The magnetic resonance in high-temperature superconductors: Evidence for an extended s-wave pairing symmetrysymmetry

0.18 MB·English

by  Guo-meng Zhao

The magnetic resonance in high-temperature superconductors: Evidence for an
extended s-wave pairing symmetry
Guo-meng Zhao∗
Department of Physics and Astronomy, California State University at Los Angeles, Los Angeles, CA 90032, USA
4
0 Wehaveidentified several important features in the neutron scattering data of cuprates, which are
0
difficult to be explained in terms of d-wave and isotropic s-wave order parameters. Alternatively,
2
we show that theneutron data are in quantitative agreement with an order parameter that has an

n extendeds-wave(A1g) symmetryand oppositesign in thebondingand antibondingelectron bands
a formedwithintheCu2O4 bilayers. Theextendeds-wavehaseightlinenodesandchangesignswhen
J
anodeiscrossed. ThisA1g pairingsymmetrymaybecompatiblewithachargefluctuationmediated
8 pairing mechanism.

] The microscopic pairing mechanism responsible for
l a
e high-temperaturesuperconductivityincopper-basedper- 800 YBa2Cu3O6.92
r- ovskite oxides is still a subject of intense debate Tc=91 K Er=41 meV
st despite tremendous experimental and theoretical ef- V) 600 Q( , )
e
at. faorrotusndfortheovreorle15of yaenatrisfe.rromThagenedteibcastpeinhaflsuccteunatteiorends 2/B T=100 K
m in high-temperature superconductivity and the sym- ( 400 T= 5 K
- metry of superconducting condensate (order parame- m
d I
ter). Extensive inelastic neutron scattering experi-
n 200
o ments have accumulated a great deal of important data
E =32 meV
c that should be sufficient to address these central is- g
[ 0
sues. Of particular interest is the magnetic resonance
8 peak that has been observed in double-layer cuprate su- 0 10 20 30 40 50 60
v perconductors such as YBa2Cu3Oy (YBCO) [1–6] and E (meV)
6
Bi2Sr2CaCu2O8+y (BSCCO) [7,8], and in a single-layer
56 compound Tl2Ba2CuO6+y (Tl-2201) [9]. A number of b YBa2Cu3O6.92
2 theoreticalmodels[10–13]havebeenproposedtoexplain 7)1000 Q=(1/2,1/2,q) T= 5K
0 themagneticresonancepeakintermsofd-wavemagnetic 2e E=40 meV
3 pairingmechanisms. Thesetheoriescanqualitativelyex- n=
0 plain some features of neutron data but are particularly (M 750
at/ difficult to account for an important feature: The mag- sity
netic resonance in optimally and overdopeddouble-layer n
m e
cuprates is much more pronounced in the odd channel nt 500
d- than in the even channel [2–4]. In order to overcome n I
o
n this difficulty, Mazin and co-worker[14] proposed an or- utr
o der parameter that has isotropic s-wave symmetry and Ne 250
c
opposite sign in the bonding and antibonding electron
: 2 4 6 8 10
v bands formed within the Cu2O4 bilayers [14]. However, q (r.l.u)
i
X this model predicts that the resonance energy is larger
r than twice the magnitude of the superconducting gap FIG. 1. a) The imaginary part of dynamic spin suscepti-
a along the Cu-O bonding direction, in disagreement with bility as a function of excitation energy for YBa2Cu3O6.92
experiment. Further, the nodeless s-wave gap symmetry in the normal and superconducting states. The figure was
isinconsistentwiththe measurementsofthe penetration reproduced from Ref. [5]. b) ql-scan at E= 40 meV in
depth,thermalconductivityandspecificheat,whichcon- YBa2Cu3O6.92 displaying a modulation typical of odd ex-
citation. The figure was reproduced from Ref. [4]. The
sistently suggest the existence of line nodes in the gap
background(lower lineand open squares) was obtained from
function of hole-doped cuprates [15,16].
q-scans across the magnetic line. The upper solid curve is a
Hereweidentifyseveralimportantfeaturesintheneu- fit to a+bF2(Q~)sin2(πzq), where F(Q~) is the Cu magnetic
l
tron scattering data of cuprates. These features are in-
form factor. The modulation is not complete as even excita-
consistent with d-wave order parameter (OP). Alterna-
tions are sizable at q= 3.5, 7 but with a magnitude 5 times
l
tively,we showthat the neutrondataareinquantitative smaller [4].
agreementwithanorderparameterthathasanextended
s-wave (A1g) symmetry [17,18] and opposite sign in the bonding and antibonding electron bands formed within
the Cu2O4 bilayers [14].
