Angle resolved photoemission on underdoped Bi2Sr2CaCu2O8 reveals that the
magnitude and d-wave anisotropy of the superconducting state energy gap are
independent of temperature all the way up to Tc. This lack of T variation of
the entire k-dependent gap is in marked contrast to mean field theory. At Tc
the point nodes of the d-wave gap abruptly expand into finite length ``Fermi
arcs''. This change occurs within the width of the resistive transition, and
thus the Fermi arcs are not simply thermally broadened nodes but rather a
unique signature of the pseudogap phase.Comment: Accepted by Phys. Rev. Let

Campuzano, J. C.

Chatterjee, U.

Kanigel, A.

Li, Z. Z.

Norman, M. R.

Raffy, H.

Randeria, M.

Shi, M.

Souma, S.

Physical Review Letters

English

arXiv

Angle resolved photoemission on underdoped Bi_2Sr_2CaCu_2O_8 reveals that the magnitude and d-wave anisotropy of the superconducting state energy gap are independent of temperature all the way up to T_c. This lack of T variation of the entire k-dependent gap is in marked contrast to mean field theory. At T_c the point nodes of the d-wave gap abruptly expand into finite length "Fermi arcs." This change occurs within the width of the resistive transition, and thus the Fermi arcs are not simply thermally broadened nodes but rather a unique signature of the pseudogap phase

