Stability analysis of large time-stepping methods for epitaxial growth models

Numerical methods for solving the continuum model of the dynamics of the molecular beam epitaxy (MBE) require very large time simulation, and therefore large time steps become necessary. The main purpose of this work is to construct and analyze highly stable time discretizations which allow much larger time steps than those of a standard implicit-explicit approach. To this end, an extra term, which is consistent with the order of the time discretization, is added to stabilize the numerical schemes. Then the stability properties of the resulting schemes are established rigorously. Numerical experiments are carried out to support the theoretical claims. The proposed methods are also applied to simulate the MBE models with large solution times. The power laws for the coarsening process are obtained and are compared with previously published results

Thin film evolution equations from (evaporating) dewetting liquid layers to epitaxial growth

In the present contribution we review basic mathematical results for three physical systems involving self-organising solid or liquid films at solid surfaces. The films may undergo a structuring process by dewetting, evaporation/condensation or epitaxial growth, respectively. We highlight similarities and differences of the three systems based on the observation that in certain limits all of them may be described using models of similar form, i.e., time evolution equations for the film thickness profile. Those equations represent gradient dynamics characterized by mobility functions and an underlying energy functional. Two basic steps of mathematical analysis are used to compare the different system. First, we discuss the linear stability of homogeneous steady states, i.e., flat films; and second the systematics of non-trivial steady states, i.e., drop/hole states for dewetting films and quantum dot states in epitaxial growth, respectively. Our aim is to illustrate that the underlying solution structure might be very complex as in the case of epitaxial growth but can be better understood when comparing to the much simpler results for the dewetting liquid film. We furthermore show that the numerical continuation techniques employed can shed some light on this structure in a more convenient way than time-stepping methods. Finally we discuss that the usage of the employed general formulation does not only relate seemingly not related physical systems mathematically, but does as well allow to discuss model extensions in a more unified way

An unconditionally energy stable finite difference scheme for a stochastic Cahn-Hilliard equation

In this work, the MMC-TDGL equation, a stochastic Cahn-Hilliard equation is solved numerically by using the finite difference method in combination with a convex splitting technique of the energy functional. For the non-stochastic case, we develop an unconditionally energy stable difference scheme which is proved to be uniquely solvable. For the stochastic case, by adopting the same splitting of the energy functional, we construct a similar and uniquely solvable difference scheme with the discretized stochastic term. The resulted schemes are nonlinear and solved by Newton iteration. For the long time simulation, an adaptive time stepping strategy is developed based on both first- and second-order derivatives of the energy. Numerical experiments are carried out to verify the energy stability, the efficiency of the adaptive time stepping and the effect of the stochastic term.Comment: This paper has been accepted for publication in SCIENCE CHINA Mathematic

A Mathematical Model for Epitaxial Semiconductor Crystal Growth from the Vapor Phase on a Masked Substrate

Certain materials used in lasers are made by a process called epitaxial semiconductor crystal growth. In this report a mathematical model is developed for this growth process which occurs on a substrate at the junction between a masked region and exposed substrate in a vapor. This new model consists of two partial differential equations; one for the surface dynamics and one for the crystal growth on the exposed substrate. An analysis of the steady state solutions is furnished. Approximate solutions for time-dependent cases are found using two numerical methods. An asymptotic analysis is also carried out to determine transient solution behavior. The undesireable "bump" structure at the mask/substrate junction which has been observed experimentally is present in the solutions found by each method