1
The magnetic resonance peak observed in opti-
mally doped and overdoped double-layer compounds s) 50 a Tc=92.5 K
r Tl Ba CuO
YBa2Cu3Oy [1–5]andBi2Sr2CaCu2O8+y [7,8]is asharp ou 2 2 6+y
collective mode that occurs at an energy of 38-43 meV h 0
and at the two-dimensional wavevector Q~AF = (π/a, r 2 I(27K)-I(99K)
e
π/a), where a is the nearest-neighbor Cu-Cu distance. p -50
Fig. 1a shows the imaginary part of the odd chan- nts
nel spin susceptibility as a function of excitation en- u-100
o
ergy for a slightly underdoped double-layer compound c
(
YBa2Cu3O6.92. The figure is reproduced from Ref. [5]. sity -150
There are several striking features in the data: (a) The n
resonance peak at Er= 41 meV appears below Tc and nte-200 Eg =38 meV
this resonance peak intensity in the superconducting I
state IS (E ) is larger than the normal-state intensity 30 35 40 45 50 55 60
odd r
IN (E ) by a factor of about 3.6, i.e., IS (E )/IN (E )
odd r odd r odd r E (meV)
=3.6;(b)AspingapfeatureisseenbelowT andthereis
c
a small shoulder that occurs at an energy slightly above
250
thespingapenergyEg =32meV.Fig.1bshowsaql-scan b
Bi Sr CaCu O
atE=40meVinYBa2Cu3O6.92displayingamodulation 200 2 2 2 8+y
)
typical of odd excitation. The figure is reproduced from rs Tc=91 K
u
Ref. [4]. From Fig. 1b, we find feature (c): The odd- o150
h
channel magnetic resonance intensity within the Cu2O4
2
bilayersislargerthantheeven-channelonebyafactorof 100
r
about5(Ref.[4]),i.e.,IoSdd(Er)/IeSven(Er)=5. Fromthe pe 50
neutronstudiesondifferentdouble-layercompoundswith s
t
n
differentdopinglevels,oneidentifiesfeature(d): Theres- u 0
o
onance energy E does not increase with doping in the
r c
overdopedrangebut is proportionalto Tc asEr/kBTc ≃ y ( -50
5.2 in both underdoped and overdoped ranges [8]. sit I(10K)-I(100K)
On the other hand, the neutron data for a single-layer en-100 Q=(0.5,0.5,-14)
Tl-2201 are different from those for double-layer com- nt
I-150
pounds. For the single-layerTl-2201,we show in Fig. 2a
the difference spectrumofthe neutronintensitiesatT =
-200
27K(<Tc)andT =99K(>Tc),atawavevectorofQ~ = 20 30 40 50 60 70
(0.5, 0.5, 12.25). The figure is reproduced from Ref. [9]. E (meV)
Although a sharp peak feature is clearly seen at E =
r
46 meV, it is remarkable that the peak intensity in the FIG. 2. a) The difference spectrum of the neutron inten-
superconducting state is close to the magnetic neutron sities of single-layer Tl-2201 crystals (with a total volume of
intensity in the normal state, that is, IS(Er)/IN(Er)≃ 0.11 cm3) at T =27 K (<Tc) and T = 99 K (>Tc), and at
1. This is in sharpcontrastwith the aboveresult for the wavevector Q~ = (0.5, 0.5, 12.25). The figure is reproduced
from Ref. [9]. The solid line is guide to the eye. The differ-
double-layer compound YBa2Cu3O6.92 where we found
IS (E )/IN (E ) = 3.6. Thus we identify feature (e) as encespectrumtendstozeroatabout56meV(about10meV
odd r odd r higher than the resonance energy). This suggests that the
IS(E )/IN(E ) ≃ 1 for the single-layer compound.
r r nonmagnetic background at 56 meV has negligible tempera-
Wewouldliketomentionthatfeature(e)identifiedfor
ture dependencebelow 100 K. b) The difference spectrum of
the single-layer Tl-2201 is valid only if the nonmagnetic
the neutron intensities of optimally doped BSCCO crystals
backgroundhasanegligibletemperaturedependencebe- (withatotalvolumeof0.06cm3)atT =10K(<Tc)andT =
low 100 K. It is known that nuclear contributions pre- 100 K (>Tc),and at wavevectorQ~ = (0.5, 0.5, -14). Thefig-
dominantly constitute the main part of the neutron sig- ureisreproducedfromRef.[7]. Thedifferencespectrumgoes
nal [3]. Any nuclear contributions (nonmagnetic back- to zero at 60 meV (about 16 meV higher than theresonance
grounds) are only expected to obey the following stan- energy). This suggests that the nonmagnetic background at
dardtemperaturedependence,1/[1−exp(−E/k T)](see 60meVhasnegligible temperaturedependencebelow100 K.
