semiconductor physics and Devices

Week 8: Lattice Vibrations (Part 2)

Lecture Topics:

1. Diatomic Lattice Vibrations

   • Extension of lattice vibration theory to a diatomic lattice (two different atoms per unit cell).

   • Derivation of the dispersion relation for a one-dimensional diatomic lattice.

   • Introduction to acoustic and optical phonons:

     • Acoustic phonons: Low-frequency vibrations where atoms in the basis oscillate in phase.

     • Optical phonons: Higher-frequency vibrations where atoms in the basis oscillate out of phase.

   • The dispersion relation for a diatomic lattice shows two distinct branches:

     • Acoustic branch: ω\omegaω approaches 0 as k→0.

     • Optical branch: Non-zero frequency at k=0.

2. Acoustic and Optical Phonons

   • Physical significance of acoustic and optical phonons in solids.

   • Acoustic phonons: Responsible for the propagation of sound waves through the solid.

   • Optical phonons: Interact strongly with electromagnetic waves, contributing to the optical properties of the material.

   • Example materials where optical phonons play a significant role in infrared absorption and Raman spectroscopy.

3. Phonon Dispersion Curves

   • Plotting and interpreting the phonon dispersion curves for both monatomic and diatomic lattices.

   • Comparison of acoustic and optical branches in the dispersion relations.

   • Understanding long-wavelength limit: For small wave vectors, the acoustic branch behaves like a linear wave, while the optical branch has a finite frequency gap.

   • Role of lattice symmetry and atomic mass in determining the characteristics of these branches.

4. Phonon Density of States (DOS)

   • Introduction to the concept of the phonon density of states, which describes the number of phonon modes per unit frequency range.

   • The relationship between the density of states and physical properties such as heat capacity.

   • Example calculation of phonon density of states for a simple lattice.

Examples:

• Derivation of the dispersion relation for a one-dimensional diatomic lattice, leading to acoustic and optical branches.

• Calculation of the phonon frequencies for specific wave vectors in a diatomic lattice.

• Graphical representation of the phonon dispersion curves and discussion of their physical interpretation.

Homework/Exercises:

1. Derive the dispersion relation for a diatomic linear chain and identify the acoustic and optical branches.

2. Plot the phonon dispersion curves for a diatomic lattice and compare them with those of a monatomic lattice.

3. Calculate the phonon density of states for a given phonon dispersion relation.

Suggested Reading:

• Charles Kittel, Introduction to Solid State Physics, Chapter 4: Phonons I: Crystal Vibrations (continued).

Key Takeaways:

• In diatomic lattices, acoustic and optical phonons exist, leading to more complex vibrational behavior than monatomic lattices.

• Acoustic phonons are related to sound propagation, while optical phonons are crucial for understanding interactions with light.

• Phonon dispersion relations provide insights into the vibrational modes of solids and their impact on material properties such as heat capacity and thermal conductivity.

This week continues the exploration of lattice vibrations, focusing on diatomic lattices and the distinctions between acoustic and optical phonons. These concepts are critical for understanding more complex solid materials' vibrational and thermal properties.