# 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

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

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

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

the displacement is given by ${\displaystyle u_{n}=Ae^{iqla-\omega t}}$

Using Fourier transform, we can define

${\displaystyle Q_{q}={\frac {1}{\sqrt {N}}}\sum _{l}u_{l}e^{-iqal}}$

and

${\displaystyle u_{l}={\frac {1}{\sqrt {N}}}\sum _{q}Q_{q}e^{iqal}}$.

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

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

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

${\displaystyle Q_{q}}$ is written as
${\displaystyle Q_{q}={\sqrt {\frac {\hbar }{2M\omega _{q}}}}(a_{-q}^{\dagger }+a_{q})}$

Then ${\displaystyle u_{l}}$ expressed as

${\displaystyle u_{l}=\sum _{q}{\sqrt {\frac {\hbar }{2MN\omega _{q}}}}(a_{q}e^{iqal}+a_{q}^{\dagger }e^{-iqal})}$

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

${\displaystyle u(r)=\sum _{q}{\sqrt {\frac {\hbar }{2MN\omega _{q}}}}e_{q}[a_{q}e^{iq\cdot r}+a_{q}^{\dagger }e^{-iq\cdot r}]}$,

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

## Interaction Hamiltonian

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

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

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

Inserting the displacement vector to the Hamiltonian results to

${\displaystyle H_{el}=D_{ac}\sum _{q}{\sqrt {\frac {\hbar }{2MN\omega _{q}}}}(ie_{q}\cdot q)[a_{q}e^{iq\cdot r}-a_{q}^{\dagger }e^{-iq\cdot r}]}$

## Scattering probability

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

${\displaystyle P(k,k')={\frac {2\pi }{\hbar }}\mid \langle k',q'|H_{el}|\ k,q\rangle \mid ^{2}\delta [\varepsilon (k')-\varepsilon (k)\mp \hbar \omega _{q}]}$
${\displaystyle ={\frac {2\pi }{\hbar }}\left|D_{ac}\sum _{q}{\sqrt {\frac {\hbar }{2MN\omega _{q}}}}(ie_{q}\cdot q){\sqrt {n_{q}+{\frac {1}{2}}\mp {\frac {1}{2}}}}\,{\frac {1}{L^{3}}}\int d^{3}r\,u_{k'}^{\ast }(r)u_{k}(r)e^{i(k-k'\pm q)\cdot r}\right|^{2}\delta [\varepsilon (k')-\varepsilon (k)\mp \hbar \omega _{q}]}$

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

${\displaystyle P(k,k')={\frac {2\pi }{\hbar }}\left(D_{ac}\sum _{q}{\sqrt {\frac {\hbar }{2MN\omega _{q}}}}|q|{\sqrt {n_{q}+{\frac {1}{2}}\mp {\frac {1}{2}}}}\,I(k,k')\delta _{k',k\pm q}\right)^{2}\delta [\varepsilon (k')-\varepsilon (k)\mp \hbar \omega _{q}],}$

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

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

## Notes

1. ^ Hamaguchi 2001, p. 208.

## References

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