Decoding Spatial Complexity in Strongly Correlated Electronic Systems
aa r X i v : . [ c ond - m a t . s t r- e l ] O c t Decoding Spatial Complexity in Strongly Correlated Electronic Systems
E. W. Carlson, S. Liu, B. Phillabaum, and K. A. Dahmen Department of Physics, Purdue University, West Lafayette, IN 47907, USA Department of Physics, University of Illinois, Urbana-Champaign, IL, 50000, USA (Dated: August 17, 2018)Inside the metals, semiconductors, and magnets of our everyday experience, electrons are uni-formly distributed throughout the material. By contrast, electrons often form clumpy patternsinside of strongly correlated electronic systems (SCES) such as colossal magnetoresistance materialsand high temperature superconductors. In copper-oxide based high temperature superconductors,scanning tunneling microscopy (STM) has detected an electron nematic on the surface of the mate-rial, in which the electrons form nanoscale structures which break the rotational symmetry of thehost crystal. These structures may hold the key to unlocking the mystery of high temperature su-perconductivity in these materials, but only if the nematic also exists throughout the entire bulk ofthe material. Using newly developed methods for decoding these surface structures, we find that thenematic indeed persists throughout the bulk of the material. We furthermore find that the intricatepattern formation is set by a delicate balance among disorder, interactions, and material anisotropy,leading to a fractal nature of the cluster pattern. The methods we have developed can be extendedto many other surface probes and materials, enabling surface probes to determine whether surfacestructures are confined only to the surface, or whether they extend throughout the material.
There is growing experimental evidence that manystrongly correlated electronic systems such as nickelates,cuprates, and manganites exhibit some degree of localinhomogeneity,[1–7] i.e. , nanoscale variations in the lo-cal electronic properties. Describing the electronic be-havior of these materials involves several degrees of free-dom, including orbital, spin, charge, and lattice degreesof freedom. Disorder only compounds the problem. Notonly can disorder destroy phase transitions, leaving merecrossovers in the wake, it can fundamentally alter groundstates, sometimes forbidding long range order. Espe-cially in systems where different physical tendencies com-pete, disorder can act as nucleation points for competingground states.One approach to disentangling disorder from the fun-damental correlations induced by strong electron interac-tions is to put resources toward developing cleaner sam-ples. While this approach is laudable and has led tomany key insights and advances in strongly correlatedelectronic systems, it is also labor intensive and expen-sive. Even the cleanest sample, when stored over time atfinite temperature, will acquire a thermodynamically re-quired concentration of defects. Moreover, in some sense,disorder is intrinsic to the correlated phases, since inmost systems the phases of interest happen upon chem-ical doping, which necessarily introduces disorder. Es-pecially in cuprates, this drive toward cleaner samplesor even toward controlling disorder in order to under-stand the intrinsic electronic states may not be neces-sary, since even “dirty” samples that have not undergonestrict preparation protocols still exhibit the salient fea-ture of superconductivity.[8] Indeed, in any high temper-ature superconductor, because the pairing scale must alsobe high, an understanding of the short-distance physics( i.e. within a few coherence lengths of the superconduc- tivity) should be sufficient to understand the origin ofpairing.[9] In this sense, long range order of a proposedpseudogap phase is neither necessary to produce super-conductivity nor is it necessary in order to understandthe superconductivity.Ultimately, the interplay between many degrees of free-dom, strong correlations, and disorder can lead to a hi-erarchy of length scales over which the resulting physicsmust be described.[1] While such electronic systems arehighly susceptible to pattern formation at the nanoscale,unfortunately most of our theoretical and experimentaltools are designed for understanding and detecting ho-mogeneous phases of matter. Therefore, there is a criti-cal need to design and develop new ways of understand-ing, detecting, and characterizing electronic pattern for-mation in strongly correlated electronic systems at thenanoscale, especially in the presence of severe disordereffects. Such theoretical guidance will enable more directcontact between theory and experiment in a number ofmaterials, and provide a path forward for understanding“disputed” regions of phase diagrams of strongly corre-lated materials.In this paper, we focus on detecting electron nematicsand other electronic phases which break the rotationalsymmetry of the host crystal. Such phases have been pro-posed and/or observed in a variety of materials and con-texts, including Sr Ru O [12], GaAs/Al x Ga − x As het-erostructures in field[13, 14], and a subset of cupratesuperconductors[11, 15–20] such as YBa Cu O x [16–18], and Bi Sr CaCu O x [11, 19, 20], as well as the ironarsenic based superconductor Ca(Fe − x Co x ) As [21].The state has been proposed to exist in many more sys-tems, such as AlAs heterostructures, the Si(111) sur-face, elemental bismuth, and both single layer and bilayergraphene.[15, 22–24] FIG. 1. Mapping of STM data to Ising nematic variables.(a) Masked image[10, 11] of R-map of Dy-Bi2212 showingthe regions of the R-map with vertically aligned nematic do-mains. (b) The complement to panel (a), showing horizontallyaligned nematic domains. (c) Mapping to the correspondingIsing geometric clusters, showing several small clusters (cir-cled in green); a smaller number of medium-sized clusters(representative clusters circled in white); and a single (or-ange) cluster which spans the entire field of view.
