The Phase Diagram of High-Tc Superconductors in the Presence of Dynamic Stripes

a r X i v :c o n d -m a t /0103519v 1 [c o n d -m a t .s u p r -c o n ] 26 M a r 2001Chemical Potential Shift in Nd 2−x Ce x CuO 4:Contrasting Behaviors of the Electron-and Hole-Doped CupratesN.Harima,J.Matsuno,and A.FujimoriDepartment of Physics and Department of Complexity Science and Engineering,University of Tokyo,Bunkyo-ku,Tokyo113-0033,JapanY.Onose,Y.Taguchi,and Y.TokuraDepartment of Applied Physics,University of Tokyo,Bunkyo-ku,Tokyo 113-8656,Japan(February 1,2008)We have studied the chemical potential shift in the electron-doped superconductor Nd 2−x Ce x CuO 4by precise measurements of core-level photoemission spectra.The result shows that the chemical potential monotonously increases with electron doping,quite differently from La 2−x Sr x CuO 4,where the shift is suppressed in the underdoped region.If the suppression of the shift in La 2−x Sr x CuO 4is attributed to strong stripe fluctuations,the monotonous increase of the chemical potential is consistent with the absence of stripe fluctuations in Nd 2−x Ce x CuO 4.The chemical potential jump between Nd 2CuO 4and La 2CuO 4is found to be much smaller than the optical band gaps.PACS numbers:79.60.-i,74.72.Jt,71.30.+h,75.50.EeHigh-T c superconductivity occurs when the parent an-tiferromagnetic (AF)insulator with the CuO 2plane is doped with holes or electrons.In the p -type materials,the long-range AF order vanishes for a slight amount of hole doping whereas in the n -type materials,the AF or-der persists up to a high doping concentration of ∼0.14electrons per Cu and the superconducting (SC)doping range is much narrower [1].The p -type materials show T -linear in-plane electrical resistivity [2]and split neu-tron peaks around q =(π,π)indicating incommensurate spin fluctuations [3]whereas the n -type materials show T 2dependence of the in-plane resistivity [4]and (π,π)commensurate spin fluctuations [5].In order to elucidate the mechanism of high-T c superconductivity,it is very important to clarify the origin of the similarities and the differences between the p -type and the n -type materials.In this Letter,we report on a study of the chemical potential shift in Nd 2−x Ce x CuO 4(NCCO)as a function of doped electron concentration.The shift can be de-duced from the core-level shifts in photoemission spectra because the binding energy of each core level is mea-sured relative to the chemical potential µ.In a previ-ous study [6],we found that in La 2−x Sr x CuO 4(LSCO)the chemical potential shift is unusually suppressed in the underdoped region and attributed this observation to the strong stripe fluctuations which exist in this system.As for the chemical potential jump between La 2CuO 4and Nd 2CuO 4,which would represent the band gap of the parent insulator,it was estimated to be at most 300meV in previous valence-band photoemission stud-ies [7,8],which is much smaller than the 1.5–2.0eV charge-transfer (CT)gap of the parent insulator esti-mated from optical studies [9].High-quality single crystals of NCCO (x =0,0.05,0.125and 0.15)were grown by the traveling-solventfloating-zone method as described elsewhere [10].Un-certainties in the Ce concentration were ±0.01.For x =0.15,both as-grown and reduced samples were mea-sured while for the other compositions only as-grown samples were measured.The as-grown samples were all antiferromagnetic and did not show superconductivity.Only the x =0.15sample showed superconductivity af-ter reduction in an Ar atmosphere and its T c was ∼25K.X-ray photoemission spectroscopy (XPS)measure-ments were performed using both the Mg Kα(h ν=1253.6eV)and Al Kα(h ν=1486.6eV)lines and a hemispherical analyzer.All the spectra were taken at liquid-nitrogen temperature (∼80K)within 40minutes after scraping.We did not observe a shoulder on the higher binding energy side of each O 1s peak,indicating the high quality of the sample surfaces free from degra-dation.Although the energy resolution was about 0.8eV for both Kαlines,we could determine the core-level shifts with an accuracy of about ±50meV because most of the spectral line shapes did not change with x .In XPS measurements,a high voltage of >1kV is used to decel-erate photoelectrons,and it is usually difficult to stabilize the high voltage with the accuracy of ≪100meV.In or-der to overcome this difficulty,we directly monitored the voltage applied to the outer hemisphere and the retard-ing fringe,and confirmed that the uncertainty could be reduced to less than 10meV.To eliminate other unex-pected causes of errors,we measured the x =0.05sample as a reference just after the measurement of each sample.Figure 1shows the XPS spectra of the O 1s ,Nd 3d 5/2and Cu 2p 3/2core levels taken with the Al Kαline.Here,the integral background has been subtracted and the in-tensity has been normalized to the peak height [11].The Nd 3d 5/2spectra are composed of the 3d 5/24f 4L ¯andRelative Binding Energy ( eV )Relative Binding Energy ( eV )Relative Binding Energy ( eV )Binding Energy (eV)Binding Energy (eV)Binding Energy (eV)FIG. 1.Core-level photoemission spectra of Nd 2−x Ce x CuO 4taken with the Al Kαline.(a),(d):O 1s ;(b),(e):Nd 3d ;(c),(f):Cu 2p .Integral background has been subtracted and the intensity has been normalized to the peak height.3d 5/24f 3final-state components,where L ¯denotes a lig-and hole,and O KLL Auger signals overlap them.The Cu 2p 3/2spectra are composed of the 2p 3/23d 10L ¯and2p 3/23d 9components,but only the 2p 3/23d 10L ¯peaks areshown in the figure.One can see the obvious doping de-pendent shifts of O 1s and Nd 3d core levels from both the displaced and overlayerd plots in Fig.1.To deduce the amount of the core-level shifts reliably,we used the peak position for the Nd 3d spectra and the mid point of the lower binding energy slope for the O 1s spectra.We used the mid-point position rather than the peak posi-tion for O 1s because the line shape on the higher binding energy side of the O 1s peak was sensitive to a slight sur-face degradation or contamination.The Cu 2p core-level line shape was not identical between different x ’s,and be-comes broader as x increases.This is because the doped electrons in the CuO 2plane produce Cu 1+sites on the otherwise Cu 2+background,which yield an overlapping chemically shifted component located on the lower bind-ing side of the Cu 2+peak.Therefore,it was difficult to uniquely determine the shift of the Cu 2p core level andwe only take its peak positions in the following.Figure 2shows the binding energy shift of each core level relative to the as-grown x =0.05sample.Here,we have assumed that the change of the electron concen-tration caused by the oxygen reduction was ∼0.04per Cu (oxygen reduction being ∼0.02)as reported previ-ously [12].One can see that the Nd 3d and O 1s levels move toward higher binding energies with electron dop-ing.The shift of Cu 2p is defined by the shift of the peak position,and is in the opposite direction to Nd 3d and O 1s because of the Cu 1+components mentioned above.We also measured the shifts of the core levels using the Mg Kαline and almost the same results were obtained as shown in Fig.2.R e l a t i v e B i n d i n g E n e r g y ( m e V )Doping Level / CuFIG.2.Binding energy shift of each core level relative to the x =0.05sample.Open and filled symbols are data taken with the Mg Kαand Al Kαradiation,respectively.While the shift of the chemical potential changes the core-level binding energy,there is another factor that could affect the binding energy,that is,the change in the Madelung potential due to Ce 4+substitution for Nd 3+.However,the identical shifts of the O 1s and Nd 3d core levels indicate that the change in the Madelung poten-tial has negligible affects on the core-level shifts because it would shift the core levels of the O 2−anion and Nd 3+cation in the opposite directions.Moreover,as the shifts of the O 1s and Nd 3d core levels toward higher binding energies with electron doping are opposite to what would be expcted from increasing core-hole screening capabil-ity with x ,excluding the core-hole screening mechanism as the main cause of the core-level shifts.Therefore,we conclude that the shifts of the O 1s and Nd 3d core levels are largely due to the chemical potential shift ∆µ.We have evaluated ∆µin NCCO by taking the average of theshifts of the two core levels.Figure3(a)shows∆µin NCCO as well as∆µin LSCO[6]as a function of electron or hole carrier con-centration.In order to obtain the jump inµbetween Nd2CuO4and La2CuO4,we also measured the O1s and Cu2p levels in LSCO as shown in Fig.4,and found that the O1s and Cu2p levels in Nd2CuO4lie at∼150 meV and∼400meV higher binding energies than those in La2CuO4,respectively.The fact that the observed jump is different between O1s and Cu2p is not surpris-ing because Nd2CuO4and La2CuO4are different mate-rials with different crystal structures.Thus the chemical potential jump between Nd2CuO4and La2CuO4cannot be uniquely determined from those data but it should be much smaller than the CT gap of about1.5eV for Nd2CuO4and2.0eV for La2CuO4estimated from the optical measurements[9].This small jump is in accor-dance with the early valence-band photoemission studies of LSCO and NCCO[7,8].Figure3(a)demonstrates the different behaviors of∆µbetween LSCO and NCCO.In LSCO,∆µis suppressed in the underdoped region x≤0.13,whereas in NCCO∆µmonotonously increases with electron doping in the whole concentration range.Figure3(c)represents the phase diagram of LSCO and NCCO drawn against the chemical potentialµand the temperature T.One can see that the µ−T phase diagram is rather symmetric between the hole doping and electron doping unlike the widely used x−T phase diagram.That is,in both the electron-and hole-doped cases,the SC region is adjecent to the AF region,as proposed by Zhang[14]based on SO(5) symmetry.The present phase diagram implies that as a function ofµ,T c increases up to the point where the superconductivity is taken over by the AF ordering.Such a behavior is reminiscent of the superfluid-solid transition in the p−T phase diagram of3He[15].The monotonous increase ofµin NCCO may be un-derstood within the simple rigid-band model.The chem-ical potential of Nd2CuO4lies near the bottom of the conduction band.When electrons are doped,µmoves upward into this band as long as the AF ordering and hence the AF band structure persist as in NCCO.This behavior is contrasted with the suppression of∆µin un-derdoped LSCO,where the AF ordering is quickly de-stroyed and the electronic structure is dramatically reor-ganized by a small amount of hole doping.It has been suggested[6]that the suppression of∆µis related to the chargefluctuations in the form of stripes in LSCO. Indeed in La2−x Sr x NiO4(LSNO),where static stripe or-der is stable at x≃0.33[16],∆µis anomalously sup-pressed in the underdoped region x≤0.33[17].In LSCO, dynamical stripefluctuations have been implied by in-elastic neutron scattering studies[3,18]whereas any sign of stripes is absent in the neutron study of NCCO[5]. In Fig.3,we compare the chemical potential shift∆µwith the incommensurabilityǫwhich was deduced from the neutron experiments both for LSCO[3]and NCCO [18].Thisfigure shows that in the region whereµdoes not move(x≤0.13in LSCO),ǫlinearly increases with x. In NCCO,where the chemical potentialµmonotonously moves upward,the incommensurability does not change with x(remains zero),in other words,static nor dynami-cal stripes do not exist.In the overdoped region of LSCO, the number of stripes saturates,doped holes overflow into the interstripe region andµmoves fast with hole doping. The smallness of the chemical potential jump between Nd2CuO4and La2CuO4indicates thatµlies within the CT gap of Nd2CuO4(∼1.5eV)and La2CuO4(∼2.0eV). This behavior was clearly observed in an angle-resolved photoemission(ARPES)study of LSCO[19],whereµis located∼0.4eV above the top of the valence band of La2CuO4.Such a behavior is quite peculiar from the view point of the rigid-band model,and therefore a dramatic change in the electronic structure should oc-cur with hole doping.In NCCO,however,the doping-induced change is not so dramatic as in LSCO as men-tioned above,and the chemical potential pinning well below the bottom of the conduction band is not very likely.Another possible cause of the small chemical po-tential jump is that the CT gap is indirect and is smaller than that estimated from the optical studies.In optical conductivity spectra,only the direct transition can be unambiguously measured,and if the gap is indirect,the gap would be estimated much larger than the CT gap. This idea is consistent with the recent resonant inelas-tic x-ray scattering study[20]and its theoretical analysis using t-t′-t′′-U model[21],which has yielded an indirect gap that is smaller than the direct one by∼0.5eV.In summary,we have experimentally determined the doping dependence of the chemical potential shift∆µin NCCO and observed a monotonous shift with doping. Comparison with LSCO indicates that the change in the electronic structure with carrier doping is more moder-ate in NCCO.The monotonous shift is consistent with the observation that spinfluctuations are commensurate in NCCO.The small chemical potential jump between the n-type and p-type materials is confirmed and is at-tributed to the indirect CT gap and the chemical poten-tial pinning within the CT gap in LSCO.The authors would like to thank S.Tesanovic,A.Ino and T.Mizokawa for enlightening discussions.Collabora-tion with G.A.Sawatzky and J.van Elp in the early stage of this work is gratefully acknowledged.This work was supported by a Grant-in-Aid for Scientific Research in Priority Area“Novel Quantum Phenomena in Transition Metal Oxides”and a Special Coordination Fund for the Promotion of Science and Technology from the Ministry of Education and Science and by New Energy and Indus-trial Technology Development Organization(NEDO).[1]H.Takagi,S.Uchida,and Y.Tokura,Phys.Rev.Lett.62,1197(1989).[2]H.Takagi,B.Batlogg,H.L.Kao,J.Kwo,R.J.Cava,J.J.Krajewski,and W.F.Peck,Jr.,Phys.Rev.Lett.69, 2975(1992).[3]K.Yamada,C.H.Lee,K.Kurahashi,J.Wada,S.Waki-moto,S.Ueki,H.Kimura,Y.Endoh,S.Hosoya,G.Shi-rane,R.J.Birgeneau,M.Greven,M.A.Kastner,and Y.J.Kim,Phys.Rev.B57,6165(1998).[4]S.J.Hagen,J.L.Peng,Z.Y.Li,and R.L.Greene,Phys.Rev.B43,13606(1991).[5]K.Yamada,K.Kurahashi,and Y.Endoh,unpublished.[6]A.Ino,T.Mizokawa, A.Fujimori,K.Tamasaku,H.Eisaki,S.Uchida,T.Kimura,T.Sasagawa,and K.Kishio,Phys.Rev.Lett.79,2101(1997).[7]H.Namatame,A.Fujimori,Y.Tokura,M.Nakamura,K.Yamaguchi,A.Misu,H.Matsubara,S.Suga,H.Eisaki, T.Ito,H.Takagi,and S.Uchida,Phys.Rev.B41,7205 (1990).[8]J.W.Allen,C.G.Olson,M.B.Maple,J.-S.Kang,L.Z.Liu,J.-H.Park,R.O.Anderson,W.P.Ellis,J.T.Markert,Y.Dalichaouch,and R.Liu,Phys.Rev.Lett.64,595(1990).[9]S.Uchida,T.Ido,H.Takagi,T.Arima,Y.Tokura,andS.Tajima,Phys.Rev.B43,7942(1991).[10]Y.Onose,Y.Taguchi,T.Ishikawa,S.Shinomori,K.Ishizaka,and Y.Tokura,Phys.Rev.Lett.82,5120 (1999).[11]We note that the background subtraction does not affectthe estimates of the shifts.[12]A.J.Schultz,J.D.Jorgensen,J.L.Peng and R.L.Greene,Phys.Rev.B53,5157(1996).[13]S.H¨u fner,Photoelectron Spectroscopy(Springer-Verlag,Berlin,1995).Chap.2,p.35.[14]S.C.Zhang,Science275,1089(1998).[15]S.Tesanovi´c,private communication.[16]J.M.Tranquada,D.J.Buttrey,V.Sachan,Phys.Rev.B54,12318(1996).[17]M.Satake,K.Kobayashi,T.Mizokawa,A.Fujimori,T.Tanabe,T.Katsufuji,and Y.Tokura,Phys.Rev.B61, 15515(2000).[18]M.Matsuda,M.Fujita,K.Yamada,R.J.Birgeneau,M.A.Kastner,H.Hiraka,Y.Endoh,S.Wakimoto and G.Shirane,Phys.Rev.B62,9148(2000).[19]A.Ino,C.Kim,M.Nakamura,T.Yoshida,T.Mizokawa,Z.-X.Shen,A.Fujimori,T.Kakeshita,H.Eisaki,and S.Uchida,Phys.Rev.B62,4137(2000).[20]M.Z.Hassan,E.D.Isaacs,Z.-X.Shen,ler,K.Tsutsui,T.Tohyama,and S.Maekawa,Science288, 1811(2000).[21]K.Tsutsui,T.Tohyama,and S.Maekawa,Phys.Rev.Lett.83,3705(1999).0.40.20.00.120.080.040.00IncommensurabilityεChemiclalPotentialShift∆µ(eV)Doping Level / CuChemical Potential µ (eV)Temperature(K)FIG.3.(a):Chemical potential shift∆µin NCCO and LSCO.(b):Incommensurabilityǫmeasured byinelastic neu-tron scattering experiments by Yamada et al.[3,5].In the hatched region,ǫvaries linearly and∆µis constant as func-tions of doping level.(c):µ−T phase diagram of NCCO and LSCO.The zero of the horizontal axis corresponds to theµof Nd2CuO4.Energy relative to µ (eV)Energy relative to µ (eV)Intensity(arb.units)FIG.4.Core-level spectra of NCCO and LSCO.(a):O1s, (b):Cu2p.。

