Tight-Binding Electron Transport in Regions with Disorder and the Dirac-Separated Infinite Square Well

Open Access
Thornton, Stephen
Area of Honors:
Bachelor of Science
Document Type:
Thesis Supervisors:
  • Jorge Osvaldo Sofo, Thesis Supervisor
  • Prof. Richard Wallace Robinett, Honors Advisor
  • tight binding
  • condensed matter
  • electron transport
  • mathematica
Electron transport in one-dimensional systems can be an analytically tractable and immensely important field of study. To first-order approximation, for instance, carbon nanotubes can be modeled as one-dimensional chains of atomic sites on which an electron can reside. I have considered electron transport in one-dimensional systems under the tight-binding model, where only nearest-neighbor ``hopping'' of electrons between sites in a discrete chain are considered. In an infinite, homogeneous, perfect chain, the eigenstates are leftward and rightward traveling plane waves that represent electron transport through the chain. Much like in the continuous case, transmission and reflection coefficients can be calculated when these plane wave states are incident on a potential step. Resonances occur when there are interactions between the wave number and the width of the step (just as in the continuous case). A more challenging problem (that can be solved computationally) is to introduce randomly distributed disorder in the region of the step, and calculate the transmission and reflection coefficients for these types of systems. We solve the problem of the transmission and reflection coefficient of this disordered region numerically. I have written code in Mathematica to calculate the transmission and reflection coefficients as a function of incident electron energy given the width of the step and the magnitude of some normally distributed disorder. I find that for slightly disordered systems, the transmission resonances are maintained, but as the disorder increases in magnitude, the transmission probability falls to $0$ regardless of the incoming electron energy. Going to a more continuum-based model, a simple toy model for an electronic state in contact with a large reservoir could be important for understanding leakage of states in quantum computers. The problem is also mathematically interesting, with a measure-valued Hamiltonian governing the time evolution of states. I calculate the probability of finding an electron in some initial state of an infinite square well separated from a larger infinite square well by a Dirac potential as a function of time, and find that the initially confined states leak into the larger reservoir well. Finally, I go back to the tight binding model and look at a discrete analogue of the continuous system studied above: two finite chains with the degree of separation of the two chains given by a different hopping parameter. The eigenstates of the system are exactly solvable (they amount to diagonalizing a finite matrix) and the decoupling is apparent when when the coupling parameter approaches zero. The time evolution of these systems can also be easily (numerically) studied. Like the continuous case, the states leak from the initial left well into the right well, and the timescale of leakage varies inversely with the magnitude of the hopping integral between the chains.