KnowledgeBank at OSU

Protected Nodes and the Collapse of Fermi Arcs in High-Tc Cuprate SuperconductorsA. Kanigel,1 U. Chatterjee,1 M. Randeria,2 M. R. Norman,3 S. Souma,1 M. Shi,1,4Z. Z. Li,5 H. Raffy,5 and J. C. Campuzano1,31Department of Physics, University of Illinois at Chicago, Chicago, Illinois 60607, USA2Department of Physics, The Ohio State University, Columbus, Ohio 43210, USA3Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA4Swiss Light Source, PSI, CH-5232 Villigen, Switzerland5Laboratorie de Physique des Solides, Universite Paris-Sud, 91405 Orsay Cedex, France(Received 11 January 2007; published 8 October 2007)Angle resolved photoemission on underdoped Bi2Sr2CaCu2O8 reveals that the magnitude and d-waveanisotropy of the superconducting state energy gap are independent of temperature all the way up to Tc.This lack of T variation of the entire k-dependent gap is in marked contrast to mean field theory. At Tc thepoint nodes of the d-wave gap abruptly expand into finite length ‘‘Fermi arcs.’’ This change occurs withinthe width of the resistive transition, and thus the Fermi arcs are not simply thermally broadened nodes butrather a unique signature of the pseudogap phase.DOI: 10.1103/PhysRevLett.99.157001 PACS numbers: 74.25.Jb, 74.72.Hs, 79.60.BmWe present in this Letter angle resolved photoemissionspectroscopy (ARPES) data on the energy gap in the super-conducting (SC) and pseudogap phases of the underdopedhigh Tc superconductor Bi2Sr2CaCu2O8 (Bi2212).Reduced Tc samples which lie in between the optimallydoped, highest Tc material and the undoped Mott insulatorare called ‘‘underdoped.’’ Such samples exhibit an unusualnormal-state pseudogap, whose signature in ARPES [1,2]is a loss of spectral weight in parts of k space, leading tolow-energy electronic excitations which live on discon-nected ‘‘Fermi arcs’’ [3]. Both the SC gap below Tc andthe pseudogap above Tc are anisotropic gaps in the single-particle excitation spectrum, but their relationship is notwell understood. Our goal is to gain insight into how theSC gap, with its d-wave anisotropy [1,2] at low T, changesas a function of temperature and evolves into the aniso-tropic pseudogap upon heating through Tc.Our first result is that the magnitude and anisotropy ofthe d-wave SC energy gap is essentially T independent forall T  Tc. This behavior is completely different from amean field description of a d-wave superconductor. Withinmean field theory the anisotropy would be T independent,but the gap magnitude, proportional to the order parameter,would be suppressed with increasing T and vanish at Tc.Remarkably, our data show that the k-dependent SC gapin underdoped cuprates does not even know about the scaleof Tc.Second, there are four point nodes at which the energygap vanishes for all T < Tc, but above Tc each point nodeabruptly expands into a gapless Fermi arc of finite extent.We show that this remarkable change occurs within thewidth of the resistive transition at Tc. The abrupt changefrom point nodes to gapless arcs is not just thermal smear-ing, but rather it is closely connected with the loss ofsuperconducting order.We present ARPES data on two underdoped Bi2212films, one near optimality (Tc  80 K) and the othermore underdoped (Tc  67 K), both of which exhibit asignificant pseudogap. The films, prepared by RF sputter-ing on a SrTiO3 substrate [4], exhibit only very weaksuperstructure replicas which simplifies the data analysis.The measurements were carried out at the SynchrotronRadiation Center, Wisconsin, using a Scienta R4000 ana-lyzer with 22 eV photons, and momentum cuts and polar-ization parallel to 0; 0 ! ; 0. The energy resolutionwas 20 meV (FWHM) with a k resolution along themultiplex direction of 0:0055 A1. The measuredARPES intensity is proportional to the spectral functiontimes the Fermi function. To determine gap values, weremove the effects of the Fermi function from the dataφNodeAnti-node(0,-π)(π,0)(π,−π)1141234567891011121314-0.1 0.0 0.1E(eV)-0.1 0.0 0.1E(eV)-0.1 0.0 0.1E(eV)-0.1 0.0 0.1E(eV)T=20K T=50K T=70K T=90K T=115K-0.1 0.0 0.1E(eV)a b c d e(π,π)(0,0)FIG. 1 (color online). Symmetrized energy distribution curves(EDCs) for the Tc  80 K sample at various temperatures. Thedifferent EDCs cover the whole Fermi surface from the node(lowest curve) up to the antinode (uppermost curve). Gaplessspectra are indicated by solid dots. The Fermi surface