Magnetic Resonant excitations in High-{$\rm T_c$} superconductors

0.17 MB·

by  Y. Sidis

phys.stat.sol.(2003)
4
0 T
Magnetic resonant excitations in High- superconductors
0 c
2
Y.Sidis1,S.Pailhe`s1, B.Keimer2,P.Bourges1,C.Ulrich2,andL.P.Regnault3
n
a 1 LaboratoireLe´onBrillouin,CEA-CNRS,CE-Saclay,91191GifsurYvette,France.
J 2 Max-Planck-Institutfu¨rFertko¨rperforschung,70569Stuttgart,Germany.

0 3 CEAGrenoble,DRFMC38054Grenoblecedex9,France.
2

Receivedzzz,revisedzzz,acceptedzzz
]
n Publishedonlinezzz
o
c
PACS 74.72.-h,78.70.Nx,75.40.Gb
-
r
p TheobservationofanunusualspinresonantexcitationinthesuperconductingstateofvariousHigh-Tccop-
u peroxidesbyinelasticneutronscatteringmeasurementsisreviewed. Thismagneticmodeisdiscussedin
s lightofafewtheoreticalmodelsandlikelycorrespondstoaspin-1collectivemode.
.
t
a
m
Morethanfifteenyearsafterthehightemperaturesuperconductivitydiscovery,antiferromagnetic(AF)
- fluctuationspairingmechanism[1]isstillhighlycontroversial.However,inelasticneutronscattering(INS)
d
n measurements have successfully brought to light the existence of unusual AF excitations that develop
o belowT andcouldbethehallmarkofanunexpectedspin-1collectivemode,tightlyboundtothesuper-
c
c conducting(SC)state. Whatevertheroleofthatmodeforsuperconductivity,ithastobederivedfromthe
[
samemicroscopicmodelusedtodiscusssuperconductivity.Weherereviewitscharacteristicfeaturesina

2 fewcupratesanddiscussitspossibleorigininlightofdifferenttheoreticalmodels.
v InoptimallydopedYBa Cu O (YBCO)(T =93K)whereithasbeendiscovered[2](Fig.1.a),the
2 3 6+x c
8
spinexcitationspectrumisdominatedintheSCstatebyasharpmagneticexcitationatanenergyof∼40
2
meVandattheplanarantiferromagneticwavevectorq =(π/a,π/a),theso-calledmagneticresonance
3 AF
1 peak[2, 3,4, 5]. Itsintensitydecreaseswith increasingtemperatureandvanishessteeplyat Tc, without
0 anysignificantshiftofitscharacteristicenergyE . Intheunderdopedregime,E monotonicallydecreases
r r
4 with decreasing hole concentration [6, 7] so that E ≃ 5 k T (Fig. 2). Besides, it is possible to vary
r B c
0
T withoutchangingthe carrierconcentrationthroughimpuritysubstitutionsof Cu in the CuO planes.
/ c 2
at ThisisthecaseinYBa2(Cu1−yNiy)3O7 (y=1%,Tc=80K),wherethemagneticresonancepeakshiftsto
m lowerenergywithapreservedEr/kBTc ratio(Fig.1.b)[8].
In optimally doped Bi Sr CaCu O (BSCO) (T =91 K), a similar magnetic resonance peak has
- 2 2 2 8+δ c
d beenobservedat43meV(Fig.1.d)[9]. Furthermore,E shiftsdownto38meVintheoverdopedregime
r
n (T =80 K) [10], preservinga constant ratio with T : E ≃ 5.4 k T (Fig. 2). Thus, whatever the hole
c c r B c
o
doping,theenergypositionofthemagneticresonancepeakalwaysscaleswithT . IncontrasttoYBCO,
c c
: wheretheresonancepeakisresolutionlimitedinenergy,theresonancepeakinBSCOexhibitsanenergy
v
widthof∼13meV.Inaddition,themomentumwidthoftheexcitationistwicebroader. Asimilarenergy
i
X andmomentumbroadeninghasbeenalsoreportedinYBa2(Cu1−yNiy)3O7[8](Fig.1.b)andcantherefore
r be ascribedto disorder,such asimpuritiesorinhomogeneities. Furthermore,the observationofa spatial
a distributionoftheSCgapinBi Sr CaCu O byScanningTunnelingMicroscopymeasurements[11]
2 2 2 8+δ
providesevidenceinfavorofanintrinsicdisorderinthissystem.
