Chapter 5: Phonons in Real Materials

1. Introduction

In previous chapters, we studied phonons from a theoretical perspective using simple models like harmonic chains and the Debye approximation. In this chapter, we bridge theory and reality by examining phonon properties in actual materials and the techniques used to study them.

Real materials exhibit rich phonon behavior that depends on their crystal structure, bonding character, and electronic properties. Understanding these phonons is essential for:

• Thermal management: Designing materials with desired thermal conductivity
• Electronics: Understanding electron-phonon scattering in semiconductors
• Superconductivity: Phonon-mediated Cooper pairing in conventional superconductors
• Thermoelectrics: Optimizing the figure of merit ZT
• Materials discovery: Predicting new materials with exceptional properties

2. Phonons in Metals

2.1 Characteristics of Metallic Phonons

Metals possess several unique features that influence their phonon properties:

• Free electrons: Conduction electrons screen ionic interactions
• Metallic bonding: Non-directional bonding leads to high coordination numbers
• High symmetry: Many metals crystallize in FCC, BCC, or HCP structures
• Electron-phonon coupling: Significant interaction between phonons and electrons

2.2 Simple Metals: Aluminum and Copper

Aluminum (FCC) and copper (FCC) are prototypical simple metals with well-studied phonon dispersions:

The phonon dispersion in FCC metals shows three acoustic branches (one longitudinal, two transverse) and no optical branches since there is only one atom per primitive cell.

2.3 Kohn Anomaly

A unique feature in metallic phonon dispersions is the Kohn anomaly, which appears as a discontinuity in the slope of the dispersion curve at specific wavevectors \(\mathbf{q}\):

where \(\mathbf{k}_F\) is the Fermi wavevector. This anomaly arises from electron-phonon coupling and screening by conduction electrons.

2.4 Effect of Electronic Screening

In metals, the free electrons screen the Coulomb interaction between ions. The effective force constant is modified by the screening:

where \(q_{TF}\) is the Thomas-Fermi screening wavevector:

This screening typically occurs over distances of a few Ångströms, making metallic force constants relatively short-ranged.

3. Phonons in Semiconductors

3.1 Covalent Bonding and Phonon Properties

Semiconductors like silicon (Si) and gallium arsenide (GaAs) have covalent or partially ionic bonding, leading to:

• Directional bonding: Tetrahedral coordination in diamond/zinc-blende structures
• Optical phonons: Two atoms per primitive cell create optical branches
• LO-TO splitting: In polar semiconductors like GaAs
• Lower thermal conductivity: Compared to metals (for intrinsic materials)

3.2 Silicon: The Prototypical Semiconductor

Silicon crystallizes in the diamond structure (FCC with two-atom basis). Key phonon properties include:

Silicon's phonon dispersion shows six branches: three acoustic (LA, TA, TA) and three optical (LO, TO, TO). The degeneracy of transverse modes reflects the cubic symmetry.

3.3 Gallium Arsenide: Polar Semiconductor

GaAs crystallizes in the zinc-blende structure and exhibits ionic character due to the electronegativity difference between Ga and As:

The LO-TO splitting arises from the long-range Coulomb interaction in polar materials. At the Γ point (\(\mathbf{q} = 0\)), the LO and TO modes have different frequencies:

where \(e^{*}\) is the effective charge, \(\epsilon_\infty\) is the high-frequency dielectric constant, \(V_0\) is the unit cell volume, and \(\mu\) is the reduced mass.

3.4 Electron-Phonon Scattering

In semiconductors, phonon scattering limits carrier mobility at room temperature. The mobility is related to the scattering rate by:

where \(\tau\) is the scattering time and \(m^*\) is the effective mass. For acoustic phonon scattering via the deformation potential:

For polar optical phonon scattering in GaAs:

where \(N(\omega)\) is the Bose-Einstein distribution and the ± corresponds to emission/absorption.

