Theory of Donor States in Silicon

By using the recently measured effective masses for $n$-type Si, $m_1=0.98\mathrm{}$ $m$ and $m_2=0.19\mathrm{}$ m, approximate solutions of the resulting effective mass Schroedinger equation are obtained. The accuracy of the solutions was tested in the limiting cases where $\frac{m_2}{m_1}=1\mathrm{} \mathrm{and} 0$ respectively. The nature of the resulting states and their degeneracy is discussed in some detail, taking into account the fact that the conduction band of Si has six equivalent minima. Experimentally measured ionization energies show that the effective mass theory is seriously in error in the case of the ground state. This error is attributed to failure of the effective mass theory near the donor nucleus, and allowance for this failure is made in the case of higher states. This leads finally to a theoretical spectrum for the electrons bound by P, As, or Sb donors.