ARTICLES

Quasiparticle interference andsuperconducting gap in Ca2minusxNaxCuO2Cl2T HANAGURI12 Y KOHSAKA3 J C DAVIS34 C LUPIEN5 I YAMADA6 M AZUMA6 M TAKANO6K OHISHI7 M ONO12 AND H TAKAGI128

1Magnetic Materials Laboratory RIKEN (The Institute of Physical and Chemical Research) Wako 351-0198 Japan2CREST Japan Science and Technology Agency Kawaguchi 332-0012 Japan3LASSP Department of Physics Cornell University Ithaca New York 14853 USA4CMPMS Department Brookhaven National Laboratory Upton New York 11973 USA5Departement de Physique Universite de Sherbrooke Sherbrooke QC J1K 2R1 Canada6Institute for Chemical Research Kyoto University Uji 601-0011 Japan7Advanced Science Research Center Japan Atomic Energy Agency Ibaraki 319-1195 Japan8Department of Advanced Materials University of Tokyo Kashiwa 277-8561 Japane-mail hanaguririkenjp

Published online 28 October 2007 doi101038nphys753

High-transition-temperature (high-Tc) superconductivity is ubiquitous in the cuprates containing CuO2 planes but each cuprate hasits own character The study of the material dependence of the d-wave superconducting gap (SG) should provide important insightsinto the mechanism of high-Tc superconductivity However because of the lsquopseudogaprsquo phenomenon it is often unclear whetherthe energy gaps observed by spectroscopic techniques really represent the SG Here we use scanning tunnelling spectroscopy toimage nearly optimally doped Ca2minusxNaxCuO2Cl2(Na-CCOC) with Tc = 25ndash28 K It enables us to observe the quasiparticle interferenceeffect in this material through which we obtain unambiguous information on the SG Our analysis of quasiparticle interference inNa-CCOC reveals that the SG dispersion near the gap node is almost identical to that of Bi2Sr2CaCu2Oy (Bi2212) at the same dopinglevel despite the Tc of Bi2212 being three times higher than that of Na-CCOC We also find that the SG in Na-CCOC is confined innarrower energy and momentum ranges than Bi2212 which explainsmdashat least in partmdashthe remarkable material dependence of Tc

Recent progress in spectroscopic techniques has enabled theelucidation of various interesting aspects of the electronic statesof high-Tc cuprates in real and momentum (k) spaces Scanningtunnelling microscopyspectroscopy (STMSTS) measurementshave provided a wide variety of real-space information1 includingvarious types of electronic inhomogeneity2ndash6 and the presenceof energy non-dispersive lsquocheckerboardrsquo (CB) modulations inthe tunnelling-conductance map at location r and energy Eg(rE) equiv dIdV (rE) where I is the tunnelling current and Vis the bias voltage7ndash10 In k space angle-resolved photoemissionspectroscopy (ARPES) on the underdoped cuprates has revealedthat coherent quasiparticle states exist only in the nodal regionaround (π2π2) (refs 11ndash14) Here the ARPES spectra awayfrom the node are very broad As a result the cylindrical Fermisurface (FS) centred around (ππ) is somehow lsquotruncatedrsquo aroundthe first Brillouin zone face and ends up with a so-called lsquoFermi arcrsquoin k space15 The Fermi arc length increases with doping and the fullFS is restored in the optimallyoverdoped region1314

It is very important to understand how superconductivityemerges from such a complicated situation Becausesuperconducting states are characterized by the SG detailedinformation on the SG is indispensable There is now consensusthat the SG in cuprates has d-wave symmetry16 and thus hasnodes in k space However direct studies of the SG structure byspectroscopic techniques are still in a pioneering stage1111416ndash19 Thereason is that whereas energy gaps can be measured in cuprates

whether such gaps represent the true SG is often unclear because ofthe presence of the lsquopseudogaprsquo20

Here we choose spectroscopic imaging scanning tunnellingmicroscopy (SI-STM) as a tool to settle this issue Of coursebecause of the electronic heterogeneity the atomic spatialresolution of SI-STM is highly advantageous Perhaps moreimportantly by using Fourier transformations we can now accessthe k-space electronic structure as well102122 So far numerousSTMSTS studies have been carried out on Bi2212 Among themthe discovery of dispersive g modulations due to quasiparticleinterference (QPI) in the d-wave superconducting state1021ndash31 andits use for simultaneous determination of the k-space regionsupporting the d-wave quasiparticles and the dispersion of theSG provided a completely new approach to studies of high-Tc superconductivity102122 Conventional spectroscopies essentiallygive us only the density-of-states (DOS) spectrum and cannotdistinguish the SG from the pseudogap By contrast when theobserved QPI pattern is fully consistent within the lsquooctetrsquo model22we can be certain that the data represent the true SG (See the nextsection) In spite of this great advantage detailed information fromQPI has been available only in Bi2212 (refs 102122) the universalcharacterization of the cuprate SG using the QPI technique has notyet been achieved

Here we report the QPI effect in nearly optimally dopedNa-CCOC with Tc = 25ndash28 K Although the CB g modulationdominates the STS data9 the full octet model of QPI is confirmed

nature physics VOL 3 DECEMBER 2007 wwwnaturecomnaturephysics 865

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ARTICLES

ndash05

0

05

ndash10

ndash05

0

05

10

0

q1

q1

q2

q2q3

q3q4

q4

q5

q5

q6

q6

q7

q7

ndash05 05ndash10 10ndash10

10

0ndash05 05ndash10 10

E

ky

kx

a b c

kx ( a0)π

π π

k y (

a0)

qx (2 a0)π

q y (2

a

0)

Figure 1 Schematic illustration of electronic states responsible for the interference effect of Bogoliubov quasiparticles in the superconducting state a Aperspective view of the expected particlendashhole-symmetric quasiparticle dispersion relation plusmn

radicε (k)2 +∆(k)2 near one of the nodes showing a banana-shaped CCE b Four

sets of CCEs (blue curves) in momentum space The underlying FS is also shown by red curves Note that the octet ends of lsquobananasrsquo track the FS Because ε (k)= 0 on theFS the SG dispersion ∆(k) can be obtained by knowing the location of octet ends of lsquobananasrsquo in momentum space Arrows denote expected scattering q vectors whichcharacterize the standing waves of quasiparticles at a specific energy Each vector connects the octet ends of lsquobananasrsquo where the DOS is high Here a0 denotes the CundashCudistance c The q-space representation of the QPI which is directly associated with the Fourier transform of the conductance map |g (q E )| Symbols represent q vectorsexpected from the octet model22 The q vectors move with energy according to the FS shown in b Thus the FS can be reconstructed and ∆(k) can be determined fromenergy-dependent q vectors

by using the contrast of spatial-phase relations between CBg modulation and QPI The SG is identified only on an lsquoarcrsquo in k spacenear the gap node No well-defined coherence peak is observed inthe tunnelling spectrum and the antinodal region therefore remainsincoherent Thus Na-CCOC shows the apparent coexistence ofnodal superconductivity with the incoherent antinodal states as forthe case of underdoped Bi2212 (ref 10) indicating that these areuniversal features of high-Tc superconductivity The SG dispersionof nearly optimally doped Na-CCOC is almost identical to that ofoptimally doped Bi2212 with Tc = 86 K (ref 22) but is confined innarrower energy and momentum ranges From these observationswe hypothesize that the energy scale which determines the Tc ofoptimally doped cuprates is not set by the superconducting gapdispersion alone but by the energy and momentum-space locationwhere this gap terminates on the Fermi arc

QPI AND THE OCTET MODEL

Originally QPI was observed on conventional metal surfaces32ndash34In the presence of elastic scattering quantum interference ofscattered electrons results in real-space electronic standing wavescharacterized by wavevectors q connecting points on the contoursof constant quasiparticle energy (CCEs) in k space This standingwave can be detected using SI-STM by mapping g(rE) whichreflects the local DOS The wavevector q changes with E accordingto the electron-energyndashdispersion relation The Fourier amplitudeof the conductance map |g(qE)| enables us to determinecharacteristic q vectors at a given E Therefore the energyndashdispersion relation can be obtained experimentally

In the superconducting state the BardeenndashCooperndashSchrieffertheory of superconductivity tells us that elementary excitationsare not bare electrons or holes but coherent admixtures ofparticle and hole states known as Bogoliubov quasiparticles35The excitation energy of a Bogoliubov quasiparticle E(k) is givenby E(k) = plusmn

radicε(k)2 +∆(k)2 where ε(k) is the normal-state

band dispersion and ∆(k) is the SG In the case of high-Tc

cuprates ∆(k) has d-wave symmetry and thus vanishes in

the (0 0)ndash(plusmnπplusmnπ) directions (nodes) Because Bogoliubovquasiparticles can be excited by either positive or negative energiesand the Fermi velocity is much larger than the nodal gap slope theexcitation spectrum around the node is a particlendashhole-symmetricthin Dirac cone with a banana-shaped CCE as illustrated inFig 1a and b QPI of Bogoliubov quasiparticles occurs in thissituation1021ndash31 The intensities of the standing waves with q vectorsconnecting the ends of lsquobananasrsquo should become large because theDOS which is inversely proportional to the slope of dispersionbecomes a maximum at the energy-dependent octet ends oflsquobananasrsquo There are seven possible vectors qi (i = 1ndash7) from oneof the octet ends as shown in Fig 1b In total 32 different qvectors (16 of them are independent) may be observed in |g(qE)|(Fig 1c) If these numerous q vectors are observed in |g(qE)|and disperse with E according to the above-mentioned lsquooctetrsquomodel22 in an internally consistent fashion the presence of coherentBogoliubov quasiparticles and the underling d-wave SG can beestablished without doubt By analysing the E dependence of qiwe can deduce the locations of octet elements in k space namely FSposition as well as the dispersion relation of the SG ∆(k) (ref 22)Although detailed calculations23ndash31 are necessary to reproduce theobserved intensity distribution in |g(qE)| the simple octet modelon the basis of the DOS argument explains the observed q vectorsin Bi2212 quite well and detailed information about the SG hasbeen obtained1022

SEARCH FOR QPI IN Na-CCOC

Bi2212 is the only material for which QPI has been detectedso far Only after identifying the SG in other cuprates by QPIcan the universal features of the superconducting electronic stateof cuprates be elucidated Even more importantly a clue to thecause of the strong variations of Tc from material to materialmay then be inferred To carry out the search for QPI wechose Na-CCOC (refs 3637) as the next likeliest cuprate BecauseNa-CCOC crystals cleave well clean and flat surfaces necessaryfor STMSTS can be obtained easily Previous STMSTS works on

866 nature physics VOL 3 DECEMBER 2007 wwwnaturecomnaturephysics

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ARTICLES

Sample bias (mV)

dId

V (n

S)

dId

V (n

S)

ndash200 0 200

Sample bias (mV)

ndash400 400 ndash40 ndash20 0 20 40

a b

c d

5 nm 5 nm

18 nS01

14

12

10

08

06

04

02

0

10

08

06

04

02

0

016 nm0

Figure 2 Spectroscopic features of nearly optimally doped Na-CCOC with Tc sim 25 K (x sim 013) a Typical topographic STM image of Na-CCOC The image wastaken at a sample-bias voltage Vs = +400mV and a tunnelling current I t = 100 pA All the spectroscopic images in this paper were taken simultaneously with similaratomic-resolution topographic images to maintain the atomic-resolution position registry b The conductance g (r E ) map at energy E= +24meV in the same field of viewApparent CB g modulation which has been found in heavily underdoped Na-CCOC (ref 9) is observed even in the nearly optimally doped sample c The spatially averagedconductance spectrum showing the V-shaped pseudogap below sim100meV with particlendashhole asymmetry at higher energies (Set-up condition Vs = +400mV andI t = 100 pA) d The spatially averaged spectrum in the low-energy region indicated by arrows in c showing a pair of kinks in the spectrum inside the pseudogap (Set-upcondition Vs = +150mV and I t = 100 pA) The central interest of this paper is in the electronic state below and around this kink energy

Na-CCOC have been carried out only in the heavily underdopedregion and revealed nanoscale electronic inhomogeneity6 and CBg modulation9 However no QPI signal was detected although theQPI signal may merely be too weak owing to the weakness ofsuperconductivity in the heavily underdoped regime

Recently nearly optimally doped Na-CCOC single crystals(Tc = 25ndash28 K) have become available and we have searched forQPI in these samples (see the Methods section) It is importantto notice that the crystal structure the atomic species in theblocking layer between CuO2 planes the number of CuO2 planesin the unit cell and the Tc at optimal doping are all differentbetween Na-CCOC and Bi2212 Because these materials are so

completely disparate in the physicschemistry of the blocking layersthrough which tunnelling occurs common QPI phenomena mustderive from the only common characteristic that is intrinsic tothe electronic structure of the CuO2 plane in which high-Tc

superconductivity takes placeFigure 2a shows a typical topographic image of the sample

with Tc sim 25 K All the spectroscopic imaging measurementsreported in this paper have been simultaneously carried out withsimilar atomic-resolution topographic images to maintain theposition registry The cleaved surfaces show a perfect square latticewith superimposed inhomogeneity similar to the topographicimages in the heavily underdoped samples69 Overall similarities

nature physics VOL 3 DECEMBER 2007 wwwnaturecomnaturephysics 867

copy 2007 Nature Publishing Group

ARTICLES

0 06 nS 0 07 nS

05 3

ndash6 meV +6 meV

10 nm

6 meV 6 meV

(01)

(10)

10 nm

10 nm

a b

c d

Figure 3 Fourier-transform STS results on Na-CCOC (Tc sim 28 K x sim 014) demonstrating the QPI in the conductance-ratio map Z (r E ) equiv g (r+E )g (rminusE )a The filled-state g (r E ) taken at minus6meV (Set-up conditions Vs = +150mV and I t = 100 pA) This map is basically dominated by the apparent CB g modulation whichconsists of components with wavelengths of 4a0 4a03 and a0 (ref 9) as manifested by |g (q E )| shown in the inset (Red dark blue and green arrows denote 4a0 4a03and a0 components respectively) The same features are also seen in g (r E ) and |g (q E )| in the empty state (E= +6meV) as shown in b Note that g (r E ) (and |g (q E )|)are very similar between filled and empty states and local conductance maximaminima occur predominantly at the same location c The conductance-ratio map Z (r E )calculated from a and b The CB g modulation almost completely disappears and a new type of modulation emerges d |Z (q E )| obtained by Fourier transformation ofZ (r E ) The q vectors which characterize Z (r E ) are clearly represented by local maxima (dark spots) in the image All of the q vectors expected from the octet model of QPIare observed in |Z (q E )| as indicated by symbols that correspond to those in Fig 1c Note that no extra q vector is detected This result strongly indicates that themodulation in Z (r E ) originates from the QPI

to the heavily underdoped samples are also seen in g(rE) andan averaged tunnelling spectrum g(rE) shows apparent CBmodulations (Fig 2b) whereas the averaged spectrum shows aV-shaped pseudogap below sim100 meV (Fig 2c)

We next focus on spectroscopic features at low energies Asshown in Fig 2d inside the V-shaped pseudogap we found a pairof kinklike features at about plusmn10 meV Thus there are at leasttwo energy scales in the system the sim100 meV pseudogap and

the sim10 meV kink It was found that the low-energy region belowthe two kinklike features tends to be filled up homogeneously overthe field of view under magnetic fields (up to 11 T) or at elevatedtemperatures (up to 20 K) suggesting that superconductivityplays a role for the suppression of low-energy QP states Theseobservations motivated us to search for QPI to directly probethe SG dispersion in this low-Tc but nearly optimally dopedhigh-Tc cuprate