Further, the magnetic resonance peak has been observed in optimally doped Tl Ba CuO (T =90
2 2 6+δ c
K) at E =47 meV (Fig. 1.c) [12]. This yields a ratio E /k T ≃ 6 slightly larger than YBa Cu O .
r r B c 2 3 6+x
Nevertheless,asinYBa Cu O ,theexcitationislimitedbytheresolutioninenergyanddisplaysamo-
2 3 7
mentumwidthof0.25A˚−1 (halfwidthathalfmaximum). Meanwhile,theenergyintegratedintensityof
Copyrightlinewillbeprovidedbythepublisher
2 Y.Sidisetal:MagneticResonantexcitationsinHigh-Tcsuperconductors
160
a ) n) 1600 b ) YBCO - E
100 mi r
s / 10 min) 50 cnts / 87 1280000 112400 BB(ASSRCCPOOE S-- E2) r∆m
Intensity (cnt 0 10 K - 100 K Intensity ( -4400000 10 K - 87 K ∆2 (meV)m18000 5.3 k T B(SSISC)O - 2 ∆m 150
-5020 30 40 50 60 20 30 40 50 E, r 60 B c
Energy (meV) Energy (meV) 100 T
40 c
urs) 50 c ) urs) 200 d ) 20 50 (K)
o o
h 0 h Underdoped Overdoped
s / 2 -50 s / 2 100 0 0
nt nt -0.1 -0.05 0 0.05 0.1
c c 0
y ( -100 y ( n -n
nsit -150 nsit -100 h opt
e e
Int -200 27 K - 99 K Int -200 10 K - 100 K Fthieg.m2agnDetoicpinrgesodnepanencedenpceeakofatth(eπe/nae,rπgy/ao)f,
30 40 50 60 30 40 50 60 Er, and of twice the maximum of the super-
Energy (meV) Energy (meV)
conducting gap, 2 ∆m. Er has been mea-
sured by INS in YBCO [2, 3, 4, 5, 6, 7],
Fig. 1 Difference spectrum of the neutron intensities at
low temperature, measured at the wave vector (π/a,π/a) and BSCO[9, 10]. 2 ∆m is determined from
and T≥ Tc: a) YBa2Cu3O6.95: Tc=93 K, V=10 cm3 angle resolved photo-emission spectroscopy
[4], b) YBa2(Cu1−yNiy)3O7: Tc=80 K, V∼2 cm3 (ARPES) [13] or superconducting-insulator-
[8], c) Tl2Ba2CuO6+δ:Tc∼90 K, V=0.11 cm3 [12], d) superconductor (SIS) tunneling [14] data per-
Btedi2Stor2aCGaCauus2sOian8+pδr;oTfilce=.91ThKe,sVo=li0d.0b6arcmin3di[c9a]t.esDtahteaaerneerfigty- fthorromuegdhinthBeSeCmOp.irTichaeldroeplaintigonle:veTlcis/Tesctmimaxa=te1d-
resolution. 82.6(nh-nopt)2[15].
themagneticresonancepeakisalmostthesameinbothsystems:0.7-0.8µ2.eV−1/CuO plane.Thus,the
B 2
magneticresonancepeakappearsasacommonexcitationtotheSCstateofallHigh-T superconductors,
c
investigated so far by INS measurements, whose maximum T can be as high as ∼90 K. Furthermore,
c
the existence of this excitation does not depend on the number of CuO planes per unit cell: one for
2
Tl Ba CuO andtwoforYBa Cu O andBi Sr CaCu O .
