnetization curve, which can be an indication of an up-up-down phase with a 1/3 magnetization plateau, further supported by our theoretical calculation. We also observe a large magnetic entropy change originated from the magnetic frustration in ${\mathrm{GdInO}}_{3}$. By considering a classical spin Hamiltonian, we establish the ground state phase diagram, which suggests that ${\mathrm{GdInO}}_{3}$ has a weak easy-axis anisotropy and is close to the equilateral triangular-lattice system. The theoretical ground-state phase diagram may be used as a reference in NMR, ESR, or $\ensuremath{\mu}\mathrm{SR}$ experiments in future.

Fingerprinting triangular-lattice antiferromagnet by excitation gaps

CeCd$_3$As$_3$ is a rare-earth triangular-lattice antiferromagnet with large inter-layer separation. Our field-dependent heat capacity measurements at dilution fridge temperatures allow us to trace the field-evolution of the spin-excitation gaps throughout the antiferromagnetic and paramagnetic regions. The distinct gap evolution places strong constraints on the microscopic pseudo-spin model, which, in return, yields a close {\it quantitative} description of the gap behavior. This analysis provides crucial insights into the nature of the magnetic state of CeCd$_3$As$_3$, with a certainty regarding its stripe order and low-energy model parameters that sets a compelling paradigm for exploring and understanding the rapidly growing family of the rare-earth-based triangular-lattice systems.

Noncoplanar multiple- Q spin textures by itinerant frustration: Effects of single-ion anisotropy and bond-dependent anisotropy

We theoretically investigate multiple-$Q$ spin textures, which are composed of superpositions of spin density waves with different wave numbers, for an effective spin model of centrosymmetric itinerant magnets. Our focus is on the interplay between biquadratic interactions arising from the spin-charge coupling and magnetic anisotropy caused by the spin-orbit coupling. Taking into account two types of the magnetic anisotropy, single-ion anisotropy and bond-dependent anisotropy, we elucidate magnetic phase diagrams for an archetypal triangular-lattice system in the absence and presence of an external magnetic field. In the case of the single-ion anisotropy, we find a plethora of multiple-$Q$ instabilities depending on the strength and the sign of the anisotropy (easy plane or easy axis), including a noncoplanar triple-$Q$ state regarded as a skyrmion crystal with topological number of two, and coplanar and noncoplanar double-$Q$ states. In an external magnetic field, we find that another noncoplanar triple-$Q$ state, a skyrmion crystal with topological number of one, is stabilized by the in-plane (out-of-plane) magnetic field under the easy-plane (easy-axis) anisotropy. A part of the results, especially for the relatively large biquadratic interaction, qualitatively reproduce those in the Kondo lattice model, which explicitly includes itinerant electrons [S. Hayami and Y. Motome, Phys. Rev. B 99, 094420 (2019)]. We also examine the stability of the field-induced skyrmion crystal by rotating the field direction. As a by-product, we show that a different triple-$Q$ state with nonzero chirality appears in the presence of the biquadratic interaction and the easy-axis anisotropy. Meanwhile, we find that the bond-dependent anisotropy also stabilizes both types of skyrmion crystals. We show that, however, for the skyrmion crystal with topological number of one, Bloch- and N\'eel-type skyrmion crystals are selectively realized depending on the sign of the bond-dependent anisotropy, since this anisotropy selects a particular set of the helicity and vorticity. Moreover, we find yet another multiple-$Q$ states with nonzero spin scalar chirality, including two types of meron crystals with the skyrmion numbers of one and two. The systematic investigation of multiple-$Q$ instabilities in triangular itinerant magnets will provide a reference to complex magnetic textures in centrosymmetric magnetic metals.

Existence of second spectrums of Timoshenko beam and Mindlin–Herrmann rod theories on the basis of atomistic studies

This paper considers a single-layer 2D triangular lattice system as an ideal higher-order rod and/or beam. The objective is to compare the solutions of these continuum models with the standard atomistic solutions. The accuracy of these higher-order rod and beam theories, in modeling several nano-systems, depends on the frequency of vibration. The existence of a second spectrum associated with these higher-order continuum theories and its contribution to the continuum solutions are studied on the basis of molecular dynamics (MD) results. It is shown that the second spectrum of a Timoshenko beam is less accurate with respect to the MD results but cannot be disregarded totally. Also, the existence of a second propagating mode associated with a Mindlin–Herrmann rod is substantiated by MD simulation results. The accuracy of these higher order rod and beam theories deteriorates beyond their cut-on frequencies. However, the use of higher-order rod and beam theories in several high-frequency dynamic and multiscale analyses is still justifiable.

