Distinguishing step relaxation mechanisms via pair correlation functions

Theoretical predictions of coupled step motion are tested by direct STM measurement of the fluctuations of near-neighbor pairs of steps on $\mathrm{Si}\left(111\right)-\sqrt{3}\times \sqrt{3}\mathrm{R}30°$ Al at $970\mathrm{K}$. The average magnitude of the pair-correlation function is within one standard deviation of zero, consistent with uncorrelated near-neighbor step fluctuations. The time dependence of the pair-correlation function shows no statistically significant agreement with the predicted $t^{1∕2}$ growth of pair correlations via rate-limiting atomic diffusion between adjacent steps. The physical considerations governing uncorrelated step fluctuations occurring via random attachment/detachment events at the step edge are discussed.

Figure 1

Schematic illustration of the two surface processes, evaporation condensation at an isolated step and diffusion across the terraces between steps, leading to $t^{1∕2}$ scaling of the temporal correlation function.

Figure 2

STM “pseudoimage” obtained at $970\mathrm{K}$ by scanning the tip repeatedly over a fixed position on the step edge $\left(y\right)$. The horizontal direction in the image is the $x$ direction and the vertical axis corresponds to the total duration of the scan. Thus, the dimensions of the image are $(100\mathrm{nm}\times 23\mathrm{s})$.

Figure 3

(a) The usual temporal correlation function averaged over four images similar to the one shown in Fig. 2 at $970\mathrm{K}$. The computed curves are only shown for early times when power law scaling is expected. (b) The pair correlation function averaged over the same four images used to measure the curve in (a) and plotted over the identical time span. Note the change in vertical scale between (a) and (b). Error bars are the standard error from the four separate measurements.

Figure 4

Comparison of the average of the pair correlation functions (open circles) with the theoretical prediction of $1∕6$ of the value of the temporal correlation function $G\left(t\right)$ (closed circles) to test the prediction for DSS from Ref. 17. The thick solid curves are the best fits to a two-parameter power law of the form shown in Eq. (2) of the text. The pair correlation function is not well-fit by a power law. Furthermore the prediction that the two quantities in this plot should be equal for DSS-limited step kinetics is clearly not satisfied. (Error bars are the same as in Fig. 3 but omitted for clarity of the fit lines.)