2 2 6+δ 2 3 6+x 2 2 2 8+δ
WhilethemagneticresonancepeakexistsincuprateswhosemaximumT isabout90K,ithasnever
c
beenobservedinthemono-layersystemLa2−xSrxCuO4withamaximumofTcof∼40K.Furthermore,
themagneticexcitationsinthatcompoundareratherstrongeveninthenormalstateandlocatedatincom-
mensurateplanarwavevectorQ = (π/a(1±δ ),π/a)and(π/a,π/a(1±δ ))[16]. Thisisina
mag inc inc
markedcontrastwith thethesystemsmentionedabove,forwhichthenormalstatemagneticfluctuations
(if observable) remain centered around (π/a,π/a). However, passing through T , the incommensurate
c
spinfluctuationsofLa2−xSrxCuO4areenhancedandbecomenarrowerinmomentumspace,inanenergy
rangewhichis about5k T [17]. Thisphenomenon,usuallyreferredto asa ”coherenceeffect”andthe
B c
resonancepeakcouldeventuallyshareacommonorigin.
InunderdopedYBa Cu O (x=0.6,T =63K,E =34meV),INSmeasurementsprovideevidencefor
2 3 6+x c r
incommensurate-likespinfluctuationsat24meVandlowtemperature(seeminglysimilartothoseobserved
La2−xSrxCuO4) [18]. These incommensurate-likespin fluctuationsare also observed at higher oxygen
concentrations: x=0.7[19], x=0.85[20]. Asafunctionoftemperatureandenergy[20],theincommensu-
rability(δ )increasesbelowT withdecreasingtemperatureanddecreasesuponapproachingE inthe
inc c r
phys.stat.sol.(2003) 3
SCstate(Fig.3.a). Thesimultaneousdisappearanceofδ atE andT indicatesthattheresonancepeak
inc r c
and the incommensurate-likespin fluctuations are intrinsic features of the SC state and that they can be
viewedascontinuouslyconnected(Fig.3.a). Inotherwords,theseresultsleadtoanunifieddescriptionof
boththeincommensuratespinexcitationsandthemagneticresonancepeakintermsofauniquecollective
spin excitation mode with a downwarddispersion[20]. Recently, the actual symmetryof this dispersion
foran optimallydopedYBCO samplehasbeenlookedat carefully[21] andwas foundbasicallycircular
within the 2D copper-oxygen plane. In addition, a second magnetic mode with much weaker intensity
is reporteddispersingupward abovethe (π/a,π/a)peak[21]. The deepunderdopedstate YBCO has
6.5
been also recently re-investigatedin partly detwinned sample with an ortho-IIstructure[22]. In contrast
to the dispersive modepicture, it is claimed[22] thatthe low energymagneticexcitationsare essentially
one-dimensionalasexpectedforhydrodynamicstripes. Therefore,thedetaileddopingdependenceofthe
spinfluctuationsneedstobeclarifiedtoreconciletheseconclusionsbystudyingfullydetwinnedsamples.
Indeed,oneneedstodeterminethespecificroleoftheCu-OchainsinYBCOforthemagneticanisotropy.
Themaindifferencebetweenmono-layerandbilayersystemsshowsupinthemomentumdependenceof
themagneticresonancepeakalongthe(001)direction.InTl Ba CuO ,theexcitationremainspurely
2 2 6+δ
bidimensional.Incontrastinbilayersystems,thetwoCuO planescorrelateantiferromagneticallywithin
2
the bilayer. ThatinterlayerAFcouplingis responsiblein insulatingparentcompoundsforbothacoustic
andopticmagnons,whosecounterpartin themetallicstate aretheodd(o)andeven(e)excitations. The
neutronscatteringcrosssectionthenreads[6]:
d2σ(Q,ω)
∝sin2(Q d/2)Im[χ (Q,ω)]+cos2(Q d/2)Im[χ (Q,ω)] (1)
z o z e
dΩdω
where Im[χ (Q,ω)] corresponds to the imaginary part of the dynamical magnetic susceptibility in
o,e
eachchannelandd(=3.3A˚)standsforthedistancebetweenCuO planeswithinthebilayer. Intuitively,
2
one could expect a splitting of the magnetic resonance peak under the interlayer AF coupling, leading
to a magnetic resonance in each channel. For a long time, the magnetic resonance peak was observed
onlyintheoddchannel. However,wecouldrecentlyobservearesonantmodeineachchannelinslightly
overdopedYBCOthrough10%substitutionofYbyCa[23].TheyoccurattwodifferentenergiesEo=36
r
meVandEe=43meVandtheevenmodeexhibitsanintensityonethirdtimeslessthantheoddone. The
r
questionwhytheevenmodeisnowsizeableinthisoverdopedregimeandnotinpreviousstudiesremains
openasitcouldbesimplyduetoimprovementofneutroninstruments. However,itmightalsoberelated
totheelectronictransportbetweencloselyspacedCuO layerswhichbecomescoherentintheoverdoped
2
regime,asdemonstratedbyrecentexperimentsshowingwell-definedbondingandantibondingbands.
