Past exam of the quantum condensed matter course of the University of Cambridge / 2007 / Section B / B1 / b

The hydrogenic donor state radius is the Bohr radius rescaled by screening and effective mass:The donor-state overlap estimate compares the mean spacing $n_D^{-1/3}$ to this radius. Taking the spacing as $a_B^{\ast }$ gives $n_D\sim (a_B^{\ast }{)}^{-3}=1.32\times 10^{25}{\mathrm{m}}^{-3}=1.32\times 10^{19}\mathrm{c}{\mathrm{m}}^{-3}$. Taking it as the orbital diameter gives a factor eight less, $1.65\times 10^{18}\mathrm{c}{\mathrm{m}}^{-3}$. Thus a geometric estimate is roughly $10^{18}$–$10^{19}\mathrm{c}{\mathrm{m}}^{-3}$, with its contact convention stated. The exponential tails do not have a sharp overlap boundary, and this estimate should not be identified with a precise metal-insulator transition threshold.