Old and New Puzzles of Gold Tellurides: Incommensurate Crystal Structure of Calaverite Aute2 and Predicted Stability of Aute

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Old and new puzzles of gold tellurides: incommensurate crystal structure of calaverite AuTe2 and predicted stability of AuTe.

Sergey V. Streltsova,b, Valerii V. Roizenc, Alexey V. Ushakova, Artem R. Oganovc,d, and Daniel I. Khomskiie

aDepartment of theory low-dimensional spin systems, Institute of Metal Physics, S. Kovalevskoy St. 18, 620990 Yekaterinburg, Russia; bDepartment of theoretical physics and applied mathematics, Ural Federal University, Mira St. 19, 620002 Yekaterinburg, Russia; cComputational materials discovery laboratory, Moscow Institute of Physics and Technology, Institutskiy per. 9, 141701 Dolgoprudny, Moscow Region, Russian Federation; dMaterials Discovery Laboratory, Skolkovo Institute of Science and Technology, Nobel St. 3, 143026, Skolkovo, Russian Federation; eII. Physikalisches Institut, Universität zu Köln, Zülpicher Straße 77, D-50937 Ko¨ln, Germany

This manuscript was compiled on July 3, 2021

Gold is a very inert element, which forms relatively few compounds. state at higher pressures. Moreover, an extensive structural Among them is a unique material - mineral calaverite, AuTe2. Be- study of different gold tellurides allowed us to predict the sides being the only compound in nature from which one can ex- existence of a novel stable compound - AuTe. We report the tract gold on an industrial scale, it is a rare example of a natural predicted crystal structure and properties of this new material. mineral with incommensurate crystal structure. Moreover, it is one of few systems based of Au, which become superconducting (at ele- 2. Old puzzle of calaverite’s crystal structure vated pressure or doped by Pd and Pt). Using ab initio calculations we theoretically explain these unusual phenomena in the picture of AuTe2 has a distorted layered CdI2-type structure (the average negative charge transfer energy and self-doping, with holes being structure has space group C2/m [6]), with triangular layers of largely in the Te 5p bands. This scenario naturally explains incom- Au with Te atoms in between. However, there is a periodic displacive modulation along [010] direction, which makes over- mensurate crystal structure of AuTe2 and it also suggests a possible mechanism of superconductivity. Ab initio evolutionary search for all crystal structure incommensurate [7]. The mechanism of incommensurability is unclear. One may argue that it can be stable compounds in the Au-Te system confirms stability of AuTe2 due a specific electronic structure, which results in a charge and AuTe3 and leads to a prediction of a new stable compound AuTe, density wave (CDW) instability, but accurate band structure which until now has not been synthesized. calculations have not found nesting of the Fermi surface at Incommensurate crystal structure | Calaverite | Superconductivity corresponding wave vectors[8, 9]. Schutte and de Boer proposed another explanation based on the formal assignment of 2+ 2− 1. Introduction valencies in Au (Te2) (in analogy with another mineral – 2+ 2− 2+ the “fool’s gold” Fe (S2) )[10]. However, whereas Fe is a t is very well known that gold is one of the least reactive stable ionic state, every chemist knows that Au2+ is extremely Ichemical elements and it is typically mined as a pure native difficult to stabilise: it exists as Au1+(d10) or Au3+ (nominally element. It also occurs in alloys but very rarely it can be found low-spin d8). If one would manage to really stabilize Au2+(d9), in the form of compounds. The only compound existing in it would be a realization of an old dream - a “magnetic gold”1.

nature from which one can extract gold on an industrial scale 1 It was actually indeed made, however not in oxides, but in systems with more ionic bonds – in is gold telluride - AuTe2, calaverite. This material is extremely Au(AuF4)2 and Au(SbF6)2[11]. interesting in many aspects. It even influenced the gold rush in Australia, where miners in gold mines first discarded calaverite as an “empty” waste and used it for paving the roads, but, after Significance Statement discovering that it contains real gold which can be extracted, very carefully scrapped all these roads. DRAFTIt is shown that the long-standing puzzle of incommensurate Another, very specific feature of AuTe2 is that it is one of crystal structure of AuTe2 can be solved, if this material is con- very few materials having in natural form an incommensurate sidered as a negative charge-transfer system. Using modern crystal structure. This at a time gave a lot of headache to min- computational methods we demonstrate that charge redistri- eralogists and crystallographers: they could not understand bution associated with incommensurate modulations of crystal peculiar faceting of calaverite crystals contradicting Haüy’s structure occurs not so much on Au, but predominantly on Te

