Contents lists available at ScienceDirect

Computational Materials Science

journal homepage: www.elsevier.com/locate/commatsci

From predicting to correlating the bonding properties of iron sulfide phases

Jinjia Liua,c, Aiju Xud, Yu Menge, Yurong Hea,c, Pengju Rena,b, Wen-Ping Guob, Qing Pengf,Yong Yanga,b, Haijun Jiaoa,g, Yongwang Lia,b, Xiao-Dong Wena,b,⁎

a State Key Laboratory of Coal Conversion, Institute of Coal Chemistry, Chinese Academy of Sciences, Taiyuan 030001, PR ChinabNational Energy Center for Coal to Clean Fuels, Synfuels China Co., Ltd, Huairou District, Beijing 101400, PR ChinacUniversity of Chinese Academy of Sciences, No. 19A Yuquan Road, Beijing 100049, PR Chinad College of Chemistry & Environmental Science, Inner Mongolia Key Laboratory of Green Catalysis, Inner Mongolia Normal University, Hohhot 010022, PR Chinae School of Chemistry and Chemical Engineering, Yulin University, No. 4 Chongwen Road, Yuyang District, Yulin, Shaanxi Province, PR ChinafNuclear Engineering and Radiological Sciences, University of Michigan, Ann Arbor, MI 48109, USAg Leibniz-Institut für Katalyse e.V. an der Universität Rostock, Albert-Einstein Strasse 29a, 18059 Rostock, Germany

A R T I C L E I N F O

Keywords:Iron sulfideDensity functional theoryBond valenceStrongly-correlated system

A B S T R A C T

Iron sulfides have emerged as a fascinating class of materials in electromagnetics and catalysis areas, whichhowever are challenging in first-principles modeling because of the strongly-correlated interactions between Fe3d and S 3p electrons. Here, we assess the performances of 14 density functionals on the structural, electronic,and magnetic properties of five iron sulfides. The PBE+U with Ueff=2.0 eV has the overall best performance.After evaluating functionals and obtaining reliable properties, our final goal is from predicting to correlating inorder to do high throughput screening for the systems since the complex structures and phases of iron sulfides, toput it in another way, to hunting a reliable descriptor for predicting their properties. In the work, we demon-strate that the crystal orbital Hamilton population (COHP) and Bader charge of Fe atoms presents a good cor-relation with the empirical bond valence. Our results open a new avenue to effectively investigate phases andproperties for various structures of iron sulfides. Indeed, the correlations between COHP/Bader charge and bondvalence can be extended to other systems.

1. Introduction

Metal sulfide minerals play a significant role in industrial and eco-nomic society since many metals such as zinc, copper, lead, gold, andplatinum are derived from sulfides ores. Among them, iron sulfides arethe most widely investigated due to their extensive distribution in theearth. They usually exhibit a wide range of physical, electronic andchemical properties which make it possible for applications in solarcells, electromagnetic devices and electro- and photo-catalysts [1–3].

Under certain temperature and pressure, iron sulfides compoundscan exhibit a variety of forms by changing the stoichiometries thatrange from the sulfur deficient mackinawite FeS1−x through iron-defi-cient pyrrhotites Fe1−xS to pyrite FeS2 [4]. Their abundant structuresbring much difficulty for experimental characterizations. With the de-velopment of density functional theory (DFT), the theoretical calcula-tion has become an effective “characterization tool” in physics, chem-istry, and material sciences. However, the DFT calculations are verycomputationally expensive. Therefore, it is of great significance anddesirable to develop an effective method or model to describe the

complicated chemical environment of Fe atoms in Fe-S compounds forthe investigation of the intrinsic properties of iron sulfides systems. Oneof the most challenge task in DFT calculations is the modeling ofstrongly-correlated interactions between Fe 3d electrons, which is a keyto predict the structural and electronic properties of ground states.Joseph et al. [5] studied the structural parameters and the band gap ofpyrite FeS2 utilizing Local Density Approximation (LDA) [6], the latticeconstants were a= b= c=5.386 Å, which are less than experimentalvalue of 5.418 Å. As for the fundamental band gap, LDA predictedpyrite as metal while the real gap is 0.95 eV by the experimental pho-toelectron spectrum. The LDA functional which is based on the homo-geneous electron gas model underestimates the structural parametersand band gaps, especially for strongly-correlated systems. GeneralizedGradient Approximation (GGA, Perdew, 1996) [7] have provided apractical method to successfully explore the structural and electronicproperties of ground states across the periodical table. Spagnoli et al.4

used GGA-PBE and PBEsol [8] approximation and present a study onstructural properties of polymorphs pyrite and marcasite, the calcula-tion results reveal that GGA could give much better lattice constants

https://doi.org/10.1016/j.commatsci.2019.04.001Received 14 January 2019; Received in revised form 30 March 2019; Accepted 1 April 2019

⁎ Corresponding author.E-mail address: wxd@sxicc.ac.cn (X.-D. Wen).

Computational Materials Science 164 (2019) 99–107

0927-0256/ © 2019 Elsevier B.V. All rights reserved.

T

respect to LDA. Whilst the band gap of pyrite is also predicted to be0 eV, suggesting that GGA also fail to capture the electronic structure ofthe ground state.

An effective method to treat with the strongly-correlated system isto add a Hubbard parameter, localized term, to LDA or GGA densityfunctional, also known as DFT+U approximation [9,10]. Choi et al.[11] reported that the predicted band gap of pyrite is 1.03 eV withGGA+U where Ueff is equal to 2.0 eV, which is very close to an ex-perimental gap of 0.95 eV. Koutti et al. [12] employed spin polarizationLSDA+U approximation to investigate the magnetic property andband gap of troilite FeS, the effect of Ueff on results are also taken intoaccount. They found that with the increasing of Ueff, the magneticmoment and band gap is increasing. When Ueff=2.0 eV, the magneticmoment is 3.66 μB and the band gap is 0.54 eV, these results are well inaccordance with experimental observation of 3.50 μB and 0.50 eV. An-other route to address strongly-correlated materials is hybrid func-tional, such as B3LYP [13], PBE0 [14], and HSE [15]. In hybrid DFT,the self-interaction error of conventional DFT is partially corrected byembedding a fixed part of exact, nonlocal Hartree-Fock (HF) exchangeinto exchange-correlation functional.

