Spatial first-passage statistics ofAl∕Si(111)−(3×3)step fluctuations

Spatial step edge fluctuations on a multicomponent surface of $\mathrm{Al}∕\mathrm{Si}(111)\text{\ensuremath{-}}(\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3})$ were measured via scanning tunneling microscopy over a temperature range of $720--1070\phantom{\rule{0.3em}{0ex}}\mathrm{K}$, for step lengths of $L=65--160\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$. Even though the time scale of fluctuations of steps on this surface varies by orders of magnitude over the indicated temperature range, measured first-passage spatial persistence and survival probabilities are temperature independent. The power law functional form for spatial persistence probabilities is confirmed and the symmetric spatial persistence exponent is measured to be $\ensuremath{\theta}=0.498\ifmmode\pm\else\textpm\fi{}0.062$ in agreement with the theoretical prediction $\ensuremath{\theta}=1∕2$. The survival probability is found to scale directly with $y∕L$, where $y$ is the distance along the step edge. The form of the survival probabilities agrees quantitatively with the theoretical prediction, which yields exponential decay in the limit of small $y∕L$. The decay constant is found experimentally to be ${y}_{s}∕L=0.076\ifmmode\pm\else\textpm\fi{}0.033$ for $y∕L\ensuremath{\le}0.2$.