Flat Band Structure Research Articles

Tunable Ambipolar Transport in a 2D Kagome Semiconductor

AbstractThe interference of electronic wavefunctions within a Kagome lattice can result in a flat band structure, where low‐energy electrons exhibit highly correlated behavior, facilitating exploration of interactions among intriguing electronic states. However, the inherent high carrier density in most known Kagome materials often hampers the manipulation of these quantum phenomena through conventional means, such as gate voltage. In this work, a unique tunability in the electrical and optoelectronic properties of exfoliated Nb3I8, a 2D semiconductor featuring a “breathing” Kagome lattice is uncovered. Characterization via temperature‐dependent ARPES confirms its semiconducting nature with a flat band structure. Experimental results from bias and gate voltage dependent transport measurements on thin film Nb3I8 transistor devices demonstrate ambipolar conductance. Notably, these findings reveal a broadband photoresponse in the device, spanning from visible to near infrared wavelengths (532 −1100 nm), with the photocurrent showing gate‐dependent characteristics that mirror the dark current's ambipolar nature. This observation marks the first instance of ambipolar transport in a Kagome semiconductor, opening up exciting new avenues for electronic and optoelectronic device development.

Manipulation of high-fidelity sidebands under Large detuning by Floquet technology: application to Multi-Mode Cooling

Abstract The Floquet technology, a powerful way to manipulate quantum states, is employed to drive sidebands transition under large detuning. Our results demonstrate that high fidelities over 99% can be achieved through optimizing suitable modulation frequencies under large detuning. We observe high-fidelity transitions within a high bandwidth by utilizing a single modulation frequency and reveal this capability is due to the emergence of a flat-band structure in the bandwidth range. The key finding of high-fidelity sideband manipulation under large detuning is experimentally confirmed in nuclear magnetic resonance platform. Finally, we propose a new parallel sideband cooling scheme that enables simultaneous cooling of multiple motional modes. This approach improves the cooling rate compared to conventional schemes with fixed laser frequency and power, and eliminates the need for mode-specific addressing. Our Floquet parallel scheme is applicable to any harmonic oscillator system and is not limited by bandwidth in theory.

Realizing multiorbital Hubbard model with commensurate fillings in the paramagnetic phase of triangular magnets (X = Mn, Co, Ni)

The multiorbital Hubbard model is renowned for its rich phase diagram, characterized by various filling-dependent Mott transitions. However, controlling electron filling to explore the complete phase diagram within a single material remains challenging. In this study, we investigate three members of a correlated magnet family: Na2BaX(PO4)2 (where X = Mn, Co, Ni). Our focus is on revealing their unusual electronic structure related to the different electron fillings. Despite differences in their effective angular momentum, our first-principles calculations reveal a strikingly similar flat-band electronic structure across all three systems. Remarkably, we found that their distinct valence configurations lead to a similar response to electronic correlations in the high-temperature paramagnetic phases, which nicely realizes Mott states with different electron concentrations. Thus, these systems can be conceptualized as multiorbital Hubbard models with differing electron fillings. Our work unifies the understanding of these structurally similar systems and opens new avenues for exploring correlated electronic structures with distinct fillings.

Interorbital antisymmetric hopping generated flat bands on kagome and pyrochlore Lattices

Flat bands are intriguing platforms for correlated and topological physics. Various methods have been developed to create flat bands utilizing lattice geometry, but the investigation of orbital symmetry in multiorbital materials is a new area of focus. Here, we introduce a site symmetry-based approach to emerging multiorbital 2D and 3D flat bands on the kagome and pyrochlore lattices. As a conceptual advance, the one-orbital flat bands are shown to originate as mutual eigenstates of isolated molecular motifs. Further developing the mutual eigenstate method for multiple orbitals transforming differently under the site symmetries, we derive interorbital hopping generated flat bands from the antisymmetric interorbital Hamiltonian and introduce group-theoretic descriptions of the flat band wavefunctions. Realizations of multiorbital flat bands in realistic materials are shown to be possible in the Slater-Koster formalism. Our findings provide new directions for exploring flat-band electronic structures for novel correlated and topological quantum states.

