## Dissertations, Theses, and Capstone Projects

#### Date of Degree

9-2022

#### Document Type

Dissertation

#### Degree Name

Ph.D.

#### Program

Physics

#### Advisor

Eugene M. Chudnovsky

#### Advisor

Dmitry A. Garanin

#### Committee Members

Vadim Oganesyan

Aditi Mitra

Alexander Punnoose

#### Subject Categories

Condensed Matter Physics

#### Keywords

Skyrmions, magnetism, nonlinear sigma-model, two-dimensional

#### Abstract

Magnetic skyrmions are whirls formed by magnetic moments in a crystal. They have attracted attention largely due to their topological protection, which provides an avenue for technology like next-generation memory storage. The idea of topologically protected solutions of a quantum field theory was originally proposed by Tony Skyrme when he developed a model to explain the stability of hadrons in particle physics. His work has extended far beyond his original intent to several areas of condensed matter physics. Here we focus on skyrmions in magnetic materials.

Skyrme's original theory modeled excitations which exist in three spatial dimensions, a requirement for the hadrons he wished to model. Belavin and Polyakov developed a solution to a two-dimensional model in a ferromagnet, bringing Skyrme's theory to the condensed matter world. The Belavin-Polyakov solution is analogous to Skyrme's one in that it has a topological charge that in theory does not change. The difference in dimensions is apparent in the mapping done to quantify topological charge. In Belavin and Polyakov's solution, the magnetic moments in a two-dimensional plane, rather than a three-dimensional one, are mapped to a 2-sphere to graphically quantify topological charge. In some texts, these two-dimensional particles are called baby-skyrmions to distinguish them from the three-dimensional analog.

This thesis focuses on two-dimensional skyrmions in magnetic systems. We use the term two-dimensional to refer both to skyrmions stabilized at the interface of thin-film heterostructures, a system that can be well approximated as a two-dimensional plane, and in thin films which host skyrmion "strings." In the latter, each layer of the material identically hosts a two-dimensional skyrmion, and the layers together form a three-dimensional "string" structure that can be described using a two-dimensional theory.

It was mentioned earlier that skyrmions are topologically protected and cannot be destroyed. This is due to the conservation of topological charge in the Hamiltonian from which Belavin and Polyakov derived the skyrmion solution. This Hamiltonian is also scale invariant, meaning that the total energy of the skyrmion does not depend on its size. In practice, skyrmions in magnetic systems live in crystals, which do not have scale invariance. The crystal introduces additional terms in the Hamiltonian, leading to the violation of both scale invariance and topological protection. The skyrmion acquires some finite lifetime in the presence of the lattice, which has inspired the work present in this thesis.

We quantify the impact of the crystal lattice by analytically exploring the reasons for the collapse of a ferromagnetic skyrmion. We begin with a model that contains only the exchange interaction, the interaction which was used to predict the skyrmions in the first place. While in practice magnetic skyrmions are stabilized by additional energy terms like the Dzyaloshinskii-Moriya, dipolar, Zeeman, or anisotropic interactions, the pure exchange model offers an interesting insight into the mechanism of skyrmion collapse on a lattice.

We further examine skyrmion stability in real systems by considering the impact of temperature, size, and material defects on skyrmions. Motivated by recent experimental progress in stabilizing skyrmions at room temperature, we explore the thermal collapse of a magnetic skyrmion. We compute the energy barrier against collapse by numerically calculating atomistic spin dynamics and comparing numerical to analytical results. Inspired by the practical push toward smaller skyrmions for nanoelectronic applications, we study the quantum stability of a ferromagnetic skyrmion. We find that the presence of the lattice leads to a finite energy barrier that the magnetic skyrmion can tunnel through to collapse. We furthermore examine the impact of a defect on the stability of a skyrmion, finding that it is stable in previously inaccessible regions of the phase diagram.

While we address ferromagnetic skyrmions in the bulk of this thesis, we also examine the stability of an antiferromagnetic skyrmion both classically and quantum-mechanically. In the classical case, we numerically solve equations of motion. In the quantum case, we derive a Hamiltonian and numerically compute transitions between eigenstates.

#### Recommended Citation

Derras-Chouk, Amel, "Stability of Two-Dimensional Magnetic Skyrmions" (2022). *CUNY Academic Works.*

https://academicworks.cuny.edu/gc_etds/5136