Fresnel incoherent correlation hologram-a reviewInvited PaperJoseph Rosen,Barak Katz1,and Gary Brooker2∗∗1Department of Electrical and Computer Engineering,Ben-Gurion University of the Negev,P.O.Box653,Beer-Sheva84105,Israel2Johns Hopkins University Microscopy Center,Montgomery County Campus,Advanced Technology Laboratory, Whiting School of Engineering,9605Medical Center Drive Suite240,Rockville,MD20850,USA∗E-mail:rosen@ee.bgu.ac.il;∗∗e-mail:gbrooker@Received July17,2009Holographic imaging offers a reliable and fast method to capture the complete three-dimensional(3D) information of the scene from a single perspective.We review our recently proposed single-channel optical system for generating digital Fresnel holograms of3D real-existing objects illuminated by incoherent light.In this motionless holographic technique,light is reflected,or emitted from a3D object,propagates througha spatial light modulator(SLM),and is recorded by a digital camera.The SLM is used as a beam-splitter of the single-channel incoherent interferometer,such that each spherical beam originated from each object point is split into two spherical beams with two different curve radii.Incoherent sum of the entire interferences between all the couples of spherical beams creates the Fresnel hologram of the observed3D object.When this hologram is reconstructed in the computer,the3D properties of the object are revealed.OCIS codes:100.6640,210.4770,180.1790.doi:10.3788/COL20090712.0000.1.IntroductionHolography is an attractive imaging technique as it offers the ability to view a complete three-dimensional (3D)volume from one image.However,holography is not widely applied to the regime of white-light imaging, because white-light is incoherent and creating holograms requires a coherent interferometer system.In this review, we describe our recently invented method of acquiring incoherent digital holograms.The term incoherent digi-tal hologram means that incoherent light beams reflected or emitted from real-existing objects interfere with each other.The resulting interferogram is recorded by a dig-ital camera and digitally processed to yield a hologram. This hologram is reconstructed in the computer so that 3D images appear on the computer screen.The oldest methods of recording incoherent holograms have made use of the property that every incoherent ob-ject is composed of many source points,each of which is self-spatial coherent and can create an interference pattern with light coming from the point’s mirrored image.Under this general principle,there are vari-ous types of holograms[1−8],including Fourier[2,6]and Fresnel holograms[3,4,8].The process of beam interfering demands high levels of light intensity,extreme stability of the optical setup,and a relatively narrow bandwidth light source.More recently,three groups of researchers have proposed computing holograms of3D incoherently illuminated objects from a set of images taken from differ-ent points of view[9−12].This method,although it shows promising prospects,is relatively slow since it is based on capturing tens of scene images from different view angles. Another method is called scanning holography[13−15],in which a pattern of Fresnel zone plates(FZPs)scans the object such that at each and every scanning position, the light intensity is integrated by a point detector.The overall process yields a Fresnel hologram obtained as a correlation between the object and FZP patterns.How-ever,the scanning process is relatively slow and is done by mechanical movements.A similar correlation is ac-tually also discussed in this review,however,unlike the case of scanning holography,our proposed system carries out a correlation without movement.2.General properties of Fresnel hologramsThis review concentrates on the technique of incoher-ent digital holography based on single-channel incoher-ent interferometers,which we have been involved in their development recently[16−19].The type of hologram dis-cussed here is the digital Fresnel hologram,which means that a hologram of a single point has the form of the well-known FZP.The axial location of the object point is encoded by the Fresnel number of the FZP,which is the technical term for the number of the FZP rings along the given radius.To understand the operation principle of any general Fresnel hologram,let us look on the difference between regular imaging and holographic systems.In classical imaging,image formation of objects at different distances from the lens results in a sharp image at the image plane for objects at only one position from the lens,as shown in Fig.1(a).The other objects at different distances from the lens are out of focus.A Fresnel holographic system,on the other hand,as depicted in Fig.1(b),1671-7694/2009/120xxx-08c 2009Chinese Optics Lettersprojects a set of rings known as the FZP onto the plane of the image for each and every point at every plane of the object being viewed.The depth of the points is en-coded by the density of the rings such that points which are closer to the system project less dense rings than distant points.Because of this encoding method,the 3D information in the volume being imaged is recorded into the recording medium.Thus once the patterns are decoded,each plane in the image space reconstructed from a Fresnel hologram is in focus at a different axial distance.The encoding is accomplished by the presence of a holographic system in the image path.At this point it should be noted that this graphical description of pro-jecting FZPs by every object point actually expresses the mathematical two-dimensional (2D)correlation (or convolution)between the object function and the FZP.In other words,the methods of creating Fresnel holo-grams are different from each other by the way they spatially correlate the FZP with the scene.Another is-sue to note is that the correlation should be done with a FZP that is somehow “sensitive”to the axial locations of the object points.Otherwise,these locations are not encoded into the hologram.The system described in this review satisfies the condition that the FZP is depen-dent on the axial distance of each and every objectpoint.parison between the Fresnel holography principle and conventional imaging.(a)Conventional imaging system;(b)fresnel holographysystem.Fig.2.Schematic of FINCH recorder [16].BS:beam splitter;L is a spherical lens with focal length f =25cm;∆λindicates a chromatic filter with a bandwidth of ∆λ=60nm.This means that indeed points,which are closer to the system,project FZP with less cycles per radial length than distant points,and by this condition the holograms can actually image the 3D scene properly.The FZP is a sum of at least three main functions,i.e.,a constant bias,a quadratic phase function and its complex conjugate.The object function is actually corre-lated with all these three functions.However,the useful information,with which the holographic imaging is real-ized,is the correlation with just one of the two quadratic phase functions.The correlation with the other quadratic phase function induces the well-known twin image.This means that the detected signal in the holographic system contains three superposed correlation functions,whereas only one of them is the required correlation between the object and the quadratic phase function.Therefore,the digital processing of the detected image should contain the ability to eliminate the two unnecessary terms.To summarize,the definition of Fresnel hologram is any hologram that contains at least a correlation (or convolu-tion)between an object function and a quadratic phase function.Moreover,the quadratic phase function must be parameterized according to the axial distance of the object points from the detection plane.In other words,the number of cycles per radial distance of each quadratic phase function in the correlation is dependent on the z distance of each object point.In the case that the object is illuminated by a coherent wave,this correlation is the complex amplitude of the electromagnetic field directly obtained,under the paraxial approximation [20],by a free propagation from the object to the detection plane.How-ever,we deal here with incoherent illumination,for which an alternative method to the free propagation should be applied.In fact,in this review we describe such method to get the desired correlation with the quadratic phase function,and this method indeed operates under inco-herent illumination.The discussed incoherent digital hologram is dubbed Fresnel incoherent correlation hologram (FINCH)[16−18].The FINCH is actually based on a single-channel on-axis incoherent interferometer.Like any Fresnel holography,in the FINCH the object is correlated with a FZP,but the correlation is carried out without any movement and without multiplexing the image of the scene.Section 3reviews the latest developments of the FINCH in the field of color holography,microscopy,and imaging with a synthetic aperture.3.Fresnel incoherent correlation holographyIn this section we describe the FINCH –a method of recording digital Fresnel holograms under incoher-ent illumination.Various aspects of the FINCH have been described in Refs.[16-19],including FINCH of re-flected white light [16],FINCH of fluorescence objects [17],a FINCH-based holographic fluorescence microscope [18],and a hologram recorder in a mode of a synthetic aperture [19].We briefly review these works in the current section.Generally,in the FINCH system the reflected incoher-ent light from a 3D object propagates through a spatial light modulator (SLM)and is recorded by a digital cam-era.One of the FINCH systems [16]is shown in Fig.2.White-light source illuminates a 3D scene,and the reflected light from the objects is captured by a charge-coupled device (CCD)camera after passing through a lens L and the SLM.In general,we regard the system as an incoherent interferometer,where the grating displayed on the SLM is considered as a beam splitter.As is com-mon in such cases,we analyze the system by following its response to an input object of a single infinitesimal point.Knowing the system’s point spread function (PSF)en-ables one to realize the system operation for any general object.Analysis of a beam originated from a narrow-band infinitesimal point source is done by using Fresnel diffraction theory [20],since such a source is coherent by definition.A Fresnel hologram of a point object is obtained when the two interfering beams are two spherical beams with different curvatures.Such a goal is achieved if the SLM’s reflection function is a sum of,for instance,constant and quadratic phase functions.When a plane wave hits the SLM,the constant term represents the reflected plane wave,and the quadratic phase term is responsible for the reflected spherical wave.A point source located at some distance from a spher-ical positive lens induces on the lens plane a diverging spherical wave.This wave is split by the SLM into two different spherical waves which propagate toward the CCD at some distance from the SLM.Consequently,in the CCD plane,the intensity of the recorded hologram is a sum of three terms:two complex-conjugated quadratic phase functions and a constant term.This result is the PSF of the holographic recording system.For a general 3D object illuminated by a narrowband incoherent illumination,the intensity of the recorded hologram is an integral of the entire PSFs,over all object intensity points.Besides a constant term,thehologramFig.3.(a)Phase distribution of the reflection masks dis-played on the SLM,with θ=0◦,(b)θ=120◦,(c)θ=240◦.(d)Enlarged portion of (a)indicating that half (randomly chosen)of the SLM’s pixels modulate light with a constant phase.(e)Magnitude and (f)phase of the final on-axis digi-tal hologram.(g)Reconstruction of the hologram of the three characters at the best focus distance of ‘O’.(h)Same recon-struction at the best focus distance of ‘S’,and (i)of ‘A’[16].expression contains two terms of correlation between an object and a quadratic phase,z -dependent,function.In order to remain with a single correlation term out of the three terms,we follow the usual procedure of on-axis digital holography [14,16−19].Three holograms of the same object are recorded with different phase con-stants.The final hologram is a superposition of the three holograms containing only the desired correlation between the object function and a single z -dependent quadratic phase.A 3D image of the object can be re-constructed from the hologram by calculating theFresnelFig.4.Schematics of the FINCH color recorder [17].L 1,L 2,L 3are spherical lenses and F 1,F 2are chromaticfilters.Fig.5.(a)Magnitude and (b)phase of the complex Fres-nel hologram of the dice.Digital reconstruction of the non-fluorescence hologram:(c)at the face of the red dots on the die,and (d)at the face of the green dots on the die.(e)Magnitude and (f)phase of the complex Fresnel hologram of the red dots.Digital reconstruction of the red fluorescence hologram:(g)at the face of the red dots on the die,and (h)at the face of the green dots on the die.(i)Magnitude and (j)phase of the complex Fresnel hologram of the green dots.Digital reconstruction of the green fluorescence hologram:(k)at the face of the red dots on the die,and (l)at the face of the green dots on the position of (c),(g),(k)and that of (d),(h),(l)are depicted in (m)and (n),respectively [17].Fig.6.FINCHSCOPE schematic in uprightfluorescence microscope[18].propagation formula.The system shown in Fig.2has been used to record the three holograms[16].The SLM has been phase-only, and as so,the desired sum of two phase functions(which is no longer a pure phase)cannot be directly displayed on this SLM.To overcome this obstacle,the quadratic phase function has been displayed randomly on only half of the SLM pixels,and the constant phase has been displayed on the other half.The randomness in distributing the two phase functions has been required because organized non-random structure produces unnecessary diffraction orders,therefore,results in lower interference efficiency. The pixels are divided equally,half to each diffractive element,to create two wavefronts with equal energy.By this method,the SLM function becomes a good approx-imation to the sum of two phase functions.The phase distributions of the three reflection masks displayed on the SLM,with phase constants of0◦,120◦and240◦,are shown in Figs.3(a),(b)and(c),respectively.Three white-on-black characters i th the same size of 2×2(mm)were located at the vicinity of rear focal point of the lens.‘O’was at z=–24mm,‘S’was at z=–48 mm,and‘A’was at z=–72mm.These characters were illuminated by a mercury arc lamp.The three holo-grams,each for a different phase constant of the SLM, were recorded by a CCD camera and processed by a computer.Thefinal hologram was calculated accord-ing to the superposition formula[14]and its magnitude and phase distributions are depicted in Figs.3(e)and (f),respectively.The hologram was reconstructed in the computer by calculating the Fresnel propagation toward various z propagation distances.Three different recon-struction planes are shown in Figs.3(g),(h),and(i).In each plane,a different character is in focus as is indeed expected from a holographic reconstruction of an object with a volume.In Ref.[17],the FINCH has been capable to record multicolor digital holograms from objects emittingfluo-rescent light.Thefluorescent light,specific to the emis-sion wavelength of variousfluorescent dyes after excita-tion of3D objects,was recorded on a digital monochrome camera after reflection from the SLM.For each wave-length offluorescent emission,the camera sequentially records three holograms reflected from the SLM,each with a different phase factor of the SLM’s function.The three holograms are again superposed in the computer to create a complex-valued Fresnel hologram of eachflu-orescent emission without the twin image problem.The holograms for eachfluorescent color are further combined in a computer to produce a multicoloredfluorescence hologram and3D color image.An experiment showing the recording of a colorfluo-rescence hologram was carried out[17]on the system in Fig. 4.The phase constants of0◦,120◦,and240◦were introduced into the three quadratic phase functions.The magnitude and phase of thefinal complex hologram,su-perposed from thefirst three holograms,are shown in Figs.5(a)and(b),respectively.The reconstruction from thefinal hologram was calculated by using the Fresnel propagation formula[20].The results are shown at the plane of the front face of the front die(Fig.5(c))and the plane of the front face of the rear die(Fig.5(d)).Note that in each plane a different die face is in focus as is indeed expected from a holographic reconstruction of an object with a volume.The second three holograms were recorded via a redfilter in the emissionfilter slider F2 which passed614–640nmfluorescent light wavelengths with a peak wavelength of626nm and a full-width at half-maximum,of11nm(FWHM).The magnitude and phase of thefinal complex hologram,superposed from the‘red’set,are shown in Figs.5(e)and(f),respectively. The reconstruction results from thisfinal hologram are shown in Figs.5(g)and(h)at the same planes as those in Figs.5(c)and(d),respectively.Finally,an additional set of three holograms was recorded with a greenfilter in emissionfilter slider F2,which passed500–532nmfluo-rescent light wavelengths with a peak wavelength of516 nm and a FWHM of16nm.The magnitude and phase of thefinal complex hologram,superposed from the‘green’set,are shown in Figs.5(i)and(j),respectively.The reconstruction results from thisfinal hologram are shown in Figs.5(k)and(l)at the same planes as those in Fig. 5(c)and(d),positions of Figs.5(c), (g),and(k)and Figs.5(d),(h),and(l)are depicted in Figs.5(m)and(n),respectively.Note that all the colors in Fig.5(colorful online)are pseudo-colors.These last results yield a complete color3D holographic image of the object including the red and greenfluorescence. While the optical arrangement in this demonstration has not been optimized for maximum resolution,it is im-portant to recognize that even with this simple optical arrangement,the resolution is good enough to image the fluorescent emissions with goodfidelity and to obtain good reflected light images of the dice.Furthermore, in the reflected light images in Figs.5(c)and(m),the system has been able to detect a specular reflection of the illumination from the edge of the front dice. Another system to be reviewed here is thefirst demon-stration of a motionless microscopy system(FINCH-SCOPE)based upon the FINCH and its use in record-ing high-resolution3Dfluorescent images of biological specimens[18].By using high numerical aperture(NA) lenses,a SLM,a CCD camera,and some simplefilters, FINCHSCOPE enables the acquisition of3D microscopic images without the need for scanning.A schematic diagram of the FINCHSCOPE for an upright microscope equipped with an arc lamp sourceFig.7.FINCHSCOPE holography of polychromatic beads.