Phase fluctuations versus Gaussian fluctuations in optimally-doped YBa$_2$Cu$_3$O$_7$

0.35 MB·English

by  Lu Li

Phase fluctuations versus Gaussian fluctuations in optimally-doped YBa Cu O .
2 3 7
Lu Li1,2, Yayu Wang1,3 and N. P. Ong1
1Department of Physics, Princeton University, Princeton, NJ 08544
2Department of Physics, University of Michigan, Ann Arbor, MI 48109
3Department of Physics, Tsinghua University, Beijing, China
(Dated: January 31, 2013)
WeanalyzerecenttorquemeasurementsofthemagnetizationM vs. magneticfieldHinoptimally
d
3 doped YBa2Cu3O7−y (OPT YBCO) to argue against a recent proposal by Rey et al. that the
1 magnetization results above Tc are consistent with Gaussian fluctuations. Wefind that, despite its
strong interlayer coupling, OPT YBCO displays an anomalous non-monotonic dependence of M
0 d
2 on H which represents direct evidence for the locking of the pair wavefunction phase θn at Tc and
the subsequent unlocking by a relatively weak H. These unusual features characterize the unusual

n natureofthetransitiontotheMeissnerstateincupratesuperconductors. Theyareabsentinlow-Tc
a superconductorstoourknowledge. Wealso stresstheimportanceofthevortexliquidstate,aswell
J
astheprofilesofthemeltingfieldHm(T)andtheuppercriticalfieldcurveHc2(T)intheT-H plane.
0 Contrary to the claims of Rey et al.,we show that the curves of the magnetization and the Nernst
3 signalillustrate theinaccessibility oftheHc2 linenearTc. Theprediction oftheHc2 linebyReyet
al. is shown to beinvalid in OPT YBCO.
]
n
o
I. INTRODUCTION doped cuprates. The collapse of its Meissner state at T
c c
- is also caused by vanishing of phase rigidity.
r In a Comment, Rey et al. [7] have fitted the magneti-
p In the mean-field Gaussian treatment of fluctuations
u zation curves for OPT YBCO in Ref. [5] to their model
(valid in low-T superconductors),the pair-wavefunction
s c and claimed that the diamagnetic signal is consistent
t. amplitude |Ψ| vanishes at the critical temperature Tc. withMFGaussianfluctuationsinalayeredsuperconduc-
a Above Tc, amplitude fluctuations about the equilibrium tor. Furthermore,theyhaveextendedtheircalculationto
m point |Ψ| = 0 may be regarded (in Schmid’s elegant de-
temperatauresT <T toinfer anupper criticalfieldH
piction [1]) as droplets of condensate of radius ξ , the c c2
- GL that rises linearly with reduced temperature (1−T/T )
d Ginzburg-Landau(GL)coherencelength. Inthecompet- c
n ing phase-disordering scenario, |Ψ| remains finite above with a slope of -3 T/K.
o Here, we show that the fluctuation signal above T is
T . ThecollapseoftheMeissnereffectaboveT iscaused c
c c c just the tip of the iceberg. Because of strong interlayer
insteadbythevanishingofphaserigidity. AboveT ,fluc-
[ c
tuationsareprimarilyofthephaseθ ofΨ,proceedingby couplinginOPTYBCO,itisnecessarytogobelowTc to

1 uncovertheclosesimilaritieswithothercuprates. InSec.
phaseslipscausedbythemotionofspontaneousvortices.
v IIwedescribethenon-monotonicvariationofthemagne-
In zero magnetic field, the net vorticity in the sample is
3 tization curve just below T , previously noted in torque
zero,sothepopulationsof“up”and“down”vorticesare c
3 measurements on OPT Bi 2212 [4]. This characteristic
3 equal. In underdoped cuprates, there is strong evidence
feature – inherent in the phase-disordering scenerio – is
7 from Nernst [2], magnetization [3–5] and other experi-
common to the cuprates investigated to date. In Sec.
