Is there a common theme behind the correlated-electron superconductivity in organic charge-transfer solids, cobaltates, spinels and fullerides?
pphysica status solidi, 8 November 2018
Is there a common theme behind thecorrelated-electron superconductivityin organic charge-transfer solids,cobaltates, spinels and fullerides?
Sumit Mazumdar *,1 , R. Torsten Clay Department of Physics, University of Arizona, Tucson, AZ 85721, USA Department of Physics and Astronomy and HPC Center for Computational Sciences, Mississippi State University, Mississippi State,MS 39762, USAReceived XXXX, revised XXXX “““““““““““““, accepted XXXXPublished online XXXX
Key words:
Strong correlations, Exotic superconductors ∗ Corresponding author: e-mail [email protected] , Phone: +001-520-621-6803, Fax: +001-520-621-4721
We posit that there exist deep and fundamental relation-ships between the above seemingly very different mate-rials. The carrier concentration-dependences of the elec-tronic behavior in the conducting organic charge-transfersolids and layered cobaltates are very similar. These de-pendences can be explained within a single theoreticalmodel, - the extended Hubbard Hamiltonian with signif-icant nearest neighbor Coulomb repulsion. Interestingly,superconductivity in the cobaltates seems to be restrictedto bandfilling exactly or close to one-quarter, as in the organics. We show that dynamic Jahn-Teller effectsand the resultant orbital ordering can lead to -filledband descriptions for both superconducting spinelsand fullerides, which show evidence for both strongelectron-electron and electron-phonon interactions. Theorbital orderings in antiferromagnetic lattice-expandedbcc M C and the superconductor are different in ourmodel. Strong correlations, quarter-filled band and lat-tice frustration are the common characteristics shared bythese unusual superconductors. Copyright line will be provided by the publisher
The field of superconductivity (SC)is facing a crisis: 25 years after the discovery of highT c SC, none of the proposed scenarios has led to amechanism that scientists agree on. Although there isnow general agreement that there exist many correlated-electron superconductors, there is no understanding howany one scenario, proposed for one class of materials,could be extended to others. In the context of organiccharge-transfer solids (CTS) alone, the natures of the in-sulating states proximate to SC can be quite different,including spin-density wave, antiferromagnetic, charge-ordered or even a so-called valence bond solid. It isunlikely that the BCS theory, designed to explain themetal-to-superconductor transition, applies to any of theseinsulator-superconducting transitions. It is equally un-likely, however, that the mechanism of SC in structurally similar CTS, with closely related molecular components,are different for different initial insulating states, as hassometimes been proposed. We believe that determinationof the characteristics shared by many seemingly differ-ent classes of superconducting materials will give the framework within which the theory of exotic SC shouldbe constructed. Our goal here is to demonstrate that sucha common framework might indeed exist for the CTS,inorganic layered cobaltates and spinels, and fullerides.In the next section we examine the counterintuitive car-rier 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 both cobal-tates and CTS can be understood within the same theoreti-cal model, within which the bandfilling is a very importantimplicit parameter. It is then noteworthy that in both fam-
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Figure 1 (Color online) Exact g ( ρ ) versus ρ for U = 10 ,and V = 0 (circles), 2 (diamonds) and 3 (triangles), forperiodic ring of 16 sites. Lines are guides to the eye.ilies SC appears to be limited to the -filled band. Otherfeatures shared by the superconducting cobaltates and or-ganics are strong Coulomb correlations and lattice frustra-tion. We show that spinels and fullerides share the featurescommon to the CTS and cobaltates. We suggest that for thestrongly correlated frustrated -filled band the Schafrothmechanism of SC [1] becomes operative. We are interested in M x CoO (M = Li, Na,K) [2], the incommensurate “misfits” [Bi A O ][CoO ] m (A = Ca, Sr, Ba) [3] as well as the hydrated superconduc-tor Na x CoO , y H O [4]. These