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