Phononic Crystals and Thermal Effects

In this work, we demonstrate a comprehensive theoretical study of one-dimensional perfect and defect phononic crystals. In our study, we investigate the elastic and shear waves with the influences of thermal effects. The numerical calculations based on the transfer matrix method (TMM) and Bloch theory are presented, where the TMM is obtained by applying the continuity conditions between two consecutive sub-cells. Also, we show that by introducing a defect layer in the perfect periodic structures (defect phononic crystals), we obtain localization modes within the band structure. These localized modes can be implemented in many applications such as impedance matching, collimation, and focusing in acoustic imaging applications. Then, we investigate the influences of the incident angle and material types on the number and intensity of the localized modes in both cases of perfect/defect crystals. In addition, we have observed that the temperature has a great effect on the wave localization phenomena in phononic band gap structures. Such effects can change the thermal properties of the PnCs structure such as thermal conductivity, and it can also control the thermal emission, which is contributed by phonons in many engineering structures.


Introduction
Phononic crystals (PnCs) are new composite materials which can interact, manipulate, trap, prohibit, and transmit the propagation of mechanical waves. Recently, great efforts have been dedicated to study these novel materials to be used in many potential engineering applications. By reference to the meaning of idiomatic, the term "phononic" was derived in analogy to the term "phonon," which is considered as a quantization of the lattice vibrations. In the previous conscious of science, these vibrations are impossible to be controlled because the atoms in the solid cannot move independently of each other because they are connected by chemical bonds. By appearing PnCs, this assumption was changed and the mechanical wave can be filtered, transmitted, stopped, and localized within specific frequencies called phononic band gaps [1,2].
Within the phononic band gaps, the mechanical waves of all types are greatly blocked. Actually, the formation of the phononic band gaps is back to the variation in the mechanical properties of the materials that build the PnC structure. Therefore, Bragg interference at the interface between each two materials can be obtained. As a result, the phononic band gaps and transmission bands are formed. Moreover, novel properties such as negative refraction and acoustic metamaterials are presented in PnC structures [3][4][5][6][7]. These novel properties of PnCs can be utilized in many industrial and engineering applications such as MEMS applications, filters, waveguides, clocking, multiplexers, and sensor applications [8][9][10][11][12].
As it is well-known, the physical origin of all crystal structures has the same idea of the design. Photonic crystals and semiconductors devices could not develop such types of the previous applications. From the scientific point of view, we should mention that the physical nature of PnCs is different from one of photonic crystals, as well as semiconductors. Generally, the various forms of waves are referred to as: electron waves as scalar waves and optical waves as vector waves, while elastic waves as tensor waves [13][14][15][16][17][18][19].
As a result, to design PnCs having a complete phononic band gap, the mechanical properties must be changed not only in one direction (i.e. x-direction) but also in all three directions of space (i.e., x, y, and z directions). Therefore, PnCs can be classified according to the periodicity into three types, i.e., the one-dimensional (1D), the two-dimensional (2D), and the three-dimensional (3D) PnCs [20][21][22][23][24]. PnCs can control the entire spectrum of phonons frequency from 1 Hz to THz range. As it is well-known, the waves that propagate through solid media are called elastic waves, while those that propagate through fluids are called acoustic waves. Also, PnCs can control the different types of mechanical waves such as elastic waves, acoustic waves, and surface waves. Therefore, unlike photonic crystals, not all 3D PnCs possess complete phononic band gaps. PnCs must possess band gaps for both elastic and acoustic waves at the same frequency region. Consequently, we have to fabricate PnC structures for both solid and fluid media [25][26][27][28]. Hence after, these different polarizations introduce more challenges to PnCs higher than other crystal, which in turn makes the theoretical manipulation of PnCs more attractive and perspective.