4. Phonons in Insulators

4.1 Ionic Insulators: Sodium Chloride

NaCl is a model ionic insulator with the rock salt structure (FCC with two-atom basis). Key features include:

• Strong ionic bonding: Large charge transfer between Na and Cl
• Large LO-TO splitting: Due to strong Coulomb interactions
• Infrared active modes: TO modes at Γ point
• Low thermal conductivity: ~6 W/m·K at 300 K

The large LO-TO splitting in NaCl reflects the strong ionic character and long-range Coulomb forces.

4.2 Covalent Insulators: Diamond

Diamond represents the extreme of covalent bonding with exceptional properties:

Diamond's exceptional thermal conductivity arises from:

1. High phonon velocities: Due to strong, stiff C-C bonds
2. High Debye temperature: Large phonon mean free path
3. Low atomic mass: High group velocities
4. Weak anharmonicity: Long phonon lifetimes

The thermal conductivity can be estimated from kinetic theory:

where \(C\) is the heat capacity, \(v\) is the average phonon velocity, and \(\ell\) is the mean free path. In diamond at 300 K, \(\ell\) can exceed 1 μm.

4.3 Lattice Dynamics and Symmetry

The number and type of phonon modes at the Γ point are determined by the space group symmetry. For the diamond structure (space group \(Fd\overline{3}m\)):

• \(T_{2g}\): Raman-active optical mode (triply degenerate)
• \(T_2\): Acoustic modes (triply degenerate)
• \(E_g\): Raman-active but silent at Γ
• \(T_{2u}\): Infrared-active but forbidden in diamond (no dipole moment)

5. Experimental Measurement Techniques

5.1 Neutron Scattering

Inelastic neutron scattering is the most direct method for measuring phonon dispersion relations across the entire Brillouin zone. The technique relies on energy and momentum conservation:

where \(\mathbf{k}_i\) and \(\mathbf{k}_f\) are the incident and scattered neutron wavevectors, \(\omega\) is the phonon frequency, \(\mathbf{q}\) is the phonon wavevector, and \(\mathbf{G}\) is a reciprocal lattice vector.

Advantages of Neutron Scattering

• Probes entire Brillouin zone (\(\mathbf{q} \neq 0\))
• No selection rules (all modes accessible)
• Direct measurement of dispersion \(\omega(\mathbf{q})\)
• Sensitive to light elements (unlike X-rays)

Limitations

• Requires large single crystals (~1 cm³)
• Access to neutron facilities (reactors or spallation sources)
• Long measurement times (hours to days)
• Limited energy resolution (~1%)

5.2 Raman Spectroscopy

Raman scattering measures optical phonon frequencies at or near the Γ point (\(\mathbf{q} \approx 0\)) through inelastic scattering of photons:

The upper sign corresponds to Stokes scattering (phonon creation) and the lower sign to anti-Stokes scattering (phonon annihilation).

Selection Rules

Only phonon modes with specific symmetries are Raman-active. For a mode to be Raman-active, it must modulate the polarizability tensor:

where \(\mathbf{e}_i\) and \(\mathbf{e}_s\) are the incident and scattered polarization vectors, and \(Q\) is the phonon normal coordinate.

Example: Silicon Raman Spectrum

Advantages of Raman Spectroscopy

• Non-destructive and non-contact
• Works on small samples (micrometer scale with confocal systems)
• High frequency resolution (~1 cm⁻¹)
• Rapid measurements (seconds to minutes)
• Ambient conditions (no vacuum required)

5.3 Infrared Spectroscopy

Infrared (IR) spectroscopy probes optical phonons that carry a dipole moment. The selection rule for IR activity is:

where \(\mathbf{E}\) is the electric field and \(\mathbf{p}\) is the dipole moment.

Reflectivity Measurements

For polar materials, IR reflectivity shows characteristic features called Reststrahlen bands between the TO and LO frequencies:

The reflectivity approaches unity in the frequency range \(\omega_{TO} < \omega < \omega_{LO}\), creating a stop band for electromagnetic radiation.

