一维分子束外延方程线性部分求解及数值模拟

THE THEORETICAL SOLUTION AND NUMERICAL SIMULATION OF THE LINEAR PART IN ONE-DIMENSION MBE EQUATION

分子束外延(molecular beam epitaxy,简称MBE)是一种在晶体基片上生长高质量的晶体薄膜的新技术,本文主要研究一维MBE方程线性部分的性质.首先,用分离变量法导出方程的理论解并证明了的解的存在性.其次,利用Fourier谱方法从数值上研究方程的解的性质,同时进行了稳定性、收敛性、误差分析.由于方程本身含有稳定项和非稳定项,且其中的未知参量决定了稳定项所占权重,故参量的大小影响着解的稳定性.数值分析结果显示,当参量较小时方程的解是不稳定的,随着时间增长,振幅最终会增大至无穷;而当参量较大时,方程的解是稳定的,随着时间增长,振幅最终趋于0.这与理论分析的结果也是一致的.

Molecular Beam Epitaxy （MBE） is a new method used to grow crystal thin films in high vacuum or ultra-high vacuum. The main purpose of this work is to study the properties of the linear part in one-dimension MBE equation. Firstly, the separation of variables is used to get theoretical solution of the equation; meanwhile the existence of the solution is proved. Next, this essay presents some of the fundamental ideas of Fourier spectral method, which are used to examine numerical solution. Then the stability, convergence and error analysis of Fourier spectral method are discussed. Since the equation includes both stable term and unstable term and the unknown term determines the weight of stable term, the unknown term determines the stability of the whole equation. The output of numerical analysis proves that, the solution is unstable when the parameter is small and as time goes by, its amplitude will increase to infinity; when the parameter is large enough, the solution is stable and as time goes by, and its amplitude gradually reduces to O. This result is also consistent with theoretical analysis.