Two Mott Insulator Theory of Superconductivity in K_3X (X: picene, .. p-terphenyl, .. C_{60})
aa r X i v : . [ c ond - m a t . s up r- c on ] A p r Two Mott Insulator Theory of Superconductivity in K X(X: picene, .. p-terphenyl, .. C ) G. Baskaran
The Institute of Mathematical Sciences, C.I.T. Campus, Chennai 600 113, India &Perimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, Canada
We look for unifying aspects behind superconductivity in aromatic hydrocarbon and fullerenefamily K X (X: picene, .. p-terphenyl, .. C ). Aromatic hydrocarbon molecules support RVBstates. Consequent stability (aromaticity) makes them reluctant electron acceptors. We argue thatX accepts only two (not all three) electrons from K and creates charged RVB’s in X − , and becomesa (molecular) Cooper pair box. A weak Josephson coupling between X − molecules creates a BoseMott insulator, a potential high Tc superconductor. Remaining lone electron in the complex (K ) occupies a suitable metal orbital hybrid. They hybridize weakly through X − molecular bridges, toform a half filled band of renormalized K atom orbitals, a Fermionic Mott insulator. An interplay ofRVB physics and charge transfer (mutual doping) or external doping leads to superconductivity inone or both Mott insulators. In our theory there is room for room temperature superconductivity. PACS numbers:
I. INTRODUCTION
Organic route to high Tc superconductivity is highlydesirable. Ever since a theoretical proposal by Little [1],of organic polymer based high Tc superconductivity, thesearch continues. Organic solids such as TTF-TCNQ anddoped polyacetylene which drew a lot of attention at thebeginning have not become superconducting. However,several aromatic hydrocarbon [2] and C molecule [3]based superconductors have been synthesized [4–6]: K X(X: picene, phenanthrene, dibenzo pentacene, coronene,... C ). Some of them are high Tc superconductors.Other low Tc organic superconductor families, Bechgardsalts and ET salts [7], continue to excite physics commu-nity from the point of view of strong correlation physics.A recent claim of superconductivity [9, 10] at an el-evated temperature of 123 K in K (p-terphenyl) is re-markable and exciting. (Three benzene rings bridged bytwo C-C bonds along a line forms para (p-) terphenyl [8],C H ). A more recent high resolution photoelectronspectroscopic evidence [11] for pairing gap in p-terphenylsystem upto 60 K adds to the excitement and makes theactivity worth pursuing. While Meissner signals and afavourable magnetic field dependence of Tc are evidencesfor superconductivity, other standard and confirmatorymeasurements are needed. Are we witnessing birth of anew revolution in high Tc superconductivity ?Aim of the present article is to suggest an unified modeland mechanism of superconductivity in the family ofalkali metal doped aromatic hydrocarbon and fullerenemolecular solids K X.Briefly, we find that K X has an unusual nominalcharge state ( K ) ( X ) − , rather than usually assumed( K ) ( X ) − . A Boson Mott insulator (even electron di-cation subsystem X − ) and Fermion Mott insulator (oddelectron dianion subsystem ( K ) ) emerge. Chargetransfer between two Mott insulators or external dop-ing can create superconductivity in one or both Mottinsulators. That is, in our theory both anion and cation subsystems could become active participants in super-conductivity. In existing theories [12–24] of supercon-ductivity, a nominal charge state ( K ) ( X ) − makes( K ) effectively an insulating spectator and ( X ) − ef-fectively becomes a (odd electron) molecular conductor(with some hybridization with K orbitals), and supportssuperconductivity based on Mott physics.Firstly, we recognize that polycyclic aromatic hydro-carbon molecules X are different from molecules withsaturated bonds. Molecule X supports a correlated manyelectron state, a p- π pool of neutral singlets (molecularRVB state). In our theory molecule X accepts only twovalence electrons, accommodates the pair in valence bondresonance and becomes a (molecular) Cooper pair boxX − . Josephson coupling between X − creates a BosonMott insulator, a potential high Tc superconductor. Fur-ther, the molecular matrix X − isolates the dication com-plex (K ) . Remaining lone electron in (K ) occupiesa suitable metal atom hybrid state. This hybrid furtherhybridizes weakly with LUMO of bridging molecules X − and forms a renormalized metal atom Mott insulator.This fermionic Mott insulator becomes a template forhigh Tc superconductivity via RVB mechanism [25, 26].Mutual charge transfer between two Mott insulators orexternal doping can