(a)Magnitude of the complex hologram 6-µm beads.Images reconstructed from the hologram at z distances of (b)34µm,(c)36µm,and (d)84µm.Line intensity profiles between the beads are shown at the bottom of panels (b)–(d).(e)Line intensity profiles along the z axis for the lower bead from reconstructed sections of a single hologram (line 1)and from a widefield stack of the same bead (28sections,line 2).Beads (6µm)excited at 640,555,and 488nm with holograms reconstructed (f)–(h)at plane (b)and (j)–(l)at plane (d).(i)and (m)are the combined RGB images for planes (b)and (d),respectively.(n)–(r)Beads (0.5µm)imaged with a 1.4-NA oil immersion objective:(n)holographic camera image;(o)magnitude of the complex hologram;(p)–(r)reconstructed image at planes 6,15,and 20µm.Scale bars indicate image size [18].Fig.8.FINCHSCOPE fluorescence sections of pollen grains and Convallaria rhizom .The arrows point to the structures in the images that are in focus at various image planes.(b)–(e)Sections reconstructed from a hologram of mixed pollen grains.(g)–(j)Sections reconstructed from a hologram of Convallaria rhizom .(a),(f)Magnitudes of the complex holograms from which the respective image planes are reconstructed.Scale bars indicate image size [18].is shown in Fig. 6.The beam of light that emerges from an infinity-corrected microscope objective trans-forms each point of the object being viewed into a plane wave,thus satisfying the first requirement of FINCH [16].A SLM and a digital camera replace the tube lens,reflec-tive mirror,and other transfer optics normally present in microscopes.Because no tube lens is required,infinity-corrected objectives from any manufacturer can be used.A filter wheel was used to select excitation wavelengths from a mercury arc lamp,and the dichroic mirror holder and the emission filter in the microscope were used to direct light to and from the specimen through an infinity-corrected objective.The ability of the FINCHSCOPE to resolve multicolor fluorescent samples was evaluated by first imaging poly-chromatic fluorescent beads.A fluorescence bead slidewith the beads separated on two separate planes was con-structed.FocalCheck polychromatic beads(6µm)were used to coat one side of a glass microscope slide and a glass coverslip.These two surfaces were juxtaposed and held together at a distance from one another of∼50µm with optical cement.The beads were sequentially excited at488-,555-,and640-nm center wavelengths(10–30nm bandwidths)with emissions recorded at515–535,585–615,and660–720nm,respectively.Figures7(a)–(d) show reconstructed image planes from6µm beads ex-cited at640nm and imaged on the FINCHSCOPE with a Zeiss PlanApo20×,0.75NA objective.Figure7(a) shows the magnitude of the complex hologram,which contains all the information about the location and in-tensity of each bead at every plane in thefield.The Fresnel reconstruction from this hologram was selected to yield49planes of the image,2-µm apart.Two beads are shown in Fig.7(b)with only the lower bead exactly in focus.Figure7(c)is2µm into thefield in the z-direction,and the upper bead is now in focus,with the lower bead slightly out of focus.The focal difference is confirmed by the line profile drawn between the beads, showing an inversion of intensity for these two beads be-tween the planes.There is another bead between these two beads,but it does not appear in Figs.7(b)or(c) (or in the intensity profile),because it is48µm from the upper bead;it instead appears in Fig.7(d)(and in the line profile),which is24sections away from the section in Fig.7(c).Notice that the beads in Figs.7(b)and(c)are no longer visible in Fig.7(d).In the complex hologram in Fig.7(a),the small circles encode the close beads and the larger circles encode the distant central bead. Figure7(e)shows that the z-resolution of the lower bead in Fig.7(b),reconstructed from sections created from a single hologram(curve1),is at least comparable to data from a widefield stack of28sections(obtained by moving the microscope objective in the z-direction)of the same field(curve2).The co-localization of thefluorescence emission was confirmed at all excitation wavelengths and at extreme z limits,as shown in Figs.7(f)–(m)for the 6-µm beads at the planes shown in Figs.7(b)((f)–(i)) and(d)((j)–(m)).In Figs.7(n)–(r),0.5-µm beads imaged with a Zeiss PlanApo×631.4NA oil-immersion objective are shown.Figure7(n)presents one of the holo-grams captured by the camera and Fig.7(o)shows the magnitude of the complex hologram.Figures7(p)–(r) show different planes(6,15,and20µm,respectively)in the bead specimen after reconstruction from the complex hologram of image slices in0.5-µm steps.Arrows show the different beads visualized in different z image planes. The computer reconstruction along the z-axis of a group offluorescently labeled pollen grains is shown in Figs. 8(b)–(e).As is expected from a holographic reconstruc-tion of a3D object with volume,any number of planes can be reconstructed.In this example,a different pollen grain was in focus in each transverse plane reconstructed from the complex hologram whose magnitude is shown in Fig.8(a).In Figs.8(b)–(e),the values of z are8,13, 20,and24µm,respectively.A similar experiment was performed with the autofluorescent Convallaria rhizom and the results are shown in Figs.8(g)–(j)at planes6, 8,11,and12µm.The most recent development in FINCH is a new lens-less incoherent holographic system operating in a syn-thetic aperture mode[19].Synthetic aperture is a well-known super-resolution technique which extends the res-olution capabilities of an imaging system beyond thetheoretical Rayleigh limit dictated by the system’s ac-tual ing this technique,several patternsacquired by an aperture-limited system,from variouslocations,are tiled together to one large pattern whichcould be captured only by a virtual system equippedwith a much wider synthetic aperture.The use of optical holography for synthetic apertureis usually restricted to coherent imaging[21−23].There-fore,the use of this technique is limited only to thoseapplications in which the observed targets can be illu-minated by a laser.Synthetic aperture carried out by acombination of several off-axis incoherent holograms inscanning holographic microscopy has been demonstratedby Indebetouw et al[24].However,this method is limitedto microscopy only,and although it is a technique ofrecording incoherent holograms,a specimen should alsobe illuminated by an interference pattern between twolaser beams.Our new scheme of holographic imaging of incoher-ently illuminated objects is dubbing a synthetic aperturewith Fresnel elements(SAFE).This holographic lens-less system contains only a SLM and a digital camera.SAFE has an extended synthetic aperture in order toimprove the transverse and axial resolutions beyond theclassic limitations.The term synthetic aperture,in thepresent context,means time(or space)multiplexing ofseveral Fresnel holographic elements captured from vari-ous viewpoints by a system with a limited real aperture.The synthetic aperture is implemented by shifting theSLM-camera set,located across thefield of view,be-tween several viewpoints.At each viewpoint,a differentmask is displayed on the SLM,and a single element ofthe Fresnel hologram is recorded(Fig.9).The variouselements,each of which is recorded by the real aperturesystem during the capturing time,are tiled together sothat thefinal mosaic hologram is effectively consideredas being captured from a single synthetic aperture,whichis much wider than the actual aperture.An example of such a system with the synthetic aper-ture three times wider than the actual aperture can beseen in Fig.9.For simplicity of the demonstration,the synthetic aperture was implemented only along thehorizontal axis.In principle,this concept can be gen-eralized for both axes and for any ratio of synthetic toactual apertures.Imaging with the synthetic apertureis necessary for the cases where the angular spectrumof the light emitted from the observed object is widerthan the NA of a given imaging system.In the SAFEshown in Fig.9,the SLM and the digital camera movein front of the object.The complete Fresnel hologramof the object,located at some distance from the SLM,isa mosaic of three holographic elements,each of which isrecorded from a different position by the system with thereal aperture of the size A x×A y.The complete hologram tiled from the three holographic Fresnel elements has thesynthetic aperture of the size3(·A x×A y)which is three times larger than the real aperture at the horizontal axis.The method to eliminate the twin image and the biasterm is the same as that has been used before[14,16−18];。

a r X i v :c o n d -m a t /0108311v 1 [c o n d -m a t .s u p r -c o n ] 20 A u g 2001CURRENT-INDUCED SUPERCONDUCTOR-INSULATOR TRANSITION INGRANULAR HIGH-T c SUPERCONDUCTORSY.Kopelevich*,C.A.M.dos Santos**,S.Moehlecke*,and A.J.S.Machado**(*)Instituto de F ´isica ”Gleb Wataghin”,Universidade Estadual de Campinas,Unicamp 13083-970,Campinas,S˜a o Paulo,Brasil(**)Departamento de Engenharia de Materiais,FAENQUIL,12600-000,Lorena,S˜a o Paulo,BrasilI.INTRODUCTIONThe occurrence of either superconducting or insulatingstate in a zero-temperature limit (T →0)under variation of system parameters such as,for instance,microscopic disorder in homogeneous thin films [1–5]or charging en-ergy/Josephson energy ratio in granular superconductors [6,7],is one of the fundamental problems in the condensed matter physics which continuously attracts an intense re-search interest.It has also been shown both theoretically [8]and experimentally that the superconductor-insulator transition in two-dimensional (2D)superconductors can be tuned by applied magnetic field.The field-tuned superconductor-insulator transition has been measured in amorphous [5,9–11]and granular [12]thin films,fabri-cated 2D Josephson-junction arrays [13,14],and bulk lay-ered quasi-2D superconductors such as high-T c cuprates [15,16].On the other hand,in granular superconductors both applied magnetic field and electrical current affect the Josephson coupling between grains [17–19].Besides,a zero-temperature current-driven dynamic transition from a vortex glass to a homogeneous flow of the vortex mat-ter is expected to occur in disordered Josephson junc-tion arrays [20],suggesting the intriguing possibility of a current-induced superconductor-insulator transition,analogous to the field-tuned transition [21].In this paper we report a systematic study of electrical current effects on the superconducting properties of gran-ular high-T c superconductors.The here presented results demonstrate the occurrence of superconductor-insulator quantum phase transition driven by the applied electri-cal current,and provide evidence that the dynamics of Josephson intergranular vortices plays a crucial role in this phenomenon.II.SAMPLES AND EXPERIMENTAL DETAILSPolycrystalline single phase Y 1−x Pr x Ba 2Cu 3O 7−δsamples were prepared using a solid state reaction method with a route similar as described in [22]and char-acterized by means of x-ray powder diffractometry,andoptical and scanning electron microscopy.The analysis showed that the samples are granular materials consist-ing of grains with an average size d ∼5µm.The su-perconducting transition was measured both resistively and with a SQUID magnetometer MPMS5(Quantum Design).Electrical transport dc measurements in ap-plied magnetic field H ≤100Oe,produced by a cop-per solenoid,were performed using standard four-probe technique with low-resistance (<1Ω)sputtered gold con-tacts.No heating effects due to current were observed for I ≤100mA,our largest measuring current.In order to eliminate thermoelectric effects,the measurements were performed inverting the applied current.Below we present the results of measurements per-formed on Y 1−x Pr x Ba 2Cu 3O 7−δsample with x =0.45close to the critical Pr concentration x c ≈0.57above which superconductivity has not been detected [22].The sample superconducting transition temperature (onset)T c 0=33K,the resistivity at the superconducting tran-sition ρN (T c 0)=20mΩcm,and dimensions l x w x t =12.6x 1.94x 1.24mm 3.III.RESULTS AND DISCUSSIONFigures 1–4present the resistance vs.temperature data obtained in a vicinity of T c 0=33K for various applied currents and magnetic fields.As Figs.1–4il-lustrate,the superconducting transition temperature on-set T c 0is both current-and field-independent.On the other hand,the zero-resistance superconducting state is destroyed by the application of both the electrical current and magnetic field.The superconducting order parameter Ψ=Ψ0e iφhas two components;a magnitude Ψ0and a phase φ.Be-cause T c 0(and hence Ψ0)remains unchanged,the low-temperature finite resistance in our sample originates from thermal and/or quantum fluctuations in the phase locking.In the context of granular superconductors,T c 0can be identified with the transition temperature of indi-vidual grains,and the phase fluctuations –with fluctuat-ing Josephson currents between grains.In the absence of external field,the critical current is that corresponding to a vortex creation and its motion,neglecting pinning.However,in granular superconductors with a very weak coupling between grains,the Earth’s magnetic field H E ∼0.5Oe (which has not been shielded in the experiments)can easily penetrate the sample.Then,the criticalvarious at the at no variousOe. various Oe.variousOe.fromtemperatures and at no appliedfield.As can be seen from Fig.5, the V(I)dependencies can be very well described by the equationV=c(T,H)(I−I th(T,H))n(T,H),(1) where I th(T,H)is the threshold current.Thus,at T=4.6K,I th=5mA and the corresponding current density(I th/cross section of the sample)j th=0.21A/cm2.T=13fitting3(T=(T=8=8K);13K).fieldH c 1J ∼8π2j c 0λL /c ∼0.2mOe,taking the maximum Josephson current densityj c 0∼10j th [23,26]and the typ-ical value of the intragranular London penetration depth λL ∼0.1µm (<<d).The obtained value of H c 1J is much smaller than the Earth’s magnetic field,indeed,i.e.our sample is deeply in the mixed state.Below the threshold current I th (H,T)the vortex mo-tion is strongly suppressed or zero.The I-V charac-teristics described by the Eq.(1)are expected in the regime where the interaction between vortices and the pinning potential dominates the vortex-vortex interac-tion [20,27–29].As the applied current increases,the V(I)curves ap-proach a linear regime of flux flow where the differential resistance R d =dV/dI is current-independent,see Fig.6.Inset in Fig.6exemplifies V(I)measured at T =13K and demonstrates that at large enough currents the V(I)can be fitted by the equationV (I )=R ff (I −I cf ).