Research Article  Open Access
Prediction of a New Phase of Cu_{x}S near Stoichiometric Composition
Abstract
Cu_{2}S is known to be a promising solar absorber material due to its suitable band gap and the abundance of its constituent elements. Cu_{2}S is known to have complex phase structures depending on the concentration of Cu vacancies. Its instability of phases is due to favorable formation of Cu vacancies and the mobility of Cu atoms within the crystal. Understanding its phase structures is of crucial important for its application as solar absorber material. In this paper, we have predicted a new crystal phase of copper sulfide (Cu_{x}S) around chemical composition of by utilizing crystal database search and density functional theory. We have shown that this new crystal phase of Cu_{x}S is more favorable than low chalcocite structure even at stoichiometric composition of . However, Cu vacancy formation probability was found to be higher in this new phase than the low chalcocite structure.
1. Introduction
Cu_{2}S is a wellknown semiconductor which has the potential to play an important role as a solar absorber material. Thin films of heterojunctions such as Cu_{2}S/CdS have shown an efficiency of 910% in the past [1–4]. However, Cu_{2}S shows complex structural behavior. Experimental results so far have shown that stoichiometric Cu_{2}S mainly exists in three forms: monoclinic phase (low chalcocite) forms at temperature below 104°C, hexagonal phase (high chalcocite) forms between 104 and 436°C, and cubic phase forms at temperature 436°C or higher [5, 6]. Theoretical calculations on three chalcocites have shown consistency with experimental facts revealing low chalcocite is favorable at 0 K [7, 8]. The low chalcocite has 96 Cu and 48 S atoms and is monoclinic in nature. A major problem in Cu_{2}S is the copper atom’s instability towards the formation of copper vacancies; this leads to the formation of different crystal structures depending on Cu vacancy concentrations [7–12]. In addition, this formation of high copper vacancies results in very high ptype doping [13] causing the material to behave like a degenerate semiconductor. Moreover, in Cu_{2}S, the positions of Cu within the sublattice of S atoms are not well defined [7, 8] as copper atoms are mobile in nature in Cu_{2}S [14, 15]. Despite being earth abundant and nontoxic, these problems hindered the further study of Cu_{2}S in photovoltaic community for some time.
Table 1 provides names of known Cu_{2}S phases those that are experimentally identified with their composition. Experimentally observed band gaps of these Cu_{x}S phases fall in the range of 1.1–1.2 eV; however, the nature of these band gaps is not well identified yet [7, 8]. Various studies have stated that studying and understanding Cu_{x}S have been a long term challenge [4–6, 16–18], especially in computational works, which is mainly due to the complex behavior of copper in Cu_{2}S. Experimental and theoretical studies have shown that Cu vacancies are inevitable in Cu_{2}S. Hence, for the effective usage as a solar absorber material an understanding and a detail map of Cu_{2}S phases based on Cu vacancy concentrations are essential. To avoid excessive ptype doping, it is desirable to stabilize a phase of Cu_{x}S near . It can be assumed that at thermodynamic equilibrium, low chalcocite structure may sustain these composition ranges near . However, this has not been tested and not reported anywhere. To the best of our knowledge, no phases have been so far identified with the chemical composition . Hence, it is important to study Cu_{x}S compounds with different possible structures with a value of near 2 to find a possible strategy to stabilize such phase. Furthermore, it is possible that the ambiguity of Cu positions in crystals may also be due to the coexistence of more than one crystal phase which are energetically close to each other. Hence, identification of thermodynamically stable new structures will shed light on this problem. Stabilization of such structures may lead to a new route for Cu_{x}S as stable solar absorber materials. In this paper, by using crystal database and density functional theory (DFT) based total energy calculations, we predicted a new stable phase for , which has not been reported before.
2. Computational Methodology
Density functional theory (DFT) as implemented in Vienna Abinitio Simulation Package (VASP) [21, 22] has been used for calculating the total energy of different possible Cu_{2}S/Cu_{x}S phases. In our calculation, the generalized gradient approximation (GGA) functional known as PerdewBurkeErnzerhof (PBE) [23] has been used with the projected augmented wave (PAW) [24, 25] method. For ionic relaxation, each ion was relaxed until the force on it is equal or less than 0.01 eV/Å, and 400 eV was used for the energy cutoff for the plane wave basis set. It is well known that DFT usually underestimates band gaps; especially for some phases of Cu_{2}S it gives negative band gap. A computationally economical way to correct such problem is to invoke DFT + method, which not only opens the band gap, but also tends to provide a d^{10} configuration for Cu in Cu_{2}S. Hence, we also calculated electronic properties, such as band gaps, of Cu_{x}S with DFT + [26, 27] with eV and eV which provides an effective (= ) of 6 eV. In DFT + the extra term is a Hubbardlike potential added to the KohnSham DFT Hamiltonian. This provides extra correlation to an orbital by introducing intraband repulsive interaction, which results in a relatively more localized band. For 3D transition metals this method has been proven to be very effective. Justification for choosing this value will be discussed in the following section.
