Dynamics of Step Bunching in Heteroepitaxial Growth on Vicinal Substrates

When a heteroepitaxial film is grown on a vicinal substrate, the terrace steps at the growth front may bunch together to relieve strain, resulting in a rough surface. On the other hand, proper manipulation of the growth kinetics may suppress the inherent bunching instability, thus preserving step-flow growth. Here we show that the step dynamics in the early stages of growth can already determine whether the bunching instability is truly suppressed, prior to bunching actually taking place in the unstable regime. We determine the critical film thickness above which steps will bunch and exploit its scaling properties and usefulness for extracting intrinsic energy parameters. Experimental studies of ${\mathrm{SrRuO}}_3$ films grown on vicinal ${\mathrm{SrTiO}}_3$ substrates clearly establish the existence of the critical film thickness in step bunching.

Figure 1

Time evolution of the relative step positions in the (a) unstable and (b) stable regime. In (c) and (d), the dashed lines are from numerical simulations using Eq. (7), while the solid lines are based on linear stability analysis. The dynamics of the deviation amplitude increases exponentially in (c) the unstable case and decreases sinusoidally in (d) the stable case. The dispersion relations obtained from the numerical simulations using Eq. (7) and from linear stability analysis are compared in the (e) unstable and (f) stable cases.

Figure 2

A schematic view of the critical thickness ${\tau }_C$ as a function of the average terrace width $L$. When $L$ is much shorter than $L^{*}$, ${\tau }_C$ scales with $L^3$. When $L$ approaches $L^{*}$, ${\tau }_C$ deviates from $C_1{L}^3$ and approaches infinity, with $C_1$ being a constant. The data points marked by the solid circles correspond to the AFM images shown in Fig. 3.

Figure 3

AFM images of ${\mathrm{SrRuO}}_3$ films of (a) 4 and (b) 10 nm grown on a vicinal ${\mathrm{SrTiO}}_3$ substrate with a terrace width of $\sim 200\mathrm{nm}$. SRO films of (c) 10 and (d) 40 nm grown on a vicinal substrate with a terrace width of $\sim 400\mathrm{nm}$. The white scale bars are ${10}^3\mathrm{nm}$ in width.