= Donor-state overlap estimate
For shallow <hydrogenic donor states> of radius $a_B^*$ and random donor <number density> $N_D$, the mean spacing is of order $N_D^{-1/3}$. Taking this spacing comparable to $a_B^*$ gives $N_D\sim(a_B^*)^{-3}$; a geometric diameter criterion instead gives $N_D\sim(2a_B^*)^{-3}$. The orbitals have exponentially decaying tails rather than hard boundaries, so there is no exact geometric onset. Appreciable overlap also limits the independent-donor ionisation model; a genuine transport transition needs additional electronic physics.
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