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

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

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

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

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

Then using Fourier transform:

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

and

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

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

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

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

${\displaystyle Q_{q}}$ is written as

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

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

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

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

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

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

Interaction Hamiltonian

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

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

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

Inserting the displacement vector to the Hamiltonian results to

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

Scattering probability

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

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

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

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

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

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

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

See also

• Phonon scattering
• Umklapp scattering