We present results from first principle, local density approximation (LDA) calculations of electronic, transport, and bulk properties of iron pyrite (FeS2). Our non-relativistic computations employed the Ceperley and Alder LDA potential and the linear combination of atomic orbitals (LCAO) formalism. The implementation of the LCAO formalism followed the Bagayoko, Zhao, and Williams (BZW) method, as enhanced by Ekuma and Franklin (BZW-EF). We discuss the electronic energy bands, total and partial densities of states, electron effective masses, and the bulk modulus. Our calculated indirect band gap of 0.959 eV (0.96), using an experimental lattice constant of 5.4166 Å, at room temperature, is in agreement with the measured indirect values, for bulk samples, ranging from 0.84 eV to 1.03 ± 0.05 eV. Our calculated bulk modulus of 147 GPa is practically in agreement with the experimental value of 145 GPa. The calculated, partial densities of states reproduced the splitting of the Fe d bands to constitute the dominant upper most valence and lower most conduction bands, separated by the generally accepted, indirect, experimental band gap of 0.95 eV.

Iron pyrite (FeS2), commonly known as “fool’s gold”, is an earth-abundant, non-toxic and most common of the sulfide minerals. It has, for a long time, been investigated as a candidate material for photo-electrochemical and photovoltaic cells.1–6 Due to its small band gap of 0.95 eV,4 very high absorption coefficient (α > 6.0×105 cm-1 for >1.3 eV),2 low material cost1,3,7 and high photocurrent quantum efficiency (>90%),3 it is being extensively used in thin film solar cells. Iron pyrites have been widely utilized in the production of sulfur dioxide and sulfuric acid, jewelry, mineral detection in radio receivers, solid state batteries and catalysis.3 

Iron pyrite has a simple cubic structure that is reminiscent of rocksalt structure. It was among the first crystal structure determined in 1914 by Bragg,8 with his new X-ray diffraction system.

Iron pyrite is becoming a material of choice for thin film photovoltaic applications. Because of its current and potential technological importance,1–6 many theoretical9–28 and experimental works1–7,29–56 have been carried out on FeS2. Bullett10 performed first principle local density approximation (LDA) calculations to study the optical properties of FeS2. The calculated, indirect band gap was 0.7 eV, for pyrite, and 0.4 eV, for marcasite structures. The marcasite structure is an orthorhombic polymorph of FeS2. This value underestimated the measured band gap by 0.25 eV for pyrite, when compared to the generally accepted, experimental, indirect band gap value of 0.95.4 Zhao et al.27 employed the self-consistent linear combination of atomic orbital (LCAO) method to study the electronic properties of FeS2. They found an indirect band gap of 0.59 eV from X to Γ and the smallest, direct gap as 0.74 eV at the zone center. They also showed results for the density of states of FeS2. Opahle et al.21,22 calculated the electronic properties of FeS2 using a LDA potential parameterized by the Perdew-Zunger (LDA-PZ). Their theoretical band gap of 0.85 eV is close to the experimental value, but it was obtained with the minimum basis set. Muscat et al.20 performed calculations of the electronic properties of FeS2 using the periodic LCAO formalism with the CRYSTAL98 package and plane wave, pseudopotential technique with the CASTEP software package. They predicted FeS2 to be a conducting material, from LDA and generalized gradient approximation (GGA) potential calculations, and a semiconductor with band gap of 11 and 2 eV, from Hartree-Fock (HF) and Becke’s three-parameter hybrid potential by Lee, Yang, and Parr (B3LYP) calculations, respectively. Guanzhou et al.23 calculated a band gap of 0.65 eV for bulk FeS2. For a nine-layer slab, they found a band gap of 0.16 eV. These authors23 performed first-principle, norm-conserving pseudopotential calculations using LDA and GGA potentials. Brian et al.19 performed fully self-consistent Green’s function (GW) and dressed Coulomb approximation calculations. They predicted a band gap of 0.83 eV for FeS2. Choi et al.12 utilized a GGA potential and the projector augmented wave (PAW) method and reported a large band gap of 2.70 eV using a hybrid functional Hyed-Scuseria-Ernzerhof (HSE06) potential. Hu et al.16,17 employed a Perdew-Burke-Ernzerhof GGA potential with Hubbard U (PBE+U) to obtain a band gap of 1.02 eV in bulk iron pyrite. They reported that the band gap of iron pyrite could be increased to 1.2–1.3 eV by replacing 10% of sulfur atoms with oxygen atoms. The above theoretical results and others are summarized in Table I below. Unlike the majority of the other ab-initio DFT calculations, the ASW28 and LMTO9 calculations produced band gaps of 0.95 and 0.9 eV, respectively, in excellent agreement experimental findings.

TABLE I.

Previous, calculated band gaps of iron pyrite (FeS2).

