Physics:Electron-longitudinal acoustic phonon interaction

The electron-longitudinal acoustic phonon interaction is an interaction that can take place between an electron and a longitudinal acoustic (LA) phonon in a material such as a semiconductor.

Displacement operator of the LA phonon

The equations of motion of the atoms of mass M which locates in the periodic lattice is

[math]\displaystyle{ M \frac {d^{2}} {dt^{2}} u_{n} = -k_{0} ( u_{n-1} + u_{n+1} -2u_{n} ) }[/math],

where [math]\displaystyle{ u_{n} }[/math] is the displacement of the nth atom from their equilibrium positions.

Defining the displacement [math]\displaystyle{ u_{\ell} }[/math] of the [math]\displaystyle{ \ell }[/math]th atom by [math]\displaystyle{ u_{\ell}= x_{\ell} - \ell a }[/math], where [math]\displaystyle{ x_{\ell} }[/math] is the coordinates of the [math]\displaystyle{ \ell }[/math]th atom and [math]\displaystyle{ a }[/math] is the lattice constant,

the displacement is given by [math]\displaystyle{ u_{l}= A e^{i ( q \ell a - \omega t)} }[/math]

Then using Fourier transform:

[math]\displaystyle{ Q_{q} = \frac {1} {\sqrt {N}} \sum_{\ell} u_{\ell} e^{- i q a \ell } }[/math]

and

[math]\displaystyle{ u_{\ell} = \frac {1} {\sqrt {N}} \sum_{q} Q_{q} e^{ i q a \ell } }[/math].

Since [math]\displaystyle{ u_{\ell} }[/math] is a Hermite operator,

[math]\displaystyle{ u_{\ell} = \frac {1} {2 \sqrt{N}} \sum_{q} (Q_{q} e^{iqa\ell} + Q^{\dagger}_{q} e^{-iqa\ell} ) }[/math]

From the definition of the creation and annihilation operator [math]\displaystyle{ 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}) }[/math]

[math]\displaystyle{ Q_{q} }[/math] is written as

[math]\displaystyle{ Q_{q} = \sqrt { \frac {\hbar} {2M\omega_{q}}}(a^{\dagger}_{-q}+a_{q}) }[/math]

Then [math]\displaystyle{ u_{\ell} }[/math] expressed as

[math]\displaystyle{ u_{\ell} = \sum_{q} \sqrt {\frac {\hbar} {2MN\omega_{q}}} (a_{q} e^{iqa\ell} + a^{\dagger}_{q} e^{-iqa\ell}) }[/math]

Hence, using the continuum model, the displacement operator for the 3-dimensional case is

[math]\displaystyle{ 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} ] }[/math],

where [math]\displaystyle{ e_{q} }[/math] is the unit vector along the displacement direction.

Interaction Hamiltonian

The electron-longitudinal acoustic phonon interaction Hamiltonian is defined as [math]\displaystyle{ H_\text{el} }[/math]

[math]\displaystyle{ H_\text{el} = D_\text{ac} \frac{\delta V}{V} = D_\text{ac} \, \mathop{\rm div} \, u(r) }[/math],

where [math]\displaystyle{ D_\text{ac} }[/math] is the deformation potential for electron scattering by acoustic phonons.

Inserting the displacement vector to the Hamiltonian results to

[math]\displaystyle{ H_\text{el} = D_\text{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} ] }[/math]

Scattering probability

The scattering probability for electrons from [math]\displaystyle{ |k \rangle }[/math] to [math]\displaystyle{ |k' \rangle }[/math] states is

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

[math]\displaystyle{ = \frac {2 \pi} {\hbar} \left| D_\text{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} ] }[/math]

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

[math]\displaystyle{ P(k,k') = \frac {2 \pi} {\hbar} \left( D_\text{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} ], }[/math]

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

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

See also

• Phonon scattering
• Umklapp scattering

References

• Hamaguchi, Chihiro (2017). Basic Semiconductor Physics. Graduate Texts in Physics (3 ed.). Springer. pp. 265–363. doi:10.1007/978-3-319-66860-4. ISBN 978-3-319-88329-8. https://www.springer.com/gp/book/9783319668598. 
• Yu, Peter Y.; Cardona, Manuel (2005). Fundamentals of Semiconductors (3rd ed.). Springer.