Vibrational Edge Modes in Finite Crystals

In addition to surfaces, real crystals have edges, corners, and steps. In this paper we present a theory of long-wavelength acoustic phonons localized at an edge of a cubic elastic medium bounded by two (100) faces. The stress-free boundary conditions on the faces of the semi-infinite medium meeting at the edge are incorporated into the equations of motion of the medium by the device of assuming position-dependent elastic constants. The equations of motion of the medium are solved by expanding each displacement component in a double series of Laguerre functions, which are orthonormal and complete in the region $x_1\ge 0$, $x_2\ge 0$. The edge modes obtained are wavelike parallel to the edge and decay rapidly with increasing distance into the medium from the edge. For the particular case of an elastically isotropic medium for which the Lamé constants $\lambda$ and $\mu$ are equal, the speed of propagation of the lowest-frequency edge mode is $0.9013c_t$, where $c_t$ is the speed of sound of bulk transverse modes and is lower than that of Rayleigh waves, which is $0.9194c_t$. A variational principle for the speeds of edge modes is also presented.