B
detaileddiscussionsinRef.[3]). Intheenergyrangeofin-
E =46meVinTl-2201,thefactor1/[1−exp(−E/k T)]
terest(E>40meV),thistemperaturedependencefactor r B
decreasesby 0.4%whenthe temperature isloweredfrom
is close to unity and nearly independent of temperature
99 K to 27 K. If we take the nonmagnetic background
forT <150K.Indeed,the normal-stateneutronintensi-
of 2500 counts per two hours [9], the nonmagnetic back-
tiesofbothYBCO[3]andBSCCO[7,8]showanegligible
ground decreases by 10 counts per two hours when the
temperature dependence for T < T < 150 K. For E =
c
2
temperature goes from 99 K to 27 K, and by 60 counts resonance energy is independent of temperature below
per two hours when the temperature goes from150 K to 100K(seeFig.1a),thenearlyzerovalueofthedifference
99K.ThevariationofthenonmagneticbackgroundatE spectrum at50 meV implies thatthe nonmagnetic back-
=46meVbelow100Kisnegligiblysmallcomparedwith groundat50meVhasnegligibletemperaturedependence
the magnetic resonance intensity (∼150 counts per two below 100 K in the case of YBCO-Ca. Even in the case
hours). Therefore, feature (e) identified for the single- ofTl-2201(seeFig.2a),thedifferencespectrumtendsto
layer Tl-2201 is well justified. zeroatabout56meV(about10meVhigherthantheres-
In order to further justify feature (e), we compare the onance energy) although one needs more data points in
ratios of the magnetic to the nonmagnetic background the vicinity of 56 meV to definitively address this issue.
signals in those neutron experiments on different com- This indicates that the nonmagnetic background at 56
pounds. From the neutron data, we find that the signal- meV has negligible temperature dependence below 100
to-background ratio is about 6% for Tl-2201 (Ref. [9]), K in the case of Tl-2201.
about 12% for an optimally doped BSCCO [7], about We would like to mention that the normal-state mag-
6% for an overdoped BSCCO [8], and about 100% for netic intensities in Fig. 4 of Ref. [9] are significantly un-
anoverdopedYBCO[3]. The signal-background-ratioin derestimated. This is because the authors assume that
theoverdopedBSCCOissimilartothatforTl-2201. But the q-width of the magnetic peak in the normal state is
the overdopedBSCCO shows a difference spectrum sim- the same as that in the superconducting state [9]. This
ilar to that for the overdoped YBCO where the signal- assumptionis unphysical. FromFig.19dandFig.19hof
background-ratio is larger than that for the overdoped Ref. [6], one can clearlysee that the q-width of the mag-
BSCCO by a factor of 20. This suggests that, if fea- netic peak in the normal state is a factor of 1.6 larger
ture (e) identified for Tl-2201were an artifact caused by thanthatin the superconductingstate in the caseofun-
a small signal-background-ratio,one would not have ob- derdoped YBa2Cu3O6.8. Further, the q-width increases
served the intrinsic difference spectra for the overdoped with the increase of doping. With a much broader mag-
BSCCO.Thefactthatthedifferencespectrafortheover- neticpeakinthenormalstate,theqrange(0.35rlu-0.65
doped BSCCO are similar to the one for YBCO sug- rlu)for the normal-stateq-scanspectrum(see Fig. 2Bof
gests that one indeed finds the intrinsic magnetic differ- Ref.[9])istoo narrowto getmeaningfulestimatesofthe
ence spectra for the overdoped BSCCO even though the nonmagneticbackgroundandthe normal-statemagnetic
signal-background-ratioisonly6%. Therearenoreasons intensity. Moreover, the normal-state q-scan spectrum
to believe that only the difference spectrum for Tl-2201 of YBa2Cu3O6.8 is nearly featureless for the same nar-
is an artifact. Therefore, the pronounced difference be- row q-range (0.35 rlu - 0.65 rlu), similar to the data of
tweenthe difference spectrumofthe single-layerTl-2201 Fig. 2B of Ref. [9]. If one would fit the normal-state q-
and that of the double-layer BSCCO (compare Fig. 2a scandataonlyinthisnarrowq-range(0.35rlu-0.65rlu)
and Fig. 2b) is due to the fact that IoSdd(Er)/IoNdd(Er) for YBa2Cu3O6.8 and assume the same q-width as that
>>1for the double-layerBSCCO whileIS(E )/IN(E ) in the superconducting state, one would find that the
r r
∼ 1 for the single-layer Tl-2201. Moreover,we will show normal-state magnetic intensity is underestimated by a
below that, for the intraband scattering (even channel) factorof4. Suchasignificantunderestimatemaybealso
in YBa2Cu3O6.92, IeSven(Er)/IeNven(Er) = 0.72, in good trueforthenormal-statemagneticintensitiesofTl-2201.
agreement with feature (e) for the single-layer Tl-2201. The authors of Ref. [9] should have extended their mea-
This consistency gives further support to the thesis that surements to a wider q range to get reliable estimates of
feature (e) is intrinsic. the nonmagnetic backgroundand the normal-state mag-
One may also argue that the glue that glues about netic intensities.