andlocation of the various EDCs are shown in the far right panel.PRL 99, 157001 (2007) P H Y S I C A L R E V I E W L E T T E R S week ending12 OCTOBER 20070031-9007=07=99(15)=157001(4) 157001-1 © 2007 The American Physical Societyfollowing two procedures: (i) symmetrization (addingspectra at E) [3] and (ii) division of the spectra by aresolution-broadened Fermi function (obtained from poly-crystalline Au in contact with the sample). Both methodsgave identical gap estimates within the error bars quotedbelow.In Fig. 1 we show symmetrized spectra for the Tc 80 K sample at various temperatures for 14 angles alongthe Fermi surface [5], indicated in the Brillouin zone (farright panel). The spectra in panels (a) (20), (b) (50) and(c) (70 K) are in the SC state. A gapless (symmetrized)spectrum has a maximum at zero energy, while gappedspectra show a minimum at zero with coherent peaks at() the gap value [6]. One can clearly see the variation ofthe SC gap along the Fermi surface. It decreases mono-tonically from its maximum value at the antinode (topcurve, point 14 on the zone edge) eventually vanishing atthe node (bottom curve, point 1 on the zone diagonal).Our main conclusions for the underdoped SC state are:(1) There are sharp quasiparticle peaks for all T < Tcwhich indicate coherent, long-lived, electronic excitationsin the SC state. (2) Both the magnitude of the SC gap andits k-variation are independent of T in the SC state. (3) Inparticular, the point node in the SC gap is protected for allT < Tc. We emphasize that these features of the SC gap areobserved all the way to 70 K in a Tc  80 K sample. Evenat T  0:87Tc, there is no indication in the k-dependentSC gap, that the system is about to lose its superconduc-tivity at Tc.To elucidate the T (in)dependence of various features indetail, we present a quantitative analysis of the data. InFig. 2 we plot the gap  as function of the angle  alongthe Fermi surface for the two samples. The gap is definedas the separation of the peak from zero energy (Fig. 1). Forthe three temperatures T < Tc, the SC gap is constant inmagnitude, being maximum at the antinode at   0 anddecreasing to zero at the node at   45, consistent withthe d-wave form j coskx  coskyj within error bars.As evident from Fig. 1, there are nontrivial line shapechanges [7] with temperature. Thus, in addition to charac-terizing the data in terms of the energy gap , we also lookat the T-dependent changes in the zero-energy intensityrelative to that at . We use a model-independent way ofcharacterizing [8] the loss of low-energy spectral weightdefined by L  1 I0=I, where I0 is the in-tensity at zero-energy and I the intensity at the peak(the gap energy). Note that for a large gap at  there is alarge loss of spectral weight, while L vanishes if thereare gapless excitations at . In Fig. 3 we plot L for thesame two samples at various T’s. Again, we see that thereare only small changes in L in the SC state with a fullydeveloped gap right up to Tc.In the pseudogap regime on the other hand, the data inFig. 1 panels (d) (90) and (e) (115 K), show that thefollowing dramatic changes occur upon heating throughTc. (1) There are no sharp peaks, implying that long-livedquasiparticle excitations do not exist above Tc. (2) Whilethe magnitude of the maximum gap on the zone boundarydepends very weakly on T, the overall anisotropy of thepseudogap above Tc shows a marked T-variation. (3) Inparticular, the point node in the SC gap below Tc suddenlyexpands into a Fermi arc of gapless excitations in thepseudogap state above Tc. The transition from point nodesto arcs is clearly visible in the raw data, with a singlegapless spectrum at the node at 70 K [Fig. 1(c)] expandinginto seven angles with gapless spectra at 90 K (1(d)], andnine angles at 115 K [1(e)]. Qualitatively similar results arefound for the Tc  67 K sample.To quantify these remarkable changes, we plot the gap inFig. 2 and the loss of spectral weight in Fig. 3. We see fromFig. 2 that the energy gap begins to acquire a significant Tdependence above Tc in the near nodal region. The pointnode below Tc expands into a Fermi arc just above Tc,5040302010050403020100NodeAnti-nodeTc=67K  20K 44K 62K 80K120Kd-wave∆)Vem(φ(deg)φ(deg)∆(meV)40302010050403020100NodeAnti-nodeTc=80K 20K 50K 70K 90K 115Kd-waveFIG. 