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{{Quantum book backlink|Condensed matter and solid-state physics}}

The '''electron–phonon interaction''' is a fundamental interaction in [[condensed matter physics]] describing how an [[electron]] couples to quantized lattice vibrations known as [[phonon]]s. In particular, 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]].

Electron–phonon interactions play a central role in determining key physical properties of solids, including [[electrical conductivity]], [[thermal conductivity]], and phenomena such as [[superconductivity]] and [[carrier scattering]]. In semiconductors, scattering of electrons by acoustic phonons is one of the dominant mechanisms limiting [[electron mobility]] at finite temperatures.

<div style="float:right; border:1px solid #e0d890; background:#fff8cc; padding:6px; margin:0 0 1em 1em; width:420px;">
[[File:Lattice wave.svg|400px]]
<div style="font-size:90%;">Propagation of a lattice vibration (phonon) through a crystal. Electron–phonon interaction arises when electrons couple to these collective oscillations.</div>
</div>

== Overview ==
In a crystalline solid, atoms are arranged in a periodic [[crystal lattice]]. Small displacements of atoms from their equilibrium positions give rise to collective vibrational modes. When these modes are quantized, they are described as phonons.

An electron moving through such a lattice interacts with these vibrations. Physically, this interaction arises because lattice distortions locally modify the potential experienced by the electron. In the case of longitudinal acoustic phonons, the interaction is associated with compressions and expansions of the lattice, leading to changes in the local electronic energy via the [[deformation potential]].

== Displacement operator of the LA phonon ==
The equations of motion of atoms of mass ''M'' in a periodic lattice are

:<math>M \frac {d^{2}} {dt^{2}} u_{n} = -k_{0} ( u_{n-1} + u_{n+1} -2u_{n} )</math>,

where <math>u_{n}</math> is the displacement of the ''n''th atom from its equilibrium position.

Defining the displacement <math>u_{\ell}</math> by
:<math>u_{\ell}= x_{\ell} - \ell a</math>,

where <math>a</math> is the lattice constant, the displacement takes the form of a wave:

:<math>u_{\ell}= A e^{i ( q \ell a - \omega t)}</math>

Using a [[Fourier transform]]:

:<math>Q_{q} = \frac {1} {\sqrt {N}} \sum_{\ell} u_{\ell} e^{- i q a \ell }</math>

:<math>u_{\ell} = \frac {1} {\sqrt {N}} \sum_{q} Q_{q} e^{ i q a \ell }</math>

Since <math>u_{\ell}</math> is Hermitian:

:<math>u_{\ell} = \frac {1} {2 \sqrt{N}} \sum_{q} (Q_{q} e^{iqa\ell} + Q^{\dagger}_{q} e^{-iqa\ell} )</math>

Introducing [[creation and annihilation operators]]:

:<math>Q_{q} = \sqrt { \frac {\hbar} {2M\omega_{q}}}(a^{\dagger}_{-q}+a_{q})</math>

the displacement becomes

:<math>u_{\ell} = \sum_{q} \sqrt {\frac {\hbar} {2MN\omega_{q}}} (a_{q} e^{iqa\ell} + a^{\dagger}_{q} e^{-iqa\ell})</math>

In three dimensions, the displacement operator is

:<math>u(r) = \sum_{q} \sqrt{ \frac {\hbar}{2M N \omega_{q} } } e_{q} [ a_{q} e^{ i q \cdot r} + a^{\dagger}_{q} e^{-i q \cdot r} ]</math>

where <math>e_{q}</math> is the polarization direction.

== Interaction Hamiltonian ==
The electron–phonon interaction Hamiltonian is given by

:<math>H_\text{el} = D_\text{ac} \, \mathrm{div}\, u(r)</math>

where <math>D_\text{ac}</math> is the deformation potential constant.<ref>
{{cite book |last=Hamaguchi |first=Chihiro |title=Basic Semiconductor Physics |year=2017 |publisher=Springer |page=292 |isbn=978-3-319-88329-8}}
</ref>

Substituting the displacement field:

:<math>H_\text{el} = D_\text{ac} \sum_{q} \sqrt{ \frac {\hbar} {2 M N \omega_{q} } } ( i e_{q} \cdot q ) [ a_{q} e^{i q \cdot r} - a^{\dagger}_{q} e^{-i q \cdot r} ]</math>

This Hamiltonian describes how electrons absorb or emit phonons while moving through the lattice.

== Scattering probability ==
The probability for an electron to scatter from state <math>|k\rangle</math> to <math>|k'\rangle</math> is

:<math>P(k,k') = \frac {2 \pi} {\hbar} \mid \langle k' , q' | H_\text{el}| k , q \rangle \mid ^ {2} \delta [ \varepsilon (k') - \varepsilon (k) \mp \hbar \omega_{q} ]</math>

which leads to

:<math>P(k,k') =
\begin{cases}
\frac {2 \pi} {\hbar} D_\text{ac}^2 \frac {\hbar} {2 M N \omega_{q} } | q |^2 n_{q} & \text{(absorption)} \\
\frac {2 \pi} {\hbar} D_\text{ac}^2 \frac {\hbar} {2 M N \omega_{q} } | q |^2 ( n_{q} + 1 ) & \text{(emission)}
\end{cases}</math>

This expression shows that scattering depends on the phonon occupation number <math>n_q</math>, linking the process directly to temperature via [[Bose–Einstein statistics]].

== Physical significance ==
Electron–phonon interaction is responsible for several important physical effects:

* '''Electrical resistance''': scattering of electrons by phonons limits conductivity
* '''Thermal transport''': phonons carry heat and interact with charge carriers
* '''Superconductivity''': electron–phonon coupling leads to [[Cooper pair]] formation
* '''Polaron formation''': electrons can become dressed by lattice distortions

At low temperatures, phonon populations decrease, reducing scattering. At higher temperatures, increased phonon density enhances electron scattering.

== See also ==
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}
* [[Phonon]]
* [[Phonon scattering]]
* [[Umklapp scattering]]
* [[Polaron]]
* [[Superconductivity]]

= References =
{{reflist|3}}

{{Author|Harold Foppele}}
{{Sourceattribution|Physics:Quantum Electron-phonon interaction|1}}

Physics:Quantum Electron-phonon interaction - Revision history

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