Electron-longitudinal acoustic phonon interaction explained

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 (d ^{2})/(dt ^{2}) u _{n} = - k _{0} ( u _{n-1} + u _{n+1} -2 u _{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}

Then using Fourier transform:

Q _{q} = (1)/(\sqrt {N) } \sum_ u_ e^

and

u _{\ell} = (1)/(\sqrt {N) } \sum_ Q_ e^.

Since

u _{\ell} is a Hermite operator,u _{\ell} = (1)/(2 \sqrt{N) } \sum_ (Q_ e^ + Q^_ e^)a ^{\dagger} _{q} = (q)/(\sqrt{ 2 M\hbar\omega _{q}) }(M\omega_Q_-iP_), \; a_ = \frac (M\omega_Q_+iP_)Q _{q} is written asQ _{q} = \sqrt { (\hbar)/(2 M\omega _{q}) }(a^_+a_)

Then

u _{\ell} expressed asu _{\ell} = \sum _{q} \sqrt { (\hbar)/(2 MN\omega _{q}) } (a_ e^ + a^_ e^)

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

u(r) = \sum _{q} \sqrt{ (\hbar)/(2 M N \omega _{q}) } e _{q} [ a _{q} e + a ^{\dagger} _{q} e ^{-i} ] ,

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} (\delta V)/(V) = D _{ac} \rmdiv 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{ (\hbar)/(2 M N \omega _{q}) } ( i e _{q} ⋅ q ) [ a _{q} e ^{i} - a ^{\dagger} _{q} e ^{-i} ]

Scattering probability

The scattering probability for electrons from

|k \rangle to |k' \rangle states isP(k,k') = (2 \pi)/(\hbar) \mid \langle k' , q' | H _{el|} k , q \rangle \mid ^{2} \delta [ \varepsilon (k') - \varepsilon (k) \mp \hbar \omega _{q} ] = (2 \pi)/(\hbar) \left| D _{ac} \sum _{q} \sqrt{ (\hbar)/(2 M N \omega _{q}) } ( i e _{q} ⋅ q ) \sqrt { n _{q} + (1)/(2) \mp (1)/(2) } (1)/(L ^{3}) \int d ^{3} r u ^{\ast} _{k'} (r) u _{k} (r) e ^{i} \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') = (2 \pi)/(\hbar) \left( D _{ac} \sum _{q} \sqrt{ (\hbar)/(2 M N \omega _{q}) } | q | \sqrt { n _{q} + (1)/(2) \mp (1)/(2) } I(k,k') \delta _{k'} \right) ^{2} \delta [ \varepsilon (k') - \varepsilon (k) \mp \hbar \omega _{q} ],

where

I(k,k') = \Omega \int _{\Omega} d ^{3} r u ^{\ast} _{k'} (r) u _{k} (r) , \Omega is the volume of a unit cell.P(k,k') = \begin{cases} (2 \pi)/(\hbar) D ^{2} _{ac} (\hbar)/(2 M N \omega _{q}) | q | ^{2} n _{q} & (k' = k + q ; absorption ), \\ (2 \pi)/(\hbar) D ^{2} _{ac} (\hbar)/(2 M N \omega _{q}) | q | ^{2} ( n _{q} + 1 ) & (k' = k - q ; emission ). \end{cases}

See also

• Phonon scattering
• Umklapp scattering

References

• Book: Hamaguchi, Chihiro . Basic Semiconductor Physics . Graduate Texts in Physics . 2017 . 3 . Springer . 265–363 . 10.1007/978-3-319-66860-4 . 978-3-319-88329-8 .
• Book: Yu, Peter Y. . Cardona, Manuel . Fundamentals of Semiconductors . 3rd . Springer . 2005.

Notes and References

1. Book: Hamaguchi, Chihiro . Basic Semiconductor Physics . Graduate Texts in Physics . 2017 . 3 . Springer . 292 . 10.1007/978-3-319-66860-4 . 978-3-319-88329-8 .