Resolution of the Scaling Exponent Puzzle for Weakly Compensated Crystalline Silicon and Germanium Metal-Insulator Systems

Using a classical theory for ionized impurity scattering, it is demonstrated that in the degenerate regime the conductivity scales as $\sqrt{{\epsilon }_F}$ where the Fermi energy is measured with respect to the mobility edge. The approach, a special case of alloy theory, explains the conductivity scaling exponent $s=\frac{1}{2}$ observed for weakly compensated, doped crystalline Si and Ge. The results explain the breadth of scaling range and suggest how to obtain a consistent picture of the scaling of the mobility, diffusion coefficient, and Hall coefficient.