Discrete differential geometry and the properties of conformal two-dimensional materials

Abstract

Two-dimensional materials were first isolated no longer than ten years ago, and a comprehensive understanding of their properties under non-planar shapes is still being developed. Strictly speaking, the theoretical study of the properties of graphene and other two-dimensional materials is the most complete for planar structures and for structures with small deformations from planarity. The opposite limit of large deformations is yet to be studied comprehensively but that limit is extremely relevant because it determines material properties near the point of failure. We are exploring uses for discrete differential geometry within the context of graphene and other two-dimensional materials, and these concepts appear promising in linking materials properties to shape regardless of how large a given material deformation is. A brief account of additional contributions arising from our group to two-dimensional materials that include graphene, stanene and phosphorene is provided towards the end of this manuscript.

Electronic and optical properties of strained graphene and other strained 2D materials: a review

“…where r l are positions of carbon atoms. The Fourier transform of this potential, denoted by ( ) k V , is obtained by integrating over the entire area S: For further reference, this transform will be labeled ( ) k V gp , which can be written as: (10) with = + G G G l h 1 2 and l h , integers. The norm of this transform is (see figure 2):…”

“…Z A S (30) Finally, the diffraction pattern of the uniform deformed lattice is a scaling of equation (10):…”

“…The bibliography in the field continues to grow. Other reviews serve as a guide through this information forest [6,[8][9][10][11][12][13][14][15], each with its own focus. The goal of the present review is to provide a didactic and basic platform to understand in simple terms representative models and results, with enough physical significance to get a grasp of the main effects of strain, the emerging consensus on the field, and its new directions.…”

“…Due to the mismatch between the graphene and substrate lattice parameters, atoms move to reduce their energy, producing a certain amount of strain too. There has been an ever increasing interest in using strain and the soft-properties of graphene to control its physical properties [6][7][8][9][10][11][12][13][14][15], and the word straintronics [16] appropriately describes this aim. Understanding how strain affects graphene's electronic and optical properties is of paramount importance, but the interest is not only of a technological nature.…”