Strained crystalline materials are found in many electronic and photonic devices, such as in integrated circuits in microelectronics, with strain resulting from the manufacturing processes, and in heteroepitaxial thin films, where strain is induced by the lattice mismatch between the film and the substrate. The resulting accumulation of elastic strain energy has a significant impact on the morphological stability of the material, triggering well-known instabilities. These include the Asaro-Tiller/Grinfeld (ATG) instability that causes surface cracking in stressed bulk materials and the Stranski-Krastanov (SK) instability that leads to formation of three-dimensional (3D) islands in heteroepitaxial films.
Numerical simulations of the surface morphological evolution of uniaxially stressed elastic crystalline solids have demonstrated that, in addition to ATG (surface cracking) instabilities, long-wavelength perturbations from the planar surface morphology can trigger a tip-splitting instability that causes formation of a pattern of secondary ripples, which cannot be explained by linear stability theory. In this presentation, we develop a weakly nonlinear stability theory, which can explain the occurrence of such secondary rippling instabilities and predict the number of secondary ripples that form on the surface as a function of perturbation wavelength. The theory shows that this type of surface pattern formation arises entirely due to the competition between surface energy and elastic strain energy, regardless of surface diffusional anisotropy or the action of externally applied fields. The origin of secondary rippling is explained through nonlinear terms included in the analysis that generate sub-harmonic ripples in the surface morphology with wave numbers that are multiples of the original surface perturbation wave number. Based on the weakly nonlinear theory, we have derived simple analytical expressions that predict the critical wavelength for the onset of secondary rippling, the increase of the number of secondary ripples with increasing perturbation wavelength, and how the onset of the secondary rippling instability and the rippled surface pattern are affected by surface diffusional anisotropy and the action of an applied electric field. The conclusions of the theory are validated by systematic comparisons with results of self-consistent dynamical simulations of surface morphological evolution. Further quantitative analysis based on a wavelength selection nonlinear theory also will be presented.
Following the same approach, we have carried out a weakly nonlinear analysis of the surface morphological evolution of a heteroepitaxial thin film on a substrate. We have found that in addition to the SK instability, long-wavelength perturbations also can trigger nonlinear secondary instabilities, resulting in the splitting of a single quantum dot (i.e., 3D island) into multiple quantum dots of smaller sizes. The theory also can predict the critical wavelength of the initial surface perturbation for the onset of the secondary nonlinear instability. A systematic protocol of self-consistent dynamical simulations based on a fully nonlinear continuum model of epitaxial film driven morphological evolution were conducted to assess the validity of the theoretical results. The theoretical predictions were found to be in good qualitative and quantitative agreement with the simulation results. Our study sets the stage for exploiting nonlinear surface phenomena toward engineering tunable-size nanoscale surface features.
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