Consideringthedifferentmodelsforthemagneticresonancepeak,wefocusonmodelswhereelectron-
electroninteractionsplaythecentralrole,despitethestillpossibleexistenceofelectron-phononcouplings
incuprates. Essentially, thereisnoindicationofaneffectofthelattice onthemagneticresonancepeak.
Secondly, we consider here models where the downward dispersion of the resonant mode would natu-
rallyemerge. Forinstance,approaches[24,25]whichassociatetheresonancepeaktoapre-existingsoft
modereminiscentofnearby(commensurateorincommensurate)AFphasewouldyieldacollectivemode
dispersingpredominantlyupward.
The existence of a spin 1-collective in d-wave superconductorssuch as High-T cuprates, is derived
c
from an itinerant description of the magnetic properties of the system and of strong correlation effects
([26, 27, 28, 29, 30, 31] and referencestherein). This leads to a particle-hole(p-h)boundstate, usually
referredasaspinexciton.Inthesestrongcouplingmodels(seealsoRefs. [32,33]intheframeworkofthe
t−J model),thegeneralizedspinsusceptibilityχ(q,ω)isexpressedasafunctionofthenoninteraction
spinsusceptibilityχ (q,ω)andthemagneticinteraction.χ(q,ω)hasanRPA-likeform:
0
χ (q,ω)
0
χ(q,ω)= (2)
1+J(q)χ (q,ω)
0
4 Y.Sidisetal:MagneticResonantexcitationsinHigh-Tcsuperconductors
50 YBa Cu O continuum
2 3 6.85 70 Imχ (q ,ω)
ω odd AF
V) 40 60 c Imχeven(qAF,ω)
Energy (me 2300 T=11 K Energy / J -12ωµ) (,).eVAFB345000 ω/t
10 T =89 K χ(q
a) c b) Im 20
0 ω
-0.4 -0.2 0 0.2 0.4 -0.4 -0.2 0 0.2 0.4 10 c continuum
δ δ
inc inc 00 0.05 0.1 0.15 0.2 0.25 0.3
ω/t
Fig. 3 a) Dispersion of the magnetic resonance peak
as a function q=(π/a(1 ± δinc),π/a), measured in Fig.4 Calculatedspinsusceptibilityforbothoddandeven
YBa2Cu3O6.85 [20],b)dispersionofthespinexciton symmetry according to Eq. 3 with an interlayer coupling
[31] (thick line) below the continuum which is repre- ofJ⊥ ≃0.1Jandwiththebandstructureparametersand
sentedbythedashedarea.Themaximumofthedashed t/J=2ofref. [31]andt=250meV.Thedottedareasshow
lines correspond to 2∆m, and their crossing defines theweakelectron-holecontinuumintensity.
