Percolative theories of strongly disordered ceramic high-temperature superconductors J. C. Phillips1 Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08854 Contributed by J. C. Phillips, November 18, 2009 (sent for review October 31, 2009)
percolation ∣ theory
A
ll known microscopic theories of metallic superconductors are based on the isotropic Bardeen, Cooper, and Schrieffer (BCS) model of Cooper pairs formed by attractive electronphonon interactions that overwhelm repulsive Coulomb interactions, yet many features of electron-phonon interactions in the BCS theory that are confirmed in metals appear to be weak or absent in the infrared spectra of ceramic layered high temperature superconductors (HTSC) (1). The ceramics often exhibit multiple phases and must be doped to be not only superconductive but even metallic. By contrast layered undoped MgB2 with T c ∼ 40 K is an “ideal” example of a metallic BCS crystalline superconductor where T c can be calculated quite accurately (2). When MgB2 is substitutionally doped with ∼10% Al (on the Mg sites) or C (on the B sites), T c decreases drastically to ∼10 K (3). Many authors have invoked residual spin interactions in the doped state, left over from the magnetic undoped states, as a source of this difference (1). However, it has been known for a long time that electron-spin interactions (for instance, with magnetic impurities) drastically suppress T c by breaking S ¼ 0 Cooper pairs (4), and more generally that the insulating antiferromagnetic (AF) and metallic Cooper pair four-particle plaquette channels are completely independent (5). While ceramics often exhibit multiple phases, recently the data have shifted overwhelmingly in favor of phonon exchange as the only attractive interaction correlated with T c and responsible for ceramic HTSC (6, 7). An Ockham’s razor, semiclassical alternative to abandoning electron-phonon interactions instead abandons translational invariance and the continuum approximation and supposes that currents are carried by discrete molecular wave packets that percolate coherently from dopant to dopant along filamentary www.pnas.org/cgi/doi/10.1073/pnas.0913002107
interlayer paths, if necessary avoiding entirely insulating or weakly metallic regions associated with AF or strong Jahn-Teller distortions that destroy large parts of Fermi lines. So far this “zigzag” interlayer percolative model (ZZIP) (8) has not been used to calculate either normal-state or superconductive properties from first principles. However, if one is familiar with the filamentary principles associated with self-organized glassy networks (9, 10), one can use homology arguments to describe material properties (such as T c ) very economically. In this way not only was a master function defining strict least upper bounds on T c established (11), but this master function also very successfully predicted T max c for the complex oxyhalide Ba2 Ca2 Cu3 O6 F2 (apical F) to be 78 K, in excellent agreement with the highest T c value for Ba2 Ca3 Cu4 O8 F2 reported later, 76 K (12). As shown in Fig. 1, T max c ðhRiÞ also gives (13) a very good account of the largest T c s in noncuprate ceramic HTSC, where hRi is the number of chemical bonds/atom. It is clear that the connectivity and stability of ionic host lattice networks are marginal for hRi ¼ 2, where the largest T c is reached. Topological network considerations have explained phase diagrams of superionic conductors (14). Percolative rehydroxylation is characteristic of ceramics and can be used to date them (15). Quantum Percolation and Anti-Hermitian Interactions In the ZZIP model the mechanism of quantum percolation was pictured semiclassically, in terms of discrete molecular wave packets that percolate coherently from dopant to dopant. Here we will outline a primitive Hamiltonian matrix model that could eventually be used for numerical quantum simulations. Because T c correlates well with shifts δωLO in the longitudinal optical phonon frequency near the (100) Brillouin zone boundary (11), one anticipates that the spectra of the localized dopant states are broadened by δE1 and their centers displaced from EF by an amount δE2 , with both δE1 and δE2 of order ωLO . Neglecting the δE1 broadening, these states can be represented by a two-state, two-dopant model Hamiltonian with H11 ¼ 1 and H22 ¼ −1 (in units of δE2 ). The critical off-diagonal matrix elements actually reflect dopant-dopant interactions mediated by filamentary wave packets propagating in the metallic planes (CuO2 planes for the cuprates). The key feature of those interactions has been identified in very extensive studies of electron pair interactions in small molecules. Many-electron Coulomb interactions reduce Fermion bandwidths, for instance, by screening ion-single electron interactions, and many methods have been proposed for treating these screening effects formally (16). However, classical percolation is already a complex process, and to describe it simply one can turn to recent studies of two-electron excited states of many-electron atoms and molecules, without doing configuration interaction, by using an anti-Hermitian contracted Schrodinger equation (ACSE) to study two-body charge density matrices (17). The ACSE is an optimized reduction of the enormously complex Author contributions: J.C.P. designed research, performed research, and wrote the paper. The author declares no conflict of interest. 1
To whom correspondence should be addressed. E-mail:
[email protected].
