[Paper Review] Theory of Twisted Bilayer Graphene Near Commensuration

Hridis K. Pal, Stephen Carter+1 more|arXiv (Cornell University)|Sep 6, 2014

Graphene research and applications3 citations

TL;DR

This paper develops a long-wavelength effective theory for twisted bilayer graphene near commensurate twist angles, extending existing frameworks beyond small angles by leveraging proximity to commensurate structures. The theory captures key commensuration effects—such as band gaps and topological order—through higher Fourier components of interlayer coupling, enabling experimental observation of these phenomena within a finite angular window around commensurate angles, not just at exact commensurability.

ABSTRACT

Incommensurately twisted graphene bilayers are described by long-wavelength theories, but to date such theories exist only at small angles of interlayer rotation. We construct a long wavelength theory without such a restriction, instead requiring nearness to commensuration. The theory inherits its energy scale from the exactly commensurate bilayer that it is close to. It is a spatial interpolation between the low-energy theories of commensurate structures with the two possible sublattice exchange (SE) symmetries: SE even and SE odd. In addition to generalizing existing theories, our theory brings into experimental reach so far elusive commensuration physics in graphene such as band gaps and nontrivial band topology.

Motivation & Objective

• To extend long-wavelength theories of twisted bilayer graphene beyond the small-angle regime where previous theories are valid.
• To enable the experimental observation of exotic commensuration-driven phenomena—such as band gaps and topological order—that were previously inaccessible due to stringent twist-angle control requirements.
• To unify the description of both incommensurate and commensurate bilayer graphene near commensurate angles using a spatial interpolation between low-energy theories of SE-even and SE-odd commensurate structures.
• To provide a framework applicable to any bilayer heterostructure with a moiré superlattice, regardless of the origin of the moiré pattern or underlying symmetry.
• To demonstrate that commensuration effects are not restricted to exactly commensurate systems but persist in a finite angular range around them, significantly relaxing experimental constraints.

Proposed method

• The theory constructs a spatially varying effective Hamiltonian by interpolating between the low-energy theories of two commensurate bilayer structures with opposite sublattice exchange (SE) symmetry—SE even and SE odd.
• It derives the effective Hamiltonian from the interlayer coupling's Fourier components, particularly emphasizing higher-order components that induce band curvature and gaps, as shown in prior commensurate studies.
• A gauge transformation is applied to align the Dirac Hamiltonians of the two layers, shifting the angular dependence into the interlayer coupling term, which is expressed in terms of Fourier components of the interlayer potential.
• The theory uses a continuum approximation around the K and K' points, with the interlayer coupling parameterized by a Gaussian potential $ V(\mathbf{r}_i, \mathbf{r}_j) = t_0 e^{-(|\mathbf{r}_i - \mathbf{r}_j|/l_0)^2} $, allowing numerical validation.
• Numerical tight-binding calculations are performed at $ \theta = 35.57^\circ $, near the commensurate angle $ \theta_c = 38.21^\circ $, to verify predictions of density oscillations and gap-like features.
• The theory's predictions are validated by matching the amplitude of density oscillations $ \delta\rho/\rho_0 $ to $ \mathcal{V}^2 $, where $ \mathcal{V} $ is derived from the commensurate case, confirming consistency without free parameters.

Experimental results

Research questions

• RQ1 Can long-wavelength theories for twisted bilayer graphene be generalized beyond the small-angle regime to include large-angle near-commensurate configurations?
• RQ2 What role do higher Fourier components of the interlayer coupling play in determining electronic structure near commensurate angles?
• RQ3 Can the effects of commensuration—such as band gaps and topological order—be observed not only at exact commensurate angles but also in a finite angular window around them?
• RQ4 How does the width of the angular window around a commensurate angle depend on the supercell size and symmetry of the commensurate structure?
• RQ5 To what extent can the electronic response near commensuration be predicted from the properties of the nearest exactly commensurate structure?

Key findings

• The theory successfully predicts density oscillations with wavevector $ \delta\mathbf{K} $ and trigonal symmetry in the local density of states at $ \theta = 35.57^\circ $, matching the predicted form $ \delta\rho(\mathbf{r})/\rho_0 \propto \mathcal{V}^2 $.
• The amplitude of density oscillations scales quadratically with the interlayer hopping strength $ t_0 $, as predicted by the theory and confirmed numerically.
• Increasing the interlayer coupling decay length $ l_0 $ suppresses the commensuration effects, with $ \delta\rho/\rho_0 \propto \mathcal{V}^2 $ decreasing exponentially, consistent with the decay of higher Fourier components.
• The $ l_0 $-dependence of $ \delta\rho/\rho_0 $ at $ \theta = 35.57^\circ $ matches exactly the $ l_0 $-dependence of $ \mathcal{V}^2 $ extracted from the commensurate structure at $ \theta_c = 38.21^\circ $, validating the theory's predictive power.
• The theory predicts a band gap of $ 0.09t_0 $ at the Dirac point for the exactly commensurate structure at $ \theta_c = 38.21^\circ $, which is absent at $ \theta = 35.57^\circ $, confirming the role of commensuration in gap generation.
• The numerically calculated density of states at $ \theta = 35.57^\circ $ agrees with the theoretical prediction based on $ \mathcal{V} $ from the commensurate case, with no free parameters, to within the theory's precision $ \delta\theta/\theta_c $.