Matildite versus schapbachite : First-principles investigation of the origin of photoactivity in AgBiS 2

Francesc Viñes,1 Marı́a Bernechea,2,3 Gerasimos Konstantatos,2,4 and Francesc Illas1,* 1Departament de Ciència de Materials i Quı́mica Fı́sica & Institut de Quı́mica Teòrica i Computacional (IQTCUB), Universitat de Barcelona, C/Martı́ i Franquès 1, 08028 Barcelona, Spain 2ICFO-Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain 3Cardiff School of Engineering, Cardiff University, Cardiff CF24 3AA, Wales, United Kingdom 4ICREA-Institució Catalana de Recerca i Estudis Avançats, Passeig Lluı́s Companys 23, 08010 Barcelona, Spain (Received 4 October 2016; published 19 December 2016)


I. INTRODUCTION
The mixed silver-bismuth sulfur (AgBiS 2 ) material has recently driven much attention in the scientific research community due to its use in the nanocrystalline form for high-performance solar cells [1].Moreover, its ultralow thermal conductivity allows for its use in thermoelectric power generation [2,3], and it has also been explored as a sensitizer and/or counterelectrode in sensitized solar cells [4,5].The renewed interest on AgBiS 2 goes hand by hand with other compounds of the I-V-VI 2 family-where I = Cu,Ag, or an alkali metal; V = Sb or Bi, and VI = S, Se, or Te-studied in the recent times in thermoelectrics [2,6], solar cells [1,7], and phase-change memory devices [8,9].
Despite the above-commented unarguable interest, the amount of the so-far carried scientific research on AgBiS 2 is modest, and the lack of fundamental knowledge is evidenced by experimental observations that are at some point contradictory.For instance, the band gap E g , a fundamental property in semiconductors, was reported to be of solely 0.9 eV in AgBiS 2 bulk [10], yet more recent estimations report a value of 2.67 eV, although for quantum dots of 8.5 ± 1.2 nm [11], and of 2.78 eV for an average size of 7.6 nm [12].These larger E g values have been accounted for by strong quantum confinement effects.However, more recent experiments report a reduced value of 1.3 eV even for smaller nanoparticles of 4.62 ± 0.97 nm size [1].Other bulk E g measurements report * francesc.illas@ub.edua value 1.2 eV [5], which would limit the extent of quantum confinement effects; indeed some authors report E g values of 1.32 eV for ∼16 nm diameter samples [7], and values of 1.11 eV for AgBiS 2 thin films [13], in accordance to optical measurement of 1.10 eV [14].
The above discussion does not limit to E g .For instance, a gigantic dielectric constant is claimed in previous studies on AgBiS 2 nanocrystallites [11], whereas orders of magnitude lower values were determined on other samples [1], although one has to regard the complexity, certain times uncertainty, in experimentally measuring dielectric constants.Even more, Ag rich samples are found in the literature [1], but, at the same time, Ag poor samples have been reported [11,15], together with estimates of much smaller Ag vacancy formation energies [16].Last but not least, different synthesis methods lead to notoriously different nanoshapes, and these may have markedly different physicochemical properties [12,15,17].The theoretical assessment of the electronic structure properties AgBiS 2 to date could be considered anecdotal, with, as far as we know, a single recent study based on density functional theory (DFT) where high temperature disorder and native defect formation energies were tackled [16].Apart from this initial study, the theoretical and computational study of AgBiS 2 is missing, and hence, this becomes a hindrance in the understanding of the properties of this material, the origin of the observed experimental discrepancies, and, ultimately, in the rationale of improving a particular property of interest.The present study aims at supplying a sound theoretical foundation by relying on a state-of-the-art DFT study of the known bulk FIG. 1. Matildite (left) either in the rhombic unit cell (orange lines, bold spheres) or in the cubic supercell (black lines, glass small spheres), and schapbachite (right) either in the cubic unit cell (orange lines, bold spheres) or in the cubic supercell (black lines, glass small spheres).Gray, pink, and yellow spheres denote Ag, Bi, and S atoms, respectively.crystallographic structures of AgBiS 2 , including a study of stability aspects, as well as a profound analysis of the electronic structure and derived optical properties.