From the previous observations of PnCs structures, several methods were developed to calculate the phononic band gaps such as Plane-Wave-expansion Method (PWM), Bloch-Floquet Method (BFM), and finite different time-domain method (FDTD) [29][30][31][32][33]. In this chapter, we will depend on the Transfer Matrix Method (TMM) [34] for calculating the reflection coefficient and the transmission coefficient of the one-dimensional PnC structure. Such method considered very suitably for the 1D structure due to its recursive nature, since it allows the continuity of the waveform at the interface between each two layers [35,36]. Also, by using the TMM, we can obtain the dispersion relations of the mechanical waves through the periodic PnC structures. PnCs and temperature are in mutual influences in several ways [37]. For example, at any temperature, mainly the huge contribution in the thermal conductivity of many materials is dominant by phonon contribution, which is a function of phonon mean free path and the Boltzmann distribution of phonons of any material. Also, the thermal conductivity is depending on the thickness of a material, where optical branches contribute with about 30% of the thermal conductivity of any material. By inhibiting the acoustic phonon population, the optical phonon relaxation is indirectly inhibited by up to 30% and hence limits their contribution to thermal conductivity. Furthermore, in silicon PnC, the thermal contribution of phonons has been reduced to less than 4% of the value for bulk silicon at room temperature [38][39][40].
Moreover, the defected PnCs structures have wonderful application in wave guiding and multiplexing. But the produced structures are different from the proposed ideal structures due to the errors and defaults in manufacturing. By removing some layers or materials from the ordered periodic structure in a PnC, we can create a point or a waveguide defects that are able to localize and bend signals. Such mechanism has been implemented similarly in photonic crystals to slow light. Consequently, the produced waveguides in PnCs structures can be used in focusing and collimation of acoustic waves in medical ultrasound applications, sensors, and MEMS applications [2,[41][42][43][44].
In the present work, we introduce the formation of phononic band gaps under the influences of many physical parameters. First, the general case of SH-wave propagation in arbitrary direction will be investigated by using TMM with calculation of the dispersion curves as well. Also, the influences of the incident angle on the band gaps are analyzed and discussed. In addition, this work focuses on correlating and comparing the results of SH-waves with in-plane waves propagating normally to the structure. For the in-plane waves, the reflection coefficients for S-and Pwaves are plotted and compared with the dispersion relations curves. Furthermore, we are demonstrating the wave localization phenomenon in PnCs and the effects of the temperature on the band structure of PnCs and the localized modes for both in-plane and SH-waves. Also, the numerical results are presented and discussed to investigate the effect of the defect layer on the wave localization modes inside the structures. Finally, the effects of the thickness and type of the defect layer material on the band gap structure had been discussed. Figure 1 shows the schematic diagram of the 1D PnC crystal structure. The proposed crystal structure has an infinite number of the periodically arranged unit cells. The unit cell may include two or more layers; here, we propose that it is made by only two materials A and B, respectively. The two materials are labeled by the subscript j ¼ 1, 2. Also, the thickness of the unit cell (the lattice constant) is a ¼ a 1 þ a 2 : The thickness, Lame' constant, shearing modulus, Poisson's ratios, mass density, and Young's modulus of the two layers are denoted by equation