materials consist of CoO layers, with the Co-ions forming a triangular lattice, sepa-rated by layers of M + ions. The Co and Co ions are intheir low-spin states because of large crystal field splitting[5]. Trigonal distortion splits the t g orbitals into low ly-ing fully occupied doubly degenerate e (cid:48) g orbitals and higherenergy a g orbitals that are occupied by one (in Co ) ortwo (in Co ) electrons [6]. Only the a g orbitals are rele-vant for transport and thermodynamics. Charge carriers are S = holes on the Co [5], and hence the carrier density ρ = 1 − x in M x CoO .The onsite repulsion Hubbard U is large in cobaltates[2]. Simplistically, large ρ < is close to the ρ = 1 Mott-Hubbard limit and should behave as strongly correlated;small ρ should behave as weakly correlated as it is close tothe ρ = 0 band semiconductor limit (all Co-ions trivalent.)Experiments probing magnetic susceptibility and thermo-electric power have, however, demonstrated that the actual ρ -dependence is exactly the opposite ; in the experimen-tally accessible range ρ = 0 . − . , small (large) ρ isstrongly (weakly) correlated [2,3]. The basic observationis the same for all M x CoO and the misfits, indicating that ρ -dependence is intrinsic to the CoO layer.A hint to the understanding of cobaltates comes fromprior observation on quasi-one-dimensional (quasi-1D)conducting CTS [7], in which ρ ranges from 0.5 to 1. CTS Figure 2 (Color online) (a) and (b) N s = 16 and 20-siteclusters investigated numerically. (c) and (d) Exact g ( ρ ) versus ρ . The circles, diamonds and triangles correspond V = 0 , 2 and 3, respectively. The boxes correspond to datapoints with total spin S > S min = 0( ) for even (odd)numbers of particles.with ρ equal to or close to 0.5 exhibit magnetic susceptibil-ity enhanced relative to the Pauli susceptibility, and latticeinstability with periodicity 4k F (where k F is the Fermiwavevector within one-electron theory), both accepted assignatures of strong correlations. In contrast, ρ between0.66 and 0.8 show weakly correlated behavior, viz., unen-hanced susceptibility and the usual 2k F Peierls instability(see Tables in reference [7]). The behavior in the CTS andthe cobaltates are then very similar.
We present here a theory of the ρ -dependentelectronic behavior, in 1D [7] and the 2D triangular lattice[8]. Consider the extended Hubbard Hamiltonian, H = − (cid:88) (cid:104) ij (cid:105) σ t ij c † iσ c jσ + U (cid:88) i n i, ↑ n i, ↓ + V (cid:88) (cid:104) ij (cid:105) n i n j (1)where c † iσ creates an electron or a hole with spin σ ( ↑ or ↓ )on site i , (cid:104) ... (cid:105) 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 ρ , 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 g ( ρ ) = (cid:104) n i, ↑ n i, ↓ (cid:105)(cid:104) n i, ↑ (cid:105)(cid:104) n i, ↓ (cid:105) (2) g ( ρ ) is clearly weakly ρ -dependent for V = 0 . For V > and ρ = 0 . , electrons are prevented from encounteringone another by V , thus reducing g ( ρ ) more than U alone. Copyright line will be provided by the publisher ss header will be provided by the publisher 3
Figure 3 (Color online) The effective -filled band natures of the supercon-ducting spinels, following the splittingof the t g orbitals (see text). The bluedotted arrows denote electron or holemotion. As ρ is increased from 0.5, more and more intersite repul-sions are generated with additions of electrons, and config-urations with a few double occupancies begin to competewith those with only singly occupied sites. Thus g ( ρ ) in-creases steeply with increase in ρ for significant V /U . As ρ increases beyond an intermediate range, the Mott-Hubbardlimit is approached and g ( ρ ) should decrease again.Exact numerical calculations of g ( ρ ) verify this conjec-ture. Our earlier 1D calculations were for relatively smallnumbers of electrons N e with varying number of sites N s [7]. Significantly larger computer capabilities now allowcalculations for different ρ with fixed N s = 16 . In Fig. 1we have shown the results of our calculations for U/ | t | =10 and V / | t | =0, 2 and 3. This value of U is considered real-istic for CTS, and there is evidence for V / | t | as large as 3.The plots are similar for U/ | t | = 6 − and V / | t | = 1 − .While CTS with ρ < . do not exist, we have includedthis smaller ρ for comparison to 2D. We conclude fromFig. 