Computational formalism and methodPotentials (DFT and others)Band gap, Eg (eV)
Xα, LCAO LDA 0.7 (indirect)10  
Linear combination of atomic orbital LDA 0.8521,22 
(LCAO) – using only the minimum basis   
LCAO LDA conductor20  
Linear combination of LDA 0.51 (indirect) 
Gaussian type orbitals (LCGO)  0.55 (direct)13  
Total-energy pseudopotential, LDA 0.16 (in 9 layer slab) 
norm-conserving pseudopotential  0.65 (in bulk)23  
Projector augmented wave (PAW) LDA 0.22 (indirect)25  
Augmented spherical wave (ASW) LDA 0.9 (indirect)14  
Linear muffin-tin orbital atomic sphere LDA 0.64 (direct)26  
approximation (LMTO-ASA)   
Augmented spherical wave (ASW) LDA 0.95 (indirect)28  
Linear muffin-tin orbital (LMTO) LDA 0.99  
band structure method   
Ab-initio pseudo potential GGA 0.48 (indirect)11  
LCAO GGA conductor20  
PAW GGA-PBE 1.02 (indirect)17  
PAW GGA-PBE 0.40 (indirect)25  
PAW AM05+U 0.72 (indirect)25  
 PBE+U 1.03 (indirect)25  
 HSE06 2.76 (indirect)25  
PAW PBE+U 0.9515  
Plane wave pseudo potential (PWPP) B3LYP 220  
LCAO exchange-correlation 0.59 (indirect) 
  0.74 (direct)27  
Quasi particle Green function approach Green’s function and dressed 0.8319  
 Coulomb Approximation (GWA) –  
 a non-DFT approach  
Electron core pseudopotential (ECP) B3LYP 1.224  
Computational formalism and methodPotentials (DFT and others)Band gap, Eg (eV)
Xα, LCAO LDA 0.7 (indirect)10  
Linear combination of atomic orbital LDA 0.8521,22 
(LCAO) – using only the minimum basis   
LCAO LDA conductor20  
Linear combination of LDA 0.51 (indirect) 
Gaussian type orbitals (LCGO)  0.55 (direct)13  
Total-energy pseudopotential, LDA 0.16 (in 9 layer slab) 
norm-conserving pseudopotential  0.65 (in bulk)23  
Projector augmented wave (PAW) LDA 0.22 (indirect)25  
Augmented spherical wave (ASW) LDA 0.9 (indirect)14  
Linear muffin-tin orbital atomic sphere LDA 0.64 (direct)26  
approximation (LMTO-ASA)   
Augmented spherical wave (ASW) LDA 0.95 (indirect)28  
Linear muffin-tin orbital (LMTO) LDA 0.99  
band structure method   
Ab-initio pseudo potential GGA 0.48 (indirect)11  
LCAO GGA conductor20  
PAW GGA-PBE 1.02 (indirect)17  
PAW GGA-PBE 0.40 (indirect)25  
PAW AM05+U 0.72 (indirect)25  
 PBE+U 1.03 (indirect)25  
 HSE06 2.76 (indirect)25  
PAW PBE+U 0.9515  
Plane wave pseudo potential (PWPP) B3LYP 220  
LCAO exchange-correlation 0.59 (indirect) 
  0.74 (direct)27  
Quasi particle Green function approach Green’s function and dressed 0.8319  
 Coulomb Approximation (GWA) –  
 a non-DFT approach  
Electron core pseudopotential (ECP) B3LYP 1.224  

Table II, shows various experimental results for the band gap of single crystals, thin films and nano-crystalline samples of iron pyrite (FeS2). A variety of experimental works1–7,29–57 have been reported for the synthesis and characterization of FeS2. Schlegel et al.50 measured the reflectivity and transmission spectrum on naturally grown single crystal of iron pyrite (FeS2). They reported that iron pyrite has a completely filled 3d t2g band separated by an indirect gap of about 0.95 eV from an empty 3d eg band, at 300 K. Kou et al.41 reported a band gap of 0.84 eV for measurements done at 297 K, on two natural single crystal pyrite samples from Peru and Spain. Karguppikar et al.56 confirmed pyrite to be an indirect band gap semiconductor from optical absorption studies and their conductivity and Hall Effect data yielded a value of 0.92 eV, at room temperature, for the energy gap. Sun et al.53 fabricated iron pyrite thin films of thickness about 500 nm by sulfurizing oxide precursor films. Their X-ray photoelectron spectroscopy (XPS) and UV-vis absorbance spectroscopy found high absorption efficiency (> 105 cm-1) and values of 0.75 and 1.19 eV for the indirect and direct band gaps, respectively. Yu et al.55 prepared thin films by a chemical bath deposition (CBD) process. They studied the effect of Mn doping in the as-prepared pyrite. They show a gradual increase of the band gap from 0.86, 0.90, and 1.30 to 1.31 eV in the samples with Mn doping of 0%, 0.54%, 1.88% and 2.15%, respectively. They also reported a rise of the absorption coefficient of the films from 2.0×105cm-1 to 3.7×105cm-1.