Theoretical analysis and numerical models 2.1 Equation of wave propagation
, respectively. The governing equation of anti-plane shear waves (SH-waves) polarized in the z-direction propagated in the xy-plane can be written in the following form [45][46][47].
where T is the temperature variation, β j is the thermal expansion coefficients, φ j x j ; y j ; t is the displacement components along the z-direction, t is the time, σ tx j ¼ ÀE j β j T= 1 À 2υ j À Á is the thermal stress, and ∇ 2 ¼ ∂=∂x 2 j þ ∂=∂y 2 j is the Laplacian operator. The solution φ j x j ; y j ; t in the jth layer with time harmonic dependence can be expressed as [48], Where, c is the velocity component of the incident wave, i 2 ¼ À1, k ¼ ω=c is the wave number, θ 0 is the angle of the incident wave, ω is the angular frequency, and ϕ j is the amplitude.
In many cases, it is more convenient to represent the thickness value by the following dimensionless coordinates: where a 1 is the average thickness value of the material A. a 1 is equal to a 1 for the periodic structures. By inserting Eq. (3) in Eqs. (1) and (2), we can obtain the following wave equations: where c j represents the wave velocity in each material, χ j ¼ σ tx j =μ j is the stress and shearing modulus ratio, α ¼ ka 1 represents the SH-waves dimensionless wave number, α j ¼ k j a 1 is the dimensionless wave number and k j ¼ ω=c j is the wave vector of materials A and B, respectively. Eq. (5) has a general dimensionless solution, and it is given in the following form, where A j and B j are unknown coefficient to be determined, and q j is a parameter and has the value q j ¼ 1 ffiffiffiffiffiffiffi . Therefore, the dimensionless solution φ j ξ j ; η j ; t with time harmonic dependence is:

Transfer matrix method
The dimensionless shear stress component is given as follows: Assuming that the PnC consists of n unit cells, the boundary conditions at the left and right sides of the two layers in the ith unit cell can be written in the following form [45], where the subscripts L and R denote the left and right sides of the two layers, and 0 ≤ ξ j ≤ ζ j ¼ a j =a 1 j ¼ 1; 2 ð Þ are the dimensionless thicknesses of materials A and B. Substituting Eqs. (7) and (8) into Eq. (9), the following matrix equation can be obtained as follows: where are the dimensionless state wave vectors at right and left sides of each unit cell and T 0 j is the 2 Â 2 transfer matrix of each unit cell. The four elements of T 0 j can be written in the following forms,

Characteristic of the dispersion relation
At the interface between the layers, the following condition is satisfied: Thus, the relationship between the right and left sides of the ith unit cell can be obtained from Eq. (10) as follows: where T i is the accumulative transfer matrix of the ith unit cell and can be written in the following form: At the interface between the right side of the unit cell and the left side of the ith unit cell, the following condition is satisfied: By equating Eqs. (13) and (15), we can obtain the relationship between the state vectors of the i À 1 ð Þth unit cells and the ith unit cell in the following form: Therefore, T i represents the transfer matrix between each two successive unit cells.
By using Floquet and Bloch theories, we can obtain the displacement and stress fields between each two neighboring unit cells (at the interface) by the following relations [46][47][48][49]: where k is the wave number representing the direction of the traveled wave across the structure. Combining the above two equations leads to the following matrix equation: By equating Eqs. (16) and (18) which leads to the following eigenvalue problem: Eq. (19) can be rewritten as follows: where λ ¼ e ika is a complex eigenvalue and V iÀ1 ð Þ 2R is a complex eigenvector.

Band structure formation
The wave number in Eqs. (17) and (18) is a complex number, so it can be a positive or negative number; in general, the wave number can be written in the following form [50]: where k real and k imaginary are the real and imaginary wave numbers, respectively. Therefore, we can deduce from this relation that the complex wave number has two forms, so it can inhibit the incident waves within the phononic band gaps. Consequently, we have two frequency ranges and they are organized as follows: From Eq. (22), we can deduce that the displacement and stress at the successive unit cells i ð Þth and i À 1 ð Þth are differ only by a phase factor e i k real j ja . At this condition, the elastic waves are allowed to propagate freely through the PnC structure with the corresponding frequencies and wave number, forming the so-called transmission bands.
2. If k ¼ Àik imaginary and k imaginary < 0 From Eq. (18), In contrast to the above case, we can deduce from Eq. (23) that the displacement and stress at successive unit cells i ð Þth and i À 1 ð Þth are the same and do not have a phase difference. Moreover, the incident wave has an exponential spatial attenuation of strength magnitude proportional to the value k imaginary . Hence after, at this case, the waves are prohibited from propagation in the PnC structure, which in turn, results in formation of the so-called phononic band gaps or stop bands.