2 (color online). The gap  as function of , the Fermisurface angle measured from the zone boundary. We show datafor two samples at five different temperatures. The dashed line isthe expected angular dependence of a d-wave gap.PRL 99, 157001 (2007) P H Y S I C A L R E V I E W L E T T E R S week ending12 OCTOBER 2007157001-2which then increases with increasing temperature, as re-ported earlier [8]. The transition from point nodes to arcs isalso clearly visible in the T dependence of the loss ofspectral weight L in Fig. 3.Close to the antinode the behavior is quite different fromthe arc region, and the pseudogap does not ‘‘close’’ withincreasing temperature, showing very little T dependenceeven above Tc. The evolution of the pseudogap here can bebest described as spectral weight ‘‘filling in’’ with increas-ing T, as evidenced by the strong T dependence of the lossfunction at small  in Fig. 3. It is useful to plot the loss ofspectral weight at the antinode L  0 as a function ofT; see Fig. 4 (upper panel). In the SC state we see a small,but systematic, decrease in L0 with increasing T, towhich both the drop in I at the peak position (evidentin the data of Fig. 1) and the filling in of I0 contribute.But at Tc there is a sharp, almost discontinuous, change ofbehavior. Above Tc, L0 is entirely dominated by thechange in I0 and the pseudogap fills in linearly withtemperature. A simple linear extrapolation of L0 tozero allows us to determine the temperature T abovewhich the pseudogap disappears (i.e., has filled in) and afull Fermi surface is recovered.Finally, we address the sharp collapse of the Fermi arcsat Tc. In the lower panel of Fig. 4 we show the length of theFermi arc as a function of the scaled temperature t T=Tx for the two samples discussed in this paper. Wealso include here all of the data from Ref. [8], where wedescribed the linear scaling of the Fermi arc length with t inthe pseudogap state. Our goal here is to see how onedeviates from the scaling line at Tc. We see that, as afunction of decreasing T at a fixed doping x, there is an1.00.80.60.40.20.0200150100500T(K)T*=170KT*=132K Tc=80K Tc=67K8060402001.00.80.60.40.20.0t=T/T* Tc=67, T*=170K Tc=80, T*=132KimreFgne L crAth (%)(I/) 0(I- 1=L∆ ) 0FIG. 4 (color online). Upper panel: loss of zero-energy spec-tral weight at the antinode as a function of temperature for thetwo samples. The lines are linear fits to the data in the pseudogapphase, the extrapolation to zero yields the value of T for eachsample. Lower panel: Fermi arc length as a function of t  T=Tfor both samples. The straight line represents the universalscaling of the arc length found in Ref. [8]. The solid curvesare the (rescaled) resistivity of the two samples.(I/)0(I-1=L∆ ) φφ(deg.)(I/)0(I-1=L∆ ) φφ(deg.)1.00.80.60.40.20.050403020100NodeAnti-nodeTc=80K 20K 50K 70K 90K 115K0.80.60.40.20.050403020100NodeAnti-nodeTc=67K 20K 44K 62K 80K 120KFIG. 3 (color online). Loss of zero-energy spectral weightL  1 I0=I as a function of , the Fermi surfaceangle measured from the zone boundary (see text). The twopanels show results at five different temperatures for twosamples.PRL 99, 157001 (2007) P H Y S I C A L R E V I E W L E T T E R S week ending12 OCTOBER 2007157001-3abrupt collapse from an arc length which lies on the scalingline above Tc to a point node below Tc. The size of thejump depends only on Tc: the higher Tc the larger the jump,with the arc length just above Tc simply determined byTc=T. We also plot the (rescaled) resistance measured onthe same samples [4], which clearly shows that the abrupt-ness of the collapse of the Fermi arcs has the same width asthe resistively measured SC transition. This implies thatthe Fermi arc is not simply due to lifetime broadening ofthe node.We now comment on the implications of our results. Theobservation of a T-independent SC gap with no sign of theimpending phase transition, even just below Tc, is veryunusual, implying that the energy gap in the underdopedSC state is not directly related to the superconducting orderparameter. As already mentioned above, this is qualita-tively different from the mean field description of d-wavesuperconductivity.For T < Tc let us ask if the essentially T-independentgap is consistent with nodal quasiparticles, whose exis-tence is well established [9,10]. We expect a T3 suppres-sion of the gap from thermally excited quasiparticles, givenby k; T  Tcoskx  cosky=2 with T=0 1 	T=03 for T  0, with  of order unity.For the Tc  80 K sample, 0  35 meV and thereforethe