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Interface states have great practical applications, therefore, searching for the existence of interface states has both scientific significance and application prospects. In this work, we tilt the structure unite of two-dimensional phononic crystal with a square lattice to construct an oblique lattice possessing linear Dirac dispersion. The Dirac dispersion gives rise to a π jump of the Zak phases of the bulk bands, so that the projected band gaps at both sides of the Dirac cone have opposite signs of surface impedance, resulting in deterministic interface states at the interface formed by the phononic crystal with a square lattice and its tilted oblique lattice system.
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Keywords:
- phononic crystal /
- linear dispersion /
- interface state /
- Zak phase
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Google Scholar
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[3] Lu M H, Zhang C, Feng L, Zhao J, Chen Y F, Mao Y W, Zi J, Zhu Y Y, Zhu S N, Ming N B 2007 Nat. Mater. 6 744
Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
[22] Rusin T M, Zawadzki W 2008 Phys. Rev. B 78 125419
Google Scholar
[23] Katsnelson M I, Novoselov K S, Geim A K 2006 Nat. Phys. 2 620
Google Scholar
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Google Scholar
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Google Scholar
[26] Yang Y, Xu T, Xu Y F, Hang Z H 2017 Opt. Lett. 42 3085
Google Scholar
[27] Guinea F, Katsnelson M I, Geim A K 2009 Nat. Phys. 6 30
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Google Scholar
[29] Chong Y D, Wen X G, Soljacic M 2008 Phys. Rev. B 77 235125
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图 1 (a) 二维正方晶格声子晶体的能带结构, 插图是原胞示意图; (b) M点附近的三维能带结构, 对应于图(a)中的虚线区域. 橡胶与水的质量密度和声速分别为:
$ \rho =1.3\times {10}^{3}\;\mathrm{k}\mathrm{g}/{\mathrm{m}}^{3} $ ,$ v=500\;\mathrm{m}/\mathrm{s} $ ;$ {\rho }_{0}=1.0\times {10}^{3}\;\mathrm{k}\mathrm{g}/{\mathrm{m}}^{3} $ ,$ {v}_{0}=1500\;\mathrm{m}/\mathrm{s} $ Figure 1. (a) Bulk band structure of a two-dimensional phononic crystal with a square lattice, consisting of a rubber cylinder in water. Inset: the unit cell. (b) 3 D bulk band structure around the M point, corresponding to the dashed region in (a). Here, the lattice constant and the radius of the cylinder are
$ a=1\;\mathrm{m} $ , and$ R=0.15 a $ , respectively. The mass densities and sound velocity of the rubber and water are:$ \rho =1.3\times {10}^{3}\;\mathrm{k}\mathrm{g}/{\mathrm{m}}^{3} $ ,$ v=500\;\mathrm{m}/\mathrm{s} $ ; and$ {\rho }_{0}=1.0\times {10}^{3}\;\mathrm{k}\mathrm{g}/{\mathrm{m}}^{3} $ ,$ {v}_{0}=1500\;\mathrm{m}/\mathrm{s} $ , respectively.
图 2 (a) 倾斜角
$ \alpha ={70}^ {\circ} $ 的斜方晶格体系的能带结构, 左插图是二维斜方晶格声子晶体的原胞, 右插图表示虚线区域的放大能带结构; (b)斜方晶格的第一布里渊区; (c) 线性狄拉克点附近的三维能带结构, 对应图(a)中的虚线区域Figure 2. (a) Bulk band structure of an oblique lattice with the tilted angle
$ \alpha ={70}^ {\circ} $ , Inset: the unit cell (left); the enlarged band structure around the Dirac point near${ {K}}_{1}$ point (right); (b) first Brillouin zone of the oblique lattice; (c) 3D bulk band structure around the Dirac point, corresponding to the dashed region in (a).