create superconductivity in one orboth Mott insulators.In building our theory we use some key quantum chem-ical and experimental facts: i) polycyclic aromatic hydro-carbon molecules have in general low electron affinities[27, 28], due to their extra stability (aromaticity) arisingfrom valence bond resonances. For example, benzene hasnegative electron affinity ≈ - 1.5 eV. Consequently elec-tron affinities of polycyclic aromatic hydrocarbons (fusedbenzene rings) are small in general. They are reluctantelectron acceptors in the solid state, ii) an ubiquitous sto-ichiometric composition K X, (X = p-terphenyl, picene,dibenzopentancene, coronene, C , ...) in all known su-perconductors and absence of superconductivity for stoi-chiometry K as well as K , iii) reduced Madelung energyfrom strong dielectric screening and iv) evidence from X-ray spectroscopy [29] for a substantial orbital hybridiza-tion of valence orbitals of K and molecular orbitals ofpicene at the fermi energy in K (picene).Our theory uses earlier and recent theoretical worksand insights: i) recognizing the molecule X as a stronglycorrelated electron system, capable of supporting chargedCooper pair inside molecule X via pair binding and[12, 13] intermolecular Josephson tunnelling [30] ii) a cluefrom a recent theoretical calculations by Naghavi andTosatti [31, 32] and by Chiappe et al. [33], which indi-cate transfer of two rather than all three electrons to Xin K X for X = picene and phenanthrene,iii) Mott in-sulating character for the stoichiometry K , [21, 23], iv)a recognition [23] that a Mott insulating reference solidK X could be important for creating superconductivity.Our paper is organized as follows. We first discuss, us-ing elementary considerations, why two electron ratherthan three electron transfer is favoured in K X. Then wediscuss how a charged dianion X − effectively becomes aCooper pair box. This is followed by discussion of a weakJosephson coupling between X − molecules and forma-tion of a Boson Mott insulator, a potential superconduc-tor. A discussion of formation of K atom based Wannierfunction and emergence of a Fermion Mott insulator isdiscussed next. A Hubbard model for the renormalizedmetal atom Mott insulator is presented. Issues of chargetransfer from Boson Mott insulator to Fermion Mott in-sulator (mutual doping), self doping or external dopingand estimation of Tc are discussed next.Before the concluding section we discuss applicabilityof our ideas to alkali metal doped superconducting ful-lerides, including Mott insulating Cs C that has A15structure. II. BOSON MOTT INSULATOR
Stability of Dianion X − in K X. To substantiateour claim of stability of dianion X − , rather than trian-ion X − we present some basic considerations of chargetransfer in molecular solid K X. Situation is more com-plex than in ionic insulator NaCl, where electron affinityof Cl, ionization energy of Na and a Madelung sum solelydetermine energetics. Electron and lattice polarizabilityaffects are small in NaCl.Situation in K X is different because of a complexmolecular and metal atom environment [28]. In our es-timate X molecule barely manages to capture two elec-trons. Our analysis is a useful first step, as we dontknow structures and real stoichiometry of many of theK X compounds experimentally.We first focus on polycyclic aromatic hydrocarbons X.Consider a benzene molecule in free space. It has nega-tive electron affinity ∼ - 1.5 eV. That is, it can not bindan extra electron in free space. Naphthalene also has anegative electron affinity. Picene and p-terphenyl havesome similarities. Picene has a small positive electron affinity E a ≈ ≈ − E a + e R ≈ + 3eV. Here R is the linear dimensions of picene molecule.Thus two electron binding is not possible in free space;i.e., a dianion (picene) − is not stable in free space, be-cause of Coulomb barrier.In the solid state environment various factors couldhelp to overcome a charging energy ∼ π bonded system, that has beeninvoked to give stable molecular charge -2e singlets andpositive pair binding energy, ii) Madelung energy gain,iii) electron, lattice, molecular orientational polarizationenergies, iv) cohesive energy in the Cooper pair Mottinsulating dianion system arising from Josephson tun-nelling. In our estimate these corrections do not add upto overcome charging energy of 3 eV per formula unit.Interestingly dication (K ) can help overcome thelarge charging energy by contributing to overall cohesionof the solid from: i) energy gain by hybridization of thelone electron with K orbitals within the complex K andwith empty LUMO orbitals of neighbouring