Topological Phonon Research Articles

Coexistence of zero-, one-, and two-dimensional degeneracy in tetragonal SnO2 phonons

Based on the dimension of degeneracy, topological electronic systems can roughly be divided into three parts: nodal point, line and surface materials corresponding to zero-, one- and two-dimensional degeneracy, respectively. In parallel to electronic systems, the concept of topology was extended to phonons, promoting the birth of topological phonons. Till date, few nodal point, line and surface phonons candidates have been predicted in solid-state materials. In this study, based on symmetry analysis and first-principles calculation, for the first time, we prove that zero-, one- and two-dimensional degeneracy co-exist in the phonon dispersion of one single realistic solid-state material SnO$_2$ with \textit{P}4$_2$/\textit{mnm} structure. In contrast to the previously reported electronic systems, the topological phonons observed in SnO$_2$ are not restricted by the Pauli exclusion principle, and they experience negligible spin-orbit coupling effect. Hence, SnO$_2$ with multiple dimensions of degeneracy phonons is a good platform for studying the entanglement among nodal point, line and surface phonons. Moreover, obvious phonon surface states are visible, which is beneficial for experimental detection.

Emergent topological fields and relativistic phonons within the thermoelectricity in topological insulators

Topological edge states are predicted to be responsible for the high efficient thermoelectric response of topological insulators, currently the best thermoelectric materials. However, to explain their figure of merit the coexistence of topological electrons, entropy and phonons can not be considered independently. In a background that puts together electrodynamics and topology, through an expression for the topological intrinsic field, we treat relativistic phonons within the topological surface showing their ability to modulate the Berry curvature of the bands and then playing a fundamental role in the thermoelectric effect. Finally, we show how the topological insulators under such relativistic thermal excitations keep time reversal symmetry allowing the observation of high figures of merit at high temperatures. The emergence of this new intrinsic topological field and other constraints are suitable to have experimental consequences opening new possibilities of improving the efficiency of this topological effect for their based technology.

Symmetry-enforced ideal lanternlike phonons in the ternary nitride Li6WN4

Condensed matter systems contain both fermionic and bosonic quasiparticles. Owing to the constraint imposed by the Fermi level, ideal material candidate for the emergent particles with higher-dimensional degeneracy manifold (i.e., nodal lines and nodal surfaces) has not been found in electronic systems. This paper demonstrates that according to the first-principle calculations and symmetry analysis, realistic ternary nitride Li$_6$WN$_4$ features ideal (nearly flat) nodal-surface and nodal-line structures in its phonon spectra. These nodal degeneracies are shaped like lanterns, and their existence is guaranteed by nonsymmorphic symmetry. The corresponding topological phonon surface state covers exactly half the surface Brillouin zone (BZ) and can thereby be distinguished from those of conventional nodal-line and nodal-surface semimetals. The results of our study demonstrate the existence of ideal lantern-like phonons in realistic materials, which enriches the classification of topological quantum phases and provides a good basis for investigating the interaction between nodal-line and nodal-surface phonons in a single material.

Lattice dynamics and topological surface phonon states in cuprous oxide Cu2O

The topological phonon state of quantum matter is an emerging field that has been attracting considerable interest. For instance, Weyl phonons in transition-metal monosilicides have been proposed theoretically and identified experimentally. However, topological phonon nodal net states are not well studied due to the lack of realistic materials. Here, based on first-principles calculations and effective model analysis, we propose an existing material---cuprous oxide---to host the nodal net phonons. The nontrivial phonon surface states and uncovered phononic arcs are clearly visible on $k$-resolved phonon spectra, which are amenable to experimental detection. Our findings offer a possible platform for realizing topologically nontrivial phonon states and their applications.

Topological gimbal phonons in T -carbon

The topological metal states in electronic systems have been extensively studied, but topological phonons were explored only in few examples so far. Here, we expose for the first time that the topological nodal gimbal phonons, type-I and type-II Weyl phonons are simultaneously present in T-carbon, a recently realized new allotrope of carbon. At about 15.2 THz, we find that there exist three mutually intersecting nodal loops (named as nodal gimbal phonons) around {\Gamma} point, and two pairs of type-I Weyl phonons on the boundary of Brillouin zone around each X point. In addition, there exist three pairs of type-II Weyl phonons at about 14.5 THz around each L point. It is shown that these exotic topological phonons are protected by corresponding symmetries, and lead to topologically nontrivial surface states. Our findings not only afford plenty of intriguing topological phonon states in a simple material like T-carbon but also provide a new platform to study novel properties of topological phonons, which would facilitate further both experimental and theoretical works in future.