arXiv:1809.08052v1 [cond-mat.str-el] 21 Sep 2018 law. Usually the stable natural facets of a crystal are those sites. This substantially reduces Coulomb energy costs for with small Miller indexes, and in calaverite everything looked creating such a unique crystal structure. The same mechanism odd, until one realised that the very crystal structure is in- also explains superconductivity of doped AuTe2. Exploring commensurate [1]. But the origin of the incommensurability different Au-Te compositions we also discovered a previously is still obscure. Last but not least, AuTe2 was found to be unknown compound AuTe, which theoretically is very stable, a superconductor at a relatively low pressure of 2.3 GPa or and we predict its crystal structure. upon Pt or Pd doping[2–5], with critical temperature ∼ 4 K. In the present paper we found that all these properties of The project was conceived by D.Kh. and S.S. The DFT calculations were performed by S.S. with help of A.U., V.R. and A.O. did study of all possible crystal structures for whole Au-Te series using AuTe2 can be naturally explained, if one takes into account USPEX. S.S. and D.Kh. analysed the results and wrote the manuscript with help of V.R. and A.O.

that it is in a negative charge transfer energy regime, which The authors declare no competing interest. drives a charge disproportionation resulting in an incommensurate crystal structure at normal conditions or superconducting 2E-mail: [email protected]

www.pnas.org/cgi/doi/10.1073/pnas.XXXXXXXXXX PNAS | July 3, 2021 | vol. XXX | no. XX | 1–6 The phenomenon of skipped valence [12] of Au2+ can lead to the possibility of charge disproportionation into Au1+ and Au3+, and it seems to naturally explain the ground state properties of AuTe2, as it works for example in Cs2Au2Cl6 [13]. The fact that CDW due to such charge disproportionation is incommensurate in AuTe2, in contrast to Cs2Au2Cl6, may be related to the triangular lattice, which Au ions form in AuTe2. This lattice is not bipartite, and the resulting frustration can lead to incommensurate modulation. While overall modulation

