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We can get a rough estimate of E$_{QD}$ , n$_{p}$ and n$_{s}$ by neglecting piezoelectricity [ P$_{inf}$ = P$_{sup}$ = P$_{0}$ (AlN), P$_{pil}$ = P$_{QD}$ = P$_{0}$ (GaN)]. We then get from equations (11) and (19b): 

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As a simple example, the pyroelectric field in a 4 nm thick QD embedded in an infinitely long nanowire ( t$_{het}$ → ∞ ) would be | E$_{QD}$ | = 7 . 03 MV/cm. In a finite heterostructure with t$_{het}$ = 20 nm, the induced charges screen this field down to | E$_{QD}$ | = 3 . 47 MV/cm. The density of the electron gas at the interface with the GaN pillar is then n$_{p}$ = 1 . 73 × 10 13 cm − $^{2}$, while the total density of charges (surface states+holes) at the top surface is n$_{s}$ = 3 . 85 × 10 13 cm − $^{2}$. Therefore, the Fermi level is actually pinned in the valence band of AlN as long as the total (donor) surface states density is lower than N + crit = 3 . 85 × 10 13 cm − $^{2}$. This critical density, although large, is yet not unreasonable for bare nanowire surfaces. We will however give further evidence in paragraph IV that the Fermi level is pinned at (or at least close to) the valence band edge of AlN. 

We have tested this simple 1D model against the numerical solution of equations (3)– (7). It gives excellent account of the electric field in the QD when t$_{het}$ < ∼ 2 R . The effects of the non-intentional doping are indeed negligible with respect to the amount of charges transferred by the pyro- and piezoelectric field. This model however tends to overestimate n$_{p}$ (as the electric field in the pillar actually decreases over tens of nanometers) and thus overestimates n$_{s}$ (by around 25% in the above example). Also, the piezoelectric polarization and field become inhomogeneous in thick heterostructures, as the strains are maximum at the interfaces and relax in between 27,28 (see Fig. 6). The 1D model above is nonetheless very helpful in understanding trends and guiding the design of nanowire heterostructures. 

## C. Discussion 

The amplitude of the electric field | E$_{QD}$ | , computed with Eqs. (3)–(7) as the difference of potential along the QD axis divided by t$_{QD}$ , is plotted in Fig. 7a as a function of t$_{inf}$ and t$_{sup}$ ( t$_{QD}$ = 4 nm). As expected from Eq. (20a), the electric field increases with the 