Complementarity with Raman

5.4 Inelastic X-ray Scattering

Inelastic X-ray scattering (IXS) is a complementary technique to neutron scattering that uses high-energy X-rays (typically 10-20 keV):

Advantages of IXS

• High momentum resolution (better than neutrons)
• Works with small samples (sub-millimeter)
• Accessible at synchrotron facilities (more available than neutron sources)
• Can probe high-pressure samples in diamond anvil cells

Challenges

• Weak scattering cross-section (~10⁷ times weaker than neutrons)
• Requires high-brilliance synchrotron sources
• Complex monochromator/analyzer systems needed for meV resolution

5.5 Comparison of Techniques

6. Computational Methods

6.1 Density Functional Theory (DFT)

Modern phonon calculations are based on density functional theory (DFT), which provides the ground-state electronic structure and forces. The basic workflow is:

1. Ground state calculation: Find the equilibrium atomic positions and unit cell
2. Force constant calculation: Compute forces from small atomic displacements
3. Dynamical matrix: Construct the dynamical matrix from force constants
4. Phonon frequencies: Diagonalize the dynamical matrix for \(\omega(\mathbf{q})\)

6.2 Common DFT Codes

Several well-established codes are used for phonon calculations:

6.3 Phonopy: Post-Processing Tool

Phonopy is a widely-used open-source tool for phonon analysis. It interfaces with various DFT codes and provides:

• Phonon dispersion and DOS calculation
• Thermal properties (heat capacity, free energy)
• Thermodynamic properties via quasi-harmonic approximation
• Visualization tools for band structures

Basic Phonopy Workflow

#!/usr/bin/env python
"""
Phonopy example: Calculate phonon dispersion for Silicon
Assumes VASP force calculations are already done
"""

from phonopy import Phonopy
from phonopy.interface.vasp import read_vasp
from phonopy.file_IO import parse_FORCE_SETS, parse_BORN
import numpy as np
import matplotlib.pyplot as plt

# Read unit cell from POSCAR
unitcell = read_vasp("POSCAR")

# Create Phonopy object with 2x2x2 supercell
phonon = Phonopy(unitcell,
         supercell_matrix=[[2, 0, 0],
                   [0, 2, 0],
                   [0, 0, 2]],
         primitive_matrix='auto')

# Read force constants from FORCE_SETS
force_sets = parse_FORCE_SETS()
phonon.dataset = force_sets

# Produce force constants
phonon.produce_force_constants()

# Optional: Read Born effective charges for polar materials
# born = parse_BORN(phonon.primitive, filename="BORN")
# phonon.nac_params = born

# Define high-symmetry path in reciprocal space
# For FCC: Γ-X-K-Γ-L
path = [[[0.0, 0.0, 0.0],  # Γ
     [0.5, 0.0, 0.5],  # X
     [0.5, 0.25, 0.75], # K (W in conventional)
     [0.0, 0.0, 0.0],  # Γ
     [0.5, 0.5, 0.5]]] # L

labels = ["$\\Gamma$", "X", "K", "$\\Gamma$", "L"]

# Get band structure along path
qpoints, connections = phonon.get_band_qpoints_and_path_connections(
  path, npoints=51)

# Calculate frequencies
phonon.run_band_structure(qpoints, path_connections=connections)
band_dict = phonon.get_band_structure_dict()

# Extract data for plotting
distances = band_dict['distances']
frequencies = band_dict['frequencies']  # Shape: (n_qpoints, n_bands)

# Convert THz to cm^-1 (common unit in spectroscopy)
# 1 THz = 33.356 cm^-1
frequencies_cm = frequencies * 33.356

# Plot phonon dispersion
fig, ax = plt.subplots(figsize=(8, 6))

# Plot all branches
for i in range(frequencies.shape[1]):
  ax.plot(distances, frequencies_cm[:, i], 'b-', linewidth=1.5)

# Add vertical lines at high-symmetry points
x_positions = [distances[0]]
for i, connection in enumerate(connections):
  if connection:
    x_positions.append(distances[connection[0]])

for x in x_positions:
  ax.axvline(x=x, color='k', linewidth=0.5, linestyle='--')