. ments in support of the phase-disordering scenario.
1 III, we discuss the upper critical field H for T close T
c2 c
0 Initially, it seemed to us that optimally-doped
in OPT YBCO. The striking inability of high-resolution
3 YBa2Cu3O7−y (OPT YBCO), which has the largest experiments to detect the H line, a corner stone of the
1 c2
interlayer coupling energy (and lowest electronic
mean-field Gaussian picture, is also a characteristic fea-
:
v anisotropy) among the known cuprates, would be most tureofthephasedisorderingmechanism. Finally,inSec.
i amenable to standard mean-field (MF) treatment, i.e.
X IVwediscussthefitstothemagnetizationaboveTc. We
its fluctuations above Tc are strictly Gaussian. In- show that the conclusions of Rey et al. [7] are not valid.
r
a deed, in the early 90s, several groups applied the con-
ventional Maki-Thompson Aslamazov-Larkin theory to
analyze fluctuation conductivity above T [6]. Sub-
c II. NON-MONOTONIC MAGNETIZATION
sequently, these mean-field “fits”, seemingly reason-
able in OPT YBCO, were found to be woefully inade-
quate in underdopedYBCO andin mostother cuprates. The curves of the diamagnetic component of the mag-
The steady accumulation of evidence from Nernst netization Md in OPT YBCO (Ref. [5]) are reproduced
and torque experiments favoring the phase-disordering in Fig 1. At temperature T above ∼100 K, Md is ini-
mechanism in underdoped hole-doped cuprates and in tially linear in H up to 6 T. As we approach Tc, the
OPT Bi Sr CaCu O (Bi 2212) has prompted a re- linear-responseregimebecomesconfinedtoprogressively
2 2 2 8+y
assessmentofthecaseforOPTYBCO.Therecenttorque smaller field ranges (for e.g. |H|< 2 T at 95 K).
magnetometry results in Ref. [5] have persuaded us that At T = 92.5 K (the present analysis shows that T is
c c
OPT YBCO is, in fact, much closer to the other hole- slightly higher than the nominal value in Ref. [5]), the
2

7 6.5 6 5 4 K NbSe2
0
Hc2
) 2
m 3
A/-500 T (K)
( 0 2 4 6 8
d
M
T)3
-1000 ( 2
H2c
1
-1500
0
H
cusp
0.0 1.0 2.0 3.0 4.0
0H (T)
FIG. 2: (color online) The magnetization curves below Tc in
NbSe2. At each T, Md(H) decrease monotonically to reach
zero at Hc2(T). Following vortex entry at Hc1 (not resolved
here), the diamagnetic screening currents are steadily weak-
enedasH suppressesthepairamplitude. Theinsetplotsthe
inferred Hc2 vs. T. (Adaptedfrom Ref. [3])
vortices causes a slight hysteresis.
This unsual non-monotonic profile – absent in low-T
c
FIG. 1: (color online) The field dependence of the diamag- superconductors to our knowledge – implies the follow-
neticcomponentofthemagnetizationMd inoptimallydoped ing picture. The collapse of the Meissner state at Hcusp
YBa2Cu3O7−y at selected temperatures T. At Tc = 92.5, a leads to a steep decrease in |M | as vortices enter the
d
weakstep-increasein|M |signalstheonsetoffullfluxexpul-
d sample. This is a consequence of field suppression of
sion. Below Tc, a field H >Hcusp causes a sharp decrease in thenascentinter-bilayerphasecoupling. (ThetwoCuO
the screening current, but at larger H, |M | resumes its in- 2
d layersbracketingthe Y ions constitute a bilayer. We are
creaseataratesimilartothataboveTc. Thisnon-monotonic
concernedwithonlythecouplingbetweenadjacentbilay-
pjaactetnetrnbsiilganyearlsstbheelofiweldT-cin.duWceedadregcuoeupthliantgtohfiθsnibsetthweeednefiadn-- erswhichareseparatedbyaspacingd=11.8˚A.Wemen-
ingmagnetization featurethatcharacterizes thetransitionin tion the intra-bilayer coupling below.) In low-Tc super-
cuprates. µ0 is the vacuum permeability. The arrows (la- conductors, the decrease invariably continues monotoni-
belled as Hc2?) are Hc2 values at 89, 90 and 91 K predicted cally to zero as H → Hc2(T) (the upper critical field at
in Ref. [7]. (Adaptedfrom Ref. [5].) temperature T). For reference, we show in Fig. 2 curves
of M vs. H in the layered superconductor NbSe [3].