(2)Here R ff is the flux-flow resistance and I cf is the so-called dynamical critical current.In this high-current regime vortices move coherently,only weakly interacting with the pinning potential.at no K (△),13K;0.215performed at various applied fields revealed that I d (H)is a decreasing function of field.Such a crossing has also been measured in Mo 77Ge 23films [30]and reproduced in numerical simulations [31]which show that the pinned fraction of vortices rapidly decreases for I >I d (H).Figure 7presents typical resistance hysteresis loops R(H)measured at T =4.2K for various currents.These measurements were performed after cooling the sample from T >T c 0to the target temperature in a zero ap-plied field.As shown in Fig.7,the resistance R cor-responding to the increasing |H |branch is larger than that corresponding to the decreasing |H |branch result-ing in “clockwise”R(H)hysteresis loops.Previous stud-ies [32–34]showed that the “clockwise”R(H)hystere-sis loops are essentially related to the granular sample structure where the resistance is determined by the mo-tion of Josephson intergranular vortices.In increasing magnetic field larger than the first critical field of the grains (H >H c 1g ),the pinning prevents Abrikosov vor-tices from entering the grains,and flux lines (Joseph-son vortices)are concentrated between grains leading to the intergrain field higher than the external applied field.When the applied field is decreased,the pinning prevents Abrikosov vortices being expelled from the grains result-ing in a lower density of Josephson intergrain vortices as compared to that in the increasing field.Thus,the data of Fig.7provide an unambiguous evidence that the resistance in our sample is governed by the motion of intergranular Josephson vortices in both low-and high-current limits.and I =60mA,demonstrating that the sample resistance is governed by the motion of intergranular Josephson vortices;arrows indicate the magnetic field direction in the measure-ments.We note further that the resistance R(T)=V(T)/I in the low-temperature limit reveals a crossover from the metallic-like (dR/dT >0)to the insulating-like (dR/dT <0)behavior as the applied current isincreased,see Figs.1-4.The results presented in Figs.1-4indi-cate also the existence of a field-dependent current I c (H),separating the metallic-like (I <I c (H))and insulating-like (I >I c (H))resistance curves,at which the resis-tance is temperature-independent.The occurrence of the temperature-independent resistance R c =(V/I)|I c can clearly be seen in Fig.8where a crossing of current-voltage I-V isotherms at I c (H)measured for two applied fields is shown.The inset in Fig.8depicts I c (H)and I d (H)which illustrates that for a given field I c (H)>I d (H).In other words,the crossing of I-V curves takes place in the regime where the interaction between Joseph-son vortices dominates that with the pinning potential.in ).Inset to to the zero-resistance (R =0)superconducting state.With in-creasing vortex density and above some critical field H c vortices delocalize and Bose condense,leading to the in-sulating state.The theory predicts a finite-temperature scaling law [2,3,8]which allows for experimental testing of the zero-temperature superconductor-insulator transition.It is assumed in this analysis that a sample is superconduc-tor or insulator at T =0when dR/dT >0or dR/dT <0,respectively.Besides,because the magnitude of the superconducting order parameter Ψ0(T c 0)remains unchanged on each grain as the intergranular resistance R(T,H,I)undergoes the transition from metallic-like to insulating-like behavior,the hard-core boson model [2,3,8]should be a good approximation in our analysisof the superconductor-insulator transition driven by the applied current.The resistance in the critical regime of the quantum transition is given by the equationR (δ,T )=R c f (|δ|/T 1/zν),(3)where R c is the resistance at the transition,f(|δ|/T 1/zν)is the scaling function such that f(0)=1,z and νare critical exponents,and δis the deviation of a variable parameter from its critical value.Assuming δ=I -I c (δ=H -H c in the case of a field-tuned transition)we plotted in Figs.9-11,R =V/I vs.|δ|/T 1/αfor three applied fields.In each case,the exponent αwas obtained from log-log plots of (dR/dI)|I c vs.T −1.The expected collapse of the resistance data onto two branches distin-guishing the I <I c from the I >I c data is very clear in Figs.9-11.scaling =16.26.7K scaling mA,(△),T the re-sults presented in Figs.9-11should be considered as empirical ones,the very good scalingfit strongly suggests the occurrence of a zero-temperature current-induced superconductor-insulator transition.Notably,the here found exponentsαagree nicely with most of the data obtained in the scaling analysis of thefield-tuned transi-tion[5,9–16].scalingmA,α),T= In conclusion,we would like to emphasize that the here studied superconductor-insulator transition driven by the electrical current is essentially related to the dynamics of Josephson intergranular vortices.Actually,as the above results demonstrate,the crossing of I-V characteristics takes place in thefluxflow regime.We stress that the crossover current I c(H)is smaller than the maximum su-percurrent I0=(2e/ )E J(E J is the Josephson coupling energy)that the Josephson junctions can support.In other words,both tunneling Cooper pairs and moving Josephson vortices are present at the I c(H)which play a dual role as discussed in the context of thefield-tuned transition[8].A self-consistent analysis would require the existence of zero-temperature“depinning”transition in the Joseph-son vortex matter driven by the applied current.The occurrence of such a transition in disordered Josephson junction arrays has theoretically been predicted,indeed [20].All these indicate that the current-induced superconductor-insulator transition can be considered as the dynamical counterpart of the magnetic-field-tuned transition.Finally,we point out that similar results were obtained on Y1−x Pr x Ba2Cu3O7−δ(x=0.5)and Bi2Sr2Ca1−x Pr x Cu2O8+δ(x=0.54)granular supercon-ductors.V.ACKNOWLEDGEMENTSThe authors would like to thank Enzo Granato for the fruitful discussions.This study was supported by FAPESP and CNPq Brazilian science agencies.New Superconductors”in H.Ehrenreich and D.Turnbull (eds.)Solid State Phys.42(1989)91.[26]C.J.Lobb,D.W.Abraham,and M.Tinkham,Phys.Rev.B27(1983)150.[27]P.Berghuis and P.H.Kes,Phys.Rev.B47(1993)262.[28]S.Bhattacharya and M.J.Higgins,Phys.Rev.Lett.70(1993)2617.[29]Y.Enomoto,K.Katsumi,and S.Maekawa,Physica C215(1993)51.[30]M.C.Hellerqvist,D.Ephron,W.R.White,M.R.Beasly,and A.Kapitulnik,Phys.Rev.Lett.76(1996)4022;M.C.Hellerqvist and A.Kapitulnik,Phys.Rev.B56(1997)5521.[31]S.Ryu,M.Hellerqvist,S.Doniach,A.Kapitulnik,andD.Stroud,Phys.Rev.Lett.77(1996)5114.[32]Y.Kopelevich,V.V.Lemanov,and V.V.Makarov,Sov.Phys.Solid State32(1990)2095;V.V.Makarov and Y.Kopelevich,Phys.Rev.B54(1996)84.[33]L.Ji,M.S.Rzchowski,N.Anand,and M.Tinkham,Phys.Rev.B47(1993)470.[34]S.Li,M.Fistul,J.Deak,P.Metcalf,and M.McElfresh,Phys.Rev.B52(1995)747.。

a r X i v :c o n d -m a t /9307055v 1 27 J u l 1993Holons in chiral spin liquids:statistics and pairingD.M.Deaven,D.S.RokhsarDepartment of Physics,University of California,Berkeley,CA 94720A.BarbieriLawrence Berkeley Laboratory,Berkeley,CA 94720(February 1,2008)We study Gutzwiller-projected variational wavefunctions for charged,spinless holon excitations in chiral spin liquids.We find that these states describe anyons,with a statistical phase Φs that is continuously adjustable between 0and π/2,depending on a variational parameter.The statistical flux attached to each holon is localized to within a lattice constant.By diagonalizing the effective Hamiltonian for charge motion in our variational basis we obtain a two-holon pair state.This two-holon state has large overlap with a two-hole state whose relative wavefunction has d -wave symmetry and breaks time-reversal and parity.I.INTRODUCTIONThe notion of a quasiparticle is a remarkably successful organizing principle for understanding many-body sys-tems.In strongly interacting Fermi systems such as 3He,for example,many-body effects simply renormalize the mass and interactions of the Landau quasiparticles;their charge,spin,and statistics the same as those of the bare atoms.1Other many-body systems,however,possess low-energy charged excitations that are dramatically differ-ent from Landau quasiparticles.In the fractional quan-tum Hall effect,for example,a large magnetic field and the Coulomb repulsion between electrons conspire to pro-duce Laughlin quasiparticles with fractional charge and statistics.2In trans-polyacetylene,a system with broken translational symmetry,charges bind to domain walls,forming soliton excitations with charge e but without spin.3In both of these cases,the novel quasiparticles have a “topological”character,because they cannot be cre-ated by local combinations of bare electron creation and destruction operators,in contrast with Landau quasipar-ticles.Nevertheless,these unconventional quasiparticles are well defined,weakly interacting objects which de-scribe the low-energy states of the many-body system.Similarly,unconventional quasiparticles have been sug-gested as the charge carriers in “resonating valence bond”theories of high temperature superconductivity.4,5,6Mo-tivated by the fact that cuprate high temperature su-perconductivity arises upon doping antiferromagnets ex-hibiting large quantum fluctuations,these theories focus on spin-1/2systems with local antiferromagnetic corre-lations but no long range order –so-called “spin liquids.”The phase diagram of the high temperature supercon-ductors motivates the study of spin liquids as a step-ping stone in a novel approach to the superconducting state.Referring to fig.1,the real materials reside inthe temperature-doping plane;an additional axis,labeled frustration,is added for theoretical convenience.When the parent insulators are doped,antiferromagnetic long range order is destroyed almost simultaneously with the onset of superconducting order.(A “spin glass”regime due to disorder may intervene.)It is appealing to try to separate the effects of loss of magnetic order from the in-stability to superconductivity.In principle,this could be accomplished theoretically by first introducing fictitious frustrating interactions in the antiferromagnetic insula-tor,thereby destroying long range order without intro-ducing charge carriers.Then,to study the appearance of superconductivity,this “spin liquid”state would be doped,so that one could consider the motion of dilute charge carriers in a spin background resembling the fluc-tuating spins in the real materials’superconducting state.Finally,to return the theory to a more realistic model of the actual copper oxide planes,the fictitious frustrat-ing interactions would be turned off–they would be no longer needed,since magnetic frustration is implicitly generated by charge carrier motion.7In this roundabout manner,one would arrive at a description of the real ma-terials’superconducting state.Assuming that no phase transition intervenes,the superconducting state achieved upon doping the spin liquid would have the same sym-metry and general features as the real superconducting state.This approach contrasts the more direct “BCS approach,”which begins from an understanding of the normal metal at high temperature (which may be prob-lematic in the cuprates)and then considers residual at-tractive interactions between quasiparticles.If properly executed,both the spin liquid and the BCS approaches should arrive at the same superconducting state,but from different directions.Unfortunately for the spin liquid approach,it has proven difficult to find suitably frustrated spin mod-FIG.1.Schematic phase diagram for high temperature superconductivity.The real materials occupy the temper-ature-doping plane;an additional axis,labeled frustration, is included for theoretical convenience.In a BCS-like ap-proach,the superconducting state is entered from a well un-derstood normal state.In the“resonating-valence-bond”or “spin liquid”approach,a disordered quantum antiferromag-net is stabilized by artificially adding magnetic frustration to the undoped parent compound.This spin liquid is then doped to form a superconductor.The condensate is formed from bound holon pairs,the charge carrying quasiparticles in a doped spin-liquid.If no phase transition intervenes as the frustration is turned offin the doped state,both the spin liquid and BCS approaches must arrive at the same super-conducting state,but from complementary physical points of view.els which destroy antiferromagnetic order without in-troducing additional unwanted broken symmetry ground states.There is no short-ranged spin model known to have a spin liquid ground state.Nevertheless,spin liq-uids may be usefully thought of as metastable phases of frustrated antiferromagnets that mimic the spin corre-lations of cuprate superconductors,even if they are not the true ground state of a particular frustrated spin sys-tem.One can still study their excitation spectra,and hope to carry out the program outlined above.In this work we consider simple variational“chiral spin liquid”states8which break time-reversal and parity symmetries but are translationally invariant.They have good vari-ational energies for simple frustrated spin Hamiltonians, and suggest natural variational states when doped.By studying the motion of a few charge carriers,we develop a scenario for understanding superconductors with broken time-reversal and parity symmetry.We consider the t-J-J′Hamiltonian H=H spin+H hop, where thefirst term is the(frustrated)spin HamiltonianH spin= ij J ij S i·S j,(1)withfirst and second neighbor antiferromagnetic cou-plings J and J′,and the second term is the nearest-neighbor hopping HamiltonianH hop=−t ij ,σc†iσc jσ.(2)In these expressions c†iσcreates a fermion at site i with spin projectionσand S i≡c†iασαβc iβis the spin operator at site i.Both Eqs.(1)and(2)are allowed to act only within the subspace of states in which no lattice site is doubly occupied.In this work we address(a)the internal structure of charged“holon”excitations in a locally stable spin-liquid and(b)the dynamics and statistics of a dilute gas of these excitations.Sections II and III contain a brief general dis-cussion of spin liquids and holon excitations and describe the specific wavefunctions we consider.Our main results are presented in sections IV and V.A brief summary is provided in section VI.II.CHIRAL SPIN LIQUIDSSpin liquids are quantum antiferromagnets whose N´e el order has been destroyed by large zero-pointfluctuations of the interacting spins.These large quantum effects pre-clude the use of a spin-wave expansion about an ordered state.Instead,we appeal to the variational principle and take advantage of qualitative similarities between spin liquid states and certain simple Slater determinant states of spin-1/2fermions.In particular,consider Slater deter-minant states|χ which(a)are translationally invariant with an average density of one fermion per site,(b)have no average magnetic moment at each site,and(c)have spin and charge correlations which decay exponentially. All three of these properties are shared by spin liquids. The main difference between|χ and a spin liquid is sim-ply one of degree–a spin liquid has no numberfluctua-tions at all,while|χ has local numberfluctuations. This distinguishing property of|χ can be eliminated by applying the complete Gutzwiller projectorP G≡Πi(1−n i↑n i↓),(3) which annihilates all configurations containing doubly-occupied sites.What remains is a superposition of spin configurations whose amplitudes are inherited from the original“pre-projected”Slater determinant.If this pre-projected state has short-ranged spin and charge correla-tions,then Gutzwiller projection is a“local”procedure.9 The correlations in the projected state then closely follow the correlations of the pre-projected determinant,up to simple renormalization factors of order unity.10,11,12It is surprisingly difficult to construct Slater determi-nants with the three properties listed above.To ob-tain a translationally invariant free-particle state with only short-range correlations,one should completelyfillFIG.2.Schematic of a tight binding model Hχwhich breaks time-reversal and parity symmetries.Circles repre-sent sites;lines represent hopping matrix elements between sites.A line with an arrow has phaseπ/4in the sense given by the arrow,a dashed line has phaseπ,and a solid line zero phase.(a)Theflux state.Each elementary plaquette hasπflux piercing it.(b)The chiral state.Each elementary triangle hasfluxπ/2piercing it.Up to a gauge transformation,these tight-binding Hamiltonians are translationally invariant.a band of single-particle states while simultaneously en-suring that a gap exists between thefilled band and the next highest energy band.For a translationally invariant system,the resulting charge density will be uniform.The difficulty arises in constructing a translationally invariant system with a one-body energy gap.The simplest way to accomplish this is to consider particles moving on a lattice in a uniform commensurate(fictitious)magnetic field which couples only to orbital motion.8,13For appro-priatefield strength,the effective unit cell is doubled, and a gap is opened at half-filling.The hopping matrix elements will now be complex;as long as the net phase factor in the product of the elements around any closed loop is invariant under translations,the model ground state will also be translationally invariant,up to an over-all gauge transformation.FIG.3.Single-particle bands of Eq.(4)in the two-site Square√2magnetic unit cell shown by the dotted line in the inset,with no background magneticfield(dashed line) and in the presence of a translationally invariant chiralfield (solid line)as illustrated infig. 2.Note that a gap equal to 4m0is opened at half-filling(filling the lower band).In particular,let|χ be the half-filled ground state of the square-lattice tight-binding Hamiltonian8Hχ= ijσχij c†iσc jσ(4)illustrated infig. 2.Hereχij is a complex link vari-able and c†iσcreates a fermion at site i with spin pro-jectionσ.We arrange for a net phaseπaround ev-ery plaquette,and phaseπ/2around every elementary triangle(encircled clockwise,say).More explicitly,the complex productχABχBCχCDχDA around any elemen-tary plaquette ABCD has phaseπ,while the product χABχBCχCA around any elementary triangle ABC has phaseπ/2.We will refer to the phase around a closed loop a“flux,”since the(Aharonov-Bohm)phase for mo-tion in a magneticfield is proportional to theflux en-closed by the path.Oneflux quantum hc/e corresponds to a phase2π.The Hamiltonian Eq.(4)explicitly breaks both time-reversal and parity symmetries,each of which inverts the flux through every closed loop.In this work we shall set χij=0for all but nearest and next-nearest neighboring sites i and j.We set|χij|=1when sites i and j are nearest neighbors,and|χij|=m0/2when sites i and j are next-nearest neighbors.This notation derives from the fact that for small values of m0and at halffilling,the noninteracting fermions whose motion is governed by Eq.