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Theory of Random Matrices and Elasticity Applied to Nanowires and Mechanical Vibrations

Grönqvist, Johan LU (2010)
Abstract
This thesis presents work in several areas relating to Random Matrix Theory and Elasticity. It contains 4 papers presenting work on different issues.



Paper I concerns correlations between eigenvalues in random matrices of real symmetric (GOE) or quaternion real (GSE) form. It presents a calculation showing that correlations between arbitrarily many eigenvalues can be obtained from averages of a two-point quantity. This result extended a previous result for Hermitian matrices (GUE), and has later been rederived in more general formulations by others.



Paper II presents a study of time evolution using a random matrix model. An experiment was performed by Weaver and Lobkis along with an analysis based on a... (More)
This thesis presents work in several areas relating to Random Matrix Theory and Elasticity. It contains 4 papers presenting work on different issues.



Paper I concerns correlations between eigenvalues in random matrices of real symmetric (GOE) or quaternion real (GSE) form. It presents a calculation showing that correlations between arbitrarily many eigenvalues can be obtained from averages of a two-point quantity. This result extended a previous result for Hermitian matrices (GUE), and has later been rederived in more general formulations by others.



Paper II presents a study of time evolution using a random matrix model. An experiment was performed by Weaver and Lobkis along with an analysis based on a discretized wave-equation. We set up a random matrix model of the experiment, and showed that the time-evolution, including the localization effect, was well reproduced by the very generic model.



Papers III and IV present work on elasticity of nanowires using an atomistic model.

In Paper III, a study of static properties is presented. We computed strain fields in nanowires using different models. The motivation was doubt that the continuum model was sufficient at the small length scales involved, and we used both continuum models, based on linear elasticity, and an atomistic model, based on the Valence Force Field potential, for both finite and infinite nanowires, to confirm that the continuum model could reproduce the effects of the atomic structure. We found that the small scale of the system was no problem for the continuum models used.

In Paper IV, we present phonon dispersion relations and phonon mode geometries obtained by the same atomistic model as used in Paper III. The deviations from bulk values are either explained using other theories, or related to the systematic errors appearing due to the simplicity of the model. We study both wurtzite and zinc blende structured cores-shell wires for core-radii varying form 0 to the full nanowire size. (Less)
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author
supervisor
opponent
  • Professor Berggren, Karl-Fredrik, Linköping University, Linköping
organization
publishing date
type
Thesis
publication status
published
subject
defense location
Lecture hall F, Department of Physics, Sölvegatan 14A, Lund University Faculty of Engineering
defense date
2010-12-06 13:30
ISBN
978-91-7473-056-2
language
English
LU publication?
yes
id
d2ca0e24-7f1a-4d47-a668-84f4bcef402d (old id 1714709)
date added to LUP
2010-11-09 09:36:38
date last changed
2016-09-19 08:45:19
@phdthesis{d2ca0e24-7f1a-4d47-a668-84f4bcef402d,
  abstract     = {This thesis presents work in several areas relating to Random Matrix Theory and Elasticity. It contains 4 papers presenting work on different issues.<br/><br>
<br/><br>
Paper I concerns correlations between eigenvalues in random matrices of real symmetric (GOE) or quaternion real (GSE) form. It presents a calculation showing that correlations between arbitrarily many eigenvalues can be obtained from averages of a two-point quantity. This result extended a previous result for Hermitian matrices (GUE), and has later been rederived in more general formulations by others.<br/><br>
<br/><br>
Paper II presents a study of time evolution using a random matrix model. An experiment was performed by Weaver and Lobkis along with an analysis based on a discretized wave-equation. We set up a random matrix model of the experiment, and showed that the time-evolution, including the localization effect, was well reproduced by the very generic model.<br/><br>
<br/><br>
Papers III and IV present work on elasticity of nanowires using an atomistic model.<br/><br>
In Paper III, a study of static properties is presented. We computed strain fields in nanowires using different models. The motivation was doubt that the continuum model was sufficient at the small length scales involved, and we used both continuum models, based on linear elasticity, and an atomistic model, based on the Valence Force Field potential, for both finite and infinite nanowires, to confirm that the continuum model could reproduce the effects of the atomic structure. We found that the small scale of the system was no problem for the continuum models used.<br/><br>
In Paper IV, we present phonon dispersion relations and phonon mode geometries obtained by the same atomistic model as used in Paper III. The deviations from bulk values are either explained using other theories, or related to the systematic errors appearing due to the simplicity of the model. We study both wurtzite and zinc blende structured cores-shell wires for core-radii varying form 0 to the full nanowire size.},
  author       = {Grönqvist, Johan},
  isbn         = {978-91-7473-056-2},
  language     = {eng},
  school       = {Lund University},
  title        = {Theory of Random Matrices and Elasticity Applied to Nanowires and Mechanical Vibrations},
  year         = {2010},
}