Zak phase and band inversion in dimerized one-dimensional locally resonant metamaterials

Weiwei Zhu, Ya-qiong Ding, Jie Ren, Yong Sun, Yunhui Li, Haitao Jiang, and Hong Chen

Phys. Rev. B 97, 195307 – Published 14 May 2018

Abstract

The Zak phase, which refers to Berry's phase picked up by a particle moving across the Brillouin zone, characterizes the topological properties of Bloch bands in a one-dimensional periodic system. Here the Zak phase in dimerized one-dimensional locally resonant metamaterials is investigated. It is found that there are some singular points in the bulk band across which the Bloch states contribute π to the Zak phase, whereas in the rest of the band the contribution is nearly zero. These singular points associated with zero reflection are caused by two different mechanisms: the dimerization-independent antiresonance of each branch and the dimerization-dependent destructive interference in multiple backscattering. The structure undergoes a topological phase-transition point in the band structure where the band inverts, and the Zak phase, which is determined by the numbers of singular points in the bulk band, changes following a shift in dimerization parameter. Finally, the interface state between two dimerized metamaterial structures with different topological properties in the first band gap is demonstrated experimentally. The quasi-one-dimensional configuration of the system allows one to explore topology-inspired new methods and applications on the subwavelength scale.

Received 3 January 2018
Revised 25 March 2018

DOI:https://doi.org/10.1103/PhysRevB.97.195307

Physics Subject Headings (PhySH)

Metamaterials Optical & microwave phenomena Topological effects in photonic systems

1-dimensional systems Interfaces

Microwave techniques Transfer matrix method

Condensed Matter, Materials & Applied PhysicsAtomic, Molecular & OpticalInterdisciplinary Physics

Authors & Affiliations

Weiwei Zhu¹, Ya-qiong Ding^1,2, Jie Ren^1,3, Yong Sun^1,*, Yunhui Li¹, Haitao Jiang¹, and Hong Chen^1,†

¹MOE Key Laboratory of Advanced Micro-Structured Materials, School of Physics Science and Engineering, Tongji University, Shanghai 200092, China
²Science College, University of Shanghai for Science and Technology, Shanghai 200093, China
³Center for Phononics and Thermal Energy Science, China-EU Joint Center for Nanophononics, Shanghai Key Laboratory of Special Artificial Microstructure Materials and Technology, Tongji University, Shanghai 200092, China

^*yongsun@tongji.edu.cn
^†hongchen@tongji.edu.cn

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Vol. 97, Iss. 19 — 15 May 2018

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Images

Figure 1
The scheme of the 1D locally resonant metamaterials made of a backbone waveguide and branch resonators. (a) One unit cell with the dimerization of $Δ = (l - L) / L$ . Here the symmetrical unit cell has a length of $Λ = 2 L$ . The distance between two identical branches in one unit cell is $l = (1 + Δ) L$ . $l_{0} = L / 2$ is the length of the branch. The backbone waveguide and branch are made of same material. (b) Structure without dimerization ( $Δ = 0$ ).
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Figure 2
Reflection coefficient for the semi-infinite systems (a) and (b) and single unit cell (c) and (d) as a function of the frequency and the dimerization. (a) and (c) are the amplitude of the reflection coefficient. (b) and (d) are the reflection phase. Four topological phase-transition points are indicated by the white circles. The states with zero reflection are marked by a group of black dashed curves.
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Figure 3
Band structures for 1D metamaterials with dimerization $Δ = - 0.8, - 0.68, - 0.5, 0, and 0.5$ . The frequency domain is divided into bulk bands and band gaps (gray area). For the convenience in expression, the $n th$ isolated passbands (band gaps) are named with green letters (brown letters). The effective constitutive parameters $ε_{eff}$ and $μ_{eff}$ of metamaterials in the band gaps are drawn with orange open circles and purple triangles, respectively. In (b) and (d), band crossing occurs, whereas in (a), (c), and (e), all of the bands considered are isolated. Symmetry of the band-edge state is indicated with the red dot for even symmetry and the pink star for odd symmetry. The Zak phases (0 or $π$ ) of the isolated bands are also given.
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Figure 4
Phase difference $δ φ_{n, q}$ between states $u_{n, q}$ and $u_{n, q + δ q}$ with $δ q = 2 π / 20$ as a function of $q$ in the first Brillouin zone under different dimerization parameters: (a) $Δ = - 0.8$ , (b) $Δ = - 0.5$ , and (c) $Δ = 0.5$ . Two types of singular points are indicated by NS-SP and MS-SP, respectively. The 1st(2nd) means the principal value (the second branch value) of the multivalued equations.
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Figure 5
Experiments of the dimerized 1D locally resonant metamaterials. (a) Photography of a sample with $Δ = - 0.5$ . (b) and (c) Amplitude and phase of reflection coefficient of the sample shown in (a). The red lines are measurements, and the blue lines are theoretical calculations from the transfer matrix. (d) and (e) The sample with $Δ = 0.5$ and corresponding results. The samples are fabricated on FR4 double-sided copper-clad boards with the geometric parameters given in the figure.
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Figure 6
Experiments of the interface state with the topological transition in the first band. (a) Photography of the sample composed by two topology-distinct 1D dimerized metamaterials, i.e., structures with $Δ = 0.5$ and $Δ = - 0.5$ . The interface is denoted by the dashed line. (c) Measured (red) and calculated (blue) transmission spectra. There is a transmission peak at 1.13 GHz emerging in the first band gap, which is represented by the gray area. (c) Electric-field distribution of the topological interface state. The measurements (red dots) are consistent with the theoretical calculations (blue lines) very well.
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