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Phononic SSH model with topological band inversion. a) 1D diatomic chain model with alternative spring force constants f + δ and f − δ represented by blue and red solid lines, respectively. The dashed box indicates the unit cell. b–d) Phononic dispersion relations for δ = −0.1f, δ = 0, and δ = 0.1f, respectively. ω0=f/m with m the atomic mass. a is the lattice constant. Red and blue color represent the out‐of‐phase vibrating mode with “+” parity and in‐phase mode with “−” parity, respectively, and the rainbow color indicates the mixing between them. The phonon dispersion of δ = 0.1f shows a topological band inversion from k = 0 to k = π/a. e) Frequencies as a function of δ/f for a finite‐size chain containing 80 atoms. The inset shows the finite‐size chain with fixed boundary condition. Red and blue lines are two in‐gap boundary modes.
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Effective manipulation of phonons is crucial to modern energy‐information science and technologies but limited by the charge neutral and spinless nature of phonons. Recently, novel quantum concepts, including Berry phase, topology, and pseudospin, are introduced to phonon systems, providing fundamentally new routes to control phonons, opening an em...
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Citations
... In recent years, topological photonics have attracted extensive attention owing to its unique characteristics to control the propagation and manipulation of light waves robustly and immune to disorder and defects [1][2][3][4][5][6] . Up to now, a diverse range of topological phenomena have been proposed in topological photonics, including topologically protected interface states [7][8][9] , corner states [10][11][12][13] , and one-way propagated edge states [14][15][16] . These discoveries have paved the way for numerous device applications, such as pseudo-spinbased light splitters [17][18][19] , high-quality factor topological interface state cavity lasers 9 , sensitivity topological optical sensors 20 , and high-speed topological electro-optic modulators 21 . ...
Topological phases in photonic systems have garnered significant attention, often relying on precise structural design for generating non-trivial topological phases. However, this dependency on fixed structures limits their adaptability. This study systematically explores incident angle-induced topological phase transitions in a one-dimensional photonic crystal (PC). Both TE and TM polarized modes undergo topological phase transitions at the same critical transition angles. Additionally, the TM-polarized mode undergoes a unique topological phase transition at the Brewster angle. When these two kinds of transition angles coincide, even if the band structure of the TM-polarized mode undergoes an open-close-reopen process, the topological properties of the corresponding bandgap remain unchanged. Based on theoretical analysis, we design the composite PCs comprising two interfaced PCs having common bandgaps but different topological properties. By tuning the incident angle, we theoretically and experimentally achieve TE-TM splitting of topological interface states in the visible region, which may have potential applications in optical communications, optical switching, photonic integrated circuits, and so on.
... Topological phenomena have attracted considerable long-term attention because of their fundamental interest and application prospects [15][16][17][18], such as the quantized information transport and robust wave propagation. Although topological orders originate from condensed matter physics, researchers from other fields, like photonics [19,20], phononics [21][22][23][24], mechanics [25][26][27], spintronics [28,29], and electrical circuits, are also interested in this topic. Among these platforms, electric circuits stand out as a competitive one for studying topological physics with simple circuit networks. ...
... Later, SSH model becomes an effective model to study topological states because of its nontrivial nature and concise structure. It provides valuable insights into the role of symmetry and topology in determining the electronic properties of materials and has contributed to the development of the broader fields in topology [24,91,92]. The bulk Hamiltonian of SSH model is expressed as [93] = ( + cos ) + sin , ...
Recently, topolectrical circuits (TECs) boom in studying the topological states of matter. The resemblance between circuit Laplacians and tight-binding models in condensed matter physics allows for the exploration of exotic topological phases on the circuit platform. In this review, we begin by presenting the basic equations for the circuit elements and units, along with the fundamentals and experimental methods for TECs. Subsequently, we retrospect the main literature in this field, encompassing the circuit realization of (higher-order) topological insulators and semimetals. Due to the abundant electrical elements and flexible connections, many unconventional topological states like the non-Hermitian, nonlinear, non-Abelian, non-periodic, non-Euclidean, and higher-dimensional topological states that are challenging to observe in conventional condensed matter physics, have been observed in circuits and summarized in this review. Furthermore, we show the capability of electrical circuits for exploring the physical phenomena in other systems, such as photonic and magnetic ones. Importantly, we highlight TEC systems are convenient for manufacture and miniaturization because of their compatibility with the traditional integrated circuits. Finally, we prospect the future directions in this exciting field, and connect the emerging TECs with the development of topology physics, (meta)material designs, and device applications.