868 nature physics VOL 3 DECEMBER 2007 wwwnaturecomnaturephysics

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ARTICLES

a bk y

( a 0

)

(meV

)

k (ordm)

100

075

050

025

0

40

30

20

10

090807060504030201001000750500250

kx ( a0)π

π

Δ

θ

θk

Bi2212 (ref 22)

Na-CCOC

Figure 4 The obtained FS and the SG dispersion for Na-CCOC (Tc sim 28 K x sim 014) a The locus of octet ends of banana-shaped CCEs in a quadrant of the first Brillouinzone which represents the underlying FS The data are symmetrized about the diagonal owing to mirror symmetry The results for two different samples are plotted in redcircles and blue squares respectively In the analysis we used four independent q4 (E )= (plusmn2kx (E )2ky (E ) ) (2ky (E )plusmn2kx (E ) ) which were clearly observed between3meVlt E lt 15meV The other qi vectors gave us consistent results but the observed energy range was narrower The error bars represent the statistical scatters b TheSG plotted as a function of the FS angle θk defined in a The two different symbols have the same meaning as in a The green line represents the SG dispersion for nearlyoptimally doped Bi2212 with Tc sim 86 K (ref 22) Even though Tc values differ by a factor of three between these two compounds the SG dispersions are very similar

QPI patterns in Bi2212 have been observed in raw g(rE)or |g(qE)| (refs 102122) However in the case of Na-CCOC(Tc sim 28 Kx sim 014) g(rE) and |g(qE)| are primarily governedby the apparent CB g modulations as shown in Fig 3ab and a clearsignature of QPI is difficult to recognize Thus we need to find anew technique to extract QPI patterns which may be hidden behindthe apparent CB g modulations

SEPARATION OF QPI FROM THE lsquoCHECKERBOARDrsquo

To separate QPI from the apparent CB g modulations we considerthe spatial-phase relations in the g modulations A comparisonbetween the positive and negative bias g(rplusmnE) shows that CBmodulations in g(r+E) and g(rminusE) are spatially quite similarand the local conductance maximaminima occur predominantlyat the same location as shown in Fig 3ab In other words theapparent CB g modulations have the same wavelength and arein phase across the Fermi energy The QPI patterns should alsohave the same wavelength in g(r+E) and g(rminusE) becauseE(k) is expected to be particlendashhole symmetric as shown inFig 1a The spatial-phase relation for QPI is not known preciselybut it is plausible that there is a phase difference betweeng(r+E) and g(rminusE) as there is a sum-rule

sumn v2

n(r)+u2n(r) = 1

between particle-like and holelike amplitudes of the Bogoliubovquasiparticle v2

n(r) and u2n(r) where n is an appropriate quantum

number38 Therefore the QPI pattern may be extracted by meansof the spatial-phase-sensitive method if they have a non-trivialphase difference

For this purpose we propose to examine the ratio mapdefined by Z(rE) equiv g(r+E)g(r minusE) and its Fouriertransform |Z(qE)| Let us introduce a basic property of|Z(qE)| If the conductance modulation amplitude at finite qgplusmn

q equiv g(qplusmnE) is much smaller than the uniform componentgplusmn

0 equiv g(q = 0plusmnE) |Z(qE)| can be approximated as (to the firstorder) |Z(qE)| asymp (g+

0 gminus

0 )|(g+

q g+

0 ) minus (gminus

q gminus

0 )| In the present

case the relation |gplusmn

q |gplusmn

0 1 holds for all the energies studiedTherefore the above approximation is well justified Note that|Z(qE)| is sensitive to the relative signs between g(q+E) andg(qminusE) (Note that gplusmn

0 gt 0) Namely taking the ratio reduces thein-phase component

In Fig 3c and d respectively we show the measuredZ(rE = 6 meV) and the Fourier transform |Z(qE = 6 meV)|It is immediately evident that the CB g modulation whichdominates g(rE) and |g(qE)| almost completely disappearsMoreover most satisfyingly another type of modulation emergesThe features of the new modulations are clearly manifested in|Z(qE)| where numerous spots are observed at incommensurateq vectors All of these q vectors are consistently explained in termsof the octet model22 of QPI as shown by symbols in Fig 3dThus the existence of the QPI effect and therefore the coherentBogoliubov quasiparticle of the d-wave superconducting state areclearly confirmed for the first time in Na-CCOC The origin ofthe enhanced QPI pattern and detailed intensity distribution in|Z(qE)| are interesting subjects for future theoretical research

We would like to note that the ratio Z has another importantfeature besides its phase sensitivity39 In general tunnellingconductance g(rE = eV ) is given by

g(reV ) =eIt N (reV )int eVs

0N (rE)dE

Here e is a unit charge It and Vs are the set-up current and biasvoltage for feedback respectively and N is a local DOS Note thatg is not proportional to N but is normalized by the integrated N which is strongly influenced by the set-up conditions through VsOn the other hand Z(r eV ) reflects the ratio of the N preciselybecause the denominator and prefactor eIt are exactly cancelledout by taking the ratio39 This fact means that observed QPI is anintrinsic phenomenon free from any set-point-related issues whichinevitably contaminate g(rE) and |g(qE)| In contrast there is a

nature physics VOL 3 DECEMBER 2007 wwwnaturecomnaturephysics 869

copy 2007 Nature Publishing Group

ARTICLES

possibility that CB g modulation is affected by the set-point effectbecause it disappears in Z at least below the kink energy where QPIis observed Therefore it is important to search for the possibleelectronic superstructure using the set-point-effect-free method

Recently ratio analyses at high energy (sim150 meV) inunderdoped Na-CCOC and Bi2212 have revealed the existenceof an electronic-cluster-glass (ECG) state characterized by thefour-CundashCu-distance-wide bond-centred unidirectional electronicdomains dispersed throughout without long-range order39Similarities between ECG and CB g modulation (for examplecharacteristic length scale of four CundashCu distances) imply closerelations between the two It is interesting to note that apparentlydifferent phenomena high-energy ECG and low-energy QPIhave been detected in similar analysis schemes depending onthe energy scale Given the fact that the ratio analysis extractsintrinsic information on the electronic states establishing therelation between ECG and QPI which may represent localizedand delocalized electronic states respectively represents a key goalfor this field

INSIGHTS FROM THE QPI ANALYSIS

To analyse the observed Na-CCOC QPI quantitatively we measuredqi at different E and determined the energy-dependent locationsof the octet ends of lsquobananasrsquo in k space The obtained locusk(E) = (kx(E)ky(E)) can be interpreted as a normal-state FS Weplot the obtained loci for two different samples in Fig 4a The resultis consistent with a cylindrical FS centred around (ππ) which isinferred from ARPES on underdoped samples12

We can determine the SG dispersion ∆(k) from k(E) as well22In Fig 4b we plot the measured gap energy as a function of FSangle θk about (ππ) For comparison SG dispersion of optimallydoped Bi2212 (ref 22) is shown by a green line Apparently SGdispersions of these two compounds are quite similar Because Tc

values are very different between Na-CCOC (Tc sim28 K) and Bi2212(Tc sim 86 K) we can conclude that the SG dispersion in the nodalregion does not set the energy scale which determines Tc alone

It is worthwhile to note that QPI in optimally dopedNa-CCOC is observed in a rather limited part of the FS near thenode (25 lt θk lt 65) This suggests that available quasiparticlestates are missing from the antinodal region40ndash42 or that theantinodal quasiparticles may not have a correlation length longenough to produce interference due to disorder43 In accord withthis well-defined coherence peaks that manifest the coherentquasiparticle states in the antinodal region1022 are not observed inthe tunnelling spectra as shown in Fig 2c and d In the ARPESdata on the x sim 012 (Tc sim 22 K) sample with slightly lowerhole concentration than the present samples the spectral weightin the antinodal region is still substantially suppressed whichwas pointed out to be linked with CB g modulations12 Here wepostulate that such incoherent antinodal states are not responsiblefor forming phase-coherent Cooper pairs If this picture is correctsuperconductivity is supported only by the coherent part of theFS around the node namely a d-wave SG opens on the so-calledFermi arc This conjecture predicts that the energy scale which setsTc is determined not only by the SG dispersion around the node butalso by the effective lsquoarc lengthrsquo Analogous proposals have also beenmade on the basis of the doping dependence of the gap amplitude44

Finally we compare the tunnelling spectrum of Na-CCOC withthat of Bi2212 In optimally doped Bi2212 well-defined coherencepeaks are ubiquitous10 and thus the entire FS may be responsiblefor superconductivity On the other hand in underdoped Bi2212a different type of spectrum characterized by the V-shaped gapwithout a clear coherence peak appears while the QPI signal is stillthere10 This spectrum is very similar to the spectrum of Na-CCOC

(refs 69) Interestingly in underdoped Bi2212 a non-dispersive CBg modulation is also seen in the region where this type of spectrumis observed10 Therefore it is reasonable to infer that the V-shapedgap without a clear coherence peak the destruction of the antinodalstate and the CB g modulation go hand in hand and are universal inhigh-Tc cuprates Another important common feature observed inboth Na-CCOC and Bi2212 is that this state is superconductingwith long-lived Bogoliubov quasiparticles which are manifestedby QPI

CONCLUSION

We report the first observation of QPI associated with d-wavesuperconductivity in Na-CCOC This provides us with firmevidence for the coexistence of d-wave superconductivity andCB g modulation presumably associated with the ECG statewhich probably represents the nodalantinodal dichotomy in kspace The striking similarity of SG dispersions between Na-CCOC(Tc sim 28 K) and Bi2212 (Tc sim 86 K) in the nodal regions stronglysuggests that the nodalantinodal dichotomy is controlling Tc frommaterial to material Further exploration of k-space physics webelieve should help identify a key ingredient to enhance Tc ofcuprate superconductors A novel analysis g-ratio or Z analysiswas developed to separate QPI signals from the CB g modulationIt is quite sensitive to the spatial-phase difference in quantumadmixtures of particle-like and holelike states and will be a powerfulprobe to explore the quasiparticle interferences in superconductorsand possibly other quantum condensates

METHODS

Na-CCOC single crystals were grown by the flux method under high pressureof the order of several gigapascals37 By optimizing the synthesis conditions wesucceeded in growing suitably sized nearly optimally doped single crystals withTc sim 25 K and 28 K which correspond to x sim 013 and 014 respectively

SI-STM measurements were carried out with a newly developedultrahigh-vacuum very-low-temperature scanning tunnelling microscope atRIKEN45 Samples were cleaved in situ at 77 K under ultrahigh vacuum andwere immediately transferred to the STM unit which was kept at lowtemperature (lt10 K) Tungsten scanning tips were sharpened andcharacterized in situ with atomic resolution by a field-ion microscope which isbuilt in to the STM chamber All the data were taken at a temperatureT sim 04 K

Received 14 February 2007 accepted 19 September 2007 published 28 October 2007

References1 Fischer Oslash Kugler M Maggio-Aprile I Berthod C amp Renner C Scanning tunneling spectroscopy

of high-temperature superconductors Rev Mod Phys 79 353ndash419 (2007)2 Cren T Roditchev D Sacks W amp Klein J Nanometer scale mapping of the density of states in an

inhomogeneous superconductor Eur Phys Lett 54 84ndash90 (2001)3 Howald C Fournier P amp Kapitulnik A Inherent inhomogeneities in tunneling spectra of

Bi2Sr2CaCu2O8minusx crystals in the superconducting state Phys Rev B 64 100504 (2001)4 Pan S H et al Microscopic electronic inhomogeneity in the high-Tc superconductor

Bi2Sr2CaCu2O8+x Nature 413 282ndash285 (2001)5 Lang K M et al Imaging the granular structure of high-Tc superconductivity in underdoped

Bi2Sr2CaCu2O8+δ Nature 415 412ndash416 (2002)6 Kohsaka Y et al Imaging nano-scale electronic inhomogeneity in lightly doped Mott insulator

Ca2minusx Nax CuO2Cl2 Phys Rev Lett 93 097004 (2004)7 Hoffman J E et al A four unit cell periodic pattern of quasi-particle states surrounding vortex cores

in Bi2Sr2CaCu2O8+δ Science 295 466ndash469 (2002)8 Vershinin M et al Local ordering in the pseudogap state of the high-Tc superconductor

Bi2Sr2CaCu2O8+δ Science 303 1995ndash1998 (2004)9 Hanaguri T et al A lsquocheckerboardrsquo electronic crystal state in lightly hole-doped Ca2minusx Nax CuO2Cl2

Nature 430 1001ndash1005 (2004)10 McElroy K et al Coincidence of checkerboard charge order and antinodal state decoherence in

strongly underdoped superconducting Bi2Sr2CaCu2O8+δ Phys Rev Lett 94 197005 (2005)11 Damascelli A Hussain Z amp Shen Z-X Angle-resolved photoemission studies of the cuprate

superconductors Rev Mod Phys 75 473ndash541 (2003)12 Shen K M et al Nodal quasiparticles and antinodal charge ordering in Ca2minusx Nax CuO2Cl2 Science

307 901ndash904 (2005)13 Yoshida T et al Low-energy electronic structure of the high-Tc cuprates La2minusx Srx CuO4 studied by

angle-resolved photoemission spectroscopy J Phys Condens Matter 19 125209 (2007)14 Tanaka K et al Distinct Fermi-momentum-dependent energy gaps in deeply underdoped Bi2212

Science 314 1910ndash1913 (2006)

870 nature physics VOL 3 DECEMBER 2007 wwwnaturecomnaturephysics

copy 2007 Nature Publishing Group

ARTICLES

15 Norman M R et al Destruction of the Fermi surface in underdoped high-Tc superconductorsNature 392 157ndash160 (1998)

16 Tsuei C C amp Kirtley J R Pairing symmetry in cuprate superconductors Rev Mod Phys 72969ndash1016 (2000)

17 Le Tacon M et al Two energy scales and two distinct quasiparticle dynamics in the superconductingstate of underdoped cuprates Nature Phys 2 537ndash543 (2006)

18 Gomes K K et al Visualizing pair formation on the atomic scale in the high-Tc superconductorBi2Sr2CaCu2O8+δ Nature 447 569ndash572 (2007)

19 Boyer M C et al Imaging the two gaps of the high-Tc superconductor Pb-Bi2Sr2CuO6+x NaturePhys published online 16 September 2007 (doi101038nphys725)

20 Timusk T amp Statt B The pseudogap in high-temperature superconductors an experimental surveyRep Prog Phys 62 61ndash122 (1999)

21 Hoffman J E et al Imaging quasiparticle interference in Bi2Sr2CaCu2O8+δ Science 2971148ndash1151 (2002)

22 McElroy K et al Relating atomic-scale electronic phenomena to wave-like quasiparticle states insuperconducting Bi2Sr2CaCu2O8+δ Nature 422 592ndash596 (2003)

23 Wang Q amp Lee D-H Quasiparticle scattering interference in high temperature superconductorsPhys Rev B 67 020511 (2003)

24 Zhang D amp Ting C S Energy-dependent modulations in the local density of states of the cupratesuperconductors Phys Rev B 67 100506 (2003)

25 Pereg-Barnea T amp Franz M Theory of quasiparticle interference patterns in the pseudogap phase ofthe cuprate superconductors Phys Rev B 68 180506 (R) (2003)

