semiconductor physics and Devices

Week 11: Free Electron Theory of Metals (Part 1)

Lecture Topics:

1. Introduction to Free Electron Theory

   • The free electron model treats conduction electrons in metals as a gas of free, non-interacting particles.

   • Importance of the free electron model in explaining the electrical and thermal properties of metals.

   • Overview of the Drude model as the classical approach to understanding electron behavior in metals.

2. Classical Drude Model

   • Assumptions of the Drude model:

     • Electrons move freely between collisions with ions in the lattice.

     • Collisions lead to resistance, and the time between collisions is characterized by the mean free time τ.

   • Electrical conductivity σ in the Drude model

• where n is the electron concentration, e is the electron charge, m is the electron mass, and τ is the mean free time.

• Ohm’s Law in the context of the Drude model.

• Limitations of the Drude model, particularly its inability to explain phenomena like temperature dependence of resistivity or the heat capacity of electrons.

Sommerfeld Model: Quantum Free Electron Theory

• Introduction to the Sommerfeld model builds on the Drude model by incorporating quantum mechanics and the Pauli exclusion principle.

• Electrons are treated as a Fermi gas: a collection of non-interacting fermions obeying the Fermi-Dirac distribution.

• Definition of the Fermi energy Ef​, the highest occupied energy level at absolute zero temperature.

• Expression for Fermi energy in a three-dimensional electron gas:

• Importance of Fermi energy in determining the electronic properties of metals.

Density of States (DOS)

• Introduction to the density of states function g(E), which describes the number of available electronic states per unit energy interval.

• Calculation of the density of states for a three-dimensional free electron gas

Physical significance of the DOS in predicting the number of electrons occupying energy states and its role in heat capacity and conductivity.

Fermi-Dirac Distribution

• The probability that an energy state is occupied by an electron is given by the Fermi-Dirac distribution:

• At absolute zero, all states below the Fermi energy are occupied, and all states above are empty.

• At finite temperatures, states near the Fermi energy are partially occupied.

Examples:

• Calculate the Fermi energy for a given electron concentration in a metal.

• Example of calculating electrical conductivity using the Drude model.

• Example of calculating the density of states for a free electron gas and relating it to the number of electrons.

Homework/Exercises:

1. Derive the expression for the electrical conductivity σ\sigmaσ using the Drude model.

2. Calculate the Fermi energy for a metal with an electron concentration of n=8.5×10^28 electrons/m^3.

3. Explain the significance of the Fermi-Dirac distribution and how it changes with temperature.

Suggested Reading:

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

Key Takeaways:

• The Drude model provides a classical explanation of electrical conductivity but has limitations that are addressed by the quantum Sommerfeld model.

• The Fermi energy and Fermi-Dirac distribution are central to understanding the behavior of conduction electrons in metals.

• The density of states function is crucial in determining the electronic and thermal properties of a metal.

This week introduces the free electron theory of metals, beginning with the classical Drude model and extending to the quantum Sommerfeld model. The concepts of Fermi energy, the density of states, and the Fermi-Dirac distribution are foundational for understanding the behavior of electrons in metals.