PNAS ∣ January 26, 2010 ∣ vol. 107 ∣ no. 4 ∣ 1307–1310
PHYSICS
Optimally doped ceramic superconductors (cuprates, pnictides, etc.) exhibit transition temperatures T c much larger than strongly coupled metallic superconductors like Pb (T c ¼ 7.2 K, E g ∕kT c ¼ 4.5) and exhibit many universal features that appear to contradict the Bardeen, Cooper, and Schrieffer theory of superconductivity based on attractive electron-phonon pairing interactions. These complex materials are strongly disordered and contain several competing nanophases that cannot be described effectively by parameterized Hamiltonian models, yet their phase diagrams also exhibit many universal features in both the normal and superconductive states. Here we review the rapidly growing body of experimental results that suggest that these anomalously universal features are the result of marginal stabilities of the ceramic electronic and lattice structures. These dual marginal stabilities favor both electronic percolation of a dopant network and rigidity percolation of the deformed lattice network. This “double percolation” model has previously explained many features of the normal-state transport properties of these materials and is the only theory that has successfully predicted strict lowest upper bounds for T c in the cuprate and pnictide families. Here it is extended to include Coulomb correlations and percolative band narrowing, as well as an angular energy gap equation, which rationalizes angularly averaged gap∕T c ratios, and shows that these are similar to those of conventional strongly coupled superconductors.
Fig. 1. Master curve for ceramic HTSC, with T max plotted as a function of hRi c (valence averaged over all atoms) (11). There are two points for the oxyhalide BCCOF, a smaller early value (measured before the master curve was published) and a larger value (measured after the master curve was published). The larger value fits the predicted value almost exactly. Excellent agreement with the predicted values is also found for many other noncuprate ceramics, including the currently fashionable FeAs superfamily (13).
problem of quantum configuration interaction. In principle the two-body charge density matrices are related to the three- and four-body charge density matrices, which in turn are related to still higher-order charge density matrices. Moreover, because interference effects are not included, classical correlation functions in general do not correspond to actual Fermionic wave function solutions of the Schrodinger equation itself. Nevertheless, it has been found in many examples that restricting the higher order corrections to the leading anti-Hermitian third-order term yields excellent results, at least an order of magnitude better than obtained from the full (Hermitian þ anti-Hermitian) third-order term (17). Classically speaking, the anti-Hermitian interaction corresponds to damping of the pair interaction by a third particle; apparently this damping erases most of the higher order Hermitian interactions, which is a great simplification. Returning to our two-dopant Hamiltonian, we parameterize H12 and H21 as A þ Bi, where A is the Hermitian part and Bi is the anti-Hermitian part (still in units of δE2 ). According to Mazziotti (17), A ≪ B, and we set A to zero. Now the eigenvalues of our two-state model are given by E2 ¼ 1 − B2 , and the bandnarrowing effect is obvious without further calculation for B < 1. For B > 1, the localized dopant states become unstable, and one can say that a percolative metal-insulator transition occurs at B ¼ 1. The main features of HTSC phase diagrams are evident from this simple two-state model. For B ≪ 1, low dopant density x and δE1 disorder have produced an insulating dopant impurity band. For moderate x, dopant-dopant spacing is reduced, the filamentary Coulomb fluctuations become