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.

Propagation of a lattice vibration (phonon) through a crystal. Electron–phonon interaction arises when electrons couple to these collective oscillations.

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

M \frac{d^{2}}{d t^{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 its equilibrium position.

Defining the displacement $u_{ℓ}$ by

u_{ℓ} = x_{ℓ} - ℓ a

,

where $a$ is the lattice constant, the displacement takes the form of a wave:

u_{ℓ} = A e^{i (q ℓ a - ω t)}

Using a Fourier transform:

Q_{q} = \frac{1}{\sqrt{N}} \sum_{ℓ} u_{ℓ} e^{- i q a ℓ}

u_{ℓ} = \frac{1}{\sqrt{N}} \sum_{q} Q_{q} e^{i q a ℓ}

Since $u_{ℓ}$ is Hermitian:

u_{ℓ} = \frac{1}{2 \sqrt{N}} \sum_{q} (Q_{q} e^{i q a ℓ} + Q_{q}^{†} e^{- i q a ℓ})

Introducing creation and annihilation operators:

Q_{q} = \sqrt{\frac{ℏ}{2 M ω_{q}}} (a_{- q}^{†} + a_{q})

the displacement becomes

u_{ℓ} = \sum_{q} \sqrt{\frac{ℏ}{2 M N ω_{q}}} (a_{q} e^{i q a ℓ} + a_{q}^{†} e^{- i q a ℓ})

In three dimensions, the displacement operator is

u (r) = \sum_{q} \sqrt{\frac{ℏ}{2 M N ω_{q}}} e_{q} [a_{q} e^{i q \cdot r} + a_{q}^{†} e^{- i q \cdot r}]

where $e_{q}$ is the polarization direction.

Interaction Hamiltonian

The electron–phonon interaction Hamiltonian is given by

H_{el} = D_{ac} d i v u (r)

where $D_{ac}$ is the deformation potential constant.^[1]

Substituting the displacement field:

H_{el} = D_{ac} \sum_{q} \sqrt{\frac{ℏ}{2 M N ω_{q}}} (i e_{q} \cdot q) [a_{q} e^{i q \cdot r} - a_{q}^{†} e^{- i q \cdot r}]

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 $| k ⟩$ to $| k^{'} ⟩$ is

P (k, k^{'}) = \frac{2 π}{ℏ} ∣ ⟨ k^{'}, q^{'} | H_{el} | k, q ⟩ ∣^{2} δ [ε (k^{'}) - ε (k) \mp ℏ ω_{q}]

which leads to

P (k, k^{'}) = {\begin{cases} \frac{2 π}{ℏ} D_{ac}^{2} \frac{ℏ}{2 M N ω_{q}} | q |^{2} n_{q} & (absorption) \\ \frac{2 π}{ℏ} D_{ac}^{2} \frac{ℏ}{2 M N ω_{q}} | q |^{2} (n_{q} + 1) & (emission) \end{cases}

This expression shows that scattering depends on the phonon occupation number $n_{q}$ , linking the process directly to temperature via Bose–Einstein statistics.

Physics:Quantum Electron-phonon interaction

Overview

Displacement operator of the LA phonon

Interaction Hamiltonian

Scattering probability

Physical significance

See also

Table of contents (185 articles)

Index

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