To have a brief view of the performances of different densityfunctional approximations on describing bulk properties of iron sul-fides, we summarize previously reported studies as shown in Table 1.Just as we know, the rational calculation method is the basis for furthercomplex investigations. Even some studies on functional benchmark foriron sulfides have been reported, they just focus on limited structuresand properties (either lattice constant or band gap), and the calculatedparameters and codes used are different, thus, the results are notcomparable. To our knowledge, the systematic study on the behavior ofdifferent exchange-correlation (XC) functionals in predicting propertiesof iron sulfides is few. In current work, we devote to give a clear profileto compare the performance of different XC-functionals in dealing withiron sulfides, furthermore, to find a valid model to describe the bondingcharacters of Fe atoms in iron sulfides.

2. Theoretical methods and computational details

To characterize magnetism, all calculations are carried out withinthe framework of the spin-polarized density functional theory using theprojected augmented wave (PAW) method [27], as implemented in theVienna Ab Initio Simulation Package (VASP) [28]. The Brillouin zone issampled using Monkhorst-Pack generated a set of k-mesh, 6×6×6 forpyrite, 7× 6×9 for marcasite and 7× 7×5 for mackinawite,11× 11×6 for NiAs-troilite and 6× 6×3 for troilite, all the k-pointsare centered with Gamma point. The plane cutoff of the energy of500 eV is used to describe the electronic wave-function. All the atomsare relaxed to their equilibrium positions when the charge in energy oneach atom between successive steps is converged to 1.0×10−5 eV/atom, the forces on each atom are converged to 0.02 eV/Å.

In our calculations, Hartree-Fock [29,30], local density approx-imation (LDA) [31], generalized gradient approximations (GGA-PBEand GGA-PBEsol) [6], meta-GGA (TPSS and SCAN) [32], Hubbard Umodel (DFT+U approach) and hybrid functionals (B3LYP, PBE0, andHSE) are used. For the GGA+U approach, in the formulation ofLichtenstein [9] and later Dudarev [10], where only a single parameter,Ueff, determines an orbital-dependent correction to the GGA calculationenergy. Here, the Ueff is equal to 2.0, 2.5, 3.5 and 4.5 eV, respectively.

In the HSE approach, the exchange–correlation part is shown in Eq.(1), where α=0.08, 0.10, 0.15, 0.20 and 0.25, ω=0.207 is used inour calculations.

= + − + +E αE ω α E ω E ω E( ) (1 ) ( ) ( )xcHSE

xHF SR

xPBE SR

xPBE LR

cPBE, , , (1)

3. Results and discussions

3.1. Iron sulfides structures

Iron and sulfur could form various structures and phases underspecific conditions. In this work, we focus on only five representativeiron sulfides including two FeS2 phases (pyrite and marcasite) and threeFeS phases (mackinawite, troilite, and NiAs-troilite), the crystal

Table 1The structural parameter (Å), magnetic moment (μB) and band gap (eV) of iron sulfides predicted with various DFT approximations.

Iron sulfides method Lattice parameter Magnetic moment Band gap

a b c

Pyrite PBE 5.403 5.403 5.403 [4] – 0 [5]PBEsol 5.315 5.315 5.315 [4] – –AM05 5.318 5.318 5.318 [4] – –WC 5.345 5.345 5.345 [4] – –HF 6.0 6.0 6.0 [4] – 11 [5]LDA 5.386 5.386 5.386 [4] – 0 [5]PBE 5.520 5.520 5.520 [4] – 0 [5]B3LYP 5.614 5.614 5.614 [4] – 2 [5]LDA 5.247 5.247 5.247 [4] – 0 [5]LDA-ASW – – – – 0.95 [16]GGA+U – – – – 1.03 [17]HSE06 – – – – 2.62 [17]Exp. 5.418 5.418 5.418 [18] 0 [19] 0.95 [20]

Marcasite PBE 3.390 4.435 5.407 [4] – –PBEsol 3.340 4.355 5.324 [4] – –AM05 3.345 4.367 5.333 [4] – –WC 3.351 4.397 5.358 [4] – –G0W0-PBE – – – – 1.06 [21]Exp. 3.386 4.449 5.432 [18] 0 [22] 0.8–1.0 [23]

Mackinawite PBE+U 3.587 3.587 4.908 [6] 0 [6] 0 [6]PBE 3.674 3.674 5.033 [6] 0 [6] –Exp. 3.674 3.674 5.033 [6] 0 [6] 0 [24]

Troilite LSDA+U – – – 3.66 [12] 0.54 [12]LSDA – – – 3.46 [12] –GGA – – – 3.69 [12] –Exp. 5.950 5.950 11.760 [25] 3.5 [25] ∼0.50 [26]

J. Liu, et al. Computational Materials Science 164 (2019) 99–107

100

structures are presented in Fig. 1. The unit cell of pyrite FeS2 is simplecubic with Pa3 space group containing 4 iron atoms and 8 sulfur atoms,as shown in Fig. 1(a). The structure of pyrite is similar to NaCl crystalwith the Fe ion occupying the Na position and the S dimer locates at theCl ion position. Each Fe atom is octahedrally coordinated with sur-rounding six S atoms where each S atom bond to three Fe atoms andone S atom. The lattice parameter is a= b= c=5.418 Å with experi-mental characterizations. Pyrite is a promising photovoltaic materialwith a suitable band gap of 0.95 eV in experiment [21].

