Electron—Phonon Interaction

Abstract

The Fröhlich Hamiltonian describes the interaction between a single electron in a solid and LO (longitudinal optical) phonons:

$$H = \sum\limits_p {\frac{{{p^2}}}{{2m}}} {c_p}^ + {c_p} + {\varpi _0}\sum\limits_q {{a_q}^ + {a_q} + } \sum\limits_{qp} {\frac{{{M_0}}}{{{v^{1/2}}}}} \frac{1}{{\left| q \right|}}c_{p + q}^ + {c_p}\left( {{a_q} + {a_q}^ + } \right)$$

$${M_0}^2\frac{{4\pi \alpha \rlap{--} h{{\left( {\rlap{--} h{\varpi _0}} \right)}^{3/2}}}}{{{{\left( {2m} \right)}^{1/2}}}}$$

((6.1.1))

$$\alpha = \frac{{{e^2}}}{{\rlap{--} h}}{\left( {\frac{m}{{2\rlap{--} h{\varpi _0}}}} \right)^{1/2}}\left( {\frac{1}{{{\varepsilon _\infty }}} - \frac{1}{{{\varepsilon _0}}}} \right)$$