The Burton-Cabrera-Frank (BCF) theory of crystal growth has been successful in describing a wide range of phenomena in surface physics. Typical crystal surfaces are slightly misoriented with respect to a facet plane; thus, the BCF theory views such systems as composed of staircase-like structures of steps separating terraces. Adsorbed atoms (adatoms), which are represented by a continuous density, diffuse on terraces, and steps move by absorbing or emitting these adatoms. Here we shed light on the microscopic origins of the BCF theory by deriving a simple, one-dimensional (1D) version of the theory from an atomistic, kinetic restricted solid-on-solid (KRSOS) model without external material deposition. We define the time-dependent adatom density and step position as appropriate ensemble averages in the KRSOS model, thereby exposing the non-equilibrium statistical mechanics origins of the BCF theory. Our analysis reveals that the BCF theory is valid in a low adatom-density regime, much in the same way that an ideal gas approximation applies to dilute gasses. We find conditions under which the surface remains in a low-density regime and discuss the microscopic origin of corrections to the BCF model.
Bibliographical noteFunding Information:
PNP was supported by the National Institute of Standards and Technology American Recovery and Reinvestment Act Measurement Science and Engineering Fellowship Program Award No. 70NANB10H026 through the University of Maryland, with ancillary support from the NSF-MRSEC under Grant No. DMR 05-20471 . This author's research was also supported by NSF-DMS 08-47587 at the University of Maryland and the Institute for Mathematics and its Applications at the University of Minnesota . TLE was supported in part by the NSF-MRSEC Grant DMR 05-20741 , and by NSF-CHE 07-50334 and 13-05892 . DM was supported by NSF-DMS 08-47587 at the University of Maryland . We acknowledge helpful interactions with A. BH. Hammouda, M. K. Hawkins, J. A. Liddle, O. Pierre-Louis, T. Schulze, M. Stiles, and J. Weare.
- Burton-Cabrera-Frank theory
- Non-equilibrium statistical mechanics
- Solid-on-solid model