26 Capriotti L Scalapino D J amp Sedgewick R D Wave-vector power spectrum of the local tunnelingdensity of states Ripples in a d-wave sea Phys Rev B 68 014508 (2003)

27 Zhu L Atkinson W A amp Hirschfeld P J Power spectrum of many impurities in a d-wavesuperconductor Phys Rev B 69 060503(R) (2004)

28 Pereg-Barnea T amp Franz M Quasiparticle interference patterns as a test for the nature of thepseudogap phase in the cuprate superconductors Int J Mod Phys B 19 731ndash761 (2005)

29 DellrsquoAnna L Lorenzana J Capone M Castellani C amp Grilli M Effect of mesoscopicinhomogeneities on local tunneling density of states in cuprates Phys Rev B 71 064518 (2005)

30 Cheng M amp Su W P Local density of states and angle-resolved photoemission spectral function ofan inhomogeneous d-wave superconductor Phys Rev B 72 094512 (2005)

31 Nunner T S et al Fourier transform spectroscopy of d-wave quasiparticles in the presence of atomicscale pairing disorder Phys Rev B 73 104511 (2006)

32 Hasegawa Y amp Avouris Ph Direct observation of standing wave formation at surface steps usingscanning tunneling spectroscopy Phys Rev Lett 71 1071ndash1074 (1993)

33 Crommie M F Lutz C P amp Eigler D M Imaging standing waves in a two-dimensional electrongas Nature 363 524ndash527 (1993)

34 Sprunger P T Petersen L Plummer E W Laeliggsgaard E amp Besenbacher F Giant Friedeloscillations on the beryllium(0001) surface Science 275 1764ndash1767 (1997)

35 Tinkham M Introduction to Superconductivity 2nd edn 43 (McGraw Hill New York 1996)36 Hiroi Z Kobayashi N amp Takano M Probable hole-doped superconductivity without apical oxygens

in (CaNa)2CuO2Cl2 Nature 371 139ndash141 (1994)37 Kohsaka Y et al Growth of Na-doped Ca2CuO2Cl2 single crystals under high pressures of several

GPa J Am Chem Soc 124 12275ndash12278 (2002)38 Balatsky A V Vekhter I amp Zhu J-X Impurity-induced states in conventional and unconventional

superconductors Rev Mod Phys 78 373ndash433 (2006)39 Kohsaka Y et al An intrinsic bond-centered electronic glass with unidirectional domains in

underdoped cuprates Science 314 1380ndash1385 (2007)40 Furukawa N Rice T M amp Salmhofer M Truncation of a two-dimensional FS due to quasiparticle

gap formation at the saddle points Phys Rev Lett 81 3195ndash3198 (1998)41 Wen X-G amp Lee P A Theory of quasiparticles in the underdoped high-Tc superconducting state

Phys Rev Lett 80 2193ndash2196 (1998)42 Li J-X Wu C-Q amp Lee D-H Checkerboard charge density wave and pseudogap of high-Tc

cuprates Phys Rev B 74 184515 (2006)43 Garg A Randeria M amp Trivedi N Strong correlations lead to protected low energy excitations in

disordered d-wave superconductors Preprint at lthttparXivorgcond-mat0609666gt (2006)44 Oda M Dipasupil R M Momono N amp Ido M Hyperbolic dependence of 210 vs Tc ratio on

hole-doping level in high-Tc cuprates Energy scale in determining Tc J Phys Soc Jpn 69983ndash984 (2000)

45 Hanaguri T Development of high-field STM and its application to the study on magnetically-tunedcriticality in Sr3Ru2O7 J Phys Conf Ser 51 514ndash521 (2006)

AcknowledgementsThe authors thank AV Balatsky C-M Ho D-H Lee K Machida A Mackenzie K McElroyT Tohyama J Zaanen and F-C Zhang for discussions They also thank J Matsuno and P Sharma forcritical readings TH MT and HT are supported by a Grant-in-Aid for Scientific Research from theMinistry of Education Culture Sports Science and Technology of Japan JCD and YK acknowledgesupport from Brookhaven National Laboratory under contract No DE-AC02-98CH1886 with the USDepartment of Energy from the US Department of Energy Award No DE-FG02-06ER46306 and fromthe US Office of Naval ResearchCorrespondence and requests for materials should be addressed to TH

Author contributionsTH was responsible for all aspects of this project except sample growth YK JCD and CLcontributed the project planning and data analysis IY MA MT and KO grew samples and MOcontributed the STM measurements HT contributed the project planning and managed thewhole project

Reprints and permission information is available online at httpnpgnaturecomreprintsandpermissions

nature physics VOL 3 DECEMBER 2007 wwwnaturecomnaturephysics 871

copy 2007 Nature Publishing Group

ARTICLES

Sample bias (mV)

dId

V (n

S)

dId

V (n

S)

ndash200 0 200

Sample bias (mV)

ndash400 400 ndash40 ndash20 0 20 40

a b

c d

5 nm 5 nm

18 nS01

14

12

10

08

06

04

02

0

10

08

06

04

02

0

016 nm0

Figure 2 Spectroscopic features of nearly optimally doped Na-CCOC with Tc sim 25 K (x sim 013) a Typical topographic STM image of Na-CCOC The image wastaken at a sample-bias voltage Vs = +400mV and a tunnelling current I t = 100 pA All the spectroscopic images in this paper were taken simultaneously with similaratomic-resolution topographic images to maintain the atomic-resolution position registry b The conductance g (r E ) map at energy E= +24meV in the same field of viewApparent CB g modulation which has been found in heavily underdoped Na-CCOC (ref 9) is observed even in the nearly optimally doped sample c The spatially averagedconductance spectrum showing the V-shaped pseudogap below sim100meV with particlendashhole asymmetry at higher energies (Set-up condition Vs = +400mV andI t = 100 pA) d The spatially averaged spectrum in the low-energy region indicated by arrows in c showing a pair of kinks in the spectrum inside the pseudogap (Set-upcondition Vs = +150mV and I t = 100 pA) The central interest of this paper is in the electronic state below and around this kink energy

Na-CCOC have been carried out only in the heavily underdopedregion and revealed nanoscale electronic inhomogeneity6 and CBg modulation9 However no QPI signal was detected although theQPI signal may merely be too weak owing to the weakness ofsuperconductivity in the heavily underdoped regime

Recently nearly optimally doped Na-CCOC single crystals(Tc = 25ndash28 K) have become available and we have searched forQPI in these samples (see the Methods section) It is importantto notice that the crystal structure the atomic species in theblocking layer between CuO2 planes the number of CuO2 planesin the unit cell and the Tc at optimal doping are all differentbetween Na-CCOC and Bi2212 Because these materials are so

completely disparate in the physicschemistry of the blocking layersthrough which tunnelling occurs common QPI phenomena mustderive from the only common characteristic that is intrinsic tothe electronic structure of the CuO2 plane in which high-Tc

superconductivity takes placeFigure 2a shows a typical topographic image of the sample

with Tc sim 25 K All the spectroscopic imaging measurementsreported in this paper have been simultaneously carried out withsimilar atomic-resolution topographic images to maintain theposition registry The cleaved surfaces show a perfect square latticewith superimposed inhomogeneity similar to the topographicimages in the heavily underdoped samples69 Overall similarities

nature physics VOL 3 DECEMBER 2007 wwwnaturecomnaturephysics 867

copy 2007 Nature Publishing Group

ARTICLES

0 06 nS 0 07 nS

05 3

ndash6 meV +6 meV

10 nm

6 meV 6 meV

(01)

(10)

10 nm

10 nm

a b

c d

Figure 3 Fourier-transform STS results on Na-CCOC (Tc sim 28 K x sim 014) demonstrating the QPI in the conductance-ratio map Z (r E ) equiv g (r+E )g (rminusE )a The filled-state g (r E ) taken at minus6meV (Set-up conditions Vs = +150mV and I t = 100 pA) This map is basically dominated by the apparent CB g modulation whichconsists of components with wavelengths of 4a0 4a03 and a0 (ref 9) as manifested by |g (q E )| shown in the inset (Red dark blue and green arrows denote 4a0 4a03and a0 components respectively) The same features are also seen in g (r E ) and |g (q E )| in the empty state (E= +6meV) as shown in b Note that g (r E ) (and |g (q E )|)are very similar between filled and empty states and local conductance maximaminima occur predominantly at the same location c The conductance-ratio map Z (r E )calculated from a and b The CB g modulation almost completely disappears and a new type of modulation emerges d |Z (q E )| obtained by Fourier transformation ofZ (r E ) The q vectors which characterize Z (r E ) are clearly represented by local maxima (dark spots) in the image All of the q vectors expected from the octet model of QPIare observed in |Z (q E )| as indicated by symbols that correspond to those in Fig 1c Note that no extra q vector is detected This result strongly indicates that themodulation in Z (r E ) originates from the QPI

to the heavily underdoped samples are also seen in g(rE) andan averaged tunnelling spectrum g(rE) shows apparent CBmodulations (Fig 2b) whereas the averaged spectrum shows aV-shaped pseudogap below sim100 meV (Fig 2c)

We next focus on spectroscopic features at low energies Asshown in Fig 2d inside the V-shaped pseudogap we found a pairof kinklike features at about plusmn10 meV Thus there are at leasttwo energy scales in the system the sim100 meV pseudogap and

the sim10 meV kink It was found that the low-energy region belowthe two kinklike features tends to be filled up homogeneously overthe field of view under magnetic fields (up to 11 T) or at elevatedtemperatures (up to 20 K) suggesting that superconductivityplays a role for the suppression of low-energy QP states Theseobservations motivated us to search for QPI to directly probethe SG dispersion in this low-Tc but nearly optimally dopedhigh-Tc cuprate

868 nature physics VOL 3 DECEMBER 2007 wwwnaturecomnaturephysics

copy 2007 Nature Publishing Group

ARTICLES

a bk y

( a 0

)

(meV

)

k (ordm)

100

075

050

025

0

40

30

20

10

090807060504030201001000750500250

kx ( a0)π

π

Δ

θ

θk

Bi2212 (ref 22)

Na-CCOC

Figure 4 The obtained FS and the SG dispersion for Na-CCOC (Tc sim 28 K x sim 014) a The locus of octet ends of banana-shaped CCEs in a quadrant of the first Brillouinzone which represents the underlying FS The data are symmetrized about the diagonal owing to mirror symmetry The results for two different samples are plotted in redcircles and blue squares respectively In the analysis we used four independent q4 (E )= (plusmn2kx (E )2ky (E ) ) (2ky (E )plusmn2kx (E ) ) which were clearly observed between3meVlt E lt 15meV The other qi vectors gave us consistent results but the observed energy range was narrower The error bars represent the statistical scatters b TheSG plotted as a function of the FS angle θk defined in a The two different symbols have the same meaning as in a The green line represents the SG dispersion for nearlyoptimally doped Bi2212 with Tc sim 86 K (ref 22) Even though Tc values differ by a factor of three between these two compounds the SG dispersions are very similar

QPI patterns in Bi2212 have been observed in raw g(rE)or |g(qE)| (refs 102122) However in the case of Na-CCOC(Tc sim 28 Kx sim 014) g(rE) and |g(qE)| are primarily governedby the apparent CB g modulations as shown in Fig 3ab and a clearsignature of QPI is difficult to recognize Thus we need to find anew technique to extract QPI patterns which may be hidden behindthe apparent CB g modulations

SEPARATION OF QPI FROM THE lsquoCHECKERBOARDrsquo

To separate QPI from the apparent CB g modulations we considerthe spatial-phase relations in the g modulations A comparisonbetween the positive and negative bias g(rplusmnE) shows that CBmodulations in g(r+E) and g(rminusE) are spatially quite similarand the local conductance maximaminima occur predominantlyat the same location as shown in Fig 3ab In other words theapparent CB g modulations have the same wavelength and arein phase across the Fermi energy The QPI patterns should alsohave the same wavelength in g(r+E) and g(rminusE) becauseE(k) is expected to be particlendashhole symmetric as shown inFig 1a The spatial-phase relation for QPI is not known preciselybut it is plausible that there is a phase difference betweeng(r+E) and g(rminusE) as there is a sum-rule

sumn v2

n(r)+u2n(r) = 1

between particle-like and holelike amplitudes of the Bogoliubovquasiparticle v2

n(r) and u2n(r) where n is an appropriate quantum

number38 Therefore the QPI pattern may be extracted by meansof the spatial-phase-sensitive method if they have a non-trivialphase difference

For this purpose we propose to examine the ratio mapdefined by Z(rE) equiv g(r+E)g(r minusE) and its Fouriertransform |Z(qE)| Let us introduce a basic property of|Z(qE)| If the conductance modulation amplitude at finite qgplusmn

q equiv g(qplusmnE) is much smaller than the uniform componentgplusmn

0 equiv g(q = 0plusmnE) |Z(qE)| can be approximated as (to the firstorder) |Z(qE)| asymp (g+

0 gminus

0 )|(g+

q g+

0 ) minus (gminus

q gminus

0 )| In the present

case the relation |gplusmn

q |gplusmn

0 1 holds for all the energies studiedTherefore the above approximation is well justified Note that|Z(qE)| is sensitive to the relative signs between g(q+E) andg(qminusE) (Note that gplusmn

0 gt 0) Namely taking the ratio reduces thein-phase component

In Fig 3c and d respectively we show the measuredZ(rE = 6 meV) and the Fourier transform |Z(qE = 6 meV)|It is immediately evident that the CB g modulation whichdominates g(rE) and |g(qE)| almost completely disappearsMoreover most satisfyingly another type of modulation emergesThe features of the new modulations are clearly manifested in|Z(qE)| where numerous spots are observed at incommensurateq vectors All of these q vectors are consistently explained in termsof the octet model22 of QPI as shown by symbols in Fig 3dThus the existence of the QPI effect and therefore the coherentBogoliubov quasiparticle of the d-wave superconducting state areclearly confirmed for the first time in Na-CCOC The origin ofthe enhanced QPI pattern and detailed intensity distribution in|Z(qE)| are interesting subjects for future theoretical research

We would like to note that the ratio Z has another importantfeature besides its phase sensitivity39 In general tunnellingconductance g(rE = eV ) is given by

g(reV ) =eIt N (reV )int eVs

0N (rE)dE

Here e is a unit charge It and Vs are the set-up current and biasvoltage for feedback respectively and N is a local DOS Note thatg is not proportional to N but is normalized by the integrated N which is strongly influenced by the set-up conditions through VsOn the other hand Z(r eV ) reflects the ratio of the N preciselybecause the denominator and prefactor eIt are exactly cancelledout by taking the ratio39 This fact means that observed QPI is anintrinsic phenomenon free from any set-point-related issues whichinevitably contaminate g(rE) and |g(qE)| In contrast there is a

nature physics VOL 3 DECEMBER 2007 wwwnaturecomnaturephysics 869

copy 2007 Nature Publishing Group

ARTICLES

possibility that CB g modulation is affected by the set-point effectbecause it disappears in Z at least below the kink energy where QPIis observed Therefore it is important to search for the possibleelectronic superstructure using the set-point-effect-free method

