Mott transition in many-valley semiconductors

The binding energy of a conduction electron bound to a positive ion of a substitutional impurity atom in a doped semiconductor through a Lindhard potential and a Hubbard-Sham potential is calculated as a function of mobile carrier concentration. A variational approach is used in which the trial wave function is the solution of the problem of an electron bound in the Hulth\'en potential. For the case of an electron in the Lindhard potential the values of the electron concentration at which the Mott transition takes place (Nc) are found to be such as to make the product aNc1/3=0.398, 0.271, and 0.263 for one-, four-, and six-valley conduction bands, respectively. Here a is the effective Bohr radius. These values are considerably larger than those obtained by Krieger and Nightingale using a hydrogenic trial wave function. For an electron in the Hubbard-Sham potential the values of aNc1/3 for one-, four-, and six-valley conduction bands are found to be 0.436, 0.299, and 0.290, respectively. These values agree very well with those obtained by Martino et al., from the numerical integration of the corresponding Schr\"odinger equation. The possibility of comparing our results with those obtained experimentally is discussed.