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

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.

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

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

Using Fourier transform, we can define

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

and

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

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

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

Q_{q}

is written as

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

Then $u_{l}$ expressed as

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

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 $H_{el}$

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

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

Inserting the displacement vector to the Hamiltonian results to

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 $|k\rangle$ to $|k'\rangle$ states is

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}]

={\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

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

P(k,k')={\begin{cases}{\frac {2\pi }{\hbar }}D_{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

↑ 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.

This article is issued from Wikipedia - version of the 5/29/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.