Recently ratio analyses at high energy (sim150 meV) inunderdoped Na-CCOC and Bi2212 have revealed the existenceof an electronic-cluster-glass (ECG) state characterized by thefour-CundashCu-distance-wide bond-centred unidirectional electronicdomains dispersed throughout without long-range order39Similarities between ECG and CB g modulation (for examplecharacteristic length scale of four CundashCu distances) imply closerelations between the two It is interesting to note that apparentlydifferent phenomena high-energy ECG and low-energy QPIhave been detected in similar analysis schemes depending onthe energy scale Given the fact that the ratio analysis extractsintrinsic information on the electronic states establishing therelation between ECG and QPI which may represent localizedand delocalized electronic states respectively represents a key goalfor this field

INSIGHTS FROM THE QPI ANALYSIS

To analyse the observed Na-CCOC QPI quantitatively we measuredqi at different E and determined the energy-dependent locationsof the octet ends of lsquobananasrsquo in k space The obtained locusk(E) = (kx(E)ky(E)) can be interpreted as a normal-state FS Weplot the obtained loci for two different samples in Fig 4a The resultis consistent with a cylindrical FS centred around (ππ) which isinferred from ARPES on underdoped samples12

We can determine the SG dispersion ∆(k) from k(E) as well22In Fig 4b we plot the measured gap energy as a function of FSangle θk about (ππ) For comparison SG dispersion of optimallydoped Bi2212 (ref 22) is shown by a green line Apparently SGdispersions of these two compounds are quite similar Because Tc

values are very different between Na-CCOC (Tc sim28 K) and Bi2212(Tc sim 86 K) we can conclude that the SG dispersion in the nodalregion does not set the energy scale which determines Tc alone

It is worthwhile to note that QPI in optimally dopedNa-CCOC is observed in a rather limited part of the FS near thenode (25 lt θk lt 65) This suggests that available quasiparticlestates are missing from the antinodal region40ndash42 or that theantinodal quasiparticles may not have a correlation length longenough to produce interference due to disorder43 In accord withthis well-defined coherence peaks that manifest the coherentquasiparticle states in the antinodal region1022 are not observed inthe tunnelling spectra as shown in Fig 2c and d In the ARPESdata on the x sim 012 (Tc sim 22 K) sample with slightly lowerhole concentration than the present samples the spectral weightin the antinodal region is still substantially suppressed whichwas pointed out to be linked with CB g modulations12 Here wepostulate that such incoherent antinodal states are not responsiblefor forming phase-coherent Cooper pairs If this picture is correctsuperconductivity is supported only by the coherent part of theFS around the node namely a d-wave SG opens on the so-calledFermi arc This conjecture predicts that the energy scale which setsTc is determined not only by the SG dispersion around the node butalso by the effective lsquoarc lengthrsquo Analogous proposals have also beenmade on the basis of the doping dependence of the gap amplitude44

Finally we compare the tunnelling spectrum of Na-CCOC withthat of Bi2212 In optimally doped Bi2212 well-defined coherencepeaks are ubiquitous10 and thus the entire FS may be responsiblefor superconductivity On the other hand in underdoped Bi2212a different type of spectrum characterized by the V-shaped gapwithout a clear coherence peak appears while the QPI signal is stillthere10 This spectrum is very similar to the spectrum of Na-CCOC

(refs 69) Interestingly in underdoped Bi2212 a non-dispersive CBg modulation is also seen in the region where this type of spectrumis observed10 Therefore it is reasonable to infer that the V-shapedgap without a clear coherence peak the destruction of the antinodalstate and the CB g modulation go hand in hand and are universal inhigh-Tc cuprates Another important common feature observed inboth Na-CCOC and Bi2212 is that this state is superconductingwith long-lived Bogoliubov quasiparticles which are manifestedby QPI

CONCLUSION

We report the first observation of QPI associated with d-wavesuperconductivity in Na-CCOC This provides us with firmevidence for the coexistence of d-wave superconductivity andCB g modulation presumably associated with the ECG statewhich probably represents the nodalantinodal dichotomy in kspace The striking similarity of SG dispersions between Na-CCOC(Tc sim 28 K) and Bi2212 (Tc sim 86 K) in the nodal regions stronglysuggests that the nodalantinodal dichotomy is controlling Tc frommaterial to material Further exploration of k-space physics webelieve should help identify a key ingredient to enhance Tc ofcuprate superconductors A novel analysis g-ratio or Z analysiswas developed to separate QPI signals from the CB g modulationIt is quite sensitive to the spatial-phase difference in quantumadmixtures of particle-like and holelike states and will be a powerfulprobe to explore the quasiparticle interferences in superconductorsand possibly other quantum condensates

METHODS

Na-CCOC single crystals were grown by the flux method under high pressureof the order of several gigapascals37 By optimizing the synthesis conditions wesucceeded in growing suitably sized nearly optimally doped single crystals withTc sim 25 K and 28 K which correspond to x sim 013 and 014 respectively

SI-STM measurements were carried out with a newly developedultrahigh-vacuum very-low-temperature scanning tunnelling microscope atRIKEN45 Samples were cleaved in situ at 77 K under ultrahigh vacuum andwere immediately transferred to the STM unit which was kept at lowtemperature (lt10 K) Tungsten scanning tips were sharpened andcharacterized in situ with atomic resolution by a field-ion microscope which isbuilt in to the STM chamber All the data were taken at a temperatureT sim 04 K

Received 14 February 2007 accepted 19 September 2007 published 28 October 2007

References1 Fischer Oslash Kugler M Maggio-Aprile I Berthod C amp Renner C Scanning tunneling spectroscopy

of high-temperature superconductors Rev Mod Phys 79 353ndash419 (2007)2 Cren T Roditchev D Sacks W amp Klein J Nanometer scale mapping of the density of states in an

inhomogeneous superconductor Eur Phys Lett 54 84ndash90 (2001)3 Howald C Fournier P amp Kapitulnik A Inherent inhomogeneities in tunneling spectra of

Bi2Sr2CaCu2O8minusx crystals in the superconducting state Phys Rev B 64 100504 (2001)4 Pan S H et al Microscopic electronic inhomogeneity in the high-Tc superconductor

Bi2Sr2CaCu2O8+x Nature 413 282ndash285 (2001)5 Lang K M et al Imaging the granular structure of high-Tc superconductivity in underdoped

Bi2Sr2CaCu2O8+δ Nature 415 412ndash416 (2002)6 Kohsaka Y et al Imaging nano-scale electronic inhomogeneity in lightly doped Mott insulator

Ca2minusx Nax CuO2Cl2 Phys Rev Lett 93 097004 (2004)7 Hoffman J E et al A four unit cell periodic pattern of quasi-particle states surrounding vortex cores

in Bi2Sr2CaCu2O8+δ Science 295 466ndash469 (2002)8 Vershinin M et al Local ordering in the pseudogap state of the high-Tc superconductor

Bi2Sr2CaCu2O8+δ Science 303 1995ndash1998 (2004)9 Hanaguri T et al A lsquocheckerboardrsquo electronic crystal state in lightly hole-doped Ca2minusx Nax CuO2Cl2

Nature 430 1001ndash1005 (2004)10 McElroy K et al Coincidence of checkerboard charge order and antinodal state decoherence in

strongly underdoped superconducting Bi2Sr2CaCu2O8+δ Phys Rev Lett 94 197005 (2005)11 Damascelli A Hussain Z amp Shen Z-X Angle-resolved photoemission studies of the cuprate

superconductors Rev Mod Phys 75 473ndash541 (2003)12 Shen K M et al Nodal quasiparticles and antinodal charge ordering in Ca2minusx Nax CuO2Cl2 Science

307 901ndash904 (2005)13 Yoshida T et al Low-energy electronic structure of the high-Tc cuprates La2minusx Srx CuO4 studied by

angle-resolved photoemission spectroscopy J Phys Condens Matter 19 125209 (2007)14 Tanaka K et al Distinct Fermi-momentum-dependent energy gaps in deeply underdoped Bi2212

Science 314 1910ndash1913 (2006)

870 nature physics VOL 3 DECEMBER 2007 wwwnaturecomnaturephysics

copy 2007 Nature Publishing Group

ARTICLES

15 Norman M R et al Destruction of the Fermi surface in underdoped high-Tc superconductorsNature 392 157ndash160 (1998)

16 Tsuei C C amp Kirtley J R Pairing symmetry in cuprate superconductors Rev Mod Phys 72969ndash1016 (2000)

17 Le Tacon M et al Two energy scales and two distinct quasiparticle dynamics in the superconductingstate of underdoped cuprates Nature Phys 2 537ndash543 (2006)

18 Gomes K K et al Visualizing pair formation on the atomic scale in the high-Tc superconductorBi2Sr2CaCu2O8+δ Nature 447 569ndash572 (2007)

19 Boyer M C et al Imaging the two gaps of the high-Tc superconductor Pb-Bi2Sr2CuO6+x NaturePhys published online 16 September 2007 (doi101038nphys725)

20 Timusk T amp Statt B The pseudogap in high-temperature superconductors an experimental surveyRep Prog Phys 62 61ndash122 (1999)

21 Hoffman J E et al Imaging quasiparticle interference in Bi2Sr2CaCu2O8+δ Science 2971148ndash1151 (2002)

22 McElroy K et al Relating atomic-scale electronic phenomena to wave-like quasiparticle states insuperconducting Bi2Sr2CaCu2O8+δ Nature 422 592ndash596 (2003)

23 Wang Q amp Lee D-H Quasiparticle scattering interference in high temperature superconductorsPhys Rev B 67 020511 (2003)

24 Zhang D amp Ting C S Energy-dependent modulations in the local density of states of the cupratesuperconductors Phys Rev B 67 100506 (2003)

25 Pereg-Barnea T amp Franz M Theory of quasiparticle interference patterns in the pseudogap phase ofthe cuprate superconductors Phys Rev B 68 180506 (R) (2003)

26 Capriotti L Scalapino D J amp Sedgewick R D Wave-vector power spectrum of the local tunnelingdensity of states Ripples in a d-wave sea Phys Rev B 68 014508 (2003)

27 Zhu L Atkinson W A amp Hirschfeld P J Power spectrum of many impurities in a d-wavesuperconductor Phys Rev B 69 060503(R) (2004)

28 Pereg-Barnea T amp Franz M Quasiparticle interference patterns as a test for the nature of thepseudogap phase in the cuprate superconductors Int J Mod Phys B 19 731ndash761 (2005)

29 DellrsquoAnna L Lorenzana J Capone M Castellani C amp Grilli M Effect of mesoscopicinhomogeneities on local tunneling density of states in cuprates Phys Rev B 71 064518 (2005)

30 Cheng M amp Su W P Local density of states and angle-resolved photoemission spectral function ofan inhomogeneous d-wave superconductor Phys Rev B 72 094512 (2005)

31 Nunner T S et al Fourier transform spectroscopy of d-wave quasiparticles in the presence of atomicscale pairing disorder Phys Rev B 73 104511 (2006)

32 Hasegawa Y amp Avouris Ph Direct observation of standing wave formation at surface steps usingscanning tunneling spectroscopy Phys Rev Lett 71 1071ndash1074 (1993)

33 Crommie M F Lutz C P amp Eigler D M Imaging standing waves in a two-dimensional electrongas Nature 363 524ndash527 (1993)

34 Sprunger P T Petersen L Plummer E W Laeliggsgaard E amp Besenbacher F Giant Friedeloscillations on the beryllium(0001) surface Science 275 1764ndash1767 (1997)

35 Tinkham M Introduction to Superconductivity 2nd edn 43 (McGraw Hill New York 1996)36 Hiroi Z Kobayashi N amp Takano M Probable hole-doped superconductivity without apical oxygens

in (CaNa)2CuO2Cl2 Nature 371 139ndash141 (1994)37 Kohsaka Y et al Growth of Na-doped Ca2CuO2Cl2 single crystals under high pressures of several

GPa J Am Chem Soc 124 12275ndash12278 (2002)38 Balatsky A V Vekhter I amp Zhu J-X Impurity-induced states in conventional and unconventional

superconductors Rev Mod Phys 78 373ndash433 (2006)39 Kohsaka Y et al An intrinsic bond-centered electronic glass with unidirectional domains in

underdoped cuprates Science 314 1380ndash1385 (2007)40 Furukawa N Rice T M amp Salmhofer M Truncation of a two-dimensional FS due to quasiparticle

gap formation at the saddle points Phys Rev Lett 81 3195ndash3198 (1998)41 Wen X-G amp Lee P A Theory of quasiparticles in the underdoped high-Tc superconducting state

Phys Rev Lett 80 2193ndash2196 (1998)42 Li J-X Wu C-Q amp Lee D-H Checkerboard charge density wave and pseudogap of high-Tc

cuprates Phys Rev B 74 184515 (2006)43 Garg A Randeria M amp Trivedi N Strong correlations lead to protected low energy excitations in

disordered d-wave superconductors Preprint at lthttparXivorgcond-mat0609666gt (2006)44 Oda M Dipasupil R M Momono N amp Ido M Hyperbolic dependence of 210 vs Tc ratio on

hole-doping level in high-Tc cuprates Energy scale in determining Tc J Phys Soc Jpn 69983ndash984 (2000)

45 Hanaguri T Development of high-field STM and its application to the study on magnetically-tunedcriticality in Sr3Ru2O7 J Phys Conf Ser 51 514ndash521 (2006)

AcknowledgementsThe authors thank AV Balatsky C-M Ho D-H Lee K Machida A Mackenzie K McElroyT Tohyama J Zaanen and F-C Zhang for discussions They also thank J Matsuno and P Sharma forcritical readings TH MT and HT are supported by a Grant-in-Aid for Scientific Research from theMinistry of Education Culture Sports Science and Technology of Japan JCD and YK acknowledgesupport from Brookhaven National Laboratory under contract No DE-AC02-98CH1886 with the USDepartment of Energy from the US Department of Energy Award No DE-FG02-06ER46306 and fromthe US Office of Naval ResearchCorrespondence and requests for materials should be addressed to TH

Author contributionsTH was responsible for all aspects of this project except sample growth YK JCD and CLcontributed the project planning and data analysis IY MA MT and KO grew samples and MOcontributed the STM measurements HT contributed the project planning and managed thewhole project

Reprints and permission information is available online at httpnpgnaturecomreprintsandpermissions

nature physics VOL 3 DECEMBER 2007 wwwnaturecomnaturephysics 871

copy 2007 Nature Publishing Group

ARTICLES

0 06 nS 0 07 nS

05 3

ndash6 meV +6 meV

10 nm

6 meV 6 meV

(01)

(10)

10 nm

10 nm

a b

c d

Figure 3 Fourier-transform STS results on Na-CCOC (Tc sim 28 K x sim 014) demonstrating the QPI in the conductance-ratio map Z (r E ) equiv g (r+E )g (rminusE )a The filled-state g (r E ) taken at minus6meV (Set-up conditions Vs = +150mV and I t = 100 pA) This map is basically dominated by the apparent CB g modulation whichconsists of components with wavelengths of 4a0 4a03 and a0 (ref 9) as manifested by |g (q E )| shown in the inset (Red dark blue and green arrows denote 4a0 4a03and a0 components respectively) The same features are also seen in g (r E ) and |g (q E )| in the empty state (E= +6meV) as shown in b Note that g (r E ) (and |g (q E )|)are very similar between filled and empty states and local conductance maximaminima occur predominantly at the same location c The conductance-ratio map Z (r E )calculated from a and b The CB g modulation almost completely disappears and a new type of modulation emerges d |Z (q E )| obtained by Fourier transformation ofZ (r E ) The q vectors which characterize Z (r E ) are clearly represented by local maxima (dark spots) in the image All of the q vectors expected from the octet model of QPIare observed in |Z (q E )| as indicated by symbols that correspond to those in Fig 1c Note that no extra q vector is detected This result strongly indicates that themodulation in Z (r E ) originates from the QPI

to the heavily underdoped samples are also seen in g(rE) andan averaged tunnelling spectrum g(rE) shows apparent CBmodulations (Fig 2b) whereas the averaged spectrum shows aV-shaped pseudogap below sim100 meV (Fig 2c)