Marcasite FeS2 is a polymorph of pyrite that attributes to Pnnmspace group and crystalizes in orthorhombic. Just as shown in Fig. 1(b),each Fe atoms is coordinated with surrounding four sulfur atoms whileeach S atoms bond with three nearby S atoms and one Fe atom. At lowtemperatures, the marcasite is metastable phase respect to pyrite, but itcan transform to pyrite when the temperature is around 700 K [33].Compared to pyrite, the corresponding studies no matter experimentalor theoretical studied on marcasite are much less. The experimentallattice constants of marcasite are a= 4.440, b=5.432 andc=3.386 Å. Jagadeesh and Seehra found that the band gap of marca-site is 0.34 eV [34], but later research has proved that the band gap ofmarcasite is no lower than pyrite and maybe larger than 0.95 eV [35].

The layered-type mackinawite is modeled in the tetragonal structure(Fig. 1(c)). Both room temperature neutron diffraction [36] andMossbauer data at 4.2 K [24] testify the absence of magnetic moment inmackinawite. The lattice parameter is a= b=3.674 and c=5.033 Åmeasured experimentally [6]. Vaughan and Ridout [24] have demon-strated that the mackinawite presents a metallic character with theelectronic states of Fe d-orbitals dominating the regions around theFermi level. According to the previous study, at ambient pressure androom temperature stoichiometric FeS crystallizes in the troilite struc-ture with a hexagonal space group p6̄2c, as shown in Fig. 1(e). At hightemperature, the Fe ions would be shifted away from their ideal posi-tions and transformed to high symmetry hexagonal p63mmc spacegroup, the so-called NiAs-type structure (Fig. 1(d)). Below TN ∼588 K,FeS undergoes an antimagnetic (AFM) phase with a narrow band gap,while a high temperature, FeS behaves as a metal [3].

3.2. XC-functionals performance for iron sulfides

We first investigate the lattice constant of five iron sulfides withdifferent functionals, the results are summarized in Table S1. The ex-perimental constants, mean absolute error (MAR) and the mean abso-lute relative error of calculated lattice constants (MARE, %) are alsopresented to compare with each other. For pyrite and marcasite, theMAR and MARE of these functionals are all within 0.1 Å and 2%, re-spectively. It means that the structural predictions of FeS2 are in-dependent of density functional approximations. Whilst for mack-inawite, GGA-PBEsol, meta-GGA (SCAN) and PBE+U (Ueff=2.0 eV)could give reasonable lattice constant when compared to experimentalvalues, the MARE are all within 2%, but for other functionals, the

MARE is beyond 4%. From Table S1, functionals (PBE, TPSS, PBE+U(Ueff=2.5, 3.5 and 4.5)) overestimate the lattice constant in c direc-tion. Dzade et al. [6] investigated the bulk property and water ad-sorption on surfaces of mackinawite with DFT-D2 approximation, theresults show that the structural parameters are consistent well withexperimental values when van deer Waals interaction is considered. ForNiAs-type troilite, only the lattice constant predicted with PBE+U(Ueff=2.0/2.5 eV) is well consistent with experimental data, the MAREof calculated lattice constant is within about 2%. For troilite, onlyPBE+U (Ueff=2.0 eV) could give accurate structure parameters.

To have a clear overview of the performance of the different func-tionals in predicting iron sulfides structural property, we plot the re-lative error (%) of predicted lattice constants in Fig. 2. It can be foundthat LDA, PBEsol, and SCAN underestimate the lattice constant for allthese five iron sulfides. PBE and TPSS overestimate the lattice constantfor mackinawite while underestimating for other four sulfides. PBE+U(Ueff=2.5/3.5/4.5 eV) overestimate the lattice parameters for thesesulfides, and with the increasing of Ueff, the MARE is also increasing.From above-computed results, we tentatively conclude that PBE+Uwith Ueff=2.0 eV could give accuracy lattice constants for iron sul-fides.

One of the important applications of iron sulfides is that they canserve as electromagnetic devices. Therefore, it is of significance to in-vestigate their magnetic and electronic properties. We calculated themagnetic orderings, magnetic moments and fundamental band gaps ofthese five iron sulfides. Table S2 summarizes the results with the

Fig. 1. The atomistic crystal structures of the five representative iron sulfides. (a) Pyrite (b) marcasite (c) mackinawite (d) NiAs-troilite (e) troilite. The yellow ballsdenote S atoms and brown balls are Fe atoms.

Fig. 2. The relative error (%) of various XC-functionals in predicting latticeconstants for five iron sulfides. DFT+U (2.0) gives the overall best perfor-mance in structure predictions.

