In ﬂ uence of external electric ﬁ eld on the electronic structure and optical properties of pyrite

Pyrite (FeS 2 ) is an attractive photovoltaic material due to its high absorption coe ﬃ cient and suitable band gap. This work focuses on the e ﬀ ect of external electric ﬁ eld on the electronic band structure and optical properties using density functional theory with Hubbard U correction (DFT+U). Calculations suggest that the application of electric ﬁ eld in x , y and z directions produce similar results both in band structure and optical properties of pyrite. The addition of external electric ﬁ eld has not changed the p-type semiconductivity but band splits signi ﬁ cantly and the gap also decreases obviously at high electric ﬁ eld, accompanied with the fracture of the crystal. The variation of dielectric function, absorption coe ﬃ cient, re ﬂ ectivity, refractive index, extinction coe ﬃ cient and conductivity are observed due to the addition of electric ﬁ eld.


Introduction
Pyrite, which is the most common sulphide mineral on earth, has received much attention due to its high light absorption coefficient (>10 À5 cm À1 ) for applications as optoelectronic and photovoltaic materials. [1][2][3][4][5] It can be made into ultrathin lm solar cells due to its high light absorption coefficient. However, its band gap of 0.95 eV is far lower than the theoretical optimal gap of $1.3 eV for good photovoltaic materials proposed by Shockley and Queisser. 6 Much fundamental research has been carried out aiming to understand the optical properties and electronic structures of pyrite and improve the photovoltaic performance, including doping/alloying with ions (cation ions such as transition metals, anion ions such as oxygen and arsenic elements) 5,7-9 varying surface stoichiometry, 10 changing the preparation process for pyrite thin lms, [11][12][13][14] etc. However, the external electric effect has not been considered in previous studies.
In practice, the addition of external electric eld has the advantage of being easy to achieve and control, and is environmentally friendly. The electric eld effects on the optical properties of a spherical quantum dot have received great interest for fundamental and application research. Applying an electric eld to the material can cause electro-optical effects and change the optical properties. It is found that the optical absorption coefficients and the optical refractive index could be affected by electric eld. 15 Studies on the nanocomposites of benzyl mercaptan-capped cadmium sulde (CdS) quantum dots found that electro-photoluminescence exhibits eld-induced quenching of photoluminescence in the presence of electric elds, and the magnitude of the photoluminescence quenching monotonically increases with increasing eld strength. 16 Beside the inuences on optical properties of materials, external electric eld also affects the adsorption of molecules on catalyst interface. [17][18][19] In order to provide an understanding on the nature of electric eld effect on optical properties of pyrite, in this paper, theoretical calculations using density functional theory (DFT) were carried out. The inuences of different electric eld directions and sizes were considered.

Computational method
Pyrite (FeS 2 ) belongs to the space group T h 6 À Pa 3, with the Fe 2+ cations on the corners and the face-centers positions of the cubic cell and the S 2 2À dimers (S1-S2) occupying the anion sites.
The unit cell of pyrite contains four Fe atoms and eight S atoms, with formula Fe 4 S 8 . Each Fe atom is coordinated to six S atoms creating a distorted octahedron, while each S atom is coordinated to three Fe atoms and one S atom in a tetrahedral conguration (see Fig. 1). External electric led was applied to x, y and z directions to obtain the structural, electronic and optical properties of pyrite. Structural optimizations and electronic and optical calculations were performed using CASTEP, GGA-PW91. 20 The crystal structure was optimized by relaxing both the cell parameters and atomic coordinates. Only the valence electrons (Fe 3d 6 4s 2 and S 3s 2 3p 4 ) were considered using ultraso pseudopotentials. 21 A plane wave cut-off energy of 350 eV was determined by tests and a Monkhorst-Pack 22,23 k-point sampling density of 4 Â 4 Â 4 mesh was used. The self-consistent eld (SCF) convergence tolerance was set to 2.0 Â 10 À6 eV per atom. The spin calculation was performed during the simulation.
The Hubbard U correction 24,25 was adopted for treatment of Fe 3d and specied by tests in this study. Table 1 shows the testing results. It is noted that the band gap of pyrite is only 0.55 eV without the addition of parameter U. With the increase of U value, the band gap of pyrite increases. When 1.2 eV of U value was specied the band gap value of 0.95 eV is the same as the experimental value, 26 and also close to the results of Zhang et al. 10 and Sun et al. 27 using GGA+U with U value of 2.0 eV which gave a band gap of 1.02 eV and 1.03 eV, respectively. Meanwhile, the error (0.56%) of cell parameter (5.3858Å) in our work calculated at this U level is small compared with the experimental value of 5.4160Å. The Wyckoff parameter for the position of the S atom (0.3844) is calculated to be very close to the experimental value of 0.3849. 28

