11-1. [Solid State Physics] Tight binding model 1

Summary

This script delves into the tight binding model, a fundamental concept in solid-state physics, contrasting it with the free electron model. It discusses the model's applicability to insulators and certain metals, emphasizing the localization of electron wave functions in atomic orbitals due to high potential energy. The script outlines three key assumptions of the tight binding model, explores the mathematical formalism, and explains how the model accounts for the periodic potential's effect on electron behavior, providing a deeper understanding of electronic properties in crystalline materials.

Takeaways

• 📚 The script discusses the Tight Binding Model, a fundamental concept in solid-state physics, particularly in Chapter 11.
• 🌀 It explains the periodic potential created by atoms in a crystal lattice, which is essential for understanding the behavior of electrons in solids.
• 🔽 The script differentiates between negative and positive potentials and their effects on electron energy levels within the context of the tight binding model.
• 🏗️ The Tight Binding Model is contrasted with the Free Electron Model, highlighting that the former assumes a high potential barrier and low kinetic energy, while the latter assumes the opposite.
• 🧲 The model is particularly useful for describing insulators, where electrons are well localized in atomic orbitals, and also applicable to certain cases in metals, such as transition metal D orbitals.
• 🤔 The script outlines three key assumptions of the Tight Binding Model: (1) the crystal Hamiltonian can be approximated by an atomic Hamiltonian, (2) bound levels are well localized, and (3) the wave function is in a stationary state over the crystal.
• 🌌 The Bloch theorem is mentioned, which states that the wave function of an electron in a periodic potential must have the form of a plane wave multiplied by a periodic function.
• 📘 The script delves into the mathematical formalism of the Tight Binding Model, including the eigenvalue problem and the expectation value calculation.
• 🔬 It explains the process of modifying atomic wave functions to account for the periodic potential in a crystal, leading to the formation of energy bands.
• ⚖️ The expectation value is used to approximate the energy of the system, considering the atomic Hamiltonian and potential energy corrections.
• 🔍 The script simplifies the complex mathematical expressions by making assumptions about the smallness of off-site contributions and the dominance of on-site terms, leading to energy eigenvalues close to atomic energies.

Q & A

What is the tight binding model discussed in Chapter 11?

The tight binding model is a theoretical approach used in solid-state physics to describe the behavior of electrons in a crystal lattice, particularly when the potential energy due to atomic interactions is high compared to the kinetic energy of the electrons.

How does the periodic potential in a crystal lattice affect the electron wave function?

The periodic potential in a crystal lattice causes the formation of energy bands due to the Bragg reflection of the electron wave function, leading to the localization of the wave function in the atomic orbitals.

What is the difference between the tight binding model and the free electron model?

The tight binding model assumes high potential energy and low kinetic energy, leading to localized electron wave functions, while the free electron model assumes low potential energy and high kinetic energy, allowing electrons to move freely without significant interaction with the potential.

Why is the tight binding model useful for describing insulators?

The tight binding model is useful for insulators because it effectively describes the localization of electronic wave functions in atomic orbitals, which is characteristic of materials with poor electrical conductivity.

In what cases can the tight binding model also be applied to metals?

The tight binding model can be applied to metals, particularly when dealing with transition metal d-orbitals or f-orbitals, where the model can describe the behavior of electrons in these specific orbitals despite the overall metallic character of the material.

What are the three main assumptions of the tight binding model?

The three main assumptions are: 1) The crystal Hamiltonian can be approximated by an atomic Hamiltonian due to the localization of the electronic wave function; 2) The bound levels of the atomic Hamiltonian are well localized, indicating high potential energy; 3) The wave function is a stationary state over the crystal, following the Bloch theorem.

How does the tight binding model deal with the potential corrections in a crystal Hamiltonian?

The tight binding model accounts for potential corrections by considering the atomic Hamiltonian plus a small potential correction term, acknowledging that while the atomic Hamiltonian is dominant, there are minor corrections due to the periodic potential.

What is the significance of the Bloch theorem in the context of the tight binding model?

The Bloch theorem is significant because it describes how the wave function of an electron in a crystal lattice must have a periodicity that matches the periodicity of the lattice itself, which is a fundamental concept in the tight binding model.

How does the expectation value calculation for the crystal Hamiltonian relate to the atomic Hamiltonian?

The expectation value calculation for the crystal Hamiltonian involves considering the atomic Hamiltonian and potential energy terms, with the understanding that the atomic wave function is well-defined and localized, leading to an expectation value that is close to the atomic energy levels.

What simplifications are made in the tight binding model when considering the interaction between atomic orbitals?

The tight binding model simplifies the interaction between atomic orbitals by assuming that the off-site orbital wave function overlap integrals are very small, effectively considering only the nearest neighbor interactions and the on-site terms.

Can you provide an example of how the tight binding model is applied to a simple case?

While the script does not provide a specific example, a simple application of the tight binding model could involve a one-dimensional lattice where the wave function and energy levels are calculated based on the atomic energy levels and the overlap integrals between neighboring atomic orbitals.