of the lattice is complex, the local distortions seems to confirm Fig. 1. (a) GGA+SOC total and partial density of states for AuAu’Te4 structure (for this skipped valence interpretation: some Au ions, say at the experimental volume), (b) charge density ρ(~r) corresponding to the top-most, partially 3 maximum of CDW, are in a linear, or dumbbell coordination filled band (isosurface corresponding to 0.003 e/Å ≈10% of maximal value of ρ(~r) is presented; the charge density was plotted using the PAW formalism). Results of (two short and four long Au-Te bonds), typical for d10 ions, the GGA+SOC calculations for AuTe2 in fully optimized “AuAgTe4” structure. Au-P 1+ here Au , whereas at the “other end”, say in the minimum of and Au-D stands for Au ions having plaquette and dumbbell surroundings. CDW, Au ions are square-coordinated – coordination typical 3+ 6 2 2 2 2 2 0 for Au (t2g(3z − r ) (x − y ) )[5]. Local surrounding for other Au interpolates between these two limits. there occurs spontaneous charge disproportionation, like 4+ 3+ 5+ This interpretation, however, was put in doubt. First, 2Fe →Fe +Fe , but occurring predominantly on lig- photoemission[14] and then x-ray absorption[15] measurements ands, i.e. this “reaction” should be rather visualised as 3+ 3+ 3+ 2 showed that apparently electronic configuration of all Au ions 2Fe L →Fe +Fe L . This process is now well established is the same, close to Au1+.2 Also ab initio calculations per- also in nickelates RNiO3 (R=Pr, Nd), where it leads to a real formed for the artificial supercell structure with four Au ions phase transition, originally interpreted as charge ordering on 3+ 2+ 4+ mimicking the small-period CDW3 does not see any difference Ni (2Ni → Ni + Ni )[22], but which is actually much 3+ 2+ 1+ 2 in occupation of d−shell for different Au ions[8]. better described by the “reaction” 2Ni → Ni + Ni L [23]. We argue that nevertheless the physics of AuTe2 is related to the eventual instability of Au2+ against charge dispropor- We claim that the same phenomenon also occurs in systems containing Au2+, such as, e.g., Cs Au Cl [13], and also in tionation, which determines main properties of AuTe2, includ- 2 2 6 ing not only incommensurate CDW, but also the tendency to calaverite AuTe2, where one can write this reaction as superconductivity. As demonstrated below, the resolution of 2Au2+(d9) → Au1+(d10) + Au3+(d8), [1] the controversy mentioned above lies in the fact that actually AuTe2 is a negative charge transfer (CT) energy system[16, 17], but in fact it should be rather visualized as with all the holes predominantly in the 5p bands of Te. The notion of CT insulators was introduced in the seminal 2Au2+ ≡ 2Au1+L → Au1+ + Au1+L2. [2] paper by Zaanen, Sawatzky, and Allen[18]. These are materials with strongly correlated electrons. However, the lowest charge Two holes (L2) in the Te p band form something like a bound excitations in them correspond not to transfer of electrons state, with the symmetry of a low-spin d8 state of Au3+ with between localized d states, dndn → dn+1dn−1, as in Mott- which it hybridises. Below we confirm this picture by the ab Hubbard insulators, but to electron transfer between TM’s initio band structure calculations. and ligands, i.e. to the processes dnp6 → dn+1p5 = dn+1L, where L stands for the ligand hole. In CT insulators this CT 3. Mechanism of incommensurability in AuTe2: Nega- n+1 5 n 6 excitation energy is positive, ∆CT = E(d p )−E(d p ) > 0, tive charge transfer energy but in principle it can be very small or even negative (naively speaking, when anion p levels lie above d levels of TM ions). It is impossible to carry out ab initio calculations for the real In this case we speak about negative CT energy. Usually incommensurate structure with the existing codes based on DRAFTdensity function theory (DFT). One needs to approximate this situation is met when the oxidation state of a metal is unusually high - e.g. 4+ for Fe or 3+ for Cu. If such states this structure by some supercell with a commensurate CDW. We borrowed an idea how to construct it from nature, taking are created by doping, as, e.g., in high-Tc cuprates the doped holes go predominantly to oxygen p states (although these initial crystal structure from the mineral sylvanite - AuAgTe4. are of course strongly hybridized with d states of Cu). But Au and Ag ions in sylvanite are ordered in stripes, Ag being 10 this situation can be realized also in undoped stoichiometric in a linearly-coordinated sites, typical for ions with d configuration, i.e. it is Ag1+ (Au-D in Fig.1b), and Au ions lie in compounds such as CaFeO3. In this case there can occur spontaneous transfer of electrons from ligands to TM ion, i.e. a square-coordinated positions, corresponding to, nominally, 3+ Fe4+ →Fe3+L. This situation can be called a self-doping[19]. Au (Au-P in Fig.1b) with its strong Jahn-Teller distortion This picture, which in physics we describe by the (negative) [6]. This structure was relaxed in the generalized gradient CT energy, in chemistry is such notions as redox reaction, approximation (GGA) taking into account the spin-orbit cou- dative bonding etc. 4 pling (SOC), then Ag was substituted by Au ion and relaxed Interestingly enough, in many systems of this class again keeping unit cell volume the same as in real AuTe2 (this structure will be labeled as AuAu’Te4 in what follows). 2 Recent, spectroscopic study however did show the existence of slightly inequivalent Au ions[4]. First of all, we found that the AuAu’Te4 structure is stable 3 Note that the structure used in Ref. [8] is somehow unnatural in a sense that four short Au-Te and there are still two differently coordinated Au. Secondly, bonds do not lie in one plane. 4 A “dictionary” helping to establish the correspondence between physical and chemical language is this structure is lower in energy than the average C2/m struc- contained in the famous book by Goodenough[20], and a very clear paper by Hoffmann[21]. ture [6] at experimental volume (by 22 meV/f.u.). Thus, one

