The equilibrium theory for a train of steps is revisited. The analysis is based on a nonlocal Langevin equation derived from the Burton-Carbrera-Frank model. The Ehrlich-Schwoebel effect, diffusion along the step edge, as well as elastic interactions between steps have been incorporated. We discuss several static correlation functions and give an improved estimate for the terrace width distribution. By exploiting the dispersion relation, the time dependence of the step fluctuations has been calculated. In the limit of well separated length scales there are several time intervals where the temporal step correlation function follows a power law with one of the exponents 1/2, 1/3, or 1/4. In the opposite situation, neither power laws nor simple scaling behaviors are obtained. We provide precise conditions on which regime must be expected in a given real situation. Moreover, it is shown that different physical mechanisms can give rise to the same exponent. This study is thus crucial for the discrimination between various physical regimes in a real experiment. The range of validity of the approximation and the crossover times are discussed for steps on Si(111).

• Received 17 October 1997

DOI:https://doi.org/10.1103/PhysRevB.58.2289