# Bachelor Theses

**Sahel Katawazi**(July 2023)*Quantum Synchronization and Entanglement calculations with the Lindblad equation (pdf, 794 kB)*

Synchronization is a phenomenon where two or more oscillators, which start off without the same frequency, eventually oscillate with the same frequency. Such a phenomenon has also been discovered in a quantum scenario. In this thesis we will describe how a simple system of two quantum interacting spins coupled to a magnon bath may evolve in an external time dependent magnetic field. We derive the corresponding Lindblad equation and analyze quantum spin synchronization and the evolution of quantum spin entanglement. Such calculations might have some interesting consequences for spintronics and quantum computing.

**Sander Leisink**(2021-2022)

Statistical physics of phase glasses (XY rotor models)*Thesis not available*

**Pim Coenders**(2021-2022)**Mapping the phase diagram of the quantum Heisenberg model with 4-spin exchange (pdf, 1,6 MB)**

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.**Samber Bastiaansen**(February 2022)*Plasmonic Excitations in Twisted bilayer Graphene (pdf, 5,9 MB)*

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.**Casper Pijnenburg**(December 2021)*Poisson solver for layered materials in inhomogeneous dielectric environments*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.

**Rens Theunissen**(November 2021)This thesis describes a minimalistic formulation for the calculation of the Hopf index for a three dimensional magnetic lattice. This formulation is achieved by approximating the lattice as a divergenceless field and calculating the necessary derivatives using the finite difference method. This calculation is only accurate but the research shown in this thesis serves as a good starting point for further development of the calculation*Computing the Hopf index for a magnetic lattice***.**(August 2021)**Adrián Sousa-Poza***Plasmonic Excitations in Mono-Layer Graphene (pdf, 7,4 MB)*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.

**Wietze Huisman**(July 2021)## Magnetism in CrI3 (pdf, 1 MB)

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.**Jesse Vos**(June 2021)## Magnetization dynamics in a honeycomb ferromagnet (pdf, 317 kB)

In this project we study several magnetic properties in a honeycomb lattice that displays antiferromagnetic properties. These magnetic properties are the conductivity, the spin-orbit torque, and the Gilbert damping. We also want to see how small perturbations to the lattice affect these three properties. This was done by constructing the Green’s functions for such a lattice, applying perturbation theory, and deriving a response matrix that contains the matrix expressions for those three properties. These expressions are very complex, and therefore we only look at the limits for these matrices in the cases of very small and very large spin-orbit and exchange interactions.

**Robin Smeets**(June 2021)## Energy Spectra: Dynamical System Approach (pdf, 1,8 MB)

Within this thesis multiple ways of determining the energy spectrum of electrons in a fractal and quasi-periodical lattice were studied. Inspired by analytical methods for certain 2D fractals, we developed a new way of determining the energy spectrum of the Schrödinger equation for any random fractal/quasi-periodical 1D potential by means of trace maps and a resulting dynamical system. This dynamical system was then studied for a few interesting specific cases and generalised for any random fractal/quasi-periodical 1D potential. This system was even further generalised by allowing the addition of a first order derivative in the Schrödinger equation.

**Femke Verheijen**(June 2021)

**Phonon Damping in Graphene**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.

**Anne Riewald**(July 2020)

**Theoretical Modeling of Plasmon Dispersions in 2D Heterostructures**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.

**Mohammad Hashem Jabr (December 2019)**## Antiferromagnetic resonance in a honeycomb lattice (pdf, 1,3 MB)

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.

**Mark Beijer**(August 2018)

## Spin textures in SrTiO3/LaAlO3 heterostructures

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.

**Lele Fang**(July 2018)

## Effect of damping on micromagnetic dynamics (pdf, 1,6 MB)

The dynamics of micromagnetic moments are described by the Landau-Lifshitz equation. This equation contains two terms: one describing the precession and the other describing damping. Through numerical simulations we studied the effects of (an)isotropic damping on the dynamics of domain wall movements in 2D ferromagnetic planes. We found that the anisotropy played no significant role on the dynamics in the systems we applied it to.

**Yann in t Veld**(July 2018)

## Using an effective hopping parameter to approximate the extended Hubbard model (pdf, 1,1 MB)

In this bachelor project the extended Hubbard model is mapped to an effective Hubbard model and applied to a graphene lattice. This was done via the Peierls-Feynman-Bogoliubov variational principle and via the Hartree-Fock approximation. Simulations on the effective Hubbard model in graphene were performed using Quantum Monte Carlo methods, which made it possible to compare the two mappings in various environments. We found that in both mappings the non-local Coulomb interactions effectively increase the hopping parameter. We also found that the Hartree-Fock approximation only worked quantitatively for U and t around zero. The mapping via the variational principle showed better quantitative results, also for higher U and t.