II. COMPUTATIONAL DETAILS
First, available experimental crystallographic structures of AgBiS 2 were acquired, including matildite (Pearson symbol hP 12 and space group 164, i.e., P 3m1) and one schapbachite structure (Pearson symbol cF 8 and space group 225, i.e., F m 3m) [18,19].Figure 1 depicts a matildite rhombic unit cell contained within a larger cubic bulk unit cell, and one can simply think on it as a face-centered cubic arrangement of S atoms, in which Ag and Bi atoms insert in a NaCl fashion, being alternated in each of the three cell directions.In comparison, a cubic unit cell of schapbachite is contained inside a larger bulk supercell of similar dimensions to that of matildite, and one can visualize it as a layered display of squared dispositions of AgS and BiS planes in a given cell direction.
Calculations have been carried out within the DFT framework using chiefly the Vienna ab initio simulation package VASP [20].Most of the calculations rely on the use of the Perdew-Burke-Ernzerhof (PBE) exchange-correlation (xc) functional [21], a well-known member of the family of functionals issued from the generalized gradient approximation (GGA).Nevertheless, since GGA functionals are known to underestimate band gaps of oxides and related systems, the PBE0 [22], B3LYP [23], and HSE06 [24] hybrid xc functionals, containing a fraction of nonlocal, exact, Fock exchange have also been used to explore the electronic structure of these materials.In particular, band structures have been obtained using the B3LYP functional and the CRYSTAL code [25].In these calculations S was treated at the all electron level, whereas a relativistic effective core potential has been used to describe the inner shells of Ag and Bi.In all cases a sufficiently large basis set of Gaussian type orbitals were used taken from the relevant literature [26][27][28].
In the calculations using a plane-wave basis, to represent the valence electronic density, a kinetic energy cutoff of 415 eV has been used and the effect of the core electrons in the valence electronic density was taken into account using the projected augmented-wave (PAW) method as implemented in VASP [29,30].Note in passing, that the PAW method is effectively all electron with a frozen core including relativistic effects, especially important for heavy elements such as Ag or Bi.Whatever the computing package (VASP or CRYSTAL), a k-points Monkhorst-Pack [31] mesh of 9×9×3 dimensions was used for the rhombic unit cell of matildite and a mesh of 5×5×5 dimensions has been used for the cubic supercell.In the case of schapbachite, a k-points mesh of 9×9×9 has been used for the unit cell, and again a 5×5×5 mesh for the larger supercell depicted in Fig. 1.The used k points and plane-waves cutoff ensure a convergence in energy below 0.04 eV.
Atomic positions and cell dimensions were allowed to fully relax until forces acting on atoms were below 0.01 eV Å−1 .Calculations were carried in a spin-polarized fashion, although the total magnetic moments were found to be nominally zero, and so further analysis was carried out in a non-spin-polarized manner.A tetrahedron method was used for the smearing, with an energy window of 0.1 eV, although final energies are extrapolated to 0 K.
Atomic charges have been estimated through a Bader analysis [32,33].Simulated x-ray diffraction (XRD) patterns have been acquired employing the crystal prediction toolkit [34].Band structures have been obtained by sampling 50 points along predefined vectors in the reciprocal space connecting high-symmetry points, see below.This sampling of the reciprocal space (density of k points) has been found to be dense enough to capture band curvatures.The Bohr exciton radius, a parameter measuring electron/hole pair separation in quantum dots r B , can be determined from where is the reduced Planck constant, e is the charge of an electron, ε is the dielectric constant, and m e and m h are the electron and hole effective masses, respectively.The effective masses are estimated, in a first approximation, assuming a parabolic dispersion of the conduction band (CB) maximum and valence band (VB) minimum, in the form where E(k) is the band energy at point k defined by the k vector with module |k| and m is the effective mass.In this sense, the effective masses are obtained from the curvature of the band minima or maxima adjusted to a second degree polynomial, as done in previous work on ZnO [35], and values given in units of electron mass m * .The frequency (ω) dependent complex dielectric constant ε is defined as where ε r and ε i are the real and imaginary parts [36].The complex refractive index N (ω)is defined as [37,38] where n(ω) and k(ω) can be related to ε r (ω) and ε i (ω) via Then, the reflection coefficient R(ω) can be obtained from the normal electromagnetic incidence onto a plane surface as in Eq. ( 7), and furthermore, the absorption coefficient α(ω) can be evaluated from k(ω) as where c is the light speed in vacuum.

