Electronic Band Structure in a 1D Crystal

Example Problem

In solid-state physics, one of the fundamental concepts is the electronic band structure.

A simple way to model this is using the tight-binding approximation for a 1D crystal with a single atomic orbital per site.

The energy dispersion relation for a one-dimensional crystal with lattice constant a and hopping energy t is given by:

E(k)=E02tcos(ka)

  • E(k) is the energy of the electron,
  • E0 is the on-site energy,
  • t is the hopping integral,
  • k is the wave vector, and
  • a is the lattice constant.

This function represents a simple tight-binding model for a one-dimensional solid.


Python Implementation

Let’s compute and visualize the electronic band structure using Python.

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import numpy as np
import matplotlib.pyplot as plt

# Define parameters
a = 1.0 # Lattice constant
E0 = 0.0 # On-site energy
t = 1.0 # Hopping integral

# Define k values in the first Brillouin zone
k = np.linspace(-np.pi/a, np.pi/a, 500)

# Compute energy band structure
E_k = E0 - 2 * t * np.cos(k * a)

# Plot the band structure
plt.figure(figsize=(8, 5))
plt.plot(k, E_k, label="$E(k) = E_0 - 2t \cos(ka)$", color='b')
plt.axhline(0, color='black', linewidth=0.5, linestyle='--')
plt.axvline(0, color='black', linewidth=0.5, linestyle='--')
plt.xlabel("Wave vector k")
plt.ylabel("Energy E(k)")
plt.title("Electronic Band Structure in 1D Crystal")
plt.legend()
plt.grid()
plt.show()

Explanation

  1. We define the lattice constant (a), on-site energy (E0), and hopping integral (t).
  2. We generate wave vectors k within the first Brillouin zone π/akπ/a.
  3. We compute the energy dispersion using E(k)=E02tcos(ka).
  4. We plot E(k) as a function of k, showing the band structure of the 1D crystal.

Interpretation

  • The energy band has a cosine shape, typical for tight-binding models.
  • The bandwidth (difference between maximum and minimum E(k) ) is 4t.
  • The band is symmetric around k=0 due to the periodic nature of the lattice.

This is a basic yet powerful model for understanding electronic band structures in solids.