Surface Dynamics and Phononic Properties of 2D Field Tunable Auxetic Crystal

Abstract

of rigid polar rods and of elastic springs. The phase speed of the transverse acoustic wave propagating parallel to the polar rods is shown to be higher than the speed of the longitudinal wave at strong flelds. An absolute stop band for the lattice waves opens in the whole Brillouin zone with increasing fleld. The surface waves and resonances at a surface parallel to the rods are studied. Inflnitely narrow resonances called exceptional surface waves are found within bulk bands at certain speciflc values of the parameters characterizing the surface. Generally, the surface layer should be signiflcantly heavier and stifier than the substrate for the phenomenon to occur. 1. The model The geometry of the model stems from the generic structure proposed by Evans [1] for auxetic materials. The model consists of rigid rods connected by elastic springs as depicted in Fig. 1. The springs fi, fl, and ∞ are supposed to take their free lengths in the equilibrium state in contrast with stretched springs of earlier models [2]. The dynamical properties of the model can be modifled or even tuned by an external fleld if the rods possess dipole moments. In the present study we assume a fleld applied in the horizontal direction that is along the axis x1 of Fig. 1. The existence of such a fleld manifests itself by an additional torque restoring the horizontal orientation of the rods. The corresponding rotational force constant is called K. This is to some extent analogous to the rotational background potential of Ref. [3]. In what follows we study the bulk and the surface properties of the model as functions of the intensity of the external fleld, i.e. of the torque force constant K. The structure shows a base centered symmetry C2mm with one rod per the primitive unit cell. Consequently, the rectangular unit cell of edges a and b = a p 3 contains two rods. Every rod has three degrees of freedom (u1;u2;`) where the vector components (u1;u2) describe the displacement of the rod’s mass center from its equilib