Theory of dynamical stability for two- and three-dimensional Lennard-Jones crystals

The dynamical stability of three-dimensional (3D) Lennard-Jones (LJ) crystals has been studied for many years. The fcc and hcp structures are dynamically stable, while the bcc structure is stable only for long-range LJ potentials that are characterized by relatively small integer pairs $(m,n)$. Here, we study the dynamical stability of two-dimensional (2D) LJ crystals, where the planar hexagonal, the buckled honeycomb, and the buckled square structures are assumed. We demonstrate that the stability property of 2D and 3D LJ crystals can be classified into four groups depending on $(m,n)$. The instabilities of the planar hexagonal, the buckled square, and the bcc structures are investigated within analytical expressions. The structure-stability relationship between the LJ crystals and the elemental metals in the periodic table is also discussed.

Figure 1

The top and side views of the 2D lattice structure: (a) HX, (b) BHC, and (c) BSQ structures. The primitive lattice vectors are indicated by arrows in the top view. $\delta$ is the buckling height.

Figure 2

The stability property for the $(m,n)$ LJ crystals in the bcc, BHC, and BSQ structures. For group D, i.e., $(m,n)=(8,3),(9,3),(10,3),(11,3)$, the LJ crystals are unstable.

Figure 3

The $R$ dependence of the $(12,6),(8,4),(6,3)$, and (9,3) LJ potentials.

Figure 4

The dispersion curves of the (12,6) LJ crystals for (a) HX, (b) BHC, (c) BSQ, (d) fcc, (e) bcc, and (f) hcp structures.

Figure 5

Same as Fig. 4, but for the (8,4) LJ crystals.

Figure 6

Same as Fig. 4, but for the (6,3) LJ crystals.

Figure 7

Same as Fig. 4, but for the (9,3) LJ crystals.