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针对半导体制造中外延层厚度的无损测量问题，本文基于红外反射光谱原理，建立了双光束及多光束干涉模型.首先，结合Drude-Lorentz模型构建了碳化硅(SiC)和硅(Si)的复折射率色散关系.针对双光束干涉情形，提出了基于反射谱波谷极值点的厚度直接计算公式，并引入三次样条插值与滤波预处理以提高极值定位精度；同时，建立了非线性最小二乘优化模型，利用Newton迭代法反演膜厚及掺杂浓度参数，两种方法所得SiC外延层厚度结果(7.544μm与7.295μm)高度一致.进一步地，推导了多光束干涉模型，提出了定量表征多光束效应强弱的指标G,揭示了高折射率差与低吸收是产生显著多光束干涉的必要条件.应用该模型分析表明，SiC样品受多光束效应影响较小，而Si样品存在显著多光束干涉.计算结果表明，修正后的多光束模型能有效消除系统误差，显著提升了高透光基底外延层厚度的测量精度.
Abstract：To address the non-destructive measurement of epitaxial layer thickness in semiconductor manufacturing, this paper establishes dual-beam and multi-beam interference models based on the principles of infrared reflection spectroscopy. First, the complex refractive index dispersion relations for silicon carbide(SiC) and silicon(Si) are constructed using the Drude-Lorentz model. For the dual-beam interference scenario, a direct thickness calculation formula based on the extrema of reflection spectrum troughs is proposed, incorporating cubic spline interpolation and filtering preprocessing to enhance the accuracy of extrema localization. Simultaneously, a non-linear least squares optimization model is established to invert film thickness and doping concentration parameters via the Newton iteration method. The thickness results for the SiC epitaxial layer obtained by the two methods(7.544 μm and 7.295 μm) show high consistency. Furthermore, a multi-beam interference model is derived, and a quantitative index G is proposed to characterize the intensity of multi-beam effects, revealing that high refractive index contrast and low absorption are necessary conditions for significant multi-beam interference. Application of the model indicates that SiC samples are minimally affected by multi-beam effects, whereas Si samples exhibit significant multi-beam interference. The results demonstrate that the modified multi-beam model effectively eliminates systematic errors and significantly improves the measurement precision of epitaxial layer thickness on highly transparent substrates.
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基本信息:
DOI：10.19943/j.2095-3070.jmmia.2026.01.09
中图分类号:TN219;TN304.24
引用信息:
[1]张新晨,徐威南,周诗贺.红外干涉法测量外延层厚度的建模分析[J].数学建模及其应用,2026,15(01):86-93.DOI:10.19943/j.2095-3070.jmmia.2026.01.09.
2026-03-15
2026-03-15