fractional change in the gap (at any k point) going from0 to 70 K is only a few percent. The effect is even smallerfor the Tc  67 K sample, because 0  42 meV islarger. Thus thermal effects of nodal quasiparticles arenot expected to influence the SC gap, since the scale ofthe gap 0 is set by T [11], which is much larger than Tcfor underdoped samples.Some authors argue that the pseudogap originates from acompeting order parameter [12,13]. This then leads to a‘‘two gap’’ scenario [14] for the SC state, in which the gapnear the antinode is associated with the non-SC orderparameter, and scales with the pseudogap T, while thegap which opens up on the arcs is associated with super-conductivity, and scales with Tc. Our results disagree withthis scenario in two major ways. First, if this were the case,there would be no reason to expect a single function of theform j coskx  coskyj to describe the gap that we observefor all T < Tc. Second, one would have expected that thesmall gap near the nodes should have knowledge of Tc, andexhibit significant T variation below Tc, which it does not;the node (and the gap anisotropy) appears to be protectedup to Tc.An alternative explanation, consistent with our under-doped results, is that Tc is controlled by the small super-fluid density [15,16] of a doped Mott insulator. This wasfirst suggested by the Uemura scaling [15] between Tc andthe T  0 superfluid density in underdoped samples [17].Thus it is fluctuations of the phase of the SC order parame-ter which destroy superconductivity, and not the (conven-tional) collapse of the SC gap. The pseudogap is then aremnant of the SC gap above Tc in a state which has nolong-range phase coherence [18]. This is also consistentwith the observation of an unusual Nernst signal [19] in arange of temperatures above Tc, but a detailed theoreticalunderstanding of the pseudogap all the way up to T islacking at the present time.In summary, we presented a complete description of thethermal evolution of the spectral gap in underdopedBi2212. The remarkable T independence of the energygap below Tc implies a nonmean field origin to the SCgap. And the sudden expansion of the d-wave nodespresent below Tc, into finite length Fermi arcs above Tcwhich scale as Tc=T, suggests that the Fermi arcs in thepseudogap regime are not simply thermally broadenednodes.This work was supported by NSF No. DMR-0606255and the U.S. DOE, Office of Science, under ContractNo. DE-AC02-06CH11357. The Synchrotron RadiationCenter is supported by NSF No. DMR-0084402.[1] J. C. Campuzano, M. R. Norman, and M. Randeria, in ThePhysics of Superconductors, edited by K. H. Bennemannand J. B. Ketterson (Springer, New York, 2004), Vol. 2.[2] A. Damascelli, Z. Hussain, and Z.-X. Shen, Rev. Mod.Phys. 75, 473 (2003).[3] M. R. Norman et al., Nature (London) 392, 157 (1998).[4] Z. Konstantinovic, Z. Z. Li, and H. Raffy, Physica(Amsterdam) 341C, 859 (2000).[5] For the determination of the Fermi momentum kF byARPES, see J. Mesot et al., Phys. Rev. B 63, 224516(2001).[6] For the determination of the superconducting gap byARPES and associated error bars, see H. Ding et al.,Phys. Rev. Lett. 74, 2784 (1995).[7] M. R. Norman et al., Phys. Rev. B 57, R11 093 (1998).[8] A. Kanigel et al., Nature Phys. 2, 447 (2006).[9] W. N. Hardy et al., Phys. Rev. Lett. 70, 3999 (1993).[10] M. Chiao et al., Phys. Rev. B 62, 3554 (2000).[11] J. C. Campuzano et al., Phys. Rev. Lett. 83, 3709 (1999).[12] S. Chakravarty et al., Phys. Rev. B 63, 094503 (2001).[13] C. M. Varma, Phys. Rev. B 73, 155113 (2006).[14] M. Le Tacon et al., Nature Phys. 2, 537 (2006);G. Deutscher, Nature (London) 397, 410 (1999).[15] Y. J. Uemura et al., Phys. Rev. Lett. 62, 2317 (1989).[16] V. J. Emery and S. A. Kivelson, Nature (London) 374, 434(1995).[17] Recent deviations [see R. Liang et al., Phys. Rev. Lett. 94,117001 (2005); Y. Zuev et al., Phys. Rev. Lett. 95, 137002(2005)] from Uemura scaling in highly underdopedsamples also find evidence for a Tc controlled by s,even though the scaling relation is nonlinear.[18] M. Randeria in Proc. Int. School of Phys. ‘‘Enrico Fermi’’on Conventional and High Tc Superconductors, edited byG. Iadonisi, J. R. Schrieffer, and M. L. Chiafalo (IOS,Amsterdam, 1998), p. 53.[19] Y. Wang, L. Li, and N. P. Ong, Phys. Rev. B 73, 024510(2006).PRL 99, 157001 (2007) P H Y S I C A L R E V I E W L E T T E R S week ending12 OCTOBER 2007157001-4

