Free-Electron Bands in a Periodic Lattice¶

Authors: Dou Du, Taylor James Baird and Giovanni Pizzi

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Source code: https://github.com/osscar-org/quantum-mechanics/blob/master/notebook/band-theory/free_electron.ipynb

The main objective of this notebook is to demonstrate the electronic bandstructure within the free-electron model for a periodic crystalline lattice of a metal.

Throughout the notebook, we employ the empty lattice (free-electron) approximation for the electrons in a periodic solid system. Using it, we compute and plot the electronic band structure for three types of Bravais lattice: simple cubic (SC), face-centered cubic (FCC) and body-centered cubic (BCC). We get the path in reciprocal space for the band structure
from the seekpath package.

Goals¶

• Familiarize yourself with the free-electron model of a metallic solid.
• Examine the electronic band structure of the free-electron model for different crystalline structures.

Background theory¶

More on the background theory.

Tasks and exercises¶

1. Can you describe the shape of the band structure in the 1st Brillouin zone?
   Solution

   In the free electron model, the dispersion relation between electronic energy and wavevector is given by $E=\frac{\hbar^2k^2}{2m}$. Accordingly, the shape of the bands is parabolic.
2. What properties of a material shall be best captured by the free-electron model?
   Solution

   As the free-electron model neglects the effect of the ionic potential on the electrons, material properties which are primarily dependent on the kinetic energy of the conduction electrons are those which shall be best described by the model.
3. Consider the simple cubic lattice, and consider the lowest energy band from Γ to the X point, from Γ to the R point, and from Γ to the M point. The curvature of those bands seem to be the same. Is this to be expected? What about the Γ-L, Γ-X, and Γ-K in the FCC case? Or the Γ-H, Γ-N, and Γ-P in the BCC case?
   Solution

   For a free electron case, the bands are isotropic (i.e., they are the same, independent of the direction in k space), and the effective mass is just the free-electron mass: $m^*=m_0$. Therefore, we expect all curvatures (of the lowest energy band starting from Γ) to be the same.

Legend¶

The 1st Brillouin zone of the selected cell is shown on the left. The path along which the band structure is calculated is indicated with blue vectors and sampled k-points are shown with red dots. The figure on the right shows the calculated band structure. We provide three kinds of cell structure: simple cubic, face-centered cubic (FCC) and body-centered cubic (BCC). Use the radio button to select the cell type.