We have proposed three approaches which, rather thanshying away from disorder, use disorder to advantagein order to detect and characterize mesoscale and multiscale order in electronic systems (such as electron ne-matics) which break the rotational symmetry of the hostcrystal: (1) Extracting critical exponents from observedmulti scale pattern formation in image data via clus-ter analyses.[7, 10]; (2) Manifestations of nonequilib-rium behavior such as hysteresis[25, 26]; and (3) Noisecharacteristics[27, 28].Method random field type (see Eqn. 1 below), hystere-sis is a prominent and robust feature, which means thathysteresis can be a good diagnostic tool for order pa- rameters which couple to material disorder via a randomfield mechanism.[28] In this case, we have proposed us-ing hysteresis in order to detect disordered electron ne-matics, even ones which never fully order but exhibitonly local nematic order. The key insight is to fieldcool in an orienting field (such as uniaxial pressure),and measure any macroscopic response function whichis sensitive to nematic order (such as anisotropic resis-tivity). Through specific field cooling and orientationalfield switching protocols as described in Ref. [25], thepresence of a disordered electron nematic can be revealedexperimentally.[26] Method D of a particular size S is power-law distributed in this image, D ( S ) ∝ S − τ ,with a power set by the Fisher exponent τ . In addi-tion, the fractal geometric structure of the clusters canbe quantified as the hull fractal dimension, d h , and thevolume (interior) fractal dimension, d v of clusters. Bystudying the orientational analogue of the spin-spin cor-relation function, the anomalous dimension η || can alsobe extracted from the image.Relating these critical exponents to a particular fixedpoint requires a model. Near a critical point, the correla-tion length grows to become the dominant length scale,and it is possible to map the real physical system to acoarse-grained model with the same universal features.Starting from the cluster map in Fig. 1(c), we assign Isingvariable σ = 1 to vertical domains, and σ = − H = − X h ij i k J || σ i σ j − X h ij i ⊥ J ⊥ σ i σ j − X i h i σ i , (1)where the sum runs over the coarse-grained regions (Isingsites) consisting of a cubic lattice, chosen with spacingcomparable to the resolution of the image(s) to be stud-ied. The tendency for neighboring regions to be of likecharacter is modeled as a nearest neighbor ferromagneticinteraction J >
0. The layered structure of the materialis captured by the in-plane coupling J || being larger thanthe coupling between planes J ⊥ . Ultimately, the critical-ity of such a quasi-two-dimensional system is controlledby a three dimensional fixed point for any finite J ⊥ . How-ever, in a strongly layered system such as the cuprateswhere J ⊥ << J || , it is possible to observe a drift fromtwo dimensional to three dimensional exponents whenobserving a finite field of view.[30]There are six critical fixed points which can arisefrom the model of Eqn. 1: In the limit of zero disor-der strength, the phase transition from disordered tolong-range ordered nematic is controlled by the two-dimensional clean Ising model (C-2D) if J ⊥ = 0, or bythe three-dimensional clean Ising model (C-3D) for anynonzero coupling between planes. Random field disorderis relevant, and so the presence of any finite amount ofrandom field disorder ∆ shifts the universality class to ei-ther the two-dimensional random field Ising model (RF-2D) or the three-dimensional random field Ising model(RF-3D).