Difference analysis model for the mismatch effect and substrate-induced lattice deformation in atomically thin materials

Atomically thin materials hold mutative electronic transition properties by virtue of the inevitable strains adding to the lattice distribution. A quantitative approach and clear systematical verification have seldom been presented to effectively illustrate the substrate-induced strain along the lattice distribution, although the substrate perturbations for monolayers have been repeatedly mentioned. Here, a model of difference analysis is introduced to reveal the structural strain of monolayers from the substrate, which is derived from the mismatch effect based on the variation of the thermal expansion coefficient between the substrate and monolayers. A coupling coefficient is proposed to quantitatively manifest the substrate-induced strain and mismatch effect. Furthermore, combined with the coupling coefficient, the lattice disorder from the mismatch effect is demonstrated, experimentally using the WS2 monolayer as reference. This study could promote broad investigations of some fundamental issues from preparation to electrical structural performances in atomically thin materials and further practical applications.