Criticality of morphological instability of a strained film growing on a patterned substrate

We show that the morphological instability of a strained film on a patterned substrate is fundamentally different from that on a flat substrate. It exhibits a film thickness ($t$) dependent critical wavelength, which takes a simple form as $\lambda_{c}=\lambda_{0}/2+\pi t$ for a very thin film, where $\lambda_{0}$ is the critical wavelength on a flat substrate. It also defines three distinct regimes of growth stability depending on the wavelength of substrate undulation ($\lambda_{s}$): for $\lambda_{s}\leq\lambda_{0}/2$, growth is stable; for $\lambda_{s}\geq\lambda_{0}$, growth is unstable; in between, growth is unstable below a critical film thickness $t_{c}$, and stable above it.