2 | www.pnas.org/cgi/doi/10.1073/pnas.XXXXXXXXXX Lead author Streltsov et al. may see that we gain a lot of energy by making distortions corresponding to the CDW (in this case a commensurate one). A close inspection of the Au 5d occupation numbers in the AuAu’Te4 structure, however, show that from the point of view of d−occupation both Au ions are 1+: corresponding occupan- cies of the d shell (as obtained within the projector augmented wave (PAW) method) are 9.90 and 9.92, so that the difference is negligible: δnAu−d = 0.02 electrons (the Bader analysis[24] gives even smaller difference, <0.01 electron), while in real sylvanite, AuAgTe4, δn ∼ 0.5 electrons, i.e. in silvanite we can indeed speak about Ag1+ and Au3+ (Au3+ again with a lot of ligand holes). This however seems to be in strong contrast with results of the lattice optimization, which gives very different local coordination for two Au: we have one linearly- (1+) and another square-coordinated (3+) Au. The difference between short and long Au-Te bonds is ∼ 0.25Å in

linear- and ∼ 0.35Å in square-coordinated Au. This is of order Fig. 2. Total energy vs. volume for different possible crystal structures (GGA+SOC of magnitude of Jahn-Teller (JT) distortions in such classical results). For each volume, optimization of the crystal structure was performed. JT systems as LaMnO3 (0.27 Å)[25] and KCuF3 (∼0.3 Å)[26]. What drives such strong lattice distortions, if not the CDW on Au sites? of structural properties of many different materials including In order to answer this question we plot in Fig.1b the those based on heavy metals[28–30]. For AuTe2 we found distribution of the charge density corresponding to the top- that the AuAu’Te4 structure with distortions reminding the most, partially filled bands illustrating a hole distribution. CDW still has the lowest total energy among hundreds of One may see that there is only a minor contribution from other structures obtained with the USPEX. There appear only the Au 5d states to the charge density corresponding to the two structures (in the interval of 100 meV/atom), which may least filled band, while the largest part comes from the Te 5p compete with it: the high pressure P 3¯m1 phase, where the orbitals. Thus, one may speak about significant contribution incommensurate superstructure disappears and all Au ions of the ligand holes to the ground state wave function. The become structurally equivalent, and a structure characterized 2 3+ symmetry of (L ) hole state around “Au ” (Au-P in Fig.1b) by the Cmmm space group, total energy of which is by 90 3+ is the same as that of a hypothetical JT active Au ion with meV/atom higher than the one of AuAu’Te4. 2 2 5 two holes on the x − y orbital , i.e. it naturally explains In the second step we carefully checked how total energies why this ion has square coordination typical for such a state. of these crystal structures depend on the volume, see Fig.2. Analysis of the DOS, shown in Fig.1a, also confirms that The AuAu’Te4 structure corresponding to the CDW is still the largest part of the holes are in the Te 5p bands and one the lowest one, while the equilibrium volume is slightly over- may speak about negative CT energy situation. The local estimated. The next one is P 3¯m1 with 5.5% smaller volume, electronic structure of Au ions in this case corresponds to 1+ the average C2/m and Cmmm structures are much higher. valence state for all Au ions (d10). These results allow us At this stage one can demonstrate a crucial role of the to reconcile the picture of charge disproportionation driven CT energy for the formation of the AuAu’Te structure with largely by skipped valence of Au2+, with the experimental 4 distortions, imitating the real structure of AuTe . For this we data [14, 15], which show that all Au ions are Au1+ from 2 performed model calculations, where the Au 5d bands were spectroscopic point of view. artificially shifted up in energy, thus increasing the CT energy Moreover, a redistribution of electrons between Te and Au and reducing the contribution of Te holes. We found that the favors strongly distorted calaverite crystal structure, reminis- DRAFTshift on only 1 eV is enough to destabilize AuAu’Te4 structure, cent of the formation of the CDW. Indeed, if the CT energy and it makes the high-pressure P 3¯m1 phase with all Au ions would be positive and there would be a real CDW with the structurally equivalent the lowest in energy: the total energy Au1+ and Au3+ ions having 5d10 and 5d8 electronic configura- difference is E ¯ − E 0 ≈ -2 meV/f.u. In the real tions, this would cost a lot of Coulomb energy (two holes on P 3m1 AuAu T e4 AuTe2, modelled by AuAu’Te4, the Au 5d states lie below Te the same d site repel each other with the energy U, which is 5p, see Fig.1a, which corresponds to a negative CT energy ∼10 eV). Redistributing a part of the charge density to ligands ∆CT . Shifting the Au 5d orbitals up leads to a decrease of we minimize the energy costs on the formation of the CDW. absolute value of ∆CT or even can make it positive. Then However, analysis of only two structures can be only quali- the charge disproportionation would have been mostly on the tative. There is no guarantee, that there is no other structures, Au sites, which leads to a drastic increase of the energy costs which would give a lower total energy. In addition the equilib- of the CDW due to Coulomb interaction, as explained above, rium volume in the DFT can be different from the experiment. and as a result AuAu’Te4 structure with inequivalent Au’s In order to overcome the first difficulty we used the USPEX becomes much higher in energy. algorithm[27] to search for all possible structures of AuTe with 2 An important question is why in real AuTe the super- all experimentally known structures included in calculation. 2 structure is incommensurate. As explained above, due to USPEX was previously successfully applied for investigation calculation limitations we had to model it by the closest com-

