Subwavelength waveguiding of surface phonons in pillars-based phononic crystal Open Access

In the last two decades, periodic materials have created wealth of studies in the phononic community due to their unusual wave properties that cannot be found in the nature.^1–7 The propagation features of the acoustic and elastic waves in these composite media are originated from the periodical repetition of resonant or non-resonant inclusions in a matrix background. Wave propagation is typically permitted or prohibited in these periodic materials for certain frequency range called band gaps. Consequently, the composite materials act as a perfect mirror in the forbidden frequency range. It is now well known that the origin of these band gaps stem from two distinct mechanisms depending if the unit cell possesses or not resonant elements.^8–10 When the unit cell is constituted by non-resonant elements, the scattering of waves by the ordered structure will induce nondestructive and destructive interferences and will create band gap at a wavelength comparable to the spatial periodicity of the structure. The forbidden frequency range is termed Bragg band gap and the composite media are called phononic crystals. Using this concept, many potential applications have been demonstrated in acoustic devices.^11–20 

The presence of resonant elements inside the unit cell will give the possibility to create locally resonant band gaps due to the hybridization between the discrete frequencies of each single resonator and incident waves. For instance, by appropriately choosing the geometrical parameters of resonator elements in the unit cell, the locally resonant band gap can be opened at lower frequencies than the Bragg gap, making the relevant wavelength much bigger than the spatial periodicity of the artificial crystal.^8,9 These locally resonant materials are belong to the family of metamaterials, which have demonstrated an astonishing wave phenomena such as superlensing, superprism and clocking effects.^21–31 

Generally, the insertion of defects inside the perfect phononic crystals will result in highly confined acoustic energy in small cavity or waveguide when the frequencies of these defect modes operate inside the frequency range of the Bragg band gap. In this case, the wavelength of the guided mode is in the same order of the period of the crystal, which can limit the range of application at low frequency domain.

On the present paper, we seek to guide acoustic waves at the subwavelength scale by using resonant defects. We focus on the propagation of surface phonons in pillars-based phononic crystal. The pillars are acting as resonators or as scatters in each unit cell. Consequently, Bragg band gap as well as a locally resonant band gap are expected to appear in the dispersion relation. The defect modes that allow a subwavelenght waveguide are obtained by altering the geometrical parameters of the pillars along a row. Especially, a careful choice of the height of the defect pillars permits to create two subwavelenght waveguiding mechanisms. A classical one based on the presence of the defect modes inside the locally resonant band gap. The seconde mechanism is based on the interaction between the phonon resonances of defect modes and the surface waves of the semi-infinite homogenous medium. The paper is arranged as follows: in Sec. I A, we describe the model of the pillars-based phononic crystals and the finite element method used for band structure calculations. In Sec. I B, we present the dispersion of the subwavelenght guiding inside and outside the band gap before giving the conclusion in Sec. I C.

The locally-resonant phononic crystal we studied in this paper is represented in Figure 1. It is a monolithic structure made of silicon and composed of cylindrical pillars arranged in a square lattice that stands on the upper surface of a semi-infinite medium. This geometry is used for the simulation of wave propagation through the surface area of the pillars. The xy plan is assumed to be parallel to the wave propagation surface, while the z axis gives the direction of the pillars. The radius r and the height h_p of the cylindrical pillars are given according to the lattice constant a of the periodic array. The height h_d of the pillars that will form the defect is also given according to a. The mass density ρ of the material involved in the calculations is equal to 2330 kg⋅m⁻³, and the surface wave velocity along the direction of propagation ΓX is 5845 m⋅s⁻¹.