As we are searching for a stable phase of Cu_{x}S near stoichiometric composition, at first DFT calculations were performed systematically on different possible stoichiometric Cu_{2}S crystal structures. The possible set of structures chosen for this study were antifluorite (AF) Cu_{2}S, acanthite like, berzelianitelike, stromeyeritelike structures, and other structures provided by material project database [28]. In addition, for comparison purposes, known Cu deficient Cu_{x}S phases have been considered as well in their stoichiometric limit. After these structures were relaxed, we compared their energetics to search for the most favorable Cu_{2}S structure. Then choosing structure with the lowest relaxed energy along with other few possible structures with comparable energies, such as low chalcocite structure, we formed copper vacant Cu_{1.98}S structures and did further calculation on them. In particular, the low chalcocite structure has been considered for comparison because we are interested in Cu_{x}S compositions near stoichiometric Cu_{2}S, and low chalcocite structure is known to be the stoichiometric structure at low temperature.
In order to have a better understanding of the electronic properties, systematic analyses of density of states (dos) plots, band structures, and Cu vacancy formation energies have been performed. The defect formation energy of Cu vacancy was obtained using the following equation: where is total energy of Cu vacancy containing cell, is total energy of cell without defect, is the number of Cu atoms removed from a defect free cell, and is the chemical potential of Cu reservoir, which was taken to be the bulk phase of copper. In addition, for the stability of Cu_{x}S, require the following condition to be satisfied:where we have calculated the formation enthalpy or heat of formation [29] of Cu_{x}S using the definitionwhere is heat of formation, is the total energy per formula unit, and and are total energy per Cu atom in bulk phase and per S atom in stable S_{8} phase, respectively. The cohesive energies of selected relaxed structures are calculated using the following definition:where and are free atomic energy of copper and sulfur, respectively. A software program called VESTA [30] was used for visualization purposes.
3. Results and Discussions
The cohesive energies of some selected stoichiometric Cu_{2}S are presented in Table 2. Among these structures, cohesive energy of the acanthite like structure was found to be −10.653 eV per formula unit and is the lowest among all structures. In comparison, the cohesive energy of low chalcocite structure is −10.637 eV per formula unit. Note that acanthite is Ag_{2}S mineral, where for the sake of the present study Ag is replaced by Cu and has been relaxed without any symmetry constrained. The resulting acanthite like structure of Cu_{x}S was found to be a very robust structure. Another closely related mineral structure stromeyerite, AgCuS, where Ag is replaced by Cu, has cohesive energy of −10.631 eV per formula unit. On the other hand, structure derived from Cu_{2}Se (berzelianite) by replacing Se by S was not very favorable with cohesive energy of −9.995 eV per formula unit.

From the above discussion it is clear that acanthite like Cu_{2}S has most favorable structure at the stoichiometric limit as predicted by DFT total energy calculations. Hence, for DFT + , we have optimized the values with respect to the acanthite structure. Note that in Cu_{2}S, Cu has d^{10} band so Cu here has +1 oxidation state (Cu^{1+}). Hence, even though the value is optimized for acanthite structure, the same value in principle can be applied to other Cu_{2}S structures as well. We have tested with several values for on Cu 3d band to calculate the band gap of Cu_{2}S and compared with the lowest known experimental gap of 1.1 eV [7]. In general, band gap increases with the value of . However, though for Cu_{2}S band gap did increase from 0.152 eV value to a value of 0.905 eV with the application of up to 7 eV, the gap became less sensitive to further increment of . Such as with eV, the band gap becomes 1.090 eV. The reason is the d^{10} nature of the Cu 3d orbital. For higher value of , the completely filled 3D band goes lower in energy within the valence band, which can only indirectly affect the band gap through S 2p hybridization. Even though eV gave band gap value closer to the known experimental range, from eV to 9 eV the band gap became indirect to direct. As Cu_{2}S is experimentally known to be an indirect gap semiconductor [31], eV would provide a picture closer to the reality. In addition, our previous study on other Cu^{1+}compounds showed improved results with similar values of [32, 33]. Hence, eV has been selected for this study as an optimal value.
