Diffusion

This problem considers the thermally driven diffusion of a dopant into a solid from a constant masked source. Parameters have been chosen to be those typically encountered in semiconductor diffusion. The PDE is just the diffusion equation:

dt(C) = div(D*grad(C)) ,

where C is the concentration and D is the diffusivity. At early times, the solution near the source can be compared to the analytic solution for 1D diffusion.

Concentration contours at time = 1 hour.

Comparison of numerical and exact solutions (C/C0) along center line.

Concentration (C/C0) vs. time at various points along the center line.

VIEW PROBLEM SCRIPT

title
'Masked Diffusion'

variables
u(threshold=0.1) { fraction of external concentration }

definitions
concs = 1.8e8 { surface concentration atom/micron^3}
D = 1.1e-2 { diffusivity micron^2/hr}
conc = concs*u
uexact1d = erfc(x/(2*sqrt(D*t))) { analytic solution to corresponding 1D problem }
cexact1d = concs*uexact1d
M = upulse(y-0.3,y-0.7) { masked surface flux multiplier }

initial values
u = 0

equations
u : div(D*grad(u)) = dt(u)

boundaries
region 1
start(0,0)
natural(u) = 0
line to (1,0) to (1,1) to (0,1)
natural(u) = 10*M*(1-u)
line to close

feature { a "gridding feature" to help localize the activity }
start (0.02,0.3) line to (0.02,0.7)

time 0 to 1 by 0.001

plots
for t=1e-5 1e-4 1e-3 1e-2 0.05 by 0.05 to 0.2 by 0.1 to endtime
contour(u)
surface(u)
elevation(u,uexact1d) from (0,0.5) to (1,0.5)
elevation(u-uexact1d) from (0,0.5) to (1,0.5)

histories
history(u) at (0.05,0.5) (0.1,0.5) (0.15,0.5) (0.2,0.5)

end

{ ============= Comment ============

{ DIFFUSION.PDE

This problem considers the thermally driven diffusion of a dopant into a solid from a constant source. Parameters have been chosen to be those typically encountered in semiconductor diffusion.

surface concentration = 1.8e20 atoms/cm^2
diffusion coefficient = 3.0e-15 cm^2/sec

The natural tendency in this type of problem is to start with zero concentration in the material, and a fixed value on the boundary. This implies an infinite curvature at the boundary, and an infinite transport velocity of the diffusing particles. It also generates over-shoot in the solution, because the Finite-Element Method tries to fit a step function with quadratics.

A better formulation is to program a large input flux, representative of the rate at which dopant can actually cross the boundary, (or approximately the molecular velocity times the concentration deficiency at the boundary).

Here we use a masked source, in order to generate a 2-dimensional pattern. This causes the result to lag a bit behind the analytical Plane-diffusion result at late times.

}

==================End Comment ========== }