12_band_structures_in_higher_dimensions.md

import numpy as np
import plotly.graph_objs as go
from matplotlib import pyplot as plt 
from math import pi

Tight binding and nearly free electrons

(based on chapter 16 of the book)

!!! success "Expected prior knowledge"

Before the start of this lecture, you should be able to:

- Apply perturbation theory to understand the band structure at crossings
- Derive the dispersion relation for a tight-binding model
- Recall the notion of the Fermi energy
- Calculate the Fermi surface, i.e. through the Fermi wavevector $k_F$, for two and three-dimensional systems
- Construct the unit cell of a crystal

!!! summary "Learning goals"

After this lecture you will be able to:

- examine 1D and 2D band structures and argue if you expect the corresponding material to be an insulator/semiconductor or a conductor.
- describe how the light absorption spectrum of a material relates to its band structure.

Band structure

In nature we see that some materials conduct electricity (conductors), some don't (insulators), and some do under specific circumstances (semiconductors). The motion of electrons is described by the band structure (dispersion relation). So far we have derived the dispersion relation for different type of models, such as the tight-binding model in which the electrons are bound by a strong potential, or when the electrons are perturbed by a small periodic potential as was the case in the NFEM. We have not yet discussed how the band structure does affect the electrical properties of a material.

For a material to be a conductor, there should be available electron states at the Fermi level. Otherwise all the states are occupied, and all the currents cancel out.

Suppose we have a band structure of a 1D material similar to one below:

We see several energy bands that may either be separated by a band gap or overlap.

When the Fermi level lies in the band gap, the material is called a semiconductor (or dielectric or insulator). When the Fermi level lies within a band, it is a conductor (metal).

??? Question "Suppose the Fermi energy lies in the band gap. What feature about the band gap determines if a material is insulating or a semiconductor?"

Its size

A simple requirement for insulators

In an insulator every single band is either completely filled or completely empty.

What determines if an energy band if fully occupied or only partly? To answer this we need to know the number of available states within an energy band, and the number of electrons in the system. We can find the number of states in a band by integrating the density of states g(E), but this is hard. Fortunately, we can easily see how many states there are in an energy band by counting the number of k-states in the first Brillouin zone.

For a single band: