semiconductor physics and Devices

Week 13: Energy Bands in Solids (Part 1)

Lecture Topics:

1. Introduction to Energy Bands

   • Transition from the free electron model to a more realistic picture of electron behavior in solids.

   • Energy bands arise from the interaction of electrons with the periodic potential of the crystal lattice.

   • Bloch’s theorem: Electrons in a periodic potential have wavefunctions of the form: 

• 1. where uk(r) has the same periodicity as the crystal lattice, and kkk is the wave vector.

  2. The concept of allowed and forbidden energy levels: electrons occupy allowed energy bands, and there are gaps (band gaps) where no energy states exist.

• Nearly Free Electron Model

  1. Introduction to the nearly free electron model, which builds on the free electron model by considering the weak periodic potential of the crystal.

  2. Energy band formation: How weak potential perturbations lead to the formation of energy gaps at the Brillouin zone boundaries.

  3. Description of the band structure: The relationship between energy EEE and wave vector kkk in a crystal.

  4. Derivation of the energy band structure in a one-dimensional crystal and explanation of Bragg reflection at the zone boundary.

• Kronig-Penney Model

  1. Introduction to the Kronig-Penney model, which is a simplified model for electrons in a periodic potential.

  2. Step-potential approximation: Using a series of square wells to approximate the periodic potential of the crystal lattice.

  3. Solution of the Schrödinger equation for the Kronig-Penney model, showing the formation of energy bands and band gaps.

  4. Graphical solution: Plotting the relationship between energy and wave vector, showing the allowed and forbidden energy regions.

• Brillouin Zones

  1. Recap of Brillouin zones from the discussion of reciprocal lattices in Week 4.

  2. Explanation of the significance of the first Brillouin zone in determining the electron energy states.

  3. Electrons in higher Brillouin zones are scattered back into the first zone by the crystal potential, leading to energy gaps at the zone boundaries.

• Distinction Between Metals, Semiconductors, and Insulators

  1. Explanation of how the energy band structure determines whether a material behaves as a metal, semiconductor, or insulator:

     1. Metals: Partially filled energy bands, allowing electrons to move freely and conduct electricity.

     2. Semiconductors: Completely filled valence band and a small energy gap between the valence and conduction bands.

     3. Insulators: Large energy gap between the valence and conduction bands, preventing significant electron conduction.

Examples:

• Use of the Kronig-Penney model to demonstrate the formation of energy bands in a one-dimensional crystal.

• Example calculation of energy gaps using the nearly free electron model.

• Explanation of how the band structure determines the electrical conductivity of metals and semiconductors.

Homework/Exercises:

1. Solve the Kronig-Penney model for a simple one-dimensional periodic potential and show the formation of energy bands.

2. Explain how Bragg's reflection leads to the formation of energy gaps in the nearly free electron model.

3. Compare the band structures of metal, semiconductor, and insulator and explain the physical significance of the band gap in each case.

Suggested Reading:

• Charles Kittel, Introduction to Solid State Physics, Chapter 7: Energy Bands.

Key Takeaways:

• Energy bands in solids arise due to the interaction between electrons and the periodic potential of the crystal lattice.

• The nearly free electron model and the Kronig-Penney model are useful for understanding the formation of energy gaps and the band structure of solids.

• The band structure of a material determines whether it behaves as a metal, semiconductor, or insulator, based on the presence and size of energy gaps.

This week introduces the concept of energy bands in solids, focusing on the nearly free electron model and the Kronig-Penney model, which explain how the periodic potential of the crystal lattice leads to the formation of energy bands and gaps. The distinction between metals, semiconductors, and insulators is also discussed based on the energy band structure.