J. Liu, et al. Computational Materials Science 164 (2019) 99–107

101

various XC-functionals. Except for functionals used in structural pre-diction, we further select Hartree-Fock and hybrid functional (B3LYP,PBE0, and HSE06) to perform a static self-consistent convergence cal-culation based on optimized structures with PBE+U (Ueff=2.0 eV).Many studies have reported that hybrid functional could be good atdescribing the magnetic property and fundamental band gaps respect toexperimental observations, especially for strongly-correlated systemssuch as transition metal oxides [37–39]. From Table S2, all thesefunctionals used in this work predict polymorphs FeS2 (pyrite andmarcasite) as nonmagnetic (NM) material, this is in coincidence withexperimental observations [18]. For three FeS cases, because themagnetic orderings are more complex than FeS2, we first calculated therelative energies (eV/unit) for antiferromagnetic (AFM), ferromagnetic(FM), and non-magnetic (NM) states with different functionals asshown in Table S3. Only the most stable states are summarized in TableS2. For mackinawite, LDA, GGA, and meta-GGA all give the predictionsthat non-magnetic states are more stable in energy, which is in line withexperimental results. As PBE+U, when Ueff=2.0 and 2.5 eV, themackinawite is predicted to be non-magnetic, when Ueff is beyond2.5 eV, the AFM state becomes more stable. In layer-type mackinawite,the adsorb-like sulfur atoms are located at hole site of quadrilateral fourFe atoms, the hybridization between Fe 3d states and S 3p states islimited, the latter calculated projected density of states also show thisphenomenon, so PBE+U (Ueff > 2.5 eV) over correct the 3d-electroninteractions. This explanation could also be proved by hybrid functionalpredictions, when the proportion of HF is beyond 15%, the energies ofAFM states are lower than NM states which are in contradict with ex-perimental results. Whilst HSE06 with 8% and 10% Hartree-Fock couldgive accurate predictions for the magnetic property of mackinawite. Inregard to two different phases of troilite, the real magnetic orderingsare antiferromagnetically determined by experiment. About theoreticalcalculations, standard DFT functionals (LDA or GGA) give wrong orunderestimated moments for these two cases. However, PBE+U,Hartree-Fock and hybrid functional could give reasonable magneticmoments respect to experimental values. According to our systematicstudy, PBE+U (Ueff=2.0 eV) is a better choice to predict magneticproperties for iron sulfides.

Using X-ray photoelectron spectroscope (XPS) and BremsstrahlungIsochromat Spectroscopy (BIS) or X-ray emission and absorptionspectra (XES & XAS), the fundamental band gaps of iron sulfides havebeen extensively investigated [40,41]. The band gaps of polymorphsFeS2 are 0.95 eV and 1.0 eV for pyrite and marcasite, respectively. Formackinawite, it’s a typical metallic material with a zero band gap.Under ambient atmosphere and room temperature, the band gap oftroilite is about 0.5 eV obtained by experimental characterization, whenthe temperature is above 588 K, troilite could transform to a highsymmetry hexagonal P63/mmc space group, namely NiAs-troilite with a

neglected band gap [3]. To cast light on the dependence of predictedband gaps on functional, we utilize different functionals to study theband gaps for five iron sulfides, the calculated results are presented inTable S2 as well as the experimental data are also listed for comparison.In pyrite case, LDA and GGA underestimate the band gaps (0.5 eV vs0.98 eV). Two meta-GGA functional predict it to be a metal other than asemiconductor. The band gap with PBE+U where Ueff=2.0 eV is0.94 eV, in excellent agreement with the experiment of 0.95 eV. Con-tinue to increase effective Hubbard U, the band gap is broadened stepby step which is in agreement with our previous study on transitionmetal oxides [42].

For Hartree-Fock approach, the band gap is much larger than realvalue, this can be ascribed to that in Hartree-Fock formula although theexchange part is exactly calculated, while the correlated part is nottaken into account, resulting in the band gaps reproduced by Hartree-Fock are deviate from the real ones.

In the case of hybrid functional, which are generally believed to begood at predicting band gaps for solid materials, B3LYP and PBE0 allmuch overestimated the band gap of pyrite. HSE06 with 8% Hartree-Fock could get an approximated band gap compared to experimentwhen the proportion is more than 8%, the predicted band gaps areobviously overestimated, the trend is in line with the study on themagnetic moment as described above. While for marcasite, the resultsare not the same. LDA, GGA, meta-GGA (SCAN) and PBE+U(Ueff=2.0 eV) could give reasonable band gaps. TPSS and HF give azero band gap, other approaches all overestimated the band gap. What’smore, it’s interesting to note that all these functionals predict mack-inawite as conductor except for PBE+U (Ueff=4.5 eV). For NiAs-troilite, PBE+U when Ueff is beyond 2.0 eV and HF give inaccurateband gap while other approaches are qualified in describing the bandgap. Go so far as to troilite, the experimental band gap is around 0.5 eV.However, in the above-involved approximations, only predictions ofPBE+U with Ueff=2.0 eV and HSE06 with 15% Hartree-Fock arecomparatively reasonable.

For the sake of providing a succinct and clear profile about differentfunctionals compartment in describing the magnetic and electronicproperties of iron sulfides, we compare the deviation between calcu-lation and experiment as plotted in Fig. 3. If the symbols fall on blacksolid lines, it indicates that the predicted results are in coincident withexperimental values. It can be perceived from Fig. 3(a) that LDA, PBE,and TPSS all underestimate magnetic moments for troilite, B3LYP,PBE0, as well as HSE06, overestimate moments for mackinawite. Inbrief, PBE+U (Ueff=2.0 eV) is an excellent choice to forecast themagnetic property of iron sulfides. In Fig. 3(b), the tendency of bandgaps is similar to magnetic moments. LDA, PBE, and TPSS all under-estimate the band gaps, conversely hybrid functional give overvaluedresults. Only the prediction utilizing PBE+U (Ueff=2.0 eV) is

Fig. 3. (a) The calculated magnetic moment (μB) and (b) band gap (eV) with different DFT approximations (horizontal axis present experimental data, vertical axisare calculated values).

J. Liu, et al. Computational Materials Science 164 (2019) 99–107

102

approximate to experimental values.According to our above systematic studies, we can tentatively draw

a conclusion that PBE+U with Ueff=2.0 eV could be qualified inpredicting the structural parameters, magnetic moments and band gapsfor iron sulfides. Thereafter, we explore the Density of States (DOS) forinvolved five iron sulfides utilizing Hubbard U approximation whereUeff is equal to 2.0 eV and then compare the DOS with experimentalphoto-electronic spectrum. The conceptually most straightforwardmethod to get band gaps is from direct and inverse X-ray photoelectronspectra (XPS and BIS) or X-ray emission and absorption spectra (XESand XAS).