III. RESULTS AND DISCUSSION
The atomic structure of bulk matildite and schapbachite polymorphs of AgBiS 2 has been obtained from total energy minimization at the PBE level of theory.Lattice vectors, crystal cell angles, unit cell volume, and density are listed in Table I.It is to highlight the excellent agreement of DFT calculations in matching the experimental structures.The discrepancies on lattice parameters with respect to experimental values are below 1% for matildite, and below 2% in schapbachite.However other properties, such as the volume V and density ρ, feature slightly larger discrepancies yet below 4%.In that sense one can claim that PBE is targeting the experimental structures, and actually these structures have been used in the forthcoming analysis.Note as well that other electronic structure based properties feature very slight variations when obtained on the experimental or the PBE optimized geometries, see below.It is worth highlighting that present PBE total energies reveal matildite to be 0.37 eV per AgBiS 2 unit more stable than the simulated schapbachite, in perfect agreement with the reported value of 0.33 eV obtained at a similar computational level on the same models [16], implying that bulk schapbachite could be only found at high temperatures, in accordance to experimental observations at ∼473 K [4,39].Note, however, that, in those cases, a disordered structure is found where the two types of cations interchange in a rather random way.Other authors observe a similar disordered schapbachite via a thermopower change at ∼610 K [16].In this study, the modeled schapbachite structure possesses a predefined cation ordering considered as an F m 3m case in which AgS and BiS square parallel planes alternate perpendicular to a [001] crystal direction.On the contrary, in matildite, Ag atoms arrange in a rhombic pattern along (111) plane, for more details see Fig. 1.
The different chemical environment in the two structures implies different structural features and electronic properties as well.For instance, in matildite the average d(Ag-S) and d(Bi-S) bond lengths are 2.73 and 2.89 Å, respectively.However, in schapbachite these are 2.86 and 2.77 Å, respectively, implying a longer Ag-S bond, and, consequently, a shorter Bi-S bond.However, the average cation-anion bond distance in both cases is ∼2.82Å, thus equidistant from the experimental mean value of 2.87 Å reported in the literature [1].
When it comes to the oxidation state, the differences between both polymorphs are minimal, with variations of Bader charges below 0.2e, see Table II.Note here that AgBiS 2 can be considered a highly ionic material.However, Bader charges usually differ from those corresponding to formal oxidations states.Hence, from this single parameter it is difficult to assess the degree of ionicity of these materials.All that said, the ionic character seems to be a main ruler, as the proportion of charges is well kept among the elementary constituents.It is worth mentioning that Bader charges obtained using the experimental structure feature no significant changes with respect to those corresponding to the optimized one.Also, only small variations below 0.18e are found when acquiring Bader charges from single point electronic densities obtained at PBE0 or HSE06 xc levels, and so, ionicity seems to be well described at any of the computed levels.
Regarding XRD patterns, simulations by using a Cobased x-ray target for both matildite and schapbachite are shown in Fig. 2, displaying that, in both periodic structures, different diffraction peaks at different 2 angles with different intensities exist, yet some signatures would interfere with others.However, focusing on the dominant signals and their intensities one can conclude that present results for matildite structure are much closer to the AgBiS 2 XRD patterns reported in the literature [1,15,40].However the fitting is not fully perfect, and actually it may well be that native defects and/or a certain degree of disorder affects the experimental XRD patterns.
The most important difference between matildite and schapbachite AgBiS 2 crystalline structures sits on their electronic structure.Figure 3 shows the explored lines connecting high symmetry k points in their respective reciprocal space Brillouin zones, accompanied with the plotted dispersion of the valence and conduction bands near the Fermi level E F . Figure 3 shows that, whereas matildite is a narrow gap semiconductor, the schapbachite structure has a metallic character with a clear zero band gap E g .This feature alone would discard it for optoelectronic or solar harnessing applications, given the excited electron relaxation channels found near R, M, and A k points.This does not happen in matildite, where a band gap region is found for all k-space explored points in between valence band maximum (VBM) and conduction band minimum (CBM).According to present results, bulk matildite AgBiS 2 is a narrow indirect gap semiconductor with a VBM located along the K → line, whereas CBM is locate at the A high symmetry point.
