Ionized-impurity scattering in the weak-screening limit

The ionized-impurity-scattering problem in free-carrier transport is considered for the regime where half the average distance between impurities, D, is much less than the screening length ${\lambda }_s$. Using a potential-cutoff approach, it is shown that the scattering by short-range potential variations due to the discrete positions of the ions becomes negligible in the limit d≡(${\lambda }_s$/D${)}^3$≳$b^{3/2}$/(8g${)}^3$ /4, where b=4$k^2$ ${\lambda }_s^2$, k is the electron wave vector, and g(b)= ln(b+1)-b/(b+1). In this region the dominant mechanism is scattering by long-range potential fluctuations due to random inhomogeneities in the impurity concentration. A random-potential-scattering theory is used to show that, for fluctuations γ small compared with the electron energy E, the momentum relaxation time reverts to the Brooks-Herring form, even though the single-site scattering picture is formally inappropriate. It is shown that the criterion γ≳E is nearly equivalent to the single-site validity criterion 〈${\tau }_D$/${\tau }_R$〉≲1 derived earlier for the small-d regime, where ${\tau }_D$ is the duration of the collisions, ${\tau }_R$ is the momentum relaxation time, and the brackets denote a weighted average over partial waves. Furthermore, at large d the linearized Thomas-Fermi approximation is valid only as long as γ≲E. Whenever γ≳E, significant discrepancies between single-site theoretical mobilities and experimental values are observed for a number of common semiconductors. It is suggested that spatial inhomogeneities in the electron density due to the fluctuating potential may be partially responsible for these discrepancies.