(4)have a gap to excitations equal to4m0(“mass,”in field-theory jargon).Figure3illustrates the gap opening due to the addition of the diagonal hopping strength m0. The similarity of the Gutzwiller-projected states P G|χ to the actual ground states of a doped antiferromagnetmay be measured by their variational energy.Consider H with J ij=J for nearest neighbor sites i and j,and J ik=J′for next-nearest neighbor sites i and k.Then for each value of J′/J there is a corresponding value of m0which minimizes the variational energyE(J′/J;m0)= χ|P G H spin(J′/J)P G|χlong-range correlations of the state,the extraπflux at the defective plaquettes should survive Gutzwiller pro-jection,resulting in localized excitations with“semion”statistics halfway between fermion and boson.Charged holons are created in this scheme by removing a site at the center of the added localizedflux.Despite scheme2’s intuitive appeal,we will not use it further,since we are interested in off-diagonal matrix ele-ments of the hopping Hamiltonian Eq.(2)between holon states at different positions.In scheme2,this would re-quire working in a mixed basis of two Slater determinants that are ground states of different pre-projected Hamil-tonians.Remarkably,it can be shown for a special value of m0that schemes1and2are essentially equivalent (details are given in the appendix).In this work,all our calculations use scheme1(Eqs.6,7,and8).IV.HOLON STATISTICSThe charge motion in a spin liquid is governed by thehopping Hamiltonian Eq.(2).The off-diagonal matrix elements of H in the holon variational basis|ij contain FIG.4.Schematic of defect Hamiltonian H d used in scheme2(described in the text)for creating spinons and holons.(a)A localizedflux is added to a plaquette,identified here by a small open circle.To do this,each link between sites crossed by the Dirac string(dashed line)originating at the addedflux is multiplied by−1.At the other end of the string another localizedflux is created(not shown).(b)A site is deleted from the system by cutting all links to it(removing all hopping matrix elements to surrounding sites).FIG.5.(a)Density of states resulting from the chiralflux Hamiltonian Hχ,Eq.(4)when m0=1(solid line).The dotted line spectrum was computed on a512-site system with periodic boundary conditions and broadened with a width 0.01.(b)Density of states resulting from the same512-site system but with a pair ofπflux defects,H d as illustrated infig. 4.Fluxes have been placed as shown in the inset. Adding thefluxes leads to the formation of midgap states which split offfrom the upper and lower band,as shown.See the appendix for further discussion of these midgap states. information regarding the kinetic energy and statistical phase of holons.In particular,these matrix elements are not simply given by the bare t ij.The effective matrix element for one holon hopping from site i to site j in the presence of another holonfixed at site k is given byT k ij=− jk|H hop|ik ,(9) where i and j are nearest-neighbor sites.We evaluate Eq.(9)using a method described previously.12Briefly,we consider a cluster of NΓsites embedded in an infinite system.Correlations within the cluster are computed exactly,while correlations outside the cluster are accounted for only approximately.The thermodynamic limit is taken by considering clusters of various shapes and sizes centered on the sites i,j,and k, and extrapolating to the infinite cluster limit.In prac-tice,this limit is reached when the cluster size exceeds the correlation lengthξχin the pre-projected Slater determi-nant|χ .This method is substantially more efficient than exact projection of Slater determinants onfinite toroidal systems(see below).The effective holon hopping matrix element T k ij con-tains information about holon statistics.17Computation of T k ij allows a direct check of the fractional statisticsinvoked in the picture of Zou et al..18The hopping Hamil-tonian Eq.(2)restricted to the holon pair basis Eq.(8) isH T=− ijk T k ij|kj ki|.(10)In general,we can separate T k ij into a magnitude and a phase,T k ij=|T k ij|exp i j i A(k,l)·d l ,(11)where the phase is interpreted as the line integral of an effective vector potential,A.This vector potential can be separated19into two distinct parts:A(k,l)=A0(l)+A S(k,l).The part of A which is independent of theposition of site k is absorbed into A0,and corresponds to a uniform background magneticflux through whichthe holons move.The remaining part A S depends on the relative holon position.Since the holon wavefunctions Eq.(8)can be madesymmetric under holon interchange,20the vector poten-tial A S contains complete information about the holonstatistics.Consider moving one holon around a part-ner which is heldfixed at site k.The net phase factorfor hopping one holon completely around anotherfixed holon is A S·d l≡2ΦS.When the moving holon path is made infinitely large,ΦS is the statistical phase for holon interchange.17To implement this program,wefirst compute the back-ground vector potential A0by calculating T k ij when k (thefixed holon location)is far from i and j(the initial andfinal moving holon locations).Wefind numerically that the backgroundflux is indistinguishable from thefic-titious uniformflux introduced in the pre-projected state. The vector potential A0corresponds to a uniformflux ofπthrough each elementary plaquette,as explicitly in-cluded in Hχ.To compute A S we again calculate21T k ij’s on a path circling thefixed partner holon,and subtract the contribution from A0.What is the“form factor”of the statisticalflux bound to each holon?One might expect that the boundflux would be distributed in a region of linear extentξabout the holon.The statistics would only be well-defined for holons separated by this distance.This situation is sim-ilar to the case of the4He atom,which is a composite of six fermions that behaves as a boson only on length scales longer than a Bohr radius,so that the internal atomic structure is essentiallyfixed.Surprisingly,wefind that for the holon wavefunctionsEq.(8)the statisticalflux around each holon is sharply peaked in its immediate vicinity,and that the radius of the boundflux tube is the lattice spacing,not the spin correlation length.Figure6shows the statisticalfield density C A S·d l where the path C encloses one plaque-tte,plotted against the plaquette’s geometrical distance from the origin.Most of the statisticalflux is concen-trated in the core region immediately surrounding the FIG.6.Statisticalfield density C A S·d l through the ele-mentary plaquettes surrounding afixed holon,when m0=1 (filled circles),m0=0.4(open squares),and m0=0.2(open circles).Monte Carlo statistical error bars are approximately the same size as the circles.The inset illustrates the positions of the plaquettes whosefield density has been plotted;each data point is accompanied by a letter corresponding to its location.holon(four plaquettes).Thus the phase shown infig. 7represents the total statistical phase for far-separated holons.Figure7shows the statisticalfluxΦS threading a small 2×2square loop around afixed holon as a function of m0.For small m0,the spin-spin correlation length tends to infinity,and the statisticalflux through this small path vanishes.For m0of order unity,the spin-spin correlation length is essentially the lattice constant,and the statis-ticalflux nearsπ/2.A slight negative contribution may be present outside the core(seefig.6,especially thefilled points repre-senting m0=1).We are unable to determine whether this contribution is present for inter-holon separations exceeding∼5lattice constants.Thus we cannot rule out the possibility that a small statisticalflux density at large distances from a holon integrates to give a signif-icant contribution toΦS.The existence of such a long-range tail to theflux distribution could make the holon statistics ill-defined,since the net phase would depend on the inter-holon separation.Neglecting this possible long-range part of theflux distribution,we conclude that the effective statistics of the excitations defined by Eq.(8) are continuously adjustable between values of zero(cor-responding to a boson)andπ/2(a“semion”).Using an analogy with the fractional quantum Hall effect,Laughlin and Zou have argued18that the holon statistics in chiral spin liquids should be quantized as semions.Although the logic of this argument seems strong,our calculations(and the numerical calculations of ref.18)find a statistical phase that is not simplyπ/2. Since here we have carefully considered extrapolation to the thermodynamic limit,this is not the source of the dis-FIG.7.Total phase accumulated by hopping one holon around another for the shaded loop shown in the inset,as a function of the mass parameter m0.This is the core contri-bution to twice the statistical phase,2ΦS.Thefilled circles are computed using the cluster method,after extrapolation to the thermodynamic limit.The open circles represent the same quantity calculated on an8×8torus,which suffers from a discrete sampling effect and is inaccurate(see text).Monte Carlo statistical error bars are approximately the same size as the circles.The lines are guides to the eye. crepancy.There are two possible resolutions:(1)there is a weak,longer-range tail to the statisticalflux distri-bution which we cannot resolve in our calculations,but which adds up to enforceΦS=π/2,or(2)the variational states we have considered are not good holon states for small m0,but only for m0of order unity.It is only for such m0that the correspondence between scheme1, scheme2,and the construction of Laughlin and Zou18 has beenfirmly established.Our embedded cluster results agree with calculations performed onfinite tori when properly extrapolated to the thermodynamic limit.While the two methods con-verge to the same result,the toroidal calculations con-verge much more slowly,and with importantfinite size effects.(This is also true for spin-spin correlations.12)To confirm this we calculated the core contribution to the statistical phase for L×L tori with L=4,6,8,10,12, 14,and16.To confront the worst-case scenario forfinite size effects we studied small m0in detail,where the cor-relation lengthξis longest.Fig.8shows the statistical phase from the nearest four plaquettes for m0=0.01and various toroidal systems.The embedded cluster method rapidly converges to zero statisticalflux,as does the the sequence of tori with L=4n+2=6,10,14,...The se-quence of tori with L=4n=4,8,12,16,...also extrap-olate to zero,but surprisingly slowly.This is likely due to the discrete Brillouin zone sampling at special symmetry points for these tori,aflaw which is not present in ourFIG.8.Extrapolation to the thermodynamic limit N→∞of the core contribution to the statisticalflux(shown infig.7)when m0=0.01.(N is the number of sites in the projected Slater determinant.)Open squares reresent4n×4n toroidal systems,solid squares(4n+2)×(4n+2)toroidal systems.The 4n×4n toriodal systems suffer from an effect resulting from the discrete sum over the single particle Brillouin zone of Hχ, Eq.(4).Open circles indicate the result obtained with N-site circular clusters embedded in an infinite system.The dashed line shows the result for a small embedded cluster(N=13), at which point convergence has already been reached for the phase information in T k ij.The Monte Carlo statistical error bars are smaller than the markers.embedded cluster method.This accounts for the appar-ent discrepancy between the8×8torus results and the embedded cluster method infig.7.(The4×4torus is a particularly extreme case:the Slater determinant ground state wavefunction of Hχis then independent of m0due to this sampling effect.)From this calculation we con-clude that thefinite tori and embedded cluster methods agree when extrapolated to the infinite system limit.V.PAIR STATESWe can now compute the pair wavefunction for two holons moving through a chiral spin liquid background by simply numerically diagonalizing H T in the two-holon basis Eq.(8).Wefind that the lowest-energy two-holon bound state is a complex d-wave state which breaks T and P symmetries.22To diagonalize H T,we note that this two-body Hamil-tonian is invariant under the same set of(magnetic)lat-tice translations as Hχ.Bloch’s theorem then states that each pair wavefunction can be labeled by a center of mass momentum K,and must have the form|Φ K= i<j e i K·(r i+r j)/2φ(ij)|ij ,(12)where r i is the position of site i,and the relative wave-functionφ(ij)depends only on the lattice translation bringing site j into the same magnetic unit cell as site i,as well as the basis indices of the two sites.TABLE I.Holon interactions.Hopping matrix element T for one holonfixed at site i and another moving from site j to site k when m0=1.In the radial case sites i,j,and k are collinear,in the azimuthal case the hopping direction(jk)is perpendicular to(ij).In each case d jk>d ij.|T i jk|/td ij radial azimuthal V ij/JFIG.10.The relative pair wavefunctionΦ(i,j).allφn are real numbers.The symmetry is a combination of d x2−y2and d xy.is then a gauge invariant description of that part of the two-holon state which resembles a singlet hole pair. How similar are two holons and a singlet hole pair? The overlap between these two states is simply an expec-tation value in the state|χ(ij)|ij =2N ij M ij χ|P G(1−n i↑)n j↑c†i↓c j↓|χ ,(16) and can be computed using the embedded cluster me-thod:The magnitude of this overlap versus inter-holon distance d ij is plotted infig.9for three different val-ues of m0.A nearest-neighbor holon pair is essentially identical to a singlet hole pair,as shown by the near-unit overlap infig.9at d ij=1in all three cases.The overlap decays quickly as sites i and j are separated,presumably since far-separated holons disrupt the system differently than holes.Holes maintain a strong spacial correlation between spin and charge,while the holon has no spin. When holons are close together,however,the composite object has charge2e and spin-0,permitting an overlap with a singlet hole pair with the same quantum numbers. The decay of the overlap with distance occurs on a length scale roughly equal to the correlation lengthξχ,as shown infig.9.Using Eq.(16)and the two-holon state|Φ obtained by exact diagonalization of H T,we can compute the“hole pair wavefunction”(the inner product Eq.(15))for inter-holon separations up to4lattice constants.Figure11 shows the radial probability density|Φ(ij)|2as a function of the inter-holon distance d ij.The amplitude is mostly concentrated near the origin,due to the kinetic resonance effect already mentioned and the short range of the holon pair/hole pair overlap.Further static local attraction due to the sharing of broken bonds only enhances this effect,FIG.11.Probability density in the two-holon wavefunctionΦ(ij)versus inter-holon separation for holons governed by H T with m0=1(filled circles),m0=0.2(open circles).In each case the wavefunction’s angular symmetry is d-wave asillustrated infig.10.without changing the angular symmetry of the state,as we verified numerically by considering additional short-range static attraction mimicking V ij.For all values of m0,the relative hole pair wavefunc-tionΦ(ij)has d-wave symmetry(i.e.,the state changes sign under90degree rotation),as illustrated infig.10. Because of the explicit breaking of time-reversal and par-ity in the chiral spin liquid,this state is a complex linear combination of d x2−y2and d xy states.(Although these states transform as different irreducible representations of the tetragonal group,they are mixed in the group of four-fold rotations that remains when reflections are re-moved from the tetragonal group.22)As shown infig.10, the relative pair wavefunction is purely real on the x and y axes,and purely imaginary along the diagonals;Φ(ij) is permitted by symmetry to be a general complex num-ber elsewhere,but wefind that it remains either purely real or imaginary in non-symmetry directions as well. Since the two-holon state|Φ has substantial overlap with a two-hole state,it is natural to consider a multi-holon state obtained by macroscopically occupying the hole pair wavefunctionΦ(ij),thus arriving at a BCS su-perconductor with a d-wave Cooper wavefunction.To show that this is the ground state of H or a similar spin Hamiltonian requires a much more elaborate calculation than we have given here.Nevertheless,if we assume that the“anyon condensate”of Laughlin et al.is obtained by a Bose condensation of holon pairs,then we may infer from our calculation that such a state will have conven-tional electron-pair off-diagonal long-range order:lim|(ij)−(i′j′)|→∞c†i↑c†j↓c i′↓c j′↑ →Ψ∗(ij)Ψ(i′j′).(17)The order parameterΨ(ij)is nothing more than the in-ternal Cooper pair wavefunction,which for dilute,non-overlapping pairs should be given simply by the relative。