[10] Note that quenched material disorder canalso give rise to randomness in the coupling strengths J ⊥ and J || , also known as random bond disorder. However,in the presence of both random bond and random fielddisorder, the critical behavior is always controlled by therandom field fixed point. For completeness, we also con-sider the possibility that the observed local orientationsare not arising from an interacting model, which corre-sponds to the percolation fixed points which occur at theinfinite temperature limit of Eqn. 1 as a function of ap-plied orienting field. These are the two-dimensional andthree-dimensional uncorrelated percolation points, P-2Dand P-3D, respectively.As shown in Fig. 1(c), there is one large spanning clus-ter, and there are several medium-sized clusters, and evenmore small clusters. By counting the number of clusters FIG. 2. An electron nematic breaks the rotational symme-try of the host crystal, in this case from C4 to C2 symmetry.The electron nematic then aligns either “vertically” or “hor-izontally.” We assign Ising variables σ = − σ = +1,respectively.FIG. 3. Scale-free behavior of electron nematic clusters inDy-Bi2212. (a) For each cluster size S (defined as the numberof sites S in the cluster), the average perimeter is plotted.Scaling is evident throughout the entire field of view. Greenline is a linear fit, yielding the ratio of fractal dimensions asdescribed in the text. (b) The probability p that two spinsa distance r apart are aligned. The blue line is a linear fit,as described in the text. In both panels, logarithmic binninghas been used, which is a standard technique for power lawanalysis.[31] D of each size S (where S is the number of Ising sites ineach cluster), one can construct the cluster size distribu-tion D ( S ). Near a critical point, this quantity exhibitspower law scaling, as D ( S ) ∝ S − τ . However, it is knownthat near criticality, the scaling function which forms theprefactor for the power law has a pronounced bump[32],at least for the 3D random field fixed point. Therefore, afinite-size field of view is expected to underestimate thetrue value of τ . In future experiments, larger fields ofview can mitigate this effect. Within the field of viewavailable, we find that a straightforward power law fityields τ = 1 . ± . P of clusters of each size S . As withthe cluster size distribution, a robust power law emergesthroughout the entire field of view. By comparing theperimeter and cluster sizes, the ratio of fractal dimen-sions P ∝ S d ∗ h /d ∗ v can be extracted, where d h and d v de-note the hull and volume fractal dimensions, respectively.(Here, the asterisk denotes the fact that only a 2D sliceof the clusters is experimentally accessible, and there- fore a corresponding geometric factor must be applied be-fore comparing directly with 3D models.[10]) The ratio offractal dimensions thus obtained is d ∗ h /d ∗ v = 0 . ± . . In Fig. 3(b), we plot the probability p ( r ) that twopseudospins σ = ± r apart are aligned.This is linearly related to the spin-spin correlation func-tion g ( r ) ∝ p ( r ). (In the physical system, the spin-spin correlation function corresponds to the orientation-orientation correlation function of the director of the elec-tron nematic.) The spin-spin correlation function be-comes power law near a critical point, g ( r ) ∝ /r d − η || .Here, we denote by η || the anomalous dimension η atthe surface of a material. As can be seen in the figure,this function is at most weakly power law in the data,with less than a decade of scaling. As such, this is theleast reliable critical exponent extracted from the clusteranalysis, yielding a value d − η || = 0 . ± . Exponent ↓ ; Model → C-2D C-3D P-2D P-3D RF-2D RF-3D Dy-Bi2212 τ . ± .