In fact, at the range of from 1.96 to up to 2, acanthite like structure was found to be even more favorable than low chalcocite structure in this study. At the stoichiometric limit, although acanthite like Cu_{2}S was found to be more favorable structure, it is more prone to form Cu vacancies even at Cu rich growth condition (will be discussed in the following). Hence, it is less likely to be stable in pristine form at the stoichiometric compositional limit. Like low chalcocite, acanthite structure is also monoclinic in nature, however with much simpler unit cell of total 12 atoms (8 Cu and 4 S atoms), whereas low chalcocite unit cell has 144 atoms.
In the following, we present results comparing both low chalcocite and acanthite structures. This choice is due to the fact that these two structures are two most favorable structures for Cu_{2}S from theory point of view. Furthermore as low chalcocite has been experimentally observed, it is important to compare properties of Cu deficient acanthite structure against it. First we created a supercell of acanthite like structure containing 96 Cu atoms and 48 S atoms similar to a low chalcocite unit cell. In order to create a composition Cu_{x}S with , we created a vacancy in both structures and relaxed them fully using DFT. We present fully optimized DFT lattice parameters for two Cu_{1.98}S structures in Table 3. Acanthite (Cu_{1.98}S) structure is slightly larger than low chalcocite (Cu_{1.98}S) as volume per formula unit of acanthite (Cu_{1.98}S) is higher than the latter. The crystal structure also shows a significant difference between the two structures which can be seen in Figure 1. Acanthite structure remained almost orthorhombic, whereas the monoclinic nature of low chalcocite is evident. In addition, Cu atoms seem to follow a particular layered structure in acanthite like (Cu_{1.98}S), in contrast to low chalcocite (Cu_{1.98}S) such layer or any Cu ordering that is absent. The average CuCu bond length is very similar, namely, 2.67 Å for acanthite (Cu_{1.98}S) and 2.69 Å for low chalcocite (Cu_{1.98}S). These calculated CuCu bond lengths are about 5% higher than metallic CuCu distance of 2.55 Å (DFT), indicating some CuCu interaction may be present in both structures. Average CuS bond length is 2.33 Å in acanthite (Cu_{1.98}S) whereas it is 2.36 Å in low chalcocite (Cu_{1.98}S). The smaller CuS bond length in acanthite compared to the low chalcocite is due to the almost planer CuS layered structure. Note that in acanthite both CuCu and CuS average bond lengths are smaller than the low chalcocite structure, even though the volume in acanthite is larger. This is a consequence of the CuS and CuCu layered structure in acanthite, where the interlayer separations are larger than these average bonds. It is important to mention here that, as opposed to djurleite or digenite structures, acanthite structure of Cu_{1.8}S cannot be derived from the low chalcocite structure by simply creating vacancies in it. Hence, acanthite and low chalcocite are structurally distinct structures.

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(b)
In order to find the probability of forming defect structures at composition we calculated the defect formation energy using (1) at two extreme cases: (i) at Curich (or Spoor) condition where Cu has the total chemical potential equal to its bulk phase and (ii) at Cupoor (or Srich) condition, where sulfur is assumed to be in thermal equilibrium with stable S_{8} phase (bulk). To form stable bulk Cu_{2}S the constraint on chemical potentials of Cu and S is given by (2). Figure 2 shows schematically the calculated Cu vacancy formation energy (by (1) and (2)) as a function of copper chemical potential for acanthite and low chalcocite structures. This figure reveals two important results: (i) Cu vacancy formation energy for low chalcocite is positive at Curich condition whereas for acanthite structure formation energies are negative at both cases and (ii) at both Curich and Cupoor conditions, the Cu vacancy formation energy of acanthite structure is lower than that of low chalcocite. These imply that formation of Cu vacancies in acanthite structure (Cu_{1.98}S) is thermodynamically more probable than low chalcocite (Cu_{1.98}S). In addition, it can be argued from Figure 2 that low chalcocite has a possibility to remain stoichiometric at a Cu rich growth condition. However, acanthite structure would form Cu vacancies spontaneously at any growth condition. Hence, even though acanthite has lower energy (i.e., higher stability) at stoichiometric composition of Cu_{2}S, low chalcocite is a relatively more probable candidate for pristine stoichiometric Cu_{2}S. However, it should be noted that at Curich condition, even though formation energy of Cu vacancy is positive for low chalcocite, it is very still very small.