1 that for realistic V /U , strong ρ -dependent electronicbehavior is warranted. Experimentally [7], all ρ = 0 . CTS(e.g., MEM(TCNQ) , Qn(TCNQ) , (TMTTF) X, etc.) ex-hibit strongly correlated behavior, while CTS with ρ =0 . − . (TTFBr . , HMTTF- and HMTSF-TCNQ with ρ = 0 . , etc.) uniformly exhibit weakly correlated be-havior. Subsequent demonstrations of strongly correlatedbehavior in 2:1 BEDT-TTF [9] or 1:2 Pd(dmit) systems[10] further support the theory.In Fig. 2 we show our results for the finite periodic2D triangular lattices with N s = 16 (Fig. 2(a)) and 20(Fig. 2(b).) We have taken t > . Elsewhere we have ar-gued that the U/ | t | and V / | t | in the cobaltates are similarto that in the CTS [8]. With the exception of a few N e ,the ground state is always in the lowest spin state (0( ) for N e =even(odd)) for t > . As seen in Figs.2(c) and (d) thebehavior of g ( ρ ) in 2D is similar to that in 1D, and explainsthe much-discussed lack of symmetry about ρ = 0 . in thecobaltates - strongly correlated behavior near ρ = andweakly correlated behavior near ρ = [2]. We point out that SC is limited to the samenarrow range of carrier concentration in both families.Given the strong role of ρ , this cannot but be significant.CTS superconductors are 2:1 cationic or 1:2 anionic ma-terials [11], with 100% charge-transfer, and consequently ρ = 0 . . The lattice structures of the superconducting CTSare not quasi-1D but anisotropic triangular, giving bothfrustration and larger bandwidth. In the hydrated superconducting Na-cobaltate, x (cid:39) . , Careful experiments have revealed however that ρ isconsiderably smaller than 0.65, as a portion of the waterenters as H O + . A significant number of investigators havedetermined that ρ in the superconductor is exactly or closeto 0.5 [12]. It is also noteworthy that x = 0 . is unique, -Hall and Seebeck coefficients change signs only at this ρ atlow temperature in both Na x CoO and Li x CoO .The limitation of SC to ρ = 0 . acquires additionalsignificance when taken together with: (a) absence of SCin the ρ = 1 triangular lattice Hubbard Hamiltonian [13],(b) the propensity at ρ = 0 . to form the paired-electroncrystal (PEC), in which spin-singlet pairs are separatedby pairs of vacant sites, in both 1D and the moderatelyfrustrated anisotropic triangular lattice [14]. We have pro-posed that with further increase in frustration the PECgives way to a superconducting paired-electron liquid [14].The proposed theory of SC is essentially the same as thatof Schafroth, within which electron-pairs form mobilemolecules (charged bosons) [1]. It is, however, differentfrom the mean-field theory of charge-fluctuation mediatedSC [15]. In the rest of this paper we show that there existother systems which likely can be understood within thesame scenario. Spinels are inorganicternary compounds AB X , where the X − 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 B sub-lattice is tetrahedral, and hence frustrated.LiTi O , CuRh S and CuRh Se are the only threespinels that have been confirmed to be superconductors; re-ports of SC in CuV S and CoCo S exist. SC in LiTi O was discovered many years before the cuprates [16], andhas been of interest ever since because of its high T c /T F ( T F is Fermi temperature), which is closer to that inthe cuprates than conventional superconductors [17]. Themechanism of SC remains controversial: strong couplingBCS theory, the RVB approach [18] and the bipolarontheory [19] of SC have all been proposed.Examination of the common valence of the B-cationsin the superconducting spinels, 3.5+, reveals remarkablesimilarities between them and the CTS and cobaltates. TheTi . ions in LiTi O have one d -electron per two Ti.Rh . and Co . in their low-spin states possess one d -hole per two metal ions. V and V have electronconfigurations 3d and 3d , respectively. In all cases staticband Jahn-Teller lattice distortions will give -filled elec- Copyright line will be provided by the publisher