Fermi Rubik’s Cube in High-pressure Induced Chlorine-riched Compounds

Abstract In the quasi-free electron model, which is widely used in modern solid-state physics, the Fermi surface spreads into a sphere in the Brillouin zone, i.e., the Fermi Sphere. The Fermi Sphere exists widely in metal systems, no matter whether the crystal is in a body-center cubic, face-center cubic, or hexagonal close-packed lattice. Here, we report a class of compounds stabilized at high pressure with a Rubik's cubic Fermi surface, in which the representative example is P m-3n-CaCl3. Our quantum-mechanical variable- composition evolutionary simulations predicted the thermal stabilities of CaCl3, and the tight-binding model revealed its unique Fermi surface originates from the quasi-one- dimensional interaction, structural symmetric protection, and particle-hole symmetry breaking. Furthermore, by its flat and steep band structure, CaCl3 has a huge span of effective mass from 9.08×103 me (super-heavy) to 5.13×10-4 me on the Fermi level, which supplies an interesting platform for quasiparticle research.

Superconductivity in pressurized trilayer La4Ni3O10-δ single crystals.

The pursuit of discovering new high-temperature superconductors that diverge from the copper-based model1-3 has profound implications for explaining mechanisms behind superconductivity and may also enable new applications4-8. Here our investigation shows that the application of pressure effectively suppresses the spin-charge order in trilayer nickelate La4Ni3O10-δ single crystals, leading to the emergence of superconductivity with a maximum critical temperature (Tc) of around 30 K at 69.0 GPa. The d.c. susceptibility measurements confirm a substantial diamagnetic response below Tc, indicating the presence of bulk superconductivity with a volume fraction exceeding 80%. In the normal state, we observe a strange metal behaviour, characterized by a linear temperature-dependent resistance extending up to 300 K. Furthermore, the layer-dependent superconductivity observed hints at a unique interlayer coupling mechanism specific to nickelates, setting them apart from cuprates in this regard. Our findings provide crucial insights into the fundamental mechanisms underpinning superconductivity, while also introducing a new material platform to explore the intricate interplay between the spin-charge order, flat band structures, interlayer coupling, strange metal behaviour and high-temperature superconductivity.

Hydrogenation-controlled band engineering of dumbbell graphene

The stability of the dumbbell structure has been confirmed by previous theory and experiment. Based on first-principles calculations, we proposed hexagonal dumbbell graphene (HDB C10) and rectangular dumbbell graphene (RDB C10) monolayers containing periodically raised C (CR) atoms. They turn out to have high mobility semiconductor properties. By adsorbing H atoms on these CR atoms, their band structures can be widely tuned from semiconductor to semimetal. When considering adsorption of two/four H atoms on the unit cell of the dumbbell structure, the bandgap can be increased, and isolated flat band structures can be obtained by further adding or removing H atoms. Remarkably, two different Dirac band structures can be found in the HDB/RDB C10H2-I monolayers. The HDB C10H2-I shows a Dirac cone with isotropic Fermi velocities, while the RDB C10H2-I monolayer exhibits a quasi-one-dimensional Dirac nodal line with varying Fermi velocities along the XS path. Tight-binding (TB) models are constructed including nearest neighbor (NN) and next NN hopping in order to understand our DFT results. These TB models are related to the Su-Schrieffer-Heeger model, and are able to explain the tunable topological properties of the RDB C10H2-I monolayer. They not only are able to explain the different kinds of Fermi velocity, but also can predict the emergence of topological edge states, providing a good platform for research on Dirac fermions. The HDB/RDB C10 monolayer exhibits more freedom of tunable band structures and more stable hydrogen storage capacity, making it superior to graphene. Finally, possible experimental synthesis paths of these DB monolayers are provided.

Excitonic Thermalization Bottleneck in Twisted TMD Heterostructures.

