Honors College Thesis
 

Building a Basic Computational Model of Surface and Impurity States in a One-dimensional Solid

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https://ir.library.oregonstate.edu/concern/honors_college_theses/gf06g481n

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  • Learning about periodic and oscillatory systems is a huge step in an undergraduate's comprehensive understanding of physics and complex systems such as solids. However, understanding how specific changes to such a system affect aspects of the entire system can be difficult. The goal of this study is to create a learning tool to see how changes to the system alter its characteristics by using a simple one-dimensional solid of length N. In order to compare how the system's energy varied for different changes to the system, a reference chain, with on-site energy a=0 and nearest neighbor coupling terms β=-1, was established and its ground state energy and wave function were calculated. The allowed states and energies of the solid were studied as the on-site energies and coupling terms at one site were modified. Surface states and impurity states were observed to localize for potentials below -1 and 0, respectively. Relationships were found between the ground-state energy and the change to the system. The deviation ΔΕ in energy from the energy of the reference matrix was found to be inversely proportional to the change in the chain length: ΔΕ ∝ Ν⁻²; inversely proportional to the decay length d as the edge potentials were varied: ΔΕ ∝ d⁻²; and quadratically related to the decrease in nearest-neighbor coupling p in the chain: ΔΕ = 1.5p² + 0.0454p. Complex systems display a similar behavior; that they should be reproduced here indicates that this program is a powerful instructional tool.
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