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Low frequency and wide band gap metamaterial with divergent shaped star units: Numerical and experimental investigations

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Nitish Kumar , Siladitya Pal; Low frequency and wide band gap metamaterial with divergent shaped star units: Numerical and experimental investigations. Appl. Phys. Lett. 16 December 2019; 115 (25): 254101. https://doi.org/10.1063/1.5119754

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Accomplishing low frequency and wide bandgaps with a single phase metamaterial remains a challenging task. In this work, we propose a single phase acoustic metamaterial having periodically arranged divergent shaped star units and performed numerical simulation to obtain the bandgaps. The simulation results demonstrate that the metamaterial exhibits an uninterrupted and extended bandgap that lies in the low frequency regime. We fabricated the structure based on numerical investigations, and using the experimental technique, elastic wave transmission characteristics have been analyzed. Experimental observations are in good agreement with numerically calculated bandgaps. Moreover, we perform a parametric study on the bandgaps with geometrical parameters like the concave angle and length of divergent ribs of the unit cell.

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Frequency-dependent superconducting states from the two-time linear response theory: Application to Sr 2 RuO 4

Olivier gingras, antoine georges, and olivier parcollet, phys. rev. b 110 , 054509 – published 12 august 2024.

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• INTRODUCTION
• BENCHMARK ON THE ATTRACTIVE HUBBARD…
• STRONTIUM RUTHENATE
• ACKNOWLEDGMENTS

We investigate the possible superconducting instabilities of strongly correlated electron materials using a generalization of linear response theory to external pairing fields depending on frequency. We compute a pairing susceptibility depending on two times, allowing us to capture dynamical pairing and in particular odd-frequency solutions. We first benchmark this method on the attractive one-band Hubbard model and then consider the superconductivity of strontium ruthenate Sr 2 RuO 4 within single-site dynamical mean-field theory, hence restricting ourselves to pairing states which are momentum independent in the orbital basis. The symmetry of the superconducting order parameter of this material is still debated, and local odd-frequency states have been proposed to explain some experimental discrepancies. In the temperature range studied, we find that the leading eigenvectors are odd-frequency intraorbital spin-triplet states, while the eigenvectors with the highest predicted transition temperature correspond to even-frequency intraorbital spin-singlet states. The latter include a state with d -wave symmetry when expressed in the band basis.

• Received 5 January 2024
• Revised 6 May 2024
• Accepted 24 July 2024

DOI: https://doi.org/10.1103/PhysRevB.110.054509

©2024 American Physical Society

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• 1 Center for Computational Quantum Physics, Flatiron Institute , 162 Fifth Avenue, New York, New York 10010, USA
• 2 Collège de France , 11 place Marcelin Berthelot, 75005 Paris, France
• 3 Centre de Physique Théorique , Ecole Polytechnique, CNRS, Institut Polytechnique de Paris , 91128 Palaiseau Cedex, France
• 4 DQMP, Université de Genève , 24 quai Ernest Ansermet, CH-1211 Genève, Switzerland
• 5 Université Paris-Saclay , CNRS, CEA, Institut de physique théorique, 91191, Gif-sur-Yvette, France
• * Contact author: [email protected]

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Dynamical pairing responses to external pairing fields for the half-filled attractive Hubbard model with U = − 3 t . The fields are proportional to Legendre polynomials P α ( τ ) that generate the purely (a) odd-frequency and (b) even-frequency finite dynamical pairing responses χ α ( τ ) . We used t = 1 ,   β = 3 , and ϕ α = 0.01 ∀ α .

Convergence of the (a) leading and (b) subleading eigenvectors with respect to the size of the Legendre basis N . The system is the half-filled attractive Hubbard model on the Bethe lattice with the same parameters used as in Fig.  1 . In the S P O T representation introduced in Appendix  pp1 and used later in the text, (a) would be classified as − S + P + O + T and (b) as + S + P + O − T . Inset: convergence of the associated eigenvalue versus N .

Imaginary-time-structure evolution of the leading superconducting eigenvector (dashed lines, right axis) into the anomalous Green's function (solid line, left axis) across the superconducting transition in the half-filled attractive Hubbard model. The same parameters as in Fig.  1 were used and the leading eigenvectors were obtained with a Legendre basis of size N = 8 .

The superconducting critical temperature in the half-filled attractive Hubbard model for U / t = − 3 . The red left axis presents the spontaneous appearance of the pairing amplitude in the superconducting state around T ∼ 1 6 . The blue right axis shows that this critical temperature is indicated from the normal state by a divergence of the dynamical pairing susceptibility. We compare two methods to obtain the inverse susceptibility: dark squares for the two-time linear response and light blue pentagons for the static method (in arbitrary units).

Examples of dynamical pairing responses to external pairing fields in strontium ruthenate. Here, β = 80 and ϕ α ; μ 1 μ 2 = 0.0002 ∀ α , μ 1 , μ 2 . In (a) and (b), the external pairing fields are odd frequency and even frequency, respectively, on the z x ; z x intra-spin-orbital component. The pair-hopping term in the interaction induces a response in all intra-spin-orbital components. In (c), the external pairing fields are even frequency on the z x ; y z inter-spin-orbital component. The spin-flip term in the interaction induces a response in the inter-spin-orbital component related by exchanging the orbital labels, here y z ; z x . The components not illustrated here are smaller than 10 − 7 for these fields.

Inverse of the dominant superconducting eigenvalues in the normal state of strontium ruthenate. The dotted lines are fits to find the temperatures at which the inverse eigenvalues diverge. Additional eigenvectors and details can be found in Appendix  pp7 . Note that odd T refers to the exchange of relative time and not time-reversal symmetry [ 4, 6 ].

Imaginary-time dependence of one of the largest components ( z x ; z x ) of the (a)  φ odd- T z x ± y z ( τ ) and (b)  φ odd- S z x ± y z ( τ ) states. These pairings take advantage of the strong retardation of the pairing interaction to avoid even-time pairing and instead utilize off-time pairing. Other components and different temperatures of all the dominant states are presented in Appendix  pp7 .

Projection on the Fermi surface of the gap functions (a)  φ odd- T z x + y z and (b)  φ odd- S z x − y z rotated to the band basis, for one flavor of pseudospin.

First eight rescaled Legendre polynomials P ( τ ) .

Convergence with respect to ϕ α = ϕ 0 ∀ α of the (a) leading and (b) subleading eigenvalues, at different inverse temperatures β . At larger β ,   ϕ 0 needs to be reduced to remain in the linear regime.

Comparison of the eigenvalues obtained using the power method (light green crosses) and diagonalizing the dynamical pairing susceptibility constructed in the Legendre basis (blue squares). The other components of this figure were presented in Fig.  4 .

Temperature dependence of the inverse of the dominant superconducting eigenvalues in the normal state of strontium ruthenate. (a) Corresponds to the odd-frequency channel and (b) to the even-frequency channel. The dotted lines are fits to find the temperatures at which the inverse eigenvalues diverge. Figure  15 of Appendix  pp7-s3 presents the temperature dependence and orbital structure of the three dominant eigenvectors in each channel. The dotted lines are fits to find the temperatures at which the inverse eigenvalues diverge. Different lines with the same color correspond to eigenvectors with the same symmetry, but orthogonal in imaginary time. An example is discussed in Appendix  pp7-s2 .

Eigenvalue convergence with respect to the Legendre basis size N at β = 80 and ϕ α ; μ 1 μ 2 = 0.0002 for all α , μ 1 , μ 2 . The three largest eigenvalues are converged at N = 4 . At larger N , new states appear corresponding to higher-frequency version of dominant eigenvectors, discussed in the next section. They can only be captured by considering higher-order polynomials. At N = 8 , the important eigenvalues are well converged.

Temperature dependence one component ( y z ; y z ) of three orthogonal states with the same hosting orbitals and S P O T classification associated to the φ odd- T z x − y z state. Once τ is rescaled by β , each state preserves it imaginary-time structure with temperature.

Inverse temperature dependence and imaginary-time structure of the nonvanishing spin-orbital components of φ odd- T x y in (a), φ odd- T z x + y z in (b), and φ odd- T z x − y z in (c), labeled as + S + P + O − T , along with φ odd- S z x + y z in (d), φ odd- S z x − y z in (e), and φ odd- S x y in (f), labeled as − S + P + O + T . The eigenvalues of these states are shown in Fig.  12 .

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Journal of Materials Chemistry A

Converting the covalent organic framework linkage from hydrazone to thiadiazole toward blue light-powered selective conversion of organic sulfides †.

