Metropolis Algorithm and Nanometer-Scale Patter...Team: 5 School: Albuquerque Academy Area of Science: Material Science, Computer Science
Interim: Metropolis Algorithm and Nanometer-Scale Pattern Formation
Definition of the Problem:
When chemicals are layered onto a surface, they form nanometer-scale patterns to reduce total free energy. This phenomenon is very important in nanotechnology because it will allow us to essentially shrink electronics to the nanometer scale, which could potentially lead to faster computers. The issue with shrinking is that it is hard to predict how circuits evolve over time under certain conditions. Such evolution can potentially be damaging. We plan on writing a computer program to simulate the evolution under certain conditions. The program will use the Metropolis Algorithm, a sort of Monte Carlo algorithm that will not only speed up calculations but will also allow us to incorporate many different conditions (e.g. different energies and temperature effects) without having to derive a new set of differential equations, which would have been the case in our previous project. Our previous project involve a set of partial-integral differential equations. In fact, given those equations, it may actually be impossible to derive any equations under more complicated conditions.
Metropolis Algorithm:
Reducing free energy is a really important part of the project and essentially governs the pattern formation. Free energy reduction is the basis of the Metropolis Algorithm. Suppose there are two states A and B with energies E1 and E2, respectively. B is a possible future state of the system. The algorithm compares these two energies. If E2 is less than E1, then the system will move to state B. If E2 is greater than E1, then a random is generated. If that random number is less than exp(-(E2-E1)/kT), where k is Boltzmann’s constant and T is the temperature, then the system will also move to state B. Otherwise, it remains at state A.
Progress:
As of right now, we have a working Metropolis Algorithm. We still plan on implementing the Parallel Metropolis Algorithm, which allows the use of multiple processors, to help speed up large simulations. We are also in the process of looking for experimental data to verify that our program does in fact produce the correct results.
Expected Results:
We expect ultimately to have a program that can qualitatively describe our system. Currently we only have a very small number of simulations and expect to get more over time. After we verify our program, we plan on simulating actual situations involving various circuits and chemical properties (e.g. what happens when you lay down dipoles?).
Citations:
Chen, L.-Q. and Shen, J., Applications of semi-implicit Fourier spectral method to phase field equations, Comp. Phys. Commun. 108 (1998) 147–158.
Lu, W., Theory and Simulation of Nanoscale Self-Assembly on Substrates. J. Comp. Theoretical Nanoscience 3.3 (2006) 342-361.
Lu, Wei, and Dongchoul Kim. “Dynamics of Nanoscale Self-Assembly of Ternary Epilayers.” Microelectronic Engineering 75 (2004): 78-84. Web. 28 Nov. 2007. .
---. “Simulation on Nanoscale Self-Assembly of Ternary-Epilayers.” Computational Materials Science 32 (2005): 20-30. Web. 8 Dec. 2007. .
Ratsch, C. and Venables J. A., Nucleation Theory and the early stages of thin film growth, J. Vac. Sci. Technol. A, Vol. 21, No. 5 (2003) S96-S109
Also, we thank Professor David Dunlap at the University of New Mexico with his help.
Team Members: Michael Wang
Jack Ingalls
Sponsoring Teacher: Jim Mims Mail the entire Team |
