Synthetic Topological Quantum Matter in Nanostructured 2D Materials for Quantum Information Processing

Abstract

In this thesis, we contribute to understanding the electronic properties of two-dimensional materials, with a strong emphasis on graphene-based nanostructures, such as twisted multilayer graphene and triangular graphene quantum dots. The thesis is organized into six chapters, including an introduction and a conclusion.

In Chapter 2, we present the theoretical methods used to carry out the calculations in this thesis. We begin by defining the geometry of twisted graphene multilayers, which serves as the basis for the methods described later. We then introduce the general many-body problem and focus on various approximations commonly used to solve it. The tight-binding model is derived, and its application to moiré materials is demonstrated, followed by a comparison with the continuum approach. We then discuss mean-field methods, specifically Density Functional Theory (DFT) and Hartree-Fock. In the final section, we introduce methods for studying electron correlations and evaluate their applicability to specific systems. These include the Configuration Interaction (CI) method, hybrid approaches combining DFT and CI, and tensor network techniques.

Chapters 3-5 include the results of our work. We first focus on twisted graphene multilayers - magic angle twisted bilayer graphene and mirror-symmetric twisted trilayer graphene. In Chapter 3, we start with a Hofstadter's butterfly spectrum for the magic angle twisted bilayer graphene obtained using an ab initio based multi-million atom tight-binding model. A nanoribbon geometry is studied, and the quantum size effects for the sample widths up to 1 μ m are analyzed. For sufficiently wide ribbons, where the role of the finite geometry is minimized, we obtain and plot the Hofstadter spectrum and identify the in-gap Chern numbers by counting the total number of chiral edge states crossing these gaps. Subsequently, we examine the Wannier diagrams to identify the insulating states at charge neutrality. We establish the presence of three types of electronic states: moiré, mixed, and conventional.

We then move on to study trilayer structures in Chapter 4. Here, the electronic properties are described by a Hubbard model with long-range tunnelling matrix elements. The electronic properties are obtained by solving the mean-field Hubbard model. We obtain the band structure with characteristic flat bands and a Dirac cone. At charge neutrality, turning on electron-electron interactions results in a metallic to antiferromagnetic phase transition, for Hubbard interaction strength considerably smaller than in other graphene multilayers. We analyze the stability of the antiferromagnetic state against the symmetry-breaking induced by hexagonal boron nitride encapsulation and mirror symmetry-breaking caused by the application of electric fields that mix the Dirac cone with the flat bands. Additionally, we explore the topological properties of the system, revealing a hidden quantum geometry.

In Chapter 5, we focus on triangular graphene quantum dots. We present a method for probing the wave functions of a degenerate shell in such dots by introducing a localized substitutional impurity. Specifically, we demonstrate this approach using a triangular graphene quantum dot containing a nitrogen impurity. Starting from the analytical solution for the degenerate states of a pristine all-carbon triangular graphene quantum dot, we predict the structure of the zero-energy shell in the presence of an impurity. We show that the impurity enables selective probing of the wave functions at the carbon site where it is located. These predictions are validated through comparison with tight-binding and ab initio calculations and experimental results. We then study a triangular graphene quantum dot with armchair edges with a nitrogen impurity. We use a technique combining the Density Functional Theory and Configuration Interactions to analyze the energy spectrum and determine the effect the nitrogen impurity has on it. Additionally, we show that including excitations lowers the ground state energy for all considered geometries. Finally, we focus on bilayer triangular graphene dots with zig-zag edges and analyze the behaviour of the degenerate zero-energy shell as a function of size and the twist angle.

Chapter 6 includes conclusions and prospects for future work.