* Corresponding authors

a Hubei Key Lab on Organic and Polymeric Optoelectronic Materials, College of Chemistry and Molecular Sciences, Wuhan University, Wuhan 430072, China E-mail: [email protected]

b Hubei Provincial Engineering Research Center of Racing Horse Detection and Application Transformation, Wuhan Business University, Wuhan 430056, China

Covalent organic frameworks (COFs) are composed of various organic linkers with dynamic covalent bonds as linkages, which, in part, establish functionality. Nevertheless, these covalent bonds are usually not robust enough under reaction conditions and, therefore, can be converted into irreversible ones because the application of COFs hinges on the characteristics of these linkages. In this work, TFPT-COF, a triazine-based COF with a hydrazone linkage, is constructed. Subsequently, the hydrazone linkage is converted to a thiadiazole linkage through thionation and oxidative cyclization at 115 °C, converting TFPT-COF to TDA-COF with a robust and irreversible thiadiazole linkage. The band gap is narrowed, and the electronic structure is altered from TFPT-COF to TDA-COF based on density functional theory calculations. Besides, experimental results suggest that TDA-COF possesses broadened light absorption and improved optoelectronic properties compared to TFPT-COF. Converting TFPT-COF to TDA-COF significantly shifts the photocatalytic activity for the selective conversion of thioanisoles to sulfoxides with oxygen (O 2 ). TDA-COF drives the blue light-powered conversion of thioanisoles to sulfoxides with O 2 , whereas TFPT-COF shows almost no activity. TDA-COF is especially accessible for converting various sulfides to sulfoxides with high selectivities, while maintaining high photocatalytic activity and stability over six cycles. This work demonstrates that converting the dynamic linkages of COFs to irreversible ones contributes positively to photocatalytic activities.

• This article is part of the themed collections: Nanomaterials for a sustainable future: From materials to devices and systems and Journal of Materials Chemistry A HOT Papers

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Converting the covalent organic framework linkage from hydrazone to thiadiazole toward blue light-powered selective conversion of organic sulfides

Y. Wang, J. Shi, X. Dong, F. Zhang and X. Lang, J. Mater. Chem. A , 2024, Advance Article , DOI: 10.1039/D4TA04548C

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• DOI: 10.1021/JP500546R
• Corpus ID: 100631708

Band Gap Engineering of SnO2 by Epitaxial Strain: Experimental and Theoretical Investigations

• Wei Zhou , Yanyu Liu , +1 author P. Wu
• Published 12 March 2014
• Physics, Materials Science, Engineering
• Journal of Physical Chemistry C

157 Citations

Epitaxial strain effect on the band gap of a ga2o3 wide bandgap material, effect of biaxial strain on sno2 bandgap: first-principles calculations, strain sensitivity in ferromagnetism and band gap of epitaxial sn0.94k0.06o2 thin films, strain-induced optical band gap variation of sno2 films, band-gap engineering of sno2, sno2 improved thermoelectric properties under compressive strain, band gap engineering of bulk and nanosheet sno: an insight into the interlayer sn-sn lone pair interactions., influence of strain on the band gap of cu2o, continuously controlled optical band gap in oxide semiconductor thin films., strain effects on the electronic structure of znsnp2 via modified becke–johnson exchange potential, 28 references, effective band gap narrowing of anatase tio2 by strain along a soft crystal direction, electronic structure and optical properties of sb-doped sno2, strain effects and band parameters in mgo, zno, and cdo, frequency shifts of the e2high raman mode due to residual stress in epitaxial zno thin films, hybrid functional investigations of band gaps and band alignments for aln, gan, inn, and ingan., structural instability of sn-doped in2o3 thin films during thermal annealing at low temperature, quasiparticle energies and uniaxial pressure effects on the properties of sno2, unusual compression behavior of anatase tio2 nanocrystals., group-v impurities in sno2 from first-principles calculations, transparent conducting f-doped sno2 thin films grown by pulsed laser deposition, related papers.

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Band gap analysis of periodic structures based on cell experimental frequency response functions (FRFs)

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An approach is proposed to estimate the transfer function of the periodic structure, which is known as an absorber due to its repetitive cells leading to the band gap phenomenon. The band gap is a frequency range in which vibration will be inhibited. A transfer function is usually performed to gain band gap. Previous scholars regard estimation of the transfer function as a forward problem assuming known cell mass and stiffness matrices. However, the estimation of band gap for irregular or complicated cells is hardly accurate because it is difficult to model the cell exactly. Therefore, we treat the estimation as an inverse problem by employing modal identification and curve fitting. A transfer matrix is then established by parameters identified through modal analysis. Both simulations and experiments have been performed. Some interesting conclusions about the relationship between modal parameters and band gap have been achieved.

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Joannopoulos, J.D., Meade, R.D., Winn, J.N.: Photoniccrystals. Princeton University Press, Princeton (1995)

Google Scholar  

Yablonovitch, E.: Inhibited spontaneous emission in solid-state physics and electronics. Phys. Rev. Lett. 58 , 2059–2062 (1987)

Article   Google Scholar  

John, S.: Strong localization of photons in certain disordered dielectric superlattices. Phys. Rev. Lett. 58 , 2486–2489 (1987)

Wang, Y.Z., Li, F.M., Kishimoto, K.: Band gaps of elastic waves in three-dimensional piezoelectric phononic crystals with initial stress. Eur. J. Mech. 29 , 182–189 (2010)

Wang, Y.Z., Li, F.M., Kishimoto, K., et al.: Wave band gaps in three-dimensional periodic piezoelectric structures. Mech. Res. Commun. 36 , 461–468 (2009)

Article   MATH   Google Scholar  

Yan, Z.Z., Wang, Y.S.: Wavelet-based method for calculating elastic band gaps of two-dimensional phononic crystals. Phys. Rev. B 74 , 224303 (2006)

Yan, Z.Z., Wang, Y.S.: Calculation of band structures for surface waves in two-dimensional phononic crystals with a wavelet-based method. Phys. Rev. B 78 , 094306 (2008)

Casadei, F., Rimoli, J.J., Ruzzene, M.: Multiscale finite element analysis of wave propagation in periodic solids. Finite Elem. Anal. Des. 108 , 81–95 (2016)

Bardi, I., Peng, G., Petersson, L.E.R.: Modeling periodic layered structures by shell elements using the finite-element method. IEEE Trans. Magn. 52 , 1–4 (2016)

Ozaki, S., Hinata, K., Senatore, C., et al.: Finite element analysis of periodic ripple formation under rigid wheels. J. Terramech. 61 , 11–22 (2015)

Zhang, Y., Shang, S., Liu, S.: A novel implementation algorithm of asymptotic homogenization for predicting the effective coefficient of thermal expansion of periodic composite materials. Acta. Mech. Sin. 33 , 368–381 (2017)

Article   MathSciNet   MATH   Google Scholar  

Wu, Z.J., Li, F.M., Zhang, C.: Vibration band-gap properties of three-dimensional kagome lattices using the spectral element method. J. Sound Vib. 341 , 162–173 (2015)

Wu, Z.J., Li, F.M., Wang, Y.Z.: Vibration band gap behaviors of sandwich panels with corrugated cores. Comput. Struct. 129 , 30–39 (2013)

Wen, S.R., Wu, Z.J., Lu, N.L.: High-precision solution to the moving load problem using an improved spectral element method. Acta Mech. Sin. 34 , 68–81 (2018)

Mead, D.J.: Wave propagation in continuous periodic structures: research contributions from southampton. J. Sound Vib. 190 , 495–524 (1996)

Gupta, G.S.: Natural frequencies of periodic skin-stringer structures using a wave approach. J. Sound Vib. 16 , 567–580 (1971)

Gupta, G.S.: Dynamics of Periodically Stiffened Structures Using a Wave Approach. University of Southampton, Southampton (1970)

Lin, Y.K., McDaniel, T.J.: Dynamics of beam-type periodic structures. J. Eng. Ind. 91 , 1133 (1969)

Ruzzene, M., Tsopelas, P.: Control of wave propagation in sandwich plate rows with periodic honeycomb core. J. Eng. Mech. 129 , 975–986 (2003)

Richards, D., Pines, D.J.: Passive reduction of gear mesh vibration using a periodic drive shaft. J. Sound Vib. 264 , 317–342 (2003)

Wang, Y.Z., Li, F.M., Huang, W.H., et al.: The propagation and localization of Rayleigh waves in disordered piezoelectric phononic crystals. J. Mech. Phys. Solids 56 , 1578–1590 (2008)

Wang, Y.Z., Li, F.M., Wang, Y.S.: Influences of active control on elastic wave propagation in a weakly nonlinear phononic crystal with a monoatomic lattice chain. Int. J. Mech. Sci. 106 , 357–362 (2016)

Wang, Y.Z., Wang, Y.S.: Active control of elastic wave propagation in nonlinear phononic crystals consisting of diatomic lattice chain. Wave Motion 78 , 1–8 (2018)

