Size-scaling exponents of current-induced step bunching on silicon surfaces

Abstract

We have experimentally determined the scaling relation ${y}_{m}{=y}_{0}{h}^{\ensuremath{\alpha}}$ between the maximum slope ${y}_{m}$ and the height h of step bunches that were created on a Si(111) surface by direct current heating. The values of the scaling exponent \ensuremath{\alpha} were $0.68\ifmmode\pm\else\textpm\fi{}0.03$ and $0.60\ifmmode\pm\else\textpm\fi{}0.04$ when the current had a step-down direction (at 1250 \ifmmode^\circ\else\textdegree\fi{}C) and a step-up direction (at 1145 \ifmmode^\circ\else\textdegree\fi{}C) respectively. The former is in agreement with the value of $\ensuremath{\alpha}=2/3,$ which was deduced from a generalized Burton-Cabrera-Frank (BCF) theory when step-step interaction energy U is inversely proportional to the square of the interstep distance l ${(U=A/l}^{2},$ where the magnitude A depends on the temperature). The latter $(\ensuremath{\alpha}=0.60\ifmmode\pm\else\textpm\fi{}0.04)$ is consistent with the value of $\ensuremath{\alpha}=3/5$ predicted for evaporation of a crystal surface, far from equilibrium, with ``transparent'' steps, i.e., when the adatoms cross the steps without attaching to a kink position. In both models (generalized BCF model and the transparent steps model) the value of the scaling exponent \ensuremath{\alpha} depends on the value of n in the step-step interaction energy ${U=A/l}^{n}.$ Our results prove $n=2$ (and definitely rule out $n=1)$ in the range $2.6\mathrm{nm}<~l<~17.4\mathrm{nm}$ covered by our experiments. The term ${y}_{0}$ in the scaling relation contains the ratio $F/A$ (F is the electric force applied to the adatoms) in the generalized BCF model, and $F/A{\ensuremath{\lambda}}_{s}$ in the model of transparent steps. Making use of the experimentally determined values of ${y}_{0}$ we calculated $F/A=2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}{\mathrm{nm}}^{\mathrm{\ensuremath{-}}2}$ at 1250 \ifmmode^\circ\else\textdegree\fi{}C and $F/A{\ensuremath{\lambda}}_{s}=1\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}8}{\mathrm{nm}}^{\mathrm{\ensuremath{-}}3}$ at 1145 \ifmmode^\circ\else\textdegree\fi{}C. Assuming $A=0.1\mathrm{eV}\mathrm{}\mathrm{nm}$ at 1250 \ifmmode^\circ\else\textdegree\fi{}C (entropic repulsion has a significant contribution to step interaction at this temperature) and making use of the value of $F/A$ (extracted from our experiments) we estimated the effective charge of the adatoms at 1250 \ifmmode^\circ\else\textdegree\fi{}C to be ${z}_{\mathrm{ef}}=0.35|e|,$ where e is the charge of the electron. Last but not least we observed two-dimensional islands, covering 2.8% of the area of the large terraces between the bunches of steps at the surface of crystals quenched from 1250 \ifmmode^\circ\else\textdegree\fi{}C, but no islands are seen on the surface of crystals quenched from 1145 \ifmmode^\circ\else\textdegree\fi{}C.