Intoduction to solid state physics

Week 12: Free Electron Theory of Metals (Part 2)

Lecture Topics:

1. Electronic Heat Capacity in Metals

   • Recap of the classical prediction from the Dulong-Petit law, which suggests that electrons should contribute significantly to the heat capacity of metals.

   • Explanation of why the actual electronic contribution to heat capacity is much smaller than predicted by classical models.

   • Derivation of the electronic heat capacity using the Sommerfeld model.

   • At low temperatures, the electronic contribution to heat capacity is proportional to temperature:

where γ is the Sommerfeld coefficient.

• Contrast with the phonon contribution, which varies as T^3 at low temperatures.

Thermal Conductivity and the Wiedemann-Franz Law

• Explanation of thermal conductivity in metals, with electrons and phonons contributing to heat transfer.

• Derivation of the Wiedemann-Franz law, which relates the electrical and thermal conductivities in metals:

• where κ is the thermal conductivity, σ\sigmaσ is the electrical conductivity, T is the temperature, and L is the Lorenz number: 

• Discussion of why this law holds in metals due to the dominant role of electrons in both electrical and thermal transport.

Electrical Resistivity of Metals

• Recap of the Drude model’s treatment of electrical resistivity, and how it depends on the mean free time τ\tauτ.

• Explanation of how resistivity increases with temperature due to increased phonon scattering (electron-phonon interactions).

• Introduction to residual resistivity, caused by impurities and defects, which is temperature-independent at low temperatures.

• Matthiessen’s rule: The total resistivity is the sum of temperature-dependent resistivity (due to phonons) and temperature-independent resistivity (due to defects):

where ρ0  is the residual resistivity, and ρphonon\rho_{\text{phonon}}ρphonon​ is the phonon contribution.

• Limitations of the Free Electron Model

  1. Overview of the successes and limitations of the free electron theory.

     1. Successes: Explains electrical and thermal conductivities and provides a reasonable estimate of the Fermi energy.

     2. Limitations: This does not account for the effects of the periodic crystal lattice, leading to an inability to explain phenomena like the band structure of semiconductors or the detailed behavior of electrons in real materials.

  2. Introduction to the nearly free electron model and the tight-binding model, which incorporate the effects of the periodic potential of the lattice and are more accurate for explaining electronic behavior in solids.

Examples:

• Calculation of the electronic heat capacity of a metal at a given temperature using the Sommerfeld model.

• Calculation of the Lorenz number from the Wiedemann-Franz law.

• Estimation of the resistivity of a metal at room temperature and low temperatures, accounting for both phonon scattering and residual resistivity.

Homework/Exercises:

1. Derive the Wiedemann-Franz law and explain its significance in relating electrical and thermal conductivities.

2. Calculate the electronic contribution to the heat capacity for a metal at 300 K, given the Fermi energy.

3. Use Matthiessen’s rule to calculate the resistivity of a metal at different temperatures, given the residual resistivity and the temperature-dependent phonon contribution.

Suggested Reading:

• Charles Kittel, Introduction to Solid State Physics, Chapter 6: Free Electron Fermi Gas (continued).

Key Takeaways:

• The Sommerfeld model provides a more accurate description of the electronic heat capacity of metals, especially at low temperatures.

• The Wiedemann-Franz law highlights the close relationship between electrical and thermal conductivities in metals.

• Electrical resistivity in metals is temperature-dependent due to phonon scattering, and impurities contribute to residual resistivity, which is temperature-independent.

This week completes the discussion of the free electron theory, with a focus on the electronic contribution to heat capacity, thermal conductivity, and resistivity. It also addresses the limitations of the free electron model and introduces more advanced approaches for understanding the behavior of electrons in solids.