Article   MathSciNet   Google Scholar  

Waki, Y.: On the Application of Finite Element Analysis to Wave Motion in One-Dimensional Waveguides. University of Southampton, Southampton (2007)

Mace, B.R., Duhamel, D., Brennan, M.J., et al.: Finite element prediction of wave motion in structural waveguides. J. Acoust. Soc. Am. 117 , 2835–2843 (2005)

Duhamel, D., Mace, B.R., Brennan, M.J.: Finite element analysis of the vibrations of waveguides and periodic structures. J. Sound Vib. 294 , 205–220 (2006)

Zhong, W.X., Williams, F.W.: On the direct solution of wave propagation for repetitive structures. J. Sound Vib. 181 , 485–501 (1995)

Zhong, W.X., Williams, F.W., Leung, A.Y.T.: Symplectic analysis for periodical electro-magnetic waveguides. J. Sound Vib. 267 , 227–244 (2003)

Hendrickx, W., Dhaene, T.: A discussion of “Rational approximation of frequency domain responses by vector fitting”. IEEE Trans. Power Syst. 21 , 441–443 (2006)

Gustavsen, B.: Relaxed vector fitting algorithm for rational approximation of frequency domain responses. In: IEEE Workshop on Signal Propagation on Interconnects, Berlin, Germany (2006)

Gustavsen, B., Semlyen, A.: Rational approximation of frequency domain responses by vector fitting. IEEE Trans. Power Deliv. 14 , 1052–1061 (1999)

Jensen, J.S.: Phononic band gaps and vibrations in one- and two-dimensional mass-spring structures. J. Sound Vib. 266 , 1053–1078 (2003)

Cremer, L., Heckl, M., Petersson, B.A.T.: Structure Borne Sound: Structural Vibrations and Sound Radiation at Audio Frequencies, 3rd edn. Springer, Berlin (2005)

Book   Google Scholar  

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 11272235).

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Wu, LJ., Song, HW. Band gap analysis of periodic structures based on cell experimental frequency response functions (FRFs). Acta Mech. Sin. 35 , 156–173 (2019). https://doi.org/10.1007/s10409-018-0781-0

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Received : 20 March 2018

Revised : 19 April 2018

Accepted : 03 May 2018

Published : 10 August 2018

Issue Date : 07 February 2019

DOI : https://doi.org/10.1007/s10409-018-0781-0

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• Published: 10 August 2024

Origin and tuning of bandgap in chiral phononic crystals

• Wei Ding 1 , 2   na1 ,
• Rui Zhang 1   na1 ,
• Tianning Chen 1 ,
• Shuai Qu 2 , 3 ,
• Dewen Yu 1 , 2 ,
• Liwei Dong 2 , 4 ,
• Jian Zhu   ORCID: orcid.org/0000-0002-4198-9009 1 ,
• Yaowen Yang   ORCID: orcid.org/0000-0002-7856-2009 2 &
• Badreddine Assouar   ORCID: orcid.org/0000-0002-5823-3320 5  

Communications Physics volume  7 , Article number:  272 ( 2024 ) Cite this article

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• Mechanical properties

The wave equation revealing the wave propagation in chiral phononic crystals, established through force equilibrium law, conceals the underlying physical information, such as the essence of the motion coupling and the inertial amplification effect. This has led to a controversy over the bandgap mechanism. In this article, we theoretically unveil the reason for this controversy, and put forward an alternative approach from wave behavior to formulate the wave equation, offering an alternative pathway to articulate the bandgap physics directly. Based on the physics revealed by our theory method, we identify the obstacles in coupled acoustic and optic branches to widen and lower the bandgap. Then we introduce an approach based on spherical hinges to decrease the barriers, for customizing the bandgap frequency and width. Finally, we validate our proposal through numerical simulation and experimental demonstration.

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Non-Hermitian chiral phononics through optomechanically induced squeezing

Introduction.

The bandgap property in phononic crystals (PnCs) is associated with extreme spatial dispersion 1 , wave guidance 2 , 3 , and thermal physics 4 . Therein, since the inertial amplification effect induced by chirality, which is beneficial for lowering the bandgap beyond the barriers constrained by mass and stiffness 5 , 6 , 7 , enables the chiral PnCs the superior performance at low-frequency regime, thus expanding its applicable scope in the elastic-wave fields 8 , 9 . The inertial amplification concept in the mechanical perspective presented in the seminal study of Yilmaz refers to a dynamic virtual inertia attached to a static mass, thus reducing the eigenfrequency of the system 10 . However, the bandgap mechanism of chiral PnCs has always been controversial 11 , 12 , 13 . The seminal theories have indicated the inertial amplification as the mechanism behind such a bandgap 14 , 15 , while different chirality assemblies have different dispersion spectrum 14 , 16 . Therefore, the mechanism has been attributed to inertial amplification and the relative orientation of adjacent chiral centers in the syndiotactic system 14 .

More recently, two explanations have been reported for a physical explanation. The first is the dimer chain 13 , where coupling longitudinal and torsional waves is similar to the coupled transverse and rotational waves in the periodic mass-spring system 17 . The study 13 concluded that a monatomic chain effect, i.e., the so-called inertial amplification method, cannot support the bandgap phenomenon. The second explanation is related to analogous Thomson scattering 12 to consolidate the inertial amplification claim 15 . In addition, the analogous Thomson scattering 12 physically detailed that inertial amplification of this chiral subunit cell is induced by coupling two or more polarizations in the same lumped mass and chirality is to achieve the secondary scattering for destructive interferences. These two theoretical interpretations are plausible because of the validation, yet they are contradictory since the debate about the existence or absence of inertial amplification.

Here, we develop a theoretical analysis based on the wave behavior in chiral PnCs, to clarify the cause of the contradiction, and then unify and refine the bandgap mechanisms. We demonstrate that the wave equation directly derived from force equilibrium law will conceal the underlying physics, e.g., inertial amplification. Our method allows to articulate bandgap physics, and calculate the transmission simultaneously. In contrast to the conventional theoretical method 6 , 10 , 18 , it allows observing the fundamental physical parameters of acoustic and optic modes under the assumption of elastic ligaments, i.e., inertial amplification coefficient, bending stiffness, stretch stiffness, and their origins and interactions. Our analysis pointed out that the rise of the inertial amplification coefficient is closely related to the bending and stretch stiffness. Consequently, the bandgap width and the reduction of the starting frequency are mutually constrained, which poses a significant challenge to the realization of wide subwavelength bandgaps (the effects of the geometrical dimensions, characterized in equivalent stiffness 19 and equivalent mass 20 , 21 , are considered in normalization). To transcend this barrier, the spherical hinges and the spiral springs are employed to partial-decouple these coupled physical parameters. The numerical and experimental results validate the correctness and the feasibility of our proposals in the theory and geometrical mode.

Theoretical observation of bandgap origin

In the chiral subunit cell (Fig.  1a ), if there is a longitudinal input \({{{{\boldsymbol{P}}}}}_{{{{\boldsymbol{I}}}}}\) on disk I ( \({{{{\boldsymbol{P}}}}}_{{{{\boldsymbol{I}}}}}{{{\boldsymbol{=}}}}{{{{\boldsymbol{A}}}}}_{{{{\bf{1}}}}}{{{{\boldsymbol{e}}}}}^{{{{\boldsymbol{-}}}}{{{\boldsymbol{i}}}}{{{\boldsymbol{(}}}}{{{\boldsymbol{wt}}}}{{{\boldsymbol{+}}}}{{{{\boldsymbol{\phi }}}}}_{{{{\bf{1}}}}}{{{\boldsymbol{)}}}}}\) , where \({{{{\boldsymbol{\phi }}}}}_{{{{\bf{1}}}}}{{{\boldsymbol{=}}}}{{{\bf{0}}}}\) ), the motion provided by \({{{{\boldsymbol{P}}}}}_{{{{\boldsymbol{I}}}}}\) will propagate in the bending deformation (Fig.  1c ) and stretch deformation (Fig.  1d ) of the ligaments simultaneously. Neglecting the local deformation of the ligaments and disks, we can observe two polarizations at disk II, i.e., longitudinal polarization \(\,{{{{\boldsymbol{P}}}}}_{{{{\boldsymbol{l}}}}}\) ( \({{{{\boldsymbol{P}}}}}_{{{{\boldsymbol{lb}}}}}{{{\boldsymbol{+}}}}{{{{\boldsymbol{P}}}}}_{{{{\boldsymbol{ls}}}}}\) ) and rotational polarization \({{{{\boldsymbol{P}}}}}_{{{{\boldsymbol{r}}}}}\) ( \({{{{\boldsymbol{P}}}}}_{{{{\boldsymbol{r}}}}{{{\boldsymbol{b}}}}}{{{\boldsymbol{+}}}}{{{{\boldsymbol{P}}}}}_{{{{\boldsymbol{rs}}}}}\) ) (where \({{{{\boldsymbol{P}}}}}_{{{{\boldsymbol{ij}}}}}\) denotes that the \({{{{\boldsymbol{j}}}}}^{{{{\boldsymbol{th}}}}}\) deformation mode of the ligaments induces the \({{{{\boldsymbol{i}}}}}^{{{{\boldsymbol{th}}}}}\) polarization of the disk. In detail, subscript \({{{\boldsymbol{i}}}}\) can be longitudinal polarization \({{{\boldsymbol{l}}}}\) or rotational polarization \({{{\boldsymbol{r}}}}\) . Subscript \({{{\boldsymbol{j}}}}\) denotes the \({{{{\boldsymbol{j}}}}}^{{{{\boldsymbol{th}}}}}\) deformation mode of the ligaments, which can be bending mode \({{{\boldsymbol{b}}}}\) or stretch mode \({{{\boldsymbol{s}}}}\) ).