We next focus on spectroscopic features at low energies Asshown in Fig 2d inside the V-shaped pseudogap we found a pairof kinklike features at about plusmn10 meV Thus there are at leasttwo energy scales in the system the sim100 meV pseudogap and

the sim10 meV kink It was found that the low-energy region belowthe two kinklike features tends to be filled up homogeneously overthe field of view under magnetic fields (up to 11 T) or at elevatedtemperatures (up to 20 K) suggesting that superconductivityplays a role for the suppression of low-energy QP states Theseobservations motivated us to search for QPI to directly probethe SG dispersion in this low-Tc but nearly optimally dopedhigh-Tc cuprate

868 nature physics VOL 3 DECEMBER 2007 wwwnaturecomnaturephysics

copy 2007 Nature Publishing Group

ARTICLES

a bk y

( a 0

)

(meV

)

k (ordm)

100

075

050

025

0

40

30

20

10

090807060504030201001000750500250

kx ( a0)π

π

Δ

θ

θk

Bi2212 (ref 22)

Na-CCOC

Figure 4 The obtained FS and the SG dispersion for Na-CCOC (Tc sim 28 K x sim 014) a The locus of octet ends of banana-shaped CCEs in a quadrant of the first Brillouinzone which represents the underlying FS The data are symmetrized about the diagonal owing to mirror symmetry The results for two different samples are plotted in redcircles and blue squares respectively In the analysis we used four independent q4 (E )= (plusmn2kx (E )2ky (E ) ) (2ky (E )plusmn2kx (E ) ) which were clearly observed between3meVlt E lt 15meV The other qi vectors gave us consistent results but the observed energy range was narrower The error bars represent the statistical scatters b TheSG plotted as a function of the FS angle θk defined in a The two different symbols have the same meaning as in a The green line represents the SG dispersion for nearlyoptimally doped Bi2212 with Tc sim 86 K (ref 22) Even though Tc values differ by a factor of three between these two compounds the SG dispersions are very similar

QPI patterns in Bi2212 have been observed in raw g(rE)or |g(qE)| (refs 102122) However in the case of Na-CCOC(Tc sim 28 Kx sim 014) g(rE) and |g(qE)| are primarily governedby the apparent CB g modulations as shown in Fig 3ab and a clearsignature of QPI is difficult to recognize Thus we need to find anew technique to extract QPI patterns which may be hidden behindthe apparent CB g modulations

SEPARATION OF QPI FROM THE lsquoCHECKERBOARDrsquo

To separate QPI from the apparent CB g modulations we considerthe spatial-phase relations in the g modulations A comparisonbetween the positive and negative bias g(rplusmnE) shows that CBmodulations in g(r+E) and g(rminusE) are spatially quite similarand the local conductance maximaminima occur predominantlyat the same location as shown in Fig 3ab In other words theapparent CB g modulations have the same wavelength and arein phase across the Fermi energy The QPI patterns should alsohave the same wavelength in g(r+E) and g(rminusE) becauseE(k) is expected to be particlendashhole symmetric as shown inFig 1a The spatial-phase relation for QPI is not known preciselybut it is plausible that there is a phase difference betweeng(r+E) and g(rminusE) as there is a sum-rule

sumn v2

n(r)+u2n(r) = 1

between particle-like and holelike amplitudes of the Bogoliubovquasiparticle v2

n(r) and u2n(r) where n is an appropriate quantum

number38 Therefore the QPI pattern may be extracted by meansof the spatial-phase-sensitive method if they have a non-trivialphase difference

For this purpose we propose to examine the ratio mapdefined by Z(rE) equiv g(r+E)g(r minusE) and its Fouriertransform |Z(qE)| Let us introduce a basic property of|Z(qE)| If the conductance modulation amplitude at finite qgplusmn

q equiv g(qplusmnE) is much smaller than the uniform componentgplusmn

0 equiv g(q = 0plusmnE) |Z(qE)| can be approximated as (to the firstorder) |Z(qE)| asymp (g+

0 gminus

0 )|(g+

q g+

0 ) minus (gminus

q gminus

0 )| In the present

case the relation |gplusmn

q |gplusmn

0 1 holds for all the energies studiedTherefore the above approximation is well justified Note that|Z(qE)| is sensitive to the relative signs between g(q+E) andg(qminusE) (Note that gplusmn

0 gt 0) Namely taking the ratio reduces thein-phase component

In Fig 3c and d respectively we show the measuredZ(rE = 6 meV) and the Fourier transform |Z(qE = 6 meV)|It is immediately evident that the CB g modulation whichdominates g(rE) and |g(qE)| almost completely disappearsMoreover most satisfyingly another type of modulation emergesThe features of the new modulations are clearly manifested in|Z(qE)| where numerous spots are observed at incommensurateq vectors All of these q vectors are consistently explained in termsof the octet model22 of QPI as shown by symbols in Fig 3dThus the existence of the QPI effect and therefore the coherentBogoliubov quasiparticle of the d-wave superconducting state areclearly confirmed for the first time in Na-CCOC The origin ofthe enhanced QPI pattern and detailed intensity distribution in|Z(qE)| are interesting subjects for future theoretical research

We would like to note that the ratio Z has another importantfeature besides its phase sensitivity39 In general tunnellingconductance g(rE = eV ) is given by

g(reV ) =eIt N (reV )int eVs

0N (rE)dE

Here e is a unit charge It and Vs are the set-up current and biasvoltage for feedback respectively and N is a local DOS Note thatg is not proportional to N but is normalized by the integrated N which is strongly influenced by the set-up conditions through VsOn the other hand Z(r eV ) reflects the ratio of the N preciselybecause the denominator and prefactor eIt are exactly cancelledout by taking the ratio39 This fact means that observed QPI is anintrinsic phenomenon free from any set-point-related issues whichinevitably contaminate g(rE) and |g(qE)| In contrast there is a

nature physics VOL 3 DECEMBER 2007 wwwnaturecomnaturephysics 869

copy 2007 Nature Publishing Group

ARTICLES

possibility that CB g modulation is affected by the set-point effectbecause it disappears in Z at least below the kink energy where QPIis observed Therefore it is important to search for the possibleelectronic superstructure using the set-point-effect-free method

Recently ratio analyses at high energy (sim150 meV) inunderdoped Na-CCOC and Bi2212 have revealed the existenceof an electronic-cluster-glass (ECG) state characterized by thefour-CundashCu-distance-wide bond-centred unidirectional electronicdomains dispersed throughout without long-range order39Similarities between ECG and CB g modulation (for examplecharacteristic length scale of four CundashCu distances) imply closerelations between the two It is interesting to note that apparentlydifferent phenomena high-energy ECG and low-energy QPIhave been detected in similar analysis schemes depending onthe energy scale Given the fact that the ratio analysis extractsintrinsic information on the electronic states establishing therelation between ECG and QPI which may represent localizedand delocalized electronic states respectively represents a key goalfor this field

INSIGHTS FROM THE QPI ANALYSIS

To analyse the observed Na-CCOC QPI quantitatively we measuredqi at different E and determined the energy-dependent locationsof the octet ends of lsquobananasrsquo in k space The obtained locusk(E) = (kx(E)ky(E)) can be interpreted as a normal-state FS Weplot the obtained loci for two different samples in Fig 4a The resultis consistent with a cylindrical FS centred around (ππ) which isinferred from ARPES on underdoped samples12

We can determine the SG dispersion ∆(k) from k(E) as well22In Fig 4b we plot the measured gap energy as a function of FSangle θk about (ππ) For comparison SG dispersion of optimallydoped Bi2212 (ref 22) is shown by a green line Apparently SGdispersions of these two compounds are quite similar Because Tc

values are very different between Na-CCOC (Tc sim28 K) and Bi2212(Tc sim 86 K) we can conclude that the SG dispersion in the nodalregion does not set the energy scale which determines Tc alone

It is worthwhile to note that QPI in optimally dopedNa-CCOC is observed in a rather limited part of the FS near thenode (25 lt θk lt 65) This suggests that available quasiparticlestates are missing from the antinodal region40ndash42 or that theantinodal quasiparticles may not have a correlation length longenough to produce interference due to disorder43 In accord withthis well-defined coherence peaks that manifest the coherentquasiparticle states in the antinodal region1022 are not observed inthe tunnelling spectra as shown in Fig 2c and d In the ARPESdata on the x sim 012 (Tc sim 22 K) sample with slightly lowerhole concentration than the present samples the spectral weightin the antinodal region is still substantially suppressed whichwas pointed out to be linked with CB g modulations12 Here wepostulate that such incoherent antinodal states are not responsiblefor forming phase-coherent Cooper pairs If this picture is correctsuperconductivity is supported only by the coherent part of theFS around the node namely a d-wave SG opens on the so-calledFermi arc This conjecture predicts that the energy scale which setsTc is determined not only by the SG dispersion around the node butalso by the effective lsquoarc lengthrsquo Analogous proposals have also beenmade on the basis of the doping dependence of the gap amplitude44

Finally we compare the tunnelling spectrum of Na-CCOC withthat of Bi2212 In optimally doped Bi2212 well-defined coherencepeaks are ubiquitous10 and thus the entire FS may be responsiblefor superconductivity On the other hand in underdoped Bi2212a different type of spectrum characterized by the V-shaped gapwithout a clear coherence peak appears while the QPI signal is stillthere10 This spectrum is very similar to the spectrum of Na-CCOC

(refs 69) Interestingly in underdoped Bi2212 a non-dispersive CBg modulation is also seen in the region where this type of spectrumis observed10 Therefore it is reasonable to infer that the V-shapedgap without a clear coherence peak the destruction of the antinodalstate and the CB g modulation go hand in hand and are universal inhigh-Tc cuprates Another important common feature observed inboth Na-CCOC and Bi2212 is that this state is superconductingwith long-lived Bogoliubov quasiparticles which are manifestedby QPI

CONCLUSION

We report the first observation of QPI associated with d-wavesuperconductivity in Na-CCOC This provides us with firmevidence for the coexistence of d-wave superconductivity andCB g modulation presumably associated with the ECG statewhich probably represents the nodalantinodal dichotomy in kspace The striking similarity of SG dispersions between Na-CCOC(Tc sim 28 K) and Bi2212 (Tc sim 86 K) in the nodal regions stronglysuggests that the nodalantinodal dichotomy is controlling Tc frommaterial to material Further exploration of k-space physics webelieve should help identify a key ingredient to enhance Tc ofcuprate superconductors A novel analysis g-ratio or Z analysiswas developed to separate QPI signals from the CB g modulationIt is quite sensitive to the spatial-phase difference in quantumadmixtures of particle-like and holelike states and will be a powerfulprobe to explore the quasiparticle interferences in superconductorsand possibly other quantum condensates

METHODS

Na-CCOC single crystals were grown by the flux method under high pressureof the order of several gigapascals37 By optimizing the synthesis conditions wesucceeded in growing suitably sized nearly optimally doped single crystals withTc sim 25 K and 28 K which correspond to x sim 013 and 014 respectively

SI-STM measurements were carried out with a newly developedultrahigh-vacuum very-low-temperature scanning tunnelling microscope atRIKEN45 Samples were cleaved in situ at 77 K under ultrahigh vacuum andwere immediately transferred to the STM unit which was kept at lowtemperature (lt10 K) Tungsten scanning tips were sharpened andcharacterized in situ with atomic resolution by a field-ion microscope which isbuilt in to the STM chamber All the data were taken at a temperatureT sim 04 K

Received 14 February 2007 accepted 19 September 2007 published 28 October 2007

References1 Fischer Oslash Kugler M Maggio-Aprile I Berthod C amp Renner C Scanning tunneling spectroscopy

of high-temperature superconductors Rev Mod Phys 79 353ndash419 (2007)2 Cren T Roditchev D Sacks W amp Klein J Nanometer scale mapping of the density of states in an

inhomogeneous superconductor Eur Phys Lett 54 84ndash90 (2001)3 Howald C Fournier P amp Kapitulnik A Inherent inhomogeneities in tunneling spectra of

Bi2Sr2CaCu2O8minusx crystals in the superconducting state Phys Rev B 64 100504 (2001)4 Pan S H et al Microscopic electronic inhomogeneity in the high-Tc superconductor

Bi2Sr2CaCu2O8+x Nature 413 282ndash285 (2001)5 Lang K M et al Imaging the granular structure of high-Tc superconductivity in underdoped

Bi2Sr2CaCu2O8+δ Nature 415 412ndash416 (2002)6 Kohsaka Y et al Imaging nano-scale electronic inhomogeneity in lightly doped Mott insulator

Ca2minusx Nax CuO2Cl2 Phys Rev Lett 93 097004 (2004)7 Hoffman J E et al A four unit cell periodic pattern of quasi-particle states surrounding vortex cores

in Bi2Sr2CaCu2O8+δ Science 295 466ndash469 (2002)8 Vershinin M et al Local ordering in the pseudogap state of the high-Tc superconductor

Bi2Sr2CaCu2O8+δ Science 303 1995ndash1998 (2004)9 Hanaguri T et al A lsquocheckerboardrsquo electronic crystal state in lightly hole-doped Ca2minusx Nax CuO2Cl2

Nature 430 1001ndash1005 (2004)10 McElroy K et al Coincidence of checkerboard charge order and antinodal state decoherence in

strongly underdoped superconducting Bi2Sr2CaCu2O8+δ Phys Rev Lett 94 197005 (2005)11 Damascelli A Hussain Z amp Shen Z-X Angle-resolved photoemission studies of the cuprate

superconductors Rev Mod Phys 75 473ndash541 (2003)12 Shen K M et al Nodal quasiparticles and antinodal charge ordering in Ca2minusx Nax CuO2Cl2 Science

307 901ndash904 (2005)13 Yoshida T et al Low-energy electronic structure of the high-Tc cuprates La2minusx Srx CuO4 studied by

angle-resolved photoemission spectroscopy J Phys Condens Matter 19 125209 (2007)14 Tanaka K et al Distinct Fermi-momentum-dependent energy gaps in deeply underdoped Bi2212

Science 314 1910ndash1913 (2006)

870 nature physics VOL 3 DECEMBER 2007 wwwnaturecomnaturephysics

copy 2007 Nature Publishing Group

ARTICLES

15 Norman M R et al Destruction of the Fermi surface in underdoped high-Tc superconductorsNature 392 157ndash160 (1998)

16 Tsuei C C amp Kirtley J R Pairing symmetry in cuprate superconductors Rev Mod Phys 72969ndash1016 (2000)

17 Le Tacon M et al Two energy scales and two distinct quasiparticle dynamics in the superconductingstate of underdoped cuprates Nature Phys 2 537ndash543 (2006)

18 Gomes K K et al Visualizing pair formation on the atomic scale in the high-Tc superconductorBi2Sr2CaCu2O8+δ Nature 447 569ndash572 (2007)

19 Boyer M C et al Imaging the two gaps of the high-Tc superconductor Pb-Bi2Sr2CuO6+x NaturePhys published online 16 September 2007 (doi101038nphys725)

