Electron-longitudinal acoustic phonon interaction

Electron-longitudinal acoustic phonon interaction is an equation concerning atoms.

Displacement operator of the longitudinal acoustic phonon

The equation of motions of the atoms of mass M which locates in the periodic lattice is

$M \frac {d^{2}} {dt^{2}} u_{n} = -k_{0} ( u_{n-1} + u_{n+1} -2u_{n} )$,

where $u_{n}$ is the displacement of the nth atom from their equilibrium positions.

If we define the displacement $u_{l}$ of the nth atom by $u_{l}= x_{l} - la$, where $x_{l}$ is the coordinates of the lth atom and a is the lattice size,

the displacement is given by $u_{n}= A e^{i q l a - \omega t}$

Using Fourier transform, we can define

$Q_{q} = \frac {1} {\sqrt {N}} \sum_{l} u_{l} e^{- i q a l }$

and

$u_{l} = \frac {1} {\sqrt {N}} \sum_{q} Q_{q} e^{ i q a l }$.

Since $u_{l}$ is a Hermite operator,

$u_{l} = \frac {1} {2 \sqrt{N}} \sum_{q} (Q_{q} e^{iqal} + Q^{\dagger}_{q} e^{-iqal} )$

From the definition of the creation and annihilation operator $a^{\dagger}_{q} = \frac {q} {\sqrt{2M\hbar\omega_{q}}}(M\omega_{q}Q_{-q}-iP_{q}), \; a_{q} = \frac {q} {\sqrt{2M\hbar\omega_{q}}}(M\omega_{q}Q_{-q}+iP_{q})$

$Q_{q}$ is written as

$Q_{q} = \sqrt { \frac {\hbar} {2M\omega_{q}}}(a^{\dagger}_{-q}+a_{q})$

Then $u_{l}$ expressed as

$u_{l} = \sum_{q} \sqrt {\frac {\hbar} {2MN\omega_{q}}} (a_{q} e^{iqal} + a^{\dagger}_{q} e^{-iqal})$

Hence, when we use continuum model, the displacement for the 3-dimensional case is

$u(r) = \sum_{q} \sqrt{ \frac {\hbar}{2M N \omega_{q} } } e_{q} [ a_{q} e^{ i q \cdot r} + a^{\dagger}_{q} e^{-i q \cdot r} ]$,

where $e_{q}$ is the unit vector along the displacement direction.

Interaction Hamiltonian

The electron-longitudinal acoustic phonon interaction Hamiltonian is defined as $H_{el}$

$H_{el} = D_{ac} \frac{\delta V}{V} = D_{ac} \, div \, u(r)$,

where $D_{ac}$ is the deformation potential for electron scattering by acoustic phonons.^[1]

Inserting the displacement vector to the Hamiltonian results to

$H_{el} = D_{ac} \sum_{q} \sqrt{ \frac {\hbar} {2 M N \omega_{q} } } ( i e_{q} \cdot q ) [ a_{q} e^{i q \cdot r} - a^{\dagger}_{q} e^{-i q \cdot r} ]$

Scattering probability

The scattering probability for electrons from $|k \rangle$ to $|k' \rangle$ states is

$P(k,k') = \frac {2 \pi} {\hbar} \mid \langle k' , q' | H_{el}| \ k , q \rangle \mid ^ {2} \delta [ \varepsilon (k') - \varepsilon (k) \mp \hbar \omega_{q} ]$

$= \frac {2 \pi} {\hbar} \left| D_{ac} \sum_{q} \sqrt{ \frac {\hbar} {2 M N \omega_{q} } } ( i e_{q} \cdot q ) \sqrt { n_{q} + \frac {1} {2} \mp \frac {1} {2} } \, \frac {1} {L^{3}} \int d^{3} r \, u^{\ast}_{k'} (r) u_{k} (r) e^{i ( k - k' \pm q ) \cdot r } \right|^2 \delta [ \varepsilon (k') - \varepsilon (k) \mp \hbar \omega_{q} ]$

Replace the integral over the whole space with a summation of unit cell integrations

$P(k,k') = \frac {2 \pi} {\hbar} \left( D_{ac} \sum_{q} \sqrt{ \frac {\hbar} {2 M N \omega_{q} } } | q | \sqrt { n_{q} + \frac {1} {2} \mp \frac {1} {2} } \, I(k,k') \delta_{k' , k \pm q } \right)^2 \delta [ \varepsilon (k') - \varepsilon (k) \mp \hbar \omega_{q} ],$

where $I(k,k') = \Omega \int_{\Omega} d^{3}r \, u^{\ast}_{k'} (r) u_{k} (r)$, $\Omega$ is the volume of a unit cell.

$P(k,k') = \begin{cases} \frac {2 \pi} {\hbar} D_{ac}^2 \frac {\hbar} {2 M N \omega_{q} } | q |^2 n_{q} & (k' = k + q ; \text{absorption}), \\ \frac {2 \pi} {\hbar} D_{ac}^2 \frac {\hbar} {2 M N \omega_{q} } | q |^2 ( n_{q} + 1 ) & (k' = k - q ; \text{emission}). \end{cases}$

Notes

1. Hamaguchi 2001, p. 208.

References

• C. Hamaguchi (2001). Basic Semiconductor Physics. Springer. pp. 183–239.
• Yu, Peter Y. and Cardona, Manuel (2005). Fundamentals of Semiconductors (3rd ed.). Springer.