Temperature influences on PnCs
When a mechanical wave propagates through a PnC structure, resultant vibrations occur which may increase or decrease phonons motions. Therefore, it can heat or cool the PnC structure. Also, the thermal expansion of the constituents materials may change, which will affect the mechanical constants of the material as well.
Since the vibrations occur very rapidly, there is no big chance to thermal energies to flow and the elastic constants measured by elastic waves propagation are changed adiabatically. The elastic constants are connected to the isothermal constants by the following relation [51][52][53][54][55], where the superscripts σ and θ indicate adiabatic and isothermal constants, β is the thermal expansion coefficient, B is the bulk modulus B ¼ λ þ 2=3μ ð Þ , C ν is the specific heat at constant volume, θ is the absolute temperature, and ρ is the mass density. From Eq. 24, we can deduce that λ σ and λ θ are not the same and the difference between them should be considered.
In addition to the above relations, if the temperature is increased, not only the wave velocities will increase but also the thickness of each layer will change by the following relation [53]: where Δa is the thickness difference of any layer, a i is the original layer thickness, and Δt is the temperature difference. Hence after, these two variables will affect the longitudinal waves speed and the stress component in SH-waves equations. Since the P-wave velocity is c P ¼ ffiffiffiffiffiffiffi ffi λþ2μ ρ q , which, in turn leads to the variation of the band structure and band gaps properties.

SH-waves results
First, from a practical point of view, the number of layers in the 1D PnC structure should take a finite number. Therefore, we consider the unit cell of the PnC structure is made from two layers. The two layers are lead and epoxy materials and denoted by the symbols A and B, respectively. Second, dispersion relations are plotted for an infinite number of unit cells because it depends on Bloch theory that manipulates the propagation of waves through infinite periodic structures. Therefore, the dispersive behavior of the periodic materials and structures with an arbitrary chosen unit cell configuration is considered as an example. The constants of the materials used in the calculations can be found in Refs. [46,49] and in Table 1.
Here, we consider the velocity c of the incident wave is 800 m/s and the angle of the incidence is θ 0 ¼ 20°. Also, the ambient temperature is proposed to be T ¼ 20°C. The dispersive properties of the inhomogeneous structures are determined by the properties ratios of the constituent materials, here we consider the two materials thicknesses ratio as 1:1. Figure 2 presents the dispersion relation of SH-waves in the first Brillouin zone, and it is plotted between the nondimensional frequency q ¼ ω a=c B and the nondimensional wave number ξ ¼ k Â a, where c B is the wave velocity in the second material (epoxy). We considered the range of the nondimensional frequency is 0 ≤ q ≤ 9 . From this figure, we can conclude that, first, there are some  frequency regions that are plotted with the white color and represent the real valued wave number and pass bands. Second, the other frequency regions corresponding to the imaginary wave number represent the phononic band gaps. Figure 3(a) and (b) show the effects of the incident angle θ 0 on the band structures of the previous perfect structure. It can be seen that the wave propagation behavior of the perfect PnC changes obviously for different incident angles. For example, the stop bands for θ 0 = 15°become a pass band for θ 0 = 45°. With increasing value of the incident angle, the same stop bands became wider in the considered frequency regions. However, in Figure 3(c), we can note a wonderful phenomenon appeared when the angle θ 0 reached the value 85°. The stop band increased to be the entire band structure of the PnC, and no propagation bands were observed. The explanation of such phenomenon is deduced from the parameter

Influence of the angle of incidence on the phononic band gap
become smaller than sin θ 0 , the displacement field will decay. Therefore, a total reflection to the elastic waves will be occurred, where all the incident wave energy is reflected back to the structure and no waves are allowed to propagate through the PnC structure.

