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
,
where
is the displacement of the nth atom from their equilibrium positions.
Defining the displacement
of the
th atom by
, where
is the coordinates of the
th atom and
is the lattice constant,
the displacement is given by
Then using Fourier transform:
![{\displaystyle Q_{q}={\frac {1}{\sqrt {N}}}\sum _{\ell }u_{\ell }e^{-iqa\ell }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/974ea0a119966e28480c8a59cc94a66b2fe6fbb6)
and
.
Since
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 })}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c6d50b302d4043f6f80f4150edb623177f9a3d17)
From the definition of the creation and annihilation operator
is written as
![{\displaystyle Q_{q}={\sqrt {\frac {\hbar }{2M\omega _{q}}}}(a_{-q}^{\dagger }+a_{q})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c93372df3dc613f55a3e24c17eb28955368b44b)
Then
expressed as
![{\displaystyle u_{\ell }=\sum _{q}{\sqrt {\frac {\hbar }{2MN\omega _{q}}}}(a_{q}e^{iqa\ell }+a_{q}^{\dagger }e^{-iqa\ell })}](https://wikimedia.org/api/rest_v1/media/math/render/svg/44ea049347f7cbb26ebe305e79ed35a95ced0831)
Hence, using the continuum model, the displacement operator for the 3-dimensional case is
,
where
is the unit vector along the displacement direction.
Interaction Hamiltonian
The electron-longitudinal acoustic phonon interaction Hamiltonian is defined as
,
where
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}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/491786bb6f58b07f6004fbaa8a9e1e53a56b1815)
Scattering probability
The scattering probability for electrons from
to
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}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5cda341c9eb0f160d3c0fcb5810503aa3e10b429)
![{\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}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d125377ab63e94b46e913a834213666654a8059)
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}],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d484d21d36fe4f7a4f6988e0655e9534caa926b)
where
,
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8fdcc86f82669d67f580b0c8f5729e49684b4621)
See also
Notes
References