Mechanical deformations provide a powerful route to engineer the electronic properties of graphene. Strain modifies the nearest-neighbor hopping parameters, an effect that can be described in tight-binding and continuum Dirac-like models by effective pseudo-gauge fields. When these gauge fields are spatially inhomogeneous, they generate effective magnetic pseudo-fields that strongly redistribute charge across the membrane and significantly alter its electronic structure [1,2].

Understanding deformed graphene is therefore essential for potential device applications, whether leveraging its intrinsic properties or employing it as an active layer or substrate in more complex structures. In this work, we investigate the relationship between the geometrical profile of a graphene membrane and its electronic properties.

To approximate realistic experimental conditions, we modeled a graphene sheet deposited on a hexagonal boron nitride (hBN) substrate patterned with an engraved periodic profile, as produced by thermal scanning probe lithography (TSPL) [3]. Using  first-principles calculations based on density functional theory,  we first studied a finite square graphene flake with zigzag edges containing 160 carbon atoms placed on a 10x10 hBN supercell exhibiting  a two-dimensional periodic deformation. After structural relaxation, the graphene layer retained a smooth corrugated profile, highlighting the dominant role  of van der Waals interactions in conforming the membrane to the patterned substrate.  

We then performed calculations for a relaxed 10×10 supercell of graphene on deformed hBN to determine the electronic properties of the periodic structure. Our results show the opening of a band gap at the original Dirac points, consistent with sublattice-symmetry-breaking induced by the alignment between the two lattices. In the aligned configuration,  one carbon atom sits above a boron atom while the other sits above a nitrogen atom, lifting the degeneracy between the two sublattices. 

Furthermore, we observe the emergence of isolated bands associated with additional gaps at both higher and lower energies. Near the neutrality point, the bands exhibit a renormalization of the Fermi velocity, indicating an incipient strain-induced band narrowing. Finally, calculations of the local density of states (LDOS) reveal the development of spatially varying sublattice polarization across the membrane, a clear signature of out-of-plane deformation and strain-induced local symmetry breaking.