ωc =2∆ks.
whereJ(q)=2J(cos(q )+cos(q ))istheintra-planeAFsuper-exchangecouplingandχ (q,ω)describes
x y 0
in an itinerant system the continuum of spin flip particle-hole (p-h) excitations, given by the Lindhard
functioninthenormalstateortheBCSfunctionintheSCstate. IntheSCstate,duetotheopeningofthe
SC gap, the continuumbecomesgaped(Fig. 3.b) below a threshold energyat ω = 2∆ , where ∆ is
c ks k
themomentumdependentsuperconductingd-waveenergygapandk istheso-calledhot-spotwavevector
s
definedasbothk andk +Q arelyingontheFermisurface. Inadditiontothep-hexcitationswithin
s s AF
the continuum,a spin tripletp-hboundstate can formbelow thecontinuumthankto the AFinteraction.
In Eq. 2, the dynamical Stoner criterion, i.e. 1 + J(q)Re[χ (q,ω)]=0, is then fulfilled for an energy
0
smallerthatthethresholdofthecontinuumatwavevectorq. Thisspin1-collectivemodeischaracterized
byadownwarddispersioncontrolledbythemomentumdependencesofthecontinuumthresholdandthe
magnetic interaction (Fig 3.b). The mode vanishes when approachingthe continuumby changingwave
vectorfrom(π/a,π/a). Themodelalsopredictedanexcitationdispersingupwardwithinthecontinuum
[29,31]whichmightcorrespondtotherecentlyobservedhighenergymode[21].
From experimentalpoint of view, a major challenge for future INS experimentswould be to observe
the magnetic continuum. So far, one can deduce the continuumthreshold, ω , from i) the measurement
c
ofthemaximumoftheSCgap,∆ (determinedbyangleresolvedphoto-emissionspectroscopy[13],by
m
the measurement of the B mode in Raman scattering[34] or by tunneling data[14], see Fig. 2) and
1g
ii) from the Fermi surface topology[35]. At optimal doping in BSCO, one can show that the magnetic
resonance peak lies well below the continuum: ∆ ≃ 35 meV, ω ≃ 1.8∆ and E ≃ 1.2∆ . From
m c m r m
optimaldoping to the overdopedregime, one may expectω to reach the limit ∼2∆ . Simultaneously,
c m
theratio2∆ /k T evolvesfrom7-8to5-6,whileaccordingtoINSdata,theratioE /k T seemstobe
m B c r B c
preserved(Fig.2). Theseevolutionssuggestthatintheoverdopedregimethebindingenergyofthespin
exciton,ω -E ,weakens,leadingtothepossibledisappearanceofthespin1-collectivemode.FurtherINS
c r
experiments,inthedeeplyoverdopedregimearerequiredtotestsuchapossibility.
Fora bilayersystem [26, 27, 36], the interlayerAF coupling,J⊥, is treatedin perturbation,suchthat
theodd(o)andeven(e)spinsusceptilitiesaregivenby:
χ(q,ω) χ(q,ω)
χ (q,ω)= and χ (q,ω)= . (3)
o e
1−J⊥χ(q,ω) 1+J⊥χ(q,ω)
phys.stat.sol.(2003) 5
InFig. 4,wecalculatedthesesusceptibilitieswitharealisticvalueofJ⊥ (∼0.1J)[37]andwithother
parametersidenticalto thoseofref. [31]. Because ofJ⊥, the oddspin excitonislikely pushedto lower
energy,whiletheevenonemergesintothecontinuum. Actually,thisexplainswhytheoddchannelmode
ismainlyobservedaswellaswhytheresonantmodeappearstoshifttolowerenergyinabilayersystem
in regardwith a mono-layersystem. Going further, their respective spectral weights W are predicted
o,e
to be approximatelyproportionalto their binding energies[27] as W ∼ ω −Eo,e. This is found in
o,e c r
thecalculatedsusceptibilitiesinFig. 4andsketchedinthefigureinset. Usingthispropertyinoverdoped
YBCO-Ca[23],onecoulddirectlyestimateofcontinuumthresholdatω (Q )≃49meV.
