Disappearance of impurity levels in silicon and germanium due to screening

We have studied the disappearance of impurity levels in silicon and germanium due to free-carrier screening of the Coulomb field of the impurity ions. The ground-state eigenfunctions and eigenvalues have been calculated for electrons described by an ellipsoidal effective-mass Hamiltonian. A two-dimensional finite-element analysis was used to obtain the solutions. Only moderate carrier densities (1019 cm−3 for silicon and 1018 cm−3 for germanium) are needed to cause the impurity levels to disappear into the conduction band, the result at high doping densities being simply a degenerate semiconductor.