Crystal structure
Electric eld tests started at 0.05 eVÅ À1 e À1 on the condition of 1.2 eV of U value. Fig. 2 shows the cell parameter changes with the increase of electric led. When electric led is less than 0.20 eVÅ À1 e À1 , the cell parameter of pyrite increases slowly with the increase of electric led in x, y and z directions. In addition, the changes are very similar in the three directions. Aer that, the differences in lattice parameters progressively become larger. The changes in y and z directions are almost the same, whereas the result in x direction is different from them. It is clear that the increase of cell parameter applying electric eld in x direction is faster than in y and z directions. The complete fracture of the crystal is found at 0.23 eVÅ À1 e À1 in x direction, while it is occurred at 0.24 eVÅ À1 e À1 in y and z directions.
The cell parameter expansion is related to the polarization effect produced by the electronic eld. Under the polarization effect produced by electric eld, the electron cloud of the ions in the lattice distribution is changed, leading to the change of the force on the atom. Consequently, the lattice atomic position varies and the cell parameter changes.
For the atomic structure calculations, we concentrated on the Wyckoff parameter for the position of the S atom. Table 2 shows the cell parameter and sulphur internal coordinate along x, y, and z axes changing with the application of electric eld at x, y, and z directions. It is shown that with the increase of electric eld applied in x-direction, the Wyckoff parameter decreases slowly but dramatically decreases at x-direction and increases at y-direction once the crystal are fractured, while in yand z-directions the Wyckoff parameters along x, y, and z axes are decreased gradually.
The detailed crystal structure changes including bond lengths in pyrite is shown in Table 3. It is clear that at x-direction the S-S and Fe-S bonds lengths increase with the increase of electric eld, and the S-S and Fe-S bonds are fractured simultaneous at 0.23 eVÅ À1 e À1 of electric eld. While the situation at y-direction is completely different that S-S bond is reduced, even at 0.25 eVÅ À1 e À1 of electric eld it is still not fractured with length of 1.922Å, in fact the crystal has been fractured at 0.24 eVÅ À1 e À1 due to the fracture of Fe-S bond (3.107Å). For electric eld applying at z-direction, both S-S and Fe-S bonds lengths increase with the increase of electric eld; however, the S-S bond length increases faster than Fe-S bond length, resulting in the fracture of S-S bond at 0.22 eVÅ À1 e À1 while Fe-S bond at 0.24 eVÅ À1 e À1 and nally the fracture of the crystal.

Electronic structures
We calculated the electronic properties, including band gap, band structures and density of states (DOS) when electric led is applied at x, y and z directions. Fig. 3 shows the band gap values as a function of electric eld. It is suggested that the band gap of pyrite decreases aer an initial slight increase. The increases of band gaps are almost the same in the three directions; however, the decrease of band gap at x direction is faster than at y and z directions. It means that at high electric eld the differences of band gaps between x direction and y and z directions are great.   In addition, at high electric eld the band gap changes are also different between y and z directions. The band gap values in y direction decrease faster than in z direction. Electronic band structures and density of states (DOS) of pyrite were plotted at electric eld of 0.15 and 0.22 eVÅ À1 e À1 , as shown in Fig. 4 and 5. It is found that when no electric eld is applied, pyrite has an indirect p-type band gap from the valence band maximum (VBM) X point to the conduction band minimum (CBD) G(G) point. At 0.15 eVÅ À1 e À1 (Fig. 4a), regardless of which direction the electric eld is applied, the indirect gap p-type is not changed. Moreover, the introduction of electrical eld in x direction does not change the positions of VBM and CBD; while electrical eld in y and z directions changes the position of VBM.
Higher electrical eld has greater inuence on the position of VBM and CBD. At 0.22 eVÅ À1 e À1 (Fig. 4b), the VBM is changed from X point to G(G) point when electric eld is applied in either direction; whereas CBD is changed from G(G) point to X point, M point and G point in x, y and z directions, respectively. It is suggested applying electric eld to z-direction causes the indirect band gap to direct type.
It is noted that no gap state is introduced in the band structure. However, it is found that no matter in which direction the electric eld is applied, the energy level is split, especially at high electrical eld in the shallow valence band and conduction band. We calculated and plotted the DOS patters shown in Fig. 5a and b. It is clear from the DOS that in valence band, nonbonding Fe 3d t 2g state locating at À1.5 to 0 eV splits from on sharp peak to two or three peaks in the three directions, especially splitting signicantly at high electric eld. In addition, the clear boundary between bonding Fe 3d e g state (À7.5 to 1.5 eV) and non-bonding t 2g state (À1.5 to 0 eV) disappears, and the same situation occurs for the two S 3p states locating at these two regions. These suggest that the delocalization of Fe 3d and S 3p electrons in valence band enhances.

