# Broadening Bandgap Width of Piezoelectric Metamaterial by Introducing Cavity
## 1 Introduction  
Metamaterials, defined as artificial structures that exhibit physical properties not available in natural material, have extraordinary capability in low-frequency sound/vibration attenuation, negative refraction, and super lenses [1; 2; 3; 4; 5; 6; 7; 8]. The acoustic metamaterials, consisting of periodically arranged unit-cells, are capable of manipulating elastic wave propagation. An acoustic metamaterial may utilize Bragg scattering or local resonance [9; 10; 11; 12; 13; 14]. Bragg bandgaps stand for the zones between incident and reflected waves, which are generated at wavelengths comparable to the spatial scale of a unit-cell [15; 16; 17]. Different from the Bragg scattering mechanism, the local resonance mechanism utilizes the sub-wavelength local resonances induced within a unit-cell. The local resonance can alter the frequency-dependent effective mass densities, and/or bulk moduli of the continuum media. In most cases, the internal resonators have highly contrasting elastic properties. Within the bandgap, the elastic wave cannot propagate, and wave energy is reflected back or temporarily stored in resonators [1; 18; 19]. Owing to their two-way electro-mechanical coupling, piezoelectric transducers have been adopted to induce or enhance bandgap behavior in metamaterials [8; 20; 21; 22]. Piezoelectric metamaterial with periodic resonant circuits can directly produce the local resonance bandgap. This is because a local resonance is created by the LC (inductor-capacitor) circuit, as the piezoelectric transducer acts electrically as a capacitor. For example, piezoelectric periodic arrays are integrated into rods for wave attenuation and localization [20]. In another example, multi-resonant shunts were adopted to generate multiple bandgaps in a piezoelectric metamaterial [21]. Theoretical and experimentalinvestigations on photonic rods and beams [23] with shunt circuits have been performed. Recently, piezoelectric metamaterials were applied in two-dimensional wave steering [24] and vibration mode customization [25]. Notably, piezoelectric metamaterial has advantages over the more conventional mechanical metamaterial in two aspects: Relatively simple configuration and adaptivity. Piezoelectric transducers are usually attached to the host structure directly, and even simple topology/geometry can yield complex dynamic phenomena locally at the unit-cell level, and globally at the metamaterial level. Moreover, the shunted piezoelectric transducers allow the online tuning of bandgap towards a desired frequency range, as one can conveniently integrate tunable circuitry elements. So far, almost all the studies on piezoelectric metamaterial are based on transfer-matrix method or finite element analysis [22; 26]. However, previous investigations concerning piezoelectric shunt design and analysis in sensing/control applications, often resort to the lumped-parameter approach, which can help in revealing the underlying physics with parametric influences [27; 28; 29]. It is worth noting that circuitry dynamics in the electrical domain is described naturally by the lumped-parameter model. The lumped-parameter approach is suitable for dealing with complex circuitry integration schemes, as well as systems with nonlinearity [30; 31]. A more common approach is to conduct continuum mechanics-based modeling of the host substrate, followed by discretization with respect to wavenumber, where an equivalent frequency-dependent Young's modulus is used to reflect the piezoelectric shunt circuit effect [22; 26]. The shunt circuit effect is included as equivalent impedance added to the original mechanical impedance of the transducer. However, such an approach may not be able to deal with complex circuitry, especially those with multiple branches or nonlinear elements. From the fundamental physics standpoint, one well-known limitation of piezoelectric metamaterial is the relatively narrow bandwidth, compared with other types of acoustic metamaterials. For example, [PERSON] et al.