5 Note that this orbital lies in the plane of Te plaquette, while central part of the charge density at mensurate structure of a silvanite, our AuAu’Te4. To check for Au-D is spherically symmetric and, thus, this band corresponds rather to 3z2 − r2 orbital. the possibility to get incommensurate structure we calculated

Lead author Streltsov et al. PNAS | July 3, 2021 | vol. XXX | no. XX | 3 Fig. 4. Diagrams illustrating (a) conventional BCS “t-”channel pairing and (b) “s- ”channel pairing proposed for AuTe2.

and AuTe2 dumbbells naturally results in dimerization of the Te atoms, which however is not a driving force but rather a consequence of the CDW formation in AuTe2. One can argue that the physics disclosed in our calcula- Fig. 3. Formation of the Te-Te dimers due to charge disproportionation on Au sites. tions, specifically the origin of the incommensurability, - the The “strength” of distortions in AuTe6 octahedra is not the same for all Au-Te bonds. There are “strongly” distorted with respect to undistorted P 3¯m1 (δ ∼ 0.45− tendency to the skipped valence and charge disproportionation Au−T e 2+ 0.55 Å) and “weakly” distorted Au-Te bonds (δAu−T e ∼ 0.15Å). Plotting (for of “Au ”, occurring in the situation with negative CT energy simplicity) only “strongly” distorted Au-Te bonds (red lines; arrows show direction of with the self-doping – is also instrumental in providing a mech- distortions) one immediately obtains Te-Te dimers (shown by blue arrows). anism of superconductivity in AuTe2 under pressure or with doping. This tendency, both on the d−levels, reaction Eq. (1), and more realistically on ligand states, reaction Eq. (2), means phonon spectrum[31] of AuTe . We indeed found that when 2 that there exists a tendency for holes to form pairs, i.e. there we start from the homogeneous high-pressure phase P 3¯m1, exists an effective attraction of these holes. some phonon frequencies became imaginary with the minimal frequency at incommensurate wave vectors q ≈ 0.41a + 0.5c The idea that the tendency to charge disproportionation (where a and c correspond to the P 3¯m1 structure), see Fig. (which actually means the local “chemical” tendency to form S1(b). Thus, the real instability of homogeneous structure pairs of electrons or holes) can be instrumental in providing would indeed lead to an incommensurate superstructure. the mechanism of Cooper pairing was first suggested by Rice Very significantly, when we shift d−levels up, as explained and Sneddon[32] in connection with the superconductivity of doped BaBiO3. This material is also known to experience above, these imaginary phonon frequencies disappear. This 4+ 3+ 5+ once again proves that the negative CT energy and corre- charge disproportionation of the type 2Bi → Bi + Bi sponding large contribution of ligand holes are crucial for the (and again with a lot of action on ligands - see, e.g., [17]). For high-Tc cuprates similar idea was proposed in [33]. It is also formation of the incommensurate structure of AuTe2. closely related to some theoretical studies of superconductivity in systems with coexisting ordinary electrons and bipolarons, 4. High pressure phase and superconductivity see e.g. [34, 35]. We suppose, by analogy with the above- Taking first derivative of E(V ) one can find that a critical mentioned papers, that the “chemical” tendency of Au2+ to 1+ 3+ pressure (Pc) required for the