300 small crystals of Tl-2201 on Al-plates would cause Since magnetic signals in both normal and supercon-
a substantial decrease of the nonmagnetic background ductingstatesstronglydependondoping,asclearlyseen
below 100 K. If this argument were relevant, one inYBCO[4],thecomparisonoftheresonancepeakinten-
would have observed a similar effect in overdoped sities and spectral weights among different compounds
Y0.9Ca0.1Ba2Cu3O7−y (YBCO-Ca) because 60 larger should be carefully made. For this Tl-2201, there is
crystals of YBCO-Ca are also glued on Al-plates [19]. some indication of a broad peak centered around Q~
AF
Because the total volume of YBCO-Ca crystals is larger even above T (Ref. [9]), as observed in underdoped
c
than that of Tl-2201 crystals by a factor of 3.2, one can YBCO [4,6]. This suggests that the Tl-2201 is slightly
readilyshowthattheamountofglueforYBCO-Cacrys- underdoped so that the magnetic intensity in the nor-
tals is comparable with or even larger than that for Tl- malstateshouldbe closeto thatfor slightlyunderdoped
2201. Thissuggeststhattheeffectofglueonthenonmag- YBa2Cu3O6.92. This is because both compounds have
netic backgroundin Tl-2201is similar to that in YBCO- a similar ratio T /T and thus a similar doping level
c cm
Ca. Fromthe data for YBCO-Ca (Fig. 2a andFig. 3aof (where T is the superconducting transitionatoptimal
cm
Ref.[19]),onecanclearlyseethatthedifferencespectrum doping). As seen from Fig. 1a, the normal-state mag-
at 50 meV is very close to zero. Because the magnetic netic intensity at 40 meV is about 100 µ2/eV per Cu in
B
signalatanenergythatisabout10meVhigherthanthe slightlyunderdopedYBa2Cu3O6.92. Theresonancepeak
3
intensity for the Tl-2201 can be estimated to be about the superconducting gap edge above the Fermi level.
120 µ2/eV per Cu from the resonance spectral weight Within the itinerant magnetism model, neutron scat-
B
of 0.7µ2 per Cu (Ref. [9]). Therefore, the resonance tering intensity at a wavevector ~q and an energy E is
B
peak intensity for the Tl-2201 is close to the normal- proportional to the imaginary part of the dynamic elec-
state magnetic intensity for the slightly underdoped tronspinsusceptibility,χ′′(~q,E). Qualitatively,withthe
YBa2Cu3O6.92. Since the normal-state magnetic inten- neglect of the Bardeen-Cooper-Schrieffer (BCS) coher-
sity for Tl-2201 should be similar to that for the slightly ence factor, the bare imaginary part of spin susceptibil-
underdoped YBa2Cu3O6.92, then IS(Er)/IN(Er) ≃ 1.2 ity,χ′◦′(~q,E),isproportionaltothejointdensityofstates
for this single-layer Tl-2201. This is in good agreement A(~q,E)= δ(E−E −E ),whereE = ǫ2 +∆2
with feature (e) deduced independently from the differ- P~k ~k+q~ ~k ~k q ~k ~k
is the quasiparticle dispersion law below T , ǫ is the
ence spectrum (Fig. 2a). Moreover,we find that the res- c ~k
electronic band dispersion, and ∆ is the order parame-
onance peak intensity for the Tl-2201 (∼120 µ2/eV per ~k
Cu)isafactorofaboutthreesmallerthanthe rBesonance ter [14]. The two particle energy E2(~k,~q) = E~k+q~+E~k
has a minimum and several saddle points for a fixed ~q.
peakintensityfortheslightlyunderdopedYBa2Cu3O6.92
(∼330 µ2/eV per Cu). Similarly, the resonance spectral The minimum defines the threshold energy, or spin gap
B energy E , which is achieved at vector~k and~k+~q such
weight for the Tl-2201 is also a factor of three smaller g
than that for the slightly underdoped YBa2Cu3O6.92. that both ~k and ~k+~q belong to the Fermi surface and
These important features we have identified above thus Eg =2∆~k (Ref. [14]).
should place strong constraints on theories for high- The joint density of states has divergences at the ex-
temperature superconductivity in cuprates. Any correct tendedsaddlepoints[14]. Thedivergentpeakinχ′◦′(~q,ω)
theories should be able to explain all the magnetic reso- occursbecauseoftransitionsbetweentheoccupiedstates
nance features in a consistent and quantitative way. In located in the extended saddle points below EF and
a more exotic approach [10], the neutron data are inter- empty quasiparticle states at the superconducting gap
preted in terms of a collective mode in the spin-triplet edgeaboveEF. ThesaddlepointsproduceVanHovesin-
particle-particle channel, which couples to the particle- gularities in the quasiparticle density of states in the su-
hole channel in the superconducting state with d-wave perconducting state at an energy of (ǫVH)2+∆2 and
q ~k ~k
OP. This model predicts that the resonance peak energy the superconducting condensate creates a sharp coher-
Er is proportionalto the doping levelp, in disagreement ence peak in the quasiparticle density of states at the
with feature (d): Er does not increase with increasing gap edge. Thus, the divergence in the joint density of
p in the overdoped range but is proportional to Tc as statesinthesuperconductingstateisthenlocatedatthe
E /k T ≃ 5.2 (Ref. [8]). Moreover, this model pre-
r B c energyE∗ =∆ + (ǫVH)2+∆2. Thissimpleexpres-
dicts [20] that E > 2∆ (where ∆ is the maximum ~k+q~ q ~k ~k
r M M
sionforE∗ hasbeenverifiedbynumericalcalculationsin
d-wave gap), which contradicts experiment. Other the-
the case of an isotropic s-wave order parameter [14].