... Topological phonons, including Weyl phonons, Dirac phonons, and nodal line phonons, have recently attracted the interest of researchers [12][13][14][15][16][17][18][19][20][21][22][23]. These types of phonons are helpful for us to realize certain phonon-based devices, such as the phonon diode effect [24,25] reported in chiral crystals, because they show little or no scattering in phonon-based transport. ...
Topological phases in kagome systems have garnered considerable interest since the introduction of the colloidal kagome lattice. Our study employs first-principle calculations and symmetry analysis to predict the existence of ideal type-I, III nodal rings (NRs), type-I, III quadratic nodal points (QNPs), and Dirac valley phonons (DVPs) in a collection of two-dimensional (2D) kagome lattices M2C3 (M = As, Bi, Cd, Hg, P, Sb, Zn). Specifically, the Dirac valley points (DVPs) can be observed at two inequivalent valleys with Berry phases of +π and −π , connected by edge arcs along the zigzag and armchair directions. Additionally, the QNP is pinned at the Γ point, and two edge states emerge from its projections. Notably, these kagome lattices also exhibit ideal type-I and III nodal rings protected by time inversion and spatial inversion symmetries. Our work examines the various categories of nodal points and nodal ring phonons within the 2D kagome systems and presents a selection of ideal candidates for investigating topological phonons in bosonic systems.
... A commonly employed topological phonon model is the phononic Su-Schrieffer-Heeger (SSH) model, featuring a two-atom chain with diverse masses and force constants, and characterized by the winding number or Zak phase [38]. The difference in atomic mass can create a topologically trivial bandgap, whereas the variance in force constants can cause a band inverse, resulting in a nontrivial gap. ...
... To display the nontrivial topological characteristics, the Berry curvatures at the corresponding k-planes, as well as the phonon surface states projected onto the surface BZs, are also calculated. More details on topological phonon theory and calculations can be found in references [30,36,38]. ...
Understanding the phonon behavior in semiconductors from a topological physics perspective provides more opportunities to uncover extraordinary physics related to phonon transport and electron-phonon interactions. While various kinds of topological phonons have been reported in different crystalline solids, their microscopic origin has not been quantitatively uncovered. In this work, four typical analytical interatomic force constant (IFC) models are employed for wurtzite GaN and AlN to help establish the relationships between phonon topology and real-space IFCs. In particular, various nearest neighbor IFCs, i.e., different levels of nonlocality, and IFC strength controlled by characteristic coefficients, can be achieved in these models. The results demonstrate that changes in the strength of both the IFCs and nonlocal interactions can induce phonon phase transitions in GaN and AlN, leading to the disappearance of existing Weyl phonons and the appearance of new Weyl phonons. These new Weyl phonons are the result of a band reversal and have a Chern number of ±1. Most of them are located in the kz=0 plane in pairs, while some of them are inside or at the boundary of the irreducible Brillouin zone. Among the various Weyl points observed, certain ones remain identical in both materials, while others exhibit variability depending on the particular case. Compared to the strength of the IFC, nonlocal interactions show much more significant effects in inducing the topological phonon phase transition, especially in cases modeled by the IFC model and SW potential. The larger number of 3NN atoms provides more space for variations in the topological phonon phase of wurtzite AlN than in GaN, resulting in a greater abundance of changes in AlN
... 10 . For example, theoretical predictions have revealed the intriguing presence of topological nodal-surface phonons in ZnTeMoO 6 , which possess great potential, including freedom from Fermi-level restrictions and exceptional thermal stability due to minimal scattering during transport processes 18 . ...