d 2
By contrast, the non-monotonicity in |M | implies that
d
the diamagnetic current in OPT YBCO turns around
only significant change is a step-increase in the magni-
tude |M | near H = 0+. Above 0.5 T, M (H) is strik- and grows stronger with H at a rate closely similar to
d d
that above T . Remarkably, if we hide the field region
ingly similar to the curves above T apart from vertical c
c
scale. We focus on the region in Fig. 1 bounded by the around Hcusp, the profiles of Md vs. H below Tc resem-
ble those above T , apart from a different vertical scale.
curves at T and 90 K (the curve at 91 K is represen- c
c
tative). At H = 0+, the step-increase in |Md| signals This implies that the transition at Tc at which flux ex-
pulsionappearsisaweak-fieldphenomenon. Beyondthis
the onset of full flux expulsion. However, a weak field
interrupts this steep rise to produce a sharp cusp at the weak-field regime, there is no hint that a transition has
occured.
fieldH (H isslightlyhigherthanthelowercritical
cusp cusp
field H because of surface pinning of vortices). When Returning to Fig. 1, we interpret the unusual non-
c1
H exceeds H , |M | falls rapidly to a broadminimum monotonic pattern(now seenin Bi2201,Bi2212,LSCO
cusp d
near 1 T, but subsequently rises to even larger values. and YBCO) [3–5] as reflecting the rapid growth of c-
This non-monotonic profile in M , appearing just below axis phase rigidity below T (in zero or weak H) and
d c
T ,isadefininghallmarkofthetransitionincuprates. In its subsequent destruction at H . In a finite interval
c cusp
this interval (90-92.5 K), M vs. H is reversible except aboveT , the pairwavefunction|Ψ |eiθn inbilayern has
d c n
in the vicinity of H , where surface-barrierpinning of a finite average amplitude h|Ψ |i. The in-plane phase-
cusp n
3
phase correlation length ζ in each bilayer, given by
a
he−iθn(0)eiθn(r)i=e−r/ζa, (1) (a) (b)
vortex
liquid Hc2
is long enough that local diamagnetic currents can be
detected by torque magnetometry. However, the c-axis H Hc2 H vlioqrutiedx
correlation length ζ ≪d, so θ is uncorrelated between
c n
adjacent bilayers. Hence Md reflects the diamagnetic re- vortex Hm
sponse of 2D supercurrents which are observable to very solid vortex Hm
solid
intense H. In Fig. 1, this 2D diamagnetic response is
represented in the field profile of M at 94 K. 0 T Tc 0 Tc TMF
d 16
Below T , ζ diverges (in zero H) to lock θ across all
c c n . V/K
bilayers. In weak H, the 3D phase rigidity produces full 3.50 m (c)
14 3.30
expulsion. However,inthe2-Kinterval90-92.5K,aweak
field H ∼Hcusp suffices to destroy the c-axis phase stiff- 12 3.00 Hc2?
