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

: $M \frac u_ = -k_ ( u_ + u_ -2u_ )$,

where $u_$ is the displacement of the ''n''th atom from their equilibrium positions.

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

the displacement is given by $u_= A e^$

Then using Fourier transform:

: $Q_ = \frac \sum_ u_ e^$

and

: $u_ = \frac \sum_ Q_ e^$.

Since $u_$ is a Hermite operator,

: $u_ = \frac \sum_ (Q_ e^ + Q^_ e^ )$

From the definition of the creation and annihilation operator $a^_ = \frac (M\omega_Q_-iP_), \; a_ = \frac (M\omega_Q_+iP_)$

: $Q_$ is written as

: $Q_ = \sqrt (a^_+a_)$

Then $u_$ expressed as

: $u_ = \sum_ \sqrt (a_ e^ + a^_ e^)$

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

: u(r) = \sum_ \sqrt e_ a_ e^ + a^_ e^ ,

where $e_$ is the unit vector along the displacement direction.

Interaction Hamiltonian

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

: $H_\text = D_\text \frac = D_\text \, \mathop \, u(r)$,

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

Inserting the displacement vector to the Hamiltonian results to

: H_\text = D_\text \sum_ \sqrt ( i e_ \cdot q ) a_ e^ - a^_ e^ /math>

Scattering probability

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

: P(k,k') = \frac \mid \langle k' , q' , H_\text, \ k , q \rangle \mid ^ \delta \varepsilon (k') - \varepsilon (k) \mp \hbar \omega_ /math>

: = \frac \left, D_\text \sum_ \sqrt ( i e_ \cdot q ) \sqrt \, \frac \int d^ r \, u^_ (r) u_ (r) e^ \^2 \delta \varepsilon (k') - \varepsilon (k) \mp \hbar \omega_

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

: P(k,k') = \frac \left( D_\text \sum_ \sqrt , q , \sqrt \, I(k,k') \delta_ \right)^2 \delta \varepsilon (k') - \varepsilon (k) \mp \hbar \omega_

where $I(k,k') = \Omega \int_ d^r \, u^_ (r) u_ (r)$, $\Omega$ is the volume of a unit cell.

: $P(k,k') = \begin \frac D_\text^2 \frac , q , ^2 n_ & (k' = k + q ; \text), \\ \frac D_\text^2 \frac , q , ^2 ( n_ + 1 ) & (k' = k - q ; \text). \end$

See also

* Phonon scattering
* Umklapp scattering

Notes

References

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{{DEFAULTSORT:Electron-Longitudinal Acoustic Phonon Interaction
Atomic physics