Electronic Correlation effects in superconducting picene from ab-initio calculations
aa r X i v : . [ c ond - m a t . s t r- e l ] A p r Electronic Correlation effects in superconducting picene from ab-initio calculations
Gianluca Giovannetti , , Massimo Capone , ISC-CNR and Dipartimento di Fisica, Universit`a di Roma “La Sapienza”, Piazzale A. Moro 5, 00185, Rome, Italy Institute for Theoretical Solid State Physics, IFW Dresden, 01171 Dresden, Germany and Democritos National Simulation Center, CNR-IOM and Scuola InternazionaleSuperiore di Studi Avanzati (SISSA), Via Bonomea 265, Trieste, Italy (Dated: August 7, 2018)We show, by means of ab-initio calculations, that electron-electron correlations play an impor-tant role in potassium-doped picene ( K x -picene), recently characterized as a superconductor with T c = 18 K . The inclusion of exchange interactions by means of hybrid functionals reproduces thecorrect gap for the undoped compound and predicts an antiferromagnetic state for x = 3, wheresuperconductivity has been observed. These calculations, which do not require to assume a valuefor the interaction strength, indirectly suggest that these materials should have a sizeable ratio be-tween the effective Coulomb repulsion U and the bandwidth. This is fully compatible with simpleestimates of this ratio. Using these values of U in a simple effective Hubbard model, an antiferro-magnetic state is indeed stabilized. Our results highlight the similarity between potassium-dopedpicene and alkali-doped fulleride superconductors. PACS numbers: 71.20.Tx,74.20.Pq,74.70.Wz,74.70.Kn
I. INTRODUCTION
The unanticipated discovery of superconductivity withcritical temperature T c = 18 K in an aromatic compoundlike potassium-doped picene opens a new path to super-conductivity in organic materials. Looking for an answerto the most natural question, namely “What is the originof the electron pairing?”, we are tempted to consider onlytwo mutually exclusive options: On one side standardelectron-phonon superconductors which are expected tohave low critical temperatures, on the other “anomalous”superconductors, dominated by electron-electron correla-tion effects. In this latter case pairing is assumed to haveelectronic origin while the critical temperature may evenexceed 100K.A challenge to the above distinction comes from a fam-ily of organic superconductors, the alkali-metal-dopedfullerides. There is wide consensus that pairing is me-diated by intramolecular phonons in these materials .Nonetheless Cs C , which has one of the largest in-tramolecular distances in the family, is an antiferro-magnetic insulator which turns into a 38K supercon-ductor under pressure . This evidence confirms a pre-vious proposal that correlations are indeed importantand helpful for superconductivity in fullerides despitethe phononic nature of the pairing ”glue”. In this con-text the discovery of relatively-high T c in a different or-ganic compound doped with potassium is exciting andbrings, among many others, the question about the roleof electron-electron correlations. This paper is devotedto a first investigation of the fingerprints of electroniccorrelations in K x -picene using a wide range of density-functional theory (DFT) methods, including in particu-lar hybrid functionals designed to include some degree ofexchange correlations.Picene is an aromatic molecule (C H ) composed byfive benzene rings arranged in an “armchair” structure which determines a high chemical stability reflected inthe band gap of 3.3 eV. Superconductivity has been dis-covered in solid picene with a monoclinic structure whenthe band insulating stoichiometric compound is dopedby alkali atoms. Indicating with K x -picene a doped solidwith x potassium atoms per picene molecule, supercon-ductivity has