TABLE II.

Measured band gaps of iron pyrite single crystals, thin films, and nanocrystals.

Experimental techniqueSample natureTransition typeEnergy gap Eg (eV)
Reflectivity and optical absorption bulk indirect 0.954,5 
Optical transmittance thin film (0.5-0.8 μm) indirect 1.12±0.003 at 300K29  
Absorbance spectroscopy nano-crystalline thin film indirect ∼0.95 
  direct ∼1.330  
TR-SPVa single-crystalline nanowire indirect 0.89±0.0532  
FTIRb at room temperature cubic crystallites  0.8734  
UV-visc spectrophotometry thin films direct 2.035  
optical absorption thin nano-films direct 1.3436  
Reflectivity and optical absorption thin film indirect ∼1.137,38 
Photoconductivity single crystal  0.87±0.02 at room temp.39  
Optical absorption natural single crystal  0.8441  
optical absorption thin films (70 nm)  0.92 
 (130 nm)  1.042  
XASd and XESe, IRf nanocrystals indirect 1.00±0.1143  
FTIR, UV-vis nanocrystals direct 2.75 at room temp.44  
UV-vis nanocrystal thin film  0.88–0.9147  
absorption coefficient thin films indirect 1.1748  
ED-X-rayg spectroscopy (room temp.) thin films indirect 0.945–1.0349  
Reflectivity single crystal  0.9550  
FTIR, UV-vis films indirect 0.85–0.87±0.0551  
 single crystal indirect 1.03±0.0551  
Optical absorption synthetic thin film indirect ∼0.7355  
Conductivity and Hall effect natural crystal indirect 0.9256  
Experimental techniqueSample natureTransition typeEnergy gap Eg (eV)
Reflectivity and optical absorption bulk indirect 0.954,5 
Optical transmittance thin film (0.5-0.8 μm) indirect 1.12±0.003 at 300K29  
Absorbance spectroscopy nano-crystalline thin film indirect ∼0.95 
  direct ∼1.330  
TR-SPVa single-crystalline nanowire indirect 0.89±0.0532  
FTIRb at room temperature cubic crystallites  0.8734  
UV-visc spectrophotometry thin films direct 2.035  
optical absorption thin nano-films direct 1.3436  
Reflectivity and optical absorption thin film indirect ∼1.137,38 
Photoconductivity single crystal  0.87±0.02 at room temp.39  
Optical absorption natural single crystal  0.8441  
optical absorption thin films (70 nm)  0.92 
 (130 nm)  1.042  
XASd and XESe, IRf nanocrystals indirect 1.00±0.1143  
FTIR, UV-vis nanocrystals direct 2.75 at room temp.44  
UV-vis nanocrystal thin film  0.88–0.9147  
absorption coefficient thin films indirect 1.1748  
ED-X-rayg spectroscopy (room temp.) thin films indirect 0.945–1.0349  
Reflectivity single crystal  0.9550  
FTIR, UV-vis films indirect 0.85–0.87±0.0551  
 single crystal indirect 1.03±0.0551  
Optical absorption synthetic thin film indirect ∼0.7355  
Conductivity and Hall effect natural crystal indirect 0.9256  
a

time resolved-surface photo voltage (TR-SPV) technique.

b

Fourier transformed infra-red (FTIR) spectroscopy.

c

UV-visible (UV-vis).

d

X-ray absorption spectroscopy (XAS).

e

X-ray emission spectroscopy (XES).

f

infra-red (IR) spectroscopy.

g

energy dispersive-X-ray (ED-X-ray).

Several other preparation processes, such as ion beam sputtering, spray pyrolysis, electro-deposition and metal-organic chemical vapor deposition (MOCVD), and aerosol assisted chemical vapor deposition (AACVD) have been reported for the synthesis of nano-wires, nano-crystals, crystallites and thin films of pure and doped iron pyrite (FeS2). The characterization of these samples led to band gaps ranging from 0.4 ± 0.1 eV to as high as 2.6 eV. Seefeld et al.51 synthesized film and bulk FeS2. Their transmittance and reflectance measurements showed an estimated indirect band gap of 0.85−0.87 ± 0.05 eV for the films and 1.03 ± 0.05 eV for the single crystals.

