Spatial First-passage Statistics of Al$/$Si(111)-($\sqrt{3}\times\sqrt{3}$) Step Fluctuations: Implications for Nanoscale Structures*

The step-edges on a multi-component surface of Al$/ $Si(111)-($\sqrt{3}\times\sqrt{3}$), observed via scanning tunneling microscopy, fluctuate in thermal equilibrium over a temperature range of 720K-1070K. For step lengths L = 65-160 nm, the measured first-passage spatial persistence and survival probabilities are found to be temperature independent and thus universally applicable. The power-law functional form for spatial persistence probabilities is confirmed, and the symmetric spatial persistence exponent is measured to be $\theta = 0.53 \pm$ 0.05, in agreement with the theoretical prediction $\theta=\frac{1}{2}$. The survival probability is found to scale with y$/$L, where y is the distance along the step edge. The functional form of the survival probabilities agrees quantitatively with the theoretical prediction, which decays exponentially as exp(-y/y$_{s}$) for small y$/$L. The experiment finds the decay constant to be y$_{s}/$L= 0.076 $\pm$ 0.033 for y$/$L $\le$ 0.2. The physical implications of these results for the predictability of nanoscale displacements and thus on device design and manufacturing will be discussed.