dianions andii) metallic cohesion in the correlated metallic state of therenormalized metal atom band.In our estimate, with help from various cohesion mech-anism one barely manages to have two electron trans-fer to the anion. Results from our simple considerationis consistent with theoretical calculations of referencesxx. When we consider three electron transfer an en-hanced charging energy overwhelms any of the contribu-tions above and we do not form a stable solid containingtransfer of all three valence electrons from K to X.Similar considerations suggests that K C family wehave stable dianions ( C ) − . Simple fcc and A15 struc-tures in this family makes this estimate more meaningful. Dianion X − as a Cooper Pair Box. Polycyclicaromatic hydrocarbon molecules support valence bondresonances. It is a correlated many electron state, a kindof molecular Mott insulator made of π electrons and welldescribed as an RVB state a la’ Pauling. This inbuiltcorrelation manifests in several ways: for example, as alarge ∼ π electron pool, get entan-gled and becomes part of the valence bond resonancecontaining two extra electrons: i.e., (n-1) bond singletpairs and two doublons. Here 2n is the number of carbonatoms (number of π electrons) in the polycyclic hydro-carbon. This charge carrying state X − is a soft statein the sense energy required to remove a Cooper pair issmaller than that in corresponding Huckel’s theory. Forp-terphenyl number of valence bonds is 9; consequentlythere are 9 neutral Cooper pairs in the box.What we have is a Cooper pair box, where electronpairs, rather than single electron, can tunnel in and outeasily, as we will see. Number of pre-existing Cooperpairs for p-terphenyl is 3 × | molecular RVB i = Y i (1 − n i ↑ n i ↓ )( X ij φ ij b † ij ) n − | i (1)where φ is a suitable short range (Cooper) pair func-tion. Doublons in this state are delocalized within themolecule in a correlated fashion and allows for possibil-ity for charge -2e pair to tunnel in and out.Polaronic effects (known in poly para-phenylene (poly-meric analogue of p-terphenyl) [34]), rather than electroncorrelation effects have been invoked for K (p-terphenyl)[9]. Strong polaronic lattice distortions will make the po-laron heavier by decreasing polaron hopping matrix ele-ment via Frank Condon overlap effects. Further, in shortmolecules, polaronic effects and lattice dimerization areless, because of quantum fluctuations arising from elec-tron correlation effects; p-terphenyl is a three benzenering system. Josephson Coupling and Boson Mott InsulatorFormation.
What is the nature of charge transfer be-tween two molecules that are correlated electron systems? Because of the large charging energy processes suchas X − X − → X − X − (one electron transfer) or X − X − → X X − (two electron transfer) are highly im-probable. We will not consider them.Physically meaningful electron transfer process are: X − X → X − X − (one electron transfer) or X − X → X X − (two electron transfer). We first focus on oneelectron tunnelling. At the level of Huckel theory thisprocess is simple. It involves hopping matrix elementst m between two resonant LUMO orbitals of neighbour-ing molecules. Since we have a strongly correlated elec-tron system we need to pull out an electron from a spinsinglet charge -2e molecular RVB state. Electron trans-fer requires a breaking a singlet bond or a doublon inthe correlated π electron pool. Further, valence bondresonance in the receiving and accepting molecule alsogets disturbed leading to a kind of electronic Frank Con-don reduction (molecular wave function renormalizationconstant, called molecular Z [35]. This single electrontransfer is strongly inhibited.Let us consider two electron transfer, X − X → X X − . Pre-existing pairing correlations in both moleculeswill enhance this matrix element. In the case of twoweakly superconductors this process is the Josephsontunneling [36] that takes advantage of the superconduct-ing coherence present. What we have is only enhancedpairing correlations within each molecule. So we can notuse Josephson or Ambagaokar Baratoff formula. We can use second order perturbation theory and getan estimate of pair tunneling matrix element t B ∼ t m E c .Here E c approximately the singlet triplet gap of themolecule.An earlier work [37] on weakly coupled large U Hub-bard clusters, studied in the context of interlayer pairtunnelling mechanism of Wheatley, Hsu and Anderson[38] is relevant for our present discussion. This studyshows that one electron hoppiing between two finite sizeHubbard clusters enable two electron transfer and a con-sequent enhancement of pairing correlation within themolecules. Using this calculation we can also give a moreaccurate estimate of t B .Charging energy U B , the energy difference between twoconfigurations X − X − and X X − , is typically largerthan Cooper pair band width ∼ B (here z is the num-ber of neighbours.Thus we have effectively a Boson Mott insulator with afinite Mott Hubbard gap. We call it Boson Mott insulatorbecause low energy charge carrying excitations above theMott Hubbard gap are charge ±