Only a few of the experimental findings in Table II concern bulk FeS2. These experimental band gaps range from 0.84 – 1.03 ± 0.05 eV. The value of 0.84 eV was obtained from natural samples whose quality is not established; further, the optical absorption method utilized to measure it has significant uncertainties due to relatively wide range of the absorption edge. Due to quantum confinement effects, measured band gaps of thin films and particularly of nano-crystals are expected to be larger than that for the single crystal. Most of the previous DFT calculations utilizing ab-initio LDA and GGA potentials underestimated the band gap of iron pyrite (FeS2) as shown in Table I. Results from hybrid functional calculations mostly overestimated the experimental gap. While the results from PBE+U calculations, i.e. 1.02 and 1.03 eV, practically agree with experiment, the ad hoc nature of the potential precludes drawing any conclusion from this agreement. The only GW calculation in Table I produced a gap of 0.83 eV, close to the generally accepted experimental value of 0.95 eV. Understandably, our discussions only concern DFT results as compared to ours. The above disagreements between theoretical band gaps from most ab-initio DFT calculations, on the one hand, and between them and the experimental values, on the other hand, are a motivation for this work. Further, the potential applications of pure and doped FeS2 add to this motivation. Once highly accurate descriptive and predictive computational capabilities are established, as we intend to do here for FeS2, then systematic and reliable calculations of properties of doped FeS2 can inform and guide device design and fabrication, including those for photovoltaic processes.

Our computational method has been described extensively in several, previous publications.58–67 We employed the Ceperley and Alder local density approximation (LDA) potential68 as parameterized by Vosko, Wilk, and Nusair69 and the linear combination of atomic orbitals (LCAO). Our implementation of the LCAO method entailed a methodical search for the absolute minima of the occupied energies, using successively larger basis sets in our self-consistent calculations with the program package from the Ames laboratory of the US Department of Energy (DOE), Ames, Iowa.70,71 As explained below, this search is necessary for any DFT electronic structure calculations without à priori knowledge of the electronic charge density for the ground state.

Indeed, Bagayoko59 showed that the second DFT theorem cannot be heeded without this search which, for close to two decades, has been carried out with the Bagayoko, Zhao, and Williams (BZW) method,61,62,65 enhanced in 2010 by Ekuma63,67 and Franklin.63,64 To search for the ground state, the method begins self-consistent calculations with a small basis set. This basis set cannot be smaller than the minimum basis set, i.e., the one just large enough to accommodate all the electrons in the system under study. This first calculation is followed by a second that employs the basis set of Calculation I, as augmented with one orbital representing an excited state. The method compares the occupied energies from Calculations I and II. In all previous cases, the occupied energies from the former were found to be higher or equal to their corresponding ones from the latter. After augmenting the basis set of Calculation II, the method requires a fully self-consistent Calculation III. We compare the occupied energies of this Calculation III to those of Calculation II. This process continues until three consecutive calculations lead to the same occupied energies, within our computational uncertainty of 5 meV. The rigorous physics and mathematics bases for the validity of our calculated band gap and of related properties rest on the following two points. The first point is that a DFT calculation, as per the second theorem, has to minimize the energy in a general fashion, using otherwise arbitrary basis sets, as explained by Bagayoko59 unless it starts with the known ground state charge density. For about fifty years, this necessary, generalized minimization of the energy has been confused with the quasi-linear minimization resulting for the attainment of self-consistency with a single basis set. The latter only produces a stationary solution, corresponding to the selected basis set, among an infinite number of such solutions. The noted general minimization is complete only when three consecutive calculations lead to the same occupied energies. This robust criterion for the attainment of the absolute minima of the occupied energies, i.e., the ground state of the system, has consistently held. We require three calculations due to the fact that two consecutive calculations can lead to the same occupied energies that can turn out to be local and not absolute minima of the occupied energies.

The second point is the following, once we reach the ground state. Let the basis set for the first of the three consecutive calculations be called the optimal basis set. There exist very many basis sets, including the optimal basis set and any large basis sets that contain it, that lead to the ground state; they also produce unoccupied energies that are generally different, except for the low laying ones for two basis sets whose difference in size is relatively small. The question then is to determine which basis set produces the DFT results. The answer follows. This optimal basis set and larger ones containing it lead to the same ground state charge density. Consequently, the Hamiltonian (not the matrix), a unique functional of ground state charge density, is the same for the noted basis sets. The first way to select the correct basis set relies on a corollary of the first DFT theorem.59 This corollary states that the spectrum of the Hamiltonian is a unique functional of the ground state charge density. Clearly, the spectrum obtained with the optimal basis set belong to the Hamiltonian. Due to the fact that this spectrum is a unique functional of the ground state charge density, any unoccupied energies from basis sets larger than the optimal one and that are different from their corresponding value produced by the optimal basis set cannot belong to the spectrum of the Hamiltonian. Consequently, the smallest basis set leading to the ground state, the optimal basis set, is the one leading to results that possess the full physical content of DFT. Years before Bagayoko work59 in 2014, we employed the following second way to select the correct basis set. It is based on the Rayleigh theorem. This theorem states that when solving the same eigenvalue equation with two basis sets 1 and 2, such that Basis set 1 is entirely included in Basis set 2, the eigenvalues obtained with the larger basis set are either lower or equal to the corresponding ones produced with the smaller basis set. From this theorem, one can see that out of the three consecutive calculations that produced the same occupied energies, the smallest basis set is the one that provides the DFT description of the material under study. Up to that basis set, the charge density changes from one calculation to the next, and so does the Hamiltonian and its spectrum. After that basis set, the Raleigh theorem strictly applies, given that the Hamiltonian keeps the same value it has with the optimal basis set. Then, a change in any unoccupied energy, while the Hamiltonian does not change, cannot be ascribed to a physical interaction; hence, it is the result of the mathematical artifact associated with the Rayleigh theorem.

