[Paper Review] Tilted Dirac cones and topological transitions in strained kagome lattices

M. A. Mojarro, Sergio E. Ulloa|arXiv (Cornell University)|Feb 28, 2023

Topological Materials and Phenomena4 citations

TL;DR

This paper demonstrates that uniaxial strain in kagome lattices with spin-orbit coupling induces tilted Dirac cones and drives topological phase transitions, with moderate strain (a few percent) tuning the system between trivial and topological insulating phases characterized by a nontrivial $Π_{2}$ index. The transition is mediated by strain-dependent bandgap closing at the M point of the strained Brillouin zone, enabling control over topological edge states and correlated phases.

ABSTRACT

We study effects of strain on the electronic properties of the kagome lattice in a tight-binding formalism with spin-orbit coupling (SOC). The degeneracy at the $Γ$ point evolves into a pair of emergent tilted Dirac cones under uniaxial strain, where the anisotropy and tilting of the bands depend on the magnitude and direction of the strain field. SOC opens gaps at the emergent Dirac points, making the flatband topological, characterized by a nontrivial $\mathbb{Z}_2$ index. Strains of a few percent drive the system into trivial or topological phases. This confirms that moderate strain can be used to engineer anisotropic Dirac bands with tunable properties to study new phases in kagome lattices.

Motivation & Objective

• To investigate how uniaxial strain modifies the electronic band structure and topological properties of kagome lattices with spin-orbit coupling.
• To determine the conditions under which strain drives topological phase transitions, particularly at 2/3 filling.
• To analyze the emergence of tilted Dirac cones and their topological characterization via the $Π_{2}$ invariant.
• To explore the role of on-site energy differences and Rashba coupling in tuning the system between trivial, topological, and semimetallic phases.

Proposed method

• A tight-binding model with nearest-neighbor hopping integrals modified by strain via the Grüneisen parameter and strain tensor $\hat{\epsilon}$.
• The strain-induced deformation is modeled through a displacement field $\mathbf{u}(\mathbf{r}) = \hat{\epsilon} \cdot \mathbf{r}$, altering nearest-neighbor vectors and hopping amplitudes.
• Spin-orbit coupling is introduced via intrinsic ($\lambda_I$) and Rashba ($\lambda_R$) terms in the Hamiltonian, breaking time-reversal symmetry and opening gaps at Dirac points.
• The $Π_{2}$ topological invariant is computed numerically to classify the system as trivial ($\nu=0$) or topological ($\nu=1$).
• Edge states are analyzed in finite-size strips to confirm topological protection and distinguish trivial from nontrivial phases.
• Phase diagrams are constructed as functions of strain magnitude and direction, with on-site energy $\varepsilon_A$ and SOC strengths as control parameters.

Experimental results

Research questions

• RQ1 How does uniaxial strain induce the formation of tilted Dirac cones in the kagome lattice with spin-orbit coupling?
• RQ2 What conditions lead to a topological phase transition in strained kagome lattices at 2/3 filling?
• RQ3 How does the direction and magnitude of strain affect the bandgap closing at the M point of the strained Brillouin zone?
• RQ4 What role does on-site energy asymmetry play in tuning the topological invariant and edge state formation?
• RQ5 How do Rashba and intrinsic spin-orbit coupling jointly influence the emergence of semimetallic and insulating phases under strain?

Key findings

• Uniaxial strain along the zigzag direction drives the system from a trivial to a topological insulator phase at 2/3 filling, with a bandgap closing at the M point of the strained Brillouin zone.
• For strain along the sawtooth direction, increasing strain drives the system from a topological to a trivial insulator phase, with a bandgap closing at the M point.
• The energy gap at the M point is given by $|\varepsilon_A - 2\sqrt{\lambda_I^2 + t_0^2(\beta\rho\epsilon + 1)^2}|$ for zigzag strain and $|\varepsilon_A - 2\sqrt{\lambda_I^2 + t_0^2(\beta\epsilon - 1)^2}|$ for sawtooth strain.
• The inclusion of Rashba coupling introduces a semimetallic phase between trivial and topological phases, with gap closing occurring at intermediate strain values.
• Topological edge states are robust in the topological phase and vanish in the trivial phase, confirming the nontrivial $Π_{2}$ invariant via edge-state analysis.
• Phase diagrams show that strain magnitude and direction, along with $\varepsilon_A$ and SOC strengths, can be tuned to access distinct topological, trivial, or semimetallic phases.