Defective mode effect on the band structure of the PnC
As shown in Figure 4, the dispersion relation of the perfect PnC structure will differ than the defected ones. We consider a defect layer from aluminum (the defect layer thickness a d = a A i.e., 1/2 of any unit cell thickness) was immersed after the second unit cell, and the material properties are mentioned in Table 1. From Figure 5, it was indicated that elastic waves can be trapped within the phononic band gap and the band structure changed significantly than the perfect ones. The width of the band gaps increased due to the increment of mismatch between the constituent materials, this back to the insertion of a second interface inside the PnC structure. Moreover, the defect layer acts as a trap inside the PnC structure, so a special wave frequency corresponding to that waveguide will propagate through the structure.
Additionally, in Figure 6(a) (a d = 2a A ) and Figure 6(b) (a d = 4a A ), we studied the effects of the defect layer thickness on the position and number of the localized modes inside the band structure of the PnC. It was shown that the number and width of the localized modes were increased by increasing the defect layer thickness. Therefore, with increasing the thickness of the defect layer inside the periodic PnC structure, the localized modes within the band gap are strongly confined within the defect layer. Not only the thickness of the defect layer has an obvious effect on wave localization but also the type of the defect layer has a significant effect on the localized modes inside the band gaps. Although we introduced two different materials in Figure 6(c) and (d) with the same thickness, the number and width of the localized peaks had greatly changed. In Figure 6(c), we used Au which has mechanical constants higher than the host materials, while in Figure 5(d), we used nylon which has mechanical constants lower than the host materials. As a result, the width and number of the transmission bands are larger in Figure 6(d) than in Figure 6(c). Actually, the nylon is a very soft material like a spring and can introduce more resonant modes inside the PnC structure.

Temperature effects on the band structure of PnC
Now we will investigate the effects of temperature elevation on the PnC structure and phononic band gaps. Two temperature degrees (T = 35 and 180°C) were considered. Figure 7 shows the response of the dispersion relations with  The dispersion curve of SH-waves propagating in the defected PnC. A defect layer from aluminum with thickness a d = a A was immersed between the two periodic unit cells (stop-bands shaded with the gray color). different temperatures at SH-waves propagation through PnCs. We can note in Figure 7(a) and (b) that, with increasing the temperature from T ¼ 35 to T ¼ 180°C, respectively, the pass/stop band width remains constant due to the opposite increment in the thermal stress, which has a negative value and maintains materials dimensions constant. Therefore, there is not any variation in the thickness  of any material, which in turn keeps the width of the band gaps constant. Finally, we can note only band edges changed slightly with increasing the temperatures. Such small shift in the band gaps is related to the different thermal expansion coefficients of the two materials. Based on this result, the effects of temperature on the PnC structure are greatly related to the type of waves that propagate through crystal structure. Hence, we will verify this result by studying the temperature effects on PnCs at plane wave propagation as well.

Plane waves results
As depicted in Figure 1, we can calculate the reflection coefficient of S-and Pwaves in the x-direction through the PnC structure. The PnC structure is proposed to be bonded between two semi-infinite materials (nylon material) at the two ends. The subscripts "0" and "e" denote the left and the right of the PnC structure, respectively.
The reflection coefficient of the displacement field through a PnC structure is given by the form [39], where U 1 is the reflected amplitude and T ij ¼ T i; j ð Þ are the elements of the total transfer matrix T ¼ T n T nÀ1 …T m …T 1 . Figure 8 shows the relation between the reflectance R versus ωa = 2Pic T (ω is considered the angular frequency in this relation and c T ¼ c SB ) for P-wave (red dashed lines) and S-wave (black solid lines) [48]. From Figure 8, we can determine the range of frequencies for which the phononic band gaps can be occurred (reflectance of P-and S-waves is high R ≈ 1). Such high reflectance within the different frequency ranges represents the frequency band gaps of P-and S-waves inside the PnC structure.
From Figure 8, it can be seen that the phononic band gaps described by the reflection coefficient are agreed with those described by the dispersion relations. The frequency regions at R ≈ 1 of P-and S-waves represent the complete band gaps of the plane waves through the PnC structure. There is only a fine difference in the band gaps edges between Figure 8 and those obtained by the dispersion relation. In Figure 8, the relation is plotted between the reflectance and ωa = 2Pic T , and c T ¼ c SB is chosen for both P-and S-waves reflectance. Therefore, we will use the reflection coefficient to describe the effects of the defect layer and temperature instead of dispersion relations in order to plot both P-and S-waves in the same graph and for the practical and industrial purpose as well.
3.2.1 Influences of the defect layer/temperature on the phononic band gaps "plane wave" In this section, we will study the effects of the defect layer and temperatures on the band structure of PnCs at the propagation of plane waves and compare with those investigated for SH-waves.