ness. The system then reverts back to the diamagnetic
2.50
response of 2D uncorrelated condensates. Consequently,
10
asH increasesfurther,the2Ddiamagneticresponsecon- 2.00
tinuestoincrease,mimickingtheprofileat94K.At14T, (cid:13) 8 1.50
itthsev2aDlureesaptoHnscueslpea(dseseto91a-vKalcuuerfvoer).|MTdh|ethwaetawkellolcekxicnegedosf H((cid:13)T(cid:13))(cid:13)0(cid:13)m 6 0.74
θn across bilayers, and the field-induced unlocking, ac- 0.37
countforthe non-monotonicityofMd aswellasthe sim- 4 0.15
ilaritiesofthehigh-fieldportionsacrossT inaphysically Hm
c
reasonable way. The juxtaposition of an extremely large 2 0.05
pairing energy scale (d-wave gap amplitude ∆∼40 mV)
and a weak c-axis phase stiffness leads to this very un- 0
60 70 80 90 100
usual phase locking-unlocking scenario which seems per-
T (K)
vasive in the hole-doped cuprates. We argue that this
situation cannot be treated by applying the mean-field
FIG.3: (coloronline)ProfilesofthemeltingfieldHm(T)and
GL approach [1] to the Lawrence-Doniach (LD) Hamil-
uppercriticalfieldHc2(T)inlow-Tcandcupratesuperconduc-
tonian [8, 9].
tors. Panel (a) shows the phase diagram for a conventional
type-II superconductor in the T-H plane. The vortex liquid
phase is wedged between the curves of Hm(T) and Hc2(T),
III. CLOSURE OF THE Hc2 CURVE both of which terminate at Tc as H → 0. In the hole-doped
cuprates [Panel (b)], the vortex liquid state dominates the
Another strong argument against the mean-field de- phasediagram. AsH →0,Hm(T)terminatesattheobserved
scription is obtained from the qualitative features of the Tc whereas Hc2(T)terminates at ahighertemperatureTMF.
Panel (c): The contour plots of the magnitude of the Nernst
phase diagramin the T-H plane (Fig. 3). A cornerstone
ofthemean-fieldGaussiandescriptionofatype-IIsuper- signal ey in OPT YBCO in the T-H plane. Values of the 10
contours are displayed on the left column. The melting field
conductor is the well-known profile of the upper critical
Hm(T)separatesthevortexsolid(blackregion)fromthevor-
field curve Hc2(T) (Fig. 3a). In the T-H plane, the Hc2 tex liquid state. The dashed line is the Hc2 line predicted in
curveisasharplydefinedboundarythatseparatesthere- Ref. [7]. (Adaptedfrom Ref. [11].)
gionwithfinitepairamplitude|Ψ|fromthenormalstate
with |Ψ|=0. The Gaussianfluctuations are fluctuations
ofthe amplitude aroundthe |Ψ|=0state aboveH (T).
c2
at the Ne´el temperature.) In an applied field H, phase
Inlow-T superconductors(forT >∼0.5T ), H (T)de-
c c c2
rigidityismaintainedifthe vorticesareinthe solidstate
creaseslinearlyin the reducedtemperature t=1−T/T
c
(and pinned). As H approaches H , the phase rigidity
to terminate at T . This profile is shown for NbSe in c2
c 2
abruptly vanishes at the melting field H which marks
the inset of Fig. 2. To stress this point, we refer to the m
thetransitiontothevortex-liquidstate(theanalogofthe
termination as “closure” of the curve of H .
c2
paramagneticstate inthe antiferromagnet). The vortex-
The closureofH reflectsananomalousaspectofthe
c2
liquid state is wedged between the curves of H (T) and
BCS transition at T . As T is elevated above T in zero m
c c
H (T) (Fig. 3a). As sketched, the anomalous feature
H, the pair condensate vanishes before it loses its phase c2
in the BCS scenario is that, as H →0, the vortex-liquid
rigidity. This implies that both the superfluidity (which
region vanishes.
dependsonfinitephaserigidity)andthephysicalentities
that manifest long-range phase coherence (the Cooper Thecompetingphase-disorderingscenariohasaqualti-
pairs) vanish at the same temperature T . (This is akin tatively different phase diagram (Fig. 3b). The vortex
c
tohavingthelocalmomentsinanantiferromagnetvanish liquid state now occupies a much larger fraction of the
4
4.0 for H (T) implies that the vortex-liquid state extends
c2
YBCO (Tc =92.0K) above Tc to TMF, and survives to H ∼Hc2.
