Past exam of the quantum condensed matter course of the University of Cambridge / 2015 / Paper 1 / 1 / c / Solution

A hydrogenic donor has the effective Bohr radius of a hydrogenic donorusing the external $a_0=0.52917721067\times 10^{-10}\mathrm{m}$.^[ref] Increasing donor density causes donor states to overlap and screens their binding, allowing a metallic impurity system rather than isolated frozen-out donors.

Using the standard approximate Mott criterion, $n_D^{1/3}{a}_B^{\ast }\simeq 0.25$, givesThe factor 0.25 is an externally documented empirical criterion, not supplied exam data or an exact universal threshold.^[ref] An order-of-magnitude overlap argument without that prefactor would give $(a_B^{\ast }{)}^{-3}\simeq 1.3\times 10^{25}{\mathrm{m}}^{-3}$. Compensation, disorder and multivalley anisotropy affect a real material's transition; this estimate uses the scalar mass and permittivity actually supplied, without substituting other germanium parameters.