Electronic and Optical Properties of Quantum Dots in Two-Dimensional Topological Insulators

Abstract

This thesis presents a theoretical study of the electronic and the optical properties of quasi two-dimensional quantum dots in topological insulators. A topological insulator is a semiconductor with an insulated bulk in which electronic states, localized at the edges of the material can be found. Two-dimensional topological insulators can be described by a simple k·p model given by the Bernevig-Hughes-Zhang (BHZ) Hamiltonian.

Topological insulators have two topologically different phases: trivial and nontrivial, and can exist in one of them. We show that the trivial versus non-trivial properties of the BHZ Hamiltonian are characterized by the different topologies that arise when mapping the in-plane wavevectors through the BHZ Hamiltonian onto a Bloch sphere. In the topologically non-trivial case, edge states are formed in the nanoribbon, disc, and square quantum dot geometries. The number of states appearing in and the size of the energy gap of these quantum dots are controlled by the spatial dimensions: width W, radius R, and lateral size a, respectively. The energy spectra of these quantum dots were found by using exact diagonalization techniques. An analytical solution for the edge states with zero-energy in the bulk energy gap of the nanoribbon was derived. From this, the decay length of the edge state into the center of the quantum dot, and its position with respect to the physical edge were determined. Furthermore, the transition from the non-trivial to the trivial phase was realized by tuning the compressive strain in the quantum dot, which resulted in the edge states disappearing from the energy gap. Thus, straining HgTe topological insulator quantum dots can be used to control the quantized spin-Hall conductance via the emergence and disappearance of edge states. These findings can be used as a design model of a quantum strain sensor based on strain-driven transitions in a HgTe quantum dot in a topological insulator. Finally, the optical response of the bulk BHZ model to circularly polarized light was studied, yielding absorption coefficients with the strength dependent on the energy of the photons. From this study, it was found that circularly polarized light is selectively absorbed depending on the spin of the topological insulator. Building on this work may allow for an understanding of the role of edge states in optical transitions, and could enable a study of excitations in HgTe in quantum dots in topological insulators.