Isbell conjugacy

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

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

where $u_n$ is the displacement of the nth atom from their equilibrium positions.

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 $u_n=A{e}^{iqla-\omega t}$

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,

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

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 $u_l$ expressed as

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

Hence, when we use continuum model, the displacement 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 $H_{el}$

$H_{el}=D_{ac}\frac{\delta V}{V}=D_{ac}divu(r)$,

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

Inserting the displacement vector to the Hamiltonian results to

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

Scattering probability

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

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

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

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

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

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.

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

Notes

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References

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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
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