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•• The donor state wave functions in silicon have small binding energies (Ebinding ≈ 0.05 eV) and large Bohr radii (ad ≈ 3 nm) for two reasons: (1) Because of the interaction between the lattice and the “nearly free” electrons in the conduction band, the electrons behave as if their mass was smaller than the free-electron mass. The “effective mass” in silicon is m* ≈ 0.2 me. (2) The polarization of the silicon lattice partially shields the +1e charge of the impurity site. In general, the force of electrostatic attraction between two charges in a polarizable medium is given by F = kq1 q2/(Kr2), where K is a dimensionless number called the dielectric constant. In silicon, K = 11.7. (a) Derive expressions for the Bohr radius and the ionization energy of the ground state of a donor wave function with effective mass m* and in a medium with dielectric constant K. (b) Compute the values of the ionization energy and Bohr radius of a donor state in silicon. (Compare your values with the known values for phosphorus in silicon of (Ebinding = 45 meV and adonor = 3.0 nm.)
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