In Fig. 4, one can well perceive that the peak positions of valenceand conduction bands by theoretical calculations are highly consistentwith experimental spectrum, this confirms that Hubbard U withUeff=2.0 eV is good in predicting the electronic properties of ironsulfides. As regards the DOS by other approximations, the detailed re-sults are plotted in Figs. S1–S5.

3.3. Electronic structure of iron sulfides

After the scientific benchmark of different calculation methods, weintend to discuss the electronic band structure and bonding property forthese five iron sulfides. The electronic band structure and corre-sponding local density of states (LDOS) of these iron sulfides com-pounds are calculated utilizing PBE+U with Ueff=2.0 eV as displayedin Fig. 5. Here the DOS is decomposed into the most significant bondingcontributions in the energy range around the Fermi level, which are S

3p and Fe 3d orbital contributions. For iron sulfide pyrite, Fig. 5(a), it’snot difficult to detect an indirect band gap of 0.95 eV. The valence bandmaximum is located between M and G K-points while the conductionband minimum is located at G K-point. From the LDOS, the most con-tributions between −4 eV to 4 eV are Fe 3d states with a non-neglectedS 3p states, it signifies that in this energy range the hybrid of Fe 3d andS 3p is strong. The covalent bond of the S dimers contributed to S 3s andS 3p orbitals are located at lower energy range which is not presentedhere.

The FeS2 marcasite is structurally closed to iron pyrite and is con-firmed that can co-exist with pyrite under certain conditions. FromFig. 5(b), one can find that marcasite has a similar band gap to pyrite,which is about 1.0 eV as corresponding to 0.95 eV for pyrite. In 1980,the optical band gap of marcasite has been reported to be 0.34 eVcharacterized by electrical resistivity measurements [34], which fardeviates from recent experimental studies and DFT calculations. Thereason might be that the stoichiometric composition is quite difficult tocontrol in experimental conditions, the S atoms are too easily escapefrom the solid surface to reproduce sulfur vacancies. The band gap isdrastically decreased when a sulfur vacancy appears. Based on ourcalculations, marcasite also exhibits an indirect band gap where thevalence band maximum is located at Y high-symmetry K-point whilethe conduction band minimum is located at U K-point.

For tetragonal FeS, namely mackinawite, from our calculated den-sity of states and electronic band structure as shown in Fig. 5(c), onecan find that the Fermi level cuts a band of Fe 3d orbital, indicating thepresence of mobile electrons carriers with metallic nature which is

Fig. 4. Comparison of theoretical total density of states (TDOS) calculated by PBE+U (Ueff=2.0 eV) method with experimental X-ray spectrum for five ironsulfides. The Fermi level εF is set to 0 eV.

J. Liu, et al. Computational Materials Science 164 (2019) 99–107

103

concordant with previous DFT and experimental studies [6]. Whencompared to above two FeS2 cases, the dominant contribution near theFermi energy level is deriving from Fe 3d states, the primary interac-tions are coming from the direct intralayer Fe-Fe bonds, the calculationresult is well consistent with the general DOS features of layer-typetetragonal iron-based compounds [43].

The iron mono-sulfide troilite transfers to NiAs-troilite when thetemperature is above the Neel temperature of 588 K. Next, we study thechange of electronic structures of FeS below and above Neel tempera-ture. Below Neel temperature, the FeS crystallizes in troilite structurewith antiferromagnetic orderings, previous literature has reported thattroilite appeared to be a semiconductor with a much-narrowed bandgap of 0.04 eV whereas behaves like metal when above Neel tempera-ture [26]. But latter photoemission and bremsstrahlung isochromatspectroscopy (BIS) spectra revealed that the real band gap of FeS shouldbe larger than 0.04 eV, the previous underestimated band gap is due tothe characterization is performed under extreme temperature andtemperature conditions. Our calculations by PBE+U where Ueff isequal to 2.0 eV reproduce a band gap of about 0.6 eV as shown inFig. 5(e). The troilite FeS is a typical semiconductor with an indirectband gap. The Fermi level is very close to the main peak of the Fe 3dband, our calculations are good agreement with previous DFT calcu-lations [12]. Last but not least, the Fermi level of NiAs-troilite cutsthrough the two electronic bands (Fig. 5(d)). In other words, it behavesa metallic feature. Compared to troilite, the shape of DOS is semblablejust the splitting at Fermi level is absent.

Moreover, in two FeS2 cases, the contributions of S 3p should not beignored no matter in valence or conduction bands below and above theFermi level, this indicates that pyrite and marcasite should fall in therealm of charge transfer insulators. While for troilite, the bands near theFermi level are dominated by Fe 3d states, suggesting that it belongs to

Mott-Hubbard insulator.

3.4. Bonding analysis

After evaluating XC-functionals and obtaining reliable data ofproperties, one can have a large space to play in predicting and cor-relating something. As known, the crystal structures and phases arevery complicated for iron sulfides for its polymorphs. As a result, nu-merous non-stoichiometric phases make a great challenge to investigatetheir properties. The intrinsic difference is the various coordinatedconditions for centered Fe atoms. The routine way to capture theproperties of these phases is to the brute-force calculations carried outone by one, which is however infeasible because the calculations aremuch computational expensive. Therefore, it is critical to developing amodel to describe the bonding character of Fe-S in different crystal, anda relationship between bonding character and other properties. To putit in another way, from a big data point of view, one can use a simplemodel or calculations to predict/calibrate more properties for variousstructures. Here, based on PBE+U (Ueff=2.0) static self-consistentcalculations, the Crystal Orbital Hamilton Population (COHP) analysisof Fe-S bonds has been performed. This COHP approach is analogous tothe Crystal Orbital Overlap Population (COOP) developed by Hoffmannin the context of extended Hückel calculations [45,46]. The purpose ofCOOP (COHP) is to extract the information of chemical bonding out ofthe standard DFT calculations, the positive and negative values ofCOOP present bonding and anti-bonding, respectively. For COHP, themeaning is just the opposite. If the COOP or COHP is zero, it suggests anon-bonding interaction.

