Fröhlich Hamiltonian for Electron-LO Phonon Coupling

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Hello! Welcome back.

In our last lesson, we derived the deformation potential Hamiltonian, which describes how electrons couple to the lattice strain created by long-wavelength acoustic phonons. This is a short-range interaction, crucial for understanding electron scattering in materials like silicon and germanium. However, this is not the whole story. In polar crystals, a much stronger, long-range interaction mechanism often dominates.

Today, we will explore this mechanism. Our goal is to derive the Fröhlich Hamiltonian, which describes the coupling between electrons and longitudinal optical (LO) phonons in polar materials. This interaction is fundamental to the electronic and optical properties of a vast class of materials, including most III-V semiconductors (like GaAs), II-VI compounds (like CdS), and ionic insulators (like NaCl). Understanding it is key to analyzing carrier mobility, hot electron relaxation, and even the nature of the charge carriers themselves.

1. The Origin of Polar Electron-Phonon Coupling

The deformation potential interaction arose from lattice strain. The Fröhlich interaction arises from electrostatics. The key ingredients are:

A Polar Crystal: A crystal with a basis of at least two non-equivalent atoms per unit cell, which carry effective opposite charges (e.g., Ga⁺ and As⁻ in Gallium Arsenide).
Optical Phonons: In these modes, the different atoms within a unit cell vibrate against each other.

When these two features combine, the out-of-phase motion of oppositely charged ions creates a net oscillating electric dipole moment in each unit cell. This cloud of microscopic dipoles gives rise to a macroscopic polarization field $P$ .

Now, consider the nature of this oscillating polarization field for different types of optical phonons:

Transverse Optical (TO) Phonons: The ions oscillate perpendicular to the phonon's propagation direction $q$ . This creates a transverse polarization wave, which can couple strongly with transverse electromagnetic waves (light), leading to absorption at the TO phonon frequency, $ω_{T O}$ .
Longitudinal Optical (LO) Phonons: The ions oscillate parallel to the propagation direction $q$ . This creates a longitudinal polarization wave.

LO Phonons, Surface Phonon Polaritons, and Fröhlich Interaction — This figure illustrates the key concepts. Panel (a) shows an LO phonon creating a polarization **P** parallel to its wavevector **k** (our **q**). This longitudinal polarization is the source of the Fröhlich interaction. Panel (c), left, shows the Feynman diagram for this interaction: an electron (e⁻) interacts by exchanging an LO phonon.

A crucial result from electromagnetism is that a spatially varying longitudinal polarization field creates a net bound charge density, $ρ_{b o u n d} = - \nabla \cdot P$ . This charge density, in turn, is the source of a long-range, macroscopic longitudinal electric field. It is this electric field that an electron moving through the crystal will feel and couple to.

2. Deriving the Interaction Potential

Our goal is to find the potential energy of an electron interacting with the electric field generated by an LO phonon. We can sketch out the derivation using a continuum model, which is highly accurate for this long-range interaction.

From Polarization to Potential: An electron with charge $- e$ in an electric potential $ϕ (r)$ has potential energy $V (r) = - e ϕ (r)$ . The potential is related to the polarization field $P (r)$ via Poisson's equation for the bound charge: $\nabla^{2} ϕ = - ρ_{b o u n d} / ϵ_{0} = (\nabla \cdot P) / ϵ_{0}$ .
Fourier Space: Working in Fourier space simplifies this. The gradient $\nabla$ becomes $i q$ and the Laplacian $\nabla^{2}$ becomes $- q^{2}$ . Poisson's equation transforms to:
$- q^{2} ϕ (q) = \frac{i q \cdot P ( q )}{ϵ _{0}}$
For a longitudinal wave, $P (q)$ is parallel to $q$ , so $q \cdot P (q) = q P (q)$ . This gives the potential in Fourier space:
$ϕ (q) = - \frac{i P ( q )}{q ϵ _{0}}$
Relating Polarization to Phonon Modes: The final step is to relate the polarization $P (q)$ to the quantum operators for the LO phonon mode. This is the most involved part of the derivation, relying on the electrostatics of a polarizable dielectric medium. The standard result, which we will take as given, connects the potential $ϕ (r)$ directly to the phonon creation and annihilation operators ( $b_{q}^{†}, b_{q}$ ).

The interaction potential $V_{e - p h} (r) = - e ϕ (r)$ can be expressed as a sum over all phonon modes $q$ :

V^_{e - p h} (r) = q \sum (V_{q} b_{q} e^{i q \cdot r} + V_{q}^{*} b_{q}^{†} e^{- i q \cdot r})

where the coupling coefficient $V_{q}$ is found to be:

V_{q} = - i \frac{e ^{2} ℏ ω _{L O}}{2 V ϵ _{0} q ^{2}} (\frac{1}{ϵ _{\infty}} - \frac{1}{ϵ _{s}})

Here, $V$ is the crystal volume, and $ω_{L O}$ is the LO phonon frequency (which we approximate as constant, independent of $q$ ).

The two dielectric constants are key:

$ϵ_{s}$ is the static (low-frequency) dielectric constant. It measures the screening response from both the fast-moving electron clouds and the slower, heavy ions.
$ϵ_{\infty}$ is the high-frequency (optical) dielectric constant. It measures the screening from only the electron clouds, as the ions are too massive to respond to high-frequency fields.

The term $(1/ ϵ_{\infty} - 1/ ϵ_{s})$ is therefore a direct measure of the strength of the ionic polarizability—precisely the physical effect we are trying to model.

Test your understanding!