ories based on spin-fermion interactions also show that
InFig.3,weplottheFermisurfaceforaslightlyunder-
E increases monotonically with increasing p [11,12], in
r
doped BSCCO, which is inferred by extending a portion
disagreement with feature (d).
of the Fermi surface determined by angle-resolved pho-
Alternatively, feature (d) is consistent with a simple
toemission spectroscopy (ARPES) [22]. There are ex-
particle-hole excitation across the superconducting gap
tended saddle points that are located near (±π, 0) and
within an itinerant magnetism model. This is because
(0, ±π), as shown by ARPES [23]. A solid line in Fig. 3
the particle-hole excitationenergy increaseswith the su-
represents a segment of the extended saddle points very
perconducting gap which in turn should be proportional
close to the Fermi surface. Other portions of the saddle
toT atleastintheoverdopedrange. Thisitinerantmag-
c
points[e.g.,alongthelinefrom(0,0)to(π,0)]aresignif-
netism model is also supported by very recent Fourier
icantlyawayfromtheFermisurface[23]andnotshownin
transform scanning tunnelling spectroscopic (FT-STS)
thefigure. Ifweonlyconsiderthemagneticexcitationsat
studies on a nearly optimally doped BSCCO [21], which
afixedwavevectorthatcorrespondsto the antiferromag-
showthatthequasiparticlesinthesuperconductingstate
netic wavevectorQ~ , only four electron wavevectorsat
exhibit particle-hole mixing similar to that of conven- AF
the Fermi surface are connected by Q~ as indicated by
tional Fermi-liquid superconductors. These FT-STS re- AF
arrow1inFig.3. Eachofthesevectorsformsanangleof
sultsthusprovideevidenceforaFermi-liquidbehaviorin
θ withrespecttotheCu-Obondingdirection. Theelec-
the superconducting state of optimally doped and over- r
tron transitions from the occupied states located at the
doped cuprates. Here we quantitatively explain allthese
extended saddle points below E to empty quasiparticle
neutron data [1–9] in terms of an order parameter (OP) F
states atthe superconducting gapedge aboveE arein-
thathasanextendeds-wavesymmetry[17,18]andoppo- F
dicated by arrow 2. Because the quasiparticle densities
site sign in the bonding and antibonding electron bands
ofstatesatthe gapedgeandthe saddlepoints arediver-
formedwithintheCu2O4 bilayers[14]. Inourmodel,the
gentatzerotemperature,suchtransitionswillproducea
neutronresonancepeak is due to excitations ofelectrons
fromtheextendedsaddlepointsbelowthe Fermilevelto
4
sharpresonancepeakatE =∆ + (ǫVH)2+∆2 sign so that the BCS coherence factor ≃ 1 and thus
r ~k+Q~AF q ~k ~k IS(E )/IN(E )>> 1 (see Ref. [13]). Therefore, the ex-
(Ref. [14]). r r
perimental observation of IS(E )/IN(E ) ∼ 1.0 in the
r r
single-layer Tl-2201 [9] rules out the d-wave order pa-
rameter. Alternatively,for a single-layercompound with
an s-wave symmetry, ∆ and ∆ have the same
~k+Q~AF ~k
sign, so the BCS coherence factor could be much less
than 1. This may lead to IS(E )/IN(E )∼ 1, in agree-
q r r
r ment with feature (e). Hence, only the intralayer (intra-
band) s-wave symmetry is compatible with feature (e):
(0, p ) IS(E )/IN(E ) ∼ 1.0. On the other hand, if extended
r r
Q saddle points are significantly below the Fermi level, the
AF
BCS coherencefactor will be substantialeven for the in-
1 tralayer (intraband) magnetic scattering in the case of
2 s-wave gap symmetry. Then IS(E )/IN(E ) could be
r r
much larger than 1 in this special case. Therefore, the
observation of IS(E )/IN(E ) >> 1 for the intralayer
r r
Saddle points
(intraband) magnetic scattering is consistent with either
(0, 0) (p , 0) s-wave or d-wave gap symmetry.
For a double-layer compound, interactions within
FIG. 3. The Fermi surface for a slightly underdoped
BSCCO with Tc = 88 K. This Fermi-surface is extrapolated Cu2O4 bilayers yield bonding and antibonding bands.
