Tight-binding theory of graphene mechanical properties

Since the atomistic mechanism against monolayer graphene deformations has not been well comprehended, modeling graphene’s mechanical properties is still an open question. The torsional stiffness associated with Gaussian curvature, in particular, is extremely difficult to understand and estimate via either experiments or simulations. In this paper, using the bond-orbital model (BOM) based on the tight-binding (TB) method, we find out that graphene can be model as the Föppl-von Karman plate with four independent mechanical parameters, and present the clear connections between the mechanical parameters and the chemical bonds for the first time. Our TB theory reveals that the Gaussian modulus only relies on the torsion of the adjacent π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi$$\end{document}-orbitals, and elucidates that the independence between out-of-plane and in-plane mechanical parameters comes from the geometrical irrelevance between the bond-formation energy of σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma$$\end{document}-bonds and that of π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi$$\end{document}-bonds. Besides, the constraints on two out-of-plane mechanical parameters are given through the thermodynamic stability requirement: the Gaussian modulus kG<0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_{G} < 0$$\end{document} and the bending modulus kB>-kG/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_{B} > - k_{G} /2$$\end{document}. The mechanical parameters obtained by our TB theory are well agreed with experiments and Quantum calculations.