c AF
Ifthespinexcitonscenarioprovidesaplausibleexplanationfortheresonancepeakaroundtheoptimal
doping,moretheoreticalworkisneededtofullyaccountforitsevolutionasafunctionofholedoping. In
particular,thereisnoexplicitrelationshipbetweentheenergyofthecollectivemodeat(π/a,π/a)andthe
valueof T . ThephenomenologicalrelationshipE /k T ≃ 5-6remainsto be explained. Furthermore,
c r B c
thesamemodelmustsimultaneouslydescribetheunusualfeaturesobservedbelowT aswellasthespin
c
excitationspectrumofthenormalstate. Intheunderdopedregime,themagnitudeandtheenergydepen-
denceofspinfluctuationsobservedbyINSarestilldifficulttoreproducequantitatively.Thus,mostlikely,
themodelneedstogobeyondapurelyitinerantpictureinordertocapturethedeepunderdopedstate.
Howeverthe spin-excitoninterpretationforthemagneticneutronresonanceandits downwarddisper-
sionisnotunique.Indeed,besidessuchanapproachthespin1-collectivemodecorrespondstoap-hbound
state,ithasalsobeenproposedthatthemagneticresonancepeakcouldbeap-ppair,theπ-excitation,in
SO(5) model(spin itinerantpicture) or a magnon-likeexcitation (spin localized picture) of a disordered
stripephase.
The SO(5) model [38] considers as a starting point that, in the AF insulating state of cuprates, there
existsa super-symmetrythatallowsthesystemtoswitch fromtheAFstate tothed-waveSCstate. This
symmetryinvolvesthe existence of a Goldstonemode, the π-excitation[39]. Thisexcitation can be de-
pictedasanexcitationthatcreatesa p-ppaircarryinga charge2e, a spinS=1 andwithtotalmomentum
(π/a,π/a). Upondopingthesymmetryisbroken,theπ-excitationsurvives,butbecomesmassive. While
itexistsalreadyinthenormalstate,itcanonlybeobservedbyINSintheSCstateduetothep-hmixing.
Its characteristic energy is roughly linear with hole doping and its intensity in the SC state scales with
|∆ |2. Moreover, a downward dispersion is obtained due to by the phase slips of the SC order param-
m
eter inducedby the propagationof the π-excitation[40]. Such an approachaccountsfor severalfeatures
observedbyINS,especiallyintheunderdopedregime,whereT andE increasewithholedoping. The
c r
π-excitation should decrease in magnitude, but remains almost at the same energy, as observed experi-
mentally. HoweverthisscenariocannotexplainthedecreaseofE intheoverdopedregime. Furthermore,
r
recentcalculationshaveshownthat,iftheπ-excitationexisted,itshouldbeobservedathighenergy,above
2∆ [41]:thiscastssomedoubtabouttheinterpretationoftheresonanceasaπ-tripletexcitation.
m
Alternatively,thestripemodelconsidersthatinaS=1/2AFHeisenbergsystem,dopedholessegregates
toformlinesofcharges,separatingAFdomainsinanti-phase.Themetallicstateisviewedasadisordered
stripe phase, where charged lines can fluctuate. While there is not a general interpretation of the mag-
netic resonancepeak in stripe models, it hasbeenfor instanceproposedthatthe resonancepeakandthe
incommensuratespinfluctuationsobservedbyINSintheSCstatecouldbeviewedasmagnon-likeexci-
tationsreminiscentofthe orderedstripephase[42, 43] or couldcorrespondto theeigenmagneticmodes
oftheliquidstripephase[44]. Magnons,developingsymmetricallyaroundthemagneticincommensurate
wave-vector,Q ,ofthestripeorderedphaseandmergingat(π/a,π/a),actuallydescribecorrectlythe
mag
spin dynamicsobserved in stripe-orderednickelates[45] as predictedin the spin-only model[42, 43]. In
cuprates,thelackofsymmetricpeaksaroundtheincommensuratewave-vectorQ (seeFig. 3)doesnot
mag
seemtovalidatetheseapproaches.Inaddition,thismodel,ifinteresting,failstoexplainwhytheresonance
peakandtheincommensuratespinfluctuationsexistbasicallyonlyintheSCstate. Independently,itcould
bealsointerestingtounderstandhow,inbilayercompounds,theadjacentCuO planessucceedinaccom-
2
modatingtheCoulombrepulsionbetweenchargedlinesandtheAFinterlayercoupling. Thisisacentral
issuetoaccountforasthemagneticresonancepeakexistsmostlyintheoddchannel.
