Phys. Status Solidi B, 1–4 (2012) / DOI 10.1002/pssb.201100723 p s sb

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basic solid state physics

Is there a common theme behind

the correlated-electron superconductivity in organiccharge-transfer solids, cobaltates, spinels, and fullerides?

Sumit Mazumdar*,1 and R. Torsten Clay2

1 Department of Physics, University of Arizona, Tucson, AZ 85721, USA2 Department of Physics and Astronomy and HPC2 Center for Computational Sciences, Mississippi State University, Mississippi State,

MS 39762, USA

Received 13 October 2011, accepted 16 November 2011

Published online 16 March 2012

Keywords exotic superconductors, strong correlations

* Corresponding author: e-mail sumit@physics.arizona.edu, Phone: þ001-520-621-6803, Fax: þ001-520-621-4721

We posit that there exist deep and fundamental relationships

between the above seemingly very different materials. The

carrier concentration-dependences of the electronic behavior

in the conducting organic charge-transfer solids and layered

cobaltates are very similar. These dependences can be

explained within a single theoretical model, the extended

Hubbard Hamiltonian with significant nearest neighbor

Coulomb repulsion. Interestingly, superconductivity in the

cobaltates seems to be restricted to bandfilling exactly or close

to one-quarter, as in the organics. We show that dynamic Jahn–

Teller effects and the resultant orbital ordering can lead to 1/4-

filled band descriptions for both superconducting spinels and

fullerides, which show evidence for both strong electron–

electron and electron–phonon interactions. The orbital order-

ings in antiferromagnetic lattice-expanded bcc M3C60 and the

superconductor are different in our model. Strong correlations,

quarter-filled band and lattice frustration are the common

characteristics shared by these unusual superconductors.

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1 Introduction The field of superconductivity (SC) isfacing a crisis: 25 years after the discovery of high Tc SC,none of the proposed scenarios has led to a mechanism thatscientists agree on. Although there is now general agreementthat there exist many correlated-electron superconductors,there is no understanding how any one scenario, proposed forone class of materials, could be extended to others. Inthe context of organic charge-transfer solids (CTS) alone, thenatures of the insulating states proximate to SC can be quitedifferent, including spin-density wave, antiferromagnetic,charge-ordered or even a so-called valence bond solid. Itis unlikely that the BCS theory, designed to explain themetal-to-superconductor transition, applies to any of theseinsulator-superconducting transitions. It is equally unlikely,however, that the mechanism of SC in structurally similarCTS, with closely related molecular components, aredifferent for different initial insulating states, as has some-times been proposed. We believe that determination of thecharacteristics shared by many seemingly different classesof superconducting materials will give the framework withinwhich the theory of exotic SC should be constructed. Our

goal here is to demonstrate that such a common frameworkmight indeed exist for the CTS, inorganic layered cobaltatesand spinels, and fullerides.

In the next section, we examine the counterintuitivecarrier density-dependent electronic behavior in the layeredcobaltates. We point out that this behavior is very similar tothat previously noted in the CTS, and show that bothcobaltates and CTS can be understood within the sametheoretical model, within which the bandfilling is a veryimportant implicit parameter. It is then noteworthy that inboth families SC appears to be limited to the 1/4-filled band.Other features shared by the superconducting cobaltates andorganics are strong Coulomb correlations and latticefrustration. We show that spinels and fullerides share thefeatures common to the CTS and cobaltates. We suggest thatfor the strongly correlated frustrated 1/4-filled band theSchafroth mechanism of SC [1] becomes operative.

2 Layered cobaltates and CTS – carrier densitydependence We are interested in MxCoO2 (M¼Li, Na, K)[2], the incommensurate ‘‘misfits’’ [Bi2A2O4][CoO2]m

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(A¼Ca, Sr, Ba) [3] as well as the hydrated superconductorNaxCoO2, yH2O [4]. These materials consist of CoO2 layers,with the Co-ions forming a triangular lattice, separated bylayers of Mþ ions. The Co3þ and Co4þ ions are in their low-spin states because of large crystal field splitting [5]. Trigonaldistortion splits the t2g orbitals into low lying fully occupieddoubly degenerate e0g orbitals and higher energy a1g orbitalsthat are occupied by one (in Co4þ) or two (in Co3þ) electrons[6]. Only the a1g orbitals are relevant for transport andthermodynamics. Charge carriers are S ¼ 1=2 holes on theCo4þ [5], and hence the carrier density r¼ 1� x in MxCoO2.