Twisted van der Waals heterostructures show intriguing interface exciton physics, including hybridization effects and emergence of moiré potentials. Recent experiments have revealed that moiré-trapped excitons exhibit remarkable dynamics, where excited states show lifetimes that are several orders of magnitude longer than in monolayers. The origin of this behavior is still under debate. Based on a microscopic many-particle approach, we investigate the phonon-driven relaxation cascade of nonequilibrium moiré excitons in the exemplary MoSe2-WSe2 heterostructure. We track exciton relaxation pathways across different moiré mini-bands and identify the phonon-scattering channels assisting the spatial redistribution of excitons into low-energy pockets of the moiré potential. We unravel a phonon bottleneck in the flat band structure at low twist angles preventing excitons from fully thermalizing into the lowest state, explaining the measured enhanced emission intensity and lifetime of excited moiré excitons. Overall, our work provides important insights into exciton relaxation dynamics in flat-band exciton materials.

Tuning the Flat Band in Bi2O2Se by Pressure to Induce Superconductivity.

The discovery of superconductivity in twisted bilayer graphene has reignited enthusiasm in the field of flat-band superconductivity. However, important challenges remain, such as constructing a flat-band structure and inducing a superconducting state in materials. Here, we successfully achieved superconductivity in Bi2O2Se by pressure-tuning the flat-band electronic structure. Experimental measurements combined with theoretical calculations reveal that the occurrence of pressure-induced superconductivity at 30 GPa is associated with a flat-band electronic structure near the Fermi level. Moreover, in Bi2O2Se, a van Hove singularity is observed at the Fermi level alongside pronounced Fermi surface nesting. These remarkable features play a crucial role in promoting strong electron-phonon interactions, thus potentially enhancing the superconducting properties of the material. These findings demonstrate that pressure offers a potential experimental strategy for precisely tuning the flat band and achieving superconductivity.

Compact gap solitons and compact edge states of exciton–polariton condensates with spin–orbit coupling in a one-dimensional flatband lattice

We propose that the compact gap solitons and compact edge states can be excited in a flatband of the incoherently-pumped exciton–polariton condensate under a one-dimensional periodic potential lattice. The combined effects of spin–orbit coupling and periodic potential depth on the flatband structures are investigated. Then how the compact gap solitons and edge states are localized and extended inside only a fraction of a single lattice site will be studied with varying pump strengths, pump spot-sizes as well as energy detuning.

Topological phase in one-dimensional momentum space lattice of ultracold atoms without chiral symmetry

Symmetry plays a crucial role in understanding topological phases in materials. In one-dimensional systems, such as the Su-Schrieffer-Heeger (SSH) model, chiral symmetry is thought to ensure the quantization of the Zak phase and the nontrivial topological phase. However, our work demonstrates that the one-dimensional lattice system with broken chiral symmetry can still possess quantized Zak phase and nontrivial topological phase. Specifically, we use a Bose-Einstein condensate of <sup>87</sup>Rb atoms in a momentum space lattice of ultracold atoms to effectively simulate a one-dimensional Zigzag model of 26 sites, which intrinsically breaks the chiral symmetry by additional next-nearest-neighbor coupling. To ensure the existence of the nontrivial topological phase, where the Zak phase can be measured from the time-averaged displacement during the system’s evolution, we need to preserve the inversion symmetry by modulating laser power so that all next-nearest-neighbor coupling strengths are equal. Furthermore, by changing the ratio of nearest-neighbor coupling strengths, we observe a topological phase transition from a nontrivial topological phase to a trivial topological phase at the point where the ratio equals 1. Our work demonstrates that the ultracold atom system provides a controllable platform for studying the symmetrical phase and topological phase, with the potential to explore nonlinear topological phenomena by increasing the interactions among atoms. In addition, our system can be used to investigate other interesting topological phenomena with more complex models, such as critical phenomena at the phase transitions and flat band structures in the extended SSH model with long-range coupling. By controlling the coupling strengths, we can also explore the influence of different symmetries on the topological properties of extended SSH models in the future. Moreover, our platform makes it possible to studythe models with more energy bands, such as the Aharonov-Bohm caging model with a three-level structure, which shows peculiar flat-band properties. This work provides opportunities for various studies in the fields of symmetry, topology, and the interaction of controllable quantum systems.

