Carbon network evolution from dimers to sheets in superconducting ytrrium dicarbide under pressure

Carbon-bearing compounds display intriguing structural diversity, due to variations in hybrid bonding of carbon. Here, first-principles calculations and unbiased structure searches on yttrium dicarbide at pressure reveal four new structures with varying carbon polymerisation, in addition to the experimentally observed high-temperature low-pressure I4/mmm dimer phase. At low pressures, a metallic C2/m phase (four-member single-chain carbide) is stable, which transforms into a Pnma phase (single-chain carbide) upon increasing pressure, with further transformation to an Immm structure (double-chain carbide) at 54 GPa and then to a P6/mmm phase (sheet carbide) at 267 GPa. Yttrium dicarbide is structurally diverse, with carbon bonded as dimers (at lowest pressure), four-member single chains, infinite single chains, double chains and eventually sheet structures on compression. Electron–phonon coupling calculations indicate that the high-pressure phases are superconducting. Our results aid the understanding and design of new superconductors and illuminate pressure-induced carbon polymerisation in carbides. Several rare earth carbides are superconducting at ambient pressure and low temperature. Here global structure searching predicts high-pressure metallic phases of yttrium dicarbide with phonon-mediated superconductivity and increased carbon network evolution from dimers to sheets with increasing pressure.

S tructural variation of the atomic arrangements in crystalline solids is the typical response to changes in chemical composition, temperature or pressure [1][2][3] . The application of varying external hydrostatic pressure can have significant impact on the structural stability of solids as their structures adapt to denser configuration on increasing pressure. In particular, pressure can induce rather surprising changes that runs counter to expectations based on traditional understandings of the chemical bond. Structures and chemical compositions may be controlled by packing considerations and optimisation of density, rather than the disposition of outer electrons that describe chemistry as it is usually understood. Hence, pressure-dependent structural variations provide a powerful method to probe the physical and chemical characteristics of materials, and especially give a route to understand in the behaviour of solids in a wide range of applications, from understanding the nature of deep planetary interiors to developing novel materials with designer properties such as super-hardness and superconductivity.
In carbon-bearing solids, the ability of carbon to form both sp 2 − and sp 3− bonding states leads to exceptional structural diversity and chemical variation, especially under non-ambient conditions. There has, for example, been long-standing interest in graphitisation of low-density arrangements, the formation of graphiterelated crystal structures, and their ultimate transformation to super-hard materials based on network polymerised structures akin to diamond 4 . Among the wide variety of such carbonbearing materials, the properties of mixed yttrium-carbon-bearing solids has attracted attention in view of its possible relationship to CaC 2 -type solids. The peculiar physical properties of compressed yttrium carbides, especially the Y-based dicarbide YC 2 , is the focus of our attention here. YC 2 , experimentally synthesised at ambient conditions, is found to be metallic and adopts a tetragonal CaC 2 -type structure (space group I4/mmm, Z = 2) 5 . The structure of YC 2 comprises strong covalently-bonded C 2 units bound together via dominantly ionic interactions to yttrium. The formal valence state of yttrium in YC 2 is of particular interest in this structure. The fact that it forms as a phase isostructural with CaC 2 6 might suggest that it is present as in YC 2 as Y 2+ , in contrast with the expected Y 3+ valence state. Indeed, within the range of organo-yttrium compounds Y has previously been observed to show variable formal oxidation state, from 0 to +3, with a variety of yttrium carbides previously recorded, including Y 3 C 7 , Y 2 C 8 , Y 2 C 3 9 , and even endohedral fullerenes 10 , as well as YC 2 . No experimental investigations have been undertaken on YC 2 at high pressure but, by analogy with other carbonbearing structures, one anticipates that changes in the carbonbonding network may occur.
Yttrium carbides have previously been demonstrated to be superconducting. As long ago as 1968, Giogri et al. determined a T c of 3.88 K in yttrium dicarbide, and indeed this was the first πreported occurrence of superconductivity in a CaC 2 -type structure 11 . Subsequently, other rare-earth-containing carbides, including Y 2 C 3 , La 2 C 3 , Y 1-x Th x C 2 , Y 1-x Ca x C 2 , and LaC 2 have been found to be superconductors at ambient pressure [12][13][14] . This family of rare-earth carbides have, however, recently attracted renewed interest owing to the discovery of superconductivity in the layered yttrium carbide halides Y 2 C 2 I 2 (T c = 9.97 K) and Y 2 C 2 Br 2 (T c = 5.04 K) 15 . Though known to be good superconductors at ambient pressure, the superconducting properties of yttrium dicarbides under pressure remain unexplored. The role of pressure is of particular interest in view of the recent discoveries of high T c superconductivity at high pressure among the rare-earth hydrides. For example, yttrium's hydrides are even predicted to be room-temperature superconductors, with a predicted T c in YH 6 of 264 K at 120 GPa and for YH 10 a T c of 305-326 K at 250 GPa [16][17][18] . This leads to the obvious need to explore potential superconductivity in compounds of yttrium with other light elements.
