Embeddings of Planar Quasimetrics into Directed ℓ 1 and Polylogarithmic Approximation for Directed Sparsest-Cut

Kawarabayashi, Ken-Ichi; Sidiropoulos, Anastasios

doi:10.1109/FOCS52979.2021.00055

Key properties of two-dimensional (2D) layered materials are highly strain tunable, arising from bond modulation and associated reconfiguration of the energy bands around the Fermi level. Approaches to locally controlling and patterning strain have included both active and passive elastic deformation via sustained loading and templating with nanostructures. Here, by float-capturing ultrathin flakes of single-crystal 2H-MoS₂ on amorphous holey silicon nitride substrates, we find that highly symmetric, high-fidelity strain patterns are formed. The hexagonally arranged holes and surface topography combine to generate highly conformal flake-substrate coverage creating patterns that match optimal centroidal Voronoi tessellation in 2D Euclidean space. Using TEM imaging and diffraction, as well as AFM topographic mapping, we determine that the substrate-driven 3D geometry of the flakes over the holes consists of symmetric, out-of-plane bowl-like deformation of up to 35 nm, with in-plane, isotropic tensile strains of up to 1.8% (measured with both selected-area diffraction and AFM). Atomistic and image simulations accurately predict spontaneous formation of the strain patterns, with van der Waals forces and substrate topography as the input parameters. These results show that predictable patterns and 3D topography can be spontaneously induced in 2D materials captured on bare, holey substrates. The method also enables electron scattering studies of precisely aligned, substrate-free strained regions in transmission mode.

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