We extract the chemical bonds with a centered Fe and plotted thetotal COHP in Fig. 6. The orange filled area presents Fe-S bonding,while the green filled area presents Fe-S anti-bonding. In Fig. 6(a)–(c),

Fig. 5. Electronic band structure and density ofstates (DOS) of five iron sulfides compounds (a)pyrite, (b) marcasite, (c) mackinawite, (d) NiAs-troilite and (e) troilite from PBE+U(Ueff=2.0 eV) calculations. The Fermi energy εF islocated at zero and the high-symmetry points of thek-path are denoted according to Bradley andCracknell [44].

J. Liu, et al. Computational Materials Science 164 (2019) 99–107

104

one can see that when the energy level is below −1 eV, there is mainlybonding contribution, whereas from −1 eV to the maximum occupa-tion energy level, namely the Fermi level, there is only anti-bondingdistribution. This indicates that the Fe atoms in these three sulfides arebinding strongly with coordinated S atoms, also suggests these phasesare stable. While for polymorphs NiAs-troilite (Fig. 6(d)) and troilite(Fig. 6(e)), the anti-bonding region are extended from Fermi level to−5 eV, this occupied anti-bonding weaken the interactions between Fe-S.

Bond valence model derived from Pauling’s rule is a robust methodto validate the chemical structure and evaluate the oxidation states ofatoms, even be used to predict other properties of complex structures.Steinfink et al. have proposed a model to describe the relationshipbetween bond valence and bond length for iron-sulfur compounds [47].The empirical bond valence is calculated based on Eq. (2)

∑==

−V S R R( / )i

CN

iN

01

0(2)

where V is the valence, Ri is the bond distance, and CN is the co-ordination number. The constants S0 is chosen as 1/3, the R0 is a hy-pothetical value for Fe2+ in an octahedral environment. The resultantequation for the iron-sulfur compounds is

∑==

−V R1/3 ( /2.515)i

CN

i1

6.81

(3)

The bond valence could reflect the bond strength, the integratedCOHP (ICOHP) could also give the information of strength for thespecific bond. If the empirical model is rational, the bond valenceshould have a liner relationship with the ICOHP. We calculate theICOHP for centered Fe atoms in five iron sulfides as presented inTable 2. The ICOHP of Fe atoms in polymorphs FeS2 marcasite andpyrite is larger than that of in other phases, indicating that the Fe in-teracts strongly with surrounded S atoms in these two polymorphs.While in two troilite phases, the ICOHP is small, suggesting that the

weak bonding for centered Fe atoms. This is well consistent with theabove COHP analysis. The ICOHP as a function of bond valence isplotted in Fig. 7(a), indicating that the ICOHP presents a good linerrelation with Fe bond valence. This correlation suggests that the em-pirical bond valence model is reasonable to describe the character ofiron-sulfur bonds in order to predict the stability of iron-sulfur phases.

Furthermore, we have calculated the Bader charge of centered Featoms in different phases (Table 2) and establish the correlation be-tween Bader charge and Fe-S bond valence, as shown in Fig. 7(b). TheBader charge is a liner function of bond valence. Because the charge ofatoms is re-distributed after bonding, the charge gathering or deletion isa signal for bonding strength. Our result indicates that this empiricalbond valence model could extend to other iron sulfides phases andexplore their properties. We know that the bonding stability or chargecharacter are important factors in screening and designing functionaland catalytic materials. In this way, such correlations can save the ex-pensive computation resources in DFT, and achieve fast-speed predic-tions.

4. Conclusions

We assess the performance of 14 XC-functional of DFT including HF,LDA, GGA, meta-GGA, DFT+U as well as hybrid functional on struc-tural, magnetic and electronic properties of two FeS2 and three FeSpolymorphs. The calculation results reveal that PBE+U withUeff=2.0 eV could give reasonable predictions for lattice constants,magnetic orderings and moments as well as fundamental band gaps.Using PBE+U (Ueff=2.0 eV) calculations, FeS2 polymorphs pyriteand marcasite both show non-magnetic, and get 0.94 eV and 0.98 eVindirect band gaps, respectively. The electronic band structures indicatethey are charge transfer semiconductors. For three FeS polymorphs,mackinawite with layer-type structure is a non-magnetic and metallicmaterial. While troilite and NiAs-troilite, they both present AFM or-derings and the band gaps are 0.64 eV and 0 eV for troilite and NiAs-troilite, respectively. At last, the COHP, as well as Bader charge ofcentered Fe atoms in various sulfides is calculated. We find that theICOHP and Bader charge has a good liner relation with bond valencefrom the empirical model. Our model could open a new routine to ef-fectively study the bonding characters and other properties for variousiron sulfides phases and structures. The correlations between bondingand electronic property will help us to achieve fast-speed predictions ofproperties and high throughput screening of interesting phases forfurther applications.

Fig. 6. The Crystal Orbital Hamilton Population analysis of Fe-S bonds in (a) pyrite (b) marcasite (c) mackinawite (d) NiAs-troilite (e) troilite. The dashed linerepresents the Fermi level which is set to zero.

Table 2The Bader charge analysis, integrated COHP (ICOHP) and bond valence of Featoms in iron sulfides.

Bader charge (e-) ICOHP Bond valence

FeS2 (pyrite) −0.67 −13.39 4.06FeS2 (marcasite) −0.68 −13.85 4.27FeS (mackinawite) −0.55 −10.56 3.78FeS (NiAs-troilite) −0.98 −7.52 2.18FeS (troilite) −0.94 −6.85 2.05

J. Liu, et al. Computational Materials Science 164 (2019) 99–107

105

CRediT authorship contribution statement

Jinjia Liu: Writing - original draft. Aiju Xu: Writing - original draft,Funding acquisition. Yu Meng: Data curation. Yurong He: Writing -review & editing. Pengju Ren: Writing - review & editing. Wen-Ping Guo: Conceptualization. Qing Peng: Writing - review & editing.Yong Yang: Supervision. Haijun Jiao: Writing - review & editing.Yongwang Li: Project administration, Supervision. Xiao-Dong Wen:Project administration, Funding acquisition, Writing - review & editing.

