A whole new class of mathematical problems occurs in nanotechnology. Part of the reason for this is that the traditional approach of using continuum models of the materials is no longer sufficient. In this traditional approach, the fields (e.g., stress and strain) are determined by solving known partial differential equations with known coefficients. Boundary conditions are understood and many computational methods have been developed to solve these difficult, but standard, problems. However, for many problems in nanotechnology, the scales are so small (sometimes just a few nm), that continuum models are no longer applicable. Hence the challenge is to bridge the information occurring at the atomic (microscopic) scale with the behavior on the macroscopic scale. This macroscopic scale can be as small as 10’s of nanometers for structures of current interest.
Our research has involved using information from either ab-initio or molecular-dynamics calculations into new continuum theories valid at this macroscopic scale. For example, my graduate student Christopher Retford, developed a molecular dynamics code to study edge energies along quantum wires. In the figure below, an example is given of a Ge quantum wire and wetting layer resting along a Si substrate. The individual atoms are shown and the reconstruction of the interface can be observed. As another example consider the problem of determining the shape of a nano-scale Ge island resting on a Si substrate, i.e., a quantum dot. Because of the lattice misfit between the Ge and Si crystal lattice, a misfit strain is developed which governs the shape of the island. At these small scales, the surface energy of the interfaces is strain dependent and it is necessary to determine the shape and evolution of the island. A multi-scale computational approach is necessary which puts information from the microscopic scale into a macroscopic scale model, i.e., the PDE’s of classical elasticity modified to account for the proper surface energy.
UNIVERSALITY AND SELF-SIMILARITY IN PINCH-OFF OF RODS BY BULK DIFFUSION
Here we examine an unexplored class of topological singularities where interface motion is controlled by the diffusion of mass through a bulk phase. We show theoretically that the dynamics are determined by a universal solution to the interface shape (which depends only on whether the high diffusivity phase is the rod or the matrix) and materials parameters. We find good agreement between theory and experimental observations of pinching liquid rods in an Al–Cu alloy. The universal solution applies to any physical system in which interfacial motion is controlled by bulk diffusion, from the break-up of rod-like reinforcing phases in eutectic composites 13–16 to topological singularities that occur during coarsening of interconnected bi-continuous structures 17–20,thus enabling the rate of topological change to be determined in a broad variety of multi-phase systems.
ROLE OF STRAIN-DEPENDENT SURFACE ENERGIES IN GE/SI(100) ISLAND FORMATION
Formation energies for Ge/Si(100) pyramidal islands are computed combining continuum calculations of strain energy with first-principles-computed strain-dependent surface energies. The strain dependence of surface energy is critically impacted by the presence of strain-induced changes in the Ge{100} surface reconstruction. The appreciable strain dependencies of rebonded-step {105} and dimer-vacancy-line-reconstructed {100} surface energies are estimated to give rise to a significant reduction in the surface contribution to island formation energies.