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<img src='content_image/354769.jpg'>

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) : 

<img src='content_image/354768.jpg'>

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 ): 

<img src='content_image/354767.jpg'>

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). 