2e singlet states ratherthan spin- electron. The effective Hamiltonian of theBoson Mott insulator involves only charge -2e or Bosondegree of freedom represented by operators b i ’s : H B = − t B X h ij i b † i b j + H.c. + U B X i ( b † i b i − (2)A chemical potential fixes total number of charge -2eBosons to be equal to total number of molecules. ThisBoson Mott insulator is a potential high Tc supercon-ductor, if only we can dope it. This superconductor willbe special, as Cooper pair size is effectively size of Xmolecule.There are theoretical evidences [20, 21, 23] that the sto-ichiometric compound K (aromatic molecule) are elec-tron Mott insulators containing two electrons per Xmolecule. According to our theory these Mott insulatorsare best viewed as charge -2e Boson Mott insulators thatalso contain superconducting phase fluctuations. Ournew insight leads to interesting consequences. III. FERMION MOTT INSULATOR — Renormalized Neutral Metal Atom and MottInsulator Formation.
In our proposal charge -2emolecular (dianionic) matrix remains insulating. Eachcation complex (K ) carries one valence electron delo-calized within the K complex. Depending on the struc-ture, not all three atoms are crystallographically equiva-lent and lone electron will be shared appropriately.What is important for us is that there is a lone elec-tron residing in isolated cation complexes (K ) . TheseK atom states will hybridize through empty LUMO ofneighbouring inert molecules and form a narrow renor-malized metal atom band containing one electron perrenormalized orbital. This is our renormalized metalatom Mott insulator described by the fermionic Hubbardmodel: H F = − t X h ijσ i c † iσ c jσ + H.c. + U X i n i ↑ n i ↓ (3)The band parameters t and U can be inferred from ex-isting band structure calculations for picene and relatedsystems. We will not present details now. Superconductivity via RVB Mechanism.
It iswell recognized now, thanks to various development inRVB approach [25, 26] that spin- single orbital Mottinsulators are seats of high Tc superconductivity for arange of doping. In real systems competing charge order,spin order and lattice distortion, when present, degradesuperconductivity by localizing valence bonds, chargesand spins.At the heart of RVB mechanism of superconductivityis use of valence bond or electron pairing correlation inspin-half Mott insulators. Doping of a Mott insulator isessential. An undoped neutral RVB state, though a co-herent quantum fluid of spin paired electrons, is a chargeincompressible state . The Mott-Hubbard gap, in the con-text of an undoped spin-half Mott insulator (a half filledband) represents charge incompressibility. Doping pro-duces a charge compressible charged RVB state, whichin general is a superconducting state.How do we get superconductivity without externaldoping in a half filled band ? Closeness of a half filledband of metal to Mott insulating state makes it a seat ofsuperconductivity. Most organic superconductors beginas Mott insulators, with a band filling of half. Pressurecauses an insulator to superconductor transition. Whatis important is that a metallic state continues to supportsuperexchange, when it is in the vicinity of a Mott insu-lating state. This is well described as a self-doped Mottinsulating state [39, 40], where, for energetic reasons, aMott insulator spontaneously generates and maintains asmall and equal density of holons (charge +e, spin-0)and doublons (charge -e, spin-0) in the background ofsuperexchange and resonating singlets.It is possible that many of the superconducting K Xsystems are at half filling and superconductivity arisesbecause of its vicinity to a Mott insulating state. In ourestimates using RVB theory we get superconducting Tcin the range of 100 K rather easily, for a reasonable choiceof band parameters and structure. Since structures arenot known precisely at the moment we will be contentedwith a ball park estimate.In the next section we will see that presence of a Bo-son Mott insulator brings new possibilities through in-teresting interplay and synergy with the Fermionic Mottinsulator.
