We consider two-dimensional phononic crystals formed from silicon and voids, and present optimized unit cell designs for (1) out-of-plane, (2) in-plane and (3) combined out-of-plane and in-plane elastic wave propagation. To feasibly search through an excessively large design space (10e40 possible realizations) we develop a specialized genetic algorithm and utilize it in conjunction with the reduced Bloch mode expansion method for fast band structure calculations. Focusing on high symmetry plain-strain square lattices, we report unit cell designs exhibiting record values of normalized band-gap size for all three categories. For the combined polarizations case, we reveal a design with a normalized band-gap size exceeding 60%.
Phononic crystals (PnCs) are periodic materials that exhibit distinct frequency characteristics such as the possibility of formation of band gaps. Within a band gap, wave propagation is effectively prohibited. This inherent dynamical phenomenon can be utilized in a broad range of technologies at different length scales. Applications of PnCs include elastic/acoustic waveguiding [1] and focusing [2], vibration minimization [3], sound collimation [4], frequency sensing [5,6], acoustic cloaking [7], acoustic rectification [8], opto-mechanical waves coupling in photonic devices [9], thermal conductivity lowering in semiconductors [10][11][12][13], among others [14].
In general, it is most advantagous to have the frequency range of a band gap maximized while pulling its midpoint as low as possible in order to keep the unit cell size to a minimum. Selecting the topological distribution of the material phases inside the unit cell provides a a powerful means towards reaching this target, and this has been the focus of numerous research studies not only on PnCs but also photonic crystals (PtCs).
The exploration for optimal unit cell designs was initiated by Cox and Dobson in 1999 [15] (in the context of PtCs). The articles by Burger et al. [16] and Jensen and Sigmund [17] provide a review of subsequent studies concerned with band-gap widening in PtCs. In the area of PnCs, the problem has been treated in a variety of settings and using several techniques. For example, unit cells have been optimized in one-dimension [18,19] and in two-dimensions (2D) [20][21][22][23][24][25], using gradient-based [21][22][23] as well as non-gradient-based [24,25] techniques. Interest in band-gap size maximation has also been treated outside the scope of the unit cell dispersion problem [21,26]. In all these optimization studies the focus has been primarily on PnCs based on an infinite thickness model and a material composition consisting of two or more solid (or solid and fluid) phases with the exception of a few investigations that considered thin-plate singlephase models [22,23]. Recognizing the practical significance of solid-and-air PnCs with relatively large crosssectional thickness, some studies considered the configuraton of a 2D solid matrix with periodic cylindrical voids -modeled under 2D plain-strain conditons [27] or as a three-dimensional continuum with free surface boundary conditions [28] -and investigated the dependence of band-gap size upon the void radius. For combined out-of-plane and in-plane waves in 2D infinite-thickness PnCs formed from silicon and a square lattice of circular voids, it has been shown that the band-gap size normalized with respect to the mid-gap frequency cannot exceed 40% [27]. In this letter we utlize a specialized optimization algorithm in pursuit of the best unit cell solid-void distribution for the 2D plain-strain problem considering high-symmetry square lattices. We cover the cases of (1) out-of-plane, (2) in-plane and (3) combined out-ofplane and in-plane elastic wave propagation. Our search methodology is also applicable to the parallel problem of 2D PtCs optimization, where transverse-electric and transverse-magnetic waves may be considered separately [29] or in combination [30].
The governing continuum equation of motion for a heterogeneous medium is
where C is the elasticity tensor, ρ is the density, u is the displacement vector, x = {x, y, z} is the position vector, ∇ is the gradient operator, and (.) T is the transpose operation. We assume the wave propagation to be confined to the x-y plane only, that is, ∂u/∂z = 0. As such, we have two independent sets of equations, one for out-ofplane motion and the other for in-plane motion. To obtain the band structure for a given PnC unit cell design we assume a Bloch solution to the governing equations in the form u(x, k; t) = ũ(x, k)e i(k.x-ωt) where ũ is the Bloch displacement vector, k is the wave vector, ω is the frequency, t is the time, and i = √ -1. Due to lattice symmetry the analysis is restricted to the first Brillouin zone. We consider square lattices and furthermore impose C 4v symmetry at the unit cell level. Subsequently design representation is needed in only a portion of the unit cell and the band structure calculation is limited to the corresponding irreducible Brillouin zone (IBZ). Furthemore, we model only the solid portion of the unit cell. The void portion is not modeled since we permit only contiguous distribution of solid material. In this manner the PnCs considered exhibit geometric periodicity (with free in-plane surfaces) and not material periodicity. In practice the voids will be either in vacuum or filled with air. Our model presents an adequate representation of both cases because the elastic waves propagating in the solid will have the dominating effect [28]. In fact, this also suggests that the results we show are practically independent of the choice of the solid material. We numerically solve the emerging e
This content is AI-processed based on open access ArXiv data.