Superconductivity in a new layered triangular-lattice system Li2IrSi2

We report on the crystal structure and superconducting properties of a novel iridium-silicide, namely Li2IrSi2. It has a Ag2NiO2-type structure (space group R-3m) with the lattice parameters a = 4.028 30(6) Å and c = 13.161 80(15) Å. The crystal structure comprises IrSi2 and double Li layers stacked alternately along the c-axis. The IrSi2 layer includes a two-dimensional Ir equilateral-triangular lattice. Electrical resistivity and static magnetic measurements revealed that Li2IrSi2 is a type-II superconductor with critical temperature (Tc) of 3.3 K. We estimated the following superconducting parameters: lower critical field Hc1(0) ∼ 42 Oe, upper critical field Hc2(0) ∼ 1.7 kOe, penetration depth λ0 ∼ 265 nm, coherence length ξ0 ∼ 44 nm, and Ginzburg–Landau parameter κGL ∼ 6.02. The specific-heat data suggested that superconductivity in Li2IrSi2 could be attributed to weak-coupling Cooper pairs.

Torque equilibrium spin wave theory study of anisotropy and Dzyaloshinskii-Moriya interaction effects on the indirect K -edge RIXS spectrum of a triangular lattice antiferromagnet

We apply the recently formulated torque equilibrium spin wave theory (TESWT) to compute the $1/S$-order interacting $K$ -edge bimagnon resonant inelastic x-ray scattering (RIXS) spectra of an anisotropic triangular lattice antiferromagnet with Dzyaloshinskii-Moriya (DM) interaction. We extend the interacting torque equilibrium formalism, incorporating the effects of DM interaction, to appropriately account for the zero-point quantum fluctuation that manifests as the emergence of spin Casimir effect in a noncollinear spin spiral state. Using inelastic neutron scattering data from Cs$_2$CuCl$_4$ we fit the 1/S corrected TESWT dispersion to extract exchange and DM interaction parameters. We use these new fit coefficients alongside other relevant model parameters to investigate, compare, and contrast the effects of spatial anisotropy and DM interaction on the RIXS spectra at various points across the magnetic Brillouin zone. We highlight the key features of the bi- and trimagnon RIXS spectrum at the two inequivalent rotonlike points, $M(0,2 \pi/\sqrt{3})$ and $M^{\prime}(\pi,\pi/\sqrt{3})$, whose behavior is quite different from an isotropic triangular lattice system. While the roton RIXS spectrum at the $M$ point undergoes a spectral downshift with increasing anisotropy, the peak at the $M^\prime$ location loses its spectral strength without any shift. With the inclusion of DM interaction the spiral phase is more stable and the peak at both $M$ and $M^\prime$ point exhibits a spectral upshift. Our calculation offers a practical example of how to calculate interacting RIXS spectra in a non-collinear quantum magnet using TESWT. Our findings provide an opportunity to experimentally test the predictions of interacting TESWT formalism using RIXS, a spectroscopic method currently in vogue.

Charge Order and Poor Glass-forming Ability of an Anisotropic Triangular-lattice System, θ-(BEDT-TTF)2TlCo(SCN)4, Investigated by NMR

The family of layered organic conductors, θ-(BEDT-TTF)2MM′(SCN)4, are anisotropic triangular-lattice systems with quarter-filled bands and exhibit charge order instability or glassy charge freezing...

