PHYSICAL REVIEW B 100, 045147 (2019) Efficient band gap prediction of semiconductors and insulators from a semilocal exchange-correlation functional Bikash Patra, 1 , * Subrata Jana, 1 , † Lucian A. Constantin, 2 , ‡ and Prasanjit Samal 1 , § 1 School of Physical Sciences, National Institute of Science Education and Research, HBNI, Bhubaneswar 752050, India 2 Center for Biomolecular Nanotechnologies @UNILE, Istituto Italiano di Tecnologia, Via Barsanti, I-73010 Arnesano, Italy (Received 5 April 2019; revised manuscript received 14 June 2019; published 29 July 2019) 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. DOI: 10.1103/PhysRevB.100.045147 I. INTRODUCTION The Kohn-Sham (KS) [1,2] formalism of the density func- tional theory (DFT) is the de facto standard method to ac- count for the electronic structure calculations of the atoms, molecules, solids, and materials. In KS-DFT, the central task is to construct the exchange-correlation (XC) energy functional or potential. Starting from the local-density ap- proximation (LDA) [2], the higher rungs of the semilocal XC functionals are represented by the generalized gradient approximation (GGAs) [3–27], and meta-GGAs [28–43]. The KS-DFT electronic structure calculations for solids are carried out mostly within the semilocal XC approximations, that are usually providing high accuracy with low computational cost. The semilocal approximations are remarkably accurate for many solid-state properties, like equilibrium lattice constants, bulk moduli, cohesive energies, work function, elastic and phonon properties, vacancy formation and surface energies [44–66]. In spite of their accuracy, the semilocal XC func- tionals actually fail in a few solid-state properties. Within those the lack of accurate prediction of band gap of the semi- conductor devices is undoubtedly one of the most important drawbacks of the semilocal approximations. Looking at the rapid development in the semiconductor devices from the technology point of view, the efficient evaluation of the band gap is necessary and still now it is an active research filed with promisingly new prospects. * bikash.patra@niser.ac.in † subrata.jana@niser.ac.in; subrata.niser@gmail.com ‡ lucian.constantin@iit.it § psamal@niser.ac.in The poor accuracy of the band gap within the semilocal XC approximations can be inferred from their underestimation of the derivative discontinuity [67–73] and modest description of the delocalization error [74,75]. In KS formalism, the band gap is defined by the difference between the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO), i.e.,  KS = ǫ LUMO − ǫ HOMO . For periodic system it becomes E g = ǫ CB − ǫ VB , where ǫ CB and ǫ VB are the lowest unoccupied and the highest occupied one-electron energies of the conduction and valence bands, respectively. However, the fundamental band gap (E g ) is the difference between ionization potential (IP) and electron affinity (EA). Note that using the variational formalism of DFT, one can obtain the relation between the fundamental band gap and KS gap as E g =  KS +  xc , where  xc is known as the XC derivative discontinuity [76–78]. It was shown in several works [70,71,73,79–82] that inclusion of  xc improves the band gap of solids computed within the LDA or GGA ap- proximations. However, very recently it was proved that meta- GGA implemented in the generalized Kohn-Sham scheme (gKS) includes some amount of the  xc [68,83]. Therefore, meta-GGA functionals may give an improvement in the band gap of solids. It was shown in Ref. [83] that the strongly constrained and appropriately normed (SCAN) [39] and meta- GGA made (very) simple (MVS) [37] improve the band gap, and for several cases the accuracy of MVS is comparable with the expensive hybrid functionals [83]. Note that the meta-GGA XC functionals, being semilocal with respect to KS orbitals, are almost as expensive as the GGA functionals. However, meta-GGA functionals hold additional features, like  xc , which make them as preferable candidates to access the band gap problem. Inspired by the appealing features of these recent meta- GGAs, in this paper, we propose Laplacian-free meta-GGA 2469-9950/2019/100(4)/045147(12) 045147-1 ©2019 American Physical Society