Abstract: This paper designs the thermal crystals composed of alloy materials with air holes and analyzes their properties of band structures, heat transmission, and flux spectra. Thermal crystals composed of Si-A (A=Ge, Sn, Pb) alloys as background materials and air holes with square array are used to construct an elastic-constant periodic structure and their high-frequency phononic band is calculated by deploying finite element methods. Moreover, this paper investigates heat transmission through a finite array of thermally excited phonons and presents the thermal crystal with maximum heat transport. The results show that a wider bandgap could be achieved by increasing the air hole radius and decreasing the lattice constant. In the alloy materials, with increasing atomic radius and thus atomic mass (Ge, Sn, Pb), the frequency range (contributed to thermal conductivity) shifts towards lower frequency. Hence, the bandgap frequencies also shift toward low frequency, but this decreasing rate is not constant or in order, so former may have a faster or slower decreasing rate than the later. Thus, the frequency range for the contribution of heat transportation overlaps with the bandgap frequency range. The development of thermal crystals is promising for managing heat and controlling the propagation of the thermal wave.

Key words: thermal crystals, phonons, finite element method, band structure, periodic structure, lattice constant

摘要： 本文设计了含气孔的合金热晶体，分析了其能带结构、传热特性和通量光谱。以Si-A (A = Ge, Sn, Pb)合金为背景材料，采用方阵气孔构造弹性常数周期结构的热晶体，采用有限元方法计算了其高频声子带。此外，本文还研究了有限热激发声子阵列的热传输，并给出了热传输最大的热晶体。结果表明，增大气孔半径和减小晶格常数可以获得更宽的带隙。在合金材料中，随着原子半径和原子量（Ge, Sn, Pb）的增加，频率范围向低频移动（有助于导热）。因此，带隙频率也向低频偏移，但这种下降率不是恒定的或有序的，因此前者相较于后者下降率可能或快或慢。因此，热输运贡献的频率范围与带隙频率范围重叠。热晶体在热管理和控制热波传播方面具有广阔的应用前景。

关键词: 热晶体；声子；有限元法；能带结构；周期结构；晶格常数