Beyond the Child-Langmuir limit.

This article presents a new formulation of the solution for fully nonlinear and unsteady planar flow of an electron beam in a diode. Using characteristic variables (i.e., variables that follow particle paths) the solution is expressed through an exact analytic, but implicit, formula for any choice of incoming velocity v(0), electric field E(0), and current J(0). For steady solutions, this approach clarifies the origin of the maximal current J(max), derived by Child and Langmuir for v(0) = 0 and by Jaffe for v(0) > 0. The implicit formulation is used to find (1) unsteady solutions having constant incoming flux J(0) > J(max), which leads to formation of a virtual cathode, and (2) time-periodic solutions whose average flux exceeds the adiabatic average of J(max).

Optimal capping layer thickness for stacked quantum dots.

We study the effect of strain on the vertical and lateral self-organization of nanoscale patterns and stacked quantum dots during epitaxial growth. The computational approach is based on the level set method in combination with an atomistic strain code. Strain changes the energetics of microscopic parameters during growth, and thus determines the nucleation sites and the growth of islands and dots. Our results show that strain can lead to vertical alignment as well as lateral organization. Moreover, our simulations suggest that there is an optimal thickness of the capping layer to get the best alignment and most uniform size distribution of stacked quantum dots, and that its variation can be used to control the formation of interesting structures.

Level set approach to reversible epitaxial growth.

We generalize the level set approach to model epitaxial growth to include thermal detachment of atoms from island edges. This means that islands do not always grow and island dissociation can occur. We make no assumptions about a critical nucleus. Excellent quantitative agreement is obtained with kinetic Monte Carlo simulations for island densities and island size distributions in the submonolayer regime.

Atomic size effects in continuum modeling.

Continuum modeling of many physical systems typically assumes that the spatial extent of an atom is small compared to the quantities of interest and can therefore be neglected. We show that this is valid only asymptotically. For many applications of practical interest, the spatial extent of a discrete atom cannot be neglected. We have developed a model for the description of epitaxial growth based on the levelset method, and find that we can accurately predict quantities such as the island densities, if we implement boundary conditions in a region with atomic width, rather than just on a line without any spatial extent. Only in the limit of very large islands and island spacings can this be neglected.

Kinetic model for a step edge in epitaxial growth.

A kinetic theory is formulated for the velocity of a step edge in epitaxial growth. The formulation involves kinetic, mean-field equations for the density of kinks and "edge adatoms" along the step edge. Equilibrium and kinetic steady states, corresponding to zero and nonzero deposition flux, respectively, are derived for a periodic sequence of step edges. The theoretical results are compared to results from kinetic Monte Carlo (KMC) simulations of a simple solid-on-solid model, and excellent agreement is obtained. This theory provides a starting point for modeling the growth of two-dimensional islands in molecular-beam epitaxy through motion of their boundaries, as an alternative to KMC simulations.