Dirac cones of an acoustic system are the foundation of most topological phase transitions and topological states and have recently become a research hotspot. Although the Dirac cones, Dirac-like cones, double Dirac cones, and semi-Dirac points are all skillfully designed, it is still indispensable to realize a tunable Dirac cone in a novel acoustic structure. This paper proposes two-dimensional acoustic metamaterials with matryoshka structure to achieve tunable Dirac cones and topological spin states. Dirac points can be obtained on the dispersion curves owing to the high symmetry. The concentric circular scattering units of the matryoshka structure are arranged in honeycomb lattices. By a rotating-scatterer mechanism to break the symmetry, the Dirac cone at K (K') is split and the topological spin states appear at the band valley. The existence of a topological transition with opposite Chern numbers as the rotating angle varies is also verified, and helical edge states are obtained along the interfaces separating the topologically opposite spin states insulators. Moreover, the frequency of the Dirac cone is tuned by rotating the inner structure in a double-layer matryoshka structure.
References
1.
Banerjee
, S.
, Singh
, R. R. P.
, Pardo
, V.
, and Pickett
, W. E.
(2009
). “Tight-binding modeling and low-energy behavior of the semi-Dirac point
,” Phys. Rev. Lett.
103
(1
), 016402
.2.
Bernevig
, B. A.
, Hughes
, T. L.
, and Zhang
, S. C.
(2006
). “Quantum spin hall effect and topological phase transition in HgTe quantum wells
,” Science
314
, 1757
–1761
.3.
Chen
, Z.-G.
, Ni
, X.
, Wu
, Y.
, He
, C.
, Sun
, X.-C.
, Zheng
, L.-Y.
, Lu
, M.-H.
, and Chen
, Y.-F.
(2014
). “Accidental degeneracy of double Dirac cones in a phononic crystal
,” Sci. Rep.
4
(1
), 4613
.4.
Dai
, H.
, Xia
, B.
, and Yu
, D.
(2017
). “Dirac cones in two-dimensional acoustic metamaterials
,” J. Appl. Phys.
122
(6
), 065103
.5.
Elford
, D. P.
, Chalmers
, L.
, Kusmartsev
, F. V.
, and Swallowe
, G. M.
(2011
). “Matryoshka locally resonant sonic crystal
,” J. Acoust. Soc. Am.
130
(5
), 2746
–2755
.6.
Fleury
, R.
, Khanikaev
, A. B.
, and Alù
, A.
(2016
). “Floquet topological insulators for sound
,” Nat. Commun.
7
, 11744
.7.
Fleury
, R.
, Sounas
, D. L.
, Sieck
, C. F.
, Haberman
, M. R.
, and Alu
, A.
(2014
). “Sound isolation and giant linear nonreciprocity in a compact acoustic circulator
,” Science
343
(6170
), 516
–519
.8.
Häusler
, W.
, and Egger
, R.
(2009
). “Artificial atoms in interacting graphene quantum dots
,” Phys. Rev. B
80
(16
), 161402
.9.
He
, C.
, Li
, Z.
, Ni
, X.
, Sun
, X.-C.
, Yu
, S.-Y.
, Lu
, M.-H.
, Liu
, X.-P.
, and Chen
, Y.-F.
(2016
). “Topological phononic states of underwater sound based on coupled ring resonators
,” Appl. Phys. Lett.
108
(3
), 031904
.10.
He
, C.
, Ni
, X.
, Ge
, H.
, Sun
, X.-C.
, Chen
, Y.-B.
, Lu
, M.-H.
, Liu
, X.-P.
, and Chen
, Y.-F.
(2016
). “Acoustic topological insulator and robust one-way sound transport
,” Nat. Phys.
12
(12
), 1124
–1129
.11.
Herring
, C.
(1937
). “Accidental degeneracy in the energy bands of crystals
,” Phys. Rev.
52
(4
), 365
–373
.12.
