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.

### 3.2.1.1 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 <sup>θ</sup><sup>0</sup> <sup>¼</sup> 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.

### 3.2.1.2 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 ωa=2π cT = 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

Figure 9. (a) Aluminum defect layer with ad = aA, (b) Aluminum defect layer with ad = 4aA.

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

The authors declare that there is no conflict of interest.

significant change.

Phononic Crystals and Thermal Effects DOI: http://dx.doi.org/10.5772/intechopen.82068

Conflict of interest

Author details

25

Arafa H. Aly\* and Ahmed Mehaney

provided the original work is properly cited.

Physics Department, Faculty of Sciences, Beni-Suef University, Egypt

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: arafaaly@aucegypt.edu

Figure 10. The reflectance R versus ωa=2π cT for P-wave (red lines) and S-waves (blue lines) at temperatures effects (a) T = 35° C and (b) T = 180° C.

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.