First-principles prediction of ideal type-II Weyl phonons in wurtzite ZnSe

Weyl materials, exhibiting topologically nontrivial touching points in band dispersion, are fascinating subjects of research, which have been extensively studied in electron-related systems. In this paper, by employing first-principles calculations of topological phonons in wurtzite-structured phases of $MX$ chalcogenides (where $M=\mathrm{Zn}$ and Cd and $X=\mathrm{O}$, S, Se, and Te), we demonstrate the existence of ideal type-II Weyl phonons in wurtzite ZnSe, a well-known II-VI semiconductor. There are in the ${\mathbf{q}}_{z}=0.0$ plane six pairs of Weyl points stemming from the inversion between the two optical branches. The nontrivial phonon surface states and surface arcs projected on the semifinite (0001) surfaces are investigated. Phonon surface arcs connecting each pair of Weyl points with opposite chirality, guaranteed to be $0.55\phantom{\rule{0.28em}{0ex}}{\AA{}}^{\ensuremath{-}1}$ and very long, are readily observable in experiment. The opposite chirality of Weyl points with quantized Berry curvature produces the Weyl phonon Hall effect. Our results propose a potential platform for future experimental study of type-II Weyl phonons in realistic materials.

Electrically Tunable Elastic Topological Insulators Using Atomically Thin Two-Dimensional Materials Pinned on Patterned Substrates

At present, functionalities, miniaturization, and tunability are useful directions in the development of practical device applications for topological photonics and phononics. Atomically thin two-dimensional (2D) materials not only have excellent mechanical properties, but also have outstanding electrical, optical, and magnetic properties, making them great candidates for tunable microdevices via electrostatic forces, magnetoacoustic, or photoacoustic coupling, etc. In this paper, we propose an elastic topological insulator based on atomically thin 2D materials pinned on honeycomb-patterned substrates. Thanks to acoustic impedance mismatch between the suspended 2D materials and the rigid substrates, both topological edge states and corner states of elastic waves inside the 2D materials can be achieved, accompanied by a display of functional components, e.g., arbitrary waveguides, splitters, resonators. Significantly, all these components for elastic waves are frequency tunable, due to the electromechanical sensitivity of the 2D material. For example, a two-port high-Q bandpass filter is estimated to achieve 7.26% operating frequency shifting by applying only 5 V of gate voltage. These kinds of elastic topological materials and corresponding devices can strongly promote the development of 2D-material-based nanoelectromechanical systems and might have immediate applications in modern wireless communication technology at radio or microwave frequencies.

Topological effects of phonons in GaN and AlGaN: A potential perspective for tuning phonon transport

Tuning thermal transport in semiconductor nanostructures is of great significance for thermal management in information and power electronics. With excellent transport properties, such as ballistic transport, immunity to point defects and disorders, and forbidden backscattering, topological phonon surface states show remarkable potential in addressing this issue. Herein, topological phonon analyses are performed on hexagonal wurtzite gallium nitride (GaN) to check the topological characteristics of phonons. And other nitrides of the same family, i.e., AlN and AlGaN crystals, are also calculated from a topological phonon phase transition perspective. With the aid of first-principle calculations and topological phonon theory, Weyl phonon states, which host surfaces states without backscattering, are investigated for all these materials. The results show that there is no nontrivial topological phonon state in GaN. However, by introducing Al atoms, i.e., in wurtzite type AlN and AlGaN, more than one Weyl phonon point is found, confirmed by obvious topological characteristics, including non-zero integer topological charges, source/sink in Berry curvature distributions, surface local density of states, and surface arcs. As AlN and AlGaN are typical materials in AlGaN/GaN heterostructure based electronics, the existence of topological phonon states in them will benefit thermal management by facilitating the design of one-way interfacial phonon transport without backscattering.

Computation and data driven discovery of topological phononic materials

The discovery of topological quantum states marks a new chapter in both condensed matter physics and materials sciences. By analogy to spin electronic system, topological concepts have been extended into phonons, boosting the birth of topological phononics (TPs). Here, we present a high-throughput screening and data-driven approach to compute and evaluate TPs among over 10,000 real materials. We have discovered 5014 TP materials and grouped them into two main classes of Weyl and nodal-line (ring) TPs. We have clarified the physical mechanism for the occurrence of single Weyl, high degenerate Weyl, individual nodal-line (ring), nodal-link, nodal-chain, and nodal-net TPs in various materials and their mutual correlations. Among the phononic systems, we have predicted the hourglass nodal net TPs in TeO3, as well as the clean and single type-I Weyl TPs between the acoustic and optical branches in half-Heusler LiCaAs. In addition, we found that different types of TPs can coexist in many materials (such as ScZn). Their potential applications and experimental detections have been discussed. This work substantially increases the amount of TP materials, which enables an in-depth investigation of their structure-property relations and opens new avenues for future device design related to TPs.

Topological phonons in oxide perovskites controlled by light.

