Two types of surface diffusion can play important roles in coverage and
deposition layers; bulk diffusion, which is the global macro-motion of the
material within the deposited layer, and surface diffusion, which relates
to the motion of metal boundaries. Here, we examine the effects of
surface diffusion on the shape of the deposition
layer.
Cale and Jain (Cale, T.S., Jain, M.K., Tracy, C.J., and Duffin, R.,
submitted for publication, J. Vac. Sci, Tech, B, 1996) have performed carefully
fit numerical experiments to match experimental evidence of surface
diffusion effects of aluminum-(1.5%)copper films. They propose
a model which contains a term which depends on the second derivative of
the curvature.
We point out the motion under the second derivative of curvature is
reminiscent of the problem of
flow under curvature.
However, the problem is delicate because this leads to a time-dependent
fourth order partial differential equation, and the presence of the fourth
derivative requires an exceedingly small time step for stability in an
explicit scheme; the linear fourth order heat equation has a stability time
step requirement of the form O(dt /h ^4); where dt is the time step and
h is the space step. While such schemes can in fact be made implicit to
allow a larger time step, the price is considerable more work to evaluate
the term.
Level Set Formulation:
The
narrow band level set method
is particularly well-suited to this sort of problem, due to the
embedded representation of the surface. Straightforward finite
difference schemes based on central differences may be used to
approximate the appropriate higher derivatives.
Results and Sample Simulations:
We show two sample simulations. First, we show the
case of straightforward isotropic deposition, modified by
a surface diffusion term which depends on the second derivative
of curvature.
More precisely, we use a speed function of the form
F = 1 + eps * d^2 K / ds^2
where eps is the diffusion coefficient, and K is the local curvature.
As the surface diffusion coefficient is
increased, the profile is rounded and muted by this effect.
Diffusion Coefficient = 0.0
Diffusion Coefficient = 0.0025
Diffusion Coefficient = 0.001
Diffusion Coefficient = 0.002
Next, we repeat the experiment, only this time we study
the interplay of ion-milling and surface diffusion.
Here, we consider a speed function
which contains isotropic deposition and an ion-milling non-convex
etch function together with a parameter study of surface diffusion.
More precisely, we use a speed function of the form
F = 1 - [1 + sin(theta)*sin(theta) ] cos(theta) + eps * d^2 K / ds^2