1. Introduction
Phononic crystals (PCs) are artificial periodic composites designed to exhibit phononic band gaps and they have been quite studied^{1}^{-}^{14}. There are no mechanical (elastic or acoustic) propagating waves in phononic band gaps, only evanescent waves. These band gaps are created by the periodically mismatch between the constituent materials. This mismatch can be considered to arise either from difference of material properties or geometry (continuum-scale theory), or from interatomic force constants and masses (atomic-scale theory).
The ability of creating phononic band gaps is similar to the electronic and photonic band gaps in semiconductors and photonic crystals^{15}^{-}^{16}, respectively. The physical origin of phononic and photonic band gaps can be understood at micro-scale using the classical wave theory to describe the Bragg and Mie resonances, respectively, based on the scattering of mechanical and electromagnetic waves propagating within the crystal^{17}.
PCs have many applications, such as vibration isolation technology^{18}^{-}^{22}, acoustic barriers/filters^{23}^{-}^{25}, noise suppression devices^{26}^{-}^{27}, surface acoustic devices^{28}, architectural design^{29}, sound shields^{30}, acoustic diodes^{31}, elastic metamaterials^{21}^{-}^{22}^{,}^{25}^{,}^{27}^{,}^{32} and thermal metamaterials^{33}^{-}^{39}.
There are also smart PCs that have been studied, such as piezoelectric^{40}^{-}^{54}, piezomagnetic^{55}^{-}^{58} and magnetoelectroelastic^{14}^{,}^{59}^{-}^{63} PCs. Among these intelligent PCs, piezoelectric PCs are sensitive to elastic and electric field. Even though the band structure behavior of piezoelectric PCs have been already investigated, to our knowledge only few studies^{46}^{-}^{49}^{,}^{51} focused on the influence of inclusion geometry and lattice on band gap formation. Wang and co-workers^{46} considered a piezoelectric PC with square lattice and different inclusion geometries (regular triangle, square, hexagon, circle and oval). They found that the largest complete band gap is obtained by selecting the inclusion with the same symmetry of lattice for the first band gap. Hsu et al.^{47} studied the band structure of a piezoelectric PC with square and triangular lattices using Mindlin-Reissner plate theory and considering only circular inclusion. Qian and co-workers^{48} studied the band structure of piezoelectric PCs with square lattice and circular and square inclusions. They revealed the existence of several very large complete band gaps in PZT rods reinforced polythene composite. Zhou et al.^{49} investigated the band structure of piezoelectric PCs consisting of rectangular inclusion in an epoxy substrate. They considered a PC with square and triangular lattices. Wang and co-workers^{51} considered a piezoelectric PC with circular inclusions and square and rectangular lattices. They verified that the largest band gap width is not always obtained for a square lattice. In some cases, rectangular lattice can produce broader gaps. In this context, we extend the studies about piezoelectric PCs^{40}^{-}^{54} considering the influence of different inclusion geometries - circular, hollow circular, square and rotated square with 45º angle of rotation with respect to the x and y axes, and different lattices - square, rectangular, triangular, honeycomb and Kagomé on the band structure.
Furthermore, researches have used PCs on the scale µm^{10}^{,}^{17}^{,}^{64}^{,}^{65} and mm, resulting in band gaps ranging from GHz and kHz to MHz, respectively. More recently, with the advance of nanomaterials fabrication, nanophononic crystals have been studied and it is possible to control wave propagation in a frequency range from hypersonic^{3}^{-}^{6}^{,}^{66}^{-}^{75} to thermal^{33}^{-}^{39}. However, nano-piezoelectric PCs have not been investigated yet even though studies about nano-piezoelectric materials have been reported^{76}^{-}^{79}.
The main purpose of this study is to investigate the elastic band structure, also known as dispersion relation, of a nano-piezoelectric phononic crystal with square, rectangular, triangular, honeycomb and Kagomé lattices, composed by a polymeric matrix and BaTiO_{3} inclusions, considering wave propagation in the xy plane and two-dimensional periodicity. We consider the following inclusions geometries - circular, hollow circular, square and rotated square with a 45º angle of rotation with respect to the x and y axes. To obtain the elastic band structure we use the semi-analytical plane wave expansion^{1}^{,}^{2} (PWE) method.
To the best of our knowledge, it is the first time that the band structure of nano-piezoelectric PCs with different inclusion geometries and different lattices is investigated.