IV. SUPERCONDUCTIVITY
Two Mott insulators and Prospects for RoomSuperconductivity.
We saw possibility of high Tc su-perconductivity in the Fermionic Mott insulator subsys-tem. In principle the Boson Mott insulator supported byX − molecular subsystem, in view of a large pairing fluc-tuations, can encourage pairing in the fermion systemthrough proximity effect. This will increase supercon-ducting Tc to some extent.There are two additional interesting possibilities. Elec-tron affinities of the Fermion Mott insulator and BosonMott insulator will be different in general. Beyond somethreshold value there is possibility of transfer of electronpairs, for example from Boson Mott insulator to FermionMott insulator. Electron transfer is more likely, as addingcharge -2e to X − is more difficult than removing charge-2e. This internal charge transfer is a mutual doping .In the present case Fermion Mott insulator is electrondoped and Boson Mott insulator is hole doped. Severalmanybody effects contribute to charge transfer and de-termine density of charge transfer n. Two major factorscontributing to them are superconducting condensationenergy gain in both systems and Madelung energy gainarising from enhanced charge fluctuation.Boson Mott insulator becomes a superconductor aftertransferring a density n of electron pairs. It is effectivelya Bose condensation of holes in a lattice of hard corebosons carrying charge -2e. When hole density n is small,superconducting Tc of doped Boson Mott insulator is wellapproximated by Bose Einstein condensation formula: k B T c ≈ . ~ n m ∗ k B (4)We will not go into details, but just point out that effec-tive mass of (hole) bosons (as estimated from pair tun-nelling matrix element) and a reasonable hole boson den-sity n gives us Tc’s in the room temperature scales. Thisis an encouraging signal worth exploring further, lookingfor optimal m ∗ and charge transfer n.Fermion Mott insulator which becomes superconduct-ing via RVB mechanism will have its own Tc, determinedby different parameters. Coupled order parameters fromtwo superconductors makes the problem richer.The synergy is interesting. Boson Mott insulator giveselectrons, Fermion Mott insulator gives holes and bothbecome superconducting. We call this shared supercon-ductivity .Second interesting possibility is existence of a com-pound K − x X, with a stoichiometry close to two. In thiscase fermion Mott insulator is not present. However, wehave a hole doped Boson Mott insulator. This systemalso can lead to high Tc superconductivity.Since we have several candidate aromatic hydrocarbonand C systems, parameter space is big and we couldrealize some of these interesting possibilities. Two Mott Insulator Theory of Superconduc-tivity in C based Systems. In the superconduct-ing compound K C , fullerene molecules form a fcc lat-tice. Fcc lattice has octahedral and tetrahedral intersti-tial sites in the ratio 1:2. Alkali atoms fill in all interstitialsites. Octahedral interstitial sites also form an fcc lattice.Tetrahedral interstitial sites form a simple cubic lattice,which is symmetrically inscribed in the fcc lattice of oc-tahedral interstitials. As interstitial lattices have differ-ent geometrical environment, distance between K atomand its nearest C molecule is closer for tetrahedral sitesthan octahedral sites.According to our hypothesis, two K atoms in tetra-hedral interstitials transfer two electrons to one C molecule. K atom from octahedral interstitial, one performula unit, retains a valence electron. Thus nominalcharge state of our reference system is K ( K ) C − .This is partly because of closeness of K atoms at tetra-hedral interstitial to four C molecules. Stability ofmolecular singlet or a positive pair binding energy anddynamical Jahn Teller distortion also help two electrontransfer and creation of a stable charge -2e spin singletmolecular state. Further Madelung energy gain decreasesbecause of dielectric screening, making charge three elec-tron transfer less likely. It is interesting to note that ina recent model calculation for pair binding in C , usingDMRG method, an orbitally symmetric and spin singletstable state with maximum pair binding has been found.According to our hypothesis single valence electron ofK atom in octahedral interstitial hybridize with LUMOof C − molecule and forms a single orbital half filled cor-related metallic state in an fcc lattice. This metal isclose to the Mott insulator boundary and support RVBmechanism based superconductivity. The well known ex-perimental