Efficient Matching Boundary Conditions of Two-dimensional Honeycomb Lattice for Atomic Simulations

Abstract

In this paper, we design a series of matching boundary conditions for a two-dimensional compound honeycomb lattice, which has an explicit and simple form, high computing efficiency and good effectiveness of suppressing boundary reflections. First, we formulate the dynamic equations and calculate the dispersion relation for the harmonic honeycomb lattice, then symmetrically choose specific atoms near the boundary to design different forms of matching boundary conditions. The boundary coefficients are determined by matching a residual function at some selected wavenumbers. Several atomic simulations are performed to test the effectiveness of matching boundary conditions in the example of a harmonic honeycomb lattice and a nonlinear honeycomb lattice with the FPU-$β$ potential. Numerical results illustrate that low-order matching boundary conditions mainly treat long waves, while the high-order matching boundary conditions can efficiently suppress short waves and long waves simultaneously. Decaying kinetic energy curves indicate the stability of matching boundary conditions in numerical simulations.

Figures (21)

• Figure 1: Left: schematic plot of the honeycomb lattice. A diamond drawn by dotted lines represents a unit cell; Right: illustration for numbering atoms. The first class of atoms are represented by blue points with the displacements denoted by $v_{n,m}$. The second class of atoms are represented by red points with the displacements denoted by $w_{n,m}$.
• Figure 2: Left: dispersion relation of the honeycomb lattice; Right: the blue domain is the reduced Brillouin zone $\Omega_1$, the whole domain is the extended Brillouin zone $\Omega$, and the yelllow domain corresponds to the optical branch $\Omega\setminus\Omega_1$.
• Figure 3: Left: vector graphic of the group velocity; Right: the group velocity along the vertical direction with $\xi_p=0$.
• Figure 4: Schematic plot of two types of boundaries. Left: the zigzag boundary; Right: the armchair boundary.
• Figure 5: Schematic plot for the bottom matching boundary conditions: (a) MBC1; (b) MBC2; (c) MBC4; (d) MBC5.