Past exam of the condensed matter physics course of the University of Oxford / 2012 / 3 / b

A substitutional group-V atom such as phosphorus has one more valence electron than germanium and acts as a donor impurity, giving an n-type semiconductor. Approximate its outer electron by a hydrogenic donor state with screened Coulomb attraction and effective mass $m_e^{\ast }$. The ionization energy is the positive binding-energy magnitudeIts effective Bohr radius of a hydrogenic donor is about $3.80\mathrm{nm}$, much larger than an atomic core, consistent with a shallow extended bound state.

At $300\mathrm{K}$, $k_{B}T=25.9\mathrm{meV}$, while the mean three-dimensional classical translational kinetic energy is $3k_{B}T/2=38.8\mathrm{meV}$. Both exceed the donor binding energy, so substantial thermal ionization is expected, unlike excitation across the much larger $0.661\mathrm{eV}$ host band gap. A precise ionized fraction also depends on the chemical potential, dopant density and donor charge-state degeneracy; comparing two energy scales alone does not prove complete ionization at arbitrary concentration.