Protected Nodes and the Collapse of Fermi Arcs in High-T_c Cuprate Superconductors

Ohio State University: Knowledge Bank

Protected Nodes and the Collapse of Fermi Arcs in High-Tc Cuprate Superconductors
A. Kanigel,1 U. Chatterjee,1 M. Randeria,2 M. R. Norman,3 S. Souma,1 M. Shi,1,4
Z. Z. Li,5 H. Raffy,5 and J. C. Campuzano1,3
1Department of Physics, University of Illinois at Chicago, Chicago, Illinois 60607, USA
2Department of Physics, The Ohio State University, Columbus, Ohio 43210, USA
3Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA
4Swiss Light Source, PSI, CH-5232 Villigen, Switzerland
5Laboratorie de Physique des Solides, Universite Paris-Sud, 91405 Orsay Cedex, France
(Received 11 January 2007; published 8 October 2007)
Angle resolved photoemission on underdoped Bi2Sr2CaCu2O8 reveals that the magnitude and d-wave
anisotropy of the superconducting state energy gap are independent of temperature all the way up to Tc.
This lack of T variation of the entire k-dependent gap is in marked contrast to mean field theory. At Tc the
point nodes of the d-wave gap abruptly expand into finite length ‘‘Fermi arcs.’’ This change occurs within
the width of the resistive transition, and thus the Fermi arcs are not simply thermally broadened nodes but
rather a unique signature of the pseudogap phase.
DOI: 10.1103/PhysRevLett.99.157001 PACS numbers: 74.25.Jb, 74.72.Hs, 79.60.Bm
We present in this Letter angle resolved photoemission
spectroscopy (ARPES) data on the energy gap in the super-
conducting (SC) and pseudogap phases of the underdoped
high Tc superconductor Bi2Sr2CaCu2O8 (Bi2212).
Reduced Tc samples which lie in between the optimally
doped, highest Tc material and the undoped Mott insulator
are called ‘‘underdoped.’’ Such samples exhibit an unusual
normal-state pseudogap, whose signature in ARPES [1,2]
is a loss of spectral weight in parts of k space, leading to
low-energy electronic excitations which live on discon-
nected ‘‘Fermi arcs’’ [3]. Both the SC gap below Tc and
the pseudogap above Tc are anisotropic gaps in the single-
particle excitation spectrum, but their relationship is not
well understood. Our goal is to gain insight into how the
SC gap, with its d-wave anisotropy [1,2] at low T, changes
as a function of temperature and evolves into the aniso-
tropic pseudogap upon heating through Tc.
Our first result is that the magnitude and anisotropy of
the d-wave SC energy gap is essentially T independent for
all T  Tc. This behavior is completely different from a
mean field description of a d-wave superconductor. Within
mean field theory the anisotropy would be T independent,
but the gap magnitude, proportional to the order parameter,
would be suppressed with increasing T and vanish at Tc.
Remarkably, our data show that the k-dependent SC gap
in underdoped cuprates does not even know about the scale
of Tc.
Second, there are four point nodes at which the energy
gap vanishes for all T < Tc, but above Tc each point node
abruptly expands into a gapless Fermi arc of finite extent.
We show that this remarkable change occurs within the
width of the resistive transition at Tc. The abrupt change
from point nodes to gapless arcs is not just thermal smear-
ing, but rather it is closely connected with the loss of
superconducting order.
We present ARPES data on two underdoped Bi2212
films, one near optimality (Tc  80 K) and the other
more underdoped (Tc  67 K), both of which exhibit a
significant pseudogap. The films, prepared by RF sputter-
ing on a SrTiO3 substrate [4], exhibit only very weak
superstructure replicas which simplifies the data analysis.
The measurements were carried out at the Synchrotron
Radiation Center, Wisconsin, using a Scienta R4000 ana-
lyzer with 22 eV photons, and momentum cuts and polar-
ization parallel to 0; 0 ! ; 0. The energy resolution
was 20 meV (FWHM) with a k resolution along the
multiplex direction of 0:0055 A1. The measured
ARPES intensity is proportional to the spectral function
times the Fermi function. To determine gap values, we
remove the effects of the Fermi function from the data
φ
Node
Anti-node
(0,-π)
(π,0)
(π,−π)
1
14
1
2
3
4
5
6
7
8
9
10
11
12
13
14
-0.1 0.0 0.1
E(eV)
-0.1 0.0 0.1
E(eV)
-0.1 0.0 0.1
E(eV)
-0.1 0.0 0.1
E(eV)
T=20K T=50K T=70K T=90K T=115K
-0.1 0.0 0.1
E(eV)
a b c d e
(π,π)
(0,0)
FIG. 1 (color online). Symmetrized energy distribution curves
(EDCs) for the Tc  80 K sample at various temperatures. The
different EDCs cover the whole Fermi surface from the node
(lowest curve) up to the antinode (uppermost curve). Gapless
spectra are indicated by solid dots. The Fermi surface and
location of the various EDCs are shown in the far right panel.
PRL 99, 157001 (2007) P H Y S I C A L R E V I E W L E T T E R S week ending12 OCTOBER 2007
0031-9007=07=99(15)=157001(4) 157001-1 © 2007 The American Physical Society
following two procedures: (i) symmetrization (adding
spectra at E) [3] and (ii) division of the spectra by a
resolution-broadened Fermi function (obtained from poly-
crystalline Au in contact with the sample). Both methods
gave identical gap estimates within the error bars quoted
below.
In Fig. 1 we show symmetrized spectra for the Tc 
80 K sample at various temperatures for 14 angles along
the Fermi surface [5], indicated in the Brillouin zone (far
right panel). The spectra in panels (a) (20), (b) (50) and
(c) (70 K) are in the SC state. A gapless (symmetrized)
spectrum has a maximum at zero energy, while gapped
spectra show a minimum at zero with coherent peaks at
