Polarizability enhancement in doped semiconductors near the metal-insulator transition

The static polarizability at $T=0\mathrm{}$ K of a doped semiconductor is evaluated in the single-bubble diagram approximation. The impurities are assumed to have a Gaussian distribution around an "averaged" regular array which is immersed in a dielectric continuum background. The effect of the randomness to the leading order in $\frac{\sigma }{d}$, $\frac{a_0}{d}$, where $d$ is the averaged lattice constant, $\sigma$ is the Gaussian width, and $a_0$ is the effective Bohr radius, is found to produce an ever increasing enhancement of the polarizability as the doping concentration is increased.