Bachelor Theses

We numerically investigate a quantum Heisenberg model with ferromagnetic 2-spin exchange and a 4-spin plaquette interaction on a square lattice and compare it to the corresponding Ising model. We use the finite temperature Lanczos method (FTLM) algorithm to approximate the lowest eigenvalues and eigenvectors and use these to calculate observables. We construct the phase diagrams for different values of the anisotropy parameter, which serves the purpose of interpolating between the quantum Heisenberg model and the Ising model. We then proceed to investigate the phases and phase transitions. We also show the spin-spin correlation in the different phases. We find that this system is very rich and non-trivial, and that the quantum Heisenberg model exhibits interesting phases not encountered in the Ising model, like a 1^st order phase transition from a ferromagnet to an antiferromagnet at zero and low temperature.

The electronic fluid in metals and semi-conductors shows a reasonable resemblance to a gas in the ionized state, which is often referred to as a plasma. The optical properties of the plasma are of ever increasing interest, due to the applicability in opto-electronics and other real-world devices. An essential property is the natural oscillation spectrum of the plasma, which is often quantised by the means of the quasi-particles called plasmons.
This work presents an extensive study on the plasmonic structure of both monolayer and (twisted) AA-stacked bilayer graphene. The real-space lattices are modelled as finite hexagonal supercells, and are reviewed within a tight-binding approximation. The Coulomb interactions are modelled according to an Ohno-interpolation, which is fitted to ab-initio data provided by earlier publications. With the application of a real-space RPA-algorithm, the full dielectric and polarisability matrices are calculated. These matrices are then used to review both the plasmonic excitations in real- and momentum space, by the means of real-space and Fourier-space electronic loss spectra, and real-space plasmonic eigenmodes.

In this thesis an algorithm is proposed to numerically solve the Poisson equation in inhomogeneous environments and approximate the boundary conditions. The Poisson equation is expressed as a system of linear equations using the finite difference method and this system of linear equations is solved using the biconjugate gradient method. The solver is tested for convergence against known exact solutions of the Poisson equation and methods to approximate the boundary conditions are discussed. A brute force way is to use logarithmically spaced grids with a large extent but this is slow and not numerically stable. A new method that approximates the boundary conditions is proposed called the nested grid method. This new method is faster and more numerically stable and converged for all systems tested.

Graphene, which is a single-layer carbon nanosheet, has shattered the world of science in the past 20 years with its incredible physical properties. Due to graphene's single-atom thin honeycomb-like structure some interesting characteristics arise, like a very strong bond between neighbouring atoms and a high electron mobility within this two-dimensional lattice. This thesis deals with the collective electron oscillations in graphene, the so-called plasmons or plasmonic excitations, and their dependence on different environments. This has been done based on first principle cRPA calculations, in order to derive a continuous model for the screened Coulomb interaction in real-space for free-standing graphene. This model has been adjusted to graphene embedded in hexagonal Boron-Nitride (h-BN) to account for environmental screening effects such that it properly corresponds to the ab initio data provided for h-BN. Using the same model, it is possible to change parameters for the dielectric environment to see how the plasmonic excitations evolve. Combining this Coulomb interaction model with a simple tight-binding Hamiltonian, it is possible to determine the dielectric function and with it the plasmonic excitations including their dependence on the environment.

In this Bachelor Thesis, we give a minimalistic description of magnetism in
monolayer CrI3. This description is constructed from two seperate exchange interactions. We combine the antiferromagnetic Kugel-Khomski direct exchange,
with the ferromagnetic Goodenough-Kanamori indirect exchange. In contrast
to the individual models, the full model exhibits a magnetic phase transition.
The near additive interplay between both exchange mechanisms allows for an
insightful qualitative description of the magnetic properties.

In this project the scattering rates,  which is equal to the inverse lifetime, of acoustic phonons was determined in graphene at 0 K. A Hamiltonian was found within a continuum elasticity description, describing the propagation of phonons and the interaction between them at large wavelength in graphene. By determining the transition probability of a phonon decay process, the rate of this phonon decay could be determined by using Fermi’s Golden Rule. With this decay rate, it could be determined if the phonon is well defined. It was found that for both longitudinal and transversal acoustic phonons decaying in two flexural acoustic phonons, the scattering rate is proportional to the initial momentum of the phonon and that both phonons are well defined at 0K, by considering only this decay process.

During this project, a theoretical model to visualize plasmonic dispersions in a 2D heterostructure was created using the WFCE approach to handle the Coulomb screening. With this, the effect of varying 2D layer height and metallic, interband, and anisotropic screening on plamonic exciations was studied. It was found that increasing the screening in any of the screening channels dampens the plasmonic excitation. Increasing the 2D layer height has the same effect. Lastly, it was found that the screening resulting from anisotropic substrate layers, can results in an anisotropic plasmonic excitation in an isotropic 2D metal.

In this project we computed the antiferromagnetic resonance frequency of a honeycomb lattice and studied the effect of electric currents on this frequency. We have done this by studying the Heisenberg model on a honeycomb lattice where neighboring spins are coupled through a negative exchange interaction. The effect of current can be included by an s-d energy term. By writing the energy of the system containing both localized spins and conducting spins, we can derive the equations of motion, called landau-lifshitz equations. By linearizing these equations one obtains the resonance frequency. The electric current gives rise to spin-orbit torques that enter the equations of motion, and thus affect the resonance frequency.

Perovskites are crystals with chemical formula ABX_3 and are normally not conductive. However, at the boundary of two perovskites, in my case, SrTiO_3 and LaAlO_3 there is a conductive layer. To investigate this phenomena I calculated the spin textures of SrTiO_3 at the boundary.

Bijbehorende publicatie:
https://journals.aps.org/prb/abstract/10.1103/PhysRevB.97.205434