Moving least-square basis for band-structure calculations of natural and artificial crystals

The chapter presents a unified mesh-free framework based on moving least-square (MLS) approximation for real-space band-structure calculations of semiconductors, photonic crystals, and phononic crystals. The cell-periodic mesh-free shape function represents the periodicity in these natural and artificial crystal structures. The MLS basis of mesh-free method lead to an efficient real-space technique that is implemented into the Schrödinger equation for calculating electronic structures, the Maxwell equations for photonic crystals, and finally the elastic wave equations for phononic crystals. The chapter describes the application of the periodic mesh-free shape function to the frequency band-structure computation of 2D homogeneous photonic crystal and examines several types of inhomogeneous photonic band-gap materials to study the performance of mesh-free method for the band-structure calculation of photonic crystals. Thus, various value-periodic problems are simulated by using periodic MLS mesh-free basis. Some of the examples are pattern formation problems, including Ginzburg-Landau equation and Swift-Hohenberg equation; the surface morphology of soft materials; quantum island formations; strain-induced nanopatterning on surface; and the periodic array of quantum heterostructures.