| Abstract |
In this work, we have examined the dynamics of an initially localized electron (or
excitation), propagating on nonlinear one-dimensional, two-dimensional square and
two-dimensional hexagonal (honeycomb) lattices. In order to do that, we solved the
Discrete Non-Linear Schrodinger (DNLS) equation numerically.
In the first chapter we introduced the Discrete Non-Linear Schrodinger (DNLS)
equation and the statistical figures that would help us investigate the dynamics of the
different lattices. The statistical figures that we used are the Mean Square Displacement (MSD), the Participation Number (Pr) and the Long-Time Average Probability at the initial site (LTAP).
In chapters two and three we used the DNLS equation in the one-dimensional lattice (chain) and the two-dimensional square lattice. First, we solved the Linear part of the DNLS analytically, we produced the solution for the Probability density of the wave-function and we found the solution for the Mean Square Displacement
(MSD) (Appendix B). Afterwards, we used Mathematica to verify the result for the MSD, and we produced the solution for the Participation Number (Pr). We compared the numerical results with the analytical ones. Next we solved the DNLS equation numerically and we obtained the value of the non-linear parameter where the phenomenon of self-trapping appears. Finally we solved the DNLS equation numerically using a term of disorder, and we estimated the value of the non-linear parameter and the region of the values that the disordered term takes, in order to accomplice absence of diffusion.
In chapter four, we solved the DNLS equation numerically in the two-dimensional
honeycomb lattice. We, first, solved the linear part of the DNLS equation, and we
calculated the Mean Square Displacement (MSD) and the Participation Number (Pr).
Then we solved the whole DNLS equation and we obtained the value of the non-linear
parameter where the phenomenon of self-trapping appears. Finally, we added a disordered
term and we estimated the value of the non-linear parameter and the region of the disordered term, and we observed absence of diffusion.
In chapter five, we compare the results that we obtained in every lattice. At first,
we compare the Mean Square Displacement (MSD) and the Participation Number
(Pr) in all three lattices. Afterwards, we compare the Long-Time Average Probability at the initial site (LTAP) in all three lattices and we compare the values where the phenomenon of self-trapping appears in each of them.
|
Numerical statistics in honeycomb lattices