Mazumdar et al.: Correlated-electron superconductivity tron (in LiTi O ) and hole (in CuRh S , CoCo S andCuV S ) bands as indicated in Fig. 3. Static lattice distor-tions should give a 3D PEC, as indeed observed in CuIr S [20] and LiRh O [21]. We propose that the superconduct-ing state is a 3D paired-electron liquid, with dynamical (asopposed to static) orbital ordering. The C − ions in su-perconducting M C form fcc lattices. Theories of ful-lerides treat each C unit as a site, with electron hoppingsbetween the triply degenerate t u antibonding molecularorbitals. Although many experiments suggest onsite pair-ing mediated by H g Jahn-Teller phonons [22], the largeT c /T F suggests a non-BCS mechanism [17]. Antiferro-magnetism in bcc Cs C also indicates strong Coulombrepulsion [23,24]. Nonzero spin gap to the lowest energyhigh-spin state in the antiferromagnet has indicated thatit is the effective -filled band Mott-Hubbard insulator ofFig. 4(a), reached after Jahn-Teller distortion [23,24].Within a dynamic mean field theory (DMFT) of thepressure-induced antiferromagnetism-to-SC in Cs C [24], pairing arises from the combined effects of Hub-bard U and Jahn-Teller interaction within the effective -filled band of Fig. 4(a), at the interface of an insulatingand a metallic phase. We recall that the DMFT approach toSC in the CTS within the effective -filled band Hubbardmodel was very similar. It is now known that SC in thislatter case was an artifact of the mean-field approximation[13]. We believe that a similar criticism applies also to theexisting theory of SC 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 sug-gest that pressure induces a new orbital ordering, with C − configurations as shown in Fig. 4(b), where two degener-ate -filled bands bands are obtained. This orbital “reorder-ing” will be driven by the lower total energy of two -filledbands compared to that of the single effective -filled band,for larger bandwidth. Each of the -filled bands can nowform its own paired-electron liquid. Our goal here was to show that thecorrelated -filled band offers a common description ofseemingly very different classes of materials as well as aplausible mechanism of SC. The key step is the realiza-tion that antiferromagnetism in the semiconducting statedoes not necessarily mean that the electronic structure ofthe superconductor is derived from the same configuration.Bandwidth-driven transition to a different state that is moreappropriately described as -filled can occur in the super-conducting 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 of electron-phonon interactions, and yet large T c /T F , 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.
References [1] M. R. Schafroth, Phys. Rev. , 463 (1955).[2] Y. Wang et al. , Nature , 425 (2003). M. L. Foo et al. ,Phys. Rev. Lett. , 247001 (2004). T. Motohashi et al. , ibid, , 195128 (2011).[3] V. Brouet et al. , Phys. Rev. B, , 100403(R) (2007).[4] K. Takada et al. , Nature (London) , 53 (2003).[5] M. Z. Hasan et al. , Phys. Rev. Lett. , 246402 (2004).[6] T. Mizokawa, New J. Phys. , 169 (2004).[7] S. Mazumdar and A. N. Bloch, Phys. Rev. Lett. , 207(1983). S. Mazumdar and S. N. Dixit, Phys. Rev. B , 3683(1986).[8] H. Li, R. T. Clay and S. Mazumdar, Phys. Rev. Lett. ,216401 (2011).[9] K. Kanoda, J. Phys. Soc. Jpn. , 051007 (2006).[10] R. Kato, Chem. Rev. , 5319 (2004).[11] G. Saito, K. Yamaji and T. Ishiguro, Organic Superconduc-tors (Springer-Verlag, 1988).[12] P. W. Barnes et al. Phys. Rev. B , 134515 (2005). H. Saku-rai et al. , ibid , , 092502 (2006). M. Ba˜nobre-L´opez et al. ,J. Am. Chem. Soc. , 9632 (2009). T. Shimojima et al. ,Phys. Rev. Lett. , 267003 (2006).[13] R. T. Clay, H. Li and S. Mazumdar, Phys. Rev. Lett. ,166403 (2008).[14] R. T. Clay et al. , see present proceedings.[15] J. Merino and R. H. McKenzie, Phys. Rev. Lett. , 237002(2001).[16] D. C. Johnston, J. Low Temp. Phys. , 145 (1976).[17] W. D. Wu et al. , Hyperfine Interactions, , 615 (1994); Y.J. Uemura, Physica C , 733 (1991).[18] P. W. Anderson et al. Phys. Rev. Lett. , 2790 (1987).[19] A. Alexandrov and J. Ranninger, Phys. Rev. B , 1164(1981).[20] P. G. Radaelli et al. , Nature
155 (2002).[21] Y. Okamoto et al. , Phys. Rev. Lett. , 086404 (2008).[22] C. M. Varma, J. Zaanen and K. Raghavachari, Science .989 (1991).[23] Y Iwasa and T Takenobu, J. Phys. Cond. Matter, R495(2003).[24] M. Capone et al. , Rev. Mod. Phys. , 943 (2009). Copyright line will be provided by the publisher ss header will be provided by the publisher 5
Figure 4 (Color online) (a) The effective -filled t u band in antiferromagnetic C − , following Jahn-Tellerdistortion. (b) The proposed doubly degenerate -filledbands in the superconductor (see text).-filledbands in the superconductor (see text).