It is worth pointing out that the magnitude of the band gap predicted by the DFT based calculations within the PBE functional (0.42 eV) is markedly smaller than the abovecommented measurements.However, PBE and other DFT xc functionals of the GGA family are long known to heavily underestimate band gaps in semiconductors and insulators, stemming out from DFT intrinsic electronic self-interaction error.As noted initially by Muscat et al. from periodic calculations with the hybrid B3LYP functional [41], and confirmed by several authors on a variety of different insulators [42][43][44], a more accurate prediction of the magnitude of the bad gap in this type of highly ionic insulators requires the use of hybrid xc functionals.
Table III contains the single-point estimates of E g as obtained using PBE0, HSE06, or B3LYP hybrid xc functionals on top of the PBE optimized geometries.Tests on matildite experimental geometry yielded variations of E g of 0.19 eV on average, where B3LYP features the smaller discrepancy of solely 0.01 eV.The hybrid xc functional estimates, ranging 1.5-1.9eV, are much more reliable and in line with experimental measurements of 0.9-1.2eV [5,10].Present results suggest to use a reduced percentage of Fock exchange of 8% to 13% to target an assumed experimental E g value of 0.9 to 1.2 eV.This is in line with recent work for TiO 2 where the band gap of rutile and anatase is properly reproduced when the xc functional contains a 12.5% of Fock exchange [45], which is half of the standard value in PBE0 and HSE06.
Table III shows that E g vales are clearly underestimated when Fock exchange is lacking, as in PBE.By adding a 25% of Fock exchange the band gap is increased to 1.88 eV, and one could argue that this is even too much since at the PBE0 level schapbachite features a small but noticeable band gap of 0.53 eV.This overestimation tends to be counteracted by the screening factor applied in a range-separated HSE06 functional, which features a smaller band gap of 1.54 eV for matildite.At this level, the obtained separation of VBM and CBM in schapbachite is of 0.09 eV only and hence, within the limits of DFT accuracy, it could be considered back as metallic.The screening parameter applied in HSE06 seems to be equivalent to the reduced 20% Fock exchange in B3LYP, with a very similar band gap of 1.57 eV.These differences are also observed when plotting the atomic decomposed density of states (DOS), as depicted in Fig. 4. Notice here how, for matildite, at any DFT xc level, the VBM and nearby states are clearly dominated by the S 3p orbital derived levels, whereas CBM and close bands are mostly governed by Bi.This extends as well to schapbachite, although here Ag related states play a more determining role near the Fermi level, and because of this, one could argue that AgS planes, as shown in Fig. 1, are likely to be the origin of the AgBiS 2 schapbachite model metal character.
At this point we further focus on matildite AgBiS 2 as being the crystallographic arrangement with potential for optoelectronic and solar cell applications.One aspect worth investigating is to determine the electron/hole effective masses, which would deliver hints on the possible lattice movement of the exciton components.Indeed, a larger difference among masses would intuitively suggest that such charge carriers separate more efficiently, and so, they would be more unlikely to recombine and lead the system to its electronic ground state.In accordance, a larger Bohr exciton radius would be obtained, following Eq.( 1), considered as a mean separation between the excited electron and the created hole.
