The downscaling of features in the semiconductor industry has continuously placed pressure on optics-based measurement methods to yield new solutions for measuring ever-smaller devices. Such measurements are desirable as optics is unique in its combination of high throughput, sensitivity, and non-destructivity. Rigorous electromagnetic modeling has already extended the utility of optical methods such as scatterometry to the measurements of dimensions well-below the conventional diffraction limit. As tolerances decrease with feature size, greater emphasis has been placed upon reducing parametric uncertainties, which can be negatively affected by parametric correlations in the theory-to-experiment fitting process. Parametric uncertainty reductions can be realized though hybrid metrology, the proper statistical treatment of additional quantitative information.
As device feature sizes are now pushing towards the sub-5 nm domain, optics-based metrology faces a daunting new challenge. Consider that a 1 nm3 volume of crystalline silicon has just 50 atoms. Although the precise number of atoms across a 5 nm-wide line depends upon the lattice orientation, at these length scales dimensions can be expressed as few as ten atoms in width. At these near-atomic scales, quantized or atomistic effects must be considered especially with respect to the existing framework of electromagnetic scattering simulations and modeling that undergirds quantitative optical measurements. This challenge affects both conventional CMOS devices and also the variety of prospective new structures, such as “gate all around” transistors, nanowire based devices, and tunnel field effect transistors.
Accurate determination of the real and imaginary dielectric constants e_1 and e_2 prove to be key to the extensibility of Maxwell’s Equations to these low dimensional structures. Several potential solutions to aspects of this measurement challenge are found in the literature, ranging from empirical determinations assuming an effective media  to density functional theory calculations of the electronic properties and the bulk dielectric tensors . We will build upon such solutions from the literature and discuss additional alternatives such as the hybridization of multiple measurements or techniques, shorter measurement wavelengths, and enhanced hardware platforms.
Figure 1 (see attachment) shows preliminary work underway in this effort. A Tauc-Lorentz fitting of scatterometry parameters (Delta) and (Psi) yields the thickness-dependent e_1 and e_2 for our initial sample set. This work will address not just the implications for thin films (i.e. two-dimensional structures) but also outline the challenges for one-dimensional and zero-dimensional structures as well.
Figure 1. (a) Schematic of an initial sample set for experimentally demonstrating changes in the dielectric constant as a function of layer thickness. Samples were prepared using atomic layer deposition.
(b) Dielectric constants e_1, e_2 for the sample set as function of photon energy. Permittivity decreases as a function of decreasing HfO2 thickness.
 P. Ebersbach, et al., “Monitoring of ion implantation in microelectronics production environment using multi-channel reflectometry,” Proc. SPIE 9778, 977812 (2016).
 P. Pusching and C. Anbrosch-Draxl, “Atomistic Modeling of Optical Properties of Thin Films,” Adv. Eng. Mat. 8, 1151-1155 (2006).