Nonlocal Effective Medium (NLEM) for Quantitative Modelling of Nanoroughness in Spectroscopic Reflectance

Abstract

Spectroscopic reflectance is a versatile optical methodology for the characterization of transparent and semi-transparent thin films in terms of thickness and refractive index. The Fresnel equations are used to interpret the measurements, but their accuracy is limited when surface roughness is present. Nanoroughness can be modelled through a discretized multi-slice and effective medium approach, but to date, this offered mostly qualitative and not quantitative accuracy. Here we introduce an adaptive and nonlocal effective medium approach, which considers the relative size and environment of each discretized slice. We develop our model using finite-difference time-domain simulation results and demonstrate its ability to predict nanoroughness size and shape with relative errors < 3% in a variety of test systems. The accuracy of the model is directly compared to the prediction capabilities of the Bruggeman and Maxwell–Garnett models, highlighting its superiority. Our model is fully parametrized and ready to use for exploring the effects of roughness on reflectance without the need for costly 3D simulations and to be integrated into the Fresnel simulator of spectroscopic reflectance tools.

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“…The absorbance data of figure 4(a) are truncated at UV = 300 nm, due to the cuvette absorbance ∼300 nm [161], making our n BP extraction qualitative, because of the finite integration range. We use the extracted BP refractive index in Fresnel equation calculations [162] to estimate the absorption of SLG/BP on Si/SiO 2 . The SLG refractive index is modelled by the Kubo conductance [163] at RT and E F = 0.38 eV, as estimated by the Raman measurements in figure 6.…”