We note that acanthite structure (Cu_{1.98}S) has the lowest heat of formation, −0.447 eV per formula unit, whereas for low chalcocite (Cu_{1.98}S) it is 27 meV per formula unit higher. Also note that, for pristine Cu_{2}S, the difference of formation enthalpies between these two structures is 16 meV. On the other hand, heat for formation for acanthite Cu_{1.98}S is lower than that of stoichiometric Cu_{2}S acanthite, which is evident from the negative formation energies of Figure 2. These indicate that Cu vacancies lower the total energy of acanthite structure even more than that of the low chalcocite structure. Hence, Cu vacancy defect formation energy and heat of formation calculation both suggest that acanthite structure is energetically more favorable as Cu_{1.98}S. However, acanthite structure has not been experimentally observed yet. It is possible that around this composition (), due to close proximity of energies, at room temperature acanthite structure can simply coexist with low chalcocite, which would make the detection even more complicated.
Next, we present electronic properties of acanthite structure of Cu_{1.98}S at both DFT and DFT + level of theory where, as mentioned earlier, eV applied to Cud band. The band structure is provided in Figure 3. As expected, without any additional term, DFT underestimates the band gap; in fact in this case the conduction and valence bands overlap implying a metal like behavior. On the other hand, DFT + opened up the gap; the calculated band gap is about 0.82 eV which is an indirect gap, along Γ to A direction. In contrast, opening of band gap in low chalcocite with DFT + is smaller; however, the gap was found to be direct. These gaps are still slightly lower than observed experimental band gaps for other Cu_{x}S which may be due to DFT related limitations. The top of the valence band for both structures is unoccupied which indicate a ptype doping scenario, as expected due to Cu vacancy.
The total density of states (DOS) and partial density of states (pDOS) calculated using DFT and DFT + are provided in Figure 4. In these plots the highest occupied band energy is scaled to 0 eV. A small peak around 0 eV is due to the presence of a hole state created by Cu vacancy. Here, it is important to provide rationale for the application of DFT + to open band gap compared to DFT. In pristine Cu_{2}S, Cu 3D bands are expected to be fully occupied (3d^{10}); hence the application of to such fully occupied band cannot open up a gap between occupied and unoccupied 3D band as a direct consequence as discussed earlier; rather it changes the energy of 3d band’s position and increases its localization. Application of in such case changes the hybridization of Cu 3d band with other bands, and hence given favorable condition can open up a band gap. We see a prominent peak at around −13.8 eV (DFT) and −13.1 eV (DFT + ) which is mainly due to low lying S sband (not shown in pDOS). This shift is due to the change in hybridization as the Cud band became more localized with the application if potential. This can be justified by the fact that the width of the valence band has been reduced by 0.9 eV due to the application of DFT + . Such results clearly indicate the localization effect of Cud electrons in Cu_{1.98}S. As the Cud band lowers in energy compared to the scenario where was not applied, the top of the valence band is no longer dominated by Cud. As an additional effect, lowering of 3D band’s energy also lowers the valence band maximum (VBM), even though the feature of the top three bands remained almost the same as seen in Figure 3. Hence an increasing value of lowers the position of VBM as well. The pdos plots show that there is more dominance of Spband at conduction bands especially after including the correlation term. The latter seems more realistic as Cudband being fully occupied has no empty band for electrons. The presence of Cudband and Spband at the valence band and conduction band is suitable for pd optical transition which is beneficial from photoconductivity point of view.
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4. Conclusions
In conclusion, a new phase of copper sulfide (Cu_{x}S) at composition around using density functional theory and crystal database has been proposed. The calculated formation energies and heat of formation indicated that acanthite like structure as a phase can occur as Cu_{1.98}S. Interestingly, acanthite (Cu_{1.98}S) is found to be structurally more preferable and thermodynamically more stable than low chalcocite structure even at the stoichiometric limit of Cu_{2}S. In addition, acanthite structure is more prone to Cu vacancy formation than low chalcocite. It is possible that near stoichiometric composition () of Cu_{x}S, both acanthite like and low chalcocite structures can coexist. For Cu_{1.98}S, CuS forms a simple layer like structure with higher cell volume than the corresponding low chalcocite structure. However, both the nearest CuCu and CuS average distances are lower in acanthite than in low chalcocite. Overall, the electronic structure near valence band is found to be similar to that of low chalcocite, though the valence band maximum of acanthite is higher in energy. Our results will provide an important insight into the complex nature of the Cu_{x}S phase space.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors are grateful to National Renewable Energy Laboratory for funding (Subcontract no. XEJ32323201). Computations were performed at the University of Texas at Arlington’s High Performance Computing Center (HPC) and at the Computing System of Texas Advanced Computing Center (TACC), Austin.
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Copyright © 2015 Prashant Khatri and Muhammad N. Huda. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.