3.5 80K A striking empirical fact in the hole-doped cuprates is
that experiments (over a 25-year period) have never ob-
82 servedanHc2 curvethatdecreaseslinearlywitht to ter-
3.0 minate at T (not counting early flux-flow resistivity ex-
c
periments that mis-identified H for H ). The absence
84 m c2
of the H curve is incompatible with the Gaussian pic-
2.5 c2
ture, but anticipated in the phase-disordering scenario.
)
K We now turn to results in OPT YBCO. Figure 3c dis-
V/ 2.0 86 playsthecontourplotsoftheNernstsignaley inanother
OPT YBCO crystal with identical T and closely com-
m( 78 c
y 1.5 76 parable quality (adapted from Ref. [11]). The Nernst
e
74 signalisdefinedase =E /|∇T|,whereE isthetrans-
88 y y y
verse electric field produced by the velocity of vortices
1.0
diffusing in the applied thermal gradient −∇T (see Ref.
[2]). In the vortex solid (H < H (T)), e is rigorously
m y
0.5 90 Hc2? zero. Just above Hm(T), ey rises very steeply reflecting
92 the steep increase in vortex velocity in the liquid state.
94 Incomparisonwithsimilarcontourplotsforunderdoped
0.0 cuprates(aswellasOPTBi2212),theinterval[T ,T ]
c MF
0 2 4 6 8 10 12 14 in OPT YBCO is smaller. Also, the curve H (T) rises
m
(cid:13) H (T) from Tc with a much steeper slope. Nevertheless, the
contour features are qualitatively similar. The contours
0
m
are very nearly vertical at T , implying that e hardly
c y
FIG.4: (coloronline)PlotsoftheNernstsignaley =Ey/|∇T| changes with H. Even exactly at Tc, there is no field
vs. H inOPTYBCOwithTc =92KatselectedT (Ey isthe scale above which the system can be described as being
Nernst E-field and −∇T the applied temperature gradient). in the “normal state” with |Ψ| = 0. In cuprates, we
At the melting field Hm, ey rises nearly vertically reflecting find it helpful to regard the segment of the H = 0 axis
the sharp increase in vortex velocity. The contour map (Fig. between T and T as the continuation of the H (T)
c MF m
3c) was derivedfrom these curves(andadditional curvesnot curve. The vortex liquid, confined between the curves
plotted). The “up” arrows denote the values for Hc2 at 88 of H and H below T , expands to occupy the entire
m c2 c
and 90 K predicted in Ref. [7]. (Adapatedfrom Ref. [10].)
region below H above T . It plays a dominant role in
c2 c
thermodynamic measurements.
InRef. [7],theH lineispredictedtoincreaselinearly
c2
region in which |Ψ|6=0. Significantly, the melting curve with t from the point (T = T ,H = 0) with a slope of
c
H (T)interceptstheH =0axisatatemperatureT (0) -3 T/K. We have drawn their H line as a dashed line
m m c2
lower than the termination point T of the H curve, (labelled as Hc2?) in Fig. 3c. Clearly,the predicted H
MF c2 c2
as sketched in Panel (b). If we increase T along the line cuts across the contours in an arbitrary way that
H = 0 axis, phase rigidity is lost at T (0) long before bears no resemblance to experiment.