a r X i v :c o n d -m a t /0402681v 2 [c o n d -m a t .s u p r -c o n ] 16 M a r 2004Superconducting phase diagram of the filled skuterrudite PrOs 4Sb 12M.-A.Measson 1,D.Braithwaite 1,J.Flouquet 1,G.Seyfarth 2,J.P.Brison 2,E.Lhotel 2,C.Paulsen 2,H.Sugawara 3,and H.Sato 31D´e partement de Recherche Fondamentale sur la Mati`e re Condens´e e,SPSMS,CEA Grenoble,38054Grenoble,France2Centre de Recherches sur les Tr`e s basses temp´e ratures,CNRS,25avenue des Martyrs,BP166,38042Grenoble CEDEX 9,France3Department of Physics,Tokyo Metropolitan University,Minami-Ohsawa 1-1,Hashioji,Tokyo 192-0397,JapanWe present new measurements of the specific heat of the heavy fermion superconductor PrOs 4Sb 12,on a sample which exhibits two sharp distinct anomalies at T c 1=1.89K and T c 2=1.72K.They are used to draw a precise magnetic field-temperature superconducting phase diagram of PrOs 4Sb 12down to 350mK.We discuss the superconducting phase diagram of PrOs 4Sb 12and its possible relation with an unconventional superconducting order parameter.We give a detailed analysis of H c 2(T ),which shows paramagnetic limitation (a support for even parity pairing)and multiband effects.PACS numbers:65.40.Ba,71.27.+a,74.25.Dw,74.25.Op,74.70.TxI.INTRODUCTIONThe first Pr-based heavy fermion (HF)superconductor PrOs 4Sb 12(T c ∼1.85K)has been recently discovered 1.Evidence for its heavy fermion behavior is provided mainly by its superconducting properties,like the height of the specific heat jump at the superconducting tran-sition or the high value of H c 2(T )relative to T c 1.PrOs 4Sb 12is cubic with T h point group symmetry 2,and has a nonmagnetic ground state,which in a single ion scheme can be either a Γ23doublet or a Γ1singlet,a ques-tion which remains a matter of controversy.Presently,most measurements in high field seem to favor a singlet ground state 3,4,5,6.In any case,whatever the degeneracy of this ground state,the first excited state (at around 6K)is low enough to allow for an induced electric quadrupo-lar moment on the Pr 3+ions 7,8,that could explain the heavy fermion properties of this system by a quadrupo-lar Kondo effect 1.Thus,while the pairing mechanism of usual HF superconducting compounds (U or Ce-based)could come from magnetic fluctuations,the supercon-ducting state of PrOs 4Sb 12could be due to quadrupolar fluctuations.Yet at present,this attractive hypothesis is backed by very few experimental facts,both as re-gards the evidence of a quadrupolar Kondo effect in the normal phase and as regards the pairing mechanism in the superconducting state.Even the question of the un-conventional nature of its superconductivity is still open.Indeed,several types of experiments have already probed the nature of this superconducting state,but with appar-ently contradictory results.Concerning the gap topology,scanning tunneling spectroscopy measurements point to a fully open gap,with some anisotropy on the Fermi surface 9.Indication of unconventional superconductivity might come from the distribution of values of the resid-ual density of states (at zero energy)on different parts of the sample surface.This could be attributed to a pair-breaking effect of disorder.The same conclusion as re-gards the gap size was reached by µSR measurements 10and NQR measurements 11,although unconventional su-perconductivity is suggested in the latter case by the ab-sence of a coherence peak below T c in 1/T 1.This should be contrasted with recent penetration depth measurements,that would indicate point nodes of the gap 12,or the angular dependence of the thermal con-ductivity which suggests an anisotropic superconducting gap with a nodal structure 13.This latter measurement also suggests multiple phases in the temperature (T )-field (H )plane,which could be connected to the dou-ble transitions observed in zero field 14,15,16.Recent µSR relaxation experiments 17detected a broadening of the internal field distribution below T c ,suggesting a multi-component order parameter or a non unitary odd parity state,with a finite magnetic moment.In this context,our results bring new insight on the question of the parity of the order parameter,and draw a definite picture of the (H,T)phase diagram as deduced from specific heat measurements.With reference to the historical case of UPt 3,we emphasize that the present status of sample quality may explain the discrepancies between the various measurements :definite claims on the nature of the superconducting state in PrOs 4Sb 12are at the very best too early,the key point being the sample quality.II.EXPERIMENTAL DETAILSWe present results on single crystals of PrOs 4Sb 12grown by the Sb flux method 18,19.These samples are aggregates of small single crystals with well developed cubic faces.They have a good RRR of about 40(between room temperature and 2K),a superconducting transition (onset of C p or ρ)at 1.887K and,they present a very sharp double superconducting transition in the specific heat.20.511.522.533.54C p/T (J /K 2.m o l )FIG.1:Specific heat of samples of the same batch including sample n ◦1asC p3C /T (a .u .)FIG.3:Field sweeps of the ac specific heat of sample n ◦1at several temperatures.An arbitrary line between the two tran-sitions was subtracted.We follow the two transitions (H c 2and H ′)down to 350mK.00.511.5µH (T )FIG.4:H −T superconducting phase diagram of PrOs 4Sb 12determined by specific heat measurements on sample n ◦1.The field dependence of T c 1and T c 2are completely similar.The lines are fits by a two-band model of the upper critical field (Section V B 2).Only T c has been changed from H c 2to H ′.IV.SAMPLE CHARACTERIZATIONThree pieces of sample n ◦1have been used for furthercharacterizations,called 1a,1b and 1c.As well as the specific heat of sample n ◦1a (2mg),we have measured the resistivity of samples n ◦1b (0.2mg)and n ◦1c ,and the ac susceptibility and dc magnetization of samples n ◦1a and n ◦1b .Concerning the specific heat,the high absolute value of C p as well as the large height of the two supercon-ducting jumps (∆(C p /T )=350mJ/mol.K 2at T c 1and ∆(C p /T )=300mJ/mol.K 2at T c 2)must be linked to the high quality of these samples (absence of Sb-flux and/or good stoichiometry).Moreover,the heights of the two steps of sample n ◦1a are quantitatively similar to those of the entire batch (7.5mg)as Vollmer has already pointed out 14.Like in previous work 1,21,we have noticed that the resistivity at 300K is very sample dependent (from 200to 900µΩ.cm).On all samples,the value of ρat 300K seems to scale with the slope at high temperature (T ≥200K),i.e.the phononic part of the resistivity,as if the discrepancies were due to an error on the geometric factor.This error could be explained by the presence of microcracks in the samples.We have taken this problem into account by normalizing all data to the slope dρ/dT at high temperature (T ≥200K)of sample n ◦1c,chosen arbitrarily.For samples n ◦1b and n ◦1c respectively (Fig.5),the RRR (ratio between 300K and 2K values)are 44and 38,the onset T c are 1.899K and 1.893K (match-ing the critical temperature obtained by specific heat),and the temperatures of vanishing resistance (T R =0)are 1.815K and 1.727K.T R =0of sample n ◦1c is equal to T c 2and this remains true under magnetic field.So,in sample n ◦1c ,the resistive superconducting transition is not complete between T c 1and T c 2.Figure 6shows the superconducting transition for sam-ples n ◦1a and n ◦1b by ac-susceptibility (H ac =0.287Oe),corrected for the demagnetization field.The onset tem-perature is the same for the two samples (1.88K).The transition is complete only at around 1.7K and two tran-sitions are visible.The field cooled dc magnetization of samples n ◦1a and n ◦1b (H dc =1Oe),shown in fig.7,gives a Meissner effect of,respectively,44%and 55%,indicat-ing (like specific heat)that the superconductivity is bulk.The two transitions are also visible.V.DISCUSSIONA.Double superconducting transitionLet us first discuss the nature (intrinsic or not)of the double transition observed in our specific heat mea-surements.The remarkable fact,compared to previous reports 14,15,is the progress on the sharpness of both tran-sitions.If for previous reports,a simple distribution of T c due to a distribution of strain in the sample could42004001000050100150300ρ(µΩ.cm)FIG.5:Resistivity of samples n◦1b and n◦1c normalized tothe slope at high temperature of sample n◦1c.The inset isa zoom on the superconducting transition.The resistivity ofsample n◦1c is zero only at T c2.-0.08-0.07-0.06-0.05-0.04-0.03-0.02-0.01χi'AC(emu/cm3)FIG.6:Real partχ′of the AC susceptibility of samples n◦1aand n◦1b measured with an AC magneticfield of0.287Oeat2.11Hz.Like in the results of resistivity measurement,thesuperconducting transition is not complete at T c1and the twotransitions are visible.have explained the transition width,it is not the caseanymore for the sharp features observed on these newsamples.Also the explanation recently proposed22thatthe lower transition at T c2would be induced by Joseph-son coupling of one sheet of the Fermi surface to anotherwith transition temperature T c1is excluded,owing to the-0.08-0.07-0.06-0.05-0.04Memu/cm3FIG.7:DC magnetization of sample n◦1a at H DC=1Oewith zerofield cooled(ZFC)andfield cooled(FC)sweeps.The Meissner effect is44%for this sample and55%for samplen◦1b(not shown).The superconductivity is bulk.sharpness of both transitions and particularly of that atT c2:see the broadening calculated by the authors22forthe Josephson induced transition.Nevertheless,some ofour results still cast serious doubts on the intrinsic na-ture of the double transition.Indeed,the susceptibilityalso shows a”double”transition,and resistivity becomeszero only at T c2.If thefirst transition at T c1was bulk-homogeneous,the resistivity should immediately sink tozero below T c1,and the susceptibility(χ)should alsoshow perfect diamagnetism far before T c2:the samplediameter is at least1000times larger than the penetra-tion depth(λ)so that the temperature dependence ofλcannot explain such a transition width ofχ(contraryto the statements of E.E.M.Chia12).So both resistiv-ity and susceptibility are indicative of remaining sampleinhomogeneities.Nevertheless,in our opinion,even the comparisonof the various characterizations of the superconductingtransition by resistivity,susceptibility and specific heaton the same sample does not allow for a definite conclu-sion.Two historical cases are worth remembering.URu2Si2showed a double transition in the specific heat in somesamples,with inhomogeneous features detected in thesusceptibility23.In that case,the authors of reference23could clearly show that it was not intrinsic(maybe aris-ing from internal strain?)because different macroscopicparts of the same sample showed one or the other tran-sition.In PrOs4Sb12,the specific heat results are repro-ducible among various samples of the same batch(seesamples n◦1and n◦1a of the present work),and such aneasy test does not work.5The second case is of course UPt3:it is now well estab-lished that the two transitions observed in zerofield are intrinsic and correspond to order parameters of different symmetries.But thefirst results on samples that were not homogeneous enough showed exactly the same be-haviour as the present one in PrOs4Sb12:two features in the susceptibility and a very broad(covering both tran-sitions)resistive transition24.It was not until the sample quality improved significantly that resistivity and suscep-tibility transitions matched the higher one25.The puz-zling result for PrOs4Sb12,compared to UPt3,is that despite the sharpness of the specific heat transitions,re-sistivity and susceptibility reveal inhomogeneities,which means that this new compound probably has unusual metallurgical specificities.Continuing the parallel with UPt3,basic measure-ments rapidly supported the intrinsic origin of the two transitions:they were probing the respectivefield and pressure dependence of both transitions.Indeed,a com-plete(H-T)phase diagram was rapidly drawn,showing that in UPt326,like in U1−x T h x Be1327,the two transi-tions observed in zerofield eventually merged under mag-neticfield,due to a substantial difference in dT c/dH.It is even more true for the pressure dependence of T c1,2,as the thermal dilation has jumps of opposite sign at the two transitions,indicating opposite variations of T c1,2under pressure(Ehrenfest relations)28.So in this compound, the two transitions could be rapidly associated with a change of the symmetry of the order parameter indicating the unconventional nature of the superconducting state. We are not so lucky in the case of PrOs4Sb12:indeed, thefield dependence of T c2seems completely similar to that of T c1(fig.4),a claim that will be made quantita-tive below.It is the same situation as in URu2Si223,and nothing in favor of an intrinsic nature of the double tran-sition can be deduced from this phase diagram.Another phase diagram has already been established by transport measurements,with a line H∗(T)separating regions of twofold and fourfold symmetry in the angular depen-dence of thermal conductivity under magneticfield13. It may seem likely13,29that this line would merge with the double transition in zerofield.From the line H′(T) drawn from our specific heat measurements(T c2(H),fig.4),we can onclude that this is not the case:H′(T)does not match the line H∗(T)drawn in reference13,unless there is an unlikely strong sample dependence of these lines.Comparison of the pressure dependence of T c1and T c2 seems more promising:contrary to case of UPt328the jump of the thermal expansion at the two superconduct-ing temperatures does not change sign16,but from the relative magnitude of these jumps and our specific heat peaks,we get a value dT c1/dp≈−0.2K/GPa,and twice as much for dT c2/dp.This supports a different origin for both transitions.The weak point is that the ther-mal expansion measurements were done on two samples mounted on top of each other,that were early samples with rather broad specific heat transitions.Thus,the question of the intrinsic nature of the double supercon-ducting transition remains open.B.Upper criticalfieldAnother quantity which has not been thoroughly dis-cussed is the upper criticalfield H c2(T).In heavy fermion superconductors,H c2(T)has always proved to be an in-teresting quantity,mainly due to the large mass enhance-ment of the quasiparticles.Indeed,the usual orbital lim-itation is very high in these systems,due to the low Fermi velocity,so that the authors of reference1could, from their H c2(T)data,confirm the implication of heavy quasiparticles in the Cooper pairs(revealed also by the specific heat jump at T c).They also gave an estimate of the heaviest mass:≈50m0,where m0is the free electron mass.But,as the orbital limitation is very high,H c2(T)is also controlled by the paramagnetic effect.A quantita-tivefit of H c2(T)easily gives the amount of both limi-tations,except that on this system,H c2(T)has an ex-tra feature:a small initial positive curvature close to T c.This feature has been systematically found,what-ever the samples and the techniques used to determine H c2(T)(ρ,χand C p)1,13,14,21.Our data,obtained byρon sample n◦1c and C p on sample n◦1,matches all other published results.So we can now consider this curvature of H c2(T)as intrinsic,and not due to some artifact of transport measurements,or coming from inhomogeneity in the sample.Such an intrinsic positive curvature also appears in MgB230or in borocarbides like YNi2B2C and LuNi2B2C31.1.Physical inputsWe propose an explanation based on different gap am-plitudes for the different sheets of the Fermi surface of PrOs4Sb12,which is made quantitative through a”two-band”model32.Microscopically,STM spectroscopy also reveals an anisotropic gap,with zero density of states at low energy(fully open gap)but a large smearing of the spectra9.Recent microwave spectroscopy measurements also discuss a two band model22,but in order to explain the double transition.We insist that our model has noth-ing to do with the double transition,which clearly in-volves heavy quasiparticles both at T c1and at T c2(see the size of the two specific heat jumps):our aim is a quan-titative understanding of H c2(T),based on the normal state properties of PrOs4Sb12,as in MgB2or borocar-bides where no double transition has ever been observed. The physical input of a multi-band model for H c2(T)is to introduce different Fermi velocities and different in-ter and intra band couplings.As a result,T c is always larger than for any of the individual bands33.The slope of H c2at T c is larger for slower Fermi velocity(heavier) bands.Positive curvature of H c2(T)is easily obtained6 if the strongest coupling is in the heaviest bands(largeH c2),with a slight T c increase due to the inter band cou-pling to the lightest bands(small initial slope)31.2.The two-band model:There are at least three sheets for the Fermi surface of PrOs4Sb12,but in the absence of a precise knowl-edge of the pairing interaction,a full model would be unrealistic,having an irrelevant number of free parame-ters.A two band model is enough to capture the physics of anisotropic pairing,although only the correspondence with band calculations becomes looser.In our model, band2would correspond to the lightest(β)band de-tected by de-Haas van Alphen measurements,and band 1would be a heavy band having most of the density of states.Indeed,the de Haas-van Alphen experiments on PrOs4Sb1234,35reveal the presence of light quasiparti-cles(bandβ)and heavier particles(bandγ).The heavi-est quasiparticles are at present only seen by thermody-namic measurements(C p or H c2).Anisotropic coupling between the quasiparticles is considered in the framework of an Eliashberg strong coupling two-band model32in the clean limit,with an Einstein phonon spectrum(charac-teristic”Debye”frequencyΩ).Let us point out that the results do not depend on(and a fortiori do not probe) the pairing mechanism,which is likely to be much more exotic than the usual electron-phonon -pared to a single band calculation,there is now a matrix of strong coupling parametersλi,j describing the diffu-sion of electrons of band i to band j by the excitations responsible for the pairing.What matters for H c2(T)is the relative weight of the λi,j,not their absolute value:we consider T c or the renor-malized Fermi velocities as experimental inputs.