a Schematics of the conventional chiral subunit cell and its macroscopic polarizations under the longitudinal input \({{{{\boldsymbol{P}}}}}_{{{{\boldsymbol{I}}}}}\) . The green arrow denotes the longitudinal input mode \({{{{\boldsymbol{P}}}}}_{{{{\boldsymbol{I}}}}}\) ; the blue arrows denote the polarizations determined by the bending deformation of the ligaments; the red arrows denote the polarizations determined by the stretch deformation of the ligaments. Therein, \({{{{\boldsymbol{P}}}}}_{{{{\boldsymbol{lb}}}}}\) is the longitudinal polarization determined by bending deformation; \({{{{\boldsymbol{P}}}}}_{{{{\boldsymbol{rb}}}}}\) is the rotational polarization determined by bending deformation; \({{{{\boldsymbol{P}}}}}_{{{{\boldsymbol{ls}}}}}\) is the longitudinal polarization determined by stretch deformation; \({{{{\boldsymbol{P}}}}}_{{{{\boldsymbol{rs}}}}}\) is the longitudinal polarization determined by stretch deformation. b The difference between the static mass and the dynamical inertia in the chiral subunit cell. The gray shadow refers to the extra inertia induced by the polarization coupling, such as \({{{{\boldsymbol{P}}}}}_{{{{\boldsymbol{lb}}}}}\)  +  \({{{{\boldsymbol{P}}}}}_{{{{\boldsymbol{rb}}}}}\) as well as \({{{{\boldsymbol{P}}}}}_{{{{\boldsymbol{ls}}}}}\)  +  \({{{{\boldsymbol{P}}}}}_{{{{\boldsymbol{rs}}}}}\) . c Polarizations of the acoustic mode determined by the bending deformation of the ligaments. The inset is the zoom of the directions of the coupled polarization ( \({{{{\boldsymbol{P}}}}}_{{{{\boldsymbol{lb}}}}}\)  +  \({{{{\boldsymbol{P}}}}}_{{{{\boldsymbol{rb}}}}}\) ). d Polarizations of the optic mode determined by the stretch deformation. The inset is the zoom of the directions of the coupled polarization ( \({{{{\boldsymbol{P}}}}}_{{{{\boldsymbol{ls}}}}}\)  +  \({{{{\boldsymbol{P}}}}}_{{{{\boldsymbol{rs}}}}}\) ). e The relation between the static mass and the dynamical inertia in the classical diatomic chain. \({{{{\boldsymbol{P}}}}}_{{{{\boldsymbol{l}}}}}\) means there is only longitudinal polarization in the classical diatomic chain. f Schematic of the classical diatomic chain. Polarization schematics of the acoustic mode ( g ) and optic mode ( h ) in the classical diatomic unit cell. i Dispersion-spectrum schematic of the classical diatomic unit cell. Therein, the shaded area indicates the bandgap.

In the scenario of Fig.  1c , based on the right-hand spiral rule, the disk II will have a longitudinal polarization along \(+z\) -axis ( \({P}_{{lb}}\) ) and a rotation polarization around \(+z\) -axis ( \({P}_{{rb}}\) ) due to the bending mode. While in the scenario denoted by Fig.  1d , the disk 2 will have \(-z\) -axis rotational polarization ( \({P}_{{rs}}\) ) in addition to the \(+z\) -axis longitudinal polarization ( \({P}_{{ls}}\) ) due to the stretch mode. In short, there must be 4 polarizations in disk II , i.e., \({P}_{{lb}}\) , \({P}_{{rb}}\) , \({P}_{{ls}}\) , and \({P}_{{rs}}\) . Therein, the longitudinal polarization \({P}_{{lb}}\) and \({P}_{{ls}}\) vibrate in the same frequency and initial directions, while the rotational polarization \({P}_{{rb}}\) and \({P}_{{rs}}\) have the same frequency but opposite initial directions.

Because \({P}_{{lb}}\) and \({P}_{{rb}}\) are resulted from the bending deformation of the ligaments, they will have the same frequency and the same phase at any time. Therefore, for the \({i}^{{th}}\) lumped mass, assuming an extremely small harmonic displacement (Otherwise, there will be nonlinearity in this chiral unit cell 5 ), \({P}_{{lb}}\) and \({P}_{{rb}}\) are linearly correlated by the inertial amplification coefficient \(p\) , as illustrated by Eq. ( 1 ).

where \(p\) denotes the conversion coefficient from longitudinal polarization \({P}_{{lb}}\) to rotational polarization \({P}_{{rb}}\) and it is characterized as the inertial amplification coefficient in the inertia matrix 12 , 15 . \({u}_{i}^{l}\) refers to longitudinal polarization induced longitudinal displacement and \({\psi }_{i}^{l}\) refers to longitudinal polarization induced rotational displacement. \({A}_{i}^{l}\) implies the translational amplitude of the \(i\) th lumped mass. The superscript \(l\) indicates that the longitudinal polarization \(u\) and rotational polarization \(\psi\) originate from the longitudinal mode. \(\varphi\) refers to the initial phase.

Like \({P}_{{lb}}\) and \({P}_{{rb}}\) , for the \({i}^{{th}}\) lumped mass, \({P}_{{rs}}\) and \({P}_{{ls}}\) satisfy

where \(q\) indicates the conversion coefficient from longitudinal polarization \({P}_{{ls}}\) to rotational polarization \({P}_{{rs}}\) . Because the sense of \(q\) is exactly opposite to that of \(p\) , \(q=\frac{1}{p}\) (see Supplementary Note  4 for more details). The superscript \(s\) indicates that the longitudinal polarization \(u\) and rotational polarization \(\psi\) originate from the stretch mode.

Because the stretch stiffness \({k}_{s}\) is different from the bending stiffness \({k}_{b}\) , \({P}_{{rb}}\) and \({P}_{{rs}}\) must have different phases, i.e., \({\varphi }^{l}\)  ≠  \({\varphi }^{s}\) , as do \({P}_{{lb}}\) and \({P}_{{ls}}\) . This means \({P}_{{lb}}+{P}_{{rb}}\) and \({P}_{{ls}}+{P}_{{rs}}\) must be two independent wave modes, although we can only see the macroscopic results of longitudinal movement \({P}_{l}\) ( \({P}_{{lb}}+{P}_{{ls}}\) ) and \({P}_{r}\) ( \({P}_{rb}+{P}_{{rs}}\) )) rotational movement rather than the results of \({P}_{{lb}}+{P}_{{rb}}\) and \({P}_{{ls}}+{P}_{{rs}}\) .

