The stability and the nonlinear elasticity of 2D hexagonal structures of Si and Ge from first-principles calculations

By using the first-principles calculations based on the density-functional theory (DFT), we study the stability and the nonlinear elasticity of two-dimensional (2D) hexagonal structures of Si and Ge. The reproduced structure optimization and phonon-dispersion curves demonstrate that Si and Ge can form stable 2D hexagonal lattices with low-buckled structures, and provide a good agreement with the previous DFT calculations. The second- and third-order elastic constants are calculated by using the method of homogeneous deformation. The present results of the linear elastic moduli agree well with the previous results. In comparison with the linear approach, the nonlinear effects really matter while strain is larger than approximately 3.5%. The force-displacement behaviors and the breaking strength of 2D hexagonal Si and Ge are discussed using the nonlinear stress-strain relationship. By using the available results of graphene, we reasonably demonstrate that the radius of the atom increases and breaking strength of this element decreases for 2D hexagonal structures of group IV-elements.