2. 2D Nano-Piezoelectric Phononic Crystal Model by PWE Method
PWE formulation of a 2D PC is presented in this section. PWE method^{1}^{,}^{2}, also known as ω(k) method, is one of the most used methods to calculate the elastic band structure of PCs and it has been applied in micro^{10} and nanophononic crystals^{71}^{,}^{75} (NPCs). PWE method is used to solve the constitutive equations of a piezoelectric material based on classical elasticity theory, ignoring nanoscopic size effects, similar to other studies^{10}^{,}^{71}^{,}^{75}.
We consider two-dimensional periodicity, transversely isotropic elastic solid and wave propagation in the xy plane. Complete band gaps between XY (longitudinal-transverse vibration) and Z (transverse vibration) modes are observed for all inclusion geometry - circular, hollow circular, square and rotated square, considering square, rectangular, triangular, honeycomb and Kagomé lattices. Piezoelectric effect on the Z modes is also investigated.
2.1 2D Phononic Crystal Model
Figure 1 (a-e) sketches the cross section of nano-piezoelectric PC unit cells taking in to account square, rectangular, triangular, honeycomb and Kagomé lattices, respectively, with arbitrary inclusion geometry. Figure 1 (f-h) represents the first Brillouin zone^{80} (FBZ) for square, rectangular, triangular, honeycomb and Kagomé lattices. Note that triangular, honeycomb and Kagomé lattices present the same FBZ shape (Figure 1 (h)). We consider four types of BaTiO_{3} inclusions: circular, hollow circular, square and rotated square with a 45º angle of rotation with respect to the x, y axes. We underline that only BaTiO_{3} inclusions are rotated for rotated square configuration, not the entire lattice.
In addition, we highlight that we consider 1/4 of the FBZ as the actual first irreducible Brillouin zone^{80} (FIBZ) for square and rotated square inclusions in triangular, honeycomb and Kagomé lattices because the symmetries of phononic crystals are reduced since the phononic crystal basis has different symmetry from the lattice^{81}. Thus, for these configurations we scan M'-Γ-X-M-X'-M'. It is very important to make this adjustment since it changes the phononic crystal band structures^{81}. For circular and hollow circular inclusions in triangular, honeycomb and Kagomé lattices, phononic crystal symmetry is not reduced thus we scan only M-Γ-X-M.
FIBZ points in Figure 1 (f-h) are Γ (0,0), X
The constitutive equations of an elastic piezoelectric material are^{41}:
where i,j,k,l = 1,2,3, σ_{ij} is the elastic stress tensor, D_{i} is the electric displacement vector, u_{i} is the elastic displacement vector, E_{l} is the electric field vector, c_{ijkl} is the elastic stiffness tensor, e_{lij} is the piezoelectric tensor and ϵ_{il} is the dielectric tensor. The standard tensor notation is used with Latin indices running from 1 to 3. They obey Einstein's summation convention when repeated.
We restricted the treatment to linear media, thus the elastic strain tensor ε_{kl} is simplified:
Furthermore, based on the quasi-static approximation, there is no electro source (i.e. since there is no voltage source, this configuration is also known as an open circuit) and the curl is zero, thus the electric field is taken as gradient of scalar potential and one can write:
The differential equations of motion in absence of body forces are given by:
where ρ is the mass density and dot denotes differentiation with respect to time. Substituting Eqs. (1-2) in Eqs. (5-6), applying the simplifications of Eqs. (3-4), considering a transversely isotropic elastic solid and for a two-dimensional problem, ∂/∂x_{3}=0, results:
where c_{66}=
In addition, considering a piezoelectric PC, one can note that c_{11} = c_{11} (r), c_{12} = c_{12} (r), c_{66} = c_{66} (r), c_{44} = c_{44} (r), e_{15} = e_{15} (r), ϵ_{11} = ϵ_{11} (r), ρ = ρ(r), because we consider two different materials - BaTiO_{3} inclusions and a polymeric matrix, and u_{i} = u_{i} (r,t). For a two-dimensional periodicity (the system has translational symmetry in z direction and material parameters depend only on the x and y coordinates), then r = xe_{1} + ye_{2} (x,y ∈ ℝ) is the two-dimensional spatial vector and e_{i} (i = 1,2) are the basis vectors in real space.
