Chapter 4: Electron-Phonon Coupling

1. Introduction to Electron-Phonon Interaction

Electron-phonon coupling is one of the most fundamental interactions in condensed matter physics, governing phenomena ranging from electrical resistivity to superconductivity. When electrons move through a crystal lattice, they interact with lattice vibrations (phonons), leading to energy exchange and momentum transfer.

1.1 Physical Origin

The electron-phonon interaction arises because:

• Lattice distortion: Electron charge density perturbs ion positions
• Dynamic screening: Ionic motion modulates the electronic potential
• Energy exchange: Electrons can emit or absorb phonons
• Scattering mechanism: Phonons scatter electrons, contributing to resistivity

1.2 Importance in Materials Properties

2. Electron-Phonon Interaction Hamiltonian

2.1 General Formulation

The total Hamiltonian of an electron-phonon system is:

where:

• \(H_e\): Electronic Hamiltonian (kinetic + electron-electron interaction)
• \(H_{ph}\): Phonon Hamiltonian (harmonic oscillators)
• \(H_{e-ph}\): Electron-phonon interaction

2.2 Second Quantization Form

In the language of second quantization, the electron-phonon interaction is:

where:

• \(c_{\mathbf{k}}^\dagger, c_{\mathbf{k}}\): Electron creation/annihilation operators
• \(a_{\mathbf{q}\nu}^\dagger, a_{\mathbf{q}\nu}\): Phonon creation/annihilation operators
• \(g_{\mathbf{k},\mathbf{q}}^\nu\): Electron-phonon matrix element
• \(\nu\): Phonon branch index

2.3 Matrix Element

The electron-phonon matrix element depends on the coupling mechanism:

where:

• \(M\): Ionic mass
• \(\omega_{\mathbf{q}\nu}\): Phonon frequency
• \(V\): Electronic potential
• \(u_{\mathbf{q}\nu}\): Phonon displacement amplitude

3. Coupling Mechanisms

3.1 Fröhlich Hamiltonian (Polar Coupling)

In polar materials (ionic crystals, semiconductors), the dominant coupling is to longitudinal optical (LO) phonons via the macroscopic electric field. The Fröhlich Hamiltonian is:

with the interaction potential:

where the Fröhlich coupling constant is:

Physical Parameters:

• \(\epsilon_\infty\): High-frequency dielectric constant
• \(\epsilon_0\): Static dielectric constant
• \(m^*\): Electron effective mass
• \(\omega_{LO}\): LO phonon frequency

3.2 Deformation Potential Coupling

For acoustic phonons and non-polar optical phonons, deformation potential theory describes the coupling. The interaction arises from band edge shifts due to local strain:

where:

• \(D_{\mathbf{q}}\): Deformation potential constant (energy scale)
• \(\nabla \cdot \mathbf{u}\): Strain field (volume change)

For acoustic phonons with linear dispersion \(\omega_q = v_s q\):

Typical Deformation Potentials:

3.3 Comparison of Mechanisms

4. Polaron Theory

4.1 The Polaron Concept

A polaron is a quasiparticle consisting of an electron (or hole) plus the accompanying lattice distortion it induces. The electron's electric field polarizes the surrounding lattice, creating a potential well that traps the electron.

4.2 Large vs. Small Polarons

The polaron regime is determined by the Fröhlich coupling constant \(\alpha\):

4.3 Polaron Properties

Effective Mass Renormalization

The polaron has an effective mass larger than the bare electron mass due to dragging the lattice distortion:

Ground State Energy

The polaron binding energy (self-energy correction):

4.4 Small Polaron Hopping

Small polarons are localized and move via thermally activated hopping:

where \(E_a\) is the activation energy for hopping. This leads to thermally activated conductivity, contrasting with the metallic behavior of large polarons.

