Sampling-time effects for persistence and survival in step structural fluctuations

Abstract

The effects of sampling rate and total measurement time have been determined for single-point measurements of step fluctuations within the context of first-passage properties. Time dependent scanning tunneling microscopy has been used to evaluate step fluctuations on Ag(111) films grown on mica as a function of temperature (300-410 K), on screw dislocations on the facets of Pb crystallites at 320 K, and on Al-terminated Si(111) over the temperature range 770-970 K. Although the fundamental time constant for step fluctuations on Ag and Al/Si varies by orders of magnitude over the temperature ranges of measurement, no dependence of the persistence amplitude on temperature is observed. Instead, the persistence probability is found to scale directly with t/Δ t where Δ t is the time interval used for sampling. Survival probabilities show a more complex scaling dependence, which includes both the sampling interval and the total measurement time t_m. Scaling with t/Δt occurs only when Δt/t_m is a constant. We show that this observation is equivalent to theoretical predictions that the survival probability will scale as Δt/L^z, where L is the effective length of a step. This implies that the survival probability for large systems, when measured with fixed values of t_m or Δt, should also show little or no temperature dependence.