# Set labels at high-symmetry points
ax.set_xticks(x_positions)
ax.set_xticklabels(labels)

# Labels and formatting
ax.set_xlabel('Wave Vector', fontsize=12)
ax.set_ylabel('Frequency (cm$^{-1}$)', fontsize=12)
ax.set_title('Silicon Phonon Dispersion', fontsize=14, fontweight='bold')
ax.set_ylim(0, 600)
ax.grid(True, alpha=0.3)

plt.tight_layout()
plt.savefig('si_phonon_dispersion.png', dpi=300)
plt.show()

# Calculate phonon DOS
phonon.run_mesh([20, 20, 20])  # Mesh for DOS calculation
phonon.run_total_dos()

dos_dict = phonon.get_total_dos_dict()
dos_frequencies = dos_dict['frequency_points']  # THz
dos = dos_dict['total_dos']

# Plot DOS
fig, ax = plt.subplots(figsize=(6, 6))
ax.plot(dos, dos_frequencies * 33.356, 'r-', linewidth=2)
ax.fill_betweenx(dos_frequencies * 33.356, 0, dos, alpha=0.3, color='red')

ax.set_xlabel('Density of States', fontsize=12)
ax.set_ylabel('Frequency (cm$^{-1}$)', fontsize=12)
ax.set_title('Silicon Phonon DOS', fontsize=14, fontweight='bold')
ax.set_ylim(0, 600)
ax.grid(True, alpha=0.3)

plt.tight_layout()
plt.savefig('si_phonon_dos.png', dpi=300)
plt.show()

# Calculate thermal properties
phonon.run_thermal_properties(t_min=0, t_max=1000, t_step=10)
tp_dict = phonon.get_thermal_properties_dict()

temperatures = tp_dict['temperatures']  # K
heat_capacity = tp_dict['heat_capacity']  # J/K/mol
entropy = tp_dict['entropy']  # J/K/mol
free_energy = tp_dict['free_energy']  # kJ/mol

# Plot heat capacity
fig, ax = plt.subplots(figsize=(8, 6))
ax.plot(temperatures, heat_capacity, 'g-', linewidth=2, label='C$_v$')
ax.axhline(y=3*8.314, color='k', linestyle='--',
       label='Dulong-Petit limit (3R)')

ax.set_xlabel('Temperature (K)', fontsize=12)
ax.set_ylabel('Heat Capacity (J/K/mol)', fontsize=12)
ax.set_title('Silicon Heat Capacity', fontsize=14, fontweight='bold')
ax.legend()
ax.grid(True, alpha=0.3)

plt.tight_layout()
plt.savefig('si_heat_capacity.png', dpi=300)
plt.show()

print("Phonon calculations complete!")
print(f"Number of q-points in dispersion: {len(distances)}")
print(f"Number of phonon branches: {frequencies.shape[1]}")
print(f"Maximum frequency: {frequencies.max():.2f} THz "
    f"({frequencies.max()*33.356:.2f} cm⁻¹)")

6.4 Finite Displacement vs. DFPT

Two main approaches exist for calculating force constants:

Finite Displacement Method

Atoms are displaced by small amounts (\(\delta \sim 0.01\) Å), and forces are calculated:

Advantages: Simple to implement, works with any DFT code
Disadvantages: Requires many supercell calculations, computationally expensive

Density Functional Perturbation Theory (DFPT)

DFPT calculates the response of the electronic system to atomic displacements directly:

Advantages: Efficient, can calculate at arbitrary q-points
Disadvantages: More complex implementation, not available in all codes

6.5 Validation and Accuracy

Computational phonon calculations should be validated against experiments. Common checks include:

• Acoustic sum rule: \(\omega(\mathbf{q} \to 0) \to 0\) for acoustic modes
• Comparison with Raman/IR: Optical phonon frequencies at Γ
• Neutron scattering data: Dispersion along high-symmetry directions
• Thermodynamic properties: Heat capacity, thermal expansion
• Convergence tests: Supercell size, k-point mesh, energy cutoff

7. Practical Applications

7.1 Thermal Conductivity

Phonons are the primary heat carriers in insulators and semiconductors. The lattice thermal conductivity is given by:

where \(C_{\mathbf{q}s}\) is the mode-specific heat capacity, \(v_{\mathbf{q}s}\) is the group velocity, and \(\tau_{\mathbf{q}s}\) is the phonon lifetime.

