semiconductor physics and Devices

Week 14: Energy Bands in Solids (Part 2)

Lecture Topics:

1. Tight-Binding Model

   • Introduction to the tight-binding model as an alternative to the nearly free electron model.

   • Assumptions of the tight-binding model:

     • Electrons are tightly bound to individual atoms, but they can "hop" to neighboring atoms.

     • The wavefunctions of electrons overlap slightly between neighboring atoms, leading to the formation of energy bands.

   • Derivation of the energy bands using the tight-binding model:

where Eo​ is the on-site energy, t is the hopping parameter, and aaa is the lattice constant.

• 1. Applications of the tight-binding model to describe materials with strongly localized electrons, such as transition metals and insulators.

Band Structure in Three Dimensions

• Extension of the nearly free electron and tight-binding models to three-dimensional crystals.

• Explanation of how the band structure becomes more complex in 3D, with multiple energy bands and energy gaps.

• Visualization of band structures in common 3D crystal lattices (e.g., FCC, BCC).

• Introduction to band structure diagrams (energy EEE vs. wavevector kkk) for three-dimensional solids.

Effective Mass of Electrons

• Introduction to the concept of the effective mass of electrons in a crystal:

• 1. Effective mass reflects the electron's behavior under an applied electric field in a periodic potential.

  2. Electrons near the bottom of a conduction band behave like free electrons but with an effective mass that can be smaller or larger than the electron mass.

• Explain why the effective mass is important for understanding charge carrier dynamics in semiconductors and metals.

Band Gaps and the Density of States

• Recap of the density of states (DOS) from earlier discussions and its importance in determining the electronic properties of solids.

• Calculation of the density of states for a 3D solid near the band edge.

• Explanation of how the density of states differs between metals, semiconductors, and insulators:

  • Metals: Continuous DOS at the Fermi level.

  • Semiconductors: Finite DOS at the conduction band edge.

  • Insulators: Large band gap with no available states in the gap.

Intrinsic and Extrinsic Semiconductors

• Introduction to intrinsic semiconductors, which have a pure band structure with no impurities.

• Temperature dependence of conductivity in intrinsic semiconductors:

• 1. where ni is the intrinsic carrier concentration and Eg​ is the band gap.

  2. Extrinsic semiconductors: Doping a semiconductor with impurities to introduce additional charge carriers.

  3. Types of doping:

     1. n-type: Donor impurities add extra electrons to the conduction band.

     2. p-type: Acceptor impurities create holes in the valence band.

Examples:

• Derivation of energy bands using the tight-binding model for a simple linear chain.

• Calculation of the effective mass of an electron at the bottom of a conduction band for a given band structure.

• Analysis of the density of states in a semiconductor and how it influences the behavior of intrinsic and extrinsic semiconductors.

Homework/Exercises:

1. Derive the energy bands using the tight-binding model for a one-dimensional solid and explain the significance of the hopping parameter t.

2. Calculate the effective mass for an electron near the band edge of a given material based on its band structure.

3. Compare the density of states for a metal and a semiconductor and explain how this difference affects their electrical conductivity.

Suggested Reading:

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

Key Takeaways:

• The tight-binding model provides a more accurate description of electron behavior in materials with localized electrons, complementing the nearly free electron model.

• The effective mass concept helps explain how electrons behave in a periodic potential and is crucial for understanding carrier dynamics in semiconductors.

• The distinction between intrinsic and extrinsic semiconductors is vital for understanding how doping affects the electrical properties of semiconductors.

This week builds on the previous discussion of energy bands, introducing the tight-binding model, effective mass, and the role of band structure in determining the properties of semiconductors. These concepts are foundational for understanding modern electronic materials and devices.