03 1 . ± . d ∗ h /d ∗ v . ± . d − η || . ± . Table I shows a comparison between the critical expo-nents derived from the observed unidirectional electronicclusters in Dy-Bi2212 and theoretical values from criticalfixed points of Eqn. 1. Note that for layered clean andrandom field Ising models, it is possible to observe a driftfrom 2D to 3D exponents in going from smaller to largerfields of view.[30] However, no such dimensional crossovermakes sense when considering uncorrelated percolation.We now compare the data-derived exponents againsttheoretical models. Note that the value of τ from thedata is lower than the theoretical value of every fixedpoint. This is expected for a finite field of view in ran-dom field models, where it is known that the cluster sizedistribution D ( S ) has a pronounced scaling bump.[32]Note also that there is not much variation in the theo-retical values of τ among the fixed points, so that whilethe presence of a robust power law in D ( S ) in the data issignificant, it is difficult in principle to determine which fixed point could be responsible for the scale-free behav- ior via this exponent.By contrast, the anomalous exponent d − η || showsa wide variation among fixed points, and can in principlebe a good value to distinguish among fixed points. Un-fortunately, the data-derived value has rather large errorbars due to a limited regime of scaling, and so yields littleinformation in this case.Solid information can be gleaned by comparing the ef-fective ratio of fractal dimensions, d ∗ h /d ∗ v . For 2Dmodels, this corresponds to the bulk fractal dimensions, d ∗ h /d ∗ v = d h /d v . For 3D models, the bulk fractal di-mensions differ from those observed on a 2D slice viageometrical factors, so that d ∗ h /d ∗ v = 3 d h / (4 d v ).[10]This ratio shows distinguishable variation among fixedpoints, and the data-derived value has small error barsand exhibits decades of scaling. All of this means thatthis exponent ratio is useful for distinguishing among thefixed points. Note that the observed ratio is inconsistentwith uncorrelated 2D percolation (P-2D), and we can ruleout this fixed point as the origin of the pattern forma-tion. Although the P-3D fixed point may appear to bea reasonable match, other considerations rule this out asthe origin of the cluster pattern. First, this fixed pointoccurs when 31% of domains point one direction, and therest point the other.[34] Such an extreme value of net ne-maticity would surely have been observed in macroscopicmeasurements on Dy-Bi2212, which is not the case. Sec-ond, while P-3D corresponds to the point at which geo-metric clusters percolate in a 3D system, this is not thesame as the point at which those clusters percolate on aslice. Rather, at the 3D percolation point when viewedon a 2D slice , there is one large spanning cluster withmany small clusters, and no robust power law behavioron the slice. So, the P-3D point can also be ruled out asthe origin of the complex pattern formation.This leaves the possibility of a dimensional crossoverfrom 2D to 3D behavior in either the clean or randomfield Ising models. The expectation in the literature isthat there should be no well-defined fractal dimensionof geometric clusters at C-3D, since in fact geometricclusters do not exhibit power law behavior at the C-3D point.[35] However, recent studies indicate that whenviewed on a 2D slice , geometric clusters do exhibit powerlaw behavior at C-3D.[36] The ratio of fractal dimensionson a 2D slice is not known in this case, and will be dis-cussed in a future publication.[37]The possibility of a dimensional crossover from 2D to3D exponents in a layered random field model is consis-tent with all data-derived exponents, and is the mostlikely source of the observed scale-free pattern forma-tion of the electron nematic. In contrast with the cleanmodel, geometric clusters do exhibit fractal dimensionsand scale-free behavior at the RF-3D fixed point. Fur-thermore, if this identification is correct, then the clearprediction is that all data-derived exponents should driftaway from the RF-2D values and closer to the RF-3Dvalues upon increasing the field of view.Other considerations also point to random field behav-ior: It has been previously shown that the slow telegraphnoise observed in transport on a YBCO nanowire in thepseudogap regime[27] is consistent with the mapping ofelectron nematic domains in a host crystal to the ran-dom field Ising model.