This feature of the BZW-EF method completes DFT in practice, inasmuch as it enables the required search and attainment of the ground state as per the above first point. With the second point, the method identifies the correct DFT results (i.e., obtained with the optimal basis set) and avoid large basis sets that are over-complete for the description of the ground state.

In the BZW method, we add the orbitals representing unoccupied energies in the order of increasing, excited state energies of the atomic or ionic species in the solid. In the BZW-EF method, we add p, d, and f orbitals, if applicable, for a given principal quantum number n, before the corresponding s orbital for that quantum number. For a given atomic site, a p, d, or f orbital is applicable if it is occupied by at least one electron in the neutral atom. This rule rests on the recognition of the primacy of the polarization of p, d, and f orbitals, for valence electrons, over the spherical symmetry of s orbitals.

Iron pyrite crystallizes in the space group Th6-Pa3¯orPa3¯Th6 with four formula units of FeS2 per unit cell and a stoichiometric lattice constant of 5.418 Å at 27 oC. Six sulfur atoms surround each iron atom octahedrally. Each sulfur atom is bonded tetrahedrally to three iron atoms and one sulfur atom.11,31,54 In the unit cell, the four iron atoms are located at positions (0, 0, 0), (0, 1/2, 1/2), (1/2 0, 1/2), and (1/2, 1/2, 0). The eight sulfur atoms are in position ± (u, u, u), ± (u+1/2, 1/2−u, −u), ± (−u, u+1/2, 1/2−u), and ± (1/2−u, −u, u+1/2) where the Wyckoff parameter u = 0.386.11,27,54

We used a room temperature experimental lattice constant of 5.4166 Å46 for the first part of our work. We first performed ab-initio, self-consistent calculation for the ionic species Fe2+ and S1- to obtain the orbitals needed for the solid state calculations. Our program package utilizes Gaussian orbital in the radial parts of the orbitals; our version of LCAO is a linear combination of Gaussian orbitals (LCGO). A set of even-tempered Gaussian functions were employed in constructing the atomic orbitals of the ionic species. The numbers of even–tempered Gaussian exponents utilized for the s, p and d orbitals of Fe2+ were 18, 18, and 16, respectively. Similarly, we employed 18, 18, 16 even-tempered exponents for s, p and d orbitals for S1-, respectively. The minimum and maximum Gaussian exponents for Fe2+ were 0.12 and 0.11×106; they were 0.1689 and 0.79×105, respectively, for S1-. The computational error in the valence charge was 0.08084499 for 112 electrons, or 7.218×10-4 per electron. We employed a mesh of 121 k-points in the irreducible Brillouin zone for our self-consistent calculations. When the difference between the values of the potentials for consecutive iterations is less than 10-5, self-consistency is reached. This self-consistency criterion was generally satisfied after 90 iterations.

We show in Table III below the successive BZW-EF basis sets we employed in our calculations. The orbitals added from one calculation to another are in bold and have a superscript zero, indicating that represent unoccupied states. Calculation IV, in the shaded row, is the first to obtain the absolute minima of the occupied energies, the ground state, as required by the second DFT theorem. The corresponding basis set is the optimal one. The indirect band gap from this calculation, 0.96 eV, is practically the same as the generally accepted, experimental, indirect band gap of 0.95 eV. We credit this excellent agreement to our strict adherence to the conditions of validity of a DFT calculation, i.e., the two DFT theorems and related corollaries. The adherence to these conditions guarantees that our results have the full, physical content of DFT. We did not invoke a self-interaction correction or a derivative discontinuity, nor did we need any ad hoc parameters. From the literature review to the content of Table III, we have placed emphasis on the band gap. The simple reason for this emphasis is rooted in the fact that once the calculated band gap is incorrect, so are the calculated optical transitions, dielectric functions, electron effective masses, and a host of other properties.

TABLE III.

Calculations showing successive addition of orbitals as per the BZW-EF method. Calculations are done at the experimental lattice constant of 5.4166 Å, at room temperature. The DFT (LDA) indirect band gap is 0.9586 or 0.96 eV.