20 Timusk T amp Statt B The pseudogap in high-temperature superconductors an experimental surveyRep Prog Phys 62 61ndash122 (1999)

21 Hoffman J E et al Imaging quasiparticle interference in Bi2Sr2CaCu2O8+δ Science 2971148ndash1151 (2002)

22 McElroy K et al Relating atomic-scale electronic phenomena to wave-like quasiparticle states insuperconducting Bi2Sr2CaCu2O8+δ Nature 422 592ndash596 (2003)

23 Wang Q amp Lee D-H Quasiparticle scattering interference in high temperature superconductorsPhys Rev B 67 020511 (2003)

24 Zhang D amp Ting C S Energy-dependent modulations in the local density of states of the cupratesuperconductors Phys Rev B 67 100506 (2003)

25 Pereg-Barnea T amp Franz M Theory of quasiparticle interference patterns in the pseudogap phase ofthe cuprate superconductors Phys Rev B 68 180506 (R) (2003)

26 Capriotti L Scalapino D J amp Sedgewick R D Wave-vector power spectrum of the local tunnelingdensity of states Ripples in a d-wave sea Phys Rev B 68 014508 (2003)

27 Zhu L Atkinson W A amp Hirschfeld P J Power spectrum of many impurities in a d-wavesuperconductor Phys Rev B 69 060503(R) (2004)

28 Pereg-Barnea T amp Franz M Quasiparticle interference patterns as a test for the nature of thepseudogap phase in the cuprate superconductors Int J Mod Phys B 19 731ndash761 (2005)

29 DellrsquoAnna L Lorenzana J Capone M Castellani C amp Grilli M Effect of mesoscopicinhomogeneities on local tunneling density of states in cuprates Phys Rev B 71 064518 (2005)

30 Cheng M amp Su W P Local density of states and angle-resolved photoemission spectral function ofan inhomogeneous d-wave superconductor Phys Rev B 72 094512 (2005)

31 Nunner T S et al Fourier transform spectroscopy of d-wave quasiparticles in the presence of atomicscale pairing disorder Phys Rev B 73 104511 (2006)

32 Hasegawa Y amp Avouris Ph Direct observation of standing wave formation at surface steps usingscanning tunneling spectroscopy Phys Rev Lett 71 1071ndash1074 (1993)

33 Crommie M F Lutz C P amp Eigler D M Imaging standing waves in a two-dimensional electrongas Nature 363 524ndash527 (1993)

34 Sprunger P T Petersen L Plummer E W Laeliggsgaard E amp Besenbacher F Giant Friedeloscillations on the beryllium(0001) surface Science 275 1764ndash1767 (1997)

35 Tinkham M Introduction to Superconductivity 2nd edn 43 (McGraw Hill New York 1996)36 Hiroi Z Kobayashi N amp Takano M Probable hole-doped superconductivity without apical oxygens

in (CaNa)2CuO2Cl2 Nature 371 139ndash141 (1994)37 Kohsaka Y et al Growth of Na-doped Ca2CuO2Cl2 single crystals under high pressures of several

GPa J Am Chem Soc 124 12275ndash12278 (2002)38 Balatsky A V Vekhter I amp Zhu J-X Impurity-induced states in conventional and unconventional

superconductors Rev Mod Phys 78 373ndash433 (2006)39 Kohsaka Y et al An intrinsic bond-centered electronic glass with unidirectional domains in

underdoped cuprates Science 314 1380ndash1385 (2007)40 Furukawa N Rice T M amp Salmhofer M Truncation of a two-dimensional FS due to quasiparticle

gap formation at the saddle points Phys Rev Lett 81 3195ndash3198 (1998)41 Wen X-G amp Lee P A Theory of quasiparticles in the underdoped high-Tc superconducting state

Phys Rev Lett 80 2193ndash2196 (1998)42 Li J-X Wu C-Q amp Lee D-H Checkerboard charge density wave and pseudogap of high-Tc

cuprates Phys Rev B 74 184515 (2006)43 Garg A Randeria M amp Trivedi N Strong correlations lead to protected low energy excitations in

disordered d-wave superconductors Preprint at lthttparXivorgcond-mat0609666gt (2006)44 Oda M Dipasupil R M Momono N amp Ido M Hyperbolic dependence of 210 vs Tc ratio on

hole-doping level in high-Tc cuprates Energy scale in determining Tc J Phys Soc Jpn 69983ndash984 (2000)

45 Hanaguri T Development of high-field STM and its application to the study on magnetically-tunedcriticality in Sr3Ru2O7 J Phys Conf Ser 51 514ndash521 (2006)

AcknowledgementsThe authors thank AV Balatsky C-M Ho D-H Lee K Machida A Mackenzie K McElroyT Tohyama J Zaanen and F-C Zhang for discussions They also thank J Matsuno and P Sharma forcritical readings TH MT and HT are supported by a Grant-in-Aid for Scientific Research from theMinistry of Education Culture Sports Science and Technology of Japan JCD and YK acknowledgesupport from Brookhaven National Laboratory under contract No DE-AC02-98CH1886 with the USDepartment of Energy from the US Department of Energy Award No DE-FG02-06ER46306 and fromthe US Office of Naval ResearchCorrespondence and requests for materials should be addressed to TH

Author contributionsTH was responsible for all aspects of this project except sample growth YK JCD and CLcontributed the project planning and data analysis IY MA MT and KO grew samples and MOcontributed the STM measurements HT contributed the project planning and managed thewhole project

Reprints and permission information is available online at httpnpgnaturecomreprintsandpermissions

nature physics VOL 3 DECEMBER 2007 wwwnaturecomnaturephysics 871

copy 2007 Nature Publishing Group

ARTICLES

a bk y

( a 0

)

(meV

)

k (ordm)

100

075

050

025

0

40

30

20

10

090807060504030201001000750500250

kx ( a0)π

π

Δ

θ

θk

Bi2212 (ref 22)

Na-CCOC

Figure 4 The obtained FS and the SG dispersion for Na-CCOC (Tc sim 28 K x sim 014) a The locus of octet ends of banana-shaped CCEs in a quadrant of the first Brillouinzone which represents the underlying FS The data are symmetrized about the diagonal owing to mirror symmetry The results for two different samples are plotted in redcircles and blue squares respectively In the analysis we used four independent q4 (E )= (plusmn2kx (E )2ky (E ) ) (2ky (E )plusmn2kx (E ) ) which were clearly observed between3meVlt E lt 15meV The other qi vectors gave us consistent results but the observed energy range was narrower The error bars represent the statistical scatters b TheSG plotted as a function of the FS angle θk defined in a The two different symbols have the same meaning as in a The green line represents the SG dispersion for nearlyoptimally doped Bi2212 with Tc sim 86 K (ref 22) Even though Tc values differ by a factor of three between these two compounds the SG dispersions are very similar

QPI patterns in Bi2212 have been observed in raw g(rE)or |g(qE)| (refs 102122) However in the case of Na-CCOC(Tc sim 28 Kx sim 014) g(rE) and |g(qE)| are primarily governedby the apparent CB g modulations as shown in Fig 3ab and a clearsignature of QPI is difficult to recognize Thus we need to find anew technique to extract QPI patterns which may be hidden behindthe apparent CB g modulations

SEPARATION OF QPI FROM THE lsquoCHECKERBOARDrsquo

To separate QPI from the apparent CB g modulations we considerthe spatial-phase relations in the g modulations A comparisonbetween the positive and negative bias g(rplusmnE) shows that CBmodulations in g(r+E) and g(rminusE) are spatially quite similarand the local conductance maximaminima occur predominantlyat the same location as shown in Fig 3ab In other words theapparent CB g modulations have the same wavelength and arein phase across the Fermi energy The QPI patterns should alsohave the same wavelength in g(r+E) and g(rminusE) becauseE(k) is expected to be particlendashhole symmetric as shown inFig 1a The spatial-phase relation for QPI is not known preciselybut it is plausible that there is a phase difference betweeng(r+E) and g(rminusE) as there is a sum-rule

sumn v2

n(r)+u2n(r) = 1

between particle-like and holelike amplitudes of the Bogoliubovquasiparticle v2

n(r) and u2n(r) where n is an appropriate quantum

number38 Therefore the QPI pattern may be extracted by meansof the spatial-phase-sensitive method if they have a non-trivialphase difference

For this purpose we propose to examine the ratio mapdefined by Z(rE) equiv g(r+E)g(r minusE) and its Fouriertransform |Z(qE)| Let us introduce a basic property of|Z(qE)| If the conductance modulation amplitude at finite qgplusmn

q equiv g(qplusmnE) is much smaller than the uniform componentgplusmn

0 equiv g(q = 0plusmnE) |Z(qE)| can be approximated as (to the firstorder) |Z(qE)| asymp (g+

0 gminus

0 )|(g+

q g+

0 ) minus (gminus

q gminus

0 )| In the present

case the relation |gplusmn

q |gplusmn

0 1 holds for all the energies studiedTherefore the above approximation is well justified Note that|Z(qE)| is sensitive to the relative signs between g(q+E) andg(qminusE) (Note that gplusmn

0 gt 0) Namely taking the ratio reduces thein-phase component

In Fig 3c and d respectively we show the measuredZ(rE = 6 meV) and the Fourier transform |Z(qE = 6 meV)|It is immediately evident that the CB g modulation whichdominates g(rE) and |g(qE)| almost completely disappearsMoreover most satisfyingly another type of modulation emergesThe features of the new modulations are clearly manifested in|Z(qE)| where numerous spots are observed at incommensurateq vectors All of these q vectors are consistently explained in termsof the octet model22 of QPI as shown by symbols in Fig 3dThus the existence of the QPI effect and therefore the coherentBogoliubov quasiparticle of the d-wave superconducting state areclearly confirmed for the first time in Na-CCOC The origin ofthe enhanced QPI pattern and detailed intensity distribution in|Z(qE)| are interesting subjects for future theoretical research

We would like to note that the ratio Z has another importantfeature besides its phase sensitivity39 In general tunnellingconductance g(rE = eV ) is given by

g(reV ) =eIt N (reV )int eVs

0N (rE)dE

Here e is a unit charge It and Vs are the set-up current and biasvoltage for feedback respectively and N is a local DOS Note thatg is not proportional to N but is normalized by the integrated N which is strongly influenced by the set-up conditions through VsOn the other hand Z(r eV ) reflects the ratio of the N preciselybecause the denominator and prefactor eIt are exactly cancelledout by taking the ratio39 This fact means that observed QPI is anintrinsic phenomenon free from any set-point-related issues whichinevitably contaminate g(rE) and |g(qE)| In contrast there is a

nature physics VOL 3 DECEMBER 2007 wwwnaturecomnaturephysics 869

copy 2007 Nature Publishing Group

ARTICLES

possibility that CB g modulation is affected by the set-point effectbecause it disappears in Z at least below the kink energy where QPIis observed Therefore it is important to search for the possibleelectronic superstructure using the set-point-effect-free method

Recently ratio analyses at high energy (sim150 meV) inunderdoped Na-CCOC and Bi2212 have revealed the existenceof an electronic-cluster-glass (ECG) state characterized by thefour-CundashCu-distance-wide bond-centred unidirectional electronicdomains dispersed throughout without long-range order39Similarities between ECG and CB g modulation (for examplecharacteristic length scale of four CundashCu distances) imply closerelations between the two It is interesting to note that apparentlydifferent phenomena high-energy ECG and low-energy QPIhave been detected in similar analysis schemes depending onthe energy scale Given the fact that the ratio analysis extractsintrinsic information on the electronic states establishing therelation between ECG and QPI which may represent localizedand delocalized electronic states respectively represents a key goalfor this field

INSIGHTS FROM THE QPI ANALYSIS

To analyse the observed Na-CCOC QPI quantitatively we measuredqi at different E and determined the energy-dependent locationsof the octet ends of lsquobananasrsquo in k space The obtained locusk(E) = (kx(E)ky(E)) can be interpreted as a normal-state FS Weplot the obtained loci for two different samples in Fig 4a The resultis consistent with a cylindrical FS centred around (ππ) which isinferred from ARPES on underdoped samples12

We can determine the SG dispersion ∆(k) from k(E) as well22In Fig 4b we plot the measured gap energy as a function of FSangle θk about (ππ) For comparison SG dispersion of optimallydoped Bi2212 (ref 22) is shown by a green line Apparently SGdispersions of these two compounds are quite similar Because Tc

values are very different between Na-CCOC (Tc sim28 K) and Bi2212(Tc sim 86 K) we can conclude that the SG dispersion in the nodalregion does not set the energy scale which determines Tc alone

It is worthwhile to note that QPI in optimally dopedNa-CCOC is observed in a rather limited part of the FS near thenode (25 lt θk lt 65) This suggests that available quasiparticlestates are missing from the antinodal region40ndash42 or that theantinodal quasiparticles may not have a correlation length longenough to produce interference due to disorder43 In accord withthis well-defined coherence peaks that manifest the coherentquasiparticle states in the antinodal region1022 are not observed inthe tunnelling spectra as shown in Fig 2c and d In the ARPESdata on the x sim 012 (Tc sim 22 K) sample with slightly lowerhole concentration than the present samples the spectral weightin the antinodal region is still substantially suppressed whichwas pointed out to be linked with CB g modulations12 Here wepostulate that such incoherent antinodal states are not responsiblefor forming phase-coherent Cooper pairs If this picture is correctsuperconductivity is supported only by the coherent part of theFS around the node namely a d-wave SG opens on the so-calledFermi arc This conjecture predicts that the energy scale which setsTc is determined not only by the SG dispersion around the node butalso by the effective lsquoarc lengthrsquo Analogous proposals have also beenmade on the basis of the doping dependence of the gap amplitude44

Finally we compare the tunnelling spectrum of Na-CCOC withthat of Bi2212 In optimally doped Bi2212 well-defined coherencepeaks are ubiquitous10 and thus the entire FS may be responsiblefor superconductivity On the other hand in underdoped Bi2212a different type of spectrum characterized by the V-shaped gapwithout a clear coherence peak appears while the QPI signal is stillthere10 This spectrum is very similar to the spectrum of Na-CCOC

(refs 69) Interestingly in underdoped Bi2212 a non-dispersive CBg modulation is also seen in the region where this type of spectrumis observed10 Therefore it is reasonable to infer that the V-shapedgap without a clear coherence peak the destruction of the antinodalstate and the CB g modulation go hand in hand and are universal inhigh-Tc cuprates Another important common feature observed inboth Na-CCOC and Bi2212 is that this state is superconductingwith long-lived Bogoliubov quasiparticles which are manifestedby QPI

CONCLUSION

We report the first observation of QPI associated with d-wavesuperconductivity in Na-CCOC This provides us with firmevidence for the coexistence of d-wave superconductivity andCB g modulation presumably associated with the ECG statewhich probably represents the nodalantinodal dichotomy in kspace The striking similarity of SG dispersions between Na-CCOC(Tc sim 28 K) and Bi2212 (Tc sim 86 K) in the nodal regions stronglysuggests that the nodalantinodal dichotomy is controlling Tc frommaterial to material Further exploration of k-space physics webelieve should help identify a key ingredient to enhance Tc ofcuprate superconductors A novel analysis g-ratio or Z analysiswas developed to separate QPI signals from the CB g modulationIt is quite sensitive to the spatial-phase difference in quantumadmixtures of particle-like and holelike states and will be a powerfulprobe to explore the quasiparticle interferences in superconductorsand possibly other quantum condensates