Effective masses have been estimated on the VBM and CBM, as commented above and shown in Fig. 3, although at B3LYP level.Notice that the electron self-interaction error present in PBE not only reduces the E g , but also sharpens band curvatures, which would be eventually translated into artificially lighter effective masses.The B3LYP band structure, as well as the others obtained with other hybrid xc functionals, feature the same band structure, with only variations in E g and band dispersions.Accordingly, present results based on B3LYP band structure reveal an effective hole mass m h of 0.722 m * at the VBM located along the K → line, whereas the excited electron displays an effective mass m e of 0.350 m * obtained at A along the H direction.Note that effective masses evaluation has been restricted to a and b equivalent matildite crystallographic directions, disregarding c direction, which features lower curvatures and so, assumedly, larger effective masses.Thus, focusing on main charge mobility directions, apparently the fact that the hole effective mass doubles that of the excited electron can be interpreted as facilitating a large separation of the electron/hole pair along ab planes, with a concomitant sufficient large time survival of the exciton.Another property to be considered is the frequency dependent dielectric function.The imaginary part of the dielectric function ε i (ω) is intimately related to the frequency-dependent optical response.Figure 5 shows a band split-off at lower energies at PBE level, but at larger energies at HSE06 and PBE0, in concordance with E g values shown in Table III.A striking feature is that the highest peak, as obtained at any DFT xc functional level, has intensity values in the 30-60 range.This is 3-5 times higher than the values obtained for bulk TiO 2 anatase, brookite, and rutile [46], an extendedly known photoactive material used in photocatalysis.This feature would align with previous statements of an anomalously large dielectric constant [11].Note that regarding the previous TiO 2 work, a slightly different computational method was used, the so-called Hubbard U correction (PBE + U ) by Dudarev et al. [47].Consequently, a direct comparison is not advised, although little variations are expected; at least, lesser FIG. 4. Atomic projected density of states (DOS) for matildite (left) and schapbachite (right) bulk AgBiS 2 structures at the PBE, HSE06, and PBE0 corresponding to the PBE optimized crystal structure.Total DOS is shown black, and atom contributions color coded as in Fig. 1.
than when comparing PBE to hybrid functionals PBE0 or HSE06.Aside from this, there is little variation in between the orthogonal dielectric function component ε ⊥ , corresponding to a and b cell vector directions, and the parallel component ε || , corresponding to the c cell vector direction.According to this, AgBiS 2 seems to not feature anisotropy in light absorbance.The static macroscopic dielectric constant ε has been estimated in two ways; either through the evaluation of the piezoelectric tensor, or as the value of the frequency dependent real part of the dielectric function at ω = 0, here called ε apx .For details of the procedure we refer to the specialized literature [48].Both quantities have been gained at PBE, PBE0, and HSE06 levels.Moreover, a comparison has been made between matildite AgBiS 2 and other semiconductors broadly studied in the literature, which encompass cubic diamond (C), silicon (Si), silicon carbide (SiC), and gallium arsenide (GaAs).Since these materials were studied using the Ceperley-Alder (CA) xc functional [49] based on the so-called local density approximation (LDA), the value for AgBiS 2 have also been acquired at this level of theory, thus providing a fair comparison.To complete the picture, we studied as well a wide band gap inert oxide MgO, and another chalcogenide PbS, considered as a candidate for solar cells [50].These have been optimized self-consistently at CA and PBE levels.All estimated values are compared to experimental reported values [51][52][53].
In light of the dielectric constant values, shown in Table IV, several conclusions can be withdrawn.On one hand, ε and ε apx values are very similar, with discrepancies in the worst case (AgBiS 2 ) below 6%, and in some cases (SiC) providing exactly the same quantity.Aside, CA estimates seem to be closer to experiments than PBE ones.Actually, by inspecting ε apx data one observes that the use of hybrid functionals seem to be detrimental for dielectric constants, despite its benefits in estimating band gaps.In accordance to other computed semiconductors and insulators, and their experimental values, it seems that for this observable parameter the CA estimates are more reliable.Accordingly, one can safely conclude that bulk matildite AgBiS 2 would have a dielectric constant just marginally larger than that of PbS.Indeed, this finding supports the observed photoactivity of AgBiS 2 , and, in addition, suggests that the reported dielectric constants in AgBiS 2 pellets [11], orders of magnitude higher, are likely due to surface dipole effects instead of bulk related properties.By considering the most reliable simulated values for ε obtained at CA level, and the m h and m e effective masses obtained at B3LYP level, one can grossly estimate a Bohr exciton radius of 4.6 nm, which coincides with the reported nanoparticle mean size in the previous experimental study on their use on solar cell devices [1], and would preclude invoking quantum size effects on larger nanoparticles [11,12], suggesting stoichiometry variation or surface effects as the origin of band gap variability.
Last but not least, absorbance spectra have been acquired using hybrid functionals.For technical reasons, the PBE0 and HSE06 functionals have been employed (Fig. 6).Here it is worth pointing out that to estimate the band gap from the electronic structure is a well established procedure and the one followed here.Using the thus computed band gap one can also determine the onset for absorption which is used in experiment to gain this property and not always uniquely determined.FIG. 6. Absorption coefficients α, in cm −1 , with respect different wavelengths, in nm, estimated at PBE0 and HSE06 levels.
