The scattering rate due to electron-plasmon interaction can be obtained using Fermi's golden rule (2.95) and the Hamiltonian
(2.130).
Plasmon scattering of electrons represents the long-range part of the electron-electron interaction [39]. Assuming a nonparabolic
and spherical analytical band structure (2.77), the scattering rate is given by:
![$\displaystyle \lambda_{plas}\left(k\right)=\frac{e^{2}\hbar} {16\pi\varepsilon\...
...pm\frac{1}{2}\right)\left(k^{2}-k_{f}^{2}\right)^{2} \ln\frac{q_{c}}{q_{\min}},$](img565.png) |
(2.131) |
The final electron energy is given as:
![$\displaystyle \epsilon_{f}=\epsilon_{i}\mp\hbar\omega_{pl},$](img566.png) |
(2.132) |
where the plasma frequency is (see Appendix B):
![$\displaystyle \omega_{pl}=\sqrt{\frac{{{e^{2}}}}{{\varepsilon}}\sum_{v}\frac{n_{v}}{(m_{d}^{*})_{v}}}.$](img567.png) |
(2.133) |
Here the summation over all possible valleys is assumed, and
stands for the contribution from valley
to the electron density.
The cut-off wave vector
is defined as follows:
![$\displaystyle q_{c}=\min\left(q_{\max},\beta_{s}\right).$](img571.png) |
(2.134) |
and
stand for the boundaries of the momentum transfer:
|
|
![$\displaystyle q_{\min}=\left\vert k-k_{f}\right\vert,$](img574.png) |
(2.135) |
|
|
![$\displaystyle q_{\max} = \left\vert k+k_{f}\right\vert.$](img575.png) |
|
is the final wave vector defined by the equation:
![$\displaystyle \epsilon(k_{f})=\epsilon_{f},$](img577.png) |
(2.136) |
is the average number of the plasmon excitations defined by the equilibrium Bose-Einstein statistics:
![$\displaystyle N\left(\omega_{pl}\right)=\frac{1}{\exp\bigl(\frac{\hbar\omega_{pl}}{k_{B}T_{L}}\bigr)-1}.$](img579.png) |
(2.137) |
S. Smirnov: