Gap properties of helical edge states in two-dimensional topological insulators with time-dependent magnetic impurities

Abstract

The quantum mechanical equivalent of the classical Hall effect can lead to interesting results in solid state physics.
A similar effect, that has received attention in recent years, occurs when large spin-orbit coupling is present in a material, the so-called quantum-spin Hall effect.
In two-dimensional materials it leads to so-called helical edge states that exhibit counter-propagating electron states located at the edge with spin-momentum locking protected by time-reversal symmetry.
That means, different spin species travel in different directions along the edge and cannot be scattered into each other unless time-reversal is broken by a magnetic contribution.
Two-dimensional materials hosting these quantum spin Hall states are commonly called two-dimensional topological insulators.

In this thesis we investigate the electronic structure of the edge states in the presence of magnetic impurities with rotating magnetic moments.
Since they break time reversal symmetry, the impurities lead to backscattering and the density of states can be altered drastically in their presence.
To calculate the time-averaged density of states, a Floquet-like approach is applied to the single-particle Green's function of the 2x2 effective edge Hamiltonian of the Bernevig-Hughes-Zhang model with impurities.
The rotation of the impurities in the x-z-plane leads to the density of states transitioning between a gapped and an ungapped state, which in turn leads to drastically different shapes of the density of states depending on the driving frequency.

A numerical model is derived and criteria for choosing reasonable numerical cut-offs are given.
The resulting density of states looks different for different driving frequencies.
Slow driving, compared to a time scale defined by the magnetic impurity strength, leads to a density of states comparable to the average over static impurities for different impurity orientations.
Fast driving effectively does not alter the low energy density of states, leaving it constant around the center, with distinct resonances at energies related to the driving frequency.
Driving with frequencies around the time scale defined by the impurities leads to different results, exhibiting additional resonances, broadened due to the impurity nature of the system.

Popular Abstract

There is a huge hype about quantum computers and other types of technologies enabled by quantum mechanics at the moment.
While experts are usually a bit more careful with claiming breakthroughs than the media, it is quite obvious that new materials are going to be needed.
One interesting class of materials are so-called two-dimensional topological insulators.
These materials exhibit dissipationless edge states, meaning they can transport electrons without losses.
This alone opens huge potential while these types materials also show other interesting effects.

We can imagine an edge in this kind of system as a highway.
First, we only allow blue and red cars on the highway.
Secondly, cars of the same color have to go in the same direction, for example red cars to the right and blue cars to the left.
If it is a perfectly straight and flat road all cars will go at the same speed and that very efficiently.
If there are some markings on the road, some people might think: “oh, maybe I’ll go slower”, while others feel like “ok, I'll look out, but in the meantime let’s speed up a bit”.
In the end, we will have a lot of cars going at slightly different speeds.
d, aligned rotating magnetic impurities (yellow) on the 1D Edge channels.

So far this model does not feel too weird, but we can now imagine that all the cars have a little arrow on the dashboard, representing the so-called spin of the electrons.
If that arrow points upwards the car is red, if it points downwards it is blue.
Now, since the car changes color if the arrow flips, it also has to change it's direction and drive on other side of the highway.
This is a simple picture of the physical term "spin-momentum-locking": the direction of the movement (color) is linked to the orientation of the spin (arrow).

But what happens if we have speed bumps instead of the markings?
Now the arrows might flip and then the car changes its color and has to change its direction.
The effect of magnetic impurities corresponds to the speed bumps possibly changing the direction of the arrow on the dashboard!
It is already rather impractical to have speed bumps on a highway, but now imagine someone decided it might be a good idea to let the bumps change their height, letting them emerge from the ground periodically.
That corresponds to rotating magnetic impurities and the effects of this are main focus of this thesis.

But why would we even need these kind of electron highways?
For example, we could need spin-filters that only let one spin-species, i.e. one color of cars, pass.
Maybe we even want a special setup of speed bumps (intentional impurities, doping) or we have some pot holes (unintentional impurities) and need to know what that does to our system.
These and more applications make two dimensional topological insulators a class of materials, we might hear a lot about in the future.

• author: Wozny, Simon
• supervisor: Martin Leijnse; Sigurður I. Erlingsson
• organization: Solid State Physics; Department of Physics
• course: FYSM60 20201
• year: 2020
• type: H2 - Master's Degree (Two Years)
• subject: Physics and Astronomy
• keywords: topological insulators, edge states, impurities, density of states, Green's function, Floquet
• language: English
• id: 9018259
• date added to LUP: 2020-06-15 13:21:51
• date last changed: 2020-06-22 03:39:11