Proximity-effect induced density limitations for electron-beam patterned planar photonic nanomaterials

Patterning of deeply subwavelength artificial nanomaterials, e.g. photonic crystals or plasmonic metamaterials, for the visible or near-infrared optical spectrum is a challenging task. Electron-beam lithography is often the method of choice thanks to its combination of flexibility, accuracy and availability in many research laboratories. In this article an analytical model for large and dense arrays of photonic nanostructures is presented which allows to predict the maximum fill ratio (radius divided by nearest neighbor distance) before the onset of resist loss between the individual elements. The model includes geometrical parameters of the design (lattice constant, lattice symmetry), the resist contrast and proximity parameters (beam broadening, backscatter range, backscatter efficiency). It is shown that the resist contrast has a significant impact on the achievable maximum fill ratio even for large nearest neighbor distances and that the beam broadening is of paramount importance. The background energy level which is determined by the backscatter efficiency and the lattice symmetry is shown to have a weaker influence on the maximum fill ratio. The derived model can be used as a guideline in the project planning stage to predict achievable fill ratios at a planned lattice constant and consequently an assessment whether a desired functionality at a certain wavelength is possible or not.