A Macroscale Perspective of Near-equilibrium Relaxation of Stepped Crystal Surfaces

Abstract

Crystal surfaces serve a crucial function as building blocks in small

electronic devices, especially for mobile communications technology and

photovoltaics. In the history of computing, for example, a crucial

innovation that hastened the demise of vacuum tube computers was the

etching of patterns on surfaces of semiconductor materials, which led to

the integrated circuit. These early procedures typically worked with

the materials at very high temperatures, where the thermally rough

surface could be modeled from the perspective of continuum

thermodynamics. More recently, with the drive towards smaller devices and

the accompanying reduction of the lifetime of surface features,

manufacturing conditions for the shaping of crystal surfaces have shifted to

lower temperatures. At these lower temperatures the surface is no longer

rough. In order to describe an evolving surface under typical experimental

conditions today, we need to consider the processes that take place at the

nanoscale.

Nanoscale descriptions of surface evolution start with the motion of

adsorbed atoms (adatoms). Because of their large numbers, the

concentration of adatoms is a meaningful object to study. Restricted to

certain bounded regions of the surface, the adatom concentration

satisfies a diffusion equation. At the boundaries between these

regions, the hopping of adatoms is governed by kinetic laws. Real-time

observation of these nanoscale processes is difficult to achieve, and

experimentalists have had to devise creative methods for inferring the

relevant energy barriers and kinetic rates. In contrast, the real-time

observation of macroscale surface evolution can be achieved with simpler

imaging techniques. Motivated by the possibility of experimental

validation, we derive an equation for the macroscale surface height,

which is consistent with the motion of adatoms. We hope to inspire future

comparison with experiments by reporting the novel results of simulating the

evolution of the macroscale surface height.

Many competing models have been proposed for the diffusion and kinetics

of adatoms. Due to the difficulty of observing adatom motion at the

nanoscale, few of the competing models can be dismissed outright for

failure to capture the observed behavior. This dissertation takes a few

of the nanoscale models and systematically derives the corresponding

macroscopic evolution laws, of which some are implemented numerically to

provide data sets for connection with experiments. For the modeling

component of this thesis, I study the effect of anisotropic adatom diffusion

at the nanoscale, the inclusion of an applied electric field, the desorption

of adatoms, and the extension of linear kinetics in the presence of step

permeability. Analytical conjectures based on the macroscale evolution

equation are presented. For the numerical component of this thesis, I

select a few representative simulations using the finite element method to

illustrate the most salient features of the surface evolution.