Perovskite oxides exhibit a rich variety of structural phases hosting different physical phenomena that generate multiple technological applications. We find that topological phonons-nodal rings, nodal lines, and Weyl points-are ubiquitous in oxide perovskites in terms of structures (tetragonal, orthorhombic, and rhombohedral), compounds (BaTiO3, PbTiO3, and SrTiO3), and external conditions (photoexcitation, strain, and temperature). In particular, in the tetragonal phase of these compounds, all types of topological phonons can simultaneously emerge when stabilized by photoexcitation, whereas the tetragonal phase stabilized by thermal fluctuations only hosts a more limited set of topological phonon states. In addition, we find that the photoexcited carrier concentration can be used to tune the topological phonon states and induce topological transitions even without associated structural phase changes. Overall, we propose oxide perovskites as a versatile platform in which to study topological phonons and their manipulation with light.

Fragile phonon topology on the honeycomb lattice with time-reversal symmetry

We use the methods of topological quantum chemistry to explore the topology of phonons on time-reversal symmetric crystals with the structure of the planar honeycomb (layer group $p6/mmm$). This approach is not tied to a particular model of atomic vibrations, but is applied to the most general dynamical matrix constrained only by the symmetries of the system. We show that four distinct fragile topological phonon phases are generically possible. Truncating the dynamical matrix to third nearest neighbors yields a model that realizes the different phonon topologies, characterized by the existence of phononic edge and corner modes and by Wilson loops with winding numbers one and two. Fitting the dynamical matrix to the DFT phonon bands shows that graphene is not very far from a topologically nontrivial phonon phase.

Unsupervised Manifold Clustering of Topological Phononics.

Classification of topological phononics is challenging due to the lack of universal topological invariants and the randomness of structure patterns. Here, we show the unsupervised manifold learning for clustering topological phononics without any apriori knowledge, neither topological invariants nor supervised trainings, even when systems are imperfect or disordered. This is achieved by exploiting the real-space projection operator about finite phononic lattices to describe the correlation between oscillators. We exemplify the efficient unsupervised manifold clustering in typical phononic systems, including a one-dimensional Su-Schrieffer-Heeger-type phononic chain with random couplings, amorphous phononic topological insulators, higher-order phononic topological states, and a non-Hermitian phononic chain with random dissipations. The results would inspire more efforts on applications of unsupervised machine learning for topological phononic devices and beyond.

Topological phonons in graphene

By means of first-principles calculations and a modeling analysis, we have predicted that traditional two-dimensional graphene hosts four types of topological phononic Dirac points (DPs) and a phononic nodal ring (PNR) in its phonon spectrum. In the phonon spectrum of graphene, there exist four types of DPs, DP1, DP2, DP3, and DP4, with both DP1 and DP2 located at the Brillouin zone (BZ) corners $K$ and ${K}^{\ensuremath{'}}$, DP3 located along the $\mathrm{\ensuremath{\Gamma}}\text{\ensuremath{-}}K$ line, and DP4 located along the $\mathrm{\ensuremath{\Gamma}}\text{\ensuremath{-}}M$ line, as well as the PNR surrounding the centered $\mathrm{\ensuremath{\Gamma}}$ point in the ${q}_{x,y}$ plane. The calculations further reveal that Berry curvatures are vanishingly zero throughout the whole BZ, except for the positions of these four Dirac phonons, at which the nonzero singular Berry curvatures appear with a Berry phase of $\ensuremath{\pi}$ or $\ensuremath{-}\ensuremath{\pi}$, confirming its topological nontrivial nature. The topologically protected nontrivial phononic edge states have been also evidenced along both the zigzag-edged and armchair-edged boundaries. These results may pave the way for further studies of the topological phononic properties of graphene, such as phononic destructive interference with a suppression of backscattering and intrinsic phononic quantum Hall-like effects.

Nonreciprocal topological phononics in optomechanical arrays

We propose a platform for robust and tunable nonreciprocal phonon transport based on arrays of optomechanical microtoroids. In our approach, time-reversal symmetry is broken by the interplay of photonic spin-orbit coupling, engineered using a state-of-the-art geometrical approach, and the optomechanical interaction. We demonstrate the topologically protected nature of this system by investigating its robustness to imperfections. This type of system could find application in phonon-based information storage and signal processing devices.