图 3 (a) 正方晶格声子晶体沿x方向的投影能带; (b) 倾斜角
$ \alpha ={70}^ {\circ} $ 的斜方晶格声子晶体沿x方向的投影能带$, \mathrm{I}\mathrm{m}\left(Z\right) $ 表示表面阻抗的虚部; (c) 由上述两个声子晶体构成的沿x方向界面的界面态色散关系, 粉色线表示界面态色散; (d)正方晶格和斜方晶格声子晶体构成的沿x方向的界面(左图), 频率为$ 937.4\;\mathrm{H}\mathrm{z} $ 的界面态本征声压场分布(右图)Figure 3. (a)Projected band structures along the
$ {k}_{x} $ direction of the phononic crystals with a square lattice; (b) projected band structures along the$ {k}_{x} $ direction of phononic crystals with an oblique lattice of$ \alpha ={70}^ {\circ} $ , Im(Z) represents the imaginary part of surface impedance; (c)interface state dispersion along the$ {k}_{x} $ direction of the interface constructed by two phononic crystals with the square and oblique lattices, the pink lines denote the interface states; (d) the interface constructed by two phononic crystals with the square and oblique lattices(left), the eigen pressure field distribution of the interface state at$ 937.4\;\mathrm{H}\mathrm{z}\left(\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}\right) $ .
图 4 倾斜角
$ {\rm{\alpha }}={70}^ {\circ} $ 的斜方晶格声子晶体在$ {k}_{x}=0.6\;{\text{π}}/a $ (a) 和$ {k}_{x}=0.85\;{\text{π}}/a $ (b)时沿$ {k}_{y} $ 方向的体能带; (c) 在$ {k}_{x}=0.85\;{\text{π}}/a $ 时, 正方晶格声子晶体沿$ {k}_{y} $ 方向的体能带, 其中红色区域和蓝色区域分别表示$ \mathrm{I}\mathrm{m}\left(Z\right) < 0 $ 和$ \mathrm{I}\mathrm{m}\left(Z\right) > 0 $ Figure 4. Bulk band structures along the
$ {k}_{y} $ direction of the phononic crystal with an oblique lattice with$ {\rm{\alpha }}={70}^ {\circ} $ for$ {k}_{x}=0.6\;{\text{π}}/a $ (a) and$ {k}_{x}=0.85\;{\text{π}}/a $ (b); (c) bulk band structures along the$ {k}_{x} $ direction of the phononic crystal with a square lattice for$ {k}_{x}=0.85\;{\text{π}}/a $ . The red and blue regions represent$ \mathrm{I}\mathrm{m}\left(Z\right) < 0 $ and$ \mathrm{I}\mathrm{m}\left(Z\right) > 0 $ , respectively.
图 5 由
$ \alpha ={50}^ {\circ} $ 斜方晶格与正方晶格声子晶体构成的沿x方向界面的界面态色散, 红色线和绿色线分别表示两个共同带隙中的界面态色散Figure 5. Interface state dispersion along the
$ {k}_{x} $ direction of the interface constructed by two phononic crystals with the square and oblique lattices with$ \alpha ={50}^ {\circ} $ , the red line and the green line represent the interface state dispersion in the two common band gaps, respectively.
