Hybrid approach to reconstruct nanoscale grating dimensions using scattering and fluorescence with soft X-rays

Scatterometry is a tested method for measuring periodic semiconductor structures. Since the sizes of modern semiconductor structures have reached the nanoscale regime, the challenge is to determine the shape of periodic nanostructures with sub-nanometer accuracy. To increase the resolution of scatterometry, short-wavelength radiation like soft X-rays can be used. But, scatterometry with soft X-rays is an inverse problem whose solutions can be ambiguous and its sensitivity should be further increased to determine the shape of even more complicated periodic nanostructures made up of different materials. To achieve unique solutions with smaller uncertainties, scatterometry can leverage the excitation of low-Z materials with soft X-rays. Additional information from soft X-ray fluorescence analysis in a hybrid measurement approach can mitigate the problem of ambiguous solutions from soft X-ray scattering and could further decrease uncertainty. In this work, the hybrid approach is utilized to perform a comparison of solutions from the inverse problem and determine the actual solution over ambiguous solutions.

💡 Research Summary

This paper presents a hybrid metrology method that simultaneously exploits soft‑X‑ray grazing‑incidence small‑angle scattering (GISAXS) and grazing‑incidence X‑ray fluorescence (GIXRF) to resolve the inverse‑problem ambiguities inherent in conventional scatterometry of nanoscale periodic structures. The authors focus on a one‑dimensional line grating composed of silicon nitride (Si₃N₄) lines capped with a silicon dioxide (SiO₂) layer on a silicon substrate. The grating has a 100 nm pitch, a nominal critical dimension (CD) of 50 nm, and a line height of 100 nm, with additional geometric parameters such as side‑wall angle, bottom and top corner radii, and SiO₂ thicknesses in the grooves and on top of the lines.

The theoretical framework combines rigorous finite‑element method (FEM) calculations of the standing‑wave electric field inside the periodic unit cell with two measurement models. For GISAXS, the diffraction efficiency of order m is proportional to the squared magnitude of the Fourier‑transformed standing‑wave field, corrected by a Debye‑Waller factor that accounts for line‑edge roughness (parameter ξ). For GIXRF, the fluorescence yield of a given elemental line is derived from a modified Sherman equation, integrating the local field intensity over each material region and weighting it by mass density, photo‑ionization cross‑section, fluorescence yield, and self‑absorption factors.

Experimentally, the hybrid setup is installed at the PTB soft‑X‑ray beamline of the BESSY II synchrotron. A monochromatic photon energy of 680 eV—well above the K‑edges of nitrogen and oxygen—is chosen to excite both Si₃N₄ and SiO₂ efficiently. An angular scan across the critical angle of total external reflection provides a regime where scattered intensity dominates (below the critical angle) and a regime where fluorescence dominates (above the critical angle). Scattered photons are recorded with a CCD detector, while fluorescence photons are collected by a silicon‑drift detector (SDD) positioned at a known solid angle.

Data analysis proceeds by defining a comprehensive geometric model with parameters (h, w, β, r_bottom, r_top, d_groove, d_line) and computing, for each trial set, both the GISAXS diffraction efficiencies and the GIXRF fluorescence intensities. A combined least‑squares objective function is built, assigning adjustable weights to the scattering and fluorescence residuals. By systematically varying these weights, the authors identify a region where the total residual is minimized and, crucially, where the previously observed multimodal solutions collapse into a single, physically plausible solution. The hybrid approach thus resolves the “multimodality” problem: parameters that are weakly constrained by scattering alone (e.g., side‑wall angle, corner radii) become tightly constrained when fluorescence data are included, while the fluorescence‑only fit benefits from the structural sensitivity of the scattering data.

The results demonstrate sub‑nanometer accuracy in reconstructing the grating dimensions, with the hybrid method outperforming standalone GISAXS or GIXRF analyses. The study also highlights the complementary nature of the two techniques: GISAXS is highly sensitive to electron‑density contrast and geometric shape, whereas GIXRF provides element‑specific mass‑distribution information, especially valuable for low‑Z materials whose absorption edges lie in the soft‑X‑ray regime. By integrating both, the method achieves non‑destructive, fast, and highly accurate metrology suitable for in‑line process control in semiconductor manufacturing.

In conclusion, the work establishes a robust framework for hybrid soft‑X‑ray scatter‑fluorescence metrology, validates it on a realistic Si₃N₄/SiO₂ grating, and shows that the combined approach can uniquely determine complex nanoscale geometries with uncertainties well below one nanometer. This paves the way for future implementation of hybrid X‑ray techniques in high‑volume manufacturing environments where precision, speed, and non‑destructiveness are paramount.