Research on the thermal expansion mechanism of self-traceable grating pitch based on thermo-mechanical coupling modeling and finite element simulation analysis

Abstract Nanometrology guarantees the dimensional accuracy in nanomanufacturing. As integrated-circuit features shrink progressively, the conventional laser-wavelength traceability system becomes highly susceptible to ambient disturbances and fails to meet in situ production requirements. Referencing a quantized optical-lattice constant, a flattened traceability chain provides higher throughput and enhanced environmental robustness by reducing the number of calibration transfer steps. Its cornerstone is the self-traceable grating. The grating’s pitch is directly traceable to the 7S 3 →7P 4 0 transition frequency of chromium (Cr) atoms, which provides picometer accuracy. However, this accuracy is vulnerable to thermal expansion. We developed a thermo-mechanical coupling model to quantify temperature effects and analyze the pitch expansion. This model was validated by finite element simulation (FES). The analysis reveals a cooperative mechanism for stress relief. This mechanism involves interfacial constraints caused by the thermal expansion coefficient (CTE) mismatch between the substrate and the atomic layer. These constraints work in concert with a buffer layer to relieve stress. Within (20 ± 5) °C, the pitch expansion of a 212.78 nm Cr grating decreases from 0.5 to 0.04 pm °C −1 when silicon substrate is replaced by zero expansion glass. This represents an order-of-magnitude improvement. Pitch expansion scales linearly with buffer layer thickness, albeit with opposite slopes for the two substrates. The proposed model and optimization rules furnish theoretical and engineering guidance for boosting nanometrological precision, and fortifying traceability reliability.