Atomic and electronic structure of monolayer ferroelectric GeS on Cu(111)

Two-dimensional (2D) ferroelectric materials are important materials for both fundamental properties and potential applications. Especially, group Ⅳ monochalcogenide possesses highest thermoelectric performance and intrinsic ferroelectric polarization properties and can sever as a model to explore ferroelectric polarization properties. However, due to the relatively large exfoliation energy, the creation of high-quality and large-size monolayer group Ⅳ monochalcogenide is not so easy, which seriously hinders the integration of these materials into the fast-developing field of 2D materials and their heterostructures. Herein, monolayer GeS is successfully fabricated on Cu(111) substrate by molecular beam epitaxy method, and the lattice structure and the electronic band structure of monolayer GeS are systematically characterized by high-resolution scanning tunneling microscopy, low-energy electron diffraction, <i>in-situ</i> X-ray photoelectron spectroscopy, Raman spectra, and angle-resolved photoelectron spectroscopy, and density functional theory calculations. All atomically resolved STM images reveal that the obtained monolayer GeS has an orthogonal lattice structure, which consists with theoretical prediction. Meanwhile, the distinct moiré pattern formed between monolayer GeS and Cu(111) substrate also confirms the orthogonal lattice structure. In order to examine the chemical composition and valence state of as-prepared monolayer GeS, <i>in-situ</i> XPS is utilized without being exposed to air. The measured spectra of XPS core levels suggest that the valence states of Ge and S elements are identified to be +2 and –2, respectively and the atomic ratio of Ge/S is 1∶1.5, which is extremely close to the stoichiometric ratio of 1∶1 for GeS. To further corroborate the quality and lattice structure of the monolayer GeS film, <i>ex-situ</i> Raman measurements are also performed for monolayer GeS on highly oriented pyrolytic graphene (HOPG) and multilayer graphene substrate. Three well-defined typical characteristic Raman peaks of GeS are observed. Finally, <i>in-situ</i> ARPES measurement are conducted to determine the electronic band structure of monolayer GeS on Cu(111). The results demonstrate that the monolayer GeS has a nearly flat band electronic band structure, consistent with our density functional theory calculation. The realization and investigation of the monolayer GeS extend the scope of 2D ferroelectric materials and make it possible to prepare high quality and large size monolayer group Ⅳ monochalcogenides, which is beneficial to the application of this main group material to the rapidly developing 2D ferroelectric materials and heterojunction research.

Influence of Building Block Symmetry on the Band Structure of Stacked 2D Polyimide Covalent Organic Frameworks.

Covalent organic frameworks (COFs) are a class of crystalline porous materials distinctively built solely from organic elements, carbon, oxygen, hydrogen, and often nitrogen or boron. They form light, mechanically rigid, and chemically stable networks that have many advantages, but their low solubility and poor processability create issues with developing large-scale films or membranes. Two-dimensional (2D) COFs possess periodic porous crystallinity, functionality, modularity, and layered one-dimensional (1D) transport channels. All of these traits, along with the semiconducting properties of selected COFs, make them interesting candidates for integration in optoelectronic devices. Therefore, it is still a challenge to explore computationally and structurally the semiconductivity of COFs and to determine their final potential. Herein, we report on the possible semiconducting properties and results of polyimide-COF materials using density functional theory calculations. Our analysis includes monolayers and multilayers (AA- and AB-stacked modes) of mellitic triimide frameworks designed from mellitic trianhydride (MTA) as the main building knot, including MTI-TAPB-COF, which was previously synthesized from the condensation reaction of MTA and 1,3,5-tris(4-aminophenyl)benzene (TAPB), and other previously unreported structures based on MTA. Respective frameworks have been selected due to the difference in building block symmetry (C3 + C2 and C3 + C3) and different chemical linkages, either by benzene or by pyridine rings. We find the polyimide multilayers to be stable and with varying electronic properties. The finite band gap exhibited by every structure (monolayer and stacked) was sensitive to atomic arrangement. Stacking introduces dispersion to an otherwise flat band structure of the materials, which appeared to be highly sensitive to stacking direction. The effect of stacking was similar for each COF, but the magnitude of band structure change was different and dependent on the symmetry of the building blocks.