METHODS

Na-CCOC single crystals were grown by the flux method under high pressureof the order of several gigapascals37 By optimizing the synthesis conditions wesucceeded in growing suitably sized nearly optimally doped single crystals withTc sim 25 K and 28 K which correspond to x sim 013 and 014 respectively

SI-STM measurements were carried out with a newly developedultrahigh-vacuum very-low-temperature scanning tunnelling microscope atRIKEN45 Samples were cleaved in situ at 77 K under ultrahigh vacuum andwere immediately transferred to the STM unit which was kept at lowtemperature (lt10 K) Tungsten scanning tips were sharpened andcharacterized in situ with atomic resolution by a field-ion microscope which isbuilt in to the STM chamber All the data were taken at a temperatureT sim 04 K

Received 14 February 2007 accepted 19 September 2007 published 28 October 2007

References1 Fischer Oslash Kugler M Maggio-Aprile I Berthod C amp Renner C Scanning tunneling spectroscopy

of high-temperature superconductors Rev Mod Phys 79 353ndash419 (2007)2 Cren T Roditchev D Sacks W amp Klein J Nanometer scale mapping of the density of states in an

inhomogeneous superconductor Eur Phys Lett 54 84ndash90 (2001)3 Howald C Fournier P amp Kapitulnik A Inherent inhomogeneities in tunneling spectra of

Bi2Sr2CaCu2O8minusx crystals in the superconducting state Phys Rev B 64 100504 (2001)4 Pan S H et al Microscopic electronic inhomogeneity in the high-Tc superconductor

Bi2Sr2CaCu2O8+x Nature 413 282ndash285 (2001)5 Lang K M et al Imaging the granular structure of high-Tc superconductivity in underdoped

Bi2Sr2CaCu2O8+δ Nature 415 412ndash416 (2002)6 Kohsaka Y et al Imaging nano-scale electronic inhomogeneity in lightly doped Mott insulator

Ca2minusx Nax CuO2Cl2 Phys Rev Lett 93 097004 (2004)7 Hoffman J E et al A four unit cell periodic pattern of quasi-particle states surrounding vortex cores

in Bi2Sr2CaCu2O8+δ Science 295 466ndash469 (2002)8 Vershinin M et al Local ordering in the pseudogap state of the high-Tc superconductor

Bi2Sr2CaCu2O8+δ Science 303 1995ndash1998 (2004)9 Hanaguri T et al A lsquocheckerboardrsquo electronic crystal state in lightly hole-doped Ca2minusx Nax CuO2Cl2

Nature 430 1001ndash1005 (2004)10 McElroy K et al Coincidence of checkerboard charge order and antinodal state decoherence in

strongly underdoped superconducting Bi2Sr2CaCu2O8+δ Phys Rev Lett 94 197005 (2005)11 Damascelli A Hussain Z amp Shen Z-X Angle-resolved photoemission studies of the cuprate

superconductors Rev Mod Phys 75 473ndash541 (2003)12 Shen K M et al Nodal quasiparticles and antinodal charge ordering in Ca2minusx Nax CuO2Cl2 Science

307 901ndash904 (2005)13 Yoshida T et al Low-energy electronic structure of the high-Tc cuprates La2minusx Srx CuO4 studied by

angle-resolved photoemission spectroscopy J Phys Condens Matter 19 125209 (2007)14 Tanaka K et al Distinct Fermi-momentum-dependent energy gaps in deeply underdoped Bi2212

Science 314 1910ndash1913 (2006)

870 nature physics VOL 3 DECEMBER 2007 wwwnaturecomnaturephysics

copy 2007 Nature Publishing Group

ARTICLES

15 Norman M R et al Destruction of the Fermi surface in underdoped high-Tc superconductorsNature 392 157ndash160 (1998)

16 Tsuei C C amp Kirtley J R Pairing symmetry in cuprate superconductors Rev Mod Phys 72969ndash1016 (2000)

17 Le Tacon M et al Two energy scales and two distinct quasiparticle dynamics in the superconductingstate of underdoped cuprates Nature Phys 2 537ndash543 (2006)

18 Gomes K K et al Visualizing pair formation on the atomic scale in the high-Tc superconductorBi2Sr2CaCu2O8+δ Nature 447 569ndash572 (2007)

19 Boyer M C et al Imaging the two gaps of the high-Tc superconductor Pb-Bi2Sr2CuO6+x NaturePhys published online 16 September 2007 (doi101038nphys725)

20 Timusk T amp Statt B The pseudogap in high-temperature superconductors an experimental surveyRep Prog Phys 62 61ndash122 (1999)

21 Hoffman J E et al Imaging quasiparticle interference in Bi2Sr2CaCu2O8+δ Science 2971148ndash1151 (2002)

22 McElroy K et al Relating atomic-scale electronic phenomena to wave-like quasiparticle states insuperconducting Bi2Sr2CaCu2O8+δ Nature 422 592ndash596 (2003)

23 Wang Q amp Lee D-H Quasiparticle scattering interference in high temperature superconductorsPhys Rev B 67 020511 (2003)

24 Zhang D amp Ting C S Energy-dependent modulations in the local density of states of the cupratesuperconductors Phys Rev B 67 100506 (2003)

25 Pereg-Barnea T amp Franz M Theory of quasiparticle interference patterns in the pseudogap phase ofthe cuprate superconductors Phys Rev B 68 180506 (R) (2003)

26 Capriotti L Scalapino D J amp Sedgewick R D Wave-vector power spectrum of the local tunnelingdensity of states Ripples in a d-wave sea Phys Rev B 68 014508 (2003)

27 Zhu L Atkinson W A amp Hirschfeld P J Power spectrum of many impurities in a d-wavesuperconductor Phys Rev B 69 060503(R) (2004)

28 Pereg-Barnea T amp Franz M Quasiparticle interference patterns as a test for the nature of thepseudogap phase in the cuprate superconductors Int J Mod Phys B 19 731ndash761 (2005)

29 DellrsquoAnna L Lorenzana J Capone M Castellani C amp Grilli M Effect of mesoscopicinhomogeneities on local tunneling density of states in cuprates Phys Rev B 71 064518 (2005)

30 Cheng M amp Su W P Local density of states and angle-resolved photoemission spectral function ofan inhomogeneous d-wave superconductor Phys Rev B 72 094512 (2005)

31 Nunner T S et al Fourier transform spectroscopy of d-wave quasiparticles in the presence of atomicscale pairing disorder Phys Rev B 73 104511 (2006)

32 Hasegawa Y amp Avouris Ph Direct observation of standing wave formation at surface steps usingscanning tunneling spectroscopy Phys Rev Lett 71 1071ndash1074 (1993)

33 Crommie M F Lutz C P amp Eigler D M Imaging standing waves in a two-dimensional electrongas Nature 363 524ndash527 (1993)

34 Sprunger P T Petersen L Plummer E W Laeliggsgaard E amp Besenbacher F Giant Friedeloscillations on the beryllium(0001) surface Science 275 1764ndash1767 (1997)

35 Tinkham M Introduction to Superconductivity 2nd edn 43 (McGraw Hill New York 1996)36 Hiroi Z Kobayashi N amp Takano M Probable hole-doped superconductivity without apical oxygens

in (CaNa)2CuO2Cl2 Nature 371 139ndash141 (1994)37 Kohsaka Y et al Growth of Na-doped Ca2CuO2Cl2 single crystals under high pressures of several

GPa J Am Chem Soc 124 12275ndash12278 (2002)38 Balatsky A V Vekhter I amp Zhu J-X Impurity-induced states in conventional and unconventional

superconductors Rev Mod Phys 78 373ndash433 (2006)39 Kohsaka Y et al An intrinsic bond-centered electronic glass with unidirectional domains in

underdoped cuprates Science 314 1380ndash1385 (2007)40 Furukawa N Rice T M amp Salmhofer M Truncation of a two-dimensional FS due to quasiparticle

gap formation at the saddle points Phys Rev Lett 81 3195ndash3198 (1998)41 Wen X-G amp Lee P A Theory of quasiparticles in the underdoped high-Tc superconducting state

Phys Rev Lett 80 2193ndash2196 (1998)42 Li J-X Wu C-Q amp Lee D-H Checkerboard charge density wave and pseudogap of high-Tc

cuprates Phys Rev B 74 184515 (2006)43 Garg A Randeria M amp Trivedi N Strong correlations lead to protected low energy excitations in

disordered d-wave superconductors Preprint at lthttparXivorgcond-mat0609666gt (2006)44 Oda M Dipasupil R M Momono N amp Ido M Hyperbolic dependence of 210 vs Tc ratio on

hole-doping level in high-Tc cuprates Energy scale in determining Tc J Phys Soc Jpn 69983ndash984 (2000)

45 Hanaguri T Development of high-field STM and its application to the study on magnetically-tunedcriticality in Sr3Ru2O7 J Phys Conf Ser 51 514ndash521 (2006)

AcknowledgementsThe authors thank AV Balatsky C-M Ho D-H Lee K Machida A Mackenzie K McElroyT Tohyama J Zaanen and F-C Zhang for discussions They also thank J Matsuno and P Sharma forcritical readings TH MT and HT are supported by a Grant-in-Aid for Scientific Research from theMinistry of Education Culture Sports Science and Technology of Japan JCD and YK acknowledgesupport from Brookhaven National Laboratory under contract No DE-AC02-98CH1886 with the USDepartment of Energy from the US Department of Energy Award No DE-FG02-06ER46306 and fromthe US Office of Naval ResearchCorrespondence and requests for materials should be addressed to TH

Author contributionsTH was responsible for all aspects of this project except sample growth YK JCD and CLcontributed the project planning and data analysis IY MA MT and KO grew samples and MOcontributed the STM measurements HT contributed the project planning and managed thewhole project

Reprints and permission information is available online at httpnpgnaturecomreprintsandpermissions

nature physics VOL 3 DECEMBER 2007 wwwnaturecomnaturephysics 871

copy 2007 Nature Publishing Group

ARTICLES

possibility that CB g modulation is affected by the set-point effectbecause it disappears in Z at least below the kink energy where QPIis observed Therefore it is important to search for the possibleelectronic superstructure using the set-point-effect-free method

Recently ratio analyses at high energy (sim150 meV) inunderdoped Na-CCOC and Bi2212 have revealed the existenceof an electronic-cluster-glass (ECG) state characterized by thefour-CundashCu-distance-wide bond-centred unidirectional electronicdomains dispersed throughout without long-range order39Similarities between ECG and CB g modulation (for examplecharacteristic length scale of four CundashCu distances) imply closerelations between the two It is interesting to note that apparentlydifferent phenomena high-energy ECG and low-energy QPIhave been detected in similar analysis schemes depending onthe energy scale Given the fact that the ratio analysis extractsintrinsic information on the electronic states establishing therelation between ECG and QPI which may represent localizedand delocalized electronic states respectively represents a key goalfor this field

INSIGHTS FROM THE QPI ANALYSIS

To analyse the observed Na-CCOC QPI quantitatively we measuredqi at different E and determined the energy-dependent locationsof the octet ends of lsquobananasrsquo in k space The obtained locusk(E) = (kx(E)ky(E)) can be interpreted as a normal-state FS Weplot the obtained loci for two different samples in Fig 4a The resultis consistent with a cylindrical FS centred around (ππ) which isinferred from ARPES on underdoped samples12

We can determine the SG dispersion ∆(k) from k(E) as well22In Fig 4b we plot the measured gap energy as a function of FSangle θk about (ππ) For comparison SG dispersion of optimallydoped Bi2212 (ref 22) is shown by a green line Apparently SGdispersions of these two compounds are quite similar Because Tc

values are very different between Na-CCOC (Tc sim28 K) and Bi2212(Tc sim 86 K) we can conclude that the SG dispersion in the nodalregion does not set the energy scale which determines Tc alone

It is worthwhile to note that QPI in optimally dopedNa-CCOC is observed in a rather limited part of the FS near thenode (25 lt θk lt 65) This suggests that available quasiparticlestates are missing from the antinodal region40ndash42 or that theantinodal quasiparticles may not have a correlation length longenough to produce interference due to disorder43 In accord withthis well-defined coherence peaks that manifest the coherentquasiparticle states in the antinodal region1022 are not observed inthe tunnelling spectra as shown in Fig 2c and d In the ARPESdata on the x sim 012 (Tc sim 22 K) sample with slightly lowerhole concentration than the present samples the spectral weightin the antinodal region is still substantially suppressed whichwas pointed out to be linked with CB g modulations12 Here wepostulate that such incoherent antinodal states are not responsiblefor forming phase-coherent Cooper pairs If this picture is correctsuperconductivity is supported only by the coherent part of theFS around the node namely a d-wave SG opens on the so-calledFermi arc This conjecture predicts that the energy scale which setsTc is determined not only by the SG dispersion around the node butalso by the effective lsquoarc lengthrsquo Analogous proposals have also beenmade on the basis of the doping dependence of the gap amplitude44

Finally we compare the tunnelling spectrum of Na-CCOC withthat of Bi2212 In optimally doped Bi2212 well-defined coherencepeaks are ubiquitous10 and thus the entire FS may be responsiblefor superconductivity On the other hand in underdoped Bi2212a different type of spectrum characterized by the V-shaped gapwithout a clear coherence peak appears while the QPI signal is stillthere10 This spectrum is very similar to the spectrum of Na-CCOC

(refs 69) Interestingly in underdoped Bi2212 a non-dispersive CBg modulation is also seen in the region where this type of spectrumis observed10 Therefore it is reasonable to infer that the V-shapedgap without a clear coherence peak the destruction of the antinodalstate and the CB g modulation go hand in hand and are universal inhigh-Tc cuprates Another important common feature observed inboth Na-CCOC and Bi2212 is that this state is superconductingwith long-lived Bogoliubov quasiparticles which are manifestedby QPI

CONCLUSION

We report the first observation of QPI associated with d-wavesuperconductivity in Na-CCOC This provides us with firmevidence for the coexistence of d-wave superconductivity andCB g modulation presumably associated with the ECG statewhich probably represents the nodalantinodal dichotomy in kspace The striking similarity of SG dispersions between Na-CCOC(Tc sim 28 K) and Bi2212 (Tc sim 86 K) in the nodal regions stronglysuggests that the nodalantinodal dichotomy is controlling Tc frommaterial to material Further exploration of k-space physics webelieve should help identify a key ingredient to enhance Tc ofcuprate superconductors A novel analysis g-ratio or Z analysiswas developed to separate QPI signals from the CB g modulationIt is quite sensitive to the spatial-phase difference in quantumadmixtures of particle-like and holelike states and will be a powerfulprobe to explore the quasiparticle interferences in superconductorsand possibly other quantum condensates

METHODS

Na-CCOC single crystals were grown by the flux method under high pressureof the order of several gigapascals37 By optimizing the synthesis conditions wesucceeded in growing suitably sized nearly optimally doped single crystals withTc sim 25 K and 28 K which correspond to x sim 013 and 014 respectively

SI-STM measurements were carried out with a newly developedultrahigh-vacuum very-low-temperature scanning tunnelling microscope atRIKEN45 Samples were cleaved in situ at 77 K under ultrahigh vacuum andwere immediately transferred to the STM unit which was kept at lowtemperature (lt10 K) Tungsten scanning tips were sharpened andcharacterized in situ with atomic resolution by a field-ion microscope which isbuilt in to the STM chamber All the data were taken at a temperatureT sim 04 K