Scattering Mechanisms

• Phonon-phonon (Umklapp): \(\tau^{-1} \propto T^2 e^{-\Theta_D/3T}\)
• Boundary scattering: \(\tau^{-1} = v/L\) (L = characteristic length)
• Point defect scattering: \(\tau^{-1} \propto \omega^4\) (Rayleigh scattering)
• Dislocation scattering: Important in heavily deformed materials

7.2 Thermoelectric Materials

The thermoelectric figure of merit depends critically on phonon properties:

where \(S\) is the Seebeck coefficient, \(\sigma\) is electrical conductivity, \(\kappa_e\) is electronic thermal conductivity, and \(\kappa_L\) is lattice thermal conductivity.

Phonon Engineering Strategies

• Nanostructuring: Introduce interfaces for boundary scattering
• Alloying: Create mass disorder for point defect scattering
• Rattler atoms: Heavy atoms in cage structures (e.g., skutterudites)
• Anharmonicity: Materials with intrinsically low \(\kappa_L\) (e.g., SnSe)

7.3 Superconductivity

In conventional superconductors, phonons mediate the attractive interaction between electrons (BCS theory). The critical temperature is:

where \(\Theta_D\) is the Debye temperature, \(\lambda\) is the electron-phonon coupling constant, and \(\mu^*\) is the Coulomb pseudopotential.

High-Tc Phonon-Mediated Superconductors

7.4 Optical Applications

Phonons play crucial roles in optical properties:

Raman Thermometry

The Raman peak position shifts with temperature due to anharmonic effects:

This enables non-contact temperature measurement in operating devices (LEDs, transistors, etc.).

Polaritons

In polar materials, photons couple with optical phonons to form phonon-polaritons:

where \(\Omega\) is the coupling strength. Applications include slow light devices and enhanced optical processes.

7.5 Mechanical Properties

Phonon properties relate to elastic constants. For example, the longitudinal sound velocity:

where \(C_{11}\) is the elastic constant and \(\rho\) is the density. The elastic constants can be derived from phonon dispersion at small \(\mathbf{q}\):

7.6 Preview: Advanced Topics

Several advanced topics build upon the foundations covered in this series:

Thermal Conductivity Calculations

• Boltzmann transport equation: Solving for \(\kappa_L\) from first principles
• Relaxation time approximation: Simplified approaches
• Phonon-phonon scattering: Three-phonon and four-phonon processes
• Tools: ShengBTE, phono3py, ALAMODE

Electron-Phonon Coupling

• Eliashberg function: \(\alpha^2F(\omega)\) characterizes coupling strength
• Temperature-dependent properties: Carrier mobility, band gap renormalization
• Superconductivity: McMillan equation and beyond
• Tools: EPW (Electron-Phonon Wannier), Quantum ESPRESSO

Anharmonic Effects

• Thermal expansion: Quasi-harmonic approximation
• Phonon-phonon interactions: Beyond harmonic approximation
• Temperature-dependent frequencies: Self-consistent phonon calculations
• Phase transitions: Soft modes and structural instabilities

Phonons in Low-Dimensional Systems

• 2D materials: Graphene, transition metal dichalcogenides
• Flexural modes: Unique to 2D systems (ZA modes)
• 1D systems: Nanotubes, nanowires
• Quantum confinement: Modified dispersion and DOS

8. Summary

This chapter bridged the gap between theoretical phonon concepts and real materials by exploring:

The combination of experimental measurements and computational predictions provides a powerful framework for understanding and designing materials with tailored phonon properties. As computational capabilities continue to advance, predictive phonon engineering will play an increasingly important role in materials discovery and optimization.