Developing materials that can conduct electricity easily, but block the motion of phonons is necessary in the applications of thermoelectric devices, which can generate electricity from temperature differences. In converse, a key requirement as chips get faster is to obtain better ways to dissipate heat. Controlling heat transfer in these crystalline materials devices -- such as silicon -- is important. The heat is actually the motion or vibration of atoms known as phonons. Finding ways to manipulate the behavior of phonons is crucial for both energy applications and the cooling of integrated circuits. A novel class of artificially periodic structured materials -- phononic crystals -- might make manipulation of thermal phonons possible. In many fields of physical sciences and engineering, acoustic wave propagation in solids attracts many researchers. Wave propagation phenomena can be analyzed by mathematically solving the acoustic wave equation. However, wave propagation in inhomogeneous media with various geometric structures is too complex to find an exact solution. Hence, the Finite Difference Time Domain method is developed to investigate these complicated problems. In this work, the Finite-Difference Time-Domain formula is derived from acoustic wave equations based on the Taylor's expansion. The numerical dispersion and stability problems are analyzed. In addition, the convergence conditions of numerical acoustic wave are stated. Based on the periodicity of phononic crystal, the Bloch's theorem is applied to fulfill the periodic boundary condition of the FDTD method. Then a wide-band input signal is used to excite various acoustic waves with different frequencies. In the beginning of the calculation process, the wave vector is chosen and fixed. By means of recording the displacement field and taking the Fourier transformation, we can obtain the eigenmodes from the resonance peaks of the spectrum and draw the dispersion relation curve of acoustic waves. With the large investment in silicon nanofabrication techniques, this makes tungsten/silicon phononic crystal a particularly attractive platform for manipulating thermal phonons. Phononic crystal makes use of the fundamental properties of waves to create band gap over which there can be no propagation of acoustic waves in the crystal. This crystal can be applied to deterministically manipulate the phonon dispersion curve affected by different crystal structures and to modify the phonon thermal conductivity accordingly. We can expect this unique metamaterial is a promising route to creating unprecedented thermal properties for highly-efficient energy harvesting and thermoelectric cooling.