[28] This identification also servesto unify several experiments, in that it offers a concreteexplanation for why certain materials display long-rangeorientational stripe order, and others do not. While truelong-range electron nematic order is possible in a real 3Dsystem, it is completely forbidden in a 2D system in thepresence of any nonzero random field disorder. Thus, ina highly layered system such as the cuprates, many sam-ples are expected to display no long-range order of theelectron nematic, although in a layered RFIM, nematicclusters can grow quite large within the plane even if truelong-range order is never achieved.[20, 30]The RF-3D fixed point is a zero temperature fixed point, which has implications for dynamics as well as fu-ture experimental tests of the critical exponents. First, itmeans that the entire finite-temperature phase transitionboundary in the layered model exhibits extreme criticalslowing down. With typical critical slowing down, therelaxation time of the system diverges as a power lawas criticality is approached, τ relax ∝ / | T − T c | − νz . How-ever, the dynamics of the 3D random field Ising model areeven more extreme near criticality, with the relaxationtime diverging exponentially as criticality is approached, τ relax ∝ exp[ ξ θ ] where ξ is the spin-spin correlation length(which here corresponds to the orientation-orientationcorrelation length of the nematic director), and θ is theviolation of hyperscaling exponent, which is nonzero atthis fixed point.[29] Second, because θ = 0 at a zero tem-perature fixed point, hyperscaling relations of critical ex-ponents (which involve the dimension of the underlyingphenomenon) must be modified.[29] Third, there is thequestion of whether fine-tuning is required to see powerlaw behavior associated with the RF-3D fixed point. Infact, partly because it is a zero temperature fixed pointwith pronounced nonequilibrium effects, there is a widecritical region associated with this fixed point. For exam-ple, in the zero temperature 3D RFIM, critical behaviorwith 2 decades of scaling can be observed even 50% awayfrom the critical point.[32]Finally, we comment on the implications of multiscalebehavior in cuprate superconductors. It is not just theelectron nematic which exhibits fractal behavior in abismuth-based cuprate, but similar behavior has beennoted in the lanthanum family of cuprates as well. Thelocal density of oxygen interstitials in LaSrCuO followsa power law at optimal doping.[7] In addition, theoreti-cal studies have shown that there is a Goldilocks type ofoptimal inhomogeneity (neither too little nor too much)which favors superconductivity in a strongly correlatedelectronic system.[38, 39] The presence of inhomogene-ity on multiple length scales, with robust power laws,in both the doping concentrations and also directly inthe electronic degrees of freedom may point to the opti-mal inhomogeneity being fractal in nature. Much like theconstruction of the Eiffel tower incorporates elements of ascale-free iron latticework in order to optimize structuralstability given a certain amount of iron to work with, hightemperature superconductors may benefit from scale-freeorganization of electronic degrees of freedom in order tooptimize the superconducting transition temperature.[40]In summary, we conclude that the complex, scale-freepattern[10] of nematic clusters observed at the surface ofDy-Bi2212 via STM[11] is controlled by the 3D randomfield Ising model fixed point. That is, the ethereal clusterstructure is due to a combination of interactions betweenclusters and quenched disorder due to material defectsthroughout the bulk of the material. As such, the pat-tern formation is not merely a surface effect. Rather, thenematic clusters form deep inside the material, and inter-sect the surface. While this analysis cannot distinguishbetween true macroscopic long-range order of the elec-tron nematic and short-range order, we can conclude thatthere is significant multiscale order in the system. In-deed, because the pairing energy scale is high, the pairingmechanism can arise from short-distance physics, and thepresence of large nematic clusters throughout the bulk ofthe material is sufficient for superconducting pairing tooriginate from the electron nematic.We thank J. Hoffman, S. Kivelson, Y. Loh, E. Main,B. Phillabaum, and C.