Calculation no.Iron (Fe2+) 4 per unit cellSulfur (S1-) 8 per unit cellNo. of functions X) energy gap (eV)
3s23p63d64p0 3s23p5 160  
II 3s23p63d64p04d0 3s23p5 200 1.033 
III 3s23p63d64p04d0 3s23p54p0 248 1.209 
IV 3s23p63d64p04d04s0 3s23p54p0 256 0.959 
3s23p63d64p04d04s0 3s23p54p04s0 272 0.954 
VI 3s23p63d64p04d04s05p0 3s23p54p04s0 296 0.936 
Calculation no.Iron (Fe2+) 4 per unit cellSulfur (S1-) 8 per unit cellNo. of functions X) energy gap (eV)
3s23p63d64p0 3s23p5 160  
II 3s23p63d64p04d0 3s23p5 200 1.033 
III 3s23p63d64p04d0 3s23p54p0 248 1.209 
IV 3s23p63d64p04d04s0 3s23p54p0 256 0.959 
3s23p63d64p04d04s0 3s23p54p04s0 272 0.954 
VI 3s23p63d64p04d04s05p0 3s23p54p04s0 296 0.936 

Figures 1(a) and (b) (for left and right, respectively), show the calculated electronic energy bands of FeS2. With an energy range of -20 to +8 eV, Figure 1(a) provides full pictures of the calculated bands for both the valence states and the low laying conduction states. This figure also shows the bands from Calculation V. As explained in the method section, Calculation V produced occupied energies identical to the corresponding ones from Calculation IV. Further, due to the fact that the actual size of the basis set per ion only increased by 2, for S1-, while it remained the same for Fe2+, in going from Calculation IV to V, the latter calculation also reproduced the low laying conduction bands obtained by Calculation IV up to +8 eV, within our computational uncertainty of 5 meV. Unlike Figure 1(a), Figure 1(b) shows a clear view of the upper-most group of valence bands. In Figure 1(b), it is apparent that the lowest, unoccupied energy is at the Γ point and the highest, occupied one is very close to the X-point in X-Γ direction. Therefore, our calculated band gap of 0.96 eV is an indirect one.

FIG. 1.

(a) (on the left). Calculated band structures of FeS2, as obtained from Calculations IV and V using the BZW-EF method, with a room temperature experimental lattice constant of 5.4166 Å. The solid lines (—) correspond to Calculation IV and the dashed lines (—) correspond to Calculation V. The horizontal line (– - – -) indicates the position of the Fermi energy (EF). (b) (on the right) provide a clear view of the upper-most group of valence bands. The valence band maximum (VBM) is between X and Γ, very close to X.

FIG. 1.

(a) (on the left). Calculated band structures of FeS2, as obtained from Calculations IV and V using the BZW-EF method, with a room temperature experimental lattice constant of 5.4166 Å. The solid lines (—) correspond to Calculation IV and the dashed lines (—) correspond to Calculation V. The horizontal line (– - – -) indicates the position of the Fermi energy (EF). (b) (on the right) provide a clear view of the upper-most group of valence bands. The valence band maximum (VBM) is between X and Γ, very close to X.

Close modal

We list, in Table IV, the calculated, electronic energies, in the range of -3.70 to +10.30 eV, at the high symmetry points in the Brillouin zone. Even though we did not find experimental data on direct transitions or the widths of the various groups of bands, this table is intended to facilitate comparisons of such future data with our findings.

TABLE IV.

Calculated eigenvalues (in eV) for iron pyrite (FeS2), as obtained with the optimal basis set of Calculation IV of the LDA-BZW-EF method, with a lattice constant 5.4166 Å at room temperature. The valence band maximum is between X and Γ, very close to X.

R-pointΓ-pointX-pointM-point
9.702 10.240 9.756 9.173 
9.702 10.240 8.663 9.173 
9.702 9.541 8.663 9.173 
3.326 3.743 3.496 3.147 
3.326 3.743 3.496 3.147 
3.326 3.262 2.673 3.147 
3.326 3.262 2.673 3.147 
2.446 3.262 2.608 2.059 
2.446 2.946 2.608 2.059 
2.446 2.946 2.420 2.059 
2.446 2.946 2.420 2.059 
1.300 1.523 1.810 1.767 
1.300 1.523 1.810 1.767 
1.300 1.523 1.668 1.767 
1.300 0.959 1.668 1.767 
-0.348 -0.270 -0.011 -0.118 
-0.348 -0.270 -0.011 -0.118 
-0.348 -0.270 -0.152 -0.118 
-0.348 -0.358 -0.152 -0.118 
-0.937 -0.358 -0.413 -0.410 
-0.937 -0.358 -0.413 -0.410 
-0.937 -0.498 -0.440 -0.410 
-0.937 -0.498 -0.440 -0.410 
-1.187 -0.919 -0.525 -0.905 
-1.187 -0.919 -0.525 -0.905 
-1.187 -0.919 -0.938 -0.905 
-1.187 -2.853 -0.938 -0.905 
-2.187 -2.853 -2.842 -2.349 
-2.187 -2.853 -2.842 -2.349 
-2.187 -3.649 -3.092 -2.349 
-2.187 -3.964 -3.092 -2.349 
R-pointΓ-pointX-pointM-point
9.702 10.240 9.756 9.173 
9.702 10.240 8.663 9.173 
9.702 9.541 8.663 9.173 
3.326 3.743 3.496 3.147 
3.326 3.743 3.496 3.147 
3.326 3.262 2.673 3.147 
3.326 3.262 2.673 3.147 
2.446 3.262 2.608 2.059 
2.446 2.946 2.608 2.059 
2.446 2.946 2.420 2.059 
2.446 2.946 2.420 2.059 
1.300 1.523 1.810 1.767 
1.300 1.523 1.810 1.767 
1.300 1.523 1.668 1.767 
1.300 0.959 1.668 1.767 
-0.348 -0.270 -0.011 -0.118 
-0.348 -0.270 -0.011 -0.118 
-0.348 -0.270 -0.152 -0.118 
-0.348 -0.358 -0.152 -0.118 
-0.937 -0.358 -0.413 -0.410 
-0.937 -0.358 -0.413 -0.410 
-0.937 -0.498 -0.440 -0.410 
-0.937 -0.498 -0.440 -0.410 
-1.187 -0.919 -0.525 -0.905 
-1.187 -0.919 -0.525 -0.905 
-1.187 -0.919 -0.938 -0.905 
-1.187 -2.853 -0.938 -0.905 
-2.187 -2.853 -2.842 -2.349 
-2.187 -2.853 -2.842 -2.349 
-2.187 -3.649 -3.092 -2.349 
-2.187 -3.964 -3.092 -2.349 