Khanikaev
, A. B.
, Fleury
, R.
, Mousavi
, S. H.
, and Alù
, A.
(2015
). “Topologically robust sound propagation in an angular-momentum-biased graphene-like resonator lattice
,” Nat. Commun.
6
(1
), 8260
.13.
Li
, Y.
, and Mei
, J.
(2015
). “Double Dirac cones in two-dimensional dielectric photonic crystals
,” Opt. Express
23
(9
), 12089
.14.
Li
, Y.
, Wu
, Y.
, and Mei
, J.
(2014
). “Double Dirac cones in phononic crystals
,” Appl. Phys. Lett.
105
(1
), 014107
.15.
Lu
, J.
, Qiu
, C.
, Xu
, S.
, Ye
, Y.
, Ke
, M.
, and Liu
, Z.
(2014
). “Dirac cones in two-dimensional artificial crystals for classical waves
,” Phys. Rev. B
89
(13
), 134302
.16.
Lu
, J.
, Qiu
, C.
, Ye
, L.
, Fan
, X.
, Ke
, M.
, Zhang
, F.
, and Liu
, Z.
(2016
). “Observation of topological valley transport of sound in sonic crystals
,” Nat. Phys.
13
(4
), 369
–374
.17.
Ma
, G.
, and Sheng
, P.
(2016
). “Acoustic metamaterials: From local resonances to broad horizons
,” Sci. Adv.
2
(2
), e1501595
.18.
Malko
, D.
, Neiss
, C.
, Viñes
, F.
, and Görling
, A.
(2012
). “Competition for graphene: Graphynes with direction-dependent Dirac cones
,” Phys. Rev. Lett.
108
(8
), 086804
.19.
Martin
, I.
, Blanter
, Y. M.
, and Morpurgo
, A. F.
(2008
). “Topological confinement in bilayer graphene
,” Phys. Rev. Lett.
100
(3
), 036804
.20.
Novoselov
, K. S.
, Geim
, A. K.
, Morozov
, S. V.
, Jiang
, D.
, Katsnelson
, M. I.
, Grigorieva
, I. V.
, Dubonos
, S. V.
, and Firsov
, A. A.
(2005
). “Two-dimensional gas of massless Dirac fermions in graphene
,” Nature
438
(7065
), 197
–200
.21.
Peng
, Y.-G.
, Qin
, C.-Z.
, Zhao
, D.-G.
, Shen
, Y.-X.
, Xu
, X.-Y.
, Bao
, M.
, Jia
, H.
, and Zhu
, X.-F.
(2016
). “Experimental demonstration of anomalous Floquet topological insulator for sound
,” Nat. Commun.
7
, 13368
.22.
Sakoda
, K.
(2012
). “Double Dirac cones in triangular-lattice metamaterials
,” Opt. Express
20
(9
), 9925
.23.
Semenoff
, G. W.
, Semenoff
, V.
, and Zhou
, F.
(2008
). “Domain walls in gapped graphene
,” Phys. Rev. Lett.
101
(8
), 087204
.24.
Torrent
, D.
, and Sánchez-Dehesa
, J.
(2012
). “Acoustic analogue of graphene: Observation of Dirac cones in acoustic surface waves
,” Phys. Rev. Lett.
108
(17
), 174301
.25.
Wang
, P.
, Casadei
, F.
, Shan
, S.
, Weaver
, J. C.
, and Bertoldi
, K.
(2014
). “Harnessing buckling to design tunable locally resonant acoustic metamaterials
,” Phys. Rev. Lett.
113
, 014301
.26.
Wang
, P.
, Lu
, L.
, and Bertoldi
, K.
(2015
). “Topological phononic crystals with one-way elastic edge waves
,” Phys. Rev. Lett.
115
, 104302
.27.
Wei
, Q.
, Tian
, Y.
, Zuo
, S.-Y.
, Cheng
, Y.
, and Liu
, X.-J.
