2.5.5.4 Second Born Correction

The additional correction applied to the ionized impurity scattering rate results from the second Born approximation. The Schwinger scattering amplitude is used to obtain a correction to the first Born approximation. The second Born correction is given by:

$$\Delta\lambda\left(k\right)=s_{0}\left(k\right)C\left(k\right) \int_{0}^{2k}\frac{q}{(q^{2}+\beta_{s}^{2})^{2}}\,dq,$$ (2.163)

where the expression for $s_{0}(k)$ is

$$s_{0}\left(k\right)=\frac{a}{1+\frac{4k^{2}}{\beta_{s}^{2}}-a},$$ (2.164)

with the parameters $a$ and $U_{0}$ defined as:

$$a=\frac{U_{0}}{\beta_{s}}\left(1-\frac{U_{0}}{4\beta_{s}}\right),$$ (2.165)

$$U_{0}=\frac{2m^{*}_{d}}{\hbar^{2}}\frac{Ze^{2}}{4\pi\varepsilon}.$$ (2.166)

The first Born approximation is valid when ${\vert U_{0}\vert}/{\beta_{s}}\ll1$ [51]. This inequality is violated at low and intermediate doping levels, where the second Born correction thus plays an important role. S. Smirnov: