Letter Cite This: J. Phys. Chem. Lett. 2018, 9, 2338−2345 pubs.acs.org/JPCL Doubly Screened Hybrid Functional: An Accurate First-Principles Approach for Both Narrow- and Wide-Gap Semiconductors Zhi-Hao Cui, Yue-Chao Wang, Min-Ye Zhang, Xi Xu, and Hong Jiang* Beijing National Laboratory for Molecular Sciences, State Key Laboratory of Rare Earth Material Chemistry and Application, Institute of Theoretical and Computational Chemistry, College of Chemistry and Molecular Engineering, Peking University, 100871 Beijing, China ABSTRACT: First-principles prediction of electronic band structures of materials is crucial for rational material design, especially in solar-energy-related materials science. Hybrid functionals that mix the Hartree−Fock exact exchange with local or semilocal density functional approximations have proven to be accurate and efficient alternatives to more sophisticated Green’s function-based many-body perturbation theory. The optimal fraction of the exact exchange, previously often treated as an empirical parameter, is closely related to the screening strength of the system under study. From a physical point of view, ϵ the screening has two extreme forms: the dielectric screening [1/ M] that is dominant in wide-gap materials and the Thomas−Fermi metallic screening [exp(−ζr) ] that is important in narrow-gap semiconductors. In this work, we have systematically investigated the performances of a nonempirical doubly screened hybrid (DSH) functional that considers both screening mechanisms and found that it excels all other existing hybrid functionals and describes the band gaps of narrow-, medium-, and wide-gap insulating systems with comparably good performances. n recent years, first-principles electronic structure theory, the band gaps of wide-gap insulators. Similar problems also − I represented by density functional theory (DFT) in the local exist for other hybrid functionals with fixed parameters.24 26 density approximation (LDA) or various generalized gradient Many-body perturbation theory in the GW approxima- approximations (GGAs), has gained tremendous popularity in tion27,28 has been systematically developed in the past decades − condensed matter physics and materials science.1 3 However, and is regarded as one of the most accurate approaches to − the widely used local/semilocal approximate functionals have electronic band structure properties of materials.4,29 31 Unlike long been plagued by the systematic underestimation of the many other methods, the GW method is able to describe the band gap of insulating materials,4,5 especially for narrow-gap band gaps of various normal insulating materials with essentially semiconductors (NGS), which are often wrongly described as comparable accuracy. One of the most important features of the metallic. Furthermore, closely related to this so-called “band GW method is its capability to accurately describe screening Downloaded via PEKING UNIV on August 7, 2019 at 07:50:41 (UTC). gap problem”, the first-principles description of a lot of other effects of different strength in different materials. The success of properties, such as the defect formation energy6 and the the hybrid functionals can also be rationalized based on the GW − 26,32,33 ionization potential of semiconducting materials,7 12 also description of screening effects. In particular, it has been − α exhibits significant errors. In contrast, hybrid functionals that noted that the fraction of the Hartree Fock exact exchange HF See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles. mix a fraction of Hartree−Fock (HF) exact exchange with is closely related to the screening strength of the system, as ϵ LDA/GGAs can provide significant improvement for those described by its macroscopic dielectric constant ( M), which is α properties as well as the band gap. Commonly used hybrid system-dependent. On the basis of such a link between HF and 13,14 ϵ functionals such as PBE0, which incorporates 1/4 of the M, several new hybrid functional approaches have been 26,32−36 Hartree−Fock exact exchange, HSE,15,16 which is similar to proposed recently. In this work, we further exploit the PBE0 but uses a screened Coulomb interaction for the exact link between the GW approach and the hybrid functionals and exchange part with an empirically determined screening emphasize the importance of considering both dielectric parameter, and the well-known B3LYP functional,17,18 have screening (dominant in insulators) and short-range metallic been widely tested in description of the band gap and screening (important for NGSs) in the hybrid functional − thermodynamic properties of semiconductors.19 23 Despite framework for accurate description of both NGSs and wide-gap fi semiconductors. their signi cantly improved performances in describing the 26,32,37 electronic band structure of typical semiconductors with respect As has been realized by many researchers, the success to LDA/GGA,23 the accuracy of each hybrid functional tends to of the hybrid functional approach to the band gap problem can be limited to a certain class of materials. For example, the HSE functional with the empirically determined screening parameter Received: March 26, 2018 performs very well in describing semiconductors with the band Accepted: April 19, 2018 gap falling in the range of 1−3 eV but tends to underestimate Published: April 19, 2018 © 2018 American Chemical Society 2338 DOI: 10.1021/acs.jpclett.8b00919 J. Phys. Chem. Lett. 2018, 9, 2338−2345 The Journal of Physical Chemistry Letters Letter be attributed to its relation to the GW approach. In particular, it different nature in the hybrid functional framework, it is crucial can be regarded as a further approximation to the Coulomb to consider both dielectric and metallic screenings with system- hole and screened exchange (COHSEX) approximation to the dependent screening parameters. For that purpose, we take a GW exchange−correlation self-energy, which can be obtained similar strategy as that in ref 34 and use a simple model by neglecting the frequency dependence of the screened dielectric function44 Coulomb interaction W in the GW self-energy28 ⎡ ⎛ ⎞2⎤−1 1 ⎢ −1 ⎜ q ⎟ ⎥ Σxc(,rr′=− ) δν()[(,)(,;0) r −′ r rr ′−W rr ′ ω = ] ϵ=+ϵ−+()q 1⎢ (M 1) α⎜ ⎟ ⎥ 2 ⎝ q ⎠ ⎣ TF ⎦ (5) Nocc − ψψ()rrrr* (′′= )W (, ; ω 0) − ∑ ii where qTF denotes the Thomas -Fermi screening parameter i=1 and α = 1.563 is an empirical parameter introduced to better describe the dielectric function of typical semiconductors ≡ΣCOH(,rr ′ ) +Σ SEX (, rr ′ ) (1) according to ref 44. The corresponding screened Coulomb The screened Coulomb potential W above is defined as potential takes the form of W(,rr′= ;ωων ) d r ″ϵ″−1 (, rr ; )( r ″′ , r ) dq 41π ∫ (2) νsc(,rr′= ) ∫ exp(iqr·−′ ( r )) (2π )32q ϵ()q − where ϵ 1(r,r″;ω) is the inverse dielectric function and ν(r″,r′) ⎛ ⎞ 11 1 exp(−q̃ |−′|rr ) denotes the bare Coulomb potential. In the following = +−⎜1 ⎟ TF ν ′ ϵ|−′|rr ⎝ ϵ ⎠ |−′|rr formalism, we use sc(r,r ) to denote the static approximation MM ν ′ ≡ ′ ω ff to W, i.e., sc(r,r ) W(r,r ; =0) . The main e ect of the (6) ff inverse dielectric function is to reduce the e ective strength of where q̃ is defined as an effective Thomas−Fermi screening electron−electron interaction, and the simplest approximation TF − parameter is to replace ϵ 1(r,r″) by a constant scaling factor equal to the ϵ 2 ⎛ ⎞ inverse of the macroscopic dielectric constant 1/ M, such that q 1 q̃ 2 ≡ TF ⎜ + 1⎟ 1 TF ⎝ ⎠ α ϵ−M 1 (7) ΣSEX(,rr′≈− ) γν(, rr′′ )(, rr ) ϵM (3) It is obvious that in eq 6 the first term corresponds to the where we introduce the first-order reduced density matrix dielectric screening, which considers the full range of a scaled Coulomb interaction, and the second term represents the Nocc γ(,rr′≡ )∑ ψψ () r* ( r′ ) metallic screening, which considers only the short-range ii contribution. To simplify the implementation, we further use i=1 (4) the complementary error function (erfc) to approximate the When approximating the local COH term by LDA or GGA, second term in eq 6 following ref 34 this gives the dielectric-dependent hybrid (DDH) func- ⎛ ⎞ tional.26,32 Marques et al.32 proposed using 1/ϵ calculated at 11 1 erfc(μ|−′|rr ) M ν (,rr′≈ ) +−⎜1 ⎟ the LDA/GGA level as the system-dependent α and obtained sc ⎝ ⎠ HF ϵ|−′|MMrr ϵ |−′|rr (8) quite good agreement with experiment for many semi- 26 conductors. Skone et al. found that a self-consistent 2qTF̃ α with μ = . Using the screened Coulomb interaction with determination of HF can further improve the agreement with 3 experiment for theoretical prediction of band gaps of typical sp both dielectric and metallic screening considered above, we − insulating materials.26 The self-consistent DDH functional has obtain the following doubly screened hybrid (DSH) exchange − been used for a variety of different systems33,38 41 and also correlation potential extended to finite systems.42 ⎛ ⎞ 1 1 From a physical point of view, the DDH approach only VVDSH(,rr′= ) HF(, rr′+ )⎜ 1 − ⎟ VHF,SR(, rr′ ;μ ) xc x ⎝ ⎠ x grasps one aspect of screenings in real solids, i.e., that of the ϵM ϵM dielectric screening, which is dominant in wide-gap insulators. ⎛ ⎞ 1 It is therefore not surprising that the most significant +−⎜1 ⎟VVPBE,LR(;rrμ )+ PBE () ⎝ ⎠ x c improvement of the DDH approach compared to PBE0 is ϵM (9) observed in systems with large band gaps. For the latter, the fi α where PBE0 approach, with a xed HF = 0.25, often underestimates the band gap.26,32 In contrast, the DDH approach uses HF,SR erfc(μ|−′|rr ) significantly larger α for those systems due to their smaller V (,rr′=−′ ;μγ ) (, rr ) HF x (10) dielectric constants and therefore predicts significantly larger |−′|rr 35 PBE,LR band gaps that are in better agreement with experiment.