Evaporation Through a Shadow Mask

What is modelled

An aluminium source is heated in vacuum, its vapour pressure sets how many atoms leave per second, and they travel in straight lines to a substrate. A small flat source emits with a cosine distribution, so the film is thickest on the axis and thinner off axis. A shadow mask held a small gap above the substrate lets atoms through only where it is open; because the source has a finite size and the rays arrive at an angle, the printed edge is blurred and displaced.

Equations

• Vapour pressure fit log10(p / Pa) = A − B / T, with B set by the heat of vaporisation.
• Hertz-Knudsen flux J = p / √(2π m k T), mass flow ṁ = m J Asource.
• Deposition rate with the cosine law dh/dt = ṁ cosθe cosθs / (π ρ r²) = ṁ H² / (π ρ (H² + x²)²) on a parallel substrate.
• Mask edge blur (full ramp) b = ds g / (H − g), 10 to 90 % width 0.8 b, lateral shift g x / (H − g) for a feature at distance x off the axis.

Checks run in the page

• Integrated film volume over the whole plane equals the evaporated volume.
• The angle form of the cosine law against the closed form.
• The 10 to 90 % edge width found by sampling the source against 0.8 b.
• Hertz-Knudsen against the wall-collision form n v / 4, and the heat of vaporisation implied by the fit.

Honest simplifications

The source is treated as a flat disc with ideal cosine emission, unit sticking, no gas scattering, no source depletion, no radiative heating of the substrate and no mask thickness or clogging. Real sources depart from a cosine law, and the vapour pressure fit here is an illustrative fit, not a calibrated aluminium evaporator, which is usually run in a boat or crucible with its own alloying problems.

• Choose Mask in contact: the edge is sharp and the shift disappears.
• Choose Big gap, big source: the edge blur and shift in the zoom view grow together.
• Choose Far source, uniform: the film is much more even across the substrate, at the cost of rate.