Time-space height correlations of thermally fluctuating 2-d systems. Application to vicinal surfaces and analysis of STM images of Cu(115)

For thermally fluctuating 2-d systems, like solid surfaces, time and space correlation of the local surface height diverge logarithmically in the rough phase, whereas saturation is obtained below the roughening transition. A 2-d Langevin formalism, allowing to recover for long times and/or large distances these asymptotic behaviors, is presented. An overall expression for correlation functions that are related to atom hopping rates and surface stiffnesses is given. Considering anisotropic systems allows describing vicinal surfaces. At finite times, time correlations cross over to power laws alpha*t^(1/n) (n = 1, 2 or 4), within limited time ranges as it was observed for isolated fluctuating steps. Limits of time ranges are related to stiffnesses and diffusion anisotropies. Application to the analysis of STM images of Cu(115) below the roughening transition is given.