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I came across the band structure colored as bellow1:

enter image description here

As you can see, each band was colored depending of the atom contribution to each band.

Googling, I found it was done using the concept of fat bands. The definition of the it is2:

The fat bands ${F_{i,n,\sigma ,\vec k}}$ are the periodic equivalent of the Mulliken population. They are defined as: $${F_{i,n,\sigma ,\vec k}} = \sum\limits_j {{C_{i,n,\sigma ,\vec k}}{C_{j,n,\sigma ,\vec k}}{S_{i,j,\vec k}}}$$ where ${{C_{i,n,\sigma ,\vec k}}}$ and $S_{i,j,\vec k}$ are the orbital coefficients and the overlap matrix elements respectively. The indices $i$ and $j$ denote basis functions, $n$ is the band index, $\sigma$ is the spin index and $\vec k$ is a reciprocal vector in the Brillouin zone.

What I understand is that now there is a projection/identification of the contribution of each atom (orbital) one each band.

My question is: why fat bands? Is there other meaning/undestanding for them?

1 Chen Gong, et al., Band Structure Engineering by Alloying for Photonics, Advanced Optical Material, 6, 1800218 (2018). dx.doi.org/10.1002/adom.201800218

2 https://www.scm.com/doc/BAND/Analysis/Band_Structure.html#definition-of-the-fat-bands

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For reasons that may be apparent from the plots you show in the question, complicated band structures are often referred to as spaghetti. It can be difficult to know what individual bands represent, and to form an intuition for them. However, if we calculate the projection of individual bands onto some orbital (often atomic s,p,d,f orbitals) and plot that, we may get a better understanding of the system. A projected density of states (PDOS) plot has essentially the same idea behind it, but for fatbands we show the projections as functions of both momentum and energy. I believe the projection onto orbitals with different angular momentum ($l=0,1,2,...$) was traditionally represented with the line width, which resulted in thick (or "fat") bands and the "fatband" moniker. However, with the reduced importance of print journals, encoding this projection information with color instead of line width seems to have become common.

For an illustration using bulk silver, and some discussion on how to calculate fatbands using the ELK and QuantumEspresso codes, see this post on Christoph Wolf's blog.

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