Acknowledgements

The authors are grateful for the financial support from the NationalNatural Science Foundation of China (No. 21473229, No. 91545121),and funding support from Synfuels China, Co. Ltd. We also acknowl-edge the Innovation Foundation of Institute of Coal Chemistry, ChineseAcademy of Sciences, Hundred-Talent Program of Chinese Academy ofSciences, Shanxi Hundred-Talent Program and National ThousandYoung Talents Program of China. We also thank the CollaborativeInnovation Center of Water Environmental Security of Inner MongoliaAutonomous Region, China (XTCX003). The computational resourcesfor the project were supplied by the Tianhe-2 in Lvliang, Shanxi.

Appendix A. Supplementary data

Supplementary data to this article can be found online at https://doi.org/10.1016/j.commatsci.2019.04.001.

References

[1] Q. Li, Q. Wei, W. Zuo, L. Huang, W. Luo, Q. An, V.O. Pelenovich, L. Mai, Q. Zhang,Fe3S4 as a new anode material for high-performance sodium-ion batteries, Chem.Sci. 8 (2017) 160–164.

[2] S.-Y. Huang, D. Sodano, T. Leonard, S. Luiso, P.S. Fedkiw, Cobalt-doped iron sulfideas an electrocatalyst for hydrogen evolution, J. Electrochem. Soc. 164 (2017)276–282.

[3] F. Ricci, E. Bousquet, Unveiling the room-temperature magnetoelectricity of troiliteFeS, Phys. Rev. Lett. 116 (2016) 227601.

[4] D. Spagnoli, K. Refson, K. Wright, J.D. Gale, Density functional theory study of therelative stability of the iron disulfide polymorphs pyrite and marcasite, Phys. Rev. B81 (2010) 094106.

[5] J. Muscat, A. Hung, S. Russo, I. Yarovsky, First-principles studies of the structuraland electronic properties of pyrite FeS2, Phys. Rev. B 65 (2002) 054107.

[6] J.P. Perdew, A. Zunger, Self-interaction correction to density-functional approx-imations for many-electron systems, Phys. Rev. B 23 (1981) 5048–5079.

[7] J.P. Perdew, M. Ernzerhof, K. Burke, Rationale for mixing exact exchange withdensity functional approximations, J. Chem. Phys. 105 (1996) 9982–9985.

[8] J.P. Perdew, A. Ruzsinszky, G.I. Csonka, O.A. Vydrov, G.E. Scuseria,L.A. Constantin, X. Zhou, K. Burke, Restoring the density-gradient expansion forexchange in solids and surfaces, Phys. Rev. Lett. 100 (2008) 136406.

[9] A.I. Liechtenstein, V.V. Anisimov, J. Zaanen, Density-functional theory and stronginteractions: Orbital ordering in Mott-Hubbard insulators, Phys. Rev. B: Condens.Matter 52 (1995) 5467–5470.

[10] S.L. Dudarev, G.A. Botton, S.Y. Savrasov, C.J. Humphreys, A.P. Sutton, Electron-energy-loss spectra and the structural stability of nickel oxide: an LSDA+U study,Phys. Rev. B 57 (1998) 1505–1509.

[11] S.G. Choi, J. Hu, L.S. Abdallah, M. Limpinsel, Y.N. Zhang, S. Zollner, R.Q. Wu,M. Law, Pseudodielectric function and critical-point energies of iron pyrite, Phys.Rev. B 86 (2012) 115207.

[12] L. Koutti, O. Bengone, J. Hugel, First-principles calculations on the electronicstructure of FeS, Comp. Mater. Sci. 17 (2000) 169–173.

[13] K. Kim, K.D. Jordan, Comparison of density functional and MP2 calculations on thewater monomer and dimer, J. Phys. Chem. 98 (1994) 10089–10094.

[14] K. Burke, John P. Perdew, M. Ernzerhof, Generalized gradient approximation madesimple, Phys. Rev. Lett. 77 (1996) 3865–3868.

[15] J. Heyd, G.E. Scuseria, M. Ernzerhof, Hybrid functionals based on a screened cou-lomb potential, J. Chem. Phys. 118 (2003) 8207.

[16] W.M. Temmerman, P.J. Durham, D.J. Vaughan, The electronic structures of thepyrite-type disulphides (MS2, where M=Mn, Fe Co, Ni, Cu, Zn) and the bulkproperties of pyrite from local density approximation (LDA) band structure calcu-lations, Phys. Chem. Miner. 20 (1993) 248–254.

[17] J. Hu, Y. Zhang, M. Law, R. Wu, First-principles studies of the electronic propertiesof native and substitutional anionic defects in bulk iron pyrite, Phys. Rev. B 85(2012) 085203.

[18] W. Paszkowicz, J.A. Leiro, Rietveld refinement study of pyrite crystals, J. Alloy.Compd. 401 (2005) 289–295.

[19] S. Ogawa, Magnetic properties of 3d transition-metal dichalcogenides with thepyrite structure, J. Appl. Phys. 50 (1979) 2308–2311.

[20] I.J. Ferrer, D.M. Nevskaia, C. de las Heras, C. Sánchez, About the band gap nature ofFeS2 as determined from optical and photoelectrochemical measurements, SolidState Commun. 74 (1990) 913–916.