Transitions between electronic states of the same type
from a part of the Fermi surface determined by ARPES [22]
using symmetry arguments. Arrow 1 indicates electron tran- (bonding-to-bondingorantibonding-to-antibonding)and
sitions from the occupied states in the superconducting gap those of opposite types are characterizedby even or odd
edge below EF to emptyquasiparticle states at thegap edge symmetry, respectively, under exchange of two adjacent
aboveEF. Arrow2markselectrontransitionsfromtheoccu- CuO2 layers. As a result, the magnetic excitations be-
piedstateslocatedintheextendedsaddlepointsbelowEF to tween different bands correspond to odd channel excita-
empty quasiparticle states in the superconducting gap edge tions while the excitations within the same bandto even
above EF. channel excitations [4]. If the order parameters in the
bonding and antibonding bands have the same sign, the
In terms of θ , both E (Q~ ) and E (Q~ ) can be
r g AF r AF magnetic resonance intensities in both channels should
rewritten as
be similar for an intraband pairing symmetry of either
E (Q~ )=2∆(θ ) (1) s-wave or d-wave. This is because the BCS coherence
g AF r
factors for both channels are the same in this case. On
and theotherhand,iftheorderparameterhasans-wavesym-
metry andopposite signinthe bonding and antibonding
Er(Q~AF)=∆(θr)+q[∆(θr)]2+(ǫVrH)2. (2) electron bands, the magnetic resonance intensity in the
odd channel will be much larger than that in the even
Here [∆(θ )]2+(ǫVH)2 is the energy of a saddle point channel due to the large difference in their BCS coher-
belowpthe Ferrmileverlalongthe θ direction. One should ence factors. Therefore, only if the order parameter has
r
note that Eq.2 is validonly if the saddle points are very an s-wave symmetry and opposite sign in the bonding
close to the Fermi surface, as is the case. andantibondingelectronbands,cantheobservedfeature
We now consider the BCS coherence factor that has (c): IoSdd(Er)/IeSven(Er) = 5 for the slightly underdoped
been ignored in the above discussions. The BCS coher- YBCO and IoSdd(Er)/IeSven(Er) >10 for the overdoped
ence factor is given by [24]: YBCO [3] be explained within the itinerant magnetism
approach.
ξ(~q,~k)= 1(1− ǫ~kǫ~k+q~+∆~k+q~∆~k). (3) Basedonthe t-J model andd-wavepairing symmetry,
2 E E Brinckmannand Lee [25]have recently attempted to ac-
~k ~k+q~
count for feature (c) using a very unrealistic parameter:
For ǫ << ∆ and ǫ << ∆ , the BCS coherence J /J =0.6(whereJ andJ aretheeffectiveintralayer
~k ~k ~k+q~ ~k+q~ ⊥ k k ⊥
factor is close to 1 when ∆ and ∆ have opposite and interlayer antiferromagnetic exchange energies, re-
~k+q~ ~k
sign, but is close to zero when ∆ and ∆ have the spectively). For undoped YBCO, neutron experiments
~k+q~ ~k
same sign. The coherence factor is not negligible for ǫ [4] show that J /J = 0.1, which is far less than 0.6
~k ⊥ k
≃ ∆ and ǫ = 0 even if ∆ and ∆ have the same used in the calculation [25]. Further, the value of J /J
~k ~k+q~ ~k+q~ ~k ⊥ k
sign. will be negligible if J remains substantial and the opti-
k
For a single-layer compound with d-wave order pa- cal magnon gap (∝ J J ) goes to zero, which should
k ⊥
p
rameter symmetry, ∆ and ∆ have opposite
~k+Q~AF ~k
5
be the case for optimally doped and overdoped YBCO. T =89K(Ref.[28]),and2∆ /k T =5.37foranover-
c M B c
Neutron data for underdoped YBCO [26] show that the dopedBSCCOwithT =80K(Ref.[29]). Itisapparent
c
optical magnon gap is about 50 meV for YBa2Cu3O6.5 that2∆M/kBTc decreasesalmostlinearlywithTc inthe
and is reduced to about 25 meV for YBa2Cu3O6.7. If overdoped range. Using the fitted curve of 2∆M/kBTc
we linearly extrapolate the optical magnon gap with the versusT , we estimate 2∆ /k T = 6.04and 2∆(θ ) =
c M B c r
oxygencontent,the gapwilltendto zeroin YBa2Cu3Oy 38.6 meV for Y0.9Ca0.1Ba2Cu3O7−y with Tc = 85.5 K.
for y > 0.9, implying that J /J << 0.1 for optimally Similarlywecanobtain2∆ /k T =5.66and2∆(θ )=
⊥ k M B c r
doped and overdoped cuprates. 35.1meVforoverdopedBSCCOwithT =83K.INSex-
c
UsingamorerealisticparameterofJ⊥ andthed-wave periment on Y0.9Ca0.1Ba2Cu3O7−y (Ref. [19]) indicates
pairing symmetry, Millis and Monien [27] appear to be Eeven = 43 meV > 2∆(θ ), in contradiction with the
r r
able to explain feature (c). They showed that the reso- theoretical model [27]. For overdoped BSCCO with T
c
nance spectral weights in the odd and even channels are = 83 K, Eodd = 38 meV (Ref. [8]) and thus Eeven = 45
r r
related to the difference between the resonance energy meV by analogywith the caseof Y0.9Ca0.1Ba2Cu3O7−y.
