7_tight_binding.md

import matplotlib
from matplotlib import pyplot

import numpy as np

from common import draw_classic_axes, configure_plotting

configure_plotting()

pi = np.pi

Electrons and phonons in 1D

(based on chapters 9.1-9.3 & 11.1-11.3 of the book)

!!! success "Expected prior knowledge"

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

- Derive Newton's equations of motion of a triatomic chain (previous lecture).
- Write down the dispersion relation of phonons in the Debye model.
- Express complex exponentials through trigonometric functions and vice versa.
- Apply Taylor expansion trigonometric functions.
- Take derivatives inverse trigonometric functions.

!!! summary "Learning goals"

After this lecture you will be able to:

- Formulate the equations of motion of electrons and phonons in 1D.
- Derive the dispersion relation from the equations of motion.
- Derive the group velocity, effective mass, and density of states from the dispersion relation.

Last lecture:

• Vibrational modes of few-atom chains (analyzed using Newton's equations)
• Orbital energies of electrons in few-atom chains (analyzed using LCAOs)

This lecture:

• Phonons and electrons in chains of infinitely many atoms.
• Main idea: use periodicity in space, similar to periodicity in time

To emphasize the similarities and the differences between electrons and phonons, we will deal with both types of particles at once.

Equations of motion

Phonons

In the Debye model, we assumed that the dispersion relation is strictly linear in k. Now is the time to revisit this assumption. To do that, let us consider a 1D homogeneous chain of atoms. We assume that the atoms in the chain interact only with their nearest neighbors through a harmonic potential, like we derived in the previous lecture. In other words, we model the atoms as point masses connected by identical springs.

We denote the displacement of atom n from equilibrium by u_n. Within this convention, Newton's equation of motion for the n-th atom is given by:

m \ddot{u}_n = -\kappa (u_n - u_{n-1}) -\kappa (u_n - u_{n+1}).

We use the periodic boundary conditions just like we did in the Sommerfield model. The boundary conditions imply that in a system of size L = Na, we have u_N = u_0.

Electrons