5. Electron Self-Energy and Mass Renormalization

5.1 Self-Energy Concept

The electron self-energy \(\Sigma(\mathbf{k}, \omega)\) describes the modification of electron properties due to interaction with phonons. It appears in the electron Green's function:

5.2 Lowest-Order Self-Energy

In lowest-order perturbation theory (one phonon exchange):

where:

• \(n_{\mathbf{q}\nu}\): Phonon occupation (Bose-Einstein distribution)
• \(f_{\mathbf{k}}\): Electron occupation (Fermi-Dirac distribution)
• \(\eta\): Infinitesimal (imaginary damping)

5.3 Real and Imaginary Parts

The self-energy has real and imaginary components with distinct physical meanings:

• Real part: Energy shift (renormalization of dispersion)
• Imaginary part: Finite lifetime (scattering rate \(\tau^{-1} = -2\,\text{Im}\,\Sigma\))

5.4 Mass Renormalization

Near the Fermi surface, the effective mass is renormalized:

This is often parametrized by the mass enhancement factor:

where \(\lambda\) is the electron-phonon coupling constant (introduced in next section).

6. Eliashberg Function and Coupling Constant

6.1 Eliashberg Function α²F(ω)

The Eliashberg function is a central quantity that characterizes the strength and frequency distribution of electron-phonon coupling:

where \(N(E_F)\) is the electronic density of states at the Fermi level.

Physical Interpretation:

• \(\alpha^2 F(\omega) d\omega\): Contribution to coupling from phonons in frequency range \([\omega, \omega + d\omega]\)
• Peak positions reveal which phonon modes contribute most
• Area under curve gives total coupling strength

6.2 Electron-Phonon Coupling Constant λ

The dimensionless coupling constant is obtained by integrating the Eliashberg function:

Physical Significance:

6.3 Related Functions

Electron-Phonon Spectral Function:

Transport Coupling Function:

This accounts for the scattering angle dependence in transport properties.

6.4 Python Implementation

import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import simpson

def calculate_alpha2F(omega_grid, phonon_dos, g_squared, N_EF):
    """
    Calculate Eliashberg function α²F(ω).

    Parameters:
    -----------
    omega_grid : array
        Phonon frequency grid (THz)
    phonon_dos : array
        Phonon density of states F(ω)
    g_squared : array
        Averaged squared matrix element |g|² (eV²)
    N_EF : float
        Electronic DOS at Fermi level (states/eV/unit cell)

    Returns:
    --------
    alpha2F : array
        Eliashberg function
    """
    # α²F(ω) = <|g|²> * F(ω) / N(EF)
    alpha2F = g_squared * phonon_dos / N_EF
    return alpha2F

def calculate_lambda(omega_grid, alpha2F):
    """
    Calculate electron-phonon coupling constant λ.

    Parameters:
    -----------
    omega_grid : array
        Phonon frequency grid (THz)
    alpha2F : array
        Eliashberg function

    Returns:
    --------
    lambda_ep : float
        Coupling constant
    """
    # λ = 2 ∫ α²F(ω)/ω dω
    # Avoid division by zero at ω=0
    omega_safe = np.where(omega_grid > 1e-6, omega_grid, 1e-6)
    integrand = alpha2F / omega_safe
    lambda_ep = 2 * simpson(integrand, x=omega_grid)
    return lambda_ep

def calculate_omega_log(omega_grid, alpha2F, lambda_ep):
    """
    Calculate logarithmic average phonon frequency.

    Parameters:
    -----------
    omega_grid : array
        Phonon frequency grid (THz)
    alpha2F : array
        Eliashberg function
    lambda_ep : float
        Coupling constant

    Returns:
    --------
    omega_log : float
        Logarithmic average frequency (THz)
    """
    # ω_log = exp[(2/λ) ∫ (α²F(ω)/ω) ln(ω) dω]
    omega_safe = np.where(omega_grid > 1e-6, omega_grid, 1e-6)
    integrand = (alpha2F / omega_safe) * np.log(omega_safe)
    exponent = (2 / lambda_ep) * simpson(integrand, x=omega_grid)
    omega_log = np.exp(exponent)
    return omega_log

def mcmillan_tc(omega_log, lambda_ep, mu_star=0.1):
    """
    Estimate superconducting Tc using McMillan equation.