λi,j de-pends both on the interaction matrix elements between bands i and j,and on thefinal density of states of band j33.In MgB2,it is claimed that electron-phonon coupling is largest within theσbands.Here,knowing nothing about the pairing mechanism,we assume con-stant inter and intra band coupling,so that the rela-tive weight of theλi,j is only governed by the density of states of band j.This density of states is itself pro-portional to the contribution of that band to the specific heat Sommerfeld coefficient:500mJ/K2.mol1for band 1,20mJ/K2.mol for band234,35.The Fermi velocity of band1(not observed by de Haas-van Alphen measure-ments)is the main adjustable parameter of thefit:we find v F1=0.0153106m/s−1,in agreement with1where the Fermi velocity has been inferred from the slope of H c2(T)at T c ignoring the initial positive curvature.All other coefficients are either arbitrary(λ1,1=1)(in agree-ment with the strong coupling superconductivity con-cluded in11),conventional values(gyromagnetic ratio for the paramagnetic limitation g=2,Coulomb repulsion parameterµ∗i,j=0.1δi,j),orfixed by experimental data (T c=1.887K=⇒Ω=21.7K,v F2=0.116.106m.s−10.511.523.5µH(T)FIG.8:Open circles show the data of H c2(T)by specific heat measurement on PrOs4Sb12(sample n◦1).The lines showfits with a two-band model(solid line),with a single-band model (dashed line),without the paramagnetic limit(g=0)(dotted line).It shows that the increase of T c due to the coupling with light qp band is rapidly suppressed in weak magnetic fields.H c2is also clearly Pauli limited,supporting a singlet superconducting state.The parameters for the solid linefit are:g=2,µ∗i,j=0.1δi,j,v F1=0.0153106m/s−1(heavy band),v F2=0.116.106m.s−1(lightest band,from de Haas-van Alphen oscillations35),T c=1.887K=⇒Ω=21.7K.from the de Haas-van Alphen data on theβband).The modelfits well the experimental data(cf.fig.4),includ-ing the small positive curvature.Before discussing the interpretation of thefit as regards the values of the pa-rameters and the parity of the superconducting order pa-rameter,let us note that we canfit the H′(T)line(fig.4) with the same parameters as for H c2exceptΩ,adjusted to give T c=1.716K.There is a good agreement with all data except at very low temperature or near T c where the curvature is reduced compared to H c2(T).However, these deviations are weak,and this is why we claim that H′(T)has the same behavior than H c2(T),which does not help to identify the second transition with a symme-try change of the superconducting order parameter.3.InterpretationShown also infig.8are the calculations of H c2(T)for a single band model:v F1,the characteristic frequency Ωandλ1,1have the same values as before,but all other λi,j coefficients have been turned down to zero,elimi-nating the effects of the light electron band.T c is then reduced(down to1.78K)and the positive curvature dis-appears.We also observe that thefit of H c2is basi-7 cally unchanged above1T,meaning that lowfields sup-press the superconductivity due to the light electron bandrestoring a”single band”superconducting state.Thisis the same effect observed more directly in MgB2withspecific heat measurements under magneticfield:thesmaller gap rapidly vanishes,leading to afinite densityof states at the Fermi level under magneticfields dueto theπband36.This suppression of the light quasi-particle superconductivity would have here an effect onspecific heat too small to be observed(contribution ofthe light quasiparticles to the specific heat of order4%of the Sommerfeld coefficient,itself buried in the largeSchottky anomaly).But it may have much larger ef-fects on transport.Let us note that the clean limit is aposteriori justified:from v F1,wefind a coherence lengthξ0∼110˚A,whereas from a residual resistivityρ0and spe-cific heat coefficientγ∼0.5J/K2.mol∼2.103J/K2.m3,3L0we get v 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IOP P UBLISHING R EPORTS ON P ROGRESS IN P HYSICS Rep.Prog.Phys.71(2008)062501(9pp)doi:10.1088/0034-4885/71/6/062501Two gaps make a high-temperature superconductor?S H¨ufner1,2,M A Hossain1,2,A Damascelli1,2and G A Sawatzky1,21AMPEL,University of British Columbia,Vancouver,British Columbia,V6T1Z4,Canada2Department of Physics and Astronomy,University of British Columbia,Vancouver,British Columbia,V6T1Z1,CanadaReceived27February2008,infinal form2April2008Published2May2008Online at /RoPP/71/062501AbstractOne of the keys to the high-temperature superconductivity puzzle is the identification of theenergy scales associated with the emergence of a coherent condensate of superconductingelectron pairs.These might provide a measure of the pairing strength and of the coherence ofthe superfluid,and ultimately reveal the nature of the elusive pairing mechanism in thesuperconducting cuprates.To this end,a great deal of effort has been devoted to investigatingthe connection between the superconducting transition temperature T c and the normal-statepseudogap crossover temperature T∗.Here we present a review of a large body ofexperimental data which suggests a coexisting two-gap scenario,i.e.superconducting gap andpseudogap,over the whole superconducting dome.We focus on spectroscopic data fromcuprate systems characterized by T maxc ∼95K,such as Bi2Sr2CaCu2O8+δ,YBa2Cu3O7−δ,Tl2Ba2CuO6+δand HgBa2CuO4+δ,with particular emphasis on the Bi-compound which has been the most extensively studied with single-particle spectroscopies.(Somefigures in this article are in colour only in the electronic version)This article was invited by Professor L Greene.Contents1.Introduction12.Emerging phenomenology32.1.Angle-resolved photoemission42.2.Tunneling52.3.Raman scattering52.4.Inelastic neutron scattering52.5.Heat conductivity63.Outlook and conclusion6 Acknowledgments6 References71.IntroductionSince their discovery[1],the copper-oxide high-T c superconductors(HTSCs)have become one of the most investigated class of solids[2–24].However,despite the intense theoretical and experimental scrutiny,an understanding of the mechanism that leads to superconductivity is still lacking.At the very basic level,what distinguishes the cuprates from the conventional superconductors is the fact that they are doped materials,the highly atomic-like Cu 3d orbitals give rise to strong electron correlations(e.g. the undoped parent compounds are antiferromagnetic Mott–Hubbard-like insulators),and the superconducting elements are weakly-coupled two-dimensional layers(i.e.the celebrated square CuO2planes).Among the properties that are unique to this class of superconducting materials,in addition to the unprecedented high superconducting T c,the normal-state gap or pseudogap is perhaps the most noteworthy.The pseudogap wasfirst detected in the temperature dependence of the spin-lattice relaxation and Knight shift in nuclear magnetic resonance and magnetic susceptibility studies[25].The Knight shift is proportional to the density of states at the Fermi energy;a gradual depletion was observed below a crossover temperature T∗,revealing the opening of the pseudogap well above T c on the underdoped side of the HTSC phase diagram(figure1).As the estimates based on thermodynamicx (c)Tx (a)x(b)T* T cT*T cT*T cFigure1.Various scenarios for the interplay of pseudogap(blue dashed line)and superconductivity(red solid line)in thetemperature-doping phase diagram of the HTSCs.While in(a)the pseudogap merges gradually with the superconducting gap in the strongly overdoped region,in(b)and(c)the pseudogap lines intersect the superconducting dome at about optimal doping(i.e.maximum T c).In most descriptions,the pseudogap line is identified with a crossover with a characteristic temperature T∗rather than a phase transition;while at all dopings T∗>T c in(a),beyond optimal doping T∗<T c in(b)and T∗does not even exist in(c).Adapted from[12].quantities are less direct than in spectroscopy we,in the course of this review,concentrate mainly on spectroscopic results; more information on other techniques can be found in the literature[5].As established by a number of spectroscopic probes, primarily angle-resolved photoemission spectroscopy,[26,27] the pseudogap manifests itself as a suppression of the normal-state electronic density of states at E F exhibiting a momentum dependence reminiscent of a d x2−y2functional form.For hole-doped cuprates,it is largest at Fermi momenta close to the antinodal region in the Brillouin zone—i.e.around (π,0)—and vanishes along the nodal direction—i.e.the(0,0) to the(π,π)line.Note however that,strictly speaking, photoemission and tunneling probe a suppression of spectral weight in the single-particle spectral function,rather than directly of density of states;to address this distinction,which is fundamental in many-body systems and will not be further discussed here,it would be very interesting to investigate the quantitative correspondence between nuclear magnetic resonance and single-particle spectroscopy results.Also,no phase information is available for the pseudogap since,unlike the case of optimally and overdoped HTSCs[28],no phase-sensitive experiments have been reported for the underdoped regime where T∗ T c.As for the doping dependence,the pseudogap T∗is much larger than the superconducting T c in underdoped samples,it smoothly decreases upon increasing the doping,and seems to merge with T c in the overdoped regime,eventually disappearing together with superconductivity at doping levels larger than x∼0.27[5–24].In order to elaborate on the connection between pseudogap and high-T c superconductivity,or in other words between the two energy scales E pg and E sc identified by T∗and T c,respectively,let us start by recalling that in conventional superconductors the onset of superconductivity is accompanied by the opening of a gap at the chemical potential in the one-electron density of states. According to the Bardeen–Cooper–Schrieffer(BCS)theory of superconductivity[29],the gap energy provides a direct measure of the binding energy of the two electrons forming a Cooper pair(the two-particle bosonic entity that characterizes the superconducting state).It therefore came as a great surprise that a gap,i.e.the pseudogap,was observed in the HTSCs not only in the superconducting state as expected from BCS, but also well above T c.Because of these properties and the hope it might reveal the mechanism for high-temperature superconductivity,the pseudogap phenomenon has been very intensely investigated.However,no general consensus has been reached yet on its origin,its role in the onset of superconductivity itself,and not even on its evolution across the HTSC phase diagram.As discussed in three recent papers on the subject [12,15,17],and here summarized infigure1,three different phase diagrams are usually considered with respect to the pseudogap line.While Millis[15]opts for a diagram like the one infigure1(a),Cho[17]prefers a situation where the pseudogap line meets the superconducting dome at x 0.16(figures1(b)and(c));Norman et al[12]provide a comprehensive discussion of the three different possibilities. One can summarize some of the key questions surrounding the pseudogap phenomenon and its relevance to high-temperature superconductivity as follows[12,15,17]:1.Which is the correct phase diagram with respect to thepseudogap line?2.Does the pseudogap connect to the insulator quasiparticlespectrum?3.Is the pseudogap the result of some one-particle bandstructure effect?4.Or,alternatively,is it a signature of a two-particle pairinginteraction?5.Is there a true order parameter defining the existence of apseudogap phase?6.Do the pseudogap and a separate superconducting gapcoexist below T c?7.Is the pseudogap a necessary ingredient for high-T csuperconductivity?In this review we revisit some of these questions,with specific emphasis on the one-versus two-gap debate.Recently,this latter aspect of the HTSCs has been discussed in great detail by Goss Levi[30],in particular based on scanning-tunneling microscopy data from various groups[31–33].Here we expand this discussion to include the plethora of experimental results available from a wide variety of techniques.We0.050.100.150.200.250408012016050100150E n e r g y (m e V )Hole doping (x)T c (K )Figure 2.Pseudogap (E pg =2 pg )and superconducting (E sc ∼5k B T c )energy scales for a number of HTSCs with T max c ∼95K(Bi2212,Y123,Tl2201and Hg1201).The datapoints were obtained,as a function of hole doping x ,by angle-resolved photoemission spectroscopy (ARPES),tunneling (STM,SIN,SIS),Andreev reflection (AR),Raman scattering (RS)and heat conductivity (HC).On the same plot we are also including the energy r of the magnetic resonance mode measured by inelastic neutron scattering (INS),which we identify with E sc because of the striking quantitative correspondence as a function of T c .The data fall on two universal curvesgiven by E pg =E max pg (0.27−x)/0.22and E sc =E max sc [1−82.6(0.16−x)2],with E maxpg =E pg (x =0.05)=152±8meV and E maxsc =E sc (x =0.16)=42±2meV (the statistical errors refer to the fit of the selected datapoints;however,the spread of all available data would be more appropriately described by ±20and ±10meV ,respectively).show that one fundamental and robust conclusion can be drawn:the HTSC phase diagram is dominated by two energy scales,the superconducting transition temperature T c and the pseudogap crossover temperature T ∗,which converge to the very same critical point at the end of the superconducting dome.Establishing whether this phenomenology can be conclusively described in terms of a coexisting two-gap scenario,and what the precise nature of the gaps would be,will require a more definite understanding of the quantities measured by the various probes.2.Emerging phenomenologyThe literature on the HTSC superconducting gap and/or pseudogap is very extensive and still growing.In this situation it seems interesting to go over the largest number of data obtained from as many experimental techniques as possible,and look for any possible systematic behavior that could be identified.This is the primary goal of this focused review.We want to emphasize right from the start that we are not aiming at providing exact quantitative estimates of superconducting and pseudogap energy scales for any specific compound or any given doping.Rather,we want to identify the general phenomenological picture emerging from the whole body of available experimental data [5,9,13,16,18,34–72].We consider some of the most direct probes of low-energy,electronic excitations and spectral gaps,such as angle-resolved photoemission (ARPES),scanning-tunneling microscopy (STM),superconductor/insulator/normal-metal(SIN)and superconductor/insulator/superconductor (SIS)tunneling,Andreev reflection tunneling (AR)and Raman scattering (RS),as well as less conventional probes such as heat conductivity (HC)and inelastic neutron scattering (INS).The emphasis in this review is on spectroscopic data because of their more direct interpretative significance;however,these will be checked against thermodynamic/transport data whenever possible.With respect to the spectroscopic data,it is important to differentiate between single-particle probes such as ARPES and STM,which directly measure the one-electron excitation energy with respect to the chemical potential (on both side of E F in STM),and two-particle probes such as Raman and inelastic neutron scattering,which instead provide information on the particle-hole excitation energy 2 .Note that the values reported here are those for the ‘full gap’2 (associated with either E sc or E pg ),while frequently only half the gap is given for instance in the ARPES literature.In doing so one implicitly assumes that the chemical potential lies half-way between the lowest-energy single-electron removal and addition states;this might not necessarily be correct but appears to be supported by the direct comparison between ARPES and STM/Raman results.A more detailed discussion of the quantities measured by the different experiments and their interpretation is provided in the following subsections.Here we would like to point out that studies of B 2g and B 1g Raman intensity [19,40,52],heat conductivity of nodal quasiparticles [70,71]and neutron magnetic resonance energy r [42]do show remarkable agreement with superconducting or pseudogap energy scales as inferred by single-particleTable1.Pseudogap E pg and superconducting E sc energy scales (2 )as inferred,for optimally doped Bi2212(T c∼90–95K),from different techniques and experiments.Abbreviations are