Therefore, generally, in the global coordinate system, for the \({i}^{{th}}\) lumped mass, the longitudinal displacement \(u\) is determined as

and the rotational displacement \(\vartheta\) is determined as

Based on the Lagrangian method (see Supplementary Eqs. ( 1 )–( 28) in Supplementary Note  1 for the derivation process), we can obtain the longitudinal displacement \({u}_{i}\) and rotational displacement \(\vartheta\) . In this way, one can see that the inertial matrix (Supplementary Equation ( 17) ) and stiffness matrix (Supplementary Eqs. ( 18 )–( 25) ) are similar but not identical to current reported results 13 . According to the above analysis, the theoretical transmission (as denoted by the red solid line in Fig.  2b ) and dispersion spectrum (Fig.  2c ) can be obtained directly. However, the inertial amplification cannot be observed in the inertial matrix (Supplementary Eq. ( 17) ). The wave equation only reveals one fact, i.e., the longitudinal polarization is coupled with torsional polarization. However, it has been demonstrated that only specific couplings (such as the syndiotactic PnCs 8 , 14 ) rather than all couplings can give rise to such a bandgap 14 , 16 . Therefore, the explanation of the coupling 13 needs to be clarified further.

a , b Theoretical and numerical transmissions of the conventional chiral PnCs. The gray line is the numerical results, and the others are the theoretical results. Therein, “no \({{{{\boldsymbol{k}}}}}_{{{{\boldsymbol{s}}}}}\) ” denotes the results of neglecting the stretch mode; the red and blue lines in ( b ) are the results of considering the stretch mode, where the word “stiffness” in braces denotes that the result is obtained based on Supplementary Eqs. ( 17 )–( 28) , and the word “inertial” denotes that the theoretical result is obtained based on Supplementary Eqs. ( 29 )–( 39) . c Theoretical and numerical dispersion spectra (see Supplementary Note  2 for the governing equation of dispersion spectrum). Therein, the gray star-shaped lines are the numerical dispersion curves and the others refers to the theoretical dispersion curves. The gray shaded area refers to the bandgap range. d The relative amplitudes of the longitudinal displacement. Therein, \({{{{\boldsymbol{u}}}}}_{{{{\boldsymbol{b}}}}}\) and \({{{{\boldsymbol{u}}}}}_{{{{\boldsymbol{s}}}}}\) refer to the relative longitudinal displacement induced by the bending and stretch modes, respectively. The relative amplitude is calculated by dividing the absolute amplitude by the input amplitude. The subscript “b” refers to the bending mode and “s” refers to be the stretch mode. e The relative amplitudes of the rotational displacement. Therein, \({{{{\boldsymbol{R}}}}}_{{{{\boldsymbol{b}}}}}\) and \({{{{\boldsymbol{R}}}}}_{{{{\boldsymbol{s}}}}}\) refer to the relative rotational displacement induced by the bending and stretch modes, respectively. The relative amplitude is calculated by dividing the absolute amplitude by the input amplitude. f Displacement contours of the boundaries of the bandgap, where \({{{{\boldsymbol{p}}}}}_{{{{\boldsymbol{l}}}}{{{\boldsymbol{1}}}}}\) , \({{{{\boldsymbol{p}}}}}_{{{{\boldsymbol{l}}}}{{{\boldsymbol{2}}}}}\) , \({{{{\boldsymbol{p}}}}}_{{{{\boldsymbol{u}}}}{{{\boldsymbol{1}}}}}\) and \({{{{\boldsymbol{p}}}}}_{{{{\boldsymbol{u}}}}{{{\boldsymbol{2}}}}}\) correspond to the frequency points marked in ( b ). Therein, \({{{{\boldsymbol{p}}}}}_{{{{\boldsymbol{l}}}}{{{\boldsymbol{1}}}}}\) and \({{{{\boldsymbol{p}}}}}_{{{{\boldsymbol{l}}}}{{{\boldsymbol{2}}}}}\) denote the lower-boundary displacement contours of the bandgap; \({{{{\boldsymbol{p}}}}}_{{{{\boldsymbol{u}}}}{{{\boldsymbol{1}}}}}\) and \({{{{\boldsymbol{p}}}}}_{{{{\boldsymbol{u}}}}{{{\boldsymbol{2}}}}}\) denote the upper-boundary displacement contours of the bandgap.

It is worth noting that, \({P}_{{lb}}+{P}_{{rb}}\) and \({P}_{{ls}}+{P}_{{rs}}\) are two independent wave modes. Therefore, it allows us to regard \({u}_{i}^{b}\) , \({u}_{i}^{s}\) , \({\psi }_{i}^{b}\) , and \({\psi }_{i}^{s}\) as the independent variables, similarly, based on the Lagrangian method (See Supplementary Eqs. ( 29 )–( 39) in Supplementary for the derivation process), as a result, the inertial matrix and stiffness matrix will be significantly different, as illustrated by Supplementary Eqs. ( 31 )–( 34 ) . Based on Supplementary Eqs. ( 31 )–( 39) and physical parameters listed in Supplementary Note  5 , we can also obtain the theoretical transmission (as denoted by the blue dashed line in Fig.  2b ), which is consistent with the numerical results. Figure  2b illustrates that both paths of establishing wave equations can yield identical transmissions to the numerical results.

However, in contrast to the former classical theory (Supplementary Eqs. ( 17 )–( 27) ), several essential information can be captured in the latter derivation method. First, as denoted by Supplementary Eqs. ( 35 )–( 37) , the stiffness matrix does not indicate the coupling effect between longitudinal polarization and rotational polarization, but the inertial matrix does. Second, the inertial matrix (Supplementary Eqs. ( 31 )–( 34) ) will reveal the existence of inertial amplification, which is derived from the coupling effect. Third, both bending and stretch modes can realize the motion coupling and thus obtain the inertial amplification effect, as denoted by \(p\) in Supplementary Equation ( 32) and \(q\) in Supplementary Eq. ( 34) . Fourth, the motion coupling guided by the bending mode is characterized by longitudinal polarization (because the primary diagonal element of \({{{{\rm{M}}}}}_{11}\) includes \({m}_{i}\) and the non-diagonal element is \({I}_{i}\) ), while the motion coupling guided by the stretch mode is characterized by the rotational mode (because the primary diagonal element of \({{{{\rm{M}}}}}_{22}\) includes \({I}_{i}\) and the secondary diagonal element is \({m}_{i}\) ).

At this point, we can learn about that the bandgap in the chiral PnCs must be accompanied by two wave modes that will truncate the bandgap range. These two wave modes are similar to the acoustic mode and optic mode of the classical diatomic chain, as shown in Fig.  1f , which together determine the bandgap range (Fig.  1i ). Therein, the two atoms of the diatomic unit cell vibrate in the same phase as an acoustic mode (as illustrated by Fig.  1g ), while the two atoms vibrate with opposite phases as an optic mode (as illustrated by Fig.  1h ). The same phenomena can be observed in the chiral PnC, as illustrated in Fig.  1 c and d . According to the left-handed feature of the subunit cell, under the longitudinal input \({P}_{i}\) , the first expected case is the longitudinal motion of the disk II accompanied by a rotation around the \(-z\) -axis (Based on the right-hand spiral rule), as shown in Fig.  1c , which corresponds to the acoustic mode in the classical diatomic chain. Similarly, for the optic mode, as shown in Fig.  1d , the longitudinal motion of the \(\,{m}_{2}\) is accompanied by a rotation around \(+z\) -axis, which corresponds to the optic mode in the classical diatomic chain. Therefore, the lower boundary of the bandgap in chiral PnCs can be named the acoustic branch (the two red pass bands in Fig.  2c ) since the vibration in the phase of adjacent atoms, and the upper boundary can be named the optic branch (the two blue pass bands in Fig.  2c ) because it is similar to that in the long-wavelength limit of an optic mode 22 . The bandgap will convert into Bragg scattering type after the optic branch 23 .

Besides, the numerical deformation contours also demonstrate the similarity between the theoretical wave modes and the classical diatomic chain, as shown in Fig.  2f . The rotational directions of \({p}_{u1}\) & \({p}_{u2}\) are opposite to that of \({p}_{l1}\) & \({p}_{l2}\) when the translation is along \(+z\) -axis, which exactly corresponds to the schematics in Fig.  1c and d , respectively. For instance, comparing Fig.  1c to the mode \({P}_{l1}\) , one can see that Fig.  1c shows a rotation around \(-z\) -axis and a translation along \(-z\) -axis since the bending along \(-z\) -axis, and \({P}_{l1}\) in Fig.  2f shows a rotation around \(+z\) -axis and a translation along \(+z\) -axis due to bending along \(-z\) -axis. The schematics of the rotation and translation phases are completely identical to those of the simulation ones. Similarly, Fig.  1d shows a rotation around \(+z\) -axis and a translation along \(-z\) -axis since the compression, and \({P}_{u1}\) in Fig.  2f shows a rotation around \(-z\) -axis and a translation along \(+z\) -axis due to stretch. These two have excellent consistency.

It is crucial to emphasize that these acoustic and optic branches essentially differ from conventional diatomic chains. In detail, the upper and lower branches in this context stem from two coupled orthogonal motions that originate from the same atom instead of from two atoms. Coincidentally, these coupled orthogonal polarizations introduce an extra control variable for bandgap modulation–inertial amplification 12 , as compared by Fig.  1 b and e . Nevertheless, the inertial amplification effect will be hidden in the wave equation if we utilize the traditional theoretical derivation directly based on force equilibrium.

