FIG. 1: (Color online) Structure of the GaN/AlN nanowire heterostructures. The radius of the nanowires is R = 15 nm. atomic positions using a conjugate gradients algorithm. The strains ε$_{αβ}$ on each atom are then calculated from the atomic positions using a method similar 33 to Ref. 34. The pyro- and piezoelectric polarization density is next computed from the strains on each cation (Ga or Al) : where z ≡ [0001], P$_{0}$ is the spontaneous polarization, and e$_{13}$ , e$_{33}$ and e$_{15}$ are the piezo- electric constants of either GaN or AlN (see Table I). Poisson’s equation for the pyro- and piezoelectric potential V$_{p}$ ( r ): is then solved on a finite difference grid 38 (see appendix I of Ref. 27 for details). κ is the dielectric constant ( κ = 9 inside the nanowire and κ = 1 outside). κ$_{0}$ ∇$_{r}$ · κ ( r ) ∇$_{r}$ V$_{p}$ ( r ) = ∇$_{r}$ · P ( r ) (2) P = P$_{0}$ z +    2 e$_{15}$ε$_{xz}$ 2 e$_{15}$ε$_{yz}$ e$_{31}$ ( ε$_{xx}$ + ε$_{yy}$ ) + e$_{33}$ε$_{zz}$    , (1) 4