Holey Substrate-Directed Strain Patterning in Bilayer MoS₂

Abstract

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-MoS 2 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 flakesubstrate 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.

Atomistically-informed continuum modeling and isogeometric analysis of 2D materials over holey substrates

“…(AFM data was provided to us by Yichao Zhang (University of Minnesota). Similar measurements are reported in [8].) AFM measurements show that the substrate has significant topography near the holes, which has a strong effect on the resulting strain in the MoS 2 layer.…”

“…There are well-established techniques to apply photolithography [22,23] and electron beam lithography [24] to Si 3 N 4 , to manufacture holey substrates in a controlled manner, as needed for strain engineering applications. A representative plot of the surface of holey substrate is shown in Figure 2, which is derived from atomic force microscopy (AFM) of a holey Si 3 N 4 substrate manufactured by Ted Pella, Inc. [8]. The Figure 1: Structure of a MoS 2 monolayer, with Mo atoms in cyan and S atoms in yellow.…”

“…This is referred to as strain engineering, and is an active field of research within nanoelectronics. A recently-developed approach to apply strains to 2D materials without the need for sustained external loading is through van der Waals (vdW) interaction with an underlying substrate [8]. This paper focuses on a single layer of MoS 2 (in the 2H phase) interacting with a silicon nitride (Si 3 N 4 ) substrate perforated by a periodic pattern of holes, i.e., a "holey substrate."…”

“…This paper focuses on a single layer of MoS 2 (in the 2H phase) interacting with a silicon nitride (Si 3 N 4 ) substrate perforated by a periodic pattern of holes, i.e., a "holey substrate." A key aspect of this problem, described in [8], is that the substrate is not flat, but rather has significant periodic topography introduced by the manufacturing process of the holes. This topography is a critical component in straining the MoS 2 layer and must be included in any modeling approach.…”

“…One approach to predicting strains in 2D materials, like MoS 2 , is to perform molecular statics (MS) simulations that directly model atomic nuclei as discrete particles interacting via empirical interatomic potentials [9]. Indeed, this was the approach taken in [8]. However, the computational cost of MS becomes prohibitive for all but the smallest physical systems.…”