An electron beam writing a resist doesn't deposit energy at a single point — scattering inside the resist and substrate spreads it out. The standard double-Gaussian point-spread function models the local absorbed-energy density at distance r from a write point as:
f(r) = 1/(1+η) · [ (1/πβf²)·e^(−r²/βf²) + (η/πβb²)·e^(−r²/βb²) ]
The first term is forward scattering: a narrow, high-intensity spot set by beam energy and resist thickness (βf ~ few nm). The second is backscattering: electrons that bounce off the substrate and re-enter the resist tens of nanometres away, spread over a much wider, lower-intensity halo (βb, tens of nm). η is the backscatter coefficient — higher for a dense substrate like gold.
Total deposited dose at any point is the sum of every write point's contribution, scaled by that point's assigned dose Di. Positive resist clears wherever the accumulated dose exceeds a threshold Eth — so pillars spaced closer than roughly βb pick up extra dose from their neighbours' backscatter halos and can bridge into a single cleared trench instead of staying separate posts. This proximity effect is exactly why fabricating the tight nanoparticle gaps needed for a plasmonic hot-spot array is hard.
Proximity-effect correction (PEC) compensates by lowering the assigned dose at densely-packed sites (Di = D0 / (1 + η·ρi), where ρi is the local pattern density seen by that site), pulling isolated and clustered dots back toward the same clearing behaviour.
- η, βf, βb — the physical scattering parameters of the beam/resist/substrate stack.
- Pitch — spacing between write points; shrink it to force neighbouring dose halos to overlap.
- D0 — nominal exposure dose before any correction.
- PEC toggle — applies the density-based dose correction above.