Supplementing rigorous electromagnetic modeling with atomistic simulations for optics-based metrology

The successful combination of electromagnetic scattering simulations and optical measurements allows for the quantification of deep-subwavelength features, including thicknesses via ellipsometry and parameterized geometries via scatterometry. Although feature size reduction has slowed in recent years, nanoelectronics still yields ever-smaller structures, thus optical measurement capabilities are ever-challenged. The critical problem is that the optical properties of materials often become thickness dependent at sub-5 nm, greatly complicating accurate fitting. These optical properties can be characterized empirically using ellipsometry and used with other prior information to reduce uncertainties via hybrid metrology, but atomistic modeling offers a unique perspective on the macroscopic optical response from features with dimensions only a few atoms in width. To illustrate the potential of such modeling, we have performed a series of density-functional theory (DFT) calculations for an ultrathin film, Si with hydrogen-terminated Si(111) surfaces. Kohn-Sham wavefunctions determined in DFT are instrumental in solving for the dielectric tensor of these configurations, as the in-plane and out-of-plane components can differ greatly with respect to incident wavelength and Si thickness. Techniques for DFT and dielectric tensor determination are reviewed, highlighting both their limitations and potential for improving optics-based metrology. The thickness- and wavelength-dependence of the resulting tensor components are parameterized using Tauc-Lorentz and Lorentz oscillators. Using an illustration from goniometric reflectometry, the quantitative effects upon dimensional metrology of employing the full thickness-dependent dielectric tensor are compared against simpler approximations of these optical properties. Reductions in parametric uncertainty in the thickness and optical constants are evaluated with a prior knowledge of the ultrathin film's thickness with uncertainties.