Since $E_G \gg k_B T$, we can only use the law of mass action.
Since $E_G \gg k_B T$, we can only use the law of mass action.
But the question offers us another piece of information - we are around $|N_D-N_A| \approx n_i$.
But the question offers us another piece of information - we are around $|N_D-N_A| \approx n_i$.
That means that we are near the transition between extrinsic and intrinsic regimes.
That means that we are near the transition between extrinsic and intrinsic regimes.
However, we also know that dopants energies are quite small such that $E_C-E_A\ll E_G$ and $E_A-E_V \ll E_G$.
However, we also know that dopants energies are quite small such that $E_C-E_D\ll E_G$ and $E_A-E_V \ll E_G$.
That means that we expect $n_i \ll n_D$ and $n_i \ll n_A$ (since both $n_i$ and dopant ionazition depends exponentially of the corresponding energy differences).
That means that we expect $n_i \ll n_D$ and $n_i \ll n_A$ (since both $n_i$ and dopant ionazition depends exponentially of the corresponding energy differences).
Therefore, we can confidently say that $N_D \gg n_D$ and $N_A \gg n_A$ so we esentially recover the dopant ionization condition.
Therefore, we can confidently say that $N_D \gg n_D$ and $N_A \gg n_A$ so we esentially recover the dopant ionization condition.
Thus, neglecting $n_D$ and $n_A$ such that they are both 0, the solutions to the charge balance are:
Thus, neglecting $n_D$ and $n_A$ such that they are both 0, the solutions to the charge balance are:
