Statistical mechanics of dislocation pileups in two dimensions

Phonon eigenfunctions of inhomogeneous lattices: Can you hear the shape of a cone?

“…We expect that the localization effect we find for inhomogeneous cones (Sections III and IV) occurs more generally in inhomogeneous lattices. Here, we illustrate this phenomenon in a relatively simple one-dimensional inhomogeneous lattice with long range interactions and only longitudinal phonons, inspired by the physics of dislocation pileups [14,15]. Specifically, we show that the normal modes of 1d dislocation pileups, a type of defect assembly embedded in two-dimensional crystals [14], exhibit quasi-localization as a function of the phonon frequency, due to a position-dependent band edge.…”

“…Dislocation pileups exemplify an intriguing class of 1d inhomogeneous lattices, whose constituents are not particles but point-like edge dislocation defects in a two-dimensional host crystal [14]. The dislocations, once emitted in response to a stress field, all have the same Burgers vector topological charge and interact via a long-ranged logarithmic potential, with an average dislocation density determined by the form of the external shear stress [15].…”

“…We find the localization domain boundaries of the longitudinal pileup phonon modes by calculating the band edge of the phonon dispersion for a uniform pileup (see Appendix A or Ref. [15]), given by the simple condition,…”

“…where L is the length of the pileup, and the average dislocation charge densities are [14,15], as plotted in Fig. 8.…”

“…In Sec. II, as a simplified warm up problem, we illustrate inhomogeneityinduced quasi-localization of lattice excitations using a onedimensional (1d) model of dislocation pileups [14,15], which exhibits the same phenomenon as the 2d conic sheet studied in the main body of the paper. In Sec.…”