been observed in the region 2 . < x < . .The first ab-initio study of K x -picene has been presentedin , where the electronic structure and Fermi surfacehave been obtained within the Local Density Approxi-mation (LDA). In this work we extend this analysis usingDFT-based methods aiming to include correlation effects.Namely we compare Perdew-Burke-Ernzerhof (PBE) implementation of generalized gradient corrections withLDA+U calculations for a tight-binding model derivedfrom PBE, and we use Heyd-Scuseria-Ernzerhof (HSE) and B3LYP hybrid functionals including different de-grees of many-body exchange.The paper is organized as follows: Section II presentsresults for undoped solid picene, while in Section III weturn to describe the doped solid picene estimating a siz-able on-site electron-electron interaction, mapping thedoped solid picene into an Hubbard model, and then dis-cussing the emergence of an AFM solution. II. BANDSTRUCTURE OF UNDOPED PICENEA. Single Picene Molecule
Given the molecular nature of solid picene, it is use-ful to recall the basis aspects of the molecular electronicstructure, which is expected is the main building blockof the bandstructure of the solid. Calculations for an iso-lated molecule are based on the GAMESS package anduse PBE, PBE0 and B3LYP functionals choosing TZVbasis set. A PBE calculation fixing the molecule in itsequilibrium geometry in the solid gives a HOMO-LUMOgap of 2.9 eV as reported in . The gap increases to 4.2(4.4) eV within B3LYP (PBE0), suggesting an impor-tant role of intramolecular exchange interactions. Sinceour focus is on the electron-doped system, the structureof the unoccupied states is of particular importance, andit is characterized by two closely lying states (LUMOand LUMO+1). The difference between these levels is0.19 (0.20) eV using PBE (B3LYP) and it can shrinkdown to 75 meV if the geometry of the molecule is op-timized, leading to a flatter structure which in turn re-duces the molecular overcrowding. On the other hand,using the geometry of the doped crystals, the LUMO andLUMO+1 gap increases to 0.36 eV.The potassium electrons will therefore populate thebands arising from the LUMO and LUMO+1, which canhost up to four electrons. We briefly anticipate two possi-ble configurations for x = 3, the potassium concentrationaround which superconductivity has been observed: • (i) the bands originating from LUMO andLUMO+1 may remain distinct with two electronsper molecule filling the LUMO band and the re-maining electron going in the LUMO+1 band,which becomes half-filled. • (ii) The two bands are not well separated, and thethree electrons partially populate both bands sothat the system behaves more like a 3/4-filled two-band system.While the former scenario would lead essentially to asingle band half-filled Hubbard model which is likelyto order antiferrogmagnetically, the latter situation isless favorable for antiferromagnetic ordering and it cangive rise to a competition between different phases in-volving magnetic and charge ordering, as shown by thephase diagrams of 3 / . Given the small molecular splittingboth situations are in principle possible. We will showthat, at least in the present DFT calculations, the firstscenario seems to be preferred, even if a more accuratetreatment of electron-electron correlations may revive thesecond possibility. B. Undoped Solid Picene: Generalized GradientApproximation and Hybrid Functionals
We now move to pristine solid picene, that we firststudy via PBE and the projector augmented wave (PAW)method using the Vienna ab-initio simulation package(VASP) . The valence pseudo-wave-functions were ex-panded in a plane-wave basis set with a cutoff energy of400 eV and all the integrations in the Brillouin zone areperformed with a Gaussian broadening σ = 0 .