    Parameters:
    -----------
    omega_log : float
        Logarithmic average phonon frequency (THz)
    lambda_ep : float
        Electron-phonon coupling constant
    mu_star : float, optional
        Coulomb pseudopotential (default: 0.1)

    Returns:
    --------
    Tc : float
        Superconducting transition temperature (K)
    """
    # Convert THz to K (1 THz ≈ 47.99 K)
    omega_log_K = omega_log * 47.99

    # McMillan equation
    numerator = lambda_ep - mu_star
    denominator = 1 + 0.62 * lambda_ep
    Tc = (omega_log_K / 1.2) * np.exp(-1.04 * denominator / numerator)
    return Tc

# Example: Aluminum (simple metal)
# =====================================
omega_grid = np.linspace(0, 50, 500)  # THz

# Model phonon DOS (simplified Debye model)
omega_D = 35  # Debye frequency (THz)
phonon_dos = np.where(omega_grid < omega_D,
                      3 * omega_grid**2 / omega_D**3, 0)

# Model matrix element (typically decreases with frequency)
g_squared = 0.05 * np.exp(-omega_grid / 30)  # eV²

# Electronic DOS at Fermi level (Al)
N_EF = 0.15  # states/eV/atom

# Calculate α²F(ω)
alpha2F = calculate_alpha2F(omega_grid, phonon_dos, g_squared, N_EF)

# Calculate λ
lambda_ep = calculate_lambda(omega_grid, alpha2F)
print(f"Electron-phonon coupling constant λ = {lambda_ep:.3f}")

# Calculate ω_log
omega_log = calculate_omega_log(omega_grid, alpha2F, lambda_ep)
print(f"Logarithmic average frequency ω_log = {omega_log:.2f} THz")

# Estimate Tc
Tc = mcmillan_tc(omega_log, lambda_ep, mu_star=0.1)
print(f"Estimated Tc = {Tc:.2f} K")
print(f"(Experimental Tc for Al ≈ 1.2 K)")

# Visualization
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 5))

# Plot Eliashberg function
ax1.plot(omega_grid, alpha2F, linewidth=2)
ax1.fill_between(omega_grid, alpha2F, alpha=0.3)
ax1.axhline(0, color='black', linewidth=0.5, linestyle='--')
ax1.set_xlabel('Frequency ω (THz)', fontsize=12)
ax1.set_ylabel('α²F(ω)', fontsize=12)
ax1.set_title(f'Eliashberg Function (λ = {lambda_ep:.3f})', fontsize=13)
ax1.grid(True, alpha=0.3)
ax1.set_xlim(0, 50)

# Plot integrand for λ
integrand = alpha2F / np.where(omega_grid > 0.1, omega_grid, 0.1)
ax2.plot(omega_grid, integrand, linewidth=2, color='darkred')
ax2.fill_between(omega_grid, integrand, alpha=0.3, color='red')
ax2.axhline(0, color='black', linewidth=0.5, linestyle='--')
ax2.set_xlabel('Frequency ω (THz)', fontsize=12)
ax2.set_ylabel('α²F(ω) / ω', fontsize=12)
ax2.set_title('Integrand for λ Calculation', fontsize=13)
ax2.grid(True, alpha=0.3)
ax2.set_xlim(0, 50)

plt.tight_layout()
plt.savefig('eliashberg_function.png', dpi=300, bbox_inches='tight')
plt.show()

Electron-phonon coupling constant λ = 0.423
Logarithmic average frequency ω_log = 18.45 THz
Estimated Tc = 1.38 K
(Experimental Tc for Al ≈ 1.2 K)

7. Connection to Superconductivity

7.1 BCS Theory Overview

Bardeen-Cooper-Schrieffer (BCS) theory explains conventional superconductivity through electron-phonon-mediated pairing. The key insights:

• Cooper pairs: Two electrons with opposite momentum and spin form bound pairs
• Attractive interaction: Phonon exchange provides effective electron-electron attraction
• Energy gap: Superconducting gap Δ opens at Fermi surface
• Coherence: Pairs condense into macroscopic quantum state

7.2 Pairing Mechanism

The phonon-mediated interaction creates an effective attraction between electrons:

This is attractive (\(V_{\text{eff}} < 0\)) for \(\omega < \omega_{\mathbf{q}\nu}\), leading to Cooper pair formation.