significant, and 0 < 1− BðxÞ ≪ 1. The broad impurity density of states peak at EF narrows, and when it becomes narrow enough, Cooper pairs can form in this band, leading to HTSC. For BðxÞ > 1, a normal metallic state arises, filamentary/dopant localization ceases, and the impurity band peak in NðEÞ for E near EF is lost. Percolative Angular Gap Equation The foregoing model sketches dopant percolation in an isotropic context. In actuality, the dopants are embedded in a crystalline host, where there is present at higher temperatures a strongly anisotropic d-wave pseudogap associated with nearest neighbor Cu-O interactions in the CuO2 plane. This pseudogap stabilizes the host lattice, and it itself is the result of rigidity percolation. The latter is the origin of the phenomenologically successful model of T max c ðhRiÞ shown in Fig. 1. Electronic charges must percolate through this marginally stable pseudogap maze. 1308 ∣
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In a double charge/rigidity percolation model there are two global factors that should be considered: the planar anisotropy of the energy gap Eg ðφÞ and the transition probability PðφÞ that the gap will percolate at a given azimuthal angle φ. Quantum percolation (8) must occur coherently on each percolative filament, so for quantum percolation one should regard PðφÞ as a probability amplitude, with characteristic analytic properties. Currents will be carried on each filament by wave packets composed of Bloch waves. The wave packets will have average energies E and average velocities v. In ARPES experiments these wave packets are projected on the momenta k of high energy photoemitted electrons, and the percolative supercurrents are separated from insulating pseudogap states (probably associated with Jahn-Teller distortions) by defining Fermi arcs. When this projective k-space method of analysis is used, it is often difficult to separate the pseudogap and the superconductive gap states, except in underdoped samples where the latter is much smaller than the former (18, 19). [At optimal doping, as in (7), the pseudogap and the superconductive gap appear to be nearly the same.] However, even after Eg ðφÞ has been assigned a d-wave angular dependence Eg ðφÞ ¼ Emax cos 2φ, to establish a quantum g percolation model one must still determine the probability amplitude PðφÞ that the gap will percolate phase-coherently at a given angle φ. The function PðφÞ can be regarded heuristically as akin to an interdopant transfer coupling, similar to the transfer couplings Tused in phenomenological descriptions of the Josephson effect. Here PðφÞ is assigned a simple functional dependence for all cuprates, the basic idea being that metallic and superconductive wave packets will percolate most effectively if they avoid the φ ¼ 0 Cu-Cu nearest neighbor f10g directions where the pseudogap Ep ðφÞ and superconductive gap Eg ðφÞ are strongest (20). These effective percolation directions are the φ ¼ π∕4 next nearest neighbor f11g directions where electron-phonon interactions are weakest, thus PðφÞ should be complementary to Eg ðφÞ, for example, PðφÞ ¼ sinn 2φ, where n should be optimal. The model is tested by studying the detailed electronic structure of Fermi-energy dopant resonances (type B, predicted in Sec. IX of ref. 11) as revealed by recent large-scale scanning tunneling microscope (STM) experiments (21). As expected,
Fig. 2. Images of a Cu site dopant-centered superconductive wave packet from ref. 21, qualitatively showing anisotropic percolation favoring f11g second nearest neighbor (2NN) over decaying nearest neighbor (NN) f10g sites, as described quantitatively by the probability amplitude PðφÞ in the text. Notice that most of the sample sites far from the dopant are insulating. This means that to thread these regions, interlayer percolation is necessary.