Advanced Materials for Heat Energy Transfer, Conversion, Storage and Utilization

Advanced Materials for Heat Energy Transfer, Conversion, Storage and Utilization

Collective Resonances of a Chain of Coupled Phononic Microresonators

We study experimentally a chain of defect resonators in a phononic crystal slab and observe its collective resonances at ultrasonic frequencies of a few megahertz. A phononic crystal of cross holes is fabricated in a thin fused-silica plate by femtosecond-laser writing followed by $\mathrm{KOH}$ etching. A chain of 17 coupled resonators is defined with no definite spatial periodicity but similar coupling strength between nearest-neighbor resonators. The full phononic band gap ensures that only evanescent waves in the crystal can tunnel between adjacent resonators in the plane. The resulting evanescent-coupling strength decreases exponentially with distance. Collective resonances are excited by a frequency-driven piezoelectric vibrator attached at one end of the chain and imaged along the chain with a laser Doppler vibrometer. A discrete spectrum of resonances is observed and explained by a model representing the chain as a phononic polymer. The theoretical analysis is supported by finite-element simulations that agree well with experimental results. Chains of evanescently coupled microresonators forming phononic polymers could find applications in ultrasonic sensing, for implementation in topological phononics, and for the design of optomechanical resonator chains.

Phononic Weyl nodal straight lines in MgB2

Based on first-principles calculations, we predict that the superconducting MgB$_2$ with a AlB$_2$-type centrosymmetric lattice host the so-called phononic topological Weyl nodal lines (PTWNLs) on its bulk phonon spectrum. These PTWNLs can be viewed as countless Weyl points (WPs) closely aligned along the straight lines in the $-$H-K-H direction within the three-dimensional Brillouin zone (BZ). Their topological non-trivial natures are confirmed by the calculated Berry curvature distributions on the planes perpendicular to these lines. These lines are highly unique, because they exactly locate at the high-symmetry boundary of the BZ protected by the mirror symmetry and, simultaneously, straightly transverse the whole BZ, in different from known classifications including nodal rings, nodal chains or nets, and nodal loops. On the (10$\bar{1}$0) crystal surface, the PTWNLs-induced drumhead-like non-trivial surface states appear within the rectangular area confined by the projected lines of the PTWNLs with opposite chirality. Moreover, when the mirror symmetry is broken, the double-degenerate PTWNLs are further lifted to form a pair WPs with opposite chirality. Our results pave the ways for future experimental study on topological phonons on MgB$_2$ and highlights similar results in a series of isostructural AlB$_2$-type metallic diborides.

Topological Phononics: From Fundamental Models to Real Materials

AbstractEffective 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 emerging field of “topological phononics.” Here, the basic concepts of Berry phase, topology, and pseudospin for phonons are introduced. Also, recent research progresses on various phononic topological states are reviewed, including phononic Su‐Schrieffer‐Heeger‐like states in 1D, quantum anomalous Hall‐like states, quantum valley Hall‐like states, and quantum spin Hall‐like states in 2D, and phononic Weyl points, nodal lines, and topological insulators in 3D. In addition to the fundamental models, material realizations and potential applications of topological phononics are comprehensively presented.

Signatures of Topological Phonons in Superisostatic Lattices.

Soft topological surface phonons in idealized ball-and-spring lattices with coordination number z=2d in d dimensions become finite-frequency surface phonons in physically realizable superisostatic lattices with z>2d. We study these finite-frequency modes in model lattices with added next-nearest-neighbor springs or bending forces at nodes with an eye to signatures of the topological surface modes that are retained in the physical lattices. Our results apply to metamaterial lattices, prepared with modern printing techniques, that closely approach isostaticity.

Phononic Weyl points and one-way topologically protected nontrivial phononic surface arc states in noncentrosymmetric WC-type materials

By means of first-principles calculations, we have predicted the existence of phononic Weyl points (WPs) along the high-symmetric K($\frac{1}{3},\frac{1}{3},0$)-H($\frac{1}{3},\frac{1}{3},\frac{1}{2}$) line in the Brillouin zone for a series of noncentrosymmetric hexagonal WC-type materials (HfS, HfSe, IrB, MoC, MoN, MoP, NbN, NbS, TaN, TaS, TiO, TiS, WN, ZrS, ZrSe, and ZrTe) and these WPs carry nonzero topological charges. In terms of symmetry analysis, we have further derived two-band $\mathbf{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{p}$ models to describe these WPs. For most WPs a first-order theory is sufficient to reproduce well the phonon spectra around these WPs. However, for some WPs a second-order theory has to be required. Particularly, when the second-order term plays a leading role, a topological charge $\ifmmode\pm\else\textpm\fi{}1\phantom{\rule{0.16em}{0ex}}\mathrm{WP}$ on the K-H line is accompanied by three nearby WPs with the opposite charges ($\ensuremath{\mp}1$) off the K-H line. These four spatially close (in the reciprocal space) WPs have a total topological charge of $\ensuremath{\mp}2$. On the crystal ($10\overline{1}0$) and ($01\overline{1}0$) surfaces, we have observed clear one-way topologically protected nontrivial phononic surface arc states connecting two WPs with opposite chirality. Our results pave the way for future experimental studies of the topological phonons for those WC-type materials.