(2017
). “Experimental demonstration of topologically protected efficient sound propagation in an acoustic waveguide network
,” Phys. Rev. B
95
(9
), 094305
.28.
Wen
, X.
, Qiu
, C.
, Lu
, J.
, He
, H.
, Ke
, M.
, and Liu
, Z.
(2018
). “Acoustic Dirac degeneracy and topological phase transitions realized by rotating scatterers
,” J. Appl. Phys.
123
(9
), 091703
.29.
Wu
, Y.
, Yang
, M.
, and Sheng
, P.
(2018
). “Perspective: Acoustic metamaterials in transition
,” J. Appl. Phys.
123
(9
), 090901
.30.
Xia
, B.
, Liu
, T.
, Huang
, G.
, Dai
, H.
, Jiao
, J.
, Zang
, X.
, Yu
, D.
, Zheng
, S.
, and Liu
, J.
(2017
). “Topological phononic insulator with robust pseudospin-dependent transport
,” Phys. Rev. B
96
, 094106
.31.
Xia
, B.
, Wang
, G.
, and Zheng
, S.
(2019
). “Robust edge states of planar phononic crystals beyond high-symmetry points of Brillouin zones
,” J. Mech. Phys. Solids
124
, 471
.32.
Xia
, B.
, Zheng
, S.
, Liu
, T.
, Jiao
, J.
, Chen
, N.
, Dai
, H. Q.
, and Liu
, J.
(2018
). “Observation of valley-like edge states of sound at a momentum away from the high-symmetry points
,” Phys. Rev. B
97
, 155124
.33.
Yang
, Z.
, Gao
, F.
, Shi
, X.
, Lin
, X.
, Gao
, Z.
, Chong
, Y.
, and Zhang
, B.
(2015
). “Topological acoustics
,” Phys. Rev. Lett.
114
(11
), 114301
.34.
Yu
, K.
, Chen
, T.
, Wang
, X.
, and Zhou
, A.
(2013
). “Large band gaps in phononic crystal slabs with rectangular cylinder inclusions parallel to the slab surfaces
,” J. Phys. Chem. Solids
74
, 1146
.35.
Yves
, S.
, Fleury
, R.
, Lemoult
, F.
, Fink
, M.
, and Lerosey
, G.
(2017
). “Topological acoustic polaritons: Robust sound manipulation at the subwavelength scale
,” New J. Phys.
19
(7
), 075003
.36.
Zhai
, F.
, Mu
, P.
, and Chang
, K.
(2011
). “Energy spectrum of Dirac electrons on the surface of a topological insulator modulated by a spiral magnetization superlattice
,” Phys. Rev. B
83
(19
), 195402
.37.
Zhang
, F.
, Jung
, J.
, Fiete
, G. A.
, Niu
, Q.
, and MacDonald
, A. H.
(2011
). “Spontaneous quantum hall states in Chirally stacked few-layer graphene systems
,” Phys. Rev. Lett.
106
(15
), 156801
.38.
Zhang
, X.
, and Liu
, Z.
(2008
). “Extremal transmission and beating effect of acoustic waves in two-dimensional sonic crystals
,” Phys. Rev. Lett.
101
(26
), 264303
.39.
Zhang
, Y.
, Tan
, Y.-W.
, Stormer
, H. L.
, and Kim
, P.
(2005
). “Experimental observation of the quantum Hall effect and Berry's phase in graphene
,” Nature
438
(7065
), 201
–204
.40.
Zhang
, Z.
, Wei
, Q.
, Cheng
, Y.
, Zhang
, T.
, Wu
, D.
, and Liu
, X.
(2017
). “Topological creation of acoustic pseudospin multipoles in a flow-free symmetry-broken metamaterial lattice
,” Phys. Rev. Lett.
118
(8
), 084303
.© 2019 Acoustical Society of America.
2019
Acoustical Society of America
You do not currently have access to this content.