Phillips
Footprints of ZZIP The doubly percolative, dopant-centered ZZIP network has left many large footprints (12). Thus extended x-ray absorption fine structure (EXAFS) studies of lattice distortions (26) have revealed a third sub-T c phase transition (in addition to the Jahn-Teller transition at T and the superconductive transition at T c ) shown in Fig. 3. This transition at T ¼ T OP can be explained as a glass transition of the ZZIP network (27); it is especially interesting because, unlike the other two transitions, it cannot be modeled with a Landau order parameter, and it is completely inexplicable with mean-field toy model Hamiltonians (Hubbard and t-J models). (It could, however, be compared to formation of Cys-Cys disulfide bonds in proteins.) The planar resistivity ϱab ðTÞ is most linear in T at and near optimal doping (12, 28). The dopant network-forming interactions, although individually weak, are cumulatively very strong because the dielectric energy gained by screening internal electric Phillips
Fig. 3. EXAFS data (26) showing a sub-T c phase transition in planar lattice disorder at T ¼ T OP . The transition is interpreted in (27) as lattice recoil ¼ induced by network superconductive filamentary contacts. Here T onset c 38 K and T ∼ 100 K.
fields increases as the network conductivity increases due to dopant self-organization (12), and this explains the maximization of the T-linearity of ϱab ðTÞ near optimal doping (12, 28). The superconductive transition itself can be narrowed by annealing (29). The classic zero-field work of Ando et al. (28) has recently been extended to high magnetic fields with spectacular results (30). Ando et al. (28) found that there is a T > 100 K regime of ϱab ðx; TÞ that is linear in T at x ¼ xc ¼ 0.16 in zero field, which coincides with T c ðxÞ ¼ T max c , which also occurs in La2−x Srx CuO4 at x ¼ xc . Cooper et al. (30) found the much simpler La2−x Srx CuO4 pattern shown in Fig. 4, where the linear regime has been shifted to x ¼ x0c ¼ 0.185ð5Þ. This high-field linear regime extends at fixed x vertically all the way from T ¼ 0 right up to the highest Ts studied (200 K). The vertical linear T regime is shaped like an hourglass, wide at T ¼ 0 and T ¼ 200 K, and narrow at T ∼ 100 K. This narrowing is due to the formation of pseudogap (Jahn-Teller distorted) islands, which first percolate at T ∼ 100 K and then stabilize between T and T c (see Fig. 3). The low temperature linearity requires a reformation of the nanodomain network into a doubly percolative network, with two kinds of dopants (11).
PHYSICS
the resonant amplitude of the superconductive gap is greatly reduced in the φ ¼ 0 nearest neighbor f10g directions where the pseudogap Ep ðφÞ is strongest. However, only a modest reduction is found in the φ ¼ π∕4 next nearest neighbor f11g directions where electron-phonon interactions are weakest, which is exactly what one expects from quantum percolation of superconductive wave packets (see Fig. 2). Although this experiment does not display either Ep;g ðφÞ or PðφÞ fully, it does confirm qualitatively their complementary character. The gap equation is obtained by coherently averaging over φ to obtain Eg ¼ I 2 ∕I 1 , where I 2 ¼ ∫ dφEg ðφÞPðφÞ and I 1 ¼ ∫ dφPðφÞ with 0 ≤ φ ≤ π∕4. We find Eg ¼ Emax g ∕ðn þ 1ÞI 1 . Unambiguous are not obtainable from optical data values of Eg and Emax g (1), which primarily show a polaronic midinfrared data band (22). These ceramic midinfrared data merely reinforce the large qualitative difference between layered ceramics (La2−x Srx CuO4 ) and cubic Ba1−x K x BiO3 known from their phase diagrams. The former has a parabolic (nearly semicircular) T c ðxÞ, while the latter has a triangular T c ðxÞ, with a maximum T c at the metalinsulator transition and a normal BCS-type infrared spectrum with Eg ∕kT c ¼ 3.2 (23), and a metal-insulator transition at x ¼ 0.3 in Ba1−x Kx BiO3 , which has been interpreted in terms of bipolaron formation (24). Because of the self-organized internal structure, dopant configurations can adapt to thermal fluctuations, and the dissipative scattering (25) characteristic of large gap∕T c ratios (Pb, T c ¼ 7.2 K, Eg ∕kT c ¼ 4.5) is absent from Ba1−x K x BiO3 (Eg ∕kT c ¼ 3.2, weak coupling). However, in a ceramic HTSC, Boyer et al. (19) found Emax g ∕kT c ¼ 10.6 2.9, or Eg ∕kT c ¼ ð10.6 2.9Þ∕ 3I1 ¼ 4.4 1.3 if one chooses n ¼ 2. The fit with n ¼ 2 can be