What insures the charge neutrality condition in the self-consistent solution of Poisson and Schrodinger equations?

Hello,

Let's say we wish to have a self-consistent Poisson-Schrodinger calculation for a nanowire in equilibrium.

The electron concentration obtained from the Schrodinger equation (which is supposed to handle the confinement problem) is calculated using the following formula:

$n = \sum_v g_v \sum_k N_{1D}\vert \psi_k(r) \vert ^ 2 \mathcal{F}_{-1/2}\(\frac{E_{Fn}-E_k}{k_BT})$

with $g_v$ being the valley degeneracy, and the states are obtained from the 2D Schrodinger equation in the transverse direction.

I was wondering what guarantees that this equation for electron concentration results in charge neutrality (i.e. $N_D = n$ for the whole device)?

Does the self-consistent solution result in charge neutrality by construction? If not, how should we take care of this problem (the problem of not having charge neutrality in our answers)?

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