Look at the expression for the coupling strength $V_\mathbf{q}$. How does the strength of the interaction depend on the phonon wavevector $q=|\mathbf{q}|$? Compare this to the deformation potential coupling for LA phonons, where the matrix element $M(\mathbf{q})$ was proportional to $\sqrt{q}$. What does this imply about the range of the two interactions?

Show answer

The Fröhlich coupling strength $V_\mathbf{q}$ is proportional to $1/q$. This means the interaction is strongest for small-$q$ (long-wavelength) phonons and diverges as $q \to 0$. This $1/q$ dependence is characteristic of the Coulomb interaction in Fourier space and signifies a **long-range** interaction. In contrast, the deformation potential coupling $M(\mathbf{q}) \propto \sqrt{q}$ vanishes for small-$q$ phonons, signifying a **short-range** interaction that is only effective when the lattice is distorted over short distances.

3. The Fröhlich Hamiltonian and the Polaron

We can now assemble the full Hamiltonian in second quantization, just as we did in the last lesson. The total Hamiltonian $H = H_{e} + H_{p h} + H_{e - p h}$ consists of three parts:

Free electrons: $H_{e} = \sum_{k} E_{k} c_{k}^{†} c_{k}$ , where $E_{k} = ℏ^{2} k^{2} /2 m_{b}$ is the electron band energy and $m_{b}$ is the band effective mass.
Free LO phonons: $H_{p h} = \sum_{q} ℏ ω_{L O} b_{q}^{†} b_{q}$ . We neglect the phonon dispersion for simplicity.
Electron-Phonon Interaction: $H_{e - p h} = \sum_{k, q} (V_{q} c_{k + q}^{†} c_{k} b_{q} + V_{q}^{*} c_{k}^{†} c_{k + q} b_{q}^{†})$ .

The interaction term describes an electron in state $k$ absorbing a phonon $q$ and scattering to $k + q$ , and the reverse process of emitting a phonon.

This interaction fundamentally changes the nature of the charge carrier. An electron moving through the lattice becomes "dressed" by a cloud of virtual phonons—a local lattice polarization that it drags along with it. This composite quasiparticle, the electron plus its polarization cloud, is called a polaron.

The strength of this interaction is conventionally captured by a single dimensionless number: the Fröhlich coupling constant, $α$ .

Lattice Polarons

The following MSc thesis chapter, 'Lattice Polarons,' introduces the Fröhlich model and provides the standard definitions for its Hamiltonian and coupling constant.

Please read section 1.1, 'Fröhlich and Holstein Polarons,' stopping at equation (1.1). The text provides context for the polaron concept and presents the final form of the Fröhlich Hamiltonian. Pay close attention to the definition of the coupling constant \alpha and how it relates the interaction strength to fundamental material properties.

As you saw in the text, the Hamiltonian can be written compactly, and the coupling constant $α$ is defined as:

α = \frac{e ^{2}}{4 π ϵ _{0} ℏ} \frac{m _{b}}{2ℏ ω _{L O}} (\frac{1}{ϵ _{\infty}} - \frac{1}{ϵ _{s}})

This constant beautifully summarizes the physics:

It's proportional to $e^{2}$ , the fundamental strength of the Coulomb interaction.
It depends on the material parameters $ω_{L O}$ , $ϵ_{\infty}$ , and $ϵ_{s}$ .
It depends on the electron's band mass $m_{b}$ . A heavier electron moves more slowly and gives the lattice more time to polarize around it, leading to stronger coupling.

The value of $α$ tells us about the nature of the polaron:

Weak-coupling ( $α ≪ 1$ ): Typical for many III-V semiconductors like GaAs ( $α \approx 0.07$ ). The electron has a light, delocalized polarization cloud. The effective mass of the polaron is slightly heavier than the band mass: $m_{p}^{*} \approx m_{b} (1 + α /6)$ .
Strong-coupling ( $α > 6$ ): Typical for ionic solids like NaCl ( $α \approx 5$ ) and alkali halides. The electron becomes heavily "self-trapped" in the deep potential well created by its own lattice distortion. The quasiparticle is localized and heavy, moving by hopping.
Intermediate-coupling ( $1 < α < 6$ ): A difficult but physically rich regime, where more advanced theoretical techniques are required.

Conclusion

Today we derived the Fröhlich Hamiltonian, which describes the long-range Coulomb interaction between electrons and longitudinal optical phonons in polar solids. This is the dominant electron-phonon scattering mechanism in many important semiconductors and insulators.

Key Takeaways:

The Fröhlich interaction is unique to polar materials and arises from the macroscopic electric field produced by the longitudinal oscillations of charged ions (LO phonons).
The interaction is long-range, with a coupling strength $V_{q} \propto 1/ q$ , making it particularly effective at scattering electrons with small momentum transfer.
The complete Fröhlich Hamiltonian describes electrons, dispersionless LO phonons, and their mutual coupling.
An electron coupled to the lattice polarization forms a new quasiparticle, the polaron.
The strength of the coupling is quantified by the dimensionless Fröhlich constant $α$ , which depends on the material's dielectric properties, LO phonon frequency, and the electron's band mass.

Preview of the Next Lesson

We have now established the two primary electron-phonon interaction Hamiltonians: the short-range deformation potential and the long-range Fröhlich interaction. Knowing the form of the interaction is the first step. The next step is to use it to calculate physical observables. In our next lesson, we will use Fermi's Golden Rule to calculate the rate at which electrons scatter from phonons, a quantity that directly determines transport properties like mobility and resistivity.