Figures 2 and 3 show the calculated, total (DOS) and partial (pDOS) densities of states, respectively. The total valence bandwidth is about 17.47 eV. As per the pDOS in Figure 3, the lowest laying group of conduction bands is mostly dominated by S-p and the Fe-d states, with a tiny contribution from S-s states. On the other hand, the upper most valence bands are mostly from Fe-d, with a significant hybridization with S-p. The group of valence bands just below the upper most one is dominated by S-p with a sizable contribution from Fe-d states; the small contributions from Fe-p and Fe-s are noticeable while those from S-s are barely so. The lowest laying group of valence bands are entirely from S-s; the group of bands immediately above it is clearly dominated by S-s, with very small contributions from Fe-p, Fe-d, Fe-s and S-p states. Future X-ray photoemission spectroscopy data are expected to confirm the above picture of the hybridization of the bands. It is gratifying to see that our calculated, partial densities of states reproduced the splitting of the Fe-d bands to constitute the dominant upper most valence band and the lower most conduction bands, separated by the generally accepted indirect gap of 0.95 eV.50 

FIG. 2.

Calculated, total density of states (DOS) of iron pyrite (FeS2), obtained from the bands in Figure 1. The dashed, vertical line shows the position of the Fermi energy (EF) set at zero.

FIG. 2.

Calculated, total density of states (DOS) of iron pyrite (FeS2), obtained from the bands in Figure 1. The dashed, vertical line shows the position of the Fermi energy (EF) set at zero.

Close modal
FIG. 3.

Calculated, partial densities of states (pDOS) of iron pyrite (FeS2), obtained from the bands in Figure 1. The dashed, vertical line indicates the position of the Fermi energy (EF) set at zero.

FIG. 3.

Calculated, partial densities of states (pDOS) of iron pyrite (FeS2), obtained from the bands in Figure 1. The dashed, vertical line indicates the position of the Fermi energy (EF) set at zero.

Close modal

The transport properties and other properties of materials, i.e., electrical conductivity and Seebeck coefficient, depend on electron and hole effective masses.18,20–23 From our calculation, the electron effective mass at the Γ point is isotropic and its value is 0.578 m0. Our calculated LDA-BZW-EF hole effective masses at the top of the valence band, along the X-Γ, X-R, and X-M directions, are 1.23 m0, 3.07 m0, and 4.07 m0, respectively. Zhao et al.27 reported the electron effective mass from their calculation to be about 0.35 m0 at Γ. Our calculated value is larger than the 0.35 m0 of Zhao et al. and the 0.49 m0 calculated by Hu et al.17 Hu et al. also reported hole effective masses ranging from 1.23 m0 to 1.98 m0. We could not find any experimental results for the electron or hole effective masses.

Our plot of the calculated total energy as a function of the lattice constant, for iron pyrite (FeS2), is shown in Figure 4 below. The total energies were calculated and plotted against the lattice constant a, ranging from 5.25 Å to 5.55 Å. The value of the lattice constant for the minimum of the total energy is the equilibrium lattice constant, determined to be 5.4082 Å. For this lattice constant, the calculated, indirect band gap (from X to Γ) is 0.961 eV. It is slightly larger, as expected, than the calculated room temperature band gap of 0.959 eV. Relatively small differences between the band gaps at room temperature and at 0 K is a common trend for indirect band gap semiconductors. Our calculated bulk modulus is 147 GPa; it agrees well with the reported experimental value of 145 GPa, overestimating it only by 1.38 %. A previous, calculated value of 185 GPa21,22 overestimated the measured value by 27.59 %.

