Efficient band gap prediction of semiconductors and insulators from a semilocal exchange-correlation functional

A semilocal exchange-correlation functional is proposed with the efficient prediction of the solid-state band gap. The underlying construction of the exchange functional is based on the modeling of the exchange hole and constructing the exchange energy functional. Being a meta-generalized gradient approximation (meta-GGA) level functional, it holds the key feature of the derivative discontinuity through the generalized Kohn-Sham (gKS) formalism. We validate our construction by demonstrating the functional performance for solid-state band gaps and comparing it with the other meta-GGA and hybrid functionals. It is shown that for the semiconductors having narrow, and moderate band gaps, as well as for layered materials the present functional performs as accurately as (or comparable to) the expensive hybrid functional, while for wide-band gap solids, it outperforms the hybrid functional. This indicates that the present functional holds promise for the multiscale modeling of materials with a very low computational cost. Also, the underlying construction is practically very useful as it can be easily implemented in any density functional platform that supports the gKS formalism.

Figure 1

(Top) The electron probability distributions $\left(4\pi r^2\rho \right)$ and the reduced gradients $\left(s_{\sigma }=\left|\nabla {\rho }_{\sigma }\right|/\left[2k_F{}_{\sigma }{\rho }_{\sigma }\right]\right)$ of the hydrogen [see Eq. (12)] and cuspless hydrogen [see Eq. (17)] densities with $\alpha =2$. (Bottom) Comparison of the mBRxH-BG and mBRxC-BG exchange enhancement factors $F_x(s,\alpha )$. Also shown, are the $\text{MGE}2\left(F_x^{\text{MGE}2}=1+0.26s^2\right)$, and the SCAN meta-GGA exchange enhancement factors. Solid and interrupted lines represent $\alpha =1$ and 10 cases, respectively.

Figure 2

Comparison of mBRxH-BG, mBRxC-BG and mBJLDA for several tests of convergence, in case of diamond. (a) Plane wave energy cutoff vs band gap convergence; (b) $k$ mesh vs band gap convergence; and (c) iteration steps vs band gap. For (a) and (c), we used the $15\times 15\times 15$ Monkhorst-Pack $k$-point mesh; and for (b) and (c), we used the 700 eV energy cutoff.

Figure 3

Experimental vs calculated band gaps for the narrow band gap (top), intermediate band gap (middle), and wide band gap (bottom) materials, using different methods. PBE values are not considered.

Figure 4

The electron pseudodensities of the (a) valence-band and (b) conduction-band minima (CBM) and valence-band maxima (VBM) are shown along [111] direction in diamond bulk, as obtained with several XC functionals.

Figure 5

Band structure of Ar solid using the mBRxC-BG and SCAN exchange-correlation functionals.