Physical mechanism on ladderphane copolymers with electron localization and phonon delocalization for capacitive energy storage

In this article, the properties of trapezoidal polymer PSBNP-co-PTNI for capacitive energy storage are investigated theoretically, stimulated by the recent experimental reports [Chen et al., Nature 615, 62 (2023)]. The flatband structures of PSBNP-co-PTNI copolymer reveal that very large effective mass of electron results in Anderson electron localization and then small electric conductance. The TNI doping significantly decreases optical absorption in an ultraviolet region, which can significantly suppress photocurrent. Raman spectrum demonstrates that phonon Debye temperature can be significantly decreased, due to the U-process of phonon scattering by TNI doping, and then results in the phonon delocalization, and then the large thermal conductivity and thermal capacity. The simultaneous regulations on decreasing electric conductivity and on increasing thermal conductivity can significantly increase the ability of capacitive energy storage. The unit-length dependent thermal electric current manifests that the thermal electric current can be significantly decreased by slightly increase in length of copolymer. Overall, our research contributes to the understanding of the impact of TNI doping on the energy storage properties of the polymer and provides a valuable reference for future research focused on polymer energy storage.

Flatband λ-Ti3O5 towards extraordinary solar steam generation.

Solar steam interfacial evaporation represents a promising strategy for seawater desalination and wastewater purification owing to its environmentally friendly character1-3. To improve the solar-to-steam generation, most previous efforts havefocused on effectively harvesting solar energy over the full solar spectrum4-7. However, the importance of tuning joint densities of states in enhancing solar absorption of photothermal materials is less emphasized. Here we propose a route to greatly elevate joint densities of states by introducing a flat-band electronic structure. Our study reveals that metallic λ-Ti3O5 powders show a high solar absorptivity of 96.4% due to Ti-Ti dimer-induced flat bands around the Fermi level. By incorporating them into three-dimensional porous hydrogel-based evaporators with a conical cavity, an unprecedentedly high evaporation rate of roughly 6.09 kilograms per square metre per hour is achieved for 3.5 weight percent saline water under 1 sun of irradiation without salt precipitation. Fundamentally, the Ti-Ti dimers and U-shaped groove structure exposed on the λ-Ti3O5 surface facilitate the dissociation of adsorbed water molecules and benefit the interfacial water evaporation in the form of small clusters. The present work highlights the crucial roles of Ti-Ti dimer-induced flat bands in enchaining solar absorption and peculiar U-shaped grooves in promoting water dissociation, offering insights into access to cost-effective solar-to-steam generation.

First-principles study on the electronic structure of Pb10−xCux(PO4)6O (x = 0, 1)

Recently, Lee et al. claimed the experimental discovery of room-temperature ambient-pressure superconductivity in a Cu-doped lead-apatite (LK-99) (arXiv:2307.12008, arXiv:2307.12037). Remarkably, the claimed superconductivity can persist up to 400 K at ambient pressure. Despite the experimental implication, the electronic structure of LK-99 has not yet been studied. Here, we investigate the electronic structures of LK-99 and its parent compound using first-principles calculations, aiming to elucidate the doping effects of Cu. Our results reveal that the parent compound Pb10(PO4)6O is an insulator, while Cu doping induces an insulator-metal transition and thus volume contraction. The band structures of LK-99 around the Fermi level are featured by a half-filled flat band and a fully-occupied flat band. These two very flat bands arise from both the 2p orbitals of 1/4-occupied O atoms and the hybridization of the 3d orbitals of Cu with the 2p orbitals of its nearest-neighboring O atoms. Interestingly, we observe four van Hove singularities on these two flat bands. Furthermore, we show that the flat band structures can be tuned by including electronic correlation effects or by doping different elements. We find that among the considered doping elements (Ni, Cu, Zn, Ag, and Au), both Ni and Zn doping result in the gap opening, whereas Au exhibits doping effects more similar to Cu than Ag. Our work establishes a foundation for future studies to investigate the role of unique electronic structures of LK-99 in its claimed superconducting properties.

