Electron-longitudinal acoustic phonon interaction
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Electron-longitudinal acoustic phonon interaction is an equation concerning atoms.
Contents
Displacement operator of the longitudinal acoustic phonon
The equation of motions of the atoms of mass M which locates in the periodic lattice is
- ,
where is the displacement of the nth atom from their equilibrium positions.
If we define the displacement of the nth atom by , where is the coordinates of the lth atom and a is the lattice size,
the displacement is given by
Using Fourier transform, we can define
and
- .
Since is a Hermite operator,
From the definition of the creation and annihilation operator
- is written as
Then expressed as
Hence, when we use continuum model, the displacement for the 3-dimensional case is
- ,
where is the unit vector along the displacement direction.
Interaction Hamiltonian
The electron-longitudinal acoustic phonon interaction Hamiltonian is defined as
- ,
where is the deformation potential for electron scattering by acoustic phonons.[1]
Inserting the displacement vector to the Hamiltonian results to
Scattering probability
The scattering probability for electrons from to states is
- Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): = \frac {2 \pi} {\hbar} \left| D_{ac} \sum_{q} \sqrt{ \frac {\hbar} {2 M N \omega_{q} } } ( i e_{q} \cdot q ) \sqrt { n_{q} + \frac {1} {2} \mp \frac {1} {2} } \, \frac {1} {L^{3}} \int d^{3} r \, u^{\ast}_{k'} (r) u_{k} (r) e^{i ( k - k' \pm q ) \cdot r } \right|^2 \delta [ \varepsilon (k') - \varepsilon (k) \mp \hbar \omega_{q} ]
Replace the integral over the whole space with a summation of unit cell integrations
where Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): I(k,k') = \Omega \int_{\Omega} d^{3}r \, u^{\ast}_{k'} (r) u_{k} (r) , is the volume of a unit cell.
- Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): P(k,k') = \begin{cases} \frac {2 \pi} {\hbar} D_{ac}^2 \frac {\hbar} {2 M N \omega_{q} } | q |^2 n_{q} & (k' = k + q ; \text{absorption}), \\ \frac {2 \pi} {\hbar} D_{ac}^2 \frac {\hbar} {2 M N \omega_{q} } | q |^2 ( n_{q} + 1 ) & (k' = k - q ; \text{emission}). \end{cases}
Notes
- ↑ Hamaguchi 2001, p. 208.
References
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