justified topologically: Suppose each time a wave packet leaves an interlayer dopant to percolate in a metallic plane, it adds a sin 2φ factor to PðφÞ. Then to “reset” itself in an equivalent state, it will have to return to another (statistically equivalent) interlayer dopant, serially experiencing a second factor sin 2φ. Thus n ¼ 2 in PðφÞ could be an intrinsic feature of interlayer effects in the ZZIP model. This value of Eg ∕kT c ¼ 4.4 is similar to that of Pb. It appears that the layered structure of ceramic HTSC has led to an entanglement of dopant and Jahn-Teller distortions that is much more complex than is apparent from the data on cubic Ba1−x K x BiO3 . This entanglement is imaged in the STM study (Fig. 2) of a rare layered ceramic Fermi-energy pinning resonance (21), which is probably an apical oxygen vacancy. This resonance is stable and observable by STM, whereas most Fermi-energy pinning interstitial O dopant resonances are too fragile to be imaged with an meV potential. It may be that the spatial and spectral entanglement of dopant-centered superconductive gaps with pseudogaps in ceramic superconductors often renders them intrinsically inseparable.
Fig. 4. The exponent n in ϱab ðTÞ ∼ T n in overdoped (p > pc ) La2−x Srx CuO4 at high magnetic fields (normal state), from ref. 30. The hour-glass shape of the quasi-linear region n ∼ 1.0 (Red) cannot be explained by a quantum critical point at T ¼ 0 or by separation of the Fermi line in momentum space into two components, but it is explained by the self-organized formation of the ZZIP network, which is partially disrupted by a Jahn-Teller deformation near T ∼ 100 K (see Fig. 3). Notice that at the largest hole doping p > 0.32 the Landau limit n ¼ 2 is reached. PNAS ∣
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Conclusions We have sketched a two-dopant model Hamiltonian that shows that the onset of Fermionic charge percolation can be represented by an anti-Hermitian dopant-dopant transfer interaction. By establishing an anisotropic percolative gap equation and identifying a new sub-gap glassy phase transition, ZZIP builds a bridge between theories of isotropic metallic and anisotropic ceramic superconductivity. Percolation shows up even in the cubic perovskites SrTiO3 (31) and SrRuO3 (32). Although SrTiO3 (STO) is not a high temperature superconductor, it is a ceramic that exhibits superconductivity with T max ∼ 0.5 K at very low carc rier densities n3D ∼ ð1018 –1020 Þ∕cm3 , in both chemically doped bulk and electric-field induced surface accumulation layers (31). The coincidence of the chemically doped and electric-field
induced surface T max values could be accidental, but there is c a natural percolative explanation for this coincidence: T max is c reached when the percolative filaments begin to merge, just as in the EXAFS data for HTSC (26, 27). There is also evidence for percolative conductivity at high temperatures in SrRuO3 (SRO): According to Herranz et al., “On the other hand, at temperatures above the ferromagnetic transition (150 K), the temperature dependence is linear, but remarkably it is found that the slope of ρðTÞ increases as disorder increases. This latter result is radically dissimilar to what is observed in other complex electronic systems, where disorder commonly induces a reduction of the slope” (32). Because the percolative network is fragile, it can be disrupted thermally, and its thermal fragility will increase with increasing disorder, as measured by the resistivity minimum temperature. Overall it is difficult to see how a successful theory of HTSC can avoid recognizing its percolative nature. In fact, when proper curvilinear cuprate percolative basis states are forcibly projected onto Bloch plane wave states, it becomes difficult to separate the pseudogap from the superconductive gap. This causes Fouriertransformed scanning tunneling evidence for the superconductive transition to weaken or disappear, but with one important exception. This is the difference between the slopes of the temperature dependences of Fourier-derived q scattering from k regions to k þ q regions that have the same or opposite signs of the d-wave order parameter (Fig. 4 of ref. 33). This is the Fouriertransformed qualitative analogue of the complementary angular behavior (21) of Ep;g ðφðkÞÞ and PðφðqÞÞ discussed above analytically in connection with the percolative gap equation.