fact [41] that intercalation with NH expandsthe lattice and converts NH K C to a Mott insulatoris not in contradiction with our hypothesis.It is also interesting that highest Tc in the alkali ful-leride family [42] is exhibited by RbCs C . This fits withour hypothesis as well, if Cs occupies the tetrahedral sitecompletely. Further, Cs has lower ionization energy com-pared to Rb and hence transfers two of its electron andmakes a Bosonic Mott insulator in the dianion molecularmatrix C − Mott insulating Cs C in the A15 structure is in-teresting [43] from the point of view of our hypothesis.Fullerene molecules form a bcc lattice. Cs atoms fill ininterstitial sites such that pairs of Cs atoms lie on threesets of parallel faces in mutually perpendicular direction.We get three sets of Cs chains. There are two possibili-ties.First one is, a stronger version of our hypothesis - thereis no charge transfer from Cs to the C sub system atall. We have essentially created an expanded lattice of Csatoms, where Mott physics begins to play an importantrole and creates a spin- renormalized Cs atom Mott in-sulator. Absence of charge transfer from K atoms in thissystem is a distinct possibility, because of an increasein distance between K atom and its nearest fullerenemolecule in this A15 structure. Pressure can induce a Mott insulator to metal transition and lead to supercon-ductivity via RVB mechanism in the vicinity of the insu-lator to metal boundary.Second intriguing possibility is creation of a BosonMott insulator as before, C − , by each fullerene moleculeaccepting just two electrons. Remaining single valenceelectron in the cation complex K( K ) forms a Mottinsulator; it can happen in few different ways. One suchpossibility is that the three sets of K atom chains alludedto earlier have a rational occupancy of and forms aWigner-Mott lattice at ambient pressure. Wigner-Mottlattice undergo metallization under pressure leading tosuperconductivity. In this case there is a lattice sym-metry breaking misfit between symmetry of the K atomsubsystem and C subsystem. It is also possible thatmutual doping, discussed earlier, creates superconduc-tivity in both Mott insulators. Superconductivity in K (p-terphenyl). Stoichio-metric K (p-terphenyl), according to our theory, has aBoson and a Fermion Mott insulator. In the absenceof detailed crystal structure and other phenomenologicalinputs it is difficult to say what is the origin of supercon-ductivity in this fascinating system, within the two Mottinsulator model. We wish to summarize the three pos-sibilities we have spelled out before: i) a charge transferfrom Boson to Fermion Mott insulator (mutual doping)and both exhibiting superconductivity, ii) Fermion partis superconducting, as it is a half filled band metal butclose to the Mott transition point; Boson part remainsessentially an insulator and iii) stoichiometry is close toK − x X and we have a hole doped Boson Mott insulator,which exhibits superconductivity.Existing phenomenology, namely a superconductinggap of about 20 to 25 meV seen in high resolution PES[11] and relatively smaller H c [9] implies that doped Bo-son Mott insulator is playing some key role in establishingsuperconductivity [44]. Pairing energy scale (pair bind-ing energy) within the molecule (p-terphenyl) − is high.It is likely that gap seen in the experiment is from induced(proximity) superconductivity in the Fermionic Mott in-sulator.As we mentioned earlier, we do not find serious theo-retical constraints on reaching higher Tc’s in these familyof compounds. V. DISCUSSION
In this article we have suggested a novel possibility offormation of a Boson and fermion Mott insulator in thealkali doped aromatic hydrocarbon and fulleride familyK X. We have discussed superconductivity in this cor-related electron system, from RVB mechanism point ofview. It is remarkable that RVB physics works in twoways: i) RVB physics inside the molecule X gives rise toa Boson Mott insulator and ii) Fermionic Mott insulatorof the metal atom leads to the standard RVB physics in-volving valence bonds living in the lattice of the metalatom complexes ( K ) − .Metal ammonia solution [45, 46] has been some whatof a mystery in condensed matter community, ever sinceOgg’s observation of Meissner signals at room tempera-ture in 1946. We find that the two Mott insulator sce-nario we have suggested for K X has an interesting roleto play [47] for metal ammonia solid solutions and relatedsystems.
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