02 usinga sampling grid of 5x7x3 k-points along lattice vectorsa,b,c of the unit cell. The lattice parameters are fixed atthe experimental values . Within the unit cell there are a ba) K -PICENE PICENE b) K PICENE AB FIG. 1. (Color online) Schematic arrangement of two inequiv-alent picene molecules (A and B) along the a , b planar molec-ular axes in a) pristine and b) doped molecular crystals. two inequivalent picene molecules (A and B) arranged inherringbone structure in the ab plane with stacking alongthe c direction (see Fig. 1).The bandstructure obtained with PBE confirms themolecular nature of the solid because the relevant bandsclearly correspond to the molecular orbitals describedabove. Our results are in good agreement with Ref. ,where the relation between the molecular orbitals andthe maximally localized Wannier orbitals of the solidhas been discussed in details. Since there are two in-equivalent picene molecules, the HOMOs of the picenemolecules hybridize giving rise to two bands just belowthe Fermi level, which are separated by a band gap of2.2eV from a manifold of four bands originating fromthe two LUMOs and two LUMO+1s. As customary,PBE underestimates the experimental value of the gap .Both the valence and the conduction bands have a band-width of around 0.5 eV. The dispersion is enhanced alongthe a ∗ ,b ∗ directions as expected from the layered crystalstructure. For a similar molecular crystal, Van der Waalsinteractions may in principle play a role in determiningthe structure. The reasonable agreement between theexperimental structure and the results of our theoreticalrelaxation suggests that this role is indeed not crucial.To overcome the limitations of PBE (and LDA) and itsshortcoming in the determination of the gap, we repeatedthe same kind of calculations using HSE and B3LYP hy-brid functionals, with the same basis set and computa-tional details mentioned above. The two approaches leadto an enhanced gap of 3.0 and 3.25 eV respectively. Theinclusion of some degree of short-range exchange interac-tions (within a Hartree-Fock treatment) is therefore es-sential to reproduce the experimental gap of 3.3 eV, sig-naling the important role of intramolecular interactions.An inspection to the density of states obtained via HSEand B3LYP clarifies the role of of the exchange interac-tions in the undoped system. The bands become signif-icantly narrower with respect to PBE as a consequenceof a more localized nature of the carriers, as expected in E F G X M L G E ( e V ) PBE DOSPBE HSE B3LYP
FIG. 2. (Color online) Results for pristine Picene: Panel (a)presents the PBE bandstructure; panel (b) shows the densityof states calculated with PBE, HSE, and B3LYP functionals.The energies are plotted along lines in the Brillouin zone con-necting the points Γ = (0 , , X = ( , , M = ( , , L = ( , , ). The zero of the energy is at the Fermi-level. a correlated system. Correspondingly, the wavefunctionsat each site become even closer to the molecular orbitals,another signature of strong correlations. This effect isparticularly strong in the unoccupied bands coming fromLUMO and LUMO+1 that the doped electrons are ex-pected to populate. III. BANDSTRUCTURE OF K x -PICENEA. PBE bandstructure We finally turn to K x -picene. The experimental resultsalready show that a rigid band picture does not hold, asshown by the difference in the measured lattice param-eters of undoped and K . -picene, which suggests thatthe dopants are not intercalated in the interlayer regionand that they affect more deeply the bandstructure, asalready discussed in Ref. . We built a unit cell withthe lattice parameters measured for K . -picene and weinserted 3 potassium dopants per picene molecule in theab plane as suggested in Ref. (see Fig. 1). We alsorepeated within PBE an optimization of the potassiumpositions starting from different configuration obtainingresults essentially equivalent to LDA optimization . Thedopant orbitals are higher in energy and very weaklyhybridized with the molecular states close to the Fermilevel. Therefore their outer s electrons are donated to thepicene bands. Indeed the four bands originating from theLUMO and LUMO+1 of the two inequivalent moleculesare filled by the six electrons donated by potassium atoms(three K per picene molecule). Within PBE (see Fig. 3a)the LUMO and LUMO+1 bands are slightly separatedso that the two low-lying bands are completely filled, −PICENE E F G X M L G E ( e V ) PBE DOSPBE HSE HSE AF
