Chiral electronic network within skyrmionic lattice on topological insulator surfaces

TL;DR

This work analyzes Dirac surface states of a topological insulator proximity-coupled to a triangular skyrmion lattice, showing that zero-mass lines around skyrmions host chiral modes whose tunneling forms electronic minibands with nontrivial topology. A Kagome network model is developed to capture the low-energy physics, but energy-independent scattering fails to reproduce the correct band topology; band reconstruction is introduced to incorporate the energy dependence of inter-skyrmion tunneling. The reconstructed bands can realize trivial, critical, and Chern phases with Chern numbers $\mathcal{C}=\\pm1$, aligning with microscopic calculations and validating the network description for this system. The results establish band reconstruction as a general necessity for network models describing electronic confinement in nanostructures and offer a framework to explore topological minibands in TI–skyrmion heterostructures with potential experimental signatures via ARPES and transport.

Abstract

We consider a proximity effect between Dirac surface states of a topological insulator and the skyrmion phase of an insulating magnet. A single skyrmion results in the surface states having a chiral gapless mode confined to the perimeter of the skyrmion. For the lattice of skyrmions, the tunneling coupling between confined states leads to the formation of low energy bands delocalized across the whole system. We show that the structure of these bands can be investigated with the help of the phenomenological chiral network model with a kagome lattice geometry. While the network model by itself can be in a chiral Floquet phase unattainable without external periodic driving, we show how to use a procedure known as band reconstruction to obtain the low energy bands of the electrons on the surface of the topological insulator for which there is no external driving. We conclude that band reconstruction is essential for the broad class of network models recently introduced to describe the electronic properties of different nanostructures.

Figures (7)

• Figure 1: (a) The spatial profile of the unit vector $\mathbf{n}(\mathbf{r})$ following the magnetization for the triangular skyrmionic lattice. (b) The Kagome network is formed by links describing chiral electrons (black arrows) circulating skyrmions (green dots) and scattering nodes (intersections of black lines) describing tunneling between chiral states trapped at adjacent skyrmions.
• Figure 2: The electronic spectrum of an isolated skyrmion, calculated numerically (blue) and analytically using semiclassical arguments (red). For the parameters considered, the skyrmion traps a pair of chiral states that are well-separated from the continuum of unbound states (yellow).
• Figure 3: Electronic minibands in the trivial phase (column 1), at the critical point (column 2), and in the Chern insulator phase (column 3) calculated using different approaches. Top row: microscopic calculations using the Dirac equation with skyrmion lattice period $L = 95, 80, 65 \; \hbox{nm}$; Bottom row: phenomenological calculations using the network band reconstruction with an energy-dependent scattering angle parametrized by its maximal value $\theta_\mathrm{max} = \pi/8, \pi/6, \pi/4$.
• Figure 4: The unit cell for the network model is denoted by the orange box. There is a single chiral state per link, and $\psi_i$ ($i=1,... ,6$) is its amplitude. Extra phases picked up at the scattering nodes are shown in the corners.
• Figure 5: Phenomenological bands calculated using the Kagome network model with an energy-independent scattering angle $\theta = \pi/8, \pi/6, \pi/4$ corresponding to the chiral Floquet phase, critical point, and Chern insulator phase respectively. In the chiral Floquet phase, the topological winding number invariant is nonzero and all bands have zero Chern number.