基于晶格动力学的硅单晶热学性质研究(III)——热膨胀系数公式推导及分析

Study on Thermal Properties of Silicon Single Crystal Based on Lattice Dynamics (III) —Derivation of and Analysis on the Formula for Thermal Expansion Coefficient of Silicon Single Crystal

在硅单晶的晶格三阶非和谐势能公式、和谐晶体晶格振动位移和哈密顿公式的基础上,利用量子力学的微扰理论和晶格动力学理论,推导了硅单晶的热膨胀系数计算公式。通过该热膨胀系数计算公式,可以对各个非线性力常数对热膨胀系数的贡献分别进行计算,从而分析各个非线性力常数和声子对热膨胀性质的影响,这对硅单晶在低温下的负热膨胀性质的物理机制探讨是非常有利的,为热膨胀系数的数值计算及硅单晶低温负热膨胀物理机制的探究做了必要的准备。最后本文还利用该公式对kz = 0时不同波矢的声子对硅单晶的热膨胀系数的贡献进行了计算,发现某些模态的声学声子对热膨胀的贡献为负,这表明本文推导的公式能从微观上解释低温下硅单晶的负热膨胀性质的物理机制。

On the basis of the formula for the third-order anharmonic lattice potential, and the formulas for the lattice vibration displacement and the Hamiltonian of harmonic silicon single crystal, the formula for thermal expansion coefficient of silicon single crystal is derived with the aid of perturbation theory of quantum mechanics and lattice dynamics theory. With this formula, the contribution of each anharmonic force constant and phonon to thermal expansion coefficient can be calculated separately, and the influence of each anharmonic force constant and phonon on thermal expansion property can be analyzed, this is very beneficial to the study on the physical mechanism of negative thermal expansion of silicon single crystal at low temperature, and is a necessary preparation for the numerical calculation of thermal expansion coefficient and the study on the physical mechanism of negative thermal expansion of silicon single crystal at low temperature. Finally, the contribution of phonons with different wave vectors to the thermal expansion coefficient of silicon single crystal is calculated by using this formula, and it is found that the contribution of acoustic phonons in some modes to thermal expansion is negative, it indicates that mechanism of the negative thermal ex-pansion properties of silicon single crystal at low temperature can be explained with the aid of the formula derived in this paper.