[21] T. Schena, G. Bihlmayer, S. Blügel, First-principles studies of FeS2 using many-bodyperturbation theory in the G0W0 approximation, Phys. Rev. B 88 (2013) 235203.

[22] P.A. Montano, M.S. Seehra, Magnetism of iron pyrite (FeS2)—a Mössbauer study inan external magnetic field, Solid State Commun. 20 (1976) 897–898.

[23] R. Sun, G. Ceder, Feasibility of band gap engineering of pyrite FeS2, Phys. Rev. B 84(2011) 245211.

[24] D.J. Vaughan, M.S. Ridout, Mössbauer studies of some sulphide minerals, J. Inorg.Nucl. Chem. 33 (1971) 741–746.

[25] P. Martin, G.D. Price, L. Vočadlo, An ab initio study of the relative stabilities andequations of state of FeS polymorphs, Mineral. Mag. 65 (2001) 181–191.

[26] J.R. Gosselin, M.G. Townsend, R.J. Tremblay, Electric anomalies at the phasetransition in FeS, Solid State Commun. 19 (1976) 799–803.

[27] P.E. Blöchl, Projector augmented-wave method, Phys. Rev. B 50 (1994)17953–17979.

[28] G. Kresse, J. Furthmüller, Efficient iterative schemes for ab initio total-energy cal-culations using a plane-wave basis set, Phys. Rev. B 54 (1996) 11169–11186.

[29] D.R. Hartree, The wave mechanics of an atom with a non-coulomb central field.Part I. Theory and methods, Math. Proc. Cambridge 24 (1928) 89.

[30] V. Fock, Naherungsmethode zur losung des quantenmechanischen mehrkorper-problems, Z. Phys. 61 (1930) 126–148.

[31] P. Hohenberg, W. Kohn, Inhomogeneous electron gas, Phys. Rev. 136 (1964)864–871.

[32] J. Sun, M. Marsman, G.I. Csonka, A. Ruzsinszky, P. Hao, Y.-S. Kim, G. Kresse,J.P. Perdew, Self-consistent meta-generalized gradient approximation within theprojector-augmented-wave method, Phys. Rev. B 84 (2011) 035117.

[33] C. Sánchez, E. Flores, M. Barawi, J.M. Clamagirand, J.R. Ares, I.J. Ferrer, Marcasiterevisited: optical absorption gap at room temperature, Solid State Commun. 230(2016) 20–24.

[34] M.S. Jagadeesh, M.S. Seehra, Electrical resistivity and band gap of marcasite (FeS2),Phys. Lett. A 80 (1980) 59–61.

[35] C.D.L. Heras, I.J. Ferrer, C. Sanchez, Temperature dependence of the optical ab-sorption edge of pyrite FeS2 thin films, J. Phys.: Condens. Matter 6 (1994)10177–10183.

[36] E.F. Bertaut, P. Burlet, J. Chappert, Sur l'absence d'ordre magnetique dans la formequadratique de FeS, Solid State Commun. 3 (1965) 335–338.

[37] X.-D. Wen, R.L. Martin, G.E. Scuseria, S.P. Rudin, E.R. Batista, A screened hybridDFT study of actinide oxides, nitrides, and carbides, J. Phys. Chem. C 117 (2013)13122–13128.

[38] Y. Meng, X.W. Liu, C.F. Huo, W.P. Guo, D.B. Cao, Q. Peng, A. Dearden, X. Gonze,Y. Yang, J. Wang, H. Jiao, Y. Li, X.D. Wen, When density functional approximationsmeet iron oxides, J. Chem. Theory Comput. 12 (2016) 5132–5144.

Fig. 7. the correlation between (a) integrated COHP (ICOHP), (b) Bader charge and bond valence of centered Fe atoms.

J. Liu, et al. Computational Materials Science 164 (2019) 99–107

106

[39] X.D. Wen, R.L. Martin, T.M. Henderson, G.E. Scuseria, Density functional theorystudies of the electronic structure of solid state actinide oxides, Chem. Rev. 113(2013) 1063–1096.

[40] A.V. Soldatov, A.N. Kravtsova, M.E. Fleet, S.L. Harmer, Electronic structure of MeS(Me=Ni Co, Fe): X-ray absorption analysis, J. Phys.: Condens. Matter 16 (2004)7545–7556.

[41] W. Folkerts, G.A. Sawatzky, C. Haas, R.A.D. Groot, F.U. Hillebrecht, Electronicstructure of some 3d transition-metal pyrites, J. Phys. C: Solid State Phys. 20 (1987)4135–4144.

[42] J.-J. Liu, Y. Meng, P. Ren, B. Zhaorigetu, W. Guo, D.-B. Cao, Y.-W. Li, H. Jiao, Z. Liu,M. Jia, Y. Yang, A. Xu, X.-D. Wen, Predicting the structural and electronic propertiesof transition metal monoxides from bulk to surface morphology, Catal. Today 282

(2017) 96–104.[43] N.Y. Dzade, A. Roldan, N.H. de Leeuw, The surface chemistry of NOx on mack-

inawite (FeS) surfaces: a DFT-D2 study, PCCP 16 (2014) 15444–15456.[44] C.J. Bradley, A.P. Cracknell, The Mathematical Theory of Symmetry in Solids,

Clarendon Press, 1972.[45] R. Hoffmann, Solids and Surfaces: A Chemist’s View of Bonding in Extended

Structures, VCH Publishers, Weinheim, 1988.[46] T. Hughbanks, R. Hoffmann, Chains of trans-edge-sharing molybdenum octahedra:

Metal-metal bonding in extended systems, J. Am. Chem. Soc. 105 (1983)3528–3537.

[47] J.T. Hoggins, H. Steinfink, Empirical bonding relationships in metal-iron-sulfidecompounds, Inorg. Chem. 15 (1976) 1682–1685.

J. Liu, et al. Computational Materials Science 164 (2019) 99–107

107