**Tom Westerhout**(August 2017)

## Plasmons in Fractals (pdf, 6,6 MB)

Recent progress in the fabrication of materials has made it possible to create arbitrary non-periodic two-dimensional structures in the quantum plasmon regime. This paves a way for exploring the quantum plasmonic properties of electron gases in complex geometries. In this work we study systems with a fractal dimension. We calculate the full dielectric functions of a Sierpinski carpet and show that the Sierpinski carpet has a dispersion comparable to a square lattice.

Bijbehorende publicatie:

https://journals.aps.org/prb/abstract/10.1103/PhysRevB.97.205434**Jacqueline Zeitler**(August 2017)

## Atomistic simulations of the shape of bubbles in graphite

Inspired by recent experimental findings, we have performed atomistic simulations of bubbles in graphene. The bubbles are formed between two sheets of graphene by trapping hydrogen molecules within them. Starting with an accurate atomistic potential and using Monte Carlo simulations we calculate the shape of bubbles formed in a graphene sheet on top of a bulk of graphite with hydrogen in between. In preliminary results we find that the aspect ratio of bubbles formed in graphene depends on the number of hydrogen molecules in the bubble but not on temperature. Further simulations will have to be carried out to substantiate these results.

**Maxime Gidding**(August 2017)

## Electrostatic screening in layered 2D materials - Applied to graphene and phosphorene (pdf, 3,5 MB)

We investigated the electrostatic screening effects in layered 2D materials using a tight-binding model. The behaviour of the electrons is calculated self-consistently using numerical techniques. This method was applied to multilayers of graphene and phosphorene. For graphene multilayers it was found that the Fermi energy is dependent on the substrate carrier densities, which is consistent with previous theoretical results. For phosphorene multilayers a similar behaviour was found, however it was found that for this material the band gap will suddenly close when the electron density in the substrate reaches a critical value. This behaviour is not observed when screening effects between layers of the material are neglected. Lastly, it was found that the value of the critical electron density can be changed by altering the dielectrics of the material or the environment.

**Luuk Coopmans**(July 2016)

## Temperature dependence of the phonons of graphene (pdf, 3,1 MB)

Monte Carlo simulations have verified the scaling behaviour of several elastic moduli of graphene which was predicted by the theory of membranes. However, the temperature dependence of the bending rigidity found in this type of calculations is not fully explained. Since the bending rigidity also determines the dispersion of out-of-plane acoustic modes, we calculated the phonon modes of graphene directly and compared them to the results of the theory of membranes. We used a new method based on Molecular Dynamics Simulations and Fourier Analysis to obtain the phonon modes of graphene at several different temperatures and extracted the temperature dependence of the bending rigidity. We can conclude that the bending rigidity increases with temperature, which confirms the results found by Monte Carlo simulations.

**Nick Brummans**(June 2015)

## Hydrogen Covered Graphene (pdf, 5,1 MB)

On room temperature graphene is rippled. Due to the favorable sp3 nature of carbon over sp2, we expected that hydrogen would bind more easily on the hills of the rippled (finite temperature) sheet than on a flat (zero temperature) sheet of graphene. To test if graphene is a possible candidate for hydrogen storage, we wanted to investigate if hydrogen molecules could be spontaneously chemisorbed at finite temperatures. Monte Carlo simulations based on the LCHBOP (Long range Carbon Hydrogen Bond Order Potential) potential were used to calculate our results. We concluded that spontaneous chemisorption of molecular hydrogen is not possible on rippled graphene and on flat graphene sheets.

**Marijn Man**(June 2015)

## Time-delay distribution as a probe of topological charge (pdf, 671 kB)

We investigate the time-delay distribution of electrons reflected from a quantum dot. The dot is generally formed by a metallic granule or a restricted region of semiconductor which is characterised by classical chaotic dynamic. Theoretically quantum dots are modelled by a Hamiltonian, which is given by a large matrix with random entities. Invariance of statistical properties of the random matrix under global rotations characterise the symmetry class. In addition the random matrices within one symmetry class may be distinguished by topological invariants. Specifically we investigate whether there is a difference in time-delay distribution between quantum dots from the symmetry class D which are characterised by different topological numbers. We find that the time-delay distribution may indeed be sensitive to the topological number provided the coupling of the dot to the electron lead is neither perfect nor too weak. The sensitivity to the topological number is most profound when there is a small number of channels in the lead.

