Physics:Quantum Electron-phonon interaction - Revision history

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	{{Quantum book backlink\|Condensed matter and solid-state physics}}		{{Quantum book backlink\|Condensed matter and solid-state physics}}
	{{Quantum article nav\|previous=Physics:Quantum Phonons\|previous label=Phonons\|next=Physics:Quantum Exchange interaction\|next label=Exchange interaction}}		{{Quantum article nav\|previous=Physics:Quantum Phonons\|previous label=Phonons\|next=Physics:Quantum Exchange interaction\|next label=Exchange interaction}}

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	{{Short description\|Quantum Collection topic on Quantum Electron-phonon interaction}}		{{Short description\|Quantum Collection topic on Quantum Electron-phonon interaction}}

	{{Quantum book backlink\|Condensed matter and solid-state physics}}		{{Quantum book backlink\|Condensed matter and solid-state physics}}
	{{Quantum article nav\|previous=Physics:Quantum Phonons\|previous label=Phonons\|next=Physics:Quantum Exchange interaction\|next label=Exchange interaction}}		{{Quantum article nav\|previous=Physics:Quantum Phonons\|previous label=Phonons\|next=Physics:Quantum Exchange interaction\|next label=Exchange interaction}}

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	{{Short description\|Quantum Collection topic on Quantum Electron-phonon interaction}}~~Book I~~		{{Short description\|Quantum Collection topic on Quantum Electron-phonon interaction}}

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	~~~~{{Short description\|Quantum Collection topic on Quantum Electron-phonon interaction}}		{{Short description\|Quantum Collection topic on Quantum Electron-phonon interaction}}Book I

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	{{Short description\|Quantum Collection topic on Quantum Electron-phonon interaction}}		{{Short description\|Quantum Collection topic on Quantum Electron-phonon interaction}}

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

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	== Overview ==		== 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.		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]].		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 ==		== Displacement operator of the LA phonon ==
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	:<math>u_{\ell} = \frac {1} {2 \sqrt{N}} \sum_{q} (Q_{q} e^{iqa\ell} + Q^{\dagger}_{q} e^{-iqa\ell} )</math>		:<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]]:		Introducing creation and annihilation operators:

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

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	{{Short description\|Quantum Collection topic on Quantum Electron-phonon interaction}}		{{Short description\|Quantum Collection topic on Quantum Electron-phonon interaction}}

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

	Using a [[Fourier transform]]:		Using a Fourier transform:

	:<math>Q_{q} = \frac {1} {\sqrt {N}} \sum_{\ell} u_{\ell} e^{- i q a \ell }</math>		:<math>Q_{q} = \frac {1} {\sqrt {N}} \sum_{\ell} u_{\ell} e^{- i q a \ell }</math>
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	\end{cases}</math>		\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]].		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 ==		== Physical significance ==
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	* '''Electrical resistance''': scattering of electrons by phonons limits conductivity		* '''Electrical resistance''': scattering of electrons by phonons limits conductivity
	* '''Thermal transport''': phonons carry heat and interact with charge carriers		* '''Thermal transport''': phonons carry heat and interact with charge carriers
	* '''Superconductivity''': electron–phonon coupling leads to [[Cooper pair]] formation		* '''Superconductivity''': electron–phonon coupling leads to Cooper pair formation
	* '''Polaron formation''': electrons can become dressed by lattice distortions		* '''Polaron formation''': electrons can become dressed by lattice distortions

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

	= References =		= References =

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	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.		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;">~~
	~~<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>~~
	</div>		</div>

	<div style="width:300px;">		<div style="width:300px;">
	[[File:Lattice wave.svg\|thumb\|280px\|~~Quantum Electron-~~phonon interaction.]]		[[File:Lattice wave.svg\|thumb\|280px\|Propagation of a lattice vibration (phonon) through a crystal. Electron–phonon interaction arises when electrons couple to these collective oscillations.]]
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			{{Short description\|Quantum Collection topic on Quantum Electron-phonon interaction}}

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

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	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]].		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]].

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	<div style="float:right; border:1px solid #e0d890; background:#fff8cc; padding:6px; margin:0 0 1em 1em; width:420px;">		<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 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>
			</div>

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			[[File:Lattice wave.svg\|thumb\|280px\|Quantum Electron-phonon interaction.]]
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	{{Quantum book backlink\|Condensed matter and solid-state physics}}		{{Quantum book backlink\|Condensed matter and solid-state physics}}
	{{Quantum article nav\|previous=Physics:Quantum Phonons\|previous label=Phonons\|next=Physics:Quantum Exchange interaction\|next label=Exchange interaction}}		{{Quantum article nav\|previous=Physics:Quantum Phonons\|previous label=Phonons\|next=Physics:Quantum Exchange interaction\|next label=Exchange interaction}}