The onsite repulsion Hubbard U is large in cobaltates [2].Simplistically, large r< 1 is close to the r¼ 1 Mott–Hubbard limit and should behave as strongly correlated;small r should behave as weakly correlated as it is close tothe r¼ 0 band semiconductor limit (all Co-ions trivalent.)Experiments probing magnetic susceptibility and thermo-electric power have, however, demonstrated that the actualr-dependence is exactly the opposite; in the experimentallyaccessible range r¼ 0.2–0.8, small (large) r is strongly(weakly) correlated [2, 3]. The basic observation is the samefor all MxCoO2 and the misfits, indicating that r-dependenceis intrinsic to the CoO2 layer.

A hint to the understanding of cobaltates comes fromprior observation on quasi-one-dimensional (quasi-1D)conducting CTS [7], in which r ranges from 0.5 to 1. CTSwith r equal to or close to 0.5 exhibit magnetic susceptibilityenhanced relative to the Pauli susceptibility, and latticeinstability with periodicity 4kF (where kF is the Fermiwavevector within one-electron theory), both accepted assignatures of strong correlations. In contrast, r between 0.66and 0.8 show weakly correlated behavior, viz., unenhancedsusceptibility and the usual 2kF Peierls instability (see Tablesin reference [7]). The behavior in the CTS and the cobaltatesare then very similar.

3 Theory of carrier density-dependentelectronic behavior We present here a theory of ther-dependent electronic behavior, in 1D [7] and the 2Dtriangular lattice [8]. Consider the extended HubbardHamiltonian,

� 20

H ¼ �X

hijistijcyiscjs þ U

X

i

ni; "ni; # þ VX

hijininj (1)

where cyis creates an electron or a hole with spin s (" or #) onsite i, h. . .i implies nearest neighbors (NN), and all otherterms have their usual meanings. Individual molecules (Co-ions) are the sites in the CTS (cobaltates).

Strongly correlated behavior, at any r, requires that theground state wavefunction has relatively small contributionfrom configurations with double occupancies. We thereforecalculate numerically the normalized probability of doubleoccupancy in the ground state

Figure 1 (online color at: www.pss-b.com) Exact g(r) versus r forU¼ 10, and V¼ 0 (circles), 2 (diamonds), and 3 (triangles), forperiodic ring of 16 sites. Lines are guides to the eye.

g ðrÞ ¼ hni; "ni; #ihni; "ihni; #i

; (2)

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g(r) is clearly weakly r-dependent for V¼ 0. For V> 0 andr¼ 0.5, electrons are prevented from encountering oneanother by V, thus reducing g(r) more than U alone. As r isincreased from 0.5, more and more intersite repulsions aregenerated with additions of electrons, and configurationswith a few double occupancies begin to compete withthose with only singly occupied sites. Thus g(r) increasessteeply with increase in r for significant V/U. As r increasesbeyond an intermediate range, the Mott-Hubbard limitis approached and g(r) should decrease again.

Exact numerical calculations of g(r) verify this con-jecture. Our earlier 1D calculations were for relatively smallnumbers of electrons Ne with varying number of sites Ns [7].Significantly larger computer capabilities now allow calcu-lations for different r with fixed Ns¼ 16. In Fig. 1, we haveshown the results of our calculations for U=jtj ¼ 10 andV=jtj ¼ 0; 2 and 3. This value of U is considered realistic forCTS, and there is evidence for V/jtj as large as 3. The plots aresimilar for U=jtj ¼ 6� 12 and V=jtj ¼ 1� 3. While CTSwith r< 0.5 do not exist, we have included this smaller r forcomparison to 2D. We conclude from Fig. 1 that for realisticV/U, strong r-dependent electronic behavior is warranted.Experimentally [7], all r¼ 0.5 CTS (e.g., MEM(TCNQ)2,Qn(TCNQ)2, (TMTTF)2X, etc.) exhibit strongly correlatedbehavior, while CTS with r¼ 0.66–0.8 (TTFBr0.74,HMTTF- and HMTSF-TCNQ with r¼ 0.75, etc.) uniformlyexhibit weakly correlated behavior. Subsequent demon-strations of strongly correlated behavior in 2:1 BEDT-TTF[9] or 1:2 Pd(dmit)2 systems [10] further support the theory.