More generally, then, many open questions remain regarding the behaviour of compressed rare-earth metal dicarbides. Among the most pressing are: (1) How do C 2 dimers evolve upon the application of increased pressure? (2) Do any new high-pressure phases display metallic properties? And finally, (3) do any new metallic high-pressure phases show superconductivity? The approach that we have adopted to investigate the structural properties of YC 2 at high pressure exploit developments in global structural searching schemes, combined with first-principles total energy calculations.
Here we show that, on increasing pressure, the dimer-based YC 2 low-pressure structure, transforms to intermediate fourmember chain structures, transforms into layered structures. Thus carbon atoms first form one-dimensional (1D) chains and then double chains, and finally two-dimensional (2D) graphene sheets separated by Y atomic layers. Of the four new high-pressure phases that we find, one has monoclinic symmetry, two have orthorhombic symmetry (Pnma and Immm) and the fourth is a MgB 2 structure-type with P6/mmm symmetry. Our electronic structure calculations demonstrate that all of these new highpressure phases are metallic in character. Finally, these highpressure phases of YC 2 show phonon-mediated superconductivity, as demonstrated by our exploration of electron-phonon coupling in these structures.

Results
Structure search. Structure search calculations on stoichiometric YC 2 performed at ambient pressure readily reproduced the experimentally-observed CaC 2 -type phase of YC 2 11 . This confirmed the reliability of our method and its application to this system (see Supplementary Table 1). Structure predictions of the most stable phases of compressed YC 2 were obtained from simulations with one to four YC 2 formula units per cell at 10, 50, 100, 200, 250, and 300 GPa. Analysis of the predicted structures provided a shortlist of candidate structures with space groups C2/ m (Z = 2), Pnma (Z = 4), Immm (Z = 4) and P6/mmm (Z = 1). Further optimisation of these structures over a pressure range of −15 to 300 GPa revealed the sequence of pressure-induced phase transitions that occur for this material (Fig. 1). Enthalpies of YC 2 for each of these structure types are also shown as a function of pressure in Fig. 1.
We found that the C2/m phase, Pnma phase and previouslyreported I4/mmm phase have very similar enthalpy at ambient pressure, and for this reason their enthalpies vs. pressure curves around ambient pressure are better depicted in the insert in Fig. 1. One can see that, for YC 2 , in addition to the previously-reported I4/mmm phase (referred as YC 2 -I), a previously-unreported C2/m structure, comprising four carbons in a kinked-linear arrangement, is stable at low pressure (YC 2 -II). On increasing pressure, a new orthorhombic Pnma phase (referred as YC 2 -III) is stable up to 54 GPa. At higher pressure, a further orthorhombic Immm phase (referred as YC 2 -IV) becomes stable and remains so over the wide pressure interval of 54-267 GPa. At the highest pressures of our simulations of 267 GPa and above, a MgB 2 -type structure with space group of P6/mmm (referred as YC 2 -V) is stable. The dynamical stability of each of the four new phases of YC 2 was confirmed by calculation of their phonon dispersion relations, and verification of the absence of imaginary frequencies, across the pressure ranges of their thermodynamic stability (Supplementary Figure 1). Furthermore, we find that (at 0 GPa and 10 GPa) YC 2 is stable against decomposition into a mixture of adjoining or end-member compositions, lying as it does on the convex hull of the Y-C system (Supplementary Figure 3).
Temperature effects. Two energetically competitive structures (C2/m and Pnma) are predicted at 0 GPa, at which pressure the I4/mmm phase, which has the higher enthalpy, was reported from experiment 11 . In view of the fact that synthesis was carried out from the melt by arc-melting followed by quenching, we further included temperature effects via quasi-harmonic Gibbs free energy calculations with phonon spectra obtained from the finitedisplacement method, for all these three phases between −12 and 15 GPa. The contribution of vibrational effects, depicted in Fig. 2, stabilises the experimentally-observed YC 2 -I phase over the C2/m phase at temperatures greater than 1500 K. This is in good agreement with the experimental results, since the melting point of I4/mmm YC 2 is reported to be 2415°C at ambient pressure 19 and the I4/mmm phase crystallises from this melt.