E and the particle-hole spin excitation energy 2∆(θ ) Both Eeven and Eodd in this overdoped BSCCO are far
r r r r
at Q~AF, i.e., Wodd/Weven = [2∆(θr)−Erodd]/[2∆(θr)− largerthan2∆(θr), incontradictionwithanytheoretical
Eeven]. Here2∆(θ )mustbelargerthanbothEeven and models based on the d-wave pairing symmetry.
r r r
Erodd (Ref.[27]). The INSdataofY0.9Ca0.1Ba2Cu3O7−y For the underdoped YBa2Cu3O6.7, the even-channel
would be consistent with this theoretical model if one magnetic intensity in the normal state is a factor of 2.0-
wouldchooseanunrealisticparameter2∆(θ )=49meV 2.5 lower than the odd-channel one for E = 40 meV,
r
(Ref. [19]). As discussed below, θ is about 15◦ for which is about 15 meV above the optical magnon gap
r
slightly overdoped cuprates so that 2∆(θ ) =1.73∆ [26]. This implies that the interlayer antiferromagnetic
r M
(where ∆ is the maximum d-wave gap). The intrinsic correlation does not influence magnetic excitations well
M
tunneling spectra,whichareunsusceptible to surfacede- above the optical magnon gap [26]. Since the opti-
terioration, can provide the most reliable determination cal magnon gaps for optimally doped and overdoped
of the bulk superconducting gap [28]. From the intrinsic YBCO are much smaller than that for the underdoped
tunneling spectra, one finds that 2∆M/kBTc = 8.09 for YBa2Cu3O6.7, one should expect that IoNdd(E)/IeNven(E)
the optimallydopedBSCCOwithT =94K(Ref.[28]), ≃ 1 for E ≃ 40 meV in optimally doped and overdoped
c
2∆ /k T =6.73foraslightlyoverdopedBSCCOwith
M B c
TABLEI. Comparison of experiments with extended s-wave, d-wave, and isotopic s-wave. In both extended and isotropic
s-wavemodels,theorderparametersareassumedtohaveoppositesignsinthebondingandantibondingelectronbandsformed
within the Cu2O4 bilayers. Here DA = definitive agreement, A = agreement, QA = qualitative agreement, PA = possible
agreement, D = disagreement and DD = definitivedisagreement.
INS data extendeds-wave d-wave isotropic s-wave
Feature (a): IS(Er)/IN(Er) = 3.6 for YBCO DA DA DA
Feature (c): IoSdd(Er)/IeSven(Er) = 5 for YBCO DA DD DA
Feature (e): IS(Er)/IN(Er) <1for Tl-2201 DA DD DA
The magnitudes of Er and Eg DA DD DD
Spin gap in La2−xSrxCuO4 DA DD DD
Otherdata extendeds-wave d-wave isotropic s-wave
Penetration depth DA QA DD
Thermal conductivity DA QA DD
Specificheat DA QA DD
Nonlinear Meissner effect DA QA DD
ARPES (underdoped) A A DD
ARPES (overdoped) DA DD DD
Quasiparticle tunneling(underdoped) PA PA DD
Quasiparticle tunneling(overdoped) DA DD DD
Raman scattering (underdoped) PA PA DD
Raman scattering (overdoped) DA DD PA
NMR/NQR A A DD
Andreevreflection A PA DD
Pb/c-axis YBCO Josephson junction A D A
c-axis BSCCO twist Josephson junction DA DD PA
Corner SQUID/Josephson junction PA PA PA
Tricrystal/Tetracrystal Josephson junctions PA PA PA
6
YBCO. Thus the observed feature bulkOPsymmetrybecausethesurfaceOPmightbedif-
(c): IS (E )/IS (E ) >> 1 for optimally doped and ferent from the bulk one [17].
odd r even r
overdoped YBCO can only be explained by an OP that For a slightly overdoped YBCO, more than six inde-
has s-wave symmetry and opposite sign in the bonding pendent experiments consistently suggest that [17] the
and antibonding electron bands. gap function is ∆(θ) = 24.5(cos4θ +0.225) meV. Sub-
For the slightly underdoped YBa2Cu3O6.92, we can stituting θr = 16◦ into the gap function, we get ∆(θr)
also deduce the value of IS (E )/IN (E ) using the = 16.3 meV and thus E = 32.6 meV, in quantitative
even r even r g
measured IS (E )/IN (E ) = 3.6, IS (E )/IS (E ) agreement with the measured one (32-33 meV).
odd r odd r odd r even r
= 5, and the inferred IN (E )/IN (E ) ≃ 1 (see above Inourmodel,thepositionoftheextendedsaddlepoint
odd r even r
discussion). From the measured IS (E )/IS (E ) along the θ direction is located at E −E /2 in the su-
odd r even r r r g
= 5, we have IS (E ) = 0.2IS (E ) and thus perconducting state (see Eqs. 1 and 2). For optimally
even r odd r
IS (E )/IN (E ) ≃ 0.2IS (E )/IN (E ) = 0.72. doped YBCO with E = 41 meV and E = 32 meV, we
even r even r odd r odd r r g
This value is close to that deduced for the single-layer find that the saddle point in the superconducting state
Tl-2201 (≃ 1). These results consistently suggest that is located at an energy of 25 meV below the Fermi level.