**Nicole Orval**(February 2014)

## Dempingsmodellen in spinsystemen (pdf, 8,6 MB)

De Landau-Lifschitz bewegingsvergelijking (LL) beschrijft de precessie van magnetisaties onder invloed van een magnetisch veld. De beweging van een domain wall in een spinsysteem kan hiermee ook worden beschreven. Naast de dempingsterm α in LL zijn er andere mogelijkheden van demping. Met behulp van simulaties is gevonden dat de plaatsing van defecten zorgt voor pinning en demping van de beweging van de domain wall.

**Timo de Ruijsscher**(Summer 2012)

## The behaviour of graphene under one-dimensional compression

**Marco Stevens**(Spring 2012)

## Wrinkled patterns in graphene

**Edo van Veen**(August 2012)

## Charge Inhomogeneities in Graphene due to Finite Size Effects (pdf, 25 MB)

In my thesis we show that finite size effects cause spatial charge distribution inhomogeneity in graphene. We investigate how the distribution of charge depends on chemical potential, sample size and shape, lattice defects and temperature. We find that finite size effects are a possible cause of experimentally found electron-hole puddles at zero chemical potential and nonzero temperature.

**Jan-Pieter Boersma**(June 2011)

## Gaining knowledge of single carbon chains (pdf, 201 kB)

We studied the convergence of energies, distances and distance alternation in carbon chains with periodic and free boundary conditions and in carbon chains with varying termination. These terminations were one, two or three hydrogens or a graphene ribbon. The bond lenghts, binding energies and charge densities were determined for these chains. From these properties we can conclude that adding termination to isolated carbon chains causes more alternation in bond lengths.

**Frank Buijnsters**(August 2010)

## Equilibrium bundles of chirally interacting particles (pdf, 21 MB)

We have introduced a highly idealised but computationally fairly cheap atomistic model of chirally interacting biopolymers, in which each monomer unit is represented as a single particle. Monte Carlo simulations are used to study either the formation of bundles from a gaseous initial state or the gradual twisting of a pre-existing bundle.

**Linde van Heeringen**(July 2010)

## De onverwachte structuur van aluminium nanokristallen: Een moleculaire dynamica simulatie (pdf, 1,1 MB)

In deze studie is de structuur van aluminiumclusters Al_{108}en Al_{1372}onderzocht met behulp van moleculaire dynamica. Er is gebruik gemaakt van een gefitte veel-lichamen potentiaal. Met behulp van verwarmen tot de vloeistoffase en koelen is voor A_{108}een amorfe structuur gevonden met een beduidend lagere energie dan een fcc-structuur (ΔE = 0.05 eV). Voor Al_{1372}verschillen beide structuren nagenoeg niet qua energie. Dit leidt tot het vermoeden dat clusters met een diameter grofweg kleiner dan 2.8 nm een stabiele amorfe structuur zullen hebben, terwijl grotere clusters de fcc-structuur behouden.

**Inka Locht**(November 2009)

## Curvature of Self-Assembled Membranes (pdf, 893 kB)

Amphiphilic molecules can self-assemble to molecular membranes. These membranes can appear in different shapes, such as vesicles and cylinders, with a radius much bigger than the length of the molecules. Self-assembled structures are of general interest in nanotechnology, but already the fact that molecules structure spontaneously is intriguing! In this report we will investigate the contribution of the solvent-tail interactions to the free energy and their in fluence on the equilibrium shapes of the membranes. There will be two starting points for this investigation: on the one hand we will start from the microscopic Flory-Huggins model to calculate the free energy of mixing. On the other hand we will assume a macroscopic approach of the geometry, shape and curvature of the membranes as two-dimensional surfaces. Special attention will be paid to the length of the molecules.

**Koen Reijnders**(August 2008)

## A non-stop journey through a graphene ribbon (pdf, 3,7 MB)

In this work we study conductance quantization in graphene nanoribbons within the adiabatic approximation. Numerical calculations are performed for the special case of a periodic ribbon, to see if one can induce a gap in the spectrum of such a ribbon.

**Jamil Hetzel**(August 2008)

## Equilibrium shapes of spontaneously bending surfaces (pdf, 2,4 MB)

For my bachelor project I have examined the equilibrium shapes of surfaces where the free energy depends on the curvature of the surface and where it is energetically favourable to be bent. It turns out that these surfaces can have non-spherical equilibrium shapes.