Evolution of magnetic correlation in an inhomogeneous square lattice

We explore the magnetic properties of a two-dimensional Hubbard model on an inhomogeneous square lattice, which provides a platform for tuning the bandwidth of the flat band. In its limit, this inhomogeneous square lattice turns into a Lieb lattice, and it exhibits abundant properties due to the flat band structure at the Fermi level. By using the determinant quantum Monte Carlo simulation, we calculate the spin susceptibility, double occupancy, magnetization, spin structure factor, and effective pairing interaction of the system. It is found that the antiferromagnetic correlation is suppressed by the inhomogeneous strength and that the ferromagnetic correlation is enhanced. Both the antiferromagnetic correlation and ferromagnetic correlation are enhanced as the interaction increases. It is also found that the effective $d$-wave pairing interaction is suppressed by the increasing inhomogeneity. In addition, we also study the thermodynamic properties of the inhomogeneous square lattice, and the calculation of specific heat provides good support for our point. Our intensive numerical results provide a rich magnetic phase diagram over both the inhomogeneity and interaction.

Tuning the Radius Ratio to Enhance Thermoelectric Properties in the Zintl Compounds AM2Sb2 (A = Ba, Sr; M = Zn, Cd)

Five novel Zintl phase solid solutions in the Ba1–xSrxZn2–yCdySb2 (0 ≤ x ≤ 0.13(1); 0 ≤ y ≤ 0.32(2)) system were successfully synthesized by the molten Pb metal-flux method, and the powder X-ray diffraction and single-crystal X-ray diffraction analyses proved that all five title compounds adopted the BaCu2S2-type phase having the orthorhombic Pnma space group (Z = 4, Pearson code oP20) with five crystallographically independent atomic sites. The previously studied BaCu2S2-type antimonides demonstrated a limited tolerance for doping in contrast to the CaAl2Si2-type antimonides. To understand the relatively narrower phase width and limited dopability of the title BaCu2S2-type phase than the CaAl2Si2-type phase in the overall Ba1–xSrxZn2–yCdySb2 system, the radius ratio of cations and anionic elements r+/r– for two structure types were thoroughly investigated. For the first time, the r+/r– ratio was identified as a critical factor for the phase selectivity: (1) r+/r– > 1 favored the BaCu2S2-type phase, and (2) r+/r– < 1 favored the CaAl2Si2-type phase. We also revealed the structural transformation mechanism from the more widely observed CaAl2Si2-type phase to the title BaCu2S2-type phase as the relatively larger cationic elements were introduced to the system. A series of DFT calculations using the three hypothetical models indicated that a resonance peak near EF in the density of states curves was descended from the relatively flat band structure at several special symmetry points rationalizing the enhanced Seebeck coefficients of Ba0.94(1)Sr0.06Zn1.86(3)Cd0.14Sb2 and Ba0.96(1)Sr0.04Zn1.68(2)Cd0.32Sb2. Electron localization function analysis rationalized the correlation between the polarity change of anionic Zn/Cd–Sb bonds and the charge carrier mobility on the anionic frameworks. Temperature-dependent thermoelectric properties were studied for the four title compounds, and the results proved that the Sr and Cd doping in the title Ba1–xSrxZn2–yCdySb2 system successfully enhanced the ZT values through the increased Seebeck coefficients and the reduced total thermal conductivities.