FIG. 3. (Color online) Results for K -Picene restricted to theenergy region close to the Fermi level: (a) shows the PBEband-structure, while (b) presents the density of states cal-culated with PBE (non magnetic), compared with HSE bothin nonmagnetic and AFM state (b). The energies are plot-ted along lines in the Brillouin zone connecting the pointsΓ = (0 , , X = ( , , M = ( , , L = ( , , ). Thezero of energy is at the Fermi-level. while the two LUMO+1 bands host an average of oneelectron per site per band, or, in other words, they arehalf-filled. The bandwidth W is of around 0.8 eV forall the relevant bands. The above described electronicstructure, with one electron per picene molecule in thehighest partially occupied band, can give rise to magneticordering of the spins of the electrons in the LUMO+1bands.Within PBE we have stabilized both a ferromag-netic solution and an antiferromagnetic (AFM) solutionin which A and B molecules have respectively paralleland opposite spins. Both solutions retain metallic char-acter even if they present a momentum of around 0.25 µ B . Their energy is comparable with the non magneticstate within the accuracy of the calculation. B. Antiferromagnetic solution within mean-field
In order to obtain a more reliable picture of the mag-netic behavior of the system we need to better accountfor electron-electron interactions, which are in princi-ple able to give rise to a Mott insulating state with orwithout magnetic ordering. A simple way is to includethe molecular Coulomb repulsion measured by the Hub-bard parameter U . It is therefore important to esti-mate the value of U , which is essentially the energeticcost to doubly occupy a molecule including the screen-ing effects of the other bands . We can compute U for a single molecule with three electrons from the en-ergies of the molecule charged with 2,3 and 4 electronsas U = E (4) − E (3) + E (2), where E ( M ) is the to-tal energy of a molecule charged with M extra electrons,and we will consider N = 3 in the following. Such anestimate for an isolated molecule needs to be correctedin order to include the screening effects in the solid. Afirst estimate can be obtained by considering the effectof the polarization of a charged molecule placed insidea cavity of an homogeneous dielectric medium charac-terized by a dielectric constant ǫ . Using typical valuesfor organic molecular crystals ǫ = 3,4, 5 and 6, we ob-tain respectively U = 1 . , . , . , .
20 eV . Even ifthis estimate can not be regarded as quantitative, giventhe bandwidth of W ≃ . eV we obtain a large value ofthe ratio U/W , very close to similar estimates for dopedC , where the signatures of the strong repulsion havebeen experimentally observed . It is therefore impor-tant to explore the consequences of electronic correlationsalso in picene-based solids. A first approach is to intro-duce the Hubbard repulsion at a mean-field level usingthe DFT bands as the bare bandstructure. Therefore webuilt a tight-binding representation of the bandstructureusing Wannier90 to compute the maximally localizedWannier orbitals starting from the PBE bandstructureobtained using Quantum Espresso . Within the tight-binding representation the system is represented as a lat-tice model, in which the lattice sites coincide with thepicene molecules. Indicating with t αβij the hopping am-plitude between sites i and j and bands α and β and with ǫ αi the local energy for an electron in the band α on sitei, our tight-binding Hamiltonian reads H = X iασ ǫ αi c † iασ c iασ + X ijαβ t αβ ( c † iασ c jβσ + h.c. ) + X i U n i , (1)where c iασ and c † iασ are creation and annihilation op-erators for electrons of spin σ on site i and orbital α ,and n i = P ασ c † iασ c iασ is the total density on site i. Inthis preliminary study we limit ourselves to onsite (in-tramolecular) interactions and we also do not considerexchange interactions between LUMO and LUMO+1. U controls both interband and intraband interactions. Inorder to implement our mean-field strategy we decouplethe interaction term in the particle-hole channel, con-sidering all possible instabilities (charge and magnetic),finding that an antiferromagnetic (AFM) solution withorder parameter m = P i ( S ziA − S ziB ) ( S z being the zcomponent of the spin on the two sublattices) is clearlyfavored for our estimated values of U . Even for the small-est estimate given above (U=1.2 eV), the system becomesAFM (the A molecules having opposite magnetizationwith respect to B molecules) with a large magnetic mo-ment of 0.96 µ B corresponding to (almost) a single spinper picene molecule. We notice in passing that a ten-dency towards a charge-ordered solution has been ob-served within this mean-field approach, even if this solu-tion is never stable. Such a tendency is intriguing becausecharge-ordering may be expected in 3/4-filled systems inthe presence of nearest-neighbor interactions, which arenot included here. Non-local interactions may stabilizethis phase, reviving the scenario in which the four bandsare democratically occupied by the six electrons leading to 3/4-filling. C. Antiferromagnetic Solution with hybridfunctionals
