sizes due to to strain relaxation and screening by the electron gas and surface charges (see the comparison between QWs, SKs and QDs in Fig. 10). 

The calculation reproduces the luminescence energies of the 1 and 2.5 nm thick QDs within the error bars, and the downward trend of the electric field with increasing QD size evidenced in the experiment. Still, the calculation underestimates the luminescence energy of the 4 nm thick QDs, where the Stark effect is the largest, by about 0 . 3 eV. Looking at Fig. 8, this suggests that the electric field in this QD is overestimated by ≃ 20 %. The reason for this discrepancy is unclear at present. Increasing the density of surface states to pin the Fermi level in the bandgap of AlN increases the electric field and ultimately charges the dots. The experimental data have, moreover, been collected at low enough excitation power to prevent screening by photogenerated multiple electron-hole pairs. 15,22,56 The calculated electric field might be affected by the electromechanical coupling 57,58 (influence of the electric field on strains), by the uncertainties in the pyro- and piezoelectric constants of GaN and especially AlN, and by their dependence on strains (non-linear piezoelectricity). 59,60 A simple 1D model however shows that the electromechanical coupling should reduce the electric field by at most ≃ 5% (see Appendix A). Although the nanowires of Ref. 20 are likely metal-polar, their polarity has not actually been assessed experimentally. 56 Interestingly, we would like to point out that a N-face polarity would give a slightly better agreement between theory and experiment. In that case, the pyro- and piezo-electric field are reversed, so that a hole gas forms at the interface between the GaN pillar and heterostructure and electrons accumulate at the top AlN surface. The difference of potential across the heterostructure [Eq. (9)] then becomes Δ V = E$_{v}$ (GaN) − E$_{c}$ (AlN) = − 5 . 45 eV, which is slightly higher in magnitude than for the metal-face polarity [Δ V = E$_{c}$ (GaN) − E$_{v}$ (AlN) = 4 . 3 eV]. As a consequence, the pyro- and piezoelectric field in the QD are better screened, so that the luminescence in the 4 nm thick QDs is raised by ≃ 150 meV. Further experiments (for different barrier and QD thicknesses), as well as detailed polarity measurements might therefore be needed to get a complete picture. 

## V. CONCLUSION 

We have modeled the quantum confined Stark effect in [0001]-oriented AlN/GaN/AlN nanowire heterostructures using a tight-binding approach. We have taken strain relaxation 

