= Solution
For a <hydrogenic donor state>, <dielectric screening> replaces $\epsilon_0$ by $\epsilon_r\epsilon_0$ and the <electron mass> by the <band effective mass>. The characteristic orbital radius is therefore
$$
a^*=\frac{4\pi\epsilon_0\epsilon_r\hbar^2}{m^*e^2}
=\epsilon_r\frac{m_e}{m^*}a_0.
$$
Using $a_0=0.0529177\,\mathrm{nm}$ gives
$$
\boxed{a^*=120a_0=6.35\,\mathrm{nm}.}
$$
A <donor-state overlap estimate> compares this radius with the mean dopant spacing $d\sim N_D^{-1/3}$. Taking $d\sim a^*$ yields
$$
\boxed{N_D\sim(a^*)^{-3}=3.91\times10^{24}\,\mathrm{m^{-3}}
=3.91\times10^{18}\,\mathrm{cm^{-3}}.}
$$
There is no sharp geometrical onset: hydrogenic <wavefunctions> have exponential tails. If “overlap” is defined by neighbouring orbital \i[diameters] touching, $d\sim2a^*$, the corresponding estimate is $4.88\times10^{23}\,\mathrm{m^{-3}}$, smaller by a factor of eight. These are conventions for the onset of appreciable <orbital overlap>, not an exact critical density for a <metal-insulator transition>.
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