In order to eliminate the factor time in Eqs. (7-10), we apply the temporal Fourier transform. Applying Floquet-Bloch's theorem^{82}^{-}^{83}, expanding Bloch wave amplitude as Fourier series in reciprocal space and considering wave propagation in the xy plane (k_{3} = 0), we can write:
where u_{i}_{k} (r) is the Bloch wave amplitude, note that u_{i}_{k} (r) = u_{i}_{k} (r + r) and u_{i}(r + r) = u_{i}(r)e^{j}^{k.r}, e^{j}^{k.r} is called Bloch periodic boundary condition, k = ub_{1} + 𝑣b_{2} is the Bloch wave vector, u, 𝑣 ∈ ℚ are the symmetry points within the FIBZ in reciprocal space, or we may write k = k_{1}
e_{1} + k_{2}
e_{2}, k_{1}, k_{2} ∈ ℝ are the point coordinates within the FIBZ in Figure 1 (f-h) for the reciprocal space. The basis vectors in reciprocal space b_{i} (i = 1,2) are defined as a_{i} ∙ b_{j} = 2πδ_{ij}, δ_{ij} = 0 if i ≠ j or δ_{ij} = 1 if i = j is the Kronecker delta, b_{1} = 2π
The lattice vector components are a_{i} = ae_{i} (i = 1,2) for square lattice, a_{1} = a_{1}
e_{1}, a_{2} = a_{2}
e_{2} for rectangular lattice, a_{1} = ae_{1}, a_{2} =
The reciprocal lattice vector is defined as
Furthermore, we may expand c_{11},c_{12},c_{66},c_{44},e_{15},ϵ_{11}, ρ in Fourier series on the reciprocal space as:
where P is one of the c_{11},c_{12},c_{66},c_{44},e_{15},ϵ_{11}, ρ and g has the same expressions of g with m, n ∈ ℤ. We use g instead of g to highlight the difference between the Fourier series expansion of material properties and displacements.
Substituting Eqs. (11) and (12) in Eqs. (7-10), with
where
The sub-matrices in Eq. (14) are given by:
The matrix M in Eq. (13) is expressed by:
The vector q in Eq. (13) is given by:
Equation (13) is an infinite system of equations, thus the Fourier series needs to be truncated. We choose m,m,n,n = [-M, ..., M] and the total number of plane waves is (2M + 1)^{2}. Equation (13) represents a generalized eigenvalue problem of ω^{2} (k) and should be solved for each k into the FIBZ for square, rectangular, triangular, honeycomb and Kagomé lattices (Figure 1(f-h)).
The Fourier coefficients are defined as:
where the indexes A and B in Eq. (25) are related to inclusion (BaTiO_{3}) and polymeric matrix, respectively, F(g) is the structure function and ƒ is the filling fraction of each type of inclusion, considering circular section of radius r̃, square section of width 2l, rotated square section of width 2l with a 45º angle of rotation with respect to x and y axes and hollow circular section with external radius R̃ and internal radius r̃, R̃ > r̃. The hollow cylinder inclusion has an internal radius r̃ of the same polymer at matrix and a thickness R̃ - r̃ of BaTiO_{3}, similar to Anjos & Arantes^{10}.
The filling fraction is defined as ƒ = S_{A}/S_{C} for square, rectangular and triangular lattices, whereas for honeycomb and Kagomé lattices it is defined as ƒ = 2S_{A}/S_{C} and ƒ = 3S_{A}/S_{C}, respectively, where S_{A} is the cross section area of the BaTiO_{3} inclusions and S_{C} = ‖a_{1} × a_{2}‖ is the cross section area of the unit cell. Note that differently from other lattices honeycomb and Kagomé lattices have two and three inclusions per unit cell, respectively.
The filling fractions for square, rectangular, triangular, honeycomb and Kagomé lattices are:
respectively.
The structure function for square, rectangular and triangular lattices F(g) is given by:
The integral in Eq. (31) is performed over the unit cell cross section. The structure functions for square, rectangular and triangular lattices are calculated by:
where g = ‖g‖ and g_{1,2} = ‖g_{1,2}‖.
The structure function for honeycomb lattice F_{H}(g) is defined as^{84}:
where ū_{1} = -ū_{2} = a(0,1/2) are vectors that define central position of the two BaTiO_{3} inclusions within the honeycomb unit cell. We choose these vectors similar to Gao et al.^{85}. Thus, the structure functions of BaTiO_{3} inclusions for honeycomb lattice are the same of Eq. (31) multiplied by cos(g ∙ u_{1}), considering ƒ from Eq. (29).
The structure function for Kagomé lattice F_{Kg} (g) is defined as:
where
where ū_{j1,2} = ‖ū_{j1,2}‖.
3. Results and Discussion
The physical parameters of BaTiO_{3} inclusion (A) and polymeric matrix (B) are listed in Table 1. We calculate the elastic band structure considering initially a fixed filling fraction, 0.335, for the four inclusion cross section geometries considered - circular, hollow circular, square and rotated square with a 45º angle of rotation with respect to the x, y axes, in a square, rectangular, triangular, honeycomb and Kagomé lattices. In the course of numerical calculations, we consider 441 plane waves for Fourier series expansion (i.e. M = 10), which results in a good convergence. Band structure plots are limited until a maximum frequency of 0.12 THz.
