Engineering Topological Bands in Strained Covalent Organic Frameworks

TL;DR

The paper investigates how to engineer topological electronic phases in covalent organic frameworks (COFs) by applying chemically feasible perturbations that mimic uniaxial strain. Using a tight-binding H-XY model on a strained honeycomb lattice and Topological Quantum Chemistry, it shows that certain linkers substitutions, such as replacing biphenyl with pyrene, can bring CTFs toward higher-order topological insulator (HOTI) regimes with obstructed atomic insulating character and corner states. Ab initio calculations on a pyrene-substituted COF predict band structures near HOTI regions, suggesting a practical route to realize TI/HOTI phases in COFs and to harness localized edge/corner states for quantum sensing or information processing. The study emphasizes exploring the full tight-binding phase diagram of realistic COF models to identify accessible topological phases via chemical design and strain engineering.

Abstract

The tunability of covalent organic frameworks (COFs) opens opportunities to engineer topological electronic phases, including topological insulators (TIs) and higher-order topological insulators (HOTIs)--materials that host in-gap states localized at their edges, hinges, or corners. Here we explore how chemically feasible perturbations can drive triazine-based COFs (CTFs) into topological regimes. Using a tight-binding model on the Honeycomb lattice inspired by the frontier electronic states of CTFs, we show that introducing an effective uniaxial strain--implemented as a modulation of electron hopping on a subset of bonds--can generate a series of distinct topological band structures. This effect can be realized in practice through chemical substitution of linkers along the strained bonds. First-principles calculations demonstrate that replacing biphenyl with pyrene linkers drives a CTF to the brink of a HOTI phase, suggesting a viable route toward topological band-structure engineering in COFs.

Figures (8)

• Figure 1: Tight-binding model description of a Honeycomb COF CTF-2 in the $\{p_{x},p_{y}\}$ basis. The dashed rhombus in (a) indicates the unit cell of CTF-2, with lattice parameters $a=b=22.06\text{\AA}$ and space group $P622$. (b) Shows the degenerate highest occupied MOs (HOMOs) of triazine. (c) Shows the unit cell of a Honeycomb lattice and illustrates hopping between degenerate orbitals of $\{p_{x},p_{y}\}$ character. (d) Displays the H-XY model (red) fitted to first-principles simulations of CTF-2, with parameters $t_{\sigma}=-0.108 {\rm meV}$, $t_{\pi}=0.637 {\rm meV}$, $E_0=-1.288 {\rm eV}$ about HOMO-3 - HOMO-6.
• Figure 2: Structure of a CTF with (a) local $D_3$ and (b) local $C_2$ symmetry at the node (triazine) sites. Both are made up of triazine cores, where the blue(red) shapes represent different functional group linkers, with associated hopping strength $t_{\gamma}$($t_{\gamma_{0}}$) $\gamma \in \{\sigma, \pi\}$. (c) displays a larger fragment of the Honeycomb lattice made up from (b).
• Figure 3: Variety of topological band structures achievable in the strained H-XY model. (a) Phase diagram showing varying $t_{\sigma_0}$ and $t_{\pi_0}$, with fixed $\delta E=0\rm meV$ and $\delta t_{\pi}=\delta t_{\sigma}=-4\rm meV$. The coloured and white regions define topologically non-trivial (topologically insulating ($TI$) and higher-order topologically insulating ($HOTI$)) and trivial regions respectively. (b) Spectrum calculated on a nanoribbon with armchair edges for a parameter set in a TI phase (indicated by star on phase diagram), verifying the presence of gapless edge states at $\pm\pi$ as shown in the inset. (c) Spectrum calculated on a quantum dot with armchair edges for parameters in the HOTI regime (indicated by diamond on phase diagram). The in-gap states in the model of the spectrum are localized as the corners of the dot, as shown in inset which shows the square of wavefunction for one particular in-gap state (indicated by the red dot).
• Figure 4: Band structure and topological phase diagram for CTF-2 with replacement of biphenyl with pyrene linkers in one direction. (a) The predicted band structure, with a fit to the strained H-XY model with parameters $t_{\sigma_0}=7.20\rm meV$ and $t_{\pi_0}=0.121\rm meV$, (b) topological phase diagram with variation of $t_{\sigma_0}$ and $t_{\pi_0}$, with fixed values of $\delta E=-7.27\rm meV$, $\delta t_{\pi}=1.58\rm meV$ and $\delta t_{\sigma}=-2.18\rm meV$. The red star indicated the best fit parameters from simulations of strained CTF-2. This parameter set lies in the topologically trivial region, but on the border of a HOTI phase.
• Figure S1: The Fundamental Domain, $\Omega$, for the first Brillouin zone of $C222$. The High Symmetry Points (HSPs) and Lines (HSLs) are labelled ${\Gamma, Y, S}$ and ${\Delta,\Sigma}$ respectively.