The generalized plane piezoelectric problem: Theoretical formulation and application to heterostructure nanowires

We present a systematic methodology for the reformulation of a broad class of three-dimensional (3D) piezoelectric problems into a two-dimensional (2D) mathematical form. The sole underlying hypothesis is that the system geometry and material properties as well as the applied loads (forces and charges) and boundary conditions are translationally invariant along some direction. This requisite holds exactly in idealized indefinite systems and to a high degree of approximation, in the sense of Saint-Venant's principle, in finite but slender systems. This class of problems is commonly denoted here as the generalized plane piezoelectric (GPP) problem. For non-piezoelectric systems, the problem becomes purely elastic and is then called the generalized plane strain (GPS) problem. The first advantage of the generalized plane problems is that they are more manageable from both analytical and computational points of view. Moreover, they are flexible enough to accommodate any geometric cross section, crystal class symmetry, axis orientation and a wide range of boundary conditions. As an illustration we present numerical simulation of indefinite lattice-mismatched core-shell nanowires made of diamond Ge/Si and zincblende piezoelectric InN/GaN materials. The remarkable agreement with exact 3D simulations of finite versions of those systems reveal the GPP approach as a reliable procedure to study accurately and with moderate computing resources the strain and electric field distribution in elongated piezoelectric systems.