In Fig. 2, we show our results for the finite periodic 2Dtriangular lattices with Ns¼ 16 (Fig. 2a) and 20 (Fig. 2b). Wehave taken t> 0. Elsewhere we have argued that the U/jtj andV/jtj in the cobaltates are similar to that in the CTS [8]. Withthe exception of a few Ne, the ground state is always in thelowest spin state (0(1/2) for Ne¼ even(odd)) for t> 0. Asseen in Fig. 2c and d, the behavior of g(r) in 2D is similarto that in 1D, and explains the much-discussed lack ofsymmetry about r¼ 0.5 in the cobaltates – strongly

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Phys. Status Solidi B (2012) 3

Original

Paper

Figure 2 (online color at: www.pss-b.com) (a) and (b) Ns¼ 16 and20-site clusters investigated numerically. (c) and (d) Exact g(r)versus r. The circles, diamonds and triangles correspond to V¼ 0, 2and 3, respectively. The boxes correspond to data points with totalspin S> Smin¼ 0 (1/2) for even (odd) numbers of particles.

correlated behavior near r¼ 1/3 and weakly correlatedbehavior near r¼ 2/3 [2].

4 Superconductivity in the CTS and layeredcobaltates We point out that SC is limited to the samenarrow range of carrier concentration in both families. Giventhe strong role of r, this cannot but be significant. CTSsuperconductors are 2:1 cationic or 1:2 anionic materials[11], with 100% charge-transfer, and consequently r¼ 0.5.The lattice structures of the superconducting CTS are notquasi-1D but anisotropic triangular, giving both frustrationand larger bandwidth.

In the hydrated superconducting Na-cobaltate, x ’ 0:35,Careful experiments have revealed however that r isconsiderably smaller than 0.65, as a portion of the waterenters as H3Oþ. A significant number of investigators havedetermined that r in the superconductor is exactly or close to0.5 [12]. It is also noteworthy that x¼ 0.5 is unique, – Halland Seebeck coefficients change signs only at this r at lowtemperature in both NaxCoO2 and LixCoO2.

The limitation of SC to r¼ 0.5 acquires additionalsignificance when taken together with: (a) absence of SC inthe r¼ 1 triangular lattice Hubbard Hamiltonian [13], (b) thepropensity at r¼ 0.5 to form the paired-electron crystal(PEC), in which spin-singlet pairs are separated by pairs ofvacant sites, in both 1D and the moderately frustratedanisotropic triangular lattice [14]. We have proposed thatwith further increase in frustration the PEC gives way to asuperconducting paired-electron liquid [14]. The proposedtheory of SC is essentially the same as that of Schafroth,

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within which electron-pairs form mobile molecules (chargedbosons) [1]. It is, however, different from the mean-fieldtheory of charge-fluctuation mediated SC [15]. In the rest ofthis paper we show that there exist other systems which likelycan be understood within the same scenario.

5 Superconducting spinels Spinels are inorganicternary compounds AB2X4, where the X2-form a close-packed structure with the A (B) sites occupying tetrahedral(octahedral) interstices. The B-cations are the active sites,with small X-mediated B-B electron hoppings. The Bsublattice is tetrahedral, and hence frustrated.

LiTi2O4, CuRh2S4 and CuRh2Se4 are the only threespinels that have been confirmed to be superconductors;reports of SC in CuV2S4 and CoCo2S4 exist. SC in LiTi2O4

was discovered many years before the cuprates [16], and hasbeen of interest ever since because of its high Tc/TF (TF isFermi temperature), which is closer to that in the cupratesthan conventional superconductors [17]. The mechanism ofSC remains controversial: strong coupling BCS theory, theRVB approach [18] and the bipolaron theory [19] of SC haveall been proposed.

