doi:10.1016/S0022-0248(00)00923-4 
Copyright © 2001 Elsevier Science B.V. All rights reserved.
A continuum model for the growth of epitaxial films
a Department of Mathematics, University of Tennessee, Knoxville, TN 37996-1300, USA
b Courant Institute of Mathematical Sciences, New York University, 251 Mercer St., New York, NY 10012, USA
Received 8 June 2000;
accepted 10 October 2000
Communicated by D.T.J. Hurle
Available online 4 January 2001.
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Abstract
The continuum equations presented here model the growth of epitaxial films in terms of a local edge density 
and surface concentration (number density) of adatoms. This model is more amenable to computations than existing models that feature discrete edges and solve continuum equations on each terrace; yet it offers a more detailed picture than continuum models that treat the surface height as the only dependent variable. This latter feature is especially important if one wishes to account for several species which may react on the surface of the film or at step edges to build complicated unit cells. The model is motivated by and compared with numerical solutions of rate equations which are derived from kinetic Monte-Carlo simulations. After introducing the model in a 1+1 dimensional setting, we extend it to a 2+1 dimensional setting assuming spatial derivatives become surface gradients. We also discuss extension for the case with multiple species.
Fig. 1. A schematic drawing of a 1+1-dimensional epitaxial film illustrating the potential barrier φ that determines the hopping rates ki±. Most sites have three nearest neighbors in the plane below the adatom, resulting in a generic hopping rate ks.
Fig. 2. These three figures illustrate typical equilibrium adatom densities for a periodic step train where (a) all of the hopping rates ki±=ks=0.5; (b) where there is an edge barrier k1±
ks but no ES barrier kn+=ks and (c) where both barriers are present and satisfy k1−<k1+<kn+<ks.
Fig. 3. The local surface current Ji as a function of i for steps with (a) and without (b) an ES barrier. Notice that the current is asymmetric in the first case.
Fig. 4. Surface average of the current as a function of time, revealing the non-equilibrium contribution to the current and fast relaxation time following a step-attachment event.
Fig. 5. A comparison of the value of S given by Eq. (11) to a value calculated from Eq. (14) and a current that is computed via the rate equations (3). The variation in these two values is shown as a function of α and kn+; in each case the remaining parameters are held fixed.
Fig. 6. (a) The quasi-equilibrium adatom density for a 200-site lattice and (b) for a 400-site lattice, all other parameters being equal. Ignoring end effects the first of these is well-approximated by a linear function; the second is linear up until around the 250th lattice site.
Fig. 7. The surface profile h(x) for imposed surface currents J=F, J=0 and J=−F. The smooth curve corresponds to the analytic solution (17); the stepped curve is the result of simulation.
Fig. 8. The surface profile h(x) for an imposed surface current J=0, 3F/4 and 3F/2. The smooth curve corresponds to the analytic solution (17); the stepped curve is the result of simulation. For the chosen parameter values, the imposed current J=3F/4 should balance the ES-barrier-induced drift, producing a linear surface.
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