Geometry/Property | Value |
---|---|
Lattice parameter (ɑ) | 30 x 10^{-9} m |
Lattice parameter (ɑ_{1}) | 30 x 10^{-9} m |
Lattice parameter (ɑ_{2}) | 25 x 10^{-9} m |
Filling fraction (ƒ) | 0.335 |
Mass density (ρ_{A},ρ_{B}) | 5730 kg/m^{3}, 1150 kg/m^{3} |
Elastic constant (c_{11A},c_{11B}) | 166 x 10^{9} N/m^{2}, 7.8 x 10^{9} N/m^{2} |
Elastic constant (c_{12A},c_{12B}) | 77 x 10^{9} N/m^{2}, 4.7 x 10^{9} N/m^{2} |
Elastic constant (c_{44A},c_{44B}) | 43 x 10^{9} N/m^{2}, 1.6 x 10^{9} N/m^{2} |
Elastic constant (c_{66A},c_{66B}) | 44.5 x 10^{9} N/m^{2}, 1.55 x 10^{9} N/m^{2} |
Piezoelectric coefficient(e_{15A},e_{15B}) | 11.6 C/m^{2}, 0 C/m^{2} |
Dielectric coefficient (𝜖_{11A},𝜖_{11B}) | 11.2 x 10^{-9} C^{2}/Nm^{2}, 0.0398 x 10^{-9} C^{2}/Nm^{2} |
Note that ais the lattice parameter for square, triangular, honeycomb and Kagomé lattices, and a_{1}, a_{2}are the lattice parameters for rectangular lattice.
Figure 2 (a-d) shows the band structures of the nano-piezoelectric PC for square lattice, considering the four types of inclusions and XY (red) and Z (blue) modes. We plot the band structures in the principal symmetry directions of FIBZ (see Figure 1 (f-h)). Plots are given in terms of frequency in Hz versus the reduced Bloch wave vector, k = kɑ/2π. The reduced Bloch wave vector for rectangular lattice is calculated taking into account
From now on the relation between parameters R̃ and r̃ for hollow circular inclusion is r̃ = 0.2R̃ and we do not investigate the influence of BaTiO_{3} thickness, i.e. R̃ - r̃, in the band structure. The thickness influence of a carbon microstructure inclusion for square lattice was investigated by Anjos & Arantes^{10}. Figure 2 (b) presents one complete band gap for hollow circular inclusion and we can observe that first branches occur in higher frequencies compared to the other inclusions.
Figure 2 (c) shows two complete band gaps for square inclusion and its first complete band gap is broader than the complete band gaps for circular and hollow circular inclusions. When these square inclusions are rotated 45º with respect to x and y axes (Figure 2 (d)) three complete gaps are opened up. The broadest complete band gap for square lattice is for rotated square inclusion with a bandwidth of 11.97 GHz (43.5 GHz is the center frequency).
Figure 3 (a-d) shows the band structures for rectangular lattice. For circular and hollow circular inclusions, the band gap behavior of rectangular lattice is similar to square lattice, however they are narrower than the complete band gaps in a square lattice. For square and rotated square inclusions, only one and two complete band gaps are opened up, respectively. Similar to square lattice, the broadest complete band gap for rectangular lattice is opened up for rotated square inclusion with a bandwidth of 8.51 GHz (44.59 GHz is the center frequency).
Figure 4 (a-d) illustrates the band structures for triangular lattice. For circular, square and rotated square inclusions, a wide complete band gap is created. The broadest complete band gap for triangular lattice is for circular inclusion with a bandwidth of 12.98 GHz (36.31 GHz is the center frequency). A narrow complete band gap is also opened up for circular inclusion in high bands. For hollow inclusion, one complete band gap is created between 58.69 GHz-49.94 GHz. Comparing the band structures of triangular lattice to square and rectangular lattices, we observe that triangular lattice with circular inclusion presents the broadest complete band gap.
In Figures 5 (a-d) and 6 (a-d), honeycomb and Kagomé lattices, respectively, the band structures of square and rotated square inclusions present similar band gap behavior. The band structure of circular inclusion in honeycomb and Kagomé lattices presents seven and five complete band gaps, respectively. For Kagomé lattice, all the complete band gaps are narrower than the broadest complete band gap from square, rectangular, triangular and honeycomb lattices. The broadest complete band gap for honeycomb lattice is for circular inclusion with a bandwidth of 9.77 GHz (31.19 GHz is the center frequency). For hollow inclusion in honeycomb and Kagomé lattices, we can observe two and one complete band gaps, respectively, and the first bands appear in higher frequencies compared to the other inclusions, similar as observed to other lattices - square, rectangular and triangular.