It should be noted that there is a discrepancy between the expected bandgap width of the dispersion spectrum and the attenuation range of the transmission. The expected bandgap covers 500 Hz–1700 Hz, while the attenuation range shown in the transmission (Fig.  2b ) only appears in 500 Hz–1400 Hz. This is due to the different boundary conditions in calculating the dispersion spectrum and the transmission. For the calculation of the dispersion spectrum, all the unit cell is free. For the calculation of the transmission, the rotation freedom of the first lumped mass is constrained. If we release this degree of freedom, the attenuation range will be 500 Hz–1700 Hz, corresponding to the expected bandgap width (Please see Supplementary Note  6 for more details.). Overall, the consistency in transmissions (Fig.  2b ), dispersion spectra Fig.  2c , as well as the deformation schematics (Figs.  1 and   2f ), can verify the correctness of our analysis.

Another essential advantage of our method is that, as illustrated by Supplementary Eqs. ( 38) , we can directly obtain the longitudinal amplitude \({u}_{i}^{b}\) determined by the bending deformation and the rotational amplitude \({\psi }_{i}^{s}\) determined by the stretch deformation. Furthermore, by substituting the results of Supplementary Equation ( 38) into Supplementary Equation ( 14) , we can obtain the rotational amplitude \({\psi }_{i}^{b}\) determined by the bending deformation and the longitudinal amplitude \({u}_{i}^{s}\) determined by the stretch deformation. In other words, we can observe the respective contributions and influences of the acoustic mode and the optic branch on the bandgap. As shown in Fig.  2d and e, the acoustic mode is dominant before the anti-resonance notch in the bandgap. After that, the optic mode will dominate the transmission coefficient. Although Fig.  2e shows that \({R}_{b}\) has almost the same relative amplitude as \({R}_{s}\) after the anti-resonance notch, it can be regarded as passive for \({R}_{b}\) to present the large amplitude according to the causal inference in Supplementary Note  1 . More simply, if we can shift the optic mode towards a higher frequency, this passive effect originates from \({R}_{s}\) to \({R}_{b}\) will be much weaker, and the frequency range dominated by \({u}_{b}\) will be broader.

In short, for chiral PnCs 24 , 25 , it is convenient and concise to characterize the dispersion spectrum and transmission properties through the wave equation established from force equilibrium, but its final formulas merely present the coupled longitudinal and rotational polarizations, thus obscuring the comprehensive physical insights. Consequently, despite the observations of the similar coupling orthogonal polarizations in two-dimensional and three-dimensional chiral structures 26 , 27 , 28 which are characterized by auxeticity in quasi-static compression 29 , and even the systematical establishment of the governing equations 30 , 31 , 32 , there have been limited discoveries of inertial amplification. In the end, many studies stagnated at the bandgap opening due to the limitations of the structural shape evolution 8 , 11 , 33 , 34 .

Coupling roots of acoustic and optic branches

As observed in the wave equation we proposed, the inertial amplification is a unique and essential advantage for the chiral PnCs. However, broadening the bandgap is extremely challenging since there must be the optic branch. To clarify the challenge, we neglect the contribution of stretch mode ( \({P}_{{ls}}\) and \({P}_{{rs}}\) ) and consider the bending mode only based on the method shown in Supplementary Eqs. ( 29 )–( 39) . The theoretical transmission (the blue line in Fig.  2a ) is still consistent with the numerical results in low frequencies, and the dynamic equation can also reveal the inertial amplification effect, as demonstrated by other report 15 . This indicates that the bending mode ( \({P}_{{lb}}\) and \({P}_{{rb}}\) ) directly determines the existence of the bandgap, and the stretch mode ( \({P}_{{ls}}\) and \({P}_{{rs}}\) ) determines the upper limit of the bandgap.

The comparison of Fig.  2 a and b might lead us to believe that the main contribution of stretch mode is only to truncate the inertial amplification-based bandgap, but that is not completely true. This bandgap formation relies on the coupled longitudinal-rotational motions of each lumped mass. In the conventional unit cell, although the longitudinal motion and rotational motion originate from the bending deformation of the ligaments, Supplementary Note  3 illustrated that, for the solid structure, the bending mode will be absent if the stretch mode does not exist because both modes are determined by the identical basic physical and geometric parameters. Therefore, it seems impossible to make the optic modes disappear completely to obtain an infinite bandgap. Therefore, it is vital to figure out the coupling between optic and acoustic modes as well as find ways to manipulate them independently.

Figure  2a illustrates that the bending mode serves to provide the stiffness \({k}_{b}\) and inertial amplification coefficient \(p\) . \({k}_{b}\) and \(p\) are critical for directly determining the acoustic branch. The comparison of Fig.  2a and b illustrates that the stretch mode determines the optic branch by the stiffness \({k}_{s}\) and \(q\) . Therefore, for a normalized low-frequency and broad bandgap, \(p\) and \({k}_{s}\) should be larger while \(q\) should be larger, and \({k}_{b}\) should be constant to provide sufficient support capacity.

However, contrary to expectations, the actual situation is unfavorable. In detail, on the one hand, Fig.  1c and d illustrated that, \({P}_{{rb}}\) and \({P}_{{rs}}\) have the opposite directions, which implies a hybridization between the rotational polarizations determined by bending and stretch modes. If there is no hybridization between \(p\) and \(q\) , (see Supplementary Note  3 and Supplementary Note  4 for more details)

The ideal inertial amplification coefficient will vary like the blue line shown in Fig.  3a . One can see that the amplified dynamic inertia \(p\) would easily exceed 100 times.

a , b Influence of \({{{\boldsymbol{\theta }}}}\) on the inertial amplification coefficient \({{{\boldsymbol{p}}}}\) , bending stiffness \({{{{\boldsymbol{k}}}}}_{{{{\boldsymbol{b}}}}}\) , and stretch stiffness \({{{{\boldsymbol{k}}}}}_{{{{\boldsymbol{s}}}}}\) of the conventional chiral PnCs. The method of the normalized stiffness is \({{{\boldsymbol{k}}}}{{{\boldsymbol{/}}}}{{{{\boldsymbol{k}}}}}_{{{{\boldsymbol{r}}}}}\) where \({{{{\boldsymbol{k}}}}}_{{{{\boldsymbol{r}}}}} = {{{\boldsymbol{1}}}}{{{\boldsymbol{e}}}}{{{\boldsymbol{5}}}}\) \({{{\bf{N}}}}\,{{{{\bf{m}}}}}^{{{{\boldsymbol{-}}}}{{{\bf{1}}}}}\) (see Fig.  S1 in Supplementary Note  3 for more details of \({{{\boldsymbol{\theta }}}}\) , and see Supplementary Eqs. ( 47 ) and ( 48) for the governing equation about \({{{{\boldsymbol{k}}}}}_{{{{\boldsymbol{b}}}}}\) and \({{{{\boldsymbol{k}}}}}_{{{{\boldsymbol{s}}}}}\) , respectively). c Bandgap variation with the different \({{{\boldsymbol{\theta }}}}{{{\boldsymbol{.}}}}\,\) Therein, the gray shaded area is the bandgap ranges.

If considering the hybridization, \(p\) is written as (see Supplementary Note  3 and Supplementary Note  4 for more details)

where \(\triangle {u}_{b}\) refers to the longitudinal displacement difference (between \({m}_{i}\) and \({m}_{i-1}\) ) caused by the bending mode under the longitudinal harmonic loads; \(\triangle {u}_{s}\) refers to the longitudinal displacement difference caused by the stretch mode under the longitudinal harmonic loads; \(\triangle {R}_{b}\) refers to the rotational displacement difference caused by the bending mode under the longitudinal harmonic loads; \(\triangle {R}_{s}\) refers to the rotational displacement difference caused by the stretch mode under the longitudinal harmonic loads.

From Eq. ( 6 ), if \(\triangle {R}_{s}\) is larger with the increase of \(\triangle {u}_{s}\) , \(p\) will be smaller, which will reduce the inertial amplification effect. This is a hybridization between \(q\) and \(p\) ((see Supplementary Eqs. ( 40 ), ( 41 ), ( 42 ), and ( 45) for more details)). Considering the hybridization, the amplified dynamic inertia \(p\) can only be at most 2.6 times.

On the other hand, as illustrated by Fig.  3b , \({k}_{b}\) will increase rapidly with the increase of \(\theta\) . Then, the difference between \({k}_{b}\) and \({k}_{s}\) will be smaller and smaller, which is not conducive to achieving a broad bandgap 13 . Ultimately, the upper boundary will approach the lower boundary of the bandgap, leading to the closure of the bandgap, as depicted in Fig.  3c . For instance, if we need the maximum inertial amplification (when \(\theta\) is about \(80^\circ\) ) to reduce the bandgap, then the stiffness difference between \({k}_{s}\) and \({k}_{b}\) is only 1.76 times. These two aspects denote a significant contradiction between the broad bandgap and the low-frequency bandgap.