6 Y.Sidisetal:MagneticResonantexcitationsinHigh-Tcsuperconductors
Finally,INSexperimentshaveshowntheexistenceofanusualenhancementofspinfluctuationsinthe
SC state around the vector (π/a,π/a) and at an energyE which is foundexperimentallyto scale with
r
T . Combined with the observation at lower energy of incommensurate spin fluctuations, that develop
c
alsobelowT ,INSdatapointtowardtheexistenceofadispersivespin1-collectivemodedeepinsidethe
c
SCstate. Theobservationofthatmode,firstdiscoveredinYBa Cu O ,hasbeenthenextendedtoother
2 3 7
systems with one or two CuO planes per unit cell, such as Bi Sr CaCu O and Tl Ba CuO .
2 2 2 2 8+δ 2 2 6+δ
This establishes the magnetic resonance peak as a generic excitation of the SC state of cuprates whose
maximum T can be as high as 90 K. In the strongly under- and overdopedregimes (T ≤ 50 K) or in
c c
othercupratefamilieswithlowermaximumTc(suchasLa2−xSrxCuO4),theobservationofthethespin
1-collectivemode(ifany)stillrequiresmoreexperimentalwork. Inanycase,theobservationofsuchan
excitation, thankto inelastic neutronscattering, is oneof the mostpersuasiveexperimentalindicationof
thecrucialroleofmagneticinteractionsinforthephysicsofHigh-T copperoxides.
c
References
[1] Seee.g.D.Scalapino,Phys.Rep.250329(1995).
[2] J.Rossat-Mignod,L.P.Regnault,C.Vettier,P.Bourges,P.Burlet,J.Bossy,etal.,PhysicaC185-189,86(1991).
[3] H.A.Mook,M.Yethiraj,G.Aeppli,T.E.Mason,andT.Armstrong,Phys.Rev.Lett.70,3490(1993).
[4] H.F.Fong,B.Keimer,P.W.Anderson,etal.,Phys.Rev.Lett.75,316(1995);Phys.Rev.B54,6708(1996).
[5] P.Bourges,L.P.Regnault,Y.Sidis,andC.Vettier,Phys.Rev.B53,876(1996).
[6] H.F.Fong,P.Bourges,Y.Sidis,L.P.Regnault,J.Bossy,A.S.Ivanov,etal.,Phys.Rev.B61,14774(2000).
[7] P.Dai,H.A.Mook,R.D.Hunt,F.Dog˘an,Phys.RevB.,63,054525(2001).
[8] Y.Sidis,P.Bourges,H.F.Fong,B.Keimer,L.P.Regnault,J.Bossy,etal.,Phys.Rev.Lett.86,4100(2001).
[9] H.F.Fong,P.Bourges,Y.Sidis,L.P.Regnault,A.S.Ivanov,G.D.Gu,etal.,Nature398,588(1999).
[10] H.He,Y.Sidis,Ph.Bourges,G.D.Gu,A.Ivanov,N.Koshizuka,B.Liang,etal.,Phys.Rev.Lett.86,1610(2001).
[11] K.M.Lang,V.Madhavan,J.E.Hoffman,E.W.Hudson,H.Eisaki,S.Uchida,J.C.Davis,Nature415,412(2002).
[12] H.He,P.Bourges,Y.Sidis,C.UlrichL.P.Regnault,S.Pailhe`s,etal.,Science295,1045(2002).
[13] J.Mesot,M.R.Norman,H.Ding,M.Randeria,J.C.Campuzano,etal.,Phys.Rev.Lett.83,840(1999).
[14] J.F.Zasadzinski,L.Ozyuzer,N.Miyakawa,etal.,Phys.Rev.Lett.87,067005(2001)(cond-mat/0102475).