Figure 2 represents the supercell model we used to calculate the dispersion diagram. This supercell contains 7 pillars including the defect pillar. The finite element method (FEM) is an efficient numerical method for band structure analysis of phononic crystal. Indeed, it permits to solve integral equations and partial differential equations (PDE) over complicated domains, thereby allowing to calculate dispersion curves and transmission spectra. We use this method to calculate the dispersion diagram of the primitive supercell repeated indefinitely along x and y directions. In this case, the Bloch-Floquet theorem states that, for a finite element of mesh that is indexed by m, the mechanical displacements u_i of the nodes on the boundary of the unit cell satisfy the following formula.

where k_x is the components of the Bloch wave vectors in the x direction.

Considering a simple harmonic motion where the mechanical fields are time dependant, the governing equation of elastic waves is given by

where ρ is the density of the material. The mechanical stresses T_jk depend on the strain as

where C_jklm are the elastic stiffness constants. The strain is related to the displacements like

The general wave motion problem with no external applied force can be written as follows

where M_uu and K_uu are the mass and stiffness matrices and u is the displacement vector at the mesh nodes. We use these formula to solve the eigenvalue problem that gives the eigenfrequency solutions in the first Brillouin zone. Then, the dispersion curves are calculated by extending these solutions over the first Brillouin zone using symmetries.

The transmission of the wave through the defect row of pillars is an important simulation to assess waveguiding features. For this purpose, we use the FEM model depicted Figure 2, which is a finite-size structure composed of pillar array of 7 × 7. The longitudinal row of pillars at the center can have a different height comparing to the other rows. In this model, the excitation source can be generated on the top of the surface by using different elements of mesh (edge, node and face). In addition, to cancel the edge effects that create the reflections of the scattering wave at the boundary limits of the structure, we use a perfectly matched layer (PML),³² that allows gradually the absorbtion of the mechanical vibrations, in all sides of the simulation domain. In this way, the structure is considered as an infinite medium, since the source is not disturbed by the border reflections. To achieve this, the governing equation (2) is written as following

where γ_j is an artificial damping at position x_j. The PML is added to reduce gradually the incoming wave in all directions of propagation. For instance, γ_x is different from 1 only in the PML subdomain and is given by the expression:

where x_l is the coordinate at the beginning of the PML subdomain. The constant σ_x is fixed to a suitable level of attenuation, such that the mechanical disturbances are absorbed before reaching the domain limits, and to ensure a very weak reflection at the interfaces between the regular domain and the PML. In our simulations, we have selected a width of PML equal to a in all the domain and σ_j equal to 10⁻¹².

We can implement the previous motion equations as partial differential equations in any free or commercial FEM software.

We have selected a perfect phononic structure consisting of a square array of cylindrical pillars with height h_p equal 0.5 × a and radius r equal 0.3 × a. Both pillars and substrate are made of Silicon. Figure 3 shows the band diagram of the propagating modes along ΓX direction for a perfect pillar-based supercell model (see Figure 2). As observed in the band diagram, this choice of the geometrical parameters ensures the emergence of a locally resonant band gap extending from 1490 m⋅Hz to 1700 m⋅Hz. In this reduced frequency range, no surface acoustic wave is allowed to propagate through the periodic array of pillars. The origin of this locally resonant band gap is well described in previous works.^8,9

After demonstrating the exitance of the locally resonant band gap in the the perfect pillar-based structure, we create a linear defect by modifying the height of the pillars along one row. In Figure 4, we present the dispersion curves of the elementary supercell including a pillar defect with a height h_d of 0.45 × a which is smaller than the pillars of the perfect crystal. The red dots represent the band structure of the perfect pillar-based crystal (h_p = 0.5 × a) compared to the band structure of the defect-based waveguide represented with blue dots. The analysis of the band diagram shows two new branches inside the locally resonant band gap. These two localized modes (M₁ and M₂) are linked to the discrete frequencies of the defect pillars and can be shifted toward high frequencies when we decrease the height of the pillars. We can also notice that the dispersion of the defect modes M₁ and M₂ is less flat comparing to lower frequency branches, which permits to transfer more efficiently the elastic energy through the defect pillars -see the zoom of these guided modes in Figure 5(a).