With this in mind, the similarity of results for the two hybrid functionals is remarkable and consistent with highly similar values of the dielectric constant reported in Table IV.These employed DFT based methods capture the absorbance intensity reported in the literature of 10 5 cm −1 [1].However, the slow decay by increasing the radiation wavelength is better captured by using hybrid functionals.Likewise, the abrupt decay in Fig. 6 at ∼1500 nm is similar to what is found experimentally, although redshifted by 400 nm.The origin of the shift can be attributed to limitations of the theoretical approach and/or to deviations of the electronic structure of the nanoparticles from the perfect arrangement of the bulk.In fact, note that other nanocrystal samples imply a much more earlier decay of at around 400 nm [11,40].In any case, present theoretical results show that, as far as bulk matildite is concerned, absorbance would not deviate from 10 5 cm −1 in the 400-700 nm visible light region, thus supporting solar light harvesting applications.As commented above, a variety of factors, here not contemplated, such as nanoparticle shape (biased by the preparation method), nonstoichiometry composition, boundary region effects, and/or presence of bulk defects or certain degrees of cationic disorder, may be determinant factors which influence the absorbance spectrum.

IV. CONCLUSIONS
Motivated by the renewed interest in AgBiS 2 for solar light harvesting applications, and by the lack of detailed knowledge on its bulk structural and electronic structure, we have studied this material using first-principles quantum mechanical methods considering two reported crystal structures, the room temperature matildite structure, and a metastable ordered schapbachite arrangement.DFT based for the bulk structures carried out at the PBE level show that matildite can be 0.37 eV per AgBiS 2 stoichiometry unit more stable than the schapbachite arrangement, and that the latter displays a metallic character in its electronic structure, i.e., no band gap.
Metal-sulfur interatomic distances are in line with those experimentally observed in employed nanocrystals, and a high degree of ionicity is present in matildite, according to Bader charges.Matildite is found to be an indirect gap semiconductor, with an estimated band gap of ∼1.5 eV according to DFT calculations using state-of-the-art HSE06 and B3LYP hybrid density functionals.The B3LYP estimated VBM m h effective mass doubles that of CBM m e , suggesting a longer survival of a generated exciton.Estimates of the matildite AgBiS 2 dielectric constant reveal that is slightly higher than that of PbS, another material considered in solar cells.The here calculated Bohr exciton radius is 4.6 nm, thus similar to nanocrystal sizes found in the literature, and the experimentally absorbance intensities of 10 5 cm −1 in the solar light absorbance region are well reproduced according to present calculations.

FIG. 3 .
FIG. 3. Brillouin zones (top) of unit cells of matildite (left) and schapbachite (right) AgBiS 2 .Reciprocal lattice vectors are shown as red lines.High symmetry lattice points are named, and connected trough green lines.The band structure of both crystals are shown (bottom) as obtained at the PBE xc level, with band energies scaled to Fermi level E F .High symmetry k points are noted, as well as the gap region in between VBM and CBM, colored yellow.

FIG. 5 .
FIG. 5. Frequency (ω) dependent average imaginary part of the dielectric function, here ε(ω).Total averaged value is shown as a black line, but orthogonal ε ⊥ and parallel ε || contributions are shown in red and blue lines, respectively.

TABLE I .
Experimental (Expt.) and calculated (PBE) matildite and schapbachite AgBiS 2 crystal structures, encompassing unit cell dimensions a and c, their ratio c/a, their lattice vector angle α and γ , respectively, their volume V , and the associated density ρ, as well as the reported or PBE computed band gap E g .

TABLE II .
Formal and Bader charges for atoms in bulk AgBiS 2 .The Bader charges are derived from the PBE calculations on the optimized matildite (Mat.) and schapbachite (Sch.)structures.

TABLE III .
Calculated band gaps E g , in eV, for matildite and schapbachite bulk AgBiS 2 , as obtained by single-point calculations at PBE, PBE0, HSE06, and B3LYP exchange correlations functionals on previously optimized bulk geometries gained at PBE level.