() the gap value [6]. One can clearly see the variation of
the SC gap along the Fermi surface. It decreases mono-
tonically from its maximum value at the antinode (top
curve, point 14 on the zone edge) eventually vanishing at
the node (bottom curve, point 1 on the zone diagonal).
Our main conclusions for the underdoped SC state are:
(1) There are sharp quasiparticle peaks for all T < Tc
which indicate coherent, long-lived, electronic excitations
in the SC state. (2) Both the magnitude of the SC gap and
its k-variation are independent of T in the SC state. (3) In
particular, the point node in the SC gap is protected for all
T < Tc. We emphasize that these features of the SC gap are
observed all the way to 70 K in a Tc  80 K sample. Even
at T  0:87Tc, there is no indication in the k-dependent
SC gap, that the system is about to lose its superconduc-
tivity at Tc.
To elucidate the T (in)dependence of various features in
detail, we present a quantitative analysis of the data. In
Fig. 2 we plot the gap  as function of the angle  along
the Fermi surface for the two samples. The gap is defined
as the separation of the peak from zero energy (Fig. 1). For
the three temperatures T < Tc, the SC gap is constant in
magnitude, being maximum at the antinode at   0 and
decreasing to zero at the node at   45, consistent with
the d-wave form j coskx  coskyj within error bars.
As evident from Fig. 1, there are nontrivial line shape
changes [7] with temperature. Thus, in addition to charac-
terizing the data in terms of the energy gap , we also look
at the T-dependent changes in the zero-energy intensity
relative to that at . We use a model-independent way of
characterizing [8] the loss of low-energy spectral weight
defined by L  1 I0=I, where I0 is the in-
tensity at zero-energy and I the intensity at the peak
(the gap energy). Note that for a large gap at  there is a
large loss of spectral weight, while L vanishes if there
are gapless excitations at . In Fig. 3 we plot L for the
same two samples at various T’s. Again, we see that there
are only small changes in L in the SC state with a fully
developed gap right up to Tc.
In the pseudogap regime on the other hand, the data in
Fig. 1 panels (d) (90) and (e) (115 K), show that the
following dramatic changes occur upon heating through
Tc. (1) There are no sharp peaks, implying that long-lived
quasiparticle excitations do not exist above Tc. (2) While
the magnitude of the maximum gap on the zone boundary
depends very weakly on T, the overall anisotropy of the
pseudogap above Tc shows a marked T-variation. (3) In
particular, the point node in the SC gap below Tc suddenly
expands into a Fermi arc of gapless excitations in the
pseudogap state above Tc. The transition from point nodes
to arcs is clearly visible in the raw data, with a single
gapless spectrum at the node at 70 K [Fig. 1(c)] expanding
into seven angles with gapless spectra at 90 K (1(d)], and
nine angles at 115 K [1(e)]. Qualitatively similar results are
found for the Tc  67 K sample.
To quantify these remarkable changes, we plot the gap in
Fig. 2 and the loss of spectral weight in Fig. 3. We see from
Fig. 2 that the energy gap begins to acquire a significant T
dependence above Tc in the near nodal region. The point
node below Tc expands into a Fermi arc just above Tc,
50
40
30
20
10
0
50403020100
NodeAnti-node
Tc=67K  20K
 44K
 62K
 80K
120K
d-wave
∆
)
Ve
m(
φ(deg)
φ(deg)
∆(
me
V)
40
30
20
10
0
50403020100
NodeAnti-node
Tc=80K
 20K
 50K
 70K
 90K
 115K
d-wave
FIG. 2 (color online). The gap  as function of , the Fermi
surface angle measured from the zone boundary. We show data
for two samples at five different temperatures. The dashed line is
the expected angular dependence of a d-wave gap.
PRL 99, 157001 (2007) P H Y S I C A L R E V I E W L E T T E R S week ending12 OCTOBER 2007
157001-2
which then increases with increasing temperature, as re-
ported earlier [8]. The transition from point nodes to arcs is
also clearly visible in the T dependence of the loss of
spectral weight L in Fig. 3.
Close to the antinode the behavior is quite different from
the arc region, and the pseudogap does not ‘‘close’’ with
increasing temperature, showing very little T dependence
even above Tc. The evolution of the pseudogap here can be
best described as spectral weight ‘‘filling in’’ with increas-
ing T, as evidenced by the strong T dependence of the loss
function at small  in Fig. 3. It is useful to plot the loss of
spectral weight at the antinode L  0 as a function of
T; see Fig. 4 (upper panel). In the SC state we see a small,
but systematic, decrease in L0 with increasing T, to
which both the drop in I at the peak position (evident
in the data of Fig. 1) and the filling in of I0 contribute.
But at Tc there is a sharp, almost discontinuous, change of
behavior. Above Tc, L0 is entirely dominated by the
change in I0 and the pseudogap fills in linearly with
temperature. A simple linear extrapolation of L0 to
zero allows us to determine the temperature T above
which the pseudogap disappears (i.e., has filled in) and a
full Fermi surface is recovered.
Finally, we address the sharp collapse of the Fermi arcs
at Tc. In the lower panel of Fig. 4 we show the length of the
Fermi arc as a function of the scaled temperature t 
T=Tx for the two samples discussed in this paper. We
also include here all of the data from Ref. [8], where we
described the linear scaling of the Fermi arc length with t in
the pseudogap state. Our goal here is to see how one