Incorporation of the effect of temperature does not change the sequence of expected phase transitions. It shifts the pressures of the phase transitions slightly, according to the Clapeyron equation. We note that the Clapeyron slope of the I-II transition is quite different from that of the II-III transition, which may be understood in terms of the structural control on ΔV at each transition: YC 2 -II and YC 2 -III are structurally similar (as chain structures) while YC 2 -I is structurally distinct. We focus on 0 K high-P calculations in the following discussions. Each of the Pinduced phase transition in YC 2 is first-order in thermodynamic character, although relationships between the structures can be deduced. The volume compressibility (Supplementary Figure 4) shows significant breaks at each phase transition, with successive high-P phases adopting increasingly dense structures.

Discussion
It is instructive to consider how the carbon-carbon-bonding changes between each of the structures with increasing P, as this yields insight into phase transformation mechanisms. The optimised structural parameters of each predicted stable phase are listed in Supplementary Table 1 and further illustrated in Fig. 3. From the low-P phase to highest-P structure, the C-C bonding shows changes due to the influence both of external P and internal chemical pre-compression associated with changes in the yttrium connectivity. In YC 2 -I, our calculations show that isolated carbon dimers are orientated along the crystallographic z axis (Fig. 3a), with (at 0 GPa) a C-C bond length of 1.303 Å. The lattice parameters match available experimental observations well, validating the structural results that we have obtained.
With increasing P, the distance between individual carbon dimers in YC 2 -I decreases, transforming to four-member carbon chains in YC 2 -II (Fig. 3b). In this structure, all yttrium atoms lie at the Wyckoff 4i site and carbon atoms occupy another two different 4i sites. Increasing P further results YC 2 -III, in which structure carbon atoms rearrange into single chains (Fig. 3c). Four yttrium atoms lie at the Wyckoff 4c site and eight carbon atoms occupy the 8 f site. As such, the carbon atoms form a onedimensional chain parallel to the y-axis.
Upon further increase in P, each carbon chain in YC 2 -III moves closer to its neighbours until carbons come together to form six-membered rings in the double chains of YC 2 -IV (Fig. 3d), akin to quasi-1D carbon ribbons, each of which lies in the centre of a 'cylinder' composed of yttrium ions. In this structure, there are four-molecules in the unit cell with all four yttrium atoms on the Wyckoff 4i site. There are two crystallographically distinct types of carbon: four lie on the Wyckoff 4 h site while the other four carbon atoms occupy Wyckoff 4 g sites.
In the double-chain YC 2 -IV phase there are two different C-C bond lengths (referred as d 1 and d 2 ) in the double chain. At 200 GPa, for example, two among the six C-C bonds (d 1 ) have a bond length of 1.418 Å, and four of them (d 2 ) have a length of 1.409 Å. Upon further compression, the difference (Δd) between d 1 and d 2 decreases smoothly (Fig. 4) up to the transition point, when d 1 and d 2 suddenly become identical at the first-order transition at 267 GPa. The double chains link up to form a hexagonal graphene-like sheet, the YC 2 -V phase (Fig. 3e). Simultaneously, the yttrium atoms become confined to the interlayer regions between the graphene sheets, and this highest-pressure structure is a layered carbide. Structurally, YC 2 -V comprises hexagonal 'honeycomb' layers (i.e. graphene sheets) of carbon atoms linked by planes of yttrium ions, with the yttrium atoms positioned above and below the centres of the carbon hexagons. Thus, the pressure-induced structural modification is akin to graphitisation of compressed YC 2 , as is observed in YC 2 -V. In YC 2 -V, there is one formula unit per unit cell and the carbon sheet is perfectly planar. Our optimisation gives equilibrium lattice parameters, at 300 GPa, of the hexagonal unit cell as a = 2.483 Å and c = 3.414 The relationship between P6/mmm YC 2 -V and Immm YC 2 -IV is supergroup-subgroup in nature, with a symmetry-breaking transition pathway from P6/mmm to Immm possible via a Cmmm intermediate. We find no evidence for the Cmmm intermediate in our calculations, but note that the behaviour of the interatomic spacings for carbon atoms (Fig. 4) is consistent with such a symmetry-breaking behaviour, with a strongly firstorder phase transition character.