图 6 铁柱子在环氧树脂中周期性排列构成二维声子晶体 (a) 二维正方晶格声子晶体的能带结构; (b) 倾斜角
$ \alpha ={70}^ {\circ} $ 的斜方晶格体系的能带结构; (c)由上述两个声子晶体构成的沿x方向界面的界面态色散关系(左), 粉色线表示界面态色散, 频率为$ 529.6\;\mathrm{H}\mathrm{z} $ 的界面态本征位移场分布(右)Figure 6. Two-dimensional phononic crystals are constructed by steel cylinders in epoxy: (a) Bulk band structure of a square lattice; (b) bulk band structure of an oblique lattice with the tilted angle
$ \alpha ={70}^ {\circ} $ ; (c) the interface state dispersion along the$ {k}_{x} $ direction of the interface constructed by these two phononic crystals(Left), the pink line denotes the interface states, the eigen displacement field distribution of the interface state at$ 529.6\;\mathrm{H}\mathrm{z}\left(\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}\right) $ . -
[1] Wu F, Liu Z, Liu Y 2002 Phys. Rev. E 66 046628
Google Scholar
[2] Ke M, Liu Z, Qiu C, Wang W, Shi J, Wen W, Sheng P 2005 Phys. Rev. B 72 064306
[3] Lu M H, Zhang C, Feng L, Zhao J, Chen Y F, Mao Y W, Zi J, Zhu Y Y, Zhu S N, Ming N B 2007 Nat. Mater. 6 744
Google Scholar
[4] Climente A, Torrent D, Sánchez-Dehesa J 2010 Appl. Phys. Lett. 97 104103
Google Scholar
[5] Hasan M Z, Kane C L 2010 Rev. Mod. Phys. 82 3045
Google Scholar
[6] Qi X L, Zhang S C 2011 Rev. Mod. Phys. 83 1057
Google Scholar
[7] Zhang X J, Xiao M, Cheng Y, Lu M H, Christensen J 2018 Commun. Phys. 1 97
Google Scholar
[8] Sigalas M, Economou E 1992 J. Sound Vib. 158 377
Google Scholar
[9] Jia D, Ge Y, Yuan S Q, Sun H X 2019 Acta Phys. Sin. 68 224301
Google Scholar
[10] Khanikaev A B, Fleury R, Mousavi S H, Alu A 2015 Nat. Commun. 6 8260
Google Scholar
[11] Ding Y J, Peng Y G, Zhu Y F, Fan X D, Yang J, Liang B, Zhu X F, Wan X G, Cheng J C 2019 Phys. Rev. Lett. 122 014302
Google Scholar
[12] Ni X, He C, Sun X C, Liu X P, Lu M H, Feng L, Chen Y F 2015 New J. Phys. 17 053016
Google Scholar
[13] He C, Ni X, Ge H, Sun X C, Chen Y B, Lu M H, Liu X P, Chen Y F 2016 Nat. Phys. 12 1124
Google Scholar
[14] Lu J Y, Qiu C Y, Ke M Z, Liu Z Y 2016 Phys. Rev. Lett. 116 093901
Google Scholar
[15] Lu J Y, Qiu C Y, Ye L P, Fan X Y, Ke M Z, Zhang F, Liu Z Y 2017 Nat. Phys. 13 369
Google Scholar
[16] Deng W Y, Huang X Q, Lu J Y, Peri V, Li F, Huber S D, Liu Z Y 2020 Nat. Commun. 11 3227
Google Scholar
[17] Xie B, Liu H, Cheng H, Liu Z, Tian J, Chen S 2020 Light Sci. Appl. 9 201
Google Scholar
[18] Xiao M, Chen W, He W, Chen C T 2015 Nat. Phys. 11 920
Google Scholar
[19] Li F, Huang X, Lu J, Ma J, Liu Z 2018 Nat. Phys. 14 30
Google Scholar
[20] Deng W, Lu J, Li F, Huang X, Yan M, Ma J, Liu Z 2019 Nat. Commun. 10 1769
Google Scholar
[21] Lu J, Huang X, Yan M, Li F, Deng W, Liu Z 2020 Phys. Rev. Appl. 13 054080
Google Scholar
[22] Rusin T M, Zawadzki W 2008 Phys. Rev. B 78 125419
Google Scholar
[23] Katsnelson M I, Novoselov K S, Geim A K 2006 Nat. Phys. 2 620
Google Scholar
[24] Huang X, Xiao M, Zhang Z Q, Chan C T 2014 Phys. Rev. B 90 075423
Google Scholar
[25] Huang X, Yang Y, Hang Z H, Zhang Z Q, Chan C T 2016 Phys. Rev. B 93 085415
Google Scholar
[26] Yang Y, Xu T, Xu Y F, Hang Z H 2017 Opt. Lett. 42 3085
Google Scholar
[27] Guinea F, Katsnelson M I, Geim A K 2009 Nat. Phys. 6 30
[28] Wen X, Qiu C, Qi Y, Ye L, Ke M, Zhang F, Liu Z 2019 Nat. Phys. 15 352
Google Scholar
[29] Chong Y D, Wen X G, Soljacic M 2008 Phys. Rev. B 77 235125
Google Scholar
[30] Xiao M, Zhang Z Q, Chan C T 2014 Phys. Rev. X 4 021017
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