given in the main text,while the original references are listed.Experiment Energy meV ReferencesARPES—(π,0)peak E pg80[34,35]Tunneling—STM…70[18,36]Tunneling—SIN…85[37]Tunneling—SIS…75[38,39]Raman—B1g…65[40]Electrodynamics…80[5,41]Neutron—(π,π) r E sc40[42]Raman—B2g…45[40]Andreev…45[43]SIS—dip…40[39]probes,or with the doping dependence of T c itself.Thus they provide,in our opinion,an additional estimate of E sc and E pg energy scales.As for the choice of the specific compounds to include in our analysis,we decided to focus on those HTSCs exhibiting a similar value of the maximum superconductingtransition temperature T maxc ,as achieved at optimal doping,so that the data could be quantitatively compared without any rescaling.We have therefore selected Bi2Sr2CaCu2O8+δ(Bi2212),YBa2Cu3O7−δ(Y123),Tl2Ba2CuO6+δ(Tl2201) and HgBa2CuO4+δ(Hg1201),which have been extensivelyinvestigated and are all characterized by T maxc ∼95K[73](with particular emphasis on Bi2212,for which the most extensive set of single-particle spectroscopy data is available). It should also be noted that while Bi2212and Y123are ‘bilayer’systems,i.e.their crystal structure contains as a key structural element sets of two adjacent CuO2layers, Tl2201and Hg1201are structurally simpler single CuO2-layer materials.Therefore,this choice of compounds ensures that our conclusions are generic to all HTSCs with a similar T c, independent of the number of CuO2layers.A compilation of experimental results for the magnitude of pseudogap(E pg=2 pg)and superconducting(E sc∼5kB T c) energy scales,as a function of carrier doping x,is presented infigure2(only some representative datapoints are shown,so as not to overload thefigure;similar compilations were also obtained by a number of other authors)[5,9,13,16,42,43,52, 57,60,70,74,75].The data for these HTSCs with comparableT max c ∼95K fall on two universal curves:a straight linefor the pseudogap energy E pg=2 pg and a parabola for the superconducting energy scale E sc∼5k B T c.The two curves converge to the same x∼0.27critical point at the end of the superconducting dome,similarly to the cartoon of figure1(a).In order to summarize the situation with respect to quantitative estimates of E pg and E sc,we have listed in table1the values as determined by the different experimental techniques on optimally doped Bi2212(with T c ranging from 90to95K).While one obtains from this compilation the average values of E pg 76meV and E sc 41meV at optimal doping,the numbers do scatter considerably.Note also that these numbers differ slightly from those given in relation to the parabolic and straight lines infigure2(e.g.E maxsc= 42meV)because the latter were inferred from afitting of superconducting and pseudogap data over the whole doping range,while those in table1were deduced from results for optimally doped Bi2212only.It is also possible to plot the pseudogap E pg and superconducting E sc energy scales as estimated simultaneously in one single experiment on the very same sample.This is done infigure3for Raman,tunneling and ARPES results from Bi2212and Hg1201,which provide evidence for the presence of two energy scales,or possibly two spectral gaps as we discuss in greater detail below,coexisting over the whole superconducting dome.2.1.Angle-resolved photoemissionThe most extensive investigation of excitation gaps in HTSCs has arguably been done by ARPES[9,10,26,27,34,35,54–66,76–80].This technique provides direct access to the one-electron removal spectrum of the many-body system;it allows,for instance in the case of a BCS superconductor[29], to measure the momentum dependence of the absolute value of the pairing amplitude2 via the excitation gap observed for single-electron removal energies,again assuming E F to be located half-way in the gap[9,10].This is the same in some tunneling experiments such as STM,which however do not provide direct momentum resolution but measure on both sides of E F[18].The gap magnitude is usually inferred from the ARPES spectra from along the normal-state Fermi surface in the antinodal region,where the d-wave gap is largest;it is estimated from the shift to high-binding energy of the quasiparticle spectral weight relative to the Fermi energy.With this approach only one gap is observed below a temperature scale that smoothly evolves from the so-called pseudogap temperature T∗in the underdoped regime,to the superconducting T c on the overdoped side.We identify this gap0.050.100.150.200.25408012016050100150Energy(meV)Hole doping (x)T c(K)Figure3.Pseudogap E pg and superconducting E sc energy scales (2 )as estimated,by a number of probes and for different compounds,in one single experiment on the very same sample. These data provide direct evidence for the simultaneous presenceof two energy scales,possibly two spectral gaps,coexisting in the superconducting state.The superconducting and pseudogap lines are defined as infigure2.with the pseudogap energy scale E pg=2 pg.This is also in agreement with recent investigations of the near-nodal ARPES spectra from single and double layer Bi-cuprates[57,76,77], which further previous studies of the underdoped cuprates’Fermi arc phenomenology[78–80].From the detailed momentum dependence of the excitation gap along the Fermi surface contour,and the different temperature trends observed in the nodal and antinodal regions,these studies suggest the coexistence of two distinct spectral gap components over the whole superconducting dome:superconducting gap and pseudogap,dominating the response in the nodal and antinodal regions,respectively,which would eventually collapse to one single energy scale in the very overdoped regime.2.2.TunnelingThe HTSCs have been investigated by a wide variety of tunneling techniques[13,18,36–39,44–51],such as SIN[38,51],SIS[37–39],STM[18,36,46],intrinsic tunneling[47–50]and Andreev reflection,which is also a tunneling experiment but involves two-particle rather than single-particle tunneling(in principle,very much like SIS) [13,43,72].All these techniques,with the exception of intrinsic tunneling3,are represented here either in thefigures or table.Similarly to what was discussed for ARPES at the antinodes,there are many STM studies that report a pseudogap E pg smoothly evolving into E sc upon overdoping[18,31]. In addition,a very recent temperature-dependent study of overdoped single-layer Bi-cuprate detected two coexisting,yet clearly distinct,energy scales in a single STM experiment[32]. In particular,while the pseudogap was clearly discernible in the differential conductance exhibiting the usual large spatial modulation,the evidence for a spatially uniform superconducting gap was obtained by normalizing the low-temperature spectra by those just above T c 15K.These values have not been included infigures2and3because T c 95K;however,this study arguably provides the most direct evidence for the coexistence of two distinct excitation gaps in the HTSCs.One can regard Andreev reflection(pair creation in addition to a hole)as the inverse of a two-particle scattering experiment such as Raman or INS.A different view is also possible:SIN tunneling goes over to AR if the insulator layer gets thinner and thinner[13];thus a SIN tunneling,as also STM,should give the same result as AR.However while SIN and STM measure the pseudogap,AR appears to be sensitive to the superconducting energy scale E sc(figure2).We can only conjecture that this has to do with the tunneling mechanisms actually being different.3The most convincing tunneling results showing two coexisting gaps were actually obtained by intrinsic tunneling[47–50],in particular from Bi2Sr2CuO6+δ(Bi2201)[48].However,because this technique suffers from systematic problems[50],and one would anyway have to scale the Bi2201data because of the lower value of T c and in turn gap energy scales,these results were not included infigures2or3.Since intrinsic tunneling is in principle a clean SIS experiment which measures pair energies through Josephson tunneling, a refinement of the technique might provide an accurate estimate of both superconducting and pseudogap simultaneously,and is thus highly desirable.SIS tunneling experiments[39]find E pg/E sc 1for Bi2212at all doping levels.There are,however,some open questions concerning the interpretation of the SIS experiments. This technique,which exploits Josephson tunneling,measures pair spectra;the magnitude of E pg can readily be obtained from the most pronounced features in the spectra[39].The signal related to E sc is seen as a‘sideband’on the E pg features;it does not seem obvious why,if the E sc signal did originate from a state of paired electrons,it would not show up more explicitly.2.3.Raman scatteringLight scattering measures a two-particle excitation spectrum providing direct insight into the total energy needed to break up a two-particle bound state or remove a pair from a condensate. Raman experiments can probe both superconducting and pseudogap energy scales,if one interprets the polarization dependent scattering intensity in terms of different momentum averages of the d-wave-like gap functions:one peaked at(π,0) in B1g geometry,and thus more sensitive to the larger E pg which dominates this region of momentum space;the other at(π/2,π/2)in B2g geometry,and provides an estimate of the slope of the gap function about the nodes,(1/¯h)(d /d k)|n, which is more sensitive to the arguably steeper functional dependence of E sc out of the nodes[19,40,52,53].One should note,however,that the signal is often riding on a high background,which might result in a considerable error and data scattering.At a more fundamental level,while the experiments in the antinodal geometry allow a straightforward determination of the gap magnitude E pg,the nodal results need a numerical analysis involving a normalization of the Raman response function over the whole Brillouin zone,a procedure based on a low-energy B2g sum rule(although also the B2g peak position leads to similar conclusions)[52].This is because a B2g Raman experiment is somewhat sensitive also to the gap in the antinodal direction,where it picks up,in particular,the contribution from the larger pseudogap.2.4.Inelastic neutron scatteringInelastic neutron scattering experiments have detected the so-called q=(π,π)resonant magnetic mode in all of the T c 95K HTSCs considered here[16].This resonance is proposed by some to be a truly collective magnetic mode that, much in the same way as phonons mediate superconductivity in the conventional BCS superconductors,might constitute the bosonic excitation mediating superconductivity in the HTSCs. The total measured intensity,however,amounts to only a small portion of what is expected based on the sum rule for the magnetic scattering from a spin1/2system[8,16,24, 42,68,69];this weakness of the magnetic response should be part of the considerations in the modeling of magnetic resonance mediated high-T c superconductivity.Alternatively, its detection below T c might be a mere consequence of the onset of superconductivity and of the corresponding suppression of quasiparticle scattering.Independently of the precise interpretation,the INS data reproduced infigure2show that the magnetic resonance energy r tracks very closely, over the whole superconducting dome,the superconductingenergy scale E sc∼5k B T c(similar behavior is observed, in the underdoped regime,also for the spin-gap at the incommensurate momentum transfer(π,π±δ)[81]).Also remarkable is the correspondence between the energy of the magnetic resonance and that of the B2g Raman peak.Note that while the q=(π,π)momentum transfer observed for the magnetic resonance in INS is a key ingredient of most proposed HTSC descriptions,Raman scattering is a q=0probe.It seems that understanding the connection between Raman and INS might reveal very important clues.2.5.Heat conductivityHeat conductivity data from Y123and Tl2201fall onto the pseudogap line.This is a somewhat puzzling result because they have been measured at very low temperatures,well into the superconducting state,and should in principle provide a measure of both gaps together if these were indeed coexisting below T c.However,similarly to the B2g Raman scattering, these experiments are only sensitive to the slope of the gap function along the Fermi surface at the nodes,(1/¯h)(d /d k)|n; the gap itself is determined through an extrapolation procedure in which only one gap was assumed.The fact that the gap values,especially for Y123,come out on the high side of the pseudogap line may be an indication that an analysis with two coexisting gaps might be more appropriate.3.Outlook and conclusionThe data infigures2and3demonstrate that there are two coexisting energy scales in the HTSCs:one associated with the superconducting T c and the other,as inferred primarily from the antinodal region properties,with the pseudogap T∗. The next most critical step is that of addressing the subtle questions concerning the nature of these energy scales and the significance of the emerging two-gap phenomenology towards the development of a microscopic description of high-T c superconductivity.As for the pseudogap,which grows upon underdoping, it seems natural to seek a connection to the physics of the insulating parent compound.Indeed,it has been pointed out that this higher energy scale might smoothly evolve,upon underdoping,into the quasiparticle dispersion observed by ARPES in the undoped antiferromagnetic insulator[82,83]. At zero doping the dispersion and quasiparticle weight in the single-hole spectral function as seen by ARPES can be very well explained in terms of a self-consistent Born approximation[84],as well as in the diagrammatic quantum Monte Carlo[85]solution to the so-called t–t –t –J model.In this model,as in the experiment[82,83],the energy difference between the top of the valence band at(π/2,π/2)and the antinodal region at(π,0)is a gap due to the quasiparticle dispersion of about250±30meV.Note that this would be a single-particle gap .For the direct comparison with the pseudogap data infigure2,we would have to consider 2 ∼500meV;this,however,is much larger than the x=0 extrapolated pseudogap value of186meV found from our analysis across the phase diagram.Thus there seems to be an important disconnection between thefinite doping pseudogapand the zero-doping quasiparticle dispersion.The fact that the pseudogap measured in ARPES and SINexperiments is only half the size of the gap in SIS,STM,B1gRaman and heat conductivity measurements,points to a pairinggap.So although the origin of the pseudogap atfinite dopingremains uncertain,we are of the opinion that it most likelyreflects a pairing energy of some sort.To this end,the trend infigure2brings additional support to the picture discussed bymany authors that the reduction in the density of states at T∗isassociated with the formation of electron pairs,well above theonset of phase coherence taking place at T c(see,e.g.[86,87]).The pseudogap energy E pg=2 pg would then be the energy needed to break up a preformed pair.To conclusively addressthis point,it would be important to study very carefully thetemperature dependence of the(π,0)response below T c;anyfurther change with the onset of superconductivity,i.e.anincrease in E pg,would confirm the two-particle pairing picture,while a lack thereof would suggest a one-particle band structureeffect as a more likely interpretation of the pseudogap.The lower energy scale connected to the superconductingT c(parabolic curve infigure2and3)has already been proposedby many authors to be associated with the condensationenergy[86–89],as well as with the magnetic resonance inINS[90].One might think of it as the energy needed totake a pair of electrons out of the condensate;however,fora condensate of charged bosons,a description in terms of acollective excitation,such as a plasmon or roton,would bemore appropriate[24].The collective excitation energy wouldthen be related to the superfluid density and in turn to T c.In thissense,this excitation would truly be a two-particle process andshould not be measurable by single-particle spectroscopies.Also,if the present interpretation is correct,this excitationwould probe predominantly the charge-response of the system;however,there must be a coupling to the spin channel,so as tomake this process neutron active(yet not as intense as predictedby the sum rule for pure spin-1/2magnetic excitations,whichis consistent with the small spectral weight observed by INS).As discussed,one aspect that needs to be addressed to validatethese conjectures is the surprising correspondence betweenq=0and q=(π,π)excitations,as probed by Raman andINS,respectively.We are led to the conclusion that the coexistence of twoenergy scales is essential for high-T c superconductivity,withthe pseudogap reflecting the pairing strength and the other,always smaller than the pseudogap,the superconductingcondensation energy.This supports the proposals thatthe HTSCs cannot be considered as classical BCSsuperconductors,but rather are smoothly evolving from theBEC into the BCS regime[91–93],as carrier doping isincreased from the underdoped to the overdoped side of thephase diagram.AcknowledgmentsSH would like to thank the University of British Columbiafor its hospitality.Helpful discussions with W N Hardy,。