Electron-longitudinal acoustic phonon interaction
# 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

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

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

Defining the displacement $u_{\ell }$ of the $\ell$th atom by $u_{\ell }=x_{\ell }-\ell a$, where $x_{\ell }$ is the coordinates of the $\ell$th atom and $a$ is the lattice constant,

the displacement is given by $u_{l}=Ae^{i(q\ell a-\omega t)}$

Then using Fourier transform:

    $Q_{q}={\frac {1}{\sqrt {N}}}\sum _{\ell }u_{\ell }e^{-iqa\ell }$

and

    $u_{\ell }={\frac {1}{\sqrt {N}}}\sum _{q}Q_{q}e^{iqa\ell }$.

Since $u_{\ell }$ is a Hermite operator,

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

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

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

Then $u_{\ell }$ expressed as

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

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

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

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

## Interaction Hamiltonian

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

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

where $D_{\text{ac}}$ is the deformation potential for electron scattering by acoustic phonons.

Inserting the displacement vector to the Hamiltonian results to

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

## Scattering probability

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

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

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

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

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

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