Lateral resolution in focused electron beam-induced deposition: scaling laws for pulsed and static exposure

Abstract

In this work, we review the single-adsorbate time-dependent continuum model for focused electron beam-induced deposition (FEBID). The differential equation for the adsorption rate will be expressed by dimensionless parameters describing the contributions of adsorption, desorption, dissociation, and the surface diffusion of the precursor adsorbates. The contributions are individually presented in order to elucidate their influence during variations in the electron beam exposure time. The findings are condensed into three new scaling laws for pulsed exposure FEBID (or FEB-induced etching) relating the lateral resolution of deposits or etch pits to surface diffusion and electron beam exposure dwell time for a given adsorbate depletion state.

Continuum models of focused electron beam induced processing

“…The resolution parameter for a stationary Gaussian beam can be obtained from the deposit shape, using a numerical solution of Equation 68, and the expression for the growth rate given by Equation 15. The numerical result is very well approximated by the -vs-scaling law of FEBIP resolution formulated in [1]: (79) The above formula shows that even a relatively small surface diffusion contribution can lead to a decreasing in the deposit and etch pit size, and thus an improved lateral resolution. Once surface diffusion overcomes irradiative depletion, the deposit or etch pit diameter will approach that of the beam ( = 1), as discussed in [1].…”

“…Introduction to continuum models of focused electron beam induced processing (FEBIP) Continuum FEBIP models enable the simulation of process rates that govern focused electron beam induced etching (FEBIE), deposition (FEBID) [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] and surface functionalization [17] techniques. They are typically used to simulate growth rates and nanostructure geometries as a function of experimental parameters, and to help elucidate the underlying growth mechanisms.…”

“…can be found by solving for the difference between the flux of molecules adsorbing from and returning to the gas phase. In the gas phase, the molecule flux is given by: (1) where P a is the pressure of the precursor gas for adsorbate 'a', m a is the mass of a gas molecule, k B is Boltzmann's constant and T g is the gas temperature. The simplest case of gas-molecule adsorption onto a substrate surface is that of physisorption, described by a single potential well at the surface as shown in Figure 2.…”

“…The decay rate depends on the effective residence time τ a,out and the irradiative depletion parameter . The resulting time-evolution of N a is given by the solution to Equation 12, which can be expressed as: (24) The corresponding vertical growth rate is proportional to N a (t) and the electron flux, and is given by: (25) The lateral size of the growing structure or etch pit will evolve in time, yielding the -vs-t law of pulsed FEBIP resolution, expressed as the lateral resolution , (neglecting surface diffusion) versus the electron beam exposure time t [1]: (26) In the limit t → 0, . In the steady state (t → ∞) this converges to the scaling law given by Equation 22.…”

“…Pulsed exposure: Equation 68 can be solved numerically, and the corresponding general scaling law of FEBIP resolution, including irradiative depletion, surface diffusion, and exposure dwell time, can be approximated by [1]:…”