Analytic Formulas for the Orientation Dependence of Step Stiffness and Line Tension: Key Ingredients

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Analytic Formulas for the Orientation Dependence
of Step Stiffness and Line Tension Key
Ingredients for Numerical Modeling T.J.
Stasevich and T.L. Einstein Department of
Physics, University of Maryland, College Park,
Maryland 20742-4111 A crucial ingredient to
characterize the fluctuations of steps is the
stiffness, essentially the energy cost of
meandering. Likewise, the line tension (free
energy per length) b of steps determines the
shape of islands as implemented in the celebrated
Wulff construction. Thus, it is important to be
able to characterize the orientation dependence
of these two quantities. Particularly the
stiffness b(q) b''(q) is often approximated as
circularly symmetric and is evaluated via the
value in a facet direction. The former is a
mediocre approximation, while the latter
procedure is poor, since there is distinctive
behavior associated with the cusp or near cusp in
b(q) in the facet directions. In earlier work,
based on the Ising model and solid-on-solid
approximation, we developed simple expressions
for the stiffness for (100) and (111) facets of
fcc lattices that agree very well with
experimental data. For the (100) it is often
important to include next-nearest neighbor
interactions as well as nearest neighbor (NN)
ones in the (111) the NN interactions typically
suffice. We have recently extended this work to
develop an analytic formula, twice differentiable
everywhere, suitable for implementation in
finite-element calculations. While no longer
simple, it is straightforward to code, and so
should have wide applicability 1. The
scheme starts with the simple expression for
inverse stiffness (or alternatively for line
tension), valid at low temperatures. Next we add
(for the (111) surface) an orientation-independent
term D which accounts for the finite-temperature
correction at the qp/6, the high-symmetry
non-facet direction, for which the solution is
know exactly. Only near the facet direction q0
does this formulation break down. In that
region, we fit to a 5th order power series. The
first 3 terms are fit to the exactly-know
solution at q0 the next 3 are adjusted to
produce a smooth joining at an angle qc
determined to optimize the fit. This procedure
is illustrated in the figure. A summary of the
results is given in the table. 1 T.J.
Stasevich and T.L. Einstein, SIAM--Multiscale
Model. Simul. 6, 90 (2007) cond-mat/0609237
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