# Why is the band structure of a supercell more dense than for simple cell?

In order to do DFT calculations of metallic alloys, the start point is a supercell, whose atoms are changed to match the desired stoichiometry. Gold and Silver, for example, both have FCC structure. In Quantum-ESPRESSO, we can both tell to the code the Bravais lattice (ibrav parameter) and specify only one atom position ( $$(0,0,0)$$ for an FCC cell ), or use a cubic lattice indicating four atoms positions ( $$(0,0,0)a$$, $$(0,0.25,0.25)a$$, $$(0.25,0,0.25)a$$, and $$(0,0.25,0.25)a$$, in terms of the lattice parameter $$a$$). In theory, the responses should be the same, and the GND energies are retrieved to be equal. However, the band structure for the four-atoms cell is way denser than the single-atom cell. Why does this occur?

What you are referring to is called band folding. Remember we are plotting the band structure in the reciprocal space. As the size of the cell in real space increases (eg: when you make a super cell), the first Brillouin zone in reciprocal space shrinks and more lines populate the band structure resulting from folding back of lines at the boundaries.

The figure$$^1$$ illustrates band folding in the super cell calculations: (a) band structure of a 2D one-band first- neighbor tight-binding model, (b) the same obtained from a 4x4 super-cell calculation, and (c) the same obtained from a 16x16 super-cell calculation. Panel (d) shows the DOS

Another figure$$^2$$ to visualize band folding in 1D

There are many tools available to unfold the band structure. Here I list a few

• BandUP: Band Unfolding code for Plane-wave based calculations available for VASP, Quantum Espresso, ABINIT and CASTEP
• pyPROCAR offers a utility to unfold bands for VASP
• Band unfolding made simple$$^3$$ by May et. al implementation for SIESTA.
• unfold-x Tool to unfold the band structure of a supercell DFT simulation to a larger Brillouin zone based on QuantumESPRESSO

References:

1. Ku, Wei, Tom Berlijn, and Chi-Cheng Lee. "Unfolding first-principles band structures." Physical review letters 104.21 (2010): 216401.
2. Yang, Shuo-Ying, et al. "Symmetry demanded topological nodal-line materials." Advances in Physics: X 3.1 (2018): 1414631.
3. Mayo, Sara G., Felix Yndurain, and Jose M. Soler. "Band unfolding made simple." Journal of Physics: Condensed Matter 32.20 (2020): 205902.
• This is a great answer. Can we send it to the wiki? May 18 '20 at 4:20
• That's best explanation I even seen recently for this band structure folding. Thanks. Oct 16 '20 at 11:44