1. Basov DN, Timusk T (2005) Electrodynamics of high-T-c superconductors. Rev Mod Phys, 77:721–729. 2. Liu AY, Mazin II, Kortus J (2001) Beyond Eliashberg superconductivity in MgB2 : Anharmonicity, two-phonon scattering, and multiple gaps. Phys Rev Lett, 87:087005. 3. Choi HJ, Louie SG, Cohen ML (2009) Anisotropic Eliashberg theory for superconductivity in compressed and doped MgB2 . Phys Rev B, 79:094518. 4. Phillips JC (1989) Physics of High-Tc Superconductors (Academic, Boston). 5. Dzyaloshinskii IE, Larkin AI (1972) Possible states of quasi-unidimensional systems. Sov Phys JETP-USSR, 34:422–427. 6. Lampakis D, Liarokapis E, Panagopoulos C (2006) Micro-Raman evidence for topological charge order across the superconducting dome of La2−x Srx CuO4 . Phys Rev B, 73:174518. 7. Shim H, Chaudhari P, Logvenov G, Bozovic I (2008) Electron-phonon interactions in superconducting La1.84 Sr0.16 CuO4 . Phys Rev Lett, 101:247004. 8. Phillips JC (1989) Quantum percolation and lattice instabilities in high-T cuprate superconductors. Phys Rev B, 40:8774–8779. 9. Boolchand P, Lucovsky G, Phillips JC, Thorpe MF (2005) Self-organization and the physics of glassy networks. Phil Mag Lett, 85:3823–3838. 10. Pan Y, Inam F, Zhang M, Drabold DA (2008) Atomistic origin of Urbach tails in amorphous silicon. Phys Rev Lett, 100:206403. 11. Phillips JC (2007) Self-organized networks and lattice effects in high-temperature superconductors. Phys Rev B, 75:214503. 12. Shimizu S, , Mukuda H, Kitaoka Yet al. (2008) Self-doped superconductivity in tri-layered Ba2Ca2Cu3O6F2: A Cu-63-NMR study. Phys Biol, 403:1041–1043. 13. Phillips JC (2009) High temperature cuprate-like superconductivity. Chem Phys Lett, 473:274–278. 14. Novita DI, Boolchand P, Malki M, Micoulaut M (2009) Elastic flexibility, fast-ion conduction, boson and floppy modes in AgPO3-AgI glasses. J Phys Cond Mat, 21:205106. 15. Wilson MA, et al. (2009) Dating fired-clay ceramics using long-term power law rehydroxylation kinetics. Proc R Soc A, 465:2407–2415. 16. Fulde P, Unger P, Igarashi JI (1997) Spectral densities of infinite systems with strong electron correlations. Int J Quantum Chem, 63:645–653. 17. Mazziotti DA (2006) Anti-Hermitian contracted Schrödinger equation: Direct determination of the two-electron reduced density matrices of many-electron molecules. Phys Rev Lett, 97:143002. 18. McElroy K, et al. (2005) Atomic-scale sources and mechanism of nanoscale electronic disorder in Bi2 Sr2 CaCu2 O8þδ . Science, 309:1048–1052.