Defect layer influences on band structure
First, we will use the same defected structure used in Section 3.1.2 with the same materials and conditions, only the angle of incidence will be maintained at θ 0 ¼ 0°. Figure 9 confirms the last results of the effects of the defect layer on the localization modes through the PnC at the plane wave propagation. In Figure 9(b), a number of the localized waves was generated inside the phononic band gaps and was increased by increasing the defect layer thickness as well.

Temperature influences on band structure
In this section, the two temperatures T = 50°C and 190°C were considered in order to illustrate the effects of temperature on the phonic band gaps. We noticed that the phononic band gaps were affected slightly by temperatures at plane wave propagation higher than SH-waves. Therefore, the propagation of elastic waves and localized modes can be affected by temperature elevation. From Figure 10(a), we can note that the reflectance of the P-wave (Red lines) is moved toward the higher frequencies (i.e., band gap at ω a=2π c T = 3.5). Such displacement in the band gap edges is quite noticeable at T = 190°C in Figure 10(b).
These temperature effects on the band gaps can be explained by two reasons. First, the P-wave velocity is increased according to Eq. (24) because the temperature has a direct effect on the elastic constants. Consequently, temperature makes a notable change in the edges of the phononic band gaps. Second, here in the plane wave case, the thermal stress is absent, so the lattice constants will increase according to Eq. (25) and can make change in the width of the phononic band gaps as well.

Conclusions
In this chapter, we used the transfer matrix method and Bloch theory to compute and study the propagation of mechanical waves through different 1D PnC structures with thermal effects. We performed numerical simulations to calculate the reflection coefficient and dispersion curves for the 1D PnC structure. We have discussed and studied the effects of many parameters on the phononic band gaps. The effects of temperature and defect mode on the properties of the PnC and on the localized modes were studied for both SH-waves and in-plane waves. The output results can be stated as follows: 1. The defect layer has a different and unique effect on the number of localized modes inside the phononic band gaps for both plane and SH-waves. The effects of the incident angle, thickness ratio on the propagation, and localization of elastic waves were studied. The number of localized modes was increased with increasing the defect layer thickness. Also, the material type of the defect layer has an obvious effect on the width and number of resonant defect modes, where the localized modes were increased by using a material with low elastic constant. Inserting a defect layer inside the perfect structure facilitates the process of localized modes generation inside the PnC structure. Also, it can be utilized in many potential applications such as acoustic filter and waveguides and can be applied to produce new types of laser called phonon laser with low thermal effects especially in the hypersonic PnCs regime.
2. Results revealed that temperature increment has a significant effect on the phononic band gaps and on the number of the localized modes inside the PnC structure, especially for plane wave, where the P-waves velocity and lattice constants are directly related to temperature increment. The contribution of phonons in the thermal conductivity of a solid can be limited by the existence of phononic band gaps, which could be very useful for the thermoelectric devices that convert thermal energy into electricity. Hence after, the performance of many devices such as Peltier thermoelectric coolers, thermocouples, sensors, and thermoelectric energy generators can be enhanced. Moreover, the thermal conductivity of many structures could experience great changes based on the temperature gradients of PnCs. When temperature is increase in PnCs, the phononic band gap experiences a significant change.