Received 14 February 2007 accepted 19 September 2007 published 28 October 2007

References1 Fischer Oslash Kugler M Maggio-Aprile I Berthod C amp Renner C Scanning tunneling spectroscopy

of high-temperature superconductors Rev Mod Phys 79 353ndash419 (2007)2 Cren T Roditchev D Sacks W amp Klein J Nanometer scale mapping of the density of states in an

inhomogeneous superconductor Eur Phys Lett 54 84ndash90 (2001)3 Howald C Fournier P amp Kapitulnik A Inherent inhomogeneities in tunneling spectra of

Bi2Sr2CaCu2O8minusx crystals in the superconducting state Phys Rev B 64 100504 (2001)4 Pan S H et al Microscopic electronic inhomogeneity in the high-Tc superconductor

Bi2Sr2CaCu2O8+x Nature 413 282ndash285 (2001)5 Lang K M et al Imaging the granular structure of high-Tc superconductivity in underdoped

Bi2Sr2CaCu2O8+δ Nature 415 412ndash416 (2002)6 Kohsaka Y et al Imaging nano-scale electronic inhomogeneity in lightly doped Mott insulator

Ca2minusx Nax CuO2Cl2 Phys Rev Lett 93 097004 (2004)7 Hoffman J E et al A four unit cell periodic pattern of quasi-particle states surrounding vortex cores

in Bi2Sr2CaCu2O8+δ Science 295 466ndash469 (2002)8 Vershinin M et al Local ordering in the pseudogap state of the high-Tc superconductor

Bi2Sr2CaCu2O8+δ Science 303 1995ndash1998 (2004)9 Hanaguri T et al A lsquocheckerboardrsquo electronic crystal state in lightly hole-doped Ca2minusx Nax CuO2Cl2

Nature 430 1001ndash1005 (2004)10 McElroy K et al Coincidence of checkerboard charge order and antinodal state decoherence in

strongly underdoped superconducting Bi2Sr2CaCu2O8+δ Phys Rev Lett 94 197005 (2005)11 Damascelli A Hussain Z amp Shen Z-X Angle-resolved photoemission studies of the cuprate

superconductors Rev Mod Phys 75 473ndash541 (2003)12 Shen K M et al Nodal quasiparticles and antinodal charge ordering in Ca2minusx Nax CuO2Cl2 Science

307 901ndash904 (2005)13 Yoshida T et al Low-energy electronic structure of the high-Tc cuprates La2minusx Srx CuO4 studied by

angle-resolved photoemission spectroscopy J Phys Condens Matter 19 125209 (2007)14 Tanaka K et al Distinct Fermi-momentum-dependent energy gaps in deeply underdoped Bi2212

Science 314 1910ndash1913 (2006)

870 nature physics VOL 3 DECEMBER 2007 wwwnaturecomnaturephysics

copy 2007 Nature Publishing Group

ARTICLES

15 Norman M R et al Destruction of the Fermi surface in underdoped high-Tc superconductorsNature 392 157ndash160 (1998)

16 Tsuei C C amp Kirtley J R Pairing symmetry in cuprate superconductors Rev Mod Phys 72969ndash1016 (2000)

17 Le Tacon M et al Two energy scales and two distinct quasiparticle dynamics in the superconductingstate of underdoped cuprates Nature Phys 2 537ndash543 (2006)

18 Gomes K K et al Visualizing pair formation on the atomic scale in the high-Tc superconductorBi2Sr2CaCu2O8+δ Nature 447 569ndash572 (2007)

19 Boyer M C et al Imaging the two gaps of the high-Tc superconductor Pb-Bi2Sr2CuO6+x NaturePhys published online 16 September 2007 (doi101038nphys725)

20 Timusk T amp Statt B The pseudogap in high-temperature superconductors an experimental surveyRep Prog Phys 62 61ndash122 (1999)

21 Hoffman J E et al Imaging quasiparticle interference in Bi2Sr2CaCu2O8+δ Science 2971148ndash1151 (2002)

22 McElroy K et al Relating atomic-scale electronic phenomena to wave-like quasiparticle states insuperconducting Bi2Sr2CaCu2O8+δ Nature 422 592ndash596 (2003)

23 Wang Q amp Lee D-H Quasiparticle scattering interference in high temperature superconductorsPhys Rev B 67 020511 (2003)

24 Zhang D amp Ting C S Energy-dependent modulations in the local density of states of the cupratesuperconductors Phys Rev B 67 100506 (2003)

25 Pereg-Barnea T amp Franz M Theory of quasiparticle interference patterns in the pseudogap phase ofthe cuprate superconductors Phys Rev B 68 180506 (R) (2003)

26 Capriotti L Scalapino D J amp Sedgewick R D Wave-vector power spectrum of the local tunnelingdensity of states Ripples in a d-wave sea Phys Rev B 68 014508 (2003)

27 Zhu L Atkinson W A amp Hirschfeld P J Power spectrum of many impurities in a d-wavesuperconductor Phys Rev B 69 060503(R) (2004)

28 Pereg-Barnea T amp Franz M Quasiparticle interference patterns as a test for the nature of thepseudogap phase in the cuprate superconductors Int J Mod Phys B 19 731ndash761 (2005)

29 DellrsquoAnna L Lorenzana J Capone M Castellani C amp Grilli M Effect of mesoscopicinhomogeneities on local tunneling density of states in cuprates Phys Rev B 71 064518 (2005)

30 Cheng M amp Su W P Local density of states and angle-resolved photoemission spectral function ofan inhomogeneous d-wave superconductor Phys Rev B 72 094512 (2005)

31 Nunner T S et al Fourier transform spectroscopy of d-wave quasiparticles in the presence of atomicscale pairing disorder Phys Rev B 73 104511 (2006)

32 Hasegawa Y amp Avouris Ph Direct observation of standing wave formation at surface steps usingscanning tunneling spectroscopy Phys Rev Lett 71 1071ndash1074 (1993)

33 Crommie M F Lutz C P amp Eigler D M Imaging standing waves in a two-dimensional electrongas Nature 363 524ndash527 (1993)

34 Sprunger P T Petersen L Plummer E W Laeliggsgaard E amp Besenbacher F Giant Friedeloscillations on the beryllium(0001) surface Science 275 1764ndash1767 (1997)

35 Tinkham M Introduction to Superconductivity 2nd edn 43 (McGraw Hill New York 1996)36 Hiroi Z Kobayashi N amp Takano M Probable hole-doped superconductivity without apical oxygens

in (CaNa)2CuO2Cl2 Nature 371 139ndash141 (1994)37 Kohsaka Y et al Growth of Na-doped Ca2CuO2Cl2 single crystals under high pressures of several

GPa J Am Chem Soc 124 12275ndash12278 (2002)38 Balatsky A V Vekhter I amp Zhu J-X Impurity-induced states in conventional and unconventional

superconductors Rev Mod Phys 78 373ndash433 (2006)39 Kohsaka Y et al An intrinsic bond-centered electronic glass with unidirectional domains in

underdoped cuprates Science 314 1380ndash1385 (2007)40 Furukawa N Rice T M amp Salmhofer M Truncation of a two-dimensional FS due to quasiparticle

gap formation at the saddle points Phys Rev Lett 81 3195ndash3198 (1998)41 Wen X-G amp Lee P A Theory of quasiparticles in the underdoped high-Tc superconducting state

Phys Rev Lett 80 2193ndash2196 (1998)42 Li J-X Wu C-Q amp Lee D-H Checkerboard charge density wave and pseudogap of high-Tc

cuprates Phys Rev B 74 184515 (2006)43 Garg A Randeria M amp Trivedi N Strong correlations lead to protected low energy excitations in

disordered d-wave superconductors Preprint at lthttparXivorgcond-mat0609666gt (2006)44 Oda M Dipasupil R M Momono N amp Ido M Hyperbolic dependence of 210 vs Tc ratio on

hole-doping level in high-Tc cuprates Energy scale in determining Tc J Phys Soc Jpn 69983ndash984 (2000)

45 Hanaguri T Development of high-field STM and its application to the study on magnetically-tunedcriticality in Sr3Ru2O7 J Phys Conf Ser 51 514ndash521 (2006)

AcknowledgementsThe authors thank AV Balatsky C-M Ho D-H Lee K Machida A Mackenzie K McElroyT Tohyama J Zaanen and F-C Zhang for discussions They also thank J Matsuno and P Sharma forcritical readings TH MT and HT are supported by a Grant-in-Aid for Scientific Research from theMinistry of Education Culture Sports Science and Technology of Japan JCD and YK acknowledgesupport from Brookhaven National Laboratory under contract No DE-AC02-98CH1886 with the USDepartment of Energy from the US Department of Energy Award No DE-FG02-06ER46306 and fromthe US Office of Naval ResearchCorrespondence and requests for materials should be addressed to TH

Author contributionsTH was responsible for all aspects of this project except sample growth YK JCD and CLcontributed the project planning and data analysis IY MA MT and KO grew samples and MOcontributed the STM measurements HT contributed the project planning and managed thewhole project

Reprints and permission information is available online at httpnpgnaturecomreprintsandpermissions

nature physics VOL 3 DECEMBER 2007 wwwnaturecomnaturephysics 871

copy 2007 Nature Publishing Group

ARTICLES

15 Norman M R et al Destruction of the Fermi surface in underdoped high-Tc superconductorsNature 392 157ndash160 (1998)

16 Tsuei C C amp Kirtley J R Pairing symmetry in cuprate superconductors Rev Mod Phys 72969ndash1016 (2000)

17 Le Tacon M et al Two energy scales and two distinct quasiparticle dynamics in the superconductingstate of underdoped cuprates Nature Phys 2 537ndash543 (2006)

18 Gomes K K et al Visualizing pair formation on the atomic scale in the high-Tc superconductorBi2Sr2CaCu2O8+δ Nature 447 569ndash572 (2007)

19 Boyer M C et al Imaging the two gaps of the high-Tc superconductor Pb-Bi2Sr2CuO6+x NaturePhys published online 16 September 2007 (doi101038nphys725)

20 Timusk T amp Statt B The pseudogap in high-temperature superconductors an experimental surveyRep Prog Phys 62 61ndash122 (1999)

21 Hoffman J E et al Imaging quasiparticle interference in Bi2Sr2CaCu2O8+δ Science 2971148ndash1151 (2002)

22 McElroy K et al Relating atomic-scale electronic phenomena to wave-like quasiparticle states insuperconducting Bi2Sr2CaCu2O8+δ Nature 422 592ndash596 (2003)

23 Wang Q amp Lee D-H Quasiparticle scattering interference in high temperature superconductorsPhys Rev B 67 020511 (2003)

24 Zhang D amp Ting C S Energy-dependent modulations in the local density of states of the cupratesuperconductors Phys Rev B 67 100506 (2003)

25 Pereg-Barnea T amp Franz M Theory of quasiparticle interference patterns in the pseudogap phase ofthe cuprate superconductors Phys Rev B 68 180506 (R) (2003)

26 Capriotti L Scalapino D J amp Sedgewick R D Wave-vector power spectrum of the local tunnelingdensity of states Ripples in a d-wave sea Phys Rev B 68 014508 (2003)

27 Zhu L Atkinson W A amp Hirschfeld P J Power spectrum of many impurities in a d-wavesuperconductor Phys Rev B 69 060503(R) (2004)

28 Pereg-Barnea T amp Franz M Quasiparticle interference patterns as a test for the nature of thepseudogap phase in the cuprate superconductors Int J Mod Phys B 19 731ndash761 (2005)

29 DellrsquoAnna L Lorenzana J Capone M Castellani C amp Grilli M Effect of mesoscopicinhomogeneities on local tunneling density of states in cuprates Phys Rev B 71 064518 (2005)

30 Cheng M amp Su W P Local density of states and angle-resolved photoemission spectral function ofan inhomogeneous d-wave superconductor Phys Rev B 72 094512 (2005)

31 Nunner T S et al Fourier transform spectroscopy of d-wave quasiparticles in the presence of atomicscale pairing disorder Phys Rev B 73 104511 (2006)

32 Hasegawa Y amp Avouris Ph Direct observation of standing wave formation at surface steps usingscanning tunneling spectroscopy Phys Rev Lett 71 1071ndash1074 (1993)

33 Crommie M F Lutz C P amp Eigler D M Imaging standing waves in a two-dimensional electrongas Nature 363 524ndash527 (1993)

34 Sprunger P T Petersen L Plummer E W Laeliggsgaard E amp Besenbacher F Giant Friedeloscillations on the beryllium(0001) surface Science 275 1764ndash1767 (1997)

35 Tinkham M Introduction to Superconductivity 2nd edn 43 (McGraw Hill New York 1996)36 Hiroi Z Kobayashi N amp Takano M Probable hole-doped superconductivity without apical oxygens

in (CaNa)2CuO2Cl2 Nature 371 139ndash141 (1994)37 Kohsaka Y et al Growth of Na-doped Ca2CuO2Cl2 single crystals under high pressures of several

GPa J Am Chem Soc 124 12275ndash12278 (2002)38 Balatsky A V Vekhter I amp Zhu J-X Impurity-induced states in conventional and unconventional

superconductors Rev Mod Phys 78 373ndash433 (2006)39 Kohsaka Y et al An intrinsic bond-centered electronic glass with unidirectional domains in

underdoped cuprates Science 314 1380ndash1385 (2007)40 Furukawa N Rice T M amp Salmhofer M Truncation of a two-dimensional FS due to quasiparticle

gap formation at the saddle points Phys Rev Lett 81 3195ndash3198 (1998)41 Wen X-G amp Lee P A Theory of quasiparticles in the underdoped high-Tc superconducting state

Phys Rev Lett 80 2193ndash2196 (1998)42 Li J-X Wu C-Q amp Lee D-H Checkerboard charge density wave and pseudogap of high-Tc

cuprates Phys Rev B 74 184515 (2006)43 Garg A Randeria M amp Trivedi N Strong correlations lead to protected low energy excitations in

disordered d-wave superconductors Preprint at lthttparXivorgcond-mat0609666gt (2006)44 Oda M Dipasupil R M Momono N amp Ido M Hyperbolic dependence of 210 vs Tc ratio on

hole-doping level in high-Tc cuprates Energy scale in determining Tc J Phys Soc Jpn 69983ndash984 (2000)

45 Hanaguri T Development of high-field STM and its application to the study on magnetically-tunedcriticality in Sr3Ru2O7 J Phys Conf Ser 51 514ndash521 (2006)

AcknowledgementsThe authors thank AV Balatsky C-M Ho D-H Lee K Machida A Mackenzie K McElroyT Tohyama J Zaanen and F-C Zhang for discussions They also thank J Matsuno and P Sharma forcritical readings TH MT and HT are supported by a Grant-in-Aid for Scientific Research from theMinistry of Education Culture Sports Science and Technology of Japan JCD and YK acknowledgesupport from Brookhaven National Laboratory under contract No DE-AC02-98CH1886 with the USDepartment of Energy from the US Department of Energy Award No DE-FG02-06ER46306 and fromthe US Office of Naval ResearchCorrespondence and requests for materials should be addressed to TH

Author contributionsTH was responsible for all aspects of this project except sample growth YK JCD and CLcontributed the project planning and data analysis IY MA MT and KO grew samples and MOcontributed the STM measurements HT contributed the project planning and managed thewhole project

Reprints and permission information is available online at httpnpgnaturecomreprintsandpermissions

nature physics VOL 3 DECEMBER 2007 wwwnaturecomnaturephysics 871

copy 2007 Nature Publishing Group