FIG. 4.

The total energy (eV) of iron pyrite (FeS2) versus the lattice constant (Å). The minimum value of the total energy is located at the equilibrium lattice constant of 5.4082 Å.

FIG. 4.

The total energy (eV) of iron pyrite (FeS2) versus the lattice constant (Å). The minimum value of the total energy is located at the equilibrium lattice constant of 5.4082 Å.

Close modal

A clarification germane to the preceding description of our method pertains to the “ground state.” We call “absolute” ground state the one attainable following a geometry optimization of a system. However, to any spatial distribution of the atomic species in a material, there corresponds an external potential that depends on the interatomic distances, i.e., the spatial configuration of the atomic species. This potential, as per the first DFT theorem, is a unique functional of the charge density. The charge density also varies with the spatial configuration of the atomic species. With a given lattice constant, the minimization of the energy leads to a ground state that could be called a “relative” ground state, i.e., one that depends on a spatial configuration other than the one resulting from geometry optimization. Our references to the ground state, in this work, are only for the “relative” ones associated with the respective, selected lattice constants. As per the first DFT theorem, given that the external potentials are different for different spatial configurations, the “absolute” ground state is attainable only with the configuration corresponding to geometry optimization.

At the end of the introductory section, we summarized our review of previous DFT calculations and experimental findings for the band gap of FeS2. While experimental findings led to indirect band gaps between 0.84 and 1.03 ± 0.05 eV, with 0.95 eV as the most accepted value at room temperature, most previous ab-initio DFT calculations underestimated the experimental findings. Understandably, the band gap from some calculations with hybrid and ad hoc potentials, i.e., GGA+U, approached or agreed with the accepted experimental value; the adjustable parameters in these potentials preclude the drawing of any conclusion from this agreement, as far as the inability of previous ab initio DFT calculations to obtain the measured band gap is concerned. In contrast to a majority of the previous, ab-initio calculations, our LDA-BZW-EF calculations led to the experimental band gap. The description of our method contains the reasons we obtained accurate results as further noted below.

The following discussions pertain only to DFT calculations that employ exclusively atomic orbitals. In particular, they do not apply to calculations in which the basis sets include plane waves. The successive additional of orbitals (including radial and angular components) is not possible with plane waves. Our LDA-BZW-EF calculations produced the correct, generally accepted, experimental, indirect band gap of 0.95 eV due to the fact that they strictly adhered to necessary conditions for the validity of DFT calculations59 using the LCAO method. Specifically, our calculations minimized the energy functional representing the Hamiltonian, as required by the second Hohenberg-Kohn theorem. We utilized progressively augmented basis sets in consecutive, self-consistent calculations. This process led to the absolute minima of the occupied energies, i.e., the ground state. We subsequently invoked the Rayleigh theorem to avoid utilizing basis sets much larger than the optimal one; such larger basis sets are generally over-complete for the description of the ground state in DFT. They therefore lead to unphysical lowering of the unoccupied energies while the occupied ones retain their values obtained with the optimal basis set – if this basis set is entirely included in them. The size of a much larger basis set, as compared to the optimal one, determines the extent of the artificial lowering of unoccupied energies. We suggest that this fact partly explains not only the failure of single-basis set DFT calculations to describe accurately the electronic properties of materials, but also the disagreements between their results which vary with the basis sets. The chances for an arbitrary, single basis set, irrespective of how judiciously it is selected, to lead to the ground state without being over-complete are very small. We recall here, however, that the ASW and the LMTO calculations, in Table I, obtained the experimental band gap for this material. Depending of the specifics about the atomic orbitals inside the spheres and the treatment of the interstitial region, these two methods tend to produce results close to experimental findings. Even if such a single basis set were to lead to the measured band gap, that would not necessarily means that the results have the full physical content of DFT. Indeed, the widths of various groups of valence bands and other pertinent, electronic properties can still be in error, as our group illustrated in the case of ZnO.64 

We performed ab-initio, self-consistent LDA calculations of electronic energy bands, total (DOS) and partial (pDOS) densities of states (pDOS), effective masses, and the bulk modulus of iron pyrite (FeS2). The distinctive feature of our calculations, as compared to most previous ab-initio and empirical DFT calculations, is the rigorous implementation of the Bagayoko, Zhao, and Williams method, as enhanced by Ekuma and Franklin (BZW-EF). The adherence of this method to the necessary conditions of validity of DFT calculations explained not only the reasons we obtained the experimental band gap and bulk modulus, but also those for the disagreements between many, previous, ab-initio DFT results and between these results and corresponding experimental ones.

This research work was funded in part by the US Department of Energy – National Nuclear Security Administration (NNSA) (Award Nos. DE-NA0001861 and DE-NA0002630), the National Science Foundation (NSF HRD-1002541), LaSPACE, and LONI-SUBR.

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