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Equation 5 · How Advanced Semiconductor Fabrication Actually Works

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dd

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the meaning a bilayer period. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

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dd

Symbol d

the meaning a bilayer period.

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describes why the stack works at all: at near-normal incidence, θ\theta is small and cos⁡\cosθ\theta is close to one, so the first-order condition reduces to roughly λ\lambda ≈\approx 2d , meaning a bilayer period d of a little under 7 nanometers should reflect strongly somewhere near 13.5–14 nanometers. Measurement bears this out closely rather than exactly: one recent study of Mo/Si multilayers built for EUV optics reports bilayer periods of 6.97 to 7.07 nanometers across fifty complete bilayers, with reflectivity near 65 percent at near-normal incidence around 10.7 degrees, interface widths of about 0.3 to 0.4 nanometers combining roughness and interdiffusion, and surface roughness held below 0.2…
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describes why the stack works at all: at near-normal incidence, θ\theta is small and cos⁡\cosθ\theta is close to one, so the first-order condition reduces to roughly λ\lambda ≈\approx 2d , meaning a bilayer period d of a little under 7 nanometers should reflect strongly somewhere near 13.5–14 nanometers. Measurement bears this out closely rather than exactly: one recent study of Mo/Si multilayers built for EUV optics reports bilayer periods of 6.97 to 7.07 nanometers across fifty complete bilayers, with reflectivity near 65 percent at near-normal incidence around 10.7 degrees, interface widths of about 0.3 to 0.4 nanometers combining roughness and interdiffusion, and surface roughness held below 0.2 nanometers root-mean-square [ 4 ] . The gap between the simple formula’s roughly-7-nanometer prediction and the measured period reflects exactly what the idealized Bragg picture leaves out — a small but real refractive-index correction at this wavelength, and the actual angle of use — which is why real multilayer engineering is a materials-science problem and not a formula lookup.

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