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

$M\frac{d^2}{d{t}^2}{u}_n=-k_0(u_{n-1}+u_{n+1}-2u_n)$,

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 $u_l=A{e}^{i(q\ell a-\omega t)}$

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,

$u_{\ell }=\frac{1}{2\sqrt{N}}\sum _q(Q_q{e}^{iqa\ell }+Q_q^{†}{e}^{-iqa\ell })$

From the definition of the creation and annihilation operator $a_q^{†}=\frac{q}{\sqrt{2M\hslash {\omega }_q}}(M{\omega }_q{Q}_{-q}-i{P}_q),a_q=\frac{q}{\sqrt{2M\hslash {\omega }_q}}(M{\omega }_q{Q}_{-q}+i{P}_q)$

$Q_q$ is written as

$Q_q=\sqrt{\frac{\hslash }{2M{\omega }_q}}(a_{-q}^{†}+a_q)$

Then $u_{\ell }$ expressed as

$u_{\ell }=\sum _q\sqrt{\frac{\hslash }{2MN{\omega }_q}}(a_q{e}^{iqa\ell }+a_q^{†}{e}^{-iqa\ell })$

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

$u(r)=\sum _q\sqrt{\frac{\hslash }{2MN{\omega }_q}}{e}_q[a_q{e}^{iq\cdot r}+a_q^{†}{e}^{-iq\cdot r}]$,

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

Interaction Hamiltonian

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

$H_{\text{el}}=D_{\text{ac}}\frac{\delta V}{V}=D_{\text{ac}}\mathrm{d}\mathrm{i}\mathrm{v}u(r)$,

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

Inserting the displacement vector to the Hamiltonian results to

$H_{\text{el}}=D_{\text{ac}}\sum _q\sqrt{\frac{\hslash }{2MN{\omega }_q}}(i{e}_q\cdot q)[a_q{e}^{iq\cdot r}-a_q^{†}{e}^{-iq\cdot r}]$

Scattering probability

The scattering probability for electrons from $|k⟩$ to $|k^{\prime }⟩$ states is

$P(k,k^{\prime })=\frac{2\pi }{\hslash }\mid ⟨k^{\prime },q^{\prime }|H_{\text{el}}|k,q⟩{\mid }^2\delta [\epsilon (k^{\prime })-\epsilon (k)\mp \hslash {\omega }_q]$

$=\frac{2\pi }{\hslash }{|D_{\text{ac}}\sum _q\sqrt{\frac{\hslash }{2MN{\omega }_q}}(i{e}_q\cdot q)\sqrt{n_q+\frac{1}{2}\mp \frac{1}{2}}\frac{1}{L^3}\int d^{3}r{u}_{k^{\prime }}^{\ast }(r)u_k(r)e^{i(k-k^{\prime }\pm q)\cdot r}|}^2\delta [\epsilon (k^{\prime })-\epsilon (k)\mp \hslash {\omega }_q]$

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

$P(k,k^{\prime })=\frac{2\pi }{\hslash }{(D_{\text{ac}}\sum _q\sqrt{\frac{\hslash }{2MN{\omega }_q}}|q|\sqrt{n_q+\frac{1}{2}\mp \frac{1}{2}}I(k,k^{\prime }){\delta }_{k^{\prime },k\pm q})}^2\delta [\epsilon (k^{\prime })-\epsilon (k)\mp \hslash {\omega }_q],$

where $I(k,k^{\prime })=\mathrm{\Omega }\underset{\mathrm{\Omega }}{\int }{d}^{3}r{u}_{k^{\prime }}^{\ast }(r)u_k(r)$, $\mathrm{\Omega }$ is the volume of a unit cell.

$P(k,k^{\prime })=\{\begin{array}{cc}\frac{2\pi }{\hslash }{D}_{\text{ac}}^2\frac{\hslash }{2MN{\omega }_q}|q{|}^2{n}_q& (k^{\prime }=k+q;\text{absorption}),\\ \frac{2\pi }{\hslash }{D}_{\text{ac}}^2\frac{\hslash }{2MN{\omega }_q}|q{|}^2(n_q+1)& (k^{\prime }=k-q;\text{emission}).\end{array}$

See also

• Phonon scattering
• Umklapp scattering

References

• Template:Cite book
• Template:Cite book