Anharmonic lattice dynamics and the origin of intrinsic ultralow thermal conductivity in AgI materials

Ionic conductors such as AgI with ultralow thermal conductivities $({\ensuremath{\kappa}}_{l})$ are of increasing interest because of their excellent thermoelectric properties. However, the origin of their intrinsic low ${\ensuremath{\kappa}}_{l}$ values remain elusive. In this study, comprehensive theoretical calculations of the lattice dynamics and the thermal transport properties of $\ensuremath{\gamma}\text{\ensuremath{-}}\mathrm{AgI}$ (zinc-blende structure) and $\ensuremath{\beta}\text{\ensuremath{-}}\mathrm{AgI}$ (wurtzite structure) as functions of temperature were carried out based on many-body perturbation theory and phonon Boltzmann transport theory. First, the mean-squared displacements (MSDs) of ${\mathrm{Ag}}^{+}$ were significantly larger than those of ${\mathrm{I}}^{\ensuremath{-}}$ in both $\ensuremath{\gamma}\text{\ensuremath{-}}$ and \ensuremath{\beta}-phases below the order-disorder phase transition temperature $({T}_{\mathrm{c}})$, which led to a characteristic ``rattling'' feature and low-frequency, nearly flat local phonon vibrations. According to our previous work [Xie et al., Phys. Rev. Lett. 125, 245901 (2020)], such nondispersive flat phonon band structures are expected to give rise to four-phonon resonance and result in a dramatic increase in the four-phonon scattering over the conventional three-phonon scattering. For $\ensuremath{\gamma}\text{\ensuremath{-}}\mathrm{AgI}$, similar four-phonon resonance behavior was also discovered for the low-lying transverse acoustic phonon branches, and it was found that their four-phonon scattering rates were an order of magnitude larger than the corresponding three-phonon scattering rates. Considering the four-phonon scattering, the theoretical ${\ensuremath{\kappa}}_{l}$ of $\ensuremath{\gamma}\text{\ensuremath{-}}\mathrm{AgI}$ was predicted to be $\ensuremath{\sim}0.32$ W/m K at 300 K, which was in good agreement with the value deduced from our experiments ($\ensuremath{\sim}0.36$ W/m K at 300 K). Compared to $\ensuremath{\gamma}\text{\ensuremath{-}}\mathrm{AgI}$, the acoustic phonons in $\ensuremath{\beta}\text{\ensuremath{-}}\mathrm{AgI}$ were more dispersive, and they intertwined with low-energy optical phonons at the zone boundaries. It was found that three-phonon resonance became as important as four-phonon resonance for the nearly flat longitudinal phonon band. The theoretical ${\ensuremath{\kappa}}_{l}$ for $\ensuremath{\beta}\text{\ensuremath{-}}\mathrm{AgI}$ was determined to be around $\ensuremath{\sim}0.32$ W/m K at room temperature, closely reproducing our measurement value $\ensuremath{\sim}0.29$ W/m K. Our results for AgI demonstrate the strong quartic anharmonicity in materials characterized by the rattling of weak bonding atoms as well as dispersionless phonon band structures. It is believed that this intimate relationship between the low-${\ensuremath{\kappa}}_{l}$ and flat phonon dispersion can be employed as a good indicator when searching for material systems with ultralow ${\ensuremath{\kappa}}_{l}$ values, e.g., cagelike rattling structures, quasi-two-dimensional structures, and chainlike structures.

Optical localization transition in a dual-periodical phase-modulated synthetic photonic lattice

We investigated optical localization and vibration in a synthetic photonic lattice (SPL) with two-phase modulators driven by different periodic voltages. By altering the periods of phase modulation, the induced phase circumfluences, band structure, and transmission behaviors of light in the SPL are studied. A transition from discrete diffraction to weak localization was observed in a small modulation period with equal modulation periodicities. For large modulation periods, the band structure exhibits a density distribution and the optical field delocalizes and vibrates. In contrast, strong localization occurs near the initial incident location with some moir\'e modulation periods of phases that correspond to a flat-band structure. This study reveals the effect of dual-periodic phase modulation on pulse propagation in discrete systems.