Electronic Structure Calculations¶
Band Structure and DOS computations details¶
We used the relaxed structure from the total energy run to perform using the same GGA functional (and U if any) a DOS and band structure run. This data is available for many entries in the materials explorer. We ran first a static run with a uniform (Monkhorst Pack or \(\Gamma\)-centered for hexagonal systems) k-point grid. From this run the DOS and the charge density were extracted. The charge density was then used to perform a band structure computation for k-points along the symmetry lines of the Brillouin zone. The symmetry lines have been determined following the methodology from Curtarolo et al.1 using the aconvasp software.
The band structure data is displayed with full lines for up spin and dashed lines for down spin. For insulators, the band gap is computed according to the band structure. The nature of the gap (direct or undirect) as well as the k-points involved in the band gap transition are displayed. The VBM and CBMs are displayed for insulators as well by red and white dots. The web site does not display the 10% highest bands as those higher energies bands tend to converge more slowly.
The DOS indicates the full DOS by default but projections along atoms or orbitals are also available through a scroll down menu. Please note that the DOS data and band structure can be slightly different as the k-point grid is not the same. For instance, the k-point grid for the DOS might not include Gamma while the band structure does.
Accuracy of Band Structures¶
Note: the term 'band gap' in this section generally refers to the fundamental gap, not the optical gap. The difference between these quantities is reported to be small in semiconductors but significant in insulators. 2
Figure 1: Experimental versus computed band gaps for 237 compounds in an internal test. The computed gaps are underestimated by an average factor of 1.6, and the residual error even after accounting for this shift is significant (MAE of 0.6 eV). We thank M. Chan for her assistance in compiling this data.
Density functional theory is formulated to calculate ground state properties. Although the band structure involves excitations of electrons to unoccupied states, the Kohn-Sham energies used in solving the DFT equations are often interpreted to correspond to electron energy levels in the solid.
The correspondence between the Kohm-Sham eigenvalues computed by DFT and true electron energies is theoretically valid only for the highest occupied electron state. The Kohn-Sham energy of this state matches the first ionization energy of the material, given an exact exchange-correlation functional. However, for other energies, there is no guarantee that Kohn-Sham eigenvalues will correspond to physical observables.
Despite the lack of a rigorous theoretical basis, the DFT band structure does provide useful information. In general, band dispersions predicted by DFT are reported to match experimental observations; one small test of band dispersion accuracy found that errors ranged from 0.1 to about 0.4 eV.3 However, predicted band gaps are usually severely underestimated. Therefore, a common way to interpret DFT band structures is to apply a 'scissor' operation whereby the conduction bands are shifted in energy by a constant amount so that the band gap matches known experimental observations.
In general, band gaps computed with common exchange-correlation functionals such as the LDA and GGA are severely underestimated. Typically the disagreement is reported to be around 50% in the literature. Some internal testing by the Materials Project supports these statements; typically, we find that band gaps are underestimated by about 40% (Figure 1). We additionally find that several known insulators are predicted to be metallic.
Origin of band gap error and improving accuracy¶
The errors in DFT band gaps obtained from calculations can be attributed to two sources: 1. Approximations employed to the exchange correlation functional 2. A derivative discontinuity term, originating from the true density functional being discontinuous with the total number of electrons in the system.
Of these contributions, (2) is generally regarded to be the larger and more important contribution to the error. It can be partly addressed by a variety of techniques such as the GW approximation but typically at high computational cost.
Strategies to improve band gap prediction at moderate to low computational cost now been developed by several groups, including Chan and Ceder (delta-sol),4 Heyd et al. (hybrid functionals) 5, and Setyawan et al. (empirical fits) 6. (These references also contain additional data regarding the accuracy of DFT band gaps.) The Materials Project may employ such methods in the future in order to more quantitatively predict band gaps. For the moment, computed band gaps should be interpreted with caution.
To cite the calculation methodology, please reference the following works:
- A. Jain, G. Hautier, C. Moore, S.P. Ong, C.C. Fischer, T. Mueller, K.A. Persson, G. Ceder., A High-Throughput Infrastructure for Density Functional Theory Calculations, Computational Materials Science, vol. 50, 2011, pp. 2295-2310. DOI:10.1016/j.commatsci.2011.02.023
- A. Jain, G. Hautier, S.P. Ong, C. Moore, C.C. Fischer, K.A. Persson, G. Ceder, Accurate Formation Enthalpies by Mixing GGA and GGA+U calculations, Physical Review B, vol. 84, 2011, p. 045115. DOI:10.1103/PhysRevB.84.045115
- Anubhav Jain
- Shyue Ping Ong
- Geoffroy Hautier
- Charles Moore
Setyawan, W.; Curtarolo, S. Computational Materials Science 2010, 49, 299-312. ↩
E.N. Brothers, A.F. Izmaylov, J.O. Normand, V. Barone, G.E. Scuseria, Accurate solid-state band gaps via screened hybrid electronic structure calculations., The Journal of Chemical Physics. 129 (2008) ↩
R. Godby, M. Schluter, L.J. Sham, Self-energy operators and exchange-correlation potentials in semiconductors, Physical Review B. 37 (1988). ↩
M. Chan, G. Ceder, Efficient Band Gap Predictions for Solids, Physical Review Letters 19 (2010) ↩
J. Heyd, J.E. Peralta, G.E. Scuseria, R.L. Martin, Energy band gaps and lattice parameters evaluated with the Heyd-Scuseria-Ernzerhof screened hybrid functional, Journal of Chemical Physics 123 (2005) ↩
W. Setyawan, R.M. Gaume, S. Lam, R. Feigelson, S. Curtarolo, High-throughput combinatorial database of electronic band structures for inorganic scintillator materials., ACS Combinatorial Science. (2011). ↩