基于晶格动力学理论,考虑原子在平面内的中心弹性力作用时,推导了平面正六边形晶 格振动的色散关系,得到第一布里渊区中沿Γ-M、M-K和Γ-K三个对称方向的色散关系表达式,沿 每一对称方向有四支格波,其中两支声学波和两支光学波。 讨论了最近邻原子间作用以及次近邻原子间作用对色散关系的影响。结果表明:只考虑最近邻原子间作用时,Γ点 两声学模频率为零、两光学模频率简并,K 点高频声学支与低频光学支频率简并,而M 点声子频率非简并;沿三个对称方向都存在一支频率为零的声学波和一支频率不变的无色散光学波,并且Γ-M 方向的声学波和光学波间具有频隙,另外两个方向的格波间无频隙。当计及次近邻原子间作用时,晶格振动的频率升高,三个对称方向的四支格波都表现出明显的色散性,沿Γ-M 方向声学支与光学支间的频隙减小,由此可见次近邻原子间作用对色散关系有显著影响,但是对Γ 点声子频率的大小和简并无影响,对K点声子频率的简并也无影响。

Based on lattice dynamics,considering the central in-plane elastic forces of the interatomic interaction,the dispersion relations of two-dimensional equilateral hexagonal lattice were derived,and the expressions of the dispersion relations along Γ-M,M-K and Γ-K three symmetry directions were obtained in the first Brillouin zone. There are four pieces of lattice waves in each direction,where two branches are acoustic waves and other two are optical waves. The effects of the nearest neighbor and secondary neighbor coupling to the dispersion relations were discussed. The results show that,in only nearest neighbor coupling,the eigen frequencies of two acoustic modes are zero and other two non-zero values which correspond to optical modes are degenerate at Γ point. One high frequency acoustic branch and one low frequency optical branch are degenerate at K point,and all four branches are nondegenerate at M point. One of the acoustic branches has totally zero frequencies and one optical branch is dispersionless over the entire symmetry directions. There is a frequency gap between acoustic branches and optical branches along Γ-M direction,and no frequency gap in other directions. With enhancement of the secondary neighbor coupling,the phonon frequencies are increased,and all four branches are displayed distinct dispersion along three directions,and the frequency gap is narrowed along Γ-M direction. The above results indicate that the dispersion relations are remarkably affected by the secondary neighbor coupling. But it has no influence on the value and degeneracy of the phonon frequency at Γ point,and it also has no effect on the degencracy of the phonon frequency at k point.