transition from AuAu’Te4 to charge disproportionate into, nominally, Au and “Au ”, P 3¯m1 is 2.6 GPa. It is striking that while our optimized which is the main ingredient of our theory and which, as we ar- structure with the commensurate CDW (AuAu’Te4) slightly gued above, plays crucial role in explaining the main properties overestimates equilibrium volume, the critical pressure for the of AuTe2, may be also instrumental in providing the mecha- transition to uniform P 3¯m1 is reproduced with good accuracy: nism, or at least helping the realization of superconductivity the experimental Pc = 2.5 GPa [5]. in AuTe2 when doped or under pressure. The high-pressure phase of AuTe2 is also very interesting One can phenomenologically describe this situation by an due to another aspect - the superconductivity, which appears effective Hamiltonian like the Anderson lattice model (where in it below Tc =2.3 K[2]. One may stabilize this phase not only 5p electrons of Te play a role of conduction electrons, while 5d by an pressure, but also by Pt doping[3], which alsoDRAFT results in electrons of Au are localised), but with an effective attraction – the stabilization of the same P 3¯m1 structure. The supercon- with negative U on localized levels. After excluding d electrons, ductivity was proposed to be induced by breaking of Te-Te we get in effect also an attraction of conduction electrons, dimers, which exist in the C2/m phase, but disappear in the which, on one hand, can provide mechanism of CDW formation high-pressure superconducting P 3¯m1 phase[3]. In particular (not even requiring nesting of the Fermi-surface, although it was speculated that the formation of Te-Te dimers modifies nesting would help). And, on the other hand, in this model electronic structure of AuTe2 though formation of bonding (σ) we have a natural mechanism of formation of Cooper pairs and antibonding (σ∗) Te 5p bands[3]. We have seen that the leading to superconductivity. On the diagramatic language, bands at the Fermi level indeed have very large contribution this mechanism of pairing is described by the Fig.4b (two of the Te 5p states, but they are strongly hybridized with electrons (or holes) of a conduction band “drop” into the Au 5d Au 5d and have the symmetry of Au 5d orbitals (see Fig.1), levels, where they experience attraction and form pairs, before while the σ−bonded Te 5p states are far away from the Fermi decaying again into conduction electrons. This situation is level (∼5.2 eV below and ∼3.2 eV above EF ). Thus, it seems reminiscent of a model with bipolarons[36], and is different that the Te-Te dimerization is not directly related to the sup- from the usual electron-phonon exchange of Fig.4a (although pression of the superconductivity. In fact, this is just one of the standard electron-phonon coupling could also contribute). the consequences of the formation of the CDW. In Fig.3 the Thus, AuTe2 may be the long-sought second example of the directions of Te atoms displacements due to the CDW are in- same physics as proposed for BaBiO3 [32], with the same dicated. One may see that the formation of AuTe4 plaquettes mechanism of both the charge disproportionation and of the

4 | www.pnas.org/cgi/doi/10.1073/pnas.XXXXXXXXXX Lead author Streltsov et al. Fig. 6. The crystal structure of novel material: AuTe. Bold solid and dashed lines correspond to short and long Au-Te bonds, respectively.