the intraband neutron intensity at E in the supercon- This is in agreementwith the ARPES studies [30]which
r
ducting state is close to that in the normal state. This suggestthattheFermilevelinthesuperconductingstate
uniqueandimportantfeaturewehaveidentifiedrulesout for optimally doped cuprates is ≤ 30 meV above the ex-
d-wave order parameter symmetry because d-wave sym- tended saddle points that have the same energy over a
metry predicts [13,25] that IS (E )/IN (E )>>1 for large momentum space [23]. Further, electronic Raman
even r even r
bilayer compounds and IS(Er)/IN(Er)>>1 for single- scattering spectra in YBa2Cu4O8 have been used to de-
layer compounds. termine the energy of the extended saddle points more
FromEqs.1 and 2,it is easy to calculate E andE if accurately [31]. At 10 K (well below T ), the energy of
g r c
oneknowsthegapfunction∆(θ) andtheθ value. From the extended saddle points is found to be 24.3 meV be-
r
the measured Fermi surface, one can readily determine lowtheFermilevel,inexcellentagreementwiththeresult
θ . For example, we find θ = 18.4◦ for a slightly un- predicted by our model from the INS data (≃ 25 meV).
r r
derdoped BSCCO from Fig. 3. For optimally doped We have identified the unique and important feature:
cuprates,we getθ ≃16.0◦. Ifwewoulduseanisotropic IS(E )/IN(E ) ≃ 1.0 for the intralayer and intraband
r r r
s-wave gap function ∆(θ) = 28 meV for a slightly over- magnetic scattering. This feature unambiguously rules
doped YBCO, we would have E = 56 meV and E > out the d-wave OP symmetry. The unambiguous deter-
g r
56 meV, which are far larger than the measured E = minationoftheintrinsicextendeds-wavepairingsymme-
g
33 meV and E = 40 meV (Ref. [3]). If we would use try for hole-doped cuprates places strong constraints on
r
a d-wave gap function ∆(θ)= 28cos2θ meV, we would the pairing mechanism for high-temperature supercon-
have E = 47.5 meV and E > 47.5 meV, which are ductivity. Arecentcalculationsuggeststhathighenergy
g r
also far largerthan the measured values. Thus, one can- Cu-O charge fluctuations can lead to an attractive in-
not quantitatively explain the neutron data in terms of teraction between conduction electrons and the pairing
d-wave and isotropic s-wave symmetries. symmetry may be of extended s-wave (A1g) [32].
Alternatively,anextendeds-wavewitheightlinenodes Indeed, precise thermal-difference reflectance spectra
(A1g symmetry) is in quantitative agreement with two- of several cuprate superconductors (Tc =105-120 K) ex-
thirds of the experiments that were designed to test the hibitpronouncedfeaturesatphotonenergiesofabout2.0
order-parameter symmetry for hole-doped cuprates [17]. eV,whichmayberelatedtotheCu-Ochargefluctuations
The remaining one-third (e.g., tricrystal grain-boundary (Ref. [33]). These features can be well described within
Josephson junction experiments) are explained qualita- Eliashberg theory with an electron-boson coupling con-
tively by Zhao [17] and by Brandow [18]. In Table 1, stant λ of about 0.40. In order to explain a supercon-
ch
we compare nearly all the experiments used to test the ductingtransitiontemperatureof105-120K,theauthors
OP symmetry with the extended s-wave, d-wave, and simulated [33]an electron-phononcoupling feature at 50
isotropic s-wave models. In both extended s-wave and meV with the coupling constant λ of about 1.0. This
ph
isotropic s-wave models, the order parameters are as- Eliashberg model’s simulated electron-phonon coupling
sumed to have opposite sign in the bonding and anti- agrees well with the results obtained from first principle
bondingelectronbandsformedwithintheCu2O4bilayers calculations [34].
[14]. The detailed comparisons with other experiments Thecontributionofthis2eVcomponentcanbeequiv-
aremadeinRef.[17]. FromTable1,onecanseethatthe alent to a negative Coulomb pseudopotential µ∗ within
neutron data alone provide a definitive answer to the in- Eliashberg theory [35]. By using a realistic electron-
trinsic, bulk OP symmetry because INS is a bulk, phase phononcouplingspectralweightdeducedfromtunneling
andanglesensitivetechnique. Otherbulkandnon-phase spectra and a µ∗ = -0.15, and taking into a polaronic
sensitiveexperiments providecomplementarysupportto effect, Zhao et al. [35] are able to explain the negligible
the present conclusions. The phase and surface sensitive isotope effect on T , and the magnitudes of T and the
c c
experiments cannot definitively determine the intrinsic superconductinggapforoptimallydoped90Ksupercon-
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8