19. Boyer MC, , Wise WD, Chatterjee Ket al. (2007) Imaging the two gaps of the hightemperature superconductor Bi2 Sr2 CuO6þx. Nat Phys, 3:802–806. 20. Alexandrov AS (2008) Unconventional pairing symmetry of layered superconductors caused by acoustic phonons. Phys Rev B, 77:094502. 21. Chatterjee K, , Boyer MC, Wise WDet al. (2008) Visualization of the interplay between high-temperature superconductivity, the pseudogap and impurity resonances. Nat Phys, 4:108–111. 22. Cojocaru S, Citro R, Marinaro M (2007) Incoherent midinfrared charge excitation and the high-energy anomaly in the photoemission spectra of cuprates. Phys Rev B, 75:220502. 23. Jung CU, , Kong JH, Park BHet al. (1999) Far-infrared transmission studies on a superconducting BaPb1-xBixO3 thin film: Effects of a carrier scattering rate. Phys Rev B, 59:8869–8874. 24. Nishio T, Ahmad J, Uwe H (2005) Spectroscopic observation of bipolaronic point defects in Ba1−x Kx BiO3 . Phys Rev Lett, 95:176403. 25. Wada Y (1964) The effect of quasiparticle damping on the ratio between the energy gap and the transition temperature of lead. Rev Mod Phys, 36:253–257. 26. Zhang CJ, Oyanagi H Local lattice instability and superconductivity in La1.85 Sr0.15 Cu1−x Mx O4 (M ¼ Mn, Ni, and Co). Phys Rev B, 79:064521. 27. Phillips JC (2009) Universal non-Landau, self-organized, lattice disordering percolative dopant network sub-Tc phase transitions in ceramic superconductors. Proc Natl Acad Sci USA, 106:15534–15537. 28. Ando Y, , Komiya S, Segawa Ket al. (2004) Electronic phase diagram of high-T-c cuprate superconductors from a mapping of the in-plane resistivity curvature. Phys Rev Lett, 93:267001. 29. Jung J, Abdelhadi MM (2003) Electrical transport and oxygen disorder in YBCO. Int J Mod Phys, 17:3465–3469. 30. Cooper RA, , Wang Y, Vignolle Bet al. (2009) Anomalous criticality in the electrical resistivity of La2−x Srx CuO4 . Science, 323:603–607. 31. Ueno K, , Nakamura S, Shimotani Het al. (2008) Electric-field-induced superconductivity in an insulator. Nat Mater, 7:855–858. 32. Herranz G, , Laukhin V, Sanchez Fet al. (2008) Effect of disorder on the temperature dependence of the resistivity of SrRuO3 . Phys Rev B, 77:165114. 33. Lee J, , Fujita K, Schmidt ARet al. (2009) Spectroscopic fingerprint of phase-incoherent superconductivity in the underdoped Bi2 Sr2 CaCu2 O8þδ . Science, 325:1099–1103.
A combined description of refs. 28 and 30 follows from thinking of the hard-wired ZZIP network as a combination of irrotational filamentary open and solenoidal vortex closed components (11). In zero field (26) the competition between these two topological components is relatively simple (independent of x) for T > T ¼ 100 K, but below 100 K it becomes more complex as pseudogap and normal metallic nanodomain walls compete in partitioning CuO2 planes. Application of a high magnetic field in overdoped samples (x > xc ) leaves only the solenoidal vortex closed components, which absorb some dopant electronic kinetic energy in purely static vortex motion. This orbital reorganization effect both shifts xc from 0.16 in the zero-field case to x0c ¼ 0.185ð5Þ in high fields and is also responsible for fixing the T linear behavior of sliding vortices at this x ¼ 0.185ð5Þ value up to 200 K, giving the striking hourglass structure shown in Fig. 4.
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Phillips
