semiconductor physics and Devices

Week 9: Thermal Properties of Solids (Part 1)

Lecture Topics:

1. Heat Capacity of Solids

   • Introduction to heat capacity and its importance in understanding how solids respond to temperature changes.

   • Distinction between specific heat at constant volume Cv​ and specific heat at constant pressure Cp.

   • Relation between heat capacity and lattice vibrations (phonons).

2. Classical Theory of Heat Capacity: Dulong-Petit Law

   • The classical theory of heat capacity based on the Dulong-Petit law.

   • Dulong-Petit Law: For a solid, the molar heat capacity at high temperatures is approximately 3R (where R is the gas constant). 

   • Cv≈3R at high temperatures

   • Limitations of the Dulong-Petit law at low temperatures, where the experimental values deviate significantly from the classical prediction.

3. Einstein Model of Heat Capacity

   • Einstein’s quantum theory of heat capacity as an improvement over the Dulong-Petit law.

   • Key assumptions of the Einstein model: All atoms vibrate independently with the same frequency (Einstein frequency).

   • Einstein heat capacity formula:

• 1. where θE​ is the Einstein temperature, related to the frequency of the atomic vibrations.

  2. Explanation of how the Einstein model successfully explains the reduction of heat capacity at low temperatures.

• Comparison of Dulong-Petit and Einstein Models

  1. Comparison between the classical Dulong-Petit law and the quantum Einstein model.

  2. Explanation of why the Einstein model provides better agreement with experimental results at low temperatures.

  3. Strengths and weaknesses of the Einstein model (e.g., inability to account for different phonon frequencies in a real lattice).

Examples:

• Calculating heat capacity using the Dulong-Petit law for a given material at high temperatures.

• Calculate the Einstein heat capacity for a solid at different temperatures.

• Graphical comparison of heat capacity predictions from the Dulong-Petit law and Einstein model.

Homework/Exercises:

1. Explain the limitations of the Dulong-Petit law at low temperatures and how the Einstein model addresses these limitations.

2. Calculate the heat capacity of a solid using the Einstein model, given the Einstein temperature θE​ and temperature T.

3. Plot the heat capacity as a function of temperature for both the Dulong-Petit and Einstein models and explain the differences.

Suggested Reading:

• Charles Kittel, Introduction to Solid State Physics, Chapter 5: Phonons II: Thermal Properties.

Key Takeaways:

• The classical Dulong-Petit law provides a simple model for heat capacity, but fails at low temperatures.

• Einstein’s quantum model successfully predicts the temperature dependence of heat capacity, particularly at low temperatures.

• Understanding heat capacity is key to analyzing how materials store and transfer thermal energy.

This week introduces the fundamental thermal properties of solids, with a focus on heat capacity and the comparison between classical and quantum models. The Einstein model of heat capacity is particularly important for explaining the behavior of solids at low temperatures.