Fig. 5. Thermodynamic convex hulls and Gibbs free energy G versus chemical potential µ for Au-Te system with different Te concentrations. corresponding to the bands at the Fermi level, shows that there are nearly equal contributions to ρ(~r) from Au 5d and Te 5p states. This may explain why USPEX did not find the solu- superconductivity. tion corresponding to charge disproportionation, as it did for calaverite (two inequivalent Au ions: in dumb-bells and plaque- 5. Novel compound: AuTe ttes): the energy costs due to the on-site Coulomb repulsion are too large in AuTe. Thus in effect AuTe should resemble Since USPEX has shown its efficiency in determining the the high-pressure phase of AuTe , with all Au equivalent, and AuTe crystal structure, we extended these calculations to a 2 2 one could expect that it could also be superconducting. whole Au1−xTex series with arbitrary x. Fig.5(a,b) shows thermodynamic convex hulls and phase diagram of the Au- Te system in the GGA and GGA+SOC approximations. A 6. Conclusions compound is thermodynamically stable if its thermodynamic The Au-Te system presents an interesting example of com- potential (e.g., the Gibbs free energy) is lower than that of any pounds of a very inert element, gold with nontrivial properties. other phase or phase assemblage of the same composition. On We found out that there exist in the Au-Te system three stable 6 a graph showing the enthalpy of formation of all compounds stoichiometric compounds: AuTe, AuTe2 and AuTe3 . The of a given system (e.g. Au-Te system) from the elements, all second and the less “popular” third compound are known points corresponding to stable compounds can be connected and studied. AuTe has not been synthesized yet, although a to form a convex hull. Height above the convex hull is a mea- similar material, mineral muthmannite AuAgTe2, is known. sure of thermodynamic instability of a compound. One may It would be very interesting to check our predictions and try notice that in addition to experimentally observed structures to synthesize and study AuTe. as AuTe2 and AuTe3[37] there appears a new one: AuTe. Much better studied, but still presenting several, until now AuTe has never been synthesized so far, but there exists min- unresolved puzzles, is calaverite AuTe2. This is the system, eral muthmannite, AuAgTe2, found in Western Romania[38], the properties of which we now explained on the basis of ab where Au and Ag ions are in 1:1 ratio. Muthmannite has a initio calculations. The picture emerging from our calcula- distorted NiAs-type structure with space group P2/m. Our tions is the following: The nominal average valence of gold calculations have shown that the C2/c structure predicted for in AuTe2 is 2+, similar to many pyrites like FeS2, MnS2 etc. AuTe by USPEX is significantly more stable (by 0.164 eV/atom and layered dichalcogenides MS2, MSe2, MTe2, where M - with SOC) than muthmannite structure. The predictedDRAFTC2/c different TM ions[40]. But this state is, first of all, chemically structure of AuTe, shown in Fig.6, can be considered as dis- unstable (only Au1+ and Au3+ are known to exist, with very torted NaCl-type structure (NiAs and NaCl structures are few exceptions). And, most importantly, both “Au2+” and 3+ relatives). The Au ions are in the strongly distorted plaquettes “Au ” in AuTe2 correspond to the situation with negative with two short (2.68 Å) and two long (2.90 Å) Au-Te bonds. charge transfer energy, i.e. practically Au2+ →Au1+L and It is worthwhile mentioning that the SOC additionally Au3+ →Au1+L2. This means that in fact all the holes go to lowers position of the Au 5d band and thus affects stability of ligand (here Te) bands (but still with significant hybridization different phases in Au-Te system. One can see from Fig.5 that with d states of Au). This is actually the situation of self- while both GGA and GGA+SOC calculations show stability doping[17, 19]. In this case there occurs a phenomenon met of the same phases and crystal structures, there are large also in several other systems: the valence, or charge dispropor- changes in stability fields. The plot of Gibbs free energy vs. tionation, which however again occurs not so much on the d chemical potential demonstrates that inclusion of the SOC shells themselves, but on ligands. I.e. corresponding dispropor- expands stability field of Au (in effect making it more inert) tionation is described not as a Eq. (1), but rather as Eq. (2). and AuTe2, at the expense of shrinking stability fields of AuTe This transition is accompanied (and is largely driven by) the and AuTe3. The relatively narrow stability field may explain change of the Au-Te bond lengths (and local coordination - lin- why AuTe is not yet known. 6 There exists also Au3Te7 with a simple cubic structure and statistical distribution of Au and Te AuTe was found to be a nonmagnetic metal in the atoms[39] and is likely a solid solution. We have not found a stable compound with such stoichiom- GGA+SOC calculations. Analysis of the charge density, ρ(~r), etry in calculations at T=0 K, which indicates that it is probably entirely entropy-stabilized.

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