In order to assess the spatial confinement of the defect modes, we plot in Figure 6 the eigenmodes of the defect branches M₁ and M₂ in the limit of the Brillouin zone where the wave vector k_x is equal to π/a. We can notice that all components u_x, u_y and u_z of the displacement field are mostly confined within the pillar defect of the waveguide for both modes. The acoustic amplitude of u_x and u_y components is magnified in the top of the pillar and is very weak at the interface with the substrate. However, the out of plane component u_z is distributed in the whole volume of the pillar and penetrate in the substrate.

To simulate the propagation of the surface phonons through the defect-based waveguide that is inside the locally resonant band gap, we built a finite size waveguide structure that is composed of 7 defect-based periods, as it is shown in Figure 7. An incident monochromatic wave operating at frequency f₁ = 1505 m⋅Hz is generated by a source placed at the entrance of the waveguide. This frequency ensures to excite exclusively the mode M₁ as it can be checked in Figure 5(a). As it is revealed in Figure 7, the defect mode M₁ is strongly confined spatially along the row of the defect pillars. The acoustic impedance mismatching between the free surface and the entrance as well as the exit of the waveguide will induce reflections at these interfaces. Thus, a constructive and destructive interferences will appear leading to an extinction of some pillars.

It is important to notice that the wavelength of the wave transmitted through the waveguide is much larger than the lattice parameter a of the artificial crystal as well as the aperture of the waveguide. Consequently, this defect mode permits a subwavelenght waveguiding.

The advantage for using the resonant defect modes is the ability to shift the waveguide frequencies by varying only their geometrical parameters. In the following, we assume that the pillar height of the defect row is equal to 0.70 × a, which is higher than those of the perfect crystal. The dispersion curves of this defect is illustrated in Figure 8 which shows a new possibility of waveguiding outside the locally resonant band gap. Indeed, this pillar defect possesses two resonant frequencies that can interact with the surface acoustic wave of the homogenous part, creating a new type of waveguiding highlighted by the branches M₃ and M₄. This waveguiding is a hybridation between a surface acoustic wave and the discrete mode of the defect pillars. We show in Figure 9 the eigenmodes of M₃ and M₄ at the limit of the Brillouin zone (k_x = π/a), which reveal that the elastic energy is strongly confined in the defect pillar, even if their frequencies are outside the band gap. The polarizations of M₃ and M₄ are very similar to those of M₁ and M₂. However, when the wave vectors of these modes are close to that of the surface acoustic waves of the free surface, an hybridation occurs between the modes allowing the energy transfer from the defect waveguide towards the surface of the substrate. To highlight the waveguiding properties of these defect-based guiding modes, we selected the frequency f₂ = 783 m⋅Hz, as depicted in Figure 10(a), and we display the out of plane displacements of mode M₃ in Figure 10(c). It is clearly observed that the incident wave propagates with a strong confinement inside the subwavelength waveguide. However, a slight amount of energy leaks out of the waveguide towards the surrounding pillars.

In this paper, we demonstrated theoretically a subwavelength waveguiding based on locally resonant defects in a monolithic structure made of silicon and composed of cylindrical pillars arranged in square lattice that stands on the upper surface of a semi-infinite medium. We used the finite element method to calculate the dispersion diagram of a primitive supercell as well as the waveguiding properties of a finite-size structure. The advantage for using the resonant defect modes is the ability to tailor the waveguide frequencies by varying only their geometrical parameters. We have shown two mechanisms of subwavelength waveguiding by changing the height of defect pillars. The first one corresponds to the presence of the defect modes inside the locally resonant band gap, while the seconde is due to the hybridation between the phonon resonances of defect modes and the surface phonons of the semi-infinite homogenous medium. These two mechanisms allow a subwavelength confinement of surface phonons through the locally resonant defects. This opens a new way to control the surface phonons and could be a potential candidate for the improvement of surface acoustic wave devices.