Customization of acoustic and optic branches

To resolve the contradiction, we propose the strategy, as shown in Fig.  4a , to achieve partial decoupling. As illustrated in Fig.  4b , the subunit cell can be divided into three components, i.e., the lumped disks, the spiral springs, and spherical hinges. Figure  4c shows a detailed schematic of the spherical hinges, and its governing equation can be found in Supplementary Note  7 . In this unit cell, the spiral springs provide \({k}_{b}\) and the spherical hinges are responsible for providing the rotational polarization while the spiral springs are compressed, thus achieving \(p\) . Therefore, \({k}_{s}\) is determined by the spherical hinges. Regarding the unit cell, its first bandgap extending from 39 Hz to 1650 Hz can be obtained in the dispersion spectrum (in Fig.  4d ) (see Supplementary Method  1 for more details of the simulation). The ratio of the lower boundary of the optic branch to the upper boundary of the acoustic branch is up to 42 times.

a Photograph of the experimental sample (See Method for experimental details). b Schematics of the subunit cell (See Supplementary Note  8 for details about geometry). c Schematic of the geometric relationship of the spherical hinges. See Supplementary S7 for the governing equations between the driven component and active component of the spherical hinges. d Normalized dispersion spectra of the chiral PnCs. The red line is the starting frequency of the unit cell without spherical hinges, and it is 0.347 (88 Hz). The shaded area indicates the bandgap range. Normalization method is \({{{{\boldsymbol{f}}}}}_{{{{\boldsymbol{n}}}}} = {{{\boldsymbol{f}}}}{{{\boldsymbol{/}}}}\left(\sqrt{{{{{\boldsymbol{k}}}}}_{{{{\boldsymbol{b}}}}}{{{\boldsymbol{/}}}}{{{{\boldsymbol{m}}}}}_{{{{\boldsymbol{e}}}}}}\right)\) , where \({{{{\boldsymbol{m}}}}}_{{{{\boldsymbol{e}}}}} = {{{\boldsymbol{0}}}}{{{\boldsymbol{.}}}}{{{\boldsymbol{6018}}}}\) kg and \({{{{\boldsymbol{k}}}}}_{{{{\boldsymbol{b}}}}} = {{{\boldsymbol{3}}}}{{{\boldsymbol{.}}}}{{{\boldsymbol{6}}}}{{{\boldsymbol{e}}}}{{{\boldsymbol{4}}}}\) \({{{\bf{N}}}}\,{{{{\bf{m}}}}}^{{{{\boldsymbol{-}}}}{{{\boldsymbol{1}}}}}\) (See Supplementary Method  2 for the reasons of the normalization method). e Numerical and experimental transmission of one unit cell. \({{{{\bf{P}}}}}_{{{{\bf{1}}}}}\) and \({{{{\bf{P}}}}}_{{{{\bf{2}}}}}\) denote the resonance peaks while \({{{{\bf{N}}}}}_{{{{\bf{1}}}}}\) and \({{{{\bf{N}}}}}_{{{{\bf{2}}}}}\) denote the anti-resonance notches.

To validate our proposed design under controlled conditions and minimize the influence of extraneous factors, thus ensuring experimental validity, one unit cell was fabricated and subjected to rigorous testing as illustrated in Fig.  4a . To avoid the local resonance modes of the lumped masses, the end of the period direction is replaced by a carbon fiber plate, which can provide a high elastic modulus with a low density (see Method for more details of the experiment). The experimental and numerical results are shown in Fig.  4d . One can see that there is an obvious attenuation after 35 Hz, and the experimental and numerical results are in satisfying agreement in 100 Hz, especially at resonance peaks ( \({{{{\rm{P}}}}}_{1}\) and \({{{{\rm{P}}}}}_{2}\) ) and anti-resonance notches ( \({{{{\rm{N}}}}}_{1}\) and \({{{{\rm{N}}}}}_{2}\) ). There are significant deviations between numerical and experimental results after 100 Hz, which might be resulted by the nonlinear collisions from the clearance in the spherical hinge 35 .

Regarding the PnC shown in Fig.  4a , the material of the spherical hinge is steel, while that of the springs is Nylon, and the springs are spiral to further decrease the equivalent stiffness \({k}_{s}\) , affording \({k}_{s}\) and \({k}_{b}\) great discrepancy. On the one hand, the discrepancy is beneficial in raising the optic branch and thus broadening the bandgap. On the other hand, because of the great discrepancy between \({k}_{s}\) and \({k}_{b}\) , the deformation ( \(\triangle {R}_{s}\) and \(\triangle {u}_{s}\) ) of the stretch mode will be much weaker, so the hybridization to \(p\) will be weakened. Therefore, the numerical inertial amplification coefficient \(p\) can be up to 13 times with the increase of the tilt angle \(\theta\) , as shown in Fig.  5a (the original coefficient is a maximum of 2.6). In this case, the lower boundary will shift to a lower frequency while the upper boundary can be almost constant, as Fig.  5b shows.

a Variation of \(p\) with different \(\theta\) . b , c Normalized bandgap width in different inertial amplification coefficients \(p\) and different stiffness ratios ( \({k}_{s}/{k}_{b}\) ). (See Supplementary Method  1 for details about the simulation).

In addition, because the functions of the spiral springs and spherical hinges are independent, the disparity between \({k}_{s}\) and \({k}_{b}\) can be magnified by variations in the material and dimensions of the spherical hinges. Consequently, with the increase in the stiffness ratio ( \({k}_{s}/{k}_{b}\) where \({k}_{b}\) is constant), the bandgap width can be expanded (Fig.  5c ), where the upper boundary will shift to a higher frequency while the lower boundary is constant.

In brief, compared to conventional unit cells, this unit cell with the spherical hinges enables the attainment of low-frequency and wide bandgaps by tuning the inclination angle \(\theta\) and the material and geometric properties of the spherical hinges, while significantly mitigating the constraints imposed by the equivalent supporting stiffness \({k}_{b}\) , equivalent density, and lattice constant. While this work showcases realization in broad and low-frequency bandgaps, it should be acknowledged that enhancing the attenuation intensity of the inertial amplification-based bandgap will be the next significant challenge 15 .

Conclusions

In summary, in this research, we have theoretically revealed that the inertial amplification effect evolves from inertia matrix to stiffness matrix, thus unifying two ostensibly conflicting explanations of the bandgap mechanism. Based on our theory, which allows to observe the comprehensive physics of acoustic and optic branches in chiral PnCs, we have clarified that the close relations between the rise of the inertial amplification coefficient and the bending and stretch stiffness, as well as the restrictions from this close relations on the creation of broad subwavelength bandgaps under boundaries constrained by the constant equivalent density, equivalent stiffness, and lattice constant. Therefore, we have used spherical hinges to achieve the partial decouple, thus releasing the mutual negative effect between the acoustic and optic boundaries. The numerical and experimental results have confirmed the effectiveness of our proposed scheme and demonstrated that the underlying physics obtained from the wave behavior is instructive for structural design. This work may be able to shield light on the discovery of the inertial amplification effects in other high-dimensional artificial structures, to realize ultra-low-frequency and ultra-broad bandgaps without the requirement of the bulky static mass and fragile static stiffness, as well as to customize the bandgap in chiral PnCs.

Experiment configuration and boundary conditions

The schematic diagram of the experimental setup is shown in Fig.  6 . The input disk is bolted to a plexiglass with a thickness of 15 mm, and the plexiglass must have approximately ten times the weight of the bottom disk to limit the freedom of rotation of the disk around the z-axis as much as possible. The shaker is excited directly on the plexiglass through the excitation bar to stimulate the harmonic excitation. Two acceleration sensors (PCB 353B15) are attached to the top and the bottom of the sample to pick up the output acceleration \({a}_{o}\) and the input acceleration \({a}_{i}\) , respectively. The experimental transmission is calculated by \({a}_{o}/{a}_{i}\) . The frequency range of the sine sweep is divided into three bands, i.e., 10 Hz–200 Hz, 200 Hz–1000 Hz, and 1000 Hz–3000 Hz, to avoid exceeding the allowable amplitude of the shaker under different voltages and to guarantee the output acceleration \({a}_{o}\) is higher than the background noise. The frequency resolution is 2 Hz, and the sweeping speed is 200 Hz/min, to guarantee the precision of experimental data.

The bottom disk of the unit cell is bolted to a plexiglass. The two yellow domains indicate the acceleration sensors. The shaker is Modelshop-K2007E01, which is bolted to the optical platform and connected to the plexiglass through an excitation bar. The foam support is used to isolate the vibration propagating from the optic platform to the plexiglass.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Code availability

The code used to generate the data used in this study is available from the corresponding author upon reasonable request.

Bossart, A. & Fleury, R. Extreme spatial dispersion in nonlocally resonant elastic metamaterials. Phys. Rev. Lett. 130 , 207201 (2023).