Examination of the common valence of the B-cations inthe superconducting spinels, 3.5þ , reveals remarkablesimilarities between them and the CTS and cobaltates. TheTi3.5þ ions in LiTi2O4 have one d-electron per two Ti. Rh3.5þ

and Co3.5þ in their low-spin states possess one d-hole per twometal ions. V3þ and V4þ have electron configurations 3d1

and 3d2, respectively. In all cases static band Jahn–Tellerlattice distortions will give 1/4-filled electron (in LiTi2O4)and hole (in CuRh2S4, CoCo2S4 and CuV2S4) bands asindicated in Fig. 3. Static lattice distortions should give a 3DPEC, as indeed observed in CuIr2S4 [20] and LiRh2O4 [21].We propose that the superconducting state is a 3D paired-electron liquid, with dynamical (as opposed to static) orbitalordering.

6 Superconducting fullerides The C3�60 ions in

superconducting M3C60 form fcc lattices. Theories offullerides treat each C60 unit as a site, with electron hoppingsbetween the triply degenerate t1u antibonding molecularorbitals. Although many experiments suggest onsite pairingmediated by Hg Jahn–Teller phonons [22], the large Tc/TF

suggests a non-BCS mechanism [17]. Antiferromagnetismin bcc Cs3C60 also indicates strong Coulomb repulsion [23,24]. Non-zero spin gap to the lowest energy high-spin state inthe antiferromagnet has indicated that it is the effective 1/2-filled band Mott–Hubbard insulator of Fig. 4a, reached afterJahn–Teller distortion [23, 24].

Within a dynamic mean field theory (DMFT) of thepressure-induced antiferromagnetism-to-SC in Cs3C60 [24],

Figure 3 (online color at: www.pss-b.com)The effective 1/4-filled band natures of thesuperconducting spinels, following the split-tingof the t2g orbitals (see text).Thebluedottedarrows denote electron or hole motion.

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Figure 4 (online color at: www.pss-b.com) (a) The effective 1/2-filled t1u band in antiferromagnetic C3�

60 , following Jahn–Tellerdistortion. (b) The proposed doubly degenerate 1/4-filled bands inthe superconductor (see text).

pairing arises from the combined effects of Hubbard U andJahn–Teller interaction within the effective 1/2-filled bandof Fig. 4a, at the interface of an insulating and a metallicphase. We recall that the DMFT approach to SC in the CTSwithin the effective 1/2-filled band Hubbard model was verysimilar. It is now known that SC in this latter case was anartifact of the mean-field approximation [13]. We believethat a similar criticism applies also to the existing theory ofSC in the fullerides.

The difficulty in arriving at a theory of SC arises fromthe bias that the orbital occupancies are the same in theantiferromagnetic and superconducting phases. We suggestthat pressure induces a new orbital ordering, with C3�

60

configurations as shown in Fig. 4b, where two degenerate1/4-filled bands are obtained. This orbital ‘‘reordering’’ willbe driven by the lower total energy of two 1/4-filled bandscompared to that of the single effective 1/2-filled band, forlarger bandwidth. Each of the 1/4-filled bands can now formits own paired-electron liquid.

7 Conclusion Our goal here was to show that thecorrelated 1/4-filled band offers a common description ofseemingly very different classes of materials as well as aplausible mechanism of SC. The key step is the realizationthat antiferromagnetism in the semiconducting state doesnot necessarily mean that the electronic structure of thesuperconductor is derived from the same configuration.Bandwidth-driven transition to a different state that ismore appropriately described as 1/4-filled can occur in thesuperconducting state [14]. This has been explicitly shown inthe context of the CTS. Work is in progress to demonstratethe same within the models of Figs. 3 and 4. The proposedmechanism offers, (a) a way to arrive at a single unifiedtheory of SC for the CTS, with different kinds of proximateinsulating states, and (b) understanding of the strong role ofelectron–phonon interactions, and yet large Tc/TF, in thespinels and the fullerides. In both spinels and fullerides,dynamic Jahn–Teller phonons will play a strong role in thecorrelated superconductor. However, this is not a signatureof BCS pairing.

Acknowledgements This work was partially supported byDOE Grant No. DE-FG02-06ER46315.

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