m
we reach TMF. Since both the Meissner effect and the To explain this better, we show in Fig. 4 the profiles
zero-resistivity state are crucially dependent on having of e vs. H at selected fixed T near T [10]. Just below
y c
long-rangephase stiffness, the Tc commonly observedby Tc (curves at 90 and 88 K), ey rises nearly vertically
these techniques is identified with Tm(0). when H exceeds Hm, reflecting the sharpincrease in the
AboveT ,wehaveavortexliquidwhoseexistencemay vortex velocity in the vortex liquid in response to the
c
be detected using the Nernst effect and torque magne- gradient −∇T . The magnitude of ey, which remains
tometry (resistivity is ineffectual). The onset tempera- large up to 14 T, implies that |Ψ| remains finite. There
tures T for the vortex Nernst and diamagnetic sig- is no experimental feature (change in slope, e.g.) that
onset
nals [2, 5] are lower bounds for TMF. Clearly, if the signals |Ψ|→0 at the values of Hc2 predicted by Rey et
interval [T ,T ] is large (underdoped cuprates), the al. (shown as “up” arrows).
c MF
curve of H (T) is nearly T independent in the inter- The same difficulty exists for the plots of M vs. H in
c2 d
val 0 > T > T ; it attains closure only at T . (The Fig. 1, where the predicted H values are indicated by
c MF c2
T-independentparthas been established inunderdoped, the arrows. Contraryto the claim in Ref. [7], the curves
single-layer Bi 2201 where H (∼50 T) is nearly acces- of M vary smoothly through the predicted field values
c2 d
sible with a 45-T magnet [5].) Hence, if we focus on at 89, 90 and 91 K. Again, there is no feature reflect-
experiments close to T , the only curve that intersects ing either a transition or crossover. Given that |M | is a
c d
the H = 0 axis is the melting curve. Even at T , H is measureofthe supercurrentdensity J surroundingeach
c c2 s
inaccessible except for Bi 2201. The absence of closure vortexinthesample(eveninthe vortexliquidstate),we
5

0 0
100 105 110 120 K
0
B =14 T
97.5
95 -1
m) -5
A/T 94 Tm)-10 5 0
H ( 93 H (A/ 6 T 1
0 92 0
M/d-10 M/d -2
91 2 T
92.5
-20
90 MF
92.3 1 T
-15
89 K
-3
0.5 T
0.25 T
Tc = 92.5 K
-20
-30
0 2 4 6 8 10 12 14
90 92 94 96 98 100
0H (T)
T (K)
FIG. 5: (color online) Plots of the susceptibility χ = M/H
FIG.6: (coloronline)TheT dependenceofthesusceptibility
dvsa.shµed0HhoirnizoOnPtaTllYinBesCiOndaictatseelethcteeldinTear(-Trces=pon9s2e.5reKgi)o.nTfhoer inOPTYBCOforselectedvaluesofH. AboveTc(92.5K),χ
is H independent below 2 T (linear response region). Below
curvesnear Tc. Tc,χbecomesstronglyH dependentatlowH becauseofthe
fieldsuppression ofphaselockinginthevicinityofHcusp. At
large H (curves at 6 and 14 T), χ varies smoothly through
would expect |M | to be considerably larger when |Ψ| is
d Tc. The dashed line is the mean-field (MF) fit to Eq. 2 with
finite (H < Hc2) compared with the Gaussian fluctua- η=3.78×10−7 and BLD =0.11.
tion regime above H . Instead, |M | hardly varies as H
c2 d
crossesthepredictedvalues. Oneconfrontstheveryawk-
ward problem of explaining why |M | retains nearly the H.
d
same value below and above “Hc2”. Even worse, close Moving to the fluctuation regime above Tc, we show
to T (curve 91 K), |M | actually increases when H ex- in Fig. 6 the plots of χ vs. T at selected values of H.
c d
ceeds 3 T (this reflects the decoupling of the bilayers as Above Tc, all values of χ below 2 T collapse to a single
discussed in Sec. II). curve, which defines the linear-response fluctuation sus-
ceptibilitythatcanbedirectlycomparedwithmean-field
theories [1, 8, 9]. For illustration, we show as a dashed
IV. FLUCTUATIONS ABOVE Tc curvethebestfittothelinear-responseMFexpression[8]
η
χ(T)=− , (2)
We comment on the fits above T reported in Ref. [7].
c p[ǫ2+ǫB ]
LD
Figure 5 shows plots of the susceptibility χ = M/H.