Article   ADS   Google Scholar  

Babaee, S., Overvelde, J. T., Chen, E. R., Tournat, V. & Bertoldi, K. Reconfigurable origami-inspired acoustic waveguides. Sci. Adv. 2 , e1601019 (2016).

Fu, Y. et al. Asymmetric generation of acoustic vortex using dual-layer metasurfaces. Phys. Rev. Lett. 128 , 104501 (2022).

Maldovan, M. Narrow low-frequency spectrum and heat management by thermocrystals. Phys. Rev. Lett. 110 , 025902 (2013).

Van Damme, B. et al. Inherent non-linear damping in resonators with inertia amplification. Appl. Phys. Lett. 119 , 061901 (2021).

Yilmaz, C. & Hulbert, G. M. Theory of phononic gaps induced by inertial amplification in finite structures. Phys. Lett. A 374 , 3576–3584 (2010).

Otlu, S. N., Acar, B., Tetik, Z. G. & Yilmaz, C. Three-dimensional ultra-wide elastic metamaterial with inertial amplification mechanisms having optimized flexure hinges. Int. J. Solids Struct. 282 , 112453 (2023).

Article   Google Scholar  

Zhao, P., Zhang, K., Hong, F. & Deng, Z. Tacticity-based one-dimensional chiral equilateral lattice for tailored wave propagation and design of elastic wave logic gate. J. Sound Vib. 521 , 116671 (2022).

Zhou, Y., Ye, L. & Chen, Y. Investigation of novel 3D-printed diatomic and local resonant metamaterials with impact mitigation capacity. Int. l J. Mech. Sci. 206 , 106632 (2021).

Yilmaz, C., Hulbert, G. M. & Kikuchi, N. Phononic band gaps induced by inertial amplification in periodic media. Phys. Rev. B 76 , 054309 (2007).

Zhao, P., Zhang, K., Qi, L. & Deng, Z. 3D chiral mechanical metamaterial for tailored band gap and manipulation of vibration isolation. Mech. Syst. Sig. Process. 180 , 109430 (2022).

Ding, W. et al. Description of Bandgaps Opening in Chiral Phononic Crystals by Analogy with Thomson scattering. N. J. Phys. 25 , 103001 (2023).

Park, J., Lee, D., Jang, Y., Lee, A. & Rho, J. Chiral trabeated metabeam for low-frequency multimode wave mitigation via dual-bandgap mechanism. Commun. Phys. 5 , 194 (2022).

Bergamini, A. et al. Tacticity in chiral phononic crystals. Nat. Commun. 10 , 4525 (2019).

Orta, A. H. & Yilmaz, C. Inertial amplification induced phononic band gaps generated by a compliant axial to rotary motion conversion mechanism. J. Sound Vib. 439 , 329–343 (2019).

Zheng, B. & Xu, J. Mechanical logic switches based on DNA-inspired acoustic metamaterials with ultrabroad low-frequency band gaps. J. Phys. D: Appl. Phys. 50 , 465601 (2017).

Oh, J. H., Qi, S., Kim, Y. Y. & Assouar, B. Elastic metamaterial insulator for broadband low-frequency flexural vibration shielding. Phys. Rev. Appl. 8 , 054034 (2017).

Zhao, C., Zhang, K., Zhao, P., Hong, F. & Deng, Z. Bandgap merging and backward wave propagation in inertial amplification metamaterials. Int. l J. Mech. Sci. 250 , 108319 (2023).

Phani, A. S., Woodhouse, J. & Fleck, N. A. Wave propagation in two-dimensional periodic lattices. J. Acous. Soc. Am. 119 , 1995–2005 (2006).

Bravo, T. & Maury, C. Causally-guided acoustic optimization of single-layer rigidly-backed micro-perforated partitions: Theory. J. Sound Vib. 520 , 116634 (2022).

Yang, M., Chen, S., Fu, C. & Sheng, P. Optimal sound-absorbing structures. Mater. Horiz. 4 , 673–680 (2017).

Patterson J. D., Bailey B. C. S olid-state Physics: Introduction to the Theory . (Springer Science & Business Media, 2007).

Yilmaz, C. & Kikuchi, N. Analysis and design of passive low-pass filter-type vibration isolators considering stiffness and mass limitations. J. Sound Vib. 293 , 171–195 (2006).

Ding, W. et al. Isotacticity in chiral phononic crystals for low-frequency bandgap. Int. l J. Mech. Sci. 261 , 108678 (2023).

Ding, W. et al. Thomson scattering-induced bandgap in planar chiral phononic crystals. Mech. Syst. Sig. Process. 186 , 109922 (2023).

Foehr, A., Bilal, O. R., Huber, S. D. & Daraio, C. Spiral-Based Phononic Plates: From Wave Beaming to Topological Insulators. Phys. Rev. Lett. 120 , 205501 (2018).

Lakes, R. Deformation mechanisms in negative Poisson’s ratio materials: structural aspects. J. Mater. Sci. 26 , 2287–2292 (1991).

Dudek, K. K., Gatt, R., Wojciechowski, K. W. & Grima, J. N. Self-induced global rotation of chiral and other mechanical metamaterials. Int. J. Solids Struct. 191-192 , 212–219 (2020).

Mizzi, L. & Spaggiari, A. Novel chiral honeycombs based on octahedral and dodecahedral Euclidean polygonal tessellations. Int. J. Solids Struct. 238 , 111428 (2022).

Liu, X. N., Huang, G. L. & Hu, G. K. Chiral effect in plane isotropic micropolar elasticity and its application to chiral lattices. J. Mech. Phys. Solids 60 , 1907–1921 (2012).

Article   ADS   MathSciNet   Google Scholar  

Spadoni, A. & Ruzzene, M. Elasto-static micropolar behavior of a chiral auxetic lattice. J. Mech. Phys. Solids 60 , 156–171 (2012).

Bacigalupo, A. & Gambarotta, L. Simplified modelling of chiral lattice materials with local resonators. Int. J. Solids Struct. 83 , 126–141 (2016).

Yin, C., Xiao, Y., Zhu, D., Wang, J. & Qin, Q.-H. Design of low-frequency 1D phononic crystals harnessing compression–twist coupling effect with large deflection angle. Thin. Wall. Struct. 179 , 109600 (2022).

Baravelli, E. & Ruzzene, M. Internally resonating lattices for bandgap generation and low-frequency vibration control. J. Sound Vib. 332 , 6562–6579 (2013).

Fang, X., Wen, J., Bonello, B., Yin, J. & Yu, D. Ultra-low and ultra-broad-band nonlinear acoustic metamaterials. Nat. Commun. 8 , 1288 (2017).

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Acknowledgements

This work was financially supported by the National Natural Science Foundation of China (No. 12002258, No. 52265015), the State Key Laboratory for Strength and Vibration of Mechanical Structures (No. SV2023-KF-08), and the China Postdoctoral Science Foundation (No. 2022M712540). Wei Ding is grateful for the support of the China Scholarship Council (No. 202206280170). Wei Ding appreciates Yuhan Hu (Hohai University) for her guidance in drawing.

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These authors contributed equally: Wei Ding, Rui Zhang.

Authors and Affiliations

School of Mechanical Engineering and State Key Laboratory of Strength & Vibration of Mechanical Structures, Xi’an Jiaotong University, Xi’an, Shaanxi, 710049, P.R. China

Wei Ding, Rui Zhang, Tianning Chen, Dewen Yu & Jian Zhu

School of Civil and Environmental Engineering, Nanyang Technological University, 50 Nanyang Avenue, 639798, Singapore, Singapore

Wei Ding, Shuai Qu, Dewen Yu, Liwei Dong & Yaowen Yang

Train and Track Research Institute, State Key Laboratory of Rail Transit Vehicle System, Southwest Jiaotong University, Chengdu, 610031, China

Institute of Rail Transit, Tongji University, Shanghai, 201804, China

Université de Lorraine, CNRS, Institut Jean Lamour, F-54000, Nancy, France

Badreddine Assouar

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W. D. and R. Z. proposed the concept and performed the theoretical analysis, numerical calculation, and experimental demonstration. T.N.C. and J.Z. provided the experimental sponsorship. W.D., R.Z, and J. Z. wrote the manuscript and the Supplementary Information. J.Z, Y.W.Y, and B.A guided the research. S.Q., D.W.Y., and L.W.D. processed the data and organized the manuscript. All the authors contributed to the discussion of the results.

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Correspondence to Jian Zhu , Yaowen Yang or Badreddine Assouar .

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Ding, W., Zhang, R., Chen, T. et al. Origin and tuning of bandgap in chiral phononic crystals. Commun Phys 7 , 272 (2024). https://doi.org/10.1038/s42005-024-01761-z

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Accepted : 31 July 2024

Published : 10 August 2024

DOI : https://doi.org/10.1038/s42005-024-01761-z

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