Comparing the band structures for honeycomb and Kagomé lattices, we observe that honeycomb lattice presents the best performance (more complete band gaps and broader complete band gaps) for all inclusions.
In Figures 7-11, we show the comparison between Z modes of the nano-piezoelectric PC with (blue asterisks) and without (e_{15A} = 0) (black circles) piezoelectricity for square, rectangular, triangular, honeycomb and Kagomé lattices, respectively. The influence of piezoelectricity is significant on the band gaps for all lattices and inclusions. This influence is more evident in high frequencies^{60} for circular, square and rotated square inclusions. However, for hollow circular inclusion in all lattices (Figures 7-11 (b)), the piezoelectricity is also significant in lower frequencies.
We illustrate in Figures 12-16 the complete band gap widths between XY and Z modes as a function of filling fraction for all inclusions and lattices. Piezoelectricity is included in this analysis.
In a square lattice (Figure 12), circular and square inclusions provide the best behavior with high complete band gap widths in a broad range of filling fraction. Considering the first band gap, square inclusion shows the highest one in a broad range of filling fraction. Furthermore, square inclusion also presents four complete band gaps. Hollow circular inclusion presents only one complete band gap, whereas rotated square inclusion does not present better performance than square inclusion.
For rectangular lattice, Figure 13, hollow circular inclusion opens up just one narrow complete band gap, whereas rotated square inclusion creates three complete band gaps, however, only the second one appear in a broad range of filling fraction. Similar as square lattice (Figure 12), circular and square inclusions present the broadest complete band gaps for rectangular lattice. Complete band gap widths in rectangular lattice are lower than in square lattice for all inclusions.
Figure 14 shows complete band gap widths for triangular lattice. Circular inclusion presents the best performance. Comparing triangular lattice (Figure 14) to square lattice (Figure 12), triangular lattice presents the best performance for circular (first complete band gap), hollow circular and rotated square inclusions. However, for square inclusion, square lattice shows better performance than triangular lattice. There are two important variables to conclude which is the best band gap performance, that is to say the band gap width (the higher, the better) and the filling fraction range in which the band gaps are opened (the broader, the better).
Comparing honeycomb (Figure 15) and Kagomé (Figure 16) lattices, the best performance is found for honeycomb lattice considering circular, hollow circular and rotated square inclusions. However, the difference among these lattices is subtle. The highest band gap width for honeycomb and Kagomé lattices is for circular inclusion. Comparing honeycomb and Kagomé lattices to the other lattices, we can observe that for all inclusions, the best behavior is found for square (Figure 12) and triangular (Figure 14) lattices. The worst performance is found for rectangular lattice with hollow circular inclusion (Figure 13 (b)).
4. Conclusion
Broad complete band gaps are obtained for a nano-piezoelectric PC, consisting of BaTiO_{3} inclusions embedded in a polymeric matrix, for different inclusion geometries and different lattices. We firstly study the elastic band structure for a fixed filling fraction of 0.335 considering XY and Z modes. For square lattice, the broadest complete band gap is found for rotated square inclusion with 11.97 GHz of bandwidth. This inclusion geometry is also the best one for rectangular lattice. Triangular lattice presents the broadest complete band gap than the other lattices for circular inclusion. The elastic band structure for honeycomb and Kagomé lattices presents several complete band gaps. However, for Kagomé lattice, all of them are narrower than the complete band gaps from square, rectangular, triangular and honeycomb lattices.
The Z mode with and without piezoelectric effect is also investigated for a fixed filling fraction of 0.335. This effect is significant on the band gaps for all lattices and inclusions, and it is more evident in higher frequencies for circular, square and rotated square inclusions. However, piezoelectricity is significant in lower frequencies for hollow circular inclusion.
We also analyze the complete band gap bandwidths as a function of filling fraction for all inclusions and lattices with piezoelectric effect. Among square, rectangular and triangular lattices, triangular lattice presents the best performance for circular inclusion (first complete band gap), hollow circular and rotated square inclusions. For square inclusion, square lattice shows the best performance. Among honeycomb and Kagomé lattices, the best performance is found for honeycomb lattice considering circular, hollow circular and rotated square inclusions.
The best band gap performance (i.e. highest band gap width and band gap opened up in a broad range of filling fraction) of the nano-piezoelectric PC investigated is for circular inclusion in a triangular lattice. However, it also depends on the application, for instance the frequency range of interest. Finally, we suggest that elastic complete band gaps in nano-piezoelectric PCs enlarge the potential applications for elastic vibration control in GHz.