The discovery of ultraconfined polaritons with extreme anisotropy in a number of van der Waals (vdW) materials has unlocked new prospects for nanophotonic and optoelectronic applications. However, the range of suitable materials for specific applications remains limited. Here we introduce tellurite molybdenum quaternary oxides—which possess non-centrosymmetric crystal structures and extraordinary nonlinear optical properties—as a highly promising vdW family of materials for tunable low-loss anisotropic polaritonics. By employing chemical flux growth and exfoliation techniques, we successfully fabricate high-quality vdW layers of various compounds, including MgTeMoO6, ZnTeMoO6, MnTeMoO6 and CdTeMoO6. We show that these quaternary vdW oxides possess two distinct types of in-plane anisotropic polaritons: slab-confined and edge-confined modes. By leveraging metal cation substitutions, we establish a systematic strategy to finely tune the in-plane polariton propagation, resulting in the selective emergence of circular, elliptical or hyperbolic polariton dispersion, accompanied by ultraslow group velocities (0.0003c) and long lifetimes (5 ps). Moreover, Reststrahlen bands of these quaternary oxides naturally overlap that of α-MoO3, providing opportunities for integration. As an example, we demonstrate that combining α-MoO3 (an in-plane hyperbolic material) with CdTeMoO6 (an in-plane isotropic material) in a heterostructure facilitates collimated, diffractionless polariton propagation. Quaternary oxides expand the family of anisotropic vdW polaritons considerably, and with it, the range of nanophotonics applications that can be envisioned.
... [1][2][3][4][5] The concepts of topological electronic states have been generalized to bosonic systems, leading to the birth of topological phonons. [6][7][8][9][10][11][12][13][14] Recently, topological phonons have received increasing research interest in both experiments and theories [14][15][16][17][18] since it is believed that the phonon can be an ideal platform for realizing topological quantum states due to its bosonic nature and hard-to-break time-reversal symmetry. 7,19 In addition, phonons, energy quanta of lattice vibration, also play an essential role in fundamental physical phenomena including heat conduction, light-matter interaction, thermoelectrics, and superconductivity. ...
... [6][7][8][9][10][11][12][13][14] Recently, topological phonons have received increasing research interest in both experiments and theories [14][15][16][17][18] since it is believed that the phonon can be an ideal platform for realizing topological quantum states due to its bosonic nature and hard-to-break time-reversal symmetry. 7,19 In addition, phonons, energy quanta of lattice vibration, also play an essential role in fundamental physical phenomena including heat conduction, light-matter interaction, thermoelectrics, and superconductivity. 12 In this context, it is of essential importance to realize and tune topological nontrivial phonon states in certain materials, which may result in novel physical properties and promising application prospects in a wide range of fields. ...
... However, limited by its charge-neutral and spinless nature, the TPT of phonons generally cannot be achieved by conventional approaches of treating electrons. 7,23 It is noted that previous studies have demonstrated TPTs of phonons induced in different approaches, including spin-lattice interactions for magnetic lattices, 29,30 magnetic field for ionic lattices, 29 Coriolis fields, 31 and optomechanical interactions. 32 In contrast, applying external pressure or strain to break the symmetry of crystals provides an alternative and universal method for achieving TPTs of phonons. ...
... Phonons, as the most fundamental bosonic quasiparticles, are responsible for the transmission of sound and heat. Inspired by the significant success of topological band theory in electronic systems, topological phonons have attracted extensive attention in the field of condensed-matter physics in recent years [16][17][18][19][20][21][22][23]. Without limitation of the Pauli exclusion principle, phonon quasiparticles can be excited to the whole frequency ranges of phonon spectrum and play an important role in thermal transports [24], electron-phonon coupling [25], or other phonon-related processes which may provide potential applications in phonon diodes [26]. ...
Multifold degenerate phonons have received much attention due to their nontrivial monopole topological charge and fascinating boundary states. Although Yu et al. recently provides a comprehensive list of all potential nodal points for systems with specific space groups [Sci. Bull. 67, 375 (2022)]. However, our understanding of the fundamental mechanisms that give rise to the formation of fourfold-degenerate (FD) phonons is still limited. In this paper, we have directed our research towards investigating the generation mechanism of these fourfold-degenerate (FD) phonons in noncentrosymmetric space groups. Using symmetry arguments and k·p model analysis, we have classified them into two categories: the first origins from the commutation/anticommutation relation of the little cogroup operations, and the second associates to the combination of threefold rotation, mirror and time-reversal symmetries. Moreover, the band dispersions of the FD phonons in the first group are required to be linear, whereas the band dispersions of the FD phonons in the second category may be quadratic. On the basis of first-principles calculations, we propose that K2Mg2O3and Na4SnSe4are representative candidates for the two categories, respectively. Furthermore, for each space group with fourfold degenerate phonons, we propose corresponding materials that must host the fourfold-degenerate points. Our work not only deepens our understanding of the mechanisms underlying the formation of these fourfold-degenerate phonons, but it also proposes practical materials for observing FD phonons in crystalline systems without inversion symmetry.
