Donor-polarizability enhancement as the insulator-metal transition is approached from the insulating side

The low-temperature static dielectric constant for $n$-type semiconductors shows a significant upward deviation from Clausius-Mossotti behavior as the donor concentration $N$ approaches the critical value $N_c$ for the onset of metallic behavior. After reviewing the possible reasons for this upward deviation and the various types of screening applied to the insulator-metal problem, a new continuum-model calculation of the donor-polarizability enhancement $\frac{{\alpha }_D\mathrm{}\left(N\right)\mathrm{}}{{\alpha }_D\mathrm{}\left(0\right)\mathrm{}}$ is presented which is based on an electrostatic donor-donor interaction potential and the Clausius-Mossotti relationship for the long-wavelength static dielectric constant $\epsilon \mathrm{}\left(N\right)\mathrm{}$. The Hassé variational approach is employed to calculate ${\alpha }_D\mathrm{}\left(N\right)\mathrm{}$. The calculated enhancement contains two contributions: (1) a size effect associated with an enlargement of the donor wave function with $N$, and (2) an increase in the average effective dielectric constant $〈\epsilon \mathrm{}(r,N)\mathrm{}〉$ with $N$ seen by the localized donor wave function. The results show $\frac{{\alpha }_D\left(N_c\right)}{{\alpha }_D\mathrm{}\left(0\right)\mathrm{}}$ ranging from about 2 for Si:Sb to about 3 for Si:As. Comparison with the experimental data for Si:As shows the calculated upward deviation from Clausius-Mossotti behavior remains smaller than the data. The values of $N_{c-\mathrm{calc}}$ [based on a polarization catastrophe, a new self-consistent Clausius-Mossotti relationship, and a self-consistent Herzfeld criterion for $N_c$ from $\epsilon (N\to N_c)\to \infty$] are between 60% and 100% larger than the experimental values for the silicon shallow donors. Possible corrections to this calculation resulting from the donor randomness, charge transfer, and corrections to the Lorentz-Lorenz local field are also discussed.