Semiconductor Photolithography & Optics
EUV Lithography Rayleigh CD & Depth of Focus Formula
Calculate Photolithography Minimum Printable Half-Pitch CD = k_1 * (lambda / NA), Rayleigh Depth of Focus DOF = k_2 * (lambda / NA^2), Minimum Pitch 2 * CD, and Transistor Area Scaling.
Master the fundamental EUV vs High-NA EUV resolution-focus tradeoff!
Two Rayleigh equations govern every ASML ArFi (193 nm immersion) and EUV (13.5 nm) semiconductor scanner on earth: (1) Minimum Single-Exposure Half-Pitch ("Critical Dimension") CD = k_1 * (lambda_nm / NA), where lambda_nm is exposure wavelength, NA is projection-lens Numerical Aperture, and k_1 >= 0.25 is the process-capability factor (limited by the 0.25 physical Abbe diffraction limit for single-exposure two-beam imaging!); and (2) Rayleigh Depth of Focus DOF = k_2 * (lambda_nm / NA^2)! Look at the quadratic trap when moving from Standard 0.33-NA EUV to High-NA 0.55-NA EUV (a 5/3 = 1.6667x jump in NA): CD shrinks by 0.60x (40% finer features!), while Depth of Focus collapses by (0.33 / 0.55)^2 = 0.36x (losing 64% of focus margin, forcing ultra-thin photoresists and planarized wafers)!
Verify on EUV_Wavelength lambda_nm = 13.5 nm (B2), Lens_Numerical_Aperture NA = 0.33 (standard EUV, B3), Process_Factor k_1 = 0.33 (B4), and Focus_Factor k_2 = 0.50 (B5): Minimum Half-Pitch CD is 0.33 * (13.5 / 0.33) = 13.50 nm (E2 -> Minimum Metal Pitch 2 * CD = 27.00 nm in E3), and Rayleigh Depth of Focus is 0.50 * (13.5 / 0.33^2) = 61.98 nm (E4)! Upgrade to High-NA EUV by setting NA = 0.55 in cell B3: Half-Pitch CD drops to 8.10 nm (16.20 nm pitch!), while Depth of Focus plunges to 22.31 nm (exact 0.36x focus budget)! Test impossible single-exposure k_1 = 0.20 (< 0.25 diffraction limit) to trigger the physical Abbe guard!
| Scanner Lithography Metric (lambda=13.5 nm EUV, k1=0.33, k2=0.50) | Formula | 0.33-NA EUV (vs 0.55 High-NA) |
|---|---|---|
| Critical Dimension Half-Pitch CD | k_1 * (lambda_nm / NA) | 13.50 nm (8.10 nm @ 0.55 NA) |
| Minimum Single-Exposure Pitch | 2 * k_1 * (lambda_nm / NA) | 27.00 nm (16.20 nm @ 0.55 NA) |
| Rayleigh Depth of Focus DOF | k_2 * (lambda_nm / NA^2) | 61.98 nm (22.31 nm @ 0.55 NA) |
| DOF-to-CD Process Latitude Ratio | DOF / CD = (k_2 / k_1) / NA | 4.5914x (2.7548x @ 0.55 NA) |
Abbe-Guarded Rayleigh Critical Dimension & Depth of Focus Formula
Enforce lambda_nm > 0, NA > 0 (note immersion 193i reaches NA = 1.35 via n_water = 1.44!), k_1 >= 0.25 (single-exposure diffraction limit), and k_2 > 0!
=LET(wl_n, B2, na_l, B3, k_1p, B4, k_2f, B5, IF(OR(wl_n <= 0, na_l <= 0, na_l > 1.5, k_1p < 0.25, k_2f <= 0), "Below Abbe Limit or Invalid Optics", ROUND(k_2f * (wl_n / (na_l ^ 2)), 2)))DOF-to-CD Aspect Latitude Ratio Formula
Monitor DOF / CD = (k_2 / k_1) / NA to verify whether resist thickness (typically <= 2.5 * CD to avoid pattern collapse) fits inside the scanner’s focal budget!
=IF(ISTEXT(E4), E4, ROUND((B5 / B4) / B3, 4))Try the example
Use the original values or change them to test your understanding.