-L. Song for helpful conversations.S.L. and E.W.C. acknowledge support from NSF GrantNo. DMR 11-06187. K.A.D. acknowledges support fromNSF Grant No. DMR 10-05209 and NSF Grant No.DMS 10-69224. [1] E. Dagotto, Science , 257 (2005).[2] K. Lai, M. Nakamura, W. Kundhikanjana, M. Kawasaki,Y. Tokura, M. Kelly, and Z. Shen, Science , 190(2010).[3] M. M. Qazilbash, M. Brehm, B.-G. Chae, P.-C. Ho, G. O.Andreev, B.-J. Kim, S. J. Yun, A. V. Balatsky, M. B.Maple, F. Keilmann, H.-T. Kim, and D. N. Basov, Sci-ence , 1750 (2007).[4] Z. Sun, J. Douglas, A. Fedorov, Y. Chuang, H. Zheng,J. Mitchell, and D. Dessau, Nature physics , 248 (2007).[5] J. M. Tranquada, B. J. Sternlieb, J. D. Axe, Y. Naka-mura, and S. Uchida, Nature , 561 (1995).[6] J. Zaanen and O. Gunnarsson, Phys. Rev. B , 7391(1989).[7] M. Fratini, N. Poccia, A. Ricci, G. Campi, M. Burgham-mer, G. Aeppli, and A. Bianconi, Nature , 841(2010).[8] R. W. Dull and H. R. Kerchner, “A teacher’s guideto superconductivity for high school students,” (1994),oRNL/M-3063/R1.[9] E. W. Carlson, V. J. Emery, S. A. Kivelson, and D. Or-gad, “Concepts in high temperature superconductiv-ity,” in The Physics of Superconductors, Vol. II , editedby J. Ketterson and K. Benneman (Springer-Verlag,2004) in The Physics of Superconductors, Vol. II, ed.J. Ketterson and K. Benneman.[10] B. Phillabaum, E. W. Carlson, and K. A. Dahmen, Na-ture Communications , 915 (2012).[11] Y. Kohsaka, C. Taylor, K. Fujita, A. Schmidt, C. Lupien,T. Hanaguri, M. Azuma, M. Takano, H. Eisaki, H. Tak-agi, S. Uchida, and J. C. Davis, Science , 1380 (2007).[12] R. Borzi, Science(Washington) , 214 (2007).[13] K. Cooper, M. Lilly, J. Eisenstein, L. Pfeiffer, andK. West, Phys. Rev. B , 241313 (2002).[14] R. Du, D. Tsui, H. Stormer, L. Pfeiffer, K. Baldwin, and K. West, Solid State Communications , 389 (1999).[15] E. Fradkin and S. Kivelson, Science , 155 (2010).[16] Y. Ando, K. Segawa, S. Komiya, and A. Lavrov, Physicalreview letters , 137005 (2002).[17] V. Hinkov, S. Pailhes, P. Bourges, Y. Sidis, A. Ivanov,A. Kulakov, C. Lin, D. Chen, C. Bernhard, andB. Keimer, Nature , 650 (2004).[18] R. Daou, J. Chang, D. Leboeuf, O. Cyr-Choiniere,F. Laliberte, N. Doiron-Leyraud, B. J. Ramshaw,R. Liang, D. A. Bonn, W. N. Hardy, and L. Taillefer,Nature , 519 (2010).[19] C. Howald, H. Eisaki, N. Kaneko, and A. Kapitulnik,Proceedings of the National Academy of Sciences of theUnited States of America , 9705 (2003).[20] M. J. Lawler, K. Fujita, J. Lee, A. R. Schmidt,Y. Kohsaka, C. K. Kim, H. Eisaki, S. Uchida, J. C. Davis,J. P. Sethna, and E.-A. Kim, Nature , 347 (2010).[21] T.-M. Chuang, M. P. Allan, J. Lee, Y. Xie, N. Ni, S. L.Bud’ko, G. S. Boebinger, P. C. Canfield, and J. C. Davis,Science , 181 (2010).[22] D. Abanin, S. Parameswaran, S. Kivelson, andS. Sondhi, Phys. Rev. B , 035428 (2010).[23] O. Vafek and K. Yang, Phys. Rev. B , 041401 (2010).[24] M. Rasolt, B. Halperin, and D. Vanderbilt, Physical re-view letters , 126 (1986), predicts Valley DegeneracyBreaking in QHE.[25] E. W. Carlson and K. A. Dahmen, “Using disorder todetect locally ordered electron nematics via hysteresis,”Submitted to Phys. Rev. Lett.[26] C. Mirri, A. Dusza, S. Bastelberger, J. H. Chu, H. H.Kuo, I. R. Fisher, and L. Degiorgi, Physical Review B , 060501 (2014).[27] J. Bonetti, D. Caplan, D. Van Harlingen, and M. Weiss-man, Physical Review Letters , 087002 (2004).[28] E. W. Carlson, K. A. Dahmen, E. Fradkin, and S. A.Kivelson, Physical Review Letters , 097003 (2006).[29] D. Fisher, Physical Review Letters , 416 (1986).[30] O. Zachar and I. Zaliznyak, Physical Review Letters ,036401 (2003).[31] M. E. J. Newman, Contemporary Physics , 323 (2005).[32] O. Perkovi´c, K. Dahmen, and J. Sethna, Physical ReviewLetters , 4528 (1995).[33] R. B. Laughlin, Phys. Rev. Lett. , 1726 (1997).[34] D. Stauffer and A. Aharony, Introduction to PercolationTheory (Taylor & Francis, Philadelphia, 1991).[35] V. Dotsenko, M. Picco, P. Windey, G. Harris, E. Mar-tinec, and E. Marinari, Nuclear Physics B , 577(1995).[36] A. A. Saberi and H. Dashti-Naserabadi, EPL (Euro-physics Letters) , 67005 (2011).[37] S. Liu, E. W. Carlson, and K. A. Dahmen, Forthcoming.[38] E. Arrigoni and S. Kivelson, Physical Review B ,180503 (2003).[39] Y. Loh and E. Carlson, Physical Review B , 132506(2007).[40] N. Poccia, A. Ricci, and A. Bianconi, Journal of Super-conductivity and Novel Magnetism24