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Register a new userCorrelations in nano-scale step fluctuations: comparison of simulation and experiments
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We analyze correlations in step-edge fluctuations using the Bortz-Kalos-Lebowitz kinetic Monte Carlo algorithm, with a 2-parameter expression for energy barriers, and compare with our VT-STM line-scan experiments on spiral steps on Pb(111). The scaling of the correlation times gives a dynamic exponent confirming the expected step-edge-diffusion rate-limiting kinetics both in the MC and in the experiments. We both calculate and measure the temperature dependence of (mass) transport properties via the characteristic hopping times and deduce therefrom the notoriously-elusive effective energy barrier for the edge fluctuations. With a careful analysis we point out the necessity of a more complex model to mimic the kinetics of a Pb(111) surface for certain parameter ranges.
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Time dependent STM has been used to evaluate step fluctuations as a function of temperature (300-450 K) on Ag(111) films grown on mica. The temporal correlation function scales as a power law in time, t^1/n with measured values of 1/n varying over a range of 0.19 pm 0.04 to 0.29 pm 0.04 with no dependence on temperature. The average value of 1/n = 0.24 pm 0.01 is consistent with step-edge diffusion limited fluctuations (n = z = 4, conserved noise). The magnitude of the time correlation function and the width of the fluctuations both scale with temperature with the same apparent activation energy of Eeff = 0.21 pm 0.02 eV, indicating that the correlation time is at most weakly temperature dependent. Direct analysis of the autocorrelation function confirms that the correlation time is at most weakly temperature dependent, and thus the apparent correlation length is strongly temperature dependent. This behavior can be reproduced by assuming that the apparent correlation length is governed by the longest wavelength of step fluctuations that can be sampled in the measurement time interval. Evaluation of the correlation time for previous measurements for Al/Si(111) (z =2) yields the same conclusion about measurement time interval. In both cases the ratio of the measurement time to the effective correlation time is on the order of 10.
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 STM 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 320K, and on Al-terminated Si(111) over the temperature range 770K - 970K. 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/Dt where Dt 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 tm. Scaling with t/Dt occurs only when Dt/tm is a constant. We show that this observation is equivalent to theoretical predictions that the survival probability will scale as Dt/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 tm or Dt should also show little or no temperature dependence.
In a recent paper [Ferrari et al., Phys. Rev. E 69, 035102(R) (2004)], the scaling law of the fluctuations of the step limiting a crystal facet has been computed as a function of the facet size. Ferrari et al. use rigorous, but physically rather obscure, arguments. Approaching the problem from a different perspective, we rederive more transparently the scaling behavior of facet edge fluctuations as a function of time. Such behavior can be scrutinized with STM experiments and with numerical simulations.
We report the results of analytic and numerical investigations of the time scale of survival or non-zero-crossing probability $S(t)$ in equilibrium step fluctuations described by Langevin equations appropriate for attachment/detachment and edge-diffusion limited kinetics. An exact relation between long-time behaviors of the survival probability and the autocorrelation function is established and numerically verified. $S(t)$ is shown to exhibit simple scaling behavior as a function of system size and sampling time. Our theoretical results are in agreement with those obtained from an analysis of experimental dynamical STM data on step fluctuations on Al/Si(111) and Ag(111) surfaces.
Results of analytic and numerical investigations of first-passage properties of equilibrium fluctuations of monatomic steps on a vicinal surface are reviewed. Both temporal and spatial persistence and survival probabilities, as well as the probability of persistent large deviations are considered. Results of experiments in which dynamical scanning tunneling microscopy is used to evaluate these first-passage properties for steps with different microscopic mechanisms of mass transport are also presented and interpreted in terms of theoretical predictions for appropriate models. Effects of discrete sampling, finite system size and finite observation time, which are important in understanding the results of experiments and simulations, are discussed.
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