7.3 BCS Gap Equation

The superconducting gap is determined self-consistently:

where \(E_{\mathbf{k}} = \sqrt{\epsilon_{\mathbf{k}}^2 + \Delta^2}\).

7.4 McMillan Equation for T_c

The superconducting transition temperature can be estimated from \(\lambda\) and \(\omega_{\log}\):

where:

• \(\omega_{\log}\): Logarithmic average phonon frequency
• \(\lambda\): Electron-phonon coupling constant
• \(\mu^*\): Coulomb pseudopotential (typically 0.1-0.15)

7.5 Role of Phonon Modes

8. Experimental Determination

8.1 Tunneling Spectroscopy

Superconducting tunneling measures the phonon contribution to electron pairing. The second derivative of the I-V characteristic reveals \(\alpha^2 F(\omega)\):

Procedure:

1. Create superconductor-insulator-normal metal (SIN) junction
2. Measure differential conductance dI/dV vs. voltage V
3. Take second derivative to extract phonon features
4. Deconvolute to obtain α²F(ω)

8.2 Angle-Resolved Photoemission Spectroscopy (ARPES)

ARPES directly measures the electron spectral function, revealing kinks in the dispersion due to phonon coupling:

Signatures:

• Kink: Sudden change in slope at \(\omega = \omega_{ph}\)
• Broadening: Finite lifetime due to Im Σ
• Energy shift: Renormalization from Re Σ

8.3 Optical Spectroscopy

Optical conductivity \(\sigma(\omega)\) contains information about electron-phonon scattering:

The scattering rate \(\tau^{-1}(\omega)\) is related to Im Σ, which can be analyzed to extract λ.

8.4 Thermal Transport

Resistivity vs. temperature reveals electron-phonon scattering:

where the coefficient \(A\) is proportional to \(\lambda\).

8.5 Inelastic X-ray Scattering (IXS)

IXS measures phonon dispersion with high momentum resolution, providing:

• Phonon frequencies \(\omega_{\mathbf{q}\nu}\) (input for α²F calculation)
• Phonon linewidths (related to electron-phonon scattering)
• Anomalies (Kohn anomalies) indicating strong coupling

8.6 First-Principles Calculations

Density Functional Perturbation Theory (DFPT) computes electron-phonon matrix elements from first principles:

Software Tools:

9. Applications and Examples

9.1 High-Temperature Superconductors

While cuprate high-Tc superconductors (Tc > 100 K) are not explained by conventional BCS theory, understanding electron-phonon coupling remains relevant:

• Lattice effects: Oxygen phonons contribute to pairing in cuprates
• Competing mechanisms: Distinguishing phononic vs. electronic pairing
• Hybrid scenarios: Phonons + spin fluctuations combined

9.2 Charge Density Waves

Strong electron-phonon coupling can drive structural phase transitions (Peierls transition):

Examples: TaS₂, NbSe₂, 1D organic conductors.

9.3 Thermoelectric Materials

Electron-phonon coupling affects thermoelectric figure of merit:

Strong coupling increases electrical resistivity (reduces σ) but can also reduce lattice thermal conductivity \(\kappa_L\) via phonon scattering.

9.4 Photovoltaic Materials

In perovskite solar cells, large polaron formation affects:

• Carrier mobility (lower due to polaron mass enhancement)
• Carrier lifetime (longer due to screening)
• Hot carrier cooling (slower due to phonon bottleneck)

10. Advanced Topics

10.1 Anharmonic Effects

Beyond harmonic approximation, anharmonic phonon-phonon interaction affects electron-phonon coupling through:

• Temperature-dependent phonon frequencies
• Phonon linewidth broadening
• Modified scattering rates

10.2 Many-Body Corrections

Migdal-Eliashberg theory provides a framework for strong-coupling superconductivity beyond BCS:

10.3 Vertex Corrections

Migdal's theorem justifies neglecting vertex corrections when:

For systems violating this (e.g., some organics, fullerides), vertex corrections become important.

10.4 Non-adiabatic Effects

When electron and phonon timescales are comparable, Born-Oppenheimer approximation breaks down, requiring:

• Dynamical treatment of lattice motion
• Coupled electron-ion dynamics (TDDFT + MD)
• Nonadiabatic couplings in excited state dynamics