deviates from the scaling line at Tc. We see that, as a
function of decreasing T at a fixed doping x, there is an
1.0
0.8
0.6
0.4
0.2
0.0
200150100500
T(K)
T*=170K
T*=132K
 Tc=80K
 Tc=67K
80
60
40
20
0
1.00.80.60.40.20.0
t=T/T*
 Tc=67, T*=170K
 Tc=80, T*=132K
i
mreF
g
ne L
 cr
A
th
 (%
)
(I/) 0(I
- 1
=L
∆ 
) 0
FIG. 4 (color online). Upper panel: loss of zero-energy spec-
tral weight at the antinode as a function of temperature for the
two samples. The lines are linear fits to the data in the pseudogap
phase, the extrapolation to zero yields the value of T for each
sample. Lower panel: Fermi arc length as a function of t  T=T
for both samples. The straight line represents the universal
scaling of the arc length found in Ref. [8]. The solid curves
are the (rescaled) resistivity of the two samples.
(I/)0(I
-1
=L
∆ 
) φ
φ(deg.)
(I/)0(I
-1
=L
∆ 
) φ
φ(deg.)
1.0
0.8
0.6
0.4
0.2
0.0
50403020100
NodeAnti-node
Tc=80K
 20K
 50K
 70K
 90K
 115K
0.8
0.6
0.4
0.2
0.0
50403020100
NodeAnti-node
Tc=67K
 20K
 44K
 62K
 80K
 120K
FIG. 3 (color online). Loss of zero-energy spectral weight
L  1 I0=I as a function of , the Fermi surface
angle measured from the zone boundary (see text). The two
panels show results at five different temperatures for two
samples.
PRL 99, 157001 (2007) P H Y S I C A L R E V I E W L E T T E R S week ending12 OCTOBER 2007
157001-3
abrupt collapse from an arc length which lies on the scaling
line above Tc to a point node below Tc. The size of the
jump depends only on Tc: the higher Tc the larger the jump,
with the arc length just above Tc simply determined by
Tc=T. We also plot the (rescaled) resistance measured on
the same samples [4], which clearly shows that the abrupt-
ness of the collapse of the Fermi arcs has the same width as
the resistively measured SC transition. This implies that
the Fermi arc is not simply due to lifetime broadening of
the node.
We now comment on the implications of our results. The
observation of a T-independent SC gap with no sign of the
impending phase transition, even just below Tc, is very
unusual, implying that the energy gap in the underdoped
SC state is not directly related to the superconducting order
parameter. As already mentioned above, this is qualita-
tively different from the mean field description of d-wave
superconductivity.
For T < Tc let us ask if the essentially T-independent
gap is consistent with nodal quasiparticles, whose exis-
tence is well established [9,10]. We expect a T3 suppres-
sion of the gap from thermally excited quasiparticles, given
by k; T  Tcoskx  cosky=2 with T=0 
1 	T=0
3 for T  0, with  of order unity.
For the Tc  80 K sample, 0  35 meV and therefore
the fractional change in the gap (at any k point) going from
0 to 70 K is only a few percent. The effect is even smaller
for the Tc  67 K sample, because 0  42 meV is
larger. Thus thermal effects of nodal quasiparticles are
not expected to influence the SC gap, since the scale of
the gap 0 is set by T [11], which is much larger than Tc
for underdoped samples.
Some authors argue that the pseudogap originates from a
competing order parameter [12,13]. This then leads to a
‘‘two gap’’ scenario [14] for the SC state, in which the gap
near the antinode is associated with the non-SC order
parameter, and scales with the pseudogap T, while the
gap which opens up on the arcs is associated with super-
conductivity, and scales with Tc. Our results disagree with
this scenario in two major ways. First, if this were the case,
there would be no reason to expect a single function of the
form j coskx  coskyj to describe the gap that we observe
for all T < Tc. Second, one would have expected that the
small gap near the nodes should have knowledge of Tc, and
exhibit significant T variation below Tc, which it does not;
the node (and the gap anisotropy) appears to be protected
up to Tc.
An alternative explanation, consistent with our under-
doped results, is that Tc is controlled by the small super-
fluid density [15,16] of a doped Mott insulator. This was
first suggested by the Uemura scaling [15] between Tc and
the T  0 superfluid density in underdoped samples [17].
Thus it is fluctuations of the phase of the SC order parame-
ter which destroy superconductivity, and not the (conven-
tional) collapse of the SC gap. The pseudogap is then a
remnant of the SC gap above Tc in a state which has no
long-range phase coherence [18]. This is also consistent
with the observation of an unusual Nernst signal [19] in a
range of temperatures above Tc, but a detailed theoretical
understanding of the pseudogap all the way up to T is
lacking at the present time.
In summary, we presented a complete description of the
thermal evolution of the spectral gap in underdoped
Bi2212. The remarkable T independence of the energy
gap below Tc implies a nonmean field origin to the SC
gap. And the sudden expansion of the d-wave nodes
present below Tc, into finite length Fermi arcs above Tc
which scale as Tc=T, suggests that the Fermi arcs in the
pseudogap regime are not simply thermally broadened
nodes.
This work was supported by NSF No. DMR-0606255
and the U.S. DOE, Office of Science, under Contract
No. DE-AC02-06CH11357. The Synchrotron Radiation
Center is supported by NSF No. DMR-0084402.
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Protected nodes and the collapse of the Fermi arcs in high Tc cuprates

Protected nodes and the collapse of the Fermi arcs in high Tc cuprates

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