The continuum step model has proved remarkably successful in bridging length scales and describing—in terms of step
stiffness, step interaction strength, and the appropriate generalized step mobility—a wide variety of phenomena on
stepped surfaces, such as step fluctuations, cluster diffusion, step kinetics, electromigration, and mound decay.
From the terrace-width (called l) distribution [TWD], one can gauge the magnitude A of the typically dominant elastic
interactions, which vary like A/l2 (like the entropic repulsion due to non-crossing of steps). Much progress can be
made by mapping the configurations of steps on vicinal surfaces into the world lines of fermions in 1-D, evolving in
time. The simplest mean field approximation is the familiar problem of a quantum mechanical problem of a particle in
a 1-D box or a parabolic well. More sophisticated analyses can be made, culminating in the use of results from
random-matrix theory, in particular the generalized Wigner surmise—a gamma distribution in l2 rather than the customary
Gaussian—to describe the TWD. The model TWDs are compared with experimental data (esp. vicinal Si and Cu), numerical
simulations (mostly Monte Carlo, some transfer matrix), and exact results. When surface states also mediate the step
interaction, the situation is more complicated. Non-equilibrium effects can severely alter the TWDs.
Work supported by NSF MRSEC at U. of Maryland, done in collaboration with H.L. Richards, O. Pierre-Louis, Hailu Gebremariam, S.D. Cohen, R.D. Schroll, N.C. Bartelt, E.D. Williams, and others at Maryland, with M. Giesen and H. Ibach at FZ-Jülich (via Humboldt Foundation), and with J.-J. Métois at Marseilles