... [3][4][5] Topology has also emerged in classical fields, including the cases of optical and acoustic waves in photonic and phononic crystals; interest in robust localized waves, known as topological boundary modes, has led to the proposal of various topological photonic/phononic crystals for achieving highly efficient energy transport. [6][7][8] Classical analogs of these systems are predominantly based on the graphene structure, a popular structure that exhibits topological effects. [9][10][11][12] Theoretical and numerical models of graphene-like mechanical structures featuring mass-spring or elastic systems have been developed, and have allowed phononic band structures with Dirac cones and edge modes to be unveiled. ...
We construct a two-dimensional mechanical wave machine based on a hexagonal lattice to investigate low-frequency flexural plate waves whose propagation mimicks a topological quantum valley Hall system. We thereby demonstrate ‘mechanical graphene’ by extension of the one-dimensional Shive wave machine. Imaging experiments, backed up by simulations, reveal the presence of boundary modes along a topological interface, despite the absence of Dirac cones in the dispersion relation under conditions of spatial inversion symmetry. This work provides an alternative route for the investigation of topological phononic crystals, and should lead to new insights in the design and observation of artificial phononic structures.
... In recent years, topological photonics have attracted extensive attention owing to its unique characteristics to control the propagation and manipulation of light waves robustly and immune to disorder and defects [1][2][3][4][5][6] . Thus far, a diverse range of topological phenomena and multifarious device applications have been proposed in topological photonics, including topologically protected interface states [7][8][9] , corner states [10][11][12][13] , one-way propagated edge states [14][15][16] , and pseudo-spin-based light splitters [17][18][19] , etc. ...
Topological phases in photonic systems have garnered significant attention, often relying on precise structural design for generating non-trivial topological phases. This study systematically explores incident angle-induced topological phase transitions in a one-dimensional photonic crystal (PC). Both TE and TM polarized modes undergo topological phase transitions at the same critical transition angles. Additionally, the TM-polarized mode undergoes a unique topological phase transition at the Brewster angle. Interestingly, when these two kinds of transition angles coincide, even the band structure of TM-polarized mode undergoes an open-close-reopen process, the topological properties of the corresponding bandgap remain unchanged. Based on theoretical analysis, we design a superlattice comprising two interfaced PCs having common bandgaps but different topological properties. By tuning the incident angle, we theoretically and experimentally achieve TE-TM splitting of topological interface states in the visible region, which may have potential applications in optical communications, optical switching, photonic integrated circuits, and so on.
... [33][34][35][36][37][38][39][40][41][42][43][44][45][46] In addition, extending these studies to phononic bosons, which share similar crystal symmetry constraints with their electronic fermion counterparts, has resulted in the emergence of topological phononics. [47][48][49][50][51][52] Furthermore, the combination of topological properties with intrinsic ferromagnetism has given rise to magnetic topological phases. [53][54][55][56][57][58][59] This exciting development highlights the interplay between topology and magnetism, opening up new avenues for exploration and application. ...
Topological states in two-dimensional materials have garnered significant research attention in recent years, particularly those with intrinsic magnetic orderings, which hold great potential for spintronic applications. Through theoretical calculations, we unveil the superior band topology of monolayer vanadium trihalides, with a specific focus on V2Cl6. These two-dimensional compounds exhibit a half-metallic ferromagnetic ground state, showcasing excellent thermodynamic and mechanical stabilities. Remarkably, clean band crossings with complete spin polarization manifest as phase transitions between Weyl semimetal states and quantum anomalous Hall states under different magnetization directions, and both topological phases yield prominent edge states. Furthermore, Monte Carlo simulations estimate a high Curie temperature of up to 381.3 K, suggesting the potential for spintronic development above room temperature. Taking a step forward, we construct two heterojunctions utilizing selected substrates, MoS2 and h-BN. These substrates not only facilitate a suitable lattice integration but also have a negligible impact on the half-metallicity and band topology. These findings lay the groundwork for exploring practical applications of two-dimensional ferromagnetic topological states. Importantly, the presented material candidates have the potential to accelerate the development of room temperature applications and integrate spintronic devices.
