Some New Epitaxial Thin Film Models With Numerical Simulations

In Chapter 1, we introduce briefly the process of establishing the phenomenolog-ical models for epitaxial thin film growth mathematically and some related numerical methods.In Chapter 2, we describe in detail the phenomenological background of our 3D continuum evolution equations. Evans, Thiel and Bartelt, in 2006, developed a better model by interpolation of the extreme cases when the gradient of the height function h is extremely large or extremely small. So it’s roughness has two different growth rates which is a very special phenomenon. The model of Evans, Thiel and Bartelt is flawed by the fact that one of the extreme cases is linear, so cannot model long-term behavior. The important contribution in this dissertation is to replace the linear equation with one which is nonlinear and more realistic. Furthermore, well-posedness of the weak solutions of the natural interpolation model and the new model we proposed is obtained.In Chapter 3, convex-concave splitting method is applied to derive their lin-ear numerical schemes. We also present the important properties of our numerical schemes such as unconditional solvability and stability, local-in-time convergence, and long time behavior of the schemes. Numerical simulation results show the mor-phological instability in the rough-smooth-rough pattern. Furthermore, roughness of the natural interpolation model and the new model we proposed shows two different growth rates. This is a special phenomenon which with-slope-selection model and without-slope-selection model do not have. Then, we give the numerical analysis of the saturation time. Numerical results show that our numerical schemes are stable and effective.In Chapter 4, we use the well-known with-slope-selection model and without- slope-selection model as an example and apply the idea of convex-concave splitting method to them to get higher-order numerical schemes.In Chapter 5, we discuss about other interesting numerical schemes applied on the with-slope-selection model by using a Lagrange multiplier to avoid managing a nolinear term and obtain linear unconditional stable and uniquely solvable alternate numerical schemes with their error estimation.In Chapter 6, we make a brief conclusion about this dissertation.