Magnetic-field-induced incommensurate to collinear spin order transition in NiBr2

The triangular spin lattice of NiBr2 is a canonical example of the frustrated helimagnet that shows a collinear commensurate antiferromagnetic to an incommensurate spin helix phase transition on cooling. Herein, we have studied a self-flux grown NiBr2 single crystal by neutron diffraction and low temperature magnetization measurements at fields up to 14 T. Experimental findings enable the deduction of the driving force responsible for the spin spiral ordering. The neutron diffraction data reveal satellite peaks representing characteristic features of an incommensurate magnetic state. The satellites develop symmetrically below TN=44.0(1)K, replacing the main magnetic reflections. Interestingly, a field-induced incommensurate to commensurate spin phase transition has been successfully demonstrated, which enforces the spin helix to restore the high temperature compensated antiferromagnetic structure. This reorientation can be described by a spin-flop in the (a–b) basal plane of a triangular spin lattice system. The findings offer a novel way for spin helix control of incommensurate phases, having immense scientific and technological implications in the next-generation data storage devices.

Exotic Magnetic Field-Induced Spin-Superstructures in a Mixed Honeycomb-Triangular Lattice System

The temperature-magnetic-field phase diagram of the mixed honeycomb triangular lattice system K$_{2}$Mn$_{3}$(VO$_{4}$)$_{2}$CO$_{3}$ is investigated by means of magnetization, heat capacity and neutron scattering measurements. The results indicate that triangular and honeycomb magnetic layers undergo sequential magnetic orderings and act as nearly independent magnetic sublattices. The honeycomb sublattice orders at about 85 K in a Ne\'{e}l-type antiferromagnetic structure, while the triangular sublattice displays two consecutive ordered states at much lower temperatures, 3 K and 2.2 K. The ground state of the triangular sublattice consists of a planar `Y' magnetic structure that emerges from an intermediate collinear `up-up-down' state. Applied magnetic fields parallel or perpendicular to the $c$-axis induce exotic ordered phases characterized by various spin-stacking sequences of triangular layers that yield bilayer, three-layer or four-layer magnetic superstructures. The observed superstructures cannot be explained in the framework of quasi-classical theory based only on nearest-neighbor interlayer coupling and point towards the presence of effective second-nearest-neighbor interactions mediated by fluctuations of the magnetic moments in the honeycomb sublattice.

Quantum magnetisms in uniform triangular lattices Li2AMo3O8 (A\u2009=\u2009In, Sc)

Molecular based spin-1/2 triangular lattice systems such as LiZn2Mo3O8 have attracted research interest. Distortions, defects, and intersite disorder are suppressed in such molecular-based magnets, and intrinsic geometrical frustration gives rise to unconventional and unexpected ground states. Li2AMo3O8 (A = In or Sc) is such a compound where spin-1/2 Mo3O13 clusters in place of Mo ions form the uniform triangular lattice. Their ground states are different according to the A site. Li2InMo3O8 undergoes conventional 120° long-range magnetic order below TN = 12 K whereas isomorphic Li2ScMo3O8 exhibits no long-range magnetic order down to 0.5 K. Here, we report exotic magnetisms in Li2InMo3O8 and Li2ScMo3O8 investigated by muon spin rotation (μSR) and inelastic neutron scattering (INS) spectroscopies using polycrystalline samples. Li2InMo3O8 and Li2ScMo3O8 show completely different behaviors observed in both μSR and INS measurements, representing their different ground states. Li2InMo3O8 exhibits spin wave excitation which is quantitatively described by the nearest neighbor anisotropic Heisenberg model based on the 120° spin structure. In contrast, Li2ScMo3O8 undergoes short-range magnetic order below 4 K with quantum-spin-liquid-like magnetic fluctuations down to the base temperature. Origin of the different ground states is discussed in terms of anisotropies of crystal structures and magnetic interactions.

Frustration and anisotropic exchange in ytterbium magnets with edge-shared octahedra

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 well as two-dimensional honeycomb and triangular lattice systems built from such edge-sharing octahedra. We find an extended regime of robust emergent weak anisotropy with dominant antiferromagnetic Heisenberg interactions as well as smaller regions with strong anisotropy. We discuss the implications of our results for known compounds with the above structures, such as the spinels AYb$_2$X$_4$ (A = Cd, Mg, X = S, Se), the triangular compound YbMgGaO$_4$, which have recently emerged as promising candidates for observing unconventional magnetic phenomena. Finally, we speculate on implications for the R$_2$M$_2$O$_7$ pyrochlore compounds and some little studied honeycomb ytterbium magnets.