Above 100 K, χ is H-independent over a large field withηanadjustablenumericalfactorandǫ=(T−T )/T
c c
region. However, as T falls below 100 K, the linear- (the dashed curve has η = 3.78× 10−7). For the LD
responseis confinedto progressivelysmallerfieldranges, parameter, we used the value proposed by Rey et al. [7]
as mentioned. Nonetheless, as T → T+, the linear- (B = 0.11).
c LD
response value of χ (dashed lines in Fig. 5) does not While the fit can account for the overallmagnitude of
diverge. Instead, the appearance of the Meissner effect χ above 100 K with reasonable parameters,as shown by
(and its subsequent suppression by H) occur at much Reyet al.,thefunctionalforminEq. 2doesnotdescribe
lower field scales as discussed. Below T , the hallmark the data trend all that well. Between 94 and 98 K, it
c
non-monotonicityofM vs. H describedaboveisharder underestimates|χ|byasmuchas15%,andoverestimates
d
to see in plots of χ vs. H because of the strong field itsvalueat92.5K.Asimilarpatternofovershootingand
variation caused by dividing a relatively flat profile by undershooting is also evident in Fig. 1(a) of Rey et al..
6
At large H (curves at 6 and 14 T), χ varies smoothly T . As a result, the corner-stone feature of the Gaussian
c
through T without detectable change in slope. This re- approach, namely an H line that terminates at T , is
c c2 c
flectsthecontinuityofthediamagneticresponseinstrong absent in OPT YBCO (and other hole-doped cuprates).
fieldsdescribedabove. Significantly,atlowH,thelinear- This contradicts a prediction of Ref. [7].
response segment of χ suddenly becomes strongly H- In the phase-disordering picture, the d-wave pairing
dependenteveninveryweakH. Aswediscussed,thisre- gap ∆ must persist high above T . Spectroscopic exper-
c
flects the non-monotonicity that appears above the field iments are increasingly able to distinguish between the
Hcusp. These features cannot be described by the Gaus- gap in the vicinity of the node from the much larger
sian approach of Rey et al. [7]. antinodal gap. The persistence of the nodal ∆ above
T has now been reported in scanning tunneling mi-
c
croscopy [12] and photoemission experiments [13, 14] on
V. SUMMARY
OPT Bi 2212, and in c-axis infrared reflectivity exper-
iments on YBCO [15]. The vanishing of the Meissner
In attempting to understand the fluctuation signals of effect above T reflects the collapse of the inter-bilayer
c
the magnetization in OPT YBCO, it is essential to view phase-coupling. Since the intra-bilayerphase coupling is
the results above and below T . We show that the non- much stronger, one might expect that the concomitant
c
monotonic curves of M vs. H just below T are con- phase rigidity can be observed experimentally above T .
d c c
sistent with the loss of long-range phase coherence in a Recently, this was detected by Dubroka et al. [16] as a
layeredsuperconductorwithextremelylargepair-binding Josephson plasma resonance that persists above T over
c
energy within each layer, but a c-axis coupling (between a broad doping range in YBCO (as high as 180 K in the
bilayers) that is suppressed by a few Teslas close to T . underdoped regime).
c
These features are incompatible with the Gaussian pic- We have benefitted from valuable discussions with
ture. In addition, when discussing the thermodynamics Zlatko Tesanovic, Oskar Vafek, Steve Kivelson, Ashvin
of hole-doped cuprates, we argue that it is vital to rec- Vishwanath,SriRaghu,AliYazdani,ChristianBernhard
ognize the elephant in the room, namely the vortex liq- and P. W. Anderson. Support from the U.S. National
uid above and below Tc. The dominant presence of the Science Foundation under MRSEC grant DMR 0819860
vortex liquid alters qualitatively the profiles of the Hc2 is gratefully acknowledged.
curve, pushing its closure to a temperature higher than
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