At a mean-field level, we stabilized an AFM state sim-ilar to the parent compound of copper oxide supercon-ductors or to the ground state of Cs C . One can sur-mise that also in K -picene superconductivity may appearclose or compete with an AFM state. It is however wellknown that the mean-field approach emphasizes the ten-dency towards antiferromagnetism. Therefore we testedthis result using HSE and B3LYP hybrid functionals,also in light of their success in determining the gap ofthe parent compound that we discussed above. Further-more, these approach does not require an estimate of U ,and represent an independent test of the importance ofelectron-electron correlations. In Fig. 3 we show resultsfor HSE, which is believed to perform better in metal-lic systems because the long-range tail of the Coulombkernel is screened , even if the results obtained withB3LYP are fully compatible. Using both approaches anAFM solution with a reduced magnetic moment of 0.4 µ B at each molecular site is indeed stabilized. WithinHSE the AFM state has a finite gap of 0.4 eV (see Fig.3b), and the gain in energy respect to the non magneticstate is 0.16 eV per unit cell which is at the limit of theaccuracy of our calculation. B3LYP produces an AFMstate with the same magnetic moment and slightly largerenergy gain (0.22 eV) and gap (0.58 eV).Therefore, using the same approaches that correctlyreproduce the gap of the parent undoped compound, thetrivalent doped system appears to be a low-spin AFM,in which less than one electron per picene molecule con-tributes to magnetism. This result seems to be close towhat we obtained by means of static mean-field, with ahalf-filled conduction band, and the smaller value of themagnetization can be ascribed either to an overestimated U in our mean-field calculation, or to the effect of quan-tum fluctuations. Ab-initio estimates of U (and possiblyof nearest neighbor repulsion and interorbital exchange)will be very helpful to understand this discrepancy, aswell as more accurate treatments of short-ranged corre-lations, such as Dynamical Mean-Field Theory. IV. CONCLUSIONS
In this manuscript, motivated by experiments showingthat for 2 . < x < . K x -picene displays superconduc-tivity with T c = 18 K , we have investigated the band-structure of this compound with a special focus on the ef-fects of electron-electron correlations. In agreement withprevious DFT calculations, we find that the potassium s-electrons are donated to four bands arising from LUMOand LUMO+1 of two inequivalent picene molecules. Sixelectrons are donated in the case of K -picene. This givesrise, within our DFT methods, to four completely filledbands arising from the LUMOs while the two bands aris-ing from the LUMO+1s are overall half-filled. Estimatesof the correlation strength pose these materials on thecorrelated side, suggesting that magnetic ordering canbe expected. While the tendency toward magnetism isvery weak in PBE, the inclusion of correlations at themean-field level gives a strong tendency towards antifer-romagnetism, which is confirmed by the accurate hybridfunctional HSE (and also by B3LYP), even if the mag-netic moment is substantially reduced. The antiferro-magnetic ordering seems quite robust, and it should notdepend on the details of the crystal structure, which isnot known experimentally for the doped system.This finding would put K x -picene in a wide class ofsuperconductors in which AFM and superconductivitycoexist and/or compete in the same phase diagram. Onthe other hand, a very recent photoemission study sug-gests that a sizable intramolecular electron-phonon cou-pling is present in K x -picene with x ≃
1, and a role of thesame kind of phonons has been proposed in a recent DFTanalysis . This is particularly interesting in light of theanalogy with fullerides. In these compounds an intra-molecular electron-phonon coupling is indeed believed toprovide pairing , and it has been shown that a simi-lar coupling can indeed be favored by strong Coulombinteraction because it does not involve charge fluctua-tions on a molecule, but rather it couples to the internal degrees of freedom, which are not frozen by Coulombinteractions. It appears therefore that this new molecu-lar superconductors shares many similarities with alkali-metal doped fullerides. In this light the AFM state thatwe find in the present study would be competing witha phonon-driven (and correlation assisted) groundstate,leading to a first-order transition as a function of the cellvolume, in close analogy to what observed in Cs C .After completion of the manuscript, we became awareof a related work by Kim et al. , arXiv:1011.2712, in whichan AFM state is found also using PBE if the volume permolecule is increased by 5%. This result is compatiblewith ours, and confirms the important role of electron-electron correlations in K x -picene. ACKNOWLEDGMENTS
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