Optical properties
CASTEP can calculate the optical properties of solids due to electronic transitions. In the present study, the frequently used quantity for expressing optical properties: dielectric function, absorption coefficient, and reectivity, were calculated and the patterns are plotted shown in Fig. 6. Here only the inuence of electric led applied to x-axis is considered due to the results for electric eld applied to y-and z-axis are similar to x-axis.  Table 3 Effects of electric field applying at different direction on pyrite bond lengths The complex dielectric function is commonly used to evaluate optical properties and then other properties can be expressed in terms of it. The real and imaginary parts of complex dielectric function are plotted as function of electric eld. The Gaussian broadening used for calculating the dielectric function is set to 0.5 eV. The real part of the dielectric function at frequency of 0 eV corresponds to the static dielectric function. It is calculated to be 17.5 without electric led, compared to 20 for experimental result. 29 We calculated the static dielectric function without Hubbard U correction and found a value of 20, consistent with the theoretical calculation result in ref. 29. It is shown that for pyrite with no electric eld, there are three dielectric peaks locating at about 3.0, 7.5 and 9.5 eV with the strongest peak occurring at about 3.0 eV. This result is close to the calculation result of Vadkhiya and Ahuja using DFT approach with the full potential augmented plane wave (FP-LAPW) method 30 and Antonov et al. using DFT approach with the fully relativistic linear-muffin-tinorbital (RLMTO) method. 31 However, the Hubbard U correction may make a shi of the critical points of dielectric function. 29 In the presence of external electric eld, the static dielectric function decrease at electric eld smaller than 0.15 eVÅ À1 e À1 but increases aer that, especially increases up to about 25 at 0.22 eVÅ À1 e À1 . For the imaginary part, peak intensity at about 3.0 eV and 9.5 eV signicantly decreases and the peak at about 9.5 eV is almost disappeared with the increase of electric eld, while the peak intensity at about 7.5 eV slightly increases. In addition, the larger the electric eld, the more obvious the whole dielectric function shis to the low energy direction.
The absorption of visible light across pyrite was investigated in terms of electric eld. It is found that the absorption coefficients of pyrite are between 0.25 and 2.0 Â 10 5 cm À1 . The application of electric eld leads to a decrease of absorption coefficient at wavelength less than 450 nm, but leads to a red shi of absorption edge, and with the increase of electric eld the absorption edge red shi increases.
The light reectivity of pyrite is plotted. The application of electric eld results in the decrease of light reectivity of pyrite in visible light region, even small electric eld is applied the change in reectivity is signicant. However, it is found that with the increase of electric eld larger than 0.20 eVÅ À1 e À1 , the light reectivity increase when wavelength is larger than 650 nm compared with no application of electric eld.

Conclusion
In conclusion, the application of electric eld in x, y and z directions produces similar results both in band structure and optical properties of pyrite. The crystal will be fractured at high electric eld, but the fracture of intrinsic Fe-S and S-S bond is different at the three directions with electric eld. The p-type semiconductivity of pyrite is not changed but band splits signicantly and the gap also decreases obviously at high electric eld. With the increase of electric led, the number of dielectric peaks is changed from 3 to 2. The absorption edge is red shied but the reectivity decreases.

Conflicts of interest
There are no conicts to declare.