[15] J.L.Tallon,C.Bernhard,H.Shaked,R.L.HittermanandJ.D.Jorgensen,Phys.Rev.B51,12911(1995).
[16] G.Aeppli,T.E.Mason,S.M.Hayden,H.A.Mook,J.Kulda, Science278,1432(1997).
[17] T.E.Mason,A.Schro¨der,G.Aeppli,H.A.Mook,andS.M.Hayden,Phys.Rev.Lett77,1604(1996).
[18] H.A.Mook,P.Dai,S.M.Hayden,G.Aeppli,T.G.Perring,andF.Dog˘an,Nature395,580(1998).
[19] M.Arai,T.Nishijima,Y.Endoh,T.Egami,S.Tajima,K.Tomimoto,etal.,Phys.Rev.Lett.83,608(1999).
[20] P.Bourges,Y.Sidis,H.F.Fong,L.P.Regnault,J.Bossy,A.Ivanov,andB.Keimer,Science288,1234(2000).
[21] D.Reznik,P.Bourges,L.Pintschovius,Y.Endoh,etal.,submittedtoPhys.Rev.Lett(cond-mat/0307591).
[22] C.Stock,W.J.L.Buyers,R.Liang,D.Peets,Z.Tun,etal.,Phys.Rev.B,(2004)(cond-mat/0308168).
[23] S.Pailhe`s,Y.Sidis,P.Bourges,C.Ulrich,etal.,Phys.Rev.Lett.91,237002(2003)(cond-mat/0308394).
[24] D.K.MorrandD.Pines,Phys.Rev.Lett.81,1086(1998).
[25] S.Sachdev,C.Buragohain,andM.Vojta,Science286,2479(1999).
[26] D.Z.Liu,Y.Zha,andK.Levin,Phys.Rev.Lett.75,4130(1995).
[27] A.J.MillisandH.Monien,Phys.Rev.B54,16172(1996).
[28] A.Abanov,andA.V.Chubukov,Phys.Rev.lett.83,1652(1999).
[29] M.N.Norman,Phys.Rev.B61,14751(2000).
[30] D.Manske,I.EreminandK.H.Bennemann,Phys.Rev.B63,054517(2001).
[31] F.Onufrieva,andP.Pfeuty,Phys.Rev.B65,014502(2002)(cond-mat/9903097).
[32] J.BrinckmannandP.A.Lee,Phys.Rev.B65,014502(2002).
[33] I.Sega,P.Prelovs˘ek,andJ.Bonc˘a,Phys.Rev.B68,054524(2003).
[34] Forarecentreview,seee.g.M.Cardona,PhysicaC318,30(1999).
[35] H.Ding,M.R.Norman,T.Yokoya,T.Takeuchi,M.Randeria,etal.,Phys.Rev.Lett.78,2628(1997).
[36] T.LiandZ.Gan,Phys.RevB60,3092(1999);T.Li,Phys.Rev.B64,012503(2001).
[37] D.Reznik,P.Bourges,H.F.Fong,L.P.Regnault,J.BossyC.Vettier,etal.,Phys.Rev.B53R14741(1996).
[38] S.C.Zhang,Science275,1089(1997).
[39] E.Demler,H.Kohno,andS.C.Zhang,Phys.Rev.B58,5719(1998).
[40] J.P.HuandS.C.Zhang,Phys.Rev.B64,100502(2001).
[41] O.Tchernychyov,M.R.NormanandA.V.Chubukov,Phys.Rev.B63,144507(2001).
[42] C.D.Bastista,G.Ortiz,andA.V.Balatsky,Phys.Rev.B65,180402(2002).
[43] F.Kru¨ger,andS.Scheidl,Phys.Rev.B67,134512(2003).
[44] N.Hasselmann,A.H.CastroNeto,C.MoraisSmith,andY.Dimashko,Phys.Rev.lett.82,2135(1999).
[45] P.Bourges,Y.Sidis,M.Braden,K.Nakajima,andJ.M.Tranquada,Phys.Rev.Lett.,90,147202(2003).