Liu Yuanbin, Qiu Jun, Liu Linhua
Opt Express. 2018 Jun 25;26(13):16560-16571. doi: 10.1364/OE.26.016560.
The effective medium approximation (EMA) has been widely applied to model the effect of a solid sample with surface roughness in spectroscopic ellipsometry (SE). There are two specific cases to utilize the EMA model. One is utilizing the EMA model to perform the inversion of the optical constants of solid samples from the SE measurements. Another is utilizing the EMA model to estimate the thickness of the rough layer at solid surface from the SE measurements under the condition in which the optical constants of samples are known. For the first case, the thickness of the rough layer is usually assumed to be the root mean square (rms) roughness as measured by atomic force microscopy (AFM). We theoretically investigate the error of the EMA model to estimate optical constants for different surface morphologies and materials. Because the EMA model only accounts for the height irregularities of rough surfaces but neglects the effect of the lateral irregularities on electromagnetic scattering from rough surfaces, it is difficult to obtain high-precision results for optical constants. Moreover, the inversion error of optical constants by using the EMA model is difficult to evaluate. In the second case, the thickness of the rough layer is estimated by using the EMA model from the SE measurements, called the EMA model roughness. We show that the EMA model roughness generally has a deviation from the rms roughness as measured by AFM. Some correlated relationships are established between the EMA model roughness and the morphological parameters of rough surfaces. It is found that these relationships have similar forms but not identical coefficients for different materials. The results from this work may facilitate a better understanding and utilization for the EMA model in SE.
有效介质近似（EMA）已被广泛应用于在光谱椭偏仪（SE）中对具有表面粗糙度的固体样品的效应进行建模。利用EMA模型有两种特定情况。一种是利用EMA模型根据SE测量结果对固体样品的光学常数进行反演。另一种是在样品光学常数已知的条件下，利用EMA模型根据SE测量结果估算固体表面粗糙层的厚度。对于第一种情况，粗糙层的厚度通常假定为通过原子力显微镜（AFM）测量的均方根（rms）粗糙度。我们从理论上研究了EMA模型在估算不同表面形态和材料的光学常数时的误差。由于EMA模型仅考虑了粗糙表面的高度不规则性，而忽略了横向不规则性对粗糙表面电磁散射的影响，因此难以获得高精度的光学常数结果。此外，使用EMA模型对光学常数进行反演的误差也难以评估。在第二种情况下，通过SE测量利用EMA模型估算粗糙层的厚度，称为EMA模型粗糙度。我们表明，EMA模型粗糙度通常与AFM测量的rms粗糙度存在偏差。在EMA模型粗糙度与粗糙表面的形态参数之间建立了一些相关关系。发现这些关系对于不同材料具有相似的形式但系数不同。这项工作的结果可能有助于更好地理解和利用SE中的EMA模型。
