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For shallow hydrogenic donor states of radius $a_B^{\ast }$ and random donor number density $N_D$, the mean spacing is of order $N_D^{-1/3}$. Taking this spacing comparable to $a_B^{\ast }$ gives $N_D\sim (a_B^{\ast }{)}^{-3}$; a geometric diameter criterion instead gives $N_D\sim (2a_B^{\ast }{)}^{-3}$. 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.

For a hydrogenic donor state, replace the bare Electron mass by $m^{\ast }=0.1m_e$ and the Coulomb permittivity by ${ϵ}_r{ϵ}_0$ with ${ϵ}_r=16$. Rescaling the Bohr model gives the positive ground-state binding magnitudeThe radius isusing the Bohr radius $a_0=5.29177\times 10^{-11}\mathrm{m}$. The supplied positive hydrogen “energy” formula should be interpreted as a binding magnitude; the bound level's energy relative to the conduction-band edge is $-E_B$.

At $300\mathrm{K}$, $k_{B}T=25.9\mathrm{meV}$, about five times the binding energy. Thus a substantial fraction, ordinarily most of the shallow donors in the dilute nondegenerate regime, will be ionised. An exact fraction also depends on the Fermi level, donor concentration and degeneracies; comparing $E_B$ with $k_{B}T$ alone does not determine it exactly. The large radius explains why a weakly bound donor extends across many unit cells.

For a hydrogenic donor state, dielectric screening replaces ${ϵ}_0$ by ${ϵ}_r{ϵ}_0$ and the electron mass by the band effective mass. The characteristic orbital radius is thereforeUsing $a_0=0.0529177\mathrm{nm}$ givesA donor-state overlap estimate compares this radius with the mean dopant spacing $d\sim N_D^{-1/3}$. Taking $d\sim a^{\ast }$ yieldsThere is no sharp geometrical onset: hydrogenic wavefunctions have exponential tails. If “overlap” is defined by neighbouring orbital diameters touching, $d\sim 2a^{\ast }$, the corresponding estimate is $4.88\times 10^{23}{\mathrm{m}}^{-3}$, 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.