Research Summary

For a more detailed description of my work, please see my Research Statement. There I describe my current research and briefly touch on future directions. For more information about specific directions I wish to take my work, see the associated Research Statement Appendix.

I work on Kinetic Monte Carlo (KMC) simulations of complex, self-assembled processes at the atomistic scale. I have developed a flexible KMC model and efficient implementation, which I have applied to several systems of interest to materials scientists. Among the systems I have simulated are:

  • Liquid Ga droplet epitaxy and crystallization by As flux to create GaAs nanostructures. Our simulations are able to capture the nanostructural dependence on temperature and As flux seen in experiments, reproducing quantum dot and nanoring formation. Simulations also predict the existence Ga/GaAs core/shell structures in high temperatures and As flux by two seperate mechanisms: first, polycrystalline nucleation at the liquid/solid interface and second, an Mullins-Sekerka instability of the GaAs growth front. We then developed an analytical model describing the exact nanostructural dependence on experimental and energetic parameters, which agrees well with our simulations.
  • Nanowire growth by the Vapor-Liquid-Solid (VLS) method. These simulations feature a three-component system representing each of the three phases in addition to catalyzed reaction events that transform the vapor phase into the solid one when inside the liquid catalyst. I studied of the effect of certain energy parameters on nanowire growth modes, exhibiting a control on nanowire growth direction by means of a parameter describing the mobility along the liquid-solid interface.

In order to simulate such complicated systems in a reasonable amount of time, efficient algorithms must be used. I have developed a fast hash table-based caching procedure that removes redundant calculations in the simulation. The hash table procedure requires the construction of an efficient hash function which was obtained through a simulated annealing search focusing on reducing the number of collisions. Optimal hash functions minimized the amount of chaining in the hash table, resulting in a 10x speed up when the full caching procedure was implemented. I have also demonstrated how the hash table may be utilized as the main data structure, resulting in good algorithmic performance.

Additionally, my advisor and I have developed a statistical mechanical theory of the Solid-on-Solid (SOS) model, developing a notion of a local equilibrium that approximates the evolution of the underlying continuous time Markov chain resulting from the KMC simulation of an SOS system. I provided numerical evidence suggesting the accuracy of the approximation.

In the future, I would like to continue my study of KMC, both in broadening the application of our model to other systems and in further developing a theoretical framework of KMC. Among the possible directions include the simulation and analysis of grain growth and systems where strain effects play a significant role. I also would like to study an algebraic/combinatoric description of the different dynamical regimes of the SOS-KMC model.

My working notes can be found here.

Publications

Presentations

Teaching

I am currently not teaching. In the past, I have taught several sections of pre-Calculus, differential Calculus, and multivariable Calculus.

For more information about my teaching philosophy, please see my Teaching Statement.

Academic Service

  • Graduate Student Representative Mathematics department computer committee Fall 2012 - present University of Michigan
  • Co-organizer Learning seminar in mathematics and materials science Fall 2012 - present University of Michigan
  • Co-founder SIAM student chapter Fall 2009 University of Michigan
  • Organizer Student applied and interdisciplinary seminar Fall 2008 - Spring 2009 University of Michigan

Contact Information

You may contact me at kgre@umich.edu. My office is at East Hall 1860. My mailing address is:

530 Church Street Ann Arbor, MI 48109-1043