Increasing polymerisation of the carbon sub-structure in YC 2 is associated with gradual changes in the nature of the Y-C bond. Our electron localisation function (ELF) calculations (Fig. 5) show that density is well-localised around carbon atoms, with doubly-bonded carbon dimers in the lowest pressure structure. In the single-chain structure, lone pair electrons are seen at every carbon, and in the double-chain structure they are seen on the 'outer' carbons, demonstrating the sp 2− like nature of bonding in these chain structures, which becomes complete in the YC 2 -V carbon planar structure. The carbon atoms accept electrons from yttrium as they populate sp 2− orbitals. We have carried out a Bader charge analysis (Supplementary Figure 5) of our structures as a function of pressure 20 . At ambient pressure, each C atom accepts 0.9 electrons from Y, with an electron loss of 1.8 electrons from each Y. This shows that carbon is isoelectronic to nitrogen in the YC 2 structure, and the polymerisation of C in the structure could, superficially, be considered relevant to polymerisation of nitrogen, a topic of significant current interest 21 . On increasing pressure, YC 2 displays gradual reduction of charge transfer between yttrium and carbon as increasingly dense structural arrangements are adopted. This is indicative of the increasing dominance of density in controlling chemical structure at high pressure. Before calculating possible superconducting properties, we analyse the electronic structures of predicted YC 2 phases (Supplementary Figure 2). The metallic nature of all the predicted phases is apparent. The projected density of states show that the Y-d orbital plays an important role in the metallisation. The YC 2 -V phase is an especially good metal with multiple bands crossing the Fermi level along several directions. It has been observed experimentally that YC 2 -I is a superconductor with a transition temperature of 3.9 K at ambient pressure [11], in good agreement with our theoretical calculation of~4.5 K based on the linear response theory. The same method was applied to explore the possible superconductivity of newly-proposed high pressure phases of YC 2 , apart from YC 2 -II, whose density of states at Fermi level of is too small for a good superconductor. The obtained electron-phonon coupling parameter (λ), logarithmic average of phonon frequency ω log and T c are given in Supplementary  Table 2. The T c of YC 2 varies in the pressure range of 0 to 300 GPa, as shown in Supplementary Figure 6. The T c is strongly structure-dependent: we found a T c of 2.9 K for YC 2 -V at 300 GPa, while the electron-coupling calculations for YC 2 -III and YC 2 -IV give relatively small T c , no greater than 2 K. The YC 2 -V structure is similar to graphene and we find that, in this system, the carbon has a significant contribution to the superconductivity.
In conclusion, we have studied the structures and properties of YC 2 as a function of pressure using first-principles electronic structure predictions and calculations. YC 2 is found to form a series of stable compounds with varying structures under pressure up to 300 GPa, on the basis of the formation enthalpies of the predicted structures relative to elemental yttrium and carbon: dimer-type carbon found in the ambient pressure structure is polymerised first into ordered arm-chair chains which polymerise further into quasi-1D well-ordered double chains and eventually form 2D graphite sheets. Our phonon calculations suggest that these YC 2 structures are also dynamically stable. Furthermore, our electron-phonon coupling calculation demonstrate that YC 2 is a good electron-phonon superconductor with a superconducting critical temperature of 2.9 K at 300 GPa. Our results reveal the role of pressure on modifying the carbon polymerization within this carbide, in particular in the gradual increase of C-C connectivity with pressure. The YC 2 system provides an unusual example of superconductivity in graphite/graphene C intercalated with metals.
Castep 46,47 was used for the AIRSS searches. The plane wave basis set was constructed using an energy cut-off of 500 eV in all CASTEP calculations, and the Brillouin zone was sampled with a k-point resolution of 2π x 0.03 Å −1 . On The Fly (OTH) potentials are adopted for both Y and C. The exchange-correlation functional was described using the Perdew-Buike-Ernzerhof form of the generalized gradient approximation, as was done for VASP calculations.  50 . Other density functionals, including LDA 51 and PBEsol 52, 53 , were also tested for comparison (see Supplementary Figure 7). Dense k-point meshes (2π X 0.03 Å −1 ) were employed to sample the first Brillion Zone for candidate structures and ensure the energies converged well.
Phonon calculations. We explored the effects of temperature using the quasiharmonic approximation. The phonon calculations were performed using the PHONOPY code 54 . Electron-phonon coupling was performed using the pseudopotential plane wave method and density-functional perturbation theory as implemented in the QUANTUM ESPRESSO package 55 . The technique for the calculation of electron-phonon coupling has been described in detail previously 18 . Convergence tests led us to determine and set the most appropriate kinetic energy cut off at 60 Ry.